Constructio curvae accessus & recessus aequabilis, ope rectificationis curvae cujusdam algebraicae: addenda nuperae solutioni mensis Junii

Construction of the curve of uniform approach and recession, by means of the rectification of a certain algebraic curve: an addendum to the recent solution of the month of June

Jacob Bernoulli · 1694 · Acta Eruditorum, September, p. 336 · ai-draft

Source: original scan (archive.org:s1id13206610 (Acta Eruditorum 1694, September, pp. 336–338; leaves 0436–0438))
"Constructio curvae accessus & recessus aequabilis, ope rectificationis curvae cujusdam algebraicae: addenda nuperae solutioni mensis Junii", Acta Eruditorum, Leipzig (September 1694), pp. 336–338 — the September sequel to the June 1694 "Solutio problematis Leibnitiani". It begins partway down p. 336 (the September p. 336, after a medical review; NOT the Opera's cited page, which lands on the July p. 336), under the running heading "Jac. B. Constructio curvae accessus …".

Significance

In this September 1694 sequel to his June solution of Leibniz's paracentric isochrone, Jacob Bernoulli gives a second construction of the curve of uniform approach and recession, this time reducing it to the rectification of an algebraic curve, $xx + yy = a\sqrt{xx - yy}$. He describes its figure-eight shape — a reclining figure of eight, "a knot of a ribbon" — and names it, for the first time, the lemniscate ("nodum fasciae, sive lemnisci"). He shows that its arc is measured by the very same integral $\displaystyle\int \frac{a^{2}\,du}{\sqrt{a^{4}-u^{4}}}$ that gives the arc of the elastic curve of the June paper, so that either curve serves for the construction, and he classifies which differential formulas reduce to the quadrature of the circle or hyperbola and which to the rectification of the lemniscate or the ellipse. 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 336

Jac. B. Constructio Curvae Accessus & Recessus aequabilis

ope rectificationis Curvae cujusdam Algebraicae, addenda nuperae solutioni Mensis Junii

Triplex praecipue modus habetur construendi Curvas Mechanicas sive Transcendentes. Primus, sed ad praxin parum idoneus, fit per quadraturas spatiorum curvilineorum. Melior est, qui instituitur per rectificationes Curvarum Algebraicarum; accuratius enim & expeditius in praxi rectificari possunt curvae, ope fili vel catenulae ipsis circumplicatae, quam quadrari spatia. Eodem loco habeo illas constructiones, quae peraguntur absque ulla rectificatione & quadratura, per solam descriptionem Curvae alicujus Mechanicae, cujus puncta, licet non omnia, infinita tamen & quantumvis proxima geometrice inveniri possunt, qualis solet esse Logarithmica, & si quae sunt ejus generis aliae. Optimus vero modus, sicubi haberi possit, ille est, qui peragitur ope alicujus Curvae, quam natura ipsa absque arte motu quodam celerrimo & quasi instantaneo ad nutum Geometrae producit; cum praecedentes modi requirant curvas, quarum delinea-

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tio sive per motum continuum sive per plurium punctorum inventionem ab Artifice instituatur, communiter vel lenta vel impedita nimis exsistit: ita constructiones illas Problematum, quae Hyperbolae quadraturam vel Logarithmicae descriptionem supponunt, caeteris paribus posthabendas censeo iis, quae ope Catenariae peraguntur, ceu curvae, quam suspensa catena sponte sua citius induerit, quam reliquis ipse describendis primam manum admoveris.

Curva Accessus & Recessus aequabilis ob solam proponentis Viri commendationem mereri videtur, ut de constructionibus secundum omnes tres modos illi adornandis simus solliciti. Constructio primi modi per quadraturas, quam in nupera solutione insuper habui ob jam dictam rationem, institui potest, sumendo in Fig. 3. Tab. VI, rectam $EN$ non, ut ibi, ipsi $Au$ sed $iu$ aequalem, iterumque curvilineo $AEN$ faciendo aequale rectangulum $AO$; hujus enim latus $AP$ nuperae curvae $AQ$ adaequabitur, sic ut sumpta ad $AB$ & nunc inventam $AP$ tertia proportionali $A\alpha$, habeatur punctum $\alpha$ in optata Curva. Tertii modi Constructio, quae fieret mediante Linea Elastica $AQ$, quam solam ibi tetigi, sine dubio foret optima, si natura alicubi tensiones viribus tendentibus simpliciter proportionales effecisset: at quia forte nullus invenitur Elater, qui hanc tensionis legem praecise observet; nec si reperiatur, illud certo constet; idcirco nec isti fidere satis tutum; praestatque recurrere ad secundum construendi modum, & quaerere Curvam Algebraicam, cujus rectificatione scopum assequamur. Talem ex voto sese sistit Curva 4 dimensionum quae hac aequatione exprimitur $xx + yy = a\sqrt{xx - yy}$, quaeque circum axem $BG\ (2a)$ constituta formam refert jacentis notae octonarii $\infty$, seu complicatae in nodum fasciae, sive lemnisci, d'un nœud de ruban Gallis. Hujus enim tum altera medietas Curvae Elasticae $AR\varphi$, tum super nodo $A$ intervallo indeterminatae $AE$ abscissa medietatis portio minor minori $AQ$, major majori Elasticae portioni $\varphi RQ$ aequatur. Unde quoque ratio constat, illam ad propositi Problematis constructionem adhibendi.

