302

AN ELEMENTARY COURSE

OF

DESCRIPTIVE GEOMETRY.

I.

HISTORICAL NOTICE.

PROBABLY there are few of our readers who have not heard of the existence of a curious branch of mathematical science, called, by Monge and the modern French writers, Géométrie Descriptive; but those who have confined their reading to English authors, have little chance of knowing much of its nature and objects. It is true that several years ago, a considerable extract was made from Monge's work in the Architectural Dictionary of Mr. Peter Nicholson; but the high price and scarcity of that work preclude most readers from the opportunity of consulting it. Besides, the work of Monge, elegant as it is, is not exactly adapted for abridgment, and still less for unconnected extracts*. Mr. Nicholson afterwards commenced a work in numbers, bearing the title of Descriptive Geometry; but the commercial casualties of the period (1825) put a stop to the undertaking before he had entered upon the essential part of his subject; so that we have no means of judging of the method in which he intended to develop it. Nothing further on this branch of science has appeared in England.

In America, however, two separate works, on the more elementary parts of descriptive geometry, have been published for the use of the United States' Military Academy, at West Point. The first was by M. Crozet, the professor of engineering, and the other by Mr. Charles Davies, professor of mathematics in that institution. M. Crozet's work is a very indifferently selected series of extracts from Monge and Hachette; but Mr. Davies's is a work of a far higher character in its mathematical composition, though still much too confined in its scope and applications. The former was published in 1821; and was superseded by the latter in 1827; and this, again, was reprinted in 1832.

These are all the attempts which have been made to supply a treatise in our own language; and the American treatises are so little known, that we have never seen more than the single copy before us of either of them. They were all intended, too, to be subservient to military engineering, and hence must, of necessity, be very deficient in their applications to the wants of engineers and architects, as well as to those of the cultivators of physical science. These desiderata we hope to supply in the course of this series of papers: and we shall especially have regard to these objects, seldom taking our examples of the application of the principles of the science from other than civil or scientific pursuits, avoiding except in rare cases, the military examples of the French, Italian, German, and American writers.

Notwithstanding the importance of Geometry in the arts, as well as in philosophy, how few there are amongst those devoted to either pursuit,

* Those extracts, we believe, were made by Mr. Webster, F.G.S., the very able geologist, who discovered the tertiary formations in the Isle of Wight, and the author of several valuable papers on different geological subjects.

who have a competent knowledge of its nature, and how few, indeed, are those whose acquirements are of sufficient extent, to be of any real use to them! It is, indeed, a source of constant complaint, especially amongst practical men, that there is no work from which they can obtain the kind of geometrical knowledge, which is adapted to their daily wants. It is true that a sufficient number of works, under various appellations, and of various degrees of merit, have been written on the geometry of rule and compass, the construction of plane curves, and such subjects; and also that we have excellent treatises on theoretical or speculative Geometry, besides that of Euclid: yet, on the construction of problems relating to space, except of a purely theoretical kind on one hand, or a set of artificial rules adapted to single cases on the other, we are, to the present hour, unsupplied with one single work,—a work in which theory and practice go hand in hand, and mutually subserve each other. The consequence is, that the small number of persons who attend to the theory, see in it only a collection of abstract propositions, connected with each other, and mentally beautiful in that connexion it is true, though still having no practical bearing on the arts and necessities of highly-civilized life: whilst the practical, who are the many that more especially require it, see only a series of isolated operations, unconnected by any common principle, and view its didactic rules as being merely so many happy contrivances discovered by accident, and resting on no other evidence than their own experience that the precepts answer their special purpose. The speculatist is satisfied with the contemplation of the truths unfolded: the practical man is satisfied so long as he finds no want beyond what his rules will help him through. Rather, we should say, this was the case, than is: as most men of science now turn their attention more or less to the utility of their inquiries; and the recent rapid strides made in every branch of the arts, shows the unavoidable and unconquerable difficulties which stand in the way of practical men, whose knowledge is not based upon theory as well as upon personal experience and traditional dogmas. There are few architects and engineers,―few even of carpenters and masons,-who have been called into an active share in executing the great undertakings that have been entered on during the last twenty years, without having felt considerably embarrassed by their want of familiarity with the higher branches of Geometry, and some degree of physical science. It is impossible to look on the great amount of labour lost, the expenses incurred, and the vexation and disappointment which have ensued, all from the want of proper mathematical acquirement in those to whom they were intrusted,-without feeling deeply anxious to prevent, as far as possible, the recurrence of such lamentable events. We believe, indeed, that there is much truth in the statement, that "Many an excellent design has been changed to suit the workman's rule for execution, instead of the rule being extended to suit the design." Skill and taste obliged to bend to the ignorance of the carpenter and the mason! It is, on the contrary, equally true that taste is too liable to outrun the laws of nature, and to violate the rules of geometrical construction; and that many designs which are beautiful on paper, are inconsistent with the principles of equilibrium, and involve impossible or incompatible geometrical conditions: and the only way by

