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Colored diff for /texts/archimedes/xml/fabri_tract_026_la_1646.xml between version 1.23 and 1.24

version 1.23, 2007/02/14 17:57:29

version 1.24, 2007/02/15 19:10:49

Line 18049

<s id="N284AF">Quintò, velocitates extremitatum, po&longs;ita diuer&longs;a longitudine manu<lb/>brij, &longs;unt vt ip&longs;æ longitudines permutando: </s>

<s id="N284B5">probatur, quia cùm &longs;it mo<lb/>tus acceleratus, &longs;patia &longs;unt vt quadrata temporum; &longs;ed velocitates <lb/>&longs;unt vt tempora, & tempora &longs;unt in ratione &longs;ubduplicata &longs;patiorum. </s>

<s id="N284BD"><lb/>id e&longs;t, vt diametri quadratorum, id e&longs;t, vt longitudines, &longs;it enim lon<lb/>gitudo AB, quæ dato tempore H decurrat &longs;patium ABF, potentia <pb pagenum="399" xlink:href="026/01/433.jpg"/>&longs;cilicet toto ni&longs;u applicata, &longs;it etiam longitudo AC dupla AB: </s>

<s id="N284C9">dico <lb/>quod eodum tempore H acquiret æquale &longs;patium &longs;cilicet CAD; </s>

<s id="N284C9">dico <lb/>quod eodem tempore H acquiret æquale &longs;patium &longs;cilicet CAD; </s>

<s id="N284CF">igitur <lb/>CAD e&longs;t 1/4 CAG, quia e&longs;t æquale BAF; </s>

<s id="N284D5">igitur CD e&longs;t 1/4 CG, &longs;ed <lb/>CG e&longs;t duplus BF; </s>

<s id="N284DB">igitur CD e&longs;t &longs;ubduplus BF; </s>

Line 18067

<s id="N28520">igitur ictus &longs;unt in ratione &longs;ubdupli<lb/>cata CAG, BAF; igitur vt ACAB. </s>

<s id="N28528">Septimò, diceret aliquis velocitatem C decur&longs;o CD, e&longs;&longs;e &longs;ubduplam <lb/>vclocitatis B decur&longs;o BF; </s>

<s id="N28528">Septimò, diceret aliquis velocitatem C decur&longs;o CD, e&longs;&longs;e &longs;ubduplam <lb/>velocitatis B decur&longs;o BF; </s>

<s id="N2852E">&longs;ed velocitas C, decur&longs;o CG, e&longs;t dupla velo<lb/>citatis eiu&longs;dem C decur&longs;o CD; </s>

<s id="N28534">igitur velocitas C, decur&longs;o CG, e&longs;t <lb/>æqualis velocitati B, decur&longs;o BF; igitur æqualis ictus. </s>

<s id="N2853A">Re&longs;p. conce&longs;&longs;a <lb/>primâ con&longs;equentiâ, vltimâ verò negatâ; </s>

<s id="N28540">quia non tantùm impetus <lb/>puncti C incutit ictum &longs;ed totius CA, qui cen&longs;etur e&longs;&longs;e collectus in <lb/>malleo in quo e&longs;t qua&longs;i centrum huius impetus, vt iam explicuimus <lb/>aliàs; &longs;ed velocitas totius CA confecto CAD e&longs;t æqualis velocitati <lb/>totius BA confecto BAF, cuius velocitas CA confecto CAG e&longs;t dupla, <lb/>vt iam probatum e&longs;t. </s>

<s id="N28550">Octanò, hinc ictus CA confecto CAD e&longs;t æqualis ictui AB con<lb/>fecto BAF, & ictus CA confecto CI duplo CD e&longs;t ad ictum CA con<lb/>fecto CD, vt radix CA ad radicem CI: </s>

<s id="N28550">Octauò, hinc ictus CA confecto CAD e&longs;t æqualis ictui AB con<lb/>fecto BAF, & ictus CA confecto CI duplo CD e&longs;t ad ictum CA con<lb/>fecto CD, vt radix CA ad radicem CI: </s>

<s id="N28558">hinc vides hunc motum con<lb/>uenire in eo cum recto, quòd &longs;cilicet ictus inflictus motu recto à mi<lb/>nori mole, &longs;it ad ictum maioris, &longs;uppo&longs;ita linea motus æquali in ratio<lb/>ne &longs;ubduplicata ponderum; quòd dicitur etiam de motu circulari duo<lb/>rum fu&longs;tium inæqualium, quorum ictus &longs;unt in ratione &longs;ubduplicata <lb/>longitudinum, a&longs;&longs;umptis duntaxat arcubus æqualibus ab extremitate <lb/>vtriu&longs;que decur&longs;is. </s>

<s id="N2856A">Nonò, cum mallei &longs;unt diucr&longs;i ponderis, & longitudinis, facilè co<lb/>gno&longs;ci poterit proportio ictuum; </s>

<s id="N2856A">Nonò, cum mallei &longs;unt diuer&longs;i ponderis, & longitudinis, facilè co<lb/>gno&longs;ci poterit proportio ictuum; </s>

<s id="N28570">e&longs;t enim compo&longs;ita ex ratione lon<lb/>gitudinum & &longs;ubduplicata ponderum v.g. &longs;it malleus A, cuius longitu<lb/>do &longs;it 2. pondus 4. &longs;it malleus B cuius longitudo &longs;it pondus; </s>

<s id="N2857A">rectè ra<lb/>tio longitudinum e&longs;t 2/3, & &longs;ubduplicata ponderum e&longs;t 2/3; </s>

<s id="N28580">ducatur vna <lb/>in aliam, vt cuadat compo&longs;ita &longs;cilicet 4/1 vel longitudo A &longs;it I, & B 2; </s>

<s id="N28580">ducatur vna <lb/>in aliam, vt euadat compo&longs;ita &longs;cilicet 4/1 vel longitudo A &longs;it I, & B 2; </s>

<s id="N28586"><lb/>habebitur ratio &longs;ubduplicata ponderum 2/<gap/>, & ratio longitudinum 2/1; </s>

<s id="N28586"><lb/>habebitur ratio &longs;ubduplicata ponderum 2/1, & ratio longitudinum 3/2; </s>

<s id="N2858D"><lb/>ducatur v<gap/>a in aliam, habebitur ratio compo&longs;ita 2/2; </s>

<s id="N2858D"><lb/>ducatur vna in aliam, habebitur ratio compo&longs;ita 2/2; </s>

<s id="N28594">igitur &longs;unt æqua-<pb pagenum="400" xlink:href="026/01/434.jpg"/>les, quæ omnia facilè intelliguntur ex dictis; </s>

<s id="N2859D">itaque habes 4. combina<lb/>tiones duorum malleorum; </s>

<s id="N285A3">vel enim e&longs;t idem pondus vtrique, & ea<lb/>dem longitudo, vel idem pondus, &longs;ed diuer&longs;a longitudo, vel eadem lon<lb/>gitudo & diuer&longs;um pondus, vel diuer&longs;um pondus & diuer&longs;a longitudo; <lb/>&longs;i verò e&longs;t diuer&longs;a longitudo &longs;imal, & diuer&longs;um pondus, vel eidem ine&longs;t <lb/>maius pondus, & maior longitudo, vel maior longitudo, & minus pou<lb/>dus, & contrà alteri minor lougitudo, & minus pondus, vel maius pon<lb/>dus, & minor longitudo, quorum omnium proportiones &longs;unt determi<lb/>natæ. </s>

<s id="N285A3">vel enim e&longs;t idem pondus vtrique, & ea<lb/>dem longitudo, vel idem pondus, &longs;ed diuer&longs;a longitudo, vel eadem lon<lb/>gitudo & diuer&longs;um pondus, vel diuer&longs;um pondus & diuer&longs;a longitudo; <lb/>&longs;i verò e&longs;t diuer&longs;a longitudo &longs;imul, & diuer&longs;um pondus, vel eidem ine&longs;t <lb/>maius pondus, & maior longitudo, vel maior longitudo, & minus pon<lb/>dus, & contrà alteri minor longitudo, & minus pondus, vel maius pon<lb/>dus, & minor longitudo, quorum omnium proportiones &longs;unt determi<lb/>natæ. </s>

<s id="N285B7">Decimò, quod &longs;pectat ad cra&longs;&longs;itudinem manubrij, illa haud dubiè <lb/>auget aliquando vim ictus, aliquando imminuit; </s>

Line 18104

<s id="N28605">&longs;i verò &longs;unt eiu&longs;dem ponderis, & cadunt <lb/>ex diuer&longs;a altitudine impetus acqui&longs;iti motu naturali, &longs;unt in ratione <lb/>&longs;ubduplicata altitudinum; </s>

<s id="N2860D">&longs;i verò &longs;unt diuer&longs;i ponderis, & cadunt ex <lb/>diuer&longs;a altitudine, &longs;unt in ratione compo&longs;ita aliquomodo ex vtraque; </s>

<s id="N28613"><lb/>dico aliquo modo, quia non e&longs;t omninò propria compo&longs;itio rationum; </s>

<s id="N28618"><lb/>pote&longs;t tamen facilè proportio ictuum inueniri, v. g. &longs;it malleus A, & <lb/>malleus B, ictns A ratione impetus impre&longs;&longs;i extrin&longs;ecxs &longs;it vt 8, ratione <lb/>ca&longs;us &longs;it vt 2; </s>

