Unpublished Appendix to 'methodus': Problem IX

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innititur: quæqꝫ magis perspicua et ornata evadet si fundamenta quædam pro more methodi syntheticæ præsternantur; qualia sunt haec.

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Axiomata.

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Ax. 1. Quæ fluxionibus æqualibus simul generantur sunt æqualia.

Ax. 2. Quæ fluxionibus in data ratione simul generantur, sunt in ratione fluxionum.

Nota {illeg} \{illeg}/, simul generari intelligo quæ tota eodem tempore generantur.

Ax. 3. Fluxio totius æquatur fluxionibus partium simul sumptis.

Ubi nota quod profluxiones affirmativè {illeg}|a|c defluxiones negativè ponendæ sint.

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Ax. \4/. Fluxio major est quæ ex minori facit æquale vel majus, aut ex æquali majus producit.

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Ax 5|4|. Fluxiones sunt ut momenta \contemporanea/ fluxionibus istis \continuè/ generata.

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Ax: 4. Momenta contemporanea sunt ut fluxiones.

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Theoremata.

Th. 1. Positis quatuor \perpetuò/ proportionalibus fluentibus quantitatibus: summa extremarum reciprocè ductaru in suas fluxiones æquatur summæ mediarum reciprocè ductarum in suas fluxiones. Sint $\mathrm{A}.\mathrm{B}\colon\colon \mathrm{C}.\mathrm{D}$ et erit $\mathrm{A}\times fl:\mathrm{D}+\mathrm{D}\times fl:\mathrm{A}=\mathrm{B}\times fl:\mathrm{C}+\mathrm{C}\times fl:\mathrm{B}$. \Id quod eodem modo demonstrari potest ac solutio Prob 1; vel etiam sic./ {Nam} esto M momentum ipsius A, et cùm momenta fluentium sint ut fluxiones, erit $\frac{fl:\mathrm{B}}{fl:\mathrm{A}}\mathrm{M}$ momentum ipsius B, $\frac{fl:\mathrm{C}}{fl:\mathrm{A}}\mathrm{M}$ momentum ipsius C, et $\frac{fl:\mathrm{D}}{fl:\mathrm{A}}\mathrm{M}$ momentum ipsius D. Quare ubi A profluendo evadit $\mathrm{A}+\mathrm{M}$, B evadet $\mathrm{B}+\frac{fl:\mathrm{B}}{fl:\mathrm{A}}\mathrm{M}$ & C evadet $\mathrm{C}+\frac{fl:\mathrm{C}}{fl:\mathrm{A}}\mathrm{M}$, ac D evadet $\mathrm{D}+\frac{fl:\mathrm{D}}{fl:\mathrm{A}}\mathrm{M}$ ubi vero \& similiter ubi/ defluendo A evadit $\mathrm{A}-\mathrm{M}$. B evadet $\mathrm{B}-\frac{fl:\mathrm{B}}{fl:\mathrm{A}}\mathrm{M}$ C evadt {sic}{illeg} $\mathrm{C}-\frac{fl:\mathrm{C}}{fl:\mathrm{A}}\mathrm{M}$ ac D evadet $\mathrm{D}-\frac{fl:\mathrm{D}}{fl:\mathrm{A}}\mathrm{M}$ hac lege scilicet ut hæ quantitates etiamnum maneant proportionales. Duc ergo extrema et media in se & provenient $\mathrm{A}\mathrm{D}+\frac{fl:\mathrm{D}}{fl:\mathrm{A}}\mathrm{A}\mathrm{M}+\mathrm{D}\mathrm{M}+\frac{fl:\mathrm{D}}{fl:\mathrm{A}}\mathrm{M}\mathrm{M}=$$\mathrm{B}\mathrm{C}+\frac{fl:\mathrm{C}}{fl:\mathrm{A}}\mathrm{B}\mathrm{M}+\frac{fl:\mathrm{B}}{fl:\mathrm{A}}\mathrm{C}\mathrm{M}+\frac{fl:\mathrm{B}\times fl:\mathrm{C}}{fl:\mathrm{A}\times fl:\mathrm{A}}\mathrm{M}\mathrm{M}$. Et $\mathrm{A}\mathrm{D}-\frac{fl:\mathrm{D}}{fl:\mathrm{A}}\mathrm{A}\mathrm{M}-\mathrm{D}\mathrm{M}+\frac{fl:\mathrm{D}}{fl:\mathrm{A}}\mathrm{M}\mathrm{M}=\mathrm{B}\mathrm{C}-\frac{fl:\mathrm{C}}{fl:\mathrm{A}}\mathrm{B}\mathrm{M}-\frac{fl:\mathrm{B}}{fl:\mathrm{A}}\mathrm{C}\mathrm{M}$$+\frac{fl:\mathrm{B}\times fl:\mathrm{C}}{fl:\mathrm{A}\times fl:\mathrm{A}}\mathrm{M}\mathrm{M}$. Aufer jam hæc posteriora æqualia \æqualia $\mathrm{A}\mathrm{D}$ & $\mathrm{B}\mathrm{C}$/ a prioribus æqualibus, & restantia æqualia \duc in $fl:\mathrm{A}$ ac/ divide per <34> /> {illeg} M. Emerget enim $\mathrm{A}\times fl:\mathrm{D}+\mathrm{D}\times fl:\mathrm{A}+fl:\mathrm{D}\times \mathrm{M}=\mathrm{B}\times fl:\mathrm{C}+\mathrm{C}\times fl:\mathrm{B}+\frac{fl:\mathrm{B}\times fl:\mathrm{C}}{fl:\mathrm{A}}\mathrm{M}$. Ubi propter infinitam parvitatem momenti M rejectis terminis per id multiplicatis{.