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Colored diff for /texts/archimedes/xml/carda_propo_015_la_1570.xml between version 1.10 and 1.15

version 1.10, 2004/02/05 10:22:16

version 1.15, 2004/02/19 17:23:10

Line 3831

<s id="id000873">Cum enim proportionis medium &longs;it latus 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 id="id000874"><arrow.to.target n="marg151"/><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"/><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"/><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"/></s>

<s id="id000874"><arrow.to.target n="marg151"/><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"/><lb/>&longs;a erunt omnia producta ex binomio in commen&longs;a reci&longs;o &longs;uo <02> nu<lb/><arrow.to.target n="marg153"/><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"/></s>

<s id="id000875"><margin.target id="marg151"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. P<emph type="italics"/>ro<lb/>po&longs;. </s>

Line 3879

<s id="id000893">Aër in &longs;ublimiore eius regione &longs;emper naturali motu fertur ex <lb/>Oriente in Occidentem, &longs;ed & infra uerum minus manife&longs;tè. </s>

<s id="id000894">At ca<lb/>&longs;u plerun que contingit, ut moueatur longè uehementius, &longs;eu ad ean<lb/>dem partem, &longs;eu aliam. </s>

<s id="id000895">Qui uerò naturalis e&longs;t, debilis <lb/><figure id="id.015.01.069.1.jpg" xlink:href="015/01/069/1.jpg"/><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. </s>

<s id="id000895">Qui uerò naturalis e&longs;t, debilis <lb/><figure id="id.015.01.069.1.jpg" xlink:href="015/01/069/1.jpg"/><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, quandoquidem fortuiti mo<lb/>tus, qui &longs;unt multo tardiores non latent nos. </s>

<s id="id000896">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 aëris <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. </s>

<s id="id000896">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 aëris <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, itaque <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">aeris</expan> e&longs;&longs;et at que continuus. </s>

<s id="id000897">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. </s>

<s id="id000898">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. </s>

<s id="id000899">A'&longs;en&longs;u quidem, quoniam nebul&etail;, <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&etail; in eo continentur, quotidieque aërem ex<lb/>periremur & nubilo&longs;um, & madidum propter mare. </s>

<s id="id000900">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. </s>

<s id="id000900">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. </s>

<s id="id000901">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. </s>

<s id="id000902">Experimento, quòd ubi hiant o&longs;tia, & ianuæ, ibi perpetuus &longs;en<lb/>titur flatus. </s>

<s id="id000903">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>

Line 3912

<s id="id000912">Dico etiam, quod tardius ad c quàm d. </s>

<s id="id000913">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. </s>

<s id="id000914">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. </s>

<s id="id000915">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. </s>

<s id="id000915">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/>tum ob cau&longs;am dictam. </s>

<s id="id000916">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>

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<s id="id000931">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 id="id000931">Ex hoc &longs;equitur, quòd graue habens partes inæquales, &longs;eu &longs;ub<lb/>&longs;tantia, &longs;eu forma, &longs;i ita excutiatur, ut pars grauior <expan abbr="nõ">non</expan> &longs;it, infrà opor<lb/>tet, ut circumuoluatur.</s>

<s id="id000932">Propo&longs;itio &longs;exage&longs;imaprima.</s>

Line 4075

<s id="id000978">Sit graue a b c alligatum funibus in d ef, dico, <lb/><figure id="id.015.01.073.1.jpg" xlink:href="015/01/073/1.jpg"/><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. </s>

<s id="id000979">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. </s>

<s id="id000979">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. </s>

<s id="id000980">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>

Line 4124

<s id="id000999"><arrow.to.target n="marg182"/><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. </s>

<s id="id001000">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"/><lb/>paruam uim habeant, & magnam repugnantiam, &longs;ed quia tran&longs;uer <lb/><arrow.to.target n="marg184"/><lb/>&longs;im impul&longs;a minus mouentur motu recto, ut &longs;upra ui&longs;um e&longs;t. </s>

<s id="id001000">Quarta cau&longs;a <lb/>e&longs;t, quod non impediatur ab aëre tran&longs;uer&longs;im moto, et à latere: ideo <lb/>leuia mobilia & magna non &longs;olum lentius de&longs;cendunt, quoniam <lb/><arrow.to.target n="marg183"/><lb/>paruam uim habeant, & magnam repugnantiam, &longs;ed quia tran&longs;uer<lb/><arrow.to.target n="marg184"/><lb/>&longs;im impul&longs;a minus mouentur motu recto, ut &longs;upra ui&longs;um e&longs;t. </s>

<s id="id001001">Por

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</row>

</table>

<s id="id001070"><arrow.to.target n="marg196"/><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"/><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"/><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"/><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"/><lb/>q <foreign lang="greek">b g</foreign> x in continua proportione per eandem. </s>

<s id="id001070"><arrow.to.target n="marg196"/><lb/>eo o p q r s in proportione b ad a, & t u x y z 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"/><lb/>n ad p duplicata ei, quæ t ad o, & x ad q triplicata ei, quæ t <lb/>ad o, & pote&longs;t etiam demon&longs;trari generaliter ultra qua<lb/><arrow.to.target n="marg198"/><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"/><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"/><lb/>q <foreign lang="greek">b g</foreign> x in continua proportione per eandem. </s>

<s id="id001071">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"/><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&etail; <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"/><lb/>ad u duplicatam ei, quæ e&longs;t o ad t. </s>

<s id="id001072">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"/><lb/>dratum a duplicata his quæ b & g ad a erunt b & g quadratorum <lb/>ad quadratum a, uelut c & h ad a. </s>

<s id="id001073">Et conuertendo qua<lb/><arrow.to.target n="table15"/><lb/>drati a ad quadratum g, ut a ad h, con&longs;tituantur ergo <lb/><figure id="id.015.01.076.2.jpg" xlink:href="015/01/076/2.jpg"/>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>

Line 4394

<s id="id001094">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. </s>

<s id="id001095">Proportio &longs;phæræ ad &longs;ectorem datum, e&longs;t duplica<lb/>ta ei, qu&etail; e&longs;t dimetientis ad lineam, quæ à uertice portionis ad lim<lb/>bum. </s>

<s id="id001096">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"/><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. </s>

<s id="id001097">At per &longs;extam harum &longs;ector &longs;phæræ e&longs;t æqua<lb/>lis cono habenti altitudinem &longs;cmidiametrum &longs;phær&etail;, ba&longs;im autem <lb/><arrow.to.target n="marg210"/><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. </s>

<s id="id001097">At per &longs;extam harum &longs;ector &longs;phæræ e&longs;t æqua<lb/>lis cono habenti altitudinem &longs;emidiametrum &longs;phær&etail;, ba&longs;im autem <lb/><arrow.to.target n="marg210"/><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. </s>

<s id="id001098">Sed proportio talium circulorum e&longs;t duplicata propor<lb/><arrow.to.target n="marg211"/><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"/><lb/>nis ad limbum duplicata. </s>

<s id="id001099">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"/><lb/>tudinem. </s>

<s id="id001100">Earum &longs;phæræ portionum, quæ æqualibus &longs;uperfi<lb/><arrow.to.target n="marg214"/><lb/>ciebus continentur medietas &longs;phæræ maxima exi&longs;tit. </s>

Line 4534

<s id="id001163">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;ectionem/><lb/>intercepta per æqualia diuidetur linea à puncto contingentis dia</s>

<s id="id001163">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;ectionem <lb/>intercepta per æqualia diuidetur linea à puncto contingentis dia</s>

<s id="id001164"><arrow.to.target n="marg238"/><lb/>metro æquidi&longs;tanti ducta. </s>

Line 4558

<pb pagenum="62" xlink:href="015/01/081.jpg"/><arrow.to.target n="marg248"/><lb/>miles, &longs;ed unam per&longs;æpe in utri&longs; que e&longs;&longs;e uult. </s>

<s id="id001179">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 axem/> coni <lb/>tran&longs;euntium in peripheriam unius circuli cadunt.</s>

<s id="id001179">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 axem coni <lb/>tran&longs;euntium in peripheriam unius circuli cadunt.</s>

Line 4630

<s id="id001203">Propo&longs;itio &longs;eptuage&longs;imaprima.</s>

<s id="id001204">Proportionem leuitatis ponderis per uirgam torcularem attra<lb/>cti ad rectam &longs;u&longs;penfionem inuenire.</s>

<s id="id001204">Proportionem leuitatis ponderis per uirgam torcularem attra<lb/>cti ad rectam &longs;u&longs;pen&longs;ionem inuenire.</s>

<s id="id001205">Sit torcularis uirga, cuius &longs;piræ a b per circui<lb/><arrow.to.target n="marg252"/><lb/>tum &longs;int centuplæ ad altitudinem a b, & axis d c <lb/><arrow.to.target n="marg253"/><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 id="id001205">Sit torcularis uirga, cuius &longs;piræ a b per circui<lb/><arrow.to.target n="marg252"/><lb/>tum &longs;int centuplæ ad altitudinem a b, & axis d c <lb/><arrow.to.target n="marg253"/><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">c rectam</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 &longs;olum particu<lb/>lam ex decem millibus recti ponderis.</s>

Line 4750

<s id="id001245"><arrow.to.target n="marg265"/><lb/>uitas ictus, ponderis, fractura, mollities eius, quodicitur, mollities <lb/>eius, quod excipit ictum, motus eiu&longs;dem, & figura lata, & inæqua<lb/>lis. </s>

<s id="id001246">Durities ergo, quatenus fracturæ opponitur, aliud e&longs;t, quam ut <lb/>molliciei: & utra que e&longs;t cau&longs;a, quæ augetictum, ut reliquæ <lb/> oppo&longs;itæ minuunt, dicemus autem de his inferius.</s>

<s id="id001246">Durities ergo, quatenus fracturæ opponitur, aliud e&longs;t, quam ut <lb/>mollitiei: & utra que e&longs;t cau&longs;a, quæ auget ictum, ut reliquæ <lb/> oppo&longs;itæ minuunt, dicemus autem de his inferius.</s>

<s id="id001248">Propo&longs;itio &longs;eptuage&longs;imaquints.</s>

<s id="id001248">Propo&longs;itio &longs;eptuage&longs;ima quinta.</s>

<s id="id001249">Proportionem immoti in aqua ad immotum in terra in excipien <lb/>do ictum inuenire.</s>

Line 4808

<s id="id001273">Sit ergo ut a b nauis tangat ro&longs;tro b c &longs;ic ut <lb/>diffugiat, erit hypomochlium c, & &longs;i tangat <lb/>e f hypomochlium e&longs;t in d dupla, ergo e&longs;t c b <lb/>ip&longs;i d e, igitur ictus duplo minor excipitur à <lb/>c b quàm ef. </s>

<s id="id001274">E&longs;t etiam tempus longè maius, <lb/>quo excipit ictum ef, quàm b c: &longs;tatim enim di&longs;cedit b c occurrit que<lb/>alijs partibus, in c f autem impingit, & angulus a d c e&longs;t longè ma<lb/>ior recto, quàm a b f: ob hæc igitur longè maior e&longs;t ictus c f quàm <lb/>b c: uocant autem hoc declinationem.</s>

<s id="id001274">E&longs;t etiam tempus longè maius, <lb/>quo excipit ictum ef, quàm b c: &longs;tatim enim di&longs;cedit b c occurritque<lb/>aliis partibus, in c f autem impingit, & angulus a d c e&longs;t longè ma<lb/>ior recto, quàm a b f: ob hæc igitur longè maior e&longs;t ictus c f quàm <lb/>b c: uocant autem hoc declinationem.</s>

<s id="id001275">Propo&longs;itio &longs;eptuage&longs;ima&longs;eptima.</s>

Line 4823

<s id="id001279">Cùm uentus fertur ad puppim rectà, naui&longs;qúe gubernaculum di

<s id="id001279">Cùm uentus fertur ad puppim rectà, naui&longs;qúe gubernaculum di<pb pagenum="67" xlink:href="015/01/086.jpg"/>rigitur, tendunturqúe uela ac expanduntur &longs;umma in parte mali, <lb/>tunc motus e&longs;t ueloci&longs;simus: fingamus autem, quod omnia ad <lb/>idem tendant præter uentum, qui non directus &longs;it ad puppim, &longs;ed <lb/>à latere, ut uides, & temo &longs;itin contrarium tantundem directus, & <lb/>&longs;upponamus pro nunc, quod uelum &longs;it &longs;olum in anteriore parte <lb/>nauis, nam &longs;ecus e&longs;&longs;et nimis magna differentia, <lb/><figure id="id.015.01.086.1.jpg" xlink:href="015/01/086/1.jpg"/><lb/>quod nauis una ageretur tribus malis alia una: <lb/>Quæritur igitur proportio motus b c ad mo<lb/>tum d e: fiat ergo c f æqualis e g, ita ut f angulus <lb/>rectus &longs;it, & manife&longs;tum e&longs;t, quod h c maior e&longs;t <lb/>c f, cum ergo angulus f rectus &longs;it, quanto maior <lb/>erit angulus h c f, tanto maior erit proportio h c <lb/>ad c f, quod e&longs;t primum a, i&nacute;de noto angulo h c f <lb/>per ea, quæ tradita &longs;unt ab A&longs;trologis de &longs;inu & <lb/>arcu erit nota proportio c h ad c f, ideo ad e g <lb/>fiat ergo c k æqualis c h, igitur c k erit maior e g, &longs;i ergo perambula<lb/>bit æqualiter c, ut c h, erit temporis motus e g ad motum e f, ut c k <lb/>ad c f, igitur cum nota &longs;it c k, e&longs;t enim æqualis c h, erit temporis ad <lb/>tempus proportio nota. </s>

<pb pagenum="67" xlink:href="015/01/086.jpg"/>rigitur, tendunturqúe uela ac expanduntur &longs;umma in parte mali, <lb/>tunc motus e&longs;t ueloci&longs;simus: fingamus autem, quod omnia ad <lb/>idem tendant præter uentum, qui non directus &longs;it ad puppim, &longs;ed <lb/>à latere, ut uides, & temo &longs;itin contrarium tantundem directus, & <lb/>&longs;upponamus pro nune, quod uelum &longs;it &longs;olum in anteriore parte <lb/>nauis, nam &longs;ecus e&longs;&longs;et nimis magna differentia, <lb/><figure id="id.015.01.086.1.jpg" xlink:href="015/01/086/1.jpg"/><lb/>quod nauis una ageretur tribus malis alia una: <lb/>Quæritur igitur proportio motus b c ad mo<lb/>tum d e: fiat ergo c f æqualis e g, ita ut f angulus <lb/>rectus &longs;it, & manife&longs;tum e&longs;t, quod h c maior e&longs;t <lb/>c f, cum ergo angulus f rectus &longs;it, quanto maior <lb/>erit angulus h c f, tanto maior erit proportio h c <lb/>ad c f, quod e&longs;t primum a, i&nacute;de noto angulo h c f <lb/>per ea, quæ tradita &longs;unt ab A&longs;trologis de &longs;inu & <lb/>arcu erit nota proportio c h ad c f, ideo ad e g <lb/>fiat ergo c k æqualis c h, igitur c k erit maior e g, &longs;i ergo perambula<lb/>bit æqualiter c, ut c h, erit temporis motus e g ad motum e f, ut c k <lb/>ad c f, igitur cum nota &longs;it c k, e&longs;t enim æqualis c h, erit temporis ad <lb/>tempus proportio nota. </s>

<s id="id001280">Quod autem in æquali tempore mouebi<lb/>tur nauis per c k & h c patet ex a&longs;&longs;umpto inferius declarando.</s>

Line 4962

Line 4959

<pb pagenum="71" xlink:href="015/01/090.jpg"/>tem &longs;ubmergeretur. </s>

<s id="id001339">Secundum ne pre&longs;&longs;a in parte anteriore dif<lb/>ficilius aquas di&longs;&longs;ecaret, & ob id longe tardiu, moueretur. </s>

<s id="id001339">Secundum ne pre&longs;&longs;a in parte anteriore dif<lb/>ficilius aquas di&longs;&longs;ecaret, & ob id longe tardius, moueretur. </s>

<s id="id001340">Pro<lb/>pter hæc duo incommoda igitur malus etiam &longs;i unicus e&longs;&longs;et <lb/>(quod uulgati&longs;simum maloribus no&longs;tris |fuit) in parte magis <lb/>proræ proxima locabatur à gubernatoribus, ut e&longs;&longs;et qua&longs;i in trien <lb/>te à ro&longs;tro in be&longs;&longs;e à puppi: Rarum fuit, & memorabile, quod nunc <lb/>pa&longs;sim habet olim Antigoni <foreign lang="greek">triame/ou&</foreign> 1, uelorum trium: quorum <lb/>po&longs;tremum Epidromus ut ip&longs;a uoce intelligamus non fui&longs;&longs;e ue<lb/>lum in malo ip&longs;o medio, &longs;ed in puppi con&longs;titutum. </s>

<s id="id001340">Pro<lb/>pter hæc duo incommoda igitur malus etiam &longs;i unicus e&longs;&longs;et <lb/>(quod uulgati&longs;simum maioribus no&longs;tris |fuit) in parte magis <lb/>proræ proxima locabatur à gubernatoribus, ut e&longs;&longs;et qua&longs;i in trien<lb/>te à ro&longs;tro in be&longs;&longs;e à puppi: Rarum fuit, & memorabile, quod nunc <lb/>pa&longs;sim habet olim Antigoni <foreign lang="greek">triame/ou&</foreign> 1, uelorum trium: quorum <lb/>po&longs;tremum Epidromus ut ip&longs;a uoce intelligamus non fui&longs;&longs;e ue<lb/>lum in malo ip&longs;o medio, &longs;ed in puppi con&longs;titutum. </s>

