AG, GB of the one are equal to the two DH, HE of the other, each to each, but the base AB greater than the base DE, the angle AGB is (I. 11.) greater than the angle D H E. But the angle AGB is double of AC B, for A G B is equal to the sum of GCB and GBC (I. 19.), which are equal to one another, because GB is equal to G C (I. 6.); and for the like reason, the angle D H E is double of DFE. Therefore (I. ax. 8.), the angle ACB is greater than the angle D FE. And hence, conversely, if A CB be greater than DFE, AB must also be greater than DE: for it cannot be equal to DE (I. 13.); neither, by what has been just demonstrated, can it be less than D E. Therefore, &c.

PROP. 9.

If a straight line be inclined to a plane, of all the angles which it makes with straight lines meeting it in that plane, the least shall be the angle of inclination; and, with respect to every other of these angles, a second angle may always be drawn which shall be equal to it, viz. upon the other side of the angle of inclination; but there cannot be drawn in the plane more than two straight lines with which the inclined straight line shall make equal angles, one upon each side of the angle of inclination.

Let the straight line A B be inclined to the plane CDE at the point B: from any point A in AB, draw A F perpendicular to the plane CDE (7.), and join B F, so that A B F may be the angle of inclination (def. 2.); and let BG be any other straight line in the plane CDĚ passing through the point B: the angle A B F shall be less than the angle A B G.

B

From the point F draw FG perpendicular to BG, and join A G. Then, because AF is perpendicular to the plane in which BG lies, and that FG is drawn from the point F perpendicular to BG, AG is likewise perpendicular to BG (4.). And, because in the rightangled triangles AFB, AGB, the hypotenuse A B is common to both, but the side A F of the first less than the side

AG (8.) of the other, the opposite angle ABF is likewise less than the angle ABG, by the foregoing Lemma.

Also, from the point B in the same plane CDE, there may be drawn a straight line BH, making an angle with A B equal to the angle AB G, viz. upon the other side of the angle AB F. For, if the angle FB H be made equal to the angle FB G, then, drawing FH perpendicular to BH, and joining AĤ, A ́H will likewise be perpendicular to BH (4.); and, because in the right-angled triangles FGB, FH B, the hypotenuse FB is common to both, and the angle FBG equal to the angle FB H, the opposite side FG is equal to FH (I. 13.): therefore, also, (8.) AG is equal to A I: and, because in the right-angled triangles AGB, AH B the hypotenuse AB is common to both, and the side A G equal to the side AH, the angle A B G is likewise equal to the angle ABH (I. 13.).

But, lastly, there cannot be drawn in the plane CDE more than two straight lines BG, B H, with which the straight line AB shall make equal angles. For, if BK be any other straight line, then, drawing FK perpendicular to B K, and joining AK, AK will likewise be perpendicular to FK (4.): and, because in the rightangled triangles FHB, FKB, the hypotenuse FB is common to both, but the angle F B H not equal to the angle FBK, the side FH is not equal to the side FK (Lemma): therefore (8.) A H is not equal to A K; and, because in the right-angled triangles AH B, AK B, the hypotenuse A B is common to both, but the side A H not equal to the side AK, the angle ABH is not equal to the angle ABK (Lemma).

Therefore, &c.

Cor. If three straight lines lie in the same plane and meet in the same point, and if a fourth straight line stand at equal angles to the three at that point; the equal angles shall be right angles, and the fourth straight line shall be

at right angles to the plane of the three.

PROP. 10.

If one straight line be parallel to another, it shall likewise be parallel to any plane which passes through that

Let the straight line AB be parallel to CD, and let EFG be any plane passing through CD: the line AB shall be parallel to the plane E F G.

C

D

For, since A B is in the plane of AB, CD (I. def. 12.), if it meet the plane EFG at all, it must meet it in the plane of AB, CD, and therefore in some point of the line CD which is the common section of the two planes. But AB cannot meet CD, being parallel to it. Therefore neither can it meet the plane EFG, that is, it is parallel to the plane EFG (def. 3.).

Next, let AB be parallel to the plane EFG, and let CD be the line in which any plane passing through A B cuts the plane EFG: A B shall be parallel to CD. For if not, it must meet it in some point. But in the same point it would meet the plane EFG, to which it is parallel: which is impossible. Therefore A B is parallel to CD.

Cor. 2. If two straight lines, which cut one another, be parallel, each of them, to the same plane; the plane of the two straight lines shall be parallel to that plane. For should the planes meet, their common section would, by the proposition, be parallel to each of the cutting straight lines; which is impossible. (I. 14. Cor. 2.)

It is possible that a plane may pass through the second straight line, and also through the first, which

is supposed to be parallel to it: in this case, it is evident that the latter is not, as is predicated in the proposition, parallel to such plane, but lies altogether in it. The enunciation must, therefore, be understood with the exception of this particular case.

