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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 babit, 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 regalar 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.
11. 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 evercise 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 falsc 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 deinonstration not only leads to certain truth, but we have also a clear view of the ground
of that certainty ; for as the series of propositions so concatenated all rest on the same basis, it is plain that one uniform ground of certainty runs through the whole. Here, then, however complicated the processes may be, every thing is perfectly clear and absolutely certain ; and the frequent exercise of the mind in such clear and sure argumentation cannot but form a habit of clear thinking and acute reasoning.
Finally: The subservience of this study to the formation of the habit of reasoning with great agility and adroitness must, after what has been already stated, be very obvious. Many mathematical truths, far from being obvious, at first seem very strange; yet they are proved to be true, by reasonings of the nature already described, and that too so clearly and unavoidably that they cannot be even fancied to be otherwise. In this manner we often arrive at the knowledge of things which seem at first not true; but when we do arrive at them, we perceive that they are just as true, and for the same reasons as the most obvious propositions, and that it would be as absurd and contradictory to suppose that they ever could, under any circumstances, be not true, as to suppose the most obvious matters not true. Now the frequent practice in such unlooked for reasonings leading to unexpected and strange conclusions must secure a facility and a dexterity both in apprehending the nature of the relations of ideas, and in drawing all the possible conclusions from all kinds of premises. Whatever pain and difficulty the persevering student may first experience in understanding and connecting the consecutive trains, he will ultimately acquire a quick, easy, and skilful perception of the relations of propositions, and the deducibleness of conclusions. The habit once acquired, the premises have only to be secn, and the inference will be almost intuitively conceived.
If the study of Mathematics is pre-eminently fitted to form the habit of attention, and to improve the reasoning powers as well as memory, its subservience to the general comprehensiveness and independence of the mind scarcely needs to be pointed out. Whatever gives vigorous exercise to the mind in any department of thought must tend to enlarge, expand, and energise it generally; and when any truth is demonstrated to us, and we know the nature of the foundation on which it rests, we are certain that it is truth, though all the world should doubt. None of the objections made against this study are applicable except to the exclusive devotement of the mind to the highest branches of Mathematics for many years, if not through life. He who pursues the study, not for itself, but for its practical uses, need apprehend no evil. Let, therefore, no prejudice, arising from inapplicable objections, or from the first impressions of its being dry and tedious, deter or repel any from the study. As they advance they will become more and more interested and delighted, and will at last be rewarded with the rich improvement we have ventured to ascribe to the study of Mathematics.
REVIEW OF BOOKS.
Indications of the Creator; Extracts, bearing greeted with all those marks of honour and
upon Theology, from the History and respect which form the well-earned meed the Philosophy of the Inductive Sciences. of toiling genius and successful devotion to By WM. WHEWELL, D.D., Master of the cause of science and of truth. The Trinity College, and Professor of Moral book before us consists of“ Extracts, bearing Philosophy in the University of Cam- upon Theology," from the professor's pubbridge. pp. 171. John W. PARKER, lished works, nearly all of which have West Strand, 1844.
