IX

No one can have studied even the elements of elementary geometry without feeling the lack of some guiding method. Every proposition has to be proved by a fresh display of ingenuity; and a science for which this is true lacks the great requisite of scientific thought, namely, method. Now the especial point of coordinate geometry is that for the first time it introduced method. The remote deductions of a mathematical science are not of primary theoretical importance. The science has not been perfected, until it consists in essence of the exhibition of great allied methods by which information, on any desired topic which falls within its scope, can easily be obtained. The growth of a science is not primarily in bulk, but in ideas; and the more the ideas grow, the fewer are the deductions which it is worth while to write down. Unfortunately, mathematics is always encumbered by the repetition in text-books of numberless subsidiary propositions, whose importance has been lost by their absorption into the role of particular cases of more general truths–-and, as we have already insisted, generality is the soul of mathematics.

Again, coordinate geometry illustrates another feature of mathematics which has already been pointed out, namely, that mathematical sciences as they develop dovetail into each other, and share the same ideas in common. It is not too much to say that the various branches of mathematics undergo a perpetual process of generalization, and that as they become generalized, they coalesce. Here again the reason springs from the very nature of the science, its generality, that is to say, from the fact that the science deals with the general truths which apply to all things in virtue of their very existence as things. In this connection the interest of coordinate geometry lies in the fact that it relates together geometry, which started as the science of space, and algebra, which has its origin in the science of number.

Let us now recall the main ideas of the two sciences, and then see how they are related by Descartes' method of coordinates. Take

algebra in the first place. We will not trouble ourselves about the imaginaries and will think merely of the real numbers with positive or negative signs. The fundamental idea is that of any number, the variable number, which is denoted by a letter and not by any definite numeral. We then proceed to the consideration of correlations between variables. For example, if $x$ and $y$ are two variables,

we may conceive them as correlated by the equations $x+y=1$, or by $x-y=1$, or in any one of an indefinite number of other ways. This at once leads to the application of the

idea of algebraic form. We think, in fact, of any correlation of some interesting type, thus rising from the initial conception of variable numbers to the secondary conception of variable correlations of numbers. Thus we generalize the correlation $x+y=1$, into the correlation $ax+by=c$. Here $a$ and $b$ and $c$, being letters, stand for any numbers and are in fact themselves variables. But they are the variables which determine the variable correlation; and the correlation, when determined, correlates the variable numbers $x$ and $y$. Variables, like $a$, $b$, and $c$ above, which are used to determine the correlation are called "constants," or parameters. The use

of the term "constant" in this connection for what is really a variable may seem at first sight to be odd; but it is really very natural. For the mathematical investigation is concerned with the relation between the variables $x$ and $y$, after $a$, $b$, $c$ are supposed to have been determined. So in a sense, relatively to $x$ and $y$, the "constants" $a$, $b$, and $c$ are constants. Thus $ax+by=c$ stands for the general example of a certain algebraic form, that is, for a variable correlation belonging to a certain class.