Solutio problematis Leibnitiani, de curva accessus & recessus aequabilis a puncto dato, mediante rectificatione curvae elasticae

Solution of Leibniz's problem, on the curve of uniform approach to and recession from a given point, by means of the rectification of the elastic curve

Jacob Bernoulli · 1694 · Acta Eruditorum, June, p. 276 · ai-draft

Source: original scan (archive.org:s1id13206610 (Acta Eruditorum 1694, June, pp. 276–280))
"Solutio problematis Leibnitiani, de curva accessus & recessus aequabilis a puncto dato, mediante rectificatione curvae elasticae", Acta Eruditorum, Leipzig (June 1694), pp. 276–280 — the closing section of Bernoulli's June 1694 elastica article, introduced by the running heading "Jac. B. Solutio problematis Leibnitiani …" (not a separately numbered paper in the Acta).

Significance

Leibniz had closed his 1689 isochrone paper by posing a harder problem: to find the paracentric isochrone, a curve of descent along which a heavy body approaches (or recedes from) a fixed point at a uniform rate. Here Jacob Bernoulli gives the first complete solution and construction — something, he notes, neither Leibniz nor anyone else had managed — reducing the problem first to a first-order differential equation and then, in its most celebrated step, to the rectification of the elastic curve (the elastica), whose arc length he expresses by an integral of the form $\displaystyle\int \frac{a^{2}\,du}{\sqrt{a^{4}-u^{4}}}$. In a sequel three months later he reduced the same construction to the rectification of a new algebraic curve he named the lemniscate. Integrals of this lemniscatic type are the ones Fagnano (1718) and then Euler (1751/1761) turned into the addition theorems at the origin of the theory of elliptic functions, and through Abel, Jacobi and Riemann into the modern theory of elliptic and Abelian functions.D. van der Lugt, "The History of the Riemann–Roch and Hirzebruch–Riemann–Roch Theorem", MSc thesis, Radboud University Nijmegen (2025) — §2, on the paracentric isochrone, the elastica and lemniscate, and the origins of elliptic-function theory.

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page 276

Jac. B. Solutio Problematis Leibnitiani

de Curva Accessus & Recessus aequabilis a puncto dato, mediante rectificatione Curvae Elasticae

page 277

Felicitatem inventi praecedentis commendare potest solutio elegantissimi Problematis Leibnitiani, de invenienda Linea, per quam descendens grave aequaliter aequalibus temporibus a dato puncto recedat, vel ad illud accedat: quod laudatissimo suo Auctori ita placuit, ut non tantum ad ejus tentamen Amicos pariter & Adversarios aliquoties in Actis provocarit (vid. Anno 1689, Apr. p. 198, & 1690. Maj. p. 229, & Jul. p. 360,) sed & ipse quoque strenue in illo desudarit, testibus nonnullis Curvae proprietatibus, quas Gallorum Diario inseri curavit, ut ex relatione Fratris habeo, qui tamen illas nominare mihi non potuit. Multus etiam in eodem fuit ipse Frater, dum Parisiis ageret, sed omni sua methodo Tangentium inversa aliud nil effecit, quam ut Problema ad hanc aequationem differentialem reduceret: $(x\,dx + y\,dy)\sqrt{y} = (x\,dy - y\,dx)\sqrt{a}$, ad quam tamen etiam nullo labore pervenitur. Plenariam vero illius solutionem, sive constructionem, nec ipse nec quisquam alius dare potuit. Nuper cum praecedens Schediasma pararem Lipsiam, partemque inventi cum figuris ipsis jamjam chartae consignassem, scrutinium hujus Lineae suscepi occasione novorum Theorematum, quae ibi dedi pro Radiis circulorum osculantium in Spiralibus (quorum tamen investigatio & ipsa incidenter tantum fiebat; praecipua enim inventi capita sub calamo demum mihi nascebantur) mox etiam post pauca conamina optatum finem consecutus sum; ubi hoc perquam singulare mihi accidit, ut absoluta analysi animadverterem, Curvam Elasticam, quae qualemcunque tantum occasionem huic tentamini dederat, illam ipsam esse, cujus rectificatione altera posset construi. Nam quanquam idem liceat exequi mediante quadratura spatii alicujus Algebraici, alterum tamen construendi modum praeferendum censeo, tum quod generaliter facilius in praxi rectificentur curvae, quam quadrentur spatia, tum praesertim quod ipsa natura (si quidem convenientem tensionis legem observet) illum praescripsisse videatur.

