But if they are not, let BAC be that angle which is not less than either of the other two, and is greater than one of them DAB; and at the point A in the straight line AB, in the plane which passes through BA, AC, make the angle BAE equal to the angle DAB; (1. 23.) and make AE equal to AD, and through E draw BEC cutting AB, AC in the points B, C, and join DB, DC. And because DA is equal to AE, and BA is common, the two DA, AB are equal to the two EA, AB each to each; and the angle DAB is equal to the angle EAB: therefore the base DB is equal to the base BE: (I. 4.) and because BD, DC are greater than CB, (1. 20.) and one of them BD has been proved equal to BE a part of CB, therefore the other DC'is greater than the remaining part EC: (1. ax. 5.) and because DA is equal to AE, and AC common, but the base DC greater than the base EC; therefore the angle DAC is greater than the angle EAC; (1.25.) and, by the construction, the angle DAB is equal to the angle BAE; wherefore the angles DAB, DAC are together greater than BAE, EAC, that is, than the angle BAC: (1. ax. 4.) but BAC is not less than either of the angles DAB, DAC: therefore BAC, with either of them, is greater than the other. Wherefore, if a solid angle, &c. Q.E.D. PROPOSITION XXI. THEOREM. Every solid angle is contained by plane angles, which together are less than four right angles. First, let the solid angle at A be contained by three plane angles BAC, CAD, DAB. These three together shall be less than four right angles. D B Take in each of the straight lines AB, AC, AD, any points B, C, D, and join BC, CD, DB. Then, because the solid angle at B is contained by the three plane angles CBA, ABD, DBC, any two of them are greater than the third; (xI. 20.) therefore the angles CBA, ABD are greater than the angle DBC: for the same reason, the angles BCA, ACD are greater than the angle DCB; and the angles CDA, ADB, greater than BDC: wherefore the six angles CBA, ABD, BCA, ACD, CDA, ADB, are greater than the three angles DBC, BCD, CDB: but the three angles DBC, BCD, CDB are equal to two right angles; (1. 32.) therefore the six angles CBA, ABD, BCA, ACD, CDA, ADB are greater than two right angles: and because the three angles of each of the triangles ABC, ACD, ADB are equal to two right angles, therefore the nine angles of these three triangles, viz. the angles CBA, BAC, ACB, ACD, CDA, DAC, ADB, DBA, BAD are equal to six right angles; of these the six angles CBA, ACB, ACD, CDA, ADB, DBA are greater than two right angles; therefore the remaining three angles BAC, CAD, DAB, which contain the solid angle at A, are less than four right angles. Next, let the solid angle at A be contained by any number of plane angles BAC, CAD, DAE, EAF, FAB. These shall together be less than four right angles. A B E D Let the planes in which the angles are, be cut by a plane, and let the common sections of it with those planes be BC, CD, DE, EF, FB. And because the solid angle at B is contained by three plane angles CBA, ABF, FBC, of which any two are greater than the third, (XI. 20.) the angles CBA, ABF, are greater than the angle FBC: for the same reason, the two plane angles at each of the points C, D, E, F, viz. those angles which are at the bases of the triangles, having the common vertex A are greater than the third angle at the same point, which is one of the angles of the polygon BCDEF: therefore all the angles at the bases of the triangles are together and because all the angles of the triangles are together equal to and that all the angles of the polygon, together with four right angles, are likewise equal to twice as many right angles as there are sides in the polygon: (1. 32. Cor. 1.) therefore all the angles of the triangles are equal to all the angles wherefore the remaining angles of the triangles, viz. those of the vertex, which contain the solid angle at A, are less than four right angles. Therefore, every solid angle, &c. Q.E.D. NOTES TO BOOK XI. THE solids considered in the eleventh and twelfth books are Geometrical solids, portions of space bounded by surfaces which are supposed capable of penetrating and intersecting one another. In the first six books, all the diagrams employed in the demonstrations, are supposed to be in the same plane which may lie in any position whatever, and be extended in every direction, and there is no difficulty in representing them roughly on any plane surface; this, however, is not the case with the diagrams employed in the demonstrations in the eleventh and twelfth books, which cannot be so intelligibly represented on a plane surface on account of the perspective. A more exact conception may be attained, by adjusting pieces of paper to represent the different planes, and drawing lines upon them as the constructions may require, and by fixing pins to represent the lines which are perpendicular to, or inclined to any planes. Any plane may be conceived to move round any fixed point in that plane, either in its own plane, or in any direction whatever; and if there be two fixed points in the plane, the plane cannot move in its own plane, but may move round the straight line which passes through the two fixed points in the plane, and may assume every possible position of the planes which pass through that line, and every different position of the plane will represent a different plane; thus, an indefinite number of planes may be conceived to pass through a straight line which will be the common intersection of all the planes. Hence, it is manifest, that though two points fix the position of a straight line in a plane, neither do two points nor a straight line fix the position of a plane in space. If, however, three points, not in the same straight line, be conceived to be fixed in the plane, it will be manifest, that the plane cannot be moved round, either in its own plane or in any other direction, and therefore is fixed. Also, any conditions which involve the consideration of three fixed points not in the same straight line, will fix the position of a plane in space; as also two straight lines which meet or intersect one another, or two parallel straight lines in the plane. Def. v. When a straight line meets a plane, it is inclined at different angles to the different lines in that plane which may meet it; and it is manifest that the inclination of the line to the plane is not determined by its meeting any line in that plane. The inclination of the line to the plane can only be