# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/DE-611-HS-861255:0101
# Permalink to this reading: https://leibnizlegible.com/page/DE-611-HS-861255:0101?run=74
# Title: LBr. 705 · Overbeck
# Folio 51r, page id DE-611-HS-861255:0101
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 50v); this page is its right half.
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=DE-611-HS-861255&page=101
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-861255/jpgs/default/00000101.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 39 lines, status machine
# Lines: 39 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.

Vir Tarissine et doctissime amice colende
tur
Desideras solutiorem aequationis Numericae, qualis est f+gx = gus
posto datosfo; et quaesitos X, y esse intsgros, et L esse minorem quam o
enquod idem est quaerithrd Aeddratas yus qui per datum 9 pitvisus restiduum relinguat f: quotiens exf
stachabelle quia X = yy-f,: g. quaratur pernods
residuorum, si quadrati ordine dividantur sper datum a. in ea: periodo appare
bit, quibus in locis residuns sit f. Talium igitur quadratorum residuum f
habentium datur series. , et horum quudratorum latera
erunt y, quotientes autem erunt X. Aliter: eb f+9X,: y = 9
Sed si quis rem a seric quadratorum traducere velit ad serdem numerorum
sumendo pro et quadralum quemvis contentur
naturalium, poterit taliue dna uti Fiat hf- h + eC, et ex aequa
y- e+
f-h+ XC, sumendo sub ee, quadram quemvissmigorem ipsointeorum minorem ipsos
tione data fint h+gX = gu-1 Ergo h +gX dividi potest per 9 +P, et per 
6b prius fiet (y+2) p = h + gx seu (y+1) p - h = gX sit (y+dp = Z fiet utique
Z-h,: g =x. Itaque exponenda est series numevorum Z, qui divisi per datum 9
habent residuum a habent datum h. quae series facile habetur, quia series na-
furalium, (seu omnium oedine numerorum) per 9 divisorum est penodica
Itorum numerorum Z, functor quivis unitate minutus haber potest pro quaesito
y quaad hanc conditionem.
Rursus ob postenus, seu quia k+9Y
dividi potest per y -l; fiet (y - 4) r = h +gX, sit (9- DY = v. fiet utique
adhibebimus
v-h, 2g =X. itaque sursusahumus seriem numerorum, v, qui divisi per
Ex his numeris e vel v, qui por g divisi dant residuum t, com-
datum g, relinquen h. 5 
parentur ill inter se ql biricce bini quivis qui dant eosdem
quotientes. At ex his combinationibus intalli inteltigat eligantur illa
ubi simul unus numerorum, ut Z, habet multiplum, qui unitate minutus
dat idem quod alter, ut Z, ubi unius et numerorum ut Z, mu lsiplus aliquis
unitate minutus, dat idem quod alterius v, multiplus aliquis
quadrati aliquo
unitate auctus. Is numems qui idem prod it utroque modo, est y
a
eodemi
quad rato
quaesitus, et quotiens in tali casu communis est X quaesitu