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It also follows immediately that,

traces of the plane, then the line, and therefore the point A, lies If a plane a is perpendicular to the horizontal plane, then every point in the plane; otherwise not. This is conveniently donc by joining in it has its horizonial projection in the horizontal trace of the plane, A: to some point p in the trace a'; this gives Pri and the point as all the rays projecting these points lie in the plane itself. where the perpendicular from p to the axis cuts the latter we join

Any plane which is perpendicular to the horizontal plane has ils to Az; this gives Do. If the vertical trace of this line lies in the verlical irace perpendicular to the axis.

vertical trace of the plane, then, and then only, does the line P, and Any plane which is perpendicular to the vertical plane has its hori; with it the point A, lie in the plane a sontal trace perpendicular to the axis and the vertical projections of all $ 8. Parallel planes have parallel traces, because parallel planes are points in the plane lie in this trace.

cut by any plane, hence also by n and by az, in parallel lines. $ 4. Representation of a Line.--A line is determined either by two Parallel lines have parallel projections, because points at infinity points in it or by two planes through it. We get accordinglý two are projected to infinity. representations of it either by projections or by traces.

Ij e line is parallel to a plane, then lines through the traces of the First. -A line a is represented by ils projections di and az on the line and parallel to the traces of the plane must meet on the axis, because two planes ai and 72. These may be any two lines, for, bringing these lines are the traces of a plane parallel to the given plane. the planes 71, 72 into their original position, the planes through these 9: To draw a plane through two intersecting lines or througk tuzo lines perpendicular to 1 and 12 respectively will intersect in some line parallel lines, we determine the traces of the lines; the lines joining a which has aı, Gas its projections.

their horizontal and vertical traces respectively will be the horizontal Secondly.-A line a is represented by ils tracesthat is, by the points and vertical traces of the plane. They will meet, at a finite point in which it culs the two planes #1, # Any two points may be taken or at infinity, on the axis if the lines do intersect. as the traces of a line in space, for it is determined when the planes To draw a plane through a line and a point without the line, we are in their original position as the line joining the two traces. This join the given point to any point in the line and determine the plane representation becomes undetermined if the two traces coincide in through this and the given line. the axis. In this case we again use a third plane, or else the pro- To draw a plane through three points which are not in a line, we jections of the line.

draw two of the lines which each join two of the given points and The fact that there are different methods of representing points draw the plane through them. If the traces of all three lines AB, and planes, and hence two methods of representing lines, suggests BC, CA be sound, these must lie in two lines which meet on the the principle of duality (section ii., Projective Geometry, § 41). It axis. is worth while to keep this in mind. It is also worth remembering 10. We have in the last example got more points, or can easily that traces of planes or lines always lie in the planes or lines which get more points, than are necessary for the determination of the they represent. Projections do not as a rule do this excepting when figure required in this case the traces of the plane. This will the point or line projected lies in one of the planes of projection. happen in a great many constructions and is of considerable im

Having now shown how to represent points, planes and lines, portance. It may happen that some of the points or lines obtained we have to state the conditions which must hold in order that these are not convenient in the actual construction, The horizontal elements may lie one in the other, or else that the figure formed by traces of the lines AB and AC may, for instance, fall very near them may possess certain metrical properties. It will be found that together, in which case the line joining them is not well defined. the former are very much simpler than the latter. Before we do this, however, we shall explain the notation used; they are practically non-existent for the construction.'

o be one op facticallyhem wekiye ball beyond the

drawing papel his that for it is of great importance to have a systematic notation. We the traces of the line BC may be used. Or, if the vertical traces of shall denote points in space by capitals A, B, C; planes in space AB and AC are both in convenient position, so that the vertical by Greek letters a, B, Y; lines in space by small letters a, b, c; trace of the required plane is sound and one of the horizontal traces horizontal projections by suffixes 1, like Ai, a; vertical projections is got, then we may join the latter to the point where the vertical

trace cuts the axis. a', a". Hence P: will be the horizontal projection of a point P in The draughtsman must remember that the lines which he drass space; a line & will have the projections di, da and the traces a' and are not mathematical lines without thickness, and therefore every al; a plane a has the traces a' and a".

drawing is affected by some errors. It is therefore very desirable $5. If a point lies in a line, the projections of the point lie in the to be able constantly to check the latter. Such checks always projections of the line.

present themselves when the same result can be obtained by different If a line lies in a plane, the traces of the line lic in the traces of the constructions, or when, as in the above case, some lines must met plane.

on the axis, or if three points must lie in a line. A carcful draughtsThese propositions follow at once from the definitions of the man will always avail himself of these checks. projections and of the traces.

