Note 10 — Chapter I · NOMENCLATURE AND THEORY.
Reading edition
Describe the semicircle BMNF. Then
OD : ON :: CH : CM;
l : ON :: v : h.
ON2 = (h2 l2) / v2. (1)
BO : ON :: ON : OF;
h + r : ON :: ON : h - r.
ON2 = h2 - r2 (2)
CF : CH :: CD : CK;
h : v :: √(r2 + l2 : l.
(h2l2/v2) = r2 + l2 (3)
∴ Eqs. (1), (2), and (3), r2 + l2 = h2 - r2.
h2 = l2 +2r2 (4)
h = l√(1 + 2(r/l)2.
Eqs. (3) and (4), (l2 + 2r2)l2 = v2(r2 + l2).
v2 = (l2 +2r2)/(1+r2/l2) = l2(1+2(r2/l2))/(1+r2/l2);
v = l(√((1+2(r/l)/(1+(r/l)2))).