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MODERN MATHEMATICS ALGEBRA 35 

To the right of are the integers and fractions, to the left of 
are the negative numbers, and to the right of stretch the 
series of positive numbers, +a coinciding with a and being 
symmetrically placed with a as regards 0. Also we deter- 
mine that the operations of what we call " addition," &c., of 
these new " numbers " must lead to the same results as the 
former operations of the same name. Thus the same symbol 
is used in different senses, and we write 

a + 6-6 = a + = ( + o) + ( + 6) + (-6)= + a=a. 
This is a remarkable sequence of quick changes. 

We have used the sign of equality, " = ". It means origi- 
nally : " is the same as." Thus 3+1 = 4. But we write, by 
the above convention, "a= +a," and so we sacrifice exactness, 
which sometimes looks rather pedantic, for the sake of keeping 
our analogy in view, and for brevity. 

Let us bear this, at first sight, puzzling but, at second sight, 
justifiable peculiarity of mathematicians in mind. It has 
always puzzled intelligent beginners and philosophers. The 
laws of calculation and convenient symbolism are the things 
a mathematician thinks of and aims at. He seems to identify 
different things if they both satisfy the same laws which are 
important to him, just as a magistrate may think that there 
is not much difference between Mr. A., who is red-haired and 
a tinker and goes to chapel, and Mr. B., who is a brown-haired 
horse-dealer and goes to church, if both have been found out 
committing petty larceny. But their respective ministers of 
religion or wives may still be able to distinguish them. 

Any two expressions connected by the signs of equality form 
an " equation." Here we must notice that the words : " Solve 
the equation a; 2 + ace = 6," means, find the value or values of 
x such that, a and 6 being given numbers, cc 2 +ax becomes 6. 
Thus, if a = 2 and 6 = 1, the solution is x = 1. 

As we saw above, Descartes fixed the custom of employing 
the letters at the beginning of the alphabet to denote known 
quantities, and those at the end of the alphabet to denote 
unknown quantities. Thus, in the above example, a and 6 are 
some numbers supposed to be given, while x is sought. The 
question is solved when x is found in terms of a and 6 and fixed 
numbers (like 1, 2, 3) ; and so, when to a and b are attributed