Ceva, Giovanni Geometria motus 1692 Bologna la cevag_geome_01_la_1692 0000000022.xml

GEOMETRIA

MOTVS.

DEF. I.

CVrrat mobile ab A in D &longs;ecundùm rectam AD, & linea BHI &longs;it naturæ illius, vt dedu­ctis ad AD perpendicularibus AB, CH, DI ex punctis quibu&longs;cunque A, C, D; veloci­tatum gradus, quos mobile &longs;ortitur in ij&longs;­dem punctis A, C, D men&longs;urentur ab ip&longs;is rectis AB, CH, CI. Figuram planam BADIHB apellabi­mus gene&longs;im motus ab A in D.

Tab. 1. Fig. 1.

DEF. II.

II&longs;dem manentibus, &longs;it etiam alia linea EFG talis natu­ræ, vt protractis rectis BA in E, HC in F, & ID in G ha­beat DG ad CF eandem reciprocè rationem, quam HC ad ID. Item &longs;it CF ad HE vt reciprocè BA ad HC, vo­cabimus figuram planam ADGIEA imaginem tempo­ris motus ab A in D iuxta gene&longs;im prædictam.

Tab. 1. Fig. 2.

DEF. III.

ADhuc po&longs;ita illa gene&longs;i, intelligatur linea PON eius naturæ, vt &longs;i &longs;it KL ad LM vt tempus lationis ab A in C ad tempus ab eodem C in D, habeat &longs;emper KP ad LO eandem rationem, quam AB ad CH; & LO ad NM eandem, quam HC ad ID: Figuram planam PKMNOP vocabimus imaginem iuxta gene&longs;im BADI motus ab A in D.

Tab. 1. Fig. 3.

Corollarium.

Patet, cum motus &longs;unt æquabiles, gene&longs;es, & imagines figu­ras eße parallelogrammas.

DEF. IV.

Tab. 1. Fig. 4.

SI &longs;int duæ gene&longs;es, aut imagines ABCD, FEG, ita vt cum gene&longs;es &longs;int, habeat AB ad FE eandem rationem, quam velocitas in A ad velocitatem in F, & cum imagines velocitatum, quarum tempora AD, FG, velocitas, quam habet mobile in&longs;tanti A ad velocitatem alterius mobilis in&longs;tanti F, &longs;it vt AB ad FE, & demum ip&longs;is figuris vt imagi­nibus temporum con&longs;ideratis habeat velocitas in A ad velocitatem in F rationem eandem, quam AB ad FE, vo­cabuntur tum gene&longs;es illæ, cum imagines inter &longs;e homo­geneæ.

DEF. V.

EAm planam Figuram, in qua ductæ quoteunque &ecedil;quidi&longs;tantes eò deinceps decre&longs;cunt, quò ad idem extremum propiores fiunt, acuminatam nuncupabimus.

DEF. VI. AX. I.

INter maximam, & minimam eiu&longs;dem imaginis veloci­tatem cadit quædam media, qua tantùm velocitate, &longs;i conciperetur motus æquabilis, nihilominùs eodem tem­pore idem &longs;patium curreretur, ac iuxta imaginem propo&longs;i­tam: eam ergo mediam velocitatem dicimus propo&longs;itæ imaginis æquatricem.

AX. II.

SPatium iuxta imaginem velocitatum quamcunque exactum, vel iuxta æquatricem imaginis e&longs;t maius eo &longs;patio, quod curreretur eodem tempore minima eiu&longs;dem imaginis velocitate; &longs;ed minus eo, quod velocitate ma­xima.

AX. III.

TEmpus, quo curritur &longs;patium iuxta quamlibet tem­poris imaginem, maius e&longs;t eo, quo idem &longs;patium curreretur maxima velocitate, &longs;ed contra minus eo altero, quo ip&longs;um &longs;patium minima velocitate exigeretur, earum videlicet, quæ &longs;unt in gene&longs;i, aut imagine velocitatum pro­po&longs;iti motus, cuius nempe illa e&longs;t imago temporis. Fit er­go, vt tempus æquale ei, quo illud ip&longs;um &longs;patium currere­tur iuxta propo&longs;itam imaginem, &longs;it inter vtrumque dicto­rum temporum maximi, & minimi.

AX. IV.

QVæcunque excogitetur figura plana, vel e&longs;t paralle­logrammum, vel acuminata figura, aut ex his com­po&longs;itum. Has tamen figuras inter binas volu­mus parallelas, ita vt vnum latus &longs;it ip&longs;as nectens normali­ter parallelas, quanquam etiam loco parallelarum po&longs;&longs;int e&longs;&longs;e puncta, nempè vbi de&longs;inunt in acuminatas pror&longs;us figuras.

PROP. I. THEOR. I.

TEmpora, quibus duo motus complentur &longs;unt in ra­

tione imaginum homogenearum ip&longs;orum temporum.

Tab. 1. Fig. 5.

Motus &longs;int primò æquabiles, curratque mobile &longs;patium AB tempore, cuius imago CAB, curratur item ab alio mo­bili &longs;patium DE tempore, cuius imago DEF, & &longs;int ip&longs;æ temporum imagines inter&longs;e homogeneæ, &longs;cilicet FD ad AC eandem habeat rationem, quam velocitas in A ad velocitatem in D. Dico, tempus per AB ad id per DE e&longs;­&longs;e vt figura ABC, ad DEF. Cum motus æquabiles &longs;int erunt figuræ dictarum imaginum rectangula, propterea il­lorum ratio componetur ex rationibus altitudinum AB ad DE, & ba&longs;ium AC ad DF, ex ij&longs;dem verò rationibus &longs;pa­tiorum &longs;cilicet, & reciproca velocitatum (&longs;unt enim ima­gines inter &longs;e homogeneæ) nectitur etiam ratio temporum, quibus percurruntur ip&longs;a &longs;patia AB, DE iuxta gene&longs;es ima­ginum ACB, DEF, ergo e&longs;t eadem ratio inter illa tempo­ra, ac inter imagines &longs;uas.

Def. 4. huius.

Cor. Def. 3. huius.

Gal. pr. S de motu æquab. Def. 4. huius.

Tab. 1. fig. 6. Def. 5. huius.

2. Sit motus vnus æquabilis, alter verò quicunque; &longs;it tamen imago huius temporis figura acuminata vt ALGE, & alterius temporis prædicti motus æquabilis, &longs;it HFM,

Cor. Def. 3. huius.

Cor. Def. 3. huius.

Ex pramiß&atail; parte.

Euang. Tor­ric. lem. 18. in libro de dim. parabolæ.

Ax. 3. huius.

3. Imagines propo&longs;itæ &longs;int duæ acuminatæ. Dico ni­hilominus, tempora iuxta illas imagines per AE, HI e&longs;&longs;e vt ip&longs;æ imagines ALGE ad HIK, quæ &longs;int inter &longs;e homoge­neæ vt &longs;emper &longs;upponetur. Nam &longs;i intelligatur alius mo­tus per MF iuxta imaginem rectangulum MFN, qui æqua­bilis erit, manife&longs;tum e&longs;t ex &longs;ecundo ca&longs;u, tempus per AE iuxta imaginem ALGE ad tempus per FM iuxta imaginem rectangulum MH, habere eandem rationem, quam imago ALGE ad imaginem rectangulum MH; & &longs;imiliter tem­pus per FM imagine rectangulum MN ad tempus per HI iuxta imaginem HKI habet eandem rationem, quam ima­go NM ad imaginem HKI, ergo ex æquali tempus per AE ad tempus per HI &longs;ecundùm imagines propo&longs;itas erit vt imago ip&longs;a ALGE ad imaginem HKI. Quod &c.

Tab. 1. Fig. 7.

Cor. Def. 3 huius.

Tab. 1 fig.

Ax. 4. huius.

Def. 4. huius.

Def: 4. huius.

Ex tertia parte huius.

Ex 2. part&etail; huius.

Corollarium.

Hinc colligitur, &longs;i prima magnitudo ad &longs;ecundam fuerit vt tertia ad quartam, item alia prima ad aliam &longs;ecundam vt alia tertia ad aliam quartam, & &longs;ic vlteriùs quoad vi&longs;u&mtail; fuerit, &longs;int præterea omnes primæ, item omnes tertiæ inter&longs;e æquales, con&longs;tat, inquam, primarum vnam ad omnes &longs;ecun­das habere eandem rationem, ac vna tertiarum ad omnes quartas.

PROP. II. THEOR. II.

SSpatia, quæ curruntur iuxta qua&longs;cunque homogeneas velocitatum imagines, &longs;unt inter&longs;e, vt eædem illæ ima­gines. Sint primùm motus æquabiles, curraturque &longs;pa­tium AB iuxta imaginem velocitatum, quæ rectangulum erit ILMK, &longs;patium verò DE tran&longs;igatur iuxta imagine&mtail; prædictæ homogeneam rectangulum FHNG (nam erunt homogeneæ ip&longs;æ imagines, &longs;i vt ex Def. 4. huius IL ad HF erit vt velocitas in&longs;tanti I ad velocitatem mobilis in&longs;tanti F) Dico &longs;patium AB ad DE e&longs;&longs;e vt imago rectangulu&mtail; ILMK ad imaginem rectangulum FHNG. Componuntur ip&longs;a illa rectangula ex ratione altitudinum IK ad FG, & ex ea ba&longs;ium IL ad FH; verùm ex ij&longs;dem, ea nempe temporum IK ad FG, atque ea velocitatum IL ad FH componitur etiam ratio &longs;patiorum AB ad DE, ergo ip&longs;a &longs;patia erunt vt propo&longs;it&ecedil; imagines.

Tab. 1. Fig. 9. Cor. Dif. 3. huius.

Gil. de motu æquabili.

Tab. 1. fig 10.

2. Sint nunc motus iuxta imagines, quarum altera acu­minata, altera rectangulum &longs;it. Dico rur&longs;us &longs;patium AB, quod curritur iuxta imaginem ABCD ad &longs;patium DE, quod curritur iuxta alteram imaginem, e&longs;&longs;e vt imago ABCD ad imaginem PHNG. Ni&longs;i ita &longs;it, erit alia magni­tudo Y maior, vel minor imagine ABCD, quæ quidem ad alteram imaginem HPGN habebit eandem rationem, quam &longs;patium AB ad DE. Sit primùm maior exce&longs;&longs;u Z. Cir­cum&longs;cribatur; vt egimus in &longs;ecunda parte primæ huius, fi­gura imagini ABCD con&longs;tans ex rectangulis æquè altis, excedatque imaginem ABCD exce&longs;&longs;u minori, quam Z; &longs;it ergo circum&longs;cripta illa AE, HF, IG, KG, quam primò fa­cilè o&longs;tendemus minorem magnitudine Y; nam hæc exce&longs;­&longs;u magis di&longs;tat ab imagine, quàm circum&longs;cripta illa. Præ­terea &longs;i intelligantur tot motus æquabiles, quot &longs;unt rectan­gula circum&longs;cripta, ij nempe, qui fierent temporibus AH, HI, IK, KD iuxta deinceps imagines ip&longs;a rectangula AE, HF, IG, KC inter&longs;e, & propo&longs;itis imaginibus homogeneas, velocitates, quibus ijdem motus con&longs;iderarentur, forent HE, IF, KG, DC, nimirum maximæ imaginum ABEH, HEFI, IFGK, KGCD; Cumque ita &longs;it, longiora &longs;patia cur­

rerentur iuxta imagines rectangula circum&longs;cripta, quam ij&longs;dem temporibus, imaginibu&longs;que po&longs;tremis, hoc e&longs;t quam tempore AD iuxta imaginem ABCD; obidque &longs;patium AB ad DE, &longs;eu magnitudo Y ad imaginem HPGN habe-bit minorem rationem, quàm omnes illæ &longs;imul imagines, &longs;eu quam circum&longs;cripta figura AE, HF, IG, KC ad ean­dem imaginem HPGN; quare Y, quæ priùs o&longs;ten&longs;a fuit maior, nunc reperitur minor eadem circum&longs;cripta, quod cum fieri nequeat, impo&longs;&longs;ibile etiam e&longs;t magnitudinem Y maiorem e&longs;&longs;e magnitudine imaginis ABCD. Sit ergo mi­nor, &longs;i etiam fieri pote&longs;t, & defectus ip&longs;ius Y &longs;upra ABCD &longs;it Z. In&longs;cribatur imagini figura ex rectangulis æquealtis, vt nempe deficiat ab imagine defectu minori Z; &longs;ic enim ip&longs;a in&longs;cripta, quæ &longs;it AB, IE, KF, DG erit magnitudine pro­pinquior imagini ABCD, quàm Y, ideoque Y minor erit dicta in&longs;cripta figura. Deinde, quoniam, &longs;i ponantur mo­tus æquabiles, quorum imagines rect angula in&longs;cripta HB, IE, KF, DG, quæque inter &longs;e, & propo&longs;itis imaginibus &longs;int homogeneæ; velocitates, quibus efficerentur dicti motus, e&longs;&longs;ent AB, IE, KF, DG, minimæ &longs;cilicet imaginum ABEH HEFI, IFGK. KGCD, & ideo &longs;patia, quæ percurrerentur temporibus HA, HI, IK, KD imaginibus illis, maiora e&longs;­&longs;ent, quàm quæ ij&longs;dem temporibus tran&longs;igerentur iuxt&atail; imagines prædictas rectangula circum&longs;cripta, hinc fit vt &longs;patium AB ad DE, &longs;eu magnitudo Y ad imagine HPGN habeat maiorem rationem, quàm in&longs;cripta figura ad ean­dem imaginem HPGN; quare Y, quæ minor erat in&longs;cripta figura, modò re&longs;ultat maior, non ergo Y minor e&longs;&longs;e pote&longs;t imagine ABCD, &longs;ed neque maior vt o&longs;tendimus, ergo &longs;pa­tium AB ad DE erit, vt imago ABCD ad imaginem PHNG. Quod &c.

Ax. 2. huius.

Cor. pr. 1. hu­ius.

Ex. 2 huius.

3. & 4. Si verò imagines acuminatæ &longs;int, aut demum quæ cumque, eodem prorsùs modo, quo prima propo&longs;itio­ne, o&longs;tendemus hoc etiam propo&longs;itum, ergo patet omne intentum.

Scholium.

Cum prorsùs geometricè o&longs;tenderimus &longs;uperiores duas pro­po&longs;itiones, vtili&longs;&longs;imum e&longs;t ob&longs;eruare, quomodo liceat vti tem­poris in&longs;tantibus, non vt punctis prorsùs geometricis, &longs;ed vt quantitatibus dicam minoribus quibu&longs;cunque datis. Hinc oritur indiui&longs;ibilium methods, quæ intelligentiam affert faciliorem, ac &longs;i rigori geometrico penitus in&longs;i&longs;teremus, quam­quam eæ tamen difficiliores Geometras mihi magis decer&etail; videantur.

PROP. III. THEOR. III.

Tab. 2, Fig. 1.

SPatia, quæ curruntur iuxta quaslibet homogeneas ve­locitatum imagines, nectuntur ex rationibus tempo­rum, ac æquatricum.

Velocitates æquatrices duorum motuum, quorum ima­gines velocitatum &longs;int ABCD, EFHI ponantur AG, EL. Dico &longs;patia, &longs;eu ip&longs;as imagines componi ex ratione tem­porum AD ad EI; & ex ea æquatricum AE ad EL. Nam &longs;i motus, qui e&longs;t iuxta imaginem ABCD per&longs;eueret velo­citate AG, e&longs;&longs;et quidem æquabilis, idemque &longs;patium illa

velocitate, & tempore AD percurreretur, ac &longs;ecundù&mtail; imaginem ABCD; Itaque exi&longs;tente rectangulo DE, quod e&longs;set imago velocitatum illius motus æquabilis, foret idem æquale imagini ABCD (nam imagines ABCD, & DG homogeneæ &longs;unt) eodem modo imago rectangulum VL æquale e&longs;set imagini EFHI. Cum ergo duæ imagines re­ctangula DE, IL componantur ex rationibus temporum AD ad EI, & ex ea æquatricum AG ad EL; ex ij&longs;de&mtail; prorsùs rationibus etiam imagines propo&longs;itæ prædictis re­ctangulis æquales nectentur. Et ideo &longs;patia, quæ propo­&longs;itis imaginibus tran&longs;iguntur, quæque ip&longs;is proportionalia &longs;unt, componentur ex rationibus temporum, & ex rationi­bus æquatricum.

Def. 6. Ax. 1.

Cor. 3. Def. 3. huius.

Pr. 2. huius.

Corollarium I.

Hinc patet &longs;i lineæ, quæ in imagine velocitatum tempus ex hibet, aplicetur rectangulum æquale propo&longs;itæ imagini ve­locitatum, fore vt latitudo eiu&longs;dem rect anguli, &longs;it velocitas æquatrix propo&longs;itæ imaginis.

Corollarium II.

Item const at, vbi tempora, vel æquatrices velocitates fue­rint æquales, rationem &longs;patiorum e&longs;&longs;e eandem, quæ æquatri­cum, vel quæ temporum.

LEMMA.

Si quælibet ratio compo&longs;ita &longs;it ex quotcumque rationibus, harum quælibet nectetur ex propo&longs;ita, & ex reliquis contra­riò &longs;umptis rationibus. Sit A ad B compo&longs;ita ex rationibus E æd F; G ad H; & I ad K. Dico quamlibet ist arum puta G ad K con&longs;tare ex rationibus A ad B, & ex reliquis reciprocè &longs;um­ptis F ad E, & I ad K. Vt E ad F, ita &longs;it A ad C, & vt D ad B &longs;ic I ad K; erit C ad D, vt G ad H; ideoque C ad D, hoc e&longs;t G adH nectetur ex C ad A, &longs;eu F ad G, & ex rationibus A ad B, B ad D, &longs;iue K ad I. Quod &c.

AECFI.KDGBH

PROP. IV. THEOR. IV.

TEmpora, quibus ab&longs;oluuntur duo motus componun­tur ex ratione &longs;patiorum, & ex reciproca æquatri­cum. Cum enim &longs;patia componantur ex ratione temporum, & ex ea velocitatum æquatricum, &longs;equitur per prædictum Lenima, quòd tempora nectantur ex rationibus &longs;patiorum, & reciproca æquatricum.

Pr. 3. huius.

Corollarium.

Manife&longs;tum e&longs;t &longs;patia, vel æquatrices velocitates, &longs;i &longs;int æquales, e&longs;&longs;e tempora in reliqua ratione reciproca æquatri­cum, vel &longs;patiorum non reciproca.

PROP. V. THEOR. V.

ÆQuatrices velocitates componuntur ex rationibus &longs;patiorum, & reciproca temporum.

Cum &longs;patia componantur ex rationibus temporum, & velocitatum æquatricum, manife&longs;tum e&longs;t ex eodem Lem­mate, velocitates ip&longs;as necti ex rationibus &longs;patiorum, & reciproca temporum.

Corollarium.

Deducitur, æquatrices velocitates e&longs;&longs;e vt tempora reciprocè &longs;umpta, vel vt &longs;patia, &longs;i altera ratio fuerit æqualitatis.

D. E F. VII.

Tab. 2. Fig. 2.

SI in gene&longs;ibus homogeneis AEC, GFK exi&longs;tente AB ad BC &longs;icut GI ad IK, habeat AE ad BD eandem ra-tionem, ac GF ad IH, motus, qui fiunt iuxta illas gene&longs;es, vocentur inter &longs;e &longs;imiles, & ip&longs;æ gene&longs;es dicentur &longs;imilium motuum; quod verò attinet ad rectas AE, BD, GF, IH apel­labimus applicatas ad homologa puncta A, B, G, I propor­tionales.

PROP. VI. THEOR. VI.

SI in imaginibus temporum homogeneis, applicatæ v­nius fuerint ad homologa puncta, proportionales ap­plicatis alterius imaginis, motus, quorum &longs;unt ip&longs;æ imagi­nes, &longs;imiles erunt.

Imagines temporum &longs;int &MLABC, &ONGIK, quæ

&longs;int homogeneæ, & cum GI ad IK &longs;it vt AB ad BC, habeat quoque AL ad BM eandem rationem, ac GN ad IO. Di­co, motus, quorum &longs;unt illæ imagines temporum inter &longs;e &longs;i­miles e&longs;&longs;e.

Tab. 2. fig. 3.

Sint apud ip&longs;as imagines eorundem motuum gene&longs;es, &longs;cilicet EAC, FGK inter&longs;e homogeneæ. Exi&longs;tente AL ad BM, vt GN ad IO, erit conuertendo BM ad AL vt IO ad GN; &longs;ed vt BM ad AL ita ob gene&longs;im EA ad DB, & vt IO ad GN, &longs;ic FG ad HI. ergo EA ad DB e&longs;t vt FG ad HI, erat autem vt AB ad BC ita etiam GI ad IK, ergo motus &longs;unt &longs;imiles, & ip&longs;æ imagines &longs;imilium motuum.

Def. 2. huius.

Def. 7. huius.

PROP. VII. THEOR. VII.

SI in imaginibus velocitatum vnius, applicate fuerint ex punctis homologè &longs;umptis proportionales applicatis alterius imaginis, motus iuxta ip&longs;as imagines erunt &longs;imi­les, ideoque ip&longs;æ imagines &longs;imilium motuum.

Tab. 2. fig. 4.

Pr. 2. huius.

PROP. VIII. THEOR. VIII.

SPatia, quæ curruntur &longs;imilibus motibus &longs;unt in ratione compo&longs;ita temporum, & homologarum velocitatum, interquas &longs;unt extremæ, aut primæ.

Tab. 2. Fig. 5

Pr. 2. huiu.

Corollarium.

Si tempora fuerint æqualia, &longs;imilium motuum &longs;patia erunt vt extremæ, vel &longs;ummæ velocitates, & contra, &longs;i i&longs;tæ æquales &longs;int, erunt &longs;patia vt tempora.

Corollarium II.

Cum &longs;patia &longs;imilium motuum nect antur ex ratione tem­porum & ex ea velocitatum &longs;ummarum, &longs;eu earum, quæ sunt ad in&longs;tantia &longs;imiliter &longs;umpta in rectis BE, GI, const at ex lem: infra cor. 2. pr. 3. huius tempora componi ex rationi­bus &longs;patiorum &longs;imilium motuum, & ex recìproca dictarumvelocitatum. Ex eadem ratione patet e&longs;&longs;e velocitates &longs;um­mas, vel homologas vti diximus in ratione compo&longs;ita dicto­rum &longs;patiorum, & ip&longs;orum temporum.

