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version 1.41, 2002/08/18 17:08:53 version 1.45, 2002/10/30 14:43:15
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 <?xml version="1.0"?> <?xml version="1.0"?>
 <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" ><archimedes>      <info> <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" >
  <archimedes xmlns:xlink="http://www.w3.org/1999/xlink">      <info>
         <author>Cardano, Girolamo</author>         <author>Cardano, Girolamo</author>
         <title>Opus novum de proportionibus</title>         <title>Opus novum de proportionibus</title>
         <date>1570</date>         <date>1570</date>
         <place>Basel</place>         <place>Basel</place>
         <translator></translator>         <translator/>
         <lang>la</lang>         <lang>la</lang>
         <cvs_file>carda_propo_01_la_1570</cvs_file>         <cvs_file>carda_propo_01_la_1570</cvs_file>
         <cvs_version></cvs_version>         <cvs_version/>
         <locator>0000000015.xml</locator>         <locator>015.xml</locator>
 </info>      <text>          <front>          </front>          <body>            <chap>        <pb/><pb/><pb/><p type="head"> </info>      <text>          <front>          </front>          <body>            <chap>        <pb/><pb/><pb/><p type="head">
  
 <s>HIERONYMI <lb/>CARDANI MEDIO <lb/>LANENSIS, CIVISQV'E BONO&shy;<lb/>NIENSIS, PHILOSOPHI, MEDICI ET <lb/>Mathematici clari&longs;simi,</s></p><p type="head"> <s>HIERONYMI <lb/>CARDANI MEDIO <lb/>LANENSIS, CIVISQV'E BONO&shy;<lb/>NIENSIS, PHILOSOPHI, MEDICI ET <lb/>Mathematici clari&longs;simi,</s></p><p type="head">
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 <s>DE ALIZA REGVLA LIBER, HOC EST, ALGEBRAICAE <lb/>logi&longs;tic&aelig; &longs;u&aelig;, numeros recondita numerandi &longs;ubtilitate, &longs;ecundum Geo&shy;<lb/>metricas quantitates inquirentis, nece&longs;&longs;aria Coronis, <lb/>nunc demum in lucem edita.</s></p><p type="head"> <s>DE ALIZA REGVLA LIBER, HOC EST, ALGEBRAICAE <lb/>logi&longs;tic&aelig; &longs;u&aelig;, numeros recondita numerandi &longs;ubtilitate, &longs;ecundum Geo&shy;<lb/>metricas quantitates inquirentis, nece&longs;&longs;aria Coronis, <lb/>nunc demum in lucem edita.</s></p><p type="head">
  
 <s>O<emph type="italics"/>pus<emph.end type="italics"/> P<emph type="italics"/>hy&longs;icis &amp;<emph.end type="italics"/> M<emph type="italics"/>athematicis imprimis <lb/>utile &amp; nece&longs;&longs;arium.<emph.end type="italics"/></s></p><figure></figure><p type="head"> <s>O<emph type="italics"/>pus<emph.end type="italics"/> P<emph type="italics"/>hy&longs;icis &amp;<emph.end type="italics"/> M<emph type="italics"/>athematicis imprimis <lb/>utile &amp; nece&longs;&longs;arium.<emph.end type="italics"/></s></p><figure/><p type="head">
  
 <s>Cum C&aelig;&longs;. </s> <s>Cum C&aelig;&longs;. </s>
  
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 <s>BASILE&AElig;.</s></p><pb/><p type="head"> <s>BASILE&AElig;.</s></p><pb/><p type="head">
  
 <s>IN LIBRVM DE <lb/>PROPORTIONIBVS HIERONYMI <lb/>CARDANI MEDIOLANENSIS, CIVISQV'E <lb/>Bononien&longs;is, Medici, Pr&aelig;fatio ad M. A. </s> <s>IN LIBRVM DE <lb/>PROPORTIONIBVS HIERONYMI <lb/>CARDANI MEDIOLANENSIS, CIVISQV'E <lb/>Bononien&longs;is, Medici, Pr&aelig;fatio ad M. A. <!-- KEEP S--></s>
  
 <s>Amulium <lb/>Venetum Card. </s> <s>Amulium <lb/>Venetum Card. </s>
  
 <s>Illu&longs;tri&longs;simum.</s></p><p type="main"> <s>Illu&longs;tri&longs;simum.</s></p><p type="main">
  
 <s>Bene Dictum e&longs;t meo iudicio &agrave; Platone M. <lb/>A. </s> <s>Bene Dictum e&longs;t meo iudicio &agrave; Platone M. <lb/>A. <!-- KEEP S--></s>
  
 <s>Amuli optime, beatas fore Re&longs;pub. </s> <s>Amuli optime, beatas fore Re&longs;pub. </s>
  
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 <s>Succedunt inde tot commoda, non &longs;olum utilia, &longs;ed pleraque<pb/>etiam nece&longs;&longs;aria, qu&aelig; nos &longs;apientia docet: huiu&longs;modi ergo omnia <lb/>c&ugrave;m libris contineantur, merit&ograve; optimus qui&longs;que librorum bono&shy;<lb/>rum perpetuitati atque in columitati fauere debet. </s> <s>Succedunt inde tot commoda, non &longs;olum utilia, &longs;ed pleraque<pb/>etiam nece&longs;&longs;aria, qu&aelig; nos &longs;apientia docet: huiu&longs;modi ergo omnia <lb/>c&ugrave;m libris contineantur, merit&ograve; optimus qui&longs;que librorum bono&shy;<lb/>rum perpetuitati atque in columitati fauere debet. </s>
  
 <s>C. </s> <s>C. <!-- KEEP S--></s>
  
 <s>Caligulam exe&shy;<lb/>cramur &longs;olum ob id quod Vergilij, &amp; T. </s> <s>Caligulam exe&shy;<lb/>cramur &longs;olum ob id quod Vergilij, &amp; T. </s>
  
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 <s>Itaque infinitis licet circumuentus <lb/>negocijs totus huic oper&aelig; in cubui, atque ade&ograve; ut pr&aelig;ter &longs;pem unius <lb/>anni pen&egrave; &longs;pacio liber ab&longs;olueretur. </s> <s>Itaque infinitis licet circumuentus <lb/>negocijs totus huic oper&aelig; in cubui, atque ade&ograve; ut pr&aelig;ter &longs;pem unius <lb/>anni pen&egrave; &longs;pacio liber ab&longs;olueretur. </s>
  
 <s>Qui cum tibi (ut dixi) iam iur&egrave; <lb/>deberetur, e&ograve; tamen magis dedicandum putaui, quod non ego &longs;o&shy;<lb/>lum quanquam id maxim&egrave;, &longs;ed communis con&longs;en&longs;us ho&shy;<lb/>minum exi&longs;timet, te &longs;ingulari uirtute omnibus <lb/>&longs;tudio&longs;is plurimum fauere, <lb/>Vale.</s></p><pb/><p type="head"> <s>Qui cum tibi (ut dixi) iam iur&egrave; <lb/>deberetur, e&ograve; tamen magis dedicandum putaui, quod non ego &longs;o&shy;<lb/>lum quanquam id maxim&egrave;, &longs;ed communis con&longs;en&longs;us ho&shy;<lb/>minum exi&longs;timet, te &longs;ingulari uirtute omnibus <lb/>&longs;tudio&longs;is plurimum fauere, <lb/>Vale.<!-- KEEP S--></s></p><pb/><p type="head">
  
