Chap. 5. PHILOSOPHICAL DISCOVERIES. 211

Let k l and L be two very near ordinates of the cycloid,meeting the generating circle in m and qj produce the chorde m till it meet the ordinate k s in p ; let q j be the perpendi-cular from o_on m p ; then draw the lines e n and m n, touchingthe circle in e and m.

Because the triangles e nm, p q_m are similar, and en=nm,therefore p q_ is equal to q_m ; and the triangle pq_m beingisosceles, the perpendicular q 0 bisects the base ? m ; so thatm p is double of ah : but, by the last proposition, l s is pa-rallel, and consequently equal, to m p, and l s is equal to2 m 0. The line l s is the increment of the Hprve e l, gene-rated in the fame time that the chord e m increases by m <?, sincee q is equal to e s, when the points q_ and m come together :Therefore the curve increases with double the velocity that thechord increases ; and since they begin, at e, to increase toge-ther, the arc of the cycloid e l will be always double of thechord em.

Corol. The femi-cycloid e l b is equal to twice the dia-meter of the generating circle, e f ; and the whole cycloidaceb is quadruple of the diameter E F.

PROPOSITION IV.

Let er be parallel to the base ab, and cr parallel to the axisof the cycloid ; and the space ecr, bounded by the arc os thecycloid e c and the lines er and rc, fall be equal to the circidararea e g k.

E e 2

Draw