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# Title: Leibniz-Handschriften zur Mathematik LH 35, 3 A 16
# Folio 2r, page id 00068229:0003
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et differentia
Invenire tres numeros tales, ut summa duorum quorumlibet
Siequ aste sit quadratus.
sint numen X, y, Z.
5s
Xry = aa X+ Z bb ytZ = cC. X. y = U X Z= mm y. Z- n
Ex c et 4 fit 2X = aa + H m bb + um et hy = aa - U = ce + nn et 2Z = bb - mm= cc- n
Ex solis aequationibus 1,2, 3 haberi potent 2X = aa + bb - cc 29 = aa - bb + cc 2Z= - aa + bb +cC
Ex solis aequationibus 4, 5, 6, nullius extribus literis X, y, Z, valor sine interventa alterius ex ipsismet haberi poest
sed loco ultima aequatione tollendo literas x et y, evanescit Z, et fit I1 + nn tum
quem vaborem ipsius rum, substicuendo in aeq. 7 et 9.  tunc ex aeq. 7. fit nn aq - bb
et ex aeq. 9. fit U = bb - cC quod si ope aequationum 14 et 15 velis tollere bb, fit I1 + nn aa- cc
quae aequatio coincidit cum aeq. g. quod est signum calculi probi. Itaque sufficinsit ad problema
solvendim sifficiunt aequationes 13, 14, 15.
pro aq. 4. scribamus 3 5G44
 6.5. 6.5 = 4A. 8:5. 8:5 ut calculum in numeris supposititiis comprobure possimus.
pro 5q. 4 fiat Et ta pro a, b, n stabit 574, 3 prob, c, tstibit 4, 815, 6i5
jam pro 14 solvenda generaliter finit: 5+ 4  (16)
(17)
pro solvenda aequatione 14 fiat a + b = dde et a - b = dec fiet
a dde + dec,: L et b = dde- dee: 2
27
similiter pro solvenda
aequatione 15 fiat b+ c C Hg et b - c = f9g. Ergo b = s9 + f9g,: 2 et c ffg- fgg. 2
(24)C
Et aequando 19, 22 fit: dd e - dec = fg + f99
sit de =hfg
et fiet eutnque valorem ipsius de substituendo in aeg. 24. fit et provenientem
aequationem dividendo, fiet:  hd = he = f +g. Ergo ex aeq. 25 et 26
fit d = hfg: C 5+g + he,: h seu hh fg ef + eg+ hee; sive
f = e9 + hee,: hhg - e. Cum ergo ex 14, 16, 17 sit n = de
et ex 15, 20, 21 sit l f9. Hinc jam solutas habemus in numeris
aequatiores 14 et 15, restatque solum solvenda aeq. B. operae pretium autem
erit solutionem aequationum 14 et 15 prius exequi, in venta sunt aequationes
 h9g +hhge: hhg- e
substiegendo valorem f in d, ft ex 29 et 27. fit 
et n = h egg + hheeg,: thg - e
et l= e99 + heeg,: chihg - e
et 2a = hegg + hhg; hhg - e h gg + hkge: hhg- e,, + e
seura = et2) hgg + hh ge+  hgg + hhge, lihg - e, ee, 1 F. htg-e
Cc hgg + hnge, hhg-c, ee,, X et Shg C
E aC
et 2b = (9 12) e9 + hee, + eg + hee, hhg - e, d9∞. M htig- e,
= a eg + hee, hhg- e, g9, W (2I, hhg-e,
et 2b. - eH Eh03 + hahC.
Et si bonus est calculus debent duo valores ipsius b coincidere, sunt autem nominatores iidem ergo
et 2C = ( d 1 eg+ hee,
coimidere ad huc debent numeratores: et siet C si gt + (eet93(hC Hg’ ee ( )
unol
+ he39 + hhgC = (eC9+ (2) h e 9  si et g L etitg + hE9 e (C g e h9)
Jeaque evolvendo omnia, fiet
16)CCh.9t +3 (2) C hg3 9e n’g e0+ (h g e heg - h’get,: () hhg- e,
quae atque quae duo revera coincidunt.
35)
2b = Ch g + ek393 + h0g + h9e,: 1 bhg - e
2a =
2CZ 2ee 93 + (3) he g + h Ct g - ek g - h3ge + e gehg: 1) chhg-e,
Habemus ergo aequationes quaesitos valores literarum a, b, C, l, n, pro  per aequa
tiones 35,3637, 33, 34. pro solvendis aequationibus 14. 15. idque quasi
semma considerari potest,  seu problema subsidiarium in principali
sit h = 2, et g = 1, et c = 1 fiet
2a = 4 + 24 + 32 - 2 - 4,: 1 4 - 1, seu a = 27:9 = 3. et 2b = 4 +8 + 2+ 4;9 seu b = 1
problemate nostro intercurrens.
n = 5: 3. l= 1
 sit h = 2. et g - 1 et C = 1.  (per 29) fi et f = 1 et (per 27) 1 = 2
et 2 = 2+6 + 4 - 4 - 8,: 9 seu c = 0
a
31) 1=1
et (per 19 vel 22) b = 1
et (per 30) n = 2
et (per 23) C20
et (per 18) a =  3 etsar cysibet