In examining the light reflected from the surface of calcareous spar, when in contact with different fluids, I observed several phenomena, both of light and colour, which led me to make the same experiments on the natural and artificial surfaces of the precious stones and other minerals. In these experiments, the intensity of the reflected pencil varied, as might have been expected, with the refractive power of the fluid; but I was not prepared for the curious fact, that when the reflective power of the surface was reduced almost to nothing, the surface was no longer able to reflect white light, but reflected pencils of different colours, depending on the approximation of the refractive power of the fluid to that of the solid. When the crystal had much double refraction, the colour of the reflected light varied with the inclination of the plane of reflection to the plane passing through the axis of the crystal.

The History of Greece, by Mr Grote, perhaps the most notable production of modern English scholarship, is characterised, amongst many great virtues, by what has always appeared to me, in a historian, a great fault—a tendency to undervalue traditional authority, and to over-rate the importance of conjectural ingenuity, in the reconstruction of the past. One of the most remarkable instances of this tendency which has fallen specially under my view, is his treatment of Lycurgus and his legislation, as it occurs near the end of his second volume. The fallacies which seem to me to lie in this treatment, it is the object of this paper shortly to set forth.

The subject of this paper is a very old one, and may to many appear to be sufficiently worn; but I venture to hope, that there are some to whom a glimpse of the successive approaches of the human mind towards the right understanding of a question of pure logic, may have an interest,—even although the problem solved be an abstract one, and the conclusion a negative conclusion, having little practical application. Like the kindred problem of the quadrature of the circle, or the metaphysical problem of “Knowing and Being,” the theory of parallels has been attacked in various directions, and although it is true that no one ever reached the goal he aimed at, yet it is not the less certain that great and positive results have followed in the history of human attainment. If no other lesson has been learnt, this at least may have been: that in reasoning it is necessary to look warily around and abroad at every step, seeing that admissions, the most obviously inadmissible, or evasions the most palpable, have foiled generations of thinkers, whilst those who have detected their errors have fallen into others of an equally destructive character.

Having determined to spend my autumn holiday in the Pyrenees, it occurred to me that I might add a link to a very interesting chain of observations which was begun by one of the Vice-Presidents of this Society in the year 1835. Twenty-eight years had elapsed since Principal Forbes made his careful observations of the temperatures of certain of the Pyrenean hot springs; and I thought that if I could repeat his experiments after so long an interval, the results might not be without some interest.

A visit to the Pyrenees in the year 1835 was so far uncommon as to make a description of the places through which the traveller passed a source of entertainment to the general as well as the scientific reader; but now such description would be tedious. Then, too, some remarks were necessary touching the geological features of the localities where the springs existed; but it is unnecessary to repeat them now, for I am not aware that the explorations into the rocks in search of new springs, or of a more abundant supply of water from the older springs, have altered the relations as described by Principal Forbes. Therefore, respecting the natural aspect of the country, beautiful though it be, and its geological features, I shall have nothing to say; and as I have no new theory of the cause of thermal springs to propound, I need not repeat the speculations with which we are familiar.

In my former paper on the subject, I selected the following problem:—

From a given square, one quarter is cut off, to divide the remaining gnomon into four such parts that they shall be capable of forming a square.

The gnomon is, I assume, incapable of being formed into a square by being cut into three parts, and consequently the number of different ways in which it can be so formed, by cutting it into four parts, must be very limited. But, to show the fertility of the method of superposition, I exhibited the solution of the problem in twelve different manners. Many of these, no doubt, have much that is in common, whilst, on the other hand, some (such as the 12th) differ in every feature from the rest. I had thoughts of following up my plea for the study of the old geometry, by exhibiting the solutions of the 47th proposition of Euclid's first book in their beautiful variety. I have indeed temptation to do so. The modification which I gave of the demonstration of this proposition in the notes to my edition of Playfair's Geometry (edition 1846, p. 273), has had the honour of being exhibited in two different mechanical forms. The first by two rotations without sliding, whereby the two squares on the sides, when placed together, are converted into the square on the hypothenuse; the second, by two transpositions (slidings) without rotation, whereby the same change is effected The former is obvious enough, and could have escaped nobody. The latter is described by Professor De-Morgan in the “Quarterly Journal of Mathematics,” vol. i. p. 236.

