April 3, 1885.] 



KNOWLEDGE 



277 



CHATS ON 



GEOMETRICAL MEASUREMENT. 



By Richard A. Proctou. 



(Continued ficiti yagf 23!>.) 



.•1. We still have to consider pyramids. I see, indeed, 

 why you spoke of pynimids as the simplest elements into 

 which plane-faced solids can be divided. Just as a recti- 

 linear area has to be divided usually into triangles, before 

 we can determine the rectangle to which it is equal, so a 

 plane-faced solid would usually have to be divided into pyra- 

 mids, before we can represent its volume by a parallelepiped. 



.1/. Precisely. But we shall have very little trouble 

 with pyramids 

 now we have 

 dealt with prisms. 

 We must first 

 show that trian- 

 gular prisms on 

 equal bases and 

 of equal altitudes 

 are equal : — 



Let A B C D 

 and abed, Fig 4, 

 be pyramids on 



Pig. 4. 



equal basis ABC, abc (B and b remote) and of the same 

 altitude, so that D rf is parallel to the plane in which ABC 

 abc both lie. Take two planes close together, and both 

 parallel to A B ca, cutting the pyramids in the thin tri- 

 angular slices P Q R, p qr. These triangular slices are 

 equal ; for (by the laws of parallels) 



PQR:ABC::(PR)=:(AC)= 

 ::(DP)'^:(DA)= 

 ■.■.(dpy-:(day- 

 :: (p r)- : (a c)- : p q r : a b c. 

 So that, axA.BC = abc,FQIl=pqr. 



This being true of each such element of the two 

 pyramids, we have finally, when their number is indefi- 

 nitely large, the sum of all the triangular slices in one 

 pyramid equal to the sum of all the triangular slices in the 

 other, or. Pyramid A D B C = pyramid a dbc. 



A. Of course, we can extend this result as we extended 

 the first result obtained in the case of prisms. 



M. Yes ; obtaining finally the general result that 

 pyramids on equal bases (of whatever number of sides, 

 equal or unequal) and of equal altitudes are equal. Thus 

 the volume of any pyramid can be represented by that of a 

 pyramid on a triangular or rectangular base (as most 

 convenient) and having one side perpendicular to the base. 



A. What is the advantage of this 1 



M. It enables us to compare any pyramidal volume with 

 the volume of either a right prism or a parallelopipedon, 

 as we may prefer. 



A. In whay way ? 



M. As follows : — 



Let A'ROd (Fig. 5) be a pyramid on the triangular 

 base ABC, its altitude being d M (perp. to the plane 

 ABC, and meeting that plane 

 in M). Draw the lines A D, 

 BE, CF at right angles to 

 A B C, and let a plane through 

 d parallel to A B C meet these 

 lines in D, E, F. Join D E, 

 EF, FD, completing the 

 upright prism A E C. 



Now join DC, D B, com- 

 pleting the ))yramid A D B C : 

 this pyramid is equal to th<; 



pyramid A B </, being on the same base and of the same 

 altitude. Join E C, conijileting the jiyramid D E F C : this 

 pyramid is equal to the pyramid ABC(/, being on an 

 equal base D 1'] F, and of the same altitude. But removing 

 the pyramids A DBC, DKFC from the prism A K C, 

 there remains only the pyramid E B C D, and tliis pyramid 

 is equal to D E F C, since these two pyrumiils are on equal 

 bases EEC, EC F, and (having a common vertex, D) arc 

 of the same altitude : therefore the pyramid K B C D is 

 equal to the pyramid A B C t /, which has been already 

 shown to be equal to the pyramid E C F D. Thus 

 prism A E C = pyr. A D C B -f- ,.yr. D K F C -f py r. E B D 



= ;{ pyramid A B C ti 

 Wherefore, A pyramid is equal in roliiiiw to one-ildrd tits 

 prism mi the same base and of the same altitude. 



A. This result, with what we have already learned, 

 enables us, I see, to compare any pyramid with any prism 

 or parallelopiped of the same altitude. 



M. Yes; we have the general relation, — A pyramid is 

 equal in volutne to one tliird of a prism trianytdar, rect- 

 angidar, quadrangular, or jiolygonal of tlie same altitude 

 and hai'ing a base of equal area. 



A. How about cases where the altitudes are different? 



M. Since the volumes of prisms and of ]>arallelopipedB 

 manifestly vary as the altitudes, so also must the volumes 

 of pyramids, so that in fine we can compare the volume of 

 any pyramid with the volume of any prism or paral- 

 lelopiped. 



A. If we wish to express the volumes of such solids 

 numerically, how must we proceed ] 



M. We may say that the volume of a prism ia the pro- 

 duct of the base by the altitude, and the volume of a 

 pyramid one-third the product of the base by the altitude, 

 if we remember that this is only a convenient way of pre- 

 senting certain numerical relations ; for in reality we can 

 no more multiply length by breadth to get an area, or an 

 area by height to get a volume, than we can multiply miles 

 by minutes. Yet just as the product of the number of 

 miles by the number of minutes occupied in traversing 

 each gives us the time (in minutes) occupied in making a 

 given journey, so the product of the number of units of 

 length in two adjacent sides of a rectangle gives ua the 

 number of units of surface in the rectangle. In like 

 manner The product of the number of itnits of surf ace in the 

 base of a prism or parallelopiped by the number of units of 

 length in its height, gives us tfie number of units of volume 

 in the pris7n or parallelopiped ; while One third the number 

 of units of surface in the base of a piyramid by the number 

 of ttnits of length in its height, gives us the number of units 

 of volume in the pyramid. 



(To he continued.) 



ELECTRO-PLATING. 



xvni. 

 By W. Slingo. 



INHERE are many interesting features pertaining to the 

 practical study of electricity, and amongst them the 

 deposition of alloys certainly takes a prominent stand. 

 Experiments in this direction will speedily disperse many 

 of the ideas of students of electricity, not the least notice- 

 able being that when a current traverses a solution con- 

 taining two or more metals, it will precipitate only one of 

 them. That this, however, is not necessarily the case may 

 be easily demonstrated. What is really essential, if we 

 wish to precipitate a single metal, is that the solution 

 employed should be pure, and such that (based on experi- 



