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theory of algebraic equations, not only in their numerical but in their most perfectly general character. Of the value of these researches, no doubt can be entertained among mathematicians; but it is not every student who will be able to profit by them, as in addition to the difficulties of the subject, it has been necessary, in order to express the very comprehensive views of the author, to adopt a new and peculiar notation. It is a proof of their value, that on an account of them being given this year to the Physical Section of the British Association, a sum of money was immediately voted, in order to have some of the prolix arithmetical operations executed, which are necessary for the full developement of the theory.
III. On the Theory and Solution of Algebraical Equations. By J. R.
YOUNG, Professor of Mathematics in Belfast College. 12mo. London,
Souter. PROFESSOR Young has been long and advantageously known to mathematical students, for a series of accurate, elegant, and perspicuous elementary treatises, forming a course of pure mathematics. The present volume supplies the only portion of the course which could be considered deficient: and is marked by the peculiarity of embodying, for the first time in an English work, the recent researches of M. Sturm, which, involving the discovery of a remarkable theorem of the utmost generality, supersede the methods of preceding algebraists, and complete the investigation of the theory of numerical equations.
IV. Minerals and Metals, with their Natural History and Uses in the Arts.
London, J. W. Parker. THIS
very small volume will be read with great interest, not only by the young, for whom it is specially intended, but by those of all ages, who have not specially made these subjects the object of their studies. have here, in a very small compass, a mass of useful information respecting those materials of all comfort and civilization, (as the mineral products of the earth may truly be called,) conveyed in an agreeable and attractive manner, and illustrated with many well-executed wood cuts. V. The Student's Cabinet Library. No. II. On the Connexion between Geology and Natural Religion.
atural Religion. By Professor HITCHCOCK, Amherst College, U. S. 12mo. Edinburgh, Clark. Though there are some few subordinate points in the author's argument which seem to us defective, yet, upon the whole, we consider this an admirable and useful tract, on a subject of the most serious importance. It stands pre-eminently distinguished by sound sense and enlightened philosophic views, amid the mass of publications on the same subject, which the present age has produced: many of which, with the best intentions, we do not doubt, on the part of their authors, from the ignorance, bad logic, and worse taste, which they display, are eminently calculated to expose to the scoffs of the sceptic, the sacred cause which they advocate, instead of affording it any rational support.
VI. Means of comparing the respective Advantages of different Lines of
Railways; and on the Use of Locomotive Engines. Translated from the French of M. NAVIER, by John MACNEILL, C. E. 12mo
96 pp. London, Roake and Varty. The discussions which took place in 1835, before Committees of the two Houses of Parliament, upon the Bill for the Great Western Railway, will ever be remembered as an epoch in the history of English civilengineering, and its professors. A fact was then elicited which may
lead to many a curious theme of inquiry and reflection as to its protracted existence, but which certainly must force those who have the direction of the expenditure of the enormous sums which are annually devoted to the construction of railways, to be better prepared for future inquiries of the same nature.
The bill above mentioned was opposed; and the most formidable source of opposition was the proposition of a rival line. A comparison of the two lines became, therefore, a very important duty of the committees. This was prosecuted successfully, without more than the usual difficulties, through most of the points of collision which arose, and one only remained which, though equal to any in importance, yet as it was a subject of mere calculation, we should not have expected, at this advanced period of railway experience, it could have raised a question among the learned engineers. The case was cleared of all questions of local interests, &c., and reduced to this simple form :-A railway for the transport of loaded carriages is desirable between two places, A and B; two lines of country are presented for this purpose, and each recommended by its respective patrons as the most eligible; the one passes from A, through a point c, to B, and the other through a point n, so that the lines are represented by ACB, and ADB. Now, by which of these lines can a passenger or ton of goods be carried cheaper or more expeditiously? Will the law-making will the scientific,—will the speculative,—will the mechanical,—will the professional world believe that no answer was given in these Committees to this question! that none could be obtained! that the bill passed into an act without the information it would have given! that in all inquiries which ever have been made into the merits of rival lines of railways by proprietors, opponents, committees,-parliamentary and others, &c. &c., the question, if asked, never was answered! and that at that time no book existed in the English language which contained the data by which the answer could be satisfactorily stated ; that is to say, not a table, formula, or statement, by which an engineer could calculate the cost of transport of a given load upon a railway of given length and given rise of inclination! It is true, that in the committee, opinions were given, that the line ACB was preferable to adB; but these were met by opinions, that the line ADB was preferable to ACB; and the committee was expected to be influenced by the weight (“ if that was weight, which weight had none,") of these conflicting opinions, in the absence of all facts and calculation founded upon accurate data. These opinions, and the skilful shuffling and mystification, which ignorance and cunning can, at times, bring successfully into play, cost the parties interested in the Great Western Railway question, a very large sum of money, without the slightest useful consequence to mitigate the loss, while the problem might have been solved, and the truth demonstrated, with mathematical precision, by a single competent witness, at the cost of about half an hour of his time!
