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# Catalogue record: 12661 — [Untersuchung der ihm von Huygens 1693-01-12 gestellten Probleme (xn+yn=nxy u. Beaunsche Aufgabe)]
# Incipit: Hugenius in literis 12 Januar 1693. ad me datis proposuit
# Date: 1693-03-20
# Akademie-Ausgabe: AA VII (assigned, not yet published)
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=12661&katnr=42079
# Work: https://leibnizlegible.com/work/00068197
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folios: Bl. 132–133 (from the shelfmark LH 35, 13, 2a Bl. 132-133): 4 pages, canvases 263–266, page ids 00068197:0263 to 00068197:0266
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068197&page=263
# Lines: 264 recognised on 4 of 4 pages, mean confidence 0.80
# 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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# After the first blank line, page by page: a "## Folio <label> — <page id>" line ("## Canvas <n>" where no folio label is recorded), "# " lines naming its source image and model, then its recognised lines in reading order, one per line.

## Folio 132r — 00068197:0263
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# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 76 lines, status machine
Hic est quadratmra mnihi ab Hugem
pro curta X+.
nxy
par3 sem eo
pagua
co
a une
Iem prbenta Bgammo propontt sili Cartescos toluo per serie ndus supponati pro habenda sere
Hugemrus in literis 12 januar 1693. ad me datis proposint mihi
quadraturam curvae cujus aequalio sit X+y= Xyn. posito n esse
constantem, et x, y esse coordinatas
Hanc sic inquiro pro
(4)  pB: 1+ p3 qunit jams quoritut a
1
ita res est reducta ad quadraturas rationales, 
quaeritur BX videamqus igi loco quisit. sxto
 pat v = e -n. pet v + n= e- 3ne + 37 e et x = n:, ce- methun, -
n:; e-ynet 3ne. quae partes sunt tractabiliores, sed placet redire ad aequ. qus
ubi p+V prodit ex = n+ v  n - nv + v0 et n - no+ ve  rursus resolvi potest
nam fiagatur vv - nV + na =0 fiet VV = nD + onn: 4 = un: 4, = na seu V - n : 2 = n f- 3: 2
seu v =  n: 2, 1+ S. 3 Ergo fit nn - nv + vDZ v - n: 2 +  n. 2 F. 3, v- n. 2 - n. 2F3
Ergo denique n + v = VAn, v- n: 2 + n. 2 F3, v- n: 2 = m. 2 F3 Sed at generalius pro
cedamus faciamus per cpendrm x =) nV: , vtn, VAP, vtq et  videam,
quomodo tale quid componi possit ex Tribus: X= b: vin, + C; vtp,, + f; vt g. quae addendo
in urum fiet: bvo + bptbgV apq + CBD+ CUrCqbenq + fDDt Sn+ BAsp,: ven vAp vtq X quae
aequatio ut conferetur aequationi I1 fiet b+ c+ f =0 seu b=O -c- f et rursus fiet lpa + cng + sup  0
et pro b substibuendo valorem, fiet ex aeq. 14 fiet: - epa - fp9 + Cnq + fnp = 0 jam quia conferendo
4, 10, 11 est p = - n: 2 + n. 2 F3 et GC =n: 2 - n: 2Y 3 fiet p9(e nn: 4 1t S.7 - 5-3 + 3 nn
et exaeq. 1b dividendo per nn, fiet - c - f  -c: 2 + c: 2 F.3 D - f: 26 f: 2 f.3 = 0 seu fiet,
C: f = - 34 53,: +3 - 5.3. Ergo fiet cif = + seu = - f sed ita sel b 6 quod satis
mirum est. Exequamur tamen: _fietque X   S:, vi in f:, v +  B- 1-, n.0
X= -f:, v +  S. 3 - 1n. 2, 45., v- F.) rr R.E (P- vf + 5i 2tiB +s f ibi  rat un
25
=fnsus, u n nn seu BC: f = (9 - 6F- 3- 3:, 9- 3=) 1 - (-3 seu C = - f+ f.3F. Ergo per 25 et 19.
