1815.] Solution of a Prollem of Col. Silas Titus. 57 



circumference. To apply this principle to the double sine of tlie 

 expression ±2 ki: \' — 1, let us divide into two equal parts the 



part 1 1 of the diameter 1 1, which is intersected between the two 



1 1 



circumferences of the ring which extends beyond the central circle ; 

 and through the point of division let us draw the dotted concentric 

 circle ; the circumference of the dotted circle will be an arithmetical 

 mean proportionat between the two extreme circumferences of the 

 ring. If we take this dotted circumference as a line of abscissa, it 

 is clear it will cut all the sections of the diameters intercepted be- 

 tween the extreme circumference into two equal parts. Each of 

 these half parts will be equal ordinates drawn on each side of the 

 circumference of the dotted circle, this circumference being taken 

 as a line of abscissa, and the two extreme circumferences will be the 

 curves described by means of these ordinates. As all these ordinates 

 are imaginary, they have only two real points, which are their two 

 extremes : one of these two extremes is a point in the dotted circle, 

 and the other is a point in one of the circles already described : 

 these three circles are then composed only of insulated points, the 

 points of the dotted circle are double, and those of the circle de- 

 scribed are simple. ± 2 w -s/ — 1 expresses the sum of the 

 points of the dotted circle ; that is to say, + 2 tt a/ — 1 is the 



sum of the points of the exterior circle, and — 2 tt \/ — i, the 

 sum of the' points of the interior circle. Resuming all this expla- 

 nation, we find ± 2 A TT \/ — 1 is the sign of the description tf 

 two concentric circles forming a ring ly assuming for a li?ie of 

 abscissa a third concentric circle jv hose circumference is a?i arith- 

 metical proportional mean between the circumferences to he described, 



the same as ± \f aa — x x is the sign of the description of a 

 simple circumference by taking its diameter for a line of the 

 abscissa. 



This granted, in order to resolve the equations (1), (2), (3), I 

 begin by multiplying their second member by the second member of 

 equation (12), wliicii gives me 



a'' + hc= IGe* '-^''^-''" = the area* of the IGth circle, 



(fig. 1) (22)' 



L* -f ac = 17 c- *'-''' ^ - ' = the area of the 17th circle , .(2;{) 



c- + a ^ = 18 e* '^ '" ^ -"^ = the area of the 18th circle . .(24) 

 1 assume for the roots of these equations, 



• Thr idea of my giiiiig one nn-.i fur tin- rncil of jinollirr nrcn, may ncrliiips he 

 ra»illcd :i( ; tun ivh<-ii \\v consider rliat llic root of tliejirca of llic square A B C 1) 

 (liR. 2) can only bf llic area of a rfciangic, >uch as A o C c or Ac UA, tlic exula- 

 Ditiiod Hill ujijx'ur cirur. 



