The First Six Books of the Elements of Euclid, with a Commentary and Geometrical Exercises

A C, must be perpendicular to the plane P A C (17, 18). Since then A D is perpendicular to the plane PA C, and AP is perpendicular to the plane E F, the inclination of those planes is that of the lines AD and A P. But PAD is a right angle, and therefore the planes are perpendicular, and the same may be proved of any plane drawn through PA.

PROPOSITION XXVI.

(51) If a plane A PC be perpendicular to another E F, and if the line P A be drawn in the plane APC perpendicular to the line of intersection BC, then PA will be perpendicular to the plane E F.

For draw A D in the plane E F and perpendicular to B C. The angle PAD is the inclination of the two planes (48), and is therefore a right angle. But PAC is a right angle by hyp. Hence the line P A being perpendicular to two lines in the plane EF is perpendicular to the plane E F (17, 18).

(52) COR.-It is evident that if from the intersection of two perpendicular planes a right line be drawn perpendicular to either, it will be entirely in the other.

PROPOSITION XXVII.

(53) If two planes be perpendicular to a third plane, their common intersection will be perpendicular to the third plane.

For if, from the point where their common intersection meets the third plane, a perpendicular to the third plane be drawn, that perpendicular must be in each of the two planes (51), and must therefore be their intersection.

PROPOSITION XXVIII.

(54) Right lines intersecting parallel planes are divided proportionally.

Let the parallel planes be E F, G H, and I K, and let the right lines which intersect them be A B and C D. Draw A D and

join the several points where the right lines meet the planes by the lines A C, LMN, 5 and B D.

cannot meet.

In the triangle B A D since L M and BD are at the same time in parallel planes, and in the same plane, they must be parallel, for they In the same way AC and M N are parallel. Hence by the two triangles we have AL: LB:: AM: MD, CN: ND::AM: MD, AL: LB::CN: ND.

therefore

PROPOSITION XXIX.

(55) If two right lines be not in the same plane, planes may be drawn through them which are parallel, and only two such planes can be drawn.

Let the lines be A B and C D. Through any point A of the line A B draw A E parallel to C D, and through any point C of the line CD draw CF parallel to AB. The planes of the angles BAE and FCD are parallel (44). It is evident that no other parallel planes can be drawn through A B and C D.

PROPOSITION XXX.

(56) If two right lines be not in the same plane, a third right line may be drawn intersecting

them at right angles, and only one such line can be drawn.

Let parallel planes be drawn through the given right lines, and also let planes be drawn through each of them at right angles to those parallel planes. The intersection of these two planes will evidently intersect the given right lines at right angles, and will be the only line which can be so drawn.

PROPOSITION XXXI.

(57) The right line which intersects perpendicularly two right lines not in the same plane, is the shortest line which can be drawn between those two right lines.

For if it is perpendicular to the parallel planes which may be drawn through them, any other line drawn between them would be oblique to these planes and therefore longer. (58) DEF.-A solid angle is formed by three or more planes which meet at the same point. Thus the planes B A C, B A D, and D A C form a solid angle.

(59) DEF.-The point A where the planes meet is called the vertex of the solid angle.

It is evident that the sides or faces of a solid angle are plane angles, and that less than three plane angles cannot form a solid angle. (60) DEF. The edges of a solid angle are the lines (A B, A C, A D) in which the plane angles intersect.

If with the point A as centre, and any distance Ab as radius, a circular arc b d be described in the plane of the angle B AD, and another b c in the plane of the angle BA C, and a third dc in the plane of the angle DAC, a triangle bd c will be formed by the three arcs, called a spherical triangle. The sides of this triangle are the measures of the plane angles which form the solid angle A, and its angles are the inclinations of the planes of these angles. The properties of solid angles are thus identified with those of spherical triangles, and they form the subject of spherical geometry and trigonometry. On this subject the student is referred to the second part of my treatise on trigonometry. If the solid angle be formed by more than three plane angles, it corresponds to a spherical figure with more than three sides. In spherical geometry the only property of a solid angle which has been borrowed from solid geometry is that which is established in the following proposition.

PROPOSITION XXXII.

(61) If a solid angle be formed by three plane angles, any two of these taken together must be greater than the third.

It is only necessary to prove that the greatest of the three plane angles is less than the sum of the other two. Let BAC be the greatest, and draw A E so that the angle CA E shall be equal to the angle C A D. On the line A D take AD equal to A E, and draw BD and D C.

In the triangles CAD and CAE the sides AD and A E are equal, A C is common, and the angles

CAD and CAE are equal, therefore the bases CE and CD

are equal. Hence BE is the difference of the sides B C and CD of the triangle B C D, and is therefore less than the base BD. In the triangles D A B and E A B the sides D A and E A are equal, and BĂ is common, and the base BD has been proved greater than BE, and therefore the angle B A D is greater than the angle BAE. To these let the equal angles CAD and CAE be added, and it follows that the sum of the angles BAD and CAD is greater than the angle B A C.

(62) All the other properties of solid angles may be deduced from the results of spherical geometry and trigonometry. Thus we find

that in a solid angle formed by several plane angles, any one of the plane angles is less than the sum of all the others. That'the sum of the plane angles which form any solid angle must be less than four right angles.' TRIG. (130).

That if a solid angle be formed by three plane angles the sum of the inclinations of the planes cannot be less than two right angles nor greater than six,' but may have any intermediate magnitude. TRIG. (140).

These and other properties too numerous to insert here will be found in the work already cited. It may, however, be worth mentioning, that of the six quantities related to a solid angle contained by three planes, viz. the three plane angles and the three inclinations of the planes, any three being given the other three can always be determined. TRIG. Part II. Sect. VII.

A C, must be perpendicular to the plane P A C (17, 18). Since then A D is perpendicular to the plane PA C, and AP is perpendicular to the plane E F, the inclination of those planes is that of the lines A D and A P. But PAD is a right angle, and therefore the planes are perpendicular, and the same may be proved of any plane drawn through PA.