Christian nation can most formally sanction, and yet assume to be a high favourite of Heaven.

On so eminent a subject, the Doctor, without incurring the least hazard of sinking the dignified specific character in which he wishes to be regarded as travelling, into the rank of the mere journey-writing reporter of curiosities, might have noticed a number of particulars, which the inquisitive reader would have been glad to know, and which would have made the picture more finished without much enlarging it: such as the dimensions of the temple, its style of building, the bulk of the idol, the number of persons requisite for drawing the car, the deportment of the persons immediately about the idol, as aparently indicative, or not, of sincere fanaticism, &c. Of the rites celebrated in the interior of the temple, called the Daily Service, he says he can report nothing of his own knowledge, not having been within.' But he unquestionably received considerable information on that subject from the principal British resident, in whose house he was entertained, and might have given it on that authority. That the resident is not likely to be quite uninformed on this matter, and indeed ought not to be uninformed, will be tolerably evident from a part of our author's statement which we will now transcribe,--and which is by several degrees more remarkable than any thing that has preceded.

6

'The Temple of Juggernaut is under the immediate controul of the English Government, who levy a tax on pilgrims as a source of revenue. See " A Regulation (by the Bengal Government) for levying a Tax on pilgrims resorting to the Temple of Juggernaut, and for the Superintendance and Management of the Temple. Passed April 3, 1806."

The province of Orissa first became subject to the British Empire under the administration of the Marquis Wellesley, who permitted the pilgrims at first to visit Juggernaut without paying tribute. It was proposed to his Lordship, soon after, to pass the above Regulation for the management of the Temple, and levying the tax; but he did not approve of it, and actually left the government without giving his sanction to the opprobrious law. When the measure was discussed by the succeeding Government, it was resisted by George Udny, Esq. one of the members of the Supreme Council, who recorded his solemn dissent on the proceedings of Govern ment, for transmission to England. The other members considered Juggernaut to be a legitimate source of revenue, on the principle, I believe, that money from other temples in Hindoostan had long been brought into the treasury. It is just that I should state, that these gentlemen, (though their opinion on this subject will differ so much from that of their countrymen at home,) are men of the most honourable principles and unimpeached integrity. Nor would any of them, I believe, (for I have the honour to know them) do any thing which he thought injurious to the honour or religion of his country. But the truth is this that those persons who go to India in early youth, and witness the Hindoo customs all their life, seeing little at the same time, of the Christian religion to coonteract the

effect, are disposed to view them with complacency, and are sometimes in danger of at length consi 'ering them even as proper or necessary.'

• Annual Expences of the Idol JUGGERNAUT, presented to the Eng lish Government.' [Extracted from the Official Accounts.]

4. Ditto of contingent expences at the different seasons of pilgrimage

5. Ditto of his elephants and horses

6. Ditto of his rutt or annual state carriage

Rupees 69,616

£8,702

• In item third, 66 wages of his servants", are included the wages of the courtesans, who are kept for the service of the temple.

Item sixth.-What is here called in the official account the "state carriage" is the same as the car or tower. Mr. Hunter [the chief Resident] informed me that the three "state carriages" were decorated this year, (in June 1806) with upwards of £200 sterling worth of English broad cloth and baize.'

These facts speak a language that cannot be misunderstood. Any comments we may incline to make upon them, together with a concise notice of the remaining portions of Dr. B's work, must be reserved to our next number.

Art. II. A Treatise on Isoperimetrical Problems, and the Calculus of Variations. By Robert Woodhouse, A. M. F. R. S. Fellow of Caius College, Cambridge. 8vo. pp. x. 154. Price 6s. Cambridge, Deighton. Black, Parry, and Kingsbury. 1810.

AMONG the various circumstances which have tended to

give a fresh impulse to the abstract sciences, at any point of their course, none seems to have been more beneficial, than the practice, which has prevailed more or less among mathematicians for the last two centuries, of proposing problems to exercise the invention and call forth the skill and ingenuity of each other. This has been remarkably exemplified in the history of the modern or infinitesimal calculus. We owe its first invention to the genius of Newton; but most of its subsequent improvements and modifications have been the result of a gradual expansion of views, occasioned by the solution of some problem or problems, the proposers of which had scarcely any conception of the amazing augmentations to science which the complete solution of their own questions would produce.

The first problem that related to a species of maxima and minima different from those usually treated by the fluxionary calculus, was proposed by Newton himself in the Principia, being that concerning the solid of least resistance. But the

peculiar doctrine on which this and various kindred problems depended was brought into discussion, in consequence of John Bernoulli's proposing, in the Leipsic Acts for June 1696, the determination of the curve of quickest descent. The solution of this problem surpassed the genius of Leibnitz; who, nevertheless, according to his usual custom, intimated that he had solved it, but begged that John Bernoulli would lengthen the period for receiving solutions, (generous man!) that other philosophers might strip him of the honour of first solving it. Accordingly, Newton gave, without the authority of his name, the first public solution in the Leipsic Acts for May 1697: Quoique l'auteur de cette construction par un exces de 'modestie ne se nomme pas, nous savons pourtant indubita'blemente, par plusieurs circonstances, que c'est le celebre 'Newton, et quand meme nous ne le saurions point d'ailleurs, ce seroit assez de le connoitre par un echantillon, comme er 'ungue Leonem. This simple and elegant construction is given at p. 232. Vol. I. Gregory's Mechanics: but that author seems to be mistaken, in supposing that it was first published in the Philosophical Transactions, No. 224.

