126 



♦ KNOWLEDGE ♦ 



[Aug. 24, 1883. 



a, side, by taking three cousocutive numbers in each of three lesser 

 squares of nine cells, horizontally or vertically, exclusive of his, 

 which must include the centre cell. There are (32 more variations 

 taking one or other of the two diagonals in each small square in 

 combination with one or other of the diagonals in two other smaller 

 squares. Numerous other variations may be made combining three 

 contiguous numbers in each of three smaller squares : thus, three 

 numbers, taken horizontally in one, vertically in another, and dia- 

 gonally in the third,— 35, 28, 33, with 20, 25, 24, and 65, 68, 71 ; or 

 12, 14^ 16, with 40, 39, 44, and 67, 68, 69, without any reference to 

 the centre cell. Again, the whole square is " tessellated," i.e., is 

 made up of nine lesser squres of nine cells each, each lesser square 

 taken separately being itself a magic square. 



2 : 



7 : 3 



G. S. concludes by remarking that a very little investigation will 

 show the principle upon which his figure is constructed. I question 

 whether this will satisfy most of your readers. My construction 

 is simple. The eight small square is Agrippa's model for all his 

 odd squares. The mean of the progressive numbers is put in 

 the centre, the first term below and the last above the centre. 

 If n represent the number of cells in a side, then the whole 

 series is divided into batches of n terms. The first batch, com- 

 mencing with 1, is placed in order downwards and diagonally. 



to the front and connected with the intercostal muscles by some 

 fine, tendonous threads. The flaps were partly distended when the 

 creature was lying on its back and immersed in water, and were 

 capable of sufficient distension to allow three of my fingers to be 

 placed in the hollow formed. Not having noticed such a construc- 

 tion in other animals, I wondered whether it was the rudiment of 

 the wing (so to speak) of the flying squirrel, which I am disposed 

 to think it is. It might be described as a huge armpit. 



Geo. Combe Willums. 

 [Mr. Williams's interesting letter was sent in May, but nn- 

 fortunately was mislaid. — E.P.] 



PARCEL POST PROBLEM. 



[905] — Determine the shape and cubic contents of the largest 

 package which can he sent by the parcel post, supposing the length 

 of a cylinder or solid of any other form to he the maximum distance 

 in a straight line between any two points on its surface. 



The shape of the parcel must be that of the volume common to a 

 sphere and a right circular cylinder, the axis of which passes through 

 the centre of the sphere. Let a; = radius of sphere 

 y = radius of cylinder. 



47r V 3 f , ,^# ) 

 Then volume common to both ='o"')* ^^ V )' i 



and 2a; -t- 2n-i/ = 



.'. 3? — (a;- — 1/-) - = a maximum. 

 Eliminating dx, d>j from the diflterentials of these equations, we get 

 TT (x' — :evx''—y^') = yVx^ — y- 



or dividing by y' 



putting - = :, and reducing 



y 



TT^l. 



■2t, 



-3 _ 





On a number of this batch leaving the bottom of the square, it 

 is transferred to a corresponding square at the top, vertically 

 above ; should it run out on the right side, it is transferred to 

 the corresponding horizontal cell on the left hand side. The first 

 of the second batch is placed two cells below the last term of the 

 first batch, and the diagonal system continued. Should the first 

 or any other of this batch fall without the square, it is brought 

 in as before explained. The first term of the third batch is placed 

 two cells below the last of the second, and so on. Thus the eightli 

 square was formed. The other squares are arranged in accordance 



with it. The whole series, through 1, 2, 3 to 79, 80, 81, is 



divided into nine sets, 1, 2, 3, &c., 9; 10, 11, &c., 18 ; 19, 20, 4c., 27, 

 &c. The first set is placed in the square corresponding with the cell 

 in the small square represented by 1, i.e., middle of bottom row. 

 The next set is put in the square corresponding with 2, i.e., the 

 right upper comer, and so on. What Agrippa calls the seal or 

 character of this square (dedicated to Saturn) is really the direc- 

 tion in which theii' numbers are placed, as in figure. J. 0. M. 



CURIOUS STRUCTURE IN THE SQUIRREL. 



[904] — After removing the skin of a common squirrel, I was 



struck by observing two thin flaps of muscular tissue extending 



from the superior part of the fore-arms to the region of the floating 



ribs ; this flap formed part of the muscle of the back, and was open 



Contents 



■■ 2-333 cubic feet. 



dy 

 dy^l 



dX TT 



Putting for tt its value, and solving the equation (which has only 

 one real root), we get 



; = 5 = 1-9625 



V 

 also X + Try = Z 

 Whence length of parcel = 2x = 2-307 feet. 

 „ girth „ = 277y = 3-693 do. 



-. ^.r'- (a;--r'l'' j =: 



= 2i do. H. F. 



[H. F. also sends the correct solution of the problem dealt with 

 at p. 76.— R. P.] 



LETTERS RECEIVED, A^"D SHORT ANSWERS. 

 A. W. Should not care to-publish such wonders till I had seen 

 them myself, and tested pretty closely. — Minnie. Saturn is in 

 Taurus. — J. D. VdS. Fear I can only offer you the Paradox 

 Column ; but I would point out, in a note, where your trisection of 

 any angle by simple geometry was invalid, or for what reason it 

 failed to solve the insoluble. — A. T. Fkasek. Prefer not to open a 

 vexed and vexing question. — A Lady Mathematicun. " Mad Tom" 

 was joking, of course. Non-Euclidean mathematics may be de- 

 scribed as mathematics based on axioms inconsistent with our con- 

 ceptions. A new arithmetic might equally well (as indeed Clifford, 

 Helmholtz, and others suggested) be based on the axiom that 2 and 

 2 make three, which in some universe unknown to us, they may do. 

 But we may wait before we give time to such new arithmetic 

 or geometry until the unknown universe to which they belong 

 begins to loom above oui- intellectual horizon. — G. G. Chisholm. 

 Cannot say ; but have great difliculty in finding space for all which 

 I should like to give in these columns. — ^W. H. B. Rannie. Quite 

 impossible to bring out the Star Maps again in Knowledge — it 

 would be unfaii' to a great number of our readers, who have 

 already had these Star Maps. The great bulk of our first sub- 

 scribers must (so far as can be judged) be with us still. — Jas. E. 

 Rattle, M. Molynettx, D. E. H. R., Renfrew, Gr.4Vesenu Doctoe, 

 and others. Think it best to insert no more prescriptions for 

 Cholera patients. — Jas. .T. Hill. The problem is insoluble in that 

 form, the velocity of a running stream depending on other matters 

 than the fall. — Bank. The star atlas you mention must be imper- 

 fect if it fails to show you to what part of the star sphere the 

 earth's axis points, southwards. There is no conspicuous star there, 

 but the place is in the constellation Octans, shown in the middle of 

 the twelfth map of my School Atlas. 