<s id="id001341">Cau&longs;a Dolonis <lb/>inferius exponetur: quod autem e&longs;&longs;et paruum, & omnium mini<lb/>mum, ut nauis &longs;acile ab eo inuerteretur. </s>

<s id="id001341">Cau&longs;a Dolonis <lb/>inferius exponetur: quod autem e&longs;&longs;et paruum, & omnium mini<lb/>mum, ut nauis facile ab eo inuerteretur. </s>

<s id="id001342">Vnde etiam nunc minus <lb/>minime habent tam quantitate, quam etiam altitudine, quod uo<lb/>cant Trinehetum, &longs;olum enim &longs;u&longs;tinet nauim, quæ à uentis, uel un<lb/>dis mergi &longs;olet: ab undis ubi humilior e&longs;t, à uentis à lateribus, et an<lb/>teriore parte. </s>

<s id="id001343">Vnde humile, & exiguum uelum efficit, ut nauis ante<lb/>riore parte leuis, nec mergatur prona à uentis, nec aquas ea exci<lb/>piat, nec tamen impelli pote&longs;t nauis in &longs;copulos, nec euerti ob cau<lb/>&longs;as dictas: ob quæ in magnis tempe&longs;tatibus hoc ip&longs;o duntaxat uti <lb/>&longs;olent. </s>

<s id="id001344">Quod et&longs;i nimium &longs;æuierint, etiam illud demittunt, & &longs;i <lb/>fieri pote&longs;t, etiam malum ip&longs;am quamuis &longs;ine uelo &longs;it. </s>

Line 4977

Line 4974

<s id="id001351">Vltimo loco, quæ qua&longs;i mediæ, ante quidem <lb/>tenues, pò&longs;t latiores ad uelocem cur&longs;um, & ferendum onera aptæ, <lb/>& humiles altis: & leui ex ligno. </s>

<s id="id001352">Sed nos de uelorum uarieta<lb/>te loquimur, non ea', quæ ad malos pertinet. </s>

<s id="id001353">Con&longs;tat enim me<lb/>dio loco plus mouere, quam in extremis, ut infrà docebi<lb/>mus. </s>

<s id="id001354">Antiquo enim tempore opus non fuit malorum mul<lb/>titudine, quoniam &longs;yderibus uias dirigebant ob id non ad <lb/>amu&longs;sim, quoniam linea dirigi non poterat maximè ob mo<lb/>tus obliquitatem in circulo ui&longs;us: ideò mali multi confu<lb/>&longs;ionem in cur&longs;u, & impedimentum in naui, maiu&longs;qúe pericu<lb/>lum attuli&longs;&longs;ent. </s>

<s id="id001354">Antiquo enim tempore opus non fuit malorum mul<lb/>titudine, quoniam &longs;ijderibus uias dirigebant ob id non ad <lb/>amu&longs;sim, quoniam linea dirigi non poterat maximè ob mo<lb/>tus obliquitatem in circulo ui&longs;us: ideò mali multi confu<lb/>&longs;ionem in cur&longs;u, & impedimentum in naui, maiu&longs;qúe pericu<lb/>lum attuli&longs;&longs;ent. </s>

<s id="id001355">At nunc inuenta pyxide, & lapidis Her

<pb pagenum="72" xlink:href="015/01/091.jpg"/>culei auxilio pluribus locis uela di&longs;po&longs;ita melius dirigunt iter, ut <lb/>qua&longs;i cra&longs;&longs;a minerua depictum, & pote&longs;tate deformatum, ad amu&longs;<lb/>&longs;im contrahant. </s>

<s id="id001356">Motus ergo magnitudo non &longs;impliciter con&longs;tat, <lb/>&longs;ed comparatione &longs;uper&longs;iciei ueli ad uelum longitudine quidem, </s>

<s id="id001356">Motus ergo magnitudo non &longs;impliciter con&longs;tat, <lb/>&longs;ed comparatione &longs;uperficiei ueli ad uelum longitudine quidem, </s>

<s id="id001357"><arrow.to.target n="marg285"/><lb/>ac latitudine conflata per multiplicationem. </s>

Line 5042

Line 5039

<s id="id001380">Non me later Ari&longs;totelem exi&longs;timare centrum mundi e&longs;&longs;e cen<lb/>trum terræ illudque proba&longs;&longs;e, quod tamen ex demon&longs;tratione no&longs;tra <lb/>mathematica apparet nunc&longs;ubijciam, & quid ad illius rationes di<lb/>cendum &longs;it, aliâs etiam dicendum erit: nam liber hic, ut mathemati<lb/>ca decet, e&longs;&longs;e debet ab omnibus contentionibus ab&longs;olutus. </s>

<s id="id001381">Con<lb/>&longs;tat &longs;anè non e&longs;&longs;e propriam uim lapidis illius, ut qui non &longs;it circum<lb/>&longs;criptus &longs;ed fru&longs;tulum quoduis id pote&longs;t, neque per &longs;e, &longs;ed in ferro & <lb/>pendulo, nec fieri pote&longs;t, ut &longs;it illius <expan abbr="tãquam">tanquam</expan> &longs;peciei unius lapidum, <lb/>&longs;ed qua&longs;i perfectæ portionis cuiu&longs;dam generis terræ, quæ ab&longs;olu<lb/>ta &longs;it, cuius indicium e&longs;t illius copia, neque enim ullibi non inuenitur, <lb/>& ubi ferrum effoditur, ut in Ilua In&longs;ula Tyrrheno mari, e&longs;t ergo fer <lb/><figure id="id.015.01.092.1.jpg" xlink:href="015/01/092/1.jpg"/><lb/>ri uis terræ maritæ, quæ perfecta in &longs;uo ge<lb/>nere, ubi uim f&oelig;cundam acceperit à ma&longs;cu<lb/>lo &longs;cilicet Herculeo lapide, quærit primum <lb/>ut de&longs;cendat, ubi hoc non po&longs;sit <expan abbr="&longs;alt&etilde;">&longs;altem</expan> quæ<lb/>rit, ut quie&longs;cere po&longs;sit. </s>

<s id="id001381">Con<lb/>&longs;tat &longs;anè non e&longs;&longs;e propriam uim lapidis illius, ut qui non &longs;it circum<lb/>&longs;criptus &longs;ed fru&longs;tulum quoduis id pote&longs;t, neque per &longs;e, &longs;ed in ferro & <lb/>pendulo, nec fieri pote&longs;t, ut &longs;it illius <expan abbr="tãquam">tanquam</expan> &longs;peciei unius lapidum, <lb/>&longs;ed qua&longs;i perfectæ portionis cuiu&longs;dam generis terræ, quæ ab&longs;olu<lb/>ta &longs;it, cuius indicium e&longs;t illius copia, neque enim ullibi non inuenitur, <lb/>& ubi ferrum effoditur, ut in Ilua In&longs;ula Tyrrheno mari, e&longs;t ergo fer <lb/><figure id="id.015.01.092.1.jpg" xlink:href="015/01/092/1.jpg"/><lb/>ri uis terræ maritæ, quæ perfecta in &longs;uo ge<lb/>nere, ubi uim fecundam acceperit à ma&longs;cu<lb/>lo &longs;cilicet Herculeo lapide, quærit primum <lb/>ut de&longs;cendat, ubi hoc non po&longs;sit <expan abbr="&longs;alt&etilde;">&longs;altem</expan> quæ<lb/>rit, ut quie&longs;cere po&longs;sit. </s>

<s id="id001382">Vt ergo quie&longs;cat à <lb/>motu c&oelig;li qui e&longs;t ab Oriente in Occiden<lb/>tem iuxta axis c&oelig;li &longs;itum &longs;e dirigit, quod <lb/>ille &longs;olus quie&longs;cat in &longs;uo motu, uel &longs;altem <lb/>tardi&longs;simè moueatur: indicio e&longs;t quod &longs;i <lb/>extra &longs;itum illum acus ferrea imbuta eo lapide ponatur, &longs;tatim tre<lb/>mit uchementer, adeò ut nec momento ullo con&longs;i&longs;tat, &longs;ed mi&longs;erè & <lb/>grauiter torqueri uideatur, non ergo quod &longs;entiat polorum locum <lb/>qui tantum abe&longs;t ab illa, ut nec ab homine perito mathematicarum, <lb/>&longs;ed quod uix illa c&oelig;li &longs;entiatur circa centrum mundi. </s>

<s id="id001382">Vt ergo quie&longs;cat à <lb/>motu c&oelig;li qui e&longs;t ab Oriente in Occiden<lb/>tem iuxta axis c&oelig;li &longs;itum &longs;e dirigit, quod <lb/>ille &longs;olus quie&longs;cat in &longs;uo motu, uel &longs;altem <lb/>tardi&longs;simè moueatur: indicio e&longs;t quod &longs;i <lb/>extra &longs;itum illum acus ferrea imbuta eo lapide ponatur, &longs;tatim tre<lb/>mit uehementer, adeò ut nec momento ullo con&longs;i&longs;tat, &longs;ed mi&longs;erè & <lb/>grauiter torqueri uideatur, non ergo quod &longs;entiat polorum locum <lb/>qui tantum abe&longs;t ab illa, ut nec ab homine perito mathematicarum, <lb/>&longs;ed quod uix illa c&oelig;li &longs;entiatur circa centrum mundi. </s>

<s id="id001383">Cuius indi<lb/>cio e&longs;t Oceani maris, aquarum fluxus & refluxus. </s>

<s id="id001384">Duos ergo ha<lb/>bet motus terra perfecta, &longs;eu ferrum lapide Herculeo <expan abbr="imbut&utilde;">imbutum</expan> &longs;ub<lb/>ordinatos imperfectum perfecto: perfectus e&longs;t, ut de&longs;cendat ad cen <lb/>trum terræ, ut ibi quie&longs;cat: imperfectum, cum à perfecto prohibe<lb/>tur, ut quie&longs;cat &longs;altem extra centrum cum in clinatione ad centrum, <lb/>et hoc fiet &longs;i &longs;ecundum longitudinem acus dirigatur per axem mun <lb/>di, cum &longs;itu tamen de&longs;cen&longs;ui ad terræ centrum proximiore, ut &longs;æpi<lb/>us &longs;uperius declarauimus, dum de motu grauium & præcipuè li<lb/>bræ, & centro grauitatis loqueremur. </s>

<s id="id001385">Quibus demon&longs;tratis tum <lb/>experimento tum ratione à Fortunio Affaytato Cremonen&longs;i Me<lb/>dico, cum per hæc po&longs;tmodum cogeretur fateri acum ad polum

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<s id="id001399">Sit aurum a, & liquor b, quæ repleant uas c, & <lb/>pondus amborum &longs;it librarum quadraginta, & <lb/><figure id="id.015.01.093.1.jpg" xlink:href="015/01/093/1.jpg"/><lb/>uas repletum liquore &longs;olo &longs;it librarum xxix, au<lb/>rum autem &longs;it ponderis librarum xij, igitur reli<lb/>quum erit ponderis xxviij, differentia ergo ua<lb/>&longs;is pleni, & non pleni liquore e&longs;t libra una, pon<lb/>dus auri e&longs;t librarum duodecim: dico quod au<lb/>ri pondus e&longs;t duodecuplum ponderi liquoris, & <pb pagenum="75" xlink:href="015/01/094.jpg"/>&longs;i fui&longs;&longs;et pondus amborum libræ xxxix, manentibus reliquis, &longs;eque <lb/>retur quod pondus liquoris e&longs;&longs;et xxvij, & quia plenum uas &longs;uppo<lb/>nitur e&longs;&longs;e librarum xxix, e&longs;&longs;et differentia libræ ij, at auri pondus e&longs;t <lb/>libræ xij, igitur proportio ponderis auri ad liquorem e&longs;&longs;et &longs;excu<lb/>pla. </s>

<pb pagenum="75" xlink:href="015/01/094.jpg"/>&longs;i fui&longs;&longs;et pondus amborum libræ xxxix, manentibus reliquis, &longs;eque <lb/>retur quod pondus liquoris e&longs;&longs;et xxvij, & quia plenum uas &longs;uppo<lb/>nitur e&longs;&longs;e librarum xxix, e&longs;&longs;et differentia libræij, at auri pondus e&longs;t <lb/>libræ xij, igitur proportio ponderis auri ad liquorem e&longs;&longs;et &longs;excu<lb/>pla. </s>

<s id="id001400">Nam &longs;i uas plenum liquore ex &longs;uppo&longs;ito e&longs;t librarum xxix, & <lb/>cum auro xl, gratia exempli, & auri pondus e&longs;t xij, igitur liquoris <lb/>pondus e&longs;t xxviij librarum: &longs;ed cum liquor &longs;it corpus &longs;imilium par<lb/>tium, igitur loci ad lo cum, ut ponderis ad pondus, ergo dum ade&longs;t <lb/>aurum, liquor occupat xxviij partes cxxxix, totius ua&longs;is igitur au<lb/>rum continet unam partem tantum, & cum aurum pondus habeat <lb/>librarum xij, & liquor unius: quia totum uas cxxxix librarum dum <lb/>e&longs;t plenum, & e&longs;t diui&longs;um in xxix partes, igitur pondus unius par<lb/>tis liquoris e&longs;t una libra, igitur pondus auri e&longs;t duode cuplum ad <lb/>pondus liquoris quod fuit propo&longs;itum.</s>

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<s id="id001493">Et ita mediæ ad minorem quintupla, & ideò maioris ad

<pb pagenum="81" xlink:href="015/01/100.jpg"/>minorem uelo citas uiginti quincupla, ut non &longs;it difformis, neque <lb/>pcriculo&longs;a, ut in rotis moletrinis, & &longs;it diui&longs;a per medium iuxta <lb/>proportionem, cum &longs;it tanto uelo cior minor media, quanto media <lb/>maiore. </s>

<pb pagenum="81" xlink:href="015/01/100.jpg"/>minorem uelo citas uiginti quincupla, ut non &longs;it difformis, neque <lb/>periculo&longs;a, ut in rotis moletrinis, & &longs;it diui&longs;a per medium iuxta <lb/>proportionem, cum &longs;it tanto uelocior minor media, quanto media <lb/>maiore. </s>

<s id="id001494">Rur&longs;us proportio partium maioris ad mediæ partes tripla <lb/>e&longs;t &longs;cilicet ccclx ad cxx, & mediæ ad <expan abbr="minor&etilde;">minorem</expan> tripla cxx ad xl, & pro<lb/>portio e&longs;t &longs;excupla, iterum igitur partes maioris ad mediam, & me<lb/>diæ ad minorem erunt in dupla proportione, utrobique, & e&longs;t pul<lb/>chrum. </s>

<s id="id001495">Ideò partes etiam minimæ rotæ erunt &longs;atis magnæ: nam <lb/>cum diameter &longs;it bes pedis, ambitus peripheriæ erit duorum pe<lb/>dum. </s>

<s id="id001496">1. unciarum uigintiquatuor: igitur diui&longs;a peripheria in xl par<lb/>ter, unaquæ que pars erit maior dimidia uncia.</s>

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<s id="id001509">Contingit hoc multis modis: primum acceptis duabus &longs;phæru<lb/>lis æqualibus ex cry&longs;tali&longs;ub&longs;tantia unaque demi&longs;&longs;a ab alti&longs;sima turri, <lb/>& men&longs;urato ictu per in&longs;trumentum præcedens, & &longs;ub totidem <lb/>momentis alia demi&longs;&longs;a in aquam, in de &longs;ub eodem tempore dimen<lb/>&longs;a altitudine, erit proportio &longs;patij ad &longs;patium, ut den&longs;itatis aquæ, ad <lb/>den&longs;itatem aëris. </s>

<s id="id001510">Item emi&longs;&longs;a &longs;phærula per in&longs;trumentum in aërem, <lb/>in de in aquam: & fumpta proportione. </s>

<s id="id001510">Item emi&longs;&longs;a &longs;phærula per in&longs;trumentum in aërem, <lb/>in de in aquam: & &longs;umpta proportione. </s>

<s id="id001511">Et uidimus &longs;corpionem, <lb/>qui <expan abbr="&longs;phærulã">&longs;phærulam</expan> creteam emittebat pedibus lxx, & in aqua per unum <lb/>& dimidium adeò, ut proportio fuerit, ut quinquaginta ad unum: <lb/>ideò e&longs;t fallax experimentum in uiolento motu: nam cum emitte<lb/>batur in aquam erat propè, & ob id in &longs;ummo robore: cùm in aë<lb/>rem, emittitur &longs;en&longs;im uis. </s>

<s id="id001512">De hoc ergo loquar.</s>

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<s id="id001544">Cubus uerò non de&longs;cendet, ni&longs;i cum di<lb/>midium d addito, quod inter cipitur inter lineam mediam, & quæ à <lb/>centro mundi ad punctum medium contactus u&longs;que quò perueniat <lb/>ad oppo&longs;itam partem, eam habuerit proportionem ad idem me<lb/>dium eadem portione detracta, quem iuncta proportioni an guli <lb/>declinationis ad re&longs;iduum recti dimidiam proportionem efficiat. <lb/></s>

