184 The Rev. E. HmcKS on the Years and Cycles 



lengthened is the diminution of the obliquity of the ecliptic. The phenomenon, 

 by which the commencement of the year was indicated, was the attainment of a 

 given length by the meridian shadow of an object ; that is, the diminution of 

 the sun's altitude beyond a given limit, or his attaining a given south declination. 

 Now, the sun being at this time In the quadrant following the equinox, he must 

 not only attain the same longitude as he had at the beginning of the year before 

 this can happen, but he must go over a small additional arc sufficient to compen- 

 sate for the decrease of obliquity. The time of his passing over this small arc 

 must be added to the mean tropical year, as well as the time of his passing over 

 the annual variation of the equation of the centre ; and the sum of all three will 

 be the tropical year, as it would have been observed by the Egyptians. I do 

 not mean to say that it would be precisely so in a single year. The lunar and 

 planetary perturbations might make it greater or less. But, taking the sum of a 

 few observed years, the effect of these perturbations would disappear, and the 

 average value of the observed year would be that which 1 have stated. 



It remains that I should ascertain the numerical value of this tropical year. 

 1 find, in the first place, that the annual precession, about the time of the chro- 



to the first powers of the variations, as well as of the eccentricity, we have by equation (3) 

 =: nS< { 1 4- 2 e cos (e — w) } — 2 Je sin (e — ■a) — 2 eSzr cos (s — w) ; 



2i5esin(£ — tsr) -|- 2eJ«rcos(£ — w). 



n 



When the mean anomaly is less than 90°, both the terms, which compose the value of St will be 

 positive. It was so, in the case we are considering, from the earliest age that can be conceived to 

 about the year 2170 B. C, when the perigee passed through the sun's place at the commencement 

 of the Egyptian year. In the next quadrant, the term depending on Se is negative ; but during the 

 greater part of the time that the perigee takes to pass through it, St will be positive, on account of 

 the variation Jm being greater than Se. If e — w be greater than 180°, but less than 270°, U will 

 consist of two negative terms ; and if it be less than 180°, but greater than 90°, its terms will be of 

 opposite signs, but the negative one will preponderate. 



In the interval between the chronological epochs of the eighteenth and third centuries before 

 Christ, the average, '^ue of s — ra- in reference to the summer solstice was about 218°. The tropi- 

 cal year commencing at that solstice was consequently less than the mean tropical year, both the 

 terms of St being negative ; and of course the cycle, formed by comparing such a year with the year 

 of 365 days, was greater than the cycle, formed by comparing the mean tropical year with the year 

 of 365 days. The latter consisted of 1505 years ; the former of 1508 years ; the coincidence of the 

 solstice with the 241st day of the year occurring in 1779 and 271 B. C. 



