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version 1.17, 2002/08/07 22:08:31 |
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| <s>definitio <lb/><expan abbr="vtriu&longs;q;">vtriu&longs;que</expan> e&longs;t a pud Euclidem ad primam propo&longs;. </s> | <s>definitio <lb/><expan abbr="vtriu&longs;q;">vtriu&longs;que</expan> e&longs;t a pud Euclidem ad primam propo&longs;. </s> |
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| <s>13. Elem. </s> | <s>13. Elem. |
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| <s>iuxta tran&longs;latio­<lb/>nem Zamb<gap/>rti, & Commandini; vbi etiam <expan abbr="quinq;">quinque</expan> priora theoremata, pri­<lb/>mò per re&longs;olutionem, deinde per compo&longs;itionem demon&longs;trantur, quæ tan­<lb/>quam per&longs;picua exempla rei propo&longs;itæ in&longs;eruire po&longs;&longs;unt. </s> | iuxta tran&longs;latio­<lb/>nem Zamb<gap/>rti, & Commandini; vbi etiam <expan abbr="quinq;">quinque</expan> priora theoremata, pri­<lb/>mò per re&longs;olutionem, deinde per compo&longs;itionem demon&longs;trantur, quæ tan­<lb/>quam per&longs;picua exempla rei propo&longs;itæ in&longs;eruire po&longs;&longs;unt. </s> |
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| <s>&longs;unt præterea fre­<lb/>quentes huiu&longs;modi re&longs;olutiones in operibus Archimedis, Apollonij, & Pap­<lb/>pi. </s> | <s>&longs;unt præterea fre­<lb/>quentes huiu&longs;modi re&longs;olutiones in operibus Archimedis, Apollonij, & Pap­<lb/>pi. </s> |
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| <s>&longs;i linea <lb/>C, men&longs;uraret A, quater &longs;umpta, ter autem &longs;umpta <lb/>non adæquaret omnino <expan abbr="lineã">lineam</expan> B, &longs;ed deficeret, vel ex­<lb/>cederet aliquantulum, <expan abbr="atq;">atque</expan> hoc fieret in quauis alia <lb/>men&longs;ura, loco ip&longs;ius C, a&longs;&longs;umpta, &longs;iue maior, &longs;iue <lb/>minor ip&longs;a C, vt <expan abbr="vtranq;">vtranque</expan> nunquam perfectè metiretur, e&longs;&longs;ent duæ illæ lineæ <lb/>incommen&longs;. </s> | <s>&longs;i linea <lb/>C, men&longs;uraret A, quater &longs;umpta, ter autem &longs;umpta <lb/>non adæquaret omnino <expan abbr="lineã">lineam</expan> B, &longs;ed deficeret, vel ex­<lb/>cederet aliquantulum, <expan abbr="atq;">atque</expan> hoc fieret in quauis alia <lb/>men&longs;ura, loco ip&longs;ius C, a&longs;&longs;umpta, &longs;iue maior, &longs;iue <lb/>minor ip&longs;a C, vt <expan abbr="vtranq;">vtranque</expan> nunquam perfectè metiretur, e&longs;&longs;ent duæ illæ lineæ <lb/>incommen&longs;. </s> |
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| <s>Extare porrò tales lineas, & &longs;uperficies, & corpora, <expan abbr="ea&qacute;">eaque</expan>; quam­<lb/>plurima, ac penè infinita ex 10. Elem. </s> | <s>Extare porrò tales lineas, & &longs;uperficies, & corpora, <expan abbr="ea&qacute;">eaque</expan>; quam­<lb/>plurima, ac penè infinita ex 10. Elem. |
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| <s>manife&longs;tum e&longs;t. </s> | manife&longs;tum e&longs;t. </s> |
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| <s>inuentum autem hu­<lb/>ius a&longs;ymmetriæ, quod Pythagoricis veteres attribuunt, mihi &longs;emper vi&longs;um <lb/>e&longs;t omni maius admiratione, cum nulla experientia, <expan abbr="nullus&qacute;">nullusque</expan>; effectus in ip­<lb/>&longs;ius cognitionem potuerit pri&longs;cos illos Geometras inducere. </s> | <s>inuentum autem hu­<lb/>ius a&longs;ymmetriæ, quod Pythagoricis veteres attribuunt, mihi &longs;emper vi&longs;um <lb/>e&longs;t omni maius admiratione, cum nulla experientia, <expan abbr="nullus&qacute;">nullusque</expan>; effectus in ip­<lb/>&longs;ius cognitionem potuerit pri&longs;cos illos Geometras inducere. </s> |
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| <s>latus B C, præcisè omnino metiatur. </s> | <s>latus B C, præcisè omnino metiatur. </s> |
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| <s>theorema <lb/>i&longs;tud demon&longs;tratur in vltima 10. Elem. </s> | <s>theorema <lb/>i&longs;tud demon&longs;tratur in vltima 10. Elem. |
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| <s>eodem me­<lb/>dio, quod ab Ari&longs;totele hic innuitur; Euclides ex <lb/>&longs;uppo&longs;itione alterius partis contradictionis ip&longs;ius <pb pagenum="38"/>propo&longs;itionis, quæfal&longs;a e&longs;t, nimirum &longs;uppo&longs;ito prædictas lineas e&longs;&longs;e comm. <lb/></s> | eodem me­<lb/>dio, quod ab Ari&longs;totele hic innuitur; Euclides ex <lb/>&longs;uppo&longs;itione alterius partis contradictionis ip&longs;ius <pb pagenum="38"/>propo&longs;itionis, quæfal&longs;a e&longs;t, nimirum &longs;uppo&longs;ito prædictas lineas e&longs;&longs;e comm. <lb/></s> |
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| <s>deducit ad impo&longs;&longs;ibile, &longs;iue, vt ait hic Ari&longs;t. | <s>deducit ad impo&longs;&longs;ibile, &longs;iue, vt ait hic Ari&longs;t. |
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| <s>1. &longs;ecti 3. lib. </s> | <s>1. &longs;ecti 3. lib. </s> |
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| <s>1. <emph type="italics"/>(Sit A, duo recti, in quo B, triangulus, in quo C, <lb/>æquicrus, ip&longs;i <expan abbr="itaq;">itaque</expan> C, ine&longs;t A. per B; ip&longs;i vero B, non amplius per aliud, per &longs;e <lb/>namque triangulus habet duos rectos)<emph.end type="italics"/> nullum aliud exemplum tam frequenter <lb/>v&longs;urpat Philo&longs;ophus, quam i&longs;tud ex Mathematicis de&longs;umptum de triangu­<lb/>lo, &longs;cilicet, omnis triangulus habet tres angulos æquales duobus rectis an­<lb/>gulis, cuius Demon&longs;tratio e&longs;t in 32. primi Elem. </s> | <s>1. <emph type="italics"/>(Sit A, duo recti, in quo B, triangulus, in quo C, <lb/>æquicrus, ip&longs;i <expan abbr="itaq;">itaque</expan> C, ine&longs;t A. per B; ip&longs;i vero B, non amplius per aliud, per &longs;e <lb/>namque triangulus habet duos rectos)<emph.end type="italics"/> nullum aliud exemplum tam frequenter <lb/>v&longs;urpat Philo&longs;ophus, quam i&longs;tud ex Mathematicis de&longs;umptum de triangu­<lb/>lo, &longs;cilicet, omnis triangulus habet tres angulos æquales duobus rectis an­<lb/>gulis, cuius Demon&longs;tratio e&longs;t in 32. primi Elem. |
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| <s>quod, vt probè intelliga­<lb/>tur, explicandum e&longs;t penes quid attendenda &longs;it æqualitas inter angulum, & <lb/>angulum, quod facile a&longs;&longs;equemur, &longs;i meminerimus angulum e&longs;&longs;e in clinatio­<lb/>nem illam, quam duæ lineæ non in directum po&longs;itæ faciunt: &longs;iue etiam (vt <lb/>melius percipiamus) angulum e&longs;&longs;e acumen illud, &longs;iue mucronem <expan abbr="illũ">illum</expan>, quem <lb/>duæ lineæ non in directum con&longs;titutæ faciunt, vt duarum linearum A B, A C, <lb/><figure id="fig7"></figure><lb/>inclinatio in puncto A, &longs;iue acumen illud, &longs;iue mucro, <lb/>e&longs;t ratio anguli. </s> | quod, vt probè intelliga­<lb/>tur, explicandum e&longs;t penes quid attendenda &longs;it æqualitas inter angulum, & <lb/>angulum, quod facile a&longs;&longs;equemur, &longs;i meminerimus angulum e&longs;&longs;e in clinatio­<lb/>nem illam, quam duæ lineæ non in directum po&longs;itæ faciunt: &longs;iue etiam (vt <lb/>melius percipiamus) angulum e&longs;&longs;e acumen illud, &longs;iue mucronem <expan abbr="illũ">illum</expan>, quem <lb/>duæ lineæ non in directum con&longs;titutæ faciunt, vt duarum linearum A B, A C, <lb/><figure id="fig7"></figure><lb/>inclinatio in puncto A, &longs;iue acumen illud, &longs;iue mucro, <lb/>e&longs;t ratio anguli. </s> |
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| <s>&longs;olum igitur duo anguli erunt æqua­<lb/>les, <expan abbr="quãdo">quando</expan> vnius acumen æquale erit acumini alterius; <lb/>etiam &longs;i lineæ con&longs;tituentes vnum angulum &longs;int lon­<lb/>giores lineis alterum angulum con&longs;tituentibus, quia <lb/>quantitas anguli non attenditur penes longitudinem <pb pagenum="40"/><expan abbr="linearũ">linearum</expan>, &longs;ed penes inclinationem, & mucronem, quem faciunt: vnde etiam&longs;i <lb/>duæ lineæ prædictæ A B, A C, productæ, &longs;iue etiam decurtatæ fuerint, dum­<lb/>modo &longs;itus, &longs;iue po&longs;itio ip&longs;arum, quam ad inuicem habent, non varietur, <lb/>erit &longs;emper eadem quantitas anguli A. </s> | <s>&longs;olum igitur duo anguli erunt æqua­<lb/>les, <expan abbr="quãdo">quando</expan> vnius acumen æquale erit acumini alterius; <lb/>etiam &longs;i lineæ con&longs;tituentes vnum angulum &longs;int lon­<lb/>giores lineis alterum angulum con&longs;tituentibus, quia <lb/>quantitas anguli non attenditur penes longitudinem <pb pagenum="40"/><expan abbr="linearũ">linearum</expan>, &longs;ed penes inclinationem, & mucronem, quem faciunt: vnde etiam&longs;i <lb/>duæ lineæ prædictæ A B, A C, productæ, &longs;iue etiam decurtatæ fuerint, dum­<lb/>modo &longs;itus, &longs;iue po&longs;itio ip&longs;arum, quam ad inuicem habent, non varietur, <lb/>erit &longs;emper eadem quantitas anguli A. </s> |
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| <s>vbi per coalternas intelligit parallelas lineas, vox <lb/>enim græca <foreign lang="greek">parallhlos,</foreign> idem &longs;ignificat, ac mutuus, & coalternus. </s> | <s>vbi per coalternas intelligit parallelas lineas, vox <lb/>enim græca <foreign lang="greek">parallhlos,</foreign> idem &longs;ignificat, ac mutuus, & coalternus. </s> |
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| <s>quoad <lb/>exempli explicationem vtor figura textibus apponi &longs;olita, quæ e&longs;t præ&longs;ens. <lb/><figure id="fig11"></figure><lb/>probat Euclides in 28. primi Elem. </s> | <s>quoad <lb/>exempli explicationem vtor figura textibus apponi &longs;olita, quæ e&longs;t præ&longs;ens. <lb/><figure id="fig11"></figure><lb/>probat Euclides in 28. primi Elem. |
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| <s>quod &longs;i <lb/>linea recta quædam, vti E F, cadens &longs;uper <lb/>duas rectas, vti &longs;unt A B, C D, fe cerit angu­<lb/>los alternos &ecedil;quales, angulos <expan abbr="nimirũ">nimirum</expan> A G H, <lb/>G H D, ij enim dicuntur alterni; &longs;iue alios <lb/>dnos, nimirum B G H, G H C, hi enim &longs;unt <lb/><expan abbr="quoq;">quoque</expan> alterni; probat inquam has duas li­<lb/>neas A B, C D, e&longs;&longs;e inuicem parallelas. </s> | quod &longs;i <lb/>linea recta quædam, vti E F, cadens &longs;uper <lb/>duas rectas, vti &longs;unt A B, C D, fe cerit angu­<lb/>los alternos &ecedil;quales, angulos <expan abbr="nimirũ">nimirum</expan> A G H, <lb/>G H D, ij enim dicuntur alterni; &longs;iue alios <lb/>dnos, nimirum B G H, G H C, hi enim &longs;unt <lb/><expan abbr="quoq;">quoque</expan> alterni; probat inquam has duas li­<lb/>neas A B, C D, e&longs;&longs;e inuicem parallelas. </s> |
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| <s>Iam &longs;i quis vellet probare, &longs;e duas <lb/>parallelas duxi&longs;&longs;e, hac ratione, quia &longs;cilicet fa ciunt prædictos angulos al­<lb/>ternos æquales; & probaret facere angulos alternos æquales, quia &longs;unt pa­<lb/>rallelæ, hic peteret principium, ide&longs;t, illud, quod principio <expan abbr="probandũ">probandum</expan> erat, <lb/>afferret pro ratione, & cau&longs;a, quod dicitur peti principium, quia tunc pe­<lb/>timus, vt concedatur nobis, id, quod principio, & primo omnium demon­<lb/>&longs;trare propo&longs;ueramus. </s> | <s>Iam &longs;i quis vellet probare, &longs;e duas <lb/>parallelas duxi&longs;&longs;e, hac ratione, quia &longs;cilicet fa ciunt prædictos angulos al­<lb/>ternos æquales; & probaret facere angulos alternos æquales, quia &longs;unt pa­<lb/>rallelæ, hic peteret principium, ide&longs;t, illud, quod principio <expan abbr="probandũ">probandum</expan> erat, <lb/>afferret pro ratione, & cau&longs;a, quod dicitur peti principium, quia tunc pe­<lb/>timus, vt concedatur nobis, id, quod principio, & primo omnium demon­<lb/>&longs;trare propo&longs;ueramus. </s> |