} Si{illeg}t $\mathrm{AB}.\mathrm{AD}\colon\colon \mathrm{AE}.\mathrm{AC}$, et erit $\mathrm{AB}\times fl:\mathrm{AC}+\mathrm{AC}\times fl:\mathrm{AB}=\mathrm{AD}\times fl:\mathrm{AE}$$+\mathrm{AE}\times fl:\mathrm{AD}$. Nam augeantur hæ lineæ momentis suis fluendo Bb, Dd, Ee, Cc fluendo, & propter perpetuam earum proportionalitatem, adeoqꝫ rectangula ab extremis et medijs constituta perpetuo æqualia \nempe $\mathrm{AF}=\mathrm{AG}$ & $\mathrm{Af}=\mathrm{Ag}$/; augmenta rectangulorum \istorum/ BFCf & DGEg æqualia erunt: hoc est $\mathrm{Ab}\times \mathrm{Cc}\left(\mathrm{Cf}\right)$$+\mathrm{AC}\times \mathrm{Bb}\left(\mathrm{bF}\right)=\mathrm{Ad}\times \mathrm{Ee}\left(\mathrm{Eg}\right)+\mathrm{AE}\times \mathrm{Dd}\left(\mathrm{dG}\right)$. Cùm autem fluxiones \{illeg}/ sint ut momenta \quantitatum ab/ istis {illeg} continuò descripta. hoc est $fl:\mathrm{AC}.\mathrm{Cc}\colon\colon fl:\mathrm{AB}.\mathrm{Bb}$ &c. Erit $\mathrm{Ab}\times fl:\mathrm{AC}.$$\mathrm{Ab}\times \mathrm{Cc}\colon\colon \mathrm{AC}\times fl\mathrm{AB}.\mathrm{AC}\times \mathrm{Bb}\colon\colon \mathrm{Ab}\times fl:\mathrm{AC}+\mathrm{AC}\times fl:\mathrm{AB}.$$\mathrm{Ab}\times \mathrm{Cc}+\mathrm{AC}\times \mathrm{Bb}$. Et eodem ratiocinio $\mathrm{AD}\times fl:\mathrm{AE}+\mathrm{AE}\times fl:\mathrm{AD}$ erit ad $\mathrm{AD}\times \mathrm{Ee}+\mathrm{AE}\times \mathrm{Dd}$ in eadem ratione. Quare $\mathrm{Ab}\times fl\mathrm{AC}+\mathrm{AC}\times fl:\mathrm{AB}$ & $\mathrm{Ad}\times fl:\mathrm{AE}+\mathrm{AE}\times fl\mathrm{AD}$, cùm sint in eadem ratione ad æqualia, etiam æqualia erunt. \*/ < insertion from the left margin > * sive $\mathrm{Ab}+\mathrm{AC}\times \frac{\mathrm{Bb}}{\mathrm{Cc}}=\mathrm{Ad}\times \frac{\mathrm{Ee}}{\mathrm{Cc}}+\mathrm{AE}\times \frac{\mathrm{Dd}}{\mathrm{Cc}}$. Adeoqꝫ cùm fluxiones sint ut momenta quantitatum ab istis continuò generata, hoc est $\frac{\mathrm{Bb}}{\mathrm{Cc}}=\frac{fl\mathrm{AB}}{fl\mathrm{AC}}${,} $\frac{\mathrm{Ee}}{\mathrm{Cc}}=\frac{fl\mathrm{AE}}{fl\mathrm{AC}}$ & $\frac{\mathrm{Dd}}{\mathrm{Cc}}=\frac{fl\mathrm{AD}}{fl\mathrm{AC}}$, erit $\mathrm{Ab}+\mathrm{AC}\times \frac{fl\mathrm{AB}}{fl\mathrm{AC}}=\mathrm{Ad}\times \frac{fl\mathrm{AE}}{fl\mathrm{AC}}+\mathrm{AE}\times \frac{fl\mathrm{AD}}{fl\mathrm{AC}}$. Sive $\mathrm{Ab}\times fl\mathrm{AC}+\mathrm{AC}\times fl\mathrm{AB}=\mathrm{Ad}\times fl\mathrm{AE}+\mathrm{AE}\times fl\mathrm{AD}$. < text from p 34 resumes > Decrescant jam rectangula Af et Ag donec in prima rectangula AF & AG redierint, & in ultimo istius infinitè parvæ defluxionis momento hoc est in primtunc Ab evadet AB atqꝫ Ad evadet AD. Quare in ultimo istius infinitè parvæ defluxionis momento, hoc est in primo momento fluxionis quadrangulorum AF et AG quando vel \incipiunt/ augeri vel diminui, erit $\mathrm{AB}\times fl:\mathrm{AC}+\mathrm{AC}\times fl\mathrm{AB}=$$\mathrm{AD}\times fl\mathrm{AE}+\mathrm{AE}\times fl\mathrm{AD}$. Q.E.D.

