42 
TRANSACTION'S OF THE TEXAS ACADEMY OF SCIENCE. 
sirable degree of suggestion in all questions of style. Or, to take an 
example more germane to our subject, it was no mean pedagogical 
achievement to conceive the idea of a comparative presentation of plane 
and spherical geometry. I wish to testify to the paradox that it is 
easier to learn plane geometry and pure spherics together than to com- 
prehend either separately. I would welcome a text-book offering a thor- 
ough-going parallel development. Both plane and sphere are surfaces 
of simple laws of being, and the advantages of a parallel development 
of the geometries of these two surfaces are too manifold to be set forth 
in this connection. It is enough to say that no other way could so em- 
phatically display the absolute and determining function of definitions 
and postulates in any mathematical discipline, and, indeed, in all or- 
ganized constructs of pure reason. 
But it is time now to show how, in my opinion, the teaching of men- 
suration can profit by generalizing comparisons. 
Waiving all questions of psychology as to the genesis of ideas of 
measurement, the measurement of any magnitude is the process of find- 
ing its ratio to another magnitude of the same kind arbitrarily chosen 
as a unit. The measure of a magnitude is this ratio — a number. Under 
the conventions of English speech the measure of any magnitude is ex- 
pressed by a phrase made up of this number and the name of the chosen 
unit. Thus, we say The length of this line is three yards/ meaning that 
the ratio of said line to the arbitrary line named “yard” is three. The 
physical fact, the spacial phenomenon, is the line, the surface, the solid; 
and the length, . the area, the volume, are numbers, viz., the ratios of the 
line, surface, and solid respectively to other magnitudes of like kind 
chosen arbitrarily as units. 
In regard to the terms length , area, and volume, writers of every rank 
might be cited as fostering by careless expression many pernicious er- 
rors. The average college graduate labors under the impression that 
his fashion of calculating areas and volumes is essential to the matter 
and arises from the very nature of things. He commonly regards the 
dangerously abbreviated statement, “the area of a rectangle is the product 
of its base and altitude,” as a proposition in the same Category, and as 
completely expressed, as that “the square on the hypothenuse equals 
the sum of the squares on the other sides.” It is true that, if you 
pressed him, he would see that the former requires the tacit understand- 
ing (always expressed, indeed, in his previous special definition of area) 
that the unit-surface be the square on the unit-line; but the idea that 
the area could possibly be anything else would be new to him. And he 
would be an extraordinary member of his class if he could explain the 
warrant for his method of calculation by pointing to the fundamental 
and truly general form of the theorem, viz., the ratio of the rectangle 
