Chapter XXVII: Book XI

S 72. In this book figures are considered which are not confined to a
plane, viz. first relations between lines and planes in space, and
afterwards properties of solids.

Of new definitions we mention those which relate to the
perpendicularity and the inclination of lines and planes.

Def. 3. _A straight line is perpendicular, or at right angles, to a
plane when it makes right angles with every straight line meeting it
in that plane_.

The definition of perpendicular planes (Def. 4) offers no difficulty.
Euclid defines the inclination of lines to planes and of planes to
planes (Defs. 5 and 6) by aid of plane angles, included by straight
lines, with which we have been made familiar in the first books.

The other important definitions are those of parallel planes, which
never meet (Def. 8), and of solid angles formed by three or more
planes meeting in a point (Def. 9).

To these we add the definition of a line parallel to a plane as a line
which does not meet the plane.

S 73. Before we investigate the contents of Book XI., it will be well
to recapitulate shortly what we know of planes and lines from the
definitions and axioms of the first book. There a plane has been
defined as a surface which has the property that every straight line
which joins two points in it lies altogether in it. This is equivalent
to saying that a straight line which has two points in a plane has all
points in the plane. Hence, a straight line which does not lie in the
plane cannot have more than one point in common with the plane. This
is virtually the same as Euclid's Prop. 1, viz.:--

Prop. 1. _One part of a straight line cannot be in a plane and another
part without it_.

It also follows, as was pointed out in S 3, in discussing the
definitions of Book I., that a plane is determined already by one
straight line and a point without it, viz. if all lines be drawn
through the point, and cutting the line, they will form a plane.

This may be stated thus:--

_A plane is determined_--

1st, _By a straight line and a point which does not lie on it;_

2nd, _By three points which do not lie in a straight line_; for if two
of these points be joined by a straight line we have case 1;

3rd, _By two intersecting straight lines_; for the point of
intersection and two other points, one in each line, give case 2;

4th, _By two parallel lines_ (Def. 35, I.).

The third case of this theorem is Euclid's

Prop. 2. _Two straight lines which cut one another are in one plane,
and three straight lines which meet one another are in one plane_.

And the fourth is Euclid's

Prop. 7. _If two straight lines be parallel, the straight line drawn
from any point in one to any point in the other is in the same plane
with the parallels_. From the definition of a plane further follows

Prop. 3. _If two planes cut one another, their common section is a
straight line_.

S 74. Whilst these propositions are virtually contained in the
definition of a plane, the next gives us a new and fundamental
property of space, showing at the same time that it is possible to
have a straight line perpendicular to a plane, according to Def. 3. It
states--

Prop. 4. _If a straight line is perpendicular to two straight lines in
a plane which it meets, then it is perpendicular to all lines in the
plane which it meets, and hence it is perpendicular to the plane_.

Def. 3 may be stated thus: If a straight line is perpendicular to a
plane, then it is perpendicular to every line in the plane which it
meets. The converse to this would be

_All straight lines which meet a given straight line in the same
point, and are perpendicular to it, lie in a plane which is
perpendicular to that line_.

This Euclid states thus:

Prop. 5. _If three straight lines meet all at one point, and a
straight line stands at right angles to each of them at that point,
the three straight lines shall be in one and the same plane_.

S 75. There follow theorems relating to the theory of parallel lines
in space, viz.:--

Prop. 6. _Any two lines which are perpendicular to the same plane are
parallel to each other;_ and conversely

Prop. 8. _If of two parallel straight lines one is perpendicular to a
plane, the other is so also._

Prop. 7. _If two straight lines are parallel, the straight line which
joins any point in one to any point in the other is in the same plane
as the parallels._ (See above, S 73.)

Prop. 9. _Two straight lines which are each of them parallel to the
same straight line, and not in the same plane with it, are parallel to
one another;_ where the words, "and not in the same plane with it,"
may be omitted, for they exclude the case of three parallels in a
plane, which has been proved before; and

Prop. 10. _If two angles in different planes have the two limits of
the one parallel to those of the other, then the angles are equal._
That their planes are parallel is shown later on in Prop. 15.

This theorem is not necessarily true, for the angles in question may
be supplementary; but then the one angle will be equal to that which
is adjacent and supplementary to the other, and this latter angle will
also have its limits parallel to those of the first.

From this theorem it follows that if we take any two straight lines in
space which do not meet, and if we draw through any point P in space
two lines parallel to them, then the angle included by these lines
will always be the same, whatever the position of the point P may be.
This angle has in modern times been called the angle between the given
lines:--

_By the angles between two not intersecting lines we understand the
angles which two intersecting lines include that are parallel
respectively to the two given lines._

S 76. It is now possible to solve the following two problems:--

_To draw a straight line perpendicular to a given plane from a given
point which lies_

1. _Not in the plane_ (Prop. 11).

