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 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 circumlocution 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 assertion 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 assert 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 assume (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 assumed to be known, just as he assumed 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 Campanus 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 assumed 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, passing 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 anticipated 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 illustration 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 classification 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 classification:

{ Straight Lines { { Circular circumferences { 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 classification not infrequently found in recent textbooks. Geminus (_ca._ 50 B.C.) is said by Proclus to have given two classifications, 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 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:

_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 quantities. 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 space inclosed, but we hardly consider it as the extremity of that space. Euclid wishes the definition before No. 14.

14. _A figure is that which is contained by any boundary or boundaries._ The definition is not satisfactory, since it excludes the unlimited straight line, the angle, an assemblage of points, and other combinations of lines and points which we should now consider as figures.

15. _A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another._

16. _And the point is called the center of the circle._

Some commentators add after "one line," definition 15, the words "which is called the circumference," but these are not in the oldest manuscripts. The Greek idea of a circle was usually that of part of a plane which is bounded by a line called in modern times the circumference, although Aristotle used "circle" as synonymous with "the bounding line." With the growth of modern mathematics, however, and particularly as a result of the development of analytic geometry, the word "circle" has come to mean the bounding line, as it did with Aristotle, a century before Euclid's time. This has grown out of the equations of the various curves, _x_^2 + _y_^2 = _r_^2 representing the circle-_line_, _a_^2_y_^2 + _b_^2_x_^2 = _a_^2_b_^2 representing the ellipse-_line_, and so on. It is natural, therefore, that circle, ellipse, parabola, and hyperbola should all be looked upon as lines. Since this is the modern use of "circle" in English, it has naturally found its way into elementary geometry, in order that students should not have to form an entirely different idea of circle on beginning analytic geometry. The general body of American teachers, therefore, at present favors using "circle" to mean the bounding line and "circumference" to mean the length of that line. This requires redefining "area of a circle," and this is done by saying that it is the area of the plane space inclosed. The matter is not of greatest consequence, but teachers will probably prefer to join in the modern American usage of the term.

17. DIAMETER. _A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle._ The word "diameter" is from two Greek words meaning a "through measurer," and it was also used by Euclid for the diagonal of a square, and more generally for the diagonal of any parallelogram. The word "diagonal" is a later term and means the "through angle." It will be noticed that Euclid adds to the usual definition the statement that a diameter bisects the circle. He does this apparently to justify his definition (18), of a semicircle (a half circle).

Thales is said to have been the first to prove that a diameter bisects the circle, this being one of three or four propositions definitely attributed to him, and it is sometimes given as a proposition to be proved. As a proposition, however, it is unsatisfactory, since the proof of what is so evident usually instills more doubt than certainty in the minds of beginners.

18. SEMICIRCLE. _A semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle._ Proclus remarked that the semicircle is the only plane figure that has its center on its perimeter. Some writers object to defining a circle as a line and then speaking of the area of a circle, showing minds that have at least one characteristic of that of Proclus. The modern definition of semicircle is "half of a circle," that is, an arc of 180°, although the term is commonly used to mean both the arc and the segment.

19. RECTILINEAR FIGURES. _Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four, straight lines._

20. _Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal._

21. _Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute._

These three definitions may properly be considered together. "Rectilinear" is from the Latin translation of the Greek _euthygrammos_, and means "right-lined," or "straight-lined." Euclid's idea of such a figure is that of the space inclosed, while the modern idea is tending to become that of the inclosing lines. In elementary geometry, however, the Euclidean idea is still held. "Trilateral" is from the Latin translation of the Greek _tripleuros_ (three-sided). In elementary geometry the word "triangle" is more commonly used, although "quadrilateral" is more common than "quadrangle." The use of these two different forms is eccentric and is merely a matter of fashion. Thus we speak of a pentagon but not of a tetragon or a trigon, although both words are correct in form. The word "multilateral" (many-sided) is a translation of the Greek _polypleuros_. Fashion has changed this to "polygonal" (many-angled), the word "multilateral" rarely being seen.