In 1855, when the systematic registration of births in Scotland was established, the schedule used exacted from the public a variety of interesting details in connection with each return,—a circumstance which gives to the registers for that year an extraordinary value. For, in consequence, I believe, of numerous complaints regarding the irksome labour of filling up the document, it was discontinued, and a much less comprehensive schedule has been in use ever since. It is from the registers of births for 1855 that I have extracted almost all the data which have yielded the results I am now about to communicate. Similar data cannot be found in the subsequent registers. The great value of these registers has been distinctly pointed out by Dr Stark, the able assistant to the Registrar-General. I must here acknowledge, with grateful thanks, the assistance and encouragement I have received from Mr Seton and other officials of the Register-House.

Crude American petroleum evolves, at ordinary temperatures, a quantity of combustible gas, which takes fire on contact with flame, and, when mixed in certain proportions with air, produces an explosive mixture. It is in consequence of this property that it has been thought necessary to pass a very stringent law, known as the Petroleum Bill, with a view of preventing accidents from the incautious storing and handling of the oil.

The more volatile liquid products obtained by distilling the crude oil are still more highly charged with combustible vapour, which, when these liquids are again distilled, escapes condensation even by the most powerful freezing mixtures.

In pursuance of an idea which occurred independently to Professor Tait and myself, a careful examination has been made of the solar autographs, taken at Kew and Cranford, under the superintendence of Mr Warren De la Rue. This was done with the view of detecting, if possible, some reference to planetary configurations in the behaviour of sun-spots, and in this undertaking, much aid was derived from a remark once made by Mr Beckley of Kew, when taking pictures of the sun, to the effect that, for a considerable period of time, he did not observe any spots in the act of breaking out on the visible disc of our luminary. A few words may not be amiss regarding the nature of the scrutiny to which the solar pictures were subjected, and also the value of this as a test of planetary action. In the first place, let us bear in mind, that by the rotation of our luminary, the different portions of his surface are successively presented to each planet in turn. Now, if the bodies of our system have any appreciable influence of this kind upon the sun, it is natural to expect that this should differ for any given portion of his surface, according as this portion is presented to the influencing body, or withdrawn by rotation, so that the sun's diameter is interposed between it and the planet. We should therefore expect to find that, for a given date, the spots should all begin to break out into visibility at or about the same ecliptical longitude. Similarly with regard to their healing up; and, generally, all spots on the sun's disc at a given date should behave in the same manner as they pass a given ecliptical longitude.

In the Transactions of the Royal Society for 1778, Mr Hunter has given an account of some experiments which he made on the freezing of the egg of the common fowl, from the results of which he inferred,—“That the fresh egg has the power of resisting heat, cold, and putrefaction, in a degree equal to many of the more imperfect animals, …. and that it is more than probable, this power arises from the same principle (a living principle) in both.”

Few departments of zoology have recently suffered more remarkable changes, both in classification and accepted views of structure, than the Polypi or Cœlenterata, and their immediate allies in the ascending scale, the Molluseoida,—greatly depending upon the more extended study of those animals of late years. We have been thus enabled to discover natural affinities which prima facie evidence would scarcely ever have indicated, as well as intrinsic differences which the same kind of evidence has hitherto been incapable of revealing to the mind.

Leading from the Protozoa to the Mollusca proper, the Cœlenterata and Molluscoida constitute an unbroken series of animals forming a considerable section of invertebrata, distinguished from the Protozoa by the development of true ova, and from the Mollusca by the property of gemmation developing compound examples of the principal types. Furthermore, the motion of the blood, or its equivalent, is effected either by ciliary action or by a propulsive organ; but when the latter occurs it is unfurnished with valves, so that the course of the circulation may be reversible in the same canals.

The whole southern part of Arran forms a field by itself, and whatever may be the deeper connections of the agencies that have fashioned the north and the south of the island, the result is a trappean area to the south, as distinct as if it lay in another hemisphere from the north, with its granitic nucleus, and encompassing rings of stratified rock.

This district is little visited, and is almost, if not quite, undescribed; it presents, however, much beautiful scenery, and for the geologist, problems of extreme difficulty and interest, which deserve more attention than they have got.

It may be divided into two belts of tilled land and moorland, above which are the hill tops. This division corresponds roughly to three regions, the lowest chiefly of sandstone, the middle where felstone prevails over the sandstone, and the highest of greenstone.

