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in Abel's theorem; and it hence appears that this celebrated result is not given in all the generality of which it is susceptible.

This is the state in which Mr. Talbot found the problem;—that is, when he came to examine what had been done by others, though he had obtained his chief results, and was in possession of his general method several years before Abeľs theorem was made public. The problem which he here proposes to solve is:—" To find the sum of a series of such integrals as S (R)dx, R being any entire polynominal, and any function whatever."

The history illustrated by examples which he gives of his own progress in these researches, is highly judicious and instructive. It is,

, however, impossible to give a condensed and intelligible account of his processes within the limits of this Magazine: nor, till the whole series is before us, would it be possible to do the method that full justice which it very obviously demands. Two methods of proceeding have already been developed,—the one of which is founded on a "change of the conditions,” shown to be necessary in the other for the solution of the problem in all its generality. The whole process is founded on the method of “integrating by parts,” a series of symmetrical functions of the assumed variables. Yet we would not have our readers think, because its first principles are known ones, that its results have ever been anticipated, or its current processes ever employed before.

This paper terminates the part; and we would earnestly recommend its attentive perusal to every mathematician who feels interested in the progress of his science. We may state, moreover, that it is not a difficult paper to read. There is none of the quackery of new symbology, or of the mystified and unsatisfactory reasonings which so much disgrace too many of “the most learned” mathematical papers of the present day, to be found in this; and no acquaintance with other writers, beyond the mere elementary ones, is required to enable the reader to comprehend it fully.

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ON THE SIGNS OF MULTIPLICATION,

DIVISION, ETC.

;

In our fifth number, (vol. i., p. 291,) we offered a conjecture respecting the origin of the signs + and -. Since then, the writer -has received from a friend, the most eminent mathematical antiquary of the present age, and a distinguished professor in the University of Oxford, another conjectural mode of deriving these symbols. An extract from his letter is placed below; and we proceed to lay before our readers the best information that is possessed respecting a few others.

The Symbol of Multiplication, X, was first used by Oughtred; and he prefaces its introduction simply with the remark :-“ Multiplicatio speciosa connectit utramque magnitudinem propositam cum notā in vel x; vel plerumque absque nota, si magnitudinis unica litera. Et si signa sint similia, producta magnitudo erit affirmata: sin diversa negata.

:

Effertur autem per in. *" It probably was used as a variation of the form of the symbol +, the operation of multiplying being merely a substitution for that of addition, in the case where all the numbers to be added are equal to one another. We had, indeed, before we were aware of this passage, imagined it to be a contracted representation of the “hand-in-hand," and to designate perfect union or amalgamation. That passage, however, does not seem compatible with such an hypothesis ; as in such case, some remark on the subject would in all probability have been made to point out the views by which he was led to it, and to enforce its adoption by others.

The Symbol of Division, - , is merely a contracted mode of designating the positions of the divisor and dividend in the old Italian mode of operating. It was employed in the first place to concentrate the matter on a printed page; as it requires two lines to express it fractionally, and only one to express it by means of this interposed symbolt.

The Sign of Equality, in its present form, =, was first employed by Recorde. He gives his reason thus:“ and to avoid the tediouse repetition of these woordes : is equalle to: I will sette as I often doe in woorke use, a paire of paralleles or gemowe lines of one lengthe, thus, =, because noe two thynges can be moare equalle.”—Whettestone of Witte, p. 105. Harriot, also, apparently used it, without any knowledge of Recorde having done so before him.-Ars Praxis Analytica, p. 10.

Before this time, and for a long period subsequently on the continent, the symbol of equality, was oc ,or 30, which is very evidently the initial diphthong æ, of æqualis .

The Symbols of Greater and Less, viz. > and <, were invented by Harriot, and first appeared in Warner's publication of the Ars Praxis Analyticæ, some years after the death of that extraordinary man $. See that work, p. 10. They are very appropriate, the point being in both cases directed towards the less quantity, and the opening towards the greater. The sign of inequality, without asigning which is the greater, viz., is of modern date, and we are not quite certain who was the first to use it. It is merely the sign of equality, “ crossed out.” Girard used ff and § for greater and less, or for > and <.