§ 11. To draw a plane through a given point parallel to e gives If a point lies in two lines its projections must lie in the projections plane a, we draw through the point two lines which are parallel to of both. Hence

the plane a, and determine the plane through them; or, as we If two lines, given by their projections, intersect, the intersection of know that the traces of the required plane are parallel to those of their plans and the intersection of their elevations must lie in a line the given one (§ 8), we need only draw one line l through the point perpendicular to the axis, because they must be the projections of parallel to the plane and find one of its traces, say the vertical trace the point common to the two lines.

l"; a line through this parallel to the vertical trace of a will be the Similarly-If two lines given by their traces lie in the same plane vertical trace Bir of the required plane B, and a line parallel to the or interseci, then the lines joining their horizontal and vertical iraces horizontal trace of a meeting B" on the axis will be the horizontal respeclively must meet on the axis, because they must be the traces trace B'. of the plane through them.

Let A, A: (fig. 42) be the given point, a'e' the given plane, a $ 6. To find the projections of a line which joins two points A, B line l, through Ai, parallel to a' and a horizontal line l through given by their projections Ar. Ag and B,, B2, we join A, B, and Az. Az will be the projections of B;; these will be the projections required. For example, the a line I through A parallel

3 traces of a line are two points in the line whose projections are to the plane, because the known or at all events easily found. They are the traces themselves horizontal plane through

! and the feet of the perpendiculars from them to the axis.

this line will cut the plane Hence if a' a' (fig. 41) are the traces of a line a, and if the per- a in a line c which has its pendiculars from them cui the axis in P and Q respectively, then the horizontal projection

line a'Q will be the horizontal and parallel to a'.
a'P the veriical projection of the $ 12. We now

the metrical properties of
Conversely, if the projections figures.
Qı, az of a line are given, and if A line is perpendicular
these cut the axis in Q and P a plane if the projec.
respectively, then the perpen- tions of the line are per.
diculars Pa' and Qa" !o the axis pendicular to the traces of the plane. We prove it for the horizontal
drawn through these points cut the projection. If a line p is perpendicular to a plane a, every plane
projections di and az in the traces ihrough p is perpendicular to a; hence also the vertical plane hich
a' and a"

projects the line p to Pi.. As this plane is perpendicular both to the To find the line of intersection of horizontal plane and to the plane a, it is also perpendicular to their two planes, we observe that this intersection--that is, to the horizontal trace of a It follows that line lies in both planes; its traces every line in this projecting plane, therefore also pr. the plan of p, is

must therefore lie in the traces perpendicular to the horizontal trace of a. of both. Hence the points where the horizontal traces of the given To draw a plane through a given point A perpendicular to es given planes meet will be the horizontal, and the point where the vertical line P, we first draw through some point Oʻin the axis lines 2.7 traces meet the vertical trace of the line required.

perpendicular respectively to the projections D and Do of the gives 9,7. To decide whether a point A, given by its projections, lies in line. These will be the traces of a plane , which is perpendicular a plane a. given by ils traces, we draw a line p by joining A to some to the given line. We next draw through the given point A a plane point in the plane a and determine its traces. If these lic in the 'parallel to the planc q; this will be the plane required.

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FIG. 42.

FIG. 41.



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FIG. 45.



Other metrical properties depend on the determination of the real point two lines perpendicular to the two planes and determine the size or shape of a figure.