 <s>TABVLA PRO&shy;<lb/>POSITIONVM DE <lb/>PROPORTIONIBVS.<lb/><arrow.to.target n="table1"></arrow.to.target></s></p><table><table.target id="table1"></table.target><row><cell>I.</cell><cell>Proportionem <emph type="italics"/>in proportionem duci, e&longs;t &longs;uperiores numeros   atque inferiores inuicem ducere.<emph.end type="italics"/></cell><cell><emph type="italics"/>pagina<emph.end type="italics"/> 6</cell></row><row><cell>II.</cell><cell>P<emph type="italics"/>roportio extremorum producitur ex intermedijs.<emph.end type="italics"/></cell><cell>7</cell></row><row><cell>III.</cell><cell>S<emph type="italics"/>i proportio ex duabus proportionibus in quatuor terminis producatur,   ip&longs;a uer&ograve; proportio inter duas alias quantitates fuerit con&longs;tituta: con&longs;urgent trecen-ti &longs;exaginta modi productionis proportionis.<emph.end type="italics"/></cell><cell>7</cell></row><row><cell>IIII.</cell><cell>S<emph type="italics"/>i fuerit proportio primi ad &longs;ecundum, producta ex proportionibus tertij ad quartum,   &amp; quinti ad &longs;extum, producetur etiam ex proportione tertij ad &longs;extum, &amp; quinti ad   quartum.<emph.end type="italics"/></cell><cell>8</cell></row><row><cell>V.</cell><cell>S<emph type="italics"/>i fuerit proportio primi ad &longs;ecundum, producta ex proportione tertij ad quartum, &amp;   quinti ad &longs;extum: erit proportio tertij ad &longs;extum, producta ex proportionibus primi   ad &longs;ecur dum, &amp; quarti ad quintum.<emph.end type="italics"/></cell><cell>8</cell></row><row><cell>VI.</cell><cell>E<emph type="italics"/>x trecentis &longs;exaginta modis producendarum proportionum triginta &longs;ex tantum e&longs;&longs;e   nece&longs;&longs;arios.<emph.end type="italics"/></cell><cell>9</cell></row><row><cell>VII.</cell><cell>I<emph type="italics"/>n modis qui nece&longs;&longs;ari&ograve; producuntur ex duabus proportionibus, cum du&aelig; quantitates ex   illis qu&aelig; modos conficiunt, &aelig;quales fuerint: proportio producta ad quatuor quanti-tates omiologas reducetur.<emph.end type="italics"/></cell><cell>10</cell></row><row><cell>VIII.</cell><cell>S<emph type="italics"/>i duarum proportionum &longs;uperiores numeri alternatim cum inferioribus multiplicen-tur atque coniungantur, erit proportio aggregati ad productum ex inferioribus in-uicem proportio, ex primis proportionibus compo&longs;ita.<emph.end type="italics"/></cell><cell>11</cell></row><row><cell>IX.</cell><cell>S<emph type="italics"/>i duarum proportionum &longs;uperiores numeri alternatim cum inferioribus multiplicen-tur, minusque productum ex maiore detrahatur, erit re&longs;idui ad productum ex in&longs;e-rioribus proportio uelut illa, qu&aelig; relinquitur detracta minore proportione ex ma-iore.<emph.end type="italics"/></cell><cell>11</cell></row><row><cell>X.</cell><cell>S<emph type="italics"/>i fuerit alicuius quantitatis ad unam partem proportio, uelut alterius partis ad &longs;ecun-dam quantitatem, erit proportio cuiu&longs;uis quantitatis eiu&longs;dem generis ad &longs;ecundam   compo&longs;ita proportio, ex proportionibus eiu&longs;dem quantitatis, a&longs;&longs;umpt&aelig; ad utranque   partem prim&aelig; quantitatis &longs;eor&longs;um.<emph.end type="italics"/></cell><cell>11</cell></row><row><cell>XI.</cell><cell>P<emph type="italics"/>roportio aggregati quarumlibet duarum quantitatum ad aggregatum duarum &aelig;qua-lium <expan abbr="quantitat&utilde;">quantitatum</expan> e&longs;t, compo&longs;ita ex proportionibus primis, &amp; diui&longs;a per duplam.<emph.end type="italics"/></cell><cell>12</cell></row><row><cell>XII.</cell><cell>P<emph type="italics"/>ropo&longs;itis duabus proportionibus unam alteri iungere ab&longs;que multiplicatione.<emph.end type="italics"/></cell><cell>12</cell></row><row><cell>XIII.</cell><cell>P<emph type="italics"/>roportio confu&longs;a aggregata prim&aelig; &amp; terti&aelig; quatuor quantitatum omiologarum ad   aggregatum &longs;ecund&aelig; &amp; quart&aelig;, e&longs;t uelut compo&longs;ita ex ei&longs;dem diui&longs;a per du-plam.<emph.end type="italics"/></cell><cell>13</cell></row><row><cell>XIIII.</cell><cell>P<emph type="italics"/>roportiones confu&longs;&aelig; &amp; coniunct&aelig; in tribus quantitatibus inuicem commutantur.<emph.end type="italics"/></cell><cell>13</cell></row><row><cell>XV.</cell><cell>S<emph type="italics"/>i fuerint quatuor quantitates proportio confu&longs;a, aggregati prim&aelig; &amp; terti&aelig;, ad aggre-gatum &longs;ecund&aelig; &amp; quart&aelig;, erit ut monadis addito prouentu, qui fit diui&longs;a differentia,   differentiarum prim&aelig; &amp; &longs;ecund&aelig;, atque quart&aelig; &amp; terti&aelig;, per aggregatum terti&aelig; &amp;   quart&aelig; ad ip&longs;am monadem.<emph.end type="italics"/></cell><cell>14</cell></row><row><cell>XVI.</cell><cell>O<emph type="italics"/>mnium quatuor quantitatum propo&longs;ita prima, qu&aelig; non minorem habet proportio-nem ad &longs;uam corre&longs;pondentem qu&agrave;m alia ad aliam, erit proportio confu&longs;a illarum,<emph.end type="italics"/></cell><cell></cell></row><pb/><row><cell></cell><cell><emph type="italics"/>ut producti ex aggregato prim&aelig; &amp; terti&aelig;, in tertiam ad productum ex iggre   gato terti&aelig; &amp; omiotat&aelig; ad &longs;ecundam in ip&longs;am quartam.<emph.end type="italics"/></cell><cell>14</cell></row><row><cell>XVII.</cell><cell>O<emph type="italics"/>mnes du&aelig; proportiones conuer&longs;&aelig; producunt &aelig;qualem proportionem.<emph.end type="italics"/></cell><cell>15</cell></row><row><cell>XVIII.</cell><cell>S<emph type="italics"/>i fuerint quotlibet quantitates in continua proportione multiplici pr&aelig;ter, <expan abbr="ultim&atilde;">ultimam</expan>   proportio uer&ograve; penultim&aelig; ad ultimam, qualis re&longs;idui prim&aelig; ad &longs;ecundam,   erit prim&aelig; ad aggregatum reliquarum, uelut penultim&aelig; ad ultimam.<emph.end type="italics"/></cell><cell>15</cell></row><row><cell>XIX.</cell><cell>S<emph type="italics"/>i fuerint aliquot quantitates arithmetic&aelig; omiolog&aelig;, quarum exce&longs;&longs;us &longs;it &aelig;qualis   minim&egrave;, omnibus autem deficientibus &longs;upplementa ad &aelig;qualitatem maxim&egrave;   adiungantur, erunt quadrata omnium quantitatum &aelig;qualium, adiecto rur&longs;us   quadrato prim&aelig; cum eo quod fit ex minima primi ordinis in aggregatum o-mnium quantitatum eiu&longs;dem, tripla aggregato quadratorum omnium quanti   tatum primi ordinis pariter acceptis.<emph.end type="italics"/></cell><cell>17</cell></row><row><cell>XX.</cell><cell>C<emph type="italics"/>um fuerint quatuor quantitates, fueritque <expan abbr="&longs;ec&utilde;da">&longs;ecunda</expan> &aelig;qualis terti&aelig;, aut prima &aelig;qualis   quart&aelig;, erit proportio prim&aelig; ad quartam, aut terti&aelig; ad &longs;ecundam, producta   ex proportionibus prim&aelig; ad &longs;ecundam &amp; terti&aelig; ad quartam.<emph.end type="italics"/></cell><cell>21</cell></row><row><cell>XXI.</cell><cell>C<emph type="italics"/>um decu&longs;&longs;atim ducta fuerit prima in quartam, &amp; &longs;ecunda in tertiam, produ-ctumque prim&aelig; in quartam, diui&longs;um fu<gap/>rit per productum &longs;ecund&aelig; in tertiam,   erit proportio prim&aelig; ad &longs;ecundam, diui&longs;a per proport&iacute;onem terti&aelig; ad quar-tam.<emph.end type="italics"/> E<emph type="italics"/>t &longs;imiliter interpo&longs;ita omiologa.<emph.end type="italics"/></cell><cell>22</cell></row><row><cell>XXII.</cell><cell>C<emph type="italics"/>um fuerit proportio prim&aelig; ad &longs;ecundam maior qu&agrave;m terti&aelig; ad quartam, erit   confu&longs;a ex his maior qu&agrave;m terti&aelig; ad quartam, minor autem qu&agrave;m prim&aelig; ad   &longs;ecundam.<emph.end type="italics"/></cell><cell>23</cell></row><row><cell>XXIII.</cell><cell>O<emph type="italics"/>mnis motus naturalis ad locum &longs;uum e&longs;t: ide&ograve; per rectam lineam fit.<emph.end type="italics"/></cell><cell>23</cell></row><row><cell>XXIIII.</cell><cell>O<emph type="italics"/>mnis motus circularis uoluntarius e&longs;t.<emph.end type="italics"/></cell><cell>23</cell></row><row><cell>XXV.</cell><cell>T<emph type="italics"/>res &longs;unt motus omnino &longs;implices naturalis, uoluntarius, &amp; uiolentus.<emph.end type="italics"/></cell><cell>24</cell></row><row><cell>XXVI.</cell><cell>M<emph type="italics"/>otus ergo compo&longs;iti quatuor nece&longs;&longs;ari&ograve; &longs;unt &longs;pecies.<emph.end type="italics"/></cell><cell>24</cell></row><row><cell>XXVII.</cell><cell>M<emph type="italics"/>otus uoluntarius e&longs;t in loco: naturalis ad locum: uiolentus ex loco.<emph.end type="italics"/></cell><cell>25</cell></row><row><cell>XXVIII.</cell><cell>M<emph type="italics"/>otus quilibet uoluntarius aut uiolentus in aliquo medio fit.<emph.end type="italics"/></cell><cell>25</cell></row><row><cell>XXIX.</cell><cell>O<emph type="italics"/>mnis motus uoluntarius &aelig;qualis e&longs;t &longs;emper: &longs;impliciter etiam quilibet alius mo-tus.<emph.end type="italics"/></cell><cell>25</cell></row><row><cell>XXX.</cell><cell>I<emph type="italics"/>n omni corpore mobili in medio partes medij re&longs;i&longs;tunt obui&aelig;, ali&aelig; impel-lunt.<emph.end type="italics"/></cell><cell>26</cell></row><row><cell>XXXI.</cell><cell>O<emph type="italics"/>mnis motus naturalis in &aelig;quali medio ualidior e&longs;t in fine qu&agrave;m in principio.<emph.end type="italics"/>V<emph type="italics"/>iolentus contr&agrave;.<emph.end type="italics"/></cell><cell>26</cell></row><row><cell>XXXII.</cell><cell>O<emph type="italics"/>mne mobile naturaliter motum &longs;eu uiolenter uelocius mouetur in medio rariore   qu&agrave;m den&longs;iore.<emph.end type="italics"/> M<emph type="italics"/>aior quoque e&longs;t proportio finis motus in corpore rariore ad   finem motus in corpore den&longs;iore qu&agrave;m principij.<emph.end type="italics"/> I<emph type="italics"/>n uiolento autem celerius   perueniret ad finem motus in corpore den&longs;iore.<emph.end type="italics"/></cell><cell>27</cell></row><row><cell>XXXIII.</cell><cell>O<emph type="italics"/>mnia duo mobilia &aelig;qualis undique magnitudinis qu&aelig; &aelig;quali in tempore &aelig;qualia   &longs;pacia pertran&longs;eunt in diuer&longs;is &longs;ub&longs;tantia medijs nece&longs;&longs;e e&longs;t, ut &longs;it ponderis ad   pondus, quem ad modum medij ad medium proportio duplicata.<emph.end type="italics"/></cell><cell>27</cell></row><row><cell>XXXIIII.</cell><cell>P<emph type="italics"/>roportio corporis cubi ad &longs;uam &longs;uperficiem quadratam, e&longs;t uelut eiu&longs;dem &longs;uperfi   ciei, ad latus eiu&longs;dem uer&ograve; ad monadem.<emph.end type="italics"/></cell><cell>28</cell></row><row><cell>XXXV.</cell><cell>V<emph type="italics"/>ocum magnitudines excre&longs;cunt in acumine, non in grauitate, finis autem e&longs;t in   utroque extremo.<emph.end type="italics"/> P<emph type="italics"/>ropter hoc minima facta uariatione in hypate acut&aelig; uix   ferunt.<emph.end type="italics"/></cell><cell>29</cell></row><row><cell>XXXVI.</cell><cell>S<emph type="italics"/>i proportio per proportionem minorem &aelig;quali ducatur, proportio minor pro-<emph.end type="italics"/></cell><cell></cell></row><pb/><row><cell></cell><cell><emph type="italics"/>ducetur.<emph.end type="italics"/> V<emph type="italics"/>nde manife&longs;tum e&longs;t duas proportiones minores &aelig;qualitate inuice<gap/> du   ctas proportionem minorem unaquaque illarum producere.<emph.end type="italics"/></cell><cell>30</cell></row><row><cell>XXXVII.</cell><cell>S<emph type="italics"/>i plures homines, quorum per &longs;e nauim mouere po&szlig;int, aut pondus ferre &longs;imul   iuncti eam moueant, aut pondus ferant, erunt ill&aelig; proportiones coniunct&aelig;   non product&aelig;.<emph.end type="italics"/></cell><cell>30</cell></row><row><cell>XXXVIII.</cell><cell>O<emph type="italics"/>mne corpus tantum re&longs;i&longs;tit motui contrario &longs;uo nat&uacute;rali, quantum mouetur oc-culto motu quie&longs;cendo.<emph.end type="italics"/></cell><cell>31</cell></row><row><cell>XXXIX.</cell><cell>A<emph type="italics"/>b &aelig;quali aut minore ui qu&agrave;m &longs;it impedimentum non fit motus.<emph.end type="italics"/></cell><cell>31</cell></row><row><cell>XL.</cell><cell>O<emph type="italics"/>mne corpus &longs;pb &aelig;ricum tangens planum in puncto mouetur ad latus per quam-cunque uim, qu&aelig; medium diuidere pote&longs;t.<emph.end type="italics"/></cell><cell>31</cell></row><row><cell>XLI.</cell><cell>S<emph type="italics"/>i fuerint du&aelig; quantitates &longs;umaturque toties <expan abbr="aggregat&utilde;">aggregatum</expan> maioris &amp; minoris, quo-ties aggregatum minoris &amp; maioris, erit proportio confu&longs;a maioris aggregati   ad minus, minor quam multiplicis maioris ad multiplex minoris.<emph.end type="italics"/></cell><cell>32</cell></row><row><cell>XLII.</cell><cell>T<emph type="italics"/>rahentium nauim, aut ferentium pondera proportiones in &longs;e inuicem, quomodo   ducere oporteat con&longs;iderare.<emph.end type="italics"/></cell><cell>32</cell></row><row><cell>XLIII.</cell><cell>P<emph type="italics"/>roductionem ad additionem retrabere.<emph.end type="italics"/></cell><cell>33</cell></row><row><cell>XLIIII.</cell><cell>S<emph type="italics"/>i fuerit proportio motoris ad id quod e&longs;t maximum non mouens, &amp; &longs;pacium &amp;   tempus, nota erit etiam reliquorum nota.<emph.end type="italics"/></cell><cell>33</cell></row><row><cell>XLV.</cell><cell>R<emph type="italics"/>ationem &longs;tater&aelig; o&longs;tendere.<emph.end type="italics"/></cell><cell>34</cell></row><row><cell>XLVI.</cell><cell>A<emph type="italics"/>n &longs;it aliqua proportio &amp; qualis inter animam &amp; uitas, &amp; &longs;ua corpora con&longs;ide-rare.<emph.end type="italics"/></cell><cell>35</cell></row><row><cell>XLVII.</cell><cell>S<emph type="italics"/>i duo mobilia &aelig;qualister in eodem circulo iuxta proprios motus moueantur, pro-ductum temporis circuituum inuicem, erit &aelig;quale producto differenti&aelig; tempo   rum circuitus duct&aelig; in tempus coniunctionis prim&aelig;.<emph.end type="italics"/></cell><cell>36</cell></row><row><cell>XLVIII.</cell><cell>S<emph type="italics"/>i tria mobilia ex eodem puncto di&longs;cedant, fuerintque duorum ac duorum coniun-ctiones in temporibus commen&longs;is, illa tria mobilia denuo coniungentur in tem   pore producto ex denominatore diui&longs;ionis temporis maioris per minus in mi-nus aut numeratore in maius.<emph.end type="italics"/></cell><cell>37</cell></row><row><cell>XLIX.</cell><cell>P<emph type="italics"/>ropofitio mobilis in circulo circuitus tempore dataque ratione di&longs;tanti&aelig; ab illo mo   bilis circuitum inuenire, quod ex <expan abbr="eod&etilde;">eodem</expan> puncto di&longs;cedens <expan abbr="c&utilde;alio">cunalio</expan> mobili in dato   puncto <expan abbr="c&otilde;ueniat">conueniat</expan> &longs;ub <expan abbr="quoc&utilde;que">quocunque</expan> numero <expan abbr="circuitu&utilde;">circuituum</expan> <expan abbr="t&etilde;pus">tempus</expan> quoque <expan abbr="c&otilde;iunctionis">coniunctionis</expan>.<emph.end type="italics"/></cell><cell>39</cell></row><row><cell>L.</cell><cell>O<emph type="italics"/>mnes circuituum portiones in ei&longs;dem temporibus repetuntur.<emph.end type="italics"/></cell><cell>40</cell></row><row><cell>LI.</cell><cell>O<emph type="italics"/>perationes dictas exemplo declarare.<emph.end type="italics"/></cell><cell>41</cell></row><row><cell>LII.</cell><cell>T<emph type="italics"/>ria mobilia coniuncta in <expan abbr="eod&etilde;">eodem</expan> puncto, quorum duo &amp; duo conueniant in partib.   incommen&longs;is inter &longs;e, in perpetuum in nullo unquam puncto conuenient.<emph.end type="italics"/></cell><cell>42</cell></row><row><cell>LIII.</cell><cell>C<emph type="italics"/>irculorum &longs;e in aduer&longs;um mouentium proportionem declarare.<emph.end type="italics"/></cell><cell>43</cell></row><row><cell>LIIII.</cell><cell>P<emph type="italics"/>roportio circuli ad &longs;uum diametrum per &longs;imilitudinem e&longs;t quarta pars periphe-ri&aelig;.<emph.end type="italics"/> R<emph type="italics"/>ur&longs;usque eiu&longs;dem circuli ad peripheriam diametri quarta pars.<emph.end type="italics"/></cell><cell>44</cell></row><row><cell>LV.</cell><cell>P<emph type="italics"/>roportionem medicamentorum per ordines &longs;up po&longs;ita &aelig;quali proportione in or-dinibus per quantitates &amp; proportiones demon&longs;trare.<emph.end type="italics"/></cell><cell>44</cell></row><row><cell>LVI.</cell><cell>P<emph type="italics"/>roportio cuiu&longs;uis binomij ad &longs;uum reci&longs;um, uel ei commen&longs;um e&longs;t duplicata ei   qu&aelig; ad numeri latus.<emph.end type="italics"/></cell><cell>49</cell></row><row><cell>LVII.</cell><cell>M<emph type="italics"/>otus rationem ad pondus inuenire.<emph.end type="italics"/></cell><cell>49</cell></row><row><cell>LVIII.</cell><cell>Q<emph type="italics"/>u&aelig; ex alto de&longs;cendunt, cur non eandem pro di&longs;tantia motus rationem in libero   a&euml;re &longs;eruent con&longs;iderare.<emph.end type="italics"/></cell><cell>49</cell></row><row><cell>LIX.</cell><cell>O<emph type="italics"/>mne mobile motum duobus motibus non ad idem tendentibus utroque &longs;eor&longs;um tar   dius mouetur &longs;imili motu.<emph.end type="italics"/></cell><cell>50</cell></row><row><cell>LX.</cell><cell>O<emph type="italics"/>mne mobile motu naturali de&longs;cendentis parte, de&longs;cendit grauiore &longs;ecundum gra-<emph.end type="italics"/></cell><cell></cell></row><pb/><row><cell></cell><cell><emph type="italics"/>uitatis centrum.<emph.end type="italics"/></cell><cell>51</cell></row><row><cell>LXI.</cell><cell>P<emph type="italics"/>roportionum ictus ad pondus rei &amp; di&longs;tantiam generaliter con&longs;iderare.<emph.end type="italics"/></cell><cell>52</cell></row><row><cell>LXII.</cell><cell>P<emph type="italics"/>roportionem motoris in plano ad motorem, qui eleuat pondus iuxta id quod   mouet, inuenire.<emph.end type="italics"/></cell><cell>53</cell></row><row><cell>LXIII.</cell><cell>O<emph type="italics"/>mne graue quanto proximius alligatum plano, tant&ograve; facilius trabitur.<emph.end type="italics"/></cell><cell>53</cell></row><row><cell>LXIIII.</cell><cell>O<emph type="italics"/>mne mobile quant&ograve; latius tanto tardius moustur in plano.<emph.end type="italics"/></cell><cell>54</cell></row><row><cell>LXV.</cell><cell>P<emph type="italics"/>roportionem duorum mobilium inter &longs;e cum auxilio medij inuenire.<emph.end type="italics"/></cell><cell>54</cell></row><row><cell>LXVI.</cell><cell>P<emph type="italics"/>roportionem laterum eptagoni, &amp; &longs;ubten&longs;arum con&longs;iderare, &amp; qu&aelig; &agrave; reflexa   proportione pendent.<emph.end type="italics"/></cell><cell>55</cell></row><row><cell>LXVII.</cell><cell>S<emph type="italics"/>i fuerint aliquot quantitates ab una quantitate ali&aelig;que totidem ab eadem analo-g&aelig;, erit proportio terti&aelig; unius ordinis ad tertiam alterius, ut &longs;ecund&aelig; ad &longs;e-cundum duplicata, &amp; quart&aelig; ad quartam triplicata, quint&aelig; ad quintam   quadruplicata, atque &longs;ic de alijs.<emph.end type="italics"/></cell><cell>57</cell></row><row><cell>LXVIII.</cell><cell>P<emph type="italics"/>ropo&longs;itio collectorum ab<emph.end type="italics"/> E<emph type="italics"/>uclide &amp;<emph.end type="italics"/> A<emph type="italics"/>rchimede.<emph.end type="italics"/></cell><cell>57</cell></row><row><cell>LXIX.</cell><cell>P<emph type="italics"/>ropo&longs;itio collectorum ex quatuor libris<emph.end type="italics"/> A<emph type="italics"/>pollonij<emph.end type="italics"/> P<emph type="italics"/>ergei &amp;<emph.end type="italics"/> <expan abbr="q.">que</expan> S<emph type="italics"/>ereni.<emph.end type="italics"/></cell><cell>59</cell></row><row><cell>LXX.</cell><cell>S<emph type="italics"/>i fuerint tres quantitates <gap/> ontinua proportione, ali&aelig;que totidem in continua   proportione poterunt con&longs;tituere tres quantitates in &aelig;quali differentia per-uer&longs;im copulat&aelig;.<emph.end type="italics"/></cell><cell>62</cell></row><row><cell>LXXI.</cell><cell>P<emph type="italics"/>roportionem leuitatis ponderis per uirgam torcularem attracti ad rectam &longs;u-&longs;pen&longs;ionem inuenire.<emph.end type="italics"/></cell><cell>63</cell></row><row><cell>LXXII.</cell><cell>P<emph type="italics"/>roportionem ponderis &longs;ph&aelig;r&aelig; pendentis ad a&longs;cendentem per accliue planum   inuenire.<emph.end type="italics"/></cell><cell>63</cell></row><row><cell>LXXIII.</cell><cell>P<emph type="italics"/>roportionem ponderum attractorum penes figuram in plano inuenire.<emph.end type="italics"/></cell><cell>64</cell></row><row><cell>LXXIIII.</cell><cell>P<emph type="italics"/>roportionem concutientis ad concu&longs;&longs;um in&longs;tabili inuenire.<emph.end type="italics"/></cell><cell>64</cell></row><row><cell>LXXV.</cell><cell>P<emph type="italics"/><expan abbr="roportion&etilde;">roportionem</expan> immoti in aqua, ad <expan abbr="immot&utilde;">immotum</expan> in terra in excipiendo <expan abbr="ict&utilde;">ictum</expan> inuenire.<emph.end type="italics"/></cell><cell>65</cell></row><row><cell>LXXVI.</cell><cell>P<emph type="italics"/>roportionem <expan abbr="duor&utilde;">duorum</expan> mobilium &longs;ibi <expan abbr="inuic&etilde;">inuicem</expan> <expan abbr="concurrenti&utilde;">concurrentium</expan> per <expan abbr="rect&atilde;">rectam</expan> inuenire.<emph.end type="italics"/></cell><cell>66</cell></row><row><cell>LXXVII.</cell><cell>P<emph type="italics"/>roportionem motus obliqui ad motum rectum in nauibus inuenire.<emph.end type="italics"/></cell><cell>66</cell></row><row><cell>LXXVIII.</cell><cell>P<emph type="italics"/>roportionem nauis ad triremes quotuis concurrentes demon&longs;trare.<emph.end type="italics"/></cell><cell>67</cell></row><row><cell>LXXIX.</cell><cell>P<emph type="italics"/>roportionem medicamentorum purgantium inuicem declarare<emph.end type="italics"/></cell><cell>68</cell></row><row><cell>LXXX.</cell><cell>P<emph type="italics"/>roportionem motus &longs;ecundum obliquum ad rectum in &longs;pacio declarare.<emph.end type="italics"/></cell><cell>69</cell></row><row><cell>LXXXI.</cell><cell>Q<emph type="italics"/>uualis &longs;it angulus, per quem pote&longs;t moueri nauis ad rectum explorare.<emph.end type="italics"/></cell><cell>70</cell></row><row><cell>LXXXII.</cell><cell>P<emph type="italics"/>roportionem uelorum indagare.<emph.end type="italics"/></cell><cell>70</cell></row><row><cell>LXXXIII.</cell><cell>P<emph type="italics"/>roportionem rece&longs;&longs;us &agrave; recta uia ad obliquitatem inue&longs;tigare.<emph.end type="italics"/></cell><cell>72</cell></row><row><cell>LXXXIIII.</cell><cell>D<emph type="italics"/><expan abbr="i&longs;tanti&atilde;">i&longs;tantiam</expan> centri terr&aelig; &agrave; centro mundi per motum lapidis<emph.end type="italics"/> H<emph type="italics"/>erculei declarare.<emph.end type="italics"/></cell><cell>73</cell></row><row><cell>LXXXV.</cell><cell>P<emph type="italics"/>roportio ponderis unius grauis ad aliud &longs;ub eadem men&longs;ura e&longs;t ueluti eiu&longs;dem   ad differentiam ponderis ua&longs;is repleti ex altero graui, &amp; ex ambobus de-tracto priore.<emph.end type="italics"/></cell><cell>74</cell></row><row><cell>LXXXVI.</cell><cell>S<emph type="italics"/>i circuli in &aelig; quales &longs;eu in &longs;ph&aelig;ra &longs;eu in plano &longs;e &longs;ecuerint, nunqu&agrave;m oppo&longs;itos   angulos &aelig;quales habent.<emph.end type="italics"/></cell><cell>77</cell></row><row><cell>LXXXVII.</cell><cell>P<emph type="italics"/>roportiones cra&szlig;itiei aqu&aelig; ad <expan abbr="a&etilde;r&etilde;">aerrem</expan> in <expan abbr="c&otilde;paratione">comparatione</expan> ad radios demon&longs;trare.<emph.end type="italics"/></cell><cell>78</cell></row><row><cell>LXXXVIII.</cell><cell>I<emph type="italics"/><expan abbr="n&longs;trument&utilde;">n&longs;trumentum</expan><emph.end type="italics"/> A<emph type="italics"/>colingen, quo momenta temporum <expan abbr="deprehend&atilde;tur">deprehendantur</expan> fabricare.<emph.end type="italics"/></cell><cell>79</cell></row><row><cell>LXXXIX.</cell><cell>P<emph type="italics"/>roportionem den&longs;itatis aqu&aelig; ad a&euml;rem per pondera inuenire.<emph.end type="italics"/></cell><cell>82</cell></row><row><cell>XC.</cell><cell>R<emph type="italics"/>ationem impetus uiolenti extra mi&szlig;i ponderis ad &aelig;qualitatem reducere.<emph.end type="italics"/></cell><cell>82</cell></row><row><cell>XCI.</cell><cell>P<emph type="italics"/>roportionem grauis cubi, &amp; &longs;ph&aelig;rici &aelig;qualium in accliui, &amp; de&longs;cen&longs;us eorum   demon&longs;trare.<emph.end type="italics"/></cell><cell>83</cell></row><row><cell>XCII.</cell><cell>P<emph type="italics"/><expan abbr="roportion&etilde;">roportionem</expan> ponderis &aelig;qualis iuxta longitudinis <expan abbr="c&otilde;paration&etilde;">comparationem</expan> demon&longs;trare.<emph.end type="italics"/></cell><cell>85</cell></row><row><cell>XCIII.</cell><cell>P<emph type="italics"/>ropter qd in <expan abbr="c&otilde;cu&szlig;ione">concu&szlig;ione</expan> <expan abbr="eti&atilde;">etiam</expan> leui nauis loco moueatar <expan abbr="o&longs;t&etilde;dere">o&longs;tendere</expan>.<emph.end type="italics"/> V<emph type="italics"/>nde manifi <expan abbr="&longs;i&utilde;">&longs;ium</expan>   e&longs;t duas naues &longs;ibi <expan abbr="inuic&etilde;">inuicem</expan> occur&longs;antes retrocedere, &amp; <expan abbr="qu&atilde;t&utilde;">quantum</expan> <expan abbr="retroced&atilde;t">retrocedant</expan> amb&aelig;.<emph.end type="italics"/></cell><cell>86</cell></row><pb/><row><cell>XCIIII.</cell><cell>S<emph type="italics"/>i <expan abbr="qu&atilde;titas">quantitas</expan> aliqua nota atque proportio erit producta, <expan abbr="qu&atilde;titas">quantitas</expan> nota &longs;imiliter.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i du&aelig;   proportiones not&aelig; fuerint, erit producta ex his atque diui&longs;a coniunctaque atque detra-cta nota.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i fuerit totius ad partem proportio nota, erit et ad aliam partem nota:   &amp; alterius partis ad <expan abbr="alter&atilde;">alteram</expan> uno minor.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i fuerit partis ad partem, erit ad to<gap/>um   monade minor atque nota.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i fuerit unius <expan abbr="qu&atilde;titatis">quantitatis</expan> ad duas <expan abbr="qu&atilde;titates">quantitates</expan> proportio   nota, erit &amp; <expan abbr="c&otilde;fu&longs;a">confu&longs;a</expan> ex eis nota.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i fuerint trium quantitatum omiologarum, aut   quatuor analogarum omnes pr&aelig;ter unam cognit&aelig;, erunt &verbar; &amp; illa alia cognita.<emph.end type="italics"/></cell><cell>87</cell></row><row><cell>XCV.</cell><cell>C<emph type="italics"/>uiu&longs;uis trigoni rectanguli, aut cuius duo auguli &longs;int in dupla proportione, aut qui   circulo in&longs;criptus &longs;it cognita quantitate unius lateris in comparatione ad dimetien   <expan abbr="t&etilde;">tem</expan>, &longs;i proportio duorum laterum cognita fuerit, <expan abbr="er&utilde;t">erunt</expan> omnia eius latera cognita.<emph.end type="italics"/></cell><cell>88</cell></row><row><cell>XCVI.</cell><cell>C<emph type="italics"/>um in <expan abbr="per&longs;picu&utilde;">per&longs;picuum</expan> den&longs;um radij lumino&longs;i inciderint, quatuor fiunt luminis genera.<emph.end type="italics"/></cell><cell>89</cell></row><row><cell>XCVII.</cell><cell>M<emph type="italics"/><expan abbr="ot&utilde;">otum</expan> inuer&longs;ionis in figuris in <expan abbr="c&otilde;paratione">comparatione</expan> ad <expan abbr="mot&utilde;">motum</expan> &longs;ph&aelig;r&aelig; in plano inue&longs;tigare.<emph.end type="italics"/></cell><cell>91</cell></row><row><cell>XCVIII.</cell><cell>P<emph type="italics"/>roportionem ponderum &aelig;qualium per differentiam angulorum inuenire.<emph.end type="italics"/></cell><cell>92</cell></row><row><cell>XCIX.</cell><cell>P<emph type="italics"/>roportionem grauitatum per multitudin<gap/> &longs;uppo&longs;itorum orbium o&longs;tendere.<emph.end type="italics"/></cell><cell>93</cell></row><row><cell>C.</cell><cell>P<emph type="italics"/><expan abbr="roportion&etilde;">roportionem</expan> grauitatis <expan abbr="ponder&utilde;">ponderum</expan> attractorum per <expan abbr="trochlear&utilde;">trochlearum</expan> <expan abbr="numer&utilde;">numerum</expan> inue&longs;tigare.<emph.end type="italics"/></cell><cell>93</cell></row><row><cell>CI.</cell><cell>P<emph type="italics"/>roportionem precij gemmarum ex tribus in eodem genere cognitis inuenire.<emph.end type="italics"/></cell><cell>94</cell></row><row><cell>CII.</cell><cell>P<emph type="italics"/>roportionem motuum inuer&longs;ionis, &amp; attractionis in plano inuenire.<emph.end type="italics"/></cell><cell>95</cell></row><row><cell>CIII.</cell><cell>P<emph type="italics"/>roportionem eorundem in accliui demon&longs;trare.<emph.end type="italics"/></cell><cell>95</cell></row><row><cell>CIIII.</cell><cell>P<emph type="italics"/>roportionem motus attractionis in decliui ad motum in plano determinare.<emph.end type="italics"/></cell><cell>95</cell></row><row><cell>CV.</cell><cell>P<emph type="italics"/>roportionem ferentium pondus in pertica inuenire.<emph.end type="italics"/></cell><cell>96</cell></row><row><cell>CVI.</cell><cell>Q<emph type="italics"/>uales proportiones angulorum doceant laterum proportiones.<emph.end type="italics"/> A<emph type="italics"/>tque uici&szlig;im deter-minare.<emph.end type="italics"/></cell><cell>97</cell></row><row><cell>CVII.</cell><cell>S<emph type="italics"/>i in circulo du&aelig; diametri ad rectum angulum &longs;e &longs;ecauerint: ali&aelig; uer&ograve; ad perpendicu-lum ex diametro exicrint ad circum ferentiam, &longs;ingul&aelig; &longs;upra diametrum erunt ma   iores portionibus reliquis diametri &longs;uperioribus, infra autem minores.<emph.end type="italics"/> D<emph type="italics"/>imidium   autem portionis &longs;uperioris re&longs;iduum ad centrum maius &longs;agitta habebit.<emph.end type="italics"/> I<emph type="italics"/>n aliqua   pr&aelig;terea portionis &longs;uperioris parte, qu&aelig; uer&longs;us diametrum tran&longs;uer&longs;um po&longs;ita   e&longs;t, maior e&longs;t differentia partis diametri ei <expan abbr="corre&longs;p&otilde;dentis">corre&longs;pondentis</expan>, <expan abbr="&qtilde;">quae</expan> line &aelig; tran&longs;uer&longs;&aelig;.<emph.end type="italics"/></cell><cell>100</cell></row><row><cell>CVIII.</cell><cell>P<emph type="italics"/>unctum &aelig;qualitatis differenti&aelig; de&longs;cen&longs;us &amp; remotionis &agrave; centro inuenire.<emph.end type="italics"/></cell><cell>100</cell></row><row><cell>CIX.</cell><cell>R<emph type="italics"/>ationem libr&aelig; expendere.<emph.end type="italics"/></cell><cell>101</cell></row><row><cell>CX.</cell><cell>S<emph type="italics"/>i du&aelig; &longs;ph&aelig;r&aelig; ex eadem materia de&longs;cendant in a&euml;re, eodem temporis momento ad   planum ueniunt.<emph.end type="italics"/></cell><cell>104</cell></row><row><cell>CXI.</cell><cell>C<emph type="italics"/>ur ex medio tela ualidiorem ictum, &amp; naues in &longs;calmo &agrave; remo ac malo recipiant in-de ex puppi explorare.<emph.end type="italics"/></cell><cell>105</cell></row><row><cell>CXII.</cell><cell>C<emph type="italics"/>ur ex imo leuia longi&ugrave;s ferantur declarare,<emph.end type="italics"/></cell><cell>106</cell></row><row><cell>CXIII.</cell><cell>C<emph type="italics"/>ur uirga longius mittatur &agrave; puero quam &agrave; uiro inueftigare.<emph.end type="italics"/></cell><cell>107</cell></row><row><cell>CXIIII.</cell><cell>C<emph type="italics"/>ircularis motus differentias quatuor e&longs;&longs;e, earumque rationem contemplari.<emph.end type="italics"/></cell><cell>108</cell></row><row><cell>CXV.</cell><cell>P<emph type="italics"/>roportionem motuum impul&longs;ionis, &amp; attractionis inter &longs;e, ab eadem ui decla-rare.<emph.end type="italics"/></cell><cell>110</cell></row><row><cell>CXVI.</cell><cell>C<emph type="italics"/>ur machin&aelig; oblong&aelig; igne&aelig; longius emittant &longs;ph&aelig;ram explorare.<emph.end type="italics"/></cell><cell>111</cell></row><row><cell>CXVII.</cell><cell>I<emph type="italics"/>n curriculis maior e&longs;t uis pulueris copio&longs;ioris ampliore in &longs;pacio, qu&agrave;m paucioris in   minore iuxta proportionem eandem.<emph.end type="italics"/></cell><cell>112</cell></row><row><cell>CXVIII.</cell><cell>Q<emph type="italics"/>uanta proportione decedat ictus in obliquum parietem ab eo qui e&longs;t ad perpendi-culum declarare.<emph.end type="italics"/></cell><cell>114</cell></row><row><cell>CXIX.</cell><cell>Q<emph type="italics"/>uantum ictus machin&aelig; procliuis ad angulum minuatur explorare.<emph.end type="italics"/></cell><cell>115</cell></row><row><cell>CXX</cell><cell>P<emph type="italics"/>roportionem partium nauis ad eundem obliquum uentum explorare.<emph.end type="italics"/></cell><cell>118</cell></row><row><cell>CXXI.</cell><cell>F<emph type="italics"/>labelli uires atque naturam declarare.<emph.end type="italics"/></cell><cell>219</cell></row><row><cell>CXXII.</cell><cell>C<emph type="italics"/>ontemptus circa<emph.end type="italics"/> S<emph type="italics"/>olis rationem in umbris declarare.<emph.end type="italics"/></cell><cell>120</cell></row><pb/><row><cell>CXXIII.</cell><cell>C<emph type="italics"/>ognita ratione umbr&aelig; ad gnomonem &longs;inum, &amp; arcum altitudinis ab horizon-te, quouis tempore digno&longs;cere.<emph.end type="italics"/></cell><cell>121</cell></row><row><cell>CXXIIII.</cell><cell>P<emph type="italics"/>roportionem umbr&aelig; uer&longs;&aelig; e&longs;&longs;e ad gnomonem, uelut gnomonis ad umbram   uer&longs;am.<emph.end type="italics"/></cell><cell>122</cell></row><row><cell>CXXV.</cell><cell>P<emph type="italics"/>roportionem dimetientis, &amp; peripheri&aelig; cuiuslibet circuli paralleli &aelig;quino-ctiali per cognitam partem magni circuli demon&longs;trare.<emph.end type="italics"/></cell><cell>123</cell></row><row><cell>CXXVI.</cell><cell>C<emph type="italics"/>irculi horarij naturam declarare.<emph.end type="italics"/></cell><cell>123</cell></row><row><cell>CXXVII.</cell><cell>D<emph type="italics"/>ata poli altitudine ortus amplitudinem demonftrare.<emph.end type="italics"/></cell><cell>124</cell></row><row><cell>CXXVIII.</cell><cell>N<emph type="italics"/>ota amplitudine ortus, cuiu&longs;que puncti arcum &longs;emidiurnum inuenire.<emph.end type="italics"/></cell><cell>124</cell></row><row><cell>CXXIX.</cell><cell>D<emph type="italics"/>ata altitudine<emph.end type="italics"/> S<emph type="italics"/>olis in quacunque regione, quacunque die di&longs;tantiam<emph.end type="italics"/> S<emph type="italics"/>olis &agrave; meri-diano cogno&longs;cere.<emph.end type="italics"/></cell><cell>124</cell></row><row><cell>CXXX.</cell><cell>D<emph type="italics"/>ata regionis altitudine, &amp; loco<emph.end type="italics"/> S<emph type="italics"/>olis proportionem gnomonis, tam ad um-bram rectam qu&agrave;m uer&longs;am, uel etiam in cylindro determinare.<emph.end type="italics"/></cell><cell>125</cell></row><row><cell>CXXXI.</cell><cell>S<emph type="italics"/>i line&aelig; alicui duplum alterius adiungatur, erit proportio d<gap/>arum ad primam   maior qu&agrave;m dupli cum prima ad primam cum una adiecta.<emph.end type="italics"/></cell><cell>126</cell></row><row><cell>CXXXII.</cell><cell>S<emph type="italics"/>i ad duas lineas quarum una alteri dupla &longs;it eadem linea addatur, erit aggrega-ti ex minore, &amp; adiecta ad ip&longs;am minorem, minor proportio qu&agrave;m aggre-gati ex maiore, &amp; adiecta ad ip&longs;am maiorem duplicata.<emph.end type="italics"/></cell><cell>126</cell></row><row><cell>CXXXIII.</cell><cell>S<emph type="italics"/>i fuerint du&aelig; quantitates, <expan abbr="quar&utilde;">quarum</expan> una alteri dupla &longs;it: minuatur &agrave; minore qu&aelig;-dam quantitas, <expan abbr="ead&etilde;que">eadenque</expan> maiori addatur, erit minoris ad re&longs;iduum maior pro-portio, qu&agrave;m aggregati ad maiorem duplicata.<emph.end type="italics"/> S<emph type="italics"/>i uer&ograve; minori addatur, &amp;   &agrave; maiore detrabatur, erit aggregati ad minorem minor proportio qu&agrave;m   maioris ad re&longs;iduum duplicata.<emph.end type="italics"/></cell><cell>127</cell></row><row><cell>CXXXIIII.</cell><cell>S<emph type="italics"/>i rectangula &longs;uperficies &longs;it, cuius pars tertia quadrata &longs;it corpus, quod ex la-tere quadrat&aelig; in re&longs;iduum &longs;uperficiei con&longs;tat, maius e&longs;t quouis corpore ex   eadem &longs;uperficies, aliter diui&longs;a con&longs;tituto.<emph.end type="italics"/></cell><cell>127</cell></row><row><cell>CXXXV.</cell><cell>S<emph type="italics"/>i linea in duas partes, quarum una fit alteri dupla diuidatur, erit quod fit ex   tertia parte in quadratum re&longs;idui parallelipedum maius omni pararalleli-pedo, quod ex diui&longs;ione eiu&longs;dem line&aelig; creari po&szlig;it.<emph.end type="italics"/></cell><cell>128</cell></row><row><cell>CXXXVI.</cell><cell>D<emph type="italics"/>enominationes in infinitum extendere.<emph.end type="italics"/></cell><cell>129</cell></row><row><cell>CXXXVII.</cell><cell>R<emph type="italics"/>ationem numerorum ex progre&szlig;ione declarare.<emph.end type="italics"/></cell><cell>131</cell></row><row><cell>CXXXVIII.</cell><cell>M<emph type="italics"/>odos u&longs;us horum numerorum declarare.<emph.end type="italics"/></cell><cell>131</cell></row><row><cell>CXXXIX.</cell><cell>R<emph type="italics"/>adices omnes &agrave; propo&longs;itis numeris extrahere.<emph.end type="italics"/></cell><cell>132</cell></row><row><cell>CXL.</cell><cell>R<emph type="italics"/>adices per numeros fractos determinare.<emph.end type="italics"/></cell><cell>133</cell></row><row><cell>CXLI.</cell><cell>N<emph type="italics"/>umeros fractos ad minores in ea <expan abbr="i&etilde;">iem</expan> proportione ualde propinqud deducere<emph.end type="italics"/></cell><cell>136</cell></row><row><cell>CXLII.</cell><cell>D<emph type="italics"/><expan abbr="enomination&utilde;">enominationum</expan> in <expan abbr="crem&etilde;ta">crementa</expan> ex extrema cognita inuenire.<emph.end type="italics"/> E<emph type="italics"/>t <expan abbr="c&otilde;uer&longs;o">conuer&longs;o</expan> modo.<emph.end type="italics"/></cell><cell>137</cell></row><row><cell>CXLIII.</cell><cell>S<emph type="italics"/>i linea in duas partes diuidatur, corpora qu&aelig; fiunt ex una parte in alterius   quadratum mutuo &aelig;qualia &longs;unt corpori, quod fit ex tota linea in &longs;uperfi-ciem unius partis in alteram.<emph.end type="italics"/></cell><cell>138</cell></row><row><cell>CXLIIII.</cell><cell>D<emph type="italics"/>uplum cubi medietatis maius e&longs;t aggregato corporum mutuorum, cuiuslibet   diui&longs;ionis quantum e&longs;t, quod fit ex tota in quadratum differenti&aelig;.<emph.end type="italics"/></cell><cell>139</cell></row><row><cell>CXLV.</cell><cell>S<emph type="italics"/>i linea in duas partes diuidatur quadrata ambarum partium detracto eo, quod   fit ex una parte in alteram, &aelig;qualia &longs;unt producto unius in alteram cum   quadrato differenti&aelig;.<emph.end type="italics"/></cell><cell>139</cell></row><row><cell>CXLVI.</cell><cell>C<emph type="italics"/>orpus quod fit ex linea diui&longs;a in &longs;uperficiem &aelig;qualem quadratis ambarum par   tium detracta &longs;uperficie unius partis in alteram, e&longs;t &aelig;quale aggregato cubo-rum ambarum partium.<emph.end type="italics"/></cell><cell>139</cell></row><row><cell>CXLVII.</cell><cell>P<emph type="italics"/>ropo&longs;ita linea diui&longs;a duas ei line as adijcere, ut proportio <expan abbr="additar&utilde;">additarum</expan> &longs;ingularium<emph.end type="italics"/></cell><cell></cell></row><pb/><row><cell></cell><cell><emph type="italics"/>&amp; partium &longs;imul iunctarum ad additas &longs;it mutua.<emph.end type="italics"/></cell><cell>148</cell></row><row><cell>CXLVIII.</cell><cell>P<emph type="italics"/>ropo&longs;itis tribus lineis primam &longs;ic diuidere, ut adiectis duabus alijs lineis, &longs;ecun-dum <expan abbr="ration&etilde;">rationem</expan> mutuam &longs;ingularum &longs;ingulis, <expan abbr="aggregat&utilde;">aggregatum</expan> ex una <expan abbr="adiectar&utilde;">adiectarum</expan>, &amp; par   te ad <expan abbr="aggregat&utilde;">aggregatum</expan> ex alia parte, &amp; adiecta &longs;e habeat, ut &longs;ecunda ad <expan abbr="terti&atilde;">tertiam</expan>.<emph.end type="italics"/></cell><cell>140</cell></row><row><cell>CXLIX.</cell><cell>D<emph type="italics"/>atam lineam &longs;ic diuidere, ut proportio quadratorum ad dupium unius partis in   alteram &longs;it, ut line&aelig; dat&aelig; ad lineam datam.<emph.end type="italics"/></cell><cell>141</cell></row><row><cell>CL.</cell><cell>P<emph type="italics"/>ropo&longs;itis duabus lineis, lineam communem utrique adiungere, ut &longs;it maioris ad ad-ditam proportio, uelut quadratorum minoris, &amp; adiect&aelig; ad duplum unius in   alteram.<emph.end type="italics"/></cell><cell>141</cell></row><row><cell>CLI.</cell><cell>P<emph type="italics"/>roportio differenti&aelig; quadratorum partium cuiu&longs;uis line&aelig;, ad quadratum diffe-renti&aelig; illarum e&longs;t, uelut totius line&aelig; ad differentiam.<emph.end type="italics"/></cell><cell>142</cell></row><row><cell>CLII.</cell><cell>S<emph type="italics"/>i linea in duas partes &aelig;quales, duasque in&aelig;quales diuidatur, fueritque proportio ag-gregati ex maiore, &amp; dimidio ad ip&longs;am maiorem, uelut ex minore, &amp; ali-qua linea ad ip&longs;am minorem, &amp; rur&longs;us aggregati ex minore, &amp; dimidio ad   ip&longs;am minorem, uelut aggregati ex maiore, &amp; alia addita ad ip&longs;am maiorem,   erit proportio dimidij ad partem unam in&aelig;qualem, uelut alterius partis in&aelig;-qualis ad &longs;uam additam mutu&ograve;, &amp; etiam proportio additarum inuicem, uelut   proportio <expan abbr="parti&utilde;">partium</expan> <expan abbr="in&aelig;quali&utilde;">in&aelig;qualium</expan> duplicata, &amp; rur&longs;us ip&longs;um <expan abbr="dimidi&utilde;">dimidium</expan> line&aelig; a&longs;&longs;um-pt&aelig; <expan abbr="medi&utilde;">medium</expan>, erit proportione inter additas.<emph.end type="italics"/> D<emph type="italics"/><expan abbr="em&utilde;">emum</expan> proportio dimidij <expan abbr="c&utilde;">cum</expan> addita   maiore ad <expan abbr="dimidi&utilde;">dimidium</expan>, cum addita minore, uelut maioris partis ad <expan abbr="minor&etilde;">minorem</expan>.<emph.end type="italics"/></cell><cell>142</cell></row><row><cell>CLIII.</cell><cell>V<emph type="italics"/>im quamcunque manus multiplicare.<emph.end type="italics"/></cell><cell>144</cell></row><row><cell>CLIIII.</cell><cell>S<emph type="italics"/>i line&aelig; dat&aelig; alia linea adiungatur, ab extremitatibus autem prioris line&aelig; du&aelig;   rect&aelig; in unum punctum concurrant proportionem habentes, quam mediam   inter tota m &amp; adiectam, &amp; adiectam erit punctus, concur&longs;us &agrave; puncto extre-mo line&aelig; adiect&aelig; di&longs;tans per lineam mediam.<emph.end type="italics"/> Q<emph type="italics"/>uod &longs;i ab extremo alicuius li-ne&aelig; &aelig;qua'is medi&aelig;, &longs;eu peripheria circuli, cuius &longs;emidiameter &longs;it media linea   du&aelig; line&aelig; ad pr&aelig;dicta puncta producantur, ip&longs;&aelig; erunt in proportione medi&aelig;   ad adiectam.<emph.end type="italics"/></cell><cell>145</cell></row><row><cell>CLV.</cell><cell>Q<emph type="italics"/>uadr atorum numerum proportionem &amp; inuentionem con&longs;iderare.<emph.end type="italics"/></cell><cell>147</cell></row><row><cell>CLVI.</cell><cell>H<emph type="italics"/>orologiorum tempus multiplicare.<emph.end type="italics"/></cell><cell>152</cell></row><row><cell>CLVII.</cell><cell>H<emph type="italics"/>orologiorum molarium rationem o&longs;tendere.<emph.end type="italics"/></cell><cell>154</cell></row><row><cell>CLVIII.</cell><cell>R<emph type="italics"/>ationem indicis mobilis cum rota, qua horarum numerus per ictus indicatur ex-plicare.<emph.end type="italics"/></cell><cell>156</cell></row><row><cell>CLIX.</cell><cell>N<emph type="italics"/>ullus angulus rectilineus &aelig;qualis e&longs;&longs;e pote&longs;t alicui angulo contento recta, &amp; cir   culi portione.<emph.end type="italics"/></cell><cell>158</cell></row><row><cell>CLX.</cell><cell>P<emph type="italics"/>ropo&longs;ita linea tribusque in ea &longs;ignis punctum inuenire, ex quo duct&aelig; tres line&aelig; ad   &longs;igna &longs;int in proportionibus datis.<emph.end type="italics"/></cell><cell>162</cell></row><row><cell>CLXI.</cell><cell>S<emph type="italics"/>i fuerint duo trianguli, quorum ba&longs;es in eadem linea &longs;int con&longs;tituti, &amp; &aelig;quales   ad unum punctum terminati, &amp; latus unum commune inter reliqua quantita-te medium nece&longs;&longs;e e&longs;t angulum &agrave; maioribus lineis <expan abbr="content&utilde;">contentum</expan> minorem e&longs;&longs;e.<emph.end type="italics"/></cell><cell>162</cell></row><row><cell>CLXII.</cell><cell>P<emph type="italics"/>roportionem duorum orbium, quorum diametrorum conuex&aelig; partis, &amp; conca-u&aelig; proportiones dat&aelig; &longs;int inue&longs;tigare.<emph.end type="italics"/></cell><cell>164</cell></row><row><cell>CLXIII.</cell><cell>P<emph type="italics"/>roportionem uirium &longs;tellarum per motus &longs;uos indagare.<emph.end type="italics"/></cell><cell>165</cell></row><row><cell>CLXIIII.</cell><cell>S<emph type="italics"/>yderum proportionem in magnitudine o&longs;tendere.<emph.end type="italics"/></cell><cell>166</cell></row><row><cell>CLXV.</cell><cell>P<emph type="italics"/>roportionem motuum omnium &longs;tellarum ad<emph.end type="italics"/> S<emph type="italics"/>olem con&longs;iderare.<emph.end type="italics"/></cell><cell>167</cell></row><row><cell>CLXVI.</cell><cell>P<emph type="italics"/>roportiones mu&longs;icas &longs;uperpartientes in eas, qu&aelig; particul&aacute; una tantum abundant   reducere.<emph.end type="italics"/></cell><cell>168</cell></row><pb/><row><cell>CLXVII.</cell><cell>P<emph type="italics"/>roportionem mu&longs;icam ad &longs;apores &amp; odores coaptare.<emph.end type="italics"/></cell><cell>176</cell></row><row><cell>CLXVIII.</cell><cell>P<emph type="italics"/>icturarum proportiones explicare.<emph.end type="italics"/></cell><cell>179</cell></row><row><cell>CLXIX.</cell><cell>P<emph type="italics"/>roportionem mu&longs;icam in in&longs;trumentis declarare iuxta compo&longs;itionis ra-tionem.<emph.end type="italics"/></cell><cell>182</cell></row><row><cell>CLXX.</cell><cell>C<emph type="italics"/>oniugationes cuiu&longs;uis numeri breuiter inuenire.<emph.end type="italics"/></cell><cell>185</cell></row><row><cell>CLXXI.</cell><cell>P<emph type="italics"/>ropo&longs;itis duobus quibuslibet numeris, quotuis alios &longs;eu in continuum &longs;eu   medios in continua proportione arithmetica, geometrica &amp; mu&longs;ica in-uenire.<emph.end type="italics"/></cell><cell>187</cell></row><row><cell>CLXXII.</cell><cell>P<emph type="italics"/>roportiones<emph.end type="italics"/> S<emph type="italics"/>tiphelij de&longs;cribere.<emph.end type="italics"/></cell><cell>191</cell></row><row><cell>CLXXIII.</cell><cell>C<emph type="italics"/>irculum &longs;uper centro &longs;uo mouere &aelig;qualiter, ita quod omnia illius puncta   per rectam lineam moueantur ultro citroque.<emph.end type="italics"/></cell><cell>192</cell></row><row><cell>CLXXIIII.</cell><cell>P<emph type="italics"/>rogre&longs;&longs;us &amp; regre&longs;&longs;us, tam &longs;ine latitudine qu&agrave;m cum latitudine in planetis   per &longs;olos concentricos circulos &aelig;qualiter motos demon&longs;trare.<emph.end type="italics"/></cell><cell>194</cell></row><row><cell>CLXXV.</cell><cell>C<emph type="italics"/>au&longs;am uarietatis diametrorum ex &longs;uppo&longs;itis concentricis demon&longs;tra-re.<emph.end type="italics"/></cell><cell>195</cell></row><row><cell>CLXXVI.</cell><cell>R<emph type="italics"/>ationem centri grauitatis declarare.<emph.end type="italics"/></cell><cell>197</cell></row><row><cell>CLXXVII.</cell><cell>S<emph type="italics"/>i proportio aliqua ex duabus proportionibus eiu&longs;dem quantitatis ad alias   duas componatur, erit proportio illarum duarum eadem proportioni   producti ex proportione in primam duarum quantitatum, detracta prio-re illa quantitate, qu&aelig; ad duas comparatur, ad eandem priorem quanti-tatem.<emph.end type="italics"/></cell><cell>198</cell></row><row><cell>CLXXVIII.</cell><cell>P<emph type="italics"/>roportionem mi&longs;tionis metallorum, maxim&egrave; auri &amp; argenti declara-re.<emph.end type="italics"/></cell><cell>199</cell></row><row><cell>CLXXIX.</cell><cell>S<emph type="italics"/>i duobus totis du&aelig; portiones &longs;imiles ab&longs;cindantur ab ei&longs;dem denu&ograve;, &amp; ab-&longs;ci&szlig;is portionibus partes e&aelig;dem auferantur, denuoque ac denu&ograve; quoties   libuerit &agrave; portionibus, &amp; &ugrave; re&longs;iduis ip&longs;arum quantitatum partes e&aelig;dem   auferantur, erit re&longs;idu&iacute; ad re&longs;iduum, ueluti totius ad totum.<emph.end type="italics"/></cell><cell>200</cell></row><row><cell>CLXXX.</cell><cell>S<emph type="italics"/>i aliqua quantitas in duas partes diuidatur, fueritque alicuius quantitatis ad   partes illas compo&longs;ita proportio, non poterit eiu&longs;dem quantitatis ad par-tes alias quantitatis diui&longs;a, aliter proportio eadem componi.<emph.end type="italics"/></cell><cell>202</cell></row><row><cell>CLXXXI.</cell><cell>C<emph type="italics"/>um fuerit aliqua proportio, compo&longs;ita ex proportionibus prim&aelig; ad &longs;ecun-dam &amp; tertiam, &amp; rur&longs;us quart&aelig; ad quintam &amp; &longs;extam: ita &longs;e habebit   proportio &longs;ecund&aelig; ad tertiam, ad proportionem quint&aelig; ad &longs;extam, uelut   producti ex proportione in &longs;ecundam detracta prima ad primam ad pro-ductum ex proportione in quintam, detracta quarta ad quartam.<emph.end type="italics"/></cell><cell>203</cell></row><row><cell>CLXXXII.</cell><cell>P<emph type="italics"/>ropo&longs;ita differentia proportionum partium &longs;imilium ad partes a&longs;&longs;umptas,   propo&longs;itaque proportione totius ad re&longs;idua eadem, differentiam propor-tionum totius ad reliquum re&longs;idui inuenire.<emph.end type="italics"/></cell><cell>203</cell></row><row><cell>CLXXXIII.</cell><cell>S<emph type="italics"/>pacium uit&aelig; naturalis per &longs;pacium uit&aelig; fortuitum declarare.<emph.end type="italics"/></cell><cell>204</cell></row><row><cell>CLXXXIIII.</cell><cell>Q<emph type="italics"/>u&aelig;cunque grauia in uorticibus aquarum, merguntur, in medio uorticis, pri-mum uer&longs;a mergantur.<emph.end type="italics"/></cell><cell>211</cell></row><row><cell>CLXXXV.</cell><cell>C<emph type="italics"/>ur homo &longs;edens quanto altius &longs;edet, &amp; quanto magis crura ad f&oelig;mora, &amp;   f&oelig;mora ad pectus reclinata habet, facilius con&longs;urgat, cum tamen h&aelig;c op-po&longs;ito modo inuicem &longs;e habeant, declarare.<emph.end type="italics"/></cell><cell>213</cell></row><row><cell>CLXXXVI.</cell><cell>S<emph type="italics"/>i fuerit proportio prim&aelig; &amp; &longs;ecund&aelig; quantitatis ad tertiam, ut prim&aelig; &amp;   quart&aelig; ad quintam, fueritque quarta &longs;ecunda maior, erit proportio quar-t&aelig; ad quintam maior qu&agrave;m &longs;ecund&aelig; ad tertiam.<emph.end type="italics"/> Q<emph type="italics"/>uod &longs;i fuerit maior<emph.end type="italics"/></cell><cell></cell></row><pb/><row><cell></cell><cell><emph type="italics"/>quart&aelig; ad quintam qu&agrave;m &longs;ecund&aelig; ad tertiam, nece&longs;&longs;e e&longs;t quartam &longs;ecunda e&longs;&longs;e   maiorem.<emph.end type="italics"/></cell><cell>214</cell></row><row><cell>CLXXXVII.</cell><cell>S<emph type="italics"/>i ei&longs;dem uiribus &amp; &lsquo;eadem&rsquo; proportione cum auxilio ponderis tertij quar-tum pondus moueatur quibus &longs;ecundum, auxilio primi nece&longs;&longs;e e&longs;t <expan abbr="quart&utilde;">quartum</expan> pon   dus tardius &amp; maiore cum difficultate moueri qu&agrave;m &longs;ecundum.<emph.end type="italics"/></cell><cell>214</cell></row><row><cell>CLXXXVIII.</cell><cell>S<emph type="italics"/>i uires aliqu&aelig; moueant cum ponderibus aliqua pondera, ut compo&longs;ita pro-portio &longs;it eadem proportioni uirium &amp; duorum ponderum mouentium ag-gregatum &aelig;quale duorum ponderum, ubi maior fuerit partium in &aelig;qual<gap/>as,   ibi erit maior difficultas.<emph.end type="italics"/></cell><cell>214</cell></row><row><cell>CLXXXIX.</cell><cell>S<emph type="italics"/>i pondus minus ad longitudinem minorem &longs;ub &aelig;quali proportione coapte-tar, facilius deor&longs;um trahetur qu&agrave;m quod maius e&longs;t &amp; propius.<emph.end type="italics"/></cell><cell>215</cell></row><row><cell>CXC.</cell><cell>S<emph type="italics"/>i fuerit primum graue minus &longs;ecundo, &amp; &longs;ecundum minus tertio, proportio   autem primi ad &longs;ecundum multo maior qu&agrave;m &longs;ecundi ad tertium, po&longs;ibile erit   propo&longs;itis uiribus ei&longs;dem addere pondus <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan>, ut ip&longs;um &amp; tertium mouea-tur facili&ugrave;s ab ei&longs;dem uiribus, &amp; primo uel &longs;ecundo qu&agrave;m antea.<emph.end type="italics"/></cell><cell>215</cell></row><row><cell>CXCL.</cell><cell>C<emph type="italics"/>um fuerint duo pondera &amp; uires, duxerisque aggregatum ex uiribus &amp; mi-nore pondere in maius, addiderisque in&longs;uper quantum e&longs;t productum dimidij ui   rium in &longs;e latus aggregati detracto dimidio uirium, dice<gap/> pondus auxiliare   &aelig;qualis proportionis.<emph.end type="italics"/></cell><cell>215</cell></row><row><cell>CXCII.</cell><cell>S<emph type="italics"/>i ex medio diametri linea ad perpendiculum erigatur ad circuli peripheri-am, ex eo puncto autem quotlibet line&aelig; ducantur &longs;eu intus ad circun ferentia<gap/>u&longs;que, &longs;eu extra ad diametrum, erit proportio totius line&aelig; ad totam uelut mu-tuo partis ad partem.<emph.end type="italics"/></cell><cell>217</cell></row><row><cell>CXCIII.</cell><cell>R<emph type="italics"/>ationem ponderis triplicem explicare.<emph.end type="italics"/></cell><cell>218</cell></row><row><cell>CXCIIII.</cell><cell>P<emph type="italics"/>roportionem ponderis longioris in medio &longs;u&longs;pen&longs;i, ad breuius illi &aelig;quale &amp; in   medio &longs;u&longs;pen&longs;um declarare.<emph.end type="italics"/></cell><cell>219</cell></row><row><cell>CXCV.</cell><cell>S<emph type="italics"/>i lectus fiat dupla longitudine ad latitudinem, melius &longs;uffulcietur re&longs;tibus   ex medio ad angulos &amp; eius &aelig;quidi&longs;tantibus qu&agrave;m &longs;ecundum longitudinem   &amp; latitudinem.<emph.end type="italics"/></cell><cell>220</cell></row><row><cell>CXCVI.</cell><cell>S<emph type="italics"/>i duo circuli &longs;uper eodem centro eodem motu trans feruntur, &aelig;quale &longs;pacium   &longs;uperant.<emph.end type="italics"/></cell><cell>221</cell></row><row><cell>CXCVII.</cell><cell>C<emph type="italics"/>ur lances ad locum &longs;uum &longs;u&longs;pen&longs;i redeant, impendentes <expan abbr="n&otilde;">non</expan>, <expan abbr="dem&otilde;&longs;trare">demon&longs;trare</expan>.<emph.end type="italics"/></cell><cell>224</cell></row><row><cell>CXCVIII.</cell><cell>C<emph type="italics"/>ur &longs;olidum quod cubus uocatur<emph.end type="italics"/> P<emph type="italics"/>yramide &longs;tabilius &longs;it o&longs;tendere.<emph.end type="italics"/></cell><cell>225</cell></row><row><cell>CXCIX.</cell><cell>R<emph type="italics"/>ationem remorum nauim impellentium inuenire.<emph.end type="italics"/></cell><cell>227</cell></row><row><cell>CC.</cell><cell>C<emph type="italics"/>ur temo cum paruus &longs;it, magnam nauim agere pote&longs;t, &amp; cur c&ugrave;m uarietas &longs;it   in prora, ip&longs;e con&longs;tituatur in puppi.<emph.end type="italics"/> E<emph type="italics"/>t cum transuer&longs;im ab aqua prematur   rect&agrave; nauim dirigat.<emph.end type="italics"/></cell><cell>228</cell></row><row><cell>CCI.</cell><cell>S<emph type="italics"/>i du&aelig; line&aelig; non &longs;ecantes circuli peripheriam in unum punctum ex ea coe-ant exterius, nece&longs;&longs;e e&longs;t illas peripheria contenta e&longs;&longs;e maiores.<emph.end type="italics"/></cell><cell>229</cell></row><row><cell>CCII.</cell><cell>R<emph type="italics"/>ationem &longs;trepitus o&longs;tendere.<emph.end type="italics"/></cell><cell>232</cell></row><row><cell>CCIII.</cell><cell>C<emph type="italics"/>ur &longs;cytalis onera portentur facili&ugrave;s, explorare.<emph.end type="italics"/></cell><cell>233</cell></row><row><cell>CCIIII.</cell><cell>C<emph type="italics"/>ur pluribus trochleis, pondera facilius eleuentur o&longs;tendere.<emph.end type="italics"/></cell><cell>233</cell></row><row><cell>CCV.</cell><cell>S<emph type="italics"/>uper uerbis<emph.end type="italics"/> P<emph type="italics"/>latonis de fine<emph.end type="italics"/> R<emph type="italics"/>eipublic&aelig;.<emph.end type="italics"/></cell><cell>234</cell></row><row><cell>CCVI.</cell><cell>R<emph type="italics"/>hombi pa&szlig;iones qua&longs;dam declarare.<emph.end type="italics"/></cell><cell>235</cell></row><row><cell>CCVII.</cell><cell>P<emph type="italics"/>roportionem agentium naturalium in tran&longs;mutatione con&longs;iderare.<emph.end type="italics"/></cell><cell>238</cell></row><row><cell>CCVIII.</cell><cell>M<emph type="italics"/>ota res &agrave; centro grauitatis per <expan abbr="prior&etilde;">priorem</expan> motum, in reditu uelocius mouetur   quam &longs;i quieuerit.<emph.end type="italics"/></cell><cell>238</cell></row><pb/><row><cell>CCIX.</cell><cell>S<emph type="italics"/>i &longs;uperficies rectangula in duas partes &aelig;quales diui&longs;a intelligatur, qu&aelig; am-b&aelig; quadrat&aelig; &longs;int, itemque in duas in&aelig;quales, erit parallelipedum ex latere   medi&aelig; partis in totam &longs;uperficiem maius aggregato parallelipedorum ex   partibus in&aelig;qualibus in latera alterius partis mutuo, in eo, quod fit ex dif   ferentia lateris minoris partis &agrave; medi&aelig; latere in differentiam maioris par-tis &longs;uperficiei &agrave; media &longs;uperficie bis, &amp; ex differentia amborum laterum   in&aelig;qualium iunctorum ad ambo latera, &aelig;qualia iuncta in minorem par-tem &longs;uperficiei.<emph.end type="italics"/></cell><cell>241</cell></row><row><cell>CCX.</cell><cell>S<emph type="italics"/>i du&aelig; line&aelig; ad &aelig;quales angulos ab eodem puncto peripheri&aelig; circuli refle-ctantur, nece&longs;&longs;e e&longs;t angulos cum dimetiente factos &aelig;quales e&longs;&longs;e.<emph.end type="italics"/> V<emph type="italics"/>nde ma-nife&longs;tum e&longs;t, protractam diametrum angulum &longs;uppo&longs;itum per &aelig;qualia di-uidere.<emph.end type="italics"/></cell><cell>242</cell></row><row><cell>CCXI.</cell><cell>S<emph type="italics"/>i du&aelig; line&aelig; ex duobus punctis peripheriam contingentes, in eandem par-tem protrahantur, &longs;emper magis di&longs;tabunt inuicem ea ex parte, &amp; nun-quam concurrent.<emph.end type="italics"/></cell><cell>243</cell></row><row><cell>CCXII.</cell><cell>S<emph type="italics"/>i ab eodem puncto ad circuli peripheriam line&aelig; quotuis ducantur, tres inue-nire lineas, qu&aelig; non in alium punctum reflectentur.<emph.end type="italics"/></cell><cell>244</cell></row><row><cell>CCXIII.</cell><cell>P<emph type="italics"/>ropo&longs;ito circulo, atque in eius peripheria puncto &longs;ignato, lineas contingentes   ultra c&iacute;traque, &amp; eam ab ip&longs;omet deducere.<emph.end type="italics"/></cell><cell>245</cell></row><row><cell>CCXIIII.</cell><cell>S<emph type="italics"/>i extra circulum duo puncta &aelig;qualiter &agrave; centro di&longs;tantia &longs;ignentur, erit pun-ctum reflexionis &aelig;qualis in medio arcus intercepti inter lineas, qu&aelig; &agrave; cen   tro ducuntur ad illa puncta.<emph.end type="italics"/> S<emph type="italics"/>i uer&ograve; unum centro proximius fuerit altero,   punctum &aelig;qualitatis in peripheria tant&ograve; longius, uer&longs;us breuiorem line-am, quant&ograve; punctum aliud &agrave; centro magis di&longs;teterit.<emph.end type="italics"/></cell><cell>245</cell></row><row><cell>CCXV.</cell><cell>P<emph type="italics"/>unctum reflexionis punctorum in&aelig;qualiter di&longs;tantium &agrave; centro, &aelig;qualiter   di&longs;tat &agrave; lineis, ductis &agrave; centro ad puncta &aelig;qualiter di&longs;tantia alterutrin-que.<emph.end type="italics"/></cell><cell>246</cell></row><row><cell>CCXVI.</cell><cell>S<emph type="italics"/>i fuerint circuli duo in&aelig;quales, &amp; extra utrunq&uacute;e punctum ad illud ex mi-nore reflex&egrave; per magnam partem minoris &agrave; maiore perueuire pote-runt.<emph.end type="italics"/></cell><cell>247</cell></row><row><cell>CCXVII.</cell><cell>O<emph type="italics"/>culus uidet partem &longs;uperficiei<emph.end type="italics"/> L<emph type="italics"/>un&aelig; illuminatam &agrave;<emph.end type="italics"/> S<emph type="italics"/>ole per radios reflexos   &agrave;<emph.end type="italics"/> S<emph type="italics"/>olis corpore: nec tamen pote&longs;t uidere imaginem ip&longs;ius in<emph.end type="italics"/> L<emph type="italics"/>una tan   quam in &longs;peculo.<emph.end type="italics"/></cell><cell>248</cell></row><row><cell>CCXVIII.</cell><cell>R<emph type="italics"/>ationem macul&aelig;<emph.end type="italics"/> L<emph type="italics"/>un&aelig; indagare.<emph.end type="italics"/></cell><cell>248</cell></row><row><cell>CCXIX.</cell><cell>R<emph type="italics"/>ationem eorum qu&aelig; apparent circa<emph.end type="italics"/> S<emph type="italics"/>olem &longs;peculo in aqua po&longs;ito decla-rare.<emph.end type="italics"/></cell><cell>150</cell></row><row><cell>CCXX.</cell><cell>C<emph type="italics"/>au&longs;am cur<emph.end type="italics"/> S<emph type="italics"/>ol &aelig;&longs;tiuis diebus exoriens umbram ad meridiem, cum in meridie   ad boream mittat, explorare.<emph.end type="italics"/></cell><cell>252</cell></row><row><cell>CCXXI.</cell><cell>M<emph type="italics"/>agnitudo<emph.end type="italics"/> L<emph type="italics"/>un&aelig; &amp; c&aelig;terorum a&longs;trorum digno&longs;citur ex proportione alio-rum ad eam iuxta di&longs;tantiam: ip&longs;ius uer&ograve; iuxta rationem pupill&aelig; ad<emph.end type="italics"/> L<emph type="italics"/>u-nam di&longs;tanti&aelig; ratione.<emph.end type="italics"/></cell><cell>354</cell></row><row><cell>CCXXII.</cell><cell>Q<emph type="italics"/>uantitates qu&aelig; &aelig;quales e&longs;&longs;e non po&longs;&longs;unt in eodem genere, maius tamen &amp;   minus recipiunt, &longs;unt in proportione pote&longs;tatis.<emph.end type="italics"/></cell><cell>255</cell></row><row><cell>CCXXIII.</cell><cell>Q<emph type="italics"/>uantitates qu&aelig; actu &aelig;quales e&longs;&longs;e non po&longs;&longs;unt, in nulla proportione actu   e&longs;&longs;e po&longs;&longs;unt.<emph.end type="italics"/></cell><cell>256</cell></row><row><cell>CCXXIIII.</cell><cell>N<emph type="italics"/>eque temporis totius, ut imaginamur, ip&longs;um e&longs;&longs;e infinitum, neque &aelig;ui ui-tarum proportio ulla e&longs;t ad tempus, quod pote&longs;tate e&longs;t, utpot&egrave; diem<emph.end type="italics"/></cell><cell></cell></row><pb/><row><cell></cell><cell><emph type="italics"/>uel men&longs;em.<emph.end type="italics"/></cell><cell>256</cell></row><row><cell>CCXXV.</cell><cell>P<emph type="italics"/>roportio media non e&longs;t ex ratione agentis, &longs;ed patientis.<emph.end type="italics"/></cell><cell>256</cell></row><row><cell>CCXXVI.</cell><cell>P<emph type="italics"/>roportio &longs;ublimis non con&longs;i&longs;tit in magnitudine, &longs;ed ordine, iuxta quem diffe-rentia e&longs;t eius quod e&longs;t ante &amp; po&longs;t.<emph.end type="italics"/></cell><cell>257</cell></row><row><cell>CCXXVII.</cell><cell>V<emph type="italics"/>it&aelig; iuxta numerum perfectionum in comparatione ad cogitationem no-&longs;tram proportionem quand am habent.<emph.end type="italics"/></cell><cell>259</cell></row><row><cell>CCXXVIII.</cell><cell>P<emph type="italics"/>roportionem &longs;cienti&aelig; futurorum &amp; c&aelig;terorum occultorum con&longs;idera-re.<emph.end type="italics"/></cell><cell>260</cell></row><row><cell>CCXXIX.</cell><cell>I<emph type="italics"/>ncorporea omnia unum &longs;unt, neque numerus e&longs;t eorum.<emph.end type="italics"/></cell><cell>261</cell></row><row><cell>CCXXX.</cell><cell>P<emph type="italics"/>roportio incorporeorum a&longs;cendentium &longs;emper maior e&longs;t.<emph.end type="italics"/></cell><cell>262</cell></row><row><cell>CCXXXI.</cell><cell>T<emph type="italics"/>res e&longs;&longs;e mundos atque inter ip&longs;os nullam e&longs;&longs;e proportionem: nec numero cos   definiri.<emph.end type="italics"/></cell><cell>263</cell></row><row><cell>CCXXXII.</cell><cell>O<emph type="italics"/>mnis motus naturalis quanto uelocior e&longs;t tanto propior e&longs;t &amp; magis &longs;imil   limus quieti.<emph.end type="italics"/></cell><cell>264</cell></row><row><cell>CCXXXIII.</cell><cell>Q<emph type="italics"/>uod e&longs;t in mundo incorporeo &aelig;ternum e&longs;t, beatum, &longs;ecurum, immutabile   &longs;ecundum locum, &longs;olum iuxta e&longs;&longs;entiam fit: iuxta quod uelut &agrave; leui &longs;u-&longs;urro aqu&aelig; &amp; aura &aelig;&longs;tiua demulcetur.<emph.end type="italics"/></cell><cell>270</cell></row></table><p type="head"> <s>TABVLA PRO&shy;<lb/>POSITIONVM DE <lb/>PROPORTIONIBVS.<lb/><arrow.to.target n="table1"/></s></p><table><table.target id="table1"/><row><cell>I.</cell><cell>Proportionem <emph type="italics"/>in proportionem duci, e&longs;t &longs;uperiores numeros   atque inferiores inuicem ducere.<emph.end type="italics"/></cell><cell><emph type="italics"/>pagina<emph.end type="italics"/> 6</cell></row><row><cell>II.</cell><cell>P<emph type="italics"/>roportio extremorum producitur ex intermedijs.<emph.end type="italics"/></cell><cell>7</cell></row><row><cell>III.</cell><cell>S<emph type="italics"/>i proportio ex duabus proportionibus in quatuor terminis producatur,   ip&longs;a uer&ograve; proportio inter duas alias quantitates fuerit con&longs;tituta: con&longs;urgent trecen-ti &longs;exaginta modi productionis proportionis.<emph.end type="italics"/></cell><cell>7</cell></row><row><cell>IIII.</cell><cell>S<emph type="italics"/>i fuerit proportio primi ad &longs;ecundum, producta ex proportionibus tertij ad quartum,   &amp; quinti ad &longs;extum, producetur etiam ex proportione tertij ad &longs;extum, &amp; quinti ad   quartum.<emph.end type="italics"/></cell><cell>8</cell></row><row><cell>V.</cell><cell>S<emph type="italics"/>i fuerit proportio primi ad &longs;ecundum, producta ex proportione tertij ad quartum, &amp;   quinti ad &longs;extum: erit proportio tertij ad &longs;extum, producta ex proportionibus primi   ad &longs;ecur dum, &amp; quarti ad quintum.<emph.end type="italics"/></cell><cell>8</cell></row><row><cell>VI.</cell><cell>E<emph type="italics"/>x trecentis &longs;exaginta modis producendarum proportionum triginta &longs;ex tantum e&longs;&longs;e   nece&longs;&longs;arios.<emph.end type="italics"/></cell><cell>9</cell></row><row><cell>VII.</cell><cell>I<emph type="italics"/>n modis qui nece&longs;&longs;ari&ograve; producuntur ex duabus proportionibus, cum du&aelig; quantitates ex   illis qu&aelig; modos conficiunt, &aelig;quales fuerint: proportio producta ad quatuor quanti-tates omiologas reducetur.<emph.end type="italics"/></cell><cell>10</cell></row><row><cell>VIII.</cell><cell>S<emph type="italics"/>i duarum proportionum &longs;uperiores numeri alternatim cum inferioribus multiplicen-tur atque coniungantur, erit proportio aggregati ad productum ex inferioribus in-uicem proportio, ex primis proportionibus compo&longs;ita.<emph.end type="italics"/></cell><cell>11</cell></row><row><cell>IX.</cell><cell>S<emph type="italics"/>i duarum proportionum &longs;uperiores numeri alternatim cum inferioribus multiplicen-tur, minusque productum ex maiore detrahatur, erit re&longs;idui ad productum ex in&longs;e-rioribus proportio uelut illa, qu&aelig; relinquitur detracta minore proportione ex ma-iore.<emph.end type="italics"/></cell><cell>11</cell></row><row><cell>X.</cell><cell>S<emph type="italics"/>i fuerit alicuius quantitatis ad unam partem proportio, uelut alterius partis ad &longs;ecun-dam quantitatem, erit proportio cuiu&longs;uis quantitatis eiu&longs;dem generis ad &longs;ecundam   compo&longs;ita proportio, ex proportionibus eiu&longs;dem quantitatis, a&longs;&longs;umpt&aelig; ad utranque   partem prim&aelig; quantitatis &longs;eor&longs;um.<emph.end type="italics"/></cell><cell>11</cell></row><row><cell>XI.</cell><cell>P<emph type="italics"/>roportio aggregati quarumlibet duarum quantitatum ad aggregatum duarum &aelig;qua-lium <expan abbr="quantitat&utilde;">quantitatum</expan> e&longs;t, compo&longs;ita ex proportionibus primis, &amp; diui&longs;a per duplam.<emph.end type="italics"/></cell><cell>12</cell></row><row><cell>XII.</cell><cell>P<emph type="italics"/>ropo&longs;itis duabus proportionibus unam alteri iungere ab&longs;que multiplicatione.<emph.end type="italics"/></cell><cell>12</cell></row><row><cell>XIII.</cell><cell>P<emph type="italics"/>roportio confu&longs;a aggregata prim&aelig; &amp; terti&aelig; quatuor quantitatum omiologarum ad   aggregatum &longs;ecund&aelig; &amp; quart&aelig;, e&longs;t uelut compo&longs;ita ex ei&longs;dem diui&longs;a per du-plam.<emph.end type="italics"/></cell><cell>13</cell></row><row><cell>XIIII.</cell><cell>P<emph type="italics"/>roportiones confu&longs;&aelig; &amp; coniunct&aelig; in tribus quantitatibus inuicem commutantur.<emph.end type="italics"/></cell><cell>13</cell></row><row><cell>XV.</cell><cell>S<emph type="italics"/>i fuerint quatuor quantitates proportio confu&longs;a, aggregati prim&aelig; &amp; terti&aelig;, ad aggre-gatum &longs;ecund&aelig; &amp; quart&aelig;, erit ut monadis addito prouentu, qui fit diui&longs;a differentia,   differentiarum prim&aelig; &amp; &longs;ecund&aelig;, atque quart&aelig; &amp; terti&aelig;, per aggregatum terti&aelig; &amp;   quart&aelig; ad ip&longs;am monadem.<emph.end type="italics"/></cell><cell>14</cell></row><row><cell>XVI.</cell><cell>O<emph type="italics"/>mnium quatuor quantitatum propo&longs;ita prima, qu&aelig; non minorem habet proportio-nem ad &longs;uam corre&longs;pondentem qu&agrave;m alia ad aliam, erit proportio confu&longs;a illarum,<emph.end type="italics"/></cell><cell/></row><pb/><row><cell/><cell><emph type="italics"/>ut producti ex aggregato prim&aelig; &amp; terti&aelig;, in tertiam ad productum ex iggre   gato terti&aelig; &amp; omiotat&aelig; ad &longs;ecundam in ip&longs;am quartam.<emph.end type="italics"/></cell><cell>14</cell></row><row><cell>XVII.</cell><cell>O<emph type="italics"/>mnes du&aelig; proportiones conuer&longs;&aelig; producunt &aelig;qualem proportionem.<emph.end type="italics"/></cell><cell>15</cell></row><row><cell>XVIII.</cell><cell>S<emph type="italics"/>i fuerint quotlibet quantitates in continua proportione multiplici pr&aelig;ter, <expan abbr="ultim&atilde;">ultimam</expan>   proportio uer&ograve; penultim&aelig; ad ultimam, qualis re&longs;idui prim&aelig; ad &longs;ecundam,   erit prim&aelig; ad aggregatum reliquarum, uelut penultim&aelig; ad ultimam.<emph.end type="italics"/></cell><cell>15</cell></row><row><cell>XIX.</cell><cell>S<emph type="italics"/>i fuerint aliquot quantitates arithmetic&aelig; omiolog&aelig;, quarum exce&longs;&longs;us &longs;it &aelig;qualis   minim&egrave;, omnibus autem deficientibus &longs;upplementa ad &aelig;qualitatem maxim&egrave;   adiungantur, erunt quadrata omnium quantitatum &aelig;qualium, adiecto rur&longs;us   quadrato prim&aelig; cum eo quod fit ex minima primi ordinis in aggregatum o-mnium quantitatum eiu&longs;dem, tripla aggregato quadratorum omnium quanti   tatum primi ordinis pariter acceptis.<emph.end type="italics"/></cell><cell>17</cell></row><row><cell>XX.</cell><cell>C<emph type="italics"/>um fuerint quatuor quantitates, fueritque <expan abbr="&longs;ec&utilde;da">&longs;ecunda</expan> &aelig;qualis terti&aelig;, aut prima &aelig;qualis   quart&aelig;, erit proportio prim&aelig; ad quartam, aut terti&aelig; ad &longs;ecundam, producta   ex proportionibus prim&aelig; ad &longs;ecundam &amp; terti&aelig; ad quartam.<emph.end type="italics"/></cell><cell>21</cell></row><row><cell>XXI.</cell><cell>C<emph type="italics"/>um decu&longs;&longs;atim ducta fuerit prima in quartam, &amp; &longs;ecunda in tertiam, produ-ctumque prim&aelig; in quartam, diui&longs;um fu<gap/>rit per productum &longs;ecund&aelig; in tertiam,   erit proportio prim&aelig; ad &longs;ecundam, diui&longs;a per proport&iacute;onem terti&aelig; ad quar-tam.<emph.end type="italics"/> E<emph type="italics"/>t &longs;imiliter interpo&longs;ita omiologa.<emph.end type="italics"/></cell><cell>22</cell></row><row><cell>XXII.</cell><cell>C<emph type="italics"/>um fuerit proportio prim&aelig; ad &longs;ecundam maior qu&agrave;m terti&aelig; ad quartam, erit   confu&longs;a ex his maior qu&agrave;m terti&aelig; ad quartam, minor autem qu&agrave;m prim&aelig; ad   &longs;ecundam.<emph.end type="italics"/></cell><cell>23</cell></row><row><cell>XXIII.</cell><cell>O<emph type="italics"/>mnis motus naturalis ad locum &longs;uum e&longs;t: ide&ograve; per rectam lineam fit.<emph.end type="italics"/></cell><cell>23</cell></row><row><cell>XXIIII.</cell><cell>O<emph type="italics"/>mnis motus circularis uoluntarius e&longs;t.<emph.end type="italics"/></cell><cell>23</cell></row><row><cell>XXV.</cell><cell>T<emph type="italics"/>res &longs;unt motus omnino &longs;implices naturalis, uoluntarius, &amp; uiolentus.<emph.end type="italics"/></cell><cell>24</cell></row><row><cell>XXVI.</cell><cell>M<emph type="italics"/>otus ergo compo&longs;iti quatuor nece&longs;&longs;ari&ograve; &longs;unt &longs;pecies.<emph.end type="italics"/></cell><cell>24</cell></row><row><cell>XXVII.</cell><cell>M<emph type="italics"/>otus uoluntarius e&longs;t in loco: naturalis ad locum: uiolentus ex loco.<emph.end type="italics"/></cell><cell>25</cell></row><row><cell>XXVIII.</cell><cell>M<emph type="italics"/>otus quilibet uoluntarius aut uiolentus in aliquo medio fit.<emph.end type="italics"/></cell><cell>25</cell></row><row><cell>XXIX.</cell><cell>O<emph type="italics"/>mnis motus uoluntarius &aelig;qualis e&longs;t &longs;emper: &longs;impliciter etiam quilibet alius mo-tus.<emph.end type="italics"/></cell><cell>25</cell></row><row><cell>XXX.</cell><cell>I<emph type="italics"/>n omni corpore mobili in medio partes medij re&longs;i&longs;tunt obui&aelig;, ali&aelig; impel-lunt.<emph.end type="italics"/></cell><cell>26</cell></row><row><cell>XXXI.</cell><cell>O<emph type="italics"/>mnis motus naturalis in &aelig;quali medio ualidior e&longs;t in fine qu&agrave;m in principio.<emph.end type="italics"/>V<emph type="italics"/>iolentus contr&agrave;.<emph.end type="italics"/></cell><cell>26</cell></row><row><cell>XXXII.</cell><cell>O<emph type="italics"/>mne mobile naturaliter motum &longs;eu uiolenter uelocius mouetur in medio rariore   qu&agrave;m den&longs;iore.<emph.end type="italics"/> M<emph type="italics"/>aior quoque e&longs;t proportio finis motus in corpore rariore ad   finem motus in corpore den&longs;iore qu&agrave;m principij.<emph.end type="italics"/> I<emph type="italics"/>n uiolento autem celerius   perueniret ad finem motus in corpore den&longs;iore.<emph.end type="italics"/></cell><cell>27</cell></row><row><cell>XXXIII.</cell><cell>O<emph type="italics"/>mnia duo mobilia &aelig;qualis undique magnitudinis qu&aelig; &aelig;quali in tempore &aelig;qualia   &longs;pacia pertran&longs;eunt in diuer&longs;is &longs;ub&longs;tantia medijs nece&longs;&longs;e e&longs;t, ut &longs;it ponderis ad   pondus, quem ad modum medij ad medium proportio duplicata.<emph.end type="italics"/></cell><cell>27</cell></row><row><cell>XXXIIII.</cell><cell>P<emph type="italics"/>roportio corporis cubi ad &longs;uam &longs;uperficiem quadratam, e&longs;t uelut eiu&longs;dem &longs;uperfi   ciei, ad latus eiu&longs;dem uer&ograve; ad monadem.<emph.end type="italics"/></cell><cell>28</cell></row><row><cell>XXXV.</cell><cell>V<emph type="italics"/>ocum magnitudines excre&longs;cunt in acumine, non in grauitate, finis autem e&longs;t in   utroque extremo.<emph.end type="italics"/> P<emph type="italics"/>ropter hoc minima facta uariatione in hypate acut&aelig; uix   ferunt.<emph.end type="italics"/></cell><cell>29</cell></row><row><cell>XXXVI.</cell><cell>S<emph type="italics"/>i proportio per proportionem minorem &aelig;quali ducatur, proportio minor pro-<emph.end type="italics"/></cell><cell/></row><pb/><row><cell/><cell><emph type="italics"/>ducetur.<emph.end type="italics"/> V<emph type="italics"/>nde manife&longs;tum e&longs;t duas proportiones minores &aelig;qualitate inuice<gap/> du   ctas proportionem minorem unaquaque illarum producere.<emph.end type="italics"/></cell><cell>30</cell></row><row><cell>XXXVII.</cell><cell>S<emph type="italics"/>i plures homines, quorum per &longs;e nauim mouere po&szlig;int, aut pondus ferre &longs;imul   iuncti eam moueant, aut pondus ferant, erunt ill&aelig; proportiones coniunct&aelig;   non product&aelig;.<emph.end type="italics"/></cell><cell>30</cell></row><row><cell>XXXVIII.</cell><cell>O<emph type="italics"/>mne corpus tantum re&longs;i&longs;tit motui contrario &longs;uo nat&uacute;rali, quantum mouetur oc-culto motu quie&longs;cendo.<emph.end type="italics"/></cell><cell>31</cell></row><row><cell>XXXIX.</cell><cell>A<emph type="italics"/>b &aelig;quali aut minore ui qu&agrave;m &longs;it impedimentum non fit motus.<emph.end type="italics"/></cell><cell>31</cell></row><row><cell>XL.</cell><cell>O<emph type="italics"/>mne corpus &longs;pb &aelig;ricum tangens planum in puncto mouetur ad latus per quam-cunque uim, qu&aelig; medium diuidere pote&longs;t.<emph.end type="italics"/></cell><cell>31</cell></row><row><cell>XLI.</cell><cell>S<emph type="italics"/>i fuerint du&aelig; quantitates &longs;umaturque toties <expan abbr="aggregat&utilde;">aggregatum</expan> maioris &amp; minoris, quo-ties aggregatum minoris &amp; maioris, erit proportio confu&longs;a maioris aggregati   ad minus, minor quam multiplicis maioris ad multiplex minoris.<emph.end type="italics"/></cell><cell>32</cell></row><row><cell>XLII.</cell><cell>T<emph type="italics"/>rahentium nauim, aut ferentium pondera proportiones in &longs;e inuicem, quomodo   ducere oporteat con&longs;iderare.<emph.end type="italics"/></cell><cell>32</cell></row><row><cell>XLIII.</cell><cell>P<emph type="italics"/>roductionem ad additionem retrabere.<emph.end type="italics"/></cell><cell>33</cell></row><row><cell>XLIIII.</cell><cell>S<emph type="italics"/>i fuerit proportio motoris ad id quod e&longs;t maximum non mouens, &amp; &longs;pacium &amp;   tempus, nota erit etiam reliquorum nota.<emph.end type="italics"/></cell><cell>33</cell></row><row><cell>XLV.</cell><cell>R<emph type="italics"/>ationem &longs;tater&aelig; o&longs;tendere.<emph.end type="italics"/></cell><cell>34</cell></row><row><cell>XLVI.</cell><cell>A<emph type="italics"/>n &longs;it aliqua proportio &amp; qualis inter animam &amp; uitas, &amp; &longs;ua corpora con&longs;ide-rare.<emph.end type="italics"/></cell><cell>35</cell></row><row><cell>XLVII.</cell><cell>S<emph type="italics"/>i duo mobilia &aelig;qualister in eodem circulo iuxta proprios motus moueantur, pro-ductum temporis circuituum inuicem, erit &aelig;quale producto differenti&aelig; tempo   rum circuitus duct&aelig; in tempus coniunctionis prim&aelig;.<emph.end type="italics"/></cell><cell>36</cell></row><row><cell>XLVIII.</cell><cell>S<emph type="italics"/>i tria mobilia ex eodem puncto di&longs;cedant, fuerintque duorum ac duorum coniun-ctiones in temporibus commen&longs;is, illa tria mobilia denuo coniungentur in tem   pore producto ex denominatore diui&longs;ionis temporis maioris per minus in mi-nus aut numeratore in maius.<emph.end type="italics"/></cell><cell>37</cell></row><row><cell>XLIX.</cell><cell>P<emph type="italics"/>ropofitio mobilis in circulo circuitus tempore dataque ratione di&longs;tanti&aelig; ab illo mo   bilis circuitum inuenire, quod ex <expan abbr="eod&etilde;">eodem</expan> puncto di&longs;cedens <expan abbr="c&utilde;alio">cunalio</expan> mobili in dato   puncto <expan abbr="c&otilde;ueniat">conueniat</expan> &longs;ub <expan abbr="quoc&utilde;que">quocunque</expan> numero <expan abbr="circuitu&utilde;">circuituum</expan> <expan abbr="t&etilde;pus">tempus</expan> quoque <expan abbr="c&otilde;iunctionis">coniunctionis</expan>.<emph.end type="italics"/></cell><cell>39</cell></row><row><cell>L.</cell><cell>O<emph type="italics"/>mnes circuituum portiones in ei&longs;dem temporibus repetuntur.<emph.end type="italics"/></cell><cell>40</cell></row><row><cell>LI.</cell><cell>O<emph type="italics"/>perationes dictas exemplo declarare.<emph.end type="italics"/></cell><cell>41</cell></row><row><cell>LII.</cell><cell>T<emph type="italics"/>ria mobilia coniuncta in <expan abbr="eod&etilde;">eodem</expan> puncto, quorum duo &amp; duo conueniant in partib.   incommen&longs;is inter &longs;e, in perpetuum in nullo unquam puncto conuenient.<emph.end type="italics"/></cell><cell>42</cell></row><row><cell>LIII.</cell><cell>C<emph type="italics"/>irculorum &longs;e in aduer&longs;um mouentium proportionem declarare.<emph.end type="italics"/></cell><cell>43</cell></row><row><cell>LIIII.</cell><cell>P<emph type="italics"/>roportio circuli ad &longs;uum diametrum per &longs;imilitudinem e&longs;t quarta pars periphe-ri&aelig;.<emph.end type="italics"/> R<emph type="italics"/>ur&longs;usque eiu&longs;dem circuli ad peripheriam diametri quarta pars.<emph.end type="italics"/></cell><cell>44</cell></row><row><cell>LV.</cell><cell>P<emph type="italics"/>roportionem medicamentorum per ordines &longs;up po&longs;ita &aelig;quali proportione in or-dinibus per quantitates &amp; proportiones demon&longs;trare.<emph.end type="italics"/></cell><cell>44</cell></row><row><cell>LVI.</cell><cell>P<emph type="italics"/>roportio cuiu&longs;uis binomij ad &longs;uum reci&longs;um, uel ei commen&longs;um e&longs;t duplicata ei   qu&aelig; ad numeri latus.<emph.end type="italics"/></cell><cell>49</cell></row><row><cell>LVII.</cell><cell>M<emph type="italics"/>otus rationem ad pondus inuenire.<emph.end type="italics"/></cell><cell>49</cell></row><row><cell>LVIII.</cell><cell>Q<emph type="italics"/>u&aelig; ex alto de&longs;cendunt, cur non eandem pro di&longs;tantia motus rationem in libero   a&euml;re &longs;eruent con&longs;iderare.<emph.end type="italics"/></cell><cell>49</cell></row><row><cell>LIX.</cell><cell>O<emph type="italics"/>mne mobile motum duobus motibus non ad idem tendentibus utroque &longs;eor&longs;um tar   dius mouetur &longs;imili motu.<emph.end type="italics"/></cell><cell>50</cell></row><row><cell>LX.</cell><cell>O<emph type="italics"/>mne mobile motu naturali de&longs;cendentis parte, de&longs;cendit grauiore &longs;ecundum gra-<emph.end type="italics"/></cell><cell/></row><pb/><row><cell/><cell><emph type="italics"/>uitatis centrum.<emph.end type="italics"/></cell><cell>51</cell></row><row><cell>LXI.</cell><cell>P<emph type="italics"/>roportionum ictus ad pondus rei &amp; di&longs;tantiam generaliter con&longs;iderare.<emph.end type="italics"/></cell><cell>52</cell></row><row><cell>LXII.</cell><cell>P<emph type="italics"/>roportionem motoris in plano ad motorem, qui eleuat pondus iuxta id quod   mouet, inuenire.<emph.end type="italics"/></cell><cell>53</cell></row><row><cell>LXIII.</cell><cell>O<emph type="italics"/>mne graue quanto proximius alligatum plano, tant&ograve; facilius trabitur.<emph.end type="italics"/></cell><cell>53</cell></row><row><cell>LXIIII.</cell><cell>O<emph type="italics"/>mne mobile quant&ograve; latius tanto tardius moustur in plano.<emph.end type="italics"/></cell><cell>54</cell></row><row><cell>LXV.</cell><cell>P<emph type="italics"/>roportionem duorum mobilium inter &longs;e cum auxilio medij inuenire.<emph.end type="italics"/></cell><cell>54</cell></row><row><cell>LXVI.</cell><cell>P<emph type="italics"/>roportionem laterum eptagoni, &amp; &longs;ubten&longs;arum con&longs;iderare, &amp; qu&aelig; &agrave; reflexa   proportione pendent.<emph.end type="italics"/></cell><cell>55</cell></row><row><cell>LXVII.</cell><cell>S<emph type="italics"/>i fuerint aliquot quantitates ab una quantitate ali&aelig;que totidem ab eadem analo-g&aelig;, erit proportio terti&aelig; unius ordinis ad tertiam alterius, ut &longs;ecund&aelig; ad &longs;e-cundum duplicata, &amp; quart&aelig; ad quartam triplicata, quint&aelig; ad quintam   quadruplicata, atque &longs;ic de alijs.<emph.end type="italics"/></cell><cell>57</cell></row><row><cell>LXVIII.</cell><cell>P<emph type="italics"/>ropo&longs;itio collectorum ab<emph.end type="italics"/> E<emph type="italics"/>uclide &amp;<emph.end type="italics"/> A<emph type="italics"/>rchimede.<emph.end type="italics"/></cell><cell>57</cell></row><row><cell>LXIX.</cell><cell>P<emph type="italics"/>ropo&longs;itio collectorum ex quatuor libris<emph.end type="italics"/> A<emph type="italics"/>pollonij<emph.end type="italics"/> P<emph type="italics"/>ergei &amp;<emph.end type="italics"/> <expan abbr="q.">que</expan> S<emph type="italics"/>ereni.<emph.end type="italics"/></cell><cell>59</cell></row><row><cell>LXX.</cell><cell>S<emph type="italics"/>i fuerint tres quantitates <gap/> ontinua proportione, ali&aelig;que totidem in continua   proportione poterunt con&longs;tituere tres quantitates in &aelig;quali differentia per-uer&longs;im copulat&aelig;.<emph.end type="italics"/></cell><cell>62</cell></row><row><cell>LXXI.</cell><cell>P<emph type="italics"/>roportionem leuitatis ponderis per uirgam torcularem attracti ad rectam &longs;u-&longs;pen&longs;ionem inuenire.<emph.end type="italics"/></cell><cell>63</cell></row><row><cell>LXXII.</cell><cell>P<emph type="italics"/>roportionem ponderis &longs;ph&aelig;r&aelig; pendentis ad a&longs;cendentem per accliue planum   inuenire.<emph.end type="italics"/></cell><cell>63</cell></row><row><cell>LXXIII.</cell><cell>P<emph type="italics"/>roportionem ponderum attractorum penes figuram in plano inuenire.<emph.end type="italics"/></cell><cell>64</cell></row><row><cell>LXXIIII.</cell><cell>P<emph type="italics"/>roportionem concutientis ad concu&longs;&longs;um in&longs;tabili inuenire.<emph.end type="italics"/></cell><cell>64</cell></row><row><cell>LXXV.</cell><cell>P<emph type="italics"/><expan abbr="roportion&etilde;">roportionem</expan> immoti in aqua, ad <expan abbr="immot&utilde;">immotum</expan> in terra in excipiendo <expan abbr="ict&utilde;">ictum</expan> inuenire.<emph.end type="italics"/></cell><cell>65</cell></row><row><cell>LXXVI.</cell><cell>P<emph type="italics"/>roportionem <expan abbr="duor&utilde;">duorum</expan> mobilium &longs;ibi <expan abbr="inuic&etilde;">inuicem</expan> <expan abbr="concurrenti&utilde;">concurrentium</expan> per <expan abbr="rect&atilde;">rectam</expan> inuenire.<emph.end type="italics"/></cell><cell>66</cell></row><row><cell>LXXVII.</cell><cell>P<emph type="italics"/>roportionem motus obliqui ad motum rectum in nauibus inuenire.<emph.end type="italics"/></cell><cell>66</cell></row><row><cell>LXXVIII.</cell><cell>P<emph type="italics"/>roportionem nauis ad triremes quotuis concurrentes demon&longs;trare.<emph.end type="italics"/></cell><cell>67</cell></row><row><cell>LXXIX.</cell><cell>P<emph type="italics"/>roportionem medicamentorum purgantium inuicem declarare<emph.end type="italics"/></cell><cell>68</cell></row><row><cell>LXXX.</cell><cell>P<emph type="italics"/>roportionem motus &longs;ecundum obliquum ad rectum in &longs;pacio declarare.<emph.end type="italics"/></cell><cell>69</cell></row><row><cell>LXXXI.</cell><cell>Q<emph type="italics"/>uualis &longs;it angulus, per quem pote&longs;t moueri nauis ad rectum explorare.<emph.end type="italics"/></cell><cell>70</cell></row><row><cell>LXXXII.</cell><cell>P<emph type="italics"/>roportionem uelorum indagare.<emph.end type="italics"/></cell><cell>70</cell></row><row><cell>LXXXIII.</cell><cell>P<emph type="italics"/>roportionem rece&longs;&longs;us &agrave; recta uia ad obliquitatem inue&longs;tigare.<emph.end type="italics"/></cell><cell>72</cell></row><row><cell>LXXXIIII.</cell><cell>D<emph type="italics"/><expan abbr="i&longs;tanti&atilde;">i&longs;tantiam</expan> centri terr&aelig; &agrave; centro mundi per motum lapidis<emph.end type="italics"/> H<emph type="italics"/>erculei declarare.<emph.end type="italics"/></cell><cell>73</cell></row><row><cell>LXXXV.</cell><cell>P<emph type="italics"/>roportio ponderis unius grauis ad aliud &longs;ub eadem men&longs;ura e&longs;t ueluti eiu&longs;dem   ad differentiam ponderis ua&longs;is repleti ex altero graui, &amp; ex ambobus de-tracto priore.<emph.end type="italics"/></cell><cell>74</cell></row><row><cell>LXXXVI.</cell><cell>S<emph type="italics"/>i circuli in &aelig; quales &longs;eu in &longs;ph&aelig;ra &longs;eu in plano &longs;e &longs;ecuerint, nunqu&agrave;m oppo&longs;itos   angulos &aelig;quales habent.<emph.end type="italics"/></cell><cell>77</cell></row><row><cell>LXXXVII.</cell><cell>P<emph type="italics"/>roportiones cra&szlig;itiei aqu&aelig; ad <expan abbr="a&etilde;r&etilde;">aerrem</expan> in <expan abbr="c&otilde;paratione">comparatione</expan> ad radios demon&longs;trare.<emph.end type="italics"/></cell><cell>78</cell></row><row><cell>LXXXVIII.</cell><cell>I<emph type="italics"/><expan abbr="n&longs;trument&utilde;">n&longs;trumentum</expan><emph.end type="italics"/> A<emph type="italics"/>colingen, quo momenta temporum <expan abbr="deprehend&atilde;tur">deprehendantur</expan> fabricare.<emph.end type="italics"/></cell><cell>79</cell></row><row><cell>LXXXIX.</cell><cell>P<emph type="italics"/>roportionem den&longs;itatis aqu&aelig; ad a&euml;rem per pondera inuenire.<emph.end type="italics"/></cell><cell>82</cell></row><row><cell>XC.</cell><cell>R<emph type="italics"/>ationem impetus uiolenti extra mi&szlig;i ponderis ad &aelig;qualitatem reducere.<emph.end type="italics"/></cell><cell>82</cell></row><row><cell>XCI.</cell><cell>P<emph type="italics"/>roportionem grauis cubi, &amp; &longs;ph&aelig;rici &aelig;qualium in accliui, &amp; de&longs;cen&longs;us eorum   demon&longs;trare.<emph.end type="italics"/></cell><cell>83</cell></row><row><cell>XCII.</cell><cell>P<emph type="italics"/><expan abbr="roportion&etilde;">roportionem</expan> ponderis &aelig;qualis iuxta longitudinis <expan abbr="c&otilde;paration&etilde;">comparationem</expan> demon&longs;trare.<emph.end type="italics"/></cell><cell>85</cell></row><row><cell>XCIII.</cell><cell>P<emph type="italics"/>ropter qd in <expan abbr="c&otilde;cu&szlig;ione">concu&szlig;ione</expan> <expan abbr="eti&atilde;">etiam</expan> leui nauis loco moueatar <expan abbr="o&longs;t&etilde;dere">o&longs;tendere</expan>.<emph.end type="italics"/> V<emph type="italics"/>nde manifi <expan abbr="&longs;i&utilde;">&longs;ium</expan>   e&longs;t duas naues &longs;ibi <expan abbr="inuic&etilde;">inuicem</expan> occur&longs;antes retrocedere, &amp; <expan abbr="qu&atilde;t&utilde;">quantum</expan> <expan abbr="retroced&atilde;t">retrocedant</expan> amb&aelig;.<emph.end type="italics"/></cell><cell>86</cell></row><pb/><row><cell>XCIIII.</cell><cell>S<emph type="italics"/>i <expan abbr="qu&atilde;titas">quantitas</expan> aliqua nota atque proportio erit producta, <expan abbr="qu&atilde;titas">quantitas</expan> nota &longs;imiliter.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i du&aelig;   proportiones not&aelig; fuerint, erit producta ex his atque diui&longs;a coniunctaque atque detra-cta nota.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i fuerit totius ad partem proportio nota, erit et ad aliam partem nota:   &amp; alterius partis ad <expan abbr="alter&atilde;">alteram</expan> uno minor.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i fuerit partis ad partem, erit ad to<gap/>um   monade minor atque nota.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i fuerit unius <expan abbr="qu&atilde;titatis">quantitatis</expan> ad duas <expan abbr="qu&atilde;titates">quantitates</expan> proportio   nota, erit &amp; <expan abbr="c&otilde;fu&longs;a">confu&longs;a</expan> ex eis nota.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i fuerint trium quantitatum omiologarum, aut   quatuor analogarum omnes pr&aelig;ter unam cognit&aelig;, erunt &verbar; &amp; illa alia cognita.<emph.end type="italics"/></cell><cell>87</cell></row><row><cell>XCV.</cell><cell>C<emph type="italics"/>uiu&longs;uis trigoni rectanguli, aut cuius duo auguli &longs;int in dupla proportione, aut qui   circulo in&longs;criptus &longs;it cognita quantitate unius lateris in comparatione ad dimetien   <expan abbr="t&etilde;">tem</expan>, &longs;i proportio duorum laterum cognita fuerit, <expan abbr="er&utilde;t">erunt</expan> omnia eius latera cognita.<emph.end type="italics"/></cell><cell>88</cell></row><row><cell>XCVI.</cell><cell>C<emph type="italics"/>um in <expan abbr="per&longs;picu&utilde;">per&longs;picuum</expan> den&longs;um radij lumino&longs;i inciderint, quatuor fiunt luminis genera.<emph.end type="italics"/></cell><cell>89</cell></row><row><cell>XCVII.</cell><cell>M<emph type="italics"/><expan abbr="ot&utilde;">otum</expan> inuer&longs;ionis in figuris in <expan abbr="c&otilde;paratione">comparatione</expan> ad <expan abbr="mot&utilde;">motum</expan> &longs;ph&aelig;r&aelig; in plano inue&longs;tigare.<emph.end type="italics"/></cell><cell>91</cell></row><row><cell>XCVIII.</cell><cell>P<emph type="italics"/>roportionem ponderum &aelig;qualium per differentiam angulorum inuenire.<emph.end type="italics"/></cell><cell>92</cell></row><row><cell>XCIX.</cell><cell>P<emph type="italics"/>roportionem grauitatum per multitudin<gap/> &longs;uppo&longs;itorum orbium o&longs;tendere.<emph.end type="italics"/></cell><cell>93</cell></row><row><cell>C.</cell><cell>P<emph type="italics"/><expan abbr="roportion&etilde;">roportionem</expan> grauitatis <expan abbr="ponder&utilde;">ponderum</expan> attractorum per <expan abbr="trochlear&utilde;">trochlearum</expan> <expan abbr="numer&utilde;">numerum</expan> inue&longs;tigare.<emph.end type="italics"/></cell><cell>93</cell></row><row><cell>CI.</cell><cell>P<emph type="italics"/>roportionem precij gemmarum ex tribus in eodem genere cognitis inuenire.<emph.end type="italics"/></cell><cell>94</cell></row><row><cell>CII.</cell><cell>P<emph type="italics"/>roportionem motuum inuer&longs;ionis, &amp; attractionis in plano inuenire.<emph.end type="italics"/></cell><cell>95</cell></row><row><cell>CIII.</cell><cell>P<emph type="italics"/>roportionem eorundem in accliui demon&longs;trare.<emph.end type="italics"/></cell><cell>95</cell></row><row><cell>CIIII.</cell><cell>P<emph type="italics"/>roportionem motus attractionis in decliui ad motum in plano determinare.<emph.end type="italics"/></cell><cell>95</cell></row><row><cell>CV.</cell><cell>P<emph type="italics"/>roportionem ferentium pondus in pertica inuenire.<emph.end type="italics"/></cell><cell>96</cell></row><row><cell>CVI.</cell><cell>Q<emph type="italics"/>uales proportiones angulorum doceant laterum proportiones.<emph.end type="italics"/> A<emph type="italics"/>tque uici&szlig;im deter-minare.<emph.end type="italics"/></cell><cell>97</cell></row><row><cell>CVII.</cell><cell>S<emph type="italics"/>i in circulo du&aelig; diametri ad rectum angulum &longs;e &longs;ecauerint: ali&aelig; uer&ograve; ad perpendicu-lum ex diametro exicrint ad circum ferentiam, &longs;ingul&aelig; &longs;upra diametrum erunt ma   iores portionibus reliquis diametri &longs;uperioribus, infra autem minores.<emph.end type="italics"/> D<emph type="italics"/>imidium   autem portionis &longs;uperioris re&longs;iduum ad centrum maius &longs;agitta habebit.<emph.end type="italics"/> I<emph type="italics"/>n aliqua   pr&aelig;terea portionis &longs;uperioris parte, qu&aelig; uer&longs;us diametrum tran&longs;uer&longs;um po&longs;ita   e&longs;t, maior e&longs;t differentia partis diametri ei <expan abbr="corre&longs;p&otilde;dentis">corre&longs;pondentis</expan>, <expan abbr="&qtilde;">quae</expan> line &aelig; tran&longs;uer&longs;&aelig;.<emph.end type="italics"/></cell><cell>100</cell></row><row><cell>CVIII.</cell><cell>P<emph type="italics"/>unctum &aelig;qualitatis differenti&aelig; de&longs;cen&longs;us &amp; remotionis &agrave; centro inuenire.<emph.end type="italics"/></cell><cell>100</cell></row><row><cell>CIX.</cell><cell>R<emph type="italics"/>ationem libr&aelig; expendere.<emph.end type="italics"/></cell><cell>101</cell></row><row><cell>CX.</cell><cell>S<emph type="italics"/>i du&aelig; &longs;ph&aelig;r&aelig; ex eadem materia de&longs;cendant in a&euml;re, eodem temporis momento ad   planum ueniunt.<emph.end type="italics"/></cell><cell>104</cell></row><row><cell>CXI.</cell><cell>C<emph type="italics"/>ur ex medio tela ualidiorem ictum, &amp; naues in &longs;calmo &agrave; remo ac malo recipiant in-de ex puppi explorare.<emph.end type="italics"/></cell><cell>105</cell></row><row><cell>CXII.</cell><cell>C<emph type="italics"/>ur ex imo leuia longi&ugrave;s ferantur declarare,<emph.end type="italics"/></cell><cell>106</cell></row><row><cell>CXIII.</cell><cell>C<emph type="italics"/>ur uirga longius mittatur &agrave; puero quam &agrave; uiro inueftigare.<emph.end type="italics"/></cell><cell>107</cell></row><row><cell>CXIIII.</cell><cell>C<emph type="italics"/>ircularis motus differentias quatuor e&longs;&longs;e, earumque rationem contemplari.<emph.end type="italics"/></cell><cell>108</cell></row><row><cell>CXV.</cell><cell>P<emph type="italics"/>roportionem motuum impul&longs;ionis, &amp; attractionis inter &longs;e, ab eadem ui decla-rare.<emph.end type="italics"/></cell><cell>110</cell></row><row><cell>CXVI.</cell><cell>C<emph type="italics"/>ur machin&aelig; oblong&aelig; igne&aelig; longius emittant &longs;ph&aelig;ram explorare.<emph.end type="italics"/></cell><cell>111</cell></row><row><cell>CXVII.</cell><cell>I<emph type="italics"/>n curriculis maior e&longs;t uis pulueris copio&longs;ioris ampliore in &longs;pacio, qu&agrave;m paucioris in   minore iuxta proportionem eandem.<emph.end type="italics"/></cell><cell>112</cell></row><row><cell>CXVIII.</cell><cell>Q<emph type="italics"/>uanta proportione decedat ictus in obliquum parietem ab eo qui e&longs;t ad perpendi-culum declarare.<emph.end type="italics"/></cell><cell>114</cell></row><row><cell>CXIX.</cell><cell>Q<emph type="italics"/>uantum ictus machin&aelig; procliuis ad angulum minuatur explorare.<emph.end type="italics"/></cell><cell>115</cell></row><row><cell>CXX</cell><cell>P<emph type="italics"/>roportionem partium nauis ad eundem obliquum uentum explorare.<emph.end type="italics"/></cell><cell>118</cell></row><row><cell>CXXI.</cell><cell>F<emph type="italics"/>labelli uires atque naturam declarare.<emph.end type="italics"/></cell><cell>219</cell></row><row><cell>CXXII.</cell><cell>C<emph type="italics"/>ontemptus circa<emph.end type="italics"/> S<emph type="italics"/>olis rationem in umbris declarare.<emph.end type="italics"/></cell><cell>120</cell></row><pb/><row><cell>CXXIII.</cell><cell>C<emph type="italics"/>ognita ratione umbr&aelig; ad gnomonem &longs;inum, &amp; arcum altitudinis ab horizon-te, quouis tempore digno&longs;cere.<emph.end type="italics"/></cell><cell>121</cell></row><row><cell>CXXIIII.</cell><cell>P<emph type="italics"/>roportionem umbr&aelig; uer&longs;&aelig; e&longs;&longs;e ad gnomonem, uelut gnomonis ad umbram   uer&longs;am.<emph.end type="italics"/></cell><cell>122</cell></row><row><cell>CXXV.</cell><cell>P<emph type="italics"/>roportionem dimetientis, &amp; peripheri&aelig; cuiuslibet circuli paralleli &aelig;quino-ctiali per cognitam partem magni circuli demon&longs;trare.<emph.end type="italics"/></cell><cell>123</cell></row><row><cell>CXXVI.</cell><cell>C<emph type="italics"/>irculi horarij naturam declarare.<emph.end type="italics"/></cell><cell>123</cell></row><row><cell>CXXVII.</cell><cell>D<emph type="italics"/>ata poli altitudine ortus amplitudinem demonftrare.<emph.end type="italics"/></cell><cell>124</cell></row><row><cell>CXXVIII.</cell><cell>N<emph type="italics"/>ota amplitudine ortus, cuiu&longs;que puncti arcum &longs;emidiurnum inuenire.<emph.end type="italics"/></cell><cell>124</cell></row><row><cell>CXXIX.</cell><cell>D<emph type="italics"/>ata altitudine<emph.end type="italics"/> S<emph type="italics"/>olis in quacunque regione, quacunque die di&longs;tantiam<emph.end type="italics"/> S<emph type="italics"/>olis &agrave; meri-diano cogno&longs;cere.<emph.end type="italics"/></cell><cell>124</cell></row><row><cell>CXXX.</cell><cell>D<emph type="italics"/>ata regionis altitudine, &amp; loco<emph.end type="italics"/> S<emph type="italics"/>olis proportionem gnomonis, tam ad um-bram rectam qu&agrave;m uer&longs;am, uel etiam in cylindro determinare.<emph.end type="italics"/></cell><cell>125</cell></row><row><cell>CXXXI.</cell><cell>S<emph type="italics"/>i line&aelig; alicui duplum alterius adiungatur, erit proportio d<gap/>arum ad primam   maior qu&agrave;m dupli cum prima ad primam cum una adiecta.<emph.end type="italics"/></cell><cell>126</cell></row><row><cell>CXXXII.</cell><cell>S<emph type="italics"/>i ad duas lineas quarum una alteri dupla &longs;it eadem linea addatur, erit aggrega-ti ex minore, &amp; adiecta ad ip&longs;am minorem, minor proportio qu&agrave;m aggre-gati ex maiore, &amp; adiecta ad ip&longs;am maiorem duplicata.<emph.end type="italics"/></cell><cell>126</cell></row><row><cell>CXXXIII.</cell><cell>S<emph type="italics"/>i fuerint du&aelig; quantitates, <expan abbr="quar&utilde;">quarum</expan> una alteri dupla &longs;it: minuatur &agrave; minore qu&aelig;-dam quantitas, <expan abbr="ead&etilde;que">eadenque</expan> maiori addatur, erit minoris ad re&longs;iduum maior pro-portio, qu&agrave;m aggregati ad maiorem duplicata.<emph.end type="italics"/> S<emph type="italics"/>i uer&ograve; minori addatur, &amp;   &agrave; maiore detrabatur, erit aggregati ad minorem minor proportio qu&agrave;m   maioris ad re&longs;iduum duplicata.<emph.end type="italics"/></cell><cell>127</cell></row><row><cell>CXXXIIII.</cell><cell>S<emph type="italics"/>i rectangula &longs;uperficies &longs;it, cuius pars tertia quadrata &longs;it corpus, quod ex la-tere quadrat&aelig; in re&longs;iduum &longs;uperficiei con&longs;tat, maius e&longs;t quouis corpore ex   eadem &longs;uperficies, aliter diui&longs;a con&longs;tituto.<emph.end type="italics"/></cell><cell>127</cell></row><row><cell>CXXXV.</cell><cell>S<emph type="italics"/>i linea in duas partes, quarum una fit alteri dupla diuidatur, erit quod fit ex   tertia parte in quadratum re&longs;idui parallelipedum maius omni pararalleli-pedo, quod ex diui&longs;ione eiu&longs;dem line&aelig; creari po&szlig;it.<emph.end type="italics"/></cell><cell>128</cell></row><row><cell>CXXXVI.</cell><cell>D<emph type="italics"/>enominationes in infinitum extendere.<emph.end type="italics"/></cell><cell>129</cell></row><row><cell>CXXXVII.</cell><cell>R<emph type="italics"/>ationem numerorum ex progre&szlig;ione declarare.<emph.end type="italics"/></cell><cell>131</cell></row><row><cell>CXXXVIII.</cell><cell>M<emph type="italics"/>odos u&longs;us horum numerorum declarare.<emph.end type="italics"/></cell><cell>131</cell></row><row><cell>CXXXIX.</cell><cell>R<emph type="italics"/>adices omnes &agrave; propo&longs;itis numeris extrahere.<emph.end type="italics"/></cell><cell>132</cell></row><row><cell>CXL.</cell><cell>R<emph type="italics"/>adices per numeros fractos determinare.<emph.end type="italics"/></cell><cell>133</cell></row><row><cell>CXLI.</cell><cell>N<emph type="italics"/>umeros fractos ad minores in ea <expan abbr="i&etilde;">iem</expan> proportione ualde propinqud deducere<emph.end type="italics"/></cell><cell>136</cell></row><row><cell>CXLII.</cell><cell>D<emph type="italics"/><expan abbr="enomination&utilde;">enominationum</expan> in <expan abbr="crem&etilde;ta">crementa</expan> ex extrema cognita inuenire.<emph.end type="italics"/> E<emph type="italics"/>t <expan abbr="c&otilde;uer&longs;o">conuer&longs;o</expan> modo.<emph.end type="italics"/></cell><cell>137</cell></row><row><cell>CXLIII.</cell><cell>S<emph type="italics"/>i linea in duas partes diuidatur, corpora qu&aelig; fiunt ex una parte in alterius   quadratum mutuo &aelig;qualia &longs;unt corpori, quod fit ex tota linea in &longs;uperfi-ciem unius partis in alteram.<emph.end type="italics"/></cell><cell>138</cell></row><row><cell>CXLIIII.</cell><cell>D<emph type="italics"/>uplum cubi medietatis maius e&longs;t aggregato corporum mutuorum, cuiuslibet   diui&longs;ionis quantum e&longs;t, quod fit ex tota in quadratum differenti&aelig;.<emph.end type="italics"/></cell><cell>139</cell></row><row><cell>CXLV.</cell><cell>S<emph type="italics"/>i linea in duas partes diuidatur quadrata ambarum partium detracto eo, quod   fit ex una parte in alteram, &aelig;qualia &longs;unt producto unius in alteram cum   quadrato differenti&aelig;.<emph.end type="italics"/></cell><cell>139</cell></row><row><cell>CXLVI.</cell><cell>C<emph type="italics"/>orpus quod fit ex linea diui&longs;a in &longs;uperficiem &aelig;qualem quadratis ambarum par   tium detracta &longs;uperficie unius partis in alteram, e&longs;t &aelig;quale aggregato cubo-rum ambarum partium.<emph.end type="italics"/></cell><cell>139</cell></row><row><cell>CXLVII.</cell><cell>P<emph type="italics"/>ropo&longs;ita linea diui&longs;a duas ei line as adijcere, ut proportio <expan abbr="additar&utilde;">additarum</expan> &longs;ingularium<emph.end type="italics"/></cell><cell/></row><pb/><row><cell/><cell><emph type="italics"/>&amp; partium &longs;imul iunctarum ad additas &longs;it mutua.<emph.end type="italics"/></cell><cell>148</cell></row><row><cell>CXLVIII.</cell><cell>P<emph type="italics"/>ropo&longs;itis tribus lineis primam &longs;ic diuidere, ut adiectis duabus alijs lineis, &longs;ecun-dum <expan abbr="ration&etilde;">rationem</expan> mutuam &longs;ingularum &longs;ingulis, <expan abbr="aggregat&utilde;">aggregatum</expan> ex una <expan abbr="adiectar&utilde;">adiectarum</expan>, &amp; par   te ad <expan abbr="aggregat&utilde;">aggregatum</expan> ex alia parte, &amp; adiecta &longs;e habeat, ut &longs;ecunda ad <expan abbr="terti&atilde;">tertiam</expan>.<emph.end type="italics"/></cell><cell>140</cell></row><row><cell>CXLIX.</cell><cell>D<emph type="italics"/>atam lineam &longs;ic diuidere, ut proportio quadratorum ad dupium unius partis in   alteram &longs;it, ut line&aelig; dat&aelig; ad lineam datam.<emph.end type="italics"/></cell><cell>141</cell></row><row><cell>CL.</cell><cell>P<emph type="italics"/>ropo&longs;itis duabus lineis, lineam communem utrique adiungere, ut &longs;it maioris ad ad-ditam proportio, uelut quadratorum minoris, &amp; adiect&aelig; ad duplum unius in   alteram.<emph.end type="italics"/></cell><cell>141</cell></row><row><cell>CLI.</cell><cell>P<emph type="italics"/>roportio differenti&aelig; quadratorum partium cuiu&longs;uis line&aelig;, ad quadratum diffe-renti&aelig; illarum e&longs;t, uelut totius line&aelig; ad differentiam.<emph.end type="italics"/></cell><cell>142</cell></row><row><cell>CLII.</cell><cell>S<emph type="italics"/>i linea in duas partes &aelig;quales, duasque in&aelig;quales diuidatur, fueritque proportio ag-gregati ex maiore, &amp; dimidio ad ip&longs;am maiorem, uelut ex minore, &amp; ali-qua linea ad ip&longs;am minorem, &amp; rur&longs;us aggregati ex minore, &amp; dimidio ad   ip&longs;am minorem, uelut aggregati ex maiore, &amp; alia addita ad ip&longs;am maiorem,   erit proportio dimidij ad partem unam in&aelig;qualem, uelut alterius partis in&aelig;-qualis ad &longs;uam additam mutu&ograve;, &amp; etiam proportio additarum inuicem, uelut   proportio <expan abbr="parti&utilde;">partium</expan> <expan abbr="in&aelig;quali&utilde;">in&aelig;qualium</expan> duplicata, &amp; rur&longs;us ip&longs;um <expan abbr="dimidi&utilde;">dimidium</expan> line&aelig; a&longs;&longs;um-pt&aelig; <expan abbr="medi&utilde;">medium</expan>, erit proportione inter additas.<emph.end type="italics"/> D<emph type="italics"/><expan abbr="em&utilde;">emum</expan> proportio dimidij <expan abbr="c&utilde;">cum</expan> addita   maiore ad <expan abbr="dimidi&utilde;">dimidium</expan>, cum addita minore, uelut maioris partis ad <expan abbr="minor&etilde;">minorem</expan>.<emph.end type="italics"/></cell><cell>142</cell></row><row><cell>CLIII.</cell><cell>V<emph type="italics"/>im quamcunque manus multiplicare.<emph.end type="italics"/></cell><cell>144</cell></row><row><cell>CLIIII.</cell><cell>S<emph type="italics"/>i line&aelig; dat&aelig; alia linea adiungatur, ab extremitatibus autem prioris line&aelig; du&aelig;   rect&aelig; in unum punctum concurrant proportionem habentes, quam mediam   inter tota m &amp; adiectam, &amp; adiectam erit punctus, concur&longs;us &agrave; puncto extre-mo line&aelig; adiect&aelig; di&longs;tans per lineam mediam.<emph.end type="italics"/> Q<emph type="italics"/>uod &longs;i ab extremo alicuius li-ne&aelig; &aelig;qua'is medi&aelig;, &longs;eu peripheria circuli, cuius &longs;emidiameter &longs;it media linea   du&aelig; line&aelig; ad pr&aelig;dicta puncta producantur, ip&longs;&aelig; erunt in proportione medi&aelig;   ad adiectam.<emph.end type="italics"/></cell><cell>145</cell></row><row><cell>CLV.</cell><cell>Q<emph type="italics"/>uadr atorum numerum proportionem &amp; inuentionem con&longs;iderare.<emph.end type="italics"/></cell><cell>147</cell></row><row><cell>CLVI.</cell><cell>H<emph type="italics"/>orologiorum tempus multiplicare.<emph.end type="italics"/></cell><cell>152</cell></row><row><cell>CLVII.</cell><cell>H<emph type="italics"/>orologiorum molarium rationem o&longs;tendere.<emph.end type="italics"/></cell><cell>154</cell></row><row><cell>CLVIII.</cell><cell>R<emph type="italics"/>ationem indicis mobilis cum rota, qua horarum numerus per ictus indicatur ex-plicare.<emph.end type="italics"/></cell><cell>156</cell></row><row><cell>CLIX.</cell><cell>N<emph type="italics"/>ullus angulus rectilineus &aelig;qualis e&longs;&longs;e pote&longs;t alicui angulo contento recta, &amp; cir   culi portione.<emph.end type="italics"/></cell><cell>158</cell></row><row><cell>CLX.</cell><cell>P<emph type="italics"/>ropo&longs;ita linea tribusque in ea &longs;ignis punctum inuenire, ex quo duct&aelig; tres line&aelig; ad   &longs;igna &longs;int in proportionibus datis.<emph.end type="italics"/></cell><cell>162</cell></row><row><cell>CLXI.</cell><cell>S<emph type="italics"/>i fuerint duo trianguli, quorum ba&longs;es in eadem linea &longs;int con&longs;tituti, &amp; &aelig;quales   ad unum punctum terminati, &amp; latus unum commune inter reliqua quantita-te medium nece&longs;&longs;e e&longs;t angulum &agrave; maioribus lineis <expan abbr="content&utilde;">contentum</expan> minorem e&longs;&longs;e.<emph.end type="italics"/></cell><cell>162</cell></row><row><cell>CLXII.</cell><cell>P<emph type="italics"/>roportionem duorum orbium, quorum diametrorum conuex&aelig; partis, &amp; conca-u&aelig; proportiones dat&aelig; &longs;int inue&longs;tigare.<emph.end type="italics"/></cell><cell>164</cell></row><row><cell>CLXIII.</cell><cell>P<emph type="italics"/>roportionem uirium &longs;tellarum per motus &longs;uos indagare.<emph.end type="italics"/></cell><cell>165</cell></row><row><cell>CLXIIII.</cell><cell>S<emph type="italics"/>yderum proportionem in magnitudine o&longs;tendere.<emph.end type="italics"/></cell><cell>166</cell></row><row><cell>CLXV.</cell><cell>P<emph type="italics"/>roportionem motuum omnium &longs;tellarum ad<emph.end type="italics"/> S<emph type="italics"/>olem con&longs;iderare.<emph.end type="italics"/></cell><cell>167</cell></row><row><cell>CLXVI.</cell><cell>P<emph type="italics"/>roportiones mu&longs;icas &longs;uperpartientes in eas, qu&aelig; particul&aacute; una tantum abundant   reducere.<emph.end type="italics"/></cell><cell>168</cell></row><pb/><row><cell>CLXVII.</cell><cell>P<emph type="italics"/>roportionem mu&longs;icam ad &longs;apores &amp; odores coaptare.<emph.end type="italics"/></cell><cell>176</cell></row><row><cell>CLXVIII.</cell><cell>P<emph type="italics"/>icturarum proportiones explicare.<emph.end type="italics"/></cell><cell>179</cell></row><row><cell>CLXIX.</cell><cell>P<emph type="italics"/>roportionem mu&longs;icam in in&longs;trumentis declarare iuxta compo&longs;itionis ra-tionem.<emph.end type="italics"/></cell><cell>182</cell></row><row><cell>CLXX.</cell><cell>C<emph type="italics"/>oniugationes cuiu&longs;uis numeri breuiter inuenire.<emph.end type="italics"/></cell><cell>185</cell></row><row><cell>CLXXI.</cell><cell>P<emph type="italics"/>ropo&longs;itis duobus quibuslibet numeris, quotuis alios &longs;eu in continuum &longs;eu   medios in continua proportione arithmetica, geometrica &amp; mu&longs;ica in-uenire.<emph.end type="italics"/></cell><cell>187</cell></row><row><cell>CLXXII.</cell><cell>P<emph type="italics"/>roportiones<emph.end type="italics"/> S<emph type="italics"/>tiphelij de&longs;cribere.<emph.end type="italics"/></cell><cell>191</cell></row><row><cell>CLXXIII.</cell><cell>C<emph type="italics"/>irculum &longs;uper centro &longs;uo mouere &aelig;qualiter, ita quod omnia illius puncta   per rectam lineam moueantur ultro citroque.<emph.end type="italics"/></cell><cell>192</cell></row><row><cell>CLXXIIII.</cell><cell>P<emph type="italics"/>rogre&longs;&longs;us &amp; regre&longs;&longs;us, tam &longs;ine latitudine qu&agrave;m cum latitudine in planetis   per &longs;olos concentricos circulos &aelig;qualiter motos demon&longs;trare.<emph.end type="italics"/></cell><cell>194</cell></row><row><cell>CLXXV.</cell><cell>C<emph type="italics"/>au&longs;am uarietatis diametrorum ex &longs;uppo&longs;itis concentricis demon&longs;tra-re.<emph.end type="italics"/></cell><cell>195</cell></row><row><cell>CLXXVI.</cell><cell>R<emph type="italics"/>ationem centri grauitatis declarare.<emph.end type="italics"/></cell><cell>197</cell></row><row><cell>CLXXVII.</cell><cell>S<emph type="italics"/>i proportio aliqua ex duabus proportionibus eiu&longs;dem quantitatis ad alias   duas componatur, erit proportio illarum duarum eadem proportioni   producti ex proportione in primam duarum quantitatum, detracta prio-re illa quantitate, qu&aelig; ad duas comparatur, ad eandem priorem quanti-tatem.<emph.end type="italics"/></cell><cell>198</cell></row><row><cell>CLXXVIII.</cell><cell>P<emph type="italics"/>roportionem mi&longs;tionis metallorum, maxim&egrave; auri &amp; argenti declara-re.<emph.end type="italics"/></cell><cell>199</cell></row><row><cell>CLXXIX.</cell><cell>S<emph type="italics"/>i duobus totis du&aelig; portiones &longs;imiles ab&longs;cindantur ab ei&longs;dem denu&ograve;, &amp; ab-&longs;ci&szlig;is portionibus partes e&aelig;dem auferantur, denuoque ac denu&ograve; quoties   libuerit &agrave; portionibus, &amp; &ugrave; re&longs;iduis ip&longs;arum quantitatum partes e&aelig;dem   auferantur, erit re&longs;idu&iacute; ad re&longs;iduum, ueluti totius ad totum.<emph.end type="italics"/></cell><cell>200</cell></row><row><cell>CLXXX.</cell><cell>S<emph type="italics"/>i aliqua quantitas in duas partes diuidatur, fueritque alicuius quantitatis ad   partes illas compo&longs;ita proportio, non poterit eiu&longs;dem quantitatis ad par-tes alias quantitatis diui&longs;a, aliter proportio eadem componi.<emph.end type="italics"/></cell><cell>202</cell></row><row><cell>CLXXXI.</cell><cell>C<emph type="italics"/>um fuerit aliqua proportio, compo&longs;ita ex proportionibus prim&aelig; ad &longs;ecun-dam &amp; tertiam, &amp; rur&longs;us quart&aelig; ad quintam &amp; &longs;extam: ita &longs;e habebit   proportio &longs;ecund&aelig; ad tertiam, ad proportionem quint&aelig; ad &longs;extam, uelut   producti ex proportione in &longs;ecundam detracta prima ad primam ad pro-ductum ex proportione in quintam, detracta quarta ad quartam.<emph.end type="italics"/></cell><cell>203</cell></row><row><cell>CLXXXII.</cell><cell>P<emph type="italics"/>ropo&longs;ita differentia proportionum partium &longs;imilium ad partes a&longs;&longs;umptas,   propo&longs;itaque proportione totius ad re&longs;idua eadem, differentiam propor-tionum totius ad reliquum re&longs;idui inuenire.<emph.end type="italics"/></cell><cell>203</cell></row><row><cell>CLXXXIII.</cell><cell>S<emph type="italics"/>pacium uit&aelig; naturalis per &longs;pacium uit&aelig; fortuitum declarare.<emph.end type="italics"/></cell><cell>204</cell></row><row><cell>CLXXXIIII.</cell><cell>Q<emph type="italics"/>u&aelig;cunque grauia in uorticibus aquarum, merguntur, in medio uorticis, pri-mum uer&longs;a mergantur.<emph.end type="italics"/></cell><cell>211</cell></row><row><cell>CLXXXV.</cell><cell>C<emph type="italics"/>ur homo &longs;edens quanto altius &longs;edet, &amp; quanto magis crura ad f&oelig;mora, &amp;   f&oelig;mora ad pectus reclinata habet, facilius con&longs;urgat, cum tamen h&aelig;c op-po&longs;ito modo inuicem &longs;e habeant, declarare.<emph.end type="italics"/></cell><cell>213</cell></row><row><cell>CLXXXVI.</cell><cell>S<emph type="italics"/>i fuerit proportio prim&aelig; &amp; &longs;ecund&aelig; quantitatis ad tertiam, ut prim&aelig; &amp;   quart&aelig; ad quintam, fueritque quarta &longs;ecunda maior, erit proportio quar-t&aelig; ad quintam maior qu&agrave;m &longs;ecund&aelig; ad tertiam.<emph.end type="italics"/> Q<emph type="italics"/>uod &longs;i fuerit maior<emph.end type="italics"/></cell><cell/></row><pb/><row><cell/><cell><emph type="italics"/>quart&aelig; ad quintam qu&agrave;m &longs;ecund&aelig; ad tertiam, nece&longs;&longs;e e&longs;t quartam &longs;ecunda e&longs;&longs;e   maiorem.<emph.end type="italics"/></cell><cell>214</cell></row><row><cell>CLXXXVII.</cell><cell>S<emph type="italics"/>i ei&longs;dem uiribus &amp; &lsquo;eadem&rsquo; proportione cum auxilio ponderis tertij quar-tum pondus moueatur quibus &longs;ecundum, auxilio primi nece&longs;&longs;e e&longs;t <expan abbr="quart&utilde;">quartum</expan> pon   dus tardius &amp; maiore cum difficultate moueri qu&agrave;m &longs;ecundum.<emph.end type="italics"/></cell><cell>214</cell></row><row><cell>CLXXXVIII.</cell><cell>S<emph type="italics"/>i uires aliqu&aelig; moueant cum ponderibus aliqua pondera, ut compo&longs;ita pro-portio &longs;it eadem proportioni uirium &amp; duorum ponderum mouentium ag-gregatum &aelig;quale duorum ponderum, ubi maior fuerit partium in &aelig;qual<gap/>as,   ibi erit maior difficultas.<emph.end type="italics"/></cell><cell>214</cell></row><row><cell>CLXXXIX.</cell><cell>S<emph type="italics"/>i pondus minus ad longitudinem minorem &longs;ub &aelig;quali proportione coapte-tar, facilius deor&longs;um trahetur qu&agrave;m quod maius e&longs;t &amp; propius.<emph.end type="italics"/></cell><cell>215</cell></row><row><cell>CXC.</cell><cell>S<emph type="italics"/>i fuerit primum graue minus &longs;ecundo, &amp; &longs;ecundum minus tertio, proportio   autem primi ad &longs;ecundum multo maior qu&agrave;m &longs;ecundi ad tertium, po&longs;ibile erit   propo&longs;itis uiribus ei&longs;dem addere pondus <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan>, ut ip&longs;um &amp; tertium mouea-tur facili&ugrave;s ab ei&longs;dem uiribus, &amp; primo uel &longs;ecundo qu&agrave;m antea.<emph.end type="italics"/></cell><cell>215</cell></row><row><cell>CXCL.</cell><cell>C<emph type="italics"/>um fuerint duo pondera &amp; uires, duxerisque aggregatum ex uiribus &amp; mi-nore pondere in maius, addiderisque in&longs;uper quantum e&longs;t productum dimidij ui   rium in &longs;e latus aggregati detracto dimidio uirium, dice<gap/> pondus auxiliare   &aelig;qualis proportionis.<emph.end type="italics"/></cell><cell>215</cell></row><row><cell>CXCII.</cell><cell>S<emph type="italics"/>i ex medio diametri linea ad perpendiculum erigatur ad circuli peripheri-am, ex eo puncto autem quotlibet line&aelig; ducantur &longs;eu intus ad circun ferentia<gap/>u&longs;que, &longs;eu extra ad diametrum, erit proportio totius line&aelig; ad totam uelut mu-tuo partis ad partem.<emph.end type="italics"/></cell><cell>217</cell></row><row><cell>CXCIII.</cell><cell>R<emph type="italics"/>ationem ponderis triplicem explicare.<emph.end type="italics"/></cell><cell>218</cell></row><row><cell>CXCIIII.</cell><cell>P<emph type="italics"/>roportionem ponderis longioris in medio &longs;u&longs;pen&longs;i, ad breuius illi &aelig;quale &amp; in   medio &longs;u&longs;pen&longs;um declarare.<emph.end type="italics"/></cell><cell>219</cell></row><row><cell>CXCV.</cell><cell>S<emph type="italics"/>i lectus fiat dupla longitudine ad latitudinem, melius &longs;uffulcietur re&longs;tibus   ex medio ad angulos &amp; eius &aelig;quidi&longs;tantibus qu&agrave;m &longs;ecundum longitudinem   &amp; latitudinem.<emph.end type="italics"/></cell><cell>220</cell></row><row><cell>CXCVI.</cell><cell>S<emph type="italics"/>i duo circuli &longs;uper eodem centro eodem motu trans feruntur, &aelig;quale &longs;pacium   &longs;uperant.<emph.end type="italics"/></cell><cell>221</cell></row><row><cell>CXCVII.</cell><cell>C<emph type="italics"/>ur lances ad locum &longs;uum &longs;u&longs;pen&longs;i redeant, impendentes <expan abbr="n&otilde;">non</expan>, <expan abbr="dem&otilde;&longs;trare">demon&longs;trare</expan>.<emph.end type="italics"/></cell><cell>224</cell></row><row><cell>CXCVIII.</cell><cell>C<emph type="italics"/>ur &longs;olidum quod cubus uocatur<emph.end type="italics"/> P<emph type="italics"/>yramide &longs;tabilius &longs;it o&longs;tendere.<emph.end type="italics"/></cell><cell>225</cell></row><row><cell>CXCIX.</cell><cell>R<emph type="italics"/>ationem remorum nauim impellentium inuenire.<emph.end type="italics"/></cell><cell>227</cell></row><row><cell>CC.</cell><cell>C<emph type="italics"/>ur temo cum paruus &longs;it, magnam nauim agere pote&longs;t, &amp; cur c&ugrave;m uarietas &longs;it   in prora, ip&longs;e con&longs;tituatur in puppi.<emph.end type="italics"/> E<emph type="italics"/>t cum transuer&longs;im ab aqua prematur   rect&agrave; nauim dirigat.<emph.end type="italics"/></cell><cell>228</cell></row><row><cell>CCI.</cell><cell>S<emph type="italics"/>i du&aelig; line&aelig; non &longs;ecantes circuli peripheriam in unum punctum ex ea coe-ant exterius, nece&longs;&longs;e e&longs;t illas peripheria contenta e&longs;&longs;e maiores.<emph.end type="italics"/></cell><cell>229</cell></row><row><cell>CCII.</cell><cell>R<emph type="italics"/>ationem &longs;trepitus o&longs;tendere.<emph.end type="italics"/></cell><cell>232</cell></row><row><cell>CCIII.</cell><cell>C<emph type="italics"/>ur &longs;cytalis onera portentur facili&ugrave;s, explorare.<emph.end type="italics"/></cell><cell>233</cell></row><row><cell>CCIIII.</cell><cell>C<emph type="italics"/>ur pluribus trochleis, pondera facilius eleuentur o&longs;tendere.<emph.end type="italics"/></cell><cell>233</cell></row><row><cell>CCV.</cell><cell>S<emph type="italics"/>uper uerbis<emph.end type="italics"/> P<emph type="italics"/>latonis de fine<emph.end type="italics"/> R<emph type="italics"/>eipublic&aelig;.<emph.end type="italics"/></cell><cell>234</cell></row><row><cell>CCVI.</cell><cell>R<emph type="italics"/>hombi pa&szlig;iones qua&longs;dam declarare.<emph.end type="italics"/></cell><cell>235</cell></row><row><cell>CCVII.</cell><cell>P<emph type="italics"/>roportionem agentium naturalium in tran&longs;mutatione con&longs;iderare.<emph.end type="italics"/></cell><cell>238</cell></row><row><cell>CCVIII.</cell><cell>M<emph type="italics"/>ota res &agrave; centro grauitatis per <expan abbr="prior&etilde;">priorem</expan> motum, in reditu uelocius mouetur   quam &longs;i quieuerit.<emph.end type="italics"/></cell><cell>238</cell></row><pb/><row><cell>CCIX.</cell><cell>S<emph type="italics"/>i &longs;uperficies rectangula in duas partes &aelig;quales diui&longs;a intelligatur, qu&aelig; am-b&aelig; quadrat&aelig; &longs;int, itemque in duas in&aelig;quales, erit parallelipedum ex latere   medi&aelig; partis in totam &longs;uperficiem maius aggregato parallelipedorum ex   partibus in&aelig;qualibus in latera alterius partis mutuo, in eo, quod fit ex dif   ferentia lateris minoris partis &agrave; medi&aelig; latere in differentiam maioris par-tis &longs;uperficiei &agrave; media &longs;uperficie bis, &amp; ex differentia amborum laterum   in&aelig;qualium iunctorum ad ambo latera, &aelig;qualia iuncta in minorem par-tem &longs;uperficiei.<emph.end type="italics"/></cell><cell>241</cell></row><row><cell>CCX.</cell><cell>S<emph type="italics"/>i du&aelig; line&aelig; ad &aelig;quales angulos ab eodem puncto peripheri&aelig; circuli refle-ctantur, nece&longs;&longs;e e&longs;t angulos cum dimetiente factos &aelig;quales e&longs;&longs;e.<emph.end type="italics"/> V<emph type="italics"/>nde ma-nife&longs;tum e&longs;t, protractam diametrum angulum &longs;uppo&longs;itum per &aelig;qualia di-uidere.<emph.end type="italics"/></cell><cell>242</cell></row><row><cell>CCXI.</cell><cell>S<emph type="italics"/>i du&aelig; line&aelig; ex duobus punctis peripheriam contingentes, in eandem par-tem protrahantur, &longs;emper magis di&longs;tabunt inuicem ea ex parte, &amp; nun-quam concurrent.<emph.end type="italics"/></cell><cell>243</cell></row><row><cell>CCXII.</cell><cell>S<emph type="italics"/>i ab eodem puncto ad circuli peripheriam line&aelig; quotuis ducantur, tres inue-nire lineas, qu&aelig; non in alium punctum reflectentur.<emph.end type="italics"/></cell><cell>244</cell></row><row><cell>CCXIII.</cell><cell>P<emph type="italics"/>ropo&longs;ito circulo, atque in eius peripheria puncto &longs;ignato, lineas contingentes   ultra c&iacute;traque, &amp; eam ab ip&longs;omet deducere.<emph.end type="italics"/></cell><cell>245</cell></row><row><cell>CCXIIII.</cell><cell>S<emph type="italics"/>i extra circulum duo puncta &aelig;qualiter &agrave; centro di&longs;tantia &longs;ignentur, erit pun-ctum reflexionis &aelig;qualis in medio arcus intercepti inter lineas, qu&aelig; &agrave; cen   tro ducuntur ad illa puncta.<emph.end type="italics"/> S<emph type="italics"/>i uer&ograve; unum centro proximius fuerit altero,   punctum &aelig;qualitatis in peripheria tant&ograve; longius, uer&longs;us breuiorem line-am, quant&ograve; punctum aliud &agrave; centro magis di&longs;teterit.<emph.end type="italics"/></cell><cell>245</cell></row><row><cell>CCXV.</cell><cell>P<emph type="italics"/>unctum reflexionis punctorum in&aelig;qualiter di&longs;tantium &agrave; centro, &aelig;qualiter   di&longs;tat &agrave; lineis, ductis &agrave; centro ad puncta &aelig;qualiter di&longs;tantia alterutrin-que.<emph.end type="italics"/></cell><cell>246</cell></row><row><cell>CCXVI.</cell><cell>S<emph type="italics"/>i fuerint circuli duo in&aelig;quales, &amp; extra utrunq&uacute;e punctum ad illud ex mi-nore reflex&egrave; per magnam partem minoris &agrave; maiore perueuire pote-runt.<emph.end type="italics"/></cell><cell>247</cell></row><row><cell>CCXVII.</cell><cell>O<emph type="italics"/>culus uidet partem &longs;uperficiei<emph.end type="italics"/> L<emph type="italics"/>un&aelig; illuminatam &agrave;<emph.end type="italics"/> S<emph type="italics"/>ole per radios reflexos   &agrave;<emph.end type="italics"/> S<emph type="italics"/>olis corpore: nec tamen pote&longs;t uidere imaginem ip&longs;ius in<emph.end type="italics"/> L<emph type="italics"/>una tan   quam in &longs;peculo.<emph.end type="italics"/></cell><cell>248</cell></row><row><cell>CCXVIII.</cell><cell>R<emph type="italics"/>ationem macul&aelig;<emph.end type="italics"/> L<emph type="italics"/>un&aelig; indagare.<emph.end type="italics"/></cell><cell>248</cell></row><row><cell>CCXIX.</cell><cell>R<emph type="italics"/>ationem eorum qu&aelig; apparent circa<emph.end type="italics"/> S<emph type="italics"/>olem &longs;peculo in aqua po&longs;ito decla-rare.<emph.end type="italics"/></cell><cell>150</cell></row><row><cell>CCXX.</cell><cell>C<emph type="italics"/>au&longs;am cur<emph.end type="italics"/> S<emph type="italics"/>ol &aelig;&longs;tiuis diebus exoriens umbram ad meridiem, cum in meridie   ad boream mittat, explorare.<emph.end type="italics"/></cell><cell>252</cell></row><row><cell>CCXXI.</cell><cell>M<emph type="italics"/>agnitudo<emph.end type="italics"/> L<emph type="italics"/>un&aelig; &amp; c&aelig;terorum a&longs;trorum digno&longs;citur ex proportione alio-rum ad eam iuxta di&longs;tantiam: ip&longs;ius uer&ograve; iuxta rationem pupill&aelig; ad<emph.end type="italics"/> L<emph type="italics"/>u-nam di&longs;tanti&aelig; ratione.<emph.end type="italics"/></cell><cell>354</cell></row><row><cell>CCXXII.</cell><cell>Q<emph type="italics"/>uantitates qu&aelig; &aelig;quales e&longs;&longs;e non po&longs;&longs;unt in eodem genere, maius tamen &amp;   minus recipiunt, &longs;unt in proportione pote&longs;tatis.<emph.end type="italics"/></cell><cell>255</cell></row><row><cell>CCXXIII.</cell><cell>Q<emph type="italics"/>uantitates qu&aelig; actu &aelig;quales e&longs;&longs;e non po&longs;&longs;unt, in nulla proportione actu   e&longs;&longs;e po&longs;&longs;unt.<emph.end type="italics"/></cell><cell>256</cell></row><row><cell>CCXXIIII.</cell><cell>N<emph type="italics"/>eque temporis totius, ut imaginamur, ip&longs;um e&longs;&longs;e infinitum, neque &aelig;ui ui-tarum proportio ulla e&longs;t ad tempus, quod pote&longs;tate e&longs;t, utpot&egrave; diem<emph.end type="italics"/></cell><cell/></row><pb/><row><cell/><cell><emph type="italics"/>uel men&longs;em.<emph.end type="italics"/></cell><cell>256</cell></row><row><cell>CCXXV.</cell><cell>P<emph type="italics"/>roportio media non e&longs;t ex ratione agentis, &longs;ed patientis.<emph.end type="italics"/></cell><cell>256</cell></row><row><cell>CCXXVI.</cell><cell>P<emph type="italics"/>roportio &longs;ublimis non con&longs;i&longs;tit in magnitudine, &longs;ed ordine, iuxta quem diffe-rentia e&longs;t eius quod e&longs;t ante &amp; po&longs;t.<emph.end type="italics"/></cell><cell>257</cell></row><row><cell>CCXXVII.</cell><cell>V<emph type="italics"/>it&aelig; iuxta numerum perfectionum in comparatione ad cogitationem no-&longs;tram proportionem quand am habent.<emph.end type="italics"/></cell><cell>259</cell></row><row><cell>CCXXVIII.</cell><cell>P<emph type="italics"/>roportionem &longs;cienti&aelig; futurorum &amp; c&aelig;terorum occultorum con&longs;idera-re.<emph.end type="italics"/></cell><cell>260</cell></row><row><cell>CCXXIX.</cell><cell>I<emph type="italics"/>ncorporea omnia unum &longs;unt, neque numerus e&longs;t eorum.<emph.end type="italics"/></cell><cell>261</cell></row><row><cell>CCXXX.</cell><cell>P<emph type="italics"/>roportio incorporeorum a&longs;cendentium &longs;emper maior e&longs;t.<emph.end type="italics"/></cell><cell>262</cell></row><row><cell>CCXXXI.</cell><cell>T<emph type="italics"/>res e&longs;&longs;e mundos atque inter ip&longs;os nullam e&longs;&longs;e proportionem: nec numero cos   definiri.<emph.end type="italics"/></cell><cell>263</cell></row><row><cell>CCXXXII.</cell><cell>O<emph type="italics"/>mnis motus naturalis quanto uelocior e&longs;t tanto propior e&longs;t &amp; magis &longs;imil   limus quieti.<emph.end type="italics"/></cell><cell>264</cell></row><row><cell>CCXXXIII.</cell><cell>Q<emph type="italics"/>uod e&longs;t in mundo incorporeo &aelig;ternum e&longs;t, beatum, &longs;ecurum, immutabile   &longs;ecundum locum, &longs;olum iuxta e&longs;&longs;entiam fit: iuxta quod uelut &agrave; leui &longs;u-&longs;urro aqu&aelig; &amp; aura &aelig;&longs;tiua demulcetur.<emph.end type="italics"/></cell><cell>270</cell></row></table><p type="head">
  
