Some minds are slow at gathering materials; yet they think vigorously. They look at facts and ideas from every possible point of view, explore their nature and relations, their content and extent, and point out their bearing upon other things by the conclusions they reach. Sometimes they go astray because they do not have sufficient data to warrant a conclusion. Their condition resembles that of the King of Siam, who did not believe that water could become solid because he had been in the nine points of his kingdom and had not seen ice.

Other men are intellectual gluttons. They keep pouring into themselves knowledge from every quarter, carry it in their minds as the overloaded stomach carries food, and end in mental dyspepsia. Better the man with few ideas, who can apply these in practical life, than the man of erudition who cannot apply his knowledge.

Too little food produces inanition and starvation; too much food brings on dyspepsia and a host of other ills and distempers. The haphazard selection of studies by inexperienced youth from the large list of electives offered by a great university is apt to result either in mental overfeeding or in intellectual starvation. The mind can be rightly formed only when it is rightly informed. To expect satisfactory thought-products when the mind lacks proper materials to act upon would be as irrational as to expect good grist from a flour-mill whose supply of grain is deficient in quality and quantity. In the process of making flour very much depends upon the instruments employed. The rude implements of antiquity, the buhr-stones of our fathers, and the improved machinery of the roller process make a difference in the product, even though the same quality of grain is used. In the elaboration of the thought-material the well-educated man uses instruments which may be likened to our modern inventions for saving labor in the domain of the mechanic arts. These instruments of thought will next claim our attention.

V

THE INSTRUMENTS OF THOUGHT

But words are things; and a small drop of ink Falling, like dew, upon thought, produces That which makes thousands, perhaps millions, think.

BYRON.

Constant thought will overflow in words unconsciously.

BYRON.

The great Lagrange specifies among the many advantages of algebraic notation that it expresses truths more general than those which were at first contemplated, so that by availing ourselves of such extensions we may develop a multitude of new truths from formulæ founded on limited truths. A glance at the history of science will show this. For example, when Kepler conceived the happy idea of infinitely great and infinitely small quantities (an idea at which common sense must have shaken its head pityingly), he devised an instrument which in expert hands may be made to reach conclusions for an infinite series of approximations without the infinite labor of going successively through these. Again, when Napier invented logarithms, even he had no suspicion of the value of this instrument. He calculated the tables merely to facilitate arithmetical computation, little dreaming that he was at the same time constructing a scale whereon to measure the density of the strata of the atmosphere, the height of the mountains, the areas of innumerable curves, and the relation of stimuli to sensations.

LEWES’S PROBLEMS OF LIFE AND MIND.

V

THE INSTRUMENTS OF THOUGHT

Of the people who, though inheriting a rich vernacular like the English, spend their lives in the routine of a farm, a trade, or a store, very few have an adequate conception of the labor-saving instruments and appliances which modern civilization places at the disposal of the thinker. The machinery by which one man does as much as a thousand hands formerly did is not a whit more wonderful than the modern appliances for reaching results in the domain of thought. Reference might be made to the machines for adding used in counting-houses, to the tables of interest used by bankers, to the tables of logarithms by which it is as easy to find the one-hundredth power as the square of a number. The last named have, so to speak, multiplied the lives of astronomers by enabling them to make in a short time calculations that formerly occupied months, and even years. It is not necessary to discuss these; their value is apparent at a glance. But the value of a rich vocabulary, the function of the symbols and formulas of chemistry, physics, mathematics, and other sciences, and the advantages derived from the use of the technical terms peculiar to every domain of thought are not so easily seen. The teacher who fails at the right time to put the pupils in possession of these instruments of thought cripples their thinking, wastes their time and effort, and seriously mars their progress. Hence it is worth while to devote a chapter or two to the consideration of instruments of thought, for the purpose of showing how, by means of them, thinking is made easier and more effective. Let some one write the amounts in a ledger column by the Roman notation, then endeavor to add them without using any figures of the Arabic notation, either in his mind or in any other way, and he will soon realize what a labor-saving device our ten digits are. Then let him face the problem of squaring the circle as it confronted Archimedes, using the obvious truth that the perimeter of an inscribed polygon is less, while the perimeter of the circumscribed polygon is greater than the circumference of the circle, and long before his calculations reach the regular polygon of ninety-six sides (which is as far as Archimedes carried it), he will realize how the great Syracusan was hampered by the lack of the arithmetical notation now in use. Next, supposing himself in possession of the Arabic method of notation, let him conceive the labor of Rudolph von Ceulen, who, before logarithms were known, computed the ratio of the circumference to the diameter to thirty-five decimal places,—an achievement considered so great that the result was inscribed upon his tombstone,—and then, turning to the calculus, let him examine the formulas by which Clausen and Dase, of Germany, computing independently of each other, carried out the value to two hundred decimal places, their results agreeing to the last figure; this will give him a conception of the superior instruments of thought invented by those who developed the calculus. His idea of the labor-saving devices introduced by the calculus will be heightened still more on learning that Mr. Shanks, of Durham, England, carried the calculation to six hundred and seven decimal places,—a result so nearly accurate that if it were correctly used in calculating the circumference of the visible universe, the possible error would be inappreciable in the most powerful microscope. On further learning that in 1882 Lindeman, of Königsberg, rigorously proved this ratio, commonly represented by the symbol π, to be incapable of representation as the root of any algebraic equation whatever with rational coefficients, he will not only refrain from joining the common herd of squarers of the circle, but no further argument will be needed to show the nature and value of the labor-saving devices introduced into the domain of thought by modern mathematics.

