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which he had planned out. One thing is particularly noteworthy, viz: the large space occupied by constructions and propositions which may be regarded more or less of a general nature, compared with the space devoted to the solution of actual examples. The same thing occurs in the lucid and valuable work of Bauschinger, “Elemente der Graphischen Statik,” in which there are sixty-two pages devoted to what may be regarded as matter of a general nature, and only thirty to the applications. In the work of M. Levy, which is the standard work of France, although he has very much reduced in the second edition the portion of purely geometrical caluculation, the whole of the first volume, of between 500 and 600 pages, is devoted to the "Principles and Applications of Pure, Graphical Statics." The same facts are derived from a study of the detailed course as set forth in the various programs of technical schools.
The conclusion is that a great deal of the matter taught under the head of “Graphical Statics” contains general principles of graphic methods of construction which might be taught apart from any applications at all, and being so taught it would be capable of application, not only in cases of statics, but in dynamics and hydromechanics. Take as an example of the proposition of the ellipse of inertia and the central ellipse, which are only applied to force, and which are given as if they referred only to the problem of the beam, but which have an equally important bearing in hydrostactics and rigid dynamics, the ellipsoid of inertia having the properties for instance of the momental ellipsoid of Cauchy, the central ellipsoid those of the equimomental ellipsoid of Legendre. Indeed, there is every indication of a gradual tendency towards the development of the science of graphical calculation quite apart from that of graphic statics. Thus we find graphical constructions originally devised and given by writers (notably Rankine) as they were needed in works of mechanics.
Next we have the first collection of graphical calculation, already referred to, of which there is remarkable examples in the little work of Cremona, “Il Calcolo Grafico"'* in the preface of the English edition of which he acknowledges the work of Culmann, but goes considerably beyond that author, particularly in adding the important chapter on “Centroids” in which the properties of the centre of gravity are treated from a purely geometrical point of view without any reference whatever to force. A still more recent work is that of Favaro, who in his “Lessons on Graphical Statics” devoted his second volume entirely to the subject of graphical calculations.
From this it is clear that a course of instruction might be given under the head of graphical methods, which might be taught in the same way as descriptive geometry, and which ought indeed to be worked in conjunction with that subject. Graphical methods should comprise the constructions of such geometrical figures as are important for graphical application, should deal with the plotting of results and the general properties of plane curves, as far as the student is able to numerically effect measurements with it, which he can check by calculation and which he should be expected to work out with accuracy and to regard the results he obtains as being sufficiently accurate for practical work, although such examples need not in the first instance be applied by him to any practical engineering problems. Thus for instance, the proposition of projective geometry so far as the null or focal system is concerned, and the projective properties of bodies, and of the pole and antipolar, might be taught, although the whole subject of projective geometry is not necessary for engineers; and, unless it is desired to study the higher branches of the subject, there is no necessity whatever of a treatment such as Culmann has given in his chapters on the “Projective Relations between the Polygon of Forces and the Funicular Polygon” and “The relation of a System of Forces with the Null system, and with Curves of the third Degree, or of the Co-linear and Reciprocal Relations of the Funicular and Force Polygon.”
* Now translated into English by Professor Hudson Beare, and published by the Clarendon Press, Oxford.
AN ELEMENTARY GENERAL COURSE. There appears to be no reason, therefore, why an elementary course of a general nature, specially arranged so as to include all that an ordinary engineering student requires to know of graphical methods, should not be introduced as a regular subject in engineering schools, and the following arguments may be brought forward in support of this view.
1. Although the time tables of an engineering department are already full, yet it will be found that a course such as that suggested really includes much of what is taught at present in a desultory way, and such 'a course would obviate some of the teaching given under the heading of "descriptive geometry," so that during one or two terms of a year, it might be taken during the same hours as are already devoted to descriptive geometry, with possibly one lecture a week for one term, in the place of the actual lectures in applied engineering, into which at present graphic methods are often obliged to be introduced for the want of proper preliminary training in the subject by a student. Moreover, the time now devoted in the engineering laboratory to the plotting of curves might much better be occupied in the drawing-room itself, in connection with the practice of the plotting and interpolation of curves as a part of the subject of graphic methods, the data obtained from the engineering laboratory affording useful information.
2. The time spent in such graphical work would be an excellent discipline in accurate drawing for a student who is often inclined to regard a sketch, roughly representing an idea, as sufficient for practical purposes. A student should learn for himself that nothing is so easily deceived as the eye. The following illustration may be cited in support of this, by which it may be proved conclusively that things which are equal to the same things are not equal to one another. That is to say, if A, B, C, D, Figures 1 and la, is a rectangle, and the line DE is drawn equal to