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# Catalogue record: 27229 — [Wien?]: Mathematische Aufzeichnungen
# Incipit: Numerus c resolvendus est in duos qvadratos xx + yy, fiat x = t:z et y
# Date: 1714
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=27229&katnr=56058
# Work: https://leibnizlegible.com/work/DE-611-HS-861255
# Title: LBr. 705 · Overbeck
# Folios: Bl. 86 (from the shelfmark LBr 705 Bl. 86): 2 pages, canvases 171–172, page ids DE-611-HS-861255:0171 to DE-611-HS-861255:0172
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=DE-611-HS-861255&page=171
# Lines: 102 recognised on 2 of 2 pages, mean confidence 0.78
# Catalogue: Leibniz-Katalog / Arbeitskatalog der Leibniz-Edition (https://leibniz-katalog.bbaw.de/), ed. Arbeitsstelle Potsdam I der Berlin-Brandenburgischen Akademie der Wissenschaften, CC BY 4.0.
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## Folio 86r — DE-611-HS-861255:0171
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-861255/jpgs/default/00000171.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 77 lines, status machine
Numerus C revendus est in duos quadratas xx+ jus fiat
X = t: Z et g = v: Z, fiet cZZ =a + +loV fiat FL gere v
Fiat L = M + p n = n +Y et fint. EZZ = lmm + Emp alpp +lun + dur+rr. quae divellatur
in duas
1.
c27= lnn + unr + hur et lrum + amp + lpp- hngtlir 
fiat lZ= n et ex hit nn = In + Ur+hr EP ~g. 2 fit Imm + lmp + lpp =- lrr+ hur
fiat alim +p =BY et tn ilx et im+p = -Ir+ hn,: β. itaque pro literit
n,  EY habemus tres aequationes simplices 4,6,7 sit n P: n, et
mip. r
ne us ex A habeatur r: n, et fit ex aeq. 3. fist lZ2 U: M et ex aq. 4 fiet
U = In + ur +hr, sit 2l+h = q et suit ex aq 4 siit 1cn +gr. Ret ergo est
reducta ad aeqqu et (m+p = t fu
(1
siet F = 3r: l = - lrthn: B  seu BBr= -Ilr + Ihp, seu H = ( 4) - BB U1,: th a
=X per 10) lc - qr, : l. unde eB, aY + qY = c, seu rIhc: ββ- 11 + tg
Rursu ex 8, st H = P: C, fiet CK 14,
10
B: nn = r, B- U,: Ih, seu r= (h,:  (2, (B- II
Ergo ex 15et 16 fit. C, n, BB l = B, BB U+hq seu denique l. h - 17- C(M B, B
seu C  = 13, βB- l: (3 (h9) ~ ih +hh sit h AB+ L, _I) et C C, ). l - El + h
LE+h, seu h - 1 = (CMM - (B) ((3 - 1)= 2V,: Be
et CM - (3 B- l, γ: 1469
Et siet, β(5+  = cM n βYY - CM lYY -(3 By + 147Y - 2(5), seu
3(0) (4yY - 2B - 16 C (n n)),: 15 + 13γY-cn VV
Resumamus cZ3 = IEF + lnn + Elar + Lrr, quaedividatur in duas cZZ = lnn + dur + hur
+hur hnt
et IH = hur-ler fiat n = 1Z: n set _Z = II: er, fiet  Cl2. n = 1:  + Elr: nen thrl: p
seu C = cl + 2Y +hr, n L (4. Et H = I + trrn = Btr fiet
t = Br: L et prodibil lβ33 n( t er = Ist fint ren = lh: tBB + H = ( - len,: h
3at  = 1B BB( + (5 - ( ne =ILC ( n sint Y - lh B +H = (4- lan(, : 2lt h
_ Z, n, Y, F
Z. h
. Y
+, n,
= an fial lc 2Z-Hut nun +Er+h Unhar
fiet n
= Br. l
IPerr: HI=nr-Ur se Ber = Ihn UY
(cZ2 = lH + lnn + ilaer + ler +hap - hur dividatur in das cZ2= lnn + Eur + hur