Since it is unreasonable to expect that every reader shall be familiar with higher mathematics, the duty of using simpler illustrations cannot be evaded. Fortunately for the purpose in hand, the book of experience furnishes these with an abundance that is almost bewildering.

A professor of chemistry was lecturing to an audience of teachers on agriculture. When he began to write upon the black-board they smiled at his spelling. Iron he wrote Fe. Water he spelled H₂O. They soon saw that he was using the instruments of thought furnished by a science with which, unfortunately, few of them were familiar. He had found that the use of these chemical symbols made his thinking as much superior to that of the ordinary man as the work of the youth upon a self-binder is superior to that of the giant working with no better instrument than the sickle of our forefathers.

The school furnishes numerous examples to illustrate this point. When the teachers of a well-known city began the use of objects to impart the ideas of number and of the fundamental rules in arithmetic, the interest of the pupils and their facility in calculation grew wonderfully. The teaching was in accordance with the laws of mental growth. For fear the pupils would manipulate the Arabic figures without corresponding ideas, collections and equal parts of objects were drawn upon the slate to illustrate addition and subtraction of integers and fractions. The plan was followed for years and carried upward through the grades. Finally the pupils were examined for admission into the high school. A problem involving the four fundamental rules in combinations which could not be illustrated by pictures of objects, or the objects themselves, was set for solution. Out of fifty-nine applicants, only ten succeeded in giving the correct answer. The same kind of problem was given three times by three different persons, and with practically the same outcome. The teachers realized that they had kept up for too long a time the thinking in things, instead of drilling the pupils upon the process of thinking in the symbols of the Arabic notation. It is, of course, possible to think number without using the Arabic digits. The Romans did so by means of their counting-boards, and the Chinese do so by devices of their own. The characters which were brought into Western Europe through Arabic influences are derived, according to Max Mueller, from the first letters of the Sanskrit words for the first ten numerals. Their use facilitated calculation to such an extent that arithmetic gradually ceased to be the prerogative of slaves and ecclesiastics; its operations began to be understood by freemen and by the nobility. If children are denied the use of objects in their early lessons in number, they resort to counting on their fingers. If they are not led from this thinking on their fingers to thinking in figures, they will never become expert in arithmetic. Sometimes the fingers no longer move, but the mind conceives pictures of the hand, and the mind’s eye runs along the fingers of hands not visible to the corporeal eye. It is equally bad if the pupils never think number except by mental pictures of blocks, sticks, balls, and the like. When the pupil sees 7 × 9, he should not conceive seven heaps of nine shoe-pegs each, and then a rearrangement into six groups of ten shoe-pegs, and three stray ones alongside of these groups; but instantaneously the symbols 7 × 9 should suggest, with unerring accuracy, the result,—63.

In the schools of another district the principal proposed concrete work in fractions. The teachers and pupils began to divide things into halves, and thirds, and fourths, and sixths. They added and subtracted by subdividing these into fractions that denoted equal parts of a unit. Whilst the charm of novelty still clung to the process, a stranger who visited the schools asked one of the teachers how the pupils and parents liked the change. “Everybody is delighted,” was the exclamation. A year later the same teacher was asked by the visitor, “How are you succeeding with your concrete work in fractions?” With a dejected air she replied, “We are disappointed with the results.” “Just as I expected,” exclaimed the visitor; “for you were making the children think on the level of barbarism, instead of teaching them to use the tools and labor-saving machinery of modern civilization.”