work, there are omissions which ought to be supplied; thus, in the chapter on tangents, he has overlooked the case in which -=

a case which will occur in the curve whose equation is x4_ay x2 + y3-0, and in various others. This is the more remarkable, because it was exhibited by an early objector to the new analysis, (M. Rolle,) as furnishing a striking exception to the universality of the fluxional method. It is, indeed, a real difficulty to a learner, though easily surmounted by the assistance of an intelligent tutor, and ought certainly to have been explained in the work before us. Under the head of tangents, too, the author should have treated the inverse problem, in which the equation expressing the nature of the curve is deduced from the analytic value of the subtangent. The examples at page 329, are too restricted to supply this deficiency.

He

Again, in the chapter on points of contrary flexure, a student will not meet with all the information which he requires. is not, for example, told that, at a certain point of a curve there will be inflexion and neither maximum nor minimum when

Y

X2

30

becomes nothing simultaneously with. Nor is he taught to distinguish between points of inflexion and regression. Regressions of x xy + ÿ ÿ ÿ―3ÿÿÿ

the second species, indicated by the formula

0 or co, certainly merited particular attention. The succeeding chapter on the radius of curvature, though excellent as far as it goes, is still defective: for here, also, the inverse problem of finding the curve from the radius of curvature is omitted, although it may be subdivided into at least four cases, viz. when the curve is referred to a focus, when it is referred to an axis, when the radius or co-radius is given in terms of the abscissa, and when it is given in a curve referred to an axis. On this part of the subject, the papers of the Riccati and of Gabriel Manfredi, in the second volume of the Bolonian Commentaries, may be consulted with advantage.

There are some very ingenious and useful propositions in the chapter on spirals; but, to have rendered it complete, the author should have noticed the spiral of Pappus, and the Loxodromic spiral; especially as the latter leads to the solution of a very in-teresting problem in navigation, a subject which Mr. Dealtry obviously does not think beneath his notice, since he has treated of Mercator's projection. But the omission, which we most regret, is that of curves of double curvature, since the consideration of their tangents, their osculatory and normal planes, is extremely interesting, and, in the usual cases, free from any difficulty which may not be easily removed.

Farther, we must notice the chapters on fluents and fluxional equations. These, as we have already intimated, are highly ingenious and valuable, and their utility is much increased by the addition of some elegant propositions from Demoivre and Cotes; but they are not altogether complete. The integration of fluxional equations involving two variable quantities is imperfectly treated; the comprehensive method by a separation of the indeterminates is scarcely adverted to, and the criterion of integrability in equations of these kinds no where exhibited. For this the reader may be referred to the works on the integral calculus by Euler, Lacroix, and Bossut; by the latter of whom this branch of the subject is treated in a very masterly manner. We lament that no English author, with whose works we are acquainted, has entered upon this particular inquiry, notwithstanding it is that to which we must look for the principal improvements in the modern analysis.

A less important circumstance, which has been left unnoticed by Mr.. Dealtry, is, that in the investigations of curves, such formulæ sometimes arise as admit of integrations which are really different, and supply us with curves of various kinds, even without the addition of any constant quantity. Thus the equation 2 x y2 y x (x-y)2.

z, may become by integration,

2

=Y,

x+y x-y

=y, &c. or

2x

2y

2x+ex-cy=fz, the flu

XY

ent varying with the assumed value of c, but being limited by certain relations of the unknown quantities.

3dly. We would give a few instances in which the solutions of particular problems might admit of improvement. And here we first turn to the investigations relative to the conchoid, where those who learn the nature of the curve from this book, will be left in ignorance as to the existence of such curves as the inferior and nodated conchoids, and of that in which there is a conjugate point. Here, too, is a solution of the problem, to find the point of contrary flexure' in a conchoid, unaccompanied by the remark that the inferior couchoid is often without any such point.

In the chapter on the maxima and minima of curves under certain conditions, we object to the solutions of the 2d and 8th examples. Thus in the problem where it is required to find the curve, which by a revolution round its axis shall generate the greatest solid under a given surface, Mr. Dealtry determines the solid to be a sphere. But this is only a particular case of the general solution: (y2-e)y for the fluxional equation to the curve is

which becomes x

У у

X [4 a2 y2—(y2-e) 2]

√(4a2—y2), an equation to a circle, only when

e-o. Again, in finding the curve of swiftest descent, when the velocity varies as the square root of the ordinate, our author determines it to be a cycloid, but does not notice the essential condition, that the curve must commence at the upper of the two given points; as was first shewn by Newton in his admirable construction of the problem, given in Phil. Transac. No. 224.

The solution of the problem in which it is required to find when that part of the equation of time which arises from the obliquity of the ecliptic, is a maximum,' is correct: but has the disadvantage of not being fluxional. Were it not that Mr. Dealtry has declined to investigate the fluxions of spherical triangles, he might have exhibited a very simple solution in a small compass. For the sun's longitude (7) will form the hypothenuse of a right angled spherical triangle, of which his right ascension (a) will be the base, and the obliquity of the ecliptic or angle between them, a constant angle. Hence, by Cagnoli's Trigonometry, page 329 and 677, we have ia sin. 21: sin. 2 a. Therefore, when ia, as it must be in the case of the maximum, sin. 27-sin. 2a. Consequently, 2l must be the supplement of 2a, or lta-90°. So that when i-a=0, or l—a—a max. that is, when this part of the equation of time is a maximum, the sum of the sun's longitude and right ascension will be 90 degrees; the sun being either in the 1st or 3d quadrant of the ecliptic. The correspondent time is about May the 7th or November the 8th.

