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14. These preliminaries over, we will pass on to the solution of a number of successive problems, which fulfil the double object of exercising us in the method of projections and make it possible for us to make further progress in descriptive geometry.

First question. Being given (Fig. 4) the projections of a point, D, d, and the projections of a line AB and ab, construct the projections of a second line passing through the given point and parallel to the first line.

Solution. The two horizontal projections of the given line and the required line must be parallel for they are the intersections of two vertical parallel planes with the same plane. Furthermore, the required line passing through the given point, its projections must be respectively pass through the same point. Therefore, if through the point D one takes EF parallel to AB and if through the point d one takes ef parallel to ab, the lines EF and ef will be the required projections.

15. Second question. Being given (Fig. 5) the traces of a plane AB, BC and the projections of a point, G, g, construct the traces of a second plane passing through the given point and parallel to the first plane.

Solution. The traces of the required plane must be parallel to the respective traces of the given plane, because these traces, considered in pairs, are the intersections of two parallel planes with the same plane. It only remains, therefore, to find for each of them the point through which the line passes. For this, through the given point conceive a horizontal line which will be in the required plane; this line will be parallel to the trace AB and it will cut the vertical plane in a point which will be one of those of the trace of the required plane on the vertical plane; and one will have its two projections leading through the point g on the horizontal gF, and through the point G of the line GI, parallel to ab. If one produces GI as far as where it meets LM, the intersection of the planes of projection, in a point I, this point will be the horizontal projection of the intersection of the horizontal line with the vertical plane. Therefore, this point of intersection will be on the vertical IF passing through the point I. But it must also be on the vertical gF; thus it will be at F, the point of intersection of the two last lines. Therefore, lastly, if through the point F one makes a parallel to  BC it will be, on the vertical plane, the trace of the required plane; and, if after producing this trace to meet LM in point E, one draws ED parallel to AB, one will have the trace of the same plane on the horizontal plane.

Instead of imagining on the required plane a horizontal line, one could imagine a parallel to the vertical plane which, by similar reasoning, will give the following construction:-

One will draw through the point G and parallel to LM, the line GD; through the point g one will take gH parallel to CB; and one will produce it to cut LM in point H, through which one makes HD perpendicular to LM; this last line will cut GD in a point , through which, if one takes a parallel to AB, one will have one of the required traces, and if, after producing this trace to meet LM in point E, one draws EF parallel to BC, one will have the trace on the vertical plane.

16. Third question. Being given (Fig. 6) the two traces of a plane AB, BC, and the projections of a point D, d, construct (i) the projections dropped perpendicularly from the line to the plane, (ii) that of the point of meeting of the line and the plane.

Solution. The perpendiculars DG, dg, dropped from the points D and d on the respective traces of the plane, will be the projections of the required line; for if through the perpendicular one imagines a vertical plane, this plane will cut the horizontal plane and the given plane in the two lines which will be both perpendicular to the common intersection of these tow planes, AB; but, the first of these lines, being the projection of the vertical plane, is also that of the perpendicular which it contains; thus, the projection of this perpendicular line passes through the point D and is perpendicular to AB.

The same demonstration holds good for the vertical projection. As to the point of meeting of the perpendicular and of the plane, it is evident that it must be on the intersection of this plane with the vertical plane taken through the perpendicular; an intersection which is projected on EF. If one has the vertical projection fe of this intersection, it will contain the required point and, because this point must also be projected on the line dg, it is at the intersection g of the two lines fe and dg. It only remains, therefore, to find the line fe; but the intersection of the given plane with the vertical plane is perpendicular with it, meeting the horizontal plane in point E; thus one will have the vertical projection e, in dropping Ee perpendicularly onto LM; and it meets the vertical plane of projection in a point of which the horizontal projection is the intersection F of the line LM with DG, produced if necessary, and of which the vertical projection must be on the vertical Ff and on the trace CB; it will, therefore, be at the point f of their intersection.

The vertical projection f of the foot of the perpendicular being found, it is easy to construct the horizontal projection; for if one drops onto LM the perpendicular gG, this line will contain the required point, but the line DF must also contain it; therefore it will be at point G, the intersection of the two lines. 

17. Fourth question. Being given (Fig. 7) the two projections of a line, AB and ab, and the projections of a point, D, d, construct the traces of the plane which passes through the point perpendicularly to the line.

Solution. One knows already, from the preceding question, that the two traces must be perpendicular to the respective projections of the two lines. It remains to find, for each of them, one of the points through which it must pass. For this, if through the given point one imagines, in the required plane, a horizontal produced to meet the vertical plane of projection, one will have its vertical projection passing through the point d of a horizontal dG, and its horizontal projection passing through point D a perpendicular DH to AB, produced to cut LM in point H, which will be the horizontal projection of the point of meeting between the horizontal with the vertical plane of projection. This meeting point, which must be found in the vertical HG and in the horizontal dG, and consequently at the point G of intersection of the two lines, will be thus one of the points in the trace of the vertical plane; therefore one will have this trace by taking the line FC through point G and perpendicular to ab; finally, if through the point C, where the first trace meets LM, one takes CE perpendicular to AB, one will have the second required trace.

