MATHEMATICAL DRAWING AND MODELLING. The necessity for geometrical drawings and models is as old as geometry itself. The figure has formed the basis of many a geometrical truth ; and demonstration by mere inspection of this has frequently to do service for more rigorous proof. So necessary is this visual representation of an idea that there is hardly a branch of mathematics which does not make use of it in the form of tables, symbols, formula:, &c. The visual method is especially important in geometry. The figure is to the geometer what the numerical example is to the algebraist - on the one hand limiting the horizon, on the other imparting life to the conception. Herein lies the didactic value of the figure, which is the more indispensable the more elementary the stage of instruction. To be able to dispense with it is a faculty acquired only after a long and special training. The power of mental picturing is a talent which can be so strengthened by use that even a slightly gifted mind may acquire the power of carrying out a series of geometrical operations without the aid of a figure, provided these do. not lead into unfamiliar regions. But each new group of ideas which the geometer would master requires a new graphic setting forth, which not even the experienced can dispense with. Drawings are sufficient in plane geometry ; but solid geometry requires models, except in specially simple cases, in which delineation by means of perspective or some conventional method may suffice. Then, again, in passing from the geometry of the plane, straight line, and point in space to that of curved surfaces, tortuous curves, Jr,c., new and distinct graphical methods are necessary. The difficulties encountered in understanding new groups of geometrical forms are best removed by a careful study of a small number of characteristic models and drawings. As a means of education, the model is lively and suggestive, forming in this way a completing factor in the course of instruction. We remember the pleasure experienced when, after a discussion which has yielded a series of hardly reconcilable properties of one and the same geometrical figure, a model is exhibited which combined these pro perties in itself ; or the striking manner in which a deformable model either of pasteboard or thread executes its transformations before the eye of the observer and scientific student. The study of the model raises new and unexpected questions, and can even do valuable service in leading to new truths.

In the more elementary departments of plane and solid geometry and descriptive geometry, models are abundant and easily obtainable ; but there are comparatively few collections of drawings and models for instruction in higher geometry. There are numerous drawings of algebraic and transcendental curves in the well-known treatises on analytical geometry of Cramer, Euler, Salmon, in Frost's Curve Tracing, &c. ; but there is still a deficiency in systematic enumerations of the forms of curves and surfaces of a given order or class. In this connexion we may mention Pliicker's System der analytischen Geometrie (curves of the third order), and Beer's Eieumeratio linearunt Ir. ordinis. A graphical representation of all the characteristics of the singular points of an algebraic curve of the fourth order is given in Zeuthen's Systemer of plane Kurver (1873). As regards tridimensional figuring, the oldest known models for instruction in the higher geometry are the thread models of skew surfaces constructed about the year 1800 under the direction of G. Monge for the Ecole Polytechnique in Paris. In 1830 Th. Olivier of Paris got the same executed in movable form. The great development in modern times of certain branches especially of the higher geometry has given a new importance to such methods of graphical representation.





Amongst the larger collections we must mention the elegant series of complex surfaces, consisting of twenty-seven items, constructed by the celebrated J. Pliicker of Bonn. After Pliicker's death copies, not very satisfactory, were made from zinc casts. The collection of plaster and thread models published by Buret of Paris (now Delagrave), and intended for instruction in descriptive geometry, contains many architectural forms. The wire models of tortuous curves by Professor Wiener of Carlsruhe, and the thread-models of developable surfaces by Professor Bjorling of Lund, merit notice amongst others. Perhaps the largest and most extensive of the collections is that of L. Brill, bookseller in Darmstadt. These represent every department of the higher and applied mathematics. The catalogue embraces some seventy numbers, with over a hundred plaster, thread, and metal models. Several series of this collection were prepared in the mathematical department of the technical college of Munich. In the preparation of these models, involving the development of a comparatively novel art, certain practical lessons were gained, especially in the working of plaster models, to which we may direct attention.

We assume that the preliminary designs are prepared with the aid of board, ruler, square, compasses, and such well-known instruments as are used by the draughtsman.

The material to be employed, whether wire or thread, interlaced pasteboard strips, or plaster, depends upon the special circumstances of each case, and upon the purpose aimed at in the construction of the model. Two bundles of parallel disks of cardboard or metal-sheeting, inclined at an adjustable angle, may be used with advantage in representing a series of different but mutually transformable surfaces. For ruled and developable surfaces the thread model is to be recommended. The surface is enclosed in a cube, or more generally in a region of space bounded by plane walls. Upon these bounding walls are marked the series of points in which they are cut by the generative lines that are to be represented by threads. Through these points the threads are drawn, and parts of the supporting walls arc then cut away so as to allow a convenient glance into the interior of the region. The more densely the threads are strung, the liker is the appearance to that of a continuous surface.

Tn the majority of cases plaster will be found the most convenient substance, being easily worked, and giving a result convenient and clear to the eye. There is the disadvantage, however, that one of the regions of space bounded by a surface is filled up. Should the boundaries of the surface be plane or capable of being turned on the lathe, the desired form is best approximated to by working wood or plaster blocks. Plaster is not easily worked on the lathe, but a plane surface is readily got by rubbing, and if not too dry it may be cut with the knife.

A surface which cannot be conveniently approximated to by the above method may be built up of strips cut to pattern, which are then filled in with some plastic material. To accomplish this, a system of sections either parallel or having a single axis is laid through the region to be filled up ; the bounding lines of these sections are calculated or obtained by geometrical construction. Strips of plate zinc are then cut to the required form and fixed securely by cross-pieces or soldered if necessary. Between the interstices of this zinc scaffolding some plastic material is filled in, such as embossing wax or damp clay ; and thus the form of the surface is rendered. The substance known in trade as plastilin is especially suitable for use in this way, as it retains its plastic property a long time. The finishing touches are given to the surface by means of a sculptor's style. From the clay model a plaster cast is formed and well dried ; and its imperfections are removed by means of plaster-files and other instruments familiar to those who work in plaster. Lines which are to be shown on the model are drawn through points already marked on the original clay model, and engraved with fine files. A gal vanoplastic copy of such a plaster model, not too deeply deposited, shows the surface even better than the plaster itself.