column; write the units of the sum under this column, and carry the tens, if any, as units to the second column. Add together the units carried and the figures in the second column; write the units of the sum under the column, and carry the tens to the third column. Proceed in this manner to the last column, under which write its entire sum.

17. Examples in Addition of whole numbers. Add together the numbers following, viz.:

39th. 78539, 31415, 86753, 57426, 96875.

40th. 153769, 476138, 297534, 472965, 187296.
41st. 475298, 351462, 299786, 163574, 582647.
42nd. 3254768, 5964783, 6927864, 8719657, 7649875.
43rd. 6471539, 4873576, 7564869, 9754768, 2518763.
44th. 57834, 198537, 27568, 4796, 758, 25389.
45th. 637, 59876, 6273, 498, 2586, 972, 37295.
46th. 1736, 298, 6571, 827, 43587, 7364, 925.
47th. 8275, 73, 8965, 584, 2986, 37746, 972865.
48th. 58, 396, 8742, 137, 65746, 975879, 568.
49th. 965, 2856, 6479, 5472, 87395, 2413, 796.
50th. 457, 98, 275, 8746, 978, 3579, 15748.
51st. 74, 853, 2961, 98, 764, 2986, 574, 835.
52nd. 839, 2567, 357, 87, 946, 578, 9727, 58047.
53rd. 58, 374, 96, 7, 5329, 695, 839, 15976.
54th. 358, 6974, 78, 536, 27981, 493, 6769, 437986.
55th, 4279, 198462, 375, 24573, 8296, 5329, 96, 7215.
56th. 6875, 34978, 549, 7294, 35281, 78, 457, 27943.
57th. 427, 3584, 79, 25347, 1973, 47625, 2886, 738.
58th. 586, 2973, 458, 59, 367, 5997, 846, 5764.
59th. 65, 347, 5824, 72573, 839, 2548.

60th. 537, 8492, 659, 43, 862, 7943, 47854, 359.

61st. 879, 1527, 3876, 8, 53492, 765, 37, 584.
62nd. 967, 3854, 7968, 53426, 851, 99, 753, 8456.
63rd. 87, 492, 3579, 69538, 5047,96, 8435768.
64th. 1479863, 653874, 74587, 4358, 674, 86, 9.
65th. 854, 7986, 3578, 42935, 647, 2569, 75.
66th. 6476, 83952, 534728, 2147, 358, 76.
67th. 8437, 98675, 6994, 35879, 849835, 6572, 978.
68th. 325, 8643, 7568, 425764, 81524, 3796, 587.
69th. 6742, 98574, 64751, 847576, 63698, 479, 6287.
70th. 93, 7581, 975, 82963, 74056, 984, 5796.
71st. 696, 3258, 46372, 5968, 257, 8435, 6875.

72nd. 35784, 35784, 35784, 35784, 35784, 35784.

73rd. 4985, 4985, 4985, 4985, 4985, 4985, 4985.

74th. 57986, 57986, 57986, 57986, 57986, 57986, 57986, 57986.
75th. 879, 879, 879, 879, 879, 879, 879, 879, 879.

76th. 986, 986, 986, 986, 986, 986, 986, 986, 986, 986.

18. The addition of general expressions of number can be indicated only: thus, if a, b, c represent different numbers, the sum of a and b is denoted by the expression a+b; and the sum of a, b, and c, by a+b+c.

Expressions like these cannot be simplified, except by giving particular values to the letters. Thus, if a=8, and b=6; a+b=8+6=14.

may be represented in the same manner, as a+a, a+a+a. But since the sum of a+a is twice a, and the sum of a+a+a is three times a, it follows that

a+a=2a.

and that a+a+a=3a.

The sum of any number of repetitions of the same letter is always expressed in this way; namely, by prefixing to the letter a figure or figures (named coefficient) expressing the number of repetitions of that letter.

In like manner the sum of 2a and 3a may be indicated by the expression 2a+3a.

