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10. The greatest number of visits to the Exhibition in any
one month was in July; the number in that month was one million three hundred and fourteen thousand one
hundred and seventy-six. 11. The total number of visits to the Exhibition, without
counting the closing day and certain private days, was six million seven thousand nine hundred and forty
12. The number of people in Great Britain and the British
Islands in 1851 was twenty million nine hundred and
thirty-six thousand four hundred and sixty-eight. (9.) From these exercises you will see that the ten figures of arithmetic are sufficient for the purpose of expressing any number, however great; and that the reason why so few are enough, is, that each figure changes its meaning as its place in a number is changed. The value of a figure in any particular place is called its local value ; thus, the local value of the 6 in 263 is sixty; and the local value of the 2 is two hundred : the local value being always ten times what it would be if the figure were in the next place on the right. This ten-fold increase in the local value of a figure, when it is advanced one place from right to left, is the reason why our system of notation in arithmetic is called the decimal system.t It may
proper to mention here, that the figures 1, 2, 3, &c. are also called digits; and that 0, or nought, is also called cipher', or zero.
(10.) It may also be further noticed, that when we have to express a very large number in words, it is convenient to separate the figures of it into periods of three figures each, by putting a comma before the last three figures, another comma before the next three, and so on : thus, the large number by which I have illustrated the Numeration Table, when the figures are divided into periods, is 375,268,436,297. The advantage of this is, that as the leading figure of each period occupies the place of hundreds,—that is, of hundreds simply, or hundreds of thousands, or hundreds of millions, &c., the number is more readily put into words. In the tables published by order of Government, relating to revenue, population, &c., this plan is always adopted.
(11.) The rule for adding a set of numbers together, so as to find the sum of them all, is called the rule for ADDITION; it is the first rule in Arithmetic, and is as follows:
* Including the six exceptional days ; namely, the opening day, the two days at one pound, the two exhibitors' days, and the closing day, the number in the 144 days was about 6,170,000.
+ The learner may be here informed, that the particular marks or symbols used in any science,-as, for instance, the notes in music, and the characters or symbols 1, 2, 3, &c. in arithmetic,-constitute the notation of that science.
Rule 1. Place the numbers to be added under one another, so that the units may all be in the first column on the right, the tens in the second column, the hundreds in the third column, and so on.
2. Add up the column of units; that is, find the sum of the units in this column; if this sum be a number of only one figure, put this figure down under the unit column; but if it be a number of more than one figure, it is not to be put down ; the last figure of it only, that is, the figure in the units' place, is to be put down, and the number expressed by the remaining figure or figures, after the one put down is rubbed out, is to be carried to the column of tens, and added up with that column. In like manner, put down the last figure of the number which is the sum of this second column, and carry the number expressed by the remaining figures, when the one put down is rubbed out, to the next column, or column of hundreds, and so on till you reach the last column, the whole amount of which is to be put down.
For example: Suppose we have to add together 327 the numbers 327, 241, and 58. Then, writing the 241 numbers under one another, so that the units may 58 all be in the first column on the right, the tens in the second column, and the hundreds in the third, 626 as in the margin, we should proceed as follows :8 and 1 are 9, and 7 are 16; therefore there are sixteen units in the first column, and as this number consists of two figures, 1 and 6, we put down only the 6, and carry the 1 to the next column, and say 1 an 5 are 6, and 4 are 10, and 2 are 12; we therefore put down 2 and carry 1 to the next column, saying 1 and 2 are 3, and 3 are 6, which we put down, and thus find the sum of the numbers to be 626; that is, six hundred and twenty-six.
Again : Suppose we have to add together the numbers 7625, 3253, 1802, and 211. Writing the numbers
7625 under one another, as before, we say, 1 and 2 are 3,
3253 and 3 are 6, and 5 are 11, 1 and carry 1; 1 and
1802 1 are 2, and 5 are 7, and 2 are 9; this being a
211 single figure, we put it down, and have nothing to carry to the next column ; 2 and 8 are 10, and 2
12891 are 12, and 6 are 18, 8 and carry 1; 1 and 1 are 2, and 3 are 5, and are 12; and having now reached the last column, the whole amount 12 is put down; so that the sum of the numbers is 12891; that is, twelve thousand eight hundred and ninety-one.
We shall go through the work of but one other example : Add together the numbers 57632, 804, 70300, 731, and 33. Writing the numbers in columns, as before, we 57632 say,
3 and 1 are 4, and 4 are 8, and 2 are 10, 804 O and carry 1; 1 and 3 are 4, and 3 are 7, and 70300 3 are 10, O and carry 1; 1 and 7 are 8, and 3 731 are 11, and 8 are 19, and 6 are 25, 5 and carry 33 2; 2 and 7 are 9; 7 and 5 are 12. So that the sum is, one hundred and twenty-nine thou- 129500 sand five hundred.