<s id="N28618"><lb/>pote&longs;t tamen facilè proportio ictuum inueniri, v. g. &longs;it malleus A, & <lb/>malleus B, ictus A ratione impetus impre&longs;&longs;i extrin&longs;ecus &longs;it vt 8, ratione <lb/>ca&longs;us &longs;it vt 2; </s>

<s id="N28625">at verò ictus B ratione impetus impre&longs;&longs;i &longs;it vt 6, ratione <lb/>ca&longs;us vt 3: </s>

<s id="N2862B">addantur 8, & 2 erunt 10; </s>

<s id="N2862F">adduntur 6, & 3 erunt 9; </s>

<s id="N28633">igitur <lb/>ictus &longs;unt in ratione (10/9), vt con&longs;tat: </s>

<s id="N28639">porrò quemadmodum nouus im<lb/>petus accedit ratione motus naturalis, cum malleus impingitur dcos<lb/>&longs;um, ita aliquid impetus de&longs;truitur cum malleus impingitur &longs;ur&longs;um, vt <lb/>patet; </s>

<s id="N28639">porrò quemadmodum nouus im<lb/>petus accedit ratione motus naturalis, cum malleus impingitur deor<lb/>&longs;um, ita aliquid impetus de&longs;truitur cum malleus impingitur &longs;ur&longs;um, vt <lb/>patet; </s>

<s id="N28643">denique, quia &longs;unt 5 termini, quos re&longs;picit ictus, &longs;cilicet pondus <lb/>mallei, longitudo manubrij, cra&longs;&longs;itudo arcus extremitatis, & linea &longs;ur<lb/>fum vel deor&longs;um, ita &longs;unt 25. combinationes ictuam; &longs;ed hoc fa<lb/>cile e&longs;t. </s>

<s id="N28643">denique, quia &longs;unt 5 termini, quos re&longs;picit ictus, &longs;cilicet pondus <lb/>mallei, longitudo manubrij, cra&longs;&longs;itudo arcus extremitatis, & linea &longs;ur<lb/>&longs;um vel deor&longs;um, ita &longs;unt 25. combinationes ictuum; &longs;ed hoc fa<lb/>cile e&longs;t. </s>

Line 18125

<s id="N28681">&longs;it enim centrum grauitatis clauæ I, in quo &longs;i <lb/>&longs;u&longs;tineatur, &longs;tabit in æquilibrio; </s>

<s id="N28687">ducatur FIE, maiorem ictum <lb/>infliget ex puncto E, quia <expan abbr="tant&utilde;dem">tantundem</expan> e&longs;t impetus in &longs;egmento <lb/>FEK quantum in &longs;egmento FEA; </s>

<s id="N28693">igitur totus impeditur impe<lb/>tus; igitur maximus erit ictus &longs;i infligat ictum motu circulari circa <lb/>aliud e&longs;t centrum percu&longs;&longs;ionis, de quo infrà. </s>

<s id="N2869B">Tertiò, hoe percu&longs;&longs;io<lb/>nis organum validum ictum infligit propter illam extremam cra&longs;<lb/>&longs;itudinem, e&longs;t enim quoddam mallei genus, & valdè periculo&longs;um; </s>

<s id="N2869B">Tertiò, hoc percu&longs;&longs;io<lb/>nis organum validum ictum infligit propter illam extremam cra&longs;<lb/>&longs;itudinem, e&longs;t enim quoddam mallei genus, & valdè periculo&longs;um; </s>

<s id="N286A3"><lb/>præ&longs;ertim &longs;i ferreis clauis armetur; </s>

<s id="N286A8">hinc vulgò tribuitur Herculi tan<lb/>quam in&longs;igne fortitudinis &longs;ymbolum; porrò tàm altè clauum infigit <lb/>&longs;ibi coniunctum, quam infigeret, &longs;i claua ip&longs;a erectum, & qua&longs;i expe<lb/>ctantem ictum feriret. </s>

<s id="N286B4">Decimoquartò, Tudicula maior reuocatur ad malleum. </s>

<s id="N286B7">Primò <lb/>faciunt ad ictum longitudo manubrij, flexibilitas, inæqualitas, mal<lb/>lei pondus, durities materiæ, arcus motus, vegetæ potentiæ vires; </s>

<s id="N286BF"><lb/>omitto ea, quæ cum malleo habet communia, quorum ratio ex <lb/>dictis conftare pote&longs;t; igitur non videntur e&longs;&longs;e repetenda. </s>

<s id="N286BF"><lb/>omitto ea, quæ cum malleo habet communia, quorum ratio ex <lb/>dictis con&longs;tare pote&longs;t; igitur non videntur e&longs;&longs;e repetenda. </s>

<s id="N286C6">Secundò, <lb/>flexibilitas manubrij auget vim ictus, tùm quia potentia diutiùs <lb/>manet applicata, cùm aliquo tempore in ip&longs;a vibratione malleus à <lb/>tergo relinquatur, tùm quia potentia illa media, de qua &longs;upra, &longs;uum <lb/>impetum, impetui alterius adiungit. </s>

<s id="N286D1">Tertiò, ita manubrium fa<lb/>bricatur, vt continua imminutione ver&longs;us malleum decre&longs;cat, quod <lb/>multum facit ad ictum, quia hæc inæqualitas inflexioni re&longs;i&longs;tit ver<lb/>&longs;us caput manubrij; </s>

<s id="N286DB">igitur initio inflectitur manubrium, non pro<lb/>cul à mallco, tùm deinde aucto impetu in partibus remotioribus, <lb/>quæ difficiliùs inflectuntur; </s>

<s id="N286DB">igitur initio inflectitur manubrium, non pro<lb/>cul à malleo, tùm deinde aucto impetu in partibus remotioribus, <lb/>quæ difficiliùs inflectuntur; </s>

<s id="N286E3">igitur inæqualiter partes illæ redeunt, <lb/>atque &longs;e&longs;e pri&longs;tino &longs;tatui re&longs;tituunt; </s>

<s id="N286E9">atqui ex illa inæqualitate diu<lb/>tiùs durat motus; </s>

<s id="N286EF">igitur inde maior cuadit: </s>

<s id="N286EF">igitur inde maior euadit: </s>

<s id="N286F3">&longs;imile quid videmus in <lb/>arcu, cuius medium cra&longs;&longs;ius e&longs;t: adde quod &longs;i æqualis &longs;it cra&longs;&longs;itudo, <lb/>incipit inflexio ver&longs;us illam extremitatem, quæ propiùs accedit ad <lb/>manum, longiùs recedit à malleo, vt patet experientiâ, in fune, <lb/>virgâ &c. </s>

<s id="N286FF">&longs;ed de arcu, ten&longs;ione, compre&longs;&longs;ione fusè agemus <lb/>tractatu &longs;ingulari: </s>

<s id="N28705">hæc tantum obiter indica&longs;&longs;e &longs;ufficiat. <pb pagenum="402" xlink:href="026/01/436.jpg"/>Quartò, maximus e&longs;t ictus, cum malleus eo in&longs;tanti attingic pilam, quo <lb/>manubrium e&longs;t rectum; </s>

<s id="N28710">tunc enim e&longs;t modum vibrationis &longs;eu reditus; <lb/>igitur maximus impetus. </s>

<s id="N28716">Quintò, &longs;i altera extremitas mallei, quæ glo<lb/>bum attingit, &longs;it obliqua, globum ip&longs;um attollit propter punctum con<lb/>tactus; quod certè clarum e&longs;t. </s>

<s id="N2871E">Sextò, durities mallei multùm facit ad <lb/>ictum; </s>

<s id="N28724">&longs;i enim cedat lignum, imminuitur impetus, vt patet; hinc ar<lb/>millâ, vel annulo ferreo armatur vtraque ba&longs;is mallei, vt finnior eua<lb/>dat. </s>

<s id="N28724">&longs;i enim cedat lignum, imminuitur impetus, vt patet; hinc ar<lb/>millâ, vel annulo ferreo armatur vtraque ba&longs;is mallei, vt firmior eua<lb/>dat. </s>

<s id="N2872C">Septimò, globi ratio multa habenda e&longs;t, cui infligitur ictus; </s>

<s id="N28730">quippe <lb/>&longs;i leuior e&longs;t ab aëre ambiente impeditur, & retinetur; </s>

<s id="N28736">&longs;i verò mollior <lb/>minor ictus infligitur, quia cedit materies; </s>

Line 18162

<s id="N28778"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 14.<emph.end type="center"/></s>

<s id="N28786"><emph type="italics"/>Ex dictis explicaris po&longs;&longs;um omnia ph&oelig;nomena, quæ ob&longs;erumtur in lu<gap/><lb/>rudis gladiatoriæ<emph.end type="italics"/>; </s>

<s id="N28786"><emph type="italics"/>Ex dictis explicaris po&longs;&longs;um omnia ph&oelig;nomena, quæ ob&longs;eruantur in ludo <lb/>rudis gladiatoriæ<emph.end type="italics"/>; </s>

<s id="N28792">Primo, tria &longs;unt in hac arte, ad quæ reliqua facilè re<lb/>ducuntur; </s>

<s id="N28798">primum e&longs;t declinatio; &longs;ecundum petitio; tertium confla<lb/>tum ex vtroque. </s>

<s id="N2879E">Secundò, pote&longs;t declinati, vel auerti ictus, &longs;eu petitio <lb/>duobus modis. </s>

