. 53. OF SIMPLE FORMS IN PARTICULAR. 45 



parallel to the sides, of the base. These axes intersect 

 each other at angles of 45. 



4. The horizontal projection is a square, equal to the 

 base, or to the square principal section. 



5. Let the side of the horizontal projection be = 1 ; the 

 axis = a, the terminal edge = x ; the lateral edge, or that 

 at the base = z ; we obtain : 



1 I n J 



cos. x = _ . cos. z = 



1 + a 2 1 + a 2 



. 53. SCALENE FOUR-BIDED PYRAMIDS. 



The scalene four- sided pyramids, Fig. 9., are con- 

 tained under eight scalene triangles. 



1. The scalene four-sided pyramids have three principal 

 sections, ABXB', ACCSIC and BC'B'C, all of which are 

 rhombs. 



2. The remaining sections, all parallel to the principal 

 ones, are likewise rhombs : the axes AX, BB', CC' there- 

 fore are of the third kind. These axes, the solid an- 

 gles through which they pass, and the sections belonging 

 to them, are called Prismatic, on account of the great 

 number and variety of oblique angular four-sided prisms 

 existing among the forms in connexion with the scalene 

 four-sided pyramid, all of which possess axes, solid angles, 

 and sections of this kind. Those denominations of axes, 

 solid angles, and sections, are likewise made use of in such 

 forms as are not connected with the scalene four-sided 

 pyramid. 



3. Any axis of the scalene four-sided pyramid can be 

 assumed as the principal one, or any solid angle can be 

 considered as the apex. After the principal axis has been 

 obtained, the subordinate axes, apices, solid angles, the base 

 and its diagonals, are thereby ascertained, and remain inva- 

 riable in all considerations of a determined form of this kind. 



