Philosophiae Naturalis Principia Mathmatica

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~~PHILOSOPHIÆ NATURALIS PRINCIPIA MATHEMATICA.~~

~~AUCTORE ISAACO NEWTONO, EQUITE A RATO.~~

~~EDITIO SECUNDA AUCTIOR ET EMBNDATIOR.~~

~~CANTABRIGIÆ, MDCCXIII.~~

~~ILLUSTRISSIMÆ SOCIETATI REGALI, A SERENISSIMO REGE CAROLO II AD PHILOSOPHIAM PROMOVENDAM FUNDATÆ, ET AUSPICIIS AUGUSTISSIMÆ REGINÆ ANNÆ FLORENTI,~~

~~TRACTATUM HUNC D.D.D.~~

~~JS. NEWTONUS.~~

~~IN VIRI PRÆSTANTISSIMI ISAACI NEWTONI OPUS HOCCE MATHEMATICO PHYSICUM~~

~~Sæculi Genti&longs;que no&longs;træ Decus egregium.~~

~~EN tibi norma Poli, & divæ libramina Molis, Computus en Jovis; & quas, dum primordia rerum.~~ ~~Conderet, omnipotens &longs;ibi Leges ip&longs;e Creator Dixerit, atque operum quæ fundamenta locarit.~~ ~~Intima panduntur victi penetralia Cæli, Nec latet, extremos quæ Vis circumrotet Orbes.~~ Sol &longs;olio re&longs;idens ad &longs;e jubet omnia prono Tendere de&longs;cen&longs;u, nec recto tramite currus Sidereos patitur va&longs;tum per inane moveri; Sed rapit immotis, &longs;e centro, &longs;ingula gyris. Hinc patet, horrificis qua &longs;it via flexa Cometis: Di&longs;cimus hinc tandem, qua cau&longs;a argentea Phœbe Pa&longs;&longs;ibus haud æquis eat, & cur &longs;ubdita nulli Hactenus A&longs;tronomo numerorum fræna recu&longs;et: Cur remeent Nodi, curque Auges progrediantur. Di&longs;cimus, & quantis refluum vaga Cynthia Pontum Viribus impellat; fe&longs;&longs;is dum fluctibus ulvam De&longs;erit, ac nautis &longs;u&longs;pectas nudat arenas; Alterni&longs;ve ruens &longs;pumantia littora pul&longs;at. Quæ toties animos veterum tor&longs;ere Sophorum, Quæque Scholas hodie rauco certamine vexant, Obvia con&longs;picimus; nubem pellente Mathe&longs;i: Quæ &longs;uperas penetrare domos, atque ardua Cæli, NEWTONI au&longs;picils, jam dat contingere Templa. ~~Surgite Mortales, terrenas mittite curas; Atque hinc cæligenæ vites cogno&longs;cite Mentis, A pecudum vita longe longeque remotæ.~~ Qui &longs;criptis primus Tabulis compe&longs;cere Cædes, Furta & Adulteria, & perjuræ crimina Fraudis; Quive vagis populis circumdare mœnibus Urbes Auctor erat; Cereri&longs;ve beavit munere gentes; Vel qui curarum lenimen pre&longs;&longs;it ab Uva; Vel qui Niliaca mon&longs;travit arundine pictos Con&longs;ociare &longs;onos, oculi&longs;que exponere Voces; Humanam &longs;ortem minus extulit; utpote pauca In commune ferens mi&longs;eræ &longs;olatia vitæ. ~~Jam vero Superis convivæ admittimur, alti Jura poli tractare licet, jamque abdita diæ Clau&longs;tra patent Naturæ, & rerum immobilis ordo; Et quæ præteritis latuere incognita &longs;æclis.~~ Talia mon&longs;trantem ju&longs;tis celebrate Camænis, Vos qui cæle&longs;ti gaudetis nectare ve&longs;ci, NEWTONUM clau&longs;i re&longs;erantem &longs;crinia Veri, NEWTONUM Mu&longs;is carum, cui pectore puro Phœbus ade&longs;t, totoque ince&longs;&longs;it Numine mentem: Nec fas e&longs;t propius Mortali attingere Divos. EDM. HALLET.

~~AUCTORIS PRÆFATIO AD LECTOREM.~~

CUM Veteres Mechanicam (uti Auctor e&longs;t Pappus) in rerum Naturalium inve&longs;tigatione maximi fecerint; & Recentiores, mi&longs;&longs;is formis &longs;ub&longs;tautialibus & qualitatibus occultis, Phænomena Naturæ ad leges Mathematicas revocare aggre&longs;&longs;i fint: Vi&longs;um e&longs;t in hoc Tractatu Mathe&longs;in excolere, quatenus ea ad Philo&longs;ophiam &longs;pectat. Mechanicam vero duplicem Veteres con&longs;tituerunt: Rationalem quæ per Demon&longs;trationes accurate procedit, & Practicam. Ad acticam &longs;pectant Artes omnes Manuales, a quibus utique Mechanica nomen mutuata e&longs;t. Cum autem Artifices parum accurate operari &longs;oleant, fit ut Mechanica omnis a Geometria ita di&longs;tinguatur, ut quicquid accuratum &longs;it ad Geometriam referatur, quicquid minus accuratum ad Mechanicam. Attamen errores non &longs;unt Artis &longs;ed Artificum. ~~Qui minus accurate operatur, imperfectior e&longs;t Mechanicus, & &longs;i quis accurati&longs;&longs;ime operari po&longs;&longs;et, hic foret Mechanicus omnium perfecti&longs;&longs;imus.~~ ~~Nam & Linearum rectarum & Circulorum de&longs;criptiones in quibus Geometria fundatur, ad Mechanicam pertinent.~~ ~~Has lineas de&longs;cribere Geometria non docet &longs;ed po&longs;tulat.~~ ~~Po&longs;tulat enim ut Tyro ea&longs;dem accurate de&longs;cribere prius didicerit quam linen atting atGeometriæ; deia, quomodo per has operationes Problemata &longs;oluantur, docet.~~ ~~Rectas & Circulos de&longs;cribere Problemata &longs;unt, &longs;ed non Geometrica.~~ ~~Ex Mechanica po&longs;lulatur horum &longs;olutio, inGeometria docetur &longs;olutorum u&longs;us.~~ ~~Ac gloriatur Geometria quod tam paucis principiis aliunde petitis tam multa præ&longs;tet.~~ ~~Fundatur igitur Geometria in praxi Mechanica, & nihil aliud e&longs;t quam Mechanicæ univer&longs;alis pars illa quæ artem men&longs;urandi accurate proponit ac demon&longs;trat.~~ ~~Cum autem artes Manuales in corporibus movendis præcipue ver&longs;entur, fit ut Geometria ad magnitudinem, Mechanica ad motum vulgo referatur.~~ Quo &longs;en&longs;u Mechanica rationalis erit Scientia Motuum qui ex viribus quibu&longs;cunque re&longs;ultant, & Virium quæ ad motus quo&longs;cunque requiruntur, accurate propo&longs;ita ac demon&longs;trata. Pars hæc Mechanicæ a Veteribus in Potentiis quinque ad artes manuales &longs;pectantibus exculta fuit, qui Gravitatem (cum potentia manualis non &longs;it) vix aliter quam in ponderibus per potentias illas movendis con&longs;iderarunt. Nos autem non Artibus &longs;ed Philo&longs;ophiæ con&longs;ulentes, deque potentiis non manualibus &longs;ed naturalibus &longs;cribentes, ea maxime tractamus quæ ad Gravitatem, Levitatem, vim Ela&longs;ticam, re&longs;i&longs;tentiam Fluidorum & eju&longs;modi vires &longs;eu attractivas &longs;eu impul&longs;ivas &longs;pectant: Et ea propter, hæc no&longs;tra tanquam Philo&longs;ophiæ principia Mathematica proponimus. ~~Omnis enim Philo&longs;ophiæ difficultas in eo ver&longs;ari videtur, ut a Phænomenis motuum inve&longs;tigemus vires Naturæ, deinde ab his viribus demon&longs;tremus phænomena liquæ.~~ ~~Et huc &longs;pectant Propo&longs;itiones generales quas Libro primo & &longs;ecundo pertractavimus.~~ ~~In Libro autem tertio Exemplum hujus rei propo&longs;uimus per explicationem Sy&longs;tematis mundaui.~~ Ibi enim, ex phænomenis cæleflibus, per Propo&longs;itiones in Libris prioribus Mathematice aemon&longs;tratas, derivantur vires Gravitatis quibus corpora ad Solem & Planetas &longs;ingulos tendunt. ~~Deinde ex his viribus per Propo&longs;itiones etiam Mathematicas, deducuntur motus Planetarum, Cometarum, Lunæ & Maris.~~ ~~Utinam cætera Naturæ phænomena ex principiis Mechanicis eodem argumentandi genere derivare liceret.~~ Nam multa me movent ut nonnihil &longs;u&longs;picer e om- nia ex viribus quibu&longs;dam pendere po&longs;&longs;e, quibus corporum particulæ per cau&longs;as nondum cognitas vel in &longs;e mutuo impelluntur & &longs;ecundum figuras regulares cohærent, vel ab invicem fugantur & recedunt: quibus viribus ignotis, Philo&longs;ophi hactenus Naturam fru&longs;tra tentarunt. ~~Spero autem quod vel huic Philo&longs;ophandi modo, vel veriori alicui, Principia hic po&longs;ita lucem aliquam præbebunt.~~

In his edendis, Vir acuti&longs;&longs;imus & in omni literarum genere eruditi&longs;&longs;imus Edmundus Halleius operam navavit, nec &longs;olum Typothetarum Sphalmata correxit & Schemata incidi curavit, &longs;ed etiam Auctor fuit ut horum editionem aggrederer. Quippe cum demon&longs;tratam a me Figuram Orbium cæle&longs;tium impetraverat, rogare non de&longs;titit ut eandem cum Societate Regali communicarem, Quæ deinde hortatibus & benignis &longs;uis au&longs;piciis effecit ut de eadem in lucem emittenda cogitare inciperem. At po&longs;tquam Motuum Lunarium inæqualitates aggre&longs;&longs;us e&longs;&longs;em, deinde etiam ælia tentare cæpi&longs;&longs;em quæ ad leges & men&longs;uras Gravitatis & aliarum virium, & Figuras a corporibus &longs;ecundum datas qua&longs;cunque leges attractis de&longs;cribendas, ad motus corporum plurium inter &longs;e, ad motus corporum in Mediis re&longs;i&longs;tentibus, ad vires, den&longs;itates & motus Mediorum, ad Orbes Cometarum & &longs;imilia &longs;pectant, editionem in aliud tempus differendam e&longs;&longs;e putavi, ut cætera rimarer & una in publicum darem. Quæ ad motus Lunares &longs;pectant, (imperfecta cum &longs;int,) in Corollariis Propo&longs;itionis LXVI. &longs;imul complexus &longs;um, ne &longs;ingula methodo prolixiore quam pro rei dignitate proponere, & &longs;igillatim demon&longs;trare tenerer, & &longs;eriem reliquarum Propo&longs;itionum interrumpere. ~~Nonnulla &longs;ero inventa locis minus idoneis in&longs;erere malui, quam numerum Propo&longs;itionum & citationes mutare.~~ ~~Ut omnia candide legantur, & defectus, in materia tam difficili non tam reprehendantur, quam novis Lectorum conatibus inve&longs;tigentur, & benigne &longs;uppleantur, enixe rogo.~~

~~Dabam Cantabrigiæ, e Collegio S. Trinitatis, Maii 8. 1686.~~

~~IS. NEWTON.~~

~~IN hac Secunda Principiorum Editione, multa &longs;par&longs;im emendantur & nonnulla adjiciuntur.~~ ~~In Libri primi Sectione II, Inventio virium quibus corpora in Orbibus datis revolvi po&longs;&longs;int, facilior redditur & amplior.~~ ~~In Libri &longs;ecundi Sectione VII, Theoria re&longs;i&longs;tentiæ Fluidorum accuratius inve&longs;tigatur & novis Experimentis confirmatur.~~ ~~In Libro tertio Theoria Lunæ & Præce&longs;&longs;io Æquinoctiorum ex Principiis &longs;uis plenius deducuntur, & Theoria Cometarum pluribus & accuratius computatis Orbium exemplis confirmatur.~~

~~Dabam Londini,Mar.~~ ~~28. 1713.~~

~~IS. NEWTON.~~

~~EDITORIS PRÆFATIO.~~

~~NEWTONIANÆ Philo&longs;ophiæ novam tibi, Lector benevole, diuque de&longs;ideratam Editionem, plurimum nunc emendatam atque auctiorem exhibemus.~~ ~~Quæ poti&longs;&longs;imum contineantur in hoc Opere celeberrimo, intelligere potes ex Indicibus adjectis: quæ vel addantur vel immutentur, ip&longs;a te fere docebit Auctoris Præfatio.~~ ~~Reliquum e&longs;t, ut adjiciantur nonnulla de Methodo hujus Philo&longs;ophiæ.~~

~~Qui Phy&longs;icam tractandam &longs;u&longs;ceperunt, ad tres fere cla&longs;&longs;es revocari po&longs;&longs;unt.~~ Extiterunt enim, qui &longs;ingulis rerum &longs;peciebus Qualitates &longs;pecificas & occultas tribuerint; ex quibus deinde corporum &longs;ingulorum operationes, ignota quadam ratione, pendere voluerunt. In hoc po&longs;ita e&longs;t &longs;umma doctrinæ Schola&longs;ticæ, ab Ari&longs;totele& Peripateticis derivatæ: Affirmant utique &longs;ingulos Effectus ex corporum &longs;ingularibus Naturis oriri; at unde &longs;int illæ Naturæ non docent; nihil itaque docent. Cumque toti &longs;int in rerum nominibus, non in ip&longs;is rebus; Sermonem quendam Philo&longs;ophicum cen&longs;endi &longs;unt adinveni&longs;&longs;e, Philo&longs;ophiam tradidi&longs;&longs;e non &longs;unt cen&longs;endi.

~~Alii ergo melioris diligentiæ laudem con&longs;equi &longs;perarunt, rejecta Vocabulorum inutili farragine.~~ Statuerunt itaque Materiam univer&longs;am homogeneam e&longs;&longs;e, omnem vero Formarum varietatem, quæ in corporibus cernitur, ex particularum componentium &longs;implici&longs;&longs;imis quibu&longs;dam & intellectu facillimis affectionibus oriri. Et recte quidem progre&longs;&longs;io in&longs;tituitur a &longs;implicioribus ad magis compo&longs;ita, &longs;i particularum primariis illis affectionibus non alios tribuunt modos, quam quos ip&longs;a tribuit Natura. Verum ubi licentiam &longs;ibi a&longs;&longs;umunt, ponendi qua&longs;cunque libet ignotas partium figuras & magnitudines, incerto&longs;que &longs;itus & motus; quin & fingendi Fluida quædam occulta, quæ corporum poros liberrime permeent, omnipotente prædita &longs;ubtilitate, motibu&longs;que occultis agitata; jam ad &longs;omnia delabuntur, neglecta rerum con&longs;titutione vera: quæ fane fru&longs;tra petenda e&longs;t ex fallacibus conjecturis, cum vix etiam per certi&longs;&longs;imas Ob&longs;ervationes inve&longs;tigari po&longs;&longs;it. Qui &longs;peculationum &longs;uarum fundamentum de&longs;umunt ab Hypothe&longs;ibus, etiam&longs;i deinde &longs;ecundum leges Mechanicas accurati&longs;&longs;ime procedant; Fabulam quidem elegantem forte & venu&longs;tam, Fabulam tamen concinnare dicendi &longs;unt.

~~Relinquitur adeo tertium genus, qui Philo&longs;ophiam &longs;cilicet Experimentalem profitentur.~~ Hi quidem ex &longs;implici&longs;&longs;imis quibus po&longs;&longs;unt principiis rerum omnium cau&longs;as derivandas e&longs;&longs;e volunt: nihil autem Principii loco a&longs;&longs;umunt, quod nondum ex Phænomenis comprobatum fuerit. ~~Hypothe&longs;es non commini&longs;cuntur, neque in Phy&longs;icam recipiunt, ni&longs;i ut Quæ&longs;tiones de quarum veritare di&longs;putetur.~~ ~~Duplici itaque Methodo incedunt, Analytica & Synthetica.~~ Naturæ vires lege&longs;que virium &longs;impliciores ex &longs;electis quibu&longs;dam Phænomenis per Analy&longs;in deducunt, ex quibus deinde per Synthe&longs;in reliquorum con&longs;titutionem tradunt. ~~Hæc illa e&longs;t Philo&longs;ophandi ratio longe optima, quam præ ceteris merito amplectendam cen&longs;uit Celeberrimus Auctor no&longs;ter.~~ ~~Hanc &longs;olam utique dignam judicavit, in qua excolenda atque adornanda operam &longs;uam collocaret.~~ ~~Hujus igitur illu&longs;tri&longs;&longs;imum dedit Exemplum, Mundani nempe Sy&longs;tematis explicationem e Theoria Gravitatis felici&longs;&longs;ime deductam.~~ Gravitatis virtutem univer&longs;is corporibus ine&longs;&longs;e, &longs;u&longs;picati &longs;unt vel finxerunt alii: primus Ille & folus ex Apparentiis demon&longs;trare potuit, & &longs;peculationibus egregiis firmi&longs;&longs;imum ponere fundamentum.

Scio equidem nonnullos magni etiam nominis Viros, præjudiciis quibu&longs;dam plus æquo occupatos, huic novo Principio ægre a&longs;&longs;entiri potui&longs;&longs;e, & certis incerta identidem prætuli&longs;&longs;e. ~~Horum famam vellicare non e&longs;t animus: Tibi potius, Benevole Lector, illa paucis exponere lubet, ex quibus Tute ip&longs;e judicium non iniquum feras.~~

Igitur ut Argumenti &longs;umatur exordium a &longs;implici&longs;&longs;imis & proximis; de&longs;piciamus pauli&longs;per qualis &longs;it in Terre&longs;tribus natura Gravitatis, ut deinde tutius progrediamur ubi ad corpora Cæle&longs;tia, longi&longs;&longs;ime a &longs;edibus no&longs;tris remota, perventum fuerit. ~~Convenit jam inter omnes Philo&longs;ophos, corpora univer&longs;a circumterre&longs;tria gravitare in Terram.~~ ~~Nulla dari corpora vere levia, jamdudum confirmavit Experientia multiplex.~~ ~~Quæ dicitur Levitas relativa, non e&longs;t vera Levitas, &longs;ed apparens &longs;olummodo; & oritur a præpollente Gravitate corporum contiguorum.~~

Porro, ut corpora univer&longs;a gravitant in Terram, ita Terra vici&longs;&longs;im in corpora æqualiter gravitat; Gravitatis enim actionem e&longs;&longs;e mutuam & utrinque æqualem, &longs;ic o&longs;tenditur. Di&longs;tinguatur Terræ totius moles in binas qua&longs;cunque partes, vel æquales vel utcunque inæquales: jam &longs;i pondera partium non e&longs;&longs;ent in &longs;e mutuo æqualia; cederet pondus minus majori, & partes conjunctæ pergerent recta moveri ad in&longs;initum, ver&longs;us plagam in quam tendit pondus majus: omnino contra Experientiam. ~~Itaque dicendum erit, pondera partium in æquilibrio e&longs;&longs;e con&longs;tituta: hoc e&longs;t, Gravitatis actionem e&longs;&longs;e mutuam & utrinque æqualem.~~

~~Pondera corporum, æqualiter a centro Terræ di&longs;tantium, &longs;unt ut quantitates materiæ in corporibus.~~ Hoc utique colligitur ex æquali acceleratione corporum omnium, e quiete per ponderum vires cadentium: nam vires quibus inæqualia corpora æqualiter accelerantur, debent e&longs;&longs;e proportionales quantitatibus materiæ movendæ. Jam vero corpora univer&longs;a cadentia æqualiter accelerari, ex eo patet, quod in Vacuo Boyliano temporibus æqualibus æqualia &longs;patia cadendo de&longs;cribunt, &longs;ublata &longs;cilicet Aeris re&longs;i&longs;tentia: accuratius autem comprobatur per Experimenta Pendulorum.

~~Vires attractivæ corporum, in æqualibus di&longs;tantiis, &longs;unt ut quantitates materiæ in corporibus.~~ Nam cum corpora in Terram & Terra vici&longs;&longs;im in corpora momentis æqualibus gravitent; Terræ pondus in unumquodque corpus, &longs;eu vis qua corpus Terram attrahit, æquabitur ponderi corporis eju&longs;dem in Terram. ~~Hoc autem pondus erat ut quantitas materiæ in corpore: itaque vis qua corpus unumquodque Terram attrahit, &longs;ive corporis vis ab&longs;oluta, erit ut eadem quantitas materiæ.~~

Oritur ergo & componitur vis attractiva corporum integrorum ex viribus attractivis partium: &longs;iquidem aucta vel diminuta mole materiæ, o&longs;ten&longs;um e&longs;t, proportionaliter augeri vel diminui ejus virtutem. Actio itaque Telluris ex conjunctis partium Actionibus conflari cen&longs;enda erit; atque adeo corpora omnia Terre&longs;tria &longs;e mutuo trahere oportet viribus ab&longs;olutis, quæ &longs;int in ratione materiæ trahentis. ~~Hæc e&longs;t natura Gravitatis apud Terram: videamus jam qualis &longs;it in Cælis.~~

Corpus omne per&longs;everare in &longs;tatu &longs;uo vel quie&longs;cendi vel movendi uniformiter in directum, ni&longs;i quatenus a viribus impre&longs;&longs;is cogitur &longs;tatum illum mutare; Naturæ lex e&longs;t ab omnibus recepta Philo&longs;ophis. ~~Inde vero &longs;equitur, corpora quæ in Curvis moventur, atque adeo de lineis rectis Orbitas &longs;uas tangentibus jugiter abeunt, Vi aliqua perpetuo agente retineri in itinere curvilineo.~~ ~~Planetis igitur in Orbibus curvis revolventibus nece&longs;&longs;ario aderit Vis aliqua, per cujus actiones repetitas inde&longs;inenter a Tangentibus deflectantur.~~

Jam illud concedi æquum e&longs;t, quod Mathematicis rationibus colligitur & certi&longs;&longs;ime demon&longs;tratur; Corpora nempe omnia, quæ moventur in linea aliqua curva in plano de&longs;cripta, quæque radio ducto ad punctum vel quie&longs;cens vel utcunque motum de&longs;cribunt areas circa punctum illud temporibus proportionales, urgeri a Viribus quæ ad idem punctum tendunt. Cum igitur in confe&longs;&longs;o &longs;it apud A&longs;tronomos, Planetas primarios circum Solem, &longs;ecundarios vero circum &longs;uos primarios, areas de&longs;cribere temporibus proportionales; con&longs;equens e&longs;t ut Vis illa, qua perpetuo detorquentur a Tangentibus rectilineis, & in Orbitis curvilineis revolvi ceguntur, ver&longs;us corpora dirigatur quæ &longs;ita &longs;unt in Orbitarum centris. Hæc itaque Vis non inepte vocari pote&longs;t, re&longs;pectu quidem corporis revolventis, Centripeta; re&longs;pectu autem corporis centralis, Attractiva; a quacunque demum cau&longs;a oriri fingatur.

Quin & hæc quoque concedenda &longs;unt, & Mathematice demon&longs;trantur: Si corpora plura motu æquabili revolvantur in Circulis concentricis, & quadrata temporum periodicorum &longs;int ut cubi di&longs;tantiarum a centro communi; Vires centripetas revolventium fore reciproce ut quadrata di&longs;tantiarum. Vel, &longs;i corpora revolvantur in Orbitis quæ &longs;unt Circulis finitimæ, & quie&longs;cant Orbitarum Ap&longs;ides; Vires centripetas revolventium fore reciproce ut quadrata di&longs;tantiarum. ~~Obtinere ca&longs;um alterutrum in Planetis univer&longs;is con&longs;entiunt A&longs;tronomi.~~ ~~Itaque Vires centripetæ Planetarum omnium &longs;unt reciproce ut quadrata di&longs;tantiarum ab Orbium centris.~~ Si quis objiciat Planetarum, & Lunæ præ&longs;ertim, Ap&longs;ides non penitus quie&longs;cere; &longs;ed motu quodam lento ferri in con&longs;equentia: re&longs;ponderi pote&longs;t, etiam&longs;i concedamus hunc motum tardi&longs;&longs;imum exinde profectum e&longs;&longs;e quod Vis centripetæ proportio aberret aliquantum a duplicata, aberrationem illam per computum Mathematicum inveniri po&longs;&longs;e & plane in&longs;en&longs;ibilem e&longs;&longs;e. Ip&longs;a enim ratio Vis centripetæ Lunaris, quæ omnium maxime turbari debet, paululum quidem duplicatam &longs;uperabit; ad hanc vero &longs;exaginta fere vicibus propius accedet quam ad triplicatam. Sed verior erit re&longs;pon&longs;io, &longs;i dicamus hanc Ap&longs;idum progre&longs;&longs;ionem, non ex aberratione a duplicata proportione, &longs;ed ex alia pror&longs;us diver&longs;a cau&longs;a oriri, quemadmodum egregie common&longs;tratur in hac Philo&longs;ophia. Re&longs;tat ergo ut Vires centripetæ, quibus Planetæ primarii tendunt ver&longs;us Solem & &longs;ecundarii ver&longs;us primarios &longs;uos, &longs;int accurate ut quadrata di&longs;tantisrum reciproce.

Ex iis quæ hactenus dicta &longs;unt, con&longs;tat Planetas in Orbitis &longs;uis retineri per Vim aliquam in ip&longs;os perpetuo agentem: con&longs;tat Vim illam dirigi &longs;emper ver&longs;us Orbitarum centra: con&longs;tat hujus efficaciam augeri in acce&longs;&longs;u ad centrum, diminui in rece&longs;&longs;n ab eodem: & augeri quidem in eadem proportione qua diminuitur quadratum di&longs;tantiæ, diminui in eadem proportione qua di&longs;tantiæ quadratum augetur. ~~Videamus jam, comparatione in&longs;tituta inter Planetarum Vires centripetas & Vim Gravitatis, annon eju&longs;dem forte &longs;int generis.~~ ~~Eju&longs;dem vero generis erunt, &longs;i deprehendantur hinc & inde leges eædem eædemque affectiones.~~ ~~Primo itaque Lunæ, quæ nobis proxima e&longs;t, Vim centripetam expendamus.~~

Spatia rectilinea, quæ a corporibus e quiete demi&longs;&longs;is dato tempore &longs;ub ip&longs;o motus initio de&longs;eribuntur, ubi a viribus quibu&longs;cunque urgentur, proportionalia &longs;unt ip&longs;is viribus: Hoc utique con&longs;equitur ex ratiociniis Mathematicis. Erit igitur Vis centripeta Lunæ in Orbita &longs;ua revolventis, ad Vim Gravitatis in &longs;uperficie Terræ, ut &longs;patium quod tempore quam minimo de&longs;criberet Luna de&longs;cendendo per Vim centripetam ver&longs;us Terram, &longs;i circulari omni motu privari fingeretur, ad &longs;patium quod eodem tempore quam minimo de&longs;cribit grave corpus in vicinia Terræ, per Vim gravitatis &longs;uæ cadendo. Horum &longs;patiorum prius æquale e&longs;t arcus a Luna per idem tempus de&longs;cripti &longs;inui ver&longs;o, quippe qui Lunæ tran&longs;lationem de Tangente, factam a Vi centripeta, metitur; atque adeo computari pote&longs;t ex datis tum Lunæ tempore periodico tum di&longs;tantia ejus a centro Terræ. Spatium po&longs;terius invenitur per Experimenta Pendulorum, quemadmodum docuit Hugenius. Inito itaque calculo, &longs;patium prius ad &longs;patium pofterius, &longs;eu vis centripeta Lunæ in Orbita &longs;ua revolventis ad vim Gravitatis in &longs;uperficie Terræ, erit ut quadratum &longs;emidiametri Terræ ad Orbitæ &longs;emidiametri quadratum. ~~Eandem habet rationem, per ea quæ &longs;uperius o&longs;tenduntur, vis centripeta Lunæ in Orbita &longs;ua revolvenris ad vim Lunæ centripetam prope Terræ &longs;uperficiem.~~ ~~Vis itaque centripeta prope Terræ &longs;uperficiem æqualis e&longs;t vi Gravitatis.~~ Non ergo diver&longs;æ &longs;unt vires, &longs;ed una atque eadem: &longs;i enim diver&longs;æ e&longs;&longs;ent, corpora viribus conjunctis duplo celerius in Terram caderent quam ex vi &longs;ola Gravitatis. Con&longs;tat igitur Vim illam centripetam, qua Luna perpetuo de Tangente vel trahitur vel impellitur & in Orbita retinetur, ip&longs;am e&longs;fe vim Gravitatis terre&longs;tris ad Lunam u&longs;que pertingentem. Et rationi quidem con&longs;entaneum e&longs;t ut ad ingentes diftantias illa &longs;e&longs;e Virtus extendat, cum nullam ejus &longs;en&longs;ibilem imminutionem, vel in alti&longs;&longs;imis montium cacuminibus, ob&longs;ervare licet. Gravitat itaque Luna in Terram: quin & actione mutua, Terra vici&longs;&longs;im in Lunam æqualiter gravitat: id quod abunde quidem confirmatur in hac Philo&longs;ophia, ubi agitur de Maris æ&longs;tu & Æquinoctiorum præce&longs;&longs;ione, ab actione tum Lunæ tum Solis in Terram oriundis. ~~Hinc & illud tandem edocemur, qua nimirum lege vis Gravitatis decre&longs;cat in majoribus a Tellure di&longs;tantiis.~~ ~~Nam cum Gravitas non diver&longs;a &longs;it a Vi centripeta Lunari, hæc vero &longs;it reciproce proportionalis quadrato di&longs;tantiæ; diminuetur & Gravitas in eadem ratione.~~

~~Progrediamur jam ad Planetas reliquos.~~ Quoniam revolutiones primariorum circa Solem & &longs;ecundariorum circa Jovem & Saturnum &longs;unt Phænomena generis eju&longs;dem ac revolutio Lunæ circa Terram, quoniam porro demon&longs;tratum e&longs;t vires centripetas primariorum dirigi ver&longs;us centrum Solis, &longs;ecundariorum ver&longs;us centra Jovis & Saturni, quemadmodum Lunæ vis centripeta ver&longs;us Terræ centrum dirigitur; adhæc, quoniam omnes illæ vires &longs;unt reciproce ut quadrata di&longs;tantiarum a centris, quemadmodum vis Lunæ e&longs;t ut quadratum di&longs;tantiæ a Terra: concludendum erit eandem e&longs;&longs;e naturam univer&longs;is. Itaque ut Luna gravitat in Terram, & Terra vici&longs;&longs;im in Lunam; &longs;ic etiam gravitabunt omnes &longs;ecundarii in primarios &longs;uos, & primarii vici&longs;&longs;im in &longs;ecundarios; &longs;ic & omnes primarii in Solem, & Sol vici&longs;&longs;im in primarios.

~~Igitur Sol in Planetas univer&longs;os gravitat & univer&longs;i in Solem.~~ ~~Nam &longs;ecundarii dum primarios &longs;uos comitantur, revolvuntur interea circum Solem una cum primariis.~~ ~~Eodem itaque argumento, utriu&longs;que generis Planetæ gravitant in Solem, & Sol in ip&longs;os.~~ Secundarios vero Planetas in Solem gravitare abunde in&longs;uper con&longs;tat ex inæqualitatibus Lunaribus; quarum accurati&longs;&longs;imam Theoriam, admiranda &longs;agacitate patefactam, in tertio hujus Operis libro expo&longs;itam habemus.

Solis virtutem attractivam quoquover&longs;um propagari ad ingentes u&longs;que di&longs;tantias, & &longs;e&longs;e diffundere ad &longs;ingulas circumjecti &longs;patii partes, aperti&longs;&longs;ime colligi pote&longs;t ex motu Cometarum; qui ab immen&longs;is intervallis profecti feruntur in viciniam Solis, & nonnunquam adeo ad ip&longs;um proxime accedunt ut Globum ejus, in Periheliis &longs;uis ver&longs;antes, tantum non contingere videantur. Horum Theoriam ab A&longs;tronomis antehac fru&longs;tra quæ&longs;itam, no&longs;tro tandem &longs;æculo feliciter inventam & per Ob&longs;ervationes certi&longs;&longs;ime demon&longs;tratam, Præ&longs;tanti&longs;&longs;imo no&longs;tro Auctori debemus. ~~Patet igitur Cometas in Sectionibus Conicis umbilicos in centro Solis habentibus moveri, & radiis ad Solem ductis areas temporibus proportionales de&longs;cribere.~~ Ex hi&longs;ce vero Phænomenis manife&longs;tum e&longs;t & Mathematice comprobatur, vires illas, quibus Cometæ retinentur in orbitis &longs;uis, re&longs;picere Solem & e&longs;&longs;e reciproce ut quadrata di&longs;tantiarum ab ip&longs;ius centro. Gravitant itaque Cometæ in Solem: atque adeo Solis vis attractiva non tantum ad corpora Planetarum in datis di&longs;tantiis & in eodem fere plano collocata, &longs;ed etiam ad Cometas in diver&longs;i&longs;&longs;imis Cælorum regionibus & in diver&longs;i&longs;&longs;imis di&longs;tantiis po&longs;itos pertingit. ~~Hæc igitur e&longs;t natura corporum gravitantium, ut vires &longs;uas edant ad omnes di&longs;tantias in omnia corpora gravitantia.~~ Inde vero &longs;equitur, Planetas & Cometas univer&longs;os &longs;e mutuo trahere, & in &longs;e mutuo graves e&longs;&longs;e: quod etiam confirmatur ex perturbatione Jovis & Saturni, A&longs;tronomis non incognita, & ab actionibus horum Planetarum in &longs;e invicem oriunda; quin & ex motu illo lenti&longs;&longs;imo Ap&longs;idum, qui &longs;upra memoratus e&longs;t, quique a cau&longs;a con&longs;imili profici&longs;citur.

~~Eo demum pervenimus ut dicendum &longs;it, & Terram & Solem & corpora omnia cæle&longs;tia, quæ Solem comitantur, &longs;e mutuo attrahere.~~ ~~Singulorum ergo particulæ quæque minimæ vires &longs;uas attractivas habebunt, pro quantitate materiæ pollentes; quemadmodum &longs;upra de Terre&longs;tribus o&longs;ten&longs;um e&longs;t.~~ In diver&longs;is autem di&longs;tantiis, erunt & harum vires in duplicata ratione di&longs;tantiarum reciproce: nam ex particulis hac lege trahentibus componi debere Globos eadem lege trahentes, Mathematice demon&longs;tratur.

Conclu&longs;iones præcedentes huic innituntur Axiomati, quod a nullis non recipitur Philo&longs;ophis; Effectuum &longs;cilicet eju&longs;dem generis, quorum nempe quæ cogno&longs;cuntur proprietates eædem &longs;unt, ea&longs;dem e&longs;&longs;e cau&longs;as & ea&longs;dem e&longs;&longs;e proprietates quæ nondum cogno&longs;cuntur. Quis enim dubitat, &longs;i Gravitas &longs;it cau&longs;a de&longs;cen&longs;us Lapidis in Europa, quin eadem &longs;it cau&longs;a de&longs;cen&longs;us in America?Si Gravitas mutua fuerit inter Lapidem & Terram in Europa; quis negabit mutuam e&longs;&longs;e in America? Si vis attractiva Lapidis & Terræ componatur, in Europa, ex viribus attractivis partium; quis negabit &longs;imilem e&longs;&longs;e compo&longs;itionem in America? Si attractio Terræ ad omnia corporum genera & ad omnes di&longs;tantias propagetur in Europa; quidni pariter propagari dicamus in America?In hac Regula fundatur omnis Philo&longs;ophia: quippe qua &longs;ublata nihil affirmare po&longs;&longs;imus de Univer&longs;is. ~~Con&longs;titutio rerum &longs;ingularum innote&longs;cit per Ob&longs;ervationes & Experimenta: inde vero non ni&longs;i per hanc Regulam de rerum univer&longs;arum natura judicamus.~~

Jam cum Gravia &longs;int omnia corpora, quæ apud Terram vel in Cælis reperiuntur, de quibus Experimenta vel Ob&longs;ervationes in&longs;tituere licet; omnino dicendum erit, Gravitatem corporibus univer&longs;is competere. ~~Et quemadmodum nulla concipi debent corpora, quæ non &longs;int Exten&longs;a, Mobilia, & Impenetrabilia; ita nulla concipi debere, quæ non &longs;int Gravia.~~ ~~Corporum Exten&longs;io, Mobilitas, & Impenetrabilitas non ni&longs;i per Experimenta innote&longs;cunt: eodem plane modo Gravitas innote&longs;cit.~~ Corpora omnia de quibus Ob&longs;ervationes habemus, Exten&longs;a &longs;unt & Mobilia & Impenetrabilia: & inde concludimus corpora univer&longs;a, etiam illa de quibus Ob&longs;ervationes non habemus, Exten&longs;a e&longs;&longs;e & Mobilia & Impenetrabilia. ~~Ita corpora omnia &longs;unt Gravia, de quibus Ob&longs;ervationes habemus: & inde concludimus corpora univer&longs;a, etiam illa de quibus Ob&longs;ervationes non habemus, Gravia e&longs;&longs;e.~~ Si quis dicat corpora Stellarum inerrantium non e&longs;&longs;e Gravia, quandoquidem eorum Gravitas nondum e&longs;t ob&longs;ervata; eodem argumento dicere licebit neque Exten&longs;a e&longs;&longs;e, nec Mobilia, nec Impenetrabilia, cum hæ Fixarum affectiones nondum &longs;int ob&longs;ervatæ. ~~Quid opus e&longs;t verbis?~~ ~~Inter primarias qualitates corporum univer&longs;orum vel Gravitas habebit locum; vel Exten&longs;io, Mobilitas, & Impenetrabilitas non habebunt.~~ ~~Et natura rerum vel recte explicabitur per corporum Gravitatem, vel non recte explicabitur per corporum Exten&longs;ionem, Mobilitatem, & Impenetrabilitatem.~~

~~Audio nonnullos hanc improbare conclu&longs;ionem, & de occultis qualitatibus ne&longs;cio quid mu&longs;&longs;itare.~~ ~~Gravitatem &longs;cilicet Occultum e&longs;&longs;e quid, perpetuo argutari &longs;olent; occultas vero cau&longs;as procul e&longs;&longs;e ablegandas a Philo&longs;ophia.~~ His autem facile re&longs;pondetur; occultas e&longs;&longs;e cau&longs;as, non illas quidem quarum exi&longs;tentia per Ob&longs;ervationes clari&longs;&longs;ime demon&longs;tratur, &longs;ed has &longs;olum quarum occulta e&longs;t & ficta exi&longs;tentia nondum vero comprobata. ~~Gravitas ergo non erit occulta cau&longs;a motuum cæle&longs;tium; &longs;iquidem ex Phænomenis o&longs;ten&longs;um e&longs;t, hanc virtutem revera exi&longs;tere.~~ Hi potius ad occultas confugiunt cau&longs;as; qui ne&longs;cio quos Vortices, materiæ cuju&longs;dam pror&longs;us fictitiæ & &longs;en&longs;ibus omnino ignotæ, motibus ii&longs;dem regendis præficiunt.

~~Ideone autem Gravitas occulta cau&longs;a dicetur, eoque nomine rejicietur e Philo&longs;ophia, quod cau&longs;a ip&longs;ius Gravitatis occulta e&longs;t & nondum inventa?~~ ~~Qui &longs;ic &longs;tatuunt, videant nequid &longs;tatu- ant ab&longs;urdi, unde totius tandem Philo&longs;ophiæ fundamenta convellantur.~~ ~~Etenim cau&longs;æ continuo nexu procedere &longs;olent a compo&longs;itis ad &longs;impliciora: ubi ad cau&longs;am &longs;implici&longs;&longs;imam perveneris, jam non licebit ulterius progredi.~~ ~~Cau&longs;æ igitur &longs;implici&longs;&longs;imæ nulla dari pote&longs;t mechanica explicatio: &longs;i daretur enim, cau&longs;a nondum e&longs;&longs;et &longs;implici&longs;&longs;ima.~~ ~~Has tu proinde cau&longs;as &longs;implici&longs;&longs;imas appellabis occultas, & exulare jubebis?~~ ~~&longs;imul vero exulabunt & ab his proxime pendentes & quæ ab illis porro pendent, u&longs;que dum a cau&longs;is omnibus vacua fuerit & probe purgata Philo&longs;ophia.~~

~~Sunt qui Gravitatem præter naturam e&longs;&longs;e dicunt, & Miraculum perpetuum vocant.~~ ~~Itaque rejiciendam e&longs;&longs;e volunt, cum in Phy&longs;ica præternaturales cau&longs;æ locum non habeant.~~ ~~Huic ineptæ pror&longs;us objectioni diluendæ, quæ & ip&longs;a Philo&longs;ophiam &longs;ubruit univer&longs;am, vix operæ pretium e&longs;t immorari.~~ Vel enim Gravitatem corporibus omnibus inditam e&longs;&longs;e negabunt, quod tamen dici non pote&longs;t: vel eo nomine præter naturam e&longs;&longs;e affirmabunt, quod ex aliis corporum affectionibus atque adeo ex cau&longs;is Mechanicis originem non habeat. ~~Dantur certe primariæ corporum affectiones; quæ, quoniam &longs;unt primariæ, non pendent ab aliis.~~ ~~Viderint igitur annon & hæ omnes &longs;int pariter præter naturam, eoque pariter rejiciendæ: viderint vero qualis &longs;it deinde futura Philo&longs;ophia.~~

~~Nonnulli &longs;unt quibus hæc tota Phy&longs;ica cæle&longs;tis vel ideo minus placet, quod cum Carte&longs;ii dogmatibus pugnare & vix conciliari po&longs;&longs;e videatur.~~ ~~His &longs;ua licebit opinione frui; ex æquo autem agant oportet: non ergo denegabunt aliis eandem libertatem quam &longs;ibi concedi po&longs;tulant.~~ ~~NEWTONIANAM itaque Philo&longs;ophiam, quæ nobis verior habetur, retinere & amplecti licebit, & cau&longs;as &longs;equi per Phænomena comprobatas, potius quam fictas & nondum comprobatas.~~ Ad veram Philo&longs;ophiam pertinet, rerum naturas ex cau&longs;is vere exi&longs;tentibus derivare: eas vero leges quærere, quibus voluit Summus opifex hunc Mundi pulcherrimum ordinem &longs;tabilire; non eas quibus potuit, &longs;i ita vi&longs;um fui&longs;&longs;et. Rationi enim con&longs;onum e&longs;t, ut a pluribus cau&longs;is, ab invicem nonnihil diver&longs;is, idem po&longs;&longs;it Effectus profici&longs;ci: hæc autem vera erit cau&longs;a, ex qua vere atque actu profici&longs;citur; reliquæ locum non habent in Philo&longs;ophia vera. ~~In Horologiis automatis idem Indicis horarii motus vel ab appen&longs;o Pondere vel ab intus conclu&longs;o Elatere oriri pote&longs;t.~~ Quod &longs;i oblatum Horologium revera &longs;it in&longs;tructum Pondere; ridebitur qui finget Elaterem, & ex Hypothe&longs;i &longs;ic præpropere conficta motum Indicis explicare &longs;u&longs;cipiet: oportuit enim internam Machinæ fabricam penitius per&longs;crutari, ut ita motus propo&longs;iti principium verum exploratum habere po&longs;&longs;et. Idem vel non ab&longs;imile feretur judicium de Philo&longs;ophis illis, qui materia quadam &longs;ubtili&longs;&longs;ima Cælos e&longs;&longs;e repletos, hanc autem in Vortices inde&longs;inenter agi voluerunt. Nam &longs;i Phænomenis vel accurati&longs;&longs;ime &longs;atisfacere po&longs;&longs;ent ex Hypothe&longs;ibus &longs;uis; veram tamen Philo&longs;ophiam tradidi&longs;&longs;e, & veras cau&longs;as motuum cæle&longs;tium inveni&longs;&longs;e nondum dicendi &longs;unt; ni&longs;i vel has revera exi&longs;tere, vel &longs;altem alias non exi&longs;tere demon&longs;traverint. Igitur &longs;i o&longs;ten&longs;um fuerit, univer&longs;orum corporum Attractionem habere verum locum in rerum natura; quinetiam o&longs;ten&longs;um fuerit, qua ratione motus omnes cæle&longs;tes abinde &longs;olutionem recipiant; vana fuerit & merito deridenda objectio, &longs;i quis dixerit eo&longs;dem motus per Vortices explicari debere, etiam&longs;i id fieri po&longs;&longs;e vel maxime conce&longs;&longs;erimus. Non autem concedimus: Nequeunt enim ullo pacto Phænomena per Vortices explicari; quod ab Auctore no&longs;tro abunde quidem & clari&longs;&longs;imis rationibus evincitur; ut &longs;omniis plus æquo indulgeant oporteat, qui inepti&longs;&longs;imo figmento re&longs;arciendo, novi&longs;que porro commentis ornando infelicem operam addicunt.

Si corpora Planetarum & Cometarum circa Solem deferantur a Vorticibus; oportet corpora delata & Vorticum partes proxime ambientes eadem velocitate eademque cur&longs;us determinatione moveri, & eandem habere den&longs;itatem vel eandem Vim inertiæ pro mole materiæ. ~~Con&longs;tat vero Planetas & Cometas, dum ver&longs;antur in ii&longs;dem regionibus Cælorum, velocitatibus variis variaque cur&longs;us determinatione moveri.~~ Nece&longs;&longs;ario itaque &longs;equitur, ut Fluidi cæle&longs;tis partes illæ, quæ &longs;unt ad ea&longs;dem di&longs;tantias a Sole, revolvantur eodem tempore in plagas diver&longs;as cum diver&longs;is velocitatibus: etenim alia opus erit directione & velocitate, ut tran&longs;ire po&longs;&longs;int Planetæ; alia, ut tran&longs;ire po&longs;&longs;int Cometæ. Quod cum explicari nequeat; vel fatendum erit, univer&longs;a corpora cæle&longs;tia non deferri a materia Vorticis; vel dicendum erit, eorundem motus repetendos e&longs;le non ab uno eodemque Vortice, &longs;ed a pluribus qui ab invicem diver&longs;i &longs;int, idemque &longs;patium Soli circumjectum pervadant.

Si plures Vortices in eodem &longs;patio contineri, & &longs;e&longs;e mutuo penetrare, motibu&longs;que diver&longs;is revolvi ponantur; quoniam hi motus debent e&longs;&longs;e conformes delatorum corporum motibus, qui &longs;unt &longs;umme regulares, & peraguntur in Sectionibus Conicis, nunc valde eccentricis, nunc ad Circulorum proxime formam accedentibus; jure quærendum erit, qui fieri po&longs;&longs;it, ut iidem integri con&longs;erventur, nec ab actionibus materiæ occur&longs;antis per tot &longs;æcula quicquam perturbentur. Sane &longs;i motus hi fictitii &longs;unt magis compo&longs;iti & difficilius explicantur, quam veri illi motus Planetarum & Cometarum; fru&longs;tra mihi videntur in Philo&longs;ophiam recipi: omnis enim Cau&longs;a debet e&longs;&longs;e E&longs;fectu &longs;uo &longs;implicior. Conce&longs;&longs;a Fabularum licentia, affirmaverit aliquis Planetas omnes & Cometas circumcingi Atmo&longs;phæris, adin&longs;tar Telluris no&longs;træ; quæ quidem Hypothe&longs;is rationi magis con&longs;entanea videbitur quam Hypothe&longs;is Vorticum. Affirmaverit deinde has Atmo&longs;phæras, ex natura &longs;ua, circa Solem moveri & Sectiones Conicas de&longs;cribere; qui &longs;ane motus multo facilius concipi pote&longs;t, quam con&longs;imilis motus Vorticum &longs;e invicem permeantium. ~~Denique Planetas ip&longs;os & Cometas circa Solem deferri ab Atmo&longs;phæris &longs;uis credendum e&longs;&longs;e &longs;tatuat, & ob repertas motuum cæle&longs;tium cau&longs;as triumphum agat.~~ Qui&longs;quis autem hanc Fabulam rejiciendam e&longs;&longs;e putet, idem & alteram Fabulam rejiciet: nam ovum non e&longs;t ovo &longs;imilius, quam Hypothe&longs;is Atmo&longs;phærarum Hypothe&longs;i Vorticum.

~~Docuit Galilæus, lapidis projecti & in Parabola moti deflexionem a cur&longs;u rectilineo oriri a Gravitate lapidis in Terram, ab occulta &longs;cilicet qualitate.~~ ~~Fieri tamen pote&longs;t ut alius aliquis, na&longs;i acutioris, Philo&longs;ophus cau&longs;am aliam commini&longs;catur.~~ Finget igitur ille materiam quandam &longs;ubtilem, quæ nec vi&longs;u, nec tactu, neque ullo &longs;en&longs;u percipitur, ver&longs;ari in regionibus quæ proxime contingunt Telluris &longs;uperficiem. ~~Hanc autem materiam, in diver&longs;as plagas, variis & plerumque contrariis motibus ferri, & lineas Parabolicas de&longs;cribere contendet.~~ ~~Deinde vero lapidis deflexionem pulchre &longs;ic expediet, & vulgi plau&longs;um merebitur.~~ ~~Lapis, inquiet, in Fluido illo &longs;ubtili natat; & cur&longs;ui ejus ob&longs;equendo, non pote&longs;t non eandem una &longs;emitam de&longs;cribere.~~ ~~Fluidum vero movetur in lineis Parabolicis; ergo lapidem in Parabola moveri nece&longs;&longs;e e&longs;t.~~ Quis nunc non mirabitur acuti&longs;&longs;imum huju&longs;ce Philo&longs;ophi ingenium, ex cau&longs;is Mechanicis, materia &longs;cilicet & motu, phænomena Naturæ ad vulgi etiam captum præclare deducentis? ~~Quis vero non &longs;ub&longs;annabit bonum illum Galilæum, qui magno molimine Mathematico qualitates occultas, e Philo&longs;ophia feliciter exclu&longs;as, denuo revocare &longs;u&longs;tinuerit?~~ ~~Sed pudet nugis diutius immorari.~~

~~Summa rei huc tandem redìt: Cometarum ingens e&longs;t numerus; motus eorum &longs;unt &longs;umme regulares, & ea&longs;dem leges cum Planetarum motibus ob&longs;ervant.~~ ~~Moventur in Orbibus Conicis, hi orbes &longs;unt valde admodum eccentrici.~~ ~~Feruntur undique in omnes Cælorum partes, & Planetarum regiones liberrime pertran&longs;eunt, & &longs;æpe contra Signorum ordinem incedunt.~~ Hæc Phænomena certi&longs;&longs;ime confirmantur ex Ob&longs;ervationibus A&longs;tronomicis: & per Vortices nequeunt explicari: Imo, ne quidem cum Vorticibus Planetarum con&longs;i&longs;tere po&longs;&longs;unt. ~~Cometarum motibus omnino locus non erit; ni&longs;i materia illa fictitia penitus e Cælis amoveatur.~~

Si enim Planetæ circum Solem a Vorticibus devehuntur; Vorticum partes, quæ proxime ambiunt unumquemque Planetam, eju&longs;dem den&longs;itatis erunt ac Planeta; uti &longs;upra dictum e&longs;t. Itaque materia illa omnis quæ contigua e&longs;t Orbis magni perimetro, parem habebit ac Tellus den&longs;itatem: quæ vero jacet intra Orbem magnum atque Orbem Saturni, vel parem vel majorem habebit. ~~Nam ut con&longs;titutio Vorticis permanere po&longs;&longs;it, debent partes minus den&longs;æ centrum occupare, magis den&longs;æ longius a centro abire.~~ ~~Cum enim Planetarum tempora periodica &longs;int in ratione &longs;e&longs;quiplicata di&longs;tantiarum a Sole, oportet partium Vorticis periodos eandem rationem &longs;ervare.~~ ~~Inde vero &longs;equitur, vires centrifugas harum partium fore reciproce ut quadrata di&longs;tantiarum.~~ Quæ igitur majore intervallo di&longs;tant a centro, nituntur ab eodem recedere minore vi: unde &longs;i minus den&longs;æ fuerint, nece&longs;&longs;e e&longs;t ut cedant vi majori, qua partes centro propiores a&longs;cendere conantur. A&longs;cendent ergo den&longs;iores, de&longs;cendent minus den&longs;æ, & locorum fiet invicem permutatio; donec ita fuerit di&longs;po&longs;ita atque ordinata materia fluida totius Vorticis, ut conquie&longs;cere jam po&longs;&longs;it in æquilibrio con&longs;tituta. Si bina Fluida, quorum diver&longs;a e&longs;t den&longs;itas, in eodem va&longs;e continentur; utique futurum e&longs;t ut Fluidum, cujus major e&longs;t den&longs;itas, majore vi Gravitatis infimum petat locum: & ratione non ab&longs;imili omnino dicendum e&longs;t, den&longs;iores Vorticis partes majore vi centrifuga petere &longs;upremum locum. Tota igitur illa & multo maxima pars Vorticis, quæ jacet extra Telluris orbem, den&longs;itatem habebit atque adeo vim inertiæ pro mole materiæ, quæ non minor erit quam den&longs;itas & vis inertiæ Telluris: inde vero Cometis trajectis orietur ingens re&longs;i&longs;tentia, & valde admodum &longs;en&longs;ibilis; ne dicam, quæ motum eorundem penitus &longs;i&longs;tere atque ab&longs;orbere po&longs;&longs;e merito videatur. Con&longs;tat autem ex motu Co- metarum pror&longs;us regulari, nullam ip&longs;os re&longs;i&longs;tentiam pati quæ vel minimum &longs;entiri pote&longs;t; atque adeo neutiquam in materiam ullam incur&longs;are, cujus aliqua &longs;it vis re&longs;i&longs;tendi, vel proinde cujus aliqua &longs;it den&longs;itas &longs;eu vis Inertiæ. ~~Nam re&longs;i&longs;tentia Mediorum oritur vel ab inertia materiæ fluidæ, vel a defectu lubricitatis.~~ ~~Quæ oritur a defectu lubricitatis, admodum exigua e&longs;t; & &longs;ane vix ob&longs;ervari pote&longs;t in Fluidis vulgo notis, ni&longs;i valde tenacia fuerint adin&longs;tar Olei & Mellis.~~ Re&longs;i&longs;tentia quæ &longs;entitur in Aere, Aqua, Hydrargyro, & huju&longs;modi Fluidis non tenacibus fere tota e&longs;t prioris generis; & minui non pote&longs;t per ulteriorem quemcunque gradum &longs;ubtilitatis, manente Fluidi den&longs;itate vel vi inertiæ, cui &longs;emper proportionalis e&longs;t hæc re&longs;i&longs;tentia; quemadmodum clari&longs;&longs;ime demon&longs;tratum e&longs;t ab Auctore no&longs;tro in peregregia Re&longs;i&longs;tentiarum Theoria, quæ paulo nunc accuratius exponitur, hac &longs;ecunda vice, & per Experimenta corporum cadentium plenius confirmatur.

~~Corpora progrediendo motum &longs;uum Fluido ambienti paulatim communicant, & communicando amittunt, amittendo autem retardantur.~~ E&longs;t itaque retardatio motui communicato proportionalis; motus vero communicatus, ubi datur corporis progredientis velocitas, e&longs;t ut Fluidi den&longs;itas; ergo retardatio &longs;eu re&longs;i&longs;tentia erit ut eadem Fluidi den&longs;itas; neque ullo pacto tolli pote&longs;t, ni&longs;i a Fluido ad partes corporis po&longs;ticas recurrente re&longs;tituatur motus ami&longs;&longs;us. Hoc autem dici non poterit, ni&longs;i impre&longs;&longs;io Fluidi in corpus ad partes po&longs;ticas æqualis fuerit impre&longs;&longs;ioni corporis in Fluidum ad partes anticas, hoc e&longs;t, ni&longs;i velocitas relativa qua Fluidum irruit in corpus a tergo, æqualis fuerit velocitati qua corpus irruit in Fluidum, id e&longs;t, ni&longs;i velocitas ab&longs;oluta Fluidi recurrentis duplo major fuerit quam velocitas ab&longs;oluta Fluidi propul&longs;i; quod fieri nequit. ~~Nullo igitur modo tolli pote&longs;t Fluidorum re&longs;i&longs;tentia, quæ oritur ab corundem den&longs;itate & vi inertiæ.~~ Itaque concludendum erit; Fluidi cæle&longs;tis nullam e&longs;&longs;e vim inertiæ, cum nulla &longs;it vis re&longs;i&longs;tendi: nullam e&longs;&longs;e vim qua motus communicetur, cum nulla &longs;it vis inertiæ: nullam e&longs;&longs;e vim qua mutatio quælibet vel corporibus &longs;ingulis vel pluribus inducatur, cum nulla &longs;it vis qua motus communicetur: nullam e&longs;&longs;e omnino efficaciam, cum nulla &longs;it facultas mutationem quamlibet inducendi. Quidni ergo hanc Hypothe&longs;in, quæ fundamento plane de&longs;tituitur, quæque naturæ rerum explicandæ ne minimum quidem in&longs;ervit, inepti&longs;&longs;imam vocare liceat & Philo&longs;opho pror- &longs;us indignam. ~~Qui Cælos materia fluida repletos e&longs;&longs;e volunt, hanc vero non inertem e&longs;&longs;e &longs;tatuunt; Hi verbis tollunt Vacuum, re ponunt.~~ ~~Nam cum huju&longs;modi materia fluida ratione nulla &longs;ecerni po&longs;&longs;it ab inani Spatio; di&longs;putatio tota fit de rerum nominibus, non de naturis.~~ ~~Quod &longs;i aliqui &longs;int adeo u&longs;que dediti Materiæ, ut Spatium a corporibus vacuui nullo pacto admittendum credere velint; videamus quo tandem oporteat illos pervenire.~~

Vel enim dicent hanc, quam confingunt, Mundi per omnia pleni con&longs;titutionem ex voluntate Dei profectam e&longs;&longs;e, propter eum finem, ut operationibus Naturæ &longs;ub&longs;idium præ&longs;ens haberi po&longs;&longs;et ab Æthere &longs;ubtili&longs;&longs;imo cuncta permeante & implente; quod tamen dici non pote&longs;t, &longs;iquidem jam o&longs;ten&longs;um e&longs;t ex Cometarum phænomenis, nullam e&longs;&longs;e hujus Ætheris efficaciam: vel dicent ex voluntate Dei profectam e&longs;&longs;e, propter finem aliquem ignotum; quod neque dici debet, &longs;iquidem diver&longs;a Mundi con&longs;titutio eodem argumento pariter &longs;tabiliri po&longs;&longs;et: vel denique non dicent ex voluntate Dei profectam e&longs;&longs;e, &longs;ed ex nece&longs;&longs;itate quadam Naturæ. ~~Tandem igitur delabi oportet in &longs;æces &longs;ordidas Gregis impuri&longs;&longs;imi.~~ Hi &longs;unt qui &longs;omniant Fato univer&longs;a regi, non Providentia; Materiam ex nece&longs;&longs;itate &longs;ua &longs;emper & ubique extiti&longs;&longs;e, infinitam e&longs;&longs;e & æternam. ~~Quibus po&longs;itis, erit etiam undiquaque uniformis: nam varietas formarum cum nece&longs;&longs;itate omnino pugnat.~~ Erit etiam immota: nam &longs;i nece&longs;&longs;ario moveatur in plagam aliquam determinatam, cum determinata aliqua velocitate; pari nece&longs;&longs;itate movebitur in plagam diver&longs;am cum diver&longs;a velocitate; in plagas autem diver&longs;as, cum diver&longs;is velocitatibus, moveri non pote&longs;t; oportet igitur immotam e&longs;&longs;e. ~~Neutiquam profecto potuit oriri Mundus, pulcherrima formarum & motuum varietate di&longs;tinctus, ni&longs;i ex liberrima voluntate cuncta providentis & gubernantis Dei.~~

Ex hoc igitur fonte promanarunt illæ omnes quæ dicuntur Naturæ leges: in quibus multa &longs;ane &longs;apienti&longs;&longs;imi con&longs;ilii, nulla nece&longs;&longs;itatis apparent ve&longs;tigia. ~~Has proinde non ab incertis conjecturis petere, &longs;ed Ob&longs;ervando atque Experiendo addi&longs;cere debemus.~~ Qui veræ Phy&longs;icæ principia Lege&longs;que rerum, &longs;ola mentis vi & interno rationis lumine fretum, invenire &longs;e po&longs;&longs;e confidit; hunc oportet vel &longs;tatuere Mundum ex nece&longs;&longs;itate fui&longs;le, Lege&longs;que propo&longs;itas ex eadem nece&longs;&longs;itate &longs;equi; vel &longs;i per voluntatem Dei con&longs;titutus &longs;it ordo Naturæ, &longs;e tamen, homuncionem mi&longs;ellum, quid optimum factu &longs;it per&longs;pectum habere. Sana omnis & vera Philo&longs;ophia fundatur in Phænomenis rerum: quæ &longs;i nos vel invitos & reluctantes ad huju&longs;modi principia deducunt, in quibus clari&longs;&longs;ime cernuntur Con&longs;ilium optimum & Dominium &longs;ummum &longs;apienti&longs;&longs;imi & potenti&longs;&longs;imi Entis; non erunt hæc ideo non admittenda principia, quod quibu&longs;dam for&longs;an hominibus minus grata &longs;int futura. His vel Miracula vel Qualitates occultæ dicantur, quæ di&longs;plicent: verum nomina malitio&longs;e indita non &longs;unt ip&longs;is rebus vitio vertenda; ni&longs;i illud fateri tandem velint, utique debere Philo&longs;ophiam in Athei&longs;mo fundari. ~~Horum hominum gratia non erit labefactanda Philo&longs;ophia, &longs;iquidem rerum ordo non vult immutari.~~

~~Obtinebit igitur apud probos & æquos Judices præ&longs;tanti&longs;&longs;ima Philo&longs;ophandi ratio, quæ fundatur in Experimentis & Ob&longs;ervationibus.~~ Huic vero, dici vix poterit, quanta lux accedat, quanta dignitas, ab hoc Opere præclaro Illu&longs;tri&longs;&longs;imi no&longs;tri Auctoris; cujus eximiam ingenii felicitatem, difficillima quæque Problemata enodantis, & ad ea porro pertingentis ad quæ nec &longs;pes erat humanam mentem a&longs;&longs;urgere potui&longs;&longs;e, merito admirantur & &longs;u&longs;piciunt quicunque paulo profundius in hi&longs;ce rebus ver&longs;ati &longs;unt. ~~Clau&longs;tris ergo re&longs;eratis, aditum Nobis aperuit ad pulcherrima rerum my&longs;teria.~~ Sy&longs;tematis Mundani compagem eleganti&longs;&longs;imam ita tandem patefecit & penitius per&longs;pectandam dedit; ut nec ip&longs;e, &longs;i nunc revivi&longs;ceret, Rex Alphon&longs;us vel &longs;implicitatem vel harmoniæ gratiam in ea de&longs;ideraret. Itaque Naturæ maje&longs;tatem propius jam licet intueri, & dulci&longs;&longs;ima contemplatione frui, Conditorem vero ac Dominum Univer&longs;orum impen&longs;ius colere & venerari, qui fructus e&longs;t Philo&longs;ophiæ multo uberrimus. Cæcum e&longs;&longs;e oportet, qui ex optimis & &longs;apienti&longs;&longs;imis rerum &longs;tructuris non &longs;tatim videat Fabricatoris Omnipotentis infinitam &longs;apientiam & bonitatem: in&longs;anum, qui profiteri nolit.

Extabit igitur Eximium NEWTONI Opus adver&longs;us Atheorum impetus muniti&longs;&longs;imum præ&longs;idium: neque enim alicunde felicius, quam ex hac pharetra, contra impiam Catervam tela depromp&longs;eris. Hoc &longs;en&longs;it pridem, & in pereruditis Concionibus Anglice Latineque editis, primus egregie demon&longs;travit Vir in omni Literarum genere præclarus idemque bonarum Artium fautor eximius RICHARDUS BENTLEIUS, Sæculi &longs;ui & Academiæ no&longs;træ magnum Ornamentum, Collegii no&longs;tri S. Trinitatis Magi&longs;ter digni&longs;&longs;imus & integerrimus. ~~Huic ego me pluribus nominibus ob&longs;trictum fateri debeo: Huic & Tuas quæ debentur gratias, Lector benevole, non denegabis.~~ Is enim, cum a longo tempore Celeberrimi Auctoris amicitia intima frueretur, (qua etiam apud Po&longs;teros cen&longs;eri non minoris æ&longs;timat, quam propriis Scriptis quæ literato orbi in deliciis &longs;unt inclare&longs;cere) Amici &longs;imul famæ & &longs;cientiarum incremento con&longs;uluit. Itaque cum Exemplaria prioris Editionis rari&longs;&longs;ima admodum & immani pretio coemenda &longs;upere&longs;&longs;ent; &longs;ua&longs;it Ille crebris efflagitationibus & tantum non objurgando perpulit denique Virum Præ&longs;tanti&longs;&longs;imum, nec mode&longs;tia minus quam eruditione &longs;umma In&longs;ignem, ut novam hanc Operis Editionem, per omnia elimatam denuo & egregiis in&longs;uper acce&longs;&longs;ionibus ditatam, &longs;uis &longs;umptibus & au&longs;piciis prodire pateretur: Mihi vero, pro jure &longs;uo, pen&longs;um non ingratum demandavit, ut quam po&longs;&longs;et emendate id fieri curarem.

~~Cantabrigiæ,Maii 12. 1713.~~

~~ROGERUS COTES Collegii S. Trinitatis Socius, A&longs;tronomiæ & Philo&longs;ophiæ Experimentalis Profe&longs;&longs;or Plumianus.~~

~~INDEX CAPITUM TOTIUS OPERIS.~~

~~PAG.~~

~~DEFINITIONES. 1~~

~~AXIOMATA, SIVE LEGES MOTUS. 12~~

~~DE MOTU CORPORUM LIBER PRIMUS.~~

~~SECT. I. DE Methodo rationum primarum & ultimarum. 24~~

~~SECT. II. De inventione Virium centripetarum. 34~~

~~SECT. III. De motu corporum in Conicis &longs;ectionibus eccentricis. 48~~

~~SECT. IV. De inventione Orbium Ellipticorum, Parabolicorum & Hyperbolicorum ex Umbilico dato. 59~~

~~SECT. V. De inventione Orbium ubi Umbilicus neuter datur. 66~~

~~SECT. VI. De inventione Motuum in Orbibus datis. 97~~

~~SECT. VII. De corporum A&longs;cen&longs;u & De&longs;cen&longs;u rectilineo. 105~~

~~SECT. VII. De inventione Orbium in quibus corpora Viribus quibu&longs;cunque centripetis agitata revolvuntur. 114~~

~~SECT. IX. De Motu corporum in Orbibus mobilibus, deque Motu Ap&longs;idum. 121~~

~~SECT. X. De Motu corporum in Superficiebus datis, deque Funependulorum Motu reciproco. 132~~

~~SECT. XI. De Motu corporum Viribus centripetis &longs;e mutuo petentium. 147~~

~~SECT. XII. De corporum Sphæricorum Viribus attractivis. 173~~

~~SECT. XIII. De corporum non Sphæricorum Viribus attractivis. 192~~

~~SECT. XIV. De Motu corporum Minimorum, quæ Veribus centripetis ad &longs;ingulas Magni alicujus corporis partes tendentibus agitantur. 203~~

~~DE MOTU CORPORUM LIBER SECUNDUS.~~

~~SECT. I. DE Motu corporum quibus re&longs;i&longs;titur in ratione Velocitatis. 211~~

~~SECT. II. De Motu corporum quibus re&longs;i&longs;titur in duplicata ratione Velocitatis. 220~~

~~SECT. III. De Motu corporum quibus re&longs;i&longs;titur partim in ratione Velocitatis, partim in eju&longs;dem ratione duplicata. 245~~

~~SECT. IV. De corporum Circulari motu in Mediis re&longs;i&longs;tentibus.253~~

~~SECT. V. De den&longs;itate & compre&longs;&longs;ione Fluidorum, deque Hydro&longs;tatica. 260~~

~~SECT. VI. De Motu & Re&longs;i&longs;tentia corporum Funependulorum.272~~

~~SECT. VII. De motu Fluidorum & re&longs;i&longs;tentia Projectilium. 294~~

~~SECT. VIII. De motu per Fluida propagato. 329~~

~~SECT. IX. De motu Circulari Fluidorum. 345~~

~~DE MUNDI SYSTEMATE LIBER TERTIUS.~~

~~REGULÆ PHILOSOPHANDI 357~~

~~PHÆNOMENA 359~~

~~PROPOSITIONES 362~~

~~SCHOLIUM GENERALE. 481~~

~~PHILOSOPHIÆ NATURALIS Principia MATHEMATICA.~~

~~DEFINITIONES.~~

~~DEFINITIO I.~~

~~Quantitas Materiæ e&longs;t men&longs;ura eju&longs;dem orta ex illius Den&longs;itate & Magnitudine conjunctim.~~

~~AER, den&longs;itate duplicata, in &longs;patio etiam duplicato fit quadruplus; in triplicato &longs;extuplus.~~ ~~Idem intellige de Nive & Pulveribus per compre&longs;&longs;ionem vel liquefactionem conden&longs;atis.~~ ~~Et par e&longs;t ratio corporum omnium, quæ per cau&longs;as qua&longs;cunque diver&longs;imode conden&longs;antur.~~ ~~Medii interea, &longs;i quod fuerit, inter&longs;titia partium libere pervadentis, hic nullam rationem habeo.~~ ~~Hanc autem Quantitatem &longs;ub nomine Corporis vel Ma&longs;&longs;æ in &longs;equentibus pa&longs;&longs;im intelligo.~~ ~~Innote&longs;cit ea per corporis cuju&longs;que Pondus.~~ ~~Nam Ponderi proportionalem e&longs;&longs;e reperi per experimenta Pendulorum accurati&longs;&longs;ime in&longs;tituta, uti po&longs;thac docebitur.~~

~~DEFINITIO II.~~

~~Quantitas Motus e&longs;t men&longs;ura eju&longs;dem orta ex Velocitate & Quantitate Materiæ conjunctim.~~

~~Motus totius e&longs;t &longs;umma motuum in partibus &longs;ingulis; adeoque in corpore duplo majore æquali cum velocitate duplus e&longs;t, & dupla cum velocitate quadruplus.~~

~~DEFINITIO III.~~

Materiæ Vis In&longs;ita e&longs;t potentia re&longs;i&longs;tendi, qua corpus unumquodque, quantum in &longs;e e&longs;t, per&longs;everat in &longs;tatu &longs;uo vel quie&longs;cendi vel movendi uniformiter in directum.

~~Hæc &longs;emper proportionalis e&longs;t &longs;uo corpori, neque differt quicquam ab Inertia ma&longs;&longs;æ, ni&longs;i in modo concipiendi.~~ ~~Per inertiam materiæ, fit ut corpus omne de &longs;tatu &longs;uo vel quie&longs;cendi vel movendi difficulter deturbetur.~~ ~~Unde etiam vis in&longs;ita nomine &longs;ignificanti&longs;&longs;imo Vis Inertiæ dici po&longs;&longs;it.~~ Exercet vero corpus hanc vim &longs;olummodo in mutatione &longs;tatus &longs;ui per vim aliam in &longs;e impre&longs;&longs;am facta; e&longs;tque exercitium ejus &longs;ub diver&longs;o re&longs;pectu & Re&longs;i&longs;tentia & Impetus: re&longs;i&longs;tentia, quatenus corpus ad con&longs;ervandum &longs;tatum &longs;uum reluctatur vi impre&longs;&longs;æ; impetus, quatenus corpus idem, vi re&longs;i&longs;tentis ob&longs;taculi difficulter cedendo, conatur &longs;tatum ejus mutare. Vulgus re&longs;i&longs;tentiam quie&longs;centibus & impetum moventibus tribuit: &longs;ed motus & quies, uti vulgo concipiuntur, re&longs;pectu &longs;olo di&longs;tinguuntur ab invicem; neque &longs;emper vere quie&longs;cunt quæ vulgo tanquam quie&longs;centia &longs;pectantur.

~~DEFINITIO IV.~~

~~Vis Impre&longs;&longs;a e&longs;t actio in corpus exercita, ad mutandum ejus &longs;tatum vel quie&longs;cendi vel movendi uniformiter in directum.~~

~~Con&longs;i&longs;tit hæc vis in actione &longs;ola, neque po&longs;t actionem permanet in corpore.~~ ~~Per&longs;everat enim corpus in &longs;tatu omni novo per &longs;olam vim inertiæ.~~ ~~E&longs;t autem vis impre&longs;&longs;a diver&longs;arum originum, ut ex Ictu, ex Pre&longs;&longs;ione, ex vi Centripeta.~~

~~DEFINITIO V.~~

~~Vis Centripeta e&longs;t, qua corpora ver&longs;us punctum aliquod tanquam ad Centrum undique trahuntur, impelluntur, vel utcunq tendunt.~~

Hujus generis e&longs;t Gravitas, qua corpora tendunt ad centrum terræ; Vis Magnetica, qua ferrum petit magnetem; & Vis illa, quæcunque &longs;it, qua Planetæ perpetuo retrahuntur a motibus rectilineis, & in lineis curvis revolvi coguntur. ~~Lapis, in funda circum- actus, a circumagente manu abire conatur; & conatu &longs;uo fundam di&longs;tendit, eoque fortius quo celerius revolvitur; &, quamprimum dimittitur, avolat.~~ ~~Vim conatui illi contrariam, qua funda lapidem in manum perpetuò retrahit & in orbe retinet, quoniam in manum ceu orbis centrum dirigitur, Centripetam appello.~~ ~~Et par e&longs;t ratio corporum omnium, quæ in gyrum aguntur.~~ Conantur ea omnia a centris orbium recedere; & ni&longs;i ad&longs;it vis aliqua conatui i&longs;ti contraria, qua cohibeantur & in orbibus retineantur, quamque ideò Centripetam appello, abibunt in rectis lineis uniformi cum motu. Projectile, &longs;i vi Gravitatis de&longs;titueretur, non deflecteretur in terram, &longs;ed in linea recta abiret in cælos; idque uniformi cum motu, &longs;i modo aeris re&longs;i&longs;tentia tolleretur. ~~Per gravitatem &longs;uam retrahitur a cur&longs;u rectilineo & in terram perpetuo flectitur, idque magis vel minus pro gravitate &longs;ua & velocitate motus.~~ ~~Quo minor erit ejus gravitas pro quantitate materiæ vel major &c.~~ ~~vel major velocitas quacum projicitur, eo minus deviabit a cur&longs;u rectilineo & longius perget.~~ Si Globus plumbeus, data cum velocitate &longs;ecundum lineam horizontalem a montis alicujus vertice vi pulveris tormentarii projectus, pergeret in linea curva ad di&longs;tantiam duorum milliarium, priu&longs;quam in terram decideret: hic dupla cum velocitate qua&longs;i duplo longius pergeret, & decupla cum velocitate qua&longs;i decuplo longius: &longs;i modo aeris re&longs;i&longs;tentia tolleretur. Et augendo velocitatem augeri po&longs;&longs;et pro lubitu di&longs;tantia in quam projiceretur, & minui curvatura lineæ quam de&longs;criberet, ita ut tandem caderet ad di&longs;tantiam graduum decem vel triginta vel nonaginta; vel eriam ut terram totam circuiret priu&longs;quam caderet; vel denique ut in terram nunquam caderet, &longs;ed in cælos abiret & motu abeundi pergeret in infinitum. Et eadem ratione, qua Projectile vi gravitatis in orbem flecti po&longs;&longs;et & terram totam circuire, pote&longs;t & Luna vel vi gravitatis, &longs;i modo gravis &longs;it, vel alia quacunque vi, qua in terram urgeatur, retrahi &longs;emper a cur&longs;u rectilineo terram ver&longs;us, & in orbem &longs;uum flecti: & ab&longs;que tali vi Luna in orbe &longs;uo retineri non pote&longs;t. Hæc vis, &longs;i ju&longs;to minor e&longs;&longs;et, non &longs;atis flecteret Lunam de cur&longs;u rectilineo: &longs;i ju&longs;to major, plus &longs;atis flecteret, ac de orbe terram ver&longs;us deduceret. Requiritur quippe, ut &longs;it ju&longs;tæ magnitudinis: & Mathematicorum e&longs;t invenire Vim, qua corpus in dato quovis orbe data cum velocitate accurate retineri po&longs;&longs;it; & vici&longs;&longs;im invenire Viam curvilineam, in quam corpus e dato quovis loco data cum velocitate egre&longs;&longs;um a data vi flectatur. ~~E&longs;t autem vis hujus centripetæ Quantitas trium generum, Ab&longs;oluta, Acceleratrix, & Motrix.~~

~~NIES.~~

~~DEFINITIO VI.~~

~~Vis centripetæ Quantitas Ab&longs;oluta e&longs;t men&longs;ura eju&longs;dem major vel minor pro Efficacia cau&longs;æ eam propagantis a centro per regiones in circuitu.~~

~~Ut vis Magnetica pro mole magnetis vel inten&longs;ione virtutis major in uno magnete, minor in alio.~~

~~DEFINITIO VII.~~

~~Vis centripetæ Quantitas Acceleratrix e&longs;t ip&longs;ius men&longs;ura Velocitati proportionalis, quam dato tempore generat.~~

Uti Virtus magnetis eju&longs;dem major in minori di&longs;tantia, minor in majori: vel vis Gravitans major in vallibus, minor in cacuminibus præaltorum montium, atque adhuc minor (ut po&longs;thac patebit) in majoribus di&longs;tantiis a globo terræ; in æqualibus autem di&longs;tantiis eadem undique, propterea quod corpora omnia cadentia (gravia an levia, magna an parva) &longs;ublata Aeris re&longs;i&longs;tentia, æqualiter accelerat.

~~DEFINITIO VIII.~~

~~Vis centripetæ Quantitas Motrix e&longs;t ip&longs;ius men&longs;ura proportionalis.~~ ~~Motui, quem dato tempore generat.~~

~~Uti Pondus majus in majore corpore, minus in minore; inque corpore eodem majus prope terram, minus in cælis.~~ Hæc Quantitas e&longs;t corporis totius centripetentia &longs;eu propen&longs;io in centrum, & (ut ita dicam) Pondus; & innote&longs;cit &longs;emper per vim ip&longs;i contrariam & æqualem, qua de&longs;cen&longs;us corporis impediri pote&longs;t.