Corollarium III.

Quare &longs;i alteræ de dubus componentibus æqualis fuerit, reliqua tantùm computanda erit.

Scholium.

Hinc emergit nis ferè doctrina grauium cum de&longs;cendunt pror&longs;us libera, aut &longs;uper planis inclinatis ad horizonte&mtail;: nec accidit veritates iam patefact as huc rur&longs;us lectoris tadio afferre, &longs;ed libcat potius, rationem metiendarum imaginum, quamuis longitudine immen&longs;arum, no&longs;tra methodo exponere.

DEF. VIII.

Tab. 2. Fig. 6.

SInt inter binas parallelas AB, GH, et IK, PQ planæ fi­guræ ABHG, IKQP, & in altera earum ducta altitudi­ne RV, &longs;int inter &longs;e ip&longs;æ figuræ talis naturæ, vt cum &longs;it GABH ad &longs;egmentum EABF factum per æquidi&longs;tantem ip&longs;i GH &longs;icut VR ad RT, verificetur &longs;emper (ducta æqui­di&longs;tanti NTO ip&longs;i PQ) e&longs;&longs;e GH ad EF vt reciprocè NO ad Pque tunc huiu&longs;modi figuras vocabimus inter &longs;e auuer&longs;as.

Corollarium.

Sequitur ex vi nunc allatæ deffi, lineam IK tunc e&longs;&longs;e in­finitam, cum AB fuerit punctum, & ideo &longs;imul con&longs;tat figu­ram IPQK immen&longs;am e&longs;&longs;e longitudine versùs K aut I, aut vtrinque, &longs;i nempe producerentur nunquam couræ lineæ QP, IK.

PROP. IX. THEOR. IX.

REctangulum &longs;ub altitudine, & ba&longs;i vnius auuer&longs;arum ad ip&longs;am auuer&longs;am figuram, eandem habet rationem, ac altera auuer&longs;a figura ad rectangulum ex ba&longs;i in altitudi­

nem eiu&longs;dem huius figuræ.

Tab. . fig. 7.

Sint auuer&longs;æ figuræ ACB, GFDEG. Dico rectangu­lum DF in DE ad figuram GFDEG, eandem habere ratio­nem ac figura ACBA ad rectangulum AB in BC. Sint pri­mùm ABC, FDE anguli recti, & ducta qualibet HI paral­lela BC, &longs;it BAC ad HIA vt DF ad KF, erit ob naturam auuer&longs;arum KL ad DE vt BC ad HI; itaque &longs;i ponatur e&longs;&longs;e quidam motus ab F in D iuxta imaginem velocitatum BAC, erit GFDEG imago temporis eiu&longs;dem motus; nam imago BAC ad imaginem HIA e&longs;t vt &longs;patium DF ad &longs;patium FK & velocitas BC ad velocitatem HI vt reciprocè KL ad DE. Sit etiam alius motus, &longs;ed æquabilis, cuius imago velocita­tum æqualis &longs;it, & homogenea ip&longs;i BAC, rectangulum nen­pe AB in BM, & ideo &longs;i fiat BM ad BC &longs;icut DE ad DN, concipiaturque rectangulum FD in DN, erit hoc imago temporis dicti motus æquabilis, homogenea, & æqualis imagini GFDEG; nam tempora, &longs;cilicet imagines GFDEG, FD in DN rectangulum componuntur ex rationibus &longs;pa­tiorum, hoc e&longs;t imaginum velocitatum inter&longs;e æqualium, ABM, ACB, & reciproca æquatricum pariter æqualium BM, BM. Cum igitur rectangulum FD in DN æquale &longs;it imagini, &longs;eu figuræ GFDEG, habebit eadem figur&atail; GFDEG ad rectangulum FD in DE eandem rationem, quam DN ad DE, hoc e&longs;t quam BC ad BM, &longs;eu quam re­ctangulum AB in BC ad rectangulum AB in BM, aut ad ei æqualem figuram ABC; & conuertendo, manife&longs;tum e&longs;t quod propo&longs;uimus, nempe rectangulum FD in DE ad fi­guram GFDEG habere eandem rationem, ac figura ACBA ad rectangulum AB in BC. quod erat demon&longs;trandum primo loco.

Def. 8. huius.

Def: 2. huius.

pr. 2. huius.

Def. 2. huius.

pr. 1. huius.

pr. 4. huius.

Cor. pr. 3. hu­ius.

2. Si verò propo&longs;itæ figuræ &longs;int quæcunque auuer&longs;æ DAE, QPLMQ poterunt hæ reuocari ad qua&longs;dam alias FKG, RSZX, quæ &longs;int inter ea&longs;dem parallelas, queis com­prehenduntur propo&longs;itæ figuræ, ad eo vt exi&longs;tentibus re­ctis angulis KFG, RXZ &longs;int ip&longs;æ binæ figuræ ab ij&longs;dem pa­rallelis interceptæ. inter &longs;e æqualiter analogæ hoc e&longs;t du­ctis æquidi&longs;tantibus, vt vi&longs;um fuerit IHBC, VTNO, &longs;int &longs;emper interiectæ lineæ IH, BC, & VT, NO æquales: hoc modo non tantùm liquet figuras FKG, DAE, nec no&ntail; RSZX, PQML æquales inter &longs;e e&longs;&longs;e, verùm etiam FKG ad IKH e&longs;&longs;e in eadem ratione, in qua QPLMQ ad QPNOQ, quamobrem ex prima parte, rectangulum ZX in RM ad figuram SRXZS, hoc e&longs;t rectangulum LM in altitudinem figuræ QPLMQ ad hanc ip&longs;am figuram habebit eandem rationem, quam figura FKG ad rectangulum KF in FG, vel quam figura DAE ad rectangulum DE in altitudinem eiu&longs;dem huius figuræ DAE; quo circa con&longs;tat omne pro­po&longs;itum.

Tab. 2. Fig. 8.

Corollarium.

Patet in prima parte repertum e&longs;&longs;e rectangulum FD i&ntail;DN æquale figuræ GFDEG, licèt hæc immen&longs;e longitudinis &longs;it versùs G, & ob id manife&longs;tum e&longs;t, quòd quamuis aliqu&atail; figura &longs;it &longs;inè fiue longa, non ideo &longs;emper magnitudinem ha­bet infinitam. Et &longs;imul illud con&longs;tat, vbi vna auuer&longs;arum, &longs;eu vbi imago velocitatum, aut temporis &longs;it magnitudine termi­nata, etiam altera auuer&longs;arum, vel imaginum erit huiu&longs;­modi &c.

Cor. pr. 18. huius.

PROP. X. THEOR. X.

Tab. 2. Fig. 9.

PROP. XI. THEOR. XI.

II&longs;dem adhuc manentibus, idem de Angelis mon&longs;trat eo­dem illo tractatu pr. 3. &longs;i quæcunque ex dictis parabo­lis &longs;ecta &longs;it qualibet recta parallela ba&longs;i BC, e&longs;&longs;e parabolam ad re&longs;ectam portionem ver&longs;us verticem, vt pote&longs;tas ba&longs;is, cuius exponens e&longs;t numerus parabolæ vnitate auctus ad &longs;imilem pote&longs;tatem ex ba&longs;i re&longs;ectæ portionis; itaque i&ntail; prima parabola e&longs;t vt quadratum ad quadratum, in &longs;ecun­da vt cubus ad cubum, & &longs;ic de cæteris. Similiter &longs;i &longs;ece­tur quodlibet ex infinitis trilineis linea GF ba&longs;i CD paral­lela, erit trilineum ad &longs;uperius &longs;ui &longs;egmentum vt pote&longs;tas ex DA, cuius exponens e&longs;t numerus trilinei vnitate auctus ad &longs;imilem pote&longs;tatem ex AF. quare trilineum primu&mtail; CAD ad GAF erit vt quadratum ex DA ad quadratum ex FA, &longs;ecundum CHAD ad &longs;egmentum HAF vt cubus ad cubum, & ita in cæteris eodem ordine.

PROP. XII. THEOR. XII.

Tab. 3. fig. 1.

SIt modò ACD angulus rectus, & linea FE talis naturæ, vt deductis ad libitum rectis AF, BE parallelis ip&longs;i CD, pote&longs;tas ex CA ad &longs;imilem pote&longs;tatem ex CB &longs;it reci­procè vt alia quædam pote&longs;tas ex BE ad &longs;imilem huic po­te&longs;tatem ex AF; patet rectas CA, CD nondum iungi cum EF, quamuis in immen&longs;um vnà producerentur. Ab hoc proprietate VValli&longs;ius & Fermatius &longs;ubtili&longs;&longs;imi authores vocauerunt curuam FE nouam hyperbolam, & eius a&longs;­&longs;ymptotos AC, CD. Omnes huiu&longs;modi hyperbolæ, quæ infinitæ numero &longs;unt, terminantur ad vnam partem ma­gnitudine, cum hyperbola communis, &longs;eu Apolloniaca &longs;it in vtranque partem magnitudine infinita. Quod ergo exi­mium e&longs;t, o&longs;tenderunt ip&longs;i authores rectangulum FA i&ntail; AC ad &longs;patium hyperbolicum quà finitum e&longs;t, licèt &longs;inè fine longum, eandem habere rationem, quam differentia exponentium pote&longs;tatum hyperbolæ ad exponentem po­te&longs;tatis minoris. Quare &longs;i in hyperbola &longs;it vt cubus CB ad cubum CA ita quadratum AF ad quadratum BE, erit prædictum rectangulum CA in AF dimidium Spatij &longs;inè fine producti A & FA; at &longs;i quadratum CB ad quadratum CA &longs;it vt recta AF ad rectam BE, rectangulum ip&longs;um CA in AF æquale erit &longs;patio A & FA, quòd &longs;i pote&longs;tas CA vel CB non fuerit altior pote&longs;tate ex BE, vel AF, tunc ip&longs;um illud &longs;patium, infinitum quoque erit magnitudine, etenim nullus exce&longs;&longs;us exponentis prædictæ pote&longs;tatis ex CA &longs;u­pra exponentem pote&longs;tatis BE, habet ad numerum expo­nentis pote&longs;tatis BE rationem infinitam.

DEMONSTRATIO.

SVpradictum propo&longs;itum habetur in commercio epi­&longs;tolico Ioannis Valli&longs;ij Epi&longs;tola quarta, quem libellum vnà cum alijs docti&longs;&longs;imis &longs;uis operibus Vincentius Viuia­nus ingens æui no&longs;tri Geometra, antequam &longs;umma cu&mtail; humanitate mi&longs;i&longs;&longs;et, eidem ip&longs;i quadraturam vnius ex di­ctis hyperbolis ex no&longs;tris principijs deductam, ac excogi­tatam, indicauimus. Cum verò po&longs;tea nobis eueni&longs;&longs;et vniuer&longs;aliorem ad alias hyperbolas (&longs;emper communi ex­cepta) accomodatam reperij&longs;&longs;e, huc debemus afferre, pri­mùm vt quendam fructum &longs;cientiæ huius; deinde cum di­ctorum authorum ip&longs;am propo&longs;itionis demon&longs;trationem non habuerimus, & demum quia ip&longs;arum hyperbolarum men&longs;ura, ac quadratura in aquarum rationibus erunt po­ti&longs;&longs;imum ex v&longs;u. Sit igitur BC vna ex infinitis hyperbolis,

quarum a&longs;lymptoti AE, EL; Sint etiam quæcunque apli­catæ AB, DC alsymptoto EL æquidi&longs;tantes, & habeat DE ad EA eandem rationem v. g. quam cubus ex AB ad cubum DC. Patet &longs;i proponeretur illi auuer&longs;a figur&atail; FGK, e&longs;&longs;etque AE ad DE vt figura GFK ad figuram IHK e&longs;&longs;e etiam FG ad IH vt DC ad AB, e&longs;t autem cubus ex DC ad cubum ex AB vt AE ad ED; ergo etiam figur&atail; FGK ad IHK (&longs;unt enim FG, IH parallel&ecedil;) habebit ean­dem rationem, ac cubus ex FG ad cubum ex IH: Itaqu&etail; GFK erit comunis parabola, hoc e&longs;t quadratica, &longs;eu &longs;ecun­da in &longs;erie infinitarum parabolarum, & ob id eadem GFK parabola ad rectangulum GF in FK erit vt 2 ad 3, in qua ratione &longs;e habebit quoque rectangulum BA in AE ad &longs;pa­tium infinitè longum & BM, et erit vt 2 ad 1; &longs;cilicet vt ex­ce&longs;&longs;us exponentis maioris pote&longs;tatis, quæ cubica e&longs;t, &longs;uper numerum exponentis, qui hoc ca&longs;u e&longs;t tantùm vnitas ra­dicis, e&longs;t ad hunc ip&longs;um exponentem, &longs;eu vnitatem lineæ indicantem, quod concordat cum propo&longs;ita dictoru&mtail; authorum.

Tab. 3. Fig. 3.

Tab. 3. fig 2.

Def. 8. huius.

Pr. 10. huius.

Pr. 9. huius.

Exemplum aliud.

Def. 8. huius.

Pr. 10. huius.

Pr. 10. huius.

Pr. 9. huius

PROP. XIII. THEOR. XIII.

SVperior demon&longs;tratio effecta fui&longs;&longs;et ampli&longs;&longs;ima, &longs;i pr&ecedil;­ponere volui&longs;&longs;emus quadraturam vt datam omnis ge­neris parabolarum, & trilineorum, verùm cum i&longs;ta pars non &longs;it plenè tradita, vt videre e&longs;t quinto libro infinitarum pa­rabolarum eiu&longs;dem de Angelis, &longs;atius ideo duximus qua­draturam hyperbolarum à VVali&longs;io, & Fermatio acuti&longs;&longs;i­mis illis viris propo&longs;itam omnino veram admittere, vt indè eam parabolarum & trilineorumvniuer&longs;alem, quam adhuc ab alijs non habemus, facillimè, compendiosèque depro­meremus. Hanc igitur ita proponimus vt &longs;ubinde o&longs;ten­damus.

Si &longs;imiles pote&longs;tates applicatarum fuerint in eadem ra­tione, ac &longs;unt inter&longs;e pote&longs;tates quædam aliæ, & eiu&longs;dem gradus diametrorum ab ip&longs;is applicatis ab&longs;ci&longs;&longs;arum v&longs;que ad verticem parabolarum, vel trilineorum; erit rectangu­lum ad parabolam &longs;ibi in&longs;criptam vt aggregatum exponen­tium vtriu&longs;que pote&longs;tatis ad exponentem altioris ip&longs;arum pote&longs;tatum parabolæ; & ad trilineum vt aggregatum ex­ponentium pote&longs;tatum trilinei ad exponentem inferioris pote&longs;tatis eiu&longs;demmet trilinei. Sic enim in expo&longs;ita figu­ra prædicta, &longs;i e&longs;&longs;et quadratum ex FG ad quadratum ex IH, &longs;icut cubus ex FK ad cubum ex IH, e&longs;&longs;et rectangulum GF in FK ad figuram GFK (quæ tunc foret trilineum, vt 5 ad 2; nam vbi pote&longs;tas ab&longs;ci&longs;&longs;arum maior e&longs;t illa applica. tarum e&longs;t &longs;emper GF trilineum. Simili modo, &longs;i &longs;it vt qua­dratum ex FK ad quadratum ex KI ita cubocubus ex FG ad cubocubum ex IH; hoc e&longs;t &longs;i &longs;it cubus ex FG ad cubum ex IH, vt linea FK ad KI (tolluntur enim vtrinque ex &longs;imi­libus &longs;imiles rationes) erit &longs;igura GFK parabola, ad quam &longs;ibi circum&longs;criptum rectangulum eandem habebit rationem, quam 4 ad 3, & &longs;ic dicendum erit de omnibus alijs para­bolis atque trilineis.

DEMONSTRATIO.

Def. 8. huius.

Pr. 12 huius.

Pr. 9. huius.

Corollarium.

Con&longs;tat &longs;i fuerit ratio A ad B eò &longs;ubmultiplicata rationis applicatarum, quoties e&longs;t numerus exponentis pote&longs;tatis ab­&longs;ci&longs;&longs;arum eiu&longs;dem parabolæ, e&longs;&longs;e ip&longs;am parabolam ad &longs;ui por­tionem in tam multiplicata ratione A ad B, ac e&longs;t numerus aggregati exponentium ambarum pote&longs;tatum parabola. Nam cum e&longs;&longs;et quadratocubus ex FG ad quadratocubum ex IH, &longs;i­cut cubus ex FK ad cubum ex IK, propo&longs;ita in&longs;uper e&longs;&longs;et A ad B. &longs;ubquindecupla alterius dictarum &longs;imilium rationum ex pote&longs;t atibus parabola, o&longs;ten&longs;um fuit rationem A ad B &longs;ubtri­plicatam ip&longs;ius GF ad IH, & &longs;ubquintuplicatam alterius FK ad KI, & tandem o&longs;tendimus parabolam GFK ad portionemeius HIK eße in octuplicata ratione eiu&longs;dem A ad B; quod idem omnino diceretur &longs;i figura GFK trilineum e&longs;&longs;et. Ratio autem A ad B dicetur impo&longs;terum logarithmica pote&longs;tatum parabolæ, &longs;eu trilinei, aut hyperbolæ.

ASSVMPTVM.

REliquum e&longs;t vt o&longs;tendamus, parabolam GFK ad portionem HIK e&longs;&longs;e vt rectangulum GF ad rectan­gulum HI in IK, &longs;cilicet e&longs;&longs;e in ratione compo&longs;ita ha&longs;ium, & altitudinum parabolarum, quod nempe &longs;ic o&longs;tendetur, Sit vt &longs;upra FGK parabola, eiu&longs;que portio IHK; exi&longs;tenti­bus verò applicatis FG, IH, fiat EG ad IE vt FK ad KI, &longs;it­que IE ba&longs;is, et K vertex parabol&ecedil; IEK &longs;imilis ip&longs;i GFK pa­tet propter &longs;imilitudinem figurarum, e&longs;&longs;e parabolam GFK ad parabolam IEK in eadem duplicata ratione FG ad IE, in qua nempe e&longs;t rectangulum GF in FK ad &longs;ibi &longs;imile re­ctangulum EI in IK, ob idque rectangulum GF in FK ad rectangulum EI in IK, cum &longs;int inter&longs;e vt parabola GFK ad parabolam EIK, hæc verò parabola ad ip&longs;am IHK habeat eandem rationem, ac IE ad IH; &longs;eu ob eandem altitudinem IK vt rectangulum EI in IK ad rectangulum HI in IK, erit ex æquali parabola GFK ad parabolam HIK vt rectangu­lum GF in FK ad rectangulum HI in IK. Quod &c.

Tab. 3. Fig. 2.

PROP. XIV. THEOR. XIV.

Tab. 2. fig. 3.

IN quacunque hyperbola (excepta &longs;emper conica) cu­ius a&longs;&longs;ymptoti EA, EM, &longs;i &longs;it pote&longs;tas applicatarum DC AB altior pote&longs;tate ab&longs;ci&longs;&longs;arum AE, ED (&longs;ic enim finit&atail; erit magnitudine &longs;ecundum eam a&longs;&longs;ymptoton, quæ appli­catis parallela e&longs;t) &longs;patium ip&longs;um hyperbolæ & BAE & ad &longs;ui portionem & CDE & habebit eandem rationem, ac rectangulum BAE ad rectangulum CDE, &longs;eu (a&longs;&longs;umpta ratione logarithmica A ad B pote&longs;tatum hyperbolæ) quam pote&longs;tas ex A, cuius exponens e&longs;t differentia exponentium pote&longs;tatum hyperbolæ ad &longs;imilem pote&longs;tatem ex B.

DEMONSTRATIO.

QVam rationem habet rectangulum BAE ad &longs;patium & BAE &, eandem habet rectangulum CDE ad

Pr. 12. huius.

PROP. XV. THEOR. XV.

SIab exponente pote&longs;tatis applicatarum hyperbol&ecedil; de­trahatur exponens minoris pote&longs;tatis ab&longs;ci&longs;&longs;arum, po­te&longs;tas reliqui exponetis erit applicatarum auuer&longs;æ figuræ, in ab&longs;ci&longs;&longs;is verò ade&longs;t vtrobique eadem pote&longs;tas. Itaque cum in &longs;uperiori hyperbola re&longs;idui exponentis pote&longs;tas quadratum e&longs;&longs;et, porrò in eius auuer&longs;a e&longs;&longs;et pote&longs;tas appli­catarum quadratica, & ab&longs;ci&longs;&longs;arum quadratocubica.

DEMONSTRATIO.

Tab. 3. Fig. 3.

ESto rur&longs;us hyperbola & BAE &, et &longs;icut dictum e&longs;t AE ad ED &longs;it in &longs;eptuplicata ratione logarithmicæ rationis A ad B, at DC ad AB in quintuplicata, videlicet quadratocubus ex AE ad quadratocubum ex DE eandem habeat rationem, ac quadratoquadratocubus ex DC ad &longs;imilem pote&longs;tatem ex AB; Dico in auuer&longs;a figura pote&longs;ta­tem aplicatarum e&longs;&longs;e quadratum, cuius exponens 2 e&longs;t dif­ferentia exponentium pote&longs;tatum hyperbolæ; pote&longs;tatem verò ab&longs;ci&longs;&longs;arum eandem e&longs;&longs;e, ab&longs;ci&longs;&longs;arum eiu&longs;dem hyper­bolæ. Sit vt &longs;upra FK ad KI vt hyperbola & BAE & ad & CDE &, hoc e&longs;t, &longs;it vt pote&longs;tas ex A, cuius exponens

e&longs;t differentia exponentium pote&longs;tatum hyperbolæ ad &longs;i­milem pote&longs;tatem ex B, & ideo FK ad KI erit duplicata ip­&longs;ius A ad B, &longs;ed DC ad AB eiu&longs;dem illius logarithmicæ quintuplicata; e&longs;tque in hac eadem ratione etiam GF ad IH; ergo cum duplicata huius &longs;it &longs;imilis quintuplicatæ KF ad KI (nam vtraque ratio continet decies A ad B) pater, quadratum ex FG ad quadratum ex IH e&longs;&longs;e eam pote&longs;ta­tem, quam propo&longs;uimus euenire in applicatis auuer&longs;æ, cum aliàs in ab&longs;ci&longs;&longs;is &longs;it vtrobique pote&longs;tas eadem, nempe qua­dratocubi. Quod &c.