But 2a=a+a, and 3a=a+a+a. Therefore 2a+3a=a+a+a+a+a=5a =(2+3) a.

If the numbers 2, 3, are replaced by the general symbols, m, n (termed literal coefficients), then ma indicates that the number a is to be repeated as often as the coefficient m contains unity; and na, that the number a is to be repeated as often as the coefficient n contains unity. Consequently, if a repeated m times is added to a repeated n times, the sum is equal to a repeated as often as m and n together contain unity. But the repetitions of unity in m and n together are indicated by m+n. Therefore, ma+na= = (m+n)a.

The sum of the coefficients is inclosed within brackets to indicate that a is repeated m+n times, and not m times only, or n times only.

b. Literal expressions, when composed of the same letters, are called like or similar; but when composed of different letters, unlike or dissimilar.

From this definition and the preceding observations, it follows that the addition of like expressions is made by adding the coefficients, and writing their sum before the common quantity, and the addition of unlike expressions, indicated by connecting them together with the symbol +.

19. Examples. Add together,

1st. a, 2a, 3a, and 4a.

2nd. 3a, 5a, 7a, and 4a.

3rd. md and nd.

Ans. 10a.

19a.

(m + n) d
(m+n+p) a.
(1+m+ n) a.

6a+8b.

7th. m+n+1; 5m + 3n+ 4 ; and 2m+ 7n + 8.

8m+11n+13.

20. In Subtraction two unequal numbers are given, and it is required to find how many units of the greater number are left when as many units as are contained in the less number have been taken from the greater.

The most elementary way of subtracting one number from another is to take I from the greater number, 1 from this remainder, I from the second remainder, and so on, till the sum of the ones taken away is equal to the less number. Thus, if it be required to subtract 5 from 9,

9-1=8,

8-1 or 9-2=7,

7-1 or 9-3=6,

6-1 or 9-4=5,

5-1 or 9-5=4.

The process consists in tracing the series of numbers downwards from the greater number until the sum of the ones taken away is equal to the less number; and it requires that the names of numbers be known in the order contrary to that of their formation.

Addition and Subtraction are, consequently, reverse operations. The first is numeration upwards, or simply numeration; the second, numeration downwards, or denumeration.

a. In Subtraction the greater number is called minuend, or number to be diminished; and the less number subtrahend, or number to be subtracted.

21. The subtraction of small numbers may be effected by taking from the minuend successive units, until the number taken away is equal to the subtrahend, a process by which the minuend is decomposed into two parts, of which the one is the subtrahend, and the other the remainder. But the subtraction of one large number from another by this method would be impracticable. In this case, the conventions of numeration, which have been employed to facilitate addition, are in the same manner rendered subsidiary to the operation of subtrac

tion; thus. Both the numbers are considered to be decomposed into collections of simple units, tens, hundreds, &c.; the units of the subtrahend are taken from the units of the minuend, the tens from the tens........ ; the units of every order in the former from the units of the corresponding order in the latter. Then, since the whole is equal to the sum of all its parts, the minuend and subtrahend are respectively equal to the sum of the units, tens, &c. into which they are decomposed. Again, all the units, tens, &c. of the subtrahend are taken from the units, tens, &c. of the minuend; that is, all the parts of the subtrahend are taken from the minuend. But to take away all the component parts of a number is to take away the whole of that number. Whence the remainder obtained by taking the units of the subtrahend from the units of the minuend, the tens from the tens, &c. is the correct expression of that number of units by which the minuend exceeds the subtrahend.

22. The process of finding the difference between any two numbers is thus reduced to that of finding the difference between two numbers expressed by one or, it may happen, two figures. The difference of such numbers may be found, at first, by means of sensible objects, or by consulting the following table, which it may be useful to commit to memory.

The horizontal column at the top of this table contains the minuends, and the left vertical column the subtrahends; the excess of any minuend over any subtrahend being placed in the square formed by the intersection of a vertical column de