battles between the Russians and French, the death of the king of Prussia, and the various other great events with which public credulity was some days ago amused, on the alledged authority of the Guzzerat Brahmins, are now universally admitted to have been atchieved by fame alone. It is particularly unfortunate for the credit of the Guzzerat Brahmins, that their speculations, in which we were told they were so very confident, should have failed in this first essay. We recommend that, in future, the authority of the Guzzerat bullocks be superadded to that of the Brahmins, an association that must give great additional weight to the testimony. We believe that no argumenis can be necessary to convince our readers, that both authorities are likely to be equally well informed respecting the political affairs of the north of Europe; and that, of the two, the quadruped is by far the least likely, either to circulate unauthorized rumour, or to indulge in any erroneous speculation. And, therefore, a report brought forward on the credit of the Brahmins, if backed by that of the bullocks, though it should not command implicit belief, must at least be intitled to the most respectful consideration, and will prove a much better apology for the publication of extraordinary papers, than any account resting on the evidence of the Brahmins alone." Is it possible, in words, at least, for human beings to be treated with greater contumely than this? When the sacred class itself is thus insulted in common newspapers, why are we imposed upon with accounts of danger from a little alledged indecorum on the part of one or two missionaries ?

There is a passage which Mr. Chatfield quotes (p. 376) from the transactions of the Baptist Missionary Society, with a view to prove the frenzy of the missionaries. We transcribe it, as a very remarkable proof of three things; 1st, of the candour and veracity of the missionaries; 2nd, of the repayment in their own coin, which the missionaries, if they were really to be guilty of coarse and violent abuse of the heathen superstitions, would be liable to receive from the Hindoos on the score of those superstitions; and 3rd, of the total ab sence of any thing like alarm on the part of the Hindoos upon the preaching of Christianity: Wherever we have gone,' say the missionaries, we have uniformly found, that so long as the people did not understand the import of our message, they appeared to listen; but the moment they understood something of it, they either became indifferent or began to ridicule. This in general has been our reception!'

There is another passage of Mr. Chatheld's, which on every view of it deserves the most indignant reprobation.

The account,' says he, of the modern conversion of the Hindoos, may remind us of the conduct of the Spaniards, in the conquest of Mexico, whose greatest glory consisted in the number of souls baptised, not in the number of those made Christians.' p. 381.

Let any one recollect the dreadful ideas which are conjured up in his mind by the mention of the conquest of Mexico, and of the horrid baptisms which attended it, ideas of cruelty, murder, and extermination, at which the blood runs cold, and then think of the attempt to associate these ideas with the peaceful efforts of the missionaries to propagate Christianity in India! If this apparent attempt to associate the two undertakings, or rather to identify them in the reader's mind, was the result of deliberate intention, and not of mere inadvertence, we must consider it as involving a stain of indelible infamy on the character of the reverend author. But laying aside all thought of this insidious comparison, as it respects the atrocities committed by the Spaniards, we must be allowed to observe, that the charge obviously intended against the missionaries of being more anxious to baptise the body than to convert the soul-is utterly false. The principles of the Baptist Missionaries, if they are intended, render the truth of such a charge peculiarly improbable: because they attribute no efficacy to the rite of baptism in the affair of human salvation, and because they deem the baptism of any who are not first 'made Christians' to be unscriptural and improper. If any one thing in the history of Missions can however be deemed certain, it is, that these missionaries have been (to say the least) exceeded by none of any age or country in the scrupulousness with which they have investigated the sincerity of professed converts, and the candour with which they have published particulars of several cases in which their utmost vigilance has eventually proved to have been ineffectual.

It only remains for us to deplore that unhappy bias of heart or subjugation of intellect, which has occasioned a clergyman of the English Church to incur the suspicion, at least, of opposing the propagation of Christianity, and recommending the persecution of its abettors.

Art. II. An Essay on the Theory of the various Orders of Logarithmis Transcendents; with an Enquiry into their Applications to the Integral Calculus and the Summation of Series. By William Spence. 4to. pr. xiv. 128. Price 7s. Murray, 1809.

MR. Spence is a mathematician, of whom few, perhaps, of

our readers have heard. He is, however, a man of no mean attainments, as an analyst: he appears to have read and thought much, and now submits part of the result of his inquiries to the consideration of the public. The principal sub

ject of this essay comprises the properties and analytical ap

x2 .23
2n

+ 場

3n

&c. This

series Mr. Spence considers as the developement of a function

n

which is denoted generally by the symbol L (1 + x); and the successive forms which it assumes when n is varied in the progression whose common difference is unity, he calls the various orders of Logarithmic Transcendents.

