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# Title: Leibniz-Handschriften zur Mathematik LH 35, 4, 13
# Folio 5v, page id 00068010:0010
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explic. a
(9) o+ a = a. seu 0
est signum nihili, quod nihil addit.
siunt aequalia. seu: si sit a = b, et b = m
erit a+b = l + m
Nam a + b = a + b (per 3). Itaque in alter
postore a +b, pro a potrendo l (ex hyp. priore)
et pro b ponendo in (ex hyp. posteriore.) quod
licet (per 2) fiet utique ex atb = a +b fiet
at b = l+ m Q. E. d
SVB TRACTIO
(11)  Explic e
+ a -a = 0 seu signum, minus,
seu signum significans minus vel subtractionem
perinde esse ac si et actum esset nihil si quid aliud adest, residuum apperatur.
significat id quod positum fuit rursus tollendo,
(12) si ab aequalibus auferas aequalia,
Dheor.
residua sunt aequalla. Seu: si sit a- l
et b = m erit a - b = l -m
Nam a = b = a - b (per 3) in posterione
pro a ponatur l (ex hyp. 1.) et pro ponatur l
(ex hyp. 2) ergo ex a - b = a - b fiet a - b = (-m
piss a quantitatibus duabus auferas
aequalia et residua sint aequalia; ipsae
quantitates sunttaequales seu: si
sit a - b = l - m et  erit b - m erit a = l
Nam si aequalibus a - b = (ex hyp.1) l- m
ddas equalia, b et ia (ex hyp. 2.) b et m, nempe
priori  posteriori m, fient (per 10) aequalia
a = b + b = l - m + m, id est (per 11) a = l
ss ab aqualibus auferas duas quan-
titates, et residua sint aequalia; erunt
14)
quantitates aequales: seu si sit etm
et sit a = l. et a - b = l- m; erit
Nam a - b = l - m (per hyp. 2). E qu addendo
b = m.
utrobique aequalia b+m et b+r fient per 10) aequalia
a -b + b + m = l -m + b + m. Ergo (per 11R junct. 7)
atm = l+b Ergo (per 11)  A quibus
aequalibus si anferantur aequalia a et l (per hyp. 1.) auferantur
fient (per 12) residua aequalia, m = b. Q. E. d
(15) si a = b + e erit a - b = e
tea
Nam  uab utroque aequalium auferendo b
 fient fient aequalia, a - b = b + e - b id est
(per u) a - b = e
Q. E. d.
(16) d  a - - b erit b = - a
Nam a = -b (ex hyp.)