<s id="id001545">Eademque ratio aliorum planorum. </s>

<s id="id001546">Dico præterea quòd motus <lb/>&longs;phæræ, & etiam corporum rectarum &longs;uperficierum in de&longs;cen&longs;u <lb/>alius e&longs;t æqualis, & alius inæqualis, & qua&longs;i à latere, uelut &longs;i angu<lb/>lus unus prolabatur, ac fiat circumuolutio: cum ergo facilius fiat <lb/>hoc, & maximè &longs;i non retineatur æqualiter, & difficile &longs;it in medio <lb/>retinere, propterea prolap&longs;us hi melius <expan abbr="retin&etilde;tur">retinentur</expan> duobus uinculis, <lb/>quàm in medio, non &longs;olum ob hanc æqualitatem, & complexum <lb/>meliorem, &longs;ed <expan abbr="etiã">etiam</expan>, quod omnes motus, omnes ponderum nixus fa <lb/>ciliùs cohibentur, & <expan abbr="deducun&ttilde;">deducuntur</expan> diui&longs;i in partes, <08> &longs;i toti contin <expan abbr="ean&ttilde;">eantur</expan>, <lb/>aut ui <expan abbr="trahãtur">trahantur</expan>. </s>

<s id="id001547">Et ideo uin cula in rami cibus duplicia dextra, & &longs;ini <lb/>&longs;tra &longs;cilicet in <expan abbr="ead&etilde;">eadem</expan> parte tamë longe &longs;unt meliora etiam ferreis, quæ <lb/>&longs;olum in medio nectantur.</s>

<s id="id001547">Et ideo uincula in rami cibus duplicia dextra, & &longs;ini<lb/>&longs;tra &longs;cilicet in <expan abbr="ead&etilde;">eadem</expan> parte tamën longe &longs;unt meliora etiam ferreis, quæ <lb/>&longs;olum in medio nectantur.</s>

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<s id="id001550">Ex hoc etiam &longs;equitur, <lb/><figure id="id.015.01.104.1.jpg" xlink:href="015/01/104/1.jpg"/><lb/>quod cùm omne graue <lb/>&longs;pontè &longs;emper appropin<lb/>quet centro mundi, & a &longs;i <lb/>moueretur per planum e, <lb/>magis remoueretur à cen<lb/>tro mundi, ut per e c per ea <lb/>quæ diximus, & quoniam <lb/>linea ex centro mundi ad <lb/>c longior e&longs;t, quàm ad e, <lb/>multò pote&longs;t enim e&longs;&longs;e, ut <lb/>in proportione diametri <lb/>quadrati ad latus eius, & <lb/>ctiam maior. </s>

<s id="id001551">ergo poterit <lb/>e&longs;&longs;e adeò parum decliuis <lb/>linea c d, ut c punctus ma<lb/>gis di&longs;ter à centro mundi, <lb/>quàm d, & tamen feretur <lb/>ex d in c motu naturali, ut demon&longs;tratum e&longs;t, ergo per purum mo<lb/>tum naturalem poterit a remoueri à centro mundi. </s>

<s id="id001552">Hoc uolui pro<lb/>ponere, ut intelligeres in plano uero c e non moueri a &longs;ponte, quia <lb/>c nece&longs;&longs;ariò altior e&longs;t d: &longs;i ergo mouebitur, non erit c e recta, &longs;ed <lb/>pars proportionis circuli &longs;uperficiei terræ, quæ &longs;en&longs;u à recta di&longs;tin<lb/>gui non poterit. </s>

<s id="id001553">Hoc ergo e&longs;t primum, ex quo &longs;equitur.</s>

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<s id="id001558">Propo&longs;itio nonage&longs;ima&longs;ecunda.</s>

<s id="id001559">Propprtionem ponderis æqualis iuxta longitu dinis compara<lb/>tionem demon&longs;trare.</s>

<s id="id001559">Proportionem ponderis æqualis iuxta longitudinis compara<lb/>tionem demon&longs;trare.</s>

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<s id="id001578"><arrow.to.target n="marg317"/><lb/>profunditatem, & aquæ den&longs;itatem, &longs;ed quanto aqua den&longs;ior e&longs;t, <lb/>tanto minus nauis de&longs;cendit, & quanto minus den&longs;a, tanto magis: <lb/>ergo pari modo fermè redduntur mobiles, & in aqua dulci & &longs;al&longs;a, <lb/>ubi naues &longs;int &longs;imiles forma, pondere, magnitudine. </s>

<s id="id001579">Quia crgo ne<lb/>ce&longs;&longs;e e&longs;t tabulam nauis e&longs;&longs;e duriorem, quam aqua ad re&longs;i&longs;tendum, <lb/>ergo pars maior ictus mouebit primo nauim, quam tabulam pe<lb/>netret, cum ergo quod facilius e&longs;t, præcedat, difficilius ergo naues

<s id="id001579">Quia ergo ne<lb/>ce&longs;&longs;e e&longs;t tabulam nauis e&longs;&longs;e duriorem, quam aqua ad re&longs;i&longs;tendum, <lb/>ergo pars maior ictus mouebit primo nauim, quam tabulam pe<lb/>netret, cum ergo quod facilius e&longs;t, præcedat, difficilius ergo naues

<pb pagenum="87" xlink:href="015/01/106.jpg"/>utrinque mouebuntur, & quia inter duos quo&longs;cunque motus contra<lb/>rios <expan abbr="nõ">non</expan> e&longs;&longs;eos, ut utar uocabulo Auerrois quinto Phy&longs;icorum, ne<lb/>ce&longs;&longs;e e&longs;t, ut intercedat quies media, & in quiete ab ictu, ut ui&longs;um e&longs;t <lb/>&longs;uperius, oportet, ut quod excipit ictum uelloco moueatur, uel ce<lb/><arrow.to.target n="marg318"/><lb/>dat, & ictus penetret, uel aër non conden&longs;etur ob tarditatem ultra <lb/>metam, nec retro cedere pote&longs;t ex &longs;uppo&longs;ito, & ictus e&longs;t magnus, <lb/>clarum e&longs;t, quod oportet, ut cedat, & &longs;i durum &longs;it confringatur. <lb/></s>

<pb pagenum="87" xlink:href="015/01/106.jpg"/>utrinque mouebuntur, & quia inter duos quo&longs;cunque motus contra<lb/>rios <expan abbr="nõ">non</expan> e&longs;&longs;e eos, ut utar uocabulo Auerrois quinto Phy&longs;icorum, ne<lb/>ce&longs;&longs;e e&longs;t, ut intercedat quies media, & in quiete ab ictu, ut ui&longs;um e&longs;t <lb/>&longs;uperius, oportet, ut quod excipit ictum uel loco moueatur, uel ce<lb/><arrow.to.target n="marg318"/><lb/>dat, & ictus penetret, uel aër non conden&longs;etur ob tarditatem ultra <lb/>metam, nec retro cedere pote&longs;t ex &longs;uppo&longs;ito, & ictus e&longs;t magnus, <lb/>clarum e&longs;t, quod oportet, ut cedat, & &longs;i durum &longs;it confringatur. <lb/></s>

<s id="id001580">Proportio ergo rece&longs;&longs;us ad ictum e&longs;t ut temporis, & magnitudinis <lb/>partis, quæ cedit, & retro ce&longs;&longs;us po&longs;ito ictu tanquam monade.</s>

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<s id="id001598"><margin.target id="marg320"/>E<emph type="italics"/>x &longs;ecunda <lb/>animi com<lb/>muni &longs;enter <lb/>tia.<emph.end type="italics"/></s>

<s id="id001598"><margin.target id="marg320"/>E<emph type="italics"/>x &longs;ecunda <lb/>animi com<lb/>muni &longs;enten<lb/>tia.<emph.end type="italics"/></s>

<s id="id001599">Sed &longs;i &longs;it pars, erit &longs;umpta &longs;ecundum partem monadis ip&longs;ius, <lb/>ideò erit cognita iuxta nomen, uelut dimidium e&longs;t dimidium mo<lb/>nadis, dimi dium tertiæ partis monadis e&longs;t cognitum, quia tertia <lb/>pars e&longs;t cognita, & &longs;cimus, quanta pars a&longs;&longs;umatur illius. </s>

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<s id="id001622"><arrow.to.target n="marg328"/><lb/>e&longs;t: &longs;it igitur primum a b c trigonus orthogonius: & &longs;it a rectus, & <lb/>proportio <expan abbr="duor&utilde;">duorum</expan> laterum cognita, dico, quod omnia latera cognita <lb/><arrow.to.target n="marg329"/><lb/><figure id="id.015.01.107.2.jpg" xlink:href="015/01/107/2.jpg"/><lb/>erunt: nam &longs;it proportio, gratia exempli, <lb/>a b ad b c, erit ergo quadrati a b ad qua<lb/>dratum b c cognita, quia duplicata: at <lb/>quadrata a b, & a c perficiunt quadratum <lb/>b c, igitur proportio quadrati a b ad a c et <lb/>e&longs;t 1 p: cognita erit, quare & a b ad a c, & <expan abbr="eod&etilde;">eodem</expan> modo a c ad b c: quod <lb/>e&longs;t primum. </s>

<s id="id001623">Exemplum, ponatur b c dupla a b, erit a b quadratum <lb/>&longs;ub quadruplum quadrato a b quare &longs;ubtriplum quadrato a cigi

<s id="id001623">Exemplum, ponatur b c dupla a b, erit a b quadratum <lb/>&longs;ub quadruplum quadrato a b quare &longs;ubtriplum quadrato a c igi<pb pagenum="89" xlink:href="015/01/108.jpg"/>tur &longs;i a b ponatur 1 b c erit 2, & a c <02> 3. Rur&longs;us ponatur angulus b <lb/>duplus angulo c quali&longs;cunque &longs;it, erit per demon&longs;trata &longs;uperius pro<lb/>portio a b b c ad a c, ut a c ad a b, &longs;i igitur nota &longs;it proportio a c ad <lb/>a b, erit nota proportio a b b c ad a b per præcedentem. </s>

<pb pagenum="89" xlink:href="015/01/108.jpg"/>tur &longs;i a b ponatur 1 b c erit 2, & a c <02> 3. Rur&longs;us ponatur angulus b <lb/>duplus angulo c quali&longs;cunque &longs;it, erit per demon&longs;trata &longs;uperius pro<lb/>portio a b b c ad a c, ut a c ad a b, &longs;i igitur nota &longs;it proportio a c ad <lb/>a b, erit nota proportio a b b c ad a b per præcedentem. </s>

<s id="id001624">Ergo per <lb/>eandem omnia nota &longs;cilicet b c ad b a, & b c ad c a. </s>

<s id="id001625">Et &longs;i e&longs;&longs;et nota <lb/>proportio a b ad b c, dico, quod e&longs;&longs;ent nota omnia, nam nota e&longs;&longs;et <lb/>a b, & b c, & quod fit ex a b in ip&longs;um aggregatum. </s>

<s id="id001626">Sed hoc e&longs;t æ<lb/><arrow.to.target n="marg330"/><lb/>quale quadrato a c, igitur notum e&longs;t quadratum a c ergo a c: igitur <lb/>proportio a b b c ad a c, & a c ad a b. </s>

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<pb pagenum="91" xlink:href="015/01/110.jpg"/>gnus &longs;plendor, referens corpus Solis. </s>

<s id="id001667">Propterea dico, quòd uel ac<lb/>eidit, quia homo non pote&longs;t ea hora intueri Solem, & etiam e&longs;t im<lb/>peditus à radijs circum&longs;tantibus, cuius indicio e&longs;t, quod in &longs;pe<lb/>culo po&longs;ito in aqua, &longs;imile uidetur &longs;tellulæ in centro Lun&etail;: & hic e&longs;t <lb/>&longs;plen dor Solis collectus in centro Lunæ. </s>

<s id="id001667">Propterea dico, quòd uel ac<lb/>cidit, quia homo non pote&longs;t ea hora intueri Solem, & etiam e&longs;t im<lb/>peditus à radijs circum&longs;tantibus, cuius indicio e&longs;t, quod in &longs;pe<lb/>culo po&longs;ito in aqua, &longs;imile uidetur &longs;tellulæ in centro Lun&etail;: & hic e&longs;t <lb/>&longs;plendor Solis collectus in centro Lunæ. </s>

<s id="id001668">po&longs;&longs;et etiam dici, quòd <lb/>Luna circa medium propter maculam non admitteret lumen, & ita <lb/>e&longs;&longs;et inæqualium partium.</s>

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<s id="id001719">E&longs;t enim in pondere utraque dupla, in precio autem <lb/>ex prima habetur tripla, ex &longs;ecunda habetur proportio maior, <lb/>quàm tredecim ad unum, propterea utendum e&longs;t proportione <lb/>propinquiori, &longs;i &longs;atis faceret. </s>

<s id="id001720">gratia exempli, in prima ad ditione fuit <lb/>unum granum, & acqui&longs;iuit proportionem triplam, in &longs;ecunda fue <lb/>runt duo grana, &longs;i ergo acqui&longs;i&longs;&longs;et &longs;olum &longs;excuplam proportio<lb/>nem, haberemus intentum. </s>

<s id="id001721">Propterea in i&longs;to ca&longs;u oportet demon<lb/>&longs;trare forma Geometrica, &longs;uppo&longs;ito, quòd &longs;it figura recta ex uno la <lb/><figure id="id.015.01.113.1.jpg" xlink:href="015/01/113/1.jpg"/><lb/>tere a b, ita ut angulus, uel minimus capiat b c æqualem a b, & ex <lb/>æquali b a c addito fiat b d tripla b c, & ex angulo b a e duplo b a d, <lb/>fiat b c d e quadragintupla a b, & iuxta rationem erit in infinitum. <lb/></s>

<s id="id001722">Siue &longs;it parabole, &longs;iue hiperbole, &longs;eu &longs;it alia coincidentium.</s>

<s id="id001722">Siue &longs;it parabole, &longs;iue hyperbole, &longs;eu &longs;it alia coincidentium.</s>

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<s id="id001728"><arrow.to.target n="marg351"/><lb/>inuer&longs;io, & quàm diuer&longs;a &longs;it rur&longs;us, & quòd attractio e&longs;t dimidium <lb/><arrow.to.target n="marg352"/><lb/>ponderis eleuati. </s>

<s id="id001729">Cum ergo con&longs;tet in inuer&longs;ione, quanta &longs;it pro<lb/>portio ponderis &longs;u&longs;pen&longs;i ad pondus inuer&longs;um, & pondus &longs;u&longs;pen&longs;i <lb/><arrow.to.target n="marg353"/><lb/>&longs;it duplum ponderi attracti, &longs;equitur, ut diuifa proportione ponde <lb/>ris &longs;u&longs;pen&longs;i ad pondus inuer&longs;um per medium cogno&longs;catur propor<lb/>tio attractionis ad inuer&longs;ionem.</s>

<s id="id001729">Cum ergo con&longs;tet in inuer&longs;ione, quanta &longs;it pro<lb/>portio ponderis &longs;u&longs;pen&longs;i ad pondus inuer&longs;um, & pondus &longs;u&longs;pen&longs;i <lb/><arrow.to.target n="marg353"/><lb/>&longs;it duplum ponderi attracti, &longs;equitur, ut diui&longs;a proportione ponde<lb/>ris &longs;u&longs;pen&longs;i ad pondus inuer&longs;um per medium cogno&longs;catur propor<lb/>tio attractionis ad inuer&longs;ionem.</s>

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<s id="id001759">Propterea dicendum e&longs;t, quòd qui ex commu<lb/>nibus propria nituntur demon&longs;trare, omnes corrumpunt di&longs;cipli<lb/>nas. </s>

<s id="id001760">Nihil deterius e&longs;t his mon&longs;tris. </s>

<s id="id001761">Nam et&longs;i hæc ratio uera e&longs;&longs;et: <lb/>non tamen reddit cau&longs;am, quia non e&longs;t ex proprijs principijs. </s>

<s id="id001762">Dico <lb/>ergo, quod &longs;i c de&longs;cendat in e, per perpendiculum de&longs;cendet, igitur <lb/>d b e&longs;t longior d a, quare angulus e a b maior e b a: igitur pondus c <lb/>plus de&longs;cendit comparatione a, quàm b, ergo plus grauat cip&longs;um a <lb/>quàm b, &longs;eu ex cau&longs;a, quod magis premat, &longs;eu ex effectu, quòd ma<lb/>gis de&longs;ce&longs;&longs;erit. </s>

<s id="id001762">Dico <lb/>ergo, quod &longs;i c de&longs;cendat in e, per perpendiculum de&longs;cendet, igitur <lb/>d b e&longs;t longior d a, quare angulus e a b maior e b a: igitur pondus c <lb/>plus de&longs;cendit comparatione a, quàm b, ergo plus grauat c ip&longs;um a <lb/>quàm b, &longs;eu ex cau&longs;a, quod magis premat, &longs;eu ex effectu, quòd ma<lb/>gis dece&longs;&longs;erit. </s>

<s id="id001763">Cau&longs;a ergo erroris e&longs;t, quod &longs;i ponatur angulus f b a <lb/>æqualis angulo f a b, & ponatur b f &etail;qualis b c, tun c in eodem tem<lb/>pore, in quo tran&longs;it dimidium c in e, tran&longs;ibit aliud dimidium c in f. <lb/></s>

<s id="id001764">quia &longs;eparat&etail; partes grauiores &longs;unt in c b, quàm c a, propter di&longs;tan<lb/>tiam ab hypomochlio, &longs;ed tunc uelo cius mouentur, & angulus fit <lb/>&etail;qualis. </s>

