For example, let it be required to plan a driveway from the gate _G_ to the porch _P_ so as to avoid a ma.s.s of rocks _R_, an arc of a circle to be taken. Of course, if we allow pupils to use the Pythagorean Theorem at this time (and for metrical purposes this is entirely proper, because they have long been familiar with it), then we may ask not only for the drawing, but we may, for example, give the length from _G_ to the point on _R_ (which we may also call _R_), and the angle _RGO_ as 60, to find the radius.
A second general line of exercises adapted to Book II is a continuation of the geometric drawing recommended as a preliminary to the work in demonstrative geometry. The copying or the making of designs requiring the describing of circles, their inscription in or circ.u.mscription about triangles, and their construction in various positions of tangency, has some value as applying the various problems studied in this book. For a number of years past, several enthusiastic teachers have made much of the designs found in Gothic windows, having their pupils make the outline drawings by the help of compa.s.ses and straightedge. While such work has its value, it is liable soon to degenerate into purposeless formalism, and hence to lose interest by taking the vigorous mind of youth from the strong study of geometry to the weak manipulation of instruments. Nevertheless its value should be appreciated and conserved, and a few ill.u.s.trations of these forms are given in order that the teacher may have examples from which to select. The best way of using this material is to offer it as supplementary work, using much or little, as may seem best, thus giving to it a freshness and interest that some have trouble in imparting to the regular book work.
The best plan is to sketch rapidly the outline of a window on the blackboard, asking the pupils to make a rough drawing, and to bring in a mathematical drawing on the following day.
[Ill.u.s.tration]
It might be said, for example, that in planning a Gothic window this drawing is needed. The arc _BC_ is drawn with _A_ as a center and _AB_ as a radius. The small arches are described with _A_, _D_, and _B_ as centers and _AD_ as a radius. The center _P_ is found by taking _A_ and _B_ as centers and _AE_ as a radius. How may the points _D_, _E_, and _F_ be found? Draw the figure. From the study of the rectilinear figures suggested by such a simple pattern the properties of the equilateral triangle may be inferred.
The Gothic window also offers some interesting possibilities in connection with the study of the square. For example, the ill.u.s.tration given on page 223 shows a number of traceries involving the construction of a square, the bisecting of angles, and the describing of circles.[72]
[Ill.u.s.tration]
The properties of the square, a figure now easily constructed by the pupils, are not numerous. What few there are may be brought out through the study of art forms, if desired. In case these forms are shown to a cla.s.s, it is important that they should be selected from good designs.
We have enough poor art in the world, so that geometry should not contribute any more. This ill.u.s.tration is a type of the best medieval Gothic parquetry.[73]
[Ill.u.s.tration: GOTHIC DESIGNS EMPLOYING CIRCLES AND BISECTED ANGLES]
Even simple designs of a semipuzzling nature have their advantage in this connection. In the following example the inner square contains all of the triangles, the letters showing where they may be fitted.[74]
Still more elaborate designs, based chiefly upon the square and circle, are shown in the window traceries on page 225, and others will be given in connection with the study of the regular polygons.
[Ill.u.s.tration]
Designs like the figure below are typical of the simple forms, based on the square and circle, that pupils may profitably incorporate in any work in art design that they may be doing at the time they are studying the circle and the problems relating to perpendiculars and squares.
[Ill.u.s.tration]
Among the applications of the problem to draw a tangent to a given circle is the case of the common tangents to two given circles. Some authors give this as a basal problem, although it is more commonly given as an exercise or a corollary. One of the most obvious applications of the idea is that relating to the transmission of circular motion by means of a band over two wheels,[75] _A_ and _B_, as shown on page 226.
[Ill.u.s.tration: GOTHIC DESIGNS EMPLOYING CIRCLES AND BISECTED ANGLES]
The band may either not be crossed (the case of the two exterior tangents), or be crossed (the interior tangents), the latter allowing the wheels to turn in opposite directions. In case the band is liable to change its length, on account of stretching or variation in heat or moisture, a third wheel, _D_, is used. We then have the case of tangents to three pairs of circles. Ill.u.s.trations of this nature make the exercise on the drawing of common tangents to two circles a.s.sume an appearance of genuine reality that is of advantage to the work.
[Ill.u.s.tration]
FOOTNOTES:
[68] This is the latest opinion. He is usually a.s.signed to the first century B.C.
[69] See page 54.
[70] A Greek philosopher and mathematician of the fifth century B.C.
[71] This ill.u.s.tration and the following two are from C. Dupin, "Mathematics Practically Applied," translated from the French by G.
Birkbeck, Halifax, 1854. This is probably the most scholarly attempt ever made at constructing a "practical geometry."
