Chapter XXXIII: Book XIII (5)

We first rabatt the plane [alpha] (fig. 46) as before so that P1 comes
to P, hence OP1 to OP. Let the given polygon in [alpha] be the figure
ABCDE. We project, not the vertices, but the sides. To project the
line AB, we produce it to cut [alpha]' in F and OP in G, and draw GG1
perpendicular to [alpha]'; then G1 corresponds to G, therefore FG1 to
FG. In the same manner we might project all the other sides, at least
those which cut OF and OP in convenient points. It will be best,
however, first to produce all the sides to cut OP and [alpha]' and
then to draw all the projecting rays through A, B, C ... perpendicular
to [alpha]', and in the same direction the lines G, G1, &c. By drawing
FG we get the points A1, B1 on the projecting ray through A and B. We
then join B to the point M where BC produced meets the trace [alpha]'.
This gives C1. So we go on till we have found E1. The line A1 E1 must
then meet AE in [alpha]', and this gives a check. If one of the sides
cuts [alpha]' or OP beyond the drawing paper this method fails, but
then we may easily find the projection of some other line, say of a
diagonal, or directly the projection of a point, by the former
methods. The diagonals may also serve to check the drawing, for two
corresponding diagonals must meet in the trace [alpha]'.

Having got the plan we easily find the elevation. The elevation of G
is above G1 in [alpha]", and that of F is at F2 in the axis. This
gives the elevation F2G2 of FG and in it we get A2B2 in the verticals
through A1 and B1. As a check we have OG = OG2. Similarly the
elevation of the other sides and vertices are found.

S 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 vertex to the base, and the point where this perpendicular cuts
the base; it is required first to develop the whole surface of the
pyramid into one plane, and second to determine its section by a plane
which cuts the plane of the base in a given 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, V1 the
plan of the vertex, then the elevations of A, B, C, D will be in the
axis at A2, B2, C2, D2, and the vertex at some point V2 above V1 at a
known distance from the axis. The lines V1A, V1B, &c., will be the
plans and the lines V2A2, V2B2, &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 each
lateral face about its lower edge into the horizontal plane by the
method used in S 14. If one face has been turned down, say ABV to ABP,
then the point Q to which the vertex of the next face BCV comes can be
got more simply by finding on the line V1Q perpendicular to BC the
point Q such that BQ = BP, for these lines represent the same edge BV
of the pyramid. Next R is found by making CR = CQ, and so on till we
have got the last vertex--in this case S. The fact that AS must equal
AP gives a convenient check.

2. The plane [alpha] whose section we have to determine has its
horizontal trace given perpendicular to the axis, and its vertical
trace makes the given angle with the axis. This determines it. To find
the section of the pyramid by this plane there are two methods
applicable: we find the sections of the plane either with the faces or
with the edges of the pyramid. We use the latter.

As the plane [alpha] is perpendicular to the vertical plane, the trace
[alpha]" contains the projection of every figure in it; the points
E2, F2, G2, H2 where this trace cuts the elevations of the edges will
therefore be the elevations of the points where the edges cut [alpha].
From these we find the plans E1, F1, G1, H1, and by joining them the
plan of the section. If from E1, F1 lines be drawn perpendicular to
AB, these will determine the points E, F on the developed face in
which the plane [alpha] cuts it; hence also the line EF. Similarly on
the other faces. Of course BF must be the same length on BP and on BQ.
If the plane [alpha] be rabatted to the plan, we get the real shape of
the section as shown in the figure in EFGH. This is done easily by
making F0F = OF2, &c. If the figure representing the development of
the pyramid, or better a copy of it, is cut out, and if the lateral
faces be bent along the lines AB, BC, &c., we get a model of the
pyramid with the section marked on its faces. This may be placed on
its plan ABCD and the plane of elevation bent about the axis x. The
pyramid stands then in front of its elevations. If next the plane
[alpha] with a hole cut out representing the true section be bent
along the trace [alpha]' till its edge coincides with [alpha]", the
edges of the hole ought to coincide with the lines EF, FG, &c., on the
faces.

S 18. Polyhedra like the pyramid in S 17 are represented by the
projections of their edges and vertices. But solids bounded by curved
surfaces, or surfaces themselves, cannot be thus represented.

For a surface we may use, as in case of the plane, its traces--that
is, the curves in which it cuts the planes of projection. We may also
project points and curves on the surface. A ray cuts the surface
generally in more than one point; hence it will happen that some of
the rays touch the surface, if two of these points coincide. The
points of contact of these rays will form some curve on the surface,
and this will appear from the centre of projection as the boundary of
the surface or of part of the surface. The outlines of all surfaces of
solids which we see about us are formed by the points at which rays
through our eye touch the surface. The projections of these contours
are therefore best adapted to give an idea of the shape of a surface.

