Of the alternative forms of the postulate, that of Proclus is generally considered the best suited to beginners. As stated by Playfair (1795), this is, "Through a given point only one parallel can be drawn to a given straight line"; and as stated by Proclus, "If a straight line intersect one of two parallels, it will intersect the other also." Playfair's form is now the common "postulate of parallels," and is the one that seems destined to endure.

Posidonius and Geminus, both Stoics of the first century B.C., gave as their alternative, "There exist straight lines everywhere equidistant from one another." One of Legendre's alternatives is, "There exists a triangle in which the sum of the three angles is equal to two right angles." One of the latest attempts to suggest a substitute is that of the Italian Ingrami (1904), “Two parallel straight lines intercept, on every transversal which passes through the mid-point of a segment included. between them, another segment the mid-point of which is the mid-point of the first."

Of course it is entirely possible to assume that through a point more than one line can be drawn parallel to a given straight line, in which case another type of geometry can be built up, equally rigorous with Euclid's. This was done at the close of the first quarter of the nineteenth century by Lobachevsky (1793-1856) and Bolyai (1802-1860), resulting in the first of several "non-Euclidean" geometries.1

1 For the early history of this movement see Engel and Stäckel, "Die Theorie der Parallellinien von Euklid bis auf Gauss," Leipzig, 1895; Bonola, Sulla teoria delle parallele e sulle geometrie noneuclidee, in his "Questioni riguardanti la geometria elementare," 1900; Karagiannides, "Die nichteuklidische Geometrie vom Alterthum bis zur Gegenwart," Berlin, 1893.

Taking the problem to be that of stating a reasonably small number of geometric assumptions that may form a basis to supplement the general axioms, that shall cover the most important matters to which the student must refer, and that shall be so simple as easily to be understood by a beginner, the following are recommended :

1. One straight line, and only one, can be drawn through two given points. This should also be stated for convenience in the form, Two points determine a straight line. From it may also be drawn this corollary, Two straight lines can intersect in only one point, since two points would determine a straight line. Such obvious restatements of or corollaries to a postulate are to be commended, since a beginner is often discouraged by having to prove what is so obvious that a demonstration fails to commend itself to his mind.

2. A straight line may be produced to any required length. This, like Postulate 1, requires the use of a straightedge for drawing the physical figure. The required length is attained by using the compasses to measure the distance. The straightedge and the compasses are the only two drawing instruments recognized in elementary geometry.1 While this involves more than Euclid's postulate, it is a better working assumption for beginners.

3. A straight line is the shortest path between two points. This is easily proved by the method of Euclid 2 for the case where the paths are broken lines, but it is needed as a postulate for the case of curve paths. It is a better statement than the common one that a straight line is the shortest distance between two points; for distance is

1 This limitation upon elementary geometry was placed by Plato (died 347 B.C.), as already stated.

2 Book I, Proposition 20.

measured on a line, but it is not itself a line. Furthermore, there are scientific objections to using the word "distance" any more than is necessary.

4. A circle may be described with any given point as a center and any given line as a radius. This involves the use of the second of the two geometric instruments, the

compasses.

5. Any figure may be moved from one place to another without altering the size or shape. This is the postulate of the homogeneity of space, and asserts that space is such that we may move a figure as we please without deformation of any kind. It is the basis of all cases of superposition.

6. All straight angles are equal. It is possible to prove this, and therefore, from the standpoint of strict logic, it is unnecessary as a postulate. On the other hand, it is poor educational policy for a beginner to attempt to prove a thing that is so obvious. The attempt leads to a loss of interest in the subject, the proposition being (to state a paradox) hard because it is so easy. It is, of course, possible to postulate that straight angles are equal, and to draw the conclusion that their halves (right angles) are equal; or to proceed in the opposite direction, and postulate that all right angles are equal, and draw the conclusion that their doubles (straight angles) are equal. Of the two the former has the advantage, since it is probably more obvious that all straight angles are equal. It is well to state the following definite corollaries to this postulate: (1) All right angles are equal; (2) From a point in a line only one perpendicular can be drawn to the line, since two perpendiculars would make the whole (right angle) equal to its part; (3) Equal angles have equal complements, equal supplements, and equal

conjugates; (4) The greater of two angles has the less complement, the less supplement, and the less conjugate. All of these four might appear as propositions, but, as already stated, they are so obvious as to be more harmful than useful to beginners when given in such form.

The postulate of parallels may properly appear in connection with that topic in Book I, and it is accordingly treated in Chapter XIV.

There is also another assumption that some writers are now trying to formulate in a simple fashion. We take, for example, a line segment AB, and describe circles with A and B respectively as centers, and with a radius AB. We say that the circles will intersect as at C and D. But how do we know that they intersect? We assume it, just as we assume that an indefinite straight line drawn from a point inclosed by a circle will, if produced far enough, cut the circle twice. Of course a pupil would not think of this if his attention was not called to it, and the harm outweighs the good in doing this with one who is beginning the study of geometry.

With axioms and with postulates, therefore, the conclusion is the same: from the standpoint of scientific geometry there is an irreducible minimum of assumptions, but from the standpoint of practical teaching this list should give place to a working set of axioms and postulates that meet the needs of the beginner.

Bibliography. Smith, Teaching of Elementary Mathematics, New York, 1900; Young, The Teaching of Mathematics, New York, 1901; Moore, On the Foundations of Mathematics, Bulletin of the American Mathematical Society, 1903, p. 402; Betz, Intuition and Logic in Geometry, The Mathematics Teacher, Vol. II, p. 3 ; Hilbert, The Foundations of Geometry, Chicago, 1902; Veblen, A System of Axioms for Geometry, Transactions of the American Mathematical Society, 1904, p. 343.

CHAPTER XII

THE DEFINITIONS OF GEOMETRY

When we consider the nature of geometry it is evident that more attention must be paid to accuracy of definitions than is the case in most of the other sciences. The essence of all geometry worthy of serious study is not. the knowledge of some fact, but the proof of that fact; and this proof is always based upon preceding proofs, assumptions (axioms or postulates), or definitions. If we are to prove that one line is perpendicular to another, it is essential that we have an exact definition of "perpendicular," else we shall not know when we have reached the conclusion of the proof.

The essential features of a definition are that the term defined shall be described in terms that are simpler than, or at least better known than, the thing itself; that this shall be done in such a way as to limit the term to the thing defined; and that the description shall not be redundant. It would not be a good definition to say that a right angle is one fourth of a perigon and one half of a straight angle, because the concept "perigon" is not so simple, and the term "perigon" is not so well known, as the term and the concept "right angle," and because the definition is redundant, containing more than is necessary..

It is evident that satisfactory definitions are not always possible; for since the number of terms is limited, there must be at least one that is at least as simple as any