This is, therefore, only a special case of the last, and is, besides,
an old acquaintance, being essentially the same problem as that
proposed in II. 11.
Prop. 30 may therefore be solved in two ways, either by aid of Prop. 29 or by aid of II. 11. Euclid gives both solutions.
§ 71. Prop. 31 (Theorem). In any right-angled triangle, any rectilineal figure described on the side subtending the right angle is equal to the similar and similarly-described figures on the sides containing the right angle,—is a pretty generalization of the theorem of Pythagoras (I. 47).
Leaving out the next proposition, which is of little interest, we come to the last in this book.
Prop. 33. In equal circles angles, whether at the centres or the circumferences, have the same ratio which the arcs on which they stand have to one another; so also have the sectors.
Of this, the part relating to angles at the centre is of special importance; it enables us to measure angles by arcs.
With this closes that part of the Elements which is devoted to the study of figures in a plane.
Book XI.
§ 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.
§ 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 § 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.
§ 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.
§ 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, § 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.
§ 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.
§ 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.
§ 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.
§ 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
