the images or vicarious symbols, about which they are conversant, being clear and simple, the deductions of these sciences are apodictic or demonstrative; that is, the possibility of the contrary is at every step seen to be excluded in the very comprehension of the terms. On the other hand, in philosophy, (with the exception of the science of logic,) and in our reasonings in general, such demonstrative certainty is rarely to be attained; probable certainty, i. e. where we are never conscious of the impossibility of the contrary, is all that can be compassed; and this also not being internally evolved from any fundamental data, must be sought for, collected, and applied from without."

"From this general contrast it will be 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. We are thus disqualified for observation either internal or external, for abstraction and generalization, and for common reasoning."

Now common reasoning we conceive to be very bad reasoning; such reasoning as fails to satisfy the man who is seeking for clear and exact views, who fears to be misled by words, and who remembers that fine phraseology teaches nothing. It may be observed

here, that whatever force or justness there is in the reviewer's general course of observation, it all lies in the word "excessive"-"an excessive study of the mathematical sciences." And it is perfectly obvious that he who is conversant only with mathematical notions and mathematical processes, may be ignorant of many other objects of human attention, which come nearer home to the business and bosoms, the pleasures and pains, of mankind at large. He who is always dwelling in circles and squares, ellipses and parabolas, differentials and integrals, may have a proportionally confined range of thought. He will not understand the feelings and thoughts of other men; and he may fancy, from the habitual association of his ideas, or from his determining everything in the same way, that he can ascertain the precise quantity of enjoyment which a company of aldermen derive from eating and drinking, by means of the differential or integral calculus, and determine the relative merits of Homer and Virgil by the rule and compasses. But what then?

Shall mathematical studies not be

valued as an essential part of the training of the youthful mind? Is Mr. Whewell's sentiment invalidated, that they are the best practical exemplification and exercise of logic? If there be one mode of studying mathematics better than another, shall not a mathematical professor discuss this question, and endeavour to settle which is best? How many sciences are there which require for their pursuit, comprehension, and enjoyment, a thorough knowledge of the higher branches of mathematics, such as astronomy, optics, dynamics, and all those which go under the name of the mixed sciences. Who would undervalue the highest mathematical attainment when applied to these branches of science; and not rather regret, when he sees the mathematician soaring in the clouds and lost in the dim distance of algebraic formulæ, his inability to follow ? "Non omnes possumus omnia." But we can all enjoy and apply those practical and simple conclusions, for the establishment of which the most profound mathematical investigations are ofttimes necessary.

If the question be, What degree of time and attention should be given up to mathematical studies in a thoroughly comprehensive course of academic education? or, How far exclusive encouragement should be given to high mathematical attainment in an university? (which the reviewer has in part raised and discussed,) this may be settled without depreciating the importance and value of mathematics for the discipline of the youthful mind. You have then to take into account the great and general purposes of education, the whole constitution of the human mind, the condition and wants of society at large, the fitness of an individual for the particular station which he is designed to occupy, and the kind of knowledge which his meditated profession may require.

It is curious to contrast the reviewer's statement of the injurious influence of mathematical science in disqualifying for observation, either internal or external, for abstraction and generalization, with the intellectual character of Sir Isaac Newton

drawn by Sir John Herschel in his Treatise on the Study of Natural Philosophy, p. 271.

"His wonderful combination of mathematical skill with physical research enabled him to invent, at pleasure, new and unheard-of methods of investigating the effects of those causes which his clear and penetrating mind detected in operation. Whatever department of science he touched, he may be said to have formed afresh. Ascending by a series of close-compacted inductive arguments to the highest axioms of dynamical science, he succeeded in applying them to the complete explanation of all the great astronomical phenomena, and many of the minuter and more enigmatical ones. In doing this, he had every thing to create; the mathematics of his age proved totally inadequate to grapple with the numerous difficulties which were to be overcome. *** Of the optical discoveries of Newton we have already spoken; and if the magnitude of the objects of his astronomical discoveries excite our admiration of the mental powers which could so familiarly grasp them, the minuteness of the researches into which he there set the first example of entering, is no less calculated to produce a corresponding impression. Whichever way we turn our view, we find ourselves compelled to bow before his genius, and to assign to the name of Newton a place in our veneration which belongs to no other in the annals of science. His era marks the accomplished