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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 l'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 admiratiori 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 beenany 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 uniVOL. II.
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 linest, 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 * This is generally called Analytical desirable to be more careful in the use of Geometry, after the French writers, who in terms. following D'Alembert in calling algebra by the name of Analysis, have created + By the term line is to be understood some confusion in the minds of young not only the straight line, but any curvestudents respecting the nature of analysis. line whatever, whose genesis is known. As if to compensate one error by another In popular language, the straight line alone equal and opposite one, all mathemati- is called in a line; but in science a line is cians have been called Geometers. It is the boundary of any surface,
must be known*. No two of these planes must be parallel, in this system t. In general they are so taken that each is perpendicular to the other two; and the projecting lines to one plane respectively parallel to the intersection of the other two.
6. Two of these planes are usually taken, the one parallel to the horizon, and the other perpendicular to it. In practice, however, it will be shown that one of these three vertical planes may be dispensed with ; and that one is always employed, whose intersection with the horizontal plane is parallel to the top and bottom of the drawing. They are the same planes which, in practical works, are employed for the plan and elevation, or section. The line of intersection is called the ground-line.
7. In the application of algebra to the projective system, the line of intersection
X above spoken of is usually marked xx!, the line of intersection perpendicular to it, in the horizontal plane, is marked yy'; and the line of intersection of the two vertical planes, is marked z z/; and the three projecting lines of the point parallel to these three, are respectively denoted by x, y, z.
We shall de
We have said known. The word given | figure different from itself, in any of these is employed by geometers in two senses, particular respects. which do not always precisely agree in signi. A point, having position but not magnification: viz., 1. to signify that which is tude, is given, when such conditions are actually given amongst the conditions of the given as shall fix its position absolutely. proposition; and 2. to signify those neces- A straight-line is said to be given in posary consequences of the conditions which sition when two points through which it are actually given, and follow from them by passes are given, and in magnitude when processes of construction, or reasoning, its length is given; and in magnitude and which have been previously established. position when the two extremities of it are In the general processes of geometrical given. analysis, this difficulty is not felt by expe- A curve-line is said to be given when its. rienced reasoners; but it creates great mode of genesis is given, and the position embarrassments to young students. In and magnitude of the quantities upon which the present case two projections must be it depends, are also given. given: but the third can at once be deduced A surface is given when its genesis is from them, and is also, in common with given, and the position and magnitude of the them, therefore known. As, however, it quantities on which they depend, are given. is sometimes difficult to avoid the use of In all these cases, it is obvious that by terms which are become part of the cur- given is meant determinable rather than rent language of science, we may probably determined--that is, fixed, rather than acinadvertently employ the word given in its tually known. Some service would be renusual signification on some occasions. We, dered to geometry by a little change in this therefore, annex the usual meanings of the language; especially as the same word, word amongst writers in general.
given, is also applied to those points, lines, The position of a point in space is a rela- or surfaces of reference, which are assumed
Till some other data be fur- as those in respect to which the others are nished in relation to which its position is compared. to be considered, we cannot say that its + We have said “ in this system,” lest position is given : but when those are given, the reader should suppose that the point the point is given in position.
was indeterminable when two of the planes A geometrical magnitude is said to be are parallel, which would be an erroneous given when such conditions respecting its supposition, as they are then perfectly defigure, magnitude, and position are given, terminate. However the projections must as shall be incompatible with any other l be upon the two which are not parallel.
note the co-ordinate axes in the same manner in our descriptive geometry.
The three lines, xx', Y Y', z zl, are in this case called axes of coordinates, axes of reference, or co-ordinate axes; the three planes are called co-ordinate planes; and the lengths, x, y, z, are called the co-ordinates of the point. In the former case, planes of projection.
8. The polar, perspective, or scenographic projection is often advantageously employed. In this case the point is projected by a line drawn through a given point, or pole, upon a given plane, and is the foundation of perspective representations. Of this we shall speak more at large hereafter, in showing the foundation of that art, and the modes of calculation by which it may also be effected.
9. It is evident that if we can project a point, we can project a second upon any planes whatever: and hence that we can project the line which joins them.