 <s>FINIS.</s></p><pb pagenum="1"/><p type="head"> <s>FINIS.<!-- KEEP S--></s></p><pb pagenum="1"/><p type="head">
  
 <s>HIERONYMI CAR <lb/>DANI MEDIOLANENSIS, CI&shy;<lb/>VI'SQVE BONONIENSIS, MEDICI&shy;<lb/>de Proportionibus, &longs;eu Ope&shy;<lb/>ris Perfecti <lb/>LIBER QVINTVS.</s></p><p type="main"> <s>HIERONYMI CAR <lb/>DANI MEDIOLANENSIS, CI&shy;<lb/>VI'SQVE BONONIENSIS, MEDICI&shy;<lb/>de Proportionibus, &longs;eu Ope&shy;<lb/>ris Perfecti <lb/>LIBER QVINTVS.</s></p><p type="main">
  
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 <s>Proportiones per &longs;imilitudinem <expan abbr="dic&utilde;tur">dicuntur</expan>, <lb/>c&ugrave;m quantitas quantitati <expan abbr="compara&ttilde;">comparatur</expan> alterius <lb/>generis, cui fingitur &aelig;qualis e&longs;&longs;e pote&longs;tate.<gap/></s></p><p type="main"> <s>Proportiones per &longs;imilitudinem <expan abbr="dic&utilde;tur">dicuntur</expan>, <lb/>c&ugrave;m quantitas quantitati <expan abbr="compara&ttilde;">comparatur</expan> alterius <lb/>generis, cui fingitur &aelig;qualis e&longs;&longs;e pote&longs;tate.<gap/></s></p><p type="main">
  
 <s>Velut &longs;i a b fingatur monas in comparatione <lb/>ad b c erit rectangulum a c &aelig;quale line&aelig; b c.</s></p><figure></figure><p type="main"> <s>Velut &longs;i a b fingatur monas in comparatione <lb/>ad b c erit rectangulum a c &aelig;quale line&aelig; b c.<!-- KEEP S--></s></p><figure/><p type="main">
  
 <s>Tertia diffinitio.</s></p><p type="main"> <s>Tertia diffinitio.</s></p><p type="main">
  
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 <s>Proportiones pote&longs;tate <expan abbr="dicun&ttilde;">dicuntur</expan>, qu&aelig;&longs;ub comparatione aliarum <lb/><expan abbr="quantitat&utilde;">quantitatum</expan> nece&longs;&longs;ariam habentium <expan abbr="c&otilde;nexionem">connexionem</expan> <expan abbr="&longs;ol&utilde;">&longs;olum</expan> <expan abbr="cogno&longs;cun&ttilde;">cogno&longs;cuntur</expan>.</s></p><p type="main"> <s>Proportiones pote&longs;tate <expan abbr="dicun&ttilde;">dicuntur</expan>, qu&aelig;&longs;ub comparatione aliarum <lb/><expan abbr="quantitat&utilde;">quantitatum</expan> nece&longs;&longs;ariam habentium <expan abbr="c&otilde;nexionem">connexionem</expan> <expan abbr="&longs;ol&utilde;">&longs;olum</expan> <expan abbr="cogno&longs;cun&ttilde;">cogno&longs;cuntur</expan>.</s></p><p type="main">
  
 <s>H&aelig; autem &longs;unt aliquando eiu&longs;dem generis, cum primis ut nu&shy;<lb/>meri: aliquand&ograve; alterius, ut linearum &amp; &longs;uperficierum, angulorum, <lb/>&amp; arcuum: aliquando eiu&longs;dem generis, &amp; diuen&longs;arum &longs;pecierum, <lb/>ut arcuum per &longs;inus, qua utuntur A&longs;tronomi.</s></p><p type="main"> <s>H&aelig; autem &longs;unt aliquando eiu&longs;dem generis, cum primis ut nu&shy;<lb/>meri: aliquand&ograve; alterius, ut linearum &amp; &longs;uperficierum, angulorum, <lb/>&amp; arcuum: aliquando eiu&longs;dem generis, &amp; diuen&longs;arum &longs;pecierum, <lb/>ut arcuum per &longs;inus, qua utuntur A&longs;tronomi.<!-- KEEP S--></s></p><p type="main">
  
 <s>Octaua diffinitio.</s></p><p type="main"> <s>Octaua diffinitio.</s></p><p type="main">
  
 <s>Proportio homonyma dicitur duarum quantitatum diuer&longs;i ge&shy;</s></p><p type="main"> <s>Proportio homonyma dicitur duarum quantitatum diuer&longs;i ge&shy;</s></p><p type="main">
  
 <s><arrow.to.target n="marg1"></arrow.to.target><lb/>neris, &longs;ed alterius a b altero dependentium, uelut motus ad tem&shy;<pb pagenum="2"/>pus. </s> <s><arrow.to.target n="marg1"/><lb/>neris, &longs;ed alterius a b altero dependentium, uelut motus ad tem&shy;<pb pagenum="2"/>pus. </s>
  
 <s>Dicimus enim motum tardum, uel uelocem in comparatione <lb/>ad tempus.</s></p><p type="margin"> <s>Dicimus enim motum tardum, uel uelocem in comparatione <lb/>ad tempus.</s></p><p type="margin">
  
 <s><margin.target id="marg1"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg1"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Nona diffinitio.</s></p><p type="main"> <s>Nona diffinitio.</s></p><p type="main">
  
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 <s>Duodecima diffinitio.</s></p><p type="main"> <s>Duodecima diffinitio.</s></p><p type="main">
  
 <s>Proportionem in proportionem duci e&longs;t, quoties recto ordine <lb/>tres quantitates in ei&longs;dem collo <expan abbr="can&ttilde;">cantur</expan>: ut &longs;int tres quan <lb/><figure id="fig1"></figure><lb/>titates a b c dicetur proportio a ad c producta ex pro <lb/>portione a ad b &amp; b ad c, &amp; &longs;imiliter proportio c ad <lb/>a producitur ex proportione b ad a, &amp; c ad b.</s></p><p type="main"> <s>Proportionem in proportionem duci e&longs;t, quoties recto ordine <lb/>tres quantitates in ei&longs;dem collo <expan abbr="can&ttilde;">cantur</expan>: ut &longs;int tres quan <lb/><figure id="fig1"/><lb/>titates a b c dicetur proportio a ad c producta ex pro <lb/>portione a ad b &amp; b ad c, &amp; &longs;imiliter proportio c ad <lb/>a producitur ex proportione b ad a, &amp; c ad b.</s></p><p type="main">
  
 <s>Tertiadecima diffinitio.</s></p><p type="main"> <s>Tertiadecima diffinitio.</s></p><p type="main">
  
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 <s>Additio proportionum intelligitur quotiens duarum quanti&shy;<lb/>tatum ad unam tertiam, proportiones per aggregatum ip&longs;arum <lb/>quantitatum ad eandem coniunguntur.</s></p><p type="main"> <s>Additio proportionum intelligitur quotiens duarum quanti&shy;<lb/>tatum ad unam tertiam, proportiones per aggregatum ip&longs;arum <lb/>quantitatum ad eandem coniunguntur.</s></p><p type="main">
  
 <s>Velut &longs;i comparentur a b &amp; b c ad d, inde tota <lb/><figure id="fig2"></figure><lb/>a c ad d dicemus proportionem, ac ad d e&longs;&longs;e con <lb/><expan abbr="iunct&atilde;">iunctam</expan> ex duabus proportionibus a b ad d &amp; b c <lb/>ad <expan abbr="eand&etilde;">eandem</expan> d. </s> <s>Velut &longs;i comparentur a b &amp; b c ad d, inde tota <lb/><figure id="fig2"/><lb/>a c ad d dicemus proportionem, ac ad d e&longs;&longs;e con <lb/><expan abbr="iunct&atilde;">iunctam</expan> ex duabus proportionibus a b ad d &amp; b c <lb/>ad <expan abbr="eand&etilde;">eandem</expan> d. <!-- KEEP S--></s>
  
 <s>Hoc &amp; duo &longs;equentes &longs;icut &amp; du&ecedil; <expan abbr="anteced&etilde;tes">antecedentes</expan> demon&shy;<lb/>&longs;trabitur e&longs;&longs;e. </s> <s>Hoc &amp; duo &longs;equentes &longs;icut &amp; du&ecedil; <expan abbr="anteced&etilde;tes">antecedentes</expan> demon&shy;<lb/>&longs;trabitur e&longs;&longs;e. </s>
  
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 <s>Detractionem proportionis &agrave; proportione intelligimus fieri <lb/>per <expan abbr="detraction&etilde;">detractionem</expan> minoris quantitatis &agrave; maiore, comparatam ad ean&shy;<lb/>dem quantitatem.</s></p><p type="main"> <s>Detractionem proportionis &agrave; proportione intelligimus fieri <lb/>per <expan abbr="detraction&etilde;">detractionem</expan> minoris quantitatis &agrave; maiore, comparatam ad ean&shy;<lb/>dem quantitatem.</s></p><p type="main">
  
 <s>Velut in exemplo &longs;uperiore detracta proportione b c ad d ex <pb pagenum="3"/>proportione a c ad d, relinquetur proportio a b ad d. </s> <s>Velut in exemplo &longs;uperiore detracta proportione b c ad d ex <pb pagenum="3"/>proportione a c ad d, relinquetur proportio a b ad d. <!-- KEEP S--></s>
  
 <s>&amp; probatur <lb/>ex conuer&longs;ione pr&aelig;cedentis.</s></p><p type="main"> <s>&amp; probatur <lb/>ex conuer&longs;ione pr&aelig;cedentis.</s></p><p type="main">
  
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 <s>Inter quantitatem, &amp; defectum minorem quantitate, cuius e&longs;t de <lb/>fectus, e&longs;t proportio, quatenus e&longs;t quantitas. </s> <s>Inter quantitatem, &amp; defectum minorem quantitate, cuius e&longs;t de <lb/>fectus, e&longs;t proportio, quatenus e&longs;t quantitas. </s>
  
 <s>Sit a b linea, &amp; detra&shy;<lb/>cta quantitas b c, non maior a b &amp; d &longs;it alia qu&aelig;uis quantitas eiu&longs;&shy;<lb/><figure id="fig3"></figure><lb/><expan abbr="d&etilde;">dem</expan> generis, dico qu&ograve;d inter d &amp; b c e&longs;t propor&shy;<lb/>tio quatenus b c e&longs;t quantitas, quia &longs;unt eiu&longs;&shy;<lb/>dem generis ideo &longs;unt in aliqua proportione <lb/>per primam diffinitionem. </s> <s>Sit a b linea, &amp; detra&shy;<lb/>cta quantitas b c, non maior a b &amp; d &longs;it alia qu&aelig;uis quantitas eiu&longs;&shy;<lb/><figure id="fig3"/><lb/><expan abbr="d&etilde;">dem</expan> generis, dico qu&ograve;d inter d &amp; b c e&longs;t propor&shy;<lb/>tio quatenus b c e&longs;t quantitas, quia &longs;unt eiu&longs;&shy;<lb/>dem generis ideo &longs;unt in aliqua proportione <lb/>per primam diffinitionem. </s>
  
 <s>Sed ut b c e&longs;t defectus, nulla e&longs;t propor&shy;<lb/>tio: quia quanto b c augetur, tanto augetur proportio d ad b c, &amp; <lb/>hoc e&longs;t contra demon&longs;trata ab Euclide.</s></p><p type="main"> <s>Sed ut b c e&longs;t defectus, nulla e&longs;t propor&shy;<lb/>tio: quia quanto b c augetur, tanto augetur proportio d ad b c, &amp; <lb/>hoc e&longs;t contra demon&longs;trata ab Euclide.<!-- KEEP S--></s></p><p type="main">
  
 <s>Quinta animi communis &longs;ententia.</s></p><p type="main"> <s>Quinta animi communis &longs;ententia.</s></p><p type="main">
  
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 <s>Decima petitio.</s></p><p type="main"> <s>Decima petitio.</s></p><p type="main">
  
 <s>In quouis genere quantitatum &longs;umere po&longs;&longs;e quantitatem, qu&aelig; <lb/><arrow.to.target n="marg2"></arrow.to.target><lb/>&longs;e habeat ad monadem in proportione data. </s> <s>In quouis genere quantitatum &longs;umere po&longs;&longs;e quantitatem, qu&aelig; <lb/><arrow.to.target n="marg2"/><lb/>&longs;e habeat ad monadem in proportione data. </s>
  
 <s>Similem huic propo&shy;<lb/>nit Euclides in lineis generaliter: nos autem contr&agrave; generaliter in <lb/>omnibus quantitatibus, &longs;ed de monade tantum.</s></p><p type="margin"> <s>Similem huic propo&shy;<lb/>nit Euclides in lineis generaliter: nos autem contr&agrave; generaliter in <lb/>omnibus quantitatibus, &longs;ed de monade tantum.</s></p><p type="margin">
  
 <s><margin.target id="marg2"></margin.target>D<emph type="italics"/>uodecima <lb/>&longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg2"/>D<emph type="italics"/>uodecima <lb/>&longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><!-- REMOVE S-->Vndecima petitio.</s></p><p type="main">
  
 <s>Vndecima petitio.</s></p><p type="main"> 
  
 <s>Monadem in quancunque quantitatem ductam &aelig;quale ip&longs;i pro&shy;<lb/>ducere. </s> <s>Monadem in quancunque quantitatem ductam &aelig;quale ip&longs;i pro&shy;<lb/>ducere. </s>
  
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 <s>Idem dico de diui&longs;ione. </s> <s>Idem dico de diui&longs;ione. </s>
  
 <s>Aequalitas etiam ducta, uel diuidens non <lb/><arrow.to.target n="marg3"></arrow.to.target><lb/>mutat proportionem: nec quantitatem ip&longs;am, igitur monas &aelig;qua&shy;<lb/>litatem refert. </s> <s>Aequalitas etiam ducta, uel diuidens non <lb/><arrow.to.target n="marg3"/><lb/>mutat proportionem: nec quantitatem ip&longs;am, igitur monas &aelig;qua&shy;<lb/>litatem refert. </s>
  
 <s>Quod etiam e&longs;t per&longs;picuum ex &longs;upradictis.</s></p><p type="margin"> <s>Quod etiam e&longs;t per&longs;picuum ex &longs;upradictis.</s></p><p type="margin">
  
 <s><margin.target id="marg3"></margin.target>S<emph type="italics"/>ecunda ani <lb/>mi <expan abbr="c&otilde;munis">communis</expan> <lb/>&longs;ententia.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg3"/>S<emph type="italics"/>ecunda ani <lb/>mi <expan abbr="c&otilde;munis">communis</expan> <lb/>&longs;ententia.<emph.end type="italics"/></s></p><p type="main">
  
 <s>Duodecima petitio.</s></p><p type="main"> <s>Duodecima petitio.</s></p><p type="main">
  
 <s>Cum fuerint quatuor quantitates &amp; ad primam, &amp; tertiam &aelig;qu&egrave; <lb/>multiplicibus a&longs;&longs;umptis, item que ad &longs;ecundam &amp; quartam, &amp; &longs;i mul&shy;<lb/>tiplex prim&aelig; maius e&longs;t multiplici &longs;ecund&aelig;, multiplex terti&aelig; &longs;it ma&shy;<lb/>ius multiplici quart&aelig;, &amp; &longs;i minus minus, &amp; &longs;i &aelig;quale &aelig;quale, idque<lb/>&longs;emper quouis modo a&longs;&longs;umptis his proportionibus ad primam &amp; <lb/>tertiam, &amp; ad &longs;ecundam &amp; quartam erit proportio prim&aelig; ad &longs;ecun<lb/>dam, ut terti&aelig; ad quartam. </s> <s>Cum fuerint quatuor quantitates &amp; ad primam, &amp; tertiam &aelig;qu&egrave; <lb/>multiplicibus a&longs;&longs;umptis, item que ad &longs;ecundam &amp; quartam, &amp; &longs;i mul&shy;<lb/>tiplex prim&aelig; maius e&longs;t multiplici &longs;ecund&aelig;, multiplex terti&aelig; &longs;it ma&shy;<lb/>ius multiplici quart&aelig;, &amp; &longs;i minus minus, &amp; &longs;i &aelig;quale &aelig;quale, idque<lb/>&longs;emper quouis modo a&longs;&longs;umptis his proportionibus ad primam &amp; <lb/>tertiam, &amp; ad &longs;ecundam &amp; quartam erit proportio prim&aelig; ad &longs;ecun<lb/>dam, ut terti&aelig; ad quartam. </s>
  
 <s>H&aelig;c etiam a&longs;&longs;umitur ab Euclide. </s> <s>H&aelig;c etiam a&longs;&longs;umitur ab Euclide. <!-- KEEP S--></s>
  
 <s>Et per <lb/><arrow.to.target n="marg4"></arrow.to.target><lb/>hanc intelligimus etiam conuer&longs;am.</s></p><p type="margin"> <s>Et per <lb/><arrow.to.target n="marg4"/><lb/>hanc intelligimus etiam conuer&longs;am.</s></p><p type="margin">
  
 <s><margin.target id="marg4"></margin.target>Q<emph type="italics"/>uinto<emph.end type="italics"/> E<emph type="italics"/>le. <lb/></s> <s><margin.target id="marg4"/>Q<emph type="italics"/>uinto<emph.end type="italics"/> E<emph type="italics"/>le. <lb/><!-- REMOVE S-->diff.<emph.end type="italics"/> 6.<!-- KEEP S--></s>
  
 <s>diff.<emph.end type="italics"/> 6.</s></p><p type="main"> </p><p type="main">
  
 <s>Tertiadecima petitio.</s></p><p type="main"> <s>Tertiadecima petitio.</s></p><p type="main">
  
 <s>Quantitates &aelig;quales, atque proportiones in qua&longs;uis quanti&shy;<lb/>tates duct&aelig; eandem &longs;eruant rationem. </s> <s>Quantitates &aelig;quales, atque proportiones in qua&longs;uis quanti&shy;<lb/>tates duct&aelig; eandem &longs;eruant rationem. </s>
  
 <s>Euclides hanc demon&longs;trat, <lb/>nos autem ad uitandum t&aelig;dium petimus concedi, &longs;ub qua in&shy;<lb/><arrow.to.target n="marg5"></arrow.to.target><lb/>cluduntur diui&longs;io etiam additio, detractio, laterum omnium in&shy;<lb/>uentio.</s></p><p type="margin"> <s>Euclides hanc demon&longs;trat, <lb/>nos autem ad uitandum t&aelig;dium petimus concedi, &longs;ub qua in&shy;<lb/><arrow.to.target n="marg5"/><lb/>cluduntur diui&longs;io etiam additio, detractio, laterum omnium in&shy;<lb/>uentio.</s></p><p type="margin">
  
 <s><margin.target id="marg5"></margin.target>Q<emph type="italics"/>uarta quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg5"/>Q<emph type="italics"/>uarta quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main">
  
 <s>Quartadecima petitio.</s></p><p type="main"> <s>Quartadecima petitio.</s></p><p type="main">
  
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 <s>Proportionem in proportionem duci e&longs;t &longs;uperiores nume&shy;<lb/>ros atque inferiores inuicem ducere.</s></p><pb pagenum="7"/><p type="main"> <s>Proportionem in proportionem duci e&longs;t &longs;uperiores nume&shy;<lb/>ros atque inferiores inuicem ducere.</s></p><pb pagenum="7"/><p type="main">
  
 <s>Sit proportio line&aelig; a ad lineam b, ut anguli cad angulum d, &longs;ta&shy;<lb/><arrow.to.target n="marg6"></arrow.to.target><lb/>tuatur e monas in genere a <lb/><figure id="fig4"></figure><lb/>b, &amp; fiat fad e, ut cad d, &amp; du <lb/><arrow.to.target n="marg7"></arrow.to.target><lb/>catur<gap/>a in f &amp; b in e, &amp; pro&shy;<lb/>ducantur g &amp; h. </s> <s>Sit proportio line&aelig; a ad lineam b, ut anguli cad angulum d, &longs;ta&shy;<lb/><arrow.to.target n="marg6"/><lb/>tuatur e monas in genere a <lb/><figure id="fig4"/><lb/>b, &amp; fiat fad e, ut cad d, &amp; du <lb/><arrow.to.target n="marg7"/><lb/>catur<gap/>a in f &amp; b in e, &amp; pro&shy;<lb/>ducantur g &amp; h. </s>
  
 <s>Quia ergo <lb/><arrow.to.target n="marg8"></arrow.to.target><lb/>fe&longs;t proportio ip&longs;a, erit g ad <lb/><arrow.to.target n="marg9"></arrow.to.target><lb/>a ut c ad d, &longs;ed h e&longs;t &aelig;qualis <lb/>b, igitur a ad h ut ad b. </s> <s>Quia ergo <lb/><arrow.to.target n="marg8"/><lb/>fe&longs;t proportio ip&longs;a, erit g ad <lb/><arrow.to.target n="marg9"/><lb/>a ut c ad d, &longs;ed h e&longs;t &aelig;qualis <lb/>b, igitur a ad h ut ad b. </s>
  
 <s>Du&shy;<lb/>cta ergo dicetur proportio a <lb/><arrow.to.target n="marg10"></arrow.to.target><lb/>ad b in proportionem c ad d <lb/>ducendo terminos proportionis, &longs;eu quantitatis recta &longs;cilicet &longs;u&shy;<lb/>periores cum &longs;uperioribus, &amp; inferiores cum inferioribus. </s> <s>Du&shy;<lb/>cta ergo dicetur proportio a <lb/><arrow.to.target n="marg10"/><lb/>ad b in proportionem c ad d <lb/>ducendo terminos proportionis, &longs;eu quantitatis recta &longs;cilicet &longs;u&shy;<lb/>periores cum &longs;uperioribus, &amp; inferiores cum inferioribus. </s>
  
 <s>Nam &longs;i <lb/><arrow.to.target n="marg11"></arrow.to.target><lb/>rur&longs;um con&longs;tituantur fad e ut a ad b c&ugrave;m f &longs;it proportio, &amp; k ad f ut <lb/><arrow.to.target n="marg12"></arrow.to.target><lb/>c ad d, erit k ad e, ut g ad h, k autem fit ex ductu proportionis a ad b, <lb/>qu&aelig; e&longs;t fin proportionem c ad d, liquet igitur propo&longs;itum.</s></p><p type="margin"> <s>Nam &longs;i <lb/><arrow.to.target n="marg11"/><lb/>rur&longs;um con&longs;tituantur fad e ut a ad b c&ugrave;m f &longs;it proportio, &amp; k ad f ut <lb/><arrow.to.target n="marg12"/><lb/>c ad d, erit k ad e, ut g ad h, k autem fit ex ductu proportionis a ad b, <lb/>qu&aelig; e&longs;t fin proportionem c ad d, liquet igitur propo&longs;itum.</s></p><p type="margin">
  
 <s><margin.target id="marg6"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="margin"> <s><margin.target id="marg6"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg7"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 9. P<emph type="italics"/>etit.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg7"/>P<emph type="italics"/>er<emph.end type="italics"/> 9. P<emph type="italics"/>etit.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg8"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg8"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg9"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 8. P<emph type="italics"/>etit.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg9"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. P<emph type="italics"/>etit.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg10"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 2. A<emph type="italics"/>ni&shy;<lb/><gap/>i &longs;entent.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg10"/>P<emph type="italics"/>er<emph.end type="italics"/> 2. A<emph type="italics"/>ni&shy;<lb/><gap/>i &longs;entent.<emph.end type="italics"/></s></p><p type="margin">
  
 <s><margin.target id="marg11"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 11. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg11"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. P<emph type="italics"/>et.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg12"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 8. P<emph type="italics"/>etit.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg12"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. P<emph type="italics"/>etit.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main">
  
 <s>Propo&longs;itio <expan abbr="&longs;ec&utilde;nda">&longs;ecunnda</expan>.</s></p><p type="main"> <s>Propo&longs;itio <expan abbr="&longs;ec&utilde;nda">&longs;ecunnda</expan>.</s></p><p type="main">
  
 <s>Proportio extremorum producitur ex intermedijs.<lb/><arrow.to.target n="marg13"></arrow.to.target></s></p><p type="margin"> <s>Proportio extremorum producitur ex intermedijs.<lb/><arrow.to.target n="marg13"/></s></p><p type="margin">
  
 <s><margin.target id="marg13"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg13"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Sint a b c quantitates dico proportio&shy;<lb/><figure id="fig5"></figure><lb/>nem a ad c, produci ex proportione a ad b </s></p><p type="main"> <s>Sint a b c quantitates dico proportio&shy;<lb/><figure id="fig5"/><lb/>nem a ad c, produci ex proportione a ad b </s></p><p type="main">
  
 <s><arrow.to.target n="marg14"></arrow.to.target><lb/>&amp; b ad c, &longs;tatuantur totidem &agrave; monade d e <lb/>f, er&uacute;ntque ex demon&longs;trantis ab Euclide in <lb/>quinto <expan abbr="Elem&etilde;torum">Elementorum</expan> in eadem proportio&shy;<lb/>ne, ftatuatur ergo d prima quantitas e &longs;e&shy;<lb/>cunda &amp; tertia f quarta. </s> <s><arrow.to.target n="marg14"/><lb/>&amp; b ad c, &longs;tatuantur totidem &agrave; monade d e <lb/>f, er&uacute;ntque ex demon&longs;trantis ab Euclide in <lb/>quinto <expan abbr="Elem&etilde;torum">Elementorum</expan> in eadem proportio&shy;<lb/>ne, ftatuatur ergo d prima quantitas e &longs;e&shy;<lb/>cunda &amp; tertia f quarta. </s>
  
 <s>eritq&uacute;e per pr&aelig;ce&shy;<lb/><arrow.to.target n="marg15"></arrow.to.target><lb/>dentem proportio productorum ex d in e <lb/>&amp; &longs;it g, &amp; in f &amp; &longs;it h, producta ex propor&shy;<lb/>tionibus d ad e &amp; e ad f, quare ex propor&shy;<lb/>tionibus a ad b &amp; b ad e, &longs;ed ex dictis cum <lb/>e &longs;it eadem, erit proportio d ad f, ut g ad h &amp; proportio, d ad f per <lb/>&aelig;quam proportionem ab Euclide demon&longs;tratam, ut a ad c, igitur <lb/><arrow.to.target n="marg16"></arrow.to.target><lb/>proportio a ad c producitur ex proportionibus a ad b &amp; b ad c, &amp; <lb/>e&longs;t proportio ip&longs;a a ad c d numerus, ut o&longs;ten&longs;um e&longs;t.</s></p><p type="margin"> <s>eritq&uacute;e per pr&aelig;ce&shy;<lb/><arrow.to.target n="marg15"/><lb/>dentem proportio productorum ex d in e <lb/>&amp; &longs;it g, &amp; in f &amp; &longs;it h, producta ex propor&shy;<lb/>tionibus d ad e &amp; e ad f, quare ex propor&shy;<lb/>tionibus a ad b &amp; b ad e, &longs;ed ex dictis cum <lb/>e &longs;it eadem, erit proportio d ad f, ut g ad h &amp; proportio, d ad f per <lb/>&aelig;quam proportionem ab Euclide demon&longs;tratam, ut a ad c, igitur <lb/><arrow.to.target n="marg16"/><lb/>proportio a ad c producitur ex proportionibus a ad b &amp; b ad c, &amp; <lb/>e&longs;t proportio ip&longs;a a ad c d numerus, ut o&longs;ten&longs;um e&longs;t.</s></p><p type="margin">
  
 <s><margin.target id="marg14"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 6. <emph type="italics"/>&amp;<emph.end type="italics"/> 9. <lb/>P<emph type="italics"/>etit.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg14"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. <emph type="italics"/>&amp;<emph.end type="italics"/> 9. <lb/>P<emph type="italics"/>etit.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg15"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 13. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg15"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. P<emph type="italics"/>et.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg16"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 13. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg16"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. P<emph type="italics"/>et.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main">
  
 <s>Ex hoc &longs;equitur, qu&ograve;d c&ugrave;m fuerit quantitas tertia monas ex pro&shy;<lb/><arrow.to.target n="marg17"></arrow.to.target><lb/>portionibus inuicem ductis producetur prima quantitas.<lb/><arrow.to.target n="marg18"></arrow.to.target></s></p><p type="margin"> <s>Ex hoc &longs;equitur, qu&ograve;d c&ugrave;m fuerit quantitas tertia monas ex pro&shy;<lb/><arrow.to.target n="marg17"/><lb/>portionibus inuicem ductis producetur prima quantitas.<lb/><arrow.to.target n="marg18"/></s></p><p type="margin">
  
 <s><margin.target id="marg17"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="margin"> <s><margin.target id="marg17"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.<!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg18"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3</s></p><p type="main"> <s><margin.target id="marg18"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3</s></p><p type="main">
  
 <s>Ex hoc &longs;equitur, qu&ograve;d conuer&longs;a proportio producitur ex con&shy;<lb/>uer&longs;is proportionibus.</s></p><p type="main"> <s>Ex hoc &longs;equitur, qu&ograve;d conuer&longs;a proportio producitur ex con&shy;<lb/>uer&longs;is proportionibus.</s></p><p type="main">
  
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 <s>Si proportio ex duabus proportionibus in quatuor terminis <lb/>producatur, ip&longs;a uer&ograve; proportio inter duas alias quantitates fue&shy;<pb pagenum="8"/>rit con&longs;tituta: con&longs;urgent trecenti &longs;exaginta modi productionis <lb/>proportionis.</s></p><p type="main"> <s>Si proportio ex duabus proportionibus in quatuor terminis <lb/>producatur, ip&longs;a uer&ograve; proportio inter duas alias quantitates fue&shy;<pb pagenum="8"/>rit con&longs;tituta: con&longs;urgent trecenti &longs;exaginta modi productionis <lb/>proportionis.</s></p><p type="main">
  
 <s><arrow.to.target n="marg19"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg19"/></s></p><p type="margin">
  
 <s><margin.target id="marg19"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg19"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>H&ecedil;c propo&longs;itio ut pr&aelig;cedens &amp; <expan abbr="&longs;equ&etilde;tes">&longs;equentes</expan> tres ab Alchindo &longs;um&shy;<lb/>pt&aelig; &longs;unt, &amp; ab eo demon&longs;trantur. </s> <s>H&ecedil;c propo&longs;itio ut pr&aelig;cedens &amp; <expan abbr="&longs;equ&etilde;tes">&longs;equentes</expan> tres ab Alchindo &longs;um&shy;<lb/>pt&aelig; &longs;unt, &amp; ab eo demon&longs;trantur. </s>
  
 <s>Sit ergo proportio a ad b, pro&shy;<lb/><arrow.to.target n="table2"></arrow.to.target><lb/>ducta ex proportione c ad d &amp; e ad f, con&longs;tat <lb/>qu&ograve;d cum &longs;int &longs;ex quantitates, qu&ograve;d fieri pote&shy;<lb/>runt quindecim coniugationes, quas po&longs;ui &agrave; la&shy;<lb/>tere facilitatis gratia, quibus re&longs;pondent totidem <lb/><arrow.to.target n="table3"></arrow.to.target><lb/>conuer&longs;&aelig;: erunt ergo triginta. </s> <s>Sit ergo proportio a ad b, pro&shy;<lb/><arrow.to.target n="table2"/><lb/>ducta ex proportione c ad d &amp; e ad f, con&longs;tat <lb/>qu&ograve;d cum &longs;int &longs;ex quantitates, qu&ograve;d fieri pote&shy;<lb/>runt quindecim coniugationes, quas po&longs;ui &agrave; la&shy;<lb/>tere facilitatis gratia, quibus re&longs;pondent totidem <lb/><arrow.to.target n="table3"/><lb/>conuer&longs;&aelig;: erunt ergo triginta. </s>
  
 <s>Singul&aelig; autem ha <lb/>rum produci po&longs;&longs;unt duodecim modis: ductis <lb/>duodecim in triginta, fiunt trecenti &longs;exaginta mo <lb/>di. </s> <s>Singul&aelig; autem ha <lb/>rum produci po&longs;&longs;unt duodecim modis: ductis <lb/>duodecim in triginta, fiunt trecenti &longs;exaginta mo <lb/>di. </s>
  
 <s>Et hoc e&longs;t clarum per&longs;e, modo <expan abbr="dem&otilde;&longs;tremus">demon&longs;tremus</expan>, <lb/>quod &longs;inguli horum modorum po&longs;sint produ&shy;<lb/>ci duodecim modis, &amp; capiamus ab primam qu&ecedil; <lb/>pote&longs;t produci ex c d &amp; e f: Item ambabus con&shy;<lb/>uer&longs;is d c &amp; fe: &amp; rur&longs;us altera recta altera con&shy;<lb/>uer&longs;a: &amp; hoc bifariam c d &amp; f e, &amp; d c &amp; e f, &longs;unt er&shy;<lb/>go iam quatuor modi. </s> <s>Et hoc e&longs;t clarum per&longs;e, modo <expan abbr="dem&otilde;&longs;tremus">demon&longs;tremus</expan>, <lb/>quod &longs;inguli horum modorum po&longs;sint produ&shy;<lb/>ci duodecim modis, &amp; capiamus ab primam qu&ecedil; <lb/>pote&longs;t produci ex c d &amp; e f: Item ambabus con&shy;<lb/>uer&longs;is d c &amp; fe: &amp; rur&longs;us altera recta altera con&shy;<lb/>uer&longs;a: &amp; hoc bifariam c d &amp; f e, &amp; d c &amp; e f, &longs;unt er&shy;<lb/>go iam quatuor modi. </s>
  
 <s>Totidem ex c e &amp; d f, toti&shy;<lb/>demque ex c f &amp; d e, igitur erunt duodecim mo&shy;<lb/>di, quibus produci po&longs;&longs;e intelligitur propor&shy;<lb/>tio a ad b.</s></p><table><table.target id="table2"></table.target><row><cell>a</cell><cell>b</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>c</cell><cell>d</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>e</cell><cell>f</cell></row><row><cell>---</cell><cell>---</cell></row></table><table><table.target id="table3"></table.target><row><cell>a b</cell><cell>b a</cell></row><row><cell>a c</cell><cell>c a</cell></row><row><cell>a d</cell><cell>d a</cell></row><row><cell>a e</cell><cell>e a</cell></row><row><cell>a f</cell><cell>f a</cell></row><row><cell>b c</cell><cell>c b</cell></row><row><cell>b d</cell><cell>d b</cell></row><row><cell>b e</cell><cell>e b</cell></row><row><cell>b f</cell><cell>f b</cell></row><row><cell>c d</cell><cell>d c</cell></row><row><cell>c e</cell><cell>e c</cell></row><row><cell>c f</cell><cell>f c</cell></row><row><cell>d e</cell><cell>e d</cell></row><row><cell>d f</cell><cell>f d</cell></row><row><cell>e f</cell><cell>f e</cell></row><row><cell>direc.</cell><cell>conuer.</cell></row></table><p type="main"> <s>Totidem ex c e &amp; d f, toti&shy;<lb/>demque ex c f &amp; d e, igitur erunt duodecim mo&shy;<lb/>di, quibus produci po&longs;&longs;e intelligitur propor&shy;<lb/>tio a ad b.</s></p><table><table.target id="table2"/><row><cell>a</cell><cell>b</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>c</cell><cell>d</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>e</cell><cell>f</cell></row><row><cell>---</cell><cell>---</cell></row></table><table><table.target id="table3"/><row><cell>a b</cell><cell>b a</cell></row><row><cell>a c</cell><cell>c a</cell></row><row><cell>a d</cell><cell>d a</cell></row><row><cell>a e</cell><cell>e a</cell></row><row><cell>a f</cell><cell>f a</cell></row><row><cell>b c</cell><cell>c b</cell></row><row><cell>b d</cell><cell>d b</cell></row><row><cell>b e</cell><cell>e b</cell></row><row><cell>b f</cell><cell>f b</cell></row><row><cell>c d</cell><cell>d c</cell></row><row><cell>c e</cell><cell>e c</cell></row><row><cell>c f</cell><cell>f c</cell></row><row><cell>d e</cell><cell>e d</cell></row><row><cell>d f</cell><cell>f d</cell></row><row><cell>e f</cell><cell>f e</cell></row><row><cell>direc.</cell><cell>conuer.</cell></row></table><p type="main">
  
 <s>Propo&longs;itio quarta.</s></p><p type="main"> <s>Propo&longs;itio quarta.</s></p><p type="main">
  
 <s>Si fuerit proportio primi ad &longs;ecundum produ&shy;<lb/>cta ex proportionibus tertij ad quartum, &amp; quin <lb/>ti ad &longs;extum, producetur etiam ex proportione <lb/>tertij ad &longs;extum, &amp; quinti ad quartum.</s></p><p type="main"> <s>Si fuerit proportio primi ad &longs;ecundum produ&shy;<lb/>cta ex proportionibus tertij ad quartum, &amp; quin <lb/>ti ad &longs;extum, producetur etiam ex proportione <lb/>tertij ad &longs;extum, &amp; quinti ad quartum.</s></p><p type="main">
  
 <s>Sit proportio a b producta ex proportioni&shy;<lb/><arrow.to.target n="table4"></arrow.to.target><lb/>bus c ad d, &amp; e ad f, dico quod etiam erit produ&shy;</s></p><table><table.target id="table4"></table.target><row><cell>a</cell><cell>b</cell><cell></cell></row><row><cell>c</cell><cell>e</cell><cell>g</cell></row><row><cell>d</cell><cell>f</cell><cell>h</cell></row><row><cell>---</cell><cell>---</cell><cell>---</cell></row><row><cell>c</cell><cell>e</cell><cell>g</cell></row><row><cell>f</cell><cell>d</cell><cell>h</cell></row></table><p type="main"> <s>Sit proportio a b producta ex proportioni&shy;<lb/><arrow.to.target n="table4"/><lb/>bus c ad d, &amp; e ad f, dico quod etiam erit produ&shy;</s></p><table><table.target id="table4"/><row><cell>a</cell><cell>b</cell><cell/></row><row><cell>c</cell><cell>e</cell><cell>g</cell></row><row><cell>d</cell><cell>f</cell><cell>h</cell></row><row><cell>---</cell><cell>---</cell><cell>---</cell></row><row><cell>c</cell><cell>e</cell><cell>g</cell></row><row><cell>f</cell><cell>d</cell><cell>h</cell></row></table><p type="main">
  
 <s><arrow.to.target n="marg20"></arrow.to.target><lb/>cta ex proportionibus c ad f, &amp; e ad d, di&longs;ponan&shy;<lb/>tur ut in figura &amp; fiat ex c in e g, &amp; ex d in fh, ergo <lb/><arrow.to.target n="marg21"></arrow.to.target><lb/>per primam harum g ad h ut a ad b, &longs;ed per pr&aelig;&shy;<lb/>&longs;uppo&longs;ita in &longs;ecunda productione etiam prode&shy;<lb/>unt g &amp; h, igitur per primam propo&longs;itionem ha&shy;<lb/>rum a ad b proportio producitur ex proportionibus c ad f terti&aelig; <lb/>&longs;cilicet ad &longs;extam, &amp; e ad d quint&ecedil; ad quartam, quod fuit <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan>.</s></p><p type="margin"> <s><arrow.to.target n="marg20"/><lb/>cta ex proportionibus c ad f, &amp; e ad d, di&longs;ponan&shy;<lb/>tur ut in figura &amp; fiat ex c in e g, &amp; ex d in fh, ergo <lb/><arrow.to.target n="marg21"/><lb/>per primam harum g ad h ut a ad b, &longs;ed per pr&aelig;&shy;<lb/>&longs;uppo&longs;ita in &longs;ecunda productione etiam prode&shy;<lb/>unt g &amp; h, igitur per primam propo&longs;itionem ha&shy;<lb/>rum a ad b proportio producitur ex proportionibus c ad f terti&aelig; <lb/>&longs;cilicet ad &longs;extam, &amp; e ad d quint&ecedil; ad quartam, quod fuit <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan>.</s></p><p type="margin">
  
 <s><margin.target id="marg20"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>petit.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg20"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>petit.<emph.end type="italics"/></s></p><p type="margin">
  
 <s><margin.target id="marg21"></margin.target>I<emph type="italics"/>n<emph.end type="italics"/> 13. <emph type="italics"/>petit.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg21"/>I<emph type="italics"/>n<emph.end type="italics"/> 13. <emph type="italics"/>petit.<emph.end type="italics"/></s></p><p type="main">
  
 <s>Propo&longs;itio quinta.</s></p><p type="main"> <s>Propo&longs;itio quinta.</s></p><p type="main">
  
 <s>Si fuerit proportio primi ad &longs;ecundum producta ex proportio&shy;<lb/>ne tertij ad quartum, &amp; quinta ad &longs;extum: erit proportio tertij ad <lb/>&longs;extum producta ex proportionibus primi ad &longs;ecundum, &amp; quar&shy;<lb/>ti ad quintum.</s></p><pb pagenum="9"/><p type="main"> <s>Si fuerit proportio primi ad &longs;ecundum producta ex proportio&shy;<lb/>ne tertij ad quartum, &amp; quinta ad &longs;extum: erit proportio tertij ad <lb/>&longs;extum producta ex proportionibus primi ad &longs;ecundum, &amp; quar&shy;<lb/>ti ad quintum.</s></p><pb pagenum="9"/><p type="main">
  
 <s>Sit proportio a ad b producta ex proportio&shy;<lb/><arrow.to.target n="marg22"></arrow.to.target><lb/><arrow.to.target n="table5"></arrow.to.target><lb/>nibus c ad d, &amp; e ad f, dico quod proportio c ad <lb/>f producitur ex proportione a ad b &amp; d ad e. </s> <s>Sit proportio a ad b producta ex proportio&shy;<lb/><arrow.to.target n="marg22"/><lb/><arrow.to.target n="table5"/><lb/>nibus c ad d, &amp; e ad f, dico quod proportio c ad <lb/>f producitur ex proportione a ad b &amp; d ad e. </s>
  
 <s>In&shy;<lb/>terponam d e inter c &amp; f, eritque ex &longs;ecunda pro&shy;<lb/>po&longs;itione repetita proportio c ad f producta ex <lb/>tribus proportionibus c ad d, d ad e, e ad f, &longs;ed <lb/>proportiones c ad d, &amp; e ad f producunt pro&shy;<lb/>portionem a ad b, igitur proportio c ad f produ <lb/>citur ex proportionibus a ad b, &amp; e ad f.<lb/><arrow.to.target n="table6"></arrow.to.target></s></p><p type="margin"> <s>In&shy;<lb/>terponam d e inter c &amp; f, eritque ex &longs;ecunda pro&shy;<lb/>po&longs;itione repetita proportio c ad f producta ex <lb/>tribus proportionibus c ad d, d ad e, e ad f, &longs;ed <lb/>proportiones c ad d, &amp; e ad f producunt pro&shy;<lb/>portionem a ad b, igitur proportio c ad f produ <lb/>citur ex proportionibus a ad b, &amp; e ad f.<lb/><arrow.to.target n="table6"/></s></p><p type="margin">
  
 <s><margin.target id="marg22"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><table><table.target id="table5"></table.target><row><cell>a</cell><cell>b</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>c</cell><cell>e</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>d</cell><cell>f</cell></row><row><cell>---</cell><cell>---</cell></row></table><table><table.target id="table6"></table.target><row><cell>c</cell></row><row><cell>-----</cell></row><row><cell>d</cell></row><row><cell>-----</cell></row><row><cell>e</cell></row><row><cell>-----</cell></row><row><cell>f</cell></row><row><cell>-----</cell></row></table><p type="main"> <s><margin.target id="marg22"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><table><table.target id="table5"/><row><cell>a</cell><cell>b</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>c</cell><cell>e</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>d</cell><cell>f</cell></row><row><cell>---</cell><cell>---</cell></row></table><table><table.target id="table6"/><row><cell>c</cell></row><row><cell>-----</cell></row><row><cell>d</cell></row><row><cell>-----</cell></row><row><cell>e</cell></row><row><cell>-----</cell></row><row><cell>f</cell></row><row><cell>-----</cell></row></table><p type="main">
  
 <s>Propo&longs;itio &longs;exta.</s></p><p type="main"> <s>Propo&longs;itio &longs;exta.</s></p><p type="main">
  
 <s>Ex trecentis &longs;exaginta modis producenda&shy;<lb/>rum proportionum triginta &longs;ex tantum e&longs;&longs;e ne&shy;<lb/>ce&longs;&longs;arios.<lb/><arrow.to.target n="table7"></arrow.to.target></s></p><table><table.target id="table7"></table.target><row><cell>c</cell><cell>p</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>a</cell><cell>d</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>b</cell><cell>e</cell></row><row><cell>---</cell><cell>---</cell></row></table><p type="main"> <s>Ex trecentis &longs;exaginta modis producenda&shy;<lb/>rum proportionum triginta &longs;ex tantum e&longs;&longs;e ne&shy;<lb/>ce&longs;&longs;arios.<lb/><arrow.to.target n="table7"/></s></p><table><table.target id="table7"/><row><cell>c</cell><cell>p</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>a</cell><cell>d</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>b</cell><cell>e</cell></row><row><cell>---</cell><cell>---</cell></row></table><p type="main">
  
 <s>Per quartam enim proportio a ad b produ&shy;<lb/><arrow.to.target n="marg23"></arrow.to.target><lb/>citur bifariam, &amp; ex c ad d, &amp; e ad f, &amp; ex c ad f, &amp; <lb/>e ad d. </s> <s>Per quartam enim proportio a ad b produ&shy;<lb/><arrow.to.target n="marg23"/><lb/>citur bifariam, &amp; ex c ad d, &amp; e ad f, &amp; ex c ad f, &amp; <lb/>e ad d. <!-- KEEP S--></s>
  
 <s>&amp; perpr&aelig; cedentem c ad f producitur ex <lb/>a ad b, &amp; d ad e, &amp; per quartam rur&longs;us ex a ad e, <lb/>&amp; d ad b. </s> <s>&amp; perpr&aelig; cedentem c ad f producitur ex <lb/>a ad b, &amp; d ad e, &amp; per quartam rur&longs;us ex a ad e, <lb/>&amp; d ad b. </s>
  
Line 465 
Line 464 
  
 <s>Quare per pr&aelig;ceden&shy;<lb/>tem c ad f ex a ad e, &amp; d ad b, &amp; ita di&longs;ponemus <lb/>hos modos in tabula. </s> <s>Quare per pr&aelig;ceden&shy;<lb/>tem c ad f ex a ad e, &amp; d ad b, &amp; ita di&longs;ponemus <lb/>hos modos in tabula. </s>
  
 <s>Vides etiam <lb/><arrow.to.target n="table8"></arrow.to.target><lb/>aliquos modos non produci, ut pri&shy;<lb/>mi ad quartum nec ad &longs;extum, &amp; li&shy;<lb/>quet, qu&ograve;d c&ugrave;m &longs;int quindecim o&shy;<lb/>mnes modi qui produci po&longs;&longs;e intelli&shy;<lb/>guntur, &amp; nouem tantum producan&shy;<lb/>tur &longs;ex e&longs;&longs;e, qui non producuntur, quos <lb/>&longs;eor&longs;um in tabula coniunxi. </s> <s>Vides etiam <lb/><arrow.to.target n="table8"/><lb/>aliquos modos non produci, ut pri&shy;<lb/>mi ad quartum nec ad &longs;extum, &amp; li&shy;<lb/>quet, qu&ograve;d c&ugrave;m &longs;int quindecim o&shy;<lb/>mnes modi qui produci po&longs;&longs;e intelli&shy;<lb/>guntur, &amp; nouem tantum producan&shy;<lb/>tur &longs;ex e&longs;&longs;e, qui non producuntur, quos <lb/>&longs;eor&longs;um in tabula coniunxi. </s>
  
 <s>Et con&shy;<lb/>&longs;tat etiam, quod totidem conuer&longs;i &longs;ci&shy;<lb/>licet decem octo <expan abbr="produc&utilde;tur">producuntur</expan>, de qui&shy;<lb/>bus diximus, ut &longs;int omnes triginta <lb/>&longs;ex, qui con&longs;tat ex duabus propo&longs;i&shy;<lb/>tionibus pr&aelig;mi&longs;sis, &amp; hac tertia, <expan abbr="qu&atilde;">quam</expan> <lb/>adiungemus &longs;cilicet, qu&ograve;d propor&shy;<lb/>tio primi ad tertium producatur ex <lb/>proportionibus <expan abbr="&longs;ec&utilde;di">&longs;ecundi</expan> ad quartum, <lb/>&amp; quinti ad <expan abbr="&longs;ext&utilde;">&longs;extum</expan>. </s> <s>Et con&shy;<lb/>&longs;tat etiam, quod totidem conuer&longs;i &longs;ci&shy;<lb/>licet decem octo <expan abbr="produc&utilde;tur">producuntur</expan>, de qui&shy;<lb/>bus diximus, ut &longs;int omnes triginta <lb/>&longs;ex, qui con&longs;tat ex duabus propo&longs;i&shy;<lb/>tionibus pr&aelig;mi&longs;sis, &amp; hac tertia, <expan abbr="qu&atilde;">quam</expan> <lb/>adiungemus &longs;cilicet, qu&ograve;d propor&shy;<lb/>tio primi ad tertium producatur ex <lb/>proportionibus <expan abbr="&longs;ec&utilde;di">&longs;ecundi</expan> ad quartum, <lb/>&amp; quinti ad <expan abbr="&longs;ext&utilde;">&longs;extum</expan>. </s>
  
 <s>Hoc enim ex pr&aelig;&shy;<lb/>cedentibus non liquet: ben&egrave; liquet <lb/>permutatis ordinibus, quod &longs;i pro&shy;<lb/>portio primi ad tertium producitur, <pb pagenum="10"/>quod etiam propor&shy;<lb/><arrow.to.target n="marg24"></arrow.to.target><lb/>tio primi ad <expan abbr="quint&utilde;">quintum</expan>. <lb/></s> <s>Hoc enim ex pr&aelig;&shy;<lb/>cedentibus non liquet: ben&egrave; liquet <lb/>permutatis ordinibus, quod &longs;i pro&shy;<lb/>portio primi ad tertium producitur, <pb pagenum="10"/>quod etiam propor&shy;<lb/><arrow.to.target n="marg24"/><lb/>tio primi ad <expan abbr="quint&utilde;">quintum</expan>. <lb/></s>
  
 <s>Nam tertium, &amp; quin <lb/>tum, item que quartum, <lb/>&amp; &longs;extum non <expan abbr="diffe-r&utilde;t">diffe&shy;<lb/>runt</expan> ni&longs;i ordine uolun <lb/>tario. </s> <s>Nam tertium, &amp; quin <lb/>tum, item que quartum, <lb/>&amp; &longs;extum non <expan abbr="diffe-r&utilde;t">diffe&shy;<lb/>runt</expan> ni&longs;i ordine uolun <lb/>tario. </s>
  
 <s>Ergo interpo&longs;i&shy;<lb/>to e inter a, &amp; c per &longs;e&shy;<lb/>cundam propo&longs;itio&shy;<lb/>nem proportio a ad c <lb/>producitur ex proportionibus a ad <lb/>e, &amp; e ad c, ut ex demon&longs;tratis in pr&aelig;&shy;<lb/>&longs;enti proportio a ad c producitur ex <lb/>c ad f &amp; b ad d. </s> <s>Ergo interpo&longs;i&shy;<lb/>to e inter a, &amp; c per &longs;e&shy;<lb/>cundam propo&longs;itio&shy;<lb/>nem proportio a ad c <lb/>producitur ex proportionibus a ad <lb/>e, &amp; e ad c, ut ex demon&longs;tratis in pr&aelig;&shy;<lb/>&longs;enti proportio a ad c producitur ex <lb/>c ad f &amp; b ad d. <!-- KEEP S--></s>
  
 <s>Proportio ergo a ad <lb/>c producitur ex proportionibus e <lb/>ad c &amp; c ad f &amp; b ad d, at e ad c &amp; c ad <lb/>f producunt eam, qu&aelig; e&longs;t e ad f per <lb/><expan abbr="&longs;ec&utilde;dam">&longs;ecundam</expan> propo&longs;itionem. </s> <s>Proportio ergo a ad <lb/>c producitur ex proportionibus e <lb/>ad c &amp; c ad f &amp; b ad d, at e ad c &amp; c ad <lb/>f producunt eam, qu&aelig; e&longs;t e ad f per <lb/><expan abbr="&longs;ec&utilde;dam">&longs;ecundam</expan> propo&longs;itionem. </s>
  
 <s>Igitur pro&shy;<lb/>portio a ad c producitur ex propor&shy;<lb/>tionibus b ad d &longs;ecundi ad quartum, <lb/>&amp; e ad f quinti ad &longs;extum. </s> <s>Igitur pro&shy;<lb/>portio a ad c producitur ex propor&shy;<lb/>tionibus b ad d &longs;ecundi ad quartum, <lb/>&amp; e ad f quinti ad &longs;extum. </s>
  
 <s>H&aelig;c Al&shy;<lb/>chindus in &longs;uo libello: &longs;ed licet inge&shy;<lb/>nio &longs;a ualde: parum <expan abbr="tam&etilde;">tamen</expan> utilia olim <lb/><expan abbr="er&atilde;t">erant</expan> nece&longs;&longs;aria ad intelligendum ma&shy;<lb/>gnam <expan abbr="c&otilde;po&longs;itionem">compo&longs;itionem</expan> Ptolem&ecedil;i, nunc <lb/>po&longs;tquam Heber has &longs;ex quantita&shy;<lb/>tes traduxit ad quatuor, pror&longs;us h&aelig;c <lb/>&longs;cientia ulli u&longs;ui e&longs;&longs;e de&longs;ijt.<lb/><arrow.to.target n="table9"></arrow.to.target></s></p><p type="margin"> <s>H&aelig;c Al&shy;<lb/>chindus in &longs;uo libello: &longs;ed licet inge&shy;<lb/>nio &longs;a ualde: parum <expan abbr="tam&etilde;">tamen</expan> utilia olim <lb/><expan abbr="er&atilde;t">erant</expan> nece&longs;&longs;aria ad intelligendum ma&shy;<lb/>gnam <expan abbr="c&otilde;po&longs;itionem">compo&longs;itionem</expan> Ptolem&ecedil;i, nunc <lb/>po&longs;tquam Heber has &longs;ex quantita&shy;<lb/>tes traduxit ad quatuor, pror&longs;us h&aelig;c <lb/>&longs;cientia ulli u&longs;ui e&longs;&longs;e de&longs;ijt.<lb/><arrow.to.target n="table9"/></s></p><p type="margin">
  
  <s><margin.target id="marg23"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="margin">
  
  <s><margin.target id="marg24"/>Modi qui <expan abbr="n&otilde;">non</expan> <lb/>producuntur <lb/>pri. <!-- REMOVE S-->ad quartu <lb/>pri. <!-- REMOVE S-->ad &longs;extum <lb/>&longs;ec. <!-- REMOVE S-->ad <expan abbr="terti&utilde;">tertium</expan> <lb/>&longs;ec. <!-- REMOVE S-->ad <expan abbr="quint&utilde;">quintum</expan> <lb/>tert. </s>
  
  
  
 <s><margin.target id="marg23"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="margin"> 
  
 <s><margin.target id="marg24"></margin.target>Modi qui <expan abbr="n&otilde;">non</expan> <lb/>producuntur <lb/>pri. </s> 
  
 <s>ad quartu <lb/>pri. </s> 
  
 <s>ad &longs;extum <lb/>&longs;ec. </s> 
  
 <s>ad <expan abbr="terti&utilde;">tertium</expan> <lb/>&longs;ec. </s> 
  
 <s>ad <expan abbr="quint&utilde;">quintum</expan> <lb/>tert. </s> 
  
 <s>ad quint. <lb/></s> <s>ad quint. <lb/></s>
  
 <s>quart. </s> <s>quart. </s>
  
 <s>ad &longs;ext.</s></p><table><table.target id="table8"></table.target><row><cell></cell><cell>Primi ad &longs;ecundum.</cell></row><row><cell>1</cell><cell>tertij ad <expan abbr="quart&utilde;">quartum</expan>, &amp; quin</cell></row><row><cell></cell><cell>ti ad &longs;extum.</cell></row><row><cell>2</cell><cell>tertij ad &longs;extum, &amp; qu<gap/>n</cell></row><row><cell></cell><cell>ti ad quartum.</cell></row><row><cell></cell><cell>Primi ad tertium.</cell></row><row><cell>3</cell><cell>&longs;ecundi ad quartum, &amp;</cell></row><row><cell></cell><cell>quinti ad &longs;extum.</cell></row><row><cell>4</cell><cell>&longs;ecundi ad &longs;extum, &amp;</cell></row><row><cell></cell><cell>quinti ad quartum.</cell></row><row><cell></cell><cell>Primi ad quintum.</cell></row><row><cell>5</cell><cell>&longs;ecundi ad <expan abbr="&longs;ext&utilde;">&longs;extum</expan>, &amp; ter-</cell></row><row><cell></cell><cell>tij ad quartum.</cell></row><row><cell>6</cell><cell>&longs;ecundi ad quartum, &amp;</cell></row><row><cell></cell><cell>tertij ad &longs;extum.</cell></row><row><cell></cell><cell>Secundi ad quartum.</cell></row><row><cell>7</cell><cell>primi ad tertium, &amp; &longs;ex</cell></row><row><cell></cell><cell>ti ad quintum.</cell></row><row><cell>8</cell><cell>primi ad quintum, et &longs;ex</cell></row><row><cell></cell><cell>ti ad tertium.</cell></row><row><cell></cell><cell>Secundi ad &longs;extum.</cell></row><row><cell>9</cell><cell>primi ad <expan abbr="quint&utilde;">quintum</expan>, &amp; quar</cell></row><row><cell></cell><cell>ti ad tertium.</cell></row><row><cell>10</cell><cell>primi ad <expan abbr="terti&utilde;">tertium</expan>, &amp; quar-</cell></row><row><cell></cell><cell>ti ad quintum.</cell></row><row><cell></cell><cell>Tertij ad quartum.</cell></row><row><cell>11</cell><cell>primi ad &longs;ecundum, &amp;</cell></row><row><cell></cell><cell>&longs;exti ad quintum.</cell></row><row><cell>12</cell><cell>primi ad quintum, &amp; &longs;ex</cell></row><row><cell></cell><cell>ti ad &longs;ecundum.</cell></row><row><cell></cell><cell>Tertij ad &longs;extum.</cell></row><row><cell>13</cell><cell>primi ad &longs;ecundum, &amp;</cell></row><row><cell></cell><cell>quarti ad quintum.</cell></row><row><cell>14</cell><cell>primi ad quintum, &amp;</cell></row><row><cell></cell><cell>quarti ad &longs;ecundum.</cell></row><row><cell></cell><cell>Quarti ad quintum.</cell></row><row><cell>15</cell><cell>&longs;ecundi ad primum, &amp;</cell></row><row><cell></cell><cell>tertij ad &longs;extum.</cell></row><row><cell>16</cell><cell>&longs;ecundi ad &longs;extum, &amp; ter</cell></row><row><cell></cell><cell>tij ad primum.</cell></row><row><cell></cell><cell>Quinti ad &longs;extum.</cell></row><row><cell>17</cell><cell>primi ad &longs;ecundum, &amp;</cell></row><row><cell></cell><cell>quarti ad tertium.</cell></row><row><cell>18</cell><cell>primi ad <expan abbr="terti&utilde;">tertium</expan>, &amp; quar-</cell></row><row><cell></cell><cell>ti ad &longs;ecundum.</cell></row></table><table><table.target id="table9"></table.target><row><cell>a</cell><cell>e c</cell><cell>a e</cell><cell>e c</cell></row><row><cell></cell><cell></cell><cell>c b</cell><cell>e</cell></row><row><cell></cell><cell></cell><cell>f d</cell><cell>c</cell></row><row><cell></cell><cell></cell><cell></cell><cell>f</cell></row></table><p type="main"> <s>ad &longs;ext.</s></p><table><table.target id="table8"/><row><cell/><cell>Primi ad &longs;ecundum.</cell></row><row><cell>1</cell><cell>tertij ad <expan abbr="quart&utilde;">quartum</expan>, &amp; quin</cell></row><row><cell/><cell>ti ad &longs;extum.</cell></row><row><cell>2</cell><cell>tertij ad &longs;extum, &amp; qu<gap/>n</cell></row><row><cell/><cell>ti ad quartum.</cell></row><row><cell/><cell>Primi ad tertium.</cell></row><row><cell>3</cell><cell>&longs;ecundi ad quartum, &amp;</cell></row><row><cell/><cell>quinti ad &longs;extum.</cell></row><row><cell>4</cell><cell>&longs;ecundi ad &longs;extum, &amp;</cell></row><row><cell/><cell>quinti ad quartum.</cell></row><row><cell/><cell>Primi ad quintum.</cell></row><row><cell>5</cell><cell>&longs;ecundi ad <expan abbr="&longs;ext&utilde;">&longs;extum</expan>, &amp; ter-</cell></row><row><cell/><cell>tij ad quartum.</cell></row><row><cell>6</cell><cell>&longs;ecundi ad quartum, &amp;</cell></row><row><cell/><cell>tertij ad &longs;extum.</cell></row><row><cell/><cell>Secundi ad quartum.</cell></row><row><cell>7</cell><cell>primi ad tertium, &amp; &longs;ex</cell></row><row><cell/><cell>ti ad quintum.</cell></row><row><cell>8</cell><cell>primi ad quintum, et &longs;ex</cell></row><row><cell/><cell>ti ad tertium.</cell></row><row><cell/><cell>Secundi ad &longs;extum.</cell></row><row><cell>9</cell><cell>primi ad <expan abbr="quint&utilde;">quintum</expan>, &amp; quar</cell></row><row><cell/><cell>ti ad tertium.</cell></row><row><cell>10</cell><cell>primi ad <expan abbr="terti&utilde;">tertium</expan>, &amp; quar-</cell></row><row><cell/><cell>ti ad quintum.</cell></row><row><cell/><cell>Tertij ad quartum.</cell></row><row><cell>11</cell><cell>primi ad &longs;ecundum, &amp;</cell></row><row><cell/><cell>&longs;exti ad quintum.</cell></row><row><cell>12</cell><cell>primi ad quintum, &amp; &longs;ex</cell></row><row><cell/><cell>ti ad &longs;ecundum.</cell></row><row><cell/><cell>Tertij ad &longs;extum.</cell></row><row><cell>13</cell><cell>primi ad &longs;ecundum, &amp;</cell></row><row><cell/><cell>quarti ad quintum.</cell></row><row><cell>14</cell><cell>primi ad quintum, &amp;</cell></row><row><cell/><cell>quarti ad &longs;ecundum.</cell></row><row><cell/><cell>Quarti ad quintum.</cell></row><row><cell>15</cell><cell>&longs;ecundi ad primum, &amp;</cell></row><row><cell/><cell>tertij ad &longs;extum.</cell></row><row><cell>16</cell><cell>&longs;ecundi ad &longs;extum, &amp; ter</cell></row><row><cell/><cell>tij ad primum.</cell></row><row><cell/><cell>Quinti ad &longs;extum.</cell></row><row><cell>17</cell><cell>primi ad &longs;ecundum, &amp;</cell></row><row><cell/><cell>quarti ad tertium.</cell></row><row><cell>18</cell><cell>primi ad <expan abbr="terti&utilde;">tertium</expan>, &amp; quar-</cell></row><row><cell/><cell>ti ad &longs;ecundum.</cell></row></table><table><table.target id="table9"/><row><cell>a</cell><cell>e c</cell><cell>a e</cell><cell>e c</cell></row><row><cell/><cell/><cell>c b</cell><cell>e</cell></row><row><cell/><cell/><cell>f d</cell><cell>c</cell></row><row><cell/><cell/><cell/><cell>f</cell></row></table><p type="main">
  
 <s>Propo&longs;itio &longs;eptima.</s></p><p type="main"> <s>Propo&longs;itio &longs;eptima.</s></p><p type="main">
  
 <s>In modis qui nece&longs;&longs;ari&ograve; produ&shy;<lb/>cuntur ex duabus proportionibus, <lb/>cum du&ecedil; quantitates ex illis, qu&ecedil; mo <lb/>dos conficiunt, &aelig;quales fuerint: pro&shy;<lb/><arrow.to.target n="table10"></arrow.to.target><lb/>portio producta ad quatuor quanti&shy;<lb/>tates omiologas reducetur.<lb/><arrow.to.target n="marg25"></arrow.to.target></s></p><p type="margin"> <s>In modis qui nece&longs;&longs;ari&ograve; produ&shy;<lb/>cuntur ex duabus proportionibus, <lb/>cum du&ecedil; quantitates ex illis, qu&ecedil; mo <lb/>dos conficiunt, &aelig;quales fuerint: pro&shy;<lb/><arrow.to.target n="table10"/><lb/>portio producta ad quatuor quanti&shy;<lb/>tates omiologas reducetur.<lb/><arrow.to.target n="marg25"/></s></p><p type="margin">
  
 <s><margin.target id="marg25"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><table><table.target id="table10"></table.target><row><cell>a</cell><cell>b</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>c</cell><cell>e</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>d</cell><cell>f</cell></row><row><cell>---</cell><cell>---</cell></row></table><p type="main"> <s><margin.target id="marg25"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><table><table.target id="table10"/><row><cell>a</cell><cell>b</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>c</cell><cell>e</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>d</cell><cell>f</cell></row><row><cell>---</cell><cell>---</cell></row></table><p type="main">
  
 <s>Sint &longs;ex quantitates a b c d e f, &amp; <lb/>producatur proportio a ad b ex pro&shy;<lb/>portione c ad d, &amp; e ad f, tu &longs;cis, qu&ograve;d <lb/>modi recepti &longs;unt prima cum &longs;ecunda, tertia uel quinta, &amp; &longs;ecunda <lb/>cum quarta, &amp; &longs;exta, &amp; tertia &longs;imiliter cum ei&longs;dem, &amp; quinta eodem <lb/>modo cum ei&longs;dem: &longs;i igitur du&ecedil; quantitates ex his, qu&ecedil; faciunt pro&shy;<pb pagenum="11"/>portionem productam inter &longs;e fuerint &aelig;quales reducetur h&aelig;c pro&shy;<lb/>portio ad quatuor quantitates omologas, &longs;cilicer abiectis amba&shy;<lb/>bus &aelig;qualibus. </s> <s>Sint &longs;ex quantitates a b c d e f, &amp; <lb/>producatur proportio a ad b ex pro&shy;<lb/>portione c ad d, &amp; e ad f, tu &longs;cis, qu&ograve;d <lb/>modi recepti &longs;unt prima cum &longs;ecunda, tertia uel quinta, &amp; &longs;ecunda <lb/>cum quarta, &amp; &longs;exta, &amp; tertia &longs;imiliter cum ei&longs;dem, &amp; quinta eodem <lb/>modo cum ei&longs;dem: &longs;i igitur du&ecedil; quantitates ex his, qu&ecedil; faciunt pro&shy;<pb pagenum="11"/>portionem productam inter &longs;e fuerint &aelig;quales reducetur h&aelig;c pro&shy;<lb/>portio ad quatuor quantitates omologas, &longs;cilicer abiectis amba&shy;<lb/>bus &aelig;qualibus. </s>
  
 <s>Sit gratia exempli prima &aelig;qualis quint&aelig;: &amp; quia <lb/>in octauo modo proportio <expan abbr="&longs;ec&utilde;di">&longs;ecundi</expan> ad quartum producitur ex pro&shy;<lb/>portione primi ad quintum, &amp; &longs;exti ad tertium, ergo per expo&longs;ita <lb/>proportio &longs;ecundi ad quartum, ut &longs;exti ad tertium, &amp; ita permutan&shy;<lb/>do, &amp; conuertendo &longs;ecundi ad &longs;extum, ut quarti ad tertium, &amp; tertij </s></p><p type="main"> <s>Sit gratia exempli prima &aelig;qualis quint&aelig;: &amp; quia <lb/>in octauo modo proportio <expan abbr="&longs;ec&utilde;di">&longs;ecundi</expan> ad quartum producitur ex pro&shy;<lb/>portione primi ad quintum, &amp; &longs;exti ad tertium, ergo per expo&longs;ita <lb/>proportio &longs;ecundi ad quartum, ut &longs;exti ad tertium, &amp; ita permutan&shy;<lb/>do, &amp; conuertendo &longs;ecundi ad &longs;extum, ut quarti ad tertium, &amp; tertij </s></p><p type="main">
  
 <s><arrow.to.target n="marg26"></arrow.to.target><lb/>ad quartum, ut &longs;exti ad &longs;ecundum.</s></p><p type="margin"> <s><arrow.to.target n="marg26"/><lb/>ad quartum, ut &longs;exti ad &longs;ecundum.</s></p><p type="margin">
  
 <s><margin.target id="marg26"></margin.target>V<emph type="italics"/>ndecima <lb/>petitione.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg26"/>V<emph type="italics"/>ndecima <lb/>petitione.<emph.end type="italics"/></s></p><p type="main">
  
 <s>Propo&longs;itio octaua.</s></p><p type="main"> <s>Propo&longs;itio octaua.</s></p><p type="main">
  
 <s>Si duarum <expan abbr="proportion&utilde;">proportionum</expan> &longs;uperiores numeri alternatim cum infe <lb/>rioribus multiplicentur, atque coniungantur: erit proportio aggre&shy;<lb/>gati ad productum ex inferioribus inuicem proportio ex primis <lb/>proportionibus compo&longs;ita.</s></p><figure></figure><p type="main"> <s>Si duarum <expan abbr="proportion&utilde;">proportionum</expan> &longs;uperiores numeri alternatim cum infe <lb/>rioribus multiplicentur, atque coniungantur: erit proportio aggre&shy;<lb/>gati ad productum ex inferioribus inuicem proportio ex primis <lb/>proportionibus compo&longs;ita.</s></p><figure/><p type="main">
  
 <s>Sit proportio una a ad b, alia c ad d, ducatur b in <lb/><arrow.to.target n="marg27"></arrow.to.target><lb/>c, fiatque e &amp; a in d, &amp; fiat f, iunganturque e &amp; f &amp; fiat h, <lb/>&amp; ducatur b in d et fiat g: dico <expan abbr="proportion&etilde;">proportionem</expan> h g com&shy;<lb/>po&longs;itam e&longs;&longs;e ex proportione a ad b, &amp; c ad d. </s> <s>Sit proportio una a ad b, alia c ad d, ducatur b in <lb/><arrow.to.target n="marg27"/><lb/>c, fiatque e &amp; a in d, &amp; fiat f, iunganturque e &amp; f &amp; fiat h, <lb/>&amp; ducatur b in d et fiat g: dico <expan abbr="proportion&etilde;">proportionem</expan> h g com&shy;<lb/>po&longs;itam e&longs;&longs;e ex proportione a ad b, &amp; c ad d. <!-- KEEP S--></s>
  
 <s>Quia <lb/><arrow.to.target n="marg28"></arrow.to.target><lb/>enim ex b in c fit e, &amp; ex b in d fit g, erit proportio e <lb/>ad g, ut c ad d, &amp; &longs;imiliter, quia ex d in a fit f, &amp; ex d in b fit g, erit f ad <lb/>g ut a ad b. </s> <s>Quia <lb/><arrow.to.target n="marg28"/><lb/>enim ex b in c fit e, &amp; ex b in d fit g, erit proportio e <lb/>ad g, ut c ad d, &amp; &longs;imiliter, quia ex d in a fit f, &amp; ex d in b fit g, erit f ad <lb/>g ut a ad b. </s>
  
 <s>Sed e &amp; f componunt h, igitur proportio h ad g e&longs;t com <lb/>po&longs;ita ex proportionibus e &amp; f ad g, igitur per communem animi <lb/>&longs;ententiam, &amp; diffinitionem compo&longs;it&aelig; proportionis, proportio h <lb/><arrow.to.target n="marg29"></arrow.to.target><lb/>ad g compo&longs;ita e&longs;t ex proportionibus a ad b, &amp; c ad d, quod e&longs;t <lb/>propo&longs;itum.</s></p><p type="margin"> <s>Sed e &amp; f componunt h, igitur proportio h ad g e&longs;t com <lb/>po&longs;ita ex proportionibus e &amp; f ad g, igitur per communem animi <lb/>&longs;ententiam, &amp; diffinitionem compo&longs;it&aelig; proportionis, proportio h <lb/><arrow.to.target n="marg29"/><lb/>ad g compo&longs;ita e&longs;t ex proportionibus a ad b, &amp; c ad d, quod e&longs;t <lb/>propo&longs;itum.</s></p><p type="margin">
  
 <s><margin.target id="marg27"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="margin"> <s><margin.target id="marg27"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg28"></margin.target>E<emph type="italics"/>x<emph.end type="italics"/> 13 <emph type="italics"/>peti&shy;<lb/>tione.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg28"/>E<emph type="italics"/>x<emph.end type="italics"/> 13 <emph type="italics"/>peti&shy;<lb/>tione.<emph.end type="italics"/></s></p><p type="margin">
  
 <s><margin.target id="marg29"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 14 <emph type="italics"/>diffi <lb/>nitionem.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg29"/>P<emph type="italics"/>er<emph.end type="italics"/> 14 <emph type="italics"/>diffi <lb/>nitionem.<emph.end type="italics"/></s></p><p type="main">
  
 <s>Propo&longs;itio nona.</s></p><p type="main"> <s>Propo&longs;itio nona.</s></p><p type="main">
  
 <s>Si duarum proportionum &longs;uperiores numeri alternatim cum <lb/>inferioribus multiplicentur, minusque productum ex maiore detra&shy;<lb/>hatur, erit re&longs;idui ad productum ex inferioribus proportio uelut <lb/>illa, qu&aelig; relinquitur detracta minore proportione ex maiore.</s></p><p type="main"> <s>Si duarum proportionum &longs;uperiores numeri alternatim cum <lb/>inferioribus multiplicentur, minusque productum ex maiore detra&shy;<lb/>hatur, erit re&longs;idui ad productum ex inferioribus proportio uelut <lb/>illa, qu&aelig; relinquitur detracta minore proportione ex maiore.</s></p><p type="main">
  
 <s>H&aelig;c eodem modo probatur, ut pr&aelig;cedens, ni&longs;i quod h fit de&shy;<lb/><arrow.to.target n="marg30"></arrow.to.target><lb/>tracto &egrave; minore: gratia exempli ex f, &amp; ita ex diffinitione patet pro&shy;<lb/>po&longs;itum.</s></p><p type="margin"> <s>H&aelig;c eodem modo probatur, ut pr&aelig;cedens, ni&longs;i quod h fit de&shy;<lb/><arrow.to.target n="marg30"/><lb/>tracto &egrave; minore: gratia exempli ex f, &amp; ita ex diffinitione patet pro&shy;<lb/>po&longs;itum.</s></p><p type="margin">
  
 <s><margin.target id="marg30"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>_{m}. <lb/>152.</s></p><p type="main"> <s><margin.target id="marg30"/>C<emph type="italics"/>or<emph.end type="italics"/>_{m}. <lb/>152.</s></p><p type="main">
  
 <s>Propo&longs;itio decima.</s></p><p type="main"> <s>Propo&longs;itio decima.</s></p><p type="main">
  
