INTRODUCTION. M Y Defign by this Introduction, is to put my Reader in Mind of fome Things in Arithmetick, which are not commonly taken Notice of in Books on that Subject; viz. that, I. There have been fome Properties attributed to Numbers, which are only the Refult of the Method of Notation now in ufe: As for Inftance, 'Tis commonly faid and allowed for Truth in the known common Way of proving Multiplication and Divifion by cafting away the Nines, that Propofition. The Number 9 has that Property, that any Number whatfoever Divided by it, fhall leave the fame Remainder, as if the Sum of the Figures compofing the faid Number were Di vided by 9. I fay, this is true in the common Method of Notation: But it is to the faid Method only, that this Fact is owing, as will appear from the following Demonftration of the Propofition above recited, and the Scholium annexed, the former of which I had from the Reverend Mr. Chinery of Middleton, near Corke, in the following Expreffions. For the clearer Demonftration of this Propofition, I fhall premife two felf evident Lemma's. Lemma Lemma 1. The Local Value of any Figure is equal to the Rectangle of its Simple Value, and the Denomination of its place: Thus 66 x 1, 60 = 6 × 10, 6006 × 100, 8r. x Lemma 2. X To Multiply or Divide one Number by an other, is in Conclufion the fame Thing with Multiplying or Dividing respectively the Sum of the Parts of the former by the latter : Thus, because 109+1; therefore 6 x 10=6x9 + 6 × 1; and, because 100=99 + 1; therefore 6 × 1c0=6 × 99f6× 1; &c. And fo 10 2/2 + 1/2/3, and &c. 100 6 99 I +능, Thefe Lemmi's premis'd, I proceed to the Demonftration of the Propos'd Propofition. confequently 3467 9 per 6 × 9+ 4 × 99 + 3 × 999 9 + 7 × 1 - - 6 × 1 + 4×1-1-3×1 (per Lemma 2.); but 6×9, 9 4× 99, 3 × 999 being Multiples of 9, their Sum must also be a Multiple thereof, and confequently Divided by it will leave no Remainder; therefore the Remainder of 3467 Divided by 9 will be equal to the Remainder of 7x1+6x I + 4x1 +3×1 Divided by 9; i. e. equal to the Remainder of 7+6 +4+3 Divided by 9. The like Demonstration may evidently be applyed as well to any other Number as to 3467. Q. E. D. See the following Notation. Scholium Scholium. From the foregoing Demonftration it is evident that any Num ber defign'd by any Method of Notation after the common one, being Divided by : 10- I thall leave the fame Remainder with that which wou'd remain if the Sum of the Figures com pofing that Number were likewife Divided by : 10 - I: : Thus, fuppofing the Method of Notation, I wou'd ufe defignable by only the five first Figures of them which are com monly us'd; viz. that 0, 1, 2, 3, 4 thall defign, as ufual, nɔthing, one, two, three, four; but 10, 11, 12, 13, 14 fhall defign five, fix, feven, eight, nine; and 20, 21, 22, 23; &c. fhall defign ten, eleven, twelve, thirteen; &c. And Suppofe 242 (or twice twenty five more four times five more two, viz. feventy two) be propos'd to be Divided by 101 which, according to this Method of Notation, : is 4: Then 4) 242 (33 three times five more three eighteen 22 o Remainder. is the Quotient: But the Remainder is o: And 2 +4 +2 (or eight) 13 being Divided by 4 is 2 exactly without any Remainder; Or 422 (or one hundred and twelve)÷4 103 or twenty eight juft. &c. II. In fome Operations in Multiplication and Divifion, it will be requifite to take Notice of the following Definitions; viz. That every Multiplication is a Proportion in which I (or Unity) is to the Multiplicator, as the Multiplicand is to the Product fought. Likewife that every Division is a Proportion in which the Divifor is to the Dividend, as 1 is to the Quotient fought: So; if it were required to Multiply 4s. 6d. by 4s. 6d. Firft, Reducing it to a Proportion, I.. 4 s. 6d. :: 4s. 6d. ..the fourth Number fought. Now, it being uncertain what Name the first Term 1 is off (viz. whether it be a Pound, Shilling, or a Penny) 'tis therefore uncertain what the Product or fourth Number fought is: Thus, if the firft Term 1 be 17. then the Product is 2216 d. =124% 3d. If the faid Term I be I s. then the Product is 2214 d. = 243d. = 20; s. And, if the faid Term be 1 d. then the Product, or fourth Number fought, in the foregoing Proportion, is 2916 d. 243 s. 121. 3 s. ood. =243 = 3s. 2916 The like is to be understood of Division. a 2 3 Tho III. Tho' the Rule of Double Pofition will not extend to Questions producing Quadratick or Superior Equations, nor even to fome fimple ones; yet, in Cafes of Approximations, it may be us'd to good Purpofe in fome Abftrufe Questions. By way of Example take the following Queftion and Solution. Two Men D and E made a Stock of 1657. D's Money was in Company for 3 Months, and E's was in for 2 Months. When they fhar'd Stock and Gain, D receiv'd for his Share 407. and E 1531. I demand D's Stock? By the Theorem in Question 3. Chap. 2. Part XI. of the Ift Book of this Treatife, you'll find D's Stock =√48169 18732,47437208 +1. But my prefent Defign is to fhew how to approach to this Stock of D's by the Help of Double Pofition. Thus, 193, and 193-165287. First 40+153 whole Gain. Next, Suppofing firft D's their Stock 207. I find it by much too little; again, Suppofing it 30. I find it ftill too little; but fuppofing it 337. I find it fomewhat too much; therefore I fuppofe it 321. then E's Stock will be 1337. = = 32 40 39.4254 5745 Error: Suppofe D's Stock to be 32.5 7. then E's Stock will be 132.5% 3 97.5 2 265.0 97.5 362.5 362.5.. 281.::97.5 .. 7.5310 + % E 40.0310 40 0310 Error? And 19.66325.6055=32.47447. — 327. 09 s. 5 3 d. +D's Stock Anfwer. But if this Answer be not thought near enough the Truth, Suppofe, again D's Stock 32.47441. and, then find the Error by the foregoing Method, and it will be found to consist in Excefs: And, for a fecond Suppofition, put D's Stock=32.4743% and proceed, in the foregoing Manner, in finding the Error, which will be found to confift in Defect. Then by these Po fitions and Errors (the Errors being purfued to a good many places in Decimals) the Anfwer will be found very near the Truth, only pursuing this one Operation by Double Pofition. ERRATA. |