The Teaching of Geometry Part 17

THEOREM. _If two sides of a quadrilateral are equal and parallel, then the other two sides are equal and parallel, and the figure is a parallelogram._

This was Euclid's first proposition on parallelograms, and Proclus speaks of it as the connecting link between the theory of parallels and that of parallelograms. The ancients, writing for mature students, did not add the words "and the figure is a parallelogram," because that follows at once from the first part and from the definition of "parallelogram," but it is helpful to younger students because it emphasizes the fact that here is a test for this kind of figure.

THEOREM. _The diagonals of a parallelogram bisect each other._

This proposition was not given in Euclid, but it is usually required in American syllabi. There is often given in connection with it the exercise in which it is proved that the diagonals of a rectangle are equal. When this is taken, it is well to state to the cla.s.s that carpenters and builders find this one of the best checks in laying out floors and other rectangles. It is frequently applied also in laying out tennis courts. If the cla.s.s is doing any work in mensuration, such as finding the area of the school grounds, it is a good plan to check a few rectangles by this method.

An interesting outdoor application of the theory of parallelograms is the following:

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Suppose you are on the side of this stream opposite to _XY_, and wish to measure the length of _XY_. Run a line _AB_ along the bank. Then take a carpenter's square, or even a large book, and walk along _AB_ until you reach _P_, a point from which you can just see _X_ and _B_ along two sides of the square. Do the same for _Y_, thus fixing _P_ and _Q_. Using the tape, bisect _PQ_ at _M_. Then walk along _YM_ produced until you reach a point _Y'_ that is exactly in line with _M_ and _Y_, and also with _P_ and _X_. Then walk along _XM_ produced until you reach a point _X'_ that is exactly in line with _M_ and _X_, and also with _Q_ and _Y_. Then measure _Y'X'_ and you have the length of _XY_. For since _YX'_ is [perp] to _PQ_, and _XY'_ is also [perp] to _PQ_, _YX'_ is || to _XY'_. And since _PM_ = _MQ_, therefore _XM_ = _MX'_ and _Y'M_ = _MY_. Therefore _Y'X'YX_ is a parallelogram.

The properties of the parallelogram are often applied to proving figures of various kinds congruent, or to constructing them so that they will be congruent.

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For example, if we draw _A'B'_ equal and parallel to _AB_, _B'C'_ equal and parallel to _BC_, and so on, it is easily proved that _ABCD_ and _A'B'C'D'_ are congruent. This may be done by ordinary superposition, or by sliding _ABCD_ along the dotted parallels.

There are many applications of this principle of parallel translation in practical construction work. The principle is more far-reaching than here intimated, however, and a few words as to its significance will now be in place.

The efforts usually made to improve the spirit of Euclid are trivial.

They ordinarily relate to some commonplace change of sequence, to some slight change in language, or to some narrow line of applications. Such attempts require no particular thought and yield no very noticeable result. But there is a possibility, remote though it may be at present, that a geometry will be developed that will be as serious as Euclid's and as effective in the education of the thinking individual. If so, it seems probable that it will not be based upon the congruence of triangles, by which so many propositions of Euclid are proved, but upon certain postulates of motion, of which one is involved in the above ill.u.s.tration,--the postulate of parallel translation. If to this we join the two postulates of rotation about an axis,[64] leading to axial symmetry; and rotation about a point,[65] leading to symmetry with respect to a center, we have a group of three motions upon which it is possible to base an extensive and rigid geometry.[66] It will be through some such effort as this, rather than through the weakening of the Euclid-Legendre style of geometry, that any improvement is likely to come. At present, in America, the important work for teachers is to vitalize the geometry they have,--an effort in which there are great possibilities,--seeing to it that geometry is not reduced to mere froth, and recognizing the possibility of another geometry that may sometime replace it,--a geometry as rigid, as thought-compelling, as logical, and as truly educational.

THEOREM. _The sum of the interior angles of a polygon is equal to two right angles, taken as many times less two as the figure has sides._

This interesting generalization of the proposition about the sum of the angles of a triangle is given by Proclus. There are several proofs, but all are based upon the possibility of dissecting the polygon into triangles. The point from which lines are drawn to the vertices is usually taken at a vertex, so that there are _n_ - 2 triangles. It may however be taken within the figure, making _n_ triangles, from the sum of the angles of which the four right angles about the point must be subtracted. The point may even be taken on one side, or outside the polygon, but the proof is not so simple. Teachers who desire to do so may suggest to particularly good students the proving of the theorem for a concave polygon, or even for a cross polygon, although the latter requires negative angles.

