274 



BECEEATIVE SCIENCE. 



an oblong or right-angled parallelogram. 

 Let us consider how many such sections can 

 be made in a cube. A little consideration 

 will show that such a section can be made in 

 six different ways. If this be not quite clear, 

 let the student have recourse to the hollow 

 cube, formed of the folded envelope, and 

 place a piece of cardboard, cut to the proper 

 size and shape, in the position of each section 

 successively. 



Let the diagonals of these parallelo- 

 grams be drawn on the piece of card- 

 board J when, if again placed in the hollow 

 cube, it will be seen that these diagonals of 

 the section are the longest lines that can be 

 drawn within the cube, and that they corre- 

 spond to the second series of axes (Fig^ 6). 



As there are six diagonal sections of the 

 cube, and two diagonal lines to each section, 

 it would appear that there should be twelve 

 such lines ; -whereas, by referring to Fig. 6, 

 it is obvious that there can be but four — an 

 apparent deception, which depends on each 

 line being common to three of the sections. 

 Possibly my readers are beginning to see 

 that there is more in a cube than a mere 

 square block. 



Having spoken of the sections of the 

 cube made by cutting through two of its 

 faces and two edges, let us now proceed to 

 consider the sections that would be obtained 

 by cutting through three faces, or, in plain 

 terms, by cutting off a solid angle. The 

 simplest experiment with a cube of soap will 

 at once prove the section to be a triangle, 

 and that it may be so varied by cutting 

 through the three faces equally, or unequally, 

 as either to have its three sides equal, two 

 only equal, or all three unequal, and thus to 

 produce an equilateral, isosceles, or scalene 

 triangle. The largest section of this kind 

 that can be obtained is formed by cutting 

 through the diagonals of the three adjacent 

 faces, as from D to B, B to c, E to d (Fig. 5). 



The sections obtained by cutting through 

 four faces of the cube are very various. If 



the section is made parallel to any of the 

 faces, it is a square equal to the faces. A 

 very interesting section is obtained by cutting 

 from one solid angle to the opposite, as from 

 B to H. If this is made equally through the 

 four faces, it cuts the edges a e and c g 

 across at the middle of each, and the section 

 is a rhomb, or diamond, the long diagonal of 

 which corresponds to one of the longest axes 

 of the cube (Fig. 8), and the shorter to one 

 of those represented in Fig. 7. Let the 

 student endeavour to discover how many 

 such sections can be made in a- cube, and the 

 number will probably surprise him. 



A plane section made through five faces 

 of the cube, will, of necessity, have five 

 edges, one corresponding to each face cut 

 through, and will therefore be a five- sided 

 plane figure, or pentagon ; but as it is im- 

 possible to cut equally through five faces, 

 the regular, or equal-sided, pentagon is 

 never produced. 



If a cube of soap, or other soft solid, 

 were given to a person, and he were asked 

 whether, with one straight cut of a knife he 

 could cut through all the six faces, he would 

 in all probability state that feat to be impos- 



JFiG. 9. 



sible ; yet, in reality, nothing is more easy. 

 To demonstrate this fact, let the student draw 

 a cube, as Fig. 1, and place a dot at the mid- 