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| <s>Cæterum Euclides propo&longs;. </s> | <s>Cæterum Euclides propo&longs;. </s> |
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| <s>28. pri­<lb/>mi Elem. </s> | <s>28. pri­<lb/>mi Elem. |
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| <s>o&longs;tendit, quod &longs;i fuerint duæ parallelæ veluti in præcedenti figura, <lb/>A B, C D, &longs;uper quas alia recta E F, incidat, nece&longs;&longs;ario faciet angulum ex­<lb/>trin&longs;ecum E G B, v. </s> | o&longs;tendit, quod &longs;i fuerint duæ parallelæ veluti in præcedenti figura, <lb/>A B, C D, &longs;uper quas alia recta E F, incidat, nece&longs;&longs;ario faciet angulum ex­<lb/>trin&longs;ecum E G B, v. </s> |
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| <s>g. </s> | <s>g. </s> |
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| <s>in probatione minoris facta e&longs;t Abductio, cum enim ip&longs;e vellet re­<lb/>ctificare circumferentiam circuli per lunulas, nec valeret, alij per lineam, <lb/>quandam quadratricem, vt e&longs;t apud Pappum <expan abbr="Alexandrinũ">Alexandrinum</expan>, & apud P. </s> | <s>in probatione minoris facta e&longs;t Abductio, cum enim ip&longs;e vellet re­<lb/>ctificare circumferentiam circuli per lunulas, nec valeret, alij per lineam, <lb/>quandam quadratricem, vt e&longs;t apud Pappum <expan abbr="Alexandrinũ">Alexandrinum</expan>, & apud P. </s> |
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| <s>Cla­<lb/>uium in fine &longs;exti Elem. </s> | <s>Cla­<lb/>uium in fine &longs;exti Elem. |
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| <s>& alij aliter fru&longs;tra conarentur, facta e&longs;t Abductio <lb/>circa probationem minoris, in qua adhuc Mathematici verfantur; quæ pro­<lb/>batio, &longs;i tandem inueniri po&longs;&longs;et, mox &longs;equeretur principale <expan abbr="propo&longs;itũ">propo&longs;itum</expan> pro­<lb/>blema, nimirum circulus quadraretur; vide quæ &longs;crip&longs;<gap/>mus in cap. </s> | & alij aliter fru&longs;tra conarentur, facta e&longs;t Abductio <lb/>circa probationem minoris, in qua adhuc Mathematici verfantur; quæ pro­<lb/>batio, &longs;i tandem inueniri po&longs;&longs;et, mox &longs;equeretur principale <expan abbr="propo&longs;itũ">propo&longs;itum</expan> pro­<lb/>blema, nimirum circulus quadraretur; vide quæ &longs;crip&longs;<gap/>mus in cap. </s> |
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| <s>3. Præ­<lb/>dicam. </s> | <s>3. Præ­<lb/>dicam. </s> |
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| <s>Tex. </s> | <s>Tex. </s> |
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| <s>13. <emph type="italics"/>(Si quis igitur mon&longs;trauerit, quod rectæ <expan abbr="nõ">non</expan> coincidunt, videbitur <expan abbr="vtiq;">vtique</expan> <lb/>buius e&longs;&longs;e demonstratio, eo quod in omnibus e&longs;t rectis; non e&longs;t autem: &longs;i quidem <lb/>non quoniam &longs;ic æquales, fit hoc, &longs;ed &longs;ecundum quod <expan abbr="quomodocunq;">quomodocunque</expan> æquales)<emph.end type="italics"/> pro­<lb/>ponit tres errores, qui circa demon&longs;trationem de vniuer&longs;ali contingunt, <lb/>quos omnes Geometricis exemplis illu&longs;trat; affert autem primo pro tertio <lb/>errore duo exempla, quorum primum in præmi&longs;&longs;is verbis continetur, <expan abbr="atq;">atque</expan> <lb/>ex 28. primi Elem. </s> | <s>13. <emph type="italics"/>(Si quis igitur mon&longs;trauerit, quod rectæ <expan abbr="nõ">non</expan> coincidunt, videbitur <expan abbr="vtiq;">vtique</expan> <lb/>buius e&longs;&longs;e demonstratio, eo quod in omnibus e&longs;t rectis; non e&longs;t autem: &longs;i quidem <lb/>non quoniam &longs;ic æquales, fit hoc, &longs;ed &longs;ecundum quod <expan abbr="quomodocunq;">quomodocunque</expan> æquales)<emph.end type="italics"/> pro­<lb/>ponit tres errores, qui circa demon&longs;trationem de vniuer&longs;ali contingunt, <lb/>quos omnes Geometricis exemplis illu&longs;trat; affert autem primo pro tertio <lb/>errore duo exempla, quorum primum in præmi&longs;&longs;is verbis continetur, <expan abbr="atq;">atque</expan> <lb/>ex 28. primi Elem. |
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| <s>de&longs;umitur, quam propterea primo loco exponendam <lb/><figure id="fig18"></figure><lb/>cen&longs;ui. </s> | de&longs;umitur, quam propterea primo loco exponendam <lb/><figure id="fig18"></figure><lb/>cen&longs;ui. </s> |
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| <s>Quando igitur duæ rectæ con&longs;titu­<lb/>tæ fuerint, vt A B, C D, in quas alia recta, <lb/>vt G F, incidens, faciat duos angulos in­<lb/>ternos, re&longs;pectu rectarum A B, C D, & ad <lb/>ea&longs;dem partes rectæ E F, vt &longs;unt ex parte <lb/>&longs;ini&longs;tra anguli A G H, C H G; exparte ve­<lb/>rò dextra B G H, D H G; &longs;i <expan abbr="inquã">inquam</expan> linea E F, <lb/>fecerit duos illos angulos ex parte &longs;ini&longs;tra &longs;imul &longs;umptos, æquales duobus <lb/>rectis angulis, vel duos ex parte dextra pariter æquales duobus rectis, pro­<lb/>bat Euclides rectas A B, C D, non concurrere, &longs;iue parallelas e&longs;&longs;e. </s> | <s>Quando igitur duæ rectæ con&longs;titu­<lb/>tæ fuerint, vt A B, C D, in quas alia recta, <lb/>vt G F, incidens, faciat duos angulos in­<lb/>ternos, re&longs;pectu rectarum A B, C D, & ad <lb/>ea&longs;dem partes rectæ E F, vt &longs;unt ex parte <lb/>&longs;ini&longs;tra anguli A G H, C H G; exparte ve­<lb/>rò dextra B G H, D H G; &longs;i <expan abbr="inquã">inquam</expan> linea E F, <lb/>fecerit duos illos angulos ex parte &longs;ini&longs;tra &longs;imul &longs;umptos, æquales duobus <lb/>rectis angulis, vel duos ex parte dextra pariter æquales duobus rectis, pro­<lb/>bat Euclides rectas A B, C D, non concurrere, &longs;iue parallelas e&longs;&longs;e. </s> |
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| <s>Ibidem <emph type="italics"/>(Propter hoc Geometriæ non licet mon&longs;trare, quod contrariorum vna <lb/>e&longs;e &longs;cientia, &longs;ed neque quod duo cubi cubus)<emph.end type="italics"/> quo ad verba illa, duo cubi cubus<gap/><lb/>quæ ad nos pertinent, vult Ari&longs;t. | <s>Ibidem <emph type="italics"/>(Propter hoc Geometriæ non licet mon&longs;trare, quod contrariorum vna <lb/>e&longs;e &longs;cientia, &longs;ed neque quod duo cubi cubus)<emph.end type="italics"/> quo ad verba illa, duo cubi cubus<gap/><lb/>quæ ad nos pertinent, vult Ari&longs;t. |
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| docere<gap/>, quod non debet Geometra o&longs;ten­<lb/>dere numerorum affectiones (per enbos enim intelligit numeros quo&longs;dam <lb/>&longs;ic dictos, vt paulo po&longs;t o&longs;tendam) vt &longs;i quis vellet geometricè o&longs;tendere id, <lb/>quod o&longs;tenditur in 4. noni Elem. </s> | docere<gap/>, quod non debet Geometra o&longs;ten­<lb/>dere numerorum affectiones (per enbos enim intelligit numeros quo&longs;dam <lb/>&longs;ic dictos, vt paulo po&longs;t o&longs;tendam) vt &longs;i quis vellet geometricè o&longs;tendere id, <lb/>quod o&longs;tenditur in 4. noni Elem. |
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| <s>&longs;cilicet, &longs;i cubus numerus cubum numerum <lb/>multiplicauerit, productus numerus erit pariter cubus. </s> | &longs;cilicet, &longs;i cubus numerus cubum numerum <lb/>multiplicauerit, productus numerus erit pariter cubus. </s> |
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| <s>nonnulli latinorum <lb/>perperam textum hunc expo&longs;uerunt putantes reperiri &longs;olummodo cubos <lb/>geometricos, at Euclides definit. </s> | <s>nonnulli latinorum <lb/>perperam textum hunc expo&longs;uerunt putantes reperiri &longs;olummodo cubos <lb/>geometricos, at Euclides definit. </s> |
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| <s>v.g. </s> | <s>v.g. </s> |
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| <s>Per&longs;pectiua dicit, ea, quæ vi­<lb/>dentur eminus videri minora, quam quæ videntur cominus, quia illa viden­<lb/>tur &longs;ub angulo minori, hæc verò &longs;ub angulo maiori, quod verò remotiora <lb/>videantur &longs;ub angulo minori, quam propinquiora cæteris paribus probat <lb/><figure id="fig24"></figure><lb/>per 21. primi Elem. </s> | <s>Per&longs;pectiua dicit, ea, quæ vi­<lb/>dentur eminus videri minora, quam quæ videntur cominus, quia illa viden­<lb/>tur &longs;ub angulo minori, hæc verò &longs;ub angulo maiori, quod verò remotiora <lb/>videantur &longs;ub angulo minori, quam propinquiora cæteris paribus probat <lb/><figure id="fig24"></figure><lb/>per 21. primi Elem. |
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| <s>&longs;it enim ma­<lb/>gnitudo vi&longs;a A B, remotior ab o­<lb/>culo in C, po&longs;ito, & vi&longs;a propin­<lb/>quior ab oculo in D. ductis lineis <lb/>vi&longs;ualibus C A, C B: D A, D B; ab <lb/>oculis C, & D, ad extremitates <lb/>&longs;pectatæ magnitudinis, erit remo­<lb/>tioris vi&longs;ionis angulus C, minor <lb/>angulo D, propinquioris, vt ex præallegata Demon&longs;tratione pater. </s> | &longs;it enim ma­<lb/>gnitudo vi&longs;a A B, remotior ab o­<lb/>culo in C, po&longs;ito, & vi&longs;a propin­<lb/>quior ab oculo in D. ductis lineis <lb/>vi&longs;ualibus C A, C B: D A, D B; ab <lb/>oculis C, & D, ad extremitates <lb/>&longs;pectatæ magnitudinis, erit remo­<lb/>tioris vi&longs;ionis angulus C, minor <lb/>angulo D, propinquioris, vt ex præallegata Demon&longs;tratione pater. </s> |
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| <s>Hine <lb/>per&longs;picuè vides, qua ratione Per&longs;pectiua Geometriæ &longs;ubalternetur, &longs;iue <lb/>quid &longs;it ip&longs;a &longs;ubalternatio, vbi medium e&longs;t Geometricum, conclu&longs;io autem <lb/>optica. </s> | <s>Hine <lb/>per&longs;picuè vides, qua ratione Per&longs;pectiua Geometriæ &longs;ubalternetur, &longs;iue <lb/>quid &longs;it ip&longs;a &longs;ubalternatio, vbi medium e&longs;t Geometricum, conclu&longs;io autem <lb/>optica. </s> |
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| <s>lib. </s> | <s>lib. </s> |
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| <s>5. <lb/>Elem. </s> | <s>5. <lb/>Elem. |
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| <s>ex quo etiam loco pauca decerpam, quæ huic loco declarando con­<lb/>ducunt. </s> | ex quo etiam loco pauca decerpam, quæ huic loco declarando con­<lb/>ducunt. </s> |
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| <s>Proportio igitur multiplex e&longs;t habitudo inter duas quantitates in­<lb/>æquales, quarum maior continet minorem, bis, vel ter, vel quater, &c. </s> | <s>Proportio igitur multiplex e&longs;t habitudo inter duas quantitates in­<lb/>æquales, quarum maior continet minorem, bis, vel ter, vel quater, &c. </s> |
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| <s>Eodem tex <emph type="italics"/>(Quando igitur cognofcimes, quod­<lb/>quatuor exteriores &longs;unt æquales, quoniam I&longs;o&longs;celes, <lb/>adhuc defseit, propier quid I&longs;o&longs;celes? </s> | <s>Eodem tex <emph type="italics"/>(Quando igitur cognofcimes, quod­<lb/>quatuor exteriores &longs;unt æquales, quoniam I&longs;o&longs;celes, <lb/>adhuc defseit, propier quid I&longs;o&longs;celes? </s> |
