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# Piece: https://leibnizlegible.com/api/records/40837/text
# Catalogue record: 40837 — Fundamenta Calculi Literalis pro Algorithmo Qvatuor Specierum et Regulae Aureae
# Incipit: Aeqvalia sunt qvae sibi substitui possunt salva qvantitate
# Date: 1678
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=40837&katnr=39611
# Work: https://leibnizlegible.com/work/00068195
# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folios: Bl. 72–73 (from the shelfmark LH 35, 12, 2 Bl. 72-73): 4 pages, canvases 143–146, page ids 00068195:0143 to 00068195:0146
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068195&page=143
# Lines: 147 recognised on 2 of 4 pages, mean confidence 0.83
# Catalogue: Leibniz-Katalog / Arbeitskatalog der Leibniz-Edition (https://leibniz-katalog.bbaw.de/), ed. Arbeitsstelle Potsdam I der Berlin-Brandenburgischen Akademie der Wissenschaften, CC BY 4.0.
# Machine transcription. Not an edition. Errors are expected: about 8% character error rate on Latin and French lines, higher on German.
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# After the first blank line, page by page: a "## Folio <label> — <page id>" line ("## Canvas <n>" where no folio label is recorded), "# " lines naming its source image and model, then its recognised lines in reading order, one per line.

## Folio 72r — 00068195:0143
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000143.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 89 lines, status machine
Bi
1
etIC =2, 21Zeu
9H =10 90C 9. 10
eonumo posapomum Fundamenta calculi Literalis
pro Algorithmo quatuor specierum, et Regulae tureae
Aequalia sunt quae sibi  substitui possunt salva quantitate
et ce designantur hoc modo a = b, id est d et b sibi aequalia sunt c
valoret 1+122
mutur 
21=8
r=13
8+- 9
Axiomata, in quibus contipetur vis characterum 10+ 10=20
2t123 31
qu = 100  
20+10 = 30
uso Doni-
=A 4+) 23.
demognatur ex sef. aaequamur
et b = a veluti V = 10 et 10 = X, 1or2- f1173.
ant
mt (2) Aequivalent a = C, et a +b = C+b. Veluti a = 2 Ergo a +3 = 2+3-5
100+1 = 101
demus
x
ajule. e
demanstratur hac ex ninalam
Aequivalent a = C et ab = cb, veluti, a = Z Ergo, ab = 2b, si b = 3
fiet a. 3 =. 2.3 = 6
(4) +at b = b + a an demons
b
1
Hrque demonstrata ex dif aequalia
ama
Et L3, 3+1 = 4 ex
Initio omitti solet +, ita e a + b, idem est a
ab = ba seu 2.3 = 3. 2 =6 vel numem
+1, a = a
a - a =0
a=
minatus
matus
B l
6) quivi)
=I a Hnc - ab =-aid
vel deter minatius
4a-b, ab
a:b, 
20
t andus tuis quibis aut etias
a+b, c = aC + bc, velut 2+3, 4 = 2. 4 + 3.9
(St a+0-a
ua
Numeri vel
postatatum a+b, ab, a- b, a:b possunt
exprinu per C, oet aliam literam,
quamanquenondum ad aliam significa
tionem determinatam
2: b = 14: b (per 8) - (per 7) a, 1: b
aliam
A, aquimmabibpor
id aequale dici potest exprimi potrest per literam nondem ad signiticationem determira
11) a: D=
tam
1. b. hoc forte ismonstres ex
Theoremata quaedam hinc demonstrata
(1) si a = b et b = c erit a = c Nam si in a = b pro b (per definitionem
12
a b
si a = l et b = m fit atb = l+m Nam F (per 2) a + b = l + = (per des.
equalium) substituas e fit a = c
3) a bc = a cb = b ac = bc a = cab = eba, Sumatur una ex his aequalitatibus
ut abc = bca, sit bc- d firt ad da quae sic demonstrabitur:
aequalium) l+ m
ab sit bc = d, erit abc = ad = da a (per 7) da = bcR. sic
et abc = cba. Nam cba = (per 7) bca = da = ad = abC. Eodemus
Et generaliter hinc celligitur
omnes ect producentium multiplicantium transpositiones esse permissas
modo et caeterae demonstrari possunt.
1)
4) C, a tb = ca + 5b Nam C, a + b = (per 7) a+b, C = (per 6) aC + bc =
sic 4, 23 = 4. 2 + 4. 3 = 8+ 12 = 20
-a, b = - a b Nam = a, b = (per 10) - 1, a, b = - 1, ab = (per 10) - ab
(per 7) ca + cb
ita - 2, 3 = -6, Hoc est quod dicunt - in + facere -, seu -1, +1 =
(6) b, - a = = ab Nam (per ) b, - a = (per 7) - a, b = (per th. 5) - ab. Hoc
est quod dicunt + in = faere =, seu +1, -1
iC

