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# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 4
# Folio 88v, page id 00068049:0178
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vdvia deo ergo
e
aequ2t Quta eteo fit co
(ax Sb.4 +BE BY
Aob: 4 =AAVV
sit do = - taeX: XX F 2rX + IV et C   
ditrum
Dco-
sit v=  ta : 2X +21.
D XFExt
6X+ 2b 2X + b er
cettaa
aabbx t abbx +aLI
ubi debet fieri b b f ab  b eethaa = 4bdr. et recidimus in idem absurdum
KKFa
retenta ergo aeq. 38) faciamus P= BX + at, EX + 2Y et erit  ob  b + ee B
9
ed exC   Ldx. βx+ eY  ta: X+Y
(31) pBX + 
Cd = 
X+x+
At
bb - BB
n  eat
X +ib YX + b2  BV (7(+4 beq
e. +2eeBX + ceaalt
b X + 2AbrX + 4abb 1Y
aaAB.aBat X  aatt
afieri bt eeaat
Ergo
orrda t
=+b reeBat
debet ee (+qeβ at
BBaat)
et fit bbaa H + PBr  +2bb r Bat. et facindo t= 0 fet ( + a bba
et habebitur rrB, at BD sc. fit bb + dB - + 2bb
4, 26
Debel
faci videamius an compendii causa ob aequationes 13, ( 15, 16. liceat facere
qu
pp: 4 = ag = 0 est autem p = 2bbr - 2Bat, :, bb- eβ fietque pp. 4 = aq2Z
B2 2bbpat + BBaatt r 4 baalt + 2bbBBbb rr abb BBa  βbF + BaC
((2 - 2bbr Bat + (BaF) ( b + ba C + Bbr BBa +) 20. quae aequatio
coincidit cum aequatione 35, quod memorabile est, si faciamus  r. b quae
aequatio propemodum caincidit cum aeq.35, et si sit + = +, seu in casu Hyperbotae
fit r = b s adeoque ex aeq. 35 fit aatt + β3bb = 2b Bat. Unde habetur at: βB = b-r
sed in casu Ellipseos, ex aeq. 35 fit bbaatt = - 2bbr Bsat - ββ . Et ex 38) = 2bbiBat - (36Y
atis
Ex quo patet in Ellipsi necessurio ZBF esse quantitatem affirmativam, et fieri β = 4bat: r. 1. bb
 quem balorem substituendae in aeq. bbaatt = 2bbr pat - br, fiet utiZ
r6 - 2rb + rb = cr 8br - 9brb - 16br seu  25b7 -10br + r - 0
seu 5bd - rr = 0 seu Y = bS5 adeoque in casu Ellipseos fit at: β = b 15
itaque tam in casu hyperbolae quam Ellipseos fit at: β = +r. Itaque ae dimensionem
X
e + LSXtS. 
a Curvarum Hyperbolae aut Ellipseos reduximus ad summ. gd7
Pelibea antams quae summa constat ex duobus membris, quorum unum quoque quodque est summabile
per reductionem ad rationales.
X+C Xx