fiet b (5 = f3f -f =) - Bif fit ergo X = -fs. :, vt n ∞-f + SE3, :, v+ (ag - 1B. 2, + f1, v- 123H M. 2
Unde sequetur legarichanam ipsius X esse lugatithocum ipsiusvanc e
e
eece
A= elog -fF. 3 log vin  -f. +f f. 3 log v + 53. rn. 2 + f log, v - F3+IN. 2 per compendum
n
30)
si sit XZ -10 log 11 + 12 log 13 + 14 log 15 fiet X- b 13 15: H X. = log, 132 151 11 foc
Eg
est paradoxum valde nitabile, qu u Nempe X: 5 erit log3, D + 5 . v- P3tihiL . v
seu Num; p.s   seu x - log, 13. . 1 15: 118. posito 10 = s. 3
eusit  coenu:v=
A Idem in fallor contingit in aliis sit XX + yy = xyn
Ergo ex aeg. 2
Examinemus rem prius in simplicioribus Xx +y = Exy haec equatio est ad rectam
fiat y = _ _ XV: n fiet nnXX +XXWD = hn XXV. et evanescentibus ex siet nutvo-bno
tyebolat
Faciamus X + yn = Xy it y - pv. n fiet: Xn + xon = XV seu nt vn =xV
 X+  = nXy sit y = XV:n ergo X n +X v = nXB seu X n + XV = nv. seu X = nv; n+V
Hujus ergo quadratura pendet ex aliquo alture qual ocurrit in curva Ellipseos.
si velim _xris servare fiet x + = n al X+y= nxq vel X   = rg .n vel xjus =Xy. n
quae estia 
si sit X + pX ∞q - 0 fit X =   + Y 4. 2 + 49. , + p er
0+ 2 = q5. l. 1+ 2.1 - 3 yy = nX - X: y
Mlemini olim ebiam incidere quadrabilia, ut y-Aptex: Saa m
4 nam doaa - Xx = a - 2x ax: Iaa - xx Hoc igitur quadrabile est: jam
FE = Y 93
sit aa - XX = a Trax +XX Wv.net a - X = a +Xx vra seu a - aex = a vv +XoV seu X - a - avv; v
et Saa = XY = a + X V. a jam a + X = (aB) + a + a  ( avD,: votaa, et sit areo
Vaa - xx = 2vaa: vo+ aa. Erso fit y = ba - (: 2va. seu sx - vrtaa. ∞ 2avdv - a- avp 2Vdg
(2votaa seu dx = ( 2av du - 2avav - 2ae vdv (= 2aV dv: vv +a seu dx - 4 a4v: (3) a + 
Ergo vax  seu bpdx: taa = 2baa aanvo: aav Hoc ergo summabile, utileque
sit  aa - ax = coco aa aa: taa- ae = y. siat
eut reducere ad lohanithmos
Paa - ax = coco eribque X =  - a - eco: a et Ax = - 2codco:a Ergo fiet
Aniter saa- ax =
ydx = ax 20 - 2 cindeo: c = -1a deo
LraL ta Taa - ax: laa tax = 7: a
 av; aator = y a sit
derum redeamus ad aeg. 4. et consideremus non quaeri a nobis
Id ergo quaeramus pag. seu
xdo, sed potius e y ax.

## Folio 132v — 00068197:0264
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# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 87 lines, status machine
s
Res ergo ab intro resumtetur
Augenius in literis cet 12 Jannar 1693 proposint, nitu quadraturam
ignae in in qua aequatio sit p +y = nxy posito X et y esse
loordinates et n esse constantem. faciamus v Xv.n fietque eque
pN + X V = nXV seu rX+ VX nv seu x nV. n+ V. sit
 quaeritur jam ydx. erit e (ex 2) = n V: n + V superest ergo ut quae-
ramus dx. side axe x near + nvav  hrevar: En+V
sit
et fiet deniqt s  EL  nev ax - 2rv dv:, Blntr
je jan
jamn ed: Bn+ = fco: co posito co n+  superest invenienda q
3V av
extma 
Litas in sodv: B (N seu fiv deo no Bxdo: BI n + V  Vdeo: c0. sed ve= co- n
ergo fit: E deo ico =  deo: a0co = = nsco: co. Atque ita res omnis ad communes
quadraturas redacta esst. Nempe - 2nvav: BrV = =n: 3 dco: cocd + 2n: 3 ndeo: c
12)
Ergo aqu. 7 explicando per P et 12 fiet ydx = redo: 8 - 2n: 3 dco: vco et fet
jam  diico = - deo: coco et d, 1icoco = -20 deo: Ct =). 2dco: co, et ideo siet.
