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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 3 (shelfmark LH 13, 3)
# Folio 107r, page id 00068200:0215
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quaentur quae sit curva in qua perpendicularisC P a cumq
ad axem ducta sit semper constans, talis curva est circulus; ubi omiu
puncta p coircidunt, quaeritur an adhuc detur alia, uubi puncta
haec diversa Y P: y: dx + dy: dx. fiet: PB:gy:: x +dus: du
M
quil
fet du aequ ev+ dus et de aequ: tus
v
saeque si jam p sit constans, aequ. 1.
D
fiot: X aequ osdedus. Unde patat hoc
problema non facile solvi posse per calculum; at idem facile solvi potest per constructio
nem, scilicet si circulus D p cujus centrum C rotetur super recta AP, et interim
centro suo C describat lineam Cc, ubi notandum patet lineam illam fleri rectam
parelletam ipsi AP. si punctum p semper idem cinea CC eris circulus. Nec
allius alterius lineae quam circuli rota tursuper recta APtalis linea describi
potest, seu ejus generatrix necessario est circulus, quia in solo circulo
aequalis est distantia omnium punctorum lib uno. Omnis autem linea
t
utique intelligi potest habere Trochoidem generatricem provolutam
Aliud problema invenire cuncam curvas duas ejus naturae, ut
si perpenticulares a puncto axis P ad unam pariter atque alteram edneatu
earum
nempe PC E si t summa earum PC + DE, aequalis rectae datae
marifestum est hapc curvage describi provolutione Ellipseos, cujus
et ipsa CE 3
duo fici seut puncta generatitia
Revotemus problema
ad calcutum sit pCoequ P, et PE aequ t - y AB aequ X. BCy
AFaego Z, 1t FErgo v. erit pp dx squ yy de Fyydy
et 26 dZ aequ on dZ + vv do BP aego = y
F aequ  - vD X + BP + PF aequ Z. seu fiet:
X + Y p  -yyt + 1o -v aequ Z. Est autem ppro
pp - yy aequ yy dy. et 1 vo aequ DVdu per l. 2.
dxva
Ergo fiet F Ydy + vdo aequ Z- x
ax
du
jam vp P + cd aequ a. fiet:
 + 
ax
15
Cdup
y  duxdu+
y Sdx +dus + V Say + do aqu. a.
d
dx crex
Et ex his aequationibus 3 et 4 determinatum est problema, verum
cx
di
quia difficultas oritur ut tolle. Caeterum opus est ad huc nova
aequatione nimirum. CE semper et constaris? Et quam
in
vocemus b. fier BF + GC aeqo EC seu Z-X + Sy- v aequ b
ol(6)
seu cyh-2ZX + XX + yy - 2yv + vv aequ bb
et differentialis
AZdZ - Z dx - x dz + X dx. + ydy - y dv - vay + V du aequ. 0.
u
Ex aequ. d etiam haec duci po: de Ldx, dy - do aequ. 0
as 2  X Sy.X  fNy-V
Ope aqu. 8 tollatur dque ex aequa 7. et _. fixtu aqu. 9. in quib restabit valor ipsiu dv.
et 1C
iteeaqu. 9 sollaturaque exaqu unde tollendo do earum diunctione,  ex g
et 10 rqu. A et squ. 11. un qua solum restatet dx. dg. Similiter open
5qu. 8. tollatur dque ex aequ. 4. fiat aequ. 12. in quasolum restabit do, vel
jungendo aequationes Z et 12. tolletur do, et fiet aequ 13. in qua solum
acstabit dx. ay. jungendo equationes 11 et 13. folletur x ax. et habetu
mine
du
qu14. jungendo i tolli poterit
lictor ipsaum g. ∞. y. v celatro,empl
du. Tollatu
10p
ce
In  habebilet dem , dus in pers 17. etr oleten p
sit