Line 18171

<s id="N287A5">Primò, &longs;i declinatio cum aliqua impactione coniungatur. </s>

<s id="N287AA">Secundò, &longs;i tantùm cum inera re&longs;i&longs;tentia, vel &longs;implici impul<lb/>&longs;ione. <lb/><arrow.to.target n="note4"/></s>

<s id="N287AA">Secundò, &longs;i tantùm cum mera re&longs;i&longs;tentia, vel &longs;implici impul<lb/>&longs;ione. <lb/><arrow.to.target n="note4"/></s>

Line 18181

<s id="N287CE">&longs;iue enim duo gladij <lb/>AC, DF, capulares pilæ AD; </s>

<s id="N287D4">&longs;it autem gladius AC declinans petitio<lb/>nem alterius DF; id certè quatuor modis præ&longs;tare pote&longs;t. </s>

<s id="N287DA">Primò, &longs;i <lb/>punctum contactus ad mucronem vtriu&longs;que propiùs accedat. </s>

<s id="N287DF">v.g. &longs;i <lb/>vterque &longs;it in &longs;itu ACDF. Secundò, &longs;i propiùs accedat ad capulum <lb/>vtriu&longs;que, talis e&longs;t &longs;itus DFCH. Tertiò, &longs;i accedat propiùs ad mu<lb/>cronem gladij petentis DF, & propiùs ad capulum declinanti<gap/>. <pb pagenum="403" xlink:href="026/01/437.jpg"/>Quartò, è contrario &longs;i accedat propiùs ad capulum petentis DF, & pro<lb/>piùs ad mucronem declinantis; addi pote&longs;t quinta combinatio, cum <lb/>&longs;cilicet contactus e&longs;t in medio vtriu&longs;que. </s>

<s id="N287DF">v.g. &longs;i <lb/>vterque &longs;it in &longs;itu ACDF. Secundò, &longs;i propiùs accedat ad capulum <lb/>vtriu&longs;que, talis e&longs;t &longs;itus DFCH. Tertiò, &longs;i accedat propiùs ad mu<lb/>cronem gladij petentis DF, & propiùs ad capulum declinantis. <pb pagenum="403" xlink:href="026/01/437.jpg"/>Quartò, è contrario &longs;i accedat propiùs ad capulum petentis DF, & pro<lb/>piùs ad mucronem declinantis; addi pote&longs;t quinta combinatio, cum <lb/>&longs;cilicet contactus e&longs;t in medio vtriu&longs;que. </s>

<s id="N287F8">Quartò, &longs;i &longs;it mera impul&longs;io &longs;ine percu&longs;&longs;ione, vel impactione, maxi<lb/>ma vis e&longs;t declinationis, cum punctum contactus accedit propiùs ad ca<lb/>pulum declinantis, & ad mucronem petentis iuxta tertiam combinatio<lb/>nem, & &longs;itum DFPE, & punctum contactus in B; </s>

Line 18215

<s id="N28888">Nonò, &longs;i primo modo declinatur ictus repul&longs;o petentis gladio maxi<lb/>ma vis erit; &longs;i punctum contactus fiat circa 2/3 de quo infrà, quod verò <lb/>&longs;pectat ad gladium, qui repellitur, eò faciliùs repellitur, quò punctum <lb/>contactus propiùs ad eius mucronem accedet. </s>

<s id="N28892">Si tertio modo, & gla<lb/>dius &longs;olus ita libraretur maxima vis e&longs;&longs;et circa centrum eius grauitatis; </s>

<s id="N28898"><lb/>in hoc enim puncto maximum ictum infligunt, quæ motu recto mo<lb/>uentur; quia verò totum &longs;egmentum brachij, qnod inter manum, & <pb pagenum="404" xlink:href="026/01/438.jpg"/>caput cubiti intercipitur, mouetur &longs;imul cum gladio motu recto, circa <lb/>capulum erit maxima vis, cùm propiùs accedat ad centrum grauitatis <lb/>totius conflati ex illo &longs;egmento brachij, & gladio. </s>

<s id="N28898"><lb/>in hoc enim puncto maximum ictum infligunt, quæ motu recto mo<lb/>uentur; quia verò totum &longs;egmentum brachij, quod inter manum, & <pb pagenum="404" xlink:href="026/01/438.jpg"/>caput cubiti intercipitur, mouetur &longs;imul cum gladio motu recto, circa <lb/>capulum erit maxima vis, cùm propiùs accedat ad centrum grauitatis <lb/>totius conflati ex illo &longs;egmento brachij, & gladio. </s>

<s id="N288AA">Decimò, denique &longs;i &longs;ecundo modo declinetur ictus, idem dicendum <lb/>e&longs;t quod de motu circulari dictum, mutato dumtaxat centro, v.g. &longs;it gla<lb/>dius declinantis RG, &longs;itque IG 1/4 totius RG circa I &longs;it motus circula<lb/>tis, centrum percu&longs;&longs;ionis erit circa 2/3 IG, vel IR. </s>

<s id="N288B7">Vndecimò, vix tamen ita acuratè hoc &longs;ecundo modo declinatur ictus, <lb/>quin tertius etiam cum &longs;ecundo coniunctus &longs;it, vt patet experientiâ; </s>

<s id="N288BD"><lb/>rotatur autem manus delcinantis vt illo qua&longs;i gyro maiorem impetum <lb/>acquirat, de quo iam &longs;uprà: immò ni&longs;i tertius modus cum &longs;ecundo e&longs;&longs;et <lb/>coniunctus, non po&longs;&longs;et delinari ictus, &longs;i contactus gladiorum fieret in <lb/>centro illius motus, vt patet. </s>

<s id="N288BD"><lb/>rotatur autem manus declinantis vt illo qua&longs;i gyro maiorem impetum <lb/>acquirat, de quo iam &longs;uprà: immò ni&longs;i tertius modus cum &longs;ecundo e&longs;&longs;et <lb/>coniunctus, non po&longs;&longs;et delinari ictus, &longs;i contactus gladiorum fieret in <lb/>centro illius motus, vt patet. </s>

<s id="N288CA">Duodecimò, quò longior e&longs;t gladius declinantis, cum iuxta mucro<lb/>nem fit contactus &longs;ine impactione e&longs;t vis debilior, quàm e&longs;&longs;et in breuio<lb/>re, patet ex vecte; &longs;i verò &longs;it impactio iuxta &longs;ecundum. </s>

<s id="N288D2">n.10. vis maior <lb/>e&longs;t cum gladius longior e&longs;t; </s>

<s id="N288D8">e&longs;t enim maior motus; </s>

<s id="N288DC">igitur maior ictus li<lb/>cèt tardior; </s>

<s id="N288E2">hinc longiore gladiore equidem fortiùs auertitur ictus quàm <lb/>breuiore, &longs;ed tardiùs; breuiore verò citiùs quàm longiore, &longs;ed debiliùs, <lb/>vt patet ex dictis. </s>

<s id="N288E2">hinc longiore gladio equidem fortiùs auertitur ictus quàm <lb/>breuiore, &longs;ed tardiùs; breuiore verò citiùs quàm longiore, &longs;ed debiliùs, <lb/>vt patet ex dictis. </s>

<s id="N288EC">Decimotertiò, longior gladius &longs;u&longs;tinetur facilè opera capularis pilæ, <lb/>quæ momentum longitudinis gladij &longs;upplet, vt con&longs;tat ex &longs;tatera, cuius <lb/>proportiones videbimus lib.&longs;eq. </s>

Line 18247

<s id="N28924">verùm etiam corpus detegitur: <lb/>adde quod ictus validior e&longs;t per lineam perpendicularem. </s>

<s id="N2892C">Decimo&longs;extò, hinc ita debet extremitas manus per hor&longs;zontalem <lb/>porrigi & brachium contractum explicari, vt maiorem lineam rectam <lb/>de&longs;cribat; </s>

<s id="N2892C">Decimo&longs;extò, hinc ita debet extremitas manus per horizontalem <lb/>porrigi & brachium contractum explicari, vt maiorem lineam rectam <lb/>de&longs;cribat; </s>

<s id="N28934">acquiritur enim maior impetus in maiori &longs;patio, quod per<lb/>curritur motu accelerato, vt con&longs;tat ex dictis, &longs;ed quò brachium con<lb/>tractius e&longs;t, cò maiorem lineam eius extremitas motu recto decurrit: <lb/>adde quod impre&longs;&longs;io totius corporis, quod in <expan abbr="eãdem">eandem</expan> partem agitur, <lb/>multùm auget vim brachij mucronem in aduer&longs;um pectus inten<lb/>tantis. </s>

<s id="N28948">Decimo&longs;eptimò, &longs;i longior e&longs;t gladius impetus, hæc videntur e&longs;&longs;e <lb/>commoda. </s>

<s id="N2894D">Primò, eius mucro longiùs producitur, & procul attingit. <pb pagenum="405" xlink:href="026/01/439.jpg"/>Secundò maiorem ictum infligit, vt iam &longs;upra dictum e&longs;t de &longs;ari&longs;&longs;a, mo<lb/>dò in eadem ratione aucta &longs;it cra&longs;&longs;itudo; non de&longs;unt tamen incommo<lb/>moda. </s>

