nomena; and that no difference in any other part can occasion the same diversity in the same phenomena," which is merely a definition of Phrenology, the very question at issue, and which still remains to be substantiated!

"We next argue the truth of Phrenology," says Mr. Smith, "in relation to memory. Upon this subject the most unphilosophical opinions were prevalent. There was nothing like classification of memory. A man was esteemed to have a good memory who could easily retain historical facts, or store up words; but phrenologists have proved that this faculty is greatly diversified." Such a statement as this is altogether absurd. Granting that the most unphilosophical opinions were prevalent in relation to memory, upon what new principle has the phrenologist proceeded to explode their fallacy? Without entering into the metaphysical parts of the statement, as to whether memory is really a distinct faculty of the mind, or the mere result of vivid impressions, we will simply quote the fundamental proposition of Mr. Stallybrass, which no system in the world can overturn, "That the science of mind can be based only on phenomena of mind, and phenomena of mind can be ascertained only by consciousness." So that all who desire to make inquiries into mental science must do so on the same footing; in fact, the organs of the brain, if there are any, are so many unmeaning excrescences, till we have first settled, by the adoption of the principles of what phrenologists call "the old school," what peculiarity each of these developments is to signify. How then can Mr. Smith assert that on Phrenological principles the faculty of memory has been proved to be greatly diversified?

We find Mr. Smith, page 106, again recurring to the results of organic structure, in the assertion that man is adapted to the circumstances under which he exists, and that were he to be transplanted to another state he would require to be adapted by an alteration in his mind to the peculiarities of that state. It is really no small tax upon our patience to recapitulate frivolities like this, which are resorted to for the purpose of proving the truth of a science; but as we are anxious not to lay ourselves open to the charge of unfairness in our criticism, we will endeavour seriously to answer them. We say, then, that although the bodily structure of man would doubtless require re-modification, yet there can be no manner of reason for supposing his mind would need alteration. We cannot think Mr. Smith is an advocate for the doctrine of innate ideas; and yet only on this ground would mental alteration be required; for while the mind has powers, and faculties essentially belonging to it, in whatever new sphere it may be called upon to operate, it will be equal to the task, and impressions, of whatever character they may be, will be received and judged of with a facility equal to that which it displays in its present situation. There is consequently no necessity for the far-fetched hypothesis that man's mind will require alteration, or that his present views and feelings will be unadapted for a future state, an hypothesis that does not deserve mention for the support of any mental system taken at its greatest worth. The physical capacities must indeed be suited to the place of existence, but the mind requires not change,-let it be transplanted to worlds infinitely

remote.

Mr. Smith has undertaken to reply to some objections which he states have been urged against Phrenology. These objections are exceedingly meagre and paltry, and easily admit of a satisfactory answer. There is,

however, one difficulty which we do not think Mr. Smith has by any means overcome, viz., "that there is no correspondence betwixt the shape of the brain and the figure of the skull;" which is an assertion, he says, quite beyond the power of proof, inasmuch as the true configuration of the brain cannot be ascertained during life, and after death it becomes collapsed. How obvious is it that this argument equally tells against the phrenological system. Is it necessary to say that, if this be true, phrenologists are equally unable to ascertain,-we use Mr. Smith's own words,-" that certain peculiarities of mental constitution are always found linked to certain peculiarities of cerebral conformation?" By the self-same process that organs are discovered can their existence be disproved; and if there is no method of disproving their existence and conformity to the skull, neither is there any method of proving them.

