texts/archimedes/xml/carda_propo_01_la_1570.xml - Colored diff

Main History Search Repository tree

Return to carda_propo_01_la_1570.xml CVS log

Up to [CVSROOT] / texts / archimedes / xml

Colored diff for /texts/archimedes/xml/Attic/carda_propo_01_la_1570.xml between version 1.1 and 1.1.1.1

version 1.1, 2002/06/18 09:37:11

version 1.1.1.1, 2002/10/28 17:54:09

Line 1

<!DOCTYPE archimedes [

<?xml version="1.0"?>

<!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" >

<author>Cardano, Girolamo</author>

<title>Opus novum de proportionibus</title>

<place>Basel</place>

<cvs_file>carda_propo_01_la_1570</cvs_file>

<cvs_version/>

</info> <text> <front> </front> <body> <chap> <pb/><pb/><pb/>

<!-- footnotes and margnotes get yanked out of their

s, repl. by

<!ELEMENT foot.target

(#PCDATA) >

<!ATTLIST foot.target

id ID #REQUIRED

n CDATA #IMPLIED >

<!ELEMENT margin.target

(#PCDATA) >

<!ATTLIST margin.target

id ID #REQUIRED

n CDATA #IMPLIED >

<!ELEMENT table.target

(#PCDATA) >

<!ATTLIST table.target

id ID #REQUIRED

n CDATA #IMPLIED >

<!ELEMENT arrow.to.target

(#PCDATA)* >

<!ATTLIST arrow.to.target

symbol CDATA #IMPLIED

id ID #IMPLIED

n CDATA #IMPLIED>

<!ELEMENT archimedes

(info, text) >

<!ATTLIST archimedes

id ID #IMPLIED

n CDATA #IMPLIED >

<!ELEMENT author

(#PCDATA) >

<!ATTLIST author

id ID #IMPLIED

n CDATA #IMPLIED >

<!ELEMENT back

( section | pb?)+ >

<!ATTLIST back

id ID #IMPLIED

n CDATA #IMPLIED

type CDATA #IMPLIED>

<!ELEMENT body

(chap, pb?)+ >

<!ATTLIST body

id ID #IMPLIED

n CDATA #IMPLIED >

<!ELEMENT chap

( p | pb | figure | table)+ >

<!ATTLIST chap

id ID #IMPLIED

n CDATA #IMPLIED

type CDATA #IMPLIED>

<!ELEMENT chunk

(#PCDATA) >

<!ATTLIST chunk

id ID #IMPLIED

n CDATA #IMPLIED

unit CDATA #IMPLIED >

<!ELEMENT date

(#PCDATA) >

<!ATTLIST date

id ID #IMPLIED

n CDATA #IMPLIED >

<!ELEMENT editor

(#PCDATA) >

<!ATTLIST editor

id ID #IMPLIED

n CDATA #IMPLIED >

<!ELEMENT emph

EMPTY>

<!ATTLIST emph

id ID #IMPLIED

n CDATA #IMPLIED>

<!ELEMENT emph.end

EMPTY>

<!ATTLIST emph.end

id ID #IMPLIED

n CDATA #IMPLIED>

<!ELEMENT expan

<!ATTLIST expan

id ID #IMPLIED

n CDATA #IMPLIED

abbr CDATA #IMPLIED

type CDATA #IMPLIED >

<!ELEMENT figure

(#PCDATA) >

<!ATTLIST figure

id ID #IMPLIED

place (margin|text) "text"

xmlns:xlink CDATA #FIXED "http://www.w3.org/1999/xlink"

xlink:type (simple) #FIXED "simple"

xlink:href CDATA #IMPLIED>

<!ELEMENT foreign

<!ATTLIST foreign

id ID #IMPLIED

n CDATA #IMPLIED

lang CDATA #IMPLIED>

<!ELEMENT front

( section | pb?)+ >

<!ATTLIST front

id ID #IMPLIED

n CDATA #IMPLIED

type CDATA #IMPLIED>

<!ELEMENT gap

EMPTY >

<!ATTLIST gap

id ID #IMPLIED

n CDATA #IMPLIED

desc CDATA #IMPLIED>

<!ELEMENT info

(author, title, date, place, editor, publisher, translator, lang, chunk,locator) >

<!ATTLIST info

id ID #IMPLIED

n CDATA #IMPLIED >

<!ELEMENT lang

(#PCDATA) >

<!ATTLIST lang

id ID #IMPLIED

n CDATA #IMPLIED >

<!ELEMENT lb EMPTY >

<!ATTLIST lb

ed CDATA #IMPLIED

id ID #IMPLIED

n CDATA #IMPLIED>

<!-- <lb>

occurs at

<s> level and at

level -->

<!-- <lb>

at end of

<s> must be placed after </s>

-->

<!ELEMENT locator (#PCDATA) >

<!ATTLIST locator

id ID #IMPLIED

n CDATA #IMPLIED >

<!ELEMENT p

(s | pb|lb|emph|emph.end|gap )+ >

<!ATTLIST p

id ID #IMPLIED

n CDATA #IMPLIED

<!ELEMENT pb

EMPTY >

<!ATTLIST pb

ed CDATA #IMPLIED

id ID #IMPLIED

n CDATA #IMPLIED

pagenum CDATA #IMPLIED>

<!-- pb can occur inside

<s> etc. and inside

and inside <chap> -->

<!--

<pb> at end of

must be placed after -->

<!ELEMENT place

(#PCDATA) >

<!ATTLIST place

id ID #IMPLIED

n CDATA #IMPLIED >

<!ELEMENT publisher

(#PCDATA)* >

<!ATTLIST publisher

id ID #IMPLIED

n CDATA #IMPLIED >

<!ELEMENT s

<!ATTLIST s

id ID #IMPLIED

n CDATA #IMPLIED >

<!ELEMENT section

( p | pb | figure)+ >

<!ATTLIST section

id ID #IMPLIED

n CDATA #IMPLIED

type CDATA #IMPLIED>

<!ELEMENT table

( table.target?, (row, pb?)+ ) >

<!-- order is now:

<table>

<table.target/>

<pb>

i.e.

<table> is equiv. to

-->

<!ATTLIST table

id ID #IMPLIED

n CDATA #IMPLIED >

<!ELEMENT row

( cell+ ) >

<!ATTLIST row

id ID #IMPLIED

n CDATA #IMPLIED >

<!ELEMENT cell

<!ATTLIST cell

id ID #IMPLIED

n CDATA #IMPLIED >

<!ELEMENT text

(pb?, front, pb?, body, pb?, back) >

<!ATTLIST text

type CDATA #IMPLIED

id ID #IMPLIED

n CDATA #IMPLIED>

<!ELEMENT title

(#PCDATA) >

<!ATTLIST title

id ID #IMPLIED

n CDATA #IMPLIED

type CDATA #IMPLIED >

<!ELEMENT translator

(#PCDATA)* >

<!ATTLIST translator

id ID #IMPLIED

n CDATA #IMPLIED >

<!ENTITY shy "[-]">

<!ENTITY AElig "[AElig]">

<!ENTITY aacute "[aacute]">

<!ENTITY acirc "[acirc]">

<!ENTITY aelig "[aelig]">

<!ENTITY agrave "[agrave]">

<!ENTITY amp "[amp]">

<!ENTITY atilde "[atilde]">

<!ENTITY ccedil "[ccedil]">

<!ENTITY dacute "[dacute]">

<!ENTITY eacute "[eacute]">

<!ENTITY ecedil "[ecedil]">

<!ENTITY ecirc "[ecirc]">

<!ENTITY egrave "[egrave]">

<!ENTITY etilde "[etilde]">

<!ENTITY euml "[euml]">

<!ENTITY eumle "[eumle]">

<!ENTITY grave "[grave]">

<!ENTITY gt "[gt]">

<!ENTITY iacute "[iacute]">

<!ENTITY itilde "[itilde]">

<!ENTITY ldquo "[ldquo]">

<!ENTITY longs "[longs]">

<!ENTITY lsquo "[lsquo]">

<!ENTITY lt "[lt]">

<!ENTITY mdash "[mdash]">

<!ENTITY mtilde "[mtilde]">

<!ENTITY nacute "[nacute]">

<!ENTITY ngrave "[ngrave]">

<!ENTITY ntilde "[ntilde]">

<!ENTITY oacute "[oacute]">

<!ENTITY oelig "[oelig]">

<!ENTITY ograve "[ograve]">

<!ENTITY otilde "[otilde]">

<!ENTITY ptilde "[ptilde]">

<!ENTITY qacute "[qacute]">

<!ENTITY qring "[qring]">

<!ENTITY qtilde "[qtilde]">

<!ENTITY rdquo "[rdquo]">

<!ENTITY rsquo "[rsquo]">

<!ENTITY shy "[shy]">

<!ENTITY szlig "[szlig]">

<!ENTITY ttilde "[ttilde]">

<!ENTITY uacute "[uacute]">

<!ENTITY ugrave "[ugrave]">

<!ENTITY utilde "[utilde]">

<!ENTITY uuml "[uuml]">

<!ENTITY verbar "[verbar]">

<!ENTITY AElig "[AElig]">

<!ENTITY aacute "[aacute]">

<!ENTITY acirc "[acirc]">

<!ENTITY aelig "[aelig]">

<!ENTITY agrave "[agrave]">

<!ENTITY amp "[amp]">

<!ENTITY atilde "[atilde]">

<!ENTITY ccedil "[ccedil]">

<!ENTITY dacute "[dacute]">

<!ENTITY eacute "[eacute]">

<!ENTITY ecedil "[ecedil]">

<!ENTITY ecirc "[ecirc]">

<!ENTITY egrave "[egrave]">

<!ENTITY etilde "[etilde]">

<!ENTITY euml "[euml]">

<!ENTITY eumle "[eumle]">

<!ENTITY grave "[grave]">

<!ENTITY gt "[gt]">

<!ENTITY iacute "[iacute]">

<!ENTITY itilde "[itilde]">

<!ENTITY lbrk "[lbrk]">

<!ENTITY ldquo "[ldquo]">

<!ENTITY longs "[longs]">

<!ENTITY lsquo "[lsquo]">

<!ENTITY lt "[lt]">

<!ENTITY mdash "[mdash]">

<!ENTITY mtilde "[mtilde]">

<!ENTITY nacute "[nacute]">

<!ENTITY ngrave "[ngrave]">

<!ENTITY ntilde "[ntilde]">

<!ENTITY oacute "[oacute]">

<!ENTITY oelig "[oelig]">

<!ENTITY ograve "[ograve]">

<!ENTITY otilde "[otilde]">

<!ENTITY ptilde "[ptilde]">

<!ENTITY qacute "[qacute]">

<!ENTITY qring "[qring]">

<!ENTITY qtilde "[qtilde]">

<!ENTITY rdquo "[rdquo]">

<!ENTITY rsquo "[rsquo]">

<!ENTITY shy "[shy]">

<!ENTITY szlig "[szlig]">

<!ENTITY ttilde "[ttilde]">

<!ENTITY uacute "[uacute]">

<!ENTITY ugrave "[ugrave]">

<!ENTITY utilde "[utilde]">

<!ENTITY uuml "[uuml]">

<!ENTITY verbar "[verbar]">

<info>

<author>Girolamo Cardano</author>

<title>De proportionibus numerorum, motuum, ponderum, sonorum, aliarumque rerum</title>

<place>Lyons</place>

</info>

<text>

<front>

</front>

<body>

<chap>

<pb/>

<s>HIERONYMI <lb/>CARDANI MEDIO <lb/>LANENSIS, CIVISQV'E BONO<lb/>NIENSIS, PHILOSOPHI, MEDICI ET <lb/>Mathematici clari&longs;simi,</s>

<s>OPVS NOVVM DE <lb/>PROPORTIONIBVS NVMERORVM, MO <lb/>TVVM, PONDERVM, SONORVM, ALIARVMQV'E RERVM <lb/>men&longs;urandarum, non &longs;olùm Geometrico more &longs;tabilitum, &longs;ed etiam <lb/>uarijs experimentis & ob&longs;eruationibus rerum in natura, &longs;olerti <lb/>demon&longs;tratione illu&longs;tratum, ad multiplices u&longs;us ac<lb/>commodatum, & in Vlibros dige&longs;tum.</s>

<s>PRAETEREA.</s>

<s>ARTIS MAGNÆ, SIVE DE REGVLIS <lb/>ALGEBRAICIS, LIBER VNVS, ABSTRVSISSIMVS <lb/>& inexhau&longs;tus plane totius Arithmeticæ the&longs;aurus, ab <lb/>authore recens multis in locis recogni<lb/>tus & auctus.</s>

<s>DE ALIZA REGVLA LIBER, HOC EST, ALGEBRAICAE <lb/>logi&longs;ticæ &longs;uæ, numeros recondita numerandi &longs;ubtilitate, &longs;ecundum Geo<lb/>metricas quantitates inquirentis, nece&longs;&longs;aria Coronis, <lb/>nunc demum in lucem edita.</s>

<s>O<emph type="italics"/>pus<emph.end type="italics"/> P<emph type="italics"/>hy&longs;icis &<emph.end type="italics"/> M<emph type="italics"/>athematicis imprimis <lb/>utile & nece&longs;&longs;arium.<emph.end type="italics"/></s>

<s>Cum Cæ&longs;. Maie&longs;t. Gratia & Priuilegio.</s>

<s>BASILEÆ.</s>

<pb/>

<s>IN LIBRVM DE <lb/>PROPORTIONIBVS HIERONYMI <lb/>CARDANI MEDIOLANENSIS, CIVISQV'E <lb/>Bononien&longs;is, Medici, Præfatio ad M. A. Amulium <lb/>Venetum Card. Illu&longs;tri&longs;simum.</s>

<s>Bene Dictum e&longs;tmeo iudicio à Platone M. <lb/>A. Amuli optime, beatas fore Re&longs;pub. &longs;i uel <lb/>illarum domini &longs;apientiæ amatores e&longs;&longs;ent, <lb/>aut qui &longs;apientiæ e&longs;&longs;ent amatores domina<lb/>rentur, hocip&longs;um clarè intelligens, &longs;tudio &longs;a <lb/>pientiæ nihil e&longs;&longs;e utilius humano generi: <lb/>quo &longs;imul & pietas, & iu&longs;titia, & mutuus <lb/>amor hominum inter &longs;e & eorum commo<lb/>da continerentur. Nempe hi&longs;ce quatuor tota no&longs;tra felicitas com<lb/>prehenditur. Si quidem pietate in Deos nihil ni&longs;i &longs;anctum, & pu<lb/>rum, & illu&longs;tre &longs;apimus: hocip&longs;o primum quod &longs;upra nos e&longs;t, intel<lb/>ligimus, Deos ueneramur, gratias agimus, timor cum ueneratione <lb/>no&longs;tros animos &longs;ubit, & de futura uita cogitamus, hæc ip&longs;a morta<lb/>lia &longs;i non negligentes &longs;altem paruifacientes. Iu&longs;titiam autem adeò <lb/>nece&longs;&longs;ariam humano generi e&longs;&longs;e &longs;cimus, ut &longs;ine illa neque e&longs;&longs;e, nedum <lb/>benè e&longs;&longs;e po&longs;símus, ut neque latronum c&oelig;tus ab&longs;que ea diu &longs;tare po&longs;<lb/>&longs;int. Porrò quid dicam de concordia, & mutua hominum beneuo<lb/>lentia, in quibus omnis uit&ecedil; human&ecedil; dulcedo repo&longs;ita e&longs;t: nec quis <lb/>&longs;u&longs;tineat uiuere, qui &longs;e omnibus odio&longs;um e&longs;&longs;e &longs;entiat. His ip&longs;is fi<lb/>lios in &longs;pem alimus, parentes fouemus, fratres tuemur, & adiuua<lb/>mus, amicis opitulamur, cum hominibus hilarem & iucundam ui<lb/>tam ducimus. Si quis &longs;erpentem in lecto haberet, nunquam &longs;om<lb/>num caperet: ita nihil mole&longs;tius e&longs;t in hac uita, quam e&longs;&longs;e cum quo <lb/>nolis, & priuari con&longs;uetudine eorum cum quibus maximè uiuere <lb/>cupias. Quid enim habent Principes præcipuum cum tota illa po<lb/>tentia quam habent, ni&longs;i hoc unum, quod &longs;uis quos amant bene fa<lb/>cere po&longs;sint: nam reliqua omnia exerceri, uenari, edere, bibere, dor<lb/>mire, iter agere, loca amæna inui&longs;ere multis alijs conce&longs;&longs;um e&longs;t, ma<lb/>ioreque commodo qui in uita priuata degunt. Si ergo principatum <lb/>cum totlaboribus, curis, periculis, & meritò omnes appetunt: nec <lb/>e&longs;t in eo quicquam præcipuum præter hoc, cui dubium e&longs;t quin <lb/>hocnon &longs;it &longs;ummum huius uitæ hominibus bonum? propter cu<lb/>ius uel dubiam &longs;pem eorum, quæ habent obliti mortales pericli<lb/>tantur. Succedunt inde tot commoda, non &longs;olum utilia, &longs;ed pleraque

<pb/>etiam nece&longs;&longs;aria, quæ nos &longs;apientia docet: huiu&longs;modi ergo omnia <lb/>cùm libris contineantur, meritò optimus qui&longs;que librorum bono<lb/>rum perpetuitati atque in columitati fauere debet. C. Caligulam exe<lb/>cramur &longs;olum ob id quod Vergilij, & T. Liuij &longs;cripta delere cogi<lb/>tauerit. Quid facturi e&longs;&longs;emus, &longs;i feci&longs;&longs;et quod cogitauerat? E&longs;t in &longs;a<lb/>pientum monumentis bonum &longs;ine malo, mens &longs;ine corporea labe: <lb/>Virtutes ab&longs;que uitijs, gratiæ & iucunditas &longs;ine &longs;orde, & immundi<lb/>tia, uoluptas &longs;ine dolore, conuer&longs;atio ab&longs;que tædio, delitiæ ab&longs;que mi&longs;e <lb/>ria nuda, omnia bona præ&longs;tant, atque laudabilia ab omnibus morta<lb/>litatis exuuijs libera, tantum commodi afferunt libri. Sed & in eo<lb/>rum electione ac &longs;tudijs modus, ac medio critas quædam &longs;eruanda <lb/>e&longs;t, quæ &longs;i quis neglexerit non leui incommodo afficietur: eam an<lb/>tiqui rationem alij proportionem appellarunt, non equidem etiam <lb/>in pertritis tam <expan abbr="facillimã">facillimam</expan>, ut rentur homines: nam in alijs rebus per<lb/>ob&longs;curam e&longs;&longs;e fatentur, ego difficillimam puto undique, & magis for <lb/>&longs;an ubi non exi&longs;timamus. Vnde plures decidere uidemus magnis <lb/>cum auxilijs, & euidenti &longs;pe: quid aliud e&longs;t in cau&longs;a quàm ignota <lb/>men&longs;ura rerum? quam tamen plerique tenere &longs;e putant. Ergo, cùm <lb/>&longs;ummum bonum in hac men&longs;ura &longs;itum e&longs;&longs;e cernerem, ut clarè o&longs;ten <lb/>dunt mu&longs;icæ uoces, quæ non ni&longs;i indiuiduo (ut ita dicam) &longs;pacio <lb/>&longs;eu loco &longs;tare po&longs;&longs;unt, ita & in figuris picturarum & &longs;tatuarum, & <lb/>diebus decretorijs, & negocijs ciuilibus oper&ecedil;precium me factu<lb/>rum exi&longs;timaui, &longs;i omnia hæc quæ latè patebant breuiter in unum <lb/>redegi&longs;&longs;em, <expan abbr="nõ">non</expan> tantum ne lectorem tædio afficerem, quàm ut quòd <lb/>aliàs do cui, breuibus tractationibus, & plura continerentur, & faci <lb/>lius docerentur. Cum uerò bona fortuna quædam effeci&longs;&longs;et, ut tibi <lb/>libellum dedica&longs;&longs;em de Prouidentia ex con&longs;titutione temporum, <lb/>longe meliore occa&longs;ione nominis tui typographi obliti &longs;int, indi<lb/>gnum fore putaui, ut non ærea (quemadmodum cum Glauco Dio <lb/>medes) cum aureis commutarem. Itaque infinitis licet circumuentus <lb/>negocijs totus huic operæ in cubui, atque adeò ut præter &longs;pem unius <lb/>anni penè &longs;pacio liber ab&longs;olueretur. Qui cum tibi (ut dixi) iam iurè <lb/>deberetur, eò tamen magis dedican dum putaui, quod non ego &longs;o<lb/>lum quanquam id maximè, &longs;ed communis con&longs;en&longs;us ho<lb/>minum exi&longs;timet, te &longs;ingulari uirtute omnibus <lb/>&longs;tudio&longs;is plurimum fauere, <lb/>Vale.</s>

<pb/>

<s>TABVLA PRO<lb/>POSITIONVM DE <lb/>PROPORTIONIBVS.<lb/>

<arrow.to.target n="table1"></arrow.to.target></s>

<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 uerò 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, & quinti ad &longs;extum, producetur etiam ex proportione tertij ad &longs;extum, & 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, & quinti ad &longs;extum: erit proportio tertij ad &longs;extum, producta ex proportionibus primi ad &longs;ecur dum, & 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;ariò producuntur ex duabus proportionibus, cum duæ quantitates ex illis quæ modos conficiunt, æquales 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, quæ 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;umptæ ad utranque partem primæ 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 æqua-lium <expan abbr="quantitat&utilde;">quantitatum</expan> e&longs;t, compo&longs;ita ex proportionibus primis, & 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 primæ & tertiæ quatuor quantitatum omiologarum ad aggregatum &longs;ecundæ & quartæ, 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;æ & coniunctæ 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 primæ & tertiæ, ad aggre-gatum &longs;ecundæ & quartæ, erit ut monadis addito prouentu, qui fit diui&longs;a differentia, differentiarum primæ & &longs;ecundæ, atque quartæ & tertiæ, per aggregatum tertiæ & quartæ 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, quæ non minorem habet proportio-nem ad &longs;uam corre&longs;pondentem quàm 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 primæ & tertiæ, in tertiam ad productum ex iggre gato tertiæ & omiotatæ 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 duæ proportiones conuer&longs;æ producunt æqualem 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 præter, <expan abbr="ultimã">ultimam</expan> proportio uerò penultimæ ad ultimam, qualis re&longs;idui primæ ad &longs;ecundam, erit primæ ad aggregatum reliquarum, uelut penultimæ ad ultimam.<emph.end type="italics"/></cell><cell>15</cell></row>

<row><cell>XIX.</cell><cell>S<emph type="italics"/>i fuerint aliquot quantitates arithmeticæ omiologæ, quarum exce&longs;&longs;us &longs;it æqualis minimè, omnibus autem deficientibus &longs;upplementa ad æqualitatem maximè adiungantur, erunt quadrata omnium quantitatum æqualium, adiecto rur&longs;us quadrato primæ 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> æqualis tertiæ, aut prima æqualis quartæ, erit proportio primæ ad quartam, aut tertiæ ad &longs;ecundam, producta ex proportionibus primæ ad &longs;ecundam & tertiæ 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, & &longs;ecunda in tertiam, produ-ctumque primæ in quartam, diui&longs;um fu<gap/>rit per productum &longs;ecundæ in tertiam, erit proportio primæ ad &longs;ecundam, diui&longs;a per proportíonem tertiæ 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 primæ ad &longs;ecundam maior quàm tertiæ ad quartam, erit confu&longs;a ex his maior quàm tertiæ ad quartam, minor autem quàm primæ 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: ideò 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, & 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;ariò &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 æqualis 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 obuiæ, aliæ impel-lunt.<emph.end type="italics"/></cell><cell>26</cell></row>

<row><cell>XXXI.</cell><cell>O<emph type="italics"/>mnis motus naturalis in æquali medio ualidior e&longs;t in fine quàm in principio.<emph.end type="italics"/>V<emph type="italics"/>iolentus contrà.<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 quàm 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 quàm 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 æqualis undique magnitudinis quæ æquali in tempore æqualia &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 uerò 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 acutæ 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 æquali 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 æqualitate 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 poßint, aut pondus ferre &longs;imul iuncti eam moueant, aut pondus ferant, erunt illæ proportiones coniunctæ non productæ.<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 natúrali, 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 æquali aut minore ui quàm &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 æricum tangens planum in puncto mouetur ad latus per quam-cunque uim, quæ 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 duæ quantitates &longs;umaturque toties <expan abbr="aggregat&utilde;">aggregatum</expan> maioris & minoris, quo-ties aggregatum minoris & 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, & &longs;pacium & 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;tateræ 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 & qualis inter animam & uitas, & &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 æqualister in eodem circulo iuxta proprios motus moueantur, pro-ductum temporis circuituum inuicem, erit æquale producto differentiæ tempo rum circuitus ductæ in tempus coniunctionis primæ.<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;tantiæ 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="cõueniat">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="cõiunctionis">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 & 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-riæ.<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 æquali proportione in or-dinibus per quantitates & 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 quæ 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"/>uæ ex alto de&longs;cendunt, cur non eandem pro di&longs;tantia motus rationem in libero aëre &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 & 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, tantò facilius trabitur.<emph.end type="italics"/></cell><cell>53</cell></row>

<row><cell>LXIIII.</cell><cell>O<emph type="italics"/>mne mobile quantò 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, & &longs;ubten&longs;arum con&longs;iderare, & quæ à 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 aliæque totidem ab eadem analo-gæ, erit proportio tertiæ unius ordinis ad tertiam alterius, ut &longs;ecundæ ad &longs;e-cundum duplicata, & quartæ ad quartam triplicata, quintæ 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 &<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 &<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, aliæque totidem in continua proportione poterunt con&longs;tituere tres quantitates in æquali differentia per-uer&longs;im copulatæ.<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;phæræ 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="rectã">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 à 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;tantiã">i&longs;tantiam</expan> centri terræ à 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, & 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 æ quales &longs;eu in &longs;phæra &longs;eu in plano &longs;e &longs;ecuerint, nunquàm oppo&longs;itos angulos æquales habent.<emph.end type="italics"/></cell><cell>77</cell></row>

<row><cell>LXXXVII.</cell><cell>P<emph type="italics"/>roportiones craßitiei aquæ ad <expan abbr="a&etilde;r&etilde;">aerrem</expan> in <expan abbr="cõparatione">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="deprehendãtur">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 aquæ ad aërem 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 mißi ponderis ad æqualitatem reducere.<emph.end type="italics"/></cell><cell>82</cell></row>

<row><cell>XCI.</cell><cell>P<emph type="italics"/>roportionem grauis cubi, & &longs;phærici æqualium in accliui, & 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 æqualis iuxta longitudinis <expan abbr="cõparation&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="cõcußione">concußione</expan> <expan abbr="etiã">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, & <expan abbr="quãt&utilde;">quantum</expan> <expan abbr="retrocedãt">retrocedant</expan> ambæ.<emph.end type="italics"/></cell><cell>86</cell></row>

<pb/>

<row><cell>XCIIII.</cell><cell>S<emph type="italics"/>i <expan abbr="quãtitas">quantitas</expan> aliqua nota atque proportio erit producta, <expan abbr="quãtitas">quantitas</expan> nota &longs;imiliter.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i duæ proportiones notæ 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: & alterius partis ad <expan abbr="alterã">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="quãtitatis">quantitatis</expan> ad duas <expan abbr="quãtitates">quantitates</expan> proportio nota, erit & <expan abbr="cõfu&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 præter unam cognitæ, erunt | & 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="cõparatione">comparatione</expan> ad <expan abbr="mot&utilde;">motum</expan> &longs;phæræ 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 æqualium 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, & 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 uicißim deter-minare.<emph.end type="italics"/></cell><cell>97</cell></row>

<row><cell>CVII.</cell><cell>S<emph type="italics"/>i in circulo duæ diametri ad rectum angulum &longs;e &longs;ecauerint: aliæ uerò ad perpendicu-lum ex diametro exicrint ad circum ferentiam, &longs;ingulæ &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 præterea portionis &longs;uperioris parte, quæ 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;põdentis">corre&longs;pondentis</expan>, <expan abbr="&qtilde;">quae</expan> line æ tran&longs;uer&longs;æ.<emph.end type="italics"/></cell><cell>100</cell></row>

<row><cell>CVIII.</cell><cell>P<emph type="italics"/>unctum æqualitatis differentiæ de&longs;cen&longs;us & remotionis à centro inuenire.<emph.end type="italics"/></cell><cell>100</cell></row>

<row><cell>CIX.</cell><cell>R<emph type="italics"/>ationem libræ expendere.<emph.end type="italics"/></cell><cell>101</cell></row>

<row><cell>CX.</cell><cell>S<emph type="italics"/>i duæ &longs;phæræ ex eadem materia de&longs;cendant in aëre, 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, & naues in &longs;calmo à 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 longiùs ferantur declarare,<emph.end type="italics"/></cell><cell>106</cell></row>

<row><cell>CXIII.</cell><cell>C<emph type="italics"/>ur uirga longius mittatur à puero quam à 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, & 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 machinæ oblongæ igneæ longius emittant &longs;phæram 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, quàm 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 machinæ 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 umbræ ad gnomonem &longs;inum, & 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 umbræ uer&longs;æ 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, & peripheriæ cuiuslibet circuli paralleli æquino-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 à 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, & loco<emph.end type="italics"/> S<emph type="italics"/>olis proportionem gnomonis, tam ad um-bram rectam quàm 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 lineæ alicui duplum alterius adiungatur, erit proportio d<gap/>arum ad primam maior quàm 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, & adiecta ad ip&longs;am minorem, minor proportio quàm aggre-gati ex maiore, & 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 duæ quantitates, <expan abbr="quar&utilde;">quarum</expan> una alteri dupla &longs;it: minuatur à minore quæ-dam quantitas, <expan abbr="ead&etilde;que">eadenque</expan> maiori addatur, erit minoris ad re&longs;iduum maior pro-portio, quàm aggregati ad maiorem duplicata.<emph.end type="italics"/> S<emph type="italics"/>i uerò minori addatur, & à maiore detrabatur, erit aggregati ad minorem minor proportio quàm 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 quadratæ 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 lineæ creari poßit.<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 progreßione 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 à 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="cõuer&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 quæ fiunt ex una parte in alterius quadratum mutuo æqualia &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 differentiæ.<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, æqualia &longs;unt producto unius in alteram cum quadrato differentiæ.<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 æqualem quadratis ambarum par tium detracta &longs;uperficie unius partis in alteram, e&longs;t æquale 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"/>& 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>, & par te ad <expan abbr="aggregat&utilde;">aggregatum</expan> ex alia parte, & adiecta &longs;e habeat, ut &longs;ecunda ad <expan abbr="tertiã">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 lineæ datæ 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, & adiectæ ad duplum unius in alteram.<emph.end type="italics"/></cell><cell>141</cell></row>

<row><cell>CLI.</cell><cell>P<emph type="italics"/>roportio differentiæ quadratorum partium cuiu&longs;uis lineæ, ad quadratum diffe-rentiæ illarum e&longs;t, uelut totius lineæ 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 æquales, duasque inæquales diuidatur, fueritque proportio ag-gregati ex maiore, & dimidio ad ip&longs;am maiorem, uelut ex minore, & ali-qua linea ad ip&longs;am minorem, & rur&longs;us aggregati ex minore, & dimidio ad ip&longs;am minorem, uelut aggregati ex maiore, & alia addita ad ip&longs;am maiorem, erit proportio dimidij ad partem unam inæqualem, uelut alterius partis inæ-qualis ad &longs;uam additam mutuò, & etiam proportio additarum inuicem, uelut proportio <expan abbr="parti&utilde;">partium</expan> <expan abbr="inæquali&utilde;">inæqualium</expan> duplicata, & rur&longs;us ip&longs;um <expan abbr="dimidi&utilde;">dimidium</expan> lineæ a&longs;&longs;um-ptæ <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 lineæ datæ alia linea adiungatur, ab extremitatibus autem prioris lineæ duæ rectæ in unum punctum concurrant proportionem habentes, quam mediam inter tota m & adiectam, & adiectam erit punctus, concur&longs;us à puncto extre-mo lineæ adiectæ di&longs;tans per lineam mediam.<emph.end type="italics"/> Q<emph type="italics"/>uod &longs;i ab extremo alicuius li-neæ æqua'is mediæ, &longs;eu peripheria circuli, cuius &longs;emidiameter &longs;it media linea duæ lineæ ad prædicta puncta producantur, ip&longs;æ erunt in proportione mediæ ad adiectam.<emph.end type="italics"/></cell><cell>145</cell></row>

<row><cell>CLV.</cell><cell>Q<emph type="italics"/>uadr atorum numerum proportionem & 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 æqualis e&longs;&longs;e pote&longs;t alicui angulo contento recta, & 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 ductæ tres lineæ 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, & æquales ad unum punctum terminati, & latus unum commune inter reliqua quantita-te medium nece&longs;&longs;e e&longs;t angulum à 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 conuexæ partis, & conca-uæ proportiones datæ &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, quæ particulá 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 & 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 & 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 æqualiter, 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 & regre&longs;&longs;us, tam &longs;ine latitudine quàm cum latitudine in planetis per &longs;olos concentricos circulos æqualiter 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, quæ 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, maximè auri & argenti declara-re.<emph.end type="italics"/></cell><cell>199</cell></row>

<row><cell>CLXXIX.</cell><cell>S<emph type="italics"/>i duobus totis duæ portiones &longs;imiles ab&longs;cindantur ab ei&longs;dem denuò, & ab-&longs;cißis portionibus partes eædem auferantur, denuoque ac denuò quoties libuerit à portionibus, & ù re&longs;iduis ip&longs;arum quantitatum partes eædem auferantur, erit re&longs;iduí 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 primæ ad &longs;ecun-dam & tertiam, & rur&longs;us quartæ ad quintam & &longs;extam: ita &longs;e habebit proportio &longs;ecundæ ad tertiam, ad proportionem quintæ 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 uitæ naturalis per &longs;pacium uitæ fortuitum declarare.<emph.end type="italics"/></cell><cell>204</cell></row>

<row><cell>CLXXXIIII.</cell><cell>Q<emph type="italics"/>uæcunque 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, & quanto magis crura ad f&oelig;mora, & f&oelig;mora ad pectus reclinata habet, facilius con&longs;urgat, cum tamen hæc 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 primæ & &longs;ecundæ quantitatis ad tertiam, ut primæ & quartæ ad quintam, fueritque quarta &longs;ecunda maior, erit proportio quar-tæ ad quintam maior quàm &longs;ecundæ 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"/>quartæ ad quintam quàm &longs;ecundæ 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 & ‘eadem’ 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 & maiore cum difficultate moueri quàm &longs;ecundum.<emph.end type="italics"/></cell><cell>214</cell></row>

