24 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
The next advance, the discovery of the most important two-term pris- 
moidal formula, occurs in a paper so little known (I read it for the first 
time March 7, 1896, in a MS. copy lent me by Prof. T. U. Taylor) that 
a translation of it is here given: 
ON THE THEORY OF THE PRISMOID. 
BT HERMANN KINKLIN, 
Teacher at the Gewerbschule in Basel. 
[Grunert’s Archiv, Vol. xxxix, 1862, pp. 181-85.] 
Suppose in space any continuous system of straights, the last return¬ 
ing into the first, then these enclose an incompletely bounded space. 
Cutting this space by two parallel planes, meeting each straight of the 
system, cuts out of it a solid called a prismoid. The two parallel plane 
cuts ai’e called bases, and the surface enclosed from the system of straights 
(a ruled surface returning into itself) is called the lateral surface. 
This lateral surface is in general curved, but may in special cases con¬ 
sist of plane pieces. 
Inversely, however, it can be said that the lateral surface in general 
consists of plane pieces, which in special cases may become an infinity 
and make a curved surface. 
Starting from this conception, in whatever follows the bases will be 
looked upon as any rectilineal polygons with relatively parallel sides [in¬ 
cluding zero sides], and the lateral surface as consisting of trapezoids 
side by side [including triangles], which immediately join the parallel 
sides of the bases. Special forms of the prismoid are, among others, the 
pyramid and cone, prism and cylinder, the frustum of a pyramid, the 
hyperboloid of one nappe, the obliquely truncated three-sided prism, the 
tent, the tetrahedron, etc. 
Altogether the prismoid is one of the most general body forms, and 
is worthy of theoretical and practical interest, the latter all the more 
as its content can be given by a very simple expression obtainable by 
most elementary means. 
The following properties appear not to have been noticed as yet. 
They concern size comparison of parallel plane cuts through the pris¬ 
moid. 
Suppose through it three equidistant plane cuts of which the two outer 
are the bases, and the middle is called the mid section. 
Designate the area of these three surfaces respectively as B 1} B 3 , M, 
and the distance between Bj and B 3 , the altitude, by a , so that M is 
distant from B l5 as also from B 3 