<s id="id001765">Sed quando pondus e&longs;t unum, & c de&longs;cendit ad e, cum de<lb/>&longs;cendat inæquali tempore, & peragat maiorem angulum compa<lb/>ratione a, quam b, &longs;equitur, ut uelo cius moueatur comparatione a <lb/>quàm b. </s>

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<s id="id001793">Sit rur&longs;us in triangulo b e d quomodolibet modo <lb/>&longs;it angulus b d e quadruplus angulo b e d, & diuidatur d per &etail;qua<lb/>lia ducta d f, erit igitur proportio f d, d e ad f e, ut e f ad f d, &longs;ed e f ad <lb/><arrow.to.target n="marg367"/><lb/>f b ut d e ad d b. </s>

<s id="id001794">igitur proportio b d, d e ad f b <expan abbr="cõpo&longs;ita">compo&longs;ita</expan> ex propor<lb/>tionibus e f ad f d, & e d ad d b. </s>

<s id="id001795">Proportio igitur b d, d e ad f b, ut <lb/>producti ex e f in e d ad productum ex d fin d b. </s>

<s id="id001796">Rur&longs;us ponamus, <lb/><arrow.to.target n="marg368"/><lb/>quod in quadrangulo a b c d primæ figuræ &longs;it a b 4 b c 3 c d 5 ad 6 <lb/>dico, quòd &longs;pacium contentum erit notum. </s>

<s id="id001797">Ductis rectis a c & b d

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<s id="id001807"><margin.target id="marg366"/>I<emph type="italics"/>n<emph.end type="italics"/> 16. <emph type="italics"/>de<emph.end type="italics"/><lb/>S<emph type="italics"/>ubtil.<emph.end type="italics"/></s>

<s id="id001808"><margin.target id="marg367"/>P<emph type="italics"/>er<emph.end type="italics"/> 3. <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>E<emph type="italics"/>Elem.<emph.end type="italics"/></s>

<s id="id001809"><margin.target id="marg368"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

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<s id="id001813">Sit modo obtu&longs;i angulus a b c, & nota latera &longs;ingula, & angu<lb/>lus a b c, & producantur latera ad perpendicu<lb/><figure id="id.015.01.118.1.jpg" xlink:href="015/01/118/1.jpg"/><lb/>lum, ut &longs;int d & e recti, & quia anguli ad a &longs;unt <lb/>æquales, erunt anguli e b a, & d e a &longs;emper æ<lb/><arrow.to.target n="marg372"/><lb/>quales. </s>

<s id="id001814">Et hoc idem contingit in acuti angulis <lb/>triangulis intus, & e&longs;t utile mechanicum: & <lb/>quia a b c notus e&longs;t, & d notus, erunt anguli tri <lb/>goni d b c noti: & &longs;i fuerit angulus a notus, <expan abbr="er&utilde;t">erunt</expan> anguli d a c & e a b <lb/>noti, & ideo anguli e b a, & d c a: & &longs;emper notum, quod fit ex b a <lb/>in a d, uel c a in a e, &longs;unt enim &etail;qualia inter &longs;e: etiam notæ ad & a c, <lb/>quoniam duplum horum e&longs;t exce&longs;&longs;us quadrati b c &longs;uper quadrata <lb/>a b, & a c. </s>

<s id="id001815">Quod uerò proponiturà Monteregio de cognitione an<lb/>gulorum in triangulis non e&longs;t intelligendum, ut uerba &longs;ignificant, <lb/><arrow.to.target n="marg373"/><lb/>&longs;ed &longs;olum de cognitione quoad u&longs;um tabularum.</s>

<s id="id001815">Quod uerò propositurà Monteregio de cognitione an<lb/>gulorum in triangulis non e&longs;t intelligendum, ut uerba &longs;ignificant, <lb/><arrow.to.target n="marg373"/><lb/>&longs;ed &longs;olum de cognitione quoad u&longs;um tabularum.</s>

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<s id="id001830">Sint du&etail; diametri a b, c d ad perpendi <lb/>culum &longs;ecantes &longs;e in centro, & <expan abbr="duc&utilde;tur">ducuntur</expan> <lb/>&longs;upr f g k h, & infra m l ad perpendicu<lb/>lum &longs;upra a b: dico f g e&longs;&longs;e maiorem f a, <lb/>& k h k a, & contrà minorem m l, quàm <lb/>m a. </s>

<s id="id001831">Per octauam enim &longs;exti, quod fit ex <lb/><arrow.to.target n="marg374"/><lb/>b f in f a æquale e&longs;t <expan abbr="&qtilde;drato">quadrato</expan> f g, &longs;ed b f e&longs;t <lb/>maior f g, quia b f e&longs;t maior c b, & ideo <lb/>e c g f, ergo f g maior e&longs;t f a, m l <expan abbr="a&utilde;t">aunt</expan> minor e&longs;t per <expan abbr="ead&etilde;">eadem</expan> e c, quare e a, <lb/>multo igitur minor m a, quod e&longs;t primum. </s>

<s id="id001831">Per octauam enim &longs;exti, quod fit ex <lb/><arrow.to.target n="marg374"/><lb/>b f in f a æquale e&longs;t <expan abbr="&qtilde;drato">quadrato</expan> f g, &longs;ed b f e&longs;t <lb/>maior f g, quia b f e&longs;t maior c b, & ideo <lb/>e c g f, ergo f g maior e&longs;t f a, m l <expan abbr="a&utilde;t">aut</expan> minor e&longs;t per <expan abbr="ead&etilde;">eadem</expan> e c, quare e a, <lb/>multo igitur minor m a, quod e&longs;t primum. </s>

<s id="id001832">Suppo&longs;ito etiam, quòd <lb/><arrow.to.target n="marg375"/><lb/>a g arcus &longs;it dimidium a c, dico a f <expan abbr="minor&etilde;">minorem</expan> e&longs;&longs;e f e, nam quadratum e <lb/><arrow.to.target n="marg376"/><lb/>g æquale e&longs;t quadratis f e, & f g, & <expan abbr="quadrat&utilde;">quadratum</expan> a g quadratis f g & f a <lb/>& e g e&longs;t &etail;qualis lateri exagoni, & a g latus octogoni, igitur e g ma<lb/><arrow.to.target n="marg377"/><lb/>ior g a, & duo quadrata e f & f g maiora duobus quadratis f g & <lb/>f a, detracto igitur communi f g quadrato, patet propo&longs;itum.<lb/><arrow.to.target n="marg378"/></s>

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<s id="id001843">Per præcedentem moto puncto a uer&longs;us c &longs;emper u&longs; que ad e, c ma <lb/><arrow.to.target n="marg380"/><lb/>gis di&longs;tat <expan abbr="p&utilde;ctum">punctum</expan> a linea a e, quàm à puncto a uer&longs;us, quia linea n h <lb/>maior e&longs;t n a, & per eandem dum appropinquat ad c cum e c fiat <lb/>&etail;qualis ea, maius fit in crementum in a e, quàm re&longs;pectu lineæ tran&longs;<lb/>uer&longs;alis. </s>

<s id="id001844">Volo ergo inuenire punctum hoc in quo fit mutatio: & <lb/>diuido arcum ac per æqualia in f, & dico illum e&longs;&longs;e punctum quæ<lb/>&longs;itum: accepto quouis puncto in e f, puta k, duco g o h p &etail;quidi&longs;tan<pb pagenum="101" xlink:href="015/01/120.jpg"/><figure id="id.015.01.120.1.jpg" xlink:href="015/01/120/1.jpg"/><lb/>tes a b, & c d: erunt que anguli q & n recti <lb/><arrow.to.target n="marg381"/><lb/>& anguli f e a, & f e c &etail;quales, igitur uter <lb/><arrow.to.target n="marg382"/><lb/>que dimidium recti: igitur per dicta in <lb/>primo Elementorum Euclidis e n &etail;qua <lb/><arrow.to.target n="marg383"/><lb/>lis n k, igitur c q æqualis e n, quare h p <lb/>æqualis g o, &longs;ed quod fit ex o k in k g e&longs;t <lb/><arrow.to.target n="marg384"/><lb/>æquale ei, quod fit ex p k in k h, igitur <lb/><arrow.to.target n="marg385"/><lb/>k h e&longs;t æqualis k g ex eisdem o&longs;tendi<lb/>tur f l m k quadratum e&longs;&longs;e. </s>

<pb pagenum="101" xlink:href="015/01/120.jpg"/><figure id="id.015.01.120.1.jpg" xlink:href="015/01/120/1.jpg"/><lb/>tes a b, & c d: erunt que anguli q & n recti <lb/><arrow.to.target n="marg381"/><lb/>& anguli f e a, & f e c &etail;quales, igitur uter <lb/><arrow.to.target n="marg382"/><lb/>que dimidium recti: igitur per dicta in <lb/>primo Elementorum Euclidis e n &etail;qua <lb/><arrow.to.target n="marg383"/><lb/>lis n k, igitur c q æqualis e n, quare h p <lb/>æqualis g o, &longs;ed quod fit ex o k in k g e&longs;t <lb/><arrow.to.target n="marg384"/><lb/>æquale ei, quod fit ex p k in k h, igitur <lb/><arrow.to.target n="marg385"/><lb/>k h e&longs;t æqualis k g ex eisdem o&longs;tendi<lb/>tur f l m k quadratum e&longs;&longs;e. </s>

<s id="id001845">Quia ergo <lb/>k h e&longs;t æqualis k g, & k l æqualis k m, erit l g æqualis m h. </s>

<s id="id001846">Er<lb/>go de&longs;cendendo ex g in f, quantum f l &longs;uperat l g, tantum de&longs;cen<lb/>dendo ex f in h, f m &longs;uperat m h per communem animi &longs;ententi<lb/>am. </s>

<s id="id001847">At f m e&longs;t de&longs;cen&longs;us f in linea a e, & m h di&longs;tantia, quæ acqui<lb/>ritur in linea f r, n m enim e&longs;t æqualis f r, igitur n h excedit f r in <lb/>h m, & ita a n excedit a r in n r &etail;quali f m. </s>

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<pb pagenum="103" xlink:href="015/01/122.jpg"/>mouebitur in h, quam in d, uelut &longs;it proportio f g ad a b dupla, ut <lb/>ergo æqualiter moueantur, &longs;i &longs;it dupla &longs;exquiquarta in d cum lan<lb/>ce ad e uacuam, erit in h &longs;exquialtera, & mouebit æquali tempore. <lb/></s>

<s id="id001876">Ergo iuxta hoc fient libræ, quæ examinabunt decimam, & uige&longs;i<lb/>mam partem grani, quod e&longs;t nece&longs;&longs;arium in precio&longs;is rebus, & me<lb/>dicamentis potentibus, & longè magis in mechanicis experimen<lb/>tis, & maximè quæ ad demon&longs;trationem pertinent magnitudinis <lb/>&longs;uperficierum, & con&longs;tat res in tribus, in longitudine, f g iungi, in le <lb/>uitate materiæ illius, & lancium, nam tanto maior redditur propor<lb/>tio ponderis exigui, & in firmitate iugi ac rectitudine. </s>

<s id="id001876">Ergo iuxta hoc fient libræ, quæ examinabunt decimam, & uige&longs;i<lb/>mam partem grani, quod e&longs;t nece&longs;&longs;arium in pretio&longs;is rebus, & me<lb/>dicamentis potentibus, & longè magis in mechanicis experimen<lb/>tis, & maximè quæ ad demon&longs;trationem pertinent magnitudinis <lb/>&longs;uperficierum, & con&longs;tat res in tribus, in longitudine, f g iungi, in le <lb/>uitate materiæ illius, & lancium, nam tanto maior redditur propor<lb/>tio ponderis exigui, & in firmitate iugi ac rectitudine. </s>

<s id="id001877">ideò debet <lb/>fieri ex chalybe purgato, durato ac tenui&longs;simo, natura que leui, & ut c <lb/>&longs;it in medio, & mobilis f g.</s>

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<s id="id001885">nam quæ huiu&longs;modi po<lb/>te&longs;t e&longs;&longs;e doctrina ni&longs;i perfecta fuerit, in omnibus etenim nece&longs;&longs;e e&longs;t <lb/>aut omnia &longs;cire, aut ignorare. </s>

<s id="id001886">In hoc igitur dico, quod h f, &longs;eu l f, <lb/>&longs;emper æquidi&longs;tant n c trutinæ, ergo cum angulus f c n in clina<lb/>to iugo fiat obtu&longs;us de&longs;cendente pondere, & n c g a&longs;cendente pon<lb/>dere fiat acutus, ergo angulus l f c tantundem fiet obtu&longs;ior, & m g c <lb/>acutior, quanto anguli ad c tales &longs;unt. </s>

<s id="id001887">Et cau&longs;a e&longs;t quia n c ratio<lb/>ne ponderis e&longs;t directa ad centrum, ergo oportet, ut pondera l, uel <lb/>h, & m, uel k, &longs;i debent tendere ad centrum, ut f l, & g m æquidi<lb/>&longs;tent n c, ni&longs;i quantum e&longs;t pro di&longs;tantia f, à puncto c, & g a b eodem, <lb/>quæ comparata ad <expan abbr="centr&utilde;">centrum</expan> terr&etail;, &longs;eu mundi, e&longs;t in&longs;en&longs;ibilis omnino. <lb/></s>

<s id="id001888">Circa hæc <expan abbr="notand&utilde;">notandum</expan> i&longs;tud mirabile fcilicet, quod ratio motus, quan<lb/>tumuis exigua &longs;ufficit ad motus <expan abbr="mod&utilde;">modum</expan>, licet uelo citas <expan abbr="p&etilde;deat">pendeat</expan> ex gra<lb/>uitate, & alijs. </s>

<s id="id001888">Circa hæc <expan abbr="notand&utilde;">notandum</expan> i&longs;tud mirabile &longs;cilicet, quod ratio motus, quan<lb/>tumuis exigua &longs;ufficit ad motus <expan abbr="mod&utilde;">modum</expan>, licet uelo citas <expan abbr="p&etilde;deat">pendeat</expan> ex gra<lb/>uitate, & alijs. </s>

<s id="id001889">Et quae graue, quod expers e&longs;t &longs;en&longs;us, debeat &longs;equi ratio <lb/>nem Geometricam uix &longs;apientibus <expan abbr="cognitã">cognitam</expan>, cau&longs;a tamen una e&longs;t, & <lb/>per&longs;picua: <expan abbr="nã">nam</expan> omne graue e&longs;t in linea à centro <expan abbr="m&utilde;di">mundi</expan>: &longs;i <expan abbr="a&utilde;t">aunt</expan> medium <lb/>grauis &longs;it extra <expan abbr="lineã">lineam</expan>, uertitur ad illam, qu&etail; e&longs;t in eo, nam <expan abbr="centr&utilde;">centrum</expan> &longs;em

<s id="id001889">Et quae graue, quod expers e&longs;t &longs;en&longs;us, debeat &longs;equi ratio <lb/>nem Geometricam uix &longs;apientibus <expan abbr="cognitã">cognitam</expan>, cau&longs;a tamen una e&longs;t, & <lb/>per&longs;picua: <expan abbr="nã">nam</expan> omne graue e&longs;t in linea à centro <expan abbr="m&utilde;di">mundi</expan>: &longs;i <expan abbr="a&utilde;t">aut</expan> medium <lb/>grauis &longs;it extra <expan abbr="lineã">lineam</expan>, uertitur ad illam, qu&etail; e&longs;t in eo, nam <expan abbr="centr&utilde;">centrum</expan> &longs;em<pb pagenum="104" xlink:href="015/01/123.jpg"/>per e&longs;t in <expan abbr="ead&etilde;">eadem</expan>. </s>

<pb pagenum="104" xlink:href="015/01/123.jpg"/>per e&longs;t in <expan abbr="ead&etilde;">eadem</expan>. </s>

<s id="id001890">Ergo &longs;ola in clinatio ad hoc ut <expan abbr="medi&utilde;">medium</expan> grauis &longs;it in li<lb/>nea <expan abbr="centror&utilde;">centrorum</expan> grauitatis & terræ, &longs;ufficit. </s>

<s id="id001891">E&longs;t ergo principium in &longs;ei<lb/>p&longs;o. </s>

<s id="id001892">In appen&longs;is &longs;imiliter. </s>

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<s id="id001898">Propo&longs;itio cente&longs;imadecima.</s>

<s id="id001899">Si duæ &longs;phæræ ex eadem materia de&longs;cendant in <expan abbr="a&etilde;">aem</expan> <lb/>re eodem temporis momento ad planum ueniunt.<lb/><figure id="id.015.01.123.1.jpg" xlink:href="015/01/123/1.jpg"/><lb/><arrow.to.target n="marg391"/></s>

<s id="id001899">Si duæ &longs;phæræ ex eadem materia de&longs;cendant in <expan abbr="a&etilde;">ae</expan> <lb/>re eodem temporis momento ad planum ueniunt.<lb/><figure id="id.015.01.123.1.jpg" xlink:href="015/01/123/1.jpg"/><lb/><arrow.to.target n="marg391"/></s>

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<s id="id001903">Sit ergo a tripla <lb/>ad b, &longs;phærula ad &longs;phærulam ex plumbo ambæ fer<lb/>ro uel lapide eiu&longs;dem generis, dico, quòd inæquali <lb/>tempore peruenient ad planum c d. </s>

<s id="id001904">Nam a propor<lb/>tionem habet ad b, ut uiginti&longs;eptem ad unum. </s>

<s id="id001905">pro<lb/>portio autem &longs;patij a ad &longs;patium b nonupla e&longs;t, & <lb/>proportio den&longs;itatis aëris ad aërem e&longs;t tripla, propterea quod den<lb/>&longs;itas illa multiplicatur propter impetus magnitudinem. </s>

