Chapter 3

In other words, we must analyze every statement which is the result of reasoning, or a statement of opinion, and see what objections, if any, can be brought against it, and then convince ourselves where the truth lies and why. The lawyer has excellent practice in doing this, for in making his own argument he is obliged to scrutinize it closely to discover what objections he would make to it, if he were the counsel on the opposite side. The lawyer, however, does not always limit himself to the discovery of the truth, but often seeks to discover and bring to bear unsound but plausible arguments to refute the other side; and by his skill in dialectics he may often deliberately "make the worse appear {34} the better reason." The student of mathematics, on the other hand, does not gain in that study much practice in weighing evidence or seeking objections to an argument, for he deals with principles which are rigid and not open to question. Professor Palmer, in his interesting book, "The Problem of Freedom," says: "Until we understand the objection to any line of thought, we do not understand that thought; nor can we feel the full force of such objections until we have them urged upon us by one who believes them." This is precisely what the advocate endeavors to do beforehand, and in the court room he is very sure to have the objections to his line of thought urged upon him and the jury by one who at all events _appears_ to believe them.

(_d_) IN STUDYING A STATEMENT, OBSERVE WHICH ARE THE NECESSARY WORDS AND WHETHER THERE ARE ANY UNNECESSARY ONES WHICH MIGHT BE OMITTED.--For instance, in the following sentence, "When a force acts upon a body, and the point of application of the force moves in the direction of the line of action of the force, the force is said to do work on the body," what is the necessity and significance of the qualifying phrase "in the direction of the line of action of {35} the force?" Are these words necessary, or could they be omitted?

Note whether another word could be substituted for one used, without rendering a statement incorrect, or whether such change would improve it and make it more accurate. For instance, in the definition "Matter is that which can occupy space" would it be proper to substitute "does" for "can" or "occupies" for "can occupy"?

Note what word or words should be emphasized in order to convey the intended meaning. In the sentence "Thou shalt not bear false witness against thy neighbor," several widely different meanings may be conveyed according to the word which is emphasized.

Students frequently seem to lack all sense of proportion and fail to acquire definite ideas because they do not see the meaning or necessity of qualifying words or phrases, or because they do not perceive where the emphasis should be placed.

(_e_) REFLECT UPON WHAT IS READ: ILLUSTRATE AND APPLY A RESULT AFTER REACHING IT, BEFORE PASSING ON TO SOMETHING ELSE.[5]--Apply it to cases entirely different from those {36} shown in the book, and try to observe how generally it is applicable. Do not leave it in the abstract. An infallible test of whether you _understand_ what you have read is your ability to _apply_ it, particularly to cases entirely different from those used in the book. An abstract idea or result not illustrated or applied concretely is like food undigested; it is not assimilated, and it soon passes from the system. In illustrating, so far as time permits, the student should use pencil and paper, 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 {37} 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, {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 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."

{40} 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.

{41}

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.

Now, this of itself is not correct reasoning, for the reason that the word "unsportsmanlike" simply means something which no sportsman should do. The conclusion, therefore, is simply a repetition of the second statement. The real thing to be proved in this case is whether Smith's act was or was not unsportsmanlike.

[1] "General ideas and great conceit are always in a fair way to bring about terrible misfortune."--_Goethe_.

[2] "I tell you earnestly and authoritatively (I know I am right in this) you must get into the habit of looking intensely at words, and assuring yourself of their meaning, syllable by syllable--nay, letter by letter."--_Ruskin: Sesame and Lilies_.

"Neither is a dictionary a bad book to read--it is full of suggestions."--_Emerson_.

Benjamin Franklin, writing to a lady who asked him to give her advice about reading said:

"I would advise you to read with a pen in your hand, and enter in a little book short hints of what you find that is curious or that may be useful ... and as many of the terms of science are such as you cannot have met with in your common reading, and may therefore be unacquainted with, I think it would be well for you to have a good dictionary at hand to consult immediately when you meet a word you do not know the precise meaning of. This may at first seem troublesome and interrupting, but it is a trouble that will daily diminish, and you will daily find less and less occasion for your dictionary, as you will become more acquainted with the terms; and in the mean time you will read with more satisfaction because with more understanding."

[3] "A man who has no acquaintance with foreign languages, knows nothing of his own."

[4] "The Principles of Argumentation" by Baker and Huntington, is another excellent book, not treating of formal logic, but discussing the general principles which should govern the preparation of a paper or argument, the principles of evidence, and the logical fallacies in reasoning. It is recommended to readers. This book is, or has been, used in the course in English at Harvard University, and similar books are used in other colleges. A thorough training in English under a good teacher is a good training in logic, for clear and logical writing requires clear and logical thinking. Nevertheless, the writer strongly advocates the study of formal logic also.

