true, but it seems to us to want at least some explanation. The student's objection is that if we are at liberty to consider the normal action at the extreme point Q of the element as coincident with the normal action at P, we might also consider the direction of the tangential actions at the two points as ultimately coincident, which he finds is not the case; and it requires a clearer insight into the doctrine of infinitesimals than the student will generally possess to see that the error in taking the directions of the normal actions as coincident will be of a higher order than that in treating the tangential actions in a similar manner, and that therefore in taking the limits the former error will disappear. Perhaps the best mode of remedying this defect would be the addition of a chapter on infinitesimals when a new edition of the Differential Calculus is called for. We have not examined the book before us with sufficient care to be able to say much as to the accuracy of the printing. One strange blunder, arising we presume from the printer, we may point out for the benefit of any of our readers taking up the book. It is at the end of article (186) where he is finding the approximate expression for the tension at the lowest point of the catenary, where in subtracting two expansions the first term of the difference is omitted. (The left hand side of each of the two last equations should be

h-k). There is also, a few lines above, a singularly careless mistake, the points of support being described as nearly in the same straight line, instead of in the same horizontal line.

Before we conclude, there is one point to which we should wish to call the attention of our mathematical readers. In the chapter on the Composition of Forces, Mr. Todhunter gives us first Duchayla's proof of the Parallelogram of Forces, (we wish he had substituted Duhamel's far more elegant demonstration) and then adds Poisson's proof which does not assume the principle of the transmissibility of force. In passing we may remark that we never could see that this was any recommendation of this class of proofs. Writers are accustomed to say that proofs such as Duchayla's will not apply to the ease of forces acting on a particle of fluid, or that the proof is imperfect because the proposition would be true even if the transmissibility of force did not hold, by which if they mean anything they must mean if no such thing as a rigid body ever existed. Such objections seem to us about equivalent to saying that a brick house cannot be built by means of a wooden scaffold. The rigid connections introduced into such proofs are purely imaginary, and when the result is established it matters not the least of what body the particle

acted upon may form a part. But to return to M. Poisson's proof, to which our attention was directed by finding it in Mr. Todhunter's book. It may sound a bold assertion to make concerning a proof published by such a man as Poisson, but we cannot help coming to the conclusion that it is a complete fallacy. We cannot give the proof at length, but the following general description of it will enable us to point out where the fallacy lies. Assuming that the direction of the resultant of two equal forces will bisect the angle between the directions of the two forces themselves, he takes two equal forces, P, inclined at an angle 2x, whose resultant is R, and assumes R=P f (x); his object being to determine the form of the function f. By resolving each of the forces P into two equal forces, Q, inclined at an angle 2 z; he arrives at the equation

ƒ (x). f(z)=ƒ (x+z) + ƒ (x—z).........(1)

This functional equation he has to solve, i.e., he has to find the most general solution, and to limit it by considerations derived from the special problem before him. This he proceeds to do as follows: "We see at once that f(x) = 2 cos cx is a solution, c being any. constant quantity. We proceed to shew that this is the only solution, and that c=1." Mr. Todhunter, perhaps, scarcely conveys Poisson's meaning here. His words are: "Or je dis que cette expression de la fonction f(x) est la seule qui satisfasse a l'équation (1), et que de plus dans la question qui nous occupe la constante e est l'unité.”

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As far as we can make out, the reasoning which follows is not intended to shew that the equation (1) admits of no other solution, (which we are required to take upon M. Poisson's assertion) but only that in the particular case before us c = 1. The steps by which it is endeavored to prove this are as follows. First, it is asserted that it is evidently true that c = 1, or that f(x) =2 cos x, when x is zero, for then the directions of the two forces P would coincide, and the resultant R would be 2P, and we must therefore have f (0) 2. Again he shews that the conditions of the problem are satisfied by assuming f(x)= 2 cos x in another particular case, viz., when x = 60° in which case the resultant R = P, which involves the assertion f (60°) = 1 which as cos 60° is satisfied by writing f(x) 2 cos x A most ingenious proof is then inserted to shew that if the relation f(x) 2 cos x is satisfied for x = 0 and for other value of x, it must be satisfied for all values of x. any proof of this assertion is derived entirely from the equation (1) itself, and inasmuch as the object in view is altogether to choose from the different solutions of the equation that one which suits the physical