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| <s>quoniain trian­<lb/>gulus: & hoc quoniam figura rectilinea, &c.)<emph.end type="italics"/> exem­<lb/>plo geometrico vult o&longs;tendere demon&longs;trationem <lb/>vniuer&longs;alem e&longs;&longs;e particulari præ&longs;tantiorem: e&longs;t <lb/>autem exemplum de pulcherrima, <expan abbr="atq;">atque</expan> admira­<lb/>bili proprietate, quæ omnibus figuris rectilineis <lb/>conuenit, e&longs;t <expan abbr="&qacute;">que</expan>; huiu&longs;modi: Omnis figuræ rectili­<lb/>neæ anguli externi omnes &longs;imul &longs;umpti, &longs;unt æqua <lb/>les quatuor rectis angulis, quæ affectio demon­<lb/>&longs;tratur in &longs;cholio 32. primi Elem. </s> | <s>quoniain trian­<lb/>gulus: & hoc quoniam figura rectilinea, &c.)<emph.end type="italics"/> exem­<lb/>plo geometrico vult o&longs;tendere demon&longs;trationem <lb/>vniuer&longs;alem e&longs;&longs;e particulari præ&longs;tantiorem: e&longs;t <lb/>autem exemplum de pulcherrima, <expan abbr="atq;">atque</expan> admira­<lb/>bili proprietate, quæ omnibus figuris rectilineis <lb/>conuenit, e&longs;t <expan abbr="&qacute;">que</expan>; huiu&longs;modi: Omnis figuræ rectili­<lb/>neæ anguli externi omnes &longs;imul &longs;umpti, &longs;unt æqua <lb/>les quatuor rectis angulis, quæ affectio demon­<lb/>&longs;tratur in &longs;cholio 32. primi Elem. |
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| <s>dicuntur autern <lb/>anguli externi, qui productis lateribus fiunt, vt in <lb/>triangulo pra&longs;enti anguli externi &longs;unt, B D C, <pb pagenum="62"/>D F E, F B A, ita vt quælibet figura tot angulos externos &longs;ortiatur, quot <lb/>habet latera; cum exproductis lateribus oriantur. </s> | dicuntur autern <lb/>anguli externi, qui productis lateribus fiunt, vt in <lb/>triangulo pra&longs;enti anguli externi &longs;unt, B D C, <pb pagenum="62"/>D F E, F B A, ita vt quælibet figura tot angulos externos &longs;ortiatur, quot <lb/>habet latera; cum exproductis lateribus oriantur. </s> |
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| <s>Vt autem propo&longs;itio ve­<lb/>rificetur, &longs;ingula latera ordinatim &longs;unt producenda, hoc e&longs;t, ver&longs;us eandem <lb/>partem, vt in figuris appo&longs;itis vides. </s> | <s>Vt autem propo&longs;itio ve­<lb/>rificetur, &longs;ingula latera ordinatim &longs;unt producenda, hoc e&longs;t, ver&longs;us eandem <lb/>partem, vt in figuris appo&longs;itis vides. </s> |
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| <s>Cæterum per prin­<lb/>cipia, ex quibus intelligit Dignitates, quia ex illis di&longs;currimus. </s> | <s>Cæterum per prin­<lb/>cipia, ex quibus intelligit Dignitates, quia ex illis di&longs;currimus. </s> |
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| <s>per princi­<lb/>pia verò circa quod, intelligit Definitiones, quibus, vt apparet apud Eucli­<lb/>dem, explicatur &longs;ubiectum, circa quod &longs;cientia ver&longs;atur; vt in definitioni­<lb/>bus primi Elem. </s> | <s>per princi­<lb/>pia verò circa quod, intelligit Definitiones, quibus, vt apparet apud Eucli­<lb/>dem, explicatur &longs;ubiectum, circa quod &longs;cientia ver&longs;atur; vt in definitioni­<lb/>bus primi Elem. |
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| <s>docemur, quid &longs;it linea, quid triangulum, quid circulus, <lb/>quid magnitudines reliquæ, quæ &longs;unt materia, circa quam Geometria &longs;pe­<lb/>culatur. </s> | docemur, quid &longs;it linea, quid triangulum, quid circulus, <lb/>quid magnitudines reliquæ, quæ &longs;unt materia, circa quam Geometria &longs;pe­<lb/>culatur. </s> |
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| <s>In &longs;eptimo verò traduntur definitiones numerorum, quid &longs;it nu­<lb/>merus, quid impar, quid compo&longs;itus, quadratus, cubus, & reliquæ nume­<lb/>rorum &longs;pecies, quæ &longs;unt materia &longs;eptimi, octaui, & noni, in quibus de Arith­<lb/>metica tractatur.</s></p><p type="main"> | <s>In &longs;eptimo verò traduntur definitiones numerorum, quid &longs;it nu­<lb/>merus, quid impar, quid compo&longs;itus, quadratus, cubus, & reliquæ nume­<lb/>rorum &longs;pecies, quæ &longs;unt materia &longs;eptimi, octaui, & noni, in quibus de Arith­<lb/>metica tractatur.</s></p><p type="main"> |
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| <s>in quo Zabarella non probatur, qui &longs;olum <lb/>ait, terram e&longs;&longs;e in medio mundi, à Phy&longs;icis demon&longs;trari. </s> | <s>in quo Zabarella non probatur, qui &longs;olum <lb/>ait, terram e&longs;&longs;e in medio mundi, à Phy&longs;icis demon&longs;trari. </s> |
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| <s><expan abbr="triãgulum">triangulum</expan> autem, <lb/>&longs;eu angulorum ip&longs;ius <expan abbr="prædicatũ">prædicatum</expan> e&longs;t æqualitas, & inæqualitas: vt cum in 32. <lb/>primi Elem. </s> | <s><expan abbr="triãgulum">triangulum</expan> autem, <lb/>&longs;eu angulorum ip&longs;ius <expan abbr="prædicatũ">prædicatum</expan> e&longs;t æqualitas, & inæqualitas: vt cum in 32. <lb/>primi Elem. |
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| <s>demon&longs;trat Euclides, omne triangulum habcre tres angulos <lb/>æquales duobus rectis.</s></p><p type="main"> | demon&longs;trat Euclides, omne triangulum habcre tres angulos <lb/>æquales duobus rectis.</s></p><p type="main"> |
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| <s><arrow.to.target n="marg64"></arrow.to.target></s></p><p type="margin"> | <s><arrow.to.target n="marg64"></arrow.to.target></s></p><p type="margin"> |
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| <s><margin.target id="marg66"></margin.target>66</s></p><p type="main"> | <s><margin.target id="marg66"></margin.target>66</s></p><p type="main"> |
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| <s>Eodem tex. <emph type="italics"/>(Definitiones verò apparent omnes &longs;upponentes, & accipientes <lb/>ip&longs;um quid e&longs;t, vt Mathematicæ, quid vnitas, quid par, & impar)<emph.end type="italics"/> alludit ad de­<lb/>finitiones 7. Elem. </s> | <s>Eodem tex. <emph type="italics"/>(Definitiones verò apparent omnes &longs;upponentes, & accipientes <lb/>ip&longs;um quid e&longs;t, vt Mathematicæ, quid vnitas, quid par, & impar)<emph.end type="italics"/> alludit ad de­<lb/>finitiones 7. Elem. |
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| <s>vbi agitur de numeris. </s> | vbi agitur de numeris. </s> |
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| <s>Quæ verò hoc loco de principijs <lb/>dicuntur, luculenti&longs;&longs;imè patent con&longs;ideranti definitiones, & axiomata, quæ <lb/>Mathematicis demon&longs;trationibus in omnibus ferè libris præmittuntur; ex <lb/>quibus &longs;tatim demon&longs;trationes deriuantur.</s></p><p type="main"> | <s>Quæ verò hoc loco de principijs <lb/>dicuntur, luculenti&longs;&longs;imè patent con&longs;ideranti definitiones, & axiomata, quæ <lb/>Mathematicis demon&longs;trationibus in omnibus ferè libris præmittuntur; ex <lb/>quibus &longs;tatim demon&longs;trationes deriuantur.</s></p><p type="main"> |
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| <s><margin.target id="marg67"></margin.target>67</s></p><p type="main"> | <s><margin.target id="marg67"></margin.target>67</s></p><p type="main"> |
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| <s>Et paulo po&longs;t <emph type="italics"/>(<expan abbr="Neq;">Neque</expan> <expan abbr="vtiq;">vtique</expan> de plano figura, non enim e&longs;t planum figura, <expan abbr="neq;">neque</expan> fi­<lb/>gura planum)<emph.end type="italics"/> alludit ad definitiones planarum figurarum, qualis e&longs;t circu­<lb/>lus, cuius definitio e&longs;t inter definitiones primi Elem. </s> | <s>Et paulo po&longs;t <emph type="italics"/>(<expan abbr="Neq;">Neque</expan> <expan abbr="vtiq;">vtique</expan> de plano figura, non enim e&longs;t planum figura, <expan abbr="neq;">neque</expan> fi­<lb/>gura planum)<emph.end type="italics"/> alludit ad definitiones planarum figurarum, qualis e&longs;t circu­<lb/>lus, cuius definitio e&longs;t inter definitiones primi Elem. |
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| <s>15. & e&longs;t huiu&longs;modi: <lb/>circulus e&longs;t figura plana, &longs;ub vnica linea comprehen&longs;a, quæ periphæria ap­<lb/>pellatur, ad quam ab vno puncto eorum, quæ intra figuram &longs;unt po&longs;ita, ca­<lb/>dentes omnes rectæ lineæ inter &longs;e &longs;unt æquales: in qua quidem definitione <lb/>non prædicatur planum de figura, nec figura de plano: <expan abbr="neq;">neque</expan> enim planum, <lb/>&longs;au plana &longs;uperficies e&longs;t figura &longs;ecundum &longs;e, ni&longs;i terminetur; <expan abbr="neq;">neque</expan> figura e&longs;t <lb/>plana &longs;uperficies, cum plurimæ &longs;int figuræ curuæ, & præterea &longs;olidæ quam­<lb/>plurimæ.</s></p><p type="main"> | 15. & e&longs;t huiu&longs;modi: <lb/>circulus e&longs;t figura plana, &longs;ub vnica linea comprehen&longs;a, quæ periphæria ap­<lb/>pellatur, ad quam ab vno puncto eorum, quæ intra figuram &longs;unt po&longs;ita, ca­<lb/>dentes omnes rectæ lineæ inter &longs;e &longs;unt æquales: in qua quidem definitione <lb/>non prædicatur planum de figura, nec figura de plano: <expan abbr="neq;">neque</expan> enim planum, <lb/>&longs;au plana &longs;uperficies e&longs;t figura &longs;ecundum &longs;e, ni&longs;i terminetur; <expan abbr="neq;">neque</expan> figura e&longs;t <lb/>plana &longs;uperficies, cum plurimæ &longs;int figuræ curuæ, & præterea &longs;olidæ quam­<lb/>plurimæ.</s></p><p type="main"> |
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| <s><arrow.to.target n="marg68"></arrow.to.target></s></p><p type="margin"> | <s><arrow.to.target n="marg68"></arrow.to.target></s></p><p type="margin"> |
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| <s><margin.target id="marg70"></margin.target>70</s></p><p type="main"> | <s><margin.target id="marg70"></margin.target>70</s></p><p type="main"> |
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| <s>Ibidem <emph type="italics"/>(Quod autem &longs;it, monstrat)<emph.end type="italics"/> vt per&longs;picuum e&longs;t in prima <expan abbr="demõ&longs;tra-tione">demon&longs;tra­<lb/>tione</expan> primi Elem. </s> | <s>Ibidem <emph type="italics"/>(Quod autem &longs;it, monstrat)<emph.end type="italics"/> vt per&longs;picuum e&longs;t in prima <expan abbr="demõ&longs;tra-tione">demon&longs;tra­<lb/>tione</expan> primi Elem. |
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| <s>vbi triangulum æquilaterum con&longs;truit, & po&longs;tea probat <lb/>illud e&longs;&longs;e triangulum æquilaterum. </s> | vbi triangulum æquilaterum con&longs;truit, & po&longs;tea probat <lb/>illud e&longs;&longs;e triangulum æquilaterum. </s> |
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| <s>Certum tamen e&longs;t, Geometram luppo­<lb/>nere triangulum in communi, cum inter definitiones ip&longs;ius contineatur, <pb pagenum="65"/>quod tamen non ob&longs;tat, quominus probare po&longs;&longs;it, aliquando po&longs;&longs;e <expan abbr="cõ&longs;trni">con&longs;trni</expan>, <lb/>& e&longs;&longs;e aliquod particulare triangulum, vt fit in prædicta demon&longs;tratione, <lb/>Euclidis.</s></p><p type="main"> | <s>Certum tamen e&longs;t, Geometram luppo­<lb/>nere triangulum in communi, cum inter definitiones ip&longs;ius contineatur, <pb pagenum="65"/>quod tamen non ob&longs;tat, quominus probare po&longs;&longs;it, aliquando po&longs;&longs;e <expan abbr="cõ&longs;trni">con&longs;trni</expan>, <lb/>& e&longs;&longs;e aliquod particulare triangulum, vt fit in prædicta demon&longs;tratione, <lb/>Euclidis.</s></p><p type="main"> |
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| <s>ab hac &longs;ententia po&longs;teriores Mu&longs;ici non recel­<lb/>&longs;erunt, vt videre e&longs;t apud Zarlinum.</s></p><p type="main"> | <s>ab hac &longs;ententia po&longs;teriores Mu&longs;ici non recel­<lb/>&longs;erunt, vt videre e&longs;t apud Zarlinum.</s></p><p type="main"> |
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| <s>In quo po&longs;tea con&longs;i&longs;tat ratio acnti anguli, explicat <expan abbr="induc&etilde;s">inducens</expan> definitionem <lb/>ip&longs;ius, quæ e&longs;t inter de&longs;initiones primi Elem. </s> | <s>In quo po&longs;tea con&longs;i&longs;tat ratio acnti anguli, explicat <expan abbr="induc&etilde;s">inducens</expan> definitionem <lb/>ip&longs;ius, quæ e&longs;t inter de&longs;initiones primi Elem. |
