2 MATHEMATICAL PROBLEM. 



the variable arc CE; and through E, fi'om the centre O* 

 draw O F, meeting the tangent C T, in F ; put C O n r, 

 O Fzzs, arc C E~x, corresponding tangent C Fzzy; then 

 by the nature of the circle, as a* : r* : : j : a-; but s* is 

 greater than r'^, therefore j) is greater than x, consequently 

 y increases faster than x: moreover, when yzrO, a + xzzd 

 + Or: a, in which case a-\-x is infinitely greater than i/, but 

 y increases faster than .r, and exceeds it infinite! v, when 

 a: — the quadrant CPD; consequently, by priu;e and 

 ultimate ratios, there is a point P betwixt C and D, wl'ich 

 cuts off an arc C P. or z, the tangent of which, C T, or 

 *— the arc A C P, i.e. a+s—t. 

 The jreometri- (C) When A C P = C T, the sector A O P = triangle 

 " biem COT; from each take the common sector COP, and the 



sector A O C ~ the space C P T; hence the problem, 

 treated geometrically, assumes tliis form ; to find a point T 

 in the tangent C F, produced if necessary, from which if 

 T O be drawn to the centre, it shall give the space C P T 

 — tlie given sector A O C, for this construction will evi- 

 dently make the tangent C T rz the arc A C P. 



When a = 0, (D) If the arc A B C = 0, the sector A O C = ; there- 



a = or is in- ^^^^ ^j^^ ^ ^ p r^ . ^^. j^^^^^^ ^^^ ^. ^ p_^ 



finitely small. . ^ . ' J \ J ' ' 



1. e. when azzo, z is evanescent; consequently, the problem 



is impossible, unless « be a finite Tiaagnitude. 

 M not restricted (E) It appears from (B). and (D), that 2 is a real, not an 

 Lgfg imaginary arc, provided a be a finite magnitude, which may 



be expressed by n Q, Q being a quadrant, and n a positive 

 number, either whole or fractional. This conclusion how- 

 ever is rejected by a celebrated mathematician, who inti- 

 mates, that n is alv ays an odd number; the passage con- 

 taining- his opinion is here quoted. 



" Invenire omi;os arcus, qui tangentlbus suis sint 

 aequales. 



** Solutio. Pr- uis arcus, hac proprietate prceditus, eat 

 infinite parVus. Tuni in secundo quadrante, quia hie tan- 

 gentes sunt negativse, datur nullus istiusmodi arcus; in 

 tertio vero quadrante dabitur unu « ^270° aliquanto minor; 

 porro dabuntur ejusmodi arcus in qumto, septimo, ixc." 



The reason assigned for n bein;',- an odd number in tliis 

 ^uotatioR is derived from the supposition, that all the tan- 



gents 