The

problem which led to it, we might a priori doubt the usefulness of such a course. In effect the reasoning is worth nothing. In the first place that f(0) = 2 may be deduced at once from equation (1) by putting z = o, and the succeeding reasoning literally gives us no information whatever. If, indeed, it could have been said that unity was the only value of c which would satisfy the conditions of the problem when x = 60°, the proposition would be established, but, unfortunately, this is not the case, for an infinite number of values might be given to c such that the conditions of the problem might be satisfied in this particular instance. For example we might put c 5, for then we should have

ƒ (60°) = 2 cos (5 x 60°) =2 cos 300 = 1,

as it should be: and the fallaciousness of Poisson's reasoning is at once apparent from this, that the very same words which he employs to shew that f (x) = 2 cos x is the proper solution of (1) might be employed to shew that f (x) = cos 5 x ought to be selected.

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We may notice that a very simple mechanical consideration will suffice for the selection of the true value of c, if it be granted that the solution of (1) is necessarily of the form f(x) = 2 cos cx. When x = O, the equal forces Pact in the same direction and the resultant is the greatest possible; when a 90°, the angle between the forces is 180°, and the resultant is zero, and it does not seem too much to assume that as x increases from 0°, to 90°, the resultant will diminish continuously. This being granted it is at once evident that c must be unity, for cos cx must vary from unity to zero continuously, as x varies from 0° to 90°. We are by no means prepared to say that this form of proof of the parallelogram of forces can be made perfect. The solution of functional equations always involves more or less of doubt and obscurity, and what is called the the "general solution" of such an equation is by no means necessarily most general that can be conceived. Certainly Mr. Todhunter deserves our thanks for giving us the classical proposition of Poisson instead of the method which had been substituted by Mr. Pratt, which is just as unsatisfactory as Poisson's and much more clumsy. We could have wished, however, that Mr. Todhunter had called attention to this singular fallacy. It seems scarcely fair to the student to put a proof in his hands, especially with such a name attached to it, without giving him so much as a hint that it contains anything unsatisfactory.

G. C. I.

SCIENTIFIC AND LITERARY NOTES.

GEOLOGY AND MINERALOGY.

GEOLOGICAL MAP OF CANADA.

The Special Correspondent of the Montreal Gazetle, writing from Paris, on the 22d of November last, remarks:-M. Elie de Beaumont, President of the Geological Society of France, considers the small edition of the Geological Map of Canada, which has been published here, so excellent, that he has requested Mr. Logan to allow it to be introduced into the bulletins of the Society. It is one of the prettiest specimens of geological chromo-lithography that has issued from the press. The scale is one-tenth of Bouchette's Map of Canada. There are twentytwo colors on the map, representing the formations, and these have required fourteen lithographic blocks to print them.

WOLFRAM.

A well-crystallized specimen of Wolfram (the manganese variety 2 [FCO, WO3] + 3 [MnO, WO3), a mineral it is believed hitherto unremarked in Canada, has been lately met with in a granitic boulder, near Orillia, C. W. A detailed notice will be given in a future number. E. J. C.

FOSSILS FROM THE ESPLANADE CUTTINGS, TORONTO.

From this spot some good casts of the following fossils may be obtained :— Chætetes lycoperdon; Glyptocrinus decadactylus (stem fragments); Modiolopsis modiolaris, Ambonychia radiata; Murchisonia gracilis, Pleurotomaria subconica; Orthoceras lamellosum, O. coralliferum (or a species of Endoceras?) It is perhaps unnecessary to state that the above belong to the Hudson River group of the Lower Silurians. E. J. C.

GEOLOGY OF SCOTLAND.

