the preface to his Graphic Statics, Culmann presupposes a knowledge of geometry of position, of the principles of which he makes frequent use. For similar reasons Favaro took the more definite course of publishing a treatise on geometry of position as an introduction to lessons on graphic statics. Before Favaro, Cremona had done much the same thing.

Thus

My own knowledge of the subject has been acquired in a rather peculiar way. Prior to reading Culmann's Graphic Statics, many years ago, I had omitted to make a previous study of geometry of position. I found myself obliged to prove independently, as occasion required, whatever theorems Culmann took for granted or acknowledged from other sources. Such a method has its merits and demerits for those who do not shrink from the labour it may entail. In the first place, it no doubt leads to original research and the discovery of new solutions, whilst it confines the attention to such parts of the subject as are really useful or required in the applied sciences. On the other hand, one is apt to lose the benefit of the labour and thought of others who may have devised shorter or more elegant solutions. Wherefore, before writing this work, I have taken the precaution to read Favaro,* whom, from the varied extent of his reading, I took for the best guide to the history of the subject. In this anticipation I was certainly not disappointed; for his work is a marvel of literary research and historical reference. But, at the same time, this very wealth of

* Translated into French by M. Paul Terrier (Gauthier Villars, 1879). Favaro borrowed largely from Reye and other authors cited in

his book.

Full and precise in his

historical reference often necessitates the exclusion of geometric proof, just where geometric proof would be most welcome and desirable. statements of fact and the descriptive solutions of problems, Favaro is all the more sparing of geometrical demonstrations. For instance, no demonstration is given of the problem how to construct a conic five of whose points are given, or of a conic five of whose tangents are known. Hence, believing that geometric proof was more needed than historical exposition, I have done my best to supply geometric demonstrations of all the problems or theorems included in this work. Some of these I have borrowed from Newton, a few from Culmann, one or two from Favaro; but the vast majority of the given solutions are original, including most of the projective demonstrations contained in chapters i. and ii., and the genesis of reciprocal figures, chapter iii.

The graphic treatment of the elastic line in single or continuous spans (ch. vi.) is due, as an original conception, to Professor Mohr. The perfected form in which it is here given embraces the further developments made by Professor W. Ritter of Zürich in his excellent little pamphlet on the subject, as well as a few embellishments in the proof made by myself.

By introducing chapters on reciprocal figures, centres of gravity, ellipses of inertia and central kerns, as well as on the elastic line, I depart from the usual arrangement with distinct advantage; for the projective relations of a stress-diagram to its frame and the determination of the centre of gravity by graphic pro

cess essentially involve geometrical problems of position; whilst the construction of ellipses of inertia, central kerns, and the elastic line of deflection under given conditions of load, ultimately depends upon the ordinary principles of the subject.

The demonstration of the perspectivity of two ranges drawn in any direction from a point on a conic (Figs. 13-17) was originally published by the author in Nature.

A great proportion of the examples is original; but for many of those set at the end of chapter ii., as well as for a few of those appended to subsequent chapters, I am indebted to the collection published with the third edition of Reye's Geometrie der Lage. By actually working out the examples I have ascertained that they can all be solved by the principles expounded in the text. The examples set at the ends of chapters i. and ii. flow naturally from the propositions established in those chapters; but subsequently, whenever an example is given requiring the aid of earlier propositions, I have taken care to add the necessary reference in brackets.

The invention of the theory of reciprocal figures has been so much disputed that it is scarcely possible for an author of a work of this kind to escape the responsibility of expressing an opinion. Culmann, on p. 10 of the preface to the second edition of his work on Graphic Statics, remarks:

"Cremona is extremely kind and polite after the Italian fashion, when he acknowledges Maxwell as the original author of reciprocal figures. The reciprocity invented by Maxwell is far from universal in the sense

that to every straight line in one figure there exists a reciprocal straight line in the other. In fact Maxwell himself admits as much in respect of Fig. 7 of his paper, published in the Philosophical Magazine for April 1864. Neither are figures 3 of the same paper reciprocal, as drawn; but they would be, if the line c were drawn through the point of intersection osr, and a through ksn. Generally speaking, reciprocal figures composed of straight lines can only be proved to be such by the help of the Nullsystem, being then regarded as limiting surfaces of a solid body. Now the introduction of the Nullsystem is due to Cremona, not to Maxwell."

This dogmatic assertion of Culmann's scarcely calls for refutation; for the very first figure in Maxwell's paper embraces the whole theory of the so-called Nullsystem; and Levy, who seems to have closely followed Cremona, bases the whole superstructure of his work on Maxwell's fundamental proposition (compare Levy's Statique Graphique, § 18, and Maxwell's Fig. 1). In Fig. 7, which Culmann says is not reciprocal, Maxwell gives us an example of what he terms the "redundant" case; and Fig. 3 is an example of the use of a "subsidiary" polygon to solve a lock-joint in the frame. Further, it would be interesting to learn whether Culmann wrote his preface before or after Art. 81 of his work on statics; for in the first paragraph of that article he states:

"The aforesaid reciprocal relations between the force and funicular polygons have been deduced without leaving the plane of the two figures; but, if the

polygons be regarded as projections of lines in space, such lines may be treated as reciprocal elements of a Nullsystem. This was first demonstrated by Maxwell in the Philosophical Magazine for April 1864 and afterwards further developed by Cremona."

Without doubt, then, Maxwell is the original author of the polar theory of reciprocal figures, which he completed in a paper contributed to the Transactions of the Royal Society of Edinburgh in 1867. In that paper the history of the subject is given in very simple and straightforward language:

"The properties of the 'triangle' and 'polygon of forces' have been long known and the diagram of forces has been used in the case of the funicular polygon, but I am not aware of any more general statement of the method of drawing diagrams of forces before Professor Rankine applied it to frames, roofs, etc. in his Applied Mechanics, p. 137. The 'polyhedron of forces,' or the proposition that forces acting on a point perpendicular and proportional to the areas of the faces. of a polyhedron are in equilibrium, has, I believe, been enunciated independently at various times; but the application of this principle to the construction of a diagram of forces in three dimensions was first made by Professor Rankine in the Philosophical Magazine for Feb. 1864. In the Philosophical Magazine for April 1864 I stated some of the properties of reciprocal figures and the conditions of their existence, and shewed that any rectilinear figure which is a perspective representation of a closed polyhedron with plane faces has a reciprocal figure. In Sept. 1867 I com