From time to time it has been asserted, that the function of impregnation of the ova of these fish is performed after the manner of that of the cartilaginous, viz., before exclusion. The instances related in proof are commonly of a vague kind, and such that little credit can be attached to them. Recently a more precise example has been adduced,—how the ova of the trout, taken from the abdomen of the parent fish, and placed in a “running stream” apart, included in a perforated box, in due time were hatched, producing young fish. The particulars of the experiment, and the result, were published in the spring of last year, and in more than one of the provincial papers; and Dr Robertson of Dunkeld was named as the institutor and reporter of the trial.

In two papers read before this Society, I have very fully described the Diatomaceæ of the Glenshira Sand, which is very remarkable both for the large number of species found in it, which is certainly more than 320, and for the circumstances in which it must have been deposited. There can be no doubt, from the nature of the locality, which I have lately visited, that this bed was formed in the bottom of the Dhu Loch, a shallow fresh-water lake, at that time extending about two miles farther up the valley than it now does, and being at a higher level. In consequence of a rise in the level of the land, or a fall in that of the sea (from which—that is, from Loch Fine, the lower end of the lake is separated by a narrow and low barrier, through which the waters of the lake pass to Loch Fine), the lake has long ago been drained, till its upper end is nearly two miles from the point it must have reached when the bed of sand was formed. The present level of the lake is considerably lower than it was then; the precise difference I had no means of ascertaining, but I believe it is about 30 feet.

In 1843 I presented a paper to the Royal Society of Edinburgh, giving an account of experiments made on the boiling point of water in the Alps, under various barometric pressures. My object was twofold: first, to describe an apparatus which I considered more practically available than those previously in use; and, secondly, to give a simple, and, as I believed, new formula for computing heights from such observations.

With reference to the second point, I became aware, some time after the publication of my paper, that Sir John Leslie had proposed to compute heights by the thermometer, assuming the change of the boiling point to be exactly in proportion to the height ascended. While cheerfully conceding to Sir John Leslie priority on this point, I submit that he did not bring forward experiments to justify its practical adoption.

Several years ago, when discussing the question of the distribution of the stars, a problem occurred to Professor Forbes, which, simple as it is, appears to have escaped notice prior to that time. Having been consulted as to its solution, I communicated my results to Professor Forbes, who has inserted one of them in his paper printed in the Philosophical Magazine for 1850, vol. xxxvii., p. 425. But for the very ingenious application which Professor Forbes has there made of it, the problem might probably not be worth recurring to. As it is, I have thought it would not be altogether uninteresting to give the complete solution.

The Problem is as follows:—There are n dice, each of which has p faces, p being not less than n; it is required to find the number of arrangements which can be formed with them, 1°, So that no two show the same face; 2°, That no three show the same face; 3°, That no four do so, and so on.

The interests connected with the natural family of the Salmonidæ, both in relation to river-sport, the pleasant recreation of angling, and to material wealth, in the instances of the migratory kinds, are so considerable and fully admitted, that I trust no apology is required for submitting to the Society any observations in the least likely to contribute to a more intimate knowledge of the species, whether as regards their structure or their habits.

It is well known to the Society, that a mineral, found in the strata of Torbanehill, not far from Bathgate, in the county of Linlithgow, has lately been the subject of a keen lawsuit. Specimens of this mineral were early in the year put into my hands by the lessee, and sometime afterwards by the proprietor of Torbanehill, for the purpose of obtaining a mineralogical opinion on the nature of the mineral. I stated at the time to both parties, that I would carefully examine it before I could presume to offer a decided opinion. In fact, the mineral was not altogether new to me; for, in the two preceding years, I had received specimens of a mineral from Torbanehill, and from Boghead, which I was unable to refer to any described species; and I was therefore determined to give my opinion with due caution, especially as I understood that large interests were involved in the question.

In approaching the subject of solar light, the first point is to endeavour to form an idea, not altogether indefinite, with respect to its quantity and intensity, as compared with some familiar standard of artificial flame. With this view were made, in the course of last summer, the observations now to be described.

After several trials, it was found, that the most convenient mode of procedure was first to compare a definite small surface, illuminated by solar light, with a like surface illuminated by the mere light of the sky, and then to compare the latter with a similar surface illuminated by the flame of a moderator lamp. The light of the sky thus affords a middle term between the extreme lights of the sun and the lamp, which are too diverse to be directly compared.