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| <s>huiu&longs;modi, Angulus acutus <lb/>e&longs;t, qui minor recto e&longs;t. </s> | huiu&longs;modi, Angulus acutus <lb/>e&longs;t, qui minor recto e&longs;t. </s> |
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| <s>Demum explicat, cur nam gladius dicatur acutus, <lb/>quia nimirum habet angulum acutum &longs;uperficialem, ide&longs;t, quem duæ &longs;uper­<lb/>ficies &longs;imul in acie gladij concurrentes efficiunt.</s></p><pb pagenum="68"/><p type="main"> | <s>Demum explicat, cur nam gladius dicatur acutus, <lb/>quia nimirum habet angulum acutum &longs;uperficialem, ide&longs;t, quem duæ &longs;uper­<lb/>ficies &longs;imul in acie gladij concurrentes efficiunt.</s></p><pb pagenum="68"/><p type="main"> |
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| <s><margin.target id="marg80"></margin.target>80</s></p><p type="main"> | <s><margin.target id="marg80"></margin.target>80</s></p><p type="main"> |
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| <s>Cap. 2. loco 32. (<emph type="italics"/>Vt qui lineam definiunt longitudmem &longs;ine latitudine e&longs;&longs;e<emph.end type="italics"/>) <lb/>&longs;uppenimus lectorem inteil exi&longs;&longs;e definitiones &longs;altem primi Elem. </s> | <s>Cap. 2. loco 32. (<emph type="italics"/>Vt qui lineam definiunt longitudmem &longs;ine latitudine e&longs;&longs;e<emph.end type="italics"/>) <lb/>&longs;uppenimus lectorem inteil exi&longs;&longs;e definitiones &longs;altem primi Elem. |
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| <s>in­<lb/>ter quas d<gap/>f<gap/>nitio lineæ e&longs;t &longs;ecunda, <expan abbr="cadem&qacute;">cademque</expan>; cum hac Ari&longs;totelis.</s></p><p type="head"> | in­<lb/>ter quas d<gap/>f<gap/>nitio lineæ e&longs;t &longs;ecunda, <expan abbr="cadem&qacute;">cademque</expan>; cum hac Ari&longs;totelis.</s></p><p type="head"> |
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| <s><emph type="italics"/>Libro Octauo.<emph.end type="italics"/></s></p><p type="main"> | <s><emph type="italics"/>Libro Octauo.<emph.end type="italics"/></s></p><p type="main"> |
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| <s>pares autem numeri, quia ne­<lb/>queunt in figuram normæ æquilateræ di&longs;poni, cum <lb/>non habeant vnitatem pro angulo, & paria po&longs;tea la­<lb/>tera, vt oportet, non merentur appellari Gnomones, vt quaternarius, &longs;i di­<lb/>&longs;ponatur &longs;ic <figure id="fig35"></figure> non refert Gnomonem, quia lateribus in&ecedil;qualibus con­<lb/>&longs;tat; <expan abbr="neq;">neque</expan> &longs;i hoc modo <figure id="fig36"></figure> quia dee&longs;t huic figuræ angularis vnitas, quæ <lb/>illi nece&longs;iaria e&longs;t. </s> | <s>pares autem numeri, quia ne­<lb/>queunt in figuram normæ æquilateræ di&longs;poni, cum <lb/>non habeant vnitatem pro angulo, & paria po&longs;tea la­<lb/>tera, vt oportet, non merentur appellari Gnomones, vt quaternarius, &longs;i di­<lb/>&longs;ponatur &longs;ic <figure id="fig35"></figure> non refert Gnomonem, quia lateribus in&ecedil;qualibus con­<lb/>&longs;tat; <expan abbr="neq;">neque</expan> &longs;i hoc modo <figure id="fig36"></figure> quia dee&longs;t huic figuræ angularis vnitas, quæ <lb/>illi nece&longs;iaria e&longs;t. </s> |
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| <s>Pythagorici igitur dicebant, numerum parem ideò e&longs;&longs;e <lb/>infinitum ip&longs;um, quia videbant ip&longs;um e&longs;&longs;e cau&longs;am perpetuæ diui&longs;ionis, cum <lb/>quælibet res quanta &longs;it diui&longs;ibilis bifariam, ide&longs;t in duo &longs;ecundum numerum <lb/>parem, & &longs;ubdiui&longs;ibilis po&longs;tea bifariam, & &longs;ic in infinitum, vt de linea pro­<lb/>blematicè probatur in 10. primi Elem. </s> | <s>Pythagorici igitur dicebant, numerum parem ideò e&longs;&longs;e <lb/>infinitum ip&longs;um, quia videbant ip&longs;um e&longs;&longs;e cau&longs;am perpetuæ diui&longs;ionis, cum <lb/>quælibet res quanta &longs;it diui&longs;ibilis bifariam, ide&longs;t in duo &longs;ecundum numerum <lb/>parem, & &longs;ubdiui&longs;ibilis po&longs;tea bifariam, & &longs;ic in infinitum, vt de linea pro­<lb/>blematicè probatur in 10. primi Elem. |
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| <s>quamuis theorematicè &longs;it axioma. <lb/></s> | quamuis theorematicè &longs;it axioma. <lb/></s> |
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| <s>hunc porrò numerum parem dicebant terminatum e&longs;&longs;e ab impari, quia ori­<lb/>tur ex diui&longs;ione cuiu&longs;uis rei, quæ vna &longs;it, &longs;umentes vnitatem pro impari. <lb/></s> | <s>hunc porrò numerum parem dicebant terminatum e&longs;&longs;e ab impari, quia ori­<lb/>tur ex diui&longs;ione cuiu&longs;uis rei, quæ vna &longs;it, &longs;umentes vnitatem pro impari. <lb/></s> |
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| <s>exponit. </s> | <s>exponit. </s> |
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| <s>alio præterea modo vtuntur infinito, vt quando &longs;upponunt data <lb/>quauis quantitate po&longs;&longs;e &longs;umi maiorem, vel etiam minorem in infinitum, vt <lb/>patet ex 6. po&longs;tulato primi Elem. </s> | <s>alio præterea modo vtuntur infinito, vt quando &longs;upponunt data <lb/>quauis quantitate po&longs;&longs;e &longs;umi maiorem, vel etiam minorem in infinitum, vt <lb/>patet ex 6. po&longs;tulato primi Elem. |
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| <s>editionis Clauianæ. </s> | editionis Clauianæ. </s> |
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| <s>numerum <expan abbr="quoq;">quoque</expan> au­<lb/>geri po&longs;&longs;e in infinitum, e&longs;t &longs;ecundum po&longs;tulatum libri 7. Elem. </s> | <s>numerum <expan abbr="quoq;">quoque</expan> au­<lb/>geri po&longs;&longs;e in infinitum, e&longs;t &longs;ecundum po&longs;tulatum libri 7. Elem. |
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| <s>vel demum <lb/>quando probant quamlibet lineam po&longs;&longs;e diuidi bifariam, quia hinc &longs;equitur <lb/>po&longs;&longs;e &longs;ub diuidi in <expan abbr="infinitũ">infinitum</expan>; his igitur modis Mathematicis <expan abbr="infinitũ">infinitum</expan> in v&longs;u e&longs;t.</s></p><p type="main"> | vel demum <lb/>quando probant quamlibet lineam po&longs;&longs;e diuidi bifariam, quia hinc &longs;equitur <lb/>po&longs;&longs;e &longs;ub diuidi in <expan abbr="infinitũ">infinitum</expan>; his igitur modis Mathematicis <expan abbr="infinitũ">infinitum</expan> in v&longs;u e&longs;t.</s></p><p type="main"> |
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| <s><arrow.to.target n="marg95"></arrow.to.target></s></p><p type="margin"> | <s><arrow.to.target n="marg95"></arrow.to.target></s></p><p type="margin"> |
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| <s>g. </s> | <s>g. </s> |
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| <s>10. primi Elem. </s> | <s>10. primi Elem. |
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| <s>quæ docet <lb/>quamlibet lineam po&longs;&longs;e diuidi bifariam nulla e&longs;&longs;et, quia linea illa, quæ con­<lb/>&longs;taret ex tribus Democriti atomis, nulla ratione bifariam &longs;ecari po&longs;&longs;et. </s> | quæ docet <lb/>quamlibet lineam po&longs;&longs;e diuidi bifariam nulla e&longs;&longs;et, quia linea illa, quæ con­<lb/>&longs;taret ex tribus Democriti atomis, nulla ratione bifariam &longs;ecari po&longs;&longs;et. </s> |
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| <s>pa­<lb/>riter totus ferè decimus liber Elem. </s> | <s>pa­<lb/>riter totus ferè decimus liber Elem. |
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| <s>deceptiuus, & nullus e&longs;&longs;et, &longs;i enim da­<lb/>rentur illæ atomi, ex quibus <expan abbr="quãtitas">quantitas</expan> conflaretur, nullæ e&longs;&longs;ent lineæ incom­<lb/>men&longs;urabiles, quandoquidem omnes communi illa, ac indiuidua, commen­<lb/>&longs;urarentur. </s> | deceptiuus, & nullus e&longs;&longs;et, &longs;i enim da­<lb/>rentur illæ atomi, ex quibus <expan abbr="quãtitas">quantitas</expan> conflaretur, nullæ e&longs;&longs;ent lineæ incom­<lb/>men&longs;urabiles, quandoquidem omnes communi illa, ac indiuidua, commen­<lb/>&longs;urarentur. </s> |
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| <s>po&longs;tulatum <expan abbr="quoq;">quoque</expan> illud, qualibet data magnitudine &longs;umi po&longs;&longs;e <lb/>minorem pror&longs;us irritum redderetur, quia data atomo, illa minor accipi <lb/>non po&longs;&longs;et.</s></p><p type="main"> | <s>po&longs;tulatum <expan abbr="quoq;">quoque</expan> illud, qualibet data magnitudine &longs;umi po&longs;&longs;e <lb/>minorem pror&longs;us irritum redderetur, quia data atomo, illa minor accipi <lb/>non po&longs;&longs;et.</s></p><p type="main"> |
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| <s>cuius demon&longs;trationem perfectam videre pote­<lb/>ris in fine commentarij P. </s> | <s>cuius demon&longs;trationem perfectam videre pote­<lb/>ris in fine commentarij P. </s> |
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| <s>Clauij &longs;uper 4. Elem. </s> | <s>Clauij &longs;uper 4. Elem. |
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| <s>nos <lb/>ea tantum attingimus, quæ percipi po&longs;&longs;int ab homi­<lb/>ne vix mathematicis tincto: &longs;ed tamen, quæ &longs;en&longs;um <lb/>Ari&longs;totelis patefaciunt. </s> | nos <lb/>ea tantum attingimus, quæ percipi po&longs;&longs;int ab homi­<lb/>ne vix mathematicis tincto: &longs;ed tamen, quæ &longs;en&longs;um <lb/>Ari&longs;totelis patefaciunt. </s> |
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| <s>Aliæ porrò figuræ replen­<lb/>tes locum planum, quibus aliquando Architectores <lb/>vtuntur, vel &longs;unt irregulares, vel ad prædictas redu­<lb/>ci po&longs;&longs;unt. </s> | <s>Aliæ porrò figuræ replen­<lb/>tes locum planum, quibus aliquando Architectores <lb/>vtuntur, vel &longs;unt irregulares, vel ad prædictas redu­<lb/>ci po&longs;&longs;unt. </s> |
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| <s>Verum enim verò plures pyramides <lb/>regulares, &longs;iue plura Tetraedra non po&longs;&longs;e replere vacuum, <expan abbr="&longs;olidum&qacute;">&longs;olidumque</expan>; con­<lb/>&longs;tituere, ex eo patet, quia &longs;i id præ&longs;tarent, conflarent nece&longs;&longs;ariò, vel vnum <lb/>ex <expan abbr="quinq;">quinque</expan> corporibus regularibus, de quibus in 13. Elemen. </s> | <s>Verum enim verò plures pyramides <lb/>regulares, &longs;iue plura Tetraedra non po&longs;&longs;e replere vacuum, <expan abbr="&longs;olidum&qacute;">&longs;olidumque</expan>; con­<lb/>&longs;tituere, ex eo patet, quia &longs;i id præ&longs;tarent, conflarent nece&longs;&longs;ariò, vel vnum <lb/>ex <expan abbr="quinq;">quinque</expan> corporibus regularibus, de quibus in 13. Elemen. </s> |
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| <s>vel aliud quod­<lb/>piam; non aliud, nam, vt patet ex &longs;cholio 13. Elem. </s> | <s>vel aliud quod­<lb/>piam; non aliud, nam, vt patet ex &longs;cholio 13. Elem. |
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| <s>non dantur, ni&longs;i illa. <lb/></s> | non dantur, ni&longs;i illa. <lb/></s> |
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| <s>quinque; <expan abbr="neq;">neque</expan> vllum ex illis, quia diameter huiu&longs;modi corporis, quod com­<lb/>poneretur ex illis pyramidibus, e&longs;&longs;et dupla lateris eiu&longs;dem, vt patet, quia <lb/>pyramides illæ omnes concurrerent ad centrum &longs;phæræ illas omnes com­<lb/>plectentis, quare latus vnius pyramidis à &longs;uperficie &longs;phæræ incipiens de&longs;i­<lb/>neret in centrum, ergo latus i&longs;tud e&longs;&longs;et &longs;emidiameter, quapropter tota dia­<lb/>meter illius &longs;ph&ecedil;ræ, & con&longs;equenter huius corporis in illa in&longs;cripti, e&longs;&longs;et du­<lb/>pla lateris eiu&longs;dem figuræ &longs;olidæ in&longs;criptæ, &longs;ed nullo talis proportio diame­<lb/>tri alicuius ex illis <expan abbr="quinq;">quinque</expan> &longs;olidis regularibus ad latus eiu&longs;dem reperitur, <lb/>quæ &longs;it nimirum dupla, vt patet ex vltimis demon&longs;trationibus 13. Elem. </s> | <s>quinque; <expan abbr="neq;">neque</expan> vllum ex illis, quia diameter huiu&longs;modi corporis, quod com­<lb/>poneretur ex illis pyramidibus, e&longs;&longs;et dupla lateris eiu&longs;dem, vt patet, quia <lb/>pyramides illæ omnes concurrerent ad centrum &longs;phæræ illas omnes com­<lb/>plectentis, quare latus vnius pyramidis à &longs;uperficie &longs;phæræ incipiens de&longs;i­<lb/>neret in centrum, ergo latus i&longs;tud e&longs;&longs;et &longs;emidiameter, quapropter tota dia­<lb/>meter illius &longs;ph&ecedil;ræ, & con&longs;equenter huius corporis in illa in&longs;cripti, e&longs;&longs;et du­<lb/>pla lateris eiu&longs;dem figuræ &longs;olidæ in&longs;criptæ, &longs;ed nullo talis proportio diame­<lb/>tri alicuius ex illis <expan abbr="quinq;">quinque</expan> &longs;olidis regularibus ad latus eiu&longs;dem reperitur, <lb/>quæ &longs;it nimirum dupla, vt patet ex vltimis demon&longs;trationibus 13. Elem. |
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| <s>ini­<lb/>tio facto à 13. demon&longs;tratione, in quibus nulla reperitur proportio dupla <lb/>inter diametrum, & latus eiu&longs;dem alicuius ex illis &longs;olidis; ex quibus mani­<lb/>fe&longs;tum e&longs;t, plures regulares pyramides quouis pacto &longs;imul vnitas nullo mo­<lb/>do replere locum &longs;olidum. </s> | ini­<lb/>tio facto à 13. demon&longs;tratione, in quibus nulla reperitur proportio dupla <lb/>inter diametrum, & latus eiu&longs;dem alicuius ex illis &longs;olidis; ex quibus mani­<lb/>fe&longs;tum e&longs;t, plures regulares pyramides quouis pacto &longs;imul vnitas nullo mo­<lb/>do replere locum &longs;olidum. </s> |
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| <s>cum igitur animaduerterem, &longs;en&longs;um Ari&longs;t. | <s>cum igitur animaduerterem, &longs;en&longs;um Ari&longs;t. |
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| <s>tu verò, &longs;i <lb/>adeo es mathematicis imbutus, con&longs;ule po&longs;tremas demon&longs;tra. </s> | <s>tu verò, &longs;i <lb/>adeo es mathematicis imbutus, con&longs;ule po&longs;tremas demon&longs;tra. </s> |
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| <s>13. Elem. </s> | <s>13. Elem. |
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| <s>& <lb/>præcipuè &longs;cholium vltimum, vbi plura de his corporibus &longs;citu digni&longs;&longs;ima, <lb/><expan abbr="atq;">atque</expan> huc &longs;pectantia reperies ex his omnibus Mathematica, quæ no&longs;træ &longs;unt <lb/>partes, per&longs;picuè &longs;atis expo&longs;uimus.</s></p><p type="main"> | & <lb/>præcipuè &longs;cholium vltimum, vbi plura de his corporibus &longs;citu digni&longs;&longs;ima, <lb/><expan abbr="atq;">atque</expan> huc &longs;pectantia reperies ex his omnibus Mathematica, quæ no&longs;træ &longs;unt <lb/>partes, per&longs;picuè &longs;atis expo&longs;uimus.</s></p><p type="main"> |
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| <s>Multo po&longs;t tempore, quàm hæc &longs;crip&longs;eram incidi fortè in cap. </s> | <s>Multo po&longs;t tempore, quàm hæc &longs;crip&longs;eram incidi fortè in cap. </s> |
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| <s>Tex. </s> | <s>Tex. </s> |
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| <s>71 <emph type="italics"/>(Deinde &longs;i terra e&longs;t cubus &c.)<emph.end type="italics"/> lege definitiones 11. Elem. </s> | <s>71 <emph type="italics"/>(Deinde &longs;i terra e&longs;t cubus &c.)<emph.end type="italics"/> lege definitiones 11. Elem. |
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| <s>quæ &longs;unt <lb/>admodum faciles, ibi reperies de&longs;initiones quinque corporum regularium, <lb/>quorum figuras Plato elementis tribuebat: qua verò id ratione faceret, ha­<lb/>bes in &longs;phæra Clau. </s> | quæ &longs;unt <lb/>admodum faciles, ibi reperies de&longs;initiones quinque corporum regularium, <lb/>quorum figuras Plato elementis tribuebat: qua verò id ratione faceret, ha­<lb/>bes in &longs;phæra Clau. </s> |
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| <s>Simpl. </s> | <s>Simpl. </s> |
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| <s>&longs;it terra in &longs;igu­<lb/>ra præ&longs;enti circulus E C D, cuius medium, &longs;ine c<gap/>­<pb pagenum="88"/>trum A. via, qua a&longs;cendit ignis &longs;it in linea A C B, quæ facit angulos in &longs;u­<lb/>perficie terræ æquales, nimirum angulos B C D, B C E. &longs;imiliter terra per <lb/>candem lineam <expan abbr="faci&etilde;s">faciens</expan> eo&longs;dem angulos æquales de&longs;cendit. </s> | <s>&longs;it terra in &longs;igu­<lb/>ra præ&longs;enti circulus E C D, cuius medium, &longs;ine c<gap/>­<pb pagenum="88"/>trum A. via, qua a&longs;cendit ignis &longs;it in linea A C B, quæ facit angulos in &longs;u­<lb/>perficie terræ æquales, nimirum angulos B C D, B C E. &longs;imiliter terra per <lb/>candem lineam <expan abbr="faci&etilde;s">faciens</expan> eo&longs;dem angulos æquales de&longs;cendit. </s> |
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| <s>linea autem, quæ <lb/>facit tales angulos tendit ad centrum &longs;phæræ A, vt patet ad &longs;en&longs;um in figu­<lb/>ra, & probari pote&longs;t geometricè ex primis tertij Elem. </s> | <s>linea autem, quæ <lb/>facit tales angulos tendit ad centrum &longs;phæræ A, vt patet ad &longs;en&longs;um in figu­<lb/>ra, & probari pote&longs;t geometricè ex primis tertij Elem. |
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| <s>ex quibus patet tam <lb/>læuia, quam grauia, quæ per talem lineam ferantur, re&longs;picere centrum A, <lb/>&longs;phæræ. </s> | ex quibus patet tam <lb/>læuia, quam grauia, quæ per talem lineam ferantur, re&longs;picere centrum A, <lb/>&longs;phæræ. </s> |
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| <s>Vtrum autem i&longs;tud centrum &longs;it idem cum <expan abbr="c&etilde;tro">centro</expan> totius mundi, alius, <lb/>inquit, e&longs;t &longs;ermo, hoc e&longs;t, ad a&longs;tronomum pertinet. </s> | <s>Vtrum autem i&longs;tud centrum &longs;it idem cum <expan abbr="c&etilde;tro">centro</expan> totius mundi, alius, <lb/>inquit, e&longs;t &longs;ermo, hoc e&longs;t, ad a&longs;tronomum pertinet. </s> |
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| <s>Tex. </s> | <s>Tex. </s> |
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| <s>11. <emph type="italics"/>(Videtur autem non &longs;olum ip&longs;um quid e&longs;t cogno&longs;cere vtile e&longs;&longs;e <lb/>ad cogno&longs;cendas cau&longs;as accidentium &longs;ub&longs;tantijs: &longs;icut in Mathemati­<lb/>cis quid rectum, & quid obliqaum, aut quid linea, & planum, ad co­<lb/>gno&longs; cendum quot rectis, trianguli anguli &longs;unt æquales)<emph.end type="italics"/> quid &longs;it <expan abbr="vnum-quodq;">vnum­<lb/>quodque</expan> ex prædictis patet tum ex definitionibus primi Elem. </s> | <s>11. <emph type="italics"/>(Videtur autem non &longs;olum ip&longs;um quid e&longs;t cogno&longs;cere vtile e&longs;&longs;e <lb/>ad cogno&longs;cendas cau&longs;as accidentium &longs;ub&longs;tantijs: &longs;icut in Mathemati­<lb/>cis quid rectum, & quid obliqaum, aut quid linea, & planum, ad co­<lb/>gno&longs; cendum quot rectis, trianguli anguli &longs;unt æquales)<emph.end type="italics"/> quid &longs;it <expan abbr="vnum-quodq;">vnum­<lb/>quodque</expan> ex prædictis patet tum ex definitionibus primi Elem. |
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| <s>tum ex com­<lb/>mentarijs ip&longs;arum; quamuis autem ibi non definiatur <expan abbr="rectũ">rectum</expan>, nec obliquum <lb/>in genere, definitur tamen linea recta, & obliqua, & plana &longs;uperficies, &longs;iue <lb/>planum, ex quibus facilè definitio recti, & obliqui colligi pote&longs;t: quæ defi­<lb/>nitiones nece&longs;&longs;ariæ &longs;unt ad cogno&longs;cendum quot rectis angulis æquales &longs;int <lb/>tres anguli cuiufuis trianguli. </s> | tum ex com­<lb/>mentarijs ip&longs;arum; quamuis autem ibi non definiatur <expan abbr="rectũ">rectum</expan>, nec obliquum <lb/>in genere, definitur tamen linea recta, & obliqua, & plana &longs;uperficies, &longs;iue <lb/>planum, ex quibus facilè definitio recti, & obliqui colligi pote&longs;t: quæ defi­<lb/>nitiones nece&longs;&longs;ariæ &longs;unt ad cogno&longs;cendum quot rectis angulis æquales &longs;int <lb/>tres anguli cuiufuis trianguli. </s> |
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| <s>vide quæ de hac æqualitate &longs;crip&longs;i lib, primo <lb/>Priorum, &longs;ecto 3. cap. </s> | <s>vide quæ de hac æqualitate &longs;crip&longs;i lib, primo <lb/>Priorum, &longs;ecto 3. cap. </s> |
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| <s>13. <emph type="italics"/>(Si igitur e&longs;t aliqua animæ operatio, aut pa&longs;&longs;io propria, continget vti­<lb/>que ip&longs;am &longs;eparari: &longs;i verò nulla e&longs;t propria ip&longs;ius non vtique erit &longs;eparabilis. </s> | <s>13. <emph type="italics"/>(Si igitur e&longs;t aliqua animæ operatio, aut pa&longs;&longs;io propria, continget vti­<lb/>que ip&longs;am &longs;eparari: &longs;i verò nulla e&longs;t propria ip&longs;ius non vtique erit &longs;eparabilis. </s> |
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| <s>&longs;ed <lb/>&longs;icut recto in quantum rectum multa accidunt, vt tangere æ<gap/>eam &longs;phæram &longs;ecun­<lb/>dum punctum, non tamen tanget hoc, rectum ip&longs;um &longs;eparatum: in&longs;eparabile enim, <lb/>&longs;i quidem cum corpore quodam &longs;emper e&longs;t)<emph.end type="italics"/> Propo&longs;itio 2. tertij Elem. </s> | <s>&longs;ed <lb/>&longs;icut recto in quantum rectum multa accidunt, vt tangere æ<gap/>eam &longs;phæram &longs;ecun­<lb/>dum punctum, non tamen tanget hoc, rectum ip&longs;um &longs;eparatum: in&longs;eparabile enim, <lb/>&longs;i quidem cum corpore quodam &longs;emper e&longs;t)<emph.end type="italics"/> Propo&longs;itio 2. tertij Elem. |
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| <s>&pacute;robat li­<lb/><figure id="fig62"></figure><lb/>neam rectam, duo quælibet puncta <expan abbr="quãtumuis">quantumuis</expan> pro­<lb/>pinqua in circuli ambitu a&longs;&longs;umpta coniungentem <lb/>cadere intra circulum. </s> | &pacute;robat li­<lb/><figure id="fig62"></figure><lb/>neam rectam, duo quælibet puncta <expan abbr="quãtumuis">quantumuis</expan> pro­<lb/>pinqua in circuli ambitu a&longs;&longs;umpta coniungentem <lb/>cadere intra circulum. </s> |
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| <s>v. </s> | <s>v. </s> |
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| <s>&longs;i quis verò dicat tetragoni&longs;mum hunc quadri­<lb/>lateri dati e&longs;&longs;e mediæ prædictæ inuentionem cau&longs;alem afferet definitionem, <lb/>cum rei cau&longs;am dicat. </s> | <s>&longs;i quis verò dicat tetragoni&longs;mum hunc quadri­<lb/>lateri dati e&longs;&longs;e mediæ prædictæ inuentionem cau&longs;alem afferet definitionem, <lb/>cum rei cau&longs;am dicat. </s> |
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| <s>Aduerte 10. Grammaticum immeritò accu&longs;are Ale­<lb/>xandrum, quod dicat quadrationem hanc per inuentionem mediæ propor­<lb/>tionalis tradi in 2. Elem. </s> | <s>Aduerte 10. Grammaticum immeritò accu&longs;are Ale­<lb/>xandrum, quod dicat quadrationem hanc per inuentionem mediæ propor­<lb/>tionalis tradi in 2. Elem. |
