seems to be that of omitting any one of the horizontal rows of figures in a second operation, and afterwards adding it to the result thus obtained, as in the following example: where 28768 is the sum: and omitting the first horizontal row of figures, we find the sum of the rest to be 19410, and to this the row 9358 omitted being now added produces 28768 the entire sum as before: whence we infer with some degree of probability, that the addition is correct: and this probability may be still further increased by repeating the operation, with the omission of any other horizontal row of figures different from the one already left out. 24. We will now place before the student a few examples for practice, some of which are properly arranged for the immediate performance of the operation, and the rest are to be first adapted for that purpose. (13) Add together 432, 8076, 458 and 5431, (14) Add together 72853, 27621, 45760, 820547 and 71425. (15) Add together 205087, 32471, 29185, 1475 and 273. (16) Find the sum of 72638594, 27836, 7805, 5271 and 1468357 and prove it to be correct by the omission of each horizontal row in succession. : (17) Find the sum of Twenty-five millions and four; Forty-seven thousand, two hundred and nine; Three hundred millions, ten thousand and one; Sixty-five thousand and eighty-seven, and Five millions and fifty: write it down in words; and apply the ordinary proof of its being correct. 25. It is usual, in many of the applications of Arithmetic, to express the operation of Addition by means of signs invented for the purpose: thus, the sum of 4 and 5 is expressed in the form, 4+ 5 = wherein the sign + between 4 and 5 denotes the addition of the latter number to the former, and is read plus or more by; and the sign between 5 and 9 expresses the result of such addition to be 9, or the equality between the sum of the numbers 4 and 5 and the number 9: so that the arithmetical expression Similarly, 2 + 3 + 7 = 12, shews the sum of the three digits 2, 3, 7, to be 12: and the same observation may be made, whatever be the numbers to be added, as in Ex. 2, of Article (21), we have 254 + 893 + 487 = 1634, expressive of the operation there performed. II. SUBTRACTION. 26. DEF. Subtraction is the second of the fundamental operations of Arithmetic, and consists in finding a number equal to the excess of one number above another, and this excess is styled the Difference or Remainder. The greater of the numbers is sometimes called the Minuend, and the less the Subtrahend. Ex. 1. Let it be required to find the difference of 7 and 2. Here it is evident that 7 units being equal to 2 units and 5 units taken together, if we withdraw the former, we shall have 5 units for the difference. The numbers and operation are usually expressed as below: Ex. 2. To subtract the number 19 from the number 37, we place the figures as in the last example, and have where the figure in the units' place of the upper line being less than that in the lower, it is manifestly impossible to substract the lower from the upper: but by considering, as on the right of the page, the 7 as 17 by taking one of the units from the 3, we find the excess of 17 above 9 to be 8, which is put in the units' place of the remainder, and then we have to take away 1 from 2 instead of 3, in consequence of having regarded the 7 as 17: hence the remainder in the tens' place will be 1, and the difference of the two numbers is therefore 18, as exhibited on the left. When the figure in the lower line is greater than that in the upper, we have borrowed ten units of the next denomination; but the same result is obtained whether we suppose to be subtracted from the upper line, or added to the lower, as the remainder will evidently be the same on both suppositions. In practice we add ten units of any denomination to both the quantities concerned; to the upper as ten of that denomination, and to the lower as one of the next superior denomination, and by this contrivance the remainder is clearly unaffected. 27. From what has been done in these examples, it will appear to be necessary to recollect for this and other purposes, the differences of every two numbers less than 20: and the reasoning here used being applicable to all other instances, the result of it may be embodied in the following rule. Rule for performing Subtraction. Place the less number under the greater, so that units may stand under units, tens under tens, and so on, as before; begin at the units' place and subtract each figure in the lower line from that in the upper, taken by itself, or increased by 10, according as it is greater or less than the said figure in the lower line, and put down the remainder, observing that whenever ten units of any denomination have been borrowed, or added to the upper line, one unit must be added to the next denomination in the lower line. 28. The operation of Subtraction being the reverse of that of Addition, it follows, that if we add together the remainder and the less of the numbers proposed, the sum thus obtained ought to be equal to the greater; and the operation of subtraction may be presumed generally to be correct when this is the case. Thus, in the following example: where the last result is the same as the greater of the numbers proposed, as it ought to be; and thence we infer that the required operation has been correctly performed. 29. The following examples, partly arranged, and partly not, are intended for practice in performing the operation of Subtraction, and also in applying the method of proof. (1) 1 4 (2) 79 (3) 4 28 (4) 7046 (8) What is the excess of 12795 above 8096? (10) Find the difference of 20470932 and 80476325. (11) How much greater is 12785462 than 1842567? (12) Required the excess of Three hundred and five millions, two hundred and four, above Seventy-five thousand, three hundred and eighty-six. 30. The operation of Subtraction, in like manner as that of Addition, is indicated or expressed by means of the sign, which is read minus or less by; thus, the excess of 7 above 3, will be expressed in the form, which is read 7-3=4, 7 minus 3 equals 4: where the sign - between 7 and 3 denotes the subtraction of the latter from the former, and the sign = between 3 and 4 shews the equality of the excess to 4. 31. DEF. Multiplication is the third of the fundamental operations of Arithmetic, and consists in finding the amount of a number, when repeated any number of times, and this amount is termed the Product. The former of these numbers is called the Multiplicand, and the latter the Multiplier. Ex. 1. To multiply the numbers 7 and 42 by the numbers 4 and 5 respectively, being to find the sums arising from the numbers 7 and 42 four and five times repeated, we may determine the products as underneath; but the operations are expressed more briefly, as follows: |