S 89. The 13th and last book of Euclid's _Elements_ is devoted to the
regular solids (see POLYHEDRON). It is shown that there are five of
them, viz.:--
1. The regular _tetrahedron_, with 4 triangular faces and 4 vertices;
2. The _cube_, with 8 vertices and 6 square faces;
3. The _octahedron_, with 6 vertices and 8 triangular faces;
4. The _dodecahedron_, with 12 pentagonal faces, 3 at each of the
20 vertices;
5. The _icosahedron_, with 20 triangular faces, 5 at each of the
12 vertices.
It is shown how to inscribe these solids in a given sphere, and how to
determine the lengths of their edges.
S 90. The 13th book, and therefore the _Elements_, conclude with the
scholium, "that no other regular solid exists besides the five ones
enumerated."
The proof is very simple. Each face is a regular polygon, hence the
angles of the faces at any vertex must be angles in equal regular
polygons, must be together less than four right angles (XI. 21), and
must be three or more in number. Each angle in a regular triangle
equals two-thirds of one right angle. Hence it is possible to form a
solid angle with three, four or five regular triangles or faces. These
give the solid angles of the tetrahedron, the octahedron and the
icosahedron. The angle in a square (the regular quadrilateral) equals
one right angle. Hence three will form a solid angle, that of the
cube, and four will not. The angle in the regular pentagon equals 6/5
of a right angle. Hence three of them equal 18/5 (i.e. less than 4)
right angles, and form the solid angle of the dodecahedron. Three
regular polygons of six or more sides cannot form a solid angle.
Therefore no other regular solids are possible. (O. H.)
II. PROJECTIVE GEOMETRY
It is difficult, at the outset, to characterize projective geometry as
compared with Euclidean. But a few examples will at least indicate the
practical differences between the two.
In Euclid's _Elements_ almost all propositions refer to the _magnitude_
of lines, angles, areas or volumes, and therefore to measurement. The
statement that an angle is right, or that two straight lines are
parallel, refers to measurement. On the other hand, the fact that a
straight line does or does not cut a circle is independent of
measurement, it being dependent only upon the mutual "position" of the
line and the circle. This difference becomes clearer if we project any
figure from one plane to another (see PROJECTION). By this the length of
lines, the magnitude of angles and areas, is altered, so that the
projection, or shadow, of a square on a plane will not be a square; it
will, however, be some quadrilateral. Again, the projection of a circle
will not be a circle, but some other curve more or less resembling a
circle. But one property may be stated at once--no straight line can cut
the projection of a circle in more than two points, because no straight
line can cut a circle in more than two points. There are, then, some
properties of figures which do not alter by projection, whilst others
do. To the latter belong nearly all properties relating to measurement,
at least in the form in which they are generally given. The others are
said to be projective properties, and their investigation forms the
subject of projective geometry.
Different as are the kinds of properties investigated in the old and the
new sciences, the methods followed differ in a still greater degree. In
Euclid each proposition stands by itself; its connexion with others is
never indicated; the leading ideas contained in its proof are not
stated; general principles do not exist. In the modern methods, on the
other hand, the greatest importance is attached to the leading thoughts
which pervade the whole; and general principles, which bring whole
groups of theorems under one aspect, are given rather than separate
propositions. The whole tendency is towards generalization. A straight
line is considered as given in its entirety, extending both ways to
infinity, while Euclid never admits anything but finite quantities. The
treatment of the infinite is in fact another fundamental difference
between the two methods: Euclid avoids it; in modern geometry it is
systematically introduced.
Of the different modern methods of geometry, we shall treat principally
of the methods of projection and correspondence which have proved to be
the most powerful. These have become independent of Euclidean Geometry,
especially through the _Geometrie der Lage_ of V. Staudt and the
_Ausdehnungslehre_ of Grassmann.
For the sake of brevity we shall presuppose a knowledge of Euclid's
_Elements_, although we shall use only a few of his propositions.
S 1. _Geometrical Elements._ We consider space as filled with points,
lines and planes, and these we call the elements out of which our
figures are to be formed, calling any combination of these elements a
"figure."
By a line we mean a straight line in its entirety, extending both ways
to infinity; and by a plane, a plane surface, extending in all
directions to infinity.
We accept the three-dimensional space of experience--the space assumed
by Euclid--which has for its properties (among others):--
Through any two points in space one and only one line may be drawn;
Through any three points which are not in a line, one and only one
plane may be placed;
The intersection of two planes is a line;
A line which has two points in common with a plane lies in the plane,
hence the intersection of a line and a plane is a single point; and
Three planes which do not meet in a line have one single point in
common.
These results may be stated differently in the following form:--
I. A plane is determined-- A point is determined--
1. By three points which do 1. By three planes which do
not lie in a line; not pass through a line;
2. By two intersecting lines; 2. By two intersecting lines;
3. By a line and a point 3. By a plane and a line
which does not lie in it. which does not lie in it.
II. A line is determined--
1. By two points; 2. By two planes.
It will be observed that not only are planes determined by points, but
also points by planes; that therefore the planes may be considered as
elements, like points; and also that in any one of the above
statements we may interchange the words point and plane, and we obtain
again a correct statement, provided that these statements themselves
are true. As they stand, we ought, in several cases, to add "if they
are not parallel," or some such words, parallel lines and planes being
evidently left altogether out of consideration. To correct this we
have to reconsider the theory of parallels.
[Illustration: FIG. 1.]
S 2. _Parallels. Point at Infinity._--Let us take in a plane a line p
(fig. 1), a point S not in this line, and a line q drawn through S.
Then this line q will meet the line p in a point A. If we turn the
line q about S towards q', its point of intersection with p will move
along p towards B, passing, on continued turning, to a greater and
greater distance, until it is moved out of our reach. If we turn q
still farther, its continuation will meet p, but now at the other side
of A. The point of intersection has disappeared to the right and
reappeared to the left. There is one intermediate position where q is
parallel to p--that is where it does not cut p. In every other
position it cuts p in some finite point. If, on the other hand, we
move the point A to an infinite distance in p, then the line q which
passes through A will be a line which does not cut p at any finite
point. Thus we are led to say: _Every_ line through S which joins it
to any point at an infinite distance in p is parallel to p. But by
Euclid's 12th axiom there is but one line parallel to p through S. The
difficulty in which we are thus involved is due to the fact that we
try to reason about infinity as if we, with our finite capabilities,
could comprehend the infinite. To overcome this difficulty, we may say
that all points at infinity in a line _appear_ to us as one, and may
be replaced by a single "ideal" point.
We may therefore now give the following definitions and axiom:--
_Definition._--Lines which meet at infinity are called parallel.
_Axiom._--All points at an infinite distance in a line may be
considered as one single point.
_Definition._--This ideal point is called the _point at infinity_ in
the line.
The axiom is equivalent to Euclid's Axiom 12, for it follows from
either that through any point only one line may be drawn parallel to a
given line.
This point at infinity in a line is reached whether we move a point in
the one or in the opposite direction of a line to infinity. A line
thus appears closed by this point, and we speak as if we could move a
point along the line from one position A to another B in two ways,
either through the point at infinity or through finite points only.
It must never be forgotten that this point at infinity is ideal; in
fact, the whole notion of "infinity" is only a mathematical
conception, and owes its introduction (as a method of research) to the
working generalizations which it permits.
S 3. _Line and Plane at Infinity._--Having arrived at the notion of
replacing all points at infinity in a line by one ideal point, there
is no difficulty in replacing all points at infinity in a plane by one
ideal line.
To make this clear, let us suppose that a line p, which cuts two fixed
lines a and b in the points A and B, moves parallel to itself to a
greater and greater distance. It will at last cut both a and b at
their points at infinity, so that a line which joins the two points at
infinity in two intersecting lines lies altogether at infinity. Every
other line in the plane will meet it therefore at infinity, and thus
it contains all points at infinity in the plane.
_All points at infinity in a plane lie in a line, which is called the_
line at infinity _in the plane._
It follows that parallel planes must be considered as planes having a
common line at infinity, for any other plane cuts them in parallel
lines which have a point at infinity in common.
If we next take two intersecting planes, then the point at infinity in
their line of intersection lies in both planes, so that their lines at
infinity meet. Hence every line at infinity meets every other line at
infinity, and they are therefore all in one plane.
_All points at infinity in space may be considered as lying in one
ideal plane, which is called the_ plane at infinity.
S 4. _Parallelism._--We have now the following definitions:--
Parallel lines are lines which meet at infinity;
Parallel planes are planes which meet at infinity;
A line is parallel to a plane if it meets it at infinity.
Theorems like this--Lines (or planes) which are parallel to a third
are parallel to each other--follow at once.
This view of parallels leads therefore to no contradiction of Euclid's
_Elements._
As immediate consequences we get the propositions:--
Every line meets a plane in one point, or it lies in it;
Every plane meets every other plane in a line;
Any two lines in the same plane meet.
S 5. _Aggregates of Geometrical Elements._--We have called points,
lines and planes the elements of geometrical figures. We also say that
an element of one kind contains one of the other if it lies in it or
passes through it.
All the elements of one kind which are contained in one or two
elements of a different kind form aggregates which have to be
enumerated. They are the following:--
I. Of one dimension.
1. The _row_, or range, _of points_ formed by all points in a line,
which is called its base.
2. The _flat pencil_ formed by all the lines through a point in a
plane. Its base is the point in the plane.
3. The _axial pencil_ formed by all planes through a line which is
called its base or axis.
II. Of two dimensions.
1. The field of points and lines--that is, a plane with all its
points and all its lines.
2. The pencil of lines and planes--that is, a point in space with
all lines and all planes through it.
III. Of three dimensions.
The space of points--that is, all points in space.
The space of planes--that is, all planes in space.
IV. Of four dimensions.
The space of lines, or all lines in space.
S 6. _Meaning of "Dimensions."_--The word dimension in the above needs
explanation. If in a plane we take a row p and a pencil with centre Q,
then through every point in p one line in the pencil will pass, and
every ray in Q will cut p in one point, so that we are entitled to say
a row contains as many points as a flat pencil lines, and, we may add,
as an axial pencil planes, because an axial pencil is cut by a plane
in a flat pencil.
The number of elements in the row, in the flat pencil, and in the
axial pencil is, of course, infinite and indefinite too, but the same
in all. This number may be denoted by [infinity]. Then a plane
contains [infinity]^2 points and as many lines. To see this, take a
flat pencil in a plane. It contains [infinity] lines, and each line
contains [infinity] points, whilst each point in the plane lies on one
of these lines. Similarly, in a plane each line cuts a fixed line in a
point. But this line is cut at each point by [infinity] lines and
contains [infinity] points; hence there are [infinity]^2 lines in a
plane.