As a conjecture respecting the symbol on placed between two quantities to signify their difference without assigning which was the greater, we have heard a very eminent scientific gentleman express his opinion that it is the letter s, employed as the initial of subtrahere: but we rather incline to think it a modification of the manuscript d. Of this the reader may easily satisfy himself by writing the small d with the

* Clavis Mathematica, p. 10, Edit. is used even by writers of our own time, Quinta, 1698.

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as a mark of

general proportion,” and + A very ample account of the history is synonymous with,

varies as.

This of arithmetical operations may be seen in use of it, probably, originated in some Leslie's Philosophy. of Arithmetic, or in fanciful “flourish” carried through the Peacock's Arithmetic in the Encyclop. four points of :: or *, which are virtually Metropol.

used to signify the same relation. # A mark very similar to this has been § Two great questions in scientific hisemployed to designate infinity, (see last tory have heen recently set at rest, in paragraph,) and precisely the same form which the name of Harriot stands con

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>

looped top, and readily trace it through its natural gradations into n. It might, possibly, have been from the Greek MS. d, as it did not make its appearance till the Greek alphabet had become familiar to mathematicians. Of course, the d would be used as the initial of the word differentia.

Albert Girard employed Recorde's sign of equality, =, for the same purpose; and it is just possible that might have been derived from this, especially if we suppose that the MS. way of writing the symbol which is printed = was continuous like our letter Z! It would then easily be rounded into 2 or y

The Symbol of a Root, w, is a slight modification of the old manuscript r, the initial of radix. The numbers designating what root were placed over it, in the earliest MSS., where it occurs precisely as in the present day.

The Symbol of Infinity, viz. Oo, is probably a rude sketch of the serpent's folds, the serpent being the familiar emblem of eternity, or endlessness, amongst the ancients. The more appropriate symbol t, (the more appropriate, because indicating the origin of its occurrence in all algebraic researches,) was first used instead of oc, by the late Baron Fourier, and is now generally employed by mathematicians*.

Sept., 1836.

Note alluded to in the beginning of this Paper.

" It agrees perfectly with the general view which I have long taken of the subject. I differ a little from you, however, in the detail. I think you will find, upon a second examination, that the line above the letters is used, not for the vowel connected with the m and n, but for those very letters themselves. Diē, for example, is diem, and tātū is tantum. This strengthens the application of the view; for here we have the symbol itself expressly for the first letter of minus.

"The hypothesis which I had framed. to myself for the other symbol, was, I what

appears to have an advantage over yours. I consider it also the first letter of the word which it had to express,-not'et,' but 'plus.' My steps analogous to yours were

+

own,

6

PA ttt

Believe me, &c.”

spicuous, by Professor Rigaud, of Oxford. graphic fac-similes of several

pages of The first, that Galileo had the priority of Harriot's own papers are faithfully given. Harriot in the discovery of Jupiter's satellites; and the second, that Harriot, * We may remark here, that this procontrary to the bold and unwarranted as- perty of the cipher, it being the reciprocal sertions of Montucla and other continental of infinity, was well understood by the historians, did fully understand the nature Indian algebraists, as is well established and management of the imaginary symbol of by Mr. Colebrooke, in his valuable and elathe roots of equations. See the supplement borate work on the Hindoo mathematical to the works of Bradley, where litho-science.

A POPULAR COURSE OF GEOLOGY.

II.

of grass

PRACTICAL APPLICATION OF GEOLOGY. But what is the use of Geology? This is a question which we have often heard asked, and to which the querists generally reply in the same breath, by denouncing it as a visionary speculation, or, at the best, laborious idleness, productive of no practical results. On this point we are prepared to join issue with these objectors, and to vindicate the utility of our science.

