May 2, 1887.] 



♦ KNOW^LEDGE ♦ 



151 



theories of the Jews. At the time when the great 

 astronomical work of the Chaldeans was compiled, and 

 when the calendar was fixed, which we must, as far 

 as evidence goes, place about B.C. 2000-2500, the bright 

 star Thuban in the constell.ition Draco was near to 

 the pole. Placed as this star Thuban, a Draconis, is, 

 it seems to me to be very possibly the star called " the 

 star of the tip of the tail."* This star was especially 

 the omen of weather changes. We read, " The star of 

 the tip of the tail a great cloud obscures. Locusts in 

 the land are," or " The star of the tip of the tail at 

 its rising, the waves of the sea rise at the beginning of 

 the month Tammuz." The connection of this constellation 

 and its bright star with storms, tempest, and sea, certainly 

 .seem to connect it with the storm-dragon, the demon who 

 figures so often in Chaldean mythology, and may, as you 

 have so truly pointed out, be a relic of astro-mythology. 

 The occurrence of these equinoxial festivals on the newly- 

 discovered inscription from Sippara serves to show that we 

 must place the rise and, to a certain extent, scientific 

 development of Babylonian astronomy further back in 

 the past than we have hitherto expected. The very 

 remarkable find of a complete library of tablets in the 

 priests' quarters of one of the oldest Babylonian temples, 

 which h:is resulted from recent explorations, may no doubt 

 throw further and more clear light upon some of the 

 obscure points in the astronomy of the ancient Chaldeans. 

 Should any such be found, I shall be glad to place them 

 liefore the readei-s of Kno« ledce, who will no doubt 

 e.stimate their scientific value better than I am able to do. 



SOLUTIONS OF PUZZLES. 



I. BUILDING A TRI.\XGUL.\R PYRAMID. 



RUBLE^[ XXV. (see last Number). To 

 arrange equal regular tetrahedrons and octa- 

 hedrons, with equal triangular faces, into 

 a single tetrahedron (or regular triangular 

 pyramid) set six of the tetrahedrons on the 

 base ABC, as shown in fig. 1, where a, h, c, 

 d, e,f, represent the apices of six tetrahedrons 

 i-tanding on the bases adf, dfg, fgk, ebh, ghl, and klc. 



A 



Fig. 1. 



It is then seen that three octahedrons will fit into the spaces 

 dosfcgJ, EftoeHfZ, and GCK_/'Le, their upper triangular faces 



* This, from what follows, seems impossible. For the " star of 

 the tip of the tail " is described as " rising." The pole star does not 

 rise. — Ed. 



(regarding abc as the base of the pyramid we are building 

 up) being, respectively, ahc, hde, and cef. Now insert a 

 seventh tetrahedron into the space hce, with an angle down- 

 wards at G. We have then the triangular surface adf. We 

 now set a tetrahedron on each of the triangles abc, lide, and 

 cpf; fit in a tetrahedron between them : on whose triangular 

 upper face we set a tetrahedron, and the regular tetrahedron 

 or triangular pyramid with equilateral faces on the equi- 

 lateral base ABC is complete. 



[The outlines of the tetrahedrons and octahedrons above 

 the lowest layer are not shown, but the reader will have no 

 difficulty in seeing how the upper layers forming a pyramid 

 on the base ad/ are formed. He may regard the part aek 

 of the figure as showing this, for it shows three tetra- 

 hedrons, ADF, DEG, FGK, One octahedron BaFcc.b, and the base 

 abc of the tetrahedron needed to complete the regular 

 tetrahedron on the base aek.] 



ScHOL. — The numbers of octahedrons and tetrahedrons 

 required to complete the successive layers of a built-up 

 tetrahedron are as follows, going downwards : — 



Xo. of Octahedrons. 







1 



[\+-2] or:5 



[l-f-2-1-3] or G 



Layer. 

 1 



2 



3 

 4 

 &c. 



n 



No, of Tetrahedrons. 



[1+2) 



[1 + 1 + 2 + 3] 



or 



or 



[1+2 + 1 +2 + 3 + 4] or ir 

 '(n—2)(n - 1 w(?i + l) 



(n—\)n 



or n- — re + 1 ) 

 Hence the total numbers of tetrahedrons and octahedrons 

 required to form a tetrahedron, each of whose edges is 

 n times as long as an edge of the smaller solids, are as 

 follows : — 

 Tetrahedrons 



n(n — 1) 



'=30'^^) 



l!i±i)=^'(,r-l) 



4 fr ' 



=s,K'r-«+ !)=;:(" +i)(2-*+i)- 



Octahedrons 



=2,l('il=^)=f^(. + l)(2,^+l)- 



Thus the ratio of the number of tetrahedrons to the number 

 of octahedrons is 



2«^ + 4 : >r-l; 

 and when n is very great this ratio becomes very nearly 

 equal to 2 : 1. 



We can now very readily see in what ways sliding is 

 possible when space has been filled up with tetrahedrons 

 and square-based pyramids, or with tetrahedrons and octa- 

 hedrons, in the manner considered above, and in Puzzles 

 XXII., XXIII., and XXIV., solved last month. 



First, note that in the loss interesting case first dealt with 

 last month, where tetrahedrons and half octahedrons are 

 used, we get layers \vluch may be shifted as layers in direc- 

 tions parallel to their plane faces. Once shifted so that the 

 triangular faces of pyramids and tetrahedrons no longer 

 coincide, there is no shifting of strips ; but when these faces 

 are brought into coincidence, as in the building up of the 

 four based pyramid, we see that longitudinal strips, formed 

 of rows of pyramids with tetrahedrons fitted in between 

 them, can be shifted in the direction of their length. There 

 are four other ways of sliding laj-ers with corresponding 

 ways of sliding strips which will be recognised from the study 

 of what follows. 



SecotuUi/, when space is built up of tetrahedrons and 

 octahedrons, in the manner indicated above in the solution 

 of Problem XXIII., as well as in that of the problem we 

 have just dealt with (both methods giving precisely the 

 same arrangement so far as the filling in of space is con- 

 cerned), we have the following ways of sliding layers. In 