Thus the tangents drawn from any finite centre to a sphere form a
right circular cone, and this will be cut by any plane in a conic. It
is often called the projection of a sphere, but it is better called
the contour-line of the sphere, as it is the boundary of the
projections of all points on the sphere.

If the centre is at infinity the tangent cone becomes a right circular
cylinder touching the sphere along a great circle, and if the
projection is, as in our case, orthographic, then the section of this
cone by a plane of projection will be a circle equal to the great
circle of the sphere. We get such a circle in the plan and another in
the elevation, their centres being plan and elevation of the centre of
the sphere.

Similarly the rays touching a cone of the second order will lie in two
planes which pass through the vertex of the cone, the contour-line of
the projection of the cone consists therefore of two lines meeting in
the projection of the vertex. These may, however, be invisible if no
real tangent rays can be drawn from the centre of projection; and this
happens when the ray projecting the centre of the vertex lies within
the cone. In this case the traces of the cone are of importance. Thus
in representing a cone of revolution with a vertical axis we get in
the plan a circular trace of the surface whose centre is the plan of
the vertex of the cone, and in the elevation the contour, consisting
of a pair of lines intersecting in the elevation of the vertex of the
cone. The circle in the plan and the pair of lines in the elevation do
not determine the surface, for an infinite number of surfaces might be
conceived which pass through the circular trace and touch two planes
through the contour lines in the vertical plane. The surface becomes
only completely defined if we write down to the figure that it shall
represent a cone. The same holds for all surfaces. Even a plane is
fully represented by its traces only under the silent understanding
that the traces are those of a plane.

S 19. Some of the simpler problems connected with the representation
of surfaces are the determination of plane sections and of the curves
of intersection of two such surfaces. The former is constantly used in
nearly all problems concerning surfaces. Its solution depends of
course on the nature of the surface.

To determine the curve of intersection of two surfaces, we take a
plane and determine its section with each of the two surfaces,
rabatting this plane if necessary. This gives two curves which lie in
the same plane and whose intersections will give us points on both
surfaces. It must here be remembered that two curves in space do not
necessarily intersect, hence that the points in which their
projections intersect are not necessarily the projections of points
common to the two curves. This will, however, be the case if the two
curves lie in a common plane. By taking then a number of plane
sections of the surfaces we can get as many points on their curve of
intersection as we like. These planes have, of course, to be selected
in such a way that the sections are curves as simple as the case
permits of, and such that they can be easily and accurately drawn.
Thus when possible the sections should be straight lines or circles.
This not only saves time in drawing but determines all points on the
sections, and therefore also the points where the two curves meet,
with equal accuracy.

S 20. We give a few examples how these sections have to be selected. A
cone is cut by every plane through the vertex in lines, and if it is a
cone of revolution by planes perpendicular to the axis in circles.

A cylinder is cut by every plane parallel to the axis in lines, and if
it is a cylinder of revolution by planes perpendicular to the axis in
circles.

A sphere is cut by every plane in a circle.

Hence in case of two cones situated anywhere in space we take sections
through both vertices. These will cut both cones in lines. Similarly
in case of two cylinders we may take sections parallel to the axis of
both. In case of a sphere and a cone of revolution with vertical axis,
horizontal sections will cut both surfaces in circles whose plans are
circles and whose elevations are lines, whilst vertical sections
through the vertex of the cone cut the latter in lines and the sphere
in circles. To avoid drawing the projections of these circles, which
would in general be ellipses, we rabatt the plane and then draw the
circles in their real shape. And so on in other cases.

Special attention should in all cases be paid to those points in which
the tangents to the projection of the curve of intersection are
parallel or perpendicular to the axis x, or where these projections
touch the contour of one of the surfaces. (O. H.)

IV. ANALYTICAL GEOMETRY

1. In the name _geometry_ there is a lasting record that the science had its origin in the knowledge that two distances may be compared by measurement, and in the idea that measurement must be effectual in the dissociation of different directions as well as in the comparison of distances in the same direction. The distance from an observer's eye of an object seen would be specified as soon as it was ascertained that a rod, straight to the eye and of length taken as known, could be given the direction of the line of vision, and had to be moved along it a certain number of times through lengths equal to its own in order to reach the object from the eye. Moreover, if a field had for two of its boundaries lines straight to the eye, one running from south to north and the other from west to east, the position of a point in the field would be specified if the rod, when directed west, had to be shifted from the point one observed number of times westward to meet the former boundary, and also, when directed south, had to be shifted another observed number of times southward to meet the latter. Comparison by measurement, the beginning of geometry, involved counting, the basis of arithmetic; and the science of number was marked out from the first as of geometrical importance.

But the arithmetic of the ancients was inadequate as a science of number. Though a length might be recognized as known when measurement certified that it was so many times a standard length, it was not every length which could be thus specified in terms of the same standard length, even by an arithmetic enriched with the notion of fractional number. The idea of possible incommensurability of lengths was introduced into Europe by Pythagoras; and the corresponding idea of irrationality of number was absent from a crude arithmetic, while there were great practical difficulties in the way of its introduction. Hence perhaps it arose that, till comparatively modern times, appeal to arithmetical aid in geometrical reasoning was in all possible ways restrained. Geometry figured rather as the helper of the more difficult science of arithmetic.