which the architect or engineer can secure himself from the disappointment and chagrin of total or partial failure, is to store his mind well with the principles upon which stability is founded, and the methods of investigating those problems by which his constructions can be executed in detail. Nature is true to her own laws,-Geometry is unvarying in her principles; and the remotest consequences of the one can be followed out by means of the other. Every man, whose profession leads him to design a structure, of whatever kind it be, is false to his own reputation, as well as false to those who repose confidence in him, if he do not, for himself, ascertain, without the experiment of success or failure, whether his design be compatible with the physical and geometrical principles that pervade alike all materials and all their thousand forms. Descriptive Geometry is one of his most essential requisites in some shape or other: and the doctrine of equilibrium and of motion, is perhaps the only one which exceeds it,—if, indeed, it do exceed it*.

The gorgeous structures of Greece rather astonish us by their magnitude, and delight us by their exquisite and inimitable beauty, than by the display of mechanical and geometrical skill: whilst those of Egypt and India, from the difference of the beau idéal in our minds and theirs, have little to recommend them to our attention besides their antiquity and their unequalled magnitude. The "Gothic," on the contrary, is full of the most ingenious contrivance, and manifests an intimate acquaintance with the principles of strength, which was totally lost with the dispersion of that singular fraternity, under whose superintendence such structures were erected; which even the most accomplished architects of the present time have been unable to develop anew; and which the most scientific amongst us dare not venture to imitate in any structure of his own. Much as we gained by the Reformation, we also lost much! It is true we have a relic of that fraternity, in the convivial band of free-masons ; and we have doubtless other relics, in the traditionary practice and rules for the simpler operations of building: but the soul is gone from that body; and can only be recalled by the assiduity of those who seek in a better spirit than has been displayed till very recently, to restore to us the knowledge which died with it.

The earliest treatise with which we are acquainted, in which any attempt beyond the most obvious geometrical operations, was made to give a body of practical instruction on subjects connected with building, was by Philibert De l'Orme, almoner to Henry the Second of France, in a work on cutting of stones, under the title of Secrets de l'Architecture, published in a folio volume, in 1642. Seven years later, the Jesuit Derande, and the architect Desargues, both of Lyons, published a more extensive work on the same subject; and in 1728, La Rue published a

* There is a multitude of excellent | had prepared a paper on oblique arches works on the first principles of statics; yet the number in which these principles are well applied to the wants of the engineer and architect are exceedingly few. We purpose to give a series of articles on different, and the most important, of the applications of the doctrines of equilibrium in our future numbers.

We

with this view, but are obliged to defer it till next month, when we shall examine, with some detail, the geometrical and statical conditions of that kind of structure. For the present, all we can say to those who are projecting such arches, is,-hesitate! beware!

collection of épures for the same purpose, accurately drawn and beautifully engraved. In 1739, Frezier, officier supérieur du génie militaire, published in three volumes, 4to, a work entitled Théorie et Pratique de la Coupe de Pierres et Bois, in which he attempted to explain, by the principles of geometry, the several combinations of lines and surfaces which arise in the cutting of wood and stones: and in 1760, another very valuable treatise on analogous subjects, under the title of Elémens de Stéréotomie, in two volumes 8vo.