<row><cell>CLXXXVIII.</cell><cell>S<emph type="italics"/>i uires aliquæ moueant cum ponderibus aliqua pondera, ut compo&longs;ita pro-portio &longs;it eadem proportioni uirium & duorum ponderum mouentium ag-gregatum æquale duorum ponderum, ubi maior fuerit partium in æqual<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 æquali proportione coapte-tar, facilius deor&longs;um trahetur quàm quod maius e&longs;t & 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, & &longs;ecundum minus tertio, proportio autem primi ad &longs;ecundum multo maior quàm &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 & tertium mouea-tur faciliùs ab ei&longs;dem uiribus, & primo uel &longs;ecundo quàm antea.<emph.end type="italics"/></cell><cell>215</cell></row>

<row><cell>CXCL.</cell><cell>C<emph type="italics"/>um fuerint duo pondera & uires, duxerisque aggregatum ex uiribus & 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 æqualis 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 lineæ ducantur &longs;eu intus ad circun ferentia<gap/>u&longs;que, &longs;eu extra ad diametrum, erit proportio totius lineæ 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 æquale & 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 & eius æquidi&longs;tantibus quàm &longs;ecundum longitudinem & 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, æquale &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="nõ">non</expan>, <expan abbr="demõ&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, & cur cùm 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 rectà nauim dirigat.<emph.end type="italics"/></cell><cell>228</cell></row>

<row><cell>CCI.</cell><cell>S<emph type="italics"/>i duæ lineæ 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 faciliùs, 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"/>eipublicæ.<emph.end type="italics"/></cell><cell>234</cell></row>

<row><cell>CCVI.</cell><cell>R<emph type="italics"/>hombi paßiones 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 à 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 æquales diui&longs;a intelligatur, quæ am-bæ quadratæ &longs;int, itemque in duas inæquales, erit parallelipedum ex latere mediæ partis in totam &longs;uperficiem maius aggregato parallelipedorum ex partibus inæqualibus in latera alterius partis mutuo, in eo, quod fit ex dif ferentia lateris minoris partis à mediæ latere in differentiam maioris par-tis &longs;uperficiei à media &longs;uperficie bis, & ex differentia amborum laterum inæqualium iunctorum ad ambo latera, æqualia 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 duæ lineæ ad æquales angulos ab eodem puncto peripheriæ circuli refle-ctantur, nece&longs;&longs;e e&longs;t angulos cum dimetiente factos æquales 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 æqualia di-uidere.<emph.end type="italics"/></cell><cell>242</cell></row>

<row><cell>CCXI.</cell><cell>S<emph type="italics"/>i duæ lineæ ex duobus punctis peripheriam contingentes, in eandem par-tem protrahantur, &longs;emper magis di&longs;tabunt inuicem ea ex parte, & 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 lineæ quotuis ducantur, tres inue-nire lineas, quæ 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 cítraque, & 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 æqualiter à centro di&longs;tantia &longs;ignentur, erit pun-ctum reflexionis æqualis in medio arcus intercepti inter lineas, quæ à cen tro ducuntur ad illa puncta.<emph.end type="italics"/> S<emph type="italics"/>i uerò unum centro proximius fuerit altero, punctum æqualitatis in peripheria tantò longius, uer&longs;us breuiorem line-am, quantò punctum aliud à 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 inæqualiter di&longs;tantium à centro, æqualiter di&longs;tat à lineis, ductis à centro ad puncta æqualiter 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 inæquales, & extra utrunqúe punctum ad illud ex mi-nore reflexè per magnam partem minoris à 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"/>unæ illuminatam à<emph.end type="italics"/> S<emph type="italics"/>ole per radios reflexos à<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 maculæ<emph.end type="italics"/> L<emph type="italics"/>unæ indagare.<emph.end type="italics"/></cell><cell>248</cell></row>

<row><cell>CCXIX.</cell><cell>R<emph type="italics"/>ationem eorum quæ 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 æ&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"/>unæ & cæterorum a&longs;trorum digno&longs;citur ex proportione alio-rum ad eam iuxta di&longs;tantiam: ip&longs;ius uerò iuxta rationem pupillæ ad<emph.end type="italics"/> L<emph type="italics"/>u-nam di&longs;tantiæ ratione.<emph.end type="italics"/></cell><cell>354</cell></row>

<row><cell>CCXXII.</cell><cell>Q<emph type="italics"/>uantitates quæ æquales e&longs;&longs;e non po&longs;&longs;unt in eodem genere, maius tamen & 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 quæ actu æquales 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 æui ui-tarum proportio ulla e&longs;t ad tempus, quod pote&longs;tate e&longs;t, utpotè 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 & po&longs;t.<emph.end type="italics"/></cell><cell>257</cell></row>

<row><cell>CCXXVII.</cell><cell>V<emph type="italics"/>itæ 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;cientiæ futurorum & cæterorum 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 & 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 æternum e&longs;t, beatum, &longs;ecurum, immutabile &longs;ecundum locum, &longs;olum iuxta e&longs;&longs;entiam fit: iuxta quod uelut à leui &longs;u-&longs;urro aquæ & aura æ&longs;tiua demulcetur.<emph.end type="italics"/></cell><cell>270</cell></row></table>

<s>FINIS.</s>

<s>HIERONYMI CAR <lb/>DANI MEDIOLANENSIS, CI<lb/>VI'SQVE BONONIENSIS, MEDICI<lb/>de Proportionibus, &longs;eu Ope<lb/>ris Perfecti <lb/>LIBER QVINTVS.</s>

<s>Prima diffinitio.</s>

<s>Proportio ab Euclide &longs;ic de&longs;cribitur, Quòd <lb/>&longs;it duarum quantitatum eiu&longs;dem generis, <lb/>quod ad magnitudinem attinet, compara<lb/>tio certa.</s>

<s>Secunda diffinitio.</s>

<s>Proportiones per &longs;imilitudinem <expan abbr="dic&utilde;tur">dicuntur</expan>, <lb/>cùm quantitas quantitati <expan abbr="compara&ttilde;">comparatur</expan> alterius <lb/>generis, cui fingitur æqualis e&longs;&longs;e pote&longs;tate.<gap/></s>

<s>Velut &longs;i a b fingatur monas in comparatione <lb/>ad b c erit rectangulum a c æquale lineæ b c.</s>

<s>Tertia diffinitio.</s>

<s>Proportio æqualis proportioni e&longs;t, cùm eodem modo termini <lb/>&longs;e habent inuicem in utraque</s>

<s>Quarta diffinitio.</s>

<s>Proportiones &longs;ecundum genus notæ dicuntur, cùm nouimus, <lb/>quòd &longs;int maiores, aut minores. Nam cùm æquales &longs;unt, &longs;imul ne<lb/>ceffe e&longs;t, ut cogno&longs;camus genus, & &longs;peciem.</s>

<s>Quinta diffinitio.</s>

<s>Datum po&longs;itione e&longs;t: quod nece&longs;&longs;ariò ex po&longs;itis certam habet <lb/>quantitatem.</s>

<s>Sexta diffinitio.</s>

<s>Datum &longs;impliciter dicitur, quod ex propo&longs;itis cogno&longs;ci pote&longs;t, <lb/>quantum &longs;it.</s>

<s>Septima diffinitio.</s>

<s>Proportiones pote&longs;tate <expan abbr="dicun&ttilde;">dicuntur</expan>, quæ&longs;ub comparatione aliarum <lb/><expan abbr="quantitat&utilde;">quantitatum</expan> nece&longs;&longs;ariam habentium <expan abbr="cõnexionem">connexionem</expan> <expan abbr="&longs;ol&utilde;">&longs;olum</expan> <expan abbr="cogno&longs;cun&ttilde;">cogno&longs;cuntur</expan>.</s>

<s>Hæ autem &longs;unt aliquando eiu&longs;dem generis, cum primis ut nu<lb/>meri: aliquandò alterius, ut linearum & &longs;uperficierum, angulorum, <lb/>& arcuum: aliquando eiu&longs;dem generis, & diuen&longs;arum &longs;pecierum, <lb/>ut arcuum per &longs;inus, qua utuntur A&longs;tronomi.</s>

<s>Octaua diffinitio.</s>

<s>Proportio homonyma dicitur duarum quantitatum diuer&longs;i ge</s>

<s>

<arrow.to.target n="marg1"></arrow.to.target><lb/>neris, &longs;ed alterius a b altero dependentium, uelut motus ad tem

<pb pagenum="2"/>pus. Dicimus enim motum tardum, uel uelocem in comparatione <lb/>ad tempus.</s>

<s>Nona diffinitio.</s>

<s>Proportionum aliæ dicuntur rhete, aliæ alogæ, rhetæ quæ &longs;unt <lb/>ut numeri ad numerum, alogæ quæ non &longs;unt numeri ad numerum.</s>

<s>Decima diffinitio</s>

<s>Proportio rhete alia æqualis, alia multiplex, uel &longs;ubmultiplex: <lb/>alia unius partis exce&longs;&longs;us, aut defectus, alia plurium, quam &longs;uper<lb/>partientem, aut &longs;upartientem uocant.</s>

<s>Vndecima diffinitio.</s>

<s>Cum diui&longs;o denominatore per numeratorem exit quantitas alo <lb/>ga, proportio dicitur aloga: &longs;i autem numerus integer, aut pars nu<lb/>meri nota dicitur rhete.</s>

<s>Duodecima diffinitio.</s>

<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/>

<arrow.to.target n="fig1"></arrow.to.target><lb/>titates a b c dicetur proportio a ad c producta ex pro <lb/>portione a ad b & b ad c, & &longs;imiliter proportio c ad <lb/>a producitur ex proportione b ad a, & c ad b.</s>

<s>Tertiadecima diffinitio.</s>

<s>Proportionem per proportionem diuidi e&longs;t, quoties ad eandem <lb/>quantitatem duæ quantitates comparantur, tunc illarum propor<lb/>tio e&longs;t, quæ prodit una per alteram diui&longs;a.</s>

<s>Sint proportiones a & b ad c & interponatur b inter a & c, dico <lb/>proportionem a ad c diui&longs;am per proportionem a ad b, & prodire <lb/>proportionem b ad c, con&longs;tat ex conuer&longs;a præcedentis.</s>

<s>Quartadecima diffinitio.</s>

<s>Additio proportionum intelligitur quotiens duarum quanti<lb/>tatum ad unam tertiam, proportiones per aggregatum ip&longs;arum <lb/>quantitatum ad eandem coniunguntur.</s>

<s>Velut &longs;i comparentur a b & b c ad d, inde tota <lb/>

<arrow.to.target n="fig2"></arrow.to.target><lb/>a c ad d dicemus proportionem, ac ad d e&longs;&longs;e con <lb/><expan abbr="iunctã">iunctam</expan> ex duabus proportionibus a b ad d & b c <lb/>ad <expan abbr="eand&etilde;">eandem</expan> d. Hoc & duo &longs;equentes &longs;icut & du&ecedil; <expan abbr="anteced&etilde;tes">antecedentes</expan> demon<lb/>&longs;trabitur e&longs;&longs;e. nunc &longs;olum quomodo <expan abbr="intelligend&utilde;">intelligendum</expan> &longs;it proponimus.</s>

<s>Quintadecima diffinitio.</s>

<s>Detractionem proportionis à proportione intelligimus fieri <lb/>per <expan abbr="detraction&etilde;">detractionem</expan> minoris quantitatis à maiore, comparatam ad ean<lb/>dem quantitatem.</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. & probatur <lb/>ex conuer&longs;ione præcedentis.</s>

<s>Sextadecima diffinitio.</s>

<s>Extractio radicum alicuius proportionis fit per extractionem <lb/>radicum quantitatum illius iuxta unam, & eandem rationem.</s>

<s>Velut quadratæ, uel cubæ, uel pronicæ, uel uniner&longs;alis, uel alte<lb/>rius modi.</s>

<s>Decima&longs;eptima diffinitio.</s>

<s>Cùm fuerint duæ proportiones &longs;imiles in tribus terminis con<lb/>tinuatæ, dicetur proportio primæ quantitatis ad tertiam ueluti <lb/>primæ ad &longs;ecundam duplicata. Et &longs;i &longs;int tres proportiones &longs;imiles <lb/>in quatuor terminis, dicetur proportio primæ quantitatis ad quar<lb/>tam triplicatà ei, quæ e&longs;t primæ ad &longs;ecundam,</s>

<s>Decimaoctaua diffinitio.</s>

<s>Confu&longs;a proportio dicitur &longs;implicis, aut compo&longs;itæ quantitatis <lb/>ad compo&longs;itam in comparatione ad proportiones ad partes.</s>

<s>Decimanona diffinitio.</s>

<s>Quantitates qu&ecedil; in continua &longs;unt proportione Analogæ <expan abbr="uocan&ttilde;">uocantur</expan>.</s>

<s>Dictum e&longs;t hoc ad fugiendum nomen barbarum, etiam ut bre<lb/>uiter tamen po&longs;&longs;emus &longs;ententiam explicare.</s>

<s>Vige&longs;ima diffinitio.</s>

<s>Reflexa proportio dicitur cum trium quantitatum aggregatum <lb/>primæ, & tertiæ &longs;e habet ad &longs;ecundam uelut &longs;ecunda ad tertiam,</s>

<s>Vige&longs;ima prima diffinitio.</s>

<s>Trium quantitatum analogarum aliæ quidem Geometricæ, <lb/>cùm proportio &longs;imilis e&longs;t: Aliæ Arithmeticæ, cum fuerit æqualis <lb/>exce&longs;&longs;us hucindè: Aliæ mu&longs;icæ cum fuerit proportio primæ ad ter <lb/>tiam multiplex, aut &longs;implex, aut compo&longs;ita exce&longs;&longs;us quæ &longs;implici <lb/>iuncta &longs;it ad multiplicis perfectionem: eadem autem &longs;it proportio <lb/>exce&longs;&longs;us primæ, & &longs;ecundæ ad exce&longs;&longs;um &longs;ecundæ &longs;upra tertiam.</s>

<s>Velut proportio 6. 4. 3. dupla e&longs;t utrinque, & 6. 3. 2 tripla. & 28. 24. <lb/>21. & 45. 40. 36. Geometrica uerò & arithmetica facilius continuan<lb/>tur in quotquot quantitatibus, &longs;ed & mu&longs;ica uelut 12. 8. 6. 4. 3. & <lb/>proportio 8 ad 5 mu&longs;ica e&longs;t: quia proportio 5 ad 4 mu&longs;ica e&longs;t, & <lb/>bene &longs;onans, igitur con&longs;titutis 8. 5. 4. cum 8 ad 4 benè &longs;onet, & 5 <lb/>ad 4, & 4 &longs;it extrema non media inde 8. & 5 benè <expan abbr="&longs;onãt">&longs;onant</expan>. nam in me<lb/>dijs <expan abbr="nõ">non</expan> e&longs;t <expan abbr="uer&utilde;">uerum</expan>, ut in 9. 6. 4 bis diapente, & 16. 12. 9 bis diate&longs;&longs;aron.</s>

<s>Vige&longs;ima &longs;ecunda diffinitio.</s>

<s>Quantitates quæ &longs;imilem habent proportionem non continua<lb/>tam, omiologæ appellantur.</s>

<s>Vige&longs;ima tertia diffinitio.</s>

<s>Prima operatione con&longs;i&longs;tere dicuntur proportiones, cùm inter <lb/>primo conflatas quantitates con&longs;titerint.</s>

<s>PRIMA Animi communis &longs;ententia.</s>

<s>Omnis Proportio e&longs;t, aut æqualitatis, aut maior inæqualis, <lb/>aut minor.</s>

<s>Secunda animi communis &longs;ententia.</s>

<s>Quilibet numerus tantus dicitur, quanta e&longs;t illius proportio ad <lb/>monadem.</s>

<s>Dicimus enim quatuor, quod monadem quater contineat. Et <lb/>duo cum dimidio cùm monadem bis & &longs;emis contineat.</s>

<s>Tertia animi communis &longs;ententia.</s>

<s>Proportionem defectus, &longs;eu detractæ quantitatis ad defectum <lb/>e&longs;&longs;e po&longs;&longs;e, ut quantitatis ad quantitatem dicuntur communes ani<lb/>mi &longs;entcntiæ, quæ ex intellectu &longs;olo terminorum, quod ueræ &longs;int, <lb/>cogno&longs;cuntur. Si ergo defectus e&longs;t quantitas, & quantitas eiu&longs;dem <lb/>&longs;peciei, quia detrahitur, & defectus non e&longs;t &longs;implicitur, &longs;ed detra<lb/>cto ergo per quartam petitionem: uel primam diffinitionem erit <lb/>proportio interillas. Sunt enim ambæ detractæ.</s>

<s>Quarta animi communis &longs;ententia.</s>

<s>Inter quantitatem, & defectum minorem quantitate, cuius e&longs;t de <lb/>fectus, e&longs;t proportio, quatenus e&longs;t quantitas. Sit a b linea, & detra<lb/>cta quantitas b c, non maior a b & d &longs;it alia quæuis quantitas eiu&longs;<lb/>

<arrow.to.target n="fig3"></arrow.to.target><lb/><expan abbr="d&etilde;">dem</expan> generis, dico quòd inter d & b c e&longs;t propor<lb/>tio quatenus b c e&longs;t quantitas, quia &longs;unt eiu&longs;<lb/>dem generis ideo &longs;unt in aliqua proportione <lb/>per primam diffinitionem. Sed ut b c e&longs;t defectus, nulla e&longs;t propor<lb/>tio: quia quanto b c augetur, tanto augetur proportio d ad b c, & <lb/>hoc e&longs;t contra demon&longs;trata ab Euclide.</s>

<s>Quinta animi communis &longs;ententia.</s>

<s>Cum proportio producitur ex proportionibus quælibet illa<lb/>rum dicetur producta diui&longs;a per alteram.</s>

<s>Sexta animi communis &longs;ententia.</s>

<s>Æqualium quantitatum &longs;eu proportionum ad tertiam compa<lb/>rabilium eadem e&longs;t proportio atque uici&longs;sim. Hæc et&longs;i demon&longs;tre<lb/>tur ab Euclide, e&longs;t tamen hic generalior: & &longs;atis per &longs;e nota. Vt &longs;it <lb/>propior animi communi &longs;ententiæ, quàm rei demon&longs;trandæ.</s>

<s>Septima animi communis &longs;ententia.</s>

<s>Ad quod quantitas proportionem habet infinitam, id in genere <lb/>illius quantitatis non comprehenditur.</s>

<s>Nam proportio e&longs;t duarum quantitatum eiu&longs;dem generis com<lb/>paratio certa: at hæc comparatio certa non e&longs;t: non igitur quantita<lb/>tes ambæ &longs;unt, aut non eiu&longs;dem generis.</s>

<s>PRIMA Petitio.</s>

<s>Si fuerit primi ad &longs;ecundum, ut tertij ad quartum, & ex primo in <lb/>&longs;ecundum producatur æquale, aut maius, aut minus primo, uel <lb/>&longs;ecundo, producetur eodem modo ex tertio in quartum &ecedil;quale aut <lb/>maius, aut minus tertio, uel quarto eadem ratione & ordine.</s>

<s>Secunda petitio.</s>

<s>Proportiones po&longs;&longs;unt duci, diuidi, iungi, & auferri, & &longs;umi radix <lb/>in eis cuiu&longs;cunque generis, atque earum quantitatis, ut libet, po&longs;&longs;e <lb/>tran&longs;ponere.</s>

<s>Tertia petitio.</s>

<s>Proportionis cuiu&longs;uis nomen à denominatore &longs;uprà &longs;cripto, & <lb/>numeratore infrà &longs;cripto &longs;umitur.</s>

<s>Quarta petitio.</s>

<s>Diui&longs;a quauis quantitate per aliam eiu&longs;dem generis, quod exit <lb/>proportio dicitur.</s>

<s>Quinta petitio.</s>

<s>Qu&ecedil;libet proportio e&longs;t uel inter duas quantitates, uel per unam <lb/>&longs;ignificatur.</s>

<s>Nam per tertiam petitionem &longs;i &longs;int duæ quantitates, quæ non hæ <lb/>beant unius rationem, nomen &longs;umit proportio à duobus numeris, <lb/>&longs;in autem &longs;it altera monas, erit per &longs;ecundam animi communem &longs;en <lb/>tentiam, proportio numerus ip&longs;e Ideò patet, quod dicitur.</s>

<s>Sexta petitio.</s>

<s>Propo&longs;ita proportione quacunque, & monade quantitatem inue <lb/>nire, quæ &longs;e habeat ad monadem in proportione propo&longs;ita.</s>

<s>Nam cùm per quartam petitionem diui&longs;a quantitate per quan<lb/>titatem exeat proportio, & numerus ad <expan abbr="monad&etilde;">monadem</expan> &longs;e habeat, ut pro<lb/>portio, ideo &longs;umpta monade &longs;ecundum illum numerum, ille nume <lb/>rus e&longs;t quantitas quæ&longs;ita.</s>

<s>Septima petitio.</s>

<s>Quamlibet quantitatem per aliam eiu&longs;dem generis diuidere <lb/>po&longs;&longs;e.</s>

<s>Octaua petitio.</s>

<s>Proportionem in proportionem ducere po&longs;&longs;e: quamuis &longs;int in<lb/>ter quantitates diuer&longs;i generis.</s>

<s>Quod dicitur de multiplicatione intelligendum e&longs;t de alijs ope<lb/>rationibus &longs;uprà enumeratis.</s>

<s>Nona petitio.</s>

<s>Monadem &longs;emper &longs;umere in quo cunque genere po&longs;&longs;e propo&longs;i<lb/>ta proportione.</s>

<s>Nam licet diuidere per &longs;eptimam petitionem quantitatem per <lb/>quantitatem proportionis: & quod exit, e&longs;t proportio per quar<lb/>tam petitionem, & per &longs;ecundam animi communem &longs;ententiam <lb/>illa proportio e&longs;t numero æqualis: ergo diui&longs;a proportione, per &longs;i<lb/>milem numerum &longs;tatuetur monas.</s>

<s>Decima petitio.</s>

<s>In quouis genere quantitatum &longs;umere po&longs;&longs;e quantitatem, quæ <lb/>

<arrow.to.target n="marg2"></arrow.to.target><lb/>&longs;e habeat ad monadem in proportione data. Similem huic propo<lb/>nit Euclides in lineis generaliter: nos autem contrà generaliter in <lb/>omnibus quantitatibus, &longs;ed de monade tantum.</s>

<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>

<s>Vndecima petitio.</s>

<s>Monadem in quancunque quantitatem ductam æquale ip&longs;i pro<lb/>ducere. Similiter & proportionem æqualem.</s>

<s>Nam cum aliqua quantitas augeat ducta aliqua minuat, nece&longs;&longs;e <lb/>e&longs;t aliquam e&longs;&longs;e, quæ nec augeat, nec minuat, & hæc e&longs;t monas. <lb/>Idem dico de diui&longs;ione. 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 æqua<lb/>litatem refert. Quod etiam e&longs;t per&longs;picuum ex &longs;upradictis.</s>

<s><margin.target id="marg3"></margin.target>S<emph type="italics"/>ecunda ani <lb/>mi <expan abbr="cõmunis">communis</expan> <lb/>&longs;ententia.<emph.end type="italics"/></s>

<s>Duodecima petitio.</s>

<s>Cum fuerint quatuor quantitates & ad primam, & tertiam æquè <lb/>multiplicibus a&longs;&longs;umptis, item que ad &longs;ecundam & quartam, & &longs;i mul<lb/>tiplex primæ maius e&longs;t multiplici &longs;ecundæ, multiplex tertiæ &longs;it ma<lb/>ius multiplici quartæ, & &longs;i minus minus, & &longs;i æquale æquale, idque<lb/>&longs;emper quouis modo a&longs;&longs;umptis his proportionibus ad primam & <lb/>tertiam, & ad &longs;ecundam & quartam erit proportio primæ ad &longs;ecun <lb/>dam, ut tertiæ ad quartam. Hæc etiam a&longs;&longs;umitur ab Euclide. Et per <lb/>

<arrow.to.target n="marg4"></arrow.to.target><lb/>hanc intelligimus etiam conuer&longs;am.</s>

<s><margin.target id="marg4"></margin.target>Q<emph type="italics"/>uinto<emph.end type="italics"/> E<emph type="italics"/>le. <lb/>diff.<emph.end type="italics"/> 6.</s>

<s>Tertiadecima petitio.</s>

<s>Quantitates æquales, atque proportiones in qua&longs;uis quanti<lb/>tates ductæ eandem &longs;eruant rationem. Euclides hanc demon&longs;trat, <lb/>nos autem ad uitandum tædium petimus concedi, &longs;ub qua in<lb/>

<arrow.to.target n="marg5"></arrow.to.target><lb/>cluduntur diui&longs;io etiam additio, detractio, laterum omnium in<lb/>uentio.</s>

<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>

<s>Quartadecima petitio.</s>

<s>Cùm termini alicuius quantitatis eandem &longs;eruant rationem in <lb/>omnibus, & firmi &longs;unt ac &longs;tabiles eiu&longs;dem rationis comparatione <lb/>contentæ partes æqualem &longs;eruant exce&longs;&longs;um, &longs;eu proportionem.</s>

<s>PROPOSITIO prima.</s>

<s>Proportionem in proportionem duci e&longs;t &longs;uperiores nume<lb/>ros atque inferiores inuicem ducere.</s>

<s>Sit proportio lineæ a ad lineam b, ut anguli cad angulum d, &longs;ta<lb/>

<arrow.to.target n="marg6"></arrow.to.target><lb/>tuatur e monas in genere a <lb/>

<arrow.to.target n="fig4"></arrow.to.target><lb/>b, & fiat fad e, ut cad d, & du <lb/>

<arrow.to.target n="marg7"></arrow.to.target><lb/>catur<gap/>a in f & b in e, & pro<lb/>ducantur g & h. 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 æqualis <lb/>b, igitur a ad h ut ad b. Du<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<lb/>periores cum &longs;uperioribus, & inferiores cum inferioribus. 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 cùm f &longs;it proportio, & 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/>quæ e&longs;t fin proportionem c ad d, liquet igitur propo&longs;itum.</s>

<s><margin.target id="marg10"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 2. A<emph type="italics"/>ni<lb/><gap/>i &longs;entent.<emph.end type="italics"/></s>

<s>Propo&longs;itio <expan abbr="&longs;ec&utilde;nda">&longs;ecunnda</expan>.</s>

<s>Proportio extremorum producitur ex intermedijs.<lb/>

<arrow.to.target n="marg13"></arrow.to.target></s>

<s>Sint a b c quantitates dico proportio<lb/>

<arrow.to.target n="fig5"></arrow.to.target><lb/>nem a ad c, produci ex proportione a ad b </s>

<s>

<arrow.to.target n="marg14"></arrow.to.target><lb/>& b ad c, &longs;tatuantur totidem à monade d e <lb/>f, erúntque ex demon&longs;trantis ab Euclide in <lb/>quinto <expan abbr="Elem&etilde;torum">Elementorum</expan> in eadem proportio<lb/>ne, ftatuatur ergo d prima quantitas e &longs;e<lb/>cunda & tertia f quarta. eritqúe per præce<lb/>

<arrow.to.target n="marg15"></arrow.to.target><lb/>dentem proportio productorum ex d in e <lb/>& &longs;it g, & in f & &longs;it h, producta ex propor<lb/>tionibus d ad e & e ad f, quare ex propor<lb/>tionibus a ad b & b ad e, &longs;ed ex dictis cum <lb/>e &longs;it eadem, erit proportio d ad f, ut g ad h & proportio, d ad f per <lb/>æquam 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 & b ad c, & <lb/>e&longs;t proportio ip&longs;a a ad c d numerus, ut o&longs;ten&longs;um e&longs;t.</s>

<s>Ex hoc &longs;equitur, quòd cùm fuerit quantitas tertia monas ex pro<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>

<s>Ex hoc &longs;equitur, quòd conuer&longs;a proportio producitur ex con<lb/>uer&longs;is proportionibus.</s>

<s>Propo&longs;itio tertia.</s>

<s>Si proportio ex duabus proportionibus in quatuor terminis <lb/>producatur, ip&longs;a uerò proportio inter duas alias quantitates fue

<pb pagenum="8"/>rit con&longs;tituta: con&longs;urgent trecenti &longs;exaginta modi productionis <lb/>proportionis.</s>

<s>

<arrow.to.target n="marg19"></arrow.to.target></s>

<s>H&ecedil;c propo&longs;itio ut præcedens & <expan abbr="&longs;equ&etilde;tes">&longs;equentes</expan> tres ab Alchindo &longs;um<lb/>ptæ &longs;unt, & ab eo demon&longs;trantur. Sit ergo proportio a ad b, pro<lb/>

<arrow.to.target n="table2"></arrow.to.target><lb/>ducta ex proportione c ad d & e ad f, con&longs;tat <lb/>quòd cum &longs;int &longs;ex quantitates, quòd fieri pote<lb/>runt quindecim coniugationes, quas po&longs;ui à la<lb/>tere facilitatis gratia, quibus re&longs;pondent totidem <lb/>

<arrow.to.target n="table3"></arrow.to.target><lb/>conuer&longs;æ: erunt ergo triginta. Singulæ autem ha <lb/>rum produci po&longs;&longs;unt duodecim modis: ductis <lb/>duodecim in triginta, fiunt trecenti &longs;exaginta mo <lb/>di. Et hoc e&longs;t clarum per&longs;e, modo <expan abbr="demõ&longs;tremus">demon&longs;tremus</expan>, <lb/>quod &longs;inguli horum modorum po&longs;sint produ<lb/>ci duodecim modis, & capiamus ab primam qu&ecedil; <lb/>pote&longs;t produci ex c d & e f: Item ambabus con<lb/>uer&longs;is d c & fe: & rur&longs;us altera recta altera con<lb/>uer&longs;a: & hoc bifariam c d & f e, & d c & e f, &longs;unt er<lb/>go iam quatuor modi. Totidem ex c e & d f, toti<lb/>demque ex c f & d e, igitur erunt duodecim mo<lb/>di, quibus produci po&longs;&longs;e intelligitur propor<lb/>tio a ad b.</s>

<table>

<table.target id="table2"></table.target>

<table>

<table.target id="table3"></table.target>

<row><cell>direc.</cell><cell>conuer.</cell></row></table>

<s>Propo&longs;itio quarta.</s>

<s>Si fuerit proportio primi ad &longs;ecundum produ<lb/>cta ex proportionibus tertij ad quartum, & quin <lb/>ti ad &longs;extum, producetur etiam ex proportione <lb/>tertij ad &longs;extum, & quinti ad quartum.</s>

<s>Sit proportio a b producta ex proportioni<lb/>

<arrow.to.target n="table4"></arrow.to.target><lb/>bus c ad d, & e ad f, dico quod etiam erit produ</s>

<table>

<table.target id="table4"></table.target>

<s>

<arrow.to.target n="marg20"></arrow.to.target><lb/>cta ex proportionibus c ad f, & e ad d, di&longs;ponan<lb/>tur ut in figura & fiat ex c in e g, & 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 præ<lb/>&longs;uppo&longs;ita in &longs;ecunda productione etiam prode<lb/>unt g & h, igitur per primam propo&longs;itionem ha<lb/>rum a ad b proportio producitur ex proportionibus c ad f tertiæ <lb/>&longs;cilicet ad &longs;extam, & e ad d quint&ecedil; ad quartam, quod fuit <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan>.</s>

<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>

<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>

<s>Propo&longs;itio quinta.</s>

<s>Si fuerit proportio primi ad &longs;ecundum producta ex proportio<lb/>ne tertij ad quartum, & quinta ad &longs;extum: erit proportio tertij ad <lb/>&longs;extum producta ex proportionibus primi ad &longs;ecundum, & quar<lb/>ti ad quintum.</s>

<s>Sit proportio a ad b producta ex proportio<lb/>

<arrow.to.target n="marg22"></arrow.to.target><lb/>

<arrow.to.target n="table5"></arrow.to.target><lb/>nibus c ad d, & e ad f, dico quod proportio c ad <lb/>f producitur ex proportione a ad b & d ad e. In<lb/>terponam d e inter c & f, eritque ex &longs;ecunda pro<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, & e ad f producunt pro<lb/>portionem a ad b, igitur proportio c ad f produ <lb/>citur ex proportionibus a ad b, & e ad f.<lb/>

<arrow.to.target n="table6"></arrow.to.target></s>

<table>

<table.target id="table5"></table.target>

<table>

<table.target id="table6"></table.target>

<s>Propo&longs;itio &longs;exta.</s>

<s>Ex trecentis &longs;exaginta modis producenda<lb/>rum proportionum triginta &longs;ex tantum e&longs;&longs;e ne<lb/>ce&longs;&longs;arios.<lb/>

<arrow.to.target n="table7"></arrow.to.target></s>

<table>

<table.target id="table7"></table.target>

<s>Per quartam enim proportio a ad b produ<lb/>

<arrow.to.target n="marg23"></arrow.to.target><lb/>citur bifariam, & ex c ad d, & e ad f, & ex c ad f, & <lb/>e ad d. & perpræ cedentem c ad f producitur ex <lb/>a ad b, & d ad e, & per quartam rur&longs;us ex a ad e, <lb/>& d ad b. Et per præcedentem rut&longs;us a ad e ex c <lb/>ad f & b ad d, igitur per quartam eadem produ<lb/>cetur ex c ad d & b ad f. Quare per præceden<lb/>tem c ad f ex a ad e, & d ad b, & ita di&longs;ponemus <lb/>hos modos in tabula. Vides etiam <lb/>

<arrow.to.target n="table8"></arrow.to.target><lb/>aliquos modos non produci, ut pri<lb/>mi ad quartum nec ad &longs;extum, & li<lb/>quet, quòd cùm &longs;int quindecim o<lb/>mnes modi qui produci po&longs;&longs;e intelli<lb/>guntur, & nouem tantum producan<lb/>tur &longs;ex e&longs;&longs;e, qui non producuntur, quos <lb/>&longs;eor&longs;um in tabula coniunxi. Et con<lb/>&longs;tat etiam, quod totidem conuer&longs;i &longs;ci<lb/>licet decem octo <expan abbr="produc&utilde;tur">producuntur</expan>, de qui<lb/>bus diximus, ut &longs;int omnes triginta <lb/>&longs;ex, qui con&longs;tat ex duabus propo&longs;i<lb/>tionibus præmi&longs;sis, & hac tertia, <expan abbr="quã">quam</expan> <lb/>adiungemus &longs;cilicet, quòd propor<lb/>tio primi ad tertium producatur ex <lb/>proportionibus <expan abbr="&longs;ec&utilde;di">&longs;ecundi</expan> ad quartum, <lb/>& quinti ad <expan abbr="&longs;ext&utilde;">&longs;extum</expan>. Hoc enim ex præ<lb/>cedentibus non liquet: benè liquet <lb/>permutatis ordinibus, quod &longs;i pro<lb/>portio primi ad tertium producitur,

