Imatges de pàgina
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and at such a length that we can do little more than exhibit the objects and results of the investigation.

In consequence of perusing Dr. Porterfield's paper on the internal motions of the eye, (Edinb. Med. Essays, vol. iv. p. 124.) Dr. Young was induced to resume this discussion; and he has in consequence made such observations as appear to him decisive in favour of his former conclusion, as far as that opinion attributed to the lens a power of changing its figure. He commences his present inquiry into the Mechanism of the Eye with a general consideration of the sense of vision; enumerating some dioptrical propositions subservient to his purpose, and describing an instrument for readily ascertaining the focal distance of the eye. Hence he passes to investigate the dimensions and refractive powers of this organ in its quiescent state; with the form and magnitude of the picture which is delineated on the retina. Next, he inquires into the extent of the changes which the eye admits; and what degree of alteration in its proportions will be necessary for these changes, on the various suppositions which are principally deserving of comparison. He then relates a variety of experiments to decide the truth of these hypotheses; and he concludes with some anatomical illustrations of the capacity of this organ in various classes of animals, for the functions attributed to them.-A series of plates accompanies the lecture; the want of which renders it impossible for us to particularize some of the most important experiments: but the principal deductions are thus recapitulated by the Doctor himself:

First, the determination of the refractive power of a variable medium, and its application to the constitution of the crystallinelens. Secondly, the construction of an instrument for ascertaining, upon inspection, the exact focal distance of every eye, and the remedy for its imperfections. Thirdly, to shew the accurate adjustment of every part of the eye, for seeing with distinctness the greatest possible extent of objects at the same instant. Fourthly, to measure the collected dispersion of coloured rays in the eye. Fifthly, by immerging the eye in water, to demonstrate that its accommodation does not depend on any change in the curvature of the cornea. Sixthly, by confining the eye at the extremities of its axis, to prove that no material alteration of its length can take place. Seventhly, to examine what inference can be drawn from the expariments hitherto made on persons deprived of the lens [as is the case in the operation for extracting cataracts, called couching] to pursue the inquiry, on the principles suggested by Dr. Porterfield, and to confirm his opinion of the utter inability of such persons to change the refractive state of the organ. Eightly, to deduce, from the aberration of the lateral rays, a decisive argument in favour of a change in the figure of the crystalline; to ascertain from the quanity of this aberration, the form into which the lens appears to be thrown

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thrown in my own eye, and the mode by which the change must be produced in that of every other person.'

Dr. Young is aware of the extreme delicacy and precaution requisite in conducting experiments on the eye, and in drawing inferences from them: but he flatters himself that he shall not be deemed too precipitate in denominating this series of experiments satisfactorily demonstrative.

Account of a Monstrous Lamb. In a Letter from Mr. Anthony Carlisle, to the Right Hon. Sir Joseph Banks, Bart., &c.-In this animal, which was born at the full period, the brain and its nerves were wanting. It is unnecessary to state the particular appearances on dissection, because similar facts have been repeatedly observed in the human foetus.

An Anatomical Description of a Male Rhinoceros. By Mr. H. Leigh Thomas, Surgeon. The principal peculiarities, observed in this dissection, consisted in the papillary shape of the processes formed by the internal coat of the intestines; and in four processes, arising from the internal and posterior portion of the sclerotic coat of the eye, and terminating in the choroid coat, at the broadest diameter of the eye. Mr. Thomas supposes that they are intended to accommodate the eye of this animal to near objects.

Account of an Elephant's Tusk, in which the Iron Head of a Spear was found embedded. By Mr. Charles Combe, of Exeter College, Oxford.-This short paper merely announces the fact mentioned in the title; nothing being known respecting the history of the animal to which the tusk belonged.

MATHEMATICAL PAPERS.

On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities. By Robert Woodhouse, A. M. Fellow of Caius College, Cambridge.-Impossible quantities employed in calculation lead to true results; that is, to the same results which may be obtained from the same principles, by a different process, acknowleged to be rigorous, and instituted for the same end: but, if the results be true in fact, they must necessarily be true; or there must be a logical process, according to which, operations with any symbols whatever that produce right conclusions are regulated. If even a compensation of errors happens, such compensation, made manifest and proved admissible, serves in part to establish the justness of the deductive process. Such is briefly the argument used by the author of the present paper, to shew that the explanation and establishment of the logical process with imaginary sym

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bols is not necessarily and inherently impracticable, but contrariwise. In the 68th vol. of the transactions, Mr. Playfair attempted to shew that operations with imaginary quantities are true, by virtue of a certain analogy existing berween the circle and hyperbola: this principle of analogy, and the mode of explanation founded on it, are examined by Mr. Woodhouse; who inquires,

• What is it that determines the nature of this analogy? or how can its several coincidences, interruptions, and limitations be ascertained, except by separate and direct investigations of the properties of the circle and hyperbola? If the analogy between the two curves depends on investigation and is limited thereby, then ail operations with imaginary expressions are perfectly nugatory: since we are not warranted to adopt a single conclusion obtained by their aid, except such conclusion be verified by a distinct and rigorous demonstration.

The author of the principle of analogy allows it to be imperfect; and I perceive no sure method of ascertaining the restrictions to which it is subject, except by the forms that result from actual in'vestigation.'

This argument seems to be conclusive against the principle of analogy; since analogy, or the similarity of the properties of two figures not existing independently of investigation, but found out by it, cannot regulate it :-to obtain a result by a process, and then to use the result in order to explain the process, is arguing, as the French express it, in a vicious circle.

After this discussion of the principle advanced by Mr. Playfair, the author gives his own mode of explanation. He begins by observing that a general demonstration, if true, must be so in every particular case that arises, on giving specific values to the symbols which enter into the calculation:—the form, for instance, of the binomial (a+b), is true when for x we put any number, whole or fractional, positive or negative: but under the general form (a+b)*, the case (a+b)* is not included, since represents no number, and is not capable of arithmetical computation; and it is impossible to prove, without previous convention or arbitrary assumption, that (a+b) ̈1+2√=1a2√/ ́ ̄ ̄1b, &c.

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in demonstration? Mr. W. says that is the abridged symbol for the above series not proved equal, but made to represent it, which it does unambiguously; since, in the form e"=1+x+, &c. proved to be true, we have only to substitute for x,-x2 for x2, &c. √ x

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Hence, according to the present author, from forms established for the functions of real quantities, we obtain a notation, or mode of representation, for other quantities; and such a notation cannot lead us into error, since the abridged symbol stands for only one determinate series.-To prove that such a notation is commodious is the next object :

Now, if e*√ be the symbol for 1+, &c.

and*

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then e*√=1 Fe√ represents 1−x2 +&c., or is the sym

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bol for cos. x; and in like manner unambiguous symbols may be obtained for other lines drawn in a circle.-What remains to be explained is the deductive process in which these symbols enter; and this, perhaps, we shall best accomplish by taking the author's instance, and his reasoning on it.

The sin. x cos. y sin. (x+y)+ sin. (x-y); and the demonstration of this form (as of all others) consists in a series of identical propositions, or symbols representing those propositions, by which the first proposition sin. x. cos. x., or its symbol, is connected with the last proposition sin. (x+j)+ž sin. (xy), or its symbol.

Now the mode of representing sin. Xcos. y, by means of ima

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These two expressions may be shewn to be equivalent, by a series of transformations, each proved lawful and equivalent to its preceding one, by executing the operations directed to be performed by the signs X,+, &c., and by referring to the series which,,,&c. are made to represent.

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If the expressions be manifested to be identical, the proposition is proved.

Thus, demonstration with these imaginary symbols is shewn to be true, without the aid of a new metaphysique, or far fetched principle; and to be conducted in a manner similar to that of demonstration, in which the signs of real quantities are employed. The remaining part of the paper is occupied in assigning the sums of such series as (sin. x)+(sin. 2x)"+, &c.; in explaining the meaning of certain expressions in which imaginary symbols enter; and in discussing the controversy between Leibnitz and others, concerning the logarithms of negative quantities. After having pointed out the erroneous notions which Euler, and even D'Alembert, admitted, Mr. W. observes;

In this controversy, the predominancy of the "Esprit Géometrique" is remarkable; if in an enquiry purely mathematical, any ambiguity or paradox presents itself, the most simple and natural method is, to recur to the original notions on which calculation has been founded. Instead of pursuing this method, the controvertists sought to derive illustration from obscure doctrines, and to discover the latent truth amidst the complex forms and involutions of analysis."

Such is the brief account of the speculations contained in this paper. Those who desire fuller matter, illustration, and a more expanded argumentation, must consult the memoir itself.

Demonstration of a Theorem, by which such Portions of the Solidity of a Sphere are assigned as admit an Algebraic Expression. By Robert Woodhouse, A. M. Fellow of Caius College.Viviani, and after him many other mathematicians, shewed that, if a sphere be pierced perpendicularly to the plane of one of its great circles by two cylinders, of which the diameters are equal to the radii of the sphere, the portion of the spherical surface taken away is such that what remains is quadrable, and equal to four times the square of the radius. By the same method of piercing the sphere, such part of the solidity is taken away that what remains is cubable, and equal to two ninths of the cube of the sphere's diameter. To demonstrate this curious property is the object of the present paper, and the demonstration is effected by the method of triple integrals; which is too much compressed to admit an analysis, yet too long and too abstruse to be inserted in our pages.

The volume concludes with the usual Meteorological Journal. The second Part for 1801 is published, and we shall report its contents as soon as other engagements will permit.

ART.

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