in that Tariffa find the nearest number to 148, which 7 7 7 4 7 6 ( 2 3 7476 (23 is leffer, and you fhall find that number to be 129, 54 87774 1977 Then double the two figures in the Quotient, and they make 46, for which number 46 make a Tariffa (as 548777476 (234 is the Second Tariffa beforegoing); and in that Tariffa 4 find the nearest number lefs than 1977, which you will there find to be 1856, againft which ftands the i 4 8 Digit 4; put 4 in the Quotient, and fubftra& 1856 from 1 29 1977, and the Remainder will be 121, to which bring down the two figures belonging to the fourth Point, namely 74, and then the Number will be 12174, and the work will stand thus, 4 1977 i 2174 4 8 129 Then double the three Figures in the Quotient, and they make 468; for which Number make a Tariffa 548777 47 6 (2342 (as is the Third Tariffa beforegoing), and in that Tariffa find the nearest Number, which is leffer,to 12174, i which I find there to be 9364, against which stands the Digit 2; put 2 in the Quotient, and substract 9364 from 12174, and the Remainder will be 2810, to which bring down the two figures belonging to the laft Point, namely 76, and then the number will be 281076, and then the work will stand thus, 1977 12174 9364 281076 8777476 This done, Double the four figures in the Quotient, and they make 4684, for which number make a T-548777 4 7 6 ( 2 3 4 2 6 riffa, as is the Fourth, in the foregoing Examples; and 4 in it find the nearest number to 281076, and in it you I 48 fhall find the very number to ftand against the Digit 129 6: Wherefore put 6 in the Quotient, and fubftract 281076, from 28 1076, and there will remain nothing; I which fhews that the given number 548777476 is an i exact Square Number, and that 23426 is the Root thereof; and fo the whole Work being ended, it will ftand as followeth. 1 8 5 6 12174 CHAP. XII. A Brief and Compendious Way of Multiplication, wherein the Product is given at once, without feveral Workings, or writing down any other Figures but the Product it felf. I. Example. L 7432 45 7432 45 4. 7432 ET it be requred to multiply 7432, by 45. 524 4.52 می Set down the Numbers as in the Margent. Then, 24 43 2 68 43 524 X 432 6368 440 7432 444 7452 45 33444 0 Example Example 524 1. Multiply as the line directs you; fay 5 times 2 is 10, place o under Unites, and carry I in mind. 2. Say 5 times 3 is 15, and 4 times 2 is 8, which added together, with 1 in mind, makes 24, fet down 4 in the next place on the left hand, and carry 2 in mind. and 4 times 3 is 12, 3. Say 5 times 4 is which added together, with 2 in mind, makes 34, place the odd 4 in the next place, and 4. Say 5 times 7 is 35, and 4 times 4 is 16, which added together, with 3 in mind, makes 54, fet down 4 in the next place, and carry 5 in mind.. Laftly, Say 4 times 7 is 28, and 3 in mind is 33, which fet down, and the work is at an end : And the Product of 7452 multiplied by 45, is 334440: As in the Margent. II. Example, Multiply, 524 by 452. 1. Set the Numbers down, as in the Margent, and fay (as the line directs) 2 times 4 is 8; fet 8 in the place of Unites under 2. 2. Say 2 times 2 is 4, and 3 times 4 is 12, which added together make 16, place 6 under 2265683, and bear 1 in mind. 3. Say 2 times 5 is 10, and 3 times 2 is 6, and 4 times 4 is 16, which added together with 1 in mind, makes 339 place 3 under 42 and bear 3 in mind. 4. Say 3 times 5 is 15, and 4 times 2 is 8, which added together, with 3 in mind, makes 26, place 6 before 3, and bear 2. in mind. Laftly, Say 4 times 5 is 20, and 2 in mind makes 22, which fet down, and the work is at an end: And the Product of 524 multiplied by 432, will be 226368, as in the Margent. СНАР. IN CHAP. XIII. Of Ceres and Virginum. fome Ancient, and in fome late Writers of Arithmetick alfo, I find a Rule called Ceres and Virginum, which teacheth only how to refolve Merry or Sporting Questions, to puzzle young Practitioners in Numbers: And in regard that this Book is fraught with matters Recreative, I shall here infert it, with fome Reflections upon it: And this Rule will be made appear beft, by the Solution of certain Questions as properly belong to it; as fuch as here follow. Question I. A Lady's Caterer bought Eight Birds of two forts, namely Geese and Hens, for 20 Shillings: The Geefe cost 4 fhillings apiece, and the Hens 2 fhillings apiece: How many did he buy of each fort? Ultiply the whole number of Birds, viz. 8, into the Leaft Price, 2 s. the Product will be 16, which take from the whole number of Birds, viz. 20, there will remain 