Fig. 3 (Tab. VI): the elastic curve $AR\varphi$, the lemniscate
looped through the node $A$, the isochronal curve of uniform approach and recession, and the
auxiliary circle, quadrants and construction lines. — Fig. 3.

Caeterum sicut infinitae Lineae Mechanicae a Curvae circularis aut parabolicae rectificatione seu Logarithmicae descriptione dependent; ita quoque praeter Curvam Accessus & Recessus infinitarum aliarum, & partim etiam ipsius Elasticae constructio (ut infra appare-

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bit) a rectificatione memoratae Algebraicae quatuor dimensionum derivatur. Et ausim asseverare, in materia constructionis mechanicarum hanc inter caeteras immediate sequi praecedentes; adeo ut constructio, quae per nullam priorum succedit, proxime vel per hujus Curvae, aut per Lineae Hyperbolicae aut Ellipticae, aut duarum simul rectificationem tentanda sit. Cujus rei ratio est, quod post differentialium formulas: $aa\,dz : \sqrt{aa - zz}$, $zz\,dz : \sqrt{aa - zz}$, $aa\,dz : \sqrt{zz - aa}$, $zz\,dz : \sqrt{zz - aa}$, quae ope quadraturae Circuli & Hyperbolae integrantur, simplicissimae fere videantur hae expressiones: $zz\,dz : \sqrt{z^{4}-a^{4}}$, $aa\,dz : \sqrt{z^{4}-a^{4}}$, $aa\,dz : \sqrt{a^{4}-z^{4}}$, $zz\,dz : \sqrt{a^{4}-z^{4}}$, & similes; quarum prima mediante Lineae Hyperbolicae, secunda & tertia Curvae nostrae lemniscatae, quarta ejusdem & Ellipticae rectificatione integratur. Etenim si indeterminata $z$ applicetur ad Hyperbolam aequilateram (cujus axis transversus est $2a$) ex ipsius centro, satisfaciet arcus vertici & applicatae interceptus pro prima formula. Et si eadem $z$ (aut tertia proportionalis ad $z$ & $a$, ubi $z$ major quam $a$) ex nodo subtendatur Curvae lemniscatae, inserviet subtensus arcus pro duabus mediis. Et si denique ex centro Ellipsis (cujus axis minor est $2a$, major $2a\sqrt{2}$) ipsa $z$ abscindatur in minore, & per sectionis terminum recta agatur majori parallela, designabitur intercepta parallelis portio Ellipticae, truncata respondente portione Curvae lemniscatae, integrale quartae formulae $zz\,dz : \sqrt{a^{4}-z^{4}}$. Unde cum eadem in cit. fig. existente $AE = z$, Elementum denotet applicatae $AP$ vel $EQ$ in Elastica, patet hanc applicatam aequari differentiae duarum portionum Curvae Ellipticae & Lemniscatae, ideoque manifestum, quomodo Elastica per rectificationem utriusque construi possit. Et quoniam Lemniscatae portio, ut supra monui, Elasticae portioni $AQ$ adaequatur, liquet etiam, Ellipticam ipsam adaequari aggregato $AQ + AP$; quae non inelegans harum Curvarum proprietas exsistit.

