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# Title: Leibniz-Handschriften zur Mathematik LH 35, 2, 1
# Folio 257v, page id 00067993:0516
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Poneem orcum quad aut
esse AV, et 63, oel A (∞) et (D) P cer ob circuli unfformire
AO, vel A(O)
 i semiare, inter de aegules
 e sive ex Extremo
vel (B (D).
Extremo A perpendicularem cagas in tangentem BD, si opus est productam, sive in an
contra ex extremo B vel (B) perpendicularem B P vel(B) B in tangentem A(D) si
opus est productam, at B P vel (B) (() aequatur ipsi AE, vel AE) sinni verso
ergo AO vel A(D) eidem aequabitur Sed et illud univer saliter patet Argulum (D) AD
esse angulo (E)B) aequalem. cum e sit M(B) paralleta AD) et () (B) parallela AD). Unde
AO Triangula AO(D), et (B) M similia, ac proinde denique AO) vel ()) ad AE) ad (E)B) ut
AD) ad ()M ad eC Quod erat ostendendum.
e
Theorema IIL
Si figurae segmentorum portioni AM X A, a Rcta MX semponiparalteta absrissae
venat Asymptoto Panallelam indefinite productam
velut eferi intelligatua per ordinatas Diametro Circuli seremmteris altitudinem figurae segmentorum
constitu referri intelligatur
pendiculare
Rectam,
Si curva figurae segmentorum, ad AY per verticem ejus A transeuntem, Asymptoto
TS. parallelam, indes per ordinatas perpendiculares referri intelligatur, sumta abscissa
velut ad Axem
ex ex Axe AY, quam vocalims V, et ordinata  V X, appellata X, erit
33 Radaque a; tenio ulicunque punctam X in cuna, vel pantum y) in axe sumatur,  erit unversaliter
 vera aequatio hacc
aty
X
CTaugentem semi arcus esse
consta Demonstratum est Theom. 2. esse AEvel AV ad
Radium ut sinus rectus ad sinum versum rectum MB. ego posito ut ante radio
Hoc ita demonstratur A Mvel YX
sina verso XX = X, sinu recto ex cognita circuli proprietate: aX X
angente semtare
eitacTangente semiarcu sdenique posita rurcus = y fiet
vel vsrax-X = Xa, Unde. 22aX - YX  X a sive
Exquae
a
2ya - YX = Xa vel 27a = = x a + xy e fit denigue. 2γa
Samr. X
a +y2
sx intem
Problem a Geometiiae Mechaniee desi Practica
 lii M aX imace habendam
irculum aut portionem ejus quamlibet quadrare, sine
Tabulis ac sine divisionibus Radicumve extractionibuus,
paucis, tantum multiplicationibus et additionibus,
Theorema
Si segmento circuli proposito, Radius, ut ante appelletur a, et Tangens Semiare,
quam localims se X,
Segmenti propositi, seu omnium appelletur b, tunc series haec:
ba etc, in infinitum continuata
sub
semi
aequabitur Differentia inter Triangulum rectangulum sinu verso totius
+30
arcus segmenti, et fangente semiarcu s; segmentumque a semiree lan gulo subtiatendum. Cicli retatione
quamlibet.
Huius ut appareat demonstratio, ad siga resumendum est, ordinatam  XY
qualis est e. g. QP, aequari aequari 2ay
modo y intelligatur esse AX, C.9 2).
Ergo a = at tam ex iis quai apud alios  inprimis doctissimum Walisium
a+ y
de Arithmetica infinitorum, et apud e ingeniosissimum Virum Nic. Mercatorem in
Logarithmotechria deripta extant constat esse  - = -  + v - v sc = 2
ac proinde X erit = -  + - . in infinitum productae. (que idemosit et
rici, 
de qualibet X  separatius quaecunque etiam AY y assumntur ido dit
g. APZA
Porro et quoniam summa omnium X, aequatur toti figurae cm
sega
. 2R QA sfignaae
fig. 1
3087