( 4 °° )

4*40

4200 > is Equal to the Solid or Product of6916 )

„ )l9, 15. 17 -t >28, 15, io.>8, »9, iz.

For the Demonstration of the Theorem Propos’d we thus argue.

I. tach Maltiplier Mukipliedby tts Remainder, ts Meafurtd or Divided,by its own Divisor, leaving such a Remainder as is Propojed. For betöre,each Multiplter was Dehn d to be a Multiplex ot ics own Divisor , plus anUnit, Whereforc Multiplyivg it by arsy Remainder, it doth only renderit a greater Muliiplex in the said Divisor, plus an Unit Multiplyedby the Re-mainder; which is no other than the Remainder it leis: but if o Remain ,that VroduB is Destroyed.

2» The Sum of the Products, Divided by each refpeffi-ve Divisor, have theRemainder i.ffigntd. For concerning the First Prcduff, it is by the FirstScction Mtaiur’d by its own Divisor, leaving the Remainder proposed ;and if we Add the rest ofthe Produth thereto, we only Add a Muliiplex ofits own Divisor , which in Division enlargeth the fsuote , but not theRemainder. Particularly the Second Multiplter is 28 x 15 x 10 x Re-tnaindir, ali which is but a Multiplex of 28. And so the 3 d. Prodidi is28 x 19 x 13 x Remainder. And what hath been said concerning theSum of the Prodatis, being Divided by the First Divisor, and leaving theRemainder thereto Aslign’d, may be said ot Each respectively.

z. The Sum ef Products, Divided by the Solid ofthe 3 Divisor«, leaves aRemainder fo £$uahfy''d as the fatJ Sum. For concerning the said Sum,'tis Evident by the Second her eos that it is no other than the First ProduB ,Increafed by addirg a just Muhiplex of the First Divisor, that thereby wedid only Enlarge the not Aster the Remainder ; By the likeReason,

the Subtrailing a just Multiplex rhereot, doth only Alter the Qiiote, not theRemainder; ButtheSv/iJ of all z Divisors , Mulciplied here by the §sucte,as there by the Remainder, is no other than a just Multiplex of the First Di -vifir. W here fore the Remainder, aster this Division is performV, is of thefame Quality as the Sum of the Produis : and Divided by the First Divis or,leaves the Remainde' proper therctoi And thehke may be said concerningeach Divsr.

As in the Method hirherto deliver’d, we requir’d the Divisors be Primi-tive to each other: so, if we take the Problem as Generally Proposed, in thePreface to Htlvicus his Chrondcgta, we are told, Common ArithmeUck Fatisin the Solurion thereof; and Tacquet Denies it to be Performahle by theRegti'a Fälst, and being Unlimiced, we must do it by Tryals. Whcrefore,Wbinany Two Divisors ivuh their Remaindtrs are Proposed, Try the Multi-plices of otte of thtm , Increafed bf its Remainder, and Divide by tbe other : Ifyou find steh Rematnders as arent for the purpofe, and that tbey are Re-peated, the Problem is Impoiitble.

Example. Divisors. ^ ^Remainders

The