An Elementary Course in Synthetic Projective Geometry | Page 4

other. Magicians are apt to assign magic properties to many of the words and symbols which they are in the habit of using, and scientists are constantly confusing objective things with the subjective formulas for them. After the physicist has set up correspondences between physical facts and mathematical formulas, the "interpretation" of these formulas is his most important and difficult task.

*3.* In mathematics, effort is constantly being made to set up one-to-one correspondences between simple notions and more complicated ones, or between the well-explored fields of research and fields less known. Thus, by means of the mechanism employed in analytic geometry, algebraic theorems are made to yield geometric ones, and vice versa. In geometry we get at the properties of the conic sections by means of the properties of the straight line, and cubic surfaces are studied by means of the plane.

[Figure 1]
FIG. 1
[Figure 2]
FIG. 2
*4. One-to-one correspondence and enumeration.* If a one-to-one correspondence has been set up between the objects of one set and the objects of another set, then the inference may usually be drawn that they have the same number of elements. If, however, there is an infinite number of individuals in each of the two sets, the notion of counting is necessarily ruled out. It may be possible, nevertheless, to set up a one-to-one correspondence between the elements of two sets even when the number is infinite. Thus, it is easy to set up such a correspondence between the points of a line an inch long and the points of a line two inches long. For let the lines (Fig. 1) be AB and _A'B'_. Join _AA'_ and _BB'_, and let these joining lines meet in S. For every point C on AB a point _C'_ may be found on _A'B'_ by joining C to S and noting the point _C'_ where CS meets _A'B'_. Similarly, a point C may be found on AB for any point _C'_ on _A'B'_. The correspondence is clearly one-to-one, but it would be absurd to infer from this that there were just as many points on AB as on _A'B'_. In fact, it would be just as reasonable to infer that there were twice as many points on _A'B'_ as on AB. For if we bend _A'B'_ into a circle with center at S (Fig. 2), we see that for every point C on AB there are two points on _A'B'_. Thus it is seen that the notion of one-to-one correspondence is more extensive than the notion of counting, and includes the notion of counting only when applied to finite assemblages.

*5. Correspondence between a part and the whole of an infinite assemblage.* In the discussion of the last paragraph the remarkable fact was brought to light that it is sometimes possible to set the elements of an assemblage into one-to-one correspondence with a part of those elements. A moment's reflection will convince one that this is never possible when there is a finite number of elements in the assemblage.--Indeed, we may take this property as our definition of an infinite assemblage, and say that an infinite assemblage is one that may be put into one-to-one correspondence with part of itself. This has the advantage of being a positive definition, as opposed to the usual negative definition of an infinite assemblage as one that cannot be counted.

*6. Infinitely distant point.* We have illustrated above a simple method of setting the points of two lines into one-to-one correspondence. The same illustration will serve also to show how it is possible to set the points on a line into one-to-one correspondence with the lines through a point. Thus, for any point C on the line AB there is a line SC through S. We must assume the line AB extended indefinitely in both directions, however, if we are to have a point on it for every line through _S_; and even with this extension there is one line through S, according to Euclid's postulate, which does not meet the line AB and which therefore has no point on AB to correspond to it. In order to smooth out this discrepancy we are accustomed to assume the existence of an infinitely distant point on the line AB and to assign this point as the corresponding point of the exceptional line of S. With this understanding, then, we may say that we have set the lines through a point and the points on a line into one-to-one correspondence. This correspondence is of such fundamental importance in the study of projective geometry that a special name is given to it. Calling the totality of points on a line a _point-row_, and the totality of lines through a point a pencil of rays, we say that the point-row and the pencil related as above are in perspective