(12.) You will easily see that the work of these examples is right: thus, looking back to the first example, you see that the column of units amounts to 16 units; that is, to one ten and six units over; the one ten is carried, as it ought to be, to the column of tens, and only the six units put in the units' place. The tens amount to 12 tens; that is, to ten tens, or one hundred, and two tens over; these two tens are therefore put in the tens' place, and the one hundred carried to the column of hundreds; the sum of this column is found to be six ; so that we have got the right number of hundreds, the right number of tens, and the right number of units, in the whole. And in the same way you may convince yourself that each of the other examples is correct, and that the rule must lead you to the true sum or amount in all cases.
(13.) There is a way of proving whether the columns are added up without error, given in most books on arithmetic; but it would cause you more trouble than to do the work over again ; you had better therefore make up your mind to perform the addition a second time, when you are not quite sure that there is no mistake: this second time you should commence at the top of each column, and add downwards ; thus, if you wish to try the correctness of example 2, above, begin at the top of the units' column, and say, 5 and 3 are 8, and 2 are 10, and I are 1l, 1 and carry 1; 1 and 2 are 3, and 5 are 8, and I are 9; and so on.
Exercises in Addition. 1. Add together the numbers 342, 165, and 34. 2. Find the sum of 87, 273, and 49. 3. Find the sum of 2860, 1723, 41, and 17. 4. Add together 5693, 482, 6297, and 13. 5. Add together 17341, 9203, 510, and 20061. 6. What is the sum of 35208, 62, 187, and 762070 ? 7. Add up 7407003, 169205, 4853, 79, and 382.
8. Add the following numbers together; namely, two thou
sand and four, seven thousand and thirty-five, one hundred and one thousand and nine, seventeen thou
sand and forty-eight, and two hundred and one. 9. On the London and Birmingham Railway, the Primrose
Hill tunnel is one thousand one hundred and twenty yards in length; the Weedon tunnel, four hundred and eighteen yards; the Kilsby tunnel, two thousand three hundred and ninety-eight yards. On the Great Western Railway, the Box tunnel is three thousand two hundred and twenty-seven yards. On the Manchester and Leeds Railway, the Littleborough tunnel is two thousand eight hundred and sixty-nine yards; and the Merstham tunnel, on the London and Brighton Railway, is one thousand seven hundred and eighty yards. What is
the sum of the lengths of these six tunnels? 10. By order of Government, all the people in this kingdom
are counted every ten years; this counting is called taking a census of the population. In 1841 and 1851 the numbers were found to be as follow :
Census of 1841. Census of 1851. In England and Wales... 15911725 17922768 Scotland
2628957 2870784 Islands in British Seas 124079 142916 What was the amount of the population in 1841 and
1851 ? 11. The number of persons visiting the Great Exhibition of
1851, during the last week, was as follows; namely,
53061 How many visits were made during the last week
altogether 12. Find how many days the Exhibition was open, and how
many visits were paid to it altogether, from the fol-
Number of Days. Number of Persons.
1133114 July 27
1023435 September 26
1155240 October 13
808237 13. The number of persons who emigrated from this king
dom,—that is, who left it to live in other countries, -
51083 What was the amount of emigration during these two
years? 14. In the year ending on the 5th of January, 1851, there
were 159 London newspapers, in which there appeared 891650 advertisements, and 222 English country newspapers, in which 875631 advertisements appeared. There were also 110 Scotch newspapers, with 249141 advertisements; and 102 Irish papers, with 236128 advertisements. Find the total number of newspapers, and the total number of advertisements.
(14.) It is proper that you should now be told, that besides the marks or symbols used in arithmetic for numbers, some other marks are also used, instead of common words, for the operations of arithmetic. The operation you have just performed, in each of the foregoing examples, is the operation of addition. Now there is a mark or sign used for addition; it is an upright cross, thus +, placed between the figures or numbers to be added, and it is called the sign of addition, and is read plus; there is also a sign for the result of any operation, as for instance, for the sum or total of a set of numbers; it is called the sign of equality, and is written thus =: whenever you see this sign, you are to understand that the words equal to are meant by it. By using these two signs, you may write down any example in addition, with the answer or result put against it, without employing any words : thus, taking the first example at page 6, you might write it so :
327 + 241 + 58 = 626; and if you were asked to read this, you would say 327 plus 241 plus 58, are equal to 626, or equals 626.
In like manner, the second example, with the answer or result, would stand thus :
7625 + 3253 + 1802 + 211 = 12891 ;