The dispute in question was continued week after week; thousand after thousand was expended; and, when a little light did once strike across the gloom, ignorance, as usual, was alarmed, and, before it could be more fully developed, she succeeded in driving it from the arena; but heads have cleared since, and inquiry has exposed more completely the deficiency in some of our professional crania and libraries.
The humiliation produced in all the honourable minds of the profession, we suppose is deep; and we are pleased to see that, by the means of M. Navier's elegant investigations, the country may be saved from similar disgrace in future. The means this gentleman presents, are the only ones which can suit the case; viz., a thorough mathematical mastery of the subject, by which the true result in every particular instance can be obtained. This will banish mere opinion from the witness-box, will shorten inquiry, furnish solid grounds for decision, and save much valuable time and labour. We confess our patriotism would have been gratified, if Mr. Macneill had presented us with an original work on a subject so eminently national, as that of railways. As it is, he has preferred to introduce to English readers the labours of a French mathematician, distinguished for his application of science to this branch of mechanics.
In some notes with which Mr. M. has prefaced his translation, he makes a suggestion, which is so in harmony with the principles upon which this Journal, and the Gallery with which it is connected, have been founded, that we give it at length.
As practical men have seldom the time, or the mathematical acquirements that are requisite for such an intricate investigation, a series of facts should be registered at all the different railway establishments in the empire. These facts should embrace, 1st, the cost of the engines, their weight, a daily journal of their repairs, and the number of miles travelled daily; the weight carried, the water and fuel consumed, the pressure of the steam on each particular part of the line, &c. &c.; if such facts were collected, and given to some of our able mathematicians, there can be no doubt of their being able to furnish formulæ that would be of the utmost importance to the country; or it might be better to employ such persons to design and carry on, under their own superintendence, sets of experiments, in any way they might think most adapted to obtain the information desired. Such persons might be easily found amongst our mathematical professors, and the expense of the experiments should be defrayed by the Railway Companies, in some certain proportion.
The perfect execution (and the imperfect, would be mischievous,) of such a proposition would be of the most extensive benefit; and the cost, if distributed as the author points out, absolutely insignificant. Our humble assistance shall be at all times ready in the furtherance of this and similar accumulations of practical information.
In the little work under consideration, the aim of M. Navier has been, to point out the greatest amount of reduction, which the establishment of a railway can effect upon the actual cost of transport, so far as can be obtained by mathematical investigation, founded upon data
furnished by experience. He supposes the material, form, &c., of the rail to be determined, the local and general interests all understood and estimated, and that there only remains the selection of the line of country, along which it can be demonstrated the transport will be the least expensive; in fact, a valuation of the effect of the several lengths and slopes of each of the rival lines, so that an accurate comparison may be made of the power which would be required to draw a given load upon each. The very problem which was the res vexata before the Great Western Railway Committee.
In the course of the investigation, M. Navier, enumerates the principal elements in the comparison of rival lines of railway; he determines the power required to draw a given train over a given railway; and also the weight of the train which can be drawn along a given railway by a locomotive engine of a given power; he examines the motion of the train on slopes, and in passing from one slope to another, and concludes, by giving instructions for summing up the effect, and comparing the advantages of the several lines under consideration.