Cor: 1. Positis tribus continuè proportionalibus, {illeg} summa extremarum reciproce ductarum in suas fluxiones æquatur duplo mediæ ductæ in suam fluxionem. Si{illeg}t $\mathrm{A}.\mathrm{B}\colon\colon \mathrm{B}.\mathrm{C}$ et erit $\mathrm{A}\times fl:\mathrm{C}+\mathrm{C}\times fl:\mathrm{A}(=$^*[1]$\mathrm{B}\times fl:\mathrm{B}+\mathrm{B}\times fl\mathrm{B})$$=2\mathrm{B}\times fl\mathrm{B}$.

Cor 2. Positis tribus continuè proportionalibus $\mathrm{A}.\mathrm{B}.\mathrm{C}$ si summa extremarum sit data quantitas, fluxio minoris extremæ erit ad fluxionem mediæ ut duplum mediæ ad differentiam extremarum. $2\mathrm{B}.\mathrm{C}-\mathrm{A}\colon\colon fl\mathrm{A}.fl\mathrm{B}$. Nam cùm $\mathrm{A}+\mathrm{C}$ ex Hypothesi non fluat, ^*[2] erit $fl:\mathrm{A}+fl:\mathrm{C}=0$, sive <35> /> $fl\mathrm{C}=-fl\mathrm{A}$. adeoqꝫ $\mathrm{A}\times fl:\mathrm{C}=-\mathrm{A}\times fl\mathrm{A}$. Quare $\stackrel{\_}{\mathrm{C}-\mathrm{A}}\times fl\mathrm{A}$$\left(=\mathrm{A}\times fl:\mathrm{C}+\mathrm{C}\times fl\mathrm{A}\right)$^*[3]$=2\mathrm{B}\times fl\mathrm{B}$, hoc est $2\mathrm{B}.\mathrm{C}-\mathrm{A}\colon\colon fl\mathrm{A}.fl\mathrm{B}$.

Cor 3{.} Sin differentia extremarum detur, fluxio alterutrius extremæ erit ad fluxionem mediæ, ut duplum mediæ ad summam extremarum. $2\mathrm{B}.\mathrm{A}+\mathrm{C}\colon\colon fl\mathrm{A}.fl\mathrm{B}$. Demonstratur ut Cor 2.

Cor 4. Quod si summa primæ et secundæ quantitatis detur, erit fluxio secundæ ad fluxionem tertiæ ut prima ad duplum secundæ auctum tertia. $\mathrm{A}.2\mathrm{B}+\mathrm{C}\colon\colon fl\mathrm{B}.fl\mathrm{C}$. Nam cùm ex Hypothesi $\mathrm{A}+\mathrm{B}$ non fluat, ^*[4] erit $fl\mathrm{A}+fl\mathrm{B}=0$ sive $fl\mathrm{B}=-fl\mathrm{A}$, adeoqꝫ $\mathrm{C}\times fl\mathrm{B}=-\mathrm{C}\times fl\mathrm{A}$. Quare $\mathrm{A}\times fl\mathrm{C}$$\left(=2\mathrm{B}\times fl\mathrm{B}-\mathrm{C}\times fl\mathrm{A}\right)=2\stackrel{\_}{\mathrm{B}+\mathrm{C}}\times fl\mathrm{B}$; hoc est $\mathrm{A}.2\mathrm{B}+\mathrm{C}\colon\colon fl:\mathrm{B}${$.$}$fl:\mathrm{C}$.

Cor: 5. Si deniqꝫ differentia primæ et secundæ datur erit fluxio primæ vel alterutrius ad fluxionem tertiæ ut prima ad duplum secundæ diminutum tertia. fl $\mathrm{A}.2\mathrm{B}-\mathrm{C}$$\colon\colon fl\mathrm{B}.fl\mathrm{C}$. Demonstratur ut Cor 4.

Cor 6. Positis quotcuncꝫ continuè proportionalibus, quarum una sit data quantitas & cæteræ fluentes: fluxiones fluentium erunt inter se ut fluentes illæ ductæ in numerum terminorum quibus distant a dato \illo/ termino. Sint $\mathrm{A}.\mathrm{B}.\mathrm{C}.\mathrm{D}.\mathrm{E}.\mathrm{F}.${illeg} continuè proportionales et si datur C, erit $-2\mathrm{A}.-\mathrm{B}.\mathrm{D}.2\mathrm{E}.3\mathrm{F}\colon\colon fl\mathrm{A}.fl:\mathrm{B}.fl\mathrm{D}.fl\mathrm{E}${$.$}$fl\mathrm{F}$. Nam propter $\mathrm{C}.\mathrm{D}.\mathrm{E}∺$, est $\mathrm{C}\times fl\mathrm{E}+\mathrm{E}\times fl\mathrm{C}=$$2\mathrm{D}\times fl\mathrm{D}$, p{illeg}|e|r Cor 1. At \ex Hypothesi/ $fl\mathrm{C}=0$. Ergo $\mathrm{C}\times fl:\mathrm{E}=2\mathrm{D}\times fl\mathrm{D}$. hoc est $fl\mathrm{D}.fl\mathrm{E}\left(\colon\colon \mathrm{C}.2\mathrm{D}\right)\colon\colon \mathrm{D}.2\mathrm{E}$.

Iterum quia $\mathrm{D}.\mathrm{E}.\mathrm{F}∺$, erit $\mathrm{D}\times fl:\mathrm{F}+\mathrm{F}\times fl:\mathrm{D}=$$\mathrm{E}\times 2fl\mathrm{E}$. Sed e jam ostensis est $\mathrm{D}fl${illeg} sive $\mathrm{D}\times fl\mathrm{F}=$$\mathrm{E}\times 2fl:\mathrm{E}-\mathrm{F}\times fl:\mathrm{D}$. Sed e jam ostensis est $\frac{\mathrm{D}\times fl\mathrm{E}}{2\mathrm{E}}=$$fl:\mathrm{D}$. Ergo $\mathrm{D}\times fl\mathrm{F}=\frac{4\mathrm{E}\mathrm{E}-\mathrm{D}\mathrm{F}}{2\mathrm{E}}\times fl\mathrm{E}$ $fl\mathrm{D}.fl\mathrm{E}\left(\colon\colon \mathrm{D}.2\mathrm{E}\right)$$\colon\colon \mathrm{E}.2\mathrm{F}$. Ergo $\mathrm{F}\times fl\mathrm{D}=\frac{1}{2}\mathrm{E}\times fl\mathrm{E}$. adeoqꝫ $\mathrm{D}\times fl\mathrm{F}=$$\frac{3}{2}\mathrm{E}\times fl\mathrm{E}$. Et $fl:\mathrm{E}.fl:\mathrm{F}\left(\colon\colon 2\mathrm{D}.3\mathrm{E}\right)\colon\colon 2\mathrm{E}.3\mathrm{F}$. Atqꝫ ita in cæteris.

