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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 223v, page id 00068196:0442
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Dala summaqquadratorum fractionum natila
lium, datur Y hiangularium et contra
a V. pendet ex quadintura Hyperbola, demonstrari
a +  potest, ex cjus Chanceri quadratrie per
regressum. Nam est Z  3 - 5 + 3 - y en
B = b + b6 - b0
Al Hyparbola numens quadrtar e ende d = 4 +  - 
et Hypebolae nunerus quadrata ep:  - L + + - +
patest ergo duplicatam esse Hyperbolum figune propositae.
Attamen vid fatendum esst ope collationis numeroum
quad vel sererum quadraticium, non nisi
totorum, ut non et sportionum respondentim
collationes institui: indicio sunt tamen At
Geometria ostendit etiam  = 67 + b6 - bd
e
b - 62
+ 5- 
Et hinc jam lux habetur praeclara, stabiliendis per
inductionem regulis de colatione totorum  inter se
ex variante in infinitum, sed regula certa partium
 perdet ex quadulum hyperbotae. fiat.
n unde ari sui 12fl p- Xy
sine le =xV. sive l a
Unde a X y.  unde 2Xly  X y
ergo - il g X. seu X lp X
si fecipsemus -xy  ifly idem
e A 9. At gg, sum omnum Hg
At summa omnium Yg aequatur figurae
summa vero omnium lu, ae quatur in
Ty- B. nostro exemplo, vid fig. C
aequatur inquam summae omnium
at summa omnium
XW ae quatur figurae complamento
et figum omniu in aluis frgurco
complemento. Ergo summa omnium Xt
aquatur figurae dimidiae. Sed haec 
ita explicanda sunt. Sumatur portio
quaedam curvae, EL. unde rectae demit
dictum est patet summam omnium lW, seu (B (en X)
aequari summae omnium y, seu spatio LHDEL.
at conta constat omnium XW summam aequari spatio
LMGEL. Ergo omnium XN summa aequabitur dimidio
spatio L MGEL., at eadem aequatur spatio LHDEL,
ergo spatium LMGEL, est duplum patii. LHDEL
proprium, et id quod utrique commune est; punctoque intersectionis
rectarum DE, et LM, appellato N, commune erit utrique satui
L NEL  quod vocabimus Ea, s et spatio
LM GEL proprium erit rectangulum NG, et spatio LHDEL
proprium erit rectangulum DL. Rata autem DH appellata)
et DE appriata e. erit rect angulam DL ,1ZVZ
Z ZC, et rectangulum NGp  e -y, + x- Z
ergo spatium LMGEL  Za F c- y, a. Z, et LHDEt,, ar Z.
Ergo. Ea, + e-y, 2X - Z , 2Ea + 2Zy, sive fa + ex - eZ - yX +7Z
H2Xa + 2ZY, sive ex = eZ VX  Ea + Zy. Et 
Et Ea B ex - ez - yX - yZ Sive rectangilum DG, demen rectanguloq
ipsam F D appellemus appellem sd
et ert Z H X - co, fietque: ex = ex + Cco - yX - yx +yco za
tantur perpendiculares in oppositas duectricus. Sit jam
EGEd D et LH, LM. aio
HM, et DM summa.
Sive eco - 2yX 4)Co Ea. p  ponamus , et pa. Ergo jungendo:   +  +  + Z0 + L0 + L + 6 etc t
e pnamus a sier: a  ad aEa addatur HM Y VX
cd itemque NGr eco- yco fier. spatium
LHFGEL. A Leco = yX. et pro es ponchdo a pro eponendo
, fier: 2a = a anferatur inde FE, seu eco, seu a
fier: - a aequais spatio LHDEL. Quod si ad va addatu B
unde si aequata fupser
fer: 28C0= 2yX, sive 24 2a  
roco esi quatu nm
sive 9+.
3+ 15. 
ergo habetur summa, umpe y+ st
  194
nde (97 3 2y -1
y-+, F y - y, 6
Horum erus. ductum
 habebimus eodem
idem aliter resolvi potest, in
Si facias ab initio
sie quod rursus in
dio resolvi potest:
vel in  - -
v. 2y41, 7 3= 2745, 3
quare si Uniue ex partibus 8 et q haberi potest
summa, etiam altrius haberi poterit
37. Y I
areorg
((+ 1)2f3 (8. vra (9.
fer  V4 Hgne
qsummabilis, dissolvatur
Sri, srgro sats ori) s 
s 2  
(19) 29 - 1
v 29+2,2541
5 N + 2y -1
m  () 13 , inde
quae quantitus dissoloi potest in hy duas:
ompsi e
Hinc patet ubi ut obiter dicam quantitatem hanc
N3 + y + y +1 vel y - y +7 = 1. duos habere
divisores pore  prist enim fit ex 3=  +,, C9 AI
posterior fit: ex 9 - 1,  g- 1