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# Title: Leibniz-Handschriften zur Mathematik LH 35, 4, 13
# Folio 10r, page id 00068010:0019
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 9v); this page is its right half.
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(e (6) 10 = -0
Nam r qu e0 +0 + 0 = 0 (per 1141) jam 0= 0- (per 2. hujas)
(6be ait  +c =
(6) si a + c = b+ c erit a = b
(6) Si a = b erit a - c = b -c. s
Nam a = c = a - C (per 1,2) in qua aequationis
 latere posteriore si pro a, ponas aequalem (ex hyp.) b
quod licet (per 1, 1) fiet a - c = b = c Z. E. d
in Lineis si ab aequalibus AC, Bd, subtrahas
eandem BC, residuae erunt aequales AB, et CD
(7) si a + C = b + C erit a = b
Nami quia a + B = b +C (ex hyp) erit (per 6 hujus)
a + C - C = b + c - c, id est (per 2 hujus) a = b L. E. D
in Lineis si AB+ BC seu AC aequ.  D C+ CB seu DB
DC 1+ m et sit b= m erit a = l
panquid al b= aolus ltin C ex hyp, erit a
8). +0 = = 0 seu nihil rejert quod nam ipsi o praefigatur signum
Nam . :0 + 0 = 0 (per I. ) et 0 = 0-0 (per 2 huius)
ergo fit 0+0 = 0-0. Unde abjiciendo utrobique o
quod licet per 7. hujus) fiet +0 = -0
a = 0 = a. seu nihilum  nihil subtrahit
Nam a = 0 = a + 0 (per 8 hujus) et a +0 = a
Aliter a = 0 = a = 0 + 0 (per 1, 4
 jam -0 +0 = 0 (per Z hujus) ergo a - 0 = a +0
seu (per 1, 4) a -0 = a
et b = m erit a - b = (-m
seu si ab aequalibus afea A) auferas aequalia (b et m) residna (aab, et lum ) erunt aequaliu
(10) si
demonstratur eodem modo ut artic. 6. capitis ujui
praecedentis jam et sit a = l erit b = m
( sam ab aequalibus; at
inferas se
et 
ualieastm  et p= s  enit B2 = 61
ei erit e a = b
Nam quia a - c = b - c  erit (per 2,5)
1 ec4
a - C + C = b - c + C, id est (per 2 hujus) a = b, Z. E. c
in lineis si EG = FG = HF - FG (id es si EF = HG) erit EG = HF
(12) Si  a - b = L m et sit a = l, erit et b = m
(12) Si a - b = l - m et sit b = m erit a = l
Nam quia a b = L m erit (per 2,6)
(per 2 hujus) a = l. 9Ed
i Leneis si EG - FG=  HK - 1K (id est Et EF = Hi) sitque FG = iK erit EG = HK
a - b + b = l - m + m Ergo
(13) e Si  a =  b erit - a  - b
Nam - a = - a (per 1, 4) Ergo in posteriore
latere hujus aequationis pro a substituendo aequalem (ex hyp.) b
quod licet (per 1, 1) fior ex = a = - a fiet - a = - b Q. E. D
(13) Si a = b - l m et sit a = l erit b = m
(13)  - a = - b erit a = b
Nam a a = b - b (per 2 hujus) ergo
13) Si a = b = l - m et sit d = l erit b = m
Q. E. d
ne snsic (per 1, 4) ergo a = 0 = a Q. E. d
Nam cauferendo utrobique aequalia  (per 10 hujus) et
ex a - b = l- m,
a et l’ex hyp)
fist   a - b - a = l - m - l  id est (per 2 huius)
sit = b = _ -m. unde (per 5 hujus) fit b = m. Q. E. D
in Lineis EG- FG = HK - IK (id est EF = HI) et F5 = IK
et EG = HK ent FG- 1K
si at b = l+  et b = m
erit a = l
= a = - b Ergo + a = + b
Nam a - a = b - b ergo utrob.que
anferendo re qualia za et = N
in Lineis si Ny = RT, et NP =R5
erit NL - N P = AT - A5, seu pq = 5F
=a = b - b Ergo +at b
Nam quia - a = - b ergo addendo
ntrobique b, fiet b a = -b + b
(per 2,5) a =b
=a + b = b + b id est (per 2 hujus
ergo -g + b = 0 Erge (per 5hujus)
eris a = b
X. Ed.
a = - b. Ergo b - - a
a a = -b ergo at a - a =
a + b = 0. Ergo
d + b( a = (0) - a