## Folio 72v — 00068195:0144
# Scan: The same scan is also registered as fol. 73r, fol. 73v; its text is given once, here.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000144.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 58 lines, status machine
1.1 = 1. Nam 1. A = 1 sit a t  n = (per AX. 8) sit a = 1 fit 1. 1 = 1
t
E 1: 1 = 1. Nam sit a =1 A aeqvalibent 1: 1 = a: a, sed aequiralet a: a  =
(per ax. 1) ergo 1: 1 = 1
0 =0. Nam O∞ t. 1 Ergo it Nam a 0-0 = 0 (per ax. 9) = 0+0 (per ax. 5)
aequival:
unde Ergo Epet ex 0-0 = 0+ 0. Ergo (per ax. 2) -0= +0
-a =a e a a et a = d aequivalent. Nam aequiralent - a = a et +a - a, = 2a
∞1
aquin (per ax. 2.) aequi. o=2a aequival. 0. 2= a aequival  t.2 = a qu 16 al. O= a
(11) 1: b, 1: d = 1: bl. Nam 1: b, 1: l, bl= 1: b, 1: l, b, l = (perl3) b, 1. b, l1. l =
per 12) b: b, l: L = (per 11) 1, 1 = Cper 8) 1)b. Ergo jungendo
prima ultimi 1: b, 1: l, bl = 1. Hinc forro (per th. 12) 1: b, 1: l, bl, : bl = 1:, bl.
Ergo (per 11) 1: b, 1: l = 1: bl
(12) a: b, l: m = al:, bm. Nam(per 12) a: b, l.m = a, 1: b, l, 1. m = (per th. 3)
al, 1: b, 1:  = (per th. 11) al, 1: bM = (per 12) al: bm
ver 12
(13) Ca: b) + (c: b) = atC : b Nam a: b = 600 a, 1: b et
c: b = c, 1. b Et (a, 1. 6) + (C, 1.b) = (per 6) atc, 1. b = (per 12) ate, : b
Aequivalent
et
(18) s  a: b = l: m erit a + b, : b = l+ m, : m. Nam
a: b = l: m aequivalet (per 2) (a: b) +1 = (L: m) + 1 A quod aequivalet
(per 11) (a: b) + b: b = (l: m) + m: m quod aequivalet (per th. 13)
a +b, : b = l + m,: m
(7) a0 = 0. Nam a0 = (per 9) a, a - a = (perth. 4) aa + (a, - a) = (per 6)
aa =aa = C per 9)0
(8) - a, - b =  t ab Nam - ab + (a, - b) = (per th. 5) 
+ (-a, b) + (-a, - b) = = a, b - b = (per 9) a0 = (per th. 7.) Osr
=(per 9) - ab + ab. Ergo prima novissimis aequando- ab + (a, - b):d
= -ib + ab Ergo (per 2) -a, - b = ab. Hoc est quod vulgo
dicunt, - in = facere +, seu = 1, -1 = 1, ita - 2, -3 =6
(9) a : b, c = ac: b Nam a : b, c = teb, ac (per 12) 1: b, a, c
=(pertZ) i ac  1:b
1: b, ac (= (per 12) ac: b Ia 2: 3, 4 = 2. 4: 3 = 8: 3
a. C:C = (per
= a. Nam ac: C = a cC (per c
bm  dt:  1: b, 1: t = p Nam 1.b, 1: l, bl = bs stib. :l, b, l = b, 1.b. l. iel a
10) aC: C
14) () a . Aequivalent a a = C et a: b = C :b. Nam
e
a: b = c: b aequivalet (per 12) ipsi a, l:b = c, 1: b qua
per 3) aequivalet ipsi a = c sic si a = 2 erit a: 3 = 2: 3
(15) (14) Aequivalent a: b = l: m et am = bl. Nam sper 3
16
aer
aequivalent a bb a: b = l: m et abm: b = _ lbm:m
id est (per th. 10) am = l b
) Aequivalent a: b = l: m et a: l = b: m. Nam (e
(por 3) aequivalent a: b, = l: m, et a . b, b.l= l.m, b il id est
th. 15
(per 10) aam = bl et (er itemque hoc et (per 11) a: l = b: n
et (per th. 15
18) Aequiasi a: b = l: m eritt a: b = a + l:, b +m. Nam, quia
(per dof. aequal.),
a: b = l: m (ex hypothesi) erit qm b (per th. 14) am = bl. Ergo (per 2) a0 + a =
ab + bl  id est, a, b+m = b. 4+ Er (per th. 14) a: b = a + l,:, b +m
17) si a. b = d l. m erit a

## Folio 73r — 00068195:0145
# The same scan as fol. 72v (00068195:0144), where its text is given.

## Folio 73v — 00068195:0146
# The same scan as fol. 72v (00068195:0144), where its text is given.