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Transactions Texas Academy of Science. 
The American Mathematical Society is now concerning itself officially 
with college entrance requirements and other allied subjects. Undoubt- 
edly the text-books of the near future will differ considerably from those 
now in use, and will make mathematical instruction far more effective. 
It is being seen more and more clearly that geometry and algebra can 
be studied effectively from grammar to graduate school. In other words, 
that portions of these subjects are sufficiently easy and important to be 
studied by the many, while other portions possess difficulties that render 
them worthy of the trained specialist. Thus the raising of any number 
to any power is a fit subject for the grammar school when by any number 
is meant any positive integer, whereas the same operation is worthy of 
study by the advanced college student when by any number is meant the 
complex variable. The obvious moral to be deduced from such facts is 
that very elementary mathematical texts must omit much and are to be 
judged almost as much by what they omit as by what they contain. 
Neither algebra nor geometry are subjects that can be given with entire 
completeness and rigor to high school or even young college students. 
Not only must much be omitted, but now and then a theorem completing 
a theory may be quoted, the proof being explicitly omitted on account of 
its difficulty. Of course, it is not for a moment to be maintained that 
what goes into an elementary text should be actually wrong, either in 
matter or logic. As far as the text goes it should be accurate and logical, 
and a further study of the subject ought to extend, but not revise, the 
knowledge furnished by it. Fortunately over-refinement is now regarded 
as almost as great an evil as an actual blunder. The good text-book will 
steer wisely between too great completeness and too great rigor on one 
hand and blunders and pseudo-proofs on the other. 
To give some idea of the changes now going on in the teaching of ele- 
mentary geometry and drawing it away from the strict Euclidean mode, 
we may note the growing popularity of models; the greater use of con- 
crete figures drawn to scale, from which the truths of theorems may be 
inferred by Baconian simple enumeration; the testing of theorems by 
drawing several figures before attempting the formal proof : the immense 
increase in the use of co-ordinate paper; the breaking down of the walls 
that divide arithmetic, algebra, and geometry from each other; the ten- 
dency to simplify proofs by explicit assumptions or omissions ; the grow- 
ing disinclination to plunge the student into abstract discussions of 
axioms, postulates and definitions. The failure of the old methods to 
teach geometry to any considerable percentage of pupils and the vast 
number it marred in making have led and will lead to changes. More 
and more will we hear of inductive geometry and mathematical labora- 
tory work, more and more will the geometric formalist and purist be 
relegated to graduate schools, rarer and rarer will become the arith- 