Constr. Descripta itaque Curva Elastica tertiae constructionis (Fig. 3) caeterisque positis ut prius, applicetur in utroque quadrante circuli $BL$, $LG$, recta $\varepsilon\zeta$ aequalis ipsi $Ag$ seu tertiae proportionali ad $AB$ & $AE$ (hic in dextro tantum quadrante ducta apparet, ut linearum confusio vitetur) & ex ducta $A\varepsilon$ (productaque si sit opus) abscindatur $A\alpha$ aequalis tertiae proportionali ad rectam $AB$ & portio-

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nem Curvae $AQ$ in sinistro, vel $\varphi RQ$ in dextro quadrante; eritque punctum $\alpha$ in Curva quadam $A\chi\omega\eta$, ita comparata, ut grave per illam latum (si prius ex altitudine $iA$ deciderit) aequalibus temporibus aequaliter ad punctum $A$ accedat, aut ab illo recedat. Cujus constructionis demonstrationem (ut Celeb. Problematis Auctorem, aliis occupatissimum, examinandi labore sublevem) hic appono.

Fig. 3: the elastic curve, the isochronal curve
$A\chi\omega\eta$, and the auxiliary circle, quadrants and construction lines used in the
demonstration. — Fig. 3.

Dem. Descriptus intelligatur centro $A$ arculus $\beta\delta$, abscindens ex applicata $A\alpha$ Elementum $\alpha\beta$, & ex Curva Elem. $\alpha\delta$, cui concipiatur in vertice portiuncula isochronos $A\omega$. Jam Ratio $\beta\delta$ ad $\alpha\beta$ composita est ex sequentibus 5 rationibus:

\[ \begin{aligned} \beta\delta : \text{Elem. periph. } B\varepsilon &:: A\alpha : A\varepsilon. \\ \text{Elem. periph. } B\varepsilon : \text{Elem. } \varepsilon\zeta &:: A\varepsilon : A\zeta, \qquad \text{ex natura Circuli.} \\ \text{Elem. } \varepsilon\zeta : \text{Elem. } AE &:: 2AE : AB, \qquad \text{per constr. \& princ. calcul. in fin. parvorum.} \\ \text{Elem. } AE : \text{Elem. } AQ\ (\varphi RQ) &:: A\zeta : AB, \qquad \text{ex nat. curvae } AQ, \text{ ut collig. ex coroll. 2. constructionis tertiae praeced.} \\ \text{Elem. } AQ\ (\varphi RQ) : \alpha\beta &:: AB : 2AQ\ (2\varphi RQ), \qquad \text{per constr. \& princ. calc. inf. parv.} \end{aligned} \]

unde facta divisione per communes rationum antecedentes & consequentes, $\beta\delta : \alpha\beta :: A\alpha : AB + AE : AQ\ (\varphi RQ,)$ & $\beta\delta^{2} : \alpha\beta^{2} :: A\alpha^{2} : AB^{2} + AE^{2} : AQ^{2}$ vel $\varphi RQ^{2}$ ($\varepsilon\zeta : A\alpha$, per constr.) $:: A\alpha^{2} \times \varepsilon\zeta : AB^{2} \times A\alpha :: A\alpha \times \varepsilon\zeta : AB^{2} ::{}$ (ob similitudinem Triangulorum $A\alpha\gamma$, $A\varepsilon\zeta$) $A\varepsilon \times \alpha\gamma : AB^{2} :: \alpha\gamma : AB\ (Ai)$, componendoque $\alpha\delta^{2} : \alpha\beta^{2} :: \alpha\gamma + Ai : Ai$; & proinde $\alpha\delta : \alpha\beta :: \sqrt{\alpha\gamma + Ai} : \sqrt{Ai} ::{}$ celeritas acquisita in $\alpha$ : celer. acquis. in $A ::{}$ (ex natura descensus gravium) $:: \alpha\delta : A\omega$ (per hyp. quia aequalibus tempusculis transiri supponuntur). Ergo $\alpha\beta = A\omega$. Ergo grave per hanc Curvam latum aequaliter aequalibus temporibus a puncto $A$ recedit. Q. E. D.

