386 G E O M Prop. 30 may therefore be solved in two wa_v.<, either by aid of Prop. 29 or by aid of ll. 11. liuclid gives both solutions. § 71. Prop. 31 (Theorem). In any right-angled triangle, any recti- linrnl figure tlcscribeil on the side subtcmling the right angle is equal in the similar and similarly-clcscribcrl _rig1n'cs on the siilrs containing the right angl/',—is a pretty generalization of the theorem of Pytha- goras (l. -17). Leaving out the next proposition, which is of little interest, we COIIIC to the last in this book. Prop. 33. In equal circles angles, whether at the centres or the cir- (‘lt))lf¢_'I'CnCCS, hare the same ratio zrhich the arcs on ‘which. they stand hare to one another ; so also hare the sectors. Of this, the part relating to angles at the centre is of special importance ; it enables us to measure angles by ares. With this closes that part of the Elements which is devoted to the study of figures in a plane. B001: XI. Q "0 3 1 -. In this book figures are considered which are not confined to a. plane, viz., first relations between lines and planes in space, and afterwards properties of solids. ()f new definitions we mention those which relate to the perpen- dicularity and the inclination of lines and planes. Def. 3. A straight line is perpendicular, or at right angles, to a plane when it 7nahcs right angles with crery straight line meeting it in that plane. The definition of perpendicular planes (Def. 4) offers no dif- liculty. Euclid defines the inclination of lines to planes and of planes to planes (Defs. 5 and 6) by aid of plane angles, included by straight lines, with which we have been made familiar in the lirst books. The other important definitions are those of parallel planes, which never meet (Def. 8), and of solid angles formed by three or more planes meeting in apoint (Def. 9). To these we add the definition of a line parallel to a plane as a line which does not meet the plane. § 73. Before we investigate the contents of Book XL, it will be well to recapitulate shortly what we know of planes and lines from the definitions and axioms of the first book. There a plane has been defined as a surface which has the property that every straight line which joins two points in it lies altogether in it. This is upli- valent to saying_ that a straight line which has two points in a plane has all points in the pl.:..e. Hence, a straight line which does not lie in the plane cannot have more than one point in common with th - p'lanc. This is virtually the same as lSuclid's Prop. 1, viz. :- l‘rop. 1. One part of a straight line cannot be in a plane and another part -without it. It also follows, as was pointed out in § 3, in discussing the dclinitions of Book I., that a plane is determined already l)y one straight line and a. point without it, viz., if all lines be drawn through the point, and cutting the line. they will form a plane. This may be stated thus :—- A plane is ¢letcrmined—— lst, By a straight line and a point which does not lie on it; 2d, By three points which do not lie in a straight line ; for if two of these points be joined by a straight line we have case 1 ; 3d, By two intersecting straight lines; for the point of intersection and two other points, one in eac.h line, give case ‘.2; 4th, By two parallel lines (Def. 35, 1.). 'l‘he third case of this theorem is l'luelid’s l’rop. *2. Two straight lines which rut one another are In one plan", anrl three straight lines which. meet one another are in one plane. And the fourth is line-lid’s Prop. 7. If taro straight lines be parallel, the straight line (lra1I.'n from any point in one to any point in the other is in the same plane with the parallels. From the definition of a plane further follows ‘ Prop. 3. If two planes cut one another, their common section is a straight line. 5‘ 74. Whilst these propositions are virtually contained in the dclinition of_a plane, the next gives us a new and fundamental pro- perty of space, showing at the same time that it is possible to have a straight line perpendicular to a plane, according to Def. 3. It states- 'l‘nr-:on1»:.t (Prop. 4).—If a straight line is perpcmliculrzr to two straight lines in a plane which it -meets, then it is perpendicular to all lines in the plane which it meets, and hence it is perpendicular to the plane. Def. 3 maybe stated thus: If a straight line is perpendicular to a plane, then it is perpendicular to every line in the plane which it meets. The converse to this would be 'l‘n l~2()I’.E.I.-—/l ll straight lines 2:-hich mat‘ a git-en. sir/Ii;/hf line in the srnn-' point, and are pcrpcnllicular to it, lie in a plane 1!.-lite/L is pcrpendiculru' to that line. This Euclid states thus -. Prop. 5. If three straight lines meet all at one point, and a straight- ETRY line stands at right anglvs to each if thrm at that point. the Mn < straight lines shall be in one and the same plane. § 75. There follov theorems relating to the theory of parallel lines in space, viz. :— 'l'nI-:o1:1-:. (l'rop. (3).