An Elementary Course in Synthetic Projective Geometry.

by Lehmer, Derrick Norman.

PREFACE

The following course is intended to give, in as simple a way as possible, the essentials of synthetic projective geometry. While, in the main, the theory is developed along the well-beaten track laid out by the great masters of the subject, it is believed that there has been a slight smoothing of the road in some places. Especially will this be observed in the chapter on Involution. The author has never felt satisfied with the usual treatment of that subject by means of circles and anharmonic ratios.

A purely projective notion ought not to be based on metrical foundations.

Metrical developments should be made there, as elsewhere in the theory, by the introduction of infinitely distant elements.

The author has departed from the century-old custom of writing in parallel columns each theorem and its dual. He has not found that it conduces to sharpness of vision to try to focus his eyes on two things at once. Those who prefer the usual method of procedure can, of course, develop the two sets of theorems side by side; the author has not found this the better plan in actual teaching.

As regards nomenclature, the author has followed the lead of the earlier writers in English, and has called the system of lines in a plane which all pa.s.s through a point a _pencil of rays_ instead of a _bundle of rays_, as later writers seem inclined to do. For a point considered as made up of all the lines and planes through it he has ventured to use the term _point system_, as being the natural dualization of the usual term _plane system_. He has also rejected the term _foci of an involution_, and has not used the customary terms for cla.s.sifying involutions-_hyperbolic involution_, _elliptic involution_ and _parabolic involution_. He has found that all these terms are very confusing to the student, who inevitably tries to connect them in some way with the conic sections.

Enough examples have been provided to give the student a clear grasp of the theory. Many are of sufficient generality to serve as a basis for individual investigation on the part of the student. Thus, the third example at the end of the first chapter will be found to be very fruitful in interesting results. A correspondence is there indicated between lines in s.p.a.ce and circles through a fixed point in s.p.a.ce. If the student will trace a few of the consequences of that correspondence, and determine what configurations of circles correspond to intersecting lines, to lines in a plane, to lines of a plane pencil, to lines cutting three skew lines, etc., he will have acquired no little practice in picturing to himself figures in s.p.a.ce.

The writer has not followed the usual practice of inserting historical notes at the foot of the page, and has tried instead, in the last chapter, to give a consecutive account of the history of pure geometry, or, at least, of as much of it as the student will be able to appreciate who has mastered the course as given in the preceding chapters. One is not apt to get a very wide view of the history of a subject by reading a hundred biographical footnotes, arranged in no sort of sequence. The writer, moreover, feels that the proper time to learn the history of a subject is after the student has some general ideas of the subject itself.

The course is not intended to furnish an ill.u.s.tration of how a subject may be developed, from the smallest possible number of fundamental a.s.sumptions. The author is aware of the importance of work of this sort, but he does not believe it is possible at the present time to write a book along such lines which shall be of much use for elementary students. For the purposes of this course the student should have a thorough grounding in ordinary elementary geometry so far as to include the study of the circle and of similar triangles. No solid geometry is needed beyond the little used in the proof of Desargues' theorem (25), and, except in certain metrical developments of the general theory, there will be no call for a knowledge of trigonometry or a.n.a.lytical geometry. Naturally the student who is equipped with these subjects as well as with the calculus will be a little more mature, and may be expected to follow the course all the more easily. The author has had no difficulty, however, in presenting it to students in the freshman cla.s.s at the University of California.

The subject of synthetic projective geometry is, in the opinion of the writer, destined shortly to force its way down into the secondary schools; and if this little book helps to accelerate the movement, he will feel amply repaid for the task of working the materials into a form available for such schools as well as for the lower cla.s.ses in the university.

The material for the course has been drawn from many sources. The author is chiefly indebted to the cla.s.sical works of Reye, Cremona, Steiner, Poncelet, and Von Staudt. Acknowledgments and thanks are also due to Professor Walter C. Eells, of the U.S. Naval Academy at Annapolis, for his searching examination and keen criticism of the ma.n.u.script; also to Professor Herbert Ellsworth Slaught, of The University of Chicago, for his many valuable suggestions, and to Professor B. M. Woods and Dr. H. N.

Wright, of the University of California, who have tried out the methods of presentation, in their own cla.s.ses.