Pr. 14. huius.

Corollarium.

Patet ex noto trilineo, vel parabola FGK e&longs;&longs;e in auuer&longs;a, &longs;cilicet in hyperbola & BAE & (quæ tunc e&longs;t &longs;emper magnitu­dine finita iuxta a&longs;symptoton EM &) pote&longs;tatem applicatarum, qua pro exponente habet &longs;ummam exponentium pote&longs;tatum parabolæ, aut trilinei; nam cum eßet in trilineo pracedentiquadratum ex FG ad quadratum ex IH vt quadratocubus ex FK ad quadratocubum ex IK, fuit equidem in hyperbol&atail; quadratoquadratocubus ex DC quadratoquadratocubum ex AB &longs;icut quadratocubus ex AE ad &longs;imilem pote&longs;tatem ex DE, &longs;cilicet inuariata pote&longs;tate ab&longs;cr&longs;arum in ambabus au­uer&longs;is. Quare ex pote&longs;tatibus notis vnius auuer&longs;arum fa­cilè inote&longs;cent pote&longs;tates alterius, atque etiam illius magnitu­do. Nunc redeamus ad motus, nouamque adhuc methodum, quam hoc loco re&longs;eruauimus, afferamus.

DEF. IX.

SIt quædam Gene&longs;is ACBH, cuius imago temporis & DCB &; item &longs;it FCBK gene&longs;is alterius motus ab eodem C in B; & actà rectà OIGE ip&longs;i AFCD parallel&atail;, ponantur CD, GE loco minimorum temporum, ita vt ten­pore CD, dum mobile ex C affectum velocitate CA, currat minimum &longs;patiolum indicatum per C, cui e&longs;t æqua­le &longs;patiolum aliud indicatum per G, quodque tran&longs;igitur tempore GE velocitate GD (nam vt e&longs;&longs;ent illa &longs;patia i&ntail; C, G æqualia, effectum fuit vt velocitas AC ad GD ean­dem reciprocè rationem haberet, ac tempus GE ad CD, id quod patet ex natura gene&longs;is ACBH, & imaginis & DCB &) et hic rur&longs;us notatu digni&longs;&longs;imum e&longs;t nulli errori obnoxium e&longs;&longs;e, quòd æquabiles in illis minimis &longs;patiolis intellexerimus motus, quamuis potius deberet videri, in ij&longs;dem interuallis reperiri innumeras, ac inæquales veloci­tates, queis nempe efficerentur motus inæquabiles, quòd gene&longs;es inæquabiles &longs;int. Cur i&longs;ta &longs;e ita habeant, hic non e&longs;t nobis di&longs;putandum, ego enim puto, non ex indiui&longs;ibili velocitates alijs &longs;uccedere, &longs;ed reuera minutulum tempo­ris con&longs;iderari debere antequam motus diuer&longs;imodè pro­cedat, nempe ac &longs;i velocitas, quæ &longs;uccedere debet priori, non ita &longs;it in promptu, aut non ita &longs;tatim mobile afficiat ad motum &longs;ibi proportionatum. Sed linquamus hæc alijs di&longs;­putanda: &longs;atis nobis &longs;it, methodum no&longs;tram, quoad no&longs;trum e&longs;t, demon&longs;trare. Ijs igitur vt &longs;upra propo&longs;itis, concipia­tur adhuc tempore CD velocitate FC &longs;patium exigi quod­dam, item aliud tempore EG, velocitateque GI, & &longs;ic per omnes qua&longs;cunque applicatas: quæritur, quod &longs;patiu&mtail; vltimò exactum e&longs;&longs;et, hoc e&longs;t quam rationem id haberet ad illud alterum &longs;patium, quod eodem tempore tran&longs;igitur iu ta gene&longs;im HACB, cuius imago temporis CD & B. I&longs;ti duo motus in exemplo e&longs;&longs;ent, &longs;i in quodam plano mo­ueretur formica, dum ip&longs;um planum vna eius extremitate immobili circumduceretur, Sic formica difficiliùs a&longs;cende­ret prout ip&longs;um planum magis ad horizontem erigeretur. Iam motus extremitatis plani circumactæ habet gene&longs;im ACBH, cuius temporis imago & DCB &, et altera gene&longs;is FCBK ribueretur motui formicæ, nam vt dictum e&longs;t varius motus formicæ pendet ex latione plani, ideò velocitates eiu&longs;dem (nam in plano immobili ponimus æquabiliter fer­ri) durant ij&longs;dem temporibus, quibus velocitates præcipuæ gene&longs;is ACBH. Sit denique LMSR imago velocitatum iuxta gene&longs;im ACBH, cuius temporis imago CD & B; pa­tet &longs;i &longs;it MP ad PS &longs;icut imago temporis CDEG ad ima­ginem & BGE &, fore LM ad PQ vt AC ad OG, & con­cepta etiam figura MNOTS inter parallelas LMN, RST ita vt &longs;it &longs;emper MN ad PO &longs;icut FC ad GI, nec non LM ad MN vt AC ad FC. (&longs;unt enim initio motuum in C, aut in&longs;tanti M, velocitates gene&longs;ium AC, CF, &longs;cilicet LM, MN; & in G, hoc e&longs;t in&longs;tanti P &longs;unt velocitates OC, GI; nimi­rum QP, PO) vocetur proinde gene&longs;is FCBK &longs;puria, ac ad&longs;tricta imaginitemporis & DCB &, cuius imago veloci­tatum MNTS pariter &longs;puria, homogenea tamen ip&longs;i legiti­æ LMSR.

Tab. 3. fig. 4.

PROP. XVI. THEOR. XVI.

SI &longs;int duo motus iuxta gene&longs;es legitimam, & &longs;puriam, erunt mobilium exacta &longs;patia, vt imagines inter&longs;e homogeneæ velocitatum, legitima ad &longs;puriam.

E&longs;to gene&longs;is legitima ACBH, cuius imago temporis & DCA &, & imago velocitatum MLRS. Sit etiam gene­&longs;is altera illi homogenea, &longs;ed &longs;puria, & ad&longs;tricta imagini temporis & DCB &, cuius imago velocitatum &longs;puria, prio­rique legitimæ homogenea NMST. Dico, &longs;patia iuxta has imagines tran&longs;acta e&longs;&longs;e vt ip&longs;æ imagines legitima LMSR ad &longs;puriam NMST. Cum temporis momenta M, P in­telligantur ex minimis temporibus, quæ proponi po&longs;&longs;unt, inter&longs;e æqualibus, & quibus æquabiliter perdurant ve­locitates, quas mobile &longs;ortitur in aduentu &longs;uo in punctis C, G, erit vt velocitas FC ad velocitatem GI &longs;ic inter&longs;e &longs;patia, quæ i&longs;tis velocitatibus, temporibu&longs;que illis æqua­libus percurrerentur, in qua ratione e&longs;t etiam NM ad OP. Deinde momento M peragerentur &longs;patia proportionalia velocitatibus FC, AC, &longs;eu rectis NM, ML, momento autem P &longs;patia proportionalia velocitatibus GI, GD, in qua ratione e&longs;t etiam OP ad PQ, & &longs;ic deinceps procedendo per &longs;ingula temporis MR momenta, adeo vt, cum &longs;patium velocitate FC exactum ad id veloci­tate CA, &longs;it vt NM ad ML, &longs;patium velocitate IG ad id exactum velocitate GD &longs;it vt OP ad PQ, & &longs;int præterea primæ inter&longs;e, hoc e&longs;t &longs;patia velocitatibus FC, GI tran­&longs;acta, proportionalia tertijs, &longs;patijs videlicet tran&longs;actis velocitatibus ML, PQ ergo vt omnes primæ ad omnes tertias quantitates, hoc e&longs;t omnia &longs;patia tran&longs;acta iuxta gene&longs;im FCBK ad omnia &longs;patia iuxta gene&longs;im ACB, ita erit &longs;umma &longs;ecundarum ad |omnes quartas, &longs;cilicet i&longs;ta erit imago NMST ad imaginem LMSR. Quod & c.

Tab. 3. Fig. 4.

Pr. 3. huius.

LIBER ALTER

DE

Motu Compo&longs;ito.

MOtum appellamus compo&longs;itum, vbi dum fer­tur mobile, con&longs;ideratur habere plures i&ntail; diuer&longs;as partes, vel etiam in eandem partem conatus, ex quibus oriatur tertia vis di&longs;tin­cta ab illis. Hunc librum, cum expleueri­mus, non pauca vnà cum priori, dicta erunt de motu, erit­que ea methodus, qua &longs;imul geometrica quædam, difficil­lima &longs;citu &longs;atis breuiter o&longs;tendemus. Nam vibrationes pendulorum exigi temporibus; quæ &longs;int in &longs;ubduplicat&atail; ratione longitudinum eorundem, planè tandem con&longs;tabit aliàs nobis di&longs;&longs;entientibus: aperiemus etiam, qua arte in­telligi queant anguli rectilinei curuilineis æquales; nec non exponemus parabolas quibu&longs;dam &longs;piralibus æquales, vt e&longs;t vulgata &longs;pirali Archimedeæ, cùm videlicet ba&longs;is para­bolæ radio circuli &longs;piralem continentis, & dimidium huius circumferentiæ circuli altitudini eiu&longs;dem parabolæ, æqua­les &longs;int.

PROP. I. THEOR. I.

Tab. 4. Fig. 1.

SI in eadem recta linea currantur &longs;patia temporibus æqualibus, & &longs;int motus &longs;implices, ac ad ea&longs;dem par­tes tendentes, eadem illa &longs;patia &longs;imul motu compo&longs;ito, ab eodemque mobili duabus illisgene&longs;ibus affecto, vnicoque ex dictis temporibus æqualibus, excurrentur.

Curratur LI iuxta imaginem velocitatum HAEF, et IO iuxta aliam dictæ homogeneam BAED. Dico LO &longs;um­mam dictorum &longs;patiorum LI, IO exactum iri vnico tem­pore AE, &longs;i nempe mobile feratur &longs;ecundum vtranque ima­ginem.

Per quodlibet punctum, &longs;eu temporis momentum M agatur recta GMC parallela HB, vel FD. Habebit mobi­le momento A, dum &longs;cilicet mouetur motu compo&longs;ito duas &longs;imul velocitates AH, AB, ide&longs;t vnicam HB. Similiter mo­mento M habebit GC, & momento E ip&longs;am FD. Itaque

erit HBDF imago velocitatum compo&longs;iti motus, qui fiet tempore AE iuxta imaginem, quæ aggregatum e&longs;t dictarum HAEF, ABDE. E&longs;t verò LI ad IO vt imago HAEF ad imaginem ABDE; ergo conuertendo, componendoqu&etail; erit vt LI ad LO, &longs;ic imago HAEF ad imaginem HBDF; propterea quemadmodum &longs;patium LI currebatur iuxt&atail; imaginem HAEF, &longs;ic LO percurretur imagine HBDF &longs;olo, eodemque tempore AE. Quod &c.

Def. 3. prima.

Corollarium.

Hinc patet graue perpendiculariter, violenterque deiectum minimè ad terram venturum aggregato virium, quarum vna e&longs;t ab impellente impreßa, altera verò à grauitate dependens. Nam ex impartita vt celerior fit ca&longs;us, quam vt graue in de­cur&longs;u &longs;uo po&longs;&longs;it ex acceler atione naturali eum gradum acqui­rere, quem certè &longs;ponte &longs;ua tantùm de&longs;cendens in fine eiu&longs;dem altitudinis adeptum e&longs;&longs;et. Hoc ita verum e&longs;t, vt aliquando minimum inter&longs;it, inter impetum ab ambabus cau&longs;is proue­nientem, & eum, qui a &longs;ola oritur grauitate, quamobrem pa­rum is proficeret, qui conaretur maiorem impetum componere in ca&longs;u grauis, illi nempe adiecta vi, mobile idem in decur&longs;u impellente, vltra natam grauitatem, quod tamen fieri haud dubiè po&longs;&longs;et, &longs;i ca&longs;us obliquus eßet.

Illud quoque hac occa&longs;ione aperiendum e&longs;t, graue naturali­ter de&longs;cendens eò concitatiùs ferri, quoad potentia re&longs;i&longs;tentis aeris (validior namque i&longs;ta fit, vbi mobilis ca&longs;us e&longs;t celerior) vi granitatis mobili inhærenti exaquatur, tunc enim cau&longs;&atail; vlterioris accelerationis adempta e&longs;t, con&longs;umiturque in lucta­tione aeris contranitentis: quare tunc grane progrederetur aquabili motu, id quòd citiùs euenire deberet &longs;i grane intr&atail; aquam de&longs;cendat.

PROP. II. THEOR. II.

SI in eadem recta duos motus &longs;ibi contrarios, fimplices, ac eodem tempore peractos intelligamus, mobile di­ferentiam illorum &longs;patiorum, &longs;i vtroque motu e&longs;&longs;et affe­ctum, percurreret.

Tab. 4. Fig. 2.

Curratur à puncto L &longs;patium LO imagine velocitatum ABFG, & codem tempore curratur etiam recta OM ex puncto altero O, &longs;cilicet contrario motu, & iuxta imaginem AHIG prædict&ecedil; homogeneam. Dico mobile, compo&longs;ito ex vtri&longs;que motu, & tempore ip&longs;o AG cur&longs;urum differentiam LM dictorum &longs;patiorum LO, OM.

Primùm intra parallelas AB, GF non &longs;e &longs;ecent lineæ

BF, HI, & ducatur quælibet DC æquidi&longs;tans AB, vel GF, quæ fecet HI in E. Manife&longs;tum e&longs;t, mobile, compo&longs;ito motu feratur habere duplicem velocitatem, vnam AB al­teram illi oppo&longs;itam AH, ob idque moueri ver&longs;us O &longs;ol&atail; velocitate HB differentia dictarum inter&longs;e pugnantium velocitatum: pariter momento D feretur mobile veloci­tate EC differentia duarum DE, DC, & in&longs;tanti G habebit differentialem IF; ex quo &longs;equitur figuram BHEIFCB, dif­ferentiam imaginum ABFG, HAGI, aptatam tempori AC imaginem e&longs;&longs;e velocitatum compo&longs;iti motus. Hoc po­&longs;ito habebit LM ad LO eandem rationem, ac BHIF ad ABFG; Propterea LM, quæ e&longs;t differentia &longs;patiorum LO, MO curretur iuxta imaginem BHIF, nempe compo&longs;ito motu, & tempore AG.

Tab. 4. Fig. 2.

Def. 3 prima.

Pr. 2. prim&atail; huius.

2. Se nunc &longs;ecent lineæ BF, HI in C. Ducatur CD pa­rallela alteri æquidi&longs;tantium AB, GF. Con&longs;tat ex prima parte, quòd mobile compo&longs;ito motu, & iuxta imaginem HBC feretur ver&longs;us O tempore AD; &longs;it ergo &longs;patium, quod curreretur illa imagine, PR, & ob id LO ad PR eande&mtail; habebit rationem quam imago ABFG ad imagine&mtail; HBC.

Tab. 4. fig. 3.

Pr. 2. prima

Similiter dum mobile mouetur tempore DG iuxta ima­gines DCIG, DCFG, feretur verè &longs;ecundùm imagine&mtail; FCI ver&longs;us L, quamobrem &longs;i &longs;patium, quod exigeretur hac imagine &longs;it RQ, habebit i&longs;tud ad LO eandem rationem, quam imago CFI ad imaginem ABFG, & ideo ex æquali QR ad PR &longs;e habebit vt imago CFI ad imaginem HBC; &longs;i igitur ponatur ABFG maior imagine AHIG, demptà co­muniter AHCFG relinquetur HBC maior imagine CEI, & ideo etiam PR maior QR: curritur verò PR versùs R tem­pore AD, & RQ versùs P tempore DG, ergo toto tempo­re AG curretur PQ differentia &longs;patiorum PR, Rque Cum verò HBC ad CFI, &longs;it vt PR ad RQ, erit diuidendo vt ex­ce&longs;&longs;us imaginis HBC &longs;upra imaginem FCI ad imagine&mtail; i&longs;tam, ita PQ ad QR, & o&longs;ten&longs;um e&longs;t QR ad LO, &longs;icut ima­go FCI ad imaginem ABFG, ergo ex æquali exce&longs;&longs;us ima­ginis HBC &longs;upra imaginem AHIG habebit eandem ratio­nem ad imaginem AHIG, ac PQ ad LO, at e&longs;t in illa eadem ratione etiam LM ad LO (e&longs;t enim LO ad MO vt imago ABFG ad imaginem AHIG) ergo PQ erit æqualis LM, atque adeo mobile dum currit vtroque motu, hoc e&longs;t iux­ta &longs;imul duas imagines propo&longs;itas contrariorum motuum, peraget &longs;patium LM versùs O &longs;ecundùm imaginem, quæ differentia e&longs;t propo&longs;itarum ABFG, AHIG, tempore AG. Quod &c.

Ex prim&atail; parte.

Pr. 2. prima.

Corollarium.

Deducìtur, mobile nullum &longs;patium emen&longs;urum, vbi ima­gines &longs;implicium motuum fuerint aquales.

PROP. III. THEOR. III.

Tab. 4. Fig. 4.

REperire eam velocitatem, eamque directionem, quæ orirentur, &longs;i mobile pluribus eodem momento velo­citatibus, &longs;eu conatibus affectum e&longs;&longs;et. Opportet autem non &longs;olum has velocitates, verùm etiam earum directio­nes manife&longs;tas e&longs;&longs;e.

Habeat mobile A, eodem momento conatum AB, quo tendat in R; AC; quo in C; & AD, quo in D. Quæritur ve­locitas, & directio, quas mobile habiturum e&longs;&longs;et in multi­plici illa affectione (Nam actu vnam velocitatem, vnam­que tantùm directionem &longs;ortiri debet) Ex duabus qui­bu&longs;que AD, AC intelligatur perfici parallelogrammum ACED, & ducta diametro AE fiat itidem aliud parallelo­grammum ABFE, cuius agatur diameter AF. Dico AF e&longs;&longs;e quæ&longs;itam velocitatem, ac directionem, quibus mobile ex illis pluribus conatibus motum &longs;uum in&longs;titueret.

Si mobili A currendum e&longs;&longs;et æquabili motu &longs;patium AE, pertran&longs;iret eodem tempore tam rectam AD, quàm

ip&longs;am AC; nam cum fertur ab A in E verè de&longs;cendit ab A in C, & ab A in D motu pariter æquabili; ergo AD ad AC, erit vt velocitas, qua curritur per AD ad velocitatem, qua curritur per AC. Itaque &longs;i mobile dum e&longs;t in A in­telligatur affectum velocitatibus AD, AC habentibus di­rectiones ip&longs;as rectas AD, AC, perinde e&longs;&longs;et, ac &longs;i &longs;ola fo­ret mobili velocitas vnâ cum directione AE. Eadem ra­tione AF velocitas habens directionem AF, æquipollebit duabus velocitatibus AB, AE iuxta directiones rectas ea&longs;-dem ABAE; hoc æquiualebit tribus AB, AC, AD. Mo­bile igitur ex affectione trium illorum conatuum, vt &longs;up­po&longs;itum fuit, nitetur &longs;ecundùm AF velocitate ip&longs;a AF Quod &c.

Cl. pr. de mo­ aquab.

DEF. I.

ACcelerationem alicuius motus, tunc intelligimus, cum velocitates, quæ &longs;ubinde mobili adueniunt, non de­lentur, &longs;ed pror&longs;us integræ, atque indelebiles mobili in ip&longs;o motu per&longs;euerant. Ex quo &longs;equitur motum &longs;implicem di­ci, cum præteritæ velocitates protinus euane&longs;cunt, illæ­que tantum con&longs;iderantur, quæ mobili &longs;ubinde oriun­tur.

PROP. IV. PROB. II.

IMaginem accelerationis cuiu&longs;cunque &longs;implicis motus exhibere.

Imago velocitatum &longs;implicis motus e&longs;to rectangulum AFDC: &longs;ic motus e&longs;t æquabilis, vt acceleretur debent in­&longs;tanti C vigere omnes velocitates in imagine AFDC con­prehen&longs;æ, & item ducta quacunque BE parallela AF, vel CD, erit mobile momento B affectum omnibus antece­dentibus velocitatibus, comprehen&longs;is nempe ab imaginis portione AFEB; quare &longs;i ponamus HLG imaginem e&longs;&longs;&etail; accelerationis, itaut nempe tempus GL æquale &longs;it tempo­ri AC; item KL æquale tempori AB, erit vt figura CAFD ad figuram BAFE, &longs;ic velocitas, qua mobile fertur momen­to G ad velocitatem, quam habet in&longs;tanti K; & ideo quia ponitur imago &longs;implicis motus rectangulum AFDC, erit rectangulum CF ad BF, hoc e&longs;t recta CA ad AB immò LG ad LK, vt GH ad KI; quamobrem GLH imago velo­citatum huiu&longs;modi motus, erit triangulum. Quod &longs;i ima-go &longs;implicis motus fui&longs;&longs;et triangulum, imago velocitatum accelerationis foret trilineum &longs;ecundum, & ita pro­portionaliter de infinitis numero accelerationibus.

Tab. 4. fig. .

Cor. def. 3. pri­mi.

Def. 1. huius.

Def. 3 primi.

Corollarium.

Hinc obiter habemus, quo pacto imago velocitatum corpo­rum naturaliter de&longs;cendentium triangulum &longs;it. Nam quo­libet momento &longs;ui ca&longs;us habet graue idem in&longs;e principiu&mtail; motus, &longs;eu grauitas, ex qua concipitur imago &longs;implicis motus &longs;i nempe priores gradus velocitatis &longs;ubinde deperirent, at quia in eius de&longs;cen&longs;u pror&longs;us per&longs;euerant (id enim &longs;upponi­tur ab&longs;trabendo ab aere) inde motus concitatur, & fit vti di­ximus imago accelerationis triangulum.

AXIOMA

QVælibet linea, vt fluxus puncti concipi po­te&longs;t.

AX. II.

VT propo&longs;ita linea ex fluxu puncti exarètur, duò tan­tùm nece&longs;&longs;aria &longs;unt, &longs;cilicet motus, & puncti di­rectio.

PROP. V. THEOR. III.

