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# Title: Leibniz-Handschriften zur Mathematik LH 35, 8, 30
# Folio 103v, page id 00068124:0206
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yy + ab
si possia tatid facere ratonalem habebo dimensionem curvae Hyperbolae. Nam si y-aaix
tiet Paxdx 1 dus ay = lat x dx. xx Habeo et meisionem ipsius TarX x seu dimensionem
curvae parabolae cubicae. Nam si y = x. zua fint dy = xX p: na et fiet  Suidus+ avax- var xiea
utraque enim summa iniri pt rationaliter, quod patet. Nam si sit S+X= y seu si fit
X. = aagy - ab seu x = Ilaagus - a b) tique pat unde si X habetur intionaliter datur summati
Mursus si sit Oaa b +x _: XF - y. seu yy X = aeb  aa X seu X N = a b:  yy-aa
qp
+rmm hpbens
ita rursus lantum opus est redd Dguaa fieri rationalem.
mo auplius videndum an hon hoc reperto omnis curva Conicasi
summam rationalem, Nempe sit  sitaq = Z2+ aC+ aecio et A aagus  adebet fieri
aequalis ipsi Z et fiel (Z + rac ZZ + 2 a 77. V + aa ee + 2ae: v + acc. vo  a2 (8
2aC 2 VV = 2ac82 + aa cevv  ab + 2ac0Dt at  seu-
+a sit ee ab et fiet 37 - 24 aceV, a cc 
tanec vo + 24e cev + acc
rahonalifatione ipsius  Saagus a b. Hinc etiam datur (gaZ. ergo datur Z2dq + aedque + (a c. V) dz
sit at.  co ergo dater codque
Eagos datur et Zduo
Z7 = 2ae + CCCOC:, 2aC + 2CC0 Ergo datur (d(raceCor cccoCO,, act 0)7
Nempe sit ay = 23 + av + ae firt argy - a b = (Z) +2)ZV + 2 aCZZ + ecvv + racev + an ée - ab = (2
22=ca0v +: 20+ 2ae
quia autem datur szclus ex posita
rationalisatione ipsius  aagus- aeb ergo dabitur et sydz. Ergo et S22dZ + Sevaque + Saedz. Ergo
videamus jam an omnis curva condea ad formam
mXX+ ad: nXX + a ,odx summari debet sit X7 = 50 et 2∞ dx = F du
Ergo et Sz d
catuur et sodZue
seu ax = fdb: 2YV, fiet  summand. ergo Sin f∞ +a;, nsv + a,  fdv: 21f0. se
ipsius  reduci possit
Jinfo +a 5fov + afV n fav: 2 si pro s0 scubatur aas.v pro do fiet: - acc: vV
et habebimus s infaa: v +a,: ns at v + af. B faa dv: vo. seu fiet
Generalius sit XX = asv Yaag,: v fiet 2Xdx = -agdv:p
seu dx = - gdv: v0): 2 NB:, 5ur a) ex et ex Y (mXX+ a) : (nXX + 2)) ax fit:
mo = max + in A3 + a  
anfv ratrusa matix
sit K
et ams
amabus
N = m in a XX + a = aagu dat _ a0 p  m  XX + n r a ausus a - m
seu XX +. 2 men a = s-
aneti
na hun
C+x B+ D.  = CC+ Vp + 2B)
s Bβ = .. . .. VVEr() -. nen
β= .
est autem is rationalis ex Hypolhesi = 8 tinc VV + 29γ - da V~ F 3+ 2 - du. 7
Ergo fit pV+2 Y, V = l
et et
yrb= mx gob = ep fiet sgs bb -mnpi aa et m pato = npaab seu p idela;
et fint yy- b0 =
mm - nn
agerentis istetqu
 mup- amno, hujus quaeditur sit amnp + axq quadratus
um np
nonel sala
Ergopo
ammpranmu ratun  aegum - aequn = oto sit am. n = W fint 2a  + aquoro- aq aus
a a o-aa eali mx
 7 +b,n P. ad ytc rp: aa Ygeb, stC= map: aa g = ump = gab, aa = n
sit p aub a 
un uocabo as pu heosbit Cmmanu). veit
yrbb minim r  i  
E ac v0 +2c
seu datur quiaantem datur  sytus - (d. gu. lb summat erd rationali  ex posita
2evn+ racv
 ah: co, fint 2Z- Ena race. co 4 adcl:, rea. co 1 cico, seu fic t
adeoue hutur et (a C: D) Azue
et. 23- et
dofall sinsvt avv: n + asv
0d
 ioast Z 6
sdaa. Erge
Vhrtae
eltis