found, then all is right; if not, care must be taken to discover and correct the Error. That any Number may be diminished, by taking another less Pumber from it. Subtraction is that Rule by which one Number is deducted or taken out of another, that fo the Remainder, Difference, or Excess may be known. As 6 taken out of 9, there remains 3. This 3 is alfo the Difference betwixt 6 and 9, or it is the Excefs of 9 above 6. Therefore the Number (or Sum) out of which Subtraction is required to be made, must be greater than (or at least equal to) the Subtrahend or Number to be fubtracted. Note, This Rule is the Converfe or Direct contrary to Addition. And here the fame Caution that was given in Addition, of placing Figures directly under thofe of the fame Value, viz. Units under Units, Tens under Tens, and Hundreds under Hundreds, &c. must be carefully obferved; alfo underneath the loweft Rank there must be drawn a Line (as before in Addition) to feparate the given Numbers from their Difference when it is found. Then having placed the leffer Number under the greater, the Operation may be thus performed. RULE. Begin at the Right Hand Figure or place of Units (as in Addition) and take or fubtract the lower Figure in that place C 2 from from the Figure that ftands over it, fetting down the Remainder or Difference underneath its 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 because the 10 thus added, was supposed to be borrowed from the next superior 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 less than really it is, or elje (which is all one, and most usual) 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 wrote down, as before directed, will ftand 2213 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 1 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 fame manner as the other 10 was, viz. either by calling the 7 in the upper Rank but 6, faying 5 from 6 there remains 1, or elfe by faying I borrowed and 5 is 6 from 7 and there remains 1, which being fet down under it's own place, all is done, and the Difference required will be 1647-7496—5849. 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 Reason 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 underftood to be the Whole, and the Subtrahend, or Number to be fubtracted, is fupposed 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 common 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 discover and correct the Error. 11827 S The Sum which is equal to the Number from 59435 whence Subtraction was made. Or Or from the abovefaid Reafon, it will be eafy to conceive how to prove the Truth of Subtraction by Subtraction. there will remain 47608 the very Number which was required to be first Subtracted. From 75643 Take 9000 Remains 66643 From 7000000 Take 986432 Remains 6013568 Sect. 4. Of Multiplication. Multiplication is a Rule by which any given Number may be speedily increafed, 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 itfelf, as there are Units in the Number multiplying; and another Number is produced, (Euclid. 7. Def. 15.) To perform Multiplication, there are 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 itfelf. 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 of the two Numbers. For inftance; fuppofe it were required to increase 6 four times, that is, to multiply 6 into or with 4. Thefe 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, Thus From hence it is evident that Multiplication is only a Compendious Way of adding any 2 -Add 5 24 given Number to itself, as many Times as may be propofed. Before any Operation can be readily performed in Multiplication, the feveral Products of two fingle Figures 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 1, it being so very eafy that any one may do it. Multiplication Table. 13×3=914×4=16|5×5=2:16×6=36|7×7=49|8×8=64| 3×4=124×5=2c5×6=3c6x7=427x8-568x972 3x5=154×6=245x7=356x8=487x9=639x9=81 3×6=184×7=285x8=4c|6×9=54] 13x9=271 I think it unneceflary to give any Explanation of this Table; for if the Signs and their Significations be well understood, (vide page 5) it must needs be eafy. Only this may be noted, that 4x3=3×4, or 7×5=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 muft be understood of all the reft in the Table. And when all these fingle Products are so 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 stands |