The next problem in the series of developement, was exhibited in James Bernoulli's famous Programma of 1697, in the following terms: Quæritur ex omnibus isoperimetris, super 'communi basi BN constitutis, illa BFN, que non ipsa quidem maximum comprehendat spatium, sed faciat, ut 'aliud curva BZN comprehensum sit maximum, cujus applicata PZ ponitur esse in ratione quavis multiplicata, vel submultiplicata, recta PF, vel arcus BF, hoc est, quæ sit quotacunque proportionalis ad datam Ag rectam PF, cur'vamve BF. This problem and its dependent inquiries became the apple of discord between the two learned brothers, John and James Bernoulli, occasioning not merely a controversy but a quarrel between them, which only terminated with the death of the latter. Indeed John gave ample proofs sixteen years after his brother's death, that he had not yet forgiven him; though, as Mr. Woodhouse remarks, that event, the lapse of time, the recollection of his brother's kindness, a zeal for a brother's fame, ought to have assuaged and laid to sleep all angry passions.' James Bernoulli's original solution was upon correct principles, though it admitted of improvement in point of brevity and perspicuity; while John's was really defective, as his brother had uniformly maintained. John Bernoulli considered only two elements of the curve; whereas it is requisite to employ three, or to introduce some equivalent condition. In problems such as that relating to * Joann. Bernoulli, Opera, tom. i. p. 197.

the curve of quickest descent, where it is simply required to fulfil the condition of the maximum or minimum, the applying of this condition to two elements of the curve is sufficient to determine its differential equation. But when, in addition to the maximum or minimum, the curve must possess a farther property, namely, that of being isoperimetrical to another, this new condition requires, that a third element of the curve shall have a certain inclination with respect to the other two and every determination founded merely on the first condition, will exhibit false results; except in those cases where a curve cannot satisfy any of the two conditions, without fulfilling the other at the same time. This essential condition of the three elements was introduced by John Bernoulli so late as 1718; and even then he had not candour enough to acknowledge, that his new solution was in substance the same as his brother's, though given in a form which considerably abridges the computation.

The consideration of this and some kindred problems, in the hands of Euler and Lagrange, led to the discovery of the Calculus of Variations-the other subject of Mr. Woodhouse's book. In this calculus, having given an expression or function of two or more variable quantities, of which the_relation is expressed by a determinate law, we can find what that function will become, when the law itself is supposed to experience any indefinitely small variation, occasioned by the variation of one or more of the terms which express it. This calculus furnishes almost the only means of resolving a multitude of problems, de maximis et minimis, whose difficulty is very far greater than in problems usually referred to the fluxionary or differential calculus. Such, for example, is the problem which requires the curve that will conduct a body falling, in virtue of its acceleration, to any given point, or right or curved line, in the shortest time. In general, every problem of this nature is reduced to the finding the maximum or the minimum of a differential formula, such as fZ dr, where Z is a function of x, or of constant quantities, or of x and y, or of x, y, z, and even still more variable quantities: indeed 2 may contain integrals, as ƒV, or integrals of integrals, as ƒVƒv, &c. and it is the manner of taking the variation of these expressions which is prescribed by the rules of this calculus.

Of this calculus, M. Lagrange is the true inventor: yet the two Bernoullis in solving the problem of which we have already spoken, and M. Euler, effected much in preparing the way for Lagrange. Euler's learned work-"Methodus Inveniendi lineas curvas proprietate maximi minimive gaudentes," &c. (1774) which is a wonderful specimen of profundity and sagacity, contains essentially all the requisite methods of solution,

and is enriched with a great variety of examples and illustrations. Yet it wants a new algorithm, a compendious method by which the theorems may be established without ambiguity and circumlocution, and an obvious principle of application to the several cases which should arise. These were supplied by the fertile genius of Lagrange, partly in the second volume of the Turin Memoirs (A. D. 1762), and more completely in the fourth volume of those Memoirs (A. D. 1767), and in his "Theorie des Fonctions Analytiques' (1797), and his Leçons sur le Calcul des Fonctions' (1806). A very perspicuous, though brief sketch of this theory is also given by M. Bossut in his Calcul Integral;' another by Lacroix in his admired performance on the same subject; and another (which several foreign mathematicians prefer to either Bossut's or Lacroix's) by M. Cousin.

From this concise history of the methods of Isoperimeters and Variations, it will be seen that no English author has attempted a treatise upon them. Some particular problems have, it is true, been considered by Maclaurin, Simpson, Emerson, and a few others; but none of these mathematicians seems to have confined his attention long enough to this interesting branch of investigation, to strike out a general theory applicable to the several cases that might occur. Simpson's seventh tract does not furnish a complete exception to this remark: for, besides that the rule he investigates applies only to isoperimetrical problems, he does not follow his own rule in some of the examples he has given. Mr. Woodhouse's, then, is the first treatise on these subjects which has yet been given in the English language, and the second distinct treatise which has been offered in any language--the first being Euler's Methodus Inveniendi, &c. before mentioned. Our author's reasons for undertaking the present work, as well as an account of the plan he pursues in it, will appear by the following quotation from his preface.

When Lagrange, in 1760, published his new method of solving problems of maxima and minima, he composed his memoir for mathematicians, familiar with its subject, and well versed in the researches of the Bernoullis and of Euler. Accordingly, he very briefly states the principles of his calculus, and enters into no explanation on the nature of the subject. His compendious method of computation, however, has been adopted; and subsequent authors have composed their treatises very much on the plan of Lagrange's memoir, with some, but slight and imperfect, preliminary explanation. These treatises, however, the student is expected to understand; that is, if the matter be fairly stated, he is expected to understand an intricate subject, with advantages much less than consummate mathematicians before him enjoyed; since there is neither proper explanation presented to him, nor is he directed, by way of paration, previously to consult the works of Euler and the Bernoullis.

pre