Of the triangles, "equilateral" means "equal-sided"; "isosceles" is from the Greek _isoskeles_, meaning "with equal legs," and "scalene" from _skalenos_, possibly from _skazo_ (to limp), or from _skolios_ (crooked). Euclid's limitation of isosceles to a triangle with two, and only two, equal sides would not now be accepted. We are at present more given to generalizing than he was, and when we have proved a proposition relating to the isosceles triangle, we wish to say that we have thereby proved it for the equilateral triangle. We therefore say that an isosceles triangle has two sides equal, leaving it possible that all three sides should be equal. The expression "equal legs" is now being discarded on the score of inelegance. In place of "right-angled triangle" modern writers speak of "right triangle," and so for the obtuse and acute triangles. The terms are briefer and are as readily understood. It may add a little interest to the subject to know that Plutarch tells us that the ancients thought that "the power of the triangle is expressive of the nature of Pluto, Bacchus, and Mars." He also states that the Pythagoreans called "the equilateral triangle the head-born Minerva and Tritogeneia (born of Triton) because it may be equally divided by the perpendicular lines drawn from each of its angles."

22. _Of quadrilateral figures a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral and not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another, but is neither equilateral nor right-angled. And let all quadrilaterals other than these be called trapezia._ In this definition Euclid also specializes in a manner not now generally approved. Thus we are more apt to-day to omit the oblong and rhomboid as unnecessary, and to define "rhombus" in such a manner as to include a square. We use "parallelogram" to cover "rhomboid," "rhombus," "oblong," and "square." For "oblong" we use "rectangle," letting it include square. Euclid's definition of "square" illustrates his freedom in stating more attributes than are necessary, in order to make sure that the concept is clear; for he might have said that it "is that which is equilateral and has one right angle." We may profit by his method, sacrificing logic to educational necessity. Euclid does not use "oblong," "rhombus," "rhomboid," and "trapezium" (_plural_, "trapezia") in his proofs, so that he might well have omitted the definitions, as we often do.

23. PARALLELS. _Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction._ This definition of parallels, simplified in its language, is the one commonly used to-day. Other definitions have been suggested, but none has been so generally used. Proclus states that Posidonius gave the definition based upon the lines always being at the same distance apart. Geminus has the same idea in his definition. There are, as Schotten has pointed out, three general types of definitions of parallels, namely:

_a._ They have no point in common. This may be expressed by saying that (1) they do not intersect, (2) they meet at infinity.

_b._ They are equidistant from one another.

_c._ They have the same direction.

Of these, the first is Euclid's, the idea of the point at infinity being suggested by Kepler (1604). The second part of this definition is, of course, unusable for beginners. Dr. (now Sir Thomas) Heath says, "It seems best, therefore, to leave to higher geometry the conception of infinitely distant points on a line and of two straight lines meeting at infinity, like imaginary points of intersection, and, for the purposes of elementary geometry, to rely on the plain distinction between 'parallel' and 'cutting,' which average human intelligence can readily grasp."

The direction definition seems to have originated with Leibnitz. It is open to the serious objection that "direction" is not easy of definition, and that it is used very loosely. If two people on different meridians travel due north, do they travel in the same direction? on parallel lines? The definition is as objectionable as that of angle as the "difference of direction" of two intersecting lines.

From these definitions of the first book of Euclid we see (1) what a small number Euclid considered as basal; (2) what a change has taken place in the generalization of concepts; (3) how the language has varied. Nevertheless we are not to be commended if we adhere to Euclid's small number, because geometry is now taught to pupils whose vocabulary is limited. It is necessary to define more terms, and to scatter the definitions through the work for use as they are needed, instead of massing them at the beginning, as in a dictionary. The most important lesson to be learned from Euclid's definitions is that only the basal ones, relatively few in number, need to be learned, and these because they are used as the foundations upon which proofs are built. It should also be noticed that Euclid explains nothing in these definitions; they are hard statements of fact, massed at the beginning of his treatise. Not always as statements, and not at all in their arrangement, are they suited to the needs of our boys and girls at present.