Ha&longs;ce virium quantitates brevitatis gratia nominare licet vires motrices, acceleratrices, & ab&longs;olutas; & di&longs;tinctionis gratia referre ad Corpora, centrum petentia, ad corporum Loca, & ad Centrum virium: nimirum vim motricem ad Corpus, tanquam conatum & propen&longs;ionem totius in centrum ex propen&longs;ionibus omnium partium compo&longs;itam; & vim acceleratricem ad Locum corporis, tanquam efficaciam quandam, de centro per loca &longs;ingula in circuitu diffu&longs;am, ad movenda corpora quæ in ip&longs;is &longs;unt; vim autem ab&longs;olutam ad Centrum, tanquam cau&longs;a aliqua præditum, &longs;ine qua vires motrices non propagantur per regiones in circuitu; &longs;ive cau&longs;a illa &longs;it corpus aliquod centrale (quale e&longs;t Magnes in centro vis magneticæ, vel Terra in centro vis gravitantis) &longs;ive alia aliqua quæ non apparet. ~~Mathematicus duntaxat e&longs;t hic conceptus.~~ ~~Nam virium cau&longs;as & &longs;edes phy&longs;icas jam non expendo.~~

~~E&longs;t igitur vis acceleratrix ad vim motricem ut celeritas ad motum.~~ ~~Oritur enim quantitas motus ex celeritate ducta in quantitatem materiæ, & vis motrix ex vi acceleratrice ducta in quantitatem eju&longs;dem materiæ.~~ ~~Nam &longs;umma actionum vis acceleratricis in &longs;ingulas corporis particulas e&longs;t vis motrix totius.~~ Unde juxta &longs;uperficiem Terræ, ubi gravitas acceleratrix &longs;eu vis gravitans in corporibus univer&longs;is eadem e&longs;t, gravitas motrix &longs;eu pondus e&longs;t ut corpus: at &longs;i in regiones a&longs;cendatur ubi gravitas acceleratrix fit minor, pondus pariter minuetur, eritque &longs;emper ut corpus in gravitatem acceleratricem ductum. ~~Sic in regionibus ubi gravitas acceleratrix duplo minor e&longs;t, pondus corporis duplo vel triplo minoris erit quadruplo vel &longs;extuplo minus.~~

~~Porro attractiones & impul&longs;us eodem &longs;en&longs;u acceleratrices & motrices nomino.~~ Voces autem Attractionis, Impul&longs;us, vel Propen&longs;ionis cuju&longs;cunque in centrum, indifferenter & pro &longs;e mutuo promi&longs;cue u&longs;urpo; has vires non Phy&longs;ice &longs;ed Mathematice tantum con&longs;iderando. Unde caveat lector, ne per huju&longs;modi voces cogitet me &longs;peciem vel modum actionis cau&longs;amve aut rationem Phy&longs;icam alicubi definire, vel centris (quæ &longs;unt puncta Mathematica) vires vere & Phy&longs;ice tribuere; &longs;i forte aut centra trahere, aut vires centrorum e&longs;&longs;e dixero.

~~Scholium.~~

~~Hactenus voces minus notas, quo &longs;en&longs;u in &longs;equentibus accipiendæ &longs;int, explicare vi&longs;um e&longs;t.~~ ~~Nam Tempus, Spatium, Locum & Motum, ut omnibus noti&longs;&longs;ima, non definio.~~ ~~Notandum tamen, quod vulgus quantitates ha&longs;ce non aliter quam ex relatione ad &longs;en&longs;ibilia concipiat.~~ ~~Et inde oriuntur præjudicia quædam, quibus tollendis convenit ea&longs;dem in ab&longs;olutas & relativas, veras & apparentes, mathematicas & vulgares di&longs;tingui.~~

I. Tempus Ab&longs;olutum, verum, & mathematicum, in &longs;e & natura &longs;ua ab&longs;que relatione ad externum quodvis, æquabiliter fluit, alioque nomine dicitur Duratio: Relativum, apparens, & vulgare e&longs;t &longs;en&longs;ibilis & externa quævis Durationis per motum men&longs;ura (&longs;eu accurata &longs;eu inæquabilis) qua vulgus vice veri temporis utitur; ut Hora, Dies, Men&longs;is, Annus.

~~NIES.~~

~~II.~~ Spatium Ab&longs;olutum, natura &longs;ua ab&longs;que relatione ad externum quodvis, &longs;emper manet &longs;imilare & immobile: Relativum e&longs;t &longs;patii hujus men&longs;ura &longs;eu dimen&longs;io quælibet mobilis, quæ a &longs;en&longs;ibus no&longs;tris per &longs;itum &longs;uum ad corpora definitur, & a vulgo pro &longs;patio immobili u&longs;urpatur: uti dimen&longs;io &longs;patii &longs;ubterranei, aerei vel cæle&longs;tis definita per &longs;itum &longs;uum ad Terram. ~~Idem &longs;unt &longs;patium ab&longs;olutum & relativum, &longs;pecie & magnitudine; &longs;ed non permanent idem &longs;emper numero.~~ Nam &longs;i Terra, verbi gratia, movetur; &longs;patium Aeris no&longs;tri, quod relative & re&longs;pectu Terræ &longs;emper manet idem, nunc erit una pars &longs;patii ab&longs;oluti in quam Aer tran&longs;it, nunc alia pars ejus; & &longs;ic ab&longs;olute mutabitur perpetuo.

~~III.~~ ~~Locus e&longs;t pars &longs;patii quam corpus occupat, e&longs;tque pro ratione &longs;patii vel Ab&longs;olutus vel Relativus.~~ ~~Pars, inquam, &longs;patii; non Situs corporis, vel Superficies ambiens.~~ Nam &longs;olidorum æqualium æquales &longs;emper &longs;unt loci; Superficies autem ob di&longs;&longs;imilitudinem figurarum ut plurimum inæquales &longs;unt; Situs vero proprie loquendo quantitatem non habent, neque tam &longs;unt loca quam affectiones locorum. Motus totius idem e&longs;t cum &longs;umma motuum partium, hoc e&longs;t, tran&longs;latio totius de &longs;uo loco eadem e&longs;t cum &longs;umma tran&longs;lationum partium de locis &longs;uis; adeoque locus totius idem cum &longs;umma locorum partium, & propterea internus & in corpore toto.

~~IV.~~ ~~Motus Ab&longs;olutus e&longs;t tran&longs;latio corporis de loco ab&longs;oluto in locum ab&longs;olutum, Relativus de relativo in relativum.~~ Sic in navi quæ velis pa&longs;&longs;is fertur, relativus corporis Locus e&longs;t navigii regio illa in qua corpus ver&longs;atur, &longs;eu cavitatis totius pars illa quam corpus implet, quæque adeo movetur una cum navi: & Quies relativa e&longs;t perman&longs;io corporis in eadem illa navis regione vel parte cavitatis. ~~At quies Vera e&longs;t perman&longs;io corporis in eadem parte &longs;patii illius immoti in qua navis ip&longs;a una cum cavitate &longs;ua & contentis univer&longs;is movetur.~~ ~~Unde &longs;i Terra vere quie&longs;cit, corpus quod relative quie&longs;cit in navi, movebitur vere & ab&longs;olute ea cum velocitate qua navis movetur in Terra.~~ Sin Terra etiam movetur; orietur verus & ab&longs;olutus corporis motus, partim ex Terræ motu vero in &longs;patio immoto, partim ex navis motu relativo in Terra: & &longs;i corpus etiam movetur relative in navi; orietur verus ejus motus, partim ex vero motu Terræ in &longs;patio immoto, partim ex relativis motibus tum navis in Terra, tum corporis in navi; & ex his motibus relativis orietur corporis motus relativus in Terra. Ut &longs;i Terræ pars illa, ubi navis ver&longs;atur, moveatur vere in orientem cum velocitate partium 10010; & velis ventoque feratur navis in occidentem cum velocitate partium decem; Nauta autem ambulet in navi ori- entem ver&longs;us cum velocitatis parte una: movebitur Nauta vere & ab&longs;olute in &longs;patio immoto cum velocitatis partibus 10001 in orientem, & relative in terra occidentem ver&longs;us cum velocitatis partibus novem.

~~Tempus Ab&longs;olutum a relativo di&longs;tinguitur in A&longs;tronomia per Æquationem temporis vulgi.~~ ~~Inæquales enim &longs;unt dies Naturales, qui vulgo tanquam æquales promen&longs;ura temporis habentur.~~ ~~Hanc inæqualitatem corrigunt A&longs;tronomi, ut ex veriore tempore men&longs;urent motus &c.~~ ~~motus cæle&longs;tes.~~ ~~Po&longs;&longs;ibile e&longs;t, ut nullus &longs;it motus æquabilis quo Tempus accurate men&longs;uretur.~~ ~~Accelerari & retardari po&longs;&longs;unt motus omnes, &longs;ed fluxus temporis Ab&longs;oluti mutari nequit.~~ Eadem e&longs;t duratio &longs;eu per&longs;everantia exi&longs;tentiæ rerum; &longs;ive motus &longs;int celeres, &longs;ive tardi, &longs;ive nulli: proinde hæc a men&longs;uris &longs;uis &longs;en&longs;ibilibus merito di&longs;tinguitur, & ex ii&longs;dem colligitur per Æquationem A&longs;tronomicam. ~~Hujus autem æquationis in determinandis Phænomenis nece&longs;&longs;itas, tum per experimentum Horologii O&longs;cillatorii, tum etiam per eclip&longs;es Satellitum Jovis evincitur.~~

~~Ut partium Temporis ordo e&longs;t immutabilis, &longs;ic etiam ordo partium Spatii.~~ ~~Moveantur hæ de locis &longs;uis, & movebuntur (ut ita dicam) de &longs;eip&longs;is.~~ ~~Nam tempora & &longs;patia &longs;unt &longs;ui ip&longs;orum & rerum omnium qua&longs;i Loca.~~ ~~In Tempore quoad ordinem &longs;ucce&longs;&longs;ionis; in Spatio quoad ordinem &longs;itus locantur univer&longs;a.~~ ~~De illorum e&longs;&longs;entia e&longs;t ut &longs;int Loca: & loca primaria moveri ab&longs;urdum e&longs;t.~~ ~~Hæc &longs;unt igitur ab&longs;oluta Loca; & &longs;olæ tran&longs;lationes de his locis &longs;unt ab&longs;oluti Motus.~~

~~Verum quoniam hæ Spatii partes videri nequeunt, & ab invicem per &longs;en&longs;us no&longs;tros di&longs;tingui; earum vice adhibemus men&longs;uras &longs;en&longs;ibiles.~~ Ex po&longs;itionibus enim & di&longs;tantiis rerum a corpore aliquo, quod &longs;pectamus ut immobile, de&longs;inimus loca univer&longs;a: deinde etiam & omnes motus æ&longs;timamus cum re&longs;pectu ad prædicta loca, quatenus corpora ab ii&longs;dem transferri concipimus. ~~Sic vice locorum & motuum ab&longs;olutorum relativis utimur; nec incommode in rebus humanis: in Philo&longs;ophicis autem ab&longs;trahendum e&longs;t a &longs;en&longs;ibus.~~ ~~Fieri etenim pote&longs;t, ut nullum revera quie&longs;cat corpus, ad quod loca motu&longs;que referantur.~~

~~Di&longs;tinguuntur autem Quies & Motus ab&longs;oluti & relativi ab invicem per Proprietates &longs;uas & Cau&longs;as & Effectus.~~ ~~Quietis proprietas e&longs;t, quod corpora vere quie&longs;centia quie&longs;cunt inter &longs;e.~~ Ideoque cum po&longs;&longs;ibile &longs;it, ut corpus aliquod in regionibus Fixarum, aut longe ultra, quie&longs;cat ab&longs;olute; &longs;ciri autem non po&longs;&longs;it ex &longs;itu corporum ad invicem in regionibus no&longs;tris, horumne aliquod ad longin-

~~quum illud datam po&longs;itionem &longs;ervet necne; quies vera ex horum &longs;itu inter &longs;e definiri nequit.~~

~~NIES.~~

~~Motus proprietas e&longs;t, quod partes, quæ datas &longs;ervant po&longs;itiones ad tota, participant motus eorundem totorum.~~ ~~Nam Gyrantium partes omnes conantur recedere ab axe motus, & Progredientium impetus oritur ex conjuncto impetu partium &longs;ingularum.~~ ~~Motis igitur corporibus ambientibus, moventur quæ in ambientibus relative quie&longs;cunt.~~ ~~Et propterea motus verus & ab&longs;olutus definiri nequit per tran&longs;lationem e vicinia corporum, quæ tanquam quie&longs;centia &longs;pectantur.~~ ~~Debent enim corpora externa non &longs;olum tanquam quie&longs;centia &longs;pectari, &longs;ed etiam vere quie&longs;cere.~~ Alioquin inclu&longs;a omnia, præter tran&longs;lationem e vicinia ambientium, participabunt etiam ambientium motus veros; & &longs;ublata illa tran&longs;latione non vere quie&longs;cent, &longs;ed tanquam quie&longs;centia &longs;olummodo &longs;pectabuntur. ~~Sunt enim ambientia ad inclu&longs;a, ut totius pars exterior ad partem interiorem, vel ut cortex ad nucleum.~~ ~~Moto autem cortice, nucleus etiam, ab&longs;que tran&longs;latione de vicinia corticis, ceu pars totius movetur.~~

~~Præcedenti proprietati affinis e&longs;t, quod moto Loco movetur una Locatum: adeoque corpus, quod de loco moto movetur, participat etiam loci &longs;ui motum.~~ Motus igitur omnes, qui de locis motis fiunt, &longs;unt partes &longs;olummodo motuum integrorum & ab&longs;olutorum: & motus omnis integer componitur ex motu corporis de loco &longs;uo primo, & motu loci hujus de loco &longs;uo, & &longs;ic deinceps; u&longs;que dum perveniatur ad locum immotum, ut in exemplo Nautæ &longs;upra memorato. ~~Unde motus integri & ab&longs;oluti non ni&longs;i per loca immota definiri po&longs;&longs;unt: & propterea hos ad loca immota, relativos ad mobilia &longs;upra retuli.~~ Loca autem immota non &longs;unt, ni&longs;i quæ omnia ab infinito in infinitum datas &longs;ervant po&longs;itiones ad invicem; atque adeo &longs;emper manent immota, &longs;patiumque con&longs;tituunt quod Immobile appello.

~~Cau&longs;æ, quibus motus veri & relativi di&longs;tinguuntur ab invicem, &longs;unt Vires in corpora impre&longs;&longs;æ ad motum generandum.~~ Motus verus nec generatur nec mutatur, ni&longs;i per vires in ip&longs;um corpus motum impre&longs;&longs;as: at motus relativus generari & mutari pote&longs;t ab&longs;que viribus impre&longs;&longs;is in hoc corpus. ~~Sufficit enim ut imprimantur in alia &longs;olum corpora ad quæ fit relatio, ut iis cedentibus mutetur relatio illa in qua hujus quies vel motus relativus con&longs;i&longs;tit.~~ ~~Rur&longs;um motus verus a viribus in corpus motum impre&longs;&longs;is &longs;emper mutatur; at motus relativus ab his viribus non mutatur nece&longs;&longs;ario.~~ Nam &longs;i eædem vires in alia etiam corpora, ad quæ &longs;it relatio, &longs;ic impri- mantur ut &longs;itus relativus con&longs;ervetur, con&longs;ervabitur relatio in qua motus relativus con&longs;i&longs;tit. Mutari igitur pote&longs;t motus omnis relativus ubi verus con&longs;ervatur, & con&longs;ervari ubi verus mutatur; & propterea motus verus in eju&longs;modi relationibus minime con&longs;i&longs;tit.

~~Effectus quibus motus ab&longs;oluti & relativi di&longs;tinguuntur ab invicem, &longs;unt vires recedendi ab axe motus circularis.~~ ~~Nam in motu circulari nude relativo hæ vires nullæ &longs;unt, in vero autem & ab&longs;oluto majores vel minores pro quantitate motus.~~ Si pendeat &longs;itula a filo prælongo, agaturque perpetuo in orbem, donec filum a contor&longs;ione admodum rige&longs;cat, dein impleatur aqua, & una cum aqua quie&longs;cat; tum vi aliqua &longs;ubitanea agatur motu contrario in orbem, & filo &longs;e relaxante, diutius per&longs;everet in hoc motu; &longs;uperficies aquæ &longs;ub initio plana erit, quemadmodum ante motum va&longs;is: at po&longs;tquam, vi in aquam paulatim impre&longs;&longs;a, effecit vas, ut hæc quoque &longs;en&longs;ibiliter revolvi incipiat; recedet ip&longs;a paulatim a medio, a&longs;cendetque ad latera va&longs;is, figuram concavam induens, (ut ip&longs;e expertus &longs;um) & incitatiore &longs;emper motu a&longs;cendet magis & magis, donec revolutiones in æqualibus cum va&longs;e temporibus peragendo, quie&longs;cat in eodem relative. Indicat hic a&longs;cen&longs;us conatum recedendi ab axe motus, & per talem conatum innote&longs;cit & men&longs;uratur motus aquæ circularis verus & ab&longs;olutus, motuique relativo hic omnino contrarius. Initio, ubi maximus erat aquæ motus relativus in va&longs;e, motus ille nullum excitabat conatum recedendi ab axe: aqua non petebat circumferentiam a&longs;cendendo ad latera va&longs;is, &longs;ed plana manebat, & propterea motus illius circularis verus nondum inceperat. Po&longs;tea vero, ubi aquæ motus relativus decrevit, a&longs;cen&longs;us ejus ad latera va&longs;is indicabat conatum recedendi ab axe; atque hic conatus mon&longs;trabat motum illius circularem verum perpetuo cre&longs;centem, ac tandem maximum factum ubi aqua quie&longs;cebat in va&longs;e relative. ~~Igitur conatus i&longs;te non pendet a tran&longs;latione aquæ re&longs;pectu corporum ambientium, & propterea motus circularis verus per tales tran&longs;lationes definiri nequit.~~ Unicus e&longs;t corporis cuju&longs;que revolventis motus vere circularis, conatui unico tanquam proprio & adæquato effectui re&longs;pondens: motus autem relativi pro variis relationibus ad externa innumeri &longs;unt; & relationum in&longs;tar, effectibus veris omnino de&longs;tituuntur, ni&longs;i quatenus verum illum & unicum motum participant. Unde & in Sy&longs;temate eorum qui Cælos no&longs;tros infra Cælos Fixarum in orbem revolvi volunt, & Planetas &longs;ecum deferre; &longs;ingulæ Cælorum partes, & Planetæ qui relative quidem in Cælis &longs;uis proximis quie&longs;cunt, moven- tur vere. Mutant enim po&longs;itiones &longs;uas ad invicem (&longs;ecus quam fit in vere quie&longs;centibus) unaque cum cælis delati participant eorum motus, & ut partes revolventium totorum, ab eorum axibus recedere conantur.

~~NIES.~~

Igitur quantitates relativæ non &longs;unt eæ ip&longs;æ quantitates, quarum nomina præ &longs;e ferunt, &longs;ed earum men&longs;uræ illæ &longs;en&longs;ibiles (veræ an errantes) quibus vulgus loco quantitatum men&longs;uratarum utitur. At &longs;i ex u&longs;u definiendæ &longs;unt verborum &longs;ignificationes; per nomina illa Temporis, Spatii, Loci & Motus proprie intelligendæ erunt hæ men&longs;uræ; & &longs;ermo erit in&longs;olens & pure Mathematicus, &longs;i quantitates men&longs;uratæ hic intelligantur. ~~Proinde vim inferunt Sacris Literis, qui voces ha&longs;ce de quantitatibus men&longs;uratis ibi interpretantur.~~ ~~Neque minus contaminant Mathe&longs;in & Philo&longs;ophiam, qui quantitates veras cum ip&longs;arum relationibus & vulgaribus menfuris confundunt.~~

~~Motus quidem veros corporum &longs;ingulorum cogno&longs;cere, & ab apparentibus actu di&longs;criminare, difficillimum.~~ ~~e&longs;t propterea quod partes &longs;patii illius immobilis, in quo corpora vere moventur, non incurrunt in &longs;en&longs;us.~~ ~~Cau&longs;a tamen non e&longs;t pror&longs;us de&longs;perata.~~ ~~Nam &longs;uppetunt argumenta, partim ex motibus apparentibus qui &longs;unt motuum verorum differentiæ, partim ex viribus quæ &longs;unt motuum verorum cau&longs;æ & effectus.~~ Ut &longs;i globi duo, ad datam ab invicem di&longs;tantiam filo intercedente connexi, revolverentur circa commune gravitatis centrum; innote&longs;ceret ex ten&longs;ione fili conatus globorum recedendi ab axe motus, & inde quantitas motus circularis computari po&longs;&longs;et. Deinde &longs;i vires quælibet æquales in alternas globorum facies ad motum circularem augendum vel minuendum &longs;imul imprimerentur, innote&longs;ceret ex aucta vel diminuta fili ten&longs;ione augmentum vel decrementum motus; & inde tandem inveniri po&longs;&longs;ent facies globorum in quas vires imprimi deberent, ut motus maxime augeretur; id e&longs;t, facies po&longs;ticæ, &longs;ive quæ in motu circulari &longs;equuntur. ~~Cognitis autem faciebus quæ &longs;equuntur, & faciebus oppo&longs;itis quæ præcedunt, cogno&longs;ceretur determinatio motus.~~ In hunc modum inveniri po&longs;&longs;et & quantitas & determinatio motus hujus circularis in vacuo quovis immen&longs;o, ubi nihil extaret externum & &longs;en&longs;ibile quocum globi conferri po&longs;&longs;ent. Si jam con&longs;tituerentur in &longs;patio illo corpora aliqua longinqua datam inter &longs;e po&longs;itionem &longs;ervantia, qualia &longs;unt Stellæ Fixæ in regionibus no&longs;tris: &longs;ciri quidem non po&longs;&longs;et ex relativa globorum tran&longs;latione inter corpora, utrum his an illis tribuendus e&longs;&longs;et motus. At &longs;i attenderetur ad filum, & deprenderetur ten&longs;ionem ejus illam ip&longs;am e&longs;&longs;e quam motus globorum requireret; concludere liceret motum e&longs;&longs;e globorum, & corpora quie&longs;cere; & tum demum ex tran&longs;latione globorum inter corpora, determinationem hujus motus colligere. Motus autem veros ex eorum cau&longs;is, effectibus, & apparentibus differentiis colligere; & contra ex motibus &longs;eu veris &longs;eu apparentibus eorum cau&longs;as & effectus, docebitur fu&longs;ius in &longs;equentibus. ~~Hunc enim in finem Tractatum &longs;equentem compo&longs;ui.~~

~~TA,~~

~~AXIOMATA, SIVE LEGES MOTUS.~~

~~LEX I.~~

~~Corpus omne per&longs;everare in &longs;tatu &longs;uo quie&longs;cendi vel movendi uniformiter in directum, ni&longs;i quatenus a viribus impre&longs;&longs;is cogitur &longs;tatum illum mutare.~~

~~PRojectilia per&longs;everant in motibus &longs;uis, ni&longs;i quatenus a re&longs;i&longs;tentia aeris retardantur, & vi gravitatis impelluntur deor&longs;um.~~ ~~Trochus, cujus partes cohærendo perpetuo retrahunt &longs;e&longs;e a motibus rectilineis, non ce&longs;&longs;at rotari, ni&longs;i quatenus ab aere retardatur.~~ ~~Majora autem Planetarum & Cometarum corpora motus &longs;uos & progre&longs;&longs;ivos & circulares in &longs;patiis minus re&longs;i&longs;tentibus factos con&longs;ervant diutius.~~

~~LEX II.~~

~~Mutationem motus proportionalem e&longs;&longs;e vi motrici impre&longs;&longs;æ, & fieri &longs;ecundum lineam rectam qua vis illa imprimitur.~~

~~Si vis aliqua motum quemvis generet; dupla duplum, tripla triplum generabit, &longs;ive &longs;imul & &longs;emel, &longs;ive gradatim & &longs;ucce&longs;&longs;ive impre&longs;&longs;a fuerit.~~ Et hic motus (quoniam in eandem &longs;emper plagam cum vi generatice determinatur) &longs;i corpus antea movebatur, motui ejus vel con&longs;piranti additur, vel contrario &longs;ubducitur, vel obliquo oblique adjicitur, & cum eo &longs;ecundum utriu&longs;que determinationem componitur.

~~LEX III.~~

~~Actioni contrariam &longs;emper & æqualem e&longs;&longs;e reactionem: &longs;ive corporum duorum actiones in &longs;e mutuo &longs;emper e&longs;&longs;e æquales & in partes contrarias dirigi.~~

~~Quicquid premit vel trahit alterum, tantundem ab eo premitur vel trahitur.~~ ~~Si quis lapidem digito premit, premitur & hujus digitus a lapide.~~ Si equus lapidem funi alligatum trahit, retrahetur etiam & equus (ut ita dicam) æqualiter in lapidem: nam funis utrinque di&longs;tentus eodem relaxandi &longs;e conatu urgebit equum ver&longs;us lapidem, ac lapidem ver&longs;us equum; tantumque impediet progre&longs;&longs;um unius quantum promovet progre&longs;&longs;um alterius. Si corpus aliquod in corpus aliud impingens, motum ejus vi &longs;ua quomodocunque mutaverit, idem quoque vici&longs;&longs;im in motu proprio eandem mutationem in partem contrariam vi alterius ob æqualitatem pre&longs;&longs;ionis mutuæ) &longs;ubibit. ~~His actionibus æquales fiunt mutationes, non velocitatum, &longs;ed motuum; &longs;cilicet in corporibus non aliunde impeditis.~~ ~~Mutationes enim velocitatum, in contrarias itidem partes factæ, quia motus æqualiter mutantur, &longs;unt corporibus reciproce proportionales.~~ ~~Obtinet etiam hæc Lex in Attractionibus, ut in Scholio proximo probabitur.~~

~~COROLLARIUM I.~~

~~Corpus viribus conjunctis diagonalem parallelogrammi eodem tempore de&longs;cribere, quo latera &longs;eparatis.~~

~~Si corpus dato tempore, vi &longs;ola~~

M in loco A impre&longs;&longs;a, ferretur uniformi cum motu ab A ad B; & vi &longs;ola N in eodem loco impre&longs;&longs;a, ferretur ab A ad C: compleatur parallelogrammum ABDC, & vi utraque feretur id eodem tempore in diagonali ab A ad D. Nam quoniam vis N agit &longs;ecundum lineam AC ip&longs;i BD parallelam, hæc vis per Legem 11 nihil mutabit velocitatem accedendi ad lineam illam BDa vi altera genitam. Accedet igitur corpus eodem tempore ad lineam BD, &longs;ive vis N imprimatur, &longs;ive non; atque adeo in fine illius temporis reperietur alicubi in linea illa BD. Eodem argumento in fine temporis eju&longs;dem reperietur alicubi in linea CD, & idcirco in utriu&longs;que lineæ concur&longs;u D reperiri nece&longs;&longs;e e&longs;t. ~~Perget autem motu rectilineo ab A ad D per Legem 1.~~

~~TA, E~~

~~COROLLARIUM II.~~

Et hinc patet compo&longs;itio vis directæ AD ex viribus quibu&longs;vis obliquis AB & BD, & vici&longs;&longs;im re&longs;olutio vis cuju&longs;vis directæAD in obliquas qua&longs;cunque AB & BD. Quæ quidem compo&longs;itio & re&longs;olutio abunde confirmatur ex Mechanica.

Ut &longs;i de rotæ alicujus centro O exeuntes radii inæquales OM, ON filis MA, NP &longs;u&longs;tineant pondera A & P, & quærantur vires ponderum ad movendam rotam: Per centrum O agatur recta KOL filis perpendiculariter occurrens in K & L, centroque O & intervallorum OK, OL majore OL

de&longs;cribatur circulus occurrens filo MA in D: & actæ rectæ OD parallela &longs;it AC, & perpendicularis DC. Quoniam nihil refert, utrum filorum puncta K, L, D affixa &longs;int an non affixa ad planum rotæ; pondera idem valebunt, ac &longs;i &longs;u&longs;penderentur a punctis K & L vel D & L.Ponderis autem A exponatur vis tota per lineam AD, & hæc re&longs;olvetur in vires AC, CD, quarum AC trahendo radium OD directe a centro nihil valet ad movendam rotam; vis autem altera DC, trahendo radium DO perpendiculariter, idem valet ac &longs;i perpendiculariter traheret radium OL ip&longs;i OD æqualem; hoc e&longs;t, idem atque pondus P, &longs;i modo pondus illud &longs;it ad pondus A ut vis DC ad vim DA, id e&longs;t (ob &longs;imilia triangula ADC, DOK,) ut OKad OD &longs;eu OL. Pondera igitur A & P, quæ &longs;unt reciproce ut radii in directum po&longs;iti OK & OL, idem pollebunt, & &longs;ic con&longs;i&longs;tent in æquilibrio: quæ e&longs;t proprietas noti&longs;&longs;ima Libræ, Vectis, & Axis in Peritrochio. ~~Sin pondus alterutrum &longs;it majus quam in hac ratione, erit vis ejus ad movendam rotam tanto major.~~

Quod &longs;i pondus p ponderi P æquale partim &longs;u&longs;pendatur filo Np,partim incumbat plano obliquo pG: agantur pH, NH, prior horizonti, po&longs;terior plano pG perpendicularis; & &longs;i vis ponderis pdeor&longs;um tendens, exponatur per lineam pH, re&longs;olvi pote&longs;t hæc in vires pN, HN. Si filo pN perpendiculare e&longs;&longs;et planum aliquod pQ, &longs;ecans planum alterum pG in linea ad horizontem parallela; & pondas p his planis pQ, pG &longs;olummodo incumberet; ur- geret illud hæc plana viribus pN, HN perpendiculariter, nimirun planum pQ vi pN, & planum pG vi HN. Ideoque &longs;i tollatur planum pQ, ut pondus tendat filum; quoniam filum &longs;u&longs;tinendo pon dus jam vicem præ&longs;tat plani &longs;ublati, tendetur illud eadem vi pN,qua planum antea urgebatur. ~~Unde ten&longs;io fili hujus obliqui erit ad ten&longs;ionem &longs;ili alterius perpendicularis PN, ut pN ad pH. Id.~~ eoque &longs;i pondus p &longs;it ad pondus A in ratione quæ componitur ratione reciproca minimarum di&longs;tantiarum &longs;uorum &longs;uorum pN, AM a centro rotæ, & ratione directa pH ad pN; pondera idem valebunt ad rotam movendam, atque adeo &longs;e mutuo &longs;u&longs;tinebunt, ut quilibet experiri pote&longs;t.

Pondus autem p, planis illis duobus obliquis incumbens, rationem habet cunei inter corporis fi&longs;&longs;i facies internas: & inde vires cunei & mallei innote&longs;cunt: utpote cum vis qua pondus p urget planum pQ &longs;it ad vim, qua idem vel gravitate &longs;ua vel ictu mallei impellitur &longs;ecundum lineam pH in plano, &c. ~~ut pN and pH; atque ad vim, qua urget planum alterum pG, ut pN ad NH. Sed & vis Cochleæ per &longs;imilem virium divi&longs;ionem colligitur; quippe quæ cuneus e&longs;t a vecte impul&longs;us.~~ U&longs;us igitur Corollarii hujus lati&longs;&longs;ime patet, & late patendo veritatem &longs;uam evincit; cum pendeat ex jam dictis Mechanica tota ab Auctoribus diver&longs;imode demon&longs;trata. Ex hi&longs;ce enim facile derivantur vires Machinarum, quæ ex Rotis, Tympanis, Trochleis, Vectibus, nervis ten&longs;is & ponderibus directe vel oblique a&longs;cendentibus, cæteri&longs;que potentiis Mechanicis componi &longs;olent, ut & vires Tendinum ad animalium o&longs;&longs;a movenda.

~~COROLLARIUM III.~~

~~Quantitas motus quæ colligitur capiendo &longs;ummam motuum factorum ad eandem partem, & differentiam factorum ad contrarias, non mutatur ab actione corporum inter &longs;e.~~

~~Etenim actio eique contraria reactio æquales &longs;unt per Legem 111, adeoque per Legem 11 æquales in motibus efficiunt mutationes ver&longs;us contrarias partes.~~ ~~Ergo &longs;i motus fiunt ad eandem partem; quicquid additur motui corporis fugientis, &longs;ubducetur motui corporis in&longs;equentis &longs;ic, ut &longs;umma maneat eadem quæ prius.~~ ~~Sin corpora obviam eant; æqualis erit &longs;ubductio de motu utriu&longs;que, adeoque differentia motuum factorum in contrarias partes manebit eadem.~~

Ut &longs;i corpus &longs;phæricum A &longs;it triplo majus corpore &longs;phærico B, habeatque duas velocitatis partes; & B &longs;equatur in eadem recta cum ve- locitatis partibus decem, adeoque motus ip&longs;ius A &longs;it ad motum ip&longs;ius B, ut &longs;ex ad decem: ponantur motus illis e&longs;&longs;e partium &longs;ex & partium decem, & &longs;umma erit partium &longs;exdecim. In corporum igitur concur&longs;u, &longs;i corpus A lucretur motus partes tres vel quatuor vel quinque, corpus B amittet partes totidem, adeoque perget corpus A po&longs;t reflexionem cum partibus novem vel decem vel undecim, & B cum partibus &longs;eptem vel &longs;ex vel quinque, exi&longs;tente &longs;emper &longs;umma partium &longs;exdecim ut prius. Si corpus A lucretur partes novem vel decem vel undecim vel duodecim, adeoque progrediatur po&longs;t concur&longs;um cum partibus quindecim vel &longs;exdecim vel &longs;eptendecim vel octodecim; corpus B, amittendo tot partes quot A lucratur, vel cum una parte progredietur ami&longs;&longs;is partibus novem, vel quie&longs;cet ami&longs;&longs;o motu &longs;uo progre&longs;&longs;ivo partium decem, vel cum una parte regredietur ami&longs;&longs;o motu &longs;uo & (ut ita dicam) una parte amplius, vel regredietur cum partibus duabus ob detractum motum progre&longs;&longs;ivum partium duodecim. ~~Atque ita &longs;ummæ motuum con&longs;pirantium 15+1 vel 16+c, & differentiæ contrariorum 17-1 & 18-2 &longs;emper erunt partium &longs;exdecim, ut ante concur&longs;um & reflexionem.~~ Cognitis autem motibus quibu&longs;cum corpora po&longs;t reflexionem pergent, invenietur cuju&longs;que velocitas, ponendo eam e&longs;&longs;e ad velocitatem ante reflexionem, ut motus po&longs;t e&longs;t ad motum ante. Ut in ca&longs;u ultimo, ubi corporis A motus erat partium &longs;ex ante reflexionem & partium octodecim po&longs;tea, & velocitas partium duarum ante reflexionem; invenietur ejus velocitas partium &longs;ex po&longs;t reflexionem, dicendo, ut motus partes &longs;ex ante reflexionem ad motus partes octodecim po&longs;tea, ita velocitatis partes duæ ante reflexionem ad velocitatis partes &longs;ex po&longs;tea.

~~TA,~~

Quod &longs;i corpora vel non Sphærica vel diver&longs;is in rectis moventia incidant in &longs;e mutuo oblique, & requirantur eorum motus po&longs;t reflexionem; cogno&longs;cendus e&longs;t &longs;itus plani a quo corpora concurrentia tanguntur in puncto concur&longs;us: dein corporis utriu&longs;que motus (per Corol.11.) di&longs;tinguendus e&longs;t in duos, unum huic plano perpendicularem, alterum eidem parallelum: motus autem paralleli, propterea quod corpora agant in &longs;e invicem &longs;ecundum lineam huic plano perpendicularem, retinendi &longs;unt iidem po&longs;t reflexionem atque antea; & motibus perpendicularibus mutationes æquales in partes contrarias tribuendæ &longs;unt &longs;ic, ut &longs;umma con&longs;pirantium & differentia contrariorum maneat eadem quæ prius. ~~Ex huju&longs;modi reflexionibus oriri etiam &longs;olent motus circulares corporum circa centra propria.~~ Sed hos ca&longs;us in &longs;equentibus non con&longs;idero, & nimis longum e&longs;&longs;et omnia huc &longs;pectantia demon&longs;trare. mutat &longs;tatum &longs;uum; & reliquorum, quibu&longs;cum actio illa non intercedit, commune gravitatis centrum nihil inde patitur; di&longs;tantia autem horum duorum centrorum dividitur a communi corporum omnium centro in partes &longs;ummis totalibus corporum quorum &longs;unt centra reciproce proportionales; adeoque centris illis duobus &longs;tatum &longs;uum movendi vel quie&longs;cendi &longs;ervantibus, commune omnium centrum &longs;ervat etiam &longs;tatum &longs;uum: manife&longs;tum e&longs;t quod commune illud omnium centrum ob actiones binorum corporum inter &longs;e nunquam mutat &longs;tatum &longs;uum quoad motum & quietem. In tali autem &longs;y&longs;temate actiones omnes corporum inter &longs;e, vel inter bina &longs;unt corpora, vel ab actionibus inter bina compo&longs;itæ; & propterea communi omnium centro mutationem in &longs;tatu motus ejus vel quietis nunquam inducunt. Quare cum centrum illud ubi corpora non agunt in &longs;e invicem, vel quie&longs;cit, vel in recta aliqua progreditur uniformiter; perget idem, non ob&longs;tantibus corporum actionibus inter &longs;e, vel &longs;emper quie&longs;cere, vel &longs;emper progredi uniformiter in directum; ni&longs;i a viribus in &longs;y&longs;tema extrin&longs;ecus impre&longs;&longs;is deturbetur de hoc &longs;tatu. ~~E&longs;t igitur &longs;y&longs;tematis corporum plurium Lex eadem quæ corporis &longs;olitarii, quoad per&longs;everantiam in &longs;tatu motus vel quietis.~~ ~~Motus enim progre&longs;&longs;ivus &longs;eu corporis &longs;olitarii &longs;eu &longs;y&longs;tematis corporum ex motu centri gravitatis æ&longs;timari &longs;emper debet.~~

~~IATA, VF.~~

~~COROLLARIUM V.~~

Corporum dato &longs;patio inclu&longs;orum iidem &longs;unt motus inter &longs;e, &longs;ive &longs;patium illud quie&longs;cat, &longs;ive moveatur idem uniformiter in directum ab&longs;que motu circulari.

Nam differentiæ motuum tendentium ad eandem partem, & &longs;ummæ tendentium ad contrarias, eædem &longs;unt &longs;ub initio in utroque ca&longs;u (ex hypothe&longs;i) & ex his &longs;ummis vel differentiis oriuntur congre&longs;&longs;us & impetus quibus corpora &longs;e mutuo feriunt. ~~Ergo per Legem 11 æquales erunt congre&longs;&longs;uum effectus in utroque ca&longs;u; & propterea manebunt motus inter &longs;e in uno ca&longs;u æquales motibus inter &longs;e in altero.~~ ~~Idem comprobatur experimento luculento.~~ ~~Motus omnes eodem modo &longs;e habent in Navi, &longs;ive ea quie&longs;cat, &longs;ive moveatur uniformiter in directum.~~

~~COROLLARIUM VI.~~

Si corpora moveantur quomodocunque inter&longs;e, & a viribus acceler atricibus æqualibus &longs;ecundum lineas parallelas urgeantur; pergent omnia eodem modo moveri inter &longs;e, ac &longs;i viribus illis non e&longs;&longs;ent incitata.

Nam vires illæ æqualiter (pro quantitatibus movendorum corpo- rum) & &longs;ecundum lineas parallelas agendo, corpora omnia æqualiter (quoad velocitatem) movebunt per Legem 11. adeoque nunquam mutabunt po&longs;itiones & motus eorum inter &longs;e.

~~Scholium.~~

~~Hactenus principia tradidi a Mathematicis recepta & experientia multiplici confirmata.~~ Per Leges duas primas & Corollaria duo prima Galilæus invenit de&longs;cen&longs;um Gravium e&longs;&longs;e in duplicata ratione temporis, & motum Projectilium fieri in Parabola; con&longs;pirante experientia, ni&longs;i quatenus motus illi per aeris re&longs;i&longs;tentiam aliquantulum retardantur. ~~Ab ii&longs;dem Legibus & Corollariis pendent demon&longs;trata de temporibus o&longs;cillantium Pendulorum, &longs;uffragante Horologiorum experientia quotidiana.~~ Ex his ii&longs;dem & Lege tertia Chri&longs;tophorus Wrennus Eques Auratus, Jobannes Walli&longs;ius S.T.D.& Chri&longs;tianus Hugenius, hujus ætatis Geometrarum facile principes, regulas congre&longs;&longs;uum & reflexionum duorum corporum &longs;eor&longs;im invenerunt, & eodem fere tempore cum Societate Regiacommunicarunt, inter &longs;e (quoad has leges) omnino con&longs;pirantes: & primus quidem Walli&longs;ius, deinde Wrennus & Hugenius inventum prodiderunt. ~~Sed & veritas comprobata e&longs;t a Wrenno coram Regia Societate per experimentum Pendulorum: quod etiam Clari&longs;&longs;imus Mariottus libro integro exponere mox dignatus e&longs;t.~~ ~~Verum, ut hoc experimentum cum Theoriis ad amu&longs;&longs;im congruat, habenda e&longs;t ratio cum re&longs;i&longs;tentiæ aeris, tum etiam vis Ela&longs;ticæ concurrentium corporum.~~ ~~Pendeant corpora A, B filis parallelis & æqualibus AC, BD, a centris C, D. His centris & intervallis de&longs;cribantur &longs;emicirculi EAF, GBH radiis CA, DB bi&longs;ecti.~~ ~~Trahatur corpus A ad arcus EAF punctum quodvis R, & (&longs;ubducto corpore B) demittatur inde, redeatque po&longs;t unam o&longs;cillationem ad punctum V. E&longs;t RV re~~

~~tardatio ex re&longs;i&longs;tentia aeris.~~ Hujus RV fiat ST pars quarta &longs;ita in medio, ita &longs;cilicet ut RS & TV æquentur, &longs;itque RS ad ST ut 3 ad 2. Et i&longs;ta ST exhibebit retardationem in de&longs;cen&longs;u ab S ad Aquam proxime. ~~Re&longs;tituatur corpus B in locum &longs;uum.~~ Cadat corpus A de puncto S, & velocitas ejus in loco reflexionis A, ab&longs;que errore &longs;en&longs;ibili, tanta erit ae &longs;i in vacuo cecidi&longs;&longs;et de loco T. Exponatur igitur hæc velocitas per chordam arcus TA. Nam velocitatem Penduli in puncto infimo e&longs;&longs;e ut chordam arcus quem cadendo de&longs;crip&longs;it, Propo&longs;itio e&longs;t e&longs;t Geometris noti&longs;&longs;ima. Po&longs;t reflexionem perveniat corpus A ad locum s, & corpus B ad locum k. Tollatur corpus B & inveniatur locus v; a quo &longs;i corpus A demittatur & po&longs;t unam o&longs;cillationem redeat ad locum r, &longs;it st pars quarta ip&longs;ius rv &longs;ita in medio, ita videlicet ut rs & tu æquentur; & per chordam arcus tA exponatur velocitas quam corpus A proxime po&longs;t reflexionem habuit in loco A. Nam t erit locus ille vcrus & correctus, ad quem corpus A, &longs;ublata aeris re&longs;i&longs;tentia, a&longs;cendere debui&longs;&longs;et: Simili methodo corrigendus erit locus k, ad quem corpus B a&longs;cendit, & inveniendus locus l, ad quem corpus illud a&longs;cendere debui&longs;&longs;et in vacuo. ~~Hoc pacto experiri licet omnia perinde ac &longs;i in vacuo con&longs;tituti e&longs;&longs;emus.~~ Tandem ducendum erit corpus A in chordam arcus TA (quæ velocitatem ejus exhibet) ut habeatur motus ejus in loco A proxime ante reflexionem; deinde in chordam arcus tA, ut habeatur motus ejus in loco A proxime po&longs;t reflexionem. ~~Et &longs;ic corpus B ducendum erit in chordam arcus Bb, ut habeatur motus ejus proxime po&longs;t reflexionem.~~ Et &longs;imili methodo, ubi corpora duo fimul demittuntur de locis diver&longs;is, inveniendi &longs;unt motus utriu&longs;que tam ante, quam po&longs;t reflexionem; & tum demum conferendi &longs;unt motus inter &longs;e & colligendi effectus reflexionis. Hoc modo in Pendulis pedum decem rem tentando, idque in corporibus tam inæqualibus quam æqualibus, & faciendo ut corpora de intervallis ampli&longs;&longs;imis, puta pedum octo vel duodecim vel &longs;exdecim, concurrerent; reperi &longs;emper &longs;ine errore trium digitorum in men&longs;uris, ubi corpora &longs;ibi mutuo directe occurrebant, quod æquales erant mutationes motuum corporibus in partes contrarias illatæ, atque adeo quod actio & reactio &longs;emper
erant æquales. Ut &longs;i corpus A incidebat in corpus B cum novem partibus motus, & ami&longs;&longs;is &longs;eptem partibus pergebat po&longs;t reflexionem cum duabus; corpus B re&longs;iliebat cum partibus i&longs;tis &longs;eptem. ~~Si corpora obviam ibant A cum duodecim partibus & B cum &longs;ex, & redibat A cum duabus; redibat B cum octo, facta detractione partium quatuordecim utrinque.~~ De motu ip&longs;ius A &longs;ubducantur partes duodecim, & re&longs;tabit nihil: &longs;ubducantur aliæ partes duæ, & fiet motus duarum partium in plagam contrariam: & &longs;ic de motu corporis B partium &longs;ex &longs;ubducendo partes quatuordecim, fient partes octo in plagam contrariam. Quod &longs;i corpora ibant ad eandam plagam, A velocius cum partibus quatuordecim, & B tardius cum partibus quinque, & po&longs;t reflexionem pergebat A cum quinque partibus; pergebat B cum quatuordecim, facta tran&longs;latione partium novem de A in B. Et &longs;ic in reliquis. ~~A congre&longs;&longs;u & colli&longs;ione corporum nunquam mutabatur quantitas motus, quæ ex &longs;umma motuum con&longs;pirantium & differentia contrariorum colligebatur.~~ ~~Nam errorem digiti unius & alterius in men&longs;uris tribuerim difficultati peragendi &longs;ingula &longs;atis accurate.~~ ~~Difficile erat, tum pendula &longs;imul demittere fic, ut corpora in &longs;e mutuo impingerent in loco infimo AB; tum loca s, k notare, ad quæ corpora a&longs;cendebant po&longs;t concur&longs;um.~~ ~~Sed & in ip&longs;is pilis inæqualis partium den&longs;itas, & textura aliis de cau&longs;is irregularis, errores inducebant.~~

~~ATA, VE~~

Porro nequis objiciat Regulam, ad quam probandam inventum e&longs;t hoc experimentum, præ&longs;upponere corpora vel ab&longs;olute dura e&longs;&longs;e, vel &longs;altem perfecte ela&longs;tica, cuju&longs;modi nulla reperiuntur in compo&longs;itionibus naturalibus; addo quod Experimenta jam de&longs;cripta &longs;uccedunt in corporibus mollibus æque ac in duris, nimirum a conditione duritiei neutiquam pendentia. ~~Nam &longs;i Regula illa in corporibus non perfecte duris tentanda e&longs;t, debebit &longs;olummodo reflexio minui in certa proportione pro quantitate vis Ela&longs;ticæ.~~ ~~In Theoria Wrenni & Hugenii corpora ab&longs;olute dura redeunt ab invicem cum velocitate congre&longs;&longs;us.~~ ~~Certius id affirmabitur de perfecte Ela&longs;ticis.~~ In imperfecte Ela&longs;ticis velocitas reditus minuenda e&longs;t &longs;imul cum vi Ela&longs;tica; propterea quod vis illa; (ni&longs;i ubi partes corporum ex congre&longs;&longs;u læduntur, vel exten&longs;ionem aliqualem qua&longs;i &longs;ub malleo patiuntur,) certa ac determinata &longs;it (quantum &longs;entio) faciatque corpora redire ab invicem cum velocitate relativa, quæ &longs;it ad relativam velocitatem concur&longs;us in data ratione. ~~Id in pilis ex lana arcte conglomerata & fortiter con&longs;tricta &longs;ic tentavi.~~ Primum demittendo Pendula & men&longs;urando reflexionem, inveni quantitatem vis Ela&longs;ticæ; deinde per hanc vim determinavi reflexiones in aliis ca&longs;ibus concur&longs;uum, & re&longs;pondebant Experimenta. ~~Redibant &longs;emper pilæ ab invicem cum velocitate relativa, quæ e&longs;&longs;et ad velocitatem relativam concur&longs;us ut 5 ad 9 circiter.~~ ~~Eadem fere cum velocitate redibant pilæ ex chalybe: aliæ ex &longs;ubere cum paulo minore: in vitreis autem proportio erat 15 ad 16 circiter.~~ ~~Atque hoc pacto Lex tertia quoad ictus & reflexiones per Theoriam comprobata e&longs;t, quæ cum experientia plane congruit.~~

~~ATA E~~

~~In Attractionibus rem &longs;ic breviter o&longs;tendo.~~ ~~Corporibus duobus quibu&longs;vis A, B &longs;e mutuo trahentibus, concipe ob&longs;taculum quodvis interponi quo congre&longs;&longs;us eorum impediatur.~~ Si corpus alterutrum A magis trahitur ver&longs;us corpus alterum B, quam illud alterum Bin prius A, ob&longs;taculum magis urgebitur pre&longs;&longs;ione corporis A quam pre&longs;&longs;ione corporis B; proindeque non manebit in æquilibrio. Prævalebit pre&longs;&longs;io fortior, facietque ut &longs;y&longs;tema corporum duorum & ob&longs;taculi moveatur in directum in partes ver&longs;us B, motuque in &longs;patiis liberis &longs;emper accelerato abeat in infinitum. ~~Quod e&longs;t ab&longs;urdum & Legi primæ contrarium.~~ Nam per Legem primam debebit &longs;y&longs;tema per&longs;everare in &longs;tatu &longs;uo quie&longs;cendi vel movendi uniformiter in directum, proindeque corpora æqualiter urgebunt ob&longs;taculum, & idcirco æqualiter trahentur in invicem. ~~Tentavi hoc in Magnete & Ferro.~~ Si hæc in va&longs;culis propriis &longs;e&longs;e contingentibus &longs;eor&longs;im po&longs;ita, in aqua &longs;tagnante juxta fluitent; neutrum propellet alterum, &longs;ed æqualitate attractionis utrinque &longs;u&longs;tinebunt conatus in &longs;e mutuos, ac tandem in æquilibrio con&longs;tituta quie&longs;cent.

~~Sic etiam gravitas inter Terram & ejus partes, mutua e&longs;t.~~ ~~Secetur Terra FI plano quovis EG in partes duas EGF & EGI:& æqualia erunt harum pondera in &longs;e mu~~

~~tuo.~~ Nam &longs;i plano alio HK quod priori EG parallelum &longs;it, pars major EGI &longs;ecetur in partes duas EGKH & HKI,quarum HKI æqualis &longs;it parti prius ab&longs;ci&longs;&longs;æ EFG: manife&longs;tum e&longs;t quod pars media EGKH pondere proprio in neutram partium extremarum propendebit, &longs;ed inter utramque in æquilibrio, ut ita dicam, &longs;u&longs;pendetur, & quie&longs;cet. Pars autem extrema HKI toto &longs;uo pondere incumbet in partem mediam, & urgebit illam in partom alteram extremam EGF; ideoque vis qua partium HKI & EGKH &longs;umma EGI tendit ver&longs;us partem tertiam EGF, æqualis e&longs;t ponderi partis HKI, id e&longs;t ponderi partis tertiæ EGF. Et propterea pondera partium duarum EGI, EGFin &longs;e mutuo &longs;unt æqualia, uti volui o&longs;tendere. ~~Et ni&longs;i pondera illa æqualia e&longs;&longs;ent, Terra tota in libero æthere fluitans ponderi majori cederet, & ab eo fugiendo abiret in infinitum.~~

Ut corpora in concur&longs;u & reflexione idem pollent, quorum velocitates &longs;unt reciproce ut vires in&longs;itæ: &longs;ic in movendis In&longs;trumentis Mechanicis agentia idem pollent & conatibus contrariis &longs;e mutuo &longs;u&longs;tinent, quorum velocitates &longs;ecundum determinationem virium æ&longs;timatæ, &longs;unt reciproce ut vires. Sie pondera æquipollent ad movenda brachia Libræ, quæ o&longs;cillante Libra &longs;unt reciproce ut eorum velocitates &longs;ur&longs;um & deor&longs;um: hoc e&longs;t, pondera, &longs;i recta a&longs;cendunt & de&longs;cendunt, æquipollent, quæ &longs;unt reciproce ut punctorum a quibus &longs;u&longs;penduntur di&longs;tantiæ ab axe Libræ; &longs;in planis obliquis alii&longs;ve admotis ob&longs;taculis impedia a&longs;cendunt vel de&longs;cendunt oblique, æquipollent quæ &longs;unt reciproce ut a&longs;cen&longs;us & de&longs;cen&longs;us, quatenus facti &longs;ecundum perpendiculum: id adeo ob determinationem gravitatis deor&longs;um. Similiter in Trochlea &longs;eu Poly&longs;pa&longs;to vis manus funem directe trahentis, quæ &longs;it ad pondus vel directe vel oblique a&longs;cendens ut velocitas a&longs;cen&longs;us perpendicularis ad velocitatem manus funem trahentis, &longs;u&longs;tinebit pondus. In Horologiis & &longs;imilibus in&longs;trumentis, quæ ex rotulis commi&longs;&longs;is con&longs;tructa &longs;unt, vires contrariæ ad motum rotularum promovendum & impediendum, &longs;i &longs;unt reciproce ut velocitates partium rotularum in quas imprimuntur, &longs;u&longs;tinebunt &longs;e mutuo. Vis Cochleæ ad premendum corpus e&longs;t ad vim manus manubrium circumagentis, ut circularis velocitas manubrii ea in parte ubi a manu urgetur, ad velocitatem progre&longs;&longs;ivam cochleæ ver&longs;us corpus pre&longs;&longs;um. Vires quibus Cuneus urget partes duas ligni fi&longs;&longs;i &longs;unt ad vim mallei in cuueum, ut progre&longs;&longs;us cunei &longs;ecundum determinationem vis a malleo in ip&longs;um impre&longs;&longs;æ, ad velocitatem qua partes gni cedunt cuneo, &longs;ecundum lineas faciebus cunei perpendiculares. ~~Et par e&longs;t ratio Machinarum omnium.~~

Harum efficacia & u&longs;us in eo &longs;olo con&longs;i&longs;tit, ut diminuendo velocitatem augeamus vim, & contra: Unde &longs;olvitur in omni aptorum in&longs;trumentorum genere Problema, Datum pondus data vi movendi, aliamve datam re&longs;i&longs;tentiam vi data &longs;uperandi. Nam &longs;i Machinæ ita formentur, ut velocitates Agentis & Re&longs;i&longs;tentis &longs;ine reciproce ut vires; Agens re&longs;i&longs;tentiam &longs;u&longs;tinebit: & majori cum velocitatum di&longs;paritate eandem vincet. Certe &longs;i tanta &longs;ic velocitatum di&longs;paritas, ut vincatur etiam re&longs;i&longs;tentia omnis, quæ tam ex contiguorum & inter &longs;e labentium corporum attritione, quam ex continuorum & ab invicem &longs;eparandorum cohæ&longs;ione & elevandorum ponderibus orirj &longs;olet; &longs;uperata omni ea re&longs;i&longs;tentia, vis redundans accelerationem motus &longs;ibi proportionalem, partim in partibus machinæ, partim in corpore re&longs;i&longs;tente producet. ~~Ceterum Mechanicam tractare non e&longs;t hujus in&longs;tituti.~~ ~~Hi&longs;ce volui tantum o&longs;tendere, quam late pateat quamque certa &longs;it Lex tertia Motus.~~ ~~Nam &longs;i æ&longs;timetur Agentis actio ex ejus vi & veloci-~~

tate conjunctim; & &longs;imiliter Re&longs;i&longs;tentis reactio æ&longs;timetur conjunctim ex ejus partium &longs;ingularum velocitatibus & viribus re&longs;i&longs;tendi ab earum attritione, cohæ&longs;ione, pondere, & acceleratione oriundis; erunt actio & reactio, in omni in&longs;trumentorum u&longs;u, &longs;ibi invicem &longs;emper æquales. ~~Et quatenus actio propagatur per in&longs;trumentum & ultimo imprimitur in corpus omne re&longs;i&longs;tens, ejus ultima determinatio determinationi reactionis &longs;emper erit contraria.~~

~~DE MOTU CORPORUM~~

~~DE MOTU CORPORUM LIBER PRIMUS.~~

~~SECTIO I.~~

~~De Methodo Rationum primarum & ultimarum, cujus ope &longs;equentia demon&longs;trantur.~~

~~LEMMA I.~~

QUantitates, ut & quantitatum rationes, quæ ad æqualitatem tempore quovis finito con&longs;tanter tendunt, & ante finem temporis illius propius ad invicem accedunt quam pro data quavis diffetia, fiunt ultimo æquales.

~~Si negas; fiant ultimò inequales, & &longs;it earum ultima differentia D. Ergo nequeunt propius ad æqualitatem accedere quam pro data differentia D: contra hypothe&longs;in.~~

~~LEMMA II.~~

Si in Figura quavis AacE, rectis Aa, AE & curva acE com preben&longs;a, in&longs;cribantur parallelogramma quotcunque Ab, Bc, Cd &c. &longs;ub ba&longs;ibus AB, BC, CD, &c. æqualibus, & lateribuBb, Cc, Dd, &c. Figuræ lateri Aa parallelis contenta; & compleantur paral-

lelogramma aKbl, bLcm, cMdn, &c. Dein boru parallelogr ammorum latitudo minuatur, & numerus augeatur in infinitum: dico quod ultimæ rationes, quas babent ad &longs;e invicem Figura in&longs;cripta AKbLcMdD, circum&longs;criptaAalbmcndoE, & curvilinea AbcdE, &longs;unt rationes æqualitatis.

Nam Figuræ in&longs;criptæ & circum&longs;criptæ differentia e&longs;t &longs;umma parallelogrammorum Kl, Lm, Mn, Do, hoc e&longs;t (ob æquales omnium ba&longs;es) rectangulum &longs;ub unius ba&longs;i Kb & altitudinum &longs;umma Aa, id e&longs;t, rectangulum ABla. Sed hoc rectangulum, eo quod latitudo ejus AB in infinitum minuitur, fit minus quovis dato. ~~Ergo (per Lemma 1) Figura in&longs;cripta & circum&longs;cripta & multo magis Figura curvilinea intermedia fiunt ultimo æquales. q.E.D.~~

~~LEMMA III.~~

~~Eædem rationes ultimæ &longs;unt etiam rationes æqualitatis, ubi par allelogr ammorum latitudines AB, BC, CD, &c. &longs;unt inæquales, & omnes minuuntur in infinitum.~~

Sit enim AF æqualis latitudini maximæ, & compleatur parallelogrammum FAaf. Hoc erit majus quam differentia Figuræ in&longs;criptæ & Figuræ circum&longs;criptæ; at latitudine &longs;ua AF in infinitum diminuta, minus fiet quam datum quodvis rectangulum. que E. D.

~~Corol. 1. Hinc &longs;umma ultima parallelogrammorum evane&longs;centium coincidit omni ex parte cum Figura curvilinea.~~

~~Corol. 2. Et multo magis Figura rectilinea, quæ chordis evane&longs;- centium arcuum ab, bc, cd, &c. comprehenditur, coincidit ultimo cum Figura curvilinea.~~

~~DE MOTU CORPORUM~~

~~Corol. 3. Ut & Figura rectilinea circum&longs;cripta quæ tangentibus eorundem arcuum comprehenditur.~~

~~Corol. 4. Et propterea hæ Figuræ ultimæ (quoad perimetros acE,) non &longs;unt rectilineæ, &longs;ed rectilinearum limites curvilinei.~~

~~LEMMA IV.~~

Si in duabus Figuris AacE, PprT, in&longs;cribantur (ut &longs;upra) duæ parallelogrammorum &longs;eries, &longs;itque idem amborum numerus, & ubi latitudines in infinitum diminuuntur, rationes ultimæ parallelogrammorum in una Figura ad parallelogramma in altera, &longs;ingulorum ad fingula, &longs;int eædem; dico quod Figuræ duæ AacE, PprT, &longs;unt ad invicem in eadem illa ratione.

Etenim ut &longs;unt parallelogramma &longs;ingula ad &longs;ingula, ita (componendo) fit &longs;umma omnium ad &longs;ummam omnium, & ita Figura ad Figuram; exi&longs;tente nimirum Figura priore (per Lemma 111) ad &longs;ummam priorem, & Figura po&longs;teriore ad &longs;ummam po&longs;teriorem in ratione æqualitatis. que E. D.

Corol. Hinc &longs;i duæ cuju&longs;cunque generis quantitates in eundem partium numerum utcunque dividantur; & partes illæ, ubi numerus earum augetur & magnitudo diminuitur in infinitum, datam obtineant rationem ad invicem, prima ad primam, &longs;ecunda ad &longs;ecundam, cæteræque &longs;uo ordine ad cæteras: erunt tota ad invicem in eadem illa data ratione. Nam &longs;i in Lemmatis hujus Figuris &longs;umantur pa- rallelogramma inter &longs;e ut partes, &longs;ummæ partium &longs;emper erunt ut &longs;ummæ parallelogrammorum; atque adeo, ubi partium & parallelogrammorum numerus augetur & magnitudo diminuitur in infinitum, in ultima ratione parallelogrammi ad parallelogrammum, id e&longs;t (per hypothe&longs;in) in ultima ratione partis ad partem.

~~LEMMA V.~~

~~Similium Figurarum latera omnia, quæ &longs;ibi mutuo re&longs;pondent, &longs;unt proportionalia, tam curvilinea quam rectilinea; & areæ &longs;unt in duplicata ratione laterum.~~

~~LEMMA VI.~~

~~Si arcus quilibet po&longs;itione datus AB &longs;ub-~~

tendatur chorda AB, & in puncto aliquo A, in medio curvaturæ continuæ, tangatur a recta utrinque productaAD; dein puncta A, B ad invicem accedant & coëant; dico quod angulusBAD, &longs;ub chorda & tangente contentus, minuetur in infinitum & ultimo evane&longs;cet.

~~Nam &longs;i angulus ille non evane&longs;cit, continebit arcus AB cum tangente AD angulum rectilineo æqualem, & propterea curvatura ad ad punctum A non erit continua, contra hypothe&longs;in.~~

~~LEMMA VII.~~

~~Ii&longs;dem po&longs;itis; dico quod ultima ratio arcus, chordæ, & tangentis ad invicem est ratio æqualitatis.~~

Nam dum punctum B ad punctum A accedit, intelligantur &longs;emper AB & AD ad puncta longinqua b ac d product, & &longs;ecanti BDparallela agatur bd. Sitque arcus Ab &longs;emper &longs;imilis arcui AB.Et punctis A, B coeuntibus, angulus dAb, per Lemma &longs;uperius, evane&longs;cet; adeoque rectæ &longs;emper &longs;initæ Ab, Ad & arcus intermedius Ab coincident, & propterea æquales erunt. ~~Unde & hi&longs;ce &longs;emper proportionales rectæ AB, AD, & arcus intermedius AB evane&longs;cent, & rationem ultimam habebunt æqualitatis. q.E.D.~~

~~DE MOTU CORPORUM~~

~~Corol. 1. Unde &longs;i per B ducatur tangenti parallela BF, rectam quamvis AF per A tran&longs;e~~

~~untem perpetuo &longs;ecans in F,hæc BF ultimo ad arcum evane&longs;centem AB rationem habebit æqualitatis, eo quod completo parallelogrammo AFBD rationem &longs;emper habet æqualitatis ad AD.~~

Corol. 2. Et &longs;i per B & A ducantur plures rectæ BE, BD, AF, AG, &longs;ecantes tangentem AD & ip&longs;ius parallelam BF; ratio ultima ab&longs;ci&longs;&longs;arum omnium AD, AE, BF, BG, chordæque & arcus AB ad invicem erit ratio æqualitatis.

~~Corol. 3. Et propterea hæ omnes lineæ, in omni de rationibus ultimis argumentatione, pro &longs;e invicem u&longs;urpari po&longs;&longs;unt.~~

~~LEMMA VIII.~~

Si rectæ datæ AR, BR cum arcu AB, chorda AB & tangenteAD, triangula tria ARB, ARB, ARD con&longs;tituunt, dein puncta A, B accedunt ad invicem: dico quod ultima forma triangulorum evane&longs;centium est &longs;imilitudinis, & ultima ratio æqualitatis.

~~Nam dum punctum B ad punctum A~~

accedit, intelligantur &longs;emper AB, AD, ARad puncta longinqua b, d & r produci, ip&longs;ique RD parallela agi rbd, & arcui AB &longs;imilis &longs;emper &longs;it arcus Ab. Et coeuntibus punctis A, B, angulus bAd evane&longs;cet, & propterea triangula tria &longs;emper finita rAb, rAb, rAd coincident, &longs;untque eo nomine &longs;imilia & æqualia. ~~Unde & hi&longs;ce &longs;emper &longs;imilia & proportionalia RAB, RAB, RAD &longs;ient ultimo &longs;ibi invicem &longs;imilia & æqualia. que E. D.~~

~~Corol. Et hinc triangula illa, in omni de rationibus ultimis argumentatione, pro &longs;e invicem u&longs;urpari po&longs;&longs;unt.~~

~~LEMMA IX.~~

Si recta AE & curva ABC po&longs;itione datæ &longs;e mutuo &longs;ecent in angulo dato A, & ad rectam illam in alio dato angulo ordinatim applicentur BD, CE, curvæ occurrentes in B, C; dein puncta B, C &longs;imul accedant ad punctum A: dico quod areæ triangulorum ABD, ACE erunt ultimo ad invicem in duplicata ratione laterum.

~~Etenim dum puncta B, C acce~~

dunt ad punctum A, intelligatur &longs;emper AD produci ad puncta longinqua d & e, ut &longs;int Ad, Ae ip&longs;is AD, AE proportionales, & erigantur ordinatæ db, ec ordinatis DB, EC parallelæ quæ occurrant ip&longs;is AB, AC productis in b & c. Duci intelligatur, tum curva Abc ip&longs;i ABC &longs;imilis, tum recta Ag, quæ tangat curvam utramque in A, & &longs;ecet ordinatim applicatas DB, EC, db, ec in F, G, f, g.Tum manente longitudine Ae coeant puncta B, C cum puncto A; & angulo cAg evane&longs;cente, coincident areæ curvilineæ Abd, Acecum rectilineis Afd, Age: adeoque (per Lemma v) erunt in duplicata ratione laterum Ad, A: Sed his areis proportionales &longs;emper &longs;unt areæ ABD, ACE, & his lateribus latera AD, AE. Ergo & areæ ABD, ACE &longs;unt ultimo in duplicata ratione laterum AD, AE. que E. D.

~~LEMMA X.~~

Spatia quæ corpus urgente quacunque Vi finita de&longs;cribit, five Vis illa determinata & immutabilis &longs;it, five eadem continuo augeatur vel continuo diminuatur, &longs;unt ip&longs;o motus initio in duplicata ratione Temporum.

Exponantur tempora per lineas AD, AE, & velocitates genitæ per ordinatas DB, EC; & &longs;patia his velocitatibus de&longs;cripta, erunt ut areæ ABD, ACE his ordinatis de&longs;criptæ, hoc e&longs;t, ip&longs;o motus initio (per Lemma IX) in duplicata ratione remporum AD, AE. que E. D.

~~DE MOTU CORPORUM~~

Corol. 1. Et hinc facile colligitur, quod corporum &longs;imiles &longs;imilium Figurarum partes temporibus proportionalibus de&longs;cribentium Errores, qui viribus quibu&longs;vis æqualibus ad corpora &longs;imiliter applicatis generantur, & men&longs;urantur per di&longs;tantias corporum a Figurarum &longs;imilium locis illis ad quæ corpora eadem temporibus ii&longs;dem proportionalibus ab&longs;que viribus i&longs;tis pervenirent, &longs;unt ut quadrata temporum in quibus generantur quam proxime.

~~Corol. 2. Errores autem qui viribus proportionalibus ad &longs;imiles Figurarum &longs;imilium partes &longs;imiliter applicatis generantur, &longs;unt ut vires & quadrata temporum conjunctim.~~

~~Corol. 3. Idem intelligendum e&longs;t de &longs;patiis quibu&longs;vis quæ corpora urgentibus diver&longs;is viribus de&longs;cribunt.~~ ~~Hæc &longs;unt, ip&longs;o motus initio, ut vires & quadrata temporum conjunctim.~~

~~Corol. 4. Ideoque vires &longs;unt ut &longs;patia, ip&longs;o motus initio, de&longs;cripta directe & quadrata temporum inver&longs;e.~~

~~Corol. 5. Et quadrata temporum &longs;unt ut de&longs;cripta &longs;patia directe & vires inver&longs;e.~~

~~Scholium.~~

Si quantitates indeterminatæ diver&longs;orum generum conferantur inter &longs;e, & earum aliqua dicatur e&longs;&longs;e ut e&longs;t alia quævis directe vel inver&longs;e: &longs;en&longs;us e&longs;t, quod prior augetur vel diminuitur in eadem ratione cum po&longs;teriore, vel cum ejus reciproca. Et &longs;i earum aliqua dicatur e&longs;&longs;e ut &longs;unt aliæ duæ vel plures directe vel inver&longs;e: &longs;en&longs;us e&longs;t, quod prima augetur vel diminuitur in ratione quæ componitur ex rationibus in quibus aliæ vel aliarum reciprocæ augentur vel diminuuntur. Ut &longs;i A dicatur e&longs;&longs;e ut B directe & C directe & D inver&longs;e: &longs;en&longs;us e&longs;t, quod A augetur vel diminuitur in eadem ratione cum BXCX1/D, hoc e&longs;t, quod A & (BC/D) &longs;unt ad invicem in ratione data.

~~LEMMA XI.~~

~~Subten&longs;a evane&longs;cens anguli contactus, in curvis omnibus curvaturam finitam ad punctum contactus habentibus, est ultimo in ratione duplicata &longs;ubten&longs;æ arcus contermini.~~

Ca&longs;. 1. Sit arcus ille AB, tangens ejus AD, &longs;ubten&longs;a anguli contactus ad tangentem perpendicularis BD, &longs;ubten&longs;a arcus AB. Huic &longs;ubten&longs;æ AB & tangenti AD perpendiculares erigantur AG, BG, concurrentes in G; dein accedant puncta D, B, G, ad puncta d, b, g,&longs;itque J inter&longs;ectio linearum BG, AG ultimo facta ubi puncta D, Baccedunt u&longs;que ad A. Manife&longs;tum e&longs;t quod di&longs;tantia GJ minor e&longs;&longs;e pote&longs;t quam a&longs;&longs;ignata quævis. ~~E&longs;t autem (ex natura circulorum per puncta ABG, Abg tran&longs;euntium) ABquad.~~

æquale AGXBD, & Ab quad. æquale AgXbd,adeoque ratio AB quad. ad Ab quad. componitur ex rationibus AG ad Ag & BD ad bd.Sed quoniam GJ a&longs;&longs;umi pote&longs;t minor longitudine quavis a&longs;&longs;ignata, fieri pote&longs;t ut ratio AGad Ag minus differat a ratione æqualitatis quam pro differentia quavis a&longs;&longs;ignata, adeoque ut ratio AB quad. ad Ab quad. minus differat a ratione BD ad bd quam pro differentia quavis a&longs;&longs;ignata. ~~E&longs;t ergo, per Lemma 1, ratio ultima AB quad. ad Ab quad. æqualis rationi ultimæ BD ad bd.~~ ~~que E. D.~~

~~Cas. 2. Inclinetur jam BD ad AD in angulo quovis dato, & eadem &longs;emper erit ratio ultima BD ad bd quæ prius, adeoque eadem ae AB quad. ad Ab quad.~~ ~~que E. D.~~

Cas. 3. Et quamvis angulus D non detur, &longs;ed recta BD ad datum punctum convergente, vel alia quacunque lege con&longs;tituatur; tamen anguli D, d communi lege con&longs;tituti ad æqualitatem &longs;emper vergent & propius accedent ad invicem quam pro differentia quavis a&longs;&longs;ignata, adeoque ultimo æquales erunt, per Lem. & propterea lineæ BD, bd &longs;unt in eadem ratione ad invicem ac prius. que E. D.

~~Corol. 1. Unde eum tangentes AD, Ad, arcus AB, Ab, & eorum &longs;inus BC, bc fiant ultimo chordis AB, Ab æquales; erunt etiam illorum quadrata ultimo ut &longs;ubten&longs;æ BD, bd.~~

~~Corol. 2. Eorundem quadrata &longs;unt etiam ultimo ut &longs;unt arcuum &longs;agittæ quæ chordas bi&longs;ecant & ad datum punctum conver gunt.~~ ~~Nam &longs;agittæ illæ &longs;unt ut &longs;ubten&longs;æ BD, bd.~~

~~Corol. 3. Ideoque &longs;agitta e&longs;t in duplicata ratione temporis quo corpus data velocitate de&longs;cribit arcum.~~

Corol. 4. Triangula rectilinea ADB, Adb &longs;unt ultimo in triplicata ratione laterum AD, Ad, inque &longs;e&longs;quiplicata laterum DB, db; utpote in compo&longs;ita ratione laterum AD, & DB, Ad & dbexi&longs;tentia. Sic & triangula ABC, Abc &longs;unt ultimo in triplicata ratione laterum BC, bc. Rationem vero Se&longs;quiplicatam voco triplicatæ &longs;ubduplicatam, quæ nempe ex &longs;implici & &longs;ubduplicata componitur, quamque alias Se&longs;quialteram dicunt.

~~DE MOTU CORPORUM~~

Corol. 5. Et quoniam DB, db &longs;unt ultimo parallelæ & in duplicata ratione ip&longs;arum AD, Ad: erunt areæ ultimæ curvilineæ ADB, Adb (ex natura Parabolæ) duæ tertiæ partes triangulorum rectilineorum ADB, Adb; & &longs;egmenta AB, Ab partes tertiæ eorundem triangulorum. ~~Et inde hæ areæ & hæc &longs;egmenta erunt in triplicata ratione tum tangentium AD, Ad; tum chordarum & arcuum AB, Ab.~~

~~Scholium.~~

Cæterum in his omnibus &longs;upponimus angulum contactus nec infinite majorem e&longs;&longs;e angulis contactuum, quos Circuli continent cum tangentibus &longs;uis, nec ii&longs;dem infinite minorem; hoc e&longs;t, curvaturam ad punctum A, nec infinite parvam e&longs;&longs;e nec infinite magnam, &longs;eu intervallum AJ finitæ e&longs;&longs;e magnitudinis. ~~Capi enim pote&longs;t DBut AD3: quo in ca&longs;u Circulus nullus per punctum A inter tangentem AD & curvam AB duci pote&longs;t, proindeque angulus contactus erit infinite minor Circularibus.~~ ~~Et &longs;imili argumento &longs;i fiat DB&longs;ucce&longs;&longs;ive ut AD4, AD5, AD6, AD7, &c.~~ ~~habebitur &longs;eries angulorum contactus pergens in infinitum, quorum quilibet po&longs;terior e&longs;t infinite minor priore.~~ ~~Et &longs;i fiat DB &longs;ucce&longs;&longs;ive ut AD2, AD3/2, AD4/3, AD5/4, AD6/5, AD7/6, &c.~~ habebitur alia &longs;eries infinita angulorum contactus, quorum primus e&longs;t eju&longs;dem generis cum Circularibus, &longs;ecundus infinite major, & quilibet po&longs;terior infinite major priore. Sed & inter duos quo&longs;vis ex his angulis pote&longs;t &longs;eries utrinque in infinitum pergens angulorum intermediorum in&longs;eri, quorum quilibet po&longs;terior erit infinite major minorve priore. ~~Ut &longs;i inter terminos AD2 & AD3 in&longs;eratur &longs;eries AD(13/6), AD(1/5), AD9/4, AD7/3, AD5/2, AD/3, AD(11/4), AD(14/5), AD(17/6), &c.~~ ~~Et rur&longs;us inter binos quo&longs;vis angulos hujus &longs;eriei in&longs;eri pote&longs;t &longs;eries nova angulorum intermediorum ab invicem infinitis intervallis differentium.~~ ~~Neque novit natura limitem.~~

~~Quæ de curvis lineis deque &longs;uperficiebus comprehen&longs;is demon&longs;trata &longs;unt, facile applicantur ad &longs;olidorum &longs;uperficies curvas & contenta.~~ ~~Præmi&longs;i vero hæc Lemmata, ut effugerem tædium deducendi perplexas demon&longs;trationes, more veterum Geometrarum, ad ab&longs;urdum.~~ ~~Contractiores enim redduntur demon&longs;trationes per methodum Indivi&longs;ibilium.~~ Sed quoniam durior e&longs;t Indivi&longs;ibilium hypothe&longs;is, & propterea methodus illa minus Geometrica cen&longs;etur; malui demon&longs;trationes rerum &longs;equentium ad ultimas quantitatum evane&longs;centium &longs;ummas & rationes, prima&longs;que na&longs;centium, id e&longs;t, ad limites &longs;ummarum & rationum deducere; & propterea limitum illorum demon&longs;trationes qua potui brevitate præmittere. ~~His enim idem præ&longs;tatur quod per methodum Indivi&longs;ibilium; & principiis demon&longs;tratis jam tutius utemur.~~ Proinde in &longs;equentibus, &longs;iquando quantitates tanquam ex particulis con&longs;tantes con&longs;ideravero, vel &longs;i pro rectis u&longs;urpavero lineolas curvas; nolim indivi&longs;ibilia, &longs;ed evane&longs;centia divi&longs;ibilia, non &longs;ummas & rationes partium determinatarum, &longs;ed &longs;ummarum & rationum limites &longs;emper intelligi; vimque talium demon&longs;trationum ad methodum præcedentium Lemmatum &longs;emper revocari.

