56 Solution of a Proliem of Col. Silas Thus. [Jan. 



ordinal number, because it is the only variable factor of the exponent 



dz 2 k X a/ — J . Thus 2 kiT represents that circumference whose 

 place is designated by k. 



± V — 1 is a sign of impomhiUly, because it expresses a 

 quantity greater than a maximum, and less than a minimum ; but 

 the area of each concentric ring is out of that circle which serves as 

 a nucleus, and it is the diameter of thai circle which h a. 7iiaximum. 

 The diameter, tlien, of the exterior circle of this ring is greater 

 tiian a maximum. The whole area of the ring which exceeds this 

 diameter is then proved imaginary, which shows that the sign 



±- J _ J belongs absolutely to it. It now remains to explain the 

 sign ±. A ring contains two circumferences of circles ; to wit, an 

 external and an internal. Now 2 ^ only expresses one ; the sign 

 ± causes it to express two ; which I thus prove : — 



Let yy = an — x x (19) 



he the eciuaiion to a circle : if we take the value of y we shall have 



y =: \/ aa —XX (20) 



Here the double sign indicates two ordinates of an equal length 

 drawn irom a?iy particular point of the diameter on each side of it. 

 The positive ordinates, designated by +, extend only to half the 

 circle; and the negative ordinates designated by — , extend to the 

 other half : in order to obtain the ordinates which extend to the 

 whole circle, we must unite the two signs, as in ±. Now when 



this sign is accompanied by ^^ — 1, it does not mean + or — , 

 but + and — ; because the. imaginary quantities always go in pairs, 

 and thev cannot be separated without an absurdity, as I will prove. 

 Thuslet 1 1 1 be the tangent to the central circle. This tangent 



I 1 



js the srealeit ordinate which can be drawn to the exterior circle 



without enterini< into the central circle 1 I 1 : its middle point is at 



111 



the same time the smallest of those which can be drawn in the 

 interior circle, since it is reduced to this point, 1, jn which the two 



n 



ordinates coincide, the two ordinates, 1 I and 1 1 having then the 



10 1 



point 1, wliicii is common to both, and are connected by that 







point. Thus they form a continued right line, which is expressed 

 bv ± V — r. If we refer this expression to the interior circum- 

 ference, we have ± \/ — 1 = 0, which is not imaginary, because 



then it is the sign \/ — 1, which ought to be considered as 0. If, 

 on the contrary, wc refer it lo the pxtreme circumference, we have 

 ^/ _ 7 = -^ A,/ — 1 — -v/ — 1, which only ceases to be 



■^ \ 



imaginary at the two points ) J , which coincide with this extreme 