 <s>Si fuerit alicuius quantitatis ad unam partem proportio uelut al <lb/>terius partis ad <expan abbr="&longs;ec&utilde;dam">&longs;ecundam</expan> quantitatem erit proportio cuiu&longs;uis quan <lb/>titatis eiu&longs;dem generis ad &longs;ecundam compo&longs;ita proportio ex pro&shy;<lb/>portionibus eiu&longs;dem quantitatis a&longs;&longs;umpt&aelig; ad utran que partem pri&shy;<lb/>m&aelig; quantitatis &longs;eor&longs;um.<lb/><arrow.to.target n="marg31"></arrow.to.target></s></p><p type="margin"> <s>Si fuerit alicuius quantitatis ad unam partem proportio uelut al <lb/>terius partis ad <expan abbr="&longs;ec&utilde;dam">&longs;ecundam</expan> quantitatem erit proportio cuiu&longs;uis quan <lb/>titatis eiu&longs;dem generis ad &longs;ecundam compo&longs;ita proportio ex pro&shy;<lb/>portionibus eiu&longs;dem quantitatis a&longs;&longs;umpt&aelig; ad utran que partem pri&shy;<lb/>m&aelig; quantitatis &longs;eor&longs;um.<lb/><arrow.to.target n="marg31"/></s></p><p type="margin">
  
 <s><margin.target id="marg31"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><figure></figure><p type="main"> <s><margin.target id="marg31"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><figure/><p type="main">
  
 <s>Sit a b quantitas diui&longs;a in c, &amp; &longs;i cut a b ad a c, <lb/>ita b c ad d: eritque iterum permutando a b ad b c, <lb/>ut a c ad d, &amp; &longs;umatur qu&aelig;dam quantitas e eiu&longs;&shy;<pb pagenum="12"/>dem tamen generis, cum illis dico qu&ograve;d proportio e ad d e&longs;t com&shy;<lb/>po&longs;ita ex proportionibus e ad a c, &amp; e ad b c. </s> <s>Sit a b quantitas diui&longs;a in c, &amp; &longs;i cut a b ad a c, <lb/>ita b c ad d: eritque iterum permutando a b ad b c, <lb/>ut a c ad d, &amp; &longs;umatur qu&aelig;dam quantitas e eiu&longs;&shy;<pb pagenum="12"/>dem tamen generis, cum illis dico qu&ograve;d proportio e ad d e&longs;t com&shy;<lb/>po&longs;ita ex proportionibus e ad a c, &amp; e ad b c. <!-- KEEP S--></s>
  
 <s>Po&longs;ita ergo e tan&lt;08&gt; &longs;u&shy;<lb/>periore numero, &amp; a c &amp; c b inferioribus, erit ex octaua propo&longs;itio&shy;<lb/>ne huius proportio productorum ex e in a c, &amp; coniunctorum, &amp; <lb/>ex con&longs;equenti per primam &longs;ecundi Elementorum producti ex e in <lb/>a b ad productum ex a c in c b compo&longs;ita ex proportionibus e ad <lb/>a c, &amp; e ad c b: at quod fit ex a c in c b, e&longs;t &aelig;quale ei quod fit ex a b in <lb/>d, eo qu&ograve;d a b, a c, c b &amp; d &longs;unt omiolog&aelig; per decimam&longs;extam &longs;exti <lb/><expan abbr="Elem&etilde;torum">Elementorum</expan>: Proportio igitur producti ex e in a b ad productum <lb/>ex d in a b e&longs;t compo&longs;ita ex proportionibus e ad a c, &amp; e ad e b: At <lb/>proportio producti ex e in a b ad productum ex d in a b, e&longs;t uelut e <lb/><arrow.to.target n="marg32"></arrow.to.target><lb/>ad d. </s> <s>Po&longs;ita ergo e tan&lt;08&gt; &longs;u&shy;<lb/>periore numero, &amp; a c &amp; c b inferioribus, erit ex octaua propo&longs;itio&shy;<lb/>ne huius proportio productorum ex e in a c, &amp; coniunctorum, &amp; <lb/>ex con&longs;equenti per primam &longs;ecundi Elementorum producti ex e in <lb/>a b ad productum ex a c in c b compo&longs;ita ex proportionibus e ad <lb/>a c, &amp; e ad c b: at quod fit ex a c in c b, e&longs;t &aelig;quale ei quod fit ex a b in <lb/>d, eo qu&ograve;d a b, a c, c b &amp; d &longs;unt omiolog&aelig; per decimam&longs;extam &longs;exti <lb/><expan abbr="Elem&etilde;torum">Elementorum</expan>: Proportio igitur producti ex e in a b ad productum <lb/>ex d in a b e&longs;t compo&longs;ita ex proportionibus e ad a c, &amp; e ad e b: At <lb/>proportio producti ex e in a b ad productum ex d in a b, e&longs;t uelut e <lb/><arrow.to.target n="marg32"/><lb/>ad d. <!-- KEEP S--></s>
  
 <s>per &longs;uppo&longs;ita igitur proportio e ad d e&longs;t compo&longs;ita ex propor<lb/>tionibus e ad a c, &amp; e ad b c, quod fuit demon&longs;trandum.</s></p><p type="margin"> <s>per &longs;uppo&longs;ita igitur proportio e ad d e&longs;t compo&longs;ita ex propor<lb/>tionibus e ad a c, &amp; e ad b c, quod fuit demon&longs;trandum.</s></p><p type="margin">
  
 <s><margin.target id="marg32"></margin.target>13. P<emph type="italics"/>etit.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg32"/>13. P<emph type="italics"/>etit.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main">
  
 <s>Propo&longs;itio undecima.</s></p><p type="main"> <s>Propo&longs;itio undecima.</s></p><p type="main">
  
 <s>Proportio aggregati quarumlibet duarum quantitatum ad ag&shy;<lb/>gregatum duarum &aelig;qualium quantitatum e&longs;t compo&longs;ita ex pro&shy;<lb/>portionibus primis, &amp; diui&longs;a per duplam.<lb/><arrow.to.target n="marg33"></arrow.to.target></s></p><p type="margin"> <s>Proportio aggregati quarumlibet duarum quantitatum ad ag&shy;<lb/>gregatum duarum &aelig;qualium quantitatum e&longs;t compo&longs;ita ex pro&shy;<lb/>portionibus primis, &amp; diui&longs;a per duplam.<lb/><arrow.to.target n="marg33"/></s></p><p type="margin">
  
 <s><margin.target id="marg33"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg33"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Sit proportio a ad c, &amp; b ad d, &amp; &longs;int c &amp; d <lb/><figure id="fig6"></figure><lb/>&aelig;quales, dico qu&ograve;d proportio a b ad c d e&longs;t <lb/>compo&longs;ita ex proportionibus a ad c, &amp; b ad <lb/>d diui&longs;o compo&longs;ito per duplam. </s> <s>Sit proportio a ad c, &amp; b ad d, &amp; &longs;int c &amp; d <lb/><figure id="fig6"/><lb/>&aelig;quales, dico qu&ograve;d proportio a b ad c d e&longs;t <lb/>compo&longs;ita ex proportionibus a ad c, &amp; b ad <lb/>d diui&longs;o compo&longs;ito per duplam. </s>
  
 <s>Quia enim </s></p><p type="main"> <s>Quia enim </s></p><p type="main">
  
 <s><arrow.to.target n="marg34"></arrow.to.target><lb/>c &amp; d &longs;unt &aelig;quales, erit b ad c, ut b ad d, qua&shy;<lb/>re ex diffinitione c&ugrave;m proportio a b ad c d <lb/><arrow.to.target n="marg35"></arrow.to.target><lb/>&longs;it compo&longs;ita ex proportionibus a ad c, &amp; b <lb/>ad c, erit etiam compo&longs;ita ex dictis ex propo&longs;itione a ad c, &amp; b ad d, <lb/><arrow.to.target n="marg36"></arrow.to.target><lb/>&longs;tatuatur ergo e &aelig;qualis c d media inter a b &amp; c. </s> <s><arrow.to.target n="marg34"/><lb/>c &amp; d &longs;unt &aelig;quales, erit b ad c, ut b ad d, qua&shy;<lb/>re ex diffinitione c&ugrave;m proportio a b ad c d <lb/><arrow.to.target n="marg35"/><lb/>&longs;it compo&longs;ita ex proportionibus a ad c, &amp; b <lb/>ad c, erit etiam compo&longs;ita ex dictis ex propo&longs;itione a ad c, &amp; b ad d, <lb/><arrow.to.target n="marg36"/><lb/>&longs;tatuatur ergo e &aelig;qualis c d media inter a b &amp; c. <!-- KEEP S--></s>
  
 <s>Et erit per &longs;ecun&shy;<lb/>dam propo&longs;itionem proportio aggregati a b ad c producta ex <lb/><arrow.to.target n="marg37"></arrow.to.target><lb/>proportione aggregati a b ad c, &amp; e ad c, igitur proportio a b ad e <lb/>erit proportio a b ad c, diui&longs;a per proportionem e ad c, &longs;ed e ad c e&longs;t <lb/><arrow.to.target n="marg38"></arrow.to.target><lb/>dupla: igitur proportio a b ad c d e&longs;t proportio a b ad c diui&longs;a per <lb/>duplam.</s></p><p type="margin"> <s>Et erit per &longs;ecun&shy;<lb/>dam propo&longs;itionem proportio aggregati a b ad c producta ex <lb/><arrow.to.target n="marg37"/><lb/>proportione aggregati a b ad c, &amp; e ad c, igitur proportio a b ad e <lb/>erit proportio a b ad c, diui&longs;a per proportionem e ad c, &longs;ed e ad c e&longs;t <lb/><arrow.to.target n="marg38"/><lb/>dupla: igitur proportio a b ad c d e&longs;t proportio a b ad c diui&longs;a per <lb/>duplam.</s></p><p type="margin">
  
  <s><margin.target id="marg34"/>E<emph type="italics"/>x &longs;exta<emph.end type="italics"/> A<emph type="italics"/>nim. <lb/><!-- REMOVE S-->com. </s>
  
 <s><margin.target id="marg34"></margin.target>E<emph type="italics"/>x &longs;exta<emph.end type="italics"/> A<emph type="italics"/>nim. <lb/></s> 
  
 <s>com. </s> 
  
 <s>&longs;ententia.<emph.end type="italics"/></s></p><p type="margin"> <s>&longs;ententia.<emph.end type="italics"/></s></p><p type="margin">
  
 <s><margin.target id="marg35"></margin.target>D<emph type="italics"/>ecimaquarta<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg35"/>D<emph type="italics"/>ecimaquarta<emph.end type="italics"/></s></p><p type="margin">
  
 <s><margin.target id="marg36"></margin.target>13. P<emph type="italics"/>etit.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg36"/>13. P<emph type="italics"/>etit.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg37"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 2. P<emph type="italics"/>etit.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg37"/>P<emph type="italics"/>er<emph.end type="italics"/> 2. P<emph type="italics"/>etit.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg38"></margin.target>P<emph type="italics"/>er quintam<emph.end type="italics"/><lb/>A<emph type="italics"/>nim. </s> <s><margin.target id="marg38"/>P<emph type="italics"/>er quintam<emph.end type="italics"/><lb/>A<emph type="italics"/>nim. </s>
  
 <s>com. </s> <s>com. </s>
  
Line 585 
Line 584 
  
 <s>Propo&longs;itio duodecima.</s></p><p type="main"> <s>Propo&longs;itio duodecima.</s></p><p type="main">
  
 <s>Propo&longs;itis duabus proportionibus unam alteri iungere ab&longs;que <lb/>multiplicatione.<lb/><arrow.to.target n="marg39"></arrow.to.target></s></p><p type="margin"> <s>Propo&longs;itis duabus proportionibus unam alteri iungere ab&longs;que <lb/>multiplicatione.<lb/><arrow.to.target n="marg39"/></s></p><p type="margin">
  
 <s><margin.target id="marg39"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. <lb/>10. P<emph type="italics"/>etit.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg39"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. <lb/>10. P<emph type="italics"/>etit.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main">
  
 <s>Sint propo&longs;it&aelig; proportiones a ad c &amp; <lb/><figure id="fig7"></figure><lb/>b ad d, &amp; a&longs;&longs;umo e ad c, iuxta ea qu&aelig; Eu&shy;<lb/>clides demon&longs;trauit, ut b ad d, erit igitur </s></p><p type="main"> <s>Sint propo&longs;it&aelig; proportiones a ad c &amp; <lb/><figure id="fig7"/><lb/>b ad d, &amp; a&longs;&longs;umo e ad c, iuxta ea qu&aelig; Eu&shy;<lb/>clides demon&longs;trauit, ut b ad d, erit igitur </s></p><p type="main">
  
 <s><arrow.to.target n="marg40"></arrow.to.target><lb/>proportio a e ad c, compo&longs;ita ex proportionibus a ad c, &amp; e ad c, <lb/>&longs;ed proportio e ad c e&longs;t, ut b ad d, igitur proportio a e ad c compo&shy;<lb/>&longs;ita e&longs;t ex proportionibus a ad c, &amp; b ad d.</s></p><p type="margin"> <s><arrow.to.target n="marg40"/><lb/>proportio a e ad c, compo&longs;ita ex proportionibus a ad c, &amp; e ad c, <lb/>&longs;ed proportio e ad c e&longs;t, ut b ad d, igitur proportio a e ad c compo&shy;<lb/>&longs;ita e&longs;t ex proportionibus a ad c, &amp; b ad d.<!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg40"></margin.target>E<emph type="italics"/>x generali <lb/>com.<emph.end type="italics"/> A<emph type="italics"/>nim. </s> <s><margin.target id="marg40"/>E<emph type="italics"/>x generali <lb/>com.<emph.end type="italics"/> A<emph type="italics"/>nim. <!-- REMOVE S-->&longs;en <lb/>tentia.<emph.end type="italics"/></s>
  
 <s>&longs;en <lb/>tentia.<emph.end type="italics"/></s></p><p type="main"> </p><p type="main">
  
 <s>Aliter ex b in c fiat fex a in d, g ex c in d h coniunctum ex f g, k.</s></p><pb pagenum="13"/><figure></figure><p type="main"> <s>Aliter ex b in c fiat fex a in d, g ex c in d h coniunctum ex f g, k.</s></p><pb pagenum="13"/><figure/><p type="main">
  
 <s>Quia ergo ex c in b fit f, ex c in d h, erit f ad h, <lb/>ut b ad d, igitur ut e ad c, &longs;ed a ad c, ut g ad h igi <lb/><arrow.to.target n="marg41"></arrow.to.target><lb/>tur a e ad c, ut k ad h, &longs;ed k ad h c&oacute;mponitur ex <lb/>proportionibus a ad c, &amp; b ad d. </s> <s>Quia ergo ex c in b fit f, ex c in d h, erit f ad h, <lb/>ut b ad d, igitur ut e ad c, &longs;ed a ad c, ut g ad h igi <lb/><arrow.to.target n="marg41"/><lb/>tur a e ad c, ut k ad h, &longs;ed k ad h c&oacute;mponitur ex <lb/>proportionibus a ad c, &amp; b ad d. <!-- KEEP S--></s>
  
 <s>Ex octaua ha <lb/>rum igitur proportio a c ad c compo&longs;ita e&longs;t ex <lb/>ei&longs;dem. </s> <s>Ex octaua ha <lb/>rum igitur proportio a c ad c compo&longs;ita e&longs;t ex <lb/>ei&longs;dem. </s>
  
 <s>For&longs;an quis dicat hanc eandem e&longs;&longs;e <lb/>octau&aelig; &longs;ed <expan abbr="n&otilde;">non</expan> e&longs;t, in illa enim proportio com&shy;<lb/>paratur ad productum, in hac ad unam ex <lb/>quantitatibus.</s></p><p type="margin"> <s>For&longs;an quis dicat hanc eandem e&longs;&longs;e <lb/>octau&aelig; &longs;ed <expan abbr="n&otilde;">non</expan> e&longs;t, in illa enim proportio com&shy;<lb/>paratur ad productum, in hac ad unam ex <lb/>quantitatibus.</s></p><p type="margin">
  
 <s><margin.target id="marg41"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 13. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg41"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. P<emph type="italics"/>et.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main">
  
 <s>Ex hoc &longs;equitur qu&ograve;d: Qu&aelig;libet du&aelig; quantitates quarum ag&shy;<lb/><arrow.to.target n="marg42"></arrow.to.target><lb/>gregatum e&longs;tidem ad eam quantitatem, componunt eandem pro&shy;<lb/>portionem.</s></p><p type="margin"> <s>Ex hoc &longs;equitur qu&ograve;d: Qu&aelig;libet du&aelig; quantitates quarum ag&shy;<lb/><arrow.to.target n="marg42"/><lb/>gregatum e&longs;tidem ad eam quantitatem, componunt eandem pro&shy;<lb/>portionem.</s></p><p type="margin">
  
 <s><margin.target id="marg42"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg42"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Propo&longs;itio tertiadecima.</s></p><p type="main"> <s>Propo&longs;itio tertiadecima.</s></p><p type="main">
  
 <s>Proportio confu&longs;a aggregati prim&aelig; &amp; terti&aelig; quatuor quantita&shy;<lb/>tum omiologarum ad <expan abbr="aggregat&utilde;">aggregatum</expan> &longs;ecund&aelig; &amp; quart&aelig;, e&longs;t uelut com <lb/>po&longs;ita ex ei&longs;dem diui&longs;a per duplam.<lb/><arrow.to.target n="marg43"></arrow.to.target></s></p><p type="margin"> <s>Proportio confu&longs;a aggregati prim&aelig; &amp; terti&aelig; quatuor quantita&shy;<lb/>tum omiologarum ad <expan abbr="aggregat&utilde;">aggregatum</expan> &longs;ecund&aelig; &amp; quart&aelig;, e&longs;t uelut com <lb/>po&longs;ita ex ei&longs;dem diui&longs;a per duplam.<lb/><arrow.to.target n="marg43"/></s></p><p type="margin">
  
 <s><margin.target id="marg43"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg43"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Sint a ad b, ut c ad d, dico, qu&ograve;d erit confu&longs;a <lb/><arrow.to.target n="table11"></arrow.to.target><lb/>proportio a c aggregati ad <expan abbr="aggregat&utilde;">aggregatum</expan> b d, com <lb/>po&longs;it&aelig; ex his proportionibus diui&longs;&aelig; per du&shy;<lb/>plam &aelig;qualis. </s> <s>Sint a ad b, ut c ad d, dico, qu&ograve;d erit confu&longs;a <lb/><arrow.to.target n="table11"/><lb/>proportio a c aggregati ad <expan abbr="aggregat&utilde;">aggregatum</expan> b d, com <lb/>po&longs;it&aelig; ex his proportionibus diui&longs;&aelig; per du&shy;<lb/>plam &aelig;qualis. </s>
  
 <s>Erit enim aggregati ex a c ad aggregatum ex b d, ue&shy;<lb/>lut a ad b per 18 quinti Elementorum. </s> <s>Erit enim aggregati ex a c ad aggregatum ex b d, ue&shy;<lb/>lut a ad b per 18 quinti Elementorum. <!-- KEEP S--></s>
  
 <s>Sed proportiones a ad b, <lb/>&amp; c ad d componunt proportionem producti a in d, &amp; c in b per <lb/>octauam harum, ad <expan abbr="product&utilde;">productum</expan> ex b in d, productum uer&ograve; ex a in d <lb/>e&longs;t &aelig;quale producto ex b in c per decimam&longs;extam &longs;exti Elemento&shy;<lb/>rum, &amp; proportio producti ex b in c ad productum ex b in d e&longs;t ue <lb/>lut c ad d, quare ut aggregati a c ad aggregatum b d, igitur propor&shy;<lb/>tio compo&longs;ita ex a ad b, &amp; c ad d, e&longs;t uelut confu&longs;a bis &longs;umpta. </s> <s>Sed proportiones a ad b, <lb/>&amp; c ad d componunt proportionem producti a in d, &amp; c in b per <lb/>octauam harum, ad <expan abbr="product&utilde;">productum</expan> ex b in d, productum uer&ograve; ex a in d <lb/>e&longs;t &aelig;quale producto ex b in c per decimam&longs;extam &longs;exti Elemento&shy;<lb/>rum, &amp; proportio producti ex b in c ad productum ex b in d e&longs;t ue <lb/>lut c ad d, quare ut aggregati a c ad aggregatum b d, igitur propor&shy;<lb/>tio compo&longs;ita ex a ad b, &amp; c ad d, e&longs;t uelut confu&longs;a bis &longs;umpta. </s>
  
 <s>Igi&shy;<lb/>tur confu&longs;a e&longs;t uelut compo&longs;ita diui&longs;a per duplam per modum un&shy;<lb/>decim&aelig; huius.</s></p><table><table.target id="table11"></table.target><row><cell>a</cell><cell>c</cell></row><row><cell>-----</cell><cell>-----</cell></row><row><cell>b</cell><cell>d</cell></row><row><cell>---</cell><cell>---</cell></row></table><p type="main"> <s>Igi&shy;<lb/>tur confu&longs;a e&longs;t uelut compo&longs;ita diui&longs;a per duplam per modum un&shy;<lb/>decim&aelig; huius.</s></p><table><table.target id="table11"/><row><cell>a</cell><cell>c</cell></row><row><cell>-----</cell><cell>-----</cell></row><row><cell>b</cell><cell>d</cell></row><row><cell>---</cell><cell>---</cell></row></table><p type="main">
  
 <s>Propo&longs;itio quartadecima.</s></p><p type="main"> <s>Propo&longs;itio quartadecima.</s></p><p type="main">
  
 <s>Proportiones confu&longs;&aelig;, &amp; coniunct&aelig; in tribus quantitatibus in&shy;<lb/>uicem commutantur.</s></p><figure></figure><p type="main"> <s>Proportiones confu&longs;&aelig;, &amp; coniunct&aelig; in tribus quantitatibus in&shy;<lb/>uicem commutantur.</s></p><figure/><p type="main">
  
 <s>Sint tres quantitates, dico, quod proportio c </s></p><p type="main"> <s>Sint tres quantitates, dico, quod proportio c </s></p><p type="main">
  
 <s><arrow.to.target n="marg44"></arrow.to.target><lb/>ad a b confu&longs;a e&longs;t, conuer&longs;a coniunct&aelig; a &amp; b ad <lb/><arrow.to.target n="marg45"></arrow.to.target><lb/>c. </s> <s><arrow.to.target n="marg44"/><lb/>ad a b confu&longs;a e&longs;t, conuer&longs;a coniunct&aelig; a &amp; b ad <lb/><arrow.to.target n="marg45"/><lb/>c. <!-- KEEP S--></s>
  
 <s>Nam per dicta proportio a b ad c efficit con&shy;<lb/>iunctam ex a b ad c: &longs;ed c ad a b conuer&longs;a e&longs;t eius, qu&aelig; e&longs;t a b ad c, &amp; <lb/>proportio c ad a b e&longs;t confu&longs;a eius, qu&aelig; e&longs;t c ad a &amp; b. </s> <s>Nam per dicta proportio a b ad c efficit con&shy;<lb/>iunctam ex a b ad c: &longs;ed c ad a b conuer&longs;a e&longs;t eius, qu&aelig; e&longs;t a b ad c, &amp; <lb/>proportio c ad a b e&longs;t confu&longs;a eius, qu&aelig; e&longs;t c ad a &amp; b. </s>
  
 <s>Igitur pro&shy;<lb/>portio confu&longs;a in tribus quantitatibus e&longs;t contraria coniunct&aelig; in <lb/>ei&longs;dem.</s></p><p type="margin"> <s>Igitur pro&shy;<lb/>portio confu&longs;a in tribus quantitatibus e&longs;t contraria coniunct&aelig; in <lb/>ei&longs;dem.</s></p><p type="margin">
  
 <s><margin.target id="marg44"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="margin"> <s><margin.target id="marg44"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg45"></margin.target>14. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg45"/>14. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="main">
  
 <s>Ex quauis ergo illarum data, data erit &amp; reliqua.<lb/><arrow.to.target n="marg46"></arrow.to.target></s></p><pb pagenum="14"/><p type="margin"> <s>Ex quauis ergo illarum data, data erit &amp; reliqua.<lb/><arrow.to.target n="marg46"/></s></p><pb pagenum="14"/><p type="margin">
  
 <s><margin.target id="marg46"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 18. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg46"/>P<emph type="italics"/>er<emph.end type="italics"/> 18. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="main">
  
 <s>Propo&longs;itio quintadecima.</s></p><p type="main"> <s>Propo&longs;itio quintadecima.</s></p><p type="main">
  
 <s>Si fuerint quatuor quantitas-proportio confu&longs;a aggregati pri&shy;<lb/>m&aelig; &amp; terti&aelig; ad aggregatum &longs;ecund&aelig;, &amp; quart&aelig; erit ut monadis <lb/>addito prouentu, qui fit diui&longs;a differentia differentiarum prim&aelig; &amp; <lb/>&longs;ecund&aelig;, atque quart&aelig; &amp; terti&aelig; per aggregatum terti&aelig;, &amp; quart&aelig; ad <lb/>ip&longs;am monadem.</s></p><figure></figure><p type="main"> <s>Si fuerint quatuor quantitas-proportio confu&longs;a aggregati pri&shy;<lb/>m&aelig; &amp; terti&aelig; ad aggregatum &longs;ecund&aelig;, &amp; quart&aelig; erit ut monadis <lb/>addito prouentu, qui fit diui&longs;a differentia differentiarum prim&aelig; &amp; <lb/>&longs;ecund&aelig;, atque quart&aelig; &amp; terti&aelig; per aggregatum terti&aelig;, &amp; quart&aelig; ad <lb/>ip&longs;am monadem.</s></p><figure/><p type="main">
  
  <s>Sint quatuor quantitates a b, c, d, e f, &amp; <lb/><arrow.to.target n="marg47"/><lb/>&longs;it a b maior cin a h, &amp; e fmaior d in f g, &amp; <lb/>differentia f g &amp; a h &longs;it a k: dico proportio&shy;<lb/>nem a b, &amp; d confu&longs;am ad c &amp; e f, e&longs;&longs;e ut mo <lb/>nadis addito prouentu, uel detracto a k diui&longs;&aelig; per aggregatum c. <lb/><!-- REMOVE S-->&amp; e f ad ip&longs;am monadem, &amp; manife&longs;tum e&longs;t, qu&ograve;d pote&longs;t continge&shy;<lb/>re pluribus modis: Primus ut a b &longs;it maior c &amp; e f minor d, &amp; tunc <lb/>differenti&aelig; coniungentur, &amp; prouentus, addetur monadi. </s>
  
 <s>Sint quatuor quantitates a b, c, d, e f, &amp; <lb/><arrow.to.target n="marg47"></arrow.to.target><lb/>&longs;it a b maior cin a h, &amp; e fmaior d in f g, &amp; <lb/>differentia f g &amp; a h &longs;it a k: dico proportio&shy;<lb/>nem a b, &amp; d confu&longs;am ad c &amp; e f, e&longs;&longs;e ut mo <lb/>nadis addito prouentu, uel detracto a k diui&longs;&aelig; per aggregatum c. <lb/></s> 
  
 <s>&amp; e f ad ip&longs;am monadem, &amp; manife&longs;tum e&longs;t, qu&ograve;d pote&longs;t continge&shy;<lb/>re pluribus modis: Primus ut a b &longs;it maior c &amp; e f minor d, &amp; tunc <lb/>differenti&aelig; coniungentur, &amp; prouentus, addetur monadi. </s> 
  
 <s>Idem fa&shy;<lb/>ciendum erit &longs;i a b &longs;it maior c, &amp; e f &longs;it minor d, &longs;ed exce&longs;&longs;us &longs;uperet <lb/>defectum. </s> <s>Idem fa&shy;<lb/>ciendum erit &longs;i a b &longs;it maior c, &amp; e f &longs;it minor d, &longs;ed exce&longs;&longs;us &longs;uperet <lb/>defectum. </s>
  
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 <s>at uer&ograve; diui&longs;a k a <lb/>per c &amp; e f fit quantum diui&longs;a eadem per b k, &amp; d, &longs;ed diui&longs;a k a per b <lb/>k, &amp; d iunctas, exit proportio a k ad aggregatum b k &amp; d: igitur di&shy;<lb/>ui&longs;a a k per aggregatum e f &amp; c, exibit eadem proportio, igitur a b <lb/>&amp; d ad aggregatum c &amp; e f e&longs;t coninncta ex monade &amp; proportio&shy;<lb/>ne a k ad aggregatum c &amp; e f, quod erat demon&longs;trandum.</s></p><p type="margin"> <s>at uer&ograve; diui&longs;a k a <lb/>per c &amp; e f fit quantum diui&longs;a eadem per b k, &amp; d, &longs;ed diui&longs;a k a per b <lb/>k, &amp; d iunctas, exit proportio a k ad aggregatum b k &amp; d: igitur di&shy;<lb/>ui&longs;a a k per aggregatum e f &amp; c, exibit eadem proportio, igitur a b <lb/>&amp; d ad aggregatum c &amp; e f e&longs;t coninncta ex monade &amp; proportio&shy;<lb/>ne a k ad aggregatum c &amp; e f, quod erat demon&longs;trandum.</s></p><p type="margin">
  
 <s><margin.target id="marg47"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><figure></figure><p type="main"> <s><margin.target id="marg47"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><figure/><p type="main">
  
 <s>Ex hoc patet quod proportionum confu&longs;io <lb/><arrow.to.target n="marg48"></arrow.to.target><lb/>fit iunctis denominatoribus numeratoris: mul&shy;<lb/>tiplicatio multiplicatis: additio multiplicatis <lb/>decu&longs;&longs;atim in numeratores ad productum ex <lb/>denominatoribus, ut in exemplis.</s></p><p type="margin"> <s>Ex hoc patet quod proportionum confu&longs;io <lb/><arrow.to.target n="marg48"/><lb/>fit iunctis denominatoribus numeratoris: mul&shy;<lb/>tiplicatio multiplicatis: additio multiplicatis <lb/>decu&longs;&longs;atim in numeratores ad productum ex <lb/>denominatoribus, ut in exemplis.</s></p><p type="margin">
  
 <s><margin.target id="marg48"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg48"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Propo&longs;itio &longs;extadecima.</s></p><p type="main"> <s>Propo&longs;itio &longs;extadecima.</s></p><p type="main">
  
 <s>Omnium quatuor quantitatum propo&longs;ita <lb/>prima, qu&aelig; non minorem habet proportionem <lb/>ad &longs;uam corre&longs;pondentem, qu&agrave;m alia ad aliam <lb/><figure id="fig8"></figure><lb/>erit proportio confu&longs;a illarum, ut pro&shy;<lb/>ducti ex aggregato prim&aelig; &amp; terti&aelig; in <pb pagenum="15"/>tertiam, ad productum ex aggregato terti&aelig; &amp; omiotat&aelig; ad &longs;ecun&shy;<lb/>dam in ip&longs;am quartam.</s></p><p type="main"> <s>Omnium quatuor quantitatum propo&longs;ita <lb/>prima, qu&aelig; non minorem habet proportionem <lb/>ad &longs;uam corre&longs;pondentem, qu&agrave;m alia ad aliam <lb/><figure id="fig8"/><lb/>erit proportio confu&longs;a illarum, ut pro&shy;<lb/>ducti ex aggregato prim&aelig; &amp; terti&aelig; in <pb pagenum="15"/>tertiam, ad productum ex aggregato terti&aelig; &amp; omiotat&aelig; ad &longs;ecun&shy;<lb/>dam in ip&longs;am quartam.</s></p><p type="main">
  
 <s>H&aelig;c magis reducit confu&longs;am proportionem ad notitiam, qu&agrave;m, <lb/>pr&aelig;cedens, quia reducit ad proportionem <expan abbr="product&atilde;">productam</expan>, qu&ecedil; operatio <lb/>e&longs;t &longs;implici&longs;sima, &longs;iue per multiplicationem quantitatum fiat, du&aelig; <lb/>&longs;unt tantum multiplicationes, &longs;iue per eundem terminum &longs;ufficit <lb/>alium addere. </s> <s>H&aelig;c magis reducit confu&longs;am proportionem ad notitiam, qu&agrave;m, <lb/>pr&aelig;cedens, quia reducit ad proportionem <expan abbr="product&atilde;">productam</expan>, qu&ecedil; operatio <lb/>e&longs;t &longs;implici&longs;sima, &longs;iue per multiplicationem quantitatum fiat, du&aelig; <lb/>&longs;unt tantum multiplicationes, &longs;iue per eundem terminum &longs;ufficit <lb/>alium addere. </s>
  
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 <s>Dico quod pro&shy;<lb/>portio confufa a b &amp; d ad c &amp; e e&longs;t uelut producti ex aggregato a b <lb/>&amp; d in d ad productum ex aggregato a f &amp; d in e. </s> <s>Dico quod pro&shy;<lb/>portio confufa a b &amp; d ad c &amp; e e&longs;t uelut producti ex aggregato a b <lb/>&amp; d in d ad productum ex aggregato a f &amp; d in e. </s>
  
 <s>Statuatur aggre&shy;<lb/><arrow.to.target n="marg49"></arrow.to.target><lb/>gatum a b &amp; d linea a d prima quantitas, &amp; aggregatum a f &amp; d, <lb/><figure id="fig9"></figure><lb/>a d &longs;ecunda quantitas, &amp; d tertia, <lb/>&amp; c quarta, &amp; ex a b in d fiat g, ex <lb/>a d in e fiat h, erit ergo per pri&shy;<lb/>mam propo&longs;itionem g ad h pro&shy;<lb/><arrow.to.target n="marg50"></arrow.to.target><lb/>ducta ex proportionibus a b d ad <lb/>a f d, &amp; d ad c. </s> <s>Statuatur aggre&shy;<lb/><arrow.to.target n="marg49"/><lb/>gatum a b &amp; d linea a d prima quantitas, &amp; aggregatum a f &amp; d, <lb/><figure id="fig9"/><lb/>a d &longs;ecunda quantitas, &amp; d tertia, <lb/>&amp; c quarta, &amp; ex a b in d fiat g, ex <lb/>a d in e fiat h, erit ergo per pri&shy;<lb/>mam propo&longs;itionem g ad h pro&shy;<lb/><arrow.to.target n="marg50"/><lb/>ducta ex proportionibus a b d ad <lb/>a f d, &amp; d ad c. <!-- KEEP S--></s>
  
 <s>Sed proportio a f d <lb/>ad aggregatum c e, e&longs;t uelut d ad <lb/>e. </s> <s>Sed proportio a f d <lb/>ad aggregatum c e, e&longs;t uelut d ad <lb/>e. </s>
  
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 <s>Ergo proportio g ad h <lb/>e&longs;t confu&longs;a ex a b ad e, &amp; d ad e, quod erat demon&longs;trandum.</s></p><p type="margin"> <s>Ergo proportio g ad h <lb/>e&longs;t confu&longs;a ex a b ad e, &amp; d ad e, quod erat demon&longs;trandum.</s></p><p type="margin">
  
 <s><margin.target id="marg49"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg49"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg50"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 13. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg50"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. P<emph type="italics"/>et.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main">
  
 <s>Propo&longs;itio decima&longs;eptima.</s></p><p type="main"> <s>Propo&longs;itio decima&longs;eptima.</s></p><p type="main">
  
 <s>Omnes du&ecedil; proportiones conuer&longs;&aelig; producunt &aelig;qualem pro&shy;<lb/>portionem.<lb/><arrow.to.target n="table12"></arrow.to.target></s></p><table><table.target id="table12"></table.target><row><cell>a</cell></row><row><cell>-----</cell></row><row><cell>b</cell></row><row><cell>---</cell></row><row><cell>c</cell></row><row><cell>----</cell></row></table><p type="main"> <s>Omnes du&ecedil; proportiones conuer&longs;&aelig; producunt &aelig;qualem pro&shy;<lb/>portionem.<lb/><arrow.to.target n="table12"/></s></p><table><table.target id="table12"/><row><cell>a</cell></row><row><cell>-----</cell></row><row><cell>b</cell></row><row><cell>---</cell></row><row><cell>c</cell></row><row><cell>----</cell></row></table><p type="main">
  
 <s>Sint du&aelig; proportiones a ad b &amp; b ad a conuer&longs;a, <lb/><arrow.to.target n="marg51"></arrow.to.target><lb/>dico, qu&ograve;d producunt proportionem &aelig;qualem. </s> <s>Sint du&aelig; proportiones a ad b &amp; b ad a conuer&longs;a, <lb/><arrow.to.target n="marg51"/><lb/>dico, qu&ograve;d producunt proportionem &aelig;qualem. </s>
  
 <s>fiat <lb/>enim b ad c, ut b ad a, erit igitur a &aelig;qualis c &amp; b c con <lb/><arrow.to.target n="marg52"></arrow.to.target><lb/>uer&longs;a eius qu&aelig; e&longs;t a ad b, &longs;ed per &longs;ecundam harum <lb/>proportiones a ad b, &amp; b ad c producunt propor&shy;<lb/>tionem a ad c, igitur proportiones etiam a ad b &amp; b ad a produ&shy;<lb/>cunt eandem.</s></p><p type="margin"> <s>fiat <lb/>enim b ad c, ut b ad a, erit igitur a &aelig;qualis c &amp; b c con <lb/><arrow.to.target n="marg52"/><lb/>uer&longs;a eius qu&aelig; e&longs;t a ad b, &longs;ed per &longs;ecundam harum <lb/>proportiones a ad b, &amp; b ad c producunt propor&shy;<lb/>tionem a ad c, igitur proportiones etiam a ad b &amp; b ad a produ&shy;<lb/>cunt eandem.</s></p><p type="margin">
  
 <s><margin.target id="marg51"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="margin"> <s><margin.target id="marg51"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg52"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 6. A<emph type="italics"/>ni&shy;<lb/>mi <expan abbr="commun&etilde;">communem</expan> <lb/>&longs;ententiam.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg52"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. A<emph type="italics"/>ni&shy;<lb/>mi <expan abbr="commun&etilde;">communem</expan> <lb/>&longs;ententiam.<emph.end type="italics"/></s></p><p type="main">
  
 <s>Propo&longs;itio decimaoctaua.</s></p><p type="main"> <s>Propo&longs;itio decimaoctaua.</s></p><p type="main">
  
 <s>Si fuerint quotlibet quantitates in continua proportione multi&shy;<lb/>plici pr&aelig;ter ultimam: proportio uer&ograve; penultim&aelig; ad ultimam qua&shy;<lb/>lis re&longs;idui prim&aelig; ad &longs;ecundam, erit prim&aelig; ad aggregatum reliqua&shy;<lb/>rum uelut penultim&aelig; ad ultimam.<pb pagenum="16"/><arrow.to.target n="marg53"></arrow.to.target></s></p><p type="margin"> <s>Si fuerint quotlibet quantitates in continua proportione multi&shy;<lb/>plici pr&aelig;ter ultimam: proportio uer&ograve; penultim&aelig; ad ultimam qua&shy;<lb/>lis re&longs;idui prim&aelig; ad &longs;ecundam, erit prim&aelig; ad aggregatum reliqua&shy;<lb/>rum uelut penultim&aelig; ad ultimam.<pb pagenum="16"/><arrow.to.target n="marg53"/></s></p><p type="margin">
  
 <s><margin.target id="marg53"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg53"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Sint quantitates a b c d in continua proportione multiplici, &longs;ed <lb/>d ad e &longs;it uelut re&longs;idui a &amp; b ad b, dico proportionem a ad b c d e <lb/>e&longs;&longs;e ut d ad e. </s> <s>Sint quantitates a b c d in continua proportione multiplici, &longs;ed <lb/>d ad e &longs;it uelut re&longs;idui a &amp; b ad b, dico proportionem a ad b c d e <lb/>e&longs;&longs;e ut d ad e. </s>
  
 <s>Quia enim e&longs;t gnomonis e ad quadratum d, ut d ad e <lb/>ex &longs;uppo&longs;ito erit per coniunctam proportionem c &amp; d ad d &amp; e, u<gap/></s></p><p type="main"> <s>Quia enim e&longs;t gnomonis e ad quadratum d, ut d ad e <lb/>ex &longs;uppo&longs;ito erit per coniunctam proportionem c &amp; d ad d &amp; e, u<gap/></s></p><p type="main">
  
 <s><arrow.to.target n="marg54"></arrow.to.target><lb/>d ad e, &longs;ed e gnomo cum quadrato d efficit qua&shy;<lb/><figure id="fig10"></figure><lb/>dratum e, igitur ut c quadrati ad d &amp; eiuncta, ita <lb/>d ad e. </s> <s><arrow.to.target n="marg54"/><lb/>d ad e, &longs;ed e gnomo cum quadrato d efficit qua&shy;<lb/><figure id="fig10"/><lb/>dratum e, igitur ut c quadrati ad d &amp; eiuncta, ita <lb/>d ad e. </s>
  
 <s>Rur&longs;us, quia b quadrati ad c quadratum, <lb/><arrow.to.target n="marg55"></arrow.to.target><lb/>ut c ad d erit gnomonis b ad quadratum c, ut <lb/>gnomonis c ad quadratum d, &amp; ita d ad e, igitur <lb/><arrow.to.target n="marg56"></arrow.to.target><lb/>gnomonum b c cum quadrato d ad aggrega&shy;<lb/>tum c d e quadratorum, ut d ad e, &longs;ed c gno&shy;<lb/>mo cum d quadrato perficit c quadratum, <lb/>&amp; c quadratum cum gnomone b perficit <lb/>quadratum b, igitur proportio quadrati b <lb/>ad quadrata c d e, ut d quadrati a d e. </s> <s>Rur&longs;us, quia b quadrati ad c quadratum, <lb/><arrow.to.target n="marg55"/><lb/>ut c ad d erit gnomonis b ad quadratum c, ut <lb/>gnomonis c ad quadratum d, &amp; ita d ad e, igitur <lb/><arrow.to.target n="marg56"/><lb/>gnomonum b c cum quadrato d ad aggrega&shy;<lb/>tum c d e quadratorum, ut d ad e, &longs;ed c gno&shy;<lb/>mo cum d quadrato perficit c quadratum, <lb/>&amp; c quadratum cum gnomone b perficit <lb/>quadratum b, igitur proportio quadrati b <lb/>ad quadrata c d e, ut d quadrati a d e. </s>
  
 <s>Et ita <lb/>repetendo de quotuis quantitatibus in infi <lb/>nitum u&longs;que. </s> <s>Et ita <lb/>repetendo de quotuis quantitatibus in infi <lb/>nitum u&longs;que. </s>
  
 <s>H&aelig;c proponitur ab Archimede in libro de quadrato <lb/>&aelig;quali parabol&aelig;, &amp; minus generaliter &amp; pluribus demon&longs;tratur. <lb/></s> <s>H&aelig;c proponitur ab Archimede in libro de quadrato <lb/>&aelig;quali parabol&aelig;, &amp; minus generaliter &amp; pluribus demon&longs;tratur. <lb/></s>
  
 <s>Ego tamen quia e&longs;t generalis, de&longs;cribam illam per corrolarium: ad&shy;<lb/>damque aliud quod ex hoc &longs;equitur.<lb/><arrow.to.target n="marg57"></arrow.to.target></s></p><p type="margin"> <s>Ego tamen quia e&longs;t generalis, de&longs;cribam illam per corrolarium: ad&shy;<lb/>damque aliud quod ex hoc &longs;equitur.<lb/><arrow.to.target n="marg57"/></s></p><p type="margin">
  
 <s><margin.target id="marg54"></margin.target>13. P<emph type="italics"/>ropo&longs;. <lb/></s> <s><margin.target id="marg54"/>13. P<emph type="italics"/>ropo&longs;. <lb/><!-- REMOVE S-->quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><!-- KEEP S--></s>
  
 <s>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin"> </p><p type="margin">
  
 <s><margin.target id="marg55"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 19. <emph type="italics"/>quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg55"/>P<emph type="italics"/>er<emph.end type="italics"/> 19. <emph type="italics"/>quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg56"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 12. <emph type="italics"/>quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg56"/>P<emph type="italics"/>er<emph.end type="italics"/> 12. <emph type="italics"/>quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg57"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main"> <s><margin.target id="marg57"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.<!-- KEEP S--></s></p><p type="main">
  
 <s>Si fuerint quotlibet <expan abbr="qu&atilde;titates">quantitates</expan> omnes analog&aelig; pr&aelig;ter ultimam, <lb/>&longs;it autem penultima ad ultimam qualis re&longs;idui prim&aelig; &amp; &longs;ecund&aelig; <lb/>ad &longs;ecundam, erit proportio prim&aelig; ad aggregatum omnium alia&shy;<lb/>rum ueluti penultim&aelig; ad ultimam.</s></p><p type="main"> <s>Si fuerint quotlibet <expan abbr="qu&atilde;titates">quantitates</expan> omnes analog&aelig; pr&aelig;ter ultimam, <lb/>&longs;it autem penultima ad ultimam qualis re&longs;idui prim&aelig; &amp; &longs;ecund&aelig; <lb/>ad &longs;ecundam, erit proportio prim&aelig; ad aggregatum omnium alia&shy;<lb/>rum ueluti penultim&aelig; ad ultimam.</s></p><p type="main">
  
 <s><arrow.to.target n="marg58"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg58"/></s></p><p type="margin">
  
 <s><margin.target id="marg58"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg58"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>H&aelig;c enim e&longs;t euidens, quia conuenit ei demon&longs;tratio propo&longs;ita. <lb/><figure id="fig11"></figure><lb/>exemplo autem in numeris &agrave; latere <lb/>po&longs;ito uides declarationem. </s> <s>H&aelig;c enim e&longs;t euidens, quia conuenit ei demon&longs;tratio propo&longs;ita. <lb/><figure id="fig11"/><lb/>exemplo autem in numeris &agrave; latere <lb/>po&longs;ito uides declarationem. </s>
  
 <s>nam <lb/>proportio 16 ad 32 e&longs;t uelut 27 re&longs;i <lb/>dui prim&aelig; &amp; &longs;ecund&aelig; ad ip&longs;am &longs;e&shy;<lb/>cundam &longs;cilicet ad 54.</s></p><p type="main"> <s>nam <lb/>proportio 16 ad 32 e&longs;t uelut 27 re&longs;i <lb/>dui prim&aelig; &amp; &longs;ecund&aelig; ad ip&longs;am &longs;e&shy;<lb/>cundam &longs;cilicet ad 54.</s></p><p type="main">
  
 <s><arrow.to.target n="marg59"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg59"/></s></p><p type="margin">
  
 <s><margin.target id="marg59"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main"> <s><margin.target id="marg59"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.<!-- KEEP S--></s></p><p type="main">
  
 <s>Ex hoc patet etiam qu&ograve;d a&longs;&longs;umptis omnibus, &longs;ub multiplicibus <lb/>analogi&aelig; u&longs;que in infinitum prima quantitas e&longs;t multiplex aggre&shy;<lb/>gati omnium reliquarum numero 1 m: quo prima e&longs;t multiplex <lb/>&longs;ecund&aelig;.</s></p><p type="main"> <s>Ex hoc patet etiam qu&ograve;d a&longs;&longs;umptis omnibus, &longs;ub multiplicibus <lb/>analogi&aelig; u&longs;que in infinitum prima quantitas e&longs;t multiplex aggre&shy;<lb/>gati omnium reliquarum numero 1 m: quo prima e&longs;t multiplex <lb/>&longs;ecund&aelig;.</s></p><p type="main">
  
 <s><arrow.to.target n="marg60"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg60"/></s></p><p type="margin">
  
 <s><margin.target id="marg60"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3.</s></p><p type="main"> <s><margin.target id="marg60"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3.<!-- KEEP S--></s></p><p type="main">
  
 <s>Si fuerint quotlibet quantitates in &longs;uper particulari proportio&shy;<lb/>ne analog&aelig;, erit proportio prim&aelig; ad aggregatum omnium in infi&shy;<lb/>nitum iuxta proportionem multiplicem conuer&longs;am illius partis.</s></p><p type="main"> <s>Si fuerint quotlibet quantitates in &longs;uper particulari proportio&shy;<lb/>ne analog&aelig;, erit proportio prim&aelig; ad aggregatum omnium in infi&shy;<lb/>nitum iuxta proportionem multiplicem conuer&longs;am illius partis.</s></p><p type="main">
  
 <s><arrow.to.target n="marg61"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg61"/></s></p><p type="margin">
  
 <s><margin.target id="marg61"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg61"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Velut collect&aelig; in &longs;e&longs;quialtera dupl&aelig; in &longs;exquitertia tripl&aelig; in <lb/>&longs;exqui&longs;eptima &longs;eptupl&aelig;. </s> <s>Velut collect&aelig; in &longs;e&longs;quialtera dupl&aelig; in &longs;exquitertia tripl&aelig; in <lb/>&longs;exqui&longs;eptima &longs;eptupl&aelig;. </s>
  
 <s>Vt capio 512 448 392 343, &amp; ita deinceps <lb/>u&longs;que in infinitum aggregatum omnium earum erit 3584. Septu&shy;<pb pagenum="17"/>plum 512, &amp; aggregatum 18. 12. 8. 5 2/3, &amp; ita deinceps in &longs;<gap/>xquialtera <lb/>erit 54 duplum 27 prim&aelig; in eo ordine.</s></p><p type="head"> <s>Vt capio 512 448 392 343, &amp; ita deinceps <lb/>u&longs;que in infinitum aggregatum omnium earum erit 3584. Septu&shy;<pb pagenum="17"/>plum 512, &amp; aggregatum 18. 12. 8. 5 2/3, &amp; ita deinceps in &longs;<gap/>xquialtera <lb/>erit 54 duplum 27 prim&aelig; in eo ordine.</s></p><p type="head">
  
 <s>SCHOLIVM.</s></p><p type="main"> <s>SCHOLIVM.<!-- KEEP S--></s></p><p type="main">
  
 <s>Ex quo patet genus demon&longs;trandi nouun &amp; pulchrum: nam <lb/>&longs;upponatur 54, aggregatum duplum 27, prim&aelig; igitur addito 27 <lb/>ad 54, cum &longs;it dimidium, &amp; addito 13 1/2, dimidio 27 ad 27, nam ex <lb/>&longs;uppo&longs;ito quantitas &longs;equens e&longs;t &longs;exquialtera ad 27, igitur 81 e&longs;t du&shy;</s></p><p type="main"> <s>Ex quo patet genus demon&longs;trandi nouun &amp; pulchrum: nam <lb/>&longs;upponatur 54, aggregatum duplum 27, prim&aelig; igitur addito 27 <lb/>ad 54, cum &longs;it dimidium, &amp; addito 13 1/2, dimidio 27 ad 27, nam ex <lb/>&longs;uppo&longs;ito quantitas &longs;equens e&longs;t &longs;exquialtera ad 27, igitur 81 e&longs;t du&shy;</s></p><p type="main">
  
 <s><arrow.to.target n="marg62"></arrow.to.target><lb/>plum ad 40 1/2. Igitur conuertendo e&longs;t proportio aggregati prioris <lb/>ad 27 e&longs;t dupla, ergo aggregatum e&longs;t 54.<lb/><arrow.to.target n="marg63"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg62"/><lb/>plum ad 40 1/2. Igitur conuertendo e&longs;t proportio aggregati prioris <lb/>ad 27 e&longs;t dupla, ergo aggregatum e&longs;t 54.<lb/><arrow.to.target n="marg63"/></s></p><p type="margin">
  
 <s><margin.target id="marg62"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 18. <emph type="italics"/>quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg62"/>P<emph type="italics"/>er<emph.end type="italics"/> 18. <emph type="italics"/>quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg63"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 4.</s></p><p type="main"> <s><margin.target id="marg63"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 4.<!-- KEEP S--></s></p><p type="main">
  
 <s>Ex hoc patet eandem generaliter quod proportio maioris quan <lb/>titatis ad aggregatum reliquarum analogarum e&longs;t, uelut eius quod <lb/>prouenit diui&longs;o quadrato maioris termini per differentiam eius, &amp; <lb/>&longs;equentis maioris in eadem proportione ad ip&longs;um maiorem.</s></p><p type="main"> <s>Ex hoc patet eandem generaliter quod proportio maioris quan <lb/>titatis ad aggregatum reliquarum analogarum e&longs;t, uelut eius quod <lb/>prouenit diui&longs;o quadrato maioris termini per differentiam eius, &amp; <lb/>&longs;equentis maioris in eadem proportione ad ip&longs;um maiorem.</s></p><p type="main">
  
 <s><arrow.to.target n="marg64"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg64"/></s></p><p type="margin">
  
 <s><margin.target id="marg64"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg64"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Exemplum &longs;it proportio augens 25 &amp; 35 duarum quintarum, uo <lb/>lo &longs;cire quantum &longs;it aggregatum omnium citra 25, maximam acci&shy;<lb/>pio 35, ulteriorem ad 25, cuius differentia a 25 e&longs;t 10, cum quo diui&shy;<lb/>do 625 quadratum, exit 62 1/2 aggregatum quantitatum. </s> <s>Exemplum &longs;it proportio augens 25 &amp; 35 duarum quintarum, uo <lb/>lo &longs;cire quantum &longs;it aggregatum omnium citra 25, maximam acci&shy;<lb/>pio 35, ulteriorem ad 25, cuius differentia a 25 e&longs;t 10, cum quo diui&shy;<lb/>do 625 quadratum, exit 62 1/2 aggregatum quantitatum. </s>
  
 <s>Et facile po&shy;</s></p><p type="main"> <s>Et facile po&shy;</s></p><p type="main">
  
 <s><arrow.to.target n="marg65"></arrow.to.target><lb/>re&longs;t demon&longs;trari. </s> <s><arrow.to.target n="marg65"/><lb/>re&longs;t demon&longs;trari. </s>
  
 <s>Si quis dicat in qua proportione &longs;unt infinit&aelig; <lb/>quantitates analog&aelig; cum 12, qu&aelig;iunct&aelig; efficiunt 10, iunge 10 cum <lb/>12 fit 22, duc 22 in 12 fit 264, diuide 264 per 10, exit 26 2/3, &amp; in ea pro&shy;<lb/>portione <expan abbr="er&utilde;t">erunt</expan> ill&aelig; quantitates, in qua &longs;unt 26 2/3 ad 12: duc per 5 fiunt <lb/>60, &amp; 132 diuide per 12, exeunt 11 &amp; 5, &amp; ita eruntin proportione 11 <lb/>ad 5 experiaris, &amp; inuenies, &amp; demon&longs;tratur ex prioribus.</s></p><p type="margin"> <s>Si quis dicat in qua proportione &longs;unt infinit&aelig; <lb/>quantitates analog&aelig; cum 12, qu&aelig;iunct&aelig; efficiunt 10, iunge 10 cum <lb/>12 fit 22, duc 22 in 12 fit 264, diuide 264 per 10, exit 26 2/3, &amp; in ea pro&shy;<lb/>portione <expan abbr="er&utilde;t">erunt</expan> ill&aelig; quantitates, in qua &longs;unt 26 2/3 ad 12: duc per 5 fiunt <lb/>60, &amp; 132 diuide per 12, exeunt 11 &amp; 5, &amp; ita eruntin proportione 11 <lb/>ad 5 experiaris, &amp; inuenies, &amp; demon&longs;tratur ex prioribus.</s></p><p type="margin">
  
 <s><margin.target id="marg65"></margin.target>Q<emph type="italics"/>u&aelig;ftio.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg65"/>Q<emph type="italics"/>u&aelig;ftio.<emph.end type="italics"/></s></p><p type="main">
  
 <s>Propo&longs;itio decimanona.</s></p><p type="main"> <s>Propo&longs;itio decimanona.</s></p><p type="main">
  
 <s>Si fu erint aliquot quantitates arithmetic&aelig; omiolog&aelig;, quarum <lb/>exce&longs;&longs;us &longs;it &aelig;qualis minim&egrave;, omnibus autem deficientibus &longs;upple&shy;<lb/>menta ad &ecedil;qualitatem maxim&egrave; adiungantur, erunt quadrata omni&shy;<lb/>um quantitatum &aelig;qualium adiecto rur&longs;us quadrato prim&aelig; cum <lb/>eo quod fit ex minima primi ordinis in <expan abbr="aggregat&utilde;">aggregatum</expan> omnium quan&shy;<lb/>titatum eiu&longs;dem tripla aggregato quadra&shy;<lb/><figure id="fig12"></figure><lb/>torum omnium quantitatum primi ordinis <lb/><arrow.to.target n="marg66"></arrow.to.target><lb/>pariter acceptis.</s></p><p type="margin"> <s>Si fu erint aliquot quantitates arithmetic&aelig; omiolog&aelig;, quarum <lb/>exce&longs;&longs;us &longs;it &aelig;qualis minim&egrave;, omnibus autem deficientibus &longs;upple&shy;<lb/>menta ad &ecedil;qualitatem maxim&egrave; adiungantur, erunt quadrata omni&shy;<lb/>um quantitatum &aelig;qualium adiecto rur&longs;us quadrato prim&aelig; cum <lb/>eo quod fit ex minima primi ordinis in <expan abbr="aggregat&utilde;">aggregatum</expan> omnium quan&shy;<lb/>titatum eiu&longs;dem tripla aggregato quadra&shy;<lb/><figure id="fig12"/><lb/>torum omnium quantitatum primi ordinis <lb/><arrow.to.target n="marg66"/><lb/>pariter acceptis.</s></p><p type="margin">
  
 <s><margin.target id="marg66"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg66"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Sint aliquot quantitates a b c d e f g h in <lb/>continua proportione. </s> <s>Sint aliquot quantitates a b c d e f g h in <lb/>continua proportione. </s>
  
Line 819 
Line 818 
  
 <s>Dico <lb/>ergo &qring;d aggregatum <expan abbr="quadrator&utilde;">quadratorum</expan> quantitatum &longs;ecundi ordinis pri <lb/>mo quadrato bis repetito, &longs;eu uno addito <expan abbr="c&utilde;">cum</expan> eo quod fit ex minima <lb/>in aggregatum quantitatum primi ordinis e&longs;t <expan abbr="tripl&utilde;">triplum</expan> aggregato ex <lb/>quadratis omnibus <expan abbr="quantitat&utilde;">quantitatum</expan> <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> primi ordinis, &amp; utres exem <lb/>plo facilius innote&longs;cat, &longs;int <expan abbr="qu&atilde;titates">quantitates</expan> primi ordinis 8. 7. 6. 5. 4. 3. 2. 1. <lb/>quorum quadrata &longs;int 64. 49. 36. 25. 16. &amp; 9.4 &amp; 1. qu&aelig; iuncta <expan abbr="faci&utilde;t">faciunt</expan> <lb/>204, dico quod &longs;i &longs;umamus quadrata omnium <expan abbr="qu&atilde;titatum">quantitatum</expan> &longs;ecundi <lb/>ordinis, qu&aelig; &longs;unt octies 64, &amp; eis addiderimus unum <expan abbr="quadrat&utilde;">quadratum</expan> ex <lb/>his, ut fiant nouies 64, &amp; erunt 556, &longs;imul iuncta &amp; eis addamus, &qring;d <lb/>fit ex 1 quantitate minima primi ordinis in 36 aggregatum quanti&shy;<lb/>tatum omnium primi ordinis, &amp; e&longs;t tale <expan abbr="product&utilde;">productum</expan> 36, ut fiat totum <lb/>612, quod tale 612 e&longs;t triplum 204, aggregati <expan abbr="quadrator&utilde;">quadratorum</expan> primi or&shy;<lb/>dinis unius demon&longs;tratio h&ecedil;c e&longs;t. </s> <s>Dico <lb/>ergo &qring;d aggregatum <expan abbr="quadrator&utilde;">quadratorum</expan> quantitatum &longs;ecundi ordinis pri <lb/>mo quadrato bis repetito, &longs;eu uno addito <expan abbr="c&utilde;">cum</expan> eo quod fit ex minima <lb/>in aggregatum quantitatum primi ordinis e&longs;t <expan abbr="tripl&utilde;">triplum</expan> aggregato ex <lb/>quadratis omnibus <expan abbr="quantitat&utilde;">quantitatum</expan> <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> primi ordinis, &amp; utres exem <lb/>plo facilius innote&longs;cat, &longs;int <expan abbr="qu&atilde;titates">quantitates</expan> primi ordinis 8. 7. 6. 5. 4. 3. 2. 1. <lb/>quorum quadrata &longs;int 64. 49. 36. 25. 16. &amp; 9.4 &amp; 1. qu&aelig; iuncta <expan abbr="faci&utilde;t">faciunt</expan> <lb/>204, dico quod &longs;i &longs;umamus quadrata omnium <expan abbr="qu&atilde;titatum">quantitatum</expan> &longs;ecundi <lb/>ordinis, qu&aelig; &longs;unt octies 64, &amp; eis addiderimus unum <expan abbr="quadrat&utilde;">quadratum</expan> ex <lb/>his, ut fiant nouies 64, &amp; erunt 556, &longs;imul iuncta &amp; eis addamus, &qring;d <lb/>fit ex 1 quantitate minima primi ordinis in 36 aggregatum quanti&shy;<lb/>tatum omnium primi ordinis, &amp; e&longs;t tale <expan abbr="product&utilde;">productum</expan> 36, ut fiat totum <lb/>612, quod tale 612 e&longs;t triplum 204, aggregati <expan abbr="quadrator&utilde;">quadratorum</expan> primi or&shy;<lb/>dinis unius demon&longs;tratio h&ecedil;c e&longs;t. </s>
  
 <s>Quia ex quarta &longs;ecundi Element. <lb/></s> <s>Quia ex quarta &longs;ecundi Element. <lb/><!-- REMOVE S-->Euclidis &longs;ingula quadrata <expan abbr="quantitat&utilde;">quantitatum</expan> <expan abbr="diui&longs;ar&utilde;">diui&longs;arum</expan> &longs;ecundi ordinis con <lb/>&longs;tant ex quatuor partibus quarum du&ecedil; &longs;unt quadrata partium, reli&shy;<lb/>qu&aelig; du&aelig; &longs;unt producta ex partibus <expan abbr="inuic&etilde;">inuicem</expan> bis, &amp; quia h fuit &aelig;qua&shy;<lb/>lis 1, &amp; p &ecedil;qualis b, quia &longs;upplementa <expan abbr="fuer&utilde;t&ecedil;qualia">fuerunt&ecedil;qualia</expan> mutu&ograve; quanti <lb/>tatibus, &amp; ita c &aelig;qualis o &amp; k &aelig;qualis g &amp; d, &aelig;qualis n &amp; l, &aelig;qualis <lb/>f, e <expan abbr="a&utilde;t">aunt</expan> &ecedil;qualis m. </s>
  
  
 <s>Euclidis &longs;ingula quadrata <expan abbr="quantitat&utilde;">quantitatum</expan> <expan abbr="diui&longs;ar&utilde;">diui&longs;arum</expan> &longs;ecundi ordinis con <lb/>&longs;tant ex quatuor partibus quarum du&ecedil; &longs;unt quadrata partium, reli&shy;<lb/>qu&aelig; du&aelig; &longs;unt producta ex partibus <expan abbr="inuic&etilde;">inuicem</expan> bis, &amp; quia h fuit &aelig;qua&shy;<lb/>lis 1, &amp; p &ecedil;qualis b, quia &longs;upplementa <expan abbr="fuer&utilde;t&ecedil;qualia">fuerunt&ecedil;qualia</expan> mutu&ograve; quanti <lb/>tatibus, &amp; ita c &aelig;qualis o &amp; k &aelig;qualis g &amp; d, &aelig;qualis n &amp; l, &aelig;qualis <lb/>f, e <expan abbr="a&utilde;t">aunt</expan> &ecedil;qualis m. </s> 
  
 <s><expan abbr="Sequi&ttilde;">Sequitur</expan> ergo quod &longs;umptis duabus quantitatibus <lb/>&longs;ecundi ordinis hab entibus <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> mutu&ograve; &aelig;qualia ip&longs;is quan <lb/>titatibus quod quadrata partium <expan abbr="er&utilde;t">erunt</expan> dupla quadratis primarum <lb/>quantitatum: ueluti capio b i &longs;ecundam &amp; h p ultimam, <expan abbr="quar&utilde;">quarum</expan> qua&shy;<pb pagenum="19"/>drata partium &longs;unt quadrata b &amp; i, &amp; h &amp; p, &longs;ed b e&longs;t &aelig;qualis p, &amp; h <lb/>&aelig;qualis i. </s> <s><expan abbr="Sequi&ttilde;">Sequitur</expan> ergo quod &longs;umptis duabus quantitatibus <lb/>&longs;ecundi ordinis hab entibus <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> mutu&ograve; &aelig;qualia ip&longs;is quan <lb/>titatibus quod quadrata partium <expan abbr="er&utilde;t">erunt</expan> dupla quadratis primarum <lb/>quantitatum: ueluti capio b i &longs;ecundam &amp; h p ultimam, <expan abbr="quar&utilde;">quarum</expan> qua&shy;<pb pagenum="19"/>drata partium &longs;unt quadrata b &amp; i, &amp; h &amp; p, &longs;ed b e&longs;t &aelig;qualis p, &amp; h <lb/>&aelig;qualis i. </s>
  
 <s>Ergo quatuor quadrata b i &amp; h p &longs;unt dupla quadratis b <lb/>&amp; h, &amp; ita <expan abbr="conclud&atilde;">concludam</expan> de omnibus ubi du&aelig; quantitates duabus com <lb/>parantur: &longs;ed in e m quia e&longs;t &longs;ola una quantitas, i&longs;tud e&longs;t etiam cla&shy;<lb/>rius, quia quadrata e &amp; m &longs;unt dupla quadrato e &longs;oli eo, quod &amp; m <lb/><arrow.to.target n="marg67"></arrow.to.target><lb/>&longs;unt &aelig;quales. </s> <s>Ergo quatuor quadrata b i &amp; h p &longs;unt dupla quadratis b <lb/>&amp; h, &amp; ita <expan abbr="conclud&atilde;">concludam</expan> de omnibus ubi du&aelig; quantitates duabus com <lb/>parantur: &longs;ed in e m quia e&longs;t &longs;ola una quantitas, i&longs;tud e&longs;t etiam cla&shy;<lb/>rius, quia quadrata e &amp; m &longs;unt dupla quadrato e &longs;oli eo, quod &amp; m <lb/><arrow.to.target n="marg67"/><lb/>&longs;unt &aelig;quales. </s>
  
 <s>Igitur per demon&longs;trata ab Euclide erit proportio o&shy;<lb/>mnium quadratorum b i, c k, d l, e m, f n, g o, h p, ad quadrata b c d e <lb/>f g h, pariter accepta proportio dupla. </s> <s>Igitur per demon&longs;trata ab Euclide erit proportio o&shy;<lb/>mnium quadratorum b i, c k, d l, e m, f n, g o, h p, ad quadrata b c d e <lb/>f g h, pariter accepta proportio dupla. </s>
  
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Line 834 
  
 <s>Re <lb/>liquum e&longs;t modo ut o&longs;tendamus dupla <expan abbr="illor&utilde;">illorum</expan> productorum, cum <lb/>eo quod fit ex minima quantitate, &longs;cilicet h in aggregatum ip&longs;arum <lb/>quantitatum primi ordinis e&longs;&longs;e &aelig;quale quadratis, <expan abbr="quantitat&utilde;">quantitatum</expan> eiu&longs;&shy;<lb/>dem primi ordinis pariter acceptis. </s> <s>Re <lb/>liquum e&longs;t modo ut o&longs;tendamus dupla <expan abbr="illor&utilde;">illorum</expan> productorum, cum <lb/>eo quod fit ex minima quantitate, &longs;cilicet h in aggregatum ip&longs;arum <lb/>quantitatum primi ordinis e&longs;&longs;e &aelig;quale quadratis, <expan abbr="quantitat&utilde;">quantitatum</expan> eiu&longs;&shy;<lb/>dem primi ordinis pariter acceptis. </s>
  
 <s>Con&longs;tatigitur, quod duplum <gap/><lb/>in b e&longs;t &aelig;quale duplo h in ip&longs;um b, quia h &amp; i &longs;unt &aelig;quales, &amp; du&shy;<lb/>plum k in ip&longs;um c, e&longs;t &aelig;quale quadruplo h in idem c, quia k e&longs;t du&shy;<lb/>pla h, &amp; &longs;imiliter duplum l in ip&longs;um d e&longs;t &aelig;quale &longs;excuplo, h in d, <lb/>quia l e&longs;t tripla h, &amp; ita procedendo erunt illa dupla producta &aelig;&shy;<lb/>qualia productis ex h in ip&longs;as quantitates toties &longs;umptis quantus <lb/>e&longs;t numerus, qui prouenit duplicato numero, &longs;ecundum <expan abbr="qu&etilde;">quem</expan> h con <lb/>tinetur in illo &longs;upplemento, exemplum uolo duplum producti lin <lb/>d bis, &longs;cio qu&ograve;d &longs;upplementum l continet h ter, duplicabo tria &amp; fi&shy;<lb/>ent &longs;ex, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="dupl&utilde;">duplum</expan> lin d &aelig;quale e&longs;t &longs;excuplo h in ip&longs;um d. </s> <s>Con&longs;tatigitur, quod duplum <gap/><lb/>in b e&longs;t &aelig;quale duplo h in ip&longs;um b, quia h &amp; i &longs;unt &aelig;quales, &amp; du&shy;<lb/>plum k in ip&longs;um c, e&longs;t &aelig;quale quadruplo h in idem c, quia k e&longs;t du&shy;<lb/>pla h, &amp; &longs;imiliter duplum l in ip&longs;um d e&longs;t &aelig;quale &longs;excuplo, h in d, <lb/>quia l e&longs;t tripla h, &amp; ita procedendo erunt illa dupla producta &aelig;&shy;<lb/>qualia productis ex h in ip&longs;as quantitates toties &longs;umptis quantus <lb/>e&longs;t numerus, qui prouenit duplicato numero, &longs;ecundum <expan abbr="qu&etilde;">quem</expan> h con <lb/>tinetur in illo &longs;upplemento, exemplum uolo duplum producti lin <lb/>d bis, &longs;cio qu&ograve;d &longs;upplementum l continet h ter, duplicabo tria &amp; fi&shy;<lb/>ent &longs;ex, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="dupl&utilde;">duplum</expan> lin d &aelig;quale e&longs;t &longs;excuplo h in ip&longs;um d. <!-- KEEP S--></s>
  
 <s>Quo con&shy;<lb/>&longs;tituto, cum &longs;uppo&longs;itum &longs;it producta illa duplicata cum producto h <lb/>in aggregatum primarum <expan abbr="qu&atilde;titatum">quantitatum</expan> e&longs;&longs;e &aelig;qualia quadratis ip&longs;a&shy;<lb/>rum quantitatum, igitur addemus <expan abbr="product&utilde;">productum</expan> ex h in &longs;ingulas quan&shy;<lb/>titates productis illis prioribus, &amp; fiet productum h in a &longs;emel, in b <lb/>ter, in c quinquies, in d &longs;epties, in e nouies, in f undecies, in g trede&shy;<lb/>cies, &amp; in h quindecies &aelig;quale duplo producti uniu&longs;cuiu&longs;que quan&shy;<lb/>titatis in &longs;uum &longs;upplementum cum producto h in <expan abbr="aggregat&utilde;">aggregatum</expan> ip&longs;a&shy;<lb/>rum quantitarum, at quadratum a e&longs;t &ecedil;quale producto ex h in eam, <lb/>qu&ecedil; talem habet proportionem ad ip&longs;um a, <expan abbr="qual&etilde;">qualem</expan> habet a ad ip&longs;um <lb/><arrow.to.target n="marg68"></arrow.to.target><lb/>h per demon&longs;trata ab Euclide, &amp; pariter de quadrato b, quod e&longs;t &ecedil;&shy;<lb/>quale ei quod fit ex h in eam qu&aelig; toties continet b, quotiens b con <lb/>tinet h, &amp; ita quadratum c &aelig;quale e&longs;t ei, quod continetur &longs;ub h, &amp; <lb/>habente proportionem ad b eandem, quam b ad h, &amp; &longs;imiliter de <lb/>quadrato c &amp; omnibus reliquis, u&longs;que ad h ip&longs;um. </s> <s>Quo con&shy;<lb/>&longs;tituto, cum &longs;uppo&longs;itum &longs;it producta illa duplicata cum producto h <lb/>in aggregatum primarum <expan abbr="qu&atilde;titatum">quantitatum</expan> e&longs;&longs;e &aelig;qualia quadratis ip&longs;a&shy;<lb/>rum quantitatum, igitur addemus <expan abbr="product&utilde;">productum</expan> ex h in &longs;ingulas quan&shy;<lb/>titates productis illis prioribus, &amp; fiet productum h in a &longs;emel, in b <lb/>ter, in c quinquies, in d &longs;epties, in e nouies, in f undecies, in g trede&shy;<lb/>cies, &amp; in h quindecies &aelig;quale duplo producti uniu&longs;cuiu&longs;que quan&shy;<lb/>titatis in &longs;uum &longs;upplementum cum producto h in <expan abbr="aggregat&utilde;">aggregatum</expan> ip&longs;a&shy;<lb/>rum quantitarum, at quadratum a e&longs;t &ecedil;quale producto ex h in eam, <lb/>qu&ecedil; talem habet proportionem ad ip&longs;um a, <expan abbr="qual&etilde;">qualem</expan> habet a ad ip&longs;um <lb/><arrow.to.target n="marg68"/><lb/>h per demon&longs;trata ab Euclide, &amp; pariter de quadrato b, quod e&longs;t &ecedil;&shy;<lb/>quale ei quod fit ex h in eam qu&aelig; toties continet b, quotiens b con <lb/>tinet h, &amp; ita quadratum c &aelig;quale e&longs;t ei, quod continetur &longs;ub h, &amp; <lb/>habente proportionem ad b eandem, quam b ad h, &amp; &longs;imiliter de <lb/>quadrato c &amp; omnibus reliquis, u&longs;que ad h ip&longs;um. </s>
  
 <s>Gratia ergo exem <pb pagenum="20"/>pli quadratum a, erit &aelig;quale producto ex h in omnes quatitates &longs;e&shy;<lb/>cundas, quia quotus e&longs;t numerus quantitatum, totus e&longs;t numerus <lb/>&longs;ecundum quem a continet h, &amp; &longs;imiliter quotus e&longs;t numerus quan <lb/>t&iacute;tatum incipiendo &agrave; b, &amp; quotus e&longs;t numerus quantitatum incipi&shy;<lb/>endo &agrave; c, toties b uel c <expan abbr="contin&etilde;t">continent</expan> h, &amp; ita de alijs, quadrata ergo om&shy;<lb/>nium quantitatum &longs;imul iuncta &longs;unt &aelig;qualia productis ex h in &longs;in&shy;<lb/>gulas illarum toties &longs;umptis, quoties ill&aelig; <expan abbr="c&otilde;tinent">continent</expan> h, &longs;eu quotus e&longs;t <lb/>numerus illius quantitatis, incipiendo ab h, &amp; <expan abbr="numer&atilde;do">numerando</expan> uer&longs;us a. <lb/></s> <s>Gratia ergo exem <pb pagenum="20"/>pli quadratum a, erit &aelig;quale producto ex h in omnes quatitates &longs;e&shy;<lb/>cundas, quia quotus e&longs;t numerus quantitatum, totus e&longs;t numerus <lb/>&longs;ecundum quem a continet h, &amp; &longs;imiliter quotus e&longs;t numerus quan <lb/>t&iacute;tatum incipiendo &agrave; b, &amp; quotus e&longs;t numerus quantitatum incipi&shy;<lb/>endo &agrave; c, toties b uel c <expan abbr="contin&etilde;t">continent</expan> h, &amp; ita de alijs, quadrata ergo om&shy;<lb/>nium quantitatum &longs;imul iuncta &longs;unt &aelig;qualia productis ex h in &longs;in&shy;<lb/>gulas illarum toties &longs;umptis, quoties ill&aelig; <expan abbr="c&otilde;tinent">continent</expan> h, &longs;eu quotus e&longs;t <lb/>numerus illius quantitatis, incipiendo ab h, &amp; <expan abbr="numer&atilde;do">numerando</expan> uer&longs;us a. <lb/></s>
  
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Line 854 
  
 <s>Sed <lb/>i&longs;ta, qu&aelig; relicta &longs;unt iam, &longs;unt manife&longs;t&egrave; &aelig;qualia, ergo etiam pri&shy;<lb/>ma aggregata ab initio fuere &aelig;qualia, ergo &amp; &aelig;qualia illis qua&shy;<lb/>drata a b c d e f g h his, qu&aelig; fiunt, ex h in ea&longs;dem quantita&shy;<lb/>tes cum duplo producti b in i, cin k, d in l, e in m, f in n, g in o, <lb/>h in p, &longs;ed iam his quadratis a b c d e f g h demon&longs;trata &longs;unt e&longs;&longs;e du&shy;<lb/>pla quadrata h p, g o, f n, e m, d l, c k, b i, cum duplo quadra&shy;<lb/>ti a, ergo quadrata omnium quantitatum &longs;ecundi ordinis cum <lb/>quadrato a rur&longs;us repetito, &amp; producto h in aggregatum quanti&shy;<lb/>tatum primi ordinis &longs;unt tripla quadratis quantitatum primi ordi&shy;<lb/>nis pariter acceptis, quod fuit propo&longs;itum, &amp; fuit Archimedis in li <lb/>bro de lineis &longs;piralibus, &amp; ego adieci hic propter modum demon <lb/>&longs;trandi, qui e&longs;t eleganti&longs;simus, &amp; procedit ex principijs arithmeti&shy;<lb/>cis, &amp; diuer&longs;is &agrave; communibus, &amp; ideo non reuoluitur, ut &longs;olentre&shy;<lb/>liqu&aelig; qu&aelig;&longs;tiones.</s></p><p type="margin"> <s>Sed <lb/>i&longs;ta, qu&aelig; relicta &longs;unt iam, &longs;unt manife&longs;t&egrave; &aelig;qualia, ergo etiam pri&shy;<lb/>ma aggregata ab initio fuere &aelig;qualia, ergo &amp; &aelig;qualia illis qua&shy;<lb/>drata a b c d e f g h his, qu&aelig; fiunt, ex h in ea&longs;dem quantita&shy;<lb/>tes cum duplo producti b in i, cin k, d in l, e in m, f in n, g in o, <lb/>h in p, &longs;ed iam his quadratis a b c d e f g h demon&longs;trata &longs;unt e&longs;&longs;e du&shy;<lb/>pla quadrata h p, g o, f n, e m, d l, c k, b i, cum duplo quadra&shy;<lb/>ti a, ergo quadrata omnium quantitatum &longs;ecundi ordinis cum <lb/>quadrato a rur&longs;us repetito, &amp; producto h in aggregatum quanti&shy;<lb/>tatum primi ordinis &longs;unt tripla quadratis quantitatum primi ordi&shy;<lb/>nis pariter acceptis, quod fuit propo&longs;itum, &amp; fuit Archimedis in li <lb/>bro de lineis &longs;piralibus, &amp; ego adieci hic propter modum demon <lb/>&longs;trandi, qui e&longs;t eleganti&longs;simus, &amp; procedit ex principijs arithmeti&shy;<lb/>cis, &amp; diuer&longs;is &agrave; communibus, &amp; ideo non reuoluitur, ut &longs;olentre&shy;<lb/>liqu&aelig; qu&aelig;&longs;tiones.</s></p><p type="margin">
  
 <s><margin.target id="marg67"></margin.target>I<emph type="italics"/>n<emph.end type="italics"/> 5. E<emph type="italics"/>l<gap/><emph.end type="italics"/><lb/>P<emph type="italics"/>rop.<emph.end type="italics"/> 12.</s></p><p type="margin"> <s><margin.target id="marg67"/>I<emph type="italics"/>n<emph.end type="italics"/> 5. E<emph type="italics"/>l<gap/><emph.end type="italics"/><lb/>P<emph type="italics"/>rop.<emph.end type="italics"/> 12.</s></p><p type="margin">
  