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| <s>nam verè in 14. 2. traditur talis inuentio, quam­<lb/>uis enim ibi nulla fiat expre&longs;&longs;a mentio huiu&longs;modi mediæ, in ip&longs;a tamen e<gap/><lb/>reperitur, ac per eam figuræ rectilineæ quadrantur: quod patet ex figura <lb/>14. prædictæ, quæ eadem e&longs;t cum figura 13. 6. qua docemur prædictam in­<lb/>uentionem.</s></p><p type="main"> | nam verè in 14. 2. traditur talis inuentio, quam­<lb/>uis enim ibi nulla fiat expre&longs;&longs;a mentio huiu&longs;modi mediæ, in ip&longs;a tamen e<gap/><lb/>reperitur, ac per eam figuræ rectilineæ quadrantur: quod patet ex figura <lb/>14. prædictæ, quæ eadem e&longs;t cum figura 13. 6. qua docemur prædictam in­<lb/>uentionem.</s></p><p type="main"> |
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| <s><arrow.to.target n="marg177"></arrow.to.target></s></p><p type="margin"> | <s><arrow.to.target n="marg177"></arrow.to.target></s></p><p type="margin"> |
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| <s>g. </s> | <s>g. </s> |
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| <s>A, &longs;unt con&longs;tituti, æquales &longs;unt <lb/>non duobus, vt e&longs;t in textu, &longs;ed quatuor rectis, <lb/>vt patet ex corollario 2. 15. primi Elem. </s> | <s>A, &longs;unt con&longs;tituti, æquales &longs;unt <lb/>non duobus, vt e&longs;t in textu, &longs;ed quatuor rectis, <lb/>vt patet ex corollario 2. 15. primi Elem. |
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| <s>quot­<lb/>quot enim anguli con&longs;tituantur ad punctum A, <lb/>omnes &longs;imul erunt æquales quatuor rectis, quos <lb/>faciunt præ&longs;entes lineæ B C, D E. vniuer&longs;i enim <lb/>illi congruent his quatuor rectis: &longs;ed Ari&longs;t. </s> | quot­<lb/>quot enim anguli con&longs;tituantur ad punctum A, <lb/>omnes &longs;imul erunt æquales quatuor rectis, quos <lb/>faciunt præ&longs;entes lineæ B C, D E. vniuer&longs;i enim <lb/>illi congruent his quatuor rectis: &longs;ed Ari&longs;t. </s> |
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| <s>&longs;en­<lb/>&longs;us e&longs;t omnes angulos ad ea&longs;dem partes con&longs;ti­<lb/>tutos, v. </s> | <s>&longs;en­<lb/>&longs;us e&longs;t omnes angulos ad ea&longs;dem partes con&longs;ti­<lb/>tutos, v. </s> |
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| <s><margin.target id="marg215"></margin.target>224</s></p><p type="main"> | <s><margin.target id="marg215"></margin.target>224</s></p><p type="main"> |
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| <s>Ibidem (<emph type="italics"/>Verum aliquid quidem, aliquid verò non, vt puta parem numerum <lb/>primum nullum e&longs;&longs;e; aut quo&longs;dam quidem, quo&longs;dam verò non<emph.end type="italics"/>) definitione 11. <lb/>7. Elem. </s> | <s>Ibidem (<emph type="italics"/>Verum aliquid quidem, aliquid verò non, vt puta parem numerum <lb/>primum nullum e&longs;&longs;e; aut quo&longs;dam quidem, quo&longs;dam verò non<emph.end type="italics"/>) definitione 11. <lb/>7. Elem. |
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| <s>&longs;ic numerus ille, qui à Mathematicis dicitur primus, definitur, pri­<lb/>mus numerus e&longs;t, quem vnitas &longs;ola metitur, vnde patet inter numeros pa­<lb/>res &longs;olum binarium e&longs;&longs;e primum, cum ip&longs;um &longs;ola vnitas bis replicata men­<lb/>&longs;uraret. </s> | &longs;ic numerus ille, qui à Mathematicis dicitur primus, definitur, pri­<lb/>mus numerus e&longs;t, quem vnitas &longs;ola metitur, vnde patet inter numeros pa­<lb/>res &longs;olum binarium e&longs;&longs;e primum, cum ip&longs;um &longs;ola vnitas bis replicata men­<lb/>&longs;uraret. </s> |
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| <s>quaternarium autem, &longs;enarium, &c. </s> | <s>quaternarium autem, &longs;enarium, &c. </s> |
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| <s>pract. <lb/></s> | <s>pract. <lb/></s> |
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| <s>&longs;i autem de ip&longs;is circulis intelligerentur fal&longs;a e&longs;&longs;ent, non enim e&longs;t circulus <lb/>ad circulum, vt diameter ad diametrum; &longs;ed circuli &longs;unt inter &longs;e, quemad­<lb/>modum à diametris ip&longs;orum quadrata per &longs;ecundam 12. Elem. </s> | <s>&longs;i autem de ip&longs;is circulis intelligerentur fal&longs;a e&longs;&longs;ent, non enim e&longs;t circulus <lb/>ad circulum, vt diameter ad diametrum; &longs;ed circuli &longs;unt inter &longs;e, quemad­<lb/>modum à diametris ip&longs;orum quadrata per &longs;ecundam 12. Elem. |
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| <s>quadrata <lb/>autem &longs;unt inter &longs;e in duplicata ratione laterum per 20. 6. <expan abbr="eiusq;">eiusque</expan> corolla­<lb/>rium; hoc e&longs;t &longs;i fiat, vt latus maioris quadrati ad latus minoris, ita latus mi­<lb/>noris ad aliam tertiam lineam, erit quadratum maius ad minus, vt latus <lb/>ip&longs;ius ad tertiam illam lineam; non autem vt ad latus minoris. </s> | quadrata <lb/>autem &longs;unt inter &longs;e in duplicata ratione laterum per 20. 6. <expan abbr="eiusq;">eiusque</expan> corolla­<lb/>rium; hoc e&longs;t &longs;i fiat, vt latus maioris quadrati ad latus minoris, ita latus mi­<lb/>noris ad aliam tertiam lineam, erit quadratum maius ad minus, vt latus <lb/>ip&longs;ius ad tertiam illam lineam; non autem vt ad latus minoris. </s> |
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| <s>cum ergo <lb/>circulus &longs;it ad circulum, vt quadratum diametri ad quadratum diametri, <lb/>& quadrata non <expan abbr="habeãt">habeant</expan> rationem laterum, &longs;eu diametrorum prædictorum, <lb/>&longs;ed illorum duplicatam, <expan abbr="neq;">neque</expan> circuli inuicem illam habere poterunt.</s></p><p type="main"> | <s>cum ergo <lb/>circulus &longs;it ad circulum, vt quadratum diametri ad quadratum diametri, <lb/>& quadrata non <expan abbr="habeãt">habeant</expan> rationem laterum, &longs;eu diametrorum prædictorum, <lb/>&longs;ed illorum duplicatam, <expan abbr="neq;">neque</expan> circuli inuicem illam habere poterunt.</s></p><p type="main"> |
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| <s><margin.target id="marg252"></margin.target>262</s></p><p type="main"> | <s><margin.target id="marg252"></margin.target>262</s></p><p type="main"> |
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| <s>Rhombus ex definitione 23. primi Elem. </s> | <s>Rhombus ex definitione 23. primi Elem. |
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| <s>e&longs;t figura æquilatera qui­<lb/><figure id="fig103"></figure><lb/>dem, &longs;ed non æquiangula, habet enim <lb/>binos oppo&longs;itos angulos acutos, & alies <lb/>binos oppo&longs;itos obtu&longs;os, talis e&longs;t præ­<lb/>&longs;ens figura A B D C. </s> | e&longs;t figura æquilatera qui­<lb/><figure id="fig103"></figure><lb/>dem, &longs;ed non æquiangula, habet enim <lb/>binos oppo&longs;itos angulos acutos, & alies <lb/>binos oppo&longs;itos obtu&longs;os, talis e&longs;t præ­<lb/>&longs;ens figura A B D C. </s> |
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| <s>In præ&longs;enti porrò quæ&longs;tione <lb/>&longs;upponitur punctum A, quod e&longs;t vnum extremum <lb/>in rhombo moueri &longs;uper latus A B, ver&longs;us B, & &longs;i­<lb/>militer interim æqua velocitate moueri alterum <lb/>extremum B, &longs;uper idem latus A B_{2} ver&longs;us A, & in­<lb/>terim dum hæc duo puncta hoc modo &longs;ibi obuiam <lb/>procedunt, moueri latus totum A B, eadem ve­<lb/>locitate, ver&longs;us latus C D, ita vt &longs;emper ip&longs;i C D, <lb/>æquidi&longs;ter, <expan abbr="de&longs;cendat&qacute;">de&longs;cendatque</expan>; per latera A C, B D, quo­<lb/>u&longs;que ip&longs;i C D, congruat.</s></p><p type="main"> | <s>In præ&longs;enti porrò quæ&longs;tione <lb/>&longs;upponitur punctum A, quod e&longs;t vnum extremum <lb/>in rhombo moueri &longs;uper latus A B, ver&longs;us B, & &longs;i­<lb/>militer interim æqua velocitate moueri alterum <lb/>extremum B, &longs;uper idem latus A B_{2} ver&longs;us A, & in­<lb/>terim dum hæc duo puncta hoc modo &longs;ibi obuiam <lb/>procedunt, moueri latus totum A B, eadem ve­<lb/>locitate, ver&longs;us latus C D, ita vt &longs;emper ip&longs;i C D, <lb/>æquidi&longs;ter, <expan abbr="de&longs;cendat&qacute;">de&longs;cendatque</expan>; per latera A C, B D, quo­<lb/>u&longs;que ip&longs;i C D, congruat.</s></p><p type="main"> |
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| <s>A lint<gap/>igentiam igitur huius operis nece&longs;&longs;arium e&longs;t noui&longs;&longs;e, quæ nam <lb/>&longs;int quantitates commen&longs;urabiles, & quæ in commen&longs;urabiles. </s> | <s>A lint<gap/>igentiam igitur huius operis nece&longs;&longs;arium e&longs;t noui&longs;&longs;e, quæ nam <lb/>&longs;int quantitates commen&longs;urabiles, & quæ in commen&longs;urabiles. </s> |
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| <s>quæ prima, <lb/>& &longs;ecunda definitione 10. Elem. </s> | <s>quæ prima, <lb/>& &longs;ecunda definitione 10. Elem. |
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| <s>explicantur; <expan abbr="ego&qacute;">egoque</expan>; eas primo Priorum oc­<lb/>ca&longs;ione a&longs;ymetriæ diametri cum co&longs;ta &longs;atis expo&longs;ui: vtrumuis locum vide­<lb/>ris præ&longs;enti nece&longs;&longs;itati con&longs;ultum erit.</s></p><p type="main"> | explicantur; <expan abbr="ego&qacute;">egoque</expan>; eas primo Priorum oc­<lb/>ca&longs;ione a&longs;ymetriæ diametri cum co&longs;ta &longs;atis expo&longs;ui: vtrumuis locum vide­<lb/>ris præ&longs;enti nece&longs;&longs;itati con&longs;ultum erit.</s></p><p type="main"> |
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| <s><arrow.to.target n="marg267"></arrow.to.target></s></p><p type="margin"> | <s><arrow.to.target n="marg267"></arrow.to.target></s></p><p type="margin"> |
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| <s>Ve­<lb/>rùm neque &longs;ecundum &longs;e aliquam definitam naturam habebunt, &longs;ed collatæ &longs;ibi ip&longs;is <lb/>tam rationales, quàm irrationales erunt omnes.<emph.end type="italics"/></s></p><p type="main"> | <s>Ve­<lb/>rùm neque &longs;ecundum &longs;e aliquam definitam naturam habebunt, &longs;ed collatæ &longs;ibi ip&longs;is <lb/>tam rationales, quàm irrationales erunt omnes.<emph.end type="italics"/></s></p><p type="main"> |
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| <s>Hæc e&longs;t alia eorumdem ratio ad idem comprobandum: quam, vt benè <lb/>percipiamus, nonnulla prius ex definitionibus 10. Elem. </s> | <s>Hæc e&longs;t alia eorumdem ratio ad idem comprobandum: quam, vt benè <lb/>percipiamus, nonnulla prius ex definitionibus 10. Elem. |
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| <s>&longs;unt explicanda: <lb/>vt quæ nam &longs;int lineæ rationales, quæ irrationales, quæ ex binis nomini­<lb/>bus, quæ Apotomæ.</s></p><p type="main"> | &longs;unt explicanda: <lb/>vt quæ nam &longs;int lineæ rationales, quæ irrationales, quæ ex binis nomini­<lb/>bus, quæ Apotomæ.</s></p><p type="main"> |
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| <s>Propo&longs;ita igitur linea quapiam, v. </s> | <s>Propo&longs;ita igitur linea quapiam, v. </s> |
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| <s>indiu duæ autem lineæ &longs;ibi ip&longs;is co<gap/>men&longs;urabiles &longs;unt longitudine, cuminter &longs;e <lb/>&longs;iat æquales; quare po<gap/>emia quoque, quod &longs;i hoc e&longs;t, diuiduum erit quadratum.<emph.end type="italics"/></s></p><pb pagenum="207"/><p type="main"> | <s>indiu duæ autem lineæ &longs;ibi ip&longs;is co<gap/>men&longs;urabiles &longs;unt longitudine, cuminter &longs;e <lb/>&longs;iat æquales; quare po<gap/>emia quoque, quod &longs;i hoc e&longs;t, diuiduum erit quadratum.<emph.end type="italics"/></s></p><pb pagenum="207"/><p type="main"> |