1

In the arrangement Mr. S. adopts, he first treats concisely of the properties of the function L (1 + x): after which he con

2

siders the function L (1 + x); investigating several of its most interesting properties, exemplifying their uses in the computation of some of its values; and annexing a small table of the integer evaluation, 1 + ≈ varying from 1 to 100. In a

3

similar manner the function L (1+x) is examined: and then the author gives some properties of the general function Ï (1 + x). The obvious transition from the series ±≈−→→

n

+.2·3

3n

&c. to the series x

x3 x+5
5n

+ &c. naturally leads Mr. Spence to examine the theory of the latter. He traces the relation between the two series, and shews that the generating function of the latter can be expressed by imaginary values of the former. The latter developement is designated by the

n

notation C (x). The properties in the case when n 1, or the function becomes C (r), are examined; and a small table of values constructed. The examination of the function C (r), arising next in the series, presents so many difficulties,

n

that the author soon passes to the general function C (x), and investigates one property of essential service in the computation of its values; thus terminating the first part of the essay. The second part is devoted to the analytical or fluxionary application of these functions. After a few preliminary ob servations, the uses of the form L (1 + x) are inquired into: Xdx

2.

Y

and here the comprehensive integral L (V) is first examined it is divided into four cases. factors of the functions Y and V are real.

1st. When all the

2ndly. When the

factors of Y are real, and some of the factors of V imaginary. 3dly. When the factors of V are real, and some of those of Y imaginary and 4thly. When some of the factors, both of Y and V, are imaginary. He next proceeds to examine the d x

1

2

functions XL(V), SPdx f Q dx, and SP d x d y.

√x2 + ax+b2

In a similar manner, several analogous integrals referable to the function L (1 + x) come under consideration; and finally, the applications of the general functions L (1+r), and C (x) are instanced in the summatim of series; such, for example,

as the serieses

2n 3m 4i

+

23

204

n

In 2m

2n 3m

3n 4m

&c.; and

x5

3r 4m 5i

n

In 2m 3i

&c.; as well as some others that are too

complex to be conveniently expressed here.

This Essay, Mr. Spence informs us, is only a part of a larger work on the theory of Analytical Functions, in which it was intended to develope that important branch of mathematics from its elementary principles; giving first an exposition of the direct theory of functions, or, in other words, of those methods by which any function may be expanded into a calculable expression; and secondly, combining some researches in the integral calculus, with an inquiry into the properties and valuation of several new functions to which that calcu lus gives rise. But the length to which a train of investigation, thus conducted, would be likely to run, and the consequent expence of publication, deterred the author from laying the whole before the world at once, and induced him rather to select the parts which constitute the present essay. Considering the performance, however, merely as an insulated member of a larger structure, we think it deserves some commendation. The researches it comprizes are, in the main, conducted with elegance and skill; they evince the extent of the author's acquaintance with the modern analysis, as it is now treated by the continental mathematicians; they shew the facility with which he often overcomes difficulties; and his aptness at exhibiting some well-known expressions in new and more convenient shapes, as well as his ingenuity in sometimes presenting a demonstration which many would render operose and tiresome, in a concise and luminous form, so as to be at once more easy to comprehend and more difficult to forget. Yet, after all, we must complain of the vagueness with which Mr. S. often refers to preceding writers, and must

confess that the essay contains but little that is really valua ble. The author does not seem to aim so much at the promulgation of new and useful truths, as at the exhibition of old ones in a new dress. Nor should we be justified in saying that the garb in which Mr. Spence's investigations and deductions appear, is always more inviting than the one which he rejects and apparently despises. For he proposes a novel kind of notation, (such at least it will be thought by the merely English reader) without ever giving a perspicuous explanation of it; he falls into the error of several of the French analysts, that of generalizing too soon; he sometimes wanders from his own principles; and, in one or two instances, he either leads us into error through the obscurity of his manner, or argues in a circle. We advert particularly to his mode of + dx deducing the differential expressions L (1 + x)

片 } (1±x) =ƒa2; ≤ 1 ± x), &c. If the author meant to af

L

x

+

firm the perfect accuracy of these equations independently of the serieses which the first members of each equation re present, he is extremely deficient in explication for there is nothing in their appearance, or in the author's statements, from which the equality of the two members could be predicated. If he means to say that the differential members of the several equations, are equivalent to the correspondent serieses of which the functions L (1 ± x), ₺ (1 ± x), are the correct, though abridged, representatives, we would wish him to consider how he can obtain those expressions independently of the well-known formulæ for the differentials of hyperbolic logarithms; and, whether it would not have been more natural,

2

1

simple, and elegant, to have assumed the expression L (1x) as a symbol for the hyperbolic logarithm of 1+ x, making this the basis of the inquiry, and pursuing it by tracing the nature of the different orders of logarithms, and their most useful analogous or dependent functions. This, however, we merely suggest for the author's consideration, should he ever be tempted to complete the task he has prescribed himself,

We shall now lay before the reader an extract from Mr. Spence's preface, and make a few remarks that have occurred to our minds on the perusal of it.

It is proper to warn the reader, that he is not here to expect any geometrical or mechanical illustrations. The plan of the work, of which this is but a small portion, excluded them from consideration; and it was not without reflection, that this plan was laid down. It is one thing to learn