Theorem 2. In triangulo quovis rectangulo summa laterum ductarum in suas fluxiones æquatur Hypothenusæ ductæ in fluxionem su{illeg}|a|m.

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Cor. 7. Si fluentes duæ quantitates se multiplicant fluxio Quoti \Facti/ compon{illeg}|i|tur ex fluxionibus factorum mul alterne ductis in factores. $Fl:\mathrm{A}\mathrm{B}=\mathrm{B}\times fl\mathrm{A}+\mathrm{A}\times fl\mathrm{B}${.} Nam $1.\mathrm{A}\colon\colon \mathrm{B}.\mathrm{A}\mathrm{B}$. Ergo per Th 1.

Cor 8. Si fluens quantitas per fluentem quantitatem divid{illeg}|i|tur: fluxio Quoti est quæ produ prodit auferendo fluxionem divisoris {illeg}|m|ultiplicatam per dividuum, a fluxione dividui multiplicata per divisore{illeg}|m| & dividendo residuum per quadratum divisoris. $Fl:\frac{\mathrm{B}}{\mathrm{A}}=\frac{\mathrm{A}\times fl:\mathrm{B}-\mathrm{B}\times fl:\mathrm{A}}{\mathrm{A}\mathrm{A}}$. Nam $\mathrm{A}.1\colon\colon \mathrm{B}.\frac{\mathrm{B}}{\mathrm{A}}$. Ergo per Th: 1, $\mathrm{A}\times fl\frac{\mathrm{B}}{\mathrm{A}}+\frac{\mathrm{B}}{\mathrm{A}}\times fl\mathrm{A}=1\times fl:\mathrm{B}$, nam $\mathrm{B}\times fl:1$ nihil est. Aufer utrobiqꝫ $\frac{\mathrm{B}}{\mathrm{A}}\times fl\mathrm{A}$ et residuum divide per A, et prodibit $fl:\frac{\mathrm{B}}{\mathrm{A}}=\frac{\mathrm{A}\times fl\mathrm{B}-\mathrm{B}\times fl\mathrm{A}}{\mathrm{A}\mathrm{A}}$.

Theor. 2. In Triangulo quovis rectangulo

Cor 9. Fluxio radicis est ad fluxionem potestatis \su/ alicujus ut radix ad potestatem illam multiplicatam per numerum dimensionum $fl:\mathrm{A}.fl{\mathrm{A}}^3$$\colon\colon \mathrm{A}.3{\mathrm{A}}^3$. vel $fl:\sqrt{}3:\mathrm{A}.fl:\mathrm{A}\colon\colon \sqrt{}3:\mathrm{A}.3\mathrm{A}$ & sic in alijs potestatibus. Patet per c|C|or: 6.

Theor. 2. In triangulo quovis \perpetim/ rectangulo \cujus latera {illeg}|quom|odocunqꝫ fluunt/ summa laterum duct{illeg}|o|rum in suas fluxiones æquatur {illeg}|h|ypotenusæ {illeg}|d|uctæ in flux{illeg}|i|onem suam. Sit $\mathrm{A}\mathrm{A}+\mathrm{B}\mathrm{B}=\mathrm{C}\mathrm{C}$ et erit $\mathrm{A}\times fl:\mathrm{A}+\mathrm{B}\times fl:\mathrm{B}=\mathrm{C}\times fl:\mathrm{C}$ Nam per Cor 9 Th 1 est $fl:\mathrm{A}.fl:\mathrm{A}\mathrm{A}\left(\colon\colon \mathrm{A}.2\mathrm{A}\mathrm{A}\right)${∷}$1.2\mathrm{A}$, adeoqꝫ $fl:\mathrm{A}\mathrm{A}=2\mathrm{A}\times fl\mathrm{A}$. Eadem ratione $fl\mathrm{B}\mathrm{B}=$$2\mathrm{B}\times fl\mathrm{B}$ & $fl\mathrm{C}\mathrm{C}=2\mathrm{C}\times fl:\mathrm{C}$. Quare cum $\mathrm{A}\mathrm{A}+\mathrm{B}\mathrm{B}$$=\mathrm{C}\mathrm{C}$ atqꝫ adeo per Ax: 1 & 3 $fl:\mathrm{A}\mathrm{A}+fl:\mathrm{B}\mathrm{B}=fl:\mathrm{C}\mathrm{C}$ \erit/ $2\mathrm{A}\times fl\mathrm{A}+2\mathrm{B}\times fl\mathrm{B}=2\mathrm{C}\times fl\mathrm{C}$. Quod dimidiatum fit $\mathrm{A}\times fl\mathrm{A}+\mathrm{B}\times fl\mathrm{B}=\mathrm{C}\times fl\mathrm{C}$. Q.E.D.

Cor 1. Si crus alterutrum sit data quantitas, erit fluxio Hypotenusa alterius cruris ad fluxionem hypotenusæ ut hypotenusa ad crus illud alterum. Detur A, et erit $\mathrm{C}.\mathrm{B}\colon\colon fl:\mathrm{B}.fl:\mathrm{C}$, \nam $\mathrm{B}\times fl\mathrm{B}=\mathrm{C}\times fl\mathrm{C}$/ propterea quòd $\mathrm{A}\times fl\mathrm{A}$ nihil sit.

Cor 2. Si hypotenusa datur, erit profluxio unius <37> /> cruris ad defluxionem alterius ut alterum illud alterum crus ad crus primum. Detur C, et erit $\mathrm{B}.\mathrm{A}\colon\colon fl:\mathrm{A}.fl:\mathrm{B}$ propterea quod $\mathrm{C}\times fl:\mathrm{C}$ nihil sit.