REcta, quæ priùs de&longs;eripta e&longs;t, pote&longs;t alijs à primis velocitatibus, rur&longs;us exarar.

Nam punctum pote&longs;t fluere &longs;ecundum quamcunque rectam, quocunque motu, ergo illam pote&longs;t etiam quibu&longs;­cunque velocitatibus affectum rur&longs;us exarare.

PROP. VI. THEOR. IV.

VT eadem recta ex fluxu puncti renouetur, opportet in quocunque illius puncto &longs;eruari pri&longs;tinas directio­nes,

Cum, vti diximus, ad de&longs;criptionem lineæ duo tantùm exigantur, nempe motus, & puncti directio; motus verò po­te&longs;t e&longs;&longs;e quilibet, &longs;equitur ergo directionem, alteram de duobus, &longs;eruari debere.

Ax. 2. buius. pr. 5. huius.

DEF. II.

LIneam dicimus curuam, in qua &longs;umptis duobus ad­libitum punctis, recta, quæ ip&longs;a puncta coniunge­ret, nullam cum propo&longs;ita linea partem &longs;it habitura com­munem.

PROP. VII. THEOR. V.

DIrectiones puncti de&longs;cribentis lineam, iuxta rectas lineas concipi debent.

Dum punctum fluere intelligimus, ine&longs;t in eo &longs;ingulis momentis certus, ac præfixus gradus velocitatis, quo tan­tùm attento, rectà, æquabilique motu in certam partem con­tenderet; at huiu&longs;modi iter, aliud non e&longs;t, quàm directio puncti, qua eius temporis momento profici&longs;citur; ergo iux­ta rectas lineas, directiones omnes con&longs;iderari opportet.

PROP. VIII. THEOR. VI.

TAngens, & directio motus in quouis curuæ puncto e&longs;t vna, atque eadem recta.

Nam in de&longs;criptione cuiu&longs;cunque rectæ procedit pun­ctum iuxta tendentias rectas, obliquatur tamen ob &longs;ub&longs;e­quentes, aliò tendentes ni&longs;us, & ob id di&longs;trahitur punctum ip&longs;um à priori tendentia, idem accidit ex alia parte &longs;i re­flaxiffet idem punctum, nempe hinc inde vnicam rectam eandemque, continuantibus oppo&longs;itis ad idem punctum directionibus, ergo directio, & tangens vna, & eadem e&longs;t recta.

Pr. 7. huius.

Corollarium.

Hinc &longs;equitur, vnicam lineam dicendam e&longs;&longs;e, cum à quo­cunque illius puncto vnica tantùm ex vtraque parte egre­ditur tangens.

DEF. III.

QVòd &longs;i ex aliquo puncto duæ tangentes hinc inde egredientes angulum efficiant; tunc propo&longs;itam li­neam inflexam dicemus, & punctum, in quo &longs;unt contactus, inflexionis appellabitur.

Corollarium I.

Ab hi&longs;ce deffinitionibus, & priori coroll. manat artificium componendi duas curuas, vel curuam & rectam, adeout vni­cam lineam efforment, nullumque angulum; nempe cum &longs;ic inuicem tuxgamus, vt taugentes ad punctum connexus, vnam tantùm rectam efficiant.

Corollarium II.

Sed & illud patet, quibus angulis inflectantur lineæ inui­cem compo&longs;it, &longs;i ad punctum inflexionis angulum tangen­tium ob&longs;eruauerimus, &longs;unt enim inter&longs;e æquales, licèt diuer­&longs;a &longs;peciei, vnus &longs;it curuilineus, & rectilineus alter.

PROP. IX. THEOR. VII.

TAngens, &longs;eu directio motus in quocunque curuæ puncto e&longs;t illa recta, quæ vtrinque &longs;tatim cadens extra curuæ conuexum ad eandem, quàm fieri pote&longs;t ex vtraque parte accedit.

Nam alia quæque recta tran&longs;iens per punctum conta­ctus ad &longs;ectionem magis accedere nequit, quin ip&longs;am illinc &longs;ecet, ob id extra conuexum eius non cadet, ab altera ve­rò parte magis à propo&longs;ita curua &longs;eparabitur, quamobrem nulla alia recta, quàm tangens poterit &longs;imul extra curuam e&longs;&longs;e, & quàm fieri pote&longs;t ad ip&longs;am accedere.

DEF. IV.

LIneæ AC, AD occurrant &longs;ibi in A, quod punctum in­telligatur transferri ab A in C vnà cum linea AD &longs;emper &longs;ibi parallela, quo tempore punctum A currat ip­&longs;am latam lineam ex A in D. Manife&longs;tum e&longs;t idip&longs;um punctum A de&longs;cripturum e&longs;&longs;e motu compo&longs;ito lineam quandam AB diagonalem &longs;uperficiei parallelogrammæ ABCD. Vocamus ergo diagonalem illam &longs;emitam com­po&longs;iti motus, & AC, AD latera illius.

Tab. 4. fig. 6.

Corollarium I.

Manife&longs;tum e&longs;t mobile dum currit AB tran&longs;ire etiam AC, AD, licèt curuæ &longs;int, nam verè transfertur illo tempore, tam ad lineam CB quam ad DB.

Corollarium II.

Præterea &longs;i ducerentur, aut&longs;int AC, CB, DA, DB, ABrectæ lineæ, efficeretur ex ijs parallelogrammum ACBD, cu­ius diameter AB; quamobrem ex datis punctis C, A, D repe­riretur &longs;tatim punctum B, &longs;cilicet extremum &longs;emitæ compo­&longs;iti motus, cuius latera ip&longs;æ curuæ, aut rectæ AC, AD —.

PROP. X. PROB. III.

EX datis quotcunque lateribus compo&longs;iti motus, huius &longs;emitæ terminum exhibere.

Tab. 4. Fig. 7.

Si latera compo&longs;iti motus e&longs;&longs;ent duo tantùm AB, AC. Facto parallelogrammo vt dictum e&longs;t, inueniretur pun­ctum E extremum motus: & quæcunque &longs;it &longs;emita, &longs;eu mo­tus, pote&longs;t idem E &longs;upponi tanquam extremum alterius la­teris, adeoque, &longs;i motus con&longs;tet ex tribus lateribus AC, AB, AD, perinde &longs;it ac &longs;i foret duorum laterum AE, AD; nam AC, AD valent &longs;imul ac &longs;olum AE; cum ita &longs;it, facto etiam parallelogrammo EADF ex datis punctis E, A, D, habebitur F extremum &longs;emitæ, cuius &longs;unt tria latera CA, AD, AB —

Corollarium.

Deducitur artificium de&longs;cribendæ &longs;emitæ AE, vel AF, &longs;i nempe a&longs;&longs;umptis partibus AG, AH, AI in dictis lateribus, quæ quidem &longs;ciantur percurri temporibus æqualibus, &longs;i per ip&longs;as &longs;ingulas mobile punctum ferretur eo modo, quo in com­po&longs;ito motu nititur per ea&longs;dem directiones; reperietur in­quam punctum K in &longs;emita AE, atque L in &longs;emita AF: qua­re hoc modo &longs;umptis alijs, atque alijs partibus in ip&longs;is lateri­bus, reperientur alia, atque alia puncta ad ip&longs;am &longs;emita&mtail; pertinentia, quorum tandem beneficio, facile erit qua&longs;itam fermè &longs;emitam exarare.

PROP. XI. PROB. IV.

EX datis imaginibus velocitatum, iuxta quas &longs;implici motu currantur latera compo&longs;iti motus; datis item tangentibus ad quæcunque puncta ip&longs;orum laterum, repe­rire &longs;emitam compo&longs;iti motus, nec non directiones, veloci­tate&longs;que puncti de&longs;cribentis ip&longs;am &longs;emitam.

Tab. 4. fig. 8.

Opportet tamen latera ip&longs;a, itemque imagines prædictas, in imperatas &longs;ecari po&longs;&longs;e rationes, quamquam nos non la­teat, in lateribus curuis hoc effici non po&longs;&longs;e, præterqua&mtail; aliquatenus in periphærijs circulorum.

Sint AB, AF latera compo&longs;iti motus, quæ quidem &longs;eor­&longs;im currantur eodem tempore QM, &longs;cilicet AB iuxta ima­ginem MNPQ, et AF iuxta imaginem alteram ei homoge­neam TMQR. Ponatur AB circuli arcus, quem tangat re­cta BC æqualis QB, at AF lineam', quæ parabola &longs;it, con­tingat recta FG æqualis Rque Reperiemus illicò punctum H extremum &longs;emitæ compo&longs;iti motus; &longs;unt enim data pun­cta A, F, B. Cum igitur mobile venerit in H. Dico, eo temporis momento velocitatem, ac directionem HL, quæ recta diameter e&longs;t parallelogrammi, cuius duo latera &longs;unt dictæ lineæ HI, HK; Iam vti diximus punctum H e&longs;t ex­tremum compo&longs;iti motus, quare eo momento, quo pun­ctum mobile e&longs;t in H, habet inibi ea&longs;dem illas velocitates, quas haberet in B, et F, dum &longs;eor&longs;im illa latera excurri&longs;&longs;et; &longs;cilicet con&longs;ideratur ip&longs;um mobile habens &longs;imul velocita­tem HI æqualem, ac æquedirectam, &longs;eu æquidi&longs;tantem ip&longs;i CB, cui e&longs;t æqualis alia QP; & velocitatem HK æqua­lem, &longs;imiliterque directam, ip&longs;i GF æquali Rque Cum ita &longs;it erit HL velocitas, & directio quæ&longs;ita momento que Eo­dem modo, &longs;i &longs;it, vel fiat vt imago PNMQ ad ONMV (ducta &longs;cilicet applicata SVO) ita BA ad AX, et ONMV ad imaginem VMTS, vt XA ad AI, percurrentur AX, AI eodem tempore MV, eritque ob id in X velocitas, & dire­ctio, tangens ip&longs;a ZX æqualis VO, & in I velocitas, & di­rectio, tangens 2 I æqualis VS; Itaque datis punctis X, I, A dabitur etiam Y extremum &longs;emitæ compo&longs;iti motus, cuius latera AX, AI, & ideo mobile dum e&longs;t in Y momento V affectum erit duplici velocitate, hoc e&longs;t Y 4 æquali ve­locitati ZX, &longs;eu VO, ac æquidi&longs;tante eidem ZX, et veloci­tate altera Y 3 æquali, & æquèdirecta ip&longs;i 2 I: quare ex datis punctis 4, Y, 3 inuenietur punctum S quartus angu­lus parallelogrammi habentis diametrum YI, quæ quidem erit directio, & velocitas mobilis currentis compo&longs;ito mo­tu in&longs;tanti V. Cumque alia quotcunque puncta eadem methodo reperire queamus, per quæ duci po&longs;&longs;it linea ferè quæ&longs;itam &longs;emitam repræ&longs;entans, atque emulans, patet idcir­co, quod propo&longs;uimus.

Pr, 10. huius.

Ex pr. 3. hu.

Pr. 2. primi huius.

Cor. 2. def. 3. huius.

Pr. 3. huius

Corollarium.

Cum verò directiones &longs;int idem, ac tangentes, liquet HLVS tangentes e&longs;&longs;e compo&longs;iti motus.

Pr. 8 huius.

PROP. XII. THEOR. VIII.

CVm imagines velocitatum, iuxta quas curruntur du&ecedil; rectæ, quæ &longs;int latera compo&longs;iti motus, &longs;unt paral. logrammum, & triangulum; tunc &longs;emita compo&longs;iti motus erit communis parabola.

Tab. 5. Fig. 1.

Tempore HM curratur latus AC iuxta imaginem velo­citatum HILM rectangulum, & latus AB iuxta imaginem triangulum HMN; erit CA ad AB, vt imago parallelogran­mum HILM ad aliam imaginem triangulum NHM. Fiat parellogrammum ACDB erit in D extremum &longs;emitæ com­po&longs;iti motus, quæ &longs;i ponatur AFC; Dico e&longs;&longs;e parabolam. Sumatur in ip&longs;a linea quoduis punctum F, ab ip&longs;o dedu-cta FE parallela AB, vti etiam FG parallela AC, erunt AE, AG latera compo&longs;iti motus, cuius &longs;emita AF: Con­cipiatur modò P momentum, quo mobile ade&longs;t in F, & ducta OPK parallela alteri HI, vel NL, erit imago MHIL ad imaginem PHIK, hoc e&longs;t MH ad HP, vt CA ad AE, &longs;eu vt BD ad GF. Pariter erit imago NHM ad imaginem OHP, hoc e&longs;t quadratum ex MH ad quadratum ex PH; immò id ex BO ad illud ex GF, vt BA ad AG; quamobrem punctum F cadet in curuam parabolicam communem, cuius diameter AB, & ba&longs;is, &longs;eu ordinatim applicata BD, &longs;cilicet AFD erit ip&longs;a curua parabolica. Quod &c.

pr. 2. primum huius.

Pr. 3. huius.

Pr. 2. huius.

Scholium.

Quoniam graue, quod iaculatur extræ perpendiculum, li­berum ab omni obice, ni&longs;i turbaretur eius motus à propri&atail; grauitate pergerct moueri æquabiliter iuxta directionem, ve­locitatemque ei traditam; habet verò coniunctam grauita­tem, qua, ni&longs;i ab impre&longs;&longs;o impetu flecteretur motus, de&longs;cen­deret iuxta perpendiculum motu naturaliter concitato, cuius imago velocitatum, triangulum e&longs;t; Hinc propterea gran&etail; vltra perpendiculum proiectum de&longs;cribit in cur&longs;u &longs;uo, motu &longs;cilicet compo&longs;ite, parabolam vulgatam. Verùm enim verò de&longs;criptionem i&longs;t am nece&longs;&longs;e aliquo pacto e&longs;t ex duabus cau&longs;is vitiari, hoc est ab aeris re&longs;i&longs;tentia, & perpendiculis non in­ter&longs;e parallelis, quippe in idem, vnumque punctum, vniuer&longs;i centrum, conuergentibus.

PROP. XIII. THEOR. IX.

SI ab a&longs;&longs;umpto hyperbolæ puncto, recta axi primo pa­rallela deducatur, quæ ad &longs;ecundam diametrum per­tingat; Quadrilineum comprehen&longs;um ab ip&longs;a curua hy­perbolica. & dictis tribus rectis, erit imago velocitatis il-lius motus de&longs;cribentis curuam parabolicam, cuius ba&longs;is ad axem eius habet eandem rationem, quam duplus axis propo&longs;itæ hyperbolæ ad ductam illam æquidi&longs;tantem inter eiu&longs;dem hyperbolæ a&longs;&longs;ymptotos interiectam.

Tab. 5. fig. 2.

Def. 7. primi & pr. 12. pri­mi huius.

Cor. pr. 4. hu.

Pr. 2. primi huius.

Ex pr. 12. hu.

Pr. 3. huius.

Pr. 11. l. 2. co­nic.

Def. 3. prima huius.

Corollarium. I.

Patet, cum latera compo&longs;iti motus &longs;int duo, & &longs;ibi ip&longs;is per­pendicularia, tunc gradum velocitatis eìu&longs;dem motus compo­&longs;iti æqualem e&longs;&longs;e potentiâ duobus &longs;imul gradibus, quos habet mobile eodem momento, ac &longs;i &longs;eorfim intelligatur in ip&longs;is ferri lateribus.

Corollarium. II.

Si verò con&longs;iderentur imagines primi &longs;ecundique Ca&longs;us inter&longs;e homogenea, erit vt quadrilineum HISM primi adquadrilineum ij&longs;dem literis notatum &longs;ecundi ca&longs;us, vt cur­ua illa parabolica ad hanc &longs;ecundi ca&longs;us parabolam.

Pr 2. prim&atail; huius.

Corollarium. III.

Illud etiam con&longs;tat, e&longs;&longs;e in vtroque ca&longs;u vt quadrilineum HIRP ad ip&longs;um PRSM, ita AF ad FD.

PROP. XIV. THEOR. X.

PRopo&longs;itis Spirali Archimedea primæ circulationis ABD, et AGF communi parabola, &longs;it FG ba&longs;is huius æqualis radio DA, et GA &longs;it dimidium circumferenti&ecedil; cir­culi AEG; erit parabola AGF axem habens GA æqualis propo&longs;itæ &longs;pirali.

Tab. 5. fig. 3.

Sit PNK communis hyperbola, cuius coniugati &longs;emia­xes &longs;int IK, IH, & a&longs;&longs;ymptotos IO. E&longs;to etiam axis hy­perbolæ huius, dupla &longs;cilicet IK, ad HO illi &ecedil;quidi&longs;tantem vt FG ad AG. Iam con&longs;tat quadrilineum IHPK fore ima­ginem velocitatum, iuxta quam curreretur parabola AGF tempore IH: &longs;i modo o&longs;tendimus hoc ip&longs;um quadrilineum e&longs;&longs;e pariter homogeneam imaginem alterius compo&longs;iti motus, quo videlicet de&longs;cribitur &longs;piralis propo&longs;ita ABD, palam erit, ip&longs;am parabolam eidem illi &longs;pirali æqualem fu­turam. Ducatur recta KL, quæ æquidi&longs;tet IH; item ex quouis puncto Q temporis IH alia deducatur recta QRMN parallela IK: erit parallelogrammum rectangulum HIKL imago velocitatum, iuxta quam curritur FG, et HIO trian­gulum imago, qua curritur AG motu grauium de&longs;cenden­tium: Verùm quia eodem tempore IH, &longs;i mobile currat æquabili motu DA æqualem FG, e&longs;t eius imago idem re­ctangulum IHKL, curriturque illo eodem tempore IH (&longs;pi­rali exigente) omnis circuli circunferentia AGEA æqua­bili etiam motu ab extremitate A radij AD circumducti in de&longs;criptione &longs;piralis; ob idque factum e&longs;t, vt IK ad HO e&longs;­&longs;et vt DA ad circunferentiam ip&longs;am AGEA; nam hoc mo-do rectangulum IH in HO e&longs;t imago velocitatum eiu&longs;­dem motus per AGEA. Ducatur nunc ex quocun­que momento Qlinea QRMN ip&longs;i IK æquidi&longs;tans, & au­&longs;picato motu ex centro D momento I, vt nempe oriatur &longs;piralis, intelligatur momento Qventum e&longs;&longs;e in B, quamo­brem ductâ DBE, erit rectangulum, &longs;eu imago QIKR ad imaginem rectangulum HIKL, ita DB ad DE, in qua ra­tione, cum propter &longs;piralem, &longs;it etiam circunferentia AGE ad circunferentiam AGEA, erit rectangulum IQ in HO imago velocitatis per AGE, e&longs;tque velocitas iuxta tangen­tem in E ad velocitatem iuxta tangentem circulum BC in B vt ED ad DB, &longs;eu vt HO ad QM; ergo cum iuxta tangen­tem in A, hoc e&longs;t in E velocitas &longs;it HO, erit &longs;ecundùm tan­gentem circulum BC in B, ip&longs;a QM velocitas; propterea­que imago triangulum HIO, quæ in parabolæ de&longs;criptio­ne erat per AG, nunc erit per omnes tangentes circulos &longs;u­binde crc&longs;centes ex D in E: &longs;cilicet momento I, erit mobi­li puncto &longs;ecundùm DA, velocitas IK; momento Q du&mtail; ade&longs;t in B, erit &longs;ecundùm BE velocitas QR, & iuxta tangen­tem in B circuli BC velocitas QM; quæ ambæ, hoc e&longs;t ve­locitates QR, QM cum &longs;int normaliter directæ, erit eidem mobili in B iuxta &longs;piralem velocitas QN potentia ip&longs;is am­babus æqualis. Similiterque momento H cum mobil&etail; fuerit in A, erit velocitas iuxta &longs;piralem, ip&longs;a HP æqualis potentiâ duabus velocitatibus HL iuxta radium, et HO iuxta tangentem; & &longs;ic omnino liquet, ip&longs;um quadrilineum HIKP e&longs;&longs;e imaginem velocitatum tam in de&longs;criptione pa­rabolæ AGF, quàm &longs;piralis Archimedeæ DBA, & cum &longs;it in ij&longs;dem de&longs;criptionibus homogenea &longs;ibi ip&longs;i, con&longs;tat ip­&longs;as curuis æquales e&longs;&longs;e. Nam vt imago illa ad &longs;e ip&longs;am ita parabola ad &longs;piralem prædictam. Quod &c.

Pr. 13. huius.

Pr. . prima.

Pr. 2. prima.

Pr. 8. huius & Cor. pr. 13.

Pr. 2. huius.

Corollarium.

Hinc aparet, &longs;piralem DB ad &longs;piralem DBG eandem habe­re rationem, quam quadrilineum QIKN ad quadrilineum HIKP; pariterque rectam DA ad eandem &longs;piralem DCB ha­bere ip&longs;am rationem, ac rectangulum HIKL ad dictum qua­drilineum HIKP. Eodem ferè modo exhiberi pißet ratio &longs;pi­ralis ad &longs;piralem, licèt plurium inter&longs;e circulationum, eritque pror&longs;us ea, quam habet vnum ad alterum eiu&longs;dem illius na­turæ, quadrilineorum.

PROP. XV. THEOR. XI.

Tab. 5. Fig. 4.

SPiralis orta ex motu naturaliter accelerato per radium circuli comprehendentis &longs;piralem ip&longs;am, & ex motu æquabili circa circumferentiam eiu&longs;dem circuli, æqualis e&longs;t ei curuæ parabolicæ natæ ex motu compo&longs;ito, cuius vnum latus curritur iuxta imaginem trianguli, nempe motu gra­uium, alterum verò latus iuxta imaginem trilinei &longs;ecundi, habebitque parabola ip&longs;a axim æqualem radio, & ba&longs;i&mtail; tertiæ parti circunferentiæ eiu&longs;dem circuli &longs;piralem com­prehendentis.

E&longs;to &longs;piralis ACB, quæ &longs;ignatur ex motu puncti A æqua biliter lati circa circumferentiam ADA, dum nempe eodem tempore IF, punctum B currit à quiete lineam BA motu grauium de&longs;cendentium; &longs;it verò imago velocitatum dicti motus æquabilis per ADA rectangulum HGFI, & alte­

Cor. pr. 4. huius.

Pr. 2. prim&atail;

Pr. 8. huius.

Cor. prop. 13. huius.

Pr. 10. primi huius.

Pr. 2. primi huius.