But the objection is obviously founded on that which constitutes the very basis of Phrenology, the assertion that the organs of the brain always cause corresponding protuberances on the skull; and therefore the difficulty ought to have been met, not by referring to the brain after death in its collapsed state, but to the interior appearance of the skull, whether there are really any concavities caused by the cerebral conformation during life, and whether these concavities do precisely correspond with the external convexities or bumps. A reply to the question, if in the negative, will at the outset completely destroy the pretensions of Phrenology, or, if in the affirmative, will correspondingly heighten its claims. We cannot do better towards the solution of this question than quote the words of that eminent physician, Dr. P. M. Roget :-"The possibility," he says, "of discovering the size and the shape of the different parts of the brain from the external examination of the head is discountenanced by anatomy. There are often considerable impressions in the interior of the skull where the corresponding exterior surface does not exhibit the slightest appearance of projection, and is sometimes even depressed; and there are frequently large prominences without, where there are no corresponding concavities within: so that when the outer surface of the bony case is compared with a mould in plaster or wax of the cavity itself they exhibit considerable differences, and, from the great variation which may take place in the thickness of the bones, this difference is not the same in any two skulls."* It would be a work of supererogation to make any lengthy comment on Mr. Smith's concluding assertion, that "Phrenology has given the most symmetrical and perfect nomenclature of the mental faculties which we possess." Our readers will find this assertion combated with considerable felicity by Mr. Stallybrass; and if what we have endeavoured to prove be correct, that Phrenology introduces no new principles for the study of mental science, the case assumes this aspect : Have the labours in mental philosophy-quite independent of physiology or anatomy-of Gull, Spurzheim, or Combe, been of greater value than those of Brown, Reid, or Stewart? We hesitate not to say that the classification of mental faculties by the former are not worthy to be compared with the labours of the latter; and for the plain reason that the sole light by which they could possibly proceed they have not only neglected but sneered at, whilst their peculiar turn of mind has induced them to treat with contempt the light of abstract truth, and to think nothing real or valuable that is not accompanied by physical appearances.

* Article Phrenology, last edition, Encyclopædia Britannica.

We are quite certain that Mr. Smith will not be displeased for the open avowal we have made of our opinion on this long agitated question, or for the manner in which we have reviewed his essay. We are not aware that we have uttered anything harsh, or capable of being construed into a disrespectful manner. Our opinion as to the merits of his paper he is possessed of, for his essay was chosen from a vast number as the most clear, concise, and elegant. We feel sure that his object as well as our own is the attainment of truth; and, if he holds any opinions that we can detect to be unsound, we are convinced he will gladly receive our criticisms in the same spirit as we shall rejoice to receive his.

We can say little of the essay of Mr. Stallybrass, for the views he takes, with few exceptions, coincide with our own. We offer him our best thanks for the care he has evidently taken in preparing his essay, and the manner in which he has succeeded does him great credit.

It is our intention on some future occasion to take a still more comprehensive view of Phrenology than our limits have now permitted.

Prize Essay.

ON THE STUDY OF MATHEMATICS.

BY T. E. STALLYBRASS, B. A.

"If a man's wits be wandering, let him study the Mathematics.”—Bacon.

THERE is so great a difference between learning Mathematics in the sense of merely committing demonstrations to memory, and studying Mathematics in the real sense of the phrase, that what may be said respecting the one cannot be at all affirmed of the other. We have known some learn off the solutions as well as the problems of Euclid by rote; as long as they were allowed to proceed without interruption, they poured them forth with a volubility almost surprising, and which, if you took the utterance for the expres sion of an intellectual process, might present a proof of a great mathematical genius. But alas! it comes out to be a mere act of memory; one question from the teacher breaks the chain of associated words, and the fluent utterer is involved as in a bog, and sticks fast as in a quagmire. Now this is not studying Mathematics at all-it is a mere committing of words to memory; and if this were the only end to be answered by this study, there is no benefit attending it in particular, because any other composition might serve quite as well, if not better, for training up the memory to words. In the study of Mathematics, however, it is not the memory (which of course is exercised), but the ratiocinating faculty that is specifically exercised. The study consists in following with a clear understanding the demonstration of mathematical truths-in perceiving how clearly and how inevitably one step succeeds another, and how all the steps lead to the conclusion-in observing how certainly and how unerringly the reasoning goes on from things perfectly self-evident, and by the smallest addition at each step, every step being as easily taken after the one before it as the first step was. This is an operation of the understanding. No one can be said to know Mathematics who has not studied them, so as to perceive how they are proved, and understand the grounds on which they rest. He may by an act of memory know their doctrines, but he has not studied them if he is unable to show why he believes them, or to prove before others that they are true.

