Bernouilli's problem in the Acta Eruditorum for October 1698 Isaac Newton c. 1,435 words The Newton Project Oxford 2020 Newton Project, University of Oxford

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 c. 1698, in Latin and English with a little Greek, c. 1,435 words, 3 ff. 3 ff.

in Latin and English with a little Greek

 MS Add. 3968, ff. 369r-371v, Cambridge University Library, Cambridge, UK UKCambridgeCambridge University Library Portsmouth Collection MS Add. 3968, ff. 369r-371v c. 1698 England English Latin Greek Isaac Newton Unknown Hand (9) Unknown Cataloguer (2) ScienceMathematics Transcription by Marie Soulier Transcription by Michelle Pfeffer Transcription continued by Robert Ralley Transcription completed by Robert Ralley. Code audited by Michael Hawkins.
369

In Actis Eruditorum pro mense Octobri Anni 1698, pag. 471, D. Iohannes Bernoullius hac scripsit. Methodum quam optaveram generalem secandi [curvas] ordinatim positione datas sive algebraicas sive trascendentales in angulo recto sive obliquo, invariabili sive data Lege variabili, tandem ex voto erui, cui Leibnitio approbatore, ne γρυ addi posset ad ulteriorem perfectionem, et vel ideo tantum quod perpetuo ad æquationem deducat in qua si interdum indeterminatæ sunt inseparabiles, Methodus non ideo imperfectior est, Non enim hujus sed alius est Methodi indeterminatas separare Rogamus igitur fratrem ut velit suas quoque vires exercere in re tanti Momenti. Suscepti Laboris Non pœnitebit si felix successus fructu jucundo compensaverit. Scio relicturum suum quem nunc fovet modum, qui in paucissimis tantum exemplis adhiberi potest.

Hi tres viri celeberrimi sese jam ab annis quatuor vel quinque circiter in solvendis hujusmodi Problematibus exercuerant. Absque spiritu divinandi eandem solutionem cum Bernoulliana tradere difficile fuerit. Sufficit quod solutio sequens sit generalis, et ad æquationem semper deducat.

Problema

Quæritur Methodus generalis inveniendi seriem Curvarum quæ Curvat in serie alia quacumque data constitutas, ad angulum vel datum, vel data Lege variabilem secabunt.

Solutio.

Natura Curvarum secandarum dat tangentes earundem ad intersectionum puncta quæcumque, et anguli intersectionum dant perpendicula Curvarum secantium, et perpendicula duo coeuntia, per concursum suum ultimum, dant centrum curvaminis Curvæ secantis ad punetum intersectionis cujuscumque. Ducatur Abscissa in situ quocumque commodo, et sit ejus fluxio unitas, et positio perpendiculi dabit fluxionem primam Ordinatæ ad Curvam quæsitam pertinentis, et curvamen hujus Curvæ dabit fluxionem secundam ejusdem Ordinatæ. Et sic Problema semper deducetur ad æquationes. Quod erat faciendum.

Scholium.

Non hujus, sed alius est methodi æquationes reducere, et indeterminabus separare. ad æqu in series convergentes, ubi opus est deducere convertere. Nam solutio Problematis per Newtoni Analysin universalem generalior det evadet. Problema hocce, cum Nullius fere sit usus, in Actis Eruditorum annos plures Neglectum, et insolutum Mansit. Et eadem de causa solutionem ejus Non ulterius prosequor.

371

If a given series of Curves drawn of the same kind

succeeding one another in an uniform manner according to any general rule be one of wchwhich is BD are to be cut by another curve CD in given angle are in right or oblique invariable or variable by any given Rule: let D be any point of intersection & the nature of the intersected Curves & the two Rules Rules will give the angle of intersection, the perpendicular to the Curve desired & the center of its Curvity. Let the Ordinate of the Curve desired be rep And The perpendicular will give the first fluxion of the Ordinate & the curvity will give the second fluxion. 2Let the Ordinate be represented by the area of another Curve upon yethe same abscissa, & the second fl first fluxion will be represented by the Ordinate of this other curve & the second fluxion by the proportion of the Ordinate to the subtangent. to the Ordinate. 1And so the Probleme is reduced to equations involving first & second fluxions. Let 3 and so the Probleme is reduced to the c area ordin the property of the tangent of a curve.