2. _In the plane_ (Prop. 12).

The second case is easily reduced to the first--viz. if by aid of the
first we have drawn any perpendicular to the plane from some point
without it, we need only draw through the given point in the plane a
line parallel to it, in order to have the required perpendicular
given. The solution of the first part is of interest in itself. It
depends upon a construction which may be expressed as a theorem.

_If from a point A without a plane a perpendicular AB be drawn to the
plane, and if from the foot B of this perpendicular another
perpendicular BC be drawn to any straight line in the plane, then the
straight line joining A to the foot C of this second perpendicular
will also be perpendicular to the line in the plane._

The theory of perpendiculars to a plane is concluded by the theorem--

Prop. 13. _Through any point in space, whether in or without a plane,
only one straight line can be drawn perpendicular to the plane._

S 77. The next four propositions treat of parallel planes. It is shown
_that planes which have a common perpendicular are parallel_ (Prop.
14); _that two planes are parallel if two intersecting straight lines
in the one are parallel respectively to two straight lines in the
other plane_ (Prop. 15); _that parallel planes are cut by any plane in
parallel straight lines_ (Prop. 16); and lastly, _that any two
straight lines are cut proportionally by a series of parallel planes_
(Prop. 17).

This theory is made more complete by adding the following theorems,
which are easy deductions from the last: _Two parallel planes have
common perpendiculars_ (converse to 14); and _Two planes which are
parallel to a third plane are parallel to each other._

It will be noted that Prop. 15 at once allows of the solution of the
problem: "Through a given point to draw a plane parallel to a given
plane." And it is also easily proved that this problem allows always
of one, and only of one, solution.

S 78. We come now to planes which are perpendicular to one another.
Two theorems relate to them.

Prop. 18. _If a straight line be at right angles to a plane, every
plane which passes through it shall be at right angles to that plane._

Prop. 19. _If two planes which cut one another be each of them
perpendicular to a third plane, their common section shall be
perpendicular to the same plane._

S 79. If three planes pass through a common point, and if they bound
each other, a solid angle of three faces, or a _trihedral_ angle, is
formed, and similarly by more planes a solid angle of more faces, or a
_polyhedral_ angle. These have many properties which are quite
analogous to those of triangles and polygons in a plane. Euclid states
some, viz.:--

Prop. 20. _If a solid angle be contained by three plane angles, any
two of them are together greater than the third._

But the next--

Prop. 21. _Every solid angle is contained by plane angles, which are
together less than four right angles_--has no analogous theorem in the
plane.

We may mention, however, that the theorems about triangles contained
in the propositions of Book I., which do not depend upon the theory of
parallels (that is all up to Prop. 27), have their corresponding
theorems about trihedral angles. The latter are formed, if for "side
of a triangle" we write "plane angle" or "face" of trihedral angle,
and for "angle of triangle" we substitute "angle between two faces"
where the planes containing the solid angle are called its _faces_. We
get, for instance, from I. 4, the theorem, _If two trihedral angles
have the angles of two faces in the one equal to the angles of two
faces in the other, and have likewise the angles included by these
faces equal, then the angles in the remaining faces are equal, and the
angles between the other faces are equal each to each, viz. those
which are opposite equal faces._ The solid angles themselves are not
necessarily equal, for they may be only symmetrical like the right
hand and the left.

The connexion indicated between triangles and trihedral angles will
also be recognized in

Prop. 22. _If every two of three plane angles be greater than the
third, and if the straight lines which contain them be all equal, a
triangle may be made of the straight lines that join the extremities
of those equal straight lines._

And Prop. 23 solves the problem, _To construct a trihedral angle
having the angles of its faces equal to three given plane angles, any
two of them being greater than the third._ It is, of course, analogous
to the problem of constructing a triangle having its sides of given
length.

Two other theorems of this kind are added by Simson in his edition of
Euclid's _Elements_.

S 80. These are the principal properties of lines and planes in space,
but before we go on to their applications it will be well to define
the word _distance_. In geometry distance means always "shortest
distance"; viz. the distance of a point from a straight line, or from
a plane, is the length of the perpendicular from the point to the line
or plane. The distance between two non-intersecting lines is the
length of their common perpendicular, there being but one. The
distance between two parallel lines or between two parallel planes is
the length of the common perpendicular between the lines or the
planes.

S 81. _Parallelepipeds_.--The rest of the book is devoted to the study
of the parallelepiped. In Prop. 24 the possibility of such a solid is
proved, viz.:--

Prop. 24. _If a solid be contained by six planes two and two of which
are parallel, the opposite planes are similar and equal
parallelograms._

Euclid calls this solid henceforth a parallelepiped, though he never
defines the word. Either face of it may be taken as _base_, and its
distance from the opposite face as _altitude_.