A pencil in space contains as many lines as a plane contains points
and as many planes as a plane contains lines, for any plane cuts the
pencil in a field of points and lines. Hence a pencil contains
[infinity]^2 lines and [infinity]^2 planes. _The field and the pencil
are of two dimensions._
To count the number of points in space we observe that each point lies
on some line in a pencil. But the pencil contains [infinity]^2 lines,
and each line [infinity] points; hence space contains [infinity]^3
points. Each plane cuts any fixed plane in a line. But a plane
contains [infinity]^2 lines, and through each pass [infinity] planes;
therefore space contains [infinity]^3 planes.
Hence space contains as many planes as points, but it contains an
infinite number of times more lines than points or planes. To count
them, notice that every line cuts a fixed plane in one point. But
[infinity]^2 lines pass through each point, and there are [infinity]^2
points in the plane. Hence there are [infinity]^4 lines in space. _The
space of points and planes is of three dimensions, but the space of
lines is of four dimensions._
A field of points or lines contains an infinite number of rows and
flat pencils; a pencil contains an infinite number of flat pencils and
of axial pencils; space contains a triple infinite number of pencils
and of fields, [infinity]^4 rows and axial pencils and [infinity]^5
flat pencils--or, in other words, each point is a centre of
[infinity]^2 flat pencils.
S 7. The above enumeration allows a classification of figures. Figures
in a row consist of groups of points only, and figures in the flat or
axial pencil consist of groups of lines or planes. In the plane we may
draw polygons; and in the pencil or in the point, solid angles, and so
on.
We may also distinguish the different measurements We have--
In the row, length of segment;
In the flat pencil, angles;
In the axial pencil, dihedral angles between two planes;
In the plane, areas;
In the pencil, solid angles;
In the space of points or planes, volumes.
SEGMENTS OF A LINE
S 8. Any two points A and B in space determine on the line through
them a finite part, which may be considered as being described by a
point moving from A to B. This we shall denote by AB, and distinguish
it from BA, which is supposed as being described by a point moving
from B to A, and hence in a direction or in a "sense" opposite to AB.
Such a finite line, which has a definite sense, we shall call a
"segment," so that AB and BA denote different segments, which are said
to be equal in length but of opposite sense. The one sense is often
called positive and the other negative.
In introducing the word "sense" for direction in a line, we have the
word direction reserved for direction of the line itself, so that
different lines have different directions, unless they be parallel,
whilst in each line we have a positive and negative sense.
We may also say, with Clifford, that AB denotes the "step" of going
from A to B.
[Illustration: FIG. 2.]
S 9. If we have three points A, B, C in a line (fig. 2), the step AB
will bring us from A to B, and the step BC from B to C. Hence both
steps are equivalent to the one step AC. This is expressed by saying
that AC is the "sum" of AB and BC; in symbols--
AB + BC = AC,
where account is to be taken of the sense.
This equation is true whatever be the position of the three points on
the line. As a special case we have
AB + BA = 0, (1)
and similarly
AB + BC + CA = 0, (2)
which again is true for any three points in a line.
We further write
AB = -BA.
where - denotes negative sense.
We can then, just as in algebra, change subtraction of segments into
addition by changing the sense, so that AB - CB is the same as AB +
(-CB) or AB + BC. A figure will at once show the truth of this. The
sense is, in fact, in every respect equivalent to the "sign" of a
number in algebra.
S 10. Of the many formulae which exist between points in a line we
shall have to use only one more, which connects the segments between
any four points A, B, C, D in a line. We have
BC = BD + DC, CA = CD + DA, AB = AD + DB;
or multiplying these by AD, BD, CD respectively, we get
BC.AD = BD.AD + DC.AD = BD.AD - CD.AD
CA.BD = CD.BD + DA.BD = CD.BD - AD.BD
AB.CD = AD.CD + DB.CD = AD.CD - BD.CD.
It will be seen that the sum of the right-hand sides vanishes, hence
that
BC.AD + CA.BD + AB.CD = 0 (3)
for any four points on a line.
[Illustration: FIG. 3.]
S 11. If C is any point in the line AB, then we say that C divides the
segment AB in the ratio AC/CB, account being taken of the sense of the
two segments AC and CB. If C lies between A and B the ratio is
positive, as AC and CB have the same sense. But if C lies without the
segment AB, i.e. if C divides AB externally, then the ratio is
negative. To see how the value of this ratio changes with C, we will
move C along the whole line (fig. 3), whilst A and B remain fixed. If
C lies at the point A, then AC = 0, hence the ratio AC : CB vanishes.
As C moves towards B, AC increases and CB decreases, so that our ratio
increases. At the middle point M of AB it assumes the value +1, and
then increases till it reaches an infinitely large value, when C
arrives at B. On passing beyond B the ratio becomes negative. If C is
at P we have AC = AP = AB + BP, hence
AC AB BP AB
-- = -- + -- = - -- - 1.
CB PB PB BP
In the last expression the ratio AB : BP is positive, has its greatest
value [infinity] when C coincides with B, and vanishes when BC becomes
infinite. Hence, as C moves from B to the right to the point at
infinity, the ratio AC : CB varies from -[infinity] to -1.
If, on the other hand, C is to the left of A, say at Q, we have AC =
AQ = AB + BQ = AB - QB, hence AC/CB = AB/QB - 1.
Here AB < QB, hence the ratio AB : QB is positive and always less than
one, so that the whole is negative and < 1. If C is at the point at
infinity it is -1, and then increases as C moves to the right, till
for C at A we get the ratio = 0. Hence--
"As C moves along the line from an infinite distance to the left to an
infinite distance at the right, the ratio always increases; it starts
with the value -1, reaches 0 at A, +1 at M, [infinity] at B, now
changes sign to -[infinity], and increases till at an infinite
distance it reaches again the value -1. _It assumes therefore all
possible values from -[infinity] to +[infinity], and each value only
once, so that not only does every position of C determine a definite
value of the ratio AC : CB, but also, conversely, to every positive or
negative value of this ratio belongs one single point in the line AB._
[Relations between segments of lines are interesting as showing an
application of algebra to geometry. The genesis of such relations
from algebraic identities is very simple. For example, if a, b, c, x
be any four quantities, then
a b
--------------------- + --------------------- +
(a - b)(a - c)(x - a) (b - c)(b - a)(x - b)
c x
--------------------- = ---------------------;
(c - a)(c - b)(x - c) (x - a)(x - b)(x - c)
this may be proved, cumbrously, by multiplying up, or, simply, by
decomposing the right-hand member of the identity into partial
fractions. Now take a line ABCDX, and let AB = a, AC = b, AD = c, AX =
x. Then obviously (a - b) = AB - AC = -BC, paying regard to signs; (a
- c) = AB - AD = DB, and so on. Substituting these values in the
identity we obtain the following relation connecting the segments
formed by five points on a line:--
AB AC AD AX
-------- + -------- + -------- = --------.
BC.BD.BX CD.CB.CX DB.DC.DX BX.CX.DX
Conversely, if a metrical relation be given, its validity may be
tested by reducing to an algebraic equation, which is an identity if
the relation be true. For example, if ABCDX be five collinear points,
prove
AD.AX BD.BX CD.CX
----- + ----- + ----- = 1.
AB.AC BC.BA CA.CB
Clearing of fractions by multiplying throughout by AB.BC.CA, we have
to prove
-AD.AX.BC - BD.BX.CA - CD.CX.AB = AB.BC.CA.
Take A as origin and let AB = a, AC = b, AD = c, AX = x. Substituting
for the segments in terms of a, b, c, x, we obtain on simplification
a^2b - ab^2 = -ab^2 + a^2b, an obvious identity.
An alternative method of testing a relation is illustrated in the
following example:-- If A, B, C, D, E, F be six collinear points, then
AE.AF BE.BF CE.CF DE.DF
-------- + -------- + -------- + -------- = 0.
AB.AC.AD BC.BD.BA CD.CA.CB DA.DB.DC
Clearing of fractions by multiplying throughout by AB.BC.CD.DA, and
reducing to a common origin O (calling OA = a, OB = b, &c.), an
equation containing the second and lower powers of OA (= a), &c., is
obtained. Calling OA = x, it is found that x = b, x = c, x = d are
solutions. Hence the quadratic has three roots; consequently it is an
identity.
The relations connecting five points which we have instanced above may
be readily deduced from the six-point relation; the first by taking D
at infinity, and the second by taking F at infinity, and then making
the obvious permutations of the points.]
PROJECTION AND CROSS-RATIOS
S 12. If we join a point A to a point S, then the point where the line
SA cuts a fixed plane [pi] is called the projection of A on the plane
[pi] from S as centre of projection. If we have two planes [pi] and
[pi]' and a point S, we may project every point A in [pi] to the other
plane. If A' is the projection of A, then A is also the projection of
A', so that the relations are reciprocal. To every figure in [pi] we
get as its projection a corresponding figure in [pi]'.
We shall determine such properties of figures as remain true for the
projection, and which are called projective properties. For this
purpose it will be sufficient to consider at first only constructions
in one plane.
[Illustration: FIG. 4.]
[Illustration: FIG. 5.]
Let us suppose we have given in a plane two lines p and p' and a
centre S (fig. 4); we may then project the points in p from S to p'.
Let A', B' ... be the projections of A, B ..., the point at infinity
in p which we shall denote by I will be projected into a finite point
I' in p', viz. into the point where the parallel to p through S cuts
p'. Similarly one point J in p will be projected into the point J' at
infinity in p'. This point J is of course the point where the parallel
to p' through S cuts p. We thus see that every point in p is projected
into a single point in p'.
Fig. 5 shows that a segment AB will be projected into a segment A'B'
which is not equal to it, at least not as a rule; and also that the
ratio AC : CB is not equal to the ratio A'C' : C'B' formed by the
projections. These ratios will become equal only if p and p' are
parallel, for in this case the triangle SAB is similar to the triangle
SA'B'. Between three points in a line and their projections there
exists therefore in general no relation. But between four points a
relation does exist.
S 13. Let A, B, C, D be four points in p, A', B', C, D' their
projections in p', then the ratio of the two ratios AC : CB and AD :
DB into which C and D divide the segment AB is equal to the
corresponding expression between A', B', C', D'. In symbols we have
AC AD A'C' A'D'
-- : -- = ---- : ----.
CB DB C'B' D'B'
This is easily proved by aid of similar triangles.
[Illustration: FIG. 6.]
Through the points A and B on p draw parallels to p', which cut the
projecting rays in C2, D2, B2 and A1, C1, D1, as indicated in fig. 6.
The two triangles ACC2 and BCC1 will be similar, as will also be the
triangles ADD2 and BDD1.
The proof is left to the reader.
This result is of fundamental importance.
The expression AC/CB : AD/DB has been called by Chasles the
"anharmonic ratio of the four points A, B, C, D." Professor Clifford
proposed the shorter name of "cross-ratio." We shall adopt the latter.