-—/Iny ta'o lin.-5 which are p::rpcn:lical(u' to the same plane are parallel to each o/hr)‘ : and (:0ll't-l'.5cly 'l'n£oIu-:.I (Prop. 8).—1f Qf(l"l) parallel straight lines on- is ner- pem/icnlar to a plane, the ollar is so also. Further, the important theorems Prop. 9. Two straight lines vrhich are each qf than paralltl lu I/'« same straight line, and not in the smnr plane with it, are parallil !; one another; where the words, "and not in the same plane with it," maybe left out, for they exclude the case of three parallv-ls in a plane, which has been proved before; and 'l'IIEOm-:.I (Prop. 10_).—1_/"two angles in ¢li_I/"crrnt pluncs hare (hr two limits of the one parallel to those of the other, than th. angles ur-' equal. That their planes are parallel is shown later on in Prop. 15. This theorem is not necessarily true, for the angles in llll(_'.~llUll may be supplementary; but then the one angle will he e-purl 1-: that which is adjacent. and supplementary to the other, and ti." latter angle will also have its limits parallel to those of the lil'.. From this theorem it follows that if we take any two straigl lines in space which do not meet, and if we draw tln'ou,-_-‘h any point P in space two lines parallel to them, then the angle in- eluded by these lines will always be the same, 'l|ate'c1'tl1e 1v()5ll.l n of the point P may be. This angle has in modern times h-.en called the angle between the given lines :—- l)1~:I-‘1.'l'r10.‘.—L’y the angles between two -not intersecting l in. - . . anclcrstanrl the angles which two intu'sccting lines incluzl-J that v r parallcl rcspectircly to the two gircn lines. § 76. It is now possible to solve the following two prul-lcnh :— 1’nonm~::I.—-To (lraw a straight lincpczpcmlicalar to a given plane from a giren point ichich lies 1. Not in the plane (Prop. 11). ‘.2. In. the plane (Prop. 1'2). The second Case is easily reduced to the fir.<t—'i/., if l-_v aid of the first we have drawn any perpendicular to the plane from sonn- point without it, we need only draw through the given point in the plane a line parallel to it, in order to have the rcrpiired 1n'1'p(-n- dicular given. The solution of the first part. is of ll1tl'l'('>t in lI~1:lf. It depends upon a construction which may he C.p1'e.<.~:-.l theorem. 'l‘nr.o1:E.I.—]f from apoint .4 without a plan!) a pm-p.-In/I. 11.3’,- AL‘ be drawn to the plane, and i/‘from. the foot L’ oft/u'~' prrp. -:- llicular another perpendicular BC be drawn to any straight linr in the plane, thrn the straight linr joining A to the foot 0 of this .SI'l.'un'l pcrpcnclicalar will also be perpendicular to the line in the plane. The theory of perpcndiculars to a plane is concluded lay tl.-» theorcn1— . Prop. 13. Through. any point in space, ‘whether in or irillwal II plane, only one straight line can be drawn pcrpcnilicnlur lu I/an plane. § 77. The next fo11r propositions treat of parallel planes. It 3: shown that planes which hare a common pcrpenilicalar are p/(rul ‘It (Prop. 14) ; that two planes are parallel if two intc:'serti'ng .strui;_u'zt lines in the one are parallel respectively to two stra.ight lines in the other plane (Prop. 15) ; that parallel planes are cut by any plane in parallel straight lines (Prop. 16) ; and lastly, that any two strain/'.I. lines are cut proportionally by a series tgfpr/rallcl plnnrs (Prop. 17!. This theory is made more eomplcte by adding the full0in-_{ theorems, which are easy deductions from the last :——T2I'o para.-" I planes hare common prrprntlicnlars (converse to 1-1); and 7'-In; plancs which are parallel to a thirrl plane are paral_b.l to each other. It will be noted that Prop. 15 at once allows of the solution of the problem: “ Tlirougli a given point to draw a plane parallel to a given plane.” And it is also easily proved that this prohl--in allows always of one, and only of one, solution. § 78. We come now to planes which are pe.rpendie'ul.u' to « .- another. Two theorems relate to them. Prop. 18. 17 a straight line be at right angles to a plane, rrrg plane which passes through it shall be at -right angles to that plan '. Prop. 19. If two planes which cut one another be each 15/ (In. L perpenrlicular to a third plane, their common section shall be 1"_l'- pcndicalar to the same plane. § 79. If three planes pass through a common point, and if the y bound each other, a solid angle of three faces, or a Irihrilral angle, is formed, and similarlyhy more planes a solid angle of more faces, or a polyhedral angle. These have many properties which are quite analogous to those of triangles and polygons in a plane. Euclid states some, viz :— l’rop. ‘.20. If a solirl angle be cont/I?'nr'rl by three plane angles, any two of them are together greater than the third. ‘nit the next- l’rop. 21. I;'rer_1/ solid angle is conminrrl by plane angles 2rlu'¢-ls are togrthcr less than four right fl7t_’/ll’S——ll{lS no analogous 1' ieorcn! in the plane. iii, CL: DI . X.
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