III

NON-EUCLIDEAN GEOMETRY

By FREDERICK S. WOODS

I. INTRODUCTION

1. The fifth postulate of Euclid reads as follows: "If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two lines, if produced indefinitely, meet on that side on which are the angles less than two right angles."

Under the term non-Euclidean geometry we shall understand a system of geometry which is built up without the use of this postulate. Strictly speaking, perhaps, the same name might be given to any geometry the basis of which differs in any essential particular from that of Euclid, but usage has decreed otherwise.

The conception of a non-Euclidean geometry came into being only after centuries of vain attempts to prove the truth of Euclid's postulate. There is no place here to review the history of such attempts.* It is sufficient to note that all inevitably failed. Some writers, however, especially Saccheri (1667-1733), Lambert (1728-77), and Legendre (1752-1833) made important contributions to what is now recognized as

*See, for example: Engel-Staeckel, Theorie der Parallellinien von Euklid bis auf Gauss, Leipzig, 1895. A shorter account is found in Bonola, Die nichteuklidische Geometrie, Vol. IV, of the series, Wissenschaft und Hypothese, Leipzig, 1908. See also the Historical Note, in Manning, Non-Euclidean Geometry, Boston, 1901; and Heath, The Thirteen Books of Euclid's Elements, Vol. I, p. 202, Cambridge, 1908.

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the non-Euclidean geometries, though each failed to see the true meaning of the results he obtained.

Finally, nearly simultaneously though quite independently, a Russian, Lobachevsky, a Hungarian, J. Bolyai, and a German, Gauss, reached the conclusion not only that the parallel postulate could not be proved, but that a logical system of geometry could be constructed without its use. The work of Gauss is only partly revealed by extracts from his correspondence and fragments of his posthumous papers. That of Lobachevsky is contained in several articles published between 1833 and 1855, and that of Bolyai in an appendix to a work of his father published in 1832-35. The system of geometry common to these three writers we shall call the Lobachevskian geometry, since Lobachevsky was the mathematician to develop it most fully." *

The Lobachevskian geometry remained for a time the sole type of a non-Euclidean geometry. In 1854, however, Riemann, working from the standpoint of the differential calculus, discovered a new type to which we shall give the name of the Riemannian geometry.

Besides the three types of geometry, the Euclidean, the Lobachevskian, and the Riemannian, there are also three methods by which the geometries may be developed. The first is by elementary methods similar to those of Euclid, and was used by Lobachevsky, Bolyai, and Gauss. The second is by use of Cayley's system of projective measurement and has been largely employed by Klein. The third is that of the calculus, and has been used by Riemann. We shall begin by employing the first method, but shall later make some reference to the other two.

It does not lie within the plan of this paper to examine the assumptions which must be made before any form of a parallel postulate can be introduced. This work has been done by

* English readers will find the simplest introduction to Lobachevsky's own work in the little book written in German and translated into English by G. B. Halsted under the title, "Geometrical researches on the theory of parallels." More complete is Engel's translation: Lobatschefsky, Zwei geometrische Abhandlungen aus dem Russischen übersetzt mit Anmerkungen and mit einer Biographie des Verfassers, Leipzig, 1879.

Professor Veblen in his paper* contained in the present collection and the results of that paper will be assumed as known and freely referred to. It is believed, however, that this paper may be easily read by any reader who prefers to start from the original definitions, common notions, and postulates, stated or implied, of Euclid.

II. PARALLEL LINES

2. We assume Euclid's fundamentals with the exception of the parallel postulate, or make Veblen's assumptions I-XII and XIV. The first twenty-eight propositions of the first book of Euclid (Veblen, VIII) are then true. We proceed to give a definition of parallel lines more general than that of Euclid.

Let PQ (Fig. 1) be any straight line and A any point not on PQ. Through A there passes a set of lines intersecting PQ,

since any point on PQ may be joined to A. It is conceivable that there may be other lines through A which do not intersect PQ. In that case, there will be lines such as AL and AK, not intersecting PQ and forming the boundaries of the set of lines which meet PQ. Such lines are said to be parallel to PQ.

Otherwise expressed: Let AB be any line through A intersecting PQ. The line AL is said to be parallel to PQ at the point A, if

(1) AL does not intersect PQ no matter how far produced.

* Monograph No. I.

(2) Any line through A in the angle opening BAL does intersect PQ.

It is evident that this definition considers only those portions of the lines AL and PQ which lie on the same side of AB. In other words, the directions of the lines are important. We shall indicate the directions of parallel lines in the usual way by the order in which the letters at these extremities are named. Thus we shall say that AL is parallel to PQ and AK is parallel to QP.

The line AB may be any line through A intersecting PQ. It is often convenient, however, to use the line AH perpendicular to PQ. We may then show that

4HAK=4HAL.

For if HAK were greater than HAL, we could draw AC meeting QP in C so that 4HACHAL. Now take C' on HQ so that HC'HC and connect A and C'. By Euclid, I, 4 (Veblen, Theorem 32) the triangles HAC and HAC" are congruent, and hence

4HAC' = 4HAC=&HAL.

This is impossible, since AL is parallel to HQ. Hence HAK cannot be greater than HAL. In like manner, HAL cannot be greater than 4HAK. Hence HAK== HAL.

The angle HAL is called the angle of parallelism for the distance AH.

In the definition, the point A plays apparently a unique role. We shall show this to be unessential by the theorem of the next section.

3. A straight line maintains the property of parallelism at all its points.

Let AK (Fig. 2) be parallel to BQ at the point A and let A be any point on AK. We wish to show that AK is parallel to BQ at the point A1.

Connect A and B and draw through A, any line AC in the angle opening BAK. Take D any point on AC and

connect D with A. The line AD prolonged will meet BQ at some point F since AK is parallel to BQ. Hence AC will meet BQ in some point between B and F (Veblen, Theorem 17). That is, any line through A, in the angle opening BAK intersects BQ. But 41K does not intersect BQ. Hence it is parallel to BQ.

The proof also holds that if A is taken on the backward extension of AK, but, in that case D must be taken on the backward extension of A1С.

We shall now show that the property of parallelism is reciprocal.

4. If a line is parallel to another line the second line is parallel to the first.

Let LK (Fig. 3) be parallel to PQ. We wish to prove that PQ is parallel to LK. From A draw a line perpendicular to LK. This perpendicular will meet PQ at some point B since LK is parallel to PQ. Draw through B any line BC in the angle opening QBA. Construct the two angles ABE and ABD so that

Hence we may draw in the angle BEK a line EF so that

BEF=BDK,