position that a creature must be absolutely dependent on its creator. All mathematical demonstration, we know, is founded upon definitions only; and to us it seems extremely clear that, when we have once established precise definitions of the words creature, dependence, and creator, the relation subsisting between these three ideas is as absolute as that which subsists between any mathematical ideas. In all sublunary things a creator can only control the creature in so far as his power is adequate, and our notions are therefore vitiated by conditions; but when we ascend to the source of all power, where adequacy is lost in omnipotence, our ideas acquire a precision which it is scarcely possible to surpass; and the relation subsisting between those of creator, dependence, and creature, is as immutable as that which subsists between those of equality,'' the sum of two right angles,' and the sum of the three acute angles in any triangle. The first principle of natural ethics,*-that all created things, inanimate as well as animate, are unconditionally dependent upon the will of the Almighty, appears to us as indisputable, as universal, and as indestructible as any mathematical truth whatsoever.

Nor, in our opinion, has Mr. Ogilvie established the solitary pre-eminence of mathematical science' in point of practical invariability. And lest we should lay ourselves open to the charge of garbling the author's phraseology by attempting to abridge it, we shall transcribe what he says on the subject, word for word. The principles of moral science (p. 14) so far as they inculcate the cardinal duties of conforming moral action to the revealed will of God, pursuing what, according to the laws of nature in every part of the universe, is intrinsically good, and avoiding what, according to the same laws, is intrinsically evil, are questionless immutable, and extend their imperial sway throughout the intellectual universe: but in the application and practice of these principles, even moral science, (so far as it depends on the pleasurable and painful, the noxious or salutary effects, which material objects produce on the external and internal organs of conscious beings, and on the social relations, that derive their origin from the varieties of this influence and action) necessarily varies with their organization.' Now we apprehend that all this may be said of mathematics also: and in order to make ourselves the better understood we must primarily inquire, what is the true object of mathematical science? This question is discussed somewhat at large by Mr. Stewart; and we had occasion to notice his reasoning on the subject in our Number for January. Our readers will there find us supporting Mr. Stewart in the observation, that hypothetical, and not absolute, truth, is the object of mathematical reasoning;-but to prevent all possibility of misapprehension, we shall extract from the Elements of the Philosophy of the Human Mind the whole passage in which this doctrine is recognized and enforced.

* We apply this term to inanimate existences, because there is no other that will answer our purpose.

† Vol. VII, p. 52.

'In mathematics, (says Mr. Stewart, Vol. II. p. 123, Boston edition) the propositions which we demonstrate only assert a connexion between certain suppositions and certain consequences. Our reasonings, therefore, in mathematics, are directed to an object essentially different from what we have in view, in any other employment of our intellectual facul ties; not to ascertain truths with respect to actual existences, but to trace the logical filiation of consequences which follow from an assumed hypo thesis. If from this hypothesis we reason with correctness, nothing, it is manifest, can be wanting to complete the evidence of the result; as this result only asserts a necessary connexion between the supposition and the conclusion.

And again (p. 165, id. vol.) he says, For the more complete illustration of this subject, it may be observed, in the first place, that although the peculiar force of that reasoning which is properly called mathematical, depends on the circumstance of its principles being hypothetical, yet if in any instance the supposition could be ascertained as actually existing, the conclusion might, with the same certainty, be applied. If I were satisfied, for example, that in a particular circle drawn on paper, all the radu were exactly equal, every property which Euclid has demonstrated of that curve might be confidently affirmed to belong to this diagram. As the thing, however, here supposed, is rendered impossible by the imperfection of our senses, the truths of geometry can never, in their practical applications, possess demonstrative evidence; but only that kind of evidence which our organs of perception enable us to maintain.'