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page 336

Jac. B. Construction of the Curve of uniform Approach & Recession

by means of the rectification of a certain algebraic Curve: an Addendum to the recent solution of the Month of June

There are chiefly three modes of constructing mechanical, or transcendental, curves. The first, but little suited to practice, is done through the quadratures of curvilinear spaces. Better is that which is set up through the rectifications of algebraic curves; for curves can be rectified in practice more accurately and expeditiously, by means of a thread or a little chain wrapped around them, than spaces can be squared. In the same rank I hold those constructions which are carried out without any rectification and quadrature, by the mere description of some mechanical curve whose points, though not all, yet infinitely many and as close together as one pleases, can be found geometrically, such as the Logarithmic curve is wont to be, and any others there may be of that kind. But the best mode, wherever it can be had, is that which is carried out by means of some curve which nature herself, without art, produces by a certain most swift and, as it were, instantaneous motion at the beck of the Geometer; whereas the preceding modes require curves whose

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delineation, whether by continuous motion or by the finding of several points, being set up by the Craftsman, commonly turns out either too slow or too much impeded: thus those constructions of Problems which suppose the quadrature of the Hyperbola or the description of the Logarithmic curve I judge, other things being equal, to be inferior to those which are carried out by means of the Catenary, that is, of the curve which a suspended chain assumes of its own accord sooner than you could set the first hand to describing the others.

The Curve of uniform Approach and Recession, for the sole recommendation of the Gentleman who proposed it, seems to deserve that we should take pains to furnish it with constructions according to all three modes. The construction of the first mode, by quadratures, which I passed over in the recent solution for the reason already stated, can be set up by taking, in Fig. 3, Tab. VI, the straight line $EN$ equal, not, as there, to $Au$ itself, but to $iu$; and again the rectangle $AO$ made equal to the curvilinear area $AEN$; for its side $AP$ will be adequate to the recent curve $AQ$, so that, taking $A\alpha$ as the third proportional to $AB$ and the now-found $AP$, the point $\alpha$ is had on the desired Curve. The construction of the third mode, which would be done by means of the Elastic line $AQ$, which alone I touched upon there, would without doubt be the best, if nature had anywhere made tensions simply proportional to the stretching forces: but since perhaps no Spring is found which observes this law of tension precisely; nor, if one were found, would that be certain; therefore it is not safe enough to trust to it either; and it is preferable to have recourse to the second mode of construction, and to seek an algebraic Curve by whose rectification we may attain our aim. Such a one, as if by wish, presents itself: a Curve of four dimensions which is expressed by this equation $xx + yy = a\sqrt{xx - yy}$, and which, set about the axis $BG\ (2a)$, refers in form to a reclining figure of the numeral eight $\infty$, or to a band folded into a knot, that is, into a lemniscus, d'un nœud de ruban as the French call it. For of this curve both the one half is [equal to] the Elastic curve $AR\varphi$, and, above the node $A$, over the indeterminate interval $AE$ cut off, the smaller portion of the half [equals] the smaller $AQ$, the larger [equals] the larger Elastic portion $\varphi RQ$. Whence too the reason is established for employing it in the construction of the proposed Problem.

For the rest, just as infinitely many mechanical lines depend on the rectification of the circular or parabolic curve, or on the logarithmic description; so too, besides the curve of approach and recession of infinitely many others, the construction of the Elastic curve itself (as will appear

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below) is in part derived from the rectification of the said algebraic curve of four dimensions. And I would dare to assert that, in the matter of the construction of mechanical curves, this one immediately follows the preceding among the rest; so much so that a construction which succeeds by none of the earlier ones is to be attempted next either by the rectification of this Curve, or of a Hyperbolic or an Elliptic line, or of two at once. The reason for this is that, after the differential formulas $aa\,dz : \sqrt{aa - zz}$, $zz\,dz : \sqrt{aa - zz}$, $aa\,dz : \sqrt{zz - aa}$, $zz\,dz : \sqrt{zz - aa}$, which are integrated by means of the quadrature of the Circle and the Hyperbola, the simplest by far seem to be these expressions, $zz\,dz : \sqrt{z^{4}-a^{4}}$, $aa\,dz : \sqrt{z^{4}-a^{4}}$, $aa\,dz : \sqrt{a^{4}-z^{4}}$, $zz\,dz : \sqrt{a^{4}-z^{4}}$, and the like; of which the first is integrated by means of a Hyperbolic line, the second and third by the rectification of our lemniscate curve, the fourth by that of the same and of an Elliptic curve. For indeed if the indeterminate $z$ be applied to an equilateral Hyperbola (whose transverse axis is $2a$) from its centre, the arc intercepted between the vertex and the applied line will satisfy the first formula. And if the same $z$ (or the third proportional to $z$ and $a$, where $z$ is greater than $a$) be subtended from the node of the lemniscate Curve, the subtended arc will serve for the two middle ones. And if, finally, from the centre of an Ellipse (whose minor axis is $2a$, the major $2a\sqrt{2}$) the same $z$ be cut off on the minor axis and through the point of section a straight line be drawn parallel to the major, the elliptic portion intercepted between the parallels will designate it, the corresponding portion of the lemniscate Curve being cut away, the integral of the fourth formula $zz\,dz : \sqrt{a^{4}-z^{4}}$. Whence, since the same line in the cited figure, $AE = z$ being given, denotes the element of the applied line $AP$ or $EQ$ in the Elastic curve, it is plain that this applied line equals the difference of two portions of the Elliptic and the Lemniscate curves, and therefore it is manifest how the Elastic curve can be constructed by the rectification of both. And since the portion of the Lemniscate, as I noted above, is adequate to the Elastic portion $AQ$, it is clear also that the Elliptic curve itself is adequate to the aggregate $AQ + AP$; which stands as a not inelegant property of these Curves.