Lydx i n:  + = n: co + nb ubi nb ascribo adEtitiam non
ariitrariam, sed postea determinandam. Porro ex aeq. 9 patet esse  x collata
cum aeg.4 patet esse co n v.X et f ex eq. 14 et 1s explicando co. fiet
Cydx = (- 2n: n v: X + Z n: n v. x + nb = 9 - = n X: v + 2 nx: v + nb. et quia
v = ny: X per aq. 2 fiet deuque sy.  Gyi yt nt e eC
16
Apparet autem ex figura (et aequaris
quod x  existente o, etiam y est o, et proinde et syax est o. Ergo
(yd = 2 X:  + 2 nX. y + nb.
eo iasu fit etium nb = 0 nam fiet  syda seu 0 -- 2 5: 8 + 2 n0: 0 + nb
nt
seu 0 = - 2 0 + 2 n0 + nb - et go, etiam b =0 evanescit . 6ie
1
et habemus denique
hB
yti
seu sydx :X
1de i ho laluntdem
13Hy-i ut  
vissen in duptam ratudende
53
pernatam
fit  2X= n - _ AD 2 AB et puto n fore BC Aagulus BAE semirectus
aut elegaatius sydx sen AEBFA est
asu quo est X=
LXX F Z nx
hinc fit hilineum AEBA B BE + st et semifitum A BFA cam sit triang.
aa
ABE dento taliner AEH,  et trang. ABE aeg. hO, fiet sol (E BE- : B) +; nn et
am. ABB
lumm inegro, auna posito B esse folie acumen, si AE- BE
pro maxima aedigatrix aminemus tangentes 3X dx + 3 dy = n. X d  
Sed hoc disculiendum
= 3X - ny, : nx =3 Hinc si ipsa ordinata sit tangens ( ex = 0 et 3y = nx quo
accura
valore ipsius x substituto in aeg. 1. por 2) y  + Bn = 3N y seu 27y = 2n
seu L y = 3R YZ et X (= = un 34. 3: n) = n  in casu maximae ordinatae
 Recta AB quae curam figuram in duo segmenta simulia e aequalia secac
debet facere sisecare angulum quen faciunt axis et duo coaves, alioqui figura
non se haberet eodem modo ad duos coaxes, quod tamen sequitur ex x et yse habentitus
remoeoptote. Referamus jam curvam ad hunc axem bisecantem duos coaxes.
Ex B puncto curvorin novum axem senurecto ad priores angulo AG, ducatur nor malis BH
et sit AH, Z et BH sit v
Et AK = Gk = BE = y +G = ys XE =GB = x - y
fiet HB, X- y: 12 = v et AG seu Z = y A2 + X: B = y. V. seu fr
X - y = vs2 et Z12 = (3y + X =y Ergo AX = v+ Z: Cr2 et y = Z- v. R
Eos valores susstibuamus in aeq. X,  = n Xy et sut. (D+ 3V Z + p 3Z B3ErLV)
WEn V ZZ. pN:2) seu fiet Z12 + 3ZV  = nZZ- nov, et fit
 sit n: s2= m et fiet Z + 3ZV = mzz - mvr et vv mZz -Z,:, n + 3C
im33 +3 m Z - 3. (t: (m +5Z seu v ZEmm + 2m ZD. 32Z:, m + 32
seu vv= mnhra unde rursus facilis ratio reducendi ad rationales nam
m+ Z +1Z in in g2- 2Z seu intq in m= Z debet aequaliquadrato ab m- Z e: a et fiet
seu v Z
Vm +Z- 422
m +sZ 28 n - q eiai seu  ma + 37a= mel - Zet seu Z= mecaa: cetsaa fietZe mt. 33
meet 3aam + 3mee (haam,: ee + haa. Ergo ex 26, 29, et m- Z n eex 3maa meetmam; persa
Ergo  ex 29,30 fit L:, m + 3. (31) in ee -qa  4 hee et ex 27 et 31 fit TmtZ- 47  a:, eerzaa