Having considered Euclid's definitions of Book I, it is proper to turn to some of those terms that have been added from time to time to his list, and are now usually incorporated in American textbooks. It will be seen that most of these were assumed by Euclid to be known by his mature readers. They need to be defined for young people, but most of them are not basal, that is, they are not used in the proofs of propositions. Some of these terms, such as magnitudes, curve line, broken line, curvilinear figure, bisector, adjacent angles, reflex angles, oblique angles and lines, and vertical angles, need merely a word of explanation so that they may be used intelligently. If they were numerous enough to make it worth the while, they could be classified in our textbooks as of minor importance, but such a course would cause more trouble than it is worth.

Other terms have come into use in modern times that are not common expressions with which students are familiar. Such a term is "straight angle," a concept not used by Euclid, but one that adds so materially to the interest and value of geometry as now to be generally recognized. There is also the word "perigon," meaning the whole angular space about a point. This was excluded by the Greeks because their idea of angle required it to be less than a straight angle. The word means "around angle," and is the best one that has been coined for the purpose. "Flat angle" and "whole angle" are among the names suggested for these two modern concepts. The terms "complement," "supplement," and "conjugate," meaning the difference between a given angle and a right angle, straight angle, and perigon respectively, have also entered our vocabulary and need defining.

There are also certain terms expressing relationship which Euclid does not define, and which have been so changed in recent times as to require careful definition at present. Chief among these are the words "equal," "congruent," and "equivalent." Euclid used the single word "equal" for all three concepts, although some of his recent editors have changed it to "identically equal" in the case of congruence. In modern speech we use the word "equal" commonly to mean "like-valued," "having the same measure," as when we say the circumference of a circle "equals" a straight line whose length is 2[pi]_r_, although it could not coincide with it. Of late, therefore, in Europe and America, and wherever European influence reaches, the word "congruent" is coming into use to mean "identically equal" in the sense of superposable. We therefore speak of congruent triangles and congruent parallelograms as being those that are superposable.

It is a little unfortunate that "equal" has come to be so loosely used in ordinary conversation that we cannot keep it to mean "congruent"; but our language will not permit it, and we are forced to use the newer word. Whenever it can be used without misunderstanding, however, it should be retained, as in the case of "equal straight lines," "equal angles," and "equal arcs of the same circle." The mathematical and educational world will never consent to use "congruent straight lines," or "congruent angles," for the reason that the terms are unnecessarily long, no misunderstanding being possible when "equal" is used.

The word "equivalent" was introduced by Legendre at the close of the eighteenth century to indicate equality of length, or of area, or of volume. Euclid had said, "Parallelograms which are on the same base and in the same parallels are equal to one another," while Legendre and his followers would modify the wording somewhat and introduce "equivalent" for "equal." This usage has been retained. Congruent polygons are therefore necessarily equivalent, but equivalent polygons are not in general congruent. Congruent polygons have mutually equal sides and mutually equal angles, while equivalent polygons have no equality save that of area.

In general, as already stated, these and other terms should be defined just before they are used instead of at the beginning of geometry. The reason for this, from the educational standpoint and considering the present position of geometry in the curriculum, is apparent.

We shall now consider the definitions of Euclid's Book III, which is usually taken as Book II in America.

1. EQUAL CIRCLES. _Equal circles are those the diameters of which are equal, or the radii of which are equal._

Manifestly this is a theorem, for it asserts that if the radii of two circles are equal, the circles may be made to coincide. In some textbooks a proof is given by superposition, and the proof is legitimate, but Euclid usually avoided superposition if possible. Nevertheless he might as well have proved this as that two triangles are congruent if two sides and the included angle of the one are respectively equal to the corresponding parts of the other, and he might as well have postulated the latter as to have substantially postulated this fact. For in reality this definition is a postulate, and it was so considered by the great Italian mathematician Tartaglia (_ca._ 1500-_ca._ 1557). The plan usually followed in America to-day is to consider this as one of many unproved propositions, too evident, indeed, for proof, accepted by intuition. The result is a loss in the logic of Euclid, but the method is thought to be better adapted to the mind of the youthful learner. It is interesting to note in this connection that the Greeks had no word for "radius," and were therefore compelled to use some such phrase as "the straight line from the center," or, briefly, "the from the center," as if "from the center" were one word.