Theor. 3|4|. Si recta circa datum punctum gyrans, secet alias duas positione datas & ad commune punctum terminatas rectas: fluxiones earum quæ positione dantur, erunt ut ipsæ illæ rectæ ductæ in conterminas partes lineæ gyrantis.      Circa datum punctum A gyret recta AC, & inter gyrandum secet ea p|r|ectas positione datas DC ac DB in punctis C et B. Dico esse $\mathrm{DB}\times \mathrm{AB}.\mathrm{DC}\times \mathrm{AC}\colon\colon$$fl\mathrm{DB}.fl\mathrm{DC}$. Sit enim Abc positio rectæ gyrantis in proximo temporis momento, hoc est Cc momentum rectæ DC et Bb contemporaneum momentum rectæ DB; et ipsi DB parallela agatur ce occurrens AC in e: et propter sim: tri: CBD, Cec, erit $\mathrm{DC}.\mathrm{DB}\colon\colon \mathrm{Cc}.\mathrm{ce}$. Dein propter sim: tri: Aec, ABb, erit $\mathrm{Ac}.\mathrm{Ab}\colon\colon \mathrm{ec}.\mathrm{Bb}$, et additis rationibus $\mathrm{DC}\times \mathrm{Ac}.\mathrm{DB}\times \mathrm{Ab}\colon\colon \mathrm{Cc}.\mathrm{Bb}\colon\colon$ (per ax 4) $fl:\mathrm{DC}.fl:\mathrm{DB}$. Coeant jam lineæ infinitè parùm distantes Ac & AC, et in momento concursus evadet $\mathrm{DC}\times \mathrm{AC}.\mathrm{DB}\times \mathrm{AB}\colon\colon fl:\mathrm{DC}.fl:\mathrm{DB}$. Q.E.D.

Cor 1{.} Iisdem positis, et ab A demissis ad DB et DC normalibus AG et AH: erit primo $\mathrm{DC}\times \mathrm{AC}.\mathrm{DB}\times \mathrm{BG}\colon\colon fl:\mathrm{DC}.$$fl:\mathrm{AB}$. Nam per Cor. 1 Th. 2, est $\mathrm{AB}.\mathrm{BG}\colon\colon fl:\mathrm{DB}.fl:\mathrm{AB}$. sive $\mathrm{DB}\times \mathrm{AB}.\mathrm{DB}\times \mathrm{BG}\colon\colon fl:\mathrm{DB}.fl\mathrm{AB}$, et supra erat $\mathrm{DC}\times \mathrm{AC}.$$\mathrm{DB}\times \mathrm{AB}\colon\colon fl:\mathrm{DC}.fl\mathrm{DB}$. {illeg}|E|rgo ex æquo $\mathrm{DC}\times \mathrm{AC}.\mathrm{DB}\times \mathrm{BG}\colon\colon$$fl:\mathrm{DC}.fl:\mathrm{AB}$.

Cor 2. Er{illeg}|i|t secundò {sic} $\mathrm{DC}\times \mathrm{CH}.\mathrm{DB}\times \mathrm{BG}\colon\colon fl\mathrm{DC}.fl\mathrm{AB}$. Nam per Cor 1 Th 2 est $\mathrm{CH}.\mathrm{AC}\left(\text{vel}\mathrm{DC}\times \mathrm{CH}.\mathrm{DC}\times \mathrm{AC}\right)\colon\colon fl\mathrm{AC}.$$fl\mathrm{DC}$. Et supra erat $\mathrm{DC}\times \mathrm{AC}.\mathrm{DB}\times \mathrm{BG}\colon\colon fl\mathrm{DC}.fl\mathrm{AB}$. Ergo ex æquo $\mathrm{DC}\times \mathrm{CH}.\mathrm{DB}\times \mathrm{BG}\colon\colon fl\mathrm{AC}.fl\mathrm{AB}$.

Theor 4 3{.} Si Trianguli alicujus Basis et longitudine et positione detur, vertex autem sit ad rectam positione datam; demisso \ab alterutro termino basis ad rectam illam positione datam/ perpendiculo, quod occurrat {casis} opposito cruri trianguli, erit fluxio ejus oppositi cruris ad fluxionem alt{illeg}|e|rius cru{illeg}|r|is |ut| illud alter{illeg}|u|m crus ad partem hujus cruris inter verticem trianguli et perpendiculum illud situm. Sit AB basis trianguli, C vertex, DE locus verticis, et <38> /> BD {sic} perpendiculariter demissa ad DC occurrat AC in F, eritqꝫ $\mathrm{BC}.\mathrm{FC}\colon\colon fl\mathrm{AC}.fl\mathrm{BC}$. Nam ab A ad DE demisso perpendiculo AD, erit (per Cor 1. Th 2) $\mathrm{BC}.\mathrm{EC}\colon\colon fl\mathrm{EC}.fl\mathrm{BC}$, et $\mathrm{DC}.\mathrm{AC}(\text{sive}$$\mathrm{EC}.\mathrm{FC})\colon\colon fl\mathrm{AC}.fl\mathrm{BC}=fl\mathrm{EC}$. Ergo ex æquo perturbatè $\mathrm{BC}.\mathrm{FC}\colon\colon fl\mathrm{AC}.fl\mathrm{BC}$. Q.E.D.

