26. The Sum of any two Digits is found after this Manner by the above Table, viz. Always find one of the Numbers on the Left-hand of the Table, and the other on the Top, and the Number ftanding in the Place, where the Rows meet, will be the Sum required. E. g. Suppose we wanted to know the Sum of 846; find 8 on the Left, and then against it, in the fixth Column, we fhall have 14, the required Number.

27. By committing the above-mentioned Table to Memory, we fhall readily know the Sum of any two Digits; and then be qualified to make Ufe of a much better Way of adding large Numbers, viz. First, take Care to place the Numbers to be added, one under another; fo, that Units may ftand under Units, Tens under Tens, Hundreds under Hundreds, &c. Then add up the Row of Units; and if the Sum be more than Ten, or two Tens, &c. write down the Overplus, and carry the Tens to the next Row, as so many Units; which (Row) add up as you did the firft, and carry the Tens of this Row, as fo many Units, to the third, or Row of Hundreds; and thus proceed 'till all the Rows are added up; and then the Tens of the last Row must be

placed

placed to the Left-hand of all the other Figures of the Sum, and then the Whole will be the Total Sum. 28. Example. What is the Sum of 5768+123 +879?

The Numbers being rightly placed will ftand thus:

57 68.

1.2.3.

879

6770 Sum of the Whole.

Operation. Add up the first Row, thus: Say (mentally) 9+3=12=(by Art. 9.) 2 above 10; make a Dot (.) for the 10, and say 2+8=10, for which make a Dot (.): Put down o under the firft Row; and then, looking on the Sum, we fee two Dots, fignifying two Tens, to be carried to the fecond Row: Therefore we fay, 2 (we carry) +7=9, +2=11, = 10+1; for the 10 make a Dot; then fay 1+6=7, which put down; then carrying 1 for the Dot, to the third Column, we fay, i (we carry) +8=9, +110, for which make a Dot; then 7 is 7, which place down: Laftly, in the fourth Row we fay, I (the Carriage from the third Column) +5=6; which, being put down, compleats the whole Sum.

29. After the young Student has been fome Time converfant in Addition, he will be able to add up, without dotting, more expeditiously, thus: In the Example, in the laft Article, fay 9+3=12, +8= 20; under the firft Row put o; and because the 2 in 20 is 2 Tens, (or, which is the fame, the 2 ftands in the Place of Tens, by Art. 9.) the said 2 must be carried to the second Row, or Row of Tens; faying 2 (you carry)+7=9, +2=11, +6=17; (but this 17, being 17 Tens, is really 170) . under the fecond Row we must put 7, and carry the 1 (viz. 100) to the Row of Hundreds, faying 1 (you carry) +8 =9, +1=10, +7=17; (but this being the Sum of the Row of Hundreds, is really 1700). under the third Row we must write 7, and carry the 1 (viz. 1000) to the Row of Thoufands, faying I (you carry)

24.

carry)+5=6; which being the Sum of the Row of Thousands, the 6 must be written in the fourth Place; which compleats the Sum, viz. 6770. From what has been faid in this Article,, the Reafon of carrying the Tens of one Row, as Units to the next higher Row, muft appear plain. Or the Reafon of this Method of performing Addition may be fhewn in a more general Manner, thus: It is plain, that, in an Operation worked by Article 27, we firft add up the Row of Units, which Sum we put down in the Place of Units, if it doth not amount to ro; but if it be 10, or more, we only put down the Excess above 10, or a compleat Number of Tens, in the Place of Units, and carry the Tens to the next Row, and add up that, &c. Now it is evident, that as we put down the Exceffes above any Number of Tens, and carry the Tens as fo many Units to the next Left-hand Row, &c. to the End, that the Sum fo found must be that required; because, that by the Nature of our Notation, 10 of 1 Row is 1 in the next Lefthand Row; and therefore, by that Method, we have taken all the Parts together, which muft* be equal to the Whole.

30. When the Columns, or Rows, to be added up, are very long, it will be proper to part them into feveral Parcels, fo that the Sum of each Row of a Parcel may not exceed an Hundred, (or there be not more than 10 Figures in any Row or Parcel ;) then, having found the Sum of each Parcel, add thefe Sums together for the Total Sum.

The

31. The best Way of proving Addition is by adding the Rows downward; by which, if they bring out the fame as when added upward, we may reasonably conclude we are right.

32. As to Literal, or Algebraical Addition, all that is neceffary to be obferved here, is, that the Quantities are added by collecting them together with their proper Signs: Thus, x added to y is x+y; x and y is x-y; alfo

x and -y is

x

-y; all this is plain from the

bare Definition of Addition. We

232

121

317

213

25 34

710

310

728
631

841

231

978

891

701

301

6322

shall only further obferve, that Total Sum 8856 when the Quantities are alike,

i. e. the Letters are all of one

Sort, the Expreffion may be fhortened, by adding them together after the fame Manner, as in common. Additions; fo, ata=2a; 3a and 5a=3a+5a=8a; alfo 3x and -2x=3x-2x=x.

33.

CHAP. IV.

Of SUBTRACTION.

SUBT

UBTRACTION, (Subtractio, from the Verb fubtrabo, Lat.) fhews to take one Number from another, and is only the Reverse of Addition.

34. Hence,

34. Hence, the Number to be fubtracted may be equal, but cannot be greater than, the Number from which it is to be taken.

35. Grant, that any Number may be leffened, by taking a leffer, or equal Number, from it. (See Art. 38.)

36. If from equal Things equal Things be taken away, the Remainders will be equal.

37. The Number from which another is to be taken, we call the Subducend (from the Latin Verb fubduco); the Number to be fubtracted the Minorand (Minor, Lat.); and their Difference, the Remainder (from the Verb remaneo, Lat.)

38. Before Children can be taught to fubtract one Number from another, they must be able to tell readily, without ftudying, the Difference between any two Digits; for which Purpose the Table in Addition will be useful; for by it the Difference between any Digit and Number, less than zo, is given by Infpection only; thus, find the Minorand-Digit in the Left-hand vertical Column, and look against it in the Table for the Subducend; then, directly over it, on the Top of the Table, will be the required Difference or Remainder. Example. Let it be required to find the Difference between 4 and 9.Here against 4 (in the Index Column) we find 9 in the Table, and directly over it, in the Top Column, is 5, their Difference.

39. To fubtract one Number from another.

Having placed Units under Units, and Tens un der Tens, &c. as we did in Addition; take each Digit (beginning with the Units Place) of the Minorand out of its correfpondent Digit of the Subducend, and put the Difference under; but if any Digit of the Minorand is greater than its correfpondent Digit of the Subducend, you may fubtract that Digit

* What we here call the Subducend, fome Authors call the Com pound Number, others the Subtrahend; and what we call the Mino rand, fome call the Subtrahend, others the Subtractor.