already taken a high rank in the literature As the author of one of the celebrated of his country. To those who have formed Bridgewater Treatises, besides several stand- some acquaintance with the leading principles ard works on the history and philosophy of of natural theology, these “ Extracts" will the inductive sciences, on various branches be a welcome addition to the earlier labours of mathematics, and on the principles of of Paley and Brougham in the same field. university education, Professor Whewell To those who are yet strangers to that is so well known, beyond the walls of that interesting study, and who, from ignorance university of which he is so distinguished of its true value and importance, have been an ornament, that any thing emanating accustomed to underrate its claims upon from his pen is sure at once to command the their attention, the work is calculated to attention of the reading public, and to be prove even yet more serviceable. The popular and often eloquent style in which he must conduct his argument, there may Dr. Whewell treats of every subject brought arise a suspicion of defect of candour and under his consideration, the elegant scholar- comprehensiveness in what he writes. It ship, deep and varied reading, and exten may be supposed that he will strain or sive acquaintance with men and things evade any thing that points away from his which he everywhere displays, cannot fail predetermined end. But a narrative of the to impress them favourably respecting his whole history of science, and an analysis of qualifications as an author on a matter of the processes by which sciences have been such wide and general scope as the natural formed, are undertakings too large, and indications of creative design, and of an their course too rigidly determined by their intelligent“ first cause of all created things,” plan, to allow them to be drawn aside by which the history and progress of science partial and irrelevant considerations. The will be found to afford. And the sound passages now extracted as having a principles of reasoning to which he conducts theological bearing will be seen, on referhis reader in the analysis and examination ence, to flow naturally from the trains of of conflicting opinions, by which he is thought with which they are combined in anxious to test the theories of others, and the original works." to which he throughout endeavours to con- This circumstance, however, gives rise to form his own determinations, appear to us a peculiarity in the work which the lover so well calculated to assist in forming and of order and method will be perhaps inmaturing the youthful judgment, to aid in clined to regret, namely, that no attempt stimulating and developing the powers of has been made to reduce these extracts into thought, and give a healthy tone to the an harmonious and connected whole. They reflective habits, that we feel the greatest seem to stand coldly aloof from each other, confidence in recommending this work to like the members of an unsocial family. the attention of young men generally, as They certainly exhibit an unmistakeable one well suited to their wants, and capable family face; but it would require considerof yielding a large amount, not only of able acuteness of observation, or a pretty sound and salutary instruction, of pleasing intimate acquaintanceship to be able to and varied information, but also of pure detect family affection. Such a want of and rational enjoyment.
any obvious coherence and mutual depenThe profuse variety of subjects, the ex- dency in the several sections of a work tended range of thought, the splendid array we think, have a tendency to circumscribe of scientific research and philosophical its usefulness, and undoubtedly detracts speculation which these extracts embrace, from the pleasure, if not from the profit, and the interest which unavoidably attaches experienced in its perusal. to many of the questions they discuss and In order, however, that our readers may aim to determine, contribute to render the be able to form some idea of the multipliwork more attractive to a young and theo- city of scientific subjects which the prorizing mind than some of the more formal fessor has laid under contribution, as well treatises which have appeared on natural as of the manner in which he has theology. It possesses also a great advan- arranged his materials, we shall endeavour tage over the manuals of preceding authori- to present a consecutive view of the conties, in having a more appreciable bearing tents of the volume. But, having already on various controversial topics that are at exceeded our accustomed limits, we must, the present time exciting a lively interest for the present, confine ourselves to a brief in the breasts of philosophers and men of notice of the commencement. The first science. As examples of this we may refer section is on Astronomy; the subjects se to the theory of analogies, or of a unity of lected for illustration being the Copernican plan in nature, as opposed to a diversity system, and the Nebular Hypothesis. Un of purpose in creation, or, as it is some der the former head we have a very elever times termed, morphology, the question of piece of writing on the character of Galileo the transmutation of species, and the and his judges, their age and country. The hypothesis of progressive tendencies, all of conclusion of this part contains such a just which are fully elucidated and commented exposition of sentiments we have long ai upon in the section on Physiology.