<pb pagenum="10"/>quod etiam propor<lb/>

<arrow.to.target n="marg24"></arrow.to.target><lb/>tio primi ad <expan abbr="quint&utilde;">quintum</expan>. <lb/>Nam tertium, & quin <lb/>tum, item que quartum, <lb/>& &longs;extum non <expan abbr="diffe-r&utilde;t">diffe<lb/>runt</expan> ni&longs;i ordine uolun <lb/>tario. Ergo interpo&longs;i<lb/>to e inter a, & c per &longs;e<lb/>cundam propo&longs;itio<lb/>nem proportio a ad c <lb/>producitur ex proportionibus a ad <lb/>e, & e ad c, ut ex demon&longs;tratis in præ<lb/>&longs;enti proportio a ad c producitur ex <lb/>c ad f & b ad d. Proportio ergo a ad <lb/>c producitur ex proportionibus e <lb/>ad c & c ad f & b ad d, at e ad c & c ad <lb/>f producunt eam, quæ e&longs;t e ad f per <lb/><expan abbr="&longs;ec&utilde;dam">&longs;ecundam</expan> propo&longs;itionem. Igitur pro<lb/>portio a ad c producitur ex propor<lb/>tionibus b ad d &longs;ecundi ad quartum, <lb/>& e ad f quinti ad &longs;extum. Hæc Al<lb/>chindus in &longs;uo libello: &longs;ed licet inge<lb/>nio &longs;a ualde: parum <expan abbr="tam&etilde;">tamen</expan> utilia olim <lb/><expan abbr="erãt">erant</expan> nece&longs;&longs;aria ad intelligendum ma<lb/>gnam <expan abbr="cõpo&longs;itionem">compo&longs;itionem</expan> Ptolem&ecedil;i, nunc <lb/>po&longs;tquam Heber has &longs;ex quantita<lb/>tes traduxit ad quatuor, pror&longs;us hæc <lb/>&longs;cientia ulli u&longs;ui e&longs;&longs;e de&longs;ijt.<lb/>

<arrow.to.target n="table9"></arrow.to.target></s>

<s><margin.target id="marg24"></margin.target>Modi qui <expan abbr="nõ">non</expan> <lb/>producuntur <lb/>pri. ad quartu <lb/>pri. ad &longs;extum <lb/>&longs;ec. ad <expan abbr="terti&utilde;">tertium</expan> <lb/>&longs;ec. ad <expan abbr="quint&utilde;">quintum</expan> <lb/>tert. ad quint. <lb/>quart. ad &longs;ext.</s>

<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>, & quin</cell></row>

<row><cell></cell><cell>ti ad &longs;extum.</cell></row>

<row><cell>2</cell><cell>tertij ad &longs;extum, & 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, &</cell></row>

<row><cell></cell><cell>quinti ad &longs;extum.</cell></row>

<row><cell>4</cell><cell>&longs;ecundi ad &longs;extum, &</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>, & ter-</cell></row>

<row><cell></cell><cell>tij ad quartum.</cell></row>

<row><cell>6</cell><cell>&longs;ecundi ad quartum, &</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, & &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>, & 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>, & 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, &</cell></row>

<row><cell></cell><cell>&longs;exti ad quintum.</cell></row>

<row><cell>12</cell><cell>primi ad quintum, & &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, &</cell></row>

<row><cell></cell><cell>quarti ad quintum.</cell></row>

<row><cell>14</cell><cell>primi ad quintum, &</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, &</cell></row>

<row><cell></cell><cell>tertij ad &longs;extum.</cell></row>

<row><cell>16</cell><cell>&longs;ecundi ad &longs;extum, & 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, &</cell></row>

<row><cell></cell><cell>quarti ad tertium.</cell></row>

<row><cell>18</cell><cell>primi ad <expan abbr="terti&utilde;">tertium</expan>, & quar-</cell></row>

<row><cell></cell><cell>ti ad &longs;ecundum.</cell></row></table>

<table>

<table.target id="table9"></table.target>

<s>Propo&longs;itio &longs;eptima.</s>

<s>In modis qui nece&longs;&longs;ariò produ<lb/>cuntur ex duabus proportionibus, <lb/>cum du&ecedil; quantitates ex illis, qu&ecedil; mo <lb/>dos conficiunt, æquales fuerint: pro<lb/>

<arrow.to.target n="table10"></arrow.to.target><lb/>portio producta ad quatuor quanti<lb/>tates omiologas reducetur.<lb/>

<arrow.to.target n="marg25"></arrow.to.target></s>

<table>

<table.target id="table10"></table.target>

<s>Sint &longs;ex quantitates a b c d e f, & <lb/>producatur proportio a ad b ex pro<lb/>portione c ad d, & e ad f, tu &longs;cis, quòd <lb/>modi recepti &longs;unt prima cum &longs;ecunda, tertia uel quinta, & &longs;ecunda <lb/>cum quarta, & &longs;exta, & tertia &longs;imiliter cum ei&longs;dem, & quinta eodem <lb/>modo cum ei&longs;dem: &longs;i igitur du&ecedil; quantitates ex his, qu&ecedil; faciunt pro

<pb pagenum="11"/>portionem productam inter &longs;e fuerint æquales reducetur hæc pro<lb/>portio ad quatuor quantitates omologas, &longs;cilicer abiectis amba<lb/>bus æqualibus. Sit gratia exempli prima æqualis quintæ: & quia <lb/>in octauo modo proportio <expan abbr="&longs;ec&utilde;di">&longs;ecundi</expan> ad quartum producitur ex pro<lb/>portione primi ad quintum, & &longs;exti ad tertium, ergo per expo&longs;ita <lb/>proportio &longs;ecundi ad quartum, ut &longs;exti ad tertium, & ita permutan<lb/>do, & conuertendo &longs;ecundi ad &longs;extum, ut quarti ad tertium, & tertij </s>

<s>

<arrow.to.target n="marg26"></arrow.to.target><lb/>ad quartum, ut &longs;exti ad &longs;ecundum.</s>

<s><margin.target id="marg26"></margin.target>V<emph type="italics"/>ndecima <lb/>petitione.<emph.end type="italics"/></s>

<s>Propo&longs;itio octaua.</s>

<s>Si duarum <expan abbr="proportion&utilde;">proportionum</expan> &longs;uperiores numeri alternatim cum infe <lb/>rioribus multiplicentur, atque coniungantur: erit proportio aggre<lb/>gati ad productum ex inferioribus inuicem proportio ex primis <lb/>proportionibus compo&longs;ita.</s>

<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 & a in d, & fiat f, iunganturque e & f & fiat h, <lb/>& ducatur b in d et fiat g: dico <expan abbr="proportion&etilde;">proportionem</expan> h g com<lb/>po&longs;itam e&longs;&longs;e ex proportione a ad b, & c ad d. Quia <lb/>

<arrow.to.target n="marg28"></arrow.to.target><lb/>enim ex b in c fit e, & ex b in d fit g, erit proportio e <lb/>ad g, ut c ad d, & &longs;imiliter, quia ex d in a fit f, & ex d in b fit g, erit f ad <lb/>g ut a ad b. Sed e & f componunt h, igitur proportio h ad g e&longs;t com <lb/>po&longs;ita ex proportionibus e & f ad g, igitur per communem animi <lb/>&longs;ententiam, & diffinitionem compo&longs;itæ 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, & c ad d, quod e&longs;t <lb/>propo&longs;itum.</s>

<s><margin.target id="marg28"></margin.target>E<emph type="italics"/>x<emph.end type="italics"/> 13 <emph type="italics"/>peti<lb/>tione.<emph.end type="italics"/></s>

<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>

<s>Propo&longs;itio nona.</s>

<s>Si duarum proportionum &longs;uperiores numeri alternatim cum <lb/>inferioribus multiplicentur, minusque productum ex maiore detra<lb/>hatur, erit re&longs;idui ad productum ex inferioribus proportio uelut <lb/>illa, quæ relinquitur detracta minore proportione ex maiore.</s>

<s>Hæc eodem modo probatur, ut præcedens, ni&longs;i quod h fit de<lb/>

<arrow.to.target n="marg30"></arrow.to.target><lb/>tracto è minore: gratia exempli ex f, & ita ex diffinitione patet pro<lb/>po&longs;itum.</s>

<s>Propo&longs;itio decima.</s>

<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<lb/>portionibus eiu&longs;dem quantitatis a&longs;&longs;umptæ ad utran que partem pri<lb/>mæ quantitatis &longs;eor&longs;um.<lb/>

<arrow.to.target n="marg31"></arrow.to.target></s>

<s>Sit a b quantitas diui&longs;a in c, & &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, & &longs;umatur quædam quantitas e eiu&longs;

<pb pagenum="12"/>dem tamen generis, cum illis dico quòd proportio e ad d e&longs;t com<lb/>po&longs;ita ex proportionibus e ad a c, & e ad b c. Po&longs;ita ergo e tan<08> &longs;u<lb/>periore numero, & a c & c b inferioribus, erit ex octaua propo&longs;itio<lb/>ne huius proportio productorum ex e in a c, & coniunctorum, & <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, & e ad c b: at quod fit ex a c in c b, e&longs;t æquale ei quod fit ex a b in <lb/>d, eo quòd a b, a c, c b & d &longs;unt omiologæ 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, & 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. per &longs;uppo&longs;ita igitur proportio e ad d e&longs;t compo&longs;ita ex propor <lb/>tionibus e ad a c, & e ad b c, quod fuit demon&longs;trandum.</s>

<s>Propo&longs;itio undecima.</s>

<s>Proportio aggregati quarumlibet duarum quantitatum ad ag<lb/>gregatum duarum æqualium quantitatum e&longs;t compo&longs;ita ex pro<lb/>portionibus primis, & diui&longs;a per duplam.<lb/>

<arrow.to.target n="marg33"></arrow.to.target></s>

<s>Sit proportio a ad c, & b ad d, & &longs;int c & d <lb/>

<arrow.to.target n="fig6"></arrow.to.target><lb/>æquales, dico quòd proportio a b ad c d e&longs;t <lb/>compo&longs;ita ex proportionibus a ad c, & b ad <lb/>d diui&longs;o compo&longs;ito per duplam. Quia enim </s>

<s>

<arrow.to.target n="marg34"></arrow.to.target><lb/>c & d &longs;unt æquales, erit b ad c, ut b ad d, qua<lb/>re ex diffinitione cùm 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, & b <lb/>ad c, erit etiam compo&longs;ita ex dictis ex propo&longs;itione a ad c, & b ad d, <lb/>

<arrow.to.target n="marg36"></arrow.to.target><lb/>&longs;tatuatur ergo e æqualis c d media inter a b & c. Et erit per &longs;ecun<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, & 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>

<s><margin.target id="marg34"></margin.target>E<emph type="italics"/>x &longs;exta<emph.end type="italics"/> A<emph type="italics"/>nim. <lb/>com. &longs;ententia.<emph.end type="italics"/></s>

<s><margin.target id="marg35"></margin.target>D<emph type="italics"/>ecimaquarta<emph.end type="italics"/></s>

<s><margin.target id="marg38"></margin.target>P<emph type="italics"/>er quintam<emph.end type="italics"/><lb/>A<emph type="italics"/>nim. com. &longs;en <lb/>tentiam.<emph.end type="italics"/></s>

<s>Propo&longs;itio duodecima.</s>

<s>Propo&longs;itis duabus proportionibus unam alteri iungere ab&longs;que <lb/>multiplicatione.<lb/>

<arrow.to.target n="marg39"></arrow.to.target></s>

<s>Sint propo&longs;itæ proportiones a ad c & <lb/>

<arrow.to.target n="fig7"></arrow.to.target><lb/>b ad d, & a&longs;&longs;umo e ad c, iuxta ea quæ Eu<lb/>clides demon&longs;trauit, ut b ad d, erit igitur </s>

<s>

<arrow.to.target n="marg40"></arrow.to.target><lb/>proportio a e ad c, compo&longs;ita ex proportionibus a ad c, & e ad c, <lb/>&longs;ed proportio e ad c e&longs;t, ut b ad d, igitur proportio a e ad c compo<lb/>&longs;ita e&longs;t ex proportionibus a ad c, & b ad d.</s>

<s><margin.target id="marg40"></margin.target>E<emph type="italics"/>x generali <lb/>com.<emph.end type="italics"/> A<emph type="italics"/>nim. &longs;en <lb/>tentia.<emph.end type="italics"/></s>

<s>Aliter ex b in c fiat fex a in d, g ex c in d h coniunctum ex f g, k.</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"></arrow.to.target><lb/>tur a e ad c, ut k ad h, &longs;ed k ad h cómponitur ex <lb/>proportionibus a ad c, & b ad d. Ex octaua ha <lb/>rum igitur proportio a c ad c compo&longs;ita e&longs;t ex <lb/>ei&longs;dem. For&longs;an quis dicat hanc eandem e&longs;&longs;e <lb/>octauæ &longs;ed <expan abbr="nõ">non</expan> e&longs;t, in illa enim proportio com<lb/>paratur ad productum, in hac ad unam ex <lb/>quantitatibus.</s>

<s>Ex hoc &longs;equitur quòd: Quælibet duæ quantitates quarum ag<lb/>

<arrow.to.target n="marg42"></arrow.to.target><lb/>gregatum e&longs;tidem ad eam quantitatem, componunt eandem pro<lb/>portionem.</s>

<s>Propo&longs;itio tertiadecima.</s>

<s>Proportio confu&longs;a aggregati primæ & tertiæ quatuor quantita<lb/>tum omiologarum ad <expan abbr="aggregat&utilde;">aggregatum</expan> &longs;ecundæ & quartæ, 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>

<s>Sint a ad b, ut c ad d, dico, quòd 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;itæ ex his proportionibus diui&longs;æ per du<lb/>plam æqualis. Erit enim aggregati ex a c ad aggregatum ex b d, ue<lb/>lut a ad b per 18 quinti Elementorum. Sed proportiones a ad b, <lb/>& c ad d componunt proportionem producti a in d, & c in b per <lb/>octauam harum, ad <expan abbr="product&utilde;">productum</expan> ex b in d, productum uerò ex a in d <lb/>e&longs;t æquale producto ex b in c per decimam&longs;extam &longs;exti Elemento<lb/>rum, & 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<lb/>tio compo&longs;ita ex a ad b, & c ad d, e&longs;t uelut confu&longs;a bis &longs;umpta. Igi<lb/>tur confu&longs;a e&longs;t uelut compo&longs;ita diui&longs;a per duplam per modum un<lb/>decimæ huius.</s>

<table>

<table.target id="table11"></table.target>

<s>Propo&longs;itio quartadecima.</s>

<s>Proportiones confu&longs;æ, & coniunctæ in tribus quantitatibus in<lb/>uicem commutantur.</s>

<s>Sint tres quantitates, dico, quod proportio c </s>

<s>

<arrow.to.target n="marg44"></arrow.to.target><lb/>ad a b confu&longs;a e&longs;t, conuer&longs;a coniunctæ a & b ad <lb/>

<arrow.to.target n="marg45"></arrow.to.target><lb/>c. Nam per dicta proportio a b ad c efficit con<lb/>iunctam ex a b ad c: &longs;ed c ad a b conuer&longs;a e&longs;t eius, quæ e&longs;t a b ad c, & <lb/>proportio c ad a b e&longs;t confu&longs;a eius, quæ e&longs;t c ad a & b. Igitur pro<lb/>portio confu&longs;a in tribus quantitatibus e&longs;t contraria coniunctæ in <lb/>ei&longs;dem.</s>

<s>Ex quauis ergo illarum data, data erit & reliqua.<lb/>

<arrow.to.target n="marg46"></arrow.to.target></s>

<s>Propo&longs;itio quintadecima.</s>

<s>Si fuerint quatuor quantitas-proportio confu&longs;a aggregati pri<lb/>mæ & tertiæ ad aggregatum &longs;ecundæ, & quartæ erit ut monadis <lb/>addito prouentu, qui fit diui&longs;a differentia differentiarum primæ & <lb/>&longs;ecundæ, atque quartæ & tertiæ per aggregatum tertiæ, & quartæ ad <lb/>ip&longs;am monadem.</s>

<s>Sint quatuor quantitates a b, c, d, e f, & <lb/>

<arrow.to.target n="marg47"></arrow.to.target><lb/>&longs;it a b maior cin a h, & e fmaior d in f g, & <lb/>differentia f g & a h &longs;it a k: dico proportio<lb/>nem a b, & d confu&longs;am ad c & e f, e&longs;&longs;e ut mo <lb/>nadis addito prouentu, uel detracto a k diui&longs;æ per aggregatum c. <lb/>& e f ad ip&longs;am monadem, & manife&longs;tum e&longs;t, quòd pote&longs;t continge<lb/>re pluribus modis: Primus ut a b &longs;it maior c & e f minor d, & tunc <lb/>differentiæ coniungentur, & prouentus, addetur monadi. Idem fa<lb/>ciendum erit &longs;i a b &longs;it maior c, & e f &longs;it minor d, &longs;ed exce&longs;&longs;us &longs;uperet <lb/>defectum. At &longs;i uel a b &longs;it minor c, & e f maior d, uel ita minor, ut c <lb/>exce&longs;&longs;us &longs;upra b a &longs;it maior defectu, detrahemus prouentum à mo<lb/>nade. Alia cautio e&longs;t quòd &longs;i fuerint utrinque exce&longs;&longs;us, aut defectus, <lb/>minuemus minorem de maiore: &longs;i autem unus &longs;it exce&longs;&longs;us alter de<lb/>fectus, iungemus illos, & po&longs;t diuidemus. uno ergo demon&longs;trato <lb/>ut pote primo intelligentur reliqui. Quia ergo b h e&longs;t æqualis c & <lb/>e g æqualis d & h k æqualis g f, erit ex communi animi &longs;ententia ag <lb/>gregatum ex d & k b æquale aggregato ex c & e f, igitur per dicta <lb/>proportio aggregati ad aggregatum e&longs;t unum. at uerò diui&longs;a k a <lb/>per c & e f fit quantum diui&longs;a eadem per b k, & d, &longs;ed diui&longs;a k a per b <lb/>k, & d iunctas, exit proportio a k ad aggregatum b k & d: igitur di<lb/>ui&longs;a a k per aggregatum e f & c, exibit eadem proportio, igitur a b <lb/>& d ad aggregatum c & e f e&longs;t coninncta ex monade & proportio<lb/>ne a k ad aggregatum c & e f, quod erat demon&longs;trandum.</s>

<s>Ex hoc patet quod proportionum confu&longs;io <lb/>

<arrow.to.target n="marg48"></arrow.to.target><lb/>fit iunctis denominatoribus numeratoris: mul<lb/>tiplicatio multiplicatis: additio multiplicatis <lb/>decu&longs;&longs;atim in numeratores ad productum ex <lb/>denominatoribus, ut in exemplis.</s>

<s>Propo&longs;itio &longs;extadecima.</s>

<s>Omnium quatuor quantitatum propo&longs;ita <lb/>prima, quæ non minorem habet proportionem <lb/>ad &longs;uam corre&longs;pondentem, quàm alia ad aliam <lb/>

<arrow.to.target n="fig8"></arrow.to.target><lb/>erit proportio confu&longs;a illarum, ut pro<lb/>ducti ex aggregato primæ & tertiæ in

<pb pagenum="15"/>tertiam, ad productum ex aggregato tertiæ & omiotatæ ad &longs;ecun<lb/>dam in ip&longs;am quartam.</s>

<s>Hæc magis reducit confu&longs;am proportionem ad notitiam, quàm, <lb/>præcedens, quia reducit ad proportionem <expan abbr="productã">productam</expan>, qu&ecedil; operatio <lb/>e&longs;t &longs;implici&longs;sima, &longs;iue per multiplicationem quantitatum fiat, duæ <lb/>&longs;unt tantum multiplicationes, &longs;iue per eundem terminum &longs;ufficit <lb/>alium addere. Summatur ergo a b, c, d & e, & non &longs;it maior propor<lb/>tio d ad e, quàm a b ad c, & &longs;tatuatur tunc prima a b, &longs;ecunda c, ter<lb/>tia d, quarta e, & po&longs;tquam non e&longs;t minor ratio a b ad c, quàm d ad <lb/>e, &longs;umatur a f ad c, ut d ad e. licet enim hoc facere. Dico quod pro<lb/>portio confufa a b & d ad c & e e&longs;t uelut producti ex aggregato a b <lb/>& d in d ad productum ex aggregato a f & d in e. Statuatur aggre<lb/>

<arrow.to.target n="marg49"></arrow.to.target><lb/>gatum a b & d linea a d prima quantitas, & aggregatum a f & d, <lb/>

<arrow.to.target n="fig9"></arrow.to.target><lb/>a d &longs;ecunda quantitas, & d tertia, <lb/>& c quarta, & ex a b in d fiat g, ex <lb/>a d in e fiat h, erit ergo per pri<lb/>mam propo&longs;itionem g ad h pro<lb/>

<arrow.to.target n="marg50"></arrow.to.target><lb/>ducta ex proportionibus a b d ad <lb/>a f d, & d ad c. Sed proportio a f d <lb/>ad aggregatum c e, e&longs;t uelut d ad <lb/>e. Proportio uerò a b d ad a f d, & <lb/>a f d ad e c producunt proportio<lb/>nem a b d ad c & e per &longs;ecundam propo&longs;itionem, harum igitur con<lb/>&longs;u&longs;a a b ad c, & d ad e, & e&longs;t proportio a b d ad c & e, producuntur <lb/>ex proportionibus a b d ad a f d, & d ad e. Ergo proportio g ad h <lb/>e&longs;t confu&longs;a ex a b ad e, & d ad e, quod erat demon&longs;trandum.</s>

<s>Propo&longs;itio decima&longs;eptima.</s>

<s>Omnes du&ecedil; proportiones conuer&longs;æ producunt æqualem pro<lb/>portionem.<lb/>

<arrow.to.target n="table12"></arrow.to.target></s>

<table>

<table.target id="table12"></table.target>

<s>Sint duæ proportiones a ad b & b ad a conuer&longs;a, <lb/>

<arrow.to.target n="marg51"></arrow.to.target><lb/>dico, quòd producunt proportionem æqualem. fiat <lb/>enim b ad c, ut b ad a, erit igitur a æqualis c & b c con <lb/>

<arrow.to.target n="marg52"></arrow.to.target><lb/>uer&longs;a eius quæ e&longs;t a ad b, &longs;ed per &longs;ecundam harum <lb/>proportiones a ad b, & b ad c producunt propor<lb/>tionem a ad c, igitur proportiones etiam a ad b & b ad a produ<lb/>cunt eandem.</s>

<s><margin.target id="marg52"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 6. A<emph type="italics"/>ni<lb/>mi <expan abbr="commun&etilde;">communem</expan> <lb/>&longs;ententiam.<emph.end type="italics"/></s>

<s>Propo&longs;itio decimaoctaua.</s>

<s>Si fuerint quotlibet quantitates in continua proportione multi<lb/>plici præter ultimam: proportio uerò penultimæ ad ultimam qua<lb/>lis re&longs;idui primæ ad &longs;ecundam, erit primæ ad aggregatum reliqua<lb/>rum uelut penultimæ ad ultimam.

<arrow.to.target n="marg53"></arrow.to.target></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 & b ad b, dico proportionem a ad b c d e <lb/>e&longs;&longs;e ut d ad e. 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 & d ad d & e, u<gap/></s>

<s>

<arrow.to.target n="marg54"></arrow.to.target><lb/>d ad e, &longs;ed e gnomo cum quadrato d efficit qua<lb/>

<arrow.to.target n="fig10"></arrow.to.target><lb/>dratum e, igitur ut c quadrati ad d & eiuncta, ita <lb/>d ad e. 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, & ita d ad e, igitur <lb/>

<arrow.to.target n="marg56"></arrow.to.target><lb/>gnomonum b c cum quadrato d ad aggrega<lb/>tum c d e quadratorum, ut d ad e, &longs;ed c gno<lb/>mo cum d quadrato perficit c quadratum, <lb/>& 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. Et ita <lb/>repetendo de quotuis quantitatibus in infi <lb/>nitum u&longs;que. Hæc proponitur ab Archimede in libro de quadrato <lb/>æquali parabolæ, & minus generaliter & pluribus demon&longs;tratur. <lb/>Ego tamen quia e&longs;t generalis, de&longs;cribam illam per corrolarium: ad<lb/>damque aliud quod ex hoc &longs;equitur.<lb/>

<arrow.to.target n="marg57"></arrow.to.target></s>

<s><margin.target id="marg54"></margin.target>13. P<emph type="italics"/>ropo&longs;. <lb/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<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>

<s>Si fuerint quotlibet <expan abbr="quãtitates">quantitates</expan> omnes analogæ præter ultimam, <lb/>&longs;it autem penultima ad ultimam qualis re&longs;idui primæ & &longs;ecundæ <lb/>ad &longs;ecundam, erit proportio primæ ad aggregatum omnium alia<lb/>rum ueluti penultimæ ad ultimam.</s>

<s>

<arrow.to.target n="marg58"></arrow.to.target></s>

<s>Hæc enim e&longs;t euidens, quia conuenit ei demon&longs;tratio propo&longs;ita. <lb/>

<arrow.to.target n="fig11"></arrow.to.target><lb/>exemplo autem in numeris à latere <lb/>po&longs;ito uides declarationem. nam <lb/>proportio 16 ad 32 e&longs;t uelut 27 re&longs;i <lb/>dui primæ & &longs;ecundæ ad ip&longs;am &longs;e<lb/>cundam &longs;cilicet ad 54.</s>

<s>

<arrow.to.target n="marg59"></arrow.to.target></s>

<s>Ex hoc patet etiam quòd a&longs;&longs;umptis omnibus, &longs;ub multiplicibus <lb/>analogiæ u&longs;que in infinitum prima quantitas e&longs;t multiplex aggre<lb/>gati omnium reliquarum numero 1 m: quo prima e&longs;t multiplex <lb/>&longs;ecundæ.</s>

<s>

<arrow.to.target n="marg60"></arrow.to.target></s>

<s>Si fuerint quotlibet quantitates in &longs;uper particulari proportio<lb/>ne analogæ, erit proportio primæ ad aggregatum omnium in infi<lb/>nitum iuxta proportionem multiplicem conuer&longs;am illius partis.</s>

<s>

<arrow.to.target n="marg61"></arrow.to.target></s>

<s>Velut collectæ in &longs;e&longs;quialtera duplæ in &longs;exquitertia triplæ in <lb/>&longs;exqui&longs;eptima &longs;eptuplæ. Vt capio 512 448 392 343, & ita deinceps <lb/>u&longs;que in infinitum aggregatum omnium earum erit 3584. Septu

<pb pagenum="17"/>plum 512, & aggregatum 18. 12. 8. 5 2/3, & ita deinceps in &longs;<gap/>xquialtera <lb/>erit 54 duplum 27 primæ in eo ordine.</s>

<s>SCHOLIVM.</s>

<s>Ex quo patet genus demon&longs;trandi nouun & pulchrum: nam <lb/>&longs;upponatur 54, aggregatum duplum 27, primæ igitur addito 27 <lb/>ad 54, cum &longs;it dimidium, & 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</s>

<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>

<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, & <lb/>&longs;equentis maioris in eadem proportione ad ip&longs;um maiorem.</s>

<s>

<arrow.to.target n="marg64"></arrow.to.target></s>

<s>Exemplum &longs;it proportio augens 25 & 35 duarum quintarum, uo <lb/>lo &longs;cire quantum &longs;it aggregatum omnium citra 25, maximam acci<lb/>pio 35, ulteriorem ad 25, cuius differentia a 25 e&longs;t 10, cum quo diui<lb/>do 625 quadratum, exit 62 1/2 aggregatum quantitatum. Et facile po</s>

<s>

<arrow.to.target n="marg65"></arrow.to.target><lb/>re&longs;t demon&longs;trari. Si quis dicat in qua proportione &longs;unt infinitæ <lb/>quantitates analogæ cum 12, quæiunctæ efficiunt 10, iunge 10 cum <lb/>12 fit 22, duc 22 in 12 fit 264, diuide 264 per 10, exit 26 2/3, & in ea pro<lb/>portione <expan abbr="er&utilde;t">erunt</expan> illæ quantitates, in qua &longs;unt 26 2/3 ad 12: duc per 5 fiunt <lb/>60, & 132 diuide per 12, exeunt 11 & 5, & ita eruntin proportione 11 <lb/>ad 5 experiaris, & inuenies, & demon&longs;tratur ex prioribus.</s>

<s><margin.target id="marg65"></margin.target>Q<emph type="italics"/>uæftio.<emph.end type="italics"/></s>

<s>Propo&longs;itio decimanona.</s>

<s>Si fu erint aliquot quantitates arithmeticæ omiologæ, quarum <lb/>exce&longs;&longs;us &longs;it æqualis minimè, omnibus autem deficientibus &longs;upple<lb/>menta ad &ecedil;qualitatem maximè adiungantur, erunt quadrata omni<lb/>um quantitatum æqualium adiecto rur&longs;us quadrato primæ cum <lb/>eo quod fit ex minima primi ordinis in <expan abbr="aggregat&utilde;">aggregatum</expan> omnium quan<lb/>titatum eiu&longs;dem tripla aggregato quadra<lb/>

<arrow.to.target n="fig12"></arrow.to.target><lb/>torum omnium quantitatum primi ordinis <lb/>

<arrow.to.target n="marg66"></arrow.to.target><lb/>pariter acceptis.</s>

<s>Sint aliquot quantitates a b c d e f g h in <lb/>continua proportione. Arithmetica di&longs;po&longs;it&ecedil; <lb/>ita ut minima <expan abbr="ear&utilde;">earum</expan> qu&ecedil; &longs;it h, &longs;it &ecedil;qualis diffe<lb/>renti&ecedil; quantitatum <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> ordinem di&longs;po <lb/><expan abbr="&longs;itar&utilde;">&longs;itarum</expan>, uelut differentia a & b, & b & c, & c & <lb/>d, et ita de alijs, addantur <expan abbr="a&utilde;t">aunt</expan> <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> &longs;in <lb/>gulis harum, quæ &longs;int i k l m n o p, ita ut <expan abbr="o&etilde;s">oens</expan> <lb/>fiant &ecedil;quales <expan abbr="c&utilde;">cum</expan> &longs;uis &longs;upplementis ip&longs;i line&ecedil; <lb/>à maiori. E&longs;tque <expan abbr="id&etilde;">idem</expan> ac &longs;i e&longs;&longs;ent aliquot quanti

<pb pagenum="18"/>tates, & <expan abbr="diuideren&ttilde;">diuiderentur</expan> &longs;ingul&ecedil; <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> numerum <expan abbr="illar&utilde;">illarum</expan>, &longs;i quatuor in <lb/>quatuor partes æquales, &longs;i quinque in quinque, &longs;i decem in decem, eara <lb/>tione ut ultima <expan abbr="diuidere&ttilde;">diuideretur</expan>, ubi e&longs;t finis primæ partis, penultima ubi <lb/>e&longs;t finis &longs;ecundæ partis, antepenultima ubi e&longs;t finis tertiæ, & &longs;ic de <lb/>alijs. Vocabo ergo primas <expan abbr="quãtitates">quantitates</expan> propo&longs;itas a b c d e f g h quan<lb/>titates primi ordinis, &longs;ed quantitates æquales quæ <expan abbr="con&longs;tãt">con&longs;tant</expan> ex quan <lb/>titatib. primi ordinis, & fupplementis, appellabo quantitates &longs;ecun <lb/>di ordinis: ex quo patet quòd prima <expan abbr="quãtitas">quantitas</expan> erit ex utro que ordine, <lb/>quia non e&longs;t diui&longs;a, reliquæ omnes differunt, quantitates uerò quas <lb/>adiunxi nominabo <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan>, & &longs;unt una minus <expan abbr="quã">quam</expan> quantitates <lb/>ordinum: ut &longs;i <expan abbr="quãtitates">quantitates</expan> ordinum &longs;int octo, erunt &longs;upplementa &longs;e<lb/>ptem, & &longs;i quantitates <expan abbr="ordin&utilde;">ordinum</expan>, e&longs;&longs;ent &longs;eptem e&longs;&longs;ent <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> &longs;ex, <lb/>quia inter &longs;upplementa <expan abbr="nõ">non</expan> <expan abbr="adnumera&ttilde;">adnumeratur</expan> quantitas indiui&longs;a. Erunt er <lb/>go &longs;upplementa i k l m n o p, quætanto erunt maiora quanto quan <lb/>titates primi ordinis &longs;unt minores, & contrà tanto maiora, quanto <lb/><expan abbr="quãtitates">quantitates</expan> primi ordinis &longs;unt maiores. quantitates <expan abbr="a&utilde;t">aunt</expan> &longs;ecundi ordi <lb/>nis <expan abbr="appellabun&ttilde;">appellabuntur</expan> a, b i, ck, dl, em, fn, go, & hp. Hæcuolui pluribus <lb/>agere, ut dilucidior e&longs;&longs;et propo&longs;itio. quæ licet <expan abbr="nõ">non</expan> &longs;it difficilis, e&longs;t <expan abbr="tam&etilde;">tamen</expan> <lb/>confu&longs;a ualde propter multitudinem <expan abbr="quantitat&utilde;">quantitatum</expan> & ordinum. 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, & utres exem <lb/>plo facilius innote&longs;cat, &longs;int <expan abbr="quãtitates">quantitates</expan> primi ordinis 8. 7. 6. 5. 4. 3. 2. 1. <lb/>quorum quadrata &longs;int 64. 49. 36. 25. 16. & 9.4 & 1. quæ iuncta <expan abbr="faci&utilde;t">faciunt</expan> <lb/>204, dico quod &longs;i &longs;umamus quadrata omnium <expan abbr="quãtitatum">quantitatum</expan> &longs;ecundi <lb/>ordinis, quæ &longs;unt octies 64, & eis addiderimus unum <expan abbr="quadrat&utilde;">quadratum</expan> ex <lb/>his, ut fiant nouies 64, & erunt 556, &longs;imul iuncta & eis addamus, &qring;d <lb/>fit ex 1 quantitate minima primi ordinis in 36 aggregatum quanti<lb/>tatum omnium primi ordinis, & 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<lb/>dinis unius demon&longs;tratio h&ecedil;c e&longs;t. Quia ex quarta &longs;ecundi Element. <lb/>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<lb/>quæ duæ &longs;unt producta ex partibus <expan abbr="inuic&etilde;">inuicem</expan> bis, & quia h fuit æqua<lb/>lis 1, & p &ecedil;qualis b, quia &longs;upplementa <expan abbr="fuer&utilde;t&ecedil;qualia">fuerunt&ecedil;qualia</expan> mutuò quanti <lb/>tatibus, & ita c æqualis o & k æqualis g & d, æqualis n & l, æqualis <lb/>f, e <expan abbr="a&utilde;t">aunt</expan> &ecedil;qualis m. <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> mutuò æqualia 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 & h p ultimam, <expan abbr="quar&utilde;">quarum</expan> qua

