Chapter XXXIV: Book XIII (6)

The applications of this theorem are very numerous; for instance, we
derive from it Pascal's theorem of the inscribed hexagon. Consider a
hexagon inscribed in a conic. The three alternate sides constitute a
cubic, and the other three alternate sides another cubic. The cubics
intersect in 9 points, being the 6 vertices of the hexagon, and the 3
Pascalian points, or intersections of the pairs of opposite sides of
the hexagon. Drawing a line through two of the Pascalian points, the
conic and this line constitute a cubic passing through 8 of the 9
points of intersection, and it therefore passes through the remaining
point of intersection--that is, the third Pascalian point; and since
obviously this does not lie on the conic, it must lie on the
line--that is, we have the theorem that the three Pascalian points (or
points of intersection of the pairs of opposite sides) lie on a line.

16. _Metrical Theory resumed. Projections and Perpendiculars._--It
is a metrical fact of fundamental importance, already used in S 8,
that, if a finite line PQ be projected on any other line OO' by
perpendiculars PP', QQ' to OO', the length of the projection P'Q' is
equal to that of PQ multiplied by the cosine of the acute angle
between the two lines. Also the algebraical sum of the projections of
the sides of any closed polygon upon any line is zero, because as a
point goes round the polygon, from any vertex A to A again, the point
which is its projection on the line passes from A' the projection of A
to A' again, i.e. traverses equal distances along the line in positive
and negative senses. If we consider the polygon as consisting of two
broken lines, each extending from the same initial to the same
terminal point, the sum of the projections of the lines which compose
the one is equal, in sign and magnitude, to the sum of the projections
of the lines composing the other. Observe that the projection on a
line of a length perpendicular to the line is zero.

Let us hence find the equation of a straight line such that the
perpendicular OD on it from the origin is of length [rho] taken as
positive, and is inclined to the axis of x at an angle xOD = [alpha],
measured counter-clockwise from Ox. Take any point P (x, y) on the
line, and construct OM and MP as in fig. 48. The sum of the
projections of OM and MP on OD is OD itself; and this gives the
equation of the line

x cos [alpha] + y sin [alpha] = [rho].

Observe that cos [alpha] and sin [alpha] here are the sin [alpha] and
-cos [alpha], or the -sin [alpha] and cos [alpha] of S 8 according to
circumstances.

We can write down an expression for the perpendicular distance from
this line of any point (x', y') which does not lie upon it. If the
parallel through (x', y') to the line meet OD in E, we have x' cos
[alpha] + y' sin [alpha] = OE, and the perpendicular distance required
is OD - OE, i.e. [rho] - x' cos [alpha] - y' sin [alpha]; it is the
perpendicular distance taken positively or negatively according as
(x', y') lies on the same side of the line as the origin or not.

The general equation Ax + By + C = 0 may be given the form x cos
[alpha] + y sin [alpha] - [rho] = 0 by dividing it by [root](A^2 +
B^3). Thus (Ax' + By' + C) / [root](A^2 + B^2) is in absolute value
the perpendicular distance of (x', y') from the line Ax + By + C = 0.
Remember, however, that there is an essential ambiguity of sign
attached to a square root. The expression found gives the distance
taken positively when (x', y') is on the origin side of the line, if
the sign of C is given to [root](A^2 + B^2).

17. _Transformation of Coordinates._--We often need to adopt new axes
of reference in place of old ones; and the above principle of
projections readily expresses the old coordinates of any point in
terms of the new.

Suppose, for instance, that we want to take for new origin the point
O' of old coordinates OA = h, AO' = k, and for new axes of X and Y
lines through O' obtained by rotating parallels to the old axes of x
and y through an angle [theta] counter-clockwise. Construct (fig. 53)
the old and new coordinates of any point P. Expressing that the
projections, first on the old axis of x and secondly on the old axis
of y, of OP are equal to the sums of the projections, on those axes
respectively, of the parts of the broken line OO'M'P, we obtain:

x = h + X cos [theta] + Y cos ([theta] + 1/2[pi]) = h + X cos [theta] -
Y sin [theta],

and

y = k + X cos (1/2[pi] - [theta]) + Y cos [theta] = k + X sin [theta] +
Y cos [theta].

Be careful to observe that these formulae do not apply to every
conceivable change of reference from one set of rectangular axes to
another. It might have been required to take O'X, O'Y' for the
positive directions of the new axes, so that the change of directions
of the axes could not be effected by rotation. We must then write -Y
for Y in the above.

Were the new axes oblique, making angles [alpha], [beta] respectively
with the old axis of x, and so inclined at the angle [beta] - [alpha],
the same method would give the formulae

x = h + X cos [alpha] + Y cos [beta], y = k + X sin [alpha] + Y sin [beta].

18. _The Conic Sections._--The conics, as they are now called, were at
first defined as curves of intersection of planes and a cone; but
Apollonius substituted a definition free from reference to space of
three dimensions. This, in effect, is that a conic is the locus of a
point the distance of which from a given point, called the focus, has
a given ratio to its distance from a given line, called the directrix
(see CONIC SECTION). If e : 1 is the ratio, e is called the
eccentricity. The distances are considered signless.

Take (h, k) for the focus, and x cos [alpha] + y sin [alpha] - p = 0
for the directrix. The absolute values of [root] {(x - h)^2 + (y -
k)^2} and p - x cos [alpha] -y sin [alpha] are to have the ratio e :
1; and this gives

(x - h)^2 + (y - k)^2 = e^2(p - x cos [alpha] - y sin [alpha])^2

as the general equation, in rectangular coordinates, of a conic.