2. It was reserved for algebra to remove the disabilities of arithmetic, and to restore the earliest ideas of the land-measurer to the position of controlling ideas in geometrical investigation. This unified science of pure number made comparatively little headway in the hands of the ancients, but began to receive due attention shortly after the revival of learning. It expresses whole classes of arithmetical facts in single statements, gives to arithmetical laws the form of equations involving symbols which may mean any known or sought numbers, and provides processes which enable us to analyse the information given by an equation and derive from that equation other equations, which express laws that are in effect consequences or causes of a law started from, but differ greatly from it in form. Above all, for present purposes, it deals not only with integral and fractional number, but with number regarded as capable of continuous growth, just as distance is capable of continuous growth. The difficulty of the arithmetical expression of irrational number, a difficulty considered by the modern school of analysts to have been at length surmounted (see FUNCTION), is not vital to it. It can call the ratio of the diagonal of a square to a side, for instance, or that of the circumference of a circle to a diameter, a number, and let a or x denote that number, just as properly as it may allow either letter to denote any rational number which may be greater or less than the ratio in question by a difference less than any minute one we choose to assign.

Counting only, and not the counting of objects, is of the essence of arithmetic, and of algebra. But it is lawful to count objects, and in particular to count equal lengths by measure. The widened idea is that even when a or x is an irrational number we may speak of a or x unit lengths by measure. We may give concrete interpretation to an algebraical equation by allowing its terms all to mean numbers of times the same unit length, or the same unit area, or &c. and in any equation lawfully derived from the first by algebraical processes we may do the same. Descartes in his _Geometrie_ (1637) was the first to systematize the application of this principle to the inherent first notions of geometry; and the methods which he instituted have become the most potent methods of all in geometrical research. It is hardly too much to say that, when known facts as to a geometrical figure have once been expressed in algebraical terms, all strictly consequential facts as to the figure can be deduced by almost mechanical processes. Some may well be unexpected consequences; and in obtaining those of which there has been suggestion beforehand the often bewildering labour of constant attention to the figure is obviated. These are the methods of what is now called _analytical_, or sometimes _algebraical_, _geometry_.

3. The modern use of the term "analytical" in geometry has obscured, but not made obsolete, an earlier use, one as old as Plato. There is nothing algebraical in this analysis, as distinguished from synthesis, of the Greeks, and of the expositors of pure geometry. It has reference to an order of ideas in demonstration, or, more frequently, in discovering means to effect the geometrical construction of a figure with an assigned special property. We have to suppose hypothetically that the construction has been performed, drawing a rough figure which exhibits it as nearly as is practicable. We then analyse or critically examine the figure, treated as correct, and ascertain other properties which it can only possess in association with the one in question. Presently one of these properties will often be found which is of such a character that the construction of a figure possessing it is simple. The means of effecting synthetically a construction such as was desired is thus brought to light by what Plato called _analysis_. Or again, being asked to prove a theorem A, we ascertain that it must be true if another theorem B is, that B must be if C is, and so on, thus eventually finding that the theorem A is the consequence, through a chain of intermediaries, of a theorem Z of which the establishment is easy. This geometrical analysis is not the subject of the present article; but in the reasoning from form to form of an equation or system of equations, with the object of basing the algebraical proof of a geometrical fact on other facts of a more obvious character, the same logic is utilized, and the name "analytical geometry" is thus in part explained.

4. In algebra real positive number was alone at first dealt with, and in geometry actual signless distance. But in algebra it became of importance to say that every equation of the first degree has a root, and the notion of negative number was introduced. The negative unit had to be defined as what can be added to the positive unit and produce the sum zero. The corresponding notion was readily at hand in geometry, where it was clear that a unit distance can be measured to the left or down from the farther end of a unit distance already measured to the right or up from a point O, with the result of reaching O again. Thus, to give full interpretation in geometry to the algebraically negative, it was only necessary to associate distinctness of sign with oppositeness of direction. Later it was discovered 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 [+-][root]-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 0 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 every root has a value included in the form a + b [root]-1, with a, b, real. The corresponding enrichment could be given to geometry, with corresponding advantages and the same absence of danger, and this was done. On a line of measurement of distance we contemplate as existing, not only an infinite continuum of points at real distances from an origin of measurement O, but a doubly infinite continuum of points, all but the singly infinite continuum of real ones imaginary, and imaginary in conjugate pairs, a conjugate pair being at imaginary distances from O, which have a real arithmetic and a real geometric mean. To geometry enriched with this conception all algebra has its application.