We have then the
FUNDAMENTAL THEOREM.--_The cross-ratio of four points in a line is
equal to the cross-ratio of their projections on any other line which
lies in the same plane with it._
S 14. Before we draw conclusions from this result, we must investigate
the meaning of a cross-ratio somewhat more fully.
If four points A, B, C, D are given, and we wish to form their
cross-ratio, we have first to divide them into two groups of two, the
points in each group being taken in a definite order. Thus, let A, B
be the first, C, D the second pair, A and C being the first points in
each pair. The cross-ratio is then the ratio AC : CB divided by AD :
DB. This will be denoted by (AB, CD), so that
AC AD
(AB, CD) = -- : --.
CB DB
This is easily remembered. In order to write it out, make first the
two lines for the fractions, and put above and below these the letters
A and B in their places, thus, A*/B : A*/B; and then fill up,
crosswise, the first by C and the other by D.
S 15. If we take the points in a different order, the value of the
cross-ratio will change. We can do this in twenty-four different ways
by forming all permutations of the letters. But of these twenty-four
cross-ratios groups of four are equal, so that there are really only
six different ones, and these six are reciprocals in pairs.
We have the following rules:--
I. If in a cross-ratio the two groups be interchanged, its value
remains unaltered, i.e.
(AB, CD) = (CD, AB) = (BA, DC) = (DC, BA).
II. If in a cross-ratio the two points belonging to one of the two
groups be interchanged, the cross-ratio changes into its reciprocal,
i.e.
(AB, CD) = 1/(AB, DC) = 1/(BA, CD) = 1/(CD, BA) = 1/(DC, AB).
From I. and II. we see that eight cross-ratios are associated with
(AB, CD).
III. If in a cross-ratio the two middle letters be interchanged, the
cross-ratio [alpha] changes into its complement 1 - [alpha], i.e. (AB,
CD) = 1 - (AC, BD).
[S 16. If [lambda] = (AB, CD), [mu] = (AC, DB), [nu] = (AD, BC), then
[lambda], [mu], [nu] and their reciprocals 1/[lambda], 1/[mu], 1/[nu]
are the values of the total number of twenty-four cross-ratios.
Moreover, [lambda], [mu], [nu] are connected by the relations
[lambda] + 1/[mu] = [mu] + 1/[nu] = [nu] + 1/[lambda] = -[lambda][mu][nu] = 1;
this proposition may be proved by substituting for [lambda], [mu],
[nu] and reducing to a common origin. There are therefore four
equations between three unknowns; hence if one cross-ratio be given,
the remaining twenty-three are determinate. Moreover, two of the
quantities [lambda], [mu], [nu] are positive, and the remaining one
negative.
The following scheme shows the twenty-four cross-ratios expressed in
terms of [lambda], [mu], [nu].]
+---------+-----------------------+---------------+---------------+
|(AB, CD) | | | |
|(BA, DC) | [lambda] | 1 - [mu] | 1/(1 - [nu]) |
|(CD, AB) | | | |
|(DC, BA) | | | |
+---------+-----------------------+---------------+---------------+
|(AC, DB) | | | |
|(BD, CA) | 1/(1 - [lambda]) | 1/[mu] |([nu] - 1)/[nu]|
|(CA, BD) | | | |
|(DB, AC) | | | |
+---------+-----------------------+---------------+---------------+
|(AB, DC) | | | |
|(BA, CD) | 1/[lambda] | 1/(1 - [mu]) | 1 - [nu] |
|(CD, BA) | | | |
|(DC, AB) | | | |
+---------+-----------------------+---------------+---------------+
|(AD, BC) | | | |
|(BC, AD) |([lambda] - 1)/[lambda]|[mu]/([mu] - 1)| [nu] |
|(CB, DA) | | | |
|(DA, CB) | | | |
+---------+-----------------------+---------------+---------------+
|(AC, BD) | | | |
|(BD, AC) | 1 - [lambda] | [mu] |[nu]/([nu] - 1)|
|(CA, DB) | | | |
|(DB, CA) | | | |
+---------+-----------------------+---------------+---------------+
|(AD, CB) | | | |
|(BC, DA) |[lambda]/([lambda] - 1)|([mu] - 1)/[mu]| 1/[nu] |
|(CB, AD) | | | |
|(DA, BC) | | | |
+---------+-----------------------+---------------+---------------+
S 17. If one of the points of which a cross-ratio is formed is the
point at infinity in the line, the cross-ratio changes into a simple
ratio. It is convenient to let the point at infinity occupy the last
place in the symbolic expression for the cross-ratio. Thus if I is a
point at infinity, we have (AB, CI) = -AC/CB, because AI : IB = -1.
Every common ratio of three points in a line may thus be expressed as
a cross-ratio, by adding the point at infinity to the group of points.
HARMONIC RANGES
S 18. If the points have special positions, the cross-ratios may have
such a value that, of the six different ones, two and two become
equal. If the first two shall be equal, we get [lambda] = 1/[lambda],
or [lambda]^2 = 1, [lambda] = [+-]1.
If we take [lambda] = +1, we have (AB, CD) = 1, or AC/CB = AD/DB; that
is, the points C and D coincide, provided that A and B are different.
If we take [lambda] = -1, so that (AB, CD) = -1, we have AC/CB =
-AD/DB. _Hence C and D divide AB internally and externally in the same
ratio._
The four points are in this case said to be _harmonic points_, and _C
and D are said to be harmonic conjugates with regard to A and B._
But we have also (CD, AB) = -1, so that A and B are harmonic
conjugates with regard to C and D.
The principal property of harmonic points is that their cross-ratio
remains unaltered if we interchange the two points belonging to one
pair, viz.
(AB, CD) = (AB, DC) = (BA, CD).
For four harmonic points the six cross-ratios become equal two and
two:
[lambda]
[lambda] = -1, 1 - [lambda] = 2, ------------ = 1/2,
[lambda] - 1
1 1 [lambda] - 1
= -------- = -1, ------------ = 1/2, ------------ = 2.
[lambda] 1 - [lambda] [lambda]
Hence if we get four points whose cross-ratio is 2 or 1/2, then they
are harmonic, but not arranged so that conjugates are paired. If this
is the case the cross-ratio = -1.
S 19. If we equate any two of the above six values of the
cross-ratios, we get either [lambda] = 1, 0, [infinity], or [lambda] =
-1, 2, 1/2, or else [lambda] becomes a root of the equation [lambda]^2
- [lambda] + 1 = 0, that is, an imaginary cube root of -1. In this
case the six values become three and three equal, so that only two
different values remain. This case, though important in the theory of
cubic curves, is for our purposes of no interest, whilst harmonic
points are all-important.
S 20. From the definition of harmonic points, and by aid of S 11, the
following properties are easily deduced.
If C and D are harmonic conjugates with regard to A and B, then one of
them lies in, the other without AB; it is impossible to move from A to
B without passing either through C or through D; the one blocks the
finite way, the other the way through infinity. This is expressed by
saying A and B are "separated" by C and D.
For every position of C there will be one and only one point D which
is its harmonic conjugate with regard to any point pair A, B.
If A and B are different points, and if C coincides with A or B, D
does. But if A and B coincide, one of the points C or D, lying between
them, coincides with them, and the other may be anywhere in the line.
It follows that, "_if of four harmonic conjugates two coincide, then a
third coincides with them, and the fourth may be any point in the
line_."
If C is the middle point between A and B, then D is the point at
infinity; for AC : CB = +1, hence AD : DB must be equal to -1. _The
harmonic conjugate of the point at infinity in a line with regard to
two points A, B is the middle point of AB._
This important property gives a first example how metric properties
are connected with projective ones.
[S 21. _Harmonic properties of the complete quadrilateral and
quadrangle._
[Illustration: FIG. 7.]
[Illustration: FIG. 8.]
A figure formed by four lines in a plane is called a _complete
quadrilateral_, or, shorter, a _four-side_. The four sides meet in six
points, named the "vertices," which may be joined by three lines
(other than the sides), named the "diagonals" or "harmonic lines." The
diagonals enclose the "harmonic triangle of the quadrilateral." In
fig. 7, A'B'C', B'AC, C'AB, CBA' are the sides, A, A', B, B', C, C'
the vertices, AA', BB', CC' the harmonic lines, and
[alpha][beta][gamma] the harmonic triangle of the quadrilateral. A
figure formed by four coplanar points is named a _complete
quadrangle_, or, shorter, a _four-point_. The four points may be
joined by six lines, named the "sides," which intersect in three other
points, termed the "diagonal or harmonic points." The harmonic points
are the vertices of the "harmonic triangle of the complete
quadrangle." In fig. 8, AA', BB' are the points, AA', BB', A'B', B'A,
AB, BA' are the sides, L, M, N are the diagonal points, and LMN is the
harmonic triangle of the quadrangle.
The harmonic property of the complete quadrilateral is: Any diagonal
or harmonic line is harmonically divided by the other two; and of a
complete quadrangle: The angle at any harmonic point is divided
harmonically by the joins to the other harmonic points. To prove the
first theorem, we have to prove (AA', [beta][gamma]), (BB',
[gamma][alpha]), (CC', [beta][alpha]) are harmonic. Consider the
cross-ratio (CC', [alpha][beta]). Then projecting from A on BB' we
have A(CC', [alpha][beta]) = A(B'B, [alpha][gamma]). Projecting from
A' on BB', A'(CC', [alpha][beta]) = A'(BB', [alpha][gamma]). Hence
(B'B, [alpha][gamma]) = (BB', [alpha][gamma]), i.e. the cross-ratio
(BB', [alpha][gamma]) equals that of its reciprocal; hence the range
is harmonic.
The second theorem states that the pencils L(BA, NM), M(B'A, LN),
N(BA, LM) are harmonic. Deferring the subject of harmonic pencils to
the next section, it will suffice to state here that any transversal
intersects an harmonic pencil in an harmonic range. Consider the
pencil L(BA, NM), then it is sufficient to prove (BA', NM') is
harmonic. This follows from the previous theorem by considering A'B as
a diagonal of the quadrilateral ALB'M.]
This property of the complete quadrilateral allows the solution of the
problem:
_To construct the harmonic conjugate D to a point C with regard to two
given points A and B._
Through A draw any two lines, and through C one cutting the former two
in G and H. Join these points to B, cutting the former two lines in E
and F. The point D where EF cuts AB will be the harmonic conjugate
required.
This remarkable construction requires nothing but the drawing of
lines, and is therefore independent of measurement. In a similar
manner the harmonic conjugate of the line VA for two lines VC, VD is
constructed with the aid of the property of the complete quadrangle.
