GEOM “'0 may mention, however, that the theorems about tri- angles contained in the propositions of Book 1., which do not depend upon the theory of parallels (that is all 11p to Prop. 27), have their corresponding theorems abo11t trihcdral angles. The latter are formed, it for “side of a triangle” we write “plane angle" or “face ” of trihedral angle, and for “ angle of triangle" we substitute “ angle between two faces” where the planes con- taining the solid angle are called its faces. We get, for in- stance, from I. 4, the theorem, If two trihedral angles hare the angles ty"tu'o faces in the one equal to the a.n_r/les of two faces in the other, and hate li/Jezeise the angles included by these faces rquu], /hen the angles in the remaining faces are equal, and the angles between. the other fares are equal each to each, ri.:., those uhich are np})I)StlC equal faces. The solid angles themselves are not necessarily equal, for they may be only syn1111etric.al like the right hand and the left. 'l‘he connexion indicated between triangles and trihedral angles will also be recognized in Prop. 22. I_/'creryt1I'o if three plane angles be greater than the thiril, and if the stra‘i_r/lit lines irhieh. contain them be all equal, a. triangle may be nzailc if the straight lines that join the e.r'treinities Q/' thus-u equal straight lin And Prop. '23. solves the problem, To eonstruet a trihedral angle hrlring the angles of its faces equal to three given plane angles, any law) of th--in bring greater than the third. It is, of course, analogous to the problem of constructing atriangle having its sides of given length. Two other theorems of this kind are added by Simson in his ulition of Euclid's Elements. sf 50. These are the principal properties of lines and planes in .):lCC‘, but‘ before we go on to their applications it will be well to deline the word distance. In geometry distance means always "shortest distance”; viz., the distance of a point from a straight line, or from a plane, is the length of the perpendicular from the point to the. line or plane. The distance between two 1101)- intersecting lines is the length of their con1n1on perpendicular, there being but one. The distance between two parallel lines or between two parallel planes is the length of the eon11non perpendi- cular between the lines or the planes. § 81. Pu/'allelepipe:ls.—-The rest of the book is devoted to the study of the parallelepiped. 111 Prop. 24 the possibility of such -.1 solid is proved, viz. :— Prop. '24. If a. soliil be eontainccl by si.2: planes two and two of iv.-Melt are parallel, the opposite planes are similar and equal p-.1:'alb logrums. I-Euclid calls this solid henceforth a parallelepiped, though he r.«_-ver tlt'llIlL'S the word. Either face of it may be taken as base, and its distance from the opposite face as altitude. Prop ‘.75. If (I soliil pru-allclepipeil be cut by a plane parallrl to I H of its opjms-itr plan-‘.9. it ilirivlrs the erholc into two soliils, the l2o.su of one of u-hit.-h shall be to the base of the other as the one solid ."" to the Jllu’/‘. This theorem corresponds to the theorem (VI. 1) that parallelo- grams between the same parallels are to one another as their bases. . similar analogy is to be observed among a number of the remaining propositions. § S2. After solving a few problems we come to Prop. '28. If a Solitl pr(i'allr'lepipI'(l be cut by a plane passing through. the diagonals of two cf the opposite planes, it shall be cut in. tu'0 equal parts. In the proof of this, as of several other propositions, Euclid neglects the ditl'erenee between solids which are symmetrical like the right hand and the left. Prop. 31. Solid parallclcpipeils, which are upon equal bases, and qr‘ the s -unr: altitude, are equal to one another. Props. '29 and 30 contain special cases of this theorem leading up to the proof of the general theorem. As consequences of this fundamental theorem we get Prop. 32. Solid parallelcpz'peils, -zehieh have the same attitude, are to one another as their bases ; and Prop. 33, .S'in1z'lar soliil parallel- tpip: {IS a. "e to one another in the tripl icatr rat in of their h.omolo_r/oussicles. If we consider, as in § 67, the ratios of lines as numbers, we may also say- The ratio (_If the volumes of similar parallrlqiipctls is equal to the ratio of the third powers of homologous sirles. Parallelepipeds which are not similar but equal are compared by aid of the theorem Prop. 34. The bases and altitudes of equal soliil parallclrpipuls urr reciprocallyproportional; and if the bases and attitudes be re- riproeal l y proportional, the solid parallclepipecls are equal. 