~~Objectio e&longs;t, quod quantitatum evane&longs;centium nulla &longs;it ultima proportio; quippe quæ, antequam evanuerunt, non e&longs;t ultima, ubi evanuerunt, nulla e&longs;t.~~ Sed & eodem argumento æque contendi po&longs;&longs;et nullam e&longs;&longs;e corporis ad certum locum pervenientis velocitatem ultimam: hanc enim, antequam corpus attingit locum, non e&longs;&longs;e ultimam, ubiattingit, nullam e&longs;&longs;e. Et re&longs;pon&longs;io facilis e&longs;t: Per velocitatem ultimam intelligi eam, qua corpus movetur neque antequam attingit locum ultimum & motus ce&longs;&longs;at, neque po&longs;tea, &longs;ed tunc cum attingit; id e&longs;t, illam ip&longs;am velocitatem quacum corpus attingit locum ultimum & quacum motus ce&longs;&longs;at. Et &longs;imiliter per ultimam rationem quantitatum evane&longs;centium, intelligendam e&longs;&longs;e rationem quantitatum non antequam evane&longs;cunt, non po&longs;tea, &longs;ed quacum evane&longs;cunt. ~~Pariter & ratio prima na&longs;centium e&longs;t ratio quacum na&longs;cuntur.~~ ~~Et &longs;umma prima & ultima e&longs;t quacum e&longs;&longs;e (vel augeri & minui) incipiunt & ce&longs;&longs;ant.~~ ~~Extat limes quem velocitas in fine motus attingere pote&longs;t, non autem tran&longs;gredi.~~ ~~Hæc e&longs;t velocitas ultima.~~ ~~Et par e&longs;t ratio limitis quantitatum & proportionum omnium incipientium & ce&longs;&longs;antium.~~ ~~Cumque hic limes &longs;it certus & definitus, Problema e&longs;t vere Geometricum eundem determinare.~~ ~~Geometrica vero omnia in aliis Geometricis determinandis ac demon&longs;trandis legitime u&longs;urpantur.~~

Contendi etiam pote&longs;t, quod &longs;i dentur ultimæ quantitatum evane&longs;centium rationes, dabuntur & ultimæ magnitudines: & &longs;ic quantitas omnis con&longs;tabit ex Indivi&longs;ibilibus, contra quam Euclides de Incommen&longs;urabilibus, in libro decimo Elementorum, demon&longs;travit. ~~Verum hæc Objectio fal&longs;æ innititur hypothe&longs;i.~~ Ultimæ rationes illæ quibu&longs;cum quantitates evane&longs;cunt, revera non &longs;unt rationes quantitatum ultimarum, &longs;ed limites ad quos quantitatum &longs;ine limite decre&longs;centium rationes &longs;emper appropinquant; & quas propius a&longs;&longs;equi po&longs;&longs;unt quam pro data quavis differentia, nunquam vero

~~tran&longs;gredi, neque prius attingere quam quantitates diminuuntur in infinitum.~~ ~~Res clarius intelligetur in infinite magnis.~~ Si quantitates duæ quarum data e&longs;t differentia auges ntur in infinitum, dabitur harum ultima ratio, nimirum ratio æqualitatis, nec tamen ideo dabuntur quantitates ultimæ &longs;eu maximæ quarum i&longs;ta e&longs;t ratio. Igitur in &longs;equentibus, &longs;iquando facili rerum conceptui con&longs;ulens dixero quantitates quam minimas, vel evane&longs;centes, vel ultimas; cave intelligas quantitates magnitudine determinatas, &longs;ed cogita &longs;emper diminuendas &longs;ine limite.

~~DE MOTU CORPORUM~~

~~SECTIO II.~~

~~De Inventione Virium Centripetarum.~~

~~PROPOSITIO I. THE OREMA I.~~

~~Areas, quas corpora in gyros acta radiis ad immobile centrum virium ductis de&longs;cribunt, & in planis immobilibus con&longs;i&longs;tere, & e&longs;&longs;e temporibus proportionales.~~

~~Dividatur tempus in partes æquales, & prima temporis parte de&longs;eribat corpus vi in&longs;ita rectam AB. Idem &longs;ecunda temporis parte, &longs;i nil impediret, recta~~

~~pergeret ad c, (per Leg.~~ 1.) de&longs;cribens lineam Bc æqualem ip&longs;i AB; adeo ut radiis AS, BS, cS ad centrum actis, confectæ forent æquales areæ ASB, BSc.Verum ubi corpus venitad B, agat vis centripeta impul&longs;u unico &longs;ed magno, efficiatque ut corpus de recta Bcdeclinet & pergat in recta BC. Ip&longs;i BS parallela agatur cC, occurens BC in C; & completa &longs;ecunda temporis parte, corpus (per Legum Corol. 1.) reperietur in C, in eodem plano cum triangulo ASB. Junge SC; & triangulum SBC,ob parallelas SB, Cc, æquale erit triangulo SBc, atque adeo etiam triangulo SAB. Simili argumento &longs;i vis centripeta &longs;ucce&longs;&longs;ive agat in C, D, E, &c. ~~faciens ut corpus &longs;ingulis temporis particulis &longs;ingulas de&longs;eribat rectas CD, DE, EF, &c.~~ ~~jacebunt hæ omnes in eodem plano; & triangulum SCD triangulo SBC, & SDE ip&longs;i SCD, & SEF ip&longs;i SDE æquale erit.~~ Æqualibus igitur temporibus æquales areæ in plano immoto de&longs;cribuntur: & componendo, &longs;unt arearum &longs;ummæ quævis SADS, SAFS inter &longs;e, ut &longs;unt tempora de&longs;criptionum. Augeatur jam numerus & minuatur latitudo triangulorum in infinitum; & eorum ultima perimeter ADF, (per Corollarium quartum Lemmatis tertii) erit linea curva: adeoque vis centripeta, qua corpus a tangente hujus curvæ perpetuo retrahitur, aget inde&longs;inenter; areæ vero quævis de&longs;criptæ SADS, SAFStemporibus de&longs;criptionum &longs;emper proportionales, erunt ii&longs;dem temporibus in hoc ca&longs;u proportionales. que E. D.

Corol. 1. Velocitas corporis in centrum immobile attracti e&longs;t in &longs;patiis non re&longs;i&longs;tentibus reciproce ut perpendiculum a centro illo in Orbis tangentem rectilineam demi&longs;&longs;um. E&longs;t enim velocitas in locis illis A, B, C, D, E, ut &longs;unt ba&longs;es æqualium triangulorum AB, BC, CD, DE, EF; & hæ ba&longs;es &longs;unt reciproce ut perpendicula in ip&longs;as demi&longs;&longs;a.

Corol. 2. Si arcuum duorum æqualibus temporibus in &longs;patiis non re&longs;i&longs;tentibus ab eodem corpore &longs;ucce&longs;&longs;ive de&longs;criptorum chordæ AB, BC compleantur in parallelogrammum ABCU, & hujus diagonalis BU in ea po&longs;itione quam ultimo habet ubi arcus illi in infinitum diminuuntur, producatur utrinque; tran&longs;ibit eadem per centrum virium.

Corol. 3. Si arcuum æqualibus temporibus in &longs;patiis non re&longs;i&longs;tentibus de&longs;criptorum chordæ AB, BC ac DE, EF compleantur in parallelogramma ABCU, DEFZ; vires in B & E &longs;unt ad invicem in ultima ratione diagonalium BU, EZ, ubi arcus i&longs;ti in infinitum diminuuntur. ~~Nam corporis motus BC & EF componuntur (per Legum Corol.~~ 1.) ex motibus Bc, BU & Ef, EZ: atqui BU & EZ, ip&longs;is Cc & Ff æquales, in Demon&longs;tratione Propo&longs;itionis hujus generabantur ab impul&longs;ibus vis centripetæ in B & E, ideoque &longs;unt his impul&longs;ibus proportionales.

Corol. 4. Vires quibus corpora quælibet in &longs;patiis non re&longs;i&longs;tentibus a motibus rectilineis retrahuntur ac detorquentur in orbes curvos &longs;unt inter &longs;e ut arcuum æqualibus temporibus de&longs;criptorum &longs;agittæ illæ quæ convergunt ad centrum virium, & chordas bi&longs;ecant ubi arcus illi in infinitum diminuuntur. ~~Nam hæ &longs;agittæ &longs;unt &longs;emi&longs;&longs;es diagonalium de quibus egimus in Corollario tertio.~~

~~DE MOTU CORPORUM~~

~~Corol. 5. Ideoque vires eædem &longs;unt ad vim gravitatis, ut hæ &longs;agittæ ad &longs;agittas horizonti perpendiculares arcuum Parabolicorum quos projectilia eodem tempore de&longs;cribunt.~~

~~Corol. 6. Eadem omnia obtinent per Legum Corol.~~ ~~IV, ubi plana in quibus corpora moventur, una cum centris virium quæ in ip&longs;is fita &longs;unt, non quie&longs;cunt, &longs;ed moventur uniformiter in directum.~~

~~PROPOSITIO II. THEOREMA II.~~

Corpus omne, quod movetur in linea aliqua curva in plano de&longs;cripta, & radio ducto ad punctum vel immobile, vel motu rectilineo uniformiter progrediens, de&longs;cribit areas circa punctum illud temporibus proportionales, urgetur a vi centripeta tendente ad idem punctum.

~~Cas. 1. Nam corpus omne quod movetur in linea curva, detorquetur de cur&longs;u rectilineo per vim aliquam in ip&longs;um agentem (per Leg.~~ ~~1.) Et vis illa qua corpus de cur&longs;u rectilineo detorquetur, & cogitur triangula quam minima SAB, SBC, SCD, &c.~~ ~~circa punctum immobile S temporibus æqualibus æqualia de&longs;cribere, agit in loco B &longs;ecundum lineam parallelam ip&longs;i cC (per Prop.~~ ~~XL, Lib.~~ ~~1 Elem.~~ ~~& Leg.~~ ~~11.) hoc e&longs;t, &longs;ecundum lineam BS; & in loco C &longs;ecundum lineam ip&longs;i dD parallelam, hoc e&longs;t, &longs;ecundum lineam SC, &c.~~ ~~Agit ergo &longs;emper &longs;ecundum lineas tendentes ad punctum illud immobile S. que E. D.~~

Cas. 2. Et, per Legum Corollarium quintum, perinde e&longs;t &longs;ive quie&longs;cat &longs;uperficies in qua corpus de&longs;cribit figuram curvilineam, &longs;ive moveatur eadem una cum corpore, figura de&longs;cripta, & puncto &longs;uo S uniformiter in directum.

Corol. 1. In Spatiis vel Mediis non re&longs;i&longs;tentibus, &longs;i areæ non &longs;unt temporibus proportionales, vires non tendunt ad concur&longs;um radiorum; &longs;ed inde declinant in con&longs;equentia &longs;eu ver&longs;us plagam in quam fit motus, &longs;i modo arearum de&longs;criptio acceleratur: &longs;in retardatur, declinant in antecedentia.

Corol. 2. In Mediis etiam re&longs;i&longs;tentibus, &longs;i arearum de&longs;criptio acceleratur, virium directiones declinant a concur&longs;u radiorum ver&longs;us plagam in quam &longs;it motus.

~~Scholium.~~

~~Urgeri pote&longs;t corpus a vi centripeta compo&longs;ita ex pluribus viribus.~~ In hoc ca&longs;u &longs;en&longs;us Propo&longs;itionis e&longs;t, quod vis illa quæ ex omnibus componitur, tendit ad punctum S. Porro &longs;i vis aliqua agat perpetuo &longs;ecundum lineam &longs;uperficiei de&longs;criptæ perpendicularem; hæc faciet ut corpus deflectatur a plano &longs;ui motus: &longs;ed quantitatem &longs;uperficiei de&longs;criptæ nec augebit nec minuet, & propterea in compo&longs;itione virium negligenda e&longs;t.

~~PROPOSITIO III. THEOREMA III.~~

Corpus omne, quod radio ad centrum corporis alterius utcunque moti ducto de&longs;cribit areas circa centrum illud temporibus proportionales, urgetur vi compo&longs;ita ex vi centripeta tendente ad corpus illud alterum, & ex vi omni acceleratrice qua corpus illud alterum urgetur.

~~Sit corpus primum L & corpus alterum T: & (per Legum Corol.~~ VI.) &longs;i vi nova, quæ æqualis & contraria &longs;it illi qua corpus alterum T urgetur, urgeatur corpus utrumque &longs;ecundum lineas parallelas; perget corpus primum L de&longs;cribere circa corpus alterum T areas ea&longs;dem ac prius: vis autem, qua corpus alterum T urgebatur, jam de&longs;truetur per vim &longs;ibi æqualem & contrariam; & propterea (per Leg. 1.) corpus illud alterum T &longs;ibimet ip&longs;i jam relictum vel quie&longs;cet vel movebitur uniformiter in directum: & corpus primum Lurgente differentia virium, id e&longs;t, urgente vi reliqua perget areas temporibus proportionales circa corpus alterum T de&longs;cribere. ~~Tendit igitur (per Theor.~~ ~~11.) differentia virium ad corpus illud alterum T ut centrum. que E. D.~~

Corol. 1. Hinc &longs;i corpus unum L radio ad alterum T ducto de&longs;cribit areas temporibus proportionales; atque de vi tota (&longs;ive &longs;implici, &longs;ive ex viribus pluribus, juxta Legum Corollarium &longs;ecundum, compo&longs;ita,) qua corpus prius L urgetur, &longs;ubducatur (per idem Legum Corollarium) vis tota acceleratrix qua corpus alterum urgetur: vis omnis reliqua qua corpus prius urgetur tendet ad corpus alterum T ut centrum.

~~Corol. 2. Et, &longs;i areæ illæ &longs;unt temporibus quamproxime proportionales, vis reliqua tendet ad corpus alterum T quamproxime.~~

~~Corol. 3. Et vice ver&longs;a, &longs;i vis reliqua tendit quamproxime ad corpus alterum T, erunt areæ illæ temporibus quamproxime proportionales.~~

~~DE MOTU CORPORUM~~

Corol. 4. Si corpus L radio ad alterum corpus T ducto de&longs;cribit areas quæ, cum temporibus collatæ, &longs;unt valde inæquales; & corpus illud alterum T vel quie&longs;cit vel movetur uniformiter in directum: actio vis centripetæ ad corpus illud alterum T tendentis, vel nulla e&longs;t, vel mi&longs;cetur & componitur cum actionibus admodum potentibus aliarum virium: Vi&longs;que tota ex omnibus, &longs;i plures &longs;unt vires, compo&longs;ita, ad aliud (&longs;ive immobile &longs;ive mobile) centrum dirigitur. ~~Idem obtinet, ubi corpus alterum motu quocunque movetur; &longs;i modo vis centripeta &longs;umatur, quæ re&longs;tat po&longs;t &longs;ubductionem vis totius in corpus illud alterum T agentis.~~

~~Scholium.~~

Quoniam æquabilis arearum de&longs;criptio Index e&longs;t Centri, quod vis illa re&longs;picit qua corpus maxime afficitur, quaque retrahitur a motu rectilineo & in orbita &longs;ua retinetur: quidni u&longs;urpemus in &longs;equentibus æquabilem arearum de&longs;criptionem, ut Indicem Centri circum quod motus omnis circularis in &longs;patiis liberis peragitur?

~~PROPOSITIO IV. THEOREMA IV.~~

Corporum, quæ diver&longs;os circulos æquabili motu de&longs;cribunt, vires centripetas ad centra eorundem circulorum tendere; & e&longs;&longs;e inter &longs;e, ut &longs;unt arcuum &longs;imul de&longs;criptorum quadrata applicata ad circulorum radios.

~~Tendunt hæ vires ad centra circulorum per Prop.II. & Corol.~~ ~~II. Prop.~~ ~~1; & &longs;unt inter &longs;e ut arcuum æqualibus temporibus quam minimis de&longs;criptorum &longs;inus ver&longs;i per Corol.~~ ~~IV. Prop.~~ ~~I; hoc e&longs;t, ut quadrata arcuum eorundem ad diametros circulorum applicata per Lem.~~ VII: & propterea, cum hi arcus &longs;int ut arcus temporibus quibu&longs;vis æqualibus de&longs;cripti, & diametri &longs;int ut eorum radii; vires erunt ut arcuum quorumvis &longs;imul de&longs;criptorum quadrata applicata ad radios circulorum. q.E.D.

Corol. 1. Igitur, cum arcus illi &longs;int ut velocitates corporum, vires centripetæ &longs;unt ut velocitatum quadrata applicata ad radios circulorum: hoc e&longs;t, ut cum Geometris loquar, vires &longs;unt in ratione compo&longs;ita ex duplicata ratione velocitatum directe & ratione &longs;implici radiorum inver&longs;e.

Corol. 2. Et, cum tempora periodica &longs;int in ratione compo&longs;ita ex ratione radiorum directe & ratione velocitatum inver&longs;e, vires centripetæ &longs;unt reciproce ut quadrata temporum periodicorum applicata ad circulorum radios; hoc e&longs;t, in ratione compo&longs;ita ex ratione radiorum directe & ratione duplicata temporum periodicorum inver&longs;e.

~~Corol. 3. Unde, &longs;i tempora periodica æquentur & propterea velocitates &longs;int ut radii; erunt etiam vires centripetæ ut radii: & contra.~~

~~Cor. 4. Si & tempora periodica & velocitates &longs;int in ratione &longs;ubduplicata radiorum; æquales erunt vires centripetæ inter &longs;e: & contra.~~

~~Corol. 5. Si tempora periodica &longs;int ut radii & propterea velocitates æquales; vires centriperæ erunt reciproce ut radii: & contra.~~

Corol. 6. Si tempora periodica &longs;int in ratione &longs;e&longs;quiplicata radiorum & propterea velocitates reciproce in radiorum ratione &longs;ubduplicata; vires centripetæ erunt reciproce ut quadrata radiorum: & contra.

Corol. 7. Et univer&longs;aliter, &longs;i tempus periodicum &longs;it ut Radii Rpote&longs;tas quælibet Rn, & propterea velocitas reciproce ut Radii pote&longs;tas Rn-1; erit vis centripeta reciproce ut Radii pote&longs;tas R2n-1:& contra.

Corol. 8. Eadem omnia de temporibus, velocitatibus, & viribus, quibus corpora &longs;imiles figurarum quarumcunque &longs;imilium, centraque in figuris illis &longs;imiliter po&longs;ita habentium, partes de&longs;cribunt, con&longs;equuntur ex Demon&longs;tratione præcedentium ad ho&longs;ce ca&longs;us applicata. ~~Applicatur autem &longs;ub&longs;tituendo æquabilem arearum de&longs;criptionem pro æquabili motu, & di&longs;tantias corporum a centris pro radiis u&longs;urpando.~~

Corol. 9. Ex eadem demon&longs;tratione con&longs;equitur etiam; quod arcus, quem corpus in circulo data vi centripeta uniformiter revolvendo tempore quovis de&longs;cribit, medius e&longs;t proportionalis inter diametrum circuli, & de&longs;cen&longs;um corporis eadem data vi eodem que tempore cadendo confectum.

~~Scholium.~~

Ca&longs;us Corollarii &longs;exti obtinet in corporibus cæle&longs;tibus, (ut &longs;eor&longs;um collegerunt etiam no&longs;trates Wrennus, Hookius & Hallæus) & propterea quæ &longs;pectant ad vim centripetam decre&longs;centem in duplicata ratione di&longs;tantiarum a centris, decrevi fu&longs;ius in &longs;equentibus exponere.

~~DE MOTU CORPORUM~~

~~Porro præcedentis propo&longs;itionis & corollariorum ejus beneficio, colligitur etiam proportio vis centripetæ ad vim quamlibet notam, qualis e&longs;t ea Gravitatis.~~ ~~Nam &longs;i corpus in circulo Terræ concentrico vi gravitatis &longs;uæ revolvatur, hæc gravitas e&longs;t ip&longs;ius vis centripeta.~~ ~~Datur autem, ex de&longs;cen&longs;u gravium, & tempus revolutionis unius, & arcus dato quovis tempore de&longs;criptus, per hujus Corol.~~ ~~IX.~~ ~~Et huju&longs;modi propo&longs;itionibus Hugenius, in eximio &longs;uo Tractatu de Horologio O&longs;cillatorio, vim gravitatis cum revolventium viribus centrifugis contulit.~~

~~Demon&longs;trari etiam po&longs;&longs;unt præcedentia in hunc modum.~~ ~~In circulo quovis de&longs;cribi intelligatur Polygonum laterum quotcunque.~~ Et &longs;i corpus, in polygoni lateribus data cum velocitate movendo, ad ejus angulos &longs;ingulos a circulo reflectatur; vis qua &longs;ingulis reflexionibus impingit in circulum erit ut ejus velocitas: adeoque &longs;umma virium in dato tempore erit ut velocitas illa & numerus reflexionum conjunctim: hoc e&longs;t (&longs;i polygonum detur &longs;pecie) ut longitudo dato illo tempore de&longs;cripta & longitudo eadem applicata ad Radium circuli; id e&longs;t, ut quadratum longitudinis illius applicatum ad Radium: adeoque, &longs;i polygonum lateribus infinite diminutis coincidat cum circulo, ut quadratum arcus dato tempore de&longs;cripti applicatum ad radium. ~~Hæc e&longs;t vis centrifuga, qua corpus urget circulum: & huic æqualis e&longs;t vis contraria, qua circulus continuo repellit corpus centrum ver&longs;us.~~

~~PROPOSITIO. V. PROBLEMA I.~~

~~Data quibu&longs;cunque in locis velocitate, qua corpus figuram datam viribus ad commune aliquod centrum tendentibus de&longs;cribit, centrum illud invenire.~~

Figuram de&longs;criptam tangant rectæ tres PT, TQV, VR in punctis totidem P, Q, R, concurrentes in T & V. Ad tangentes erigantur perpendicula PA, QB, RC, velocitatibus corporis in punctis illis P, Q, R a quibus eriguntur reciproce proportionalia; id e&longs;t, ita ut &longs;it PA ad QB ut velocitas in Q ad velocitatem in P, & QB ad RC ut velocitas in R ad velocitatem in que Per perpendiculorum terminos A, B, C ad angulos rectos ducantur AD, DBE, EC concurrentes in D & E: Et actæ TD, VE concurrent in centro q&longs;ito S.

~~Nam perpendicula a centro Sin tangentes PT, QT demi&longs;&longs;a (per Corol.~~ ~~1. Prop.I.) &longs;unt reciproce ut velocitates corporis in punctis P & V; &c.~~ ~~adeoque per con&longs;tructionem ut perpendicula AP, BQ directe, id e&longs;t ut perpendicula a puncto D in tangentes demi&longs;&longs;a.~~ ~~Unde facile colligitur quod puncta S, D, T, &longs;unt in una recta.~~ ~~Et &longs;imili argumento puncta S, E, V &longs;unt etiam in una recta; & propterea centrum S in concur&longs;u rectarum TD, VEver&longs;atur. q.E.D.~~

~~PROPOSITIO VI. THEOREMA V.~~

Si corpus in &longs;patio non re&longs;i&longs;tente circa centrum immobile in Orbe quocunque revolvatur, & arcum quemvis jamjam na&longs;centem tempore quàm minimo de&longs;cribat, & &longs;agitta arcus duci intelligatur quæ chor dam bi&longs;ecet, & producta tran&longs;eat per centrum virium: erit vis centripeta in medio arcus, ut &longs;agitta directe & tempus bis inver&longs;e.

Nam &longs;agitta dato tempore e&longs;t ut vis (per Corol.4 Prop.I,) & augendo tempus in ratione quavis, ob auctum arcum in eadem ratione &longs;agitta augetur in ratione illa duplicata (per Corol. ~~2 & 3, Lem.~~ ~~XI,) adeoque e&longs;t ut vis &longs;emel & tempus bis.~~ ~~Subducatur duplicata ratio temporis utrinque, & fiet vis ut &longs;agitta directe & tempus bis inver&longs;e. q.E.D.~~

~~Idem facile demon&longs;tratur etiam per Corol.~~ ~~4 Lem.~~ X.

~~Corol. 1. Si corpus P revolvendo~~

circa centrum S de&longs;cribat lineam curvam APQ, tangat verò recta ZPR curvam illam in puncto quovis P, & ad tangentem ab alio quovis Curvæ puncto Q agatur QR di&longs;tantiæ SP parallela, ac demittatur QT perpendicularis ad di&longs;tantiam illam SP: vis centripeta erit reciproce ut &longs;olidum (SP quad.XQT quad./QR) &longs;i modo &longs;olidi illius ea &longs;emper &longs;umatur quantitas, quæ ultimò fit ubi coeunt puncta P & que Nam QR æqualis

e&longs;t &longs;agittæ dupli arcus QP, in cujus medio e&longs;t P, & duplum trianguli SQP &longs;ive SPXQT, tempori quo arcus i&longs;te duplus de&longs;cribitur proportionale e&longs;t, ideoque pro temporis exponente &longs;cribi pote&longs;t.

~~DE MOTU CORPORUM~~

~~Corol. 2. Eodem argumento vis centripeta e&longs;t reciprocè ut &longs;olidum (SYqXQPq/QR), &longs;i modo SY perpendiculum &longs;it a centro virium in Orbis tangentem PR demi&longs;&longs;um.~~ ~~Nam rectangula SYXQP & SPXQTæquantur.~~

Corol. 3. Si Orbis vel circulus e&longs;t, vel angulum contactus cum circulo quam minimum continet, eandem habens curvaturam eundemque radium curvaturæ ad punctum contactus P; & &longs;i PV chorda &longs;it circuli hujus a corpore per centrum virium acta: erit vis centripeta reciproce ut &longs;olidum SYqXPV. Nam PV e&longs;t (QPq/QR).

~~Corol. 4. Ii&longs;dem po&longs;itis, e&longs;t vis centripeta ut velocitas bis directe, & chorda illa inver&longs;e.~~ ~~Nam velocitas e&longs;t reciproce ut perpendiculum SY per Corol.~~ ~~I Prop.~~ I.

Corol. 5. Hinc &longs;i detur figura quævis curvilinea APQ, & in ea detur etiam punctum S ad quod vis centripeta perpetuo dirigitur, inveniri pote&longs;t lex vis centripetæ, qua corpus quodvis P a cur&longs;u rectilineo perpetuò retractum in figuræ illius perimetro detinebitur eamque revolvendo de&longs;cribet. ~~Nimirum computandum e&longs;t vel &longs;olidum (SPqXQTq/QR) vel &longs;olidum SYqXPV huic vi reciproce proportionale.~~ ~~Ejus rei dabimus exempla in Problematis &longs;equentibus.~~

~~PROPOSITIO VII. PROBLEMA II.~~

~~Gyretur corpus in circumferentia Circuli, requiritur Lex vis centripetæ tendentis ad punctum quodcunque datum.~~

~~E&longs;to Circuli circumferentia~~

VQPA, punctum datum ad quod vis ceu ad centrum &longs;uum tendit S, corpus in circumferentia latum P, locus proximus in quem movebitur Q, & circuli tangens ad locum priorem PRZ. Per punctum S ducatur chorda PV,& acta circuli diametro VA jungatur AP, & ad SP demittatur perpendiculum QT, quod productum occurrat tangenti PR in Z, ac denique per punctum Q agatur LR quæ ip&longs;i SP parallela &longs;it & occurrat tum circulo in L tum tangenti PZ in R. Et ob &longs;imilia triangula ZQR, ZTP, VPA; erit RP quad. hoc e&longs;t QRL ad QT quad. ut AV quad. ad PV quad. Ideoque (QRLXPV quad./AV quad.) æquatur QT quad. Ducantur hæc æqualia in (SP quad./QR) &, punctis P & Q coeuntibus, &longs;cribatur PV pro RL.Sic fiet (SP quad.XPV cub./AV quad.) æquale (SP quad.XQT quad./QR) Ergo (per Corol.1 & 5 Prop.VI.) vis centripeta e&longs;t reciproce ut (SPqXPV cub./AV quad) id e&longs;t, (ob datum AV quad.) reciproce ut quadratum di&longs;tantiæ &longs;eu altitudinis SP & cubus chordæ PV conjunctim. q.E.I.

~~Idem aliter.~~

Ad tangentem PR productam demittatur perpendiculum SY,& ob &longs;imilia triangula SYP, VPA; erit AV ad PV ut SP ad SY, ideoque (SPXPV/AV) æquale SY, & (SP quad.XPV cub./AV quad.) æquale SY quad.XPV. Et propterea (per Corol.3 & 5 Prop.VI.) vis centripeta e&longs;t reciproce ut (SPqXPV cub./AVq) hoc e&longs;t, ob datam AV, reciproce ut SPqXPV cub. ~~que E. I.~~

Corol. 1. Hinc &longs;i punctum datum S ad quod vis centripeta &longs;emper tendit, locetur in circumferentia hujus circuli, puta ad V; erit vis centripeta reciproce ut quadrato cubus altitudinis SP.

~~Corol. 2. Vis qua corpus P in cir~~

culo APTV circum virium centrum S revolvitur, e&longs;t ad vim qua corpus idem P in eodem circulo & eodem tempore periodico circum aliud quodvis virium centrum R revolvi pote&longs;t, ut RP quad.XSP ad cubum rectæ SGquæ a primo virium centro S ad orbis tangentem PG ducitur, & di&longs;tantiæ corporis a &longs;ecundo virium centro parallela e&longs;t. Nam, per con&longs;tructionem hujus Propo&longs;itionis, vis prior e&longs;t ad vim po&longs;teriorem, ut RPqXPT cub. ad SPqXPV cub. id e&longs;t, ut SPXRPq ad (SP cub.XPV cub/PT cub.) &longs;ive (ob &longs;imilia triangula PSG, TPV) ad SG cub.

~~DE MOTU CORPORUM~~

Corol. 3. Vis, qua corpus P in Orbe quocunque circum virium centrum S revolvitur, e&longs;t ad vim qua corpus idem P in eodem orbe eodemque tempore periodico circum aliud quodvis virium centrum R revolvi pote&longs;t, ut SPXRPq contentum utique &longs;ub di&longs;tantia corporis a primo virium centro S & quadrato di&longs;tantiæ ejus a &longs;ecundo virium centro R ad cubum rectæ SG quæ a primo virium centro S ad orbis tangentem PG ducitur, & corporis a &longs;ecundo virium centro di&longs;tantiæ RP parallela e&longs;t. ~~Nam vires in hoc Orbe, ad ejus punctum quodvis P, eædem &longs;unt ac in Circulo eju&longs;dem curvaturæ.~~

~~PROPOSITIO. VIII. PROBLEMA. III.~~

~~Moveatur corpus in Circulo PQA: ad hunc effectum requiritur Lex vis centripetæ tendentis ad punctum adeo longinquum S, ut lineæ omnes PS, RS ad id ductæ, pro parallelis haberi po&longs;&longs;int.~~

~~A Circuli centro C agatur &longs;emidiameter CA parallelas i&longs;tas perpendiculariter &longs;ecans in M &~~

N, & jungatur CP. Ob &longs;imilia triangula CPM, PZT & RZQe&longs;t CPq ad PMq ut PRq ad QTq & ex natura Circuli PRqæquale e&longs;t rectangulo QRX√RN+QN &c. &longs;ive coeuntibus punctis P, Q rectangulo QRX2PM. Ergo e&longs;t CPq ad PM quad. ut QRX2PMad QT quad. adeoque (QT quad./QR) æquale (2PM cub./CP quad.), & (QT quad.XSP quad./QR) æquale (2PM cub.XSP qu./CP quad.) E&longs;t ergo (per Corol. ~~1 & 5 Prop.~~ ~~VI.) vis centripeta reciproce ut (2PMcub.XSP quad./CP quad.) hoc e&longs;t (neglecta ratione determinata (2SP quad./CP quad.)) reciproce ut PM cub.~~ ~~que E. I.~~

~~Idem facile colligitur etiam ex Propo&longs;itione præcedente.~~

~~Scholium.~~

Et &longs;imili argumento corpus movebitur in Ellip&longs;i vel etiam in Hyperbola vel Parabola, vi centripeta quæ &longs;it reciproce ut cubus ordinatim applicatæ ad centrum virium maxime longinquum tendentis.

~~PROPOSITIO IX. PROBLEMA IV.~~

~~Gyretur corpus in Spirali PQS &longs;ecante radios omnes SP, SQ, &c.~~

~~in angulo dato: requiritur Lex vis centripetæ tendentis ad centrum Spiralis.~~

Detur angulus indefinite parvus PSQ, & ob datos omnes angulos dabitur &longs;pecie figura SPQRT. Ergo datur ratio (QT/QR), e&longs;tque (QT quad./QR) ut QT, hoc e&longs;t ut SP. Mutetur jam uteunque angulus PSQ,& recta QR angulum contactus QPR &longs;ubtendens mutabitur (per Lemma XI.) in duplicata ratione ip&longs;ius PR vel QT. Ergo manebit (QT quad./QR) eadem quæ prius, hoc e&longs;t ut SP. Quare (QTq.XSPq/QR) e&longs;t ut SP cub. adeoque (per Corol. ~~1 & 5 Prop.~~ ~~VI.) vis centripeta e&longs;t reciproce ut cubus di&longs;tantiæ SP. que E. I.~~

~~Idem aliter.~~

Perpendiculum SY in tangentem demi&longs;&longs;um, & circuli Spiralem tangentis chorda PV &longs;unt ad altitudinem SP in datis rationibus; ideoque SP cub. e&longs;t ut SYqXPV, hoc e&longs;t (per Corol. ~~3 & 5 Prop.VI.) reciproce ut vis centripeta.~~

~~LEMMA XII.~~

~~Parallelogramma omnia, circa datæ Ellip&longs;eos vel Hyperbolæ diametros qua&longs;vis conjugatas de&longs;cripta, e&longs;&longs;e inter &longs;e æqualia.~~

~~Con&longs;tat ex Conicis.~~

~~DE MOTU CORPORUM~~

~~PROPOSITIO X. PROBLEMA. V.~~

~~Gyretur corpus in Ellip&longs;i: requiritur lex vis centripetæ tendentis ad centrum Ellip&longs;eos.~~

~~Sunto CA, CB &longs;emiaxes Ellip&longs;eos; GP, DK diametri conjugatæ; PF, Qt perpendicula ad diametros; Qv ordinatim applicata ad diametrum~~

GP; & &longs;i compleatur parallelogrammum QvPR, erit (ex Conicis) PvG ad Qv quad.ut PC quad. ad CD quad. & (ob &longs;imilia triangula Qvt, PCF) Qv quad. e&longs;t ad Qt quad. ut PC quad. ad PF quad. & conjunctis rationibus, PvGad Qt quad. ut PC quad. ad CD quad.& PC quad. ad PF quad. id e&longs;t, vG ad (Qt quad./Pv) ut PC quad.ad (CDqXPFq/PCq). Scribe QR pro Pv, & (per Lemma XII.) BCXCApro CDXPF, nec non, punctis P & Q coeuntibus, 2PC pro vG, & ductis extremis & mediis in &longs;e mutuo, fiet (Qt quad.XPCq/QR) æquale (2BCqXCAq/PC). E&longs;t ergo (per Corol. ~~5 Prop.~~ ~~VI.) vis centripeta reciproce ut (2BCqXGAq;/PC) id e&longs;t (ob datum 2BCqXCAq) reciproce ut (1/PC); hoc e&longs;t, directe ut di&longs;tantia PC. que E. I.~~

~~Idem aliter.~~

In PG ab altera parte puncti t po&longs;ita intelligatur tu æqualis ip&longs;i tv; deinde cape uV quæ &longs;it ad vG ut e&longs;t DC quad. ad PC quad.Et quoniam ex Conicis Qv quad. ad PvG, ut DC quad. ad PC quad: erit Qv quad. æquale PvXuV. Unde quadratum chor- dæ arcus PQ erit æquale rectangulo VPv; adeoque Circulus qui tangit Sectionem Conicam in P & tran&longs;it per punctum Q, tran&longs;ibit etiam per punctum V. Coeant puncta P & Q, & hic circulus eju&longs;dem erit curvaturæ cum &longs;ectione conica in P, & PV æqualis erit (2DCq/PC). Proinde vis qua corpus P in Ellip&longs;i revolvitur, erit reciproce ut (2DCq/PC) in PFq (per Corol. ~~3 Prop.~~ ~~VI.) hoc e&longs;t (ob datum 2DCq in PFq) directe ut PC. que E. I.~~

~~LIBER PRIMUS.~~

Corol. 1. E&longs;t igitur vis ut di&longs;tantia corporis a centro Ellip&longs;eos: & vici&longs;&longs;im, &longs;i vis &longs;it ut di&longs;tantia, movebitur corpus in Ellip&longs;i centrum habente in centro virium, aut forte in Circulo, in quem utique Ellip&longs;is migrare pote&longs;t.

~~Corol. 2. Et æqualia erunt revolutionum in Ellip&longs;ibus univer&longs;is circum centrum idem factarum periodica tempora.~~ ~~Nam tempora illa in Ellip&longs;ibus &longs;imilibus æqualia &longs;unt per Corol.~~ ~~3 & 8, Prop.~~ IV: in Ellip&longs;ibus autem communem habentibus axem majorem, &longs;unt ad invicem ut Ellip&longs;eon areæ totæ directe & arearum particulæ &longs;imul de&longs;criptæ inver&longs;e; id e&longs;t, ut axes minores directe & corporum velocitates in verticibus principalibus inver&longs;e; hoc e&longs;t, ut axes illi minores directe & ordinatim applicatæ ad axes alteros inver&longs;e; & propterea (ob æqualitatem rationum directarum & inver&longs;arum) in ratione æqualitatis.

~~Scholium.~~

~~Si Ellip&longs;is, centro in infinitum abeunte vertatur in Parabolam, corpus movebitur in hac Parabola; & vis ad centrum infinite di&longs;tans jam tendens evadet æquabilis.~~ Hoc e&longs;t Theorema Galilæi.Et &longs;i coni &longs;ectio Parabolica, inclinatione plani ad conum &longs;ectum mutata, vertatur in Hyperbolam, movebitur corpus in hujus perimetro, vi centripeta in centrifugam ver&longs;a. Et quemadmodum in Circulo vel Ellip&longs;i, &longs;i vires tendunt ad centrum figuræ in Ab&longs;ci&longs;&longs;a po&longs;itum, hæ vires augendo vel diminuendo Ordinatas in ratione quacunque data, vel etiam mutando angulum inclinationis Ordinatarum ad Ab&longs;ci&longs;&longs;am, &longs;emper augentur vel diminuuntur in ratione di&longs;tantiarum a centro, &longs;i modo tempora periodica maneant æqualia: &longs;ic etiam in figuris univer&longs;is, &longs;i Ordinatæ augeantur vel diminuantur in ratione quacunque data, vel angulus ordinationis utcunque mutetur, manente tempore periodico; vires ad centrum quodcunque in Ab&longs;ci&longs;&longs;a po&longs;itum tendentes a binis quibu&longs;vis figurarum locis, ad quæ terminantur Ordinatæ corre&longs;pondentibus Ab&longs;ci&longs;&longs;arum punctis in&longs;i&longs;tentes, augentur vel &c. ~~augentur vel diminuuntur in ratione di&longs;tantiarum a centro.~~

~~DE MOTU CORPORUM~~

~~SECTIO III.~~

~~De motu Corporum in Conicis Sectionibus excentricis.~~

~~PROPOSITIO XI. PROBLEMA VI.~~

~~Revolvatur corpus in Ellip&longs;i: requiritur Lex vis centripetæ tendentis ad umbilicum Ellip&longs;eos.~~

E&longs;to Ellip&longs;eos umbilicus S. Agatur SP &longs;ecans Ellip&longs;eos tum diametrum DK in E, tum ordinatim applicatam Qv in x, & compleatur parallelogrammum QxPR. Patet EP æqualem e&longs;&longs;e &longs;emiaxi ma

jori AC, eo quod acta ab altero Ellip&longs;eos umbilico H linea HI ip&longs;i EC parallela, (ob æquales CS, CH) æquentur ES, EI, adeo ut EP&longs;emi&longs;umma &longs;it ip&longs;arum PS, PI, id e&longs;t (ob parallelas HI, PR & angulos æquales IPR, HPZ) ip&longs;arum PS, PH,quæ conjunctim axem totum 2AC adæquant. Ad SP demittatur perpendicularis QT, & Ellip&longs;eos latere recto principali (&longs;eu (2BC quad./AC)) dicto L, erit LXQR ad LXPv ut QR ad Pv, id e&longs;t ut PE &longs;eu AC ad PC; & LXPv ad GvP ut L ad Gv; & GvP ad Qv quad. ut PC quad. ad CD quad; & (per Corol. ~~2 Lem.~~ VII.) Qv quad. ad Qx quad, punctis Q & P coeuntibus, e&longs;t ratio æqualitatis; & Qx quad. &longs;eu Qv quad. e&longs;t ad QT quad.ut EP quad. ad PF quad, id e&longs;t ut CA quad. ad PF quad. &longs;ive (per Lem XII.) ut CD quad. ad CB quad. Et conjunctis his omnibus rationibus, LXQR fit ad QT quad. ut ACXLXPCq.XCDque &longs;eu 2CBque XPCq.XCDque ad PCXGvXCDq.XCBque &longs;ive ut 2PC ad Gv. Sed, punctis Q & P coeuntibus, æquantur 2PC & Gv. Ergo & his pro portionalia LXQR & QT quad. æquantur. ~~Ducantur hæc æqualia in (SPq/QR) & fiet LXSPque æquale (SPq.XQTq/QR). Ergo (per Corol.~~ ~~1 & 5 Prop.~~ ~~VI.) vis centripeta reciproce e&longs;t ut LXSPque id e&longs;t, reciproce in ratione duplicata di&longs;tantiæ SP. q.E.I.~~

~~LIBER PRIMUS.~~

~~Idem aliter.~~

~~Cum vis ad centrum Ellip&longs;eos tendens, qua corpus P in Ellip&longs;i illa revolvi pote&longs;t, &longs;it (per Corol.~~ ~~I Prop.~~ X) ut CP di&longs;tantia corporis ab Ellip&longs;eos centro C; ducatur CE parallela Ellip&longs;eos tangenti PR: & vis qua corpus idem P, circum aliud quodvis Ellip&longs;eos punctum S revolvi pote&longs;t, &longs;i CE & PS concurrant in E, erit ut (PE cub./SPq) (per Corol. ~~3 Prop.~~ ~~VII,) hoc e&longs;t, &longs;i punctum S &longs;it umbilicus Ellip&longs;eos, adeoque PE detur, ut SPq reciproce. q.E.I.~~

Eadem brevitate qua traduximus Problema quintum ad Parabolam, & Hyperbolam, liceret idem hic facere: verum ob dignitatem Problematis & u&longs;um ejus in &longs;equentibus, non pigebit ca&longs;us ceteros demon&longs;tratione confirmare.

~~PROPOSITIO XII. PROBLEMA. VII.~~

~~Moveatur corpus in Hyperbola: requiritur Lex vis centripetæ tendentis ad umbilicum figuræ.~~

Sunto CA, CB &longs;emi-axes Hyperbolæ; PG, KD diametri conjugatæ; PF, Qt perpendicula ad diametros; & Qv ordinatim applicata ad diametrum GP. Agatur SP &longs;ecans cum diametrum DK in E, tum ordinatim applicatam Qv in x, & compleatur parallelogrammum QRPx. Patet EP æqualem e&longs;&longs;e &longs;emiaxi tran&longs;ver&longs;o AC, eo quod, acta ab altero Hyperbolæ umbilico H linea HI ip&longs;i EC parallela, ob æquales CS, CH, æquentur ES, EI; adeo ut EP &longs;emidifferentia &longs;it ip&longs;arum PS, PI, id e&longs;t (ob parallelas IH, PR & angulos æquales IPR, HPZ) ip&longs;arum PS, PH, quarum differentia axem totum 2AC adæquat. Ad SP demittatur perpendicularis QT. Et Hyperbolæ latere recto principali (&longs;eu (2BCq/AC)) dicto L, erit LXQR ad LXPv ut QR ad Pv,id e&longs;t, ut PE &longs;eu AC ad PC; Et LXPv ad GvP ut L ad Gv; & GvP ad Qv quad. ut PCque ad CDq; & (per Corol. ~~2. Lem.~~ ~~VII.) Qv quad. ad Qx quad. punctis Q & P coeuntibus fit ratio æqualitatis; & Qx quad. &longs;eu Qv quad. e&longs;t ad QTque ut EPquead PFq, id e&longs;t ut CAq, ad PFq, &longs;ive (per Lem.~~ XII.) ut CDq,ad CBq: & conjunctis his omnibus rationibus LXQR fit ad QTque ut ACXLXPCqXCDq &longs;eu 2CBqXPCqXCDq ad PCXGvXCDqXCB quad. &longs;ive ut 2PC ad Gv. Sed punctis P & Q cocuntibus æquantur 2PC & Gv. Ergo & his proportionalia LXQR & QTque æquantur. ~~Ducantur hæc æqualia in (SPq/QR). & fiet LXSPque æquale (SPqXQTq/QR). Ergo (per Corol.~~ I

~~& 5 Prop.~~ ~~VI.) vis centripeta reciproce e&longs;t ut LXSPq, id e&longs;t reciproce in ratione duplicata di&longs;tantiæ SP. que E. I.~~

~~DE MOTU CORPORUM~~

~~LIBER PRIMUS.~~

~~Idem aliter.~~

~~Inveniatur vis quæ tendit ab Hyperbolæ centro C. Prodibit hæc di&longs;tantiæ CP proportionalis.~~ ~~Inde vero (per Corol.~~ ~~3 Prop.~~ ~~VII.) vis ad umbilicum S tendens erit ut (PEcub/SPq), hoc e&longs;t, ob datam PE,reciproce ut SPque q.E.I.~~

~~Eodem modo demon&longs;tratur quod corpus, hac vi centripeta in centrifugam ver&longs;a, movebitur in Hyperbola conjugata.~~

~~LEMMA XIII.~~

~~Latus rectum Parabolæ ad verticem quemvis pertinens, e&longs;t quadruplum di&longs;tantiæ verticis illius ab umbilico figuræ. Patet ex Conicis.~~

~~LEMMA XIV.~~

~~Perpendiculum quod ab umbilico Parabolæ ad tangentem ejus demittitur, medium e&longs;t proportionale inter di&longs;tantias umbilici a puncto contactus & a vertice principali figuræ.~~

~~Sit enim AQP Parabola, S umbilicus ejus, A vertex principalis P punctum~~

~~contactus, POordinatim applicata ad diametrum principalem, PMtangens diametro principali occurrens in M, & SN,linea perpendicularis ab umbilico in tangentem.~~ Jungatur AN, & ob æquales MS & SP, MN & NP, MA & AO, parallelæ erunt rectæ AN & OP, & inde triangulum SAN rectangulum erit ad A & &longs;imile triangulis æqualibus SNM, SNP: Ergo PS e&longs;t ad SN,ut SN ad SA. q.E.D.

~~Corol. 1. PSque e&longs;t ad SNque ut PS ad SA.~~

~~Corol. 2. Et ob datam SA, e&longs;t SNque ut PS.~~

~~DE MOTU CORPORUM~~

~~Corol. 3. Et concur&longs;us tangentis cuju&longs;vis PM cum recta SN,quæ ab umbilico in ip&longs;am perpendicularis e&longs;t, incidit in rectam AN,quæ Parabolam tangit in vertice principali.~~

~~PROPOSITIO. XIII. PROBLEMA VIII.~~

~~Moveatur corpus in perimetro Parabolæ: requiritur Lex vis centripetæ tendentis ad umbilicum hujus figuræ.~~

Maneat con&longs;tructio Lemmatis, &longs;itque P corpus in perimetro Parabolæ, & a loco Q in quem corpus proxime movetur, age ip&longs;i SPparallelam QR & perpendicularem QT, necnon Qv tangenti parallelam & occurrentem tum diametro YPG in v, tum di&longs;tantiæ SP in x. Jam ob &longs;imilia triangula Pxv, SPM & æqualia unius latera SM, SP, æqualia &longs;unt alterius latera Px &longs;eu QR & Pv.Sed, ex Conicis, quadratum ordinatæ Qv æquale e&longs;t rectangulo &longs;ub latere recto & &longs;egmento diametri Pv, id e&longs;t (per Lem. ~~XIII.) rectangulo 4 PSXPv, &longs;eu 4 PSXQR; & punctis P & Q coeuntibus, ratio Qv ad Qx per (per Corol.~~ ~~2 Lem.~~ ~~VII.) fit ratio æqualitatis.~~ ~~Ergo Qxquad. eo~~

~~in ca&longs;u, æquale e&longs;t rectangulo 4 PSXQR.E&longs;t autem (ob &longs;imilia triangula QxT, SPN) Qxquead QTque ut PSque ad SNquehoc e&longs;t (per Corol.~~ ~~1. Lem.~~ ~~XIV.) ut PS ad SA, id e&longs;t ut 4 PSXQRad 4SAXQR, & inde (per Prop.~~ ~~IX. Lib.~~ ~~v. Elem.) QTque & 4SAXQR æquantur.~~ ~~Ducantur hæc æqualia in (SPq./QR), & fiet (SPq.XQTq./QR) æquale SPq.X4SA: & propterea (per Corol.~~ ~~1 & 5 Prop.~~ ~~VI.) vis centripeta e&longs;t reciproce ut SPq.X4SA, id e&longs;t, ob datam 4SA, reciproce in duplicata ratione di&longs;tantiæ SP. q.E.I.~~

~~Corol. 1. Ex tribus novi&longs;&longs;imis Propo&longs;itionibus con&longs;equens e&longs;t, quod~~

&longs;i corpus quodvis P, &longs;ecundum lineam quamvis rectam PR, quacunque cum velocitate exeat de loco P, & vi centripeta quæ &longs;it reciproce proportionalis quadrato di&longs;tantiæ locorum a centro, &longs;imul agitetur; movebitur hoc corpus in aliqua &longs;ectionum Conicarum umbilicum habente in centro virium; & contra. ~~Nam datis umbilico & puncto contactus & po&longs;itione tangentis, de&longs;cribi pote&longs;t &longs;ectio Conica quæ curvaturam datam ad punctum illud habebit.~~ ~~Datur autem curvatura ex data vi centripeta: & Orbes duo &longs;e mutuo tangentes, eadem vi centripeta de&longs;cribi non po&longs;&longs;unt.~~

~~LIBER PRIMUS.~~

Corol. 2. Si velocitas, quacum corpus exit de loco &longs;uo P, ea &longs;it, qua lineola PR in minima aliqua temporis particula de&longs;cribi po&longs;&longs;it, & vis centripeta potis &longs;it eodem tempore corpus idem movere per &longs;patium QR: movebitur hoc corpus in Conica aliqua &longs;ectione, cujus latus rectum principale e&longs;t quantitas illa (QTq./QR) quæ ultimo fit ubi lineolæ PR, QR in infinitum diminuuntur. ~~Circulum in his Corollariis refero ad Ellip&longs;in, & ca&longs;um excipio ubi corpus recta de&longs;cendit ad centrum.~~

~~PROPOSITIO XIV. THEOREMA VI.~~

Si corpora plura revolvantur circa centrum commune, & vis centripeta &longs;it reciproce in duplicata ratione di&longs;tantiæ locorum a centro; dico quod Orbium Latera recta principalia &longs;unt in duplicata ratioone arearum quas corpora, radiis ad centrum ductis, eodem tempore de&longs;cribunt.

~~Nam, per Corol.~~ ~~2. Prop.~~ XIII, Latus rectum L æquale e&longs;t quantitati (QTq./QR) quæ ultimo fit ubi coeunt puncta P & que Sed linea minima QR, dato tempore, e&longs;t ut vis centripeta generans, hoc e&longs;t (per Hypothe&longs;in) reciproce ut SPque Ergo (QTq./QR) e&longs;t ut QTq.XSPque hoc e&longs;t, latus rectum L in duplicata ratione areæ QTXSP. q.E.D.

~~DE MOTU CORPORUM~~

Corol. Hinc Ellip&longs;eos area tota, eique proportionale rectangulum &longs;ub axibus, e&longs;t in ratione compo&longs;ita ex &longs;ubduplicata ratione lateris recti & ratione temporis periodici. ~~Namque area tota e&longs;t ut area QTXSP, quæ dato tempore de&longs;cribitur, ducta in &c.~~ ~~ducta in tempus periodicum.~~

~~PROPOSITIO XV. THEOREMA VII.~~

~~Ii&longs;dem po&longs;itis, dico quod Tempora periodica in Ellip&longs;ibus &longs;unt in ratione &longs;e&longs;quiplicata majorum axium.~~

Namque axis minor e&longs;t medius proportionalis inter axem majorem & latus rectum, atque adeo rectangulum &longs;ub axibus e&longs;t in ratione compo&longs;ita ex &longs;ubduplicata ratione lateris recti & &longs;e&longs;quiplicata ratione axis majoris. ~~Sed hoc rectangulum, per Corollarium Prop.~~ ~~XIV. e&longs;t in ratione compo&longs;ita ex &longs;ubduplicata ratione lateris recti & ratione periodici temporis.~~ ~~Dematur utrobique &longs;ubduplicata ratio lateris recti, & manebit &longs;e&longs;quiplicata ratio majoris axis æqualis rationi periodici temporis. q.E.D.~~

~~Corol. Sunt igitur tempora periodica in Ellip&longs;ibus eadem ac in Circulis, quorum diametri æquantur majoribus axibus Ellip&longs;eon.~~

~~PROPOSITIO XVI. THEOREMA VIII.~~

Ii&longs;dem po&longs;itis, & actis ad corpora lineis rectis, quæ ibidem tangant Orbitas, demi&longs;&longs;i&longs;que ab umbilico communi ad has tangentes perpendicularibus: dico quod Velocitates corporum &longs;unt in ratione compo&longs;ita ex ratione perpendiculorum inver&longs;e & &longs;ubduplicata ratione laterum rectorum principalium directe.

Ab umbilico S ad tangentem PR demitte perpendiculum SY& velocitas corporis P erit reciproce in &longs;ubduplicata ratione quantitatis (SYq/L). Nam velocitas illa e&longs;t ut arcus quam minimus PQin data temporis particula de&longs;criptus, hoc e&longs;t (per Lem. VII.) ut tangens PR, id e&longs;t (ob proportionales PR ad QT & SP ad SY) ut (SPXQT/SY), &longs;ive ut SY reciproce & SPXQT directe; e&longs;tque SPXQT ut area dato tempore de&longs;cripta, id e&longs;t, per Prop. ~~XIV.~~

~~in &longs;ubduplicata ratione lateris recti. q.E.D.~~

~~LIBER PRIMUS.~~

~~Corol. 1. Latera recta principalia &longs;unt in ratione compo&longs;ita ex duplicata ratione perpendiculorum & duplicata ratione velocitatum.~~

Corol. 2. Velocitates corporum in maximis & minimis ab umbilico communi di&longs;tantiis, &longs;unt in ratione compo&longs;ita ex ratione di&longs;tantiarum inver&longs;e & &longs;ubduplicata ratione laterum rectorum principalium directe. ~~Nam perpendicula jam &longs;unt ip&longs;æ di&longs;tantiæ.~~

Corol. 3. Ideoque velocitas in Conica &longs;ectione, in maxima vel minima ab umbilico di&longs;tantia, e&longs;t ad velocitatem in Circulo in eadem à centro di&longs;tantia, in &longs;ubduplicata ratione lateris recti principalis ad duplam illam di&longs;tantiam.

Corol. 4. Corporum in Ellip&longs;ibus gyrantium velocitates in mediocribus di&longs;tantus ab umbilico communi &longs;unt eædem quæ corporum gyrantium in Circulis ad ea&longs;dem di&longs;tantias; hoc e&longs;t (per Corol 6. Prop. ~~IV.) reciproce in &longs;ubduplicata ratione di&longs;tantiarum.~~ ~~Nam perpendicula jam &longs;unt &longs;emi-axes minores; & hi &longs;unt ut mediæ proportionales inter di&longs;tantias & latera recta.~~ ~~Componatur hæc ratio inver&longs;e cum &longs;ubduplicata ratione laterum rectorum directe, & fiet ratio &longs;ubduplicata di&longs;tantiarum inver&longs;e.~~

Corol. 5. In eadem figura, vel etiam in figuris diver&longs;is, quarum latera recta principalia &longs;unt æqualia, velocitas corporis e&longs;t reciproce ut perpendiculum demi&longs;&longs;um ab umbilico ad tangentem.

~~DE MOTU CORPORUM~~

Corol. 6. In Parabola, velocitas e&longs;t reciproce in &longs;ubduplicata ratione di&longs;tantiæ corporis ab umbilico figuræ; in Ellip&longs;i magis variaur, in Hyperbola minus, quam in hac ratione. ~~Nam (per Corol.~~ ~~2. Lem.~~ ~~XIV.) perpendiculum demi&longs;&longs;um ab umbilico ad tangentem Parabolæ e&longs;t in &longs;ubduplicata ratione di&longs;tantiæ.~~ ~~In Hyperbola perpendiculum minus variatur, in Ellip&longs;i magis.~~

Corol. 7. In Parabola, velocitas corporis ad quamvis ab umbilico di&longs;tantiam, e&longs;t ad velocitatem corporis revolventis in Circulo ad eandem a centro di&longs;tantiam, in &longs;ubduplicata ratione numeri binarii ad unitatem; in Ellip&longs;i minor e&longs;t, in Hyperbola major quam in hac ratione. Nam per hujus Corollarium &longs;ecundum, velocitas in vertice Parabolæ e&longs;t in hac ratione, & per Corollaria &longs;exta hujus & Propo&longs;itionis quartæ, &longs;ervatur eadem proportio in omnibus di&longs;tantiis. ~~Hinc etiam in Parabola velocitas ubique æqualis e&longs;t velocitati corporis revolventis in Circulo ad dimidiam di&longs;tantiam, in Ellip&longs;i minor e&longs;t, in Hyperbola major.~~

Corol. 8. Velocitas gyrantis in Sectione quavis Conica e&longs;t ad velocitatem gyrantis in Circulo in di&longs;tantia dimidii lateris recti principalis Sectionis, ut di&longs;tantia illa ad perpendiculum ab umbilico in tangentem Sectionis demi&longs;&longs;um. ~~Patet per Corollarium quintum.~~

~~Corol. 9. Unde cum (per Corol.~~ ~~6. Prop.~~ IV.) velocitas gyrantis in hoc Circulo &longs;it ad velocitatem gyrantis in Circulo quovis alio, reciproce in &longs;ubduplicata ratione di&longs;tantiarum; fiet ex æquo velocitas gyrantis in Conica &longs;ectione ad velocitatem gyrantis in Circulo in eadem di&longs;tantia, ut media proportionalis inter di&longs;tantiam illam communem & &longs;emi&longs;&longs;em principalis lateris recti &longs;ectionis, ad perpendiculum ab umbilico communi in tangentem &longs;ectionis demi&longs;&longs;um.

~~PROPOSITIO XVII. PROBLEMA. IX.~~

Po&longs;ito quod vis centripeta &longs;it reciproce proportionalis quadrato di&longs;tan&longs;tantiæ locorum a centro, & quod vis illius quantitas ab&longs;oluta &longs;it cognita; requiritur Linea quam corpus de&longs;cribit, de loco dato, cum data velocitate, &longs;ecundum datam rectam egrediens.

Vis centripeta tendens ad punctum S ea &longs;it qua corpus p in orbita quavis data pq gyretur, & cogno&longs;catur hujus velocitas in loco p. De loco P, &longs;ecundum lineam PR, exeat corpus P, cum data velo citate, & mox inde, cogente vi centripeta, deflectat illud in Coni&longs;ectionem Pque Hanc igitur recta PR tanget in P. Tangat itidem recta aliqua pr Orbitam pq in p, & &longs;i ab S ad eas tangentes demitti intelligantur perpendicula, erit (per Corol. ~~1. Prop.~~ XVI.) latus rectum principale Coni&longs;ectionis ad latus rectum principale Orbitæ, in ratione compo&longs;ita ex duplicata ratione perpendiculorum & duplicata ratione velocitatum, atque adeo datur. ~~Sit i&longs;tud L. Datur præterea Coni&longs;e~~

~~ctionis umbilicus S.Anguli RPS complementum ad duos rectos fiat angulus RPH, & dabitur po&longs;itione linea PH, in qua umbilicus alter H locatur.~~ Demi&longs;&longs;o ad PH perpendiculo SK, erigi intelligatur &longs;emiaxis conjugatus BC, & erit SPq.-2KPH+PHq.=SHq.=4CHq.=4BHq-4BCq.= ―SP+PH: quad. -LX―SP+PH=SPq.+2SPH+PHq. -LX―SP+PH. Addantur utrobique 2KPH-SPq-PHq +LX―SP+PH, & fiet LX―SP+PH=2SPH+2KPH,&longs;eu SP+PH, ad PH, ut 2SP+2KP ad L. Unde datur PHtam longitudine quam po&longs;itione. ~~Nimirum &longs;i ea fit corporis &c.~~ in Pvelocitas, ut latus rectum L minus fuerit quam 2 SP+2KP,jacebit PH ad eandem partem tangentis PR cum linea PS,adeoque figura erit Ellip&longs;is, & ex datis umbilicis S, H, & axe principali SP+PH, dabitur: Sin tanta &longs;it corporis velocitas ut latus rectum L æquale fuerit 2 SP+2KP, longitudo PH infinita erit, & propterea figura erit Parabola axem habens SH parallelum lineæ PK, & inde dabitur. Quod &longs;i corpus majori adhuc cum velocitate de loco &longs;uo P exeat, capienda erit longitudo PHad alteram partem tangentis, adeoque tangente inter umbilicos pergente, figura erit Hyperbola axem habens principalem æqualem differentiæ linearum SP & PH, & inde dabitur. q.E.I.

~~LIBER PRIMUS.~~

Corol. 1. Hinc in omni Coni&longs;ectione ex dato vertice principali D,latere recto L, & umbilico S, datur umbilicus alter H capiendo DH,ad DS ut e&longs;t latus rectum ad differentiam inter latus rectum & 4 DS. Nam proportio SP+PH ad PH ut 2 SP+2KP ad L, in ca&longs;u hujus Corollarii, &longs;it DS+DH ad DH ut 4 DS ad L, & divi&longs;im DS ad DH ut 4 DS-L ad L.

~~DE MOTU CORPORUM~~

Corol. 2. Unde &longs;i datur corporis velocitas in vertice principali D,invenietur Orbita expedite, capiendo &longs;cilicet latus rectum ejus, ad duplam di&longs;tantiam DS, in duplicata ratione velocitatis hujus datæ ad velocitatem corporis in Circulo, ad di&longs;tantiam DS, gyrantis (per Corol. ~~3. Prop.~~ ~~XVI.) dein DH ad DS ut latus rectum ad differentiam inter latus rectum & 4 DS.~~

Corol. 3. Hinc etiam &longs;i corpus moveatur in Sectione quacunque Conica, & ex Orbe &longs;uo impul&longs;u quocunque exturbetur; cogno&longs;ci pote&longs;t Orbis in quo po&longs;tea cur&longs;um &longs;uum peraget. Nam componendo proprium corporis motum cum motu illo quem impul&longs;us &longs;olus generaret, habebitur motus quocum corpus de dato impul&longs;us loco, &longs;ecundum rectam po&longs;itione datam, exibit.

Corol. 4. Et &longs;i corpus illud vi aliqua extrin&longs;ecus impre&longs;&longs;a continuo perturbetur, innote&longs;cet cur&longs;us quam proxime, colligendo mutationes quas vis illa in punctis quibu&longs;dam inducit, & ex &longs;eriei analogia mutationes continuas in locis intermediis æ&longs;timando.

~~Scholium.~~

~~Si corpus P vi centripeta ad~~

punctum quodcunque datum Rtendente moveatur in perimetro datæ cuju&longs;cunque Sectionis conicæ cujus centrum &longs;it C, & requiratur Lex vis centripetæ: ducatur CG radio RP parallela, & Orbis tangenti PG occurrens in G; & vis illa (per Corol. ~~1 & Schol.~~ ~~Prop.~~ ~~X, & Corol.~~ ~~3 Prop.~~ ~~VII.) erit ut (CG cub./RP quad.)~~

~~LIBER PRIMUS.~~

~~SECTIO IV.~~

~~De Inventione Orbium Ellipticorum, Parabolicorum & Hyperbolicorum ex umbilico dato.~~

~~LEMMA XV.~~

Si ab Ellip&longs;eos vel Hyperbolæ cuju&longs;vis umbilicis duobus S, H, ad punctum quodvis tertium V inflectantur rectæ duæ SV, HV, quarum una HV æqualis &longs;it axi principali figuræ, altera SV a perpendiculo TR in &longs;e demi&longs;&longs;o bi-

~~&longs;ecetur in T; perpendiculum illudTR &longs;ectionem Conicam alicubi tanget: & contra, &longs;i tangit, erit HV æqualis axi principali figuræ.~~

Secet enim perpendiculum TR rectam HV productam, &longs;i opus fuerit, in R; & jungatur SR. Ob æquales TS, TV, æquales erunt & rectæ SR, VR & anguli TRS, TRV.Unde punctum R erit ad Sectionem Conicam, & perpendiculum TR tanget eandem: & contra. q.E.D.

~~PROPOSITIO XVIII. PROBLEMA X.~~

~~Datis umbilico & axibus principalibus de&longs;cribere Trajectorias Ellipticas & Hyperbolicas, quæ tran&longs;ibunt per puncta data, & rectas po&longs;itione datas contingent.~~

~~Sit S communis umbilicus figurarum; AB longitudo axis principalis Trajectoriæ cuju&longs;vis; P punctum per quod Trajectoria debet tran&longs;ire; & TR recta quam debet tangere.~~ Centro P intervallo AB-SP, &longs;i orbita &longs;it Ellip&longs;is, vel AB+SP, &longs;i ea &longs;it Hyperbola, de&longs;cribatur circulus HG. Ad tangentem TR demittatur perpendiculum ST, & producatur idem ad V, ut &longs;it TV æqualis ST; centroque V & intervallo AB de&longs;cribatur circulus FH. Hac methodo &longs;ive dentur duo puncta P, p, &longs;ive duæ tangentes TR, tr, &longs;ive punctum P & tangens

~~TR, de&longs;cribendi &longs;unt circuli duo.~~ ~~Sit H eorum inter&longs;ectio communis, & umbilicis S, H, axe illo dato de&longs;cribatur Trajectoria.~~ ~~Dico factum.~~ Nam Trajectoctoria de&longs;cripta (eo quod PH +SP in Ellip&longs;i, & PH-SPin Hyperbola æquatur axi) tran&longs;ibit per punctum P, & (per Lemma &longs;uperius) tanget rectam TR. Et eodem argumento vel tran&longs;ibit eadem per puncta duo P, p, vel tanget rectas duas TR, tr. ~~q.E.F.~~

~~DE MOTU CORPORUM~~

~~PROPOSITIO XIX. PROBLEMA XI.~~

~~Circa datum umbilicum Trajectoriam Parabolicam de&longs;cribere, quæ tran&longs;ibit per puncta data, & rectas po&longs;itione datas continget.~~

~~Sit S umbilicus, P punctum & TR tangens Trajectoriæ de&longs;cribendæ.~~ ~~Centro P, intervallo PS de&longs;cribe cir~~

culum FG. Ab umbilico ad tangentem demitte perpendicularem ST, & produc eam ad V,ut &longs;it TV æqualis ST. Eodem modo de&longs;cribendus e&longs;t alter circulus fg, &longs;i datur alterum punctum p; vel inveniendum alterum punctum v, &longs;i datur altera tangens tr; dein ducenda rea IF quæ tangat duos circulos FG, fg &longs;i dantur duo puncta P, p, vel tran&longs;eat per duo puncta V, v, &longs;i dantur duæ tangentes TR, tr, vel tangat circulum FG & tran&longs;eat per punctum V,&longs;i datur punctum P & tangens TR. Ad FI demitte perpendicularem SI, eamque bi&longs;eca in K; & axe SK, vertice principali K de&longs;cribatur Parabola. ~~Dico factum.~~ ~~Nam Parabola, ob æquales SK & IK, SP & FP, tran&longs;ibit per punctum P; & (per Lemmatis XIV. Corol.~~ ~~3.) ob æquales ST & TV & angulum rectum STR, tanget rectam TR. q.E.F.~~

~~LIBER PRIMUS.~~

~~PROPOSITIO XX. PROBLEMA XII.~~

~~Circa datum umbilicum Trajectoriam quamvis &longs;pecie datam de&longs;cribere, quæ per data puncta tran&longs;ibit & rectas tanget pofitione datas.~~

~~Cas. 1. Dato umbilico S, de&longs;cribenda &longs;it Trajectoria ABC per puncta duo B, C. Quoniam Trajectoria datur &longs;pecie, dabitur ratio axis principalis ad di&longs;tantiam~~

~~umbilicorum.~~ In ea ratione cape KB ad BS, & LC ad CS. Centris B, C, intervallis BK, CL, de&longs;cribe circulos duos, & ad rectam KL, quæ tangat eo&longs;dem in K & L, demitte perpendiculum SG, idemque &longs;eca in A & a, ita ut &longs;it GA ad AS & Ga ad aS, ut e&longs;t KB ad BS, & axe &c. Aa, verticibus A, a, de&longs;cribatur Trajectoria. ~~Dico factum.~~ ~~Sit enim H umbilicus alter Figuræ de&longs;criptæ, & cum &longs;it GA ad AS ut Ga ad aS, erit divi&longs;im Ga-GA &longs;eu Aa ad aS-AS &longs;eu SH in eadem &c.~~ ratione, adeoque in ratione quam habet axis principalis Figuræ de&longs;cribendæ ad di&longs;tantiam umbilicorum ejus; & propterea Figura de&longs;cripta e&longs;t eju&longs;dem &longs;peciei cum de&longs;cribenda. ~~Cumque &longs;int KB ad BS & LCad CS in eadem ratione, tran&longs;ibit hæc Figura per puncta B, C, ut ex Conicis manife&longs;tum e&longs;t.~~

~~Cas. 2. Dato umbilico S, de&longs;cribenda &longs;it Trajectoria quæ rectas duas TR, tr alicubi contingat.~~ ~~Ab umbilico in tangentes demitte perpendicula ST, St & produc ea~~

dem ad V, v, ut &longs;int TV, tv æquales TS, tS. Bi&longs;eca Vv in O,& erige perpendiculum infinitum OH, rectamque VS infinite productam &longs;eca in K & k ita, ut &longs;it VK ad KS & Vk ad kS ut e&longs;t Trajectoriæ de&longs;cribendæ axis principalis ad umbilicorum di&longs;tantiam. ~~Super diametro Kk de&longs;cribatur circulus &longs;ecans OH in H; & umbilicis S, H, axe principali ip&longs;am VH æquante, de&longs;cribatur Trajectoria.~~ ~~Dico factum.~~ Nam bi&longs;eca Kk in X, & junge HX, HS, HV, Hv. Quoniam e&longs;t VK ad KSut Vk ad kS; & compofite ut VK+Vk ad KS+kS; divi&longs;imque ut Vk-VK ad kS-KS, id e&longs;t ut 2 VX ad 2 KX & 2 KX ad 2 SX, adeoque ut VX ad HX & HX ad SX, &longs;imilia erunt triangula VXH, HXS, & propterea VH erit ad SH ut VX ad XH,adeoque ut VK ad KS. Habet igitur Trajectoriæ de&longs;criptæ axis principalis VH eam rationem ad ip&longs;ius umbilicorum di&longs;tantiam SH,quam habet Trajectoriæ de&longs;cribendæ axis principalis ad ip&longs;ius umbilicorum di&longs;tantiam, & propterea eju&longs;dem e&longs;t &longs;peciei. ~~In&longs;uper cum VH, vH æquentur axi principali, & VS, vS a rectis TR, trperpendiculariter bi&longs;ecentur, liquet, ex Lemmate XV, rectas illas Trajectoriam de&longs;criptam tangere. q.E.F.~~

~~DE MOTU CORPORUM~~

Cas. 3. Dato umbilico S de&longs;cribenda &longs;it Trajectoria quæ rectam TR tanget in puncto dato R. In rectam TR demitte perpendicularem ST, & produc eandem ad V, ut &longs;it TV æqualis ST. Junge VR, & rectam VS infinite productam &longs;eca in K & k, ita ut &longs;it VK ad SK & Vk ad Sk ut Ellip&longs;eos de&longs;cribendæ axis principalis ad di&longs;tantiam umbilicorum; circuloque &longs;uper diametro Kk de&longs;cripto, &longs;ecetur producta recta VR in H, & umbilicis S, H, axe principali rectam VH æquante, de&longs;cribatur Trajectoria. ~~Dico factum.~~ ~~Namque VH e&longs;&longs;e ad~~

SH ut VK ad SK, atque adeo ut axis principalis Trajectoriæ de&longs;cribendæ ad di&longs;tantiam umbilicorum ejus, patet ex demon&longs;lratis in Ca&longs;u &longs;ecundo, & propterea Trajectoriam de&longs;criptam eju&longs;dem e&longs;&longs;e &longs;peciei cum de&longs;cribenda; rectam vero TR qua angulus VRS bi&longs;ecatur, tangere Trajectoriam in puncto R, patet ex Conicis. q.E.F.

Cas. 4. Circa umbilicum S de&longs;cribenda jam &longs;it Trajectoria APB,quæ tangat rectam TR, tran&longs;eatque per punctum quodvis P extra tangentem datum, quæque &longs;imilis &longs;it Figuræ apb, axe principali ab & umbilicis s, h de&longs;criptæ. In tangentem TR demitte perpendiculum ST, & produc idem ad V, ut &longs;it TV æqualis ST. Angulis autem VSP, SVP fac angulos hsq, shq æquales; centroque q & intervallo quod &longs;it ad ab ut SP ad VS de&longs;cribe circulum &longs;ecantem Figuram apb in p. Junge sp & age SH quæ &longs;it ad sh ut e&longs;t SP ad sp, quæque angulum PSH angulo psh & angulum VSH angulo psq æquales con&longs;tituat. ~~Denique umbilicis S, H,& axe principali AB di&longs;tantiam VH æquante, de&longs;cribatur &longs;ectio Conica.~~ ~~Dico factum.~~ Nam &longs;i agatur sv quæ &longs;it ad sp ut e&longs;t sh ad sq, quæque con&longs;tituat angulum vsp angulo hsq & angulum vsh angulo psq æquales, triangula svh, spq erunt &longs;imilia, & propterea vh erit ad pq ut e&longs;t sh ad sq, id e&longs;t (ob &longs;imilia triangula

VSP, hsq) ut e&longs;t VS ad SP &longs;eu ab ad pque Æquantur ergo vh & ab. Porro ob &longs;imilia triangula VSH. vsh, e&longs;t VH ad SH ut vh ad sh, id e&longs;t, axis Conicæ &longs;ectionis jam de&longs;criptæ ad illius umbilicorum intervallum, ut axis ab ad umbilicorum intervallum sh; & propterea Figura jam de&longs;eripta &longs;imilis e&longs;t Figuræ apb. Tran&longs;it autem hæc Figura per punctum P, eo quod triangulum PSH &longs;imile &longs;it triangulo psh; & quia VH æquatur ip&longs;ius axi & VS bi&longs;ecatur perpendiculariter a recta TR, tangit eadem rectam TR. q.E.F.

~~LIBER PRIMUS.~~

~~LEMMA XVI.~~

~~A datis tribus punctis ad quartum non datum inflectere tres rectas quarum differentiæ vel dantur vel nullæ &longs;unt.~~

Cas. 1. Sunto puncta illa data A, B, C & punctum quartum Z,quod invenire oportet; Ob datam differentiam linearum AZ, BZ,locabitur punctum Z in Hyperbola cujus umbilici &longs;unt A & B, & principalis axis differentia illa data. Sit axis ille MN. Cape PM. ad MA ut e&longs;t MN ad AB, & erecta PR perpendiculari ad AB,demi&longs;&longs;aque ZR perpendiculari ad PR; erit, ex natura hujus Hyperbolæ, ZR ad AZ ut e&longs;t MN ad AB. Simili di&longs;cur&longs;u punctum Z locabitur in alia Hyperbola, cujus umbilici &longs;unt A, C & principalis axis differentia inter AZ & CZ, ducique pote&longs;t QS ip&longs;i ACperpendicularis, ad quam &longs;i ab Hyperbolæ hujus puncto quovis Zdemittatur normalis ZS, hæc fuerit ad AZ ut e&longs;t differentia inter AZ & CZ ad AC. Dantur ergo rationes ip&longs;arum ZR & ZSad AZ, & idcirco datur earun

~~dem ZR & ZS ratio ad invicem; ideoque &longs;i rectæ RP, SQ concurrant in T, & agatur TZ, figura TRZS, dabitur &longs;pecie, & recta TZ in qua punctum Z alicubi locatur, dabitur po&longs;itione.~~ ~~Eadem methodo per Hyperbolam tertiam, cujus umbilici &longs;unt B & C& axis principalis differentia rectarum BZ, CZ, inveniri pote&longs;t alia recta in qua punctum Z locatur.~~ ~~Habitis autem duobus Locis rectilineis, habetur punctum quæ&longs;itum Z in eorum inter&longs;ectione. q.E.I.~~

~~DE MOTU CORPORUM~~

~~Cas. 2. Si duæ ex tribus lineis, puta AZ & BZ æquantur, punctum Z locabitur in perpendiculo bi&longs;ecante di&longs;tantiam AB, & locus alius rectilineus invenietur ut &longs;upra. q.E.I.~~

~~Cas. 3. Si omnes tres æquantur, locabitur punctum Z in centro Circuli per puncta A, B, C tran&longs;euntis. q.E.I.~~

~~Solvitur etiam hoc Lemma problematicum per Librum Tactionum Apollonii a Vieta re&longs;titutum.~~

~~PROPOSITIO XXI. PROBLEMA XIII.~~

~~Trajectoriam circa datum umbilicum de&longs;cribere, quæ tran&longs;ibit per puncta data & rectas po&longs;itione datas continget.~~

Detur umbilicus S, punctum P, & tangens TR, & inveniendus &longs;it umbilicus alter H. Ad tangentem demitte perpendiculum ST, & produc idem ad Y, ut &longs;it TY æqualis ST, & erit YH æqualis axi principali. ~~Junge SP, HP, & erit SP differentia inter HP & axem principalem.~~ ~~Hoc modo &longs;i dentur plures tangen- tes TR, vel plura puncta P, devenietur &longs;emper ad lineas totidem YH, vel PH, a dictis punctis Y vel~~

P ad umbilicum H ductas, quæ vel æquantur axibus, vel datis longitudinibus SP differunt ab ii&longs;dem, atque adeo quæ vel æquantur &longs;ibi invicem, vel datas habent differentias; & inde, per Lemma &longs;uperius, datur umbilicus ille alter H. Habitis autem umbilicis una cum axis longitudine (quæ vel e&longs;t YH; vel, &longs;i Trajectoria Ellip&longs;is e&longs;t, PH+SP; &longs;in Hyperbola, PH-SP) habetur Trajectoria. q.E.I.

