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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folio 169v, page id 00068199:0338
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Formula l xx + lmaxalnaa est ita aequanda quadrato fi  yy, ut fiat
a 3022+3) a Z + 3raa
3027+31a2+32aa
dprehendi posse fieri 22: 2 = 20g 39: 30 = 43: 40
et ita minui aequationes comparatitias seclvocalento
suppono praeterea Denominatorem 302Z + 31aZ + 32aa esse quadratum a 2130 + a Y32, unde fit 
302Z+ 24Z130. 32 + 32aa et fit 31 = 2 130. 32 et 31: 4 = 30. 32 seu 32 =31:4.30. Ergofiet
22: 20 = 31:4. 30 = 4240
 H, 20. 20 Z7 +2. 20.21 aZ
lumm 20.30.. + 1. 20. 31..
1. 21. 30,
ln, 30.30. +2. 30. 31..
l, - 40. 40.. 82. 4091..
(1) aqu. 0 (2) aequ. 0
aaZZ
+ 2. 20. 22
+ 1. 21. 21 Z + 2. 21. 22 a3 Z +, 22. 2c
+ 1. 20. 32..
1. 21,31..
1. 22. 30
+ 2. 30.32.
+ 1. 31. 31..
8- 2. 40.42.
 -1. 41. 41
(3) aequ. 0
+ 1. 21.32.
1. 22,31. + 1. 22.32..
42. 31. 32, +132. 32
82. 91. 42.. A1. 92. 42
(4) aqu.0 (5) aequ0
si po 22 et 3 cer substituantur valores, coincident aeqq. 4 et 5, cum aequ. 1 e1Z
seu aequationes natae ex destructione termini seu ultimi et eta de
atimimpenaltim et altimi coincident cum aequationibus ortis ex destnctione lepmini primi et
secundi. Nam ex aq. 5; fit 20 32H+ 29. 3 20. 327 + 26. 30. 3234 +30, 32. 32 40. 327-0
unde divisis omnibus per 32 prodit tadem aequatgo quae ex repmino primo; idem
Eodem mode terminus pepceltumus reducitur ad secundum. C sufficiunt ergo aequatione
1, 2, 3, nascentes ex destiuctione Termini primi, secundi, tertus.
Exprimo
20l + 20. 30lm + 30 In = 90 l. Habetur ego 20 per 20 et 30
(2. 20l. = 1. 30m) 21 + (1. 20m + 2.30n) 31 = 2. 40. 4 1su comperdio _ 7o l2 + vol 31 = 2. 40. til
seu 70. 21 + 80. 31 = 2. 40. 41 u et quadrando 70. 21+ 2. 70. 8D. 21. 31 + 88. 31 = 40. 91
Exsecurto
As 20 (4. 90 - 78) 21. 21  + (. 40 lm - 2. 70. 8oll) 21.316 + (4. 40lh - 9dU) 31  = 0 5V
quarum una non habeat i altera habeat, quae erunt:
2 20.22l + 20.32 lin + 2230lm 12. 3032ln= 2.4042 e (n sit EP
vellare int dua
usiderari debat, ut involvens 1)
sit 42.42 = 32. 32 In. f 42 = 321 nol)
et firt (20. 32( + 22.30) lm ~ 2. 4042
e 32 = 22l =0 seu 32=_- 22l: m et fiet 2. 21. 22l( - 22(: m) 21M L 22. 3,m = 2.3, 2 l: m) n
2. 4V4C2. 4132 S(n: 1) seu 21. (23lm F (22) 3imm = 2. 31 (22)In (== 2. 41. 32lm Y(n: 1)
= - 2. 41 (27)1(:. n) Un) s(n. l) C. =- 2. 41l AIn 42 = 3+(4. 30) In: 1)
fiet 4. 3P: 30 = - 22. l. m seu 22 = - ~31 m: 4. 30l
4. 2oUm = 2. 70.8013= (4. 20 Um (F 4) ~. 3olmm t 4. 3o lmm. 2l 30 2es Elgoki r3oik Eli holion ihrutryu - 20. 42 = (+) H
quid si Hoc ejqu omisso videamus an liceat aequationem destructitiam Termini medii
24= 31,4.3
aequ. (2. 40. 317+ 4. 30. 417 =) 2. 20. 31 U+ (2, 20. 30.3 1 F 2. 30. 31 b) . (30. 7. r 2 30. 70. 80. 21311 38 863, (ll. 46).
4. 40= 76 = (4. 20+ 4. 2030 lin  4. 30 br (. 4. 28l 4, 20. hol m) 3omm = 50 (atm - mm) - 30 pl
4 40 170.8o, = 4. 20 H+ 4. 2ogolm 1 F. 3D In 52. 2olhom 2 2. 2ol. sn 2 3on- 2. 3om. 2om- 2. 3m. 2. gon
4. 40- 7  = (4. 20 I(+ 4. 20. 30lm f, 4. 30 ln (= 4. 20U 4. 20. 3olm 30mm = 20 (4ln - mm) = 30 pl
4. 40Un -80 1 = 4. 20Un (F 4. 20. 30 lm,  4. 30 lnn) - 20mm (-4. 20. 30m 4. 30 l nn)
30. 21. 21B = 2. 20. 30. 2.31-+ 20.35=0 seu 30. 21 = 20. 71. quod es non licet
Ergo rec licet facere 20 22:20 = 32, 30 = 42:40
etiam
Ergo ex aeq. 5 fat mattilicat pert, fiet 
et fiat
Veniamus jam ad terminum 3tium seu medium quidaf ubi pro 32 ponendo 31:  4. 30, et pro 22 20. 31: 43 8 d2, 90. 2: 430
2202 2 0l + 1. 4. 30 . 20. 31P. 4. 30. 25 (+ 30. 3i, ( 4. 30. 2. 3lm (4 30. 3 m 2. 3d. 31  438 dinx
et divisis, omnibus per 30 fit (4ac 70) cessu ta q  90) 2t q + (4 40 30) diln
= 2. 20. 30l (mm - 4 ln) = - 2. 20. 30pl. Et tandem explicemus
= 2(20 l (4 ln = mm) = 20 pl.
multilicata per l, et divisa per pl, fiet: