in crystallized glass.' Dr. B. farther concludes that the structure of the crystalline lens in fishes is not symmetrical, as has hitherto been supposed, consisting merely of a number of coats of different densities; but it has a distinct relation to that diameter of the sphere which is the axis of vision.' He deems it probable that the final cause of this peculiar structure is to correct the spherical aberration.

Some farther Account of the fossil Remains of an Animal, of which a Description was given to the Society in 1814. By Sir Ev. Home, Bart. V.P.R.S. The animal in question was found at Lyme, in the summer of 1814, and was dug out of the cliff on the beach. From the parts which have been obtained, it seems evidently to have been a fish, but of a nondescript genus; thus affording an additional evidence of the general fact, that a race of animals lived on the earth in some of its former states that were different from any which now exist on it.

Farther Observations on the Feet of Animals whose progressive Motion can be carried on against Gravity. By the Same.

Mr. Bauer examined the feet of these animals with high magnifiers, and has given drawings of the appearances which he saw, whence we learn very fully the nature of their mechanism. The toes of the different kinds of flies are furnished with suckers, connected with muscles under the power of the will, which vary both in their form and number in different species, but which all agree in the mode of their operation. The paper concludes with an account of a peculiar apparatus, like a cushion, attached to the feet of certain animals that leap to great distances, for the purpose of diminishing the shock which they would otherwise experience when they alight on the ground.

MATHEMATICS.

An Essay towards the Calculus of Functions, Part II. By C. Babbage, Esq. At the conclusion of his former paper, Mr. B. gave us reason to expect some farther communications from him relative to his calculus of functions. In the preceding memoir, his views were principally directed towards the solution of functional equations containing only one variable: but it appears that subsequent inquiries led him to several new methods, and to the resolution of functional equations of a much more complicated form; which, he says, has convinced him of the importance of the calculus, particularly as an instrument of discovery in the more difficult branches of analysis.

• Nor

Nor is it only in the recesses of this abstract science that its advantages will be felt: it is peculiarly adapted to the discovery of those laws of action by which one particle of matter attracts or repels another of the same or of a different species; consequently, it may be applied to every branch of natural philosophy, where the object is to discover, by calculation from the results of experiment, the laws which regulate the action of the ultimate particles of bodies. To the accomplishment of these desirable purposes it must be confessed that it is in its present state unequal; but should the labours of future enquirers give to it that perfection, which other methods of investigation have attained, it is not too much to hope, that its maturer age shall unveil the hidden laws which govern the phenomena of magnetic, electric, or even of chemical action.'

Without entertaining ideas so sanguine as those of Mr. Babbage on this subject, we can perceive that an immense advantage would be gained, and an almost indefinite extent given to the analytical sciences, if the calculus to which he here alludes could be brought to perfection: but, unfortunately, we see little prospect of this being effected. Mr. B.'s two memoirs occupy a considerable number of pages, and we must say that few advances are made towards the completion of his views. We think that, by this time, he must be well aware of the great difference between forming a project and carrying it into execution; and he must have discovered that it is one thing to be convinced of the utility of a calculus, and another to reduce it to a manageable shape, to fix its limits, and to investigate and establish its rules.

In our remarks on this gentleman's former paper, we expressed some apprehension of a complicated notation and an artificial solution; and we conceive that no person, who looks even slightly at the pages of the present article, will say that our apprehensions were groundless. Every case seems to require distinct symbols; and new characters start up in nearly every problem, with which the mind becomes bewildered, embarrassed, and fatigued. If any tolerably simple means could have been devised for the solution of problems of the kind proposed by Mr. B., it would, as we before observed, have been an important improvement in analysis; and we were willing to entertain hopes when we reported his last paper, that in the present article, which was then promised, we should have seen a direct application of his calculus to the solution of some simple and obvious examples: but we do not think that those which are given can be considered as falling under this class. Of the problems, the second is perhaps the simplest, as well as the nearest in character to those in the former memoir; and it may therefore be understood by those

E 4

who

who have attended to the definitions there given. We consequently select it for the perusal of our readers:

• Given the same equation

↓ (x, y) = ↓ (α x, By)

Suppose one particular solution of this equation is known, let it be f (x, y),

then take (x, y) = f (x, y), being perfectly arbitrary, and the given equation becomes f(x, y) = ƒ (ax, By),

of

which is evidently satisfied since ƒ (x, y) =ƒ (ax, By) by the hypothesis.

one particular solution of this equation is ƒ (x, y) = x hence the general solution is

log.y

Ex. 2. Given the equation (x, y) = (x", ") a particular

case is ƒ (x, y) =

log.x

Tog.y

hence the general solution is

+ (x, y) = 4 (log⋅ x

log.y

Ex. 3. Given the equation ↓ (x, y) = ↓ (x", y′′)

In order to get a particular case, let us put

ƒ (x, y) = log.' x + a log.2 y

by substituting this value we shall find that it is a particular solu

tion of the equation, if a =

log.n
log.m

log. 3

:)

as in the last example,

Ifnm we have ↓ (x, y) = 4

I

and if m =

n

we have the same solution as in the first.'

In these examples, it will be observed that one case is always supposed known, by which means it appears to us that the greatest part of the difficulty is avoided; it is sufficiently obvious, when pointed out, that in example 1. ƒ (x, y) = x log.y; log,x &c. but how should we

and in example 2. ƒ (x, y) = log⋅y

arrive at this conclusion if the case had not been exhibited? It is true that we might by chance have stumbled on the solution but the matter required is a direct process, which Mr. Babbage does not appear to have acquired.

A New Demonstration of the binomial Theorem. By Thomas Knight, Esq.-The author begins this memoir by observing

that

It is somewhat remarkable that, amongst the various and far-fetched methods and artifices by which the binomial theorem has been obtained, no one should once have thought of the only course which seems obvious and natural. The equation

(a + x)m × (a + y)TM = { (a + x) (a + y)}" expresses the general property of powers, whether m be positive or negative, whole or fractional, and from this equation, without the help of any artifice, the series in question is deduced.

Some investigations have been found fault with, as drawn from principles allied to the method of fluxions; whilst, on the other hand, a demonstration, taken from the "Théorie des Fonctions," has been represented as perfect; but I cannot help thinking that it is as much connected with the fluxional calculus as any of the rest; for it seems to make no difference whether, in (a + x)" we substitute x+u for x, and take the co-efficient of u, or substitute x+x, and take the co-efficient of x. The former substitution was made because it was known to be equivalent to the other, and has so little apparent connection with the subject, that a student would hardly understand why it was made. The demonstration of M. La Croix in the Introduction to his "Calcul Différentiel," is liable of course to the same objection. If we multiply a +x, continually, first by itself, and then by the powers successively arising, we easily see that the second term of each succeeding product is of the form na x, n being the exponent of the power: this does not require a more formal proof, and I assume it in what follows. Nor is it more difficult to perceive, that, generally m being positive or negative, whole or fractional, the following form may be assumed.

(a + x)” = a” + ́Aam

Aam-3 x3+ where A, "A, "A, &c. are expressions depending on m alone.'

In the same manner,

m-I

(a + y)" = a" + 'A a"-1y + &c.

and therefore, by comparing (a + x)TM × (a + y)", with (a2 + ax + ay + xy)" or with a" (a + x + xy)”, by first making = 1 + and retaining only the first power of y

a

in both expansions, the forms or values of the co-efficients 'A, "A, &c. become determined.

We

We have only to observe that the difference between this investigation, and that which has been given by Manning, Woodhouse, and Bonnycastle, and particularly that of Mr. Spence, appears to us too small to justify Mr. Knight in calling it a new demonstration of the binomial theorem.'

On the Fluents of Irrational Functions. By E. F. Bromhead, Esq. M.A. This is an attempt to generalize and systematize the methods of determining the fluents of irrational functions; and to shew that all known forms result from other forms of the greatest extent, not depending on particular functions, but on the properties of all rational functions whatever.

We must beg, however, to refrain from giving any specimen of the author's operations; because, with all the advantages derived from Mr. Bromhead's notes and his numerous examples, we are still doubtful whether we exactly understand what it is that he intends to perform; and at all events we are certain that it would be useless to attempt to illustrate it in the space within which it is necessary for us to confine this article.

We have at different times endeavoured to expose the absurdity of perpetually introducing new symbols, new terms, and new significations of old terms, which seems to be the principal aim of some of our modern analysts: but in no instance have we seen it carried to so great a length as in the papers which we have just examined.

ART. V. Redemption; a Poem in Twenty Books. By George Woodley, Author of "The Church-Yard," and other Poems. 2 Vols. 8vo. 16s. Boards. Law and Whittaker. 1816.

WE E wish most cordially that it were possible to devise some legitimate means of putting an effectual stop to such libels on our age and country, as are constituted by the poem of Redemption,' and by innumerable others of the same poetically prosaic and impiously pious description. What earthly purpose can be answered by works which only serve to waste time, money, and all other good things, we cannot conceive: but when, to these indefensible objects, is added the seemingly studied purpose of degrading the Bible to the level of a prolix and vulgar Christmas carol, it would be difficult to describe our mingled feelings. Happily for our readers and for ourselves, we need not attempt so unpleasing a task, since half-a-dozen lines, selected from any portion of either of these closely printed volumes, will speak for themselves: but we shall make conviction doubly clear by a fuller trial.