 <s><margin.target id="marg68"></margin.target>L<emph type="italics"/>ib.<emph.end type="italics"/> 6. E<emph type="italics"/>l<gap/>.<emph.end type="italics"/><lb/>P<emph type="italics"/>rop.<emph.end type="italics"/> 17.</s></p><p type="main"> <s><margin.target id="marg68"/>L<emph type="italics"/>ib.<emph.end type="italics"/> 6. E<emph type="italics"/>l<gap/>.<emph.end type="italics"/><lb/>P<emph type="italics"/>rop.<emph.end type="italics"/> 17.</s></p><p type="main">
  
 <s>Propo&longs;itio uige&longs;ima.</s></p><p type="main"> <s>Propo&longs;itio uige&longs;ima.</s></p><p type="main">
  
 <s>C&ugrave;m fuerint quatuor quantitates, fueritque &longs;ecunda &aelig;qualis ter&shy;<lb/>ti&aelig;, aut prim&aelig; &aelig;qualis quart&aelig;, erit proportio prim&aelig; ad quartam, <lb/>aut terti&aelig; ad &longs;ecundam producta ex proportionibus prim&aelig; ad &longs;e&shy;<lb/>cundam, &amp; terti&aelig; ad quartam.<lb/><arrow.to.target n="marg69"></arrow.to.target></s></p><p type="margin"> <s>C&ugrave;m fuerint quatuor quantitates, fueritque &longs;ecunda &aelig;qualis ter&shy;<lb/>ti&aelig;, aut prim&aelig; &aelig;qualis quart&aelig;, erit proportio prim&aelig; ad quartam, <lb/>aut terti&aelig; ad &longs;ecundam producta ex proportionibus prim&aelig; ad &longs;e&shy;<lb/>cundam, &amp; terti&aelig; ad quartam.<lb/><arrow.to.target n="marg69"/></s></p><p type="margin">
  
 <s><margin.target id="marg69"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg69"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>C&ugrave;m enim quantitates h&aelig; non fuerint &ecedil;quales, <expan abbr="c&otilde;&longs;tat">con&longs;tat</expan> per &longs;ecun&shy;<pb pagenum="22"/>dam harum, quod proportio prim&aelig; ad <expan abbr="quart&atilde;">quartam</expan> producitur ex pro&shy;<lb/>portione prim&aelig; ad &longs;ecundam, &longs;ecund&ecedil; ad tertiam, &amp; terti&ecedil; ad quar <lb/>tam: ergo non ex &longs;olis proportionibus prim&aelig; ad &longs;ecundam, &amp; ter&shy;<lb/>ti&aelig; ad quartam, &amp; &longs;imiliter ex prima harum proportio prim&ecedil; ad &longs;e&shy;<lb/>cundam, &amp; terti&aelig; ad quartam producunt proportionem producti <lb/>prim&aelig; in &longs;ecundam ad productum terti&aelig; in quartam. </s> <s>C&ugrave;m enim quantitates h&aelig; non fuerint &ecedil;quales, <expan abbr="c&otilde;&longs;tat">con&longs;tat</expan> per &longs;ecun&shy;<pb pagenum="22"/>dam harum, quod proportio prim&aelig; ad <expan abbr="quart&atilde;">quartam</expan> producitur ex pro&shy;<lb/>portione prim&aelig; ad &longs;ecundam, &longs;ecund&ecedil; ad tertiam, &amp; terti&ecedil; ad quar <lb/>tam: ergo non ex &longs;olis proportionibus prim&aelig; ad &longs;ecundam, &amp; ter&shy;<lb/>ti&aelig; ad quartam, &amp; &longs;imiliter ex prima harum proportio prim&ecedil; ad &longs;e&shy;<lb/>cundam, &amp; terti&aelig; ad quartam producunt proportionem producti <lb/>prim&aelig; in &longs;ecundam ad productum terti&aelig; in quartam. </s>
  
 <s>Et in multi&shy;<lb/>plicatione proportio, qu&aelig; &longs;olet e&longs;&longs;e inter producta illa, &amp; e&longs;t qua&longs;i <lb/>duplicata e&longs;t inter ip&longs;as quantitates. </s> <s>Et in multi&shy;<lb/>plicatione proportio, qu&aelig; &longs;olet e&longs;&longs;e inter producta illa, &amp; e&longs;t qua&longs;i <lb/>duplicata e&longs;t inter ip&longs;as quantitates. </s>
  
 <s>Sint igitur quantitates a b c d, <lb/>&amp; &longs;it b &aelig;qualis c, ponantur ergo recto ordine a b c d, eritque propor <lb/><figure id="fig13"></figure><lb/>tio a ad d producta ex proportioni&shy;<lb/>bus a ad b, b ad c, &amp; c ad d, producan&shy;<lb/>tur igitur ex proportionibus a ad b, c <lb/>ad d. </s> <s>Sint igitur quantitates a b c d, <lb/>&amp; &longs;it b &aelig;qualis c, ponantur ergo recto ordine a b c d, eritque propor <lb/><figure id="fig13"/><lb/>tio a ad d producta ex proportioni&shy;<lb/>bus a ad b, b ad c, &amp; c ad d, producan&shy;<lb/>tur igitur ex proportionibus a ad b, c <lb/>ad d. <!-- KEEP S--></s>
  
 <s>proportio c ad f, erit igitur pro&shy;<lb/>portio e ad f, &longs;i multiplicetur per pro&shy;<lb/>portionem b ad c eadem qu&aelig; prius, &amp; </s></p><p type="main"> <s>proportio c ad f, erit igitur pro&shy;<lb/>portio e ad f, &longs;i multiplicetur per pro&shy;<lb/>portionem b ad c eadem qu&aelig; prius, &amp; </s></p><p type="main">
  
 <s><arrow.to.target n="marg70"></arrow.to.target><lb/>producta iam e&longs;t eadem ei, qu&aelig; e&longs;t a <lb/>ad d, ergo proportio a ad d erit producta ex proportionibus a ad <lb/>b, c ad d per primam propo&longs;itionem. </s> <s><arrow.to.target n="marg70"/><lb/>producta iam e&longs;t eadem ei, qu&aelig; e&longs;t a <lb/>ad d, ergo proportio a ad d erit producta ex proportionibus a ad <lb/>b, c ad d per primam propo&longs;itionem. </s>
  
 <s>Quod uer&ograve; diximus de pri&shy;<lb/>ma &amp; quarta &longs;i &longs;int &aelig;quales, manife&longs;tum e&longs;t, qu&ograve;d res redit ad idem <lb/>&longs;olum tran&longs;mutato ordine, ut tertia, &amp; quarta pr&aelig;mittantur prim&ecedil;, <lb/>&amp; &longs;ecund&aelig;. </s> <s>Quod uer&ograve; diximus de pri&shy;<lb/>ma &amp; quarta &longs;i &longs;int &aelig;quales, manife&longs;tum e&longs;t, qu&ograve;d res redit ad idem <lb/>&longs;olum tran&longs;mutato ordine, ut tertia, &amp; quarta pr&aelig;mittantur prim&ecedil;, <lb/>&amp; &longs;ecund&aelig;. </s>
  
 <s>H&aelig;cigitur propo&longs;itio nihil aliud innuit, qu&agrave;m quod <lb/>in hoc ca&longs;u productio, qu&aelig;&longs;olet fieri ex tribus proportionibus fiat <lb/>ex duabus tantum.</s></p><p type="margin"> <s>H&aelig;cigitur propo&longs;itio nihil aliud innuit, qu&agrave;m quod <lb/>in hoc ca&longs;u productio, qu&aelig;&longs;olet fieri ex tribus proportionibus fiat <lb/>ex duabus tantum.</s></p><p type="margin">
  
 <s><margin.target id="marg70"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 16. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg70"/>P<emph type="italics"/>er<emph.end type="italics"/> 16. P<emph type="italics"/>et.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main">
  
 <s>Propo&longs;itio uige&longs;imaprima.</s></p><p type="main"> <s>Propo&longs;itio uige&longs;imaprima.</s></p><p type="main">
  
 <s>C&ugrave;m decu&longs;&longs;atim ducta fuerit prima in quartam, &amp; &longs;ecunda in ter <lb/>tiam; productumque prim&aelig; in quartam diui&longs;um fuerit per produ&shy;<lb/>ctum &longs;ecund&aelig; in tertiam erit proportio prim&aelig; ad &longs;ecundam diui&shy;<lb/>&longs;a per proportionem terti&aelig; ad quartam. </s> <s>C&ugrave;m decu&longs;&longs;atim ducta fuerit prima in quartam, &amp; &longs;ecunda in ter <lb/>tiam; productumque prim&aelig; in quartam diui&longs;um fuerit per produ&shy;<lb/>ctum &longs;ecund&aelig; in tertiam erit proportio prim&aelig; ad &longs;ecundam diui&shy;<lb/>&longs;a per proportionem terti&aelig; ad quartam. </s>
  
 <s>Et &longs;imiliter interpo&longs;ita <lb/>omiologa.<lb/><arrow.to.target n="marg71"></arrow.to.target></s></p><p type="margin"> <s>Et &longs;imiliter interpo&longs;ita <lb/>omiologa.<lb/><arrow.to.target n="marg71"/></s></p><p type="margin">
  
 <s><margin.target id="marg71"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><figure></figure><p type="main"> <s><margin.target id="marg71"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><figure/><p type="main">
  
 <s>Primum exponamus &longs;ecundam partem, &longs;it <lb/>proportio a ad b, quam uolo diuidere per <lb/>proportionem c ad d, facio e ad b, ut c ad d, erit <lb/><arrow.to.target n="marg72"></arrow.to.target><lb/>ergo per <expan abbr="&longs;ec&utilde;dam">&longs;ecundam</expan> harum proportio ad b pro&shy;<lb/>ducta ex proportione a ad e, &amp; e ad b, quare ex a ad e, &amp; c ad d, ergo <lb/>diui&longs;a proportione a ad b per proportionem c ad d exit proportio <lb/>a ad e, &amp; hic e&longs;t &longs;ecundus modus. </s> <s>Primum exponamus &longs;ecundam partem, &longs;it <lb/>proportio a ad b, quam uolo diuidere per <lb/>proportionem c ad d, facio e ad b, ut c ad d, erit <lb/><arrow.to.target n="marg72"/><lb/>ergo per <expan abbr="&longs;ec&utilde;dam">&longs;ecundam</expan> harum proportio ad b pro&shy;<lb/>ducta ex proportione a ad e, &amp; e ad b, quare ex a ad e, &amp; c ad d, ergo <lb/>diui&longs;a proportione a ad b per proportionem c ad d exit proportio <lb/>a ad e, &amp; hic e&longs;t &longs;ecundus modus. </s>
  
 <s>Primus autem modus ducatur a <lb/>in d &amp; fiat f, &amp; b in c &amp; fiat g, dico proportione f ad g e&longs;&longs;e prouen&shy;<lb/>tum proportionis a ad b, diuide per proportionem c ad d, ducatur <lb/>igitur c in f &amp; fiat h, &amp; d in g &amp; fiat k, quia igitur h producitur ex c <lb/>in f, &amp; f producitur ex a in d, ergo h producetur ex producto c in d, <lb/>in a, &amp; &longs;imiliter quia k producitur ex d in g, &amp; g producitur ex b in <pb pagenum="23"/>c, ergo k producetur ex c d in b, ergo ex c d in a fit h, ex c d in b fit k. <lb/></s> <s>Primus autem modus ducatur a <lb/>in d &amp; fiat f, &amp; b in c &amp; fiat g, dico proportione f ad g e&longs;&longs;e prouen&shy;<lb/>tum proportionis a ad b, diuide per proportionem c ad d, ducatur <lb/>igitur c in f &amp; fiat h, &amp; d in g &amp; fiat k, quia igitur h producitur ex c <lb/>in f, &amp; f producitur ex a in d, ergo h producetur ex producto c in d, <lb/>in a, &amp; &longs;imiliter quia k producitur ex d in g, &amp; g producitur ex b in <pb pagenum="23"/>c, ergo k producetur ex c d in b, ergo ex c d in a fit h, ex c d in b fit k. <lb/></s>
  
 <s>erit a ad b ut h ad k, igitur ex prima harum cum ex c in f producatur <lb/>h, &amp; ex d in g k, &amp; dicatur produci proportio h ad k ex proportio&shy;<lb/>ne c ad d, &amp; f ad g, &amp; proportio h ad k &longs;it eadem, qu&aelig; a ad b, ergo <lb/>proportio a ad b producitur ex c ad d, &amp; f ad g, ergo diui&longs;a propor&shy;<lb/>tione a ad b prodibit proportio f ad g, quod fuit propo&longs;itum.</s></p><p type="margin"> <s>erit a ad b ut h ad k, igitur ex prima harum cum ex c in f producatur <lb/>h, &amp; ex d in g k, &amp; dicatur produci proportio h ad k ex proportio&shy;<lb/>ne c ad d, &amp; f ad g, &amp; proportio h ad k &longs;it eadem, qu&aelig; a ad b, ergo <lb/>proportio a ad b producitur ex c ad d, &amp; f ad g, ergo diui&longs;a propor&shy;<lb/>tione a ad b prodibit proportio f ad g, quod fuit propo&longs;itum.</s></p><p type="margin">
  
 <s><margin.target id="marg72"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg72"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main">
  
 <s>Propo&longs;itio uige&longs;ima&longs;ecunda.</s></p><p type="main"> <s>Propo&longs;itio uige&longs;ima&longs;ecunda.</s></p><p type="main">
  
 <s>C&ugrave;m fuerit proportio prim&aelig; ad &longs;ecundam maior, qu&agrave;m terti&aelig; <lb/>ad quartam, erit confu&longs;a ex his maior qu&agrave;m terti&aelig; ad quartam, mi&shy;<lb/>nor autem qu&agrave;m prim&aelig; ad &longs;ecundam.</s></p><figure></figure><p type="main"> <s>C&ugrave;m fuerit proportio prim&aelig; ad &longs;ecundam maior, qu&agrave;m terti&aelig; <lb/>ad quartam, erit confu&longs;a ex his maior qu&agrave;m terti&aelig; ad quartam, mi&shy;<lb/>nor autem qu&agrave;m prim&aelig; ad &longs;ecundam.</s></p><figure/><p type="main">
  
 <s>Sit proportio a ad b maior qu&agrave;m c <lb/><arrow.to.target n="marg73"></arrow.to.target><lb/>ad d, dico, quod confu&longs;a ex a c ad b d <lb/>e&longs;t maior, qu&agrave;m c ad d, et minor qu&agrave;m <lb/>a ad b, ut enim c ad d ita fiat e ad b, erit que per tertiamdecimam ha&shy;<lb/><arrow.to.target n="marg74"></arrow.to.target><lb/>rum e c ad b d confu&longs;a minor qu&agrave;m a c ad b d, nam e e&longs;t minor a, <lb/>quia proportionem habent minorem ad b quam a eo qu&ograve;d e ha&shy;<lb/>bet proportionem ad b, quam c ad d, qu&aelig; <expan abbr="aut&etilde;">autem</expan> c ad d minor, qu&aacute;m <lb/>a ad b, ut &longs;uppo&longs;itum e&longs;t, igitur e c ad b d minor, qu&agrave;m a b ad c d, e b <lb/>autem ad c d e&longs;t, ut demon&longs;tratum e&longs;t qualis c ad d, ergo c ad d mi&shy;<lb/>nor, qu&agrave;m confu&longs;a a b ad c d, quod e&longs;t &longs;ecundum per idem proba&shy;<lb/>bitur, &amp; primum po&longs;ita f ad d, ut a ad b, eritque a maior c, igitur ma&shy;<lb/>ior proportio a f ad b d, qu&agrave;m a c ad b d, &longs;ed a f ad b d, ut a ad b per <lb/>candem tertiamdecimam huius ergo proportio confu&longs;a a b ad c d <lb/>e&longs;t minor, qu&agrave;m a ad b.</s></p><p type="margin"> <s>Sit proportio a ad b maior qu&agrave;m c <lb/><arrow.to.target n="marg73"/><lb/>ad d, dico, quod confu&longs;a ex a c ad b d <lb/>e&longs;t maior, qu&agrave;m c ad d, et minor qu&agrave;m <lb/>a ad b, ut enim c ad d ita fiat e ad b, erit que per tertiamdecimam ha&shy;<lb/><arrow.to.target n="marg74"/><lb/>rum e c ad b d confu&longs;a minor qu&agrave;m a c ad b d, nam e e&longs;t minor a, <lb/>quia proportionem habent minorem ad b quam a eo qu&ograve;d e ha&shy;<lb/>bet proportionem ad b, quam c ad d, qu&aelig; <expan abbr="aut&etilde;">autem</expan> c ad d minor, qu&aacute;m <lb/>a ad b, ut &longs;uppo&longs;itum e&longs;t, igitur e c ad b d minor, qu&agrave;m a b ad c d, e b <lb/>autem ad c d e&longs;t, ut demon&longs;tratum e&longs;t qualis c ad d, ergo c ad d mi&shy;<lb/>nor, qu&agrave;m confu&longs;a a b ad c d, quod e&longs;t &longs;ecundum per idem proba&shy;<lb/>bitur, &amp; primum po&longs;ita f ad d, ut a ad b, eritque a maior c, igitur ma&shy;<lb/>ior proportio a f ad b d, qu&agrave;m a c ad b d, &longs;ed a f ad b d, ut a ad b per <lb/>candem tertiamdecimam huius ergo proportio confu&longs;a a b ad c d <lb/>e&longs;t minor, qu&agrave;m a ad b.</s></p><p type="margin">
  
 <s><margin.target id="marg73"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin"> <s><margin.target id="marg73"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg74"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg74"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main">
  
 <s>Propo&longs;itio uige&longs;imatertia.</s></p><p type="main"> <s>Propo&longs;itio uige&longs;imatertia.</s></p><p type="main">
  
 <s>Omnis motus naturalis ad locum &longs;uum e&longs;t: ideo per rectam li&shy;<lb/>neam fit.<lb/><arrow.to.target n="marg75"></arrow.to.target></s></p><p type="margin"> <s>Omnis motus naturalis ad locum &longs;uum e&longs;t: ideo per rectam li&shy;<lb/>neam fit.<lb/><arrow.to.target n="marg75"/></s></p><p type="margin">
  
 <s><margin.target id="marg75"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg75"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Motus naturalis e&longs;t, ut con&longs;eruetur corpus, &amp; conueniat locus <lb/>corpori, igitur fit ad &longs;uum locum. </s> <s>Motus naturalis e&longs;t, ut con&longs;eruetur corpus, &amp; conueniat locus <lb/>corpori, igitur fit ad &longs;uum locum. </s>
  
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 <s>Et quia quanto natura celerius <lb/>&longs;uum finem pote&longs;t a&longs;&longs;equi (quia finis bonus e&longs;t aliter non illum ap&shy;<lb/>peteret) eum qu&aelig;rit, c&ugrave;m &longs;it &longs;apienti&longs;sim&aelig; uit&aelig; mini&longs;tra: at linea re&shy;</s></p><p type="main"> <s>Et quia quanto natura celerius <lb/>&longs;uum finem pote&longs;t a&longs;&longs;equi (quia finis bonus e&longs;t aliter non illum ap&shy;<lb/>peteret) eum qu&aelig;rit, c&ugrave;m &longs;it &longs;apienti&longs;sim&aelig; uit&aelig; mini&longs;tra: at linea re&shy;</s></p><p type="main">
  
 <s><arrow.to.target n="marg76"></arrow.to.target><lb/>cta breui&longs;sima e&longs;t Euclide te&longs;te &agrave; puncto ad punctum, igitur omnis <lb/>motus naturalis e&longs;t &longs;ur&longs;um aut deor&longs;um per rectam lineam.</s></p><p type="margin"> <s><arrow.to.target n="marg76"/><lb/>cta breui&longs;sima e&longs;t Euclide te&longs;te &agrave; puncto ad punctum, igitur omnis <lb/>motus naturalis e&longs;t &longs;ur&longs;um aut deor&longs;um per rectam lineam.</s></p><p type="margin">
  
 <s><margin.target id="marg76"></margin.target>D<emph type="italics"/>i&longs;t. </s> <s><margin.target id="marg76"/>D<emph type="italics"/>i&longs;t. <!-- REMOVE S-->tertia <lb/>primi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><!-- KEEP S--></s>
  
 <s>tertia <lb/>primi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main"> </p><p type="main">
  
 <s>Propo&longs;itio uige&longs;imaquarta.</s></p><p type="main"> <s>Propo&longs;itio uige&longs;imaquarta.</s></p><p type="main">
  
 <s>Omnis motus circularis uoluntarius e&longs;t.</s></p><p type="main"> <s>Omnis motus circularis uoluntarius e&longs;t.</s></p><p type="main">
  
 <s>Sit motus in circulo &longs;eu per circulum in orbe cuius &longs;it centrum, <lb/>&longs;it c mundi centrum: igitur ex diffinitione circuli tantum di&longs;tabit a, <lb/>quantum b ab ip&longs;o c: &longs;ed in motu naturali per pr&ecedil;cedentem nece&longs;&longs;e <lb/>e&longs;t, ut recta feratur ad c, uel recedat, igitur motus a e&longs;t uoluntarius, <pb pagenum="24"/><figure id="fig14"></figure><lb/>non naturalis. </s> <s>Sit motus in circulo &longs;eu per circulum in orbe cuius &longs;it centrum, <lb/>&longs;it c mundi centrum: igitur ex diffinitione circuli tantum di&longs;tabit a, <lb/>quantum b ab ip&longs;o c: &longs;ed in motu naturali per pr&ecedil;cedentem nece&longs;&longs;e <lb/>e&longs;t, ut recta feratur ad c, uel recedat, igitur motus a e&longs;t uoluntarius, <pb pagenum="24"/><figure id="fig14"/><lb/>non naturalis. </s>
  
 <s>nam &longs;i uiolentus e&longs;&longs;et, non <lb/>e&longs;&longs;et perpetuus. </s> <s>nam &longs;i uiolentus e&longs;&longs;et, non <lb/>e&longs;&longs;et perpetuus. </s>
  
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 <s>Propo&longs;itio uige&longs;imaquinta.</s></p><p type="main"> <s>Propo&longs;itio uige&longs;imaquinta.</s></p><p type="main">
  
 <s>Tres &longs;unt motus omnino &longs;implices naturalis, uoluntarius &amp; <lb/>uiolentus.<lb/><arrow.to.target n="marg77"></arrow.to.target></s></p><p type="margin"> <s>Tres &longs;unt motus omnino &longs;implices naturalis, uoluntarius &amp; <lb/>uiolentus.<lb/><arrow.to.target n="marg77"/></s></p><p type="margin">
  
 <s><margin.target id="marg77"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg77"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Tres &longs;unt modi, quibus po&longs;&longs;unt moueri in comparatione ad cen <lb/>trum &longs;cilicet uel recta cum centro, uel &aelig;quidi&longs;tando &agrave; centro, uel <lb/>neutro modo, igitur tres motus. </s> <s>Tres &longs;unt modi, quibus po&longs;&longs;unt moueri in comparatione ad cen <lb/>trum &longs;cilicet uel recta cum centro, uel &aelig;quidi&longs;tando &agrave; centro, uel <lb/>neutro modo, igitur tres motus. </s>
  
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 <s>Sunt enim quatuor genera mo&shy;</s></p><p type="main"> <s>Sunt enim quatuor genera mo&shy;</s></p><p type="main">
  
 <s><arrow.to.target n="marg78"></arrow.to.target><lb/>tus uiolenti ab Ari&longs;totele po&longs;ita, uectio, tractio, pul&longs;io, &amp; uolutio: <lb/>quanquam his non opus &longs;it in demon&longs;tratiua &longs;cientia. </s> <s><arrow.to.target n="marg78"/><lb/>tus uiolenti ab Ari&longs;totele po&longs;ita, uectio, tractio, pul&longs;io, &amp; uolutio: <lb/>quanquam his non opus &longs;it in demon&longs;tratiua &longs;cientia. </s>
  
 <s><expan abbr="c&otilde;&longs;tat">con&longs;tat</expan> enim <lb/>uolutionem ex tractione, &amp; pul&longs;ione apud illum con&longs;i&longs;tere.</s></p><p type="margin"> <s><expan abbr="c&otilde;&longs;tat">con&longs;tat</expan> enim <lb/>uolutionem ex tractione, &amp; pul&longs;ione apud illum con&longs;i&longs;tere.</s></p><p type="margin">
  
 <s><margin.target id="marg78"></margin.target>7. P<emph type="italics"/>hy&longs;. <lb/></s> <s><margin.target id="marg78"/>7. P<emph type="italics"/>hy&longs;. <lb/><!-- REMOVE S-->cap.<emph.end type="italics"/> 2.<!-- KEEP S--></s>
  
 <s>cap.<emph.end type="italics"/> 2.</s></p><p type="main"> </p><p type="main">
  
 <s>Propo&longs;itio uige&longs;ima.</s></p><p type="main"> <s>Propo&longs;itio uige&longs;ima.</s></p><p type="main">
  
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 <s>Propo&longs;itio uige&longs;imaoctaua.</s></p><p type="main"> <s>Propo&longs;itio uige&longs;imaoctaua.</s></p><p type="main">
  
 <s>Motus quilibet naturalis aut uiolentus in aliquo medio fit.<lb/><arrow.to.target n="marg79"></arrow.to.target></s></p><p type="margin"> <s>Motus quilibet naturalis aut uiolentus in aliquo medio fit.<lb/><arrow.to.target n="marg79"/></s></p><p type="margin">
  
 <s><margin.target id="marg79"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg79"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>C&ugrave;m uacuum non detur, &amp; omnis motus naturalis &longs;it ad locum, <lb/>et uiolentus ex loco per pr&aelig;cedentem, igitur c&ugrave;m non &longs;it in medio, <lb/>uacuum erit in aliquo corpore, uelut aere, aqua, igne, ligno.</s></p><p type="main"> <s>C&ugrave;m uacuum non detur, &amp; omnis motus naturalis &longs;it ad locum, <lb/>et uiolentus ex loco per pr&aelig;cedentem, igitur c&ugrave;m non &longs;it in medio, <lb/>uacuum erit in aliquo corpore, uelut aere, aqua, igne, ligno.</s></p><p type="main">
  
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 <s>Omnis motus uoluntarius &aelig;qualis e&longs;t &longs;emper: &longs;impliciter etiam <lb/>quilibet alius motus.</s></p><pb pagenum="26"/><p type="main"> <s>Omnis motus uoluntarius &aelig;qualis e&longs;t &longs;emper: &longs;impliciter etiam <lb/>quilibet alius motus.</s></p><pb pagenum="26"/><p type="main">
  
 <s><arrow.to.target n="marg80"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg80"/></s></p><p type="margin">
  
 <s><margin.target id="marg80"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main"> <s><margin.target id="marg80"/>C<emph type="italics"/>o<emph.end type="italics"/>m.<!-- KEEP S--></s></p><p type="main">
  
 <s>Motus uoluntarius non habet, qu&ograve;d fatiget, &amp; &longs;umma perfectio <lb/>e&longs;t &aelig;qualitas, &amp; natura qu&aelig; mouet non debilitatur, igitur perpe&shy;<lb/>tuo per&longs;euerat &aelig;qualis. </s> <s>Motus uoluntarius non habet, qu&ograve;d fatiget, &amp; &longs;umma perfectio <lb/>e&longs;t &aelig;qualitas, &amp; natura qu&aelig; mouet non debilitatur, igitur perpe&shy;<lb/>tuo per&longs;euerat &aelig;qualis. </s>
  
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 <s>In omni corpore mobili in medio, partes medij re&longs;i&longs;tunt obui&aelig;, <lb/>ali&aelig; impellunt.</s></p><p type="main"> <s>In omni corpore mobili in medio, partes medij re&longs;i&longs;tunt obui&aelig;, <lb/>ali&aelig; impellunt.</s></p><p type="main">
  
 <s><arrow.to.target n="marg81"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg81"/></s></p><p type="margin">
  
 <s><margin.target id="marg81"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg81"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Sit mobile a cui partes &longs;ubiaceant direct&aelig; b, &amp; &longs;it graue. </s> <s>Sit mobile a cui partes &longs;ubiaceant direct&aelig; b, &amp; &longs;it graue. </s>
  
 <s>Et pa&shy;<lb/>tet ne diuidatur b re&longs;i&longs;tere, cum autem &longs;uperauerit, partes b de&longs;cen&shy;<lb/>dunt ante a, &amp; trahunt partes c &amp; d adh&ecedil;rentes &longs;ecum, atque ita e c d f <lb/><figure id="fig15"></figure><lb/>adiuuant ad de&longs;cen&longs;um partes etiam laterales <lb/>g &amp; h cum a tran&longs;it in b, ne detur uacuum, tran&shy;<lb/>&longs;eunt in k uelo ci motu, ergo propellunt a maio<lb/>reimpetu inferius.</s></p><p type="main"> <s>Et pa&shy;<lb/>tet ne diuidatur b re&longs;i&longs;tere, cum autem &longs;uperauerit, partes b de&longs;cen&shy;<lb/>dunt ante a, &amp; trahunt partes c &amp; d adh&ecedil;rentes &longs;ecum, atque ita e c d f <lb/><figure id="fig15"/><lb/>adiuuant ad de&longs;cen&longs;um partes etiam laterales <lb/>g &amp; h cum a tran&longs;it in b, ne detur uacuum, tran&shy;<lb/>&longs;eunt in k uelo ci motu, ergo propellunt a maio<lb/>reimpetu inferius.</s></p><p type="main">
  
 <s><arrow.to.target n="marg82"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg82"/></s></p><p type="margin">
  
 <s><margin.target id="marg82"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg82"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Ex quo patet, quod in omni motu naturali, <lb/>uel uiolento fit augumentum uelocitatis ab initio &longs;altem u&longs;que <lb/>ad aliquid.</s></p><p type="main"> <s>Ex quo patet, quod in omni motu naturali, <lb/>uel uiolento fit augumentum uelocitatis ab initio &longs;altem u&longs;que <lb/>ad aliquid.</s></p><p type="main">
  
 <s><arrow.to.target n="marg83"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg83"/></s></p><p type="margin">
  
 <s><margin.target id="marg83"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg83"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Et ide&ograve; etiam bellic&aelig; machin&aelig; cuiu&longs;cunque generis certam exi&shy;<lb/>gunt di&longs;tantiam, ut uiolentius feriant.</s></p><p type="main"> <s>Et ide&ograve; etiam bellic&aelig; machin&aelig; cuiu&longs;cunque generis certam exi&shy;<lb/>gunt di&longs;tantiam, ut uiolentius feriant.</s></p><p type="main">
  
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 <s>Omnis motus naturalis in &aelig;quali medio ualidior e&longs;t in fine, <lb/>qu&agrave;m in principio: uiolentus contr&agrave;.</s></p><p type="main"> <s>Omnis motus naturalis in &aelig;quali medio ualidior e&longs;t in fine, <lb/>qu&agrave;m in principio: uiolentus contr&agrave;.</s></p><p type="main">
  
 <s><arrow.to.target n="marg84"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg84"/></s></p><p type="margin">
  
 <s><margin.target id="marg84"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg84"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>C&ugrave;m enim ex pr&aelig;cedenti augeantur &longs;emper ob medium, &amp; cau&shy;<lb/>fa, qu&aelig; mouet, &longs;it perpetua, &amp; &agrave; principio &aelig;terno, quod per dict&aelig; <lb/>&aelig;qualiter mouet, igitur motus ille fiet uelo cior in fine qu&agrave;m in alia <lb/>parte temporis. </s> <s>C&ugrave;m enim ex pr&aelig;cedenti augeantur &longs;emper ob medium, &amp; cau&shy;<lb/>fa, qu&aelig; mouet, &longs;it perpetua, &amp; &agrave; principio &aelig;terno, quod per dict&aelig; <lb/>&aelig;qualiter mouet, igitur motus ille fiet uelo cior in fine qu&agrave;m in alia <lb/>parte temporis. </s>
  
 <s>In uiolento autem, c&ugrave;m perueniat ad finem de&longs;init </s></p><p type="main"> <s>In uiolento autem, c&ugrave;m perueniat ad finem de&longs;init </s></p><p type="main">
  
 <s><arrow.to.target n="marg85"></arrow.to.target><lb/>uis illa nece&longs;&longs;ari&ograve;, qu&aelig; mouet, &amp; &longs;uperatur &agrave; ui naturali, qu&aelig; mo&shy;<lb/>uet in contrarium, igitur antequam ce&longs;&longs;et motus fiet tardi&longs;simus <lb/>in fine.</s></p><p type="margin"> <s><arrow.to.target n="marg85"/><lb/>uis illa nece&longs;&longs;ari&ograve;, qu&aelig; mouet, &amp; &longs;uperatur &agrave; ui naturali, qu&aelig; mo&shy;<lb/>uet in contrarium, igitur antequam ce&longs;&longs;et motus fiet tardi&longs;simus <lb/>in fine.</s></p><p type="margin">
  
 <s><margin.target id="marg85"></margin.target><gap/> 29. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg85"/><gap/> 29. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main">
  
 <s>Ex quo patet, qu&ograve;d motus quadrifariam mi&longs;ti dicuntur, aut &longs;pe&shy;<lb/><arrow.to.target n="marg86"></arrow.to.target><lb/>cie, ut c&ugrave;m quis iacit lapidem &egrave; turri: uel ex occulto naturali, &amp; uio&shy;<lb/>lento manife&longs;to: uelut c&ugrave;m quis iacit lapidem, &amp; de&longs;cendit po&longs;tmo <lb/><figure id="fig16"></figure><lb/>dum ex b in c motu utroque manife&longs;to, &longs;ed ex a <lb/>in b motu uiolento manife&longs;to, &amp; naturali oc&shy;<lb/>culto: uel ratione medij, &amp; hoc modo omnis <lb/>motus naturalis etiam non &longs;olum uiolentus e&longs;t <lb/>mi&longs;tus ex proportione uirtutis mouentis, cum motu medij, ad me&shy;<lb/>dium ip&longs;um, uel &longs;i uiolentus &longs;it ex proportione uirtutis mouentis, <pb pagenum="27"/>&amp; medij ad mobile, ac medium, quod re&longs;i&longs;tit. </s> <s>Ex quo patet, qu&ograve;d motus quadrifariam mi&longs;ti dicuntur, aut &longs;pe&shy;<lb/><arrow.to.target n="marg86"/><lb/>cie, ut c&ugrave;m quis iacit lapidem &egrave; turri: uel ex occulto naturali, &amp; uio&shy;<lb/>lento manife&longs;to: uelut c&ugrave;m quis iacit lapidem, &amp; de&longs;cendit po&longs;tmo <lb/><figure id="fig16"/><lb/>dum ex b in c motu utroque manife&longs;to, &longs;ed ex a <lb/>in b motu uiolento manife&longs;to, &amp; naturali oc&shy;<lb/>culto: uel ratione medij, &amp; hoc modo omnis <lb/>motus naturalis etiam non &longs;olum uiolentus e&longs;t <lb/>mi&longs;tus ex proportione uirtutis mouentis, cum motu medij, ad me&shy;<lb/>dium ip&longs;um, uel &longs;i uiolentus &longs;it ex proportione uirtutis mouentis, <pb pagenum="27"/>&amp; medij ad mobile, ac medium, quod re&longs;i&longs;tit. </s>
  
 <s>Quarto ex motibus <lb/>imperfectis natura &longs;ua, &amp; non e&longs;t uera mi&longs;tio, &amp; hoc apparet in mo&shy;<lb/>tibus uoluntarijs animalium, qui non &longs;unt neque &aelig;quales, neque perfe <lb/>ct&egrave; circa medium: &longs;ed &longs;unt potius &longs;imiles uoluntarijs. </s> <s>Quarto ex motibus <lb/>imperfectis natura &longs;ua, &amp; non e&longs;t uera mi&longs;tio, &amp; hoc apparet in mo&shy;<lb/>tibus uoluntarijs animalium, qui non &longs;unt neque &aelig;quales, neque perfe <lb/>ct&egrave; circa medium: &longs;ed &longs;unt potius &longs;imiles uoluntarijs. </s>
  
 <s>Etideo de&shy;<lb/>mon&longs;trationes ill&aelig; Ari&longs;totelis quoad u&longs;um nihil iuuant nos.</s></p><p type="margin"> <s>Etideo de&shy;<lb/>mon&longs;trationes ill&aelig; Ari&longs;totelis quoad u&longs;um nihil iuuant nos.</s></p><p type="margin">
  
 <s><margin.target id="marg86"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg86"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Propo&longs;itio trige&longs;ima&longs;ecunda.</s></p><p type="main"> <s>Propo&longs;itio trige&longs;ima&longs;ecunda.</s></p><p type="main">
  
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 <s>Maior quoque e&longs;t proportio fi&shy;<lb/>nis motus in corpore rariore ad finem motus in corpore den&longs;iore, <lb/>qu&agrave;m principij. </s> <s>Maior quoque e&longs;t proportio fi&shy;<lb/>nis motus in corpore rariore ad finem motus in corpore den&longs;iore, <lb/>qu&agrave;m principij. </s>
  
 <s>In uiolento autem celeri&ugrave;s perueniet ad finem mo<lb/>tus in corpore den&longs;iore.</s></p><figure></figure><p type="main"> <s>In uiolento autem celeri&ugrave;s perueniet ad finem mo<lb/>tus in corpore den&longs;iore.</s></p><figure/><p type="main">
  
 <s>A mobile moueatur in b medio rariore, &amp; in c den&longs;io&shy;<lb/><arrow.to.target n="marg87"></arrow.to.target><lb/>re, igitur b minus re&longs;i&longs;tit, qu&agrave;m c &amp; magis adiuuat, quia <lb/>ueloci&ugrave;s mouetur: igitur duplici de cau&longs;a a mouebitur <lb/>ueloci&ugrave;s in b qu&agrave;m in c: &amp; quia per corrolarium trige&longs;i&shy;<lb/>m&aelig;, &amp; pr&aelig;cedentis proportio finis (ubi &aelig;qualiter moueantur) ad <lb/>&longs;ua principia maior erit in d, qu&agrave;m in e: ergo per <expan abbr="dem&otilde;&longs;trata">demon&longs;trata</expan> &agrave; Cam <lb/>pano po&longs;ita d prima, b &longs;ecunda, e tertia, c quarta, maior erit propor&shy;<lb/>tio d ad e, qu&agrave;m b ad c quod fuit propo&longs;itum in naturali.</s></p><p type="margin"> <s>A mobile moueatur in b medio rariore, &amp; in c den&longs;io&shy;<lb/><arrow.to.target n="marg87"/><lb/>re, igitur b minus re&longs;i&longs;tit, qu&agrave;m c &amp; magis adiuuat, quia <lb/>ueloci&ugrave;s mouetur: igitur duplici de cau&longs;a a mouebitur <lb/>ueloci&ugrave;s in b qu&agrave;m in c: &amp; quia per corrolarium trige&longs;i&shy;<lb/>m&aelig;, &amp; pr&aelig;cedentis proportio finis (ubi &aelig;qualiter moueantur) ad <lb/>&longs;ua principia maior erit in d, qu&agrave;m in e: ergo per <expan abbr="dem&otilde;&longs;trata">demon&longs;trata</expan> &agrave; Cam <lb/>pano po&longs;ita d prima, b &longs;ecunda, e tertia, c quarta, maior erit propor&shy;<lb/>tio d ad e, qu&agrave;m b ad c quod fuit propo&longs;itum in naturali.</s></p><p type="margin">
  
 <s><margin.target id="marg87"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg87"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Propo&longs;itio trige&longs;imatertia.</s></p><p type="main"> <s>Propo&longs;itio trige&longs;imatertia.</s></p><p type="main">
  
 <s>Omnia duo mobilia &aelig;qualis undique magnitudinis, qu&aelig; &aelig;quali <lb/>in tempore &aelig;qualia &longs;patia pertran&longs;eunt in diuer&longs;is &longs;ub&longs;tantia me&shy;<lb/>dijs, nece&longs;&longs;e e&longs;t, ut &longs;it ponderis ad pondus, quemadmodum medij <lb/>ad medium, proportio duplicata.<lb/><arrow.to.target n="marg88"></arrow.to.target></s></p><p type="margin"> <s>Omnia duo mobilia &aelig;qualis undique magnitudinis, qu&aelig; &aelig;quali <lb/>in tempore &aelig;qualia &longs;patia pertran&longs;eunt in diuer&longs;is &longs;ub&longs;tantia me&shy;<lb/>dijs, nece&longs;&longs;e e&longs;t, ut &longs;it ponderis ad pondus, quemadmodum medij <lb/>ad medium, proportio duplicata.<lb/><arrow.to.target n="marg88"/></s></p><p type="margin">
  
 <s><margin.target id="marg88"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg88"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Sint duo mobilia a &amp; b magnitudine, &amp; forma omnino paria, <lb/>&amp; &longs;int media c &amp; d, exempli gratia: &amp; pertran&longs;eant &aelig;quale &longs;patium <lb/>in utroque in eodem tempore, e dico proportionem ponderis b ad <lb/>pondus a e&longs;&longs;e duplicatam ei qu&aelig; e&longs;t raritatis c ad raritatem d. </s> <s>Sint duo mobilia a &amp; b magnitudine, &amp; forma omnino paria, <lb/>&amp; &longs;int media c &amp; d, exempli gratia: &amp; pertran&longs;eant &aelig;quale &longs;patium <lb/>in utroque in eodem tempore, e dico proportionem ponderis b ad <lb/>pondus a e&longs;&longs;e duplicatam ei qu&aelig; e&longs;t raritatis c ad raritatem d. <!-- KEEP S--></s>
  
 <s>Quia <lb/>enim feruntur &aelig;qualiter, nam in &aelig;quali tem&shy;<lb/><figure id="fig17"></figure><lb/>pore, &longs;eu eodem &aelig;qualia &longs;patia pertran&longs;e&shy;<lb/>unt, erit proportio potenti&aelig; a cum &longs;uo auxi&shy;<lb/>lio ad id, quod re&longs;i&longs;tit ex c ut b cum &longs;uo au&shy;<lb/>xilio ad id, quod re&longs;i&longs;tit ex d, permutando igi <lb/>tur d ad c, ut b ad a, &longs;ed c ad d proportio rari&shy;<lb/>tatis duplicat actionem, tum minus re&longs;i&longs;ten&shy;<lb/>do, tum adiuuando motum a, igitur proportio differenti&aelig; motus <lb/>e&longs;t duplicata proportioni raritatis: &longs;ed proportio motus e&longs;t &aelig;qua&shy;<lb/>lis proportioni ponderis uici&longs;sim per uige&longs;imam&longs;extam &longs;exti Ele&shy;<lb/>mentorum b ad a, igitur proportio b ad a ponderis e&longs;t duplicata ei, <lb/>qu&aelig; e&longs;t raritatis c ad raritatem d.</s></p><pb pagenum="28"/><p type="head"> <s>Quia <lb/>enim feruntur &aelig;qualiter, nam in &aelig;quali tem&shy;<lb/><figure id="fig17"/><lb/>pore, &longs;eu eodem &aelig;qualia &longs;patia pertran&longs;e&shy;<lb/>unt, erit proportio potenti&aelig; a cum &longs;uo auxi&shy;<lb/>lio ad id, quod re&longs;i&longs;tit ex c ut b cum &longs;uo au&shy;<lb/>xilio ad id, quod re&longs;i&longs;tit ex d, permutando igi <lb/>tur d ad c, ut b ad a, &longs;ed c ad d proportio rari&shy;<lb/>tatis duplicat actionem, tum minus re&longs;i&longs;ten&shy;<lb/>do, tum adiuuando motum a, igitur proportio differenti&aelig; motus <lb/>e&longs;t duplicata proportioni raritatis: &longs;ed proportio motus e&longs;t &aelig;qua&shy;<lb/>lis proportioni ponderis uici&longs;sim per uige&longs;imam&longs;extam &longs;exti Ele&shy;<lb/>mentorum b ad a, igitur proportio b ad a ponderis e&longs;t duplicata ei, <lb/>qu&aelig; e&longs;t raritatis c ad raritatem d.<!-- KEEP S--></s></p><pb pagenum="28"/><p type="head">
  
 <s>SCHOLIVM PRIMVM.</s></p><p type="main"> <s>SCHOLIVM PRIMVM.<!-- KEEP S--></s></p><p type="main">
  
 <s>Ne tamen &longs;ine exemplo intelligas hanc duplicatam rationem, <lb/>proponatur craritas quatuor, d unum, a pondus duodecim libra&shy;<lb/><figure id="fig18"></figure><lb/>rum, tunc c re&longs;i&longs;tit &longs;olum ex quarta parte, &amp; effi&shy;<lb/>cit a quadruplo maioris actionis, &longs;cilicet ut qua&shy;<lb/>draginta octo, tota igitur proportio, qua mo&shy;<lb/>uebitur a in c, erit centum nonaginta duorum, &amp; hoc diuidemus <lb/>per d, quod e&longs;t unum, exibit <expan abbr="p&otilde;dus">pondus</expan> b centum nonaginta duo. </s> <s>Ne tamen &longs;ine exemplo intelligas hanc duplicatam rationem, <lb/>proponatur craritas quatuor, d unum, a pondus duodecim libra&shy;<lb/><figure id="fig18"/><lb/>rum, tunc c re&longs;i&longs;tit &longs;olum ex quarta parte, &amp; effi&shy;<lb/>cit a quadruplo maioris actionis, &longs;cilicet ut qua&shy;<lb/>draginta octo, tota igitur proportio, qua mo&shy;<lb/>uebitur a in c, erit centum nonaginta duorum, &amp; hoc diuidemus <lb/>per d, quod e&longs;t unum, exibit <expan abbr="p&otilde;dus">pondus</expan> b centum nonaginta duo. </s>
  
 <s>Pro&shy;<lb/>portio igitur b ad a e&longs;t &longs;exde cupla, &amp; h&aelig;c e&longs;t duplicata quadrupl&aelig; <lb/>raritatis c ad raritatem d.</s></p><p type="main"> <s>Pro&shy;<lb/>portio igitur b ad a e&longs;t &longs;exde cupla, &amp; h&aelig;c e&longs;t duplicata quadrupl&aelig; <lb/>raritatis c ad raritatem d.<!-- KEEP S--></s></p><p type="main">
  
 <s>Qu&ograve;d &longs;i quis neget tantundem augere c actionem a, quanto mi&shy;<lb/>nus re&longs;i&longs;tit, &longs;ed aut magis aut minus, &amp; &longs;it proportio b ad a dupli&shy;<lb/>cata ip&longs;i f, dico fe&longs;&longs;e proportionem c ad d, nam proportio b ad a <lb/>e&longs;t uelut actionis c ad d per decimam&longs;extam &longs;exti Elementorum, <lb/>ergo ex auxilio c in proportionem a ad c fit proportio b ad a, &longs;ed ex <lb/>fin &longs;e fit proportio b ad a ex diffinitione proportionis duplicat&aelig;. <lb/></s> <s>Qu&ograve;d &longs;i quis neget tantundem augere c actionem a, quanto mi&shy;<lb/>nus re&longs;i&longs;tit, &longs;ed aut magis aut minus, &amp; &longs;it proportio b ad a dupli&shy;<lb/>cata ip&longs;i f, dico fe&longs;&longs;e proportionem c ad d, nam proportio b ad a <lb/>e&longs;t uelut actionis c ad d per decimam&longs;extam &longs;exti Elementorum, <lb/>ergo ex auxilio c in proportionem a ad c fit proportio b ad a, &longs;ed ex <lb/>fin &longs;e fit proportio b ad a ex diffinitione proportionis duplicat&aelig;. <lb/></s>
  
 <s>Sed ex duabus proportionibus a ad c, &amp; actionis ex c ad a produ&shy;<lb/>citur proportio b ad a, igitur per <expan abbr="decimam&longs;eptim&atilde;">decimam&longs;eptimam</expan> &longs;exti Elemento&shy;<lb/>rum proportio c ad d e&longs;t media inter proportiones a ad c, &amp; actio&shy;<lb/>nis a in c, quare &aelig;qualis f, igitur proportio b ad a duplicata ei, qu&aelig; <lb/>e&longs;t c ad d quod erat demon&longs;trandum.</s></p><p type="head"> <s>Sed ex duabus proportionibus a ad c, &amp; actionis ex c ad a produ&shy;<lb/>citur proportio b ad a, igitur per <expan abbr="decimam&longs;eptim&atilde;">decimam&longs;eptimam</expan> &longs;exti Elemento&shy;<lb/>rum proportio c ad d e&longs;t media inter proportiones a ad c, &amp; actio&shy;<lb/>nis a in c, quare &aelig;qualis f, igitur proportio b ad a duplicata ei, qu&aelig; <lb/>e&longs;t c ad d quod erat demon&longs;trandum.</s></p><p type="head">
  
 <s>SCHOLIVM SECVNDVM.</s></p><p type="main"> <s>SCHOLIVM SECVNDVM.<!-- KEEP S--></s></p><p type="main">
  
 <s>Si autem media fuerint diuer&longs;arum rationum, ut aqua, &amp; a&euml;r non <lb/>demon&longs;trat argumentum, quia pondera inter &longs;e non &longs;eruant ratio&shy;<lb/>nem. </s> <s>Si autem media fuerint diuer&longs;arum rationum, ut aqua, &amp; a&euml;r non <lb/>demon&longs;trat argumentum, quia pondera inter &longs;e non &longs;eruant ratio&shy;<lb/>nem. </s>
  
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 <s>Proportio corporis cubi ad &longs;uam &longs;uperficiem quadratam, e&longs;t ue&shy;<lb/>lut eiu&longs;dem &longs;uperficiei ad latus, eiu&longs;dem uer&ograve; ad monadem.</s></p><p type="main"> <s>Proportio corporis cubi ad &longs;uam &longs;uperficiem quadratam, e&longs;t ue&shy;<lb/>lut eiu&longs;dem &longs;uperficiei ad latus, eiu&longs;dem uer&ograve; ad monadem.</s></p><p type="main">
  
 <s><arrow.to.target n="marg89"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg89"/></s></p><p type="margin">
  
 <s><margin.target id="marg89"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg89"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Sit cubus a b c eius quadrata, &longs;uperficies a <lb/><figure id="fig19"></figure><lb/>c, latus a b, monas d, dico eas e&longs;&longs;e inuicem <lb/>analogas. </s> <s>Sit cubus a b c eius quadrata, &longs;uperficies a <lb/><figure id="fig19"/><lb/>c, latus a b, monas d, dico eas e&longs;&longs;e inuicem <lb/>analogas. </s>
  
 <s>Quia enim proportio a b c ad a c <lb/>e&longs;t, ut quoties a&longs;&longs;umitur a c in a b c, &amp; toties <lb/>ctiam a&longs;&longs;umitur a b in a c ex diffinitione Eucli </s></p><p type="main"> <s>Quia enim proportio a b c ad a c <lb/>e&longs;t, ut quoties a&longs;&longs;umitur a c in a b c, &amp; toties <lb/>ctiam a&longs;&longs;umitur a b in a c ex diffinitione Eucli </s></p><p type="main">
  
 <s><arrow.to.target n="marg90"></arrow.to.target><lb/>dis &longs;ecundo Elementorum, &longs;i ergo monas e&longs;t <lb/>in continua proportione, habeo intentum: &longs;i <lb/>non ponatur e media inter a e &amp; d, erit ergo <lb/>per decimam noni Elementorum elatus a c, <lb/>ergo &aelig;qualis a b, igitur cum a c, e &amp; d &longs;int analog&aelig;, erunt &amp; a b c, <lb/>a b, &amp; d analog&aelig;, quod fuit demon&longs;trandum.</s></p><pb pagenum="29"/><p type="margin"> <s><arrow.to.target n="marg90"/><lb/>dis &longs;ecundo Elementorum, &longs;i ergo monas e&longs;t <lb/>in continua proportione, habeo intentum: &longs;i <lb/>non ponatur e media inter a e &amp; d, erit ergo <lb/>per decimam noni Elementorum elatus a c, <lb/>ergo &aelig;qualis a b, igitur cum a c, e &amp; d &longs;int analog&aelig;, erunt &amp; a b c, <lb/>a b, &amp; d analog&aelig;, quod fuit demon&longs;trandum.</s></p><pb pagenum="29"/><p type="margin">
  
 <s><margin.target id="marg90"></margin.target>P<emph type="italics"/>rima ex<emph.end type="italics"/><lb/>C<emph type="italics"/>ampano.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg90"/>P<emph type="italics"/>rima ex<emph.end type="italics"/><lb/>C<emph type="italics"/>ampano.<emph.end type="italics"/></s></p><p type="main">
  
 <s>Propo&longs;itio trige&longs;imaquinta.</s></p><p type="main"> <s>Propo&longs;itio trige&longs;imaquinta.</s></p><p type="main">
  
 <s>Vocum magnitudines excre&longs;cunt in acumine non in grauitate, <lb/>finis autem e&longs;t in utroque extremo, propter hoc minima facta uaria&shy;<lb/>tione in hypate acut&aelig; uix ferunt.<lb/><arrow.to.target n="marg91"></arrow.to.target></s></p><p type="margin"> <s>Vocum magnitudines excre&longs;cunt in acumine non in grauitate, <lb/>finis autem e&longs;t in utroque extremo, propter hoc minima facta uaria&shy;<lb/>tione in hypate acut&aelig; uix ferunt.<lb/><arrow.to.target n="marg91"/></s></p><p type="margin">
  
 <s><margin.target id="marg91"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main"> <s><margin.target id="marg91"/>C<emph type="italics"/>o<emph.end type="italics"/>m.<!-- KEEP S--></s></p><p type="main">
  
 <s>Quoniam facta uariatione in hypate, qu&aelig; e&longs;t <lb/>in Diapa&longs;on, uel bis D&iacute;apa&longs;on maiore interual&shy;<lb/><figure id="fig20"></figure><lb/>lo di&longs;tat, uelut ex a in b in grauiore, maius e&longs;t in&shy;<lb/>teruallum ex c in d, igitur maior e&longs;t b d, qu&agrave;m a c <lb/>ergo &longs;ingul&aelig; uoces inter b &amp; d magis di&longs;tant, <lb/><figure id="fig21"></figure><lb/>qu&agrave;m inter a &amp; c, &amp; quanto magis appropin&shy;<lb/>quant ad d, igitur d maius e&longs;t qu&agrave;m b. </s> <s>Quoniam facta uariatione in hypate, qu&aelig; e&longs;t <lb/>in Diapa&longs;on, uel bis D&iacute;apa&longs;on maiore interual&shy;<lb/><figure id="fig20"/><lb/>lo di&longs;tat, uelut ex a in b in grauiore, maius e&longs;t in&shy;<lb/>teruallum ex c in d, igitur maior e&longs;t b d, qu&agrave;m a c <lb/>ergo &longs;ingul&aelig; uoces inter b &amp; d magis di&longs;tant, <lb/><figure id="fig21"/><lb/>qu&agrave;m inter a &amp; c, &amp; quanto magis appropin&shy;<lb/>quant ad d, igitur d maius e&longs;t qu&agrave;m b. </s>
  
 <s>Ergo magnitudo e&longs;t ratione <lb/>acuitatis, non grauitatis, cum &longs;uppo&longs;uerimus d e&longs;&longs;e acutiorem b &amp; <lb/>cip&longs;o a. </s> <s>Ergo magnitudo e&longs;t ratione <lb/>acuitatis, non grauitatis, cum &longs;uppo&longs;uerimus d e&longs;&longs;e acutiorem b &amp; <lb/>cip&longs;o a. </s>
  
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 <s>At in uelo ci&longs;simo motu oportet <lb/>uel fidem uel arteriam contrahi, &amp; non contrahitur ni&longs;i per mu&longs;cu&shy;<lb/>los, igitur contentio illa finem habet. </s> <s>At in uelo ci&longs;simo motu oportet <lb/>uel fidem uel arteriam contrahi, &amp; non contrahitur ni&longs;i per mu&longs;cu&shy;<lb/>los, igitur contentio illa finem habet. </s>
  
 <s>Si autem non &longs;it nece&longs;&longs;arium <lb/>habere, uel ualde procul po&longs;sit extendi contentio, ut in machinis <lb/>igneis &longs;trepitus fit maximus, nam motus, ut motus e&longs;t etiam in a&euml;re <lb/>nullum finem per &longs;e habet ni&longs;i ratione in&longs;trumenti, ergo &longs;trepitus <lb/>tantus e&longs;&longs;e pote&longs;t, ut ferm&egrave; ob&longs;urde&longs;cant, qui audierint, ut ferunt de <lb/>Nili cataractis.</s></p><figure></figure><p type="main"> <s>Si autem non &longs;it nece&longs;&longs;arium <lb/>habere, uel ualde procul po&longs;sit extendi contentio, ut in machinis <lb/>igneis &longs;trepitus fit maximus, nam motus, ut motus e&longs;t etiam in a&euml;re <lb/>nullum finem per &longs;e habet ni&longs;i ratione in&longs;trumenti, ergo &longs;trepitus <lb/>tantus e&longs;&longs;e pote&longs;t, ut ferm&egrave; ob&longs;urde&longs;cant, qui audierint, ut ferunt de <lb/>Nili cataractis.</s></p><figure/><p type="main">
  
 <s>Tertium &longs;ic &longs;it a b humi&shy;<lb/>lior uox, qu&aelig; excre&longs;cat &longs;e&shy;<lb/>mitonio minore &longs;olum in <lb/>c, &amp; &longs;it d e dupla ad ab &longs;e&shy;<lb/><figure id="fig22"></figure><lb/>cundum naturam, ut in uo&shy;<lb/>cibus medijs fiet, ut &longs;i e debeat excre&longs;cere &longs;emitonio minore per de&shy;<lb/>cimamnonam quinti <expan abbr="Elem&etilde;torum">Elementorum</expan> fe dupla c b, &amp; in acutis ubi ex&shy;<lb/>creuerit ad diapa&longs;on quadrupla: pueri autem uox, qu&aelig; iam diapa&shy;<lb/>&longs;on altior e&longs;t d e, erit bis diapa&longs;on, &amp; ide&ograve; quadrupla b c, &longs;ed in acu&shy;<lb/>tioribus erit dupla, nullus enim puer e&longs;t adeo fract&aelig; uocis, qui&longs;u&shy;<lb/>pra humillimam non a&longs;cendat per diapa&longs;on, igitur interuallum uo&shy;<lb/>cum erit octuplum a d, b c, &longs;ed communiter a&longs;cen dunt ad bis diapa <lb/>&longs;on, igitur interuallum unius uocis etiam cum &longs;emitonio propor&shy;<lb/>tionem habentis e&longs;t &aelig;quale ferm&egrave; toti a b, cum autem in diapa&longs;on <lb/>&longs;int duodecim &longs;emitonia, &amp; duo comata, manife&longs;tum e&longs;t, quod ex&shy;<lb/>ten&longs;io illa erit maxima in <expan abbr="c&otilde;parat&iacute;one">comparat&iacute;one</expan> grauioris uo cis a b. </s> <s>Tertium &longs;ic &longs;it a b humi&shy;<lb/>lior uox, qu&aelig; excre&longs;cat &longs;e&shy;<lb/>mitonio minore &longs;olum in <lb/>c, &amp; &longs;it d e dupla ad ab &longs;e&shy;<lb/><figure id="fig22"/><lb/>cundum naturam, ut in uo&shy;<lb/>cibus medijs fiet, ut &longs;i e debeat excre&longs;cere &longs;emitonio minore per de&shy;<lb/>cimamnonam quinti <expan abbr="Elem&etilde;torum">Elementorum</expan> fe dupla c b, &amp; in acutis ubi ex&shy;<lb/>creuerit ad diapa&longs;on quadrupla: pueri autem uox, qu&aelig; iam diapa&shy;<lb/>&longs;on altior e&longs;t d e, erit bis diapa&longs;on, &amp; ide&ograve; quadrupla b c, &longs;ed in acu&shy;<lb/>tioribus erit dupla, nullus enim puer e&longs;t adeo fract&aelig; uocis, qui&longs;u&shy;<lb/>pra humillimam non a&longs;cendat per diapa&longs;on, igitur interuallum uo&shy;<lb/>cum erit octuplum a d, b c, &longs;ed communiter a&longs;cen dunt ad bis diapa <lb/>&longs;on, igitur interuallum unius uocis etiam cum &longs;emitonio propor&shy;<lb/>tionem habentis e&longs;t &aelig;quale ferm&egrave; toti a b, cum autem in diapa&longs;on <lb/>&longs;int duodecim &longs;emitonia, &amp; duo comata, manife&longs;tum e&longs;t, quod ex&shy;<lb/>ten&longs;io illa erit maxima in <expan abbr="c&otilde;parat&iacute;one">comparat&iacute;one</expan> grauioris uo cis a b. </s>
  
 <s>Etide&ograve; <lb/>minimum in crementum in humilioribus uocibus, ubi quis coga&shy;<pb pagenum="30"/>tur a&longs;cendere, maximum e&longs;&longs;e uidetur, ade&ograve; ut &aelig;gr&egrave; &agrave; pluribus fera&shy;<lb/>tur, &agrave; quibu&longs;dam non omnino feratur.</s></p><p type="head"> <s>Etide&ograve; <lb/>minimum in crementum in humilioribus uocibus, ubi quis coga&shy;<pb pagenum="30"/>tur a&longs;cendere, maximum e&longs;&longs;e uidetur, ade&ograve; ut &aelig;gr&egrave; &agrave; pluribus fera&shy;<lb/>tur, &agrave; quibu&longs;dam non omnino feratur.</s></p><p type="head">
  
 <s>SCHOLIVM.</s></p><p type="main"> <s>SCHOLIVM.<!-- KEEP S--></s></p><p type="main">
  
 <s>Ob hoc natura fecit, ut non quemadmodum in fidibus uoces ex <lb/>breuitate intenderentur, &longs;ed ex con&longs;trictione ligul&aelig;, ut dicunt, &longs;u&shy;<lb/>per a&longs;peram arteriam uox ad diapa&longs;on acueretur addito impetu <lb/>proportione, ut ex con&longs;trictione, &amp; impetu <expan abbr="c&otilde;&longs;urgeret">con&longs;urgeret</expan> dupla pro&shy;<lb/>portio. </s> <s>Ob hoc natura fecit, ut non quemadmodum in fidibus uoces ex <lb/>breuitate intenderentur, &longs;ed ex con&longs;trictione ligul&aelig;, ut dicunt, &longs;u&shy;<lb/>per a&longs;peram arteriam uox ad diapa&longs;on acueretur addito impetu <lb/>proportione, ut ex con&longs;trictione, &amp; impetu <expan abbr="c&otilde;&longs;urgeret">con&longs;urgeret</expan> dupla pro&shy;<lb/>portio. </s>
  
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 <s>Vnde manife&longs;tum e&longs;t duas proportio&shy;<lb/>nes minores &aelig;qualitate inuicem ductas proportionem minorem <lb/>unaquaque illarum producere.</s></p><p type="main"> <s>Vnde manife&longs;tum e&longs;t duas proportio&shy;<lb/>nes minores &aelig;qualitate inuicem ductas proportionem minorem <lb/>unaquaque illarum producere.</s></p><p type="main">
  
 <s><arrow.to.target n="marg92"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg92"/></s></p><p type="margin">
  
  <s><margin.target id="marg92"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><figure/><p type="main">
  
 <s><margin.target id="marg92"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><figure></figure><p type="main"> <s>Proportio a b ad c, quali&longs;cunque &longs;it, duca&shy;<lb/>tur in proportionem minorem &aelig;qualitate <lb/>fad g, dico quod producta proportio erit <lb/>minor ea, qu&aelig; e&longs;t a b ad c fiat d ad a b, ut f <lb/>ad g, et erit per &longs;ecundam huius d ad c pro&shy;<lb/>ducta ex proportionibus a b ad c, &amp; f g. <!-- KEEP S--></s>
  
 <s>Proportio a b ad c, quali&longs;cunque &longs;it, duca&shy;<lb/>tur in proportionem minorem &aelig;qualitate <lb/>fad g, dico quod producta proportio erit <lb/>minor ea, qu&aelig; e&longs;t a b ad c fiat d ad a b, ut f <lb/>ad g, et erit per &longs;ecundam huius d ad c pro&shy;<lb/>ducta ex proportionibus a b ad c, &amp; f g. </s> <s>Itemque per decimamquar&shy;<lb/><arrow.to.target n="marg93"/><lb/>tam quinti <expan abbr="Elementor&utilde;">Elementorum</expan> erit d minor a b, igitur maior a b ad c, qu&agrave;m <lb/>d ad c. <!-- REMOVE S-->igitur qu&agrave;m proportio a b ad c in proportionem f ad g. <!-- KEEP S--></s>
  
 <s>Itemque per decimamquar&shy;<lb/><arrow.to.target n="marg93"></arrow.to.target><lb/>tam quinti <expan abbr="Elementor&utilde;">Elementorum</expan> erit d minor a b, igitur maior a b ad c, qu&agrave;m <lb/>d ad c. </s> 
  
 <s>igitur qu&agrave;m proportio a b ad c in proportionem f ad g. </s> 
  
 <s>Sit <lb/>autem utraque minor &aelig;qualitate ea, qu&aelig; a b ad c, &amp; ea qu&aelig; f ad g, di&shy;<lb/>co productam unaquaque earum e&longs;&longs;e minorem. </s> <s>Sit <lb/>autem utraque minor &aelig;qualitate ea, qu&aelig; a b ad c, &amp; ea qu&aelig; f ad g, di&shy;<lb/>co productam unaquaque earum e&longs;&longs;e minorem. </s>
  
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 <s>Quia enim mi&shy;<lb/>nor e&longs;t a b ad c, &aelig;qualitate erit a b minor c, fiat ergo h &aelig;qualis a b, <lb/>erit ergo d ad h, ut d ad a b per &longs;eptimam quinti Elementorum, at d <lb/>ad c minor qu&agrave;m d ad h per octauam eiu&longs;dem, igitur minor d ad c, <lb/>qu&agrave;m d ad a b, igitur patet propo&longs;itum.</s></p><p type="margin"> <s>Quia enim mi&shy;<lb/>nor e&longs;t a b ad c, &aelig;qualitate erit a b minor c, fiat ergo h &aelig;qualis a b, <lb/>erit ergo d ad h, ut d ad a b per &longs;eptimam quinti Elementorum, at d <lb/>ad c minor qu&agrave;m d ad h per octauam eiu&longs;dem, igitur minor d ad c, <lb/>qu&agrave;m d ad a b, igitur patet propo&longs;itum.</s></p><p type="margin">
  
 <s><margin.target id="marg93"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 1 <gap/>. </s> <s><margin.target id="marg93"/>P<emph type="italics"/>er<emph.end type="italics"/> 1 <gap/>. </s>
  
 <s>P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="main"> <s>P<emph type="italics"/>et.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main">
  
 <s>Propo&longs;itio trige&longs;ima&longs;eptima.</s></p><p type="main"> <s>Propo&longs;itio trige&longs;ima&longs;eptima.</s></p><p type="main">
  
 <s>Si plures homines, quorum nulli per &longs;e nauim mouere po&longs;sint, <lb/>aut pondus ferre &longs;imul iuncti eam moueant, aut pondus ferant, <lb/>erunt ill&aelig; proportiones coniunct&aelig; non product&aelig;.<lb/><arrow.to.target n="marg94"></arrow.to.target></s></p><p type="margin"> <s>Si plures homines, quorum nulli per &longs;e nauim mouere po&longs;sint, <lb/>aut pondus ferre &longs;imul iuncti eam moueant, aut pondus ferant, <lb/>erunt ill&aelig; proportiones coniunct&aelig; non product&aelig;.<lb/><arrow.to.target n="marg94"/></s></p><p type="margin">
  
 <s><margin.target id="marg94"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg94"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>C&ugrave;m enim primus non po&longs;sit mouere nec &longs;ecundus, erunt pro&shy;<lb/>portiones minores &aelig;qualitate, Ide&ograve; per &longs;ecundam partem pr&aelig;ce&shy;<lb/>dentis multo minus mouerent duo, qu&agrave;m unus. </s> <s>C&ugrave;m enim primus non po&longs;sit mouere nec &longs;ecundus, erunt pro&shy;<lb/>portiones minores &aelig;qualitate, Ide&ograve; per &longs;ecundam partem pr&aelig;ce&shy;<lb/>dentis multo minus mouerent duo, qu&agrave;m unus. </s>
  
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 <s>Omne corpus tant&ugrave;m re&longs;i&longs;tit motui contrario &longs;uo naturali quan <lb/>cum mouetur occulto motu quie&longs;cendo.</s></p><p type="main"> <s>Omne corpus tant&ugrave;m re&longs;i&longs;tit motui contrario &longs;uo naturali quan <lb/>cum mouetur occulto motu quie&longs;cendo.</s></p><p type="main">
  
 <s><arrow.to.target n="marg95"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg95"/></s></p><p type="margin">
  
 <s><margin.target id="marg95"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main"> <s><margin.target id="marg95"/>C<emph type="italics"/>o<emph.end type="italics"/>m.<!-- KEEP S--></s></p><p type="main">
  
 <s>Sit a corpus quie&longs;cens in pauimento b, &amp; mouetur in eo occul&shy;</s></p><p type="main"> <s>Sit a corpus quie&longs;cens in pauimento b, &amp; mouetur in eo occul&shy;</s></p><p type="main">
  
 <s><arrow.to.target n="marg96"></arrow.to.target><lb/>to motu uer&longs;us centrum, ut &longs;upr&agrave; ui&longs;um e&longs;t, contra&shy;<lb/><figure id="fig23"></figure><lb/>rius illi &longs;it motus ad c, &longs;i ergo a quie&longs;ceret in c moue&shy;<lb/>retur ad b occulto motu certa ui, ergo eadem re&longs;titit, <lb/>ne traheretur ad c. </s> <s><arrow.to.target n="marg96"/><lb/>to motu uer&longs;us centrum, ut &longs;upr&agrave; ui&longs;um e&longs;t, contra&shy;<lb/><figure id="fig23"/><lb/>rius illi &longs;it motus ad c, &longs;i ergo a quie&longs;ceret in c moue&shy;<lb/>retur ad b occulto motu certa ui, ergo eadem re&longs;titit, <lb/>ne traheretur ad c. <!-- KEEP S--></s>
  
 <s>Manife&longs;tum e&longs;t autem, quod hic <lb/><arrow.to.target n="marg97"></arrow.to.target><lb/>motus occultus e&longs;t minor manife&longs;to.<lb/><arrow.to.target n="marg98"></arrow.to.target></s></p><p type="margin"> <s>Manife&longs;tum e&longs;t autem, quod hic <lb/><arrow.to.target n="marg97"/><lb/>motus occultus e&longs;t minor manife&longs;to.<lb/><arrow.to.target n="marg98"/></s></p><p type="margin">
  
 <s><margin.target id="marg96"></margin.target>I<emph type="italics"/>n commen.<emph.end type="italics"/><lb/>26. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg96"/>I<emph type="italics"/>n commen.<emph.end type="italics"/><lb/>26. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg97"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 30. P<emph type="italics"/>ro <lb/>po&longs;.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg97"/>P<emph type="italics"/>er<emph.end type="italics"/> 30. P<emph type="italics"/>ro <lb/>po&longs;.<emph.end type="italics"/></s></p><p type="margin">
  
 <s><margin.target id="marg98"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg98"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Ex hoc patet cur naues &amp; currus ab initio tard&egrave; &amp; difficulter mo<lb/>ueantur, ubi moueri c&oelig;perint motus augetur: quoniam re&longs;i&longs;tunt </s></p><p type="main"> <s>Ex hoc patet cur naues &amp; currus ab initio tard&egrave; &amp; difficulter mo<lb/>ueantur, ubi moueri c&oelig;perint motus augetur: quoniam re&longs;i&longs;tunt </s></p><p type="main">
  
 <s><arrow.to.target n="marg99"></arrow.to.target><lb/>per motum occultum naturalem qui maximus e&longs;t dum quie&longs;cunt, <lb/>ut etiam do cebat philo&longs;ophus in mechanicis, nam motus ille natu&shy;<lb/>ralis e&longs;t, &amp; ide&ograve; contrarius uiolento: Ergo cum iam mouetur uio&shy;<lb/>lenter minus, mouetur naturaliter, igitur minus re&longs;i&longs;tit. </s> <s><arrow.to.target n="marg99"/><lb/>per motum occultum naturalem qui maximus e&longs;t dum quie&longs;cunt, <lb/>ut etiam do cebat philo&longs;ophus in mechanicis, nam motus ille natu&shy;<lb/>ralis e&longs;t, &amp; ide&ograve; contrarius uiolento: Ergo cum iam mouetur uio&shy;<lb/>lenter minus, mouetur naturaliter, igitur minus re&longs;i&longs;tit. </s>
  
 <s>Declarabi&shy;<lb/>tur enim infr&agrave; qu&ograve;d omne quod mouetur duobus motibus tanto <lb/><arrow.to.target n="marg100"></arrow.to.target><lb/>minus uno mouetur quanto magis altero.</s></p><p type="margin"> <s>Declarabi&shy;<lb/>tur enim infr&agrave; qu&ograve;d omne quod mouetur duobus motibus tanto <lb/><arrow.to.target n="marg100"/><lb/>minus uno mouetur quanto magis altero.</s></p><p type="margin">
  
 <s><margin.target id="marg99"></margin.target>Q<emph type="italics"/>ue&longs;t.<emph.end type="italics"/> 31.</s></p><p type="margin"> <s><margin.target id="marg99"/>Q<emph type="italics"/>ue&longs;t.<emph.end type="italics"/> 31.</s></p><p type="margin">
  
 <s><margin.target id="marg100"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 59.</s></p><p type="main"> <s><margin.target id="marg100"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 59.</s></p><p type="main">
  
 <s>Propo&longs;itio trige&longs;imanona.</s></p><p type="main"> <s>Propo&longs;itio trige&longs;imanona.</s></p><p type="main">
  
 <s>Ab &aelig;quali aut minore ui, qu&agrave;m &longs;it <expan abbr="impediment&utilde;">impedimentum</expan>, non fit motus.</s></p><p type="main"> <s>Ab &aelig;quali aut minore ui, qu&agrave;m &longs;it <expan abbr="impediment&utilde;">impedimentum</expan>, non fit motus.</s></p><p type="main">
  
 <s>Sit a quod re&longs;i&longs;tat, ne &longs;ur&longs;um trahatur per decem, dico, quod <expan abbr="n&otilde;">non</expan> <lb/><arrow.to.target n="marg101"></arrow.to.target><lb/>&longs;ur&longs;um trahetur neque &agrave; decem, neque minore: nam &longs;i impedimen&shy;<lb/>tum non e&longs;&longs;et, moueretur infra ut decem, ergo &longs;i traheretur &longs;ur&longs;um <lb/>per decem tantum moueretur &longs;ur&longs;um, <expan abbr="quant&utilde;">quantum</expan> deor&longs;um, ergo quie&shy;<lb/>&longs;ceret. </s> <s>Sit a quod re&longs;i&longs;tat, ne &longs;ur&longs;um trahatur per decem, dico, quod <expan abbr="n&otilde;">non</expan> <lb/><arrow.to.target n="marg101"/><lb/>&longs;ur&longs;um trahetur neque &agrave; decem, neque minore: nam &longs;i impedimen&shy;<lb/>tum non e&longs;&longs;et, moueretur infra ut decem, ergo &longs;i traheretur &longs;ur&longs;um <lb/>per decem tantum moueretur &longs;ur&longs;um, <expan abbr="quant&utilde;">quantum</expan> deor&longs;um, ergo quie&shy;<lb/>&longs;ceret. </s>
  
 <s>Si uer&ograve; &agrave; minore moueretur &agrave; maiore ui deor&longs;um, quam &longs;ur&shy;<lb/>&longs;um, ergo deor&longs;um &longs;impliciter non &longs;ur&longs;um.</s></p><p type="margin"> <s>Si uer&ograve; &agrave; minore moueretur &agrave; maiore ui deor&longs;um, quam &longs;ur&shy;<lb/>&longs;um, ergo deor&longs;um &longs;impliciter non &longs;ur&longs;um.</s></p><p type="margin">
  
 <s><margin.target id="marg101"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg101"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Propo&longs;itio quadrage&longs;ima.</s></p><p type="main"> <s>Propo&longs;itio quadrage&longs;ima.</s></p><p type="main">
  
 <s>Omne corpus &longs;ph&aelig;ricum tangens planum in puncto mouetur <lb/>ad latus per quancunque uim, qu&aelig; medium diuidere pote&longs;t.</s></p><figure></figure><p type="main"> <s>Omne corpus &longs;ph&aelig;ricum tangens planum in puncto mouetur <lb/>ad latus per quancunque uim, qu&aelig; medium diuidere pote&longs;t.</s></p><figure/><p type="main">
  
 <s>Sit corpus ad unguem &longs;ph&aelig;ricum a tan&shy;<lb/><arrow.to.target n="marg102"></arrow.to.target><lb/>gens planum b in puncto c (e&longs;t enim hoc <lb/>nece&longs;&longs;arium ex demon&longs;tratis ab Euclide in <lb/>decima&longs;exta Propo&longs;itione tertij Elemento&shy;<lb/>rum) dico, quod mouebitur &agrave; ui, qu&aelig; pote&longs;t <lb/>&longs;cindere a&euml;rem. </s> <s>Sit corpus ad unguem &longs;ph&aelig;ricum a tan&shy;<lb/><arrow.to.target n="marg102"/><lb/>gens planum b in puncto c (e&longs;t enim hoc <lb/>nece&longs;&longs;arium ex demon&longs;tratis ab Euclide in <lb/>decima&longs;exta Propo&longs;itione tertij Elemento&shy;<lb/>rum) dico, quod mouebitur &agrave; ui, qu&aelig; pote&longs;t <lb/>&longs;cindere a&euml;rem. </s>
  
 <s>Nam cum non a&longs;cendat, nec <lb/>de&longs;cendat, &longs;ed qua&longs;i in circulo ad centrum <lb/>mundi moueatur, pondus non affert. </s> <s>Nam cum non a&longs;cendat, nec <lb/>de&longs;cendat, &longs;ed qua&longs;i in circulo ad centrum <lb/>mundi moueatur, pondus non affert. </s>
  
 <s>Neque<lb/>ratione magnitudinis contactus, cum &longs;it in <lb/>puncto &longs;olo, igitur remanet &longs;olum a&euml;ris impedimentum.<pb pagenum="32"/><arrow.to.target n="marg103"></arrow.to.target></s></p><p type="margin"> <s>Neque<lb/>ratione magnitudinis contactus, cum &longs;it in <lb/>puncto &longs;olo, igitur remanet &longs;olum a&euml;ris impedimentum.<pb pagenum="32"/><arrow.to.target n="marg103"/></s></p><p type="margin">
  
 <s><margin.target id="marg102"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin"> <s><margin.target id="marg102"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg103"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main"> <s><margin.target id="marg103"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.<!-- KEEP S--></s></p><p type="main">
  
 <s>Ex hoc liquet, quod oportet b planum e&longs;&longs;e ex duri&longs;sima mate&shy;<lb/>ria, qu&aelig; nullo modo cedat, aliter tanget plu&longs;qu&agrave;m in puncto.</s></p><p type="main"> <s>Ex hoc liquet, quod oportet b planum e&longs;&longs;e ex duri&longs;sima mate&shy;<lb/>ria, qu&aelig; nullo modo cedat, aliter tanget plu&longs;qu&agrave;m in puncto.</s></p><p type="main">
  
 <s><arrow.to.target n="marg104"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg104"/></s></p><p type="margin">
  
 <s><margin.target id="marg104"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main"> <s><margin.target id="marg104"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.<!-- KEEP S--></s></p><p type="main">
  
 <s>Vix fieri pote&longs;t, utin elementaribus &longs;ph&aelig;ra tangat planum in <lb/>puncto. </s> <s>Vix fieri pote&longs;t, utin elementaribus &longs;ph&aelig;ra tangat planum in <lb/>puncto. </s>
  
Line 1279 
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 <s>Si fuerint du&aelig; quantitates &longs;umaturque totius aggregatum maio&shy;<lb/>ris &amp; minoris, quoties aggregatum minoris, &amp; maioris, erit pro&shy;<lb/>portio confu&longs;a maioris aggregati ad minus, minor qu&agrave;m multipli&shy;<lb/>cis maioris ad multiplex minoris.</s></p><p type="main"> <s>Si fuerint du&aelig; quantitates &longs;umaturque totius aggregatum maio&shy;<lb/>ris &amp; minoris, quoties aggregatum minoris, &amp; maioris, erit pro&shy;<lb/>portio confu&longs;a maioris aggregati ad minus, minor qu&agrave;m multipli&shy;<lb/>cis maioris ad multiplex minoris.</s></p><p type="main">
  
 <s><arrow.to.target n="marg105"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg105"/></s></p><p type="margin">
  
 <s><margin.target id="marg105"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg105"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Sint du&aelig; magnitudines a &amp; b, &amp; &longs;it a maior <lb/><figure id="fig24"></figure><lb/>b, &amp; &longs;umatur exempli gratia a quater cum b &longs;e&shy;<lb/>mel, &amp; b quater cum a &longs;emel, dico, quod propor<lb/>tio (quam confu&longs;am e&longs;&longs;e liquet) aggregati primi ad &longs;ecundum, e&longs;t </s></p><p type="main"> <s>Sint du&aelig; magnitudines a &amp; b, &amp; &longs;it a maior <lb/><figure id="fig24"/><lb/>b, &amp; &longs;umatur exempli gratia a quater cum b &longs;e&shy;<lb/>mel, &amp; b quater cum a &longs;emel, dico, quod propor<lb/>tio (quam confu&longs;am e&longs;&longs;e liquet) aggregati primi ad &longs;ecundum, e&longs;t </s></p><p type="main">
  
 <s><arrow.to.target n="marg106"></arrow.to.target><lb/>minor qu&agrave;m quadrupla. </s> <s><arrow.to.target n="marg106"/><lb/>minor qu&agrave;m quadrupla. </s>
  
 <s>Con&longs;tat enim quod proportio quadru&shy;<lb/>pli a ad a e&longs;t maior, quam b ad quadruplum b, cum una &longs;it quadru&shy;<lb/>pla, alia &longs;ub quadrupla, igitur per uige&longs;imam&longs;ecundam huius ag&shy;<lb/>gregati quadrupli a cum b &longs;emel, ad quadruplum b cum a &longs;emel mi <lb/><arrow.to.target n="marg107"></arrow.to.target><lb/>nor, qu&agrave;m quadrupli a ad a, &amp; maior qu&agrave;m b ad quadruplum b, &amp; <lb/>e&longs;t pro intellectu Archimedis.</s></p><p type="margin"> <s>Con&longs;tat enim quod proportio quadru&shy;<lb/>pli a ad a e&longs;t maior, quam b ad quadruplum b, cum una &longs;it quadru&shy;<lb/>pla, alia &longs;ub quadrupla, igitur per uige&longs;imam&longs;ecundam huius ag&shy;<lb/>gregati quadrupli a cum b &longs;emel, ad quadruplum b cum a &longs;emel mi <lb/><arrow.to.target n="marg107"/><lb/>nor, qu&agrave;m quadrupli a ad a, &amp; maior qu&agrave;m b ad quadruplum b, &amp; <lb/>e&longs;t pro intellectu Archimedis.<!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg106"></margin.target>E<emph type="italics"/>x<emph.end type="italics"/> 18. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg106"/>E<emph type="italics"/>x<emph.end type="italics"/> 18. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="margin">
  
 <s><margin.target id="marg107"></margin.target>I<emph type="italics"/>n<emph.end type="italics"/> 2. <emph type="italics"/>lib.  <s><margin.target id="marg107"/>I<emph type="italics"/>n<emph.end type="italics"/> 2. <emph type="italics"/>lib.
  
 de<emph.end type="italics"/><lb/>A<emph type="italics"/>tqui pon&shy;<lb/>deran.<emph.end type="italics"/><lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 10.</s></p><p type="main"> de<emph.end type="italics"/><lb/>A<emph type="italics"/>tqui pon&shy;<lb/>deran.<emph.end type="italics"/><lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 10.</s></p><p type="main">
  
 <s>Propo&longs;itio quadrage&longs;ima&longs;ecunda.</s></p><p type="main"> <s>Propo&longs;itio quadrage&longs;ima&longs;ecunda.</s></p><p type="main">
  
 <s>Trahentium nauim, ut ferentium pondera proportiones in &longs;e in&shy;<lb/>uicem, quomodo ducere oporteat con&longs;iderare.<lb/><arrow.to.target n="marg108"></arrow.to.target></s></p><p type="margin"> <s>Trahentium nauim, ut ferentium pondera proportiones in &longs;e in&shy;<lb/>uicem, quomodo ducere oporteat con&longs;iderare.<lb/><arrow.to.target n="marg108"/></s></p><p type="margin">
  
 <s><margin.target id="marg108"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg108"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Hoc quomodo non po&longs;sit fieri &longs;upr&agrave; docuimus, nunc etiam ge&shy;</s></p><p type="main"> <s>Hoc quomodo non po&longs;sit fieri &longs;upr&agrave; docuimus, nunc etiam ge&shy;</s></p><p type="main">
  
 <s><arrow.to.target n="marg109"></arrow.to.target><lb/>neraliter dicam, cum con&longs;i&longs;tant h&aelig;c in duobus terminis, productio <lb/>uer&ograve; pr&aelig;&longs;upponit quatuor terminos, ut in prima propo&longs;itione, aut <lb/>&longs;altem tres, atque in his medius habet rationem mouentis, &amp; moti, <lb/>ergo cum in huiu&longs;modi <expan abbr="n&otilde;">non</expan> &longs;int quatuor termini, nec tres, &egrave; quibus <lb/>unus &longs;it mouens, &amp; motum proportio non poterit produci. </s> <s><arrow.to.target n="marg109"/><lb/>neraliter dicam, cum con&longs;i&longs;tant h&aelig;c in duobus terminis, productio <lb/>uer&ograve; pr&aelig;&longs;upponit quatuor terminos, ut in prima propo&longs;itione, aut <lb/>&longs;altem tres, atque in his medius habet rationem mouentis, &amp; moti, <lb/>ergo cum in huiu&longs;modi <expan abbr="n&otilde;">non</expan> &longs;int quatuor termini, nec tres, &egrave; quibus <lb/>unus &longs;it mouens, &amp; motum proportio non poterit produci. </s>
  
 <s>Illud <lb/>etiam patet exemplo, nam &longs;i e&longs;&longs;et lapis, aut nauis ob&longs;i&longs;tens ut &longs;ex, &amp; <lb/>e&longs;&longs;ent homines uiribus &longs;inguli, ut quatuor cum dimidio, tres mo&shy;<lb/>uerent in proportione dupla &longs;exquiquarta perdicta &longs;uperius eo&shy;<lb/>dem loco, at &longs;i proportio duci po&longs;&longs;et aliquorum hominum nume&shy;<lb/>rus po&longs;&longs;et mouere in duplicata proportione ad unguem &longs;cilicet <lb/>5 1/16 ut e&longs;&longs;et uix hominum collectorum 30 3/8 at nullus e&longs;t numerus ho <lb/>minum qui collectus faciat hunc numerum, nam &longs;ex homines ex&shy;<lb/>plentnumerum 27, &amp; &longs;eptem 31 1/2, &amp; ide&ograve; non pote&longs;t duci propor&shy;<lb/>tio. </s> <s>Illud <lb/>etiam patet exemplo, nam &longs;i e&longs;&longs;et lapis, aut nauis ob&longs;i&longs;tens ut &longs;ex, &amp; <lb/>e&longs;&longs;ent homines uiribus &longs;inguli, ut quatuor cum dimidio, tres mo&shy;<lb/>uerent in proportione dupla &longs;exquiquarta perdicta &longs;uperius eo&shy;<lb/>dem loco, at &longs;i proportio duci po&longs;&longs;et aliquorum hominum nume&shy;<lb/>rus po&longs;&longs;et mouere in duplicata proportione ad unguem &longs;cilicet <lb/>5 1/16 ut e&longs;&longs;et uix hominum collectorum 30 3/8 at nullus e&longs;t numerus ho <lb/>minum qui collectus faciat hunc numerum, nam &longs;ex homines ex&shy;<lb/>plentnumerum 27, &amp; &longs;eptem 31 1/2, &amp; ide&ograve; non pote&longs;t duci propor&shy;<lb/>tio. </s>
  
Line 1317 
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 <s>Vnde &longs;i duo moueant in proportione &longs;ex&shy;<lb/>quialtera, &amp; &longs;ex in proportione quadrupla cum dimidia, &amp; iungan <lb/>tur, ut fiant octo, non oportebit ducere &longs;exquialteram, in quadru&shy;<lb/>plam &longs;exquialteram, &longs;ed cum octo ad duo &longs;it in proportione qua&shy;<lb/>drupla, &longs;umemus quadruplam ad &longs;exquialteram, qu&ecedil; erit &longs;excupla, <lb/>&amp; octo mouebunt, aut pondus gerentin proportione &longs;excupla.</s></p><p type="margin"> <s>Vnde &longs;i duo moueant in proportione &longs;ex&shy;<lb/>quialtera, &amp; &longs;ex in proportione quadrupla cum dimidia, &amp; iungan <lb/>tur, ut fiant octo, non oportebit ducere &longs;exquialteram, in quadru&shy;<lb/>plam &longs;exquialteram, &longs;ed cum octo ad duo &longs;it in proportione qua&shy;<lb/>drupla, &longs;umemus quadruplam ad &longs;exquialteram, qu&ecedil; erit &longs;excupla, <lb/>&amp; octo mouebunt, aut pondus gerentin proportione &longs;excupla.</s></p><p type="margin">
  
 <s><margin.target id="marg109"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 37.</s></p><p type="main"> <s><margin.target id="marg109"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 37.</s></p><p type="main">
  
 <s>Propo&longs;itio quadrage&longs;imatertia.</s></p><p type="main"> <s>Propo&longs;itio quadrage&longs;imatertia.</s></p><p type="main">
  
 <s>Productionem ad additionem retrahere.<lb/><arrow.to.target n="marg110"></arrow.to.target></s></p><p type="margin"> <s>Productionem ad additionem retrahere.<lb/><arrow.to.target n="marg110"/></s></p><p type="margin">
  
 <s><margin.target id="marg110"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><figure></figure><p type="main"> <s><margin.target id="marg110"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><figure/><p type="main">
  
 <s>Sit proportio a ad b dupla pote&longs;tate li&shy;<lb/>cet &longs;int quinque homines, &amp; &longs;int quindecim <lb/>homines c, &amp; habebunt ad b &longs;excuplam <lb/>proportionem per pr&aelig;cedentem. </s> <s>Sit proportio a ad b dupla pote&longs;tate li&shy;<lb/>cet &longs;int quinque homines, &amp; &longs;int quindecim <lb/>homines c, &amp; habebunt ad b &longs;excuplam <lb/>proportionem per pr&aelig;cedentem. </s>
  
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 <s>S&aelig;pe contingit, ut quinque homines moueant nauim, &amp; &longs;patium <lb/>ad tempus notum, &amp; etiam cognitum maximum, quod mouere <lb/>non pote&longs;t. </s> <s>S&aelig;pe contingit, ut quinque homines moueant nauim, &amp; &longs;patium <lb/>ad tempus notum, &amp; etiam cognitum maximum, quod mouere <lb/>non pote&longs;t. </s>
  
 <s>Sit ergo a numerus hominum, b na&shy;<lb/><figure id="fig25"></figure><lb/>uis, c maximum, quod non mouere pote&longs;t, d <lb/>tempus, e &longs;patium, f motor alius &longs;iue numerus <lb/>hominum notus, &amp; g tempus, dico, quod h &longs;patium notum erit, &longs;eu <lb/><expan abbr="not&utilde;">notum</expan> g tempus, &amp; h &longs;patium, dico, quod erit f motor, &longs;eu numerus <pb pagenum="34"/>hominum notus. </s> <s>Sit ergo a numerus hominum, b na&shy;<lb/><figure id="fig25"/><lb/>uis, c maximum, quod non mouere pote&longs;t, d <lb/>tempus, e &longs;patium, f motor alius &longs;iue numerus <lb/>hominum notus, &amp; g tempus, dico, quod h &longs;patium notum erit, &longs;eu <lb/><expan abbr="not&utilde;">notum</expan> g tempus, &amp; h &longs;patium, dico, quod erit f motor, &longs;eu numerus <pb pagenum="34"/>hominum notus. </s>
  
 <s>Quoniam ergo notum e&longs;t a &amp; c, quia e&longs;t &aelig;quale <lb/>b, igitur proportio a ad b nota e&longs;t: &longs;ed iuxta illam a mouet b in d <lb/>tempore per e &longs;patium, igitur per pr&aelig;cedentem, ut f ad a ita &longs;patij <lb/>ad e in d tempore. </s> <s>Quoniam ergo notum e&longs;t a &amp; c, quia e&longs;t &aelig;quale <lb/>b, igitur proportio a ad b nota e&longs;t: &longs;ed iuxta illam a mouet b in d <lb/>tempore per e &longs;patium, igitur per pr&aelig;cedentem, ut f ad a ita &longs;patij <lb/>ad e in d tempore. </s>
  
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 <s>Propo&longs;itio quadrage&longs;imaquinta.</s></p><p type="main"> <s>Propo&longs;itio quadrage&longs;imaquinta.</s></p><p type="main">
  
 <s>Rationem &longs;tater&aelig; o&longs;tendere.<lb/><arrow.to.target n="marg111"></arrow.to.target></s></p><p type="margin"> <s>Rationem &longs;tater&aelig; o&longs;tendere.<lb/><arrow.to.target n="marg111"/></s></p><p type="margin">
  
 <s><margin.target id="marg111"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg111"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Archimedes nititur huic fundamento, quod pondera, qu&aelig; pro&shy;<lb/>portionem mutuam habent, ut di&longs;tanti&aelig; &agrave; libella a, qu&aelig; &longs;u&longs;pen&shy;<lb/>duntur, &aelig;qualiter ponderant, &longs;it ergo libella a b, &amp; &longs;u&longs;pen&longs;a in a cen <lb/>trum mundi c, ad quod dirigitur pondus, &amp; liquet, quod ip&longs;um <lb/>non &longs;e inclin abit ex uige&longs;imatertia propo&longs;itione. </s> <s>Archimedes nititur huic fundamento, quod pondera, qu&aelig; pro&shy;<lb/>portionem mutuam habent, ut di&longs;tanti&aelig; &agrave; libella a, qu&aelig; &longs;u&longs;pen&shy;<lb/>duntur, &aelig;qualiter ponderant, &longs;it ergo libella a b, &amp; &longs;u&longs;pen&longs;a in a cen <lb/>trum mundi c, ad quod dirigitur pondus, &amp; liquet, quod ip&longs;um <lb/>non &longs;e inclin abit ex uige&longs;imatertia propo&longs;itione. </s>
  
 <s>Si ergo ponantur <lb/>lo co line&aelig; b d in e &amp; f, &amp; &longs;it proportio e b <lb/><figure id="fig26"></figure><lb/>ad b f, ut g ad h, dico, qu&ograve;d erit &aelig;quili&shy;<lb/>brium, per eandem enim h mouebitur in k, <lb/>&longs;cilicet ut perueniat in rectam a d, &longs;i enim <lb/>non e&longs;&longs;et &verbar;&longs;u&longs;pen&longs;um h, moueretur in re&shy;<lb/>cta e h per eandem, quia ergo retinetur, mo&shy;<lb/>uetur per obliquam h k, &amp; &longs;umatur in pro&shy;<lb/>pin quum punctum in b e, &amp; n in &aelig;quali di&shy;<lb/>&longs;tantia in e f, quia ergo e b totum mouetur <lb/>eadem ui in &longs;ingulis partibus, quia a pon&shy;<lb/>dere h, &amp; in h mouetur per h k in m per m <lb/>p, ergo qualis e&longs;t proportio magnitudinis h k ad m p, talis e&longs;t uis <lb/>in m p ad uim in h k, &amp; ita in b erit pen&egrave; infinita: quia quanta ui ex&shy;<lb/>tenditur ex h in k tanta puncta b, &longs;e circumuertit ergo propor&shy;<lb/>tio hypomochlij ad &longs;patium, uelut roboris ad robur, at eadem n o <lb/>ad h k, e&longs;t enim n o &aelig;qualis m p, &amp; n b, &amp; b m &aelig;quales, ut uer&ograve; g ad <lb/>h, ita e b ad b f: ergo ut e b ad b f, ita uirium n o ad h k, ut igitur g ad <lb/>h, ita uirium m p ad h k: ut etiam g l ad n o, ita uirium f b ad n b. <lb/></s> <s>Si ergo ponantur <lb/>lo co line&aelig; b d in e &amp; f, &amp; &longs;it proportio e b <lb/><figure id="fig26"/><lb/>ad b f, ut g ad h, dico, qu&ograve;d erit &aelig;quili&shy;<lb/>brium, per eandem enim h mouebitur in k, <lb/>&longs;cilicet ut perueniat in rectam a d, &longs;i enim <lb/>non e&longs;&longs;et &verbar;&longs;u&longs;pen&longs;um h, moueretur in re&shy;<lb/>cta e h per eandem, quia ergo retinetur, mo&shy;<lb/>uetur per obliquam h k, &amp; &longs;umatur in pro&shy;<lb/>pin quum punctum in b e, &amp; n in &aelig;quali di&shy;<lb/>&longs;tantia in e f, quia ergo e b totum mouetur <lb/>eadem ui in &longs;ingulis partibus, quia a pon&shy;<lb/>dere h, &amp; in h mouetur per h k in m per m <lb/>p, ergo qualis e&longs;t proportio magnitudinis h k ad m p, talis e&longs;t uis <lb/>in m p ad uim in h k, &amp; ita in b erit pen&egrave; infinita: quia quanta ui ex&shy;<lb/>tenditur ex h in k tanta puncta b, &longs;e circumuertit ergo propor&shy;<lb/>tio hypomochlij ad &longs;patium, uelut roboris ad robur, at eadem n o <lb/>ad h k, e&longs;t enim n o &aelig;qualis m p, &amp; n b, &amp; b m &aelig;quales, ut uer&ograve; g ad <lb/>h, ita e b ad b f: ergo ut e b ad b f, ita uirium n o ad h k, ut igitur g ad <lb/>h, ita uirium m p ad h k: ut etiam g l ad n o, ita uirium f b ad n b. <lb/></s>
  
 <s>nam idem pondus &longs;cilicet g mouet totam b f, igitur ut g &longs;e habet </s></p><p type="main"> <s>nam idem pondus &longs;cilicet g mouet totam b f, igitur ut g &longs;e habet </s></p><p type="main">
  
 <s><arrow.to.target n="marg112"></arrow.to.target><lb/>ad n o, ita h ad m p, &longs;ed m p &amp; n o &longs;unt &aelig;quales, ergo tanta e&longs;t uis g <lb/>in f, quanta h in e.<lb/><arrow.to.target n="marg113"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg112"/><lb/>ad n o, ita h ad m p, &longs;ed m p &amp; n o &longs;unt &aelig;quales, ergo tanta e&longs;t uis g <lb/>in f, quanta h in e.<lb/><arrow.to.target n="marg113"/></s></p><p type="margin">
  
 <s><margin.target id="marg112"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 9. <emph type="italics"/>quin&shy;<lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg112"/>P<emph type="italics"/>er<emph.end type="italics"/> 9. <emph type="italics"/>quin&shy;<lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg113"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main"> <s><margin.target id="marg113"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.<!-- KEEP S--></s></p><p type="main">
  
 <s>Ex quo patet, quod hypomo chlion moueretur infinita ui, &longs;i po&longs;&shy;<lb/>&longs;et e&longs;&longs;e punctus: &longs;ed quia in extrema &longs;uperficie cylindri, ide&ograve; pote&longs;t <lb/>aliqua ui retineri.</s></p><p type="main"> <s>Ex quo patet, quod hypomo chlion moueretur infinita ui, &longs;i po&longs;&shy;<lb/>&longs;et e&longs;&longs;e punctus: &longs;ed quia in extrema &longs;uperficie cylindri, ide&ograve; pote&longs;t <lb/>aliqua ui retineri.</s></p><p type="main">
  
 <s><arrow.to.target n="marg114"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg114"/></s></p><p type="margin">
  
 <s><margin.target id="marg114"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main"> <s><margin.target id="marg114"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.<!-- KEEP S--></s></p><p type="main">
  
 <s>Et &longs;i quis po&longs;&longs;et capere ha&longs;tam in extremo puncto, non po&longs;&longs;et <lb/>eam mouere, etiam quod haberet robur infinitum, quia ab &aelig;quali <lb/>non fit motus per trige&longs;imamnonam propo&longs;itionem.</s></p><p type="main"> <s>Et &longs;i quis po&longs;&longs;et capere ha&longs;tam in extremo puncto, non po&longs;&longs;et <lb/>eam mouere, etiam quod haberet robur infinitum, quia ab &aelig;quali <lb/>non fit motus per trige&longs;imamnonam propo&longs;itionem.</s></p><p type="main">
  
 <s><arrow.to.target n="marg115"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg115"/></s></p><p type="margin">
  
 <s><margin.target id="marg115"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3.</s></p><p type="main"> <s><margin.target id="marg115"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3.<!-- KEEP S--></s></p><p type="main">
  
 <s>Et libella nihil retinet ni&longs;i quantum e&longs;t pondus eius quod cu&shy;<pb pagenum="35"/>pit ad centrum peruenire, &amp; pondus ei appen&longs;um non prohi&shy;<lb/>bet motum, etiam &longs;i e&longs;&longs;et infinitum, ni&longs;i quatenus non uult recede&shy;<lb/>re ex directo centri mundi: &amp; ut grauat hypomochlion faciens im&shy;<lb/>pre&longs;sionem.</s></p><p type="main"> <s>Et libella nihil retinet ni&longs;i quantum e&longs;t pondus eius quod cu&shy;<pb pagenum="35"/>pit ad centrum peruenire, &amp; pondus ei appen&longs;um non prohi&shy;<lb/>bet motum, etiam &longs;i e&longs;&longs;et infinitum, ni&longs;i quatenus non uult recede&shy;<lb/>re ex directo centri mundi: &amp; ut grauat hypomochlion faciens im&shy;<lb/>pre&longs;sionem.</s></p><p type="main">
  
 <s><arrow.to.target n="marg116"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg116"/></s></p><p type="margin">
  
 <s><margin.target id="marg116"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 4.</s></p><p type="main"> <s><margin.target id="marg116"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 4.<!-- KEEP S--></s></p><p type="main">
  
 <s>Et &longs;i terra tota e&longs;&longs;et appen&longs;a polo, moueretur magna ui: quoni&shy;<lb/>am uis eadem e&longs;t in polo, qu&aelig; in circulo toto &aelig;quinoctij.</s></p><p type="main"> <s>Et &longs;i terra tota e&longs;&longs;et appen&longs;a polo, moueretur magna ui: quoni&shy;<lb/>am uis eadem e&longs;t in polo, qu&aelig; in circulo toto &aelig;quinoctij.</s></p><p type="main">
  
 <s><arrow.to.target n="marg117"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg117"/></s></p><p type="margin">
  
 <s><margin.target id="marg117"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 5.</s></p><p type="main"> <s><margin.target id="marg117"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 5.<!-- KEEP S--></s></p><p type="main">
  
 <s>Etrota, quanto uelocius mouetur in ambitu, tanto mi<lb/>norem habet uim: &longs;ed propter a&euml;rem, qui &longs;ecum circum&shy;<lb/><figure id="fig27"></figure><lb/>fertur, mouetur magno impetu, &amp; magnas facit l&aelig;&longs;iones. <lb/></s> <s>Etrota, quanto uelocius mouetur in ambitu, tanto mi<lb/>norem habet uim: &longs;ed propter a&euml;rem, qui &longs;ecum circum&shy;<lb/><figure id="fig27"/><lb/>fertur, mouetur magno impetu, &amp; magnas facit l&aelig;&longs;iones. <lb/></s>
  
 <s>Ide&ograve; hoc in cono non accidit.</s></p><p type="main"> <s>Ide&ograve; hoc in cono non accidit.</s></p><p type="main">
  
 <s><arrow.to.target n="marg118"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg118"/></s></p><p type="margin">
  
 <s><margin.target id="marg118"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 6.</s></p><p type="main"> <s><margin.target id="marg118"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 6.<!-- KEEP S--></s></p><p type="main">
  
 <s>Ex quo patet ratio eleuandi pondera magna per tra&shy;<lb/>bem, ut &agrave; latere uides.</s></p><p type="main"> <s>Ex quo patet ratio eleuandi pondera magna per tra&shy;<lb/>bem, ut &agrave; latere uides.</s></p><p type="main">
  
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 <s>An &longs;it aliqua proportio, &amp; qualis inter animam, &amp; ui&shy;<lb/>tas, &amp; &longs;ua corpora con&longs;iderare.</s></p><p type="main"> <s>An &longs;it aliqua proportio, &amp; qualis inter animam, &amp; ui&shy;<lb/>tas, &amp; &longs;ua corpora con&longs;iderare.</s></p><p type="main">
  
 <s><arrow.to.target n="marg119"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg119"/></s></p><p type="margin">
  
 <s><margin.target id="marg119"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg119"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Declarauimus motum c&oelig;li e&longs;&longs;e uoluntarium, ob&longs;equente c&oelig;&shy;<lb/>lo per uirtutem in eo infu&longs;am. </s> <s>Declarauimus motum c&oelig;li e&longs;&longs;e uoluntarium, ob&longs;equente c&oelig;&shy;<lb/>lo per uirtutem in eo infu&longs;am. </s>
  
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 <s>Certum </s></p><p type="main"> <s>Certum </s></p><p type="main">
  
 <s><arrow.to.target n="marg120"></arrow.to.target><lb/>tamen e&longs;t plen&egrave; ob&longs;equi c&oelig;lum uit&aelig;, nec pror&longs;us repugnare. </s> <s><arrow.to.target n="marg120"/><lb/>tamen e&longs;t plen&egrave; ob&longs;equi c&oelig;lum uit&aelig;, nec pror&longs;us repugnare. </s>
  
 <s>So&shy;<lb/>let Ari&longs;toteli imponi, qu&ograve;d &longs;i adderetur a&longs;trum c&oelig;lo, qu&ograve;d c&oelig;lum <lb/>aut quie&longs;ceret, aut tardius moueretur: quod e&longs;t, ac &longs;i diceremus, <lb/>qu&ograve;d homo paruus &longs;i fieret maior, non e&longs;&longs;et ade&ograve; agilis, tanquam <lb/>motus ille e&longs;&longs;et ab externa cau&longs;a. </s> <s>So&shy;<lb/>let Ari&longs;toteli imponi, qu&ograve;d &longs;i adderetur a&longs;trum c&oelig;lo, qu&ograve;d c&oelig;lum <lb/>aut quie&longs;ceret, aut tardius moueretur: quod e&longs;t, ac &longs;i diceremus, <lb/>qu&ograve;d homo paruus &longs;i fieret maior, non e&longs;&longs;et ade&ograve; agilis, tanquam <lb/>motus ille e&longs;&longs;et ab externa cau&longs;a. </s>
  
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 <s>Et ita &longs;i Ari&longs;toteles illud dixi&longs;&longs;et, o&longs;tendi&longs;&longs;et magnam <lb/>imperitiam. </s> <s>Et ita &longs;i Ari&longs;toteles illud dixi&longs;&longs;et, o&longs;tendi&longs;&longs;et magnam <lb/>imperitiam. </s>
  
 <s>Ide&ograve; quale iudicium debemus facere de Alexandro, &amp; <lb/><arrow.to.target n="marg121"></arrow.to.target><lb/>Aueroe, qui hoc ei tribuunt. </s> <s>Ide&ograve; quale iudicium debemus facere de Alexandro, &amp; <lb/><arrow.to.target n="marg121"/><lb/>Aueroe, qui hoc ei tribuunt. </s>
  
 <s><expan abbr="legi&ttilde;">legitur</expan> enim in textu Arabico tale quip&shy;<lb/>piam. </s> <s><expan abbr="legi&ttilde;">legitur</expan> enim in textu Arabico tale quip&shy;<lb/>piam. </s>
  
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 <s>Dico ergo proportionem e&longs;&longs;e infinitam: nam &longs;i corpus e&longs;&shy;<lb/>&longs;et infinitum &amp; optim&egrave; di&longs;po&longs;itum infinita ui moueretur &amp; agili&shy;<lb/>tate, ut enim maius e&longs;t eo maiores uires habet.</s></p><p type="margin"> <s>Dico ergo proportionem e&longs;&longs;e infinitam: nam &longs;i corpus e&longs;&shy;<lb/>&longs;et infinitum &amp; optim&egrave; di&longs;po&longs;itum infinita ui moueretur &amp; agili&shy;<lb/>tate, ut enim maius e&longs;t eo maiores uires habet.</s></p><p type="margin">
  
 <s><margin.target id="marg120"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 27.</s></p><p type="margin"> <s><margin.target id="marg120"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 27.</s></p><p type="margin">
  
 <s><margin.target id="marg121"></margin.target>T<emph type="italics"/>ex.<emph.end type="italics"/> 71. <lb/>2. <emph type="italics"/>de<emph.end type="italics"/> C<emph type="italics"/><gap/>.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg121"/>T<emph type="italics"/>ex.<emph.end type="italics"/> 71. <lb/>2. <emph type="italics"/>de<emph.end type="italics"/> C<emph type="italics"/><gap/>.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main">
  
 <s>Propo&longs;itio quadrage&longs;ima&longs;eptima.</s></p><p type="main"> <s>Propo&longs;itio quadrage&longs;ima&longs;eptima.</s></p><p type="main">
  
 <s>Si duo mobilia &aelig;qualiter in eodem circulo iuxta proprios mo&shy;<lb/>tus moueantur, productum temporis circuituum inuicem erit &aelig;&shy;<lb/>quale producto differenti&aelig; temporum circuitus duct&aelig; in tempus <lb/>coniunctionis prim&aelig;.<lb/><arrow.to.target n="marg122"></arrow.to.target></s></p><p type="margin"> <s>Si duo mobilia &aelig;qualiter in eodem circulo iuxta proprios mo&shy;<lb/>tus moueantur, productum temporis circuituum inuicem erit &aelig;&shy;<lb/>quale producto differenti&aelig; temporum circuitus duct&aelig; in tempus <lb/>coniunctionis prim&aelig;.<lb/><arrow.to.target n="marg122"/></s></p><p type="margin">
  
 <s><margin.target id="marg122"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg122"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Sint duo mobilia a &amp; b in eodem pun&shy;<lb/><figure id="fig28"></figure><lb/>cto, qu&aelig; &aelig;qualiter uer&longs;us eandem partem <lb/>moueantur &aelig;qualibus in temporibus, inui <lb/>cem tamen in &aelig;qualiter, ita quod a in f &amp; b <lb/>in g temporibus ab&longs;oluant circulum, &amp; ho <lb/>rum differentia &longs;it h. </s> <s>Sint duo mobilia a &amp; b in eodem pun&shy;<lb/><figure id="fig28"/><lb/>cto, qu&aelig; &aelig;qualiter uer&longs;us eandem partem <lb/>moueantur &aelig;qualibus in temporibus, inui <lb/>cem tamen in &aelig;qualiter, ita quod a in f &amp; b <lb/>in g temporibus ab&longs;oluant circulum, &amp; ho <lb/>rum differentia &longs;it h. </s>
  
 <s>Dum itaque a perficit <lb/>circulum b perueniat in c, igitur c d b e&longs;t dif <lb/>ferentia, qu&aelig; &longs;uperanda e&longs;t, &amp; proportio <lb/>circuli ad b c ut g ad f, quare reliqui ad reli&shy;<lb/>quum, ut re&longs;idui ad re&longs;iduum, &longs;cilicet circu&shy;<lb/>li ad c d b, ut g ad h, &amp; b c ad c d b ut f ad h, coniungantur igitur in k <lb/>tempore, eruntque k f g h omiologa, ut productum ex circulo in b c <lb/>diui&longs;o per certam quantitatem &amp; cum circulo &amp; b c &amp; c d b diffe&shy;<lb/>rentia, &amp; &longs;it &longs;productum exfin g, dico quod diui&longs;a &longs; per h exibit k <lb/>tempus coniunctionis prim&aelig;, &longs;it itaque d locus coniunctionis, dico <lb/>igitur quod differentia &longs;patij pertran&longs;iti a b, a &amp; a, b in reditu ex con <lb/>iunctione prima ad d e&longs;t unus circulus completus, non enim po&longs;&shy;<lb/>&longs;unt e&longs;&longs;e plures, nam &longs;equeretur, qu&ograve;d a aliquando pertran&longs;i&longs;&longs;et b, <lb/>et &longs;ic non e&longs;&longs;et prima coniunctio, nec pote&longs;t e&longs;&longs;e minus, nam &longs;ic cum <lb/>a &amp; b &longs;int in d ultra perfectas circulationes uterque eorum pertran <lb/>&longs;iuit arcum b c, igitur nullo modo differentia pote&longs;t e&longs;&longs;e minor cir&shy;<lb/>culo, neque maior, ut declaratum e&longs;t, igitur e&longs;t unus circulus ad un&shy;<pb pagenum="37"/>guem. </s> <s>Dum itaque a perficit <lb/>circulum b perueniat in c, igitur c d b e&longs;t dif <lb/>ferentia, qu&aelig; &longs;uperanda e&longs;t, &amp; proportio <lb/>circuli ad b c ut g ad f, quare reliqui ad reli&shy;<lb/>quum, ut re&longs;idui ad re&longs;iduum, &longs;cilicet circu&shy;<lb/>li ad c d b, ut g ad h, &amp; b c ad c d b ut f ad h, coniungantur igitur in k <lb/>tempore, eruntque k f g h omiologa, ut productum ex circulo in b c <lb/>diui&longs;o per certam quantitatem &amp; cum circulo &amp; b c &amp; c d b diffe&shy;<lb/>rentia, &amp; &longs;it &longs;productum exfin g, dico quod diui&longs;a &longs; per h exibit k <lb/>tempus coniunctionis prim&aelig;, &longs;it itaque d locus coniunctionis, dico <lb/>igitur quod differentia &longs;patij pertran&longs;iti a b, a &amp; a, b in reditu ex con <lb/>iunctione prima ad d e&longs;t unus circulus completus, non enim po&longs;&shy;<lb/>&longs;unt e&longs;&longs;e plures, nam &longs;equeretur, qu&ograve;d a aliquando pertran&longs;i&longs;&longs;et b, <lb/>et &longs;ic non e&longs;&longs;et prima coniunctio, nec pote&longs;t e&longs;&longs;e minus, nam &longs;ic cum <lb/>a &amp; b &longs;int in d ultra perfectas circulationes uterque eorum pertran <lb/>&longs;iuit arcum b c, igitur nullo modo differentia pote&longs;t e&longs;&longs;e minor cir&shy;<lb/>culo, neque maior, ut declaratum e&longs;t, igitur e&longs;t unus circulus ad un&shy;<pb pagenum="37"/>guem. </s>
  
 <s>Hoc declarato ponatur m &longs;patium compofitum ex circulis <lb/>pertran&longs;itis a b a cum &longs;patio b d, etenim &longs;patium, quod pertran&longs;it <lb/>b a coniunctione in a, ad coniunctionem primam in d, &amp; erit ex de&shy;<lb/>mon&longs;tratis horum differentia circulus qui uocetur o, &amp; &longs;it p &longs;pa&shy;<lb/>tium, quod pertran&longs;it b in tempore eodem, in quo a pertran&longs;it o, &amp; <lb/>&longs;it q differentia o, &amp; p qu&ecedil; in circulo e&longs;t c d l b, quia igitur in eodem <lb/>tempore a pertran&longs;it m &amp; b, n, erit m ad n, ut a ad b, &amp; eadem ratio&shy;<lb/>ne a ad b, ut o ad p, igitur ex undecima quinti Euclidis m ad n, ut o <lb/>ad p, quare cum o &longs;it differentia m &amp; n, &amp; q, differentia o &amp; p erit ex <lb/>decimanona quinti Euclidis, m ad o, ut o ad q, &amp; ita circulus e&longs;t ana <lb/>logus inter &longs;patium pertran&longs;itum &agrave; motore uelociori, &amp; inter diffe&shy;<lb/>rentiam &longs;patij qu&aelig; accidit, dum uelocior motor pertran&longs;it circu&shy;<lb/>lum, id e&longs;t qu&ograve;d circulus a c d e&longs;t analogus inter c d l b, &amp; circulos <lb/>pertran&longs;itos a b a cum portione b d. </s> <s>Hoc declarato ponatur m &longs;patium compofitum ex circulis <lb/>pertran&longs;itis a b a cum &longs;patio b d, etenim &longs;patium, quod pertran&longs;it <lb/>b a coniunctione in a, ad coniunctionem primam in d, &amp; erit ex de&shy;<lb/>mon&longs;tratis horum differentia circulus qui uocetur o, &amp; &longs;it p &longs;pa&shy;<lb/>tium, quod pertran&longs;it b in tempore eodem, in quo a pertran&longs;it o, &amp; <lb/>&longs;it q differentia o, &amp; p qu&ecedil; in circulo e&longs;t c d l b, quia igitur in eodem <lb/>tempore a pertran&longs;it m &amp; b, n, erit m ad n, ut a ad b, &amp; eadem ratio&shy;<lb/>ne a ad b, ut o ad p, igitur ex undecima quinti Euclidis m ad n, ut o <lb/>ad p, quare cum o &longs;it differentia m &amp; n, &amp; q, differentia o &amp; p erit ex <lb/>decimanona quinti Euclidis, m ad o, ut o ad q, &amp; ita circulus e&longs;t ana <lb/>logus inter &longs;patium pertran&longs;itum &agrave; motore uelociori, &amp; inter diffe&shy;<lb/>rentiam &longs;patij qu&aelig; accidit, dum uelocior motor pertran&longs;it circu&shy;<lb/>lum, id e&longs;t qu&ograve;d circulus a c d e&longs;t analogus inter c d l b, &amp; circulos <lb/>pertran&longs;itos a b a cum portione b d. <!-- KEEP S--></s>
  
 <s>Reuertor igitur ad propo&longs;i&shy;<lb/>tum, cum &longs;it m ad o, ut o ad q, &amp; m ad o, ut n ad p, ex &longs;extadecima <lb/>quinti Euclidis, erit ex undecima eiu&longs;dem n ad p, ut o ad q, quare ex <lb/>&longs;extadecima &longs;exti Elementorum ducto o, id e&longs;t circulo, &longs;eu maiore <lb/>numero in p &longs;patium pertran&longs;itum a b, &longs;eu ducto fin g, &amp; diui&longs;o per <lb/>q differentiam &longs;patiorum, &longs;eu per h exibit n, &longs;eu &longs;patium quod <lb/>pertran&longs;it b ab una coniunctione ad aliam quod erat demon&shy;<lb/>&longs;trandum.</s></p><p type="main"> <s>Reuertor igitur ad propo&longs;i&shy;<lb/>tum, cum &longs;it m ad o, ut o ad q, &amp; m ad o, ut n ad p, ex &longs;extadecima <lb/>quinti Euclidis, erit ex undecima eiu&longs;dem n ad p, ut o ad q, quare ex <lb/>&longs;extadecima &longs;exti Elementorum ducto o, id e&longs;t circulo, &longs;eu maiore <lb/>numero in p &longs;patium pertran&longs;itum a b, &longs;eu ducto fin g, &amp; diui&longs;o per <lb/>q differentiam &longs;patiorum, &longs;eu per h exibit n, &longs;eu &longs;patium quod <lb/>pertran&longs;it b ab una coniunctione ad aliam quod erat demon&shy;<lb/>&longs;trandum.</s></p><p type="main">
  
 <s><arrow.to.target n="marg123"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg123"/></s></p><p type="margin">
  
 <s><margin.target id="marg123"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg123"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Ex hoc patet, quod proportio temporis coniunctionis ad tem&shy;<lb/>pus tardioris motus circuitionis e&longs;t ueluti temporis circuitus uelo <lb/>cioris motoris ad differentiam temporis motus tardioris, &amp; uelo&shy;<lb/>cioris motoris in uno circuitu.</s></p><p type="main"> <s>Ex hoc patet, quod proportio temporis coniunctionis ad tem&shy;<lb/>pus tardioris motus circuitionis e&longs;t ueluti temporis circuitus uelo <lb/>cioris motoris ad differentiam temporis motus tardioris, &amp; uelo&shy;<lb/>cioris motoris in uno circuitu.</s></p><p type="main">
  
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 <s>Si tria mobilia ex eodem puncto di&longs;cedant, fuerintque duorum, ac <lb/>duorum coniunctiones in temporibus commen&longs;is illa tria mobi&shy;<lb/>lia denu&ograve; coniungentur in tempore producto ex denominatore di <lb/>ui&longs;ionis temporis maioris per minus in minus, aut numeratore <lb/>in maius.</s></p><p type="main"> <s>Si tria mobilia ex eodem puncto di&longs;cedant, fuerintque duorum, ac <lb/>duorum coniunctiones in temporibus commen&longs;is illa tria mobi&shy;<lb/>lia denu&ograve; coniungentur in tempore producto ex denominatore di <lb/>ui&longs;ionis temporis maioris per minus in minus, aut numeratore <lb/>in maius.</s></p><p type="main">
  
 <s><arrow.to.target n="marg124"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg124"/></s></p><p type="margin">
  
 <s><margin.target id="marg124"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg124"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Sint tria mobilia a, quod circuat in duobus annis b in quinque, <lb/>c in &longs;eptem. </s> <s>Sint tria mobilia a, quod circuat in duobus annis b in quinque, <lb/>c in &longs;eptem. </s>
  
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 <s>Nam in &longs;eptuaginta annis a perficiet tri&shy;<lb/>gintaquinque reuolutiones b quatuordecim, c decem: ergo <expan abbr="redib&utilde;t">redibunt</expan> <lb/>per perfectos circuitus ad idem punctum. </s> <s>Nam in &longs;eptuaginta annis a perficiet tri&shy;<lb/>gintaquinque reuolutiones b quatuordecim, c decem: ergo <expan abbr="redib&utilde;t">redibunt</expan> <lb/>per perfectos circuitus ad idem punctum. </s>
  
 <s>O&longs;tendo modo quod <lb/>non ante: nam &longs;i &longs;ic: &longs;it, ut in trigintaquinque annis igitur b &amp; c per&shy;<lb/>ficient perfectos circuitus, ergo <expan abbr="redib&utilde;t">redibunt</expan> ad idem punctum, a autem <lb/>non redibit, quoniam eius circuitus non numerat trigintaquinque<lb/>aliter non fui&longs;&longs;et &longs;eptuaginta minimus numeratus ab a b c, cum <pb pagenum="38"/>ergo iam &longs;upponatur numerari a b &amp; c non numerabitur a b a, er&shy;<lb/>go a non perficiet circuitus, ergo non redibit ad primum <expan abbr="loc&utilde;">locum</expan>, ergo <lb/>non erit iunctus cum b &amp; c. </s> <s>O&longs;tendo modo quod <lb/>non ante: nam &longs;i &longs;ic: &longs;it, ut in trigintaquinque annis igitur b &amp; c per&shy;<lb/>ficient perfectos circuitus, ergo <expan abbr="redib&utilde;t">redibunt</expan> ad idem punctum, a autem <lb/>non redibit, quoniam eius circuitus non numerat trigintaquinque<lb/>aliter non fui&longs;&longs;et &longs;eptuaginta minimus numeratus ab a b c, cum <pb pagenum="38"/>ergo iam &longs;upponatur numerari a b &amp; c non numerabitur a b a, er&shy;<lb/>go a non perficiet circuitus, ergo non redibit ad primum <expan abbr="loc&utilde;">locum</expan>, ergo <lb/>non erit iunctus cum b &amp; c. <!-- KEEP S--></s>
  
 <s>Quod &longs;i dicas a b c coniungi in decem <lb/>&longs;eptem annis numero non numerato ab ali <lb/><figure id="fig29"></figure><lb/>quo illorum temporum, auferantur perfe&shy;<lb/>ct&aelig; circulationes, &amp; <expan abbr="remaneb&utilde;t">remanebunt</expan> dimidium <lb/>ex a, du&aelig; quint&aelig; ex b, tres &longs;eptim&aelig; ex c, igi&shy;<lb/>tur oportebit ut h&aelig; portiones &longs;int &aelig;qua&shy;<lb/>les, ut po&longs;t perfectas circulationes in idem <lb/>punctum, <expan abbr="c&otilde;ueniant">conueniant</expan>, ergo 1/2 &amp; 2/5 &amp; 3/7 &aelig;qui&shy;<lb/>ualebunt, quare proportio 7 ad 3 &amp; 5 ad 2 <lb/>&amp; 2 ad 1, e&longs;t una, quare permutando 3 ad 2 <lb/>ut 7 ad 5, &longs;ed 7 &amp; 5 &longs;unt contra &longs;e primi, ergo in &longs;ua proportione mi <lb/>nimi per dicta in &longs;eptimo Elementorum: ergo tria, &amp; duo non &longs;unt <lb/>in eadem proportione. </s> <s>Quod &longs;i dicas a b c coniungi in decem <lb/>&longs;eptem annis numero non numerato ab ali <lb/><figure id="fig29"/><lb/>quo illorum temporum, auferantur perfe&shy;<lb/>ct&aelig; circulationes, &amp; <expan abbr="remaneb&utilde;t">remanebunt</expan> dimidium <lb/>ex a, du&aelig; quint&aelig; ex b, tres &longs;eptim&aelig; ex c, igi&shy;<lb/>tur oportebit ut h&aelig; portiones &longs;int &aelig;qua&shy;<lb/>les, ut po&longs;t perfectas circulationes in idem <lb/>punctum, <expan abbr="c&otilde;ueniant">conueniant</expan>, ergo 1/2 &amp; 2/5 &amp; 3/7 &aelig;qui&shy;<lb/>ualebunt, quare proportio 7 ad 3 &amp; 5 ad 2 <lb/>&amp; 2 ad 1, e&longs;t una, quare permutando 3 ad 2 <lb/>ut 7 ad 5, &longs;ed 7 &amp; 5 &longs;unt contra &longs;e primi, ergo in &longs;ua proportione mi <lb/>nimi per dicta in &longs;eptimo Elementorum: ergo tria, &amp; duo non &longs;unt <lb/>in eadem proportione. </s>
  
 <s>Rur&longs;us dicantur conuenire in annis qua&shy;</s></p><p type="main"> <s>Rur&longs;us dicantur conuenire in annis qua&shy;</s></p><p type="main">
  
 <s><arrow.to.target n="marg125"></arrow.to.target><lb/>tuordecim cum dimidio, ergo in uiginti nouem conuenient ite&shy;<lb/>rum: ergo per &longs;ecundam partem erit &longs;eptem ad unum, ut duo ad <lb/>unum, igitur permutando unius ad unum, ut &longs;eptem ad duo, &longs;ed <lb/>unum e&longs;t &aelig;quale uni, ergo duo erunt &aelig;qualia &longs;eptem. </s> <s><arrow.to.target n="marg125"/><lb/>tuordecim cum dimidio, ergo in uiginti nouem conuenient ite&shy;<lb/>rum: ergo per &longs;ecundam partem erit &longs;eptem ad unum, ut duo ad <lb/>unum, igitur permutando unius ad unum, ut &longs;eptem ad duo, &longs;ed <lb/>unum e&longs;t &aelig;quale uni, ergo duo erunt &aelig;qualia &longs;eptem. </s>
  
 <s>Rur&longs;us dica&shy;<lb/>mus, quod in tempore annorum &lt;02&gt; quadrata decem &longs;imiliter aufe&shy;<lb/>ram integras reuolutiones, quas potero, &amp; erunt &lt;02&gt; 2 1/2 m: 1, &amp; &lt;02&gt; 2/5 &amp; <lb/>&lt;02&gt; 10/49 &aelig;qualia. </s> <s>Rur&longs;us dica&shy;<lb/>mus, quod in tempore annorum &lt;02&gt; quadrata decem &longs;imiliter aufe&shy;<lb/>ram integras reuolutiones, quas potero, &amp; erunt &lt;02&gt; 2 1/2 m: 1, &amp; &lt;02&gt; 2/5 &amp; <lb/>&lt;02&gt; 10/49 &aelig;qualia. </s>
  
 <s>Hic uides infinita &longs;equi in conuenientia, qu&aelig; longum <lb/>e&longs;&longs;et numerare, nam &longs;eptem e&longs;&longs;et &aelig;quale quinque, &amp; proportio reci&longs;i <lb/>ad potentia rethe, ut numeri ad numerum. </s> <s>Hic uides infinita &longs;equi in conuenientia, qu&aelig; longum <lb/>e&longs;&longs;et numerare, nam &longs;eptem e&longs;&longs;et &aelig;quale quinque, &amp; proportio reci&longs;i <lb/>ad potentia rethe, ut numeri ad numerum. </s>
  
 <s>Igitur non conueniunt <lb/>ante &longs;eptuaginta annos.<lb/><arrow.to.target n="marg126"></arrow.to.target></s></p><p type="margin"> <s>Igitur non conueniunt <lb/>ante &longs;eptuaginta annos.<lb/><arrow.to.target n="marg126"/></s></p><p type="margin">
  
 <s><margin.target id="marg125"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 23</s></p><p type="margin"> <s><margin.target id="marg125"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 23</s></p><p type="margin">
  
 <s><margin.target id="marg126"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main"> <s><margin.target id="marg126"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.<!-- KEEP S--></s></p><p type="main">
  
 <s>Ex hoc &longs;equitur, qu&ograve;d nullibi conuenient pr&aelig;terqu&agrave;m in eo&shy;<lb/>dem puncto, &longs;cilicet in quo ab initio coniuncti fuerunt.</s></p><p type="main"> <s>Ex hoc &longs;equitur, qu&ograve;d nullibi conuenient pr&aelig;terqu&agrave;m in eo&shy;<lb/>dem puncto, &longs;cilicet in quo ab initio coniuncti fuerunt.</s></p><p type="main">
  
 <s><arrow.to.target n="marg127"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg127"/></s></p><p type="margin">
  
 <s><margin.target id="marg127"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>m. </s> <s><margin.target id="marg127"/>C<emph type="italics"/>or<emph.end type="italics"/>m. <!-- KEEP S--></s>
  
 <s>2.</s></p><p type="main"> <s>2.<!-- KEEP S--></s></p><p type="main">
  
 <s>Sequitur denuo ex propo&longs;itione ip&longs;a repetita, &amp; primo corrola&shy;<lb/>rio, quod nullibi alibi conuenient qu&agrave;m in dato primo puncto, in <lb/>quo coniuncti fuerant ab initio etiam u&longs;que in &aelig;ternum.</s></p><p type="main"> <s>Sequitur denuo ex propo&longs;itione ip&longs;a repetita, &amp; primo corrola&shy;<lb/>rio, quod nullibi alibi conuenient qu&agrave;m in dato primo puncto, in <lb/>quo coniuncti fuerant ab initio etiam u&longs;que in &aelig;ternum.</s></p><p type="main">
  
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 <s>Propo&longs;ito mobilis in circulo circuitus tempore, dataque ratione <lb/>di&longs;tanti&aelig; ab illo mobilis circuitum inuenire, quod ex eodem pun&shy;<lb/>cto di&longs;cedens cum alio mobili in dato puncto conueniat &longs;ub quo&shy;<lb/>cunque numero circuituum tempus quoque coniunctionis.</s></p><p type="main"> <s>Propo&longs;ito mobilis in circulo circuitus tempore, dataque ratione <lb/>di&longs;tanti&aelig; ab illo mobilis circuitum inuenire, quod ex eodem pun&shy;<lb/>cto di&longs;cedens cum alio mobili in dato puncto conueniat &longs;ub quo&shy;<lb/>cunque numero circuituum tempus quoque coniunctionis.</s></p><p type="main">
  
 <s><arrow.to.target n="marg128"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg128"/></s></p><p type="margin">
  
 <s><margin.target id="marg128"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><figure></figure><p type="main"> <s><margin.target id="marg128"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><figure/><p type="main">
  
 <s>Sit in circuli peripheria a <expan abbr="p&utilde;ctus">punctus</expan>, qui cir <lb/>cuat &aelig;quali motu (hoc enim &longs;emper intel&shy;<lb/>ligitur) in b tempore: &amp; &longs;it datus punctus c <lb/>in quo di&longs;cedens e mobile ex coniunctio&shy;<lb/>ne cum a po&longs;t certos circuitus proprios, <lb/>aut etiam. </s> <s>Sit in circuli peripheria a <expan abbr="p&utilde;ctus">punctus</expan>, qui cir <lb/>cuat &aelig;quali motu (hoc enim &longs;emper intel&shy;<lb/>ligitur) in b tempore: &amp; &longs;it datus punctus c <lb/>in quo di&longs;cedens e mobile ex coniunctio&shy;<lb/>ne cum a po&longs;t certos circuitus proprios, <lb/>aut etiam. </s>
  
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 <s>nam &longs;i a o c <lb/>currit e in prima circuitione ip&longs;ius e, igitur a mouetur uelocius <lb/>qu&agrave;m e, cum ergo debeat attingere ip&longs;um e, nece&longs;&longs;e e&longs;t ut a pertran&shy;<lb/>&longs;eat prius per punctum ex quo di&longs;ce&longs;sit antequam redeat ad con&shy;<lb/>iunctionem e: ergo perficiet &longs;altem unam circuitionem. </s> <s>nam &longs;i a o c <lb/>currit e in prima circuitione ip&longs;ius e, igitur a mouetur uelocius <lb/>qu&agrave;m e, cum ergo debeat attingere ip&longs;um e, nece&longs;&longs;e e&longs;t ut a pertran&shy;<lb/>&longs;eat prius per punctum ex quo di&longs;ce&longs;sit antequam redeat ad con&shy;<lb/>iunctionem e: ergo perficiet &longs;altem unam circuitionem. </s>
  
 <s>Ducemus <lb/>ergo f in b, &amp; fiet g tempus circuitus aut circuituum a, &amp; quia &longs;pa&shy;<lb/>tium a c datum e&longs;t, &longs;it b temporis circuitus a ad h, uelut circuli to&shy;<lb/><arrow.to.target n="marg129"></arrow.to.target><lb/>tius ad a c, &amp; iungatur g cum h &amp; fiat k. </s> <s>Ducemus <lb/>ergo f in b, &amp; fiet g tempus circuitus aut circuituum a, &amp; quia &longs;pa&shy;<lb/>tium a c datum e&longs;t, &longs;it b temporis circuitus a ad h, uelut circuli to&shy;<lb/><arrow.to.target n="marg129"/><lb/>tius ad a c, &amp; iungatur g cum h &amp; fiat k. </s>
  
 <s>Fiat quoque, ut monadis <lb/>ad h, ita l ad monadem, &amp; ducatur l in k, &amp; fiat m: dico m e&longs;&longs;e tem&shy;<lb/>pus circuitus e. </s> <s>Fiat quoque, ut monadis <lb/>ad h, ita l ad monadem, &amp; ducatur l in k, &amp; fiat m: dico m e&longs;&longs;e tem&shy;<lb/>pus circuitus e. </s>
  
 <s>Con&longs;tat enim ex &longs;uppo&longs;ito, quod k e&longs;t tempus to&shy;<lb/>tum in quo a peruenit po&longs;t b circuitiones in c, &longs;i ergo e moueretur <lb/>per m tempus totum ex &longs;uppo&longs;ito perficeret circuitum, at quia cir&shy;<lb/>cuitus ad a c, ut monadis ad h, igitur etiam ut l ad monadem, ergo <lb/>proportio circuitus ad a c, ut m ad monadem: ergo &longs;i in m tran&longs;it to <lb/>tum circuitum in monade tran&longs;it a c: &longs;ed monas ducta in k facit k, <lb/>igitur e in tempore k perueniet in c, quod erat demon&longs;trandum. <lb/></s> <s>Con&longs;tat enim ex &longs;uppo&longs;ito, quod k e&longs;t tempus to&shy;<lb/>tum in quo a peruenit po&longs;t b circuitiones in c, &longs;i ergo e moueretur <lb/>per m tempus totum ex &longs;uppo&longs;ito perficeret circuitum, at quia cir&shy;<lb/>cuitus ad a c, ut monadis ad h, igitur etiam ut l ad monadem, ergo <lb/>proportio circuitus ad a c, ut m ad monadem: ergo &longs;i in m tran&longs;it to <lb/>tum circuitum in monade tran&longs;it a c: &longs;ed monas ducta in k facit k, <lb/>igitur e in tempore k perueniet in c, quod erat demon&longs;trandum. <lb/></s>
  
 <s>Proponatur modo tempus reuolutionum e ip&longs;um d: eodem mo&shy;<lb/><arrow.to.target n="marg130"></arrow.to.target><lb/>do agemus ducendo fin b fit g, addatur h &amp; fiat k, diuidatur k per <lb/>aggregatum d &amp; a e, &amp; exeat m, (idem enim e&longs;t diuidere per aggre&shy;<lb/>gatum d &amp; h, &amp; multiplicare per l) dico ergo ut in demon&longs;tratione <lb/>priore, quod m e&longs;t tempus circuitus e. </s> <s>Proponatur modo tempus reuolutionum e ip&longs;um d: eodem mo&shy;<lb/><arrow.to.target n="marg130"/><lb/>do agemus ducendo fin b fit g, addatur h &amp; fiat k, diuidatur k per <lb/>aggregatum d &amp; a e, &amp; exeat m, (idem enim e&longs;t diuidere per aggre&shy;<lb/>gatum d &amp; h, &amp; multiplicare per l) dico ergo ut in demon&longs;tratione <lb/>priore, quod m e&longs;t tempus circuitus e. </s>
  
 <s>Nam cum k &longs;it tempus, in <lb/>quo a po&longs;t circuitus f peruenit ad c, ergo diui&longs;o ip&longs;o toto tempore <pb pagenum="40"/>per numerum reuolutionum d, &amp; partem reuolutionis exibit tem&shy;<lb/>pus unius reuolutionis.</s></p><p type="margin"> <s>Nam cum k &longs;it tempus, in <lb/>quo a po&longs;t circuitus f peruenit ad c, ergo diui&longs;o ip&longs;o toto tempore <pb pagenum="40"/>per numerum reuolutionum d, &amp; partem reuolutionis exibit tem&shy;<lb/>pus unius reuolutionis.</s></p><p type="margin">
  
 <s><margin.target id="marg129"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg129"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg130"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 11. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg130"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. P<emph type="italics"/>et.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main">
  
 <s>Exemplum primi in repaul&ograve; ob&longs;curiore: &longs;it f 4 &amp; b 2 1/2 &amp; a c 4/5, du <lb/>cemus 4 in 2 1/2 fit 10, adde 4/5 6 quod e&longs;t 2 fit 12, diuide per 4/5 &longs;eu mul&shy;<lb/>tiplica per 5/4 quod idem e&longs;t, fit 15 circuitus e, in quatuor ergo circui&shy;<lb/>tibus, &amp; 4/5 qui &longs;unt duo decim anni perueniet a ad c, &amp; in duodecim <lb/>annis e perueniet ad c, nam 12 &longs;unt 4/5 ip&longs;ius 15. Similiter in &longs;ecundo <lb/>ca&longs;u &longs;it f 4 ut prius b 2 1/3 a c 1/7, ducemus 4 in 2 1/3 fit 9 1/3, addemusque h <lb/>portionem b qualis a c e&longs;t totius circuitus, id e&longs;t 1/7, e&longs;t autem 1/7 2 1/3, 1/3 <lb/>fient 9 1/3, &longs;imiliter ponatur d 5, &amp; quia a c e&longs;t 1/7 erunt 36/7, diuide ergo <lb/>9 2/3 id e&longs;t 29/3 per 36/7 exeunt 203/108 tempus reuolutionis e. </s> <s>Exemplum primi in repaul&ograve; ob&longs;curiore: &longs;it f 4 &amp; b 2 1/2 &amp; a c 4/5, du <lb/>cemus 4 in 2 1/2 fit 10, adde 4/5 6 quod e&longs;t 2 fit 12, diuide per 4/5 &longs;eu mul&shy;<lb/>tiplica per 5/4 quod idem e&longs;t, fit 15 circuitus e, in quatuor ergo circui&shy;<lb/>tibus, &amp; 4/5 qui &longs;unt duo decim anni perueniet a ad c, &amp; in duodecim <lb/>annis e perueniet ad c, nam 12 &longs;unt 4/5 ip&longs;ius 15. Similiter in &longs;ecundo <lb/>ca&longs;u &longs;it f 4 ut prius b 2 1/3 a c 1/7, ducemus 4 in 2 1/3 fit 9 1/3, addemusque h <lb/>portionem b qualis a c e&longs;t totius circuitus, id e&longs;t 1/7, e&longs;t autem 1/7 2 1/3, 1/3 <lb/>fient 9 1/3, &longs;imiliter ponatur d 5, &amp; quia a c e&longs;t 1/7 erunt 36/7, diuide ergo <lb/>9 2/3 id e&longs;t 29/3 per 36/7 exeunt 203/108 tempus reuolutionis e. </s>
  
 <s>Quin que ergo <lb/>reuolutiones e erunt 1015/108 addita &longs;eptima parte, qu&aelig; e&longs;t 29/108 fient 2044/108 <lb/>&longs;eu 261/27, &amp; &longs;unt anni 9 18/27 &longs;eu 9 2/3, ergo in tanto tempore a faciet qua&shy;<lb/>tuor circuitus, &amp; &longs;eptimam partem, &amp; e quinque circuitus, &amp; &longs;e&shy;<lb/>ptimam.<lb/><arrow.to.target n="marg131"></arrow.to.target></s></p><p type="margin"> <s>Quin que ergo <lb/>reuolutiones e erunt 1015/108 addita &longs;eptima parte, qu&aelig; e&longs;t 29/108 fient 2044/108 <lb/>&longs;eu 261/27, &amp; &longs;unt anni 9 18/27 &longs;eu 9 2/3, ergo in tanto tempore a faciet qua&shy;<lb/>tuor circuitus, &amp; &longs;eptimam partem, &amp; e quinque circuitus, &amp; &longs;e&shy;<lb/>ptimam.<lb/><arrow.to.target n="marg131"/></s></p><p type="margin">
  
 <s><margin.target id="marg131"></margin.target>C<emph type="italics"/><gap/><emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg131"/>C<emph type="italics"/><gap/><emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Ex hoc patet, quod non coniungentur in alio loco, neque alio tem <lb/>pore ante pr&aelig;dictum tempus.</s></p><p type="main"> <s>Ex hoc patet, quod non coniungentur in alio loco, neque alio tem <lb/>pore ante pr&aelig;dictum tempus.</s></p><p type="main">
  
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 <s>Omnes circuituum portiones in eiu&longs;dem temporibus <expan abbr="repetun&ttilde;">repetuntur</expan>.</s></p><p type="main"> <s>Omnes circuituum portiones in eiu&longs;dem temporibus <expan abbr="repetun&ttilde;">repetuntur</expan>.</s></p><p type="main">
  
 <s>Sint in circulo a b c d e f g: a &amp; b iuncta, &amp; in primo congre&longs;&longs;u <lb/>iungantur in c, in &longs;ecundo in d, in tertio in e, in quarto in f, in quinto <lb/>in g, in &longs;exto in h, in &longs;eptimo in k, in octauo in l. </s> <s>Sint in circulo a b c d e f g: a &amp; b iuncta, &amp; in primo congre&longs;&longs;u <lb/>iungantur in c, in &longs;ecundo in d, in tertio in e, in quarto in f, in quinto <lb/>in g, in &longs;exto in h, in &longs;eptimo in k, in octauo in l. <!-- KEEP S--></s>
  
 <s>Et &longs;ic deinceps <expan abbr="c&utilde;quetempora">cunque<lb/>tempora</expan> &longs;int &aelig;qualia, erunt &amp; circuitus totidem numero, &amp; exce&longs;&shy;<lb/>&longs;us &aelig;quales etiam a c, c d, d e, e f, f g, g h, h k, <lb/><figure id="fig30"></figure><lb/>k l. </s> <s>Et &longs;ic deinceps <expan abbr="c&utilde;quetempora">cunque<lb/>tempora</expan> &longs;int &aelig;qualia, erunt &amp; circuitus totidem numero, &amp; exce&longs;&shy;<lb/>&longs;us &aelig;quales etiam a c, c d, d e, e f, f g, g h, h k, <lb/><figure id="fig30"/><lb/>k l. <!-- KEEP S--></s>
  
 <s>Et &longs;i aggregatum a &longs;cilicet circulorum, <lb/>&amp; portionis fuerit commen&longs;um circulo, &amp; <lb/>ita de b erunt omnia <expan abbr="c&otilde;men&longs;a">commen&longs;a</expan> ad circulum, </s></p><p type="main"> <s>Et &longs;i aggregatum a &longs;cilicet circulorum, <lb/>&amp; portionis fuerit commen&longs;um circulo, &amp; <lb/>ita de b erunt omnia <expan abbr="c&otilde;men&longs;a">commen&longs;a</expan> ad circulum, </s></p><p type="main">
  
 <s><arrow.to.target n="marg132"></arrow.to.target><lb/>&amp; etiam inter &longs;e. </s> <s><arrow.to.target n="marg132"/><lb/>&amp; etiam inter &longs;e. </s>
  
 <s>Et &longs;i inter &longs;e aggregata, uel <lb/>portiones erunt, &amp; eodem modo reliqua. <lb/></s> <s>Et &longs;i inter &longs;e aggregata, uel <lb/>portiones erunt, &amp; eodem modo reliqua. <lb/></s>
  
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 <s>Erunt tamen incommen&longs;a nece&longs;&longs;ari&ograve;, &longs;i partes fuerint <lb/>incommen&longs;&aelig; toti. </s> <s>Erunt tamen incommen&longs;a nece&longs;&longs;ari&ograve;, &longs;i partes fuerint <lb/>incommen&longs;&aelig; toti. </s>
  
 <s>Ponatur a c incommen&longs;a toti circulo dico, quod <lb/>a k <expan abbr="eti&atilde;">etiam</expan> e&longs;t incommen&longs;a toti circulo: &amp; <expan abbr="eti&atilde;">etiam</expan> a k, &amp; k c. </s> <s>Ponatur a c incommen&longs;a toti circulo dico, quod <lb/>a k <expan abbr="eti&atilde;">etiam</expan> e&longs;t incommen&longs;a toti circulo: &amp; <expan abbr="eti&atilde;">etiam</expan> a k, &amp; k c. <!-- KEEP S--></s>
  
 <s>Quia enim a c <lb/>e&longs;t incommen&longs;a circulo, &amp; k a cum toto circulo &longs;emel e&longs;t commen&shy;<pb pagenum="41"/>&longs;a a c, quia multiplex ei. </s> <s>Quia enim a c <lb/>e&longs;t incommen&longs;a circulo, &amp; k a cum toto circulo &longs;emel e&longs;t commen&shy;<pb pagenum="41"/>&longs;a a c, quia multiplex ei. </s>
  
 <s>igitur cum circulus, &amp; a k diuidantur in cir&shy;<lb/><arrow.to.target n="marg133"></arrow.to.target><lb/>culum et a k, &amp; circulus &longs;it incommen&longs;us circulo, cum a k erit aggre. <lb/></s> <s>igitur cum circulus, &amp; a k diuidantur in cir&shy;<lb/><arrow.to.target n="marg133"/><lb/>culum et a k, &amp; circulus &longs;it incommen&longs;us circulo, cum a k erit aggre. <lb/></s>
  
 <s>gatum ex circulo, &amp; a k incommen&longs;um ip&longs;i a k, &amp; a k pariter incom <lb/><arrow.to.target n="marg134"></arrow.to.target><lb/>men&longs;a circulo. </s> <s>gatum ex circulo, &amp; a k incommen&longs;um ip&longs;i a k, &amp; a k pariter incom <lb/><arrow.to.target n="marg134"/><lb/>men&longs;a circulo. </s>
  
 <s>Rur&longs;us quia a k e&longs;t incommen&longs;a circulo cum a k, &amp; <lb/>circulus cum a k &longs;it multiplex ad a c, erit a k incommen&longs;a a c, quare <lb/><arrow.to.target n="marg135"></arrow.to.target><lb/>erit c k incommen&longs;a a k &amp; a c, &amp; circulo ad dita a k. </s> <s>Rur&longs;us quia a k e&longs;t incommen&longs;a circulo cum a k, &amp; <lb/>circulus cum a k &longs;it multiplex ad a c, erit a k incommen&longs;a a c, quare <lb/><arrow.to.target n="marg135"/><lb/>erit c k incommen&longs;a a k &amp; a c, &amp; circulo ad dita a k. </s>
  
 <s>Si ergo a c &longs;it <lb/>commen&longs;a circulo, erunt omnes portiones e genere numeri, &amp; &longs;i <lb/><arrow.to.target n="marg136"></arrow.to.target><lb/>potentia rhete erunt omnes, uel potentia rhete, uel circulis detra&shy;<lb/>ctis, ut a k &amp; a l reci&longs;a: &amp; a c &longs;it potentia &longs;ecunda rhete, id e&longs;t radix cu <lb/>bica erunt omnes c d, d e, e f, potentia &longs;ecunda rhete, et radices cubi&shy;<lb/>c&aelig; numeri, &longs;eu latera corporum rhete, a k uero &amp; a l, &amp; huiu&longs;modi <lb/>in infinitum reci&longs;a potentia rhete.<lb/><arrow.to.target n="marg137"></arrow.to.target></s></p><p type="margin"> <s>Si ergo a c &longs;it <lb/>commen&longs;a circulo, erunt omnes portiones e genere numeri, &amp; &longs;i <lb/><arrow.to.target n="marg136"/><lb/>potentia rhete erunt omnes, uel potentia rhete, uel circulis detra&shy;<lb/>ctis, ut a k &amp; a l reci&longs;a: &amp; a c &longs;it potentia &longs;ecunda rhete, id e&longs;t radix cu <lb/>bica erunt omnes c d, d e, e f, potentia &longs;ecunda rhete, et radices cubi&shy;<lb/>c&aelig; numeri, &longs;eu latera corporum rhete, a k uero &amp; a l, &amp; huiu&longs;modi <lb/>in infinitum reci&longs;a potentia rhete.<lb/><arrow.to.target n="marg137"/></s></p><p type="margin">
  
 <s><margin.target id="marg132"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> C<emph type="italics"/>or<emph.end type="italics"/>^{m}. <lb/><emph type="italics"/>pr&aelig;cedentis.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg132"/>P<emph type="italics"/>er<emph.end type="italics"/> C<emph type="italics"/>or<emph.end type="italics"/>^{m}. <lb/><emph type="italics"/>pr&aelig;cedentis.<emph.end type="italics"/></s></p><p type="margin">
  
 <s><margin.target id="marg133"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 14. <emph type="italics"/>deci <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg133"/>P<emph type="italics"/>er<emph.end type="italics"/> 14. <emph type="italics"/>deci <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg134"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 17. <lb/><emph type="italics"/>eiu&longs;dem.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg134"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <lb/><emph type="italics"/>eiu&longs;dem.<emph.end type="italics"/></s></p><p type="margin">
  
 <s><margin.target id="marg135"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 14. <lb/><emph type="italics"/>rur&longs;us.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg135"/>P<emph type="italics"/>er<emph.end type="italics"/> 14. <lb/><emph type="italics"/>rur&longs;us.<emph.end type="italics"/></s></p><p type="margin">
  
 <s><margin.target id="marg136"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 17. <lb/><emph type="italics"/>rur&longs;us.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg136"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <lb/><emph type="italics"/>rur&longs;us.<emph.end type="italics"/></s></p><p type="margin">
  
 <s><margin.target id="marg137"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg137"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Ex hoc patet, quod cum circulus po&longs;sit diuidi in infinita gene&shy;</s></p><p type="main"> <s>Ex hoc patet, quod cum circulus po&longs;sit diuidi in infinita gene&shy;</s></p><p type="main">
  
 <s><arrow.to.target n="marg138"></arrow.to.target><lb/>ra quantitatum, qu&aelig; non &longs;unt inuicem commen&longs;&aelig; cumque coniun&shy;<lb/>ctiones h&aelig; &longs;emper in eodem genere maneant, quod infinita pun&shy;<lb/>cta, &amp; infinitis in &longs;peciebus quantitatum remanebunt in quibus a <lb/>&amp; b in perpetuum nunquam conuenient. </s> <s><arrow.to.target n="marg138"/><lb/>ra quantitatum, qu&aelig; non &longs;unt inuicem commen&longs;&aelig; cumque coniun&shy;<lb/>ctiones h&aelig; &longs;emper in eodem genere maneant, quod infinita pun&shy;<lb/>cta, &amp; infinitis in &longs;peciebus quantitatum remanebunt in quibus a <lb/>&amp; b in perpetuum nunquam conuenient. </s>
  
 <s>Velut &longs;i coniunctio pri&shy;<lb/>ma fiat in &lt;02&gt; cu. </s> <s>Velut &longs;i coniunctio pri&shy;<lb/>ma fiat in &lt;02&gt; cu. </s>
  
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 <s>4, &amp; &longs;ic <lb/>de alijs.</s></p><p type="margin"> <s>4, &amp; &longs;ic <lb/>de alijs.</s></p><p type="margin">
  
 <s><margin.target id="marg138"></margin.target>P<emph type="italics"/>er penulti&shy;<lb/>mam uige&longs;i&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg138"/>P<emph type="italics"/>er penulti&shy;<lb/>mam uige&longs;i&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main">
  
 <s>Propo&longs;itio quinquage&longs;imaprima.</s></p><p type="main"> <s>Propo&longs;itio quinquage&longs;imaprima.</s></p><p type="main">
  
 <s>Operationes dictas exemplo declarare.<lb/><arrow.to.target n="marg139"></arrow.to.target></s></p><p type="margin"> <s>Operationes dictas exemplo declarare.<lb/><arrow.to.target n="marg139"/></s></p><p type="margin">
  
 <s><margin.target id="marg139"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg139"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Supponamus in circulo pr&aelig;dicto a c &lt;02&gt; 7 con&longs;tat, quod e&longs;&longs;e non <lb/>pote&longs;t, quia &lt;02&gt; 7 e&longs;t maior monade, ideo toto circulo, quare non po<lb/>terit e&longs;&longs;e pars circuli, &longs;ed referetur ad <expan abbr="quantitat&etilde;">quantitatem</expan> certam, uelut quod <lb/>circulus &longs;it 10. &longs;emper ergo diuidemus &lt;02&gt; 7, &longs;eu eam portionem per <lb/>10 quantitatem circuli &amp; exibit &lt;02&gt; 7/100, &amp; h&aelig;c erit portio circuli, &amp; ita <lb/>&longs;i portio &longs;it &lt;02&gt; cub. </s> <s>Supponamus in circulo pr&aelig;dicto a c &lt;02&gt; 7 con&longs;tat, quod e&longs;&longs;e non <lb/>pote&longs;t, quia &lt;02&gt; 7 e&longs;t maior monade, ideo toto circulo, quare non po<lb/>terit e&longs;&longs;e pars circuli, &longs;ed referetur ad <expan abbr="quantitat&etilde;">quantitatem</expan> certam, uelut quod <lb/>circulus &longs;it 10. &longs;emper ergo diuidemus &lt;02&gt; 7, &longs;eu eam portionem per <lb/>10 quantitatem circuli &amp; exibit &lt;02&gt; 7/100, &amp; h&aelig;c erit portio circuli, &amp; ita <lb/>&longs;i portio &longs;it &lt;02&gt; cub. </s>
  
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 <s>Velut &lt;02&gt; 7/100 quater &longs;um&shy;<lb/>pta efficit &lt;02&gt; 112/100. Et hoc e&longs;t potentia rhete, &longs;ed &longs;i quis auferat mona&shy;<lb/>dem fiet &lt;02&gt; 112/100 m: 1, &amp; hoc e&longs;t reci&longs;um 1, &longs;cilicet 1 p: &lt;02&gt; v: 23/25 m: &lt;02&gt; 28/25, &longs;ed ta <lb/>men uer&egrave; e&longs;t linea media.</s></p><p type="main"> <s>Velut &lt;02&gt; 7/100 quater &longs;um&shy;<lb/>pta efficit &lt;02&gt; 112/100. Et hoc e&longs;t potentia rhete, &longs;ed &longs;i quis auferat mona&shy;<lb/>dem fiet &lt;02&gt; 112/100 m: 1, &amp; hoc e&longs;t reci&longs;um 1, &longs;cilicet 1 p: &lt;02&gt; v: 23/25 m: &lt;02&gt; 28/25, &longs;ed ta <lb/>men uer&egrave; e&longs;t linea media.</s></p><p type="main">
  
 <s>Quod uer&ograve; non contingat coniungi in alio loco, neque tem&shy;<lb/>pore &longs;it, ut a b iungantur in c, &amp; &longs;it reuolutio a triplex integra, &amp; b <pb pagenum="42"/>&longs;excuplex, &amp; tempus totum decem annorum: ita ut a c &longs;it tertia <lb/>pars circuitus, &amp; a circuitus tres anni, &amp; quia circuitus b &longs;unt fex <lb/>cum tertia, diuidemus decem per 6 1/3 exit <lb/>1 11/29, dico quod non prius, neque in alio <lb/><figure id="fig31"></figure><lb/>puncto. </s> <s>Quod uer&ograve; non contingat coniungi in alio loco, neque tem&shy;<lb/>pore &longs;it, ut a b iungantur in c, &amp; &longs;it reuolutio a triplex integra, &amp; b <pb pagenum="42"/>&longs;excuplex, &amp; tempus totum decem annorum: ita ut a c &longs;it tertia <lb/>pars circuitus, &amp; a circuitus tres anni, &amp; quia circuitus b &longs;unt fex <lb/>cum tertia, diuidemus decem per 6 1/3 exit <lb/>1 11/29, dico quod non prius, neque in alio <lb/><figure id="fig31"/><lb/>puncto. </s>
  
 <s>Si enim prim&ugrave;m in eodem pun&shy;<lb/>cto, &amp;, gratia exempli, in quatuor annis <lb/>congruit enim, &amp; b dicamus quod per&shy;<lb/>egerit duas reuolutiones cum tertia, hoc <lb/>enim e&longs;t nece&longs;&longs;arium, &longs;i debet perueni&shy;<lb/>re ad c, &amp; erunt anni tres, &amp; 23/19, non ergo <lb/>anni quatuor. </s> <s>Si enim prim&ugrave;m in eodem pun&shy;<lb/>cto, &amp;, gratia exempli, in quatuor annis <lb/>congruit enim, &amp; b dicamus quod per&shy;<lb/>egerit duas reuolutiones cum tertia, hoc <lb/>enim e&longs;t nece&longs;&longs;arium, &longs;i debet perueni&shy;<lb/>re ad c, &amp; erunt anni tres, &amp; 23/19, non ergo <lb/>anni quatuor. </s>
  