Among other useful demonstrations, M. Navier proves that slopes on railways do not lengthen the time, nor increase the cost of transit, provided they do not exceed an inclination of 1 in 250, and that the points of departure and arrival are upon the same horizontal line. Hence, for example, on a railway, say of 50 miles in length, and in which the above conditions exist, the load, during the transit, may be carried up slopes, even over a summit of 500 feet, as cheaply and as expeditiously, as if the line had been horizontal throughout its whole length.
That if the slopes do exceed, but by a little, the above inclination, then the balance between the power expended in the up-slope, and that regained in the down-one is destroyed, and there must, consequently, be a wasteful expenditure in the cost of transport along such a railway. It is, perhaps, the ignorance of this limit, that has produced some of the confusion which seems to prevail in minds otherwise tolerably clear on this subject.
In short, the formulæ of M. Navier, are general, and by them, the cost of the transit over any given railway may be accurately calculated from data easily attainable in all cases.
M. Navier mentions, that his work is merely an extract from a course of lectures, delivered to the students of the Board of Bridges and Highways (Ponts et Chaussées) of France. How enviable appears the position of the French engineer-student to those of our own country. The one has a profusion of gratuitous instruction by the greatest masters of mathematical science, and is distinguished by honours in proportion to his acquirements; the other can scarcely procure a solitary guide at any cost, and is seldom gratified by receiving distinction, however merited, until age or exertion has made him indifferent to its stimulus.
VII. Principles of the Differential and Integral Calculus, familiarly
illustrated, and applied to a variety of useful Purposes; designed for the Instruction of Youth. By the Rev. W. RITCHIE, L.L.D., F.R.S., Professor of Natural Philosophy, &c.; 12mo., 174 PP., woodcuts.
London, Taylor This will be found to be an inestimable little work by those who are about to seek in good earnest for information in this important department of mathematics, with the intention of applying it to practical purposes. Before its appearance, the English student might have searched in vain for such a help. We can already speak, from actual observation, of the delight and satisfaction it has excited in more than one young mind, by the clearness and intelligibility of the definitions, and the rules for acquiring correct notions of the terms, &c., used. In this respect, it surpasses even the estimable French work of M. Boucharlat.
Take, for example, an extract from page 3, explaining what is really meant, when it is said, that 1 divided by 0, is infinite. Here the following series of leading questions is proposed.
How much is I divided by 1 ?-Answer, 1.
How much is 1 divided by a fraction, having 1 for its numerator; and 1, with as many cyphers as would reach to one of the fixed stars for its denominator ?-Answer, 10,000,000, &c., to the fixed star.
Hence, as the divisor approaches 0, what does the quotient approach ?Answer, Infinity
Hence, by an extension of reasoning, when the divisor becomes nothing, what is the quotient or value of the fraction ?-Answer, Infinite.
Professor Ritchie, still continues the use of the usual or continental notation. He agrees with Mr. Woolhouse, that the signs recently introduced by the Cambridge mathematicians, "have no recommendation whatever, over those already established and incorporated in all the most valuable works of science.”
He states, that his aim has been to simplify and illustrate, by familiar examples, one of the most elegant and useful branches of mathematical science, “and that his plan is founded on the same process of thought, by which we arrive at actual discovery, namely, by proceeding step by step from the simplest particular examples, till the principle unfolds itself in all its generality.” Again, in this arrangement, the differential calculus, and the integral calculus,“ are made to travel hand in hand, till they arrive at the point where the former naturally stops, and the latter advances, without requiring further aid from its companion.” The author has, in general, successfully executed the task he proposed to himself, and the student who steadily follows his steps, and fairly attains the end of the work, diminutive as it is in its dimensions, will have mastered difficulties which have cost many minds, though assisted by the best works that existed, months, and even years to overcome. He may then proceed, immediately, to the valuable work on the same subject, recently published by Mr. Hall, fully prepared to open the mine it contains, and possess himself of its wealth.