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conclusion; whereas the ancient geometrical construction, they contend, leads him to the end, more circuitously, indeed, but by his own exertion, and with a clear consciousness of every step in the procedure. Others, on the contrary, disgusted with the tedious and complex operations of geometry, recommend the algebraic process as that most favorable to the powers of generalization and reasoning; for, concentrating into the narrowest compass the greatest complement of meaning, it obviates, they maintain, all irrelevant distractions, and enables the intellect to operate, for a longer continuance, more energetically, securely, and effectively. The arguments in favor of the study thus neutralize each other, and the reasoning of those who deny it more than a subordinate and partial utility, stands not only uncontroverted, but untouched—not only untouched, but admitted."
In claiming that the arguments in favor of the two methods neutralize each other, Hamilton draws his conclusions from the unfavorable opinion which the advocates of each method hold respecting the other.
Now, these views are negative rather than positive. It is far more rational to draw conclusions from the positive knowledge which the advocates of each method possess. The advocates of the analytical method find beauties and advantages in that method; hence, these beauties and advantages are there. The advocates of the synthetic method find clearness and discipline in that method; hence, clearness and discipline are there. The two methods, therefore, when both are together studied, instead of neutralizing each other, must, by their combination, result in a higher beauty, clearness, and discipline. Some have complained of the encroachment of modern analysis upon the synthetic method of the ancients; but this is the inevitable result of the progress of the science. Synthetic geometry will always have its place; but in the higher investigations, and the more difficult applications of mathematics, to abandon the analytic method for the synthetic would be analagous to abandoning the railroad, the telegraph, and all the inventions which characterize the present age, and going back to the primitive customs of our fathers.
It is a part of the business of education to furnish the mind with the facilities for investigation; and no more effective
instrument for this purpose has ever been discovered than the modern mathematical analysis.
We have always admired the opening paragraph of the first article written by the “ Autocrat of the Breakfast-table" for the Atlantic Monthly. He says: “I was just going to say, when I was interrupted, that one of the many ways of classifying minds is under the heads of arithmetical and algebraic intellects. All economical and practical wisdom is an extension or variation of the formula, 2 + 2 = 4. Every philosophical proposition has the more general character of the expression, a +b=c. We are mere operatives, empirics, and egotists, until we learn to think in letters instead of figures. We cannot, however, give our unqualified approval of the following quotation from the same popular author: “Given certain factors, and a sound brain should always evolve the same fixed product with the certainty of Babbage's calculating machine. What a satire, by the way, is that machine on the mere mathematician! A Frankenstein-monster, a thing without brains and without heart, too stupid to make a blunder, that turns out results like a corn-sheller, and never grows any wiser or better, though it grinds a thousand bushels of them. I have an immense respect for a man of talents plus the mathematics. But the calculating power alone should seem to be the least human of qualities, and to have the smallest amount of reason in it; since a machine can be made to do the work of three or four calculators, and better than any one of them. Sometimes I have been troubled that I had not a deeper intuitive apprehension of the relation of numbers. But the triumph of the ciphering hand-organ has consoled me. The power of dealing with numbers is a kind of detached level arrangement, which may be put into a mighty poor watch. I suppose it is about as common as the power of moving the ears voluntarily, which is a moderately rare endowment." The mere calculator is not a mathematician; but when a machine is invented which will work itself, and not only calculate, but develop formulas, conduct demonstrations, originate methods, it will then do to exclaim, What a satire on the mathematician!
The calculating power, though by no means the highest mathematical faculty, is nevertheless useful, and not to be despised.
It is the tendency of the human mind lightly to esteem what it cannot possess, and to draw consolation for its deficiencies from the consideration that what it does not possess is not worth possessing. This characteristic of humanity has been well satirized by the fable of "the fox and the grapes."
The mind adapted to excel in mathematical pursuits is not the mere calculator, nor the sluggish intellect, but is of that class which, as a general thing, succeeds best in language, in philosophy, and in the higher metaphysical speculations.
The mathematical sciences have given us some of our noblest thoughts. How have our conceptions of the perfections of God, and the vastness and grandeur of his empire, been exalted by the revelations of astronomy! By the discovery of the law
! of universal gravitation, mathematical science has demonstrated that all worlds are linked together in mutual dependence, constituting a universe. From the unity of creation we infer the unity of the Creator, a truth of the highest importance in theology, and which can, in no other way, be so satisfactorily determined.
