702 PHILOSOPHICAL TRANSACTIONS. ANNO 1780, 



Hence the lines oca and ic(3, traced by the points v and k, are at right angles, 

 and the ruler mvEK moves exactly in the same manner as if it was guided by the 

 points V and k sliding on the lines acaiand /•Cf3, at right angles to each other; 

 just in the same manner as the ruler msl is guided by the points s and l, sliding 

 on the lines ac and bc. Therefore if any point m be assumed in the line Kvm, 

 as. a describing point, the figure described will be an ellipse, the position of whose 

 principal axes are the lines acaand Z'C(3 ; the centre of the ellipse being still in c 

 as before. If niK. be taken equal to ml, the ellipse thus described by the point 

 7« will be the same with that described by the point m, only in another position : 

 its greater semi-axis «c making an angle with ac, the greater semi-axis of the 

 former ellipse, equal to half mew, the angle which the rulers or lines me and 

 WE make with each other. 



Scholium. — This proposition is demonstrated in Schooten's Exercitationes, &c. 

 p. 305 ; but he makes 12 cases of it : had he made use of the 20th of the 3d 

 El. they might have been all comprehended under one. In the turning of ovals, 

 the top of the rest which supports the tool is always made to pass through s and 

 L (fig. 2,) the two centres round which the oval engine turns ; and in this case 

 the ruler or line msel represents the top of the rest. If the tool be held on any 

 part of the rest between the workman and the nearest centre, as at m, an oval 

 will be turned having its longer axis Aa (in one position of the work) coinciding 

 with the top of the rest. As the tool is removed towards s, the oval will grow 

 narrower, and at s become a right line. Beyond s towards e it will grow 

 rounder, and at e become a circle ; beyond e it will grow narrower, and at l be- 

 come a right line at right angles to the right line described when the tool was at 

 s. If the tool be removed beyond l, it will describe an oval again, whose longer 

 axis is at right angles to the longer axis of the oval first described when the tool 

 was at m. It may be very convenient to mark the points s and l, and also their 

 middle point e, on the top or face of the rest that supports the tool. If any 

 thing be interposed between the tool and the top of the rest, so as to raise the 

 tool above the line passing through the centres s and l, an oval will yet be des- 

 cribed, whose centre will be the same with that of the oval first described when 

 the tool was at m ; but its principal axis will cross the principal axis of that oval 

 (fig. 3 and 4.) Draw right lines, both from m the old place of the tool, and 

 from m the new place of the tool, to the point e marked on the rest. Half the 

 angle which these two lines make with each other will be the angle which the 

 principal axis of the new oval makes with the principal axis of the old one. 



It is well known, that when the oval engine is set in order for working, there 

 is a part which slides back, and is then fixed, which separates the 2 centres of 



motion, and gives the eccentricity; for the difference between the 1st and 2d 



semi-axes, will be just as much as the centres are thus separated : call the dis- 