~~LIBER PRIMUS.~~

~~Scholium.~~

~~Ca&longs;us ubi dantur tria puncta &longs;ic &longs;olvitur expeditius.~~ Dentur puncta B, C, D. Junctas BC, CD produc ad E, F, ut &longs;it EB ad EC ut SB ad SC, & FC ad FD ut SC ad SD. Ad EF ductam & productam demitte normales SG, BH, inque GS infinite producta cape GA ad AS & Ga ad aS ut e&longs;t HB ad BS; & erit A vertex, & Aa axis principalis Trajectoriæ: quæ, perinde ut GAmajor, æqualis, vel minor fuerit quam AS, erit Ellip&longs;is, Parabola vel Hyperbola; pun

cto a in primo ca&longs;u cadente ad eandem partem lineæ GFcum puncto A; in &longs;ecundo ca&longs;u abeunte in infinitum; in tertio cadente ad contrariam partem lineæ GF.Nam &longs;i demittantur ad GF perpendicula CI, DK; erit IC ad HB ut EC ad EB, hoc e&longs;t, ut SC ad SB; & vici&longs;&longs;im IC ad SC ut HB ad SB &longs;ive ut GA ad SA. Et &longs;imili argumento probabitur e&longs;&longs;e KD ad SD in eadem ratione. Jacent ergo puncta B, C, D in Coni&longs;ectione circa umbilicum S ita de&longs;cripta, ut rectæ omnes ab umbilico S ad &longs;ingula Sectionis puncta ductæ, &longs;int ad perpendicula a punctis ii&longs;dem ad rectam GF demi&longs;&longs;a in data illa ratione.

~~Methodo haud multum di&longs;&longs;imili hujus problematis &longs;olutionem tradit Clari&longs;&longs;imus Geometra de la Hire, Conicorum &longs;uorum Lib.~~ ~~VIII. Prop.~~ ~~XXV.~~

~~DE MOTU CORPORUM~~

~~SECTIO V.~~

~~Inventio Orbium ubi umbilicus neuter datur.~~

~~LEMMA XVII.~~

Si a datæ Conicæ Sectionis puncto quovis P, ad Trapezii alicujusABDC, in Conica illa &longs;ectione in&longs;cripti, latera quatuor infinite producta AB, CD, AC, DB, totidem rectæ PQ, PR, PS, PT in datis angulis ducantur, &longs;ingulæ ad &longs;ingula: rectangulum ductarum ad oppo&longs;ita duo latera PQXPR, erit ad rectangulum ductarum ad alia duo latera oppo&longs;ita PSXPT in data ratione.

~~Cas. 1. Ponamus primo lineas ad~~

oppo&longs;ita latera ductas parallelas e&longs;&longs;e alterutri reliquorum laterum, puta PQ & PR lateri AC, & PSac PT lateri AB. Sintque in&longs;uper latera duo ex oppo&longs;itis, puta AC& BD, &longs;ibi invicem parallela. Et recta quæ bi&longs;ecat parallela illa latera erit una ex diametris Conicæ &longs;ectionis, & bi&longs;ecabit etiam Rque Sit O punctum in quo RQ bi&longs;ecatur, & erit PO ordinatim applicata ad diametrum illam. ~~Produc PO ad K ut &longs;it OK æqualis PO, & erit OK ordinatim applicata ad contrarias partes diametri.~~ ~~Cum igitur puncta A, B, P & K &longs;int ad Conicam &longs;ectionem, & PK &longs;ecet AB in dato angulo, erit (per Prop.17 & 18 Lib.~~ ~~III Conicorum Apollonii) rectangulum PQK ad rectangulum AQB in data ratione.~~ Sed QK & PRæquales &longs;unt, utpote æqualium OK, OP, & OQ, OR differentiæ, & inde etiam rectangula PQK & PQXPR æqualia &longs;unt; atque adeo rectangulum PQXPR e&longs;t ad rectangulum AQB, hoc e&longs;t ad rectangulum PSXPT in data ratione. q.E.D.

~~Cas. 2. Ponamus jam Trapezii latera oppo&longs;ita AC & BD non e&longs;&longs;e parallela.~~ ~~Age Bd parallelam AC & occurrentem tum rectæ ST in t, tum Conicæ &longs;ectioni in d. Junge Cd &longs;ecantem PQ in r,& ip&longs;i PQ parallelam age DM~~

&longs;ecantem Cd in M & AB in N.Jam ob &longs;imilia triangula BTt, DBN; e&longs;t Bt &longs;eu PQ ad Tt ut DN ad NB. Sic & Rr e&longs;t ad AQ &longs;eu PS ut DM ad AN.Ergo, ducendo antecedentes in antecedentes & con&longs;equentes in con&longs;equentes, ut rectangulum PQin Rr e&longs;t ad rectangulum PS in Tt, ita rectangulum NDM e&longs;t ad rectangulum ANB, & (per Ca&longs;.1) ita rectangulum PQ in Pr e&longs;t ad rectangulum PS in Pt, ac divi&longs;im ita rectangulum PQXPRe&longs;t ad rectangulum PSXPT. q.E.D.

~~LIBIR PRIMUS.~~

~~Cas. 3. Ponamus denique lineas~~

~~quatuor PQ, PR, PS, PT non e&longs;&longs;e parallelas lateribus AC, AB,&longs;ed ad ea utcunque inclinatas.~~ Earum vice age Pq, Pr parallelas ip&longs;i AC; & Ps, Pt parallelas ip&longs;i AB; & propter datos angulos triangulorum PQq, PRr, PSs, PTt, dabuntur rationes PQ ad Pq, PR ad Pr, PSad Ps, & PT ad Pt; atque adeo rationes compo&longs;itæ PQXPRad PqXPr, & PSXPT ad PsXPt. Sed, per &longs;uperius demon&longs;trata, ratio PqXPr ad PsXPt data e&longs;t: Ergo & ratio PQXPR ad PSXPT. q.E.D.

~~LEMMA XVIII.~~

Ii&longs;dem po&longs;itis, &longs;i rectangulum ductarum ad oppo&longs;ita duo latera Trapezii PQXPR &longs;it adrectangulum ductarum ad reliqua duo latera PSXPT in data ratione; punctum P, a quo lineæ ducuntur, tanget Conicam &longs;ectionem circa Trapezium de&longs;criptam.

~~Per puncta A, B, C, D & aliquod infinitorum punctorum P, pu ta p, concipe Conicam &longs;ectionem de&longs;cribi: dico punctum P hanc &longs;emper tangere.~~ ~~Si negas,~~

junge AP &longs;ecantem hanc Conicam &longs;ectionem alibi quam in P, &longs;i fieri pote&longs;t, puta in b. Ergo &longs;i ab his punctis p & b ducantur in datis angulis ad latera Trapezii rectæ pq, pr, ps, pt& bk, br, b&longs;, bd; erit ut bkXbr ad b&longs;Xbd ita (per Lem. XVII) pqXprad psXpt, & ita (per Hypoth.) PQXPR ad PSXPT. E&longs;t & propter &longs;imilitudinem Trapeziorum bkA&longs;, PQAS, ut bk ad b&longs; ita PQ ad PS. Quare, applicando terminos prioris proportionis ad terminos corre&longs;pondentes hujus, erit br ad bd ut PR ad PT. Ergo Trapezia æquiangula Dr bd, DRPT &longs;imilia &longs;unt, & eorum diagonales Db, DP propterea coincidunt. ~~Incidit itaque b in inter&longs;ectionem rectarum AP, DP adeoque coincidit cum puncto P. Quare punctum P, ubicunque &longs;umatur, incidit in a&longs;&longs;ignatam Conicam &longs;ectionem. q.E.D.~~

~~DE MOTU CORPORUM~~

Corol. Hinc &longs;i rectæ tres PQ, PR, PS a puncto communi Pad alias totidem po&longs;itione datas rectas AB, CD, AC, &longs;ingulæ ad &longs;ingulas, in datis angulis ducantur, &longs;itque rectangulum &longs;ub duabus ductis PQXPR ad quadratum tertiæ PS quad. in data ratione: punctum P, a quibus rectæ ducuntur, locabitur in &longs;ectione Conica quæ tangit lineas AB, CD in A & C; & contra. Nam coeat linea BD cum linea AC manente po&longs;itione trium AB, CD, AC; dein coeat etiam linea PT cum linea PS: & rectangulum PSXPTevadet PS quad. rectæque AB, CD quæ curvam in punctis A & B, C & D &longs;ecabant, jam Curvam in punctis illis coeuntibus non amplius &longs;ecare po&longs;&longs;unt &longs;ed tantum tangent.

~~Scholium.~~

~~Nomen Conicæ &longs;ectionis in hoc Lemmate late &longs;umitur, ita ut &longs;ectio tam Rectilinea per verticem Coni tran&longs;iens, quam Circularis ba&longs;i parallela includatur.~~ Nam &longs;i punctum p incidit in rectam, qua quævis ex punctis quatuor A, B, C, D junguntur, Conica &longs;ectio vertetur in geminas Rectas, quarum una e&longs;t recta illa in quam pun ctum p incidit, & altera e&longs;t recta qua alia duo ex punctis quatuor junguntur. Si Trapezii anguli duo oppo&longs;iti &longs;imul &longs;umpti æquentur duobus rectis, & lineæ quatuor PQ, PR, PS, PT ducantur ad latera ejus vel perpendiculariter vel in angulis quibu&longs;vis æqualibus, &longs;itque rectangulum &longs;ub duabus ductis PQXPR æquale rectangulo &longs;ub duabus aliis PSXPT, Sectio conica evadet Circulus. Idem fiet &longs;i lineæ quatuor ducantur in angulis quibu&longs;vis & rectangulum &longs;ub duabus ductis PQXPR &longs;it ad rectangulum &longs;ub aliis duabus PSXPT ut rectangulum &longs;ub &longs;inubus angulorum S, T, in quibus duæ ultimæ PS, PT ducuntur, ad rectangulum &longs;ub &longs;inubus angulorum Q, R, in quibus duæ primæ PQ, PR ducuntur. ~~Cæteris in ca&longs;ibus Locus puncti P erit aliqua trium figurarum quæ vulgo nominantur Sectiones Conicæ.~~ ~~Vice autem Trapezii ABCD &longs;ub&longs;titui pote&longs;t Quadrilaterum cujus latera duo oppo&longs;ita &longs;e mutuo in&longs;tar diagonalium decu&longs;&longs;ant.~~ Sed & e punctis quatuor A, B, C, Dpo&longs;&longs;unt unum vel duo abire ad infinitum, eoque pacto latera figuræ quæ ad puncta illa convergunt, evadere parallela: quo in ca&longs;u Sectio Conica tran&longs;ibit per cætera puncta, & in plagas parallelarum abibit in infinitum.

~~LIBER PRIMUS.~~

~~LEMMA XIX.~~

~~Invenire punctum P, a quo &longs;irectæ~~

quatuor PQ, PR, PS, PT, ad alias totidem po&longs;itione da tas rectas AB, CD, AC, BD, &longs;ingulæ ad &longs;ingulas in datis angulis ducantur, rectangulum &longs;ub duabus ductis, PQXPR, &longs;it ad rectangulum &longs;ub aliis duabus, PSXPT, in data ratione.

Lineæ AB, CD, ad quas rectæ duæ PQ, PR, unum rectangulorum continentes ducuntur, conveniant cum aliis duabus po&longs;itione datis lineis in punctis A, B, C, D. Ab eorum aliquo A age rectam quamlibet AH, in qua velis punctum P reperiri. ~~Secet ea lineas oppo&longs;itas BD, CD, nimirum BD in H & CD in I, & ob datos omnes angulos figuræ, dabuntur rationes PQ ad PA & PA ad PS, adeoque ratio PQ ad~~

~~PS. Auferendo hanca data ratione PQXPR ad PSXPT,dabitur ratio PR ad PT, & addendo datas rationes PI ad PR, & PT ad PH dabitur ratio PI ad PH atque adeo punctum P. q.E.I.~~

~~DE MOTU CORPORUM~~

~~Corol. 1. Hinc etiam ad Loci punctorum infinitorum P punctum quodvis D tangens duci pote&longs;t.~~ ~~Nam chorda PD ubi puncta P ac D conveniunt, hoc e&longs;t, ubi AH ducitur per punctum D, tangens evadit.~~ ~~Quo in ca&longs;u, ultima ratio evane&longs;centium IP & PH invenietur ut &longs;upra.~~ Ip&longs;i igitur AD due parallelam CF, occurrentem BD in F, & in ea ultima ratione &longs;ectam in E, & DE tangens erit, propterea quod CF& evane&longs;cens IH parallelæ &longs;unt, & in E & P fimiliter &longs;ectæ.

~~Corol. 2. Hinc etiam Locus punctorum omnium P definiri pote&longs;t.~~ ~~Per quodvis punctorum A, B, C, D, puta A, duc Loci tangentem AE & per aliud quodvis punctum B duc tangenti parallelam BFoccurrentem Loco in F. Invenie~~

~~tur autem punctum F per Lem.~~ ~~XIX.~~ ~~Bi&longs;eca BF in G, & acta indefinita AG erit po&longs;itio diametri ad quam BG & FG ordinatim applicantur.~~ Hæc AG occurrat Loco in H, & erit AH diameter &longs;ive latus tran&longs;ver&longs;um, ad quod latus rectum erit ut BGque ad AGH. Si AG nullibi occurrit Loco, linea AH exi&longs;tente infinita, Locus erit Parabola & larum rectum ejus ad diametrum AGpertinens erit (BGq./AG) Sin ea alicubi occurrit, Locus Hyperbola erit ubi puncta A & H &longs;ita &longs;unt ad ea&longs;dem partes ip&longs;ius G: & Ellip&longs;is, ubi G intermedium e&longs;t, ni&longs;i forte angulus AGB rectus &longs;it & in&longs;uper BG quad. æquale rectangulo AGH, quo in ca&longs;u Circulus habebitur.

Atque ita Problematis Veterum de quatuor lineis ab Euclide incæpti & ab Apollonio continuati non calculus, &longs;ed compo&longs;itio Geometrica, qualem Veteres quærebant, in hoc Corollario exhibetur.

~~LIBER PRIMUS.~~

~~LEMMA XX.~~

Si Parallelogr ammum quodvis ASPQ angulis duobus oppo&longs;itis A &P tangit &longs;ectionem quamvis Conicam in punctis A & P; &, lateribus unius angulorum illorum infinite productis AQ, AS, occurrit eidem &longs;ectioni Conicæ in B & C; a punctis autem occur&longs;uum B &C ad quintum quodvis &longs;ectionis Conicæ punctum D agantur rectæ duæ BD, CD occurrentes alteris duobus infinite productis parallelogrammi lateribus PS, PQ in T & R: erunt &longs;emper ab&longs;ci&longs;&longs;æ laterum partes PR & PT adinvicem in data ratione. ~~Et contra, &longs;i partes illæ ab&longs;ci&longs;&longs;æ &longs;unt ad invicem in data ratione, punctum D tanget Sectionem Conicam per puncta quatuor A, B, C, P tran&longs;euntem.~~

~~Cas. 1. Jungantur BP, CP & a puncto D agantur rectæ duæ DG, DE, quarum prior~~

DG ip&longs;i AB parallela &longs;it & occurrat PB, PQ, CA in H, I, G; altera DE parallela &longs;it ipfi AC & occurrat PC, PS, AB in F, K, E:& erit (per Lemma XVII.) rectangulum DEXDF ad rectangulum DGXDH in ratione data. Sed e&longs;t PQ ad DE (&longs;eu IQ) ut PB ad HB,adeoque ut PT ad DH; & vici&longs;&longs;im PQ ad PT ut DE ad DH. E&longs;t & PR ad DF ut RCad DC, adeoque ut (IG vel) PS ad DG, & vici&longs;&longs;im PR ad PSut DF ad DG; & conjunctis rationibus fit rectangulum PQXPRad rectangulum PSXPT ut rectangulum DEXDF ad rectangulum DGXDH, atque adeo in data ratione. ~~Sed dantur PQ& PS & propterea ratio PR ad PT datur. q.E.D.~~

Cas. 2. Quod &longs;i PR & PT ponantur in data ratione ad invicem, tum &longs;imili ratiocinio regrediendo, &longs;equetur e&longs;&longs;e rectangulum DEXDF ad rectangulum DGXDH in ratione data, adeoque punctum D (per Lemma XVIII.) contingere Conicam &longs;ectionem tran&longs;euntem per puncta A, B, C, P. q.E.D.

~~DE MOTU CORPORUM~~

Corol. 1. Hinc &longs;i agatur BC &longs;ecans PQ in r, & in PT capiatur Pt in ratione ad Pr quam habet PT ad PR: erit Bt tangens Conicæ &longs;ectionis ad punctum B. Nam concipe punctum D coire cum puncto B ita ut, chorda BD evane&longs;cente, BT tangens evadat; & CD ac BT coincident cum CB & Bt.

~~Corol. 2. Et vice ver&longs;a &longs;i~~

Bt fit tangens, & ad quodvis Conicæ &longs;ectionis punctum D conveniant BD, CD; erit PR ad PT ut ut Pr ad Pt. Et contra, &longs;i &longs;it PR ad PT ut Pr ad Pt: convenient BD, CDad Conicæ Sectionis puncum aliquod D.

~~Corol. 3. Conica &longs;ectio non &longs;ecat Conicam &longs;ectionem in punctis pluribus quam quatuor.~~ Nam, &longs;i fieri pote&longs;t, tran&longs;eant duæ Conicæ &longs;ectiones per quinque puncta A, B, C, P, O; ea&longs;que &longs;ecet recta BD in punctis D, d, & ip&longs;am PQ &longs;ecet recta Cdin r. ~~Ergo PR e&longs;t ad PT ut Pr ad PT; unde PR & Pr &longs;ibi invicem æquantur, contra Hypothe&longs;in.~~

~~LEMMA XXI.~~

Si rectæ duæ mobiles & infinitæ BM, CM per data puncta B, C, ceu polos ductæ, concur&longs;u &longs;uo M de&longs;cribant tertiam po&longs;itione datam rectam MN; & aliæ duæ infinitæ rectæ BD, CD cum prioribus duabus ad puncta illa data B, C datos angulosMBD, MCD efficientes ducantur; dico quod hæ duæ BD, CD concur&longs;u &longs;uo D de&longs;cribent &longs;ectionem Conicam per punctaB, C tran&longs;euntem. Et vice ver&longs;a, &longs;i rectæ BD, CD concur&longs;u &longs;uo D de&longs;cribant Sectionem Conicam per data puncta B, C, A tran&longs;euntem, & &longs;it angulus DBM &longs;emper æqualis angulo datoABC, angulu&longs;que DCM &longs;emper æqualis angulo dato ACB: punctum M continget rectam po&longs;itione datam.

~~LIBER PRIMUS.~~

~~Nam in recta MN detur punctum N, & ubi punctum mobile M incidit in immotum N, incidat punctum mobile D in immotum P, Junge CN, BN,~~

CP, BP, & a puncto P age rectas PT, PRoccurrentes ip&longs;is BD, CD in T & R, & facientes angulum BPTæqualem angulo dato BNM, & angulum CPR æqualem angugulo dato CNM. Cum ergo (ex Hypothe&longs;i) æquales &longs;int anguli MBD, NBP, ut & anguli MCD, NCP; aufer communes NBD& NCD, & re&longs;tabunt æquales NBM & PBT, NCM & PCR: adeoque triangula NBM, PBT &longs;imilia &longs;unt, ut & triangula NCM, PCR. Quare PT e&longs;t ad NM ut PB ad NB, & PR ad NM ut PC ad NC. Sunt autem puncta B, C, N, Pimmobilia. Ergo PT & PR datam habent rationem ad NM, proindeque datam rationem inter &longs;e; atque adeo, per Lemma xx, punctum D (perpetuus rectarum mobilium BT & CR concur&longs;us) contingit &longs;ectionem Conicam, per puncta B, C, P tran&longs;euntem. q.E.D.

Et contra, &longs;i punctum mobile D contingat &longs;ectionem Conicam tran&longs;euntem per data puncta B, C, A, & &longs;it angulus DBM &longs;emper æqualis angulo dato ABC, & angulus DCM &longs;emper æqualis angulo dato ACB, & ubi punctum D incidit &longs;ucce&longs;&longs;ive in duo quævis &longs;ectionis puncta immobilia p, P, punctum mobile M incidat &longs;ucce&longs;&longs;ive in puncta duo immobilia n, N: per eadem n, N agatur Recta n N,& hæc erit Locus perpetuus puncti illius mobilis M. Nam, &longs;i fieri pote&longs;t, ver&longs;etur punctum M in linea aliqua Curva. ~~Tanget ergo punctum D &longs;ectionem Conicam per puncta quinque B, CA, p, P,tran&longs;euntem, ubi punctum M perpetuo tangit lineam Curvam.~~ ~~Sed & ex jam demon&longs;tratis tanget etiam punctum D &longs;ectionem Conicam per eadem quinque puncta B, C, A, p, P tran&longs;euntem, ubi pun-~~

~~ctum M perpetuo tangit lineam Rectam.~~ ~~Ergo duæ &longs;ectiones Conicæ tran&longs;ibunt per eadem quinque puncta, contra Corol.~~ ~~3. Lem.~~ ~~xx.~~ ~~Igitur punctum M ver&longs;ari in linea Curva ab&longs;urdum e&longs;t. q.E.D.~~

~~DE MOTU CORPORUM~~

~~PROPOSITIO XXII. PROBLEMA. XIV.~~

~~Trajectoriam per data quinque puncta de&longs;cribere.~~

~~Dentur puncta quinque A, B, C, P, D. Ab eorum aliquo A ad alia duo quævis B, C, quæ poli nominentur, age rectas AB, AC,~~

hi&longs;que parallelas TPS, PRQ per punctum quartum P. Deinde a polis duobus B, C age per punctum quintum D infinitas duas BDT, CRD, novi&longs;&longs;ime ductis TPS, PRQ (priorem priori & po&longs;teriorem po&longs;teriori) occurrentes in T & R. Denique de rectis PT, PR, acta recta tr ip&longs;i TR parallela, ab&longs;cinde qua&longs;vis Pt, Pr ip&longs;is PT, PR proportionales; & &longs;i per earum terminos t, r & polos B, C actæ Bt, Cr concurrant in d, locabitur punctum illud d in Trajectoria quæ&longs;ita. Nam punctum illud d (per Lemma xx) ver&longs;atur in Conica Sectione per puncta quatuor A, B, C, P tran&longs;eunte; &, lineis Rr, Tt evane&longs;centibus, coit punctum d cum puncto D. Tran&longs;it ergo &longs;ectio Conica per puncta quinque A, B, C, P, D. q.E.D.

~~Idem aliter.~~

~~LIBER PRIMUS.~~

~~E punctis datis junge tria quævis A, B, C; &, circum duo eorum B, C ceu polos, rotando angulos magnitudine datos ABC, ACB, applicentur cru~~

~~ra BA, CA primo ad punctum D, deinde ad punctum P, & notentur puncta M, N in quibus altera crura BL, CL ca&longs;u utroque &longs;e decu&longs;&longs;ant.~~ Agatur recta infinita MN, & rotentur anguli illi mobiles circum polos &longs;uos B, C, ea lege ut crurum BL, CL vel BM, CM inter&longs;ectio quæ jam &longs;it m incidat &longs;emper in rectam illam infinitam MN & crurum BA, CA, vel BD, CD inter&longs;ectio, quæ jam &longs;it d, Trajectoriam quæ&longs;itam PAD dB delineabit. ~~Nam punctum d, per Lem.~~ XXI, continget &longs;ectionem Conicam per puncta B, C tran&longs;euntem; & ubi punctum m accedit ad puncta L, M, N, punctum d (per con&longs;tructionem) accedet ad puncta A, D, P. De&longs;cribetur itaque &longs;ectio Conica tran&longs;iens per puncta quinque A, B, C, P, D. q.E.F.

~~Corol. 1. Hinc recta expedite duci pote&longs;t quæ Trajectoriam quæ&longs;itam, in puncto quovis dato B, continget.~~ ~~Accedat punctum d ad punctum B, & recta Bd evadet tangens quæ&longs;ita.~~

~~Corol. 2. Unde etiam Trajectoriarum Centra, Diametri & Latera recta inveniri po&longs;&longs;unt, ut in Corollario &longs;ecundo Lemmatis XIX.~~

~~Scholium.~~

Con&longs;tructio prior evadet paulo &longs;implicior jungendo BP, & in ea, &longs;i opus e&longs;t, producta capiendo Bp ad BP ut e&longs;t PR ad PT; & per p agendo rectam infinitam pd ip&longs;i SPT parallelam, inque ea capiendo &longs;emper pd æqualem Pr; & agendo rectas Bd, Cr concurrentes in d. Nam cum &longs;int Pr ad Pt, PR ad PT, pB ad PB, pd ad Pt in eadem ratione; erunt pd & Pr &longs;emper æqua- les. ~~Hac methodo puncta Trajectoriæ inveniuntur expediti&longs;&longs;ime,~~

~~ni&longs;i mavis Curvam, ut in con&longs;tructione &longs;ecunda, de&longs;eribere Mechanice.~~

~~DE MOTU CORPORUM~~

~~PROPOSITIO XXIII. PROBLEMA XV.~~

~~Trajectoriam de&longs;cribere quæ per data quatuor puncta tran&longs;ibit, & rectam continget po&longs;itione datam.~~

~~Cas. 1. Dentur tangens HB, punctum contactus B, & alia tria puncta C, D, P. Junge BC, & agendo PS parallelam BH,& PQ parallelam BC, comple parallelogrammum BSPque~~

Age BD &longs;ecantem SP in T, & CD &longs;ecantem PQ in R. Denique, agendo quamvis tr ip&longs;i TR parallelam, de PQ, PSab&longs;cinde Pr, Pt ip&longs;is PR, PT proportionales re&longs;pective; & actarum Cr, Bt concur&longs;us d (per Lem. ~~xx) incidet &longs;emper in Trajectoriam de&longs;cribendam.~~

~~LIBER PRIMUS.~~

~~Idem aliter.~~

Revolvatur tum angulus magnitudine datus CBH circa polum B, tum radius quilibet rectilineus & utrinque productus DC circa polum C. Notentur puncta M, N in quibus anguli crus BC&longs;ecat radium illum ubi crus alterum BH concurrit cum eodem radio in punctis P & D. Deinde ad actam infinitam MN con

~~currant perpetuo radius ille CP vel CD & anguli crus BC, & cruris alterius BH concur&longs;us cum radio delineabit Trajectoriam quæ&longs;itam.~~

Nam &longs;i in con&longs;tructionibus Problematis &longs;uperioris accedat punctum A ad punctum B, lineæ CA & CB coincident, & linea AB in ultimo &longs;uo &longs;itu fiet tangens BH, atque adeo con&longs;tructiones ibi po&longs;itæ evadent eædem cum con&longs;tructionibus hic de&longs;criptis. ~~Delineabit igitur cruris BH concur&longs;us cum radio &longs;ectionem Conicam per puncta C, D, P tran&longs;euntem, & rectam BH tangentem in puncto B. q.E.F.~~

~~Cas. 2. Dentur puncta quatuor B, C, D, P extra tangentem HI &longs;ita.~~ ~~Junge bina lineis BD, CP concurrentibus in G, tangen- tique occurrentibus in H & I. Secetur tangens in A, ita ut &longs;it HA ad AI, ut e&longs;t rectan~~

gulum &longs;ub media proportionali inter CG & GP & media proportionali inter BH & HD, ad rectangulum &longs;ub media proportionali inter DG & GB & media proportionali inter PI & IC; & erit A punctum contactus. Nam &longs;i rectæ PI parallela HX Trajectoriam &longs;ecet in punctis quibu&longs;vis X & Y: erit (ex Conicis) punctum A ita locandum, ut fuerit HA quad. ad AI quad. in ratione compo&longs;ita ex ratione rectanguli XHY ad rectangulum BHD&longs;eu rectanguli CGP ad rectangulum DGB & ex ratione rectanguli BHD ad rectangulum PIC. Invento autem contactus puncto A, de&longs;cribetur Trajectoria ut in ca&longs;u primo. q.E.F.Capi autem pote&longs;t punctum A vel inter puncta H & I, vel extra; & perinde Trajectoria dupliciter de&longs;cribi.

~~DE MOTU CORPORUM~~

~~PROPOSITIO XXIV. PROBLEMA XVI.~~

~~Trajectoriam de&longs;cribere quæ tran&longs;ibit per data tria puncta & rectas duas po&longs;itione datas continget.~~

~~Dentur tangentes HI, KL &~~

puncta B, C, D. Per punctorum duo quævis B, D age rectam infinitam BD tangentibus occurrentem in punctis H, K. Deinde etiam per alia duo quævis C, Dage infinitam CD tangentibus occurrentem in punctis I, L. Actas ita &longs;eca in R & S, ut &longs;it HR ad KR ut e&longs;t media proportionalis inter BH & HD ad mediam proportionalem inter BK & KD; & IS ad LS ut e&longs;t media proportionalis inter CI & ID ad mediam proportionalem inter CL & LD. Seca autem pro lubitu vel inter puncta K & H, I & L, vel extra eadem: dein age RS &longs;ecantem tangentes in A& P, & erunt A & P puncta contactuum. Nam &longs;i A & P&longs;upponantur e&longs;&longs;e puncta contactuum alicubi in tangentibus &longs;ita; & per punctorum H, I, K, L quodvis I, in tangente alterutra HI &longs;itum, agatur recta IY tangenti alteri KL parallela, quæ occurrat curvæ in X & Y, & in ea &longs;umatur IZ media proportionalis inter IX & IY: erit, ex Conicis, rectangulum XIY &longs;eu IZ quad. ad LP quad. ut rectangulum CID ad rectangulum CLD, id e&longs;t (per con&longs;tructionem) ut SI quad. ad SL quad: atque adeo IZ ad LP ut SI ad SL. Jacent ergo puncta S, P, Z in una recta. Porro tangentibus concurrentibus in G, erit (ex Conicis) rectangulum XIY &longs;eu IZ quad. ad IA quad. ut GP quad ad GA quad: adeoque IZ & IA ut GP ad GA. Jacent ergo puncta P, Z & A in una recta, adeoque puncta S, P & A&longs;unt in una recta. ~~Et eodem argumento probabitur quod puncta R, P & A &longs;unt in una recta.~~ ~~Jacent igitur puncta contactuum A& P in recta RS. Hi&longs;ce autem inventis, Trajectoria de&longs;eribetur ut in ca&longs;u primo Problematis &longs;uperioris. q.E.F.~~

~~LIBER PRIMUS.~~

~~LEMMA XXII.~~

~~Figuras in alias eju&longs;dem generis figur as mutare.~~

~~Tran&longs;mutanda &longs;it figura quævis HGI. Ducantur pro lubitu rectæ duæ parallelæ AO, BL tertiam quamvis po&longs;itione datam AB &longs;ecantes in A & B,~~

~~& a figuræ puncto quovis G, ad rectam ABducatur quævis GD,ip&longs;i OA parallela.~~ Deinde a puncto aliquo O,in linea OA dato, ad punctum D ducatur recta OD, ip&longs;i BL occurrens in d, & a puncto occur&longs;us erigatur recta dg datum quemvis angulum cum recta BL continens, atque eam habens rationem ad Od quam habet DG ad OD; & erit g punctum in figura nova hgi puncto G re&longs;pondens. Eadem ratione puncta &longs;ingula figuræ primæ dabunt puncta totidem figura novæ. Concipe igitur punctum G motu continuo percurrere puncta omnia figuræ primæ, & punctum g motu itidem continuo percurret puncta omnia figuræ novæ & eandem de&longs;cribet. Di&longs;tinctionis gratia nominemus DG ordinatam primam, dg ordinatam novam; AD ab&longs;ci&longs;&longs;am primam, ad ab&longs;ci&longs;&longs;am novam; O polum, OD radium ab&longs;cidentem, OA radium ordinatum primum, & Oa (qno parallelogrammum OABa completur) radium ordinatum novum.

~~DE MOTU CORPORUM~~

~~Dico jam quod, &longs;i punctum G tangit rectam Lineam po&longs;itione datam, punctum g tanget etiam Lineam rectam po&longs;itione datam.~~ ~~Si punctum G tangit Conicam &longs;ectionem, punctum g tanget etiam Conicam &longs;ectionem.~~ ~~Conicis &longs;ectionibus hic Circulum annumero.~~ ~~Porro &longs;i punctum G tan~~

~~git Lineam tertii ordinis Analytici, punctum gtanget Lineam tertii itidem ordinis; & &longs;ic de curvis lineis &longs;uperiorum ordinum.~~ ~~Lineæ duæ erunt eju&longs;dem &longs;emper ordinis Analytici quas puncta G, g tangunt.~~ Etenim ut e&longs;t ad ad OAita &longs;unt Od ad OD, dg ad DG, & AB ad AD; adeoque ADæqualis e&longs;t (OAXAB/ad), & DG æqualis e&longs;t (OAXdg/ad). Jam &longs;i punctum G tangit rectam Lineam, atque adeo in æquatione quavis, qua relatio inter ab&longs;ci&longs;&longs;am AD & ordinatam DG habetur, indeterminatæ illæ AD & DG ad unicam tantum dimen&longs;ionem a&longs;cendunt, &longs;cribendo in hac æquatione (OAXAB/ad) pro AD, & (OAXdg/ad) pro DG, producetur æquatio nova, in qua ab&longs;ci&longs;&longs;a nova ad & ordinata nova dg ad unicam tantum dimen&longs;ionem a&longs;cendent, atque adeo quæ de&longs;ignat Lineam rectam. ~~Sin AD & DG(vel earum alterutra) a&longs;cendebant ad duas dimen&longs;iones in æquatione prima, a&longs;cendent itidem ad & dg ad duas in æquatione &longs;ecunda.~~ ~~Et &longs;ic de tribus vel pluribus dimen&longs;ionibus.~~ Indeterminatæ ad, dg in æquatione &longs;ecunda & AD, DG in prima a&longs;cendent &longs;emper ad eundem dimen&longs;ionum numerum, & propterea Lineæ, quas puncta G, g tangunt, &longs;unt eju&longs;dem ordinis Analytici.

~~Dico præterea quod &longs;i recta aliqua tangat lineam curvam in fi gura prima; hæc recta eodem modo cum curva in figuram novam tran&longs;lata tanget lineam illam curvam in figura nova: & contra.~~ Nam &longs;i Curvæ puncta quævis duo accedunt ad invicem & coeunt in figura prima, puncta eadem tran&longs;lata accedent ad invicem & coibunt in figura nova, atque adeo rectæ, quibus hæc puncta junguntur, &longs;imul evadent curvarum tangentes in figura utraque. ~~Componi po&longs;&longs;ent harum a&longs;&longs;ertionum Demon&longs;trationes more magis Geometrico.~~ ~~Sed brevitati con&longs;ulo.~~

~~LIBER PRIMUS.~~

Igitur &longs;i figura rectilinea in aliam tran&longs;mutanda e&longs;t, &longs;ufficit rectarum a quibus conflatur inter&longs;ectiones transferre, & per ea&longs;dem in figura nova lineas rectas ducere. ~~Sin curvilineam tran&longs;mutare oportet, transferenda &longs;unt puncta, tangentes & aliæ rectæ quarum ope curva linea definitur.~~ ~~In&longs;ervit autem hoc Lemma &longs;olutioni difficiliorum Problematum, tran&longs;mutando figuras propo&longs;itas in &longs;impliciores.~~ Nam rectæ quævis convergentes tran&longs;mutantur in parallelas, adhibendo pro radio ordinato primo, lineam quamvis rectam quæ per concur&longs;um convergentium tran&longs;it: id adeo quia concur&longs;us ille hoc pacto abit in infinitum, lineæ autem parallelæ &longs;unt quæ ad punctum infinite di&longs;tans tendunt. ~~Po&longs;tquam autem Problema &longs;olvitur in figura nova, &longs;i per inver&longs;as operationes tran&longs;mutetur hæc figura in figuram primam, habebitur &longs;olutio quæ&longs;ita.~~

~~Utile e&longs;t etiam hoc Lemma in &longs;olutione Solidorum Problematum.~~ Nam quoties duæ &longs;ectiones Conicæ obvenerint, quarum inter&longs;ectione Problema &longs;olvi pote&longs;t, tran&longs;mutare licet earum alterutram, &longs;i Hyperbola &longs;it vel Parabola, in Ellip&longs;in: deinde Ellip&longs;is facile mutatur in Circulum. ~~Recta item & &longs;ectio Conica, in con&longs;tructione Planorum Problematum, vertuntur in Rectam & Circulum.~~

~~PROPOSITIO XXV. PROBLEMA XVII.~~

~~Trajectoriam de&longs;cribere qua per data duo puncta tran&longs;ibit & rectas tres continget po&longs;itione datas.~~

Per concur&longs;um tangentium quarumvis duarum cum &longs;e invicem, & concur&longs;um tangentis tertiæ cum recta illa, quæ per puncta duo data tran&longs;it, age rectam infinitam; eaque adhibita pro radio ordinato primo, tran&longs;mutetur figura, per Lemma &longs;uperius, in figuram novam. ~~In hac figura tangentes illæ duæ evadent &longs;ibi invicem parallelæ, & tangens tertia fiet parallela rectæ per~~

~~puncta duo data tran&longs;eunti.~~ Sunto hi, kl tangentes illæ duæ parallelæ, ik tangens tertia, & hl recta huic parallela tran&longs;iens per puncta illa a, b, per quæ Conica &longs;ectio in hac figura nova tran&longs;ire debet, & parallelogrammum hikl complens. Secentur rectæ hi, ik, kl in c, d, e,ita ut &longs;it hc ad latus quadratum rectanguli ahb, ic ad id, & kead kd ut e&longs;t &longs;umma rectarum hi& kl ad &longs;ummam trium linearum quarum prima e&longs;t recta ik, & alteræ duæ &longs;unt latera quadrata rectangulorum ahb & alb & erunt c, d, e puncta contactuum. Etenim, ex Conicis, &longs;unt hc quadratum ad rectangulum ahb, & ic quadratum ad id quadratum, & ke quadratum ad kd quadratum, & el quadratum ad rectangulum alb in eadem ratione; & propterea hc ad latus quadratum ip&longs;ius ahb, ic ad id, ke ad kd, & el ad latus quadratum ip&longs;ius alb &longs;unt in &longs;ubduplicata illa ratione, & compo&longs;ite, in data ratione omnium antecedentium hi & kl ad omnes con&longs;equentes, quæ &longs;unt latus quadratum rectanguli ahb & recta ik & latus quadratum rectanguli alb. Habentur igitur ex data illa ratione puncta contactuum c, d, e, in figura nova. ~~Per inver&longs;as operationes Lemmatis novi&longs;&longs;imi transferantur hæc puncta in figuram primam & ibi, per Probl.~~ ~~XIV, de&longs;cribetur Trajectoria. q.E.F. Ceterum perinde ut puncta a, b jacent vel inter puncta h, l, vel extra, debent puncta c, d, e vel inter puncta h, i, k, l capi, vel extra.~~ ~~Si punctorum a, b alterutrum cadit inter puncta h, l, & alterum extra, Problema impo&longs;&longs;ibile e&longs;t.~~

~~DE MOTU CORPORUM~~

~~PROPOSITIO XXVI. PROBLEMA XVIII.~~

~~Trajectoriam de&longs;cribere quæ tran&longs;ibit per punctum datum & rectas quatuor po&longs;itione datas continget.~~

Ab inter&longs;ectione communi duarum quarumlibet tangentium ad inter&longs;ectionem communem reliquarum duarum agatur recta infini- ta, & eadem pro radio ordinato primo adhibita, tran&longs;mutetur fi gura (per Lem. ~~XXII) in figuram novam, & tangentes binæ, quæ ad radium ordinatum primum concurrebant, jam evadent parallelæ.~~ ~~Sunto illæ hi & kl, ik & hl continentes parallelogrammum hikl. Sitque p punctum in hac nova figura, puncto in figura prima dato re&longs;pondens.~~ ~~Per figuræ centrum O agatur pq, & exi&longs;tente Oq æquali Op, erit q punctum alterum per quod &longs;ectio Conica in hac figura nova tran&longs;ire debet.~~ ~~Per Lemmatis XXII operationem inver&longs;am transferatur hoc punctum in figuram primam, & ibi habebuntur puncta duo per quæ Trajectoria de&longs;cribenda e&longs;t.~~ ~~Per eadem vero de&longs;cribi pote&longs;t Trajectoria illa per Prob.~~ ~~XVII. q.E.F.~~

~~LIBER PRIMUS.~~

~~LEMMA XXIII.~~

Si rectæ duæ po&longs;itione datæ AC, BD ad data puncta A, B, terminentur, datamque habeant rationem ad invicem, & rectaCD, qua puncta indeterminata C, D junguntur, &longs;ecetur in ratione data in K: dico quod punctum K locabitur in recta po&longs;itione data.

~~Concurrant enim rectæ AC,~~

BD in E, & in BE capiatur BGad AE ut e&longs;t BD ad AC, &longs;itque FD &longs;emper æqualis datæ EG; & erit ex con&longs;tructione EC ad GD, hoc e&longs;t, ad EF ut AC ad BD, adeoque in ratione data, & propterea dabitur &longs;pecie triangulum EFC. Secetur CFin L ut &longs;it CL ad CF in ratione CK ad CD; &, ob datam illam rationem, dabitur etiam &longs;pecie triangulum EFL; proindeque punctum L locabitur in recta EL po&longs;itione data. ~~Junge LK, & &longs;imilia erunt triangula CLK, CFD; &, ob datam FD & datam rationem LK ad FD, dabitur LK. Huic æqualis capiatur EH,& erit &longs;emper ELKH parallelogrammum.~~ ~~Locatur igitur punctum K in parallelogrammi illius latere po&longs;itione dato HK. q.E.D.~~

~~DE MOTU CORPORUM~~

~~LEMMA XXIV.~~

Si rectæ tres tangant quamcunque Coni&longs;ectionem, quarum duæ parallelæ &longs;int ac dentur po&longs;itione; dico quod Sectionis &longs;emidiameter hi&longs;ce duabus parallela, &longs;it media proportionalis inter harum &longs;egmenta, punctis contactuum & tangenti tertiæ interjecta.

~~Sunto AF, GB pa~~

rallelæ duæ Coni&longs;ectionem ADB tangentes in A & B; EFrecta tertia Coni&longs;ectionem tangens in I,& occurrens prioribus tangentibus in F & G; &longs;itque CD &longs;emidiameter Figuræ tangentibus parallela: Dico quod AF, CD, BG&longs;unt continue proportionales.

Nam &longs;i diametri conjugatæ AB, DM tangenti FG occurrant in E & H, &longs;eque mutuo &longs;ecent in C, & compleatur parallelogrammum IKCL; erit, ex natura Sectionum Conicarum, ut EC ad CA ita CA ad CL, & ita divi&longs;im EC-CA ad CA-CL, &longs;eu EA ad AL, & compo&longs;ite EA ad EA+AL &longs;eu EL ut EC ad EC+CA &longs;eu EB; adeoque (ob &longs;imilitudinem triangulorum EAF, ELI, ECH, EBG) AF ad LI ut CH ad BG. E&longs;t itidem, ex natura Sectionum Conicarum, LI (&longs;eu CK) ad CD ut CD ad CH; atque, adeo ex æquo perturbate, AF ad CD ut CD ad BG. q.E.D.

Corol. 1. Hinc &longs;i tangentes duæ FG, PQ tangentibus parallelis AF, BG occurrant in F & G, P & Q, &longs;eque mutuo &longs;ecent in O; erit (ex æquo perturbate) AF ad BQ ut AP ad BG, & divi&longs;im ut FP ad GQ, atque adeo ut FO ad OG.

~~Corol. 2. Unde etiam rectæ duæ PG, FQ per puncta P & G, F & Q ductæ, concurrent ad rectam ACB per centrum Figuræ & puncta contactuum A, B tran&longs;euntem.~~

~~LIBER PRIMUS.~~

~~LEMMA XXV.~~

Si parallelogrammi latera quatuor infinite producta tangant Sectionem quamcunque Conicam, & ab&longs;cindantur ad tangentem quamvis quintam; &longs;umantur autem laterum quorumvis duorum conterminorum ab&longs;ci&longs;&longs;æ terminatæ ad angulos oppo&longs;itos parallelogrammi: dico quod ab&longs;ci&longs;&longs;a alterutra &longs;it ad latus illud a quo est ab&longs;ci&longs;&longs;a, ut pars lateris alterius contermini inter punctum contactus & latus tertium, est ad ab&longs;ci&longs;&longs;arum alteram.

~~Tangant parallelogrammi MLIK latera quatuor ML, IK, KL, MI &longs;ectionem Conicam in A, B, C, D, & &longs;ecet tangens quinta FQhæc latera in F, Q, H~~

& E; &longs;umantur autem laterum MI, KI ab&longs;ci&longs;&longs;æ ME, KQ, vel laterum KL, ML ab&longs;ci&longs;&longs;æ KH, MF: dico quod &longs;it ME ad MI ut BK ad KQ; & KH ad KL ut AM ad MF. Nam per Corollarium &longs;ecundum Lemmatis &longs;uperioris, e&longs;t ME ad EI ut (AM &longs;eu) BK ad BQ, & componendo ME ad MI ut BK ad Kque q.E.D.Item KH ad HL ut (BK &longs;eu) AM ad AF, & dividendo KH ad KL ut AM ad MF. q.E.D.

~~Corol. 1. Hinc &longs;i datur parallelogramum IKLM, circa datam Sectionem Conicam de&longs;eriptum, dabitur rectangulum KQXME, ut & huic æquale rectangulum KHXMF.~~

~~Corol. 2. Et &longs;i &longs;exta ducatur tangens eq tangentibus KI, MIoccurrens in q & e; rectangulum KQXME æquabitur rectangulo KqXMe; eritque KQ ad Me ut Kq ad ME, & divi&longs;im ut Qq ad Ee.~~

~~Corol. 3. Unde etiam &longs;i Eq, eQ jungantur & bi&longs;ecentur, & recta per puncta bi&longs;ectionum agatur, tran&longs;ibit hæc per centrum Sectionis Conicæ.~~ ~~Nam cum &longs;it Qq ad Ee ut KQ ad Me, tran&longs;ibit ea- dem recta per medium omnium Eq, eQ, MK; (per Lem.~~ ~~XXIII) & medium rectæ MK e&longs;t centrum Sectionis.~~

~~DE MOTU CORPORUM~~

~~PROPOSITIO XXVII. PROBLEMA XIX.~~

~~Trajectoriam de&longs;cribere quæ rectas quinque po&longs;itione datas continget.~~

~~Dentur pofitione tangentes ABG, BCF, GCD, FDE, EA.Figuræ quadrilateræ &longs;ub quatuor quibu&longs;vis contentæ ABFE diagonales AF, BE bi&longs;eca, & (per Corol.~~ ~~3. Lem.~~ ~~XXV) recta MNper puncta bi&longs;ectionum acta tran&longs;ibit per centrum Trajectoriæ.~~ ~~Rur&longs;us Figuræ quadrilateræ BGDF, &longs;ub aliis quibu&longs;vis quatuor~~

~~tangentibus contentæ, diagonales (ut ita dicam) BD, GF bi&longs;eca in P & Q: & recta PQ per puncta bi&longs;ectionum acta tran&longs;ibit per centrum Trajectoriæ.~~ ~~Dabitur ergo centrum in concur&longs;u bi&longs;ecantium.~~ ~~Sit illud O. Tangenti cuivis BC parallelam age KL,ad eam di&longs;tantiam ut centrum O in medio inter parallelas locetur, & acta KL tanget Trajectoriam de&longs;cribendam.~~ Secet hæc tan- gentes alias qua&longs;vis duas GCD, FDE in L & K. Per harum tangentium non parallelarum CL, FK cum parallelis CF, KLconcur&longs;us C & K, F & L age CK, FL concurrentes in R, & recta OR ducta & producta &longs;ecabit tangentes parallelas CF, KL in punctis contactuum. ~~Patet hoc per Corol.~~ ~~2. Lem.~~ ~~XXIV.~~ ~~Eadem methodo invenire licet alia contactuum puncta, & tum demum per Probl.~~ ~~XIV. &c.~~ ~~Trajectoriam de&longs;cribere. q.E.F.~~

~~LIBER PRIMUS.~~

~~Scholium.~~

~~Problemata, ubi dantur Trajectoriarum vel centra vel A&longs;ymptoti, includuntnr in præcedentibus.~~ ~~Nam datis punctis & tangentibus una cum centro, dantur alia totidem puncta aliæque tangentes a centro ex altera ejus parte æqualiter di&longs;tantes.~~ ~~A&longs;ymptotos autem pro tangente habenda e&longs;t, & ejus terminus infinite di&longs;tans (&longs;i ita loqui fas &longs;it) pro puncto contactus.~~ Concipe tangentis cuju&longs;vis punctum contactus abire in infinitum, & tangens vertetur in A&longs;ymptoton, atque con&longs;tructiones Problematis XIV & Ca&longs;us primi Problematis XV vertentur in con&longs;tructiones Problematum ubi A&longs;ymptoti dantur.

~~Po&longs;tquam Trajectoria de&longs;cripta e&longs;t, invenire licet axes & umbilicos ejus hac methodo.~~ ~~In con&longs;tructione & figura Lemmatis XXI, fac ut angulorum mobi~~

lium PBN, PCN crura BP, CP, quorum concur&longs;u Trajectoria de&longs;cribebatur, &longs;int &longs;ibi invicem parallela, eumque &longs;ervantia &longs;itum revolvantur circa polos &longs;uos B, Cin figura illa. Interea vero de&longs;cribant altera angulorum illorum crura CN, BN, concur&longs;u &longs;uo K vel k, Circulum IBKGC. Sit Circuli hujus centrum O. Ab hoc centro ad Regulam MN, ad quam altera illa crura CN, BN interea concurrebant dum Trajectoria de&longs;cribebatur, demitte normalem OH Circulo occurrentem in K & L. Et ubi crura illa altera CK, BK concurrunt ad punctum illud K quod Regulæ propius e&longs;t, crura prima CP, BP parallela erunt axi majori, & perpendicularia minori; & contrarium eveniet &longs;i crura eadem concurrunt ad punctum remotius L. Unde &longs;i detur Trajectoriæ centrum, dabuntur axes. ~~Hi&longs;ce autem datis, umbilici &longs;unt in promptu.~~

~~DE MOTU CORPORUM~~

~~Axium vero quadrata &longs;unt ad invicem ut KH ad LH, & inde facile e&longs;t Trajectoriam~~

~~&longs;pecie datam per data quatuor puncta de&longs;cribere.~~ Nam &longs;i duo ex punctis datis con&longs;tituantur poli C, B, tertium dabit angulos mobiles PCK, PBK; his autem datis de&longs;cribi pote&longs;t Circulus IBKGC.Tum ob datam &longs;pecie Trajectoriam, dabitur ratio OH ad OK, adeoque ip&longs;a OH. Centro O & intervallo OHde&longs;cribe alium circulum, & recta quæ tangit hunc circulum, & tran&longs;it per concur&longs;um crurum CK, BK, ubi crura prima CP, BP concurrunt ad quartum datum punctum erit Regula illa MN cujus ope Trajectoria de&longs;cribetur. ~~Unde etiam vici&longs;&longs;im Trapezium &longs;pecie datum (&longs;i ca&longs;us quidam impo&longs;&longs;ibiles excipiantur) in data quavis Sectione Conica in&longs;cribi pote&longs;t.~~

~~Sunt & alia Lemata quorum ope Trajectoriæ &longs;pecie datæ, datis punctis & tangentibus, de&longs;cribi po&longs;&longs;unt.~~ Ejus generis e&longs;t quod, &longs;i recta linea per punctum quodvis po&longs;itione datum ducatur, quæ datam Coni&longs;ectionem in punctis duobus inter&longs;ecet, & inter&longs;ectionum intervallum bi&longs;ecetur, punctum bi&longs;ectionis tanget aliam Coni&longs;ectionem eju&longs;dem &longs;peciei cum priore, atque axes habentem prioris axibus parallelos. ~~Sed propero ad magis utilia.~~

~~LIBER PRIMUS.~~

~~LEMMA XXVI.~~

~~Trianguli &longs;pecie & magnitudine dati tres angulos ad rectas totidem po&longs;itione datas, quæ non &longs;unt omnes parallelæ, &longs;ingulos ad &longs;ingulas ponere.~~

~~Dantur po&longs;itione tres rectæ infinitæ AB, AC, BC, & oportet triangulum DEF ita locare, ut angulus ejus D lineam AB,angulus E lineam AC,~~

~~& angulus F lineam BC tangat.~~ ~~Super DE, DF & EF de&longs;cribe tria circulorum &longs;egmenta DRE, DGF, EMF, quæ capiant angulos angulis BAC, ABC, ACB æquales re&longs;pective.~~ De&longs;cribantur autem hæc &longs;egmenta ad eas partes linearum DE, DF, EF ut literæ DRED eodem ordine cum literis BACB, literæ DGFDeodem cum literis ABCA, & literæ EMFE eodem cum literis ACBA in orbem redeant; deinde compleantur hæc &longs;egmenta in circulos integros. Secent circuli duo priores &longs;e mutuo in G, &longs;intque centra eorum P & que Junctis GP, PQ,cape Ga ad AB ut e&longs;t GP ad PQ, & centro G, intervallo Gade&longs;cribe circulum, qui &longs;ecet circulum primum DGE in a. Jungatur tum aD &longs;ecans circulum &longs;ecundum DFG in b, tum aE &longs;ecans cir- culum tertium EMF in c. Et compleatur Figura ABC def &longs;imilis & æqualis Figuræ abcDEF. Dico factum.

~~DE MOTU CORPORUM~~

~~Agatur enim Fc ip&longs;i aD occurrens in n, & jungantur aG, bG, QG, QD, PD. Ex con&longs;tructione e&longs;t angulus EaD æqualis angulo CAB, & angulus~~

~~acF æqualis angulo ACB, adeoque triangulum anc triangulo ABC æquiangulum.~~ Ergo angulus anc &longs;eu FnD angulo ABC,adeoque angulo FbDæqualis e&longs;t; & propterea punctum n incidit in punctum b. Porro angulus GPQ, qui dimidius e&longs;t anguli ad centrum GPD, æqualis e&longs;t angulo ad circumferentiam GaD; & angulus GQP, qui dimidius e&longs;t anguli ad centrum GQD, æqualis e&longs;t complemento ad duos rectos anguli ad circumferentiam GbD, adeoque æqualis angulo Gba; funtque ideo triangula GPQ, Gab &longs;imilia; & Ga e&longs;t ad abut GP ad PQ; id e&longs;t (ex con&longs;tructione) ut Ga ad AB. Æquantur itaque ab & AB; & propterea triangula abc, ABC, quæ modo &longs;imilia e&longs;&longs;e probavimus, &longs;unt etiam æqualia. Unde, cum tangant in&longs;uper trianguli DEF anguli D, E, F trianguli abc latera ab, ac, bc re&longs;pective, compleri pote&longs;t Figura ABCdef Figuræ abc DEF &longs;imilis & æqualis, atque eam complendo &longs;olvetur Problema. q.E.F.

~~Corol. Hinc recta duci pote&longs;t cujus partes longitudine datæ rectis tribus po&longs;itione datis interjacebunt.~~ Concipe Triangulum DEF,puncto D ad latus EF accedente, & lateribus DE, DF in directum po&longs;itis, mutari in lineam rectam, cujus pars data DE rectis po&longs;itione datis AB, AC, & pars data DF rectis po&longs;itione datis AB, BC interponi debet; & applicando con&longs;tructionem præcedentem ad hunc ca&longs;um &longs;olvetur Problema.

~~LIBER PRIMUS.~~

~~PROPOSITIO XXVIII. PROBLEMA XX.~~

~~Trajectoriam &longs;pecie & magnitudine datam de&longs;cribere, cujus partes datæ rectis tribus po&longs;itione datis interjacebunt.~~

~~De&longs;cribenda &longs;it Trajectoria quæ &longs;it &longs;imilis & æqualis Lineæ curvæ DEF, quæque a rectis tribus AB, AC, BC po&longs;itione datis, in~~

~~partes datis hujus partibus DE & EF &longs;imiles & æquales &longs;ecabitur.~~

~~Age rectas DE, EF, DF, & trianguli hujus DEF pone anlos D, E, F ad rectas illas po&longs;itione datas (per Lem.~~ ~~XXVI) Dein circa triangulum de&longs;cribe Trajectoriam Curvæ DEF &longs;imilem & æqualem. q.E.F.~~

~~DE MOTU CORPORUM~~

~~LEMMA XXVII.~~

Trapezium &longs;pecie datum de&longs;cribere cujus anguli ad rectas quatuor po&longs;itione datas, quæ neque omnes parallelæ &longs;unt, neque ad commune punctum convergunt, &longs;inguli ad &longs;ingulas con&longs;i&longs;tent.

Dentur po&longs;itione rectæ quatuor ABC, AD, BD, CE, quarum prima &longs;ecet &longs;ecundam in A, tertiam in B, & quartam in C:& de&longs;cribendum &longs;it Trapezium fghi quod &longs;it Trapezio FGHI

~~&longs;imile, & cujus angulus f, angulo dato F æqualis, tangat rectam ABC, cæterique anguli g, h, i, cæteris angulis datis G, H, I æquales, tangant cæteras lineas AD, BD, CE re&longs;pective.~~ Jungatur FH & &longs;uper FG, FH, FI de&longs;cribantur totidem circulorum &longs;egmenta FSG, FTH, FVI; quorum primum FSG capiat angu- lum æqualem angulo BAD, &longs;ecundum FTH capiat angulum æ qualem angulo CBD, ac tertium FVI capiat angulum æqualem angulo ACE. De&longs;cribi autem debent &longs;egmenta ad eas partes linearum FG, FH, FI, ut literarum FSGF idem &longs;it ordo circularis qui literarum BADB, utque literæ FTHF eodem ordine cum literis CBDC, & literæ FVIF eodem cum literis ACEA in orbem redeant. Compleantur &longs;egmenta in circulos integros, &longs;itque Pcentrum circuli primi FSG, & Q centrum &longs;ecundi FTH. Jungatur & utrinque producatur PQ, & in ea capiatur QR in ea ratione ad PQ quam habet BC ad AB. Capiatur autem QR ad eas partes puncti Q ut literarum P, Q, R idem &longs;it ordo atque literarum A, B, C: centroque R & intervallo RF de&longs;cribatur circulus quartus FNc &longs;ecans circulum tertium FVI in c. Jungatur Fc &longs;ecans circulum primum in a & &longs;ecundum in b. Agantur a G, b H, c I, & Figuræ abc FGHI &longs;imilis con&longs;tituatur Figura ABCfghi: Eritque Trapezium fghi illud ip&longs;um quod con&longs;tituere oportebat.

~~LIBER PRIMUS.~~

Secent enim circuli duo primi FSG, FTH &longs;e mutuo in K.Jungantur PK, QK, RK, a K, b K, c K, & producatur QP ad L.Anguli ad circumferentias FaK, FbK, FcK &longs;unt &longs;emi&longs;&longs;es angulorum FPK, FQK, FRK ad centra, adeoque angulorum illorum dimidiis LPK, LQK, LRK æquales. E&longs;t ergo Figura PQRK Figuræ abcK æquiangula & &longs;imilis, & propterea ab e&longs;t ad bc ut PQ ad QR, id e&longs;t, ut AB ad BC. Angulis in&longs;uper FaG, FbH, FcI æquantur fAg, fBh, fCi per con&longs;tructionem. Ergo Figuræ abcFGHI Figura &longs;imilis ABCfghi compleri pote&longs;t Quo facto Trapezium fghi con&longs;tituetur &longs;imile Trapezio FGHI& angulis &longs;uis f, g, h, i tanget rectas ABC, AD, BD, CE q.E.F.

~~Corol. Hinc recta duci pote&longs;t cujus partes, rectis quatuor po&longs;itione datis dato ordine interjectæ, datam habebunt proportionem ad invicem.~~ Augeantur anguli FGH, GHI u&longs;que eo, ut rectæ FG, GH, HI in directum jaceant, & in hoc ca&longs;u con&longs;truendo Problema, ducetur recta fghi cujus partes fg, gh, hi, rectis quatuor po&longs;itione datis AB & AD, AD & BD, BD & CE interjectæ, erunt ad invicem ut lineæ FG, GH, HI, eundemque &longs;ervabunt ordinem inter &longs;e. ~~Idem vero &longs;ic fit expeditius.~~

~~DE MOTU CORPORUM~~

Producantur AB ad K, & BD ad L, ut &longs;it BK ad AB ut HI ad GH; & DL ad BD ut GI ad FG; & jungatur KLoccurrens rectæ CE in i. Producatur iL ad M, ut &longs;it LM ad iLut GH ad HI, & agatur tum MQ ip&longs;i LB parallela rectæque AD occurrens in g, tum gi &longs;ecans AB, BD in f, h. Dico factum.

Secet enim Mg rectam AB in Q, & AD rectam KL in S, & agatur AP quæ &longs;it ip&longs;i BD parallela & occurrat iL in P, & erunt gM ad Lh (gi ad hi, Mi ad Li, GI ad HI, AK ad BK) & AP ad BL in eadem ratione. ~~Secetur DL in R ut &longs;it~~

DL ad RL in eadem illa ratione, & ob proportionales gS ad gM, AS ad AP, & DS ad DL; erit, ex æquo, ut gS ad Lh ita AS ad BL & DS ad RL; & mixtim, BL-RL ad Lh-BLut AS-DS ad gS-AS. Id e&longs;t BR ad Bh ut AD ad Ag adeoque ut BD ad gque Et vici&longs;&longs;im BR ad BD ut Bh ad gQ, &longs;eu fh ad fg. Sed ex con&longs;tructione linea RL eadem ratione &longs;ecta fuit in D & R atque linea FI in G & H: ideoque e&longs;t BR ad BDut FH ad FG. Ergo fh e&longs;t ad fg ut FH ad FG. Cum igitur &longs;it etiam gi ad hi ut Mi ad Li, id e&longs;t, ut GI ad HI, patet lineas FI, fi in g & h, G & H &longs;imiliter &longs;ectas e&longs;&longs;e. q.E.F.

~~In con&longs;tructione Corollarii hujus po&longs;tquam ducitur LK &longs;ecans~~

CE in i, producere licet iE ad V, ut &longs;it EV ad Ei ut FH ad HI,& agere Vf parallelam ip&longs;i BD. Eodem recidit &longs;i centro i, intervallo IH, de&longs;cribatur circulus &longs;ecans BD in X, & producatur iX ad Y, ut &longs;it iY æqualis IF, & agatur Yf ip&longs;i BD parallela.

~~LIBER PRIMUS.~~

~~Problematis hujus &longs;olutiones alias Wrennus & Walli&longs;ius olim excogitarunt.~~

~~PROPOSITIO XXIX. PROBLEMA XXI.~~

~~Trajectoriam &longs;pecie datam de&longs;cribere, quæ a rectis quatuor po&longs;itione datis in partes &longs;ecabitur, ordine, &longs;pecie & proportione datas.~~

~~De&longs;cribenda &longs;it Trajectoria~~

fghi, quæ &longs;imilis &longs;it Lincæ curvæ FGHI, & cujus partes fg, gh, hiillius partibus FG, GH, HI &longs;imiles & proportionales, rectis AB & AD, AD & BD, BD& CE po&longs;itione datis, prima primis, &longs;ecunda &longs;ecundis, tertia tertiis interjaceant. ~~Actis rectis FG, GH, HI, FI, de&longs;cribatur (per Lem.~~ ~~XXVII.) Trapezium fghiquod &longs;it Trapezio FGHI &longs;imile & cujus anguli f, g, h, i tangant rectas illas po&longs;itione datas AB, AD, BD, CE, &longs;inguli &longs;ingulas dicto ordine.~~ ~~Dein circa hoc Trapezium de&longs;cribatur Trajectoria curvæ Lineæ FGHI con&longs;imilis.~~

~~DE MOTU CORPORUM~~

~~Scholium.~~

~~Con&longs;trui etiam pote&longs;t hoc Problema ut &longs;equitur.~~ ~~Junctis FG, GH, HI, FI produc GF ad V, jungeque FH, IG, & angulis FGH, VFH fac angulos CAK, DAL æquales.~~ Concurrant AK, AL cum recta BD in K & L, & inde agantur KM, LN,quarum KM con&longs;tituat angulum AKM æqualem angulo GHI,&longs;itque ad AK ut e&longs;t HI ad GH; & LN con&longs;tituat angulum ALN æqualem angulo FHI, &longs;itque ad AL ut HI ad FH. Ducantur autem AK, KM, AL, LN ad eas partes linearum AD, AK, AL, ut literæ CAKMC, ALKA, DALND eodem ordine cum literis FGHIF in orbem redeant; & act MN occurrat rectæ CE in i. Fac angulum iEP æqualem angulo IGF,

&longs;itque PE ad Ei ut FG ad GI; & per P agatur PQf, quæ cum recta ADE contineat angulum PQE æqualem angulo FIG, rectæque AB occurrat in f, & jungatur fi. Agantur aurem PE & PQ ad eas partes linearum CE, PE, ut literarum PEiP & PEQP idem &longs;it ordo circularis qui literarum FGHIF,& &longs;i &longs;uper linea fi eodem quoque literarum ordine con&longs;tituatur Trapezium fghi Trapezio FGHI &longs;imile, & circum&longs;cribatur Trajectoria &longs;pecie data, &longs;olvetur Problema.

~~Hactenus de Orbibus inveniendis.~~ ~~Supere&longs;t ut Motus corporum in Orbibus inventis determinemus.~~

~~LIBER PRIMUS.~~

~~SECTIO VI.~~

~~De Inventione Motuum in Orbibus datis.~~

~~PROPOSITIO XXX. PROBLEMA XXII.~~

~~Corporis in data Trajectoria Parabolica moti invenire locum ad tempus a&longs;&longs;ignatum.~~

~~Sit S umbilicus & A vertex principa~~

lis Parabolæ, &longs;itque 4 ASXM æquale areæ Parabolicæ ab&longs;cindendæ APS,quæ radio SP, vel po&longs;t exce&longs;&longs;um corporis de vertice de&longs;cripta fuit, vel ante appul&longs;um ejus ad verticem de&longs;cribenda e&longs;t. ~~Innote&longs;cit quantitas areæ illius ab&longs;cindendæ ex tempore ip&longs;i proportionali.~~ Bi&longs;eca AS in G, erigeque perpendiculum GH æquale 3 M, & Circulus centro H, intervallo HSde&longs;criptus &longs;ecabit Parabolam in loco quæ&longs;ito P. Nam, demi&longs;&longs;a ad axem perpendiculari PO & ducta PH, e&longs;t AGq+GHq (=HP q=―AO-AG: quad.+―PO-GH: quad.)= AOq+POq-2 GAO-2GHXPO+AGq+GHque Unde 2 GHXPO (=AOq+POq-2GAO)=AOq+1/4 POquePro AOq &longs;cribe (AOXPOq/4AS); &, applicatis terminis omnibus ad 3PO ducti&longs;que in 2AS, fiet 4/3 GHXAS(=1/6AOXPO+1/2 ASXPO =(AO+3AS/6)XPO=(4AO-3SO/6)XPO=areæ ―APO-SPO)=areæ APS. Sed GH erat 3 M, & inde 4/3 GHXAS e&longs;t 4 ASXM. ~~Ergo area ab&longs;ci&longs;&longs;a APS æqualis e&longs;t ab&longs;cindendæ 4ASXM. q.E.D.~~

Corol. 1. Hinc GH e&longs;t ad AS, ut tempus quo corpùs de&longs;crip&longs;it arcum AP ad tempus quo corpus de&longs;crip&longs;it arcum inter verticem A & perpendiculum ad axem ab umbilico S erectum.

Corol. 2. Et Circulo ASP per corpus motum P perpetuo tran&longs;eunte, velocitas puncti H e&longs;t ad velocitatem quam corpus habuit in vertice A, ut 3 ad 8; adeoque in ea etiam ratione e&longs;t linea GHad lineam rectam quam corpus tempore motus &longs;ui ab A ad P, ea cum velocitate quam habuit in vertice A, de&longs;cribere po&longs;&longs;et.

~~DE MOTU CORPORUM~~

Corol. 3. Hinc etiam vice ver&longs;a inveniri pote&longs;t tempus quo corpus de&longs;crip&longs;it arcum quemvis a&longs;&longs;ignatum AP. Junge AP & ad medium ejus punctum erige perpendiculum rectæ GH occurrens in H.

~~LEMMA XXVIII.~~

~~Nulla extat Figura Ovalis cujus area, rectis pro lubitu ab&longs;ci&longs;&longs;a, po&longs;&longs;it per æquationes numero terminorum ac dimen&longs;ionum finitas generaliter inveniri.~~

Intra Ovalem detur punctum quodvis, circa quod ceu polum revolvatur perpetuo linea recta, uniformi cum motu, & interea in recta illa exeat punctum mobile de polo, pergatque &longs;emper ea cum velocitate, quæ &longs;it ut rectæ illius intra Ovalem quadratum. ~~Hoc motu punctum illud de&longs;cribet Spiralem gyris infinitis.~~ Jam &longs;i areæ Ovalis a recta illa ab&longs;ci&longs;&longs;æ incrementum per finitam æquationem inveniri pote&longs;t, invenietur etiam per eandem æquationem di&longs;tantia puncti a polo, quæ huic areæ proportionalis e&longs;t, adeoque omnia Spiralis puncta per æquationem finitam inveniri po&longs;&longs;unt: & propterea rectæ cuju&longs;vis po&longs;itione datæ inter&longs;ectio cum Spirali inveniri etiam pote&longs;t per æquationem finitam. Atqui recta omnis infinite producta Spiralem &longs;ecat in punctis numero infinitis, & æquatio, qua inter&longs;ectio aliqua duarum linearum invenitur, exhibet earum inter&longs;ectiones omnes radicibus totidem, adeoque a&longs;cendit ad rot dimen&longs;iones quot &longs;unt inter&longs;ectiones. Quoniam Circuli duo &longs;e mutuo &longs;ecant in punctis duobus, inter&longs;ectio una non invenietur ni&longs;i per æquationem duarum dimen&longs;ionum, qua inter&longs;ectio altera etiam inveniatur. Quoniam duarum &longs;ectionum Conicarum quatuor e&longs;&longs;e po&longs;&longs;unt inter&longs;ectiones, non pote&longs;t aliqua earum generaliter inveniri ni&longs;i per æquationem quatuor dimen&longs;ionum, qua omnes &longs;imul inveniantur. Nam &longs;i inter&longs;ectiones illæ &longs;eor&longs;im quærantur, quoniam eadem e&longs;t omnium lex & conditio, idem erit calculus in ca&longs;u unoquoque & propterea eadem &longs;emper conclu&longs;io, quæ igitur debet omnes inter&longs;ectiones &longs;imul complecti & indifferenter exhibere. Unde etiam inter&longs;ectiones Sectionum Conicarum & Curvarum ter tiæ pote&longs;tatis, eo quod &longs;ex e&longs;&longs;e po&longs;&longs;unt, &longs;imul prodeunt per æquationes &longs;ex dimen&longs;ionum, & inter&longs;ectiones duarum Curvarum tertiæ pote&longs;tatis, quia novem e&longs;&longs;e po&longs;&longs;unt, &longs;imul prodeunt per æquationes dimen&longs;ionum novem. ~~Id ni&longs;i nece&longs;&longs;ario fieret, reducere liccret Problemata omnia Solida ad Plana, & plu&longs;quam Solida ad Solida.~~ ~~Loquor hic de Curvis pote&longs;tate irreducibilibus.~~ Nam &longs;i æquatio per quam Curva definitur, ad inferiorem pote&longs;tatem reduci po&longs;&longs;it: Curva non erit unica, &longs;ed ex duabus vel pluribus compo&longs;ita, quarum inter&longs;ectiones per calculos diver&longs;os &longs;eor&longs;im inveniri po&longs;&longs;unt. Ad eundem modum inter&longs;ectiones binæ rectarum & &longs;ectionum Conicarum prodeunt &longs;emper per æquationes duarum dimen&longs;ionum; ternæ rectarum & Curvarum irreducibilium tertiæ pote&longs;tatis per æquationes trium, quaternæ rectarum & Curvarvm irreducibilium quartæ pote&longs;tatis per æquationes dimen&longs;ionum quatuor, & &longs;ic in infinitum. Ergo rectæ & Spiralis inter&longs;ectiones numero infinitæ, cum Curva hæc &longs;it &longs;implex & in Curvas plures irreducibilis, requirunt æquationes numero dimen&longs;ionum & radicum infinitas, quibus omnes po&longs;&longs;unt &longs;imul exhiberi. ~~E&longs;t enim eadem omnium lex & idem calculus.~~ Nam &longs;i a polo in rectam illam &longs;ecantem demittatur perpendiculum, & perpendiculum illud una cum &longs;ecante revolvatur circa polum, inter&longs;ectiones Spiralis tran&longs;ibunt in &longs;e mutuo, quæque prima erat &longs;eu proxima, po&longs;t unam revolutionem &longs;ecunda erit, po&longs;t duas tertia, & &longs;ic deinceps: nec interea mutabitur æquatio ni&longs;i pro mutata magnitudine quantitatum per quas po&longs;itio &longs;ecantis determinatur. Unde cum quantitates illæ po&longs;t &longs;ingulas revolutiones redeunt ad magnitudines primas, æquatio redibit ad formam primam, adeoque una eademque exhibebit inter&longs;ectiones omnes, & propterea radices habebit numero infinitas, quibus omnes exhiberi po&longs;&longs;unt. Nequit ergo inter&longs;ectio rectæ & Spiralis per æquationem finitam generaliter inveniri, & idcirco nulla extat Ovalis cujus area, rectis imperatis ab&longs;ci&longs;&longs;a, po&longs;&longs;it per talem æquationem generaliter exhiberi.