Pr. 2. prima.

Scholium.

Exemplo traditarum curuarum, po&longs;&longs;unt innumeræ &longs;pira­les &longs;uis parabolis æquales excogitari, nec ideo res minùs de­mon&longs;trabitur, &longs;i loco rectarum, &longs;eu laterum OT, OP compo&longs;iti motus, &longs;ub&longs;tituantur circuli, aut circulorum arcus, qui ad re­ctos angulos &longs;e &longs;ecent, &longs;cilicet cum tangentes ad punctum infle­xionis, &longs;eu occur&longs;us ip&longs;arum curuarum &longs;ibi ip&longs;is perpendicu­res fuerint. Quòd &longs;i ip&longs;a curua latera ad rectos angulos non &longs;e &longs;ecent curuæ nihilominus ab ip&longs;o compo&longs;ito motu na&longs;cen­tes poterunt exhiberi curuas parabolicas exequantes, quarum itidem latera &longs;int rectæ eundem angulum, quem pradictæ tan­gentes, comprehendentes. Sed de his &longs;atis, nunc dicamus ea tempora, quibus duorum pendulorum &longs;imiles vibrationes ab­&longs;oluuntur, hoc e&longs;t Galilei &longs;ententiam demon&longs;trabimus, quam quondam haud ruditer decepti fal&longs;am credidimus.

Vincentius Viuianus eximius no&longs;tri æui Geometra vt tue­retur Galilei &longs;ententiam, cuius digni&longs;&longs;imè &longs;e fui&longs;&longs;e di&longs;ipu­lum profitetur, tradidit mihi per admodum Reuerendum, at-que culti&longs;&longs;imum Patrem Io&longs;eph Ferronum è Societate Ie&longs;u, de­mon&longs;trationem &longs;uam verè pulcherrimam, ac di&longs;erti&longs;&longs;imè exaratam, qua vna potui&longs;&longs;em de Galilei aßerto &longs;atisfactus e&longs;&longs;e; eam demon&longs;trationem, ij&longs;dem pror&longs;us verbis, ac figuris, quibus ad me peruenit hic duxi reponendam, ne gloria&mtail;, quam Vir tantus meretur, ip&longs;i videremur no&longs;tra, quam inde &longs;ubdemus, demon&longs;tratione, &longs;ubripere.

Inquit ergo.

TEmpora naturalium de cur&longs;uum &longs;phærarum grauium per &longs;imiles, &longs;imiliterque ad horizontem inclinatos arcus curuarum linearum in planis, aut verticalibus, aut ad horizontem æqualiter inclinatis de&longs;criptarum, & quæ totæ &longs;int ad ea&longs;dem partes cauæ, inter&longs;e &longs;unt in &longs;ubdupli­cata ratione chordarum eorundem arcuum homologè &longs;umptarum.

Tab. 6. fig. 1. 2 3. 4.

Ex puncto A ad curuam lineam BCD extra ip&longs;am i&ntail; plano po&longs;itam, & in totum ad ea&longs;dem partes cauam, quæ­cunque ea &longs;it (vel nimirum pars aliqua circumferentiæ circuli, vel alicuius ex infinitis ellip&longs;ibus, aut parabolis, aut hyperbolis, aut &longs;piralibus, aut cycloidibus, vel concoidis, vel ci&longs;oidis, &longs;eu alterius cuiu&longs;cumque ex notis, vel ignotis curuis educantur omnes rectæ AB, AC, AD &c. quæ à punctis E, F, C, vel intra, vel extra eas &longs;umptis proportio­nalibus &longs;ecentur, ita vt &longs;it AB ad AE, &longs;icut AC ad AF, & &longs;icut AD ad AG &c. & hoc &longs;emper. Sic enim dubio pro­cul apparet, prout facillimum e&longs;t o&longs;tendere, lineam EFG tran&longs;euntem per &longs;ingula puncta E, F, G &longs;ic inuenta, cur­uam quoque e&longs;&longs;e, & eiu&longs;dem penitus naturæ, ac data BCD eique &longs;imilem, &longs;imiliterque cum ip&longs;a po&longs;itam, atque in to­tum cauam ad ea&longs;dem partes, ad quas ponitur caua ip&longs;&atail; BCD. Concipiatur modò planum, in quo manent huiu&longs;­modi &longs;imilium curuarum &longs;imiles arcus BCD, EFG, vel e&longs;&longs;e ad horizontem erectum, nempè verticale, vel ad ip&longs;u&mtail; horizontem inclinatum iuxta curuitates ip&longs;orum arcuum BCD, EFG inflexas e&longs;&longs;e &longs;uperficies eidem plano erectas, ita tamen, vt &longs;uper has po&longs;itis grauibus &longs;phæris in A, E per ip&longs;as &longs;ic inflexas &longs;uperficies eædem &longs;phæræ naturaliter decurrere queant; id quod &longs;anè accidet, cum arcus BCD totus fuerit infra horizontalem IL ex arcus &longs;ubli­miori puncto B ductam, fuerintque ab hac continuati re­ce&longs;&longs;us, ac totus ad vnam partem perpendiculi BH: nam &longs;ic talis quoque erit alter arcus EFG illi BCD &longs;imilis, &longs;imili­terque po&longs;itus. His omnibus &longs;ic manentibus: Dico tem­pus decur&longs;us &longs;phæræ grauis E per &longs;imilem, &longs;imiliterque po­&longs;itum arcum EFG, e&longs;&longs;e in &longs;ubduplicata ratione chordarum BO, EG arcus ip&longs;os &longs;ubtendentium. Secto enim bfariam angulo BAD per rectam AC arcum BD &longs;ecantem in C, atque arcum EFG in F, iungantur chordæ BC, CD, et EF, FG, quæ ex huiu&longs;modi curuarum natura cadent totæ intra ip&longs;os arcus, &longs;ed in prima, & &longs;ecunda figura ad partes poli A, in tertia verò, & quarta ad oppo&longs;itas.

Corollarium.

Ex modò sten&longs;is &longs;uper prima, ac &longs;ecunda figura, manife­&longs;tum fit celeberrimum illud magni Galilei pronuntiatum, quòd videlicet, ratio temporum &longs;imilium vibrationum pen­dulorum &longs;it &longs;ubduplicata rationis longitudinum filorum ho­mologè &longs;umptorum, non tantum verum e&longs;&longs;e de vibrationibus pendulorum per arcus &longs;imiles, &longs;imiliterque po&longs;itos, &longs;umptos ex circulorum quadrantibus ad perpendiculum v&longs;que termi­nantes, &longs;ed etiam de vibrationibus per arcus quo&longs;cumque &longs;i­miles quadrantum à perpendiculo &longs;eiunctos: dummodo ip&longs;i &longs;imiles arcus &longs;int quoque &longs;imiliter po&longs;iti: quales nimirùm ap­parent in figuris prima, ac &longs;ecunda arcus BCD, EFG, dum grauia B, E ex filis, aut ha&longs;tulis AB, AE circa punctum A conuertibilibus appen&longs;a concipiantur.

Scholium.

Si curua BCD, EFG in prima, & &longs;ecunda figura fuerint &longs;imiles arcus ex circulis commune centrum A habentibus; ac in verticali plano po&longs;itis, & in prima figura recta AB, AE fuerint fila aut ha&longs;tulæ quædam circa clauum A conuertibi­les, in &longs;ecunda verò recta AB, AE concipiantur, vt ha&longs;tulæ inflexibiles, volubile&longs;que circa imum punctum E, atque ex huiu&longs;odi filorum, aut ha&longs;tularum terminis B, E pendeant graues &longs;phæræ B, E (cum eadem &longs;int tempora prout a&longs;&longs;umi­tur quoque ab ip&longs;o met Ceua) tempora inquam decur&longs;uum liberorum granium B, E per arcus BCD, EFG, ac tempor&atail;de&longs;cen&longs;uum ip&longs;orum grauium per eo&longs;dem arcus (vel hac à filis pendeant, vel ab hastulis &longs;ustineantur) erit quoque tem­pus de&longs;cen&longs;us, &longs;eu vibrationis penduli B per arcum BCD ad tempus de&longs;cen&longs;us, &longs;eu vibrationis penduli E per arcum EFD in &longs;ubduplicata ratione chordæ BD ad chordam EG; &longs;ed hæc ratio chordarum BD, EG eadem e&longs;t, ac ratio filorum, aut ha­&longs;tularum AB, AE; Ergo tempus vibrationis penduli AB per arcum BCD ad tempus vibrationis penduli AE per arcum il­li &longs;imilem, &longs;imiliterque po&longs;itum EFG est quoque in &longs;ubdupli­cata ratione longitudinum, vel filorum, aut ha&longs;tularum, ex quibus eadem grauia pendula &longs;imiles vibrationes ab&longs;oluunt BCD, EFG.

Scholium.

Cæterùm non me latet con&longs;tructionem, ac demonstratio-nem à nobis &longs;uperiùs allatam nonnullis euidentiorem forta&longs;&longs;e eua&longs;uram, &longs;i ommi&longs;&longs;a illa continua bi&longs;ectione angulorum &longs;i­miles, &longs;imiliterque po&longs;itos arcus ab&longs;cindentium ex &longs;imilibus curuis ibidem de&longs;criptis; atque ommi&longs;&longs;a pariter continua co­niunctione chordarum, vt ibi factum fuit, horum vice, vt in quinta figura, ex punctis B, D binæ tangentes curuam BCD ducantur BH, DH, quæ omninò mutuò &longs;e &longs;ecabunt in puncto H (ob conditiones in ip&longs;a Theorematis expo&longs;itione vltimo lo­co po&longs;itas) atque ex E, G ip&longs;is BH, DH agantur æquidistan­tes, quæ iunctæ, AH &longs;imul occurrent in I, curuamque EFG contingent pariter ad E, G (quæ omnia &longs;i opus fuerit, facilè demon&longs;trabuntur) ac in&longs;uper, &longs;i à puncto C, in quo iunct&atail; AH &longs;ecat arcum BCD, agatur tangens LM primas BH, DH &longs;ecans in LM; Per F verò, in quo AICH &longs;ecat arcum EFG agatur NO parallela tangenti LM, quæ curuam pariter EFG tanget ad F, ac tangentes EI, GI &longs;ecabit ad NO: & &longs;i iunctis in&longs;uper AL, AM, eadem, quam nunc explicauimus, continue­tur con&longs;tructio per alias, atque alias tangentes, ac parallelas&c. &longs;ic enim vnicuique harum curuarum circum&longs;cribetur rectilineum, primò ex binis tangentibus, &longs;ecundò ex tribus, tertiò ex quinque, quartò ex &longs;eptem, & &longs;ic vlteriùs iuxta re­liquos impares numeros &longs;ucce&longs;&longs;iuè &longs;umptos; atque omnia pa­ria talium æquidi&longs;tantium tangentium eam &longs;emper inter &longs;e rationem &longs;eruabunt, quam habent chorda BD, EG, &longs;en quam habent rectæ BA, EA, eruntque inter&longs;e æqualiter inclinatæ; adeoque tempora decur&longs;uum grauium B, E tam per &longs;ummas binarum tangentium BH, HD, EI, IG, quàm per minores &longs;ummas, ex quinque &longs;imul chordis vtrinque &longs;umptas, aut quàm per alias &longs;emper minores &longs;ummas huiu&longs;modi tangen­tium iuxta quantumuis maiorem numerum imparem æquè multipliciter &longs;umptarum, erunt perpetuò proportionalia tem­poribus decur&longs;uum per chordas BD, EG; & hoc &longs;emper; etiam­&longs;i per huiu&longs;modi decrementa aggregaterum ex tangentibus vtrinque æquèmultipliciter &longs;umptis, deueniatur ad vltims, ac breui&longs;&longs;imas ip&longs;is arcubus circum&longs;criptiones polgonorum ex lateribus numero innumerabiliter aquèmultiplicibus, hoc e&longs;t ad ip&longs;os &longs;imiles, &longs;imilitorque po&longs;itos arcus BCD, EFG, quorum &longs;ingula homologorum laterum, &longs;eu punctorum paria, vt B, & E; C et F; D, et G &c. haberi poßunt tanquam paria parallelarum, ac proportionalium tangentium ip&longs;os &longs;i­miles, ac &longs;imiliter po&longs;itos arcus con&longs;tituentia. Quapropter ratio quoque temporum decur&longs;uum per ip&longs;os arcus, &longs;imilis erit rationi temporum decur&longs;uum per chordas; &longs;ed horum decur­&longs;uum ratio &longs;ubdupla e&longs;t rationis inter ip&longs;as chordas. Quare, & alia hac methodo con&longs;taret propo&longs;itum.

Tab. 6 fig. 5.

Hactenus graui&longs;&longs;imus Vir; &longs;upere&longs;t modò, vt quemadmo­dum annuimus, veritatem eandem no&longs;tra quoque methodo, confirmemus, vt ijs, quibus &longs;atis probat demon&longs;tratio allata, &longs;it nostra, quam afferemus, in experimentum traditarum hùc v&longs;que rérum; & quibus &longs;ecùs acciderit ex aliqua dubitatione, hæc per demon&longs;trationes no&longs;tras pror&longs;us, &longs;atimq tollatur. Illud etiam admoneo, eam rem non tantum me o&longs;ten&longs;urum,vt pulcherrima, vtilimaque veritas pluribus demon&longs;trationi­bus aperiatur; verùm potius vt ampli&longs;&longs;ima Methodus, qua tum vtemur, aliorum motuum demon&longs;trandorum in exemplum veniat.

PROP. XVI. THEOR. XII.

IN eadem recta CD coeant duæ planæ, inter&longs;eque &longs;imiles, ac pror&longs;us æquales figuræ ADCA, BDCB, & quidem ita, vt ab eodem puncto M &longs;i ducatur MH parallela CA, et ML ip&longs;i CB, &longs;it &longs;emper MH æqualis ML, quemadmo­dum æquales &longs;unt inter&longs;e CA, CB. Dico (&longs;i concipiatur &longs;olidum eius indolis, vt ductis rectis BA, LH cadant i&longs;tæ omninò in &longs;olidi i&longs;tius &longs;uperficie; ip&longs;um verò &longs;olidum, quod &longs;it BADC, &longs;ecetur plano quolibet æquidi&longs;tante figuræ BCD) fore, vt &longs;ectio i&longs;ta KFEIK, &longs;it pror&longs;us &longs;imilis, æqua­li&longs;que alteri conterminæ AEI; &longs;ed opportet, vt palam e&longs;t, coeuntes illæ figuræ non in eodem plano reperiantur.

Tab. 6. fig. 7.

Cum duo plana inuicem parallela KIE, BCD &longs;ecent alia duo inter&longs;e item parallela ACB, HML, erunt commu­nes &longs;ectiones, inter&longs;e omnes æquidi&longs;tantes rectæ lineæ KI, GF, ML, CB. Cum verò ob naturam &longs;olidi, &longs;ectiones BAC, IHM triangula &longs;int rectilinea, erit vt BC ad CA, ita KI ad IA. Sunt autem priores inter&longs;e æquales, ergo & po&longs;tremæ KI, AI inter&longs;e æquabuntur. Eademque ratione &longs;unt æquales HG, GF: & quoniam ob fimilitudinem figu­rarum angulus BCD æquatur angulo ACD, & angulus BCD æqualis angulo KIE (nam etiam CD, IE &longs;unt rectæ æquidi&longs;tantes, cum nempe &longs;int communes &longs;ectiones plani DCA &longs;ecantis duo æquidi&longs;tantia KIE, BCD) ergo cu&mtail; angulus pariter ACD æquet angulum AIE, erunt anguli KIE, AIE, et FGE, HGF æquales. Quod &c.

PROP. XVII. THEOR. XIII.

II&longs;dem manentibus. Dico triangula ACB, LHM e&longs;&longs;&etail; &longs;imilia. Sunt enim parallelæ &c. inter&longs;e tam rectæ CB, ML, quàm CA, MH; ideo anguli ACB, HML inter&longs;e æquabuntur, & &longs;unt circa eos proportionalia latera, nem. pe BC ad CA, vt LM, MH; ergo con&longs;tat propo&longs;itum.

Corollarium.

Simul con&longs;tat rectas AB, LH inter&longs;e æquidi&longs;tare.

PROP. XVIII. THEOR. XIV.

II&longs;dem vt &longs;upra manentibus, ita tamen vt ACD &longs;it an­gulus rectus (&longs;ic enim DC perpendicularis erit duabus AC, CB) Dico &longs;olidum huiu&longs;modi ad pri&longs;ma, cuius ba&longs;is ABC, & altitudo CD eandem habere rationem, quam &longs;o­lidum rotundum ortum ex rotatione figuræ CAD circ&atail; axem CD ad cylindrum genitum ex conuer&longs;ione rectan­guli AC in CD circa eundem axem.

Tab. 6. Fig. 8.

Compleatur ip&longs;um pri&longs;ma, & &longs;it quidem AQDPBC, quod &longs;ecetur vnà cum propo&longs;ito &longs;olido per quoduis pla­num ba&longs;i ACB æquidi&longs;tans: fiet in pri&longs;mate &longs;ectio trian­gulum OMN &longs;imile, æqualeque ip&longs;i ACB, & in altero &longs;o­lido triangulum LHM eidem ACB &longs;imile. Triangulum ACB pri&longs;matis ad triangulum idem &longs;olido propo&longs;ito com­mune, e&longs;t vt circulus radio CA de&longs;criptus ad circulum eundem; Item triangulum NOM &longs;ectio pri&longs;matis e&longs;t ad triangulum LHM &longs;ectionem propo&longs;iti &longs;olidi, vt circulus ex radio MO de&longs;criptus ad circulum radio MH. Cum dein­de idem dicatur de alijs omnibus &longs;ectionibus pri&longs;matis, & propo&longs;iti &longs;olidi erunt omnes &longs;imul primæ, quæ inter&longs;&etail;

æquales &longs;unt, ad omnes &longs;imul &longs;ecundas vt omnes tertiæ, his partibus inter&longs;e æqualibus, ad omnes quartas; &longs;cilicet erunt omnia triangula pri&longs;matis, &longs;eu ip&longs;um pri&longs;ma ad om­nia triangula propo&longs;iti &longs;olidi, &longs;eu ad ip&longs;um &longs;olidum, vt om­nes circuli eius cylindri, qui oritur ex conuer&longs;ione figuræ ADCA circa axem CD, hoc e&longs;t vt ip&longs;um &longs;olidum rotun­dum, &longs;eu cylindrus ad omnes &longs;imul circulos &longs;olidi rotundi geniti ex rotatione figuræ AHDCA circa axem ipsum CD, &longs;eu ad ip&longs;um propo&longs;itum &longs;olidum. Quod &c.

Imma 18. in libro de dim. parab. Euang. Trricl.

PROP. XIX. THEOR. XV.

ET rur&longs;us ip&longs;a manente figura patet, &longs;i ducantur HR, LS parallelæ MD, fore non &longs;olum figuram AHDPA, &longs;imilem, ac æqualem BLDQB; verùm etiam APRHA ip&longs;i BLSQB: Cum ita &longs;it, aio, eundem cylindrum ad &longs;oli­dum rotundum genitum, ex volutatione figuræ APD cir­ca eundem axem CD eandem rationem habere, ac pri&longs;ma prædictum, cuius ba&longs;is ACB, altitudo AP ad &longs;olidum, quod &longs;upere&longs;t ex ip&longs;o pri&longs;mate, dempto &longs;olido ACBLDHA.

Nam ex præterita propo&longs;itione nouimus, dictum pri&longs;ma ad &longs;olidum eius partem ACBLDHA e&longs;&longs;e vt cylindrus or­tus ex conuer&longs;ione rectanguli CP circa axem CD ad par­tem eius rotundum circa axem eundem CD conuer&longs;a fi­gura ADC, ergo per conuer&longs;ionem rationis, erit id quod propo&longs;uimus.

DEF. IV.

QVodcunque ex dictis propo&longs;itis &longs;olidis vocetur ab ea figura, iuxta quam intelligitur ortum. Scilicet ACBLDHA dicatur à figura AHDCA, & alte­rum, quod fuit re&longs;iduum prædictum dicatur à figura AH­DPA.

PROP. XX. THEOR. XVI.

SI à quibu&longs;cunque figuris fuerint duo &longs;olida, hæc inter­&longs;e erunt vt &longs;olida alia genita ex conuer&longs;ione illarum figurarum circa communem &longs;ectionem &longs;imilium, æqua­lium, ac inter&longs;e coeuntium figurarum.

Tab. 6. Fig. 9.

Solidum à figura ABC &longs;it CAFDBC, & quod e&longs;t à fi­gura GLH e&longs;to HGILH. Dico illud ad hoc &longs;olidum e&longs;&longs;e vt rotundum natum ex conuer&longs;ione figuræ ABC circ&atail; axem CE ad rotundum ortum ex conuer&longs;ione figuræ GLH circa axem HL. Opportet tamen angulos ACF, GHI æquales e&longs;&longs;e. Intelligantur pri&longs;mata triangularia, quorum ba&longs;es ACF, GHI, & altitudines CE, HL; hoc e&longs;t &longs;int ip&longs;a &longs;olida pri&longs;matica AFCEBD, GIHLMK. Solidum à figu­ra ABC ad pri&longs;ma AFCEBD habet eandem rationem, quam &longs;olidum rotundum ortum ex conuer&longs;ione &longs;iguræ ABC circa axem CE ad cylindrum natum ex rotatione ABEC circa eundem axem CE; hic verò cylindrus ad cy­lindrum alium natum ex rotatione rectanguli GMLH cir­ca axem HL e&longs;t vt pri&longs;ma, cuius ba&longs;is ACF, altitudineque CE ad alterum pri&longs;ma ba&longs;em habens GHI &longs;imilem ip&longs;i CF (nam circa angulos æquales H, C &longs;unt latera etiam pro­portionalia, nempe æqualia) & altitudinem HL. Solidum præterea, hoc e&longs;t pri&longs;ma GKHM ad &longs;olidum, quod e&longs;t à plano GLH habet eandem rationem, ac cylindrus, qui fit ex conuer&longs;ione rectanguli HM circa axem HL ad &longs;olidum rotundum ortum ex circumactione figuræ GLH circa ip­&longs;um axem HL, ergo ex æquali erit &longs;olidum à figura ABC ad &longs;olidum à figura GLH, vt rotundum ex rotatione figu­ræ ABC circa axem CE ad rotundum alterum ex conuer­&longs;ione alterius figuræ GLH circa axem HL. Quod &c.

