VOL. XXII.] PHILOSOPHICAL TRANSACTIONS. 463 



or unbend ; or so that the same line becomes successively a less and less arc of 

 a reciprocally greater circle. 



2. Let AHKA, fig. 8, pi. U, be the periphery of a circle; ae a tangent to 

 the point A. Let this circular line be supposed cut or divided at a, and then 

 to unbend like a spring, its upper end remaining fixed to its tangent ae, while 

 the other parts equally evolve or extend themselves through all the degrees of 

 less curvature, as in abd, amc, &c. till they become straight in coincidence 

 with the tangent ae, 



3. Let amc be the evolving curve, in any middle position between its first 

 and last. Join the fixed end a, and the moving end c, by the chord-line ac, 

 intersecting the first circle at h. That amc is a like segment to Ann, cut 

 off in the first circle, by the chord ah. For, by the supposition amc is the arc 

 of a circle, having ae a tangent common both to it and Ann, and both arcs 

 are terminated in the same right line ac 



4. Hence the curve adce, described by the moving end of the periphery in 

 its evolution, may be thus constructed. Let the circle ahka be by bisections 

 divided into any number of equal parts. Let h be one of the points of such 

 division. Then say, as the number of equal parts in the arc aua: is to the 

 number of parts in the whole periphery ahka : : so is the chord ah : to a fourth 

 line, which let be ac in ah produced. So is c a point in the curve adce. 



5. Dein. Upon AC describe amc, an arc like to the arc auh. Whence 

 AH : AC :: auh : amc. But by construction, ah : AC :: Ann : periphery ahka, 

 therefore is the arc amc equal to the whole periphery ahka, and similar to the 

 arc AUH. Consequently amc represents the evolving periphery, in a position 

 similar to the arc auh, and c is the describing point. 



6. After the same manner may be found other points through which the 

 curve may be drawn. But here, as in the old quadratix of Dinostratus, the 

 point E cannot be precisely determined, but the curve may be brought so near 

 it, that its flexure or tendency will so lead to the point E, that ae shall be near 

 enough to the truth for common uses. 



7. Supposing the point e found, a tangent to any point of the curve may 

 be drawn: and supposing a tangent drawn, the point e may be determined; 

 the property of the tangent being this, that supposing rt a tangent to the 

 point c, and CA, ce, drawn from c to each end of the rectified circle, the 

 angle act (the lesser angle that ac makes with the tangent) is equal to the 

 tangent made by the two lines drawn from c. 



8. Let c be a point in the quadratix indefinitely near to c ; and draw ac 

 intersecting ahka in h, and amc in o. To ac as a chord, draw the arc Amc 

 like to the arc Anh. To the point c of the arc amc draw the tangent cl 