~~LIBER PRIMUS.~~

Eodem argumento, &longs;i intervallum poli & puncti, quo Spiralis de&longs;cribitur, capiatur Ovalis perimetro ab&longs;ci&longs;&longs;æ proportionale, probari pote&longs;t quod longitudo perimetri nequit per finitam æquationem generaliter exhiberi. ~~De Ovalibus autem hic loquor quæ non tanguntur a figuris conjugatis in infinitum pergentibus.~~

~~DE MOTU CORPORUM~~

~~Corollarium.~~

Hinc area Ellip&longs;eos, quæ radio ab umbilico ad corpus mobile ducto de&longs;cribitur, non prodit ex dato tempore, per æquationem finitam; & propterea per de&longs;criptionem Curvarum Geometrice rationalium determinari nequit. Curvas Geometrice rationales appello quarum puncta omnia per longitudines æquationibus definitas, id e&longs;t, per longitudinum rationes complicatas, determinari po&longs;&longs;unt; cætera&longs;que (ut Spirales, Quadratrices, Trochoides) Geometrice irrationales. ~~Nam longitudines quæ &longs;unt vel non &longs;unt ut numerus ad numerum (quemadmodum in decimo Elementorum) &longs;unt Arithmetice rationales vel irrationales.~~ ~~Aream igitur Ellip&longs;eos tempori proportionalem ab&longs;cindo per Curvam Geometrice irrationalem ut &longs;equitur.~~

~~PROPOSITIO XXXI. PROBLEMA XXIII.~~

~~Corporis in data Trajectoria Elliptica moti invenire locum ad tempus a&longs;&longs;ignatum.~~

~~Ellip&longs;eos APB &longs;it A vertex principalis, S umbilicus, & Ocentrum, &longs;itque P corporis locus inveniendus.~~ ~~Produc OA ad G,ut &longs;it OG ad OA ut OA ad OS. Erige perpendiculum GH, centroque~~

O & intervallo OG de&longs;cribe circulum EFG, & &longs;uper regula GH,ceu fundo, progrediatur Rota GEF revolvendo circa axem &longs;uum, & interea puncto &longs;uo A de&longs;cribendo Trochoidem ALI. Quo facto, cape GK in ratione ad Rotæ perimetrum GEFG, ut e&longs;t tempus quo corpus progrediendo ab A de&longs;crip&longs;it arcum AP, ad tempus revolutionis unius in Ellip&longs;i. ~~Erigatur perpendiculum KLoccurrens Trochoidi in L, & acta LP ip&longs;i KG parallela occurret Ellip&longs;i in corporis loco quæ&longs;ito P.~~

~~LIBER PRIMUS.~~

Nam centro O, intervallo OA de&longs;cribatur &longs;emicirculus AQB,& arcui AQ occurrat LP producta in Q, junganturque SQ, OqueArcui EFG occurrat OQ in F, & in eandem OQ demittatur perpendiculum SR. Area APS e&longs;t ut area AQS, id e&longs;t, ut differentia inter &longs;ectorem OQA & triangulum OQS, &longs;ive ut differentia rectangulorum 1/2 OQXAQ & 1/2 OQXSR, hoc e&longs;t, ob datam 1/2 OQ, ut differentia inter arcum AQ & rectam SR, adeoque (ob æqualitatem datarum rationum SR ad &longs;inum arcus AQ, OS ad OA, OA ad OG, AQ ad GF, & divi&longs;im AQ-SR ad GF-&longs;in. ~~arc. AQ) ut GK differentia inter arcum GF & &longs;inum arcus Aque que E. D.~~

~~Scholium.~~

~~Cæterum, cum difficilis &longs;it hujus Curvæ de&longs;criptio, præ&longs;tat &longs;olutionem vero proximam adhibere.~~ Inveniatur tum angulus quidam B, qui &longs;it ad angulum graduum 57,29578, quem arcus radio æqualis &longs;ubtendit, ut e&longs;t umbilicorum di&longs;tantia SH ad Ellip&longs;eos diametrum AB; tum etiam longitudo quædam L, quæ &longs;it ad radium in eadem ratione inver&longs;e. ~~Quibus &longs;emel inventis, Problema deinceps confit per &longs;equentem Analy&longs;in.~~ ~~Per con&longs;tructionem quamvis (vel.~~ ~~utcunque conjec~~

turam faciendo) cogno&longs;catur corporis locus P proximus vero ejus loco p. Demi&longs;&longs;aque ad axem Ellip&longs;eos ordinatim applicata PR, ex proportione diametrorum Ellip&longs;eos, dabitur Circuli circum&longs;cripti AQB ordinatim applicata RQ, quæ &longs;inus e&longs;t anguli AOQ exi&longs;tente AO radio. ~~Sufficit angulum illum rudi calculo in numeris proximis invenire.~~ Cogno&longs;catur etiam angulus tempori propor- tionalis, id e&longs;t, qui &longs;it ad quatuor rectos, ut e&longs;t tempus quo corpus de&longs;crip&longs;it arcum Ap, ad tempus revolutionis unius in Ellip&longs;i. ~~Sit angulus i&longs;te N.~~ Tum capiatur & angulus D ad angulum B, ut e&longs;t &longs;inus i&longs;te anguli AOQ ad radium, & angulus E ad angulum N-AOQ+D, ut e&longs;t longitudo L ad longitudinem eandem L co&longs;inu anguli AOQ diminutam, ubi angulus i&longs;te recto minor e&longs;t, auctam ubi major. Po&longs;tea capiatur tum angulus F ad angulum B, ut e&longs;t &longs;inus anguli AOQ+E ad radium, tum angulus G ad angulum N-AOQ-E+F ut e&longs;t longitudo L ad longitudinem eandem co&longs;inu anguli AOQ+E diminutam ubi angulus i&longs;te recto minor e&longs;t, auctam ubi major. Tertia vice capiatur angulus H ad angulum B, ut e&longs;t &longs;inus anguli AOQ+E+G ad radium; & angulus I ad angulum N-AOQ-E-G+H, ut e&longs;t longitudo L ad eandem longitudinem co&longs;inu anguli AOQ+E+G diminutam, ubi angulus i&longs;te re
cto minor e&longs;t, auctam ubi major. ~~Et &longs;ic pergere licet in infinitum.~~ ~~Denique capiatur angulus AOq æqualis angulo AOQ+E +G+I+&c.~~ e t ex co&longs;inu ejus Or& ordinata pr, quæ e&longs;t ad &longs;inum ejus qr ut Ellip&longs;eos axis minor ad axem majorem, habebitur corporis locus correctus p. Si quando angulus N-AOQ+D negativus e&longs;t, debet &longs;ignum+ip&longs;ius E ubique mutari in-, & &longs;ignum-in+. Idem intelligendum e&longs;t de &longs;ignis ip&longs;orum G & I, ubi anguli N-AOQ-E+F, & N-AOQ-E-G+H negativi prodeunt. ~~Convergit autem &longs;eries infinita AOQ+E+G+I+&c.~~ ~~quam celerrime, adeo ut vix unquam opus fuerit ultra progredi quam ad terminum &longs;ecundum E.~~ ~~Et fundatur calculus in hoc Theoremate, quod area APS &longs;it ut differentia inter arcum AQ & rectam ab umbilico S in Radium OQ perpendiculariter demi&longs;&longs;am.~~

~~DE MOTU CORPORUM~~

~~Non di&longs;&longs;imili calculo conficitur Problema in Hyperbola.~~ ~~Sit ejus Centrum O, Vertex A, Umbilicus S & A&longs;ymptotos OK. Cog- no&longs;catur quantitas areæ ab&longs;cindendæ tempori proportionalis.~~ ~~Sit ea A, & fiat conjectura de po&longs;itione rectæ SP, quæ aream APSab&longs;cindat veræ proximam.~~ ~~Jun~~

gatur OP, & ab A & P ad A&longs;ymptoton agantur AI, PKA&longs;ymptoto alteri parallelæ, & per Tabulam Logarithmorum dabitur Area AIKP, eique æqualis area OPA, quæ &longs;ubducta de triangulo OPS relinquet aream ab&longs;ci&longs;&longs;am APS. Applicando areæ ab&longs;cindendæ A & ab&longs;ci&longs;&longs;æ APSdifferentiam duplam 2 APS-2 A vel 2 A-2 APS ad lineam SN, quæ ab umbilico S in tangentem PT perpendicularis e&longs;t, orietur longitudo chordæ Pque In&longs;cribatur autem chorda illa PQ inter A & P, &longs;i area ab&longs;ci&longs;&longs;a APSmajor &longs;it area ab&longs;cindenda A, &longs;ecus ad puncti P contrarias partes: & punctum Q erit locus corporis accuratior. ~~Et computatione repetita invenietur idem accuratior in perpetuum.~~

~~LIBER PRIMUS.~~

~~Atque his calculis Problema generaliter confit Analytice.~~ ~~Verum u&longs;ibus A&longs;tronomicis accommodatior e&longs;t calculus particularis qui &longs;equitur.~~ Exi&longs;tentibus AO, OB, OD &longs;emiaxibus Ellip&longs;eos, & L ip&longs;ius latere recto, ac D differentia inter &longs;emiaxem minorem OD& lateris recti &longs;emi&longs;&longs;em 1/2 L; quære tum angulum Y, cujus &longs;inus &longs;it ad Radium ut e&longs;t rectangu

lum &longs;ub differentia illa D, & &longs;emi&longs;umma axium AO+ODad quadratum axis majoris AB; tum angulum Z, cujus &longs;inus &longs;it ad Radium ut e&longs;t duplum rectangulum &longs;ub umbilicorum di&longs;tantia SH & differentia illa D ad triplum quadratum &longs;emiaxis majoris AO. His angulis &longs;emel inventis; locus corporis &longs;ic deinceps determinabitur. Sume angulum T proportionalem tempori quo arcus BP de&longs;criptus e&longs;t, &longs;cu motui medio (ut loquuntur) æqualem; & angulum V (primam medii motus æquationem) ad angulum Y (æquationem maximam primam) ut e&longs;t &longs;inus dupli anguli T ad Radium; atque angulum X (æquationem &longs;ecundam) ad angulum Z (æquationem maximam &longs;ecundam) ut e&longs;t cubus &longs;inus anguli T ad cubum Radii. Angulorum T, V, X vel &longs;ummæ T+X+V, &longs;i angulus T recto minor e&longs;t, vel differentiæ T+X-V, &longs;i is recto major e&longs;t recti&longs;que duobus minor, æqualem cape angulum BHP (motum medium æquatum;) &, &longs;i HP occurrat Ellip&longs;i in P, acta SP ab&longs;cindet aream BSP tempori proportionalem quamproxime. Hæc Praxis &longs;atis expedita videtur,
propterea quod angulorum perexiguorum V & X (in minutis &longs;ecundis, &longs;i placet, po&longs;itorum) figuras duas ter&longs;ve primas invenire &longs;ufficit. ~~Sed & &longs;atis accurata e&longs;t ad Theoriam Planetarum.~~ ~~Nam in Orbe vel Martis ip&longs;ius, cujus Æquatio centri maxima e&longs;t graduum decem, error vix &longs;uperabit minutum unum &longs;ecundum.~~ ~~Invento autem angulo motus medii æquati BHP, angulus veri motus BSP & di&longs;tantia SP in promptu &longs;unt per Wardi methodum noti&longs;&longs;imam.~~

~~DE MOTU CORPORUM~~

~~Hactenus de Motu corporum in lineis Curvis.~~ ~~Fieri autem pote&longs;t ut mobile recta de&longs;cendat vel recta a&longs;cendat, & quæ ad i&longs;tiu&longs;modi Motus &longs;pectant, pergo jam exponere.~~

~~LIBER PRIMUS.~~

~~SECTIO VII.~~

~~De Corporum A&longs;cen&longs;u & De&longs;cen&longs;u Rectilineo.~~

~~PROPOSITIO XXXII. PROBLEMA XXIV.~~

~~Po&longs;ito quod Vis centripeta &longs;it reciproce proportionalis quadrato di&longs;tantiæ locorum a centro, Spatia definire quæ corpus recta cadendo datis temporibus de&longs;cribit.~~

~~Cas. 1. Si Corpus non cadit perpendicu~~

~~lariter de&longs;cribet id, per Corol.~~ ~~1. Prop.~~ ~~XIII, Sectionem aliquam Conicam cujus umbilicus congruit cum centro virium.~~ Sit Sectio illa Conica ARPB & umbilicus ejus S.Et primo &longs;i Figura Ellip&longs;is e&longs;t, &longs;uper hujus axe majore AB de&longs;cribatur Semicirculus ADB, & per corpus decidens tran&longs;eat recta DPC perpendicularis ad axem; acti&longs;que DS, PS erit area ASD areæ ASP atque adeo etiam tempori proportionalis. ~~Manente axe AB minuatur perpetuo latitudo Ellip&longs;eos, & &longs;emper manebit area ASDtempori proportionalis.~~ ~~Minuatur latitudo illa in infinitum: &, Orbe APB jam coincidente cum axe AB & umbilico S cum axis termino B, de&longs;cendet corpus in recta AC, & area ABD evadet tempori proportionalis.~~ Dabitur itaque Spatium AC,quod corpus de loco A perpendiculariter cadendo tempore dato de&longs;cribit, &longs;i modo tempori proportionalis capiatur area ABD, & a puncto D ad rectam AB demittatur perpendicularis DC. que E. I.

~~DE MOTU CORPORUM~~

Cas. 2. Si Figura illa RPB Hyperbola e&longs;t, de&longs;cribatur ad eandem diametrum principalem AB Hyperbola rectangula BED:& quoniam areæ CSP, CBfP, SPfB &longs;unt ad areas CSD, CBED, SDEB, &longs;ingulæ ad &longs;ingulas, in data ratione altitudinum CP, CD; & area SPfB

~~proportionalis e&longs;t tempori quo corpus P movebitur per arcum PfB; erit etiam area SDEB eidem tempori proportionalis.~~ Minuatur latus rectum Hyperbolæ RPB in infinitum manente latere tran&longs;ver&longs;o, & coibit arcus PB cum recta CB & umbilicus S cum vertice B & recta SD cum recta BD. Proinde area BDEB proportionalis erit tempori quo corpus C recto de&longs;cen&longs;u de&longs;cribit lineam CB. que E. I.

Cas. 3. Et &longs;imili argumento &longs;i Figura RPB Parabola e&longs;t, & eodem vertice principali B de&longs;cribatur alia Parabola BED,quæ &longs;emper maneat data interea dum Parabola prior in cujus perimetro corpus P movetur, diminuto & in nihilum redacto ejus latere recto, conveniat cum linea CB; fiet &longs;egmentum Parabolicum BDEB proportionale tempori quo corpus illud P vel C de&longs;cendet ad centrum S vel B. que E. I.

~~PROPOSITIO XXXIII. THEOREMA IX.~~

Po&longs;itis jam inventis, dico quod corporis cadentis Velocitas in loco quovis C est ad velocitatem corporis centro B intervallo BC Circulum de&longs;cribentis, in &longs;ubduplicata ratione quam AC, di&longs;tantia corporis a Circuli vel Hyperbolæ rect angulæ vertice ulteriore A, habet ad Figuræ &longs;emidiametrum principalem 1/2 AB.

Bi&longs;ecetur AB, communis utriu&longs;que Figuræ RPB, DEB diameter, in O; & agatur recta PT quæ tangat Figuram RPB in P, atque etiam &longs;ecet communem illam diametrum AB (&longs;i opus e&longs;t productam)

~~in T; &longs;itque SY ad hanc rectam, & BQ ad~~

~~hanc diametrum perpendicularis, atque Figuræ RPB latus rectum ponatur L.~~ ~~Con&longs;tat per Cor.~~ ~~9. Prop.~~ XVI, quod corporis in linea RPB circa centrum S moventis velocitas in loco quovis P &longs;it ad velocitatem corporis intervallo SP circa idem centrum Circulum de&longs;cribentis in &longs;ubduplicata ratione rectanguli 1/2 LXSP ad SY quadratum. ~~E&longs;t autem ex Conicis ACB ad CPq ut 2 AO ad L, adeoque (2CPqXAO/ACB) æquale L.~~ Ergo velocitates illæ &longs;unt ad invicem in &longs;ubduplicata ratione (CPqXAOXSP/ACB) ad SY quad. Porro ex Conicis e&longs;t CO ad BO ut BO ad TO,& compo&longs;ite vel divi&longs;im ut CB ad BT.Unde vel dividendo vel componendo fit BO-vel+CO ad BO ut CT ad BT, id e&longs;t AC ad AO ut CP ad BQ; indeque (CPqXAOXSP/ACB) æquale e&longs;t (BQqXACXSP/AOXBC.) Minuatur jam in infinitum Figuræ RPB latitudo CP, &longs;ic ut punctum P coeat cum puncto C, punctumque S cum puncto B, & linea SP cum linea BC, lineaque SY cum linea BQ; & corporis jam recta de&longs;cendentis in linea CB velocitas fiet ad velocitatem corporis centro B intervallo BC Circulum de&longs;cribentis, in &longs;ubduplicata ratione ip&longs;ius (BQqXACXSP/AOXBC) ad SYq, hoc e&longs;t (neglectis æqualitatis rationibus SP ad BC & BQq ad SYq) in &longs;ubduplicata ratione AC ad AO &longs;ive 1/2 AB. que E. D.

~~LIBER PRIMUS.~~

~~Corol. 1. Punctis B & S coeuntibus, fit TC ad TS ut ACad AO.~~

~~Corol. 2. Corpus ad datam a centro di&longs;tantiam in Circulo quovis revolvens, motu &longs;uo &longs;ur&longs;um ver&longs;o a&longs;cendet ad duplam &longs;uam a centro di&longs;tantiam.~~

~~DE MOTU CORPORUM~~

~~PROPOSITIO XXXIV. THEOREMA X.~~

~~Si Figura BED Parabola e&longs;t, dico~~

~~quod corporis cadentis Velocitas in loco quovis C æqualis e&longs;t velocitati qua corpus centro B dimidio intervalli &longs;ui BC Circulum uniformiter de&longs;cribere potest.~~

~~Nam corporis Parabolam RPB circa centrum S de&longs;cribentis velocitas in loco quovis P (per Corol.~~ ~~7. Prop.~~ ~~XVI) æqualis e&longs;t velocitati corporis dimidio intervalli SP Circulum circa idem centrum S uniformiter de&longs;cribentis.~~ Minuatur Parabolæ latitudo CP in infinitum eo, ut arcus Parabolicus PfB cum recta CB, centrum S cum vertice B,& intervallum SP cum intervallo BC coincidat, & con&longs;tabit Propo&longs;itio. que E. D.

~~PROPOSITIO XXXV. THEOREMA XI.~~

Ii&longs;dem po&longs;itis, dico quod area Figuræ DES, radio indefinito SD de&longs;cripta, æqualis &longs;it areæ quam corpus, radio dimidium lateris recti Figuræ DES æquante, circa centrum S uniformiter gyrando, eodem tempore de&longs;cribere potest.

~~Nam concipe corpus C quam minima temporis particula lineolam Cc cadendo de&longs;cribere, & interea corpus aliud K, uniformiter in Circulo OKk circa centrum S gyrando, arcum Kk de&longs;cribere.~~ ~~Erigantur perpendicula CD, cd occurrentia Figuræ DESin D, d. Jungantur SD, Sd, SK, Sk & ducatur Dd axi AS ocrens in T, & ad eam demittatur perpendiculum SY.~~

~~Ca&longs;. 1. Jam &longs;i Figura DES Circulus e&longs;t vel Hyperbola, bi&longs;ece tur ejus tran&longs;ver&longs;a diameter AS in O, & erit~~

~~SO dimidium lateris recti.~~ ~~Et quoniam e&longs;t TC ad TD ut Cc ad Dd, & TD ad TS ut CD ad SY, erit ex æquo TC ad TS ut CDXCc ad SYXDd. Sed per Corol.~~ ~~1. Prop.~~ ~~XXXIII, e&longs;t TC ad TS ut AC ad AO, puta &longs;i in coitu punctorum D, d capiantur linearum rationes ultimæ.~~ Ergo AC e&longs;t ad (AO &longs;eu) SKut CDXCc ad SYXDd. Porro corporis de&longs;cendentis velocitas in C e&longs;t ad velocitatem corporis Circulum intervallo SC circa centrum S de&longs;cribentis in &longs;ubduplicata ratione AC ad (AO vel) SK (per Prop. ~~XXXIII.) Et hæc velocitas ad velocitatem corporis de&longs;cribentis Circulum OKk in &longs;ubduplicata ratione SK ad SC per Cor.~~ ~~6. Prop.~~ IV, & ex æquo velocitas prima ad ultimam, hoc e&longs;t lineola Cc ad arcum Kk in &longs;ubduplicata ratione AC ad SC,id e&longs;t in ratione AC ad CD. Quare e&longs;t CDXCcæquale ACXKk, & propterea AC ad SK ut ACXKk ad SYXDd, indeque SKXKk æquale SYXDd, & 1/2 SKXKk æquale 1/2 SYXDd,id e&longs;t area KSk æqualis areæ SDd. Singulis igitur temporis particulis generantur arearum duarum particulæ KSk, & SDd, quæ, &longs;i magnitudo earum minuatur & numerus augeatur in infinitum, rationem obtinent æqualitatis, & propterea (per Corollarium Lemmatis IV) areæ totæ &longs;imul genitæ &longs;unt &longs;emper æquales, que E. D.

~~LIBER PRIMUS.~~

Ca&longs;. 2. Quod &longs;i Figura DES Parabola &longs;it, invenietur e&longs;&longs;e ut &longs;upra CDXCc ad SYXDd ut TC ad TS, hoc e&longs;t ut 2 ad 1, adeoque 1/4 CDXCc æquale e&longs;&longs;e 1/2 SYXDd. Sed corporis cadentis velocitas in C æqualis e&longs;t velocitati qua Circulus intervallo 1/2 SCuniformiter de&longs;cribi po&longs;&longs;it (per Prop. ~~XXXIV) Et hæc velocitas ad velocitatem qua Circulus radio SK de&longs;cribi po&longs;&longs;it, hoc e&longs;t, lineola Cc ad arcum Kk (per Corol.~~ ~~6. Prop.~~ IV) e&longs;t in &longs;ubduplicata ratione SK ad 1/2 SC, id e&longs;t, in ratione SK ad 1/2 CD. Quare e&longs;t 1/2 SKXKkæquale 1/4 CDXCc, adeoque æquale 1/2 SYXDd, hoc e&longs;t, area KSkæqualis areæ SDd, ut &longs;upra. que E. D.

~~DE MOTU CORPORUM~~

~~PROPOSITIO XXXVI. PROBLEMA XXV.~~

~~Corporis de loco dato A cadentis determinare Tempora de&longs;cen&longs;us.~~

Super diametro AS (di&longs;tantia corporis a centro &longs;ub initio) de&longs;cribe Semicirculum ADS, ut & huic æqualem Semicirculum OKH circa centrum S. De corporis loco quovis C erige ordinatim applicatam CD. Junge SD, & areæ ASD æqualem con&longs;titue &longs;ectorem OSK. Patet per PropXXXV, quod corpus cadendo de&longs;cribet &longs;patium ACeodem Tempore quo corpus aliud uniformiter circa centrum S gyrando, de&longs;cribere pote&longs;t arcum OK. que E. F.

~~PROPOSITIO XXXVII. PROBLEMA XXVI.~~

~~Corporis de loco dato &longs;ur&longs;um vel deor&longs;um projecti definire Tempora a&longs;cen&longs;us vel de&longs;cen&longs;us.~~

~~Exeat corpus de loco dato G &longs;ecundum~~

~~lineam ASG cum velocitate quacunque.~~ In duplicata ratione hujus velocitatis ad uniformem in Circulo velocitatem, qua corpus ad intervallum datum SG circa centrum S revolvi po&longs;&longs;et, cape GA ad 1/2 AS.Si ratio illa e&longs;t numeri binarii ad unitatem, punctum A infinite di&longs;tat, quo ca&longs;u Parabola vertice S, axe SC, latere quovis recto de&longs;cribenda e&longs;t. ~~Patet hoc per Prop.~~ ~~XXXIV.~~ ~~Sin ratio illa minor vel major e&longs;t quam 2 ad 1, priore ca&longs;u Circulus, po&longs;teriore Hyperbola rectangula &longs;uper diametro SA de&longs;cribi debet.~~ ~~Patet per Prop.~~ ~~XXXIII.~~ Tum centro S, intervallo æquante dimidium lateris recti, de&longs;cribatur Circulus HKk, & ad corporis a&longs;cendentis vel de&longs;cendentis loca duo quævis G, C,erigantur perpendicula GI, CD occurrentia Conicæ Sectioni vel Circulo in I ac D. Dein junctis SI, SD, fiant &longs;egmentis SEIS, SEDS, &longs;ec tores HSK, HSk æquales, & per Prop. ~~XXXV, corpus G de&longs;cribet &longs;patium GC eodem Tempore quo corpus K de&longs;cribere pote&longs;t arcum Kk.~~ ~~que E. F.~~

~~LIBER PRIMUS.~~

~~PROPOSITIO XXXVIII. THEOREMA XII.~~

Po&longs;ito quod Vis centripeta proportionalis &longs;it altitudini &longs;eu di&longs;tantiæ locorum a centro, dico quod cadentium Tempora, Velocitates & Spatia de&longs;cripta &longs;unt arcubus, arcuumque finibus rectis & &longs;inibus ver&longs;is re&longs;pective proportionalia.

~~Cadat corpus de loco quovis A &longs;ecun~~

dum rectam AS; & centro virium S, intervallo AS, de&longs;cribatur Circuli quadrans AE, &longs;itque CD &longs;inus rectus arcus cuju&longs;vis AD; & corpus A, Tempore AD, cadendo de&longs;cribet Spatium AC, inque loco C acquiret Velocitatem CD.

~~Demon&longs;tratur eodem modo ex Propo&longs;itione X, quo Propo&longs;itio XXXII, ex Propo&longs;itione XI demon&longs;trata fuit.~~

~~Corol. 1. Hinc æqualia &longs;unt Tempora quibus corpus unum de loco A cadendo pervenit ad centrum S, & corpus aliud revolvendo de&longs;cribit arcum quadrantalem ADE.~~

~~Corol. 2. Proinde æqualia &longs;unt Tempora omnia quibus corpora de locis quibu&longs;vis ad u&longs;que centrum cadunt.~~ ~~Nam revolventium tempora omnia periodica (per Corol.~~ ~~3. Prop.~~ ~~IV.) æquantur.~~

~~DE MOTU CORPORUM~~

~~PROPOSITIO XXXIX. PROBLEMA XXVII.~~

Po&longs;ita cuju&longs;cunque generis Vi centripeta, & conce&longs;&longs;is figurarum curvilinearum quadraturis, requiritu, corporis recta a&longs;cendentis vel de&longs;cendentis tum Velocitas in locis &longs;ingulis, tum Tempus quo corpus ad locum quemvis perveniet: Et contra.

~~De loco quovis A in recta ADEC cadat corpus E, deque loco ejus E erigatur &longs;emper perpendicularis EG, vi centripetæ in loco illo ad centrum C tendenti proportio~~

~~nalis: Sitque BFG linea curva quam punctum G perpetuo tangit.~~ ~~Coincidat autem EG ip&longs;o motus initio cum perpendiculari AB, & erit corporis Velocitas in loco quovis E ut areæ curvilineæ ABGE latus quadratum. que E. I.~~

In EG capiatur EM lateri quadrato areæ ABGE reciproce proportionalis, & &longs;it ALM linea curva quam punctum Mperpetuotangit, & erit Tempus quo corpus cadendo de&longs;cribit lineam AE ut area curvilinea ALME. que E. I.

Etenim in recta AE capiatur linea quam minima DE datæ longitudinis, &longs;itque DLF locus lineæ EMG ubi corpus ver&longs;abatur in D; & &longs;i ea &longs;it vis centripeta, ut areæ ABGElatus quadratum &longs;it ut de&longs;cendentis velocitas, erit area ip&longs;a in duplicata ratione velocitatis, id e&longs;t, &longs;i pro velocitatibus in D & E&longs;cribantur V & V+I, erit area ABFD ut VV, & area ABGE ut VV+2 VI+II, & divi&longs;im area DFGE ut 2 VI+II, adeoque (DFGE/DE) ut (2VI+II/DE), id e&longs;t, &longs;i primæ quantitatum na&longs;centium rationes &longs;umantur, longitudo DF ut quantitas (2VI/DE), adeoque etiam ut quantitatis hujus dimidium (IXV/DE). E&longs;t autem tempus quo corpus cadendo de&longs;cribit lineolam DE, ut lineola illa directe & velocitas V inver&longs;e, e&longs;tque vis ut velocitatis incrementum I directe & tempus inver&longs;e, adeoque &longs;i primæ na&longs;centium rationes &longs;umantur, ut (IXV/DE), hoc e&longs;t, ut longitudo DF. Ergo Vis ip&longs;i DF vel EGproportionalis facit ut corpus ea cum Velocitate de&longs;cendat quæ &longs;it ut areæ ABGE latus quadratum. que E. D.

~~LIBER PRIMUS.~~

Porro cum tempus, quo quælibet longitudinis datæ lineola DEde&longs;cribatur, &longs;it ut velocitas inver&longs;e adeoque ut areæ ABFD latus quadratum inver&longs;e; &longs;itque DL, atque adeo area na&longs;cens DLME,ut idem latus quadratum inver&longs;e: erit tempus ut area DLME, & &longs;umma omnium temporum ut &longs;umma omnium arearum, hoc e&longs;t (per Corol. ~~Lem.~~ ~~IV) Tempus totum quo linea AE de&longs;cribitur ut area tota AME. que E. D.~~

Corol. 1. Si P &longs;it locus de quo corpus cadere debet, ut, urgente aliqua uniformi vi centripeta nota (qualis vulgo &longs;upponitur Gravitas) velocitatem acquirat in loco D æqualem velocitati quam corpus aliud vi quacunque cadens acqui&longs;ivit eodem loco D,& in perpendiculari DF capiatur DR, quæ &longs;it ad DF ut vis illa uniformis ad vim alteram in loco D, & compleatur rectangulum PDRQ, eique æqualis ab&longs;cindatur area ABFD; erit A locus de quo corpus alterum cecidit. Namque completo rectangulo DRSE, cum &longs;it area ABFD ad aream DFGE ut VV ad 2VI, adeoque ut 1/2 V ad I, id e&longs;t, ut &longs;emi&longs;&longs;is velocitatis totius ad incrementum velocitatis corporis vi inæquabili cadentis; & &longs;imiliter area PQRD ad aream DRSE ut &longs;emi&longs;&longs;is velocitatis totius ad incrementum velocitatis corporis uniformi vi cadentis; &longs;intque incrementa illa (ob æqualitatem temporum na&longs;centium) ut vires generatrices, id e&longs;t, ut ordinatim applicatæ DF, DR,adeoque ut areæ na&longs;centes DFGE, DRSE; erunt (ex æquo) areæ totæ ABFD, PQRD ad invicem ut &longs;emi&longs;&longs;es totarum velocitatum, & propterea (ob æqualitatem velocitatum) æquantur.

Corol. 2. Unde &longs;i corpus quodlibet de loco quocunque D data cum velocitate vel &longs;ur&longs;um vel deor&longs;um projiciatur, & detur lex vis centripetæ, invenietur velocitas ejus in alio quovis loco e, erigendo ordinatam eg, & capiendo velocitatem illam ad velocitatem in loco D ut e&longs;t latus quadratum rectanguli PQRD area curvilinea DFge vel aucti, &longs;i locus e e&longs;t loco D inferior, vel diminuti, &longs;i is &longs;uperior e&longs;t, ad latus quadratum rectanguli &longs;olius PQRD, id e&longs;t, ut √PQRD+vel-DFge ad √PQRD.

~~DE MOTU CORPORUM~~

Corol. 3. Tempus quoque innote&longs;cet erigendo ordinatam em reciproce proportionalem lateri quadrato ex PQRD+vel-DFge,& capiendo tempus quo corpus de&longs;crip&longs;it lineam De ad tempus quo corpus alterum vi uniformi cecidit a P & cadendo pervenit ad D, ut area curvilinea DLme ad rectangulum 2PDXDL. Namque tempus quo corpus vi uniformi de&longs;cendens de&longs;crip&longs;it lineam PD e&longs;t ad tempus quo corpus idem de&longs;crip&longs;it lineam PE in &longs;ubduplicata ratione PD ad PE, id e&longs;t (lineola DE jamjam na&longs;cente) in ratione PD ad PD+1/2 DE &longs;eu 2PD ad 2PD+DE,& divi&longs;im, ad tempus quo corpus idem de&longs;crip&longs;it lineolam DEut 2PD ad DE, adeoque ut rectangulum 2PDXDL ad aream DLME; e&longs;tque tempus quo corpus utrumque de&longs;crip&longs;it lineolam DE ad tempus quo corpus alterum inæquabili motu de&longs;crip&longs;it lineam De ut area DLME ad aream DLme, & ex æquo tempus primum ad tempus ultimum ut rectangulum 2PDXDLad aream DLme.

~~SECTIO VIII.~~

~~De Inventione Orbium in quibus corpora Viribus quibu&longs;cunque centripetis agitata revolvuntur.~~

~~PROPOSITIO XL. THEOREMA XIII.~~

Si corpus, cogente Vi quacunque centripeta, moveatur utcunque, & corpus aliud recta a&longs;cendat vel de&longs;cendat, &longs;intque eorum Velocitates in aliquo æqualium altitudinum ca&longs;u æquales, Velocitates eorum in omnibus æqualibus altitudinibus erunt æquales.

De&longs;cendat corpus aliquod ab A per D, E, ad centrum C, & moveatur corpus aliud a V in linea curva VIKk, Centro C intervallis quibu&longs;vis de&longs;cribantur circuli concentrici DI, EK rectæ AC in D & E, curvæque VIK in I & K occurrentes. Jungatur IC occurrens ip&longs;i KE in N; & in IK demittatur perpendiculum NT; &longs;itque circumferentiarum circulorum intervallum DEvel IN quam minimum, & habeant corpora in D & I velocita- tes æquales. ~~Quoniam di&longs;tantiæ CD, CI æquantur, erunt vi~~

~~res centripetæ in D & I æquales.~~ ~~Exponantur hæ vires per æquales lineolas DE, IN; & &longs;i vis una IN (per Legum Corol.~~ 2.) re&longs;olvatur in duas NT & IT, vis NT, agendo &longs;ecundum lineam NT corporis cur&longs;ui ITK perpendicularem, nil mutabit velocitatem corporis in cur&longs;u illo, &longs;ed retrahet &longs;olummodo corpus a cur&longs;u rectilineo, facietque ip&longs;um de Orbis tangente perpetuo deflectere, inque via curvilinea ITKk progredi. In hoc effectu producendo vis illa tota con&longs;umetur: vis autem altera IT, &longs;ecundum corporis cur&longs;um agendo, tota accelerabit illud, ac dato tempore quam minimo accelerationem generabit &longs;ibi ip&longs;i proportionalem. Proinde corporum in D & I accelerationes æqualibus temporibus factæ (&longs;i &longs;umantur linearum na&longs;centium DE, IN, IK, IT, NT rationes primæ) &longs;unt ut lineæ DE, IT: temporibus autem inæqualibus ut lineæ illæ & tempora conjunctim. ~~Tempora autem quibus DE & IK de&longs;cribuntur, ob æqualitatem velocita~~

tum &longs;unt ut viæ de&longs;criptæ DE & IK, adeoque accelerationes, in cur&longs;u corporum per lineas DE & IK, funt ut DE & IT, DE & IK conjunctim, id e&longs;t ut DE quad & ITXIK rectangulum. Sed rectangulum ITXIK æquale e&longs;t IN quadrato, hoc e&longs;t, æquale DE quadrato; & propterea accelerationes in tran&longs;itu corporum a D & I ad E & K æquales gcnerantur. ~~Æquales igitur &longs;unt cor- porum velocitates in E & K & eodem argumento &longs;emper reperientur æquales in &longs;ub&longs;equentibus æqualibus di&longs;tantiis. que E. D.~~

~~LIBER PRIMUS.~~

~~DE MOTU CORPORUM~~

~~Sed & eodem argumento corpora æquivelocia & æqualiter a centro di&longs;tantia, in a&longs;cen&longs;u ad æquales di&longs;tantias æqualiter retardabuntur. q.E.D.~~

Corol. 1. Hinc &longs;i corpus vel funipendulum o&longs;cilletur, vel impedimento quovis politi&longs;&longs;imo & perfecte lubrico cogatur in linea curva moveri, & corpus aliud recta a&longs;cendat vel de&longs;cendat, &longs;intque velocitates eorum in eadem quacunque altitudine æquales: erunt velocitates eorum in aliis quibu&longs;cunque æqualibus altitudinibus æquales. Namque impedimento va&longs;is ab&longs;olute lubrici idem præ&longs;tatur quod vi tran&longs;ver&longs;a NT. Corpus eo non retardatur, non acceleratur, &longs;ed tantum cogitur de cur&longs;u rectilineo di&longs;cedere.

Corol. 2. Hinc etiam &longs;i quantitas P &longs;it mazima a centro di&longs;tantia, ad quam corpus vel o&longs;cillans vel in Trajectoria quacunque revolvens, deque quovis Trajectoriæ puncto, ea quam ibi habet velocitate &longs;ur&longs;um projectum a&longs;cendere po&longs;&longs;it; &longs;itque quantitas A di&longs;tantia corporis a centro in alio quovis Orbitæ puncto, & vis centripeta &longs;emper &longs;it ut ip&longs;ius A dignitas quælibet An-1, cujus Index n-1 e&longs;t numerus quilibet n unitate diminutus; velocitas corporis in omni altitudine A erit ut √Pn-An, atque adeo datur. ~~Namque velocitas recta a&longs;cendentis ac de&longs;cendentis (per Prop.~~ ~~XXXIX) e&longs;t in hac ip&longs;a ratione.~~

~~LIBER PRIMUS.~~

~~PROPOSITIO XLI. PROBLEMA XXVIII.~~

Po&longs;ita cuju&longs;cunque generis Vi centripeta & conce&longs;&longs;is Figurarum curvilinearum quadraturis, requiruntur tum Trajectoriæ in quibus corpora movebuntur, tum Tempora motuum in Trajectoriis inventis.

Tendat vis quælibet ad centrum C & invenienda &longs;it Trajectoria VITKk. Detur Circulus VXY centro C intervallo quovis CVde&longs;criptus, centroque eodem de&longs;cribantur alii quivis circuli ID, KE Trajectoriam &longs;ecantes in I & K rectamque CV in D & E.Age tum rectam CNIX &longs;ecantem circulos KE, VY in N & X,tum rectam CKY occurrentem circulo VXY in Y. Sint autem puncta I & K &longs;ibi invicem vicini&longs;&longs;ima, & pergat corpus ab V per I, T & K ad k; &longs;itque punctum A locus ille de quo corpus aliud cadere debet ut in loco D velocitatem acquirat æqualem velocitati corporis prioris in I; & &longs;tantibus quæ in Propo&longs;itione XXXIX, lineola IK, dato tempore quam minimo de&longs;cripta, erit ut velocitas atque adeo ut latus quadratum areæ ABFD, & triangulum ICK tempori proportionale dabitur, adeoque KN erit reciproce ut altitudo IC, id e&longs;t, &longs;i detur quantitas aliqua Q, & altitudo IC nominetur A, ut Q/A. Hanc quantitatem Q/A nominemus Z, & ponamus eam e&longs;&longs;e magnitudinem ip&longs;ius Q ut &longs;it in aliquo ca&longs;u √ ABFD ad Z ut e&longs;t IK ad KN, & erit in omni ca&longs;u √ABFD ad Z ut IK ad KN, & ABFD ad ZZ ut IKque ad KNque& divi&longs;im ABFD-ZZ ad ZZ ut IN quad ad KN quad, adeoque √ABFD-ZZ ad (Z &longs;eu)Q/A ut IN ad KN, & propterea AXKN æquale (QXIN/√ABFD-ZZ). Unde cum YXXXC &longs;it ad AXKN ut CXq ad AA, erit rectangulum YXXXC æquale (QXINXCX quad./AA√ABFD-ZZ). Igitur &longs;i in perpendiculo DF capiantur &longs;emper Db, Dc ip&longs;is (Q/2√ABFD-ZZ) & (QXCX quad./2AA√ABFD-ZZ) æquales re&longs;pective, & de&longs;cribantur curvæ lineæ ab, cd quas puncta b, c perpetuo tangunt; deque puncto V ad lineam AC erigatur perpendiculum Vad ab&longs;cindens areas curvilineas VDba, VDcd, & erigantur etiam ordinatæ Ez, Ex: quoniam rectangulum DbXIN &longs;eu DbzE æquale e&longs;t dimidio rectanguli AXKN, &longs;eu triangulo ICK; & rectangulum DcXIN &longs;eu DcxE æquale e&longs;t dimidio rectanguli YXXXC, &longs;eu triangulo XCY; hoc e&longs;t, quoniam arearum VDba, VIC æquales &longs;emper &longs;unt na&longs;centes particulæ DbzE, ICK, & arearum VDcd, VCX æquales &longs;emper &longs;unt na&longs;centes particulæ DcxE, XCY,erit area genita VDba æqualis areæ genitæ VIC, adeoque tempori proportionalis, & area genita VDcd æqualis Sectori genito VCX. Dato igitur tempore quovis ex quo corpus di&longs;ce&longs;&longs;it de loco V, dabitur area ip&longs;i proportionalis VDba, & inde dabitur corporis altitudo CD vel CI; & area VDcd, eique æqualis Sector VCX una cum ejus angulo VCI. Datis autem angulo VCI & altitudine CI datur locus I, in quo corpus completo illo tempore reperietur. q.E.I.

~~DE MOTU CORPORUM~~

~~Corol. 1. Hinc maximæ minimæque corporum altitudines, id e&longs;t Ap&longs;ides Trajectoriarum expedite inveniri po&longs;&longs;unt.~~ Sunt enim Ap&longs;ides puncta illa in quibus recta IC per centrum ducta incidit perpendiculariter in Trajectoriam VIK: id quod &longs;it ubi rectæ IK& NK æquantur, adeoque ubi area ABFD æqualis e&longs;t ZZ.

Corol. 2. Sed & angulus KIN, in quo Trajectoria alibi &longs;ecat lineam illam IC, ex data corporis altitudine IC expedite invenitur; nimirum capiendo &longs;inum ejus ad radium ut KN ad IK, id e&longs;t, ut Z ad latus quadratum areæ ABFD.

Corol. 3. Si centro C & vertice principali V de&longs;cribatur Sectio quælibet Conica VRS, & a quovis ejus puncto R agatur Tangens RToccurrens axi infinite producto CV in puncto T; dein juncta CRducatur recta CP, quæ æqualis &longs;it ab&longs;ci&longs;&longs;æ CT, angulumque VCPSectori VCR proportionalem con&longs;tituat; tendat autem ad centrum CVis centripeta Cubo di&longs;tantiæ locorum a centro reciproce proportionalis, & exeat corpus de loco V ju&longs;ta cum Velocitate &longs;ecundum lineam rectæ CV perpendicularem: progredietur corpus illud in Trajectoria quam punctum P perpetuo tangit; adeoque &longs;i Conica &longs;ectio CVRS Hyperbola &longs;it, de&longs;cendet idem ad centrum: Sin ea Ellip&longs;is &longs;it, a&longs;cendet illud perpetuo & abibit in infinitum. Et contra, &longs;i corpus quacunque cum Velocitate exeat de loco V, & perinde ut incæperit vel oblique de&longs;cendere ad centrum, vel ab eo ob- lique a&longs;cendere, Figura CVRS vel Hyperbola &longs;it vel Ellip&longs;is, in veniri pote&longs;t Trajectoria augendo vel minuendo angulum VCPin data aliqua ratione. ~~Sed &, Vi centripeta in centrifugam ver&longs;a,~~

~~a&longs;cendet corpus oblique in Trajectoria VPQ quæ invenitur capiendo angulum VCP Sectori Elliptico CVRC proportionalem, & longitudinem CP longitudini CT æqualem ut &longs;upra.~~ ~~Con&longs;equuntur hæc omnia ex Propo&longs;itione præcedente, per Curvæ cuju&longs;dam quadraturam, cujus inventionem, ut &longs;atis facilem, brevitatis gratia mi&longs;&longs;am facio.~~

~~LIBER PRIMUS.~~

~~PROPOSITIO XLII. PROBLEMA XXIX.~~

~~Data lege Vis centripetæ, requiritur motus corporis de loco dato data cum Velocitate &longs;ecundum datam rectam egre&longs;&longs;i.~~

Stantibus quæ in tribus Propo&longs;itionibus præcedentibus: exeat corpus de loco I &longs;ecundum lineolam IT, ea cum Velocitate quam corpus aliud, vi aliqua uniformi centripeta, de loco P cadendo acquirere po&longs;&longs;et in D: &longs;itque hæc vis uniformis ad vim qua corpus primum urgetur in I, ut DR ad DF. Pergat autem corpus ver&longs;us k; centroque C & intervallo Ck de&longs;cribatur circulus ke occurrens rectæ PD in e, & erigantur curvarum ALMm, BFGg, abzv, dcxw

ordinatim applicatæ em, eg, ev, ew. Ex dato rectangulo PDRQ,dataque lege vis centripetæ qua corpus primum agitatur, dantur curvæ lineæ BFGg, ALMm, per con&longs;tructionem Problematis XXVII, & ejus Corol. ~~1. Deinde ex dato angulo CIT datur proportio na&longs;centium IK, KN, & inde, per con&longs;tructionem Prob.~~ XXVIII, datur quantitas Q, una cum curvis lineis abzv, dcxw: adeoque completo tempore quovis Dbve, datur tum corporis altitudo Ce vel Ck,tum area Dcwe, eique æqualis Sector XCy, angulu&longs;que ICk & locus k in quo corpus tunc ver&longs;abitur. q.E.I.

~~DE MOTU CORPORUM~~

Supponimus autem in his Propo&longs;itionibus Vim centripetam in rece&longs;&longs;u quidem a centro variari &longs;ecundum legem quamcunque quam quis imaginari pote&longs;t, in æqualibus autem a centro di&longs;tantiis e&longs;&longs;e undeque eandem. ~~Atque hactenus Motum corporum in Orbibus immobilibus con&longs;ideravimus.~~ ~~Supere&longs;t ut de Motu eorum in Orbibus qui circa centrum virium revolvuntur adjiciamus pauca.~~

~~LIBER PRIMUS.~~

~~SECTIO IX.~~

~~De Motu Corporum in Orbibus mobilibus, deque motu Apfidum.~~

~~PROPOSITIO XLIII. PROBLEMA XXX.~~

~~Efficiendum est ut corpus in Trajectoria quacunque circa centrum Virium revolvente perinde moveri po&longs;&longs;it, atque corpus aliud in eadem Trajectoria quie&longs;cente.~~

~~In Orbe VPK po~~

&longs;itione dato revolvatur corpus P pergendo a V ver&longs;us K. A centro C agatur &longs;emper Cp,quæ &longs;it ip&longs;i CP æqualis, angulumque VCp angulo VCP proportionalem con&longs;tituat; & area quam linea Cp de&longs;cribit erit ad aream VCP quam linea CP&longs;imul de&longs;cribit, ut velocitas lineæ de&longs;cribentis Cp ad velocitatem lineæ de&longs;cribentis CP; hoc e&longs;t, ut angulus VCp ad angulum VCP, adeoque in data ratione, & propterea tempori proportionalis. Cum area tempori proportionalis &longs;it quam linea Cp in plano immobili de&longs;cribit, manife&longs;tum e&longs;t quod corpus, cogente ju&longs;tæ quantitatis Vi centripeta, revolvi po&longs;&longs;it una cum puncto p in Curva illa linea quam punctum idem p ratione jam expo&longs;ita de&longs;cribit in plano immobili. Fiat angulus VCu angulo PCp, & linea Cu lineæ CV, atque Figura uCp Figuræ VCP æqualis, & corpus in p &longs;emper exi&longs;tens movebitur in perimetro Figuræ revolventis uCp, eodemque tempore de&longs;cribet arcum ejus up quo corpus aliud P arcum ip&longs;i &longs;imilem & æqualem VP in Figura quie&longs;cente VPK de&longs;cribere pote&longs;t. Quæratur igitur, per Corollarium quintum propo&longs;itionis VI, Vis centripeta qua corpus revolvi po&longs;&longs;it in Curva illa linea quam punctum p de&longs;cribit in plano immobili, & &longs;olvetur Problema. q.E.F.

~~DE MOTU CORPORUM~~

~~PROPOSITIO XLIV. THEOREMA XIV.~~

Differentia Virium, quibus corpus in Orbe quie&longs;cente, & corpus aliud in eodem Orbe revolvente æqualiter moveri po&longs;&longs;unt, est in triplicata ratione communis altitudinis inver&longs;e.

~~Partibus Orbis quie~~

~~&longs;centis VP, PK &longs;unto &longs;imiles & æquales Orbis revolventis partes up, pk; & punctorum P, K di&longs;tantia intelligatur e&longs;&longs;e quam minima.~~ A puncto k in rectam pC demitte perpendiculum kr, idemque produc ad m, ut &longs;it mr ad kr ut angulus VCp ad angulum VCP.Quoniam corporum altitudines PC & pC, KC& kC &longs;emper æquantur, manife&longs;tum e&longs;t quod linearum PC & pC incrementa vel decrementa &longs;emper &longs;int æqualia, ideoque &longs;i corporum in locis P & p exi&longs;tentium diftinguantur motus &longs;inguli (per Legum Corol. 2.) in binos, quorum hi ver&longs;us centrum, &longs;ive &longs;ecundum lineas PC, pC determinentur, & alteri prioribus tran&longs;ver&longs;i &longs;int, & &longs;ecundum lineas ip&longs;is PC, pC perpendiculares directionem habeant; motus ver&longs;us centrum erunt æquales, & motus tran&longs;ver&longs;us corporis p erit ad motum tran&longs;ver&longs;um corporis P, ut motus angularis lineæ pC, ad motum angularem lineæ PC, id e&longs;t, ut angulus VCp ad angulum VCP. Igitur eodem tempore quo corpus P motu &longs;uo utroque pervenit ad punctum K, corpus p æquali in centrum motu æqualiter movebitur a p ver&longs;us C, adeoque completo illo tempore reperietur alicubi in linea mkr, quæ per punctum k in lineam pC perpendicularis e&longs;t; & motu tran&longs;ver&longs;o acquiret di&longs;tantiam a linea pC, quæ &longs;it ad di&longs;tantiam quam corpus alterum P acquirit a linea PC, ut e&longs;t motus tran&longs;ver&longs;us corporis p ad motum tran&longs;ver&longs;um corporis alterius P. Quare cum kr æqualis &longs;it di&longs;tantiæ quam corpus P acquirit a linea PC, &longs;itque mr ad kr ut angulus VCp ad angulum VCP, hoc e&longs;t, ut motus tran&longs;ver&longs;us corporis p ad motum tran&longs;ver&longs;um corporis P, manife&longs;tum e&longs;t quod corpus p completo illo tempore reperietur in loco m. Hæc ita &longs;e habebunt ubi corpora p & P æqualiter &longs;ecundum lineas pC & PC moventur, adeoque æqualibus Viribus &longs;ecundum lineas illas urgentur. Capiatur autem angulum pCn ad angulum pCk ut e&longs;t angulus VCp ad angulus VCP, &longs;itque nC æqualis kC, & corpus p completo illo tempore revera reperietur in n; adeoque Vi majore urgetur quam corpus P, &longs;i modo angulus mCpangulo kCp major e&longs;t, id e&longs;t &longs;i Orbis upk vel movetur in con&longs;equentia, vel movetur in antecedentia majore celeritate quam &longs;it dupla ejus qua linea CP in con&longs;equentia fertur; & Vi minore &longs;i Orbis tardius movetur in antecedentia. ~~E&longs;tque Virium differentia ut locorum intervallum mn, per quod corpus illud pip&longs;ius actione, dato illo temporis &longs;patio, transferri debet.~~ Centro C intervallo Cn vel Ck de&longs;cribi intelligatur Circulus &longs;ecans lineas mr, mn productas in s & t, & erit rectangulum mnXmt æquale rectangulo mkXms, adeoque mn æquale (mkXms/mt). Cum autem triangula pCk, pCn dentur magnitudine, &longs;unt kr & mr,earumque differentia mk & &longs;umma ms reciproce ut altitudo pC,adeoque rectangulum mkXms e&longs;t reciproce ut quadratum altitudinis pC. E&longs;t & mt directe ut 1/2 mt, id e&longs;t, ut altitudo pC. Hæ &longs;unt primæ rationes linearum na&longs;centium; & hinc fit (mkXms/mt), id e&longs;t lineola na&longs;cens mn, eique proportionalis Virium differentia reciproce ut cubus altitudinis pC. que E. D.

~~LIBER PRIMUS.~~

Corol. 1. Hinc differentia virium in locis P & p vel K & k, e&longs;t ad vim qua corpus motu Circulari revolvi po&longs;&longs;it ab R ad K eodem tempore quo corpus P in Orbe immobili de&longs;cribit arcum PK, ut lineola na&longs;cens mn ad &longs;inum ver&longs;um arcus na&longs;centis RK, id e&longs;t ut (mkXms/mt) ad (rkq/2kC), vel ut mkXms ad rk quadratum; hoc e&longs;t, &longs;i capiantur datæ quantitates F, G in ea ratione ad invicem quam habet angulus VCP ad angulum VCp, ut GG-FF ad FF. Et propterea, &longs;i centro C intervallo quovis CP vel Cp de&longs;cribatur Sector circularis æqualis areæ toti VPC, quam corpus P tempore quovis in Orbe immobili revolvens radio ad centrum ducto de&longs;crip &longs;it: differentia virium, quibus corpus P in Orbe immobili & corpus p in Orbe mobili revolvuntur, erit ad vim centripetam qua corpus aliquod radio ad centrum ducto Sectorem illum, eodem tempore quo de&longs;cripta &longs;it area VPC uniformiter de&longs;eribere potui&longs;&longs;et, ut GG-FF ad FF. ~~Namque Sector ille & area pCk &longs;unt ad invicem ut tempora quibus de&longs;cribuntur.~~

~~DE MOTU CORPORUM~~

Corol. 2. Si Orbis VPK Ellip&longs;is &longs;it umbilicum habens C & Ap&longs;idem &longs;ummam V; eique &longs;imilis & æqualis ponatur Ellip&longs;is upk,ita ut &longs;it &longs;emper pC æqualis PC, & angulus VCp &longs;it ad angulum VCP in data ratione G ad F; pro altitudine autem PC vel pC&longs;cribatur A, & pro Ellip&longs;eos latere recto ponatur 2 R: erit vis qua corpus in Ellip&longs;i mobili revolvi pote&longs;t, ut (FF/AA)+(RGG-RFF/A cub.) & contra. Exponatur enim vis qua corpus revolvatur in immota Ellip&longs;i per quantitatem (FF/AA), & vis in V erit (FF/CV quad.). Vis autem qua corpus in Circulo ad di&longs;tantiam CV ea cum velocitate revolvi po&longs;&longs;et quam corpus in Ellip&longs;i revolvens habet in V,e&longs;t ad vim qua corpus in Ellip&longs;i revolvens urgetur in Ap&longs;ide V,ut dimidium lateris recti Ellip&longs;eos. ~~ad Circuli &longs;emidiametrum CV,adeoque valet (RFF/CV cub.): & vis quæ &longs;it ad hanc ut GG-FF ad FF, valet (RGG-RFF/CV cub.): e&longs;tque hæc vis (per hujus Corol.~~ ~~1.) differentia virium in V quibus corpus P in Ellipfi immota VPK,& corpus p in Ellip&longs;i mobili upk revolvuntur.~~ ~~Unde cum (per hanc Prop.) differentia illa in alia quavis altitudine A &longs;it ad &longs;eip&longs;am in altitudine CV ut (1/A cub.) ad (1/CV cub.), eadem differentia in omni altitudine.~~ A valebit (RGG-RFF/A cub.). Igitur ad vim (FF/AA) qua corpus revolvi pote&longs;t in Ellip&longs;i immobili VPK, addatur exce&longs;&longs;us (RGG-RFF/A cub.) & componetur vis tota (FF/AA)+(RGG-RFF/A cub.) qua corpus in Ellip&longs;i mobili upk ii&longs;dem temporibus revolvi po&longs;&longs;it.

~~LIBER PRIMUS.~~

Corol. 3. Ad eundem modum colligetur quod, &longs;i Orbis immobilis VPK Ellip&longs;is &longs;it centrum habens in virium centro C; eique &longs;imilis, æqualis & concentrica ponatur Ellip&longs;is mobilis upk;&longs;itque 2 R Ellip&longs;eos hujus latus rectum principale, & 2T latus tran&longs;ver&longs;um &longs;ive axis major, atque angulus VCp &longs;emper &longs;it ad angulum VCP ut G ad F; vires quibus corpora in Ellip&longs;i immobili & mobili temporibus æqualibus revolvi po&longs;&longs;unt, erunt ut (FFA/T cub.) & (FFA/T cub.)+(RGG-RFF/A cub.) re&longs;pective.

Corol. 4. Et univer&longs;aliter, &longs;i corporis altitudo maxima CV nominetur T, & radius curvaturæ quam Orbis VPK habet in V, id e&longs;t radius Circuli æqualiter curvi, nominetur R, & vis centripeta qua corpus in Trajectoria quacunque immobili VPK revolvi pote&longs;t, in loco V dicatur (VFF/TT), atque aliis in locis P indefinite dicatur X, altitudine CP nominata A, & capiatur G ad F in data ratione anguli VCp ad angulum VCP: erit vis centripeta qua corpus idem eo&longs;dem motus in eadem Trajectoria upk circulariter mota temporibus ii&longs;dem peragere pote&longs;t, ut &longs;umma virium X+(VRGG-VRFF/A cub.).

Corol. 5. Dato igitur motu corporis in Orbe quocunque immobili, augeri vel minui pote&longs;t ejus motus angularis circa centrum virium in ratione data, & inde inveniri novi Orbes immobiles in quibus corpora novis viribus centripetis gyrentur.

~~Corol. 6. Igitur &longs;i ad rectam CV po~~

&longs;itione datam erigatur perpendiculum VP longitudinis indeterminatæ, jungaturque CP, & ip&longs;i æqualis agatur Cp, con&longs;tituens angulum VCp, qui &longs;it ad angulum VCP in data ratione; vis qua corpus gyrari pote&longs;t in Curva illa Vpk quam punctum p perpetuo tangit, erit reciproce ut cubus altitudinis Cp. Nam corpus P, per vim inertiæ, nulla alia vi urgente, uniformiter progredi pote&longs;t in recta VP. Addatur vis in centrum C, cubo altitudinis CP vel Cp reciproce proportionalis, & (per jam demon&longs;trata) detorquebitur motus ille rectilineus in lineam curvam Vpk. E&longs;t autem hæc Curva Vpk eadem cum Curva illa VPQ in Corol. ~~3. Prop.~~ ~~XLI inventa, in qua ibi diximus corpora huju&longs;modi viribus attracta oblique a&longs;cendere.~~

~~DE MOTU CORPORUM~~

~~PROPOSITIO XLV. PROBLEMA XXXI.~~

~~Orbium qui &longs;unt Circulis maxime &longs;initimi requiruntur motus Ap&longs;idum.~~

~~Problema &longs;olvitur Arithmetice faciendo ut Orbis, quem corpus in Ellip&longs;i mobili (ut in Propo&longs;itionis &longs;uperioris Corol.~~ ~~2, vel 3) revolvens de&longs;cribit in plano immobili, accedat ad formam Orbis cujus Ap&longs;ides requiruatur, & quærendo Ap&longs;ides Orbis quem corpus illud in plano immobili de&longs;cribit.~~ ~~Orbes autem eandem acquirent formam, &longs;i vires centripetæ quibus de&longs;cribuntur, inter &longs;e collatæ, in æqualibus altitudinibus reddantur proportionales.~~ Sit punctum V Ap&longs;is &longs;umma, & &longs;cribantur T pro altitudine maxima CV, A pro altitudine quavis alia CP vel Cp, & X pro altititudinum differentia CV-CP; & vis qua corpus in Ellip&longs;i circa umbilicum &longs;uum C (ut in Corollario 2.) revolvente movetur, quæque in Corollario 2. erat ut (FF/AA)+(RGG-RFF/A cub.), id e&longs;t ut (FFA+RGG-RFF/A cub.), &longs;ub&longs;tituendo T-X pro A, erit ut (RGG-RFF+TFF-FFX/A cub.). Reducenda &longs;imiliter e&longs;t vis alia quævis centripeta ad fractionem cujus denominator &longs;it A cub., & numeratores, facta homologorum terminorum collatione, &longs;tatuendi &longs;unt analogi. ~~Res Exemplis patebit.~~

Exempl. 1. Ponamus vim centripetam uniformem e&longs;&longs;e, adeoque ut (A cub./A cub.), &longs;ive (&longs;cribendo T-X pro A in Numeratore) ut (T cub.-3TTX+3TXX-X cub./A cub.); & collatis Numeratorum terminis corre&longs;pondentibus, nimirum datis cum datis & non datis cum non datis, fiet RGG-RFF+TFF ad T cub. ut-FFX ad -3TTX+3TXX-Xcub. &longs;ive ut-FF ad-3TT+3TX -XX. Jam cum Orbis ponatur Circulo quam maxime finitimus, coeat Orbis cum Circulo; & ob factas R, T æquales, atque X in infi- nitum diminutam, rationes ultimæ erunt RGG ad T cub. ut-FF ad-3TT &longs;eu GG ad TT ut FF ad 3TT & vici&longs;&longs;im GG ad FF ut TT ad 3 TT id e&longs;t, ut 1 ad 3; adeoque G ad F, hoc e&longs;t angulus VCp ad angulum VCP, ut 1 ad √3. Ergo cum corpus in Ellip&longs;i immobili, ab Ap&longs;ide &longs;umma ad Ap&longs;idem imam de&longs;cendendo conficiat angulum VCP (ut ita dicam) gradum 180; corpus aliud in Ellip&longs;i mobili, atque adeo in Orbe immobili de quo agimus, ab Ap&longs;ide &longs;umma ad Ap&longs;idem imam de&longs;cendendo conficiet angulum VCp gradum (180/√3): id adeo ob &longs;imilitudinem Orbis hujus, quem corpus agente uniformi vi centripeta de&longs;cribit, & Orbis illius quem corpus in Ellip&longs;i revolvente gyros peragens de&longs;cribit in plano quie&longs;cente. ~~Per &longs;uperiorem terminorum collationem &longs;imiles redduntur hi Orbes, non univer&longs;aliter, &longs;ed tunc cum ad formam circularem quam maxime appropinquant.~~ Corpus igitur uniformi cum vi centripeta in Orbe propemodum circulari revolvens, inter Ap&longs;idem &longs;ummam & Ap&longs;idem imam conficiet &longs;emper angulum (180/√3) graduum, &longs;eu 103 gr. 55 m. 23 &longs;ec. ad centrum; perveniens ab Ap&longs;ide &longs;umma ad Ap&longs;idem imam ubi &longs;emel confecit hunc angulum, & inde ad Ap&longs;idem &longs;ummam rediens ubi iterum confecit eundem angulum; & &longs;ic deinceps in infinitum.

~~LIBER PRIMUS.~~

Exempl. 2. Ponamus vim centripetam e&longs;&longs;e ut altitudinis A dignitas quælibet An-3 &longs;eu (An/A3): ubi n-3 & n &longs;ignificant dignitatum indices quo&longs;cunque integros vel fractos, rationales vel irrationales, affirmativos vel negativos. ~~Numerator ille An &longs;eu ―T-Xnin &longs;eriem indeterminatam per Methodum no&longs;tram Serierum convergentium reducta, evadit Tn-nXTn-1+(nn-n/2)XXTn-2 &c.~~ ~~Et collatis hujus terminis cum terminis Numeratoris alterius RGG-RFF+TFF-FFX, fit RGG-RFF+TFF ad Tnut-FF ad-nTn-1+(nn-n/2)XTn-2 &c.~~ Et &longs;umendo rationes ultimas ubi Orbes ad formam circularem accedunt, fit RGG ad Tn ut-FF ad-nTn-1, &longs;eu GG ad Tn-1 ut FF ad nTn-1, & vici&longs;&longs;im GG ad FF ut Tn-1 ad nTn-1 id e&longs;t ut 1 ad n; adeoque G ad F, id e&longs;t angulus VCp ad angulum VCP, ut 1 ad √n. Quare cum angulus VCP, in de&longs;cen&longs;u corporis ab Ap&longs;ide &longs;umma ad Ap&longs;idem imam in Ellip&longs;i confectus, &longs;it graduum 180; conficietur angulus VCp, in de&longs;cen&longs;u corporis ab Ap&longs;ide &longs;umma ad Ap&longs;idem imam, in Orbe propemodum Circulari quem corpus quodvis vi centripeta dignitati An-3 proportionali de&longs;cribit, æqualis angulo graduum (180/√n); & hoc angulo repetito corpus redibit ab Ap&longs;ide ima ad Ap&longs;idem &longs;ummam, & &longs;ic deinceps in infinitum. Ut &longs;i vis centripeta &longs;it ut di&longs;tantia corporis a centro, id e&longs;t, ut A &longs;eu (A4/A3), erit n æqualis 4 & √n æqualis 2; adeoque angulus inter Ap&longs;idem &longs;ummam & Ap&longs;idem imam æqualis (180/2) gr. &longs;eu 90 gr. Completa igitur quarta parte revolutionis unius corpus perveniet ad Ap&longs;idem imam, & completa alia quarta parte ad Ap&longs;idem &longs;ummam, & &longs;ic deinceps per vices in infinitum. ~~Id quod etiam ex Propo&longs;itione x.~~ ~~manife&longs;tum e&longs;t.~~ ~~Nam corpus urgente hac vi centripeta revolvetur in Ellip&longs;i immobili, cujus centrum e&longs;t in centro virium.~~ Quod &longs;i vis centripeta &longs;it reciproce ut di&longs;tantia, id e&longs;t directe ut 1/A &longs;eu (A2/A3), erit n æqualis 2, adeoque inter Ap&longs;idem &longs;ummam & imam angulus erit graduum (180/√2) &longs;eu 127 gr. 16 m. 45 &longs;ec. & propterea corpus tali vi revolvens, perpetua anguli hujus repetitione, vicibus alternis ab Ap&longs;ide &longs;umma ad imam & ab ima ad &longs;ummam perveniet in æternum. Porro &longs;i vis centripeta &longs;it reciproce ut latus quadrato-quadratum undecimæ dignitatis altitudinis, id e&longs;t reciproce ut A (11/4), adeoque directe ut (1/A11/4) &longs;eu ut (A1/4/A3) erit n æqualis 1/4, & (180/√n) gr. æqualis 360 gr. & propterea corpus de Ap&longs;ide &longs;umma di&longs;cedens & &longs;ubinde perpetuo de&longs;cendens, perveniet ad Ap&longs;idem imam ubi complevit revolutionem integram, dein perpetuo a&longs;cen&longs;u complendo aliam revolutionem inregram, redibit ad Ap&longs;idem &longs;ummam: & &longs;ic per vices in æternum.

~~DE MOTU CORPORUM~~

Exempl. 3. A&longs;&longs;umentes m & n pro quibu&longs;vis indicibus dignitatum Altitudinis, & b, c pro numeris quibu&longs;vis datis, ponamus vim centripetam e&longs;&longs;e ut (bAm+cAn/A cub.), id e&longs;t, ut (b in ―T-Xm+c in ―T-Xn/A cub.) &longs;eu (per eandem Methodum no&longs;tram Serierum convergentium) ut (bTm+cTn-mbXTm-1-ncXTn-1+(mm-mb/2)XXTm-2+(nn-nc/2)XXTn-2 &c./A cub.) & collatis numeratorum terminis, fiet RGG-RFF+TFF ad bTm+cTn, ut -FF ad -mbTm-1-ncTn-1+(mm-m/2)bXTm-2+(nn-n/2)cXTn-2 &c. Et &longs;umendo rationes ultimas quæ prodeunt ubi Orbes ad formam circularem accedunt, fit GG ad bTm-1+cTn-1, ut FF ad mbTm-1+ncTn-1, & vici&longs;&longs;im GG ad FF ut bTm-1+cTn-1 ad mbTm-1+ncTn-1. Quæ proportio, exponendo altitudinem maximam CV &longs;eu T Arithmetice per Unitatem, fit GG ad FF ut b+c ad mb+nc, adeoque ut 1 ad (mb+nc/b+c). Unde e&longs;t G ad F, id e&longs;t angulus VCp ad angulum VCP, ut 1 ad √(mb+nc/b+c). Et propterea cum angulus VCP inter Ap&longs;idem &longs;ummam & Ap&longs;idem imam in Ellip&longs;i immobili &longs;it 180 gr.erit angulus VCp inter ea&longs;dem Ap&longs;ides, in Orbe quem corpus vi centripeta quantitati (bAm+cAn/A cub.) proportionali de&longs;cribit, æqualis angulo graduum 180 √(b+c/mb+nc). Et eodem argumento &longs;i vis centripeta &longs;it ut (bAm-cAn/A cub.), angulus inter Ap&longs;ides invenietur graduum 180 √(b-c/mb-nc). Nec &longs;ecus re&longs;olvetur Problema in ca&longs;ibus difficilioribus. Quantitas cui vis centripeta proportionalis e&longs;t, re&longs;olvi &longs;emper debet in Series convergentes denominatorem habentes A cub. Dein pars data numeratoris qui ex illa operatione provenit ad ip&longs;ius partem alteram non datam, & pars data numeratoris hujus RGG-RFF+TFF-FFX ad ip&longs;ius partem alteram non datam in eadem ratione ponendæ &longs;unt: Et quantitates &longs;uperfluas delendo, &longs;cribendoque Unitatem pro T, obtinebitur proportio G ad F.

~~LIBER PRIMUS.~~

~~Corol. 1. Hinc &longs;i vis centripeta &longs;it ut aliqua altitudinis dignitas, inveniri pote&longs;t dignitas illa ex motu Ap&longs;idum; & contra.~~ Nimirum &longs;i motus totus angularis, quo corpus redit ad Ap&longs;idem eandem, &longs;it ad motum angularem revolutionis unius, &longs;eu graduum 360, ut numerus aliquis m ad numerum alium n, & altitudo nominetur A: erit vis ut altitudinis dignitas illa A(nn/mm)-3, cujus In- dex e&longs;t (nn/mm)-3. Id quod per Exempla &longs;ecunda manife&longs;tum e&longs;t. Unde liquet vim illam in majore quam triplicata altitudinis ratione, in rece&longs;&longs;u a centro, decre&longs;cere non po&longs;&longs;e: Corpus tali vi revolvens deque Ap&longs;ide di&longs;cedens, &longs;i cæperit de&longs;cendere nunquam perveniet ad Ap&longs;idem imam &longs;eu altitudinem minimam, &longs;ed de&longs;cendet u&longs;que ad centrum, de&longs;cribens Curvam illam lineam de qua egimus in Cor. ~~3. Prop.~~ ~~XLI.~~ ~~Sin cæperit illud, de Ap&longs;ide di&longs;cedens, vel minimum a&longs;cendere; a&longs;cendet in infinitum, neque unquam perveniet ad Ap&longs;idem &longs;ummam.~~ ~~De&longs;cribet enim Curvam illam lineam de qua actum e&longs;t in eodem Corol.~~ ~~& in Corol.~~ ~~6, Prop.~~ ~~XLIV.~~ Sic & ubi vis, in rece&longs;&longs;u a centro, decre&longs;cit in majore quam triplicata ratione altitudinis, corpus de Ap&longs;ide di&longs;cedens, perinde ut cæperit de&longs;cendere vel a&longs;cendere, vel de&longs;cendet ad centrum u&longs;que vel a&longs;cendet in infinitum. At &longs;i vis, in rece&longs;&longs;u a centro, vel decre&longs;cat in minore quam triplicata ratione altitudinis, vel crefcat in altitudinis ratione quacunque; corpus nunquam de&longs;cendet ad centrum u&longs;que, &longs;ed ad Ap&longs;idem imam aliquando perveniet: & contra, &longs;i corpus de Ap&longs;ide ad Ap&longs;idem alternis vicibus de&longs;cendens & a&longs;cendens nunquam appellat ad centrum; vis in rece&longs;&longs;u a centro aut augebitur, aut in minore quam triplicata altitudinis ratione decre&longs;cet: & quo citius corpus de Ap&longs;ide ad Ap&longs;idem redierit, eo longius ratio virium recedet a ratione illa triplicata. Ut &longs;i corpus revolutionibus 8 vel 4 vel 2 vel 1 1/2 de Ap&longs;ide &longs;umma ad Ap&longs;idem &longs;ummam alterno de&longs;cen&longs;u & a&longs;cen&longs;u redierit; hoc e&longs;t, &longs;i fuerit m ad n ut 8 vel 4 vel 2 vel 1 1/2 ad 1, adeoque (nn/mm)-3 valeat (1/64)-3 vel (1/16) -3 vel 1/4-3 vel 4/9-3: erit vis ut A(1/64)-3 vel A(1/16)-3 vel A1/4-3 vel A4/9-3, id e&longs;t, reciproce ut A3-(1/64) vel A3-(1/16) vel A3-1/4 vel A3-4/9. Si corpus &longs;ingulis revolutionibus redierit ad Ap&longs;idem eandem immotam; erit m ad n ut 1 ad 1, adeoque A (nn/mm)-3 æqualis A-2 &longs;eu (1/AA) & propterea decrementum virium in ratione duplicata altitudinis, ut in præcedentibus demon&longs;tratum e&longs;t. Si corpus partibus revolutionis unius vel tribus quartis, vel duabus tertiis, vel una tertia, vel una quarta, ad Ap&longs;idem eandem redierit; erit m ad n ut 1/4 vel 2/3 vel 1/3 vel 1/4 ad 1, adeoque A(nn/mm)-3 æqualis A(16/9)-3 vel A9/4-3 vel A9-3 vel A16-3; & propterea vis aut reciproce ut A(11/9) vel A1/4, aut directe ut A6 vel A 13. Denique &longs;i corpus pergendo ab Ap&longs;ide &longs;umma ad Ap&longs;idem &longs;ummam confecerit revolutionem integram, & præterea gradus tres, adeoque Ap&longs;is illa &longs;ingulis corporis revolutionibus confecerit in con&longs;equentia gradus tres; erit m ad n ut 363 gr. ad 360gr. &longs;ive ut 121 ad 120, adeoque A(nn/mm)-3 erit æquale A-(29523/14641); & propterea vis centripeta reciproce ut A (29523/14641) &longs;eu reciproce ut A2 (4/2+3) proxime. ~~Decre&longs;cit igitur vis centripeta in ratione paulo majore quam duplicata, &longs;ed quæ vicibus 59 3/4 propius ad duplicatam quam ad triplicatam accedit.~~

~~DE MOTU CORPORUM~~

~~LIBER PRIMUS.~~

Corol. 2. Hinc etiam &longs;i corpus, vi centripeta quæ &longs;it reciproce ut quadratum altitudinis, revolvatur in Ellip&longs;i umbilicum habente in centro virium, & huic vi centripetæ addatur vel auferatur vis alia quævis extranea; cogno&longs;ci pote&longs;t (per Exempla tertia) motus Ap&longs;idum qui ex vi illa extranea orietur: & contra. Ut &longs;i vis qua corpus revolvitur in Ellip&longs;i &longs;it ut (1/AA), & vis extranea ablata ut c A, adeoque vis reliqua ut (A-c A4/A cub.); erit (in Exemplis tertiis) b æqualis 1, m æqualis 1, n æqualis 4, adeoque angulus revolutionis inter Ap&longs;ides æqualis angulo graduum 180 √(1-c/1-4c). Ponatur vim illam extraneam e&longs;&longs;e 357,45 partibus minorem quam vis altera qua corpus revolvitur in Ellip&longs;i, id e&longs;t c e&longs;&longs;e (100/35745), exi&longs;tente A vel T æquali 1; & 180 √(1-c/1-4c) evadet 180 √(35645/35345), &longs;eu 180, 7623, id e&longs;t, 180 gr. 45 m. 44 &longs;. Igitur corpus de Ap&longs;ide &longs;umma di&longs;cedens, motu angulari 180 gr. 45 m. 44. &longs;. perveniet ad Ap&longs;idem imam, & hoc motu duplicato ad Ap&longs;idem &longs;ummam redibit: adeoque Ap&longs;is &longs;umma &longs;ingulis revolutionibus progrediendo conficiet 1 gr. 31 m. 28 &longs;ec.

~~Hactenus de Motu corporum in Orbibus quorum plana per centrum Virium tran&longs;eunt.~~ ~~Supere&longs;t ut Motus etiam determinemus in planis excentricis.~~ Nam Scriptores qui Motum gravium tractant, con&longs;iderare &longs;olent a&longs;cen&longs;us & de&longs;cen&longs;us ponderum, tam obliquos in planis quibu&longs;cunque datis, quam perpendiculares: & pari jure Motus corporum Viribus quibu&longs;cunque cen- tra petentium, & planis excentricis innitentium hic con&longs;iderandus venit. ~~Plana autem &longs;upponimus e&longs;&longs;e politi&longs;&longs;ima & ab&longs;olute lubrica ne corpora retardent.~~ Quinimo, in his demon&longs;trationibus, vice planorum quibus corpora incumbunt quæque tangunt incumbendo, u&longs;urpamus plana his parallela, in quibus centra corporum moventur & Orbitas movendo de&longs;cribunt. ~~Et eadem lege Motus corporum in &longs;uperficiebus Curvis peractos &longs;ubinde determinamus.~~

~~DE MOTU CORPORUM~~

~~SECTIO X.~~

~~De Motu Corporum in Superficiebus datis, deque Funipendulorum Motu reciproco.~~

~~PROPOSITIO XLVI. PROBLEMA XXXII.~~

Po&longs;ita cuju&longs;cunque generis Vi centripeta, datoque tum Virium centro tum Plano quocunque in quo corpus revolvitur, & conce&longs;&longs;is Figurarum curvilinearum quadraturis: requiritur Motus corporis de loco dato, data cum Velocitate, &longs;ecundum rectam in Plano illo datam egre&longs;&longs;i.

Sit S centrum Virium, SC di&longs;tantia minima centri hujus a Plano dato, P corpus de loco P &longs;ecundum rectam PZ egrediens, Qcorpus idem in Trajectoria &longs;ua revolvens, & PQR Trajectoria illa, in Plano dato de&longs;cripta, quam invenire oportet. Jungantur CQ QS, & &longs;i in QS capiatur SV proportionalis vi centripetæ qua corpus trahitur ver&longs;us centrum S, & agatur VT quæ fit parallela CQ & occurrat SC in T: Vis SV re&longs;olvetur (per Legum Corol. ~~2.) in vires ST, TV; quarum ST trahendo corpus &longs;ecundum lineam plano perpendicularem, nil mutat motum ejus in hoc plano.~~ Vis autem altera TV, agendo &longs;ecundum po&longs;itionem plani, trahit corpus directe ver&longs;us punctum C in plano datum, adeoque facit illud in hoc plano perinde moveri ac &longs;i vis ST tolleretur, & corpus vi &longs;ola TV revolveretur circa centrum C in &longs;patio libero. ~~Data autem vi centripeta TV qua corpus Q in &longs;patio libero circa centrum datum C revolvitur, datur per Prop.~~ ~~XLII, tum Trajectoria PQRquam corpus de&longs;cribit, tum locus Q in quo corpus ad datum quodvis tempus ver&longs;abitur, tum denique velocitas corporis in loco illo Q; & contra. que E. I.~~

~~LIBER PRIMUS.~~

~~PROPOSITIO XLVII. THEOREMA XV.~~

Po&longs;ito quod Vis centripeta proportionalis &longs;it di&longs;tantiæ corporis a centro; corpora omnia in planis quibu&longs;cunque revolventia de&longs;cribent Ellip&longs;es, & revolutiones Temporibus æqualibus peragent; quæque moventur in lineis rectis, ultro citroque di&longs;currendo, &longs;ingulas eundi & redeundi periodos ii&longs;dem Temporibus ab&longs;olvent.

~~Nam, &longs;tantibus quæ~~

in &longs;uperiore Propo&longs;itione, vis SV qua corpus Q in plano quovis PQRrevolvens trahitur ver&longs;us centrum S e&longs;t ut diftantia Sque atque adeo ob proportionales SV& SQ, TV & CQ, vis TV qua corpus trahitur ver&longs;us punctum Cin Orbis plano datum, e&longs;t ut di&longs;tantia Cque Vires igitur, quibus corpora in plano PQRver&longs;antia trahuntur ver&longs;us punctum C, &longs;unt pro ratione di&longs;tantiarum æquales viribus quibus corpora undiquaque trahuntur ver&longs;us centrum S; & propterea corpora movebuntur ii&longs;dem Temporibus, in ii&longs;dem Figuris, in plano quovis PQR circa punctum C, atque in &longs;patiis liberis circa centrum S; adeoque (per Corol. ~~2. Prop.~~ ~~X, & Corol.~~ ~~2. Prop.~~ XXXVIII) Temporibus &longs;emper æqualibus, vel de&longs;cribent Ellip&longs;es in plano illo circa centrum C,vel periodos movendi ultro citroque in lineis rectis per centrum Cin plano illo ductis, complebunt. que E. D.

~~DE MOTU CORPORUM~~

~~Scholium.~~

~~His affines &longs;unt a&longs;cen&longs;us ac de&longs;cen&longs;us corporum in &longs;uperficiebus curvis.~~ Concipe lineas curvas in plano de&longs;cribi, dein circa axes quo&longs;vis datos per centrum Virium tran&longs;euntes revolvi, & ea revolutione &longs;uperficies curvas de&longs;cribere; tum corpora ita moveri ut eorum centra in his &longs;uperficiebus perpetuo reperiantur. Si corpora illa oblique a&longs;cendendo & de&longs;cendendo currant ultro citroque peragentur corum motus in planis per axem tran&longs;euntibus, atque adeo in lineis curvis quarum revolutione curvæ illæ &longs;uperficies genitæ &longs;unt. ~~I&longs;tis igitur in ca&longs;ibus &longs;ufficit motum in his lineis curvis con&longs;iderare.~~

~~PROPOSITIO XLVIII. THEOREMA XVI.~~

Si Rota Globo extrin&longs;ecus ad angulos rectos in&longs;i&longs;tat, & more rotarum revolvendo progrediatur in circulo maximo; longitudo Itineris curvilinei, quod punctum quodvis in Rotæ perimetro datum, ex quo Globum tetigit, confecit, (quodque Cycloidem vel Epicycloidem nominare licet) erit ad duplicatum &longs;inum ver&longs;um arcus dimidii qui Globum ex eo tempore inter eundum tetigit, ut &longs;umma diametrorum Globi & Rotæ ad &longs;emidiametrum Globi.