It is of the second degree, and is the general equation of that
degree. If, in fact, we multiply it by an unknown [lambda], we can, by
solving six simultaneous equations in the six unknowns [lambda], h, k,
e, p, [alpha], so choose values for these as to make the coefficients
in the equation equal to those in any equation of the second degree
which may be given. There is no failure of this statement in the
special case when the given equation represents two straight lines, as
in S 10, but there is speciality: if the two lines intersect, the
intersection and either bisector of the angle between them are a focus
and directrix; if they are united in one line, any point on the line
and a perpendicular to it through the point are: if they are parallel,
the case is a limiting one in which e and h^2 + k^2 have become
infinite while e^(-2)(h^2 + k^2) remains finite. In the case (S 9) of
an equation such as represents a circle there is another instance of
proceeding to a limit: e has to become 0, while ep remains finite:
moreover [alpha] is indeterminate. The centre of a circle is its
focus, and its directrix has gone to infinity, having no special
direction. This last fact illustrates the necessity, which is also
forced on plane geometry by three-dimensional considerations, of
treating all points at infinity in a plane as lying on a single
straight line.

Sometimes, in reducing an equation to the above focus and directrix
form, we find for h, k, e, p, tan [alpha], or some of them, only
imaginary values, as quadratic equations have to be solved; and we
have in fact to contemplate the existence of entirely imaginary
conics. For instance, no real values of x and y satisfy x^2 + 2y^2 + 3
= 0. Even when the locus represented is real, we obtain, as a rule,
four sets of values of h, k, e, p, of which two sets are imaginary; a
real conic has, besides two real foci and corresponding directrices,
two others that are imaginary.

In oblique as well as rectangular coordinates equations of the second
degree represent conics.

19. _The three Species of Conics._--A real conic, which does not
degenerate into straight lines, is called an ellipse, parabola or
hyperbola according as e <, = , or > 1. To trace the three forms it is best so to
choose the axes of reference as to simplify their equations.

In the case of a parabola, let 2c be the distance between the given
focus and directrix, and take axes referred to which these are the
point (c, 0) and the line x = - c. The equation becomes (x - c)^2 +
y^2 = (x + c)^2, i.e. y^2 = 4cx.

In the other cases, take a such that a(e ~ e^(-1)) is the distance of
focus from directrix, and so choose axes that these are (ae, 0) and x
= ae^(-1), thus getting the equation(x - ae)^2 + y^2 = e^2(x -
ae^(-1))^2, i.e. (1 - e^2)x^2 + y^2 = a^2(1 - e^2). When e < 1, i.e.
in the case of an ellipse, this may be written x^2/a^2 + y^2/b^2 = 1,
where b^2 = a^2(1 - e^2); and when e > 1, i.e. in the case of an
hyperbola, x^2/a^2 - y^2/b^2 = 1, where b^2 = a^2(e^2 - 1). The axes
thus chosen for the ellipse and hyperbola are called the principal
axes.

In figs. 54, 55, 56 in order, conics of the three species, thus
referred, are depicted.

The oblique straight lines in fig. 56 are the _asymptotes_ x/a =
[+-]y/b of the hyperbola, lines to which the curve tends with
unlimited closeness as it goes to infinity. The hyperbola would have
an equation of the form xy = c if referred to its asymptotes as axes,
the coordinates being then oblique, unless a = b, in which case the
hyperbola is called rectangular. An ellipse has two imaginary
asymptotes. In particular a circle x^2 + y^2 = a^2, a particular
ellipse, has for asymptotes the imaginary lines x = [+-]y [root]-1.
These run from the centre to the so-called circular points at
infinity.

20. _Tangents and Curvature._--Let (x', y') and (x' + h, y' + k) be
two neighbouring points P, P' on a curve. The equation of the line on
which both lie is h(y - y') = k(x - x'). Now keep P fixed, and let P'
move towards coincidence with it along the curve. The connecting line
will tend towards a limiting position, to which it can never attain as
long as P and P' are distinct. The line which occupies this limiting
position is the tangent at P. Now if we subtract the equation of the
curve, with (x', y') for the coordinates in it, from the like equation
in (x' + h, y' + k), we obtain a relation in h and k, which will, as a
rule, be of the form 0 = Ah + Bk + terms of higher degrees in h and k,
where A, B and the other coefficients involve x' and y'. This gives
k/h = -A/B + terms which tend to vanish as h and k do, so that -A : B
is the limiting value tended to by k : h. Hence the equation of the
tangent is B(y - y') + A(x - x') = 0.

The _normal_ at (x', y') is the line through it at right angles to the
tangent, and its equation is A(y - y') - B(x - x') = 0.

In the case of the conic (a, b, c, f, g, h) (x, y, 1)^2 = 0 we find
that A/B = (ax' + hy' + g)/(hx' + by' + f).

We can obtain the coordinates of Q, the intersection of the normals
QP, QP' at (x', y') and (x' + h, y' + k), and then, using the limiting
value of k : h, deduce those of its limiting position as P' moves up
to P. This is the _centre of curvature_ of the curve at P (x', y'),
and is so called because it is the centre of the circle of closest
contact with the curve at that point. That it is so follows from the
facts that the closest circle is the limit tended to by the circle
which touches the curve at P and passes through P', and that the arc
from P to P' of this circle lies between the circles of centre Q and
radii QP, QP', which circles tend, not to different limits as P' moves
up to P, but to one. The distance from P to the centre of curvature is
the _radius of curvature_.

21. _Differential Plane Geometry._--The language and notation of the
differential calculus are very useful in the study of tangents and
curvature. Denoting by ([xi], [eta]) the current coordinates, we find,
as above, that the tangent at a point (x, y) of a curve is [eta] - y =
([xi] - x)dy/dx, where dy/dx is found from the equation of the curve.
If this be f(x, y) = 0 the tangent is ([xi] - x) (dPf/dPx) + ([eta] -
y) (dPf/dPy) = 0. If [rho] and ([alpha], [beta]) are the radius and
centre of curvature at (x, y), we find that q([alpha] - x) = -p(1 +
p^2), q([beta] - y) = 1 + p^2, q^2[rho]^2 = (1 + p^2)^3, where p, q
denote dy/dx, d^2y/dx^2 respectively. (See INFINITESIMAL CALCULUS.)

