ncrete
impression upon the minds of most students. They do not understand
exactly what it means, and they can easily be trapped into misapplying
it. To study it, the student should ask himself what each word of the
statement means, and whether all are necessary. Can the word
"independent" be omitted? If not, why not? What does this word really
mean in this connection? Must each equation contain all the unknown
quantities? May some of these equations contain none of the unknown
quantities? What would be the condition of things if there were fewer
equations than unknown quantities? What if there were more equations
than unknown quantities?

This problem too, affords a good illustration of the advantage of
translation into other terms? What, for instance, is an equation
anyway? Is it merely a combination of letters with signs between? The
student should translate, and perceive that an equation is really an
intelligible sentence, expressing some statement of fact, {38} in which
the terms are merely represented by letters. An equation tells us
something. Let the student state what it tells in ordinary
non-mathematical language. Then again, a certain combination of
equations, taken together, may express some single fact or conclusion
which may be stated entirely independent of the terms of the equations.
Thus, in mechanics the three equations _[sigma]H_=0; _[sigma]V_=0;
_[sigma]M_=0; taken together, merely say, in English, that a certain
set of forces is in equilibrium; they are the mathematical statement of
that simple fact. If the equations are fulfilled, the forces are in
equilibrium; if not fulfilled, the forces are not in equilibrium.

Following this farther, the student should perceive, in
non-mathematical language, that an equation is independent of other
equations if the fact that it expresses is not expressed by any of the
others, and cannot be deduced from the facts expressed in the others.

The benefit of translation into common, everyday language, may be shown
by another mathematical illustration. Every student of Algebra learns
the binomial theorem, or expression for the square of the sum of two
quantities; but he does not reflect upon it, illustrate it, or perceive
{39} its every-day applications, and if asked to give the square of 21,
will fail to see that he should be able to give the answer instantly
without pencil or paper, by mental arithmetic alone. Any student who
_fully grasps_ the b