, C 50 
Fluxiocurvae facilius interdum repentur per comparatio- 
nem inter Triangula fimilia C E D, C B O, inftitai enim po- 
reft haec proportio, C B, C O : : C E, CD, hoc eft, pro 
. rx 
circulo, \/zrx — xx, r:: x , 
V 2r X — XX, 
Curva Cycloidis eadcm opera cognofci poterit. Sit ALK 
femicyclois cujus circulus genitor ADL, Affumpto in diame- 
tro AL quovis pun6to B, ducatur B j parallela bafi LK^ pe- 
ripherise circuli in pun6lo D occurrens j compleatur rsdan- 
gulura AE j B ducacurque FH rc&x E j parallela^ eidemque 
infinite vicina^B j produ<9:am fecans in G^curvamqae A K in 
H; ponatur AL = </, AB = E j==;c,GH=i; Notum eft 
redam B G effe ubique aggregatum areas AD 6c finus redi 
B D, hinc manifeftum eft fluxioncm J G effe aggregatum flu- 
xionum arcus A D & finus redi B D. Porro fluxio arcus A D 
reperta eft = -] --^^ ^ , fluxio autem finus redi B D re- pi. 
^ ydx—xx 
d'x — zx'x , . d'x — Xx 
perietur = — 77 , igitur j G = — ideoque 
zVdx — XX' — 
d dxx — dxxx 
jH^ = jG^ + GHf= 4j^^xx — ^^QH^niobrem jH[= 
x\/d d—dx xVd ,f— I. ,f f 
7- =^ K a: , proindeque A J = z ^ 
s/ dx — XX y 
= 2V dx — zh'D, 
Hsec conclufio minimo cum laborc deduci poteft ex nota 
proprietate Tangentis, cum enim illius portiuncula j H lemper 
fit parallela chordae A D, fit ut Triangula j G H, A B D fint 
fimilia, unde AB, AD::GH/jH, hoc c^x^V dxxx'x , 
x\/dx . . ._. i\/dx 
— ^ , igitur ] H = =^d „ X 
Sed nihil vetat quominus adhibito fluxionis j H auxiliO;, ip- 
fam Cycloidis aream inveftigemus. Fluxio Areas A E j eft 
d " x^ X 
rcftangulumEiG^ — = ^ ^ -z^'xs^dx—xx fed flu- 
Vdx — XX . 
xlo portionis A B D noi alia eft ab ilia : Icaqae Aral S"E!f ^ 
correfpondenrque circuli portio A B D femper font a^qiiA!^. ' 
Efto AB curva Parabola cujus Axis AF, parainecer a-i ; ;. 
ponatur AE = ;e,EB=7, ABa:,, BD^i-, DCr^;/'" 
B C=i^, affumpta aequatione Parabolas naturam conftituenr- 
K a vide- 