Cor. 1{.} Si a {sic} D demittatur K perpendicularis ad BC erit $\mathrm{AC}.\mathrm{KC}\colon\colon fl\mathrm{AC}.fl\mathrm{BC}$ quovis \rectæ DC/ puncto R rectæ DE demittantur ad AC et BC perpendicula RS et RT: erit {illeg} $\mathrm{SC}.\mathrm{TC}\colon\colon$$fl\mathrm{AC}.fl\mathrm{BC}$. Nam $\mathrm{RS}.\mathrm{RT}\colon\colon \mathrm{BC}.\mathrm{FC}$ $\mathrm{SC}.\mathrm{TC}\colon\colon \mathrm{BC}.\mathrm{FC}$. |Est $fl:\mathrm{AC}$ ad $fl:\mathrm{BC}$ ut {illeg}|c|osinus anguli ACD ad cosin: ang. BCD. Nam cosinus isti sunt ut BC ad FC.|

C{illeg}|o|r 2. Si punctum A infinitè distet a B, hoc est si AC sit ipsi AB parallela, age quamvis {illeg} RQ occurrenten {sic} AC in Q, sitqꝫ RQ positione data, et demisso ad BC normali RT, erit $\mathrm{TC}.\mathrm{QC}\colon\colon fl:\mathrm{BC}.fl\mathrm{QC}$. {illeg} Nam demisso insuper ad QC normali RS, erit per Cor. 1, $\mathrm{TC}.\mathrm{SC}\colon\colon fl:\mathrm{BC}.fl\mathrm{AC}=fl:\mathrm{SC}$, & propter datam rationem SC ad QC erit per ax 2 $\mathrm{SC}.\mathrm{QC}\colon\colon fl:\mathrm{SC}.fl:\mathrm{QC}$. Ergo additis rationibus $\mathrm{TC}.$$\mathrm{QC}\colon\colon fl:\mathrm{BC}.fl:\mathrm{QC}$. Q.E.D.

Theor 5

Hujusmodi alia T{illeg}|h|eoremata non inutilia proponi possent: sed ad fluxiones superficierum festinamus.

Theor. 5. Si recta quævis motu parallelo per aream aliquam, a duabus parallelis & positione datis rectis terminatam transferatur: erit fluxio areæ ut fluxio alterutrius rectæ parallelæ. Sint AB, DC rectæ parallelæ, & BC recta per spatium interjectum ADCB in data inclinati{illeg}|o|ne ABC translata et AD terminus a quo incipit transferri; et erit $fl:\mathrm{ADCB}$ ut $fl\mathrm{AB}$. Nam area BD est ut longitudo AB Quare per Ax: 2 $fl:\mathrm{BD}$ ut $fl:\mathrm{AB}$.

Schol.. Si angulus ABC rectus sit, tum quemadmodum statui solet $\mathrm{BD}=\mathrm{AB}\times \mathrm{BC}$, sic nos statuemus <39> /> $fl\mathrm{BD}=fl\mathrm{AB}\times \mathrm{BC}$ hoc est (per Cor 7 Th 1) $=\mathrm{BC}\times fl\mathrm{AB}$. Sed hic sicut per $\mathrm{AB}\times \mathrm{BC}$ non intelligitur linea sed productum arithmeticum quod exprimit numerum unitatum in area BD ex Hypothesi quod unitas superficialis sit quadratum cujus latera sunt unitates lineares: sic in hoc Scholio per $\mathrm{BC}\times fl\mathrm{AB}$ non intelligitur fluxio linearis sed fluxio generans productum arithmeticum quod \exprimit/ numerum unitatum superficialium in BD exprimit ex {hac} Hypothesi quod {BD} momentum basis fluentis ductum in datam al{illeg}titudinem parallelogrammi facit momentum parallelogrammi.

Theor 6. Si recta quævis motu parallelo transferatur per aream alijs duabus {illeg} ad {illeg} positione datis et non parallelis rectis terminatam, erit fluxio areæ ut fluxio alterutrius rectæ positione datæ ducta in rectam mobilem. Transferatur BC per spatium CAB rectis AC AB positione datis terminatum: et erit $fl:\mathrm{CAB}$ ut $\mathrm{BC}\times fl:\mathrm{AB}$. Etenim triangulum CAB est ut ${\mathrm{AB}}^q$. ergo, per Ax. 2, fl: tri: CAB est ut $fl{\mathrm{AB}}^q$. Sed per Cor: 6 & 9, Th: 1, $fl{\mathrm{AB}}^q$ est ut $2\mathrm{AB}\times fl\mathrm{AB}$ Ergo hoc est cùm $2\mathrm{AB}$ et BC sint in data ratione, ut $\mathrm{BC}\times fl\mathrm{AB}$.

Schol. Si ang ABC rectus sit, potest juxta Schol præcedens, poni $fl\mathrm{CAB}=\frac{1}{2}\mathrm{BC}\times fl:\mathrm{AB}$.

Theor 7{.} Si recta circa datum punctum gyrans, continuò terminetur ad aliam rectam positione datam: erit fluxio spatij a gyrante recta descripti, ut fluxio alterius rectæ. Gyret recta CB circa punctum C sitqꝫ CA terminus a quo incipit gyrare, et AB recta ad quam terminatur, et erit area ABC ut recta AB; adeoqꝫ (per Ax 2) $fl\mathrm{ABC}$ ut $fl\mathrm{AB}$.

Schol. Demisso ad AB normali CD, erit (juxta Schol Theor 5) $\frac{1}{2}\mathrm{CD}\times fl:\mathrm{AB}=fl\mathrm{ABC}$, quia $\frac{1}{2}\mathrm{CD}\times \mathrm{AB}=\mathrm{ABC}$.

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Theor 8. Iisdem positis, si ab alio insuper quovis dato puncto recta perpetim ducatur ad concursum priorum rectarum, et a primo puncto ad hanc rectam agatur demittatur \linea/ perpendicularis terminatam ad rectam positione datam: erit fluxio hujus novæ rectæ ducta {illeg} in lineam perpendicularem, ut fluxio areæ a prima recta descriptæ. A da

A dato E agatur EB, et ad hanc perpendicularis {illeg} CH occurrens AB in H, eritqꝫ $\mathrm{CH}\times fl:\mathrm{EB}$, ut $fl:\mathrm{ABC}$. Nam ad demisso ad AB normali EF, erit per Cor: 1 Th: 2, $fl\mathrm{AB}\left(fl:\mathrm{FB}\right).fl:\mathrm{EB}\colon\colon \mathrm{EB}.\mathrm{FB}$. (hoc est propter sim. tri. EBF, HCD) $\colon\colon \mathrm{HC}.\mathrm{CD}$. Ergo $\mathrm{CD}\times fl:\mathrm{AB}=\mathrm{HC}\times fl:\mathrm{EB}$. Quare cùm CD datum sit, adeoqꝫ $\mathrm{CD}\times fl:\mathrm{AB}$ ut $fl:\mathrm{AB}$, sitqꝫ etiam (per Th 7) $fl\mathrm{AB}$ ut $fl\mathrm{ABC}$, erit $\mathrm{HC}\times fl\mathrm{EB}$ ut $fl\mathrm{ABC}$.

