( ) 
tfaie. The Semi-diameter of the little Wheel muft be 
about the Sum of the thicknefles of the two Rulars 
EF and A B, that it may touch the Paper. Alfo it 
will be convenient that fts edge be thin, and a little 
rough, that it may not Hide flat-ways, and that it 
may leave a vifible irapreffion. 
From this Conftrudion the following Properties are 
demonftrable. 
I. It is evident frosa the Conftrudion, that the Sum ef 
the Tangent and SubtaUgent is every where equal to the 
fame given Line = MR + Ri = TW.) for the String 
( firft ftraight at T W, afterwards making an Angle at 
R ) being every where the fame, the Line Ri ("or R p 
_+ p i ) is always the Tangent, and the Remainder RM 
the Subtangent 5 the Contad of the Wheel with the 
Plain, being the point of the Curve to which they be- 
long. 
II. It hence follows, that any affignable part of the 
Curve i« Rectifable, or equal to any affignable ftraight 
Line; In Fig. a. Let F A E be a part of the Curve, its 
Vertex F. HD d is the Line defcribed by the motion of 
the Pin R ( in Fig. 1. ) and may be (hewn to be Af- 
fymptote to the Curve. F H a perpendicular to H D. 
Let A be given point in the Curve, A D the Tangent, 
and B D the Subtangent to the fame point A. Let a be 
another point in the Curve infinitely near to A. to which 
let ad be the Tangent, and b d the Subtangent. Draw 
A G, a g perpendicular to F H and A B, a b perpendicu- 
lar to H D. By the Conftrudion A D + D B — ad + 
d b. Let a / be made equal to a D, and draw D s. Then 
becaufe a d -V b d — AD + DB. Subtrad b D and 
a D C or a S ) from both Sums ( Equals from Equals ) 
there remains S d + d D = A a + B b ( or C a ) A a C, 
13 x 3 a 
