VOL. II.] PHILOSOPHICAL TRANSACTIONS. 20Q 



For demonstration of this rule we argue thus : 



1. Each multiplier multiplied by its remainder, is measured or divided by its 

 own divisor, leaving such a remainder as is proposed. For each multiplier was 

 defined above to be a multiple of its own divisor, plus an unit. Therefore 

 multiplying it by any remainder, it only renders it a greater multiple in the said 

 divisor, plus an unit, multiplied by the remainder ; which is no other than the 

 remainder itself; but if remains that product is destroyed. 



2. The sum of the products divided by each respective divisor, leaves the re- 

 mainder assigned. For concerning the first product, it is by the first section 

 measured by its own divisor, leaving the remainder proposed ; and if we add 

 the rest of the products, we only add a multiple of its own divisor, which in 

 division enlarges the quotient, but not the remainder. In particular the second 

 multiplier is 28 X 15 X 10 X remainder, all which is but a multiple of 28. 

 And so the third product is 28 X IQ X 13 X remainder. 



And what has been said concerning the sum of the products, being divided 

 by the first divisor, and leaving the remainder thereto assigned, may be said of 

 each respectively. 



3. The sum of the products divided by the solid of the three divisors, leaves 

 a remainder so qualified as the said sum. For that sum, by the 2d article, is 

 the first product increased by adding a just multiple of the first divisor, that 

 thereby we only enlarge the quotient, not alter the remainder. By the like 

 reason, the subtracting a just multiple, only alters the quotient, not the re- 

 mainder; but the solid of all three divisors multiplied here by the quotient, as 

 there by the remainder, is only a just multiple of the first divisor. Wherefore 

 the remainder, after this division is performed, is of the same quality as the sum 

 of the products, and divided by the first divisor, leaves the remainder proper 

 thereto : and the like may be said concerning each divisor. 



To find the year of the Julian period for any year of our Lord proposed : 

 It is necessary to know the sun's cycle, the prime number, and the number of 

 the Roman indiction, which Mr. Street performs by the following verse ; 



When 1, 9, 3, to the year have added been. 

 Divide by IQ, 28, fifteen. 



The remainders are the numbers sought, or the cycles and indiction re- 

 quired ; as we found them for the year l668, in the foregoing example. 



The use of the prime is to find the epact, and thereby the moon's age, the 

 time of high water, &c. A farther use of the sun's cycle is, to find the domi- 

 nical letter, and thereby to know the day of the week on which any day of any 



VOL. I. Db 



