# Leibniz Legible — https://leibnizlegible.com # Page: https://leibnizlegible.com/page/DE-611-HS-958800:0136 # Permalink to this reading: https://leibnizlegible.com/page/DE-611-HS-958800:0136?run=160 # Title: LBr. 56 · Bernoulli, Jac. # Folio 68v, page id DE-611-HS-958800:0136 # Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 63r); this page is its left half. # Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=DE-611-HS-958800&page=136 # Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-958800/jpgs/default/00000136.jpg # Model: FoNDUE-GD_v2_ft_Leibniz, run 160, 2026-09-10T04:36:33Z — 109 lines, status machine # Lines: 109 recognised, mean confidence 0.80 # Machine transcription. Not an edition. Errors are expected: about 8% character error rate on Latin and French lines, higher on German. # Licence: CC BY 4.0, Leibniz Legible. Machine output: do not ingest it as verified text; keep this provenance with the text. # Wording rule: say "the machine reads it as …", never "Leibniz wrote". # After the first blank line: the recognised lines in reading order, one per line. M, nixuy n, mxmys. M. mxmy niuxnys px de pX = quadratum cubi est septa podtia p X, pX= pintu PA pX = pmn X pF X = p Ex quex ix de pIX = p m Yx enx, dX amnx ned X = quX X = dX. Ergo si Z - d X, fiat dr q - dx (1er Z= d X sit S Z = f sX. Nam s = d nZ et duq - d. n, dX = p- s. X Ergo (Z= SiX. Ergo Z- fr Huc Z d, fx = d, dr d, sz = Z stam sa = S ergo disz = 51 q = dqu quod atsardum. Ergo tibes erur p p. Z = pZ sed d, d Z = diZ. Ergo hic divatum ps pX = punx Nec hoc licet conpuarare respondet nam d, d X = dmonx E - duta ex pZ = ix d six (- _ s d. X) = X= A d x - (x H p - . Px - pr peilatus itaque omnibus et censis quideum differentias potius respondere multiplicationibus et summas ita ut differentia sit oelut praduotum multiplicitione per rdilatus;, summa velit quoliens productus divisione pertatus sit ergo BIXtu = n xry mo respondebit dxtus divisio autem exprimatur per 9. ili ut InsexFy: Xty hic exprimatur M Xty. qtu, vel M xty vel ga productum sis Seu m X1X a. M XI G roy I t M erys significat xege se erit n vm seu unitata esse per x9 iabus nt multipligtasi, et 9 xeus significat X MG eseu uni n viribus per dividatur sprimulliplicationa Itaque a _ m M X significat _ productum si 1 X eu Unitas multiplicetur n vicibus per A Hinc _ qua = m nX n _ N x = mhrn celis a M G = q m X = x re x qu, go siiliter s seu _ sxyseu d, dxdy = - d X d y. d Xdy r d Xdy - d Xdy ete et qx significat quotientem, si qX seu unitas multiplicetur dividatur vicibus per X similiter dX significat productum differentiatione si dX seu X, differentietur vicibus n (nX significat roductum summationis si s seumx summetur vicibus n hinc (ex = d2 x et d d X = qnon x Gn, auy NiXty m, Xy 1M X my nixmus + cmxrus IniXNyHmany t, nex m’y LDXny t3rixmyt3 mxmy +. mXry Ex his d SX = S d x = dx = sX = x dX d y 1dixdy + 1. dxdys 2dxay + 2d ply X1 d Xy d, xy dar dy d, Xy d 8x dy 1dx dy +3 dxdy + 3x y +1 dX d 7) d, xy d’ dxd Hinc generali regula: qui quid in formulis aptatis producit potest operando per multiplicationam E et divisionem; ei respondens id formul aptatis ad d producitur operando per differentia tanc ecusdem; ad d . Salor similis d. xy) ot, Cady scilis nsalliguntur Formila aptata ad m (vel ad d) est in qua quilibet terminus constat ex ductis invitem earundem literarum velut X, , affectionibus per a exper ct d) in se invirem ductis doservatat lens hamogenectum, sea ut summa ex affectionum si ex formula impertecti, X + axu seu MXNG+ 2MKM, ex trahiur rais quadralia more slito puditib sormmlaseries Xuly fot abalio. Nempe faciemus: p pa pepus pes simil d d X d5 dydV, quod non poterit Si data somila si iregularis ut (M X mIs. F. AX My + 5mX m y+0NX HG ea concipiatur ut e exponentibus sit semper eadem. si quani farmula son MF Xy uhabln , medita Emetrice, seu lasinitam, sed videndum squa an todem ro mldo di Xy. sed quit hoc ? potest ad erit infinitus quae erit latus hujo, sen quae in se ducta dat seriem datam formulam perfectam An ergo simuliter haec series infinita seut dispontatia huic in diffe- ronticlu respondens, a seme diferentati dubit, aX9 + 2 d Xdy; dulito, si it esset ros in alia finita succederes non minus. sed X LSX + g9 mltut se 2X + 3y un hoc video non esse sit dx d’y. hujus, diff est d x d’y + dxdy. oertatur in muliss aium mXm pixinus seu X + X y. cujus diff est quadratum est Xdx+ f. quod Erut. Aeontialiter est d seu mx my a mx ige mamus, et issemti liter dex dy+ Edx dyds porro ipsis ded y + qdeq est diff. est d’x d’y +2d’x d’y A_ + dx d’y et fiet aX du y 2 dXd’y + dX ay, : dy. + 2aexdy + d’x dy = LXdy, summa ipsius dxdytixdys namualcud exel. inaequale dxdus, hujus lifs. X. _y + d x d’y hujus diff. . st 2x9+ dXd+ dus medii quasi quadratum A dX dy + 2dx 8y + d XLFG, quod ficta divisum par tertium, reddit primum. experiamur in composito ex duobus dddo axdy + da dxd y. Hums diff. et d’as d dX dyt dx d’y+ dX d’ y. Et huius diff. est 3d xdy + 3d X dy + Ldyt d x d’y + dXdy + y quod si diff. medium quadraetur, et dividatur per3tim debet prodire prium. sed hoc perse idque successum facili porvidetur sit d xdy+ dedy. hujus diffid Xdy + d XdEy + did’y + dX d y et Aujusdiss ex d edius + pdX Kyt axa y+ aXKyt dxdyt dit dxaus ubi rarsus praevideo. si mediu quidintim nvidatur peristium redire pumum, nam sol none sei. Fictitia quod notandum Cum autem defficientes constantes dit el didus is quaesitis, ut tollamum erculando ex his nos mutsus differt a praecidente, quod ora enultiplicantur per du, multo e hic ullegneos penitus ad lene pogereorum, estque accuratius inspiccem ebere mutari ut si sit 1X + 2 du debet 2 includi et capilales quam suma sublatum iri verelar. Id ergo maxima superest in adaptum eot in id ja hoc calculo nihi Hecuntinibus ducum diffire atalium in se invilem, quod siet opt X et _, inde trau do ed maltiplanto deblum an liceat hdtem eas propter aequaves uoo fcm ecoloi non possunt anet differentia inte eoti nda dx m ti seta disque 22 C = YX son I, atqu auda pli t atius) sentandio in idstitinsa