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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folio 132v, page id 00068197:0264
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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