Some observations are lastly added concerning the nature and situation of the ciliary processes in various animals; also on the nature of the marsupium nigrum of birds, and the horseshoe-like appearance in the choroid of fishes; both which have improperly been termed muscular,-the former being a mere duplicature of a membrane which may be unfolded; and in the latter the whole mass being evidently of an uniform texture, the fibrous appearance which has misled some former observers being the effect of transverse fissures, or cracks, which may easily be mistaken for filaments.

The lecture concludes with a few observations on the bony scales of the eyes of birds, to which the author denies any concern in changing the focus of the eye; and on a cavity observable in the eyes of some insects which has been supposed to be in some measure subservient to this purpose.

On the necessary Truth of certain Conclusions obtained by Means of Imaginary Quantities. By Robert Woodhouse, A.M. Fellow of Caius College. Communicated by the Rev. S. Vince, A.M. Plumian Professor of Astronomy in the University of Cambridge. Read January 8, 1801. [Phil. Trans. 1801, p. 89.]

The object of this paper is to show, that we may be assured of the justness and accuracy of conclusions obtained by means of imaginary quantities, without verifying such conclusions by separate investigations, or without inferring their truth from analogy. In the first part the author premises at some length certain arguments, to show that the operations with impossible quantities must have a logic equally strict and certain with the logic that appertains to real quantities, and that the aid obtained by these quantities would be perfectly useless if such conclusions rested only on the frail basis of analogy.

The author proceeds next to show that operations with imaginary quantities are by no means mechanical, but that they are conducted according to the rules of strict and rigorous logic; and that, although strictly speaking no proposition concerning them can be true or false, yet, after the demonstrations of certain formulæ for real quantities, demonstrations with impossible quantities may be legitimately and logically conducted. The series, for instance, for the development of an exponential, when the exponent is an impossible quantity, can never, independently of certain arbitrary assumptions, be duly established; and yet, when the exponent is the sign of a real quantity, the formula for the development may be rigorously proved. With regard to demonstration, it is shown, as in the case of real quantities, it actually proceeds by a series of transformation, each proved to be the same as the foregoing, not by any arguments grounded on the properties of real quantities, but by reference to the forms certain abridged symbols are made to represent, and to the nature of the operations directed to be performed with certain signs.

After thus establishing the principle by which operations with imaginary characters are regulated, the author shows its efficacy and

the use of imaginary characters in the summation of series, proceeding according to the powers of the series and cosines of arcs in arithmetical progression. He likewise shows, that according to his mode of explanation, certain ambiguous expressions that occur in analysis are perfectly intelligible, and that in the controversy concerning the logarithms of negative quantities, carried on formerly between Leibnitz, Bernoulli, Euler, and Dalembert, all paradox and ambiguity may be made to disappear, by referring to the origin and real import of the impossible exponential quantities.

Although the principal object of this paper is to vindicate the indubitable justness of the operation conducted with imaginary characters, yet in the latter part some arguments are likewise offered in favour of the commodiousness of imaginary expressions for facilitating calculations. And, lastly, it is contended, that in the present state of analysis, these expressions are particularly useful in deducing certain conclusions, which without their aid could not be obtained without much difficulty.

On the Production of Artificial Cold by Means of Muriate of Lime. By Mr. Richard Walker. Communicated by Henry Cavendish, Esq. F.R.S. Read January 22, 1801. [Phil. Trans. 1801, p. 120.]

Mr. Walker, since his late communications to the Society on the best means of producing artificial cold, received intelligence that Mr. Lowitz, Professor of Chemistry at Petersburgh, had made some experiments, in which a neutral salt different from those he had himself used, and which is but little known or attended to, produced effects which exceeded his expectations. The salt is the muriated lime; which, mixed with snow in the proportion of about 3 to 2, at the temperature of +27°, produced a refrigeration which sunk the thermometer to 55°; and with this mixture the Professor in one experiment froze no less than 35lbs of quicksilver.

Mr. Walker repeated the experiment with success; but finding that it can only be made during a freezing atmosphere, he resolved to try the effect of this salt, reduced to such a strength by evaporation as to endure being kept in a solid state throughout the year. After describing the expedients used for this purpose, he enumerates the results of two sets of experiments; the first made with the muriated lime prepared so as to be used in winter only, that is, of the specific strength of 1-450; and the other made with the salt prepared so as to be kept for use at any time, the strength of which was 1.490. The apparatus here used (though somewhat improved) is not unlike that described in Mr. Walker's former communication, nor is the process materially different.

The paper concludes with a general view of the different frigorific mixtures-1st, those composed of chemical substances with ice; and 2nd, those in which the use of ice is dispensed with. In a postscript the author suggests a method of obtaining transparent ice, fit for optical purposes, which is effected merely by immersing a vessel

containing a frigorific mixture in water: by this means he has frequently obtained a pellucid coating of ice on the outside of the vessels, of considerable thickness, and, by adapting the form of the vessel, of any figure that might be required.

Account of a monstrous Lamb. In a Letter from Mr. Anthony Carlisle to the Right Honourable Sir Joseph Banks, Bart. K.B. P.R.S. Read January 29, 1801. [Phil. Trans. 1801, p. 139.]

The head of this animal, or rather fœtus, for it was not born alive, was disproportionately small, and had no resemblance to the natural form except in the external ears, which were contiguous, and placed on the front part of the head. Between them was an opening, which proved to be the common passage to both the oesophagus and the trachea. All the organs which are usually found on the face were here wanting; there being neither eyes, nose, nor any of the apparatus belonging to the mouth: the cranium was formed into a hard bone, bearing a near resemblance to the head of a tortoise, and about the size of a plover's egg.

On dissecting this singular production, it was found that the whole cerebrum and all its nerves were wanting. It is hence inferred that the formation and growth of animals in the uterus are independent of any influence from those parts of the brain which properly belong to sensation. The author regrets that this animal did not live to show the phænomena of volition directed to its limbs and other parts, without that intelligence from the organs of the senses which regulate the actions of perfect animals. A careful observance of such circumstances, he thinks, might lead to discoveries of the greatest importance in that part of physiology which is still enveloped in much obscurity.

An Anatomical Description of a male Rhinoceros. By Mr. H. Leigh Thomas, Surgeon. Communicated by George Fordyce, M.D. F.R.S. Read January 29, 1801. [Phil. Trans. 1801, p. 145.]

An opportunity having lately offered of examining a living rhinoceros, and of dissecting it after death, Mr. Thomas availed himself of the favourable incident; and in this paper affords us all the information he could gather concerning that curious animal.

Dr. James Parsons having, upwards of fifty years ago, laid before the Society some account of the external parts of a rhinoceros, a recapitulation is here given of what is contained in that paper; but on the other hand, the description of the internal parts, and of some of the organs, is the more ample, and, together with some observations on its habits, will probably prove equally satisfactory to the anatomist and the physiologist.

Without entering into the technical part of this description, we shall only notice here certain peculiarities concerning the eye, in

which a structure has been observed, of which no other animal is as yet known to partake.

In cutting through the eye, four processes were met with, arising by distinct tendons from the internal or posterior portion of the sclerotic, which, passing forward, gradually became broader, and insensibly lost themselves in and formed a part of the choroid. These processes had a muscular appearance. The ciliary processes were affixed to the crystalline lens, which was nearly spherical, with the anterior surface somewhat flattened.

Concerning the use of this structure, especially of the processes, the author observes, that as the natural unwieldiness of this animal probably unfits it for quickly directing its sight to objects which for its own preservation it ought to be able to examine easily and minutely, nature seems to have supplied it with an apparatus calculated to remedy this imperfection. The change in the eye, which adapts it for distinct vision at different distances, the author ascribes in a great measure to the four above-mentioned processes, which upon contracting will shorten the axis of the eye, and produce the desired effect for near objects, it is observed that this animal has the eyes placed much nearer the mouth than any other, whence, without any adaptation of the eye, it is capable of performing the most essential of its functions, viz. that of examining the food necessary for its subsistence.

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, Cambridge. Communicated by Joseph Planta, Esq. Sec.R.S. Read February 12, 1801. [Phil. Trans. 1801, p. 153.]

In the second volume of the Memoirs of the National Institute, M. Bossut announces a theorem relative to the solidity of a sphere similar to Viviani's, by which quadrable portions of a hemispherical vault are assigned. M. Bossut withholds the analysis that led him to his result, but mentions that it involves an integration much more complicated than that which occurs in Viviani's problem. In the present paper, Mr. Woodhouse furnishes the analysis that leads to the result announced in Bossut's theorem, and, by a transformation of the co-ordinates of the sphere, arrives at a differential expression, the integration of which does not appear more complicated than that employed in the solution of Viviani's problem.

Account of the Discovery of Silver in Herland Copper Mine. By John Hawkins, Esq. Communicated by the Right Hon. Sir Joseph Banks, Bart. K.B. P.R.S. Read February 12, 1801. [Phil. Trans. 1801, p. 159.]

This ore was found in the Herland mine, about six miles from St. Michael's Mount in Cornwall. It consists chiefly of lodes of

copper, of which several run nearly in a parallel direction from east to west.

The peculiarity here is, that there are several cross courses which run north and south, the greatest part of which contain no metal : these meeting with the lodes of copper ore, interrupt their continuity, or, as is expressed by the miners, heave them out of their direction, so that at the place of intersection the copper lodes seem to have been forced aside eighteen or twenty inches. One of these cross courses has of late been discovered to yield silver in no inconsiderable quantity; but with this particular circumstance, that at and near the place of intersection the ores both of silver and copper are much less productive than at some distance from it. The silver ore consists of a mixture of galena, native bismuth, gray cobalt, vitreous silver, and native silver chiefly in a capillary form.

Account of an Elephant's Tusk, in which the Iron Head of a Spear was found imbedded. By Mr. Charles Combe, of Exeter College, Oxford. In a Letter to the Right Hon. Sir Joseph Banks, Bart. K.B. P.R.S. Read February 19, 1801. [Phil. Trans. 1801, p. 165.] This tusk, which weighed fifty pounds, and measured about six feet in length, is supposed to have been imported from Africa, and was purchased at Liverpool by a manufacturer of Birmingham. shaking it, a rattling noise was heard near the middle part, which, on cutting the tooth transversely, was found to be occasioned by an iron spear-head, about six inches and a half long, which lay in the longitudinal direction of the tooth, with the point foremost, and was considerably corroded. It is conjectured that the spear had entered at the basis of the trunk, between the interior angle of the eye and the proboscis, the cavity of the tusk being placed immediately beneath this part. From the quantity of bony matter that had been formed round this extraneous body, it is inferred that the animal must have lived a considerable time after it had received the wound. -The spear-head and the part of the tooth in which it had been imbedded were exhibited to the Society at the close of the Meeting.

Description of the Arseniates of Copper, and of Iron, from the County of Cornwall. By the Count de Bournon. Communicated by the Right Hon. Sir Joseph Banks, Bart.K.B.P.R.S. Read February 19, 1801. [Phil. Trans. 1801, p. 169.]

In the mine called Huel Gorland, in the parish of Gwennap in Cornwall, has lately been discovered a species of ore, consisting of different combinations of the arsenic acid with copper and iron, which, though some mention have been made of it by German writers as having been found in Silesia, appears yet, from the great abundance afforded by the above-mentioned mine, to be almost peculiar to this country. After an historical account of the little that has hitherto been known concerning this substance, Count Bournon