5. Actual geometry is one, two or three-dimensional, i.e. lineal, plane or solid. In one-dimensional geometry positions and measurements in a single line only are admitted. Now descriptive constructions for points in a line are impossible without going out of the line. It has therefore been held that there is a sense in which no science of geometry strictly confined to one dimension exists. But an algebra of one variable can be applied to the study of distances along a line measured from a chosen point on it, so that the idea of construction as distinct from measurement is not essential to a one-dimensional geometry aided by algebra. In geometry of two dimensions, the flat of the land-measurer, the passage from one point O to any other point, can be effected by two successive marches, one east or west and one north or south, and, as will be seen, an algebra of two variables suffices for geometrical exploitation. In geometry of three dimensions, that of space, any point can be reached from a chosen one by three marches, one east or west, one north or south, and one up or down; and we shall see that an algebra of three variables is all that is necessary. With three dimensions actual geometry stops; but algebra can supply any number of variables. Four or more variables have been used in ways analogous to those in which one, two and three variables are used for the purposes of one, two and three-dimensional geometry, and the results have been expressed in quasi-geometrical language on the supposition that a higher space can be conceived of, though not realized, in which four independent directions exist, such that no succession of marches along three of them can effect the same displacement of a point as a march along the fourth; and similarly for higher numbers than four. Thus analytical, though not actual, geometries exist for four and more dimensions. They are in fact algebras furnished with nomenclature of a geometrical cast, suggested by convenient forms of expression which actual geometry has, in return for benefits received, conferred on algebras of one, two and three variables.

We will confine ourselves to the dimensions of actual geometry, and will devote no space to the one-dimensional, except incidentally as existing within the two-dimensional. The analytical method will now be explained for the cases of two and three dimensions in succession. The form of it originated by Descartes, and thence known as Cartesian, will alone be considered in much detail.

I. _Plane Analytical Geometry._

6. _Coordinates._--It is assumed that the points, lines and figures
considered lie in one and the same plane, which plane therefore need
not be in any way referred to. In the plane a point O, and two lines
x'Ox, y'Oy, intersecting in O, are taken once for all, and regarded as
fixed. O is called the origin, and x'Ox, y'Oy the axes of x and y
respectively. Other positions in the plane are specified in relation
to this fixed origin and these fixed axes. From any point P we suppose
PM drawn parallel to the axis of y to meet the axis of x in M, and may
also suppose PN drawn parallel to the axis of x to meet 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). If OM is x times the unit of a
scale of measurement chosen at pleasure, and MP is y times the unit,
so that x and y have numerical values, we call x and y the (Cartesian)
coordinates of P. To distinguish 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
according as the point P is on one side or the other of the axis of y,
and y one sign or the other according as P is on one side or the other
of the axis of x. Using the letters N, E, S, W, as in a map, and
considering the plane as divided into four quadrants by the axes, the
signs are usually taken to be:

x y For quadrant

+ + N E
+ - S E
- + N W
- - S W

A point is referred to as the point (a, b), when its coordinates are x
= a, y = b. A point may be fixed, or it may be variable, i.e. be
regarded for the time being as free to move in the plane. The
coordinates (x, y) of a variable point are algebraic variables, and
are said to be "current coordinates."

The axes of x and y are usually (as in fig. 48) taken at right angles
to one another, and we then speak of them as rectangular axes, and of
x and y as "rectangular coordinates" of a point P; OMPN is then a
rectangle. Sometimes, however, it is convenient to use axes which are
oblique to one another, so that (as in fig. 49) the angle xOy between
their positive directions is some known angle [omega] distinct from a
right angle, and OMPN is always an oblique parallelogram with given
angles; and we then speak of x and y as "oblique coordinates." The
coordinates are as a rule taken to be rectangular in what follows.

7. _Equations and loci._ If (x, y) is the point P, and if we are given
that x = 0, we are told that, in fig. 48 or fig. 49, the point M lies
at O, whatever value y may have, i.e. we are told the one fact that P
lies on the axis of y. Conversely, if P lies anywhere on the axis of
y, we have always OM = 0, i.e. x = 0. Thus the equation x = 0 is one
satisfied by the coordinates (x, y) of every point in the axis of y,
and not by those of any other point. We say that x = 0 is the equation
of the axis of y, and that the axis of y is the locus represented by
the equation x = 0. Similarly y = 0 is the equation of the axis of x.
An equation x = a, where a is a constant, expresses that P lies on a
parallel to the axis of y through a point M on the axis of x such that
OM = a. Every line parallel to the axis of y has an equation of this
form. Similarly, every line parallel to the axis of x has an equation
of the form y = b, where b is some definite constant.

