54 Solution of a Problem of Col. Silas Tittis. [Jak. 



unit is the quotient of a number divided by itself, while the real 

 unit is the product of a numbei- multiplied by the inverse of that 

 number. Let n be any number, n divided by n is a speculative unit, 



and n x - is a real unit, as a rectangle would be. If, then, - be 



the supposed unit, it is necessary, in order to obtain a real unit, to 



multiply - by » ; consequently, the nearer we approach one side, 



the farther we recede on the other. 



The following solution, which for the first time is given of Pell's 

 problem, is the only one exempt from this absurdity : — 



Problem. 

 The following equations being proposed, viz. 



a^ -^ b c = \G (1) 



^2 + ac = 17 (2) 



c= + c Z/ «B 1 8 . . . . (3) 



To find c, I, c, (See Wallis's Algebra, chap. CO, 62, &c.) let 

 tbere be a series of concentric circles, (Plate XXVII. fig. I,) 

 1 1 1, 1 1 1, 1 1 1 1 1 1 1 1 1 1 1 1, and let 



111 222 13 13 13 17 17 17 21 21 21 



them be so described that we have 01 = 11 = 11 = 11 = 



1 13 2 3 



&c. = 1. 

 If we make a = the sector O 1 1 1 (4) 



13 13 IS 



I = the sector O 1 1 1 (5) 



17 17 17 



c — the sector O 1 1 1 (6) 



21 <l 21 



Then if the areas of these sectors be substituted for a, b, c, in 

 the equation (1), (2), (3), according to the following method, their 

 differences will reduce these equations to identical ones. 



Demonstration. 



This demonstration is founded upon a remarkable property of the 

 concentric circles of this figure. This property is, that the areas of 

 each of the rings intercepted between two consecutive circum- 

 ferences are equal to the area of the central circle. If we take the 

 area of the central circle for an unit, the areas of each of the rings 

 ivill be = 1. 



To prove this, let O 1 be the radius of the central circle, we may 







easily perceive that the radii of the successive circles are the hypo- 

 theneuses of right angled triangles, whose sides are, 



1. The radius of the preceding circle. 2. A constant tangent 

 equal to the radius of the central circle. The series of radii will 



be then expressed by O 1 ( -v' 1, V 2, V 3, \/ 4, &c.) ... .(7) 



■ ' 