Some schools have transit instruments for the use of their cla.s.ses in trigonometry. In such a case it is a good plan to measure the angles in some piece of land so as to verify the proposition, as well as show the care that must be taken in reading angles. In the absence of this exercise it is well to take any irregular polygon and measure the angles by the help of a protractor, and thus accomplish the same results.

THEOREM. _The sum of the exterior angles of a polygon, made by producing each of its sides in succession, is equal to four right angles._

This is also a proposition not given by the ancient writers. We have, however, no more valuable theorem for the purpose of showing the nature and significance of the negative angle; and teachers may arouse a great deal of interest in the negative quant.i.ty by showing to a cla.s.s that when an interior angle becomes 180 the exterior angle becomes 0, and when the polygon becomes concave the exterior angle becomes negative, the theorem holding for all these cases. We have few better ill.u.s.trations of the significance of the negative quant.i.ty, and few better opportunities to use the knowledge of this kind of quant.i.ty already acquired in algebra.

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In the hilly and mountainous parts of America, where irregular-shaped fields are more common than in the more level portions, a common test for a survey is that of finding the exterior angles when the transit instrument is set at the corners. In this field these angles are given, and it will be seen that the sum is 360. In the absence of any outdoor work a protractor may be used to measure the exterior angles of a polygon drawn on paper. If there is an irregular piece of land near the school, the exterior angles can be fairly well measured by an ordinary paper protractor.

The idea of locus is usually introduced at the end of Book I. It is too abstract to be introduced successfully any earlier, although authors repeat the attempt from time to time, unmindful of the fact that all experience is opposed to it. The loci propositions are not ancient. The Greeks used the word "locus" (in Greek, _topos_), however. Proclus, for example, says, "I call those locus theorems in which the same property is found to exist on the whole of some locus." Teachers should be careful to have the pupils recognize the necessity for proving two things with respect to any locus: (1) that any point on the supposed locus satisfies the condition; (2) that any point outside the supposed locus does not satisfy the given condition. The first of these is called the "sufficient condition," and the second the "necessary condition."

Thus in the case of the locus of points in a plane equidistant from two given points, it is _sufficient_ that the point be on the perpendicular bisector of the line joining the given points, and this is the first part of the proof; it is also _necessary_ that it be on this line, i.e.

it cannot be outside this line, and this is the second part of the proof. The proof of loci cases, therefore, involves a consideration of "the necessary and sufficient condition" that is so often spoken of in higher mathematics. This expression might well be incorporated into elementary geometry, and when it becomes better understood by teachers, it probably will be more often used.

In teaching loci it is helpful to call attention to loci in s.p.a.ce (meaning thereby the s.p.a.ce of three dimensions), without stopping to prove the proposition involved. Indeed, it is desirable all through plane geometry to refer incidentally to solid geometry. In the mensuration of plane figures, which may be boundaries of solid figures, this is particularly true.

It is a great defect in most school courses in geometry that they are entirely confined to two dimensions. Even if solid geometry in the usual sense is not attempted, every occasion should be taken to liberate boys' minds from what becomes the tyranny of paper. Thus the questions: "What is the locus of a point equidistant from two given points; at a constant distance from a given straight line or from a given point?" should be extended to s.p.a.ce.[67]

The two loci problems usually given at this time, referring to a point equidistant from the extremities of a given line, and to a point equidistant from two intersecting lines, both permit of an interesting extension to three dimensions without any formal proof. It is possible to give other loci at this point, but it is preferable merely to introduce the subject in Book I, reserving the further discussion until after the circle has been studied.

It is well, in speaking of loci, to remember that it is entirely proper to speak of the "locus of a point" or the "locus of points." Thus the locus of a _point_ so moving in a plane as constantly to be at a given distance from a fixed point in the plane is a circle. In a.n.a.lytic geometry we usually speak of the locus of a _point_, thinking of the point as being anywhere on the locus. Some teachers of elementary geometry, however, prefer to speak of the locus of _points_, or the locus of _all points_, thus tending to make the language of elementary geometry differ from that of a.n.a.lytic geometry. Since it is a trivial matter of phraseology, it is better to recognize both forms of expression and to let pupils use the two interchangeably.

FOOTNOTES:

[57] Address at Brussels, August, 1910.