<pb pagenum="19"/>drata partium &longs;unt quadrata b & i, & h & p, &longs;ed b e&longs;t æqualis p, & h <lb/>æqualis i. Ergo quatuor quadrata b i & h p &longs;unt dupla quadratis b <lb/>& h, & ita <expan abbr="concludã">concludam</expan> de omnibus ubi duæ 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<lb/>rius, quia quadrata e & m &longs;unt dupla quadrato e &longs;oli eo, quod & m <lb/>

<arrow.to.target n="marg67"></arrow.to.target><lb/>&longs;unt æquales. Igitur per demon&longs;trata ab Euclide erit proportio o<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. atuerò addito quadrato a <lb/>quadratis b c d e f g h, & erunt quadrata omnium quantitatum, & <lb/>quadratis b i, c k, d l, e m, f n, g o, h p, duplo qu<gap/>drati a &longs;cilicet &longs;emel, <lb/>quia a e&longs;t ex &longs;ecundo ordine quantitatum, & &longs;emel, quia hoc fuit a&longs;<lb/>&longs;umptum in Problemate. Sequitur ut quadrata omnia <expan abbr="quãtitatum">quantitatum</expan> <lb/>&longs;ecundi ordinis, prout &longs;unt diui&longs;a in partes addito quadrato a, &longs;int <lb/>dupla quadratis primarum quantítatum, &longs;imul pariter acceptis. 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 æquale quadratis, <expan abbr="quantitat&utilde;">quantitatum</expan> eiu&longs;<lb/>dem primi ordinis pariter acceptis. Con&longs;tatigitur, quod duplum <gap/><lb/>in b e&longs;t æquale duplo h in ip&longs;um b, quia h & i &longs;unt æquales, & du<lb/>plum k in ip&longs;um c, e&longs;t æquale quadruplo h in idem c, quia k e&longs;t du<lb/>pla h, & &longs;imiliter duplum l in ip&longs;um d e&longs;t æquale &longs;excuplo, h in d, <lb/>quia l e&longs;t tripla h, & ita procedendo erunt illa dupla producta æ<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 quòd &longs;upplementum l continet h ter, duplicabo tria & fi<lb/>ent &longs;ex, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="dupl&utilde;">duplum</expan> lin d æquale e&longs;t &longs;excuplo h in ip&longs;um d. Quo con<lb/>&longs;tituto, cum &longs;uppo&longs;itum &longs;it producta illa duplicata cum producto h <lb/>in aggregatum primarum <expan abbr="quãtitatum">quantitatum</expan> e&longs;&longs;e æqualia quadratis ip&longs;a<lb/>rum quantitatum, igitur addemus <expan abbr="product&utilde;">productum</expan> ex h in &longs;ingulas quan<lb/>titates productis illis prioribus, & 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<lb/>cies, & in h quindecies æquale duplo producti uniu&longs;cuiu&longs;que quan<lb/>titatis in &longs;uum &longs;upplementum cum producto h in <expan abbr="aggregat&utilde;">aggregatum</expan> ip&longs;a<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, & pariter de quadrato b, quod e&longs;t &ecedil;<lb/>quale ei quod fit ex h in eam quæ toties continet b, quotiens b con <lb/>tinet h, & ita quadratum c æquale e&longs;t ei, quod continetur &longs;ub h, & <lb/>habente proportionem ad b eandem, quam b ad h, & &longs;imiliter de <lb/>quadrato c & omnibus reliquis, u&longs;que ad h ip&longs;um. Gratia ergo exem

<pb pagenum="20"/>pli quadratum a, erit æquale producto ex h in omnes quatitates &longs;e<lb/>cundas, quia quotus e&longs;t numerus quantitatum, totus e&longs;t numerus <lb/>&longs;ecundum quem a continet h, & &longs;imiliter quotus e&longs;t numerus quan <lb/>títatum incipiendo à b, & quotus e&longs;t numerus quantitatum incipi<lb/>endo à c, toties b uel c <expan abbr="contin&etilde;t">continent</expan> h, & ita de alijs, quadrata ergo om<lb/>nium quantitatum &longs;imul iuncta &longs;unt æqualia productis ex h in &longs;in<lb/>gulas illarum toties &longs;umptis, quoties illæ <expan abbr="cõtinent">continent</expan> h, &longs;eu quotus e&longs;t <lb/>numerus illius quantitatis, incipiendo ab h, & <expan abbr="numerãdo">numerando</expan> uer&longs;us a. <lb/>Rur&longs;us dico, quod productum multiplicis cuiuslibet <expan abbr="quãtitatis">quantitatis</expan> in <lb/>minimam, &longs;eu quadratum eiu&longs;dem quantitatis &ecedil;quale e&longs;t producto <lb/>eiu&longs;dem quantitatis, & dupli omnium &longs;equentium primi ordinis in <lb/>ip&longs;am minimam quantitatem, uelut quadratum a e&longs;t æquale produ <lb/>cto ex h in a, & in duplum b c d e f g h, hoc <expan abbr="aut&etilde;">autem</expan> facile e&longs;t probare in <lb/>his quantitatibus, quia &longs;i quadratum a e&longs;t æquale producto h in o<lb/>mnes quantitates &longs;ecundi ordinis, & omnes quantitates &longs;ecundi or <lb/>dinis &longs;imul &longs;umptæ &longs;unt &ecedil;quales ip&longs;i a, & duplo <expan abbr="reliquar&utilde;">reliquarum</expan> primi or <lb/>dinis, quia tales quantitates &longs;unt æquales &longs;uis &longs;upplementis uici&longs;<lb/>&longs;im, ut h cum i, k cum g, f cum l, e <expan abbr="c&utilde;">cum</expan> m, ergo tam &longs;upplementa, quàm <lb/>quantitates primi ordinis &longs;unt dimidium quantitatum &longs;ecundi or<lb/>dinis, ergo duplum quantitatum primi ordinis e&longs;t dimidium quan <lb/>titatum &longs;ecundi ordinis, uerùm de b dico idem accidere, quia qua<lb/>dratum b e&longs;t &ecedil;quale producto ex h in b, & in duplum reliquarum à <lb/>b, &longs;cilicet duplum c d e f g h, & hoc e&longs;t o&longs;tendere, quod i&longs;t&ecedil; quantita <lb/>tes &longs;unt dimidium totidem quantitatum æqualium b, nam c e&longs;t mi<lb/>nor b in h, & &longs;upplementum p quod e&longs;t æquale ip&longs;i b, &longs;i tota h p fiat <lb/>æqualis ip&longs;i b, ut pote h q erit ip&longs;a q dempta h æqualis ip&longs;i c, ergo <lb/>quantitates primi ordinis &longs;emper &longs;unt æquales &longs;upplementis non <lb/>ueris, &longs;ed prioris quantitatis a&longs;&longs;umptæ, &longs;eu in comparatione ad il<lb/>lam, quadratum igitur b e&longs;t æquale producto ex h in b, & in duplum <lb/>c d e f g h, & &longs;imiliter per eadem, quadratum c e&longs;t æquale producto <lb/>ex h in c, & in duplum d e f g h, & &longs;ic de alijs. Habemus ergo, quod <lb/>quadrata a b c d e f g h &longs;imul iuncta &longs;unt æqualia producto ex h in <lb/>a, & in duplum reliquarum, & ex h in b, & in duplum reliquarum <lb/>&longs;equentium, & producto ex h in c &longs;emel, & in duplum &longs;equentium <lb/>u&longs;que ad h, & ita de reliquis. hoc enim e&longs;t, quod nuper demon&longs;traui<lb/>mus. Antea quo que <expan abbr="demõ&longs;tratum">demon&longs;tratum</expan> e&longs;t, quod duplum b in i, c in k, d in <lb/>l, e in m, f in n, g in o, h in p, <expan abbr="c&utilde;">cum</expan> producto h in <expan abbr="aggregat&utilde;">aggregatum</expan> a b c d e f g h <lb/>erat &ecedil;quale productis ex h in a &longs;emel, & in b ter, & in c quinquies, in <lb/>d &longs;epties, in e nouies, in fundecies, in g tredecies, in &longs;eip&longs;am h quin<lb/>decies, detractis ergo p <expan abbr="ordin&etilde;">ordinem</expan>, &qring;d fit ex h in a ab utro que aggregato, <lb/>& ex h in b c d e f g h bis <expan abbr="relinque&ttilde;">relinquetur</expan> ex una parte, quae fit ex h in b &longs;emel

<pb pagenum="21"/>cum &longs;uis duplicatis &longs;equentibus, & in c, & in d, & in reliquis pa<lb/>riter conduplicatis &longs;uis &longs;equentibus ex altera, quod fit ex h in b &longs;e<lb/>mel, in c ter, in d quinquies, in e &longs;epties, in f nouies, in g undecies, <lb/>in h tredecies, detractis ergo rur&longs;us quod fit ex h in b &longs;emel, & ex <lb/>h in c d e f g h bis relinquetur, quod fit ex h in c, & duplo &longs;equen<lb/>tium, & d & duplo &longs;equentium, & e & aliarum pariter: & ex alia <lb/>parte, quod fit ex h in c &longs;emel, & in d ter, & in e quinquies, in f &longs;e<lb/>pties, in g nouies, in h undecies. Ab his rur&longs;us detractis, quòd fit <lb/>ex h in c &longs;emel, & in &longs;equentes bis, relinquetur h in d &longs;emel cum &longs;uis <lb/>&longs;equentibus bis, & in e &longs;emel cum &longs;uis &longs;equentibus & in f, & in g & <lb/>in h pariter, & ex alia parte, quod fit ex h in d &longs;emel, in e ter, f quin<lb/>quies, g &longs;epties, h nouies, ab his rur&longs;us detraho, quod fit ex h in d <lb/>&longs;emel, & in &longs;equentes bis, relinquetur ex una parte, quod fit ex h <lb/>in e f g h cum duplo &longs;equentium ex alia, quod fit ex h in e &longs;e<lb/>mel, f ter, g quinquies, h &longs;epties, & &longs;imiliter ab his detractis, quod <lb/>fit ex h in e &longs;emel, & bis in &longs;equentes, relinquetur ex una par<lb/>te; quod fit ex h in f &longs;emel, & in g h bis, & in g &longs;emel, & in h bis, <lb/>& in h &longs;emel, & ex alia, quod fit ex h in f &longs;emel, in g ter, in h quin<lb/>quies. Iterum detractis, quod fit ex h in f &longs;emel, & in g h bis com<lb/>muniter relin quetur, quod fit ex h in g &longs;emel, & in h bis, & in h &longs;e<lb/>mel, & ex alia parte quod fit ex h in g &longs;emel, & ex h in h ter. Sed <lb/>i&longs;ta, quæ relicta &longs;unt iam, &longs;unt manife&longs;tè æqualia, ergo etiam pri<lb/>ma aggregata ab initio fuere æqualia, ergo & æqualia illis qua<lb/>drata a b c d e f g h his, quæ fiunt, ex h in ea&longs;dem quantita<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<lb/>pla quadrata h p, g o, f n, e m, d l, c k, b i, cum duplo quadra<lb/>ti a, ergo quadrata omnium quantitatum &longs;ecundi ordinis cum <lb/>quadrato a rur&longs;us repetito, & producto h in aggregatum quanti<lb/>tatum primi ordinis &longs;unt tripla quadratis quantitatum primi ordi<lb/>nis pariter acceptis, quod fuit propo&longs;itum, & fuit Archimedis in li <lb/>bro de lineis &longs;piralibus, & ego adieci hic propter modum demon <lb/>&longs;trandi, qui e&longs;t eleganti&longs;simus, & procedit ex principijs arithmeti<lb/>cis, & diuer&longs;is à communibus, & ideo non reuoluitur, ut &longs;olentre<lb/>liquæ quæ&longs;tiones.</s>

<s>Propo&longs;itio uige&longs;ima.</s>

<s>Cùm fuerint quatuor quantitates, fueritque &longs;ecunda æqualis ter<lb/>tiæ, aut primæ æqualis quartæ, erit proportio primæ ad quartam, <lb/>aut tertiæ ad &longs;ecundam producta ex proportionibus primæ ad &longs;e<lb/>cundam, & tertiæ ad quartam.<lb/>

<arrow.to.target n="marg69"></arrow.to.target></s>

<s>Cùm enim quantitates hæ non fuerint &ecedil;quales, <expan abbr="cõ&longs;tat">con&longs;tat</expan> per &longs;ecun

<pb pagenum="22"/>dam harum, quod proportio primæ ad <expan abbr="quartã">quartam</expan> producitur ex pro<lb/>portione primæ ad &longs;ecundam, &longs;ecund&ecedil; ad tertiam, & terti&ecedil; ad quar <lb/>tam: ergo non ex &longs;olis proportionibus primæ ad &longs;ecundam, & ter<lb/>tiæ ad quartam, & &longs;imiliter ex prima harum proportio prim&ecedil; ad &longs;e<lb/>cundam, & tertiæ ad quartam producunt proportionem producti <lb/>primæ in &longs;ecundam ad productum tertiæ in quartam. Et in multi<lb/>plicatione proportio, quæ &longs;olet e&longs;&longs;e inter producta illa, & e&longs;t qua&longs;i <lb/>duplicata e&longs;t inter ip&longs;as quantitates. Sint igitur quantitates a b c d, <lb/>& &longs;it b æqualis c, ponantur ergo recto ordine a b c d, eritque propor <lb/>

<arrow.to.target n="fig13"></arrow.to.target><lb/>tio a ad d producta ex proportioni<lb/>bus a ad b, b ad c, & c ad d, producan<lb/>tur igitur ex proportionibus a ad b, c <lb/>ad d. proportio c ad f, erit igitur pro<lb/>portio e ad f, &longs;i multiplicetur per pro<lb/>portionem b ad c eadem quæ prius, & </s>

<s>

<arrow.to.target n="marg70"></arrow.to.target><lb/>producta iam e&longs;t eadem ei, quæ 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. Quod uerò diximus de pri<lb/>ma & quarta &longs;i &longs;int æquales, manife&longs;tum e&longs;t, quòd res redit ad idem <lb/>&longs;olum tran&longs;mutato ordine, ut tertia, & quarta præmittantur prim&ecedil;, <lb/>& &longs;ecundæ. Hæcigitur propo&longs;itio nihil aliud innuit, quàm quod <lb/>in hoc ca&longs;u productio, quæ&longs;olet fieri ex tribus proportionibus fiat <lb/>ex duabus tantum.</s>

<s>Propo&longs;itio uige&longs;imaprima.</s>

<s>Cùm decu&longs;&longs;atim ducta fuerit prima in quartam, & &longs;ecunda in ter <lb/>tiam; productumque primæ in quartam diui&longs;um fuerit per produ<lb/>ctum &longs;ecundæ in tertiam erit proportio primæ ad &longs;ecundam diui<lb/>&longs;a per proportionem tertiæ ad quartam. Et &longs;imiliter interpo&longs;ita <lb/>omiologa.<lb/>

<arrow.to.target n="marg71"></arrow.to.target></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"></arrow.to.target><lb/>ergo per <expan abbr="&longs;ec&utilde;dam">&longs;ecundam</expan> harum proportio ad b pro<lb/>ducta ex proportione a ad e, & e ad b, quare ex a ad e, & c ad d, ergo <lb/>diui&longs;a proportione a ad b per proportionem c ad d exit proportio <lb/>a ad e, & hic e&longs;t &longs;ecundus modus. Primus autem modus ducatur a <lb/>in d & fiat f, & b in c & fiat g, dico proportione f ad g e&longs;&longs;e prouen<lb/>tum proportionis a ad b, diuide per proportionem c ad d, ducatur <lb/>igitur c in f & fiat h, & d in g & fiat k, quia igitur h producitur ex c <lb/>in f, & f producitur ex a in d, ergo h producetur ex producto c in d, <lb/>in a, & &longs;imiliter quia k producitur ex d in g, & 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/>erit a ad b ut h ad k, igitur ex prima harum cum ex c in f producatur <lb/>h, & ex d in g k, & dicatur produci proportio h ad k ex proportio<lb/>ne c ad d, & f ad g, & proportio h ad k &longs;it eadem, quæ a ad b, ergo <lb/>proportio a ad b producitur ex c ad d, & f ad g, ergo diui&longs;a propor<lb/>tione a ad b prodibit proportio f ad g, quod fuit propo&longs;itum.</s>

<s>Propo&longs;itio uige&longs;ima&longs;ecunda.</s>

<s>Cùm fuerit proportio primæ ad &longs;ecundam maior, quàm tertiæ <lb/>ad quartam, erit confu&longs;a ex his maior quàm tertiæ ad quartam, mi<lb/>nor autem quàm primæ ad &longs;ecundam.</s>

<s>Sit proportio a ad b maior quàm 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, quàm c ad d, et minor quàm <lb/>a ad b, ut enim c ad d ita fiat e ad b, erit que per tertiamdecimam ha<lb/>

<arrow.to.target n="marg74"></arrow.to.target><lb/>rum e c ad b d confu&longs;a minor quàm a c ad b d, nam e e&longs;t minor a, <lb/>quia proportionem habent minorem ad b quam a eo quòd e ha<lb/>bet proportionem ad b, quam c ad d, quæ <expan abbr="aut&etilde;">autem</expan> c ad d minor, quám <lb/>a ad b, ut &longs;uppo&longs;itum e&longs;t, igitur e c ad b d minor, quàm 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<lb/>nor, quàm confu&longs;a a b ad c d, quod e&longs;t &longs;ecundum per idem proba<lb/>bitur, & primum po&longs;ita f ad d, ut a ad b, eritque a maior c, igitur ma<lb/>ior proportio a f ad b d, quàm 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, quàm a ad b.</s>

<s>Propo&longs;itio uige&longs;imatertia.</s>

<s>Omnis motus naturalis ad locum &longs;uum e&longs;t: ideo per rectam li<lb/>neam fit.<lb/>

<arrow.to.target n="marg75"></arrow.to.target></s>

<s>Motus naturalis e&longs;t, ut con&longs;eruetur corpus, & conueniat locus <lb/>corpori, igitur fit ad &longs;uum locum. Locus autem dicitur in compara <lb/>tione ad uniuer&longs;um. ideo omnis motus naturalis e&longs;t à centro mun<lb/>di &longs;ur&longs;um, uel ad centrum deor&longs;um. 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<lb/>peteret) eum quærit, cùm &longs;it &longs;apienti&longs;simæ uitæ mini&longs;tra: at linea re</s>

<s>

<arrow.to.target n="marg76"></arrow.to.target><lb/>cta breui&longs;sima e&longs;t Euclide te&longs;te à puncto ad punctum, igitur omnis <lb/>motus naturalis e&longs;t &longs;ur&longs;um aut deor&longs;um per rectam lineam.</s>

<s><margin.target id="marg76"></margin.target>D<emph type="italics"/>i&longs;t. tertia <lb/>primi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s>Propo&longs;itio uige&longs;imaquarta.</s>

<s>Omnis motus circularis uoluntarius e&longs;t.</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,

<arrow.to.target n="fig14"></arrow.to.target><lb/>non naturalis. nam &longs;i uiolentus e&longs;&longs;et, non <lb/>e&longs;&longs;et perpetuus. Omnia ergo a&longs;tra feruntur <lb/>circa centrum mundi. Sit modo rota e f g, di <lb/>co enon moueri motu circulari nam linea <lb/>e <expan abbr="clõgior">clongior</expan> e&longs;t g c, ergo recta mouetur ad cen <lb/>trum non circa centrum. Indicio etiamid <lb/>e&longs;t: quòd &longs;i in e ponatur fru&longs;tum aliquod <lb/>in&longs;igne plumbi in motu ad g per f de&longs;cen<lb/>det raptim: at dum ex g in e magna cum dif<lb/>ficultate, igitur motus hic non e&longs;t naturalis, <lb/>nec circularis. nihil etiam hoc modo &longs;ponte mouetur. Sed cum non <lb/>moueatur per rectam naturaliter, nec æquidi&longs;tans à centro per cir<lb/>culum relinquitur, ut moueatur motu uiolento, aut mi&longs;to, &longs;ed non <lb/>ex uoluntario, cum nullo modo moueatur æquidi&longs;tans à centro, <lb/>&longs;ed &longs;emper ab e lineæ ad centrum fiant breuiores, liquet e&longs;&longs;e mo<lb/>tum uiolentum: aut mi&longs;tum ex naturali, & uiolento.</s>

<s>Propo&longs;itio uige&longs;imaquinta.</s>

<s>Tres &longs;unt motus omnino &longs;implices naturalis, uoluntarius & <lb/>uiolentus.<lb/>

<arrow.to.target n="marg77"></arrow.to.target></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 æquidi&longs;tando à centro, uel <lb/>neutro modo, igitur tres motus. Rur&longs;us uel à principio interiore <lb/>non intelligente, & e&longs;t naturalis, uel intelligente & e&longs;t uoluntarius: <lb/>uel exteriore & e&longs;t uiolentus. Hæc autem diui&longs;io e&longs;t &longs;olum propria <lb/>non prima. Nam e&longs;t uiolentus in recta ad centrum: ideo omnis, qui <lb/>non e&longs;t in recta ad centrum, nec æquidi&longs;tat, uiolentus e&longs;t: non ta<lb/>men omnis uiolentus e&longs;t extra rectam. Attractio autem, quæ fit ob <lb/>raritatem corporum, &longs;eu, ut dicunt, à uacuo, uiolenta e&longs;t non natu<lb/>ralis ni&longs;i ratione finis, non agentis. Sunt enim quatuor genera mo</s>

<s>

<arrow.to.target n="marg78"></arrow.to.target><lb/>tus uiolenti ab Ari&longs;totele po&longs;ita, uectio, tractio, pul&longs;io, & uolutio: <lb/>quanquam his non opus &longs;it in demon&longs;tratiua &longs;cientia. <expan abbr="cõ&longs;tat">con&longs;tat</expan> enim <lb/>uolutionem ex tractione, & pul&longs;ione apud illum con&longs;i&longs;tere.</s>

<s><margin.target id="marg78"></margin.target>7. P<emph type="italics"/>hy&longs;. <lb/>cap.<emph.end type="italics"/> 2.</s>

<s>Propo&longs;itio uige&longs;ima.</s>

<s>Motus ergo compo&longs;iti quatuor nece&longs;&longs;ariò &longs;unt &longs;pecies.</s>

<s>Si tantum &longs;unt tres &longs;pecies &longs;implicium, con&longs;tat ratione arithme<lb/>tica quatuor e&longs;&longs;e compo&longs;itorum. Di&longs;quiramus ergo an &longs;int natura<lb/>liter tot &longs;pecies, for&longs;an enim repugnabit aliquis alicui. Porrò uidea<lb/>mus primò, quot &longs;int uiolentorum &longs;pecies: Prima erit cum non &longs;e<lb/>cundum rectam lineam fuerit: nec à centro æquidi&longs;tantem. Secun<lb/>da cum fuerit &longs;ecundum rectam, &longs;ed non ad centrum. Tertia cum <lb/>fuerit in recta ad centrum, &longs;ed contrario modo, uelut terræ &longs;ur&longs;um.

<pb pagenum="25"/>Quarta cùm in recta ad centrum, &longs;ecundum naturam, &longs;ed <expan abbr="nõ">non</expan> à prin <lb/>cipio naturali. Velut cum quis proij cit lapidem rectà in terram è <lb/>turri uiolentius, quàm ille &longs;ua grauitate de&longs;cen&longs;urus e&longs;&longs;et. Hic igi<lb/>tur motus e&longs;t compo&longs;itus ex naturali, & uiolento. Animalium au<lb/>tem motus uoluntarius e&longs;t, cum &longs;it à principio interiore cogno&longs;cen <lb/>te: & &longs;it quatenus à principio in linea circulari æqualiter di&longs;tante à <lb/>centro: &longs;ed quia ob&longs;tat grauitas, ideò mi&longs;tus e&longs;t ex naturali, & uo<lb/>luntario. Sed circularis, & uiolentus &longs;oli e&longs;&longs;e non po&longs;&longs;unt: nam uio <lb/>lentus e&longs;t nece&longs;&longs;ariò in corpore graui aut leui: &longs;ed omne corpus gra <lb/>ue aut leue, cùm mouetur, naturaliter mouetur &longs;altem in fine: & per <lb/>totum motum, motu ócculto, qui maximè in hoc libro dignus e&longs;t <lb/>con&longs;ideratione, igitur motus uoluntarius, & uiolentus non po&longs;<lb/>&longs;unt e&longs;&longs;e &longs;imul &longs;oli. Eruntergo &longs;ecundum naturam tantùm tres &longs;pe<lb/>cies. Velut cùm quis &longs;candit, aut&longs;alit: E&longs;t enim motus naturalis &longs;al<lb/>tem in fine, & uoluntarius, & uiolentus. Si quis autem uelit uiolen<lb/>tum cum uoluntario copulare dicemus con&longs;tare eam compo&longs;itio<lb/>nem in initio &longs;aliendi. Motum autem occultum uocamus grauita<lb/>tem aut leuitatem.</s>

<s>Propo&longs;itio uige&longs;ima&longs;eptima.</s>

<s>Motus uoluntarius e&longs;t in loco: naturalis ad locum: uiolentus <lb/>exloco.</s>

<s>Hæc e&longs;t tertia differentia primarum &longs;pecierum motuum uolun<lb/>tarius fit manente corpore toto in eodem loco, ideo proprius e&longs;t <lb/>c&oelig;lo, corpora autem animalium in eodem loco feruntur: quia in <lb/>eodem orbe nata redire ad proprium locum. Et ideò, ut dixi, e&longs;t mo <lb/>tus mi&longs;tus ex naturali, & uoluntario, qui &longs;i per &longs;e fieret, non fatiga<lb/>ret mobile, cùm ex utroque principio ab interiore ui procedat. Sed <lb/>quia fit per mu&longs;culos, qui trahuntur: hic autem motus e&longs;t uiolen<lb/>tus, ideò per con&longs;equentiam fatigat. Qui uerò naturalis, e&longs;t ut re<lb/>deat corpus ad &longs;uum locum, igitur naturalis e&longs;t ad locum. Sed <lb/>uiolenti finis e&longs;t, ut protrudatur ex loco in quo e&longs;t, non habens cer<lb/>tum finem. licet enim qui trahit, ad &longs;uum locum trabat, non tamen <lb/>ad locum mobilis.</s>

<s>Propo&longs;itio uige&longs;imaoctaua.</s>

<s>Motus quilibet naturalis aut uiolentus in aliquo medio fit.<lb/>

<arrow.to.target n="marg79"></arrow.to.target></s>

<s>Cùm uacuum non detur, & omnis motus naturalis &longs;it ad locum, <lb/>et uiolentus ex loco per præcedentem, igitur cùm non &longs;it in medio, <lb/>uacuum erit in aliquo corpore, uelut aere, aqua, igne, ligno.</s>

<s>Propo&longs;itio uige&longs;imanona.</s>

<s>Omnis motus uoluntarius æqualis e&longs;t &longs;emper: &longs;impliciter etiam <lb/>quilibet alius motus.</s>

<s>

<arrow.to.target n="marg80"></arrow.to.target></s>

<s>Motus uoluntarius non habet, quòd fatiget, & &longs;umma perfectio <lb/>e&longs;t æqualitas, & natura quæ mouet non debilitatur, igitur perpe<lb/>tuo per&longs;euerat æqualis. neque enim e&longs;t, ut dixi, per medium corpus. <lb/>Naturalis quoque, & uiolentus cum ratione proportionis mouentis <lb/>&longs;upra mobile per&longs;e non uarientur, & ab &ecedil;quali proportione &ecedil;qua<lb/>lis uelo citas proueniat, igitur natura tales motus &longs;unt &ecedil;quales, nam <lb/>in utroque mouens, mouet &longs;ecundum ultimam &longs;uam uim.</s>

<s>Propo&longs;itio trige&longs;ima.</s>

<s>In omni corpore mobili in medio, partes medij re&longs;i&longs;tunt obuiæ, <lb/>aliæ impellunt.</s>

<s>

<arrow.to.target n="marg81"></arrow.to.target></s>

<s>Sit mobile a cui partes &longs;ubiaceant directæ b, & &longs;it graue. Et pa<lb/>tet ne diuidatur b re&longs;i&longs;tere, cum autem &longs;uperauerit, partes b de&longs;cen<lb/>dunt ante a, & trahunt partes c & d adh&ecedil;rentes &longs;ecum, atque ita e c d f <lb/>

<arrow.to.target n="fig15"></arrow.to.target><lb/>adiuuant ad de&longs;cen&longs;um partes etiam laterales <lb/>g & h cum a tran&longs;it in b, ne detur uacuum, tran<lb/>&longs;eunt in k uelo ci motu, ergo propellunt a maio <lb/>reimpetu inferius.</s>

<s>

<arrow.to.target n="marg82"></arrow.to.target></s>

<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>

<s>

<arrow.to.target n="marg83"></arrow.to.target></s>

<s>Et ideò etiam bellicæ machinæ cuiu&longs;cunque generis certam exi<lb/>gunt di&longs;tantiam, ut uiolentius feriant.</s>

<s>Propo&longs;itio trige&longs;imaprima.</s>

<s>Omnis motus naturalis in æquali medio ualidior e&longs;t in fine, <lb/>quàm in principio: uiolentus contrà.</s>

<s>

<arrow.to.target n="marg84"></arrow.to.target></s>

<s>Cùm enim ex præcedenti augeantur &longs;emper ob medium, & cau<lb/>fa, quæ mouet, &longs;it perpetua, & à principio æterno, quod per dictæ <lb/>æqualiter mouet, igitur motus ille fiet uelo cior in fine quàm in alia <lb/>parte temporis. In uiolento autem, cùm perueniat ad finem de&longs;init </s>

<s>

<arrow.to.target n="marg85"></arrow.to.target><lb/>uis illa nece&longs;&longs;ariò, quæ mouet, & &longs;uperatur à ui naturali, quæ mo<lb/>uet in contrarium, igitur antequam ce&longs;&longs;et motus fiet tardi&longs;simus <lb/>in fine.</s>

<s><margin.target id="marg85"></margin.target><gap/> 29. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>

<s>Ex quo patet, quòd motus quadrifariam mi&longs;ti dicuntur, aut &longs;pe<lb/>

<arrow.to.target n="marg86"></arrow.to.target><lb/>cie, ut cùm quis iacit lapidem è turri: uel ex occulto naturali, & uio<lb/>lento manife&longs;to: uelut cùm quis iacit lapidem, & de&longs;cendit po&longs;tmo <lb/>

<arrow.to.target n="fig16"></arrow.to.target><lb/>dum ex b in c motu utroque manife&longs;to, &longs;ed ex a <lb/>in b motu uiolento manife&longs;to, & naturali oc<lb/>culto: uel ratione medij, & 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<lb/>dium ip&longs;um, uel &longs;i uiolentus &longs;it ex proportione uirtutis mouentis,

<pb pagenum="27"/>& medij ad mobile, ac medium, quod re&longs;i&longs;tit. Quarto ex motibus <lb/>imperfectis natura &longs;ua, & non e&longs;t uera mi&longs;tio, & hoc apparet in mo<lb/>tibus uoluntarijs animalium, qui non &longs;unt neque æquales, neque perfe <lb/>ctè circa medium: &longs;ed &longs;unt potius &longs;imiles uoluntarijs. Etideo de<lb/>mon&longs;trationes illæ Ari&longs;totelis quoad u&longs;um nihil iuuant nos.</s>

<s>Propo&longs;itio trige&longs;ima&longs;ecunda.</s>

<s>Omne mobile naturaliter motum, &longs;eu uiolenter uelo cius moue<lb/>tur in medio rariore, quàm den&longs;iore. Maior quoque e&longs;t proportio fi<lb/>nis motus in corpore rariore ad finem motus in corpore den&longs;iore, <lb/>quàm principij. In uiolento autem celeriùs perueniet ad finem mo <lb/>tus in corpore den&longs;iore.</s>

<s>A mobile moueatur in b medio rariore, & in c den&longs;io<lb/>

<arrow.to.target n="marg87"></arrow.to.target><lb/>re, igitur b minus re&longs;i&longs;tit, quàm c & magis adiuuat, quia <lb/>uelociùs mouetur: igitur duplici de cau&longs;a a mouebitur <lb/>uelociùs in b quàm in c: & quia per corrolarium trige&longs;i<lb/>mæ, & præcedentis proportio finis (ubi æqualiter moueantur) ad <lb/>&longs;ua principia maior erit in d, quàm in e: ergo per <expan abbr="demõ&longs;trata">demon&longs;trata</expan> à Cam <lb/>pano po&longs;ita d prima, b &longs;ecunda, e tertia, c quarta, maior erit propor<lb/>tio d ad e, quàm b ad c quod fuit propo&longs;itum in naturali.</s>

<s>Propo&longs;itio trige&longs;imatertia.</s>

<s>Omnia duo mobilia æqualis undique magnitudinis, quæ æquali <lb/>in tempore æqualia &longs;patia pertran&longs;eunt in diuer&longs;is &longs;ub&longs;tantia me<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>

<s>Sint duo mobilia a & b magnitudine, & forma omnino paria, <lb/>& &longs;int media c & d, exempli gratia: & pertran&longs;eant æquale &longs;patium <lb/>in utroque in eodem tempore, e dico proportionem ponderis b ad <lb/>pondus a e&longs;&longs;e duplicatam ei quæ e&longs;t raritatis c ad raritatem d. Quia <lb/>enim feruntur æqualiter, nam in æquali tem<lb/>

