We agree with Mr. Locke, that there is no study better suited to exercise and strengthen the reasoning powers than that of the mathematical sciences; but our reasons for this preference would by no means lead us to exclude the algebraic mode of investigation. We assign this exalted rank to mathematics, first, because there is no other branch of science which affords such scope to long and, at the same time, accurate trains of reasoning; secondly, because it gives full, and at the same time, safe play, to the two principal mental powers employed in the discovery of truth and the detection of error, namely, the powers of invention and of perceiving relations; and thirdly, because in mathematics we are less influenced in our reasonings by authority or by prejudice of such kind as would give a false bias to the judgement, than in any other region of human inquiry. But none of these reasons restrict the advantages of mathematics to geometry. There is another reason assigned by Lord Kames; which is this. In mathematics the reasoning process is shortened by the invention of signs which, by a single dash of the pen, express clearly what would require many words. By that means a very long chain of reasoning is expressed by a ' few symbols; a method that contributes greatly to readiness of comprehension. If in such reasonings words were necessary, the mind, embarrassed with their multiplicity, would have great difficulty to follow any long chain of reasoning. A line drawn upon paper represents an ideal line, and a few simple characters represent the abstract ideas of number.' But neither does this confine the advantages to geometry. It rather tends to show, contrary to the obvious opinion of Mr. Creswell, but corresponding, we believe, with the experience of most mathematical teachers, that even geometrical demonstrations are much better comprehended by students when they are brought into short compass by the use of symbols, as in Barrow's Euclid, than when they are drawn out in words at length, as in the edition of Dr. Simson. Let us now, however, attend to some of the observations of Mr. Creswell: we dare not descant upon them all.

In the first of the paragraphs above cited, the language which we have first printed in Italics seems to imply that, in his estimation, the energies of the mind are best exerted when the memory is most oppressed; but leaving this, we object to the representation of the algebraic process, which says that, the conclusion is to be admitted independently of every step but the last.' There is often, in the geometric and algebraic methods of obtaining results, so close an analogy, that it is astonishing it should have escaped the notice of Mr. Creswell in the way it bx+ab 24 b seems to have done. Suppose the equation

13

=

13

were

proposed in order to determine the unknown quantity x. What would be the process? First, because the members on each side the symbol of equality are equal, and because by a fundamental axiom, when equal quantities are multiplied by the same quantity the results are equal, it follows that when both sides of the equation are multiplied by 13, equality will still subsist between the results, that is, bab24b. Again, because, by another axiom, when equal quantities are divided by the same quantity, the results will be equal, it will follow that if the last step be divided by the quantity b, the result, that is, a + a =24, will still be a proper equation. But, in this last, since the first member evidently exceeds the quantity a by the quantity a, the second necessarily does the same; and hence, by another axiom, if the quantity a be taken away from both members, the results will again be equal, that is, x=24-a. Now, every one will perceive, that, in this example, the value of x is obtained by the genuine and received rules of the algebraic method. Let our author compare it with the demonstration of a proposition in Simson's Euclid, say the 20th in the first book, and see if there be any other difference in the train of reasoning, than what is occasioned by the contemplation of a geometrical, and of another quantity; and let him farther ask himself, if, in this solution of an equation, he can affirm that the conclusion is independent of every step but the preceding, in any sense but that in which he might say that the demonstration of the 48th proposition of Euclid's first book, is independent of all but the 47th and the 8th; that is, in any but a very incorrect sense.

In the second passage we have quoted, our author affirms that 'pure geometry is always precise and logical.' This, to be at all applicable to the discussion upon which he has entered, must mean, that when geometers arrive at true results it must always be by a precise and logical process. Yet, this is far from the case. Let Mr. Creswell examine Cavalerius's demonstration of the proposition, 'every sphere is two-thirds of its circumscribing cylinder,' and he will find that though it be in a certain sense elegant and even beautiful, it is far from precise and logical. Let him examine the treatises on geometry by Malton and Leslie, and he will find that many of their propositions are demonstrated in the most loose and illogical way imaginable. Let him look also at the solid geometry of Bonnycastle, and he will find several propositions demonstrated (as the author doubtless supposed) either by taking for granted a particular case of the thing to be proved, or by a palpable contradiction. Let him turn even to R. Simson, the most precise and logical' of modern geometers, and he will find that in the attempt to establish the 12th axiom, he has committed several paralogisms, and especially in the fifth proposition of that demonstration has

assumed the truth of a particular case of the general theorem he was aiming to confirm. Such examples, and we could add greatly to their number, show that 'pure geometry' is not always,' any more than algebra, precise and logical.'

In the third of our quotations, Mr. Creswell is pleased to speak of the absurdities which have been published with a view of explaining the rule for algebraic multiplication:' but he surely cannot mean to say that nothing but absurdities' have been advanced on that point; for unanswerable proofs of the truth of the operation as it respects the change of signs, given by Mr. Jones and Mr. Sheepshanks when they were tutors in his own college, must have been well known to this gentleman. But, supposing we were to admit all that he requires, and allow that nothing but what was absurd or unsatisfactory had yet been offered in reference to common measures, the binomial theorem, and some other points brought under notice in treatises of algebra; still this does not prove that algebra is useless or unfavourable as a species of mental discipline. Has our author never heard of the dreams and whimsies which have been advanced by geometers under pretence of squaring the circle, trisecting an angle, or doubling the cube? Is he aware that even Euclid himself, demonstrates the 4th proposition of his first book, by imagining the motion of something which cannot be a triangle, and yet must be thought one; that is, by a process which is a fiction of an impossibility? And is he not farther aware, that among the numerous precise and logical' geometers who have edited Euclid, not one has noticed this unsatisfactory demonstration of a proposition which lies nearly at the foundation of the Elements? How then does it happen that he did not push his reasoning farther, and instead of employing it to depreciate algebraic investigation unduly, employ it (as he might have done with equal propriety) to depreciate mathematical studies altogether?

