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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 275r, page id 00068196:0545
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Hyperbolae curva mesuram recipet per summationes
Enrum hujus
XX + a: XX+ b. ubi in fallor signa inensi
in nominatore sint negaliva, in numeratore sunt afformativa itacredo
Aogleipsis videanis
sies meretur ut calculum reapse exequamur, quia tandem
viam reperimus sit aequatio pro Ellipsi foet Hyperbolas
fZ - f: C ZZ = yy ubi flatus rectum, c latus transversum, seu axis
directus, cui scilicet applicantur e ordinalim applicatae normales. ubi si c sit quantitas
ponamus
+C-x. fiet (ex aequ. lper
(3)
(4)
Exle  sec Z
negativa, ex  Ellipsi fit hyperbola.
et: ZfC = fx - f: C ZCC - Cx + XX _ (pel 121) = yy = (ex 3) = se - fX - 1 fC r Gp - f. CX)
Haere ut ad lineas
seu destinctis destinendis ex aeq. 4. fit yy =  fc -  f: CXX
applicemus erit AB sint vertices A, V foci FC ordinata ad Focum sit FE
i quaevis alia ordinata ad ellipsiu sit BC, occurrepa BL, occurrens
axi in B curvae in L, ef et REC _f FE erit Zf (ut a me
demonstratum est) AV erit C Medium axeos punctum seu E centrumfigurae C
CB  AB erit Z et CB erit X ubi
notandum crescentibus et BL, erit y. Ubi notandum crescentibus y
et C AC erit LC
14 X =  c. F y- 2cc jam sit d + (sax + du = arcui Ellipseos et _ ex aequ. 5. fit
(5
c. f gix crescere Z, sed decrescere X.
y ay = f:c X dx. Ergo ly = fiC X: y dx. et d dy = f ff. cC XX: ( 3sC. F a
(y
et 8 et 5
ergo fil ex 6 per 9 fit (ax tra  s   x
(11)
10)
cc At i3 X
arcus Ellipseos = (dx Y 1 + f: C XK:  CC - XX = sax +
2)
2CC = XX
jam f: c =1 = f - c, :c  fc - ce: cc jam in Ellipsi est  chitres
est fC = hh, posito polito n esse daptam ipsius lats axem conjugatum
seu duplum ipsius CG. (nam E AC in FE = CG) itaque fiet rursus
  est autem ubique  ee major C seu AV major quam h seu bis CG. ergo
et ec major quam hh seu fc, utique quia c majost quam f, in Ellipsi in Hyperbola
la Bcetarundea E
vero contra. Cumelgol Itaque aeen fiet Arcus Eltipseos 
Arcus Ellipseos GEL = (ax 2s hcc-F xx quia X sen et crosnte id
Unde cum amba
tam numerator, quam nominator habeant  suas omnes radices
c
ZCC = XX
vo
C
Et quia in Ayperbol
Ellaipse elique curop honca res succetit
dimenset in H perbola
no Ellipsi
pro Hyperbola non potest reperiri punctum G. Ergo pro ipsis X substituamue
et.
antas
ipsus  quae scilicet crescunt inde ab A, substituamus ipsas
12
2) l14
inde ab A, ut habeatur arcus AL,,  est autem X = 2C - Z et dx = - dZ
(17) (per 16
fiet (ex 5) feet XX =  cc -  cZ + Z2 et L cC - XX =  ce Ece + 1Z- 22
et (per 16)  f: CXX = 2 fC - fZ + f: CZ2. Ergo LCC - =F Xx ( CC - XX + f. C XX)
(18)
=(per  ( ex 29 per 17 et 18) ce + cZ -22 + + fc - 5 + f: CZ, seu ce
20
i cC - == F xX  C +f e C= fZ 4 Z-  - f: C ZZ
(22)
(43
et fit
SC 1 L C Z - f: C22. posito
sed redcamus ad aequationem  quippe breviorem, et eu ea jam odaramus solutionem
Haplibola, vam Ellipsi arcuem Ata
cc + cZ - -
e fcouentiiorem affirmativn sin Hyp.
tam
nempe ita transformando formulas per x expressaancta vel miuuta X ut nominator
in Ellips
nohinator habeant divisorem communeus sit x- v+ e et est fiet
= 12cC -  ee, trev - c5 vv- Ee p Numeraloge
2cC  C -  eV - VV 
+ cc -  XX 
(24
10
ero Rominator
divilatur Numentor per - F. et nominator per - 1 itaque debent vV =Lev + ec = C. 2Cf, et vDFe
et
CC  = XX
habere divijorem communem, videamus jam quando duae formulae x vv + m V+n et vV F pB +9 debent
habere divisorem communem, quomodop i inveniatur dividatur scilicet una per aliam, residuque debet
esse divisor communis, qui invenietur actu dividendo vDmtn + 1. sit residuns m-pl n-9
hoc iudatur prior diviso, fa L + vV + pV +4) _ v + p- 7
n- P
+ mmq +49 etvisi binibus pe
e l ranus
(empu +3) 4
= vp - 
vo
map
desetne pb.
Ergo fit -n- 90
Inn + pr + 4) 
= vo - pP - 9
 + q= 0 se q.  qe
im- p - a +q seu
seu fit qa 9 m-p = n.
seu qui rampl qus = nem- lxp- nule - 1m p) quD Cqu 19
fici
= dx =
y
.=
lrgg. 2s
Xx = f: C 97 - = sC
+appn. + nn = nip A