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.
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.
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.
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.
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.
All things which are true of some numbers of one unit are true of
the same numbers of any other unit. Thus, 15 pebbles and 7 pebbles
together make 22 pebbles; 15 acres and 7 acres together make 22 acres,
and so on. From this we come to say that 15 and 7 make 22, meaning that
15 things of the same kind, and 7 more of the same kind as the first,
together make 22 of that kind, whether the kind mentioned be pebbles,
horsemen, acres of land, or any other. For these it is but necessary to
say, once for all, that 15 and 7 make 22. Therefore, in future, on this
part of the subject I shall cease to talk of any particular units, such
as pebbles or acres, and speak of numbers only. A number, considered
without intending to allude to any particular things, is called an
abstract number: and it then merely signifies repetitions of a
unit, or the number of times a unit is repeated.
I will now repeat the principal things which have been mentioned in
this chapter.