In any given case we can, at all events in theory, eliminate x, y
between the above equations for [alpha] - x and [beta] - y, and the
equation of the curve. The resulting equation in ([alpha], [beta])
represents the locus of the centre of curvature. This is the _evolute_
of the curve.

22. _Polar Coordinates._--In plane geometry the distance of any point
P from a fixed origin (or pole) O, and the inclination xOP of OP to a
fixed line Ox, determine the point: r, the numerical measure of OP,
the _radius vector_, and [theta], the circular measure of xOP, the
_inclination_, are called polar coordinates of P. The formulae x = r
cos [theta], y = r sin [theta] connect Cartesian and polar
coordinates, and make transition from either system to the other easy.
In polar coordinates the equations of a circle through O, and of a
conic with O as focus, take the simple forms r = 2a cos
([theta]-[alpha]), r {1 - e cos ([theta]-[alpha])} = l. The use of
polar coordinates is very convenient in discussing curves which have
properties of symmetry akin to that of a regular polygon, such curves
for instance as r = a cos m [theta], with m integral, and also the
curves called spirals, which have equations giving r as functions of
[theta] itself, and not merely of sin [theta] and cos [theta]. In the
geometry of motion under central forces the advantage of working with
polar coordinates is great.

23. _Trilinear and Areal Coordinates._--Consider a fixed triangle ABC,
and regard its sides as produced without limit. Denote, as in
trigonometry, by a, b, c the positive numbers of units of a chosen
scale contained in the lengths BC, CA, AB, by A, B, C the angles, and
by [Delta] the area, of the triangle. We might, as in S 6, take CA, CB
as axes of x and y, inclined at an angle C. Any point P (x, y) in the
plane is at perpendicular distances y sin C and x sin C from CA and
CB. Call these [beta] and [alpha] respectively. The signs of [beta]
and [alpha] are those of y and x, i.e. [beta] is positive or negative
according as P lies on the same side of CA as B does or the opposite,
and similarly for [alpha]. An equation in (x, y) of any degree may,
upon replacing in it x and y by [alpha] cosec C and [beta] cosec C, be
written as one of the same degree in ([alpha], [beta]). Now let
[gamma] be the perpendicular distance of P from the third side AB,
taken as positive or negative as P is on the C side of AB or not. The
geometry of the figure tells us that a[alpha] + b[beta] + c[gamma] =
2[Delta]. By means of this relation in [alpha], [beta], [gamma] we can
give an equation considered countless other forms, involving two or
all of [alpha], [beta], [gamma]. In particular we may make it
_homogeneous_ in [alpha], [beta], [gamma]: to do this we have only to
multiply the terms of every degree less than the highest present in
the equation by a power of (a[alpha] + b[beta] + c[gamma])/2[Delta]
just sufficient to raise them, in each case, to the highest degree.

We call ([alpha], [beta], [gamma]) _trilinear coordinates_, and an
equation in them the trilinear equation of the locus represented.
Trilinear equations are, as a rule, dealt with in their homogeneous
forms. An advantage thus gained is that we need not mean by ([alpha],
[beta], [gamma]) the actual measures of the perpendicular distances,
but any properly signed numbers which have the same ratio two and two
as these distances.

In place of [alpha], [beta], [gamma] it is lawful to use, as
coordinates specifying the position of a point in the plane of a
triangle of reference ABC, any given multiples of these. For instance,
we may use x = a[alpha]/2[Delta], y = b[beta]/2[Delta], z =
c[gamma]/2[Delta], the properly signed ratios of the triangular areas
PBC, PCA, PAB to the triangular area ABC. These are called the _areal_
coordinates of P. In areal coordinates the relation which enables us
to make any equation homogeneous takes the simple form x + y + z = 1;
and, as before, we need mean by x, y, z, in a homogeneous equation,
only signed numbers in the right ratios.

Straight lines and conics are represented in trilinear and in areal,
because in Cartesian, coordinates by equations of the first and second
degrees respectively, and these degrees are preserved when the
equations are made homogeneous. What must be said about points
infinitely far off in order to make universal the statement, to which
there is no exception as long as finite distances alone are
considered, that _every_ homogeneous equation of the first degree
represents a straight line? Let the point of areal coordinates (x',
y', z') move infinitely far off, and mean by x, y, z finite quantities
in the ratios which x', y', z' tend to assume as they become infinite.
The relation x' + y' + z' = 1 gives that the limiting state of things
tended to is expressed by x + y + z = 0. This particular equation of
the first degree is satisfied by no point at a finite distance; but we
see the propriety of saying that it has to be taken as satisfied by
all the points conceived of as actually at infinity. Accordingly the
special property of these points is expressed by saying that they lie
on a special straight line, of which the areal equation is x + y + z =
0. In trilinear coordinates this _line at infinity_ has for equation
a[alpha] + b[beta] + c[gamma] = 0.

On the one special line at infinity parallel lines are treated as
meeting. There are on it two special (imaginary) points, the circular
points at infinity of S 19, through which all circles pass in the same
sense. In fact if S = O be one circle, in areal coordinates, S + (x +
y + z)(lx + my + nz) = 0 may, by proper choice of l, m, n, be made any
other; since the added terms are once lx + my + nz, and have the
generality of any expression like a'x + b'y + c' in Cartesian
coordinates. Now these two circles intersect in the two points where
either meets x + y + z = 0 as well as in two points on the radical
axis lx + my + nz = 0.

24. Let us consider the perpendicular distance of a point ([alpha]',
[beta]', [gamma]') from a line l[alpha] + m[beta] + n[gamma]. We can
take rectangular axes of Cartesian coordinates (for clearness as to
equalities of angle it is best to choose an origin inside ABC), and
refer to them, by putting expressions p - x cos[theta] - y sin[theta],
&c., for [alpha] &c.; we can then apply S 16 to get the perpendicular
distance; and finally revert to the trilinear notation. The result is
to find that the required distance is

(l[alpha]' + m[beta]' + n[gamma]')/{l, m, n},

where {l, m, n}^2 = l^2 + m^2 + n^2 - 2mn cos A - 2nl cos B - 2lm cos C.