tertained respecting the relations of science Another advantage is thus referred to by to Scripture, and to which we have already the author in his preface: “ Perhaps also given expression in the pages of The Str there may be some recommendation of these dent, that we make no further apology for Indications of the Creator in their being the the following somewhat lengthy quotation, result of researches and reasonings under and with this we close the present notice: taken with no purpose of bringing such “But there remains something more to be indications into view, but with objects of attended to in the case of Galileo ; for, quite another kind. For when an author though the See of Rome might exaggemte writes with a theological conclusion set the claims of religious authority, there is a before him from the first, as that to which question, of no small real difficulty, which the progress of science often brings into were with the soundest religious views ; notice, as it did then. The revelation on and the world then looks back with surwhich our religion is founded seems to de- prise at the error of those who thought clare, or to take for granted, opinions on that the essence of revelation was involved points on which science also gives her de in their own arbitrary version of some colcision: and we then come to this dilemma, lateral circumstance. At the present day, --that doctrines, established by a scientific we can hardly conceive how reasonable use of reason, may seem to contradict the men should have imagined that religious declarations of revelation, according to our reflections on the stability of the earth, and view of its meaning ; and yet that we can- the beauty and use of the luminaries which not, in consistency with our religious views, revolve round it, would be interfered with make reason a judge of the truth of revealed by its being acknowledged that this rest doctrines. In the case of astronomy, on and motion are apparent only. which Galileo was called in question, the “In the next place, we may observe, that general sense of cultivated and sober-minded those who thus adhere tenaciously to the men has long ago drawn the distinction be- traditionary or abitrary mode of undertween religious and physical tenets which standing scriptural expressions of physical is necessary to solve this dilemma. On this events are always strongly condemned by point it is reasonably held, that the phrases succeeding generations. They are looked which are employed in Scripture respecting upon with contempt by the world at large, astronomical facts are not to be made use who cannot enter into the obsolete difficulof to guide our scientific opinions; they may ties with which they encumbered thembe supposed to answer their end, if they selves, and with pity by the more consifall in with common notions, and are thus derate and serious, who know how much effectually subservient to the moral and sagacity and right-mindedness are requisite religious import of revelation. But the for the conduct of philosophers and religious establishment of this distinction was not men on such occasions, but who know also accomplished without long and distressing how weak and vain is the attempt to get controversies. Nor, if we wish to include rid of the difficulty by merely denouncing all cases in which the same dilemma may the new tenets as inconsistent with religiagain come into play, is it easy to lay down ous belief, and by visiting the promulgators an adequate canon for the purpose. For of them with severity such as the state of we can hardly see beforehand what part of opinions and institutions may allow. The the past history of the universe may even- prosecutors of Galileo are still held up to tually be found to come within the domain the scorn and aversion of mankind; although, of science, or what bearing the tenets which as we have seen, they did not act till it science establishes may have upon our view seemed that their position compelled them of the providential and revealed govern- to do so, and then proceeded with all the ment of the world. But, without attempt gentleness and moderation which were coming here to generalize on this subject, there patible with judicial forms." are two reflections which may be worth our notice. They are supported by what Report of Lectures on the Natural History took place in reference to astronomy, on of Plants yielding Food. By EDWIN the occasion of which we are speaking, and LANKESTER, M.D., F.L.S., pp. 48 may, at other periods, be applicable to London: John CHURCHILL, 46, Princes other sciences.
Street, Soho. “ In the first place, the meaning which “These Lectures," the author tells us, any generation puts upon the phrases of “were delivered at the Manchester Royal Scripture, depends, more than is at first Institution, and reported at the time in the sight supposed, upon the received philoso- Manchester Guardian. They were afterphy of the time. Hence, while men ima- wards published in The Institute." gine that they are contending for revela- After a few preliminary observations tion, they are in fact contending for their upon the three kingdoms of nature, Dr. own interpretation of revelation, unconsci- Lankester introduces the subject of the ously adapted to what they believe to be vegetable secretions, which he divides into rationally probable ; and the new inter- two great classes,-the medicinal and the pretation, which the new philosophy re- alimentary. Under the former he ranks quires, and which appears to the older the alkaloids ; of which he notices theine, school to be a fatal violence done to the caffeine, and the obromine, as contained in authority of religion, is accepted by their tea, coffee, and chocolate, The identity of successors without the dangerous results theine and caffeine is alluded to, and also that were apprehended. When the lan- their remarkable affinity of composition to guage of Scripture, invested with its new taurine,-an important nitrogenous principle meaning, has become familiar to men, it is of the bill, resulting from the metamorphofound that the ideas which it calls up are sis of the tissues, the degree of which is quite as reconcileable as the former ones chiefly regulated by the production of force