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# Title: Leibniz-Handschriften zur Mathematik LH 35, 2, 1
# Folio 31v, page id 00067993:0062
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propositions expositam, lri simplicitate elegantia et fructu vix alteri
cessuram, reperi. Methodus autem qua ad eam perveni ex asse mea est, 
eum levemalin cue cinvecrttem cesti pepbilit. ec nec praeter notam
omnibus quadraturam paraboloidum  supponit. e
mie eustite et si einuenta diffichnthate vianl vet usa astimandua vant, eorateanti
non veretr. Hinc jam similem regulam pro bs  regressu 
Trigonometrico, seu inventione latorum ex augulis datis, demonstrare
difficile non fuit; quam nunc exponam.
PROPosiriO XL VII.
 Si arcus sit a, radius r. sinus xeetus  ; erit
Quaequal. a a + a
er  eS enr ec.
i sit linea curva A. B, C)C, cujus abscissae  AB AB et c.
appellentur a. et ordinatae  ut BC appellentur . V sitq  parameter
seu recta constans DR, quam vocabimus sitque sequatio curvae
naturam explicans NVaequ.
c. qualem dixi, ajo curvam esse
Lineam sinuum e qualem explicabo, id est si recta AB aequetur arcui R
quadrimat Et, centisradis BR descripti, tunc recta ibC. aequabiturassuni rato, ejus. arcus ,  nempe rectae R.G vele 1
aequatur radio DR
respondenti HHoc ita demonstrabimus.
(1) sit autu ei p A3 B,  a, erit utique 2B, C a a
una scilicet infinitesima parte atus
et
yr etc7. sa
Inter ordinatam 3BC, et aliam minorem, B.C, sit differentia 2L, et inter
sum inter ordinatas 3B3C, 2B2C, 1B1C, dic. differentiae designentur:
3L3C, n2L2C, etc. idque ponatur usque ad A continuatum esse, easque
differentia
ipsa scilicet 1B2B, vel 2B3B semper constante
a
a C
 quoniam omnium
Euismodi diffecentiarum ut L3C, + 2L 2C + _C. dat rltimam ordinatarum 3B3C
et summa omniumsite BC auscaatibus,  diffecentris ipparene ,)
t reddit 
use rente etimlulimum ordinatan esst
sula tem dum   (C ductari, see sunt crcclare elorine
, ec ei se
summa autem omnium  seu Ac
esc. dat ad ab
cujus radius A3B, e sit a
opa  cadioediitum lier ig C
et seu summa omnium a
infecatae affeetu eu
quem supra in venimus valorem ultimae diffe sie diater si fe e refiet.. Ergo area figura, ditisae penee
seu spais a 3B3C A divisa per r quadration ab a DR, de sutracta
subtracta ab A,B B sine =  ud idem  relinquet 3L3C.
ac AB eA=
coilgenda hoer
sive generaliter loquendo: a ABC q. LC
Nunc id ostendam fieri in Linea singum qualem descripsi; quoniquo
EIF, EIH, EyF arcus (rectis AIB, AB, A3B, aequales sunt, erunt F, F
2R,F, ipsis 1B, B, 2638 aequales, adeoque constanti quartitati 2., socilicet
uni infinitesima arcus circuli, loco arius F F+F, ad hibeatur polysgo
Nunc id ostendam fieriem linea sinuum qualem descripsi. Arcu circuli
AFF qsas e in infinitas partes aequales diviso, vel ejus loco
 et in  per pendicalures MG
H aequale rectangulu, M. M
adhibito polygono ce regulari infinitorum laterum circumscripto, cujus latera
sint , A, _M M M, nMM ec. quorum puncta media
 me N, et ex punctis M demittantur in DE
1N, 2N. ducantur radii
perpendiculares MH, in quibus sumantur M+ P, LM2P, dtc. aquales ipsis
IG2G, 2G3G.; et ex punctis N, in DR, demissis  per pendicularibus N
patet Triangula, AM. P2M et 1N, 8 D esse similia sob angulum DNM rectum)
D. E. Ete adeoque esse ARP rectanu. 21 P in D, N ( t  1S1H2 1 quia H e qual
Rabin  aequal
o ar 
quae si prius divisa per addtur
2, sive A3BE B arcat A. B in unitate
3B2B ducte peradiu divisen, redibit
art er c
Spries lerita lpet Y, agenitur Postens