determined by its inclination to some fixed line in the plane. If a point be taken in the line different from that point where the line meets the plane, and a perpendicular be drawn to meet the plane in another point; then these two points in the plane will fix the position of the line which passes through them in that plane, and the angle contained by this line and the given line, will measure the inclination of the line to the plane; and it will be found to be the least angle which can be formed with the given line and any other straight line in the plane. If two perpendiculars be drawn upon a plane from the extremities of a straight line which is inclined to that plane, the straight line in the plane intercepted between the perpendiculars is called the projection of the line on that plane; and it is obvious that the inclination of a straight line to a plane is equal to the inclination of the straight line to its projection on the plane. If, however, the line be parallel to the plane, the projection of the line is of the same length as the line itself; in all other cases the projection of the line is less than the line, being the base of a right-angled triangle, the hypotenuse of which is the line itself. The inclination of two lines to each other, which do not meet, is measured by the angle contained by two lines drawn through the same point and parallel to the two given lines. Def. vi. Planes are distinguished from one another by their inclinations, and the inclinations of two planes to one another will be found to be measured by the acute angle formed by two straight lines drawn in the planes, and perpendicular to the straight line which is the common intersection of the two planes. It is also obvious that the inclination of one plane to another will be measured by the angle contained between two straight lines drawn from the same point, and perpendicular, one on each of the two planes. The intersection of two planes suggests a new conception of the straight line. Def. ix. Στερεὰ γωνία ἐστὶν ἡ ὑπὸ πλειόνων ἢ δύο γωνιών ἐπιπέδων περιεχομένη, μὴ οὐσῶν ἐν τῷ αὐτῷ ἐπιπέδῳ πρὸς ἑνὶ σημείῳ συνισταμένων. The rendering of this definition by Simson may be slightly amended. The word TEPLEXoμévn is rather comprehended or contained than made: and ovvioraμévwv means joined and fitted together, not meeting. "A solid angle is that which is contained by more than two plane angles joined together at one point, (but) which are not in the same plane.' When a solid angle is contained by three plane angles, each plane which contains one plane angle, is fixed by the position of the other two, and consequently, only one solid angle can be formed by three plane angles. But when a solid angle is formed by more than three plane angles, if one of the planes be considered fixed in position, there are no conditions which fix the position of the rest of the planes which contain the solid angle, and hence, an indefinite number of solid angles, unequal to one another, may be formed by the same plane angles, when the number of plane angles is more than three. Def. A. Parallelopipeds are solid figures in some respects analogous to parallelograms, and remarks might be made on parallelopipeds similar to those which were made on rectangular parallelograms in the notes to Book II., p. 99; and every right-angled parallelopiped may be said to be contained by any three of the straight lines which contain the three right angles by which any one of the solid angles of the figure is formed; or more briefly, by the three adjacent edges of the parallelopiped. As all lines are measured by lines, and all surfaces by surfaces, so all solids are measured by solids. The cube is the figure assumed as the measure of solids or volumes, and the unit of volume is that cube, the edge of which is one unit in length. If the edges of a rectangular parallelopiped can be divided into units of the same length, a numerical expression for the number of cubic units in the parallelopiped may be found, by a process similar to that by which a numerical expression for the area of a rectangle was found. Let AB, AC, AD be the adjacent edges of a rectangular parallelopiped AG, and let AB contain 5 units, AC, 4 units, and AD, 3 units in length. Then if through the points of division of AB, AC, AD, planes be drawn parallel to the faces BG, BD, AE respectively, the parallelopiped will be divided into cubic units, all equal to one another. And since the rectangle ABEC contains 5 × 4 square units, (Book II, note, p. 100) and that for every linear unit in AD there is a layer of 5 x 4 cubic units corresponding to it; consequently, there are 5 x 4 x 3 cubic units in the whole parallelopiped AG. That is, the product of the three numbers which express the number of linear units in the three edges, will give the number of cubic units in the parallelopiped, and therefore will be the arithmetical representation of its volume. And generally, if AB, AC, AD; instead of 5, 4 and 3, consisted of a, b, and c linear units, it may be shewn, in a similar manner, that the volume of the parallelopiped would contain abc cubic units, and the product abc would be a proper representation of the volume of the parallelopiped. If the three sides of the figure were equal to one another, or b and c each equal to a, the figure would become a cube, and its volume would be represented by a aa, or a3. It may easily be shewn algebraically that the volumes of similar rectangular parallelopipeds are proportional to the cubes of their homologous edges. Let the adjacent edges of two similar parallelopipeds contain a, b, c, and a', b', c', units respectively. Also let V, V', denote their volumes. Then Vabc, and V' a'b'c. = a b = α' a f's b' = = In a similar manner, it may be shewn that the volumes of all similar solid figures bounded by planes, are proportional to the cubes of their homologous edges. Prop. vi. From the diagram, the following important construction may be made. If from B a perpendicular BF be drawn to the opposite side DE of the triangle DBE, and AF be joined; then AF shall be perpendicular to DE, and the angle AFB measures the inclination of the planes AED and BED. Prop. XIX. It is also obvious, that if three planes intersect one another; and if the first be perpendicular to the second, and the second be perpendicular to the third; the first shall be perpendicular to the third; also the intersections of every two shall be perpendicular to one another. |