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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 54r, page id 00068196:0107
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in curocilliptica, habemus perpendicalarem aream momentm tam exaxe conpagato
primarum AC, quam ex axc conjugato tCB, ita positione datur recta per C et G
posito C esse centrum grav curvae, sed patet ham nunquam posse langere curram
centiorum
Esto Hyperbolae centrum C, versex V, axis CVL, ordinata, normalis LM
 vn
Assumatur diam eter ali CN P per V dacatur parlela ipsi LM, diametro
occurrens in N et cum dala sit Cr assumtu re determinabit diametrum.
jam producaturL M dum diametro occurrat in P, et ex M
agatur My normalis ad CN. ob triangula CVN, et CL
similia sit LP: et = YN. CV, et LP = et. VN. CV, rugus
obtritnguls cD et map similia, fit Mp: G MG = mp. CV. CN
mo-+ (Am La MQ  ALF(Y) LM, CV: CN = et ( FCL. vN)Im. 1 C
a et P = ON C+ CL. CN: CY et yP
et QP = Mp. C: VN et fit t jam CQ = X p - ZB, fiet CZ = EL VN. CN - Mp. CN. PV
CV. VN
crar po
LV. ter e
tet Cq = CLx a. CN.  Vi AB) YN. C1. ex () LM. CY, CH,: ev
similia nt triangula CYN, et B, MQP., quaeruntur CZ et qM ex CL et M
Sit Hyperbola V. L, vertice Y, centroC, axis rectus CHitrnus versus CM per V, trans-
versus CN, pu coordinatae ad hosaxes LM, LN Ducantur alia binae diametri
quaeritur
inter se normales Cp et CQ occurrentes tangenti verticis V, in tGG et F
LP 1 LQ, ex LM, LN F ipsa cor et ad has coordiratae hormales LP, LZ
quaeruntur hae ex LM LN. priores coodinatae secent novos axes, nempe LM
ipsam c in Z, et CF in si et LN ipsam  in T, B CF, et CF in W
iem d  tot= HF jam loq -Wot CV. C AL CM et et N CH Eit simolite
Wg.g L = CV. Y Wg = ZL. CV: VF. aD = ML. CF: FV
 TB PL FY 2HC RA RP = PL. CV: VF CR = NL. CF: FV
jam  CW + Wq = PL = 3L. CY + ML CF et CR - PR = NL. cF  = PL. CY 1  2L
ML et NL, nam foet
ope harum duarum aequationum
PL = ML + AL C PL. C
M. Cr. (V + NL. CKFL, : (t V eY
et 2FV haberi potest PL et QL, pe
Ct. seu et=M
 Ht. Fs EV
(FV FVFCV. CV =) FC. FC, : FLLFV
et _g - at PL = ML. CF + HL. CF. CV = PL. CV. CY. crgo B = ML CF. FV+ N1. CF Cv  v 
seu  PL = ML FV, AI CY B B = 17 + Nu N ( Ego (FV. FY + CV. CH) PL M. cerN  
CE RREYEY
FV. FV
seu E. CF (CD = ML (CF EV + NL (CD CV. seu PL = 
Et 2L = NL. 1 = 2t 2  ML 
D + M2. Cx - np a ve
Ergo (EV. FV + CY CY) GL = NE (CD) FV- M L (D CY
et Fit 9L = NL. F  = ML . CV = Lβ - Nd
perbordinatas LM, LN
sit punctum L, relatum ad Condirectrices normales CM, CN,
et rursus relatum ad Condalias condirectrices normales eque pri eodem
vesdinatas
NF0
C p,C9
Generaliter
is conddirerium  per N agatur parallela ipsi CQ. ipsis CP, LZ, productis si opuis, occurrenso
quo priones puncto concurren
in punctis R et S. ita habebimus Lp et LQ per LM et LN.. Nam est
it data in CM datar
= SNERA NNF NS+NR et Lq = LS = CR. Cumque trianus ila L5N, NRC sut somilia
Tlangulo FVC
id est LP = LN. EY + LM. EV et Lg = H LN. EY - CM C
Si XL et XM. rad
normalem
porro ut CN est extra, con intra  alterns condiretrices; ita Cp est extra, Cq intra. Et ut duximus VF uni directri-
GreotasL
peminia
Cium lateriorum CM, oecurrentem alteri cq; ita possumus diaere CG normalem uni ipsicq occurrentem ipsi CM,
et LM = Lp. FY - LG ci
ut per i dox imus paratelam ipsi _B, interceptam inter ep et tq ictaper  ducamus patalletam ipsi C M, inde
et fiet
LN = LP. CV + Lg FV
ita per P ducamus  parallelam ipsi cm, interceptam nter NN et LM
et LM = L -
fietque LN = WP. _ + CP
Sit jam L punctum curve Conicae cujuscentrum axis directus seu per verticem, M, conjugatus CN, si esset VM, X. olats rectum ,
transversum 9, foret, gy = 2rX + YxX = 4, XX + 2gx +99 - 4q = F 7 -99 = FqZ - rq = gy, et fiet
ydu = XZdque. et dydy = Zd7 3, XZ2- 1 =  v d 7dZ ZZ-4qB et a bementum Curvae conicae seu
+Caxdx + dydy) erit L (a ZaZ 22 49 + 722: , ZZ 49) seu dz  (6+2) 23-99: 72 94), huius
ut ex alio ffim
eoctum Eementum napu
nilio
Ducatur  faxdx + dydys) in L
teducitur ad quadraturas ratiorales
seu in L-VM, sed utrsue modo res rede
Qm +yH. Aliud et P
si quae ramus momentum curva conicae ex axe ep, seu elementum ejus
ta po curva Hperbolica, sed certo tantum nigarcu
itatis
et caut iatra cepujus
Eagretitunum aete s au posia i ut aieo sepius
enic