4 for a Dividend, which must be divided by the difference of the Prices of the two forts of Birds, namely, by 2, and the Quotient will be 2 for the number of Geefe; then must the Hens be 6. And fo "There were One and Twenty perfons in Company, fome Men, fome Women, and fome Children; and amongst them they spent 26 Shillings, and fo, that every Man Spent 2 Shillings, every Woman I Shilling, and every Child 6 Pence: How many must there be of each fort? To refolve this or the like, this is the RULE: Multiply the Number of Perfons by the leaft Expence, and take the Product of it from the whole Expence, the rest shall be the Dividend; which divided by the difference between the Greatest and Leaft particular Expence, the Quotient is a Number, which the number of Men (or they which spend most) comes near to, but cannot exceed. Or if the faid Dividend be divided by the Sum of the Greatest and Leaft Expence, the Quotient is a Number, than which the number of men, (or those which spend most) cannot be much less. in this Question, 21 Perfons multiplied by 6 Pence (the least Expence), the Product is 10, which taken from 26 (the whole Expence) the Remainder is 15 for the Dividend; and then taking from there refts for the Divifor, and the Quotient is ", which is fomething more than 10; the number of men therefore must be but 9. 2, Then turn the Dividend and the Divifor both into whole Numbers, by multiplying them into the common Denominator 2, fo they being reduced, will be 31 and 3, as before is feen in the Quotient. G. 2 Mul Multiply the Divifor 3 by 9 (which is the number of Men) the Product is 27, which taken from 31 (which is the reduced Dividend) the remainder is 4 for the Number of Women; and then the Children muft be 8; as in the first Example. But the number of Men may be alfo but 8, which multiplied by the reduced Divifor 3, it produceth 24, which taken from 31, the remainder is 7 for the Women; and then the Children must be 6: As in this fecond Example. Or the number of Men may be 7, which multiplied by 3, produceth 21, which taken from 31, there remains 10 for the Women, and then the Children must be 4, as in this third Example. By what is here delivered in this fecond Queftion, it is plain that it is capable of feveral Solutions, and all of them true. But further, The number of Men may be 10, and not more; for if you put them 11, that multiplied by 3 produceth 33, which is greater than 31 from whence it fhould be taken: But they may be ro, and then there is only One Woman and 10 Children; and this confirms the former part of the Rule. Now for the later part of it: If the Dividend 31, be divided by the Sum of the two Extreams (reduced by doubling as the Dividend is) 4, the Quotient will be 7, and the Men may be 7 as hath been fhewed; but they may be alfo but 6, and fewer they cannot be: As 6 Men, 13 Women, and 2 Children: For if you put the Men Five, that multiplied by 3, produceth 15, which taken from 31, there remains 16 for the Women, and fo there fhould be no Children, which is contrary to the Suppofition. And further, Because the Quotient was 7, the number of Men might be fo (if pure Arithmetical Divifion be only regarded.) And then the number of Women are 7, and the Children 5, as in this Example. 7 Men, There is 900 1. to be diftributed by the Will of a Deceafed Friend, to 30 Perfons of three feveral Qualifications, as Minifters, Lame Soldiers, and Poor Tradefmen; fo that each Minifter may have 60 1. each Soldier 40 1. and each Tradefman 20 1. How many Perfons of each Qualification muft (or may) there be? M Ultiply (according to the Rule) 30, the Number of Perfons, by 20, the leaft Share, the Product is 600, which taken from 900, the whole Legacy, there remains 300 for the Dividend; and 20, the leaft fhare, taken from 60, the greatest share, leaves 40 for a Divifor; and fo the Quotient will be 7, and more the Minifters cannot be: Alfo add 60 and 20 (the greatest and leaft Shares together) their Sum is 80, by which divide 300, and the Quotient will be 3, and much fewer the Minifters cannot be. Not to stand upon the Fractions (in the dividing of Men) That the Minifters cannot (in whole less than 3, may thus be proved. times 40 is 320 which is more than 300, — Numbers) be more than 7 or First, Let them be 8, then 8 from which it should be taken. Secondly, Let them be 2, then 2 times 40 is 80, out of 300 there remains 220, which divided by 20, gives in the Quotient 11 for the Soldiers; fo the Minifters and Soldiers being 13, the Tradesmen muft be 17. |