The parallels, which join the corresponding angles of the two polygons, are called the principal edges of the prism: the polygons are called bases; the parallelograms, sides; and the surfaces of the parallelograms together constitute what is called the lateral or convex surface of the prism.

The altitude of a prism is the perpendicular distance between its two bases.

15. A prism is said to be triangular, or quadrilateral, or pentagonal, &c., according as its bases are triangles, or quadrilaterals, or pentagons, &c.

A prism is also said to be right or oblique, according as the principal edges are perpendicular to the bases, or inclined to them.

16. A regular prism is a right prism, which has for its bases two regular polygons; and the straight line which joins their centres is called the axis of the prism.

17. (Euc. xi. def. 12.) A pyramid is a solid figure, having any number of faces, one of which is a triangle or other rectilineal figure, and the rest triangles which have a common vertex, and for their bases the sides of the first triangle or rectilineal figure.

Such a figure may be formed by drawing straight lines from the angles of any rectilineal figure A B CDE to any point V which is not in the same plane with it.

V

E

The straight lines VA, V B, &c., which are the sides of the triangles, are called the principal edges of the pyramid: the first triangle or rectilineal figure is called the base, the other triangles the sides, and their common vertex, the vertex or summit: the surfaces of the latter triangles also constitute what is called the lateral or convex surface of the pyramid.

The altitude of a pyramid is the perpendicular distance of the vertex from the base or the base produced.

18. A pyramid is said to be triangular, or quadrilateral or pentagonal, &c. according as its base is a triangle, or a quadrilateral, or a pentagon, &c.

19. A regular pyramid is that which has for its base a regular polygon, and the straight line which is drawn from the vertex to the centre of the base perpendicular to the base; and the line so drawn is called the axis of the pyramid.

20. If a pyramid be divided into two parts by a plane parallel to its base, the part next the base is called a frustum of a pyramid, or sometimes a truncated pyramid.

21. (Euc. xi. def. 14.) A sphere is a solid figure, every point in the surface of which is at the same distance from a certain point within the figure, which is called the centre. The distance from the centre to the surface is called the radius, or sometimes the semidiameter of the sphere, because it is the half of a straight line which passes through the centre, and is terminated both ways by the surface, which straight line is called a diameter.

B

Such a figure may be conceived to be generated by the revolution of a semi- D circle ADB about its diameter A B ;that is to say, if the semicircle be made to revolve round its diameter AB, its plane will, in the course of the revolution, pass through the whole solid space about the line AB produced, and the semicircular portion ADB will pass through the whole spherical space upon the diameter A B, so that there shall not be a point within that space with which some point or other of the semicircle will not have coincided, but the same cannot be said of any point without the sphere. By the word " generate," it is intended to convey the idea that the parts of the solid start into existence as they are successively traversed by the generating plane.

PROP. 1. (EUc. xi. 2.)

A plane, and one only, may be made to pass through a given straight line and a given point without it, or through three given points which are not in the same straight line.

Let A B be a given straight line, and C a given point without it: a plane, and only one plane, may be made to pass through the straight line AB and the point C.

A

Ꭱ

For a plane may be made to pass through AB, and this plane may be turned about A B until it pass through C. Now, let any other plane be made to pass through the same straight line AB and the same point C; and let P be any point taken in it: P shall likewise be a point in the first plane. For, if in AB any two points A, B be taken, and CA, CB be joined, the straight lines CA, C B will lie in each of the planes (I. def. 7.). And because P is a point in the same plane with CA and CB, through P there may be drawn PQ parallel to A B to meet C A in some point Q and CB in some point R (I. 14. Cor. 3). Then, because the lines CA, CB are in the first plane, the points Q, R are likewise in that plane, and therefore the straight line PQR, which passes through them, and the point P of that straight line, are in the same plane. Therefore, there is no point in either of the two planes which is not also in the other plane, that is, they are one and the same plane.

Again, any plane which passes through the straight line AB and the point C without it, passes also through the three points A, B, C, which are not in the same straight line; and reversely. Therefore, since it has been shown that a plane, and one only, may be made to pass through the straight line A B and the point C, it follows that a plane, and one only, may be made to pass through the three points A, B, C.

Therefore, &c.

Cor. 1. A plane, and one only, may be made to pass through the sides of a given rectilineal angle, or through two given parallels.

Cor. 2. Any number of parallels through which the same straight line passes are in one and the same plane.

Cor. 3. It follows from the preceding corollary, that a plane may be conceived to be generated by a straight line which moves along a given straight line so as always to continue parallel to another given straight line.

Cor. 4. Any number of planes may be made to pass through the same straight line.

a perpendicular may be drawn to it in each of these planes from the point A, the case supposed in the proposition is evidently possible.