S 22. _Harmonic Pencils._--The theory of cross-ratios may be extended
from points in a row to lines in a flat pencil and to planes in an
axial pencil. We have seen (S 13) that if the lines which join four
points A, B, C, D to any point S be cut by any other line in A', B',
C', D', then (AB, CD) = (A'B', C'D'). In other words, four lines in a
flat pencil are cut by every other line in four points whose
cross-ratio is constant.
_Definition._--By the cross-ratio of four rays in a flat pencil is
meant the cross-ratio of the four points in which the rays are cut by
any line. If a, b, c, d be the lines, then this cross-ratio is denoted
by (ab, cd).
_Definition._--By the cross-ratio of four planes in an axial pencil is
understood the cross-ratio of the four points in which any line cuts
the planes, or, what is the same thing, the cross-ratio of the four
rays in which any plane cuts the four planes.
In order that this definition may have a meaning, it has to be proved
that all lines cut the pencil in points which have the same
cross-ratio. This is seen at once for two intersecting lines, as their
plane cuts the axial pencil in a flat pencil, which is itself cut by
the two lines. The cross-ratio of the four points on one line is
therefore equal to that on the other, and equal to that of the four
rays in the flat pencil.
If two non-intersecting lines p and q cut the four planes in A, B, C,
D and A', B', C', D', draw a line r to meet both p and q, and let this
line cut the planes in A", B", C", D". Then (AB, CD) = (A'B', C'D'),
for each is equal to (A"B", C"D").
S 23. We may now also extend the notion of harmonic elements, viz.
_Definition._--Four rays in a flat pencil and four planes in an axial
pencil are said to be harmonic if their cross-ratio equals -1, that
is, if they are cut by a line in four harmonic points.
If we understand by a "median line" of a triangle a line which joins a
vertex to the middle point of the opposite side, and by a "median
line" of a parallelogram a line joining middle points of opposite
sides, we get as special cases of the last theorem:
_The diagonals and median lines of a parallelogram form an harmonic
pencil_; and
_At a vertex of any triangle, the two sides, the median line, and the
line parallel to the base form an harmonic pencil._
Taking the parallelogram a rectangle, or the triangle isosceles, we
get:
_Any two lines and the bisections of their angles form an harmonic
pencil._ Or:
_In an harmonic pencil, if two conjugate rays are perpendicular, then
the other two are equally inclined to them_; and, conversely, _if one
ray bisects the angle between conjugate rays, it is perpendicular to
its conjugate_.
This connects perpendicularity and bisection of angles with projective
properties.
S 24. We add a few theorems and problems which are easily proved or
solved by aid of harmonics.
An harmonic pencil is cut by a line parallel to one of its rays in
three equidistant points.
Through a given point to draw a line such that the segment determined
on it by a given angle is bisected at that point.
Having given two parallel lines, to bisect on either any given segment
without using a pair of compasses.
Having given in a line a segment and its middle point, to draw through
any given point in the plane a line parallel to the given line.
To draw a line which joins a given point to the intersection of two
given lines which meet off the drawing paper (by aid of S 21).
CORRESPONDENCE. HOMOGRAPHIC AND PERSPECTIVE RANGES
S 25. Two rows, p and p', which are one the projection of the other
(as in fig. 5), stand in a definite relation to each other,
characterized by the following properties.
1. _To each point in either corresponds one point in the other_; that
is, those points are said to correspond which are projections of one
another.
2. _The cross-ratio of any four points in one equals that of the
corresponding points in the other._
3. _The lines joining corresponding points all pass through the same
point._
If we suppose corresponding points marked, and the rows brought into
any other position, then the lines joining corresponding points will
no longer meet in a common point, and hence the third of the above
properties will not hold any longer; but we have still a
correspondence between the points in the two rows possessing the first
two properties. Such a correspondence has been called a _one-one
correspondence_, whilst the two rows between which such correspondence
has been established are said to be _projective_ or _homographic_. Two
rows which are each the projection of the other are therefore
_projective_. We shall presently see, also, that any two projective
rows may always be placed in such a position that one appears as the
projection of the other. If they are in such a position the rows are
said to be in _perspective position_, or simply to be in
_perspective_.
S 26. The notion of a one-one correspondence between rows may be
extended to flat and axial pencils, viz. a flat pencil will be said to
be projective to a flat pencil if to each ray in the first corresponds
one ray in the second, and if the cross-ratio of four rays in one
equals that of the corresponding rays in the second.
Similarly an axial pencil may be projective to an axial pencil. But a
flat pencil may also be projective to an axial pencil, or either
pencil may be projective to a row. The definition is the same in each
case: there is a one-one correspondence between the elements, and four
elements have the same cross-ratio as the corresponding ones.
S 27. There is also in each case a special position which is called
_perspective_, viz.
1. Two projective rows are perspective if they lie in the same plane,
and if the one row is a projection of the other.
2. Two projective flat pencils are perspective--(1) if they lie in the
same plane, and have a row as a common section; (2) if they lie in the
same pencil (in space), and are both sections of the same axial
pencil; (3) if they are in space and have a row as common section, or
are both sections of the same axial pencil, one of the conditions
involving the other.
3. Two projective axial pencils, if their axes meet, and if they have
a flat pencil as a common section.
4. A row and a projective flat pencil, if the row is a section of the
pencil, each point lying in its corresponding line.
5. A row and a projective axial pencil, if the row is a section of the
pencil, each point lying in its corresponding line.
6. A flat and a projective axial pencil, if the former is a section of
the other, each ray lying in its corresponding plane.
That in each case the correspondence established by the position
indicated is such as has been called projective follows at once from
the definition. It is not so evident that the perspective position may
always be obtained. We shall show in S 30 this for the first three
cases. First, however, we shall give a few theorems which relate to
the general correspondence, not to the perspective position.
S 28. _Two rows or pencils, flat or axial, which are projective to a
third are projective to each other_; this follows at once from the
definitions.
S 29. _If two rows, or two pencils, either flat or axial, or a row and
a pencil, be projective, we may assume to any three elements in the
one the three corresponding elements in the other, and then the
correspondence is uniquely determined._
For if in two projective rows we assume that the points A, B, C in the
first correspond to the given points A', B', C' in the second, then to
any fourth point D in the first will correspond a point D' in the
second, so that
(AB, CD) = (A'B', C'D').
But there is only one point, D', which makes the cross-ratio (A'B',
C'D') equal to the given number (AB, CD).
The same reasoning holds in the other cases.
S 30. If two rows are perspective, then the lines joining
corresponding points all meet in a point, the centre of projection;
and the point in which the two bases of the rows intersect as a point
in the first row coincides with its corresponding point in the second.
This follows from the definition. The converse also holds, viz.
_If two projective rows have such a position that one point in the one
coincides with its corresponding point in the other, then they are
perspective, that is, the lines joining corresponding points all pass
through a common point, and form a flat pencil._
For let A, B, C, D ... be points in the one, and A', B', C', D' ...
the corresponding points in the other row, and let A be made to
coincide with its corresponding point A'. Let S be the point where the
lines BB' and CC' meet, and let us join S to the point D in the first
row. This line will cut the second row in a point D", so that A, B, C,
D are projected from S into the points A, B', C', D". The cross-ratio
(AB, CD) is therefore equal to (AB', C'D"), and by hypothesis it is
equal to (A'B', C'D'). Hence (A'B', C'D") = (A'B', C'D'), that is, D"
is the same point as D'.
S 31. If two projected flat pencils in the same plane are in
perspective, then the intersections of corresponding lines form a row,
and the line joining the two centres as a line in the first pencil
corresponds to the same line as a line in the second. And conversely,
_If two projective pencils in the same plane, but with different
centres, have one line in the one coincident with its corresponding
line in the other, then the two pencils are perspective, that is, the
intersection of corresponding lines lie in a line._
The proof is the same as in S 30.
S 32. If two projective flat pencils in the same point (pencil in
space), but not in the same plane, are perspective, then the planes
joining corresponding rays all pass through a line (they form an axial
pencil), and the line common to the two pencils (in which their planes
intersect) corresponds to itself. And conversely:--
If two flat pencils which have a common centre, but do not lie in a
common plane, are placed so that one ray in the one coincides with its
corresponding ray in the other, then they are perspective, that is,
the planes joining corresponding lines all pass through a line.
S 33. If two projective axial pencils are perspective, then the
intersection of corresponding planes lie in a plane, and the plane
common to the two pencils (in which the two axes lie) corresponds to
itself. And conversely:--
If two projective axial pencils are placed in such a position that a
plane in the one coincides with its corresponding plane, then the two
pencils are perspective, that is, corresponding planes meet in lines
which lie in a plane.
The proof again is the same as in S 30.
S 34. These theorems relating to perspective position become illusory
if the projective rows of pencils have a common base. We then have:--
In two projective rows on the same line--and also in two projective
and concentric flat pencils in the same plane, or in two projective
axial pencils with a common axis--every element in the one coincides
with its corresponding element in the other as soon as three elements
in the one coincide with their corresponding elements in the other.
_Proof_ (in case of two rows).--Between four elements A, B, C, D and
their corresponding elements A', B', C', D' exists the relation (ABCD)
= (A'B'C'D'). If now A', B', C' coincide respectively with A, B, C, we
get (AB, CD) = (AB, CD'), hence D and D' coincide.
The last theorem may also be stated thus:--
In two projective rows or pencils, which have a common base but are
not identical, not more than two elements in the one can coincide with
their corresponding elements in the other.
Thus two projective rows on the same line cannot have more than two
pairs of coincident points unless every point coincides with its
corresponding point.
It is easy to construct two projective rows on the same line, which
have two pairs of corresponding points coincident. Let the points A,
B, C as points belonging to the one row correspond to A, B, and C' as
points in the second. Then A and B coincide with their corresponding
points, but C does not. It is, however, not necessary that two such
rows have twice a point coincident with its corresponding point; it is
possible that this happens only once or not at all. Of this we shall
see examples later.
[Illustration: FIG. 9.]
S 35. If two projective rows or pencils are in perspective position,
we know at once which element in one corresponds to any given element
in the other. If p and q (fig. 9) are two projective rows, so that K
corresponds to itself, and if we know that to A and B in p correspond
A' and B' in q, then the point S, where AA' meets BB', is the centre
of projection, and hence, in order to find the point C' corresponding
to C, we have only to join C to S; the point C', where this line cuts
q, is the point required.
[Illustration: FIG. 10.]
If two flat pencils, S1 and S2, in a plane are perspective (fig. 10),
we need only to know two pairs, a, a' and b, b', of corresponding rays
in order to find the axis s of projection. This being known, a ray c'
in S2, corresponding to a given ray c in S1, is found by joining S2 to
the point where c cuts the axis s.
A similar construction holds in the other cases of perspective
figures.
On this depends the solution of the following general problem.
S 36. Three pairs of corresponding elements in two projective rows or
pencils being given, to determine for any element in one the
corresponding element in the other.