Quae hic praecipue observanda veniunt, sunt sequentia:

1. Si grave altius vel humilius, quam ex $i$, decidere supponatur, puta ex $\tau$, non opus est nova Elastica: Sufficit haec una omnibus Curvis isochronis describendis. Omnes enim inter se similes sunt, determinanturque, faciendo tantum, ut $Ai$ ad $A\tau$, sic $A\alpha$ ad aliam $A\alpha$ abscindendam ex eadem $A\varepsilon$. Estque ipsa curva Schematis nostri constructa pro gravi tanquam ex $\tau$, non $i$, decidente.

2. Sed & infinitae dantur aliae, quas grave ex eadem altitudine

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$\tau A$ delapsum ita permeare potest, ut aequabiliter ad punctum aliquod accedat vel recedat; at illud continuo diversum ab $A$; nec nisi per unicam curvam ex eadem altitudine, respectu ejusdem puncti, accessus vel recessus aequabilis effici potest.

3. Curva nostra Isochronos initium capit in ipso puncto $A$, cujus respectu motus aequabilis est, & terminatur in puncto $\eta$ ejusdem cum illo altitudinis; utraque vero extremitate tangit horizontalem rectam $A\eta$. Unde flexum contrarium habere constat. Quod si ex altera parte statuatur huic similis & aequalis curva $A\lambda\theta\psi$, & grave dimittatur ex $\eta$ cum celeritate, quam acquirere potest cadendo ex altitudine aequali ipsi $\tau A$, illud primo descendendo ad $\omega$, mox reascendendo, aequabiliter ad $A$ accedet, indeque pergendo continue per alteram $A\lambda\theta\psi$ ab eodem puncto $A$ eadem ratione elongabitur.

4. Tangens Curvae in locis intermediis, velut $\alpha$, reperitur, si ducta $A\mu$ perpendiculari ad $A\alpha$, fiat, ut $\sqrt{A\tau}$ ad $\sqrt{\alpha\gamma}$; ita $A\alpha$ ad $A\mu$; ducta enim $\alpha\mu$ curvam tanget: quod vel ex usu, quem praestare debet curva, facile infertur.

5. Recta $A\eta$ quadrupla est rectae $A\theta$.

6. Si in Elastica punctum $Q$ assumatur tale, ut tertia proportionalis ad subtangentem $Pp$ & rectam $AB$ aequetur curvae $AQ$, invenietur ejus ope punctum $\chi$ portionis $A\chi\theta$, quod omnium maxime distat a perpendiculari $A\theta$.

7. Sin punctum $Q$ ponatur tale, ut tangens $Qp$ aequetur curvae $\varphi RQ$, illo mediante habebitur in altera punctum $\omega$, omnium infimum & remotissimum ab horizantali $A\eta$.

8. Tandem & hoc advertendum est, quod ex propositi Problematis constructione proclive jam colligere sit, quales suppositiones faciendae, ad reducendam aequationem $(x\,dx + y\,dy)\sqrt{y} = (x\,dy - y\,dx)\sqrt{a}$. Nam si loco $x$ & $y$ assumantur duae aliae indeterminatae $t$ & $z$, ponendo $ay = tz$, & $ax = t\sqrt{aa - zz}$; prodibit aequatio, in qua separari possunt indeterminatae $t$ & $z$ cum suis differentialibus a se invicem; quod sufficit ad constructionem deinde, saltem per quadraturas, expediendam. Duo enim in hoc calculo adhuc praecipue desiderari videntur, quae si semper fieri possent, omnia reperta essent; unum ut differentialia secundi altiorumve generum ad differentialia primi reducantur; alterum, ut in aequationibus differentialibus primi

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generis indeterminatae, si invicem permixtae sint, a se mutuo separentur, ut unaquaeque cum sua differentiali peculiarem aequationis partem constituat, h. e. ut reperiatur methodus inversa tangentium. Pro utroque dedi Regulas quasdam (pluresque sine fine dare possem) similes illis, quas Frater Parisiis apud Marchionem Hospitalium deposuit, quibusque methodum inesse initio existimarat. At statim sensi, illas non continere nisi artificia quaedam particularia, quae methodum appellare non ausim, utpote qualem non magis dari posse arbitror, quam dari potest universalis methodus pro construendis Aequationibus Algebraicis quorumvis promiscue graduum. Illa enim (cujus olim in Actis mentio facta est) qua pono indefinite $0 = a + bx + cy + exy$, &c. & quaero deinde ejus ope lineae tangentem, vix aliter quam in speculatione succedere videtur. Quod dictis fidem faciat, hoc esto, quod praesens Problema, quanquam ut apparet haud admodum difficile, nullius regulae methodive hactenus repertae ope solvi a quoquam potuerit.

