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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 397r, page id 00068196:0789
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  a
ordinatim positione datarum
Determinari potest curva per concursum rectarum ecurarum positione datarum iI, es
quae determinantur punctis commodissime punctis  et g, respondentibus in rectis A1,
et A8 datis hanc ad rem autem tantum D opus est dari relationem inter AV et AG, respon
dentes. AG sit  et A AV sit Z, opusque est dari relationem inter Z et v. Concurrat
autem 1V. 8, ipsi (21628  in 1Y; et 2428 ipsi 3P28 in 2Y, formabitur curva Y YY
Ex eY demittatur perpendicularis in AG, perpendicularis 1Y1X, _ et AX vocetur X, XI ucequ
Xes y:, v - X = Z. V, seu X+vy =  , et diffecentiamdo fit -ydo - = dv + vaque - X dZ
Et differentiando  ydo + XdZ = Zdo + vdZue, ubi do est negativum eo casu, quo
crescente q decrescit, B, ut in ca su figurae adjectae. Sed inter sectiones seu puncta
Y caderent brans axem A8, do maneret quantitas afformativa. Ns interiar
pelinquimus indifferentem, satva accommodatione, ad casum. Porro si ponamu
y dari ex Z, eoanesint explicatione, counescit du in cu aequatione 2da, et
proinde utraque tam prima quam secunda aequatio est extra sine differentralis,
etrumque ope sollentio Z et v, habetur selatio inter Z X et y
sit D = Z fiet d0 = d E IZ et ex ∞9. 2 fiet ydZ + X = Z dq + H Z
Sed idem proponemus aliter, considerando p esse tangentem
curvae YY. fiet dy. Ax _ A Y - y t nrt K A. 1Y = 1X Y+ 1V, 1
W  Y.  A1X = ay. dx seu 1V  = X duyi dx ergo A. 1 = y + X dugsiadx =Z A, 4 = A1X + 1X1 4
et 1X, ∞: 1X. 7 = ax dus fit 1X. P = ydx. dy et A. G = X + y dx: dus =v Ergo
cum datur telatio inter y + xdy. dx et X + ydx: dus proprietat tangentium
inveniri curon potest per Geo metriam ordinariam curva potest, cujus haec est
proprietas tangentiuu oliter  quia dy: dx = Z. V, hinc fit g +XZ. P = Z et XFyv. L. ∞
Et tit y = Z = XZ. B = v - X Z. V fit VZ = XZ = VZ- FZ uinde hae duae aequationes revera
si esset 1 (yt xdg: dx) = ax + aydx. dus fet gy + 2 yx ag. x  + ex desduys. drdx - ax + a lx dus
coincidunt ut patet.
seu 22 - a0
por si esset aa = Zv, fieret Xy + XXdy lx + yy ax: dus + Xy = aa
seu 2Xy + XX dy: ax + qy ax: dy = aa seu dydus + 2 XG dy.   - aa dy + 9y
ascenditur ad dy: dst
Ex
dydus + n= aa du + 2 rregus -4aaxy + a = Gr X G9 - 4 aa Xy + a C 4 XX qg)
3
A
xx
ex
4 X X
Hoc casu 5
si velimus scite an aquatio aliqua differentialis sit reducibilis hac via, t faciamus y. + Xdy. dx = 2
axts
et X  yaxidu - v et videa mus an ope aequationis datae O, et harum ouarum, tollendo
x
X)
binas ex his ay: do, x, y simul tollatur tertia, et ita habebitur relatio inter Z e V
pro constructione habermus has binas aequationesy:, v. X = Z. 
et dyettids = vih
et ydo L XaZ - 2 orva
si aequatio fuisset 13 pro
pro puoblemate tangentium habemus. y. 1d X = Z. B