If the question was to find the meeting point of the planes with the line, one woud operate as in the preceding question.

Lastly, if it is necessary to drop a perpendicular from the given point to the line, one will construct, as we have just said, the meeting of the line with the plane taken through the given point and which is perpendicular to it; and one will have, for each of the two projections of the required perpendicular, two points through which it must pass.

18. Fifth question. The position of two planes being given (Fig. 8) by means of their traces, AB and Ab for one, and CD and Cd for the other, construct the projections of the line in which they intersect.

Solution. All the points of the trace AB are on the first of the two given planes, and all these of the trace CD are on the second. E, the point of intersection of the two traces, is evidently on both planes; it is, consequently, one of the points on the required line. One seeks, likewise, F, the point of intersection of the two traces in the vertical plane, which is another point on this line. The intersection of the two planes is, therefore, located, in that it meets the horizontal plane in E and the vertical plane in F.

Thus, if one projects point F to the horizontal plane, by dropping the perpendicular Ff onto LM, and if one draws the line fE, this will be the horizontal projection of the intersection of the two planes. Likewise, if one projects the point E onto the vertical plane, by dropping the perpendicular Ee onto LM, and if one draws the line eF, this will be the vertical projection of the same intersection.

19. Sixth question. Two planes (Fig. 9) being given by their traces, AB and Ab for the first, and CD and Cd for the second, construct the angle formed between them.

Solution. After having constructed, as in the preceding question, the horizontal projection Ef of the intersection of the two planes, if one imagines a third plane which is perpendicular to them, and which is consequently perpendicular to their common intersection, this third plane will cut the two given planes in two lines, which will contain between them an angle equal to the required angle.

Furthermore, the horizontal trace of this third plane will be perpendicular to the projection Ef of the intersection of the two given planes, and it will form with the two other lines a triangle, the angle of which, opposite to the horizontal side, will be the angle required. It is only necessary to construct this triangle.

But it is immaterial through which point in the intersection of the two first planes the third plane passes; one can thus assume its trace at pleasure on the horizontal plane, provided that it is perpendicular Ef. Taking, therefore, any line GH, perpendicular to Ef, terminated at G and H on the traces of the two given planes and which meet Ef in a point T, this line will be the base of the triangle to be constructed. Actually, let us imagine that the plane of the triangle turns about its base GH, like a hinge, into the horizontal plane; in this movement, its apex, which is originally in the intersection of the two planes, does not leave the vertical plane taken through this intersection, because the vertical plane is perpendicular to GH; and when the plane of the triangle is revolved flat, this apex is on the one of the points of line Ef. Thus, it only remains to find the height of the tirnalge or the length of the perpendicular dropped from point I to the intersection of the two planes.

But this perpendicular is comprised of Ef in the verical plane. If one imagines that this plane is turned about the vertical fF and is applied to the vertical plane of projection, and if one carries fE from f to e, fI from f to I, the line eF will be the length of the part of the intersection between the two planes of projection; and if from the point i, one drops to this line a perpendicular ik, this will be the height of the triangle required.

Therefore, finally, carrying ik from I to K and achieving the triangle GKH, the angle at K will be equal to the angle formed by the two planes.

20. Seventh Question. Two lines which intersect in space (Fig. 10) being given by their horizontal projections AB and AC, and by their vertical projections ab and ac, construct the true angle between them.

Before proceeding to the solution, we will notice that since the two given lines are supposed to intersect, the point A of their horizontal projections and the point a of their vertical projections will be the point in which they cut and will lie, consequently, in the same line aGA perpendicular to LM. If the two points A and a were not on the same perpendicular to LM, the given lines could not intersect and so could not be in the same plane.

Solution. One will imagine the two given lines produced to meet the horizontal plane, each in a point, and will construct these two points of meeting. To do this one will produce the lines ab, ac until they cut LM in two points d, e, which will be the vertical projections of these two points; through the points d,e, one will take, in the horizontal plane and perpendicular to LM, two lines dD, eE, which by passing each through one of these points, will determine their positions through their intersections D, E with the respective horizontal projections AB, AC, produced if necessary.

This done, if one leads the line DE, this line and the two parts of the given lines between their point of intersection and the points D, E, form a triangle, whose angle opposite to DE will be the required angle. Thus it is only necessary to construct this triangle. For this, after having dropped from point A a perpendicular, AF to DE, if one iagines that the plane of the triangle is turned about its base DE, like a hinge, until it is laid down ontot he horizontal plane, the apex of this triangle, during its movement, will not emerge from the vertical plane taken through AF, and will lie on FA produced to H, the distance of which from the base DE remains to be found.

But the horizontal projection fo this distance ti the line AF, and the vertical height of one of its extremities above the other is equal to aG. Therefore, by virtue of Fig. 3, if on LM one carries AF from G to f, and if one takes the hypotenuse af, this will be the required distance. Thus, lastly, if one transfers af from F to H, and if through H one takes the two lines HD, HE, the triangle is constructed and angle DHE will be the angle required.

21. Eighth Question. Being given the projections of a line and the traces of a plane, find the angle formed between the line and the plane.

Solution. If through a point taken on the given line, one imagines a perpendicular to the given plane, the angle formed between this perpendicular and the given line will be the complement of the required angle, and it is sufficient to find this angle to resolve the question.

But, if on the two projections of the line, one takes two points which will lie in the same perpendicular to the intersection of the two planes of projection, and if through these two points one takes perpendiculars to the respective traces of the given plane, one will have the horizontal and vertical projections of the second line. The question will thus be reduced to constructing the angle formed by two lines which intersect, and will return to the preceding case.

22. When one proposes to map a country one usually imagines a series of lines through particular points which form a number of triangles, and then one reproduces these triangles on a chart to a very small scale, placing the triangles in the same order as those they represent. The operations which are necessary on the land itself consist principally of the measurement of the angles of these triangles, and so that these angles can be reproduced on the chart, they must be angles in the horizontal plane, i.e. parallel to those of the chart. If the plane of the angle is oblique to the horizon, it is not the angle itself which must be reproduced, but its horizontal projection; and it is always possible to find this projection when, after having measured the angle itself, one has moreover measured the angles which the two sides form with the horizontal. This is given in the following operation and is known as the reduction of an angle to the horizontal.

Ninth Question. Being given the angle formed by two lines and the angles which each form with the horizontal plane, construct the horizontal projection of the first of these angles.

Being given the angle formed by two lines and the angles which each form with the horizontal plane, construct the horizontal projection fo the first of these angles.

Solution. Let A’ (Fig. 11) be the horizontal projection of the apex of the angle required, and AB be that of one of its sides, so that the other side AE can be drawn later. One will imagine that the vertical plane of projection passes through AB, and having taken through point A a vertical Aa of indefinite length one will take on it, at pleasure, a point d, which can be regarded as the vertical projection of the apex of the observed angle. That done, if through the point d one takes the line dB which makes with the horizontal an angle dBA equal to that of the first side with the horizon, B will be the meeting point of the side with the horizontal plane. Likewise, if through the point d one takes the line dC, which makes with the horizontal an angle dCA equal to that of the second side with the horizon, and if from the point A as centre, with the radius AC, one describes an arc of a circle CEF, the second side can meet the horizontal plane only in one of the points of the arc CEF. It is necessary only to find the distance of this point from some other point such as B.

But this last distance is in the plane of the angle observed. If therefore, one takes the line dD, such that the angle DdB will be equal to the observed angle, and if one transfers dC from d to D, the line DB will be equal to this distance.

Thus, if with the point B as centre, and radius equal to BD, one describes an arc, the point E where it cuts the first arc CEF will be the point at which the second line meets the horizontal plane; therefore, the line AE will be the horizontal projection of this side and the angle BAE, that of the observed angle.

The nine preceding questions are sufficient to give an idea of the method of projections; they cannot demonstrate all its uses. But to appraise these we will rise to considerations more general, and we will see the operations which will be more appropriate to fulfill this object.

11. The convention which serves as the basis of the method of projections is fit to express the position of a point in space, to express that of a straight line of finite or indefinite length and, in consequence, to represent the form and position of a body bounded by plane faces, by rectlineal edges, and by apexes of solid angles because, in this case, the body is entirely known when one knows the position of all its edges and all its angles. But if the body were bounded by a curved surface of which all the points were subjected to the same law, as in the case of a sphere, or by a discontinuous assemblage of parts with different curved surfaces, as in the case of a body fashioned on a lathe, this convention will not only be inconvenient, impractical and disadvantageous, but will be incapable of further development.

In the first place, it is easy to see that the convention which we have adopted will be inconvenient and impractical taken on its own, for to express the positions of all the points of a curved surface it is not only necessary that each of them shall be indicated by its horizontal and vertical projections, but that the two projections of the same point shall be joined to one another, in order that one will not confuse the horizontal projection of one point with the vertical projection of another; and the simplest means of connecting these two projections being to join them by a line perpendicular to the line of intersection of the two planes of projection, one will crowd drawings with a prodigious number of lines which will cause confusion, the greater as one approaches greater exactness. We shall see below that this method is insufficient and lacking in development potential.

Amongst the infinite number of different curved surfaces, there exist some which only extend to a finite part of space, and whose projections have a limited extent in all directions; the sphere, for example, is in this class. The extent of its projection onto a plane is reduced to a circle of the same radius as the sphere, and one can ensure that the plane upon which one must make the projection has dimensions large enough to take it. But all the cylindrical surfaces are indefinite in one direction – that of the line which serves as their generator. The plane itself, the simplest of surfaces, is unlimited in both directions. Lastly there exist a large number of surfaces which extend into all regions of space. But the planes upon which one executes projections are necessarily of limited extent. If one has no other means of making known the nature of a curved surface but the two projections of all the points, this method will only be applicable to those parts of the surfaces which correspond to the extent of the projections; all those points which lie beyond can never be expressed or known; the method is, therefore, insufficient. Lastly, it lacks elegance, because one can deduce nothing about planes tangent to the surfaces, about their normals, about the curvatures at each point, or indeed about any of the conditions it is necessary to consider when one wishes to investigate a curved surface.

It is necessary therefore to have recourse to a new convention which is compatible with the first, but which can supply that in which it is deficient. It is this new convention which we shall now introduce.

