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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 287r, page id 00068196:0569
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Tolabilia hic obiter corcr–
uet modos a
selo trlses
de Reductione tangentium inversarum ad Quadraturas
et notas eo utiles
per X: y. seu rationem x ad y. Nisul hic ab solutu
 sit _ aequatio in qua x dy: de, seu in qua ratio velocitatum
eranti elgenerantium dantur ex ratione relili generatorum Ajo eam
reductam esse ad quadraturas, seus trn velicitatum  exalteratio
n gerentiu dare exe alteritoegent et earum ope duri antioem EEE
Fretationem inter generata.
sit Nam e d y. k = x dy, - ydX,. XXE E. C
 Asi dyrx - Gix dx vocemus gixesit sit compendii causa y: X = Z
 fiete dZ y cujus differentialis dZ =  Xdy  - ydX , 1 Xx = dy. X - Zdx 13 Ergo 
ex 2 et 4edisidendo dividatur aq. 5 per dx. et ila quia ex dito proposita aequa
fiet dy de Dedz. x 4Z dy: dx = Z = dz. dx. j sit jam
 aequatio proposita, aqu. 1. vel 3= ay: dx unde valor dy: dx substituatur
 solulum est problema per quadraturas.
Z: Z-Z + a et
Videamus jam an omnem aequationem inerementalem mutatis quantitalibus
6d eam formam revocare possimus, in qua ratione ratio incrementorum
detur ex ratione quantitatum. sit verb. gr. XAA+Xy, Cjy  X _ = ay: ax - t
Xa +XG,: gy + ya = dy: dx = t  fiat na = ZZ et ya on
e  debet tulis assimi debent tales assumi aequationes novae, ut, sublata a,
B simul tollantur X, et y, sit ergo na +xy = VaZR et yy+ya = na Habemus
F2ZEl vo Zetc rctu Z: v = +
jam ex aeq. 3. Pldece
fit. a = =Xy;:  Z. X. vet rursus a = yey: v-y,. quos duosvalores
 ipsius a aequando, habemus . X: ZX = y: v -y
 Sit jam du: dZ = Y et
A res jam eo redit ut inveniamus aequationem per solas Z, v, Y cui non interveniala
est pnveniendus est valor ipsius + per Z. V, Y. H Ex aequ b haberi potest
poqet aequalius differentiales sine a, rempe ex aqu. b. fit Xdy + ydx Z- X - xy Z- dx, =0
= haber posse
D L. et ex ag. 7. fit 27 dy 50 .y= yy do- dy. opeaeq. 8 tolli potest X ex aeq. 9.
 habenus aq. (11) in qua extabunt  3, dy, dx, dZ, dv. et per aequ. 10 et 11 tollendo y
 habemus aequ. 12). in qua extabunt v, Z, dv, sZ, dx, dy. e seu v, q ndx, sz. d
seu B, Z dv: dx dz. ax nam dy: dx tollitur ex Z: v  5g. 5. et 2 sed non video
quomddo tollatur ex.
Cnsidero Quoniam ergo nihil referte utram
inveniamus Y, per  v. Z solam, an vero ra
Videamus erquam
E liceat uti aequetaenibus 6 et 7.  in 9 et 10 Nempe per aqu. 47 lollatur
 ex aeq 1e, Ret aeq. 13) per V. dv. dy. et tollendo y ex aq.  per aeq. 7.
habetur aequ. 14 inter a ex ag. 6 tollamus y per aeq. 7.. sit aeq. (13) in qua extabunt
, X, Z, v, ejus quaeratur dafferentialis (14) sine a, at in ea extabunt Z, X, v, aZ, dx, du
seu habetur de per Z. X, V, dz, du.
Et quia in eq, 10. habetur dy per V, y, dv, fiet
ideo ex 10 et 14 scet aqu. (15) in qua dyi dx seu t habetur per Y, _v, Z, X, y
quam jangendo cum aeq. 5. habetur aequ. inter Y, D, Z, X, y. Unde Colli pt
sed non apparet X per 19. 10 sed si velimus et y tolbere redeundum ad a.
an sic assumatur aequalio nova pro a,
verb. gr. in aequ. 1 qio ce
Ta +  Xy scribamus Ra + XX = xX + ya
et faciams Xa +X Za
et _ ya - Xx = va et yy+ ya = ea, et fiet ∞+ Z: e = t. Exaeg.8 fit
fit a = XX. Z.X
x =1  a =m b-n
dl- dm 2 dn-o
1= me 1 = n fit H a dm = tn
An sic: Xa +Xy
 Ex aeg. 19 sit
sn sic: Karry  + _ dueditur Ractio plinyam fierya, foe l G. F. g +  ( C
(3) 274ya
sit y: a =  e Ae v: X, fet _vAx: v.X F1 = +
fiet WX+XX, 1v+X = F, sed hac non suffieit
ot.  dividatur per yanifel multiplielsur utramque
reducendo g +Xy  tyy. fiat a A Ergo Z = v. X
X +Xy,: yy+ ga = +. dividendo per ga, fractionem per ya ft X:y+ Mculliplicando
nurinatorem per X, et numeratorem per y, fiet Rya + XN3,: Xy + xya usi loluus qui lollere quidem
t s e po qe tla quo vsut
di e in E. X
 12s 1