Schol. Est et per Schol. sup. $\frac{1}{2}\mathrm{HC}\times fl\mathrm{EB}=fl\mathrm{ABC}$$\left(=\frac{1}{2}\mathrm{CD}\times fl\mathrm{AB}\right)=fl\mathrm{ABC}$. Et insuper in eo temporis momento quo contingit angulum EBC rectum esse, est $\frac{1}{2}\mathrm{BC}\times fl:\mathrm{EB}$$=fl:\mathrm{ABC}$ quia tunc HC et BC coincidunt.

Theor 9. Si recta circa datum punctum gyrans, secet alias duas positione datas rectas: superficie{illeg}|r|u inter datum punctum et rectas positione dat{illeg}|a|s isto motu generataru \fluxiones/ erunt ut quadrata longitudinum generantium. Sit A datum punctum circa quod AC gyrat, sintqꝫ BD et CD rectæ positione datæ, & AD principium a quo AC incipit gyrare, et erit $fl:\mathrm{ADB}.fl\mathrm{ADC}\colon\colon {\mathrm{AB}}^q.{\mathrm{AC}}^q$. Sit enim {illeg} Abc positio rectæ gyrantis in proximo temporis momento, et triangula infinitè parva ABb, ACc erunt momenta superficierum ADB, ADC, adeoqꝫ ut ipsarum fluxiones. Sed per 15. 6. Elem. ista triangula sunt ut $\mathrm{AB}\times \mathrm{Ab}$ ad $\mathrm{AC}\times \mathrm{Ac}$: Quæ ratio, si Abc retro volvatur donec redeat in AC, in ultimo ejus regressûs momento, hoc est in primo momento progressûs ubi AC incipit <41> pergere ad Ac evadit ${\mathrm{AB}}^q$ ad ${\mathrm{AC}}^q$. Q.E.D.

Theor: 10. Si recta positione data tangat curva positione datam et utraqꝫ secentur ab alia utcunqꝫ motâ rectâ: fluxiones curvæ illius & tangentis ejus in eo temporis momento æquales erunt, quo mota illa linea secat utramqꝫ in puncto contactû;s. Esto curva RS, Tangens ejus AB punctum contactûs C et linea mobilis DE: dico fluxiones linearum RC et AC æquales evadere quando DE pertingit ad C. Nam in RC sumatur arcus infinitè parvus Cc, & cum hæc juxta Hypothesin Archimedeam pro recta haberi possit, produc eam utrinqꝫ in directum, sitqꝫ ea producta AB propterea quod ipsa AB tantùm tangat curvam. RS itaqꝫ et AB commune habent momentum Cc, & proinde eandem fluxionem dum DE {illeg}|t|ransit per illud momentum. Q.E.D.

Cor. Hinc omnia quæ in Theor: 3 et 4 de fluxionibus rectarum positione datarum demonstrata sunt, conveniunt etiam fluxionibus curvarum quas rectæ illæ tangunt in intersectione cum linea mobili.

Theor. 11|2|. Si recta illa DE circa datum punctum D convoluta, describat duas superficies quarum una DRC terminatur ad curvam RC, altera DAC ad tangentem curvæ AC: fluxiones illarum superficierum æquales erunt in eo temporis momento quo recta circumacta transit per punctum contactus C {illeg}. Nempe $fl\mathrm{DRC}=fl\mathrm{DAC}$ quia tunc commune est utriusqꝫ momentum CDc.

Cor:{.} Hinc omnia quæ in Theor 7, 8 & 9 de fluxionibus superficierum rectis positione datis terminatarum demonstrata sunt, conveniunt etiam fluxionibus superficierum terminatarum curvis positione datis quas rectæ illæ tangunt.

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Theor 12|3|. Si recta mobilis perpetuò tangat curvam, punctum contactûs in si|om|ni temporis momento erit centru{illeg}|m| circa quod recta in illo momento volvitur. Concipe Curvam TV lineolis parvitate et multitudine infinitis constare quarum duæ sunto AB & BC. Hasce produc utrinqꝫ in directum, nempe AB ad D et E et BC ad d et e, et manifestum est quod \tangens mobilis/ in eo temporis momento quo tangens mobilis volvitur de loco DE in locum de, convertitur circa punctum contactus B, propterea quod istud B sit communis intersectio locorum DE ac de.

Cor 1. Hinc omnia quæ in Theor 3, 4, 7, 8, 9, 10, 11 de recta circa datum punctum \ceu centrum/ volvente demonstrata sunt, conveniunt etiam rectæ perpetuò tangente|i| curvam lineam positione datam, si modò punctum contactûs pro centro habeatur circa quod recta illa in momento contactûs illius convolvitur, vicem centri dati gerere concipiatur. Et proinde sigillatim{.}

Cor 2. S

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cruris ad defluxionem alterius ut illud alterum crus ad crus primum. Detur C, et erit $\mathrm{B}.\mathrm{A}\colon\colon fl:\mathrm{A}.fl\mathrm{B}$ propterea quod $\mathrm{C}\times fl:\mathrm{C}$ nihil sit.