Notwithstanding the progress made of late years in animal chemistry in connection with comparative anatomy, I am not aware of any observations that have yet been published on the urinary secretion of fishes. The neglect of this inquiry probably has arisen from several circumstances,—the nature of the element inhabited, the peculiarities of the urinary organs, the difficulty of collecting the matter voided, and its having no well-marked distinctive qualities obvious to the senses.

For some years, as leisure and opportunities offered, I have given attention to the subject, and in the paper which now I have the honour to submit to the Society, I beg to communicate the observations I have made. Few and imperfect as these are, they are given mainly with the hope of attracting notice to the inquiry and of inducing others more favourably situated to engage in its prosecution.

(1.) The inquiry which, with its results, I propose to lay before the Society, was suggested by the following passage in the very popular Treatise on Logic by Dr Whately, now Archbishop of Dublin.

“As in the case of two probable premises, the conclusion is not established except upon the supposition of their being both true, so in the case of two (and the like holds good with any number) distinct and independent indications of the truth of some proposition, unless both of them fail, the proposition must be true: we therefore multiply together the fractions indicating the probability of the failure of each—the chances against it—and the result being the total chances against the establishment of the conclusion by these arguments, this fraction being deducted from unity, the remainder gives the probability for it.

It has been long known that contractile tissue presents itself in the human body in two forms, one composed of fibres of considerable magnitude, and therefore readily visible under a low magnifying power, and marked very characteristically with transverse lines at short intervals, the other consisting of fibres much more minute, of exceedingly soft and delicate aspect, and destitute of transverse striæ. The former variety constitutes the muscles of the limbs, and of all parts whose movements are under the dominion of the will; while the latter forms the contractile element of organs, such as the intestines, which are placed beyond the control of volition. There are, however, some exceptions to this general rule, the principal of which is the heart, whose fibres are a variety of the striped kind.

How convenient soever the wet bulb thermometer and the various organic hygrometers may be for giving indications, by simple inspection, regarding the relative states of dryness and humidity of the atmosphere, it is scarcely possible, in conducting meteorological observations, to dispense altogether with instruments calculated for affording more direct information respecting the amount of aqueous vapour present in the air at any particular time.

The general identity, in idea and design, of the ancient monuments of Southern and Western Europe, with those of Hindostan, is so marked, as to appear to justify the inference that races of Asiatics proceeded westward at different ages, and established themselves along the shores of the Baltic and Mediterranean Seas, and part of the Atlantic Ocean; along which route they have left characteristic monuments, which resemble those of their original country.

The subject which I propose to discuss in this paper is the value of the method of demonstration by superposition. I am satisfied that it has been much under-rated, and in some cases misunderstood. It may be stated, that the essential characteristic of this method of demonstration, is the mental comparison of two magnitudes, by placing one of them upon the other. Euclid's axiom of equality (which, perhaps, is rather a definition) is this: “Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal.” Accordingly, in his first four books, Euclid never regards two magnitudes as equal, except under circumstances wherein it can be shown that this condition is satisfied. Only in one proposition has he avoided the labour which a strict attention to this requirement necessarily imposes; and perhaps, even in that case, it is hypercritical to object to what he has done. All that he assumes is this: it being admitted that when A fills the same space as B, A is equal to B; it must therefore be admitted, that when A and C together fill the same space as B and C together, A is also equal to B.

To those who study the progress of exact science, the common spinning-top is a symbol of the labours and the perplexities of men who had successfully threaded the mazes of the planetary motions. The mathematicians of the last age, searching through nature for problems worthy of their analysis, found in this toy of their youth, ample occupation for their highest mathematical powers.

No illustration of astronomical precession can be devised more perfect than that presented by a properly balanced top, but yet the motion of rotation has intricacies far exceeding those of the theory of precession.

The memoir which I have now the honour of presenting to the Society contains part of a continued investigation, of which the first results were published in a paper read, last year, before the Chemical Society of London. The nature of the materials experimented on is such, and the scope of the subject so large, as to afford nothing more to all I have at present brought to light than the name of a bare and partial outline; yet the facts I have now to adduce, of some interest in themselves, will increase that of those already made known, and prove of additional service in aiding to complete that desirable object, a knowledge of the chemical constitution of the natural bases. In the paper alluded to, I shewed that, by the action of iodide of ethyl and of methyl upon morphia and codeine, the two most prominent alkaloids of opium, new salts are produced, in which the basic molecules appear to assimilate by their chemical characters to the hypothetical metal ammonium, or its oxide rather; so that, in the system of Hofmann, these peculiar results of natural, and as yet inimitable agencies, would take rank among the nitryle bases.