These are simple cases of the fact that a single equation in the
current coordinates of a variable point (x, y) imposes one limitation
on the freedom of that point to vary. The coordinates of a point taken
at random in the plane will, as a rule, not satisfy the equation, but
infinitely many points, and in most cases infinitely many real ones,
have coordinates which do satisfy it, and these points are exactly
those which lie upon some locus of one dimension, a straight line or
more frequently a curve, which is said to be represented by the
equation. Take, for instance, the equation y = mx, where m is a given
constant. It is satisfied by the coordinates of every point P, which
is such that, in fig. 48, the distance MP, with its proper sign, is m
times the distance OM, with its proper sign, i.e. by the coordinates
of every point in the straight line through O which we arrive at by
making a line, originally coincident with x'Ox, revolve about O in the
direction opposite to that of the hands of a watch through an angle of
which m is the tangent, and by those of no other points. That line is
the locus which it represents. Take, more generally, the equation y =
[phi](x), where [phi](x) is any given non-ambiguous function of x.
Choosing any point M on x'Ox in fig. 1, and giving to x the value of
the numerical measure of OM, the equation determines a single
corresponding y, and so determines a single point P on the line
through M parallel to y'Oy. This is one point whose coordinates
satisfy the equation. Now let M move from the extreme left to the
extreme right of the line x'Ox, regarded as extended both ways as far
as we like, i.e. let x take all real values from -[oo] to [oo]. With
every value goes a point P, as above, on the parallel to y'Oy through
the corresponding M; and we thus find that there is a path from the
extreme left to the extreme right of the figure, all points P along
which are distinguished from other points by the exceptional property
of satisfying the equation by their coordinates. This path is a locus;
and the equation y = [phi](x) represents it. More generally still,
take an equation f(x, y) = 0 which involves both x and y under a
functional form. Any particular value given to x in it produces from
it an equation for the determination of a value or values of y, which
go with that value of x in specifying a point or points (x, y), of
which the coordinates satisfy the equation f(x, y) = 0. Here again, as
x takes all values, the point or points describe a path or paths,
which constitute a locus represented by the equation. Except when y
enters to the first degree only in f(x, y), it is not to be expected
that all the values of y, determined as going with a chosen value of
x, will be necessarily real; indeed it is not uncommon for all to be
imaginary for some ranges of values of x. The locus may largely
consist of continua of imaginary points; but the real parts of it
constitute a real curve or real curves. Note that we have to allow x
to admit of all imaginary, as well as of all real, values, in order to
obtain all imaginary parts of the locus.

A locus or curve may be algebraically specified in another way; viz.
we may be given two equations x = f([theta]), y = F([theta]), which
express the coordinates of any point of it as two functions of the
same variable parameter [theta] to which all values are open. As
[theta] takes all values in turn, the point (x, y) traverses the
curve.

It is a good exercise to trace a number of curves, taken as defined by
the equations which represent them. This, in simple cases, can be done
approximately by plotting the values of y given by the equation of a
curve as going with a considerable number of values of x, and
connecting the various points (x, y) thus obtained. But methods exist
for diminishing the labour of this tentative process.

Another problem, which will be more attended to here, is that of
determining the equations of curves of known interest, taken as
defined by geometrical properties. It is not a matter for surprise
that the curves which have been most and longest studied geometrically
are among those represented by equations of the simplest character.

8. _The Straight Line._--This is the simplest type of locus. Also the
simplest type of equation in x and y is Ax + By + C = 0, one of the
first degree. Here the coefficients A, B, C are constants. They are,
like the current coordinates, x, y, numerical. But, in giving
interpretation to such an equation, we must of course refer to numbers
Ax, By, C of unit magnitudes of the same kind, of units of counting
for instance, or unit lengths or unit squares. It will now be seen
that every straight line has an equation of the first degree, and that
every equation of the first degree represents a straight line.

It has been seen (S 7) that lines parallel to the axes have equations
of the first degree, free from one of the variables. Take now a
straight line ABC inclined to both axes. Let it make a given angle
[alpha] with the positive direction of the axis of x, i.e. in fig. 50
let this be the angle through which Ax must be revolved
counter-clockwise about A in order to be made coincident with the
line. Let C, of coordinates (h, k), be a fixed point on the line, and
P(x, y) any other point upon it. Draw the ordinates CD, PM of C and P,
and let the parallel to the axis of x through C meet PM, produced if
necessary, in R. The right-angled triangle CRP tells us that, with the
signs appropriate to their directions attached to CR and RP,

RP = CR tan [alpha], i.e. MP - DC = (OM - OD) tan [alpha],

and this gives that

y - k = tan [alpha] (x - h),

an equation of the first degree satisfied by x and y. No point not on
the line satisfies the same equation; for the line from C to any point
off the line would make with CR some angle [beta] different from
[alpha], and the point in question would satisfy an equation y - k =
tan [beta](x - h), which is inconsistent with the above equation.

The equation of the line may also be written y = mx + b, where m = tan
[alpha], and b = k - h tan [alpha]. Here b is the value obtained for y
from the equation when 0 is put for x, i.e. it is the numerical
measure, with proper sign, of OB, the intercept made by the line on
the axis of y, measured from the origin. For different straight lines,
m and b may have any constant values we like.

Now the general equation of the first degree Ax + By + C = 0 may be
written y = -(A/B)x - C/B, unless B = 0, in which case it represents a
line parallel to the axis of y; and -A/B, -C/B are values which can be
given to m and b, so that every equation of the first degree
represents a straight line. It is important to notice that the general
equation, which in appearance contains three constants A, B, C, in
effect depends on two only, the ratios of two of them to the third. In
virtue of this last remark, we see that two distinct conditions
suffice to determine a straight line. For instance, it is easy from
the above to see that

x y
-- + -- = 1
a b

is the equation of a straight line determined by the two conditions
that it makes intercepts OA, OB on the two axes, of which a and b are
the numerical measures with proper signs: note that in fig. 50 a is
negative. Again,

y2 - y1
y - y1 = ------- (x - x1),
x2 - x1

i.e.

(y1 - y2)x - (x1 - x2)y + x1y2 - x2y1 = 0,

represents the line determined by the data that it passes through two
given points (x1, y1) and (x2, y2). To prove this find m in the
equation y - y1 = m(x - x1) of a line through (x1, y1), from the
condition that (x2, y2) lies on the line.

In this paragraph the coordinates have been assumed rectangular. Had
they been oblique, the doctrine of similar triangles would have given
the same results, except that in the forms of equation y - k = m(x -
h), y = mx + b, we should not have had m = tan [alpha].

9. _The Circle._--It is easy to write down the equation of a given
circle. Let (h, k) be its given centre C, and [rho] the numerical
measure of its given radius. Take P (x, y) any point on its
circumference, and construct the triangle CRP, in fig. 50 as above.
The fact that this is right-angled tells us that

CR^2 + RP^2 = CP^2,

and this at once gives the equation

(x - h)^2 + (y - k)^2 = [rho]^2.

A point not upon the circumference of the particular circle is at some
distance from (h, k) different from [rho], and satisfies an equation
inconsistent with this one; which accordingly represents the
circumference, or, as we say, the circle.

The equation is of the form

x^2 + y^2 + 2Ax + 2By + C = 0.

Conversely every equation of this form represents a circle: we have
only to take -A, -B, A^2 + B^2 - C for h, k, [rho]^2 respectively, to
obtain its centre and radius. But this statement must appear too
unrestricted. Ought we not to require A^2 + B^2 - C to be positive?
Certainly, if by circle we are only to mean the visible round
circumference of the geometrical definition. Yet, analytically, we
contemplate altogether imaginary circles, for which [rho]^2 is
negative, and circles, for which [rho] = 0, with all their reality
condensed into their centres. Even when [rho]^2 is positive, so that a
visible round circumference exists, we do not regard this as
constituting the whole of the circle. Giving to x any value whatever
in (x - h)^2 + (y - k)^2 = [rho]^2, we obtain two values of y, real,
coincident or imaginary, each of which goes with the abscissa x as the
ordinate of a point, real or imaginary, on what is represented by the
equation of the circle.

The doctrine of the imaginary on a circle, and in geometry generally,
is of purely algebraical inception; but it has been in its entirety
accepted by modern pure geometers, and signal success has attended the
efforts of those who, like K.G.C. von Staudt, have striven to base its
conclusions on principles not at all algebraical in form, though of
course cognate to those adopted in introducing the imaginary into
algebra.

A circle with its centre at the origin has an equation x^2 + y^2 =
[rho]^2.

In oblique coordinates the general equation of a circle is x^2 + 2xy
cos [omega] + y^2 + 2Ax + 2By + C = 0.

10. The conic sections are the next simplest loci; and it will be seen
later that they are the loci represented by equations of the second
degree. Circles are particular cases of conic sections; and they have
just been seen to have for their equations a particular class of
equations of the second degree. Another particular class of such
equations is that included in the form (Ax + By + C)(A'x + B'y + C') =
0, which represents two straight lines, because the product on the
left vanishes if, and only if, one of the two factors does, i.e. if,
and only if, (x, y) lies on one or other of two straight lines. The
condition that ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0, which is often
written (a, b, c, f, g, h)(x, y, I)^2 = 0, takes this form is abc +
2fgh-af^2-bg^2 - ch^2 = 0. Note that the two lines may, in particular
cases, be parallel or coincident.

Any equation like F1(x, y) F2(x, y) ... F_n(x, y) = 0, of which the
left-hand side breaks up into factors, represents all the loci
separately represented by F1(x, y) = 0, F2(x, y) = 0, ... F_n(x, y) =
0. In particular an equation of degree n which is free from x
represents n straight lines parallel to the axis of x, and one of
degree n which is homogeneous in x and y, i.e. one which upon division
by x^n, becomes an equation in the ratio y/x, represents n straight
lines through the origin.

Curves represented by equations of the third degree are called cubic
curves. The general equation of this degree will be written (*)(x, y,
I)^3 = 0.