<s id="id001906">nam &longs;i ro<lb/>bur, ut decem percutiat baculo lato, ut quatuor ictus erit maior du<lb/>plo, quàm &longs;it robur, ut quinque percutiat baculo, ut duo: propter <lb/>den&longs;itatem ergo maiorem aëris in a, quam in b: & quoniam &longs;i &longs;ub <lb/>maiore impetu mouetur <expan abbr="a&etilde;r">aerr</expan> &longs;ub a, quam &longs;ub b, igitur proportio <lb/>erit comparanda longitudini à centro a ad longitudinem a centro <lb/>b, quæ e&longs;t tripla. </s>

<s id="id001906">nam &longs;i ro<lb/>bur, ut decem percutiat baculo lato, ut quatuor ictus erit maior du<lb/>plo, quàm &longs;it robur, ut quinque percutiat baculo, ut duo: propter <lb/>den&longs;itatem ergo maiorem aëris in a, quam in b: & quoniam &longs;i &longs;ub <lb/>maiore impetu mouetur <expan abbr="a&etilde;r">aer</expan> &longs;ub a, quam &longs;ub b, igitur proportio <lb/>erit comparanda longitudini à centro a ad longitudinem a centro <lb/>b, quæ e&longs;t tripla. </s>

<s id="id001907">Si ergo &longs;ubtripla e&longs;t ratio motus b ad a, quod <lb/>ad medium attinet, tripla autem propter uelo citatem di&longs;ce&longs;&longs;us aë<lb/>ris à medio grauitatis, quod e&longs;t in &longs;uperficie e regione centri graui<lb/>tatis in linea ad centrum mundi, ut dictum e&longs;t in præcedenti: mani<lb/>fe&longs;tum e&longs;t, quod a, & b inæquali tempore peruenient ad &longs;ubie<lb/>ctum planum, & æquidi&longs;tans centris eorum. </s>

<s id="id001908">Similiter & in aqua:

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<pb pagenum="106" xlink:href="015/01/125.jpg"/>cilius &longs;it in anteriore parte propinquius comprehen&longs;um calci, & &longs;i <lb/>cra&longs;sius, uel grauius propius cu&longs;pidi. </s>

<s id="id001922">Semper igitur ob hanc cau<lb/>&longs;am mota ex medio grauitatis, &longs;eu uelo, &longs;eu ramo, &longs;eu manu uelo<lb/>cius mouentur, quàm ex alijs partibus. </s>

<s id="id001923">In remo etiam pote&longs;t acce<lb/>dere illud commodum, cuius meminit Ari&longs;tcteles. </s>

<s id="id001923">In remo etiam pote&longs;t acce<lb/>dere illud commodum, cuius meminit Ari&longs;toteles. </s>

<s id="id001924">Propter hoc igi <lb/>tur, qui malum in naui collo carunt tantùm unum, in medio fermè <lb/>eum collocarunt, ut antiqui: & qui duos aut tres, maiorem cra&longs;sio<lb/><arrow.to.target n="marg396"/><lb/>rem &longs;cilicet, & altiorem in medio con&longs;tituerunt.</s>

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<s id="id001941">Nec tamen maiore impetu, quia obliquè fertur, & quæ obliquè <lb/><expan abbr="feri&utilde;t">feriunt</expan>, minore cum impetu feriunt: at que eo magis &longs;i leuia fuerint: ab <lb/>aëre enim circumambiente perturbantur, & in incertum trudun<lb/>tur. </s>

<s id="id001942">Quæ ergo grauia &longs;unt ex medio emi&longs;&longs;a, & ad æquidi&longs;tantem <lb/>longius feruntur, & maiore cum impetu, quia magis directè: leuia <lb/>autem longius ex imo, &longs;ed minore cum impetu, &longs;i aliqua cau&longs;a à re<lb/>cto, & æquidi&longs;tante declinauerint. </s>

<s id="id001943">At &longs;i à &longs;uprema parte, & iuxta <lb/>cu&longs;pidem, neque procul feruntur, neque cum impetu ob cau&longs;as di<lb/>ctas. </s>

<s id="id001944">Eadem quoque ratio e&longs;t omnium machinarum: ideò oblon<lb/>g&etail;longius eiaculantur, quoniam proportionem &longs;eruant ad cana<lb/><arrow.to.target n="marg400"/><lb/>iem. </s>

<s id="id001944">Eadem quoque ratio e&longs;t omnium machinarum: ideò ob lon<lb/>g&etail; longius eiaculantur, quoniam proportionem &longs;eruant ad cana<lb/><arrow.to.target n="marg400"/><lb/>lem. </s>

<s id="id001945">Sed de hoc inferius agetur.</s>

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<pb pagenum="109" xlink:href="015/01/128.jpg"/>axungia. </s>

<s id="id001979">Et quae rota magna &longs;it: quoniam cum <expan abbr="nõ">non</expan> rota, &longs;ed axis traha<lb/>tur in æquali tempore & magna, & parua trahitur: utra que uerò una <lb/>conuer&longs;ione tantam <expan abbr="lineã">lineam</expan> rectam &longs;uperat, quanta e&longs;t rotæ periphe<lb/>ria. </s>

<s id="id001980">Quod &longs;i plures &longs;int rotæ celerius feruntur, quia axis minus tan<lb/>to <expan abbr="rotã">rotam</expan> premit. </s>

<s id="id001981">Et &longs;i rectus &longs;it axis, & bene rotundus, & foramen ro <lb/>tundum, & latius, & è duri&longs;simo ligno, ut non po&longs;sit in clinari: & <lb/>rota ip&longs;a in ambitu æqualis, omnia hæc faciunt ad motus uelo cita<lb/>tem, unde Homerus.<lb/><arrow.to.target n="marg405"/></s>

<s id="id001981">Et &longs;i rectus &longs;it axis, & bene rotundus, & foramen ro<lb/>tundum, & latius, & è duri&longs;simo ligno, ut non po&longs;sit in clinari: & <lb/>rota ip&longs;a in ambitu æqualis, omnia hæc faciunt ad motus uelo cita<lb/>tem, unde Homerus.<lb/><arrow.to.target n="marg405"/></s>

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<s id="id001985"><foreign lang="greek">I)/xnia tu/pte w_o/dessi w_a/r & ko/nin a)|mfi xuqu_nai</foreign>.</s>

<s id="id001985"><foreign lang="greek">i)/xnia tu/pte o/deosi a/r & ko/nin a)mfixuqῡnai</foreign>.</s>

<s id="id001986">Id e&longs;t, ue&longs;tigia per cu&longs;sit pedibus, ante que illa puluis pedibus ex<lb/>cu&longs;&longs;us (ue&longs;tigia &longs;cilicet relinquentibus) ingrederetur. </s>

<s id="id001987">Principalis <lb/>autem cau&longs;a uelo citatis e&longs;t agens, uelut equi. </s>

<s id="id001988">Sed inter <expan abbr="h&utilde;c">hunc</expan> motum <lb/>& priorem medius e&longs;t Scitalæ uocatæ, nam ut in primo axis proci<lb/>dit & rotundum à &longs;uperficie circumagitur, licet axis etiam circum<lb/>ducatur, ut axis, & rota, aut &longs;phæra duplici motu moueantur, fci<lb/>licet antror&longs;um, & circumcirca, in rota currus duo ijdem motus <lb/>&longs;int, axis quo que antror&longs;um moueatur, &longs;ed non circumagatur: unde <lb/>impeditior e&longs;t hic motus: ita in Scytala utrun que utro que motu mo<lb/>uetur, & circumcirca, & antror&longs;um, at que id commune e&longs;t, cum pri<lb/>mo ita axis mouet rotas, non rotæ axem, quòd &longs;ecundo motui ro<lb/>tarum in curru proprium e&longs;t, ut tantum degenerent à primo motu, <lb/>quanto leuius uertuntur, quàm in &longs;ecundo motu. </s>

<s id="id001988">Sed inter <expan abbr="h&utilde;c">hunc</expan> motum <lb/>& priorem medius e&longs;t Scitalæ uocatæ, nam ut in primo axis proci<lb/>dit & rotundum à &longs;uperficie circumagitur, licet axis etiam circum<lb/>ducatur, ut axis, & rota, aut &longs;phæra duplici motu moueantur, &longs;ci<lb/>licet antror&longs;um, & circumcirca, in rota currus duo ijdem motus <lb/>&longs;int, axis quo que antror&longs;um moueatur, &longs;ed non circumagatur: unde <lb/>impeditior e&longs;t hic motus: ita in Scytala utrunque utro que motu mo<lb/>uetur, & circumcirca, & antror&longs;um, at que id commune e&longs;t, cum pri<lb/>mo ita axis mouet rotas, non rotæ axem, quòd &longs;ecundo motui ro<lb/>tarum in curru proprium e&longs;t, ut tantum degenerent à primo motu, <lb/>quanto leuius uertuntur, quàm in &longs;ecundo motu. </s>

<s id="id001989">Trahitur ergo <lb/><figure id="id.015.01.128.1.jpg" xlink:href="015/01/128/1.jpg"/><lb/>iugum in &longs;citala, uelut in rotis currus, <lb/>&longs;ed e&longs;t annexum rotis non in curri<lb/>bus. </s>

<s id="id001990">Propterea in primo motu trahi<lb/>tur, uel impellitur à &longs;uperficie: in &longs;e<lb/>cundo a b axe, &longs;ed non affixo rotis, unde ægrè trahuntur in &longs;cyta<lb/>la ab axe affixo rot&etail;. </s>

<s id="id001991">Quare leuius quàm in curru, difficilius quàm <lb/>in rota uel &longs;phæra à &longs;uperficie extima circumacta. </s>

<s id="id001992">Quartus modus <lb/>e&longs;t, ut dixi, circumuecta rota ab axe, quum non progreditur, ut in <lb/>moletrinis, & rotis, quibus ferrum exacuitur. </s>

<s id="id001993">E&longs;t enim hic &longs;imilior <lb/>primo, quia contrarius, in primo enim procedit rota, & uertitur à <lb/>circumferentia, hic quie&longs;cit rota, & mouetur ab axe. </s>

<s id="id001994">Proximus huic <lb/>e&longs;t, qui fit in &longs;ucculis ob firmitatem axis: nam axis e&longs;t coniunctus <lb/>rotæ. </s>

<s id="id001995">Vltimus e&longs;t trochlearum, qui & difficillimus: &longs;it enim à cir<lb/>cunferentia, & axis di&longs;iunctus e&longs;t à trochlea: quod ad dit difficulta<lb/>tem. </s>

<s id="id001995">Vltimus e&longs;t trochlearum, qui & difficillimus: &longs;it enim à cir<lb/>cumferentia, & axis di&longs;iunctus e&longs;t à trochlea: quod ad dit difficulta<lb/>tem. </s>

<s id="id001996">Sed & trochlea caret colloppibus. </s>

<s id="id001997">Ergo uerum e&longs;t, quod o<lb/>mnia rotunda facilius circumaguntur, &longs;ed uaria ratione: nam plus <lb/>mota &longs;uper aliquo plano, ut in plau&longs;tris & &longs;cytalis: minus in &longs;uccu<lb/>lis, & rotis acuentibus ferrum, & molis: nam & &longs;i rotun ditatem iu<lb/>uet ob æqualitatem ad conuer&longs;ionem, non tamen in his e&longs;t ad eò

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<s id="id002021"><arrow.to.target n="marg408"/><lb/>&longs;is (uocant cerbatanas) non pote&longs;t &longs;atisfacere, cum tamen idem &longs;e<lb/>quatur in his, ut in illis uidetur, qua&longs;i uis e&longs;&longs;e in &longs;phærula &longs;ic emi&longs;<lb/>&longs;a, & non in aëre, quemadmodum dicebamus, coniuncto e&longs;&longs;e. </s>

<s id="id002022">Ex <lb/>quo nece&longs;&longs;e e&longs;&longs;et, ut quod longius ferretur, etiam ualidiores ictus <lb/><figure id="id.015.01.130.1.jpg" xlink:href="015/01/130/1.jpg"/><lb/>inferret, hoc autem <lb/>non ita &longs;e habet, &longs;ed <lb/>ictus magnitud o <lb/>ex robore machi<lb/>narum tam ignea<lb/>rum, quam &longs;corpio <lb/>num pendet, nam <lb/>&longs;it a &longs;corpio ma<lb/>gnus, &longs;ed tenuis, ex <lb/>hòc palam e&longs;t lon<lb/>gius mittere &longs;agit<lb/>tam, quòd à parua, <lb/>& breui, quantun<lb/>uis cra&longs;&longs;a non lon<lb/>ge mittitur: at uerò <lb/>quod b cra&longs;&longs;us & paruus maiore cum impetu mittat o&longs;tenditur <lb/>nam ea pondera &longs;agittæ mouet, quæ non pote&longs;t mouere a, igitur b <lb/>ualidiore robore mouet, quam a. </s>

<s id="id002023">Prætera illud o&longs;ten dit iugum fu<lb/>nis arcus cra&longs;siora duriora, quæ maioribus uiribus <expan abbr="indig&etilde;t">indigent</expan>, quam <lb/>a, qui à puero tendi poterit. </s>

<s id="id002023">Præterea illud o&longs;tendit iugum fu<lb/>nis arcus cra&longs;siora duriora, quæ maioribus uiribus <expan abbr="indig&etilde;t">indigent</expan>, quam <lb/>a, qui à puero tendi poterit. </s>

<s id="id002024">Non e&longs;t ergo eadem ratio mittendi <lb/>longius, & ualidiore cum robore. </s>

<s id="id002025">Eadem ergo cum ratio &longs;it in <lb/>machinis igneis, cra&longs;siores enim, & latiores ac breuiores magis <lb/>concutiunt, quam longiores tenuiores minoris &longs;phæræ capaces: <lb/>non &longs;olum ob mag nitudinem &longs;phæræ magis illæ concutiunt, &longs;ed, <lb/>ut dixi, ob maiorem impetus uim: cau&longs;a ergo e&longs;t manife&longs;ta in his, <lb/>&longs;ed non cau&longs;a, qua longius ferantur in longiore canali. </s>

<s id="id002026">Sed uide

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<s id="id002032">E&longs;t enim ictus robur <lb/>non à uelo citate &longs;olum, &longs;ed maiore ex ponderis grauitate, quæ &longs;ola <lb/>premit, urget, & frangit etiam &longs;ine motu. </s>

<s id="id002033">Solum ergo id re&longs;tat du<lb/>bium, cur &longs;i grauius e&longs;t, moueatur eodem ferm é impetu: nam quo <lb/>maiore impetu fertur, eo longius fertur, non tamen magis ferit, con <lb/>cutit, aut qua&longs;&longs;at, &longs;ed grauitas ad hoc plus facit impetu. </s>

<s id="id002034">Palea maxi<lb/>mo impetu demi&longs;&longs;a non ferit, non ledit, & celerius de&longs;cendit, fer<lb/>rum &longs;ola grauitate actum, imò etiam temperato ictu lædit graui<lb/>ter, qua&longs;&longs;at, & frangit: itaque f maiore indiget quantitate pyrij pulue<lb/>ris, quàm e: &longs;iquidem tertia parte ponderis &longs;uæ &longs;phæræ: at maius <lb/>e&longs;t pondus f quam e, ergo maius pondus pulueris f quàm e, ergo <lb/>maior uehementia ictus, &longs;iquidem ea &longs;equitur, robur cau&longs;æ mouen <lb/>tis &longs;im pliciter: ut concludamus longitudinem ictus &longs;equi propor<lb/>tionem motoris ad motum, &longs;ed uehementia robur motoris: nam &longs;i <lb/>ex portione mouet æquale pondus maiore cum impetu mouet, <lb/>quoniam maior e&longs;t proportio: &longs;i minore igitur pondus maius e&longs;t, <lb/>&, ut dixi plus facit magnitudo ponderis cum leui ictu, quàm ma<lb/>gnitudo ictus cum leui pondere. </s>

<s id="id002035">Quæ ergo feruntur per longio<lb/>res canales maiore impetu feruntur, & &longs;ocietatem <expan abbr="hab&etilde;t">habent</expan> aëris moti <lb/>per longius <expan abbr="&longs;pati&utilde;">&longs;patium</expan>, ut tardius remittatur, quia longiore tempore <expan abbr="u&itilde;s">uins</expan> <lb/>motus confirmata e&longs;t, & proportio eius, quòd mouet, maior e&longs;t ad id, <lb/>quod <expan abbr="moue&ttilde;">mouetur</expan>, quia minus extenditur, at uerò f <expan abbr="mot&utilde;">motum</expan> minore propor<lb/>tione <expan abbr="ict&utilde;">ictum</expan> facit <expan abbr="maior&etilde;">maiorem</expan>, proa, ut dixi, <expan abbr="tãto">tanto</expan> grauius, e&longs;t quod ferit. </s>

<s id="id002035">Quæ ergo feruntur per longio<lb/>res canales maiore impetu feruntur, & &longs;ocietatem <expan abbr="hab&etilde;t">habent</expan> aëris moti <lb/>per longius <expan abbr="&longs;pati&utilde;">&longs;patium</expan>, ut tardius remittatur, quia longiore tempore <expan abbr="u&itilde;s">uis</expan><lb/>motus confirmata e&longs;t, & proportio eius, quòd mouet, maior e&longs;t ad id, <lb/>quod <expan abbr="moue&ttilde;">mouetur</expan>, quia minus extenditur, at uerò f <expan abbr="mot&utilde;">motum</expan> minore propor<lb/>tione <expan abbr="ict&utilde;">ictum</expan> facit <expan abbr="maior&etilde;">maiorem</expan>, quia, ut dixi, <expan abbr="tãto">tanto</expan> grauius, e&longs;t quod ferit. </s>

