[ 445 1 
I The Conftru^ion of a Quadratrix to the Circle, 
ing the Curv defcribed by its Equable Evolution. 
I. T) Y the Equable Evolution of a Circle, I meanfuch a gra- 
j3 dual approach of its Periferie to ReBitude, as that all its 
parts do together^ and equally evolve or unbend : or fo that the 
fame Line becomes fucceffively a lefs and lefs Arc of a reci- 
procally greater Circle. 
2. Let A H K A (Fig. 6.) be the Periferie of a Circle. A E 
a Tangent to the point A. Let this Circular Line be luppos'd 
cut or divided at A, and then to unbend (VikQ a Sfring) its^ 
upper end remaining fixt to its Tangent AE, whilft the other 
parts do Equally Evohe or extend themlelves thorough all the 
degrees of lefs Curvature fas in ABD, AMC, &c, ) till they 
become ftraight in coincidence with the Tangent AE. 
Let AMC be t\\Q Evolving Curv in any middle pofuion 
between its firft and laft. Joyn the fixt end A, and the mo- 
ving end C» by the Chord-line AC, interfering the firft Cir- 
cle at H. I fay that AMC is a like Segment to A n cut 
off in the firft Circle by the Chord A H. For, by the fuppo- 
fition of AMC is the Arc of a Circle , having AE a Tangent 
common both to it and A nH, and both Arcs are terminated 
in the fame Right line AC; 
4. Hence the Curv ADCE fdefcrib'd by the moving end 
of the Periferie in its Evolution) may be thus conftrucSled, 
Let the Circle AHKA be by bifedions divided into any num- 
ber of equal parts. Let H be one of the points of fuch divi-^ 
fion. Then fay, as the number of equal parts in the Arc 
An A: is to the number of parts in the whole Periferie 
A H K A : : fo is the Chord AH; to a fourth Line, which 
let be A C in A H produc'd. So is C a point in the Curve 
ADCE. 
Dem. Upon A C defcribe AMC, an Arc like to the 
Arc An H. Whence— AH: AC:: AnH: AMC. 
But by conftruaion, A H : A C : : A n H : perif ; AHKA, 
therefore is the Arc AMC equal to the whole Periferie AHKA 
and 