18. huius.

PROP. XXI. THEOR. XVI.

PRopo&longs;itis ij&longs;dem &longs;olidis, erunt inter &longs;e, vt momenta fi­gurarum a quibus &longs;unt, quæ tamen figuræ &longs;u&longs;pen&longs;æ &longs;int ex longitudinibus deductis ab ip&longs;arum grauitatu&mtail; centris v&longs;que ad coeuntium figurarum communes illas &longs;e­ctiones.

Figuræ, à quibus &longs;unt &longs;olida, ponantur ABC, GLH, cen­tra grauitatum illarum M, N; axes, &longs;iue communes &longs;ectio­nes coeuntium binarum inter&longs;e &longs;imilium, ac æqualium fi­gurarum à quibus dicuntur ip&longs;a &longs;olida; & demum MO, NP perpendiculares &longs;int ab ip&longs;is centris ad illas communes &longs;e­ctiones deductæ CE, HL. Dico, &longs;olidum à plana figur&atail; ABC ad &longs;olidum a plana GHL eandem habere rationem, ac momentum figuræ ABC pendentis ex MO ad momen­tum alterius figuræ &longs;u&longs;pen&longs;æ ex NP, &longs;unt enim hæc &longs;oli­da inter&longs;e, vt rotunda, quorum genetrices figuræ ABC, GLH circa axes CE, HL, huiu&longs;modi verò &longs;olida &longs;unt vt momenta propo&longs;ita; ergo &longs;olidum à plana figura ABC ad &longs;olidum à plana GLH, erit vt momentum figuræ ABC &longs;u&longs;pen&longs;æ ex MO ad momentum GLH pendentis ex NP. Quod &c.

Tab. 6. fig. 10.

pr. 20. huius.

Ter. lem. 31. in libro di­. p ab

Corollarium.

Cum ip&longs;a illa momenta nectantur ex rationibus figurarum ABC, GLH, & ex longitudinibus, ex quibus pendent ip&longs;æ fi­gura (nam habentur vt grauia) ex ij&longs;dem etiam rationibus componentur &longs;olida, qua &longs;unt ab ip&longs;is figuris—

Ex mechani­cis,

PROP. XXII. THEOR. XVII.

Tab. 7. fig. 1.

IMagines velocitatum, &longs;eu &longs;patia, quæ curruntur accele­ratis motibus, &longs;unt vt &longs;olida ab imaginibus &longs;implicium motuum, ex quibus ip&longs;i gignuntur accelerati.

Sint imagines &longs;implicium motuum ABC, GLH, & &longs;oli­da ab ip&longs;is imaginibus (angulis ACQ, GHD &longs;emper re­ctis, aut &longs;altem æqualibus) intelligantur ABCRQ, GLHD. Dico, vt &longs;unt inter&longs;e i&longs;ta &longs;olida, &longs;ic e&longs;&longs;e homologè &longs;patium exactum tempore AC motu accelerato ex &longs;implici motu imaginis ABC ad &longs;patium tran&longs;actum tempore GH motu item accelerato ex &longs;implici imagine priori homogene&atail; GLH: &longs;ecetur &longs;olidum ABCRQ plano æquidi&longs;tanti QCR,

quod faciat in &longs;olido ip&longs;o &longs;ectionem TSVX: erit hæc figu­ra pror&longs;us &longs;imilis, ac æqualis conterminæ ABVI; quare cum in accelerato motu velocitas, quæ habetur momen­to C ad velocitatem momento S &longs;it vt imago ABC &longs;im­plex ad &longs;egmentum eius ABVS: erit etiam QCR æqualis ABC ad &longs;ectionem &longs;olidi TSVX, quæ æquatur ABVS, vt illa eadem velocitas momento C mobili inhærens ad ve­locitatem momento S alterius accelerati motus. E&longs;t au­tem &longs;ectio TSVX ad libitum &longs;umpta; ergo &longs;olidum ABC­QR pote&longs;t &longs;umi merito vt imago velocitatum accelerati motus, cuius &longs;implex imago ABC: & eodem modo &longs;oli­dum alterum vicem geret imaginis velocitatum alterius motus ex &longs;implici imagine GLH, itaque erit ob homoge­neitatem &longs;patium tran&longs;actum motu accelerato iuxta &longs;im­plicem imaginem ABC ad &longs;patium tran&longs;actum motu ac­celerato iuxta &longs;implicem imaginem GLH, temporibus AC, GH, vt &longs;olidum ABCQR ad ALHD,

16. huius.

4. huius,

16. huius.

Def. .3. primi huius.

& Def. 1. hu­ius vnà cum pr. 4. huius.

PROP. XXII. THEOR. XVIII.

SInt nunc CE, HL communes &longs;ectiones imaginum &longs;im­plicium ABC, GLH, &longs;i extenderentur cum &longs;ujs æqua­libus, ac &longs;imilibus coeuntibus figuris. E&longs;to pariter M cen­trum grauitatis imaginis ABC, et N grauitatis alterius ima­ginis GLH; actis demùm MO, NP perpendicularibus ad ip&longs;as CE, HL. Dico, &longs;patium accelerati motus ab imagine &longs;implici ABC ad &longs;patium accelerati alterius motus ab ima­gine &longs;implici GLH componi ex ratione imaginis ABC ad imaginem GLH, & ex ea perpendicularis MO ad perpen­dicularem NP. Cum hæc ip&longs;a &longs;patia &longs;int o&longs;ten&longs;a, vt &longs;oli­da à figuris ABC, GLH; hæc verò &longs;unt vt momenta ip&longs;a­rum figurarum &longs;u&longs;pen&longs;arum ex MO, NP. Ergo quemad­modum momenta i&longs;ta nectuntur ex rationibus figurarum tanquam magnitudinum ABC ad LGH, & di&longs;tantiarum MO ad NP, ita pariter ex his nectentur propo&longs;ita &longs;patia.

Tab. 7. Fig. 2.

21. huius.

20. huius.

Ex mechani­cis.

Corollarium.

Patet communes &longs;ectiones CE, HL e&longs;&longs;e æquidi&longs;tantes ap­plicatis AB, HL, quæ in imaginibus &longs;umuntur perpendicula­res rectis AC, GH. nam HL est recta, in quam coeunt figuraplanæ &longs;imiles, ac æquales.

Pr 2. prim&atail; huius.

PROP. XXIV. THEOR. XIX.

SI imagines &longs;implicium motuum fuerint &longs;imiles, &longs;imili­terque &longs;u&longs;pen&longs;æ, imagines velocitatum accelerato­rum motuum erunt in triplicata ratione temporum &longs;impli­cium motuum, aut in triplicata homologarum, vel extre­marum velocitatum eorundem &longs;implicium motuum.

Cum centra grauitatum &longs;imilium imaginum, &longs;eu figu­rarum, &longs;int puncta in ij&longs;dem figuris &longs;imiliter po&longs;ita, ponun­tur verò imagines &longs;imiliter &longs;u&longs;pen&longs;æ, ergo &longs;equitur ip&longs;as longitudines e&longs;&longs;e vt latera homologa dictarum imaginum, &longs;cilicet vt tempus AC ad tempus FG, vel vt extremæ ve­locitates BC ad KE. Quamobrem imagines ip&longs;æ, cum &longs;int in duplicata ratione laterum homologorum, &longs;i huic dupli­catæ addatur alia ratio &longs;imilis rationi longitudinum, fiet ratio imaginum velocitatum, &longs;eu &longs;patiorum acceleratorum motuum ex &longs;implicibus illis deriuantium triplicata tempo­rum, vel extremarum velocitatum &longs;implicium motuum.

Tab. 7. Fig. 3.

PROP. XXV. THEOR. XX.

SI verò &longs;implices motus extiterint &longs;imiles, æqualibu&longs;que temporibus ab&longs;oluantur, imagines acceleratorum motuum erunt in &longs;ola ratione amplitudinum imaginum &longs;implicium.

Tab. 7. fig. 4.

Sint imagines &longs;imilium, ac &longs;implicium motuum BAC, KFG, quarum grauitatis centra D, H, erunt ex hypothe&longs;i tempora AC, FG æqualia; & ideo &longs;patia, &longs;cilicet imagines velocitatum BAC, KFG habebunt eandem rationem, quam &longs;ummæ, aut extremæ motuum &longs;implicium velocita­tes, &longs;cilicet, quam amplitudines imaginum, &longs;eu gene&longs;um: &longs;unt verò di&longs;tantiæ DE, HI pariter æquales, quia AC, FG æquales &longs;unt; ergo cum &longs;patia acceleratorum motuum ne­ctantur ex imaginibus &longs;implicium motuum ABC, KFG, & ex di&longs;tantijs DE ad HI, liquet ip&longs;a &longs;patia e&longs;&longs;e in vnica, &longs;o­laque ratione amplitudinum BC, KG, aut amplitudinum gene&longs;um.

8 primi huius

2 primi huius

23. huius.

PROP. XXVI. THEOR. XX.

AT&longs;i &longs;implicium, &longs;imiliumque motuum fuerint imagi­nes æquè amplæ, imagines acceleratorum motuum, &longs;iue tempora erunt in duplicata ratione temporum i&longs;to­rum, vel illorum motuum.

Tab. 7. fig. 5.

Def. 7. primi huius.

23. huius.

PROP. XXVII. THEOR. XXI.

Tab. 7. fig. 6.

DEmùm &longs;i &longs;int imagines, quæcunque velocitatnm &longs;im­plicium, &longs;imiliumque motuum, imagines accelera­torum motuum, &longs;eu &longs;patia ijs motibus exacta componen­tur ex duplicata temporum ratione, & ex ea amplitudi­num, vel applicatarum homologarum earundem imagi­num.

Imagines &longs;imilium, &longs;impliciumque motuum &longs;int BAC, KFG. Dico, imagines acceleratorum motuum ab illis &longs;im­plicibus deriuantium habere rationem compo&longs;itam ex du­plicata temporum AC ad FG, & amplitudinum imaginum dictarum, vel gene&longs;um. Intelligatur alius &longs;imilis motus, cuius velocitatum imago &longs;it DFG æquèampla, ac homo­genea ip&longs;i BCA; nimirum &longs;it DG æqualis BC. Quoniam imago accelerati motus ex &longs;implici imagine BA ad imagi­nem accelerati ex &longs;implici imagine KFG componitur ex ratione imaginis accelerati motus, cuius &longs;implex imago BAC ad imaginem accelerati motus ex &longs;implici DFG, & ex imagine huius accelerati motus ad accelerati imaginem à &longs;implici KFG; e&longs;t autem prior ratio imaginum, &longs;eu &longs;pa­tiorum acceleratis motibus percur&longs;orum ip&longs;a temporum

duplicata AC ad FG, & altera dictarum imaginum, &longs;eu &longs;patiorum item acceleratis motibus confectorum, & quo­rum &longs;implices imagines &longs;unt DFG, KFG, e&longs;t eadem, ac ra­tio amplitudinum DG, &longs;eu BC ad KG. Ergo cum i&longs;tæ amplitudines &longs;int eædem, ac illæ gene&longs;um, con&longs;tar propo­&longs;itam rationem acceleratorum motuum ex &longs;implicibus imaginibus BAC, KFG habere rationem compo&longs;itam ex duplicata temporum AC ad FG, & ex ea amplitudinum imaginum &longs;implicium BC ad KG, &longs;eu amplitudinum gene­&longs;um. Quod &c.

Pr. 26 huius.

5. huius.

PROP. XXVIII. THEOR. XXII.

SI gene&longs;es &longs;imilium, &longs;impliciumque motuum fuerint æquèamplæ, imagines acceleratorum motuum erunt in duplicata ratione temporum, vel altitudinum ip&longs;arum gene&longs;um.

Gene&longs;es &longs;imilium, ac &longs;implicium motuum &longs;unto ABC, DEF, quarum amplitudines æquales &longs;int AC, DF. Dico, imagines, &longs;iue &longs;patia acceleratorum motuum e&longs;&longs;e in dupli­cata ratione temporum, vel altitudinum BC ad EF. Cum AC, DF &longs;int gradus velocitatum in extremitatibus &longs;impli­cium decur&longs;uum, etiam imagines velocitatum, iuxta ip&longs;as gene&longs;es, quæ &longs;int inter&longs;e homogeneæ, erunt æquèamplæ, & &longs;unt &longs;imilium motuum; ergo imagines acceleratorum motuum, iuxta &longs;implices illas gene&longs;es, aut imagines æquè­amplas erunt in duplicata ratione temporum: &longs;unt autem imagines velocitatum æquèamplæ, &longs;imiliumque motuum, hoc e&longs;t &longs;patia BC ad EF vt ip&longs;a tempora; ergo &longs;patia acce­leratorum, propo&longs;itorumque motuum erunt in ratione du­plicata altitudinum BC, EF &longs;implicium gene&longs;um, ABC, DEF. Quod &c.

Tab. 7. Fig. 7.

26. huius.

26. huius.

PROP. XXIX. THEOR. XXIII.

SI gene&longs;es &longs;imilium, &longs;impliciumque motuum fuerint æquèaltæ, imagines, &longs;iue &longs;patia, acceleratorum mo­tuum erunt vt tempora, vel reciprocè vt amplitudines ge­ne&longs;um ip&longs;orum &longs;implicium motuum.

Gene&longs;es &longs;imilium, &longs;impliciumque motuum, ac inter&longs;e homogeneæ &longs;int BAC, DEF, quæ habeant altitudines AC, EF æquales. Dico, imagines acceleratorum motuum e&longs;&longs;e inter &longs;e, vt tempora dictorum &longs;implicium motuum, vel reciprocè vt amplitudines ip&longs;arum gene&longs;um. Concipian-tur imagines velocitatum &longs;implicium motuum, &longs;cilicet GHI iuxta gene&longs;im BAC, et MKL iuxta alteram gene&longs;im DEF, & quia, vtpotè homogene&ecedil;, &longs;unt inter &longs;e vt &longs;patia &ecedil;qualia AC ad EF, erunt ip&longs;æ imagines &ecedil;quales inter &longs;e, cum verò ob &longs;imili tudinem motuum eæ ip&longs;æ imagines nectantur ex rationibus GI ad ML, & ex ea, quam habet HI ad KL, &longs;equitur e&longs;&longs;e GI ad ML, vt KL ad IH, & demum quia acceleratorum motuum &longs;patia à &longs;implicibus imaginibus GHI, MKL ne­ctuntur ex duplicata temporum HI ad KL, & ex ea an pli­tudinum GI ad ML, &longs;iue ex ea, quam habet KL ad HI, re­linquitur, &longs;patia acceleratis illis motibus confecta e&longs; i&ntail; &longs;ola, vnicaque ratione temporum HI ad KL, vel in ei &ecedil;qua­li ratione, reciproca amplitudinum imaginum ML ad GI, vel gene&longs;um DF ad BC. Quod &c,

Tab. 7. fig. 8.

27. huius.

PROP. XXX. THEOR. XXIV.

QVæcunque fuerint gene&longs;es &longs;imilium, &longs;impliciumque motuum, dum inter&longs;e homogeneæ, &longs;patia accelera­tis motibus ex illis &longs;implicibus exacta nectentur ex duplicata ratione altitudinum, & reciproca amplitudi­num earundem &longs;implicium gene&longs;um,

Sint quæcunque &longs;imilium motuum gene&longs;es BAC, KFG. Dico, &longs;patia acceleratorum motuum, ab ijs &longs;implicibus de­riuantium, componi ex duplicata ratione altitudinum AC ad FG, & ex ratione extremarum velocitatum, &longs;eu ampli­tudinum reciprocè &longs;umptarum ip&longs;arum gene&longs;um: e&longs;to alia gene&longs;is DFG illis homogenea, & motu pariter &longs;imilis cum ij&longs;dem gene&longs;ibus. Eadem &longs;it amplitudine æqualis BAC, & altitudo eius &longs;it FG, &longs;patia acceleratorum motuum ex &longs;implicibus gene&longs;ibus æquales amplitudines habenbus, & &longs;imilium motuum BAC, DFG &longs;unt in duplicata 'ratione rectarum, &longs;eu altitudinum AC ad FG, & &longs;patia accelera­torum motuum ex &longs;implicibus gene&longs;ibus, quæ &longs;int in ea­dem altitudine DFG, KFG, &longs;unt in reciproca ratione am­plitudinum, &longs;eu primarum vel ocitatum KG ad DG, vel BC; ex æquali igitur &longs;patia acceleratorum motuum ex propo&longs;itis &longs;implicibus gene&longs;ibus BAC, KFG nectentur ex ratione duplicata altitudinum AC ad FG, & reciproca amplitudinum KG ad BC earundem gene&longs;um BAC, KFG. Quod &c.

Tab. 7. fig. 6.

28. huius.

29. huius.

Scholium.

At quia in &longs;patijs, quæ accelerato motu peraguntur; non &longs;eruatur ratio altitudinum gene&longs;um &longs;implicium, ex quo ori­tur in hac methodo quædam percipiendi dificultas; ideo &longs;e­quenti problemate, alij&longs;que iam notis veritatibus, rem planè illu&longs;trabimus, ac &longs;imul doctrina v&longs;um trademus.

PROP. XXXI. PROB. VI.

EX datis &longs;patijs accelerato motu confectis, cogniti&longs;­que primis, aut po&longs;tremis &longs;imilium, &longs;impliciumque motuum velocitatibus, reperire tempora ip&longs;orum de­cur&longs;uum.

Spatia motibus acceleratis exacta &longs;unt C, D, & velo­tates, &longs;eu amplitudines gene&longs;um ponantur e&longs;&longs;e A, B, &longs;cili­cet A principio motus per C, & B initio motus per D, quæ­ritur ratio temporum, quibus exiguntur propo&longs;ita &longs;patia. Vt A ad B, ita fiat C ad E, & inter E, et D &longs;umatur F me­dia proportionalis. Dico ip&longs;a tempora e&longs;&longs;e vt E ad F. Componuntur &longs;patia acceleratis motibus exacta ex ratio­ne quadratorum temporum, & ex ea amplitudinum, &longs;eu homologarum velocitatum in &longs;implicibus motibus, &longs;imili­bu&longs;que &longs;umptarum; & ideo temporum quadrata necten­tur ex ratione &longs;patiorum C ad D, & ex reciproca ampli-tudinum E ad C; temporum igitur quadrata erunt vt E ad D, ip&longs;a verò tempora vt E ad F. Quod &c.

Tab. 8. Fig. 1.

27. huiuij

lem. pr. 3. pri­mi huius.

PROP. XXXII. PROB. VII.

EXdatis &longs;patijs accelerato motu tran&longs;actis, datis item primis velocitatibus &longs;imilium, &longs;impliciumque mo­tuum, inuenire altitudines &longs;implicium gene&longs;um, ex quibus propo&longs;ita &longs;patia effecta &longs;unt.

Tab. 7. fig. 1. 30. huius.

Spatia &longs;int E, D reliquis, vt &longs;upra, manentibus: quoniam &longs;patia accelerato motu tran&longs;acta componuntur ex ratio­nibus amplitudinum gene&longs;um &longs;implicium, &longs;imiliumqu&etail; motuum reciprocè &longs;umptarum B ad A, &longs;iue E ad C, & ex ea quadratorum altitudinum ip&longs;arum gene&longs;um; erit ratio dictarum altitudinum duplicata C ad D; quare F, &longs;i &longs;it me­dia proportionalis, non inter E, & D (vt antea po&longs;uimus) &longs;ed inter C ad D; erit &longs;anè C ad F ratio altitudinum gene­&longs;um &longs;implicium, &longs;imiliumque motuum, quam quereba­mus.

Exemplum primum.

SI idem graue naturaliter cadens percurrerit à quiete duo &longs;patia; tempora erunt in ratione &longs;ubduplicat&atail; corundem &longs;patiorum.

Ex Cor. pr: 4. huius con&longs;tat rectangula e&longs;&longs;e gene&longs;es &longs;im­plicium motuum grauium naturaliter de&longs;cendentium, & ex def. 7. primi liquet ea&longs;dem gene&longs;es e&longs;&longs;e motuum &longs;imi­lium. Cumque eiu&longs;dem mobilis naturaliter cadentis ve­locitas à quiete &longs;it vna, eademque; &longs;implices motus erunt ij, vt gene&longs;um &longs;imilium, &longs;impliciumque motuum amplitu­dines æquales &longs;int, proptereaque, vt in figura præcedentis propo&longs;itionis æquales erunt C, E, atque adeo &longs;patiu&mtail; C, &longs;iue E ad D erit in duplicata ratione temporum E ad F.

Exemplum II.

PROP. XXXIV. THEOR. XXVII.

TEmpora &longs;imilium vibrationum &longs;unt in &longs;ubduplicata ratione arcuum exactorum, &longs;eu longitudinum pen­dulorum, quorum &longs;unt vibrationes. Sint grauia pendula LA, LF, quæ ab eadem recta LF di&longs;cedentia currant &longs;u&longs;­pen&longs;a ex L duos &longs;imiles arcus circulares FI, AC. Dico tempora horum de&longs;cen&longs;uum e&longs;&longs;e in ratione &longs;ubduplicat&atail; arcuum FI, AC, &longs;eu longitudinum filorum, aut ha&longs;tularum FA, LA. Ducamus quamcumque rectam LBG, erit AB ad BC, vt FG ad GI, & cum præterea velocitates pendu­lorum a quiete in A, F &longs;int æquales, pariterque velocita­tes æquales a quiete in B, G; erit velocitas in A ad veloci­tatem in B, vt velocitas in F ad velocitatem in G, quare con&longs;ideratis arcubus ABC, FGI, vt altitudines rect&ecedil;, (quæ item forent in B, G proportionaliter &longs;ect&ecedil;) gene&longs;um &longs;imi­lium &longs;impliciumque motuum, quarum amplitudines æqua les &longs;unt, erunt &longs;patia in acceleratis decur&longs;ubus per FI, AC in ratione duplicata temporum, &longs;cilicet ip&longs;i arcus, aut lon­gitudines LF, LA erunt in ratione duplicata temporu&mtail;. Quod &c.

Tab. 8. fig. 2.

Def. 7. pri

Idem demon&longs;tratum e&longs;&longs;et beneficio imaginum, quæ vt­pote eorundem illorum motuum &longs;implicium, forent etiam &longs;imilium, & &longs;unt amplitudines æquales, etenim eæde&mtail; &longs;unt, ac gene&longs;um ergo rur&longs;us &longs;patia, hoc e&longs;t arcus ABC, FGI, nempe longitudines filorum IF, AC erunt in ratione duplicata temporum. Quod &c.