12. Every curved surface can be regarded as generated by the movement of a curved line, either constant in form while its position changes, or variable in form and position in space at the same time. As this proposition may be difficult to understand, because of its generality, we are going to explain it by taking a few examples with which we are already familiar.

Cylindrical surfaces may be generated in two principal ways; by the movement of a straight line which always remains parallel to a given straight line while it moves so as to pass through all the points of a given curve, or by the movement of the curve which in the first case served as a guide for the straight line, in such away that, one of its points moving along a given straight line, all the other points may describe lines parallel to this line. In either of these methods the generating line, which is a straight line in the first case and a curve in the second, is constant in form; it is constant in form and only changes its position in space.

Conical surfaces have the same two principal generations.

One can firstly regard them as produced by a line of indefinite length which, being subject to passing always through a given point, is placed such that it supports itself constantly on a given curve which guides it in its movement. The unique point through which the line passes is the centre of the surface; it is improperly that which is called the apex. In this generation, the generating line is again constant in form; it is always a straight line.

One can also produce conical surfaces in another way which, for greater simplicity, we will apply here only to the case of those which have circular bases. These surfaces can be regarded as the paths of the circumference of a circle placed with its plane always parallel to itself and with its centre always on the line running to the apex, its radiuses in each instant of movement being proportional to the distance of its centre from the apex. One sees that if, in its movement, the plane of the circle tends to approach the apex, the circle radius decreases to nothing when the plane passes through the apex, and that this radius changes its sense to increase indefinitely when the plane, having passed the apex, continues further. In this second generation, not only the circle circumference, which is the surface generator, changes position, but it changes its form at each instant of movement, since it changes its radius and, consequently, its curvature.

Let us quote a third example.

A surface of revolution can be produced by the movement of a plane curve which moves about a straight line positioned somewhere in the plane. Considering this method, the generating curve is constant in form and is only variable in position. But one can also regard it as produced by the circumference of a circle with its centre always on the axis, and its plane being always perpendicular to that axis, its radius being at each instant equal to the point where the plane of the circle cuts the axis and that where it cuts some given curve in space. Then the generating curve changes both in form and position.

These three examples must suffice to make clear that all curved surfaces can be produced by the movement of certain curved lines, and that there are none of which the form and position cannot be entirely determined by the exact and complete definition of its generation. It is this new conception which forms the complement to the method of projections. We will often have occasion later to assure ourselves of its simplicity and its potential.

It is not therefore, by giving the projections of individual points on a curved surface that one determines its form and position, but by being able to construct the generating curve through any point according to the form and position which belong to it in passing through this point. Here it should be observed, first, that each curved surface can be produced in an infinite number of ways and that the ingenuity and skill of the constructor are shown in choosing that which uses the simplest curve and requires the least trouble; second, that long usage has shown that in place of considering each curved surface as generated in one way only, which requires a study of the law of movement and a study of the change of form of the generator, it is often simpler to consider simultaneously two different generations, and to indicate for each point the constructions of two curved generators.

Thus, in descriptive geometry, to express the form and position of a curved surface it suffices, for any point on this surface and of which one of the porojections can be taken at plaeasure, to give the method of construction of the horizontal and vertical projections of two different generators which pass through this point.

13. Let us actually apply the above generalities to the plane, which is the simplest of surfaces and the one most frequently employed.

The plane is produced by a primary line whose first position is given, and which moves such that all its points describe lines parallel to a second given straight line. If the second line is itself in the plane considered, one can say also that this plane is produced by the second line, which moves such that all its points describe straight lines parallel to the first line.

One has, therefore, and idea of the position of a plane through the consideration of two straight lines, both of which can be regarded as the generator. The position of these two lines in the plane they produce is of no consequence. It is only necessary, then, in the method of projections, to choose those which require the simplest construction. And, for this reason, in descriptive geometry, one indicates the position of a plane by giving the two straight lines in which the plane cuts the planes of projection. It is easy to see that the two lines must meet at the same point on the intersection of the two planes of projection and that, consequentially, this point is where they must meet each other.

As we will very frequently have planes to consider, for shortness we will give the name traces to the lines where the plane cuts the planes of projection and which serve to indicate its position.

10. This would appear to be the place to indicate the manner of constructing the projections of solids bounded by planes and rectilinear edges, but for this operation there is no general rule. One perceives, for instance, that, according to the way in which the position of the apexes of solid angles of a solid are defined, the construction of their projections can be more or less simple, and that the nature of the work must depend on the method of definition. In this respect it is like algebra, in which there is no general procedure for solving a problem in equations. In each particular case, the step depends upon the way in which the relation between the quantities given and those unknown is expressed; and it is only through varied examples that one is able to accustom beginners to see these relations and write them into equations. It is much the same for descriptive geometry. It is through numerous examples and through the use of the straight edge and compass in the classroom that one can acquire the habits of the constructions and can accustom oneself to the choice of the simplest and most elegant methods in each particular case. But also, as in analysis, when a problem is put into an equation, procedures exist for treating these equations and for deducing the unknown quantities; in the same way, in descriptive geometry, when the projections are produced, general methods exist for constructing all which results from the form and the position of bodies.