In enumerating the advantages to be derived from it, we shall begin with its economical importance, because the majority of mankind is composed of those who refer all things to this standard. And here we must confess that, as regards utility, as well as the loftiness of their speculations, geologists must be contented to yield the first place to Astronomy. We pretend not to guide the sailor across the deep, and to enable him, by measuring the distance of the moon from some of the fixed stars, to ascertain, within five miles, his situation on the pathless ocean, after he has been months without seeing land; but, upon our own element, the land, we can confer upon mankind benefits of no mean order. We can assist the farmer to fertilize the surface of the earth, so that two blades

shall

grow where one grew before; and we can impart system to the labours of the miner, so that, no longer groping his way in the dark, or trusting to dreams, to omens, and the divining-rod, he may prosecute, with confidence, and with an approach to certainty, those costly operations which are necessary in order to extract from the earth the treasures which have there been stored up for our use. These treasures exist in sufficient abundance to afford a rich reward to our toils, and, at the same time, they have wisely and beneficently been rendered sufficiently difficult of access, to stimulate industry, and call forth all our energies.

The mineral wealth of the earth has not been distributed through it at random; but each formation, as geologists call a group of strata, is, over extensive areas at least, the peculiar receptacle of certain minerals. Thus, tin is found only in granitic districts, and copper is most abundant in those and the adjoining schistose rocks. That thick formation of limestone, to which the name of Carboniferous has been given, because the great body of the coal-measures rest upon it, is, in England, the chief depository of lead. These metals, with silver, and some others, occur in veins, traversing the strata. Gold, on the contrary, is rarely met with in veins, but is disseminated in small quantities through those rocks in which it occurs, and the principal supplies of it are derived from alluvial gravel, which has resulted from the destruction of those rocks. Platinum and diamonds are likewise found in alluvial gravel. Iron, to which the name of precious might with more propriety be applied than to gold or silver, occurs in the greatest abundance interstratified with coal; so that, by an admirable arrangement of Providence, the bulky ore of this useful metal is found in juxtaposition with the fuel and the limestone necessary for its reduction to the metallic state.

In the present state of our knowledge, it is too much to affirm that these general rules prevail over the whole earth; but they hold good over extensive portions of the earth's surface, though, even within those areas, there are exceptions to the rule; and the study of the rule and the exception is alike profitable.

Let us take, for example, the case of coal. Though raised from great depths below the surface, coal is vegetable matter, accumulated during the earlier ages of the world; so that we are warming our houses, and lighting our streets, and spinning our cotton, and propelling our steam-vessels, and we may be shot along a railway at the rate of sixty miles an hour, by means of fuel derived from the wreck of forests that flourished myriads, perhaps, of years before the existence of the human

race.

Deposits of vegetable matter occur in formations of all ages, but they occur only in thin seams, and in small quantities. The great coaldeposit lies between two formations known by the names of the old and the new red sand-stones, closely resembling each other in mineral composition, though very different in their zoological characters. A few years ago we should have said that this carboniferous limestone was the base of the coal-measures; but Professor Sedgwick has shown that in the range of this formation towards the North of England it becomes a complex deposit, containing beds of shale and sand-stone, with seams of coal, which on the borders of Scotland are so largely developed as to be worked for the supply of the metropolis. It may, therefore, be said, that the great mass of bituminous coal, capable of being profitably worked, lies above the old and below the new red sand-stone. The coal-fields of Brora, in Scotland, and of Whitby, in Yorkshire, can scarcely be called exceptions, though they are situated in a newer group of rocks, called the pölitic. For they afford an inferior kind of non-bituminous coal, worked only to supply a local demand.

In communicating these facts we are obliged to anticipate the knowledge of our readers as to the order in which the strata succeed each other,

Things by their names we call, though yet unnamed, and therefore, perhaps, they can scarcely perceive the full import of these remarks. In that case, we must request them to return to the subject when they shall have become familiar with the names of the formations, and with their order of superposition.

Now, the practical results to be derived from a knowledge of this general rule and its exceptions are these: That searches for coal ought only to be undertaken with the greatest caution, and under peculiar circumstances, in strata beyond the limits of the regular coal-strata, but that in certain situations, where dislocations of the strata have brought to the surface portions of these anomalous coal-measures, and when the carbonaceous matter is seen in force, the working of them may sometimes be attended with success in districts which are ill supplied with coal from the coal-measures properly so called, The same remark applies to the lignite, or wood-coal, of the tertiary strata, which is wood partially carbonized, affording a very inferior fuel, which would never be used where the produce of the coal-fields of Newcastle, or Staffordshire, or South Wales,

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