. 103. OF THE CONNEXION OF FORMS. 103 



point of the produced axis, the straight lines A'a, A"3, 

 &c., X% X'% &c. These lines, and accordingly their 

 intersections S, S', &c. will be situated in planes, which are 

 perpendicular to the faces of the isosceles four-sided pyra- 

 mids ; and the lines BS, C'S, &c. are therefore equal to 

 each other. Thus likewise the triangles B3S, SaC', &c. 

 are equal and similar to each other, and the form obtained 

 by the derivation is a simple one, namely, the scalene 

 eight-sided pyramid a'BSC'S'B'S'"CS'% 



The designation of the scalene eight-sided pyramids is 

 (P -f n) ra , in agreement with . 92. It comprehends as it 

 were at once the two pyramids (P + n) m and (P + n) m of the 

 mentioned paragraph, which forms, in the present case, would 

 be equal and similar. The axis of the scalene eight-sided 



pyramid is = 2 5 . m. a ; where 2 5 . a is the axis of P + n. 



The relative length of the axis of the eight-sided and the 

 four-sided pyramids, is expressed by the number m, as in 

 . 92. The only values of this number yet ascertained by 

 observation relative to the pyramids, are 3, 4, and 5 ; and 

 although it cannot be determined in general, yet it must 

 always be rational, positive, and greater than 1 + *J 2 

 (. 56. 5.). This supposition is necessary for making it possible 

 to determine the position of scalene eight-sided pyramids 

 among themselves, and towards isosceles four-sided pyra- 

 mids. If m is equal to 1 + >J 2, the eight-sided pyramid 

 is isosceles ; if it is less than 1 + ^/2, the acute terminal 

 edges are transformed into the pbtuse ones, and vice versa. 

 And since every scalene four-sided pyramid, derivable from 

 P + n, according to a certain m less than 1 + ^/2, can like- 

 wise be derived according to another m greater than l+y^/2, 

 from another more obtuse isosceles four-sided pyramid P -|- n' 

 connected with P ; this supposition, by excluding the above 

 mentioned values of m, produces at once simplicity and clear* 

 ness in the consideration of these forms. By the supposition 

 of m being greater than 1 + fj 2, it is also possible to avoid 

 a double designation of the same kind as that mentioned in 

 . 95. These considerations, however, yield a formula 

 for changing any pyramid (P + n) m , in which m is less 



