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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 401r, page id 00068196:0797
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sectionis lonicae
Si g sit tatus transversum et Y latus rectum C et Z sit
abscissa ex axe inde a centro, et y ordinata ad axem, et hn = ra
=qu. 4 + 4. r yy vel yy- + h5F ih 77
492
et firt dx + du = dy sut atrgy: Siz +2y)  + 414,  trg,. squati
Ute
conicae
H
E.
Ut igitur Elementum curvae revocemus ad
quantitatem rationorem, tentemus
generalitur aequare quodrato  alicquamlibet ex his duabus
formulis: b + cZZ et b + ez7 aequare quadrato  per
calculum indefinitum, ita ut indefinita non ingrediatur vinclum.
Non possunt b et C aut b et e simul esse negativae, nam tota
formula esset negativa unde radix foret impossibilis. Ergo si
b est negativn necessario c et e sunt affirmativa; Etsi c est
negativa, b est affirmativai et si  e est negativa, b est affirmativa
Si utamur TCS y erit b semper affirmativa. Si T6Z, erit b
affirmativa in Ellipsi, negativa in Hyperbola. Itaque in Ellipsi
b semper est affirmatisa, sive Ted y sivo Tu Zutamur. Videamus
aftirmativam,
sC
inc ina L  utamu sive pro Z sive proy.
1b
 2  yV +ZZVV seu pro , ponendo n et divedendo
B
siet CZ V Et sit B + cZZ= b + 2 Sb ZV +ZZVY, seu pro 2 Vb ponendo n et
et sitb+ CZ
dividendo per Z, fiet CZ = nv + ZvV. seu Z= nv: c- vV atque isa formula
b+ c2Z aequata est quadrato  quod sufficeret si sola staret, sed
formmua b+ ezZ etiam aequanda est quadrato, et eodem modo procedendo
fiet Z = ncd: e= coco. Et aequationes 3 et 4 aequandae sunt inter se
(67
quod ut fiat facilius, ponatus co XV fiet 1. C vv xic xxrv. seu
C C XKov LEXC, XDD seu VD= XC- e; X- XX
C11Z
10)
vel ponamus XC -e = l et X = p , et X- 1 = (t fiet
cra
5. 6
vv = 1. pp son  v = 1. (+) p8 jam per 10 et, et. fit =0
p251 (t et per 9 et 10 fit pSC C t fint rursus vD= M XX
(14)1
et aequando B et 9 fiet: XC - C: X  = mmiX sen CXX- ex = nimX-um
quae aequatio simplicior est prioribus, ascendit enim incogni indetermina ter
qantilas mnex ad solidum, quae prius ad quadrato quadratum. Erqo
Mm  CXX - CX. X~1 sed video nos nihil lumatos. Redeamus ad
aequatines 3 et 4 fiet enV - nVCoCd = cn Cd - nCOVV. explicentur
Y et Cd per  m et p. it destruatur  p3 et mo et p
sit V=  m+daK edF 2M +2P+4 et c = 5m+ 6p +7
avcoco
am 23. b74 2. 6 +3. 3.6R np +64p el
i
3. 6. bise 401
lego rns at als sete ao  a iandti eis a itum aseo vita i ba