<arrow.to.target n="fig17"></arrow.to.target><lb/>pore, &longs;eu eodem æqualia &longs;patia pertran&longs;e<lb/>unt, erit proportio potentiæ a cum &longs;uo auxi<lb/>lio ad id, quod re&longs;i&longs;tit ex c ut b cum &longs;uo au<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<lb/>tatis duplicat actionem, tum minus re&longs;i&longs;ten<lb/>do, tum adiuuando motum a, igitur proportio differentiæ motus <lb/>e&longs;t duplicata proportioni raritatis: &longs;ed proportio motus e&longs;t æqua<lb/>lis proportioni ponderis uici&longs;sim per uige&longs;imam&longs;extam &longs;exti Ele<lb/>mentorum b ad a, igitur proportio b ad a ponderis e&longs;t duplicata ei, <lb/>quæ e&longs;t raritatis c ad raritatem d.</s>

<s>SCHOLIVM PRIMVM.</s>

<s>Ne tamen &longs;ine exemplo intelligas hanc duplicatam rationem, <lb/>proponatur craritas quatuor, d unum, a pondus duodecim libra<lb/>

<arrow.to.target n="fig18"></arrow.to.target><lb/>rum, tunc c re&longs;i&longs;tit &longs;olum ex quarta parte, & effi<lb/>cit a quadruplo maioris actionis, &longs;cilicet ut qua<lb/>draginta octo, tota igitur proportio, qua mo<lb/>uebitur a in c, erit centum nonaginta duorum, & hoc diuidemus <lb/>per d, quod e&longs;t unum, exibit <expan abbr="põdus">pondus</expan> b centum nonaginta duo. Pro<lb/>portio igitur b ad a e&longs;t &longs;exde cupla, & hæc e&longs;t duplicata quadruplæ <lb/>raritatis c ad raritatem d.</s>

<s>Quòd &longs;i quis neget tantundem augere c actionem a, quanto mi<lb/>nus re&longs;i&longs;tit, &longs;ed aut magis aut minus, & &longs;it proportio b ad a dupli<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 duplicatæ. <lb/>Sed ex duabus proportionibus a ad c, & actionis ex c ad a produ<lb/>citur proportio b ad a, igitur per <expan abbr="decimam&longs;eptimã">decimam&longs;eptimam</expan> &longs;exti Elemento<lb/>rum proportio c ad d e&longs;t media inter proportiones a ad c, & actio<lb/>nis a in c, quare æqualis f, igitur proportio b ad a duplicata ei, quæ <lb/>e&longs;t c ad d quod erat demon&longs;trandum.</s>

<s>SCHOLIVM SECVNDVM.</s>

<s>Si autem media fuerint diuer&longs;arum rationum, ut aqua, & aër non <lb/>demon&longs;trat argumentum, quia pondera inter &longs;e non &longs;eruant ratio<lb/>nem. Nam lignum centum librarum ex &longs;alicis arbore, non magis <lb/>de&longs;cendit, quàm lignum libræ unius. Ideò nec in comparatione ad <lb/>medium aëris.</s>

<s>Propo&longs;itio trige&longs;imaquarta.</s>

<s>Proportio corporis cubi ad &longs;uam &longs;uperficiem quadratam, e&longs;t ue<lb/>lut eiu&longs;dem &longs;uperficiei ad latus, eiu&longs;dem uerò ad monadem.</s>

<s>

<arrow.to.target n="marg89"></arrow.to.target></s>

<s>Sit cubus a b c eius quadrata, &longs;uperficies a <lb/>

<arrow.to.target n="fig19"></arrow.to.target><lb/>c, latus a b, monas d, dico eas e&longs;&longs;e inuicem <lb/>analogas. 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, & toties <lb/>ctiam a&longs;&longs;umitur a b in a c ex diffinitione Eucli </s>

<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 & d, erit ergo <lb/>per decimam noni Elementorum elatus a c, <lb/>ergo æqualis a b, igitur cum a c, e & d &longs;int analogæ, erunt & a b c, <lb/>a b, & d analogæ, quod fuit demon&longs;trandum.</s>

<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>

<s>Propo&longs;itio trige&longs;imaquinta.</s>

<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<lb/>tione in hypate acutæ uix ferunt.<lb/>

<arrow.to.target n="marg91"></arrow.to.target></s>

<s>Quoniam facta uariatione in hypate, quæ e&longs;t <lb/>in Diapa&longs;on, uel bis Díapa&longs;on maiore interual<lb/>

<arrow.to.target n="fig20"></arrow.to.target><lb/>lo di&longs;tat, uelut ex a in b in grauiore, maius e&longs;t in<lb/>teruallum ex c in d, igitur maior e&longs;t b d, quàm a c <lb/>ergo &longs;ingulæ uoces inter b & d magis di&longs;tant, <lb/>

<arrow.to.target n="fig21"></arrow.to.target><lb/>quàm inter a & c, & quanto magis appropin<lb/>quant ad d, igitur d maius e&longs;t quàm b. Ergo magnitudo e&longs;t ratione <lb/>acuitatis, non grauitatis, cum &longs;uppo&longs;uerimus d e&longs;&longs;e acutiorem b & <lb/>cip&longs;o a. O&longs;tenditur etiam idem quia uox grauis fit ex priuatione <lb/>motus &longs;icut acuta ex uehementia. Motus autem e&longs;t res, quies, <lb/>priuatio.</s>

<s>Secundum &longs;ic: nam remi&longs;sio mota non feriet aurem, ideò &longs;onum <lb/>non pariet ob nimiam tarditatem. At in uelo ci&longs;simo motu oportet <lb/>uel fidem uel arteriam contrahi, & non contrahitur ni&longs;i per mu&longs;cu<lb/>los, igitur contentio illa finem habet. 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 aëre <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 fermè ob&longs;urde&longs;cant, qui audierint, ut ferunt de <lb/>Nili cataractis.</s>

<s>Tertium &longs;ic &longs;it a b humi<lb/>lior uox, quæ excre&longs;cat &longs;e<lb/>mitonio minore &longs;olum in <lb/>c, & &longs;it d e dupla ad ab &longs;e<lb/>

<arrow.to.target n="fig22"></arrow.to.target><lb/>cundum naturam, ut in uo<lb/>cibus medijs fiet, ut &longs;i e debeat excre&longs;cere &longs;emitonio minore per de<lb/>cimamnonam quinti <expan abbr="Elem&etilde;torum">Elementorum</expan> fe dupla c b, & in acutis ubi ex<lb/>creuerit ad diapa&longs;on quadrupla: pueri autem uox, quæ iam diapa<lb/>&longs;on altior e&longs;t d e, erit bis diapa&longs;on, & ideò quadrupla b c, &longs;ed in acu<lb/>tioribus erit dupla, nullus enim puer e&longs;t adeo fractæ uocis, qui&longs;u<lb/>pra humillimam non a&longs;cendat per diapa&longs;on, igitur interuallum uo<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<lb/>tionem habentis e&longs;t æquale fermè toti a b, cum autem in diapa&longs;on <lb/>&longs;int duodecim &longs;emitonia, & duo comata, manife&longs;tum e&longs;t, quod ex<lb/>ten&longs;io illa erit maxima in <expan abbr="cõparatíone">comparatíone</expan> grauioris uo cis a b. Etideò <lb/>minimum in crementum in humilioribus uocibus, ubi quis coga

<pb pagenum="30"/>tur a&longs;cendere, maximum e&longs;&longs;e uidetur, adeò ut ægrè à pluribus fera<lb/>tur, à quibu&longs;dam non omnino feratur.</s>

<s>SCHOLIVM.</s>

<s>Ob hoc natura fecit, ut non quemadmodum in fidibus uoces ex <lb/>breuitate intenderentur, &longs;ed ex con&longs;trictione ligulæ, ut dicunt, &longs;u<lb/>per a&longs;peram arteriam uox ad diapa&longs;on acueretur addito impetu <lb/>proportione, ut ex con&longs;trictione, & impetu <expan abbr="cõ&longs;urgeret">con&longs;urgeret</expan> dupla pro<lb/>portio. Hoc autem manife&longs;tè experimur in elymis in quibus nullæ <lb/>pror&longs;us facta mutatione in&longs;trumenti con&longs;tantibus digitis omni<lb/>bus præter pollicem &longs;ini&longs;træ uocem exacuimus ad diapa&longs;on, inde <lb/>etiam ad bis diapa&longs;on: &longs;icut declarauimus in commentarijs Epi<lb/>demiorum.</s>

<s>Propo&longs;itio trige&longs;ima&longs;exta.</s>

<s>Si proportio per proportionem minorem æquali ducatur, pro<lb/>portio minor producetur. Vnde manife&longs;tum e&longs;t duas proportio<lb/>nes minores æqualitate inuicem ductas proportionem minorem <lb/>unaquaque illarum producere.</s>

<s>

<arrow.to.target n="marg92"></arrow.to.target></s>

<s>Proportio a b ad c, quali&longs;cunque &longs;it, duca<lb/>tur in proportionem minorem æqualitate <lb/>fad g, dico quod producta proportio erit <lb/>minor ea, quæ 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<lb/>ducta ex proportionibus a b ad c, & f g. Itemque per decimamquar<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, quàm <lb/>d ad c. igitur quàm proportio a b ad c in proportionem f ad g. Sit <lb/>autem utraque minor æqualitate ea, quæ a b ad c, & ea quæ f ad g, di<lb/>co productam unaquaque earum e&longs;&longs;e minorem. Quod enim (manen <lb/>tibus his, quæ dicta &longs;unt) minor &longs;it d ad c, quam a b ad c ex prima <lb/>parte o&longs;ten&longs;um e&longs;t. Quòd uerò etiam minor &longs;it d ad c, quàm d ad <lb/>a b, & ex con&longs;equenti quàm f ad g demon&longs;tratur &longs;ic. Quia enim mi<lb/>nor e&longs;t a b ad c, æqualitate erit a b minor c, fiat ergo h æqualis a b, <lb/>erit ergo d ad h, ut d ad a b per &longs;eptimam quinti Elementorum, at d <lb/>ad c minor quàm d ad h per octauam eiu&longs;dem, igitur minor d ad c, <lb/>quàm d ad a b, igitur patet propo&longs;itum.</s>

<s>Propo&longs;itio trige&longs;ima&longs;eptima.</s>

<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 illæ proportiones coniunctæ non productæ.<lb/>

<arrow.to.target n="marg94"></arrow.to.target></s>

<s>Cùm enim primus non po&longs;sit mouere nec &longs;ecundus, erunt pro<lb/>portiones minores æqualitate, Ideò per &longs;ecundam partem præce<lb/>dentis multo minus mouerent duo, quàm unus. Et &longs;i quatuor mo

<pb pagenum="31"/>uerent unusque per &longs;e mouere non po&longs;&longs;et, adderetur &longs;i proportio <lb/>produceretur, fieret minor, ergo minus mouerent quinque quàm <lb/>quatuor ex ij&longs;dem, quod e&longs;t ab&longs;urdum.</s>

<s>Propo&longs;itio trige&longs;imao ctaua.</s>

<s>Omne corpus tantùm re&longs;i&longs;tit motui contrario &longs;uo naturali quan <lb/>cum mouetur occulto motu quie&longs;cendo.</s>

<s>

<arrow.to.target n="marg95"></arrow.to.target></s>

<s>Sit a corpus quie&longs;cens in pauimento b, & mouetur in eo occul</s>

<s>

<arrow.to.target n="marg96"></arrow.to.target><lb/>to motu uer&longs;us centrum, ut &longs;uprà ui&longs;um e&longs;t, contra<lb/>

<arrow.to.target n="fig23"></arrow.to.target><lb/>rius illi &longs;it motus ad c, &longs;i ergo a quie&longs;ceret in c moue<lb/>retur ad b occulto motu certa ui, ergo eadem re&longs;titit, <lb/>ne traheretur ad c. 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>

<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>

<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>

<s>Ex hoc patet cur naues & currus ab initio tardè & difficulter mo <lb/>ueantur, ubi moueri c&oelig;perint motus augetur: quoniam re&longs;i&longs;tunt </s>

<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<lb/>ralis e&longs;t, & ideò contrarius uiolento: Ergo cum iam mouetur uio<lb/>lenter minus, mouetur naturaliter, igitur minus re&longs;i&longs;tit. Declarabi<lb/>tur enim infrà quòd omne quod mouetur duobus motibus tanto <lb/>

<arrow.to.target n="marg100"></arrow.to.target><lb/>minus uno mouetur quanto magis altero.</s>

<s><margin.target id="marg99"></margin.target>Q<emph type="italics"/>ue&longs;t.<emph.end type="italics"/> 31.</s>

<s><margin.target id="marg100"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 59.</s>

<s>Propo&longs;itio trige&longs;imanona.</s>

<s>Ab æquali aut minore ui, quàm &longs;it <expan abbr="impediment&utilde;">impedimentum</expan>, non fit motus.</s>

<s>Sit a quod re&longs;i&longs;tat, ne &longs;ur&longs;um trahatur per decem, dico, quod <expan abbr="nõ">non</expan> <lb/>

<arrow.to.target n="marg101"></arrow.to.target><lb/>&longs;ur&longs;um trahetur neque à decem, neque minore: nam &longs;i impedimen<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<lb/>&longs;ceret. Si uerò à minore moueretur à maiore ui deor&longs;um, quam &longs;ur<lb/>&longs;um, ergo deor&longs;um &longs;impliciter non &longs;ur&longs;um.</s>

<s>Propo&longs;itio quadrage&longs;ima.</s>

<s>Omne corpus &longs;phæricum tangens planum in puncto mouetur <lb/>ad latus per quancunque uim, quæ medium diuidere pote&longs;t.</s>

<s>Sit corpus ad unguem &longs;phæricum a tan<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<lb/>rum) dico, quod mouebitur à ui, quæ pote&longs;t <lb/>&longs;cindere aërem. 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. Neque<lb/>ratione magnitudinis contactus, cum &longs;it in <lb/>puncto &longs;olo, igitur remanet &longs;olum aëris impedimentum.

<arrow.to.target n="marg103"></arrow.to.target></s>

<s>Ex hoc liquet, quod oportet b planum e&longs;&longs;e ex duri&longs;sima mate<lb/>ria, quæ nullo modo cedat, aliter tanget plu&longs;quàm in puncto.</s>

<s>

<arrow.to.target n="marg104"></arrow.to.target></s>

<s>Vix fieri pote&longs;t, utin elementaribus &longs;phæra tangat planum in <lb/>puncto. Vel quia planum non erit exactè rectum, uel non durum, <lb/>ut pror&longs;us non cedat, uel non ad æquilibrium po&longs;itum, uel &longs;phæra <lb/>non erit exactè rotunda.</s>

<s>Propo&longs;itio quadrage&longs;imaprima.</s>

<s>Si fuerint duæ quantitates &longs;umaturque totius aggregatum maio<lb/>ris & minoris, quoties aggregatum minoris, & maioris, erit pro<lb/>portio confu&longs;a maioris aggregati ad minus, minor quàm multipli<lb/>cis maioris ad multiplex minoris.</s>

<s>

<arrow.to.target n="marg105"></arrow.to.target></s>

<s>Sint duæ magnitudines a & b, & &longs;it a maior <lb/>

<arrow.to.target n="fig24"></arrow.to.target><lb/>b, & &longs;umatur exempli gratia a quater cum b &longs;e<lb/>mel, & 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>

<s>

<arrow.to.target n="marg106"></arrow.to.target><lb/>minor quàm quadrupla. Con&longs;tat enim quod proportio quadru<lb/>pli a ad a e&longs;t maior, quam b ad quadruplum b, cum una &longs;it quadru<lb/>pla, alia &longs;ub quadrupla, igitur per uige&longs;imam&longs;ecundam huius ag<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, quàm quadrupli a ad a, & maior quàm b ad quadruplum b, & <lb/>e&longs;t pro intellectu Archimedis.</s>

<s><margin.target id="marg107"></margin.target>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<lb/>deran.<emph.end type="italics"/><lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 10.</s>

<s>Propo&longs;itio quadrage&longs;ima&longs;ecunda.</s>

<s>Trahentium nauim, ut ferentium pondera proportiones in &longs;e in<lb/>uicem, quomodo ducere oporteat con&longs;iderare.<lb/>

<arrow.to.target n="marg108"></arrow.to.target></s>

<s>Hoc quomodo non po&longs;sit fieri &longs;uprà docuimus, nunc etiam ge</s>

<s>

<arrow.to.target n="marg109"></arrow.to.target><lb/>neraliter dicam, cum con&longs;i&longs;tant hæc in duobus terminis, productio <lb/>uerò præ&longs;upponit quatuor terminos, ut in prima propo&longs;itione, aut <lb/>&longs;altem tres, atque in his medius habet rationem mouentis, & moti, <lb/>ergo cum in huiu&longs;modi <expan abbr="nõ">non</expan> &longs;int quatuor termini, nec tres, è quibus <lb/>unus &longs;it mouens, & motum proportio non poterit produci. Illud <lb/>etiam patet exemplo, nam &longs;i e&longs;&longs;et lapis, aut nauis ob&longs;i&longs;tens ut &longs;ex, & <lb/>e&longs;&longs;ent homines uiribus &longs;inguli, ut quatuor cum dimidio, tres mo<lb/>uerent in proportione dupla &longs;exquiquarta perdicta &longs;uperius eo<lb/>dem loco, at &longs;i proportio duci po&longs;&longs;et aliquorum hominum nume<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<lb/>plentnumerum 27, & &longs;eptem 31 1/2, & ideò non pote&longs;t duci propor<lb/>tio. Et ideò maximus e&longs;t error dicendo decem homines mouent na <lb/>uim proportione tripla, ergo triginta alij additis illis &longs;imiles robo<lb/>re mouebunt à proportione uiginti &longs;eptupla &longs;cilicet ducta nonu

<pb pagenum="33"/>pla in triplam. Sed &longs;umpta proportione alio modo producitur. Ve <lb/>lut &longs;i dicam, homines decem mouent nauim, aut <expan abbr="fer&utilde;t">ferunt</expan> pondus pro<lb/>portione tripla, igitur quadraginta homines idem facient propor<lb/>tione duodecupla &longs;cilicet quadrupla in triplam ducta. Cum ergo <lb/>addo triginta homines, qui mouent in proportione nonupla, non <lb/>oportet ducere nonuplam in triplam, &longs;ed totum numerum accipe<lb/>re, & quam proportionem habet ad partem, tandem habet uis mo<lb/>uens ad uim <expan abbr="mou&etilde;tem">mouentem</expan>. Vnde &longs;i duo moueant in proportione &longs;ex<lb/>quialtera, & &longs;ex in proportione quadrupla cum dimidia, & iungan <lb/>tur, ut fiant octo, non oportebit ducere &longs;exquialteram, in quadru<lb/>plam &longs;exquialteram, &longs;ed cum octo ad duo &longs;it in proportione qua<lb/>drupla, &longs;umemus quadruplam ad &longs;exquialteram, qu&ecedil; erit &longs;excupla, <lb/>& octo mouebunt, aut pondus gerentin proportione &longs;excupla.</s>

<s><margin.target id="marg109"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 37.</s>

<s>Propo&longs;itio quadrage&longs;imatertia.</s>

<s>Productionem ad additionem retrahere.<lb/>

<arrow.to.target n="marg110"></arrow.to.target></s>

<s>Sit proportio a ad b dupla pote&longs;tate li<lb/>cet &longs;int quinque homines, & &longs;int quindecim <lb/>homines c, & habebunt ad b &longs;excuplam <lb/>proportionem per præcedentem. Iuncta <lb/>ergo a, & c per octauam huius <expan abbr="moueb&utilde;t">mouebunt</expan> <lb/>b proportione octupla, dico, quod &longs;i du<lb/>xeris <expan abbr="proportion&etilde;">proportionem</expan> c ad a plus uno. i. qua<lb/>druplam in proportionem a ad b, quæ e&longs;t dupla, proueniet eadem <lb/>octupla. Nam quia in coniunctione &longs;ufficit iungere c cum a, & &longs;u<lb/>mitur &longs;ecundum proportionem a ad b, igitur cum proportio a ad <lb/>b co mparata ad proportionem c & a ad b &longs;it, &longs;icut proportio c & a <lb/>ad a, & proportio c & a ad a &longs;it, &longs;icut proportio c ad a, & a ad a, & <lb/>proportio a ad a habet rationem unius, igitur proportio aggregati <lb/>c a ad b e&longs;t producta ex proportione c ad a plus monade in propor <lb/>tionem a ad b, quod erat demon&longs;trandum.</s>

<s>Propo&longs;itio quadrage&longs;imaquarta.</s>

<s>Si fuerit proportio motoris ad id, quod e&longs;t maximum non mo<lb/>uens & &longs;patium, & tempus, nota erit etiam reliquorum nota.</s>

<s>Sæpe contingit, ut quinque homines moueant nauim, & &longs;patium <lb/>ad tempus notum, & etiam cognitum maximum, quod mouere <lb/>non pote&longs;t. Sit ergo a numerus hominum, b na<lb/>

<arrow.to.target n="fig25"></arrow.to.target><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, & g tempus, dico, quod h &longs;patium notum erit, &longs;eu <lb/><expan abbr="not&utilde;">notum</expan> g tempus, & h &longs;patium, dico, quod erit f motor, &longs;eu numerus

<pb pagenum="34"/>hominum notus. Quoniam ergo notum e&longs;t a & c, quia e&longs;t æquale <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 præcedentem, ut f ad a ita &longs;patij <lb/>ad e in d tempore. Sed per eadem ut temporis d ad &longs;patium illud, <lb/>ita g ad h, ergo cum nota &longs;int d e f g erit etiam h, & ita conuertendo.</s>

<s>Propo&longs;itio quadrage&longs;imaquinta.</s>

<s>Rationem &longs;tateræ o&longs;tendere.<lb/>

<arrow.to.target n="marg111"></arrow.to.target></s>

<s>Archimedes nititur huic fundamento, quod pondera, quæ pro<lb/>portionem mutuam habent, ut di&longs;tantiæ à libella a, quæ &longs;u&longs;pen<lb/>duntur, æqualiter ponderant, &longs;it ergo libella a b, & &longs;u&longs;pen&longs;a in a cen <lb/>trum mundi c, ad quod dirigitur pondus, & liquet, quod ip&longs;um <lb/>non &longs;e inclin abit ex uige&longs;imatertia propo&longs;itione. Si ergo ponantur <lb/>lo co lineæ b d in e & f, & &longs;it proportio e b <lb/>

<arrow.to.target n="fig26"></arrow.to.target><lb/>ad b f, ut g ad h, dico, quòd erit æquili<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 |&longs;u&longs;pen&longs;um h, moueretur in re<lb/>cta e h per eandem, quia ergo retinetur, mo<lb/>uetur per obliquam h k, & &longs;umatur in pro<lb/>pin quum punctum in b e, & n in æquali di<lb/>&longs;tantia in e f, quia ergo e b totum mouetur <lb/>eadem ui in &longs;ingulis partibus, quia a pon<lb/>dere h, & 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, & ita in b erit penè infinita: quia quanta ui ex<lb/>tenditur ex h in k tanta puncta b, &longs;e circumuertit ergo propor<lb/>tio hypomochlij ad &longs;patium, uelut roboris ad robur, at eadem n o <lb/>ad h k, e&longs;t enim n o æqualis m p, & n b, & b m æquales, ut uerò 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/>nam idem pondus &longs;cilicet g mouet totam b f, igitur ut g &longs;e habet </s>

<s>

<arrow.to.target n="marg112"></arrow.to.target><lb/>ad n o, ita h ad m p, &longs;ed m p & n o &longs;unt æquales, ergo tanta e&longs;t uis g <lb/>in f, quanta h in e.<lb/>

<arrow.to.target n="marg113"></arrow.to.target></s>

<s>Ex quo patet, quod hypomo chlion moueretur infinita ui, &longs;i po&longs;<lb/>&longs;et e&longs;&longs;e punctus: &longs;ed quia in extrema &longs;uperficie cylindri, ideò pote&longs;t <lb/>aliqua ui retineri.</s>

<s>

<arrow.to.target n="marg114"></arrow.to.target></s>

<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 æquali <lb/>non fit motus per trige&longs;imamnonam propo&longs;itionem.</s>

<s>

<arrow.to.target n="marg115"></arrow.to.target></s>

<s>Et libella nihil retinet ni&longs;i quantum e&longs;t pondus eius quod cu

<pb pagenum="35"/>pit ad centrum peruenire, & pondus ei appen&longs;um non prohi<lb/>bet motum, etiam &longs;i e&longs;&longs;et infinitum, ni&longs;i quatenus non uult recede<lb/>re ex directo centri mundi: & ut grauat hypomochlion faciens im<lb/>pre&longs;sionem.</s>

<s>

<arrow.to.target n="marg116"></arrow.to.target></s>

<s>Et &longs;i terra tota e&longs;&longs;et appen&longs;a polo, moueretur magna ui: quoni<lb/>am uis eadem e&longs;t in polo, quæ in circulo toto æquinoctij.</s>

<s>

<arrow.to.target n="marg117"></arrow.to.target></s>

<s>Etrota, quanto uelocius mouetur in ambitu, tanto mi <lb/>norem habet uim: &longs;ed propter aërem, qui &longs;ecum circum<lb/>

<arrow.to.target n="fig27"></arrow.to.target><lb/>fertur, mouetur magno impetu, & magnas facit læ&longs;iones. <lb/>Ideò hoc in cono non accidit.</s>

<s>

<arrow.to.target n="marg118"></arrow.to.target></s>

<s>Ex quo patet ratio eleuandi pondera magna per tra<lb/>bem, ut à latere uides.</s>

<s>Propo&longs;itio quadrage&longs;ima&longs;exta.</s>

<s>An &longs;it aliqua proportio, & qualis inter animam, & ui<lb/>tas, & &longs;ua corpora con&longs;iderare.</s>

<s>

<arrow.to.target n="marg119"></arrow.to.target></s>

<s>Declarauimus motum c&oelig;li e&longs;&longs;e uoluntarium, ob&longs;equente c&oelig;<lb/>lo per uirtutem in eo infu&longs;am. In animalibus autem, & præcipuè <lb/>in homine notius e&longs;t hoc experientibus nobis in ip&longs;is: &longs;ed motus <lb/>hic, ut dixi &longs;upra, mi&longs;tus e&longs;t, ille uerò c&oelig;le&longs;tis ignotior e&longs;t. Certum </s>

<s>

<arrow.to.target n="marg120"></arrow.to.target><lb/>tamen e&longs;t plenè ob&longs;equi c&oelig;lum uitæ, nec pror&longs;us repugnare. So<lb/>let Ari&longs;toteli imponi, quòd &longs;i adderetur a&longs;trum c&oelig;lo, quòd c&oelig;lum <lb/>aut quie&longs;ceret, aut tardius moueretur: quod e&longs;t, ac &longs;i diceremus, <lb/>quòd homo paruus &longs;i fieret maior, non e&longs;&longs;et adeò agilis, tanquam <lb/>motus ille e&longs;&longs;et ab externa cau&longs;a. Imò perinde e&longs;&longs;et, ac&longs;i quis dice<lb/>ret, quod lapides magni minus uelociter de&longs;cenderent, quam par<lb/>ui. Quin potius ut lapis magnus uelociùs mouetur: quàm par<lb/>uus naturali motu, & tardius præternaturali, ita c&oelig;lum motu uo<lb/>luntario, &longs;i ita dici po&longs;&longs;et æqualius & maiore cum efficacia, quan<lb/>to den&longs;ius. Et ita &longs;i Ari&longs;toteles illud dixi&longs;&longs;et, o&longs;tendi&longs;&longs;et magnam <lb/>imperitiam. Ideò quale iudicium debemus facere de Alexandro, & <lb/>

<arrow.to.target n="marg121"></arrow.to.target><lb/>Aueroe, qui hoc ei tribuunt. <expan abbr="legi&ttilde;">legitur</expan> enim in textu Arabico tale quip<lb/>piam. De Animalibus for&longs;an po&longs;&longs;et hoc dici, <expan abbr="quoniã">quoniam</expan>, ut &longs;uprà dixi<lb/>mus, motus ille mi&longs;tus e&longs;t. Remanet ergo difficultas, <expan abbr="quoniã">quoniam</expan> &longs;i mo<lb/>tus i&longs;te non à proportione fit, quare non e&longs;t infinitus? & dico quae in <lb/>animalibus tres &longs;unt cau&longs;æ, una, quia e&longs;t mi&longs;tus, & habet repugnan <lb/>tiam: &longs;ecunda, quia e&longs;t de loco ad locum, motus autem c&oelig;li e&longs;t in lo <lb/>co: tertia e&longs;t communis etiam c&oelig;lo, et e&longs;t, <expan abbr="quoniã">quoniam</expan> non e&longs;t ratio finis. <lb/>Natura enim diuina non appetit mouere <expan abbr="tã">tam</expan> celeriter. Quid e&longs;t ergo <lb/>proportio, <expan abbr="c&utilde;">cum</expan> &longs;it <expan abbr="ultim&utilde;">ultimum</expan> uoluntatis uit&ecedil;, ut obtemperet primæ cau&longs;æ, <lb/>ideo illud e&longs;t <expan abbr="ultim&utilde;">ultimum</expan>, &qring; mouet. E&longs;t <expan abbr="a&utilde;t">aunt</expan> idem uelle, & po&longs;&longs;e. In natura

<pb pagenum="36"/>enim c&oelig;li e&longs;t ille appetitus, cuius prin cipium e&longs;t uita: & eíus uolun <lb/>tatis bonum ip&longs;um. Et ideo hæc proportio <expan abbr="nõ">non</expan> diuiditur. In anima<lb/>libus autem non e&longs;t uis illa ni&longs;i, cum proportione, quia primum in<lb/>&longs;trumentum, quod recipit, & e&longs;t &longs;piritus uim habet determinatam, <lb/>cum &longs;it uirtus in materia: ideo <expan abbr="nõ">non</expan> mouet ni&longs;i cum certa proportio<lb/>ne, uelut lumen in medio in &longs;e non habet proportionem ni&longs;i ad lu<lb/>cem, &longs;ed ut e&longs;t in illo, pote&longs;t e&longs;&longs;e remi&longs;&longs;um, <expan abbr="ob&longs;cur&utilde;">ob&longs;curum</expan> & hebes. Quæ<lb/>ritur ergo quantitas illius? &longs;i dicas, quòd e&longs;t à luce: quæro quanti<lb/>tas lucis, unde &longs;it? for&longs;an dicendum, quòd uelutin motibus, quanto <lb/>den&longs;iora &longs;unt corpora tanto <expan abbr="mouen&ttilde;">mouentur</expan> maiore nixu, & robore. Nam <lb/>calor in materia augetur iuxta illius quantitatem: idem in luce, & <lb/>reliquis. Dico ergo proportionem e&longs;&longs;e infinitam: nam &longs;i corpus e&longs;<lb/>&longs;et infinitum & optimè di&longs;po&longs;itum infinita ui moueretur & agili<lb/>tate, ut enim maius e&longs;t eo maiores uires habet.</s>

<s><margin.target id="marg120"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 27.</s>

<s>Propo&longs;itio quadrage&longs;ima&longs;eptima.</s>

<s>Si duo mobilia æqualiter in eodem circulo iuxta proprios mo<lb/>tus moueantur, productum temporis circuituum inuicem erit æ<lb/>quale producto differentiæ temporum circuitus ductæ in tempus <lb/>coniunctionis primæ.<lb/>

<arrow.to.target n="marg122"></arrow.to.target></s>

<s>Sint duo mobilia a & b in eodem pun<lb/>

<arrow.to.target n="fig28"></arrow.to.target><lb/>cto, quæ æqualiter uer&longs;us candem partem <lb/>moueantur æqualibus in temporibus, inui <lb/>cem tamen in æqualiter, ita quod a in f & b <lb/>in g temporibus ab&longs;oluant circulum, & ho <lb/>rum differentia &longs;it h. Dum ita que a perficit <lb/>circulum b perueniat in c, igitur c d b e&longs;t dif <lb/>ferentia, quæ &longs;uperanda e&longs;t, & proportio <lb/>circuli ad b c ut g ad f, quare reliqui ad reli<lb/>quum, ut re&longs;idui ad re&longs;iduum, &longs;cilicet circu<lb/>li ad c d b, ut g ad h, & 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 & cum circulo & b c & c d b diffe<lb/>rentia, & &longs;it &longs;productum exfin g, dico quod diui&longs;a &longs; per h exibit k <lb/>tempus coniunctionis primæ, &longs;it itaque d locus coniunctionis, dico <lb/>igitur quod differentia &longs;patij pertran&longs;iti a b, a & a, b in reditu ex con <lb/>iunctione prima ad d e&longs;t unus circulus completus, non enim po&longs;<lb/>&longs;unt e&longs;&longs;e plures, nam &longs;equeretur, quòd 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 & 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<lb/>culo, neque maior, ut declaratum e&longs;t, igitur e&longs;t unus circulus ad un

<pb pagenum="37"/>guem. 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, & erit ex de<lb/>mon&longs;tratis horum differentia circulus qui uocetur o, & &longs;it p &longs;pa<lb/>tium, quod pertran&longs;it b in tempore eodem, in quo a pertran&longs;it o, & <lb/>&longs;it q differentia o, & p qu&ecedil; in circulo e&longs;t c d l b, quia igitur in eodem <lb/>tempore a pertran&longs;it m & b, n, erit m ad n, ut a ad b, & eadem ratio<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 & n, & q, differentia o & p erit ex <lb/>decimanona quinti Euclidis, m ad o, ut o ad q, & ita circulus e&longs;t ana <lb/>logus inter &longs;patium pertran&longs;itum à motore uelociori, & inter diffe<lb/>rentiam &longs;patij quæ accidit, dum uelocior motor pertran&longs;it circu<lb/>lum, id e&longs;t quòd circulus a c d e&longs;t analogus inter c d l b, & circulos <lb/>pertran&longs;itos a b a cum portione b d. Reuertor igitur ad propo&longs;i<lb/>tum, cum &longs;it m ad o, ut o ad q, & 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, & 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<lb/>&longs;trandum.</s>

<s>

<arrow.to.target n="marg123"></arrow.to.target></s>

<s>Ex hoc patet, quod proportio temporis coniunctionis ad tem<lb/>pus tardioris motus circuitionis e&longs;t ueluti temporis circuitus uelo <lb/>cioris motoris ad differentiam temporis motus tardioris, & uelo<lb/>cioris motoris in uno circuitu.</s>

<s>Propo&longs;itio quadrage&longs;imao ctaua.</s>

<s>Si tria mobilia ex eodem puncto di&longs;cedant, fuerintque duorum, ac <lb/>duorum coniunctiones in temporibus commen&longs;is illa tria mobi<lb/>lia denuò coniungentur in tempore producto ex denominatore di <lb/>ui&longs;ionis temporis maioris per minus in minus, aut numeratore <lb/>in maius.</s>

