[Presented to the Texas Academy of Science , April 5 , 1895.1 
THE CRITERION FOR TWO-TERM PRISMOIDAL FORMULAS. 
BY DR. GEORGE BRUCE HALSTED. 
I. 
The word Prismoid is a good old mathematical term, and has always 
had and kept the meaning recognized, for example, by Charles Hutton 
in his Mathematical Dictionary, the new edition of which was published 
in London in 1815. There, under the word, you read as follows: “Pris¬ 
moid ... Its ends are any dissimilar parallel plane figures of the same 
number of sides; the upright sides being trapezoids. If the ends of the 
prismoid be bounded by dissimilar curves, it is sometimes called a cylin- 
droid.” This meaning the word maintained down to my own college 
days at Princeton, where I remember it in the text-books of Loomis, and 
still maintains, See for example the Mensuration , in the latest edition of 
the Encyclopaedia Britannica. 
But the formula usually called by the name of this solid, but herein 
to be called Newton’s three-term Prismoidal formula, went far beyond 
the prismoid in its exact applicability. 
Newton (Methodus Differentialis, published 1711; further carried out 
by Cotes, on Newton’s Meth. Dif. in the works collected posthumously, 
1722) showed how an area or volume could be evaluated approximately 
from parallel cross-sections, and especially that from three cross-sections, 
following at the same distance apart, we get approximately the enclosed 
segment if we add the outer sections to four times the mid section and 
multiply the sum by a sixth of the distance between the outer sections. 
Maclaurin (1742, Fluxions, No. 848), referring both to Newton and 
Cotes, made additions which indicate that this special rule of Newton’s, 
the Old Prismoidal Formula, gives the content exactly when every sec¬ 
tion parallel to the base is a function of its distance from it of a degree 
not higher than the third, 
/(»)— n o +' »i n+n z ® :2 + n 3 • 
After a century of applications to areas and volumes, in 1842 Steiner 
conquered it by elementary geometry, and indicated its applicability to 
warped or ruled surfaces. 
But, in seeming ignorance of all this, American engineers began and 
fl9) 