In demonstrating a property of the lever, for example, we make no allowances for weight or size,-but confine our reasoning to an inflexible mathematical line; whereas in all actual existences both these considerations must have place; and our conclusions will be found to vary according to the size and weight of every individual lever which we may have occasion to examine. To mathematics, therefore, we may apply the same language which our author uses in relation to moral science. It is equally true of both that, while the principles themselves are absolutely immutable, the practice of those principles must necessarily vary with the organization' of the subjects to which they are applied. The only thing absolutely immutable about either, is the relation subsisting between abstract ideas,-between existences which are altogether unconnected with the grossness and corruptibility of matter, and which therefore seem to have a fair prospect of sharing the immortality of mind.

In the foregoing remarks, we must not be considered as placing moral science upon the same footing with mathematics, in the demonstrative clearness and precision of its reasoning:-nor, on the other hand, must we be supposed to concede, in the full extent of the proposition, that its inferiority in this particular is ascribable to any necessary incapability of such precision and clearness. Fortunately mankind can reason upon mathematical problems with unbiassed deliberation,-for the interests of no individual seem to be particularly at stake in the conclusions which may be deduced: but when the propositions of morality are to be examined, every human being reads his own fate in the truths which

may be discovered; and some shrink from the task without proceeding a step, while others go just so far and admit just so much as will insure the safety of their single selves.' Perhaps a majority will acknowledge the existence of an omnipotent Creator, and the consequent dependence of all created things upon his single will; but then there still remains sufficient room for scepticism in determining how and where that will is manifested. Some will conclude that their own reason is only to be consulted, yet neglect to exercise that reason; while others will confess that a revelation must be had, yet deny the authenticity of that which we already possess; and thus, in some way or another, mankind will contrive to get rid of a question, which so deeply involves their own welfare in the present world, and so completely determines the complexion of their destiny in that which is to come. Had all this been at stake in verifying the proposition, that the square of the hypothenuse of a right-angled triangle is equal to the sum of the squares of the other two sides, we somewhat question whether Pythagoras, or any body else, would have ventured upon the demonstration.

With the remainder of Mr. Ogilvie's remarks upon the study of mathematical science our own opinion very nearly coincides; and we think he is particularly happy in demonstrating the effect which it produces upon the intellectual habits of the student. We shall not have space for much detail; but the whole of his reasoning is compendiously stated in the following paragraph:

The study of mathematical science then ought, he conceives, to enter extensively into every course of liberal education, because it has a strong and peculiar tendency to exercise the governing faculty of the mind, the understanding; because it communicates, and because from this source only we can derive, an accurate knowledge of immutable truths, susceptible of practical applications infinitely diversified, and imparting to every subject to which they are applied, all the distinctness and precision of thought, which the human mind is capable of reaching; and because the study of mathematical science has a stronger tendency to establish habits of composure, recollectedness, dispassionate inquiry, intense reflection, and patient investigation, than any other study that can engage the attention of youth.' p. 18.

There are one or two topics upon which we wish Mr. Ogilvie had dwelt with more emphasis; and as we sympathize with him in a desire of recommending the study of mathematics, we shall record a few of our own reflections upon what we consider is an all-important subject. Perhaps the greatest benefit derivable from the intellectual discipline of mathematics, is the habit and disposition which it gives the mind of resting contented with nothing far short of demonstration in moral and political science. To use a parallelism, of which Mr. Ogilvie will particularly recognize the cogency, the mind that has become accustomed to mathematics is like the stomach which has been addicted to opium; in neither case will any ordinary stimulus produce the satisfaction for which our artificial habitude has given us a craving; and we naturally

seek to make up in quantity what we perceive to be deficient in kind. The person, therefore, who has been well educated in ma thematics will never terminate his research on moral subjects, till by the multiplication, if not by the cogency, of his proofs, the proposition he may be examining is reduced to something like de monstrative certainty. When a theorem of morality is proposed to such a person, the mathematical habit of his mind induces him to conclude at once, that, like all the other propositions with which he has been conversant, the one before him is capable of complete demonstration; and thus he acquires a faith which, we are told both in sacred and in profane writings, is about equivalent to power itself: potest, quia posse videtur. This is unquestionably one of the senses in which mathematical studies are said to 'strengthen' the natural powers of the mind. Exertion is the soul of ability; and any intellectual regimen which furnishes a provocative to the mind is, in effect, the creator of just so much power as it is the means of bringing into play.

