The Teaching of Geometry

by David Eugene Smith

3. The extremities of a line are points. This is not a definition in the sense of its two predecessors. A modern writer would put it as a note under the definition of line. Euclid did not wish to define a point as the extremity of a line, for Aristotle had asserted that this was not scientific; so he defined point and line, and then added this statement to show the relation of one to the other. Aristotle had improved upon this by stating that the "division" of a line, as well as an extremity, is a point, as is also the intersection of two lines. These statements, if they had been made by Euclid, would have avoided the objection made by Proclus, that some lines have no extremities, as, for example, a circle, and also a straight line extending infinitely in both directions.

4. Straight Line. A straight line is that which lies evenly with respect to the points on itself. This is the least satisfactory of all of the definitions of Euclid, and emphasizes the fact that the straight line is the most difficult to define of the elementary concepts of geometry. What is meant by "lies evenly"? Who would know what a straight line is, from this definition, if he did not know in advance?

The ancients suggested many definitions of straight line, and it is well to consider a few in order to appreciate the difficulties involved. Plato spoke of it as "that of which the middle covers the ends," meaning that if looked at endways, the middle would make it impossible to see the remote end. This is often modified to read that "a straight line when looked at endways appears as a point,"—an idea that involves the postulate that our line of sight is straight. Archimedes made the statement that "of all the lines which have the same extremities, the straight line is the least," and this has been modified by later writers into the statement that "a straight line is the shortest distance between two points." This is open to two objections as a definition: (1) a line is not distance, but distance is the length of a line,—it is measured on a line; (2) it is merely stating a property of a straight line to say that "a straight line is the shortest path between two points,"—a proper postulate but not a good definition. Equally objectionable is one of the definitions suggested by both Heron and Proclus, that "a straight line is a line that is stretched to its uttermost"; for even then it is reasonable to think of it as a catenary, although Proclus doubtless had in mind the Archimedes statement. He also stated that "a straight line is a line such that if any part of it is in a plane, the whole of it is in the plane,"—a definition that runs in a circle, since plane is defined by means of straight line. Proclus also defines it as "a uniform line, capable of sliding along itself," but this is also true of a circle.

Of the various definitions two of the best go back to Heron, about the beginning of our era. Proclus gives one of them in this form, "That line which, when its ends remain fixed, itself remains fixed." Heron proposed to add, "when it is, as it were, turned round in the same plane." This has been modified into "that which does not change its position when it is turned about its extremities as poles," and appears in substantially this form in the works of Leibnitz and Gauss. The definition of a straight line as "such a line as, with another straight line, does not inclose space," is only a modification of this one. The other definition of Heron states that in a straight line "all its parts fit on all in all ways," and this in its modern form is perhaps the most satisfactory of all. In this modern form it may be stated, "A line such that any part, placed with its ends on any other part, must lie wholly in the line, is called a straight line," in which the force of the word "must" should be noted. This whole historical discussion goes to show how futile it is to attempt to define a straight line. What is needed is that we should explain what is meant by a straight line, that we should illustrate it, and that pupils should then read the definition understandingly.

5. Surface. A surface is that which has length and breadth. This is substantially the common definition of our modern textbooks. As with line, so with surface, the definition is not entirely satisfactory, and the chief consideration is that the meaning of the term should be made clear by explanations and illustrations. The shadow cast on a table top is a good illustration, since all idea of thickness is wanting. It adds to the understanding of the concept to introduce Aristotle's statement that a surface is generated by a moving line, modified by saying that it may be so generated, since the line might slide along its own trace, or, as is commonly said in mathematics, along itself.

6. The extremities of a surface are lines. This is open to the same explanation and objection as definition 3, and is not usually given in modern textbooks. Proclus calls attention to the fact that the statement is hardly true for a complete spherical surface.

7. Plane. A plane surface is a surface which lies evenly with the straight lines on itself. Euclid here follows his definition of straight line, with a result that is equally unsatisfactory. For teaching purposes the translation from the Greek is not clear to a beginner, since "lies evenly" is a term not simpler than the one defined. As with the definition of a straight line, so with that of a plane, numerous efforts at improvement have been made. Proclus, following a hint of Heron's, defines it as "the surface which is stretched to the utmost," and also, this time influenced by Archimedes's assumption concerning a straight line, as "the least surface among all those which have the same extremities." Heron gave one of the best definitions, "A surface all the parts of which have the property of fitting on [each other]." The definition that has met with the widest acceptance, however, is a modification of one due to Proclus, "A surface such that a straight line fits on all parts of it." Proclus elsewhere says, "[A plane surface is] such that the straight line fits on it all ways," and Heron gives it in this form, "[A plane surface is] such that, if a straight line pass through two points on it, the line coincides with it at every spot, all ways." In modern form this appears as follows: "A surface such that a straight line joining any two of its points lies wholly in the surface is called a plane," and for teaching purposes we have no better definition. It is often known as Simson's definition, having been given by Robert Simson in 1756.

The French mathematician, Fourier, proposed to define a plane as formed by the aggregate of all the straight lines which, passing through one point on a straight line in space, are perpendicular to that line. This is clear, but it is not so usable for beginners as Simson's definition. It appears as a theorem in many recent geometries. The German mathematician, Crelle, defined a plane as a surface containing all the straight lines (throughout their whole length) passing through a fixed point and also intersecting a straight line in space, but of course this intersected straight line must not pass through the fixed point. Crelle's definition is occasionally seen in modern textbooks, but it is not so clear to the pupil as Simson's. Of the various ultrascientific definitions of a plane that have been suggested of late it is hardly of use to speak in a book concerned primarily with practical teaching. No one of them is adapted to the needs and the comprehension of the beginner, and it seems that we are not likely to improve upon the so-called Simson form.

8. Plane Angle. A plane angle is the inclination to each other of two lines in a plane which meet each other and do not lie in a straight line. This definition, it will be noticed, includes curvilinear angles, and the expression "and do not lie in a straight line" states that the lines must not be continuous one with the other, that is, that zero and straight angles are excluded. Since Euclid does not use the curvilinear angle, and it is only the rectilinear angle with which we are concerned, we will pass to the next definition and consider this one in connection therewith.

9. Rectilinear Angle. When the lines containing the angle are straight, the angle is called rectilinear. This definition, taken with the preceding one, has always been a subject of criticism. In the first place it expressly excludes the straight angle, and, indeed, the angles of Euclid are always less than 180°, contrary to our modern concept. In the second place it defines angle by means of the word "inclination," which is itself as difficult to define as angle. To remedy these defects many substitutes have been proposed. Apollonius defined angle as "a contracting of a surface or a solid at one point under a broken line or surface." Another of the Greeks defined it as "a quantity, namely, a distance between the lines or surfaces containing it." Schotten[56] says that the definitions of angle generally fall into three groups: