if the case demands, draw sketches where applicable, write out the statement arrived at in language different from that used by the author, study each word and the best method of expression, and practise to be concise and to omit everything unnecessary to the exact meaning. Herndon in his "Life of Lincoln" says of that great man, "He studied to see the subject matter clearly and to express it truly and strongly; I have known him to study for hours the best way of three to express an idea." This kind of practice inevitably leads to a thorough grasp of a subject.

Some of these principles may be illustrated by considering the study of the algebraical conditions under which a certain number of unknown quantities may be found from a number of equations. The student will perhaps find the necessary condition expressed by the statement that "the number of independent equations must equal the number of unknown quantities." Now this statement makes little or no concrete 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, 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 ΣH=0; ΣV=0; ΣM=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 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 binomial theorem can give (without hesitation) the square of 21, or of 21.5, or any similar quantity. With practice and reflection, results which seem astonishing may be attained.

(f) KEEP THE MIND ACTIVE AND ALERT.—Do not simply sit and gaze upon a book, expecting to have ideas come to you, but exert the mind. Study is active and intelligent, not dreamy. By this is not meant that haste is to be practised. On the contrary, what might perhaps be called a sort of dreamy thinking often gives time and opportunity for ideas to clarify and take shape and proportion in the mind. We often learn most in hours of comparative idleness, meditating without strenuous mental activity upon what we have read. Such meditation is of the greatest value, but it is very different from the mental indolence of which the poet speaks when he says:

"'Tis thus the imagination takes repose
In indolent vacuity of thought,
And rests and is refreshed."

This is beneficial to the proper extent; but it is rest, not study.

(g) WHEN YOU MEET WITH DIFFERENCES OF OPINION UPON A SUBJECT, REFLECT UPON THE REASONS WHICH MAY CAUSE INTELLIGENT MEN TO ARRIVE AT DIFFERENT CONCLUSIONS.—These reasons are:

1. One or both may fail to grasp all the pertinent facts, or even the problem itself, or may assume, as true, facts or principles which are really erroneous. This should easily be ascertainable.

2. One or both may reason incorrectly even from accurate premises. This also should be discoverable.

3. One or both may see facts out of proportion—may lack a true mental balance or perspective.

4. One or both may illustrate the inherent stubbornness or imperviousness of the human mind.

Whether the student can discover the last two sources of error will depend upon his own mental characteristics. He must not forget, however, that on many matters no definite demonstrable conclusion is possible, and that the result must remain more or less a matter of opinion.

(h) REMEMBER THAT A STATEMENT IS NOT A PROOF. MANY STUDENTS THINK THEY PROVE A STATEMENT BY MERELY REPEATING IT IN DIFFERENT WORDS. YOU DO NOT UNDERSTAND A CONCLUSION UNLESS YOU CAN SEE THE STEPS IN ITS LOGICAL DEMONSTRATION.

It is quite surprising how many students commit this error. For instance, if I am asked why can I see through glass and I reply, because it is transparent, I am giving no reason at all, for transparent means what can be seen through, so I am simply saying that I can see through glass because I can see through glass. The same error often occurs in arguments or syllogisms. For instance, suppose I make the following statements:

No unsportsmanlike act should be done;
Smith's act was unsportsmanlike;
Therefore, Smith's act should not have been done.