A recent paper read by Sir R. Murchison to the Geological Society, announces the discovery of Upper Silurian fossils, in the parish of Lesmahagow, in Lanarkshire. The fossils were first found by Mr. Sliman, a native of the district, which has since been visited by Sir Roderick and Professor Ramsay. The succession of rocks from the coal and mountain limestone downwards is traced in Nethan and Logan waters, which are branches of the Clyde flowing north-eastward from the borders of Ayrshire. The rocks mentioned are followed by conglomerates and flagstones representing the old red sandstone, under which are dark gray, slightly micaceous, flag-like schists, containing crustaceans of the genera of Pterygotus and Eurypterus, with the Lingula cornea and Trochus helicites (shells). On the ground of these fossils, Sir Roderick considers the flag-like schists as the equivalents of the upper Ludlow rock, or tilestones of England. In the geological map of Scotland, therefore, a track of conntry about ten miles broad, colored as old red and coal by Dr. M'Culloch, must now be added to the Silurians. C. M.

COMPOSITION AND FORMATION OF STEEL.

At a recent meeting of the Boston Society of Natural History, Dr. Jackson gave an account of some researches into the composition and manner of formation of different kinds of steel.

As commonly known, steel is a combination of carbon and iron, made by heating flat bars of pure iron, in combination with charcoal. The carbon is first converted into oxide of carbon, and then unites with the iron as carburet. The result of this process is known as blistered steel, from the bubbles generated by gases upon its surface. Shear steel consists of parallel plates of pure iron and steel, welded by folding and uniting the bars of blistered steel. Cast steel is fused in pots of the most refractory material, and differs from cast iron which likewise contains carbon, in this respect, that cast iron is a mixture of coarsely aggregated matters, graphite and iron, whilst cast steel is a chemical combination of carbon and iron.

From the researches of Berthier, it is known that manganese will form an alloy with iron. When iron is mingled with a considerable portion of manganese, a brittle compound results; but when combined with a very small proportion of manganese, a steel of very fine quality is obtained, which has this advantage over carbon steel: carbon steel becomes coarse when tempered in thick masses, from segregation of the particles of carbon; but no such trouble arises with magnesian steel. Parties in England have lately introduced excellent wire for piano-forte strings, made of this kind of steel, as well as for cutting instruments, and other purposes. In the wire, Dr. Jackson has found 1.12 per cent. of manganese, and has established the fact that it resists, to a very remarkable degree, the action of hydrochloric acid. Sixteen years since, Franklinite iron was manufactured by Mr. Osborn into very hard and fine steel. This steel required tempering at a lower heat than carbon steel. Many of our manganesian irons might be manufactured into steel, by the simple process of fusion, and a steel of uniform character might be made without previous cementation with carbon.

Dr. Jackson explained the reduction of iron in blast and reverberatory furnaces. Manganesian iron ore is reduced to pure iron, or "comes to nature," in the language of the workmen, with much greater rapidity than carbon iron; hence the two metals are often mixed to "come to nature" at a good time, requiring less care and watchfulness on the part of the workman. Manganesian iron makes the best bar iron.

PHYSIOLOGY AND NATURAL HISTORY.

FREAKS OF NATURE.

The following singular illustration of the tendency of wild animals, when domesicated, to change their uniform natural color, is exhibited in a way both curious and unusual. A writer in the "Scottish Press" says:-Mr. Souter, of Boxgrove, has a game fowl which, four years since, was perfectly black, the second year it was brown, the third white, and at the present time it is speckled black and white. Though more in accordance with ordinary operations of nature, the following example of animals changing their color with the season of the year, is interesting as occurring in our own vicinity. The Rev. Thomas Schreiber remarks in a note to the Editor :-Is the following circumstance a freak of nature, or is it a happy dispensation of Providence, mindful for every contingency to provide for the safety of the animal creation? Last summer several rabbits, black and grey in color, were turned out on the grounds about the Homewood, Toronto; during the autumn their progeny were of the same color: since the snow has covered the ground two litters have shewn themselves, one litter of seven completely snow white, the