13. It is important to recollect that this way of writing numbers, which has become so familiar as to seem the _natural_ method, is not more natural than any other. For example, we might agree to signify one ten by the figure of one with an accent, thus, 1′; twenty or two tens by 2′; and so on: one hundred or ten tens by 1″; two hundred by 2″; one thousand by 1‴; and so on: putting Roman figures for accents when they become too many to write with convenience. The fourth number in the table would then be written 2‴ 3′ 4′ 8, which might also be expressed by 8 4′ 3″ 2‴, 4′ 8 3″ 2‴; or the order of the figures might be changed in any way, because their meaning depends upon the accents which are attached to them, and not upon the place in which they stand. Hence, a cipher would never be necessary; for 104 would be distinguished from 14 by writing for the first 1″ 4, and for the second 1′ 4. The common method is preferred, not because it is more exact than this, but because it is more simple. 14. The distinction between our method of numeration and that of the ancients, is in the meaning of each figure depending partly upon the place in which it stands. Thus, in 44444 each four stands for four of _something_; but in the first column on the right it signifies only four of the pebbles which are counted; in the second, it means four collections of ten pebbles each; in the third, four of one hundred each; and so on. 15. The things measured in (11) were yards of cloth. In this case one yard of cloth is called the _unit_. The first figure on the right is said to be in the _units’ place_, because it only stands for so many units as are in the number that is written under it in (6). The second figure is said to be in the _tens’_ place, because it stands for a number of tens of units. The third, fourth, and fifth figures are in the places of the _hundreds_, _thousands_, and _tens of thousands_, for a similar reason. 16. If the quantity measured had been acres of land, an acre of land would have been called the _unit_, for the unit is _one_ of the things which are measured. Quantities are of two sorts; those which contain an exact number of units, as 47 yards, and those which do not, as 47 yards and a half. Of these, for the present, we only consider the first. 17. In most parts of arithmetic, all quantities must have the same unit. You cannot say that 2 yards and 3 feet make 5 _yards_ or 5 _feet_, because 2 and 3 make 5; yet you may say that 2 _yards_ and 3 _yards_ make 5 _yards_, and that 2 _feet_ and 3 _feet_ make 5 _feet_. It would be absurd to try to measure a quantity of one kind with a unit which is a quantity of another kind; for example, to attempt to tell how many yards there are in a gallon, or how many bushels of corn there are in a barrel of wine.