~~PROPOSITIO XLIX. THEOREMA XVII.~~

Si Rota Globo concavo ad rectos angulos intrin&longs;ecus in&longs;i&longs;tat & revolvendo progrediatur in circulo maximo; longitudo Itineris curvilinei quod punctum quodvis in Rotæ perimetro datum, ex quo Globum tetigit, confecit, erit ad duplicatum &longs;inum ver&longs;um arcus dimidii qui Globum toto hoc tempore inter eundum tetigit, ut differentia diametrorum Globi & Rotæ ad &longs;emidiametrum Globi.

~~Sit ABL Globus, C centrum ejus, BPV Rota ei in&longs;i&longs;tens, E centrum Rotæ, B punctum contactus, & P punctum datum in perimetro Rotæ.~~ Concipe hanc Rotam pergere in circulo maximo ABL ab A per B ver&longs;us L, & inter cundum ita revolvi ut arcus AB, PB &longs;ibi invicem &longs;emper æquentur, atque punctum illud P in perimetro Rotæ datum interea de&longs;cribere Viam curvilineam AP. Sit autem AP Via tota curvilinea de&longs;cripta ex quo Rota Globum tetigit in A, & erit Viæ hujus longitudo AP ad duplum

&longs;inum ver&longs;um arcus 1/2 PB, ut 2 CE ad CB. Nam recta CE (&longs;i opus e&longs;t producta) occurrat Rotæ in V, junganturque CP, BP, EP, VP, & in CP productam demittatur normalis VF. Tangant PH, VH Circulum in P & V concurrentes in H, &longs;ecetque PH ip&longs;am VF in G, & ad VP demittantur normales GI, HK. Centro item C & intervallo quovis de&longs;cribatur circulus nom &longs;ecans rectam CP in n, Rotæ perimetrum BP &c. ~~in o, & Viam curvilineam AP in m; centroque V & intervallo Vo de&longs;cribatur circulus &longs;ecans VP productam in que~~

~~LIBER PRIMUS.~~

~~DE MOTU CORPORUM~~

~~Quoniam Rota eundo &longs;emper revolvitur circa punctum contactus B, manife&longs;tum e&longs;t quod recta BP perpendicularis e&longs;t ad~~

lineam illam curvam AP quam Rotæ punctum P de&longs;cribit, atque adeo quod recta VP tanget hanc curvam in puncto P. Circuli nom radius &longs;en&longs;im auctus vel diminutus æquetur tandem di&longs;tantiæ CP; &, ob &longs;imilitudinem Figuræ evane&longs;centis Pnomq & Figuræ PFGVI, ratio ultima lineolarum evane&longs;centium Pm, Pn, Po, Pq, id e&longs;t, ratio mutationum momentanearum curvæ AP, rectæ CP, arcus circularis BP, ac rectæ VP, eadem erit quæ linearum PV, PF, PG, PI re&longs;pective. Cum autem VF ad CF & VH ad CV perpendiculares &longs;unt, angulique HVG, VCF propterea æquales; & angulus VHG (ob angulos quadrilateri HVEPad V & P rectos) angulo CEP æqualis e&longs;t, &longs;imilia erunt triangula VHG, CEP; & inde fiet ut EP ad CE ita HG ad HV&longs;eu HP & ita KI ad KP, & compo&longs;ite vel divi&longs;im ut CB ad CE ita PI ad PK, & duplicatis con&longs;equentibus ut CB ad 2 CEita PI ad PV, atque ita adeo Pq ad Pm. E&longs;t igitur decrementum lineæ VP, id e&longs;t, incrementum lineæ BV-VP ad incrementum lineæ curvæ AP in data ratione CB ad 2 CE, & propterea (per Corol. ~~Lem.~~ ~~IV.) longitudines BV-VP & AP, incrementis illis genitæ, &longs;unt in eadem ratione.~~ Sed, exi&longs;tente BV radio, e&longs;t VP co-&longs;inus anguli BVP &longs;eu 1/2 BEP, adeoque BV-VP&longs;inus ver&longs;us eju&longs;dem anguli; & propterea in hac Rota, cujus radius e&longs;t 1/2 BV, erit BV-VP duplus &longs;inus ver&longs;us arcus 1/2 BP. Ergo AP e&longs;t ad duplum &longs;inum ver&longs;um arcus 1/2 BP ut 2 CE ad CB. que E. D.

~~LIBER PRIMUS.~~

~~Lineam autem AP in Propo&longs;itione priore Cycloidem extra Globum, alteram in po&longs;teriore Cycloidem intra Globum di&longs;tinctionis gratia nominabimus.~~

Corol. 1. Hinc &longs;i de&longs;cribatur Cyclois integra ASL & bi&longs;ecetur ea in S, erit longitudo partis PS ad longitudinem VP (quæ duplus e&longs;t &longs;inus anguli VBP, exi&longs;tente EB radio) ut 2 CE ad CB,atque adeo in ratione data.

~~Corol. 2. Et longitudo &longs;emiperimetri Cycloidis AS æquabitur lineæ rectæ quæ e&longs;t ad Rotæ diametrum BV, ut 2 CE ad CB.~~

~~PROPOSITIO L. PROBLEMA XXXIII.~~

~~Facere ut Corpus pendulum o&longs;cilletur in Cycloide data.~~

~~Intra Globum QVS, centro C de&longs;criptum, detur Cyclois QRSbi&longs;ecta in R & punctis &longs;uis extremis Q & S &longs;uperficiei Globi hinc inde occurrens.~~ Agatur CR bi&longs;ecans arcum QS in O, & producatur ea ad A, ut &longs;it CA ad CO ut CO ad CR. Centro C in- tervallo CA de&longs;eribatur Globus exterior ABD, & intra hunc Globum a Rota, cujus diameter &longs;it AO, de&longs;cribantur duæ Semicycloides AQ, AS, quæ Globum interiorem tangant in Q & S & Globo exteriori occurrant in A. A puncto illo A, Filo APT longitudinem AR æquante, pendeat corpus T, & ita intra Semicycloides AQ, AS o&longs;cilletur, ut quoties pendulum digreditur a perpendiculo AR,

Filum parte &longs;ui &longs;uperiore AP applicetur ad Semicycloidem illam APS ver&longs;us quam peragitur motus, & circum eam ceu ob&longs;taculum flectatur, parteque reliqua PT cui Semicyclois nondum objicitur, protendatur in lineam rectam; & pondus T o&longs;cillabitur in Cycloide data QRS. que E. F.

~~DE MOTU CORPORUM~~

Occurrat enim Filum PT tum Cycloidi QRS in T, tum circulo QOS in V, agaturque CV; & ad Fili partem rectam PT, e punctis extremis P ac T, erigantur perpendicula PB, TW, occurrentia rectæ CV in B & W. Patet, ex con&longs;tructione & gene&longs;i &longs;imilium Figurarum AS, SR, perpendicula illa PB, TW ab&longs;cindere de CV longitudines VB, VW Rotarum diametris OA, OR æquales. E&longs;t igitur TP ad VP (duplum &longs;inum anguli VBP exi&longs;tente 1/2 BV ra- dio) ut BW ad BV, &longs;eu AO+OR ad AO, id e&longs;t (cum &longs;int CA ad CO, CO ad CR & divi&longs;im AO ad OR proportionales,) ut CA+CO ad CA vel, &longs;i bi&longs;ecetur BV in E, ut 2 CE ad CB.Proinde, per Corol. ~~1. Prop.~~ ~~XLIX, longitudo partis rectæ Fili PTæquatur &longs;emper Cycloidis arcui PS, & Filum totum APT æquatur &longs;emper Cycloidis arcui dimidio APS, hoc e&longs;t (per Corol.~~ ~~2. Prop.~~ ~~XLIX) longitudini AR. Et propterea vici&longs;&longs;im &longs;i Filum manet &longs;emper æquale longitudini AR movebitur punctum T in Cycloide data QRS. que E. D.~~

~~LIBER PRIMUS.~~

~~Corol. Filum AR æquatur Semicycloidi AS, adeoque ad &longs;emidiametrum AC eandem habet rationem quam &longs;imilis illi Semicyclois SR habet ad &longs;emidiametrum CO.~~

~~PROPOSITIO LI. THEOREMA XVIII.~~

Si Vis centripeta tendens undique ad Globi centrum C &longs;it in locis &longs;ingulis ut di&longs;tantia loci cuju&longs;que a centro, & hac &longs;ola Vi agente corpus T o&longs;cilletur (modo jam de&longs;cripto) in perimetro Cycloidis QRS: dico quod o&longs;cillationum utcunque inæqualium æqualia erunt Tempora.

Nam in Cycloidis tangentem TW infinite productam cadat perpendiculum CX & jungatur CT. Quoniam vis centripeta qua corpus T impellitur ver&longs;us C e&longs;t ut di&longs;tantia CT, atque hæc (per Legum Corol. 2.) re&longs;olvitur in partes CX, TX, quarum CX impellendo corpus directe a P di&longs;tendit filum PT & per ejus re&longs;i&longs;tentiam tota ce&longs;&longs;at, nullum alium edens effectum; pars autem altera TX,urgendo corpus tran&longs;ver&longs;im &longs;eu ver&longs;us X, directe accelerat motum ejus in Cycloide; manife&longs;tum e&longs;t quod corporis acceleratio, huic vi acceleratrici proportionalis, &longs;it &longs;ingulis momentis ut longitudo TX, id e&longs;t, (ob datas CV, WV ii&longs;que proportionales TX, TW,) ut longitudo TW, hoc e&longs;t (per Corol. ~~1. Prop.~~ XLIX,) ut longitudo arcus Cycloidis TR. Pendulis igitur duobus APT, Apt de perpendiculo AR inæqualiter deductis & &longs;imul dimi&longs;&longs;is, accelerationes eorum &longs;emper erunt ut arcus de&longs;cribendi TR, tR. Sunt autem partes &longs;ub initio de&longs;criptæ ut accelerationes, hoc e&longs;t, ut totæ &longs;ub initio de&longs;cribendæ, & propterea partes quæ manent de&longs;criben- dæ & accelerationes &longs;ub&longs;equentes, his partibus proportionales, &longs;unt etiam ut totæ; & &longs;ic deinceps. Sunt igitur accelerationes atque adeo velocitates genitæ & partes his velocitatibus de&longs;criptæ parte&longs;que de&longs;cribendæ, &longs;emper ut totæ; & propterea partes de&longs;cribendæ datam &longs;ervantes rationem ad invicem &longs;imul evane&longs;cent, id e&longs;t, corpora duo o&longs;cillantia &longs;imul pervenient ad perpendiculum AR.Cumque vici&longs;&longs;im a&longs;cen&longs;us perpendiculorum de loco in&longs;imo R, per eo&longs;dem arcus Cycloidales motu retrogrado facti, retardentur in locis &longs;ingulis a viribus ii&longs;dem a quibus de&longs;cen&longs;us accelerabantur, patet velocitates a&longs;cen&longs;uum ac de&longs;cen&longs;uum per eo&longs;dem arcus factorum æquales e&longs;&longs;e, atque adeo temporibus æqualibus fieri; & propterea, cum Cycloidis partes duæ RS & RQ ad utrumque perpendiculi latus jacentes &longs;int &longs;imiles & æquales, pendula duo o&longs;cillationes &longs;uas tam totas quam dimidias ii&longs;dem temporibus &longs;emper peragent. que E. D.

~~DE MOTU CORPORUM~~

Corol. Vis qua corpus T in loco quovis T acceleratur vel retartur in Cycloide, e&longs;t ad totum corporis eju&longs;dem Pondus in loco alti&longs;&longs;imo S vel Q, ut Cycloidis arcus TR ad eju&longs;dem arcum SRvel QR.

~~PROPOSITIO LII. PROBLEMA XXXIV.~~

~~Definire & Velocitates Pendulorum in locis &longs;ingulis, & Tempora quibus tum o&longs;cillationes totæ, tum &longs;ingulæ o&longs;cillationum partes peraguntur.~~

~~Centro quovis G, intervallo GH Cycloidis arcum RS æquante, de&longs;cribe &longs;emicirculum HKMG &longs;emidiametro GK bi&longs;ectum.~~ ~~Et &longs;i vis centripeta, di&longs;tantiis locorum a centro proportionalis, tendat ad centrum G, &longs;itque ea in perimetro HIK æqualis vi centripetæ in perimetro Globi QOS (Vide Fig.~~ Prop. L.) ad ip&longs;ius centrum tendenti; & eodem tempore quo pendulum T dimittitur e loco &longs;upremo S, cadat corpus aliquod L ab H ad G: quoniam vires quibus corpora urgentur &longs;unt æquales &longs;ub initio & &longs;patiis de&longs;cribendis TR, LG &longs;emper proportionales, atque adeo, &longs;i æquantur TR & LG, æquales in locis T & L; patet corpora illa de&longs;cribere &longs;patia ST, HL æqualia &longs;ub initio, adeoque &longs;ubinde pergere æqualiter urgeri, & æqualia &longs;patia de&longs;cribere. ~~Quare, per Prop.~~ XXXVIII, tempus quo corpus de&longs;cribit arcum ST e&longs;t ad tempus o&longs;cillationis unius, ut arcus HI (tempus quo corpus H perveniet ad L) ad &longs;emiperipheriam HKM (tempus quo corpus H perveniet ad M.) Et velocitas corporis penduli in loco T e&longs;t ad veloc tatem ipfius in loco infimo R, (hoc e&longs;t, velocitas corporis H in loco L ad velocitatem ejus in loco G, &longs;eu incrementum momentaneum lineæ HL ad incrementum momentaneum lineæ HG, arcubus HI, HK æquabili fluxu cre&longs;centibus) ut ordinatim applicata LI ad radium GK, &longs;ive ut √SRq.-TRque ad SR. Unde cum, in o&longs;cillationibus inæqualibus, de&longs;cribantur æqualibus temporibus arcus totis o&longs;cillationum arcubus proportionales; habentur, ex datis temporibus, & velocitates & arcus de&longs;cripti in o&longs;cillationibus univer&longs;is. ~~Quæ erant primo invenienda.~~

~~LIBER PRIMUS.~~

~~O&longs;cillentur jam Funipendula~~

corpora in Cycloidibus diver&longs;is intra Globos diver&longs;os, quorum diver&longs;æ &longs;unt etiam Vires ab&longs;olutæ, de&longs;criptis: &, &longs;i Vis ab&longs;oluta Globi cuju&longs;vis QOS dicatur V, Vis acceleratrix qua Pendulum urgetur in circumferentia hujus Globi, ubi incipit directe ver&longs;us centrum ejus moveri, erit ut di&longs;tantia Corporis penduli a centro illo & Vis ab&longs;oluta Globi conjunctim, hoc e&longs;t, ut COXV. Itaque lineola HY, quæ &longs;it ut hæc Vis acceleratrix COXV, de&longs;cribetur dato tempore; &, &longs;i erigatur normalis YZcircumferentiæ occurrens in Z, arcus na&longs;cens HZ denotabit datum illud tempus. ~~E&longs;t autem arcus hic na&longs;cens HZ in &longs;ubduplicata ratione rectanguli GHY, adeoque ut √GHXCOXV.~~ Unde Tempus o&longs;cillationis integræ in Cycloide QRS (cum &longs;it ut &longs;emiperipheria HKM, quæ o&longs;cillationem illam integram denotat, directe, utque arcus HZ, qui datum tempus &longs;imiliter denotat, inver&longs;e) fiet ut GH directe & √GHXCOXV inver&longs;e, hoc e&longs;t, ob æquales GH& SR, ut √(SR/COXV), &longs;ive (per Corol. ~~Prop.~~ L) ut √(AR/ACXV). Itaque O&longs;cillationes in Globis & Cycloidibus omnibus, quibu&longs;cunque cum Viribus ab&longs;olutis factæ, &longs;unt in ratione quæ componitur ex &longs;ubduplicata ratione longitudinis Fili directe, & &longs;ubduplicata ratione di&longs;tantiæ inter punctum &longs;u&longs;pen&longs;ionis & centrum Globi inver&longs;e, & &longs;ubduplicata ratione Vis ab&longs;olutæ Globi etiam inver&longs;e. que E. I.

~~DE MOTU CORPORUM~~

~~Corol. 1. Hinc etiam O&longs;cillantium, Cadentium & Revolventium corporum tempora po&longs;&longs;unt inter &longs;e conferri.~~ Nam &longs;i Rotæ, qua Cyclois intra globum de&longs;cribitur, diameter con&longs;tituatur æqualis &longs;emidiametro globi, Cyclois evadet Linea recta per centrum globi tran&longs;iens, & O&longs;cillatio jam erit de&longs;cen&longs;us & &longs;ub&longs;equens a&longs;cen&longs;us in hac recta. Unde datur tum tempus de&longs;cen&longs;us de loco quovis ad centrum, tum tempus huic æquale quo corpus uniformiter circa centrum globi ad di&longs;tantiam quamvis revolvendo arcum quadrantalem de&longs;cribit. ~~E&longs;t enim hoc tempus (per Ca&longs;um &longs;ecundum) ad tempus &longs;emio&longs;cillationis in Cycloide quavis QRS ut 1 ad √(AR/AC).~~

~~Corol. 2. Hinc etiam con&longs;ectantur quæ Wrennus & Hugenius de Cycloide vulgari adinvenerunt.~~ Nam &longs;i Globi diameter augeatur in infinitum: mutabitur ejus &longs;uperficies &longs;phærica in planum, Vi&longs;que centripeta aget uniformiter &longs;ecundum lineas huic plano perpendiculares, & Cyclois no&longs;tra abibit in Cycloidem vulgi. I&longs;to autem in ca&longs;u longitudo arcus Cycloidis, inter planum illud & punctum de&longs;cribens, æqualis evadet quadruplicato &longs;inui ver&longs;o dimidii arcus Rotæ inter idem planum & punctum de&longs;cribens; ut invenit Wrennus: Et Pendulum inter duas eju&longs;modi Cycloides in &longs;imili & æquali Cycloide temporibus æqualibus O&longs;cillabitur, ut demon&longs;travit Hugenius. Sed & De&longs;cen&longs;us gravium, tempore O&longs;cillationis unius, is erit quem Hugenius indicavit.

Aptantur autem Propo&longs;itiones a nobis demon&longs;tratæ ad veram con&longs;titutionem Terræ, quatenus Rotæ eundo in ejus circulis maximis de&longs;cribunt motu Clavorum, perimetris &longs;uis infixorum, Cycloides extra globum; & Pendula inferius in fodinis & cavernis Terra &longs;u&longs;pen&longs;a, in Cycloidibus intra globos O&longs;cillari debent, ut O&longs;cillationes omnes evadant I&longs;ochronæ. Nam Gravitas (ut in Libro tertio docebitur) decre&longs;cit in progre&longs;&longs;u a &longs;uperficie Terræ, &longs;ur&longs;um quidem in duplicata ratione di&longs;tantiarum a centro ejus, de or&longs;um vero in ratione &longs;implici.

~~LIBER PRIMUS.~~

~~PROPOSITIO LIII. PROBLEMA XXXV.~~

~~Conce&longs;&longs;is Figurarum curvilinearum quadraturis, invenire Vires quibus corpora in datis curvis lineis O&longs;cillationes &longs;emper I&longs;ochronas peragent.~~

O&longs;cilletur corpus T in curva quavis linea STRQ, cujus axis &longs;it OR tran&longs;iens per virium centrum C. Agatur TX quæ curvam illam in corporis loco quovis T contingat, inque hac tangente TX

~~capiatur TY æqualis arcui TR. Nam longitudo arcus illius ex Figurarum quadraturis (per Methodos vulgares) innote&longs;cit.~~ ~~De puncto Y educatur recta YZ tangenti perpendicularis.~~ ~~Agatur CT perpendiculari illi occurrens in Z, & erit Vis centripeta proportionalis rectæ TZ. que E. I.~~

~~DE MOTU CORPORUM~~

Nam &longs;i vis, qua corpus trahitur de T ver&longs;us C, exponatur per rectam TZ captam ip&longs;i proportionalem, re&longs;olvetur hæc in vires TY, YZ; quarum YZ trahendo corpus &longs;ecundum longitudinem Fili PT, motum ejus nil mutat, vis autem altera TY motum ejus in curva STRQ directe accelerat vel directe retardat. Proinde cum hæc &longs;it ut via de&longs;cribenda TR, accelerationes corporis vel retardationes in O&longs;cillationum duarum (majoris & minoris) partibus proportionalibus de&longs;cribendis, erunt &longs;emper ut partes illæ, & propterea facient ut partes illæ &longs;imul de&longs;cribantur. ~~Corpora autem quæ partes totis &longs;emper proportionales &longs;imul de&longs;cribunt, &longs;imul de&longs;cribent totas. que E. D.~~

Corol. 1. Hinc &longs;i corpus T Filo rectilineo AT a centro A pendens, de&longs;cribat arcum circularem STRQ, & interea urgeatur &longs;ecundum lineas parallelas deor&longs;um a vi aliqua, quæ &longs;it ad vim uniformem Gravitatis, ut arcus TR ad ejus &longs;inum TN: æqualia erunt O&longs;cillationum &longs;ingularum tempora. Etenim ob parallelas TZ, AR, &longs;imilia erunt triangula ATN, ZTY; & propterea TZ erit ad AT ut TY ad TN; hoc e&longs;t, (&longs;i Gravitatis vis uniformis exponatur per longitudinem datam AT) vis TZ, qua O&longs;cillationes evadent I&longs;ochronæ, erit ad vim Gravitatis AT, ut arcus TR ip&longs;i TY æqualis ad arcus illius &longs;inum TN.

Corol. 2. Igitur in Horologiis, &longs;i vires a Machina in Pendulum ad motum con&longs;ervandum impre&longs;&longs;æ ita cum vi Gravitatis componi po&longs;&longs;int, ut vis tota deor&longs;um &longs;emper &longs;it ut linea quæ oritur applicando rectangulum &longs;ub arcu TR & radio AR ad &longs;inum TN,O&longs;cillationes omnes erunt I&longs;ochronæ.

~~PROPOSITIO LIV. PROBLEMA XXXVI.~~

Conce&longs;&longs;is Figurarum curvilinearum quadraturis, invenire Tempora quibus corpora Vi qualibet centripeta in lineis quibu&longs;cunque curvis, in plano per centrum Virium tran&longs;eunte de&longs;criptis, de&longs;cendent & a&longs;cendent.

~~De&longs;cendat corpus de loco quovis S per lineam quamvis curvam STtR, in plano per virium centrum C tran&longs;eunte datam.~~ ~~Jungatur CS & dividatur eadem in partes innumeras æquales, &longs;itque Dd partium illarum aliqua.~~ ~~Centro C, intervallis CD, Cd de&longs;criban~~

~~tur circuli DT, dt, lineæ curvæ STtR occurrentes in T & t. Et ex data tum lege vis centripetæ, tum~~

~~altitudine CS de qua corpus cecidit; dabitur velocitas corporis in alia quavis altitudine CT, per Prop.~~ ~~XXXIX.~~ ~~Tempus autem, quo corpus de&longs;cribit lineolam Tt, e&longs;t ut lineolæ hujus longitudo (id e&longs;t ut &longs;ecans anguli tTC) directe, & velocitas inver&longs;e.~~ ~~Tempori huic proportionalis &longs;it ordinatim applicata DN ad rectam CS per punctum D perpendicularis, & ob datam Dderit rectangulum DdXDN, hoc e&longs;t area DNnd, eidem tempori proportionale.~~ Ergo &longs;i SNn &longs;it curva illa linea quam punctum N perpetuo tangit, erit area SNDS proportionalis tempori quo corpus de&longs;cendendo de&longs;crip&longs;it lineam ST; proindeque ex inventa illa area dabitur Tempus. q.E.I.

~~LIBER PRIMUS.~~

~~PROPOSITIO LV. THEOREMA XIX.~~

Si corpus movetur in &longs;uperficie quacunque curva, cujus axis per centrum Virium tran&longs;it, & a corpore in axem demittatur perpendicularis, eique parallela & æqualis ab axis puncto quovis dato ducatur: dico quod parallela illa aream tempori proportionalem de&longs;cribet.

Sit BSKL &longs;uperficies curva, T corpus in ea revolvens, STtRTrajectoria quam corpus in eadem de&longs;cribit, S initium Trajectoriæ, OMNK axis &longs;uperficiei curvæ, TN recta a corpore in axem perpendicularis, OP huic parallela & æqualis a puncto O quod in axe datur educta, AP ve&longs;tigium Trajectoriæ a puncto P in lineæ volubilis OP plano AOP de&longs;criptum, A ve&longs;tigii initium puncto Sre&longs;pondens, TC recta a corpore ad centrum ducta; TG pars ejus vi centripetæ qua corpus urgetur in centrum C proportionalis; TM recta ad &longs;uperficiem cutvam perpendicularis, TI pars ejus vi pre&longs;&longs;ionis, qua corpus urget &longs;uperficiem vici&longs;&longs;imque urgetur ver&longs;us M a &longs;uperficie, proportiona

~~lis; PHTF recta axi parallela per corpus tran&longs;iens, & GF, IH rectæ a punctis G & I in parallelam illam PHTFperpendiculariter demi&longs;&longs;æ.~~ ~~Dico jam quod area AOP, radio OP ab initio motus de&longs;cripta, &longs;it tempori proportionalis.~~ ~~Nam vis TG (per Legum Corol.~~ 2.) re&longs;olvitur in vires TF, FG; & vis TI in vires TH, HI:Vires autem TF, THagendo &longs;ecundum lineam PF plano AOP perpendicularem mutant &longs;olummodo motum corporis quatenus huic plano perpendicularem. Ideoque motus ejus quatenus &longs;ecundum po&longs;itionem plani factus, hoc e&longs;t, motus puncti P quo Trajectoriæ ve&longs;tigium AP in hoc plano de&longs;cribitur, idem e&longs;t ac &longs;i vires TF, TH tollerentur, & corpus &longs;olis viribus FG, HI agitaretur; hoc e&longs;t, idem ac &longs;i corpus in plano AOP, vi centripeta ad centrum O tendente & &longs;ummam virium FG & HI æquante, de&longs;criberet curvam AP. Sed vi tali de&longs;cribitur area AOP (per Prop. ~~1.) tempori proportionalis. q.E.D.~~

~~DE MOTU CORPORUM~~

Corol. Eodem argumento &longs;i corpus a viribus agitatum ad centra duo vel plura in eadem quavis recta CO data tendentibus, de&longs;criberet in &longs;patio libero lineam quamcunque curvam ST; foret area AOP tempori &longs;emper proportionalis.

~~PROPOSITIO LVI. PROBLEMA XXXVII.~~

Conce&longs;&longs;is Figurarum curvilinearum quadraturis, dati&longs;que tum lege Vis centripetæ ad centrum datum tendentis, tum &longs;uperficie curva cujus axis per centrum illud træn&longs;it; invenieuda est Trajectoria quam corpus in eadem &longs;uperficie de&longs;cribet, de loco dato, data cum Velocitate, ver&longs;us plagam in &longs;uperficie illa datam egre&longs;&longs;um.

Stantibus quæ in &longs;uperiore Propo&longs;itione con&longs;tructa &longs;unt, exeat corpus de loco S in Trajectoriam inveniendam STtR; &, ex data ejus velocitate in altitudine SC, dabitur ejus velocitas in alia quavis altitudine TC. Ea cum velocitate, dato tempore quam minimo, de&longs;cribat corpus Trajectoriæ &longs;uæ particulam Tt, &longs;itque Pp ve&longs;tigium ejus in plano AOP de&longs;criptum. ~~Jungatur Op, & Circelli centro T intervallo Tt in &longs;uperficie curva de&longs;cripti &longs;it PpQve&longs;tigium Ellipticum in eodem plano OAPp de&longs;criptum.~~ Et ob datum magnitudine & po&longs;itione Circellum, dabitur Ellip&longs;is illa Ppque Cumque area POp &longs;it tempori proportionalis, atque adeo ex dato tempore detur, dabitur Op po&longs;itione, & inde dabitur communis ejus & Ellip&longs;eos inter&longs;ectio p, una cum angulo OPp,in quo Trajectoriæ ve&longs;tigium APp &longs;ecat lineam OP. Inde autem invenietur Trajectoriæ ve&longs;tigium illud APp, eadem methodo qua curva linea VIKk, in Propo&longs;itione XLI, ex &longs;imilibus datis inventa fuit. ~~Tum ex &longs;ingulis ve&longs;tigii punctis P erigendo ad planum AOP perpendicula PT &longs;uperficiei curvæ occurrentia in T,dabuntur &longs;ingula Trajectoriæ puncta T. q.E.I.~~

~~LIBER PRIMUS.~~

~~SECTIO XI.~~

~~De Motu Corporum Viribus centripetis &longs;e mutuo petentium.~~

~~Hactenus expo&longs;ui Motus corporum attractorum ad centrum Immobile, quale tamen vix extat in rerum natura.~~ Attractiones enim fieri &longs;olent ad corpora; & corporum trahentium & attractorum actiones &longs;emper mutuæ &longs;unt & æquales, per Legem tertiam: adeo ut neque attrahens po&longs;&longs;it quie&longs;cere neque attractum, &longs;i duo &longs;int corpora, &longs;ed ambo (per Legum Corollarium quartum) qua&longs;i attractione mutua, circum gravitatis centrum commune revolvantur: & &longs;i plura &longs;int corpora (quæ vel ab unico attrahantur vel omnia &longs;e mutuo attrahant) hæc ita inter &longs;e moveri debeant, ut gravitatis centrum commune vel quie&longs;cat vel uniformiter moveatur in directum. Qua de cau&longs;a jam pergo Motum exponere corporum &longs;e mutuo trahentium, con&longs;iderando Vires centripetas tanquam Attractiones, quamvis forta&longs;&longs;e, &longs;i phy&longs;ice loquamur, verius dicantur Impul&longs;us. In Mathematicis enim jam ver&longs;amur, & propterea mi&longs;&longs;is di&longs;putationibus Phy&longs;icis, familiari utimur &longs;ermone, quo po&longs;&longs;imus a Lectoribus Mathematicis facilius intelligi.

~~DE MOTU CORPORUM~~

~~PROPOSITIO LVII. THEOREMA XX.~~

~~Corpora duo &longs;e invicem trahentia de&longs;cribunt, & circum commune centrum gravitatis, & circum &longs;e mutuo, Figuras &longs;imiles.~~

Sunt enim di&longs;tantiæ a communi gravitatis centro reciproce proportionales corporibus, atque adeo in data ratione ad invicem, & componendo, in data ratione ad di&longs;tantiam totam inter corpora. ~~Feruntur autem hæ di&longs;tantiæ circum terminos &longs;uos communi motu angulari, propterea quod in directum &longs;emper jacentes non mutant inclinationem ad &longs;e mutuo.~~ Lineæ autem rectæ, quæ &longs;unt in data ratione ad invicem, & æquali motu angulari circum terminos &longs;uos feruntur, Figuras circum eo&longs;dem terminos (in planis quæ una cum his terminis vel quie&longs;cunt vel motu quovis non angulari moventur) de&longs;cribunt omnino &longs;imiles. ~~Proinde &longs;imiles &longs;unt Figuræ quæ his di&longs;tantiis circumactis de&longs;cribuntur. q.E.D.~~

~~PROPOSITIO LVIII. THEOREMA XXI.~~

Si corpora duo Viribus quibu&longs;vis &longs;e mutuo trahunt, & interea revolvuntur circa gravitatis centrum commune: dico quod Figuris, quas corpora &longs;ic mota de&longs;cribunt circum &longs;e mutuo, potest Figura &longs;imilis & æqualis, circum corpus alterutrum immotum, Viribus ii&longs;dem de&longs;eribi.

~~Revolvantur corpora S, P circa commune gravitatis centrum C, pergendo de S ad T deque P ad que A dato puncto s ip&longs;is~~

SP, TQ æquales & parallelæ ducantur &longs;emper sp, sq; & Curva pqv quam punctum p, revolvendo circum punctum immotum s, de&longs;cribit, erit &longs;imilis & æqualis Curvis quas corpora S, P de&longs;cri bunt circum &longs;e mutuo: proindeque (per Theor. ~~XX) &longs;imilis Curvis ST & PQV, quas eadem corpora de&longs;cribunt circum commune gravitatis centrum C: id adeo quia proportiones linearum SC, CP& SP vel. sp ad invicem dantur.~~

~~LIBER PRIMUS.~~

~~Cas. 1. Commune illud gravitatis centrum C, per Legum Corollarium quartum, vel quie&longs;cit vel movetur uniformiter in directum.~~ ~~Ponamus primo quod id quie&longs;cit, inque s & p locentur corpora duo, immobile in s, mobile in p, corporibus S & P &longs;imilia & æqualia.~~ ~~Dein tangant rectæ PR & pr Curvas PQ & pq in P & p, & producantur CQ & sq ad R & r. Et, ob &longs;imilitudinem Figurarum CPRQ, sprq, erit RQ ad rq ut CP ad sp, adeoque in data ratione.~~ Proinde &longs;i vis qua corpus P ver&longs;us corpus S, atque adeo ver&longs;us centrum intermedium C attrahitur, e&longs;&longs;et ad vim qua corpus p ver&longs;us centrum s attrahitur in eadem illa ratione data; hæ vires æqualibus temporibus attraherent &longs;emper corpora de tangentibus PR, pr ad arcus PQ, pq, per intervalla ip&longs;is proportionalia RQ, rque adeoque vis po&longs;terior efficeret ut corpus p gyraretur in Curva pqv, quæ &longs;imilis e&longs;&longs;et Curvæ PQV, in qua vis prior efficit ut corpus P gyretur, & revolutiones ii&longs;dem temporibus complerentur. At quoniam vires illæ non &longs;unt ad invicem in ratione CP ad sp, &longs;ed (ob &longs;imilitudinem & æqualitatem corporum S & s, P & p, æqualitatem di&longs;tantiarum SP, sp) &longs;ibi mutuo æquales; corpora æqualibus temporibus æqualiter trahentur de tangentibus: & propterea, ut corpus po&longs;terius p trahatur per intervallum majus rq, requiritur tempus majus, idque in &longs;ubduplicata ratione intervallorum; propterea quod (per Lemma decimum) &longs;patia, ip&longs;o motus initio de&longs;cripta, &longs;unt in duplicata ratione temporum. Ponatur igitur velocitas corporis p e&longs;&longs;e ad velocitatem corporis P in &longs;ubduplicata ratione di&longs;tantiæ sp ad di&longs;tantiam CP, eo ut temporibus quæ &longs;int in eadem &longs;ubduplicata ratione de&longs;cribantur arcus pq, PQ, qui &longs;unt in ratione integra: Et corpora P, p viribus æqualibus &longs;emper attracta de&longs;cribent circum centra quie&longs;centia C & s Figuras &longs;imiles PQV, pqv, quarum po&longs;terior pqv &longs;imilis e&longs;t & æqualis Figuræ quam corpus P circum corpus mobile S de&longs;cribit. q.E.D.

Cas. 2. Ponamus jam quod commune gravitatis centrum, una cum &longs;patio in quo corpora moventur inter &longs;e, progreditur uniformiter in directum; &, per Legum Corollarium &longs;extum, motus omnes in hoc &longs;patio peragentur ut prius, adeoque corpora de&longs;cri- bent circum &longs;e mutuo Figuras ea&longs;dem ac prius, & propterea Figuræ pqv &longs;imiles & æquales. q.E.D.

~~DE MOTU CORPORUM~~

~~Corol. 1. Hinc corpora duo Viribus di&longs;tantiæ &longs;uæ proportionalibus &longs;e mutuo trahentia, de&longs;cribunt (per Prop.~~ X,) & circum commune gravitatis centrum, & circum &longs;e mutuo, Ellip&longs;es concentricas: & vice ver&longs;a, &longs;i tales Figuræ de&longs;cribuntur, &longs;unt Vires di&longs;tantiæ proportionales.

~~Corol. 2. Et corpora duo Viribus quadrato di&longs;tantiæ &longs;uæ reciproce proportionalibus de&longs;cribunt (per Prop.~~ ~~XI, XII, XIII) & circum commune gravitatis centrum, & circum &longs;e mutuo, Sectiones conicas umbilicum habentes in centro circum quod Figuræ de&longs;cribuntur.~~ ~~Et vice ver&longs;a, &longs;i tales Figuræ de&longs;cribuntur, Vires centripetæ &longs;unt quadrato di&longs;tantiæ reciproce proportionales.~~

~~Corol. 3. Corpora duo quævis cirum gravitatis centrum commune gyrantia, radiis & ad centrum illud & ad &longs;e mutuo ductis, de&longs;cribunt areas temporibus proportionales.~~

~~PROPOSITIO LIX. THEOREMA XXII.~~

Corporum duorum S & P circa commune gravitatis centrum C revolventium Tempus periodicum e&longs;&longs;e ad Tempus periodicum corporis alterutrius P, circa alterum immotum S gyrantis & Figuris quæ corpora circum &longs;e mutuo de&longs;cribunt Figuram &longs;imilem & æqualem de&longs;cribentis, in &longs;ubduplicata ratione corporis alterins S, ad &longs;ummam corporum S+P.

Namque, ex demon&longs;tratione &longs;uperioris Propo&longs;itionis, tempora quibus arcus quivis &longs;imiles PQ & pq de&longs;cribuntur, &longs;unt in &longs;ubduplicata ratione di&longs;tantiarum CP & SP vel sp, hoc e&longs;t, in &longs;ubduplicata ratione corporis S ad &longs;ummam corporum S+P. Et componendo, &longs;ummæ temporum quibus arcus omnes &longs;imiles PQ & pqde&longs;cribuntur, hoc e&longs;t, tempora tota quibus Figuræ totæ &longs;imiles de&longs;cribuntur, &longs;unt in eadem &longs;ubduplicata ratione. q.E.D.

~~PROPOSITIO LX. THEOREMA XXIII.~~

~~LIBER PRIMUS.~~

St corpora duo S & P, Viribus quadrato di&longs;tantiæ &longs;uæ reciproee proportionalibus &longs;e mutuo trahentia, revalvuntur circa gravitatis centrum commune: dico quod Ellip&longs;eos, quam corpus alterutrum P hoc motu circa alterum S de&longs;cribit, Axis principalis erit ad Axem principalem Ellip&longs;eos, quam corpus idem P circa alterum quie&longs;cens S eodem tempore periodico de&longs;cribere po&longs;&longs;et, ut &longs;umma corporum duorum S+P ad primam duarum medie proportionalium inter hanc &longs;ummam & corpus illud alterum S.

Nam &longs;i de&longs;criptæ Ellip&longs;es e&longs;&longs;ent &longs;ibi invicem æquales, tempora periodica (per Theorema &longs;uperius) forent in &longs;ubduplicata ratione corporis S ad &longs;ummam corporum S+P. Minuatur in hac ratione tempus periodicum in Ellip&longs;i po&longs;teriore, & tempora periodica evadent æqualia; Ellip&longs;eos autem axis prineipalis (per Prop. XV.) minuetur in ratione cujus hæc e&longs;t &longs;e&longs;quiplicata, id e&longs;t in ratione, cujus ratio S ad S+P e&longs;t triplicata; adeoque erit ad axem principalem Ellip&longs;eos alterius, ut prima duarum medie proportionalium inter S+P & S ad S+P. Et inver&longs;e, axis principalis Ellip&longs;eos circa corpus mobile de&longs;criptæ erit ad axem principalem de&longs;criptæ circa immobile, ut S+P ad primam duarum medie proportionalium inter S+P & S. q.E.D.

~~PROPOSITIO LXI. THEOREMA XXIV.~~

Si corpora duo Viribus quibu&longs;vis &longs;e mutuo trabentia, neque alias agitata vel impedita, quomodocunque moveantur; motus eorum perinde &longs;e habebunt ac &longs;i non traherent &longs;e mutuo, &longs;ed utrumque a corpore tertio in communi gravitatis centro con&longs;tituto Viribus ii&longs;dem traberetur: Et Virium trahentium eadem erit Lex re&longs;pectu di&longs;tantiæ corporum a centro illo communi atque re&longs;pectu di&longs;tantiæ totius inter corpora.

~~Nam vires illæ, quibus corpora &longs;e mutuo trahunt, tendendo ad corpora, tendunt ad commune gravitatis centrum interme-~~

~~dium, adeoque eædem &longs;unt ac &longs;i a corpore intermedio manarent. q.E.D.~~

~~DE MOTU CORPORUM~~

Et quoniam data e&longs;t ratio di&longs;tantiæ corporis utriu&longs;vis a centro illo communi ad di&longs;tantiam corporis eju&longs;dem a corpore altero, dabitur ratio cuju&longs;vis pote&longs;tatis di&longs;tantiæ unius ad eandem pote&longs;tatem di&longs;tantiæ alterius; ut & ratio quantitatis cuju&longs;vis, quæ ex una di&longs;tantia & quantitatibus datis utcunque derivatur, ad quantitatem aliam, quæ ex altera di&longs;tantia & quantitatibus totidem datis datamque illam di&longs;tantiarum rationem ad priores habentibus &longs;imiliter derivatur. Proinde &longs;i vis, qua corpus unum ab altero trahitur, &longs;it directe vel inver&longs;e ut di&longs;tantia corporum ab invicem; vel ut quælibet hujus di&longs;tantiæ pote&longs;tas; vel denique ut quantitas quævis ex hac di&longs;tantia & quantitatibus datis quomodocunque derivata: erit eadem vis, qua corpus idem ad commune gravitatis centrum trahitur, directe itidem vel inver&longs;e ut corporis attracti di&longs;tantia a centro illo communi, vel ut eadem di&longs;tantiæ hujus pote&longs;tas, vel denique ut quantitas ex hac di&longs;tantia & analogis quantitatibus datis &longs;imiliter derivata. ~~Hoc e&longs;t, Vis trahentis eadem erit Lex re&longs;pectu di&longs;tantiæ utriu&longs;que. q.E.D.~~

~~PROPOSITIO LXII. PROBLEMA XXXVIII.~~

~~Corporum duorum quæ Viribus quadrato di&longs;tantiæ &longs;uæ reciproce proportionalibus &longs;e mutuo trahunt, ac de locis datis demittuntur, determinare Motus.~~

Corpora (per Theorema novi&longs;&longs;imum) perinde movebuntur ac &longs;i a corpore tertio, in communi gravitatis centro con&longs;tituto, traherentur; & centrum illud ip&longs;o motus initio quie&longs;cet per Hypothe&longs;in; & propterea (per Legum Corol. ~~4.) &longs;emper quie&longs;cet.~~ ~~Determinandi &longs;unt igitur motus corporum (per Prob.~~ ~~XXV,) perinde ac &longs;i a viribus ad centrum illud tendentibus urgerentur, & habebuntur motus corporum &longs;e mutuo trahentium. q.E.I.~~

~~PROPOSITIO LXIII. PROBLEMA XXXIX.~~

Corporum duorum quæ Viribus quadrato di&longs;tantiæ &longs;uæ reciproce proproportionalibus &longs;e mutuo trahunt, deque locis datis, &longs;ecundum datas rectas, datis cum Velocitatibus exeunt, determinare Motus.

Ex datis corporum motibus &longs;ub initio, datur uniformis motus centri communis gravitatis, ut & motus &longs;patii quod una cum hoc centro movetur uniformiter in directum, nec non corporum motus initiales re&longs;pectu hujus &longs;patii. Motus autem &longs;ub&longs;equentes (per Legum Corollarium quintum, & Theorema novi&longs;&longs;imum) perinde fiunt in hoc &longs;patio, ac &longs;i &longs;patium ip&longs;um una cum communi illo gravitatis centro quie&longs;ceret, & corpora non traherent &longs;e mutuo, &longs;ed a corpore tertio &longs;ito in centro illo traherentur. Corporis igitur alterutrius in hoc &longs;patio mobili, de loco dato, &longs;ecundum datam rectam, data cum velocitate exeuntis, & vi centripeta ad centrum illud tendente correpti, determinandus e&longs;t motus per Problema nonum & vice&longs;imum &longs;extum: & habebitur &longs;imul motus corporis alterius e regione. Cum hoc motu componendus e&longs;t uniformis ille Sy&longs;tematis &longs;patii & corporum in eo gyrantium motus progre&longs;&longs;ivus &longs;upra inventus, & habebitur motus ab&longs;olutus corporum in &longs;patio immobili. q.E.I.

~~LIBER PRIMUS.~~

~~PROPOSITIO LXIV. PROBLEMA XL.~~

~~Viribus quibus Corpora &longs;e mutuo trahunt cre&longs;centibus in &longs;implici ratione di&longs;tantiarum centris: requiruntur Motus plurium Corporum inter &longs;e.~~

Ponantur primo corpora duo T & L commune habentia gravitatis centrum D. De&longs;cribent hæc (per Corollarium primum Theorematis XXI) Ellip&longs;es centra habentes in D, quarum magnitudo ex Problemate V, innote&longs;cit.

~~Trahat jam corpus tertium~~

~~S priora duo T & L viribus acceleratricibus ST, SL,& ab ip&longs;is vici&longs;&longs;im trahatur.~~ ~~Vis ST (per Legum Cor.~~ 2.) re&longs;olvitur in vires SD, DT; & vis SL in vires SD, DL.Vires autem DT, DL, quæ &longs;unt ut ip&longs;arum &longs;umma TL,atque adeo ut vires acceleratrices quibus corpora T & L &longs;e mutuo trahunt, additæ his viribus corporum T & L, prior priori & po&longs;terior po&longs;teriori, componunt vires di&longs;tantiis DT ac DL proportionales, ut prius, &longs;ed viribus prioribus majores; adeoque (per Corol. ~~1. Prop.~~ ~~X. & Corol.~~ ~~1 & 8. Prop, IV) efficiunt ut corpora illa de&longs;cribant Ellip&longs;es ut prius, &longs;ed motu celeriore.~~ Vires reliquæ acceleratrices SD & SD, actionibus motricibus SDXT & SDXL, quæ &longs;unt ut corpora, trahendo corpora illa æqualiter & &longs;ecundum lineas TI, LK, ip&longs;i DSparallelas, nil mutant &longs;itus eorum ad invicem, &longs;ed faciunt ut ip&longs;a æqualiter accedant ad lineam IK; quam ductam concipe per medium corporis S, & lineæ DS perpendicularem. Impedietur autem i&longs;te ad lineam IK acce&longs;&longs;us faciendo ut Sy&longs;tema corporum T & Lex una parte, & corpus S ex altera, ju&longs;tis cum velocitatibus, gyrentur circa commune gravitatis centrum C. Tali motu corpus S(eo quod &longs;umma virium motricium SDXT & SDXL, di&longs;tantiæ CS proportionalium, tendit ver&longs;us centrum C) de&longs;cribit Ellip&longs;in circa idem C; & punctum D, ob proportionales CS, CD,de&longs;cribet Ellip&longs;in con&longs;imilem e regione. Corpora autem T & Lviribus motricibus SDXT
& SDXL, (prius priore, po&longs;terius po&longs;teriore) æqualiter & &longs;ecundum lineas parallelas TI & LK (ut dictum e&longs;t) attracta, pergent (per Legum Corollarium quintum & &longs;extum) circa centrum mobile D Ellip&longs;es &longs;uas de&longs;cribere, ut prius. q.E.I.

~~DE MOTU CORPORUM~~

Addatur jam corpus quartum V, & &longs;imili argumento concludetur hoc & punctum C Ellip&longs;es circa omnium commune centrum gravitatis B de&longs;cribere; manentibus motibus priorum corporum T, L & S circa centra D & C, &longs;ed paulo acceleratis. ~~Et eadem methodo corpora plura adjungere licebit. q.E.I.~~

~~Hæc ita &longs;e habent ubi corpora T & L trahunt &longs;e mutuo viribus acceleratricibus majoribus vel minoribus quam quibus trahunt corpora reliqua pro ratione di&longs;tantiarum.~~ Sunto mutuæ omnium attractiones acceleratrices ad invicem ut di&longs;tantiæ ductæ in corpora trahentia, & ex præcedentibus facile deducetur quod corpora omnia æqualibus temporibus periodicis Ellip&longs;es varias, circa omnium commune gravitatis centrum B, in plano immobili de&longs;cribunt. q.E.I.

~~PROPOSITIO LXV. THEOREMA XXV.~~

~~LIBER PRIMUS.~~

Corpora plura, quorum Vires decre&longs;cunt in duplicata ratione di&longs;tantiarum ab eorundem centris, moveri po&longs;&longs;e inter &longs;e in Ellip&longs;ibus; & radiis ad umbilicos ductis areas de&longs;cribere temporibus proportionales quam proxime.

~~In Propo&longs;itione &longs;uperiore demon&longs;tratus e&longs;t ca&longs;us ubi motus plures peraguntur in Ellip&longs;ibus accurate.~~ Quo magis recedit Lex virium a Lege ibi po&longs;ita, eo magis corpora perturbabunt mutuos motus; neque fieri pote&longs;t ut corpora, &longs;ecundum Legem hic po&longs;itam &longs;e mutuo trahentia, moveantur in Ellip&longs;ibus accurate, ni&longs;i &longs;ervando certam proportionem di&longs;tantiarum ab invicem. ~~In &longs;equentibus autem ca&longs;ibus non multum ab Ellip&longs;ibus errabitur.~~

~~Cas. 1. Pone corpora plura minora circa maximum aliquod ad varias ab eo di&longs;tantias revolvi, tendantque ad &longs;ingula vires ab&longs;olutæ proportionales ii&longs;dem corporibus.~~ ~~Et quoniam omnium commune gravitatis centrum (per Legum Corol.~~ quartum) vel quie&longs;cit vel movetur uniformiter in directum, fingamus corpora minora tam parva e&longs;&longs;e, ut corpus maximum nunquam di&longs;tet &longs;en&longs;ibiliter ab hoc centro: & maximum illud vel quie&longs;cet vel movebitur uniformiter in directum, ab&longs;que errore &longs;en&longs;ibili; minora autem revolventur circa hoc maximum in Ellip&longs;ibus, atque radiis ad idem ductis de&longs;cribent areas temporibus proportionales; ni&longs;i quatenus errores inducuntur, vel per errorem maximi a communi illo gravitatis centro, vel per actiones minorum corporum in &longs;e mutuo. Diminui autem po&longs;&longs;unt corpora minora u&longs;que donec error i&longs;te & actiones mutuæ &longs;int datis quibu&longs;vis minores, atque adeo donec Orbes cum Ellip&longs;ibus quadrent, & areæ re&longs;pondeant temporibus, ab&longs;que errore qui non &longs;it minor quovis dato. q.E.O.

Cas. 2. Fingamus jam Sy&longs;tema corporum minorum modo jam de&longs;cripto circa maximum revolventium, aliudve quodvis duorum circum &longs;e mutuo revolventium corporum Sy&longs;tema progredi uniformiter in directum, & interea vi corporis alterius longe maximi & ad magnam di&longs;tantiam &longs;iti urgeri ad latus. Et quoniam æquales vires acceleratrices, quibus corpora &longs;ecundum lineas parallelas urgentur, non mutant &longs;itus corporum ad invicem, &longs;ed ut Sy&longs;tema totum, &longs;ervatis partium motibus inter &longs;e, &longs;imul transferatur efficiunt: manife&longs;tum e&longs;t quod, ex attractionibus in corpus maximum,

nulla pror&longs;us orietur mutatio motus attractorum inter &longs;e, ni&longs;i vel ex attractionum acceleratricum inæqualitate, vel ex inclinatione linearum ad invicem, &longs;ecundum quas attractiones fiunt. Pone ergo attractiones omnes acceleratrices in corpus maximum e&longs;&longs;e inter &longs;e reciproce ut quadrata di&longs;tantiarum; &, augendo corporis maximi di&longs;tantiam, donec rectarum ab hoc ad reliqua ductarum differentiæ re&longs;pectu earum longitudinis & inclinationes ad invicem minores &longs;int quam datæ quævis, per&longs;everabunt motus partium Sy&longs;tematis inter &longs;e ab&longs;que erroribus qui non &longs;int quibu&longs;vis datis minores. Et quoniam, ob exiguam partium illarum ab invicem di&longs;tantiam, Sy&longs;tema totum ad modum corporis unius attrahitur; movebitur idem hac attractione ad modum corporis unius; hoc e&longs;t, centro &longs;uo gravitatis de&longs;cribet circa corpus maximum Sectionem aliquam Conicam (viz. Hyperbolam vel Parabolam attractione languida, Ellip&longs;in fortiore,) & Radio ad maximum ducto de&longs;cribet areas temporibus proportionales, ab&longs;que ullis erroribus, ni&longs;i quas partium di&longs;tantiæ (perexiguæ &longs;ane & pro lubitu minuendæ) valeant efficere. q.E.O.

~~DE MOTU CORPORUM~~

~~Simili argumento pergere licet ad ca&longs;us magis compo&longs;itos in infinitum.~~

Corol. 1. In ca&longs;u &longs;ecundo; quo propius accedit corpus omnium maximum ad Sy&longs;tema duorum vel plurium, eo magis turbabuntur motus partium Sy&longs;tematis inter &longs;e; propterea quod linearum a corpore maximo ad has ductarum jam major e&longs;t inclinatio ad invicem, majorque proportionis inæqualitas.

Corol. 2. Maxime autem turbabuntur, ponendo quod attractiones acceleratrices partium Sy&longs;tematis ver&longs;us corpus omnium maximum, non &longs;int ad invicem reciproce ut quadrata di&longs;tantiarum a corpore illo maximo; præ&longs;ertim &longs;i proportionis hujus inæqualitas major &longs;it quam inæqualitas proportionis di&longs;tantiarum a corpore maximo: Nam &longs;i vis acceleratrix, æqualiter & &longs;ecundum lineas parallelas agendo, nil perturbat motus inter &longs;e, nece&longs;&longs;e e&longs;t ut ex actionis inæqualitate perturbatio oriatur, majorque &longs;it vel minor pro majore vel minore inæqualitate. ~~Exce&longs;&longs;us impul&longs;uum majorum, agendo in aliqua corpora & non agendo in alia, nece&longs;&longs;ario mutabunt &longs;itum eorum inter &longs;e.~~ ~~Et hæc perturbatio, addita perturbationi quæ ex linearum inclinatione & inæqualitate oritur, majorem reddet perturbationem totam.~~

Corol. 3. Unde &longs;i Sy&longs;tematis hujus partes in Ellip&longs;ibus vel Circulis &longs;ine perturbatione in&longs;igni moveantur; manife&longs;tum e&longs;t, quod eædem a viribus acceleratricibus ad alia corpora tendentibus, aut non urgentur ni&longs;i levi&longs;&longs;ime, aut urgentur æqualiter & &longs;ecundum lineas parallelas quamproxime.

~~LIBER PRIMUS.~~

~~PROPOSITIO LXVI. THEOREMA XXVI.~~

Si Corpora tria, quorum Vires decre&longs;cunt in duplicata ratione di&longs;tantiarum, &longs;e mutuo trahant, & attractiones acceleratrices binorum quorumcunque in tertium &longs;int inter &longs;e reciproce ut quadrata di&longs;tantiarum; minora autem circa maximum revolvantur: Dico quod interius circa intimum & maximum, radiis ad ip&longs;um ductis, de&longs;cribet areas temporibus magis proportionales, & Figuram ad formam Ellip&longs;eos umbilicum in concur&longs;u radiorum habentis magis accedentem, &longs;i corpus maximum his attractionibus agitetur, quam &longs;i maximum illud vel a minoribus non attractum quie&longs;cat, vel multo minus vel multo magis attractum aut multo minus aut multo magis agitetur.

~~Liquet fere ex demon&longs;tratione Corollarii &longs;ecundi Propo&longs;itionis præcedentis; &longs;ed argumento magis di&longs;tincto & latius cogente &longs;ie evincitur.~~

~~Cas. 1. Revolvantur~~

corpora minora P & Sin eodem plano circa maximum T, quorum P de&longs;cribat Orbem interiorem PAB, & Sexteriorem SE. Sit SK mediocris di&longs;tantia corporum P & S; & corporis P ver&longs;us S attractio acceleratrix in mediocri illa di&longs;tantia exponatur per eandem. In duplicata ratione SK ad SP capiatur SL ad SK, & erit SL attractio acceleratrix corporis P ver&longs;us S in di&longs;tantia quavis SP. Junge PT, eique parallelam age LM occurrentem ST in M,& attractio SL re&longs;olvetur (per Legum Corol 2.) in attractiones SM, LM. Et &longs;ic urgebitur corpus P vi acceleratrice triplici: una tendente ad T & oriunda a mutua attractione corporum T & P.Hac vi &longs;ola corpus P circum corpus T, &longs;ive immotum &longs;ive hac attractione agitatum, de&longs;cribere deberet & areas, radio PT, temporibus proportionales, & Ellip&longs;in cui umbilicus e&longs;t in centro corporis T. Patet hoc per Prop. ~~XI. & Corollaria 2 & 3 Theor.~~ ~~XXI.~~ Vis altera e&longs;t attractionis LM, quæ quoniam tendit a P ad T, &longs;uperaddita vi priori coincidet cum ip&longs;a, & &longs;ic faciet ut areæ etiamnum temporibus proportionales de&longs;cribantur per Corol. ~~3. Theor.~~ ~~XXI.~~ At quoniam non e&longs;t quadrato di&longs;tantiæ PT reciproce proportionalis, componet ea cum vi priore vim ab hac proportione aberrantem, idque eo magis quo major e&longs;t proportio hujus vis ad vim priorem, cæteris paribus. ~~Proinde cum (per Prop.~~ ~~XI, & per Corol.~~ ~~2. Theor.~~ XXI) vis qua Ellip&longs;is circa umbilicum T de&longs;cribitur tendere debeat ad umbilicum illum, & e&longs;&longs;e quadrato di&longs;tantiæ PT reciproce proportionalis; vis illa
compo&longs;ita, aberrando ab hac proportione, faciet ut Orbis PABaberret a forma Ellip&longs;eos umbilicum habentis in S; idque eo magis quo major e&longs;t aberratio ab hac proportione; atque adeo etiam quo major e&longs;t proportio vis &longs;ecundæ LM ad vim primam, cæteris paribus. Jam vero vis tertia SM, trahendo corpus P &longs;ecundum lineam ip&longs;i ST parallelam, componet cum viribus prioribus vim quæ non amplius dirigitur a P in T, quæque ab hac determinatione tanto magis aberrat, quanto major e&longs;t proportio hujus tertiæ vis ad vires priores, cæteris paribus; atque adeo quæ faciet ut corpus P, radio TP, areas non amplius temporibus proportionales de&longs;cribat, atque aberratio ab hac proportionalitate ut tanto major &longs;it, quanto major e&longs;t proportio vis hujus tertiæ ad vires cæteras. Orbis vero PAB aberrationem a forma Elliptica præfata hæcvis tertia duplici de cau&longs;a adaugebit, tum quod non dirigatur a Pad T, tum etiam quod non &longs;it proportionalis quadrato di&longs;tantiæ PT.Quibus intellectis, manife&longs;tum e&longs;t quod areæ temporibus tum maxime fiunt proportionales, ubi vis tertia, manentibus viribus cæteris, fit minima; & quod Orbis PAB tum maxime accedit ad præfatam formam Ellipticam, ubi vis tam &longs;ecunda quam tertia, &longs;ed præcipue vis tertia, fit minima, vi prima manente.

~~DE MOTU CORPORUM~~

Exponatur corporis T attractio acceleratrix ver&longs;us S per lineam SN; & &longs;i attractiones acceleratrices SM, SN æquales e&longs;&longs;ent; hæ, trahendo corpora T & P æqualiter & &longs;ecundum lineas parallelas, nil mutarent &longs;itum eorum ad invicem. ~~Iidem jam forent corporum illorum motus inter &longs;e (per Legum Corol.~~ ~~6.) ac &longs;i hæ attractiones tollerentur.~~ Et pari ratione &longs;i attractio SN minor e&longs;&longs;et attractione SM, tolleret ip&longs;a attractionis SM partem SN, & maneret pars &longs;ola MN, qua temporum & arearum proportionalitas & Orbitæ forma illa Elliptica perturbaretur. ~~Et &longs;imiliter &longs;i attractio SN major e&longs;&longs;et attractione SM, oriretur ex differentia &longs;ola MN perturbatio proportionalitatis & Orbitæ.~~ Sic per attractionem SN reducitur &longs;emper attractio tertia &longs;uperior SM ad attractionem MN, attractione prima & &longs;ecunda manentibus pror&longs;us immutatis: & propterea areæ ac tempora ad proportionalitatem, & Orbita PAB ad formam præfatam Ellipticam tum maxime accedunt, ubi attractio MN vel nulla e&longs;t, vel quam fieri po&longs;&longs;it minima; hoc e&longs;t, ubi corporum P & T attractiones acceleratrices, factæ ver&longs;us corpus S, accedunt quantum fieri pote&longs;t ad æqualitatem; id e&longs;t, ubi attractio SN non e&longs;t nulla, neque minor minima attractionum omnium SM, &longs;ed inter attractionum omnium SMmaximam & minimam qua&longs;i mediocris, hoc e&longs;t, non multo major neque multo minor attractione SK. q.E.D.

~~LIBER PRIMUS.~~

Cas. 2. Revolvantur jam corpora minora P, S circa maximum Tin planis diver&longs;is; & vis LM, agendo &longs;ecundum lineam PT in plano Orbitæ PAB &longs;itam, eundem habebit effectum ac prius, neque corpus P de plano Orbitæ &longs;uæ deturbabit. At vis altera NM,agendo &longs;ecundum lineam quæ ip&longs;i ST parallela e&longs;t, (atque adco, quando corpus S ver&longs;atur extra lineam Nodorum, inclinatur ad planum Orbitæ PAB;) præter perturbationem motus in Longitudinem jam ante expo&longs;itam, inducet perturbationem motus in Latitudinem, trahendo corpus P de plano &longs;uæ Orbitæ. Et hæc perturbatio, in dato quovis corporum P & T ad invicem &longs;itu, erit ut vis illa generans MN, adeoque minima evadet ubi MN e&longs;t minima, hoc e&longs;t (uti jam expo&longs;ui) ubi attractio SN non e&longs;t multo major, neque multo minor attractione SK. q.E.D.

~~Corol. 1. Ex his facile colligitur quod, &longs;i corpora plura minora P, S, R, &c.~~ revolvantur circa maximum T, motus corporis intimi P minime perturbabitur attractionibus exteriorum, ubi corpus maximum T pariter a cæteris, pro ratione virium acceleratricum, attrahitur & agitatur atque cætera a &longs;e mutuo.

~~DE MOTU CORPORUM~~

Corol. 2. In Sy&longs;temate vero trium corporum T, P, S, &longs;i attractiones acceleratrices binorum quorumcunque in tertium &longs;int ad invicem reciproce ut quadrata di&longs;tantiarum; corpus P, radio PT, aream circa corpus T velocius de&longs;cribet prope Conjunctionem A & Oppo&longs;itionem B, quam prope Quadraturas C, D. Namque vis omnis qua corpus P urgetur & corpus T non urgetur, quæque non agit &longs;ecundum lineam PT accelerat vel retardat de&longs;criptionem areæ, perinde ut ip&longs;a in con&longs;equentia vel in antecedentia dirigitur. Talis e&longs;t vis NM. Hæc in tran&longs;itu corporis P a C ad A tendit in con&longs;equentia, motumque accelerat; dein u&longs;que ad D in antecedentia, & motum retardat; tum in con&longs;equentia u&longs;que ad B, & ultimo in antecedentia tran&longs;eundo a B ad C.

~~Corol. 3. Et eodem argumento patet quod corpus P, cæteris paribus, velocius movetur in Conjunctione & Oppo&longs;itione quam in Quadraturis.~~

~~Corol. 4. Orbita corporis P, cæteris paribus, curvior e&longs;t in Quadraturis quam in Conjunctione & Oppo&longs;itione.~~ ~~Nam corpora velociora minus deflec~~

~~tunt a recto tramite.~~ Et præterea vis KL vel NM, in Conjunctione & Oppo&longs;itione, contraria e&longs;t vi qua corpus T trahit corpus P,adeoque vim illam minuit; corpus autem Pminus deflectet a recto tramite, ubi minus urgetur in corpus T.

~~Corol. 5. Unde corpus P, cæteris paribus, longius recedet a corpore T in Quadraturis, quam in Conjunctione & Oppo&longs;itione.~~ ~~Hæc ita &longs;e habent exclu&longs;o motu Excentricitatis.~~ ~~Nam &longs;i Orbita corporis P excentrica &longs;it: Excentricitas ejus (ut mox in hujus Corol.~~ 9. o&longs;tendetur) evadet maxima ubi Ap&longs;ides &longs;unt in Syzygiis; indeque fieri pote&longs;t ut corpus P, ad Ap&longs;idem &longs;ummam appellans, ab&longs;it longius a corpore T in Syzygiis quam in Quadraturis.

Corol. 6. Quoniam vis centripeta corporis centralis T, qua corpus P retinetur in Orbe &longs;uo, augetur in Quadraturis per additionem vis LM, ac diminuitur in Syzygiis per ablationem vis KL, & ob magnitudinem vis KL, magis diminuitur quam augetur; e&longs;t autem vis illa centripeta (per Corol. ~~2, Prop.~~ ~~IV.) in ratione compo&longs;ita ex ratione &longs;implici radii TP directe & ratione duplicata tempo- ris periodici inver&longs;e: patet hanc rationem compo&longs;itam diminui per~~

actionem vis KL, adeoque tempus periodicum, &longs;i maneat Orbis radius TP, augeri, idque in &longs;ubduplicata ratione qua vis illa centripeta diminuitur: auctoque adeo vel diminuto hoc Radio, tempus periodicum augeri magis, vel diminui minus quam in Radii hujus ratione &longs;e&longs;quiplicata, per Corol. ~~6. Prop.~~ ~~IV.~~ Si vis illa corporis centralis paulatim langue&longs;ceret, corpus P minus &longs;emper & minus attractum perpetuo recederet longius a centro T; & contra, &longs;i vis illa augeretur, accederet propius. Ergo &longs;i actio corporis longinqui S, qua vis illa diminuitur, augeatur ac diminuatur per vices; augebitur &longs;imul ac diminuetur Radius TP per vices, & tempus periodicum augebitur ac diminuetur in ratione compo&longs;ita ex ratione &longs;e&longs;quiplicata Radii & ratione &longs;ubduplicata qua vis illa centripeta corporis centralis T, per incrementum vel decrementum actionis corporis longinqui S, diminuitur vel augetur.

~~LIBER PRIMUS.~~

Corol. 7. Ex præmi&longs;&longs;is con&longs;equitur etiam quod Ellip&longs;eos a corpore P de&longs;criptæ Axis, &longs;eu Ap&longs;idum linea, quoad motum angularem progreditur & regreditur per vices, &longs;ed magis tamen progreditur, & in &longs;ingulis corporis revolutionibus per exce&longs;&longs;um progre&longs;&longs;ionis fertur in con&longs;equentia. Nam vis qua corpus P urgetur in corpus T in Quadraturis, ubi vis MN evanuit, componitur ex vi LM & vi centripeta qua corpus T trahit corpus P. Vis prior LM,&longs;i augeatur di&longs;tantia PT, augetur in eadem fere ratione cum hac di&longs;tantia, & vis po&longs;terior decre&longs;cit in duplicata illa ratione, adeoque &longs;umma harum virium decre&longs;cit in minore quam duplicata ratione di&longs;tantiæ PT, & propterea (per Corol. ~~1. Prop.~~ ~~XLV) efficit ut Aux, &longs;eu Ap&longs;is &longs;umma, regrediatur.~~ In Conjunctione vero & Oppo&longs;itione, vis qua corpus P urgetur in corpus T differentia e&longs;t inter vim qua corpus T trahit corpus P & vim KL; & differentia illa, propterea quod vis KL augetur quamproxime in ratione di&longs;tantiæ PT, decre&longs;cit in majore quam duplicata ratione di&longs;tantiæ PT, adeoque (per Corol. ~~1. Prop.XLV) efficit ut Aux progrediatur.~~ ~~In locis inter Syzygias & Quadraturas pendet motus Augis ex cau&longs;a utraque conjunctim, adeo ut pro hujus vel alterius exce&longs;&longs;u progrediatur ip&longs;a vel regrediatur.~~ Unde cum vis KL in Syzygiis &longs;it qua&longs;i duplo major quam vis LM in Quadraturis, exce&longs;&longs;us in tota revolutione erit penes vim KL, transferetque Augem &longs;ingulis revolutionibus in con&longs;equentia. Veritas autem hujus & præcedentis Corollarii facilius intelligetur concipiendo Sy&longs;tema corporum duorum T, P corporibus pluribus S, S, S, &c, in Orbe ESE con&longs;i&longs;tentibus, undique cingi. ~~Namque horum actioni- bus actio ip&longs;ius T minuetur undique, decre&longs;cetque in ratione plu&longs;quam duplicata di&longs;tantiæ.~~

~~DE MOTU CORPORUM~~

Corol. 8. Cum autem pendeat Ap&longs;idum progre&longs;&longs;us vel regre&longs;&longs;us a decremento vis centripetæ facto in majori vel minori quam duplicata ratione di&longs;tantiæ TP, in tran&longs;itu corporis ab Ap&longs;ide ima ad Ap&longs;idem &longs;ummam; ut & a &longs;imili incremento in reditu ad Ap&longs;idem imam; atque adeo maximus &longs;it ubi proportio vis in Ap&longs;ide &longs;umma ad vim in Ap&longs;ide ima maxime recedit a duplicata ratione di&longs;tantiarum inver&longs;a: manife&longs;tum e&longs;t quod Ap&longs;ides in Syzygiis &longs;uis, per vim ablatitiam KL &longs;eu NM-LM, progredientur velocius, inque Quadraturis &longs;uis tardius recedent per vim addititiam LM. Ob diuturnitatem vero temporis quo velocitas progre&longs;&longs;us vel tarditas regre&longs;&longs;us continuatur, fit hæc inæqualitas longe maxima.

Corol. 9. Si corpus aliquod vi reciproce proportionali quadrato di&longs;tantiæ &longs;uæ a centro, revolveretur circa hoc centrum in Ellip&longs;i, & mox, in de&longs;cen&longs;u ab Ap&longs;ide &longs;umma &longs;eu Auge ad Ap&longs;idem imam, vis illa per acce&longs;&longs;um perpetuum vis novæ augeretur in ratione plu&longs;quam dupli

cata di&longs;tantiæ diminutæ: manife&longs;tum e&longs;t quod corpus, perpetuo acce&longs;&longs;u vis illius novæ impul&longs;um &longs;emper in centrum, magis vergeret in hoc centrum, quam &longs;i urgeretur vi &longs;ola cre&longs;cente in duplicata ratione di&longs;tantiæ diminutæ, adeoque Orbem de&longs;criberet Orbe Elliptico interiorem, & in Ap&longs;ide ima propius accederet ad centrum quam prius. ~~Orbis igitur, acce&longs;&longs;u hujus vis novæ, fiet magis excentricus.~~ Si jam vis, in rece&longs;&longs;u corporis ab Ap&longs;ide ima ad Ap&longs;idem &longs;ummam, decre&longs;ceret ii&longs;dem gradibus quibus ante creverat, rediret corpus ad di&longs;tantiam priorem, adeoque &longs;i vis decre&longs;cat in majori ratione, corpus jam minus attractum a&longs;cendet ad di&longs;tantiam majorem & &longs;ic Orbis Excentricitas adhuc magis augebitur. Igitur &longs;i ratio incrementi & decrementi vis centripetæ &longs;ingulis revolutionibus augeatur, augebitur &longs;emper Excentricitas; & e contra, diminuetur eadem &longs;i ratio illa decre&longs;cat. Jam vero in Sy&longs;temate corporum T, P, S, ubi Ap&longs;ides Orbis PAB&longs;unt in Quadraturis, ratio illa incrementi ac decrementi minima e&longs;t, & maxima fit ubi Ap&longs;ides &longs;unt in Syzygiis. Si Ap&longs;ides con&longs;tituan tur in Quadraturis, ratio prope Ap&longs;ides minor e&longs;t & prope Syzygias major quam duplicata di&longs;tantiarum, & ex ratione illa majori oritur Augis motus veloci&longs;&longs;imus, uti jam dictum e&longs;t. ~~At &longs;i con&longs;ideretur ratio incrementi vel decrementi totius in progre&longs;&longs;u inter Ap&longs;ides, hæc minor e&longs;t quam duplicata di&longs;tantiarum.~~ Vis in Ap&longs;ide ima e&longs;t ad vim in Ap&longs;ide &longs;umma in minore quam duplicata ratione di&longs;tantiæ Ap&longs;idis &longs;ummæ ab umbilico Ellip&longs;eos ad di&longs;tantiam Ap&longs;idis imæ ab eodem umbilico: & e contra, ubi Ap&longs;ides con&longs;tituuntur in Syzygiis, vis in Ap&longs;ide ima e&longs;t ad vim in Ap&longs;ide &longs;umma in majore quam duplicata ratione di&longs;tantiarum. ~~Nam vires LM in Quadraturis additæ viribus corporis T componunt vires in ratione minore, & vires KL in Syzygiis &longs;ubductæ viribus corporis T relinquunt vires in ratione majore.~~ E&longs;t igitur ratio decrementi & incrementi totius, in tran&longs;itu inter Ap&longs;ides, minima in Quadraturis, maxima in Syzygiis: & propterea in tran&longs;itu Ap&longs;idum a Quadraturis ad Syzygias perpetuo augetur, augetque Excentricitatem Ellip&longs;eos; inque tran&longs;itu a Syzygiis ad Quadraturas perpetuo diminuitur, & Excentricitatem diminuit.

~~LIBER PRIMUS.~~

Corol. 10. Ut rationem ineamus errorum in Latitudinem, fingamus planum Orbis EST immobile manere; & ex errorum expo&longs;ita cau&longs;a manife&longs;tum e&longs;t quod, ex viribus NM, ML, quæ &longs;unt cau&longs;a illa tota, vis ML agendo &longs;emper &longs;ecundum planum Orbis PAB, nunquam perturbat motus in Latitudinem; quodque vis NM,ubi Nodi &longs;unt in Syzygiis, agendo etiam &longs;ecundum idem Orbis planum, non perturbat hos motus; ubi vero &longs;unt in Quadraturis eos maxime perturbat, corpu&longs;que P de plano Orbis &longs;ui perpetuo trahendo, minuit inclinationem plani in tran&longs;itu corporis a Quadraturis ad Syzygias, augetque vici&longs;&longs;im eandem in tran&longs;itu a Syzygiis ad Quadraturas. ~~Unde fit ut corpore in Syzygiis exi&longs;tente inclinatio evadat omnium minima, redeatque ad priorem magnitudinem circiter, ubi corpus ad Nodum proximum accedit.~~ At &longs;i Nodi con&longs;tituantur in Octantibus po&longs;t Quadraturas, id e&longs;t, inter C & A, D & B, intelligetur ex modo expo&longs;itis quod, in tran&longs;itu corporis P a Nodo alterutro ad gradum inde nonage&longs;imum, inclinatio plani perpetuo minuitur; deinde in tran&longs;itu per proximos 45 gradus, u&longs;que ad Quadraturam proximam, inclinatio augetur, & po&longs;tea denuo in tran&longs;itu per alios 45 gradus, u&longs;que ad Nodum proximum, diminuitur. ~~Magis itaque diminuitur inclinatio quam augetur, & propterea minor e&longs;t &longs;emper in Nodo &longs;ub&longs;equente quam in præce- dente.~~ Et &longs;imili ratiocinio, inclinatio magis augetur quam diminuitur ubi Nodi &longs;unt in Octantibus alteris inter A & D, B & C. Inclinatio igitur ubi Nodi &longs;unt in Syzygiis e&longs;t omnium maxima. In tran&longs;itu corum a Syzygiis ad Quadraturas, in &longs;ingulis corporis ad Nodos appul&longs;ibus, diminuitur, fitque omnium minima ubi Nodi &longs;unt in Quadraturis & corpus in Syzygiis: dein cre&longs;cit ii&longs;dem gradibus quibus antea decreverat, Nodi&longs;que ad Syzygias proximas appul&longs;is ad magnitudinem primam revertitur.

~~DE MOTU CORPORUM~~

Corol. 11. Quoniam corpus P ubi Nodi &longs;unt in Quadraturis perpetuo trahitur de plano Orbis &longs;ui, idque in partem ver&longs;us S, in tran&longs;itu &longs;uo a Nodo C per Conjunctionem A ad Nodum D; & in contrariam partem in tran&longs;itu a Nodo D per Oppo&longs;itionem B ad Nodum C; manife&longs;tum e&longs;t quod in motu &longs;uo a Nodo C, corpus perpetuo recedit ab Orbis &longs;ui plano primo CD, u&longs;que dum perventum e&longs;t ad Nodum proximum; adeoque in hoc Nodo, longi&longs;&longs;ime di&longs;tans a plano illo primo CD, tran&longs;it per planum Orbis ESTnon in plani illius Nodo altero D, &longs;ed in puncto quod inde vergit ad partes corporis S, quodque proinde novus e&longs;t Nodi locus in anteriora vergens. ~~Et &longs;imili argumento pergent Nodi recedere in tran&longs;itu corporis de hoc Nodo in Nodum proximum.~~ Nodi igitur in Quadraturis con&longs;tituti perpetuo recedunt; in Syzygiis (ubi motus in Latitudinem nil perturbatur) quie&longs;cunt; in locis intermediis, conditionis utriu&longs;que participes, recedunt tardius; adeoque, &longs;emper vel retrogradi vel &longs;tationarii, &longs;ingulis revolutionibus feruntur in antecedentia.

