20. If we take out of a number, as 16785, any of those figures which come together, as 67, and ask, what does this sixty-seven mean? of what is it sixty-seven? the answer is, sixty-seven of the same collections as the 7, when it was in the number; that is, 67 hundreds. For the 6 is 6 thousands, or 6 ten hundreds, or sixty hundreds; which, with the 7, or 7 hundreds, is 67 hundreds: similarly, the 678 is 678 tens. This number may then be expressed either as 1 ten thousand 6 thousands 7 hundreds 8 tens and 5; or 16 thousands 78 tens and 5; or 1 ten thousand 678 tens and 5; or 167 hundreds 8 tens and 5; or 1678 tens and 5, and so on. 21. EXERCISES. I. Write down the signs for--four hundred and seventy-six; two thousand and ninety-seven; sixty-four thousand three hundred and fifty; two millions seven hundred and four; five hundred and seventy-eight millions of millions. II. Write at full length 53, 1805, 1830, 66707, 180917324, 66713721, 90976390, 25000000. III. What alteration takes place in a number made up entirely of nines, such as 99999, by adding one to it? IV. Shew that a number which has five figures in it must be greater than one which has four, though the first have none but small figures in it, and the second none but large ones. For example, that 10111 is greater than 9879. 22. You now see that the convenience of our method of numeration arises from a few simple signs being made to change their value as they change the column in which they are placed. The same advantage arises from counting in a similar way all the articles which are used in every-day life. For example, we count money by dividing it into pounds, shillings, and pence, of which a shilling is 12 pence, and a pound 20 shillings, or 240 pence. We write a number of pounds, shillings, and pence in three columns, generally placing points between the columns. Thus, 263 pence would not be written as 263, but as £1. 1. 11, where £ shews that the 1 in the first column is a pound. Here is a _system of numeration_ in which a number in the second column on the right means 12 times as much as the same number in the first; and one in the third column is twenty times as great as the same in the second, or 240 times as great as the same in the first. In each of the tables of measures which you will hereafter meet with, you will see a separate system of numeration, but the methods of calculation for all will be the same. 23. In order to make the language of arithmetic shorter, some other signs are used. They are as follow: I. 15 + 38 means that 38 is to be added to 15, and is the same thing as 53. This is the _sum_ of 15 and 38, and is read fifteen _plus_ thirty-eight (_plus_ is the Latin for _more_). II. 64-12 means that 12 is to be taken away from 64, and is the same thing as 52. This is the _difference_ of 64 and 12, and is read sixty-four _minus_ twelve (_minus_ is the Latin for _less_).