We solve this in the two cases of two projective rows and of two
projective flat pencils in a plane.
_Problem_ I.--Let A, B, C be _Problem_ II.--Let a, b, c be
three points in a row s, A', B', three rays in a pencil S, a',
C' the corresponding points in a b', c' the corresponding rays in
projective row s', both being in a projective pencil S', both
a plane; it is required to find being in the same plane; it is
for any point D in s the required to find for any ray d
corresponding point D' in s'. in S the corresponding ray d' in
S'.
The solution is made to depend on the construction of an auxiliary row
or pencil which is perspective to both the given ones. This is found
as follows:--
[Illustration: FIG. 11.]
_Solution of Problem_ I.--On the line joining two corresponding
points, say AA' (fig. 11), take any two points, S and S', as centres
of auxiliary pencils. Join the intersection B1 of SB and S'B' to the
intersection C1 of SC and S'C' by the line s1. Then a row on s1 will
be perspective to s with S as centre of projection, and to s' with S'
as centre. To find now the point D' on s' corresponding to a point D
on s we have only to determine the point D1, where the line SD cuts
s1, and to draw S'D1; the point where this line cuts s' will be the
required point D'.
_Proof._--The rows s and s' are both perspective to the row s1, hence
they are projective to one another. To A, B, C, D on s correspond A1,
B1, C1, D1 on s1, and to these correspond A', B', C', D' on s'; so
that D and D' are corresponding points as required.
[Illustration: FIG. 12.]
_Solution of Problem_ II.--Through the intersection A of two
corresponding rays a and a' (fig. 12), take two lines, s and s', as
bases of auxiliary rows. Let S1 be the point where the line b1, which
joins B and B', cuts the line c1, which joins C and C'. Then a pencil
S1 will be perspective to S with s as axis of projection. To find the
ray d' in S' corresponding to a given ray d in S, cut d by s at D;
project this point from S1 to D' on s' and join D' to S'. This will be
the required ray.
_Proof._--That the pencil S1 is perspective to S and also to S'
follows from construction. To the lines a1, b1, c1, d1 in S1
correspond the lines a, b, c, d in S and the lines a', b', c', d' in
S', so that d and d' are corresponding rays.
In the first solution the two centres, S, S', are _any_ two points on
a line joining any two corresponding points, so that the solution of
the problem allows of a great many different constructions. _But
whatever construction be used, the point D', corresponding to D, must
be always the same_, according to the theorem in S 29. This gives rise
to a number of theorems, into which, however, we shall not enter. The
same remarks hold for the second problem.
S 37. _Homological Triangles._--As a further application of the
theorems about perspective rows and pencils we shall prove the
following important theorem.
_Theorem._--If ABC and A'B'C' (fig. 13) be two triangles, such that
the lines AA', BB', CC' meet in a point S, then the intersections of
BC and B'C', of CA and C'A', and of AB and A'B' will lie in a line.
Such triangles are said to be homological, or in perspective. The
triangles are "co-axial" in virtue of the property that the meets of
corresponding sides are collinear and copolar, since the lines joining
corresponding vertices are concurrent.
_Proof._--Let a, b, c denote the lines AA', BB', CC', which meet at S.
Then these may be taken as bases of projective rows, so that A, A', S
on a correspond to B, B', S on b, and to C, C', S on c. As the point S
is common to all, any two of these rows will be perspective.
If S1 be the centre of projection of rows b and c,
S2 " " " c and a,
S3 " " " a and b,
and if the line S1S2 cuts a in A1, and b in B1, and c in C1, then A1,
B1 will be corresponding points in a and b, both corresponding to C1
in c. But a and b are perspective, therefore the line A1B1, that is
S1S2, joining corresponding points must pass through the centre of
projection S3 of a and b. In other words, S1, S2, S3 lie in a line.
This is Desargues' celebrated theorem if we state it thus:--
[Illustration: FIG. 13.]
_Theorem of Desargues._--If each of two triangles has one vertex on
each of three concurrent lines, then the intersections of
corresponding sides lie in a line, those sides being called
corresponding which are opposite to vertices on the same line.
The converse theorem holds also, viz.
_Theorem._--If the sides of one triangle meet those of another in
three points which lie in a line, then the vertices lie on three lines
which meet in a point.
The proof is almost the same as before.
S 38. _Metrical Relations between Projective Rows._--Every row
contains one point which is distinguished from all others, viz. the
point at infinity. In two projective rows, to the point I at infinity
in one corresponds a point I' in the other, and to the point J' at
infinity in the second corresponds a point J in the first. The points
I' and J are in general finite. If now A and B are any two points in
the one, A', B' the corresponding points in the other row, then
(AB, JI) = (A'B', J'I'),
or
AJ/JB : AI/IB = A'J'/J'B' : A'I'/I'B'.
But, by S 17,
AI/IB = A'J'/J'B' = -1;
therefore the last equation changes into
AJ.A'I' = BJ.B'I',
that is to say--
_Theorem._--The product of the distances of any two corresponding
points in two projective rows from the points which correspond to the
points at infinity in the other is constant, viz. AJ.A'I' = k.
Steiner has called this number k the _Power of the correspondence_.
[The relation AJ . A'I' = k shows that if J, I' be given then the
point A' corresponding to a specified point A is readily found; hence
A, A' generate homographic ranges of which I and J' correspond to the
points at infinity on the ranges. If we take any two origins O, O', on
the ranges and reduce the expression AJ . A'I' = k to its algebraic
equivalent, we derive an equation of the form [alpha]xx' + [beta]x +
[gamma]x' + [delta] = 0. Conversely, if a relation of this nature
holds, then points corresponding to solutions in x, x' form
homographic ranges.]
S 39. _Similar Rows._--If the points at infinity in two projective
rows correspond so that I' and J are at infinity, this result loses
its meaning. But if A, B, C be any three points in one, A', B', C' the
corresponding ones on the other row, we have
(AB, CI) = (A'B', C'I'),
which reduces to
AC/CB = A'C'/C'B' or AC/A'C' = BC/B'C',
that is, corresponding segments are proportional. Conversely, if
corresponding segments are proportional, then to the point at infinity
in one corresponds the point at infinity in the other. If we call such
rows _similar_, we may state the result thus--
_Theorem._--Two projective rows are similar if to the point at
infinity in one corresponds the point at infinity in the other, and
conversely, if two rows are similar then they are projective, and the
points at infinity are corresponding points.
From this the well-known propositions follow:--
Two lines are cut proportionally (in similar rows) by a series of
parallels. The rows are perspective, with centre of projection at
infinity.
If two similar rows are placed parallel, then the lines joining
homologous points pass through a common point.
S 40. If two flat pencils be projective, then there exists in either,
one single pair of lines at right angles to one another, such that the
corresponding lines in the other pencil are again at right angles.
[Illustration: FIG. 14.]
To prove this, we place the pencils in perspective position (fig. 14)
by making one ray coincident with its corresponding ray. Corresponding
rays meet then on a line p. And now we draw the circle which has its
centre O on p, and which passes through the centres S and S' of the
two pencils. This circle cuts p in two points H and K. The two pairs
of rays, h, k, and h', k', joining these points to S and S' will be
pairs of corresponding rays at right angles. The construction gives in
general but one circle, but if the line p is the perpendicular
bisector of SS', there exists an infinite number, and _to every right
angle in the one pencil corresponds a right angle in the other_.
PRINCIPLE OF DUALITY
S 41. It has been stated in S 1 that not only points, but also planes
and lines, are taken as elements out of which figures are built up. We
shall now see that the construction of one figure which possesses
certain properties gives rise in many cases to the construction of
another figure, by replacing, according to definite rules, elements of
one kind by those of another. The new figure thus obtained will then
possess properties which may be stated as soon as those of the
original figure are known.
We obtain thus a principle, known as the _principle of duality_ or of
_reciprocity_, which enables us to construct to any figure not
containing any measurement in its construction a _reciprocal_ figure,
as it is called, and to deduce from any theorem a _reciprocal_
theorem, for which no further proof is needed.
It is convenient to print reciprocal propositions on opposite sides of
a page broken into two columns, and this plan will occasionally be
adopted.
We begin by repeating in this form a few of our former statements:--
Two points determine a line. Two planes determine a line.
Three points which are not in a Three planes which do not pass
line determine a plane. through a line determine a point.
A line and a point without it A line and a plane not through
determine a plane. it determine a point.
Two lines in a plane determine Two lines through a point
a point. determine a plane.
These propositions show that it will be possible, when any figure is
given, to construct a second figure by taking planes instead of
points, and points instead of planes, but lines where we had lines.
For instance, if in the first figure we take a plane and three points
in it, we have to take in the second figure a point and three planes
through it. The three points in the first, together with the three
lines joining them two and two, form a triangle; the three planes in
the second and their three lines of intersection form a trihedral
angle. A triangle and a trihedral angle are therefore reciprocal
figures.
Similarly, to any figure in a plane consisting of points and lines
will correspond a figure consisting of planes and lines passing
through a point S, and hence belonging to the pencil which has S as
centre.
The figure reciprocal to four points in space which do not lie in a
plane will consist of four planes which do not meet in a point. In
this case each figure forms a tetrahedron.
S 42. As other examples we have the following:--
To a row is reciprocal an axial pencil,
" a flat pencil " a flat pencil,
" a field of points and lines " a pencil of planes and lines,
" the space of points " the space of planes.
For the row consists of a line and all the points in it, reciprocal to
it therefore will be a line with all planes through it, that is, an
axial pencil; and so for the other cases.
This correspondence of reciprocity breaks down, however, if we take
figures which contain measurement in their construction. For instance,
there is no figure reciprocal to two planes at _right angles_, because
there is no segment in a row which has a magnitude as definite as a
right angle.
We add a few examples of reciprocal propositions which are easily
proved.
_Theorem._--If A, B, C, D are _Theorem._--If [alpha], [beta],
any four points in space, and if [gamma], [delta] are four planes
the lines AB and CD meet, then in space, and if the lines
all four points lie in a plane, [alpha][beta] and [gamma][delta]
hence also AC and BD, as well meet, then all four planes lie
as AD and BC, meet. in a point (pencil), hence also
[alpha][gamma] and [beta][delta],
well as [alpha][delta] and
as [beta][gamma], meet.
Theorem.--_If of any number of lines every one meets every other,
whilst all do not_
_lie in a point, then all lie in _lie in a plane, then all lie in
a plane._ a point (pencil)._
S 43. Reciprocal figures as explained lie both in space of three
dimensions. If the one is confined to a plane (is formed of elements
which lie in a plane), then the reciprocal figure is confined to a
pencil (is formed of elements which pass through a point).