Let the straight line AP, therefore, stand at right angles to each of the straight lines A B, A C, at their point of intersection A, and let AD be any other straight line in the plane A B C, which passes through the same point A: AP shall be at right angles to A D.

In AB, AC, take any points whatever B, C ; and in BA, CA produced make Ab equal to AB, and Ac equal to AC: join B ̊C, bc, and let A D and DA produced cut B C and b c respectively in D and d: take any point P in AP, and join PB, PC, PD, Pb, Pc, Pd. Then, because in the triangles A B C, Abc, the two sides A B, A C are equal to the two A b, A c, each to each, and the included angles (I. 3.) equal to one another, the base B C is equal to the base bc, and the angle ABC to the angle Abc (I. 4.); and because in the tri

angles ABD, Abd the side AB is equal to the side A b, and the angles A BD, BAD (I.3.) equal to the angles Abd, b Ad, each to each, the side AD is equal to the side A d (I. 5.), and B D to bd. Again, the triangles PA C, PA c have the two sides PA, AC of the one equal to the two PA, Ac of the other, each to each, and the included angles right angles; therefore the base PC is equal to the base Pc (I.4.): and for the like reason, PB is equal to Pb; and it was before shown that B C is equal to bc: therefore, the triangles PBC, Pbc have the three sides of the one equal to the three sides of the other, each to each; therefore, also, the angle PBC is equal to the angle Pbc (I. 7.). And, because the triangles PBD, Pb d have two sides PB, BD of the one equal to two sides Pb, bd of the other, each to each, and the included angles PBD, P bd equal to one another, the bases PD, Pd are likewise equal (I.4.). Lastly, therefore, because the triangles PAD, PAd have the three sides of the one equal to the three sides of the other, each to each, the angles PAD, PAd are equal to one another, and (I def. 6.) PA is at right angles to A D. And because PA is at right angles to every straight line AD, which meets it in the plane BAC, it is at right angles to that plane (def. 1.)

Therefore, &c.

Cor. 1. (Euc. xi. 5.) Any number of straight lines which are drawn at right angles to the same straight line from the same point of it, lie all of them in the plane which is perpendicular to the straight line at that point.

Cor. 2. Hence, if the plane of a right angle be made to revolve about one of its legs, the other leg will describe a plane at right angles to the first leg.

and from B let BH be drawn perpendicular to FG: take any point A in AB, and join AH: A H`shall be perpendicular to F G.

In HF take any point F, make HG equal to HF, and join A F, AG, BF, BG. Then in the triangles B HF, BHG, because the two sides B H, H F are equal to the two BH, HG, each to each, and the included angles right angles, B F is equal to B G (I. 4.). Again, because AB is perpendicular to the plane CDE, the angles ABF, ABG are right angles (def. 1.): and because in the triangles A B F, AB G, the two sides AB, BF are equal to the two AB, BG, each to each, and the included angles right angles, AF is equal to AĞ (I. 4.). Therefore, lastly, because the triangles AHF, AHG have the three sides of the one equal to the three sides of the other, each to each, the angle AHF is equal to the angle AHG (I. 7.); and they are adjacent angles; therefore, each of them is a right angle (I. def. 10.), and A H is at right angles to F G.

Therefore, &c.

Cor. Hence, also, if a straight line be perpendicular to a plane, and if from any point of it a perpendicular be drawn to a straight line taken in the plane, the straight line which joins the feet of the perpendiculars shall likewise be perpendicular to the straight line taken in the plane.

to DH, because it is perpendicular to the plane EFG (def. 1.): therefore, DB, DA, and D C lie (3. Cor. 1.) in one and the same plane. But BA lies in the plane of DB and D A: therefore, B A and D C lie in the same plane. Again, because AB and C D are perpendicular each of them to the plane EFG, they are perpendicular each of them to the straight line BD (def. 1.). And because they are in the same plane and perpendicular to the same straight line, they are parallels (I. 14.).

Next, let AB, CD be parallels, meeting the plane E FG in the points B, D, respectively, and let A B be perpendicular to the plane EFG: CD shall likewise be perpendicular to it.

Join B D, and draw, as before, in the plane EFG, DH perpendicular to B D; and, taking any point A in A B, join AD. Then, as before, DA is (4.) perpendicular to DH. Therefore, D H is perpendicular to the plane of AD, DB (3.). But CD is in that plane; because it is parallel to AB, and therefore in the same plane with AB and the point D (1.Cor.1.). Therefore CD is likewise perpendicular to DH (3.). Again, because AB, CD are parallel, and that A B (being perpendicular to the plane E FG) is perpendicular to B D, (def. 1.) CD is likewise perpendicular to BD (I. 14.). Therefore, CD is perpendicular to each of the straight lines B D, DH, that is, it is perpendicular to the plane BDH (3.) or EFG.

