why he takes the trouble of writing this treatise against such an opinion, as actually held, and held by a whole "school of mathematics?" Perhaps, he means by "any mathematician" -any mathematician worthy of the name. But then if this "school of mathematics" be so contemptible, why write, and that so seriously, against them? This, we may observe, is not the only contradiction in the pamphlet we have been wholly unable to reconcile.

But, in the fifth place, the contrast of the mathematician and metaphysician is itself in error.-In regard to the exculpation of the mathematicians, we need look no farther than to the late Sir John Leslie for its disproof. "Geometry" (says that original thinker, and he surely was a mathematician), " is thus founded likewise on observation; but of a kind so familiar and obvious, that the primary notions which it furnishes might seem intuitive.”—As to the inculpation of the metaphysicians-why was Locke not mentioned in place of Hume? If Hume did advance such a doctrine, he only skeptically took up what Locke dogmatically laid down. But Locke himself received this opinion from a mathematician; for this part of his philosophy he borrows from Gassendi: and, what is curious, he here deserts the schoolman from whom he may appear to have adopted, as the basis of his philosophy, the twofold origin of knowledge-Sense and Reflection; for the unacknowledged master maintains on this, as on many other questions, opinions far more profound than those of his disciple. But in regard to Hume, Mr. Whewell is wholly wrong. So far is this philosopher from holding "that geometrical truths are learnt by experience," that, while rating mathematical science, as a study, at a very low account, he was all too acute to countenance so crude an opinion in regard to its foundation; and, in fact, is celebrated for maintaining one precisely the reverse. On this point Hume was neither sensualist nor skeptic, but deserted Aenesidemus and Locke to encamp with Descartes and Leibnitz.

In the sixth place, the quality of necessity is correctly stated by Mr. Whewell as the criterion of a pure or a priori knowledge. So far, however, from this being a truism always familiar to mathematicians, it only shows that Mr. Whewell has himself been recently dipping into the Kantian philosophy; of which he here

1 Rudiments of Plane Geometry, p. 18; and more fully in Elements of Geometry and of Geometrical Analysis, p. 453.

adduces a famous principle and one of the most ordinary illustrations. The principle was indeed enounced by Leibnitz, in whom mathematics may assert a share; but that philosopher failed to carry it out to its most important applications. In his philosophy, our conceptions of Space and Time are derived from experience. We can trace it also obscurely in Descartes, and several of the older metaphysicians; but assuredly it was nothing “palpable," nothing to which the mathematicians can lay claim. On this principle, as first evolved-at least, first signalized by Kant, Space and Time are merely modifications of mind, and mathematics thus only conversant about necessary thoughts-thoughts which can even make no pretension to truth and objective reality. Are the foundations of the science thus better laid ?-But to more important matters.

It is an ancient and universal observation, that different studies cultivate the mind to a different development; and as the end of a liberal education is the general and harmonious evolution of its faculties and capacities in their relative subordination, the folly has accordingly been long and generally denounced, which would attempt to accomplish this result, by the partial application of certain partial studies. And not only has the effect of a one-sided discipline been remarked upon the mind in general, in the disproportioned development of one power at the expense of others; it has been equally observed in the exclusive cultivation of the same power to some special energy, or in relation to some particular class of objects. Of this no one had a clearer perception than Aristotle; and no one has better illustrated the evil effects of such a cultivation of the mind, on all and each of its faculties. He says:

"The capacity of receiving knowledge is modified by the habits of the recipient mind. For, as we have been habituated to learn, do we deem that every thing ought to be taught; and the same object presented in an unfamiliar manner, strikes us, not only as unlike itself, but, from want of custom, as comparatively strange and unknown. For the accustomed is the better known. How great, indeed, is the influence of custom, is manifested in the laws; for here the fabulous and puerile exert a stronger influence through habit, than, through knowledge, do the true and the expedient. Some, therefore (who have been over much accustomed to mathematical studies), will only listen to one who demonstrates like a mathematician; others (who have exclusively cultivated analogical reasoning), require the employment of examples; while others, again (whose imagination has been exercised at the expense of judgment), deem it sufficient to adduce the testimony of a poet. Some are satisfied only with an exact treatment of every subject; to others, again, from a trifling disposi

tion, or an impotence of continued thought, the exact treatinent of any becomes irksome. We ought, therefore, to be educated to the different modes and amount of evidence, which the different objects of our knowl edge admit."

