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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 398v, page id 00068196:0792
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si esset p FgEp. + arp. + a t =o sit pBdt aptq a ad at  per pro fit B Bs  per P +G
fitpt + s + p + appat + a Pd
+ 9PP +agpar + ag at
a
= p = ApP - arp
et
fit g = - a
et fit p= +a dta
cujus coordinatae x et y.)
conat et
eat + a t
lgdq
de
aa at - ar - aa a
C1
vifferentiando secundum P, et fit et . 1. 11 +  2. 12 p + 3. 13pp + etc 0 quarum duarum
ter varians, et
a9u. ope tollendo p, habetur aequatio,  (3) quae est sine p, parametio variante ad curvam
relationem exprimentem inter X, et y, seu ad curvam concursu formatam. Sed
sia quia curva concursibus formata, tangitur a formante, hinc prapri formatam
et formantem eadem recta tinget, et quia aequ. 1. dat propdietatem curvae formantis,
hinc potiu derivabimus et proprietatem tangentium curvae formantis, quae est
aEt quia 10 f dependet ex X et y, hinc dio v explicetur per d 10 ax +  o
(4
et ita in cateris, fietque aequatio A1oLx+ pE x + p disX + p  13  L0 quae est
10 dy + pd udys App = 12 dys + p  13 dy
aequatio tangentialis curvae formantis Hinc si ope 1 et 4 tollantur p, habl tur equatio
aequ. (5) quae est tongentialis cuvae quaesitae, seu per concursus formatae. Itaque si aequ.
Hinc si ag. 3 o differentietur, ut inde fiat aeq. (6.) in qua erunt
erit dy: dx, et ope aeqqu. 4 et 6 tollatur dy: dx, debet oriri aquatio (7) coincidens
ipsi aquationi 3, vel saltem cum ex ens per eam divisibilis aut ei condivi-
Artificium foret ex aeq. 4 per regressum posse invenire aequationem 1
Ex 1 posset dari B, saltem irratio naliter Et feri aequatio p= O posito c esse
sibilis.
quantitatem utcunque pationaliter vel irrationaliter ex pressam ex X, y. sed ex tuli aequatione
non posset duci aeq. 2 quia, quia omnia utroque utrobique exanescerent. posset tamen inde deduci
equatio qualis 4 nam fieret  d0 dx + +O dus 20 itaque artis foret invenire quanti-
tatem rationalem vel irrationalem C ita ut dO sit data,  et d0 sit etiam data
Aliter: potest concipi CpD O fiet pd D dx + p d3dus= d dx + dC) dus, unde tollendo
vaiore
fit +C +5 dx c= + D d0dy unde ruisus d  AD - D de et dato)
alore
40. 0 +7
quaeruntur C et
pro aeq. 89. tentemns O = M seu log. O =  n logm = sd: 5) = n samim et d0C) -  ndm:m + dustn
-5 +O dx -O + D dy
-
m nam + m in sam obi estenda ad huc foret sam, et nova aequitu pis bis
sed quia per haec difficulter dabitur
quaerenda secundum dx, et secundum du
m
Sed quia per hoc difficulter dabituo alitus redeamus ad aegus, 2 et 4.
aditus, ideo ve
generali theoremate ex 1 et 2, dedutamus aequationem 3
Et primum
simul eadem opera
Generali theoremate Ex aeq. 1 et 4 habetimus aeq. 5. Si ope hujus thecr moitis ex 5 posset
p
a Xxigy  ax + Cx + ey + si
ep 4 2 p (Cx F eyra5) + ce XX + eegus + ff an 4 2 Cexy + 2a sCX + 2acfy = aa xX + aagus + ax
restitui 1. haberemus quaesitum.
Ex priore 0 edx + edus + Xex, tydy+  adx
six tyg t av)
(d10 ex + 5 d 10 dy = 10
posito 10 = xy