angle between the latter as above. In general the projection of a figure differs both in size and shape In special cases it is simpler to determine at once the angle between from the figure itself. But figures in a plane parallel to a plane the two planes by taking a plane section perpendicular to the interof projection will be identical with their projections, and will thus section of the two planes and rabatt this. This is especially the be given in their true dimensions. In other cases there is the case if one of the planes is the horizontal or vertical plane of proproblem, constantly recurring, either to find the true shape and jection. size of a plane figure when plan and elevation are given, or, con- Thus in fig. 45 the angle PQR is the angle which the plane a versely, to find the latter from the known true shape of the figure makes with the horizontal plane. itself. To do this, the plane is turned about one of its traces till it 15. We return to the general. case of rabatting a plane a of is laid down into that plane of projection to which the trace belongs. which the traces d'a' are given. This is technically called rabatring the plane respectively into the Here it will be convenient to determine first the position which plane of the plan or the elevation. As there is no difference in the the tracc a'-which is a line in a-assumes when rabatted. Points treatment of the two cases, we shall consider only the case of rabatt in this line coincide with their clevations. Hence it is given in ing a plane a into the plane of the plan. The plan of the figure is its true dimension, and we can measure off along it the true distance a parallel (orthographic) projection of the figure itself. The results between two points in it. If therefore.(fig. 45) P is any point in a' of parallel projection (see PROJECTION, $& 17 and 18) may there originally. coincident with fore now be used. The trace a' will hereby take the place of what its elevation Pz, and if o formerly was called the axis of projection. Hence we see that corre- is the point where a cuts sponding points in the plan and in the rabatted plane are joined by the axis xy, so that is lines which are perpendicular to the trace a' and that corresponding also in a', then the point P lines meet on this trace. We also see that the correspondence is will after rabatting the completely determined if we know for one point or one line in the plane assume such a posiplan the corresponding point or line in the rabatted plane.

tion that OP=OPz. At Before, however, we treat of this we consider some special cases. the same time the plan is

$13. To determine the distance between two points A, B given by their an orthographic projection projections A, B, and Az, B2, or, in other words, to delermine the true

of the plane a.

Hence the length of a line the plan and elevation of which are given.

line joining P to the plan Solution. The two points A, B in space lie vertically above their P, will after rabatting be plans A., B. (fig. 43) and AJA=AA:, BB=B.Bs. The four points perpendicular to. a'. But

A, B, A, B, therefore form a plane P, is known; it is the foot
quadrilateral on the base A,B, and of the perpendicular from
having right angles at the base. P, to the axis xy. We
This plane we rabatt about A,B, draw therefore, to find P,
by drawing AA and B.B per. from P, a perpendicular Pill to a' and find on it a point P such that
pendicular to A, B, and making OP=OP. Then the line OP will be the position of a' when

AA = A A2, B,B=B.Bz. Then rabatted. This line corresponds therefore to the plan of c'—that y AB will give the length required. is, to the axis xy, corresponding points on these lines being those

The construction might have which lie on a perpendicular to a'. been performed in the elevation We have thus one pair of corresponding lines and can now find by making AzA= AA, and for any point B, in the plan the corresponding point B in the rabatted B B = B.B, on lines perpendicular planc. We draw a line through Bi, say B.P., cutting a' in C. To it to A,B2. Of course AB must have corresponds the line CP, and the point where this is cut by the project. the same length in both cases. ing ray through Bu, perpendicular to a', is the required point B.

This figure may be turned into Similarly any figure in the rabatted plane can be found when the a model.

Cut the paper, along plan is known; but this is usually found in a different manner AA, AB and BB1, and fold the without any reference to the general theory of parallel projection.

piece A,ABB, over along A,B, till As this method and the reasoning employed for it have their peculiar it stands upright at right angles to the horizontal plane. The points advantages, we give it also. A, B will then be in their true position in space relative to . Simi- Supposing the planes a and F to be in their positions in space larly if B:BAA, be cut out and turned along A,B, through a right perpendicular to each other, we take a section of the whole figure angle we shall get AB in its true position relative to the plane by a plane perpendicular to the trace a' about which we are going

Lastly we lold the whole plane of the paper along the axis x to rabatt the plane a. Let this section pass through the point Q in till the plane 2 is at right angles to... In this position the two a'.. Its traces will then be the lines QP, and PP, (fig. 9). These sets of points AB will coincide if the drawing has been accurate. will be at right angles, and will therefore, together with the section

Models of this kind can be made in many cases and their con QP2 of the plane a, form a right-angled triangle QP,P, with the struction cannot be too highly recommended in order to realize right angle at Pu, and having the sides P Q and Pip, which both orthographic projection.

are given in their true lengths. This triangle we rabatt about its $14. To find the angle between two given lines a, b of which the base PQ, making P, R =P,Pz. The line QR

will then give the true projections an. b. and az, bq are given.

length of the line QP in space. If now the plane a be turned about Solution.-Let a, b, (fig. 44) meet in P1, 02, ba in T, then is the line a' the point P will describe a circle about Q as centre with radius P.T is not perpendicular to the axis the two lines will not meet. In QP-QR, in a plane perpendicular to the trace a'. Hence when the

this case we draw a line parallel plane a has been rabatted into the horizontal plane the point P will
to b to meet the line a. This is lie in the perpendicular P Q to a', so that QP=QR.
casiest done by drawing first the : If A; is the plan of a point A in the plane a, and if A, lies in QP1,
line P P2 perpendicular to the then the point A will lie vertically above A, in the line QP. On
axis to meet d: in Ps, and then turning down the triangle QP.Pz. the point A will come to Ao, the
drawing through P, a line cz line A A, being perpendicular to QP. Hence A will be a point in
parallel to ba; then bı, cz will be QP such that OA =QA.
the projections of a line c which If B, is the plan of another point, but such that A, B, is parallel
is parallel to b and meets a in P. to a', then the corresponding line AB will also be parallel to a'.
The plane a which these two Hence, is through A a line AB be drawn parallel to a', and B. B
lines determine we rabatt to the perpendicular to a', then their intersection gives the point B. Thus
plan. We determine the traces of any point given in plan the real position in the plane a, when
a' and c' of the lines a and c; rabatted, can be found by this second method. This is the one
then a'd' is the trace a' of their most generally given in books on geometrical drawing. The first
plane. On rabatting the point method explained is, however, in most cases preferable as it gives
P comes to a point S on the line the draughtsman a greater variety of constructions. It requires a

P,Q perpendicular to a'c', so somewhat greater amount of theoretical knowledge. that QS=QP. But QP is the hypotenuse of a triangle PP Q with If instead of our knowing the plan of a figure the latter is itself a right angle P This we construct by making QR =P

P,; then given, then the process of finding the plan is the reverse of the P.R

= PQ. The lines a'S and c'S will therefore include angles equal above and needs little explanation. We give an example. to those made by the given lines. It is to be remembered that two $16. It is required lo draw the plan and elevation of a polygon of lines include two angles which are supplementary. Which of these which the real shape and position in a given plane a are known. is to be taken in any special case depends upon the circumstances. We first rabatt the plane a (fig. 46) as before so that P, comes to

To determine the angle between a line and a plane, we draw through P. hence OP, to OP. Let the given polygon in a be the figure any point in the line a perpendicular to the plane (812) and determine ABCDE. We project, not the vertices, but the sides. To project the angle between it and the given line. The complement of this the line AB, we produce it to cut a' in F and OP in G, and draw GG angle is the required one.

perpendicular to a'; then G, corresponds to G, therefore FG, to FG. To delermine the angle between two planes, we draw through any In the same manner we might project all the other sides, at least


FIG. 43.

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FIG. 44.










those which cut oF and OP in convenient points. It will be best, 1 making FoF = OFs, &c. If the figure representing the development however, first to produce all the sides to cut OP and a' and then to of the pyramid, or better a copy of it, is cut out, and if the lateral draw all the projecting rays through A, B, C ..., perpendicular to faces be bent along the lines AB, BC, &c., we get a model of the pyra

a', and in the same mid with the section marked on its faces. This may be placed on
direction the lines its plan ABCD and the plane of elevation bent about the axis .
G. G. &c. By The pyramid stands then in front of its elevations. If next the plane
drawing FG a with a hole cut out representing the true section be bent along the
get the points As, trace a' till its edge coincides with a', the edges of the hole ought to
B, on the project coincide with the lines EF, FG, &c., on the faces.
ing ray through A $.18. Polyhedra like the pyramid in § 17 are represented by the
and B. We then projections of their edges and vertices. But solids bounded by
join B to the point curved surfaces, or surfaces themselves, cannot be thus represented.
M where BC pro- For a surface we may use, as in case of the plane, its traces--that
duced meets the is, the curves in which it cuts the planes of projection. We may

a'. This also project points and curves on the surface. A ray cuts the gives C. So we surface generally in more than one point; hence it will happen go on till we have that some of the rays touch the surface, if two of these points coincide. Yound Ei. The The points of contact of these rays will form some curve on the surface, line A, E, must and this will appear from the centre of projection as the boundary then meet AE in of the surface or of part of the surface. The outlines of all surfaces a', and this gives of solids which we see about us are formed by the points at which a check.

If one

rays through our eye touch the surface. The projections of these of the sides cuts contours are therefore best adapted to give an idea of the shape of a a' or OP beyond surface. the drawing, paper Thus the tangents drawn from any finite centre to a sphere form this method fails, a right circular cone, and this will be cut by any plane in a conic. but then we may easily find the projection of other line, say of

diagonal, directly the projection of a point,

the former

methods. The FIG. 46.

diagonals may

also serve to check the drawing, for two corresponding diagonals must meet in the trace a'.