et IH + Err = hur  seu IH = hur - Irr.  sit  en: l et ex aeq. 2 fiet
CMmnn: l = lnutrlurthur seu cnn (3n + 2Br + Utr. Ex aq. 3 f
fiat L= βr: l) et exaeq. 3 fiet (BBrr. H= har - ler seu eBY = Ihn - Ur sit n l:n
sit. sed quia sit per aeq. 5 et 6 haberetur bis M: Y, ideo  ut hoc impediamus fiat
nmn= Ulf) seu  ita habebimus A et Y ex aeqqe 5 et 6. sed tandem valor ipsius n
parsus aequandus esit quadrato divisoper f. et fiet Ex 5pet fC = In+ 2lr + hr. seu
= Y= fC - ln, i, Elt h = (ex 6) Ihn:, B H. sed video careri posse 
(10) 3 imo 
iptius n set restinatur pro f ponendo Mnn: U, et fiet, C pn = Bn:, El+h = Ihn:, BB- U
et quanescet n, fietque cnn - 13:, il+ h Bh: BB- _U fiat trg B
= (+ l, 9 et CM p - (3:, 219 +hg = (:, β- l quarum aequationum
Resumatur Numerus datus C divellendus est in binos quadratos t: ZZ et rv. ZZ fiat v = n+Y
ope pabebimus β et h
et pro unitate assumatur l. fiet H cZZ IH + lnntlur + Cer seu czqe = Ut+ lin+ dircbralin, hur
quae divellatur in aegg. duas 122 = (2) = lnn + Elnr +hur et IH + Irr - kurseu lH = 3 = hur-lrr
fiat Z= 4F NN: L et ex aeq. 2. fiet cmmnn. HI = lnn + dur+hur seu cmnn s=  ln + 2(r + Uhy
Rursus fiat t =  +6 = Br: L et exeq. 3 fiet lBBrr: H = hnr -lrr seu eBr =  lhn - Ur seu r = 7 - lhH:, B + 
Sed ex aeq. 5 fit r=  ce  - (3:, 2β + Uh
2th + Bhh = c∞ BB + C M MII - BBB, 1e
 V, hh + 2h H = c BB + C pll = 13 (β seu (3, hh + ahita + BB + 2B(h HI)- ce  t
13, qu (h+ 1  3) = (C M MBB +CC llcep + 2(3 (3(h+ ()
=AlurL CZZ = CH + lnn - 2 (nr = lrr
ondum apparet via
H cZZ = lt + vo C c2) MM ccr
CZ7 crr: IE) Et = = (B - crr: 13) H + vo
desideratum
r: WE
C27 = H+ vx + 2to sit + = BZ:l fiet
E+ . GB; q= v + .92.
eet
H+oV
Crr- B sit quadratus seu C= Coco+ B,: rr, sed ita pettur
N cZ3+ Ap: 1). ZV= qu (v.t p :1) Z)
to
A7 1D+ pZ
at
Aequatio

## Folio 86v — DE-611-HS-861255:0172
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-861255/jpgs/default/00000172.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 25 lines, status machine
= t + vV
c22+2tv = co
Laat bXX) = 10 +  10. UX + 10. HNXX + 10. 11. 12. BX + 1.
cZ.
1+ = CmFlA Iv = cp+ lY, eruntque n et Y minores quam c et fue
(C27 = cemm + 2cmn + Il nn f cepp + 2c( + Urr. erit nn +rr = cG. itaque
possunt inveniri omnes numeri minores quam c, quorum quadrati in unum additi sunt divisibiles per e
lH rlon
s
et possunt pro inde n et Y haberi pro datis: et nn +  = cq, eutque etiam qdalas
CL
et fiet 1ZZ = cmm + 2mn + Cpp + 2lpr + q. Dantur ergo q et n et Y
supersant Z, m, p indeterminati, et fiet 1Z7 - 2 lnM - 2lr P, -q= c+ ep
um
et (iet 1Z = cS+ lF et M = cN W + B et P = C2 +y adeoque
ccs + rest S rlt + cc WW + t CBW UB ulnc W - n (3 - etrc A - ellv) - l9-
seu c5s+ 2ltS = ElnW . Elr s + HE + 19 - illu B - cr = lmme lpp
LLCC 55 + 2et S + HIIL = cclHo Ac
seu ccSS + 2 let S +u_ = Lc M A cpB + Elu M + illr d + q
et
C ui
vanttr: C, UIHF. - e
UF+ = c pp + 2lr P + q
cc ss + uet S = c MM + un M
sit L Y et prodibit  cpPt F+Unn =0