Therefore, &c.

Cor. If from different points in the same straight line perpendiculars be drawn to the same plane, these perpendiculars shall lie in one plane, and their feet in one and the same straight line: for the perpendiculars, being parallel and passing through the same straight line, lie in one plane (1 Cor. 2.); and the common section of this plane with the first is a straight line (2.).,

PROP. 6. (EUc. xi. 9.)

Straight lines which are parallel to the same straight line, though not both of them in one plane with it, are parallel

to one another.

Let the straight lines AB, CD be each of them parallel to the straight line EF, and not in one plane with

it: A B shall be parallel to CD.

-B

In E F take any point G, and from G, in the plane of Å B, E F, draw G H at right angles to E F; and from the same point G, in the plane of C D, E F, draw G K at right angles to E F. Then, because E F is at right angles to each of the lines G H, GK, it is at right angles to the plane HGK (3.). But AB and CD are each of them parallel to EF. Therefore, A B and C D are also at right an gles to the plane H G K (5.): and because they are at right angles to the same plane, they are parallel (5.). Therefore, &c.

PROP. 7. (EUc. xi. 11, 12, and 13.)

A straight line may be drawn perpendicular to a given plane of indefinite extent from any given point, whether the given point be without or in the plane; be drawn more than one perpendicular but from the same point there cannot to the same plane.

Let A be a point without the plane BCD: a perpendicular, and one only, may be drawn from the point A to the plane B CD.

B

K

E

G

F

In the plane BCD draw any line E F, and from the point A draw AG perpendicular to EF: from G draw, in the plane BCD, GH perpendicular to E F; and from A draw AH perpendicular to GH (I. 45.). Through H draw KL parallel to EF (I. 48.), and therefore lying in the same plane with H and E F, that is, in the plane B CD. Then, because E F is at right angles to each of the straight lines AG, GH, it is at right angles to the plane AGH (3.); but KL is parallel to EF; therefore KL is at right angles to the plane AG H, and the angle A H K is a right angle (5.). And because AH is at right angles to each of the straight lines KH, HG, it is at right angles to the plane KH G_(3.), that is, to the plane BCD. But from the same point A, there cannot be drawn any other straight line which is at right angles to the plane B CD; for if we suppose AG to be any other straight line drawn from A and meeting the plane BCD in G, and if H G be joined, the angle AH G will be a right angle, and therefore (I. 8.) the angle A G H less than a right angle; so that A G cannot (def. 1.) be at right angles to the plane BCD.,

through A, AH be drawn parallel to A' H' (I. 48.), AH will likewise (5.) be perpendicular to the plane. And from the same point A, there cannot be drawn any other straight line perpendicular to the plane B CD: for, if AG be any other straight line drawn from A, and if the plane HAG cut BCD in the line KL (2.), the angle HAK will be a right angle, and therefore GAK will not be a right angle; so that AG cannot be at right angles (def. 1.) to the plane BCD. Therefore, &c.

equal to AF, and the straight line AH which is greater than AF; and join EF, EG, EH: the perpendicular A E shall be less than the straight line AF; the distance EG shall be equal to the distance E F, and the distance E H greater than the distance E F.

For, in the first place, because AE is perpendicular to the plane B CD, the angle A E F is a right angle; wherefore AFE is less than a right angle (I. 8.), and in the triangle A E F (I. 9.) the side A E is less than the side AF. Next, because the angles A E F, A E G are both of them right angles (def. 1.), AEF and AEG are right-angled triangles which have the hypotenuse AF equal to the hypotenuse A G, and the side A E com

mon to both; therefore (I. 13.) the remaining sides E F, EG are equal to one another. Lastly, because A EH is a right angle, the square of AH is equal to the squares of AE, EH (I. 36.); and for the like reason the square of AF is equal to the squares of AE, EF: but the square of AH is greater than the square of AF, because AH is greater than AF: therefore, the squares of AE, EH are together greater than the squares of A E, E F; therefore the square of EH is greater than the square of E F, and EH is greater than EF.

And hence, conversely, if the distances EF, E G be equal to one another, the line AF must be equal to the line A G; for, if not, the distances EF, E G would be unequal: and in like manner, if the distance EH be greater than the distance E F, the line AH must be greater than the line AF.

Bisect A C and DF in the points G and H respectively (I. 43.): then, if G B and H E be joined, they will be equal respectively to the halves of A C and D F (I. 19. Cor. 4.), and therefore (I. ax. 5. equal to one another. And, because in the triangles GAB, HDE, the two sides