Having got the plan we easily find the elevation. The elevation of G is above G, in a', and that of F is at F, in the axis. This gives the elevation FG7 of FG and in it we get A,B, in the verticals through A, and B.. As a check we have OG=OGz. Similarly the elevation of the other sides and vertices are found.

$ 17. We proceed to give some applications of the above principles to the representation of solids and of the solution of problems connected with them.

of a pyramid are given its base, the length of the perpendicular from the verlex to the base, and the point where this perpendicular cuis the base; it is required first to develop the whole surface of the pyramid into one plane, and second to determine its section by é plane which culs the plane of the base in a gwen line and makes a given angle with it.

1. As the planes of projection are not given we can take them as we like, and we select them in such a manner that the solution becomes as simple as possible. We take the plane of the base as the horizontal plane and the vertical plane perpendicular to the plane of the section, Let then (fig: 47) ABCD be the base of the pyramid, V, the plan of the vertex, then the elevations of A, B, C, D will be in the axis at As, B2, C2, D3, and the vertex at some point Va above V, at a known distance from the axis. The lines ViA, VB, &c., will be the plans and the lines V.A., V,Ba, &c., the elevations of the edges of the pyramid, of which thus plan and elevation are known.

We develop the surface into the plane of the base by turning It is often called the projection of a sphere, but it is better called each lateral face about its lower edge into the horizontal plane by the contour-line of the sphere, as it is the boundary of the projections the method used in $ 14. If one face has been turned down, say of all points on the sphere. ABV to ABP, then the point Q to which the vertex of the next If the centre is at infinity the tangent cone becomes a right face BCV comes can be got more simply by finding on the line circular cylinder touching the sphere along a great circle, and it ViQ perpendicular to BC the point Q such that BQ=BP, for these the projection is, as in our case, orthographic, then the section of lines represent the same edge BV of the pyraniid. Next R is this cone by a plane of projection will be a circle equal to the great sound by making CR=CQ, and so on till we have got the last vertex circle of the sphere. We get such a circle in the plan and another in --in this case S. The fact that AS must equal AP gives a convenient the elevation, their centres being plan and elevation of the centre of check.

the sphere. 2. The plane a whose section we have to determine has its hori. Similarly the rays touching a cone of the second order will lie zontal trace given perpendicular to the axis, and its vertical trace in two planes which pass through the vertex of the cone, the contour. makes the given angle with the axis. This determines it. To find line of the projection of the cone consists therefore of two lines the section of the pyramid by this plane there are two methods meeting in the projection of the vertex. These may, however, applicable: we find the sections of the plane either with the faces be invisible is no real tangent rays can be drawn from the centre of or with the edges of the pyramid. We use the latter.

projection; and this happens when the ray projecting the centre As the plane a is perpendicular to the vertical plane, the trace of the vertex lies within the cone. In this case the traces of the a' contains the projection of every figure in it; the points Es, Fa, cone are of importance. Thus in representing a cone of revolution G2, H, where this trace cuts the elevations of the edges will therefore with a vertical axis we get in the plan a circular trace of the surface be the elevations of the points where the edges cut a. From these whose centre is the plan of the vertex of the cone, and in the elevation we find the plans E. F1, G, H., and by joining them the plan the contour, consisting of a pair of lines intersecting in the elevation of the section. If from E, F, lines be drawn perpendicular to AB, of the vertex of the cone. The circle in the plan and the pair of lines these will determine the points E, F on the developed face in which in the elevation do not determine the surface, for an infinite number the plane a cuts it; hence also the line EF. Similarly on the other of surfaces might be conceived which pass through the circular trace faces. Of course BF must be the same length on-BP and on BQ. and touch two planes through the contour lines in the vertical plane. If the plane a be rabatted to the plan,

we get the real shape of the The surface becomes only completely defined if we write down to section as shown in the figure in EFGH. This is done easily by 'the figure that it shall represent a cone. The same holds for all



FIG. 47

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surfaces. Even a plane is fully represented by its traces only under | possible ways restrained. Geometry figured rather as the helper the silent understanding that the traces are those of a plane.

of the more difficult science of arithmetic. $ 19. Some of the simpler problems connected with the repre. sentation of surfaces are the determination of plane sections and of 2. It was reserved for algebra to remove the disabilities of the curves of intersection of two such surfaces. The former is arithmetic, and to restore the earliest ideas of the land-measurer constantly used in nearly all problems concerning surfaces. Its to the position of controlling ideas in geometrical investigation. solution depends of course on the nature of the surface.