Prop. 25. _If a solid parallelepiped be cut by a plane parallel to two
of its opposite planes, it divides the whole into two solids, the base
of one of which shall be to the base of the other as the one solid is
to the other_.

This theorem corresponds to the theorem (VI. 1) that parallelograms
between the same parallels are to one another as their bases. A
similar analogy is to be observed among a number of the remaining
propositions.

S 82. After solving a few problems we come to

Prop. 28. _If a solid parallelepiped be cut by a plane passing through
the diagonals of two of the opposite planes, it shall be cut in two
equal parts._

In the proof of this, as of several other propositions, Euclid
neglects the difference between solids which are symmetrical like the
right hand and the left.

Prop. 31. _Solid parallelepipeds, which are upon equal bases, and of
the same altitude, are equal to one another._

Props. 29 and 30 contain special cases of this theorem leading up to
the proof of the general theorem.

As consequences of this fundamental theorem we get

Prop. 32. _Solid parallelepipeds, which have the same altitude, are to
one another as their bases;_ and

Prop. 33. _Similar solid parallelepipeds are to one another in the
triplicate ratio of their homologous sides._

If we consider, as in S 67, the ratios of lines as numbers, we may
also say--

_The ratio of the volumes of similar parallelepipeds is equal to the
ratio of the third powers of homologous sides._

Parallelepipeds which are not similar but equal are compared by aid of
the theorem

Prop. 34. _The bases and altitudes of equal solid parallelepipeds are
reciprocally proportional; and if the bases and altitudes be
reciprocally proportional, the solid parallelepipeds are equal._

S 83. Of the following propositions the 37th and 40th are of special
interest.

Prop. 37. _If four straight lines be proportionals, the similar solid
parallelepipeds, similarly described from them, shall also be
proportionals; and if the similar parallelepipeds similarly described
from four straight lines be proportionals, the straight lines shall be
proportionals._

In symbols it says--

If a : b = c : d, then a^3 : b^3 = c^3 : d^3.

Prop. 40 teaches how to compare the volumes of triangular prisms with
those of parallelepipeds, by proving _that a triangular prism is equal
in volume to a parallelepiped, which has its altitude and its base
equal to the altitude and the base of the triangular prism._

S 84. From these propositions follow all results relating to the
mensuration of volumes. We shall state these as we did in the case of
areas. The starting-point is the "rectangular" parallelepiped, which
has every edge perpendicular to the planes it meets, and which takes
the place of the rectangle in the plane. If this has all its edges
equal we obtain the "cube."

If we take a certain line u as unit length, then the square on u is
the unit of area, and the cube on u the unit of volume, that is to
say, if we wish to measure a volume we have to determine how many unit
cubes it contains.

A rectangular parallelepiped has, as a rule, the three edges unequal,
which meet at a point. Every other edge is equal to one of them. If a,
b, c be the three edges meeting at a point, then we may take the
rectangle contained by two of them, say by b and c, as base and the
third as altitude. Let V be its volume, V' that of another rectangular
parallelepiped which has the edges a', b, c, hence the same base as
the first. It follows then easily, from Prop. 25 or 32, that V : V' =
a : a'; or in words,

_Rectangular parallelepipeds on equal bases are proportional to their
altitudes._

If we have two rectangular parallelepipeds, of which the first has the
volume V and the edges a, b, c, and the second, the volume V' and the
edges a', b', c', we may compare them by aid of two new ones which
have respectively the edges a', b, c and a', b', c, and the volumes V1
and V2. We then have

V : V1 = a : a'; V1 : V2 = b : b', V2 : V' = c : c'.

Compounding these, we have

V : V' = (a : a')(b : b')(c : c'),

or

V a b c
-- = -- . -- . --.
V' a' b' c'

Hence, as a special case, making V' equal to the unit cube U on u we
get

V a b c
-- = -- . -- . -- = [alpha].[beta].[gamma],
U u u u

where [alpha], [beta], [gamma] are the numerical values of a, b, c;
that is, _The number of unit cubes in a rectangular parallelepiped_ is
equal to the product of the numerical values of its three edges. This
is generally expressed by saying the volume of a rectangular
parallelepiped is measured by the product of its sides, or by the
product of its base into its altitude, which in this case is the same.

Prop. 31 allows us to extend this to any parallelepipeds, and Props.
28 or 40, to triangular prisms.

_The volume of any parallelepiped, or of any triangular prism, is
measured by the product of base and altitude._

The consideration that any polygonal prism may be divided into a
number of triangular prisms, which have the same altitude and the sum
of their bases equal to the base of the polygonal prism, shows further
that the same holds for any prism whatever.