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version 1.11, 2004/02/05 17:04:23 |
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| <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="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> |
| </p> | </p> |
| <p type="main"> | <p type="main"> |
| <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> |
| </p> | </p> |
| <p type="margin"> | <p type="margin"> |
| <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> | <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> |
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| <p type="main"> | <p type="main"> |
| <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="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="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="ictuũ">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="ictuũ">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="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="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="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 & cœlum naturaliter perpetuò mouentur, quare etiam <lb/>aër. </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 & cœlum 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="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> | <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> |
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| <s id="id000912">Dico etiam, quod tardius ad c quàm d. </s> | <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="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="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> | <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> |
| </p> | </p> |
| <p type="margin"> | <p type="margin"> |
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| <s id="id000930"><margin.target id="marg163"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s> | <s id="id000930"><margin.target id="marg163"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s> |
| </p> | </p> |
| <p type="main"> | <p type="main"> |
| <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> |
| </p> | </p> |
| <p type="main"> | <p type="main"> |
| <s id="id000932">Propo&longs;itio &longs;exage&longs;imaprima.</s> | <s id="id000932">Propo&longs;itio &longs;exage&longs;imaprima.</s> |
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| </p> | </p> |
| <p type="main"> | <p type="main"> |
| <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="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> | <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> |
| </p> | </p> |
| <p type="main"> | <p type="main"> |
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| </p> | </p> |
| <p type="main"> | <p type="main"> |
| <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="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 | <s id="id001001">Por­ |
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| </row> | </row> |
| </table> | </table> |
| <p type="main"> | <p type="main"> |
| <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="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="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="quadratũ">quadratum</expan> g, ita c ad h: &longs;ed qua­<lb/>drati b ad quadratum g, ut b ad g proportio duplicata <lb/>igitur e ad h, ut b ad g duplicata.</s> | <s 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="quadratũ">quadratum</expan> g, ita c ad h: &longs;ed qua­<lb/>drati b ad quadratum g, ut b ad g proportio duplicata <lb/>igitur e ad h, ut b ad g duplicata.</s> |
| |
| <s 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="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="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="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="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="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> | <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> |
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