In areal coordinates the perpendicular distance from (x', y', z') to
lx + my + nz = 0 is 2[Delta](lx' + my' + nz')/{al, bm, cn}. In both
cases the coordinates are of course actual values.

Now let [xi], [eta], [zeta] be the perpendiculars on the line from the
vertices A, B, C, i.e. the points (1, 0, 0), (0, 1, 0), (0, 0, 1),
with signs in accord with a convention that oppositeness of sign
implies distinction between one side of the line and the other. Three
applications of the result above give

[xi]/l = 2[Delta]/{al, bm, cn} = [eta]/m = [zeta]/n;

and we thus have the important fact that [xi]x' + [eta]y' + [zeta]z'
is the perpendicular distance between a point of areal coordinates
(x'y'z') and a line on which the perpendiculars from A, B, C are [xi],
[eta], [zeta] respectively. We have also that [xi]x + [eta]y + [zeta]z
= 0 is the areal equation of the line on which the perpendiculars are
[xi], [eta], [zeta]; and, by equating the two expressions for the
perpendiculars from (x', y', z') on the line, that in all cases
{a[xi], b[eta], c[zeta]}^2 = 4[Delta]^2.

25. _Line-coordinates. Duality._--A quite different order of ideas
may be followed in applying analysis to geometry. The notion of a
straight line specified may precede that of a point, and points may be
dealt with as the intersections of lines. The specification of a line
may be by means of coordinates, and that of a point by an equation,
satisfied by the coordinates of lines which pass through it. Systems
of _line-coordinates_ will here be only briefly considered. Every such
system is allied to some system of point-coordinates; and space will
be saved by giving prominence to this fact, and not recommencing _ab
initio_.

Suppose that any particular system of point-coordinates, in which lx +
my + nz = 0 may represent any straight line, is before us: notice that
not only are trilinear and areal coordinates such systems, but
Cartesian coordinates also, since we may write x/z, y/z for the
Cartesian x, y, and multiply through by z. The line is exactly
assigned if l, m, n, or their mutual ratios, are known. Call (l, m, n)
the _coordinates_ of the line. Now keep x, y, z constant, and let the
coordinates of the line vary, but always so as to satisfy the
equation. This equation, which we now write xl + ym + zn = 0, is
satisfied by the coordinates of every line through a certain fixed
point, and by those of no other line; it is the equation of that point
in the line-coordinates l, m, n.

Line-coordinates are also called _tangential_ coordinates. A curve is
the envelope of lines which touch it, as well as the locus of points
which lie on it. A homogeneous equation of degree above the first in
l, m, n is a relation connecting the coordinates of every line which
touches some curve, and represents that curve, regarded as an
envelope. For instance, the condition that the line of coordinates (l,
m, n), i.e. the line of which the allied point-coordinate equation is
lx + my + nz = 0, may touch a conic (a, b, c, f, g, h) (x, y, z)^2 =
0, is readily found to be of the form (A, B, C, F, G, H) (l, m, n)^2 =
0, i.e. to be of the second degree in the line-coordinates. It is not
hard to show that the _general_ equation of the second degree in l, m,
n thus represents a conic; but the degenerate conics of
line-coordinates are not line-pairs, as in point-coordinates, but
point-pairs.

The degree of the point-coordinate equation of a curve is the _order_
of the curve, the number of points in which it cuts a straight line.
That of the line-coordinate equation is its _class_, the number of
tangents to it from a point. The order and class of a curve are
generally different when either exceeds two.

26. The system of line-coordinates allied to the areal system of
point-coordinates has special interest.

The l, m, n of this system are the perpendiculars [xi], [eta], [zeta]
of S 24; and x'[xi] + y'[eta] + z'[zeta] = 0 is the equation of the
point of areal coordinates (x', y', z'), i.e. is a relation which the
perpendiculars from the vertices of the triangle of reference on every
line through the point, but no other line, satisfy. Notice that a
non-homogeneous equation of the first degree in [xi], [eta], [zeta]
does not, as a homogeneous one does, represent a point, but a circle.
In fact x'[xi] + y'[eta] + z'[zeta] = R expresses the constancy of the
perpendicular distance of the fixed point x'[xi] + y'[eta] + z'[zeta]
= 0 from the variable line ([xi], [eta], [zeta]), i.e. the fact that
([xi], [eta], [zeta]) touches a circle with the fixed point for
centre. The relation in any [xi], [eta], [zeta] which enables us to
make an equation homogeneous is not linear, as in point-coordinates,
but quadratic, viz. it is the relation {a[xi], b[eta], c[zeta]}^2 =
4[Delta]^2 of S 24. Accordingly the homogeneous equation of the above
circle is

4[Delta]^2(x'[xi] + y'[eta] + z'[zeta])^2 = R^2{a[xi], b[eta], c[zeta]}^2.

Every circle has an equation of this form in the present system of
line-coordinates. Notice that the equation of any circle is satisfied
by those coordinates of lines which satisfy both x'[xi] + y'[eta] +
z'[zeta] = 0, the equation of its centre, and {a[xi], b[eta],
c[zeta]}^2 = 0. This last equation, of which the left-hand side
satisfies the condition for breaking up into two factors, represents
the two imaginary circular points at infinity, through which all
circles and their asymptotes pass.

There is strict duality in descriptive geometry between
point-line-locus and line-point-envelope theorems. But in metrical
geometry duality is encumbered by the fact that there is in a plane
one special line only, associated with distance, while of special
points, associated with direction, there are two: moreover the line is
real, and the points both imaginary.