<s id="N2895A">Primò ratione vectis maius e&longs;t illius pondus. </s>

<s id="N2895D">Secundò faciliùs de<lb/>clinatur ictus propter <expan abbr="eãdem">eandem</expan> ratiodem. </s>

<s id="N2895D">Secundò faciliùs de<lb/>clinatur ictus propter <expan abbr="eãdem">eandem</expan> rationem. </s>

<s id="N28966">Tertiò, &longs;i tantillùm deflecte<lb/>tur, corpus omninò detegit propter maiorem cum, &longs;unt enim arcus <lb/>vt radij, vel longitudines. </s>

<s id="N2896D">Quartò, hinc pugiles faciliùs decu&longs;&longs;atis gla<lb/>dijs &longs;e&longs;e mutuò præhendunt, & luctâ decernunt. </s>

Line 18282

<s id="N289D0"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 23.<emph.end type="center"/></s>

<s id="N289DE"><emph type="italics"/>Explieari po&longs;&longs;unt omnia ph&oelig;nomena percu&longs;&longs;ionis, quæ infligitur à <lb/>serpore graui deor&longs;um &longs;ua &longs;ponte cadente motu naturaliter accele<lb/>rato.<emph.end type="italics"/></s>

<s id="N289DE"><emph type="italics"/>Explicari po&longs;&longs;unt omnia ph&oelig;nomena percu&longs;&longs;ionis, quæ infligitur à <lb/>corpore graui deor&longs;um &longs;ua &longs;ponte cadente motu naturaliter accele<lb/>rato.<emph.end type="italics"/></s>

<s id="N289EB">Primò, corpus graue cadens ex maiore altitudine fortiùs ferit: ratio <lb/>e&longs;t; </s>

Line 18294

<s id="N28A03">Secundò, &longs;unt 4. combinationes grauium; </s>

<s id="N28A07">vel enim e&longs;t idem pondus <lb/>e&longs;t altitudo; </s>

<s id="N28A0D">vel idem pondus, diuer&longs;a altitudo; </s>

<s id="N28A11">vel eadem altitudo di<lb/>uer&longs;um poudus; </s>

<s id="N28A11">vel eadem altitudo di<lb/>uer&longs;um pondus; </s>

<s id="N28A17">vel diuer&longs;um pondus & diuer&longs;a altitudo; addi pote&longs;t <lb/>diuer&longs;us incidentiæ angulus, immò diuer&longs;a figura corporis cadentis, quæ <lb/>omnia infrà demon&longs;trabimus. </s>

<s id="N28A21">Tertiò, &longs;i &longs;it æquale pondus, & æqualis altitudo &longs;uppo&longs;ito ca&longs;u <pb pagenum="406" xlink:href="026/01/440.jpg"/>perpendiculari æquales &longs;unt ictus, patet; quia eadem cau&longs;a <expan abbr="e&utilde;dem">eundem</expan> h<gap/><lb/>bet eflectum. </s>

<s id="N28A33">Quartò, &longs;i &longs;it æquale pondus, & inæqualis altitudo, ictus &longs;unt in ra<lb/>tione &longs;ubduplicata altitudinum v.g. &longs;it altitudo 4. cubitorum, & altera <lb/>tautum cubitalis; </s>

<s id="N28A3D">certè cùm acquirantur æqualibus temporibus æqua<lb/>lia velocitatis momenta, velocitates acqui&longs;itæ &longs;unt vt tempora, impetus <lb/>vt velocitates, ictus vt impetus; </s>

<s id="N28A45">&longs;ed tempora &longs;unt in ratione &longs;ubdupli<lb/>cata &longs;patiorum vel altitudinem; </s>

<s id="N28A4B">igitur & ictus; igitur ictus inflictus à <lb/>corpore cadente ex altitudine 4. cubitorum e&longs;t duplus ictus eiu&longs;dem <lb/>corporis cadentis ex altitudine cubitali. </s>

Line 18312

<s id="N28A67">Sextò, &longs;i &longs;int infinita in&longs;tantia, e&longs;t infinita proportio inter actum in<lb/>flictum à corpore cadente, & vim grauitationis eiu&longs;dem; </s>

<s id="N28A6D">quia dato quo<lb/>cunque tempore po&longs;&longs;et dari minnus, & minus; igitur dato puocunque <lb/>ictu po&longs;&longs;et dari minor, & minor in infinitum, quod ex illa hypothe&longs;i <lb/>nece&longs;&longs;ariò con&longs;equitur. </s>

<s id="N28A6D">quia dato quo<lb/>cunque tempore po&longs;&longs;et dari minus, & minus; igitur dato quocunque <lb/>ictu po&longs;&longs;et dari minor, & minor in infinitum, quod ex illa hypothe&longs;i <lb/>nece&longs;&longs;ariò con&longs;equitur. </s>

<s id="N28A79">Septimò, immò &longs;i &longs;int infinita in&longs;tantia, &longs;ique infinita proportio in<lb/>ter ictum inflictum à corpore cadente, & vim grauitationis eiu&longs;dem, e&longs;t <lb/>etiam infinita proportio inter <expan abbr="eumd&etilde;ictum">eumdenictum</expan>, & vim grauitationis cuiu&longs;<lb/>libet alterius corporis quantumuis immen&longs;i, inter duas grauitationes <lb/>duorum corporum datur proportio, vt con&longs;tat; </s>

<s id="N28A79">Septimò, immò &longs;i &longs;int infinita in&longs;tantia, &longs;ique infinita proportio in<lb/>ter ictum inflictum à corpore cadente, & vim grauitationis eiu&longs;dem, e&longs;t <lb/>etiam infinita proportio inter <expan abbr="eumd&etilde;">eumdem</expan> ictum, & vim grauitationis cuiu&longs;<lb/>libet alterius corporis quantumuis immen&longs;i, inter duas grauitationes <lb/>duorum corporum datur proportio, vt con&longs;tat; </s>

<s id="N28A89">&longs;unt enim vt pondera; <lb/>igitur &longs;i nullam habet proportionem cum ictu corporis grauis cadentis, <lb/>nullam etiam habebit altera, vt patet ex elementis. </s>

Line 18326

<s id="N28AA9">Nonò, &longs;i altitudo &longs;it diuer&longs;a, & pondus diuer&longs;um, ictus &longs;unt in ratione <lb/>compo&longs;ita ex ratione ponderum, & &longs;ubduplicata altitudinum, patet ex <lb/>dictis. </s>

<s id="N28AB2">Decimò, &longs;i &longs;int infinita in&longs;tantia dato ictu cuiu&longs;libet corporis caden<lb/>tis ex quacunque altitudine, non pote&longs;t dari vlla corporis mo<gap/>s, qua <lb/>&longs;uo pondere id præ&longs;tat, quod illud præ&longs;titit &longs;uo ca&longs;u. </s>

<s id="N28AB2">Decimò, &longs;i &longs;int infinita in&longs;tantia dato ictu cuiu&longs;libet corporis caden<lb/>tis ex quacunque altitudine, non pote&longs;t dari vlla corporis moles, qua <lb/>&longs;uo pondere id præ&longs;tat, quod illud præ&longs;titit &longs;uo ca&longs;u. </s>

<s id="N28ABB">Probatur ex n. </s>

<s id="N28ABE">7. <lb/>hinc fru&longs;trà proponitur hæc quæ&longs;tio ab ijs, qui agno&longs;cunt infinitos tar<lb/>ditatis gradus, per quos propagatur motus; nam reuerâ ex hac hypothe&longs;i <lb/>e&longs;t infinita proportio inter ictum, & vim grauitationis. </s>

<s id="N28ACA">Vndecimò, &longs;i tamen ponantur finita in&longs;tantia; </s>

<s id="N28ACE">haud dubiè hæcpro<lb/>po&longs;itio non e&longs;t infinita; </s>

<s id="N28AD4">&longs;it enim quodlibet corpus cadens ex quacun<lb/>que data altitudine per 100. in&longs;tantia, &longs;cu partes temporis æquales pri<lb/>mo in&longs;tanti quo mouetur; </s>

<s id="N28AD4">&longs;it enim quodlibet corpus cadens ex quacun<lb/>que data altitudine per 100. in&longs;tantia, &longs;eu partes temporis æquales pri<lb/>mo in&longs;tanti quo mouetur; </s>

<s id="N28ADC">haud dubiè ictus ab eo inflictus cadendo e&longs;t <pb pagenum="407" xlink:href="026/01/441.jpg"/>ad vim grauitationis eiu&longs;dem vt 1001. ad 1. cùm enim &longs;ingulis in&longs;tan<lb/>tibus æqualibus acquirantur æqualia velocitatis momenta, &longs;eu æqualis <lb/>impetus; </s>

<s id="N28AE9">certè 1000. in&longs;tantibus, quibus mouetur acqui&longs;iuit 1000. gra<lb/>uus impetus æquales primo, quem habebat in ptima grauitatione; </s>

<s id="N28AE9">certè 1000. in&longs;tantibus, quibus mouetur acqui&longs;iuit 1000. gra<lb/>dus impetus æquales primo, quem habebat in prima grauitatione; </s>

<s id="N28AEF">& <lb/>qui fuit cau&longs;a motus primi in&longs;tantis; </s>

<s id="N28AF5">igitur &longs;i hic addatur 1000. erunt <lb/>1001. hinc &longs;i corpus moueatur tantùm vno in&longs;tanti, ictus erit duplus <lb/>tantùm grauitationis: &longs;uppono autem nullam e&longs;&longs;e medij re&longs;i&longs;tentiam, <lb/>ictumque infligi per lineam directam. </s>