<s id="id002036">Quod <lb/><expan abbr="aut&etilde;">autem</expan> minus <expan abbr="ext&etilde;datur">extendatur</expan> machina a b quam c d, <expan abbr="n&utilde;c">nunc</expan> <expan abbr="o&longs;t&etilde;dere">o&longs;tendere</expan> oporter.</s>

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<s id="id002044">Nam <lb/>qui &longs;uccutiunt, ubi pondus, aut coniunctio maior &longs;it, quàm ut di<lb/>&longs;trahere po&longs;sint, conden&longs;ant partes proximiores, & rimas faciunt, <lb/>per quas exhalat ignis aut omnino extinguitur, aut conden&longs;atur. <lb/></s>

<s id="id002045">At ergo in bellicis machinis, minus dilatat puluis, cum fuerit in lon <lb/>go canali, ob id ergo maiore impetu feruntur per illas, quàm per <lb/>breuiores, etiam quòd minor &longs;it puluis, minor &longs;it ignis. </s>

<s id="id002046">Experimen <lb/>tum facies in canali, ubi &longs;ambuci medulla pro globulo flatu impel<lb/>lente expellitur ab&longs; que periculo: nam quanto minor fuerit canalis <lb/>ambitu ac longior eo maiore impetu pellitur. </s>

<s id="id002047">For&longs;an qui&longs;piam nos <lb/>meritò poterit uideri <expan abbr="repreh&etilde;di&longs;&longs;e">reprehendi&longs;&longs;e</expan>, quòd inanis gloriæ &longs;tudio per<lb/>nitio&longs;a humano generi do ceam. </s>

<s id="id002048">Quibus re&longs;pondeo, me nihil do cu <lb/>i&longs;&longs;e, quod ín humani generis detrimentum cedat, huiu&longs;mo di que pr&etail;<lb/>cepta iam ob&longs;cura&longs;&longs;e, ut ne quid mali accidere po&longs;&longs;et hominibus ex <lb/>his: <expan abbr="nã">nam</expan> quòd ad ea, quæ declarata, &longs;unt, cau&longs;as &longs;olùm retuli, effectus <lb/>ip&longs;imodi artis <expan abbr="nimi&utilde;">nimium</expan> feruntur, ac nimio plu&longs;quam <expan abbr="uell&etilde;">uellem</expan> in telligun<lb/>tur. </s>

<s id="id002049">Vt cum ad copiam, ad magnitudinem, ad coacta imperia mi&longs;e<lb/>rorum re&longs;picio, nihil plus po&longs;sit addi. </s>

<s id="id002050">Omnia enim hucu&longs; que <expan abbr="&longs;pectãt">&longs;pectant</expan> <lb/>ad potentiorum in crementa. </s>

<s id="id002050">Omnia enim huiu&longs;que <expan abbr="&longs;pectãt">&longs;pectant</expan> <lb/>ad potentiorum in crementa. </s>

<s id="id002051">An ergo &longs;uccurrere afflictis, ob&longs;e&longs;sis, <lb/>cinctis, æquare <expan abbr="condition&etilde;">conditionem</expan>, liberare à &longs;eruitute etiam rebelles <expan abbr="nõ">non</expan> li<lb/>cebit? </s>

<s id="id002052">Ab initio fuimus omnes liberi: excogitata fuit regni ratio ad <lb/>commodum hominum, ea uer&longs;a e&longs;t per uim in <expan abbr="Tyrannid&etilde;">Tyrannidem</expan>. </s>

<s id="id002053">Subtili <lb/>ergo ratione <expan abbr="occurrend&utilde;">occurrendum</expan> e&longs;t imbecillioribus: <expan abbr="nã">nam</expan> reliqua omnia ni<lb/>mis, ut dixi, qu&etail; ad cuniculos ad <expan abbr="magnitudin&etilde;">magnitudinem</expan> <expan abbr="machinar&utilde;">machinarum</expan> ad rectos <lb/>ictus ad <expan abbr="libram&etilde;ta">libramenta</expan> ad longitudinem &longs;pacij, per quos globus ille de<lb/>fertur, nota &longs;unt improbis illis artificibus, nec no&longs;trum e&longs;t &longs;pectare, <lb/>cur id licuerit, po&longs;tquam Deus hanc uiolentiam e&longs;&longs;e uoluit. </s>

<s id="id002053">Subtili <lb/>ergo ratione <expan abbr="occurrend&utilde;">occurrendum</expan> e&longs;t imbecillioribus: <expan abbr="nã">nam</expan> reliqua omnia ni<lb/>mis, ut dixi, qu&etail; ad cuniculos ad <expan abbr="magnitudin&etilde;">magnitudinem</expan> <expan abbr="machinar&utilde;">machinarum</expan> ad rectos <lb/>ictus ad <expan abbr="libram&etilde;ta">libramenta</expan> ad longitudinem &longs;patij, per quos globus ille de<lb/>fertur, nota &longs;unt improbis illis artificibus, nec no&longs;trum e&longs;t &longs;pectare, <lb/>cur id licuerit, po&longs;tquam Deus hanc uiolentiam e&longs;&longs;e uoluit. </s>

<s id="id002054">Multa <lb/>damnamus, <expan abbr="&qtilde;">quae</expan> Deus e&longs;&longs;e uult: boni uiri e&longs;t <expan abbr="nõ">non</expan> ni&longs;i opitulari homini<lb/>bus, <expan abbr="etiã">etiam</expan> malis modo bonis futuri <expan abbr="nõ">non</expan> &longs;int <expan abbr="impedim&etilde;to">impedimento</expan>: <expan abbr="quamobr&etilde;">quamobrem</expan>

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<s id="id002059">Sit paries b d e, ex a <expan abbr="fera&ttilde;">feratur</expan> in dictus, qui &longs;i <lb/><arrow.to.target n="marg410"/><lb/>e&longs;&longs;et in c d <expan abbr="pariet&etilde;">parietem</expan> e&longs;&longs;e ad perpendiculum, & <lb/>ualidi&longs;simus, &longs;in uero in f g abraderet, & <expan abbr="nõ">non</expan> <lb/><expan abbr="cõqua&longs;&longs;aret">conqua&longs;&longs;aret</expan>. </s>

<s id="id002060">Quæritur ergo ex b d e muro <lb/>qualis excipietur? </s>

<s id="id002061">erit ergo proportio anguli c d a ad <expan abbr="angul&utilde;">angulum</expan> b d a, <lb/>ueluti ictus a d in d c ad <expan abbr="ict&utilde;">ictum</expan> in b d, <expan abbr="manife&longs;t&utilde;">manife&longs;tum</expan> e&longs;t <expan abbr="a&utilde;t">aunt</expan> &longs;equi proportio<lb/>nem, <expan abbr="quoniã">quoniam</expan> maxima uarietate <expan abbr="cõ&longs;tat">con&longs;tat</expan> dum ex angulo b d a acuto fit <lb/>acutior, <expan abbr="quoniã">quoniam</expan> &longs;i b d c &longs;it <expan abbr="&qtilde;druplus">quadruplus</expan> b d a erit re&longs;iduus ad <expan abbr="dimidi&utilde;">dimidium</expan> b <lb/>d a nonuplus ip&longs;i dimidio, & ad <expan abbr="quartã">quartam</expan> <expan abbr="part&etilde;">partem</expan> habebit proportionem <lb/><expan abbr="decemnou&etilde;">decemnouem</expan> ad <expan abbr="un&utilde;">unum</expan>. </s>

<s id="id002061">erit ergo proportio anguli c d a ad <expan abbr="angul&utilde;">angulum</expan> b d a, <lb/>ueluti ictus a d in d c ad <expan abbr="ict&utilde;">ictum</expan> in b d, <expan abbr="manife&longs;t&utilde;">manife&longs;tum</expan> e&longs;t <expan abbr="a&utilde;t">aut</expan> &longs;equi proportio<lb/>nem, <expan abbr="quoniã">quoniam</expan> maxima uarietate <expan abbr="cõ&longs;tat">con&longs;tat</expan> dum ex angulo b d a acuto fit <lb/>acutior, <expan abbr="quoniã">quoniam</expan> &longs;i b d c &longs;it <expan abbr="&qtilde;druplus">quadruplus</expan> b d a erit re&longs;iduus ad <expan abbr="dimidi&utilde;">dimidium</expan> b <lb/>d a nonuplus ip&longs;i dimidio, & ad <expan abbr="quartã">quartam</expan> <expan abbr="part&etilde;">partem</expan> habebit proportionem <lb/><expan abbr="decemnou&etilde;">decem nouem</expan> ad <expan abbr="un&utilde;">unum</expan>. </s>

<s id="id002062">Si ergo <expan abbr="etiã">etiam</expan> in <expan abbr="id&etilde;">idem</expan> tenderent, <expan abbr="nõ">non</expan> efficerent mille <lb/>ictus &qring;d tres, cuius demon&longs;tratio h&etail;c e&longs;t. </s>

<s id="id002063">Supponamus <expan abbr="proportion&etilde;">proportionem</expan> <lb/>b d c ad <expan abbr="&qtilde;rtam">quartam</expan> <expan abbr="part&etilde;">partem</expan> a d b ad dito re&longs;iduo ad b d c e&longs;&longs;e <expan abbr="&longs;ol&utilde;">&longs;olum</expan> <expan abbr="decuplã">decuplam</expan>: <lb/><expan abbr="t&utilde;c">tunc</expan> ex duob. </s>

<s id="id002064">ictibus centupla erit in d c ad <expan abbr="eã">eam</expan>, qu&etail; in b e, <expan abbr="etiã">etiam</expan> tribus <lb/>millecupla: nam <expan abbr="cõqua&longs;&longs;ata">conqua&longs;&longs;ata</expan> turri in primo ictu, id d decuplo magis <lb/>ad perpendiculum <08> in b d e <expan abbr="&longs;uma&ttilde;">&longs;umatur</expan> decima pars in ambitu d, & illa <lb/>erit ergo <expan abbr="tã">tam</expan> di&longs;&longs;oluta, & infirma ex &longs;uppo&longs;ito, <08> e&longs;t tota b e: &longs;ed ex &longs;e <lb/>cundo ictu decuplo magis <expan abbr="cõqua&longs;&longs;abi&ttilde;">conqua&longs;&longs;abitur</expan> illa pars, <08> b e ergo tota d c <lb/>centuplo magis <expan abbr="qua&longs;&longs;abi&ttilde;">qua&longs;&longs;abitur</expan> ex duob. </s>

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<s id="id002075">Quantum ictus machin&etail; procliuis ad <expan abbr="angul&utilde;">angulum</expan> <expan abbr="minua&ttilde;">minuatur</expan> explorare.</s>

<s id="id002076">Huiu&longs;ce cau&longs;a <expan abbr="excogitar&utilde;t">excogitarunt</expan>, ut ictus ad <expan abbr="perpendicul&utilde;">perpendiculum</expan> <expan abbr="dirigere&ttilde;">dirigeretur</expan>, & <lb/><arrow.to.target n="marg411"/><lb/><expan abbr="quanquã">quanquam</expan> angulus d e f &longs;it &etail;quali angulo a b c, longè <expan abbr="tñ">tnm</expan> maior e&longs;t uis <lb/>a b <08> d e duplici cau&longs;a, & <expan abbr="quoniã">quoniam</expan> a b e&longs;t <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> nat uram impetus <lb/><figure id="id.015.01.134.1.jpg" xlink:href="015/01/134/1.jpg"/><lb/>ignis, & <expan abbr="etiã">etiam</expan> <expan abbr="eor&utilde;">eorum</expan>, qu&etail; <expan abbr="emittun&ttilde;">emittuntur</expan> in altum: & &qring;d pars <lb/>&longs;uperior in b retineat <expan abbr="ict&utilde;">ictum</expan>, in e non retineat. </s>

<s id="id002076">Huiu&longs;ce cau&longs;a <expan abbr="excogitar&utilde;t">excogitarunt</expan>, ut ictus ad <expan abbr="perpendicul&utilde;">perpendiculum</expan> <expan abbr="dirigere&ttilde;">dirigeretur</expan>, & <lb/><arrow.to.target n="marg411"/><lb/><expan abbr="quanquã">quanquam</expan> angulus d e f &longs;it &etail;quali angulo a b c, longè <expan abbr="tñ">tamen</expan> maior e&longs;t uis <lb/>a b <08> d e duplici cau&longs;a, & <expan abbr="quoniã">quoniam</expan> a b e&longs;t <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> nat uram impetus <lb/><figure id="id.015.01.134.1.jpg" xlink:href="015/01/134/1.jpg"/><lb/>ignis, & <expan abbr="etiã">etiam</expan> <expan abbr="eor&utilde;">eorum</expan>, qu&etail; <expan abbr="emittun&ttilde;">emittuntur</expan> in altum: & &qring;d pars <lb/>&longs;uperior in b retineat <expan abbr="ict&utilde;">ictum</expan>, in e non retineat. </s>

<s id="id002077">Sed caui <lb/>tas fiat maior in inferiore parte: cuius <expan abbr="experim&etilde;tum">experimentum</expan> <lb/>quiliber facere pote&longs;t <expan abbr="c&utilde;">cum</expan> ha&longs;ta. </s>

<s id="id002078">Huic ergo &longs;olertiæ, <expan abbr="&qtilde;">quae</expan> <lb/>tormenta iubet altius collocare ob&longs;tat <expan abbr="prim&utilde;">primum</expan>, quod <lb/>ictus ex decliui &longs;itu periculo&longs;ior e&longs;t pro machina, & ma <lb/>ximè &qring;d retro impellit, quae ex retro ce&longs;&longs;a, po&longs;t <08> exone <lb/>rata e&longs;t, <expan abbr="digno&longs;ci&ttilde;">digno&longs;citur</expan>, & ad <expan abbr="collimand&utilde;">collimandum</expan> decedit parte <expan abbr="uiri&utilde;">ui<lb/>rium</expan> &longs;uarum, &qring;d et&longs;i <expan abbr="paru&utilde;">paruum</expan> &longs;it in ductu <expan abbr="tñ">tnm</expan>, & <expan abbr="ictu&utilde;">ictuum</expan> mul <lb/>tiplicatione <expan abbr="magn&utilde;">magnum</expan> affert di&longs;crimen. </s>

<s id="id002078">Huic ergo &longs;olertiæ, <expan abbr="&qtilde;">quae</expan> <lb/>tormenta iubet altius collocare ob&longs;tat <expan abbr="prim&utilde;">primum</expan>, quod <lb/>ictus ex decliui &longs;itu periculo&longs;ior e&longs;t pro machina, & ma<lb/>ximè &qring;d retro impellit, quae ex retro ce&longs;&longs;a, po&longs;t <08> exone<lb/>rata e&longs;t, <expan abbr="digno&longs;ci&ttilde;">digno&longs;citur</expan>, & ad <expan abbr="collimand&utilde;">collimandum</expan> decedit parte <expan abbr="uiri&utilde;">ui<lb/>rium</expan> &longs;uarum, &qring;d et&longs;i <expan abbr="paru&utilde;">paruum</expan> &longs;it in ductu <expan abbr="tñ">tamen</expan>, & <expan abbr="ictu&utilde;">ictuum</expan> mul<lb/>tiplicatione <expan abbr="magn&utilde;">magnum</expan> affert di&longs;crimen. </s>

<s id="id002079">Habet & <expan abbr="cõmo">commo</expan> <lb/>dum &longs;itus muri accliuis <expan abbr="terrã">terram</expan> <expan abbr="&longs;uppo&longs;itã">&longs;uppo&longs;itam</expan> ad perpendiculum, <expan abbr="&qtilde;">quae</expan> ictum <lb/>&longs;u&longs;tinet: adeò ut omnib. </s>

<s id="id002080"><expan abbr="inuic&etilde;">inuicem</expan> collectis, perinde &longs;it ac &longs;i ex perpen<lb/>diculo, et &etail;quidi&longs;tanti ad <expan abbr="&longs;ol&utilde;">&longs;olum</expan> <expan abbr="feria&ttilde;">feriatur</expan>. </s>

<s id="id002081">Venetus. </s>

<s id="id002082">S. aliter Patauij cauit, <lb/>uidetur que, quae &longs;apienti&longs;simus &longs;it, & eandem &longs;equatur ubi que normam, <lb/>po&longs;t <08> in <expan abbr="rotundã">rotundam</expan> figuram <expan abbr="tot&utilde;">totum</expan> urbis ambitum formauit, & fo&longs;&longs;a la <lb/>ta, ac pro fundi&longs;sima aqua que perenni muniuit, & <expan abbr="&longs;ummã">&longs;ummam</expan> muri partem <lb/><expan abbr="rotundã">rotundam</expan> in hunc <expan abbr="mod&utilde;">modum</expan> effecit <expan abbr="cauã">cauam</expan> que interius undi que, ne cuniculis <lb/><figure id="id.015.01.134.2.jpg" xlink:href="015/01/134/2.jpg"/><lb/>po&longs;&longs;et euerti, à lateribus uerò humiles, ac cra&longs;si&longs;simas turres, ut nul <lb/>la ui po&longs;&longs;ent dirui, eas que tormentis bellicis, undi que latera lu&longs;trantib. <lb/></s>

