30 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
Theorem,” I also very naturally but incorrectly attributed to Koppe. 
Professor Taylor having procured a copy of Koppe’s book, pointed out 
that no mention or even hint of the Associate Theorem occurs therein. 
i 
Koppe’s own demonstration of his theorem is singularly elementary, 
but utterly different from the one I gave in Scientiae Baccalaureus. His 
theorem as he states it is as follows: 
On page 5 of “ Ein neuer Lehrsatz der Stereometrie,” by Karl Koppe, 
1843, §5. “Theorem: 
“ Each obelisk is equal to the sum of a prism and a pyramid, both of like 
altitude with the obelisk, and whose bases correspond in the angles with 
the bases of the obelisk, while the sides of the base of the prism equal the 
half sum and the sides of the base of the pyramid the half difference of 
the like-lying sides of the two bases of the obelisk.’’ 
§6, Art. 7, p. 20.—“ The application of our theorem extends in general 
to all bodies with two parallel bases and any curved lateral surface on 
which straight lines can be drawn, whatever the curve lines be which 
bound the bases. 
“As example may serve a solid whose two bases are ellipses.’’ 
§ 8, p. 25.—“ If in a trapezoid the smaller parallel decreases to zero, the 
trapezoid passes over into a triangle; one may therefore consider the tri¬ 
angle as a special case of the trapezoid, which arises if the smaller parallel 
vanishes. Our theorem consequently must still hold also for such solids 
as have parallel bases, and for lateral surfaces part’y trapezoids, partly 
triangles.’’ 
Instead of the older and better name prismoid, Koppe used the word 
Obelisk, suggested by a friend of his, an official in Berlin. There are 
articles on these Koppe Obelisks in Crelle, Vols. 18, 23, 25, and Grunert’s 
Archiv, Vols. 9 and 11. 
What Prof. T. U. Taylor has called the “Associate Theorem ’’ is given 
with an elementary proof in the well known “ Elemente der Mathematik” 
of Dr. Richard Baltzer, second edition, 1867, where it is thus stated: 
“The segment of any ruled surface contained between parallel planes can 
be expressed by two cylinders and a cone. Construct between the par¬ 
allel planes both the cylinders whose bases are those surfaces of the seg¬ 
ment lying on the parallel planes, and also the associate cone of the ruled 
surface, whose vertex lies on one of the parallel planes, and whose ele¬ 
ments in order lie parallel to the straights of the ruled surface. Twice 
the solid-segment is the sum of the two cylinders, less the associate cone.” 
V=a|(B 1 +B 3 )-aiB c . 