SECTION 2.-Of Planes which are parallel, or inclined, or perpendicular to other Planes.

PROP. 11. (EUc. xi. 14.)

Planes, to which the same straight line is perpendicular, are parallel: and, conversely, if two planes be parallel, and if one of them be perpendicular to a straight line, the other shall be perpendicular to the same straight line.

Let the straight line A B be perpendicular to each of the planes CDE, FGH: the plane CDE shall be parallel to the plane F G H.

For, if they are not parallel, let them meet one another, and let K be any point of the Join KA, KB. Then, because A B is percommon section. pendicular to the plane C D E, the

angle K AB is a right angle (def. 1.); and, because the same AB is perpendicular to the plane FG H, the angle KBA is a right angle: therefore, two angles of the triangle KAB are together equal to two right angles; which (I. 8.) is impossible. Therefore, the planes do not meet one another, to whatever extent they may be produced, that is, (def. 6.) they are parallel to one another.

Next, let the plane C D E be parallel to the plane FGH, and from any point A. of the first let A B be drawn perpendicular to the other plane F GH: AB shall likewise be perpendicular to CDE. For, if not, (def. 1.) there must be some line in the plane CDE which meets the line A B, and does not make a right angle with it: let KA be such a line, and let the plane KAB cut the plane FGH in the straight line B L (2.). Then, because AB is at right angles to the plane FG H, the angle A B L is a right angle: but BA K is not a right angle: therefore the straight lines AK and BL will meet, if produced, in some point (I. 15. Cor. 4.) which will be common to both the planes; and, because the planes meet one another in this point, they cannot be parallel, which is contrary to the supposition. Therefore the straight line A B makes a right angle with every

straight line meeting it in the plane CDE, that is, it is (def. 1.) at right angles to the plane C D E. Therefore, &c.

Cor. 1. Through any given point a plane may be drawn, and one only, which shall be parallel to a given plane. For a perpendicular A B may be drawn from the given point A to the given plane F G H (7.); and from the same point A there may be drawn in two different planes straight lines at right angles to this perpendicular; the plane of which straight lines (3.), and evidently none other which passes through the given point, is perpendicular to the straight line A B, and therefore parallel to the given plane FG H.

Cor. 2. Planes, which are parallel to the same plane, are parallel to one another.

PROP. 12. (Euc. xi. 16.)

If parallel planes be cut by the same plane, their common sections with it shall be parallels.

For, these common sections lying one of them in one of the planes, and the other in the other, cannot meet one another, unless the planes meet one another; which they do not, because they are parallel: also, the common sections lie

in the same (viz the cutting) plane: therefore they are parallels (I. def. 12.). Therefore, &c.

Cor. If two planes which cut one another be parallel to other two which

PROP. 13.

If two parallel straight lines be cut by two parallel planes, the parts of the straight lines which are intercepted between the planes shall be equal to one another.

For, if the plane of the parallels be drawn to cut the two parallel planes, the common sections will be parallel (12.), and will therefore include, together with the parts in question, a parallelogram, of which the parts in question are opposite sides, and therefore are equal to one another (I. 22.). Therefore, &c.

Cor. 1. Parallel planes are every where equidistant (5.).

Cor. 2. If, from any number of points in the same plane, there be drawn without the plane as many equal and parallel straight lines, the other extremities of these straight lines shall lie in a second plane parallel to the first.

respectively: AE shall be to EB as CF to FD.

Join AD, and let it cut the plane KL in O: and join O E, OF, AC, BD. Then, because the parallel planes K L, M N are cut by the plane of the triangle ABD, the common sections E O, B D are parallel (12.): therefore (II. 29.) AE is to EB as AO to OD. Again, because the parallel planes GH, K L are cut by the plane of the triangle DAC, the common sections AC, OF are parallel; and therefore (II. 29.) C F is to FD as AO to O D. Therefore (I. 12.) AE is to E B as CFFD.

Therefore, &c.

Cor. If straight lines be drawn to a plane from any point without it, and if each of these straight lines, or each of them produced, be divided in the same ratio towards the same parts; the points of division shall all lie in a second plane parallel to the first.

PROP. 15. (EUc. xi. 10 and 15.) | If two straight lines, which meet one another, be parallel respectively to two other straight lines, which meet one another, but are not in the same plane with the first two; the contained angles shall be equal, and their planes parallel.