The object of intellectual education may be said to consist in aiding the development of the mind in the three general departments of memory, reason, and imagination. Before, however, any of these states can be properly unfolded, there is a certain habit of mind

which must be acquired, and in proportion to the strength of which will generally be the development of the faculties. This habit is attention-the concentration or continuous fixedness of the mind on the object, whether of memory, reason, or imagination. Without it, it is impossible for any of the faculties to be developed to any extent; and we shall show that the study of Mathematics is one of the best means for the acquirement of this habit, as well as peculiarly fitted for the improvement of the reasoning powers.

1. The study of Mathematics cannot fail to form the habit of close attention.—It is clear that attention is a mere habit of mind, consisting of the continuous fixedness of the mind on some one particular subject, and resulting from a combination of what is at the time the prevalent desire, or, in other words, the will, with the faculty corresponding to the particular subject of consideration. It is the sovereign power of the will, controling and directing the operation of the faculties, the possession of which, with its results, form the distinction between the mind in its states of childhood and of manhood. In early youth we have not this governing capacity, but are the subjects of a variety of instinctive tendencies, desires, and passions, each of which blindly seeks its own satisfaction. The mental faculties act without the will's impressing upon them any direction, and under the sole impulse of these tendencies. The movement of the mental machinery is then instinctive, not voluntary. The passion strongest at the time sways the rest, and all our faculties take the direction which it prescribes; but the instant another yet stronger passion rises up, our faculties quit the first direction and follow a new one. This vacillation of mind--this ceaseless fluctuation of feeling is observable in the conduct of children, and is the consequence of the speedy exhaustion of prevailing passions and the difficulties attending their complete satisfaction-difficulties which cannot be overcome as long as the mind is carried forward by mere impulses, and its faculties act without any precise direction. On arriving to years of reflection, we understand this; reflection furnishes us with a purpose, and the will concentrates all the faculties to overcome every obstacle in its way; their united power is brought to bear on this one point where they have encountered difficulty. Here we have the first consciousness of the power of the will-the power of controling our faculties, and of concentrating the forces before diffused. Attention is this very capacity of governing our minds at pleasure, or at the dictate of reason and conscience. Perhaps few things are of such difficult acquisition. In the first attempts, we feel pain, the physical as well as mental effort is most fatiguing, and a thousand objects intrude upon us to divert our faculties from the direction we would prescribe; but when the habit is acquired, like every other habit, it becomes easy, pleasant, and natural.

Now perhaps nothing is better adapted than the study of Mathematics for the attainment of this habit of attention. In the solution of a problem, every thing is conducted in a regular consecutive gradation-every step in the series is closely and clearly connected, until the conclusion is arrived at. Each step is in itself easily understood, as also its relation to either the preceding or the following step. Each of these parts, separately considered, may be understood without taxing the attention much. But to contemplate in one continuous view the entire train of parts so closely connected-to perceive how clearly and inevitably one part follows another, and all the parts lead to the conclusionto understand how unerringly the reasoning proceeds from things perfectly self-evidentthis process of argumentation cannot be carried on without the intensest attention. The regular exercise of attention in this manner will induce the habit of close attention to whatever the mind may be engaged in. The faculties, though at first they are restless and indomitable, because unaccustomed to government, will gradually be trained to a complete submission to the sovereignty of the will. If such, then, is to be the reward of the study of Mathematics-if the habit of attention cannot fail to be formed by this mental application-he who perseveres in it will indeed be well repaid for all his troubles.

II. The study of Mathematics is peculiarly adapted for the improvement of the reasoning faculties. What is true of the powers of man's physical constitution in a sound condition, is equally true of the faculties of his mental constitution-that exercise enlarges and strengthens them. As the volume and power of the muscles is increased by their continual and vigorous exertion, so the compass and strength of the mental faculties is augmented by their constant and vigorous exercise. The ratiocinating powers consist in the deduction of legitimate conclusions from, sound premises; they are of great value, not merely to the dialectitian, but equally to the man who has to perform no more than the ordinary duties of every-day life. It is much to be regretted that so little pains are now bestowed on the art of reasoning in the course of a liberal education. For want of the means of improvement, the capacity, where naturally possessed, may lie dormant through life; and many, doubtless, whose reasoning talents have been buried by never having been put to use, if they had received the advantages which the greatest philosophers have enjoyed, for their cultivation, might have displayed as good faculties as any of them.

