Encyclopaedia Britannica, 11th Edition, "Geodesy" to "Geometry"

Chapter XXIX: Book XIII (1)

Text size

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.

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.

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.

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.

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.

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].]

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._

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.

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.

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.

Previous Next

Comments

Encyclopaedia Britannica, 11th Edition, "Geodesy" to "Geometry"Chapter XXIX: Book XIII (1)

0%34 min left in chapter