MATHEMATICS. PLANE GEOMETRY. [BOOK xi. PBOF. t rr. 



CHAPTER VII. 

 PLANES : 



BEIXO THB FIRST TWBXTT-OXI PROPOSITIOX8 OF THB ELEVEXTH BOOK OF PCLtD's OEOMETRT. 



or 



1 NTRODCCTIOX. The figures, lines, angles, Ac., the 



properties of which form the subject of the First Six 



Books of Euclid's Geometry, are supposed to lie in one 



! imith plane, or to be in space of two dimensions. * 



and WMdth. fiie following pages contain a few elementary 

 propositions on the relations between lines, angles, <ta, 

 which do not lie in one plane, but are in solid space, or 

 space of three dimensions.-^ The student 

 . will find the following propositions very 

 easy, when once he has distinctly conceived the 

 meaning of their enunciations. The figures which are 

 given to each proposition cannot represent the proposi- 

 tion to the eye so perfectly as in the former books, in 

 consequence of their having to be drawn in perspective. 

 It is hoped, however, that the shading introduced into 

 the diagrams will aid the student in conceiving the pro- 

 position they belong to. 



It is to be added, that, as in Plane Geometry, we are 

 allowed to draw lines in any direction, and to produce 

 them to any extent, so in solid Geometry we are allowed 

 to draw planes in any direction, and to produce them to 

 any extent. Moreover, two lines intersect in a point ; 

 in like manner it will be shown that two planes intersect 

 in a line. Also, as we may suppose, a fine to revolve 

 round a point till it comes to a point on its plane, so we 

 may suppose a plane to revolve round a given line until 

 it comes to a given point situated anywhere in space. 



DEFINITIONS. 



I. 



A straight line is perpen- 

 dicular, or at right angles to a 

 plane, when it makes right 

 angles with every straight line 

 in that plane which meets it. 



Thus, if B D be a plane, PA 

 a line perpendicular to it. n 



Through A draw any lines A B, A C, A D .... in that 

 plane, then will PA B, PAC, PAD, <tc., be right 

 angles. 



n. 



A plane is perpendicular to a plane, when a straight 

 line drawn in one plane perpendicular to the intersection 

 of the planes is at right angles 

 to the other plane. 



Thus, letABD, ABC, be 

 two planes, let the former be 

 perpendicular to the latter, 

 and let A B bo the line of in- ~^ : \ 



tersection of the planes in 

 the plane A B D, draw P N at 



right angles to A B. Then is P N at right angles to the 

 plane ABC. 



in. 



The inclination of a straight line to a plane, is the 

 acute angle contained by that straight line, and another 

 drawn from the point in which the first lino meets the 

 plane, to the point in which a 

 perpendicular to the plane P 



drawn from any point of the 

 first line above the plane meets 

 the same plane. 



Thus, let A N B be a plane, 

 A P a lino meeting the plane 

 in A ; from P draw P N per- 

 pendicular to the plane, and meeting the plane in N. 

 Join A X, then the angle P A N is the inclination of the 

 liue P A to the plane A N B. 



IV. 



The inclination of a plane to a plane, is the acute angle 



-''""1 



contained by two straight lines drawn from any one point 

 of their common section at right angles to it, one upon 

 one plane, the other upon the other. 



Let PAC, PBC be two 

 planes intersecting in the line 

 PC. From P in the former 

 plane, draw P A at right angles 

 to P C ; and from the same 

 point P on the latter plane, 

 draw P n at right angles to 

 1' C. Then if M P A be an 

 acute angle, this is the inclina- 

 tion of the planes to each other. 



v. 



Two planes have the same inclination to one another 

 which two other planes have, when the said angles of in- 

 clination are equal to one another. 



VT. 



Parallel planes are such as do not intersect, though 

 produced ever so far in all directions. 



vi r. 



A solid angle is that which is made by the meeting of 

 more than two plane angles, which are not in the same 

 plane, in one point. 



PROPOSITION I. THEOREM. 

 One part (AB) of a straight line (ABC) cannot be in a 



plane, and another part (BC) be above it. 



For let us suppose this possible, then since the straight 



line A B is in the plane, it can be produced in that plllM : 



let it be produced to D. Now, suppose a plane to pass 



through the straight line AD, and let 



it be turned round that line, till it 



comes to the point C. Then because 



B and C are in the plane, the straight 



6 Df. i. line B C is in it :* . ' . there are two straight 



lines A BC, AB D in the same plane, having a common 



+ Cor. 11 1. segment A B, which ia impossible, t Q- E. D. 



PROPOSITION II. THEOREM. 



Two ttraifiht lines (A B, C D) which cut one another (in the 

 point E) are in one plane. And three straight lines 

 (13 C, C E, E B) which meet one another, are in one 

 plane. 



Let any plane pass through E B, and let the piano be 

 turned about E B produced if necessary, 

 until it pass through the point C. _ Then 

 because the points C and E are in this 

 e Def. i. plane, the line C E is in it. * 

 For the same reason the straight line 

 B C ia in the same plane, and by the 

 hypothesis E B is in it ; .'. tho three 

 straight lines B C, C E, E H are in one 

 plane But A B is in the same plane as E B, and I) C 

 1 1 XI. as E C.f Also A B and D C are in the same 

 piano. Q. E. D. 



PROPOSITION III. THEOREM. 

 If two planes (A B, B C) cut one another, their common 



section (D B) is a straight fine. 



For, if not, sine* D, B are points in the plane A B, 

 draw the straight line D F B in 

 that plane, and similarly draw 

 the line DEB in the plane B C. 

 Then because these two straight 

 lines have the same extremities, 

 they enclose a space, which U 

 i 3 I. AX. 1. absurd j* .'. the com- 

 mon section B D cannot but be a 

 straight line. Q. E.D. 