 <s>Cum enim tempora di&shy;<lb/>uer&longs;a diuiduntur per numeros haben&shy;<lb/>tes proportionem erunt, qui prodeunt <lb/><arrow.to.target n="table13"></arrow.to.target><lb/>numeri in eadem ratione. </s> <s>Cum enim tempora di&shy;<lb/>uer&longs;a diuiduntur per numeros haben&shy;<lb/>tes proportionem erunt, qui prodeunt <lb/><arrow.to.target n="table13"/><lb/>numeri in eadem ratione. </s>
  
 <s>Diui&longs;o ergo <lb/>10 per 1 11/19 exit 6 2/3, &amp; diui&longs;o 4 per 1 11/19 exit <lb/>2 8/15, igitur 6 1/3 ad 2 8/15, ut 10 ad 4, igitur 8/25 <lb/>non pote&longs;t e&longs;&longs;e &aelig;quale 1/3. Si enim per <lb/>pr&aelig;cedentem repetuntur, ergo non po&longs;&shy;<lb/>&longs;unt redire, doneciterum coniung antur in ip&longs;o a. </s> <s>Diui&longs;o ergo <lb/>10 per 1 11/19 exit 6 2/3, &amp; diui&longs;o 4 per 1 11/19 exit <lb/>2 8/15, igitur 6 1/3 ad 2 8/15, ut 10 ad 4, igitur 8/25 <lb/>non pote&longs;t e&longs;&longs;e &aelig;quale 1/3. Si enim per <lb/>pr&aelig;cedentem repetuntur, ergo non po&longs;&shy;<lb/>&longs;unt redire, doneciterum coniung antur in ip&longs;o a. </s>
  
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 <s>Sin uer&ograve; coniunctio fiat in d, igitur per pr&aelig;cedentem d e e&longs;t <lb/>pars a c &longs;ubmultiplex quomodolibet, quare non fuerunt a&longs;&longs;um&shy;<lb/>pti primi numeri. </s> <s>Sin uer&ograve; coniunctio fiat in d, igitur per pr&aelig;cedentem d e e&longs;t <lb/>pars a c &longs;ubmultiplex quomodolibet, quare non fuerunt a&longs;&longs;um&shy;<lb/>pti primi numeri. </s>
  
 <s>Veluti in exemplo con&longs;tituimus, quod a, &amp; b <lb/>conueniunt in c in decem annis, &amp; a c e&longs;t tertia pars circuitus: er&shy;<lb/>go in triginta annis conueniunt in a, &amp; in quadraginta rur&longs;us in c. <lb/></s> <s>Veluti in exemplo con&longs;tituimus, quod a, &amp; b <lb/>conueniunt in c in decem annis, &amp; a c e&longs;t tertia pars circuitus: er&shy;<lb/>go in triginta annis conueniunt in a, &amp; in quadraginta rur&longs;us in c. <lb/><!-- REMOVE S-->&longs;i ergo quis a&longs;&longs;ump&longs;i&longs;&longs;et quadraginta annos ab initio pro con&shy;<lb/>gre&longs;&longs;u, &amp; diui&longs;i&longs;&longs;et per 1 12/19 exiret 25 1/3, &amp; &longs;i per 3 exiret 13 1/3, &amp; mani&shy;<lb/>fe&longs;tum e&longs;t, quod uterque numerus pote&longs;t diuidi per eundem nu&shy;<lb/>merum, utpote 4 &amp; exit numerus cum eadem parte &longs;cilicet 6 1/3 &amp; <lb/>3 1/3 ergo conuenient ante, non ergo a&longs;&longs;ump&longs;i&longs;ti minimos in ea pro&shy;<lb/>portione. </s>
  
 <s>&longs;i ergo quis a&longs;&longs;ump&longs;i&longs;&longs;et quadraginta annos ab initio pro con&shy;<lb/>gre&longs;&longs;u, &amp; diui&longs;i&longs;&longs;et per 1 12/19 exiret 25 1/3, &amp; &longs;i per 3 exiret 13 1/3, &amp; mani&shy;<lb/>fe&longs;tum e&longs;t, quod uterque numerus pote&longs;t diuidi per eundem nu&shy;<lb/>merum, utpote 4 &amp; exit numerus cum eadem parte &longs;cilicet 6 1/3 &amp; <lb/>3 1/3 ergo conuenient ante, non ergo a&longs;&longs;ump&longs;i&longs;ti minimos in ea pro&shy;<lb/>portione. </s> 
  
 <s>Illi autem nequaquam amplius diuidi non po&longs;&longs;unt eo&shy;<lb/>dem modo.</s></p><table><table.target id="table13"></table.target><row><cell>Decem</cell><cell></cell><cell>Quatuor</cell><cell></cell></row><row><cell>3</cell><cell>3 1/3</cell><cell>1 11/19</cell><cell>2/(<gap/>/2<gap/>)</cell></row><row><cell>1 11/19</cell><cell>6 1/3</cell><cell></cell><cell></cell></row></table><p type="main"> 
  <s>Illi autem nequaquam amplius diuidi non po&longs;&longs;unt eo&shy;<lb/>dem modo.</s></p><table><table.target id="table13"/><row><cell>Decem</cell><cell/><cell>Quatuor</cell><cell/></row><row><cell>3</cell><cell>3 1/3</cell><cell>1 11/19</cell><cell>2/(<gap/>/2<gap/>)</cell></row><row><cell>1 11/19</cell><cell>6 1/3</cell><cell/><cell/></row></table><p type="main">
  
 <s>Propo&longs;itio quinquage&longs;ima&longs;ecunda.</s></p><p type="main"> <s>Propo&longs;itio quinquage&longs;ima&longs;ecunda.</s></p><p type="main">
  
 <s>Tria mobilia coniuncta in eodem puncto, quorum duo, &amp; duo <lb/>conueniant in partibus in commen&longs;is inter &longs;e, in perpetuum in nul&shy;<lb/>lo unquam puncto conuenient.</s></p><p type="main"> <s>Tria mobilia coniuncta in eodem puncto, quorum duo, &amp; duo <lb/>conueniant in partibus in commen&longs;is inter &longs;e, in perpetuum in nul&shy;<lb/>lo unquam puncto conuenient.</s></p><p type="main">
  
 <s><arrow.to.target n="marg140"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg140"/></s></p><p type="margin">
  
 <s><margin.target id="marg140"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg140"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Sint a b c iuncta, &amp; primo iungantur a &amp; b, iterum in d &amp; b, &amp; <lb/>c in e, &amp; &longs;int a d, a e inconimen&longs;&aelig;, dico qu&ograve;d a b c nunquam con&shy;<lb/>uenient in aliquo puncto, &longs;eu primo, &longs;eu alio &agrave; prim o: &longs;i non con&shy;<pb pagenum="43"/><figure id="fig32"></figure><lb/>ueniant in f, erunt ergo in g tempore re&shy;<lb/>uolutiones integr&aelig;, &amp; portio a f in&longs;uper. <lb/></s> <s>Sint a b c iuncta, &amp; primo iungantur a &amp; b, iterum in d &amp; b, &amp; <lb/>c in e, &amp; &longs;int a d, a e inconimen&longs;&aelig;, dico qu&ograve;d a b c nunquam con&shy;<lb/>uenient in aliquo puncto, &longs;eu primo, &longs;eu alio &agrave; prim o: &longs;i non con&shy;<pb pagenum="43"/><figure id="fig32"/><lb/>ueniant in f, erunt ergo in g tempore re&shy;<lb/>uolutiones integr&aelig;, &amp; portio a f in&longs;uper. <lb/></s>
  
 <s>Et quia h&aelig; con&longs;tituuntur per congre&longs;&longs;us <lb/>b cum a, &amp; &longs;unt &longs;patia a d, &amp; b cum c, &amp; <lb/>&longs;unt &longs;patia e f, igitur &longs;patium a f erit ex ge&shy;<lb/>nere quantitatis a d, &amp; a e per quinqua&shy;<lb/>ge&longs;imam, harum ergo erunt commen&longs;&aelig;: <lb/>quod e&longs;t contra &longs;uppo&longs;itum. </s> <s>Et quia h&aelig; con&longs;tituuntur per congre&longs;&longs;us <lb/>b cum a, &amp; &longs;unt &longs;patia a d, &amp; b cum c, &amp; <lb/>&longs;unt &longs;patia e f, igitur &longs;patium a f erit ex ge&shy;<lb/>nere quantitatis a d, &amp; a e per quinqua&shy;<lb/>ge&longs;imam, harum ergo erunt commen&longs;&aelig;: <lb/>quod e&longs;t contra &longs;uppo&longs;itum. </s>
  
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 <s><expan abbr="Circulor&utilde;">Circulorum</expan> &longs;e in aduer&longs;um mouentium proportionem declarare.</s></p><p type="main"> <s><expan abbr="Circulor&utilde;">Circulorum</expan> &longs;e in aduer&longs;um mouentium proportionem declarare.</s></p><p type="main">
  
 <s><arrow.to.target n="marg141"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg141"/></s></p><p type="margin">
  
 <s><margin.target id="marg141"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg141"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Sit orbis a b cuius cen&shy;<lb/><figure id="fig33"></figure><lb/>centrum c, manubrium c <lb/>d f e, &longs;eu uero tangat circu <lb/>lum g, &longs;eu more gemmas <lb/>&longs;culpentium aligetur al&shy;<lb/>teri orbi funiculo a l b, &amp; <lb/>&longs;it in uertice axis k m or&shy;<lb/>biculus &longs;olidus aut &longs;emi&shy;<lb/>circulari forma m, dico <lb/>quod proportio motus a <lb/>b ad motum m e&longs;t produ <lb/>cta ex duabus proportio&shy;<lb/>nibus c n <expan abbr="&longs;emidimeti&etilde;tis">&longs;emidimetientis</expan>, <lb/>&amp; &longs;emidimetientis m ad k <lb/>o, quare ut rectanguli c n <lb/>in dimidium dimetientis <lb/>m ad quadratum o, ut enim a b ad ol orbem, id e&longs;t <expan abbr="peripheriar&utilde;">peripheriarum</expan> ita <lb/>c n ad o k, quoniam o l mouetur toties in una circuitione a b, quo&shy;<lb/>ties <expan abbr="peripheri&atilde;">peripheriam</expan> o l <expan abbr="contine&ttilde;">continetur</expan> in peripheria a b, ergo quoties o k con&shy;<lb/>tinetur in c n toties in una circuitione a b o l circumuertitur, &longs;ed <lb/>quoties circumuertitur ol, toties etiam m, quia uterque mouetur eo&shy;<lb/>dem circuitu k m axis, ergo quoties m circumducitur in circuitu a <lb/>b toties o k continetur in c n, ergo &longs;i fiat comparatio &longs;emidiametri <lb/>m ad c n, erit product a proportio circuitus a b ad circuitum m ex <lb/>proportione c n ad o k, et &longs;emidimetientis m ad <expan abbr="id&etilde;">idem</expan> o k, ergo per 26 <lb/>proportio numeri circuitus unius p <expan abbr="alter&utilde;">alterum</expan> e&longs;t, ut rectanguli &longs;ub c n, <lb/>&amp; &longs;emidimetiente m ad quadratum k o, quod erat <expan abbr="demon&longs;trand&utilde;">demon&longs;trandum</expan>.</s></p><p type="main"> <s>Sit orbis a b cuius cen&shy;<lb/><figure id="fig33"/><lb/>centrum c, manubrium c <lb/>d f e, &longs;eu uero tangat circu <lb/>lum g, &longs;eu more gemmas <lb/>&longs;culpentium aligetur al&shy;<lb/>teri orbi funiculo a l b, &amp; <lb/>&longs;it in uertice axis k m or&shy;<lb/>biculus &longs;olidus aut &longs;emi&shy;<lb/>circulari forma m, dico <lb/>quod proportio motus a <lb/>b ad motum m e&longs;t produ <lb/>cta ex duabus proportio&shy;<lb/>nibus c n <expan abbr="&longs;emidimeti&etilde;tis">&longs;emidimetientis</expan>, <lb/>&amp; &longs;emidimetientis m ad k <lb/>o, quare ut rectanguli c n <lb/>in dimidium dimetientis <lb/>m ad quadratum o, ut enim a b ad ol orbem, id e&longs;t <expan abbr="peripheriar&utilde;">peripheriarum</expan> ita <lb/>c n ad o k, quoniam o l mouetur toties in una circuitione a b, quo&shy;<lb/>ties <expan abbr="peripheri&atilde;">peripheriam</expan> o l <expan abbr="contine&ttilde;">continetur</expan> in peripheria a b, ergo quoties o k con&shy;<lb/>tinetur in c n toties in una circuitione a b o l circumuertitur, &longs;ed <lb/>quoties circumuertitur ol, toties etiam m, quia uterque mouetur eo&shy;<lb/>dem circuitu k m axis, ergo quoties m circumducitur in circuitu a <lb/>b toties o k continetur in c n, ergo &longs;i fiat comparatio &longs;emidiametri <lb/>m ad c n, erit product a proportio circuitus a b ad circuitum m ex <lb/>proportione c n ad o k, et &longs;emidimetientis m ad <expan abbr="id&etilde;">idem</expan> o k, ergo per 26 <lb/>proportio numeri circuitus unius p <expan abbr="alter&utilde;">alterum</expan> e&longs;t, ut rectanguli &longs;ub c n, <lb/>&amp; &longs;emidimetiente m ad quadratum k o, quod erat <expan abbr="demon&longs;trand&utilde;">demon&longs;trandum</expan>.</s></p><p type="main">
  
 <s>Manife&longs;tum e&longs;t autem ex ip&longs;a &longs;ola con&longs;titutione, quod &longs;i a b mo&shy;</s></p><p type="main"> <s>Manife&longs;tum e&longs;t autem ex ip&longs;a &longs;ola con&longs;titutione, quod &longs;i a b mo&shy;</s></p><p type="main">
  
 <s><arrow.to.target n="marg142"></arrow.to.target><pb pagenum="44"/>uetur &longs;ur&longs;um &agrave; dextro in &longs;ini&longs;trum in inferiore parte, mouebitur &agrave; <lb/>&longs;ini&longs;tro in dextrum, &amp; uterque circulorum g &amp; k in &longs;uperiore parte, <lb/>&amp; in inferiore mouebitur contrario motu, &longs;cilicet in &longs;uperiore &agrave; &longs;ini <lb/>&longs;tro in dextrum, &amp; inferiore &agrave; dextro in &longs;ini&longs;trum, illi uer&ograve; duo or&shy;<lb/>bes &longs;imili motu mouebuntur tam in parte &longs;uperiore, qu&agrave;m inferio&shy;<lb/>re, &amp; proportio motuum eorum inter &longs;e erit uelut dimetientium <lb/>corundem.<lb/><arrow.to.target n="marg143"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg142"/><pb pagenum="44"/>uetur &longs;ur&longs;um &agrave; dextro in &longs;ini&longs;trum in inferiore parte, mouebitur &agrave; <lb/>&longs;ini&longs;tro in dextrum, &amp; uterque circulorum g &amp; k in &longs;uperiore parte, <lb/>&amp; in inferiore mouebitur contrario motu, &longs;cilicet in &longs;uperiore &agrave; &longs;ini <lb/>&longs;tro in dextrum, &amp; inferiore &agrave; dextro in &longs;ini&longs;trum, illi uer&ograve; duo or&shy;<lb/>bes &longs;imili motu mouebuntur tam in parte &longs;uperiore, qu&agrave;m inferio&shy;<lb/>re, &amp; proportio motuum eorum inter &longs;e erit uelut dimetientium <lb/>corundem.<lb/><arrow.to.target n="marg143"/></s></p><p type="margin">
  
 <s><margin.target id="marg142"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="margin"> <s><margin.target id="marg142"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.<!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg143"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main"> <s><margin.target id="marg143"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.<!-- KEEP S--></s></p><p type="main">
  
 <s>Rur&longs;us cum a b circumuertatur cum manubrio c d f e, tanto uelo <lb/>cius circumuertetur, &amp; in ea proportione, qua d f continetur in c n, <lb/>&amp; in eodem tempore, in quo manubrium circumuertitur in eodem <lb/>axis circumuertitur, &amp; orbis, ut dictum e&longs;t, ergo in eodem tempo&shy;<lb/>re, in quo axis circumuertitur in eodem orbis: ergo tanto tardius <lb/>uidebitur moueri axis ip&longs;o orbe, quanta e&longs;t proportio minoris in <lb/>&aelig;qualitatis ip&longs;ius axis, &longs;eu ambitus, &longs;eu &longs;emidimetientis ad ambi&shy;<lb/>tum, &longs;eu &longs;emidimetientem orbis.</s></p><p type="main"> <s>Rur&longs;us cum a b circumuertatur cum manubrio c d f e, tanto uelo <lb/>cius circumuertetur, &amp; in ea proportione, qua d f continetur in c n, <lb/>&amp; in eodem tempore, in quo manubrium circumuertitur in eodem <lb/>axis circumuertitur, &amp; orbis, ut dictum e&longs;t, ergo in eodem tempo&shy;<lb/>re, in quo axis circumuertitur in eodem orbis: ergo tanto tardius <lb/>uidebitur moueri axis ip&longs;o orbe, quanta e&longs;t proportio minoris in <lb/>&aelig;qualitatis ip&longs;ius axis, &longs;eu ambitus, &longs;eu &longs;emidimetientis ad ambi&shy;<lb/>tum, &longs;eu &longs;emidimetientem orbis.</s></p><p type="main">
  
Line 1703 
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 <s>Rur&longs;usque eiu&longs;dem circuli ad peripheriam diame<lb/>tri quarta pars.</s></p><p type="main"> <s>Rur&longs;usque eiu&longs;dem circuli ad peripheriam diame<lb/>tri quarta pars.</s></p><p type="main">
  
 <s><arrow.to.target n="marg144"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg144"/></s></p><p type="margin">
  
 <s><margin.target id="marg144"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg144"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Quoniam enim &longs;uperficies circuli, ut ab <lb/><figure id="fig34"></figure><lb/>Archimede demon&longs;tratum e&longs;t, fit ex dimi&shy;</s></p><p type="main"> <s>Quoniam enim &longs;uperficies circuli, ut ab <lb/><figure id="fig34"/><lb/>Archimede demon&longs;tratum e&longs;t, fit ex dimi&shy;</s></p><p type="main">
  
 <s><arrow.to.target n="marg145"></arrow.to.target><lb/>dio diametri in <expan abbr="dimidi&utilde;">dimidium</expan> peripheri&aelig; erit, ut <lb/>eadem fiat ex tota peripheria in <expan abbr="quart&atilde;">quartam</expan> par <lb/>tem diametri, &amp; ex tota diametro in quar&shy;<lb/>tam <expan abbr="part&etilde;">partem</expan> peripheri&ecedil;. </s> <s><arrow.to.target n="marg145"/><lb/>dio diametri in <expan abbr="dimidi&utilde;">dimidium</expan> peripheri&aelig; erit, ut <lb/>eadem fiat ex tota peripheria in <expan abbr="quart&atilde;">quartam</expan> par <lb/>tem diametri, &amp; ex tota diametro in quar&shy;<lb/>tam <expan abbr="part&etilde;">partem</expan> peripheri&ecedil;. </s>
  
 <s>ergo proportio are&ecedil; <lb/>circuli ad diametrum per &longs;imilitudinem <lb/><arrow.to.target n="marg146"></arrow.to.target><lb/>e&longs;t quarta pars peripheri&ecedil;, &amp; proportio are&ecedil; <lb/>ad <expan abbr="peripheri&atilde;">peripheriam</expan> e&longs;t quarta pars dimetientis, quod erat probandum.</s></p><p type="margin"> <s>ergo proportio are&ecedil; <lb/>circuli ad diametrum per &longs;imilitudinem <lb/><arrow.to.target n="marg146"/><lb/>e&longs;t quarta pars peripheri&ecedil;, &amp; proportio are&ecedil; <lb/>ad <expan abbr="peripheri&atilde;">peripheriam</expan> e&longs;t quarta pars dimetientis, quod erat probandum.</s></p><p type="margin">
  
 <s><margin.target id="marg145"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 16. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg145"/>P<emph type="italics"/>er<emph.end type="italics"/> 16. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg146"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 2. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg146"/>P<emph type="italics"/>er<emph.end type="italics"/> 2. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="main">
  
 <s>Propo&longs;itio quinquage&longs;imaquinta.</s></p><p type="main"> <s>Propo&longs;itio quinquage&longs;imaquinta.</s></p><p type="main">
  
 <s>Proportionem medicamentorum per ordines &longs;uppo&longs;ita &aelig;quali <lb/>proportione in ordinibus per quantitates, &amp; proportiones de&shy;<lb/>mon&longs;trare.<lb/><arrow.to.target n="marg147"></arrow.to.target></s></p><p type="margin"> <s>Proportionem medicamentorum per ordines &longs;uppo&longs;ita &aelig;quali <lb/>proportione in ordinibus per quantitates, &amp; proportiones de&shy;<lb/>mon&longs;trare.<lb/><arrow.to.target n="marg147"/></s></p><p type="margin">
  
 <s><margin.target id="marg147"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg147"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Galenus libro quinto de Simplicibus medicamentis, quem &longs;e&shy;</s></p><p type="main"> <s>Galenus libro quinto de Simplicibus medicamentis, quem &longs;e&shy;</s></p><p type="main">
  
 <s><arrow.to.target n="marg148"></arrow.to.target><lb/>quuti &longs;unt alij medici, ponit quatuor ordines <expan abbr="medicamentor&utilde;">medicamentorum</expan> iux&shy;<lb/>ta qualitates calidi, frigidi, &longs;icci, &amp; humidi, &amp; primus e&longs;t cum <expan abbr="medi-cament&utilde;">medi&shy;<lb/>camentum</expan> non &longs;entitur quale &longs;it licet operetur, uelut cam&ecedil;melon, ab&shy;<lb/>&longs;ynthium, &amp; oriza: &longs;ecundus e&longs;t, cum &longs;entitur, &longs;ed non l&aelig;dit, ut nux <lb/>myri&longs;tica, &longs;aluia, ozimum: tertius e&longs;t cum &longs;entitur, &amp; l&aelig;dit, &longs;ed <lb/>non de&longs;truit, neque corrumpit corpus, uelut a&longs;&longs;arum apium &longs;ta&shy;<lb/>phi&longs;agria, cappares, myrrha, ruta: quartus e&longs;t, cum de&longs;truit ue&shy;<lb/>lut pyretrum, piper, euphorbium c&aelig;pe aggre&longs;te, &amp; &longs;inapis, cina&shy;<pb pagenum="45"/>momum autem, &amp; gingiber numerantur inter medicinas cal&iacute;das <lb/>tertij gradus, &amp; hoc opus comparatur ad corpus &longs;icut dicit Gale&shy;<lb/>nus, &amp; Serapio non ad linguam, ut medici no&longs;tri temporis interpre <lb/>tantur. </s> <s><arrow.to.target n="marg148"/><lb/>quuti &longs;unt alij medici, ponit quatuor ordines <expan abbr="medicamentor&utilde;">medicamentorum</expan> iux&shy;<lb/>ta qualitates calidi, frigidi, &longs;icci, &amp; humidi, &amp; primus e&longs;t cum <expan abbr="medi-cament&utilde;">medi&shy;<lb/>camentum</expan> non &longs;entitur quale &longs;it licet operetur, uelut cam&ecedil;melon, ab&shy;<lb/>&longs;ynthium, &amp; oriza: &longs;ecundus e&longs;t, cum &longs;entitur, &longs;ed non l&aelig;dit, ut nux <lb/>myri&longs;tica, &longs;aluia, ozimum: tertius e&longs;t cum &longs;entitur, &amp; l&aelig;dit, &longs;ed <lb/>non de&longs;truit, neque corrumpit corpus, uelut a&longs;&longs;arum apium &longs;ta&shy;<lb/>phi&longs;agria, cappares, myrrha, ruta: quartus e&longs;t, cum de&longs;truit ue&shy;<lb/>lut pyretrum, piper, euphorbium c&aelig;pe aggre&longs;te, &amp; &longs;inapis, cina&shy;<pb pagenum="45"/>momum autem, &amp; gingiber numerantur inter medicinas cal&iacute;das <lb/>tertij gradus, &amp; hoc opus comparatur ad corpus &longs;icut dicit Gale&shy;<lb/>nus, &amp; Serapio non ad linguam, ut medici no&longs;tri temporis interpre <lb/>tantur. </s>
  
 <s>Ex quo patet, quod aliqua medicina poterit e&longs;&longs;e quarti ordl <lb/>nis, &amp; non l&aelig;dere linguam in gu&longs;tu, &amp; alia tertij ordinis, qu&aelig; non <lb/>&longs;olum l&aelig;det linguam, &longs;ed &longs;en&longs;um eius corrumpet, et de&longs;truet, quod <lb/>contingit propter &longs;ub&longs;tantiam tenuem cra&longs;&longs;&aelig; mi&longs;tam cum &longs;iccitate <lb/>pari ip&longs;i calori. </s> <s>Ex quo patet, quod aliqua medicina poterit e&longs;&longs;e quarti ordl <lb/>nis, &amp; non l&aelig;dere linguam in gu&longs;tu, &amp; alia tertij ordinis, qu&aelig; non <lb/>&longs;olum l&aelig;det linguam, &longs;ed &longs;en&longs;um eius corrumpet, et de&longs;truet, quod <lb/>contingit propter &longs;ub&longs;tantiam tenuem cra&longs;&longs;&aelig; mi&longs;tam cum &longs;iccitate <lb/>pari ip&longs;i calori. </s>
  
Line 1733 
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 <s>Nam de &longs;icco, <lb/>&amp; humido, cum &longs;int long&egrave; minoris actionis, qu&agrave;m calidum, &amp; fri&shy;<lb/>gidum, &amp; pr&aelig;cipu&egrave; humidum, non uideo quomodo po&longs;sit Gale&shy;<lb/>nus &longs;tatuere medicinam humidam tertij gradus, nedum quarti, <lb/>cum non po&longs;sit inueniri medicina, qu&aelig; de&longs;truat corpus no&longs;trum <lb/>propter humidam qualitatem. </s> <s>Nam de &longs;icco, <lb/>&amp; humido, cum &longs;int long&egrave; minoris actionis, qu&agrave;m calidum, &amp; fri&shy;<lb/>gidum, &amp; pr&aelig;cipu&egrave; humidum, non uideo quomodo po&longs;sit Gale&shy;<lb/>nus &longs;tatuere medicinam humidam tertij gradus, nedum quarti, <lb/>cum non po&longs;sit inueniri medicina, qu&aelig; de&longs;truat corpus no&longs;trum <lb/>propter humidam qualitatem. </s>
  
 <s>Et licet Serapio po&longs;uerit gingiber <lb/><arrow.to.target n="marg149"></arrow.to.target><lb/>&amp; enulam &amp; zelim in tertio ordine calidorum &amp; humidorum: &amp; <lb/>inter frigidas, &amp; humidas in tertio portulacam, aizoum, &amp; uirgam <lb/>pa&longs;toris, &amp; fungos. </s> <s>Et licet Serapio po&longs;uerit gingiber <lb/><arrow.to.target n="marg149"/><lb/>&amp; enulam &amp; zelim in tertio ordine calidorum &amp; humidorum: &amp; <lb/>inter frigidas, &amp; humidas in tertio portulacam, aizoum, &amp; uirgam <lb/>pa&longs;toris, &amp; fungos. </s>
  
 <s>Primum non au&longs;us e&longs;t ponere medicinas ullas <lb/>calidas, aut frigidas in quarto ordine, qu&ecedil; &longs;int humid&aelig;. </s> <s>Primum non au&longs;us e&longs;t ponere medicinas ullas <lb/>calidas, aut frigidas in quarto ordine, qu&ecedil; &longs;int humid&aelig;. </s>
  
Line 1743 
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 <s>Clarius etiam de frigidis &amp; humidis, nam por&shy;<lb/>tula cam dicit e&longs;&longs;e frigidam in tertio, humidam in &longs;ecundo, &amp; quod <lb/>maius, e&longs;t cum collo ca&longs;&longs;et aizoum inter medicinas frigidas, &amp; hu&shy;<lb/>midas in tertio ordine, dicit, quod e&longs;t frigidum in tertio ordine, ad&shy;<lb/>ijcit, quod e&longs;t &longs;iccum parum, &amp; de uirga pa&longs;toris nihil dicit de hu&shy;<lb/>mido, &longs;ed dicit, quod a&longs;tringit, ex quo concludo, quod &longs;ecun&shy;<lb/>dum mentem Serapionis nulla e&longs;t medicina humidior portulaca, <lb/>etiam uidetur innuere de fungis, &longs;atis e&longs;t quod non excedunt &longs;ecun<lb/>dum ordinem in humido neque calida neque frigida, &longs;ed frigida &longs;unt <lb/>humidiora, ut fungi, &amp; portulaca, quia frigiditas in generatione <lb/>humidum magis admittit, qu&agrave;m caliditas, &amp; calida magis hu&shy;<pb pagenum="46"/>mectant, quia magis penetrat uis medicamenti, &amp; h&aelig;c regula de <lb/>humido, &amp; &longs;icco e&longs;t generalis apud Serapionem, quod non intelli&shy;<lb/>gitur ordo in pa&longs;siuis, ni&longs;i &longs;pecialiter exprimatur, nam de &longs;iccitate <lb/>non nego, quin inueniantur medicin&aelig; &longs;icc&aelig; in tertio, &amp; for&longs;an in <lb/>quarto ordine, &longs;ed de hac Galeni o&longs;citantia, qu&aelig; in illo peculiaris <lb/>e&longs;t dum uult &longs;equi &longs;uas methodos &longs;ine alio di&longs;crimine, medicis con <lb/>&longs;i derandum relinquo.</s></p><p type="margin"> <s>Clarius etiam de frigidis &amp; humidis, nam por&shy;<lb/>tula cam dicit e&longs;&longs;e frigidam in tertio, humidam in &longs;ecundo, &amp; quod <lb/>maius, e&longs;t cum collo ca&longs;&longs;et aizoum inter medicinas frigidas, &amp; hu&shy;<lb/>midas in tertio ordine, dicit, quod e&longs;t frigidum in tertio ordine, ad&shy;<lb/>ijcit, quod e&longs;t &longs;iccum parum, &amp; de uirga pa&longs;toris nihil dicit de hu&shy;<lb/>mido, &longs;ed dicit, quod a&longs;tringit, ex quo concludo, quod &longs;ecun&shy;<lb/>dum mentem Serapionis nulla e&longs;t medicina humidior portulaca, <lb/>etiam uidetur innuere de fungis, &longs;atis e&longs;t quod non excedunt &longs;ecun<lb/>dum ordinem in humido neque calida neque frigida, &longs;ed frigida &longs;unt <lb/>humidiora, ut fungi, &amp; portulaca, quia frigiditas in generatione <lb/>humidum magis admittit, qu&agrave;m caliditas, &amp; calida magis hu&shy;<pb pagenum="46"/>mectant, quia magis penetrat uis medicamenti, &amp; h&aelig;c regula de <lb/>humido, &amp; &longs;icco e&longs;t generalis apud Serapionem, quod non intelli&shy;<lb/>gitur ordo in pa&longs;siuis, ni&longs;i &longs;pecialiter exprimatur, nam de &longs;iccitate <lb/>non nego, quin inueniantur medicin&aelig; &longs;icc&aelig; in tertio, &amp; for&longs;an in <lb/>quarto ordine, &longs;ed de hac Galeni o&longs;citantia, qu&aelig; in illo peculiaris <lb/>e&longs;t dum uult &longs;equi &longs;uas methodos &longs;ine alio di&longs;crimine, medicis con <lb/>&longs;i derandum relinquo.</s></p><p type="margin">
  
 <s><margin.target id="marg148"></margin.target>C<emph type="italics"/>ap. </s> <s><margin.target id="marg148"/>C<emph type="italics"/>ap. </s>
  
 <s>ult.<emph.end type="italics"/></s></p><p type="margin"> <s>ult.<emph.end type="italics"/></s></p><p type="margin">
  
 <s><margin.target id="marg149"></margin.target>C<emph type="italics"/>ap.<emph.end type="italics"/> 336. <lb/>337. &amp; <lb/>338.</s></p><p type="main"> <s><margin.target id="marg149"/>C<emph type="italics"/>ap.<emph.end type="italics"/> 336. <lb/>337. &amp; <lb/>338.</s></p><p type="main">
  
 <s>Secunda difficultas e&longs;t maior, &amp; magis pertinet ad nos, &amp; e&longs;t, <lb/>qu&ograve;d non declarauit an i&longs;ti ordines inter &longs;e <expan abbr="aliqu&atilde;">aliquam</expan> proportionem <lb/>&longs;eruarent, an omnino nullam, &longs;i enim nulla proportio &longs;eruatur, fieri <lb/>nullo modo pote&longs;t, ut per cognitionem temperatur&aelig; &longs;implicium <lb/>medicamentorum cogno&longs;camus temperaturam compo&longs;itorum ex <lb/>illis ratione ulla, &longs;ed oportebit &longs;olum experiri. </s> <s>Secunda difficultas e&longs;t maior, &amp; magis pertinet ad nos, &amp; e&longs;t, <lb/>qu&ograve;d non declarauit an i&longs;ti ordines inter &longs;e <expan abbr="aliqu&atilde;">aliquam</expan> proportionem <lb/>&longs;eruarent, an omnino nullam, &longs;i enim nulla proportio &longs;eruatur, fieri <lb/>nullo modo pote&longs;t, ut per cognitionem temperatur&aelig; &longs;implicium <lb/>medicamentorum cogno&longs;camus temperaturam compo&longs;itorum ex <lb/>illis ratione ulla, &longs;ed oportebit &longs;olum experiri. </s>
  
Line 1759 
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 <s>Ex quo &longs;e&shy;<lb/>quitur, quod Aueroes qu&agrave;m o&longs;citanter tractauerit in quinto &longs;uo&shy;<lb/>rum collectaneorum de hoc, &amp; non di&longs;tinguit, neque docet pri&shy;<lb/>mum an &longs;it aliqua proportio, deinde &longs;i qua &longs;it, cuius generis &longs;it, &amp; <lb/>cum in re tam clara pugnet pror&longs;us, ut c&oelig;cus ictus maximos eden&shy;<lb/>do, &longs;ed in ca&longs;&longs;um plero&longs;que, qu&agrave;m mal&egrave; agant qui ei in arduis tan&shy;<lb/>tum tribuunt fidei, &amp; authoritatis, &longs;ed h&aelig;c e&longs;t infelicitas no&longs;tra, &amp; <lb/>ira Deorum. </s> <s>Ex quo &longs;e&shy;<lb/>quitur, quod Aueroes qu&agrave;m o&longs;citanter tractauerit in quinto &longs;uo&shy;<lb/>rum collectaneorum de hoc, &amp; non di&longs;tinguit, neque docet pri&shy;<lb/>mum an &longs;it aliqua proportio, deinde &longs;i qua &longs;it, cuius generis &longs;it, &amp; <lb/>cum in re tam clara pugnet pror&longs;us, ut c&oelig;cus ictus maximos eden&shy;<lb/>do, &longs;ed in ca&longs;&longs;um plero&longs;que, qu&agrave;m mal&egrave; agant qui ei in arduis tan&shy;<lb/>tum tribuunt fidei, &amp; authoritatis, &longs;ed h&aelig;c e&longs;t infelicitas no&longs;tra, &amp; <lb/>ira Deorum. </s>
  
 <s>Suppo&longs;ito ergo quod prim&ograve; ordines di&longs;tinguantur <lb/>per proportionem arithmeticam, &longs;it &longs;uperficies a b pro quantitate, <lb/><figure id="fig35"></figure><lb/>&amp; a &longs;it calida in primo gradu, &amp; b in ter&shy;<lb/>tio, erit ergo perinde ac &longs;i duo corpora <lb/>e&longs;&longs;ent unum altitudinis unius cum ba&longs;i <lb/>quadrilatera rectangula a, aliud altitu&shy;<lb/>dinis trium, ba&longs;i autem quadrilatera &longs;u&shy;<lb/>perficie rectangula b, hoc igitur erit to&shy;<lb/>tum mi&longs;tum, &amp; quia quantitas medicamenti non mutatur qu&aelig; e&longs;t <lb/>a, b, ergo talia corpora &aelig;quantur uni corpori, cuius ba&longs;is e&longs;t a b, <lb/>cum ergo talia corpora producantur ex a in unum, &amp; b in tria, ergo <pb pagenum="47"/>diui&longs;o aggregato per a b prodibit altitudo, &longs;eu ordo qualitatis to&shy;<lb/>tius medicamenti, iuxta quod con&longs;tituitur regula prima libri artis <lb/>medendi paru&aelig; huiu&longs;modi, &amp; reliqu&aelig;, traduxi autem illas ad hunc <lb/>locuin, &ldquo;quia pendent ex demon&longs;tratione hac: &ldquo;duc numerum ordi&shy;<lb/>nis &longs;ingulorum medicamentorum in numerum quantitatis, &longs;imilia <lb/>iunge, di&longs;similia detrahe, quod fit, diuide per aggregatum, quanti&shy;<lb/>tatum, exibit numerus ordinis compo&longs;iti. </s> <s>Suppo&longs;ito ergo quod prim&ograve; ordines di&longs;tinguantur <lb/>per proportionem arithmeticam, &longs;it &longs;uperficies a b pro quantitate, <lb/><figure id="fig35"/><lb/>&amp; a &longs;it calida in primo gradu, &amp; b in ter&shy;<lb/>tio, erit ergo perinde ac &longs;i duo corpora <lb/>e&longs;&longs;ent unum altitudinis unius cum ba&longs;i <lb/>quadrilatera rectangula a, aliud altitu&shy;<lb/>dinis trium, ba&longs;i autem quadrilatera &longs;u&shy;<lb/>perficie rectangula b, hoc igitur erit to&shy;<lb/>tum mi&longs;tum, &amp; quia quantitas medicamenti non mutatur qu&aelig; e&longs;t <lb/>a, b, ergo talia corpora &aelig;quantur uni corpori, cuius ba&longs;is e&longs;t a b, <lb/>cum ergo talia corpora producantur ex a in unum, &amp; b in tria, ergo <pb pagenum="47"/>diui&longs;o aggregato per a b prodibit altitudo, &longs;eu ordo qualitatis to&shy;<lb/>tius medicamenti, iuxta quod con&longs;tituitur regula prima libri artis <lb/>medendi paru&aelig; huiu&longs;modi, &amp; reliqu&aelig;, traduxi autem illas ad hunc <lb/>locuin, &ldquo;quia pendent ex demon&longs;tratione hac: &ldquo;duc numerum ordi&shy;<lb/>nis &longs;ingulorum medicamentorum in numerum quantitatis, &longs;imilia <lb/>iunge, di&longs;similia detrahe, quod fit, diuide per aggregatum, quanti&shy;<lb/>tatum, exibit numerus ordinis compo&longs;iti. </s>
  
 <s>Sic mi&longs;cendo calidum in <lb/>&longs;ecundo ordine cum duplo pondere temperati conflabit calidum <lb/>in be&longs;&longs;e. </s> <s>Sic mi&longs;cendo calidum in <lb/>&longs;ecundo ordine cum duplo pondere temperati conflabit calidum <lb/>in be&longs;&longs;e. </s>
  
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 <s>Qu&aelig; autem &longs;ub mi<lb/>nore quantitate exhibentur non &longs;ub remi&longs;siore ordine agant, &longs;ed <lb/>uel facilius impediuntur, uel minorem corporis partem, uel leuius <lb/>immutant.</s></p><p type="main"> <s>Qu&aelig; autem &longs;ub mi<lb/>nore quantitate exhibentur non &longs;ub remi&longs;siore ordine agant, &longs;ed <lb/>uel facilius impediuntur, uel minorem corporis partem, uel leuius <lb/>immutant.</s></p><p type="main">
  
 <s>Quod &longs;i &longs;tatuamus proportionem e&longs;&longs;e Geometricam, modus <lb/>erit idem in omnibus, &amp; quo ad numerum etiam in primo, &amp; &longs;ecun<lb/>do ordine, quia in proportione dupla Geometrica &longs;ecundus ordo <lb/>tantundem di&longs;tat &agrave; primo, quantum primus ab &aelig;qualitate, quia <lb/>unum &amp; duo &longs;eruant proportionem, &amp; &aelig;qualem di&longs;tantiam, &longs;ed in <lb/>c&aelig;teris ordinibus non ita erit, quia qui e&longs;&longs;et trium in Arithmetica, <lb/>&longs;cilicet totius ordo e&longs;t, quatuor in Geometrica, &amp; quartus ordo, <lb/>qui e&longs;&longs;et quatuor in Arithmetica, e&longs;&longs;et octo in Geometrica, ideo <lb/><figure id="fig36"></figure><lb/>&longs;cribemus ordines hoc modo, &amp; operabimur cum <lb/>numeris loco ordinum, exemplum ergo primum <lb/>&longs;it medicina calida in tertio ordine quatuor uncia&shy;<lb/>rum, &amp; medicina frigida in <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan> ordine duarum <lb/>unciarum, duco quatuor in tria, &longs;i proportio &longs;it Arithmetica, fit <lb/>duodecim, duco duo in duo fit quatuor, detraho quatuor in duo&shy;<lb/>decim, quia omnis medicina tantum retondit de contrario, &longs;eu mi&shy;<lb/>nuit relin quuntur octo &longs;cilicet caliditatis, diuido per &longs;ex ag&shy;<pb pagenum="48"/>gregatum unciarum exit unum, &amp; tertia, ergo erit calida in princi&shy;<lb/>pio &longs;ecundi ordinis. </s> <s>Quod &longs;i &longs;tatuamus proportionem e&longs;&longs;e Geometricam, modus <lb/>erit idem in omnibus, &amp; quo ad numerum etiam in primo, &amp; &longs;ecun<lb/>do ordine, quia in proportione dupla Geometrica &longs;ecundus ordo <lb/>tantundem di&longs;tat &agrave; primo, quantum primus ab &aelig;qualitate, quia <lb/>unum &amp; duo &longs;eruant proportionem, &amp; &aelig;qualem di&longs;tantiam, &longs;ed in <lb/>c&aelig;teris ordinibus non ita erit, quia qui e&longs;&longs;et trium in Arithmetica, <lb/>&longs;cilicet totius ordo e&longs;t, quatuor in Geometrica, &amp; quartus ordo, <lb/>qui e&longs;&longs;et quatuor in Arithmetica, e&longs;&longs;et octo in Geometrica, ideo <lb/><figure id="fig36"/><lb/>&longs;cribemus ordines hoc modo, &amp; operabimur cum <lb/>numeris loco ordinum, exemplum ergo primum <lb/>&longs;it medicina calida in tertio ordine quatuor uncia&shy;<lb/>rum, &amp; medicina frigida in <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan> ordine duarum <lb/>unciarum, duco quatuor in tria, &longs;i proportio &longs;it Arithmetica, fit <lb/>duodecim, duco duo in duo fit quatuor, detraho quatuor in duo&shy;<lb/>decim, quia omnis medicina tantum retondit de contrario, &longs;eu mi&shy;<lb/>nuit relin quuntur octo &longs;cilicet caliditatis, diuido per &longs;ex ag&shy;<pb pagenum="48"/>gregatum unciarum exit unum, &amp; tertia, ergo erit calida in princi&shy;<lb/>pio &longs;ecundi ordinis. </s>
  
 <s>Secundum exemplum &longs;int e&aelig;dem medicin&aelig;, <lb/>&amp; &longs;it proportio Geometrica, ducemus ergo quatuor in quatuor, &amp; <lb/>fiunt &longs;exdecim, &amp; duo in duo fiunt quatuor, detrahe quatuor ex &longs;ex <lb/>decim, &amp; remanent duodecim, diuide per &longs;ex, ut prius, exeunt duo, <lb/>ergo erit calida in fine &longs;ecund i gradus uides ergo di&longs;crimen. </s> <s>Secundum exemplum &longs;int e&aelig;dem medicin&aelig;, <lb/>&amp; &longs;it proportio Geometrica, ducemus ergo quatuor in quatuor, &amp; <lb/>fiunt &longs;exdecim, &amp; duo in duo fiunt quatuor, detrahe quatuor ex &longs;ex <lb/>decim, &amp; remanent duodecim, diuide per &longs;ex, ut prius, exeunt duo, <lb/>ergo erit calida in fine &longs;ecund i gradus uides ergo di&longs;crimen. </s>
  
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 <s>Propo&longs;itio quinquage&longs;ima&longs;exta.</s></p><p type="main"> <s>Propo&longs;itio quinquage&longs;ima&longs;exta.</s></p><p type="main">
  
 <s>Proportio cuiu&longs;uis binomij ad &longs;uum reci&longs;um, uel ei commen&shy;<lb/>&longs;um e&longs;t duplicata ei, qu&aelig; ad numeri latus.<lb/><arrow.to.target n="marg150"></arrow.to.target></s></p><p type="margin"> <s>Proportio cuiu&longs;uis binomij ad &longs;uum reci&longs;um, uel ei commen&shy;<lb/>&longs;um e&longs;t duplicata ei, qu&aelig; ad numeri latus.<lb/><arrow.to.target n="marg150"/></s></p><p type="margin">
  
 <s><margin.target id="marg150"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main"> <s><margin.target id="marg150"/>C<emph type="italics"/>o<emph.end type="italics"/>m.<!-- KEEP S--></s></p><p type="main">
  
 <s>Cum enim proportionis medium &longs;it latus numeri eo quod ex bi <lb/>nomio in reci&longs;um &longs;uum fit numerus ex his, qu&aelig; demon&longs;trata &longs;unt <lb/>generaliter in tertio Arithmetic&aelig; de omnibus binomijs cum &longs;uis </s></p><p type="main"> <s>Cum enim proportionis medium &longs;it latus numeri eo quod ex bi <lb/>nomio in reci&longs;um &longs;uum fit numerus ex his, qu&aelig; demon&longs;trata &longs;unt <lb/>generaliter in tertio Arithmetic&aelig; de omnibus binomijs cum &longs;uis </s></p><p type="main">
  
 <s><arrow.to.target n="marg151"></arrow.to.target><lb/>reci&longs;is, uel in quadratis lateribus erit &lt;02&gt; numeri media proportione <lb/>inter binomium, &amp; &longs;uum reci&longs;um, igitur cum proportio producto&shy;<lb/>rum ex binomio in commen&longs;a reci&longs;o &longs;it, ut commen&longs;orum ad reci&shy;<lb/><arrow.to.target n="marg152"></arrow.to.target><lb/>&longs;a crunt omnia producta ex binomio in commen&longs;a reci&longs;o &longs;uo &lt;02&gt; nu <lb/><arrow.to.target n="marg153"></arrow.to.target><lb/>meri, igitur proportio binomij ad reci&longs;um &longs;uum, &amp; omnia com&shy;<lb/>men&longs;a illi, e&longs;t duplicata ei qu&aelig; ad &lt;02&gt; numeri.<lb/><arrow.to.target n="marg154"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg151"/><lb/>reci&longs;is, uel in quadratis lateribus erit &lt;02&gt; numeri media proportione <lb/>inter binomium, &amp; &longs;uum reci&longs;um, igitur cum proportio producto&shy;<lb/>rum ex binomio in commen&longs;a reci&longs;o &longs;it, ut commen&longs;orum ad reci&shy;<lb/><arrow.to.target n="marg152"/><lb/>&longs;a crunt omnia producta ex binomio in commen&longs;a reci&longs;o &longs;uo &lt;02&gt; nu <lb/><arrow.to.target n="marg153"/><lb/>meri, igitur proportio binomij ad reci&longs;um &longs;uum, &amp; omnia com&shy;<lb/>men&longs;a illi, e&longs;t duplicata ei qu&aelig; ad &lt;02&gt; numeri.<lb/><arrow.to.target n="marg154"/></s></p><p type="margin">
  
 <s><margin.target id="marg151"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 6. P<emph type="italics"/>ro&shy;<lb/>po&longs;. </s> <s><margin.target id="marg151"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. P<emph type="italics"/>ro&shy;<lb/>po&longs;. </s>
  
 <s>lib.  <s>lib.
  
 de<emph.end type="italics"/><lb/>A<emph type="italics"/>liza.<emph.end type="italics"/></s></p><p type="margin"> de<emph.end type="italics"/><lb/>A<emph type="italics"/>liza.<emph.end type="italics"/></s></p><p type="margin">
  
 <s><margin.target id="marg152"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg152"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg153"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 17. <lb/><emph type="italics"/>&longs;eptimi <lb/>eiu&longs;dem.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg153"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <lb/><emph type="italics"/>&longs;eptimi <lb/>eiu&longs;dem.<emph.end type="italics"/></s></p><p type="margin">
  
 <s><margin.target id="marg154"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 6. <emph type="italics"/>deci&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement:<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg154"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. <emph type="italics"/>deci&shy;<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement:<emph.end type="italics"/></s></p><p type="main">
  
 <s>Propo&longs;itio quinquage&longs;ima&longs;eptima.</s></p><p type="main"> <s>Propo&longs;itio quinquage&longs;ima&longs;eptima.</s></p><p type="main">
  
 <s>Motus rationem ad pondus inuenire.</s></p><p type="main"> <s>Motus rationem ad pondus inuenire.</s></p><p type="main">
  
 <s><arrow.to.target n="marg155"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg155"/></s></p><p type="margin">
  
 <s><margin.target id="marg155"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg155"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>O&longs;ten&longs;um e&longs;t antea, quod motus naturalis uelocior fit in fine, ac <lb/>magis augetur ob a&euml;ris motum, ubi uer&ograve; h&aelig;ret e&longs;t ac &longs;i quie&longs;cat. <lb/></s> <s>O&longs;ten&longs;um e&longs;t antea, quod motus naturalis uelocior fit in fine, ac <lb/>magis augetur ob a&euml;ris motum, ubi uer&ograve; h&aelig;ret e&longs;t ac &longs;i quie&longs;cat. <lb/></s>
  
 <s>Eadem autem e&longs;t ratio in motis uiolenter, &amp; naturaliter dum &ecedil;qua&shy;<lb/>li impetu feruntur. </s> <s>Eadem autem e&longs;t ratio in motis uiolenter, &amp; naturaliter dum &ecedil;qua&shy;<lb/>li impetu feruntur. </s>
  
 <s>Sed &longs;ubit&ograve; po&longs;t etiam, quod motus &aelig;qualiter <lb/>augerentur minus tamen cre&longs;cit proportio uiolenti &longs;cilicet ob im&shy;<lb/><figure id="fig37"></figure><lb/>pedimentum naturale. </s> <s>Sed &longs;ubit&ograve; po&longs;t etiam, quod motus &aelig;qualiter <lb/>augerentur minus tamen cre&longs;cit proportio uiolenti &longs;cilicet ob im&shy;<lb/><figure id="fig37"/><lb/>pedimentum naturale. </s>
  
 <s>Sed &longs;i uis mouens fuerit <lb/>ade&ograve; ualida ut proportio incrementi ex a&euml;re &longs;it <lb/>maior, qu&agrave;m impedimentum, &amp; in crementum al <lb/>terius mobilis naturaliter moti, motus ille uelo&shy;<lb/>cior fiet naturali, ut in &longs;ph&aelig;ris ferreis ex machina <lb/>igne excu&longs;sis, quod ergo attinet ad pr&aelig;&longs;entem <lb/>motum ratio e&longs;t eadem. </s> <s>Sed &longs;i uis mouens fuerit <lb/>ade&ograve; ualida ut proportio incrementi ex a&euml;re &longs;it <lb/>maior, qu&agrave;m impedimentum, &amp; in crementum al <lb/>terius mobilis naturaliter moti, motus ille uelo&shy;<lb/>cior fiet naturali, ut in &longs;ph&aelig;ris ferreis ex machina <lb/>igne excu&longs;sis, quod ergo attinet ad pr&aelig;&longs;entem <lb/>motum ratio e&longs;t eadem. </s>
  
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 <s>At ca&shy;<lb/>&longs;u plerun que contingit, ut moueatur long&egrave; uehementius, &longs;eu ad ean&shy;<lb/>dem partem, &longs;eu aliam. </s> <s>At ca&shy;<lb/>&longs;u plerun que contingit, ut moueatur long&egrave; uehementius, &longs;eu ad ean&shy;<lb/>dem partem, &longs;eu aliam. </s>
  
 <s>Qui uer&ograve; naturalis e&longs;t, debilis <lb/><figure id="fig38"></figure><lb/>e&longs;t, quoniam in tenui ualde &longs;ub&longs;tantia e&longs;t: nec <expan abbr="c&otilde;tinuus">continuus</expan> <lb/>&longs;ed in&longs;tar motus aqu&aelig; maris fluit ac refluit: aliter ne&shy;<lb/>ce&longs;&longs;e e&longs;&longs;et, ut &longs;ingulis horis per mille milliaria procede&shy;<lb/>ret, ut &longs;ic ne que latere po&longs;&longs;et, quarndoquidem fortuiti mo<lb/>tus, qui &longs;unt multo tardiores non latentnos. </s> <s>Qui uer&ograve; naturalis e&longs;t, debilis <lb/><figure id="fig38"/><lb/>e&longs;t, quoniam in tenui ualde &longs;ub&longs;tantia e&longs;t: nec <expan abbr="c&otilde;tinuus">continuus</expan> <lb/>&longs;ed in&longs;tar motus aqu&aelig; maris fluit ac refluit: aliter ne&shy;<lb/>ce&longs;&longs;e e&longs;&longs;et, ut &longs;ingulis horis per mille milliaria procede&shy;<lb/>ret, ut &longs;ic ne que latere po&longs;&longs;et, quarndoquidem fortuiti mo<lb/>tus, qui &longs;unt multo tardiores non latentnos. </s>
  
 <s>Nam tardiores illos <lb/>e&longs;&longs;e <expan abbr="c&otilde;&longs;tat">con&longs;tat</expan>, cum in hora &longs;int pul&longs;us arteriarum, quatuor millia <expan abbr="ictu&utilde;">ictuum</expan> <lb/>in homine prope temperamentum: &longs;i igitur motus naturalis a&euml;ris <lb/>e&longs;&longs;et continuus, in hora a&euml;r procederet ob ambitum terr&aelig; millies <lb/>mille pa&longs;&longs;us, <expan abbr="igi&ttilde;">igitur</expan> in ictu pul&longs;us &longs;uperaret pa&longs;&longs;us 250. At experimur <lb/>nullum uentum aut procellam &longs;uperare quinquaginta pa&longs;&longs;us, cum <lb/>etiam continuus e&longs;&longs;e nunquam &longs;oleat, im&ograve; ne po&longs;sit quidem, ita que<lb/>cum hic multo tardior etiam in &longs;ublimi, dum e&longs;t, nos latere non <lb/>queat, multo minus po&longs;&longs;et naturalis latere, &longs;i ade&ograve; uelox &amp; in ea&shy;<lb/>dem parte <expan abbr="a&etilde;ris">aerris</expan> e&longs;&longs;et at que continuus. </s> <s>Nam tardiores illos <lb/>e&longs;&longs;e <expan abbr="c&otilde;&longs;tat">con&longs;tat</expan>, cum in hora &longs;int pul&longs;us arteriarum, quatuor millia <expan abbr="ictu&utilde;">ictuum</expan> <lb/>in homine prope temperamentum: &longs;i igitur motus naturalis a&euml;ris <lb/>e&longs;&longs;et continuus, in hora a&euml;r procederet ob ambitum terr&aelig; millies <lb/>mille pa&longs;&longs;us, <expan abbr="igi&ttilde;">igitur</expan> in ictu pul&longs;us &longs;uperaret pa&longs;&longs;us 250. At experimur <lb/>nullum uentum aut procellam &longs;uperare quinquaginta pa&longs;&longs;us, cum <lb/>etiam continuus e&longs;&longs;e nunquam &longs;oleat, im&ograve; ne po&longs;sit quidem, ita que<lb/>cum hic multo tardior etiam in &longs;ublimi, dum e&longs;t, nos latere non <lb/>queat, multo minus po&longs;&longs;et naturalis latere, &longs;i ade&ograve; uelox &amp; in ea&shy;<lb/>dem parte <expan abbr="a&etilde;ris">aerris</expan> e&longs;&longs;et at que continuus. </s>
  
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 <s>Propo&longs;itio quin quage&longs;imanona.</s></p><p type="main"> <s>Propo&longs;itio quin quage&longs;imanona.</s></p><p type="main">
  
 <s><arrow.to.target n="marg156"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg156"/></s></p><p type="margin">
  
 <s><margin.target id="marg156"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main"> <s><margin.target id="marg156"/>C<emph type="italics"/>o<emph.end type="italics"/>m.<!-- KEEP S--></s></p><p type="main">
  
 <s>Omne mobile motum duobus motibus non ad idem tendenti&shy;<lb/>bus, utro que &longs;eor&longs;um tardius mouetur &longs;imili motu.</s></p><pb pagenum="51"/><p type="main"> <s>Omne mobile motum duobus motibus non ad idem tendenti&shy;<lb/>bus, utro que &longs;eor&longs;um tardius mouetur &longs;imili motu.</s></p><pb pagenum="51"/><p type="main">
  
 <s>Sit a mobile, quod moueatur per a b c impul&longs;u uenti aut uiolen&shy;</s></p><p type="main"> <s>Sit a mobile, quod moueatur per a b c impul&longs;u uenti aut uiolen&shy;</s></p><p type="main">
  
 <s><arrow.to.target n="marg157"></arrow.to.target><lb/><figure id="fig39"></figure><lb/>to cum naturali coniuncto: &amp; &longs;it terminus naturalis e, <lb/><arrow.to.target n="marg158"></arrow.to.target><lb/>&amp; uiolenti d: uter que in directo c, dico, quod tardius per&shy;<lb/>ueniet ad c quam d, uel e. </s> <s><arrow.to.target n="marg157"/><lb/><figure id="fig39"/><lb/>to cum naturali coniuncto: &amp; &longs;it terminus naturalis e, <lb/><arrow.to.target n="marg158"/><lb/>&amp; uiolenti d: uter que in directo c, dico, quod tardius per&shy;<lb/>ueniet ad c quam d, uel e. </s>
  
 <s>De e manife&longs;tum e&longs;t, quoniam <lb/>motus a&euml;ris, qui intendit motum a, diu&iacute;ditur in partem, <lb/>qu&aelig; iuuat motum ad d, &amp; partem, qu&aelig; mouetur ad e, <lb/>igitur fit minor adiectio. </s> <s>De e manife&longs;tum e&longs;t, quoniam <lb/>motus a&euml;ris, qui intendit motum a, diu&iacute;ditur in partem, <lb/>qu&aelig; iuuat motum ad d, &amp; partem, qu&aelig; mouetur ad e, <lb/>igitur fit minor adiectio. </s>
  
 <s>Et etiam quia a c e&longs;t longior <lb/>a e ex diffinitione rect&aelig;: quare tardius perueniet ad c qu&agrave;m ad e du <lb/>plici ratione. </s> <s>Et etiam quia a c e&longs;t longior <lb/>a e ex diffinitione rect&aelig;: quare tardius perueniet ad c qu&agrave;m ad e du <lb/>plici ratione. </s>
  
 <s>Dico etiam, quod tardius ad c qu&agrave;m d. </s> <s>Dico etiam, quod tardius ad c qu&agrave;m d. <!-- KEEP S--></s>
  
 <s>Quia enim <lb/>uis, qu&aelig; fert ad d repugnat ei, qu&aelig; fert ad e, &amp; uis, qu&aelig; fert ad e, re&shy;<lb/>pugnat ei qu&aelig; fert ad d, igitur tardius perueniet ad c, qu&agrave;m d. </s> <s>Quia enim <lb/>uis, qu&aelig; fert ad d repugnat ei, qu&aelig; fert ad e, &amp; uis, qu&aelig; fert ad e, re&shy;<lb/>pugnat ei qu&aelig; fert ad d, igitur tardius perueniet ad c, qu&agrave;m d. <!-- KEEP S--></s>
  
 <s>Nec <lb/>potes dicere, qu&ograve;d uis, qu&aelig; fert ad c adiuuet ad motum &egrave; regione <lb/>d, nam cum unus motus non po&longs;sit perfici &longs;ine altero, igitur quan&shy;<lb/>tum motus ad eretar dabit motum ad d, tanto motus a c erit tard&iacute;&shy;<lb/>or ab&longs;olut&egrave; motu ad d. </s> <s>Nec <lb/>potes dicere, qu&ograve;d uis, qu&aelig; fert ad c adiuuet ad motum &egrave; regione <lb/>d, nam cum unus motus non po&longs;sit perfici &longs;ine altero, igitur quan&shy;<lb/>tum motus ad eretar dabit motum ad d, tanto motus a c erit tard&iacute;&shy;<lb/>or ab&longs;olut&egrave; motu ad d. <!-- KEEP S--></s>
  
 <s>Verum etiam e&longs;t, quod c e breuior erit a d, <lb/>quia motus ad e &longs;emper contrahit motum ad d naturalis uiolen&shy;<lb/>rum ob cau&longs;am dictam. </s> <s>Verum etiam e&longs;t, quod c e breuior erit a d, <lb/>quia motus ad e &longs;emper contrahit motum ad d naturalis uiolen&shy;<lb/>rum ob cau&longs;am dictam. </s>
  
 <s>Vtr&ugrave;m uer&ograve; motus ad c ab&longs;olut&egrave; &longs;it tardi&shy;<lb/>or, qu&agrave;m ad d, non &longs;uppo&longs;ito, quod c e &longs;it &aelig;qualis a d, &longs;ed minor, <lb/>nunc non e&longs;t locus determinandi.</s></p><p type="margin"> <s>Vtr&ugrave;m uer&ograve; motus ad c ab&longs;olut&egrave; &longs;it tardi&shy;<lb/>or, qu&agrave;m ad d, non &longs;uppo&longs;ito, quod c e &longs;it &aelig;qualis a d, &longs;ed minor, <lb/>nunc non e&longs;t locus determinandi.</s></p><p type="margin">
  
 <s><margin.target id="marg157"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin"> <s><margin.target id="marg157"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg158"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>bu-ius.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg158"/>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>bu-ius.<emph.end type="italics"/></s></p><p type="main">
  
 <s>Ex hoc patet, quod motus &aelig;quidi&longs;tantis mobilis, finis e&longs;t mini&shy;<lb/><arrow.to.target n="marg159"></arrow.to.target><lb/>mus omnium: quoniam mobile qua&longs;i quie&longs;cit in illo. </s> <s>Ex hoc patet, quod motus &aelig;quidi&longs;tantis mobilis, finis e&longs;t mini&shy;<lb/><arrow.to.target n="marg159"/><lb/>mus omnium: quoniam mobile qua&longs;i quie&longs;cit in illo. </s>
  
 <s>Velut &longs;i a mo<lb/>ueatur ad b, inde deflectat ad c minimus motus erit in b, ubi incipit <lb/>naturalis: nam cum incipiat, erit debili&longs;simus, quia non <lb/><figure id="fig40"></figure><lb/>e&longs;t motus actu: uiolentus autem &aelig;qualis e&longs;t naturali, <lb/>dum minimus e&longs;t: ergo cum ex di&longs;tantia medij palmi <lb/>duplicetur, naturalis erit motus in b minimus, ni&longs;i b c <lb/><arrow.to.target n="marg160"></arrow.to.target><lb/>e&longs;&longs;et minor dimidio palmi. </s> <s>Velut &longs;i a mo<lb/>ueatur ad b, inde deflectat ad c minimus motus erit in b, ubi incipit <lb/>naturalis: nam cum incipiat, erit debili&longs;simus, quia non <lb/><figure id="fig40"/><lb/>e&longs;t motus actu: uiolentus autem &aelig;qualis e&longs;t naturali, <lb/>dum minimus e&longs;t: ergo cum ex di&longs;tantia medij palmi <lb/>duplicetur, naturalis erit motus in b minimus, ni&longs;i b c <lb/><arrow.to.target n="marg160"/><lb/>e&longs;&longs;et minor dimidio palmi. </s>
  
 <s>Et etiam qu&ograve;d e&longs;&longs;et minor, quia ut di&shy;<lb/>ctum e&longs;t, uter que &longs;imul iunctus e&longs;t &aelig;qualis uni eorum non impedito <lb/>uel minor.</s></p><p type="margin"> <s>Et etiam qu&ograve;d e&longs;&longs;et minor, quia ut di&shy;<lb/>ctum e&longs;t, uter que &longs;imul iunctus e&longs;t &aelig;qualis uni eorum non impedito <lb/>uel minor.</s></p><p type="margin">
  
 <s><margin.target id="marg159"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin"> <s><margin.target id="marg159"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg160"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 57. <emph type="italics"/>bu-ius.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg160"/>P<emph type="italics"/>er<emph.end type="italics"/> 57. <emph type="italics"/>bu-ius.<emph.end type="italics"/></s></p><p type="main">
  
 <s>Propo&longs;itio &longs;exage&longs;ima.</s></p><p type="main"> <s>Propo&longs;itio &longs;exage&longs;ima.</s></p><p type="main">
  
 <s>Omne mobile motu naturali de&longs;cendens parte, de&longs;cendit gra&shy;<lb/>uiore &longs;ecundum grauitatis centrum.</s></p><p type="main"> <s>Omne mobile motu naturali de&longs;cendens parte, de&longs;cendit gra&shy;<lb/>uiore &longs;ecundum grauitatis centrum.</s></p><p type="main">
  
 <s>Sit a mobile, grauitatis centrum b, cuius pars ei pro&shy;<lb/><arrow.to.target n="marg161"></arrow.to.target><lb/><figure id="fig41"></figure><lb/>ximior &longs;it c a, dico quod de&longs;cendat motu naturali c a, <lb/>parte tangendo terram, quia enim totum a non pote&longs;t <lb/>de&longs;cendere ad centrum de&longs;cendit b, quia eadem e&longs;t na&shy;<lb/>tura partis, &amp; totius: totius autem terr&aelig; natura e&longs;t ut <lb/>centrum, totius &longs;it centrum grauitatis, quare b breuiore uia fertur <lb/><arrow.to.target n="marg162"></arrow.to.target><lb/>ad centrum, ergo per c d proximiorem partem ip&longs;i b. </s> <s>Sit a mobile, grauitatis centrum b, cuius pars ei pro&shy;<lb/><arrow.to.target n="marg161"/><lb/><figure id="fig41"/><lb/>ximior &longs;it c a, dico quod de&longs;cendat motu naturali c a, <lb/>parte tangendo terram, quia enim totum a non pote&longs;t <lb/>de&longs;cendere ad centrum de&longs;cendit b, quia eadem e&longs;t na&shy;<lb/>tura partis, &amp; totius: totius autem terr&aelig; natura e&longs;t ut <lb/>centrum, totius &longs;it centrum grauitatis, quare b breuiore uia fertur <lb/><arrow.to.target n="marg162"/><lb/>ad centrum, ergo per c d proximiorem partem ip&longs;i b. </s>
  
 <s>Sed pars pro&shy;<lb/>ximior nece&longs;&longs;ari&ograve; e&longs;t grauior, quia centrum e&longs;t in medio grauita&shy;<pb pagenum="52"/>tis, ergo omne mobile de&longs;cendit motu naturali per &longs;ui grauio&shy;<lb/>rem partem.<lb/><arrow.to.target n="marg163"></arrow.to.target></s></p><p type="margin"> <s>Sed pars pro&shy;<lb/>ximior nece&longs;&longs;ari&ograve; e&longs;t grauior, quia centrum e&longs;t in medio grauita&shy;<pb pagenum="52"/>tis, ergo omne mobile de&longs;cendit motu naturali per &longs;ui grauio&shy;<lb/>rem partem.<lb/><arrow.to.target n="marg163"/></s></p><p type="margin">
  
 <s><margin.target id="marg161"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin"> <s><margin.target id="marg161"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg162"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>bu-ius.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg162"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>bu-ius.<emph.end type="italics"/></s></p><p type="margin">
  
 <s><margin.target id="marg163"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg163"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Ex hoc &longs;equitur, qu&ograve;d graue habens partes in&aelig;quales, &longs;eu &longs;ub&shy;<lb/>&longs;tantia, &longs;cu forma, &longs;i ita excutiatur, ut pars grauior <expan abbr="n&otilde;">non</expan> &longs;it, infr&agrave; opor&shy;<lb/>tet, ut circumuoluatur.</s></p><p type="main"> <s>Ex hoc &longs;equitur, qu&ograve;d graue habens partes in&aelig;quales, &longs;eu &longs;ub&shy;<lb/>&longs;tantia, &longs;cu forma, &longs;i ita excutiatur, ut pars grauior <expan abbr="n&otilde;">non</expan> &longs;it, infr&agrave; opor&shy;<lb/>tet, ut circumuoluatur.</s></p><p type="main">
  
Line 1911 
Line 1910 
  
 <s>Proportionem ictus ad pondus rei, &amp; di&longs;tantiam generaliter <lb/>con&longs;iderare.</s></p><p type="main"> <s>Proportionem ictus ad pondus rei, &amp; di&longs;tantiam generaliter <lb/>con&longs;iderare.</s></p><p type="main">
  
 <s><arrow.to.target n="marg164"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg164"/></s></p><p type="margin">
  
 <s><margin.target id="marg164"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg164"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Dictum e&longs;t &longs;uperius de proportione de&longs;cen&longs;us ad grauitatem: </s></p><p type="main"> <s>Dictum e&longs;t &longs;uperius de proportione de&longs;cen&longs;us ad grauitatem: </s></p><p type="main">
  
 <s><arrow.to.target n="marg165"></arrow.to.target><lb/>&amp; qu&ograve;d &longs;i graue de&longs;cendat ex alto impeditur &agrave; motu a&euml;ris: &amp; qu&ograve;d <lb/><arrow.to.target n="marg166"></arrow.to.target><lb/>res, qu&aelig; mouetur duobus motibus non ad idem tendentibus tar&shy;<lb/><arrow.to.target n="marg167"></arrow.to.target><lb/>dius mouetur, quam motus &longs;it unu&longs;qui&longs;que. </s> <s><arrow.to.target n="marg165"/><lb/>&amp; qu&ograve;d &longs;i graue de&longs;cendat ex alto impeditur &agrave; motu a&euml;ris: &amp; qu&ograve;d <lb/><arrow.to.target n="marg166"/><lb/>res, qu&aelig; mouetur duobus motibus non ad idem tendentibus tar&shy;<lb/><arrow.to.target n="marg167"/><lb/>dius mouetur, quam motus &longs;it unu&longs;qui&longs;que. </s>
  
 <s>Dem&ugrave;m qu&ograve;d graue <lb/><arrow.to.target n="marg168"></arrow.to.target><lb/>de&longs;cendens circumuoluitur, &longs;i pars grauior non &longs;it, deor&longs;um: &amp; an&shy;<lb/>tea ubi egimus de proportione motus ad grauitatem, quod h&ecedil;cin&shy;<lb/>telligenda &longs;unt prout po&longs;&longs;unt intelligi de motu etiam uiolento. <lb/></s> <s>Dem&ugrave;m qu&ograve;d graue <lb/><arrow.to.target n="marg168"/><lb/>de&longs;cendens circumuoluitur, &longs;i pars grauior non &longs;it, deor&longs;um: &amp; an&shy;<lb/>tea ubi egimus de proportione motus ad grauitatem, quod h&ecedil;cin&shy;<lb/>telligenda &longs;unt prout po&longs;&longs;unt intelligi de motu etiam uiolento. <lb/></s>
  
 <s>Cum ergo uideamus duo h&aelig;c, quod res acuta frangit caput, &longs;i ex <lb/>alto incidat, &longs;ed non concutit, lata concutit, &longs;ed non diuidit, premit <lb/>tamen carnem &longs;ubiectam: nec hoc accidit merito ponderis: nam ut <lb/>ui&longs;um e&longs;t &longs;emilibra lapidis, uel ferri cadens ex alto contundit caput, <lb/>&amp; uulnerat, &amp; non eleuat in &aelig;quilibrio, ut pot&egrave; ex alto cadens loco <lb/>per &longs;patium octo palmorum pondus &longs;exdecim librarum, &amp; a pon&shy;<lb/>dere &longs;exdecim librarum homo non l&aelig;ditur, nec uulneratur, ergo id <lb/>accidit ex alia cau&longs;a, &amp; e&longs;t, quod a&euml;r interceptus inter graue, &amp; cor&shy;<lb/>pus no&longs;trum non pote&longs;t dilabi tam cit&ograve;, ergo ne corpus penetret, <lb/>cogitur ingredi locum, cui e&longs;t obuius, at que ita concutere, &amp; diuide&shy;<lb/>re. </s> <s>Cum ergo uideamus duo h&aelig;c, quod res acuta frangit caput, &longs;i ex <lb/>alto incidat, &longs;ed non concutit, lata concutit, &longs;ed non diuidit, premit <lb/>tamen carnem &longs;ubiectam: nec hoc accidit merito ponderis: nam ut <lb/>ui&longs;um e&longs;t &longs;emilibra lapidis, uel ferri cadens ex alto contundit caput, <lb/>&amp; uulnerat, &amp; non eleuat in &aelig;quilibrio, ut pot&egrave; ex alto cadens loco <lb/>per &longs;patium octo palmorum pondus &longs;exdecim librarum, &amp; a pon&shy;<lb/>dere &longs;exdecim librarum homo non l&aelig;ditur, nec uulneratur, ergo id <lb/>accidit ex alia cau&longs;a, &amp; e&longs;t, quod a&euml;r interceptus inter graue, &amp; cor&shy;<lb/>pus no&longs;trum non pote&longs;t dilabi tam cit&ograve;, ergo ne corpus penetret, <lb/>cogitur ingredi locum, cui e&longs;t obuius, at que ita concutere, &amp; diuide&shy;<lb/>re. </s>
  
 <s>Ex quibus &longs;equuntur omnia h&aelig;c.<lb/><arrow.to.target n="marg169"></arrow.to.target></s></p><p type="margin"> <s>Ex quibus &longs;equuntur omnia h&aelig;c.<lb/><arrow.to.target n="marg169"/></s></p><p type="margin">
  
 <s><margin.target id="marg165"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 57.</s></p><p type="margin"> <s><margin.target id="marg165"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 57.</s></p><p type="margin">
  
 <s><margin.target id="marg166"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 58.</s></p><p type="margin"> <s><margin.target id="marg166"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 58.</s></p><p type="margin">
  
 <s><margin.target id="marg167"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 59.</s></p><p type="margin"> <s><margin.target id="marg167"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 59.</s></p><p type="margin">
  
 <s><margin.target id="marg168"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 60.</s></p><p type="margin"> <s><margin.target id="marg168"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 60.</s></p><p type="margin">
  
 <s><margin.target id="marg169"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg169"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Prim&ugrave;m &longs;i quod incidit, molle fuerit, non uulneratur caput, uel <lb/>pars &longs;ubiecta, quia re&longs;ilit in corpus molle: nec &agrave; molli, quia retundi&shy;<lb/>tur, pote&longs;t uulnerari: ergo nullo modo. </s> <s>Prim&ugrave;m &longs;i quod incidit, molle fuerit, non uulneratur caput, uel <lb/>pars &longs;ubiecta, quia re&longs;ilit in corpus molle: nec &agrave; molli, quia retundi&shy;<lb/>tur, pote&longs;t uulnerari: ergo nullo modo. </s>
  
 <s>Sed neque ade&ograve; concutit, <lb/>quia a&euml;r rediens, &amp; receptus in molli corpore pro parte, non uer&shy;<lb/>berat locum.</s></p><p type="main"> <s>Sed neque ade&ograve; concutit, <lb/>quia a&euml;r rediens, &amp; receptus in molli corpore pro parte, non uer&shy;<lb/>berat locum.</s></p><p type="main">
  
 <s><arrow.to.target n="marg170"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg170"/></s></p><p type="margin">
  
 <s><margin.target id="marg170"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg170"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Secundum in omni colli&longs;ione &longs;eu duri, &longs;eu mollis, &longs;ed magis du&shy;<lb/>ri, dilabuntur partes a&euml;ris ad latera, ideo quod partes medi&aelig; pre&shy;<lb/>muntur. </s> <s>Secundum in omni colli&longs;ione &longs;eu duri, &longs;eu mollis, &longs;ed magis du&shy;<lb/>ri, dilabuntur partes a&euml;ris ad latera, ideo quod partes medi&aelig; pre&shy;<lb/>muntur. </s>
  
 <s>Et quanto motus e&longs;t tardior.</s></p><p type="main"> <s>Et quanto motus e&longs;t tardior.</s></p><p type="main">
  
 <s><arrow.to.target n="marg171"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg171"/></s></p><p type="margin">
  
 <s><margin.target id="marg171"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg171"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Tertium in motu uelo ci fit maior ictus &amp; l&aelig;&longs;io, &amp; maiora omnia <lb/>quam proproportione motus: quoniam ob uelo <expan abbr="citat&etilde;">citatem</expan> minus diffu <lb/>git a&euml;ris. </s> <s>Tertium in motu uelo ci fit maior ictus &amp; l&aelig;&longs;io, &amp; maiora omnia <lb/>quam proproportione motus: quoniam ob uelo <expan abbr="citat&etilde;">citatem</expan> minus diffu <lb/>git a&euml;ris. </s>
  
 <s>Et ide&ograve; fiunt grauia uulnera ex modico incremento uelo&shy;<lb/>citatis motus.</s></p><p type="main"> <s>Et ide&ograve; fiunt grauia uulnera ex modico incremento uelo&shy;<lb/>citatis motus.</s></p><p type="main">
  
 <s><arrow.to.target n="marg172"></arrow.to.target></s></p><p type="margin"> <s><arrow.to.target n="marg172"/></s></p><p type="margin">
  
 <s><margin.target id="marg172"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg172"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Quartum res lat&aelig;, dur&aelig; concutiunt, &amp; non uulnerant ni&longs;i &longs;int <lb/>cum magno impetu, aut ualde graues: acut&aelig; autem uulnerant, &longs;ed <lb/>non concutiunt, ni&longs;i parti acut&aelig; lata &longs;uccedat.</s></p><pb pagenum="53"/><p type="main"> <s>Quartum res lat&aelig;, dur&aelig; concutiunt, &amp; non uulnerant ni&longs;i &longs;int <lb/>cum magno impetu, aut ualde graues: acut&aelig; autem uulnerant, &longs;ed <lb/>non concutiunt, ni&longs;i parti acut&aelig; lata &longs;uccedat.</s></p><pb pagenum="53"/><p type="main">
  
 <s>Quintum, corpora dura magis l&aelig;duntur &agrave; latis, quia &longs;cindun&shy;</s></p><p type="main"> <s>Quintum, corpora dura magis l&aelig;duntur &agrave; latis, quia &longs;cindun&shy;</s></p><p type="main">
  
 <s><arrow.to.target n="marg173"></arrow.to.target><lb/>tur, mollia autem &agrave; tenuibus, quia diuiduntur: nam mollitie excipi&shy;<lb/>unt a&euml;rem, &amp; ita &agrave; latis non ade&ograve; patiuntur, &amp; etiam, quoniam nec <lb/>franguntur, nec &longs;ponte &longs;cinduntur.</s></p><p type="margin"> <s><arrow.to.target n="marg173"/><lb/>tur, mollia autem &agrave; tenuibus, quia diuiduntur: nam mollitie excipi&shy;<lb/>unt a&euml;rem, &amp; ita &agrave; latis non ade&ograve; patiuntur, &amp; etiam, quoniam nec <lb/>franguntur, nec &longs;ponte &longs;cinduntur.</s></p><p type="margin">
  