[72] This ill.u.s.tration and others of the same type used in this work are from the excellent drawings by R. W. Billings, in "The Infinity of Geometric Design Exemplified," London, 1849.
[73] From H. Kolb, "Der Ornamentenschatz ... aus allen Kunst-Epochen,"
Stuttgart, 1883. The original is in the Church of Saint Anastasia in Verona.
[74] From J. Bennett, "The Arcanum ... A Concise Theory of Practicable Geometry," London, 1838, one of the many books that have a.s.sumed to revolutionize geometry by making it practical.
[75] The figures are from Dupin, loc. cit.
CHAPTER XVI
THE LEADING PROPOSITIONS OF BOOK III
In the American textbooks Book III is usually a.s.signed to proportion. It is therefore necessary at the beginning of this discussion to consider what is meant by ratio and proportion, and to compare the ancient and the modern theories. The subject is treated by Euclid in his Book V, and an anonymous commentator has told us that it "is the discovery of Eudoxus, the teacher of Plato." Now proportion had been known long before the time of Eudoxus (408-355 B.C.), but it was numerical proportion, and as such it had been studied by the Pythagoreans. They were also the first to study seriously the incommensurable number, and with this study the treatment of proportion from the standpoint of rational numbers lost its scientific position with respect to geometry.
It was because of this that Eudoxus worked out a theory of geometric proportion that was independent of number as an expression of ratio.
The following four definitions from Euclid are the basal ones of the ancient theory:
A ratio is a sort of relation in respect of size between two magnitudes of the same kind.
Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.
Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.
Let magnitudes which have the same ratio be called proportional.[76]
Of these, the first is so loose in statement as often to have been thought to be an interpolation of some later writer. It was probably, however, put into the original for the sake of completeness, to have some kind of statement concerning ratio as a preliminary to the important definition of quant.i.ties in the same ratio. Like the definition of "straight line," it was not intended to be taken seriously as a mathematical statement.
The second definition is intended to exclude zero and infinite magnitudes, and to show that incommensurable magnitudes are included.
The third definition is the essential one of the ancient theory. It defines what is meant by saying that magnitudes are in the same ratio; in other words, it defines a proportion. Into the merits of the definition it is not proposed to enter, for the reason that it is no longer met in teaching in America, and is practically abandoned even where the rest of Euclid"s work is in use. It should be said, however, that it is scientifically correct, that it covers the case of incommensurable magnitudes as well as that of commensurable ones, and that it is the Greek forerunner of the modern theories of irrational numbers.
As compared with the above treatment, the one now given in textbooks is unscientific. We define ratio as "the quotient of the numerical measures of two quant.i.ties of the same kind," and proportion as "an equality of ratios."
But what do we mean by the quotient, say of [sqrt]2 by [sqrt]3? And when we multiply a ratio by [sqrt]5, what is the meaning of this operation?
If we say that [sqrt]2 : [sqrt]3 means a quotient, what meaning shall we a.s.sign to "quotient"? If it is the number that shows how many times one number is contained in another, how many _times_ is [sqrt]3 contained in [sqrt]2? If to multiply is to take a number a certain number of times, how many times do we take it when we multiply by [sqrt]5? We certainly take it more than 2 times and less than 3 times, but what meaning can we a.s.sign to [sqrt]5 times? It will thus be seen that our treatment of proportion a.s.sumes that we already know the theory of irrationals and can apply it to geometric magnitudes, while the ancient treatment is independent of this theory.
Educationally, however, we are forced to proceed as we do. Just as Dedekind"s theory of numbers is a simple one for college students, so is the ancient theory of proportion; but as the former is not suited to pupils in the high school, so the latter must be relegated to the college cla.s.ses. And in this we merely harmonize educational progress with world progress, for the numerical theory of proportion long preceded the theory of Eudoxus.
The ancients made much of such terms as duplicate, triplicate, alternate, and inverse ratio, and also such as composition, separation, and conversion of ratio. These entered into such propositions as, "If four magnitudes are proportional, they will also be proportional alternately." In later works they appear in the form of "proportion by composition," "by division," and "by composition and division." None of these is to-day of much importance, since modern symbolism has greatly simplified the ancient expressions, and in particular the proposition concerning "composition and division" is no longer a basal theorem in geometry. Indeed, if our course of study were properly arranged, we might well relegate the whole theory of proportion to algebra, allowing this to precede the work in geometry.
We shall now consider a few of the princ.i.p.al propositions of Book III.
THEOREM. _If a line is drawn through two sides of a triangle parallel to the third side, it divides those sides proportionally._
In addition to the usual proof it is instructive to consider in cla.s.s the cases in which the parallel is drawn through the two sides produced, either below the base or above the vertex, and also in which the parallel is drawn through the vertex.