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# Title: Ad majorem Dei gloriam. Theorema Arithmeticum (shelfmarks Leibn. Marg. 88, Stück 5; Leibn. Marg. 88)
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em ae et
du dem Borito Luibuitemm C3. itie des d. Sibor ses
Inhaet
1. H. Ratem, Laerche Alribra, Furiat. 1654.
s acili Namirnm perfectur up (618.
S. oiit Nova arithmetica quadicae, lamleua 1bbt
J. Remmelint Namciaa Aigurataa Niuc Arithontica
alytica... Fealhaberi ara in refatatue, e. b. It s
5. Mcorcua enthmelicum c. Rosponsat I. ctencoler
in aoponlad) Dranacm
er ACC
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Ad majorem Dei gloriam:
  n L a L d Vhergola
ARITITMETTODM.
Non est possibile invenire tres numeros, quorum
differentia quadrati; e quorum differentia
quadratorum, quadrati.
Pro Colutione Problematis propotiri (ub dic 27. Fei
bruarii 1674. Cub his verbis:
Invenire tres quadratos, ut duorum quorumlibet intervallum
sit quadratus, es intervallum laterum ex duobus quibus-
cumque sit etiam quadratus.
Respondit PETRVS MENOOLUS PriOE S. Magd.
Mechanicus Bononien.
Cui relpondit (ub fnem  A C O BU S O Z A N A M
Dombenlis, Protess. Machems dic 18. Aprilis 1674.
etfacren
av
ad   t 
MI. D C. L Z X IV:

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sI
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DERSÄCHSISCHE LANDESBIB
atur: Leibr. Marg. 88, Suick 5.
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## Folio 3 — 827668708:0005
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bensoben sobesobe ddben
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ARITIIMETTOOM
Macese
on est posibile, tres inaquales numeros assi-
enare: quorum differentia, tres quadrati;
eg differentiae quadratorum, tres quadrati.
s
e
ne
OSSTBTLE CA enim, duos inaquales nume
10s allignare, quorum differentia quadrato.
cum, quadracus: ec primum duos, quorum
differentia quadratorum quadratus impai.
ADumatur enim quodcumque Lacusa: e du-
plum lacus 2 a: ex augeatur rnicare, ve hat 2 a F r. APumatux
ejuldem laceris rriansulus a 2F a (2): e quadruplus criangu
LuS2a2F2a: d mulciplicetur per quemdam imparem nume-
rum b, ve Aac 2 a 2 b F2 a b, numerus minor: qui augeatur iplo
impare b, ve frac 2 a 2b F 2 ab Fb, numerus major. FcaTuman-
cur duorum hujusmodi numerorum quadract, 4 a 4 b 2 + 8 a
b2 + 4a 2b2, minor quadratuS; ec 4 a 4 b 2 F8a 3b 2 + 8 a
2b2t 4 ab 2Fb2, major quadratus: quorum differentia + a
2b2+ 4 ab 2 + b2, quadratus a radice 2 a b t b: quxe producieur
ex 2 a FF, in b.
Irem possibile est duos assignare numeros, quorum difte-
rentia quadtacorum quadracus PaI.
ANumatur enim quodcumque lacus a: ejusque quadratus
a 2: Qui vnITaTe MINUAEIT, VE AAC a 2 = f: de per quemeum
que numerum b, mulciplicetur; ve fiac productus a 2 b - b.
ex minor numerus: cui addaTUF 2b, PaF, ve AaE majOF nu-
merus a 2b F b. Ouorum AanC quodraEt, a A6Z = 2 a 2 b 2 1
r
A
D
2
e
A