In solving the mechanical problems in which the effects of friction will be very considerable, it might have been advisable to shew how those effects are to be estimated or brought into the calculus upon any assumed hypothesis: though if substances were perfectly smooth, or chains, cords, &c. perfectly flexible, the process of Mr. Dealtry would be strictly correct. Here too we would remark that, in prob. 107, where the time is to be computed in which a chain will run off a pulley, the length of the chain being L, the difference in the length of its two ends at the commencement of the motion 2a, and m=161 feet, Mr. Dealtry's final expression for the time t is

4m

£
hyp. log. L+V ( a L−2 a2 )+({ L−a j2

a

but this manifestly reduces to the simpler and more convenient expression,

L+√(L2-4a2).

2a

Prob. 108, is Suppose a weight suspended by a cord passing over a fixed pulley, to be uniformly drawn up: required the number of vibrations which the weight would make before it reaches the

pulley? It is demonstrated by a fluxional process, t that the number of vibrations made by such a variable pendulum is twice the number that would be made in the same time by a common pendulum whose length is a, the primitive length of the variable pendulum. The fluxional solution was certainly the only one open to our author; but the mathematical student will be aware that the problem may be solved more easily without fluxions: for an answer may be obtained by merely summing a series of fractions, whose numerators are equal, and whose denominators are square roots, whose sides are single powers, decreasing from a giving term in a given arithmetical progression. A very elegant solution, to a far more general problem, is given by Dr. Hutton, at page 195 of his Select Exercises.'

We have now obtruded on the patience of our readers, and the candour of the author, a considerable number of objections; but the truth is, that this is one of those works which can endure objection, and of which, therefore, it is a more useful task to point out the defects than the merits. We are sensible that, to the exceptions we have taken, Mr. Dealtry may have an answer which, in point of legal strictness at least, would be in a good measure available. He may allege that he has, in his preface, expressly disclaimed the purpose of writing a complete treatise; that his object, as there stated, was merely to collect so much of analytical knowledge as might suffice for the illustration of the chief propositions of Newton's Principia;-and that he has in terms protested against all demands exceeding this limit. It is difficult, we acknowledge, to draw the exact line in such cases; and perhaps most of what we have described as the desiderata of Mr. Dealtry's publication, may have been omitted by him from deliberation, not from inadvertence. At the same time, we should more easily allow to this writer the benefit of the plea in question, if he had executed less well that which he has actually attempted; and we have so favourable an opinion of his performance that we cannot help wishing it were as complete as it is excellent. Indeed we know not where to look for a work which might so securely be recommended to that class of persons whom the author avows himself to have had principally in view;-academical students of the mathematics. Nor, amidst the other and more peculiarly appropriate merits which we have already ascribed to it, can we forget to mention a quality, in which some mathematical compositions of considerable eminence have been greatly defective-the unaffected language and unpretending manner in which its principles and results are developed.

This work is handsomely and, in general, correctly printed. There appears to us, however, to be nothing, either in the quantity

of matter introduced, or in the length and structure of the analytical expressions, which could call for the royal octavo size; the only effect of which is, that the volume is rendered unnecessarily cumbersome and expensive.

ART. V. The State of the Established Church, in a Series of Letters to the Right Honourable Spencer Perceval, Chancellor of the Exchequer, &c. Second Edition, corrected and enlarged, With an Appendix of Official Documents. pp. 151. Stockdale, Pall-Mall, London. 1810.

N Established Church has for its end the maintenance of reli

Avion, and it pursues that end by the appointment of ministers,

supported from public funds, whose business it is to perform religious offices, convey religious instruction, and promote, by precept and example, consistent practice. It must at all times, therefore, be desirable to discuss, in what degree the means employed effect the end proposed: and, if any plans are suggested by which greater efficacy may be given to the means, it is important that they should be fairly stated and considered.

ness.

But, in proportion as the subject involves considerations of deep and serious concern, it is essentially requisite that the task of discussing it should not be lightly undertaken. The person who comes forward for this purpose may reasonably be expected, to have previously examined his competency to the busiHe should be satisfied that he is not led away by a fondness for finding faults, and amusing the public with plans of fancied perfection, never to be realized in any establishments in which human beings are concerned. Above all, he should be cautious not to bring to the discussion a mind soured by spleen, or perverted by prejudice; a disposition to give exaggerated statements of e existing imperfections; and to set off facts and characters so as to convey, on the whole, a most unfair and false representation. Should he be unfortunately deficient in discretion, candour, or good temper, it is highly probable, that, whatever be his desire of doing good to the church, he may inflict upon it serious and positive injury.

The anonymous author of the work before us, who has thought proper to volunteer his services for the benefit of the establishment, is unquestionably very far from possessing these sterling qualities. In many of his assertions respecting particular circumstances of the church, and the character and conduct of the clergy, he violates every law of justice and decorum. His invectives are