IQQ CHRONOLOGICAL OBSERVATIONS 



divide the circle into 360 degrees (= 6 X 6 x 10, or the square of 

 6 X 10) ; and the quarter of the day into the same number of minutes: 

 making 1440 (= sq. 6 x 40) minutes for the entire day. 



The fractional quantity which would join the two ends of the cycle 

 of time, has been given above, as £ d — J 7 d = / TJ 3 g d =:341|f m . These 

 minutes, when expressed as a whole number (26,280 seventy-sevenths 

 m.), are divisible only by 360. 



The Proportion of "113 : 355" is well known as an approximation 

 to the ratio of the diameter to the circumference ; and is commonly 

 supposed to have been discovered by Archimedes. The first of these 

 numbers proves to be the counterpart of the last employed fraction 

 added to a quarter of a day, for i d + || d = 410i° m = 4520 elevenths 

 m. = 40x 113. 



Here, we again find the "sacred number" 40; and if we multiply 

 together the squares of these two fractional quantities (sq. 341§f m x sq. 

 410} £ m ), we shall include the square of 1440, or of the number of 

 minutes in a day. 



On recurring to the Proportion which gives the irregularity in the 

 revolutions of Jupiter's satellites, one of the portions of the quarter of 

 a day (i d — 19 /^y" =) 340/jyy", is found to be divisible by 360; 

 and also by 11 ; while the complemental portion, if left unreduced 

 (19^~^fo m ) and expressed as a whole number, is itself the square 

 of 1440. On attempting to divide the square of 1440 by 11, the re- 

 mainder is found to be T ] T ; and it may also be remarked, that the sum 

 of the two portions (=360 x 1337 x 80) is divisible by 7. 



Multiplying this unreduced quarter by 1440 divided by the first mem- 

 ber of the Proportion, gives 360 x 1337 x 80 x 79{|J = 3,073,593,600 

 = sq. 36 x sq. 1540 =sq. 24 x sq. 2310; and the same result is obtained 

 from all fractions, not exceeding i, that can be expressed in minutes 

 or 77ths of a minute. The unconformable fractions require multi- 

 plying by the first denominator ; thus, 



1420/2 1440 ^40 . 1R . ftv1! , w 



— — mi. : : , gives sq. 36 x sq. 1540 x 1337. 



77 77 77 & * ^ 



14. A standard measure of time should follow the properties of 

 number: and it may already have been perceived, that lh Egyptian 

 Great Years (= 2310 = 2 x 3 x 5 x 7 x 11) are composed of the in- 

 divisible elementary numbers multiplied together. 



It now appears, that 2310 is a universal number ; divisible by each of 

 its component elements, and by every combination of two or more of 