<s>

<arrow.to.target n="marg124"></arrow.to.target></s>

<s>Sint tria mobilia a, quod circuat in duobus annis b in quinque, <lb/>c in &longs;eptem. Dico quod primum redibunt in numero producto ex <lb/>&longs;eptem quinque & duobus, qui &longs;unt numeri primi, & erit ille nume<lb/>rus &longs;eptuaginta annorum. Nam in &longs;eptuaginta annis a perficiet tri<lb/>gintaquinque reuolutiones b quatuordecim, c decem: ergo <expan abbr="redib&utilde;t">redibunt</expan> <lb/>per perfectos circuitus ad idem punctum. O&longs;tendo modo quod <lb/>non ante: nam &longs;i &longs;ic: &longs;it, ut in trigintaquinque annis igitur b & c per<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 & c non numerabitur a b a, er<lb/>go a non perficiet circuitus, ergo non redibit ad primum <expan abbr="loc&utilde;">locum</expan>, ergo <lb/>non erit iunctus cum b & c. Quod &longs;i dicas a b c coniungi in decem <lb/>&longs;eptem annis numero non numerato ab ali <lb/>

<arrow.to.target n="fig29"></arrow.to.target><lb/>quo illorum temporum, auferantur perfe<lb/>ctæ circulationes, & <expan abbr="remaneb&utilde;t">remanebunt</expan> dimidium <lb/>ex a, duæ quintæ ex b, tres &longs;eptimæ ex c, igi<lb/>tur oportebit ut hæ portiones &longs;int æqua<lb/>les, ut po&longs;t perfectas circulationes in idem <lb/>punctum, <expan abbr="cõueniant">conueniant</expan>, ergo 1/2 & 2/5 & 3/7 æqui<lb/>ualebunt, quare proportio 7 ad 3 & 5 ad 2 <lb/>& 2 ad 1, e&longs;t una, quare permutando 3 ad 2 <lb/>ut 7 ad 5, &longs;ed 7 & 5 &longs;unt contra &longs;e primi, ergo in &longs;ua proportione mi <lb/>nimi per dicta in &longs;eptimo Elementorum: ergo tria, & duo non &longs;unt <lb/>in eadem proportione. Rur&longs;us dicantur conuenire in annis qua</s>

<s>

<arrow.to.target n="marg125"></arrow.to.target><lb/>tuordecim cum dimidio, ergo in uiginti nouem conuenient ite<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 æquale uni, ergo duo erunt æqualia &longs;eptem. Rur&longs;us dica<lb/>mus, quod in tempore annorum <02> quadrata decem &longs;imiliter aufe<lb/>ram integras reuolutiones, quas potero, & erunt <02> 2 1/2 m: 1, & <02> 2/5 & <lb/><02> 10/49 æqualia. Hic uides infinita &longs;equi in conuenientia, quæ longum <lb/>e&longs;&longs;et numerare, nam &longs;eptem e&longs;&longs;et æquale quinque, & proportio reci&longs;i <lb/>ad potentia rethe, ut numeri ad numerum. Igitur non conueniunt <lb/>ante &longs;eptuaginta annos.<lb/>

<arrow.to.target n="marg126"></arrow.to.target></s>

<s><margin.target id="marg125"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 23</s>

<s>Ex hoc &longs;equitur, quòd nullibi conuenient præterquàm in eo<lb/>dem puncto, &longs;cilicet in quo ab initio coniuncti fuerunt.</s>

<s>

<arrow.to.target n="marg127"></arrow.to.target></s>

<s>Sequitur denuo ex propo&longs;itione ip&longs;a repetita, & primo corrola<lb/>rio, quod nullibi alibi conuenient quàm in dato primo puncto, in <lb/>quo coniuncti fuerant ab initio etiam u&longs;que in æternum.</s>

<s>Sit rur&longs;us ut a circuat in annis duobus cum dimidio, b in tribus <lb/>cum tertia parte, cin quatuor cum quarta parte ducam per &longs;uos <lb/>denominatores, & erit ut a in quinque annis. b in decem, c in decem<lb/>&longs;eptem circuant, & redeant ad idem punctum, & quia quin que nu<lb/>merat decem, & decem, & decem&longs;eptem &longs;unt numeri inuicem pri<lb/>mi, ducam decem in decem&longs;eptem fiunt centum &longs;eptuaginta. Con<lb/>&longs;tat igitur c quadragíes, b quinquagies &longs;emel, a &longs;exagies octies cir<lb/>cumuerti, & redire ad idem punctum: ergo rur&longs;us coibunt po&longs;t tot <lb/>annos in eo, dico modo, quod non ante: nam &longs;i non &longs;it, ut in trigin<lb/>ta tribus annis. gratia exempli, aufero <expan abbr="decem&longs;ept&etilde;">decem&longs;eptem</expan>, decem, & quin<lb/>que, & relinquentur &longs;exdecim tria & tria, & rur&longs;us ex &longs;exde cim tres

<pb pagenum="39"/>cir cuitus c, & relinquentur 3 3/4 &longs;equetur igitur, ut &longs;it proportio 17 ad <lb/>13, & 2 1/2 ad 1/2 & 3 1/3 ad 3 eadem, & ita 17/13, 5/2 & 10/9 eadem &longs;i iam &longs;uppo<gap/><lb/>mus 17 & 10 e&longs;&longs;e primos inuicem, ut in &longs;ecunda demon&longs;tratione<gap/><lb/>Igitur &longs;equuntur eadem corrolaria, quæ dicta &longs;unt.</s>

<s>Propo&longs;itio quadrage&longs;imanona.</s>

<s>Propo&longs;ito mobilis in circulo circuitus tempore, dataque ratione <lb/>di&longs;tantiæ ab illo mobilis circuitum inuenire, quod ex eodem pun<lb/>cto di&longs;cedens cum alio mobili in dato puncto conueniat &longs;ub quo<lb/>cunque numero circuituum tempus quoque coniunctionis.</s>

<s>

<arrow.to.target n="marg128"></arrow.to.target></s>

<s>Sit in circuli peripheria a <expan abbr="p&utilde;ctus">punctus</expan>, qui cir <lb/>cuat æquali motu (hocenim &longs;emper intel<lb/>ligitur) in b tempore: & &longs;it datus punctus c <lb/>in quo di&longs;cedens e mobile ex coniunctio<lb/>ne cum a po&longs;t certos circuitus proprios, <lb/>aut etiam. &longs;ine ulla circuitione perfecta de<lb/>beat conuenire. Volo &longs;cire tempus circui<lb/>tionis e: & etiam tempus coniunctionis. <lb/>Sit ergo primum ut ab&longs;que circuitione ulla e, a debeat comprehen<lb/>dere e in c po&longs;t numerum circuitionum ip&longs;ius a, qui &longs;it f. nam &longs;i a o c <lb/>currit e in prima circuitione ip&longs;ius e, igitur a mouetur uelocius <lb/>quàm e, cum ergo debeat attingere ip&longs;um e, nece&longs;&longs;e e&longs;t ut a pertran<lb/>&longs;eat prius per punctum ex quo di&longs;ce&longs;sit antequam redeat ad con<lb/>iunctionem e: ergo perficiet &longs;altem unam circuitionem. Ducemus <lb/>ergo f in b, & fiet g tempus circuitus aut circuituum a, & quia &longs;pa<lb/>tium a c datum e&longs;t, &longs;it b temporis circuitus a ad h, uelut circuli to<lb/>

<arrow.to.target n="marg129"></arrow.to.target><lb/>tius ad a c, & iungatur g cum h & fiat k. Fiat quoque, ut monadis <lb/>ad h, ita l ad monadem, & ducatur l in k, & fiat m: dico m e&longs;&longs;e tem<lb/>pus circuitus e. Con&longs;tat enim ex &longs;uppo&longs;ito, quod k e&longs;t tempus to<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<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/>Proponatur modo tempus reuolutionum e ip&longs;um d: eodem mo<lb/>

<arrow.to.target n="marg130"></arrow.to.target><lb/>do agemus ducendo fin b fit g, addatur h & fiat k, diuidatur k per <lb/>aggregatum d & a e, & exeat m, (idem enim e&longs;t diuidere per aggre<lb/>gatum d & h, & multiplicare per l) dico ergo ut in demon&longs;tratione <lb/>priore, quod m e&longs;t tempus circuitus e. 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, & partem reuolutionis exibit tem<lb/>pus unius reuolutionis.</s>

<s>Exemplum primi in repaulò ob&longs;curiore: &longs;it f 4 & b 2 1/2 & 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<lb/>tiplica per 5/4 quod idem e&longs;t, fit 15 circuitus e, in quatuor ergo circui<lb/>tibus, & 4/5 qui &longs;unt duo decim anni perueniet a ad c, & 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, & 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. Quin que ergo <lb/>reuolutiones e erunt 1015/108 addita &longs;eptima parte, quæ e&longs;t 29/108 fient 2044/108 <lb/>&longs;eu 261/27, & &longs;unt anni 9 18/27 &longs;eu 9 2/3, ergo in tanto tempore a faciet qua<lb/>tuor circuitus, & &longs;eptimam partem, & e quinque circuitus, & &longs;e<lb/>ptimam.<lb/>

<arrow.to.target n="marg131"></arrow.to.target></s>

<s>Ex hoc patet, quod non coniungentur in alio loco, neque alio tem <lb/>pore ante prædictum tempus.</s>

<s>Propo&longs;itio quinquage&longs;ima.</s>

<s>Omnes circuituum portiones in eiu&longs;dem temporibus <expan abbr="repetun&ttilde;">repetuntur</expan>.</s>

<s>Sint in circulo a b c d e f g: a & b iuncta, & 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. Et &longs;ic deinceps <expan abbr="c&utilde;quetempora">cunque<lb/>tempora</expan> &longs;int æqualia, erunt & circuitus totidem numero, & exce&longs;<lb/>&longs;us æquales etiam a c, c d, d e, e f, f g, g h, h k, <lb/>

<arrow.to.target n="fig30"></arrow.to.target><lb/>k l. Et &longs;i aggregatum a &longs;cilicet circulorum, <lb/>& portionis fuerit commen&longs;um circulo, & <lb/>ita de b erunt omnia <expan abbr="cõmen&longs;a">commen&longs;a</expan> ad circulum, </s>

<s>

<arrow.to.target n="marg132"></arrow.to.target><lb/>& etiam inter &longs;e. Et &longs;i inter &longs;e aggregata, uel <lb/>portiones erunt, & eodem modo reliqua. <lb/>Et quoniam circuli circulis commen&longs;i &longs;unt: <lb/>&longs;i portiones erunt inuicem commen&longs;æ <expan abbr="er&utilde;t">erunt</expan>, <lb/>& toti circuitus cum partibus commen&longs;i, & <lb/>&longs;i non commen&longs;i, neque erunt inter &longs;e, neque ad circulum. Et &longs;i totum <lb/>&longs;patium cum circuitibus erit unius generis, erunt duplicata, & tri<lb/>plicata, & quadruplicata eiu&longs;dem generis: quare cum &longs;patia ip&longs;a <lb/>detractis circuitibus uelut rhete habeant naturam reci&longs;i, & &longs;patia <lb/>ip&longs;a tota &longs;int eiu&longs;dem generis, erunt &longs;patia, quæ relinquuntur eiu&longs;<lb/>dem generis. Erunt tamen incommen&longs;a nece&longs;&longs;ariò, &longs;i partes fuerint <lb/>incommen&longs;æ toti. Ponatur a c incommen&longs;a toti circulo dico, quod <lb/>a k <expan abbr="etiã">etiam</expan> e&longs;t incommen&longs;a toti circulo: & <expan abbr="etiã">etiam</expan> a k, & k c. Quia enim a c <lb/>e&longs;t incommen&longs;a circulo, & k a cum toto circulo &longs;emel e&longs;t commen

<pb pagenum="41"/>&longs;a a c, quia multiplex ei. igitur cum circulus, & a k diuidantur in cir<lb/>

<arrow.to.target n="marg133"></arrow.to.target><lb/>culum et a k, & circulus &longs;it incommen&longs;us circulo, cum a k erit aggre. <lb/>gatum ex circulo, & a k incommen&longs;um ip&longs;i a k, & a k pariter incom <lb/>

<arrow.to.target n="marg134"></arrow.to.target><lb/>men&longs;a circulo. Rur&longs;us quia a k e&longs;t incommen&longs;a circulo cum a k, & <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 & a c, & circulo ad dita a k. Si ergo a c &longs;it <lb/>commen&longs;a circulo, erunt omnes portiones e genere numeri, & &longs;i <lb/>

<arrow.to.target n="marg136"></arrow.to.target><lb/>potentia rhete erunt omnes, uel potentia rhete, uel circulis detra<lb/>ctis, ut a k & a l reci&longs;a: & 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<lb/>cæ numeri, &longs;eu latera corporum rhete, a k uero & a l, & huiu&longs;modi <lb/>in infinitum reci&longs;a potentia rhete.<lb/>

<arrow.to.target n="marg137"></arrow.to.target></s>

<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"/>præcedentis.<emph.end type="italics"/></s>

<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>

<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>

<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>

<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>

<s>Ex hoc patet, quod cum circulus po&longs;sit diuidi in infinita gene</s>

<s>

<arrow.to.target n="marg138"></arrow.to.target><lb/>ra quantitatum, quæ non &longs;unt inuicem commen&longs;æ cumque coniun<lb/>ctiones hæ &longs;emper in eodem genere maneant, quod infinita pun<lb/>cta, & infinitis in &longs;peciebus quantitatum remanebunt in quibus a <lb/>& b in perpetuum nunquam conuenient. Velut &longs;i coniunctio pri<lb/>ma fiat in <02> cu. 1/2 alicuius circuli, nunquam conuenient, neque in me<lb/>dietate, neque in quarta parte, nec octaua, nec tertia, nec &longs;exta, nec no<lb/>na, nec quinta, nec decima, & &longs;ic de &longs;ingulis in genere commen&longs;a<lb/>rum toti circulo. Neque in <02> quadrata 1/2 uel 1/3 uel 1/5 neque <02> 1/6 uel 1/20, <lb/>neque in <02> 3 m: 1, nec 2 m: <02> 3 nec in <02> <02> 2 aut 3 aut 7 nec in <02> rela<lb/>ta alicuius numeri, nec in 2 m: <02> <02> cub. 3 nec 2 m: <02> cub. 4, & &longs;ic <lb/>de alijs.</s>

<s><margin.target id="marg138"></margin.target>P<emph type="italics"/>er penulti<lb/>mam uige&longs;i<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>

<s>Propo&longs;itio quinquage&longs;imaprima.</s>

<s>Operationes dictas exemplo declarare.<lb/>

<arrow.to.target n="marg139"></arrow.to.target></s>

<s>Supponamus in circulo prædicto a c <02> 7 con&longs;tat, quod e&longs;&longs;e non <lb/>pote&longs;t, quia <02> 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 <02> 7, &longs;eu eam portionem per <lb/>10 quantitatem circuli & exibit <02> 7/100, & hæc erit portio circuli, & ita <lb/>&longs;i portio &longs;it <02> cub. 16, diuidemus <02> cub. 16 per 10 exibit <02> cu 2/125, & <lb/>ita de alijs.</s>

<s>Sed cum ex repetitione cre&longs;cat portio illa, donec exuperet mo<lb/>nadem, aut aliquem quemuis numerum detracta monade aut nu<lb/>mero circuituum habebit rationem reci&longs;i. Velut <02> 7/100 quater &longs;um<lb/>pta efficit <02> 112/100. Et hoc e&longs;t potentia rhete, &longs;ed &longs;i quis auferat mona<lb/>dem fiet <02> 112/100 m: 1, & hoc e&longs;t reci&longs;um 1, &longs;cilicet 1 p: <02> v: 23/25 m: <02> 28/25, &longs;ed ta <lb/>men uerè e&longs;t linea media.</s>

<s>Quod uerò non contingat coniungi in alio loco, neque tem<lb/>pore &longs;it, ut a b iungantur in c, & &longs;it reuolutio a triplex integra, & b

<pb pagenum="42"/>&longs;excuplex, & tempus totum decem annorum: ita ut a c &longs;it tertia <lb/>pars circuitus, & a circuitus tres anni, & quia circuitus b funt fex <lb/>cum tertia, diuidemus decem per 6 1/3 exit <lb/>1 11/29, dico quod non prius, neque in alio <lb/>

<arrow.to.target n="fig31"></arrow.to.target><lb/>puncto. Si enim primùm in eodem pun<lb/>cto, &, gratia exempli, in quatuor annis <lb/>congruit enim, & b dicamus quod per<lb/>egerit duas reuolutiones cum tertia, hoc <lb/>enim e&longs;t nece&longs;&longs;arium, &longs;i debet perueni<lb/>re ad c, & erunt anni tres, & 23/19, non ergo <lb/>anni quatuor. Cum enim tempora di<lb/>uer&longs;a diuiduntur per numeros haben<lb/>tes proportionem erunt, qui prodeunt <lb/>

<arrow.to.target n="table13"></arrow.to.target><lb/>numeri in eadem ratione. Diui&longs;o ergo <lb/>10 per 1 11/19 exit 6 2/3, & 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 æquale 1/3. Si enim per <lb/>præcedentem repetuntur, ergo non po&longs;<lb/>&longs;unt redire, doneciterum coniung antur in ip&longs;o a. Si enim aliter &longs;it <lb/>ut ex e, igitur e c e&longs;t æqualis a c pars toti, quod contingere non po<lb/>te&longs;t. Sin uerò coniunctio fiat in d, igitur per præcedentem d e e&longs;t <lb/>pars a c &longs;ubmultiplex quomodolibet, quare non fuerunt a&longs;&longs;um<lb/>pti primi numeri. Veluti in exemplo con&longs;tituimus, quod a, & b <lb/>conueniunt in c in decem annis, & a c e&longs;t tertia pars circuitus: er<lb/>go in triginta annis conueniunt in a, & in quadraginta rur&longs;us in c. <lb/>&longs;i ergo quis a&longs;&longs;ump&longs;i&longs;&longs;et quadraginta annos ab initio pro con<lb/>gre&longs;&longs;u, & diui&longs;i&longs;&longs;et per 1 12/19 exiret 25 1/3, & &longs;i per 3 exiret 13 1/3, & mani<lb/>fe&longs;tum e&longs;t, quod uterque numerus pote&longs;t diuidi per eundem nu<lb/>merum, utpote 4 & exit numerus cum eadem parte &longs;cilicet 6 1/3 & <lb/>3 1/3 ergo conuenient ante, non ergo a&longs;&longs;ump&longs;i&longs;ti minimos in ea pro<lb/>portione. Illi autem nequaquam amplius diuidi non po&longs;&longs;unt eo<lb/>dem modo.</s>

<table>

<table.target id="table13"></table.target>

<row><cell>Decem</cell><cell></cell><cell>Quatuor</cell><cell></cell></row>

<s>Propo&longs;itio quinquage&longs;ima&longs;ecunda.</s>

<s>Tria mobilia coniuncta in eodem puncto, quorum duo, & duo <lb/>conueniant in partibus in commen&longs;is inter &longs;e, in perpetuum in nul<lb/>lo unquam puncto conuenient.</s>

<s>

<arrow.to.target n="marg140"></arrow.to.target></s>

<s>Sint a b c iuncta, & primo iungantur a & b, iterum in d & b, & <lb/>c in e, & &longs;int a d, a e inconimen&longs;æ, dico quòd a b c nunquam con<lb/>uenient in aliquo puncto, &longs;eu primo, &longs;eu alio à prim o: &longs;i non con

<arrow.to.target n="fig32"></arrow.to.target><lb/>ueniant in f, erunt ergo in g tempore re<lb/>uolutiones integræ, & portio a f in&longs;uper. <lb/>Et quia hæ con&longs;tituuntur per congre&longs;&longs;us <lb/>b cum a, & &longs;unt &longs;patia a d, & b cum c, & <lb/>&longs;unt &longs;patia e f, igitur &longs;patium a f erit ex ge<lb/>nere quantitatis a d, & a e per quinqua<lb/>ge&longs;imam, harum ergo erunt commen&longs;æ: <lb/>quod e&longs;t contra &longs;uppo&longs;itum. Et harum <lb/>propo&longs;itionum principium e&longs;t traditum <lb/>à Campano Nouarien&longs;i Euclidis expo&longs;itore, in quodam libello <lb/>non edito qui diligentia patris mei Facij ad me peruenit.</s>

<s>Propo&longs;itio quinquage&longs;imatertia.</s>

<s><expan abbr="Circulor&utilde;">Circulorum</expan> &longs;e in aduer&longs;um mouentium proportionem declarare.</s>

<s>

<arrow.to.target n="marg141"></arrow.to.target></s>

<s>Sit orbis a b cuius cen<lb/>

<arrow.to.target n="fig33"></arrow.to.target><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<lb/>teri orbi funiculo a l b, & <lb/>&longs;it in uertice axis k m or<lb/>biculus &longs;olidus aut &longs;emi<lb/>circulari forma m, dico <lb/>quod proportio motus a <lb/>b ad motum m e&longs;t produ <lb/>cta ex duabus proportio<lb/>nibus c n <expan abbr="&longs;emidimeti&etilde;tis">&longs;emidimetientis</expan>, <lb/>& &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<lb/>ties <expan abbr="peripheriã">peripheriam</expan> o l <expan abbr="contine&ttilde;">continetur</expan> in peripheria a b, ergo quoties o k con<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<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/>& &longs;emidimetiente m ad quadratum k o, quod erat <expan abbr="demon&longs;trand&utilde;">demon&longs;trandum</expan>.</s>

<s>Manife&longs;tum e&longs;t autem ex ip&longs;a &longs;ola con&longs;titutione, quod &longs;i a b mo</s>

<s>

<arrow.to.target n="marg142"></arrow.to.target>

<pb pagenum="44"/>uetur &longs;ur&longs;um à dextro in &longs;ini&longs;trum in inferiore parte, mouebitur à <lb/>&longs;ini&longs;tro in dextrum, & uterque circulorum g & k in &longs;uperiore parte, <lb/>& in inferiore mouebitur contrario motu, &longs;cilicet in &longs;uperiore à &longs;ini <lb/>&longs;tro in dextrum, & inferiore à dextro in &longs;ini&longs;trum, illi uerò duo or<lb/>bes &longs;imili motu mouebuntur tam in parte &longs;uperiore, quàm inferio<lb/>re, & proportio motuum eorum inter &longs;e erit uelut dimetientium <lb/>corundem.<lb/>

<arrow.to.target n="marg143"></arrow.to.target></s>

<s>Rur&longs;us cum a b circumuertatur cum manubrio c d f e, tanto uelo <lb/>cius circumuertetur, & in ea proportione, qua d f continetur in c n, <lb/>& in eodem tempore, in quo manubrium circumuertitur in eodem <lb/>axis circumuertitur, & orbis, ut dictum e&longs;t, ergo in eodem tempo<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/>æqualitatis ip&longs;ius axis, &longs;eu ambitus, &longs;eu &longs;emidimetientis ad ambi<lb/>tum, &longs;eu &longs;emidimetientem orbis.</s>

<s>Propo&longs;itio quinquage&longs;imaquarta.</s>

<s>Proportio circuli ad &longs;uum diametrum per <expan abbr="&longs;imilitudin&etilde;">&longs;imilitudinem</expan> e&longs;t quar<lb/>ta pars peripheriæ. Rur&longs;usque eiu&longs;dem circuli ad peripheriam diame <lb/>tri quarta pars.</s>

<s>

<arrow.to.target n="marg144"></arrow.to.target></s>

<s>Quoniam enim &longs;uperficies circuli, ut ab <lb/>

<arrow.to.target n="fig34"></arrow.to.target><lb/>Archimede demon&longs;tratum e&longs;t, fit ex dimi</s>

<s>

<arrow.to.target n="marg145"></arrow.to.target><lb/>dio diametri in <expan abbr="dimidi&utilde;">dimidium</expan> peripheriæ erit, ut <lb/>eadem fiat ex tota peripheria in <expan abbr="quartã">quartam</expan> par <lb/>tem diametri, & ex tota diametro in quar<lb/>tam <expan abbr="part&etilde;">partem</expan> peripheri&ecedil;. 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;, & proportio are&ecedil; <lb/>ad <expan abbr="peripheriã">peripheriam</expan> e&longs;t quarta pars dimetientis, quod erat probandum.</s>

<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>

<s>Propo&longs;itio quinquage&longs;imaquinta.</s>

<s>Proportionem medicamentorum per ordines &longs;uppo&longs;ita æquali <lb/>proportione in ordinibus per quantitates, & proportiones de<lb/>mon&longs;trare.<lb/>

<arrow.to.target n="marg147"></arrow.to.target></s>

<s>Galenus libro quinto de Simplicibus medicamentis, quem &longs;e</s>

<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<lb/>ta qualitates calidi, frigidi, &longs;icci, & humidi, & primus e&longs;t cum <expan abbr="medi-cament&utilde;">medi<lb/>camentum</expan> non &longs;entitur quale &longs;it licet operetur, uelut cam&ecedil;melon, ab<lb/>&longs;ynthium, & oriza: &longs;ecundus e&longs;t, cum &longs;entitur, &longs;ed non lædit, ut nux <lb/>myri&longs;tica, &longs;aluia, ozimum: tertius e&longs;t cum &longs;entitur, & lædit, &longs;ed <lb/>non de&longs;truit, neque corrumpit corpus, uelut a&longs;&longs;arum apium &longs;ta<lb/>phi&longs;agria, cappares, myrrha, ruta: quartus e&longs;t, cum de&longs;truit ue<lb/>lut pyretrum, piper, euphorbium cæpe aggre&longs;te, & &longs;inapis, cina

<pb pagenum="45"/>momum autem, & gingiber numerantur inter medicinas calídas <lb/>tertij gradus, & hoc opus comparatur ad corpus &longs;icut dicit Gale<lb/>nus, & Serapio non ad linguam, ut medici no&longs;tri temporis interpre <lb/>tantur. Ex quo patet, quod aliqua medicina poterit e&longs;&longs;e quarti ordl <lb/>nis, & non lædere linguam in gu&longs;tu, & alia tertij ordinis, quæ non <lb/>&longs;olum lædet 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;æ mi&longs;tam cum &longs;iccitate <lb/>pari ip&longs;i calori. Sed non oportet h&ecedil;c nunc tractar, enon &longs;olum quia <lb/>non &longs;it locus, &longs;ed etiam quòd con&longs;u&longs;a &longs;it per &longs;eip&longs;a materia ab&longs;que <lb/>eo, quod difficultatem difficultati addamus, &longs;olum ergo eas dubita <lb/>tiones adiungemus, quas <expan abbr="uol&etilde;tes">uolentes</expan> declarare propo&longs;itionem præ&longs;en <lb/>tem, neque &longs;uperfugere, neque declinare po&longs;&longs;umus. Nam de &longs;icco, <lb/>& humido, cum &longs;int longè minoris actionis, quàm calidum, & fri<lb/>gidum, & præcipuè humidum, non uideo quomodo po&longs;sit Gale<lb/>nus &longs;tatuere medicinam humidam tertij gradus, nedum quarti, <lb/>cum non po&longs;sit inueniri medicina, quæ de&longs;truat corpus no&longs;trum <lb/>propter humidam qualitatem. Et licet Serapio po&longs;uerit gingiber <lb/>

<arrow.to.target n="marg149"></arrow.to.target><lb/>& enulam & zelim in tertio ordine calidorum & humidorum: & <lb/>inter frigidas, & humidas in tertio portulacam, aizoum, & uirgam <lb/>pa&longs;toris, & fungos. Primum non au&longs;us e&longs;t ponere medicinas ullas <lb/>calidas, aut frigidas in quarto ordine, qu&ecedil; &longs;int humidæ. &longs;ecundum, <lb/>quando dicit medicinas calídas, aut frigidas, atque humídas in ter<lb/>tio ordine, intelligit &longs;olum de qualitate actiua &longs;cilicet caliditate, uel <lb/>frigiditate, & non de humida qualitate, quod o&longs;tendit de gingibe<lb/>re, & enula, dicens, quod &longs;unt calidæ in tertio ordine, & humidæ <lb/>humido crudo, non au&longs;us addere ordinem, quia non uídit ratio<lb/>nem, qua po&longs;&longs;ent dici humidæ in tertio. Et clarius in capite de zei<lb/>len, quem &longs;tatuerat inter medicinas calidas, & humidas in tertio, di <lb/>cit quod e&longs;t calida in tertio, & humida in primo, ergo non intelligit <lb/>per medicinas calidas & humidas in tertio ordine, quod &longs;int humi<lb/>dæ in tertio ordine. Clarius etiam de frigidis & humidis, nam por<lb/>tula cam dicit e&longs;&longs;e frigidam in tertio, humidam in &longs;ecundo, & quod <lb/>maius, e&longs;t cum collo ca&longs;&longs;et aizoum inter medicinas frigidas, & hu<lb/>midas in tertio ordine, dicit, quod e&longs;t frigidum in tertio ordine, ad<lb/>ijcit, quod e&longs;t &longs;iccum parum, & de uirga pa&longs;toris nihil dicit de hu<lb/>mido, &longs;ed dicit, quod a&longs;tringit, ex quo concludo, quod &longs;ecun<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, & portulaca, quia frigiditas in generatione <lb/>humidum magis admittit, quàm caliditas, & calida magis hu

<pb pagenum="46"/>mectant, quia magis penetrat uis medicamenti, & hæc regula de <lb/>humido, & &longs;icco e&longs;t generalis apud Serapionem, quod non intelli<lb/>gitur ordo in pa&longs;siuis, ni&longs;i &longs;pecialiter exprimatur, nam de &longs;iccitate <lb/>non nego, quin inueniantur medicinæ &longs;iccæ in tertio, & for&longs;an in <lb/>quarto ordine, &longs;ed de hac Galeni o&longs;citantia, quæ 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>

<s>Secunda difficultas e&longs;t maior, & magis pertinet ad nos, & e&longs;t, <lb/>quòd non declarauit an i&longs;ti ordines inter &longs;e <expan abbr="aliquã">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 temperaturæ &longs;implicium <lb/>medicamentorum cogno &longs;camus temperaturam compo&longs;itorum ex <lb/>illis ratione ulla, &longs;ed oportebit &longs;olum experiri. Sed &longs;i ordines &longs;er<lb/>uant proportionem, adhuc relinquitur dubium, an illa proportio <lb/>&longs;it Arithmetica, uel Geometrica, uel Mu&longs;ica, & nihil mirum e&longs;&longs;et, <lb/>quod e&longs;&longs;et Mu&longs;ica, ut aliâs docuimus, ubitractauimus de differen<lb/>tia inter &longs;en&longs;um auditus, et ui&longs;us. Sed quia de hac nullus medicus ui <lb/>detur intellexi&longs;&longs;e, omittam hanc tractationem. Et quanquàm Gale<lb/>nus po&longs;sit uideri non exi&longs;tima&longs;&longs;e, quòd hi ordines non &longs;eruent <lb/>proportionem ullam, quia non au&longs;us e&longs;t tractare de temperamen<lb/>to medicamentorum compo&longs;itorum per rationem temperamen<lb/>ti &longs;implicium, nihilominus &longs;uppo&longs;ito quod ita e&longs;&longs;et, quod &longs;eruetur <lb/>altera proportionum, uolo o&longs;tendere rationem componendi in <lb/>utraque proportione & Arithmetica, & Geometrica. Ex quo &longs;e<lb/>quitur, quod Aueroes quàm o&longs;citanter tractauerit in quinto &longs;uo<lb/>rum collectaneorum de hoc, & non di&longs;tinguit, neque docet pri<lb/>mum an &longs;it aliqua proportio, deinde &longs;i qua &longs;it, cuius generis &longs;it, & <lb/>cum in re tam clara pugnet pror&longs;us, ut c&oelig;cus ictus maximos eden<lb/>do, &longs;ed in ca&longs;&longs;um plero&longs;que, quàm malè agant qui ei in arduis tan<lb/>tum tribuunt fidei, & authoritatis, &longs;ed hæc e&longs;t infelicitas no&longs;tra, & <lb/>ira Deorum. Suppo&longs;ito ergo quod primò ordines di&longs;tinguantur <lb/>per proportionem arithmeticam, &longs;it &longs;uperficies a b pro quantitate, <lb/>

<arrow.to.target n="fig35"></arrow.to.target><lb/>& a &longs;it calida in primo gradu, & b in ter<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<lb/>dinis trium, ba&longs;i autem quadrilatera &longs;u<lb/>perficie rectangula b, hoc igitur erit to<lb/>tum mi&longs;tum, & quia quantitas medicamenti non mutatur quæ e&longs;t <lb/>a, b, ergo talia corpora æquantur uni corpori, cuius ba&longs;is e&longs;t a b, <lb/>cum ergo talia corpora producantur ex a in unum, & b in tria, ergo

<pb pagenum="47"/>diui&longs;o aggregato per a b prodibit altitudo, &longs;eu ordo qualitatis to<lb/>tius medicamenti, iuxta quod con&longs;tituitur regula prima libri artis <lb/>medendi paruæ huiu&longs;modi, & reliquæ, traduxi autem illas ad hunc <lb/>locuin, “quia pendent ex demon&longs;tratione hac: “duc numerum ordi<lb/>nis &longs;ingulorum medicamentorum in numerum quantitatis, &longs;imilia <lb/>iunge, di&longs;similia detrahe, quod fit, diuide per aggregatum, quanti<lb/>tatum, exibit numerus ordinis compo&longs;iti. Sic mi&longs;cendo calidum in <lb/>&longs;ecundo ordine cum duplo pondere temperati conflabit calidum <lb/>in be&longs;&longs;e. Secunda &longs;i ex pluribus diuer&longs;arum, qualitatum, & ordi<lb/>num temperatum efficere uelis, duc quæ &longs;unt eiu&longs;dem qualitatis in <lb/>&longs;uas quantitates, & iunge, quod fit, diuide per numerum or dinis <lb/>medicamenti contrarij, exibit quantitas illius, &longs;ub qua &longs;i iungatur, <lb/>fiet medicamentum temperatum. Tertia cum nolueris ex tempera<lb/>to, & alio cuiu&longs;cunque ordinis medicamen conficere ordinis re<lb/>mi&longs;sionis, detrahe numerum ordinis eius, quod conficere uis ex nu <lb/>mero ordinis eius, quod habes, & cum re&longs;iduo diuide numerum <lb/>medicaminis, quod conficere uis, quod exit e&longs;t numerus quantita<lb/>tis medicamenti non temperati in comparatione ad temperatum.” <lb/>Ex his potes propo&longs;itis quibu&longs;cunque medicamentis conficere <lb/>antidotum &longs;ub quo cunque ordine remi&longs;siore potenti&longs;simo ex il<lb/>lis. Quarta in compo&longs;itione, quæ non fermente&longs;cit calida, calidis <lb/>iuncta &longs;emper opus augent, ut mel cum pipere. Quæ 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>