Twenty-two years have now elapsed, since Runge first published his remark, able experiments on coal naphtha, and it would, perhaps, be difficult to instance any chemical investigation which has formed the point of departure of a greater number of researches. When we consider the vast quantity of bodies which have, first and last, been obtained from coal-tar, it might appear that little more remained to be done,—that the mine was exhausted,—but so far from this being the case, the discovery of one substance has only served to pave the way for the isolation of others.

Among the bodies examined by Runge, there was one which apparently possessed comparatively few features of interest; indeed its very name (the first syllable derived from λευχός was intended to express its supposed inability to produce coloured reactions, a feature which, in the chemistry of the time, militated greatly against its claims to notice. I have used the expression “supposed inability,” because I shall show further on, that this substance is capable, under certain conditions, of affording extremely brilliant colorations. Eventually, Gerhardt, by acting on quinine, cinchonine, and strychnine, with hydrate of potash, obtained the same body. The first chemist who succeeded in procuring any of its compounds in a state of tolerable purity was Hofmann, whose analysis of the platinum salt is very nearly exact. But, at the time of that analysis, he was of opinion that the products obtained from coal and chinoline were essentially different, an opinion which he subsequently retracted. In the mean time, the alkaloid, as obtained from cinchonine was examined by Bromeis and Laurent, their results, however, not elucidating the composition of the basic fluid obtained in the manner alluded to.

It is well known that no satisfactory demonstration has ever been given of Fermat's celebrated theorem, which asserts that the equation an = bn + cn is impossible, if a, b, c, are whole numbers, and n is any whole number greater than 2. In Legendre's Théorie des Nombres, he demonstrates the cases of n = 3, n = 4, and n = 5, the latter only in his Second Supplement. In Crelle's Mathematical Journal, ix. 390, M. Dirichlet, a mathematician of Berlin, has demonstrated the case of n = 14, but I am not aware whether his demonstration is considered successful. Legendre informs us (Second Supplement, p. 3) that the Academy of Sciences, with the view of doing honour to the memory of Fermat, proposed, as the subject of one of its mathematical prizes, the demonstration of this theorem; but the Concourse, though prolonged beyond the usual term, produced no result.

In a paper on pyromeconic acid read before this Society, and since published in the Philosophical Magazine for September 1852, I have detailed the preparation and properties of a compound obtained by the substitution of an equivalent of bromine for an equivalent of hydrogen in that acid. Having observed that this substitution was very easily effected, I was induced to attempt the formation of an iodopyromeconic acid, in the hope of adding one to the very few instances in which the direct substitution of iodine for hydrogen has been found possible. For this purpose, I digested pyromeconic acid with tincture of iodine, but no success attended the experiment, the acid remaining entirely unchanged. The failure of this attempt led me to speculate as to its cause, and to contrive a method of producing the required substitution which has proved entirely successful, and has the further advantage of being perfectly general, so that its application will enable chemists to obtain iodine substitution compounds in cases in which they have hitherto failed.

Owing to the great length of time over which the investigation of the products of the destructive distillation of animal substances has stretched, and various circumstances which it is unnecessary to detail, the inquiry has been pursued in a somewhat fragmentary manner, and with less continuity than might have been desired. The difficulties attending many of the experiments, and the occasional exhaustion of materials prepared by laborious processes, extending in many instances over considerable periods, have occasioned long intervals in the regular course of the inquiry which it became necessary to occupy with the examination of such matters as could be taken up at the moment. In this way a number of facts required to complete the history of the bases already described have gradually been accumulated, some of the products of their decomposition examined, and the pyrrol so frequently adverted to in the previous parts of this paper has been subjected to a full investigation. The details of these experiments form the subject of the present communication.

The object of the following communication is to describe a method by which every variety of visible colour may be exhibited to the eye in such a form as to admit of accurate comparison; to show how experiments so made may be registered numerically; and to deduce from these numerical results certain laws of vision.

The different tints are produced by means of a combination of discs of paper, painted with the pigments commonly used in the arts, and arranged round an axis, so that a sector of any required angular magnitude of each colour may be exposed. When this system of discs is set in rapid rotation, the sectors of the different colours become indistinguishable, and the whole appears of one uniform tint. The resultant tints of two different combinations of colours may be compared by using a second set of discs of a smaller size, and placing these over the centre of the first set, so as to leave the outer portion of the larger discs exposed. The resultant tint of the first combination will then appear in a ring round that of the second, and may be very carefully compared with it.