In the fourth paragraph, we are told that the greatest exertion of intellect is in translating the conditions of the question into algebraic language, and the rest is merely mechanical. To rectify our author's notions in this respect, we request him to attend to the celebrated problem proposed by Colonel Titus to Dr. Wallis, in which there are given three sums; that of the square of one quantity added to the product of two others; that of the square of the second, added to the product of the first and third; and that of the square of the third added to the product of the first and second; to find the three numbers separately. Let him first translate the conditions of the problem into algebraic language, and then find the numbers. And if after this he ask himself, first, whether the solution of the pro

blem does not require a much greater exertion of intellect' than the conversion of it into the language of algebra; and next whether the discovery of the numbers does not give as fine a play to the inventive faculties, call into exercise as many expedients, and require as close and deep thinking, as the investigation of any of Dr. Stewart's general theorems; we have no doubt that to both questions his answer will be affirmative. How then, can a lover of geometry, when recommending his favourite study, reason so loosely, and rest his case upon such disputable positions?

Geometry has its advantages, and striking ones too; but they are not such as make mere geometers the best reasoners. Dr. Simson and Dr. Matthew Stewart, confined their attention almost exclusively to matters of pure geometry. Maclaurin and the late Professor Robison, were also excellent geometricians; but, in their investigation of mathematical truths, they did not think that geometry was the only instrument wh ch ought to be employed. And yet we are persuaded that no person who is acquainted with the various writings of these four eminent men, will maintain that Simson is a better writer or a closer reasoner than Maclaurin, or Stewart than Robison.

The truth is, we have no doubt, that a man who confines his attention altogether to geometry, will become very accurate in his notions, as far as they go, but very contracted in his views; just, as one who accustomed himself to poring upon an object at the distance of four inches from his nose would become shortsighted. Besides, that by the inevitable constitution of our nature, it is absolutely necessary that a student should indulge to a certain extent in variety of views, diversity of pursuits, and different modes of enquiry, to prevent his becoming a bigot or a pedant. Let it also be borne in mind that the term of human life is short, and the time of a collegiate or academical course still shorter; and it will appear very unwise to confine the general student too long to mere geometry. Suppose him to spend his weeks, and months, and years, (and years would be required according to this supposition), in reading Euclid, and Archimedes, and Apollonius, and Pappus, Hugo D'Omerique, and Lawson, and Stewart; and, if it be not intended to make him a consummate geometrician, by the sacrifice of other branches of human knowledge, but to make him a sound and able reasoner, where will be the benefit? We reply in the words of Mr. Locke: It is but like a 'monkey shifting his oyster from one hand to the other; and, had he but words, might no doubt have said, oyster in right hand is subject, aud oyster in left hand is predicate: and so might have 'made a self-evident proposition of oyster, i. e. oyster is oyster; and yet with all this, not have been one whit the

.

'wiser or more knowing: and that way of handling the matter, would much at one have satisfied the monkey's hunger,, or a 'man's understanding; and they two would have improved in 'knowledge and in bulk together.'*

It will be seen, from the preceding observations, that we are no friends to the manufacture of intellectual Fakirs, who by holding the limbs of their understanding (if we may so express ourselves) long in one direction, become unable to move them into another. Yet we are far from depreciating mathematical studies. The "accurate sciences," by their tendency to improve the arts and manufactures of a country, to direct the power of the various agents, animated and inanimate, employed in machinery most advantageously, and by enabling philosophical inquirers to attain the sublimest heights in their pursuits, to penetrate the mists which hang about the top of the mountain of physical knowledge, and to "look through nature up to Nature's God," are of inestimable value. Nor are they of small moment considered in reference to mental discipline. They furnish innumerable trophies of the victories gained by human intellect in the pursuit of truth; and present a more copious repository of important facts and indisputable propositions than can be supplied by all other regions of unassisted human inquiry taken together. We agree with the author who remarked, in the beginning of the last century, that in the 'search of truth, an imitation of the method of geometers will carry a man farther than all the dialectical rules. Their ana

lysis is the proper model we ought to form ourselves upon, and imitate in the regular disposition, and gradual progress of our inquiries.' But while we admit this fully, we must also observe, that in the practice of generalizing results, and forming universal propositions, a student cannot do better than take for his model the method of the algebraists, who arrive at theorems the most comprehensive and exact, by processes which are, or may be, at every step accompanied by decisive marks of their complete agreement with truth, according to the principles assumed.

We have stood so long arguing with Mr. Creswell upon the very threshold of his edifice, that we have scarcely time to look into the apartments he has prepared for our examination. Yet it is due to him and to our readers that we inspect them, however hastily.

The first division of the publication is purely geometrical, and easy application, for the most part, of the Elements of Euclid.'

an

The propositions of the first and second sections of this first part, form a distinct and important subject: they lead to results *Hum. Und, book iv. ch. 8.