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PREFACE.

The present work is intended for those students who are occupied in graphically representing the forms of bodies, and the delineations of machines. To such a class the advantage of having general methods by which the position of points, lines, and surfaces, may be determined with exactness and precision, is very obvious. Descriptive Geometry supplies this want. Invented by the genius of Monge, and pursued with ardour and success by the most eminent French Geometricians, it is now taught in almost all the universities and in the principal schools of the continent. In England it was unknown, as a branch of instruction, until lectures were given upon it by Mr. Bradley, in the Engineering Department of King's College; and the present work has been undertaken to supply the students with a text book, that by it they might the more profitably attend to what they heard in the lecture room : and as an elementary book was necessary for beginners, it has been thought expedient to place before the students, in an English dress, one which has stood the test of experience. The treatise on Descriptive Geometry, by Mr. Lefebure de Fourcy has, therefore, been

selected, and the following pages are, for the most part, translated from it.

And now a few words on the subject itself. It is obvious that, in the progress of the arts, there is a constant necessity of transmitting to others the knowledge that is possessed of the forms of bodies; either for the purpose of exhibiting the geometrical relations that have been discovered, or of guiding the workman engaged in the construction of objects whose dimensions are given. To effect this, two methods may be used ; 1°, a Model ; 2°, a Drawing or Plan. The first method is commonly impracticable; not indeed of itself, but from the difficulty and expense of execution, and the time which would be spent in making the model. To the second method, the same objections do not apply with the same force as to the first, but another and peculiar difficulty belongs to it, which it is the special object of Descriptive Geometry to remove. For, the bodies to which reference has been made, the buildings which the workman has to erect, the machines, parts of which the drawings represent, are in different planes, while the sheet of paper which contains their graphical descriptions is in one plane. Means, therefore, must be found, by which the points in space may be referred to a single plane, and which, in every problem, shall express with precision the given and required quantities. It is this exactness which constitutes the important difference between ordinary and Descriptive Geometry. In the former, the figures, often drawn

vaguely, serve to guide the mind in the series of reasonings necessary to establish the truth of a theorem or the construction of a problem. In the latter, the steadiness of the hand and eye is also requisite: one line carelessly or inaccurately drawn, vitiates the whole result.

It has been mentioned, that Descriptive Geometry owes its origin to Monge. He first conceived, that bodies might be represented by figures drawn in a single plane, by the aid of projections on two different planes, commonly perpendicular to each other, and which we may consider as respectively horizontal and vertical. One of these, as, for instance, the vertical, is supposed to turn round their common intersection until it coincides with the prolongation of the other. The vertical projections are thus reduced to the horizontal plane, and their actual distances from the line of intersection always preserved. Considering points and lines to be determined by their projections, and surfaces by the projections of their generatrices, Monge ascertained that the various problems of the geometry of three dimensions, which had been solved before his time by a variety of processes, could be reduced to a small number of abstract problems, and formed simple and entirely general methods for their graphical solution. For the development and full application of these principles, the reader is referred to the works of this great Geometer, and to those of the other French mathematicians, who have enriched the subject with a great variety of problems.

· It remains for me to express my acknowledgments to Mr. Bradley, of King's College, for the kind assistance he afforded me, during the progress of this work. Without his help, I should not have ventured to undertake it: he carefully revised all that I had written, and when I was unable to give my attention to it, prepared a considerable portion of the book for the press.

SS.

T. G. HALL.

King's College, London,

October, 1841.

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