84 



♦ KNOWLEDGE ♦ 



[February 1, 1887. 



rows of oranges, the rows coataining alternately 6 and 5 

 oranges ; so that each layer contains 3 times 11, or 33 

 oranges. 



We must next determine how many layers we can get 

 within the depth of 11 [*V inches. For this we must find 

 how much higher the centre of an orange e is above the 

 level in which lie the centres a, h, c. But for convenience 

 (so that tig, 2, b, may still suit us) we may suppose an 

 orange set in the space above o, fig. 2, h ; for clearly we shall 

 have the same height in one case as the other. The 

 centre will lie in the perpendicular from o, and at a distance 

 of 2 inches from b, as in the other case, illustrated in fig. 1, b. 

 Thus if we draw od perpendicular to OB in fig. 2, b, and 

 with B.I (=:2 inches) as radius describe the arc ad around B 

 as centre cutting od in D, we clearly have od equal to the 

 height we require, by which the centres of the oranges in 

 one layer are above those in the next below [or OB^o6, 

 BD:=2 inches, and bod is a right angle). But since 



B0=0A=#.v/3, and bd=2. 



oD- 



1:=';, and 



OD 



=2N/|=§N/tj = § (24495) = l-633 inch. By this amount 

 each layer rises above the layer next below; and as the 

 lowest is 2 inches high, and the total height is 11 '8 inches, 

 we divide 9'8 inches by 1'633, getting C as the number of 

 layers above the lowest, the 7 layers reaching to a height of 

 2 + 6 X l'633=:ir79S inches, or falling just within the 

 depth of the box. 



The total number of oranges, since each layer contains 33, 

 is 7 X 33, or 231, as required. 



Puzzle XVIII. Box No. 3 is cubical — the inside length, 

 breadth, and depth being 11 /'jj inches — and in this box 

 256 oranges are to be packed. 



Here the arranffement adopted must be that shown in 

 fig. 3, where the dark and light circles are to be understood 



Fig. 3 



as before. Here the rows range from each other as the 

 layers do in the first arrangement, so that, as shown by aid 

 of fig. 1, J, in considering that case, 8 rows occupy a breadth 

 of 11 '89 inches. (We may regard (/, b, and c in fig. 3 as 

 corresponding to «, b, and c in fig. 1, a.) INIoreover, we 

 observe that the arrangement of oranges in each layer of 

 this third box is, in a sense, the same as in the layers of box 

 No. 1, the centres of four adjacent oranges forming a square 

 2 inches in the side, as in that case. Hence the layers 

 lange in height in the third case precisely as in the fir.st, or 

 there are 8 layers. 



Since, then, there are 8 layers, and 32 in each, as fig. 3 

 show.s, there ai-e in all 256 oranges, as required. 



It will be readily seen that the arrangement in the solution 

 of each puzzle is the best for that special case. If we try in 

 box No. 1 the arrangement used in bos No. 2, we get in 

 i layers of 30 and 3 of 25, or onlj' 195 oranges ; if the 

 arrangement used in box No. 3, we get in i layers of 25 and 

 i of 24, or only 196. Again, if in box No. 2 we employ 

 airangement No. 1, we get in i layers of 30 and 3 layers of 

 26, or only 195 oranges ; and if in box No. 2 we employ 

 arrangement No. 3, we get in 8 layers of 28, or only 221 

 oranges ; lastly, if in box No. 3 we employ arrangement 

 No. 1, we get in 8 layers of 25, or only 200 ; while if we 

 employ arrangement No. 2, we get 7 layers of 30, or only 

 210 oranges. 



As regards closeness of packing, the methods are in one 

 sense identical, a dozen or so of oranges in the middle of 

 any box being arranged relatively to each other precisely like 

 a set of as many which can be taken from the middle of 

 any other box. But, considered with reference to the several 

 boxes, the methods of packing are not equally close. We 

 may clearly represent the closeness of packing for No. 1, 

 No. 2, and No. 3 by the number in each box divided by the 

 cubical content of the box, getting the respective expres.sions 



200 



231 



-, and 



256 



On deductinu 



(ll)-xll y 12x1 1-23 X 11-8' """■ (119)'' 

 the values of these expressions, which can be very easily 

 done by logarithms, we obtain the following jiroportion : — 

 Packing No. 1 : Packing No. 2 : Packing No. 3 : : 1389 

 : 1453 : 1516. 

 Thus the packing is considerably closer in box No. 2 than 

 in box No. 1 , and in box No. 3 than in box No. 2. 



PAST AND PRESENT VOLCANOES. 



AIN great volcanic disturbances remind 

 us of the energies which our earth once 

 possessed. For they err who imagine 

 that the uniformitarian theory, which 

 has replaced among the geologists of 

 our da}' the catastrophic theory of former 

 times, implies forces of disturbance as great 

 now as they were during past ages of the 

 earth's volcanic history. The processes of upheaval and 

 down-sinking which affect the earth's crust proceed uniformly 

 now, the catastrophic action witnessed in earthquakes and 

 volcanic eruptions being as nothing compared with the 

 steady but irresistible movements all the time going on. 

 Nay, one may almost say that eruptions and earthquakes 

 indicate rather the interruptions of the earth's vulcanian 

 work than its true progress. But the steady, as well as the 

 catastrophic, action of the earth's internal forces must be 

 recognised as fai- weaker now than it was in former ages. 

 The two forms of force are doubtless related to each other 

 in a nearly constant proportion, so that one may be inferred 

 when the other is known. Hence, though we cannot tell 

 from an}- direct evidence the energy of stead}' upheaval and 

 contraction possessed by the earth in past ages of her 

 history, for we have full evidence only as to work done and 

 no sutEcient evidence .as to the time occupied in doing the 

 work, we can safely infer what that energy was by noting 

 the evidence of the tremendous energy with which the 

 interruptions to that steady work went on. 



Unquestionably the extrusion of matter in volcanic erup- 

 tions was a much more important work in past ages than 

 now. We cannot go back, indeed, to the beginning. We 

 cannot even form an opinion as to the volcanic energies of 

 the earth in Cambrian and Silurian times, which were by 