Scholium.

Vides, quàm breuiter rei di&longs;ficillimæ demon&longs;tr ationem at­tulimus, nec dubium, quin illa extendi queat ad qua&longs;cum­que lineas decur&longs;uum, dummodo &longs;imiles, ac &longs;imiliter po&longs;itas in ij &longs;dem, vel æqualibus ab horizonte planis elenatis, quemad­modum Dominus Viuianus pulcherrimè propo&longs;uit.

Exemplum III.

PROP. XXXV. THEOR. XXVIII.

TEmpora lationum à quiete per plana eandem eleua­tionem habentia &longs;unt homologè vt longitudines planorum.

Tab. 8. fig. 3.

Sint plana AB, AC eandem eleuationem AD habentia. Dico tempus lationis per AC ad id per AB e&longs;&longs;e vt AC ad AB. (hæc Torricellij propo&longs;itio, expo&longs;itioque e&longs;t, hancque eandem veritatem ex no&longs;tris principijs demon&longs;trare visum e&longs;t, non vt de re illa dubitemus, immò contrà, quòd de e&atail; plenè &longs;atisfacti &longs;imus, ex eo rur&longs;us demon&longs;trandam &longs;u&longs;ce­pimus, vt exinde methodus no&longs;tra, quàm vera &longs;it, eluce&longs;­cat) Momentum de&longs;cen&longs;us inplano AC ad id de&longs;cen&longs;us &longs;u­per plano AB e&longs;t vt AB ad AC; &longs;unt autem de&longs;cendentium grauium, etiam &longs;uper planis inclinatis motus, quos &longs;impli­ces appellamus, inter &longs;e &longs;imiles, nempe quorum gene&longs;es &longs;unt rectangula; ergo habebimus &longs;implices gene&longs;es, vnam, cuius altitudo AC amplitudoque AB; alteram, cuius am­plitudo AC, altitudo autem AB; itaque propo&longs;itis &longs;patijs AC, AB, primi&longs;que velocitatibus AB, AC, &longs;i fiat AB ad AC vt CA ad EA, erit EA ad AB duplicata temporum, & ideo ratio temporum per AC, AB erit CA ad AB. Quod &c.

Tor. pr. 2. de motu grauium.

Cor pr. 4. huius.

31. i 27. hu.

Exemplum IV.

PROP. XXXVI. THEOR. XXIX.

II&longs;dem pror&longs;us manentibus demon&longs;trarunt Gallileus, ac Torricellius, gradus velocitatum acqui&longs;itos in B, et C eiu&longs;dem mobilis de&longs;cendentis à quiete in A pares e&longs;&longs;e; idip&longs;um nos o&longs;tendemus.

Cum tempora &longs;int vt AC ad AB, & velocitates à quie­te in ratione reciproca temporum, &longs;cilicet vt AB ad AC, &longs;int deinde velocitates eæ vt amplitudines imaginum &longs;im­plicium, &longs;imiliumque illorum motuum (nam amplitudines imaginum velocitatum &longs;unt pror&longs;us eædem, ac illæ gene­&longs;um) erunt ip&longs;æ imagines &longs;implicium motuum æquales; nam tempora, quæ &longs;ummuntur vt altitudines imaginum reciprocantur, vt dictum e&longs;t, amplitudinibus, &longs;eu primis à quiete velocit atibus, at in motibus acceleratis ip&longs;æ inte­græ imagines &longs;implicium motuum &longs;unt loco graduum ve­locitatum in extremo &longs;patiorum acqui&longs;itorum; ergo in B, et C gradus velocitatum æquales erunt.

Tab. 8. fig. 4. 33. huius.

4. huius.

PROP. XXXVII. THEOR. XXX.

SI æqualia pondera, &longs;u&longs;pen&longs;a &longs;int ex filis, quorum par­tes inter&longs;e æquales, præ tractione æqualiter elongen­ter tempora in reditu ip&longs;orum filorum, cum ab ip&longs;is graui­bus &longs;tatim liberantur, æqualia erunt. Hoc primùm demon­&longs;trabimus alia via, tum methodo no&longs;tra, vt de ea aliud exemplum tradamus. Sint funiculi AB, DC, & ex ijs pendeant æqualia grauia B, C, adeo vt &longs;umptis hinc indè partibus æqualibus eorundem &longs;uniculorum, con&longs;tet ip&longs;as æqualiter ab ip&longs;is grauibus trahi, atque produci. Dico, &longs;i elongationes &longs;int HB, GC, & omnibus &longs;ic &longs;tantibus pon-dera &longs;ubmoueantur ex B, et C funiculis eæ&longs;is, fore vt eæ­dem extremitates re&longs;tituantur in H, et G æqualibus tem­poribus. Sit AE æqualis DC, erit porrò elongatio facta per idem graue B, quæ &longs;it EF, æqualis GC; propterea li­beratis funiculis ad B, et C, eodem tempore re&longs;tituetur C in G, ac E in F, quo tempore etiam B in H re&longs;titutum fue­rit; nam vno puncto in primum &longs;uum locum redito, etiam alia &longs;ingula in &longs;uum locum perueni&longs;&longs;e, opportebit.

Tab. 8. fig. 5.

Exemplum.

HAc occa&longs;ione de funiculis erit non iniucunda di&longs;er­tatio, remque &longs;ic adhuc intactam promouebimus, &longs;imulque demon&longs;trabimus.

Idip&longs;um propo&longs;itum no&longs;tris principijs &longs;ic demon&longs;tra­mus.

Sint eadem, quæ &longs;upra, &longs;cilicet conceptis in filo AB quot­libet partibus inter&longs;e æqualibus, longitudinenque totam im­plentibus, hæ &longs;ingulæ æqualiter à pondere B trahentur, eritque BH &longs;umma omnium dictarum partium elongatio­num, & eodem pacto EF erit &longs;umma elongationum partium omnium in AE contentarum, ab eodemque pondere effe­ctarum; propterea vt AB ad BH, ita erit AE ad EF; quamo brem velocitas etiam puncti B &longs;ublato pondere B erit ad velocit atem puncti E ob eandem detractionem, vt BH ad EF, vel BA ad EA (nam quot &longs;unt partes concept&ecedil; i&ntail; vtraque fili longitudine, totidem &longs;unt etiam impetus inter &longs;e æquales) idem o&longs;tenderemus &longs;i loco ponderis B, minus quodcumque &longs;u&longs;penderemus, vt &longs;cilicet puncta B, et E ad quemuis locum &longs;uperius remanerent, librarenturque cum re&longs;i&longs;tentijs partium eò elongatarum, ergo tran&longs;itus ex B in H, & puncti E in F &longs;ubducto pondere B erunt motus &longs;imilium &longs;impliciumque; &longs;ed motus ex C in G exempto pondere C e&longs;t pror&longs;us idem, ac motus E in F, ergo motus &longs;imiles, ac &longs;implices ex B in H, & ex C in G, ex quibus fiunt accele­rati, gene&longs;es habebunt, quarum primæ velocitates, &longs;eu am­plitudines proportionales &longs;unt altitudinibus earundem, &longs;patijs nimirum CG, BH accelerato motu exigendis; qua­mobrem componentur ex ratione ip&longs;arum velocitatum, &longs;eu amplitudinum CG ad BH, & ex ea quadratorum tem­porum, quæ proinde æqualitatis erit; itaque etiam huius &longs;ubduplicata; hoc e&longs;t tempora in tran&longs;itibus accelarato motu exactis, erunt paria.

pr. 4. huius.

Corollarium.

Hinc patet, vbi æquè cra&longs;&longs;is filis eiu&longs;demque materiei vel cedentiæ &longs;u&longs;pen&longs;a &longs;int æqualia pondera, tunc primas velocita­tes, &longs;ubductis ponderibus, fore in eadem ratione elongationum, vel longitudinum filorum.

PROP. XXXVIII. THEOR. XXXI.

SI extremitatibus funiculorum ex vna parte firmatorum, ac eandem cra&longs;&longs;itiem habentium, nec non eiu&longs;dem cædentiæ exi&longs;tentium, fuerint &longs;u&longs;pen&longs;a æqualia pondera, quæ inde ij&longs;dem longitudinibus &longs;eruatis, quomodo opor­tet tollantur, erunt &longs;patia recur&longs;uum, temporibus &longs;impli­cium motuum exacta in ratione longitudinum pendulo­rum.

Sit funiculus AC æquè cra&longs;&longs;us ac BD, & &longs;u&longs;pen&longs;is hinc inde ponderibus æqualibus, elongatio primi funiculi &longs;it CE, & alterius &longs;it DF. Dico &longs;patia temporibus &longs;impli­cium imaginum, ab extremitatibus &longs;olutis exacta, fore i&ntail; ratione longitudinum ip&longs;orum funiculorum.

Iam con&longs;tat CE ad DF e&longs;&longs;e, vt AC ad BD, in qua ratione &longs;unt etiam velocitates à quiete, dum pondera &longs;ubduceren­tur ex E, et F, vel ex alijs punctis quibu&longs;cunque &longs;i æqualia pondera &longs;u&longs;pen&longs;a fui&longs;&longs;ent maioris, vel minoris ponderis, &longs;ic enim concipiuntur gene&longs;es &longs;imilium, &longs;impliciumque motuum, quarum altitudines æquantur elongationibus funiculorum; propterea &longs;patia recur&longs;uum temporibus &longs;im­plicium motuum exacta, nectentur ex rationibus duplicata CE ad DF, hoc e&longs;t AC ad BD, & ex reciproca filorum, &longs;cilicet BD ad AC, quæ ratio, vti diximus, e&longs;t reciproc&atail; primarum velocitatum, &longs;eu amplitudinum gene&longs;um &longs;impli­cium, ergo ip&longs;a &longs;patia in reditu filorum ab extremitatibus &longs;olutis exacta, erunt vt AC ad BF, &longs;eu vt CE ad DF. Quod &c.

PROP. XXXIX. THEOR. XXXI.

TEmpora &longs;implicium, &longs;imiliumque dictorum motuum &longs;unt æqualia.

Nam cor. 2. pr. 8. huius primi demon&longs;tratum e&longs;t, tem­pora &longs;implicium, &longs;imiliumque motuum componi ex ratio­ne &longs;patiorum, &longs;eu altitudinum gene&longs;um, & reciproca pri­marum, aut extremarum velocitatum, &longs;eu amplitudinum gene&longs;um: &longs;unt autem altitudines gene&longs;um tractiones, &longs;eu elongationes funiculorum, quæ &longs;unt vt longitudines funi­culorum, ergo tempora æqualia erunt.

Corollarium.

Con&longs;tat, tempora a &longs;implicium gene&longs;um in tractionibus fu­niculorum, e&longs;&longs;e compo&longs;ita ex ratione elongationum funiculo­rum, & ex reciproca primarum velocitatum.

Scholium.

Superioris propo&longs;itionis veritas concordat cum prop. 37. bu­ius, in eo tantùm variatur, quod ibi ponuntur data &longs;pati&atail; elongitiones funiculorum, hic verò tempora &longs;impliciu&mtail; motuum, & quia elungationes o&longs;ten&longs;æ &longs;unt proportionales &longs;pa tijs nunc exactis, manife&longs;tum e&longs;t, no&longs;tri iuris e&longs;&longs;e modò &longs;patia acceleratis motibus exact a ex temporibus &longs;implicium motuum datis concludere, modò contrà, ex &longs;patijs altitudinibus gene­&longs;um proportionalibus, qua item data &longs;unt, tempora inuenire, qua proinde methodus mihi videtur ampli&longs;&longs;ima.

PROP. XXXX. THEOR. XXXIII.

SI eiu&longs;dem cra&longs;&longs;itiei funiculis pondera dependeant, qu&ecedil; &longs;int in ratione reciproca longitudinum ip&longs;orum funi­culorum, &longs;patia temporibus gene&longs;um &longs;implicium motuum exacta erunt in ratione duplicata elongationum.

Tab. 8. Fig. 6.

Nam &longs;i &longs;it pondus E ad F &longs;icuti longitudo DB ad CA, & &longs;int, cra&longs;sities funiculorum æquales erit &longs;anè ratio, quæ componi­tur ex ratione funiculorum, & ex ea ponderum, æqualitatis; ob idque gene&longs;es &longs;implicium motuum, quarum altitudines CE, DF habebunt amplitudines, nempe primas velocitates inter&longs;e&ecedil;qua les (nam cum pondera erant æqualia, primæ velocitates proportionabantur longitudinibus funiculorum, ideo, cum

pondera reciprocantur longitudinibus ij&longs;dem, &longs;eu viribus funiculorum, fit vt primæ velocitates æquales reddantur) cum ergo ita &longs;it, &longs;patia recur&longs;uum temporibus imaginu&mtail; &longs;implicium & accelerato motu confecta erunt in ratione duplicata elongationum.

28. huius.

Corollarium.

Cum ex eadem pr. 28. huius, eadem &longs;patia &longs;int vt quadra­ta temporum, erunt ip&longs;a tempera in ratione &longs;ubduplicat&atail; elongationum.

PROP. XXXXI THEOR. XXXIV.

SI funiculis æqualem cra&longs;&longs;itiem habentibus fuerint &longs;u&longs;­pen&longs;a inæqualia pondera, &longs;patia, quæ acceleratis mo­tibus, ac temporibus gene&longs;um &longs;implicium recurruntur ne­ctentur ex ratione duplicata elongationum, & ex duabus reciprocè &longs;umptis rationibus, nempe longitudinum prima­rum funiculorum, antequam pondera &longs;u&longs;penderentur; & ip&longs;orum ponderum.

Tab. 8. Fig. 6.

In antecedenti figura illud primum &longs;atis patet, quòd &longs;i loco ponderis F &longs;u&longs;pen&longs;um fui&longs;&longs;et pondus aliud grauius, aut leuius, prior velocitas in a&longs;cen&longs;u fili, &longs;eu funiculi, aut chordæ aucta, vel imminuta fui&longs;&longs;et pro magnitudine pon­deris &longs;ub&longs;tituti; quamobrem priores velocitates ex inæqua litate ponderum eidem chordæ &longs;u&longs;pen&longs;orum dependentes forent, vt ip&longs;a pondera; verùm cum &longs;uppo&longs;itis funiculis æqualia pondera &longs;u&longs;pen&longs;a veniunt, primæ velocitates &longs;unt vt longitudines funiculorum, ergo velocitates primæ, cum inæqualia &longs;unt pondera, quæ &longs;ubtrahuntur, nectentur ex ratione longitudinum funiculorum, & ex ea ponderum inæqualium: quæcumque igitur &longs;it tractio DF, gene&longs;es ha­bebimus &longs;imilium &longs;impliciumque motuum, vnam, cuius al­titudo CE, & alteram habentem altitudinem DF, & &longs;unt earundem gene&longs;um amplitudines, &longs;eu primæ velocitates in ratione compo&longs;ita funiculorum AC ad BD, & ponderis pendentis ex E ad pondus &longs;u&longs;pen&longs;um in F; ergo &longs;patia ac­celeratis motibus tran&longs;acta temporibus gene&longs;um &longs;implicium nectentur ex ratione dublicata elongationum, &longs;iue altitu­dinum gene&longs;um, & ex duabus rationibus reciprocè &longs;um­ptis funiculorum AC ad BD, & ponderum E ad F. Quod &c.

Cor. pr. 37. huius.

30. huius.

PROP. XXXXII. THEOR. XXXV.

II&longs;dem po&longs;itis, &longs;i &longs;patia recur&longs;uum erunt ip&longs;æ elongatio­nes, tempora, quibus ab extremitatibus &longs;olutis recur­runtur, erunt in ratione &longs;ubduplicata eorundem. Nam cum gene&longs;es &longs;imilium, &longs;impliciumque motuum &longs;int æquè am­plæ, erunt, tempora in ratione &longs;ubduplicata imaginum, &longs;eu &longs;patiorum acceleratorum motuum, &longs;unt verò &longs;patia ip&longs;æ elongationes; ergo &c.

PROP. XXXXIII. THEOR. XXXVI.

CHordæ non eiu&longs;dem cra&longs;&longs;itiei, eiu&longs;dem tamen mate­riæ, ac longitudinis, tunc æquè trahentur bi &longs;u&longs;pen­&longs;a pondera cra&longs;&longs;itut inibus proportionalia fuerint. Nam cra&longs;&longs;ior chorda pote&longs;t concipi compo&longs;ita ex funiculis eiu&longs; dem cra&longs;&longs;itiei alterius chordæ, &longs;i illa huius fuerit multiplex, & &longs;i partes exilior funiculus fuerit alterius cra&longs;&longs;ioris, erit cra&longs;&longs;ities alicuius alterius funiculi, quæ pluries accept&atail; con&longs;tituere poterit vtranque cra&longs;&longs;itiem funiculorum pro­po&longs;itorum (hìc enim non accidit enumerare cra&longs;&longs;ities in­ter&longs;e irrationales, quippe quia, quod de iam dictis o&longs;ten­derimus, de his quoque facilè e&longs;t iudicare, &longs;ecùs e&longs;&longs;emus longi, quam par e&longs;t, poti&longs;&longs;imùm cum hæc præter in&longs;titutum adijciantur, & quidem vt con&longs;tet, quomodo methodus i&longs;ta no&longs;tra facilis &longs;it, ac vtili&longs;&longs;ima) quapropter &longs;i cuique acce­ptarum æqualium chordarum, pondera æqualia &longs;u&longs;pen&longs;a &longs;int, porrò hæc omnes æquè trahentur ab ip&longs;is æqualibus ponderibus, & &longs;ic etiam compo&longs;ita, nempe choidæ pro-po&longs;itæ; &longs;untque ita pondera in eadem ratione cra&longs;&longs;itierum, &longs;icut propo&longs;uimus; ergo patet propo&longs;itum.

PROP. XXXXIV. THEOR. XXXVII.

SI fuerint eiu&longs;dem materiæ funiculi, & &longs;int illis &longs;u&longs;pen&longs;a pondera cra&longs;&longs;itiebus proportionalia, ratio &longs;patiorum in reditibus accelerato motu exactorum, temporibus &longs;im­plicium gene&longs;um, erit eadem ac funiculorum.

Tab. 8. fig. 6. 42. huius.

Nam, vt in præcedenti figura, erit tractio CE ad DF ita AC ad BD, vel AE ad BF, &longs;unt autem primæ velocitates, &longs;eu amplitudines gene&longs;um &longs;implicium, &longs;imiliumque motuum in ratione funiculorum, ergo decur&longs;uum &longs;patia motibus acceleratis exacta nectentur ex ratione duplicata altitu­dinum gene&longs;um &longs;implicium, nempe duplicata funiculorum, & reciproca amplitudinum, &longs;untque ip&longs;æ amplitudines homologè vt longitudines funiculorum, ergo relinquitur vt ip&longs;a &longs;patia &longs;int in vnica ratione longitudinum funicu­lorum.

Cor. pr. 37. huius.

27. huius.

Quòd &longs;i &longs;patia recur&longs;uum ponantur ip&longs;æ tractiones, vel longitudines funiculorum, o&longs;tendetur tempora e&longs;&longs;e æqua­lia, quemadmodum æqualia &longs;unt tempora &longs;uperius pro­po&longs;ita &longs;implicium gene&longs;um.

PROP. XXXXV. THEOR. XXXVIII.

Tab. 8. fig. 7.

SI eiu&longs;dem materiei quibu&longs;cunque funiculis aligentur quæcunque pondera, ijs &longs;ublatis a&longs;cen&longs;uum &longs;patia ab extremitatibus &longs;olutis exacta temporibus gene&longs;um &longs;impli­cium, ijs nempe quæ impenderentur in motibus iuxta &longs;im­plices gene&longs;es, erunt in ratione compo&longs;ita quadratorum elongationum chordarum, ex ea cra&longs;&longs;itierum, & ex duabus reciprocè &longs;umptis rationibus, nempe longitudinum fu­niculorum antequam traherentur; & &longs;u&longs;pen&longs;orum ponde­rum.

Funiculi AB, GH trahantur à ponderibus quibu&longs;cunque C, I in C, et I. Dico &longs;i exempta &longs;int pondera, fore, vt &longs;patia quæ acceleratis motibus exiguntur ab extremitatibus &longs;o­lutis C, I &longs;int in ratione compo&longs;ita ex duplicata IH ad BC, cra&longs;&longs;itudinis ad cra&longs;&longs;itudinem funiculorum AB, GH; dein­de ex funiculi longitudine HG ad longitudinem AB, pon­deri&longs;que I ad pondus C. Intelligatur funiculus, &longs;eu chor­da, æque cra&longs;&longs;a, ac &longs;imiliter cedens, quàm GH (id quod &longs;emper intelligimus quoties funiculi, inter&longs;e comparantur) &longs;ed æquè longa, ac AB, &longs;itque illi pondus F adiectum, ad quod C eandem habeat rationem, ac cra&longs;&longs;ities AB ad cra&longs;­&longs;itiem DE, con&longs;tat elongationem EF æqualem fieri ip&longs;i CB, & cum primæ velocitates, &longs;eu amplitudines æquè al­tarum gene&longs;um &longs;imilium, &longs;impliciumque motuum &longs;int etiam æquales, &longs;patia decur&longs;uum acceleratis motibus exacta erunt pror&longs;us æqualia; &longs;unt verò funiculi DE, GH eiu&longs;dem cra&longs;­&longs;itiei, ei&longs;que &longs;unt &longs;u&longs;pen&longs;a duo'pondera inæqualia F, I; ergo decur&longs;uum &longs;patia ab extremitatibus &longs;olutis exacta necten­tur ex ratione duplicata elongationum FE, &longs;eu CB ad IH, ex ratione, quam habent longitudines funiculorum HG ad DE, &longs;eu AB, & ex ea ponderum I ad F; verùm pondera I ad F nectuntur ex rationibus ponderum I ad C et C ad F, quæ po&longs;trema e&longs;t ratio cra&longs;&longs;itiei funiculi AB ad craffitiem funiculi DE, &longs;eu GH; ergo vt propo&longs;uimus &longs;patia accele­ratis motibus exacta, nectentur ex rationibus quadratorum CB ad HI; cra&longs;&longs;itudinum funiculorum AB, GH; ponderum I ad C, & longitudinum HG ad AB. Quod &c.