And again:

"It is the part of a well-educated man to require that measure of accu· racy in every discussion, which the nature of its object-matter allows; for it would not be more absurd to tolerate a persuasive mathematician, than to astrict an orator to demonstration. But every one judges competently in the matters with which he is conversant. Of these, therefore, he is a good judge of each, he who has been disciplined in each, absolutely, he who has been disciplined in all.”

But the difference between different studies, in their contracting influence, is great. Some exercise, and consequently develope perhaps, one faculty on a single phasis, or to a low degree; while others, from the variety of objects and of relations which they present, calling into strong and unexclusive activity the whole circle of the higher powers, may almost pretend to accomplish alone the work of Catholic education.

If we consult reason, experience, and the common testimony of ancient and modern times, none of our intellectual studies tend to cultivate a smaller number of the faculties, in a more partial or feeble manner, than mathematics. This is acknowledged by every writer on education of the least pretension to judgment and experience; nor is it denied, even by those who are the most decidedly opposed to their total banishment from the sphere of a liberal instruction. Germany is the country which has far distanced every other in the theory and practice of education; and the three following testimonies may represent the actual state of opinion in the three kingdoms of the Germanic union which stand the highest in point of intelligence-Prussia, Bavaria, and Wirtemberg.

The first authority is that of:-Bernhardi, one of the most intelligent and experienced authorities on education to be found in Prussia.

"It is asked-Do mathematics awaken the judgment, the reasoning faculty, and the understanding in general to an all-sided activity? We

1. Metaph. 1. ii. ("Aλpa тò ëλatтov) c. 3, text. 14.

2 Eth. Nicom. 1. i. c. 3. The text universally received ("Exaσros dè kpiveɩ kadôs ἅ γινώσκει καὶ τούτων ἐστὶν ἀγαθὸς κριτής· καθ ̓ ἕκαστον ἄρα ὁ πεπαιδευμένος ἁπλῶς δὲ ὁ περὶ πᾶν πεπαιδευμένος), is at once defective and tautological. The cause of the corruption is manifest; the emendation simple and, we think, certain. Εκαστος δὲ κρίνει καλῶς ἃ γινώσκει, τούτων ἆρ ̓ ἐστὶν ἀγαθὸς κριτής· καθ ̓ ἕκαστον, ὁ καθ ̓ ἕκαστον πεπαιδευμένος, ἁπλῶς δὲ, ὁ περὶ πᾶν πεπαιδευμένος.

are compelled to answer-No. For they do this only in relation to a knowledge of quantity, neglecting altogether that of quality.-Further, is this mathematical evidence, is this coincidence of theory and practice actually found to hold in the other branches of our knowledge? The slightest survey of the sciences proves the very reverse; and teaches us that mathematics tend necessarily to induce that numb rigidity into our intellectual life, which, pressing obstinately straight onward to the end in view, takes no heed or account of the means by which, in different subjects, it must be differently attained."

The second authority we quote, is that of the distinguished philosopher who has long so beneficially presided over the Royal Institute of Studies in Munich-Von Weiller :

“Mathematics and Grammar differ essentially from each other, in respect to their efficiency, as general means of intellectual cultivation.2 The former have to do only with the intuitions of space and time, and are, therefore, even in their foundation, limited to a special department of our being; whereas the latter, occupied with the primary notions of our intellectual life in general, is co-extensive with its universal empire. On this account, the grammatical exercise of mind must, if beneficially applied precede the mathematical. And thus are we to explain why the efficiency of the latter does not stretch so widely over our intellectual territory; why it never develops the mind on so many sides; and why, also, it never penetrates so profoundly. By mathematics, the powers of thought are less stirred up in their inner essence, than drilled to outward order and severity; and, consequently, manifest their education more by a certain formal precision, than through their fertility and depth. This truth is even signally confirmed by the experience of our own institution. The best of our former Real scholars, when brought into collation with the Latin scholars could, in general, hardly compete with the most middling of these-not merely in matters of language, but in every thing which demanded a more developed faculty of thought."s