[58] For a recent discussion of this general subject, see Professor Hobson on "The Tendencies of Modern Mathematics," in the _Educational Review_, New York, 1910, Vol. XL, p. 524.

[59] A more extended list of applications is given later in this work.

[60] Ab[=u]'l-'Abb[=a]s al-Fadl ibn H[=a]tim al-Nair[=i]z[=i], so called from his birthplace, Nair[=i]z, was a well-known Arab writer. He died about 922 A.D. He wrote a commentary on Euclid.

[61] This ill.u.s.tration, taken from a book in the author's library, appeared in a valuable monograph by W. E. Stark, "Measuring Instruments of Long Ago," published in _School Science and Mathematics_, Vol. X, pp.

48, 126. With others of the same nature it is here reproduced by the courtesy of Princ.i.p.al Stark and of the editors of the journal in which it appeared.

[62] In speaking of two congruent triangles it is somewhat easier to follow the congruence if the two are read in the same order, even though the relatively unimportant counterclockwise reading is neglected. No one should be a slave to such a formalism, but should follow the plan when convenient.

[63] Stark, loc. cit.

[64] Of which so much was made by Professor Olaus Henrici in his "Congruent Figures," London, 1879,--a book that every teacher of geometry should own.

[65] Much is made of this in the excellent work by Henrici and Treutlein, "Lehrbuch der Geometrie," Leipzig, 1881.

[66] Meray did much for this movement in France, and the recent works of Bourlet and Borel have brought it to the front in that country.

[67] W. N. Bruce, "Teaching of Geometry and Graphic Algebra in Secondary Schools," Board of Education circular (No. 711), p. 8, London, 1909.

CHAPTER XV

THE LEADING PROPOSITIONS OF BOOK II

Having taken up all of the propositions usually given in Book I, it seems unnecessary to consider as specifically all those in subsequent books. It is therefore proposed to select certain ones that have some special interest, either from the standpoint of mathematics or from that of history or application, and to discuss them as fully as the circ.u.mstances seem to warrant.

THEOREMS. _In the same circle or in equal circles equal central angles intercept equal arcs; and of two unequal central angles the greater intercepts the greater arc_, and conversely for both of these cases.

Euclid made these the twenty-sixth and twenty-seventh propositions of his Book III, but he limited them as follows: "In equal circles equal angles stand on equal circ.u.mferences, whether they stand at the centers or at the circ.u.mferences, and conversely." He therefore included two of our present theorems in one, thus making the proposition doubly hard for a beginner. After these two propositions the Law of Converse, already mentioned on page 190, may properly be introduced.

THEOREMS. _In the same circle or in equal circles, if two arcs are equal, they are subtended by equal chords; and if two arcs are unequal, the greater is subtended by the greater chord_, and conversely.

Euclid dismisses all this with the simple theorem, "In equal circles equal circ.u.mferences are subtended by equal straight lines." It will therefore be noticed that he has no special word for "chord" and none for "arc," and that the word "circ.u.mference," which some teachers are so anxious to retain, is used to mean both the whole circle and any arc. It cannot be doubted that later writers have greatly improved the language of geometry by the use of these modern terms. The word "arc" is the same, etymologically, as "arch," each being derived from the Latin _arcus_ (a bow). "Chord" is from the Greek, meaning "the string of a musical instrument." "Subtend" is from the Latin _sub_ (under), and _tendere_ (to stretch).

It should be noticed that Euclid speaks of "equal circles," while we speak of "the same circle or equal circles," confining our proofs to the latter, on the supposition that this sufficiently covers the former.

THEOREM. _A line through the center of a circle perpendicular to a chord bisects the chord and the arcs subtended by it._

This is an improvement on Euclid, III, 3: "If in a circle a straight line through the center bisects a straight line not through the center, it also cuts it at right angles; and if it cuts it at right angles, it also bisects it." It is a very important proposition, theoretically and practically, for it enables us to find the center of a circle if we know any part of its arc. A civil engineer, for example, who wishes to find the center of the circle of which some curve (like that on a running track, on a railroad, or in a park) is an arc, takes two chords, say of one hundred feet each, and erects perpendicular bisectors. It is well to ask a cla.s.s why, in practice, it is better to take these chords some distance apart. Engineers often check their work by taking three chords, the perpendicular bisectors of the three pa.s.sing through a single point. Ill.u.s.trations of this kind of work are given later in this chapter.