 <s><margin.target id="marg173"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg173"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Sextum, etiam in duris penetrat aliquid a&euml;ris, aliter tota frange&shy;<lb/><arrow.to.target n="marg174"></arrow.to.target><lb/>rentur. </s> <s>Sextum, etiam in duris penetrat aliquid a&euml;ris, aliter tota frange&shy;<lb/><arrow.to.target n="marg174"/><lb/>rentur. </s>
  
 <s>Con&longs;tat etiam omnem lapidem marmoreum, aut &longs;iliceum <lb/>e&longs;&longs;e poro&longs;um, ut dicunt. </s> <s>Con&longs;tat etiam omnem lapidem marmoreum, aut &longs;iliceum <lb/>e&longs;&longs;e poro&longs;um, ut dicunt. </s>
  
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 <s>Hoc etiam uidetur &longs;en&longs;i&longs;&longs;e Philo <lb/>&longs;ophus, qui uult, qu&ograve;d res franguntur ob poros.</s></p><p type="margin"> <s>Hoc etiam uidetur &longs;en&longs;i&longs;&longs;e Philo <lb/>&longs;ophus, qui uult, qu&ograve;d res franguntur ob poros.</s></p><p type="margin">
  
 <s><margin.target id="marg174"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg174"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Propo&longs;itio &longs;exage&longs;ima&longs;ecunda.</s></p><p type="main"> <s>Propo&longs;itio &longs;exage&longs;ima&longs;ecunda.</s></p><p type="main">
  
 <s>Proportionem motoris in plano ad motorem, qui eleuat pon&shy;<lb/>dus iuxta id, quod mouet inuenire.</s></p><p type="main"> <s>Proportionem motoris in plano ad motorem, qui eleuat pon&shy;<lb/>dus iuxta id, quod mouet inuenire.</s></p><p type="main">
  
 <s>Con&longs;titutum e&longs;t inuenire proportionem uirium, qu&aelig; eleuant <lb/><arrow.to.target n="marg175"></arrow.to.target><lb/>pondus ad uires, qu&aelig; ip&longs;um in plano leui trahere po&longs;&shy;<lb/><figure id="fig42"></figure><lb/>&longs;unt. </s> <s>Con&longs;titutum e&longs;t inuenire proportionem uirium, qu&aelig; eleuant <lb/><arrow.to.target n="marg175"/><lb/>pondus ad uires, qu&aelig; ip&longs;um in plano leui trahere po&longs;&shy;<lb/><figure id="fig42"/><lb/>&longs;unt. </s>
  
 <s>Vires enim, qu&aelig; eleuant pondus a &longs;unt e&aelig;dem <lb/>puta b, qu&aelig; uero trahunt c, &longs;ed h&aelig; po&longs;&longs;unt uariari, nam <lb/>quanto uinculum altius, aut decliuis locus magis, aut <lb/>a&longs;pera &longs;uperficies &longs;eu ponderis &longs;eu plani, tanto difficilius trahitur, <lb/>&amp; maiores expo&longs;cit uires: hoc enim experimento deprehenditur. <lb/></s> <s>Vires enim, qu&aelig; eleuant pondus a &longs;unt e&aelig;dem <lb/>puta b, qu&aelig; uero trahunt c, &longs;ed h&aelig; po&longs;&longs;unt uariari, nam <lb/>quanto uinculum altius, aut decliuis locus magis, aut <lb/>a&longs;pera &longs;uperficies &longs;eu ponderis &longs;eu plani, tanto difficilius trahitur, <lb/>&amp; maiores expo&longs;cit uires: hoc enim experimento deprehenditur. <lb/></s>
  
 <s>Du&aelig; uer&ograve; po&longs;trem&aelig; cau&longs;&aelig; etiam per &longs;e per&longs;picu&aelig; &longs;unt, nec demon <lb/>&longs;tratione indigent: ni&longs;i quod &longs;i planum &longs;it duri&longs;simum, ac leui&longs;si&shy;<lb/>mum, quod e&longs;t a&longs;perum facilius trahitur, quia minore &longs;ui parte pla&shy;<lb/>num tangit. </s> <s>Du&aelig; uer&ograve; po&longs;trem&aelig; cau&longs;&aelig; etiam per &longs;e per&longs;picu&aelig; &longs;unt, nec demon <lb/>&longs;tratione indigent: ni&longs;i quod &longs;i planum &longs;it duri&longs;simum, ac leui&longs;si&shy;<lb/>mum, quod e&longs;t a&longs;perum facilius trahitur, quia minore &longs;ui parte pla&shy;<lb/>num tangit. </s>
  
 <s>Nos pr&aelig;terea &longs;upponimus planum &aelig;quale undique <lb/>leue durum, &amp; corpus undique &longs;ibi &longs;imile, id e&longs;t cubi formam refe&shy;<lb/>rens, &amp; uinculum in imo: Demon&longs;trare igitur expedit primum, <lb/>qu&ograve;d in hoc ca&longs;u b e&longs;t duplum ad c. </s> <s>Nos pr&aelig;terea &longs;upponimus planum &aelig;quale undique <lb/>leue durum, &amp; corpus undique &longs;ibi &longs;imile, id e&longs;t cubi formam refe&shy;<lb/>rens, &amp; uinculum in imo: Demon&longs;trare igitur expedit primum, <lb/>qu&ograve;d in hoc ca&longs;u b e&longs;t duplum ad c. <!-- KEEP S--></s>
  
 <s>Quia enim cum a eleuatur b ui <lb/>res &longs;uperant motum ob&longs;curum &longs;eu occultum, &longs;eu pondus a, &amp; &longs;i <lb/>permitteretur &longs;ine eo, quod &longs;u&longs;tineret, de&longs;cenderet iuxta pondus <lb/>&longs;uum, quod &longs;it d: nititur ergo per pondus d, at quia trahendo duci&shy;<lb/>tur circa medium, nam plana &longs;uperficies parum differt &agrave; rotunda <lb/>terr&aelig; ob terr&aelig; magnitudinem, media erit repugnantia: in eo enim <lb/>quod mouetur, grauitatem habet d in eo, quod <expan abbr="n&otilde;">non</expan> remouetur nul&shy;<lb/>lam habet grauitatem, mediam ergo retinet grauitatem, quare ut b <lb/>ad d, ita c ad dimidium, grauitatis a, at b e&longs;t primum, quod pote&longs;t <lb/>mouere d, igitur c e&longs;t primum, quod pote&longs;t mouere dimidium a, ut <lb/>ergo dimidium a ad d, ita c ad b, e&longs;t igitur c dimidium b.</s></p><p type="margin"> <s>Quia enim cum a eleuatur b ui <lb/>res &longs;uperant motum ob&longs;curum &longs;eu occultum, &longs;eu pondus a, &amp; &longs;i <lb/>permitteretur &longs;ine eo, quod &longs;u&longs;tineret, de&longs;cenderet iuxta pondus <lb/>&longs;uum, quod &longs;it d: nititur ergo per pondus d, at quia trahendo duci&shy;<lb/>tur circa medium, nam plana &longs;uperficies parum differt &agrave; rotunda <lb/>terr&aelig; ob terr&aelig; magnitudinem, media erit repugnantia: in eo enim <lb/>quod mouetur, grauitatem habet d in eo, quod <expan abbr="n&otilde;">non</expan> remouetur nul&shy;<lb/>lam habet grauitatem, mediam ergo retinet grauitatem, quare ut b <lb/>ad d, ita c ad dimidium, grauitatis a, at b e&longs;t primum, quod pote&longs;t <lb/>mouere d, igitur c e&longs;t primum, quod pote&longs;t mouere dimidium a, ut <lb/>ergo dimidium a ad d, ita c ad b, e&longs;t igitur c dimidium b.</s></p><p type="margin">
  
 <s><margin.target id="marg175"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg175"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Propo&longs;itio &longs;exage&longs;imatertia.</s></p><p type="main"> <s>Propo&longs;itio &longs;exage&longs;imatertia.</s></p><p type="main">
  
 <s>Omne graue quanto proximius alligatum plano, tanto faci&shy;<lb/>lius trahitur.<pb pagenum="54"/><arrow.to.target n="marg176"></arrow.to.target></s></p><p type="margin"> <s>Omne graue quanto proximius alligatum plano, tanto faci&shy;<lb/>lius trahitur.<pb pagenum="54"/><arrow.to.target n="marg176"/></s></p><p type="margin">
  
 <s><margin.target id="marg176"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg176"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Sit graue a b c alligatum funibus in d ef, dico, <lb/><figure id="fig43"></figure><lb/>qu&ograve;d facilius trahetur per fe qu&agrave;m c b &amp; e b, qu&agrave;m <lb/>d a, quia &longs;i debet trahi ex a uel b, aut cadet, aut uis ex <lb/>a &amp; b communicabitur c, igitur erit minor qu&agrave;m in <lb/>c, &amp; hoc naturaliter. </s> <s>Sit graue a b c alligatum funibus in d ef, dico, <lb/><figure id="fig43"/><lb/>qu&ograve;d facilius trahetur per fe qu&agrave;m c b &amp; e b, qu&agrave;m <lb/>d a, quia &longs;i debet trahi ex a uel b, aut cadet, aut uis ex <lb/>a &amp; b communicabitur c, igitur erit minor qu&agrave;m in <lb/>c, &amp; hoc naturaliter. </s>
  
 <s>Mathematica autem ratione quoniam ex a tra&shy;<lb/>hetur c, qua&longs;i per lineam d c: at attractio recta e&longs;t ualidior obliqua&shy;<lb/>igitur attractio c per d e&longs;t debilior, qu&agrave;m per f. </s> <s>Mathematica autem ratione quoniam ex a tra&shy;<lb/>hetur c, qua&longs;i per lineam d c: at attractio recta e&longs;t ualidior obliqua&shy;<lb/>igitur attractio c per d e&longs;t debilior, qu&agrave;m per f. </s>
  
 <s>Rur&longs;us &longs;i e trahitur <lb/>per d c&ugrave;m a peruenerit in d, erit perinde ac, &longs;i attractum e&longs;&longs;et per li&shy;<lb/>neam c d, &longs;ed linea c d mouet duobus motibus, uno ad &longs;uperiora, al </s></p><p type="main"> <s>Rur&longs;us &longs;i e trahitur <lb/>per d c&ugrave;m a peruenerit in d, erit perinde ac, &longs;i attractum e&longs;&longs;et per li&shy;<lb/>neam c d, &longs;ed linea c d mouet duobus motibus, uno ad &longs;uperiora, al </s></p><p type="main">
  
 <s><arrow.to.target n="marg177"></arrow.to.target><lb/>tero ad latus, ergo lentius ad f per d c qu&agrave;m f c, quod erat demon&shy;<lb/>&longs;trandum.</s></p><p type="margin"> <s><arrow.to.target n="marg177"/><lb/>tero ad latus, ergo lentius ad f per d c qu&agrave;m f c, quod erat demon&shy;<lb/>&longs;trandum.</s></p><p type="margin">
  
 <s><margin.target id="marg177"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 59. <emph type="italics"/>bu-ius.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg177"/>P<emph type="italics"/>er<emph.end type="italics"/> 59. <emph type="italics"/>bu-ius.<emph.end type="italics"/></s></p><p type="main">
  
 <s>Propo&longs;itio &longs;exage&longs;imaquarta.</s></p><p type="main"> <s>Propo&longs;itio &longs;exage&longs;imaquarta.</s></p><p type="main">
  
 <s>Omne mobile quanto latius tanto tardius mouetur in plano.<lb/><arrow.to.target n="marg178"></arrow.to.target></s></p><p type="margin"> <s>Omne mobile quanto latius tanto tardius mouetur in plano.<lb/><arrow.to.target n="marg178"/></s></p><p type="margin">
  
 <s><margin.target id="marg178"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg178"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Demon&longs;tratum e&longs;t &longs;uperius qu&ograve;d &longs;i mobile &longs;it &longs;ph&ecedil;ricum, &amp; tan </s></p><p type="main"> <s>Demon&longs;tratum e&longs;t &longs;uperius qu&ograve;d &longs;i mobile &longs;it &longs;ph&ecedil;ricum, &amp; tan </s></p><p type="main">
  
 <s><arrow.to.target n="marg179"></arrow.to.target><lb/>gat planum in puncto, qu&ograve;d mouetur per quancunque uim aptam <lb/>diuidere medium. </s> <s><arrow.to.target n="marg179"/><lb/>gat planum in puncto, qu&ograve;d mouetur per quancunque uim aptam <lb/>diuidere medium. </s>
  
 <s>Quia ergo &longs;i tangat in puncto facillime moue&shy;<lb/>tur, &longs;i in linea paul&ograve; difficilius, &longs;i per &longs;uperficiem adhuc difficilius, <lb/>igitur cum fiat attritio in motu quanto latius e&longs;t mobile eo diffici&shy;<lb/>lius mouetur. </s> <s>Quia ergo &longs;i tangat in puncto facillime moue&shy;<lb/>tur, &longs;i in linea paul&ograve; difficilius, &longs;i per &longs;uperficiem adhuc difficilius, <lb/>igitur cum fiat attritio in motu quanto latius e&longs;t mobile eo diffici&shy;<lb/>lius mouetur. </s>
  
 <s>Sit ergo mobile a b, quod moueatur uer&longs;us c, &amp; quia <lb/>pars b &longs;eu dimidium mouetur iuxta rationem me&shy;<lb/><figure id="fig44"></figure><lb/>dietatis, &amp; pars a eodem modo, ergo conduplicata <lb/>difficultate, quia medietas b impedit medietatem, a <lb/>quanto latius e&longs;t, &amp; longius a b, tanto difficilius <lb/><arrow.to.target n="marg180"></arrow.to.target><lb/>mouetur. </s> <s>Sit ergo mobile a b, quod moueatur uer&longs;us c, &amp; quia <lb/>pars b &longs;eu dimidium mouetur iuxta rationem me&shy;<lb/><figure id="fig44"/><lb/>dietatis, &amp; pars a eodem modo, ergo conduplicata <lb/>difficultate, quia medietas b impedit medietatem, a <lb/>quanto latius e&longs;t, &amp; longius a b, tanto difficilius <lb/><arrow.to.target n="marg180"/><lb/>mouetur. </s>
  
 <s>Et hoc intelligitur de corporibus ualde <lb/>latis propter dicta &longs;uperius.</s></p><p type="margin"> <s>Et hoc intelligitur de corporibus ualde <lb/>latis propter dicta &longs;uperius.</s></p><p type="margin">
  
 <s><margin.target id="marg179"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 40.</s></p><p type="margin"> <s><margin.target id="marg179"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 40.</s></p><p type="margin">
  
 <s><margin.target id="marg180"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 62</s></p><p type="main"> <s><margin.target id="marg180"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 62</s></p><p type="main">
  
 <s>Propo&longs;itio &longs;exage&longs;imaquinta.</s></p><p type="main"> <s>Propo&longs;itio &longs;exage&longs;imaquinta.</s></p><p type="main">
  
 <s>Proportionem duorum mobilium inter &longs;e cum auxilio medij <lb/>inuenire.<lb/><arrow.to.target n="marg181"></arrow.to.target></s></p><p type="margin"> <s>Proportionem duorum mobilium inter &longs;e cum auxilio medij <lb/>inuenire.<lb/><arrow.to.target n="marg181"/></s></p><p type="margin">
  
 <s><margin.target id="marg181"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main"> <s><margin.target id="marg181"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Graue de&longs;cendit naturaliter quatuor cau&longs;is: prima e&longs;t ponderis <lb/>magnitudo, unde quod grauius e&longs;t celerius de&longs;cendit. </s> <s>Graue de&longs;cendit naturaliter quatuor cau&longs;is: prima e&longs;t ponderis <lb/>magnitudo, unde quod grauius e&longs;t celerius de&longs;cendit. </s>
  
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 <s>Ter&shy;<lb/>ti&ograve; ob impetum a&euml;ris &longs;ub &longs;equentis: &amp; ideo mobile qu&ograve;d ex eadem </s></p><p type="main"> <s>Ter&shy;<lb/>ti&ograve; ob impetum a&euml;ris &longs;ub &longs;equentis: &amp; ideo mobile qu&ograve;d ex eadem </s></p><p type="main">
  
 <s><arrow.to.target n="marg182"></arrow.to.target><lb/>materia con&longs;tat, &longs;emper de&longs;cendit parte acutiore &longs;uprapo&longs;ita, ne a&euml;r <lb/>cogatur celerius ferri: &amp; quanto diutius de&longs;cendit, tanto magis in&shy;<lb/>tenditur motus, at que augetur, ut &longs;upr&agrave; de claratum e&longs;t. </s> <s><arrow.to.target n="marg182"/><lb/>materia con&longs;tat, &longs;emper de&longs;cendit parte acutiore &longs;uprapo&longs;ita, ne a&euml;r <lb/>cogatur celerius ferri: &amp; quanto diutius de&longs;cendit, tanto magis in&shy;<lb/>tenditur motus, at que augetur, ut &longs;upr&agrave; de claratum e&longs;t. </s>
  
 <s>Quarta cau&longs;a <lb/>e&longs;t, quod non impediatur ab a&euml;re tran&longs;uerfim moto, et &agrave; latere: ideo <lb/>leuia mobilia &amp; magna non &longs;olum lentius de&longs;cendunt, quoniam <lb/><arrow.to.target n="marg183"></arrow.to.target><lb/>paruam uim habeant, &amp; magnam repugnantiam, &longs;ed quia tran&longs;uer <lb/><arrow.to.target n="marg184"></arrow.to.target><lb/>&longs;im impul&longs;a minus mouentur motu recto, ut &longs;upra ui&longs;um e&longs;t. </s> <s>Quarta cau&longs;a <lb/>e&longs;t, quod non impediatur ab a&euml;re tran&longs;uerfim moto, et &agrave; latere: ideo <lb/>leuia mobilia &amp; magna non &longs;olum lentius de&longs;cendunt, quoniam <lb/><arrow.to.target n="marg183"/><lb/>paruam uim habeant, &amp; magnam repugnantiam, &longs;ed quia tran&longs;uer <lb/><arrow.to.target n="marg184"/><lb/>&longs;im impul&longs;a minus mouentur motu recto, ut &longs;upra ui&longs;um e&longs;t. </s>
  
 <s>Por&shy;<pb pagenum="55"/>r&ograve; proportio ratione de&longs;cen&longs;us aucta, declarata e&longs;t paulo ant&egrave;, <lb/>quare cum medium &longs;upponatur eiu&longs;dem generis, &amp; figura non <lb/>eiu&longs;modi, nec leuitas, ut pror&longs;us non impellat, nedum ut moueat la <lb/>tus: figura quo que eadem ambobus relinquetur proportio motus <lb/>ad motum producta ex proportionibus incrementi in proportio&shy;<lb/><arrow.to.target n="marg185"></arrow.to.target><lb/>nem ponderum, &amp; iam habuimus proportionem incrementi ex <lb/><arrow.to.target n="marg186"></arrow.to.target><lb/>motu a&euml;ris, ergo proportio unius motus producti ad alteram no&shy;<lb/>ta erit.</s></p><p type="margin"> <s>Por&shy;<pb pagenum="55"/>r&ograve; proportio ratione de&longs;cen&longs;us aucta, declarata e&longs;t paulo ant&egrave;, <lb/>quare cum medium &longs;upponatur eiu&longs;dem generis, &amp; figura non <lb/>eiu&longs;modi, nec leuitas, ut pror&longs;us non impellat, nedum ut moueat la <lb/>tus: figura quo que eadem ambobus relinquetur proportio motus <lb/>ad motum producta ex proportionibus incrementi in proportio&shy;<lb/><arrow.to.target n="marg185"/><lb/>nem ponderum, &amp; iam habuimus proportionem incrementi ex <lb/><arrow.to.target n="marg186"/><lb/>motu a&euml;ris, ergo proportio unius motus producti ad alteram no&shy;<lb/>ta erit.</s></p><p type="margin">
  
 <s><margin.target id="marg182"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 30.</s></p><p type="margin"> <s><margin.target id="marg182"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 30.</s></p><p type="margin">
  
 <s><margin.target id="marg183"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 59.</s></p><p type="margin"> <s><margin.target id="marg183"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 59.</s></p><p type="margin">
  
 <s><margin.target id="marg184"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 62.</s></p><p type="margin"> <s><margin.target id="marg184"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 62.</s></p><p type="margin">
  
 <s><margin.target id="marg185"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 42. <emph type="italics"/>ha-rum.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg185"/>P<emph type="italics"/>er<emph.end type="italics"/> 42. <emph type="italics"/>ha-rum.<emph.end type="italics"/></s></p><p type="margin">
  
 <s><margin.target id="marg186"></margin.target>I<emph type="italics"/>n<emph.end type="italics"/> 61. <emph type="italics"/>ha-rum.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg186"/>I<emph type="italics"/>n<emph.end type="italics"/> 61. <emph type="italics"/>ha-rum.<emph.end type="italics"/></s></p><p type="main">
  
 <s>Propo&longs;itio &longs;exage&longs;ima&longs;exta.</s></p><p type="main"> <s>Propo&longs;itio &longs;exage&longs;ima&longs;exta.</s></p><p type="main">
  
 <s>Proportionem laterum eptagoni, &amp; &longs;ubten&longs;arum con&longs;iderare, <lb/>&amp; qu&aelig; &agrave; reflexa proportione pendent.<lb/><arrow.to.target n="marg187"></arrow.to.target></s></p><p type="margin"> <s>Proportionem laterum eptagoni, &amp; &longs;ubten&longs;arum con&longs;iderare, <lb/>&amp; qu&aelig; &agrave; reflexa proportione pendent.<lb/><arrow.to.target n="marg187"/></s></p><p type="margin">
  
 <s><margin.target id="marg187"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main"> <s><margin.target id="marg187"/>C<emph type="italics"/>o<emph.end type="italics"/>m.<!-- KEEP S--></s></p><p type="main">
  
 <s>Sit eptagonus a b d f g e c, &amp; &longs;ubten&longs;&aelig; b <lb/><figure id="fig45"></figure><lb/>c, &amp; f e duobus lateribus, tribus autem d c <lb/>d e, &amp; erunt (quia intelligitur eptagono &aelig;&shy;<lb/>quilatero, &amp; &aelig;quiangulo) b c &amp; e finuicem <lb/>&aelig;quales: &amp; item d c, &amp; d e &aelig;quales: &amp; &longs;i du&shy;<lb/>cerentur b e &amp; c f inuicem &aelig;quales: &amp; ad a c <lb/>&amp; d g: quare cum angulus cb d con&longs;i&longs;tatin </s></p><p type="main"> <s>Sit eptagonus a b d f g e c, &amp; &longs;ubten&longs;&aelig; b <lb/><figure id="fig45"/><lb/>c, &amp; f e duobus lateribus, tribus autem d c <lb/>d e, &amp; erunt (quia intelligitur eptagono &aelig;&shy;<lb/>quilatero, &amp; &aelig;quiangulo) b c &amp; e finuicem <lb/>&aelig;quales: &amp; item d c, &amp; d e &aelig;quales: &amp; &longs;i du&shy;<lb/>cerentur b e &amp; c f inuicem &aelig;quales: &amp; ad a c <lb/>&amp; d g: quare cum angulus cb d con&longs;i&longs;tatin </s></p><p type="main">
  
 <s><arrow.to.target n="marg188"></arrow.to.target><lb/>arcu c e g f d, &amp; angulus b d c in arcu b a c, <lb/>&amp; angulus b c d in arcu b d; &amp; &longs;it arcus c e g <lb/>f d duplus arcus b a c, quia c e g f d &longs;ubtendit quatuor latera epta&shy;<lb/>goni, &amp; arcus b a c duo, &amp; ita arcus etiam b a c duplus arcui b d <lb/>erit angulus d b e duplus angulo c d b, &amp; angulus c d b duplus an&shy;<lb/><arrow.to.target n="marg189"></arrow.to.target><lb/>gulo b c d, quare per demon&longs;trata &agrave; nobis proportio laterum b d, <lb/>b c, c d, e&longs;t reflexa, igitur proportio d b &amp; b c, ad d c, ut d e ad b c, &amp; <lb/><arrow.to.target n="marg190"></arrow.to.target><lb/>rur&longs;us proportio b d &amp; d e ad b e, ut b e ad b d. </s> <s><arrow.to.target n="marg188"/><lb/>arcu c e g f d, &amp; angulus b d c in arcu b a c, <lb/>&amp; angulus b c d in arcu b d; &amp; &longs;it arcus c e g <lb/>f d duplus arcus b a c, quia c e g f d &longs;ubtendit quatuor latera epta&shy;<lb/>goni, &amp; arcus b a c duo, &amp; ita arcus etiam b a c duplus arcui b d <lb/>erit angulus d b e duplus angulo c d b, &amp; angulus c d b duplus an&shy;<lb/><arrow.to.target n="marg189"/><lb/>gulo b c d, quare per demon&longs;trata &agrave; nobis proportio laterum b d, <lb/>b c, c d, e&longs;t reflexa, igitur proportio d b &amp; b c, ad d c, ut d e ad b c, &amp; <lb/><arrow.to.target n="marg190"/><lb/>rur&longs;us proportio b d &amp; d e ad b e, ut b e ad b d. <!-- KEEP S--></s>
  
 <s>Quare &longs;uppo&longs;ita <lb/>d b 1, b c 1 po&longs;itione, erit d c latus 1 quad. </s> <s>Quare &longs;uppo&longs;ita <lb/>d b 1, b c 1 po&longs;itione, erit d c latus 1 quad. </s>
  
 <s>p: 1 po&longs;itione. </s> <s>p: 1 po&longs;itione. </s>
  
 <s>Proportio <lb/><arrow.to.target n="marg191"></arrow.to.target><lb/>uer&ograve;, ut dictum e&longs;t b d &amp; d c ad b c, id e&longs;t p: &lt;02&gt; 1 quad. </s> <s>Proportio <lb/><arrow.to.target n="marg191"/><lb/>uer&ograve;, ut dictum e&longs;t b d &amp; d c ad b c, id e&longs;t p: &lt;02&gt; 1 quad. </s>
  
 <s>p: 1 pos, ad 1 <lb/>pos e&longs;t, ut b c ad b d, id e&longs;t 1 pos ad 1, igitur 1 p: &lt;02&gt; v: 1 quad. </s> <s>p: 1 pos, ad 1 <lb/>pos e&longs;t, ut b c ad b d, id e&longs;t 1 pos ad 1, igitur 1 p: &lt;02&gt; v: 1 quad. </s>
  
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 <s>&aelig;qualem 1 3/4 pos p: 7/8.</s></p><p type="margin"> <s>&aelig;qualem 1 3/4 pos p: 7/8.</s></p><p type="margin">
  
 <s><margin.target id="marg188"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 28. &amp; 29. <emph type="italics"/>tertij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg188"/>P<emph type="italics"/>er<emph.end type="italics"/> 28. &amp; 29. <emph type="italics"/>tertij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg189"></margin.target>P<emph type="italics"/>er ult. </s> <s><margin.target id="marg189"/>P<emph type="italics"/>er ult. </s>
  
 <s>&longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin"> <s>&longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg190"></margin.target>D<emph type="italics"/>e<emph.end type="italics"/> S<emph type="italics"/>uh. </s> <s><margin.target id="marg190"/>D<emph type="italics"/>e<emph.end type="italics"/> S<emph type="italics"/>uh. <!-- REMOVE S-->lib.<emph.end type="italics"/> 16.</s>
  
 <s>lib.<emph.end type="italics"/> 16.</s></p><p type="margin"> </p><p type="margin">
  
 <s><margin.target id="marg191"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg191"/>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="main">
  
 <s>Aliter &longs;tante &longs;uppo&longs;itione ut Ludouicus Ferrarius ex demon&shy;<lb/>&longs;tratis &agrave; Ptolem&aelig;o quadratum b c, &amp; e&longs;t 1 quad e&longs;t &aelig;quale produ&shy;<lb/>cto ex b d in c e, quod e&longs;t 1, &amp; a b in d c, igitur detracto 1, produ&shy;<lb/>cto b d in c e ex 1 quad. </s> <s>Aliter &longs;tante &longs;uppo&longs;itione ut Ludouicus Ferrarius ex demon&shy;<lb/>&longs;tratis &agrave; Ptolem&aelig;o quadratum b c, &amp; e&longs;t 1 quad e&longs;t &aelig;quale produ&shy;<lb/>cto ex b d in c e, quod e&longs;t 1, &amp; a b in d c, igitur detracto 1, produ&shy;<lb/>cto b d in c e ex 1 quad. </s>
  
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 <s>p: 2 pos.</s></p><p type="main"> <s>p: 2 pos.</s></p><p type="main">
  
 <s>Aliter ut Pacciolus, concurrant latera eptagoni b d, c e in a, &amp; du <lb/>cantur perpendiculares a f, d g &amp; c d, &amp; &longs;it c e i ca 1 pos, &amp; quia ut <lb/><arrow.to.target n="marg192"></arrow.to.target><lb/>a e ad a c, ita d e ad b c, erit ergo b c (1 posp: 1)/(1 pos) quare b f (1/2 pos 1/2,)/(2 pos) &amp; <lb/>quia d h e&longs;t dimidium d e, erit d h, &amp; g f <lb/><figure id="fig46"></figure><lb/>1/2, cum ergo b f &longs;it (1/2 pos p: 1/2)/pos erit ergo di&shy;<lb/>ui&longs;a 1/2 pos per 1 pos, &amp; exit 1/2, b f 1/2p: 1/2/pos <lb/>igitur detracta g f relinquetur g b 1/2/(1 pos). <lb/>&amp; eius quadratum 1/4/(1 quad). igitur cum qua&shy;<lb/>dratum b d &longs;it 1, erit quadratum g d 1 m: <lb/>2/4/(2 quad)g c autem e&longs;t compo&longs;ita ex e f, qu&aelig; <lb/>e&longs;t 1/2p: 1/2/(1 pos) &amp; f g qu&aelig; e&longs;t 1/2, erit igitur c <lb/>g 1 p: 1/2/(1 pos), &amp; <expan abbr="quadrat&utilde;">quadratum</expan> eius 1 p: 1/pos e&longs;t 1/4/(1 quad.) quare <expan abbr="&qtilde;drat&utilde;">quadratum</expan> e d &qring;d e&longs;t <lb/><arrow.to.target n="marg193"></arrow.to.target><lb/>compo&longs;itum ex quadratis c g &amp; g d erit 2 p: 1/pos c a uer&ograve; e&longs;t &aelig;qua&shy;<lb/>lis c d, quia, ut demon&longs;tratum e&longs;t angulus d c e e&longs;t &longs;eptima pars <lb/>duorum rectorum, &amp; angulus b c e ei duplus, quare cum c f a &longs;it re&shy;<lb/>ctus erit ex trige&longs;ima&longs;ecunda primi Elementorum f a c tres &longs;epti&shy;<lb/>m&aelig; unius recti, ergo d a c 6/7 unius recti, d c a uer&ograve; 2/7 unius recti, quia <lb/><arrow.to.target n="marg194"></arrow.to.target><lb/>e&longs;t &longs;eptima pars duorum rectorum, &iacute;gitur a d c e&longs;t 6/7 unius recti: igi&shy;<lb/>tur c d e&longs;t &aelig;qualis c a, ergo quadratum quadrato: igitur 1 quad. </s> <s>Aliter ut Pacciolus, concurrant latera eptagoni b d, c e in a, &amp; du <lb/>cantur perpendiculares a f, d g &amp; c d, &amp; &longs;it c e i ca 1 pos, &amp; quia ut <lb/><arrow.to.target n="marg192"/><lb/>a e ad a c, ita d e ad b c, erit ergo b c (1 posp: 1)/(1 pos) quare b f (1/2 pos 1/2,)/(2 pos) &amp; <lb/>quia d h e&longs;t dimidium d e, erit d h, &amp; g f <lb/><figure id="fig46"/><lb/>1/2, cum ergo b f &longs;it (1/2 pos p: 1/2)/pos erit ergo di&shy;<lb/>ui&longs;a 1/2 pos per 1 pos, &amp; exit 1/2, b f 1/2p: 1/2/pos <lb/>igitur detracta g f relinquetur g b 1/2/(1 pos). <lb/>&amp; eius quadratum 1/4/(1 quad). igitur cum qua&shy;<lb/>dratum b d &longs;it 1, erit quadratum g d 1 m: <lb/>2/4/(2 quad)g c autem e&longs;t compo&longs;ita ex e f, qu&aelig; <lb/>e&longs;t 1/2p: 1/2/(1 pos) &amp; f g qu&aelig; e&longs;t 1/2, erit igitur c <lb/>g 1 p: 1/2/(1 pos), &amp; <expan abbr="quadrat&utilde;">quadratum</expan> eius 1 p: 1/pos e&longs;t 1/4/(1 quad.) quare <expan abbr="&qtilde;drat&utilde;">quadratum</expan> e d &qring;d e&longs;t <lb/><arrow.to.target n="marg193"/><lb/>compo&longs;itum ex quadratis c g &amp; g d erit 2 p: 1/pos c a uer&ograve; e&longs;t &aelig;qua&shy;<lb/>lis c d, quia, ut demon&longs;tratum e&longs;t angulus d c e e&longs;t &longs;eptima pars <lb/>duorum rectorum, &amp; angulus b c e ei duplus, quare cum c f a &longs;it re&shy;<lb/>ctus erit ex trige&longs;ima&longs;ecunda primi Elementorum f a c tres &longs;epti&shy;<lb/>m&aelig; unius recti, ergo d a c 6/7 unius recti, d c a uer&ograve; 2/7 unius recti, quia <lb/><arrow.to.target n="marg194"/><lb/>e&longs;t &longs;eptima pars duorum rectorum, &iacute;gitur a d c e&longs;t 6/7 unius recti: igi&shy;<lb/>tur c d e&longs;t &aelig;qualis c a, ergo quadratum quadrato: igitur 1 quad. </s>
  
 <s>p: 2 <lb/>pos p: 1, &aelig;quatur 2 p: 1/(1 pos) igitur 1 quad. </s> <s>p: 2 <lb/>pos p: 1, &aelig;quatur 2 p: 1/(1 pos) igitur 1 quad. </s>
  
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 <s>p: 2 quad. </s> <s>p: 2 quad. </s>
  
 <s>&aelig;quatur 1 pos p: 1. <lb/><figure id="fig47"></figure><lb/>Sit etiam angulus a duplus b, &amp; b c dupla <lb/>b a: &amp; erit per eadem proportio a c, &amp; a b <lb/>ad c b, ut c b ad c a. </s> <s>&aelig;quatur 1 pos p: 1. <lb/><figure id="fig47"/><lb/>Sit etiam angulus a duplus b, &amp; b c dupla <lb/>b a: &amp; erit per eadem proportio a c, &amp; a b <lb/>ad c b, ut c b ad c a. </s>
  
 <s>Ponamus ergo ab 1, erit <lb/>b c 2, &amp; a c 1 pos, &amp; a c, a b 1 pos p: 1, &amp; du&shy;<lb/>cta in a c fit 1 quad. </s> <s>Ponamus ergo ab 1, erit <lb/>b c 2, &amp; a c 1 pos, &amp; a c, a b 1 pos p: 1, &amp; du&shy;<lb/>cta in a c fit 1 quad. </s>
  
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 <s>Igitur a c e&longs;t &lt;02&gt; 4 1/4 m: 1/2, &amp; ita <lb/>de alijs.</s></p><p type="margin"> <s>Igitur a c e&longs;t &lt;02&gt; 4 1/4 m: 1/2, &amp; ita <lb/>de alijs.</s></p><p type="margin">
  
 <s><margin.target id="marg192"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 42. <emph type="italics"/>pri mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg192"/>P<emph type="italics"/>er<emph.end type="italics"/> 42. <emph type="italics"/>pri mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg193"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 32. <emph type="italics"/>pri mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg193"/>P<emph type="italics"/>er<emph.end type="italics"/> 32. <emph type="italics"/>pri mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg194"></margin.target>P<emph type="italics"/>er &longs;extam eiu&longs;dem.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="marg194"/>P<emph type="italics"/>er &longs;extam eiu&longs;dem.<emph.end type="italics"/></s></p><p type="main">
  
 <s>Propo&longs;itio &longs;exage&longs;ima&longs;eptima.</s></p><p type="main"> <s>Propo&longs;itio &longs;exage&longs;ima&longs;eptima.</s></p><p type="main">
  
 <s>Si fuerint aliquot quantitates ab una quantitate, ali&aelig;que totidem <pb pagenum="57"/>ab eadem analo g&aelig;, erit proportio terti&aelig; unius ordinis ad tertiam <lb/>alterius, ut &longs;ecund&aelig; ad &longs;ecundam duplicata, &amp; quart&aelig; ad quartam <lb/>triplicata, quint&aelig; ad quintam quadruplicata, at que &longs;ic de alijs.<lb/><arrow.to.target n="marg195"></arrow.to.target></s></p><p type="margin"> <s>Si fuerint aliquot quantitates ab una quantitate, ali&aelig;que totidem <pb pagenum="57"/>ab eadem analo g&aelig;, erit proportio terti&aelig; unius ordinis ad tertiam <lb/>alterius, ut &longs;ecund&aelig; ad &longs;ecundam duplicata, &amp; quart&aelig; ad quartam <lb/>triplicata, quint&aelig; ad quintam quadruplicata, at que &longs;ic de alijs.<lb/><arrow.to.target n="marg195"/></s></p><p type="margin">
  
 <s><margin.target id="marg195"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="main"> <s><margin.target id="marg195"/>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.<!-- KEEP S--></s></p><p type="main">
  
 <s>Sint quantitates b c d e f, ab a in continua proportio&shy;<lb/><arrow.to.target n="table14"></arrow.to.target><lb/>ne, &amp; ali&aelig; totidem g h k l m, dico quod proportio h c e&longs;t <lb/>duplicata ei, qu&aelig; e&longs;t g ad b, &amp; k ad d triplicata, &amp; l ad e <lb/>quadruplicata, &amp; &longs;ic deinceps, &longs;umatur enim unum, &amp; ab </s></p><table><table.target id="table14"></table.target><row><cell></cell><cell>a</cell><cell></cell></row><row><cell>b</cell><cell></cell><cell>g</cell></row><row><cell>c</cell><cell></cell><cell>h</cell></row><row><cell>d</cell><cell></cell><cell>k</cell></row><row><cell>e</cell><cell></cell><cell>l</cell></row><row><cell>f</cell><cell></cell><cell>m</cell></row><row><cell></cell><cell>n</cell><cell></cell></row><row><cell>o</cell><cell></cell><cell>t</cell></row><row><cell>p</cell><cell><foreign lang="greek">a</foreign></cell><cell>u</cell></row><row><cell>q</cell><cell><foreign lang="greek">b g</foreign></cell><cell>x</cell></row><row><cell>z</cell><cell></cell><cell>y</cell></row><row><cell>s</cell><cell></cell><cell>z</cell></row></table><p type="main"> <s>Sint quantitates b c d e f, ab a in continua proportio&shy;<lb/><arrow.to.target n="table14"/><lb/>ne, &amp; ali&aelig; totidem g h k l m, dico quod proportio h c e&longs;t <lb/>duplicata ei, qu&aelig; e&longs;t g ad b, &amp; k ad d triplicata, &amp; l ad e <lb/>quadruplicata, &amp; &longs;ic deinceps, &longs;umatur enim unum, &amp; ab </s></p><table><table.target id="table14"/><row><cell/><cell>a</cell><cell/></row><row><cell>b</cell><cell/><cell>g</cell></row><row><cell>c</cell><cell/><cell>h</cell></row><row><cell>d</cell><cell/><cell>k</cell></row><row><cell>e</cell><cell/><cell>l</cell></row><row><cell>f</cell><cell/><cell>m</cell></row><row><cell/><cell>n</cell><cell/></row><row><cell>o</cell><cell/><cell>t</cell></row><row><cell>p</cell><cell><foreign lang="greek">a</foreign></cell><cell>u</cell></row><row><cell>q</cell><cell><foreign lang="greek">b g</foreign></cell><cell>x</cell></row><row><cell>z</cell><cell/><cell>y</cell></row><row><cell>s</cell><cell/><cell>z</cell></row></table><p type="main">
  
 <s><arrow.to.target n="marg196"></arrow.to.target><lb/>co o p q r s in proportione b ad a, &amp; tuxyz in propor&shy;<lb/>tione g ad a, erit igitur p quadratum o, &amp; u quadratum t, <lb/>&amp; q cubus o, &amp; x cubus t, &amp; ita de alijs: ergo proportio <lb/><arrow.to.target n="marg197"></arrow.to.target><lb/>n ad p duplicata ei, qu&aelig; t ad o, &amp; x ad q triplicata ei, qu&aelig;t <lb/>ad o, &amp; pote&longs;t etiam demon&longs;trari generaliter ultra qua&shy;<lb/><arrow.to.target n="marg198"></arrow.to.target><lb/>dratum, &amp; cubum: nam &longs;i ducatur t in o, fiat que <foreign lang="greek">a</foreign> erit, pro&shy;<lb/>portio enim ad <foreign lang="greek">a</foreign> eadem qu&aelig; t ad o, &amp; proportio a ad p, <lb/>ut t ad o, igitur per diffinitionem proportionis duplicat&aelig; <lb/><arrow.to.target n="marg199"></arrow.to.target><lb/>po&longs;itam in quinto libro ab Euclide u ad p duplicata ei, <lb/>qu&aelig; t ad o, &amp; &longs;imiliter ex t in p fit <foreign lang="greek">b</foreign> ex o in u, <foreign lang="greek">g</foreign> eruntque<lb/><arrow.to.target n="marg200"></arrow.to.target><lb/>q <foreign lang="greek">b g</foreign> x in continua proportione per eandem. </s> <s><arrow.to.target n="marg196"/><lb/>co o p q r s in proportione b ad a, &amp; tuxyz in propor&shy;<lb/>tione g ad a, erit igitur p quadratum o, &amp; u quadratum t, <lb/>&amp; q cubus o, &amp; x cubus t, &amp; ita de alijs: ergo proportio <lb/><arrow.to.target n="marg197"/><lb/>n ad p duplicata ei, qu&aelig; t ad o, &amp; x ad q triplicata ei, qu&aelig;t <lb/>ad o, &amp; pote&longs;t etiam demon&longs;trari generaliter ultra qua&shy;<lb/><arrow.to.target n="marg198"/><lb/>dratum, &amp; cubum: nam &longs;i ducatur t in o, fiat que <foreign lang="greek">a</foreign> erit, pro&shy;<lb/>portio enim ad <foreign lang="greek">a</foreign> eadem qu&aelig; t ad o, &amp; proportio a ad p, <lb/>ut t ad o, igitur per diffinitionem proportionis duplicat&aelig; <lb/><arrow.to.target n="marg199"/><lb/>po&longs;itam in quinto libro ab Euclide u ad p duplicata ei, <lb/>qu&aelig; t ad o, &amp; &longs;imiliter ex t in p fit <foreign lang="greek">b</foreign> ex o in u, <foreign lang="greek">g</foreign> eruntque<lb/><arrow.to.target n="marg200"/><lb/>q <foreign lang="greek">b g</foreign> x in continua proportione per eandem. </s>
  
 <s>Quia ergo propor&shy;<lb/>tio q ad <foreign lang="greek">b</foreign> e&longs;t ut o ad t, patet, quod x ad q e&longs;t triplicata ei, qu&aelig; e&longs;t t ad <lb/>o, &amp; ita de reliquis, cum ergo proportio p ad o &longs;it, ut e ad b, &amp; o ad <lb/><arrow.to.target n="marg201"></arrow.to.target><lb/>n, ut b ad a, &amp; n ad t, ut a ad g, &amp; t ad u, ut g ad h, &longs;equitur ut &longs;it t ad a, <lb/>ut g ad b, &amp; u ad p, ut h ad c, igitur cum &longs;it ut u ad p duplicata ei, qu&ecedil; <lb/>e&longs;t t ad o erit h ad e, duplicata ei qu&aelig; e&longs;t g ad b, &amp; ita de reliquis, &amp; <lb/>no&ngrave; refert, &longs;eu dicas u ad p duplicatam ei, qu&aelig; e&longs;t t ad o, &longs;eu dicas p <lb/><arrow.to.target n="marg202"></arrow.to.target><lb/>ad u duplicatam ei, qu&aelig; e&longs;t o ad t. </s> <s>Quia ergo propor&shy;<lb/>tio q ad <foreign lang="greek">b</foreign> e&longs;t ut o ad t, patet, quod x ad q e&longs;t triplicata ei, qu&aelig; e&longs;t t ad <lb/>o, &amp; ita de reliquis, cum ergo proportio p ad o &longs;it, ut e ad b, &amp; o ad <lb/><arrow.to.target n="marg201"/><lb/>n, ut b ad a, &amp; n ad t, ut a ad g, &amp; t ad u, ut g ad h, &longs;equitur ut &longs;it t ad a, <lb/>ut g ad b, &amp; u ad p, ut h ad c, igitur cum &longs;it ut u ad p duplicata ei, qu&ecedil; <lb/>e&longs;t t ad o erit h ad e, duplicata ei qu&aelig; e&longs;t g ad b, &amp; ita de reliquis, &amp; <lb/>no&ngrave; refert, &longs;eu dicas u ad p duplicatam ei, qu&aelig; e&longs;t t ad o, &longs;eu dicas p <lb/><arrow.to.target n="marg202"/><lb/>ad u duplicatam ei, qu&aelig; e&longs;t o ad t. </s>
  
 <s>Aliter &amp; euidentius in duabus <lb/>&longs;oleo demon&longs;trare: cum enim &longs;it e &amp; h duplicata ei qu&aelig; e&longs;t b &amp; g <lb/>ad a, ut &longs;upra, &amp; quadrati b ad quadratum a, &amp; quadrati g ad qua&shy;<lb/><arrow.to.target n="marg203"></arrow.to.target><lb/>dratum a duplicata his qu&aelig; b &amp; g ad a erunt b &amp; g quadratorum <lb/>ad quadratum a, uelut c &amp; h ad a. </s> <s>Aliter &amp; euidentius in duabus <lb/>&longs;oleo demon&longs;trare: cum enim &longs;it e &amp; h duplicata ei qu&aelig; e&longs;t b &amp; g <lb/>ad a, ut &longs;upra, &amp; quadrati b ad quadratum a, &amp; quadrati g ad qua&shy;<lb/><arrow.to.target n="marg203"/><lb/>dratum a duplicata his qu&aelig; b &amp; g ad a erunt b &amp; g quadratorum <lb/>ad quadratum a, uelut c &amp; h ad a. </s>
  
 <s>Et conuertendo qua&shy;<lb/><arrow.to.target n="table15"></arrow.to.target><lb/>drati a ad quadratum g, ut a ad h, con&longs;tituantur ergo <lb/>hic &amp; erit quadrati b ad <expan abbr="quadrat&utilde;">quadratum</expan> g, ita c ad h: &longs;ed qua&shy;<lb/>drati b ad quadratum g, ut b ad g proportio duplicata <lb/>igitur e ad h, ut b ad g duplicata.</s></p><p type="margin"> <s>Et conuertendo qua&shy;<lb/><arrow.to.target n="table15"/><lb/>drati a ad quadratum g, ut a ad h, con&longs;tituantur ergo <lb/>hic &amp; erit quadrati b ad <expan abbr="quadrat&utilde;">quadratum</expan> g, ita c ad h: &longs;ed qua&shy;<lb/>drati b ad quadratum g, ut b ad g proportio duplicata <lb/>igitur e ad h, ut b ad g duplicata.</s></p><p type="margin">
  
 <s><margin.target id="marg196"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>non<gap/><emph.end type="italics"/> E<emph type="italics"/>le.<emph.end type="italics"/> &amp; 22. &amp; 23. <emph type="italics"/>octa ui.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg196"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>non<gap/><emph.end type="italics"/> E<emph type="italics"/>le.<emph.end type="italics"/> &amp; 22. &amp; 23. <emph type="italics"/>octa ui.<emph.end type="italics"/></s></p><p type="margin">
  
 <s><margin.target id="marg197"></margin.target>V<emph type="italics"/>ide per<emph.end type="italics"/> 23. P<emph type="italics"/>etit.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg197"/>V<emph type="italics"/>ide per<emph.end type="italics"/> 23. P<emph type="italics"/>etit.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg198"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>&longs;ex ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/> &amp; 33. <emph type="italics"/>undeci-mi.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg198"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>&longs;ex ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/> &amp; 33. <emph type="italics"/>undeci-mi.<emph.end type="italics"/></s></p><p type="margin">
  
 <s><margin.target id="marg199"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>&longs;e-ptimi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg199"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>&longs;e-ptimi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg200"></margin.target>D<emph type="italics"/>iff.<emph.end type="italics"/> 10.</s></p><p type="margin"> <s><margin.target id="marg200"/>D<emph type="italics"/>iff.<emph.end type="italics"/> 10.</s></p><p type="margin">
  
 <s><margin.target id="marg201"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 24. <emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg201"/>P<emph type="italics"/>er<emph.end type="italics"/> 24. <emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg202"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 10 <emph type="italics"/>diff. </s> <s><margin.target id="marg202"/>P<emph type="italics"/>er<emph.end type="italics"/> 10 <emph type="italics"/>diff. </s>
  
 <s>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin"> <s>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg203"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>&longs;ex ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><table><table.target id="table15"></table.target><row><cell><expan abbr="&qtilde;d">quad</expan>.</cell><cell>b</cell><cell>e</cell></row><row><cell><expan abbr="&qtilde;d">quad</expan>.</cell><cell>a</cell><cell>a</cell></row><row><cell><expan abbr="&qtilde;d">quad</expan>.</cell><cell>g</cell><cell>h</cell></row></table><p type="main"> <s><margin.target id="marg203"/>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>&longs;ex ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/><!-- KEEP S--></s></p><table><table.target id="table15"/><row><cell><expan abbr="&qtilde;d">quad</expan>.</cell><cell>b</cell><cell>e</cell></row><row><cell><expan abbr="&qtilde;d">quad</expan>.</cell><cell>a</cell><cell>a</cell></row><row><cell><expan abbr="&qtilde;d">quad</expan>.</cell><cell>g</cell><cell>h</cell></row></table><p type="main">
  
 <s>Propo&longs;itio &longs;exage&longs;imaoctaua, collectorum ab Euclide <lb/>&amp; Archimede.</s></p><p type="main"> <s>Propo&longs;itio &longs;exage&longs;imaoctaua, collectorum ab Euclide <lb/>&amp; Archimede.</s></p><p type="main">
  
 <s>Omnis cylindrus cono habenti ba&longs;im, &amp; altitudinem eandem <lb/><arrow.to.target n="marg204"></arrow.to.target><lb/>triplus e&longs;t. </s> <s>Omnis cylindrus cono habenti ba&longs;im, &amp; altitudinem eandem <lb/><arrow.to.target n="marg204"/><lb/>triplus e&longs;t. </s>
  
 <s>Omnis cylindrus &longs;ph&aelig;r&aelig; habenti eundem magnum <lb/><arrow.to.target n="marg205"></arrow.to.target><lb/>circulum, &amp; altitudinem &longs;exquialter e&longs;t. </s> <s>Omnis cylindrus &longs;ph&aelig;r&aelig; habenti eundem magnum <lb/><arrow.to.target n="marg205"/><lb/>circulum, &amp; altitudinem &longs;exquialter e&longs;t. </s>
  
 <s>Omnis &longs;ph&aelig;ra dupla e&longs;t <lb/><arrow.to.target n="marg206"></arrow.to.target><lb/>cono, cuius ba&longs;is e&longs;t eius circulus magnus, &amp; altitudo eadem, qu&aelig; <lb/>&longs;ph&aelig;r&aelig; ip&longs;ius. </s> <s>Omnis &longs;ph&aelig;ra dupla e&longs;t <lb/><arrow.to.target n="marg206"/><lb/>cono, cuius ba&longs;is e&longs;t eius circulus magnus, &amp; altitudo eadem, qu&aelig; <lb/>&longs;ph&aelig;r&aelig; ip&longs;ius. </s>
  
 <s>Omnis &longs;uperficies &longs;ph&aelig;r&aelig; quadrupla e&longs;t maiori <lb/><arrow.to.target n="marg207"></arrow.to.target><lb/>&longs;uo circulo. </s> <s>Omnis &longs;uperficies &longs;ph&aelig;r&aelig; quadrupla e&longs;t maiori <lb/><arrow.to.target n="marg207"/><lb/>&longs;uo circulo. </s>
  
 <s>Superficies portionis &longs;ph&aelig;r&aelig; e&longs;t &aelig;qualis circulo, cu <lb/><arrow.to.target n="marg208"></arrow.to.target><pb pagenum="58"/>ius &longs;emidiameter e&longs;t linea ducta &agrave; uertice portionis ad finem illius.</s></p><p type="margin"> <s>Superficies portionis &longs;ph&aelig;r&aelig; e&longs;t &aelig;qualis circulo, cu <lb/><arrow.to.target n="marg208"/><pb pagenum="58"/>ius &longs;emidiameter e&longs;t linea ducta &agrave; uertice portionis ad finem illius.</s></p><p type="margin">
  
 <s><margin.target id="marg204"></margin.target>1</s></p><p type="margin"> <s><margin.target id="marg204"/>1</s></p><p type="margin">
  
 <s><margin.target id="marg205"></margin.target>2</s></p><p type="margin"> <s><margin.target id="marg205"/>2</s></p><p type="margin">
  
 <s><margin.target id="marg206"></margin.target>3</s></p><p type="margin"> <s><margin.target id="marg206"/>3</s></p><p type="margin">
  
 <s><margin.target id="marg207"></margin.target>4</s></p><p type="margin"> <s><margin.target id="marg207"/>4</s></p><p type="margin">
  
 <s><margin.target id="marg208"></margin.target>5</s></p><p type="main"> <s><margin.target id="marg208"/>5</s></p><p type="main">
  
 <s>Quilibet &longs;ector &longs;ph&aelig;r&aelig; &aelig;qualis e&longs;t cono, cuius ba&longs;is e&longs;t circu&shy;<lb/>lus &aelig;qualis &longs;uperficiei eiu&longs;dem portionis, altitudo uer&ograve; &longs;ph&aelig;r&aelig; &longs;e&shy;<lb/>midiameter. </s> <s>Quilibet &longs;ector &longs;ph&aelig;r&aelig; &aelig;qualis e&longs;t cono, cuius ba&longs;is e&longs;t circu&shy;<lb/>lus &aelig;qualis &longs;uperficiei eiu&longs;dem portionis, altitudo uer&ograve; &longs;ph&aelig;r&aelig; &longs;e&shy;<lb/>midiameter. </s>
  
 <s>Proportio &longs;ph&aelig;r&aelig; ad &longs;ectorem datum, e&longs;t duplica&shy;<lb/>ta ei, qu&ecedil; e&longs;t dimetientis ad lineam, qu&aelig; &agrave; uertice portionis ad lim&shy;<lb/>bum. </s> <s>Proportio &longs;ph&aelig;r&aelig; ad &longs;ectorem datum, e&longs;t duplica&shy;<lb/>ta ei, qu&ecedil; e&longs;t dimetientis ad lineam, qu&aelig; &agrave; uertice portionis ad lim&shy;<lb/>bum. </s>
  
 <s>Cum enim &longs;ph&aelig;ra &longs;it &aelig;qualis cono, cuius ba&longs;is e&longs;t maior cir&shy;<lb/>culus, altitudo uer&ograve; dupla dimetienti per tertiam harum, qu&aelig; hic <lb/><arrow.to.target n="marg209"></arrow.to.target><lb/>proponuntur: erit &longs;ph&aelig;ra &aelig;qualis cono ba&longs;im habenti circulum, <lb/>cuius &longs;emidiameter &longs;it &aelig;qualis diametro &longs;ph&aelig;r&aelig;, altitudo uer&ograve; &longs;e&shy;<lb/>midiameter &longs;ph&aelig;r&aelig;. </s> <s>Cum enim &longs;ph&aelig;ra &longs;it &aelig;qualis cono, cuius ba&longs;is e&longs;t maior cir&shy;<lb/>culus, altitudo uer&ograve; dupla dimetienti per tertiam harum, qu&aelig; hic <lb/><arrow.to.target n="marg209"/><lb/>proponuntur: erit &longs;ph&aelig;ra &aelig;qualis cono ba&longs;im habenti circulum, <lb/>cuius &longs;emidiameter &longs;it &aelig;qualis diametro &longs;ph&aelig;r&aelig;, altitudo uer&ograve; &longs;e&shy;<lb/>midiameter &longs;ph&aelig;r&aelig;. </s>
  
 <s>At per &longs;extam harum &longs;ector &longs;ph&aelig;r&aelig; e&longs;t &aelig;qua&shy;<lb/>lis cono habenti altitudinem &longs;cmidiametrum &longs;ph&aelig;r&ecedil;, ba&longs;im autem <lb/><arrow.to.target n="marg210"></arrow.to.target><lb/>ip&longs;am portionis &longs;uperficiem: igitur proportio &longs;ph&aelig;r&aelig; ad &longs;ecto&shy;<lb/>rem, uelut circuli cuius diameter e&longs;t dupla dimetienti &longs;ph&aelig;r&aelig; ad <lb/>c&iacute;rculum &aelig;qualem &longs;uperficiei portionis: at &longs;uperficies portionis <lb/>per quintam harum e&longs;t &aelig;qualis circulo, cuius &longs;emidiameter e&longs;t li&shy;<lb/>nea &agrave; uertice portionis ad limbum eiu&longs;dem: ergo proportio &longs;ph&aelig;&shy;<lb/>r&aelig; ad &longs;uum &longs;ectorem e&longs;t uelut circuli, cuius dimetiens e&longs;t duplus di <lb/>metienti &longs;ph&aelig;r&aelig;, aut &longs;emidimetiens e&longs;t &aelig;qualis dimetienti &longs;ph&aelig;r&aelig; <lb/>ad circulum, cuius &longs;emidimetiens e&longs;t linea &agrave; uertice portionis ad <lb/>limbum. </s> <s>At per &longs;extam harum &longs;ector &longs;ph&aelig;r&aelig; e&longs;t &aelig;qua&shy;<lb/>lis cono habenti altitudinem &longs;cmidiametrum &longs;ph&aelig;r&ecedil;, ba&longs;im autem <lb/><arrow.to.target n="marg210"/><lb/>ip&longs;am portionis &longs;uperficiem: igitur proportio &longs;ph&aelig;r&aelig; ad &longs;ecto&shy;<lb/>rem, uelut circuli cuius diameter e&longs;t dupla dimetienti &longs;ph&aelig;r&aelig; ad <lb/>c&iacute;rculum &aelig;qualem &longs;uperficiei portionis: at &longs;uperficies portionis <lb/>per quintam harum e&longs;t &aelig;qualis circulo, cuius &longs;emidiameter e&longs;t li&shy;<lb/>nea &agrave; uertice portionis ad limbum eiu&longs;dem: ergo proportio &longs;ph&aelig;&shy;<lb/>r&aelig; ad &longs;uum &longs;ectorem e&longs;t uelut circuli, cuius dimetiens e&longs;t duplus di <lb/>metienti &longs;ph&aelig;r&aelig;, aut &longs;emidimetiens e&longs;t &aelig;qualis dimetienti &longs;ph&aelig;r&aelig; <lb/>ad circulum, cuius &longs;emidimetiens e&longs;t linea &agrave; uertice portionis ad <lb/>limbum. </s>
  
 <s>Sed proportio talium circulorum e&longs;t duplicata propor&shy;<lb/><arrow.to.target n="marg211"></arrow.to.target><lb/>tioni &longs;emidimetientium, igitur proportio &longs;ph&aelig;r&aelig; ad &longs;uum &longs;ecto&shy;<lb/>rem e&longs;t ueluti dimetientis &longs;ph&aelig;r&aelig; ad lineam, qu&aelig; &aacute; uertice portio&shy;<lb/><arrow.to.target n="marg212"></arrow.to.target><lb/>nis ad limbum duplicata. </s> <s>Sed proportio talium circulorum e&longs;t duplicata propor&shy;<lb/><arrow.to.target n="marg211"/><lb/>tioni &longs;emidimetientium, igitur proportio &longs;ph&aelig;r&aelig; ad &longs;uum &longs;ecto&shy;<lb/>rem e&longs;t ueluti dimetientis &longs;ph&aelig;r&aelig; ad lineam, qu&aelig; &aacute; uertice portio&shy;<lb/><arrow.to.target n="marg212"/><lb/>nis ad limbum duplicata. </s>
  
 <s>Cuicunque portioni &longs;ph&aelig;r&aelig; conus ille <lb/>habetur &aelig;qualis, qui ba&longs;im hab eat eandem cum portione, altitudi&shy;<lb/>nem uer&ograve; lineam rectam, qu&aelig; ad altitudinem portionis eandem <lb/>habeat proportionem, quam &longs;emidiametros &longs;ph&aelig;r&aelig; un&agrave; cum alti&shy;<lb/>tudine reliqu&aelig; portionis habet ad eandem reliqu&aelig; portionis alti&shy;<lb/><arrow.to.target n="marg213"></arrow.to.target><lb/>tudinem. </s> <s>Cuicunque portioni &longs;ph&aelig;r&aelig; conus ille <lb/>habetur &aelig;qualis, qui ba&longs;im hab eat eandem cum portione, altitudi&shy;<lb/>nem uer&ograve; lineam rectam, qu&aelig; ad altitudinem portionis eandem <lb/>habeat proportionem, quam &longs;emidiametros &longs;ph&aelig;r&aelig; un&agrave; cum alti&shy;<lb/>tudine reliqu&aelig; portionis habet ad eandem reliqu&aelig; portionis alti&shy;<lb/><arrow.to.target n="marg213"/><lb/>tudinem. </s>
  
 <s>Earum &longs;ph&aelig;r&aelig; portionum, qu&aelig; &aelig;qualibus &longs;uperfi&shy;<lb/><arrow.to.target n="marg214"></arrow.to.target><lb/>ciebus continentur medietas &longs;ph&aelig;r&aelig; maxima exi&longs;tit. </s> <s>Earum &longs;ph&aelig;r&aelig; portionum, qu&aelig; &aelig;qualibus &longs;uperfi&shy;<lb/><arrow.to.target n="marg214"/><lb/>ciebus continentur medietas &longs;ph&aelig;r&aelig; maxima exi&longs;tit. </s>
  
 <s>Proportio <lb/>&longs;uperficiei &longs;ph&aelig;r&aelig; plano diui&longs;&aelig; ad reliqu&aelig; portionis &longs;uperficiem, <lb/>&amp; re&longs;idui &longs;ectoris ad &longs;ectorem, e&longs;t uelut quadratorum duarum li&shy;<lb/>nearum qu&aelig; &agrave; uerticulis &longs;ectionum ad communem &longs;uperficiem <lb/>plani portiones &longs;ecantis de&longs;cendunt: nam &longs;ectorem &longs;ph&aelig;r&aelig;, dico <lb/><arrow.to.target n="marg215"></arrow.to.target><lb/>corpus compo&longs;itum ex portione, &amp; cono illo. </s> <s>Proportio <lb/>&longs;uperficiei &longs;ph&aelig;r&aelig; plano diui&longs;&aelig; ad reliqu&aelig; portionis &longs;uperficiem, <lb/>&amp; re&longs;idui &longs;ectoris ad &longs;ectorem, e&longs;t uelut quadratorum duarum li&shy;<lb/>nearum qu&aelig; &agrave; uerticulis &longs;ectionum ad communem &longs;uperficiem <lb/>plani portiones &longs;ecantis de&longs;cendunt: nam &longs;ectorem &longs;ph&aelig;r&aelig;, dico <lb/><arrow.to.target n="marg215"/><lb/>corpus compo&longs;itum ex portione, &amp; cono illo. </s>
  
 <s>Ille idem etiam defi&shy;<lb/>nit Ellip&longs;im coni a cuti anguli &longs;ectionem, quam dicit etiam fieri &longs;e&shy;<lb/><arrow.to.target n="marg216"></arrow.to.target><lb/>cto cylindro per planum non ad angulos rectos &longs;tante &longs;uper cylin&shy;<lb/>dri axem. </s> <s>Ille idem etiam defi&shy;<lb/>nit Ellip&longs;im coni a cuti anguli &longs;ectionem, quam dicit etiam fieri &longs;e&shy;<lb/><arrow.to.target n="marg216"/><lb/>cto cylindro per planum non ad angulos rectos &longs;tante &longs;uper cylin&shy;<lb/>dri axem. </s>
  
 <s>Ab hac igitur coni acuti anguli &longs;ectione &longs;eu ellip&longs;i cir&shy;<lb/><arrow.to.target n="marg217"></arrow.to.target><lb/>cumacta figura &longs;ph&aelig;roides corpus quod ba&longs;im rotundam habet, <lb/>uocat: id que duplex ob longum, quod fit diametro longiore quie&shy;<lb/>&longs;cente, &amp; prolatum quod fit quie&longs;cente breuiore: &longs;icut reliquam &longs;ci <lb/>licet parabolen aut hyperbolen, quia inferius non e&longs;t terminata, <pb pagenum="59"/>in cono rectangulo uocat rectanguli coni &longs;ectionem: ex qua cir&shy;<lb/>cumacta fit conoidale, quia planam habet ba&longs;im. </s> <s>Ab hac igitur coni acuti anguli &longs;ectione &longs;eu ellip&longs;i cir&shy;<lb/><arrow.to.target n="marg217"/><lb/>cumacta figura &longs;ph&aelig;roides corpus quod ba&longs;im rotundam habet, <lb/>uocat: id que duplex ob longum, quod fit diametro longiore quie&shy;<lb/>&longs;cente, &amp; prolatum quod fit quie&longs;cente breuiore: &longs;icut reliquam &longs;ci <lb/>licet parabolen aut hyperbolen, quia inferius non e&longs;t terminata, <pb pagenum="59"/>in cono rectangulo uocat rectanguli coni &longs;ectionem: ex qua cir&shy;<lb/>cumacta fit conoidale, quia planam habet ba&longs;im. </s>
  
 <s>Si ergo in ea&shy;<lb/><arrow.to.target n="marg218"></arrow.to.target><lb/>dem rectanguli coni &longs;ectione &agrave; plano portiones &aelig;quales habentes <lb/>diametros ab&longs;cindantur, ill&aelig; portiones erunt &aelig;quales. </s> <s>Si ergo in ea&shy;<lb/><arrow.to.target n="marg218"/><lb/>dem rectanguli coni &longs;ectione &agrave; plano portiones &aelig;quales habentes <lb/>diametros ab&longs;cindantur, ill&aelig; portiones erunt &aelig;quales. </s>
  
 <s>Et triangu&shy;<lb/>li in ei&longs;dem portionibus in&longs;cripti &aelig;quales erunt. </s> <s>Et triangu&shy;<lb/>li in ei&longs;dem portionibus in&longs;cripti &aelig;quales erunt. </s>
  
 <s>Diametrum uo&shy;<lb/>cat in <expan abbr="quacunq&utilde;e">quacunqune</expan> portione lineam, qu&aelig; omnes lineas ba&longs;i &aelig;quidi&shy;<lb/>&longs;tantes per &aelig;qualia diuidit. </s> <s>Diametrum uo&shy;<lb/>cat in <expan abbr="quacunq&utilde;e">quacunqune</expan> portione lineam, qu&aelig; omnes lineas ba&longs;i &aelig;quidi&shy;<lb/>&longs;tantes per &aelig;qualia diuidit. </s>
  
 <s>Omnis circuli cuius diameter e&longs;t ma <lb/><arrow.to.target n="marg219"></arrow.to.target><lb/>ior diameter ellip&longs;is proportio ad ellip&longs;im e&longs;t uelut direct&egrave; diame&shy;<lb/>tri ellip&longs;is ad diametrum tran&longs;uer&longs;am. </s> <s>Omnis circuli cuius diameter e&longs;t ma <lb/><arrow.to.target n="marg219"/><lb/>ior diameter ellip&longs;is proportio ad ellip&longs;im e&longs;t uelut direct&egrave; diame&shy;<lb/>tri ellip&longs;is ad diametrum tran&longs;uer&longs;am. </s>
  
 <s>Ex quo patet quod pro&shy;<lb/><arrow.to.target n="marg220"></arrow.to.target><lb/>portio cuiuslibet circuli ad ellip&longs;im e&longs;t uelut quadrati &longs;u&aelig; diame&shy;<lb/>tri ad rectangulum recta, &amp; tran&longs;uer&longs;a diametro ellip&longs;is compre&shy;<lb/>hen&longs;um. </s> <s>Ex quo patet quod pro&shy;<lb/><arrow.to.target n="marg220"/><lb/>portio cuiuslibet circuli ad ellip&longs;im e&longs;t uelut quadrati &longs;u&aelig; diame&shy;<lb/>tri ad rectangulum recta, &amp; tran&longs;uer&longs;a diametro ellip&longs;is compre&shy;<lb/>hen&longs;um. </s>
  
 <s>Ex hoc rur&longs;us &longs;equitur quod ellip&longs;is ad ellip&longs;im, ut re&shy;<lb/><arrow.to.target n="marg221"></arrow.to.target><lb/>ctanguli ex diametris unius ad rectangulum ex diametris alterius.</s></p><p type="margin"> <s>Ex hoc rur&longs;us &longs;equitur quod ellip&longs;is ad ellip&longs;im, ut re&shy;<lb/><arrow.to.target n="marg221"/><lb/>ctanguli ex diametris unius ad rectangulum ex diametris alterius.</s></p><p type="margin">
  
 <s><margin.target id="marg209"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 14. &amp; 15. <emph type="italics"/>duodeci mi<emph.end type="italics"/> E<emph type="italics"/>le.<emph.end type="italics"/> E<emph type="italics"/>ucl.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg209"/>P<emph type="italics"/>er<emph.end type="italics"/> 14. &amp; 15. <emph type="italics"/>duodeci mi<emph.end type="italics"/> E<emph type="italics"/>le.<emph.end type="italics"/> E<emph type="italics"/>ucl.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg210"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>duo decimi<emph.end type="italics"/> E<emph type="italics"/>le.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg210"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>duo decimi<emph.end type="italics"/> E<emph type="italics"/>le.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg211"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 2. <emph type="italics"/>duode cimi<emph.end type="italics"/>, &amp; 20. <emph type="italics"/>&longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg211"/>P<emph type="italics"/>er<emph.end type="italics"/> 2. <emph type="italics"/>duode cimi<emph.end type="italics"/>, &amp; 20. <emph type="italics"/>&longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg212"></margin.target>8</s></p><p type="margin"> <s><margin.target id="marg212"/>8</s></p><p type="margin">
  
 <s><margin.target id="marg213"></margin.target>9</s></p><p type="margin"> <s><margin.target id="marg213"/>9</s></p><p type="margin">
  
 <s><margin.target id="marg214"></margin.target>10</s></p><p type="margin"> <s><margin.target id="marg214"/>10</s></p><p type="margin">
  
 <s><margin.target id="marg215"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 22. <emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg215"/>P<emph type="italics"/>er<emph.end type="italics"/> 22. <emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg216"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>&longs;ex ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg216"/>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>&longs;ex ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg217"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg217"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin">
  
 <s><margin.target id="marg218"></margin.target>11</s></p><p type="margin"> <s><margin.target id="marg218"/>11</s></p><p type="margin">
  
 <s><margin.target id="marg219"></margin.target>12</s></p><p type="margin"> <s><margin.target id="marg219"/>12</s></p><p type="margin">
  
 <s><margin.target id="marg220"></margin.target>13</s></p><p type="margin"> <s><margin.target id="marg220"/>13</s></p><p type="margin">
  
 <s><margin.target id="marg221"></margin.target>14</s></p><p type="main"> <s><margin.target id="marg221"/>14</s></p><p type="main">
  
 <s>Si conoides &amp; &longs;ph&aelig;roides &longs;ecet plano &aelig;quidi&longs;tanti axi fiet &longs;e&shy;<lb/><arrow.to.target n="marg222"></arrow.to.target><lb/>ctio conoidalis &longs;imilis ei &agrave; qua conoides &longs;eu &longs;ph&aelig;roides de&longs;cri&shy;<lb/>ptum e&longs;t. </s> <s>Si conoides &amp; &longs;ph&aelig;roides &longs;ecet plano &aelig;quidi&longs;tanti axi fiet &longs;e&shy;<lb/><arrow.to.target n="marg222"/><lb/>ctio conoidalis &longs;imilis ei &agrave; qua conoides &longs;eu &longs;ph&aelig;roides de&longs;cri&shy;<lb/>ptum e&longs;t. </s>
  
 <s>Sin autem &longs;upra axem plano ad perpendiculum erecto <lb/>&longs;ectio circulus erit. </s> <s>Sin autem &longs;upra axem plano ad perpendiculum erecto <lb/>&longs;ectio circulus erit. </s>
  
 <s>Et &longs;i &longs;ecentur obliqu&egrave; fiet ellip&longs;is, modo omnia <lb/>latera comprehendat. </s> <s>Et &longs;i &longs;ecentur obliqu&egrave; fiet ellip&longs;is, modo omnia <lb/>latera comprehendat. </s>
  
 <s>Omnis portio conoidalis rectanguli, quam <lb/><arrow.to.target n="marg223"></arrow.to.target><lb/>planum &longs;ecat, &longs;exquialtera e&longs;t, cono qui ba&longs;im &amp; axem eandem ha&shy;<lb/>bet. </s> <s>Omnis portio conoidalis rectanguli, quam <lb/><arrow.to.target n="marg223"/><lb/>planum &longs;ecat, &longs;exquialtera e&longs;t, cono qui ba&longs;im &amp; axem eandem ha&shy;<lb/>bet. </s>
  
 <s>Ex quo patet, quod &longs;i portio conoidalis rectanguli &amp; &longs;ph&aelig;&shy;<lb/><arrow.to.target n="marg224"></arrow.to.target><lb/>r&aelig; medietas eandem ba&longs;im habeant &amp; axem eundem, medietas <lb/>&longs;ph&aelig;r&aelig; &longs;exquitertia erit conoidali portioni. </s> <s>Ex quo patet, quod &longs;i portio conoidalis rectanguli &amp; &longs;ph&aelig;&shy;<lb/><arrow.to.target n="marg224"/><lb/>r&aelig; medietas eandem ba&longs;im habeant &amp; axem eundem, medietas <lb/>&longs;ph&aelig;r&aelig; &longs;exquitertia erit conoidali portioni. </s>
  
 <s>Et &longs;i eiu&longs;dem rectan <lb/><arrow.to.target n="marg225"></arrow.to.target><lb/>guli conoidalis portiones ab&longs;cin dantur erit portionum propor&shy;<lb/>tio uelut quadratorum axium. </s> <s>Et &longs;i eiu&longs;dem rectan <lb/><arrow.to.target n="marg225"/><lb/>guli conoidalis portiones ab&longs;cin dantur erit portionum propor&shy;<lb/>tio uelut quadratorum axium. </s>
  
 <s>Cuiuslibet &longs;ph&aelig;roidis pars pla&shy;<lb/><arrow.to.target n="marg226"></arrow.to.target><lb/>no per centrum ab&longs;ci&longs;&longs;a dupla e&longs;t cono ba&longs;im &amp; axem eadem ha&shy;<lb/>benti. </s> <s>Cuiuslibet &longs;ph&aelig;roidis pars pla&shy;<lb/><arrow.to.target n="marg226"/><lb/>no per centrum ab&longs;ci&longs;&longs;a dupla e&longs;t cono ba&longs;im &amp; axem eadem ha&shy;<lb/>benti. </s>
  
 <s>Si autem non &longs;uper centrum erit proportio earum ad co&shy;<lb/><arrow.to.target n="marg227"></arrow.to.target><lb/>num ba&longs;im, &amp; axem eandem habentem uelut coniunct&aelig; ex axe al&shy;<lb/>terius partis &amp; dimidio axis &longs;ph&aelig;roidis ad axem alterius partis.</s></p><p type="margin"> <s>Si autem non &longs;uper centrum erit proportio earum ad co&shy;<lb/><arrow.to.target n="marg227"/><lb/>num ba&longs;im, &amp; axem eandem habentem uelut coniunct&aelig; ex axe al&shy;<lb/>terius partis &amp; dimidio axis &longs;ph&aelig;roidis ad axem alterius partis.</s></p><p type="margin">
  
 <s><margin.target id="marg222"></margin.target>15</s></p><p type="margin"> <s><margin.target id="marg222"/>15</s></p><p type="margin">
  
 <s><margin.target id="marg223"></margin.target>16</s></p><p type="margin"> <s><margin.target id="marg223"/>16</s></p><p type="margin">
  
 <s><margin.target id="marg224"></margin.target>17</s></p><p type="margin"> <s><margin.target id="marg224"/>17</s></p><p type="margin">
  
 <s><margin.target id="marg225"></margin.target>18</s></p><p type="margin"> <s><margin.target id="marg225"/>18</s></p><p type="margin">
  
 <s><margin.target id="marg226"></margin.target>19</s></p><p type="margin"> <s><margin.target id="marg226"/>19</s></p><p type="margin">
  
 <s><margin.target id="marg227"></margin.target>20</s></p><p type="main"> <s><margin.target id="marg227"/>20</s></p><p type="main">
  
 <s>Demum proportio partis conoidis obtu&longs;i anguli plano ab&longs;ci&longs;&shy;<lb/><arrow.to.target n="marg228"></arrow.to.target><lb/>&longs;&aelig; ad conum, ba&longs;im &amp; axem eadem habentem e&longs;t ueluti line&aelig;, com <lb/>po&longs;it&aelig; ex axe portionis &amp; triplo adiect&aelig; ad compo&longs;itum ex axe <lb/>portionis &amp; duplo eiu&longs;dem adiect&aelig;. </s> <s>Demum proportio partis conoidis obtu&longs;i anguli plano ab&longs;ci&longs;&shy;<lb/><arrow.to.target n="marg228"/><lb/>&longs;&aelig; ad conum, ba&longs;im &amp; axem eadem habentem e&longs;t ueluti line&aelig;, com <lb/>po&longs;it&aelig; ex axe portionis &amp; triplo adiect&aelig; ad compo&longs;itum ex axe <lb/>portionis &amp; duplo eiu&longs;dem adiect&aelig;. </s>
  
 <s>Adiectam uocat hyperbolis <lb/>tran&longs;uer&longs;am. </s> <s>Adiectam uocat hyperbolis <lb/>tran&longs;uer&longs;am. </s>
  
 <s>Omnis cylindrus cono triplus e&longs;t habenti eandem <lb/><arrow.to.target n="marg229"></arrow.to.target><lb/>ba&longs;im &amp; altitudinem. </s> <s>Omnis cylindrus cono triplus e&longs;t habenti eandem <lb/><arrow.to.target n="marg229"/><lb/>ba&longs;im &amp; altitudinem. </s>
  
 <s>Omnes cylindri coni &longs;ph&aelig;r&aelig; &longs;unt in pro&shy;<lb/><arrow.to.target n="marg230"></arrow.to.target><lb/>portione corporum &longs;imilium planis &longs;uperficiebus contentarum.</s></p><p type="margin"> <s>Omnes cylindri coni &longs;ph&aelig;r&aelig; &longs;unt in pro&shy;<lb/><arrow.to.target n="marg230"/><lb/>portione corporum &longs;imilium planis &longs;uperficiebus contentarum.</s></p><p type="margin">
  
 <s><margin.target id="marg228"></margin.target>21</s></p><p type="margin"> <s><margin.target id="marg228"/>21</s></p><p type="margin">