[pi] = 2^_n_[sqrt](2 - [sqrt](2 + [sqrt](2 + [sqrt](2 + [sqrt](2...))))).
Students of elementary geometry are not prepared to appreciate it, but teachers will be interested in the remarkable formula discovered by Euler (1707-1783), the great Swiss mathematician, namely, 1 + _e_^{_i_[pi]} = 0. In this relation are included the five most interesting quant.i.ties in mathematics,--zero, the unit, the base of the so-called Napierian logarithms, _i_ = [sqrt](-1), and [pi]. It was by means of this relation that the transcendence of _e_ was proved by the French mathematician Hermite, and the transcendence of [pi] by the German Lindemann.
[Ill.u.s.tration]
There should be introduced at this time, if it has not already been done, the proposition of the lunes of Hippocrates (_ca._ 470 B.C.), who proved a theorem that a.s.serts, in somewhat more general form, that if three semicircles be described on the sides of a right triangle as diameters, as shown, the lunes _L_ + _L"_ are together equivalent to the triangle _T_.
[Ill.u.s.tration]
In the use of the circle in design one of the simplest forms suggested by Book V is the trefoil (three-leaf), as here shown, with the necessary construction lines. This is a very common ornament in architecture, both with rounded ends and with the ends slightly pointed.
The trefoil is closely connected with hexagonal designs, since the regular hexagon is formed from the inscribed equilateral triangle by doubling the number of sides. The following are designs that are easily made:
[Ill.u.s.tration]
It is not very profitable, because it is manifestly unreal, to measure the parts of such figures, but it offers plenty of practice in numerical work.
[Ill.u.s.tration: CHOIR OF LINCOLN CATHEDRAL]
[Ill.u.s.tration: PORCH OF LINCOLN CATHEDRAL]
In the ill.u.s.trations of the Gothic windows given in Chapter XV only the square and circle were generally involved. Teachers who feel it necessary or advisable to go outside the regular work of geometry for the purpose of increasing the pupil"s interest or of training his hand in the drawing of figures will find plenty of designs given in any pictures of Gothic cathedrals. For example, this picture of the n.o.ble window in the choir of Lincoln Cathedral shows the use of the square, hexagon, and pentagon. In the porch of the same cathedral, shown in the next ill.u.s.tration, the architect has made use of the triangle, square, and pentagon in planning his ornamental stonework. It is possible to add to the work in pure geometry some work in the mensuration of the curvilinear figures shown in these designs. This form of mensuration is not of much value, however, since it places before the pupil a problem that he sees at once is fict.i.tious, and that has no human interest.
[Ill.u.s.tration: GOTHIC DESIGNS EMPLOYING CIRCLES AND BISECTED ANGLES]
[Ill.u.s.tration: GOTHIC DESIGNS EMPLOYING CIRCLES AND SQUARES]
[Ill.u.s.tration: GOTHIC DESIGNS EMPLOYING CIRCLES AND THE EQUILATERAL TRIANGLE]
[Ill.u.s.tration: GOTHIC DESIGNS EMPLOYING CIRCLES AND THE REGULAR HEXAGON]
The designs given on page 283 involve chiefly the square as a basis, but it will be seen from one of the figures that the equilateral triangle and the hexagon also enter. The possibilities of endless variation of a single design are shown in the ill.u.s.tration on page 284, the basis in this case being the square. The variations in the use of the triangle and hexagon have been the object of study of many designers of Gothic windows, and some examples of these forms are shown on page 285. In more simple form this ringing of the changes on elementary figures is shown on page 286. Some teachers have used color work with such designs for the purpose of increasing the interest of their pupils, but the danger of thus using the time with no serious end in view will be apparent.
[Ill.u.s.tration]
In the matter of the mensuration of the circle the annexed design has some interest. The figure is not uncommon in decoration, and it is interesting to show, as a matter of pure geometry, that the area of the circle is divided into three equal portions by means of the four interior semicircles.
[Ill.u.s.tration]
An important application of the formula _a_ = [pi]_r_^2 is seen in the area of the annulus, or ring, the formula being _a_ = [pi]_r_^2 - [pi]_r"_^2 = [pi](_r_^2 - _r"_^2) = [pi](_r_ + _r"_)(_r_ - _r"_).
It is used in finding the area of the cross section of pipes, and this is needed when we wish to compute the volume of the iron used.
Another excellent application is that of finding the area of the surface of a cylinder, there being no reason why such simple cases from solid geometry should not furnish working material for plane geometry, particularly as they have already been met by the pupils in arithmetic.
A little problem that always has some interest for pupils is one that Napoleon is said to have suggested to his staff on his voyage to Egypt: To divide a circle into four equal parts by the use of circles alone.
[Ill.u.s.tration]
Here the circles _B_ are tangent to the circle _A_ at the points of division. Furthermore, considering areas, and taking _r_ as the radius of _A_, we have _A_ = [pi]_r_^2, and _B_ = [pi](_r_/2)^2. Hence _B_ = 1/4_A_, or the sum of the areas of the four circles _B_ equals the area of _A_. Hence the four _D"_s must equal the four _C"_s, and _D_ = _C_. The rest of the argument is evident. The problem has some interest to pupils aside from the original question suggested by Napoleon.
At the close of plane geometry teachers may find it helpful to have the cla.s.s make a list of the propositions that are actually used in proving other propositions, and to have it appear what ones are proved by them.
This forms a kind of genealogical tree that serves to fix the parent propositions in mind. Such a work may also be carried on at the close of each book, if desired. It should be understood, however, that certain propositions are used in the exercises, even though they are not referred to in subsequent propositions, so that their omission must not be construed to mean that they are not important.
