Page:Philosophical Transactions - Volume 002.djvu/169

(570)

is 285, which multiply by all numbers successively, and divide by 28, till you find the Remainder required. Thus twice 285 is 570, which divided by 28, the remainder is 10, also thrice 285 is 855, which divided by 28, the remainder is 15. Thus if you try on successively, you'l find, that 17 times 285, which is 4845, is the Number required, the which divided by 28, the Remainder is an Unit. Hence then we shall find, that

4845	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$
4200
6916

is equal to the Solid or Product of

$\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}$	19, 15, 17
	28, 15, 10
	28, 19, 13

More easie ways of performing this postulatum, are to be found in Van Schooten's Miscellanies, and Tacquet's Arithmetick, which perchance are not so obvious to every understanding.

For illustration of the Rule proposed, take this Example.

In the year 1668.	Cyclus Solis	25	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$	The Multipliers	$\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}$	4845	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$	Products	$\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}$	121125
	Cyclus Lunæ	16				4200				67200
	Indictio	6				6916				41496
			The Sum of the Products							229821

The which divided by 7980, the remainder is 6381, for the year of the Julian Period; from which subtracting 709, there remains 5622, for the Age of the World, according to Archbishop Usher.

For Demonstration of this Rule we thus argue:

1. Each Multiplier multiplied by its Remainder, is measured or divided by its own Divisor, leaving such a Remainder as is proposed.

For before, each Multiplier was defined to be a Multiplex of its own Divisor, plus an Unit: Wherefore multiplying it by any Remainder, it doth only render it a greater Multiplex in the said Divisor, plus an Unit, multiplied by the Remainder, which is no other than the Remainder it self; but if 0 remain, that Product is destroyed.

2. The Sum of the Products, divided by each respective Divisor, have the Remainder assigned.

For concerning the first Product, it is by the first Section mea-

sur'd