1/2 Of the Division of Ike Circle into 



Circle. S. „ , ^ 



^^tSyfiod' ^!?"°"';."'}l 25 49 0-8491040724 

 Do. O'sanom. ". . 29 6 19-25S74710?8 

 Do. ©to the D's £3 1 40 13-872 146564(J 



Hence a Chaldean period of 223 lunations 



Days. hrs. min. sec. 



contains 6585 7 42 24-4 



Nineteen rev. of the O a' ]) 's 8 = G585 18 39 o8-4 



Circles. S. <> n 



Motion of the })'s anom. 238 11 27 10 9-3502 

 Do. O's anom. IS 10 29 34-7006 



Dist. of the O ft' ]) 's e IS 11 29 31 33-4887 

 Whence thecomp. of the motion of the © to the ]) 's S3 in 

 each Chaldean period amounts to 28' 26"-5113, as I before 

 stated. 



XXV. Ow the Division of the Circle into seventeen equal Parts, 



To Mr. Tilloch. 



Sir, — ix curious discovery lately made in pure mathematics, 

 we owe to M. Gauss of Gijttingen, who has shown, contrary to 

 the opinion that has prevailed from the most ancient times, that 

 a regular polygon of seventeen sides may be inscribed in a circle, 

 without having recourse to any other principles than those ad- 

 mitted in the plane geometry. As the author's own solution of 

 this problem is a part of a peculiar and very abstruse and recon- 

 dite theory, the following communication, in which the problem 

 is solved without any reference to ^that theory, may not be un- 

 acceptable. 



Let a = — = 21°-y. If we set about solving the problem 



in the usual way, by seeking a value of cos. n, we shall obtain 

 an equation of sixteen dimensions; because, in this mode of pro- 

 ceeding, the cosines of the several arcs, a, 2ci, 3a, Sec, sixteen in 

 miinber, are not distinguished from one another, and are all com- 

 prehended indiscriminately in the sania result. The equation 

 thus obtained, although of sixteen dimensions, is however only 

 of the eighth order, every two arcs that together make the 

 whole circumference iiaving the same cosine, viz. cos a = cos 16o, 

 cos 2a = 15«, cos 3a = cos I4fir, &c. 



Supposing n to denote any whole number, let f = 4na ; then,, 



4f + -f =; « X 17tt = n X 360", (1) 



wlicrc- 