30t
sed quaeritur dq, ut
Ergo e ex 26 (per 32, 3) fit v 4) mace-ma: C t zaae
habectur vdz. e d fil dq (2 (et Emead) + haa2mede  mee 2ede + maazede,: 1) cetzaa et
ila fiet: Vdque 2 dm e 8amC de -8am de,:, 1l ée+ 3aa. ee + 344 2 ha rede  dha
et fiet pdZ  m The dua dh - Z a m dh.: Lstara:, na quae quomodo summabilis reddi possit
 eredte  a psicabiu s ae . 8 se
ida sit astectre us  aequatis ha  abentus By lxt p. 18. sect
M + 3Z in m - Z
dab ne + 3uZ m Z. 3

## Folio 133r — 00068197:0265
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# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 75 lines, status machine
(4
140)
seu trilineum AE BF A ag trap ei AHBE. trilim AHBFA. sed Trap AABC= Emper 15B Ex hilamus 4G
et traper AGBE =   volanus AB _ (seu Xy) - trianus AGH (seu gy: 2) et trianus BHG p Z7:2
Ergo fiet denum  Tuilin. AEBF A6q. Xy - gy: 2 + 72. 2 - frilin. AHBFA et pro trilineo AEBFA substituendo
valorem ex aq. 18 fiet pilin. AABFA = 2 nX . y - = X: yy - xy + 9y: 2-27:2 est autem
Xy C (per 20. 21) +6177- vv, : 2 ergo = xy = 22. 2 (= _ - 7Z + vv. 2 et yy, = 2 1V + vV. 2. adeoque
=Xy + 9y: 2 - Z2: 2 2) (- 77 + vr: 2 + Z7: 4 - Zv: 2 + vr: 4 -) 322: 4 + 3vv. 4 - Zv. 2
Z AXy8- ZX: gy = 2n L. v EtV: iq, +    Z∞ + 6Zv  gZV + vF: 8, : Z. yvrvv; 2
seu 2nxy- Lx : gu ~ 1en Z7. vr Z+V + =84ZV+6ZV +ZV+ V,: 4. 22. 230 + W Ergo
trilineum AHBFA fief  16n 97. vv. ZFD + Z4 4ZV+ BZEBVH V, V B. 3 ( WV L 2V - 2ZV _1ZV,: , 4 - v
examinis probationis causa
Sed nunc his omissis saltem calculum ese prioem insistendo ipsis er reassemamus
(4
u
352
sed hoc prolxius ut verear ne in calculo sit eiratum.
X+ G = nxy sit G = xv:n fiet X + xV. D = n Xxv:n seu rXeX = nV seu x = nv: n+V
et  per 2 et 4 fiet y =  (v: n NV:, n’ + V =) nV:, n + v. et dx =  v + N, Ndv, - n v 3Vdo,: EV
= ( PaV) + n dB = dv: ( v + M= dx. Ego per Set 6 fil Gax = nd v a - 2 nv v7:, (L V + _. Cujusqueritur
summa. Quod ut fiat commodius, ponatur V4+ 13 = co et pier 3Vd a seu v co- n et 3Vav deo seu
v du = deo: 3. Ergo per 9 et 11 ex ag. 7 fiet : ydx = = n deo - 2 n co - 13 de, :  co = 3 n deo C - Z  clleo + 31 la,: c0
= nd dco: C5 - 2ndeo: 3coco= ydx jam (ac: coco  Fisco  Ergo  Et pro co in aeq. B substituendo
adesque pro deo, ponendo 2do, fit S2odo: ct = - 1: 0c adeoque Sdco: a = - 1: 2cdc. Ergo ex aeqe 12
fit (ydx A = nb A- n: 2c0c + 2n. 3co Assumta nb tanquam adjectione
summationis ex propessu determinanda. set jam cosseu n3+ v) = n v: X ex aeg. +. et v Z ny: x ex .
Ergo ex 16e117 fit co = nt y: Xx. Unde ex aep. s fit syx = ab - n X. e y +2n X: 3nty se tbdem
seu sydx nb - X: 2yy _ + 2nXX: 3y. jam si X = 0 fit etiam y -0 ex ag. 1. et Sydx = 0.