II. _Solid Analytical Geometry._

27. Any point in space may be specified by three coordinates. We
consider three fixed planes of reference, and generally, as in all
that follows, three which are at right angles two and two. They
intersect, two and two, in lines x'Ox, y'Oy, z'Oz, called the axes of
x, y, z respectively, and divide all space into eight parts called
octants. If from any point P in space we draw PN parallel to zOz' to
meet the plane xOy in N, and then from N draw NM parallel to yOy' to
meet x'Ox in M, the coordinates (x, y, z) of P are the numerical
measures of OM, MN, NP; in the case of rectangular coordinates these
are the perpendicular distances of P from the three planes of
reference. The sign of each coordinate is positive or negative as P
lies on one side or the other of the corresponding plane. In the
octant delineated the signs are taken all positive.

In fig. 57 the delineation is on a plane of the paper taken parallel
to the plane zOx, the points of a solid figure being projected on that
plane by parallels to some chosen line through O in the positive
octant. Sometimes it is clearer to delineate, as in fig. 58, by
projection parallel to that line in the octant which is equally
inclined to Ox, Oy, Oz upon a plane of the paper perpendicular to it.
It is possible by parallel projection to delineate equal scales along
Ox, Oy, Oz by scales having any ratios we like along lines in a plane
having any mutual inclinations we like.

For the delineation of a surface of simple form it frequently suffices
to delineate the sections by the coordinate planes; and, in
particular, when the surface has symmetry about each coordinate plane,
to delineate the quarter-sections belonging to a single octant. Thus
fig. 59 conveniently represents an octant of the wave surface, which
cuts each coordinate plane in a circle and an ellipse. Or we may
delineate a series of contour lines, i.e. sections by planes parallel
to xOy, or some other chosen plane; of course other sections may be
indicated too for greater clearness. For the delineation of a curve a
good method is to represent, as above, a series of points P thereof,
each accompanied by its ordinate PN, which serves to refer it to the
plane of xy. The employment of stereographic projection is also
interesting.

28. In plane geometry, reckoning the line as a curve of the first
order, we have only the point and the curve. In solid geometry,
reckoning a line as a curve of the first order, and the plane as a
surface of the first order, we have the point, the curve and the
surface; but the increase of complexity is far greater than would
hence at first sight appear. In plane geometry a curve is considered
in connexion with lines (its tangents); but in solid geometry the
curve is considered in connexion with lines and planes (its tangents
and osculating planes), and the surface also in connexion with lines
and planes (its tangent lines and tangent planes); there are surfaces
arising out of the line--cones, skew surfaces, developables, doubly
and triply infinite systems of lines, and whole classes of theories
which have nothing analogous to them in plane geometry: it is thus a
very small part indeed of the subject which can be even referred to in
the present article.

In the case of a surface we have between the coordinates (x, y, z) a
single, or say a onefold relation, which can be represented by a
single relation [f](x, y, z) = 0; or we may consider the coordinates
expressed each of them as a given function of two variable parameters
p, q; the form z = [f](x, y) is a particular case of each of these
modes of representation; in other words, we have in the first mode
[f](x, y, z) = z - [f](x, y), and in the second mode x = p, y = q for
the expression of two of the coordinates in terms of the parameters.

In the case of a curve we have between the coordinates (x, y, z) a
twofold relation: two equations [f](x, y, z) = 0, [phi](x, y, z) = 0
give such a relation; i.e. the curve is here considered as the
intersection of two surfaces (but the curve is not always the complete
intersection of two surfaces, and there are hence difficulties); or,
again, the coordinates may be given each of them as a function of a
single variable parameter. The form y = [phi](x), z = [psi](x), where
two of the coordinates are given in terms of the third, is a
particular case of each of these modes of representation.

29. The remarks under plane geometry as to descriptive and metrical
propositions, and as to the non-metrical character of the method of
coordinates when used for the proof of a descriptive proposition,
apply also to solid geometry; and they might be illustrated in like
manner by the instance of the theorem of the radical centre of four
spheres. The proof is obtained from the consideration that S and S'
being each of them a function of the form x^2 + y^2 + z^2 + ax + by +
cz + d, the difference S-S' is a mere linear function of the
coordinates, and consequently that S-S' = 0 is the equation of the
plane containing the circle of intersection of the two spheres S = 0
and S' = 0.

30. _Metrical Theory._--The foundation in solid geometry of the
metrical theory is in fact the before-mentioned theorem that if a
finite right line PQ be projected upon any other line OO' by lines
perpendicular to OO', then the length of the projection P'Q' is equal
to the length of PQ into the cosine of its inclination to P'Q'--or (in
the form in which it is now convenient to state the theorem) the
perpendicular distance P'Q' of two parallel planes is equal to the
inclined distance PQ into the cosine of the inclination. The principle
of S 16, that the algebraical sum of the projections of the sides of
any closed polygon on any line is zero, or that the two sets of sides
of the polygon which connect a vertex A and a vertex B have the same
sum of projections on the line, in sign and magnitude, as we pass from
A to B, is applicable when the sides do not all lie in one plane.

31. Consider the skew quadrilateral QMNP, the sides QM, MN, NP being
respectively parallel to the three rectangular axes Ox, Oy, Oz; let
the lengths of these sides be [xi], [eta], [zeta], and that of the
side QP be = [rho]; and let the cosines of the inclinations (or say
the cosine-inclinations) of [rho] to the three axes be [alpha],
[beta], [gamma]; then projecting successively on the three sides and
on QP we have

[xi], [eta], [zeta] = [rho][alpha], [rho][beta], [rho][gamma],

and

[rho] = [alpha][xi] + [beta][eta] + [gamma][zeta],

whence [rho]^2 = [xi]^2 + [eta]^2 + [zeta]^2, which is the relation
between a distance [rho] and its projections [xi], [eta], [zeta] upon
three rectangular axes. And from the same equations we obtain
[alpha]^2 + [beta]^2 + [gamma]^2 = 1, which is a relation connecting
the cosine-inclinations of a line to three rectangular axes.

Suppose we have through Q any other line QT, and let the
cosine-inclinations of this to the axes be [alpha]', [beta]',
[gamma]', and [delta] be its cosine-inclination to QP; also let [rho]
be the length of the projection of QP upon QT; then projecting on QT
we have

[rho] = [alpha]'[xi] + [beta]'[eta] + [gamma]'[zeta] = [rho][delta].