That mathematical studies, when properly pursued, call forth and develop the powers of the mind, stands forth a demonstrated fact. Even the analytical method, which some, while adınitting its wonderful perfection as an instrument of investigation, regard as of little value as a means of education, is, when properly employed, in the highest degree efficient as an educational agency. One may, indeed, passively follow, with little apparent profit, the tranformations of an equation, through a variety of forms, till he reach the conclusion that a projectile describes a parabola. But no little effort is required to originate the demonstration, or even to ascertain the reasons for the successive steps.
There is a most beautiful philosophy in the analytical method which renders it a most profitable subject for study; and the application of this method, in investigating the laws of natural phenomena, presents an inexhaustible field for the exercise of the highest intellectual powers.
We do not advocate the exclusive study of mathematics, the tendency of which would be to neglect observation and experiment, and to attempt to deduce the facts, principles, and truths of
science from a few fundamental axioms. But even this result is chargeable, not to a knowledge of mathematics, but to a neglect of observation and experiment. Let this evil be corrected, not by neglecting mathematics, but by widening the range of thought, by enlarging our views from a comprehensive survey of the vast field of human knowledge. Because air is necessary to sustain life, shall we, therefore, neglect food and drink, and live on air? Let us have the air, by all means; but let us have the bread, beef, and water also.
Let suficient time be given in our colleges for the mastery of the calculus, and we shall not witness those ridiculous exhibitions of the celebration of its obsequies ; but our students would go out with their minds not only sharpened and invigorated by its acquisition, but furnished with the most powerful instrument for investigation.
We would respectfully suggest that our college courses are overcrowded. So much is attempted that but a superficial scholarship is secured ; and, what is still more deplorable, bad habits of study are induced. Let the languages, the mathematics, rhetoric, logic, mental philosophy, and the principal physical sciences be thoroughly studied, and the scholarship of our students is secured. Other studies, deemed important, might be made a requisition for the second degree.
ART. V.-THE BIBLE BETTER THAN THE ECUMEN.
THE civilized world is watching the great Council at Rome. Prelates, priests, and rulers may there take counsel together against the Lord and his Anointed; but “He that sitteth in the heavens shall laugh; the Lord shall have them in derision." Long before the Council met, a higher decree than it can make had been declared. A higher Sovereign than the Pope is set upon the holy hill of Zion. “ Blessed are all they that put their trust in Him."
In a certain professed religious community, calling itself the “liberal Church,” we are pointed to the advanced social and moral condition of humanity, and are exhorted to believe that it is the result of the movements of human agencies and civilization. In their view, the kingdom of God advances as the enterprise of men opens the way: through the cleft mountain, across the bridged river, over valleys exalted, crooked ways straightened and rough places made plain, the elements of an improved social condition are marching, and thus the glory of the Lord is revealed! The Bible, and what the Bible plainly teaches, is forgotten or ignored by such inculcation. Effect is mistaken for cause. The history of civilization would never have been written had there been no Bible studied and followed.
In another professed religious community, calling itself "the Catholic Church," ecclesiastical despotism attempts to shut up the Bible, and to smother the truths of revelation with the traditions and opinions of men. They would have the human mind renounce its freedom, and deny the supreme authority of the Holy Scriptures.
Every Protestant press and every Protestant pulpit should unite to expose these great defections from true Christianity, and now, more than ever, proclaim the grand truth, that the Bible, divinely inspired, is the only and sufficient rule of faith and practice. To its mandates alone should human belief and human conscience bow. Every other gospel is a spiritual deceit, though it wear the livery of the schools, or canonize its dogmas in councils. The happiness which society enjoys in civilized life, and the character which fits men for happiness in the life to come, must be attributed alone to the Bible. What. ever adorns the history of humanity, its government or laws, its civil or religious liberty, its social institutions or moral life, emanates from the influence of that divine book. “The vision is shut up-the testimony is sealed—the word of the Lord is ended, and this solitary volume, with its chapters and verses, is the sum total of all for which the chariot of heaven made so many visits to the earth, and the Son of God himself tabernacled and dwelt among us." Beyond what it reveals, the mysteries of eternity are unknown. Omniscient Wisdoin has stored therein marvelous truths which otherwise had dwelt, unrevealed to men, in the bosom of God. It is the emanation from the divine mind, and is replete with the treasures of heavenly wisdom. Perfection has here attained its end; for
Fourth SERIES, VOL. XXII.-6