GEORGE BRUCE HALSTED-TWO-TERM PRISMOIDAL FORMULAS. 
27 
which gives for V the expression, 
V = i a [2(Bi+B 3 +S)+(S—B x )a 2 / a t +(S—B 2 )a 1 / a 2 . . (4). 
With what precedes, many other considerations are connected, of which 
some will be mentioned, since they lead to remarkable results. Consider 
first the relation to one another of two sections made at respectively 
equal distances from the bases. 
Let the sections be S, and S„; their distances from the B, base a, and 
1 <5 7 11 
a 8 , then from (3) follows, 
a , S 1 = (aJ—a 1 a 3 )B 1 +(a®—a 1 a 3 )B 3 +4a 1 a 3 M, 
a a S 3 = (a®—a 1 a 3 )B 1 +(a 3 —a 1 a 8 '>B 8 +4a 1 a 3 M. 
Hence follows by subtraction, since a=a 1 +a 2 ^ l 
Sj—S 8 = (B 1 —B 3 ) (a 3 — a x )/a . (5); 
that is the difference of two sections at equal distances from the bases is to 
the difference of the bases as their distance apart is to the whole altitude. 
If the two just mentioned sections divide the altitude a of the prismoid 
into three equal parts, they may be called one-third sections, and if S x be 
the further from B x , then a^fa, and a 3 = ^a; consequently, 
S.-S 1 =*(B 1 —B.); 
and the formula (4) for volume becomes 
V=ia(B,+3S 1 ). (6). 
This last expression is remarkable for this, that to get by it the volume 
of the prismoid are necessary only the altitude and two parallel sections, 
namely, the lower base B 2 and the upper one-third section S 1? or the 
upper base B, and the lower one-third section S 8 . 
Taking B 1 +3S 1 as sum of four magnitudes, the last equation gives: 
The prismoid is equivalent to a prism of equal height whose base is the 
arithmetical mean between the lower base and triple the upper one-third 
section of the prismoid.” 
While working on this subject, in May, 1895, Prof. T. U. Taylor dis¬ 
covered that in the third edition of “ Haupsatze der Elementar-Mathe- 
matik,” by F. G. Meliler, Berlin, 1864, page 121, after showing (as on 
page 122 of Halsted’s Mensuration) that Newton’s rule holds for the 
volume of solids whose cross-section is a cubic function of its altitude; 
for which therefore the volume 
V=ah+l i bh 2 +tch 3 +ldh i ...... (7), 
the author adds: “ Remark.—For d = o we can put (7) in the form 
V=i/j[a+3(a+5 fM-c(|7i) 2 )], 
consequently if z is the section parallel to a at the distance §7i, 
V=|7i(a+3z).” 
In 1878, in Schlbmilch’s Zeitschrift, Becker published what, not having 
access to the original, I will quote from Halsted’s Mensuration, 1881: 
