VIII. Of the Digit 8. To multiply any number by 8, double the number given, and subftra&t that double from the given number, (a Cypher being added to the given number), that Remainder fhall be the Product of the given number, multiplied by 8. So if you would multiply 365 by 8, the double of 365 is 730, a Cypher added to 365 is 3650, from which fubftract the double of 365, namely, 730, and the remainder will be 2920, the Product of 365 multiplied by 8.-In like manner, if you would divide any number by 8, take half the number given fucceffively three times, the third half fhall be the Quotient of the given number divided by 8. So if 2920 were a number given to be divided by 8, the half of it is 1460, the half of that is 730, and the half of that is 365, which is the Quotient of 2920 divided by 8. And this Digit 8 hath another Property in it, for no Square Number can terminate with it. IX. Of the Digit 9. On the To multiply any number by 9, add a Cypher to the given number, and from that number fubftract the given number, the Remainder fhall be the Produ&. So if you would multiply 365 by 9, add a Cypher to it, it makes 3650, from which fubftract 365, and the remainder will be 3285, equal to the Product of 365 being multiplied by 9. contrary, if you would divide any number by 9, take the third the given number twice fucceffively, the fecond remainder fhall be the Quotient: So if you would divide 3285 by 9, one third of 3285 is 1095, and one third of 1095, is 365, which is the Quotient of 3285 divided by 9. III. Obfervation. part of This Digit 9 hath a Privilege above all the other Digits; for if you take any number, the Nines taken out of the grofs fum of that number, or of all the parts thereof feverally, the remaining Digit will be fill the fame: Example; In the number 36 there are 4 Nines contained therein; and fo if you multiply 9 by 4, it will produce 36; and if you take the Nines out of 36, they are four alfo. In like manner, if you take the Nines out of this number 248, it is all one as if fhould take the Nines out of the fimple figures 2, 4, and 8, you which do make 14, from which 9 being taken, there will remain the Digit 5; and alfo if you divide 248 by 9, the Quotient will be 27, which fhews that there are 27 times 9 in 248, and the Digit 5 remaining. X. How the Nine Digits may be difpofed in fuch Order, that the Number 15 shall be accounted in Rank, File, and Diagonally, 32 times. In the Nine Digits there are Four Even, and Five Odd fet the four Even Digits, 2, 4, 6, and 8, at the four corners of a Square, ; now if you and 5 (which is the middlemoft of the five odd Digits) in the Center or or middle of the Square, as is done in the Margent then between every two even Digits put fuch an odd Digit as fhall make the number in that line both in Rank and File 15; as between 2 and 4, put 9, which together make 15, and fo the reft; the which 9 Digits thus difpofed, 15 may be accounted in Rank, File, and Diagonally, 32 times: Example, There is a notable harmony between fome Numbers, as particularly in these two, 220 and 284: For the addition of the Aliquot Parts of the one, do make up the Sum of the other: So XI. Of the Article Numbers, 10, 100, 1000. If you would multiply any number by 10, 100, or 1000, &c. it is done by adding of fo many Cyphers towards the right hand of the given number, as there are Cyphers in the Article number by which you are to multiply: So On the contrary, any number may be divided by an Article Number, by cutting off from the given number fo many figures towards the right hand, as there are Cyphers in the Article Number by which you are to divide: So XI. Of the Numbers 11, 12, 13, 14, 15, 16, 17, 18, and 19. you 86 If would multiply any number by any of these numbers that hath a Unite in the Tenth place, obferve the method following, where you may set down the Product in one fingle line; as in the Margent ; 786 12 9432 fet Let 786 be a number given to be multiplied by 12; them down as in the Margent, and fay, Two times 6 is 12; fet down 2, and keep 1 in mind: -Then, 2 times 8 is 16, and I in mind is 17, and 8 over the 1, is 23; fet down 3, and bear 2 in mind: Then, 2 times 7 is 14, and 2 in mind is 16, and 8 over 1 is 24: fet down 4, and bear 2 in mind; which 2 add to the last figure 7, it makes 9; so that 786 multiplied by 12, produces 9432. And fo of others; as in Examples, XII. Of Square and Cube Numbers. It is obferved before, that no Square Number, can terminate in any of these four Digits 2, 3, 7, or 8: Notwithstanding, there are many fubtle Properties in Square Numbers: As, 1. If from an Unite you do fucceffively add the next odd digit number, the fum of those two Numbers fhall produce Square Numbers: -As, if to you add 3, the next odd number, the Sum is 4, a Square Number; to which if add the next odd Digit, it makes 9, a Square Number; to which if you add 7, the next odd Digit, it makes 16, a Square Number; and fo on, as in this little Table. you 5, 3 4? 4 5 9 9 16 7 16 9 25 For 25 and 11 makes 3 36 13 49 80 6400 A Square Number. 2. If a Series of Eleven Square Numbers be orderly fet one under another, the terminating figure of the firft, and the terminating figure of the laft fhall be the fame. And fo if the terminating figure of the firft number be a Cypher, the second shall be 1, the third 4, the fourth 9, the fifth 6, the fixth (or middlemoft) 5, the feventh 6, the eighth 9, the ninth 4, the tenth 1, the eleventh o, as in this Table. So likewife if Cube Numbers be fuc ceffively added from Unity, those Numbers shall alfo be Square Numbers. 16561 6724 983 6889 84 7056 85 7225 86 7396 87 7569 |