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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 3 (shelfmark LH 13, 3)
# Folio 144v, page id 00068200:0290
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Nunc ad Ellipsin transeamus. Generaliter Elementum curvae tam Circuli quam
Ellipseos est
sive Elementum Ellipseos erit:
rax + a X A + 2a) y
Adeoque eui
+2ay39 y
sive pro X pionendofet. y
+ Y 79
Ean f aX E tunc pro raX f+ ax, fier.  2ay + 2a4, + 1a5
auc fier 04aq + ay
+eazy + ag2
pro xponatur V + g sumendo 3. et d. pro signis fiet
+ 2a59 + afy
+292
delet antem esse ipi) Y + )a osses 170.
erit ergo dN + 1.
adeoque XN (7 Y + 9, fietue rax + a X
+a9e fay
quod si jam qBa et +1p) = 1 ut in circulo, erit
Iu
a 1I
S, +Y
ay
Multisticetur L1 + V elementum curvae circuli pers
y
quod fiet ad aliam curram Ellis ordinala quae fiet erit ad aliam
onal dm Elementis curvae Circuli.
mo F 9. fiet: Y2 + d
quarum ordinatarum habentur summae.
ac
b25
Sed haec ut demonstrentur sit theorema:
2
au
sive
sit,  BC, EBHC
si spatium comprehensum rectae BD Curva C() (C)
axe BD duabus ordinatis BC, DG et curva qualibet
C(C) G et axem si opus est productum
alicubi secet ala in A alia recta AF
producanturque Pordiriatae CB, (C) (B) etc dum
ipsi AH occurrant in punctis
E, (E) etc. H
et ex punctis E, (E), Hi redlae EF (E) (F) etc. HL
ipsis BC, (B) (C), aequales, ad rectam AY.
ordi velut axem orditientur, et curva F(F) L per earum
extrema ducatur, erit spatium ipsa EH velut are,  duabus ordinatis extremuss
EF, HL, et curva E F (F) L comprehensum; ad spatium prius CBDG (C) C
ut est recta AE ad rectam AB. Quod ita demonstro
sive quod idem est, si iisdem positis duarum figurarum ordinatis,
abscissae suit proportionales, erunt figurae in ratione abscissarum.
compleatur rectangula AEHLM et ADGD Et complementa duorum
spatiorum quadrilineorum, erunt quinquelinersque q quinque rectis et una
curva comprehensa _ FEA NLG _ CB A P GC ipsorum
quae erunt etiam in ratione abscistarum
spatiorum complemeta tad rectangale.
I EH, BD, sive E (E), B(B). Quod sijam ostendam spatiorum ipsorum complementu
rectogula circum scripta, esse rectungalis sine his duobus rectanigulis, sive quod idem
inter se repoindentibus proportionalia, sive esse FMLF ad CPGC, ut EL a
abscissus
bG
sive ut recta EH ad BD. sequitur ipsa spatia eodem modo se habere. quod dixi de complementi
sic ostendo: ex punctis ( dem (C). demittantur in CP. ordinatae (C) R) CCDRN et ex
punctis (F) ((FBen FM, ordimatae (F) (N) ((FDerD erc. Pater odasissi E(N). EN
id et BCR).  et E(N) (ad est (B) (C) et (E) (F):) semper existentibus aequalibus
ordinatas , (C) (R) esse ut  (F) (N) esse in ratione rectangulorum abscissarum
(quispe illis aequalium) B (B) ad E (E) id est ut BD ad EH. sive ut rectangulum BG ad EL.
Ai spatia quorum abscissae eaedem ordinatae autem proportionales, sunt in ratione ordina–
tarum ut constat.. Ergo complementa CPGC, FMLF, adeoque et spatia ipsa
CB, DGC, FEHLF rectangulis  BG, EL, id est surs abscissis B(B), E (2) proportionalia
sunt.
Hinc sequitur Corollariumi si figurarum duarum pariter et abscissae
abscissis, et ordinatae ordmatis sint proportionales erunt figurae in ratione composita
ordinatarum et abscissarum. ut si in ipsis B C (B) (C) et o sumantur puncta
ad spatium ZBDSG ad
spatium FEHLF, in composita ratione Bq ad vel D5 ad HL er BD ad EH, id est
erunt duo spatia, ut rectangula circumscripta BDS, EHL.
Pursus pro  pon
Eodem modo
fiet
heorema.