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30 THE NATURE OF MATHEMATICS 

arithmetic, and does not always learn to distinguish rightly 
between the two. Of the operations of arithmetic only addition 
and subtraction can be performed with concrete numbers, and 
without speaking of more than one sort of 1. Miles can be 
added to miles, or taken from miles. Multiplication involves 
a new sort of 1, 2, 3, &c., standing for repetitions (or times, 
as they are called). Take 6 miles 5 times. Here are two 
kinds of units, 1 mile and 1 time. In multiplication, one of 
the units must be a number of repetitions or times, and to talk 
of multiplying 6 feet by 3 feet would be absurd. What notion 
can be formed of 6 feet taken " 3 feet " times ? In solving the 
following question, " If 1 yard cost 5 shillings how much will 
12 yards cost?" we do not multiply the 12 yards by the 5 
shillings ; the process we go through is the following : Since 
each yard costs 5 shillings, the buyer must put down 5 
shillings as often (as many times) as the seller uses a one-yard 
measure; that is, 5 shillings is taken 12 times. In division 
we must have the idea either of repetition or of partition, that 
is, of cutting a quantity into a number of equal parts. " Divide 
18 miles by 3 miles," means, find out how many times 3 miles 
must be repeated to give 18 miles: but "divide 18 miles by 
3 " means, cut 18 miles into 3 equal parts, and find how many 
miles are in each part. 

The symbols of arithmetic have a determinate connection; 
for instance, 4 is always 2 + 2, whatever the things mentioned 
may be, miles, feet, acres, &c. In algebra we take symbols 
for numbers which have no determinate connection. As in 
arithmetic we draw conclusions about 1, 2, 3, &c., which are 
equally true of 1 foot, 2 feet, &c., 1 minute, 2 minutes, &c. ; 
so in algebra we reason upon numbers in general, and draw 
conclusions which are equally true of all numbers. It is true 
that we also use, in kinds of algebra which have been developed 
within the last century, letters to represent things other than 
numbers for example, classes of individuals with a certain 
property, such as "horned animals," for logical purposes; or 
certain geometrical or physical things with directions in space, 
such as "forces" and signs like " + " and "-" to represent 
ways of combination of the things, which are analogous to, 
but not identical with, addition and subtraction. If " a " 
denotes " the class of horned animals " and " 6 " denotes " the