Book II offers two general lines of application that may be introduced

to advantage, preferably as additions to the textbook work. One of these has reference to topographical drawing and related subjects, and the other to geometric design. As long as these can be introduced to the pupil with an air of reality, they serve a good purpose, but if made a part of textbook work, they soon come to have less interest than the exercises of a more abstract character. If a teacher can relate the problems in topographical drawing to the pupil's home town, and can occasionally set some outdoor work of the nature here suggested, the results are usually salutary; but if he reiterates only a half-dozen simple propositions time after time, with only slight changes in the nature of the application, then the results will not lead to a cultivation of power in geometry,--a point which the writers on applied geometry usually fail to recognize.

One of the simple applications of this book relates to the rounding of corners in laying out streets in some of our modern towns where there is a desire to depart from the conventional square corner. It is also used in laying out park walks and drives.

The figure in the middle of the page represents two streets, _AP_ and _BQ_, that would, if prolonged, intersect at _C_. It is required to construct an arc so that they shall begin to curve at _P_ and _Q_, where _CP_ = _CQ_, and hence the "center of curvature" _O_ must be found.

The problem is a common one in railroad work, only here _AP_ is usually oblique to _BQ_ if they are produced to meet at _C_, as in the second figure on page 218. It is required to construct an arc so that the tracks shall begin to curve at _P_ and _Q_, where _CP_ = _CQ_.

The problem becomes a little more complicated, and correspondingly more interesting, when we have to find the center of curvature for a street railway track that must turn a corner in such a way as to allow, say, exactly 5 feet from the point _P_, on account of a sidewalk.

The problem becomes still more difficult if we have two roads of different widths that we wish to join on a curve. Here the two centers of curvature are not the same, and the one road narrows to the other on the curve. The solutions will be understood from a study of the figures.

The number of problems of this kind that can easily be made is limitless, and it is well to avoid the danger of hobby riding on this or any similar topic. Therefore a single one will suffice to close this group.

If a road _AB_ on an arc described about _O_, is to be joined to road _CD_, described about _O'_, the arc _BC_ should evidently be internally tangent to _AB_ and externally tangent to _CD_. Hence the center is on _BOX_ and _O'CY_, and is therefore at _P_. The problem becomes more real if we give some width to the roads in making the drawing, and imagine them in a park that is being laid out with drives.

It will be noticed that the above problems require the erecting of perpendiculars, the bisecting of angles, and the application of the propositions on tangents.

A somewhat different line of problems is that relating to the passing of a circle through three given points. It is very easy to manufacture problems of this kind that have a semblance of reality.

For example, let it be required to plan a driveway from the gate _G_ to the porch _P_ so as to avoid a mass 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 circumscription 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 compasses 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 illustrations 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.

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 illustration 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]

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 class, 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 illustration is a type of the best medieval Gothic parquetry.[73]

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.

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.

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.

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. Illustrations of this nature make the exercise on the drawing of common tangents to two circles assume an appearance of genuine reality that is of advantage to the work.

FOOTNOTES:

[68] This is the latest opinion. He is usually assigned to the first century B.C.

[69] See page 54.

[70] A Greek philosopher and mathematician of the fifth century B.C.

[71] This illustration 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 illustration 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 assumed to revolutionize geometry by making it practical.

[75] The figures are from Dupin, loc. cit.