~~Corol. 12. Omnes illi in his Corollariis de&longs;cripti Errores &longs;unt paulo majores in Conjunctione corporum P, S quam in eorum Oppo&longs;itione, idque ob majores vires generantes NM & ML.~~

Corol. 13. Cumque rationes horum Corollariorum non pendeant a magnitudine corporis S, obtinent præcedentia omnia, ubi corporis S tanta &longs;tatuitur magnitudo ut circa ip&longs;um revolvatur corporum duorum T & P Sy&longs;tema. Et ex aucto corpore S auctaque adeo ip&longs;ius vi centripeta, a qua errores corporis P oriuntur, evadent errores illi omnes (paribus di&longs;tantiis) majores in hoc ca&longs;u quam in altero, ubi corpus S circum Sy&longs;tema corporum P & T revolvitur.

Corol. 14. Cum autem vires NM, ML, ubi corpus S longinquum e&longs;t, &longs;int quamproxime ut vis SK & ratio PT ad ST conjunctim, hoc e&longs;t, &longs;i detur tum di&longs;tantia PT, tum corporis S vis ab&longs;oluta, ut ST cub. reciproce; &longs;int autem vires illæ NM, MLcau&longs;æ errorum & effectuum omnium de quibus actum e&longs;t in præce- dentibus Corollariis: manife&longs;tum e&longs;t quod effectus illi omnes, &longs;tan te corporum T & P Sy&longs;temate, & mutatis tantum di&longs;tantia ST & vi ab&longs;oluta corporis S, &longs;int quamproxime in ratione compo&longs;ita ex ratione directa vis ab&longs;olutæ corporis S & ratione triplicata inver&longs;a di&longs;tantiæ ST. Unde &longs;i Sy&longs;tema corporum T & P revolvatur circa corpus longinquum S, vires illæ NM, ML & earum effectus erunt (per Corol. ~~2. & 6. Prop.~~ ~~IV.) reciproce in duplicata ratione temporis periodici.~~ Et inde etiam, &longs;i magnitudo corporis S proportionalis &longs;it ip&longs;ius vi ab&longs;olutæ, erunt vires illæ NM, ML & earum effectus directe ut cubus diametri apparentis longinqui corporis S e corpore T &longs;pectati, & vice ver&longs;a. ~~Namque hæ rationes eædem &longs;unt atque ratio &longs;uperior compo&longs;ita.~~

~~LIBER PRIMUS.~~

Corol. 15. Et quoniam &longs;i, manentibus Orbium ESE & PABforma, proportionibus & inclinatione ad invicem, mutetur eorum magnitudo, & &longs;i corporum S & T vel maneant vel mutentur vires in data quavis ratio

ne, hæ vires (hoc e&longs;t, vis corporis T qua corpus P de recto tramite in Orbitam PABdeflectere, & vis corporis S qua corpus idem P de Orbita illa deviare cogitur) agunt &longs;emper eodem modo & eadem proportione: nece&longs;&longs;e e&longs;t ut &longs;imiles & proportionales &longs;int effectus omnes & proportionalia effectuum tempora; hoc e&longs;t, ut errores omnes lineares &longs;int ut Orbium diametri, angulares vero iidem qui prius, & errorum linearium &longs;imilium vel angularium æqualium tempora ut Orbium tempora periodica.

Corol. 16. Unde, &longs;i dentur Orbium formæ & inclinatio ad invicem, & mutentur utcunque corporum magnitudines, vires & di&longs;tantiæ; ex datis erroribus & errorum temporibus in uno Ca&longs;u, colligi po&longs;&longs;unt errores & errorum tempora in alio quovis, quam proxime: Sed brevius hac Methodo. ~~Vires NM, ML, cæteris &longs;tantibus, &longs;unt ut Radius TP, & harum effectus periodici (per Corol.2, Lem.~~ ~~X) ut vires & quadratum temporis periodici corporis P conjunctim.~~ Hi &longs;unt errores lineares corporis P; & hinc errores angulares e centro T &longs;pectati (id e&longs;t, tam motus Augis & Nodorum, quam omnes in Longitudinem & Latitudinem errores apparentes) &longs;unt, in qualibet revolutione corporis P, ut quadratum temporis revolutionis quam proxime. Conjungantur hæ rationes cum rationibus Corollarii 14, & in quolibet corporum T, P, S Sy&longs;temate, ubi P circum T &longs;ibi propinquum, & T circum S longinquum revolvitur, errores angulares corporis P, de centro T apparentes, erunt, in &longs;ingulis revolutionibus corporis illius P, ut quadratum temporis periodici corporis P directe & quadratum temporis periodici corporis T inver&longs;e. ~~Et inde motus medius Augis erit in data ratione ad motum medium Nodorum; & motus uterque erit ut tempus periodicum corporis &c.~~ ~~quadratum temporis periodici corporis P directe & quadratum temporis periodici corporis T inver&longs;e.~~ ~~Augendo vel minuendo Excentricitatem & Inclinationem Orbis PAB non mutantur motus Augis & Nodorum &longs;en&longs;ibiliter, ni&longs;i ubi eædem &longs;unt nimis magnæ.~~

~~DE MOTU CORPORUM~~

~~Corol. 17. Cum autem linea LM nunc major &longs;it nunc minor quam radius PT, exponatur vis mediocris LM per radium illum PT; & erit hæc ad~~

vim mediocrem SKvel SN (quam exponere licet per ST) ut longitudo PT ad longitudinem ST. E&longs;t autem vis mediocris SNvel ST, qua corpus Tretinetur in Orbe &longs;uo circum S, ad vim qua corpus P retinetur in Orbe &longs;uo circum T, in ratione compo&longs;ita ex ratione radii ST ad radium PT, & ratione duplicata temporis periodici corporis P circum T ad tempus periodicum corporis Tcircum S. Et ex æquo, vis mediocris LM, ad vim qua corpus P retinetur in Orbe &longs;uo circum T (quave corpus idem P, eodem tempore periodico, circum punctum quodvis immobile T ad di&longs;tantiam PT revolvi po&longs;&longs;et) e&longs;t in ratione illa duplicata periodicorum temporum. ~~Datis igitur temporibus periodicis una cum di&longs;tantia PT, datur vis mediocris LM; & ea data, datur etiam vis MN quamproxime per analogiam linearum PT, MN.~~

Corol. 18. Ii&longs;dem legibus quibus corpus P circum corpus T revolvitur, fingamus corpora plura fluida circum idem T ad æquales ab ip&longs;o di&longs;tantias moveri; deinde ex his contiguis factis conflari Annulum fluidum, rotundum ac corpori T concentricum; & &longs;ingulæ Annuli partes, motus &longs;uos omnes ad legem corporis P per- agendo, propius accedent ad corpus T, & celerius movebuntur in Conjunctione & Oppo&longs;itione ip&longs;arum & corporis S, quam in Quadraturis. Et Nodi Annuli hujus &longs;eu inter&longs;ectiones ejus cum plano Orbitæ corporis S vel T, quie&longs;cent in Syzygiis; extra Syzygias vero movebuntur in anteccdentia, & veloci&longs;&longs;ime quidem in Quadraturis, tardius aliis in locis. Annuli quoque inclinatio variabitur, & axis ejus &longs;ingulis revolutionibus o&longs;cillabitur, completaque revolutione ad pri&longs;tinum &longs;itum redibit, ni&longs;i quatenus per præce&longs;&longs;ionem Nodorum circumfertur.

~~LIBER PRIMUS.~~

Corol. 19. Fingas jam Globum corporis T, ex materia non fluida con&longs;tantem, ampliari & extendi u&longs;que ad hunc Annulum, & alveo per circuitum excavato continere Aquam, motuque eodem periodico circa axem &longs;uum uniformiter revolvi. Hic liquor per vices acceleratus & retardatus (ut in &longs;uperiore Corollario) in Syzygiis velocior erit, in Quadraturis tardior quam &longs;uperficies Globi, & &longs;ic fluet in alveo refluet que ad modum Maris. ~~Aqua revolvendo circa Globi centrum quie&longs;cens, &longs;i tollatur attractio corporis S nullum acquiret motum fluxus & refluxus.~~ ~~Par e&longs;t ratio Globi uniformiter progredientis in directum & interea revolventis circa centrum &longs;uum (per Legum Corol.~~ ~~5.) ut & Globi de cur&longs;u rectilineo uniformiter tracti, per Legum Corol.~~ ~~6. Accedat autem corpus S,& ab ip&longs;ius inæquabili attractione mox turbabitur Aqua.~~ ~~Etenim major erit attractio aquæ propioris, minor ea remotioris.~~ Vis autem LM trahet aquam deor&longs;um in Quadraturis, facietque ip&longs;am de&longs;cendere u&longs;que ad Syzygias; & vis KL trahet eandem &longs;ur&longs;um in Syzygiis, &longs;i&longs;tetque de&longs;cen&longs;um ejus & faciet ip&longs;am a&longs;cendere u&longs;que ad Quadraturas.

~~Corol. 20. Si Annulus jam rigeat & minuatur Globus, ce&longs;&longs;abit motus fluendi & refluendi; &longs;ed O&longs;cillatorius ille inclinationis motus & præce&longs;&longs;io Nodorum manebunt.~~ Habeat Globus eundem axem cum Annulo, gyro&longs;que compleat ii&longs;dem temporibus, & &longs;uperficie &longs;ua contingat ip&longs;um interius, eique inhæreat; & participando motum ejus, compages utriu&longs;que O&longs;cillabitur & Nodi regredientur. ~~Nam Globus, ut mox dicetur, ad &longs;u&longs;cipiendas impre&longs;&longs;iones omnes indifferens e&longs;t.~~ ~~Annuli Globo orbati maximus inclinationis angulus e&longs;t ubi Nodi &longs;unt in Syzygiis.~~ ~~Inde in progre&longs;&longs;u Nodorum ad Quadraturas conatur is inclinationem &longs;uam minuere, & i&longs;to conatu motum imprimit Globo toti.~~ Retinet Globus motum impre&longs;&longs;um u&longs;que dum Annulus conatu contrario motum hunc tollat, imprimatque motum novum in contrariam partem: Atque hac ra- tione maximus decre&longs;centis inclinationis motus fit in Quadraturis Nodorum, & minimus inclinationis angulus in Octantibus po&longs;t Quadraturas; dein maximus reclinationis motus in Syzygiis, & maximus angulus in Octantibus proximis. ~~Et eadem e&longs;t ratio Globi Annulo nudati, qui in regionibus æquatoris vel altior e&longs;t paulo quam juxta polos, vel con&longs;tat ex nateria paulo den&longs;iore.~~ ~~Supplet enim vicem Annuli i&longs;te materiæ in æquatoris regionibus exce&longs;&longs;us.~~ Et quanquam, aucta utcunque Globi hujus vi centripeta, tendere &longs;upponantur omnes ejus partes deor&longs;um, ad modum gravitantium partium telluris, tamen Phænomena hujus & præcedentis Corollarii vix inde mutabuntur.

~~DE MOTU CORPORUM~~

Corol. 21. Eadem ratione qua materia Globi juxta æquatorem redundans efficit ut Nodi regrediantur, atque adeo per hujus incrementum augetur i&longs;te regre&longs;&longs;us, p diminutionem vero diminuitur & per ablationem tollitur; &longs;i materia plu&longs;quam redundans tollatur, hoc e&longs;t, &longs;i Globus juxta æquatorem vel depre&longs;&longs;ior reddatur vel rarior quam juxta polos, orietur motus Nodorum in con&longs;equentia.

~~Corol. 22. Et inde vici&longs;&longs;im, ex motu Nodorum innote&longs;cit con&longs;titutio Globi.~~ ~~Nimirum &longs;i Globus polos eo&longs;dem con&longs;tanter &longs;ervat, & motus fit in antecedentia, materia juxta æquatorem redundat; &longs;i in con&longs;equentia, deficit.~~ Pone Globum uniformem & perfecte circinatum in &longs;patiis liberis primo quie&longs;cere; dein impetu quocunque oblique in &longs;uperficiem &longs;uam facto propelli, & motum inde concipere partim circularem, partim in directum. Quoniam Globus i&longs;te ad axes omnes per centrum &longs;uum tran&longs;euntes indifferenter &longs;e habet, neque propen&longs;ior e&longs;t in unum axem, unumve axis &longs;itum, quam in alium quemvis; per&longs;picuum e&longs;t quod is axem &longs;uum axi&longs;que inclinationem vi propria nunquam mutabit. Impellatur jam Globus oblique, in eadem illa &longs;uperficiei parte qua prius, impul&longs;u quocunque novo; & cum citior vel &longs;erior impul&longs;us effectum nil mutet, manife&longs;tum e&longs;t quod hi duo impul&longs;us &longs;ucce&longs;&longs;ive impre&longs;&longs;i eundem producent motum ac &longs;i &longs;imul impre&longs;&longs;i fui&longs;&longs;ent, hoc e&longs;t, eundem ac &longs;i Globus vi &longs;implici ex utroque (per Legum Corol. ~~2.) compo&longs;ita impul&longs;us fui&longs;&longs;et, atque adeo &longs;implicem, circa axem inclinatione datum.~~ Et par e&longs;t ratio impul&longs;us &longs;ecundi facti in locum alium quemvis in æquatore motus primi; ut & impul&longs;us primi facti in locum quemvis in æquatore motus, quem impul&longs;us &longs;ecundus ab&longs;que primo generaret; atque adeo impul&longs;uum amborum factorum in loca quæcunque: Generabunt hi eundem motum cir- cularem ac &longs;i &longs;imul & &longs;emel in locum inter&longs;ectionis æquatorum motuum illorum, quos &longs;eor&longs;im generarent, fui&longs;&longs;ent impre&longs;&longs;i. Globus igitur homogeneus & perfectus non retinet motus plures di&longs;tinctos, &longs;ed impre&longs;&longs;os omnes componit & ad unum reducit, & quatenus in &longs;e e&longs;t, gyratur &longs;emper motu &longs;implici & uniformi circa axem unicum, inclinatione &longs;emper invariabili datum. ~~Sed nec vis centripeta inclinationem axis, aut rotationis velocitatem mutare pote&longs;t.~~ Si Globus plano quocunque, per centrum &longs;uum & centrum in quod vis dirigitur tran&longs;eunte, dividi intelligatur in duo hemi&longs;phæria; urgebit &longs;emper vis illa utrumque hemi&longs;phærium æqualiter, & propterea Globum, quoad motum rotationis, nullam in partem inclinabit. Addatur vero alicubi inter polum & æquatorem materia nova in formam montis cumulata, & hæc, perpetuo conatu recedendi a centro &longs;ui motus, turbabit motum Globi, facietque polos ejus errare per ip&longs;ius &longs;uperficiem, & circulos circum &longs;e punctumque &longs;ibi oppo&longs;itum perpetuo de&longs;cribere. ~~Neque corrigetur i&longs;ta vagationis enormitas, ni&longs;i locando montem illum vel in polo alterutro, quo in Ca&longs;u (per Corol.~~ ~~21) Nodi æquatoris progredientur; vel in æquatore, qua ratione (per Corol.~~ 20) Nodi regredientur; vel denique ex altera axis parte addendo materiam novam, qua mons inter movendum libretur, & hoc pacto Nodi vel progredientur, vel recedent, perinde ut mons & hæcce nova materia &longs;unt vel polo vel æquatori propiores.

~~LIBER PRIMUS.~~

~~PROPOSITIO LXVII. THEOREMA XXVII.~~

Po&longs;itis ii&longs;dem attractionum legibus, dico quod corpus exterius S, circa interiorum P, T commune gravitatis centrum C, radiis ad centrum illud ductis, de&longs;cribit areas temporibus magis proportionales & Orbem ad formam Ellip&longs;eos umbilicum in centro eodem habentis magis accedentem, quam circa corpus intimum & maximum T, radiis ad ip&longs;um ductis, de&longs;cribere potest.

Nam corporis S attractiones ver&longs;us T & P componunt ip&longs;ius attractionem ab&longs;olutam, quæ magis dirigitur in corporum T & P commune gravitatis centrum C, quam in corpus maximum T, quæque quadrato di&longs;tantiæ SC magis e&longs;t proportionalis reciproce, quam quadrato di&longs;tantiæ ST: ut rem perpendenti facile con&longs;tabit.

~~DE MOTU CORPORUM~~

~~PROPOSITIO LXVIII. THEOREMA XXVIII.~~

Po&longs;itis ii&longs;dem attractionum legibus, dico quod corpus exterius S, circa interiorum P & T commune gravitatis centrum C, radiis ad centrum illud ductis, de&longs;cribit areas temporibus magis proportionales, & Orbem ad formam Ellip&longs;eos umbilicum in centro eodem habentis magis accedentem, &longs;i corpus intimum & maximum his attractionibus perinde atque cætera agitetur, quam &longs;i id vel non attractum quie&longs;cat, vel multo magis aut multo minus attractum aut multo magis aut multo minus agitetur.

~~Demon&longs;tratur eo~~

~~dem fere modo cum Prop.~~ ~~LXVI, &longs;ed argumento prolixiore, quod ideo prætereo.~~ ~~Suffecerit rem &longs;ic æ&longs;timare.~~ ~~Ex demon&longs;tratione Propo&longs;itionis novi&longs;&longs;imæ liquet centrum in quod corpus S conjunctis viribus urgetur, proximum e&longs;&longs;e communi centro gravitatis duorum illorum.~~ Si coincideret hoc centrum cum centro illo communi, & quie&longs;ceret commune centrum gravitatis corporum trium; de&longs;criberent corpus S ex una parte, & commune centrum aliorum duorum ex altera parte, circa commune omnium centrum quie&longs;cens, Ellip&longs;es accuratas. ~~Liquet hoc per Corollarium &longs;ecundum Propo&longs;itionis LVIII collatum cum demon&longs;tratis in Propo&longs;.~~ ~~LXIV & LXV.~~ ~~Perturbatur i&longs;te motus Ellipticus aliquantulum per di&longs;tantiam centri duorum a centro in quod tertium S attrahitur.~~ ~~Detur præterea motus communi trium centro, & augebitur perturbatio.~~ Proinde minima e&longs;t perturbatio ubi commune trium centrum quie&longs;cit, hoc e&longs;t, ubi corpus intimum & maximum T lege cæterorum attrahitur: fitque major &longs;emper ubi trium commune illud centrum, minuendo motum corporis T, moveri incipit & magis deinceps magi&longs;que agitatur.

Corol. Et hinc, &longs;i corpora plura minora revolvantur circa maxi mum, colligere licet quod Orbitæ de&longs;criptæ propius accedent ad Ellipticas, & arearum de&longs;criptiones fient magis æquabiles, &longs;i corpora omnia viribus acceleratricibus, quæ &longs;unt ut eorum vires ab&longs;olutæ directe & quadrata di&longs;tantiarum inver&longs;e, &longs;e mutuo trahant agitentque, & Orbitæ cuju&longs;que umbilicus collocetur in communi centro gravitatis corporum omnium interiorum (nimirum umbilicus Orbitæ primæ & intimæ in centro gravitatis corporis maximi & intimi; ille Orbitæ &longs;ecundæ, in communi centro gravitatis corporum duorum intimorum; i&longs;te tertiæ, in communi centro gravitatis trium interiorum; & &longs;ic deinceps) quam &longs;i corpus intimum quie&longs;cat & &longs;tatuatur communis umbilicus Orbitarum omnium.

~~LIBER PRIMUS.~~

~~PROPOSITIO LXIX. THEOREMA XXIX.~~

~~In Sy&longs;temate corporum plurium A, B, C, D, &c.~~ ~~&longs;i corpus aliquodA trahit cætera omnia B, C, D, &c.~~ ~~viribus acceler atricibus quæ &longs;unt reciproce ut quadrata di&longs;tantiarum a trahente; & corpus aliud B trahit etiam cætera A, C, D, &c.~~ viribus quæ &longs;unt reciproce ut quadrata di&longs;tantiarum a trahente: erunt Ab&longs;olutæ corporum trahentium A, B vires ad invicem, ut &longs;unt ip&longs;a corpora A, B, quorum &longs;unt vires.

Nam attractiones acceleratrices corporum omnium B, C, D ver&longs;us A, paribus di&longs;tantiis, &longs;ibi invicem æquantur ex Hypothe&longs;i; & &longs;imiliter attractiones acceleratrices corporum omnium ver&longs;us B,paribus di&longs;tantiis, &longs;ibi invicem æquantur. E&longs;t autem ab&longs;oluta vis attractiva corporis A ad vim ab&longs;olutam attractivam corporis B, ut attractio acceleratrix corporum omnium ver&longs;us A ad attractionem acceleratricem corporum omnium ver&longs;us B, paribus di&longs;tantiis; & ita e&longs;t attractio acceleratrix corporis B ver&longs;us A, ad attractionem acceleratricem corporis A ver&longs;us B. Sed attractio acceleratrix corporis B ver&longs;us A e&longs;t ad attractionem acceleratricem corporis Aver&longs;us B, ut ma&longs;&longs;a corporis A ad ma&longs;&longs;am corporis B; propterea quod vires motrices, quæ (per Definitionem &longs;ecundam, &longs;eptimam & octavam) ex viribus acceleratricibus in corpora attracta ductis oriuntur, &longs;unt (per motus Legem tertiam) &longs;ibi invicem æqua- les. ~~Ergo ab&longs;oluta vis attractiva corporis A e&longs;t ad ab&longs;olutam vim attractivam corporis B, ut ma&longs;&longs;a corporis A ad ma&longs;&longs;am corporis B. q.E.D.~~

~~DE MOTU CORPORUM~~

Corol. 1. Hinc &longs;i &longs;ingula Sy&longs;tematis corpora A, B, C, D, &c. &longs;eor&longs;im &longs;pectata trahant cætera omnia viribus acceleratricibus quæ &longs;unt reciproce ut quadrata di&longs;tantiarum a trahente; erunt corporum illorum omnium vires ab&longs;olutæ ad invicem ut &longs;unt ip&longs;a corpora.

~~Corol. 2. Eodem argumento, &longs;i &longs;ingula Sy&longs;tematis corpora A, B, C, D, &c.~~ &longs;eor&longs;im &longs;pectata trahant cætera omnia viribus acceleratricibus quæ &longs;unt vel reciproce vel directe in ratione dignitatis cuju&longs;cunque di&longs;tantiarum a trahente, quæve &longs;ecundum Legem quamcunque communem ex di&longs;tantiis ab unoquoque trahente definiuntur; con&longs;tat quod corporum illorum vires ab&longs;olutæ &longs;unt ut corpora.

Corol. 3. In Sy&longs;temate corporum, quorum vires decre&longs;cunt in ratione duplicata di&longs;tantiarum, &longs;i minora circa maximum in Ellip&longs;ibus umbilicum communem in maximi illius centro habentibus quam fieri pote&longs;t accurati&longs;&longs;imis revolvantur, & radiis ad maximum illud ductis de&longs;cribant areas temporibus quam maxime proportionales: erunt corporum illorum vires ab&longs;olutæ ad invicem, aut accurate aut quamproxime in ratione corporum; & contra. ~~Patet per Corol.~~ ~~Prop.~~ ~~LXVIII collatum cum hujus Corol.~~ 1.

~~Scholium.~~

~~His Propo&longs;itionibus manuducimur ad analogiam inter vires centripetas & corpora centralia, ad quæ vires illæ dirigi &longs;olent.~~ ~~Rationi enim con&longs;entaneum e&longs;t, ut vires quæ ad corpora diriguntur pendeant ab eorundem natura & quantitate, ut fit in Magneticis.~~ ~~Et quoties huju&longs;modi ca&longs;us incidunt, æ&longs;timandæ erunt corporum attractiones, a&longs;&longs;ignando &longs;ingulis eorum particulis vires proprias, & colligendo &longs;ummas virium.~~ Vocem Attractionis hic generaliter u&longs;urpo pro corporum conatu quocunque accedendi ad invicem; &longs;ive conatus i&longs;te fiat ab actione corporum, vel &longs;e mutuo petentium, vel per Spiritus emi&longs;&longs;os &longs;e invicem agitantium, &longs;ive is ab actione Ætheris, aut Aeris, Mediive cuju&longs;cunque &longs;eu corporei &longs;eu incorporei oriatur corpora innatantia in &longs;e invicem utcunque impellentis. Eodem &longs;en&longs;u generali u&longs;urpo vocem Impul&longs;us, non &longs;pecies virium & qualitates Phy&longs;icas, &longs;ed quantitates & proportiones Mathema ticas in hoc Tractatu expendens, ut in Definitionibus explicui. In Mathe&longs;i inve&longs;tigandæ &longs;unt virium quantitates & rationes illæ, quæ ex conditionibus quibu&longs;cunque po&longs;itis con&longs;equentur: deinde, ubi in Phy&longs;icam de&longs;cenditur, conferendæ &longs;unt hæ rationes cum Phænomenis, ut innote&longs;cat quænam virium conditiones &longs;ingulis corporum attractivorum generibus competant. ~~Et tum demum de virium &longs;peciebus, cau&longs;is & rationibus Phy&longs;icis tutius di&longs;putare licebit.~~ ~~Videamus igitur quibus viribus corpora Sphærica, ex particulis modo jam expo&longs;ito attractivis con&longs;tantia, debeant in &longs;e mutuoagere, & quales motus inde con&longs;equantur.~~

~~LIBER PRIMUS.~~

~~SECTIO XII.~~

~~De Corporum Sphæriccrum Viribus attractivis.~~

~~PROPOSITIO LXX. THEOREMA XXX.~~

Si ad Sphæricæ &longs;uperficiei puncta &longs;ingula tendant vires æquales centripetæ decre&longs;centes in duplicata ratione di&longs;tantiarum a punctis: dico quod corpu&longs;culum in&longs;uperficiem con&longs;titutum his viribus nullam in partem attrahitur.

~~Sit HIKL &longs;uperficies illa Sphæri~~

~~ca, & P corpu&longs;culum intus con&longs;titutum.~~ ~~Per P agantur ad hanc &longs;uperficiem lineæ duæ HK, IL, arcus quam minimos HI, KL intercipientes; &, ob triangula HPI, LPK(per Corol.~~ ~~3. Lem.~~ VII) &longs;imilia, arcus illi erunt di&longs;tantiis HP, LP proportionales; & &longs;uperficiei Sphæricæ particulæ quævis ad HI & KL, rectis per punctum P tran&longs;euntibus undique terminatæ, erunt in duplicata illa ratione. ~~Ergo vires harum particularum in corpus P exercitæ &longs;unt inter &longs;e æquales.~~ ~~Sunt enim ut particulæ directe & quadrata di&longs;tantiarum inver&longs;e.~~ ~~Et hæ duæ rationes componunt rationem æqualitatis.~~ ~~Attractiones igitur, in contrarias partes æqualiter factæ, &longs;e mutuo de&longs;truunt.~~ ~~Et &longs;imili argumento, attractiones omnes per totam Sphæricam &longs;uperficiem a contrariis attractionibus de&longs;truuntur.~~ ~~Proinde corpus P nullam in partem his attractionibus impellitur. q.E.D.~~

~~DE MOTU CORPORUM~~

~~PROPOSITIO LXXI. THEOREMA XXXI.~~

Ii&longs;dem po&longs;itis, dico quod corpu&longs;culum extra Sphæricam &longs;uperficiem con&longs;titutum attrahitur ad centrum Sphæræ, vi reciproce proportionali quadrato di&longs;tantiæ &longs;uæ ab eodem centro.

~~Sint AHKB, ahkb æquales duæ &longs;uperficies Sphæricæ, centris S, s, diametris AB, ab de&longs;criptæ, & P, p corpu&longs;cula &longs;ita extrin&longs;ecus in diametris illis productis.~~ ~~Agantur a corpu&longs;culis lineæ~~

PHK, PIL, phk, pil, auferentes a circulis maximis AHB, ahb, æquales arcus HK, hk & IL, il: Et ad eas demittantur perpendicula SD, sd; SE, se; IR, ir; quorum SD, sd &longs;ecent PL, pl in F & f: Demittantur etiam ad diametros perpendicula IQ, ique Evane&longs;cant anguli DPE, dpe: & (ob æquales DS & ds, ES & es,) lineæ PE, PF & pe, pf& lineolæ DF, df pro æqualibus habeantur; quippe quarum ratio ultima, angulis illis DPE, dpe &longs;imul evane&longs;centibus, e&longs;t æqualitatis. ~~His itaque con&longs;titutis, erit PI ad PF ut RI ad DF,& pf ad pi ut df vel DF ad ri; & ex æquo PIXpf ad PFXpiut RI ad ri, hoc e&longs;t (per Corol.~~ ~~3. Lem.~~ VII,) ut arcus IH ad arcum ih. Rur&longs;us PI ad PS ut IQ ad SE, & ps and pi ut sevel SE ad ique & ex æquo PIXps ad PSXpi ut IQ ad ique ET conjunctis rationibus PI quad.XpfXps ad pi quad.XPFXPS,ut IHXIQ ad ihXique hoc e&longs;t, ut &longs;uperficies circularis, quam arcus IH convolutione &longs;emicirculi AKB circa diametrum AB de&longs;cribet, ad &longs;uperficiem circularem, quam arcus ih convolutione &longs;emicirculi akb circa diametrum ab de&longs;cribet. Et vires, quibus hæ &longs;uperficies &longs;ecundum lineas ad &longs;e tendentes attrahunt corpu&longs;cula P & p, &longs;unt (per Hypothe&longs;in) ut ip&longs;æ &longs;uperficies applicatæ ad quadrata di&longs;tantiarum &longs;uarum a corporibus, hoc e&longs;t, ut pfXpsad PFXPS. Suntque hæ vires ad ip&longs;arum partes obliquas quæ (facta per Legum Corol. 2. re&longs;olutione virium) &longs;ecundum lineas PS, ps ad centra tendunt, ut PI ad PQ, & pi ad pque id e&longs;t (ob &longs;imilia triangula PIQ & PSF, piq & psf) ut PS ad PF & ps ad pf. Unde, ex æquo, fit attractio corpu&longs;culi hujus Pver&longs;us S ad attractionem corpu&longs;culi p ver&longs;us s, ut (PFXpfXps/PS) ad (pfXPFXPS/ps), hoc e&longs;t, ut ps quad. ad PS quad. Et &longs;imili argumento vires, quibus &longs;uperficies convolutione arcuum KL, kl de&longs;criptæ trahunt corpu&longs;cula, erunt ut ps quad. ad PS quad.; inque eadem ratione erunt vires &longs;uperficierum omnium circularium in quas utraque &longs;uperficies Sphærica, capiendo &longs;emper sd æqualem SD & se æqualem SE, di&longs;tingui pote&longs;t. ~~Et, per compo&longs;itionem, vires totarum &longs;uperficierum Sphæricarum in corpu&longs;cula exercitæ erunt in eadem ratione. que E. D.~~

~~LIBER PRIMUS.~~

~~PROPOSITIO LXXII. THEOREMA XXXII.~~

Si ad Sphæræ cuju&longs;vis puncta &longs;ingula tendant vires æquales centripetæ decre&longs;centes in duplicata ratione di&longs;tantiarum a punctis, ac detur tum Sphæræ den&longs;itas, tum ratio diametri Sphæræ ad di&longs;tantiam corpu&longs;culi a centro ejus; dico quod vis qua corpu&longs;culum attrahitur proportionalis erit &longs;emidiametro Sphæræ.

Nam concipe corpu&longs;cula duo &longs;eor&longs;im a Sphæris duabus attrahi, unum ab una & alterum ab altera, & di&longs;tantias eorum a Sphærarum centris proportionales e&longs;&longs;e diametris Sphærarum re&longs;pective, Sphæras autem re&longs;olvi in particulas &longs;imiles & &longs;imiliter po&longs;itas ad corpu&longs;cula. Et attractiones corpu&longs;culi unius, factæ ver&longs;us &longs;ingulas particulas Sphæræ unius, erunt ad attractiones alterius ver&longs;us analogas totidem particulas Sphæræ alterius, in ratione compo&longs;ita ex ratione particularum directe & ratione duplicata di&longs;tantiarum in- ver&longs;e. Sed particulæ &longs;unt ut Sphæræ, hoc e&longs;t, in ratione triplicata diametrorum, & di&longs;tantiæ &longs;unt ut diametri, & ratio prior directe una cum ratione po&longs;teriore bis inver&longs;e e&longs;t ratio diametri ad diametrum. que E. D.

~~DE MOTU CORPORUM~~

Corol. 1. Hinc &longs;i corpu&longs;cula in Circulis, circa Sphæras ex materia æqualiter attractiva con&longs;tantes, revolvantur; &longs;intque di&longs;tantiæ a centris Sphærarum proportionales earundem diametris: Tempora periodica erunt æqualia.

~~Corol. 2. Et vice ver&longs;a, &longs;i Tempora periodica &longs;unt æqualia; di&longs;tantiæ erunt proportionales diametris.~~ ~~Con&longs;tant hæc duo per Corol.~~ ~~3. Prop.~~ ~~IV.~~

Corol. 3. Si ad Solidorum durorum quorumvis &longs;imilium & æqualiter den&longs;orum puncta &longs;ingula tendant vires æquales centripetæ decre&longs;centes in duplicata ratione di&longs;tantiarum a punctis: vires quibus corpu&longs;cula, ad Solida illa duo &longs;imiliter &longs;ita, attrahentur ab ii&longs;dem, erunt ad invicem ut diametri Solidorum.

~~PROPOSITIO LXXIII. THEOREMA XXXIII.~~

Si ad Sphæræ alicujus datæ puncta &longs;ingula tendant æquales vires centripetæ decre&longs;centes in duplicata ratione di&longs;tantiarum a punctis: dico quod corpu&longs;culum intra Sphæram con&longs;titutum attrabitur vi proportionali di&longs;tantiæ &longs;uæ ab ip&longs;ius centro.

~~In Sphæra ABCD, centro S de&longs;cripta,~~

~~locetur corpu&longs;culum P; & centro eodem S,intervallo SP, concipe Sphæram interiorem PEQF de&longs;cribi.~~ ~~Manife&longs;tum e&longs;t, per Prop.~~ LXX, quod Sphæricæ &longs;uperficies concentricæ ex quibus Sphærarum differentia AEBFcomponitur, attractionibus per attractiones contrarias de&longs;tructis, nil agunt in corpus P. Re&longs;tat &longs;ola attractio Sphæræ interioris PEQF. Et per Prop. ~~LXXII, hæc e&longs;t ut di&longs;tantia PS. que E. D.~~

~~Scholium.~~

Superficies ex quibus &longs;olida componuntur, hic non &longs;unt pure Mathematicæ, &longs;ed Orbes adeo tenues ut eorum cra&longs;&longs;itudo in&longs;tar nihili &longs;it; nimirum Orbes evane&longs;centes ex quibus Sphæra ultimo con&longs;tat, ubi Orbium illorum numerus augetur & cra&longs;&longs;itudo minuitur in infinitum. ~~Similiter per Puncta, ex quibus lineæ, &longs;uperficies & &longs;olida componi dicuntur, intelligendæ &longs;unt particulæ æquales magnitudinis contemnendæ.~~

~~LIBER PRIMUS.~~

~~PROPOSITIO LXXIV. THEOREMA XXXIV.~~

~~Ii&longs;dem po&longs;itis, dico quod corpu&longs;culum extra Sphæram con&longs;titutum attrabitur vi reciproce proportionali quadrato di&longs;tantiæ &longs;uæ ab ip&longs;ius centro.~~

Nam di&longs;tinguatur Sphæra in &longs;uperficies Sphæricas innumeras concentricas, & attractiones corpu&longs;culi a &longs;ingulis &longs;uperficiebus oriundæ erunt reciproce proportionales quadrato di&longs;tantiæ corpu&longs;culi a centro, per Prop. ~~LXXI.~~ ~~Et componendo, fiet &longs;umma attractionum, hoc e&longs;t attractio corpu&longs;culi in Sphæram totam, in eadem ratione. que E. D.~~

~~Corol. 1. Hinc in æqualibus di&longs;tantiis a centris homogenearum Sphærarum, attractiones &longs;unt ut Sphæræ.~~ ~~Nam per Prop.~~ ~~LXXII, &longs;i di&longs;tantiæ &longs;unt proportionales diametris Sphærarum, vires erunt ut diametri.~~ Minuatur di&longs;tantia major in illa ratione; &, di&longs;tantiis jam factis æqualibus, augebitur attractio in duplicata illa ratione, adeoque erit ad attractionem alteram in triplicata illa ratione, hoc e&longs;t, in ratione Sphærarum.

~~Corol. 2. In di&longs;tantiis quibu&longs;vis attractiones &longs;unt ut Sphæræ applicatæ ad quadrata di&longs;tantiarum.~~

Corol. 3. Si corpu&longs;culum, extra Sphæram homogeneam po&longs;itum, trahitur vi reciproce proportionali quadrato di&longs;tantiæ &longs;uæ ab ip&longs;ius centro, con&longs;tet autem Sphæra ex particulis attractivis; decre&longs;cet vis particulæ cuju&longs;que in duplicata ratione di&longs;tantiæ a particula.

~~PROPOSITIO LXXV. THEOREMA XXXV.~~

Si ad Sphæræ datæ puncta &longs;ingula tendant vires æquales centripetæ, decre&longs;centes in duplicata ratione di&longs;tantiarum a punctis; dico quod Sphæra quævis alia &longs;imilaris ab eadem attrahitur vi reciproce proportionali quadrato di&longs;tantiæ centrorum.

~~Nam particulæ cuju&longs;vis attractio e&longs;t reciproce ut quadratum di&longs;tantiæ &longs;uæ a centro Sphæræ trahentis, (per Prop.~~ ~~LXXIV) & prop- terea eadem e&longs;t ac &longs;i vis tota attrahens manaret de corpu&longs;culo unico &longs;ito in centro hujus Sphæræ.~~ Hæc autem attractio tanta e&longs;t quanta foret vici&longs;&longs;im attractio corpu&longs;culi eju&longs;dem, &longs;i modo illud a &longs;ingulis Sphæræ attractæ particulis eadem vi traheretur qua ip&longs;as attrahit. ~~Foret autem illa corpu&longs;culi attractio (per Prop.~~ ~~LXXIV) reciproce proportionalis quadrato di&longs;tantiæ &longs;uæ a centro Sphæræ; adeoque huic æqualis attractio Sphæræ e&longs;t in eadem ratione. que E. D.~~

~~DE MOTU CORPORUM~~

Corol. 1. Attractiones Sphærarum, ver&longs;us alias Sphæras homogeneas, &longs;unt ut Sphæræ trahentes applicatæ ad quadrata di&longs;tantiarum centrorum &longs;uorum a centris earum quas attrahunt.

~~Corol. 2. Idem valet ubi Sphæra attracta etiam attrahit.~~ Namque hujus puncta &longs;ingula trahent &longs;ingula alterius, eadem vi qua ab ip&longs;is vici&longs;&longs;im trahuntur, adeoque cum in omni attractione urgeatur (per Legem III) tam punctum attrahens, quam punctum attractum, geminabitur vis attractionis mutuæ, con&longs;ervatis proportionibus.

Corol. 3. Eadem omnia, quæ &longs;uperius de motu corporum circa umbilicum Conicarum Sectionum demon&longs;trata &longs;unt, obtinent ubi Sphæra attrahens locatur in umbilico & corpora moventur extra Sphæram.

~~Corol. 4. Ea vero quæ de motu corporum circa centrum Conicarum Sectionum demon&longs;trantur, obtinent ubi motus peraguntur intra Sphæram.~~

~~PROPOSITIO LXXVI. THEOREMA XXXVI.~~

Si Sphæræ in progre&longs;&longs;u a centro ad circumferentiam (quoad materiæ den&longs;itatem & vim attractivam) utcunque di&longs;&longs;imilares, in progre&longs;&longs;u vero per circuitum ad datam omnem a centro di&longs;tantiam &longs;unt undique &longs;imilares, & vis attractiva puncti cuju&longs;que decre&longs;cit in duplicata ratione di&longs;tantiæ corporis attracti: dico quod vis tota qua huju&longs;modi Sphæra una attrahit aliam &longs;it reciproce proportionalis quadrato di&longs;tantiæ centrorum.

~~Sunto Sphæræ quotcunque concentricæ &longs;imilares AB, CD, EF,&c.~~ ~~quarum interiores additæ exterioribus componant materiam den&longs;iorem ver&longs;us centrum, vel &longs;ubductæ relinquant tenuiorem; & hæ (per Prop.~~ ~~LXXV) trahent Sphæras alias quotcunque concentricas &longs;imilares GH, IK, LM, &c.~~ &longs;ingulæ &longs;ingulas, viribus reciproce proportionalibus quadrato di&longs;tantiæ SP. Et componendo vel dividendo, &longs;umma virium illarum omnium, vel exce&longs;&longs;us aliquarum &longs;upra alias, hoc e&longs;t, vis quas Sphæra tota ex concentricis quibu&longs;cunque vel concentricarum differentiis compo&longs;ita AB,trahit totam ex concentricis quibu&longs;cunque vel concentricarum differentiis compo&longs;itam GH, erit in eadem ratione. Augeatur numerus Sphærarum concentricarum in infinitum &longs;ic, ut materiæ den&longs;itas una cum vi attractiva, in progre&longs;&longs;u a circumferentia ad centrum, &longs;ecundum Legem quamcunque cre&longs;cat vel decre&longs;cat: &, ad

dita materia non attractiva, compleatur ubivis den&longs;itas deficiens, eo ut Sphæræ acquirant formam quamvis optatam; & vis qua harum una attrahet alteram erit etiamnum (per argumentum &longs;uperius) in eadem illa di&longs;tantiæ quadratæ ratione inver&longs;a. que E. D.

~~LIBER PRIMUS.~~

Corol. 1. Hinc &longs;i eju&longs;modi Sphæræ complures, &longs;ibi invicem per omnia &longs;imiles, &longs;e mutuo trahant; attractiones acceleratrices &longs;ingularum in &longs;ingulas erunt, in æqualibus quibu&longs;vis centrorum di&longs;tantiis, ut Sphæræ attrahentes.

~~Corol. 2. Inque di&longs;tantiis quibu&longs;vis inæqualibus, ut Sphæræ attrahentes applicatæ ad quadrata di&longs;tantiarum inter centra.~~

Corol. 3. Attractiones vero motrices, &longs;eu pondera Sphærarum in Sphæras erunt, in æqualibus centrorum di&longs;tantiis, ut Sphæræ attrahentes & attractæ conjunctim, id e&longs;t, ut contenta &longs;ub Sphæris per multiplicationem producta.

~~Corol. 4. Inque di&longs;tantiis inæqualibus, ut contenta illa applicata ad quadrata di&longs;tantiarum inter centra.~~

~~DE MOTU CORPORUM~~

~~Corol. 5. Eadem valent ubi attractio oritur a Sphæræ utriu&longs;que virtute attractiva, mutuo exercita in Sphæram alteram.~~ ~~Nam viribus ambabus geminatur attractio, proportione &longs;ervata.~~

Corol. 6. Si huju&longs;modi Sphæræ aliquæ circa alias quie&longs;centes revolvantur, &longs;ingulæ circa &longs;ingulas, &longs;intque di&longs;tantiæ inter centra revolventium & quie&longs;centium proportionales quie&longs;centium diametris; æqualia erunt Tempora periodica.

~~Corol. 7. Et vici&longs;&longs;im, &longs;i Tempora periodica &longs;unt æqualia; di&longs;tantiæ erunt proportionales diametris.~~

Corol. 8. Eadem omnia, quæ &longs;uperius de motu corporum circa umbilicos Conicarum Sectionum demon&longs;trata &longs;unt, obtinent ubi Sphæra attrahens, formæ & conditionis cuju&longs;vis jam de&longs;criptæ, locatur in umbilico.

~~Corol. 9. Ut & ubi gyrantia &longs;unt etiam Sphæræ attrahentes, conditionis cuju&longs;vis jam de&longs;criptæ.~~

~~PROPOSITIO LXXVII. THEOREMA XXXVII.~~

Si ad &longs;ingula Sphærarum puncta tendant vires centripetæ, propertionales di&longs;tantiis punctorum a corporibus attractis: dico quod vis compo&longs;ita, qua Sphæræ duæ &longs;e mutuo trahent, est ut di&longs;tantia inter centra Sphærarum.

~~Cas. 1. Sit AEBF Sphæra, S~~

centrum ejus, P corpu&longs;culum attractum, PASB axis Sphæræ per centrum corpu&longs;culi tran&longs;iens, EF, ef plana duo quibus Sphæra &longs;ecatur, huic axi perpendicularia & hinc inde æqualiter di&longs;tantia a centro Sphæræ; G, g inter&longs;ectiones planorum & axis, & H punctum quodvis in plano EF. Puncti H vis centripeta in corpu&longs;culum P, &longs;ecundum lineam PH exercita, e&longs;t ut di&longs;tantia PH; & (per Legum Corol. 2.) &longs;ecundum lineam PG, &longs;eu ver&longs;us centrum S, ut longitudo PG. Igitur punctorum omnium in plano EF, hoc e&longs;t plani totius vis, qua corpu&longs;culum P trahitur ver&longs;us centrum S, e&longs;t ut numerus punctorum ductus in di&longs;tantiam PG: id e&longs;t, ut contentum &longs;ub plano ip&longs;o EF& di&longs;tantia illa PG. Et &longs;imiliter vis plani ef, qua corpu&longs;culum P trahitur ver&longs;us centrum S, e&longs;t ut planum illud ductum in di&longs;tantiam

&longs;uam Pg, &longs;ive ut huic æquale planum EF ductum in di&longs;tantiam illam Pg; & &longs;umma virium plani utriu&longs;que ut planum EF ductum in &longs;ummam di&longs;tantiarum PG+Pg, id e&longs;t, ut planum illud ductum in duplam centri & corpu&longs;culi di&longs;tantiam PS, hoc e&longs;t, ut duplum planum EF ductum in di&longs;tantiam PS, vel ut &longs;umma æqualium planorum EF+ef ducta in di&longs;tantiam eandem. Et &longs;imili argumento, vires omnium planorum in Sphæra tota, hinc inde æqualiter a centro Sphæræ di&longs;tantium, &longs;unt ut &longs;umma planorum ducta in di&longs;tantiam PS, hoc e&longs;t, ut Sphæra tota ducta in di&longs;tantiam centri &longs;ui S a corpu&longs;culo P. que E. D.

~~LIBER PRIMUS.~~

~~Cas. 2. Trahat jam corpu&longs;culum P Sphæram AEBF. Et eodem argumento probabitur quod vis, qua Sphæra illa trahitur, erit: ut di&longs;tantia PS. que E. D.~~

Cas. 3. Componatur jam Sphæra altera ex corpu&longs;culis innumeris P; & quoniam vis; qua corpu&longs;culum unumquodque trahitur, e&longs;t ut di&longs;tantia corpu&longs;culi a centro Sphæræ primæ ducta in Sphæram eandem, atque adeo eadem e&longs;t ac &longs;i prodiret tota de corpu&longs;culo unico in centro Sphæræ; vis tota qua corpu&longs;cula omnia in Sphæra &longs;ecunda trahuntur, hoc e&longs;t, qua Sphæra illa tota trahitur, eadem erit ac &longs;i Sphæra illa traheretur vi prodeunte de corpu&longs;culo unico in centro Sphæræ primæ, & propterea proportionalis e&longs;t di&longs;tantiæ inter centra Sphærarum. que E. D.

~~Cas. 4. Trahant Sphæræ &longs;e mutuo, & vis geminata proportionem priorem &longs;ervabit. que E. D.~~

Cas. 5. Locetur jam corpu&longs;culum p intra Sphæram AEBF; & quoniam vis plani ef in corpu&longs;culum e&longs;t ut contentum &longs;ub plano illo & di&longs;tantia pg; & vis contraria plani EF ut contentum &longs;ub plano illo & di&longs;tantia pG; erit vis ex utraque compo&longs;ita ut differentia contentorum, hoc e&longs;t, ut &longs;umma æqualium planorum ducta in &longs;emi&longs;&longs;em differentiæ di&longs;tantiarum, id e&longs;t, ut &longs;umma illa ducta in pS di&longs;tantiam corpu&longs;culi a centro Sphæræ. Et &longs;imili argumento, attractio planorum omnium EF, ef in Sphæra tota, hoc e&longs;t, attractio Sphæræ totius, e&longs;t ut &longs;umma planorum omnium, &longs;eu Sphæra tota, ducta in pS di&longs;tantiam corpu&longs;culi a centro Sphæræ. que E. D.

Cas. 6. Et &longs;i ex corpu&longs;culis innumeris p componatur Sphæra nova, intra Sphæram priorem AEBF &longs;ita; probabitur ut prius quod attractio, &longs;ive &longs;implex Sphæræ unius in alteram, &longs;ive mutua utriu&longs;que in &longs;e invicem, erit ut di&longs;tantia centrorum pS. q.E.D.

~~DE MOTU CORPORUM~~

~~PROPOSITIO LXXVIII. THEOREMA XXXVIII.~~

Si Sphæræ in progre&longs;&longs;u a centro ad circumferentiam &longs;int utcunque di&longs;&longs;imilares & inæquabiles, in progre&longs;&longs;u vero per circuitum ad datam omnem a centro di&longs;tantiam &longs;int undique &longs;imilares; & vis attractiva puncti cuju&longs;que &longs;it ut di&longs;tantia corporis attracti: dico quod vis tota qua huju&longs;modi Sphæræ duæ &longs;e mutuo trahunt &longs;it proportionalis di&longs;tantiæ inter centra Sphærarum.

~~Demon&longs;tratur ex Propo&longs;itione præcedente, eodem modo quo Propo&longs;itio LXXVI ex Propo&longs;itione LXXV demon&longs;trata fuit.~~

Corol. Quæ &longs;uperius in Propo&longs;itionibus X & LXIV de motu corporum circa centra Conicarum Sectionum demon&longs;trata &longs;unt, valent ubi attractiones omnes fiunt vi Corporum Sphæricorum conditionis jam de&longs;criptæ, &longs;untque corpora attracta Sphæræ conditionis eju&longs;dem.

~~Scholium.~~

Attractionum Ca&longs;us duos in&longs;igniores jam dedi expo&longs;itos; nimirum ubi Vires centripetæ decre&longs;cunt in duplicata di&longs;tantiarum ratione, vel cre&longs;cunt in di&longs;tantiarum ratione &longs;implici; efficientes in utroque Ca&longs;u ut corpora gyrentur in Conicis Sectionibus, & componentes corporum Sphæricorum Vires centripetas eadem Lege, in rece&longs;&longs;u a centro, decre&longs;centes vel cre&longs;centes cum &longs;eip&longs;is: Quod e&longs;t notatu dignum. ~~Ca&longs;us cæteros, qui conclu&longs;iones minus elegantes exhibent, &longs;igillatim percurrere longum e&longs;&longs;et.~~ ~~Malim cunctos methodo generali &longs;imul comprehendere ac determinare, ut &longs;equitur.~~

~~LEMMA XXIX.~~

Si de&longs;cribantur centro S circulus quilibet AEB, & centro P circuli duo EF, ef, &longs;ecantes priorem in E, e, lineamque PS inF, f; & ad PS demittantur perpendicula ED, ed: dico quod, fi di&longs;tantia arcuum EF, ef in infinitum minui intelligatur, ratio ultima lineæ evane&longs;centis Dd ad lineam evane&longs;centem Ff ea &longs;it, quæ lineæ PE ad lineam PS.

Nam &longs;i linea Pe &longs;ecet arcum EF in q; & recta Ee, quæ cum arcu evane&longs;cente Ee coincidit, producta occurrat rectæ PS in T; & ab S demittatur in PE normalis SG: ob &longs;imilia triangula DTE, dTe, DES; erit Dd ad Ee, ut DT ad TE, &longs;eu DE ad

~~ES; & ob triangula Eeq, ESG (per Lem.~~ ~~VIII, & Corol.~~ ~~3. Lem.~~ ~~VII) &longs;imilia, erit Ee ad eq &longs;eu Ff, ut ES ad SG; & ex æquo, Dd ad Ff ut DE ad SG; hoc e&longs;t (ob &longs;imilia triangula PDE, PGS) ut PE ad PS. que E. D.~~

~~LIBER PRIMUS.~~

~~PROPOSITIO LXXIX. THEOREMA XXXIX.~~

Si &longs;uperficies ob latitudinem infinite diminutam jamjam evane&longs;censEF fe, convolutione &longs;ui circa axem PS, de&longs;cribat &longs;olidum Sphæricum concavo convexum, ad cujus particulas &longs;ingulas æquales tendant æquales vires centripetæ: dico quod Vis, qua &longs;olidum illud trahit corpu&longs;culum &longs;itum in P, est in ratione compota ex ratione &longs;olidi DEqXFf & ratione vis qua particula data in loco Ff traheret idem corpu&longs;culum.

Nam &longs;i primo con&longs;ideremus vim &longs;uperficiei Sphæricæ FE, quæ convolutione arcus FE generatur, & a linea de ubivis &longs;ecatur in r; erit &longs;uperficiei pars annularis, convolutione arcus rE genita, ut lineola Dd, manente Sphæræ radio PE, (uti demon&longs;travit Archimedes in Lib. de Sphæra & Cylindro.) Et hujus vis &longs;ecundum lineas PE vel Pr undique in &longs;uperficie conica &longs;itas exercita, ut hæc ip&longs;a &longs;uperficiei pars annularis; hoc e&longs;t, ut lineola Dd vel, quod perinde e&longs;t, ut rectangulum &longs;ub dato Sphæræ radio PE & lineola illa Dd: at &longs;ecundum lineam PS ad centrum S tendentem minor, in ratione PD ad PE, adeoque ut PDXDd. Dividi jam intelligatur linea DF in particulas innumeras æquales, quæ &longs;ingulæ nominentur Dd; & &longs;uperficies FE dividetur in totidem æquales annulos, quorum vires erunt ut &longs;umma omnium PDXDd,hoc e&longs;t, ut 1/2 PFq-1/2PDq, adeoque ut DE quad. Ducatur

jam &longs;uperficies FE in altitudinem Ef; & fiet &longs;olidi EFfe vis exercita in corpu&longs;culum P ut DEqXFf: puta &longs;i detur vis quam particula aliqua data Ff in di&longs;tantia PF exercet in corpu&longs;culum P. At &longs;i vis illa non detur, fiet vis &longs;olidi EFfe ut &longs;olidum DEqXFf & vis illa non data conjunctim. que E. D.

~~DE MOTU CORPORUM~~

~~PROPOSITIO LXXX. THEOREMA XL.~~

Si ad Sphæræ alicujus ABE, centro S de&longs;criptæ, particulas &longs;ingulas æquales tendant æquales vires centripetæ, & ad Sphæræ axem AB, in quo corpu&longs;culum aliquod P locatur, erigantur de punctis &longs;ingulis D perpendicula DE, Sphæræ occurrentia in E, & in ip&longs;is capiantur longitudines DN, quæ &longs;int ut quantitas(DEqXPS/PE) & vis quam Sphæræ particula &longs;ita in axe ad di&longs;tantiam PE exercet in corpu&longs;culum P conjunctim: dico quod Vis tota, qua corpu&longs;culum P trahitur ver&longs;us Sphæram, est ut area comprehen&longs;a &longs;ub axe Sphæræ AB & linea curva ANB, quam punctum N perpetuo tangit.

Etenim &longs;tantibus quæ in Lemmate & Theoremate novi&longs;&longs;imo con&longs;tructa &longs;unt, concipe axem Sphæræ AB dividi in particulas innumeras æquales Dd, & Sphæram totam dividi in totidem laminas Sphæricas concavo-convexas EFfe; & erigatur perpendiculum dn. Per Theorema &longs;uperius, vis qua lamina EFfetrahit corpu&longs;culum P e&longs;t ut DEqXFf & vis particulæ unius ad di&longs;tantiam PE vel PF exercita conjunctim. E&longs;t autem per Lemma novi&longs;&longs;imum, Dd ad Ff ut PE ad PS, & inde Ff æqualis (PSXDd/PE); & DEqXFf æquale Dd in (DEqXPS/PE), & propterea vis laminæ EFfe e&longs;t ut Dd in (DEqXPS/PE) & vis particulæ ad di&longs;tantiam PF exercita conjunctim, hoc e&longs;t (ex Hypothe&longs;i) ut DNXDd, &longs;eu area evane&longs;cens DNnd. Sunt igitur laminarum omnium vires in corpus P exercitæ, ut areæ omnes DNnd, hoc e&longs;t, Sphæræ vis tota ut area tota ABNA. q.E.D.

~~LIBER PRIMUS.~~

Corol. 1. Hinc &longs;i vis centripeta, ad particulas &longs;ingulas tendens, eadem &longs;emper maneat in omnibus di&longs;tantiis, & fiat DN ut (DEqXPS/PE): erit vis tota qua corpu&longs;culum a Sphæra attrahitur, ut area ABNA.

Corol. 2. Si particularum vis centripeta &longs;it reciproce ut di&longs;tantia corpu&longs;culi a &longs;e attracti, & fiat DN ut (DEqXPS/PEq): erit vis qua corpu&longs;culum P a Sphæra tota attrahitur ut area ABNA.

Corol. 3. Si particularum vis centripeta &longs;it reciproce ut cubus di&longs;tantiæ corpu&longs;culi a &longs;e attracti, & fiat DN ut (DEqXPS/PEqq): erit vis qua corpu&longs;culum a tota Sphæra attrahitur ut area ABNA.

Corol. 4. Et univer&longs;aliter &longs;i vis centripeta ad &longs;ingulas Sphæræ particulas tendens ponatur e&longs;&longs;e reciproce ut quantitas V, fiat autem DN ut (DEqXPS/PEXV); erit vis qua corpu&longs;culum a Sphæra tota attrahitur ut area ABNA.

~~DE MOTU CORPORUM~~

~~PROPOSITIO LXXXI. PROBLEMA XLI.~~

~~Stantibus jam po&longs;itis, men&longs;uranda est Area ABNA.~~

~~A puncto P ducatur recta PH Sphæram tangens in H, & ad axem PAB demi&longs;&longs;a normali HI, bi&longs;ecetur PI in L; & erit (per Prop.~~ ~~12, Lib.~~ 2. Elem.) PEq æquale PSq + SEq + 2PSD. E&longs;t autem SEq &longs;eu SHq (ob &longs;imilitudinem triangulorum SPH, SHI) æquale rectangulo PSI. Ergo PEq æquale e&longs;t contento &longs;ub PS & PS+SI+2SD, hoc e&longs;t, &longs;ub PS & 2LS+2SD, id e&longs;t, &longs;ub PS & 2LD. Porro DE quad æquale e&longs;t SEq-SDq, &longs;eu SEq -LSq+2SLD-LDq, id e&longs;t, 2SLD-LDq-ALB. Nam LSq-SEq &longs;eu LSq-SAq

~~(per Prop.~~ ~~6, Lib.~~ 2. Elem.) æquatur rectangulo ALB. Scribatur itaque 2SLD -LDq -ALB pro DEq; & quantitas (DEqXPS/PEXV), quæ &longs;ecundum Corollarium quartum Propo&longs;itionis præcedentis e&longs;t ut longitudo ordinatim applicatæ DN, re&longs;olvet &longs;e&longs;e in tres partes (2SLDXPS/PEXV)-(LDqXPS/PEXV)-(ALBXPS/PEXV): ubi &longs;i pro V &longs;cribatur ratio inver&longs;a vis centripetæ, & pro PE medium proportionale inter PS & 2LD; tres illæ partes evadent ordinatim applicatæ linearum totidem curvarum, quarum areæ per Methodos vulgatas innote&longs;cunt. que E. F.

~~LIBER PRIMUS.~~

Exempl. 1. Si vis centripeta ad &longs;ingulas Sphæræ particulas tendens &longs;it reciproce ut di&longs;tantia; pro V &longs;cribe di&longs;tantiam PE; dein 2PSXLD pro PEq, & fiet DN ut SL-1/2LD-(ALB/2LD).Pone DN æqualem duplo ejus 2SL-LD-(ALB/LD): & ordinatæ pars data 2SL ducta in longitudinem AB de&longs;cribet aream rectangulam 2SLXAB; & pars indefinita LD ducta normaliter in eandem longitudinem per motum continuum, ea lege ut inter movendum cre&longs;cendo vel decre&longs;cendo æquetur &longs;emper longitudini LD, de&longs;cribet aream (LBq-LAq/2), id e&longs;t, aream SLXAB; quæ &longs;ubducta de area priore 2SLXAB relinquit aream SLXAB.Pars autem tertia (ALB/LD) ducta itidem per motum localem normaliter in eandem longitudinem, de&longs;cribet

~~aream Hyperbolicam; quæ &longs;ubducta de area SLXAB relinquet aream quæ&longs;itam ABNA. Unde talis emergit Problematis con&longs;tructio.~~ ~~Ad puncta L, A, Berige perpendicula Ll, Aa, Bb, quorum Aa ip&longs;i LB, & Bb ip&longs;i LA æquetur.~~ ~~A&longs;ymptotis Ll, LB, per puncta a, b de&longs;cribatur Hyperbola ab. Et acta chorda ba claudet aream aba areæ quæ&longs;itæ ABNA æqualem.~~

Exempl. 2. Si vis centripeta ad &longs;ingulas Sphæræ particulas tendens &longs;it reciproce ut cubus di&longs;tantiæ, vel (quod perinde e&longs;t) ut cubus ille applicatus ad planum quodvis datum; &longs;cribe (PEcub/2ASq) pro V, dein 2PSXLD pro PEq; & fiet DN ut (SLXASq/PSXLD)-(ASq/2PS) -(ALBXASq/2PSXLDq), id e&longs;t (ob continue proportionales PS, AS, SI) ut (LSI/LD)-1/2SI-(ALBXSI/2LDq). Si ducantur hujus partes tres in longitudinem AB, prima (LSI/LD) generabit aream Hyper-

~~bolicam; &longs;ecunda 1/2SI aream 1/2ABXSI; tertia (ALBXSI/2LDq) aream (ALBXSI/2LA)-(ALBXSI/2LB), id e&longs;t 1/2ABXSI. De prima &longs;ubducatur &longs;umma &longs;ecundæ & tertiæ, &~~

~~manebit area quæ&longs;ita ABNA. Unde talis emergit Problematis con&longs;tructio.~~ Ad puncta L, A, S, B erige perpendicula Ll, Aa, Ss, Bb, quorum Ss ip&longs;i SI æquetur, perque punctum s A&longs;ymptotis Ll, LB de&longs;cribatur Hyperbola asb occurrens perpendiculis Aa, Bb in a & b; & rectangulum 2ASI &longs;ubductum de area Hyperbolica AasbB reliquet aream quæ&longs;itam ABNA.

~~DE MOTU CORPORUM~~

Exempl. 3. Si Vis centripeta, ad &longs;ingulas Sphæræ particulas tendens, decre&longs;cit in quadruplicata ratione di&longs;tantiæ a particulis; &longs;cribe (PEqq/2AScub) pro V, dein √2PSXLD pro PE, & fiet DN ut (SIqXSL/√2SI)X(1/√LDc),-(SIq/2√2SI)X(1/√LD),-(SIqXALB/2√2SI)X(1/√LDqc).Cujus tres partes ductæ in longitudinem AB, producunt areas totidem, viz. (2SIqXSL/√2SI) in (1/√LA)-(1/√LB); (SIq/√2SI) in √LB-√LA; & (SIqXALB/3√2SI) in (1/√LAcub)-(1/√LBcub). Et hæ po&longs;t debitam reductionem fiunt (2SIqXSL/LI), SIq, & SIq+(2SIcub/3LI). Hæ vero, &longs;ubctis po&longs;terioribus de priore, evadunt (4SIcub/3LI). Igitur vis tota, qua corpu&longs;culum P in Sphæræ centrum trahitur, e&longs;t ut (SIcub/PI), id e&longs;t, reciproce ut PS cubXPI. que E. I.

~~Eadem Methodo determinari pote&longs;t Attractio corpu&longs;culi &longs;iti intra Sphæram, &longs;ed expeditius per Theorema &longs;equens.~~

~~LIBER PRIMUS.~~

~~PROPOSITIO LXXXII. THEOREMA XLI.~~

In Sphæra centro S intervallo SA de&longs;cripta, &longs;i capiantur SI, SA, SP continue proportionales: dico quod corpu&longs;culi intra Sphæram in loco quovis I attractio est ad attractionem ip&longs;ius extra Sphæram in loco P, in ratione compo&longs;ita ex &longs;ubduplicata ratione di&longs;tantiarum a centro IS, PS & &longs;ubduplicata ratione virium centripetarum, in locis illis P & I, ad centrum tendentium.

Ut &longs;i vires centripetæ particularum Sphæræ &longs;int reciproce ut di&longs;tantiæ corpu&longs;culi a &longs;e attracti; vis, qua corpu&longs;culum &longs;itum in Itrahitur a Sphæra tota, erit ad vim qua trahitur in P, in ratione

compo&longs;ita ex &longs;ubduplicata ratione di&longs;tantiæ SI ad di&longs;tantiam SP& ratione &longs;ubduplicata vis centripetæ in loco I, a particula aliqua in centro oriundæ, ad vim centripetam in loco P ab eadem in centro particula oriundam, id e&longs;t, ratione &longs;ubduplicata di&longs;tantiarum SI, SP ad invicem reciproce. ~~Hæ duæ rationes &longs;ubduplicatæ componunt rationem æqualitatis, & propterea attractiones in I & Pa Sphæra tota factæ æquantur.~~ Simili computo, &longs;i vires particularum Sphæræ &longs;unt reciproce in duplicata ratione di&longs;tantiarum, colligetur quod attractio in I &longs;it ad attractionem in P, ut di&longs;tantia SPad Sphæræ &longs;emidiametrum SA: Si vires illæ &longs;unt reciproce in trplicata ratione di&longs;tantiarum, attractiones in I & P erunt ad invi- cem ut SP quad ad SA quad: Si in quadruplicata, ut SP cub ad SA cub. Unde cum attractio in P, in hoc ultimo ca&longs;u, inventa fuit reciproce ut PS cubXPI, attractio in I erit reciproce ut SA cubXPI, id e&longs;t (ob datum SA cub) reciproce ut PI. Et &longs;imilis e&longs;t progre&longs;&longs;us in infinitum. ~~Theorema vero &longs;ic demon&longs;tratur.~~

~~DE MOTU CORPORUM~~

Stantibus jam ante con&longs;tructis, & exi&longs;tente corpore in loco quovis P, ordinatim applicata DN inventa fuit ut (DEqXPS/PEXV). Ergo &longs;i agatur IE, ordinata illa ad alium quemvis locum I, mutatis mutandis, evadet ut (DEqXIS/IEXV). Pone vires centripetas, e Sphæræ puncto quovis E manantes, e&longs;&longs;e ad invicem in di&longs;tantiis IE, PE, ut PEn ad IEn, (ubi numerus n de&longs;ignet indicem pote&longs;tatum PE & IE) & ordinatæ illæ fient ut (DEqXPS/PEXPEn) & (DEqXIS/IEXIEn), quarum ratio ad invicem e&longs;t ut PSXIEXIEn ad ISXPEXPEn. Quoniam ob &longs;imilia triangula SPE, SEI, fit IE ad PE ut IS ad SE vel SA; pro ratione IE ad PE &longs;cribe rationem IS ad SA; & ordinatarum ratio evadet PSXIEn ad SAXPEn. Sed PS ad SA &longs;ubduplicata e&longs;t ratio di&longs;tantiarum PS, SI; & IEn ad PEn &longs;ubduplicata e&longs;t ratio virium in di&longs;tantiis PS, IS. Ergo ordinatæ, & propterea areæ quas ordinatæ de&longs;cribunt, hi&longs;que proportionales attractiones, &longs;unt in ratione compo&longs;ita ex &longs;ubduplicatis illis rationibus. que E. D.

~~PROPOSITIO LXXXIII. PROBLEMA XLII.~~

~~Invenire vim qua corpu&longs;culum in centro Sphæræ locatum ad ejus Segmentum quodcunque attrahitur.~~

~~Sit P corpus in centro Sphæræ, & RBSD Segmentum ejus plano RDS & &longs;uperficie Sphærica RBS contentum.~~ Superficie Sphærica EFG centro P de&longs;cripta &longs;ecetur DB in F, ac di&longs;tinguatur Segmentum in partes BREFGS, FEDG. Sit autem &longs;uperficies illa non pure Mathematica, &longs;ed Phy&longs;ica, profunditatem habens quam minimam. ~~Nominetur i&longs;ta profundi- tas O, & erit hæc &longs;uperficies (per de~~

mon&longs;trata Archimedis) ut PFXDFXO.Ponamus præterea vires attractivas particularum Sphæræ e&longs;&longs;e reciproce ut di&longs;tantiarum dignitas illa cujus Index e&longs;t n; & vis qua &longs;uperficies FE trahit corpus P erit ut (DFXO/PFn-1). Huic proportionale &longs;it perpendiculum FN ductum in O; & area curvilinea BDLIB,quam ordinatim applicata FN in longitudinem DB per motum continuum ducta de&longs;cribit, erit ut vis tota qua Segmentum totum RBSD trahit corpus P. que E. I.

~~LIBER PRIMUS.~~

~~PROPOSITIO LXXXIV. PROBLEMA XLIII.~~

~~Invenire vim qua corpu&longs;culum, extra centrum Sphæræ in axe Segmenti cuju&longs;vis locatum, attrahitur ab eodem Segmento.~~

~~A Segmento EBK trahatur corpus P (Vide Fig.~~ ~~Prop.~~ ~~LXXIX, LXXX, LXXXI) in ejus axe ADB locatum.~~ ~~Centro P intervallo PE de&longs;cribatur &longs;uperficies Sphærica EFK, qua di&longs;tinguatur Segmentum in partes duas EBKF & EFKD. Quæratur vis partis prioris per Prop.~~ ~~LXXXI, & vis partis po&longs;terioris per Prop.~~ ~~LXXXIII; & &longs;umma virium erit vis Segmenti totius EBKD. que E. I.~~

~~Scholium.~~

Explicatis attractionibus corporum Sphæricorum, jam pergere liceret ad Leges attractionum aliorum quorundam ex particulis attractivis &longs;imiliter con&longs;tantium corporum; &longs;ed i&longs;ta particulatim tractare minus ad in&longs;titutum &longs;pectat. Suffecerit Propo&longs;itiones qua&longs;dam generaliores de viribus huju&longs;modi corporum, deque motibus inde oriundis, ob earum in rebus Philo&longs;ophicis aliqualem u&longs;um, &longs;ubjungere.

~~DE MOTU CORPORUM~~

~~SECTIO XIII.~~

~~De Corporum non Sphæricorum viribus attactivis.~~

~~PROPOSITIO LXXXV. THEOREMA XLII.~~

Si corporis attracti, ubi attrahenti contiguum est, attractio longe fortior &longs;it, quam cum vel minimo intervallo &longs;eparantur ab invicem: vires particularum trahentis, in rece&longs;&longs;u corporis attracti, decre&longs;cunt in ratione plu&longs;quam duplicata di&longs;tantiarum a particulis.

~~Nam &longs;i vires decre&longs;cunt in ratione duplicata di&longs;tantiarum a particulis; attractio ver&longs;us corpus Sphæricum, propterea quod (per Prop.~~ LXXIV) &longs;it reciproce ut quadratum di&longs;tantiæ attracti corporis a centro Sphæræ, haud &longs;en&longs;ibiliter augebitur ex contactu; atque adhuc minus augebitur ex contactu, &longs;i attractio in rece&longs;&longs;u corporis attracti decre&longs;cat in ratione minore. ~~Patet igitur Propo&longs;itio de Sphæris attractivis.~~ ~~Et par e&longs;t ratio Orbium Sphæricorum concavorum corpora externa trahentium.~~ ~~Et multo magis res con&longs;tat in Orbibus corpora interius con&longs;tituta trahentibus, cum attractiones pa&longs;&longs;im per Orbium cavitates ab attractionibus contrariis (per Prop.~~ ~~LXX) tollantur, ideoque vel in ip&longs;o contactu nullæ &longs;unt.~~ Quod &longs;i Sphæris hi&longs;ce Orbibu&longs;que Sphæricis partes quælibet a loco contactus remotæ auferantur, & partes novæ ubivis addantur: mutari po&longs;&longs;unt figuræ horum corporum attractivorum pro lubitu, nec tamen partes additæ vel &longs;ubductæ, cum &longs;int a loco contactus remotæ, augebunt notabiliter attractionis exce&longs;&longs;um qui ex contactu oritur. ~~Con&longs;tat igitur Propo&longs;itio de corporibus Figurarum omnium. que E. D.~~

~~LIBER PRIMUS.~~

~~PROPOSITIO LXXXVI. THEOREMA XLIII.~~

Si particularum, ex quibus corpus attractivum componitur, vires in rece&longs;&longs;u corporis attracti decre&longs;cunt in triplicata vel plu&longs;quam triplicata ratione di&longs;tantiarum a particulis: attractio longe fortior erit in contactu, quam cum attrahens & attractum intervallo vel minimo &longs;eparantur ab invicem.

Nam attractionem in acce&longs;&longs;u attracti corpu&longs;culi ad huju&longs;modi Sphæram trahentem augeri in infinitum, con&longs;tat per &longs;olutionem Problematis XLI, in Exemplo &longs;ecundo ac tertio exhibitam. Idem, per Exempla illa & Theorema XLI inter &longs;e collata, facile colligitur de attractionibus corporum ver&longs;us Orbes concavo-convexos, &longs;ive corpora attracta collocentur extra Orbes, &longs;ive intra in eorum cavitatibus. Sed & addendo vel auferendo his Sphæris & Orbibus ubivis extra locum contactus materiam quamlibet attractivam, eo ut corpora attractiva induant figuram quamvis a&longs;&longs;ignatam, con&longs;tabit Propo&longs;itio de corporibus univer&longs;is. que E. D.

~~PROPOSITIO LXXXVII. THEOREMA XLIV.~~

Si corpora duo &longs;ibi invicem &longs;imilia, & ex materia æqualiter attractiva con&longs;tantia, &longs;eor&longs;im attrahant corpu&longs;cula &longs;ibi ip&longs;is proportionalia & ad &longs;e &longs;imiliter po&longs;ita: attractiones acceleratrices corpu&longs;culorum in corpora tota erunt ut attractiones acceleratrices corpu&longs;culorum in eorum particulas totis proportionales & in totis &longs;imiliter po&longs;itas.

Nam &longs;i corpora di&longs;tinguantur in particulas, quæ &longs;int totis proportionales & in totis &longs;imiliter &longs;itæ; erit, ut attractio in particulam quamlibet unius corporis ad attractionem in particulam corre&longs;pondentem in corpore altero, ita attractiones in particulas &longs;ingulas primi corporis ad attractiones in alterius particulas &longs;ingulas corre&longs;pondentes; & componendo, ita attractio in totum primum corpus ad attractionem in totum &longs;ecundum. que E. D.

Corol. 1. Ergo &longs;i vires attractivæ particularum, augendo di&longs;tantias corpu&longs;culorum attractorum, decre&longs;cant in ratione dignitatis cuju&longs;vis di&longs;tantiarum: attractiones acceleratrices in corpora tota erunt ut corpora directe & di&longs;tantiarum dignitates illæ inver&longs;e. Ut &longs;i vires particularum decre&longs;cant in ratione duplicata di&longs;tantiarum a corpu&longs;culis attractis, corpora autem &longs;int ut A cub. & B cub. adeoque tum corporum latera cubica, tum corpu&longs;culorum attractorum di&longs;tantiæ a corporibus, ut A & B: attractiones acceleratrices in corpora erunt ut (Acub./Aquad.) & (Bcub./Bquad.) id e&longs;t, ut corporum latera illa cubica A & B. Si vires particularum decre&longs;cant in ratione triplicata di&longs;tantiarum a corpu&longs;culis attractis; attractiones acceleratrices in corpora tota erunt ut (Acub./Acub.) & (Bcub./Bcub.), id e&longs;t, æquales. ~~Si vires decre&longs;cant in ratione quadruplicata; attractiones in corpora erunt ut (Acub./Aqq.) & (Bcub./Bqq.) id e&longs;t, reciproce ut latera cubica A & B. Et &longs;ic in cæteris.~~

~~DE MOTU CORPORUM~~

Corol. 2. Unde vici&longs;&longs;im, ex viribus quibus corpora &longs;imilia trahunt corpu&longs;cula ad &longs;e &longs;imiliter po&longs;ita, colligi pote&longs;t ratio decrementi virium particularum attractivarum in rece&longs;&longs;u corpu&longs;culi attracti; &longs;i modo decrementum illud &longs;it directe vel inver&longs;e in ratione aliqua di&longs;tantiarum.

~~PROPOSITIO LXXXVIII. THEOREMA XLV.~~

Si particularum æqualium Corporis cuju&longs;cunque vires attractivæ &longs;int ut di&longs;tantiæ locorum a particulis: vis corporis totius tendet ad ip&longs;ius centrum gravitatis; & eadem erit cum vi Globi ex materia con&longs;imili & æquali con&longs;tantis & centrum habentis in ejus centro gravitatis.

~~Corporis RSTV particulæ A, B trahant corpu&longs;culum aliquod~~

Z viribus quæ, &longs;i particulæ æquantur inter &longs;e, &longs;int ut di&longs;tantiæ AZ, BZ; &longs;in particulæ &longs;tatuantur inæquales, &longs;int ut hæ particulæ in di&longs;tantias &longs;uas AZ, BZre&longs;pective ductæ. Et exponantur hæ vires per contenta illa AXAZ & BXBZ. Jungatur AB,& &longs;ecetur ea in G ut &longs;it AG ad BG ut particula B ad particulam A; & erit G commune centrum gravitatis particularum A & B. Vis AXAZ (per Legum Corol.2.) re&longs;olvitur in vires AXGZ & AXAG& vis BXBZ in vires BXGZ & BXBG. Vires autem AXAG& BXBG, ob proportionales A ad B & BG ad AG, æquantur; adeoque cum dirigantur in partes contrarias, &longs;e mutuo de&longs;truunt. Re&longs;tant vires AXGZ & BXGZ. Tendunt hæ ab Z ver&longs;us centrum G, & vim ―A+BXGZ componunt; hoc e&longs;t, vim eandem ac &longs;i particulæ attractivæ A & B con&longs;i&longs;terent in eorum communi gravitatis centro G, Globum ibi componentes.

~~LIBER PRIMUS.~~

Eodem argumento, &longs;i adjungatur particula tertia C, & componatur hujus vis cum vi ―A+BXGZ tendente ad centrum G; vis inde oriunda tendet ad commune centrum gravitatis Globi illius G& particulæ C; hoc e&longs;t, ad commune centrum gravitatis trium particularum A, B, C; & eadem erit ac &longs;i Globus & particula C con&longs;i&longs;terent in centro illo communi, Globum majorem ibi componentes. ~~Et &longs;ic pergitur in infinitum.~~ ~~Eadem e&longs;t igitur vis tota particularum omnium corporis cuju&longs;cunque RSTV ac &longs;i corpus illud, &longs;ervato gravitatis centro, figuram Globi indueret. que E. D.~~

Corol. Hinc motus corporis attracti Z idem erit ac &longs;i corpus attrahens RSTV e&longs;&longs;et Sphæricum: & propterea &longs;i corpus illud attrahens vel quie&longs;cat, vel progrediatur uniformiter in directum; corpus attractum movebitur in Ellip&longs;i centrum habente in attrahentis centro gravitatis.

