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# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 27
# Folio 12v, page id 00068072:0024
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Resumamus calculum
X: a = (1) SddZ: dZu et y: a (1) dv: dZ et dx: a = (3) d d Z: d7
et dy: a = (4). dq ddv - dvddz,: dz dZ. sit jam aequatio tangentialis
ad resolvendum data dy. Unde dy: dx = d Zddv - d vddz,: (dZ d2:
d ddz (: az =  ddv: ddz - dvZ dv: dZ = (5) dy: ax.
sit jam aequatio tangentialis ad resolvendum data dy: dx = (5) exryy: aa
fet Unde per 1,2, 5. fiet adv: dedzue - du: dz = (7) qu sal Z: dz + dudo: dzdZ, quam multiplicand
pe C per dZue fiet dZddv: ddz -dv = dz qu (dd Z: d + dudo: d2. sit jam
dZ ddv: ddz - dv dv:dz = (8) deo constansi, et ex praecedenti aquatione fiet
et ex proxima aequatione, fiet deo =  (8) dque qu sddz:dz + du, quam
differentiando, prodibit erit 0 = 2 dZ (sdaz. d2) aa (idZ + ddzue qu saaz: a +adv seu
Ju Sddzidz + 2 Sddz. dz + dav: daz = 0. Et extrahendo radicem ex hac aequa–
tione, fiet fddz: dz L = 1 + Y(1 = ddv: daZ)  C unde
qu. Sdaz: dz = (11) 2 - ddv: ddz + 2 + (. - ddv: dd Z) per 2 = (12) ddv: ddz - dv. 4Z = dvdv: did
 seu - 2 Ysi - dav: daz) = Y dv: dZ + dpav: di dZ + 2 - 2 ddv: ddZ, et
compendio fiat  dv: dz tdvdv: dzdz +2 = (13) 2d: de et ex proxima
aequatione fiet - L + - ddv: ddZue) =4 fd: du = Zddv: daz, et quadrando
1 =- ddv: ddque  dip d: deodeo - x dr 3dip ddx: dZ deo dd Z + ddvddv: de Z dd Z
itaque ut extrahamus sl deda: deda Lc
iel
davat A- 3g) ddv + a () (1 - 2 de) = 1 - 4. lex + 4 d d.s) + 4 - 4 dis d) : 4
da t ad Z
o daz
deo
deo duodeo)
dodco
unde cum  dip: deo detur
et prodibit ddv = d - = + L ((5 - 448)
per do: dz ex aeg. 13, hinc ettam t hinc res redacta est ad qadraturas;, sed
des
hanc tentemus reddere rationalem, et 5 4dpido oetur e seu (per 13
144)  - 2dv: dz - dedv. d7dZ (, aequetur quadrateo (17) 1+. dvdg. d  t dode: dZ
ab 1 = dvdg: dzdZ, it fiet (d) - 2du idque - Av dv a2dZ = (1 - 2 dvdes: dy dz + dvardisle: dz
et divisis omnibas per dv: dz fiet + 2 v: due L  d: de - dv dedg: d ZdZdZ, atque adeo
(v: dque (17) 2 d8 2Z,  dz dZ des d et quia t5- 4+4. duo) - 1 dvds : did
ibi substituto varore ipsius deidq modibit B 4duido- didz des (Ed9+ 2de dZ:de
27 (def. dZ, 1): (de df: dadZ) t1) 27 d8: dz =1, id de: didz +
u du 2 dessie E A d2 : ds19
seu du: dz d   9aZ- dzdZ,: d d9 + did, quo valore
substituto in (5- 4dib), seu in 5 _+ seu Cer) in + 2dvidque d
a2 42
deo
1- dv des: dzdZ, fiet di d dq 2td ed edZ 1 d2 - = d9  d3 + 2d didz
sive dZ aZ - dv des, : d7dZ = (dZ d Z de d9 e d7 d Z + Ed( d73;d2 + dz u des l
yl17)
dyd - dvda ee ed 1 =2do: d3 = dodv: 47dz aequant debet quadrato ab
I= avda: dZdz, seu ad dZ dz = 2d0 d2- dudo aequari debet quadrato  a
ipsi de et dedz- d 2dvdg  + evdr d8s: dydzue, et
fiat - 2 dZ = dv = 2d9 + dv
1 5 ++, , 4 daq  a d 1Z - 4 das dg d 7. 7 - 3dg a 3 + 4 dZ
disdes + dea
4s + dq d3 (2 ( d5 + d3 dZ)
M
seu dest d tdatus ereque +ded ( a di d d9des adid + 4 d  adodestdsses
 itaque ((5 - 4 d p. 10) debet aequari quadrato seu per aquis a debet
ac aequari quodrato AtA) - 2dv: dZ - do dv: a2 dz (4), aequari quadrato
itaque 1 = 2dv: dz dvdr: dzdz aequetur quadrato ab 1- odG: dydzue  tvardfdes. d
erit S5 = + d 1: aco) = 1 - dxdg: dzdz. et erit (1) 2dv: dq Edvdv: d 7dZ = (1) - 2 d vdg. d7 d2
+ duav dedg: d3 et divisis omnibus per du fiat -2 -, dv: dZ = - 2 de: d3e + dxdis de: d3
seu du: dz = (19) (dg. d _  2, de: dz - 1,: d8 d8: d2d7 +2 His rntor substituatu
i 2d23,; d9 des +az d7
 seu du 
seu do = (20) dy dZ de - d33: ddgdyd Hic vator substituatur in 1- 2dr: d2
ia aqu. 17. fint
-ErdridZdZ (it Ac 4d8dBL  Zdc. = l3d82+ dZ + d3dZ d ed5 dyaz de des
s () dz  4 did de de + la) dides Edzat ttidzi didz rajar ds des + Mxt d des djurte.
ndrat dic +4ax dg  142):,  (lesdes + 2dide)
At seu
avid Z2
e 1zde d7d
ds +d2dZ