Page:Philosophical Transactions - Volume 002.djvu/170

 by its own Divisor, leaving the remainder proposed; and if we add the rest of the Products thereto, we only add a Multiplex of its own Divisor, which in Division enlargeth the Quote, but not the Remainder.

Particularly the second Multiplier is 28 + 15 + 10 + Remainder, all which is but a Multiplex of 28.

And so the third Product is 28 + 19 + 13 + Remainder.

And what hath been said concerning the sum of the Products, being divided by the first Divisor, and leaving the Remainder thereto assign'd, 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 concerning the said Sum, 'tis evident by the second hereof, that it is no other than the first Product; increas'd by adding a just Multiplex of the first Divisor, that thereby we did only enlarge the Quote, not alter the Remainder. By the like reason, the subtracting a just Multiplex thereof, doth only alter the Quote, not the Remainder; but the Solid of all three Divisors, multiplied here by the Quote, as there by the Remainder, is no other than a just Multiplex of the first Divisor. Wherefore the Remainder, after this Division is perform'd, 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.

in the Method hitherto deliver'd, we requir'd the Divisors be Primitive to each other; so, if we take the Problem as generally proposed, in the Preface to Helvicus his Chronologia, we are told, common Arithmetick fails in the solution thereof, and Tacquet denies it to be performable by the Regula Falsi, and being unlimited, we must do it by Tryals. Wherefore,

When any two Divisors with their Remainders are proposed, try the Multiplices of one of them, increased by its Remainder, and divide by the other: If you find such Remainders as are not for the purpose, and that they are repeated, the Problem is impossible.

Example.