And in the last equation substituting for [xi], [eta], [zeta] their
values [rho][alpha], [rho][beta], [rho][gamma] we find

[delta] = [alpha][alpha]' + [beta][beta]' + [gamma][gamma]',

which is an expression for the mutual cosine-inclination of two lines,
the cosine-inclinations of which to the axes are [alpha], [beta],
[gamma] and [alpha]', [beta]', [gamma]' respectively. We have of
course [alpha]^2 + [beta]^2 + [gamma]^2 = 1 and [alpha]'^2 + [beta]'^2
+ [gamma]'^2 = 1; and hence also

1 - [delta]^2 = ([alpha]^2 + [beta]^2 + [gamma]^2)([alpha]'^2 + [beta]'^2 + [gamma]'^2)
- ([alpha][alpha]' + [beta][beta]' + [gamma][gamma]')^2,

= ([beta][gamma]' - [beta]'[gamma])^2 + ([gamma][alpha]' - [gamma]'[alpha])^2 +
([alpha][beta]' - [alpha]'[beta])^2;

so that the sine of the inclination can only be expressed as a square
root. These formulae are the foundation of spherical trigonometry.

32. _Straight Lines, Planes and Spheres._--The foregoing formulae give
at once the equations of these loci.

For first, taking Q to be a fixed point, coordinates (a, b, c), and
the cosine-inclinations ([alpha], [beta], [gamma]) to be constant,
then P will be a point in the line through Q in the direction thus
determined; or, taking (x, y, z) for its coordinates, these will be
the current coordinates of a point in the line. The values of [xi],
[eta], [zeta] then are x - a, y - b, z - c, and we thus have

x - a y - b z - c
------- = ----- = ------- (= [rho]),
[alpha] [beta] [gamma]

which (omitting the last equation, = [rho]) are the equations of the
line through the point (a, b, c), the cosine-inclinations to the axes
being [alpha], [beta], [gamma], and these quantities being connected
by the relation [alpha]^2 + [beta]^2 + [gamma]^2 = 1. This equation
may be omitted, and then [alpha], [beta], [gamma], instead of being
equal, will only be proportional, to the cosine-inclinations.

Using the last equation, and writing

x, y, z = a + [alpha][rho], b + [beta][rho], c + [gamma][rho],

these are expressions for the current coordinates in terms of a
parameter [rho], which is in fact the distance from the fixed point
(a, b, c).

It is easy to see that, if the coordinates (x, y, z) are connected by
any two linear equations, these equations can always be brought into
the foregoing form, and hence that the two linear equations represent
a line.

Secondly, taking for greater simplicity the point Q to be coincident
with the origin, and [alpha]', [beta]', [gamma]', p to be constant,
then p is the perpendicular distance of a plane from the origin, and
[alpha]', [beta]', [gamma]' are the cosine-inclinations of this
distance to the axes ([alpha]'^2 + [beta]'^2 + [gamma]'^2 = 1). P is
any point in this plane, and taking its coordinates to be (x, y, z)
then ([xi], [eta], [zeta]) are = (x, y, z), and the foregoing equation
p = [alpha]'[xi] + [beta]'[eta] + [gamma]'[zeta] becomes

[alpha]'x + [beta]'y + [gamma]'z = p,

which is the equation of the plane in question.

If, more generally, Q is not coincident with the origin, then, taking
its coordinates to be (a, b, c), and writing p1 instead of p, the
equation is

[alpha]'(x - a) + [beta]'(y - b) + [gamma]'(z - c) = p1;

and we thence have p1 = p - (a[alpha]' + b[beta]' + c[gamma]'), which
is an expression for the perpendicular distance of the point (a, b, c)
from the plane in question.

It is obvious that any linear equation Ax + By + Cz + D = O between
the coordinates can always be brought into the foregoing form, and
hence that such an equation represents a plane.

Thirdly, supposing Q to be a fixed point, coordinates (a, b, c), and
the distance QP = [rho], to be constant, say this is = d, then, as
before, the values of [xi], [eta], [zeta] are x - a, y - b, z - c, and
the equation [xi]^2 + [eta]^2 + [zeta]^2 = [rho]^2 becomes

(x - a)^2 + (y - b)^2 + (z - c)^2 = d^2,

which is the equation of the sphere, coordinates of the centre = (a,
b, c), and radius = d.

A quadric equation wherein the terms of the second order are x^2 + y^2
+ z^2, viz. an equation

x^2 + y^2 + z^2 + Ax + By + Cz + D = 0,

can always, it is clear, be brought into the foregoing form; and it
thus appears that this is the equation of a sphere, coordinates of the
centre -1/2A, -1/2B, -1/2C, and squared radius = 1/4(A^2 + B^2 + C^2)
- D.

33. _Cylinders, Cones, ruled Surfaces._--If the two equations of a
straight line involve a parameter to which any value may be given, we
have a singly infinite system of lines. They cover a surface, and the
equation of the surface is obtained by eliminating the parameter
between the two equations.

If the lines all pass through a given point, then the surface is a
cone; and, in particular, if the lines are all parallel to a given
line, then the surface is a cylinder.

Beginning with this last case, suppose the lines are parallel to the
line x = mz, y = nz, the equations of a line of the system are x = mz
+ a, y = nz + b,--where a, b are supposed to be functions of the
variable parameter, or, what is the same thing, there is between them
a relation f(a, b) = 0: we have a = x - mz, b = y - nz, and the result
of the elimination of the parameter therefore is [f](x - mz, y - nz) =
0, which is thus the general equation of the cylinder the generating
lines whereof are parallel to the line x = mz, y = nz. The equation of
the section by the plane z = 0 is [f](x, y) = 0, and conversely if the
cylinder be determined by means of its curve of intersection with the
plane z = 0, then, taking the equation of this curve to be f(x, y) =
0, the equation of the cylinder is [f](x - mz, y - nz) = 0. Thus, if
the curve of intersection be the circle (x - [alpha])^2 + (y -
[beta])^2 = [gamma]^2, we have (x - mz - [alpha])^2 + (y - nz -
[beta])^2 = [gamma]^2 as the equation of an oblique cylinder on this
base, and thus also (x - [alpha])^2 + (y - [beta])^2 = [gamma]^2 as
the equation of the right cylinder.