But there is also a more special principle of duality, according to
which figures are reciprocal which lie both in a plane or both in a
pencil. In the plane we take points and lines as reciprocal elements,
for they have this fundamental property in common, that two elements
of one kind determine one of the other. In the pencil, on the other
hand, lines and planes have to be taken as reciprocal, and here it
holds again that two lines or planes determine one plane or line.
Thus, to one plane figure we can construct one reciprocal figure in
the plane, and to each one reciprocal figure in a pencil. We mention a
few of these. At first we explain a few names:--
A figure consisting of n points A figure consisting of n lines
in a plane will be called an in a plane will be called an
n-point. n-side.
A figure consisting of n planes A figure consisting of n lines
in a pencil will be called an in a pencil will be called an
n-flat. n-edge.
It will be understood that an n-side is different from a polygon of n
sides. The latter has sides of finite length and n vertices, the
former has sides all of infinite extension, and every point where two
of the sides meet will be a vertex. A similar difference exists
between a solid angle and an n-edge or an n-flat. We notice
particularly--
A four-point has six sides, of A four-side has six vertices, of
which two and two are opposite, which two and two are opposite,
and three diagonal points, which and three diagonals, which join
are intersections of opposite opposite vertices.
sides.
A four-flat has six edges, of A four-edge has six faces, of
which two and two are opposite, which two and two are opposite,
and three diagonal planes, which and three diagonal edges, which
pass through opposite edges. are intersections of opposite
faces.
A four-side is usually called a complete quadrilateral, and a
four-point a complete quadrangle. The above notation, however, seems
better adapted for the statement of reciprocal propositions.
S 44.
If a point moves in a plane it If a line moves in a plane it
describes a plane curve. envelopes a plane curve (fig. 15).
If a plane moves in a pencil it If a line moves in a pencil it
envelopes a cone. describes a cone.
A curve thus appears as generated either by points, and then we call
it a "locus," or by lines, and then we call it an "envelope." In the
same manner a cone, which means here a surface, appears either as the
locus of lines passing through a fixed point, the "vertex" of the
cone, or as the envelope of planes passing through the same point.
[Illustration: FIG. 15.]
To a surface as locus of points corresponds, in the same manner, a
surface as envelope of planes; and to a curve in space as locus of
points corresponds a developable surface as envelope of planes.
It will be seen from the above that we may, by aid of the principle of
duality, construct for every figure a reciprocal figure, and that to
any property of the one a reciprocal property of the other will exist,
as long as we consider only properties which depend upon nothing but
the positions and intersections of the different elements and not upon
measurement.
For such propositions it will therefore be unnecessary to prove more
than one of two reciprocal theorems.
GENERATION OF CURVES AND CONES OF SECOND ORDER OR SECOND CLASS
S 45. _Conics._--If we have two projective pencils in a plane,
corresponding rays will meet, and their point of intersection will
constitute some locus which we have to investigate. Reciprocally, if
two projective rows in a plane are given, then the lines which join
corresponding points will envelope some curve. We prove first:--
_Theorem._--If two projective _Theorem._--If two projective
flat pencils lie in a plane, but rows lie in a plane, but are
are neither in perspective nor neither in perspective nor on a
concentric, then the locus of common base, then the envelope
intersections of corresponding of lines joining corresponding
rays is a curve of the second points is a curve of the second
order, that is, no line contains class, that is, through no point
more than two points of the pass more than two of the
locus. enveloping lines.
Proof.--We draw any line t. _Proof._--We take any point T
This cuts each of the pencils in and join it to all points in each
a row, so that we have on t two row. This gives two concentric
rows, and these are projective pencils, which are projective
because the pencils are because the rows are projective.
projective. If corresponding rays If a line joining corresponding
of the two pencils meet on the points in the two rows passes
line t, their intersection will through T, it will be a line in
be a point in the one row which the one pencil which coincides
coincides with its corresponding with its corresponding line in
point in the other. But two the other. But two projective
projective rows on the same base concentric flat pencils in the
cannot have more than two same plane cannot have more than
points of one coincident with two lines of one coincident with
their corresponding points in their corresponding line in the
the other (S 34). other (S 34).
It will be seen that the proofs are reciprocal, so that the one may be
copied from the other by simply interchanging the words point and
line, locus and envelope, row and pencil, and so on. We shall
therefore in future prove seldom more than one of two reciprocal
theorems, and often state one theorem only, the reader being
recommended to go through the reciprocal proof by himself, and to
supply the reciprocal theorems when not given.
S 46. We state the theorems in the pencil reciprocal to the last,
without proving them:--
_Theorem._--If two projective _Theorem._--If two projective
flat pencils are concentric, but axial pencils lie in the same
are neither perspective nor pencil (their axes meet in a
coplanar, then the envelope of point), but are neither perspective
the planes joining corresponding nor co-axial, then the locus
rays is a cone of the second of lines joining corresponding
class; that is, no line through planes is a cone of the second
the common centre contains more order; that is, no plane in the
than two of the enveloping pencil contains more than two
planes.
S 47. Of theorems about cones of second order and cones of second
class we shall state only very few. We point out, however, the
following connexion between the curves and cones under consideration:
The lines which join any point Every plane section of a cone
in space to the points on a curve of the second order is a curve of
of the second order form a cone the second order.
of the second order.
The planes which join any Every plane section of a cone
point in space to the lines of the second class is a curve of
enveloping a curve of the the second class.
second class envelope themselves
a cone of the second class.
By its aid, or by the principle of duality, it will be easy to obtain
theorems about them from the theorems about the curves.
We prove the first. A curve of the second order is generated by two
projective pencils. These pencils, when joined to the point in space,
give rise to two projective axial pencils, which generate the cone in
question as the locus of the lines where corresponding planes meet.
S48.
_Theorem._--The curve of second _Theorem._--The envelope of
order which is generated by two second class which is generated
projective flat pencils passes by two projective rows contains
through the centres of the two the bases of these rows as
pencils. enveloping lines or tangents.
_Proof._--If S and S' are the _Proof._--If s and s' are the
two pencils, then to the ray SS' two rows, then to the point ss'
or p' in the pencil S' or P' as a point in s'
corresponds in the pencil S a corresponds in s a point P,
ray p, which is different from which is not coincident with P',
p', for the pencils are not for the rows are not
perspective. But p and p' meet perspective. But P and P' are
at S, so that S is a point on joined by s, so that s is one of
the curve, and similarly S'. the enveloping lines, and
similarly s'.
It follows that every line in one of the two pencils cuts the curve in
two points, viz. once at the centre S of the pencil, and once where it
cuts its corresponding ray in the other pencil. These two points,
however, coincide, if the line is cut by its corresponding line at S
itself. The line p in S, which corresponds to the line SS' in S', is
therefore the only line through S which has but one point in common
with the curve, or which cuts the curve in two coincident points. Such
a line is called a _tangent_ to the curve, touching the latter at the
point S, which is called the "point of contact."
In the same manner we get in the reciprocal investigation the result
that through every point in one of the rows, say in s, two tangents
may be drawn to the curve, the one being s, the other the line joining
the point to its corresponding point in s'. There is, however, one
point P in s for which these two lines coincide. Such a point in one
of the tangents is called the "point of contact" of the tangent. We
thus get--
_Theorem._--To the line joining _Theorem._--To the point of
the centres of the projective intersection of the bases of two
pencils as a line in one pencil projective rows as a point in
corresponds in the other the one row corresponds in the other
tangent at its centre. the _point of contact_ of its
base.
S 49. Two projective pencils are determined if three pairs of
corresponding lines are given. Hence if a1, b1, c1 are three lines in
a pencil S1, and a2, b2, c2 the corresponding lines in a projective
pencil S2, the correspondence and therefore the curve of the second
order generated by the points of intersection of corresponding rays is
determined. Of this curve we know the two centres S1 and S2, and the
three points a1a2, b1b2, c1c2, hence five points in all. This and the
reciprocal considerations enable us to solve the following two
problems:
_Problem._--To construct a curve _Problem._--To construct a curve
of the second order, of which of the second class, of which
five points S1, S2, A, B, C are five tangents u1, u2, a, b, c
given. are given.
In order to solve the left-hand problem, we take two of the given
points, say S1 and S2, as centres of pencils. These we make projective
by taking the rays a1, b1, c1, which join S1 to A, B, C respectively,
as corresponding to the rays a2, b2, c2, which join S2 to A, B, C
respectively, so that three rays meet their corresponding rays at the
given points A, B, C. This determines the correspondence of the
pencils which will generate a curve of the second order passing
through A, B, C and through the centres S1 and S2, hence through the
five given points. To find more points on the curve we have to
construct for any ray in S1 the corresponding ray in S2. This has been
done in S 36. But we repeat the construction in order to deduce
further properties from it. We also solve the right-hand problem. Here
we select two, viz. u1, u2 of the five given lines, u1, u2, a, b, c,
as bases of two rows, and the points A1, B1, C1 where a, b, c cut u1
as corresponding to the points A2, B2, C2 where a, b, c cut u2.
We get then the following solutions of the two problems:
_Solution._--Through the point A _Solution._--In the line a take
draw any two lines, u1 and u2 any two points S1 and S2 as
(fig. 16), the first u1 to cut centres of pencils (fig. 17),
the pencil S1 in a row AB1C1, the first S1 (A1B1C1) to project
the other u2 to cut the pencil the row u1, the other S2
S2 in a row AB2C2. These two (A2B2C2) to project the row u2.
rows will be perspective, as the These two pencils will be
point A corresponds to itself, perspective, the line S1A1 being
and the centre of projection the same as the corresponding
will be the point S, where the line S2A2, and the axis of
lines B1B2 and C1C2 meet. To projection will be the line u,
find now for any ray d1 in S1 which joins the intersection B
its corresponding ray d2 in S2, of S1B1 and S2B2 to the
we determine the point D1 where intersection C of S1C1 and S2C2.
d1 cuts u1, project this point To find now for any point D1 in
from S to D2 on u2 and join S2 u1 the corresponding point D2 in
to D2. This will be the required u2, we draw S1D1 and project the
ray d2 which cuts d1 at some point D where this line cuts u
point D on the curve. from S2 to u2. This will give
the required point D2, and the
line d joining D1 to D2 will be
a new tangent to the curve.
S 50. These constructions prove, when rightly interpreted, very
important properties of the curves in question.
[Illustration: FIG. 16.]
If in fig. 16 we draw in the pencil S1 the ray k1 which passes through
the auxiliary centre S, it will be found that the corresponding ray k2
cuts it on u2. Hence--
_Theorem._--In the above _Theorem._--In the above
construction the bases of the construction (fig. 17) the
auxiliary rows u1 and u2 cut the tangents to the curve from the
curve where they cut the rays centres of the auxiliary pencils
S2S and S1S respectively. S1 and S2 are the lines which
pass through u2u and u1u
respectively.