Produced by claude-opus-4-8, effort: high (2026-07-30)

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Jac. B. Solution of Leibniz's Problem

on the Curve of uniform Approach to and Recession from a given point, by means of the rectification of the Elastic Curve

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The felicity of the preceding discovery may be commended by the solution of that most elegant Leibnizian Problem, of finding the line along which a descending heavy body recedes uniformly, in equal times, from a given point, or approaches it: which so pleased its most praised Author that he not only several times challenged Friends and Adversaries alike, in the Acta, to attempt it (see Anno 1689, Apr. p. 198, and 1690. May p. 229, and Jul. p. 360,) but himself also strenuously toiled at it, as witnessed by several properties of the Curve which he had had inserted in the French Journal, as I have from the report of my Brother, who nevertheless could not name them to me. My Brother himself, too, was much occupied with the same thing while he was staying at Paris, but with all his inverse Method of Tangents he accomplished nothing else than to reduce the Problem to this differential equation: $(x\,dx + y\,dy)\sqrt{y} = (x\,dy - y\,dx)\sqrt{a}$, to which, however, one arrives even with no labour. But its full solution, or construction, neither he nor anyone else was able to give. Recently, while I was preparing the preceding essay for Leipzig, and had already committed to paper part of the discovery together with the figures themselves, I undertook the scrutiny of this Line, on the occasion of the new Theorems which I gave there for the Radii of the osculating circles in Spirals (whose investigation, however, was itself made only incidentally; for the chief points of the discovery were only coming to birth under my pen) soon after a few attempts I attained the wished-for end; whereupon this most singular thing befell me, that, the analysis being fully worked out, I noticed that the Elastic curve, which had given only some chance occasion to this attempt, was that very one by whose rectification the other could be constructed. For although the same might be accomplished by means of the quadrature of some algebraic space, I nevertheless judge the other mode of construction to be preferred, both because in general curves are more easily rectified in practice than spaces are squared, and especially because nature herself (provided she observes a suitable law of tension) seems to have prescribed that one.

Constr. The Elastic curve of the third construction (Fig. 3) having therefore been described, and the rest being placed as before, let there be applied, in each quadrant of the circle $BL$, $LG$, the straight line $\varepsilon\zeta$ equal to $Ag$ itself, that is, to the third proportional to $AB$ and $AE$ (here it appears drawn only in the right quadrant, so that confusion of the lines may be avoided) and from the drawn $A\varepsilon$ (produced, if there be need) let there be cut off $A\alpha$ equal to the third proportional to the straight line $AB$ and the

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portion of the curve $AQ$ in the left, or $\varphi RQ$ in the right quadrant; and the point $\alpha$ will be on a certain Curve $A\chi\omega\eta$, so contrived that a heavy body carried along it (if it has first fallen from the height $iA$) approaches the point $A$ uniformly in equal times, or recedes from it. The demonstration of this construction (so that I may relieve the most Celebrated Author of the Problem, most busied with other matters, of the labour of examining it) I here append.

Dem. Let a little circle $\beta\delta$ be understood as described with centre $A$, cutting off from the applied line $A\alpha$ the Element $\alpha\beta$, and from the Curve the Element $\alpha\delta$, at whose vertex let the isochronous little portion $A\omega$ be conceived. Now the Ratio of $\beta\delta$ to $\alpha\beta$ is compounded of the following 5 ratios:

