Solutio Problematis. A Clariss. viro D. Jo. Bernoulli in Diario Gallico Febr. 1403. Propositi. Quam D. G. Cheynaeo Communicavit Jo. Craig

were so divided, that they never fell to the Earth, but were exhaled up into the Clouds. In the said small Particles of Water are conveyed the above-mentioned small Animalcula far up into the Land, and when the Ground becomes dry, they contract themselves into an oval Figure, and the Pores of their Skin are so well clos'd, that they do not perspire at all, whereby they preserve themselves till it Rains, upon which they open their Bodies and enjoy the moisture. And thus, in my poor opinion, it happens that we find these Animalcula in every Meadow of our Country, none of which are very remote from the Sea or Water Canals. II. Solutio Problematis. A Clariss. viro D. Jo. Bernoulli in Diario Gallico Febr. 1403. Propositi. Quam D. G. Cheynæo communicavit Jo. Craig. Problema. Propositæ Curvæ Geometricæ alias innumeræ Longitudine æquales invenire. Solutio. Sint \( w, s \), co-ordinatae Curvæ datae; & Curvæ quaestæ sint co-ordinatae \( x, y \); tum ex conditione Problematis erit \( dw^2 + ds^2 = dx^2 + dy^2 \). Ponatur \( dx = dw - m dz \), unde erit \( dy = \sqrt{ds^2 + 2m dw dz - m^2 dz^2} \); in hac pro \( ds \) substituatur ejus valor per \( w, dw \) & determinatas expressus: & pro \( dz \) assumatur talis valor ex \( w, dw \) & determinatis compositus, ut valores quantitatum \( dx, dy \) sint summabiles: Et sic habentur \( x \) ac \( y \) Co-ordinatae Curvæ quaestæ. Q. E. J. Exemplum 1. Invenire Curvam æqualem Lineæ Parabolicæ. Sit \( z \) latus rectum Parabolæ; adeoque \( z \) \[ ds = w^2 \quad \text{unde } ds^2 = a^2 - w^2 \quad \text{adque } dy = \sqrt{a^2 - w^2} \quad dw + 2m \quad dw \quad dz - m^2 \quad dz^2; \quad \text{ut hæc sit summabilis assumatur } mdz = \frac{w^2}{a^2} \quad dw; \quad \text{unde } dx = dw - a^2 \quad w^2 \quad dw; \quad dy = dw \quad \sqrt{3a^2 - w^2} - a^4 \quad w^4 \quad \text{quarum integrales per Methodos dudum cognitas inveniuntur } x = w - \frac{w^3}{3a^2}, y = \frac{w^2 - 3a^2}{3a^2} \quad \sqrt{3a^2 - w^2}. \] Exemp. 2. Invenire Curvam æqualem Circulari. Sit a radius Circuli; tum \( s = \sqrt{a^2 - w^2} \); unde \( ds^2 = \frac{w^2}{a^2} \quad dw^2 \); & proinde erit \( dy = \sqrt{\frac{w^2}{a^2 - w^2}} + 2m \quad dw \quad dz - m^2 \quad dz^2; \quad \text{ut hæc sit summabilis, assumatur } mdz = \frac{4w^2}{a^2} \quad dw, \quad \text{adcoq; } dx = dw - \frac{4w^2}{a^2} \quad dw; \quad dy = -\frac{3a^2w + 4w^3}{a^2 \sqrt{a^2 - w^2}} \quad dw. \quad \text{Quarum integrales per communes Methodos inveniuntur } x = w - \frac{4w^3}{3a^2}, y = \frac{a^2 - 4w^2}{3a^2} \quad \sqrt{a^2 - w^2}: Exemp. 3. Invenire Curvam æqualem Ellipticæ. Sit \( r \) latus rectum, \( 2a \) latus transversum, tum \( s = \frac{r \sqrt{a^2 - w^2}}{a} \), unde erit \( ds^2 = \frac{r^2}{a^2} \quad w^2 \quad dw^2 \), adque \( dy = \sqrt{\frac{r^2}{a^2 - a^2w^2}} + 2m \quad dw \quad dz - m^2 \quad dz^2; \quad \text{ut hæc sit summabilis assumatur } mdz = \frac{2a + 2r}{a^2} \quad w^2 \quad dw : \quad \text{unde } dx = dw - \frac{2a + 2r}{a^2} \quad w^2 \quad dw. \quad dy = dw \quad \sqrt{\frac{r^2}{a^2 - a^2w^2}} + \frac{2a + 4r}{a^2} \quad w^2 + \frac{2a + 2r}{a^2} \quad w^4; \quad \text{quarum Inte-} tegrales per Methodos novissimos inveniuntur \( x = \frac{2a^3 - ra^2}{3a^2} \) \( y = \frac{2a^3 - ra^2 - 2aw^2 + 2w^3}{3a^2} \) Exemp. 4. Invenire Curvam æqualem Parabœæ Cubicæ cujus æquatio sit \( 3a^2 s = w^3 \). Unde \( ds^2 = \frac{w^4 dw^2}{a^2} \) & proinde \( dy = \sqrt{a^4 w^4 dw^2 + 2m dw dz - m^2 dz^2} \); Ut hæc sit summabilis assumatur \( mdz = \frac{w^2 dw}{2a^2} \). Unde \( dx = dw - \frac{w^2 dw}{2a^2} \sqrt{3w^2 + 4a^2} \). Quorum integrales per Methodos vulgo notas sunt \( x = w - \frac{w^3}{6a^2}, y = \frac{2}{9} + \frac{1}{3} w^2 + 4a^2 \). Ex aliis infinitis valoribus quantitatis \( m dz \) debité assumptis infinitas invenias Curvas dataæ æquales. Tu vero, vir Eruditissime, facilè percipias hoc Problema aliquam habere cum Problemate quodam Diophantœ affinitatem: Problema Dio- phanti est, dividere summam duorum Quadratorum in duo alia quadrata, quorum latera sint rationalia; & Problema Bernoullii est, dividere summam duorum Quadratorum in alia duo Quadrata, quorum latera sint summabilia. Sic ut Problematis Diophantœi solutio a vulgari tantum Algebra de- pendet, sic Bernoulliani Problematis solutio communes tan- tum Fluxionum Methodos inversas requirit: utriusq; artefi- cium in debito laterum quaestiorum summatione consistit; sed. Diophantœum ut sint rationalia, Bernoullianum ut sint sum- mabilia. III. Part