<s id="id002083">reple&longs;&longs;et, illud diligenti&longs;sime cauit, ne murus humilior e&longs;&longs;et aduer&longs;a <lb/>ripa, &longs;ed ad <expan abbr="libellã">libellam</expan> tamen depre&longs;&longs;us, ut <expan abbr="etiã">etiam</expan> machinis in terram exten <lb/>&longs;is &longs;ph&etail;rulæ non tangerent <expan abbr="mur&utilde;">murum</expan>: nam <expan abbr="c&utilde;">cum</expan> fo&longs;&longs;a &longs;it quadraginta pa&longs;<lb/>&longs;uum, excedat <expan abbr="a&utilde;t">aunt</expan> murus <expan abbr="exterior&etilde;">exteriorem</expan> aggerem uno pa&longs;&longs;u, ut quicquid <lb/>in ambitu e&longs;t uno ictu oculi cogno&longs;ci po&longs;sit, & aggeris angulus ma <lb/>ior &longs;it uno pa&longs;&longs;u, <expan abbr="t&utilde;">tum</expan> magis adiecta cra&longs;sitie machin&etail; fieri non pote&longs;t, <lb/>utictus in <expan abbr="mur&utilde;">murum</expan> dirigatur. </s>

<s id="id002083">reple&longs;&longs;et, illud diligenti&longs;sime cauit, ne murus humilior e&longs;&longs;et aduer&longs;a <lb/>ripa, &longs;ed ad <expan abbr="libellã">libellam</expan> tamen depre&longs;&longs;us, ut <expan abbr="etiã">etiam</expan> machinis in terram exten <lb/>&longs;is &longs;ph&etail;rulæ non tangerent <expan abbr="mur&utilde;">murum</expan>: nam <expan abbr="c&utilde;">cum</expan> fo&longs;&longs;a &longs;it quadraginta pa&longs;<lb/>&longs;uum, excedat <expan abbr="a&utilde;t">aut</expan> murus <expan abbr="exterior&etilde;">exteriorem</expan> aggerem uno pa&longs;&longs;u, ut quicquid <lb/>in ambitu e&longs;t uno ictu oculi cogno&longs;ci po&longs;sit, & aggeris angulus ma<lb/>ior &longs;it uno pa&longs;&longs;u, <expan abbr="t&utilde;">tum</expan> magis adiecta cra&longs;sitie machin&etail; fieri non pote&longs;t, <lb/>ut ictus in <expan abbr="mur&utilde;">murum</expan> dirigatur. </s>

<s id="id002084">Eam ob cau&longs;am <expan abbr="etiã">etiam</expan> cauit, ne <expan abbr="&etail;difici&utilde;">&etail;dificium</expan> ul<lb/><figure id="id.015.01.134.3.jpg" xlink:href="015/01/134/3.jpg"/><lb/>lum, aut planta, uel colliculus e&longs;&longs;et cir<lb/>cum circa <expan abbr="urb&etilde;">urbem</expan> ad tria M. P. laborat hoc <lb/>periculo h&etail;c urbs, ne tota &etail;dificijs euer<lb/>&longs;is concidat. </s>

<s id="id002085"><expan abbr="Turcar&utilde;">Turcarum</expan> enim Princeps di<lb/>dicit, ut in Nouo ca&longs;tro in Melit&etail; In&longs;ul&etail; <lb/>arce S. </s>

<s id="id002086">Elmi appellata plu&longs; <08> mille icti<lb/>bus in &longs;ingulos dies imo M D obtundere

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<s id="id002097">Exercitus magnus di&longs;&longs;oluitur, & &longs;emetip&longs;um con&longs;umit, &longs;i nulla fiat <lb/>acce&longs;sio aut exigua quomodo &longs;tabit: &longs;i magna auxilia omnia cor<lb/>rumpuntur. </s>

<s id="id002098">Contrà ob&longs;e&longs;sis auxilia &longs;i ueniant lu&longs;trata, & munita, et <lb/>omnibus nece&longs;&longs;arijs ornata uiri integri <expan abbr="cõtra">contra</expan> fatigatos, & fe&longs;&longs;os cor <lb/>pore, armati contra inermes, alacres contra torpidos &longs;uperueniunt. <lb/></s>

<s id="id002099">Ob id præcipuum e&longs;t auxilium pr&etail;ter h&etail;c his, qui oppugnantur co <lb/>pia militum, qui per initia nun <08> quie&longs;cant diu noctu que, <expan abbr="uer&utilde;">uerum</expan> noctu <lb/>duo tubicines per&longs;æpe <expan abbr="exercit&utilde;">exercitum</expan> <expan abbr="in&longs;omn&etilde;">in&longs;omnem</expan> in armis tota nocte <expan abbr="cõtine">contine</expan> <lb/><expan abbr="b&utilde;t">bunt</expan>. </s>

<s id="id002100">Serio <expan abbr="a&utilde;t">aunt</expan> die pugnare, & noctu <expan abbr="c&utilde;">cum</expan> minimè id <expan abbr="&longs;perãt">&longs;perant</expan>, & fatigati <lb/>&longs;unt: mira euenire &longs;olent in his in&longs;peratis, ac audacibus eruptionib. <lb/></s>

<s id="id002100">Serio <expan abbr="a&utilde;t">aut</expan> die pugnare, & noctu <expan abbr="c&utilde;">cum</expan> minimè id <expan abbr="&longs;perãt">&longs;perant</expan>, & fatigati <lb/>&longs;unt: mira euenire &longs;olent in his in&longs;peratis, ac audacibus eruptionib. <lb/></s>

<s id="id002101">per&longs;&etail;pe <expan abbr="etiã">etiam</expan> omnino &longs;upra <expan abbr="fid&etilde;">fidem</expan>. </s>

<s id="id002102">Ita <expan abbr="nõ">non</expan> conquie&longs;cere oportet donec, <lb/>uel omnino à cepto de&longs;inat ho&longs;tis, aut <expan abbr="loc&utilde;">locum</expan> occupet &longs;ibi <expan abbr="relict&utilde;">relictum</expan> po<lb/>tius <08> <expan abbr="qu&etilde;">quem</expan> elegerit. </s>

<s id="id002102">Ita <expan abbr="nõ">non</expan> conquie&longs;cere oportet donec, <lb/>uel omnino à capto de&longs;inat ho&longs;tis, aut <expan abbr="loc&utilde;">locum</expan> occupet &longs;ibi <expan abbr="relict&utilde;">relictum</expan> po<lb/>tius <08> <expan abbr="qu&etilde;">quem</expan> elegerit. </s>

<s id="id002103">nam <expan abbr="experiment&utilde;">experimentum</expan> frequens do cuit, ubi illæ ma <lb/>gn&etail; uires &longs;uo arbitrio <expan abbr="loc&utilde;">locum</expan>, <expan abbr="qu&etilde;">quem</expan> <expan abbr="eleger&utilde;t">elegerunt</expan> obtinere potuerint, <expan abbr="tand&etilde;">tandem</expan> <lb/>potiri locis <expan abbr="quãtumuis">quantumuis</expan> munitis in hoc &qring;d diximus <expan abbr="cõtra">contra</expan> <expan abbr="oppona&ttilde;">opponatur</expan>. <lb/></s>

<s id="id002104">Etenim <expan abbr="&longs;ept&etilde;">&longs;eptem</expan> modis <expan abbr="c&utilde;">cum</expan> urbes, at que arces <expan abbr="capian&ttilde;">capiantur</expan>, <expan abbr="quor&utilde;">quorum</expan> duo &longs;unt ex <lb/>tra <expan abbr="&ptilde;&longs;ent&etilde;">pn&longs;entem</expan> <expan abbr="con&longs;ideration&etilde;">con&longs;iderationem</expan> ob&longs;idio, <expan abbr="&qtilde;">quae</expan> magnitudine ambitus loci <expan abbr="tolli&ttilde;">tol<lb/>litur</expan>, & proditio, <expan abbr="&qtilde;">quae</expan> cu&longs;to <expan abbr="d&utilde;">dum</expan> <expan abbr="uigilãtia">uigilantia</expan>, cuniculi, euer&longs;io &longs;uperioris muri, <lb/>euer&longs;io ab imo per machinas, cuniculi, &longs;eu &longs;uffo&longs;sio, urbis euer&longs;io, &longs;eu <lb/><expan abbr="&etail;dificior&utilde;">&etail;dificiorum</expan>: & <expan abbr="&qtilde;uo">quauo</expan> cant aggre&longs;sio, &longs;eu oppugnatio per &longs;calas, & crates <lb/><expan abbr="c&utilde;">cum</expan> &longs;agittarijs: his omnib. </s>

<s id="id002105"><expan abbr="&longs;atisfact&utilde;">&longs;atisfactum</expan> puto, pr&etail;ter <08> oppugnationi pro<lb/>pter <expan abbr="humilitat&etilde;">humilitatem</expan> <expan abbr="muror&utilde;">murorum</expan>: <expan abbr="nã">nam</expan> lignis <expan abbr="opplen&ttilde;">opplentur</expan>, at que fa&longs;ciculis, terra que fo&longs; <lb/>&longs;&etail;: nihil. </s>

<s id="id002107">re&longs;i&longs;tit immen&longs;&etail; illi pote&longs;tati, & crudelitati <expan abbr="&longs;&etail;ui&longs;simor&utilde;">&longs;&etail;ui&longs;simorum</expan> ty <lb/><expan abbr="rãnor&utilde;">rannorum</expan>. </s>

<s id="id002108"><expan abbr="Ver&utilde;">Verum</expan>, ut dixi, terra noctu <expan abbr="effodi&ttilde;">effoditur</expan>, ligna artificio&longs;is ignib. </s>

<s id="id002109">eru

<s id="id002109">eru<pb pagenum="117" xlink:href="015/01/136.jpg"/>untur. </s>

<pb pagenum="117" xlink:href="015/01/136.jpg"/>untur. </s>

<s id="id002110">Et longum e&longs;t opus &longs;iue per paucos, &longs;iue per multos quis ef<lb/>ficere conetur: ut non minus exigat temporis, quàm ob&longs;idio: nam <lb/>multitudine unus alterum impedit, & mortui uiuos, ut omnino res <lb/>&longs;it non &longs;peranda ni&longs;i aduer&longs;us inerti&longs;simos. </s>

<s id="id002111">Pontes euertunt machi <lb/>næ, ignes que. </s>

<s id="id002112">Sed ubi etiam muros obtinuerint ob rotunditatem in <lb/>illis con&longs;i&longs;tere non po&longs;&longs;unt. </s>

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<s id="id002119">Di <lb/>ces, h&etail;c quid ad proportionem? </s>

<s id="id002120">Imò uelut machina ad <expan abbr="perpendicul&utilde;">perpendiculum</expan> <lb/>librata pauculo illo puluere Pyrio <expan abbr="urb&etilde;">urbem</expan> euertit, ita &longs;cintilla illa infer <lb/>ni ignis &longs;emini magni tyranni indita euertit at que di&longs;&longs;oluit totum re<lb/>gnum &longs;ine machinis, ut dixi, uel exercitibus ullis, & quod maius e&longs;t <lb/>remedio nullo. </s>

<s id="id002121">Sed puerulo indito luxus, ignauiæ, crudelitatis at que<lb/>&longs;tultiti&etail; fontibus, mirabile dictu &longs;anè, & ad proportionem diuino<lb/>rum <expan abbr="in&longs;trumentor&utilde;">in&longs;trumentorum</expan> pertinens. </s>

<s id="id002122">Sed redeamus ad in&longs;titutum: Video <lb/>enim, quid po&longs;sit obijci, &longs;cilicet muros cra&longs;&longs;os, et altiores tueri <expan abbr="urb&etilde;">urbem</expan> <lb/>& ædificia illius po&longs;&longs;e ab&longs;que aggeris erectione, & &longs;i <expan abbr="diruan&ttilde;">diruantur</expan> manere <lb/>etiam nihilominus imo magis, quod e&longs;t terram, u&longs;que <expan abbr="quoniã">quoniam</expan> eadem <lb/>ratione manet, quia concuti non po&longs;sit à machinis: nec ho&longs;tes id cu <lb/>raturos, &longs;perantes hoc <expan abbr="&longs;ol&utilde;">&longs;olum</expan> &longs;ufficere, &qring;d m&oelig;nia &longs;olo <expan abbr="æquen&ttilde;">æquentur</expan>, at que id <lb/><expan abbr="fact&utilde;">factum</expan> e&longs;t Mediolani, & in arce eius, <expan abbr="t&utilde;">tum</expan> Papi&etail; & in Cremonen&longs;i arce. <lb/></s>

<s id="id002122">Sed redeamus ad in&longs;titutum: Video <lb/>enim, quid po&longs;sit obijci, &longs;cilicet muros cra&longs;&longs;os, et altiores tueri <expan abbr="urb&etilde;">urbem</expan> <lb/>& ædificia illius po&longs;&longs;e ab&longs;que aggeris erectione, & &longs;i <expan abbr="diruan&ttilde;">diruantur</expan> manere <lb/>etiam nihilominus imo magis, quod e&longs;t terram, u&longs;que <expan abbr="quoniã">quoniam</expan> eadem <lb/>ratione manet, quia concuti non po&longs;sit à machinis: nec ho&longs;tes id cu<lb/>raturos, &longs;perantes hoc <expan abbr="&longs;ol&utilde;">&longs;olum</expan> &longs;ufficere, quod m&oelig;nia &longs;olo <expan abbr="æquen&ttilde;">æquentur</expan>, at que id <lb/><expan abbr="fact&utilde;">factum</expan> e&longs;t Mediolani, & in arce eius, <expan abbr="t&utilde;">tum</expan> Papi&etail; & in Cremonen&longs;i arce. <lb/></s>

<s id="id002123">Verùm ni fallor, ut paruis arcibus à tanta ui tormentorum nullum <lb/>e&longs;t <expan abbr="præ&longs;idi&utilde;">præ&longs;idium</expan>, aut &longs;alutis &longs;pes, ita neque <expan abbr="cõuenit">conuenit</expan>, ut muris humilibus ag <lb/>geri confidant, nam & pauci homines tanto labori non &longs;ufficerent, <lb/>& agger cum fo&longs;&longs;a effo&longs;&longs;a &longs;cilicet terra defen&longs;ores nimis in <expan abbr="angu&longs;t&utilde;">angu&longs;tum</expan> <lb/>cogeret. </s>

<s id="id002124">At in urbibus contra eueniet: muris enim erectis altius ma <lb/>chinæ lapidum fru&longs;tis hominem <expan abbr="occid&etilde;t">occident</expan>: an percu&longs;&longs;a &longs;uperiore par <lb/>te ob coniunctionem inferior concutitur, & in de <expan abbr="tot&utilde;">totum</expan> &longs;imul cadit, <lb/>ut uidimus Papi&etail;, quo <expan abbr="cad&etilde;te">cadente</expan>, & fo&longs;&longs;a impletur, & <foreign lang="greek">tEIkole/tois</foreign> facilior <lb/>aditus ad &longs;ubruendum reliquas partes <expan abbr="pr&etail;be&ttilde;">pr&etail;betur</expan>: imò percul&longs;i defen

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<pb pagenum="119" xlink:href="015/01/138.jpg"/>o a, ad o c, igitur clauus e&longs;t longè potentior in comparatione ueli <lb/>d, quam f, ergo uelum d minus agit nauim, quam f. </s>

<s id="id002135">Sed ut extrema <lb/>&longs;e habent, ita medium eorum comparatione, igitur malus b e uali<lb/>dior e&longs;t, multo d a, & infirmior c f. </s>

<s id="id002136">Verùm, ut dixi, ob &longs;itum &longs;impli<lb/>citer ualidius e&longs;t, uelum e quam f, & etiam quia, ut dixi, altior & <lb/>era&longs;sior &longs;olet e&longs;&longs;e, ideo multo ualidior tribus his cau&longs;is, quàm e f: <lb/>adde quartam quòd uelum habet maius, antiquo tempore uoca<lb/>tum acatius. </s>

<s id="id002136">Verùm, ut dixi, ob &longs;itum &longs;impli<lb/>citer ualidius e&longs;t, uelum e quam f, & etiam quia, ut dixi, altior & <lb/>cra&longs;sior &longs;olet e&longs;&longs;e, ideo multo ualidior tribus his cau&longs;is, quàm e f: <lb/>adde quartam quòd uelum habet maius, antiquo tempore uoca<lb/>tum acatius. </s>

<s id="id002137">At ut etiam docui c b non e&longs;t in medio, nec æquidi&longs;tat <lb/>ab a d & c f, &longs;ed in clinatur ad proram ideoque imbecillior: cum ergo <lb/>&longs;it æqualium, & paulo maiorum uirium, quàm c f, & tutior, & me<lb/>lius agatur per <expan abbr="clau&utilde;">clauum</expan> quàm c f, & &longs;it a d nimis iu&longs;to imbecillis, pro<lb/>pterea b e mali, & ueli maximus e&longs;t u&longs;us: adeò mali nomen per an<lb/>tonoma&longs;iam de ip&longs;o &longs;impliciter intelligatur.</s>

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<s id="id002140">Sit flabellum a b c appen&longs;um, ut &longs;olet, in a, & moueatur motu </s>

