28. There is no process in arithmetic which does not consist entirely in the increase or diminution of numbers. There is then nothing which might not be done with collections of pebbles. Probably, at first, either these or the fingers were used. Our word _calculation_ is derived from the Latin word _calculus_, which means a pebble. Shorter ways of counting have been invented, by which many calculations, which would require long and tedious reckoning if pebbles were used, are made at once with very little trouble. The four great methods are, Addition, Subtraction, Multiplication, and Division; of which, the last two are only ways of doing several of the first and second at once. 29. When one number is increased by others, the number which is as large as all the numbers together is called their _sum_. The process of finding the sum of two or more numbers is called ADDITION, and, as was said before, is denoted by placing a cross (+) between the numbers which are to be added together. Suppose it required to find the sum of 1834 and 2799. In order to add these numbers, take them to pieces, dividing each into its units, tens, hundreds, and thousands: 1834 is 1 thous. 8 hund. 3 tens and 4; 2799 is 2 thous. 7 hund. 9 tens and 9. Each number is thus broken up into four parts. If to each part of the first you add the part of the second which is under it, and then put together what you get from these additions, you will have added 1834 and 2799. In the first number are 4 units, and in the second 9: these will, when the numbers are added together, contribute 13 units to the sum. Again, the 3 tens in the first and the 9 tens in the second will contribute 12 tens to the sum. The 8 hundreds in the first and the 7 hundreds in the second will add 15 hundreds to the sum; and the thousand in the first with the 2 thousands in the second will contribute 3 thousands to the sum; therefore the sum required is 3 thousands, 15 hundreds, 12 tens, and 13 units. To simplify this result, you must recollect that-- 13 units are 1 ten and 3 units. 12 tens are 1 hund. and 2 tens. 15 hund. are 1 thous. and 5 hund. 3 thous. are 3 thous. Now collect the numbers on the right hand side together, as was done before, and this will give, as the sum of 1834 and 2799, 4 thousands, 6 hundreds, 3 tens, and 3 units, which (19) is written 4633. 30. The former process, written with the signs of (23) is as follows: 1834 = 1 × 1000 + 8 × 100 + 3 × 10 + 4 2799 = 2 × 1000 + 7 × 100 + 9 × 10 + 9 Therefore, 1834 + 2799 = 3 × 1000 + 15 × 100 + 12 × 10 + 13 But 13 = 1 × 10 + 3 12 × 10 = 1 × 100 + 2 × 10 15 × 100 = 1 × 1000 + 5 × 100 3 × 1000 = 3 × 1000 Therefore, 1834 + 2799 = 4 × 1000 + 6 × 100 + 3 × 10 + 3 = 4633.