for turning Ovals, 381 
the former at the center, the latter at the circumference; 
therefore the former is double the latter. Now as this 
holds in every pofition of the rulers during their joint 
motion, it is manifeft, the points v and k will each de- 
fcribe right lines, namely, a&x. and <5c/3, palling through 
c and making the angles ^ca and bci, ( = bc/S) each equal 
to half me m. j 
Hence the lines acy. and &c(3, traced by the points v 
and k are at right angles, and the ruler m \ ek moves 
exadlly in the fame manner as if it was guided by the 
the points v and k Aiding on the lines aca and heft, at 
right angles to each other ; juft in the fame manner as 
the ruler msl is guided by the points s and l, Hiding 
I 
on the lines ac and bc. Therefore if any point m be 
affumed in the line k vm as a defcribing point, the figure 
defcribed will be an ellipfe, the pofition of whofe princi- 
pal axes are the lines ac% and be (3; the center of the 
ellipfe being ftill in c as before. If m'L is taken equal to 
ml, the ellipfe thus defcribed by the point m will be the 
fame with that defcribed by the point m, only in another 
pofition : its greater femi-axis ac making an angle with 
ac, the greater femi-axis of the former ellipfe, equal to 
half me hif the angle which the rulers or lines me and 
m e make with each other. 
E e e 2 Scholium. 