Line 18347

<s id="N28B21">Decimotertiò, hinc non pote&longs;t determinari proportio corporis ca<lb/>dentis, & grauitantis, ni&longs;i ex hypothe&longs;i; </s>

<s id="N28B27">quia nemo &longs;cit quot fluxerint <lb/>iu&longs;tantia in dato motu; </s>

<s id="N28B27">quia nemo &longs;cit quot fluxerint <lb/>in&longs;tantia in dato motu; </s>

<s id="N28B2D">qaod reuerâ &longs;ciri po&longs;&longs;et &longs;i po&longs;&longs;et aliqua arte in<lb/>ueniri corpus, cuius grauitatio haberet <expan abbr="effect&utilde;">effectum</expan>, quem habet alteriusictus, <lb/>qnod nec &longs;ciri pote&longs;t per depre&longs;&longs;um cylindrum cereum vel <expan abbr="plumbe&utilde;">plumbeum</expan>, vel <lb/>alterius mollioris materiæ, quia æqualis depre&longs;&longs;io accuratè cogno&longs;ci non <lb/>pote&longs;t; &longs;i quis enim diceret dee&longs;&longs;e, vel &longs;upere&longs;&longs;e 1000. &longs;uperficies, quà <lb/>ratione conuinci po&longs;&longs;et? </s>

<s id="N28B2D">quoad reuerâ &longs;ciri po&longs;&longs;et &longs;i po&longs;&longs;et aliqua arte in<lb/>ueniri corpus, cuius grauitatio haberet <expan abbr="effect&utilde;">effectum</expan>, quem habet alterius ictus, <lb/>quod nec &longs;ciri pote&longs;t per depre&longs;&longs;um cylindrum cereum vel <expan abbr="plumbe&utilde;">plumbeum</expan>, vel <lb/>alterius mollioris materiæ, quia æqualis depre&longs;&longs;io accuratè cogno&longs;ci non <lb/>pote&longs;t; &longs;i quis enim diceret dee&longs;&longs;e, vel &longs;upere&longs;&longs;e 1000. &longs;uperficies, quà <lb/>ratione conuinci po&longs;&longs;et? </s>

<s id="N28B43">non pote&longs;t etiam &longs;ciri operâ libræ, in cuius al<lb/>terum brachium cadat mobile, quia &longs;unt ferè infiniti motus in&longs;en&longs;ibiles, <lb/>vt con&longs;ideranti patebit; igitur proportio hæc tantùm, determinari pote&longs;t <lb/>ex hypothe&longs;i data, vt clari&longs;&longs;imè con&longs;tat ex dictis. </s>

<s id="N28B4F">Decimoquartò, hinc maxima e&longs;t proportio inter ictum, & grauita<lb/>tionem; </s>

<s id="N28B55">cùm modicus malleoli ca&longs;us eum effectum præ&longs;tet, quem in<lb/>gens corporis moles &longs;ua grauitatione præ&longs;tare non po&longs;&longs;et; </s>

<s id="N28B5B">non e&longs;t tamen <lb/>infinita proportio, quia pote&longs;t tanta e&longs;&longs;e moles grauitatis, & tam par<lb/>uum corporis cadentis pondus, vt illa præualeat, vt con&longs;tat experientiâ, <lb/>quæ nobis cuidenti&longs;&longs;imam &longs;uggerit rationem; </s>

<s id="N28B5B">non e&longs;t tamen <lb/>infinita proportio, quia pote&longs;t tanta e&longs;&longs;e moles grauitatis, & tam par<lb/>uum corporis cadentis pondus, vt illa præualeat, vt con&longs;tat experientiâ, <lb/>quæ nobis euidenti&longs;&longs;imam &longs;uggerit rationem; </s>

<s id="N28B65">quia reiicimus infinitos <lb/>illos tarditatis gradus, quos a&longs;&longs;ump&longs;it Galilæus ad probandam &longs;uam <lb/>hypothe&longs;im de motu accelerato, & infinita &longs;iu&longs;dem & aliorum multo<lb/>rum in&longs;tantia, de quibus alibi in Metaphy&longs;icâ; </s>

<s id="N28B65">quia reiicimus infinitos <lb/>illos tarditatis gradus, quos a&longs;&longs;ump&longs;it Galilæus ad probandam &longs;uam <lb/>hypothe&longs;im de motu accelerato, & infinita eiu&longs;dem & aliorum multo<lb/>rum in&longs;tantia, de quibus alibi in Metaphy&longs;icâ; </s>

<s id="N28B6F">e&longs;t tamen maxima illa <lb/>proportio, vt dixi; </s>

<s id="N28B75">quia perexigua temporis pars infinitis ferè in&longs;tanti<lb/>tibus con&longs;tat; </s>

<s id="N28B75">quia perexigua temporis pars infinitis ferè in&longs;tanti<lb/>bus con&longs;tat; </s>

<s id="N28B7B">quorum certè numerum recen&longs;ere po&longs;&longs;emus, &longs;i quis mo<lb/>dum inueniat, quo po&longs;&longs;it ab&longs;olutè adæquare grauitationis dati corporis <lb/>effectum cum effectu ictus alterius cadentis: quod meo iudicio non <lb/>modo geometricè, vetùm etiam mechanicè, &longs;altem accuratè fieri non <lb/>pote&longs;t. </s>

<s id="N28B7B">quorum certè numerum recen&longs;ere po&longs;&longs;emus, &longs;i quis mo<lb/>dum inueniat, quo po&longs;&longs;it ab&longs;olutè adæquare grauitationis dati corporis <lb/>effectum cum effectu ictus alterius cadentis: quod meo iudicio non <lb/>modo geometricè, verùm etiam mechanicè, &longs;altem accuratè fieri non <lb/>pote&longs;t. </s>

<s id="N28B89">Decimoquintò, nec illud, quod habet Dominus Hobs apud Mer&longs;en<lb/>num, in ph&oelig;nom. </s>

Line 18368

<s id="N28B9D">Secundò, quia dicit cona-<pb pagenum="408" xlink:href="026/01/442.jpg"/>tum primum puncti AC, & totius axis AB, quamdiu de&longs;cendit vterque, <lb/>e&longs;&longs;e æqualem; </s>

<s id="N28BA8">quod tamen dici non pote&longs;t, quia conatus &longs;ingulorum <lb/>punctorum &longs;eor&longs;im &longs;unt æquales; </s>

<s id="N28BAE">&longs;ed conatus omnium coniunctim e&longs;t <lb/>maior conatu &longs;ingulorum; </s>

<s id="N28BB4">nam &longs;ingula habent &longs;uum impetum; verum <lb/>e&longs;t quidem moueri motu æquali, quia &longs;ingula æquali impetu moueutur. </s>

<s id="N28BB4">nam &longs;ingula habent &longs;uum impetum; verum <lb/>e&longs;t quidem moueri motu æquali, quia &longs;ingula æquali impetu mouentur. </s>

<s id="N28BBA"><lb/>Tertiò, quia vult po&longs;ito cylindro &longs;upra ba&longs;im 4. illam immediatè premi <lb/>à puncto EB, hoc verò punctum à puncto DE, & hoc ab CD, & hoc ab <lb/>AC; </s>

<s id="N28BC3">quod tamen dici non pote&longs;t; quis enim dicat granulum &longs;uperpo&longs;i<lb/>tum rupi in illam grauitare? </s>

<s id="N28BC9">Equidem cum illa grauitat grauitatione <lb/>communi, vt dictum e&longs;t &longs;uprà, non tamen in illam. </s>

<s id="N28BCE">Quartò, quia dicit <lb/>pumum B cum conatu totius cylinèri incubantis eo tempore, quo pun<lb/>ctum AC conficeret AC, conficere AB, quod repugnat progre&longs;&longs;ioni <lb/>Galilei, quam &longs;equitur ip&longs;e; </s>

<s id="N28BD8">puia conatus &longs;unt, vt velocitates; </s>

<s id="N28BD8">quia conatus &longs;unt, vt velocitates; </s>

<s id="N28BDC">hæ verò <lb/>vt tempora; &longs;ed &longs;patia in ratione duplicata temporum. </s>

Line 18407

<s id="N28C6C">e&longs;t enim &longs;ubduplum pa<lb/>rallelipedi, cuius ba&longs;is &longs;it quadratum AC, & altitudo RE; </s>

<s id="N28C72">&longs;i enim trian<lb/>gulum ADE ducatur in latus AB vel EF habebitur &longs;olidum cunci, vt <lb/>con&longs;tat, vnde cunei eiu&longs;dem latitudinis &longs;unt, vt triangula, v.g. cuneus A <lb/>F ad cumdem YF; </s>

<s id="N28C7E">vt triangulum ADE ad triangulum YHE: </s>

<s id="N28C82">hoc po<lb/>&longs;ito &longs;it triangulum MKN æqualis ADF, & primo ictu tota EI vel N <lb/>Z &longs;ecundo ictu defigitur, non quidem æquali altitudinc, &longs;ed æquali &longs;oli<lb/>do; </s>

<s id="N28C8C">cùm autem triangulum XZN &longs;it &longs;ubquadruplum trianguli QON <lb/>&longs;it media proportionalis Ninter NZNO, triangulum N<foreign lang="greek">b</foreign> Y e&longs;t du<lb/>plum NZX; </s>

