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# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 1
# Folio 14r, page id 00067956:0027
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Quaeritur an langentium inversae possint reduci ad motum, reluti
si data relatione inter  dy: dx, X, et y, haberi posset relatio
inter AT et TC. sit AT, f
TC, Z. ABX, BC, y.
(primem) ZZ = gy +XX + 2X F + tt et sen(2dc) t+X, :y = dx: dy
stertio
sit jam Fex, Ege dy: dx = co: a posito co dari utcunque
ex y et X, et fiet  (+t) dy x ++ X, : y = a: c. Quod si jam
ope aegq. 1 et 4 simul tolli possent x et y haberetur desideratum
sed videamus an non liceat dy: dx adhibere ad huc semel obtinere
Et ne quid laba titubemus. pro recta tangente TC, ad hibeamus secantem
in duabus punctis. Patet puncta quidem A et T manere simpla, sed B et C, geminari
ita ut ex B fiant P et (p) ex C fiant K et (X) et fiet. A P sit X, L
et A (P) sit (X); PK sit y et P(K) sit (7); TK sit Z, et T(X) sit (Z), demque
sola ex quantitalibus variabilibus simpla erit AT seu T. Ubi consi-
vel ata
derandum in aequatione 3tia qualorem ipsius co sumi, ex G
loco ex non sumi essum
posse um
item y vel (y), sed Cex X + (X):  et ex y+ (7): 2 tanquam medio
amborum arithnetico. Esto autem
P (P), dx; et q(K) est dus
et K (K) est dz = (5t0) idxx+ dydus sit jam quadranda
aequatio prima differentianda fit ZdZue = (6to) ydy + xdx + tdx. Quae
aeqie differentiatio intelligenda est, is eodem nanente
puncto T. Ergo ex 5 et 6 fit Z Fdxdxtdyaque = (7mo) ydy+ xdx + + dx
itaque ex aq. 3 et 7 fit  Z 1 + (0c0: aa) = y (co: a) + X + F
se Ex aeq. V
tollamus F, ope aequ. 2. firt Zd3= ydyt Xde + dxdx ydus dxyfdx dys) seu
XZ seu Z: y = dydus Fdxdx, dydz unde fiet
GZ: gy- (dy dy+ dxdx): dydys d3dZ fiet Z
fiet ZdZ = ydy + xdx + y (dx: dy) dx. et ex
Cn
Z7 = yy + gy dxdx: dydy)  -1
seu Z2 dZdZmo et se cunda aeq. fiet
jam Z7ddZ (qydis t XXdxdx + 2 yx dydx (+ 2 yy dx dX) + 2X y d X : dy)
(+ yy (dxdx: dydy) dxdx ex et
yy + yy dxdx: dy dy) dxdx + dydy) =.
yys dxdx F yy dydy + yydx: dydyset qydx dx
sed Z - Idy dy + dx de dy dy + dxdx
 dy
quodu non ex psa
consequatur
de dy
adx + et ex oq. fi
et 22dZLZ = fi et
Z+q = ydy + ylx: dy) de
su 2 = dy dy + dx dx
dz dy
Ergo dz= dydy +dx dx
E itaque aequ. 5. nil dat novi
differentiatione aequationis 1 et valore psiust
Sed etsi hoc modo non sit in costra potestate reducere problema tangentium inversum ad
tractoriam constrnctionem, aliam tamen ejus reirationem excogitavi; sane semper
judicari nihi esse conslinctume tractoria aptius ad hunc scopum, quia ibi lantur ferngentes
ex ipsa constructione seu natura motus; adeo primus exeagrtavi rationem et in aotis
eruditorum ibidi de quadraturis omnibus per notum construendis quam deinde prose intus est
du jac Bernoullius. sed de inde pogitavi  fandem rationea omnia priblemata tangen