As A is any given point on the curve, and u1 any line through it, we
have solved the problems:
_Problem._--To find the second _Problem._--To find the second
point in which any line through tangent which can be drawn from
a known point on the curve cuts any point in a given tangent to
the curve. the curve.
If we determine in S1 (fig. 16) the ray corresponding to the ray S2S1
in S2, we get the tangent at S1. Similarly, we can determine the point
of contact of the tangents u1 or u2 in fig. 17.
[Illustration: FIG. 17.]
S 51. If five points are given, of which not three are in a line, then
we can, as has just been shown, always draw a curve of the second
order through them; we select two of the points as centres of
projective pencils, and then one such curve is determined. It will be
presently shown that we get always the same curve if two other points
are taken as centres of pencils, that therefore five points
_determine_ one curve of the second order, and reciprocally, that five
tangents determine one curve of the second class. Six points taken at
random will therefore not lie on a curve of the second order. In order
that this may be the case a certain condition has to be satisfied, and
this condition is easily obtained from the construction in S 49, fig.
16. If we consider the conic determined by the five points A, S1, S2,
K, L, then the point D will be on the curve if, and only if, the
points on D1, S, D2 be in a line.
[Illustration: FIG. 18.]
This may be stated differently if we take AKS1DS2L (figs. 16 and 18)
as a hexagon inscribed in the conic, then AK and DS2 will be opposite
sides, so will be KS1 and S2L, as well as S1D and LA. The first two
meet in D2, the others in S and D1 respectively. We may therefore
state the required condition, together with the reciprocal one, as
follows:--
_Pascal's Theorem._--If a hexagon _Brianchon's Theorem._--If a
be inscribed in a curve of the hexagon be circumscribed about
second order, then the a curve of the second class, then
intersectionsof opposite sides the lines joining opposite vertices
are three points in a line. are three lines meeting in a point.
These celebrated theorems, which are known by the names of their
discoverers, are perhaps the most fruitful in the whole theory of
conics. Before we go over to their applications we have to show that
we obtain the same curve if we take, instead of S1, S2, any two other
points on the curve as centres of projective pencils.
S 52. We know that the curve depends only upon the correspondence
between the pencils S1 and S2, and not upon the special construction
used for finding new points on the curve. The point A (fig. 16 or 18),
through which the two auxiliary rows u1, u2 were drawn, may therefore
be changed to any other point on the curve. Let us now suppose the
curve drawn, and keep the points S1, S2, K, L and D, and hence also
the point S fixed, whilst we move A along the curve. Then the line AL
will describe a pencil about L as centre, and the point D1 a row on
S1D perspective to the pencil L. At the same time AK describes a
pencil about K and D2 a row perspective to it on S2D. But by Pascal's
theorem D1 and D2 will always lie in a line with S, so that the rows
described by D1 and D2 are perspective. It follows that the pencils K
and L will themselves be projective, corresponding rays meeting on the
curve. This proves that we get the same curve whatever pair of the
five given points we take as centres of projective pencils. Hence--
Only one curve of the second Only one curve of the second
order can be drawn which passes class can be drawn which touches
through five given points. five given lines.
We have seen that if on a curve of the second order two points
coincide at A, the line joining them becomes the tangent at A. If,
therefore, a point on the curve and its tangent are given, this will
be equivalent to having given two points on the curve. Similarly, if
on the curve of second class a tangent and its point of contact are
given, this will be equivalent to two given tangents.
We may therefore extend the last theorem:
Only one curve of the second Only one curve of the second
order can be drawn, of which class can be drawn, of which four
four points and the tangent at tangents and the point of contact
oneof them, or three points at one of them, or three tangents
and the tangents at two of and the points of contact at two
them, are given. of them, are given.
S 53. At the same time it has been proved:
If all points on a curve of the All tangents to a curve of second
second order be joined to any class are cut by any two of
two of them, then the two them in projective rows, those
pencils thus formed are being corresponding points which
projective, those rays being lie on the same tangent. Hence--
corresponding which meet on the
curve. Hence--
The cross-ratio of four rays The cross-ratio of the four
joining a point S on a curve of points in which any tangent u is
second order to four fixed cut by four fixed tangents a, b, c,
points A, B, C, D in the curve d is independent of the position of
is independent of the position u, and is called the cross-ratio of
of S, and is called the cross- the four tangents a, b, c, d.
ratio of the four points A, B,
C, D.
If this cross-ratio equals -1 If this cross-ratio equals -1
the four points are said to be the four tangents are said to be
four harmonic points. four harmonic tangents.
We have seen that a curve of second order, as generated by projective
pencils, has at the centre of each pencil one tangent; and further,
that any point on the curve may be taken as centre of such pencil.
Hence--
A curve of second order has A curve of second class has on
at every point one tangent. every tangent a point of contact.
S 54. We return to Pascal's and Brianchon's theorems and their
applications, and shall, as before, state the results both for curves
of the second order and curves of the second class, but prove them
only for the former.
Pascal's theorem may be used when five points are given to find more
points on the curve, viz. it enables us to find the point where any
line through one of the given points cuts the curve again. It is
convenient, in making use of Pascal's theorem, to number the points,
to indicate the order in which they are to be taken in forming a
hexagon, which, by the way, may be done in 60 different ways. It will
be seen that 1 2 (leaving out 3) 4 5 are opposite sides, so are 2 3
and (leaving out 4) 5 6, and also 3 4 and (leaving out 5) 6 1.
If the points 1 2 3 4 5 are given, and we want a 6th point on a line
drawn through 1, we know all the sides of the hexagon with the
exception of 5 6, and this is found by Pascal's theorem.
If this line should happen to pass through 1, then 6 and 1 coincide,
or the line 6 1 is the tangent at 1. And always if two consecutive
vertices of the hexagon approach nearer and nearer, then the side
joining them will ultimately become a tangent.
We may therefore consider a pentagon inscribed in a curve of second
order and the tangent at one of its vertices as a hexagon, and thus
get the theorem:
Every pentagon inscribed in a Every pentagon circumscribed
curve of second order has the about a curve of the second class
property that the intersections has the property that the lines
of two pairs of non-consecutive which join two pairs of non-
sides lie in a line with the consecutive vertices meet on that
point where the fifth side cuts line which joins the fifth vertex
the tangent at the opposite to the point of contact of the
vertex. opposite side.
This enables us also to solve the following problems.
Given five points on a curve of Given five tangents to a curve
second order to construct the of second class to construct the
tangent at any one of them. point of contact of any one of
them.
[Illustration: FIG. 19.]
If two pairs of adjacent vertices coincide, the hexagon becomes a
quadrilateral, with tangents at two vertices. These we take to be
opposite, and get the following theorems:
If a quadrilateral be inscribed If a quadrilateral be circumscribed
in a curve of second order, the about a curve of second
intersections of opposite sides, class, the lines joining opposite
and also the intersections of vertices, and also the lines joining
the tangents at opposite points of contact of opposite
vertices, lie in a line (fig. sides, meet in a point.
19).
[Illustration: FIG. 20.]
If we consider the hexagon made up of a triangle and the tangents at
its vertices, we get--
If a triangle is inscribed in a If a triangle be circumscribed
curve of the second order, the about a curve of second class,
points in which the sides are the lines which join the vertices
cut by the tangents at the to the points of contact of the
opposite vertices meet in a opposite sides meet in a point
point. (fig. 20).
S 55. Of these theorems, those about the quadrilateral give rise to a
number of others. Four points A, B, C, D may in three different ways
be formed into a quadrilateral, for we may take them in the order
ABCD, or ACBD, or ACDB, so that either of the points B, C, D may be
taken as the vertex opposite to A. Accordingly we may apply the
theorem in three different ways.
Let A, B, C, D be four points on a curve of second order (fig. 21),
and let us take them as forming a quadrilateral by taking the points
in the order ABCD, so that A, C and also B, D are pairs of opposite
vertices. Then P, Q will be the points where opposite sides meet, and
E, F the intersections of tangents at opposite vertices. The four
points P, Q, E, F lie therefore in a line. The quadrilateral ACBD
gives us in the same way the four points Q, R, G, H in a line, and the
quadrilateral ABDC a line containing the four points R, P, I, K. These
three lines form a triangle PQR.
The relation between the points and lines in this figure may be
expressed more clearly if we consider ABCD as a four-point inscribed
in a conic, and the tangents at these points as a four-side
circumscribed about it,--viz. it will be seen that P, Q, R are the
diagonal points of the four-point ABCD, whilst the sides of the
triangle PQR are the diagonals of the circumscribing four-side. Hence
the theorem--
_Any four-point on a curve of the second order and the four-side
formed by the tangents at these points stand in this relation that the
diagonal points of the four-point lie in the diagonals of the
four-side._ And conversely,
_If a four-point and a circumscribed four-side stand in the above
relation, then a curve of the second order may be described which
passes through the four points and touches there the four sides of
these figures._
That the last part of the theorem is true follows from the fact that
the four points A, B, C, D and the line a, as tangent at A, determine
a curve of the second order, and the tangents to this curve at the
other points B, C, D are given by the construction which leads to fig.
21.
[Illustration: FIG. 21.]
The theorem reciprocal to the last is--
_Any four-side circumscribed about a curve of second class and the
four-point formed by the points of contact stand in this relation that
the diagonals of the four-side pass through the diagonal points of the
four-point._ And conversely,
_If a four-side and an inscribed four-point stand in the above
relation, then a curve of the second class may be described which
touches the sides of the four-side at the points of the four-point._
S 56. The four-point and the four-side in the two reciprocal theorems
are alike. Hence if we have a four-point ABCD and a four-side abcd
related in the manner described, then not only may a curve of the
second order be drawn, but also a curve of the second class, which
both touch the lines a, b, c, d at the points A, B, C, D.
The curve of second order is already more than determined by the
points A, B, C and the tangents a, b, c at A, B and C. The point D may
therefore be _any_ point on this curve, and d any tangent to the
curve. On the other hand the curve of the second class is more than
determined by the three tangents a, b, c and their points of contact
A, B, C, so that d is any tangent to this curve. It follows that every
tangent to the curve of second order is a tangent of a curve of the
second class having the same point of contact. In other words, the
curve of second order is a curve of second class, and _vice versa_.
Hence the important theorems--
_Every curve of second order is _Every curve of second class is a
a curve of second class._ curve of second order._
The curves of second order and of second class, having thus been
proved to be identical, shall henceforth be called by the common name
of _Conics_.
For these curves hold, therefore, all properties which have been
proved for curves of second order or of second class. We may therefore
now state Pascal's and Brianchon's theorem thus--
_Pascal's Theorem._--If a hexagon be inscribed in a conic, then the
intersections of opposite sides lie in a line.
_Brianchon's Theorem._--If a hexagon be circumscribed about a conic,
then the diagonals forming opposite centres meet in a point.
