# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/DE-611-HS-957479:0005
# Permalink to this reading: https://leibnizlegible.com/page/DE-611-HS-957479:0005?run=82
# Title: LBr. 454 · Augustin Thomas à S.Josepho
# Folio 3r, page id DE-611-HS-957479:0005
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=DE-611-HS-957479&page=5
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-957479/jpgs/default/00000005.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 82, 2026-09-09T18:17:48Z — 44 lines, status machine
# Lines: 44 recognised, mean confidence 0.91
# Machine transcription. Not an edition. Errors are expected: about 8% character error rate on Latin and French lines, higher on German.
# Licence: CC BY 4.0, Leibniz Legible. Machine output: do not ingest it as verified text; keep this provenance with the text.
# Wording rule: say "the machine reads it as …", never "Leibniz wrote".
# After the first blank line: the recognised lines in reading order, one per line.

me
Senillustris Die Patrone Col.
Terpulchram speculationem Verill. D. Tua mihi refricuit jam olim
habitam occasione Problt itriu numeror talud, ut tam suma, quam diffa
binoru quorumvis sit eta propositi olim A. 1676 Parisiis a D. Dzanam
Profe ibidem famosissimo; cumque illud considerarem solvi non pose, nisi per
duplicatas aequalitates Diophantis, et in nodum incidere inextricabilem,
vix non in opinionem impossibilitatis cu Vetro Mengolo incidi. It a autem
S. 
in casu tantu particulari ilud fueram aggressus. Proposuera mihi duos
DEtos X+ 4X +4 et q + 4X +1 et utrinque exemi 4X. habuiqus tales tres numeros
X+ 4. 4X. et 4X F1. quoru medius a X hutrique extremor sive adjectus, sive detractus
quatuor aeta constituit X +4X+ 4. X = 4X + 4. ) 4X + 4X F 1. et demud 4X= 4X + 1.
reliqua erat, ut etiam extremoru tam suma quam diffa esset ta
Lrat autem Suma 5X + 5 = 1to
diffa 3X2 -3 = Uto. cumqus nullus ex hs terminis
esset quadratus, judicaveram primitus hanc duplicata aeglitate ee inenodabilem.
s
Nam inspectu notis Bacheti et Termatii in 24. Prop. Lib. 6. Dioph. reperi tantu
6. modos resolvendi duplic. aglit. inter quos tiu noster praesens casus non erat;
quinimo rejectu legi ab utroque, eo quod nullus terminus esset quadratus. Quid agere.
quo me verterem? nolui Probl. tam raru, quam jucundu dimitlere. Tandem incidit
methodus, qua uti solemus nonnungua in simplici aequatione per substitutione ad eliciendu
os
etu. Consideravi ergo supiores B. 5X +5 et 3X = 3 et deprehendi, si _4 ducat in 5
et producto 20 addat 5, nasci utu 25. Similiter, si idem 4 ducatu in 3, et producto 12 dema
tur 3, etiam fieri utu 9. quod mihi jam satis erat; itaque ex 4 radicem 2 arripui
atque ita substitur X = y -2. erit X = y = 4y +4
5XZ = 5y = 207 +20. Ac demid 5X + 5 = 5y 20y +25
3X = 3y= 12y+ 12.
3X=3 = 37 =12y + 9
Ducatu Suxior terminus per g. inferior  25. et habebimus novos terminos
75y = 3007 +225 = Eto
45Y = 1807 + 225 = ato
Diffa 30y - 120y. Hujus producentes ita quaesivi:
Ex 225 dupla rad. 30 dividens 120y profert 4y. et 4y dividens tam 30y quam 120y
profert 130 sumt ergo duo producentes 15730 et 4y
Horud suma 237 -30. dimidia 23y - XC
4. 52972-3457 +225 = 75y = 300y +225
et fit y = 2940
ideoque y - 2 = 698 = X. Per hunc valorem
resoluta supiores Positiones x + 1. X + 4. 4X. extibent Comisso communi Denominatore)
in inlegru tres quastids 18. mnimos 2399057. 2286168. 1873432.
Htunc in Moduta pexta alias Vositiones elicient, alii in infinitu H. satisfacientes.