3' 8-3. Of the following propositions the 37th a11d 40th are of special interest. Prop. 37. If four straight lines be proportionals, the sinzilar solid parallelepipcils, similarly clescribeil from them, shall also be pro- portion/tls ; and if the similar parallrlepipeds sivnilarly 0’I'seribe(l from four straight lines be proportionals, the straight lines shall be proportio/Lats. " BOOKS xi. xu.] E T R Y 111 symbols it says- lfa :b=c :11, then a" :b"=e” :(l3. Prop. 40 teaches how to compare the volumes of tliangular prisms with those of parallelepipeds, by proving that a. triangular prism is equal in rolunze to a paratlelrpipecl, uhich has its altitude amt its base equal to the altitude and the base of the triangular prism. §8-4. From these propositions follow all results relating to the mensuration of volumes. 'e shall state these as we did in the case of areas. The starting-point is the “rectangular” parallelepiped, which has every edge perpendicular to the planes it meets, and which takes the place of the rectangle in the plane. If this has all its edges equal we obtain the “cube. " If we take a certain line u as unit length, then the square c-n 4' is the unit of area, and the cube on u the unit of volume, that is to say, if we wish to measure a volume we have to determine how many unit cubes it contains. A rectangular parallelepiped has, as a rule, the three edges unequal, which meet at a point. Every other edge is equal to one of them. If a, b, c be. the three edges meeting at a point, tlun we may take the rectangle contained by two of them, say by b and c, as base and the third as altitude. Let V be its volume, V’ that of another rectangular parallelepiped which has the edges a’, b, e, hence the same base as the tiist. It follows then casily, from Prop. 25 or 32, that V : V'=a :a.' ; or in words, Itrctangular parallelcpipecls on equal bases are proportional to their altitudes. If we have two rectangular parallelepipeds, of which the first has the volume V and the edges a, b,"e, and the second,the volume. " and the edges a’, b’, e’, we may compare them by aid of two new ones which have respectively th.- edges a’, b, e, and a’, b’, c, and the volumes V, and ‘'2. 'e then have V : V1=a :a'; V, :'2=b :b', V2 : "=e: c’. Compounding these, we have V : V’=(a :a') (b : b’) (c :e'), V a b c °" v"=;u'z7"c" Ilenee, as a special case, making V’ equal to the unit cube U on u we "ct CI ., U where a, B, 'y are the numerical values of a, b, c; that is, The number of unit cubes in a. rectangular parallelepiped is equal to the product of the numerical values of its three edges. This is generally expressed by saying the volume of a rectangular parallelepiped is measured by the product of its sides, or by the product of its base. into its altitude, which in this case is the same. Prop. 31 allows us to extend this to any parallelepipcds, and Props. 28 or 40, to triangular prisms. ’l‘n1:or.I:n.——Th.e roluine of any parallelepiped, or of any tri- angular prism, is eneasurctl by the product of base and altitude. The consideration that any polygonal prism may be divided into a number of triangular prisms, which have the same altitude and the sum of their bases equal to the base of the polygonal prism, shows further that the same holds forany prism whatever. 387 b c —°1—l=a. 3.7, u 11. BOOK XII. § 85. In the last part of Book XI. we have learnt how to com- pare the volumes of parallelepipeds and of prisms. In order to determine the vol11n1e of any solid bounded by plane faces we must determine the volume of pyramids, for every such solid may be decomposed into a number of pyramids. As every pyramid may again be decomposed into triangular pyramids, it becomes only necessary to determine their volume. This is done by the Theorem.——Every triangular pyramid is equal in volume to one third of a triangular prism having the same base and the same altitude as the pyramid. This is an immediate consequence of Euclid’s Prop. 7. Every prism hating a triangular base may be iliriihtl into three pyramicls that hare triangular bases, an.rl are equal to 4. tr another. The proof of this theorem is diflicult, because the three trian.-_—;ular pyramids into which the prism is divided are by no means Cqual in shape, and cannot be made to coincide. It has first to be proved that two triangular pyramids have equal volumes, if they have equal bases and equal altitudes. This Euclid does in the following manner. He first shows (Prop. 3) that a triangular pyrannd may be divided into four parts, of which two are equal triangular pyra- mids similar to the whole pyramid, whilst the other two are equal triangular prisms, and further, that these two prisms togcthc-1_' are greater than the two pyramids, hence more than half the given
pyramid. He next shows (Prop. 4) that if two triangular pyra-