238 



♦ KNOWLEDGE 



[March 20, 1885. 



they stand on equal bases, and lie between the same 

 parallels. 



JL True. But observe that owing to their being very 

 thin, — in fact, of indefinite thinness, — each may be 

 regarded as a rectangular or upright prism. The volume 

 lying between the enclosing ]>lanes of P Q K, differs from 

 the volume of a rectangular prism on the lower triangle 

 P Q R and between the same planes, only by three 

 exceedingly thin triangular prisms, which become in- 

 definitely s-mall ultimately compared with either. About 

 the equality of two rectangular prisms on equal bases and 

 of the same height there can, of course, be no doubt, since 

 each can be divided into the same (indefinitely large) 

 number of indefinitely small parallelopipeds, of the same 

 height as tlie prisms, but of exceedingly small cross section, 

 and all of nearly the same volume. 



A. The slices P Q R, p q r are then equal. 



M. Yes ; and as the two prisms may be divided thus 

 into an indefinite number of slices, each slice of one being 

 equal to the corresponding slice of the other, it follows that 

 the prism A E C is equal to the prism a ec. 



A. Of course it follows froai this demonstration that 

 Triangular prisms on llin same base and between the same 

 ■parallels are equal. 



M. Yes ; and also it is obvious tliat Triangular prisms 

 on equal (triangular) bases and of the same altitude, or 

 between equidistant parallels, are equal. 



A. Do we next pass to p;irallelopipeds? 



.1/. We might prove, in just the same way, that Paralleh- 

 ji/ptds on equal bases and betireen tlie same or equidistant 

 parallels or of the same altitude, are equal ; but we may 

 conveniently make this a corollary from what has just been 

 shown about prisms. 



Fig. 2. 



A. How so ? 



M. Let A F D a c/d (Fig. 2), be parallelopipeds on 

 equal bases ABCD, abed (BC, be, remote from the 

 eye), and between the same parallel planes E g, A c. Then 

 if we draw B D, F H, b d, fh, we manifestly divide each 

 rectangle into two prisms on triangular bases ; and since 

 ABD = BCD ~ ab d = b dc, and the prisms on these 

 equal bases are between the same parallel.", they are all 

 four equal. Thus the parallelopiped A G is equal to the 

 parallelopiped a g, these pirallelopipeds being double the 

 equal triangular prisms on bases B A.D, b a d respectively. 



A. How about prisms on quadrangular but not rectan- 

 gular bases'! 



M. You can show directly, in the same way as with 

 triangular prisms, that quadrangular prisms on equal bases 

 and between the same parallels are equal — for, in Fig. 1, 

 the equality of the strips P Q R, p q r does not depend on 

 their being triangular. But you may employ what has 

 been already shown respecting triangular prisms. 



Thus let ABCD, abed ( Fig. 3) be the equal quadran- 

 gu'ar bases of the prisms \G,ag. Then drawing K, 

 parallel to B D, to meet A D, produced, in K, we have the 

 triangle B K D equd to the triangle C B D, and therefore 



the prism on base B D K equal to the prism of same alti- 

 tude on base BCD, or — adding the prism on base A B D 

 to both — the prism on base A B K is equal to prism of 

 same altitude on base ABCD. So making triangle F L E 



- / \ 



Fig. 



equal in all respects to triangle B K A, we have thc 

 triangular prism A F K equal to the quadrangular prism 

 A G. Make in like manner the triangles a b k, efl equal to 

 the quadrangle abc d, and we get the prism afh equal to the 

 quadrangular prism a g. But the triangles A B K, abk are 

 equal, being severally equal to the equal quadrangles 

 ABCD, abed; hence prism A F K=prism afk ; where- 

 fore prism A G = prism a g. 



A. Clearly this method can be extended to prisms having 

 any number of angles. 



M. Yes ; we obviously obtain this general result. Prisma 

 of equal altitude and on equal bases, triangular, quad- 

 rangular, or polygonal, are equal. And obviously it 

 matters not whether the prisms have the same or a 

 different number of angles. If their bases are equal and 

 their altitudes equal, their volumes are equal. 



A. If then we can determine the area of the base of any 

 prism, and the altitude of the prism, we know the volume 

 of the prism. 



J/. Yes ; it is equal to the volume of a rectangular 

 parallelopiped on an equal (rectangular) base, and of the 

 same altitude. 



A. Can we not represent the volume of any prism by a 

 cube, as we can represent the area of any rectilinear figtire 

 by a square 1 



J/. No : the general problem " Determine the side of a 

 cube icldclt shall be equal in volume to a given parallelo- 

 piped " is not solvable by the geometry of the line and 

 circle. 



(To le contimied.) 



At a meeting of the Anthropological Institute, on March 10, Mr. 

 James G. Fraser read a paper on " Certain Burial Customs, as 

 illustrative of the Primitive Theory of the Soul." The Eomans 

 had a custom that when a man who had been reported to have died 

 abroad returned home alive, he should enter his house, not by th& 

 door, but over the roof. This custom (which is still observed in 

 Persia) owed its origin to certain primitive beliefs and customs 

 with regard to the dead. The ghost of an unburied man was sup- 

 posed to haunt and molest the living, especially his relatives. 

 Hence the importance attached to the burial of the dead, and various 

 precautions were taken that the ghost should not return. When the 

 body of a dead man could not be found, he was buried in effigy, and 

 this fictitious burial was held to be sufficient to lay the wandering 

 ghost, for it is a principle of primitive thought that what is done 

 to the effigy of a man is done to the man himself. The Director 

 read a paper by Admiral F. S. Treralett, on the " Sculptured 

 Dolmens of the Morbihan " (Xorth-west Prance). About eighty 

 sculptures had been found, invariably on the interior surfaces of 

 tlic cap-stones and their supports. It is remarkable that they are 

 confined within a distance of about twelve miles, and are all 

 situated near the sea-coast, beyond which, although the megaliths 

 are numerous, there is a complete absence of sculptures. The 

 sculptures vary in intricacy, from simple wave-lines and cup- 

 markings, to some that have been compared to the t.attooing of the 

 New Zealanders. 



