The Teaching of Geometry Part 10

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

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

[52] Book I, Proposition 20.

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, a.s.sumptions (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 other, and this cannot be described in terms simpler than itself. Such, for example, is the term "angle." We can easily explain the meaning of this word, and we can make the concept clear, but this must be done by a certain amount of circ.u.mlocution and explanation, not by a concise and perfect definition. Unless a beginner in geometry knows what an angle is before he reads the definition in a textbook, he will not know from the definition. This fact of the impossibility of defining some of the fundamental concepts will be evident when we come to consider certain attempts that have been made in this direction.

It should also be understood in this connection that a definition makes no a.s.sertion as to the existence of the thing defined. If we say that a tangent to a circle is an unlimited straight line that touches the circle in one point, and only one, we do not a.s.sert that it is possible to have such a line; that is a matter for proof. Not in all cases, however, can this proof be given, as in the existence of the simplest concepts. We cannot, for example, prove that a point or a straight line exists after we have defined these concepts. We therefore tacitly or explicitly a.s.sume (postulate) the existence of these fundamentals of geometry. On the other hand, we can prove that a tangent exists, and this may properly be considered a legitimate proposition or corollary of elementary geometry. In relation to geometric proof it is necessary to bear in mind, therefore, that we are permitted to define any term we please; for example, "a seven-edged polyhedron" or Leibnitz's "ten-faced regular polyhedron," neither of which exists; but, strictly speaking, we have no right to make use of a definition in a proof until we have shown or postulated that the thing defined has an existence. This is one of the strong features of Euclid's textbook. Not being able to prove that a point, a straight line, and a circle exists, he practically postulates these facts; but he uses no other definition in a proof without showing that the thing defined exists, and this is his reason for mingling his problems with his theorems. At the present time we confessedly sacrifice his logic in this respect for the reason that we teach geometry to pupils who are too young to appreciate that logic.

It was pointed out by Aristotle, long before Euclid, that it is not a satisfactory procedure to define a thing by means of terms that are strictly not prior to it, as when we attempt to define something by means of its opposite. Thus to define a curve as "a line, no part of which is straight," would be a bad definition unless "straight" had already been explicitly defined; and to define "bad" as "not good" is unsatisfactory for the reason that "bad" and "good" are concepts that are evolved simultaneously. But all this is only a detail under the general principle that a definition must employ terms that are better understood than the one defined.

It should be understood that some definitions are much more important than others, considered from the point of view of the logic of geometry.

Those that enter into geometric proofs are basal; those that form part of the conversational language of geometry are not. Euclid gave twenty-three definitions in Book I, and did not make use of even all of these terms. Other terms, those not employed in his proofs, he a.s.sumed to be known, just as he a.s.sumed a knowledge of any other words in his language. Such procedure would not be satisfactory under modern conditions, but it is of great importance that the teacher should recognize that certain definitions are basal, while others are merely informational.

It is now proposed to consider the basal definitions of geometry, first, that the teacher may know what ones are to be emphasized and learned; and second, that he may know that the idea that the standard definitions can easily be improved is incorrect. It is hoped that the result will be the bringing into prominence of the basal concepts, and the discouraging of attempts to change in unimportant respects the definitions in the textbook used by the pupil.

In order to have a systematic basis for work, the definitions of two books of Euclid will first be considered.[53]

1. POINT. _A point is that which has no part._ This was incorrectly translated by Capella in the fifth century, "Punctum est cuius pars nihil est" (a point is that of which a part is nothing), which is as much as to say that the point itself is nothing. It generally appears, however, as in the Campa.n.u.s edition,[54] "Punctus est cuius pars non est," which is substantially Euclid's wording. Aristotle tells of the definitions of point, line, and surface that prevailed in his time, saying that they all defined the prior by means of the posterior.[55]

Thus a point was defined as "an extremity of a line," a line as "the extremity of a surface," and a surface as "the extremity of a solid,"--definitions still in use and not without their value. For it must not be a.s.sumed that scientific priority is necessarily priority in fact; a child knows of "solid" before he knows of "point," so that it may be a very good way to explain, if not to define, by beginning with solid, pa.s.sing thence to surface, thence to line, and thence to point.

The first definition of point of which Proclus could learn is attributed by him to the Pythagoreans, namely, "a monad having position," the early form of our present popular definition of a point as "position without magnitude." Plato defined it as "the beginning of a line," thus presupposing the definition of "line"; and, strangely enough, he antic.i.p.ated by two thousand years Cavalieri, the Italian geometer, by speaking of points as "indivisible lines." To Aristotle, who protested against Plato's definitions, is due the definition of a point as "something indivisible but having position."