If the lines all pass through a given point (a, b, c), then the
equations of a line are x - a = [alpha](z - c), y - b = [beta](z - c),
where [alpha], [beta] are functions of the variable parameter, or,
what is the same thing, there exists between them an equation
f([alpha], [beta]) = 0; the elimination of the parameter gives,
therefore, f[(x - a)/(x - c'), (y - b)/(z - c)] = 0; and this
equation, or, what is the same thing, any homogeneous equation f(x -
a, y - b, z - c) = 0, or, taking f to be a rational and integral
function of the order n, say (*)(x - a, y - b, z - c)^n = 0, is the
general equation of the cone having the point (a, b, c) for its
vertex. Taking the vertex to be at the origin, the equation is (*)(x,
y, z)^n = 0; and, in particular, (*)(x, y, z)^2 = 0 is the equation of
a cone of the second order, or quadricone, having the origin for its
vertex.

34. In the general case of a singly infinite system of lines, the
locus is a ruled surface (or _regulus_). Now, when a line is changing
its position in space, it may be looked upon as in a state of turning
about some point in itself, while that point is, as a rule, in a state
of moving out of the plane in which the turning takes place. If
instantaneously it is only in a state of turning, it is usual, though
not strictly accurate, to say that it intersects its consecutive
position. A regulus such that consecutive lines on it do not
intersect, in this sense, is called a skew surface, or _scroll_; one
on which they do is called a developable surface or _torse_.

Suppose, for instance, that the equations of a line (depending on the
variable parameter [theta]) are x/a + y/c = [theta] (1 + y/b), x/a -
z/c = 1/[theta] (1 - y/b); then, eliminating [theta] we have x^2/a^2 -
z^2/c^2 = 1 - y^2/b^2, or say, x^2/a^2 + z^2/b^2 - z^2/c^2 = 1, the
equation of a quadric surface, afterwards called the hyperboloid of
one sheet; this surface is consequently a scroll. It is to be remarked
that we have upon the surface a second singly infinite series of
lines; the equations of a line of this second system (depending on the
variable parameter [phi]) are

x z / y \ x z 1 / y \
-- + -- = [phi]( 1 - -- ), -- - -- = ----- ( 1 + -- ).
a c \ b / a c [phi] \ b /

It is easily shown that any line of the one system intersects every
line of the other system.

Considering any curve (of double curvature) whatever, the tangent
lines of the curve form a singly infinite system of lines, each line
intersecting the consecutive line of the system,--that is, they form a
developable, or torse; the curve and torse are thus inseparably
connected together, forming a single geometrical figure. An osculating
plane of the curve (see S 38 below) is a tangent plane of the torse
all along a generating line.

35. _Transformation of Coordinates._--There is no difficulty in
changing the origin, and it is for brevity assumed that the origin
remains unaltered. We have, then, two sets of rectangular axes, Ox,
Oy, Oz, and Ox1, Oy1, Ozx1, the mutual cosine-inclinations being shown
by the diagram--

| x | y | z |
----+---------+--------+---------+
x1 | [alpha] | [beta] | [gamma] |
----+---------+--------+---------+
y1 | [alpha] | [beta]'| [gamma]'|
----+---------+--------+---------+
z1 | [alpha]"| [beta]"| [gamma]"|
----+---------+--------+---------+

that is, [alpha], [beta], [gamma] are the cosine-inclinations of Ox1
to Ox, Oy, Oz; [alpha]', [beta]', [gamma]' those of Oy1, &c.

And this diagram gives also the linear expressions of the coordinates
(x1, y1, z1) or (x, y, z) of either set in terms of those of the other
set; we thus have

x1 = [alpha] x + [beta] y + [gamma] z,
x = [alpha]x1 + [alpha]'y1 + [alpha]"z1,

y1 = [alpha]'x + [beta]'y + [gamma]'z,
y = [beta]x1 + [beta]'y1 + [beta]"z1,

z1 = [alpha]"x + [beta]"y + [gamma]"z,
z = [gamma]x1 + [gamma]'y1 + [gamma]"z1,

which are obtained by projection, as above explained. Each of these
equations is, in fact, nothing else than the before-mentioned equation
p = [alpha]'[xi] + [beta]'[eta] + [gamma]'[zeta], adapted to the
problem in hand.

But we have to consider the relations between the nine coefficients.
By what precedes, or by the consideration that we must have
identically x^2 + y^2 + z^2 = x1^2 + y1^2 + z1^2, it appears that
these satisfy the relations--

a^2 + [beta]^2 + [gamma]^2 = 1,
[alpha]^2 + [alpha]'^2 + [alpha]"^2 = 1,

[alpha]'^2 + [beta]'^2 + [gamma]'^2 = 1,
[beta]^2 + [beta]'^2 + [beta]"^2 = 1,

[alpha]"^2 + [beta]"^2 + [gamma]"^2 = 1,
[gamma]^2 + [gamma]'^2 + [gamma]"^2 = 1,

a'a" + [beta]'[beta]" + [gamma]'[gamma]" = 0,
[beta][gamma] +[beta]'[gamma]' + [beta]"[gamma]" = 0,

[alpha]"[alpha] + [beta]"[beta] + [gamma]"[gamma] = 0,
[gamma][alpha] + [gamma]'[alpha]' + [gamma]"[alpha]" = 0,

[alpha][alpha]' + [beta][beta]' + [gamma][gamma]' = 0,
[alpha][beta] +[alpha]'[beta]' + [alpha]"[beta]" = 0,

either set of six equations being implied in the other set.

