The Teaching of Geometry Part 11

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 a.s.semblage 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 circ.u.mference," but these are not in the oldest ma.n.u.scripts. 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 circ.u.mference, 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 a.n.a.lytic 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 a.n.a.lytic geometry. The general body of American teachers, therefore, at present favors using "circle" to mean the bounding line and "circ.u.mference" 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 s.p.a.ce 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 circ.u.mference 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 circ.u.mference 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 s.p.a.ce 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 fas.h.i.+on. 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_. Fas.h.i.+on 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" ill.u.s.trates 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 ma.s.sing 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, ma.s.sed 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 a.s.sumed 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 cla.s.sified 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 s.p.a.ce 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 relations.h.i.+p 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 circ.u.mference 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 a.s.serts 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 "circ.u.mference" 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 pa.s.sing 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 circ.u.mference 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 "circ.u.mference" to mean "arc."

7. ANGLE OF A SEGMENT. _An angle of a segment is that contained by a straight line and a circ.u.mference 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 circ.u.mference 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 "circ.u.mference of the segment" would not be used.