This unified science of pure number made comparatively litile To determine the curve of intersection of two surfaces, we take a plane and determine its section with each of the two surfaces, headway in the hands of the ancients, but began to receive rabatting this plane if necessary. This gives two curves which lie due attention shortly after the revival of learning. It expresses in the same plane and whose intersections will give us points on whole classes of arithmetical facts in single statements, gives surfaces. It must here be remembered that tw curves in

to arithmetical laws the form of equations involving symbols space do not necessarily intersect, hence that the points in which their projections intersect are not necessarily the projections of which may mean any known or sought numbers, and provides points common to the two curves. This will, however, be the case processes which enable us to analyse the information given by an is the two curves lie in a common plane. By taking then a number equation and derive from that equation other equations, which curve of intersection as we like: These planes have, of course, to express laws that are in effect consequences or causes of a law be selected in such a way that the sections are curves as simple as started from, but differ greatly from it in form. Above all

, for the case permits of, and such that they can be easily and accurately present purposes, it deals not only with integral and fractional drawn. Thus when possible the sections should be straight lines number, but with number regarded as capable of continuous or circles. This not only saves time in drawing but determines all growth, just as distance is capable of continuous growth. The points on the sections, and therefore also the points where the two difficulty of the arithmetical expression of irrational number, curves meet, with equal accuracy.

$ 20. We give a few examples how these sections have to be a difficulty considered by the modern school of analysts to have selected. A cone is cut by every plane through the vertex in lines, been at length surmounted (see FUNCTION), is not vital to it. and if it is a cone of revolution by planes perpendicular to the It can call the ratio of the diagonal of a square to a side, for axis in circles.

A cylinder is cut by every plane parallel to the axis in lines, and instance, or that of the circumference of a circle to a diameter, if it is a cylinder of revolution by planes perpendicular to the axis a number, and let a or x denote that number, just as properly in circles.

as it may allow either letter to denote any rational number A sphere is cut by every plane in a circle.

Hence in case of two cones situated anywhere in space we take which may be greater or less than the ratio in question by a sections through both vertices. These will cut both cones in lines. difference less than any minute one ye choose to assign. Similarly in case of two cylinders we may take sections parallel to Counting only, and not the counting of objects, is of the essence the axis of both. In case of a sphere and a cone of revolution with of arithmetic, and of algebra. But it is lawful to count objects, vertical axis, horizontal sections will cut both surfaces in circles and in particular to count equal lengths by measure. whose plans are circles and whose elevations are lines, whilst vertical sections through the vertex of the cone cut the latter in lines and widened idea is that even when a or x is an irrational number the sphere in circles. To avoid drawing the projections of these we may speak of a or x unit lengths by measure. We may give circles, which would in general be ellipses, we rabatt the plane and concrete interpretation to an algebraical equation by allowing then draw the circles in their real shape. And so on in other cases. its terms all to mean numbers of times the same unit length,

Special attention should in all cases be paid to those points in which the tangents to the projection of the curve of intersection are

or the same unit area, or &c. and in any equation lawfully parallel or perpendicular to the axis x, or where these projections derived from the first by algebraical processes we may do the touch the contour of one of the surfaces.

(O. H.) same. Descartes in his Géométrie (1637) was the first to system

atize the application of this principle to the inherent first IV. ANALYTICAL GEOMETRY

notions of geometry; and the methods which he instituted have 1. In the name geometry there is a lasting record that the become the most potent methods of all in geometrical research. science had its origin in the knowledge that two distances may It is hardly too much to say that, when known facts as to a be compared by measurement, and in the idea that measurement geometrical figure have once been expressed in algebraical must be efíectual in the dissociation of different directions as well terms, all strictly consequential facts as to the figure can be as in the comparison of distances in the same direction. The deduced by almost mechanical processes. Some may well be distance from an observer's eye of an object seen would be unexpected consequences; and in obtaining those of which specified as soon as it was ascertained that a rod, straight to the there has been suggestion beforehand the often bewildering eye and of length taken as known, could be given the direction labour of constant attention to the figure is obviated. These of the line of vision, and had to be moved along it a certain are the methods of what is now called analytical, or sometimes number of times through lengths equal to its own in order to algebraical, geometry. reach the object from the eye. Moreover, if a field had for two 3. The modern use of the term "analytical " in geometry has of its boundaries lines straight to the eye, one running from south obscured, but not made obsolete, an earlier use, one as old as to north and the other from west to east, the position of a point Plato. There is nothing algebraical in this analysis, as disin the field would be specified if the rod, when directed west, tinguished from synthesis, of the Greeks, and of the expositors had to be shifted from the point one observed number of times of pure geometry. It has reference to an order of ideas in westward to meet the former boundary, and also, when directed demonstration, or, more frequently, in discovering means to south, had to be shifted another observed number of times effect the geometrical construction of a figure with an assigned southward to meet the latter. Comparison by measurement, special property. We have to suppose hypothetically that the the beginning of geometry, involved counting, the basis of arith- construction has been performed, drawing a rough figure which metic; and the science of number was marked out from the as nearly as is practicable. We then analyse or first as of geometrical importance.

