34 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
formula published is not the one that moderns have ascribed to him (see 
article *7). In 1843 this “ New Theorem in Stereometry” was issued in 
a small pamphlet of 35 pages. 
In the Journal of the Franklin Institute, December, 1857, page 372, 
W. M. Gillespie showed that the Newtonian formula applies to the volume 
of earthwork when the upper surface is generated by a straight line mov¬ 
ing on two longitudinal straight lines, thus making the upper surface a 
hyperbolic paraboloid. This proof was based on elementary principles, 
but it had realty been given long before. It can be readily shown that 
any cross-section is a quadratic function of its distance from either base, 
and therefore the Newtonian formula applies. Still the method of Gil¬ 
lespie’s proof was elegant, and it has been widely copied.* In 1864 
Mehler, in his “Elemental’ Mathematik,” page 121, established the two- 
term formula where the volume is found by multiplying one-fourth of 
the altitude by the sum of either base and three times a section at two- 
thirds the altitude from such base. The method of his proof was as 
follows: 
If S x =a-|-bx-)-cx 2 -]-dx 3 , 
We have by Newton’s formula 
V=?|(6a+3bH+2cH 8 +fdH 3 ). 
If we make x=|H, we get the section G at two-thirds the altitude. 
... G = a+|bH+fcH 8 + f 8 T dH 3 . 
But V=j(4a+2bH+fcH 3 +dH 3 ) 
II dlFM 
= T [a+3(a+fbH+fcH 8 +-3-J . 
If d = 0, we have 
V=|(a+3G). 
Professor Beman, of Ann Arbor, has called m}^ attention to the fact 
this two-term formula was published by Hermann Kinklin in 1862, in 
Grunert’s Archiv der Mathematik und Physik, Yol. XXXIX. Acting 
upon this reference hint, I secured a copy of Kinklin’s proof of his two- 
term formula in May, 1895. Since the reading of this paper before the 
Texas Academy of Science on March 6, 1896, this MS. copy of Kinklin’s 
work was furnished Dr. Halstead, and he has kindly printed a translation 
of it in his pamphlet on the criterion for two-term prismoidal formulae 
just issued, where those interested can find it. 
In 1894 Professor Echols, by the use of Elliot’s Extension of Hold- 
itch’s Theorem (Annals of Mathematics, November, 1894), showed that 
there is an infinite number of two-term formulae. The present paper was 
primarily undertaken to establish this result by elementary mathematics. 
* See Gillespie’s “Roads and Railroads.” 
