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# Title: Leibniz-Handschriften zur Mathematik LH 35, 3 A 23
# Folio 6r, page id 00068236:0011
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G
Scheda, 1II. de quotientibus replicatis
hoc loco-
Quoniam divisio nihil aliud est quam subtractu, quoties fieri potest
et numerus subtractionum semper debet esse integer seu aggregatum
unitatum; hinc pro investigandis quotientibus replicatis orietur talis
progressio. Sit aequalio gradus secundi generalis: H
Sbb + t ab + vaa aequ. o. ponendo. s. t. v esse numeros integros cognitos
et ut et ab. et signa, eorum esse qualiacunque. Facimmus
jam + a + b aequ. c. ita ut C sit differentia inter a et b. et
4 significet +  si a sit major, et et - si b sit major; et
 + contra. erit b aequ =c + a. Ergo substituendo hunc valorem
in aequ. 1. fiet.
Sec +25 a C + saa aequ. 0.
4 + a c + + aa
ad instarae qu. 1
+ v aa
cujus aequationis 4 loco si scribamus: (5) CC + (+) ac + (V) aa aequ. v.
16
amparando aequationes 4 et 5. fiet: (5) aequ. s. (t) aequ. 4 25  ++
Unde ja jam porro poni potest si subtracto adhac continuari potest
eodem modo, seu s si jam ponamus eem a esse
majorem quam b tunc fuit +a - b aequ. c. Et jam 
videndum an ab a adhuc semel subtrahi possit b. seu an
scribi possit aC - baequ. d. an vero contra ponendo C
esse minorem quam b: scribendum sit = c +d aequ. d. sed jam
video satius esse sine discrimine cogniti et incogniti incipere
semper a maximo, sit ergo a b aequ. c. subflitende
seu a equ C +d. substituendo in aequ. 1. in locum ipsius a.
fies: sit ergo: sed ita non sutticeret explicari c deberet
etiam explicari a, quare sic optime opinor proodemue:
saa + + ab + vbliiquo + a + b aequ. c. seu a aequ b+c. b +C
(4
fiet :+ s bb  + 25bC + SCC aequ. 0 seu 15) bb + (+) bc + (8) ce aqu. 0
+ +
+ V
+ + bc
ergo (5) aequ S + L + B. (+) aqu f 25 +F et (Bsoqu.s.
(∞).
Rursus videamus an possimus scribi possit ce b +C aequ. d.
Soed difficultas progressumo in eo nascitur, quod, si b sit
ominor sive subtrahendus, tunc a in nova aequationem inter
C et d ingredi debet, si vero sit major, tunc b potius.
itaque resumendum in fallor, semperque positoque a esse cognitam
b incognitam semper servanda a, et proquediemur hoc modo:
(2s
saa+ tab + v bb. aequ. 0. bequ. + a = ba qu c. seu baequ +a +c
fiet: + fua  ef 425 ac + cc + Saa 4 ac
+taa
ac
+ aa
+v aa
 tac
+ 2V
vago s+ t +B