It follows that the square of the determinant

|[alpha], [beta], [gamma] |
| |
|[alpha]', [beta]', [gamma]'|
| |
|[alpha]", [beta]", [gamma]"|

is = 1; and hence that the determinant itself is = [+-] 1. The
distinction of the two cases is an important one: if the determinant
is = + 1, then the axes Ox1, Oy1, Oz1 are such that they can by a
rotation about O be brought to coincide with Ox, Oy, Oz respectively;
if it is = -1, then they cannot. But in the latter case, by measuring
x1, y1, z1 in the opposite directions we change the signs of all the
coefficients and so make the determinant to be = + 1; hence the former
case need alone be considered, and it is accordingly assumed that the
determinant is = + 1. This being so, it is found that we have the
equality [alpha] = [beta]'[gamma]" - [beta]"[gamma]', and eight like
ones, obtained from this by cyclical interchanges of the letters
[alpha], [beta], [gamma], and of unaccented, singly and doubly
accented letters.

36. The nine cosine-inclinations above are, as has been seen,
connected by six equations. It ought then to be possible to express
them all in terms of three parameters. An elegant means of doing this
has been given by Rodrigues, who has shown that the tabular expression
of the formulae of transformation may be written

| x | y | z |
----+--------------------------------+--------------------------------+--------------------------------+
x1 |1 + [lambda]^2 - [mu]^2 - [nu]^2| 2([lambda][mu] - [nu]) | 2([nu][lambda] + [mu]) |
----+--------------------------------+--------------------------+-----+--------------------------------+
y1 | 2([lambda][mu] + [nu]) |1 - [lambda]^2 + [mu]^2 - [nu]^2| 2([mu][nu] + [lambda]) |
----+--------------------------------+--------------------------+-----+--------------------------------+
z1 | 2([nu][lambda] - [mu]) | 2([mu][nu] + [lambda]) |1 - [lambda]^2 - [mu]^2 + [nu]^2|
----+--------------------------------+--------------------------------+--------------------------------+
/(1 + [lambda]^2 + [mu]^2 + [nu]^2),

the meaning being that the coefficients in the transformation are
fractions, with numerators expressed as in the table, and the common
denominator.

37. _The Species of Quadric Surfaces_.--Surfaces represented by
equations of the second degree are called _quadric_ surfaces. Quadric
surfaces are either _proper_ or _special_. The special ones arise when
the coefficients in the general equation are limited to satisfy
certain special equations; they comprise (1) plane-pairs, including in
particular one plane twice repeated, and (2) cones, including in
particular cylinders; there is but one form of cone, but cylinders may
be elliptic, parabolic or hyperbolic.

A discussion of the general equation of the second degree shows that
the _proper_ quadric surfaces are of five kinds, represented
respectively, when referred to the most convenient axes of reference,
by equations of the five types (a and b positive):

x^2 y^2
(1) z = --- + ---, elliptic paraboloid.
2a 2b

x^2 y^2
(2) z = --- - ---, hyperbolic paraboloid.
2a 2b

x^2 y^2 z^2
(3) --- + --- + --- = 1, ellipsoid.
a^2 b^2 c^2

x^2 y^2 z^2
(4) --- + --- - --- = 1, hyperboloid of one sheet.
a^2 b^2 c^2

x^2 y^2 z^2
(5) --- + --- - --- = -1, hyperboloid of two sheets.
a^2 b^2 c^2

It is at once seen that these are distinct surfaces; and the equations
also show very readily the general form and mode of generation of the
several surfaces.

In the elliptic paraboloid (fig. 61) the sections by the planes of zx
and zy are the parabolas

x^2 y^2
z = ---, z = ---
2a 2b

having the common axes Oz; and the section by any plane z = [gamma]
parallel to that of xy is the ellipse

x^2 y^2
[gamma] = --- + ---;
2a 2b

so that the surface is generated by a variable ellipse moving parallel
to itself along the parabolas as directrices.

In the hyperbolic paraboloid (figs. 62 and 63) the sections by the
planes of zx, zy are the parabolas z = x^2/2a, z = - y^2/2b, having
the opposite axes Oz, Oz', and the section by a plane z = [gamma]
parallel to that of xy is the hyperbola [gamma] = x^2/2a - y^2/2b,
which has its transverse axis parallel to Ox or Oy according as
[gamma] is positive or negative. The surface is thus generated by a
variable hyperbola moving parallel to itself along the parabolas as
directrices. The form is best seen from fig. 63, which represents the
sections by planes parallel to the plane of xy, or say the contour
lines; the continuous lines are the sections above the plane of xy,
and the dotted lines the sections below this plane. The form is, in
fact, that of a saddle.

In the ellipsoid (fig. 64) the sections by the planes of zx, zy, and
xy are each of them an ellipse, and the section by any parallel plane
is also an ellipse. The surface may be considered as generated by an
ellipse moving parallel to itself along two ellipses as directrices.

In the hyperboloid of one sheet (fig. 65), the sections by the planes
of zx, zy are the hyperbolas

x^2 z^2 y^2 z^2
--- - --- = 1, --- - --- = 1,
c^2 c^2 b^2 c^2

having a common conjugate axis zOz'; the section by the plane of x, y,
and that by any parallel plane, is an ellipse; and the surface may be
considered as generated by a variable ellipse moving parallel to
itself along the two hyperbolas as directrices. If we imagine two
equal and parallel circular disks, their points connected by strings
of equal lengths, so that these are the generators of a right circular
cylinder, and if we turn one of the disks about its centre through an
angle in its plane, the strings in their new positions will be one
system of generators of a hyperboloid of one sheet, for which a = b;
and if we turn it through the same angle in the opposite direction, we
get in like manner the generators of the other system; there will be
the same general configuration when a = | b. The hyperbolic paraboloid
is also covered by two systems of rectilinear generators as a method
like that used in S 34 establishes without difficulty. The figures
should be studied to see how they can lie.

In the hyperboloid of two sheets (fig. 66) the sections by the planes
of zx and zy are the hyperbolas