critically examine the figure, treated as correct, and ascertain But the arithmetic of the ancients was inadequate as a science other properties which it can only possess in association with of number. Though a length might be recognized as known the one in question. Presently one of these properties will often when measurement certified that it was so many times a standard be found which is of such a character that the construction of length, it was not every length which could be thus specified a figure possessing it is simple. The means of effecting syntheticin terms of the same standard length, even by an arithmetically a construction such as was desired is thus brought to light by enriched with the notion of fractional number. The idea of what Plato called analysis. Or again, being asked to prove a possible incommensurability of lengths was introduced into theorem A, we ascertain that it must be true if another theorem Europe by Pythagoras; and the corresponding idea of irration- B is, that B must be if C is, and so on, thus eventually finding ality of number was absent from a crude arithmetic, while there that the theorem A is the consequence, through a chain of interwere great practical difficulties in the way of its introduction. mediaries, of a theorem Z of which the establishment is easy. Hence perhaps it arose that, till comparatively modern times, This geometrical analysis is not the subject of the present article; appeal to arithmetical aid in geometrical reasoning was in all I but in the reasoning from form to form of an equation or system



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of equations, with the object of basing the algebraical proof forms of expression which actual geometry has, in return for of a geometrical fact on other facts of a more obvious character, benefits received, conferred on algebras of one, two and three the same logic is utilized, and the name "analytical geometry variables. is thus in part explained.

We will confine ourselves to the dimensions of actual geometry, 4. In algebra rcal positive number was alone at first dealt and will devote no space to the one-dimensional, except incidentwith, and in geometry actual signless distance. But in algebra. ally as existing within the two-dimensional. The analytical it became of importance to say that every equation of the first method will now be explained for the cases of two and three degree has a root, and the notion of negative number was intro- dimensions in succession. The form of it originated by Descartes, duced. The negative unit had to be defined as what can be and thence known as Cartesian, will alone be considered in much added to the positive unit and produce the sum zero. The detail. corresponding notion was readily at hand in geometry, where

I. Plane Analytical Geometry. it was clear that a unit distance can be measured to the left 6. Coordinales.-It is assumed that the points, lines and figures or down from the farther end of a unit distance already measured considered lie in one and the same plane, which plane therefore need to the right or up from a point 0, with the result of reaching

not be in any way referred to. In the plane a point 0, and two lines

x'Ox, y'Oy, intersecting in O, are taken once for all, and regarded as again. Thus, to give full interpretation in geometry to the fixed. Oʻis called the origin, and x'Ox, yOy the axes of x and y algebraically negative, it was only necessary to associate distinct respectivelyOther positions in the plane are specified in relation ness of sign with oppositeness of direction. Later it was discovered to this fixed origin and these fixed axes. From any point P we that algebraical reasoning would be much facilitated, and that conclusions as to the real would retain all their soundness, if a pair of imaginary units #V-1 of what might be called number were allowed to be contemplated, the pair being defined, though not separately, by the two properties of having the real sum o and the real product 1. Only in these two real combinations do they enter in conclusions as to the real. An advantage gained was that every quadratic equation, and not some quadratics only, could be spoken of as having two roots. These admissions of new units into algebra were final, as it admitted of proof that all equations of degrees higher than two have the full numbers of roots possible for their respective degrees in any case, and that

FIG. 48.

