VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. \>j 



Sometimes the fluxion of the curve is more easily found by comparing toge- 

 ther the similar triangles c d e, c b o, making this proportion cb:co::ce:cd, 



that is, for the circle, ^^ 1 r x — x x : r:: i 



'J 2 r X — X X 



The curve of the cycloid may be known in the same manner. Let a k l be 

 a semicycloid, whose generating circle is a d l (fig. 3). Any point b being as- 

 sumed in the diameter a l; let b i be drawn parallel to the base k l, meeting the 

 circle in the point d ; complete the rectangle a e i b ; also draw f h parallel and 

 infinitely near, to e i, cutting b i produced in g, and the curve a k in h. Then, 

 putting A L = <f, A B = E I = a:, G H = i', it is known that the right line b i is 

 everywhere the aggregate of the arc A d and the right sine b d, and hence it is 

 manifest that the fluxion i g is the sum of the fluxions of the arc a d and of the 



sine B D. But the fluxion of the arc a d is found to be = - ^ " -, and the 



■vd X — X X 



nuxion or the sme b d is = — , , therefore i g = , ; conse- 



2 Vd X ~ X X Vd X — X X 



.1 o . o ddxxdxxx , X Vd d — d X x ,Jd 



quently i h' = i g^ 4- g h" = — .,_,_ — , and i h = ,, = ^ = 



a X —~ X X /^ ^ X ^^ X X V^ 



S x~^ X ; therefore a i = 2 cJ* or* = 2 ^d a? ^ 2 A d. 



This same conclusion may be deduced with very little trouble from the known 

 property of the tangent. For since its particle i h is always parallel to the chord 

 A D, it makes the two triangles i g h, a b d to be similar; whence a b : a d :: g h : 

 H I, that is, X : ^/ dx :'. X •.w.i=: = d^ x^ x, as before. 



Now by the help of the fluxion i h we may find the area of the cycloid. For 

 the fluxion of the area a e i is the rectangle e i g := ~ " . = .r \^d x — x x ; 



Vd X —XI 



but the fluxion of the portion a b d is the same; therefore the area A e i and the 

 corresponding portion a b d, of the circle, are always equal. 



Let A B (fig. 4) be the curve of a parabola, its axis A p, and parameter a: 

 Put AE=:.r, EB:=7/, AB = z, BD = .r, DC:=_y, BC = i; then assuming the 

 equation expressing the nature of the parabola, s'\z.ax-=.yy, it gives ax='lyy, 



and hence x = -^ ; but b c^ = b d^ + d c^, that \s, z z = x x + ii y =. -1-12 



a I J f -^ •-' a a 



yy — .J-jLL iA therefore z = — ^-^^ — y = " ^ ' y; if therefore 



■ 7^-^ — y be transformed into an infinite series, the curve a b may thence be 



easily known. 



Now it will also appear that if the hyperbolic space were given, this would be 

 given also, and vice versa. For s. az = y */ y^ ■\- \o.a, and hence -i- a z = the 



VOL, IV. D 



