Mr. ludlam on the 'Engine for turning Ovals. 379 
Let a a and b b (fig. 1 .) be two indefinite lines, inter- 
iedting each other at right angles in c. Let lsm be the 
beam, or a rigid right line, in which aflume two fixed 
. 
points l and s at pleafure. If the fixed point l be kept 
always Hiding upon the line bc£, and the other point s 
always Hiding upon the line ac^; I fay then, that any 
point m in the line ls, or that line produced, will defcribe 
an ellipfe. 
Bifect ls in e, and through c and e draw the indefi- 
nite right line ceh. Upon ls as a diameter with the 
center e defcribe. a femi-circle, and becaufe lcs is a right 
angle, it will pafs through c, and ec = el. Through m 
draw mph perpendicular to ac meeting ce produced in 
h; and becaufe mh is parallel to cl, th~ triangles meh 
and cel are fimilar, and he = me, and he + ec = me + el, 
or ch-lm. The point h therefore always falls in the 
circumference of the circle had a defcribed with the cen- 
ter c and radius ch=lm. Now the fimilar triangles chp 
and smp give cH : sm :: ph : pm. But when l arrives at 
c, then lm (=ch) coincides with ca ; and when s arrives 
at c, then sm coincides with cb ; therefore ca : cb :: ph : pm, 
and ca 2 :cb 2 ::ph 2 :pm% or ca 2 :cb 2 :: apxp<2:pm% which 
is the property of an ellipfe, whofe firft femi-axe is ca 
or lm, and fecond femi-axe is cb=sm. 
Vol. LXX. E e e Produce 