D. N. LEHMER

BERKELEY, CALIFORNIA

CHAPTER I - ONE-TO-ONE CORRESPONDENCE

*1. Definition of one-to-one correspondence.* Given any two sets of individuals, if it is possible to set up such a correspondence between the two sets that to any individual in one set corresponds one and only one individual in the other, then the two sets are said to be in _one-to-one correspondence_ with each other. This notion, simple as it is, is of fundamental importance in all branches of science. The process of counting is nothing but a setting up of a one-to-one correspondence between the objects to be counted and certain words, 'one,' 'two,' 'three,' etc., in the mind. Many savage peoples have discovered no better method of counting than by setting up a one-to-one correspondence between the objects to be counted and their fingers. The scientist who busies himself with naming and cla.s.sifying the objects of nature is only setting up a one-to-one correspondence between the objects and certain words which serve, not as a means of counting the objects, but of listing them in a convenient way.

Thus he may be able to marshal and array his material in such a way as to bring to light relations that may exist between the objects themselves.

Indeed, the whole notion of language springs from this idea of one-to-one correspondence.

*2. Consequences of one-to-one correspondence.* The most useful and interesting problem that may arise in connection with any one-to-one correspondence is to determine just what relations existing between the individuals of one a.s.semblage may be carried over to another a.s.semblage in one-to-one correspondence with it. It is a favorite error to a.s.sume that whatever holds for one set must also hold for the other. Magicians are apt to a.s.sign 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 a.n.a.lytic 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 a.s.semblages.

*5. Correspondence between a part and the whole of an infinite a.s.semblage.* 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 a.s.semblage 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 a.s.semblage.-Indeed, we may take this property as our definition of an infinite a.s.semblage, and say that an infinite a.s.semblage 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 a.s.semblage as one that cannot be counted.

*6. Infinitely distant point.* We have ill.u.s.trated above a simple method of setting the points of two lines into one-to-one correspondence. The same ill.u.s.tration 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 a.s.sume 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 a.s.sume the existence of an _infinitely distant_ point on the line _AB_ and to a.s.sign 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 position_, or that they are _perspectively related_.

*7. Axial pencil; fundamental forms.* A similar correspondence may be set up between the points on a line and the planes through another line which does not meet the first. Such a system of planes is called an _axial pencil_, and the three a.s.semblages-the point-row, the pencil of rays, and the axial pencil-are called _fundamental forms_. The fact that they may all be set into one-to-one correspondence with each other is expressed by saying that they are of the same order. It is usual also to speak of them as of the first order. We shall see presently that there are other a.s.semblages which cannot be put into this sort of one-to-one correspondence with the points on a line, and that they will very reasonably be said to be of a higher order.

*8. Perspective position.* We have said that a point-row and a pencil of rays are in perspective position if each ray of the pencil goes through the point of the point-row which corresponds to it. Two pencils of rays are also said to be in perspective position if corresponding rays meet on a straight line which is called the axis of perspectivity. Also, two point-rows are said to be in perspective position if corresponding points lie on straight lines through a point which is called the center of perspectivity. A point-row and an axial pencil are in perspective position if each plane of the pencil goes through the point on the point-row which corresponds to it, and an axial pencil and a pencil of rays are in perspective position if each ray lies in the plane which corresponds to it; and, finally, two axial pencils are perspectively related if corresponding planes meet in a plane.

*9. Projective relation.* It is easy to imagine a more general correspondence between the points of two point-rows than the one just described. If we take two perspective pencils, _A_ and _S_, then a point-row _a_ perspective to _A_ will be in one-to-one correspondence with a point-row _b_ perspective to _B_, but corresponding points will not, in general, lie on lines which all pa.s.s through a point. Two such point-rows are said to be _projectively related_, or simply projective to each other.

Similarly, two pencils of rays, or of planes, are projectively related to each other if they are perspective to two perspective point-rows. This idea will be generalized later on. It is important to note that between the elements of two projective fundamental forms there is a one-to-one correspondence, and also that this correspondence is in general _continuous_; that is, by taking two elements of one form sufficiently close to each other, the two corresponding elements in the other form may be made to approach each other arbitrarily close. In the case of point-rows this continuity is subject to exception in the neighborhood of the point "at infinity."

*10. Infinity-to-one correspondence.* It might be inferred that any infinite a.s.semblage could be put into one-to-one correspondence with any other. Such is not the case, however, if the correspondence is to be continuous, between the points on a line and the points on a plane.

Consider two lines which lie in different planes, and take _m_ points on one and _n_ points on the other. The number of lines joining the _m_ points of one to the _n_ points jof the other is clearly _mn_. If we symbolize the totality of points on a line by [infinity], then a reasonable symbol for the totality of lines drawn to cut two lines would be [infinity]2. Clearly, for every point on one line there are [infinity]

lines cutting across the other, so that the correspondence might be called [infinity]-to-one. Thus the a.s.semblage of lines cutting across two lines is of higher order than the a.s.semblage of points on a line; and as we have called the point-row an a.s.semblage of the first order, the system of lines cutting across two lines ought to be called of the second order.