Chap. 2. from 9 and there remains 4, to be fet down under it's own place EXAMPLE 3. Remains 89408 By this Example you may perceive that Cyphers in the Subtrahend, viz. in the Numbers to be fubtracted, do not diminish the Number from whence Subtraction is made. See Page 4. These Three Examples, I prefume, may be fufficient to fhew the young Learner the Method of Subtracting whole Numbers; as for the Reafon thereof it is the fame with that of Addition, Page 10, viz. of the Whole being Equal to all it's Parts taken together. That is, in this Rule the Number from which Subtraction is required to be made, is understood to be the Whole, and the Subtrabend, or Number to be fubtracted, is fuppofed to be a Part of that Whole; confequéntly, if that Part be taken from the Whole, the Remainder will be the other part. ..From hence is deduced the commmon Method of proving Subtraction, by adding together the Subtrahend and the Remainder. For if the Sum of thofe Two (which are here called Parts) be equal to the Number from whence Subtraction was made (which is here called the Whole) then the Work is right; if not, care muft be taken to difcover and correct the Error. EXAMPLE. From 59435 Take 47608 Add The Sum which is equal to the Number from whence Subtraction was made. Or Or from the abovefaid Reason, it will be eafy to conceive how to prove the Truth of Subtraction by Subtraction. W Sect. 4. Of Multiplication. Multiplication is a Rule by which any given Number may be fpeedily increased, according to any propofed Number of Times. That is, One Number is faid to Multiply another, when the Number multiplied is so often added to itself, as there are Units in the Number multiplying; and another Number is produced, (Euclid. 7. Def. 15.) To perform Multiplication, there is required two given Numbers, called Factors. The First is the Number to be multiplied, which is generally put the greater of the Two Numbers, and is commonly called the Multiplicand. The other is that Number by which the Firft is to be multiplied, and is ufually called the Multiplicator or Multiplier; and this denotes the Number of Times that the Multiplicand is required to be added to itself. For so many Units as are contained in the Multiplier, fo many times will the Multiplicand be really added to itself (as per Euclid above). And from thence will arife a Third Number, called the Product. But in Geometrical Operations it is called the Rectangle or Plain. For instance; fuppofe it were required to increase 6 four times, that is, to multiply 6 into or with 4. These two Numbers are to be fet (or placed) down as in Addition or Subtraction, Thus Product 24 viz. 4 times 6 is 24, as plainly appears by Addition, viz. By fetting down 6 four times, and then adding them together into one Sum, 116 26 Add Thus 3 6 24 From hence it is evident that Multiplication is only a Concife or Compendious Way of ad 'ding any given Number to itself, fo often as any Number of Times may be propofed. Before any Operation can be readily performed in Multiplication, the feveral Products of the fingle Figures one into another must be perfectly learned by Heart, viz. That 2 times 2 is 4, that 3 times 3 is 9, and 3 times 6 is 18, &c. According as they are expreffed in the following Table; wherein I have omitted multiplying with 2, it being fo very eafy that any one may do it. Multiplication Table. 19x9=81 |3×3=914×4=1615×5=25|6×6=36|7×7=49]8×8=64| 3x4 12 4x5=2015×6=30[6x7=42 7x8=5618x 3x5=15 4x6 24 5x7=35 6x8-487x9=63 3×6=184x7=285x8=40 6x9=54 3x7=214x8=32 5x9=45 3x8=244x9=36 3x9=271 I think it needlefs to give any Explanation of this Table for if the Signs and their Significations be well understood, (vide page 5) it muft needs be eafy. Only this may be noted, that 4x3 3x4, or 7x5=5×7, &c. That is, 3 times 4 is the fame with 4 times 3, or 5 times 7 is the fame with 7 times 5, &c. The like must be understood of all the reft in the Table. And when all these fingle Products are fo perfectly learned by Heart, as to be faid without paufing; you may then proceed (but not till then) to the Bufinefs of Multiplication; which will be found very eafy, if the following Rule (and Examples) be carefully