2. TANGENT. _A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle._

Teachers who prefer to use "circumference" instead of "circle" for the line should notice how often such phrases as "cut the circle" and "intersecting circle" are used,--phrases that signify nothing unless "circle" is taken to mean the line. So Aristotle uses an expression meaning that the locus of a certain point is a circle, and he speaks of a circle as passing through "all the angles." Our word "touch" is from the Latin _tangere_, from which comes "tangent," and also "tag," an old touching game.

3. TANGENT CIRCLES. _Circles are said to touch one another which, meeting one another, do not cut one another._

The definition has not been looked upon as entirely satisfactory, even aside from its unfortunate phraseology. It is not certain, for instance, whether Euclid meant that the circles could not cut at some other point than that of tangency. Furthermore, no distinction is made between external and internal contact, although both forms are used in the propositions. Modern textbook makers find it convenient to define tangent circles as those that are tangent to the same straight line at the same point, and to define external and internal tangency by reference to their position with respect to the line, although this may be characterized as open to about the same objection as Euclid's.

4. DISTANCE. _In a circle straight lines are said to be equally distant from the center, when the perpendiculars drawn to them from the center are equal._

It is now customary to define "distance" from a point to a line as the length of the perpendicular from the point to the line, and to do this in Book I. In higher mathematics it is found that distance is not a satisfactory term to use, but the objections to it have no particular significance in elementary geometry.

5. GREATER DISTANCE. _And that straight line is said to be at a greater distance on which the greater perpendicular falls._

Such a definition is not thought essential at the present time.

6. SEGMENT. _A segment of a circle is the figure contained by a straight line and the circumference of a circle._

The word "segment" is from the Latin root _sect_, meaning "cut." So we have "sector" (a cutter), "section" (a cut), "intersect," and so on. The word is not limited to a circle; we have long spoken of a spherical segment, and it is common to-day to speak of a line segment, to which some would apply a new name "sect." There is little confusion in the matter, however, for the context shows what kind of a segment is to be understood, so that the word "sect" is rather pedantic than important. It will be noticed that Euclid here uses "circumference" to mean "arc."

7. ANGLE OF A SEGMENT. _An angle of a segment is that contained by a straight line and a circumference of a circle._

This term has entirely dropped out of geometry, and few teachers would know what it meant if they should hear it used. Proclus called such angles "mixed."

8. ANGLE IN A SEGMENT. _An angle in a segment is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it to the extremities of the straight line which is the base of the segment, is contained by the straight lines so joined._

Such an involved definition would not be usable to-day. Moreover, the words "circumference of the segment" would not be used.

9. _And when the straight lines containing the angle cut off a circumference, the angle is said to stand upon that circumference._

10. SECTOR. _A sector of a circle is the figure which, when an angle is constructed at the center of the circle, is contained by the straight lines containing the angle and the circumference cut off by them._

There is no reason for such an extended definition, our modern phraseology being both more exact (as seen in the above use of "circumference" for "arc") and more intelligible. The Greek word for "sector" is "knife" (_tomeus_), "sector" being the Latin translation. A sector is supposed to resemble a shoemaker's knife, and hence the significance of the term. Euclid followed this by a definition of similar sectors, a term now generally abandoned as unnecessary.

It will be noticed that Euclid did not use or define the word "polygon." He uses "rectilinear figure" instead. Polygon may be defined to be a bounding line, as a circle is now defined, or as the space inclosed by a broken line, or as a figure formed by a broken line, thus including both the limited plane and its boundary. It is not of any great consequence geometrically which of these ideas is adopted, so that the usual definition of a portion of a plane bounded by a broken line may be taken as sufficient for elementary purposes. It is proper to call attention, however, to the fact that we may have cross polygons of various types, and that the line that "bounds" the polygon must be continuous, as the definition states. That is, in the second of these figures the shaded portion is not considered a polygon. Such special cases are not liable to arise, but if questions relating to them are suggested, the teacher should be prepared to answer them. If suggested to a class, a note of this kind should come out only incidentally as a bit of interest, and should not occupy much time nor be unduly emphasized.