## Folio 4 — 827668708:0006
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# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 38 lines, status machine
b 4, minor, dc a 4b2 F2 a 2b2 t b 4, major: quorum difte-
rentia 4 a 2 b2, quadratus; a radice 2 a b, producta ex a in 2b.
Porro duorum numerorum a, b, differentia quadeaeorum
a 2 = b 2, eA productus ex summa iplorum a lb, de diffe-
Tentia a -- b.
Deinde omnis productus duorum numerorum, in luOS
primosrelolutorum, est ex iildem primis productus: puta ex
a, b, primis productus est a b. Ec ex producentibus in pri-
mosrelolutis, puta ex ab, e c d e, eAt productus a b c d c.
Et exproducentibus a 2 b, _ a b 2, est productus a 3 b 3: d
SC deinceps.
Ouare omnis quadratus in suos primos relolurus, ex se
cundis porectatibus primorum eAt productus: a 2, ex a in a:
a 2b2, Cx a2 in b 2; vel ex abinab; vel ex a b2 ina; vel ex
a 2 b in b: ec non aliter.
Cum itaque duorum numerorum differentia quadratorum
est quadeatus, si differentia lacerum ex primis quibusdam
numeris componitur a b, necesse est, quod summa laterum
non ex iisdem Polum componatur, non enim eset major
summa, quam differentia: Ped quod ex iildem, et alio quo-
PIaM COMPONatuT, non Cemel, Sed bIS ACCePTO; puta CX ab5
2, VE ita Aat a 2b2C2, quadratus. Si enim non ex iildem,
Ted Polum ex bc2 Componeretur, productus ex a b, differen-
tia, ex Cx bc2 Tumma, a b 2c2, non eNsec quadratus. Irem
si ex iisdem, ex alio quopiam, sed non quadeato compone-
retur, productus ex ab differentia, ec ex a b c lumma, a 2 b
2C, non eNec quadratus. Ohare summa ad differentiam est
mulciplex, per quadratum; a b c 2, ad a b, PeF C 2, qua-
deatum.
Toitur e converso, s duo inxquales numeri tuerine asS-
snaci, quorum differentia quadratorum, quadeatus; pocerit
ipsa differentia quadeatorum, in suos primos resolvi, ee in
secundas porestares primorum: crieque summa lacerum, ad
differentiam, mulciplex, per humerum quadratum: crieque
radix differentix quadratorum, mumerus ex iildem primis
Composicus, et ad lacerum differentiam multiplex.
Si ergo duorum numerorum differentia quadratorum est
impar: eric ee lacerum differentia impar, quxe dicatur b; erie

## Folio 5 — 827668708:0007
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quoque radix dicterentia quadcatorum impar, multiplex ad
b: vnde sublato impare b, relinquetur par, mulciplex ad
b, cujus dimidium dicatur a: exieque 2 a b F b, radix diffe-
rentix quadratorum: e erit differentia quadratorum 4 a
2b2 F 4 a b 2 7 b2: e Cumma laterum 4 a 2 b F 4 ab F b: cui
addica differentia b, Facit duplum majus latus + a 2b f 4 ab F 2b:
e majus lacus 2 a2 b F2 a b + b, impar: _ minus latuS 2 a 2b-
Y2 ab, par; mulciplex ad differentiam b, per 2a 2 Y 2a, qua-
druplum criangulum a latere a.
Icem s duorum numerorum differentia quadratorum est
par, erit lacerum differentia par: cujus dimidium dicatur b.
Frit quoque radix differentia quadratorum par, mulciplex
ad differentiam 2b, per numerum, qui dicatur a.  2rC2 ab.
radix erit differentix quadcatorum: e eric ipla differentia
quadratorum 4 a 2b2, & Cumma laterum 2 a 2b: ex ex Cum-
ma, e differentia laterum, duplum majus latus 2 a 2 b f 2b:
iplumque latus majus a 2bF b: d minus latus a 2b - b, mul-
ciplex ad b, per a 2 =r, quadrarum laceris a, rnitate inul.
TaEUM.
Teaque Si tres numeri assionari fuerine inxquales, aximus
a, medius b, minimus c: quorum differentia a -- b, b - c
a =c, quadraci; ex differentia quadratorum a 2 - b2, b2
= c2, a2 =c2, quadraci: cum differentix quadratorum
ad differentias laterum sinc mulciplices per Cummas, ee ve
quadeacus numerus ad quadeacum, erunt mulciplices per qua-
deatos: iplaeque Cummae laterum erunE quadrati a Fb, b + c.
a +C
Porro a, b, c, alsignati numeri, vel lunt incer se primi,
vel non. Si lunc primi, non lunt omnes pares: led eorum
aliguis impaF.
Deindenon sunc eres impares; immo neque duo. Si enim
essent duo impares a, b, esset eorum differentia a = b, qua-
dracus par, diriduus per 4: idcirco lemissis differentix eNec
Par numerus, per quem numerus b, eset multiplex ad quem-
dam quadeacum ruitate multatum: atque ita eRet PaF, COn-
era hypochesim. Non ergo a, b, lune impares. Similiter
neque b, C, TUnE iMparCS, nequC a, C, IMPATCS.
Irem a, non est par. Si enim a cet par, eset b, vel cim-