obferved. RULE Always begin with that Figure which fands in the Units place of the Multiplier, and with it multiply the Figure, which flands. in the Units place of the Multiplicand; if their Product be less than Ten, fet it down underneath it's own place of Units, and proceed to the next Figure of the Multiplicand. But if their Product be above Ten (or Tens) then fet down the Overplus only (or odd Figure, as in Addition) and bear (or carry) the faid Ten or Tens in mind until you have multiplied the next Figure of the Multiplicand, with the fame Figure of the Multiplier; then to their Product add the Ten or Tens carried in mind, fetting down the Overplus of their Sum above the Tens, as before: and fo proceed on in the very fame manner, until all the Figures of the Multiplicand are multiplied with that Figure of the Multiplier. EXAMPLE I. Suppose it were required to multiply 3213 into or with 3. Product 9639 1 Beginning at the Units place, fay 3 times 3 is 9, which, because it is less than Ten, fet down underneath it's own own place, and proceed to the next place of Tens, faying 3 times 1 is 3, which fet down underneath it's own place; then to the next place, viz. of Hundreds, faying 3 times 2 is 6, which fet down, as before; laftly, at the place of Thousands, fay 3 times 3 is 9, which being fet down underneath it's own place, the Operation is finished; and the true Product is 9639=3213×3, as was required. EXAMPLE 2. Let it be required to multiply 8569 into 8. Set down these Numbers as before, 7 68552 Beginning at the Units place, fay 8 times 9 is 72, fet down the 2 underneath it's own place of Units, and bear the 70, or Tens in mind, and proceed to the next Figure of the Multiplicand (at which place the 7 Tens are only 7) faying 8 times" 6 is 48, and the 7 carried in mind is 55; fet down the odd underneath it's own place of Tens, and carry the 50 (which is really 500) to the next place (viz. of Hundreds) at which place it is only 5, where fay, 8 times 5 is 40, and the 5 carried in mind is 45; fet down the 5 underneath it's own place, and carry the 40 or 4 Tens (which is really 4000) to the ་་ . next next place, viz. of Thousands, faving, 8 times 8 is 64, and 4 carried in mind is 68. (Now this being the laft Place or Figure to be multiplied) Set down the whole Product 68, and the Work is done. So that, 8569x868552, the Product required. Now the Reafon of this and all other the like Operations, may be eafily conceived from this which follows. 8 5 6 8 { The fame Factors as before. Here 8 times 9 is 72, as before, because the 9 72 {ftands in the Units place. Now here it is not really 8 times 6=48, but it is 4808 times 60-480, because the 6 ftands in the place of Tens. And here it is not 8 times 5=40, but it is really Jololo 8 times 500-4000, because the 5 ftands in the place of Hundreds. Laftly, because the 8 in the Multiplicand stands 64000 in the place of the Thousands, it is therefore 8 times 8000 64000, and not 8 times 8=64. S The Sum of the particular Products, which gives 68552 {the true Product, as before. By what hath been already faid, with a little Confideration bad to the Examples, I prefume the Learner may easily underftand how to multiply whole Numbers with any fingle Figure. And when it is required to multiply with more than one; then fo many Figures as there are in the Multiplier, fo many particular Products there must be. That is, all the Figures of the Multiplicand, must be multiplied with every fingle Figure of the Multiplier, as if there were but one fingle Figure: and the Sum of all thofe particular Produals, will be the true Product required. But in those Operations, great Care must be taken in fetting down the particular Products (which arife by each multiplying Figure) in their proper places. Which will be eafily done, if the following Directions be carefully observed. Always place the first Figure (or Cypher) of every Viz.particular Product, directly underneath the multiplying Figure. Or thus: The First Figure (or Cypher) of the Second particular Product must stand directly under the fecond Figure (or place) of the First Product; and the First Figure (or Cypher) of the Third D particular |