<s>Quod &longs;i &longs;tatuamus proportionem e&longs;&longs;e Geometricam, modus <lb/>erit idem in omnibus, & quo ad numerum etiam in primo, & &longs;ecun <lb/>do ordine, quia in proportione dupla Geometrica &longs;ecundus ordo <lb/>tantundem di&longs;tat à primo, quantum primus ab æqualitate, quia <lb/>unum & duo &longs;eruant proportionem, & æqualem di&longs;tantiam, &longs;ed in <lb/>cæteris ordinibus non ita erit, quia qui e&longs;&longs;et trium in Arithmetica, <lb/>&longs;cilicet totius ordo e&longs;t, quatuor in Geometrica, & quartus ordo, <lb/>qui e&longs;&longs;et quatuor in Arithmetica, e&longs;&longs;et octo in Geometrica, ideo <lb/>

<arrow.to.target n="fig36"></arrow.to.target><lb/>&longs;cribemus ordines hoc modo, & operabimur cum <lb/>numeris loco ordinum, exemplum ergo primum <lb/>&longs;it medicina calida in tertio ordine quatuor uncia<lb/>rum, & 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<lb/>decim, quia omnis medicina tantum retondit de contrario, &longs;eu mi<lb/>nuit relin quuntur octo &longs;cilicet caliditatis, diuido per &longs;ex ag

<pb pagenum="48"/>gregatum unciarum exit unum, & tertia, ergo erit calida in princi<lb/>pio &longs;ecundi ordinis. Secundum exemplum &longs;int eædem medicinæ, <lb/>& &longs;it proportio Geometrica, ducemus ergo quatuor in quatuor, & <lb/>fiunt &longs;exdecim, & duo in duo fiunt quatuor, detrahe quatuor ex &longs;ex <lb/>decim, & 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. rur&longs;us <lb/>&longs;int ambæ medicinæ calidæ, & ducemus, ut prius in tertio exem<lb/>plo, ubi proportio &longs;it Arithmetica iungendo duodecim cum qua<lb/>tuor, & fient &longs;exdecim, diuide per &longs;ex, exeunt duo, & duæ tertiæ, er<lb/>go erit calida in medio tertij gradus, rur&longs;us in quarto exemplo iun <lb/>gemus &longs;edecim cum quatuor, & fient uiginti, diuide per &longs;ex exi<lb/>bunt tria & tertia, & ita erit in medio tertij gradus, ut prius, &longs;ed &longs;i <lb/>ille quatuor unciæ e&longs;&longs;ent calidæ in quarto gradu, & illæ duæ unciæ <lb/>in &longs;ecundo gradu, ut prius ducendo quatuor in quatuor fiunt &longs;ex<lb/>decim, & duo in duo fiunt quatuor, iunge, & fient uiginti, diuide <lb/>per &longs;ex exeunt tria cum tertia, ergo erit calida in principio quarti <lb/>gradus &longs;ecundum proportionem Arithmeticam, &longs;ed &longs;ecundum <lb/>Geometricam duc quatuor in octo, fiunt triginta duo, adde qua<lb/>tuor ut prius, &longs;cilicet productum duorum in duo fiunt triginta &longs;ex, <lb/>diuide per &longs;ex, exeunt &longs;ex, & quia &longs;ex ad quatuor maiorem habent <lb/>proportionem, quàm octo ad &longs;ex ideo hæc medicina erit calida ul<lb/>tra medium quarti gradus, iam ergo uides rationem, & differen<lb/>tiam horum.</s>

<s>Quod &longs;i quis dicat, an debeat attendi Geometrica proportio in <lb/>medicamentis, an Arithmetica, re&longs;pondeo, quòd ueri&longs;imilius e&longs;t de <lb/>Arithmetica, quia illa proportio etiam quod &longs;it minor quatuor ad <lb/>trium, quàm trium ad duo, & multò minor quàm duo ad unum ni<lb/>hilominus longè plus operatur, quia tertius ordo iam incipit e&longs;&longs;e <lb/>præter naturam, & uidemus, quod læ&longs;io facta in uulnerato, etiam <lb/>quòd &longs;it quadruplo minor, plus nocet longè, quàm in &longs;ano qua<lb/>druplo maior: quia termini præter naturam &longs;unt ualdè angu&longs;ti in <lb/>comparatione ad latitudinem naturalem, &longs;icut etiam uidemus in<lb/>tendendis chordis &longs;corpionum, quod ultima pars e&longs;t breuis & ta<lb/>men homini tantam difficultatem adijcit. Notandum e&longs;t etiam, <lb/>quòd ob hoc diui&longs;erunt ordines in tres partes, uelut gingiber e&longs;t <lb/>calidum in fine tertij ordinis, origanum in medio, cinamomum in <lb/>principio, & ita euphorbium e&longs;t calidum in principio quarti gra<lb/>dus, &longs;ed in fine principij piper, in prin cipio principij aqua &longs;epara<lb/>tionis in medio quarti ordinis, &longs;ed oleum chalcanthi factum ea ar<lb/>te, ut exurat paleas, &longs;icut ignis e&longs;t calidum in fine quarti ordinis, & <lb/>ita &longs;ufficiet diuidere propter eandem cau&longs;am primum, & &longs;ecun

<pb pagenum="49"/>dum ordinem in duas tantum partes non ratione latitudinis, quæ <lb/>e&longs;t æqualis, uel etiam for&longs;an maior, &longs;ed ratione uarietatis operatio<lb/>nis quæ minus &longs;entitur, & maximè in primo ordine.</s>

<s>Propo&longs;itio quinquage&longs;ima&longs;exta.</s>

<s>Proportio cuiu&longs;uis binomij ad &longs;uum reci&longs;um, uel ei commen<lb/>&longs;um e&longs;t duplicata ei, quæ ad numeri latus.<lb/>

<arrow.to.target n="marg150"></arrow.to.target></s>

<s>Cum enim proportionis medium &longs;itlatus numeri eo quod ex bi <lb/>nomio in reci&longs;um &longs;uum fit numerus ex his, quæ demon&longs;trata &longs;unt <lb/>generaliter in tertio Arithmeticæ de omnibus binomijs cum &longs;uis </s>

<s>

<arrow.to.target n="marg151"></arrow.to.target><lb/>reci&longs;is, uel in quadratis lateribus erit <02> numeri media proportione <lb/>inter binomium, & &longs;uum reci&longs;um, igitur cum proportio producto<lb/>rum ex binomio in commen&longs;a reci&longs;o &longs;it, ut commen&longs;orum ad reci<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 <02> nu <lb/>

<arrow.to.target n="marg153"></arrow.to.target><lb/>meri, igitur proportio binomij ad reci&longs;um &longs;uum, & omnia com<lb/>men&longs;a illi, e&longs;t duplicata ei quæ ad <02> numeri.<lb/>

<arrow.to.target n="marg154"></arrow.to.target></s>

<s><margin.target id="marg151"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 6. P<emph type="italics"/>ro<lb/>po&longs;. lib. de<emph.end type="italics"/><lb/>A<emph type="italics"/>liza.<emph.end type="italics"/></s>

<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>

<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>

<s><margin.target id="marg154"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 6. <emph type="italics"/>deci<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement:<emph.end type="italics"/></s>

<s>Propo&longs;itio quinquage&longs;ima&longs;eptima.</s>

<s>Motus rationem ad pondus inuenire.</s>

<s>

<arrow.to.target n="marg155"></arrow.to.target></s>

<s>O&longs;ten&longs;um e&longs;t antea, quod motus naturalis uelocior fit in fine, ac <lb/>magis augetur ob aëris motum, ubi uerò hæret e&longs;t ac &longs;i quie&longs;cat. <lb/>Eadem autem e&longs;t ratio in motis uiolenter, & naturaliter dum &ecedil;qua<lb/>li impetu feruntur. Sed &longs;ubitò po&longs;t etiam, quod motus æqualiter <lb/>augerentur minus tamen cre&longs;cit proportio uiolenti &longs;cilicet ob im<lb/>

<arrow.to.target n="fig37"></arrow.to.target><lb/>pedimentum naturale. Sed &longs;i uis mouens fuerit <lb/>adeò ualida ut proportio incrementi ex aëre &longs;it <lb/>maior, quàm impedimentum, & in crementum al <lb/>terius mobilis naturaliter moti, motus ille uelo<lb/>cior fiet naturali, ut in &longs;phæris ferreis ex machina <lb/>igne excu&longs;sis, quod ergo attinet ad præ&longs;entem <lb/>motum ratio e&longs;t eadem. Quicun que ergo motus <lb/>minoris grauis cogit de&longs;cendere lancem ex ad<lb/>uer&longs;o proportionem habet eandem ad &longs;uum mo <lb/>bile quam habet graue æquiponderans. Sit ergo <lb/>ut a ex b, c, d, e, eleuet eodem ordine pondera e, f, <lb/>g, h, erit ergo ponderum h, g, f, e, ad &longs;e inuicem, & ad a qualis mo<lb/>tuum ob di&longs;tantiam intentorum. Experimentum ergo docet, quòd <lb/>dimidium ponderis æquilibrium facit ex palmo minoris dimidio <lb/>motum manife&longs;tum, & ex palmo quarta pars ponderis, ergo &longs;e ha<lb/>bent prope portionem.</s>

<s>Propo&longs;itio quinquage&longs;imaoctaua.</s>

<s>Qu&ecedil; ex alto de&longs;cendunt cur non eandem pro di&longs;tantia motus ra <lb/>tionem in libero aëre &longs;eruent con&longs;iderare.</s>

<s>Aër in &longs;ublimiore eius regione &longs;emper naturali motu fertur ex <lb/>Oriente in Occidentem, &longs;ed & infra uerum minus manife&longs;tè. At ca<lb/>&longs;u plerun que contingit, ut moueatur longè uehementius, &longs;eu ad ean<lb/>dem partem, &longs;eu aliam. Qui uerò naturalis e&longs;t, debilis <lb/>

<arrow.to.target n="fig38"></arrow.to.target><lb/>e&longs;t, quoniam in tenui ualde &longs;ub&longs;tantia e&longs;t: nec <expan abbr="cõtinuus">continuus</expan> <lb/>&longs;ed in&longs;tar motus aquæ maris fluit ac refluit: aliter ne<lb/>ce&longs;&longs;e e&longs;&longs;et, ut &longs;ingulis horis per mille milliaria procede<lb/>ret, ut &longs;ic ne que latere po&longs;&longs;et, quarndoquidem fortuiti mo <lb/>tus, qui &longs;unt multo tardiores non latentnos. Nam tardiores illos <lb/>e&longs;&longs;e <expan abbr="cõ&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&eumle;ris <lb/>e&longs;&longs;et continuus, in hora aër procederet ob ambitum terræ 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, imò 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 adeò uelox & in ea<lb/>dem parte <expan abbr="a&etilde;ris">aerris</expan> e&longs;&longs;et at que continuus. Præterea tantus impetus nun<lb/>quam à minore motu, aut cau&longs;a &longs;uperaretur, adeò ut &longs;emper flatum <lb/>aëris orientalem &longs;entiremus. Quotidie etiam aduenire ad nos aë<lb/>rem ex Illyrico, Macedonia, My&longs;ia, Ponto, Bythínia, Capado cia, Sy <lb/>ria, Babylonia, Hyrcanomarí, Bactrianis, Sacís, Scythis, ac Seris, to<lb/>to præterea Oceano orientali tam ua&longs;to, & Gallica noua, terra que flo <lb/>rida non &longs;olum res e&longs;t admirabilis', & incredibilis, &longs;ed etiam aliena <lb/>à &longs;en&longs;u, & ab his, quæ eueniunt. A'&longs;en&longs;u quidem, quoniam nebul&ecedil;, <lb/>quæ in aëre mouentur, primùm non in eandem partem &longs;emper mo <lb/>uentur: nun quam autem adeò celeriter: at &longs;i aër &longs;ic circumuoluere<lb/>tur, mouerentur & illa, qu&ecedil; in eo continentur, quotidieque aërem ex<lb/>periremur & nubilo&longs;um, & madidum propter mare. Nechis, quæ <lb/>eueniunt hoc &longs;atis re&longs;pondet, nec nobis id contingeret, ut &longs;i pe&longs;ti<lb/>aliqua in regione no&longs;tra directa &longs;æuiret, ut aër &longs;ingulis diebus la<lb/>be ea infectus ad nos deferretur. Moueri uerò aërem &longs;emper mani<lb/>fe&longs;ti&longs;simum e&longs;t tum experimento, tum ratione: ratione &longs;iquidem, <lb/>quod aqua & c&oelig;lum naturaliter perpetuò mouentur, quare etiam <lb/>aër. Experimento, quòd ubi hiant o&longs;tia, & ianuæ, ibi perpetuus &longs;en<lb/>titur flatus. Ergo &longs;i a pondus de&longs;cendat in c, ex alto fertur rectà, &longs;ed <lb/>&longs;i ex &longs;ublimi transferetur in b, & indirecta, & ad latus, unde ex <lb/>hoc &longs;equitur.</s>

<s>Propo&longs;itio quin quage&longs;imanona.</s>

<s>

<arrow.to.target n="marg156"></arrow.to.target></s>

<s>Omne mobile motum duobus motibus non ad idem tendenti<lb/>bus, utro que &longs;eor&longs;um tardius mouetur &longs;imili motu.</s>

<s>Sit a mobile, quod moueatur per a b c impul&longs;u uenti aut uiolen</s>

<s>

<arrow.to.target n="marg157"></arrow.to.target><lb/>

<arrow.to.target n="fig39"></arrow.to.target><lb/>to cum naturali coniuncto: & &longs;it terminus naturalis e, <lb/>

<arrow.to.target n="marg158"></arrow.to.target><lb/>& uiolenti d: uter que in directo c, dico, quod tardius per<lb/>ueniet ad c quam d, uel e. De e manife&longs;tum e&longs;t, quoniam <lb/>motus aëris, qui intendit motum a, diuíditur in partem, <lb/>quæ iuuat motum ad d, & partem, quæ mouetur ad e, <lb/>igitur fit minor adiectio. Et etiam quia a c e&longs;t longior <lb/>a e ex diffinitione rectæ: quare tardius perueniet ad c quàm ad e du <lb/>plici ratione. Dico etiam, quod tardius ad c quàm d. Quia enim <lb/>uis, quæ fert ad d repugnat ei, quæ fert ad e, & uis, quæ fert ad e, re<lb/>pugnat ei quæ fert ad d, igitur tardius perueniet ad c, quàm d. Nec <lb/>potes dicere, quòd uis, quæ fert ad c adiuuet ad motum è regione <lb/>d, nam cum unus motus non po&longs;sit perfici &longs;ine altero, igitur quan<lb/>tum motus ad eretar dabit motum ad d, tanto motus a c erit tardí<lb/>or ab&longs;olutè motu ad d. 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<lb/>rum ob cau&longs;am dictam. Vtrùm uerò motus ad c ab&longs;olutè &longs;it tardi<lb/>or, quàm ad d, non &longs;uppo&longs;ito, quod c e &longs;it æqualis a d, &longs;ed minor, <lb/>nunc non e&longs;t locus determinandi.</s>

<s>Ex hoc patet, quod motus æquidi&longs;tantis mobilis, finis e&longs;t mini<lb/>

<arrow.to.target n="marg159"></arrow.to.target><lb/>mus omnium: quoniam mobile qua&longs;i quie&longs;cit in illo. 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/>

<arrow.to.target n="fig40"></arrow.to.target><lb/>e&longs;t motus actu: uiolentus autem æqualis 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. Et etiam quòd e&longs;&longs;et minor, quia ut di<lb/>ctum e&longs;t, uter que &longs;imul iunctus e&longs;t æqualis uni eorum non impedito <lb/>uel minor.</s>

<s>Propo&longs;itio &longs;exage&longs;ima.</s>

<s>Omne mobile motu naturali de&longs;cendens parte, de&longs;cendit gra<lb/>uiore &longs;ecundum grauitatis centrum.</s>

<s>Sit a mobile, grauitatis centrum b, cuius pars ei pro<lb/>

<arrow.to.target n="marg161"></arrow.to.target><lb/>

<arrow.to.target n="fig41"></arrow.to.target><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<lb/>tura partis, & totius: totius autem terræ 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. Sed pars pro<lb/>ximior nece&longs;&longs;ariò e&longs;t grauior, quia centrum e&longs;t in medio grauita

<pb pagenum="52"/>tis, ergo omne mobile de&longs;cendit motu naturali per &longs;ui grauio<lb/>rem partem.<lb/>

<arrow.to.target n="marg163"></arrow.to.target></s>

<s>Ex hoc &longs;equitur, quòd graue habens partes inæquales, &longs;eu &longs;ub<lb/>&longs;tantia, &longs;cu forma, &longs;i ita excutiatur, ut pars grauior <expan abbr="nõ">non</expan> &longs;it, infrà opor<lb/>tet, ut circumuoluatur.</s>

<s>Propo&longs;itio &longs;exage&longs;imaprima.</s>

<s>Proportionem ictus ad pondus rei, & di&longs;tantiam generaliter <lb/>con&longs;iderare.</s>

<s>

<arrow.to.target n="marg164"></arrow.to.target></s>

<s>Dictum e&longs;t &longs;uperius de proportione de&longs;cenfus ad grauitatem: </s>

<s>

<arrow.to.target n="marg165"></arrow.to.target><lb/>& quòd &longs;i graue de&longs;cendat ex alto impeditur à motu aëris: & quòd <lb/>

<arrow.to.target n="marg166"></arrow.to.target><lb/>res, quæ mouetur duobus motibus non ad idem tendentibus tar<lb/>

<arrow.to.target n="marg167"></arrow.to.target><lb/>dius mouetur, quam motus &longs;it unu&longs;qui&longs;que. Demùm quòd 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: & an<lb/>tea ubi egimus de proportione motus ad grauitatem, quod h&ecedil;cin<lb/>telligenda &longs;unt prout po&longs;&longs;unt intelligi de motu etiam uiolento. <lb/>Cum ergo uideamus duo hæc, quodres 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/>& uulnerat, & non eleuat in æquilibrio, ut potè ex alto cadens loco <lb/>per &longs;patium octo palmorum pondus &longs;exdecim librarum, & a pon<lb/>dere &longs;exdecim librarum homo non læditur, nec uulneratur, ergo id <lb/>accidit ex alia cau&longs;a, & e&longs;t, quod aër interceptus inter graue, & cor<lb/>pus no&longs;trum non pote&longs;t dilabi tam citò, ergo ne corpus penetret, <lb/>cogitur ingredi locum, cui e&longs;t obuius, at que ita concutere, & diuide<lb/>re. Ex quibus &longs;equuntur omnia hæc.<lb/>

<arrow.to.target n="marg169"></arrow.to.target></s>

<s><margin.target id="marg165"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 57.</s>

<s><margin.target id="marg166"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 58.</s>

<s><margin.target id="marg167"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 59.</s>

<s><margin.target id="marg168"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 60.</s>

<s>Primùm &longs;i quod incidit, molle fuerit, non uulneratur caput, uel <lb/>pars &longs;ubiecta, quia re&longs;ilit in corpus molle: nec à molli, quia retundi<lb/>tur, pote&longs;t uulnerari: ergo nullo modo. Sed neque adeò concutit, <lb/>quia aër rediens, & receptus in molli corpore pro parte, non uer<lb/>berat locum.</s>

<s>

<arrow.to.target n="marg170"></arrow.to.target></s>

<s>Secundum in omni colli&longs;ione &longs;eu duri, &longs;eu mollis, &longs;ed magis du<lb/>ri, dilabuntur partes aëris ad latera, ideo quod partes mediæ pre<lb/>muntur. Et quanto motus e&longs;t tardior.</s>

<s>

<arrow.to.target n="marg171"></arrow.to.target></s>

<s>Tertium in motu uelo ci fit maior ictus & læ&longs;io, & maiora omnia <lb/>quam proproportione motus: quoniam ob uelo <expan abbr="citat&etilde;">citatem</expan> minus diffu <lb/>git aëris. Et ideò fiunt grauia uulnera ex modico incremento uelo<lb/>citatis motus.</s>

<s>

<arrow.to.target n="marg172"></arrow.to.target></s>

<s>Quartum res latæ, duræ concutiunt, & non uulnerant ni&longs;i &longs;int <lb/>cum magno impetu, aut ualde graues: acutæ autem uulnerant, &longs;ed <lb/>non concutiunt, ni&longs;i parti acutæ lata &longs;uccedat.</s>

<s>Quintum, corpora dura magis læduntur à latis, quia &longs;cindun</s>

<s>

<arrow.to.target n="marg173"></arrow.to.target><lb/>tur, mollia autem à tenuibus, quia diuiduntur: nam mollitie excipi<lb/>unt aërem, & ita à latis non adeò patiuntur, & etiam, quoniam nec <lb/>franguntur, nec &longs;ponte &longs;cinduntur.</s>

<s>Sextum, etiam in duris penetrat aliquid aëris, aliter tota frange<lb/>

<arrow.to.target n="marg174"></arrow.to.target><lb/>rentur. Con&longs;tat etiam omnem lapidem marmoreum, aut &longs;iliceum <lb/>e&longs;&longs;e poro&longs;um, ut dicunt. Et etiam quia recipitur in mollioribus, er<lb/>go etiam in durioribus & in duri&longs;simis: quod &longs;i non recipiant ut ui <lb/>trum, & gemmæ tota franguntur. Hoc etiam uidetur &longs;en&longs;i&longs;&longs;e Philo <lb/>&longs;ophus, qui uult, quòd res franguntur ob poros.</s>

<s>Propo&longs;itio &longs;exage&longs;ima&longs;ecunda.</s>

<s>Proportionem motoris in plano ad motorem, qui eleuat pon<lb/>dus iuxta id, quod mouet inuenire.</s>

<s>Con&longs;titutum e&longs;t inuenire proportionem uirium, quæ eleuant <lb/>

<arrow.to.target n="marg175"></arrow.to.target><lb/>pondus ad uires, quæ ip&longs;um in plano leui trahere po&longs;<lb/>

<arrow.to.target n="fig42"></arrow.to.target><lb/>&longs;unt. Vires enim, quæ eleuant pondus a &longs;unt eædem <lb/>puta b, quæ uero trahunt c, &longs;ed hæ 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/>& maiores expo&longs;cit uires: hoc enim experimento deprehenditur. <lb/>Duæ uerò po&longs;tremæ cau&longs;æ etiam per &longs;e per&longs;picuæ &longs;unt, nec demon <lb/>&longs;tratione indigent: ni&longs;i quod &longs;i planum &longs;it duri&longs;simum, ac leui&longs;si<lb/>mum, quod e&longs;t a&longs;perum facilius trahitur, quia minore &longs;ui parte pla<lb/>num tangit. Nos præterea &longs;upponimus planum æquale undique <lb/>leue durum, & corpus undique &longs;ibi &longs;imile, id e&longs;t cubi formam refe<lb/>rens, & uinculum in imo: Demon&longs;trare igitur expedit primum, <lb/>quòd in hoc ca&longs;u b e&longs;t duplum ad c. Quia enim cum a eleuatur b ui <lb/>res &longs;uperant motum ob&longs;curum &longs;eu occultum, &longs;eu pondus a, & &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<lb/>tur circa medium, nam plana &longs;uperficies parum differt à rotunda <lb/>terræ ob terræ magnitudinem, media erit repugnantia: in eo enim <lb/>quod mouetur, grauitatem habet d in eo, quod <expan abbr="nõ">non</expan> remouetur nul<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>

<s>Propo&longs;itio &longs;exage&longs;imatertia.</s>

<s>Omne graue quanto proximius alligatum plano, tanto faci<lb/>lius trahitur.

<arrow.to.target n="marg176"></arrow.to.target></s>

<s>Sit graue a b c alligatum funibus in d ef, dico, <lb/>

<arrow.to.target n="fig43"></arrow.to.target><lb/>quòd facilius trahetur per fe quàm c b & e b, quàm <lb/>d a, quia &longs;i debet trahi ex a uel b, aut cadet, aut uis ex <lb/>a & b communicabitur c, igitur erit minor quàm in <lb/>c, & hoc naturaliter. Mathematica autem ratione quoniam ex a tra<lb/>hetur c, qua&longs;i per lineam d c: at attractio recta e&longs;t ualidior obliqua<lb/>igitur attractio c per d e&longs;t debilior, quàm per f. Rur&longs;us &longs;i e trahitur <lb/>per d cùm a peruenerit in d, erit perinde ac, &longs;i attractum e&longs;&longs;et per li<lb/>neam c d, &longs;ed linea c d mouet duobus motibus, uno ad &longs;uperiora, al </s>

<s>

<arrow.to.target n="marg177"></arrow.to.target><lb/>tero ad latus, ergo lentius ad f per d c quàm f c, quod erat demon<lb/>&longs;trandum.</s>

<s>Propo&longs;itio &longs;exage&longs;imaquarta.</s>

<s>Omne mobile quanto latius tanto tardius mouetur in plano.<lb/>

<arrow.to.target n="marg178"></arrow.to.target></s>

<s>Demon&longs;tratum e&longs;t &longs;uperius quòd &longs;i mobile &longs;it &longs;ph&ecedil;ricum, & tan </s>

<s>

<arrow.to.target n="marg179"></arrow.to.target><lb/>gat planum in puncto, quòd mouetur per quancunque uim aptam <lb/>diuidere medium. Quia ergo &longs;i tangat in puncto facillime moue<lb/>tur, &longs;i in linea paulò difficilius, &longs;i per &longs;uperficiem adhuc difficilius, <lb/>igitur cum fiat attritio in motu quanto latius e&longs;t mobile eo diffici<lb/>lius mouetur. Sit ergo mobile a b, quod moueatur uer&longs;us c, & quia <lb/>pars b &longs;eu dimidium mouetur iuxta rationem me<lb/>

<arrow.to.target n="fig44"></arrow.to.target><lb/>dietatis, & pars a eodem modo, ergo conduplicata <lb/>difficultate, quia medietas b impedit medietatem, a <lb/>quanto latius e&longs;t, & longius a b, tanto difficilius <lb/>

<arrow.to.target n="marg180"></arrow.to.target><lb/>mouetur. Et hoc intelligitur de corporibus ualde <lb/>latis propter dicta &longs;uperius.</s>

<s><margin.target id="marg179"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 40.</s>

<s><margin.target id="marg180"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 62</s>

<s>Propo&longs;itio &longs;exage&longs;imaquinta.</s>

<s>Proportionem duorum mobilium inter &longs;e cum auxilio medij <lb/>inuenire.<lb/>

<arrow.to.target n="marg181"></arrow.to.target></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. Secundò ob <lb/>paruam medij repugnantiam, ideo quanto medium e&longs;t rarius & <lb/>mobile tenuius, tanto celerius de&longs;cendit: contrà uerò tardius. Ter<lb/>tiò ob impetum aëris &longs;ub &longs;equentis: & ideo mobile quòd ex eadem </s>

<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 aër <lb/>cogatur celerius ferri: & quanto diutius de&longs;cendit, tanto magis in<lb/>tenditur motus, at que augetur, ut &longs;uprà de claratum e&longs;t. Quarta cau&longs;a <lb/>e&longs;t, quod non impediatur ab aëre tran&longs;uerfim moto, et à latere: ideo <lb/>leuia mobilia & magna non &longs;olum lentius de&longs;cendunt, quoniam <lb/>

<arrow.to.target n="marg183"></arrow.to.target><lb/>paruam uim habeant, & 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. Por

<pb pagenum="55"/>rò proportio ratione de&longs;cen&longs;us aucta, declarata e&longs;t paulo antè, <lb/>quare cum medium &longs;upponatur eiu&longs;dem generis, & 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<lb/>

<arrow.to.target n="marg185"></arrow.to.target><lb/>nem ponderum, & iam habuimus proportionem incrementi ex <lb/>

<arrow.to.target n="marg186"></arrow.to.target><lb/>motu aëris, ergo proportio unius motus producti ad alteram no<lb/>ta erit.</s>

<s><margin.target id="marg182"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 30.</s>

<s><margin.target id="marg183"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 59.</s>

<s><margin.target id="marg184"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 62.</s>

<s>Propo&longs;itio &longs;exage&longs;ima&longs;exta.</s>

<s>Proportionem laterum eptagoni, & &longs;ubten&longs;arum con&longs;iderare, <lb/>& quæ à reflexa proportione pendent.<lb/>

<arrow.to.target n="marg187"></arrow.to.target></s>

<s>Sit eptagonus a b d f g e c, & &longs;ubten&longs;æ b <lb/>

<arrow.to.target n="fig45"></arrow.to.target><lb/>c, & f e duobus lateribus, tribus autem d c <lb/>d e, & erunt (quia intelligitur eptagono æ<lb/>quilatero, & æquiangulo) b c & e finuicem <lb/>æquales: & item d c, & d e æquales: & &longs;i du<lb/>cerentur b e & c f inuicem æquales: & ad a c <lb/>& d g: quare cum angulus cb d con&longs;i&longs;tatin </s>

<s>

<arrow.to.target n="marg188"></arrow.to.target><lb/>arcu c e g f d, & angulus b d c in arcu b a c, <lb/>& angulus b c d in arcu b d; & &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<lb/>goni, & arcus b a c duo, & ita arcus etiam b a c duplus arcui b d <lb/>erit angulus d b e duplus angulo c d b, & angulus c d b duplus an<lb/>

<arrow.to.target n="marg189"></arrow.to.target><lb/>gulo b c d, quare per demon&longs;trata à nobis proportio laterum b d, <lb/>b c, c d, e&longs;t reflexa, igitur proportio d b & b c, ad d c, ut d e ad b c, & <lb/>

<arrow.to.target n="marg190"></arrow.to.target><lb/>rur&longs;us proportio b d & d e ad b e, ut b e ad b d. Quare &longs;uppo&longs;ita <lb/>d b 1, b c 1 po&longs;itione, erit d c latus 1 quad. p: 1 po&longs;itione. Proportio <lb/>

<arrow.to.target n="marg191"></arrow.to.target><lb/>uerò, ut dictum e&longs;t b d & d c ad b c, id e&longs;t p: <02> 1 quad. 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: <02> v: 1 quad. p: 1 pos <lb/>æquatur quadrato b c, quod e&longs;t 1 quad. igitur 1 quad. m: 1 æquatur <lb/><02> v: 1 quad. p: 1 pos quare 1 quad. quad. m: 2, quad. p: 1 æquatur 1 <lb/>quad. p: 1 pos. Additis igitur communiter quatuor quadratis fient <lb/>1 quad. quad. p: 2 quad. p: 1 æqualia 5 quad. p: 1 pos. Et reducitur ad <lb/>1 cu. æqualem 1 3/4 pos p: 7/8.</s>

<s><margin.target id="marg188"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 28. & 29. <emph type="italics"/>tertij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s><margin.target id="marg189"></margin.target>P<emph type="italics"/>er ult. &longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s>Aliter &longs;tante &longs;uppo&longs;itione ut Ludouicus Ferrarius ex demon<lb/>&longs;tratis à Ptolemæo quadratum b c, & e&longs;t 1 quad e&longs;t æquale produ<lb/>cto ex b d in c e, quod e&longs;t 1, & a b in d c, igitur detracto 1, produ<lb/>cto b d in c e ex 1 quad. quadrato c b, relinquitur productum ex <lb/>a b in c d 1 quad. m: 1, ergo diui&longs;o co per a b, quæ e&longs;t 1, relinquitur <lb/>c d 1 quad. m: 1 huius uerò quadratum per <expan abbr="ead&etilde;">eadem</expan> demon&longs;trata à Pto

<pb pagenum="56"/>lemæo, &ecedil;quale e&longs;t rectangulis ex b c in de, & b d in c e, igitur 1 quad. <lb/>quad. m: 2 quad. p: 1 e&longs;t æquale 1 producto b d in c e, & producto b <lb/>cin d e detracto 1 communi, relin quetur productum ex b c in d e 1 <lb/>quad. quad. m: 2 quad. igitur diui&longs;o 1 quad. quad. m: 2 quad. per 1 <lb/>pos, exit 1 cu. m: 2 pos æqualia d e, & d e e&longs;t æqualis d c, ut ab initio <lb/>demon&longs;trauimus, & d c fuit 1 quad. m: 1, igitur 1 cu. m: 2 æquantur 1 <lb/>quad. m: 1, igitur 1 cu. p: 1 æquantur 1 quad. p: 2 pos.</s>

<s>Aliter ut Pacciolus, concurrant latera eptagoni b d, c e in a, & du <lb/>cantur perpendiculares a f, d g & c d, & &longs;it c e i ca 1 pos, & 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) & <lb/>quia d h e&longs;t dimidium d e, erit d h, & g f <lb/>

<arrow.to.target n="fig46"></arrow.to.target><lb/>1/2, cum ergo b f &longs;it (1/2 pos p: 1/2)/pos erit ergo di<lb/>ui&longs;a 1/2 pos per 1 pos, & exit 1/2, b f 1/2p: 1/2/pos <lb/>igitur detracta g f relinquetur g b 1/2/(1 pos). <lb/>& eius quadratum 1/4/(1 quad). igitur cum qua<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, quæ <lb/>e&longs;t 1/2p: 1/2/(1 pos) & f g quæ e&longs;t 1/2, erit igitur c <lb/>g 1 p: 1/2/(1 pos), & <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 & g d erit 2 p: 1/pos c a uerò e&longs;t æqua<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, & angulus b c e ei duplus, quare cum c f a &longs;it re<lb/>ctus erit ex trige&longs;ima&longs;ecunda primi Elementorum f a c tres &longs;epti<lb/>mæ unius recti, ergo d a c 6/7 unius recti, d c a uerò 2/7 unius recti, quia <lb/>

<arrow.to.target n="marg194"></arrow.to.target><lb/>e&longs;t &longs;eptima pars duorum rectorum, ígitur a d c e&longs;t 6/7 unius recti: igi<lb/>tur c d e&longs;t æqualis c a, ergo quadratum quadrato: igitur 1 quad. p: 2 <lb/>pos p: 1, æquatur 2 p: 1/(1 pos) igitur 1 quad. p: 2 pos, æquantur 1 p: 1/(1 pos). <lb/>Quare 1 cub. p: 2 quad. æquatur 1 pos p: 1. <lb/>

