HOW THE NEW MATHEMATICS INTERPRETS THE OLD. 
By George Bruce Halsted, A. M., Ph. D., Member of the London Mathe¬ 
matical Society. Professor of Mathematics in the 
University of Texas. 
Bead March 4, 1893. 
Sylvester told a story of C. G. J. Jacobi, that, having accepted the 
profuse hospitality of Cambridge University, having at dinner consumed 
astounding quantities of Trinity Audit Ale, the renowned man was 
asked: “Who are the greatest living English mathematicians?” and 
roared out: “ There are none! ” 
This happened about half a century ago, a decade before Rowan Ham¬ 
ilton published the first of his two monumental works on Quaternions; 
when Sylvester, Cayley, Boole, were youths; now they are immortals, 
one at least outranking the greatest of the Jacobis. 
I will not recall the creation of the vast Invariantive Algebra, or the 
Algebra of Matrices, or Quaternions, or even the Algebras of Logic. 
But Clifford attributes to Sylvester and Cayley the basis for assigning 
the true nature of arithmetical propositions. However that may be, we 
now know that it is objectionable to say, “And the units and the meas¬ 
ured things are conci'ete numbers,” as is said in the most ambitious of 
American algebras; or to say, “And the whole answer to the question 
how many is twelve apples, or twelve and two-thirds yards. Such num¬ 
bers are concrete numbers; and concrete numbers may be defined as 
measured quantities,” as is said, 1892, in Jones’ Drill-Book in Algebra 
(Cornell). 
’ We now know that “ such numbers ” are no numbers at all. 
Number is of essence abstract. Number is primarily a characteristic or 
quality of a collection or group. A number-symbol may be used to repre¬ 
sent an operation which obeys exactly the same formal laws as the number, 
yet no number is primarily an operation. Still that imperfection of the 
past hangs on as far as Byerly’s Integral Calculus, 1891, though Gauss 
made all the world familiar with j/ —1 or i as a number , an independent 
number-unit like the number 1, I call it neomon, and such numbers 
neomonic . 
In nature, each distinct thing is perceived as an individual. Each 
distinct thing is a whole by itself, a unit. The individual thing is the 
only whole, or distinct object, which exists in nature. 
But the human mind takes like individuals together and makes of 
them a single whole and names it. Thus we have made the concept a 
flock, a herd, a bevy, a covey, a drove, a family, a genus, a species. 
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