ON GEOMETRY 
617 

representation, and therefore should remain unchanged when we pass from one frame of reference to another. Any characteristic which is changed by such a transformation of our systems of reference is clearly an extrinsic characteristic injected by the form of representation and not belonging to our subject; and so the transformation of coordinates is precisely the test we need and use.
Let us take for instance the line segment, as in Fig. 3. We may refer
P1P2 to a system 0, or to a system 0'. Obviously the length of the line is independent of the axes of reference used, and the formula for the length of a line is not altered, although the values for the x's and y's are different in the two systems. In other words, the sum of the squares of the differences of the 







used.
A great step forward in the formulation of methods which lead to invariant and intrinsic formulations was made in the invention of what is called the vector calculus and its extension in the modern tensor calculus. A few explanations of this principle will be of interest.
A vector is roughly a directed segment of a straight line on which we distinguish the initial and the terminal points. A vector has thus magnitude and direction. In practice we deal with two kinds of entities; some are purely 
