Mathematical Models

Mathematical models need not be accurate representations of a function in the same way, as, say, logarithmic tables or scales. They are not to be considered in the same category as graphs or nomograms. (See NOMOGRAPHY.) But they need to be constructed with reasonable care, and of suitable materials. It is sufficient if they enable the student to visualise the problem and follow the algebraic analysis involved. Mathematical models serve not to prove propositions but to demonstrate problems.

Materials for

mathematically plane surface has for example no counterpart in practice, but thin sheet metal or cardboard suffices for many purposes although in certain cases transparent celluloid or glass is to be preferred, whilst in others, strings, elastic, silk or cotton cords—which may be of different colours,—arranged closely together and parallel may be employed as where, for instance, it may be desired to demonstrate and variably warp or deform a surface; or to illustrate the con tinuously intersecting planes of descriptive geometry or the dis criminant surfaces involved in algebraic equations of the 4th or 5th degree, etc.

Descriptive

the study of descriptive geom etry (q.v.), perspective (q.v.), etc., a useful device is found in a pair of planes hinged together and possibly provided with a third plane of reference. Such folding planes if perforated allow of the setting up of problems in situ and the elucidation of the problems of orthogonal projection. The models introduced by Prof. Osborne Reynolds and G. Cussons of Manchester (1876) (P1. I fig. 3) and the more recent developments by Mr. Andrew H. Miller of Glas gow University (Pl. I. fig. 4), are interesting examples of this class. In the former type the problems are permanently drawn out, in the latter they may be built up before the eyes of the student, pre caution being taken in the design to avoid distracting the students' attention from the mathematics to the mechanism thus enabling the solution of the problems to be demonstrated in proper se quence step by step. In this class may be included the design (Pl. I., fig. 5), of Mr. H. G. Green of Nottingham University College and described in the Mathematical Gazette (London) No. 174. as A Model for Figures in Three Dimensions, which is partic ularly useful for three dimensional and trigonometrical studies. Posts may readily be fixed in the holes of the double base of the apparatus and cords as "Lines of vision," etc., serve to illustrate questions of the man and flagstaff type, subtended angles, etc.

Models of Wood.—Solid models of wood may be sectioned to elucidate many problems, an impressive example being shown in Plate I. fig. 6, a cube is cut into four different tetrahedra of equal volume, without making new corners. One face common to all four is that of half of the face of the cube; the sides being a face diagonal and two edges of the cube, the combination eluci dating the problems relating to square root, etc.

A further example is the well known model of the Binomial Cube, i.e., a cube built up of small cubes and prisms whose length of edge is represented by arbitrary value of a and b, and an entirely new and of course larger cube being formable by a combination of blocks equalling The study of conic sections so frequently treated analytically, is much simplified by the use of a model, such as the right cir cular cone, in which plane sections are made I) parallel to the base; 2) parallel to a generating line of the cone; 3) inclined to the axis at an angle greater than the semi-vertical angle of the cone; 4) inclined to the axis at an angle less than the semi-vertical angle of the cone, giving respectively the circle, parabola, ellipse, and hyperbola (one branch), while a combination of solid, wire and plane model allows demonstration (as in fig. i) of such solutions as the deter mination of the slope of a line by the method of inscribed spheres.

Problems concerned with the toroid (anchor ring) and cylinder, and interpenetration generally, can be most satisfactorily illus trated by wooden models since the common element is produced in the course of manufacture and its shape may be separately examined (Pl. I., fig. 7).

An interesting series of models is presented by the development of the higher species from the forms of the regular solids by cut ting off corners and edges and/or producing the faces until they meet again. Kepler (1619) appears to have discussed the species and it is known that they received attention at the hands of Meister (1771) although definite records are lost; but they were rediscovered by Poinsot in 1809 and have since been widely treated in particular by Cauchy, Bertrand, Cayley and Wiener. Since in the tetrahedron the faces already cut one another, it will be evident that it cannot have any higher species. Producing the faces of the cube we get a group of three intersecting square prisms, the faces of which may intersect again at infinity.