The military problem of Defilement was actively discussed by the successive professors in the Military Institution at Mézières; by Millet de Moreau in 1749, Dubuat in 1768, and Meusnier in 1777. About 1764, Monge was appointed (at the age of twenty) professor of military drawing in that celebrated institution, and he entered ardently upon the investigation of the same problem: and before he left that establishment (1784), he had published no less than seventeen mémoires on different subjects connected with the geometry of space, several of which were of great interest and importance, and had also framed the chief part of the work which was to give a new aspect and an unprecedented efficiency to geometrical science. It was first made public in his lectures (1794) while professor of geometry in Ecole Centrale de Travaux Publics (since so celebrated as l'Ecole Polytechnique), and printed for the use of the élèves of that school. The extreme simplicity of its mode of research, and the striking results to which it directly led, immediately gained for it not only the universal admiration of the mathematicians of that period, but the more enduring advantage of general study. In that most fickle period of all the versatile periods of French history, it might, however, have soon been thrown aside to make way for the next novelty, but for the extraordinary perseverance of the amiable and indefatigable Hachette (who had been appointed Monge's professeur adjoint, and became ultimately his successor alike in the professorial chair, and as a victim of Bourbon oppression), who extended its application to civil as well as military purposes, and likewise to a great number of physical inquiries. Its cultivators increased so as to include all, or nearly all, who have been any way distinguished in mathematical and physical science or the arts of life in France for the last thirty years. There is not a periodical journal on the continent, devoted to mathematics, which does not bear witness to this; and if other proof be still required, we would point to the works written on the subject by the French alone, leaving out of the account those of Italy and Germany. Hachette's own works are extremely tasteful and diversified and those of Vallée, Léroy (to say nothing of several smaller ones, on the pure science, and those of Dupin, and a hundred others on its applications to physics and the arts,) bear ample testimony to our statement. It was introduced into Germany in 1804, if not earlier, and into Italy some years before that period: and the languages of those countries are now enriched with works of great merit on every part of the science. Even in Russia, it has, for twenty years at least, formed a standard branch of military and of a liberal education.

:

Still, England is to the present hour without a single work on the subject! Not even in her military institutions, though professedly a military subject, is a single lecture given on it! Not even in her uni

VOL. II.

X

10

versities, so strictly a branch of pure science as it is, is the very name familiar; nor does one single question occur either on the College or Senate House "Papers," that has the least reference to it! Not even amongst her practical men, whose best interests are so deeply involved in it, is its very nature and character known! Nor is this the less remarkable, when we reflect that in military works the operations of the method perpetually occur,-that in the geometry of co-ordinates we represent algebraically the data, and the conditions of the problems of descriptive geometry,-and that in every branch of the arts of life, we have occasion to employ the very processes of which the theoretical part of this science furnish the method and the demonstrations.

As the Geometry of Co-ordinates* is often a convenient mode of investigating the problems of Descriptive Geometry, we purpose to carry the two systems on together; and after having shown the geometrical construction, deduce also the same conclusions by means of the literal calculus. This course will, therefore, render intelligible to the minds of those who are familiar with co-ordinates the geometrical constructions of the objects with which those co-ordinate expressions are conversant; and to the minds of those who are in some degree familiar with "lines and curves in the solid," the significations of those equations by which the same ideas are expressible in symbols.

II.

FUNDAMENTAL PRINCIPLES AND DEFINITIONS, MODES OF REPRESENTATION AND NOTATION.

1. The peculiar character of the descriptive geometry consists in this: that whilst its reasonings are respecting geometrical magnitudes, any how situated in space, all its constructions are effected by lines †, situated in one plane, and having specific relations to the given magnitudes. To effect this, the method of projection is employed; both the orthographic, and the scenographic or perspective.

2. A point is said to be projected on a plane, when a line is drawn through it, in a specified manner, to cut that plane; and the point of intersection is called the projection of that point; and the line itself is called the projecting line.

3. When the line is perpendicular to the plane, the system of projection is called the orthographic or rectangular. In descriptive geometry, the rectangular is more frequently used than any other inclination of the projecting line.

4. The plane upon which the point is projected is called the plane of projection.

5. To fix the position of point, its projections upon three planes desirable to be more careful in the use of terms.

This is generally called Analytical Geometry, after the French writers, who in following D'Alembert in calling algebra by the name of Analysis, have created some confusion in the minds of young students respecting the nature of analysis. As if to compensate one error by another equal and opposite one, all mathematicians have been called Geometers. It is

By the term line is to be understood not only the straight line, but any curveline whatever, whose genesis is known. In popular language, the straight line alone is called in a line; but in science a line is the boundary of any surface.