It is not without reason that we compare here descriptive geometry with algebra; the two sciences have very close resemblances. There are no constructions of descriptive geometry which cannot be translated into analysis; and when the questions have not more than three unknowns, each analytical operation can be regarded as the portrayal of an act in geometry.

It is desirable that the two sciences should be cultivated together; in the most complicated analytical operations descriptive geometry will carry that clarity which is its very nature and, in its turn, analysis will bring to geometry that generality which is fit and proper.

2. The surfaces of all natural bodies can be considered as made up of points, and the first step we are going to make in this treatise must be to indicate how one can express the position of a point in space.

Space is without limit: all parts of space are alike: there is nothing characteristic about any particular part so that it can serve as a reference for indicating the position of a particular point.

Thus, to define the position of a point in space it is necessary to refer this position to those of other objects which are of known position in some distinctive part of space, the number of objects being as many as are required to define the point; and for the process to be amenable for easy and everyday use, it is necessary that these objects should be the simplest possible so that their positions can be most easily imagined.

3. Amongst all the simple objects, we will investigate which present the most facility in determining the position of a point; and firstly, because geometry offers nothing more simple than a point, we will examine what kind of considerations are involved if, to define the position of a point, one refers it to a certain number of other points whose positions are known; for the sake of clarity in this exposition, we will designate these known points by the letters A, B, C, etc.

Suppose that in defining the position of a point we say we begin with that it is one metre from the point A.

Everyone knows that it is a property of the sphere that every point on its surface is of equal distance from its centre. Thus the definition given above satisfies this property; that is, the point to be found could be any of those lying in the surface of a sphere with centre at A and of radius one metre. The points on the surface of this sphere are the only ones in all space which have this required property, for all the points in space which are outside this sphere are further than one methre from A and all those which are between the surface and the centre are, contrariwise, nearer than one metre. Therefore, the points on the surface of the sphere are the only ones which possess the property stated in the proposition. Finally, therefore, this proposition expresses that the point required is one of those on the surface of a sphere with centre A and radius one metre. This makes the point distincitive from those in an infinity of other places in space, surface, and other conditions are necessary if the required point is to be recognised amongst them.

Suppose that, in defining the position of this point, we say that it must also be two metres from a second known point, B: it is clear that the reasoning for this second condition is as for the first. The point must be one of those on the surface of a sphere with centre at B and of radius two metres. This point, finding itself simultaneously on the surfaces of two spheres, can only be confused with those others which are common to the two spheres’ surfaces and which lie in the spheres’ common intersection. Those who are familiar with geometrical concepts will know that the intersection of two spherical surfaces is the circumference of a ircle, whose centre lies on the straight line joining the centres of the two spheres and whose plane is perpendicular to this line. So by virtue of the two conditions stated together, the point searched for is distinguished from those generally on the surfaces fo the two spheres and is one lying on the circumference of the circle which only satisfies both conditions. It is necessary therefore, to stipulate a third condition to absolutely determine the required point.

Suppose, finally, that this point must also be three metres from a third point C. This third condition places the point amongst all those on the surface of a third sphere with centre at C and of three metres radius: and because we have seen that it must lie on the circumference of a circle of known position, to satisfy also the third condition it must be one of the points common to the surface of the third sphere and the circumference of the circle. But it is known that the circumference of a circle and the surface of a sphere can only meet in two points: therefore, by virtue of the three conditions, the point is distinguished from all those in space and can only be one of the two points found. If one further indicates on which side it lies of the plane passing through the three centres of the spheres, i.e. points, A, B, and C, the point is absolutely determined and cannot be confused with any other.

One sees that determining the position of a point in space by referring it to known points, of which the number is necessarily three, involves one in considerations not simple enough for everyday use.

4. Let us see what will actually be the result if, instead of referring the position of a point to three other known points, it is referred to three lines of given position.

A line need not be considered to be of finite length but can always be indefinitely produced in one direction or the other. To simplify, we will label the lines we will be obliged to use successively, A, B, C, etc.

If, in defining the position of a point, we say that it must be found, for example, at a distance of one metre from the first known line, A, we are saying that this point is one of those in the surface of a cylinder of circular base with the line A as axis and of radius one metre, and which is indefinitely produced in both directions: for all the points on this surface possess the property stated in the definition and are the only ones which possess it. In this way, the point is distinguished from others in space which are outside or inside of the cylinder, and it can only be confused with those in the surface of the cylinder, amongst which one cannot distinguish it by means of the new condition.

Suppose, therefore, that the point sought is also to be placed at two metres from a second line B, one sees likewise that one places this point on the surface of a second cylinder, whose axis is in the line B and shoe radius is two metres. But it is confused with all the other points on this cylinder surface if only this second condition is considered. Through uniting these two conditions the point must be simultaneously on the first cylindrical surface and on the second: therefore, it can only be one of the common points of these two surfaces, i.e. one at their common intersection. This line, on which the point must lie, has the curvature of both the surfaces of the first and second cylinders and is, in general, known as a curve of double curvature.

To distinguish the point from all those on this line it is necessary to resort to a third condition.