Fig. 49. every root has a value included in the form a+b V-1, with a, b, real. The corresponding enrichment could be given to geometry, I suppose PM drawn parallel to the axis of y to meet the axis of r in with corresponding advantages and the same absence of danger, M, and may also suppose PN drawn parallel to the axis of x to meet and this was done. On a line of measurement of distance we

the axis of 'y in N, so that OMPN is a parallelogram. The position

of P is determined when we know OM (=NP) and MP (=ON). contemplate as existing, not only an infinite continuum of points 11 OM is x times the unit of a scale of measurement chosen at pleasure, at real distances from an origin of measurement O, but a doubly and MP is y times the unit, so that x and y have numerical values, infinite continuum of points, all but the singly infinite continuum

we call x and y the (Cartesian) coordinates of P. To distinguish of real ones imaginary, and imaginary in conjugate pairs, a

them we often speak of y as the ordinate, and of x as the abscissa.

It is necessary to attend to signs; x has one sign or the other conjugate pair being at imaginary distances from 0, which have according as the point P is on one side or the other of the axis of y, a real arithmetic and a real geometric mean. To geometry and y one sign or the other according as P is on one side or the other enriched with this conception all algebra has its application.

of the axis of x. Using the letters N, E, S, W, as in a map, and 5. Actual geometry is one, two or three-dimensional, i.e. considering the plane as divided into four quadrants by the axes, lincal, plane or solid. In one-dimensional geometry positions

the signs are usually taken to be: and measurements in a single line only are admitted. Now

For quadrant

NE descriptive constructions for points in a line are impossible

S E without going out of the line. It has therefore been held that

NW there is a sense in which no science of geometry strictly confined

SW to one dimension exists. But an algebra of one variable can be a point is referred to as the point (a, b), when its coordinates are applied to the study of distances along a line measured from a x=a, y=b. A point may be fixed, or it may be variable, i.e. be chosen point on it, so that the idea of construction as distinct regarded for the time being as free to move in the plane. The from measurement is not essential to a one-dimensional geo- said to be current coordinates.'

coordinates (x, y) of a variable point are algebraic variables, and are metry aided by algebra. In geometry of two dimensions, the The axes of x and y are usually (as in fig: 48) taken at right angles flat of the land-measurer, the passage from one point o to any

to one another, and we then speak of them as rectangular axes, other point, can be effected by two successive marches, one cast

and of x and y as " rectangular coordinates" of a point P; OMPN or west and one north or south, and, as will be seen, an algebra axes which are oblique to one another, so that (as in fig. 49) the angle

is then a rectangle. Sometimes, however, it is convenient to use of two variables suffices for geometrical exploitation. In 20y between their positive directions is some known angle w geometry of three dimensions, that of space, any point can be distinct from a right angle, and OMPN is always an oblique paralleloreached from a chosen one by three marches, one east or west,

gram with given angles; and we then speak of x and y as "oblique one north or south, and one up or down; and we shall see that I in what follows.

coordinates.' The coordinates are as a rule taken to be rectangular an algebra of three variables is all that is necessary. With 7. Equations and loci. If (x, y) is the point P, and if we are three dimensions actual geometry stops; but algebra can supply given that x=0, we are told that, in fig. 48 or fig. 49, the point M lies any number of variables. Four or more variables have been

at 0, whatever value y may have, i.e. we are told the one fact that used in ways analogous to those in which one, two and three of y, we have always OM =0, i.e. x = 0. Thus the equation x = o is

P lies on the axis of y. Conversely, if P lies anywhere on the axis variables are used for the purposes of one, two and three- one satished by the coordinates (x, y) of every point in the axis of y. dimensional geometry, and the results have been expressed in and not by those of any other point. We say that tro is the quasi-geometrical language on the supposition that a higher equation of the axis of y, and that the axis of ý is the locus represpace can be conceived of, though not realized, in which four axis of x. An cquation x=0, where a is a constant, expresses that

sented by the equation x=0. Similarly, y=o is the equation of the independent directions exist, such that no succession of marches P lies on a parallel to the axis of y through a point M on the axis along three of them can effect the same displacement of a point of x such that OM =a. Every line parallel to the axis of y has an as a march along the fourth; and similarly for higher numbers çquation of this form. Similarly, every line parallel to the axis of x than four. Thus analytical, though not actual, geometries exist has an equation of the form y=b, where b is some definite constant. for four and more dimensions. They are in fact algebrasfurnished current coordinates of a variable point (x, y) imposes one limitation

These are simple cases of the fact that a single equation in the with nomenclature of a geometrical cast, suggested by conveniert on the freedom of that point to vary. The coordinates of a point


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