doctrine." We do not quote the foregoing to disparage metaphysics; for, equally with mathematics, they have been our study and our delight. But it shows how easily, and with apparent truth, allegations can be made against the noblest of the sciences.

The history of science will attest that every great step in the progress of science in general, and in the intellectual elevation of the human race, was preceded by renewed activity in respect to metaphysical inquiry, as is evinced in the epochs of Socrates, Abelard, and Bacon.

Again, Hamilton quotes, with apparent approval: "The mathematician is either a beggar, a dunce, or a visionary, or the three in one." This is verified by those illustrious beggars, dunces, and visionaries-Newton, Leibnitz, and La Place!

Again, he quotes: "A great genius cannot be a great mathematician." Compare this statement with Hamilton's own account of Descartes : "The greatest mathematician of his age, and in spite of his mathematics, also its greatest philosopher."

The assertion, "A great genius cannot be a great mathematician," harmonizes beautifully with the following: "We are far from meaning hereby to disparage the mathematical genius which invents new methods and formulas, or new and felicitous applications of the old." It also harmonizes, even more beautifully, with the following quotation made by Hamilton: "There is, no doubt, a point at which the mathematics themselves require that luminous power of invention, without which it is impossible to penetrate into the secrets of nature. At the summit of thought, the imaginations of Homer and Newton seem to unite."

Again, Hamilton says: "The principles of mathematics are self-evident; and every transition, every successive step in their evolution, is equally self-evident. But the mere act of intellect, which an intuitive proposition determines, is, of all mental energies, the easiest-the nearest, in fact, to a negation of thought altogether. But as every step in mathematical demonstration is intuitive, every step in mathematical demonstration calls forth an absolute minimum of thought; and as a faculty is always evolved in proportion to its competent degree of exercise, consequently mathematics, in determining reason to its

feeblest energy, determines reason to its most limited development." What in this quotation is true of mathematics, is equally true of all reasoning. The passage from the premises to the conclusion of any argument is always self-evident; for if not, as the premises are the only warrant for the conclusion, the step is unauthorized, and the conclusion unwarranted. But the true test is found in discovering and arranging the premises; in originating, not in following, a demonstration. Granting that some equations, when stated, can be readily solved by rule, yet others require the exercise of the greatest ingenuity to place them under the form to which the rule is directly applicable. Again, the equations which express the relations of the known and unknown quantities of problems, frequently cannot be found without the severest exercise of the mind.

It would afford us great delight to test those who so flippantly assert that "mathematics call forth an absolute minimum of thought," in originating demonstrations, transforming equations, stating problems, and overcoming difficulties which, as yet, have baffled the ablest mathematicians. They might find that more than a minimum of thought would be required to meet the test.

The assertion, "It requires, indeed, a most ingenious stupidity to go wrong, where it is far more easy to keep right," is contradicted by the experience of every teacher of mathematics. Even in geometry, where the demonstrations are usually given in the text-book, in which case the above assertion might be expected to prove true, if at all, a perfect demonstration is the exception, not the rule.

In reference to the class of faculties cultivated by the mathematics, one of Hamilton's authorities says: "We shall, first of all, admit that mathematics only cultivate the mind. on a single phasis. . . . So, likewise, on the other hand, the memory and imagination remain in a great measure unemployed; so that, strictly speaking, the understanding alone remains to them, and even this is cultivated and pointed only in one special direction."

Another of his authorities says: "Persons of an oblivious memory are likewise disqualified; for if the previous steps be forgotten, not a hundredth of the others can be retained-such, in these sciences, is the series and continuous concate

nation of the proofs." This looks as if the memory is called into exercise by mathematical study, as well as the understanding.

Another of his authorities says: "Some delight to investigate the causes and substances of things, and these are the philosophers properly so called. Others again, inquiring into the relations of certain accidents, are chiefly occupied about these, such as numbers and figures, and, in general, quantities. These latter are principally potent in the faculty of the imagination, and in that part of brain which lies toward its center; this, therefore, they have hot and capacious, and excellently conservative. Hence, they imagine well how things stand in their wholes and in relation to each other. But we have said that every one finds pleasure in those functions which he is capable of performing well. Wherefore, these principally delight in that knowledge which is situate in the imagination, and they are denominated mathematicians." This looks as if the imagination was exercised as well as the understanding and the memory.

