GEORGE BRUCE HALSTED-THE NEW MATHEMATICS. 
93 
Thus what Descartes really did was to make what we would now call a 
new algebra, the third algebra, calling the algebra of discrete number the 
first algebra, and that geometric algebra the second in which the product 
of two sects was a rectangle, the product of three sects was a cuboid, 
and in which we might now call the product of four sects a tesseroid. 
In Descartes’ algebra the product of two sects is a sect, of three sects still 
a sect, etc. 
In the second algebra, add to each sect the compound quality direc¬ 
tion, and Grassmann’s algebra emerges, in which the product of two 
directed sects, or vectors, is the parallelogram they determine, and the 
square of a vector is zero. 
Add direction to each sect in Descartes’ algebra, and you have Qua¬ 
ternions, or else a sixth algebra, according as the square of a unit vector 
is taken equal to minus or plus the number one. 
Thus notice that even before the conception of continuous number had 
come into anyone’s head, the world had multiplication of magnitudes, 
which certainly were not abstract numbers; and it is wholly from studies 
of such multiplication that our present knowledge of the operation has 
come. 
Yet our Cornell friend, Mr. Jones, on page 6 of his Drill-Book, tells 
us, “ The product of two concrete numbers is an absurdity.” On page 
2 he has given us as an example of a concrete number 12f yards. 
Now 12| yards multiplied by itself gives as a product in the second 
algebra a rectangle containing 160-| square yards; in Descartes’ algebra, 
a sect, 16fff yards; in Grassmann, zero; in Quaternions, —160|q in the 
sixth algebra, -f 160f. 
An Algebra is an artificial language composed of symbols, with their 
laws of combination, and possessed of peculiar advantages in giving of 
actual relations representations which can be manipulated according to 
rules of operation and procedure, experimented upon to give new 
knowledge, according to organized processes. The first algebra was 
slowly formed throughout centuries, to investigate the properties of 
numbers. The digital numerals differ in origin from the operational 
signs of algebra, in that the numbers are contracted pictures of the 
quality they now represent; while the operation signs were gradually 
introduced for words. Modern algebras embrace all organized systems 
of symbols combining according to definite laws, and so extend to sub¬ 
ject-matter of which number constitutes but one element, or in which 
this element is entirely absent. 
Notice the thought wrongly attributed to Descartes, “A mere number 
or ratio.” Now “number” was never thought of by anyone as syn¬ 
onymous with ratio until long after 1637. 
This radical innovation, the creation of the epoch-marking paradox, 