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| <s>Pergit adhuc nouis rationibus aduer&longs;arios refellere, dicens, &longs;i extarent <lb/>huiu&longs;modi indiuiduæ lineæ, &longs;equeretur omnes omninò lineas e&longs;&longs;e commen­<lb/>&longs;urabiles, quod e&longs;t contra demon&longs;trata in 10. Elem. </s> | <s>Pergit adhuc nouis rationibus aduer&longs;arios refellere, dicens, &longs;i extarent <lb/>huiu&longs;modi indiuiduæ lineæ, &longs;equeretur omnes omninò lineas e&longs;&longs;e commen­<lb/>&longs;urabiles, quod e&longs;t contra demon&longs;trata in 10. Elem. |
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| <s>quia cum omnes lineæ <lb/><expan abbr="con&longs;t&etilde;t">con&longs;tent</expan> per ip&longs;os ex lineis atomis, i&longs;tæ atomæ e&longs;&longs;ent omnium linearum com­<lb/>munes men&longs;uræ, vnde & illæ, quæ dicuntur potentia tantum commen&longs;ura­<lb/>biles, vt &longs;upra explicaui, erunt etiam commen&longs;urabiles longitudine. </s> | quia cum omnes lineæ <lb/><expan abbr="con&longs;t&etilde;t">con&longs;tent</expan> per ip&longs;os ex lineis atomis, i&longs;tæ atomæ e&longs;&longs;ent omnium linearum com­<lb/>munes men&longs;uræ, vnde & illæ, quæ dicuntur potentia tantum commen&longs;ura­<lb/>biles, vt &longs;upra explicaui, erunt etiam commen&longs;urabiles longitudine. </s> |
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| <s>indiui­<lb/>duæ verò ip&longs;æ, cum &longs;int inuicem æquales, erunt ip&longs;æ <expan abbr="quoq;">quoque</expan> commen&longs;urabi­<lb/>les longitudine, quare & potentia, omnes enim longitudine commen&longs;ura­<lb/>biles, &longs;unt etiam potentia commen&longs;urabiles, ex 9. 10. vnde &longs;equitur qua­<lb/>drata earum omnia e&longs;&longs;e <expan abbr="quoq;">quoque</expan> commen&longs;urabilia: <expan abbr="atq;">atque</expan> hinc con&longs;equitur, in­<lb/>quit, ea e&longs;&longs;e <expan abbr="quoq;">quoque</expan> diuidua (quam con&longs;ecutionem probat infra num. </s> | <s>indiui­<lb/>duæ verò ip&longs;æ, cum &longs;int inuicem æquales, erunt ip&longs;æ <expan abbr="quoq;">quoque</expan> commen&longs;urabi­<lb/>les longitudine, quare & potentia, omnes enim longitudine commen&longs;ura­<lb/>biles, &longs;unt etiam potentia commen&longs;urabiles, ex 9. 10. vnde &longs;equitur qua­<lb/>drata earum omnia e&longs;&longs;e <expan abbr="quoq;">quoque</expan> commen&longs;urabilia: <expan abbr="atq;">atque</expan> hinc con&longs;equitur, in­<lb/>quit, ea e&longs;&longs;e <expan abbr="quoq;">quoque</expan> diuidua (quam con&longs;ecutionem probat infra num. </s> |
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| <s>in omni autem æquilatero perpendicularis <lb/>in mediam ba&longs;im mcidit, quare, & in medium indiuiduæ.<emph.end type="italics"/></s></p><p type="main"> | <s>in omni autem æquilatero perpendicularis <lb/>in mediam ba&longs;im mcidit, quare, & in medium indiuiduæ.<emph.end type="italics"/></s></p><p type="main"> |
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| <s>Ex 22. primi Elem. </s> | <s>Ex 22. primi Elem. |
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| <s>ex tribus datis lineis, quarum quælibet duæ &longs;int, re­<lb/>liqua maiores pote&longs;t con&longs;titui triangulum: poterit igitur ex tribus indiui­<lb/><figure id="fig122"></figure><lb/>duis con&longs;titui <expan abbr="triãgulum">triangulum</expan>, <expan abbr="illud&qacute;">illudque</expan>; æquilaterum, cum omnes in­<lb/>diuiduæ lineæ &longs;int æquales. </s> | ex tribus datis lineis, quarum quælibet duæ &longs;int, re­<lb/>liqua maiores pote&longs;t con&longs;titui triangulum: poterit igitur ex tribus indiui­<lb/><figure id="fig122"></figure><lb/>duis con&longs;titui <expan abbr="triãgulum">triangulum</expan>, <expan abbr="illud&qacute;">illudque</expan>; æquilaterum, cum omnes in­<lb/>diuiduæ lineæ &longs;int æquales. </s> |
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| <s>&longs;it igitur ex eis triangulum A B C, <lb/>&longs;i igitur ab angulo A, ducatur perpendicularis A D, ad ba&longs;im <lb/>B C, eam bif<gap/>riam &longs;ecabit ex &longs;cholio 26. primi, erit igitur li­<lb/>nea B C, &longs;ecabilis, contra quam aduer&longs;arij opinantur.</s></p><p type="main"> | <s>&longs;it igitur ex eis triangulum A B C, <lb/>&longs;i igitur ab angulo A, ducatur perpendicularis A D, ad ba&longs;im <lb/>B C, eam bif<gap/>riam &longs;ecabit ex &longs;cholio 26. primi, erit igitur li­<lb/>nea B C, &longs;ecabilis, contra quam aduer&longs;arij opinantur.</s></p><p type="main"> |
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| <s>&longs;e rectos. </s> | <s>&longs;e rectos. </s> |
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| <s>contemplatur præterea Geometra omnes <lb/>angulos rectos e&longs;&longs;e inter &longs;e æquales, vt in 12. axiomate primi Elem. </s> | <s>contemplatur præterea Geometra omnes <lb/>angulos rectos e&longs;&longs;e inter &longs;e æquales, vt in 12. axiomate primi Elem. |
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| <s>ponitur, <lb/>& &longs;imilia plura alia, quorum con&longs;iderationem Faber omninò negligit.</s></p><p type="main"> | ponitur, <lb/>& &longs;imilia plura alia, quorum con&longs;iderationem Faber omninò negligit.</s></p><p type="main"> |
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| <s><arrow.to.target n="marg293"></arrow.to.target></s></p><p type="margin"> | <s><arrow.to.target n="marg293"></arrow.to.target></s></p><p type="margin"> |
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| <s>Ibidem <emph type="italics"/>(Nim proportio æqualitas e&longs;t rationum)<emph.end type="italics"/> Per proportionem hoc lo­<lb/>co intelligenda e&longs;t illa, quam nunc appellant proportionalitatem, quæ e&longs;t <lb/>duarum rationum, &longs;eu proportionum &longs;imilitudo, &longs;iue æqualitas, vt manife­<lb/>&longs;tum e&longs;t ex 4. definit. </s> | <s>Ibidem <emph type="italics"/>(Nim proportio æqualitas e&longs;t rationum)<emph.end type="italics"/> Per proportionem hoc lo­<lb/>co intelligenda e&longs;t illa, quam nunc appellant proportionalitatem, quæ e&longs;t <lb/>duarum rationum, &longs;eu proportionum &longs;imilitudo, &longs;iue æqualitas, vt manife­<lb/>&longs;tum e&longs;t ex 4. definit. </s> |
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| <s>5. Elem. </s> | <s>5. Elem. |
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| <s>v. </s> | v. </s> |
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| <s>g. </s> | <s>g. </s> |
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| <s>vt par, impar, quadratum, quadrantale, lon­<lb/>gum, planum, primum, compo&longs;itum)<emph.end type="italics"/> Cur omnes nationes miro quodam con­<lb/>&longs;en&longs;u &longs;uos numeros in denas, veluti in gradus quo&longs;dam diuidant, Ari&longs;toteles <lb/>cau&longs;am indagaturus, re&longs;pondet primò id fortè accidi&longs;&longs;e ob denarij numeri <lb/>perfectionem: cuius perfectionis hoe e&longs;t indicium, quod denarius continea<gap/><lb/>omnes numerorum &longs;pecies. </s> | <s>vt par, impar, quadratum, quadrantale, lon­<lb/>gum, planum, primum, compo&longs;itum)<emph.end type="italics"/> Cur omnes nationes miro quodam con­<lb/>&longs;en&longs;u &longs;uos numeros in denas, veluti in gradus quo&longs;dam diuidant, Ari&longs;toteles <lb/>cau&longs;am indagaturus, re&longs;pondet primò id fortè accidi&longs;&longs;e ob denarij numeri <lb/>perfectionem: cuius perfectionis hoe e&longs;t indicium, quod denarius continea<gap/><lb/>omnes numerorum &longs;pecies. </s> |
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| <s>quæ quidem omnes numerorum &longs;pecies in defi­<lb/>nitionibus 7. Elem. </s> | <s>quæ quidem omnes numerorum &longs;pecies in defi­<lb/>nitionibus 7. Elem. |
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| <s>exponuntur, quas con&longs;ulere debes. </s> | exponuntur, quas con&longs;ulere debes. </s> |
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| <s>in denario numero <lb/>contineri numeros pares, ac impares, per &longs;e patet. </s> | <s>in denario numero <lb/>contineri numeros pares, ac impares, per &longs;e patet. </s> |
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| <s>in denario autem <expan abbr="cõtinetur">continetur</expan> etiam <lb/>hic numerus, e&longs;t enim octonarius numerus cubus, fit enim ex binario ter in <lb/>&longs;e ip&longs;um multiplicato, hoc modo; duo bis faciunt quatuor: rur&longs;us duo qua­<lb/>ter faciunt octo; quem ex definitione numeri cubi, con&longs;tat e&longs;&longs;e cubum. </s> | <s>in denario autem <expan abbr="cõtinetur">continetur</expan> etiam <lb/>hic numerus, e&longs;t enim octonarius numerus cubus, fit enim ex binario ter in <lb/>&longs;e ip&longs;um multiplicato, hoc modo; duo bis faciunt quatuor: rur&longs;us duo qua­<lb/>ter faciunt octo; quem ex definitione numeri cubi, con&longs;tat e&longs;&longs;e cubum. </s> |
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| <s>qua <lb/>ratione deinde reliqui numeri, longus, planus, primus, compo&longs;itus, in de­<lb/>nario exi&longs;tant, facilè e&longs;t cogno&longs;cere, dummodo eorum definitiones tenean­<lb/>tur, quæ initio 7. Elem. </s> | <s>qua <lb/>ratione deinde reliqui numeri, longus, planus, primus, compo&longs;itus, in de­<lb/>nario exi&longs;tant, facilè e&longs;t cogno&longs;cere, dummodo eorum definitiones tenean­<lb/>tur, quæ initio 7. Elem. |
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| <s>traduntur.</s></p><p type="main"> | traduntur.</s></p><p type="main"> |
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| <s><arrow.to.target n="marg332"></arrow.to.target></s></p><p type="margin"> | <s><arrow.to.target n="marg332"></arrow.to.target></s></p><p type="margin"> |
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| <s>comprobare, <expan abbr="atq;">atque</expan> intelligere po&longs;­<lb/>&longs;umus. </s> | <s>comprobare, <expan abbr="atq;">atque</expan> intelligere po&longs;­<lb/>&longs;umus. </s> |
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| <s>e&longs;t autem 8. 9. Elem. </s> | <s>e&longs;t autem 8. 9. Elem. |
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| <s>propo&longs;itio hæc: &longs;i decem numeri in eadem pro­<pb pagenum="230"/>portione progrediantur ab vnitate incipientes, erunt ex illis quatuor cubi, <lb/>v.g. </s> | propo&longs;itio hæc: &longs;i decem numeri in eadem pro­<pb pagenum="230"/>portione progrediantur ab vnitate incipientes, erunt ex illis quatuor cubi, <lb/>v.g. </s> |
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| <s>in &longs;erie duplæ proportionis progrediantur hi decem termini: 1. 2. 4. 8. <lb/>16. 32. 64. 128. 256. 512. ex his decem numeris &longs;unt quatuor cubi, nimi­<lb/>rum hi 1. 8. 64. 512. numerus cubus e&longs;t, qui fit ex tribus æqualibus numeris <lb/>in &longs;e multiplicatis. </s> | <s>in &longs;erie duplæ proportionis progrediantur hi decem termini: 1. 2. 4. 8. <lb/>16. 32. 64. 128. 256. 512. ex his decem numeris &longs;unt quatuor cubi, nimi­<lb/>rum hi 1. 8. 64. 512. numerus cubus e&longs;t, qui fit ex tribus æqualibus numeris <lb/>in &longs;e multiplicatis. </s> |
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| <s><expan abbr="manife&longs;tũ">manife&longs;tum</expan> eft enim, quò lon­<lb/>gius radij C F, C H, producti fuerint, eò <lb/>maiorem fore <expan abbr="diametrũ">diametrum</expan> illuminationis F H. <lb/>euadet enim L O, & &longs;imiliter ex productio­<lb/>ne radiorum D K, D E, diameter alterius <lb/>illuminationis K E, augebitur, & fiet N M. <lb/>& con&longs;equenter duæ ip&longs;arum peripheriæ &longs;i­<lb/>mul maiores fient, ac proinde ad vnius cir­<lb/>culi &longs;imilitudinem ex <expan abbr="&longs;ecũda">&longs;ecunda</expan> notatione per­<lb/>uenient. </s> | <s><expan abbr="manife&longs;tũ">manife&longs;tum</expan> eft enim, quò lon­<lb/>gius radij C F, C H, producti fuerint, eò <lb/>maiorem fore <expan abbr="diametrũ">diametrum</expan> illuminationis F H. <lb/>euadet enim L O, & &longs;imiliter ex productio­<lb/>ne radiorum D K, D E, diameter alterius <lb/>illuminationis K E, augebitur, & fiet N M. <lb/>& con&longs;equenter duæ ip&longs;arum peripheriæ &longs;i­<lb/>mul maiores fient, ac proinde ad vnius cir­<lb/>culi &longs;imilitudinem ex <expan abbr="&longs;ecũda">&longs;ecunda</expan> notatione per­<lb/>uenient. </s> |
