52. On Mr. Levy’s Property of the regular Octahedron. 
is some wit in his concluding paragraph; but wit is a poor 
substitute for argument. Yet it is, perhaps, the best resource 
in the absence of the latter, as it frequently makes a man ap- 
pear, on quitting the field, equal, though seldom superior to 
his adversary. I remain, sir, your obedient servant, 
Gloucester Place, Hackney Road, JAMES Burns. 
January 4, 1826. 
. < : c = c 
Errata in the formule, page 345 : For sin. rE read sin.2 as and for 
sin. 3 y, read sin.? 4 y. 
VII. Demonstration of Mr. Levy’s Property of the regular 
Octahedron ;—with a Postscript on P. Q’s Defence of Mr. 
Herapatn’s Demonstration. By T.S. Daviss, Esq. 
"Pus very neat but simple A 
theorem was given by its 
discoverer (unaccompanied how- 
ever by the demonstration) to 
Mr. Brooke. The latter gen- 
tleman’s proof (Crystallography, 
pp- 317, 318) is unnecessarily 
complicated; and is, besides, 
effected by means not strictly 
mathematical. The following 
one, it is presumed, is liable to 
neither of these objections. 
Theorem.—Let ABCD be a 
plane cutting off one of the solid 
angles E of a regular octahe- 
dron ; then 
l i 1 1 
ae + De =p + Ee’ 
Demonstration.— We assume the truth of the following 
well-known elementary properties: 
1. The diagonals AD, BC of the plane of section intersect 
in some point F in that diameter of the octahedron which 
passes through E. 
2. The angles AED, BEC are right angles. 
$. The line EF bisects these right angles. 
Then, if 2 EDA = ¢, we have, by trigonometry, 
BB+ 5{ sing cia EGF} |i _s{=e [eos 1g 
DE 7 singS°+tp@ EA cin45°+o° 
EF EF i ‘ _— 
whence abl a ORME ERED Ut Ah Ang 
DE AE: sin 45° 25 Q 
the lower sien, ** —,” referring to the position D!A’. 
5 8 p I 
n 