It is, moreover, easy to perceive that the projection of a straight-line upon any plane, is the same with the line which joins the projection of its extremities. For, conceive two points, situated above the plane represented by the paper, and PP', Qe', the projecting
lines, and p', a', their projections themŚ
selves. Now the two lines, pp' and
Qel, being perpendicular to the same plane, are parallel to one another. (Hutton's Course*, vol. i., cor. pr. 103, p. 339.)
But being parallel, they are in the same plane, by the definition of parallels. Hence that plane will intersect the plane of projection in a straight line, (Course i., pr. 89,) or the line joining P'd' is the intersection of the projecting plane with the plane passing through the projecting lines of the extremities of that line. Now the plane e pa e passes through the perpendicular p p', and is hence perpendicular to the plane of projection. (Course, pr. 110.) Hence we deduce the fact that the projection of a given line upon a given plane is the intersection of that plane made by another plane which passes through the given line perpendicularly to the former one.
io. When the projections of a point upon two planes are given, the position of the point itself is known.
For when the projections are given, the projecting lines are also known; and as they both pass through the projected point, their intersection determines that point.
* We employ Hutton's Course of Mathe- | whom this course of papers is more immematics as our book of reference for the diately intended-especially a comprehentheorems respecting the intersections of sive and familiar treatise on the numerical lines and planes, for several reasons; but solution of equations by continuous apchiefly that it is more concise than Euclid's, proximation, invented by Mr. Horner, and and equally satisfactory and rigid in its drawn up in the most simple form of which demonstrations. It, moreover, contains it appears to be susceptible. We refer many other matters which will be inter- exclusively to the 11th edition, edited by esting and useful to the class of readers for | Dr. Gregory.
11. When two projections of a line are given, the position of the line itself is known.
For when the projections are given, the projecting planes are known, and as they both pass through the projected line, their intersection determines that line.
12. Straight-lines and planes admit of indefinite extension. They are said to be given in position in this case; but in magnitude and position, when their extremities are known, given, or determinable. When the lengths or boundaries only are known, they are said to be given in magnitude. (See note on Art. 5.)
13. When a line, whose projection on a plane is given, intersects that plane, the part before the plane, if a vertical one, or above the plane, if a horizontal one, is visible; and the parts of the pro
-R jection which represent the visible is traced in full line; whilst the parts of the projection which represent the concealed parts of the line, are marked in dotted or broken lines. Thus, PR or P'R' represents the projection of the visible, and P Q or pa that of the hidden, part of the line, whose projection is Q R or q' r'.
14. We have shown in Art. 10 that two projections are sufficient for the determination of a point or straight-line, and from this it may be inferred at once, that two projections are sufficient for the determination of any curve-line, however it be situated in space, since it is sufficient for the determination of any point taken arbitrarily in that curve, and hence for all of them, or for the whole curve which is their path or locus.
15. If a plane be drawn through the two projecting lines of any point, it will be perpendicular to the intersection of the planes of projection.
For these lines are, by the method of projection, perpendicular to the planes of projection, and hence the plane passing through them is perpendicular to those planes (Course, pr. 110); and hence again, perpendicular to their common section (pr. 110, cor. 3), or in other words, to the ground-line. (Art. 6.)
16. This line is called the projecting plane of the point, since it is the plane in which the projecting lines of that point are situated ; and its intersection with the planes of projection are called the projecting parallèls, since they are respectively parallel to the projecting lines, that on the vertical plane to the vertical projecting line, and that on the horizontal to the horizontal. They are called, therefore, the vertical projecting parallel, and the horizontal projecting parallel. Sometimes, simply the vertical parallel, and the horizontal parallel. They are both perpendicular to the ground-line. (Course, pr. 110, cor. 3.)
17. If the vertical plane of projection be made to revolve about the ground-line till it coincides with either the visible or hidden portion of the horizontal plane of projection; or the horizontal plane revolves about the ground-line till it coincides with the vertical one; then in both cases the vertical projecting parallel will coincide either by superposition or continuation, as the case may be, with the horizontal projecting parallel, and they will form one straight-line, perpendicular to the ground-line.