PROP. XXXXVI. THEOR. XXXIX.

TEmpora gene&longs;um &longs;implicium, dum chordis &longs;u&longs;pen­duntur quæcunque grauia, nectuntur, ex ratione elongationum funiculorum, & ex contrariè &longs;umptis ratio nibus, cra&longs;&longs;itudinum, longitudinumque funiculorum, nec non ponderum funiculis &longs;u&longs;pen&longs;orum.

Nam Cor: 2. pr. 8. primi demon&longs;tratum e&longs;t, tempor&atail; &longs;implicium &longs;imiliumque motuum componi ex ratione &longs;pa­tiorum, &longs;eu altitudinum gene&longs;um, & reciproca primarum velocitatum, &longs;eu amplitudinum gene&longs;um, &longs;unt autem alti­tudines gene&longs;um tractiones, &longs;eu elongationes funiculorum; velocitatesverò primæ nectuntur ex rationibus cra&longs;&longs;itudi­num, & ex ea longitudinum funiculorum antequam tra­herentur (hoc enim &longs;ubinde o&longs;tendemus) ergo tempora propo&longs;ita &longs;implicium gene&longs;um, dum chordis alligantur qu&ecedil;­cunque inæqualia pondera, componentur ex rationibus elongationum funiculorum, & ex contrariè &longs;umptis cra&longs;&longs;i­tudinum, longitudinumque funiculorum, & ponderum.

Aßumptum.

Tab. 8. fig. 7.

VErùm primæ velocitates in ij&longs;dem chordis componi ex ratione cra&longs;&longs;itudinum, longitudinum funiculorum, & &longs;u&longs;pen&longs;orum ponderum, &longs;ic o&longs;tendemus,

Quoniam in eadem po&longs;trema figura velocitas, qua&mtail; haberet funiculus AB ex liberatione ponderis e&longs;t æqualis velocitati, quam haberet alius funiculus, vbi hic etiam li­

beraretur à pondere, &longs;cilicet cum pondera cra&longs;&longs;itiebus fu­niculorum proportionalia &longs;unt, & ip&longs;i funiculi æquè longi; velocitas funiculi DE à pondere F ad velocitatem eiu&longs;dem funiculi, &longs;i loco ponderis F &longs;ub&longs;titutum e&longs;&longs;et aliud æquale ip&longs;i I, e&longs;&longs;et vt pondus F ad &longs;ub&longs;titutum, &longs;eu ad I, e&longs;t autem velocitas eiu&longs;dem funiculi DE, dum fui&longs;&longs;et pondus ei &longs;u&longs;­pen&longs;um æquale I ad velocitatem funiculi GH a pondere I vt longitudo DE ad GH; ergo patet propo&longs;itum.

43. huius.

Ex 41. huius.

Scholium.

Quod hucu&longs;que ostendimus in funiculis ponderibus de­grauatis, non ab&longs;imili modo præst abimus in chordis ad vtran­que extremitatem firmatis, & adductis, hoc tantum di&longs;crimi­ne, vt &longs;i in ijs pondere &longs;ublato, motus extremitatis &longs;olutæ at­tendebatur, hìc media parte attractâ chordâ, & &longs;ubinde &longs;ui iuris relictâ, vibrationem eius ob&longs;eruamus, & equidem illa omnia in hunc finem o&longs;tendimus, quippe ab hac re, plurima vtili&longs;&longs;imæque veritates manere po&longs;&longs;unt. Nam de arcubus po&longs;&longs;es pulcherrima in&longs;titui ratio, & qui vellet armonicorum &longs;ono­rum, vel vocum per chordarum vibrationes editarum, tempo­ra, cum &longs;oni ad aures perueniunt, inue&longs;tigare, reor non aliam viam, quàm hanc ingredi nos debere, atque indè con&longs;onantia­rum forta&longs;&longs;e naturam pertipere po&longs;&longs;e, vt primus ui Gal­lileus quamquam vibrationes ten&longs;arum chordarum differant ab ijs pendulorum.

FINIS.

SPIEGATIONE

di vna nuoua &longs;pecie di Bale&longs;tra.

Tab. 9.

IN que&longs;ta figura &longs;i e&longs;prime vna nuoua inuen­tione di Bale&longs;tra, la quale, ma&longs;&longs;imamente in grande, per tirar granate, ò &longs;a&longs;&longs;i può e&longs;­&longs;ere di gran con&longs;eguenze nella militare, co­me dimo&longs;trera&longs;&longs;i.

Dalle &longs;ue parti &longs;i verrà in cognitione del modo di fabbricarla, e &longs;ono le &longs;eguenti.

AM, MN &longs;ono amendue le braccia. Il punto M è il cen­tro della machina. Per la cauità M deue pa&longs;lar la pall&atail; &longs;cagliata dalla corda; e per di &longs;otto M &longs;i ferma & inca&longs;lra nel manico, al modo delle bale&longs;tre communi. Ai due capi, ò &longs;iano e&longs;tremità A, N &longs;i annette la fune. I punti A, E, F, G, I, K &longs;ono in vna linea retta. Gl' interualli AE, EF, FG, GI, IK, &longs;ono, benche non di nece&longs;&longs;ità, eguali. Le altezze, ò comme&longs;&longs;ure KL, IH, GD, FC, EB perpendiculari, nell' incuruar&longs;i dell' arco, &longs;i aprono intorno a' centri K, I, G, F, E. Donde ne &longs;iegue, che prendendo la corda dal &longs;uo mez­zo, e tirandola ver&longs;o O; amendue le braccia &longs;i aprono nel­le predette comme&longs;&longs;ure, come compare nell' vno d'e&longs;&longs;i &longs;e­gnato a punti con le lettere corri&longs;pondenti. Cia&longs;cuna delle predette comme&longs;&longs;ure viene &longs;trettamente rin&longs;errat&atail; da vna molla, come &longs;i vede in L, H, D, C, B; e que&longs;te mol­le, quanto più &longs;i auuicinano al centro M, deuono e&longs;&longs;ere più grandi e più ma&longs;&longs;iccie, in modo che, per cagione della gran­dezza opportuna, vengano ad aprir&longs;i con egual facilità dell'altre, e per cagione della gro&longs;&longs;ezza, habbiano nel &longs;er­rar&longs;i maggior forza, ò &longs;ia momento, per la ragione, che &longs;ot­to &longs;i dirà.

Ciò pre&longs;uppo&longs;to, è facil co&longs;a dimo&longs;trare i vantaggi di que&longs;ta machina &longs;opra le ordinatie.

Primieramente nel triangolo ALK, e&longs;&longs;endo le altezz&etail; EB, FC, GD, IH, KL perpendicolari, e perciò paralelle; ne &longs;iegue che le proportioni di AE ad EB, di AF ad FC, di AG a GD, di AI ad IH, di AK a KL &longs;ieno tutte eguali; e douendo e&longs;&longs;ere parimente eguali le re&longs;i&longs;tenze delle molle in B, C, D, H, L, che &longs;i &longs;uppongono di egual neruo nell' aprir&longs;i; ne &longs;iegue (&longs;econdo i principij della Meccanica) che attraendo&longs;i con la fune l'e&longs;tremità A, nel mede&longs;imo tempo e con la mede&longs;ima facilità vincera&longs;&longs;i l'equilibrio di tutte le molle; la re&longs;i&longs;tenza delle quali &longs;i con&longs;idera in ragione di pe&longs;o, &longs;i come le linee AE, EB; AF, FC; AG, GD; &c. &longs;i con­&longs;iderano come vetti, ò lieue, che hanno i loro ippomoclij, ò &longs;iano centri in E, F, G, I, K, e la potenza in A, la quale è comune a tutte.

In &longs;econdo luogo, hauendo il braccio AE al braccio EB (il &longs;imile dica&longs;i degli altri) hauendo, dico, gran proportio­ne, re&longs;terà molto ageuolato il moto.

Terzo e&longs;&longs;endo molte le molle, e a prendo&longs;i tutte, ne deue &longs;eguire vn notabile incuruamento d'amendue le braccia; onde la&longs;ciando l'arco in libertà, e chiudendo&longs;i tutte le &longs;u­det te molle nel mede&longs;imo tempo, cioè qua&longs;i in vn'attimo; dourà la corda, che era tirata ver&longs;o O, pa&longs;&longs;are qua&longs;i in i&longs;tan­te ver&longs;o M; il che non potendo&longs;i fare &longs;e non con &longs;omma ve­locità, per la grandezza dello &longs;patio; e a que&longs;ta corri&longs;pon­dendo la forza, ne &longs;eguirà vn colpo molto con&longs;iderabile, e vantaggio&longs;o, come cia&longs;cuno può arguire.

Re&longs;tano hora a &longs;eior&longs;i alcune difficoltà. La prima è, che, quantunque &longs;ia vero, che quella forza ba&longs;tante in A per vincer l'equilibrio della molla B, quella mede&longs;ima al­tre&longs;i &longs;ia &longs;ufficiente a vincer l' equilibrio di tutte l'altre, per e&longs;&longs;ere eguali le proportioni delle vetti; ciò non o&longs;tante, con­&longs;iderando&longs;i il braccio incuruato, come &longs;i vede nell' arco KLA &longs;egnato a punti, le proportioni rie&longs;cono alterate; do-uendo&longs;i prendere per le lunghezze delle vetti &longs;udette, non più le lunghezze di prima, ma bensi le applicate di detto arco, cioe af, ag, ai, ak; delle quali aK, e l' altre a lei più vicine &longs;i abbreuiano molto più quando l' arco è incurua­to, che quando non è: Onde per tal ragione dourebbero le parti più vicine al centro M aprir&longs;i meno dell'altre più vicine alle e&longs;tremità. A ciò &longs;i ri&longs;ponde, che per e&longs;&longs;er la corda a o più obliqua alla lunghezza a e di quel che &longs;ia all' altre più vicine al centro M, quindi ne &longs;iegue, che per quel' altra cagione s' aprono più ageuolmente le parti vicine al centro; onde, temperata vna ragione con l' altra (quando l' arco non &longs;ia e&longs;tremamente incuruato) &longs;i con&longs;egui&longs;ce vno &longs;tato d'apertura opportuna.

La &longs;econda difficoltà è che cia&longs;cuna molla nel &longs;uo re­&longs;tringer&longs;i, par che cagioni qualche effetto contrario all'in­tento. Imperoche, per e&longs;empio, nella molla B il mezzo anello, che ri&longs;guarda l'e&longs;tremità A, nello &longs;tringer&longs;i fà ben&longs;i il &longs;uo douere, perche il &longs;uo moto è ver&longs;o il centro M; ma l' altra metà, che ri&longs;guarda il &longs;udetto centro M, nello &longs;trin­ger&longs;i, hauendo il &longs;uo moto ver&longs;o A, &longs;i oppone al chiudi­mento della molla &longs;eguente C; e il &longs;imile dica&longs;i dell' altre. A ciò &longs;i è po&longs;to rimedio col far più grandi, e più mafficcie le molle più vicine al centro M, accre&longs;cendole, e ingro&longs;&longs;an­dole di mano in mano opportunamente. Quindi ne &longs;egue che per la maggior grandezza con&longs;entono egualmente all' aprir&longs;i con facilità; ma all' incontro nel &longs;errar&longs;i, per e&longs;&longs;ere più ma&longs;&longs;iccie, e di maggior corpo, vengono ad hauere maggior momento delle men corpulenti, &longs;uperando co&ntail; ciò non &longs;olo il detto moto oppo&longs;to, ma etiandio impri­mendo maggior moto al ferro dell'arco, con cui &longs;i acco­muna il moto.

Auuerta&longs;i, che quanto &longs;aranno di maggior numero le comme&longs;lure, le molle di maggior pe&longs;o, e l'arco più pouero di corpo, tanto riu&longs;cirà il colpo a di&longs;mi&longs;ura maggiore, per l' incuruamento notabile delle braccia, e per il maggior mo­mento delle molle; e ciò con adoperare la mede&longs;ima forza.

Auuerta&longs;i parimente, che il braccio AE, è il &longs;uo corri&longs;­pondente deuono e&longs;&longs;ere alquanto più corti, cioè A vna delle e&longs;tremità dell'arco deue e&longs;&longs;ere più ver&longs;o il centro di quel che &longs;ia il concor&longs;o delle linee LB, KE, come pure dall' altra parte; perche &longs;i vede che aprendo&longs;i meno le parti vi­cine ad A, l'altre molle fanno miglior effetto.

Finalmente la &longs;perienza ha mo&longs;trato, che e&longs;&longs;endo&longs;i la­uorata vna tal machina con pochi&longs;&longs;imi nodi, ageuoli&longs;&longs;ima ad aprir&longs;i, e &longs;enza hauer ingrandite e ingro&longs;late le molle, che più &longs;i vanno auuicinando al centro M, come &longs;i è det­to; con tutto ciò l' ordigno è riu&longs;cito di forza molto &longs;upe­riore a vna bale&longs;tra grande, e difficilif&longs;ima a inarcar&longs;i. On­de non dubito, che, facendo&longs;i con tutte ie regole accenna­te, non debba riu&longs;cire vna machina di effetto marauiglio&longs;o aggiungendo che per tirar granate dourebbero i bracci e&longs;&longs;er di legno, armati di ferro &longs;ol doue &longs;i richiede.

Nouum genus Bali&longs;tæ Explicatio.

Tab. 9.

IN hac figura exprimitur nouum genus Bali­&longs;tæ, quæ machina præ&longs;ertim in mole maio­ri, non parum vtilitatis afferre pote&longs;t rei mi­litari ad eiaculanda mi&longs;&longs;ilia, vt demon&longs;tra­bitur. Ex eius verò partibus, quas &longs;ubinde recen&longs;eo, etiam modus &longs;tructuræ apparebit.

AM, MX &longs;unt brachia. Punctum M centrum machi­næ. Per auitatem M tran&longs;it telum emi&longs;&longs;um. Infra M in­&longs;eritur manubrium, vt in bali&longs;tis vulgaribus. Extremis capitibus A, N adnectitur funis. Puncta A, E, F, G, I, K &longs;unt in linea recta. Interualla AE, EF, FG, GI, IK &longs;unt (li­cèt non nece&longs;&longs;ariò) æqualia. Altitudinis, &longs;eu commi&longs;&longs;uræ KL, IH, GD, FC, EB &longs;unt perpendiculares rectæ occultæ KA. Singulæ autem, dum curuatur arcus, aperiuntur cer­ca centra K, I, G, F, E. Hinc &longs;equitur vt funis ex medio dum attrahitur in O, aperiantur prædictæ commi&longs;&longs;uræ, &longs;eu nodi, & curuentur vtraque brachia, vt in eorum altero ap­paret punctis notato. Quilibet ex his nodis arcti&longs;limè &longs;trin­gitur &longs;upernè, a &longs;uo elaterio, vt videre e&longs;t in L, H, D, C, B. Elateria autem quò propinquiora centro M tanto maiora, & cra&longs;&longs;iora debent e&longs;&longs;e remotioribus: Hinc fit vt, propter molem opportunè auctam, æquè facilè aperiantur, ac cæ­tera; & vice ver&longs;a, propter cra&longs;&longs;itiem maiorem, &longs;ibi relicta validiùs re&longs;tringantur. Cuius rei paulo infra rationem dabimus.

His po&longs;itis facile e&longs;t o&longs;tendere, quantum præ&longs;tet hu­iu&longs;cemodi machina vulgaribus & communibus bali&longs;tis.

Primùm, in Triangulo ALK cùm altitudines EB, FC, GD, IH, KL &longs;int perpendiculares, ideoque parallelæ, hinc &longs;it vtrationes AE ad EB, AF ad FC, AG ad GD, AI ad IH, AK ad KL &longs;int æquales. Sunt pariter æquales re&longs;i­&longs;tentiæ elateriorum in B, C, D, H, L (po&longs;uimus enim ela­teria ita opportunè aucta vt æquè facile &longs;ingula aperian­tur) ergo (ex primis principijs mecanicorum) dum attra­huntur fune extrema capita A, N, eodem tempore, eadem­que facili ate vincetur æquilibrium omnium elateriorum, quorum re&longs;i&longs;tentia in &longs;ingulis con&longs;ideratur in ratione pon­deris, quemadmodum lineæ AE, EB; AF, FC; AG, GD &c. con&longs;iderantur vt vectes, quorum hippomoclia &longs;eu cen­tra &longs;unt in E, F, G, I, K, potentia autem con&longs;ideratur in A communis omnibus.

Scundò, cùm AE ad EB (idem die de cæteris) habeant magnam proportionem, facilè aperientur nodi, & curuabi­tur arcus; quantumuis augeatur numerus nodorum.

Tertiò Cum &longs;int plures nodi, atque omnes aperiantur, nece&longs;&longs;e e&longs;t vt brachia arcus valdè incuruentur; quamobrem &longs;i idem arcus &longs;ibi relinquatur, prædicti nodi omnes, vi ela­teriorum, ictu oculi claudentur; e odemque puncto tempo­ris corda ex O percurret totum &longs;patium v&longs;que ad M: Quòd cùm fieri nequeat ni&longs;i &longs;umma velocitate, propter magni­tudinem prædicti &longs;patij, & velocitati re&longs;pondent vis, atque impetus, nece&longs;&longs;e e&longs;t vt hinc &longs;equatur ictus valde notabilis, vt facilè e&longs;t vnicuique conijcere.

Super &longs;unt nunc difficultates nonnullæ &longs;oluendæ. Prima e&longs;t, quòd licèt vis &longs;ufficiens in A ad vincendum æquilibrium elaterij B, illa eadem quoque &longs;ufficiat ad vincendum æqui­librium cæterorum, propter æquales proportiones vectium; his tamen non ob&longs;tantibus, &longs;i con&longs;ideretur brachium iam incuruatum, vt apparet in KLA punctis notato, proportio­nes illæ cernuntur notabiliter variatæ. Neque enim pro longitudinibus vectium &longs;umi po&longs;&longs;unt longitudines priores, &longs;ed loco ip&longs;arum accipiendæ &longs;unt applicatæ arcus, videli­cet af, ag, ai, ak quarum ak, eidemque propinquiores, quan-do arcus incuruatur, breuiores fiunt, quàm e&longs;&longs;ent ante&atail;. Re&longs;pondeo, quòd corda ao cùm &longs;it obliquior re&longs;pectu longitudinis ae, quàm re&longs;pectu cæterarum centro propin­quiorum, hinc fit vt, quantùm e&longs;t ex hac ratione, faciliùs aperiantur partes propinquiores centro; quamobrem, vtra­que ratione inuicem temperata, dummodo arcus non &longs;it &longs;ummè incuruatus omnes partes aperientur, quantum &longs;a­tis e&longs;t ad intentum.

Altera difficultas e&longs;t, quod elaterium quodlibet dum re&longs;tringitur videtur ob&longs;tare motui elaterij &longs;equentis. Nam, exempli gratia, in elaterio B &longs;emiannulus re&longs;piciens extre­mum A, dum &longs;tringitur, optim è præ&longs;tat &longs;uum effectum, cum eius motus &longs;it versùs centrum M At è contrario reliqua pars, &longs;eu &longs;emiannulus re&longs;piciens prædictum centrum M, cum habeat &longs;uum motum ver&longs;us A videtur ob&longs;tare, quo minus liberè claudatur &longs;equens elaterium C. Aque idem de cæte­ris dicendum. Huic incommodo con&longs;ultum e&longs;t augendo magnitudinem, & cra&longs;&longs;itiem elateriorum, quò magis acce­dunt ad centrum M. Hinc enim &longs;equitur vt propter ma­gnitudinem facilè con&longs;entiant arcui, dum incuruatur; at dum idem arcus liberè &longs;ibi relinquitur, cum &longs;int corpulen­tiora & cra&longs;&longs;iora habent maius momentum, quàm cætera graciliora, ideoque non modo vincunt motum illum op­po&longs;itum, &longs;ed etiam imprimunt maiorem motum ferro ar­cus, cui ille motus communicatur.

Aduerte quod commi&longs;&longs;uræ &longs;eu nodi, quò plures fuerint, elateria autem maioris ponderis, arcus denique corporis gracilioris equæ expeditioris, tanto ictus longat præ&longs;tan­tior &longs;equetur, tum propter notabilem curuaturam brachio­rum, tum propter momentum maius elateriorum, & quidem po&longs;ita eadem potentia, aut etiam minori, pro vt longitudi­nes vectium &longs;tatuuntur.

Aduerte etiam, longitudinem brachij AE, eiu&longs;demqu&etail; corre&longs;pondentis debere cæteris paribus nonnihil imminui, quod fiet &longs;i A, alterum extremum arcus, &longs;it propriùs cen­tro M, quàm &longs;it concur&longs;us linearum LB, KE. Idem dicen­dum de altero extremo N. Nam cùm minus aperiantur partes propinquiores puncto A, cætera elateria, vt com­pertum e&longs;t, meliorem effectum præ&longs;tant.

Fauet denique experientia. Nam huiu&longs;cemodi machi­na pauci&longs;&longs;imis nodis con&longs;tructa, facillimæ curuaturæ, cum elaterijs eiu&longs;dem pror&longs;us molis & cra&longs;&longs;itudinis; nihilomi­nus longè &longs;uperauit vim bali&longs;tæ communis maximæ, & dif ficillimæ flexionis. Quamobrem non dubito quin, &longs;i præ­cepta &longs;uperiùs data exactè &longs;eruentur, elaborari po&longs;&longs;it ma­china miræ vtilitatis. Adde po&longs;tremo ad iacienda quædam mi&longs;&longs;ilia, vt e&longs;t genus quoddam bolidum, vulgo granate, op­portuniora e&longs;&longs;e brachia lignea, tantummodo, vbi nece&longs;&longs;i­tas po&longs;tulat, armata ferro.

FINIS.

Vid. D. Bernardus Marchellus Re­ctor Pœnitent. in Metropol. Bonon. pro Illu&longs;tri&longs;s. & Reverendi&longs;s. Domino D. Iacobo Boncompagno Archiepi&longs;­copo, & Principe.

Vid. Silue&longs;ter Bonfiliolus Inqui&longs;itionis reui&longs;or, & imprimi po&longs;&longs;e cen&longs;uit.

Stante atte&longs;tatione.

Imprimatur.

F. Io&longs;eph Maria Agudius Vicarius Sancti Offi c ij Bononiæ.

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BVLA I.