158 ANNUAL OF SCIENTIFIC DISCO VEUY. 



therefore, more convex) than the mean radius of the moon by about 

 eighteen parts in 1,000, and, of which, the centre is situated nearer to 

 the earth than that of the whole moon, by seventy-three thousandths 

 of such mean radius (seventy-nine English miles). The portion of the 

 moon, then, turned toward the earth may be considered as a continu- 

 ous mountain mass, in the form of a meniscus lens, capping the sphere 

 of the moon, and rising in its middle to a height of about seventy-nine 

 English miles above the general level of its figure of equilibrium. 



THE MATHEMATICS OF THE BEEHIVE. 



In the Annals of Natural History (London), 1863, will be found an 

 analysis of the mathematics of the beehive, by the Rev. S. Houghton, 

 in which the theory of the bee's forming hexagonal cells is completely 

 overthrown. Lord Brougham, in his treatise Dialogues on Instinct, 

 remarks : " There is no bee in the world that ever made cylindrical 

 cells ; " and the fact of the existence of hexagonal cells in the honey- 

 comb is generally quoted as a wonderful example of instinctive com- 

 bination of means to ends in a low form of animal existence. Mr. 

 Houghton, however, shows that the bee makes only cylindrical cells, 

 and that the hexagonal and rhomboidal cells are alike the result of 

 pressure, and represent the angles of equilibrium between the pressure 

 and the resistance, just as the orbits of the planets are the midway lines 

 between centrifugal and centripetal forces ; the bee is not, therefore, 

 such a mathematician as has been generally supposed. The alleged 

 economy of material resulting from the bee's method of working is also 

 shown to be fallacious. Several mathematicians have carefully inves- 

 tigated the relation of expenditure of material to the mathematical re- 

 quirements of connected cells of given dimensions, and of a form adapt- 

 ed to the uses to which they are to be put. L'Hullier, in 1781, arrived 

 at the conclusion that the economy of wax referred to the total expen- 

 diture is -g^st, so that the bees can make fifty-one cells instead of- 

 fifty by the adoption of the rhombic dodecahedron. He also showed 

 that mathematicians can make cells of the same form as those of the 

 bees, which, instead of using only a minimum of wax, would use the 

 minimum minimorum, so that five cells could be made of less wax 

 than that which now makes only four, instead of fifty-one out of fifty. 

 The humble-bee, moreover, in the construction of its cells, uses propor- 

 tionably more than three times the amount of material that is used by 

 the hive-bee. 



SUBSTITUTE FOR WOOD-ENGRAVING. 



The London Art Journal gives the following description of a process 

 invented by Mr. Schulze, a German architect, for producing blocks 

 for type-printing, to be used as a substitute for wood-engraving. " The 

 material on which the drawing is to be made may be of glass or any 

 other hard and smooth surface. The drawing is produced with a pen, 

 and ink composed of pure gum-arabic dissolved in water, with the ad- 

 dition of sufficient sugar to prevent it cracking when dry; lamp-black, 

 or any other color, is mixed with the gum solution to render the work 

 visible. When the drawing is completed, it is covered with a coat of 

 bees-wax, aspbalturn, resin, and linseed oil. The thickness of the cov- 

 ering depends on the kind of work adopted by the artist ; if the lines 