Euclid's definition is essentially that of Aristotle, and is followed by most modern textbook writers, except as to its omission of the reference to position. It has been criticized as being negative, "which has _no_ part"; but it is generally admitted that a negative definition is admissible in the case of the most elementary concepts. For example, "blind" must be defined in terms of a negation.

At present not much attention is given to the definition of "point,"

since the term is not used as the basis of a proof, but every effort is made to have the concept clear. It is the custom to start from a small solid, conceive it to decrease in size, and think of the point as the limit to which it is approaching, using these terms in their usual sense without further explanation.

2. LINE. _A line is breadthless length._ This is usually modified in modern textbooks by saying that "a line is that which has length without breadth or thickness," a statement that is better understood by beginners. Euclid's definition is thought to be due to Plato, and is only one of many definitions that have been suggested. The Pythagoreans having spoken of the point as a monad naturally were led to speak of the line as dyadic, or related to two. Proclus speaks of another definition, "magnitude in one dimension," and he gives an excellent ill.u.s.tration of line as "the edge of a shadow," thus making it real but not material.

Aristotle speaks of a line as a magnitude "divisible in one way only,"

as contrasted with a surface which is divisible in two ways, and with a solid which is divisible in three ways. Proclus also gives another definition as the "flux of a point," which is sometimes rendered as the path of a moving point. Aristotle had suggested the idea when he wrote, "They say that a line by its motion produces a surface, and a point by its motion a line."

Euclid did not deem it necessary to attempt a cla.s.sification of lines, contenting himself with defining only a straight line and a circle, and these are really the only lines needed in elementary geometry. His commentators, however, made the attempt. For example. Heron (first century A.D.) probably followed his definition of line by this cla.s.sification:

{ Straight Lines { { Circular circ.u.mferences { Not straight { Spiral shaped { Curved (generally)

Proclus relates that both Plato and Aristotle divided lines into "straight," "circular," and "a mixture of the two," a statement which is not quite exact, but which shows the origin of a cla.s.sification not infrequently found in recent textbooks. Geminus (_ca._ 50 B.C.) is said by Proclus to have given two cla.s.sifications, of which one will suffice for our purposes:

{ Composite (broken line forming an angle) { Lines { { Forming a figure, or determinate. (Circle, { { ellipse, cissoid.) { Incomposite { Not forming a figure, or indeterminate and { extending without a limit. (Straight { line, parabola, hyperbola, conchoid.)

Of course his view of the cissoid, the curve represented by the equation _y_^2(_a_ + _x_) = (_a_ - _x_)^3, is not the modern view.

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 a.s.serted 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 s.p.a.ce," 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 ill.u.s.trate 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 ill.u.s.trations. The shadow cast on a table top is a good ill.u.s.tration, 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 a.s.sumption 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 pa.s.s 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, pa.s.sing through one point on a straight line in s.p.a.ce, 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) pa.s.sing through a fixed point and also intersecting a straight line in s.p.a.ce, but of course this intersected straight line must not pa.s.s 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 pa.s.s 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 subst.i.tutes 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 quant.i.ty, namely, a distance between the lines or surfaces containing it." Schotten[56] says that the definitions of angle generally fall into three groups:

_a._ An angle is the difference of direction between two lines that meet. This is no better than Euclid's, since "difference of direction"

is as difficult to define as "inclination."

_b._ An angle is the amount of turning necessary to bring one side to the position of the other side.

_c._ An angle is the portion of the plane included between its sides.

Of these, _b_ is given by way of explanation in most modern textbooks.

Indeed, we cannot do better than simply to define an angle as the opening between two lines which meet, and then explain what is meant by size, through the bringing in of the idea of rotation. This is a simple presentation, it is easily understood, and it is sufficiently accurate for the real purpose in mind, namely, the grasping of the concept. We should frankly acknowledge that the concept of angle is such a simple one that a satisfactory definition is impossible, and we should therefore confine our attention to having the concept understood.

10. _When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands._ We at present separate these definitions and simplify the language.

11. _An obtuse angle is an angle greater than a right angle._

12. _An acute angle is an angle less than a right angle._

The question sometimes asked as to whether an angle of 200 is obtuse, and whether a negative angle, say -90, is acute, is answered by saying that Euclid did not conceive of angles equal to or greater than 180 and had no notion of negative quant.i.ties. Generally to-day we define an obtuse angle as "greater than one and less than two right angles." An acute angle is defined as "an angle less than a right angle," and is considered as positive under the general understanding that all geometric magnitudes are positive unless the contrary is stated.

13. _A boundary is that which is an extremity of anything._ The definition is not exactly satisfactory, for a circle is the boundary of the s.p.a.ce inclosed, but we hardly consider it as the extremity of that s.p.a.ce. Euclid wishes the definition before No. 14.