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# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folio 123r, page id 00068195:0245
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curvie semiparabolicae
Opera pretium videfur, quae de intervallis Tangentium parabolae a
puncto in ejus applica axia altitudinis applicataeque aequalium a
communi, ratiocinari coepimus, absolvere, nam si ad progressionem
queundam simmiabilem reverari posset eorum series,  curvae
parabolicae rectam aequalem geometrice descriptam, ac proinde
perbu
Geometricam, non ut hactenus mechanicam tantum parabolae
quadraturam haberemus. Esto altitudo parabolae AB. aequalis applicatae
BC = a, DB = 2a ejus B. 4a. DBC = a. summa quadratorum
Sa Et RqSa = DC. Triangulum DBC. = 2a a2 = a. dividatu
per k5a, et productum duplcetur fiet 1a, RqSa 2=
Ra -Ta2
sa
4a 2a = BE. Intervalli Tangulis De.
A75. A95.
Assumatuo nuncalia altitudo minor quam
a puncto B.
nempe AF. cdjus ratio ad AB sit quaecunque (X, erit AF = (a). et Cumque rat
 quadrata applicatarum sint ut altitudines, et applicata prior BC.
fuerit a. erit ejus quadratum a multiplicetur per  fiet a
c et R4 = = (a erit applicata FG. Ejgs F-  addatur Y. kF: 4C
H ~
a eqe
fet a 4a- aa + 4a a= a a + 4a cum bqus cit. (19 a a +  sG
Triangulum DHG erit 24 a, 2. =  a) divit atm per HG
2
a
Rga
pudidumqu duplicetmr fiet: a seput
Bq + 4a
RgaX
(e90)
gae
e
_agat Rg. da +
Dividatur per HG productumque duplicetur, fiet: Aga
2a
eq
a
= FI Cumque sit ut HR ad AB. ita FI ad BK
Rga
ita stabit regula proportunum HF- 2a dat HB = (a + 4) quid datit FD
A9 a f4a
a ea 1a
L
pq 
R92
2a
Rga +4C =
Rq a + 4
(aa + a ) BK
Rja +44
Ergo si ratio altiludinum sit  1. Ratio intero allommeit.
24 aa+a. - 2 Rq. a + + a
nas aqu +a Rsat R45
si jam ponatur altitudo continue decrescere uniformiter, erit a= infinito e
reliqua infinit - 1X infinit - 2X. infin. 3X. etc et si partes sint
funtae erit esto potent esse a = 10 X. reliqua _9X. 8X. 7X etc.
ergo rationes ea obunt:  3o Z =  crgo 1o - a
Erit ergo a. modo 
1  10 etc. Nole sumum labissener ita inititur
1+ = + 1Z +1+ 2, +1+ Z etc
10d+ a
49 4 + 4
0a + a
A4  +
de
2