Again, we find it quoted: "It is an observation which all the world can verify, that there is nothing so deplorable as the conduct of some celebrated mathematicians in their own affairs, nor any thing so absurd as their opinions on the sciences not within their jurisdiction. I have seen of them those who ruined. themselves in groundless lawsuits; who built extravagantly; who embarked in undertakings of which every one foresaw the ill success; who quaked for terror at the pettiest accident of life; who formed only chimeras in politics; and who had no more of our civilization than if born among the Hurons or the Iroquois. Hence, sir, you may form some judgment of how far algebra conduces to common sense." In reply to this quotation, it may safely be said that the author of it is an illustration of the fact that mathematicians are not the only persons destitute of common sense. Take lawyers, physicians, divines, farmers, mechanics, merchants, statesmen, or philosophers, and there will be found, even among distinguished names, those who entertain the most absurd notions on subjects not within their jurisdiction. Hence, sir, you may form some judgment of how far any of the pursuits of life conduces to common sense!

As to the influence of mathematics on religious belief, we find the quotations: "To cultivate astronomy and geometry is to abandon the cause of salvation and to follow that of error." "It infects them with fatalism, spiritual insensibility, brutalism, disbelief, and an almost incurable presumption." This harmonizes beautifully with the quotation from Voltaire: "Mathematics leave the intellect as they find it."

The necessary connection between mathematics and skepticism is illustrated in the case of Newton and Leibnitz. That its neglect is conducive to a correct estimate of moral evidence and to piety is illustrated in the case of Gibbon, whom Hamilton quotes thus: "As soon as I understood the principles, I relinquished forever the pursuit of mathematics; nor can I lament that I desisted before my mind was hardened by the habit of rigid demonstration, so destructive of the finer feelings of moral evidence, which must, however, determine the actions and opinions of our lives."

Seriously, is it necessary to state that a man's religious belief does not depend on his mathematics, when mathematicians, as well as the rest of mankind, are divided in reference to matters of religious faith? Mathematical study, so far from inducing skepticism in the present writer's own mind, has induced faith. Perplexed with the uncertainty attending political, moral, and religious questions, he had almost concluded that, to the human mind, truth must forever remain unknown, or, at least, its certainty be but probable; and, in his despair of ever finding it, was in danger of lapsing into universal skepticism. But a study of mathematics revealed to him the fact that there is truth which can be certainly known and positively demonstrated. Confidence was restored as to the reality of truth, and faith inspired in the ability of the human mind to succeed in its discovery and demonstration, and this confidence and faith have been carried into other departments of thought.

Again, Hamilton says: "It will be easily seen how an excessive study of the mathematical sciences not only does not prepare, but absolutely incapacitates, the mind for those intellectual energies which philosophy and life require." But is it not evident that the disqualification arises, not from a knowledge of mathematics, but from an ignorance of other things? Should it be said that an exclusive study of mathematics leads

to an ignorance of other things, we reply, no one advocates such exclusive study.

Again, he says: "Mathematics afford no assistance, either in conquering the difficulties, or in avoiding the dangers which we encounter in the great field of probabilities wherein we live and move." This objection has force only on the condition that it can be shown that it is the exclusive business of life to deal with probabilities, and that, in dealing with probabilities, the mathematics render no assistance. But is it the exclusive business of life to deal with probabilities? Many of the ordinary affairs of life, as well as the great commercial transactions of the world, require exact calculation. The arts of the surveyor, the navigator, the engineer, and the architect, so essential to the welfare of mankind, depend upon exact science. Even the problems which "we encounter in the great field of probabilities wherein we live and move," have their exact parts whose determination requires rigid deduction. Let it be remembered that there is a mathematical theory of probabilities which is applied to life assurance, which involves the interests of such vast and increasing numbers of our people.

In the ordinary forms of probability, the mathematics are not chargeable with the delinquencies and failures so frequently witnessed. To prepare the mind to grapple successfully with such forms of probabilities is not the province of mathematics. They make no such pretentions. A steamship is not to be condemned because it cannot accomplish a voyage overland across the continent. Let every thing stand or fall on its own merits, and let not one thing be chargeable with the deficiencies of another. Observation, experiment, and an enlarged experience, together with common sense, are essential to success in the various callings of life, and their absence is attended with failure; but let not these failures be charged upon mathematics which are in no wise responsible.

Again, he says, in reference to the comparative utility of the analytic and synthetic departments of mathematics as affording discipline for the mind: "Some are willing to surrender the modern analysis as a gymnastic of the mind. They confess that its very perfection, as an instrument of discovery, unfits it for an instrument of mental cultivation; its formulas mechanically transporting the student, with closed eyes, to the