It may also be mentioned to a class at some convenient time that the old idea of a polygon was that of a convex figure, and that the modern idea, which is met in higher mathematics, leads to a modification of earlier concepts. For example, here is a quadrilateral with one of its diagonals, _BD_, _outside_ the figure. Furthermore, if we consider a quadrilateral as a figure formed by four intersecting lines, _AC_, _CF_, _BE_, and _EA_, it is apparent that this _general quadrilateral_ has six vertices, _A_, _B_, _C_, _D_, _E_, _F_, and three diagonals, _AD_, _BF_, and _CE_. Such broader ideas of geometry form the basis of what is called modern elementary geometry.

The other definitions of plane geometry need not be discussed, since all that have any historical interest have been considered. On the whole it may be said that our definitions to-day are not in general so carefully considered as those of Euclid, who weighed each word with greatest skill, but they are more teachable to beginners, and are, on the whole, more satisfactory from the educational standpoint. The greatest lesson to be learned from this discussion is that the number of basal definitions to be learned for subsequent use is very small.

Since teachers are occasionally disturbed over the form in which definitions are stated, it is well to say a few words upon this subject. There are several standard types that may be used. (1) We may use the dictionary form, putting the word defined first, thus: "_Right triangle_. A triangle that has one of its angles a right angle." This is scientifically correct, but it is not a complete sentence, and hence it is not easily repeated when it has to be quoted as an authority. (2) We may put the word defined at the end, thus: "A triangle that has one of its angles a right angle is called a right triangle." This is more satisfactory. (3) We may combine (1) and (2), thus: "_Right triangle_. A triangle that has one of its angles a right angle is called a right triangle." This is still better, for it has the catchword at the beginning of the paragraph.

There is occasionally some mental agitation over the trivial things of a definition, such as the use of the words "is called." It would not be a very serious matter if they were omitted, but it is better to have them there. The reason is that they mark the statement at once as a definition. For example, suppose we say that "a triangle that has one of its angles a right angle is a right triangle." We have also the fact that "a triangle whose base is the diameter of a semicircle and whose vertex lies on the semicircle is a right triangle." The style of statement is the same, and we have nothing in the phraseology to show that the first is a definition and the second a theorem. This may happen with most of the definitions, and hence the most careful writers have not consented to omit the distinctive words in question.

Apropos of the definitions of geometry, the great French philosopher and mathematician, Pascal, set forth certain rules relating to this subject, as also to the axioms employed, and these may properly sum up this chapter.

1. Do not attempt to define terms so well known in themselves that there are no simpler terms by which to express them.

2. Admit no obscure or equivocal terms without defining them.

3. Use in the definitions only terms that are perfectly understood or are there explained.

4. Omit no necessary principles without general agreement, however clear and evident they may be.

5. Set forth in the axioms only those things that are in themselves perfectly evident.

6. Do not attempt to demonstrate anything that is so evident in itself that there is nothing more simple by which to prove it.

7. Prove whatever is in the least obscure, using in the demonstration only axioms that are perfectly evident in themselves, or propositions already demonstrated or allowed.

8. In case of any uncertainty arising from a term employed, always substitute mentally the definition for the term itself.

=Bibliography.= Heath, Euclid, as cited; Frankland, The First Book of Euclid, as cited; Smith, Teaching of Elementary Mathematics, p. 257, New York, 1900; Young, Teaching of Mathematics, p. 189, New York, 1907; Veblen, On Definitions, in the _Monist_, 1903, p. 303.

FOOTNOTES:

[53] Free use has been made of W. B. Frankland, "The First Book of Euclid's 'Elements,'" Cambridge, 1905; T. L. Heath, "The Thirteen Books of Euclid's 'Elements,'" Cambridge, 1908; H. Schotten, "Inhalt und Methode des planimetrischen Unterrichts," Leipzig, 1893; M. Simon, "Euclid und die sechs planimetrischen Bücher," Leipzig, 1901.

[54] For a facsimile of a thirteenth-century MS. containing this definition, see the author's "Rara Arithmetica," Plate IV, Boston, 1909.

[55] Our slang expression "The cart before the horse" is suggestive of this procedure.

[56] Loc. cit., Vol. II, p. 94.