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# Title: Leibniz-Handschriften zur Mathematik LH 35, 15, 5
# Folio 15r, page id 00068207:0027
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Ad Hospitalii Analys. inf. pay. 75
APX
PMy C centrum curvedinis ME, Z Bedu CE, est 1)
eu dzz pro hoc casu seu RM, = dy
ipse radius
curvedinis MC, sen rA B. M n sit dm
fiet r = y dm: dx jam dr = 0 hoc loco fiet utique
0 = 7 d(dm: dx) F dydm: dx. adeoque dy: Z = - dlm: dy J: d m: dx)
adem (dy: Z = loq (dx: dm) velt q = dy:  d log(X: dm)
ret
vit e Y vel Z a est ad constantem, et plenrer Elementum ithenerati ordinatae ad
quem habat stunt consinus mcecivitatis ate radtum
elementum loganthmi rationis infer radium et sinum dedivitatiet. declivitatem autem
pt consinum declivitatis Est autum declvitates, angulus quem facit tangens
curvae ad horizontem, posito axe verticaliter erecto. Declivitatis anim rescit
facitu
angulo hoc crescende. Est ZC _sc: dn)a (den: aX) c. Sed rursus esst
Y= qdm. dx Gudu fiet Y= du damidx) porro radius curvedinis est ad
ut de eo plementum curvae Etem rudius ad sim consinum declivitatis. Ergo
u
Ergo radius curvedinis est ad nonstantem in ratione composita  Elementi ordinatae
ex ratione ad radii admsinum dealivitatis, et rationem quem habet Elementum
ordinata ad Elementum logarithmi quem habet consinus  declivitatis eragradium.
exponentis rationem
=  dysam: dx) bg d log axedm) vel Y = (dm: dx) dy: (d (dm : dx) : (dm: dx)
sen
Y = -(dmdm: dxdx du d (dm: dX) seu radies pentet Ratio quam habet radius osculi
ad constantem est in ratione composita Hinc  (dy:r = (b: a) 4 - dm: dx
per MR: RM: Mm = ME: EC: CM rdy = Vam
cip = (X dZ - 1
r = 1 dmidus 0 = AX Cdm: des) + Ip d Cdm: du). Ergo Y= e- dx (d m: du): d (dm:du), adeoque
du
do
p = d X(dmam: dydys): dHm: dy) = du (dmam: dx): d (dm: dx) jat
et d(dm:dx) = dx dam- amddx,: dxdx. Erg.o tandem -r e dmam:, dyddm- dmady
si dn sit constans
ut clome arine qudineterum est si curaea
Et vodem modo -r = dydmam:, axdam- am ddx
itaque crescat uniforuiter, fiet elementum nnius coordinatae, ad alteram coordinatam, ut elementum
ipt exam: ddu = r = dydm: ddx
sed generaliter
dy: Y = d logdx: dm seu radius est ad elementum coordinata, sut elementum rationis
culvae ad radium curvedinis e du
aserius alem coordinatae elemensum habet ad curva
se mavis radius est ad anitutem, ut elementum ordinata
cupq
bam 11
cirvedinis
ad steauto es lagari ad e elementum Logartum rationis quam habest
= dy: d log (dx: dm)
logarithmi rationem exprimentis, 
elemenum curvae. Vel
o
r. dus 
elemente
serius coordinatae Y curiae