That the Predications of Geometry are of the same nature with those of Arithmetic, is a truth of the greatest importance, and capable of being established by very obvious reasoning. It is well known, that all reasoning about quantity can be expressed in the form of algebraic equations. But the two sides of an algebraic equation are of necessity two marks or two names for the same thing; of which the one on the right-hand side is more distinct, at least to the present purpose of the inquirer, than the one on the left-hand side; and the whole purpose of an algebraic investigation, which is a mere series of changes of names, is to obtain, at last, a distinct name, a name the marking power of which is perfectly known to us, on the right-hand side of the equation. The language of geometry { 191} itself, in the more simple cases, makes manifest the same observation. The amount of the three angles of a triangle, is twice a right angle. I arrive at this conclusion, as it is called, by a process of reasoning: that is to say, I find out a name "twice a right angle," which much more distinctly points out to me a certain quantity, than my first name, "amount of the three angles of a triangle;" and the process by which I arrive at this name is a successive change of names, and nothing more; as any one may prove to himself by merely observing the steps of the demonstration.[57]
[Editor's footnote 57: I cannot see any propriety in the expression that when we infer the sum of the three angles of a triangle to be twice a right angle, the operation consists in finding a second name which more distinctly points out the quantity than the first name. When we assent to the proof of this theorem, we do much more than obtain a new and more expressive name for a known fact; we learn a fact previously unknown. It is true that one result of our knowledge of this theorem is to give us a name for the sum of the three angles, "the marking power of which is perfectly known to us:" but it was not for want of knowing the marking power of the phrase "sum of the three angles of a triangle" that we did not know what that sum amounted to. We knew perfectly what the expression "sum of the three angles" was appointed to mark. What we have obtained, that we did not previously possess, is not a better mark for the same thing, but an additional fact to mark the fact which is marked by predicating of that sum, the phrase "twice a right angle."--_Ed._]