whence, the division being made by the common antecedents and consequents of the ratios, $\beta\delta : \alpha\beta :: A\alpha : AB + AE : AQ\ (\varphi RQ,)$ and $\beta\delta^{2} : \alpha\beta^{2} :: A\alpha^{2} : AB^{2} + AE^{2} : AQ^{2}$ or $\varphi RQ^{2}$ ($\varepsilon\zeta : A\alpha$, by constr.) $:: A\alpha^{2} \times \varepsilon\zeta : AB^{2} \times A\alpha :: A\alpha \times \varepsilon\zeta : AB^{2} ::{}$ (on account of the similarity of the Triangles $A\alpha\gamma$, $A\varepsilon\zeta$) $A\varepsilon \times \alpha\gamma : AB^{2} :: \alpha\gamma : AB\ (Ai)$, and, by composition, $\alpha\delta^{2} : \alpha\beta^{2} :: \alpha\gamma + Ai : Ai$; and accordingly $\alpha\delta : \alpha\beta :: \sqrt{\alpha\gamma + Ai} : \sqrt{Ai} ::{}$ the velocity acquired at $\alpha$ : the velocity acquired at $A ::{}$ (from the nature of the descent of heavy bodies) $:: \alpha\delta : A\omega$ (by hypothesis, since they are supposed to be traversed in equal little times). Therefore $\alpha\beta = A\omega$. Therefore a heavy body carried along this Curve recedes uniformly, in equal times, from the point $A$. Q. E. D.

The things chiefly to be observed here are the following:

1. If the heavy body be supposed to fall from higher or lower than from $i$, say from $\tau$, there is no need of a new Elastic curve: this one alone suffices for describing all the isochronous Curves. For they are all similar to one another, and are determined merely by making, as $Ai$ to $A\tau$, so $A\alpha$ to another $A\alpha$ to be cut off from the same $A\varepsilon$. And the curve of our Figure is itself constructed for a heavy body falling as though from $\tau$, not from $i$.

2. But there are also infinitely many others which a heavy body, having slipped down from the same height

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$\tau A$, can traverse so as to approach or recede uniformly from some point; but that point continually different from $A$; nor, except by a single curve from the same height, with respect to the same point, can uniform approach or recession be brought about.

3. Our Isochronous Curve takes its beginning at the very point $A$, with respect to which the motion is uniform, and is terminated at the point $\eta$ of the same height as it; each extremity indeed touches the horizontal straight line $A\eta$. Whence it is established to have a contrary flexure. But if on the other side there be set up a curve $A\lambda\theta\psi$ similar and equal to this, and the heavy body be released from $\eta$ with the velocity which it can acquire by falling from a height equal to $\tau A$ itself, then, first by descending to $\omega$, then by re-ascending, it will approach $A$ uniformly, and thence, going on continually through the other $A\lambda\theta\psi$, it will be drawn away from the same point $A$ by the same law.

4. The tangent of the Curve at intermediate places, such as $\alpha$, is found if, having drawn $A\mu$ perpendicular to $A\alpha$, one makes, as $\sqrt{A\tau}$ to $\sqrt{\alpha\gamma}$, so $A\alpha$ to $A\mu$; for $\alpha\mu$ being drawn will touch the curve: which is easily inferred even from the use which the curve ought to serve.

5. The straight line $A\eta$ is quadruple the straight line $A\theta$.

6. If on the Elastic curve a point $Q$ be taken such that the third proportional to the subtangent $Pp$ and the straight line $AB$ equals the curve $AQ$, there will be found by its means the point $\chi$ of the portion $A\chi\theta$ which of all is the most distant from the perpendicular $A\theta$.

7. But if the point $Q$ be placed such that the tangent $Qp$ equals the curve $\varphi RQ$, by its means there will be had, on the other curve, the point $\omega$, the lowest of all, and the most remote from the horizontal $A\eta$.

8. Finally, this too is to be noted, that from the construction of the proposed Problem it is now easy to gather what suppositions are to be made in order to reduce the equation $(x\,dx + y\,dy)\sqrt{y} = (x\,dy - y\,dx)\sqrt{a}$. For if in place of $x$ and $y$ there be assumed two other indeterminates $t$ and $z$, by putting $ay = tz$, and $ax = t\sqrt{aa - zz}$; there will come forth an equation in which the indeterminates $t$ and $z$, with their differentials, can be separated from one another; which suffices for then expediting the construction, at least by quadratures. For two things in this calculus seem still chiefly to be wanting, which, if they could always be done, everything would be found out; one, that differentials of the second or higher kinds be reduced to differentials of the first; the other, that in differential equations of the first

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kind the indeterminates, if they be mixed together, be mutually separated from one another, so that each with its own differential may constitute a particular part of the equation; that is, that the inverse method of tangents be found. For each I have given certain Rules (and I could give more without end), similar to those which my Brother deposited at Paris with the Marquis de l'Hospital, in which he had at first supposed a method to reside. But I at once perceived that they contained nothing but certain particular artifices, which I would not dare to call a method; inasmuch as I judge that no such thing can be given, any more than a universal method can be given for constructing algebraic Equations of any degrees whatsoever indiscriminately. For that one (of which mention was once made in the Acta), by which I posit indefinitely $0 = a + bx + cy + exy$, etc. and then seek by its means the tangent of the line, seems scarcely to succeed otherwise than in speculation. That this may lend credence to what has been said, let it be this: that the present Problem, although as it appears not very difficult, could be solved by no one by the aid of any rule or method hitherto discovered.