SrSir

Mr Iohn Bernoulli in the Acta Eruditorum for October 1698. pag. 471 wriotes thus in this manner.. Methodum quam optaveram generalem secandi [curvas] ordinatim positione datas sive algebraicas sive mechanicas trascendentes, in angulo recto, sive obliquo, invariabili, sive data lege variabili, tandem ex voto erui, cui Leibnitio approbatore, ne γρυ addi posset ad ulteriorem perfectionem, et vel ideo tantum quod perpetuo ad æquationem deducat, in qua si interdum indeterminatæ sunt inseparabiles, methodus non ideo imperfectior est, non enim hujus sed alius est methodi indeterminatas separare. Rogamus itque fratrem ut velit suas quoque vires exercere in re tanti momenti. Suscepti laboris non penitebit si fœlix successus fructu jucundo compensaverit. To give the Scio relicturum suum quem nunc fovet modum qui in paucissimus tantum exemplis adhibere potest. These Gentlemen had been four or five years about Problemes of this kind, & to give the very same solution with that here mentioned might require a spirit of divination. But the Probleme may be generally solved after the following manner.

The Probleme

If aA series of Curves of being given of one & the same kind, succeeding one another (in forme & position) in an uniform manner according to any general Rule are to be cut it being given & find another Curve is to be found wchwhich shall cut all the Curves in the said Series, in any angle right, or oblique, invariable, or variable according to any Rule given assigned Rule assigned.

The method of Solution

Let BD be any one of the Curves in the Series, CD the Curve wchwhich is to cut it, D the point of intersection & AE the common Abscissa of the two Curves & ED the common ordinate of the two Curves: & the two Rules will give the [the angle of intersection at D the point D] the perpendicular to the Curve CD & the center radius & radius of its curvity at the point D, & the position of the perpendicular will give the first fluxion & the curvity the second fluxion of it the ordinate of the same curve D. And so the Probleme is will be reduced to equations involving fluxions And how to manage these Equations is not the business of this Probleme & by Separating or extracting the fluents will be resolved.

Mr Leibnitz in the Acta Eruditorum for Ma

When Mr Fatio asserted the method of fluxions to Mr Newton Mr Leibnits challenged him to solve this Problem in the Acta Eruditorum for May 1700 pag 204, challenged Mr Fatio to solve this Probleme, Invenire Curvam aut saltem proprietatem tangentium Curvæ quæ Curvas etiam transcendentes ordinatim datas secet ad angulos rectos. Let ED the Ordinate of the Curve CD be represented by the area of another Curve upon the same Abscissa CD AE & the first fluxion of this Ordinate will be represented by the Ordinate of thise other Curve & the second fluxion by the proportion of this last Ordinate to yethe subtangent of the other curve. And so the property of the Tangent of the other Curve is given.

Let the Ordinate of the Curve desired be represented by the area of another another curve of upon the same Abscissa & the first fluxions will be represented by thise Ordinate of theis other Curve, & the second fluxion by the proportion of thise Ordinate to the subtangent. &And so the Probleme is reduced to the property of all of the Tangent of this other Curve & to the Quadrature thereof of a Tangent

and the first Rule will give the tangent of the Curve BD at the point D, & the second Rule will give the angle of intersection & tangent of the other Curve CD at yethe same point D, & both the Rules together will give the Radius of its curvity & the fluxion of & both the Rules together will give the Radius of the curvity of the other Curve CD at the same point D. Let the Abscissa AE flow uniformly & its fluxion be called 1, & the position of the Tangent of yethe Curve CD will give the first fluxion of th its Ordinate ED, & the Curvity of the same Curve CD at the point D will give the second fluxion of the same Ordinate And so the Problem will be reduced to equations involving the first & second fluxions of the Ordinate of the Curve desired, & by reducing the Equations & extracting or separtating the fluents will (wchwhich is not the business of this Probleme method) will be resolved.

There may be some Art in chusing the Ordinat Abscissa & Ordinate or other Fluents to which the invention of yethe Curve CD may be best referred & in reducing the Equations after the best manner.. But nothing more is here desired then a general method of resolving the Probleme without entring into particular cases.

The curvity of the intersecting Curve CD at yethe point D is found by taking in the tangent of that intersecting Curve CD another point d infinitely near to yethe point D, & finding the tangent at yethe point d of the Curve in the series wchwhich passes through that point, d & also the tangent of the intersecting Curve intersecting it at yethe same point d; & at upon the two intersecting curves at yethe points D & d of intersection D & d erecting perpendiculars. For these perpendiculars shall intersect one another at yethe Center of the curvity of the intersecting Curves.