<s id="N28C98">igitur &longs;ecundo ictu defigetur N<foreign lang="greek">b</foreign>: </s>

<s id="N28CA0">&longs;imiliter &longs;i vt NZ ad N <lb/><foreign lang="greek">b</foreign>, ita N<foreign lang="greek">b</foreign> ad N. Tertio, ita defigetur NT, & quarto NO dupla NI: ra<lb/>tio e&longs;t, quia æquales ictus æquales habent effectus. </s>

<s id="N28CB1">Vige&longs;imò, &longs;i æquales accipiantur altitudines &longs;ingulis ictibus, ictus <lb/>&longs;unt in ratione duplicata altitudinum, &longs;uppo&longs;itâ prædicta hypothe&longs;i cunei <lb/>v.g.&longs;i dato ictu defigatur NZ, & altero NO, &longs;ecundus e&longs;t ictus quadruplus <lb/>primi; </s>

<s id="N28CBB">&longs;i verò tertio ictu defigatur N<foreign lang="greek">q</foreign> tripla NZ, ictus e&longs;t ad primum <lb/>in ratione 9/1. &longs;i denique dato ictu defigatur NM, ictus e&longs;t ad primum <lb/>in ratione (26/<gap/>), vt patet ex dictis; &longs;i verò primo ictu defigatur NZ, &longs;ecundo <lb/>ZO, tertio O<foreign lang="greek">q</foreign>, quarto <foreign lang="greek">q</foreign>M, ictus &longs;unt, vt numeri impares 1. <lb/>3. 7. 9. </s>

<s id="N28CBB">&longs;i verò tertio ictu defigatur N<foreign lang="greek">q</foreign> tripla NZ, ictus e&longs;t ad primum <lb/>in ratione 9/1. &longs;i denique dato ictu defigatur NM, ictus e&longs;t ad primum <lb/>in ratione 36/3, vt patet ex dictis; &longs;i verò primo ictu defigatur NZ, &longs;ecundo <lb/>ZO, tertio O <foreign lang="greek">q</foreign>, quarto <foreign lang="greek">q</foreign> M, ictus &longs;unt, vt numeri impares 1. <lb/>3. 7. 9. </s>

<s id="N28CD7">Vige&longs;imoprimò, hinc &longs;i dentur duo ictus, & eorum proportio deter<lb/>minari, vt pote&longs;t proportio altitudinum, quæ defiguntur, quæ &longs;unt in <lb/>ratione &longs;ubduplicata ictuum, &longs;uppo&longs;ito cuneo: </s>

<s id="N28CDF">&longs;imiliter, &longs;i dentur alti<lb/>tudines, carumque proportio, determinari pote&longs;t proportio ictum; </s>

<s id="N28CE5">&longs;unt <lb/>enim in ratione duplicata, vt patet ex dictis; porrò vtrumque pote&longs;t <lb/>con&longs;iderari duobus modis. </s>

<s id="N28CED">Primò, coniunctim, &longs;i &longs;ecundus ictus &longs;ucce<lb/>d<gap/> primo, & eius altitudinem augeat. </s>

<s id="N28CED">Primò, coniunctim, &longs;i &longs;ecundus ictus &longs;ucce<lb/>dat primo, & eius altitudinem augeat. </s>

<s id="N28CF4">Secundò, &longs;i &longs;eor&longs;im vterque <lb/>con&longs;ideretur, &c. </s>

Line 18436

<s id="N28D2C">nam primò deptimitur prima &longs;uperficies <lb/>cylindri, & extenditur; quia cùm materia. </s>

<s id="N28D32">&longs;it mollior, prematurque a <lb/>duobus corporibus duris vtrinque, &longs;cilicet ab vtraque ba&longs;i, cedit & di<lb/>latatur propter humorem in cauitatibus contentum. </s>

<s id="N28D39">Secundò, aliquan<lb/>do totus cylindrus deprimitur &longs;eruatà &longs;emper cylindri licet cra&longs;&longs;io<lb/>ris figurâ, quod vt fiat, molli&longs;&longs;imam materiam e&longs;&longs;e nece&longs;&longs;e e&longs;t. </s>

<s id="N28D40">Ter-<pb pagenum="410" xlink:href="026/01/444.jpg"/>tiò, aliquando primæ tantùm &longs;uperficies extenduntur, vt videmus in <lb/>capite, &longs;eu ba&longs;i cuncorum; quia materies durior multùm re&longs;i&longs;tit. </s>

<s id="N28D40">Ter-<pb pagenum="410" xlink:href="026/01/444.jpg"/>tiò, aliquando primæ tantùm &longs;uperficies extenduntur, vt videmus in <lb/>capite, &longs;eu ba&longs;i cuneorum; quia materies durior multùm re&longs;i&longs;tit. </s>

<s id="N28D4B">Quartò, <lb/>limbus ba&longs;is dilatatæ contrahitur deinde, &longs;eu retorquetur deor&longs;um; </s>

<s id="N28D51">quia <lb/>cùm interiores circuli dilatentur, deberet facere limbus ille maiorem <lb/>circulum; quod cùm fieri non po&longs;&longs;it, contrahitur &longs;eu incuruatur deor<lb/>&longs;um, quod facilè &longs;ine figura intelligi pote&longs;t. </s>

<s id="N28D5B">Quintò, pote&longs;t deter<lb/>minari proportio ictuum, quibus deprimuntur cyiindri; </s>

<s id="N28D61">&longs;i enim &longs;up<lb/>ponatur eadem altitudo, &longs;eu linea depre&longs;&longs;ionis, & diuer&longs;a cra&longs;&longs;i<lb/>tudo cylindrorum ictus, erunt vt ba&longs;es; </s>

<s id="N28D69">nam quò plures partes de<lb/>primendæ &longs;unt, maiore ictu opus e&longs;t, &longs;i opponatur eadem cra&longs;&longs;itudo <lb/>vtriu&longs;que cylindri &longs;ed diuer&longs;a depre&longs;&longs;ionis linea vel altitudo, ictus <lb/>erunt vt altitudines; </s>

<s id="N28D73">&longs;i vtraque &longs;upponitur diuer&longs;a, ictus erunt in ra<lb/>rione compo&longs;ita ex ratione ba&longs;ium, & altitudinum; quæ omnia con&longs;tant <lb/>ex dictis. </s>

<s id="N28D73">&longs;i vtraque &longs;upponitur diuer&longs;a, ictus erunt in ra<lb/>tione compo&longs;ita ex ratione ba&longs;ium, & altitudinum; quæ omnia con&longs;tant <lb/>ex dictis. </s>

<s id="N28D7D">Ob&longs;eruabis tamen cre&longs;cere re&longs;i&longs;tentiam ex duplici capite. </s>

<s id="N28D80">Primò, <lb/>ex eo quod aliquæ vacuitates occupentur à partibus depre&longs;&longs;is, ac proin<lb/>de cylindrus induretur; &longs;ic incus durior cuadit &longs;ub mallco, & & pila <lb/>lignea &longs;ub ictibus. </s>

<s id="N28D80">Primò, <lb/>ex eo quod aliquæ vacuitates occupentur à partibus depre&longs;&longs;is, ac proin<lb/>de cylindrus induretur; &longs;ic intus durior euadit &longs;ub malleo, & & pila <lb/>lignea &longs;ub ictibus. </s>

<s id="N28D8A">Secundò, latiorem illam &longs;uperficiem impedire di<lb/>latationem aliarum partium: </s>

<s id="N28D90">hinc variè di&longs;cerpitur eius limbus, vt <lb/>videre e&longs;t in cuneo ferreo: </s>

<s id="N28D96">atqui in explicandis &longs;uprà ictuum propor<lb/>tionibus, hoc geminum re&longs;i&longs;tentiæ caput nullo modo con&longs;iderauimus: </s>

<s id="N28D9C"><lb/>&longs;extò, quærunt aliqui dato ictu, quo deprimitur cylindrus data alti<lb/>tudine, quantum pondus e&longs;&longs;e debeat, quod &longs;ua grauitatione cum<lb/>dem præ&longs;tet effectum; &longs;ed profectò id nemo vnquam determinauit, <lb/>ni&longs;i primò inueniat pondus, cuius ca&longs;u prædictus cylindrus codem <lb/>modo deprimatur. </s>

<s id="N28D9C"><lb/>&longs;extò, quærunt aliqui dato ictu, quo deprimitur cylindrus data alti<lb/>tudine, quantum pondus e&longs;&longs;e debeat, quod &longs;ua grauitatione eum<lb/>dem præ&longs;tet effectum; &longs;ed profectò id nemo vnquam determinauit, <lb/>ni&longs;i primò inueniat pondus, cuius ca&longs;u prædictus cylindrus codem <lb/>modo deprimatur. </s>

<s id="N28DA9">Secundò, ni&longs;i &longs;ciat quot in&longs;tantibus de&longs;cendat, vt <lb/>patet ex his quæ diximus &longs;uprà; vt autem comparetur ictus inflictus <lb/>à brachio cum ictu inflicto à pondere cadente, debet con&longs;uli diuer&longs;a <lb/>depre&longs;&longs;io, vel defixio. </s>

Line 18513

<s id="N28EC1">ratio e&longs;t, quia maior impetus imprimitur Tudiculæ à potentia <lb/>diutiùs applicata; diutiùs autem applicatur maiori, quia tardiùs moue<lb/>tur, vt &longs;uprà diximus. </s>