Suppose, finally, that the definition states that the point must also be at three metres from a third line C.

This new condition states that it is one of those points on a third cylinder of which the third line will be the axis and which will have a radius of three metres. Therefore, in taking the three conditions together, the sought point can only be one of those which are common to the third cylinder’s surface and to the curve of double curvature – the intersection of the first two cylinders. But this curve can be cut, in general, by the third cylindrical surface in eight points, and amongst these the point can be distinguished by circumstances, similar to those detailed in the previous case.

One sees that the considerations for determining the position of a point in space by recognition of its distances from three known straight lines are less simple than those in which the distances are given from three points, and they are thus less able to serve as a basic method for everyday use.

5. Among the simple objects which geometry considers, it is necessary to notice principally, first, the point which has no dimensions, secondly the line which has one, and thirdly the plane which has two. Let us investigate whether it is not more simple to determine the position of a point by recognising its distances from known planes, instead of using its distances from points or straight lines.

Suppose we have non-parallel planes of known position in space, which we will designate successively, A, B, C, D, etc.

If, in defining the position of a point, we say it must be, for example, one metre from the first plane A, without stating on which side of the plane, we are saying that it must be one of those points on two planes parallel to A, placed one either side of plane A, and both one metre from it, for all the points on both these parallel planes satisfy the expressed condition and are, in all space, the only ones which satisfy it.

To distinguish amongst all the points of these two planes that which is in the required position, it is necessary again to have recourse to other conditions.

Suppose, secondly, that the point sought must be two metres from a second plane, B, then one places it on two planes parallel to plane B, both at two metres distance, and one on either side. To satisfy at the same time the two conditions it is necessary that the point should be on one of the two planes parallel to plane A and on one of the two planes parallel to plane B; consequently it is one of the points in the common intersection of these four planes. But the intersection of four planes of known position is a group of four straight lines equally of known position. Therefore, in considering simultaneously both conditions, the point is no longer confused with all those in space, neither likewise with all those in the four planes, but only with those on four straight lines. Finally, if the point must also be three metres from a third plane C, one expresses that it must be on one of the two other planes parallel to C, placed one on either side at three metres distance. So, by virtue of three conditions, it must be simultaneously on one of the two last planes and on one of the four straight lines. But as each of the two planes has a common point with each of the four straight lines, there are eight points in space which satisfy the three conditions; therefore, by these three conditions jointly the point required can only be one of the eight determined points, and amongst these one can distinguish which by means of particular circumstances.

For example, if one indicates the distance of the point from the first plane A, one expresses also in what sense, with respect to this plane, the distance is to be taken; instead of two planes parallel to plane A, there is only one which needs to be considered; it is that one which is situated on the side towards which the distance is normally measured. Likewise, if one indicates the general sense in which distances from the second plane are to be measured, the point is no longer on the four lines of intersection of four parallel planes, but only on the intersection of two planes, that is to say, on a straight line of known position. Finally, if one indicates also the sense in which the point is placed in relation to the third plane its position will in consequence be entirely determined.

One can see, therefore, that although the plane is an object less simple than the line which has only one dimension and the point which has none, referring to planes provides an easier system for the determination of points in space than to points or lines. It is this procedure which we will ordinarily employ in the application of algebra to geometry, or for finding the position of a point – the principle of relating its distances to three planes of known position.

However, in descriptive geometry, which has been pratised for a long time by a large number of people and by many to whom time was precious, the process can again be simplified and, instead of considering three planes, we find that, by means of projections, we only have need for two of these.

6. The projection of a point on a plane may be defined as the foot of the perpendicular lowered form the point to the plane.

It follows that if on two planes of known position in space one is given on each of these planes the projection of the point whose position one wishes to define, this point will be perfectly determined.

In effect, if from the projection on the first plane one constructs a perpendicular to the plane, it is evident that it will pass through the point defined. Likewise, if from its projection on the second plane one constructs a perpendicular to the plane, it also passes through the point defined. Therefore the point will be simultaneously on two lines of known position in space; therefore it will be uniquely at their intersection and is, accordingly, perfectly determined.

7.

If, from all the points on a straight line of indefinite length, AB, oriented in any direction in space, one can imagine perpendiculars dropped to a plane, LMNO, in some given position, all the points at the meeting of these perpendicular with the plane will lie on another straight line of indefinite length, ab; for they will all lie in the plane passing through AB lying perpendicular to the plane LMNO, and they will only be able to meet the latter at the common intersection of two planes, which, as one knows, is a straight line.

The line ab on the plan LMNO, which is formed by the projection of all the points from another line AB, is called the projection of the line AB onto the plane.

Since two points are sufficient to fix the position of a  straight line, to construct the projection of a straight line it is only necessary to project these two points, the projection of the line passing through the two points where the projectors meet the plane.

Being given on two non-parallel planes LMNO and LMPQ, the projections ab and a’b’ of the line AB, the projection of the line AB is fully determined; for if through one of the projections ab one imagines a plane perpendicular to LMNO, this plane of known position must necessarily pass through line AB; likewise, if through the other projection a’b’ one imagines a plane perpendicular to LMPQ, this plane of known position also passes through the line AB. The position of this line, which is simultaneously on two known planes, is consequently at their common intersection and its position is, therefore, absolutely determined.

