from the Figure that ftands over it, fetting down the Remainder or Difference underneath it's own place. If the Two Figures chance to be Equal, fet down a Cypher: But if the upper Figure be less than the lower Figure, then you must add 10 to the upper Figure, or mentally call it 10 more than it is, and from that Sum fubtract the lower Figure, fetting down the Remainder (as before directed). Now becaufe the 10 thus added, was fuppofed to be borrowed from the next fuperior place (viz. of Tens) in the upper Figures, therefore you must either call the upper Figure in that place from whence the 10 was borrowed, one lefs than really it is, or elfe (which is all one, and moft ufual) you must call the lower Figure in that place one more than it really is, and then proceed to Subtraction in that place, as in the former; and fo gradually on from one Row of Figures to another until all be done. Let it be required to find the Difference between 6785, and 4572. That is, let 4572 be fubtracted from 6785. Thefe Numbers being placed down, as before directed, will ftand Beginning at the place of Units, take 2 from 5 and there will remain 3 which must be fet down underneath it's own place, and then proceed to the place of Tens, taking 7 from 8, and there will remain 1, to be fet down underneath it's own place; again, at the place of Hundreds, take 5 from 7, and there remains 2, which fet down, as before; laftly, take 4 from 6 and there will remain 2, which being fet down underneath it's own place, the Work is finished, and the Difference fo found will be 2213-6785-4572, as was required. EXAMPLE 2. 7496 5849 The Difference between 5849, and 7496 is required. Having placed the Numbers as in the Margin, begin at the place of Units (as before) and fay 9 from 6 cannot be, but 9 from 16 and there remains 7, to be fet down under it's own place; next proceed to the place of Tens, where you must now pay the 10 that was borrowed to make the 6, 16, by accounting the upper Figure 9 in that place one lefs than it is, faying 4 from 8 and there remains 4, or elfe (which is the most practifed) fay I borrowed and 4 is 5 1647 from from 9 and there remains 4, to be fet down under it's own place (as before); again, at the place of Hundreds, fay 8 from 4 that cannot be, but 8 from 14 there will remain 6 to be fet down; and here I have borrowed 10 (as before) which must be paid in the same manner as the other 10 was, viz. either by calling the 7 in the upper Rank but 6, faying 5 from 6 there remains I, or elfe by faying I borrowed and 5 is 6 from 7 and there remains I, which being fet down under it's own place all is done, and the Difference required will be 1647-7496–5849. EXAMPLE 3. Take 741068 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 Subtrahend, or Number to be fubtracted, is fuppofed to be a Part of that Whole; confequently, 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 must be taken to difcover and correct the Error. EXAMPLE. The Sum which is equal to the Number from whence Subtraction was made. Or Or from the abovefaid Reafon, it will be easy to conceive how to prove the Truth of Subtraction by Subtraction. 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 fo 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 fo 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 inftance; 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 itfelf, 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. 3×3 Multiplication Table. 914×4=16|5×5=25|6×6=36|7x7=49]8x8=64 3x4 12 4x5 20 5x6=306x7=42 7x8=5618x9=72 3x5=154x6-245x7=35 6x8-48 7x9=639x9=81 3×6=184x7=285x8=40 6x9=54| 3x7=214x8=325×9=45| 3x8=24 4×9=36| 3×9=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=3×4, 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 Business of Multiplication; which will be found very eafy, if the following Rule (and Examples) be carefully obferved. RULE. Always begin with that Figure which flands in the Units place of the Multiplier, and with it multiply the Figure which fands in 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 Beginning at the Units place, fay 3 times 3 is 9, which, be cause 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, Thus { 856g 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 7 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 5 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 |