In the straight lines AB, AC take any points B, C: make D E equal to AB and DF equal to AC, and join BC, EF, AD, BE, CF. Then, because the straight lines AB, DE are equal and parallel, BE is equal to AD and parallel to it (I. 21.): and for the like reason CF is equal to AD and parallel to it: therefore also C F, BE are equal and parallel (I. ax. i. and 6), and consequently (I. 21.) BC is equal and parallel to E F. Therefore, because the triangles A B C, DEF have the three sides of the one equal

It will be observed that this proposition is an extension of I. 18. to the case in which the sides of the one angle are not in the same plane with the sides of the other. In order to exclude the case in which the angles would be supplementary, not equal, to one another, it was required, in the enunciation of I. 18., that the parallel sides should be "in the same order" or direction from one another; and a similar limitation for the same purpose is obviously necessary in the proposition before us. In the former case, a very simple criterion is afforded by the position of the angles relatively to the joining line BE (see the figure of I. 18.); for, in order that the angles may be equal, the sides which are parallel ought to be, each pair, upon the same side, or each pair upon opposite sides, of the joining line BE; for, if one pair lie towards the same parts, and the other towards opposite parts, the angles will be, not equal, but supplementary. And the present case admits of a criterion equally simple; for, in the case of equality, the parallel sides lie, each

pair, upon the same side, or each pair upon opposite sides, of any plane which passes through BE; whereas, when the angles are supplementary to one another, one pair of parallel sides lie towards the same parts of any such plane, and the other pair towards opposite parts.

BC, and let the two E FG, HFG cut one another in the straight line FG; also, let the angle ABD which is contained by perpendiculars to BC drawn in the two first planes from the point B, be equal to the angle E FH which is contained by perpendiculars to FG drawn from the point F in the two others: the dihedral angle AB CD shall be equal to the dihedral angle E F G H.

For, if the point B be made to coincide with the point F, the straight line BC with FG, and the plane ABC with the plane E FG, the straight line AB will coincide with EF, because the angles A B C, EFG are right angles (I. 1.). Also, because BD is perpendicular to B C, it will be (3 Cor. 1) in the plane EFH which (3.) is at right angles to F G, and therefore BD will coincide with FH, because the angle ABD is equal to the angle EFH. Therefore the plane

as in the last proposition: the dihedral angles shall be to one another as the angles A B D, E FH.

For, if the angle EFH be divided into any number of equal angles by the straight lines Fe, &c., which straight lines, being in the plane of the angle EFH, are all of them perpendicular to the common section FG (3.), the dihedral angle EFGH will be divided into the same number of equal dihedral angles by the planes G Fe, &c. (16.). And, if the angle E Fe be contained in A B D any number of times with a remainder less than E Fe, the dihedral angle EFGe will be contained in A B C D the same number of times with a remainder less than EF Ge(16.). And this will be the case, how great soever be the number of parts into which the angle E FH is divided. Therefore (II. def. 7.), the dihedral angle ABCD is to the dihedral angle E F G H as the angle ABD to the angle EF H.

Therefore, &c.

Cor. If one plane be at right angles to another, the perpendiculars to the common section, which are drawn in the two planes from the same point of the common section, will be at right angles to one another; and conversely.

Scholium.

Hence a dihedral angle is said to be measured by the plane angle of the per

pendiculars. And, in thus measuring the dihedral angle, it is indifferent from what point of the common section the perpendiculars are drawn; for, if any two points be taken, the planes of the two pairs of perpendiculars drawn from these points will be (3.) perpendicular to the common section, and therefore parallel (11.); and it has been seen (15.Cor.) that, if a dihedral angle be cut by parallel planes, the intercepted angles will be equal to one another.

It has been already stated (17.Cor.) that a dihedral right angle is measured by a plane right angle: it follows that a dihedral obtuse angle is measured by a plane obtuse angle, and a dihedral acute angle by a plane acute angle.

Many propositions with regard to cutting planes are readily derived from this mode of measuring dihedral angles. Of these the following only need here be mentioned:

1. If two planes cut one another, the vertical dihedral angles will be equal.

2. If a plane fall upon two parallel planes, it shall make the alternate dihedral angles equal to one another, the exterior dihedral angle equal to the interior and opposite upon the same side, and the two interior dihedral angles upon the same side together equal to two right angles.

3. Of two dihedral angles, if the planes of the one be parallel to the planes of the other, or perpendicular to the planes of the other respectively, the two dihedral angles shall be equal.*

4. If two planes cut one another, and if perpendiculars be drawn to them from any the same point, the adjacent angles contained by the perpendiculars shall measure respectively the adjacent dihedral angles contained by the two planes.

PROP. 18. (EUc. xi. 18.)

If one plane be perpendicular to another, any straight line which is drawn in the first plane at right angles to their common section shall be perpendicular to the other plane: and, conversely, if a straight line be perpendicular to a plane, any plane which passes through it shall be perpendicular to the same plane.

It being provided also, that the parallel planes lie, each pair, upon the same side, or each pair upon opposite sides of the plane which passes through the common sections (A B, C D in the figure of 12. Cor.); for, if one pair lie towards the same parts, and the other pair towards opposite parts of that plane, the dihedral angles will be supplementary, not equal, to one another. See the note at Prop. 15.