The third witness whom we call, is one, be it remarked, with

1 Ansichten, &c., i. e. Thoughts on the Organization of Learned Schools, by A. F. Bernhardi, Doctor of Philosophy, Director and Professor of the Frederician Gymnasium, in Berlin, and Member of the Consistorial Council, 1818.

2 Vide Morgensterni Orat. De Litteris Humanioribus, p. 11.

3 From a Dissertation accompanying the Annual Report of the Royal Institute of Studies, in Munich, for the year 1822, by its Director, Cajetan von Weiller, Privy Counselor, Perpetual Secretary of the Royal Academy of Sciences, &c. This testimony is worthy of attention, not merely on account of the high talent, knowledge, and experience of the witness, but because it hints at the result of a disastrous experiment made by authority of Government throughout the schools of an extensive kingdom ;an experiment of which certain empirics would recommend a repetition among ourselves. But the experiment, which in schools organized and controlled like those of Bavaria, could be at once arrested when its evil tendency was sufficiently apparent, would, in schools circumstanced like ours, end only, either in their ruin, or in their conversion from inadequate instruments of a higher cultivation to effective engines of a disguised barbarism. We may endeavor, erelong, to prevent the experience of other nations from being altogether unprofitable to ourselves.

"Felix quem faciunt aliena pericula cautum.”

a stronger bias to realism, in the higher instruction, than is of late, after the experience of the past, easily to be found in Germany. Professor Klumpp observes:

"We shall first of all admit, that mathematics only cultivate the mind on a single phasis. Their object is merely form and quantity. They thus remain, as it were, only on the surface of things without reaching their essential qualities, or their internal and far more important relations -to the feelings, namely, and the will-and consequently without determining the higher faculties to activity. 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. To a many-sided culture-to an all-sided harmonious excitation and development of the many various powers, they can make no pretension. This, too, is strongly confirmed by experience, inasmuch as many mere mathematicians, however learned and estimable they may be, are still notorious for a certain one-sidedness of mind, and for a want of practical tact. If, therefore, mathematical instruction is to operate beneficially as a mean of mental cultivation, the chasms which it leaves must be filled up by other objects of study, and that harmonious evolution of the faculties procured, which our learned schools are bound to propose as their necessary end."

To the same general fact, we shall add the testimony of one of the shrewdest of human observers, we mean Goethe, who in a letter to Zelter thus speaks:

"This also shows me more and more distinctly, what I have long in secret been aware of, that the cultivation afforded by the Mathematics is, in the highest degree, one-sided and contracted. Nay, Voltaire does not hesitate somewhere to affirm, "j'ai toujours remarqué que la géometrie laisse l'esprit ou elle le trouve. Franklin, also, has clearly and explicitly enounced his particular aversion for mathematicians; as he found them, in the intercourse of society, insupportable from their trifling and captious spirit." 2

Even D'Alembert, the mathematician, and professed encomiast of the mathematics, can not deny the charge that they freeze and parch the mind: but he endeavors to evade it.

"We shall content ourselves with the remark, that if mathematics (as is asserted with sufficient reason) only make straight the minds which

1 Die Gelehrten Schulen, &c., i. e. Learned Schools, according to the principles of a genuine humanism, and the demands of the age. By F. W. Klumpp, Professor in the Royal Gymnasium of Stuttgart. 1829, vol. ii. p. 41. An interesting account of the seminary established on Klumpp's principles, by the King of Wirtemberg, at his pleasure palace of Stetten, in 1831, is to be found in the Conversations Lexicon für neuesten Zeit, i. p. 727.

2 Briefwechsel zwischen Goethe und Zelter, 1833, i. p. 430.