<s id="N28ECB">Sextò, pila emi&longs;&longs;a veloci&longs;&longs;imè mouetur eo in&longs;tanti, quo vltimo tan<lb/>gitur à Tudicula; quia deinceps nihil pror&longs;us impetus accedit, ac proin<lb/>de continuò &longs;en&longs;im de&longs;truitur ab eo inftanti. </s>

<s id="N28ED5">Septimò, nunquam mouetur pila emi&longs;&longs;a velociùs ip&longs;a Tudiculâ, cum <lb/>&longs;cilicet emi&longs;&longs;io fit per meram impul&longs;ionem; </s>

Line 18543

<s id="N28F40"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 17.<emph.end type="center"/></s>

<s id="N28F4E"><emph type="italics"/>Explicari poffunt omnia ph&oelig;nomena emi&longs;&longs;ionum, quæ fiunt cum perco&longs;&longs;ione.<emph.end type="italics"/></s>

<s id="N28F4E"><emph type="italics"/>Explicari po&longs;&longs;unt omnia ph&oelig;nomena emi&longs;&longs;ionum, quæ fiunt cum percu&longs;&longs;ione.<emph.end type="italics"/></s>

<s id="N28F57">Primò, &longs;it percu&longs;&longs;io minoris Tudiculæ v.g. eo maior e&longs;t, quò Tudi<lb/>cula maior e&longs;t; rationem iam attulimus &longs;uprà num.5.Th.16. </s>

Line 18556

<s id="N28F73">Quartò, quod &longs;pectat ad Tudiculam maiorem, iam &longs;uprà explicuimus <lb/>cuncta illius ph&oelig;nomena, cum de malleo: certum e&longs;t enim primò ma<lb/>iorem à maiore ictum infligi, cæteris partibus, quàm à minore propter <lb/>prædictum rationem. </s>

<s id="N28F7D">Secundò, certum e&longs;t longitudinem manubrij fle<lb/>xibilitatem, inæqnalitatem, materiem, duritiem mallei, æqualitatem ba&longs;is <lb/>&c. </s>

<s id="N28F7D">Secundò, certum e&longs;t longitudinem manubrij fle<lb/>xibilitatem, inæqualitatem, materiem, duritiem mallei, æqualitatem ba&longs;is <lb/>&c. </s>

<s id="N28F84">multùm conferre ad maiorem cùm ictus. </s>

<s id="N28F87">Tertiò certum e&longs;t mino<lb/>rem globum, in quem impingitur Tudicula, citiùs moueri, inaiorem tar<lb/>diùs, cæteris paribus. </s>

<s id="N28F8E">Quartò, globus maior in alium impactus Tudiculâ <lb/>maiorem ictum infligit, vt con&longs;tat experientiâ; ratî; </s>

<s id="N28F94">e&longs;t, quia tardiùs <lb/>mouetur; </s>

<s id="N28F9A">igitur diutiùs applicatur: Equidem globus proiectus in alium <lb/>fortiorem ictum infligit ex duplici capite, vt dicam infrà. </s>

<s id="N28FA0">1&degree;. </s>

<s id="N28FA3">Quia ma<lb/>iorem impetum à potentia diutiùs applicata.2&degree;.Quia diutiùs applicatur <lb/>globo in quem impingitur; at verò quando impigitur Tudiculâ maiore, <lb/>ex duplici quoque capite cre&longs;cit ictus.1&degree;.quia globus globo diutiùs ma<lb/>net applicatus, cùm tardior motus dicat plùs temporis. </s>

<s id="N28FA3">Quia ma<lb/>iorem impetum à potentia diutiùs applicata.2&degree;.Quia diutiùs applicatur <lb/>globo in quem impingitur; at verò quando impingitur Tudiculâ maiore, <lb/>ex duplici quoque capite cre&longs;cit ictus.1&degree;.quia globus globo diutiùs ma<lb/>net applicatus, cùm tardior motus dicat plùs temporis. </s>

<s id="N28FAF">2&degree;. </s>

<s id="N28FB2">quia malleus <lb/>tardiorem motum imprimis globo; </s>

<s id="N28FB8">igitur diutiùs manet applicatus: e&longs;t <lb/>en<gap/>m hæc abta lex agentium, vt longiore tempore maior effectus produ<lb/>catur, minor verò minore, reliqua ex dictis facilè intelligentur. </s>

<s id="N28FB8">igitur diutiùs manet applicatus: e&longs;t <lb/>enim hæc abta lex agentium, vt longiore tempore maior effectus produ<lb/>catur, minor verò minore, reliqua ex dictis facilè intelligentur. </s>

<s id="N28FC4"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 18.<emph.end type="center"/></s>

Line 18597

<s id="N2903B">Quartò, leui&longs;&longs;imi lapides vix iaciuntur ad modicam di&longs;tantiam v. g. <lb/>granula &longs;abuli; ratio e&longs;t, 1&degree;. </s>

<s id="N29044">quia accipiunt minùs impetus, quia citiùs <lb/>&longs;eparantur à iaciente manu, vt patet. </s>

<s id="N29049">2&degree;. </s>

<s id="N2904C">quia mouetur initio velociùs i<gap/><lb/>aëre; </s>

<s id="N2904C">quia mouetur initio velociùs in <lb/>aëre; </s>

<s id="N29053">igitur &longs;ingulis in&longs;tantibus plùs impetus de&longs;truitur, vt con&longs;tat; nam <lb/>in maiori &longs;patio aëris e&longs;t maior re&longs;i&longs;tentia. </s>

<s id="N29059">3&degree;.quia cùm aër perpetuo <lb/>motu agitetur, vt certum e&longs;t, in leuiori corpore impetum imprimit; igi<lb/>tur aliam &longs;i&longs;tit vel deflectit. </s>

<s id="N29061">4&degree;.quia manu non pote&longs;t rectè prehendi ia<lb/>ciendus lapillus &c. </s>

<s id="N29068">Quintò, grauior lapis ad modicam tantùm diftantiam iacitur; ratio <lb/>e&longs;t 1&degree;.quia producitur remi&longs;&longs;ior impetus, cùm &longs;cilicet pluribus partibus <lb/>&longs;ubiecti di&longs;tribuatur. </s>

<s id="N29068">Quintò, grauior lapis ad modicam tantùm di&longs;tantiam iacitur; ratio <lb/>e&longs;t 1&degree;.quia producitur remi&longs;&longs;ior impetus, cùm &longs;cilicet pluribus partibus <lb/>&longs;ubiecti di&longs;tribuatur. </s>

<s id="N29070">2&degree;.quia impetus grauitationis citiùs de&longs;truit impe<lb/>tum extrin&longs;ecus aduenientem. </s>

Line 18619

<s id="N2909D"><lb/>&longs;ic vulgò iaciuntur grauiora &longs;axa; </s>

<s id="N290A2">ad iactum autem conferunt vires po<lb/>tentiæ, brachium longiùs, longior arcus, Tangens, per quam emittitur di<lb/>mi&longs;&longs;um &longs;axum, quæ debet facere cum horizontali angulum grad.45. ma<lb/>nus &longs;imul explicata; </s>

<s id="N290AC">&longs;i enim vna pars ante aliam dimittatur, retinetur <lb/>iactus, vt vulgò dicitur, figura, & moles lapidis; </s>

<s id="N290B2">&longs;i enim maior e&longs;t, non <lb/>procul emittitur præuia brachij gyratio, quia impetus augetur: denique <lb/>impre&longs;&longs;us toti corpori impetus, quæ omnia mirificè maiorem iactum ef<lb/><gap/>ciunt, vt con&longs;tat ex dictis &longs;uprà. </s>

<s id="N290B2">&longs;i enim maior e&longs;t, non <lb/>procul emittitur præuia brachij gyratio, quia impetus augetur: denique <lb/>impre&longs;&longs;us toti corpori impetus, quæ omnia mirificè maiorem iactum ef<lb/>ficiunt, vt con&longs;tat ex dictis &longs;uprà. </s>

<s id="N290BD">2&degree;.iacitur lapis rotato quidem deor&longs;um <lb/>brachio, &longs;ed non &longs;iue aliqua eiu&longs;dem brachij contractione, & aliquot <pb pagenum="414" xlink:href="026/01/448.jpg"/>gyris: &longs;ic vulgò iaciuntur &longs;axa minora, tuncque præ&longs;ertim contentis ner<lb/>uis toti corpori impetus accedit, qui deinde ad augendam iactum in <lb/>ip&longs;um brachium qua&longs;i refunditur.3&degree;. </s>

<s id="N290CC">iacitur lapis negligenti qua&longs;i ni&longs;u, <lb/>&longs;eu reiectione circumacta manu horizonti parallela, & contracto tan<lb/>tillùm brachio. </s>

<s id="N290D3">4&degree;. </s>

Line 18646

<s id="N29145">Vndecimò, funda vulgare e&longs;t organum iactus, cuius ph&oelig;nomena fa<lb/>cilè explicari po&longs;&longs;unt.1&degree;. </s>

