Letter to Charles Montagu describing the solution to the mathematical problems proposed by John Bernoulli Isaac Newton c. 522 words The Newton Project Falmer 2014 Newton Project, University of Sussex

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 30 January 1697, in Latin, c. 545 words, 2 p. 2 p.

in Latin

Published in Philosophical Transactions of the Royal Society, No. 224 (January 1696/7), pp. 384-9.

 EL/N1/61a, Royal Society Library, London, UK UKLondonRoyal Society Library EL/N1/61a 30 January 1697 England Latin French Holograph Hans Sloane Unknown Hand (1) Unknown Hand (2) Unknown Hand (3) Unknown Hand (4) Unknown Hand (5) Unknown Hand (6) Unknown Hand (7) Unknown Cataloguer (1) Unknown Cataloguer (2) MathematicsCorrespondence Transcription by Daniele Cassisa Code audited by Michael Hawkins
61aNewton 21.61b

Ian. 30. 16967.

3. Epistola, præhonorabili viro D. Carolo Mountague Armig. Scaccarij Cancellario & S. R. Præsidi, inscripta qua solvuntur duo problemata Mathematica a Johanne Bernoullo Mathematico celeberrimo proposita. Ian. 30. 16967. Accepi, Vir Amplissime, ex Gallia hesterno die duo Problematum a Joanne Bernoullo Mathematicorum acutissimo propositorum exemplaria Groningæ edita in hæc verba. Acutissimis qui toto Orbe florent Mathematicis S. P. D. Ioannes Bernoulli Math. P. P. Cum compertum habeamus &c ........ eruendam relinquimus. Dabam Groningæ ipsis Cal. Ian. 1697. Hactenus Bernoullus: Problematum verò solutiones sunt hujusmodi Probl.Problema I. Investiganda est curva Linea ADB in qua grave a dato quovis puncto A ad datum quodvis punctum B vi gravitatis suæ citissimè descendet. Solutio. A dato puncto A ducatur recta infinita APCZ horizonti parallela et super eadem recta describatur tum Cyclois quæcunqꝫue AQP rectæ AB (ductæ et si opus est productæ) occurrens in puncto Q, tum Cyclois alia ADC cujus basis et altitudo sit ad prioris basem et altitudinem respectivè ut AB ad AQ. Et hæc Cyclois novissima transibit per punctum B et erit Curva illa linea in qua grave a puncto A ad punctum vi gravitatis suæ citissime perveniet. Q.E.I. Mmm TrnaTerna fol. 389.Prob.Problema II. Problema alterum, si recte intellexi, (nam quæ in Actis Lips. ab Auctore citantur ad id spectantia, nondum vidi,) sic proponi potest. Quæritur Curva KIL ea lege ut si recta PKL a dato quodam puncto P, ceu Polo, utcunqꝫue ducatur, et eidem Curvæ in punctis duobus K et L occurrat, potestates duorum ejus segmentorum PK et PL a dato illo puncto P ad occursus illos ductorum, si sint æque altæ (id est vel quadrata, vel cubi vel quadrato-quadrata &c) datam summam PKq+PLq vel PKcub+PLcub &c (in omni rectæ illius positione) conficiant. Solutio. Per datum quodvis punctum A ducatur recta quævis infinita positione data ADB rectæ mobili PKL occurrens in D, et nominentur AD x et PK vel PL y, sintqꝫue Q et R quantitates ex quantitatibus quibuscunqꝫue datis et quantitate x quomodocunqꝫue constantes et relatio inter x et y definiatur per hanc æquationem y⁢y+Q⁢y+R=0. Et si R sit quantitas data, Rectangulum sub segmentis PK et PL dabitur. Si Q sit quantitas data summa segmentorum illorum (sub signis proprijs conjunctorum) dabitur. Si Q⁢Q−2⁢R datur, summa quadratorum PKq+PLq dabitur. Si Q3−3⁢Q⁢R data sit quantitas, summa cuborum PKcub+PLcub dabitur. Si Q4−4⁢Q⁢Q⁢R+2⁢R⁢R data sit quantitas summa quadrato-quadratorum PKqq+PLqq dabitur. Et sic deinceps in infinitum. Efficiatur itaqꝫue ut R, Q, Q⁢Q−2⁢R, Q3−3⁢Q⁢R &c datæ sint quantitates & Problema solvetur. Q.E.F. Ad eundem modum Curvæ inveniri possunt quæ tria vel plura abscindent segmenta similes proprietates habentia. Sit æquatio y3+Q⁢y⁢y+R⁢y+S=0 ubi Q, R et S quantitates significant ex quantitatibus quibuscunqꝫue datis et quantitate x utcunqꝫue constantes; et Curva abscindet segmenta tria. Et si S data sit quantitas contentum solidum illorum trium dabitur [Si Q sit quantitas data, summa trium illorum dabitur]. Si Q⁢Q−2⁢R sit data quantitas, summa quadratorum ex tribus illis dabitur. 61– A Solution of Bernoulli's Problemes. PubldPublished in Ph: Tr. V. L. Abr. V 1. p. 551. Newton B. 2. 53. N P For the RtRight HonbleHonourable Cha: Montagu Esq. Chancellour of the Exchequer Read Febr: 24: 1696. Phil. Trans: 224.