In the paper which I had the honour to read before the Society last winter, I stated the reasons, drawn from history and from comparative philology, which exist for concluding that the Brahmanical Indians belong to the same race as the Greek, the Latin, the Teutonic, and other nations of Europe. If this conclusion be well-founded, it is evident that at the time when the several branches of the great Indo-European family separated to commence their migrations in the direction of their future homes, they must have possessed in common a large stock of religious and mythological conceptions. This common mythology would, in the natural course of events, and from the action of various causes, undergo a gradual modification analogous to that undergone by the common language which had originally been spoken by all these tribes during the period of their union; and, in the one case as in the other, this modification would assume in the different races a varying character, corresponding to the diversity of the influences to which they were severally subjected.

It is certain that the unities which constitute aeriform media, when they have been fully separated from each other by an adequate temperature, and relieved from excessive pressure, are all equal to each other in volume, whatever the aeriform may be, either singly or in couples, or in pairs of couples, double pairs of couples, &c., giving such ratios as 1: ½: 4: 8, &c.

The Memoir now submitted to the Society, while it details the chief events in the life of a man who occupied a distinguished place in the literature of Scotland, at a period when it had attained a high reputation, cannot claim to be so complete as might be desired. His life was prolonged for several years after nearly all of his early friends had passed away; and since his death many papers have been destroyed or have fallen aside, which would now be of the greatest interest.

Whilst in this way much has been lost that might have given greater completeness to these pages, still, the recent publication of the Diary of his friend Dr Carlyle of Inveresk, has furnished many additional details, and afforded further evidence of the estimation in which he was held by his literary associates.

Several letters selected from the lives of his distinguished friends, and from the Manuscript Collection of the University, in addition to information derived from the short notices of his life already printed, have afforded the materials for preparing this sketch of one, whose career was more varied, while his public labours and literary connections were not less important and extensive, than those of any of his contemporaries.

In the year 1859, a book of remarkable power and originality was published by Mr John Taylor, of London, entitled “The Great Pyramid, why was it built?”

On first looking into it, I was unfortunately inclined to fear that its results were unlikely to be very sound; though merely because they seemed to bear, in a hitherto nearly barren, or difficult, and certainly most mysterious field, such a remarkably large crop of rich and promising-looking fruit. But considering afterwards, that that was not the proper frame of mind to be indulged in connection with, and certainly not in place of, strict scientific investigation into the merits of the case,—I read the book carefully, and then searched for the data required to test it, both in the original authors appealed to, and in some others.

In the following remarks I intend to confine myself to the consideration of those compounds which have not only the same composition per cent., and the same molecular weight, but also the same constitutional formula. Such compounds may be termed absolutely isomeric. As the constitutional formula of few substances is fully known, this class is of course a small one, or rather there are few substances of which we can certainly say that they belong to this class.

The general proposition in the theory of commensurables is to determine the conditions under which lines, surfaces, or solidities, connected with prescribed figures or forms, may have their ratios expressible by integer numbers.

The attention of geometers must have been drawn to this subject by the contemplation of incommensurable lines: the altitude of an equilateral trigon is incommensurable with the base; the diagonal of a square incommensurable with the side, and so on. And, when the two sides of a right angle are expressed by two numbers, the hypotenuse is, in the great multitude of cases, incommensurable with the sides: thus, if the sides be 5 and 7 inches respectively, the length of the subtense cannot be accurately expressed either in integers or fractions. However, when the sides are 3 and 4 units, the hypotenuse is exactly 5 of the same units.

The rapid advances made of late in the diagnosis of cardiac and other diseases affecting the organs of circulation, render the present inquiry into the normal or healthy condition of the valves of the vascular system, not more important anatomically, than medically. As the nature and composition of the parts in which valves are found in some instances materially influence their action, I have deemed it necessary to advert briefly to the properties and structure of the veins and arteries, when describing the venous and arterial or semilunar valves; and to the arrangement of the muscular fibres in the ventricles, when pointing out the peculiarities of the auriculo-ventricular ones. As, moreover, much information is to be obtained by comparing analogous structures, I have, in the present instance, not confined my observations to any particular form of valve, but have examined in succession the entire valvular arrangements of the fish, the reptile, the bird, and the mammal; my object being to arrive, if possible, at a correct knowledge of the more elaborate varieties as they exist in man, and in the higher mammalia.