132 



♦ KNOW^LEDGE ♦ 



[April 1, 1887. 



diamond faces. The figure of the rhombus may be defined 

 as having its larger angle equal to that subtended liy an 

 edge of a regular tetrahedron from the tetrahedron's centre 

 of figure. Thus, let abcd, fig. 2, be a regular tetrahedron ; 

 BE perpendicular to and bisecting AC ; df perpendicular to 

 BE (ef one-third of be) ; fg one-fourth of df (so that G is 

 the centre of gravity, and of figure, of the tetrahedron 

 ABCD. Join AG, GB, BC. Then agc, bgc, bga, are severally 



Fig. 3. 



halves of the rhombus required ; and agbc is one of the 

 three-planed solid angles of the solid required. It is easy 

 to obtain D(; by a geometrical construction; or arithmetically 

 thus, putting the edge of the tetrahedron equal to 2« : — 

 dg=|i>f=Jv/db- — BF-; 

 also DB''=4n^; BF^=|BE^=|a* ; 



^ _- re _ , 49« 



wherefore DG=Jv/^«==a v/S or ^x/6 = l-225a nearly=^^ 



nearly. 

 Thus dg=ag=bg=cg=aw forty-nine eightieths of AC, 

 ab, bc, or BD. 



The diamond faces for our solid can be readily constituted 

 from this known proportion. They have the shape and form 

 in the twelve diamonds of fig. 3, which represents what is 



called the '' net " for the required solid. The student will 

 find it interesting to cut out such a " net " in card, dividing 

 half through the card along ab, bi, hi, (fee, and then bending 

 the card into the form of the solid shown, in two aspects, in 

 figs. 4 and .'j. 



MATHEMATICAL RECREATIONS. 



OUR PUZZLES. 



OR this month I suggest three puzzles con- 

 nected with the last three, and also with 

 those which preceded them in March, 

 February, and even January (when the 

 orange-packing puzzles appeared). Any 

 one who studies out the geometrical and 



arithmetical relations involved in these 



puzzles will find that he has gone through 



instructive exercises. 



Puzzle XXVIII. tihow how to build up n rer/ular 



fetrnhedron of smnller ietrahedroiis and octahedrons having 



equal triaiufular faces and edges commensurable in length 

 with those of the hitlt U}} tetrahedron. (Simplest form of 

 the problem is to build four tetrahedrons and one octahedron 

 into a tetrahedron.) 



Puzzle XXIX. A number of equal spherical shells, of 

 highly elastic skin-like material, are filled with a gas ivhich 

 expands energetically when its temperature is raised. They 

 are piiled up in pyramids like cannon-balls, and the pyramids 

 enclosed in rigid casings ; or they are packed in boxes like 

 the oranges in the January piuzzles, remaining all globukir 

 and equal. Being theyi exposed to a constantly increccsitig 

 temperature, the shells expand, losiiig their globrdar form, 

 and eventually becoming everywhere flat-faced. Determine 

 the shapes the shells assume in (he middle of the heap, where 

 the pressures around each may be assumed to be uniform. 

 i^Near the enclosing casings, of course, the pressures are not 

 uniform.) 



Puzzle XXX. Six Alpine tourists, three of them men, 

 are nearing the place where they propose to take their midday 

 meal, when one of the ladies asks what they are to do for 

 seats. One of the men, of the kind defined by Dickens as 

 " ((n ingenious beast," propounds to the other two a phn by 

 which the six aljienstocks of the party can be converted, in a 

 few 7ninutes, into a comfortable and symmetrical structiire 

 on tvhich the ichole company may be seated. The party have 

 with them plenty of rope {being prudent mountaineers) and 

 a good supply of stout cord. Within less than ten minutes 

 after the arrival of the tourists at their resting-jjlace all six 

 are comfort nhly seated. What v^as the ingenious beast's plan? 

 {The alpenstocks are each six feet long.) 



Note. — For reasons not belonging to the domain of 

 mathematics, the three masculine tourists considered it 

 especially desirable that the party sho'ild be seated in pairs, 

 each pair in some degree apart from the others. The in- 

 genious one secui-ed this result also. 



Race Characteristics of the Jews. — Dr. A. Neubauer 

 read a paper recently, before the British Anthropological 

 Institute, on '• Race Types of the Jews," the purport of 

 which was to show tliat there had been considerable inter- 

 mixtures in the Hebrew race from the time of Abraham 

 down. Jo.seph married an Egyptian and Moses a Midianite; 

 David was descended from a Moabitess, and Solomon was 

 the son of a Hittite woman. So we read of the non-Jewish 

 women in contact with the Israelites, and undoubtedly the 

 proselytes increased the mixture of races by marrying Jewish 

 women. INIoreover, some quite marked differences prevailed 

 in the Middle Ages, and .still exist, between the Jews 

 residing in different nations. Mr. J. Jacobs, in a paper 

 " On the Racial Characteristics of INIodern Jews," took a 

 different view. Regai-ding only the Askenasian Jews, who 

 form more than nine-tenths of the whole number, he pointed 

 out as among their characteiistics fertility, short stature as 

 compared with Europeans, and narrow chests, brachycephalic 

 skulls, darker hair and eyes than those of any nation in 

 Northern Europe (though nearly one-fifth of the Jews have 

 blue eyes, and they have nearly twice as many red-haired 

 individuals as the inhabitants of the Continent), and a 

 peculiar cast of countenance. He pointed out that the 

 pui'ity of the race depended on the number of proselytes 

 made by the Jews in ancient and Mediaeval times. The 

 earlier proselytes, before the foundation of Christianity, 

 were mostly fellow- Semites, and would not affect the type, 

 while the numbers made afterward were too small to modify 

 the race. A considerable number of Jews, the Cohens, were 

 not allowed to marry proselytes, and must consequently be 

 tolerably pure. Mr. Jacobs's general conclusion was there- 

 fore in favour of the purity of the Jewish race. 