[5] "It is not enough to know, we must also apply; it is not enough to will, we must also do."--_Goethe_.

{42}

III

THE THIRD ESSENTIAL FOR A PROPER METHOD OF STUDY IS SYSTEM

(_a_) DISCOVER THE FUNDAMENTAL IDEA OF THE SUBJECT.--Strip off the detail and get down to the root of the thing. See the really important point. Then, after this has been clearly perceived and mastered, arrange the details in their proper relations to the fundamentals. The subject will thus have a skeleton, and upon this the details will be placed. A subject of study thus viewed may be compared to the human body, with its bony skeleton or framework, and all the various organs and parts supported by it; or to a tree, with its trunk, branches and leaves. Thus to consider the relative importance of facts, to sift out the essential ones, will train the power of mental discrimination and cultivate the judgment.

When this is done, subsequent facts relating to the subject can be correlated with what is already known, and will in this way be easily retained by the memory. Remember and observe Jacotot's maxim, "Learn something accurately, and refer {43} the rest to that." Unessential facts, or those of secondary importance, may be passed over in the first reading, and left for a second or later reading, for a proper method of study _always involves re-reading_, perhaps many times.

You cannot possibly know everything even of a single subject, hence the importance of knowing the fundamental things about it and knowing them thoroughly. Even if you gain but an elementary knowledge of a subject, that knowledge may be thorough and should include fundamentals. Thorough elementary knowledge must not be confused with _a smattering_. The latter is worse than useless, and is marked by vagueness, uncertainty, and failure to grasp fundamentals. But elementary knowledge, if clear and definite as far as it goes, is valuable, and the first step toward more complete knowledge. Many students deceive themselves and others into thinking that they know something of a subject, because they have looked into it, while their knowledge may be entirely superficial and valueless.

When the fundamental principle or fact is perceived, study this carefully until it is thoroughly mastered. One who knows how to study properly will thus pick out the sentence or the paragraph which contains the key to the {44} subject--the fundamental fact or principle--and will read and re-read this many times until its full meaning is clearly grasped. When this is done it is sometimes remarkable how quickly the rest of the chapter or subject may be mastered, for it will often be found to consist of discussions or illustrations, which will be obvious once the fundamentals are clearly in the mind. The ordinary student, however, does not do this. He does not see the fundamental principle, and each illustration is like a separate problem, different from the others, which has to be studied by itself, and is never fully mastered, because the underlying fundamental principle is not grasped.

(_b_) BEFORE YOU BEGIN TO STUDY A SUBJECT, THINK IT OVER CAREFULLY AND FIND OUT WHAT YOU ALREADY KNOW OF IT OR WHAT YOU CAN ARRIVE AT BY YOUR UNAIDED EFFORTS.--Try also to perceive what you expect to get out of the study of the subject, and how it is related with what you have already studied, and how it is to find application.[1] The historian, Edward Gibbon, states in his autobiography that before reading any book, he made it a rule to reflect {45} upon the subject, arranging and classifying what he already knew of it.

This method may be followed to different degrees, depending on the subject. A student beginning the study of a new science which he has never studied before, can do comparatively little; but at least he can insist upon getting a clear idea of what the subject or problem is, its extent, what its objects and methods are, how it is related to other subjects, what its uses are, and how other studies will find their application in it.

(_c_) CLASSIFY AND ARRANGE WHAT YOU HAVE LEARNED.--When you have finished part of a subject, stop and think over the ground that has been covered, and arrange the various points made. Draw up a topical index and compare it with the table of contents. Note the correlation or interdependence of facts and link them together. By the principle of association the retention of facts and principles in the memory will be much facilitated. Note down concisely the steps of an argument in your own words, and see if the conclusion is justified. Close the book from time to time and go over in your mind what you have learned.

The importance of systematic classification is very great. The minds of many students are {46} like a library without arrangement or catalogue; the books may be there, but cannot be found when wanted, and so are valueless for use.[2]

[1] "We must keep carefully that rule of Aristotle which teaches that the best way to learn anything well which has to be done after it is learned, is always to be a-doing while we are a-learning."--_Richard Mulcaster_.

[2] "There's a vast difference between having a carload of miscellaneous facts sloshing around loose in your head and getting all mixed up in transit, and carrying the same assortment properly boxed and crated for convenient handling and immediate delivery."--_Lorimer: Letters from a Self-made Merchant to his Son at College_.