S 57. If we suppose in fig. 21 that the point D together with the
tangent d moves along the curve, whilst A, B, C and their tangents a,
b, c remain fixed, then the ray DA will describe a pencil about A, the
point Q a projective row on the fixed line BC, the point F the row b,
and the ray EF a pencil about E. But EF passes always through Q. Hence
the pencil described by AD is projective to the pencil described by
EF, and therefore to the row described by F on b. At the same time the
line BD describes a pencil about B projective to that described by AD
(S 53). Therefore the pencil BD and the row F on b are projective.
Hence--
_If on a conic a point A be taken and the tangent a at this point,
then the cross-ratio of the four rays which join A to any four points
on the curve is equal to the cross-ratio of the points in which the
tangents at these points cut the tangent at A._
S 58. There are theorems about cones of second order and second class
in a pencil which are reciprocal to the above, according to S 43. We
mention only a few of the more important ones.
The locus of intersections of corresponding planes in two projective
axial pencils whose axes meet is a cone of the second order.
The envelope of planes which join corresponding lines in two
projective flat pencils, not in the same plane, is a cone of the
second class.
Cones of second order and cones of second class are identical.
Every plane cuts a cone of the second order in a conic.
_A cone of second order is uniquely determined by five of its edges or
by five of its tangent planes, or by four edges and the tangent plane
at one of them, &c. &c._
_Pascal's Theorem._--If a solid angle of six faces be inscribed in a
cone of the second order, then the intersections of opposite faces are
three lines in a plane.
_Brianchon's Theorem._--If a solid angle of six edges be circumscribed
about a cone of the second order, then the planes through opposite
edges meet in a line.
Each of the other theorems about conics may be stated for cones of the
second order.
S 59. _Projective Definitions of the Conics._--We now consider the
shape of the conics. We know that any line in the plane of the conic,
and hence that the line at infinity, either has no point in common
with the curve, or one (counting for two coincident points) or two
distinct points. If the line at infinity has no point on the curve the
latter is altogether finite, and is called an _Ellipse_ (fig. 21). If
the line at infinity has only one point in common with the conic, the
latter extends to infinity, and has the line at infinity a tangent. It
is called a _Parabola_ (fig. 22). If, lastly, the line at infinity
cuts the curve in two points, it consists of two separate parts which
each extend in two branches to the points at infinity where they meet.
The curve is in this case called an _Hyperbola_ (see fig. 20). The
tangents at the two points at infinity are finite because the line at
infinity is not a tangent. They are called _Asymptotes_. The branches
of the hyperbola approach these lines indefinitely as a point on the
curves moves to infinity.
[Illustration: FIG. 22.]
S 60. That the circle belongs to the curves of the second order is
seen at once if we state in a slightly different form the theorem that
in a circle all angles at the circumference standing upon the same arc
are equal. If two points S1, S2 on a circle be joined to any other two
points A and B on the circle, then the angle included by the rays S1A
and S1B is equal to that between the rays S2A and S2B, so that as A
moves along the circumference the rays S1A and S2A describe equal and
therefore projective pencils. The circle can thus be generated by two
projective pencils, and is a curve of the second order.
If we join a point in space to all points on a circle, we get a
(circular) cone of the second order (S 43). Every plane section of
this cone is a conic. This conic will be an ellipse, a parabola, or an
hyperbola, according as the line at infinity in the plane has no, one
or two points in common with the conic in which the plane at infinity
cuts the cone. It follows that our curves of second order may be
obtained as sections of a circular cone, and that they are identical
with the "Conic Sections" of the Greek mathematicians.
S 61. Any two tangents to a parabola are cut by all others in
projective rows; but the line at infinity being one of the tangents,
the points at infinity on the rows are corresponding points, and the
rows therefore similar. Hence the theorem--
_The tangents to a parabola cut each other proportionally._
POLE AND POLAR
S 62. We return once again to fig. 21, which we obtained in S 55.
If a four-side be circumscribed about and a four-point inscribed in a
conic, so that the vertices of the second are the points of contact of
the sides of the first, then the triangle formed by the diagonals of
the first is the same as that formed by the diagonal points of the
other.
Such a triangle will be called a _polar-triangle_ of the conic, so
that PQR in fig. 21 is a polar-triangle. It has the property that on
the side p opposite P meet the tangents at A and B, and also those at
C and D. From the harmonic properties of four-points and four-sides it
follows further that the points L, M, where it cuts the lines AB and
CD, are harmonic conjugates with regard to AB and CD respectively.
If the point P is given, and we draw a line through it, cutting the
conic in A and B, then the point Q harmonic conjugate to P with regard
to AB, and the point H where the tangents at A and B meet, are
determined. But they lie both on p, and therefore this line is
determined. If we now draw a second line through P, cutting the conic
in C and D, then the point M harmonic conjugate to P with regard to
CD, and the point G where the tangents at C and D meet, must also lie
on p. As the first line through P already determines p, the second may
be any line through P. Now every two lines through P determine a
four-point ABCD on the conic, and therefore a polar-triangle which has
one vertex at P and its opposite side at p. This result, together with
its reciprocal, gives the theorems--
_All polar-triangles which have one vertex in common have also the
opposite side in common._
_All polar-triangles which have one side in common have also the
opposite vertex in common._
S 63. To any point P in the plane of, but not on, a conic corresponds
thus one line p as the side opposite to P in all polar-triangles which
have one vertex at P, and reciprocally to every line p corresponds one
point P as the vertex opposite to p in all triangles which have p as
one side.
We call the line p the _polar_ of P, and the point P the _pole_ of the
line p with regard to the conic.
If a point lies on the conic, we call the tangent at that point its
polar; and reciprocally we call the point of contact the pole of
tangent.
S 64. From these definitions and former results follow--
The polar of any point P not The pole of any line p not a
on the conic is a line p, which tangent to the conic is a point
has the following properties:-- P, which has the following
properties:--
1. On every line through P 1. Of all lines through a point
which cuts the conic, the polar on p from which two tangents
of P contains the harmonic may be drawn to the conic, the
conjugate of P with regard to pole P contains the line which is
those points on the conic. harmonic conjugate to p, with
regard to the two tangents.
2. If tangents can be drawn 2. If p cuts the conic, the
from P, their points of contact tangents at the intersections
lie on p. meet at P.
3. Tangents drawn at the 3. The point of contact of
points where any line through P tangents drawn from any point
cuts the conic meet on p; and on p to the conic lie in a line
conversely, with P; and conversely,
4. If from any point on p, 4. Tangents drawn at points
tangents be drawn, their points where any line through P cuts the
of contact will lie in a line conic meet on p.
with P.
5. Any four-point on the conic 5. Any four-side circumscribed
which has one diagonal point at about a conic which has one
P has the other two lying on p. diagonal on p has the other two
meeting at P.
The truth of 2 follows from 1. If T be a point where p cuts the conic,
then one of the points where PT cuts the conic, and which are harmonic
conjugates with regard to PT, coincides with T; hence the other
does--that is, PT touches the curve at T.
That 4 is true follows thus: If we draw from a point H on the polar
one tangent a to the conic, join its point of contact A to the pole P,
determine the second point of intersection B of this line with the
conic, and draw the tangent at B, it will pass through H, and will
therefore be the second tangent which may be drawn from H to the
curve.
S 65. The second property of the polar or pole gives rise to the
theorem--
From a point in the plane of a A line in the plane of a conic
conic, two, one or no tangents has two, one or no points in
may be drawn to the conic, common with the conic, according
as its polar has two, as two, one or no tangents
one, or no points in common can be drawn from its pole to the
with the curve. conic.
Of any point in the plane of a conic we say that it was _without_, on
or _within_ the curve according as two, one or no tangents to the
curve pass through it. The points on the conic separate those within
the conic from those without. That this is true for a circle is known
from elementary geometry. That it also holds for other conics follows
from the fact that every conic may be considered as the projection of
a circle, which will be proved later on.
The fifth property of pole and polar stated in S 64 shows how to find
the polar of any point and the pole of any line by aid of the
straight-edge only. Practically it is often convenient to draw three
secants through the pole, and to determine only one of the diagonal
points for two of the four-points formed by pairs of these lines and
the conic (fig. 22).
These constructions also solve the problem--
From a point without a conic, to draw the two tangents to the conic by
aid of the straight-edge only.
For we need only draw the polar of the point in order to find the
points of contact.
S 66. The property of a polar-triangle may now be stated thus--
In a polar-triangle each side is the polar of the opposite vertex, and
each vertex is the pole of the opposite side.
[Illustration: FIG. 23.]
If P is one vertex of a polar-triangle, then the other vertices, Q and
R, lie on the polar p of P. One of these vertices we may choose
arbitrarily. For if from any point Q on the polar a secant be drawn
cutting the conic in A and D (fig. 23), and if the lines joining these
points to P cut the conic again at B and C, then the line BC will pass
through Q. Hence P and Q are two of the vertices on the polar-triangle
which is determined by the four-point ABCD. The third vertex R lies
also on the line p. It follows, therefore, also--
_If Q is a point on the polar of P, then P is a point on the polar of
Q_; and reciprocally,
_If q is a line through the pole of p, then p is a line through the
pole of q._
This is a very important theorem. It may also be stated thus--
_If a point moves along a line describing a row, its polar turns about
the pole of the line describing a pencil._
_This pencil is projective to the row, so that the cross-ratio of four
poles in a row equals the cross-ratio of its four polars, which pass
through the pole of the row._
To prove the last part, let us suppose that P, A and B in fig. 23
remain fixed, whilst Q moves along the polar p of P. This will make CD
turn about P and move R along p, whilst QD and RD describe projective
pencils about A and B. Hence Q and R describe projective rows, and
hence PR, which is the polar of Q, describes a pencil projective to
either.
S 67. Two points, of which one, and therefore each, lies on the polar
of the other, are said to be _conjugate with regard to the conic_; and
two lines, of which one, and therefore each, passes through the pole
of the other, are said to be _conjugate with regard to the conic_.
Hence all points conjugate to a point P lie on the polar of P; all
lines conjugate to a line p pass through the pole of p.
If the line joining two conjugate poles cuts the conic, then the poles
are harmonic conjugates with regard to the points of intersection;
hence one lies within the other without the conic, and all points
conjugate to a point within a conic lie without it.
Of a polar-triangle any two vertices are conjugate poles, any two
sides conjugate lines. If, therefore, one side cuts a conic, then one
of the two vertices which lie on this side is within and the other
without the conic. The vertex opposite this side lies also without,
for it is the pole of a line which cuts the curve. In this case
therefore one vertex lies within, the other two without. If, on the
other hand, we begin with a side which does not cut the conic, then
its pole lies within and the other vertices without. Hence--
Every polar-triangle has one and only one vertex within the conic.
We add, without a proof, the theorem--