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| <s>& quamuis ex radiorum produ­<lb/>ctione augeantur non &longs;olum prædictæ dia­<lb/>metri illuminationum, &longs;ed etiam earum <lb/>differentiæ F K, & H E; hæ tamen differen­<lb/>tiæ re&longs;pectu illorum nihil, quod &longs;en&longs;ibile &longs;it <lb/>augentur; quod inde oritur, quia angulus F C H, maior e&longs;t angulo F B K, <lb/>per 16. primi Elem. </s> | <s>& quamuis ex radiorum produ­<lb/>ctione augeantur non &longs;olum prædictæ dia­<lb/>metri illuminationum, &longs;ed etiam earum <lb/>differentiæ F K, & H E; hæ tamen differen­<lb/>tiæ re&longs;pectu illorum nihil, quod &longs;en&longs;ibile &longs;it <lb/>augentur; quod inde oritur, quia angulus F C H, maior e&longs;t angulo F B K, <lb/>per 16. primi Elem. |
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| <s>& ideò crura F C, H C, magis dilatata &longs;unt quàm cru­<lb/>ra F B, K B, & ideò &longs;i producantur, multò magis cre&longs;cit F H, dum euadit <lb/>M N, quàm F K, dum euadit M O. eodem modo magis cre&longs;cit K E, dum fit <lb/>O L, quàm H E, dum fit K L. quare ex &longs;ecunda notatione earum periphe­<lb/>riæ ad vnius orbis figuram tandem concurrere videbuntur. </s> | & ideò crura F C, H C, magis dilatata &longs;unt quàm cru­<lb/>ra F B, K B, & ideò &longs;i producantur, multò magis cre&longs;cit F H, dum euadit <lb/>M N, quàm F K, dum euadit M O. eodem modo magis cre&longs;cit K E, dum fit <lb/>O L, quàm H E, dum fit K L. quare ex &longs;ecunda notatione earum periphe­<lb/>riæ ad vnius orbis figuram tandem concurrere videbuntur. </s> |
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| <s>multò autem <lb/>euidentius ad rotunditatem accederent, &longs;i tertia illuminatio per tertium <lb/>aliquod punctum tran&longs;iens, &longs;ic perueniret; & quo plures, eò etiam perfectius, <lb/>omnes enim rotundæ e&longs;&longs;ent, & ex radiorum proce&longs;&longs;u augerentur, atque ad <lb/>vnius orbis formam &longs;e &longs;e reciperent.</s></p><p type="main"> | <s>multò autem <lb/>euidentius ad rotunditatem accederent, &longs;i tertia illuminatio per tertium <lb/>aliquod punctum tran&longs;iens, &longs;ic perueniret; & quo plures, eò etiam perfectius, <lb/>omnes enim rotundæ e&longs;&longs;ent, & ex radiorum proce&longs;&longs;u augerentur, atque ad <lb/>vnius orbis formam &longs;e &longs;e reciperent.</s></p><p type="main"> |
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| <s>quapropter optimè intelliges textum hunc, &longs;i huiu&longs;modi <lb/>volumen bis &longs;ecueris, primo quidem &longs;ectione ba&longs;i voluminis parallela; &longs;e­<lb/>cundo verò &longs;ectione tran&longs;uer&longs;ali, &longs;eu obliqua ad ba&longs;im: nam explicata pri­<lb/>ma &longs;ectione apparebit eam e&longs;&longs;e lineam rectam: euoluta verò <expan abbr="&longs;ecũda">&longs;ecunda</expan> &longs;ectio­<lb/>ne apparebit eam e&longs;&longs;e tortuo&longs;am, & flexuo&longs;am; Arift. </s> | <s>quapropter optimè intelliges textum hunc, &longs;i huiu&longs;modi <lb/>volumen bis &longs;ecueris, primo quidem &longs;ectione ba&longs;i voluminis parallela; &longs;e­<lb/>cundo verò &longs;ectione tran&longs;uer&longs;ali, &longs;eu obliqua ad ba&longs;im: nam explicata pri­<lb/>ma &longs;ectione apparebit eam e&longs;&longs;e lineam rectam: euoluta verò <expan abbr="&longs;ecũda">&longs;ecunda</expan> &longs;ectio­<lb/>ne apparebit eam e&longs;&longs;e tortuo&longs;am, & flexuo&longs;am; Arift. </s> |
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| <s>reddens rationem, <lb/>cur hæc &longs;it tortuo&longs;a, ait id e&longs;&longs;e, quia &longs;ectione obliqua exi&longs;tente, ide&longs;t ex vna <lb/>parte depre&longs;&longs;iori, & ex altera altiori, &longs;equitur, quod circuli, qui ex tali &longs;e­<lb/>ctione oriuntur non remanent in eodem plano, dum euoluuntur; quare <expan abbr="neq;">neque</expan> <lb/>linea, ex qua illi circuli con&longs;tant, poterit e&longs;&longs;e in eodem plano, & ideo <expan abbr="neq;">neque</expan> <lb/>recta e&longs;&longs;e poterit, quia fieri nequit, vt eiu&longs;dem lineæ pars &longs;it in plano vno, <lb/>pars verò in altero; quod o&longs;tenditur in prima 11. Elem. </s> | <s>reddens rationem, <lb/>cur hæc &longs;it tortuo&longs;a, ait id e&longs;&longs;e, quia &longs;ectione obliqua exi&longs;tente, ide&longs;t ex vna <lb/>parte depre&longs;&longs;iori, & ex altera altiori, &longs;equitur, quod circuli, qui ex tali &longs;e­<lb/>ctione oriuntur non remanent in eodem plano, dum euoluuntur; quare <expan abbr="neq;">neque</expan> <lb/>linea, ex qua illi circuli con&longs;tant, poterit e&longs;&longs;e in eodem plano, & ideo <expan abbr="neq;">neque</expan> <lb/>recta e&longs;&longs;e poterit, quia fieri nequit, vt eiu&longs;dem lineæ pars &longs;it in plano vno, <lb/>pars verò in altero; quod o&longs;tenditur in prima 11. Elem. |
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| <s>quæ e&longs;t hæc; rectæ <lb/>lineæ pars quædam non e&longs;t in &longs;ubiecto plano, pars verò in &longs;ublimi.</s></p><p type="main"> | quæ e&longs;t hæc; rectæ <lb/>lineæ pars quædam non e&longs;t in &longs;ubiecto plano, pars verò in &longs;ublimi.</s></p><p type="main"> |
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| <s><arrow.to.target n="marg349"></arrow.to.target></s></p><p type="margin"> | <s><arrow.to.target n="marg349"></arrow.to.target></s></p><p type="margin"> |
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| <s>In 2. problema. </s> | <s>In 2. problema. </s> |
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| <s>In verba illa <emph type="italics"/>(Sed quemadmodum linea bipedalis non<emph.end type="italics"/><lb/><figure id="fig143"></figure><lb/><emph type="italics"/>duplum, &longs;ed quadruplum quoddam de&longs;cribit, <lb/>&longs;ic, &c.)<emph.end type="italics"/> ide&longs;t, quemadmodum linea bi­<lb/>pedalis, quæ quamuis &longs;it dupla lineæ pedalis non <lb/>tamen de&longs;cribit quadratum duplum quadrati il­<lb/>lius, &longs;ed quadruplum: vt probatur in &longs;cholio 4. <lb/>2. Elem. </s> | <s>In verba illa <emph type="italics"/>(Sed quemadmodum linea bipedalis non<emph.end type="italics"/><lb/><figure id="fig143"></figure><lb/><emph type="italics"/>duplum, &longs;ed quadruplum quoddam de&longs;cribit, <lb/>&longs;ic, &c.)<emph.end type="italics"/> ide&longs;t, quemadmodum linea bi­<lb/>pedalis, quæ quamuis &longs;it dupla lineæ pedalis non <lb/>tamen de&longs;cribit quadratum duplum quadrati il­<lb/>lius, &longs;ed quadruplum: vt probatur in &longs;cholio 4. <lb/>2. Elem. |
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| <s>& videre e&longs;t in hac figura, vbi linea A B, <lb/>e&longs;t dupla lineæ A C. quadratum verò lineæ A B, <lb/>&longs;cilicet quadratum A B D E, e&longs;t <expan abbr="quadruplũ">quadruplum</expan> qua­<lb/>drati lineæ A C, quadrati nimirum A C F G. re­<lb/>liqua huius textus manife&longs;ta &longs;unt</s></p><p type="main"> | & videre e&longs;t in hac figura, vbi linea A B, <lb/>e&longs;t dupla lineæ A C. quadratum verò lineæ A B, <lb/>&longs;cilicet quadratum A B D E, e&longs;t <expan abbr="quadruplũ">quadruplum</expan> qua­<lb/>drati lineæ A C, quadrati nimirum A C F G. re­<lb/>liqua huius textus manife&longs;ta &longs;unt</s></p><p type="main"> |
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| <s>Scias Lector, me nullum, horum de Mu&longs;ica Problematum (quemadmo­<lb/>dum & in pluribus alijs mathematicis locis accidit) vidi&longs;&longs;e expo&longs;itorem, <lb/>præter vnum Petrum Aponentem, quem tamen tanquam omninò his rebus <lb/>elucidandis ineptum, reieci.</s></p><pb pagenum="247"/><p type="main"> | <s>Scias Lector, me nullum, horum de Mu&longs;ica Problematum (quemadmo­<lb/>dum & in pluribus alijs mathematicis locis accidit) vidi&longs;&longs;e expo&longs;itorem, <lb/>præter vnum Petrum Aponentem, quem tamen tanquam omninò his rebus <lb/>elucidandis ineptum, reieci.</s></p><pb pagenum="247"/><p type="main"> |
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| <s>quàm admira­<lb/>bilis e&longs;t 47. primi, pro cuius inuentione Pythagoras Mu&longs;is Hecatombas <lb/>immolauit? </s> | <s>quàm admira­<lb/>bilis e&longs;t 47. primi, pro cuius inuentione Pythagoras Mu&longs;is Hecatombas <lb/>immolauit? </s> |
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| <s>In &longs;ecundò deinde Elem. </s> | <s>In &longs;ecundò deinde Elem. |
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| <s>lib. </s> | lib. </s> |
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| <s>quàm &longs;ubtilis e&longs;t 14. quæ rectili­<lb/>neo cuiuis quadratum exhibet æquale. </s> | <s>quàm &longs;ubtilis e&longs;t 14. quæ rectili­<lb/>neo cuiuis quadratum exhibet æquale. </s> |
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| <s>omne triangulum habere tres, &c. </s> | <s>omne triangulum habere tres, &c. </s> |
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| <s>quæ e&longs;t 32. primi Elem. </s> | <s>quæ e&longs;t 32. primi Elem. |
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| <s>primi <lb/>Pythag. </s> | primi <lb/>Pythag. </s> |
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| <s>demon&longs;trarunt. </s> | <s>demon&longs;trarunt. </s> |
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| <s>47. primi Elem. </s> | <s>47. primi Elem. |
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| <s>reperit, pro qua Mu&longs;is Hecatombas <lb/>immolauit. </s> | reperit, pro qua Mu&longs;is Hecatombas <lb/>immolauit. </s> |
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| <s>primus Mathematicæ ludum aperuit. </s> | <s>primus Mathematicæ ludum aperuit. </s> |
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| <s>Diog, <gap/> aert.</s></p><p type="main"> | <s>Diog, <gap/> aert.</s></p><p type="main"> |
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| <s>OENIPEDES CHIVS Democriti &longs;yncronus, inuenit 12. & 23. <lb/>primi Elem. </s> | <s>OENIPEDES CHIVS Democriti &longs;yncronus, inuenit 12. & 23. <lb/>primi Elem. |
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| <s>huius di&longs;cipulus fuit Zenodorus.</s></p><p type="main"> | huius di&longs;cipulus fuit Zenodorus.</s></p><p type="main"> |
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| <s>ZENODORVS auctor tractatus de figuris I&longs;operimetris, qui e&longs;t <lb/>apud Clauium in &longs;phæra, & in Geometric. </s> | <s>ZENODORVS auctor tractatus de figuris I&longs;operimetris, qui e&longs;t <lb/>apud Clauium in &longs;phæra, & in Geometric. </s> |
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| <s>Vitr.</s></p><p type="main"> | <s>Vitr.</s></p><p type="main"> |
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| <s>ISIDORVS Philo&longs;ophus, Hyp&longs;iclis Alexandrini præceptor; nam <lb/>Hyp&longs;icles in 15. Elem. </s> | <s>ISIDORVS Philo&longs;ophus, Hyp&longs;iclis Alexandrini præceptor; nam <lb/>Hyp&longs;icles in 15. Elem. |
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| <s>vbi ponit inclinationes <expan abbr="quinq;">quinque</expan> <expan abbr="corporũ">corporum</expan> regularium, <lb/>ait &longs;e eas ab I&longs;idoro Magno præceptore &longs;uo accepi&longs;&longs;e. </s> | vbi ponit inclinationes <expan abbr="quinq;">quinque</expan> <expan abbr="corporũ">corporum</expan> regularium, <lb/>ait &longs;e eas ab I&longs;idoro Magno præceptore &longs;uo accepi&longs;&longs;e. </s> |
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| <s>Plinius eum citat in <lb/>Geographicis. </s> | <s>Plinius eum citat in <lb/>Geographicis. </s> |
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