{47}

IV

MENTAL INITIATIVE

It will become evident from the foregoing that a fourth essential for proper study is mental initiative. The student must have a definite purpose, and must do what is the proper thing without it being suggested to him. He must not simply do as he is told. If he have not initiative and cannot develop it, he will probably never study intelligently, nor gain a thorough understanding of what he reads, but will merely memorize.

Memory is a most important faculty; it is not, however, a _substitute_ for thought, but should be based upon it. Thinking is essential in order to decide what to memorize. Memory, however, is often made the sole factor in study. Fundamental principles should frequently be memorized, so that by numberless repetitions they may be permanently impressed upon the consciousness, and can be repeated verbatim as a guide in any concrete case where they are to be applied.

Some suggestions may be useful as to the use and cultivation of mental initiative.

{48}

(_a_) CULTIVATE AN INTEREST IN WHAT YOU ARE STUDYING, AND SOME IDEA OF WHAT IT LEADS TO.--Without interest your study will be perfunctory and of little use to you. Make yourself believe that for you, at that time, it is the most important thing in the world. It is of course true that in most schools students are required to study definite subjects according to a curriculum arranged by the faculty. In some of these subjects a student may take little interest; indeed they may be so foreign to his natural tastes that he is not able to cultivate any interest in them. In such a case his study of them will be of little value to him. If, relying upon the judgment of those who prescribe the curriculum as necessary or desirable for the object which he has in view, he cannot persuade himself that they have value for him or make himself take an interest in them, it would probably be better for him to drop them even though he may thereby become a special student in the school or lose his degree. A degree which simply means slipshod, unintelligent and uninterested study of a considerable number of subjects embraced in the curriculum, is verily a "scrap of paper" not worth having. If you wish to concentrate your entire attention upon certain subjects in which {49} you take an active interest you may become proficient in those, but you may become very narrow minded and altogether lacking in that all-around breadth of view which comes from the cultivation of other subjects which well informed men consider necessary.

(_b_) INSIST UPON FIRST CLEARLY FORMULATING THE PROBLEM, IF ONE IS BEFORE YOU.--Many students literally do not know what they are doing, because they neglect this injunction, which is a necessary corollary of the necessity of forming definite ideas. Do not proceed to endeavor to solve the problem until it is clearly formulated, no matter how long it may take. See what the data of the problem are, whether definite or not, and what is required. See also how variations of the data, if indefinite, would affect the result.

(_c_) WORK INDEPENDENTLY OF OTHERS.--Solve your own difficulties and welcome them. Do not expect things to be easy. You will never gain strength by being shown, but only by the exercise of your own unaided powers. Therefore, do everything for yourself, so far as possible. Seek only _suggestions_ from your teacher, when you need help, except in regard to mere matters of fact, which you could not be expected to {50} reason out. Let the suggestions be as slight as possible.

If you have problems assigned, solve them entirely by yourself, even if you make mistakes. Then, when those mistakes are pointed out, consider them with great care and discover the causes for them, and _remedy them_, so that you will not again make the same mistake or one analogous to it. You should delight in discovering difficulties which give you an opportunity to test and increase your strength and so avoid future errors. In the same way, examinations should be welcomed, not dreaded. The teacher does not mark you--you mark yourself; the teacher merely records the mark. Even if you fail in the examination, that should indicate to you what you lack, and so be a benefit. Indeed, it is better to fail than to scrape through.[1] There must be a line somewhere. The man just above the line passes, and the man just below the line fails. The former may not be as capable as the latter, but, having passed, he does not remedy his faults; while the man who has failed is required to remedy his. Huxley said that the next best thing to being right is to be completely and wholesomely wrong.

{51}

(_d_) DRAW YOUR OWN CONCLUSIONS, WHENEVER POSSIBLE, BEFORE YOU KNOW THOSE OF THE WRITER You ARE STUDYING.--When you read, "From the above it is evident," stop, close the book, and see if you can state what is evident. When you have written this down, compare with the result reached by the writer. Practise such exercises in whatever form they present themselves. If your conclusions are different from those of the writer, in kind or in character, see which is right, or whether both are right. If you are right, why did the writer not reach your conclusion? Was it because it was not pertinent to his problem? Is it simply a difference of expression?

The process of investigating any subject is a process of question and answer. The student must first propound to himself a question, and it must be the proper question. He must be able to perceive what the proper question is, under the circumstances. Then he must give to himself the proper answer out of all the possible answers that are verbally correct, namely, the answer that affords a new vantage ground from which another question may be asked; and so the problem may be gradually unravelled.

Then again, many questions are indefinite, and {52} can only be answered indefinitely; but to all questions a correct answer can be given, and the student must give the most definite answer the case admits of, and must gain the ability to qualify his answer or classify possible cases in such manner as may be necessary.