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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folio 44r, page id 00068197:0087
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pac
Exitus non hcabetur vid pag. Seq. ubi NNG. notanda tane reductio ad infinitum
AB. AC. BC = AA BBCC. sit AB = Xt C AC = x +m  BC = X + n
fiat A = 11 1OQ + 12V3E
B = 21104 + 2217C
C = 31 YO8 +32 194
P ABL 5. . 2
a S 1-00.60 al
(AAC
31 OG
- gt) oc ob a in
+ 11. 219. 9D
12.2
+ I1. 328797
12. 31
(+(6) -. ( + 31 3 69 + 31. 22 8 13)
a
Lm = V H -m = X l- n = (n 00 21  b0 - 2 0s a 32. 21
 identificitur primus qualor ipsius e 
pone
et tertius Secundo
et Brim
Habebimus binas aequs ex irrationalibus.  tertuis
et aliaste ... rationalibus
videot une settiomsia seu
si
istum
rersan
ets
pano 00.
In Errationalibus ex valore primo et altimo fit a 11+ 12 = 31 + 32 Barsus vecnet 3t 13D)  VH ∞12900 (1+ 12) = 11. 32 + 12. 31
fit 32= UH st = aC per compendium sit 11+ 12 = 1 13 fiat 32X13 - 31 = 00. 13 - 12. 31, : 11. Unde u. 13 - 1. X = 00. 13 -12,9. f
sist 12 = ∞ + (00 1) 13 r H. 3L, 3 quo valore substituto in equ.  13 - 31 + 2 et fit 31 = (1 - 0d 13, :, 11 - 12, quo valor e
substituto in valore ipsus ipsius 32 fit 32 x H.lB. (1. 13) - 12. 13 = 11. 13) +00. 13,:, 11 - 12 le 32 = (12 -00) 13,:, 10- (1. qui bini valores sunt
harmonici, permutantur eneu il et 2 quod si malimus 12 et 32. fint  1Z  32 = 11+ 12 - 3, = 00 (11+ 2) - 12. 31, : B
inde HI. 11 + 11. 12 = 11. 31 = 00. 11 400. (2 = 1251, fit 12 = Ba (00 -11 + 3111 :,  - 0d + 11 + 31 quo ip siy
12 valore sabstituto i
equate
(00. 11= 1I. 1D F 21. 31 =- c. 32-00. 31 (+00. 11) EN 1. 32 (t 11, 3(- 1I. 19 + 2. 32+ 31. 31(, IC. 3D
fiet
da
et fiet 32 = (00 - 31 = 11) 31 p : =00 + 31 + 11 seu 12  00 -1+21
 vasermutande
s datau ipsiusX a
An a0IOV RK
00 aV
11 (od 21) 412 (00+2
11(00 31) O + 12. (00- 37)∞
11
= x=
105 duos ipsius Q valors aequando, fit e 11 (00 - 31)  C = 432 (00 - 12) Y 5
= 31 (00 - 11) 1 .
= Ca 21 A
250.31 41 C
10+ 131 3 003+i
0as
31 (00 -1) C + 32(00 - 12)D
permatentur L. et 3. dimulu aer seu si
ordinatim,
2FF quae aequatio harmonica est, nam permanet si (1. 3. A, V. cum 32; l., V, A, permutentur
1 ∞=E ed quia ratio constans inter C et D habeni non debet, ideo digellenda est haec aequatio in duas
(4
dut evaneseant C et D, fiatque +1 (00 - 31) 5t - 31 (00 -1)V 20 et 32(00 - 12) Y 12 (00 -32) A  0
Quarum aegq. una fit ex altera, si permutentur et et 12. compender fint von A - 15 00- 31.Z
ed 12 15 00 3  36 ubi t. dut B t 1: 3c dab 3: Et et dat 5 est
00 Ooni - et Oo 312 031 00 12 MM 00n 3= 032 L00n m- 
 Sitt _  1 est A . 3L C4Y = (1) = 0 d 30 fe0 12) V = 12 (00- 3LX), _) = 0
E et ab valeres pmotes 17 B) e  3l + el
e
en
A -m - L+ M
29000 t. 4 200ull
M= vorH  f. Z oHV
= p- l = = 
Habemus ergo aequationes quatuor quas notavimus per 1, 2, 3,4. prima dat
31  uH
secunda valorem ipsius 32 sine 13. qe tertia et quarta
2 et B32 Sinc 11 ~ t 31
 sunt binae novissimae. Tertia sine (2 et 32, quarta habet hos numeros. Itaque si i
Avalorem ipsius 12 sine 32  11 +t 31
Ein quarta pro 12 et 32 substituamus eorum valores ex prima et secunda
in aprima et secunda compandii causa  pro eo et faciumus ou et pro 00. s sit 00 n e
i  Ex quarta sit 00. 32Y = 12. 32V = 00. 12X f12. 32 X = 0 seu 32:12 = 00M : 100V - 12Y + 12X
∞o et quia A - Y = = n fiet 32: 12 = 00:, 00V _ - 12 (n. Ex aqu. 1 t 2 fit n Eadem ratio
F2.
