VOL. XXV.] PHILOSOPHICAL TRANSACTIONS. 303 



and radius M, the same with the small wheel. The design of this piece is, that 

 in the several positions of the ruler ef, the circular limb kl always touching and 

 sliding by the edge of the ruler ab, the centre of the wheel may be always in a 

 line, im, parallel to the ruler ab. In the ruler cd make mb = ih or ik, and at 

 M fasten a small pin, and another to the ruler ep near the wheel^ as at p. To 

 these two pins let be fastened the two ends of a string mr, so that its whole 

 length, from pin to pin, + ^h b^ equal to the intended axis, tw, of the curve. 



The instrument being thus prepared, let a strong ruler so, be held fast on 

 the paper or plain that the curve is to be drawn on. Lay the ruler ef from m 

 towards a, and parallel to ab, so that the string lie straight along the edge 

 of the ruler ef from m to p, the point h of the quadrantal piece of wood rest- 

 ing on the edge of the ruler ab. Then with a small pin at m keeping the string 

 close to the edge of the ruler ep, and with the other hand on the end e, keep- 

 ing the wheel tight to the paper or plane, move the pin, string, and ruler ep, 

 from M towards o, the ruler cd sliding along by the fastened ruler so, in a right 

 line; the wheel h will by its motion describe the desired curve tv. 



Note. — ^The semidiameter of the Hltle wheel must be about the sum of the 

 thicknesses of the two rulers ef and ab, that it may touch the paper. Also it 

 will be convenient that its edge be thin, and a little rough, that it may not slide 

 flatwise, and that it may leave a visible impression. 



From this construction the following properties are demonstrable : 



I. It is evident from the construction, that the sum of the tangent and sub- 

 tangent is every where equal to the same given line = mr + k? = tw. For the 

 string, first straight at tw, afterwards making an angle at r, being every where 

 the same, the line r/, or rp -f- vi, is always the tangent, and the remainder rm 

 the subtangent ; the contact of the wheel with the plane being the point of the 

 curve to which they belong. 



II. It hence follows, that any assignable part of the curve is rectifiable, or 

 equal to eny assignable straight line. In fig. Q, let pae be a part of the curve, 

 its vertex p. hdc? is the line described by the motion of the pin r in fig. 8, and 

 may be shown to be an asymptote to the curve, ph a perpendicular to hd. 

 Let A be a given point in the curve, ad the tangent, and bd the subtangent, to 

 the same point a. Let a be another point in the curve, infinitely near to a, 

 to which let ad be the tangent, and bd the subtangent. Draw ag, ag, perpen- 

 dicular to PH, and AB, ab perpendicular to hd. By the construction, ad -j- db 

 z= ad ■\- db. Let aS be made equal to gd, and draw DiJ*. Then because ad -{- 

 ^d = ad + DB ; subtract bn and «d, or ai^ from both sums, and there remains 

 Sd ■\- c?d = Aa -j- b/', or ca . Aac, nd^ are similar triangles, therefore ca (b^) : 

 A.a :: Sd'.nd; and compounding, Bb -\- Aa : Aa :: Sd -{- T)d : J)d ; alternating, ^k 