Theor. 3. Si recta utcunqꝫ moveatur per superficiem aliquam: fluxio superficiei isto motu generatæ erit ut fluxio alterius alicujus rectæ a dato puncto ad medium primæ rectæ demissæ, ducta in primam rectam. Describat recta AB superficiem ABC motu quocunqꝫ, & ad medium ejus punctum D erigatur perpendiculum DE ad datum quodvis punctum E terminatum: et fluxio superficiei ABC erit ut $\mathrm{AB}\times fl:\mathrm{ED}$. Sit enim adb, positio r in momento temporis proximè sequenti, positio rectæ AB motu quocunqꝫ prolabentis, et erit ABβ ABba momentum superficiei ABC, et Dd mo momentum contemporaneum momentum rectæ ED. Quare cùm ABba æquetur $\mathrm{AB}\times \mathrm{Dd}$, & fluxiones sint ut momenta, erit fluxio generans ABba ut AB in fluxionem generantem Dd. Q.E.D. Quod autem dixerim esse $\mathrm{AB}\times \mathrm{Dd}=$ $\mathrm{ABba}=\mathrm{AB}\times \mathrm{Dd}$: ad A et B erige ipsi AB perpendicula AF, BG occurentia Aα, Bβ occurentia ab in α & β, et erit trapezium $\mathrm{AB\beta \alpha }\left(=\mathrm{AB}\times \stackrel{\_}{\frac{1}{2}\mathrm{B\beta }+\frac{1}{2}\mathrm{A\alpha }}\right)=\mathrm{AB}\times \mathrm{Dd}$. Et hoc trapezium non differt a momento ABβα {sic} nisi in spatijs Bβb & Aαa, quæ \ex Hypothesi/ sunt infinitè minora dicto trapezio & ad eam collata instar nihili, vel puncti ad lineam.

Schol. Dixi fluxionem superficiei ACB esse ut $\mathrm{AB}\times fl:\mathrm{ED}$: sed quemadmodum Geometræ statuere solent parallelogrammum æquale esse facto ex lateribus, quod (veriùs loquendo) est ut factum ex lateribus; sic eg{illeg} ego ob usum commodiorem Theorematis hujus dicam in sequentibus esse $fl:\mathrm{ACB}$$=\mathrm{AB}\times fl:\mathrm{ED}$.

Cor. Si in semicirculo ACB recta AC generet segmentum ACE gyrando circa termi/num\ <46> diametri A, turn ab altero diametri termino B demisso BC normali ad AC, erit fluxio segmenti ACE ut dimidium fluxionis perpendiculi BC ductum in AC: vel juxta Scholium $fl:\mathrm{ACE}=\frac{1}{2}\mathrm{AC}\times fl\mathrm{BC}$.

Jactis hisce demonstrationum fundamentis, methodus tenendi demonstrationes, uno et altero exemplo constabit. Proponatur itaqꝫ constructio in Exemplo 2^do demonstranda. Per Cor. 2 Th. 1, est $fl:\mathrm{ID}$ ad $fl:\mathrm{IP}$ {illeg} ut AI ad ID. Estqꝫ AI ad ID {illeg}|ut| ID ad CE ex natura Curvæ AGE. Et proinde $\mathrm{CE}\times fl:\mathrm{ID}=\mathrm{ID}\times fl:\mathrm{IP}$. Sed \(per Schol. Th. 3)/ $\mathrm{CE}\times fl\mathrm{ID}=$ fluxioni areæ ACEG, et $\mathrm{ID}\times fl:\mathrm{IP}=$ fluxioni areæ PDI. et proinde areæ illæ per Ax: 1 æquantur. Q.E.D.

Proponatur denuò constructio qua Cissoidis area in Exemplo 3 determinatur. Ad hanc autem demonstrandam, lineæ punctim notatæ deleantur in schemate deleantur et aga\n/tur DQ, \AE/ et Cissoidis Asymptoton QR. Iam propter $\mathrm{AQ},\mathrm{DQ},\mathrm{CQ}∺$, est (per Cor 2 Th 1) $fl\mathrm{DQ}.$$fl\mathrm{CQ}\colon\colon \mathrm{DQ}.2\mathrm{CQ}$. Et propter sim. tri. QDC, DEA, est $\mathrm{DQ}.2\mathrm{CQ}\colon\colon \mathrm{ED}.2\mathrm{AD}$. Ergo $fl\mathrm{DQ}.fl\mathrm{CQ}\colon\colon \mathrm{ED}.2\mathrm{AD}$. & $\mathrm{ED}\times fl\mathrm{CQ}=2\mathrm{AD}\times fl:\mathrm{DQ}$ sive = $4\mathrm{AD}\times \frac{1}{2}fl\mathrm{DQ}$ {illeg} $4\times \frac{1}{2}\mathrm{AD}\times fl\mathrm{DQ}$. Sed per Cor. 1, Theor. 3, est $\frac{1}{2}\mathrm{AD}\times fl\mathrm{DQ}$= fluxioni generanti aream ADOQ, est et ejus quadruplum $\mathrm{ED}\times fl\mathrm{CQ}=$ fluxioni generanti Cissoidalem aream QREDO. Et proinde \per Ax: 2/ area illa infinite longa QREDO generatur quadrupla alterius ADOQ. Q.E.D.

Deniqꝫ ad demonstrandam constructionem areæ Conchoidalis in Exempl: 4; Ubi demonstraveris ut supra quod sit $\mathrm{AG}.\mathrm{AP}\colon\colon \mathrm{DE}.\mathrm{MK}$, sic procede. Est autem (per Cor. 2, Th. 3) $\mathrm{MK}\times fl\mathrm{PC}=fl:\mathrm{are\ae \; PKC}$ & $\mathrm{DE}\times fl\mathrm{PC}$$=fl\mathrm{are\ae \; DPE}$. Ergo $fl\mathrm{PKC}.fl\mathrm{DPE}\left(\colon\colon \mathrm{MK}.\mathrm{DE}\right)\colon\colon$$\mathrm{AP}.\mathrm{AG}$. Adeoqꝫ per Ax. 2 Areæ PKC et DPE sunt in eadem ratione.