There is another indirect sense in which we say the mind is strengthened by being versed in mathematical reasoning. Perhaps it is not so much in the actual increase, as in the due application of intellectual strength, that the benefit of such studies can with propriety be said to consist. Great native vigour, without some artificial skill, is only superior to absolute imbecility in the means which it possesses of defeating its own purposes. If it makes a false or clumsy trip, its own innate power will often bring it to the ground. But, on the other hand, even comparative impotency acquires prodigious power by learning to apply itself in the right season and at the right spot; insomuch that the greatest natural strength is often obliged to suffer the mortification of being overthrown by an opponent, of which almost the only power seems to be that of proper application. Nothing is better calculated to give the mind this kind of power, than the study of mathematical science. In demonstration of all sorts we learn by sad experience that we must patiently labour from the beginning to the end,and that it will never do to anticipate some steps and pass hastily over others. Every successive part must wait for its turn; and to begin in the middle, or hurry on to the end, is soon found to be a misapplication both of time and of ability. When this experience generates a habit and the mind acquires skill in handling its faculties, we are enabled to discuss moral or political science with a success which untutored intellect can never hope to attain.

It is acknowledged, moreover, that the mind actually acquires strength by mathematical exercise; and on this part of the subject we adopt in its full extent the observations of the essayist before us. The understanding (says he) is as naturally and necessarily invigorated by a study of this sort, as our arm, or any other limb, by the gymnastic exercises that call into frequent and vigorous action, the muscles that actuate it; or, any organ of sense by frequent and concentrated attention to the class of sensi

ble objects, to the perception of which it is exclusively adapted.'-These we consider as the leading benefits derivable from mathematical pursuits. Our author has not entered so deeply as we expected he would into that part of the subject which was to point out their bearing upon the acquisition of ability and skill, in oratory;' a circumstance which is, we think, the more to be regretted, because his own experience has been peculiarly favourable to such an examination. We can readily perceive, however, that the same mathematical habits which would assist us in all kinds of moral reasoning, must of course be advantageous in composing or in delivering the argumentative part of an oration.

"The study of mathematical science has a strong tendency to imbue the mind with impartiality and candour in estimating the strength of reasoning; to weaken the influence of every sort of prejudice; to render the mind less accessible to the perturbations of passion, even in deliberating on a subject peculiarly calculated to excite and inflame passion; to enable the orator to exert an habitual recollectedness, a dignified self-possession, a philosophical composure of temper, even amidst the turbulence, and strife, and rancorous contentions of popular assemblies, vested with supreme political power, and debating on measures of the most momentous consequence to the community. The study of mathematical science, has also a peculiar tendency to train and prepare the mind, to investigate with patient and persevering attention any subject, (how novel, complicated and tedious soever) the investigation of which, may be necessary to the successful exertion of oratorical skill.' p. 23.

This Essay is closed with an examination of the effects produced upon the mind by an exclusive employment on mathematical subjects. As our reasonings are entirely confined to ideas which are purely hypothetical, and can never be strictly applied to actual things, it is pretty plain that a mind exclusively employed in this manner may, in no great length of time, become so completely heremetical in the body as to lose all inclination and capacity to participate in concerns of the world. This will appear to be more especially true, when we consider how completely some persons become fascinated with the study, and how thoroughly it stifles and suppresses every faculty of the soul-except the understanding. This absolute predominance of the strongest intellectual power is analagous to usurpations of every other sort:The stability of empire can only be secured either by utterly exterminating all the other powers, or by reducing them to one uniform level of obedient insignificance. Hence the exclusive mathematician is incapable of perceiving the beauty of outward objects,of distinguishing the boundaries between the shades of moral reasoning, or of sympathizing with the feelings of his fellow crea

tures.

'The eye of the mere mathematician is open, and the pictures of extermal objects are optically delineated on his retina, but he is insensible to the beauty or deformity, and often unconscious of the existence of the