94 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
continuous number, is due to Newton. Newton makes this vast step ex¬ 
plicitly and consciously. The lectures which he delivered as Lucasian 
professor at Cambridge were published under the title “Aritlimetica 
Universalis.” 
At the beginning of his “Aritlimetica Universalis,” he says: 
“Per Numernm non tam multitudinem unitatum quam abstractam 
quantitatis cujusvis ad aliam ejusdem generis quantitatem quae pro 
unitate habetur rationem intelligimus. Estque triplex; integer, fractus, 
et surdus: Integer quem unitas metitur, Fractus quern unitatis pars 
submultiplex metitur, et Surdus eui unitas est incommensurabilis. * * 
* * Quantitates vel Affirmative sunt seu majores nihilo , vel Negative 
seu nihilo minores . ” 
Here we have at once the whole continuous system of real number, 
containing not only the absolute negative, but the general irrational, for 
notice that here a “ surd ” is not a “ root,” but the abstract ratio to the 
unit sect of any possible sect incommensurable with the unit sect. 
In a proof-sheet recently sent me by that sound geometer Hayward of 
Harrow, I read: “Number is essentially discrete or discontinuous , pro¬ 
ceeding from one value to the next by a finite increment or jump, and 
so can not, except in the way of a limit, represent, relatively to a given 
unit, a continuous magnitude, for which the passage from one value to 
another may always be conceived as a growth through every intermedi¬ 
ate value.” 
But the moment we accept Newton’s definition of number it is no more 
discrete than is a line-segment or sect. 
After Newton it was possible to use “ number or ratio ” as synonymous, 
perhaps up to the time of Gauss. But what the English language still 
needs is any adequate explication of this very idea “ continuous.” 
Mathematicians who write in English really attempt no definition of 
continuous; adopting with Clerk-Maxwell (Matter & Motion, Art. XXV) 
the holding up of a scrap supposed to have the quality, or quoting Aris¬ 
totle’s inadequate “ the boundary separating two contiguous parts is 
common to both.” [Can a number have a boundary?] Then they 
hasten to say that the geometric magnitudes are continuous; and then 
the entire system of real numbers or ratios, inasmuch as it contains an in¬ 
dividual number to correspond to every individual point in the continu¬ 
ous series of points forming a straight line, is continuous . 
This word “ continuous” is even introduced at the very beginning of 
some elementary geometries (W. B. Smith, Introductory Geometry, p. 3), 
though no adequate idea can be attached to it before the demonstration 
of incommensurability. Most people live and die without ever knowing 
or conceiving that lines, even pieces of the straight line, exist, which can 
have no common unit of measure, try units as small as you choose. 