11. _Descriptive Geometry._--A geometrical proposition is either
descriptive or metrical: in the former case the statement of it is
independent of the idea of magnitude (length, inclination, &c.), and
in the latter it has reference to this idea. The method of coordinates
seems to be by its inception essentially metrical. Yet in dealing by
this method with descriptive propositions we are eminently free from
metrical considerations, because of our power to use general
equations, and to avoid all assumption that measurements implied are
any particular measurements.

12. It is worth while to illustrate this by the instance of the
well-known theorem of the radical centre of three circles. The theorem
is that, given any three circles A, B, C (fig. 51), the common chords
[alpha][alpha]', [beta][beta]', [gamma][gamma]' of the three pairs of
circles meet in a point.

The geometrical proof is metrical throughout:--

Take O the point of intersection of [alpha][alpha]', [beta][beta]',
and joining this with [gamma]', suppose that [gamma]'O does not pass
through [gamma], but that it meets the circles A, B in two distinct
points [gamma]2, [gamma]1 respectively. We have then the known
metrical property of intersecting chords of a circle; viz. in circle
C, where [alpha][alpha]', [beta][beta]', are chords meeting at a point
O,

O[alpha].O[alpha]' = O[beta].O[beta]',

where, as well as in what immediately follows, O[alpha], &c. denote,
of course, _lengths_ or _distances_.

Similarly in circle A,

O[beta].O[beta]' = O[gamma]2.O[gamma]',

and in circle B,

O[alpha].O[alpha]' = O[gamma]1.O[gamma]'.

Consequently O[gamma]1.O[gamma]' = O[gamma]2.O[gamma]', that is,
O[gamma]1 = O[gamma]2, or the points [gamma]1 and [gamma]2 coincide;
that is, they each coincide with [gamma].

We contrast this with the analytical method:--

Here it only requires to be known that an equation Ax + By + C = 0
represents a line, and an equation x^2 + y^2 + Ax + By + C = 0
represents a circle. A, B, C have, in the two cases respectively,
metrical significations; but these we are not concerned with. Using S
to denote the function x^2 + y^2 + Ax + By + C, the equation of a
circle is S = o. Let the equation of any other circle be S', = x^2 +
y^2 + A'x + B'y + C' = 0; the equation S - S' = 0 is a linear equation
(S - S' is in fact = (A - A')x + (B - B')y + C - C), and it thus
represents a line; this equation is satisfied by the coordinates of
each of the points of intersection of the two circles (for at each of
these points S = 0 and S' = 0, therefore also S - S' = 0); hence the
equation S - S' = 0 is that of the line joining the two points of
intersection of the two circles, or say it is the equation of the
common chord of the two circles. Considering then a third circle S", =
x^2 + y^2 + A"x + B"y + C" = 0, the equations of the common chords are
S-S' = 0, S - S" = 0, S' - S" = 0 (each of these a linear equation);
at the intersection of the first and second of these lines S = S' and
S = S", therefore also S' = S", or the equation of the third line is
satisfied by the coordinates of the point in question; that is, the
three chords intersect in a point O, the coordinates of which are
determined by the equations S = S' = S".

It further appears that if the two circles S = 0, S' = 0 do not
intersect in any real points, they must be regarded as intersecting in
two imaginary points, such that the line joining them is the real line
represented by the equation S - S' = 0; or that two circles, whether
their intersections be real or imaginary, have always a real common
chord (or radical axis), and that for _any_ three circles the common
chords intersect in a point (of course real) which is the radical
centre. And by this very theorem, given two circles with imaginary
intersections, we can, by drawing circles which meet each of them in
real points, construct the radical axis of the first-mentioned two
circles.

13. The principle employed in showing that the equation of the common
chord of two circles is S - S' = 0 is one of very extensive
application, and some more illustrations of it may be given.

Suppose S = 0, S' = 0 are lines (that is, let S, S' now denote linear
functions Ax + By + C, A'x + B'y + C'), then S - kS' = 0 (k an
arbitrary constant) is the equation of any line passing through the
point of intersection of the two given lines. Such a line may be made
to pass through any given point, say the point (x0, y0); if S0, S'0
are what S, S' respectively become on writing for (x, y) the values
(x0, y0), then the value of k is k = S0 : S'0. The equation in fact is
SS'0 - S0S' = 0; and starting from this equation we at once verify it
_a posteriori_; the equation is a linear equation satisfied by the
values of (x, y) which make S = 0, S' = 0; and satisfied also by the
values (x0, y0); and it is thus the equation of the line in question.

If, as before, S = 0, S' = 0 represent circles, then (k being
arbitrary) S - kS' = 0 is the equation of any circle passing through
the two points of intersection of the two circles; and to make this
pass through a given point (x0, y0) we have again k = S0 : S'0. In the
particular case k = 1, the circle becomes the common chord (more
accurately it becomes the common chord together with the line
infinity; see S 23 below).