d
5.
e
et ex aeg9. 1 et 2, fiet 32: 12 = 24 -319 31 :, 23+31, I1  quos duos valores aequando, fiet:
32 ad 12 quaeratur ex aegg. 1 et  t13
15 00. 31(24- 31) Y - 00. 11 (23B,) A
00. 11 (23. + 31) A = 00() 00. 3 (24 -3) V = 12 .31 (24-31) n. et fiet 12-
et
00. 00. 31, 00. 11.3)1- 00. 31. 311 00. 60. 11) + 00. 11 1(5 = 00. 1. 31)31(24-3)
29. 31 )  00. 31 4 11. 31 = 31.31 A M
sigo, d crant sed valores per ectum per Nume ratores
 patitur
Levenit ut in 32 et 12 namerato
t sir us st
per ipsum distingui debent
Quaeramus valorem ipsus 32 eodem modo. Ex aequ. 4 est 12 : 32 = 001:, 903 - 32 + 32V seu 121 32 = 00r3, 00A + 32n
tnator hani permatation
ubi in nuaitaon
permurantur 1L. H. f H1. 31. A. V. 1B
=(23+31) 11, : (24 - 31)31 et fit 00. 31 (24- 3)Y. = 00. 11 (23+1) R + 32 (23+3D)M, unde sit 
 ubi patet eundem esse
3(h4-3)) en seu 00. 31 + 1. 31 31. 31, 
11 (23+31
et
a instituatur
I2 cum 3l. H. V. A. MC
alore, 3t et 32 e et
= 00. 11(2311) X, a 1
quaeramus valorem ipsuis 32 eodem in
dunc est =11 (23+31) n seu D  00. 11 = 11. 11 F _ 11. 31, n. Unde permutando signa
et
signa sipei
Nm Numeratorem qui ante. Nomnator autem qur prius cerat
une fit 32 = 00. 31 (2 ed pernutania in
in numeratore, firret in tnitore 00. 11 - 11. 11 FH.31,-in, ut oportet
= 00 H + 3C, 1  11, 90 -1 +3 (, 39 00- 31+1, 3
Hinc jam ex duobus valoribus ejus dem humerutoris, erunt Valentia reciproce ut
diffent a priore ae tH = 21, 3L. : 00- 1 4 31, H fit ergo resunula compendio 11 (23+ 31): 31 (24- 3) = (24 3) 3t : (23+19)11
= H 00, I1 = H. II 4 H. 3l, a,00. 31=31,3(+11. 31
et fiet H. _ qu e DE II. H qu (23+ 31) = 3l. 31 (29- 31) seu 1 (23+31) = (31 (24 - 31), seu explicando
00HH. H f H. 3t = L vo 3t  13C 4 313) f00. 3t 2 H.3t f3l. 31 si esset f = + fieret
2 10
1). H31. 3I 100. 3t a0 et = 0 et ferei H2L. 1+ 3P - 00  0. quod si faciamus + esse
qua riatio neso tipat
00. 11 = 0 seu 11-3C, H+ 3 200 = Quod si factimus + esse
fiet 11. N F3C, 3I 2. 1. 31 00. 3t 00 H = 0 sei  ias quae aqualia daterrationalem
SF + 4 H H 4u00.t 10000- 4. H. 1 400 11 +0000- 4 H. H F 4 ep. 1)
Et siet 3C = 2.H + 00l4) 00 51 fit 31 = lvet 31- 11+ 00
irademms obgo t. sombres Hl 2g. 13  1tt 0vuli- 10
a leum nucvent
siet H. H H. H 31 31 (00. 3C
vat agut 3t. 3t - 08
c3
1d =
EI
vi