<s id="id002141"><arrow.to.target n="marg413"/><lb/>qua&longs;i circa axem p a q in parte inferiore, & aër comprehen&longs;us &longs;ub <lb/>b h k, & &longs;patium &longs;it 1 m figuræ nauicularis, quæ con&longs;tat e&longs;&longs;e par<lb/>tem cylindri inanis ex formatione ab Euclide &longs;cripta: nam &longs;i pro<lb/>poneretur p a q ad perpendiculum &longs;uper&longs;tans plano, fieret circum<lb/>ducta a b c &longs;uperficie, quæ e&longs;&longs;et lata &longs;uperius, &longs;icut etiam inferius <lb/><arrow.to.target n="marg414"/><lb/>cylindrus: at &longs;uperius a b tenuis e&longs;t, & angu&longs;ta, ergo fiet pars cy<lb/>lindri inanis: quia non circunuoluitur, donecredeat. </s>

<s id="id002142">Ergo per di<lb/>cta &longs;uperius &longs;ectio illius p r q s per axem e&longs;t pars cuiu&longs;dam elly<lb/><arrow.to.target n="marg415"/><lb/>p&longs;is. </s>

<s id="id002143">Et &longs;ectio quæuis planæ &longs;uperficiei æquidi&longs;tans a b cuelut tu, <lb/>item que æquidi&longs;tans axi p a q e&longs;t &longs;uperficies rectangula, quarum <lb/>una e&longs;t &longs;imilis, & æqualis b h k, e&longs;t in una &longs;uperficie cum axe p a q <lb/>alia uerò e&longs;t æquidi&longs;tans eidem axi maior aut minor æquidi&longs;tanti<lb/>um, & ip&longs;a laterum, at que rectangula ac &longs;i cylindrus &longs;tans axi plano <lb/>æquidi&longs;tanti &longs;ecaretur iuxta longitudinem &longs;eu altitudinem &longs;uam: <lb/>& manife&longs;tum e&longs;t, quod i&longs;ta duo plana, & eorum &longs;uperficies &longs;ecant <lb/>&longs;e mutuò ad rectos angulos.</s>

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<s id="id002147">Quibus con&longs;titutis, qui &longs;tabunt iuxta l, & m longitudines aëris <lb/>moti, & loci, per quem tran&longs;it flabellum, &longs;entient magnum uentum, <lb/>quoniam cum corpus m x l ab extremis partibus &longs;it elatius a b ex<lb/>tremis, &longs;tantes, & alti tangentur à uento agitato. </s>

<s id="id002148">Si uero &longs;edeant, aer <lb/>primum non attinget illos, ut etiam quia &longs;ur&longs;um pellitur non per<lb/>ueniet ad illos, imò diffugiet, ergo non refrigerabuntur. </s>

<s id="id002149">Qui uerò <lb/>à lateribus l x m <expan abbr="&longs;tab&utilde;t">&longs;tabunt</expan> hiccinde, uelut in f g, &longs;i &longs;teterint, <expan abbr="nõ">non</expan> refrigeræ <lb/><expan abbr="b&utilde;tur">buntur</expan>, quia <expan abbr="quãdo">quando</expan> flabellum erit in l, uel m aer de&longs;cendet, ergo fugi <lb/>et ab illis, cum autem fuerit in x, erit in loco humiliori, & mouebi

<s id="id002149">Qui uerò <lb/>à lateribus l x m <expan abbr="&longs;tab&utilde;t">&longs;tabunt</expan> hiccinde, uelut in f g, &longs;i &longs;teterint, <expan abbr="nõ">non</expan> refrigera<lb/><expan abbr="b&utilde;tur">buntur</expan>, quia <expan abbr="quãdo">quando</expan> flabellum erit in l, uel m aer de&longs;cendet, ergo fugi <lb/>et ab illis, cum autem fuerit in x, erit in loco humiliori, & mouebi

<pb pagenum="120" xlink:href="015/01/139.jpg"/>tur diuer&longs;a ratione, quippe ab f in h, & non ad latera, ergo ne que <lb/><figure id="id.015.01.139.1.jpg" xlink:href="015/01/139/1.jpg"/><lb/>contactu, neque motu, qui <lb/>fiet per æquidi&longs;tantem f, <lb/>& g non poterunt refrige<lb/>rari. </s>

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<s id="id002186">Nam exempli gratia, cum &longs;ol e&longs;t in initio Capricorni, & in C&oelig;li <lb/>medio, minima e&longs;t umbra eius diei, & totius anni. </s>

<s id="id002187">Cum ergo fuerit <lb/>ante meridiem, uel po&longs;t, erit umbra maior ex &longs;uppo&longs;ito &longs;ecudo um<lb/>bra meridiei: at ei æqualis poterit e&longs;&longs;e umbra meridiei alterius diei <lb/>ex primo &longs;uppo&longs;ito, ergo umbræ æquales diuer&longs;orum dierum fi<lb/>unt &longs;ub diuer&longs;o &longs;itu &longs;olis, quo æd circulum meridiei, quod erat de<lb/>mon&longs;trandum.</s>

<s id="id002187">Cum ergo fuerit <lb/>ante meridiem, uel po&longs;t, erit umbra maior ex &longs;uppo&longs;ito &longs;ecudo um<lb/>bra meridiei: at ei æqualis poterit e&longs;&longs;e umbra meridiei alterius diei <lb/>ex primo &longs;uppo&longs;ito, ergo umbræ æquales diuer&longs;orum dierum fi<lb/>unt &longs;ub diuer&longs;o &longs;itu &longs;olis, quo ad circulum meridiei, quod erat de<lb/>mon&longs;trandum.</s>

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<s id="id002260">Data regionis altitudine, & loco &longs;olis proportionem gnomo<lb/>nis tam ad umbram rectam, quam uer&longs;am, uel etiam in cylindro de<lb/>terminare.</s>

<s id="id002261">H&etail;c e&longs;t propo&longs;itio illa pulcherrima, quam tot ambagibus tradi<lb/><arrow.to.target n="marg445"/><lb/>dere antiqui cum &longs;uis analematibus, & &longs;cioteris, nec tamen demon <lb/>&longs;trationem, nec rationem exactam in&longs;trumenortum con&longs;tructio<lb/>nem, qua po&longs;&longs;emus per umbras rectas uer&longs;as, & cylindricas &longs;cire ad <lb/>unguem, qualis hora, & minutum, & &longs;ecundum diei e&longs;&longs;et quocun<lb/>que anni tempore. </s>

<s id="id002261">H&etail;c e&longs;t propo&longs;itio illa pulcherrima, quam tot ambagibus tradi<lb/><arrow.to.target n="marg445"/><lb/>dere antiqui cum &longs;uis analematibus, & &longs;cioteris, nec tamen demon <lb/>&longs;trationem, nec rationem exactam in&longs;trumenorum con&longs;tructio<lb/>nem, qua po&longs;&longs;emus per umbras rectas uer&longs;as, & cylindricas &longs;cire ad <lb/>unguem, qualis hora, & minutum, & &longs;ecundum diei e&longs;&longs;et quocun<lb/>que anni tempore. </s>

<s id="id002262">Plerique autem tam laborio&longs;è id conati &longs;unt de<lb/>mon&longs;trare, ut &longs;tudio&longs;os deterruerint ab opere: res autem ip&longs;a facil<lb/>lima e&longs;t. </s>

<s id="id002263">Propo&longs;ita ergo Poli exacta altitudine &longs;olis in Meridie <lb/>declinatione addita uel detracta, habebis re&longs;iduum eius ad qua<lb/>drantem f g, & &longs;imiliter habebis ex declinatione nota loci &longs;olis de<lb/>tracta à quadrante f h & iuxta horam tuam, & minutum multi<lb/><arrow.to.target n="marg446"/><lb/>plicatum per quindecim arcum k e quare angulum f, ex quo arcum <lb/>g h, quare re&longs;iduum h l, igitur punctum umbr&etail; rect&etail;, uel uer&longs;&etail; ip&longs;i<lb/>us gnomonis ad unguem, & ita con&longs;titues horologium exacti&longs;si<lb/>mum &longs;ecundum ea, quæ dixi in Corrolarijs &longs;upradictis, & quia ho<lb/><arrow.to.target n="marg447"/><lb/>rizon a b c d &longs;ecat æquinoctialem in <expan abbr="c&etilde;tro">centro</expan> terræ ducta g h k, erunt <lb/>anguli b h g, & k h l &etail;quales. </s>

<s id="id002264">Igitur po&longs;ito g ortu puncti eclypti<lb/>cæ, erit g b ortus amplitudo nota, & ideò angulus b h g, & k h l

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<s id="id002291">Sint duæ line&etail; a b, & c d. </s>

<s id="id002292">& &longs;it c d dupla ad a b, ad datur <expan abbr="cõmunis">communis</expan> <lb/><figure id="id.015.01.145.3.jpg" xlink:href="015/01/145/3.jpg"/><lb/>b e, & uo cetur iuncta c d, d f dico, <lb/>quod proportio e a ad a b, e&longs;t mi<lb/>nor duplicata f c ad c d, adij cia<lb/>tur d f æqualis g f, quia ergo g d <lb/>e&longs;t dupla ad f d, ideo ad e b c d autem e&longs;t du pla ad a b, tota igitur

<s id="id002292">& &longs;it c d dupla ad a b, ad datur <expan abbr="cõmunis">communis</expan> <lb/><figure id="id.015.01.145.3.jpg" xlink:href="015/01/145/3.jpg"/><lb/>b e, & uocetur iuncta c d, d f dico, <lb/>quod proportio e a ad a b, e&longs;t mi<lb/>nor duplicata f c ad c d, adijcia<lb/>tur d f æqualis g f, quia ergo g d <lb/>e&longs;t dupla ad f d, ideo ad e b c d autem e&longs;t dupla ad a b, tota igitur <pb pagenum="127" xlink:href="015/01/146.jpg"/>g c dupla toti e a. </s>

<pb pagenum="127" xlink:href="015/01/146.jpg"/>g c duplatoti e a. </s>

<s id="id002293">quare ut g c ad g d ut e a ad e b <expan abbr="permutãdo">permutando</expan>, & per <lb/>euer&longs;am ut e a ad a b, ita g c ad c d, ut g c ad c d <expan abbr="cõponitur">componitur</expan> ex g e ad <lb/>f e, & f c ad c d, igitur e a ad c b componitur ex ei&longs;dem. </s>

<s id="id002294">Proportio <lb/>autem g c ad f c e&longs;t minor, quam f c ad c d, igitur minor quàm du<lb/>plicata f c ad c d. </s>

<s id="id002295">con&longs;tat uerò ex ei&longs;dem, quod proportio c a ad a b <lb/>maior e&longs;t duplicata g c ad f c.</s>

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<s id="id002405">Pro relata prima ergo ponamus, ut ue<lb/>lim <02> relatam <expan abbr="primã">primam</expan> 25, accipio 5 <02> 25 cuius <02> e&longs;t, ut ui&longs;um e&longs;t, 2 104/441 <lb/>&longs;imiliter <02> cu: 5 fuit 1 69/98 igitur reducam ad unam denominationem, <lb/>& inueniam quatuor numeros in <expan abbr="cõtinua">continua</expan> proportione inter illos, <lb/>& &longs;ecundus po&longs;t minimum ex illis erit <02> relata prima propinqui&longs;<lb/>&longs;ima 25. Quomodo uerò inueniantur facillimè illi termini, do<lb/>cui in &longs;exto libro operis perfecti.</s>

<s id="id002406">Quarta regula e&longs;t utilior, licet minus uideatur nobilis, & e&longs;t &longs;un<lb/>data in hoc, quod &longs;i a b &longs;it maior c & eis ad dantur b e, & d f æqua<lb/>les dico, quod erit minor proportio a c ad c f, quam a b ad c d, & ex <lb/>con&longs;equenti per <expan abbr="uiã">uiam</expan> fracti maior pars unius erit c fip&longs;ius a e, quàm

<s id="id002406">Quarta regula e&longs;t utilior, licet minus uideatur nobilis, & e&longs;t fun<lb/>data in hoc, quod &longs;i a b &longs;it maior c & eis ad dantur b e, & d f æqua<lb/>les dico, quod erit minor proportio a c ad c f, quam a b ad c d, & ex <lb/>con&longs;equenti per <expan abbr="uiã">uiam</expan> fracti maior pars unius erit c f ip&longs;ius a e, quàm <pb pagenum="135" xlink:href="015/01/154.jpg"/>c d ip&longs;ius a f ex Euclide. </s>

<pb pagenum="135" xlink:href="015/01/154.jpg"/>c d ip&longs;ius a f ex Euclide. </s>

<s id="id002407">Dico ergo quod maior e&longs;t proportio a b <lb/><figure id="id.015.01.154.1.jpg" xlink:href="015/01/154/1.jpg"/><lb/>ad c d, quàm a e ad e f, fiat d g ad quam &longs;it b c ut <lb/><arrow.to.target n="marg467"/><lb/>a b ad c d, eritque a e ad c g ut a b ad c d, minor au<lb/>tem e&longs;t a e ad c f, quam ad c g, igitur minor a e ad <lb/>c f quàm a b ad c d quod fuit propo&longs;itum. </s>

<s id="id002408">Simili <lb/>ter &longs;i fuerint duæ quantitates, a b & c d, quarum a b &longs;it maiore, c d <lb/>autem eadem e minor, dico, quòd dimidium aggregati a b & c d <lb/>maiorem habebit proportionem ad e, quàm c d & minor, nam iun<lb/>cta b f æquali d e ad a b, ita ut f g &longs;it dimidium totius a f, qùia ergo <lb/><figure id="id.015.01.154.2.jpg" xlink:href="015/01/154/2.jpg"/><lb/>f g e&longs;t dimidium f a & fb e&longs;t minor dimidio <lb/><arrow.to.target n="marg468"/><lb/>f a cum &longs;it minor b a, & &longs;imiliter f g e&longs;t mi<lb/>nor a b, quia a b e&longs;t maior dimidio a f, quia <lb/>e&longs;t maior b f, ergo proportio g f ad c e&longs;t ma <lb/>ior quam b f ad e, ita quam c d ad e, & mi<lb/><arrow.to.target n="marg469"/><lb/>nor quàm a b ad e, quod fuit propo&longs;itum. </s>

<s id="id002409">Quo ui&longs;o uolo <02> 1000 <lb/>quadratam, & quòd de quadrata dico, dico etiam de alijs radici<lb/>bus & erit ex &longs;ecunda regula harum 31 39/62 & quadratum erit 1000 <lb/>1521/3844. Iuxta ergo primam partem regulæ 31 38/61 erit minus, & in ueritate <lb/>in eo, quod fit ducendo, ut uides, & hoc e&longs;t pro<lb/><figure id="id.015.01.154.3.jpg" xlink:href="015/01/154/3.jpg"/><lb/>ximum ad 11/160, multiplico igitur duplum 31 39/62, <lb/>quod e&longs;t fermè 63 1/4 in 1/160 fient 63/160 detrahe ex <lb/>1521/3844 hoc modo, diuide 3844 per 160 exit 24 /40 <lb/>diuide 1521 per 24, exit 63 3/8, habes igitur quod <lb/>1521/3844 &longs;unt 63/160, igitur detracto 63/160 ex 63/160 nihil relinquitur, & erit <02> exa<lb/>cta ualde 1000 hoc 31 38/61 cuius quadratum 1000 41/3421 uides breuita <lb/>tem, & propinquitatem in producto differentia e&longs;t 1/100 aut parum <lb/>maius quod ad radicem comparatum cum debeat diuidi per du<lb/>plum eius erit paulo maius 1/6300. Vnde facilior e&longs;t, & breuior hæc <lb/>uia quàm per 00 ad ditus. </s>

<s id="id002410">Rur&longs;us uolo aliquid <expan abbr="adi&mtilde;ere">adimnere</expan> & cum pro <lb/>pinquitate ita facio. </s>

<s id="id002410">Rur&longs;us uolo aliquid <expan abbr="adi&mtilde;ere">adimere</expan> & cum pro<lb/>pinquitate ita facio. </s>

<s id="id002411">Con&longs;idero quòd 31 38/61 e&longs;t maius 1/6300 radice, di<lb/>uido 6300 per 62 exit 103 fermè, neque enim curo in hoc fractiones, <lb/>multiplico ergo 103 in 38/61 & habeo 3914/6283 hic denominator e&longs;t proxi<lb/>mus 6300, aufero ergo 1 ex 3914, habebo ualde proximam <02> 1000, <lb/>31 3913/6283 cuius quadratum e&longs;t 1000 minus 1/1048 hoc ut dixi diui&longs;um <lb/>per duplum <02> quod e&longs;t 63 e&longs;t omnino in&longs;en&longs;ile in radice.</s>

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<s id="id002425">Numeros fractos ad minores in <expan abbr="ead&etilde;">eadem</expan> proportione ualde propinqua</s>

<s id="id002426">Cum plerunque numeri fracti hab cantur per radices, ut aliquan<lb/><arrow.to.target n="marg470"/><lb/>do maiores &longs;int, aut minores eo fit, ut po&longs;sint reduci ad mino<lb/>res numeros, ut melius intelligi po&longs;sint & facilius tractari, &

<s id="id002426">Cum plerunque numeri fracti habeantur per radices, ut aliquan<lb/><arrow.to.target n="marg470"/><lb/>do maiores &longs;int, aut minores eo fit, ut po&longs;sint reduci ad mino<lb/>res numeros, ut melius intelligi po&longs;sint & facilius tractari, &

<pb pagenum="137" xlink:href="015/01/156.jpg"/>cum hoc &longs;it exactior illa pars exemplum, ergo habeo 2 3829/12566, quem <lb/>uolo certa ratione ad minores diui&longs;iones deducere. </s>

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