## Folio 6 — 827668708:0008
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paT. Fsob impar. Fric ergo differentia a-b, impar: e dit-
Ferentia quadracorum a 2 -- - b2, impar: e minoris quadraci
lacus b, par, conera hypochelim. Non ergo a, est par, sed
impar: _ reliqui b, c, LUnE ParCS.
EA ergo a --- b, differentia impar, ex quadeacus: cujus
radix dicatur d: critque d2, ipla differentia a - - - b. Pric etiam
a 2, IImp2T, _b 2, Par: _ differentia quadratorum a 2 - - - b 23
ipar, e ipla quadratus: cujus radix impar, ex mulciplex
2d d 2: vnde sublato impare d 2, relinquetur par mulciplex
ad d 2: cujus dimidium dicatur e: exirque lacus a, idem ac-
que 2c2 d2 +2ed 2 + d 2: e minus latus b, idem atque 2 c 2 d
2Y2 ed2, mulciplex ad differentiam d2, Per 2 e2 + 2e, qua
druplum criangulum a latere c.
Acqui b - - - c, est differentia par, e quadratus par: cujus
radix par, de radicis dimidium dicatur f. Prit ergo b - - -c,
idem acque 4f2. Fric quoque radix differentix quadratorum
b2 -- - c2, par, mulciplex ad differentiam 4f2, per numc-
rum, qui dicacur o. Onare 45f2, radix erit disterentix qua-
dratorum: ec eric ipla differentia quadratorum 1682f4: 8
Cumma lacerum 492f2: _ duplum majus latus 48 2f2 F 4f2:
ec majus ipsum lacus b, erir idem atque 28 2 f2 F2 f2: 8t
minuslatus c, idem acque 292f2 ---2f2, mulciplex ad 2f23
per 52--- +, quadracum lateris e, vnicate mulcatum.
Sunt ergo eidem b, cadem, e incer le eadem, 2c2d2 7
2cd2, _ 292f2 72f2: ex est a, idem acque 2 c2 d2 7 2 c d 2 7
d2: ergo a, est idem acque 252f2 F 2f2 + d2: ec idem ac-
que 282f2-- - 2f2: corumque differentia eA 4f2 7 d 2.
Es ec a 2, idem acque 15 +f4 F882f 4 Y 4f4 F 482f2
d2 F 4f2d2 + d4: e c2, idem atque 45 4f4 = 882f 4 Y
4f4: e differentia quadratorum a 2 =c2, eadem acque
1682f47 452f2 d2Y +f2 d2 + d 4: acque ipla est quadratus
2 radice 4f2 7 d 2.
EK ergo radix difterentix quadratorum a2 = c2, ad dit-
ferenciam lacerum a -c, aequalis, e non mulciplex. Atque
ita differentia quadrarorum a 2 = c2, ad differentiam lace-
Tum a =c, CAVE a = c, ad rnitacem: sed eriam est ve a fc,
ad rnitatem: quare a -c. 8 a tc, Cunc aequales. EF hoc est
abCurdum.

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Non ergo possibile est inrenire rres numeros primos incer
se, quorum differencix, quadrari, ec differentix quadtato
rum, quadrari.
Sed Aa, b, c, tuerine assignati non primi, accipiatur eorum
maxima communis mensura: gux profecto difterentiarum
a = b, b = C, a =- C, COMMUNIS mentura eA quadratorum,
iplaque quadratus numerus est. Ideo enim a, b, non lunt
primi, quia a, relolutus in primos, re hac hpg, ec brelolutus
in primos, re tiac hps, habent aliquem componentem com-
munem, putahp: ec differentia a - b, est hp g -- hp5: Ouxe
non porest esse quadratus, nisi ec hp, SEquadtatUS, e 9 - 5,
sIE quadracus: alcerum enim sinc alrero non (ufficit. FC CirO
iplorum a, b, c, communis maxima mentura quadeatus, qux
dicaeur h2.
Ec dividaneur a, b, c, per h2, ve hanc primi numeril, m,
n: ec differentia, quadeari a = b, b = c, a = c, dividantur
Per h2, VE Aant l = m, m - n, l = n, paricer quadrari nu-
Mmeri: item a2, b2, c2, diridantur per h4, ve tiane quadraci
l2, m 2, n 2: e differentia quadratorum, quadraci a 2 - b2,
b2 = C2, a2 = c2, diridantur per h4, Ve Aanc l2 = M 2,
m 2= n 2, l2 = n 2, quadraci numeri.
Sed hujusmodi priminumeri l, m, n, alsignari non possunt,
ve ostenlum est: ergo nullomodo poruerunt astignaria, b, c5
quorum differentix a b, b - c, a = c, quadrati; e quorum
differentiae quadratorum a 2 = b2, b2 = C2, a 2 = C2, qu2-
drati. OMare, ec.
LVAN ZDM sic erronea propositi nostri Problematis allata
solutio, quaesiti tres quadrati sequentes indicabunt.
52357727962243 35097474586243 5755474489249.
Latera 2288168
1873432
2399057
Intervallum primi es secundi quadrati est 1725965337600.
cujus quadrata radix est 1313760.
intervallum secundi es tercii quadrati est 2245727030625,
cujus radix 1498575.

## Folio 8 — 827668708:0010
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8
Incervallum certii ex primi quadrati est 529 7 61693025
cujus radix 720945.
Intervallum primi es secundi lateris est 4Y47 36, cujus ra-
dix 644.
intervallum secundi es tertii lateris est 525625, cujus radix
125.
Incervallum tereii ex primi lateris est 220889, cujus radix
3.
Mremee enjun
tn

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## Folio  -  — 827668708:0013
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