<arrow.to.target n="fig47"></arrow.to.target><lb/>Sit etiam angulus a duplus b, & b c dupla <lb/>b a: & erit per eadem proportio a c, & a b <lb/>ad c b, ut c b ad c a. Ponamus ergo ab 1, erit <lb/>b c 2, & a c 1 pos, & a c, a b 1 pos p: 1, & du<lb/>cta in a c fit 1 quad. p: 1 pos, & hoc e&longs;t æquale 4 quadrato b c per re<lb/>flexæ proportionis diffinitionem. Igitur a c e&longs;t <02> 4 1/4 m: 1/2, & ita <lb/>de alijs.</s>

<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>

<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>

<s><margin.target id="marg194"></margin.target>P<emph type="italics"/>er &longs;extam eiu&longs;dem.<emph.end type="italics"/></s>

<s>Propo&longs;itio &longs;exage&longs;ima&longs;eptima.</s>

<s>Si fuerint aliquot quantitates ab una quantitate, aliæque totidem

<pb pagenum="57"/>ab eadem analo gæ, erit proportio tertiæ unius ordinis ad tertiam <lb/>alterius, ut &longs;ecundæ ad &longs;ecundam duplicata, & quartæ ad quartam <lb/>triplicata, quintæ ad quintam quadruplicata, at que &longs;ic de alijs.<lb/>

<arrow.to.target n="marg195"></arrow.to.target></s>

<s>Sint quantitates b c d e f, ab a in continua proportio<lb/>

<arrow.to.target n="table14"></arrow.to.target><lb/>ne, & aliæ totidem g h k l m, dico quod proportio h c e&longs;t <lb/>duplicata ei, quæ e&longs;t g ad b, & k ad d triplicata, & l ad e <lb/>quadruplicata, & &longs;ic deinceps, &longs;umatur enim unum, & ab </s>

<table>

<table.target id="table14"></table.target>

<s>

<arrow.to.target n="marg196"></arrow.to.target><lb/>co o p q r s in proportione b ad a, & tuxyz in propor<lb/>tione g ad a, erit igitur p quadratum o, & u quadratum t, <lb/>& q cubus o, & x cubus t, & ita de alijs: ergo proportio <lb/>

<arrow.to.target n="marg197"></arrow.to.target><lb/>n ad p duplicata ei, quæ t ad o, & x ad q triplicata ei, quæt <lb/>ad o, & pote&longs;t etiam demon&longs;trari generaliter ultra qua<lb/>

<arrow.to.target n="marg198"></arrow.to.target><lb/>dratum, & cubum: nam &longs;i ducatur t in o, fiat que <foreign lang="greek">a</foreign> erit, pro<lb/>portio enim ad <foreign lang="greek">a</foreign> eadem quæ t ad o, & proportio a ad p, <lb/>ut t ad o, igitur per diffinitionem proportionis duplicatæ <lb/>

<arrow.to.target n="marg199"></arrow.to.target><lb/>po&longs;itam in quinto libro ab Euclide u ad p duplicata ei, <lb/>quæ t ad o, & &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. Quia ergo propor<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, quæ e&longs;t t ad <lb/>o, & ita de reliquis, cum ergo proportio p ad o &longs;it, ut e ad b, & o ad <lb/>

<arrow.to.target n="marg201"></arrow.to.target><lb/>n, ut b ad a, & n ad t, ut a ad g, & t ad u, ut g ad h, &longs;equitur ut &longs;it t ad a, <lb/>ut g ad b, & 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 quæ e&longs;t g ad b, & ita de reliquis, & <lb/>no&ngrave; refert, &longs;eu dicas u ad p duplicatam ei, quæ e&longs;t t ad o, &longs;eu dicas p <lb/>

<arrow.to.target n="marg202"></arrow.to.target><lb/>ad u duplicatam ei, quæ e&longs;t o ad t. Aliter & euidentius in duabus <lb/>&longs;oleo demon&longs;trare: cum enim &longs;it e & h duplicata ei quæ e&longs;t b & g <lb/>ad a, ut &longs;upra, & quadrati b ad quadratum a, & quadrati g ad qua<lb/>

<arrow.to.target n="marg203"></arrow.to.target><lb/>dratum a duplicata his quæ b & g ad a erunt b & g quadratorum <lb/>ad quadratum a, uelut c & h ad a. Et conuertendo qua<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 & erit quadrati b ad <expan abbr="quadrat&utilde;">quadratum</expan> g, ita c ad h: &longs;ed qua<lb/>drati b ad quadratum g, ut b ad g proportio duplicata <lb/>igitur e ad h, ut b ad g duplicata.</s>

<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"/> & 33. <emph type="italics"/>undeci-mi.<emph.end type="italics"/></s>

<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>

<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>

<s><margin.target id="marg202"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 10 <emph type="italics"/>diff. quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<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>

<table>

<table.target id="table15"></table.target>

<s>Propo&longs;itio &longs;exage&longs;imaoctaua, collectorum ab Euclide <lb/>& Archimede.</s>

<s>Omnis cylindrus cono habenti ba&longs;im, & altitudinem eandem <lb/>

<arrow.to.target n="marg204"></arrow.to.target><lb/>triplus e&longs;t. Omnis cylindrus &longs;phæræ habenti eundem magnum <lb/>

<arrow.to.target n="marg205"></arrow.to.target><lb/>circulum, & altitudinem &longs;exquialter e&longs;t. Omnis &longs;phæra 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, & altitudo eadem, quæ <lb/>&longs;phæræ ip&longs;ius. Omnis &longs;uperficies &longs;phæræ quadrupla e&longs;t maiori <lb/>

<arrow.to.target n="marg207"></arrow.to.target><lb/>&longs;uo circulo. Superficies portionis &longs;phæræ e&longs;t æqualis circulo, cu <lb/>

<arrow.to.target n="marg208"></arrow.to.target>

<pb pagenum="58"/>ius &longs;emidiameter e&longs;t linea ducta à uertice portionis ad finem illius.</s>

<s>Quilibet &longs;ector &longs;phæræ æqualis e&longs;t cono, cuius ba&longs;is e&longs;t circu<lb/>lus æqualis &longs;uperficiei eiu&longs;dem portionis, altitudo uerò &longs;phæræ &longs;e<lb/>midiameter. Proportio &longs;phæræ ad &longs;ectorem datum, e&longs;t duplica<lb/>ta ei, qu&ecedil; e&longs;t dimetientis ad lineam, quæ à uertice portionis ad lim<lb/>bum. Cum enim &longs;phæra &longs;it æqualis cono, cuius ba&longs;is e&longs;t maior cir<lb/>culus, altitudo uerò dupla dimetienti per tertiam harum, quæ hic <lb/>

<arrow.to.target n="marg209"></arrow.to.target><lb/>proponuntur: erit &longs;phæra æqualis cono ba&longs;im habenti circulum, <lb/>cuius &longs;emidiameter &longs;it æqualis diametro &longs;phæræ, altitudo uerò &longs;e<lb/>midiameter &longs;phæræ. At per &longs;extam harum &longs;ector &longs;phæræ e&longs;t æqua<lb/>lis cono habenti altitudinem &longs;cmidiametrum &longs;phær&ecedil;, ba&longs;im autem <lb/>

<arrow.to.target n="marg210"></arrow.to.target><lb/>ip&longs;am portionis &longs;uperficiem: igitur proportio &longs;phæræ ad &longs;ecto<lb/>rem, uelut circuli cuius diameter e&longs;t dupla dimetienti &longs;phæræ ad <lb/>círculum æqualem &longs;uperficiei portionis: at &longs;uperficies portionis <lb/>per quintam harum e&longs;t æqualis circulo, cuius &longs;emidiameter e&longs;t li<lb/>nea à uertice portionis ad limbum eiu&longs;dem: ergo proportio &longs;phæ<lb/>ræ ad &longs;uum &longs;ectorem e&longs;t uelut circuli, cuius dimetiens e&longs;t duplus di <lb/>metienti &longs;phæræ, aut &longs;emidimetiens e&longs;t æqualis dimetienti &longs;phæræ <lb/>ad circulum, cuius &longs;emidimetiens e&longs;t linea à uertice portionis ad <lb/>limbum. Sed proportio talium circulorum e&longs;t duplicata propor<lb/>

<arrow.to.target n="marg211"></arrow.to.target><lb/>tioni &longs;emidimetientium, igitur proportio &longs;phæræ ad &longs;uum &longs;ecto<lb/>rem e&longs;t ueluti dimetientis &longs;phæræ ad lineam, quæ á uertice portio<lb/>

<arrow.to.target n="marg212"></arrow.to.target><lb/>nis ad limbum duplicata. Cuicunque portioni &longs;phæræ conus ille <lb/>habetur æqualis, qui ba&longs;im hab eat eandem cum portione, altitudi<lb/>nem uerò lineam rectam, quæ ad altitudinem portionis eandem <lb/>habeat proportionem, quam &longs;emidiametros &longs;phæræ unà cum alti<lb/>tudine reliquæ portionis habet ad eandem reliquæ portionis alti<lb/>

<arrow.to.target n="marg213"></arrow.to.target><lb/>tudinem. Earum &longs;phæræ portionum, quæ æqualibus &longs;uperfi<lb/>

<arrow.to.target n="marg214"></arrow.to.target><lb/>ciebus continentur medietas &longs;phæræ maxima exi&longs;tit. Proportio <lb/>&longs;uperficiei &longs;phæræ plano diui&longs;æ ad reliquæ portionis &longs;uperficiem, <lb/>& re&longs;idui &longs;ectoris ad &longs;ectorem, e&longs;t uelut quadratorum duarum li<lb/>nearum quæ à uerticulis &longs;ectionum ad communem &longs;uperficiem <lb/>plani portiones &longs;ecantis de&longs;cendunt: nam &longs;ectorem &longs;phæræ, dico <lb/>

<arrow.to.target n="marg215"></arrow.to.target><lb/>corpus compo&longs;itum ex portione, & cono illo. Ille idem etiam defi<lb/>nit Ellip&longs;im coni a cuti anguli &longs;ectionem, quam dicit etiam fieri &longs;e<lb/>

<arrow.to.target n="marg216"></arrow.to.target><lb/>cto cylindro per planum non ad angulos rectos &longs;tante &longs;uper cylin<lb/>dri axem. Ab hac igitur coni acuti anguli &longs;ectione &longs;eu ellip&longs;i cir<lb/>

<arrow.to.target n="marg217"></arrow.to.target><lb/>cumacta figura &longs;phæroides corpus quod ba&longs;im rotundam habet, <lb/>uocat: id que duplex ob longum, quod fit diametro longiore quie<lb/>&longs;cente, & 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<lb/>cumacta fit conoidale, quia planam habet ba&longs;im. Si ergo in ea<lb/>

<arrow.to.target n="marg218"></arrow.to.target><lb/>dem rectanguli coni &longs;ectione à plano portiones æquales habentes <lb/>diametros ab&longs;cindantur, illæ portiones erunt æquales. Et triangu<lb/>li in ei&longs;dem portionibus in&longs;cripti æquales erunt. Diametrum uo<lb/>cat in <expan abbr="quacunq&utilde;e">quacunqune</expan> portione lineam, quæ omnes lineas ba&longs;i æquidi<lb/>&longs;tantes per æqualia diuidit. 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 directè diame<lb/>tri ellip&longs;is ad diametrum tran&longs;uer&longs;am. Ex quo patet quod pro<lb/>

<arrow.to.target n="marg220"></arrow.to.target><lb/>portio cuiuslibet circuli ad ellip&longs;im e&longs;t uelut quadrati &longs;uæ diame<lb/>tri ad rectangulum recta, & tran&longs;uer&longs;a diametro ellip&longs;is compre<lb/>hen&longs;um. Ex hoc rur&longs;us &longs;equitur quod ellip&longs;is ad ellip&longs;im, ut re<lb/>

<arrow.to.target n="marg221"></arrow.to.target><lb/>ctanguli ex diametris unius ad rectangulum ex diametris alterius.</s>

<s><margin.target id="marg209"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 14. & 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>

<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>

<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"/>, & 20. <emph type="italics"/>&longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<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>

<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>

<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>

<s>Si conoides & &longs;phæroides &longs;ecet plano æquidi&longs;tanti axi fiet &longs;e<lb/>

<arrow.to.target n="marg222"></arrow.to.target><lb/>ctio conoidalis &longs;imilis ei à qua conoides &longs;eu &longs;phæroides de&longs;cri<lb/>ptum e&longs;t. Sin autem &longs;upra axem plano ad perpendiculum erecto <lb/>&longs;ectio circulus erit. Et &longs;i &longs;ecentur obliquè fiet ellip&longs;is, modo omnia <lb/>latera comprehendat. 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 & axem eandem ha<lb/>bet. Ex quo patet, quod &longs;i portio conoidalis rectanguli & &longs;phæ<lb/>

<arrow.to.target n="marg224"></arrow.to.target><lb/>ræ medietas eandem ba&longs;im habeant & axem eundem, medietas <lb/>&longs;phæræ &longs;exquitertia erit conoidali portioni. 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<lb/>tio uelut quadratorum axium. Cuiuslibet &longs;phæroidis pars pla<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 & axem eadem ha<lb/>benti. Si autem non &longs;uper centrum erit proportio earum ad co<lb/>

<arrow.to.target n="marg227"></arrow.to.target><lb/>num ba&longs;im, & axem eandem habentem uelut coniunctæ ex axe al<lb/>terius partis & dimidio axis &longs;phæroidis ad axem alterius partis.</s>

<s>Demum proportio partis conoidis obtu&longs;i anguli plano ab&longs;ci&longs;<lb/>

<arrow.to.target n="marg228"></arrow.to.target><lb/>&longs;æ ad conum, ba&longs;im & axem eadem habentem e&longs;t ueluti lineæ, com <lb/>po&longs;itæ ex axe portionis & triplo adiectæ ad compo&longs;itum ex axe <lb/>portionis & duplo eiu&longs;dem adiectæ. Adiectam uocat hyperbolis <lb/>tran&longs;uer&longs;am. Omnis cylindrus cono triplus e&longs;t habenti eandem <lb/>

<arrow.to.target n="marg229"></arrow.to.target><lb/>ba&longs;im & altitudinem. Omnes cylindri coni &longs;phæræ &longs;unt in pro<lb/>

<arrow.to.target n="marg230"></arrow.to.target><lb/>portione corporum &longs;imilium planis &longs;uperficiebus contentarum.</s>

<s>Propo&longs;itio &longs;exage&longs;imanona, collectorum ex quatuor libris <lb/>Apollonij Pergei & <expan abbr="q.">que</expan> Sereni.</s>

<s>Si fuerit linea bifariam diui&longs;a, eique in longum alia addita, & rur<lb/>

<arrow.to.target n="marg231"></arrow.to.target><lb/>&longs;us alia detracta, fueritque totius cum addita ad eam, quæ addita e&longs;t <lb/>ueluti re&longs;idui ad detractam erit lineæ com<lb/>

<arrow.to.target n="fig48"></arrow.to.target><lb/>po&longs;itæ ex addita, & dimidia ad dimidiam

<pb pagenum="60"/>ip&longs;am uelut dimidiæ ad differentiam eius, & detractæ. Rur&longs;usque li<lb/>neæ compo&longs;itæ ex dimidio & re&longs;iduo dimidiæ ac detractæ ad li<lb/>neam compo&longs;itam ex addita & detracta ut re&longs;idui dimidiæ, & de<lb/>tractæ ad partem detractam. Et rur&longs;us totius compo&longs;itæ ad com<lb/>po&longs;itam ex dimidia & addita, uelut compo&longs;itæ ex addita, & diffe<lb/>rentia ad ip&longs;am additam. Velut &longs;it propo&longs;ita a b per æqualia diui&longs;a <lb/>in c, addita b d, & detracta b e, &longs;it proportio a d ad d b, ut a e ad e b, <lb/>dico e&longs;&longs;e, ut c d ad cb, ita ab ad c e. Et ut a e ad e d ut c e ad e b. Etite<lb/>

<arrow.to.target n="marg232"></arrow.to.target><lb/>rum ut a d ad c d uelut e d ad d b. In parabole proportio partium <lb/>diametri ad uerticem terminantium duplicata e&longs;t proportioni li<lb/>nearum ab ei&longs;dem punctis ordinatim ductarum ad ip&longs;am &longs;ectio<lb/>

<arrow.to.target n="marg233"></arrow.to.target><lb/>nem. In hyperbole autem & ellip&longs;i & circuli circumferentia erit <lb/>quadratorum linearum ordinatim ductarum inter &longs;e uelut rectan<lb/>

<arrow.to.target n="marg234"></arrow.to.target><lb/>gulorum partium diametri ad eadem puncta terminantium. Et in <lb/>ei&longs;dem &longs;i à puncto peripheriæ contingens ad diametrum ducatur, <lb/>& ab eodem ordinata, erit ut partis diametri intercept&ecedil; inter extre<lb/>mum, & ordinatam ad partem inter ordinatam & peripheriam, ue<lb/>lut interceptæ inter extremum & contingentem ad interceptam <lb/>

<arrow.to.target n="marg235"></arrow.to.target><lb/>exterius inter finem contingentis & peripheriam. Et in ei&longs;dem <lb/>quadratum &longs;emidiametri æquale e&longs;&longs;e rectangulo ex intercepta in<lb/>ter centrum & ca&longs;um contingentis in inter ceptam inter centrum & <lb/>

<arrow.to.target n="marg236"></arrow.to.target><lb/>ca&longs;um ordinatæ à loco contactus productæ. Si parabolen recta <lb/>linea contingens ad diametrum perueniat, &longs;umptoque puncto alio <lb/>in &longs;ectione æquidi&longs;tans ab eo ducatur contingenti: & ab utroque <lb/>etiam ad diametrum ordinatæ, demum à uertice æquidi&longs;tans illis, <lb/>& à priore puncto diametro æquidi&longs;tans donec concurrant, erit <lb/>triangulus ex ordinata, & æquidi&longs;tante à &longs;ecundo puncto, & dia<lb/>metri parte contentus rectangulo ex prima ordinata & parte dia<lb/>metri inter uerticem & &longs;ecundam ordinatam contento æqualis.<lb/>

<arrow.to.target n="marg237"></arrow.to.target></s>

<s>Si in parabole contingente ad diametrum ducta ex alio puncto <lb/>ei æquidi&longs;tans ducatur ex ip&longs;a &longs;ectione, ubi iterum &longs;ecat &longs;ectione<gap/><lb/>intercepta per æqualia diuidetur linea à puncto contingentis dia</s>

<s>

<arrow.to.target n="marg238"></arrow.to.target><lb/>metro æquidi&longs;tanti ducta. Idem uerò fermè continget ducta li<lb/>nea à centro in locum contactus, &longs;ecabit enim omnes contingenti <lb/>

<arrow.to.target n="marg239"></arrow.to.target><lb/>æquidi&longs;tantes in hyperbole, ellip&longs;i at que circulo. E&longs;t autem omne <lb/>centrum in medio diametri: diameter autem in circulo & ellip&longs;i il<lb/>las per æqualia diuidit intus enim e&longs;t: in contrapo&longs;itis inter uerti<lb/>cem, & uerticem po&longs;ita e&longs;t exterius utriu&longs;que contingenti ad per<lb/>pendiculum in&longs;i&longs;tens. In hyperbole autem exterius etiam adiacet, <lb/>ut in contrapo&longs;itis eadem & tran&longs;uer&longs;a uo catur: cuius terminus e&longs;t <lb/>punctus concur&longs;us cum latere trianguli, qui conum per axem diui

<pb pagenum="61"/>dit: linea uerò tangens uerticem hyperbolis ad quam ordinatæ <lb/>

<arrow.to.target n="marg240"></arrow.to.target><lb/>po&longs;&longs;unt, Recta appellabitur. Datarecta linea po&longs;itione, aliaque ma <lb/>gnitudine data & angülo parabolen, & hyperbolen, & ellip&longs;im, <lb/>& contrapo&longs;itas circa datam po&longs;itione tanquàm diametrum de<lb/>&longs;cribere tanquàm cono erecto, ut angulus ad uerticem &longs;ectionis <lb/>comprehen&longs;us &longs;it, & per rectam rectangulum æquale comprehen<lb/>datur quadrato datæ lineæ magnitudine. Si linea in duas partes <lb/>

<arrow.to.target n="marg241"></arrow.to.target><lb/>diuidatur, eique utrinque æquales lineæ adiun<lb/>

<arrow.to.target n="fig49"></arrow.to.target><lb/>gantur erit rectangulum ex partibus totius æ<lb/>quale rectangulis partium prioris lineæ, & ex <lb/>priore linea cum una adiecta in eam, quæ adiecta e&longs;t. Si hyperbo <lb/>

<arrow.to.target n="marg242"></arrow.to.target><lb/>len recta linea in uertice contingat, & utrinque ab&longs;cindatur, quan<lb/>tum e&longs;t, quod pote&longs;t in quartam partem rectanguli ex diametro <lb/>tran&longs;uer&longs;a hyperbolis, quæ exterius adiacetin eam, quæ recta dici<lb/>tur, ad quam, quæ ordinatim ducuntur, &longs;unt æquidi&longs;tantes lineæ, <lb/>quæ à &longs;ectionis centro ad terminos contingentis ducuntur &longs;emper <lb/>ip&longs;i &longs;ectioni magis appropinquabunt, nec unquam conuenient: & <lb/>ob id a&longs;ymptoton appellantur. Nec ullæ aliæ intra <expan abbr="angul&utilde;">angulum</expan> illum <lb/>

<arrow.to.target n="marg243"></arrow.to.target><lb/>inueniri poterunt. Vnde etiam intra <expan abbr="dat&utilde;">datum</expan> angulum de&longs;cribere do<lb/>cemur hyperbolen cuius anguli latera &longs;int a&longs;ymptota. A&longs;ymptotis <lb/>

<arrow.to.target n="marg244"></arrow.to.target><lb/>duabus propo&longs;itis uni hyperboli, in finitas alías eidem a&longs;ymptotas <lb/>inuenire. Duabus rectis a&longs;ymptotis infinitas &longs;ubijci po&longs;&longs;e hyperbo <lb/>les illis rectis, & inter &longs;e a&longs;ymptotas. Cum in duabus &longs;uperficie<lb/>

<arrow.to.target n="marg245"></arrow.to.target><lb/>bus æquidi&longs;tantibus duo circuli æquales, quorum linea per cen<lb/>tra non e&longs;t ad perpendiculum earum infinitis planis &longs;ecantur, fiunt <lb/>in ip&longs;is lineæ à peripheria in peripheriam rectæ quæ corpus cylin<lb/>dricum claudunt quod &longs;calenus cylindrus appellatur: longè alius <lb/>ab eo, qui fit recto cylindro per duo plana æquidi&longs;tantia, &longs;ed non <lb/>ad perpendiculum po&longs;ita di&longs;&longs;ecto. nam eius extremæ &longs;uperficies <lb/>non circuli, &longs;ed ellip&longs;es &longs;unt. Si &longs;calenus cylindrus plano non æ<lb/>

<arrow.to.target n="marg246"></arrow.to.target><lb/>quidi&longs;tanti ba&longs;i, &longs;ed ita ut angulos interiores æquales faciat angu<lb/>lis ba&longs;is &longs;ectio circulus erit: uo caturque hæc&longs;ectio &longs;ubcontraria: nec <lb/>ulla præter hanc & ba&longs;i æquidi&longs;tantem &longs;ectio circulus e&longs;&longs;e pote&longs;t: <lb/>&longs;ed &longs;unt ellip&longs;es. Super eundem circulum, & &longs;ub eadem altitudi<lb/>

<arrow.to.target n="marg247"></arrow.to.target><lb/>ne ellip&longs;es &longs;imiles in cono & cylindro e&longs;&longs;e po&longs;&longs;unt, quæ ab eodem <lb/>plano fiant, docetque uel ba&longs;i uel cono uel cylindro, aut cono pro<lb/>po&longs;ito reliqua facere, quod e&longs;t ualde admirabile: cum ellip&longs;is cylin<lb/>drica &longs;emper æqualis &longs;it in utraque parte à diametro tran&longs;uer&longs;a <lb/>utrinque æqualiter di&longs;tante, conica uerò minor nece&longs;&longs;ariò &longs;it in &longs;u<lb/>periore parte uer&longs;us coni uerticem latior in inferiore, ubi partes a <lb/>diametro tran&longs;uer&longs;a æqualiter di&longs;teterint: ip&longs;&ecedil; autem non &longs;olum &longs;i

<arrow.to.target n="marg248"></arrow.to.target><lb/>miles, &longs;ed unam per&longs;æpe in utri&longs; que e&longs;&longs;e uult. Sed & hoc Archime<lb/>des dicere uidetur: lineæ ductæ à uertice coni&longs;caleni ad perpendi<lb/>culum &longs;uper ba&longs;es &longs;ingulas omnium triangulorum per axe<gap/> coni <lb/>tran&longs;euntium in peripheriam unius circuli cadunt.</s>

<s>Propo&longs;itio &longs;eptuage&longs;ima.</s>

<s>Si fuerint tres quantitates in continua proportione, aliæque toti<lb/>dem in continua proportione, poterunt con&longs;tituere tres quantita<lb/>tes in æquali differentia peruer&longs;im copulatæ.<lb/>

<arrow.to.target n="marg249"></arrow.to.target></s>

<s>Velut &longs;int a b c primi ordi<lb/>

<arrow.to.target n="fig50"></arrow.to.target><lb/>nis, & d ef &longs;ecundi, & &longs;it 28, </s>

<s>

<arrow.to.target n="marg250"></arrow.to.target><lb/>b 4, c 2, & d 2 1/4, e 1 1/2, f 1, tunc <lb/>iunctis a & e fit 9 1/2, & b & d b <lb/>1/4, & e cum f 3, at 3 & 6 1/4 & 9 1/2 <lb/>æqualiter di&longs;tant, nam diffe<lb/>rentia e&longs;t 3 1/4. At &longs;i iungatur <lb/>cum e, & b cum f, & c cum d <lb/>idem poterit contingere: ut in <lb/>figura uides, nam a e e&longs;t 8 1/2, <lb/>p: <02> 1 1/<gap/>4, & b f 7, & c d 5 1/2, m: <02> 1 1/4, & differentia b f ab utro que com<lb/>po&longs;ito, e&longs;t 1 1/2 p: <02> 1 1/4, qua excedit & exceditur. Dico modo, qua&longs;i <lb/>ex ordine coniungantur quale&longs;cun que proportiones fuerint, modo <lb/>non &longs;int ambæ æqualitatis 1, ut b iungatur cum c, & reliquæ ut li<lb/>bet, uelut a cum d, & c cum f, uel a cum f, & e cum d, nunquam fient <lb/>

<arrow.to.target n="marg251"></arrow.to.target><lb/>æquales exce&longs;&longs;us, nam de primo e&longs;t clarum: nam &longs;i a cum diun<lb/>gatur, & ambæ fuerint maximæ, maior e&longs;t differentia a ad b, quàm <lb/>b ad c, & maior etiam d ad e quàm e ad f, ideo maior erit differentia <lb/>a & d ad b e quàm b e ad c f, quod erat probandum. Eodem modo <lb/>&longs;ed laborio&longs;ius demon&longs;tratur reliquus modus &longs;cilicet, quod con<lb/>iunctio a f ad b e e&longs;t maior aut minor quàm b e ad c d, ex hoc&longs;e<lb/>quuntur corrolaria.</s>

<s>Primum, tres æquales quantitates non po&longs;&longs;unt diuidi in tres, & <lb/>tres quantitates in continua proportione ordinatè, ut dixi, ni&longs;i u<lb/>triu&longs;que ordinis tres, ac tres inuicem &longs;int æquales.</s>

<s>Secundum, tres quantitates in æquali exce&longs;&longs;u ordinate, ut dixi, <lb/>non po&longs;&longs;unt diuidi in tres, & tres quantitates, quæ &longs;int in eadem <lb/>proportione quantumcun que proportiones illæ duorum ordinum <lb/>fint diuer &longs;æ.</s>

<s>Tertium, tres quantitates, quæ &longs;intin eadem proportione non <lb/>po&longs;&longs;unt diuidi ordinate in tres ac tres, quæ &longs;int in continua propor <lb/>tione ni&longs;i &longs;int ambæ proportiones eædem cum proportione ip&longs;a<lb/>rum quantitatum.</s>

<s>Propo&longs;itio &longs;eptuage&longs;imaprima.</s>

<s>Proportionem leuitatis ponderis per uirgam torcularem attra<lb/>cti ad rectam &longs;u&longs;penfionem inuenire.</s>

<s>Sit torcularis uirga, cuius &longs;piræ a b per circui<lb/>

<arrow.to.target n="marg252"></arrow.to.target><lb/>tum &longs;int centuplæ ad altitudinem a b, & axis d c <lb/>

<arrow.to.target n="marg253"></arrow.to.target><lb/>&longs;emidiametro b c centupla, & quoniam per &longs;upe<lb/>rius a&longs;&longs;umpta, qualis e&longs;t proportio &longs;patij ad &longs;pa<lb/>tium, talis leuitatis ad <expan abbr="leuitat&etilde;">leuitatem</expan>, <expan abbr="igi&ttilde;">igitur</expan> e pondus a&longs;cen <lb/>dens per a b leuius quam per b <expan abbr="crectã">crectam</expan> centuplo, et <lb/>&longs;imiliter cum circuitus b c, & d c &longs;int in eodem tem <lb/>pore, & circuitus d c, &longs;it centuplus ad &longs;piralem b c <lb/>per demon&longs;trata ab Euclide, ergo e erit centuplo <lb/>leuius circum ductum per d quàm b, &longs;ed per b circumductum cen<lb/>tuplo leuius e&longs;t, quàm per rectam, igitur e ponderat folum particu<lb/>lam ex decem millibus recti ponderis.</s>

<s><margin.target id="marg253"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 45.</s>

<s>Propo&longs;itio &longs;eptuage&longs;ima&longs;ecunda.</s>

<s>Proportionem ponde<gap/>is &longs;ph&ecedil;ræ pendentis ad a&longs;cendentem per <lb/>accliue planum inueni<gap/></s>

<s>Sit &longs;phæra æqualis ponderig in pun<lb/>

<arrow.to.target n="marg254"></arrow.to.target><lb/>cto b, quæ debeat trahi &longs;uper b c accli<lb/>ue planum b e ad perpendiculum pla<lb/>

<arrow.to.target n="marg255"></arrow.to.target><lb/>ni b f. Quia ergo in b e mouetur a, qua<lb/>uis modica ui per dicta &longs;uperius, erit per <lb/>communem animi &longs;ententiam uis, quæ <lb/>mouebit a per e b nulla: per dicta uerò <lb/>a mouebitur ad f &longs;emper, a con&longs;tanti ui <lb/>æquali g, & per b c a con&longs;tanti ui æqua<lb/>li k, &longs;icut per b d a con&longs;tanti æquali h, ergo per ultimam petitio<lb/>nem, cum termini &longs;eruent, quo ad partes eandem rationem &longs;in<lb/>guli per &longs;e, & motus per b e &longs;it a nulla ui, erit proportio g ad k, ue<lb/>lut proportio uis, quæ mouet per b f ad uim, quæ mouet per <lb/>b c, & uelut anguli per e b f recti ad angulum e b c, & ita uis, <lb/>quæ mouet a per b f, & e&longs;t, ut dictum e&longs;t, g ad uim, quæ mouet <lb/>per b d, & e&longs;t h ex &longs;uppo&longs;ito, ut c b f ad e b d, igitur proportio dif<lb/>ficultatis motus a per b d ad idem a per b c, e&longs;t uelut h ad k, quod <lb/>erat demon&longs;trandum.</s>

<s><margin.target id="marg255"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 40. 7</s>

<s>Propo&longs;itio &longs;eptuage&longs;imatertia.</s>

<s>Proportionem ponderum attractorum penes figuram in pla<lb/>no inuenire.<lb/>

<arrow.to.target n="marg256"></arrow.to.target></s>

<s>Sint duo pondera æqualia in plano a & b, & &longs;it <lb/>

<arrow.to.target n="fig51"></arrow.to.target><lb/>a &longs;uperficies qua planum tangit dupla b &longs;uperfi<lb/>ciei, qua planum tangit: dico quod &longs;i trahantur ab <lb/>imo, quod erunt æqualia: &longs;u&longs;pendantur, & erunt <lb/>æqualia ex &longs;uppo&longs;ito, &longs;ed a quie&longs;cens in plano e&longs;t <lb/>dimidium a &longs;u&longs;pen&longs;i, & b quie&longs;cens in plano e&longs;t di <lb/>midium b &longs;u&longs;pen&longs;i ex demon&longs;tratis &longs;uperius, igi<lb/>tur per communem animi &longs;ententiam a & b in pla<lb/>no &longs;unt æqualia.</s>

<s>

<arrow.to.target n="marg257"></arrow.to.target></s>

<s>Ex hoc manife&longs;tum e&longs;t, quod proportio uirium trahentium pon <lb/>dera in plano eadem e&longs;t, quæ ip&longs;orum ponderum dum &longs;u&longs;pendun<lb/>tur. Vbiplanum æquale &longs;it, & &longs;olidum.</s>

<s>

<arrow.to.target n="marg258"></arrow.to.target></s>

<s><margin.target id="marg258"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 62.</s>

<s>Propo&longs;itio &longs;eptuage&longs;imaquarta.</s>

<s>Proportionem concutientis ad concu&longs;&longs;um &longs;tabili inuenire.</s>

<s>

<arrow.to.target n="marg259"></arrow.to.target></s>

<s>Intelligo concutiens e&longs;&longs;e &longs;olidum, quod non frangitur, idque gra<lb/>uitate, & impetu concutere, nam de duritie &longs;upponitur, & grauitas, <lb/>ut demon&longs;trabitur in corrolario e&longs;t iuxta &longs;uperficiem inferiorem <lb/>ponderi comparatam. Cum ergo motus concu&longs;sionis magnitudo <lb/>con&longs;tet ex grauitate, impetu & figura, concu&longs;si autem ex pondere <lb/>& connexione: multiplicatis inuicem partibus productorum pro<lb/>portio, erit proportio concu&longs;sionis: ut &longs;it grauitas decem, impetus <lb/>quadraginta: pondus icti centum connexio ut duo, ducemus qua<lb/>dragintain decem, & fient quadringenta, et duo in centum, fient du <lb/>centa, igitur concu&longs;sio erit dupla.</s>

<s>

<arrow.to.target n="marg260"></arrow.to.target></s>

<s>Cum fuerit figura rotunda, concu&longs;sio erit integra in puncto: <lb/>quia &longs;phæra iacens in plano totum pondus in punctum cogit.</s>

<s>

<arrow.to.target n="marg261"></arrow.to.target></s>