Page:Philosophical Transactions - Volume 002.djvu/168

 and Records) can never happen alike. And these Remarques being given, the year of the Julian Period is by the former Rule infallibly found.

This Period is used by the said Archbishop in his Annals, and is by him accounted to exceed the Age of the World 709 years. Those that desire further satisfaction about Æra's, Epocha's, and Periods, may repair to many Authors, and among them to Gregory's Posthuma in English, Helvici Chronologia, Ægidii Strauchii Breviarium Chronologicum, who is one of the latest Authors.

Now as to the Problem it self, it may be thus proposed:

Any number of Divisors, together with their Remainders after Division, being proposed, to find the Dividend.

This thus generally proposed is no new Problem, and was resolved long since, by John Geysius, by the help of particular Multipliers, such as those above-mentioned, and publish'd by Alstedius in his Encyclopedia in Ann. 1630. and by Van Schooten his Miscellanies.

We shall clear up, what Authors have omitted concerning the Definition and Demonstration of such fixed Multipliers, &c. And therefore say, that each Multiplier is relative to the Divisor, to which it belongs, and thus define it;

It is such a Number, as divided by the rest of the Divisors, or their Product, the Remainder is 0; but divided by its own Divisor, the Remainder is an Unit.

We require the Divisors proposed to be Primitive each to other, i.e. that no two or more of them can be reduced to lesser terms by any common Divisor: For if so, the Question may be possible in it self, but not revolvable by help of such Multipliers, such being impossible to be found. The reason is, because the Product of an Odd and Even Number is always Even, and that divided by an Even Number, leaves either Nothing, or an Even Number.

Divisors The Multipliers relative thereto are

The Definition affords light enough for the discovery of these Numbers. To instance in the first: The Product of 19 and 15 is