~~PROPOSITIO LXXXIX. THEOREMA XLVI.~~

Si Corpora &longs;int plura ex particulis æqualibus con&longs;tantia, quarum vires &longs;unt ut di&longs;tantiæ locorum a &longs;ingulis: vis ex omnium viribus compo&longs;ita, qua corpu&longs;culum quodcunque trahitur, tendet ad trahentium commune centrum gravitatis, & eadem erit ac &longs;i trahentia illa, &longs;ervato gravitatis centro communi, coirent & in Globum formarentur.

~~Demon&longs;tratur eodem modo, atque Propo&longs;itio &longs;uperior.~~

~~Corol. Ergo motus corporis attracti idem erit ac &longs;i corpora trahentia, &longs;ervato communi gravitatis centro, coirent & in Globum formarentur.~~ Ideoque &longs;i corporum trahentium commune gravitatis centrum vel quie&longs;cit, vel progreditur uniformiter in linea recta: corpus attractum movebitur in Ellip&longs;i, centrum habente in communi illo trahentium centro gravitatis.

~~DE MOTU CORPORUM~~

~~PROPOSITIO XC. PROBLEMA XLIV.~~

Si ad &longs;ingula Circuli cuju&longs;cunque puncta tendant vires æquales centripetæ, decre&longs;centes in quacunque di&longs;tantiarum ratione invenire vim qua corpu&longs;culum attrahitur ubivis po&longs;itum in recta quæ plano Circuli ad centrum ejus perpendiculariter in&longs;i&longs;tit.

~~Centro A intervallo quovis AD, in plano cui recta AP perpendicularis e&longs;t, de&longs;cribi intelligatur Circulus; & invenienda &longs;it vis qua corpu&longs;culum quodvis P in eundem attrahitur.~~ ~~A Circuli puncto quovis E ad corpu&longs;culum attractum P agatur recta PE: In recta PA capiatur PF ip&longs;i PE æ~~

~~qualis, & erigatur normalis FK,quæ &longs;it ut vis qua punctum E trahit corpu&longs;culum P. Sitque IKLcurva linea quam punctum K perpetuo tangit.~~ Occurrat eadem Circuli plano in L. In PA capiatur PH æqualis PD, & erigatur perpendiculum HI curvæ prædictæ occurrens in I; & erit corpu&longs;culi P attractio in Circulum ut area AHIL ducta in altitudinem AP. que E. I.

~~Etenim in AE capiatur linea quam minima Ee. Jungatur Pe,& in PE, PA capiantur PC, Pf ip&longs;i Pe æquales.~~ Et quoniam vis, qua annuli punctum quodvis E trahit ad &longs;e corpus P, ponitur e&longs;&longs;e ut FK, & inde vis qua punctum illud trahit corpus P ver&longs;us A e&longs;t ut (APXFK/PE), & vis qua annulus totus trahit corpus P ver&longs;us A, ut annulus & (APXFK/PE) conjunctim; annulus autem i&longs;te e&longs;t ut rectangulum &longs;ub radio AE & latitudine Ee, & hoc rectangulum (ob proportionales PE & AE, Ee & CE) æquatur rectangulo PEXCE&longs;eu PEXFf; erit vis qua annulus i&longs;te trahit corpus P ver&longs;us A, ut PEXFf & (APXFK/PE) conjunctim, id e&longs;t, ut contentum FfXFKXAP, &longs;ive ut area FKkf ducta in AP. Et propterea &longs;umma virium, quibus annuli omnes in Circulo, qui centro A & in- tervallo AD de&longs;cribitur, trahunt corpus P ver&longs;us A, e&longs;t ut area tota AHIKL ducta in AP. que E. D.

~~LIBER PRIMUS.~~

Corol. 1. Hinc &longs;i vires punctorum decre&longs;cunt in duplicata di&longs;tantiarum ratione, hoc e&longs;t, &longs;i &longs;it FK ut (1/PFquad.), atque adeo area AHIKL ut (1/PA-1/PH); erit attractio corpu&longs;culi P in Circulum ut (1-PA/PH), id e&longs;t, ut (AH/PH).

Corol. 2. Et univer&longs;aliter, &longs;i vires punctorum ad di&longs;tantias D &longs;int reciproce ut di&longs;tantiarum dignitas quælibet Dn, hoc e&longs;t, &longs;i &longs;it FKut (1/Dn), adeoque area AHIKL ut (1/PAn-1-1/PHn-1); erit attractio corpu&longs;culi P in Circulum ut (1/PAn-2-PA/PHn-1).

Corol 3. Et &longs;i diameter Circuli augeatur in infinitum, & numerus n &longs;it unitate major; attractio corpu&longs;culi P in planum totum infinitum erit reciproce ut PAn-2, propterea quod terminus alter (PA/PHn-1) evane&longs;cet.

~~PROPOSITIO XCI. PROBLEMA XLV.~~

~~Invenire attractionem corpu&longs;culi &longs;iti in axe Solidi rotundi, ad cujus puncta &longs;ingula tendunt vires æquales centripetæ in quacunque di&longs;tantiarum ratione decre&longs;centes.~~

~~In Solidum ADEFG tra~~

hatur corpu&longs;culum P, &longs;itum in ejus axe AB. Circulo quolibet RFS ad hunc axem perpendiculari &longs;ecetur hoc Solidum, & in ejus diametro FS, in plano aliquo PALKB per axem tran&longs;eunte, capiatur (per Prop. ~~XC) longitudo FK vi qua corpu&longs;culum P in circulum illum attrahitur proportionalis.~~ ~~Tangat autem punctum K curvam lineam LKI, planis extimorum circulorum AL & BI occurrentem in L & I; & erit attractio corpu&longs;culi P in Solidum ut area LABI. que E. I.~~

~~DE MOTU CORPORUM~~

~~Corol. 1. Unde &longs;i Solidum~~

Cylindrus &longs;it, parallelogrammo ADEB circa axem AB revoluto de&longs;criptus, & vires centripetæ in &longs;ingula ejus puncta tendentes &longs;int reciproce ut quadrata di&longs;tantiarum a punctis: erit attractio corpu&longs;culi P in hunc Cylindrum ut AB-PE+PD.Nam ordinatim applicata FK(per Corol. ~~1. Prop.~~ XC) erit ut 1-(PF/PR). Hujus pars 1 ducta in longitudinem AB, de&longs;cribit aream 1XAB; & pars altera (PF/PR) ducta in longitudinem PB, de&longs;cribit aream 1 in ―(PE-AD) (id quod ex curvæ LIK quadratura facile o&longs;tendi pote&longs;t:) & &longs;imiliter pars eadem ducta in longitudinem PA de&longs;cribit aream 1 in ―(PD-AD), ductaque in ip&longs;arum PB, PA differentiam AB de&longs;cribit arearum differentiam 1 in ―(PE-PD). De contento primo 1XAB auferatur contentum po&longs;tremum 1 in ―(PE-PD), & re&longs;tabit area LABIæqualis 1 in ―(AB-PE+PD). Ergo vis, huic areæ proportionalis, e&longs;t ut AB-PE+PD.

~~Corol. 2. Hinc etiam~~

~~vis innote&longs;cit qua Sphærois AGBCD attrahit corpus quodvis P, exterius in axe &longs;uo AB &longs;itum.~~ Sit NKRM Sectio Conica cujus ordinatim applicata ER, ip&longs;i PE perpendicularis, æquetur &longs;emper longitudini PD, quæ ducitur ad punctum illud D, in quo applicata i&longs;ta Sphæroidem &longs;ecat. A Sphæroidis verticibus A, Bad ejus axem AB erigantur perpendicula AK, BM ip&longs;is AP, BPæqualia re&longs;pective, & propterea Sectioni Conicæ occurrentia in K& M; & jungatur KM auferens ab eadem &longs;egmentum KMRK.Sit autem Sphæroidis centrum S & &longs;emidiameter maxima SC: & vis qua Sphærois trahit corpus P erit ad vim qua Sphæra, diametro AB

~~de&longs;cripta, trahit idem corpus, ut (ASXCSq-PSXKMRK/PSq+CSq-ASq) ad (AS cub/3PS quad). Et eodem computandi fundamento invenire licet vires &longs;egmentorum Sphæroidis.~~

~~LIBER PRIMUS.~~

~~Corol. 3. Quod &longs;i corpu&longs;culum intra Sphæroidem, in data quavis eju&longs;dem diametro, collocetur; attractio erit ut ip&longs;ius di&longs;tantia a centro.~~ ~~Id quod facilius colligetur hoc argumento.~~ ~~Sit AGOFSphærois attrahens, S centrum ejus & P corpus attractum.~~ Per corpus illud P agantur tum &longs;emidiameter SPA, tum rectæ duæ quævis DE, FG Sphæroidi hinc inde occurrentes in D & E, F& G: Sintque PCM, HLN &longs;uperficies Sphæroidum duarum interiorum, exteriori &longs;imilium & concentricarum, quarum prior tran&longs;eat per corpus P & &longs;ecet rectas DE & FG in B & C, po&longs;terior &longs;ecet ea&longs;dem rectas in H, I & K, L. Habeant autem Sphæroides omnes axem communem, & erunt rect

arum partes hinc inde interceptæ DP& BE, FP & CG, DH & IE, FK& LG &longs;ibi mutuo æquales; propterea quod rectæ DE, PB & HI bi&longs;ecantur in eodem puncto, ut & rectæ FG, PC & KL. Concipe jam DPF, EPG de&longs;ignare Conos oppo&longs;itos, angulis verticalibus DPF, EPG infinite parvis de&longs;criptos, & lineas etiam DH, EI infinite parvas e&longs;&longs;e; & Conorum particulæ Sphæroidum &longs;uperficiebus ab&longs;ci&longs;&longs;æ DHKF, GLIE, ob æqualitatem linearum DH, EI, erunt ad invicem ut quadrata di&longs;tantiarum &longs;uarum a corpu&longs;culo P, & propterea corpu&longs;culum illud æqualiter trahent. Et pari ratione, &longs;i &longs;uperficiebus Sphæroidum innumerarum &longs;imilium concentricarum & axem communem habentium dividantur &longs;patia DPF, EGCB in particulas, hæ omnes utrinque æqualiter trahent corpus P in partes contrarias. ~~Æquales igitur &longs;unt vires Coni DPF & &longs;egmenti Conici EGCB, & per contrarietatem &longs;e mutuo de&longs;truunt.~~ ~~Et par e&longs;t ratio virium materiæ omnis extra Sphæroidem intimam PCBM. Trahitur igitur corpus P a &longs;ola Sphæroide intima PCBM, & propterea (per Corol.~~ ~~3. Prop.~~ ~~LXXII) attractio ejus e&longs;t ad vim, qua corpus A trahitur a Sphæroide tota AGOD, ut di&longs;tantia PS ad di&longs;tantiam AS. que E. D.~~

~~DE MOTU CORPORUM~~

~~PROPOSITIO XCII. PROBLEMA XLVI.~~

~~Dato Corpore attractivo, invenire rationem decrementi virium centripetarum in ejus puncta &longs;ingula tendentium.~~

~~E Corpore dato formanda e&longs;t Sphæra vel Cylindrus aliave figura regularis, cujus lex attractionis, cuivis decrementi rationi congruens (per Prop.~~ ~~LXXX, LXXXI, & XCI) inveniri pote&longs;t.~~ Dein factis experimentis invenienda e&longs;t vis attractionis in diver&longs;is di&longs;tantiis, & lex attractionis in totum inde patefacta dabit rationem decrementi virium partium &longs;ingularum, quam invenire oportuit.

~~PROPOSITIO XCIII. THEOREMA XLVII.~~

Si Solidum ex una parte planum, ex reliquis autem partibus infinitum, con&longs;tet ex particulis æqualibus æqualiter attractivis, quarum vires in rece&longs;&longs;u a Solido decre&longs;cunt in ratione pote&longs;tatis cuju&longs;vis di&longs;tantiarum plu&longs;quam quadraticæ, & vi Solidi totius corpu&longs;culum ad utramvis plani partem con&longs;titutum trahatur: dico quod Solidi vis illa attractiva, in rece&longs;&longs;u ab ejus &longs;uperficie plana, decre&longs;cet in ratione pote&longs;tatis, cujus latus est di&longs;tantia corpu&longs;culi a plano, & Index ternario minor quam Index pote&longs;tatis di&longs;tantiarum.

~~Cas. 1. Sit LGl planum~~

~~quo Solidum terminatur.~~ ~~Jaceat Solidum autem ex parte plani hujus ver&longs;us I, inque plana innumera mHM, nIN, &c.~~ ~~ip&longs;i GLparallela re&longs;olvatur.~~ ~~Et primo collocetur corpus attractum C extra Solidum.~~ Agatur autem CGHI planis illis innumeris perpendicularis, & decre&longs;cant vires attractivæ punctorum Solidi in ratione pote&longs;tatis di&longs;tantiarum, cujus index &longs;it numerus n ternario non minor. ~~Ergo (per Corol.~~ ~~3. Prop.~~ XC) vis qua planum quodvis mHM trahit punctum C e&longs;t reciproce ut CHn-2. In plano mHM capiatur longitudo HM ip&longs;i CHn-2 reciproce proportionalis, & erit vis illa ut HM. Similiter in planis &longs;ingulis lGL, nIN, oKO, &c. ~~capiantur longitudines GL, IN, KO, &c.~~ ~~ip&longs;is CGn-2, CIn-2, CKn-2, &c.~~ reciproce proportionales; & vires planorum eorundem erunt ut longitudines captæ, adeoque &longs;umma virium ut &longs;umma longitudinum, hoc e&longs;t, vis Solidi totius ut area GLOK in infinitum ver&longs;us OK producta. ~~Sed area illa (per notas quadraturarum methodos) e&longs;t reciproce ut CGn-3, & propterea vis Solidi totius e&longs;t reciproce ut CGn-3.~~ ~~que E. D.~~

~~LIBER PRIMUS.~~

Cas. 2. Collocetur jam corpu&longs;culum C ex parte plani lGL intra Solidum, & capiatur di&longs;tantia CK æqualis di&longs;tantiæ CG. Et Solidi pars LGloKO, planis parallelis lGL, oKO terminata, corpu&longs;culum C in medio &longs;itum nullam in partem trahet, contrariis oppo&longs;itorum punctorum actionibus &longs;e mutuo per æqualitatem tollentibus. ~~Proinde corpu&longs;culum C &longs;ola vi Solidi ultra planum OK &longs;iti trahitur.~~ ~~Hæc autem vis (per Ca&longs;um primum) e&longs;t reciproce ut CKn-3,hoc e&longs;t (ob æquales CG, CK) reciproce ut CGn-3.~~ ~~que E. D.~~

Corol. 1. Hinc &longs;i Solidum LGIN planis duobus infinitis parallelis LG, IN utrinque terminetur; innote&longs;cit ejus vis attractiva, &longs;ubducendo de vi attractiva Solidi totius infiniti LGKOvim attractivam partis ulterioris NICO, in infinitum ver&longs;us KOproductæ.

Corol. 2. Si Solidi hujus infiniti pars ulterior, quando attractio ejus collata cum attractione partis citerioris nullius pene e&longs;t momenti, rejiciatur: attractio partis illius citerioris augendo di&longs;tantiam decre&longs;cet quam proxime in ratione pote&longs;tatis CGn-3.

Corol. 3. Et hinc &longs;i corpus quodvis finitum & ex una parte planum trahat corpu&longs;culum e regione medii illius plani, & di&longs;tantia inter corpu&longs;culum & planum collata cum dimen&longs;ionibus corporis attrahentis perexigua &longs;it, con&longs;tet autem corpus attrahens ex particulis homogeneis, quarum vires attractivæ decre&longs;cunt in ratione pote&longs;tatis cuju&longs;vis plu&longs;quam quadruplicatæ di&longs;tantiarum; vis attractiva corporis totius decre&longs;cet quamproxime in ratione pote&longs;tatis, cujus latus &longs;it di&longs;tantia illa perexigua, & Index ternario minor quam Index pote&longs;tatis prioris. De corpore ex particulis con&longs;tante, quarum vires attractivæ decre&longs;cunt in ratione pote&longs;tatis triplicatæ di&longs;tantiarum, a&longs;&longs;ertio non valet; propterea quod, in hoc ca&longs;u, attractio partis illius ulterioris corporis infiniti in Corollario &longs;ecundo, &longs;emper e&longs;t infinite major quam attractio partis citerioris.

~~DE MOTU CORPORUM~~

~~Scholium.~~

~~Si corpus aliquod perpendiculariter ver&longs;us planum datum trahatur, & ex data lege attractionis quæratur motus corporis: Solvetur Problema quærendo (per Prop.~~ ~~XXXIX) motum corporis recta de&longs;cendentis ad hoc planum, & (per Legum Corol.~~ ~~2.) componendo motum i&longs;tum cum uniformi motu, &longs;ecundum lineas eidem plano parallelas facto.~~ Et contra, &longs;i quæratur Lex attractionis in planum &longs;ecundum lineas perpendiculares factæ, ea conditione ut corpus attractum in data quacunque curva linea moveatur, &longs;olvetur Problema operando ad exemplum Problematis tertii.

~~Operationes autem contrahi &longs;olent re&longs;olvendo ordinatim applicatas in Series convergentes.~~ Ut &longs;i ad ba&longs;em A in angulo quovis dato ordinatim applicetur longitudo B, quæ &longs;it ut ba&longs;is dignitas quælibet Am/n; & quæratur vis qua corpus, &longs;ecundum po&longs;itionem ordinatim applicatæ, vel in ba&longs;em attractum vel a ba&longs;i fugatum, moveri po&longs;&longs;it in curva linea quam ordinatim applicata termino &longs;uo &longs;uperiore &longs;emper attingit: Suppono ba&longs;em augeri parte quam minima O, & ordinatim applicatam ―(A+O)m/n re&longs;olvo in Seriem infinitam Am/n+m/n OA(m-n/n)+(mm-mn/2nn) OOA(m-2n/n) &c. ~~atque hujus termino in quo O duarum e&longs;t dimen&longs;ionum, id e&longs;t, termino (mm-mn/2nn) OOA(m-2n/n) vim proportionalem e&longs;&longs;e &longs;uppono.~~ ~~E&longs;t igitur vis quæ&longs;ita ut (mm-mn/nn)A(m-2n/n), vel quod perinde e&longs;t, ut (mm-mn/nn)B(m-2n/m).~~ ~~Ut &longs;i ordinatim applicata Parabolam attingat, exi&longs;tente m=2, & n=1: fiet vis ut data 2B°, adeoque dabitur.~~ ~~Data igitur vi corpus movebitur in Parabola, quemadmodum Galilæus demon&longs;travit.~~ Quod &longs;i ordinatim applicata Hyperbolam attingat, exi&longs;tente m=o-1, & n=1; fiet vis ut 2A-3 &longs;eu 2B3: adeoque vi, quæ &longs;it ut cubus ordinatim applicatæ, corpus movebitur in Hyperbola. ~~Sed mi&longs;&longs;is huju&longs;modi Propo&longs;itionibus, pergo ad alias qua&longs;dam de Motu, quas nondum attigi.~~

~~LIBER PRIMUS.~~

~~SECTIO XIV.~~

~~De Motu corporum minimorum, quæ Viribus centripetis ad &longs;ingulas magni alicujus corporis partes tendentibus agitantur.~~

~~PROPOSITIO XCIV. THEOREMA XLVIII.~~

Si Media duo &longs;imilaria, &longs;patio planis parallelis utrinque terminato, di&longs;tinguantur ab invicem, & corpus in tran&longs;itu per hoc &longs;patium attrahatur vel impellatur perpendiculariter ver&longs;us Medium alterutrum, neque ulla alia vi agitetur vel impediatur: Sit autem attractio, in æqualibus ab utroque plano di&longs;tantiis ad eandem ip&longs;ius partem captis, ubique eadem: dico quod &longs;inus incidentiæ in planum alterutrum erit ad &longs;inum emergentiæ ex plano altero in ratione data.

~~Cas. 1. Sunto Aa, Bb~~

~~plana duo parallela.~~ Incidat corpus in planum prius Aa &longs;ecundum lineam GH, ac toto &longs;uo per &longs;patium intermedium tran&longs;itu attrahatur vel impellatur ver&longs;us Medium incidentiæ, eaque actione de&longs;cribat lineam curvam HI, & emergat &longs;ecundum lineam IK. Ad planum emergentiæ Bb erigatur perpendiculum IM, occurrens tum lineæ incidentiæ GH productæ in M,tum plano incidentiæ Aa in R; & linea emergentiæ KI producta occurrat HM in L. Centro L intervallo LI de&longs;cribatur Circulus, &longs;ecans tam HM in P & Q, quam MI productam in N, & primo &longs;i attractio vel impul&longs;us ponatur uniformis, erit (ex demon&longs;tratis Galilæi) curva HI Parabola, cujus hæc e&longs;t proprietas, ut rectangulum &longs;ub dato latere recto & linea IM æquale &longs;it HM quadrato; &longs;ed & linea HM bi&longs;ecabitur in L. Unde &longs;i ad MI demittatur perpendiculum LO, æ
quales erunt MO, OR; & additis æqualibus ON, OI, fient totæ æquales MN, IR. Proinde cum IR detur, datur etiam MN; e&longs;tque rectangulum NMI ad rectangulum &longs;ub latere recto & IM, hoc e&longs;t, ad HMq,in data ratione. Sed rectangulum NMI æquale e&longs;t rectangulo PMQ, id e&longs;t, differentiæ quadratorum MLq, & PLq &longs;eu LIq; & HMq datam rationem habet ad &longs;ui ip&longs;ius quartam partem MLq: ergo datur ratio MLq-LIq ad MLq, & divi&longs;im, ratio LIq ad MLq, & ratio dimidiata LI ad ML. Sed in omni triangulo LMI, &longs;inus angulorum &longs;unt proportionales lateribus oppo&longs;itis. ~~Ergo datur ratio &longs;inus anguli incidentiæ LMR ad &longs;inum anguli emergentiæ LIR. que E. D.~~

~~DE MOTU CORPORUM~~

~~Cas. 2. Tran&longs;eat jam corpus &longs;ucce&longs;&longs;ive per &longs;patia plura parallelis planis terminata, AabB, BbcC, &c.~~ & agitetur vi quæ &longs;it in &longs;ingulis &longs;eparatim uniformis, at in diver&longs;is diver&longs;a; & per jam de mon&longs;trata, &longs;inus incidentiæ in planum primum Aa erit ad &longs;inum emergentiæ ex plano &longs;ecundo Bb, in data ratione; & hic &longs;inus, qui e&longs;t &longs;inus incidentiæ in planum &longs;ecundum Bb, erit ad &longs;inum emergentiæ ex plano tertio Cc, in data ratione; & hic &longs;inus ad &longs;inum emergentiæ ex plano quarto Dd, in data ratione; & &longs;ic in infinitum: & ex æquo, &longs;inus incidentiæ in planum primum ad &longs;inum emergentiæ ex plano ultimo in data ratione. Minuantur jam planorum intervalla & augeatur numerus in infinitum, eo ut attractionis vel impul&longs;us actio, &longs;ecundum legem quamcunque a&longs;&longs;ignatam, continua reddatur; & ratio &longs;inus incidentiæ in planum primum ad &longs;inum emergentiæ ex plano ultimo, &longs;emper data exi&longs;tens, etiamnum dabitur. que E. D.

~~LIBER PRIMUS.~~

~~PROPOSITIO XCV. THEOREMA XLIX.~~

~~Ii&longs;dem po&longs;itis; dico quod velocitas corporis ante incidentiam ect ad ejus velocitatem poct emergentiam, ut &longs;inus emergentiæ ad &longs;inum incidentiæ.~~

Capiantur AH, Id æquales, & erigantur perpendicula AG, dKoccurrentia lineis incidentiæ & emergentiæ GH, IK, in G & K.In GH capiatur TH æqualis IK, & ad planum Aa demittatur normaliter Tv. Et (per Legum Corol. ~~2) di&longs;tinguatur motus corporis in duos, unum planis Aa, Bb, Cc, &c.~~ ~~perpendicularem, alterum ii&longs;dem parallelum.~~ Vis attractionis vel impul&longs;us, agendo &longs;ecundum lineas perpendiculares, nil mutat motum &longs;ecundum parallelas, & propterea corpus hoc motu conficiet æqualibus temporibus æqualia illa &longs;ecundum parallelas intervalla, quæ &longs;unt inter lineam AG & punctum H, interque punctum I & lineam dK; hoc e&longs;t, æqualibus temporibus de&longs;cribet lineas GH, IK. Proinde velocitas ante incidentiam e&longs;t ad velocitatem po&longs;t emergentiam, ut GH ad IK vel TH, id e&longs;t, ut AH vel Id ad vH, hoc e&longs;t (re&longs;pectu radii TH vel IK) ut &longs;inus emergentiæ ad &longs;inum incidentiæ. que E. D.

~~DE MOTU CORPORUM~~

~~PROPOSITIO XCVI. THEOREMA L.~~

Ii&longs;dem po&longs;itis & quod motus ante incidentiam velocior &longs;it quam po&longs;tea: dico quod corpus, inclinando lineam incidentiæ, reflectetur tandem, & angulus reflexionis fiet æqualis angulo incidentiæ.

~~Nam concipe corpus inter parallela plana Aa, Bb, Cc, &c.~~ ~~de&longs;cribere arcus Parabolicos, ut &longs;upra; &longs;intque arcus illi HP, PQ, QR, &c.~~ Et &longs;it ea lineæ incidentiæ GH obliquitas ad planum primum Aa, ut &longs;inus incidentiæ &longs;it ad radium circuli, cujus e&longs;t &longs;inus, in ea ratione quam habet idem &longs;inus incidentiæ ad &longs;inum emergentiæ ex plano Dd, in &longs;patium DdeE: & ob &longs;inum emergentiæ jam factum æqualem radio, angulus emergentiæ erit rectus, adeoque linea emergentiæ coincidet cum plano Dd. Perveniat corpus ad hoc planum in puncto R; & quoniam linea emergentiæ coincidit cum eodem

plano, per&longs;picuum e&longs;t quod corpus non pote&longs;t ultra pergere ver&longs;us planum Ee. Sed nec pote&longs;t idem pergere in linea emergentiæ Rd, propterea quod perpetuo attrahitur vel impellitur ver&longs;us Medium incidentiæ. Revertetur itaque inter plana Cc, Dd, de&longs;cribendo arcum Parabolæ QRq, cujus vertex principalis (juxta demon&longs;trata Galilæi) e&longs;t in R; &longs;ecabit planum Cc in eodem angulo in q, ac prius in Q; dein pergendo in arcubus parabolicis qp, ph, &c. ~~arcubus prioribus QP, PH &longs;imilibus & æqualibus, &longs;ecabit reliqua plana in ii&longs;dem angulis in p, h, &c.~~ ~~ac prius in P, H, &c.~~ ~~emergetque tandem eadem obliquitate in h, qua incidit in H. Concipe jam planorum Aa, Bb, Cc, Dd, Ee, &c.~~ intervalla in infinitum minui & numerum augeri, eo ut actio attractionis vel impul&longs;us &longs;ecundum legem quamcunque a&longs;&longs;ignatam continua reddatur; & angulus cmergentiæ &longs;emper angulo incidentiæ æqualis exi&longs;tens, eidem etiamnum manebit æqualis. que E. D.

~~LIBER PRIMUS.~~

~~Scholium.~~

Harum attractionum haud multum di&longs;&longs;imiles &longs;unt Lucis reflexiones & refractiones, factæ &longs;ecundum datam Secantium rationem, ut invenit Snellius, & per con&longs;equens &longs;ecundum datam Sinuum rationem, ut expo&longs;uit Carte&longs;ius. Namque Lucem &longs;ucce&longs;&longs;ive propagari & &longs;patio qua&longs;i &longs;eptem vel octo minutorum primorum a Sole ad Terram venire, jam con&longs;tat per Phænomena Satellitum Jovis, Ob&longs;ervationibus diver&longs;orum A&longs;tronomorum confirmata. Radii autem in aere exi&longs;tentes (uti dudum Grimaldus, luce per foramen in tenebro&longs;um cubiculum admi&longs;&longs;a, invenit, & ip&longs;e quoque expertus &longs;um) in tran&longs;itu &longs;uo prope corporum vel opacorum vel per&longs;picuorum angulos (quales &longs;unt nummorum ex auro, argento & ære cu&longs;orum termini rectanguli circulares, & cultrorum, lapidum aut fractorum vitrorum acies) incurvantur circum corpora, qua&longs;i attracti in eadem; & ex his radiis, qui in tran&longs;itu illo propius accedunt ad corpora incurvantur magis, qua

~~&longs;i magis attracti, ut ip&longs;e etiam diligenter ob&longs;ervavi.~~ In figura de&longs;ignat s aciem cultri vel cunei cuju&longs;vis AsB; & gowog, fnunf, emtme, dlsld, &longs;unt radii, arcubus owo, nun, mtm, lsl ver&longs;us cultrum incurvati; idque magis vel minus pro di&longs;tantia eorum a cultro. ~~Cum autem talis incurvatio radiorum fiat in aere extra cultrum, debebunt etiam radii, qui incidunt in cultrum, prius incurvari in aere quam cultrum attingunt.~~ ~~Et par e&longs;t ratio incidentium in vitrum.~~ Fit igitur refractio, non in puncto incidentiæ, &longs;ed paulatim per continuam incurvationem radiorum, factam partim in aere antequam attingunt vitrum, partim (ni fallor) in vitro, po&longs;tquam illud ingre&longs;&longs;i &longs;unt: uti in radiis ckzkc, biyib, ahxha incidentibus ad r, q, p, & inter k & z, i & y, h & x incurvatis, delineatum e&longs;t. Igitur ob analogiam quæ e&longs;t inter prop-gationem radiorum lucis & progre&longs;&longs;um corporum, vi&longs;um e&longs;t Propo&longs;itiones &longs;equentes in u&longs;us Opticos &longs;ubjungere; interea de natura radiorum (utrum &longs;int corpora necne) nihil omnino di&longs;putans, &longs;ed Trajectorias corporum Trajectoriis radiorum per&longs;imiles &longs;olummodo determinans.

~~DE MOTU CORPORUM~~

~~PROPOSITIO XCVII. PROBLEMA XLVII.~~

Po&longs;ito quod &longs;inus incidentiæ in &longs;uperficiem aliquam &longs;it ad &longs;inum emergentiæ in data ratione, quodque incurvatio viæ corporum juxta &longs;uperficiem illam fiat in &longs;patio brevi&longs;&longs;imo, quod ut punctum con&longs;iderari po&longs;&longs;it; determinare &longs;uperficiem quæ corpu&longs;cula omnia de loco dato &longs;ucce&longs;&longs;ive manantia convergere faciat ad alium locum datum.

Sit A locus a quo corpu&longs;cula divergunt; B locus in quem convergere debent; CDE curva linea quæ circa axem AB revoluta de&longs;cribat &longs;uperficiem quæ&longs;itam; D, E curvæ illius puncta duo quævis; & EF, EG perpendicula in corporis vias AD, DB demi&longs;&longs;a. ~~Accedat punctum D ad punctum E; & lineæ DF qua AD augetur, ad lineam DG qua DB diminuitur, ratio ultima erit eadem quæ &longs;inus incidentiæ ad &longs;inum emergentiæ.~~ ~~Datur ergo ratio~~

incrementi lineæ AD ad decrementum lineæ DB; & propterea &longs;i in axe AB &longs;umatur ubivis punctum C, per quod curva CDEtran&longs;ire debet, & capiatur ip&longs;ius AC incrementum CM, ad ip&longs;ius BC decrementum CN in data illa ratione; centri&longs;que A, B, & intervallis AM, BN de&longs;cribantur circuli duo &longs;e mutuo &longs;ecantes in D: punctum illud D tanget curvam quæ&longs;itam CDE, eandemque ubivis tangendo determinabit. que E. I.

Corol. 1. Faciendo autem ut punctum A vel B nunc abeat in infinitum, nunc migret ad alteras partes puncti C, habebuntur Figuræ illæ omnes quas Carte&longs;ius in Optica & Geometria ad Refractiones expo&longs;uit. ~~Quarum inventionem cum Carte&longs;ius maximi fecerit & &longs;tudio&longs;e celaverit, vi&longs;um fuit hac propo&longs;itione exponere.~~

~~Corol. 2. Si corpus in &longs;uperficiem quamvis CD, &longs;ecundum lineam rectam AD lege quavis ductam incidens, emergat &longs;ecundum aliam quamvis rectam DK,~~

& a puncto C duci intelligantur Lineæ curvæ CP, CQ ip&longs;is AD, DK&longs;emper perpendiculares: erunt incrementa linearum PD, QD, atque adeo lineæ ip&longs;æ PD, QD,incrementis i&longs;tis genitæ, ut &longs;inus incidentiæ & emergentiæ ad invicem: & contra.

~~LIBER PRIMUS.~~

~~PROPOSITIO XCVIII. PROBLEMA XLVIII.~~

Ii&longs;dem po&longs;itis, & circa axem AB de&longs;cripta &longs;uperficie quacunque attractiva CD, regulari vel irregulari, per quam corpora de loco dato A exeuntia tran&longs;ire debent: invenire &longs;uperficiem &longs;ecundam attractivam EF, quæ corpora illa ad locum datum B convergere faciat.

~~Juncta AB &longs;ecet &longs;uperficiem primam in C & &longs;ecundam in E,puncto D utcunque a&longs;&longs;umpto.~~ Et po&longs;ito &longs;inu incidentiæ in &longs;uperficiem primam ad &longs;inum emergentiæ ex eadem, & &longs;inu emergentiæ e &longs;uperficie &longs;ecunda ad &longs;inum incidentiæ in eandem, ut quantitas aliqua data M ad aliam datam N; produc tum AB ad G ut &longs;it BGad CE ut M-N ad N, tum AD ad H ut &longs;it AH æqualis AG, tum etiam DF ad K ut &longs;it DK ad DH ut N ad M. Junge KB, & centro D intervallo DH de&longs;cribe circulum occurrentem KB productæ in L, ip&longs;ique DL parallelam age BF: & punctum F tanget Lineam EF, quæ circa axem AB revoluta de&longs;cribet &longs;uperficiem quæ&longs;itam. que E. F.

Nam concipe Lineas CP, CQ ip&longs;is AD, DF re&longs;pective, & Lineas ER, ES ip&longs;is FB, FD ubique perpendiculares e&longs;&longs;e, adeoque QS ip&longs;i CE &longs;emper æqualem; & erit (per Corol. ~~2. Prop.~~ XCVII) PD ad QD ut M ad N, adeoque ut DL ad DK vel FB ad FK; & divi&longs;im ut DL-FP &longs;eu PH-PD-FB ad FD &longs;eu FQ-QD; & compo&longs;ite ut PH-FB ad FQ, id e&longs;t (ob æquales PH& CG, QS & CE)

~~CE+BG-FR ad CE-FS. Verum (ob proportionales BG ad CE & M-N ad N) e&longs;t etiam CE+BG ad CE ut M ad N: adeoque divi&longs;im FR ad FS ut M ad N, & propterea per Corol.~~ ~~2. Prop.~~ ~~XCVII, &longs;uperficies EF cogit corpus, in ip&longs;am &longs;ecundum lineam DF incidens, pergere in linea FRad locum B. que E. D.~~

~~DE MOTU CORPORUM~~

~~Scholium.~~

~~Eadem methodo pergere liceret ad &longs;uperficies tres vel plures.~~ ~~Ad u&longs;us autem Opticos maxime accommodatæ &longs;unt figuræ Sphæricæ.~~ Si Per&longs;picillorum vitra Objectiva ex vitris duobus Sphærice figuratis & Aquam inter &longs;e claudentibus conflentur; fieri pote&longs;t ut a refractionibus Aquæ errores refractionum, quæ fiunt in vitrorum &longs;uperficiebus extremis, &longs;atis accurate corrigantur. Talia autem vitra Objectiva vitris Ellipticis & Hyperbolicis præferenda &longs;unt, non &longs;olum quod facilius & accuratius formari po&longs;&longs;int, &longs;ed etiam quod Penicillos radiorum extra axem vitri &longs;itos accurativs refringant. ~~Verum tamen diver&longs;a diver&longs;orum radiorum Refrangibilitas impedimento e&longs;t, quo minus Optica per Figuras vel Sphæricas vel alias qua&longs;cunque perfici po&longs;&longs;it.~~ ~~Ni&longs;i corrigi po&longs;&longs;int errores illinc oriundi, labor omnis in cæteris corrigendis imperite collocabitur.~~

~~LIBER SECUNDUS.~~

~~DE MOTU CORPORUM LIBER SECUNDUS.~~

~~SECTIO I.~~

~~De Motu Corporum quibus re&longs;i&longs;titur in ratione Velocitatis.~~

~~PROPOSITIO I. THEOREMA I.~~

~~Corporis, cui re&longs;i&longs;titur in ratione velocitatis, motus ex re&longs;i&longs;tentia ami&longs;&longs;us ect ut &longs;patium movendo confectum.~~

NAm cum motus &longs;ingulis temporis particulis æqualibus ami&longs;&longs;us &longs;it ut velocitas, hoc e&longs;t, ut itineris confecti particula: erit, componendo, motus toto tempore ami&longs;&longs;us ut iter totum. q.E.D.

Corol. Igitur &longs;i corpus, gravitate omni de&longs;titutum, in &longs;patiis liberis &longs;ola vi in&longs;ita moveatur; ac detur tum motus totus &longs;ub initio, tum etiam motus reliquus po&longs;t &longs;patium aliquod confectum: dabitur &longs;patium totum quod corpus infinito tempore de&longs;cribere pote&longs;t. ~~Erit enim &longs;patium illud ad &longs;patium jam de&longs;criptum, ut motus totus &longs;ub initio ad motus illius partem ami&longs;&longs;am.~~

~~LEMMA I.~~

~~Quantitates differentiis &longs;uis proportionales, &longs;unt continue propor tionales.~~

~~Sit A ad A-B ut B ad B-C & C ad C-D, &c.~~ ~~& dividendo fiet A ad B ut B ad C & C ad D, &c. q.E.D.~~

~~DE MOTU CORPORUM~~

~~PROPOSITIO II. THEOREMA II.~~

Si Corpori re&longs;i&longs;titur in ratione velocitatis, & idem &longs;ola vi in&longs;ita per Medium &longs;imilare moveatur, &longs;umantur autem tempora æqualia: velocitates in principiis &longs;ingulorum temporum &longs;unt in progre&longs;&longs;ione Geometrica, & &longs;patia &longs;ingulis temporibus de&longs;cripta &longs;unt ut velocitates.

Cas. 1. Dividatur tempus in particulas æquales; & &longs;i ip&longs;is particularum initiis agat vis re&longs;i&longs;tentiæ impul&longs;o unico, quæ &longs;it ut velocitas: erit decrementum velocitatis &longs;ingulis temporis particulis ut eadem velocitas. ~~Sunt ergo velocitates differentiis &longs;uis proportionales, & propterea (per Lem.~~ ~~I. Lib.~~ ~~II.) continue proportionales.~~ Proinde &longs;i ex æquali particularum numero componantur tempora quælibet æqualia, erunt velocitates ip&longs;is temporum initiis, ut termini in progre&longs;&longs;ione continua, qui per &longs;altum capiuntur, omi&longs;&longs;o pa&longs;&longs;im æquali terminorum intermediorum numero. ~~Componuntur autem horum terminorum rationes ex æqualibus rationibus terminorum intermediorum æqualiter repetiti & propterea &longs;unt æquales.~~ ~~Igitur velocitates, his terminis proportionales, &longs;unt in progre&longs;&longs;ione Geometrica.~~ Minuantur jam æquales illæ temporum particulæ, & augeatur earum numerus in infinitum, eo ut re&longs;i&longs;tentiæ impul&longs;us reddatur continuus; & velocitates in principiis æqualium temporum, &longs;emper continue proportionales, erunt in hoc etiam ca&longs;u continue proportionales. q.E.D.

Cas. 2. Et divi&longs;im velocitatum differentiæ, hoc e&longs;t, earum partes &longs;ingulis temporibus ami&longs;&longs;æ, &longs;unt ut totæ: Spatia autem &longs;ingulis temporibus de&longs;cripta &longs;unt ut velocitatum partes ami&longs;&longs;æ, (per Prop. I. ~~Lib II.) & propterea etiam ut totæ. que E. D.~~

Corol. Hinc &longs;i A&longs;ymptotis rectangulis ADC, CH de&longs;cribatur Hyperbola BG, &longs;intque AB, DG ad A&longs;ymptoton AC perpendiculares, & exponatur tum corporis velocitas tum re&longs;i&longs;tentia Medii, ip&longs;o motus initio, per lineam quam

vis datam AC, elap&longs;o autem tempore aliquo per lineam indefinitam DC: exponi pote&longs;t tempus per aream ABGD, & &longs;patium eo tempore de&longs;criptum per lineam AD. Nam &longs;i area illa per motum puncti D augeatur uniformiter ad modum tempo- ris, decre&longs;cet recta DC in ratione Geometrica ad modum veloci tatis, & partes rectæ AC æqualibus temporibus de&longs;criptæ decre&longs;cent in eadem ratione.

~~LIBER SECUNDUS.~~

~~PROPOSITIO III. PROBLEMA I.~~

~~Corporis, cui dum in Medio &longs;imilari recta a&longs;cendit vel de&longs;cendit, re&longs;i&longs;titur in ratione velocitatis, quodque ab uniformi gravitate urgetur, definire motum.~~

~~Corpore a&longs;cendente, ex~~

~~ponatur gravitas per datum quodvis rectangulum BC, & re&longs;i&longs;tentia Medii initio a&longs;cen&longs;us per rectangulum BD&longs;umptum ad contrarias partes.~~ A&longs;ymptotis rectangulis AC, CH, per punctum B de&longs;cribatur Hyperbola &longs;ecans perpendicula DE, de in G, g; & corpus a&longs;cendendo, tempore DGgd, de&longs;cribet &longs;patium EGge, tempore DGBA &longs;patium a&longs;cen&longs;us totius EGB; tempore AB2G2D&longs;patium de&longs;cen&longs;us BF2G, atque tempore 2D2G2g2d &longs;patium de&longs;cen&longs;us 2GF2e2g: & velocitates corporis (re&longs;i&longs;tentiæ Medii proportionales) in horum temporum periodis erunt ABED, ABed, nulla, ABF2D, AB2e2d re&longs;pective; atque maxima velocitas, quam corpus de&longs;cendendo pote&longs;t acquirere, erit BC.

~~Re&longs;olvatur enim rectan~~

~~gulum AH in rectangula innumera Ak, Kl, Lm, Mn,&c.~~ ~~quæ &longs;int ut incrementa velocitatum æqualibus totidem temporibus facta; & erunt nihil, Ak, Al, Am, An,&c.~~ ~~ut velocitates totæ, atque adeo (per Hypothe&longs;in) ut re&longs;i&longs;tentiæ Medii principio &longs;ingulorum temporum æqualium.~~ Fiat AC ad AK vel ABHC ad ABkK, ut vis gravitatis ad re&longs;i&longs;tentiam in principio temporis &longs;ecundi, deque vi gravi- tatis &longs;ubducantur re&longs;i&longs;tentiæ, & manebunt ABHC, KkHC, LlHC, NnHC, &c. ut vires ab&longs;olutæ quibus corpus in principio &longs;ingulorum temporum urgetur, atque adeo (per motus Legem 11) ut incrementa velocitatum, id e&longs;t, ut rectangula Ak, Kl, Lm, Mn, &c; & propterea (per Lem. ~~I. Lib.~~ ~~II) in progre&longs;&longs;ione Geometrica.~~ ~~Quare &longs;i rectæ Kk, Ll, Mm, Nn, &c.~~ ~~productæ occurrant Hyperbolæ in q, r, s, t, &c.~~ ~~erunt areæ ABqK, KqrL, LrsM, MstN, &c. æquales, adeoque tum temporibus tum viribus gravitatis &longs;emper æqualibus analogæ.~~ ~~E&longs;t autem area ABqK (per Corol.~~ ~~3. Lem.~~ ~~VII, & Lem.~~ ~~VIII, Lib.~~ ~~I) ad aream Bkq ut Kq ad 1/2 kq &longs;eu AC ad 1/2 AK,hoc e&longs;t, ut vis gravitatis ad re&longs;i&longs;tentiam in medio temporis primi.~~ Et &longs;imili argumento areæ
qKLr, rLMs, sMNt, &c. &longs;unt ad areas qklr, rlms, smnt, &c. ~~ut vires gravitatis ad re&longs;i&longs;tentias in medio temporis &longs;ecundi, tertii, quarti, &c.~~ ~~Proinde cum areæ æquales BAKq, qKLr, rLMs, sMNt, &c.~~ ~~&longs;int viribus gravitatis analogæ, erunt areæ Bkq, qklr, rlms, smnt, &c.~~ ~~re&longs;i&longs;tentiis in mediis &longs;ingulorum temporum, hoc e&longs;t (per Hypothe&longs;in) velocitatibus, atque adeo de&longs;criptis &longs;patiis analogæ.~~ ~~Sumantur analogarum &longs;ummæ, & erunt areæ Bkq, Blr, Bms, Bnt,&c.~~ ~~&longs;patiis totis de&longs;criptis analogæ; necnon areæ ABqK, ABrL, ABsM, ABtN, &c.~~ ~~temporibus.~~ ~~Corpus igitur inter de&longs;cendendum, tempore quovis ABrL, de&longs;cribit &longs;patium Blr, & tempore LrtN &longs;patium rlnt.~~ ~~q.E.D. Et &longs;imilis e&longs;t demon&longs;tratio motus expo&longs;iti in a&longs;cen&longs;u. q.E.D.~~

~~DE MOTU CORPORUM~~

Corol. 1. Igitur velocitas maxima, quam corpus cadendo pote&longs;t acquirere, e&longs;t ad velocitatem dato quovis tempore acqui&longs;itam, uvis data gravitatis qua perpetuo urgetur, ad vim re&longs;i&longs;tentiæ qua ifine temporis illius impeditur.

Corol. 2. Tempore autem aucto in progre&longs;&longs;ione Arithmetica, &longs;ummvelocitatis illius maximæ ac velocitatis in a&longs;cen&longs;u (atque etiam earun dem differentia in de&longs;cen&longs;u) decre&longs;cit in progre&longs;&longs;ione Geometrica.

~~Corol. 3. Sed & differentiæ &longs;patiorum, quæ in æqualibus tempo rum differentiis de&longs;cribuntur, decre&longs;cunt in eadem progre&longs;&longs;ion Geometrica.~~

Corol. 4. Spatium vero a corpore de&longs;criptum differentia e&longs;t duo rum &longs;patiorum, quorum alterum e&longs;t ut tempus &longs;umptum ab initio de&longs;cen&longs;us, & alterum ut velocitas, quæ etiam ip&longs;o de&longs;cen&longs;us initio æquantur inter &longs;e.

~~LIBER SECUNDUS.~~

~~PROPOSITIO IV. PROBLEMA II.~~

Po&longs;ito quod vis gravitatis in Medio aliquo &longs;imilari uniformis &longs;it, ac tendat perpendiculariter ad planum Horizontis; definire motum Projectilis in eodem, re&longs;i&longs;tentiam velocitati proportionalem patientis.

~~Eloco quovis D egrediatur Pro~~

~~jectile &longs;ecundum lineam quamvis rectam DP, & per longitudinem DP exponatur eju&longs;dem velocitas &longs;ub initio motus.~~ A puncto P ad lineam Horizontalem DC demittatur perpendiculum PC, & &longs;ecetur DC in Aut &longs;it DA ad AC ut re&longs;i&longs;tentia Medii, ex motu in altitudinem &longs;ub initio orta, ad vim gravitatis; vel (quod perinde e&longs;t) ut &longs;it rectangulum &longs;ub DA & DPad rectangulum &longs;ub AC & CPut re&longs;i&longs;tentia tota &longs;ub initio motus ad vim gravitatis. A&longs;ymptotis DC, CP, de&longs;cribatur Hyperbola quævis GTBS &longs;ecans perpendicula DG, AB in G & B; & compleatur parallelogrammum DGKC, cujus latus GK &longs;ecet AB in que Capiatur linea N in ratione ad QB qua DC &longs;it ad CP; & ad rectæ DC punctum quodvis R erecto perpendiculo RT, quod Hyperbolæ in T, & rectis EH, GK, DPin I, t & V occurrat; in eo cape Vr æqualem (tGT/N), vel quod per- inde e&longs;t, cape Rr æqualem (GTIE/N); & Projectile tempore DRTGperveniet ad punctum r, de&longs;cribens curvam lineam DraF, quam punctum r &longs;emper tangit, perveniens autem ad maximam altitudinem a in perpendiculo AB, & po&longs;tea &longs;emper appropinquans ad A&longs;ymptoton PLC. E&longs;tque velocitas ejus in puncto quovis r ut Curvæ Tangens rL. que E. I.

~~DE MOTU CORPORUN~~

E&longs;t enim N ad QB ut DC ad CP &longs;eu DR ad RV, adeoque RVæqualis (DRXQB/N), & Rr (id e&longs;t RV-Vr &longs;eu (DRXQB-tGT/N)) æqualis (DRXAB-RDGT/N). Exponatur jam tempus per aream RDGT, & (per Legum

~~Corol.~~ ~~2.) di&longs;tinguatur motus corporis in duos, unum a&longs;cen&longs;us, alterum ad latus.~~ Et cum re&longs;i&longs;tentia &longs;it ut motus, di&longs;tinguetur etiam hæc in partes duas partibus motus proportionales & contrarias: ideoque longitudo, a motu ad latus de&longs;cripta, erit (per Prop. ~~11. hujus) ut linea DR, altitudo vero (per Prop.~~ 111. hujus) ut area DRXAB -RDGT, hoc e&longs;t, ut linea Rr.Ip&longs;o autem motus initio area RDGT æqualis e&longs;t rectangulo DRXAQ, ideoque linea illa Rr(&longs;eu (DRXAB-DRXAQ/N)) tunc e&longs;t ad DR ut AB-AQ&longs;eu QB ad N, id e&longs;t, ut CPad DC; atque adeo ut motus in altitudinem ad motum in longitudinem &longs;ub initio. Cum igitur Rr &longs;emper &longs;it ut altitudo, ac DR &longs;emper ut longitudo, atque Rr ad DR &longs;ub initio ut altitudo ad longitudinem: nece&longs;&longs;e e&longs;t ut Rr &longs;emper &longs;it ad DR ut altitudo ad longitudinem, & propterea ut corpus moveatur in linea DraF, quam punctum r perpetuo tangit. q.E.D.

Corol. 1. E&longs;t igitur Rr æqualis (DRXAB/N)-(RDGT/N), ideoque &longs;i producatur RT ad X ut &longs;it RX æqualis (DRXAB/N), (id e&longs;t, &longs;i compleatur parallelogrammum ACPY, jungatur DY &longs;ecans CPin Z, & producatur RT donec occurrat DY in X;) erit Xr æqualis (RDGT/N), & propterea tempori proportionalis.

~~LIBER SECUNDUS.~~

~~Corol. 2. Unde &longs;i capiantur innumeræ CR vel, quod perinde e&longs;t, innumeræ ZX, in progre&longs;&longs;ione Geometrica; erunt totidem Xr in progre&longs;&longs;ione Arithmetica.~~ ~~Et hinc Curva DraF per tabulam Logarithmorum facile delineatur.~~

Corol. 3. Si vertice D, diametro DE deor&longs;um producta, & Latere recto quod &longs;it ad 2DP ut re&longs;i&longs;tentia tota, ip&longs;o motus initio, ad vim gravitatis, Parabola con&longs;truatur: velocitas quacum corpus exire debet de loco D &longs;ecundum rectam DP, ut in Medio uniformi re&longs;i&longs;tente de&longs;cribat Curvam DraF, ea ip&longs;a erit quacum exire debet de eodem loco D, &longs;ecundum eandem rectam DP, ut in &longs;patio non re&longs;i&longs;tente de&longs;cribat Parabolam. Nam Latus rectum Parabolæ hujus, ip&longs;o motus initio, e&longs;t (DVquad./Vr) & Vre&longs;t (tGT/N) &longs;eu (DRXTt/2N). Recta autem quæ, &longs;i duceretur, Hyperbolam GTB tangeret in G, parallela e&longs;t ip&longs;i DK, ideoque Tt e&longs;t (CKXDR/DC) & N erat (QBXDC/CP). Et propterea Vr e&longs;t (DRqXCKXCP/2DCqXQB), id e&longs;t, (ob proportionales DR & DC, DV& DP) (DVqXCKXCP/2DPqXQB), & Latus rectum (DVquad./Vr) prodit (2DPqXQB/CKXCP), id e&longs;t (ob proportionales QB & CK, DA & AC) (2DPqXDA/ACXCP), adeoque ad 2 DP, ut DPXDA ad CPXAC; hoc e&longs;t, ut re&longs;i&longs;tentia ad gravitatem. q.E.D.

Corol. 4. Unde &longs;i corpus de loco quovis D, data cum velocitate, &longs;ecundum rectam quamvis po&longs;itione datam DP projiciatur; & re&longs;i&longs;tentia Medii ip&longs;o motus initio detur: inveniri pote&longs;t Curva DraF, quam corpus idem de&longs;cribet. ~~Nam ex data velocitate datur latus rectum Parabolæ, ut notum e&longs;t.~~ Et &longs;umendo 2DPad latus illud rectum, ut e&longs;t vis gravitatis ad vim re&longs;i&longs;tentiæ, datur DP. Dein &longs;ecando DCin A, ut &longs;it CPXAC ad DPXDA in eadem illa ratione gravitatis ad re&longs;i&longs;tentiam, dabitur punctum A. Et inde datur Curva DraF.

~~DE MOTU CORPORUM~~

~~Corol. 5. Et contra, &longs;i datur~~

Curva DraF, dabitur & velocitas corporis & re&longs;i&longs;tentia Medii in locis fingulis r. Nam ex data ratione CPXAC ad DPXDA, datur tum re&longs;i&longs;tentia Medii &longs;ub initio motus, tum latus rectum Parabolæ: & inde datur etiam velocitas &longs;ub initio motus. ~~Deinde ex longitudine tangentis rL, datur & huic proportionalis velocitas, & velocitati proportionalis re&longs;i&longs;tentia in loco quovis r.~~

Corol. 6. Cum autem longitudo DP &longs;it ad latus rectum Parabolæ ut gravitas ad re&longs;i&longs;tentiam in D; & ex aucta velocitate angeatur re&longs;i&longs;tentia in eadem ratione, at latus rectum Parabolæ augeatur in ratione illa duplicata: patet longitudinem 2DP augeri in ratione illa &longs;implici, adeoque velocitati &longs;emper proportionalem e&longs;&longs;e, neque ex angulo CDP mutato augeri vel minui, ni&longs;i mutetur quoque velocitas.

~~Corol. 7. Unde liquet methodus determinandi Curvam DraFex Phænomenis quamproxime, & inde colligendi re&longs;i&longs;tentiam & velocitatem quacum corpus projicitur.~~ Projiciantur corpora duo &longs;imilia & æqualia eadem cum veloci, de loco D, &longs;ecundum angulos diver&longs;os GDP, cD (mino&longs; literarum locis fubintellectis) & cogno&longs;ca loc F, f, abi incidunt in horizontale plum DC. Tum, a&longs;&longs;u unque longitudine pro Dvel Dp, fingatur qod re&longs; in D &longs;ad gravitem in ra- tione qualibet, & exponatur ratio illa per longitudinem quamvis SM. Deinde per computationem, ex longitudine illa a&longs;&longs;umpta DP, inveniantur longitudines DF, Df, ac de ratione (Ef/DF) per calculum inventa, auferatur ratio eadem

per expe inveta, & exptur differentia per perpendiculum MN. Idem fac iterum ac tertio, a&longs;&longs;umendo &longs;emper novam re&longs;i&longs;tentiæ ad gravitatem rationem SM, & colligendo novam differentiam MN. Ducantur autem differentiæ affirmativæ ad unam partem rectæ SM, & negativæ ad alteram; & per puncta N, N, N agatur ourva regularis NNN &longs;ecans rectam SMMM in X, & erit SXvera ratio re&longs;i&longs;tentiæ ad gravitatem, quam invenire oportuit. Ex hac ratione colligenda e&longs;t longitudo DF per calculum; & longitudo quæ &longs;it ad a&longs;&longs;umptam longitudinem DP, at longitudo DFper experimentum cognita ad longitudinem DF modo inventam, erit vera longitudo DP. Qua inventa, habetur tum Curva linea DraF quam corpus de&longs;cribit, tum corporis velocitas & re&longs;i&longs;tentia in locis &longs;ingulis.

~~LIBER SECUNDUS.~~

~~Scholium.~~

Cæterum, re&longs;i&longs;tentiam corporum e&longs;&longs;e in ratione velocitatis Hypothe&longs;is e&longs;t magis Mathematica quam Naturalis Obtinet hæc ratio quamproxime ubi corpora in Mediis rigore aliquo præditis tardi&longs;&longs;ime moventur. ~~In Mediis antem quæ rigore omni vacant re&longs;i&longs;tentiæ corporum &longs;unt in duplicata ratione velocitatum.~~ Etenim actione corporis velocioris communicatur eidem Medii quantitati, tempore minore, motus major in ratione majoris velocitatis; adeoque tempore æquali (ob majorem Medii quantitatem perturbatam) communicatur motus in duplicata ratione major; e&longs;t que re&longs;i&longs;tentia (per motus Legem II & III) ut motus communicatus. ~~Videamus igitur quades oriantur motus ex hac lege Re&longs;i&longs;tentiæ.~~

~~DE MOTU CORPORUM~~

~~SECTIO II.~~

~~De motu Corporum quibus re&longs;i&longs;titur in duplicata ratione Velocitatum.~~

~~PROPOSITIO V. THEOREMA III.~~

Si Corpori re&longs;i&longs;iitur in velocitatis ratione duplicata, & idem &longs;ola vi in&longs;ita per Medium &longs;imilare movetur; tempora vero &longs;umantur in progre&longs;&longs;ione Geometrica a minoribus terminis ad majores pergente: dico quod velocitates initio &longs;ingulorum temporum &longs;unt in eadem progre&longs;&longs;ione Geometrica inver&longs;e, & quod &longs;patia &longs;unt æqualia quæ &longs;ingulis temporibus de&longs;cribuntur.

~~Nam quoniam quadrato velocita~~

tis proportionalis e&longs;t re&longs;i&longs;tentia Medii, & re&longs;i&longs;tentiæ proportionale e&longs;t decrementum velocitatis; &longs;i tempus in particulas innumeras æquales dividatur, quadrata velocitatum &longs;ingulis temporum initiis erunt velocitatum earundem differentiis proportionalia. ~~Sunto temporis particulæ illæ AK, KL, LM, &c.~~ ~~in recta CD &longs;umptæ, & erigantur perpendicula AB, Kk, Ll, Mm, &c.~~ ~~Hyperbolæ BklmG,centro C A&longs;ymptotis rectangulis CD, CH de&longs;criptæ, occurrentia in B, k, t, m, &c.~~ & erit AB ad Kk ut CK ad CA, & divi&longs;im AB-Kk ad Kk ut AK ad CA, & vici&longs;&longs;im AB-Kk ad AKut Kk ad CA, adeoque ut ABXKk ad ABXCA. Unde, cum AK & ABXCA dentur, erit AB-Kk ut ABXKk; & ultimo, ubi coeunt AB & Kk, ut ABque Et &longs;imili argumento erunt Kk-Ll, Ll-Mm, &c. ~~ut Kkq, Llq, &c.~~ Linearum igitur AB, Kk, Ll, Mm quadrata &longs;unt ut earundem differentiæ; & idcirco cum quadrata ve locitatum fuerint etiam ut ip&longs;arum differentiæ, &longs;imilis erit ambarum progre&longs;&longs;io. Quo demon&longs;trato, con&longs;equens e&longs;t etiam ut areæ his lineis de&longs;criptæ &longs;int in progre&longs;&longs;ione con&longs;imili cum &longs;patiis quæ velocitatibus de&longs;cribuntur. Ergo &longs;i velocitas initio primi temporis AK exponatur per lineam AB, & velocitas initio &longs;ecundi KLper lineam Kk, & longitudo primo tempore de&longs;cripta per aream AKkB; velocitates omnes &longs;ub&longs;equentes exponentur per lineas &longs;ub&longs;equentes Ll, Mm, &c. ~~& longitudines de&longs;criptæ per areas Kl, Lm, &c.~~ Et compo&longs;ite, &longs;i tempus totum exponatur per &longs;ummam partium &longs;uarum AM, longitudo tota de&longs;cripta exponetur per &longs;ummam partium &longs;uarum AMmB. Concipe jam tempus AM ita dividi in partes AK, KL, LM, &c. ~~ut &longs;int CA, CK, CL, CM,&c.~~ ~~in progre&longs;&longs;ione Geometrica; & erunt partes illæ in eadem progre&longs;&longs;ione, & velocitates AB, Kk, Ll, Mm, &c.~~ ~~in progre&longs;&longs;ione eadem inver&longs;a, atque &longs;patia de&longs;cripta Ak, Kl, Lm, &c.~~ ~~æqualia. q.E.D.~~

~~LIBER SECUNDUS.~~

Corol. 1. Pater ergo quod, &longs;i tempus exponatur per A&longs;ymptoti partem quamvis AD, & velocitas in principio temporis per ordinatim applicatam AB; velocitas in fine temporis exponetur per ordinatam DG, & &longs;patium totum de&longs;criptum per aream Hyperbolicam adjacentem ABGD; necnon &longs;patium quod corpus aliquod eodem tempore AD, velocitate prima AB, in Medio non re&longs;i&longs;tente de&longs;cribere po&longs;&longs;et, per rectangulum ABXAD.

Corol. 2. Unde datur &longs;patium in Medio re&longs;i&longs;tente de&longs;criptum, capiendo illud ad &longs;patium quod velocitate uniformi AB in medio non re&longs;i&longs;tente &longs;imul de&longs;cribi po&longs;&longs;et, ut e&longs;t area Hyperbolica ABGDad rectangulum ABXAD.

Corol. 3. Datur etiam re&longs;i&longs;tentia Medii, &longs;tatuendo eam ip&longs;o motus initio æqualem e&longs;&longs;e vi uniformi centripetæ, quæ in cadente corpore, tempore AC, in Medio non re&longs;i&longs;tente, generare po&longs;&longs;et velocitatem AB. Nam &longs;i ducatur BT quæ tangat Hyperbolam in B,& occurrat A&longs;ymptoto in T; recta AT æqualis erit ip&longs;i AC, & tempus exponet quo re&longs;i&longs;tentia prima uniformiter continuata tollere po&longs;&longs;et velocitatem totam AB.

~~Corol 4. Et inde datur etiam proportio hujus re&longs;i&longs;tentiæ ad vim gravitatis, aliamve quamvis datam vim centripetam.~~

Corol. 5. Et vicever&longs;a, &longs;i datur proportio re&longs;i&longs;tentiæ ad datam quamvis vim centripetam; datur tempus AC, quo vis centripeta re&longs;i&longs;tentiæ æqualis generare po&longs;&longs;it velocitatem quamvis AB; & in- de datur punctum B per quod Hyperbola, A&longs;ymptoris CH, CD,de&longs;cribi debet; ut & &longs;patium ABGD, quod corpus incipiende motum &longs;uum cum vcitate illa AB, tempore quovis AD, in Medio &longs;imilari re&longs;i&longs;tente de&longs;cribere pote&longs;t.

~~DE MOTU CORPORUM~~

~~PROPOSITIO VI. THEOREMA IV.~~

Corpora Spherica homogemea & æqualia, re&longs;i&longs;tentiis in duplicata ratione velocitatum impodita, & &longs;olis viribus in&longs;itis incitata, temporibus quæ &longs;unt reciproce ut velocitates &longs;ub initio, de&longs;cribunt &longs;emper æqualia &longs;patia, & annittunt partes velocitatuns propertionales totis.

~~A&longs;ymptotis rectangulis CD,~~

CH de&longs;cripta Hyperbola quavis BbEe &longs;ecante perpendicula AB, ab, DE, de, in B, b, E, e,exponantur velocitates initiales per perpendicula AB, DE, & tempora per lineas Aa, Dd. E&longs;t ergo ut Aa ad Dd ita (per Hypothe&longs;in) DEad AB, & ita (ex natura Hyperbolæ) CA ad CD; & componendo, ita Ca ad Cd. Ergo areæ ABba, DEed, hoc e&longs;t, &longs;patia de&longs;cripta æquamtur inter &longs;e, & velocitates primæ AB, DE &longs;unt ultimis ab, de, & propterea (dividendo) partibus etiam &longs;uis ami&longs;&longs;is AB-ab, DE-de proportionales. q.E.D.

~~PROPOSITIO VII. THEOREMA V.~~

Corpora Sphærica quibus re&longs;i&longs;titur in duplicata ratione velocitatum, temporibus quæ &longs;unt ut motus primi directe & re&longs;i&longs;tentiæ primæ inver&longs;e, ainittent partes motuum proportionales totis, & &longs;patia de&longs;cribent temporibus i&longs;tis in velocitates primas ductis proportionalia.

~~Namque motuum partes ami&longs;&longs;æ &longs;unt ut re&longs;i&longs;tentiæ & tempora conjunctim.~~ ~~Igitur ut partes illæ &longs;int totis proportionales, debe bit re&longs;i&longs;tentia & tempus conjunctim e&longs;&longs;e ut motus.~~ ~~Proinde tempus erit ut motus directe & re&longs;i&longs;tentia inver&longs;e.~~ ~~Quare temporam particulis in ea ratione &longs;umptis, corpora amittem &longs;emper particulas motuum proportionales totis, adeoque retinebunt velocitates in ratione prima.~~ ~~Et ob datam velocitatum rationem, de&longs;cribent &longs;emper &longs;patia quæ &longs;unt ut velocitates primæ & tempora conjunctim. q.E.D.~~

~~LIBER SECUNDUS.~~

Corol. 1. Igitur &longs;i æquivelocibus corporibus re&longs;i&longs;titur in duplicata ratione diametrorum: Globi homogenei quibu&longs;cunque cum velocitatibus moti, de&longs;cribendo &longs;patia diametris &longs;uis proportionalia, amittent partes motuum proportionales totis. Motus enim Globi cuju&longs;que erit ut ejus velocitas & Ma&longs;&longs;a conjunctim, id e&longs;t, ut velocitas & cubus diametri; re&longs;i&longs;tentia (per Hypothe&longs;in) erit ut quadratum diametri & quadratum velocitatis conjunctim; & tempus (per hanc Propo&longs;itionem) e&longs;t in ratione priore directe & ratione po&longs;teriore inver&longs;e, id e&longs;t, ut diameter directe & velocitas inver&longs;e; adeoque &longs;patium (tempori & velocitati proportionale) e&longs;t ut diameter.

Corol. 2. Si æquivelocibus corporibus re&longs;i&longs;titur in ratione &longs;e&longs;quialtera diametrorum: Globi homogenei quibu&longs;cunque cum velocitatibus moti, de&longs;cribendo &longs;patia in &longs;e&longs;quialtera ratione diametrorum, amittent partes motuum proportionales totis.

Corol. 3. Et univer&longs;aliter, &longs;i æquivelocibus corporibus re&longs;i&longs;titur in ratione dignitatis cuju&longs;cunque diametrorum: &longs;patia quibus Globi homogenei, quibu&longs;cunque cum velocitatibus moti, amittent partes motuum proportionales totis, erunt ut cubi diametrorum ad dignitatem illam applicati. Sunto diametri D & E; & &longs;i re&longs;i&longs;tentiæ, ubi velocitates æquales ponuntur, &longs;int ut Dn & En: &longs;putia quibus Globi quibu&longs;cunque cum velocitatibus moti, amitteus partes motuum proportionales totis, erunt ut D3-n & E3-n. ~~Igitur de&longs;cribendo &longs;patia ip&longs;is D3-n & E3-n proportionalia, retinebunt velocitates in eadem ratione ad invicem ac &longs;ub initio.~~

~~Corol. 4. Quod &longs;i Globi non &longs;int homogenei, &longs;patium a Globo den&longs;iore de&longs;eriptum augeri debet in ratione den&longs;itatis.~~ Motus enim, &longs;ub pari velocitare, major e&longs;t in ratione den&longs;itatis, & tempus (per lianc Propo&longs;itionem) augetur in ratione motus directe, ac &longs;patium de&longs;criptum in ratione temporis.

~~DE MOTU CORPORUM~~

~~Corol. 5. Et &longs;i Globi moveantur in Mediis diver&longs;is; &longs;patium in Medio, quod cæteris paribus magis re&longs;i&longs;tit, diminuendum erit in ratione majoris re&longs;i&longs;tentiæ.~~ ~~Tempus enim (per hanc Propo&longs;itionem) diminuetur in ratione re&longs;i&longs;tentiæ auctæ, & &longs;patium in ratione temporis.~~

~~LEMMA II.~~

~~Momentum Genitæ æquatur Momentis laterum &longs;ingulorum generantium in eorundem laterum indices dignitatum & coefficientia continue ductis.~~

Genitam voco quantitatem omnem quæ ex lateribus vel terminis quibu&longs;cunque, in Arithmetica per multiplicationem, divi&longs;ionem, & extractionem radicum; in Geometria per inventionem vel contentorum & laterum, vel extremarum & mediarum proportionalium, ab&longs;que additione & &longs;ubductione generatur. ~~Eju&longs;modi quantitates &longs;unt Facti, Quoti, Radices, Rectangula, Quadrata, Cubi, Latera quadrata, Latera cubica, & &longs;imiles.~~ Has quantitates ut indeterminatas & in&longs;tabiles, & qua&longs;i motu fluxuve perpetuo cre&longs;centes vel decre&longs;centes, hic con&longs;idero; & earum incrementa vel decrementa momentanea &longs;ub nomine Momentorum intelligo: ita ut incrementa pro momentis addititiis &longs;eu affirmativis, ac decrementa pro &longs;ubductitiis &longs;eu negativis habeantur. ~~Cave tamen intellexeris particulas finitas.~~ ~~Particulæ finitæ non &longs;unt momenta, &longs;ed quantitates ip&longs;æ ex momentis genitæ.~~ ~~Intelligenda &longs;unt principia jamjam na&longs;centia finitarum magnitudinum.~~ ~~Neque enim &longs;pectatur in hoc Lemmate magnitudo momentorum, &longs;ed prima na&longs;centium proportio.~~ Eodem recidit &longs;i loco momentorum u&longs;urpentur vel velocitates incrementorum ac decrementorum, (quas etiam motus, mutationes & fluxiones quantitatum nominare licet) vel finitæ quævis quantitates velocitatibus hi&longs;ce proportionales. ~~Lateris autem cuju&longs;que generantis Coefficiens e&longs;t quantitas, quæ oritur applicando Genitam ad hoc latus.~~

~~Igitur &longs;en&longs;us Lemmatis e&longs;t, ut, &longs;i quantitatum quarumcunque perpetuo motu cre&longs;centium vel decre&longs;centium A, B, C, &c.~~ ~~momenta, vel mutationum velocitates dicantur a, b, c, &c.~~ ~~momentum vel mutatio geniti rectanguli AB fuerit aB+bA, & geniti contenti ABC momentum fuerit aBC+bAC+cAB: & genitarum dignitatum A3, A3, A4, A1/2, A1/3, A1/3, A2/3, A-1, A-2, & A-1/2 momenta~~

~~2aA, 3aA2, 4aA3, 1/2aA-1/2, 3/2aA1/2, 1/3aA-2/3, 2/3aA-1/3, -aA-2, -2aA-3, & -1/2aA-1/2 re&longs;pective.~~ ~~Et generaliter, ut dignitatis cuju&longs;cunque An/m momentum fuerit n/m aA(n-m/m).~~ ~~Item ut Genitæ A2B momentum fuerit 2aAB+bA2; & Genitæ A3B4C2 momentum 3aA2B4C2+4bA3B3C2+2cA3B4C; & Genitæ (A3/B2) &longs;ive A3B-2 momentum 3aA2B-2-2bA3B-3: & &longs;ic in cæteris.~~ ~~Demon&longs;tratur vero Lemma in hunc modum.~~

~~LIBER SECUNDUS.~~

Cas. 1. Rectangulum quodvis motu perpetuo auctum AB, ubi de lateribus A & B deerant momentorum dimidia 1/2a & 1/2b,fuit A-1/2a in B-1/2b, &longs;eu AB-1/2a B-1/2b A+1/4ab; & quam primum latera A & B alteris momentorum dimidiis aucta &longs;unt, evadit A+1/2a in B+1/2b &longs;eu AB+1/2a B+1/2b A+1/4ab. De hoc rectangulo &longs;ubducatur rectangulum prius, & manebit exce&longs;&longs;us aB+bA. ~~Igitur laterum incrementis totis a & b generatur rectanguli incrementum aB+bA. q.E.D.~~

~~Cas. 2. Ponatur AB &longs;emper æquale G, & contenti ABC &longs;eu GC momentum (per Cas.~~ ~~1.) erit gC+cG, id e&longs;t (&longs;i pro G & g&longs;cribantur AB & aB+bA) aBC+bAC+cAB.~~ ~~Et par e&longs;t ratio contenti &longs;ub lateribus quotcunque. q.E.D.~~

Cas. 3. Ponantur latera A, B, C &longs;ibi mutuo &longs;emper æqualia; & ip&longs;ius A2, id e&longs;t rectanguli AB, momentum aB+bA erit 2aA, ip&longs;ius autem A3, id e&longs;t contenti ABC, momentum aBC+bAC +cAB erit 3aA2. ~~Et eodem argumento momentum dignitatis cuju&longs;cunque An e&longs;t naAn-1. q.E.D.~~

~~Cas. 4. Unde cum 1/A in A &longs;it 1, momentum ip&longs;ius 1/A ductum in A, una cum 1/A ducto in a erit momentum ip&longs;ius 1, id e&longs;t, nihil.~~ Proinde momentum ip&longs;ius 1/A &longs;eu ip&longs;ius A-1 e&longs;t (-a/A2). Et generaliter cum (1/An) in An &longs;it 1, momentum ip&longs;ius (1/An) ductum in A una cum (1/An) in naAn-1 erit nihil. ~~Et propterea momentum ip&longs;ius (1/An) &longs;eu A-n erit-(na/An+1). q.ED.~~

~~DE MOTU CORPORUM~~

~~Cas. 5. Et cum A1/2 in A1/2 &longs;it A, momentum ip&longs;ius A1/2 ductum in 2A1/2 erit a, per Cas.~~ ~~3: ideoque momentum ip&longs;ius A1/2 erit (a/2A 1/2) &longs;ive 1/2aA-1/2.~~ Et generaliter &longs;i ponatur Am/n æquale B, erit Am æquale Bn, ideoque maAm-1 æquale nbBn-1, & maA-1 æquale nbB-1 &longs;eu nbA-m/n, adeoque m/n aA(m-n/n) æquale b, id e&longs;t, æquale momento ip&longs;ius Am/n, q.E.D.

Cas. 6. Igitur Genitæ cuju&longs;eunque AmBn momentum e&longs;t momentum ip&longs;ius Am ductum in Bn, una cum momento ip&longs;ius Bn ducto in Am, id e&longs;t maAm-1Bn+nbBn-1Am; idque &longs;ive dignitatum indices m & n &longs;int integri numeri vel fracti, &longs;ive affirmativi vel negativi. ~~Et par e&longs;t ratio contenti &longs;ub pluribus dignitatibus. q.E.D.~~

Corol. 1. Hinc in continue proportionalibus, &longs;i terminus unus datur, momenta terminorum reliquorum erunt ut iidem termini multiplicati per numerum intervallorum inter ip&longs;os & terminum datum. ~~Sunto A, B, C, D, E, F continue proportionales; & &longs;i detur terminus C, momenta reliquorum terminorum erunt inter &longs;e ut-2A, -B, D, 2E, 3F.~~

~~Corol. 2. Et &longs;i in quatuor proportionalibus duæ mediæ dentur, momenta extremarum erunt ut eædem extremæ.~~ ~~Idem intelligendum e&longs;t de lateribus rectanguli cuju&longs;cunque dati.~~

~~Corol. 3. Et &longs;i &longs;umma vel differentia duorum quadratorum detur, momenta laterum erunt reciproce ut latera.~~

~~Scholium.~~

In literis quæ mihi cum Geometra periti&longs;&longs;imo G.G. Leibnitio annis abhinc decem intercedebant, cum &longs;ignificarem me compotem e&longs;&longs;e methodi determinandi Maximas & Minimas, ducendi Tangentes, & &longs;imilia peragendi, quæ in terminis &longs;urdis æque ac in rationalibus procederet, & literis tran&longs;po&longs;itis hanc &longs;ententiam involven- tibus [Data Æquatione quotcunque Fluentes quantitates invelven- te, Fluxiones invenire, & vice ver&longs;a] eandem celarem: re&longs;crip&longs;it Vir Clari&longs;&longs;imus &longs;e quoque in eju&longs;modi methodum incidi&longs;&longs;e, & methodum &longs;uam communicavit a mea vix abludentem præterquam in verborum & notarum formulis, & Idea generationis quantitatum. ~~Utriu&longs;que fundamentum continetur in hoc Lemmate.~~

~~LIBER SECUNDUS.~~

~~PROPOSITIO VIII. THEOREMA VI.~~

Si corpus in Medio uniformi, Gravitate uniformiter agente, recta a&longs;cendat vel de&longs;cendat, & &longs;patium totum de&longs;criptum di&longs;tinguatur in partes æquales, inque principiis &longs;ingularum partium (addendo refi&longs;tentiam Medii ad vim gravitatis, quando corpus a&longs;cendit, vel &longs;ubducendo ip&longs;am quando corpus de&longs;cendit) colligantur vires ab&longs;olutæ; dico quod vires illæ ab&longs;olutæ &longs;unt in progre&longs;&longs;ione Geometrica.

~~Exponatur enim vis gravitatis per datam lineam~~