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# Title: Leibniz-Handschriften zur Mathematik LH 35, 10, 7
# Folio 3r, page id 00068160:0005
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A _ 5: v - B s L  - 42
Trax - a XX = qu
2F syqq
1
Elliphin
Parallelogramma omnia circa eandem seltiotam conicam conscripta
se sunt inter se aequalio
placet
sit curu A Est Gei Elli 
demonstratum est, sed patet idem investigare calculo  sit centrum C, vertex
set primarius A, oppositus V. a set rectagla ABV, esse proportionalia quadratis BG.
ergo ABY ad AB sit X, BGy.  a essentia latus rectum a, transversum
seu bi _G b q patot esse 2aX + a x - ys et adx 4 axdx = ydy.
seu dy: Ax:: a + ax,: y::  A: BGse  Ergp BP = a + 2 X et
CF sitCA jam CA = 4 fit pC = q 4d jacd PF: PC:: PB: P
 pG= 2ax fa xx + a + 2a x + a X. et pe qu sit qu a = e. fiet.
a.
Ex et pC Ceaq . q et PC= C)
+5ax +3
et p a 4 2aqX + a: qu. X. Ergo +5 Nmet B a Ergo PF = a + x. e + ax: +a +sax+
et p + GF = p PG+ PF  EA Fit autem parllelomamni Area ductr
ijus t quartae partis parallelogramum circumscripti integri, seu parallalasiammi GeK
irea exductu ipsius GF in CK. Hanc ergo CK quaeramus. Et primum quidem CM et MX
jam ob triangula similia  GBP et CBK CMK sit CM: MK:: GBGBseu
sit AM m, et NX, Mk, _ K fiet ram + a M = K: et q M: K:: C
ergo k q gm + n O = ram + a m, et m- Et 9 - m = OK. etm-
et m = 4 - BK. Ergo fiet 2a9 - +OK + 2 q - 29CK + O2X = X hoc
est ft: quia + = - firt +4aq +2O2K2 = k. seu K = Aaq: 1 ∞
∞ jam CM = q - m Ergo = OK. Et firt CK = O K + K = K (1+ O at H
GB continuata ipsi Koc
Ergo CH = ageq Y aq. 1+ C: 1 - 0
fint GA fiet BM BA. BC:: O  G GB continuata ipsi Ck occurxat in N
siet Gs BA. BC: O Be CB: BA.: O: 1. seu BA - CB: O
GB: BP:: O: 1. Ergo GB + BY e seu PG: BD:: O + 1,: 1. seu pG = BP (0+1
e e PC: PF:: PG : PB  Erg:O2+ 1: 1. Ergo pF = PC. 10+1
 BC scribatur Y = a + a x.  erit O = y: v. et PC = _ v - BP.
it E   GF = pG+ PF = BP (O +1 + PC: 1B2+,
seu pG + PF = BP. O2+1 + PC F seu = BP, O + BC: O = GF
 jam Bp = a + a X = v et BC = 9 - X. et O2 = Y: v
Et BPO = 5 + y: v = rax F a x7: a + ax et DPO BC = (2a2 (+ a X) + ag ( av f a x ( 9X). 
postat ut inveniamus
D F 1-O = rax + a + rax + X - 2aX + qX,: v
Parallelogramma ommnia  circa  eandem Ellipsin descripta esse inter se aequalia
ex scriptoribus constat, placet autem veritatem ejus theorematis investigare, calculo.
sit Ellipseos AGEVK vertex A, e oppositus V. centruum C. latus rectum a, transversam
seu AC, 4 acissac ABC Voidiuata BC . diametri conjugatae GG, Hk, ex puncto G et ex
puncto K in axem AV ducantur ordinatae A, GB, XM, eut AG, X. BG,y. es AM m, MK, K
a)2
+ quin GY perpen-
Ex puncto G sit perpendicularis GF in coningatam diametrum ek, secans AC in RP
ptt Y. patet parallelogram mum AC X, fieri ex ductu GF in CK. investigemus ergo utramque
quia  BR edX. GB seu y: BP:: dr: dy seu C: 1 et aequatio ellipleos est rax  a X = 9,
dicularis ad tan
ventem in G
erit ejus differentialis adx - a xax = s y. seu du dy.: y:  a- ax: q ergo fint BP = a gr. q.  
tan u sit Aet = m et MK = X. firrq. m: K :: (3: 1 Kusu Ta Shra et ram - amg = KK
A, + CM, qum
Ex aeq. S est m= q. Op  quo substituto in aeg. 5 fit: 2KO = aq-  d. 9 R ∞( qost  HK
(2aq( 2aOB - a: 9 RD ( 29OV +8K = K seu  aH+ a K = a9
seu K = ag: Saq + aa 6 = CK. Rursus ut inveniamus GF, patet esse
O2+1: 1.
CM +H = 1
12 1
est GF =GP + PF, ex GR. BP jam quia GB: BP:: O: 1 fine GB + BP,: BY
seu GP Bp O1. Bursus DC = sit BP, v. fiet GP = V (O2 1-BC sit C
euit pC = c - v. rursqus, pPCA: PF:: GB. BPs: VOH: 1 fiet pF - C. v: 50+1 = Cv. Syrd.  = 1. 2
Ergo GF =  v.O +1 + c= B: YO +1. seu v O (+∞) + cCY,: (O+I et explicando
erit GF = _ y + vC. Ly + v et CK ~ aq. Saqet aa
seu GF = GP + PF = sy + v + c v v: ss +0 = y+ V) + Cv Cv; sy  v. - G1
at K ela seu CK - aqu: SaquV + aa Est autem y + = 2aX = aX: 9 + a -2a x. 9 + a . 99
seu a + 2ac = aX. g:  1- a. 9
LH XXXV X 7
BL. 3
X= 1: V2
im 21 = 1.V
e