1893.] Elasticity of a Crystal according to Boscovich. 7 



perpendicular to it from the opposite angle. The planes through this 

 perpendicular, and the angles of the triangle, contain the perpen- 

 diculars from these angles to the opposite faces of the tetrahedron, 

 and therefore cut the opposite sides of the triangle perpendicularly. 

 (Thus, parenthetically, we have a proof of the known theorem of 

 elementary geometry that the perpendiculars from the three angles 

 of a triangle to the opposite sides intersect in one point.) Let ABC 



be the chosen triangle and S the point in which it is cut by the per- 

 pendicular from 0, the opposite corner of the tetrahedron. AS, BS,, 

 CS, produced through S, cut the opposite sides perpendicularly, and 

 therefore we find the point S by drawing two of these perpendiculars 

 and taking their point of intersection. The tetrahedron is then found 

 by drawing through S a line SO of the given length perpendicular 

 to the plane of ABC. (We have, again parenthetically, an interesting^ 

 geometrical theorem. The perpendiculars from A, B, C to the planes 

 of OBC, OCA, OAB cut OS in the same point; SO being of any 

 arbitrarily chosen length.) 



33. I wish now to show how an incompressible homogeneous, 

 solid of wholly oblique crystalline configuration can be constructed 

 without going beyond Boscovich for material. Consider, in any 

 scalene assemblage, the plane of the line A'OA through any point O 

 and its nearest neighbours, and the line B'OB through the same 

 point and its next-nearest neighbours. To fix the ideas, and avoid 

 circumlocutions, we shall suppose this plane to be horizontal. Con- 

 sider the two parallel planes of points nearest to the plane above it 

 and below it. The corner C of the acute-angled tetrahedron OABC, 

 which we have been considering, is one of the points in one of the two 

 nearest parallel planes, that above AOB we^ shall suppose. And the- 

 corner C' of the equal and dichirally similar tetrahedron OA'B'C' is 

 one of the points in the nearest parallel plane below. All the points 

 in the plane through C are corners of equal tetrahedrons chirally 

 similar to OABC, and standing on the horizontal triangles oriented 



