YOL. XVI.] PHILOSOPHICAL TRANSACTIONS. SQS 



ful ; wherefore they, as from a more certain sera, choose to begin their annals 

 with the third king Hoam-ti, and the year before Christ 2697. This Hoam-ti 

 is said to have instituted the sexagenary cycles, or periods of 60 years, accord- 

 ing to which this chronology is adjusted. Since this institution, there are now 

 73 periods elapsed, and the 74th is current ; in which time they account that 

 there has been 234 kings of China, sprung from no less than 22 several royal fa- 

 milies; the king now reigning being the second of the race of the Tartars, who 

 within these 50 years have thoroughly subjected China. 



The third king, Chuen-hio, is said to be the author of the Chinese calendar, 

 and to have appointed the beginning of the year to be on the new moon next 

 the beginning of the spring, which the Chinese account to be when the sun is 

 in 5" of Aquarius ; this account is still in use, though instituted 2500 years 

 before Christ. About 700 years after, the king Chim-tam reduced the begin- 

 ning of the year to the winter solstice, but the former was restored about J 00 

 years before Christ, and still continues. 



The years of this account are luni-solar, or consisting of 12 lunar months, 

 half of 30 days, and the rest of 29 days, with the intercalation of 7 months in 

 19 years ; so that 7 years in each cycle have 13 months. This distribution of 

 months was ordained by king Yao, about 2300 years before Christ, and is, if 

 rightly intercalated, a more exact measure of the celestial motions than our 

 Julian account, or old style, for that fails a day in 1 3 1 years ; whereas this ac- 

 count of the Chinese (which is nearly the same with the Jewish) fails but a day 

 in 225 years, or 4 days in 900 years. It is here said, that the famous wall of 

 China, extending above 400 leagues, was begun by king Xi-Hoam-ti, about the 

 year before Christ 210, to hinder the incursions of the Tartars, which in all ages 

 have infested this country. 



On the Numbers and Limits of the Roots of Cubic and Biquadratic Equations. 

 By Mr. E. Halley. N° 19O, p. 387. Translated from the Latin. 

 It appears that, both in cubic and biquadratic equations, the roots may be 

 expounded by perpendiculars let fail upon the axis, or a given diameter of a 

 given parabola, from the intersections of that curve with the circle ; and since a 

 circle, cutting a parabola, must necessarily intersect it either in four or in two 

 points, it is manifest that in biquadratics there always are either two or four 

 true roots, affirmative or negative; unless the circle happen to touch it, in 

 which case the equality of two roots, of the same sign, is concluded. But in 

 cubic equations, because one of the intersections is required for the construc- 

 tion, therefore either only one, or the three remaining roots denote one or 

 three, as in the case of contact ; whence it appears, that there are found two 

 equal roots, and that the problem, from whence the equation results, is really 



3 E 2 