If S denote the general quadric function,

S = ax^2 +2hxy + by^2 + 2fy + 2gx + c,

then the equation S = 0 represents a conic; assuming this, then, if S'
= 0 represents another conic, the equation S - kS' = 0 represents
_any_ conic through the four points of intersection of the two conics.

14. The object still being to illustrate the mode of working with
coordinates for descriptive purposes, we consider the theorem of the
polar of a point in regard to a circle. Given a circle and a point O
(fig. 52), we draw through O any two lines meeting the circle in the
points A, A' and B, B' respectively, and then taking Q as the
intersection of the lines AB' and A'B, the theorem is that the locus
of the point Q is a right line depending only upon O and the circle,
but independent of the particular lines OAA' and OBB'.

Taking O as the origin, and for the axes any two lines through O at
right angles to each other, the equation of the circle will be

x^2 + y^2 + 2Ax + 2By + C = 0;

and if the equation of the line OAA' is taken to be y = mx, then the
points A, A' are found as the intersections of the straight line with
the circle; or to determine x we have

x^2(1 + m^2) + 2x(A + Bm) + C = 0.

If(x1, y1) are the coordinates of A, and (x2, y2) of A', then the
roots of this equation are x1, x2, whence easily

1 1 A + Bm
-- + -- = -2 ------.
x1 x2 C

And similarly, if the equation of the line OBB' is taken to be y =
m'x1 and the coordinates of B, B' to be (x3, y3) and (x4, y4)
respectively, then

1 1 A + Bm'
-- + -- = -2 -------.
x3 x4 C'

We have then by S 8

x(y1 - y4) - y(x1 - x4) + x1y4 - x4y1 = 0,

x(y2 - y3) - y(x2 - x3) + x2y3 - x3y2 = 0,

as the equations of the lines AB' and A'B respectively. Reducing by
means of the relations y1 - mx1 = 0, y2 - mx2 = 0, y3 - m'x3 = 0, y4 -
m'x4 = 0, the two equations become

x(mx1 - m'x4) - y(x1 - x4) + (m'- m)x1x4 = 0,

x(mx2 - m'x3) - y(x2 - x3) + (m'- m)x2x3 = 0,

and if we divide the first of these equations by x1x4, and the second
by x2x3 and then add, we obtain
_ _ _ _
| / 1 1 \ / 1 1 \ | | 1 1 / 1 1 \ |
x| m( -- + -- ) - m'( -- + -- ) | - y| -- + -- - ( -- + -- ) |
|_ \ x3 x4 / \ x1 x2/ _| |_ x3 x4 \ x1 x2/ _|

+ 2m' - 2m = 0,

or, what is the same thing,

/ 1 1 \ / 1 1 \
( -- + -- )(y - m'x) - ( -- + -- )(y - mx) + 2m' - 2m = 0,
\ x1 x2 / \ x3 x4 /

which by what precedes is the equation of a line through the point Q.
Substituting herein for 1/x1 + 1/x2, 1/x3 + 1/x4 their foregoing
values, the equation becomes

-(A + Bm)(y - m'x) + (A + Bm')(y - mx) + C(m' - m) = 0;

that is,

(m - m')(Ax + By + C) = 0;

or finally it is Ax + By + C = 0, showing that the point Q lies in a
line the position of which is independent of the particular lines
OAA', OBB' used in the construction. It is proper to notice that there
is no correspondence to each other of the points A, A' and B, B'; the
grouping might as well have been A, A' and B', B; and it thence
appears that the line Ax + By + C = 0 just obtained is in fact the
line joining the point Q with the point R which is the intersection of
AB and A'B'.

15. In S 8 it has been seen that two conditions determine the equation
of a straight line, because in Ax + By + C = 0 one of the coefficients
may be divided out, leaving only two parameters to be determined.
Similarly five conditions instead of six determine an equation of the
second degree (a, b, c, f, g, h)(x, y, 1)^2 = 0, and nine instead of
ten determine a cubic (*)(x, y, 1)^3 = 0. It thus appears that a cubic
can be made to pass through 9 given points, and that the cubic so
passing through 9 given points is completely determined. There is,
however, a remarkable exception. Considering two given cubic curves S
= 0, S' = 0, these intersect in 9 points, and through these 9 points
we have the whole series of cubics S - kS' = 0, where k is an
arbitrary constant: k may be determined so that the cubic shall pass
through a given tenth point (k = S0 : S'0, if the coordinates are (x0,
y0), and S0, S'0 denote the corresponding values of S, S'). The
resulting curve SS'0 - S'S0 = 0 may be regarded as the cubic
determined by the conditions of passing through 8 of the 9 points and
through the given point (x0, y0); and from the equation it thence
appears that the curve passes through the remaining one of the 9
points. In other words, we thus have the theorem, any cubic curve
which passes through 8 of the 9 intersections of two given cubic
curves passes through the 9th intersection.