used hy the Ancient Egyptians. 165 



It appears from Inspection of this table, in the first place, that the year 3285 

 B. C. ought not to have been selected as the year of coincidence between the 

 solstice and the first day of Pachon, but rather 3287 or 3286. It appears also 

 that about this period the summer solstice fell regularly on the 21st of July of 

 the proleptic Julian year, and was only beginning to fall on the 20th in leap 

 years. M. Biot, instead of directly determining the coincidence of the solstice 

 and the heliacal rising of Sirlus, which would be a purely mathematical problem, 

 independent of any artificial divisions of time,* uses the 20th of July of the 



* From the definition of heliacal rising, the sun's longitude at the time of the star's rising must 

 exceed 6y, but by a less quantity than the space which it passes over in a day. Its average excess 

 over it in a period of four years may be estimated at half this space, say 29'. 30". Consequently, 

 when the heliacal rising coincides with the solstice, 9y must be equal to the difference between 90° 

 and this last-mentioned quantity, or to 89°. 30'. 30". If A, /x, and to be calculated for any epoch, 

 their values for other years may be expressed by series of the form a„ -|- a, < -|- Aj<', &c., t being the 



number of years after the epoch; and by the formulas of the preceding note, ^ 4" ''> K' ^'^^> "'*'" 

 mately, 9y may be expressed in similar series. The value of t, which will satisfy the equation 

 fly=r89°. 30'. 30", will give the precise number of years after the assumed epoch, at which the 

 required coincidence took place. In order to simplify the calculation, the epoch for which A, jw,, 

 and ui are calculated should not be far removed from the epoch of coincidence. In that case, we 

 may confine ourselves to the terms in the above series which are independent of t, or contain only 

 its first power. These terms will at any rate give a first approximation ; and we may then calculate 

 the values of A, ^, and w for the year so found as a new epoch. The great practical difficulty arises 

 from the uncertainty which there is as to the proper motion of Sirius, and as to the precession, and 

 change of the obliquity of the ecliptic. According to the best data that I have been able to procure, 

 (namely, the values of the precession and obliquity given by M. Biot from Laplace's formulas, and the 

 proper motion givenin the catalogue of the stars in the Encyclopaedia Metropolitana,) I make the right 

 ascension of Sirius in 3285 B. C. to have been 43°. 47'. 15", and his dechnation 23°. 37'. 45". The 

 former was increasing at the rate of 38",43 a year, the latter diminishing at the rate of 13", 64 a 

 year. The obliquity was 24°. 6'. 30", and its diminution annually 0",33. Substituting these 

 values for A, /x, and oi in the formulas of the preceding note, and making a = 30° and y =: 11° (the 

 values assumed by M. Biot) I find 



/x 4- )/=58°. 25'. 6" + 28", 45 1 ; 

 e<, = 74o. 7'.43" + 28",36<; 

 fly =88°. 32'. 1" 4- 25", 95 <. 



In these expressions, the rates of increase are much more to be depended on for accuracy than the 

 values at the epoch. 

 From the equation 



