306 Mr DAVIES on the Equations of Loci 



2. This, indeed, is one of the solutions given by JAMES BERNOULLI 

 to the celebrated Enigma of VIVIANI proposed to the Analysts of his day, 

 in 1682. Nor does our construction differ in any particular worthy of men- 

 tion from that given by the illustrious Professor of Basle *. The construc- 

 tion of VIVIANI himself, which he published without demonstration, and 

 which has usually (by writers implicitly following the statements of GRAN- 

 m and MONTUCLA) been characterised as the most elegant given of that 

 problem. But it will be presently shown that the methods of VIVIANI and 

 JAMES BERNOULLI are identical f. I shall here, however, remark, that 

 the second and third solutions of BERNOULLI are inaccurate, as professing 

 to leave quadrable spaces on the surface of the sphere. The second of these 

 is given by the condition 



m sin (p = n sin 6 t, 

 and the elementary area is, 



d A = (1 cos d>) d sin~ J sin <b 



n 



m ,, ,. cosd>d(b 

 = -(1 cos0). r r = (3.) 



n /-, m IA. 



V 1 ^ sm V 



which a little management will transform into elliptic, but not into circular 

 functions. Hence the problem does not receive a solution from that pro- 

 cess. 



It is easy to discover the cause of this oversight of BERNOULLI, 

 and his commentator. They forgot that it was a residue that was to an- 

 swer the question. Had it been the area of the figure itself that was qua- 

 drable, then, of course, any other which had a rational ratio to it would 

 have been quadrable too : but as the area of the figure was required to be a 



* See his Works, collected by Cramer, vol. ii. p. 512. 



f See Art. XX. Indeed this has been done by BERNOULLI himself, very simply 

 and elegantly, Op. torn. ii. p. 744. 



$ Bern. Op. ii. p. 513. 