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Revision history (6)

2026-07-31 — Removed the 1744 Opera's editorial footnotes (a)–(m) and their in-text reference marks from original.tex and en.tex, to keep the text faithful to Bernoulli's original 1694 Acta Eruditorum publication.
2026-07-31 — Normalized the author's parenthetical insertions and value-substitutions to round parentheses ( ) throughout (they had been set with square brackets), and set the two section labels as the original's italic abbreviations (\emph{Constr.}, \emph{Dem.}) rather than expanded bold words — matching the original's typography.
2026-07-31 — Re-transcribed from the ORIGINAL 1694 Acta Eruditorum (June, pp. 276–280, Internet Archive s1id13206610) after a full page-by-page collation showed the earlier Opera-based text differed pervasively. Adopted the Acta framing (running heading "Jac. B. Solutio problematis Leibnitiani …"; \origpage now 276–280, not Opera 601–607) and folded in every divergence: corrected "Ge"→"Be" and dropped Cramer's "(el)"/"(ek)" cross-refs in the five ratios; restored the proportion sign "::"; fixed observation 8's differential equation to the Acta's single consistent form $(x\,dx+y\,dy)\sqrt{y}=(x\,dy-y\,dx)\sqrt{a}$ (the Opera variant was a misprint); restored dropped words (quoque, vero, -que in "quibusque"), "effecit" (not efficit), "altiorumve" (not aliorumve), "arculus" (not circulus), and the reading "absoluta analysi" (the earlier "absoluta soluta" was a mangled hyphenation); and conformed the Acta's fuller abbreviations/capitalisation throughout. Applied identically to original.tex and en.tex.
2026-07-30 — Translation (en) added (AI draft).
2026-07-30 — Note (i) reading "Ai [a] :" confirmed by reviewer; \uncertain{} removed in both files.
2026-07-29 — Transcription added (AI draft), pp. 601–607 (No. LIX) from the 1744 Opera scan.

@article{Bernoulli1694Solutio,
  author = {Bernoulli, Jacob},
  title = {Solutio problematis Leibnitiani, de curva accessus & recessus aequabilis a puncto dato, mediante rectificatione curvae elasticae},
  journal = {Acta Eruditorum},
  pages = {276},
  year = {1694},
  url = {https://archive.org/details/s1id13206610/page/276/mode/1up}
}

@misc{Bernoulli1694Solutio-rtm,
  author = {Bernoulli, Jacob},
  title = {Solutio problematis Leibnitiani, de curva accessus & recessus aequabilis a puncto dato, mediante rectificatione curvae elasticae --- LaTeX transcription},
  howpublished = {ReadTheMasters, \url{https://readthemasters.org/works/jacob-bernoulli-1694-isochrona-elastica/}},
  note = {AI transcription (claude-opus-4-8); status: ai-draft. Original: Acta Eruditorum 1694},
  year = {2026}
}

@misc{Bernoulli1694Solutio-en,
  author = {Bernoulli, Jacob},
  title = {Solution of Leibniz's problem, on the curve of uniform approach to and recession from a given point, by means of the rectification of the elastic curve --- English translation},
  howpublished = {ReadTheMasters, \url{https://readthemasters.org/works/jacob-bernoulli-1694-isochrona-elastica/}},
  note = {English translation (AI, claude-opus-4-8); status: ai-draft},
  language = {English},
  year = {2026}
}

Solutio problematis Leibnitiani, de curva accessus & recessus aequabilis a puncto dato, mediante rectificatione curvae elasticae

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Original (Latin)

Jac. B. Solutio Problematis Leibnitiani

de Curva Accessus & Recessus aequabilis a puncto dato, mediante rectificatione Curvae Elasticae

English translation

Jac. B. Solution of Leibniz's Problem

on the Curve of uniform Approach to and Recession from a given point, by means of the rectification of the Elastic Curve