8. What has been said above is independent of the position of the planes of projection and equally of the angle between the planes; but if the angle formed by the two planes of projection is very obtuse, the angle formed between the perpendiculars to these planes will be very acute, and any small drawing errors will cause considerable error in determining the position of the line AB. In order to avoid this cause of inaccuracy, unless it is otherwise necessary for ease of presentment, the planes of projection are always made to be perpendicular to one another. As the majority of draughtsmen who will practise this method are already familiar with the position of a horizontal plane and the direction of a plumbline, they will be quite used to supposing that of the two planes of projection, one is horizontal and the other vertical.

The need for making the drawings of the two projections on a single sheet and for carrying out the operation in the same area, again calls for the draughtsmen to imagine that the vertical plane is turned about its intersection with the horizontal plane, like a hinge, to lie flat in the horizontal plane and form with it one continuous plane; and it is in this state that he will construct his projections.

Thus the vertical projection is always drawn in the horizontal plane and it is necessary to imagine that it is raised up and put back into place by means of a quarter revolution about the intersection of the horizontal and vertical planes. It is necessary, accordingly, that this intersection line is made so that it can be clearly seen on the drawing.

Thus, in Fig. 2, the projection a’b’ of the line AB is not executed on a plane which is really vertical; one imagines that the plane is turned about the axis LM to the position LMP’Q’, and it is in this position of the plane that one carries out the vertical projection a’’b’’.

Apart from the ease of execution which this arrangement allows, it has also the advantage of minimising the work of making projections. For instance, let us suppose that the points a, a’ are the horizontal and vertical projections of point A; the plane carried through the lines Aa, Aa’ will be at the same time perpendicular to the two planes of projection, since it passes through lines which are perpendicular to them; it will be then, also perpendicular to their common intersection LM, and the lines aC, a’C, at which it cuts the two planes, will be themselves perpendicular to LM.

But, when the vertical plane is turned about LM as a hinge, the line a’C does not cease, through this movement, to be perpendicular to LM, and it is still perpendicular to it when the vertical plane is laid down to give the position Ca’’. Therefore, the two lines aC, Ca’’, both passing through the point C and both being perpendicular to LM, are in one straight line; it is the same with the lines bD, Db’’ by resemblance to any other point such as B. From which it follows that, if one has the horizontal projection of a point, the projection of the same point on the vertical plane supposed laid down, will be in the line taken through the horizontal projection perpendicular to the intersection, LM, of the two planes of projection, and vice versa.

This result is of very great use in practice.

9. Up to now we have considered the line AB (Fig. 1) to be of indefinite length, and we have occupied ourselves only with its direction; but it is possible for this line to be considered terminated by the two points, A and B, and one may need to know its length. We are going to see how one can deduce this from its two projections.

When a straight line is parallel to one of the two planes upon which it is projected, its length is equal to that of its projection on this plane; for the line and its projection, being both terminated by two perpendiculars to the plane of projection are parallel to each other and fall between parallel lines. Thus, in this case the projection being given, the length of the line which is equal to it is also given.

One knows that a line is parallel to one of the two planes of projection when its projection onto the other plane is parallel to the intersection of the two planes.

If the line is oblique to both of the two planes, its length is greater than that of either of its projections, but may be deduced through a very simple construction.

Fig. 2. Let AB be the straight line, whose two projections ab and a’b’ are given, and whose length is to be found. If through one of its extremities A, and in the vertical plane which passes through the line, one constructs a horizontal AE, produced as far as to meet at E the vertical dropped from the other extremity, one will form a right-angled triangle AEB, which is to be constructed to find the length of AB, the hypotenuse. But, in this triangle, as well as the right angle one knows the side AE, which is equal to the projection ab. Furthermore, if in the vertical plane one takes through the point a’ a horizontal a’e, which will be the projection of AE, it will cut the b’D in a point e, which will be the projection of point E. thus b’e will be the vertical projection of BE and will be, in consequence, of the same length. Therefore, knowing the two sides of the right-angled triangle, it may easily be constructed, and its hypotenuse will give the length of AB.

Fig. 2, being in perspective, has no resemblance to the construction used in the method of projections; we are here going to give the construction of this first question in all its simplicity.

Fig. 3. The line LM, being supposed to be the intersection of the two planes of projection, and the lines ab and a’’b’’ being the given projections of a straight line, to find the length of this line one takes through the point a’’ the horizontal He, which will cut the line bb’’ in a point e, and upon this horizontal one will transfer ab from e to H. One will then take the hypotenuse Hb’’ and the length of this hypotenuse will be that of the line required.

As the two planes are at right angles, the operation which has been made on one of the planes could just as well be made on the other and would give the same result.

After the above, one sees that if one has the two projections of a body terminated by plane faces, by rectilineal edges and by solid angles, the projections of which become a system of lines, it will be easy to find the length of any dimension one may wish; for such a dimension will be parallel to one of the two planes of projections or it will be oblique to both. In the first case the length required will be equal to its projection; in the second, one will deduce it from these two projections through the procedure described above.

