[Read before the Texas Academy of Science March 6 , IS96.] 
PRISMOIDAL FORMULAE: WITH SPECIAL DERIVATION 
OF TWO-TERM FORMULAE. 
BY THOMAS U. TAYLOR, 
C. E., University of Virginia; M. C. E., Cornell University; Associate Professor of Ap¬ 
plied Mathematics University of Texas; Associate Member American Society 
of Civil Enginers; Member of American Mathematical Society; 
and Fellow of Texas Academy of Science. 
INTRODUCTION. 
The well known formula which bears the name of Sir Isaac Newton has 
so long been called the “ prismoidal formula” that confusion has arisen 
in many text-books as to its proper application. It is usually established 
for the prismoid, and its allied forms—the prismatoid and cylindroid— 
and its more general application to the volume of any solid whose cross- 
sections are cubic functions of their distance from one base is entirely 
omitted, notwithstanding the fact that its application to these higher 
solids can be established without the use of the calculus. The first real 
prismoidal formula was established by Karl Koppe in 1838, and it is a 
two-term formula, as can be seen by referring to these pages. I defer 
further remarks upon special formula till the next article (2). 
Articles 8, 9, 10, 15, and 16 were read before the Texas Academy of 
Science May 3, 1895; articles 11, 12, 13, and 14 were read before the 
Cosmos Club of the University of Texas on May 4, 1895. All of articles 
8-20 are extracts from my thesis at Cornell on “ Prismoidal Formulae,” 
June, 1895. 
THOS. U. TAYLOR. 
Austin, Texas, March 6, 1896. 
HISTORICAL NOTES. 
Reference has already been made to the range of the Newtonian form¬ 
ula, and no allusion will be made to the various proofs of it, except that 
of Steiner. In 1842 Steiner, by elementary geometry, proved that the 
Newtonian formula gives the exact contents of a prismoid in its general 
sense. Four years before, in 1838, Karl Koppe published his celebrated 
formula in Crelle’s Journal. I here call attention to the fact that the 
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