When we consider reason, not only as the gift of God, but as designed for and tending to our preservation from false views, prejudices, and superstition, and our obtaining correct ideas and a clear insight into every subject of thought or deliberation, thus elevating us in the rank of intelligence, its improvement to the best of our ability must be of momentous importance.

It is easy to show how the study of Mathematics is pre-eminently fitted to exercise, and therefore to improve the reasoning powers. It is a mistake to suppose that Logic teaches us how to reason. The proper province of Logic, in which it is of great value, is to explain the laws of argumentation, and to furnish the reasoner with rules by which he is enlightened in his practice, so that he can work with more assurance, and by which he is enabled to correct his own errors, and to detect those of others. But then it presup poses the practice of reasoning. Its business is, not to furnish us with the practice, but to guide us in it. Hence Locke's council in his Thoughts on Education, "If you would have your son to reason well, let him read Chillingworth," solely on the principle that our improvement in reasoning is to be expected much more from an intimate acquaintance with authors who reason best, than from studying all the systems of Logic. Now there are no specimens of reasoning better than what are to be found in the mathematical sciences. They form the noblest praxis of Logic, because they are purely ratiocinative. In them we may perceive how the stated forms of syllogism are exemplified on one subject, viz, the predicament of quantity; and, by marking the manner in which they are there applied, we are enabled to apply them on any other subject, and the mathematical student may thus become an acute reasoner in all the subjects of the science of deliberation.

We may further show how this study is peculiarly fitted to exercise and improve the reasoning powers in their different characteristics of strength, clearness, and quickness, or dexterity.

First: There is no other branch of learning that gives such scope to long and accurate trains of reasoning, or in which there is so little room for authority or prejudice of any kind to give a false bias to the judgment. Here syllogism takes up the process from the commencement; starting from the simplest positions, it advances by a sublime intellectual motion to the most complicated speculations. Each step in the progress is syllogistic. When a youth begins to study, every thing is new to him: his apprehension is unsteady, his judgment feeble, and rests partly on the evidence of the thing, and partly on the authority of his teacher. He finds his reason unequal to the task of grappling with such demonstrations. But every time he goes over the elementary propositions, more light breaks in upon him, and more strength is infused into him; and as he advances, the road of demonstration becomes easy and smooth-he can walk on it firmly and with wider steps, till at last, by the continual stretch on which his mind is kept in those intricate processes of argumentation, he acquires the habit not only of comprehending a demonstration, but also of discovering and demonstrating mathematical truths. The mental exertion or "contention" which is necessitated by the study, cannot but strengthen and invigorate the reasoning powers. It is hence a fact, not at all surprising, that good mathe maticians have always been men of strong minds.

Again: In few other departments of learning does truth preside with such a bold and resistless conviction as it does in the Mathematics. In all their processes of ratiocination every thing is light and certainty. They assume nothing; and assumption of things, without examination of the grounds on which they rest, is the great fault and weakness of unpractised reasoners. The object of Mathematics is demonstration, which implies the removal of every possibility of doubt or error; and whatever is not demonstration is below or beyond the mathematician's regard. In the inductive method of reasoning, there may be moral certainty, but not demonstration. Absolute certainty can be had only in those sciences which are conversant about ideas and their relations, where probability is unknown, and every thing is certainly what it appears to be. The proof of a proposition is a syllogism, or a series of syllogisms, collecting that proposition from evident truths. The conclusion of the first syllogism becomes a premiss in the second, and the conclusion of the second a premiss of the third, and so on, till the conclusion of the last syllogism gives the proposition to be proved. A great number of syllogisms are sometimes thus linked together; but they are clear and certain, being primarily founded, either on definitions in which the description of an idea is connected with its designation, and as to the truth of which there can be no dispute,—or on self-evident propositions, about which there can be no uncertainty. Propositions before established are conclusions obtained from such definitions or axioms, and they are therefore clearly and necessarily true; and this being the case, the last conclusion must be so too. Thus mathematical demonstration not only leads to certain truth, but we have also a clear view of the ground