1. Descriptive Geometry has two objects: the first to give the methods for representing on a sheet of paper which has but two dimensions, length and breadth, all the bodies of nature which have three, length, breadth, and depth, provided that these bodies are able to be rigorously defined.

The second object is to give a means of knowing, through an exact description, the forms of bodies and to deduce all the resulting truths from their forms and their positions respectively.

We are going to indicate firstly the procedure, which has come to light through long experience, for fulfilling the first of these two objects. After this we shall give the methods for fulfilling the second.

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I

Nos of Articles

1 – The Object of Descriptive Geometry

2 – 9 Considerations of the Determination of the Position of a Point in Space. The Methods of Projections. (Fig. 1-3)

10 – Solids bounded by planes and rectilinear edges

11-13   Convention fit to Express the Forms and Positions of Surfaces. Application to the Plane

14-22   Solutions of some Elementary Questions relative to the Straight Line and the Plane (Fig. 4-11)

In order to raise the French nation from the position of dependence on foreign industry, in which it has continued to the present time, it is necessary in the first place to direct national education towards an acquaintance with matters which demand exactness, a study which hitherto has been totally neglected; and to accustom the hands of our artificers to the handling of tools of all kinds, which serve to give precision to workmanship, and for estimating its different degrees of excellence. Then the consumer, appreciating exactness, will be able to insist upon it in the various types of workmanship and to fix its proper price; and our craftsmen, accustomed to it from an early age, will be capable of attaining it.

It is necessary in the second place to make popular a recognition of a number of natural phenomena indispensable for the progress of industry, and to exploit, through the advancement of the general instruction of the nation, the fortunate condition in which it finds itself of having at its command the principal resources which are necessary.

Finally, it is necessary to disseminate among our craftsmen the knowledge of the processes used in the crafts and in machines which have for their object either the diminuation of manual labour or the imparting of more uniformity and precision to the results of workmanship; and in this respect it must be admitted that we have much to learn from foreign nations.

All these objectives can only be reached by gaining a new direction to national education.

This is to be done in the first place by familiarising all young persons of intelligence with descriptive geometry, that those of independent fortune may be someday in a position to use their capital usefully for themselves and the state, as well as those who have no fortune but their education, that they may someday be able to impart a higher value to their workmanship.

This art has two principal objects.

The first is to represent with exactness upon drawings which have only two dimensions such objects as have three and which are susceptible of rigorous definition.

From this point of view it is a language necessary to a man of genius, who conceives a project, to those who are obliged to direct its execution, and finally to the craftsmen who are obliged to make the different parts.

The second object of descriptive geometry is to deduce from the exact description of bodies all which necessarily follows from their forms and respective positions. In this sense it is a means of investigating truth; it perpetually offers examples of passing from the known to the unknown; and since it is always applied to objects with the most elementary shapes, it is necessary to introduce it into the plan of national education. It is not only fitted to exercise the intellectual faculties of a great people, and to contribute thereby to perfecting the human species, but, moreover, it is indispensable to all workmen whose aim is to give bodies certain defined forms; and it is principally because the methods of this art have up to the present time been too little disseminated, or even almost entirely neglected, that the progress of our industry has been so slow.

It will be a contribution, therefore, towards giving an advantageous direction to national education, if we familiarise our young craftsmen with the application of descriptive geometry to the graphical constructions which are necessary in a great many of the arts and crafts, and make use of this geometry for the representation and determination of the elements of machines by which man, controlling the forces of nature, reserves for himself, so to speak, no other labour in his work but that of intelligence.

It is no less advantageous to disseminate a knowledge of the phenomena of nature which can be turned to account in the arts and crafts.

The charm which accompanies these studies will conquer the repugnance which men have in general for intense thought and make them find pleasure in that exercise of their intellect which almost all regard as painful and irksome.

Accordingly, there must be a Course of Descriptive Geometry at the Ecole Normale.

But as we have no elementary, well-executed book upon this art, either because hitherto scientific men have taken too little interest in the subject, or because it has been practised in an obscure manner by persons whose education has not been sufficiently attended to and who do not know how to communicate the results of their meditations, a simple oral course would be absolutely useless.

It is necessary for the Course of Descriptive Geometry to associate practice and execution with the listening to a description of the method. Therefore the students ought to exercise themselves in the graphical constructions of descriptive geometry. The graphical arts have general methods with which it is only possible to become familiar by the use of the straight edge and compass.

Among the different applications which can be made of descriptive geometry there are two which are remarkable, both by their generality and by the ingenuity which attaches to them; these are the constructions of perspective and the rigorous determination of shadows in drawings. These two parts can be considered as the complement of the art of graphically describing objects. Here one exercises those persons who, being destined one day to teach the procedures of descriptive geometry, will have to be cognisant with all its ramifications.

It follows that the methods of projections will be applied to the graphical constructions necessary in the greater number of crafts, such as the work of stone-cutters, carpenters, and so on.

Finally the rest of the duration of the course will be employed firstly on the description of machine elements and then the study of their forms and effects, followed by that of complete machines, of which it is most important to spread knowledge; so that machines will have as their object the giving of more precision and uniformity to work and the employing in production of the forces of nature to augment the national power.