Itaque eo casu ex aeq. 19 fil 0 =  nb - 6 00: 2 _ + 2n0: 3 seu 0 = nb. Ergo b = adeoque connesis,
et filt denique sydx = 2nXX: 3y - X. 2yy Ut supra. Huic ergo calculo fide potest
vider non esse necesse po C Ctolli Epli
Generalior fieri potest inquisitio. sit, 22+CV  Inxy posito c esse numerum quentum
sit  xo: n fiei  nX + C Vx = n v. seu x = nEr: n’+ CV fiei y = nV., ut cv
et et dx =  (Cp + n3, nav , =no3Cvdv (2 P + h =) ndo - 2 Vdo,: (2 ((V + R. Ergo
sit Cv+ n= co  fit v = co:c - n:C
 et 3Cv dv = deo fit v du  deo: 3C et ex org. 7 fit n deo: 3c - 2C n. . C - n C devis
yIx Z n vao - 2en v d, : (3)  CV + n3
= ydx = (n deo: 3Cco = 2n: 3cct +2n: 3cc =) ndeo: cc , - 2n: 3cc ade oq fiet
jam co = nV: X et v= hy: v. Ergo  co= nt y: XX
adeoque syax = e, 2nX: 3Cy = X. 27y Hinc si c- 2 fei sydx = nN: 3y - x: 4y
(12
Cydx=
-n3: 2C c02 +2n5: 3CC
Aluud problema iihi ab Hugeuio in eadem Epistota propositum
.cuius
invenire lineam cartesio a Beaunio sropositam Ept. 79. voluminis 3 artesii
subtangens sit yy -xy: a, aut dx: du = + posito t: y = dx: du ~ y - xB: a. fiet
dy = 1DdqZe F u dv et Fydy = _ 10 2dZ f u v dB F 10. 1 Zdv. + 10. 1. vd z 410. na d Z +. 1. 2av 117)
adx = ydy = x dy sit y = 10~ + 11 V + 1a
- xdy = - 10. 20Z 4Z - 11. 21 viv - 11. 202 .1v - 10. 21 vtZ 11o. 22 a d z - 11. 2a dv
c
= 20 adz -21 a dv
alatu fiat 10. 11 - 11. 20 = 10. 11 = 10. 21 et summabilis crit aquatio 7. Ex fit 11. 20 = 10. 2)
=adx=
C
non procedit, quic fieret X = 21y - 1a + 22a quod absurdum. jam enim haberetur lineu
ub si fiat H = 1 et 20 = 1. 2t. fieret y = 10Z + v 4 12
ante applicaoem ad problema. Hinc nec licet  tollere Zdo et naqu nam et sic earum coetticem
ferent aequales inte, se, aclae solicet ipsi a. Tollemus ergi si licet terminos in quibus sunt es
sne y vel V. et fiet 6. 12 - 10. 22 - 20=0 et 11. 12 - 1122 -21 20
Erg
20 = 10. 12 - 10. 22, et jam 22  12 - 21: 1  et fiet 20 = to. 12, 10. 1112 + 10. 21, : 11
e Et et ex 10 et 11, fet 202 (10. 12 - 10. 12) + 10. 21: 1I et fiet y = 10Z + IIV+ 12a et f- 1B9. 27 + 21B + 2. 1
sit = 1 et  =- 1 et 13 = 0 et fiet sed vi sed video rarsus rem non procedere nam fit
sit X = _ 10 + u y + 121 + 13 13 + 14 1t fiet ad . = a1  + 2. 1298 + 3. 127 4 4. 140 el sit 10-8 et l -0
a
-
1 - 12 - 13 et
fit X = 29 22 217: 11 = 1 2a + 12-
yd=
14 = 1: 23. 4 eu
oidutu it ie te, st sti  ee su desteti
4d
et fiet X= _  y +  21 + 2740
Edgat

## Folio 133v — 00068197:0266
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e
si AE sit numerus. BF est logarithmus
AB a
BE -7
 EF seu 9 - aa = sl C a - x + X - H exc.
a+)
et B Fiasco (yi  -  X + X - X exc. = _ec y
(ag. ang = 4y seu 1Y. ( = a+X: a sit jam
a +  = 0 + , y + n 5 + 137 , 14 y) etc
a dx = 1a +2. 12, 9 + 3. B3y + 4. qu et
fit 1 = , 1= 1: 2a
13 = 1:, 2. 34 1421:, 23. 7a
Ergo si sit aequatio s X = y + Z + Y, +  etc. erit
2a 23a 234a
1+X numerus, et y logarithmus. Sed si detur aequatio
hat a 2ita et.
fiet X = y+Z, ft seu
a + X = a + y +Z erit a + y +Z lonavithm numerus posito
y esse logarithmum. Hinc conclusio habetur tulis:
numer
quantilas Y + Y etc aequatur rogariEmi ipsius y ex cessui
super super a+y seu y + g + y ex = Num. de y - a - y
ra 23a 234a
 seu est differentia inter Nantium et ipsius siqibhmu
umerum quantitate a minulum, et ipsius loai numer
logan huum