An exercise of distinctly less value is the cla.s.sification of the definitions. For example, the cla.s.sification of polygons or of quadrilaterals, once so popular in textbook making, has generally been abandoned as tending to create or perpetuate unnecessary terms. Such work is therefore not recommended.
FOOTNOTES:
[83] Bosanquet and Sayre, "The Babylonian Astronomy," _Monthly Notices of the Royal Asiatic Society_, Vol. XL, p. 108.
[84] This and the three ill.u.s.trations following are from Kolb, loc. cit.
[85] This was in five colors of marble.
[86] The proof is too involved to be given here. The writer has set it forth in a chapter on the transcendency of [pi] in a work soon to be published by Professor Young of The University of Chicago.
CHAPTER XIX
THE LEADING PROPOSITIONS OF BOOK VI
There have been numerous suggestions with respect to solid geometry, to the effect that it should be more closely connected with plane geometry.
The attempt has been made, notably by Meray in France and de Paolis in Italy, to treat the corresponding propositions of plane and solid geometry together; as, for example, those relating to parallelograms and parallelepipeds, and those relating to plane and spherical triangles.
Whatever the merits of this plan, it is not feasible in America at present, partly because of the nature of the college-entrance requirements. While it is true that to a boy or girl a solid is more concrete than a plane, it is not true that a geometric solid is more concrete than a geometric plane. Just as the world developed its solid geometry, as a science, long after it had developed its plane geometry, so the human mind grasps the ideas of plane figures earlier than those of the geometric solid.
There is, however, every reason for referring to the corresponding proposition of plane geometry when any given proposition of solid geometry is under consideration, and frequent references of this kind will be made in speaking of the propositions in this and the two succeeding chapters. Such reference has value in the apperception of the various laws of solid geometry, and it also adds an interest to the subject and creates some approach to power in the discovery of new facts in relation to figures of three dimensions.
The introduction to solid geometry should be made slowly. The pupil has been accustomed to seeing only plane figures, and therefore the drawing of a solid figure in the flat is confusing. The best way for the teacher to antic.i.p.ate this difficulty is to have a few pieces of cardboard, a few knitting needles filed to sharp points, a pine board about a foot square, and some small corks. With the cardboard he can ill.u.s.trate planes, whether alone, intersecting obliquely or at right angles, or parallel, and he can easily ill.u.s.trate the figures given in the textbook in use. There are models of this kind for sale, but the simple ones made in a few seconds by the teacher or the pupil have much more meaning. The knitting needles may be stuck in the board to ill.u.s.trate perpendicular or oblique lines, and if two or more are to meet in a point, they may be held together by sticking them in one of the small corks. Such homely apparatus, costing almost nothing, to be put together in cla.s.s, seems much more real and is much more satisfactory than the German models.[87]
An extensive use of models is, however, unwise. The pupil must learn very early how to visualize a solid from the flat outline picture, just as a builder or a mechanic learns to read his working drawings. To have a model for each proposition, or even to have a photograph or a stereoscopic picture, is a very poor educational policy. A textbook may properly ill.u.s.trate a few propositions by photographic aids, but after that the pupil should use the kind of figures that he must meet in his mathematical work. A child should not be kept in a perambulator all his life,--he must learn to walk if he is to be strong and grow to maturity; and it is so with a pupil in the use of models in solid geometry.[88]
The case is somewhat similar with respect to colored crayons. They have their value and their proper place, but they also have their strict limitations. It is difficult to keep their use within bounds; pupils come to use them to make pleasing pictures, and teachers unconsciously fall into the same habit. The value of colored crayons is two-fold: (1) they sometimes make two planes stand out more clearly, or they serve to differentiate some line that is under consideration from others that are not; (2) they enable a cla.s.s to follow a demonstration more easily by hearing of "the red plane perpendicular to the blue one," instead of "the plane _MN_ perpendicular to the plane _PQ_." But it should always be borne in mind that in practical work we do not have colored ink or colored pencils commonly at hand, nor do we generally have colored crayons. Pupils should therefore become accustomed to the pencil and the white crayon as the regulation tools, and in general they should use them. The figures may not be as striking, but they are more quickly made and they are more practical.
The definition of "plane" has already been discussed in Chapter XII, and the other definitions of Book VI are not of enough interest to call for special remark. The axioms are the same as in plane geometry, but there is at least one postulate that needs to be added, although it would be possible to state various a.n.a.logues of the postulates of plane geometry if we cared unnecessarily to enlarge the number.
The most important postulate of solid geometry is as follows: _One plane, and only one, can be pa.s.sed through two intersecting straight lines._ This is easily ill.u.s.trated, as in most textbooks, as also are three important corollaries derived from it:
1. _A straight line and a point not in the line determine a plane._ Of course this may be made the postulate, as may also the next one, the postulate being placed among the corollaries, but the arrangement here adopted is probably the most satisfactory for educational purposes.
2. _Three points not in a straight line determine a plane._ The common question as to why a three-legged stool stands firmly, while a four-legged table often does not, will add some interest at this point.
3. _Two parallel lines determine a plane._ This requires a slight but informal proof to show that it properly follows as a corollary from the postulate, but a single sentence suffices.
While studying this book questions of the following nature may arise with an advanced cla.s.s, or may be suggested to those who have had higher algebra:
How many straight lines are in general (that is, at the most) determined by _n_ points in s.p.a.ce? Two points determine 1 line, a third point adds (in general, in all these cases) 2 more, a fourth point adds 3 more, and an _n_th point _n_ - 1 more. Hence the maximum is 1 + 2 + 3 + ... + (_n_ - 1), or _n_(_n_-1)/2, which the pupil will understand if he has studied arithmetical progression. The maximum number of intersection points of _n_ straight lines in the same plane is also _n_(_n_ - 1)/2.