<s id="N2914A">rotatur vt maiorem impetum acquirat ad mo<lb/>tus recticulo lapis, 2&degree;.quò longior e&longs;t funda, longiùs lapis abigitur, quia <lb/>diutiùs manet applicatus, cùm maiorem arcum decurrat, 3&degree;.lapis in reti<lb/>culo fundæ retinetur; quia cùm per Tangentem lineam &longs;ingulis in&longs;tanti<lb/>bus determinetur, vt con&longs;tat ex dictis &longs;uprà, impeditur & retinetur à re<lb/>ticulo, per quod Tangens illa duci tantùm pote&longs;t, e&longs;t eadem ratio, quæ <lb/>orbis rotati, de quo Th.3.num.10. 4&degree;. </s>

<s id="N2914A">rotatur vt maiorem impetum acquirat ad mo<lb/>tus reticulo lapis, 2&degree;.quò longior e&longs;t funda, longiùs lapis abigitur, quia <lb/>diutiùs manet applicatus, cùm maiorem arcum decurrat, 3&degree;.lapis in reti<lb/>culo fundæ retinetur; quia cùm per Tangentem lineam &longs;ingulis in&longs;tanti<lb/>bus determinetur, vt con&longs;tat ex dictis &longs;uprà, impeditur & retinetur à re<lb/>ticulo, per quod Tangens illa duci tantùm pote&longs;t, e&longs;t eadem ratio, quæ <lb/>orbis rotati, de quo Th.3.num.10. 4&degree;. </s>

<s id="N2915A">hinc demi&longs;&longs;o altero fundæ funi<lb/>culo lapis iacitur, quia nihil e&longs;t à quo retineri ampliùs queat. </s>

<s id="N2915F">5&degree;. </s>

<s id="N29162">quò <lb/>maior e&longs;t lapis cæteris paribus, tardiùs rotatur funda, at maior impetus <lb/>lapidi imprimitur; quia diutiùs manet applicatus. </s>

<s id="N2916A">6&degree;. </s>

<s id="N2916D">tenditur conti<lb/>nuò rota, quantumuis rotetur; quia &longs;cilicet non quidem à pondere <pb pagenum="415" xlink:href="026/01/449.jpg"/>lapidis, &longs;ed ab eius impetu ad Tangentem determinato eò trahitur. </s>

<s id="N29178"><lb/>Septimò, quod autem ad Tangentem continuò determinetm <gap/>ea mo<lb/>tus, patet ex dictis, cum de motu circulari. </s>

<s id="N29178"><lb/>Septimò, quod autem ad Tangentem continuò determinetur linea mo<lb/>tus, patet ex dictis, cum de motu circulari. </s>

<s id="N29180">Octauò, longi&longs;&longs;imus erit ia<lb/>ctus, &longs;i Tangens, ad quam motus lapidis determinatur, eo in&longs;tanti, quo <lb/>demittitur faciat angulum 45. grad. cum horizontali. </s>

<s id="N29189">Nonò, vt rectè <lb/>collimetur, &longs;eu dirigatur lapis ad propo&longs;itum fcopum, egregium artifi<lb/>cium e&longs;&longs;e pote&longs;t; quod totum in eo po&longs;itum e&longs;t, vt inueniatur illa Tan<lb/>gens, quæ ducitur ad &longs;copum. </s>

<s id="N29189">Nonò, vt rectè <lb/>collimetur, &longs;eu dirigatur lapis ad propo&longs;itum &longs;copum, egregium artifi<lb/>cium e&longs;&longs;e pote&longs;t; quod totum in eo po&longs;itum e&longs;t, vt inueniatur illa Tan<lb/>gens, quæ ducitur ad &longs;copum. </s>

<s id="N29193">Decimò, ad fundam reuocari pote&longs;t, li<lb/>neailla fi&longs;&longs;i baculi furca, cui &longs;i lapis in&longs;eratur, facilè deinde emittitur; </s>

<s id="N29199"><lb/>&longs;it enim linea furca AB; </s>

<s id="N2919E">&longs;it lapis in&longs;ertus B, &longs;i rotetur maximo ni&longs;u furca <lb/>AB circa centrum A, vel circa centrum humeri; </s>

Line 18665

<s id="N291BB">Duodecimò, claua pilaris, &longs;eu reticulum notum e&longs;t omnibus or<lb/>ganum, cuius ph&oelig;nomena clari&longs;&longs;ima &longs;unt. </s>

<s id="N291C0">Primò, reticulo longiùs <lb/>emittitur pila, quàm clauulâ, propter ten&longs;ionem & reditum chordarum. </s>

<s id="N291C0">Primò, reticulo longiùs <lb/>emittitur pila, quàm clauiculâ, propter ten&longs;ionem & reditum chordarum. </s>

<s id="N291C5"><lb/>Secundò, quò longiùs e&longs;t clauulæ manubrium, longiùs abigitur pila. </s>

<s id="N291C9"><lb/>Tertiò, vt &longs;u&longs;tineatur ictus breui manubrio, reticulo opus e&longs;t. </s>

<s id="N291CD">Quartò, <lb/>auer&longs;a manu impacto reticulo, pila longiùs emittitur. </s>

Line 18685

<s id="N29223"><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 19.<emph.end type="center"/></s>

<s id="N29231"><emph type="italics"/>Aliæ &longs;unt plurimæ motionum &longs;pecies, quas in hoc Theoremate exponi<lb/>miu.<emph.end type="italics"/></s>

<s id="N29231"><emph type="italics"/>Aliæ &longs;unt plurimæ motionum &longs;pecies, quas in hoc Theoremate exponi<lb/>mus.<emph.end type="italics"/></s>

Line 18704

<s id="N29275">ad intru&longs;ionem & extru&longs;ionem reuocari <lb/>pote&longs;t ductus auri, vel argenti, vel alterius ductilis materiæ; &longs;ed hunc <lb/>rei ductilis &longs;tatum Tomo quinto explicabimus cum alijs corporeum &longs;ta<lb/>tibus. </s>

<s id="N29281">Tertiò, di&longs;po&longs;itio fit per ciaculationem, vel minimarum partium, quæ <lb/>&longs;imul omnes vno iactu demittuntur manu; </s>

<s id="N29281">Tertiò, di&longs;po&longs;itio fit per eiaculationem, vel minimarum partium, quæ <lb/>&longs;imul omnes vno iactu demittuntur manu; </s>

<s id="N29287">&longs;it plura grana tritici vel <lb/>arenæ iaciuntur, vel alicuius corporis, cuius partes &longs;eparantur in ip&longs;o <lb/>iactu; cur verò vna per hanc lineam, alia per aliam feratur, determi<lb/>natur vel à concur&longs;u cum alia parte, vel à &longs;itu, quem &longs;ingulæ in iacien<lb/>tis manu habebant priùs, vel ab ordine, quo &longs;ingulæ proce&longs;&longs;erunt. </s>

Line 18714

<s id="N292A3">&longs;ic adducitur clauus, vel <lb/>truncus, vel per circuitionem &longs;implicem, &longs;ic adducitur rotati baculi ex<lb/>tremitas; vel per circuitionem mixtam: </s>

<s id="N292AB">&longs;ic adducitur extremitas funis <lb/>flagelli; vel cum aliquo iactu; </s>

<s id="N292B1">&longs;ic adducitur pulmentum vt in va&longs;e <lb/>optimè commi&longs;ceatur v.g. &longs;ic coqui adducunt frixum & inuertunt, por<lb/>recto tantillùm, tùm deinde rotato &longs;artaginis manubrio: </s>

<s id="N292BB">&longs;i enim e&longs;&longs;et <lb/>vera rotatio, &longs;rixum per Tangentem erit; at verò propter motum rectum <lb/>po&longs;t inuer&longs;ionem ab ip&longs;a &longs;artagine minimè recedit. </s>

<s id="N292BB">&longs;i enim e&longs;&longs;et <lb/>vera rotatio, frixum per Tangentem erit; at verò propter motum rectum <lb/>po&longs;t inuer&longs;ionem ab ip&longs;a &longs;artagine minimè recedit. </s>

<s id="N292C5">Quintò, ventilatio e&longs;t motio, quâ frumentum excernitur vanno; </s>

Line 18728

<s id="N292F7">Sextò, remigatio fit pellendo, trahendoque, de qua iam &longs;uprà Th. 6. <lb/>16.longior & latior remus maiorem vim aquæ impellit; </s>

<s id="N292FD">difficiliùs taman <lb/>mouetur, quò maior e&longs;t illius portio à centro motus ver&longs;us manum re<lb/>migantis, faciliùs mouetur propter rationem vectis; </s>

<s id="N29305">faciliùs mouetur, &longs;i <lb/>aduer&longs;o flumine feratur nauis: </s>

<s id="N2930B">ratio e&longs;t, quia aqua pul&longs;a ver&longs;us eam <lb/>partem, in quam fluir minùs re&longs;i&longs;tit, quando eundem remum tractant, <pb pagenum="417" xlink:href="026/01/451.jpg"/>ille plus confert, qui ad extremiiatem propiùs accedit; ratio clara e&longs;t: </s>

<s id="N2930B">ratio e&longs;t, quia aqua pul&longs;a ver&longs;us eam <lb/>partem, in quam fluit minùs re&longs;i&longs;tit, quando eundem remum tractant, <pb pagenum="417" xlink:href="026/01/451.jpg"/>ille plus confert, qui ad extremitatem propiùs accedit; ratio clara e&longs;t: </s>

<s id="N29316"><lb/>&longs;ed de re nautica aliàs; vide interim locum citatum. </s>

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