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Revision history (5)

2026-07-31 — Removed the 1744 Opera's editorial footnotes (a)–(f) 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 original.tex and en.tex, matching the original's typography (they had been set with square brackets).
2026-07-31 — Re-transcribed from the ORIGINAL 1694 Acta Eruditorum (September, pp. 336–338, Internet Archive s1id13206610, leaves 0436–0438) after locating the paper — it is on the September p. 336 (after a medical review), not where the Opera's "Sept. p. 336" citation lands (the July p. 336). Adopted the Acta framing (running heading "Jac. B. Constructio curvae accessus …"; \origpage now 336–338, not Opera 608–612) and folded in every divergence: corrected "per curvaturas"→"per quadraturas", restored dropped "in praxi" and "quoque" ("ita quoque praeter"), word order ("solet esse", "forte nullus"), "sunt" (not sint), "ceu" (not seu), "foret" (not fieret, 2nd), "construi" (not confici), the numeral "Curva 4 dimensionum", the plate ref "Fig. 3. Tab. VI", and the third differential formula $aa\,dz : \sqrt{zz-aa}$ (the Opera's $\sqrt{aa+zz}$ was an Opera reading). Removed the Opera's closing "Caeterum quantitas …" paragraph (absent from the Acta). Applied identically to original.tex and en.tex.
2026-07-30 — Transcription added (AI draft), pp. 608–612 (No. LX) from the 1744 Opera scan.
2026-07-30 — Translation (en) added (AI draft).

@article{Bernoulli1694Constructio,
  author = {Bernoulli, Jacob},
  title = {Constructio curvae accessus & recessus aequabilis, ope rectificationis curvae cujusdam algebraicae: addenda nuperae solutioni mensis Junii},
  journal = {Acta Eruditorum},
  pages = {336},
  year = {1694},
  url = {https://archive.org/details/s1id13206610/page/336/mode/1up}
}

@misc{Bernoulli1694Constructio-rtm,
  author = {Bernoulli, Jacob},
  title = {Constructio curvae accessus & recessus aequabilis, ope rectificationis curvae cujusdam algebraicae: addenda nuperae solutioni mensis Junii --- LaTeX transcription},
  howpublished = {ReadTheMasters, \url{https://readthemasters.org/works/jacob-bernoulli-1694-constructio-lemniscata/}},
  note = {AI transcription (claude-opus-4-8); status: ai-draft. Original: Acta Eruditorum 1694},
  year = {2026}
}

@misc{Bernoulli1694Constructio-en,
  author = {Bernoulli, Jacob},
  title = {Construction of the curve of uniform approach and recession, by means of the rectification of a certain algebraic curve: an addendum to the recent solution of the month of June --- English translation},
  howpublished = {ReadTheMasters, \url{https://readthemasters.org/works/jacob-bernoulli-1694-constructio-lemniscata/}},
  note = {English translation (AI, claude-opus-4-8); status: ai-draft},
  language = {English},
  year = {2026}
}

Constructio curvae accessus & recessus aequabilis, ope rectificationis curvae cujusdam algebraicae: addenda nuperae solutioni mensis Junii

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Original (Latin)

Jac. B. Constructio Curvae Accessus & Recessus aequabilis

ope rectificationis Curvae cujusdam Algebraicae, addenda nuperae solutioni Mensis Junii

English translation

Jac. B. Construction of the Curve of uniform Approach & Recession

by means of the rectification of a certain algebraic Curve: an Addendum to the recent solution of the Month of June