Notwithstanding the rapid progress of zoology in other departments, the Heteropoda still remain imperfectly known, if one may form a judgment from the scantiness of definite information respecting them to be found in systematic works. Nearly all the available space is usually occupied with an exposition of the errors and doubts of the great men who gave us the first outlines of the order, while comparatively little is done to improve the subject, or make it intelligible to the student.

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.

My first precise views of the structure of the Tunicata were formed by the perusal of Mr Huxley's masterly papers dealing with the anatomy of Salpa, Pyrosoma, Doliolum, and Appendicularia, in the Phil. Trans. for 1851, Part II.; and I have since had abundant opportunity of verifying all the important facts, made known by that original observer, in the papers to which I have alluded.

Having thus acknowledged my guide in this field of research, I can scarcely claim much originality as relates to pure anatomy; but I hope that the method here adopted, in opening up this interesting subject, will be found in keeping with nature, as it is the result of much study and practical investigation.

Great credit is due to those observers who have been enabled to give to science both clear and comprehensive views of the anatomy of creatures which have only been presented to them in a spirit-preserved, opaque, brittle, and contracted state; for it is certain that their penetration in this respect could not be successfully brought into exercise without the aid of much knowledge, both bibliographical and practical. Yet there are many whole animals, and, in particular, parts of animals, which must be seen in the living state, to be at all comprehended by even the most brilliant mind; and this fact has induced me to make drawings and notes of many interesting matters connected with the pelagic mollusca, when the living animals fell casually under my own observation. In the present communication, however, I shall confine myself to the genus Clio.

1. It is a well-known property of simple harmonic functions, that the sum of any two or more of them having the same period, is itself a simple harmonic function having the same period as its components. The same thing must be true of their mean, since this is equal to the sum divided by a constant; and it will still be true when the number of components is indefinitely great, and the mean becomes an integral.

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.

Nearly a hundred years ago, Dr Heberden of London made the following experiment:—Having prepared three exactly similar rain-gauges, he placed one of them on the roof of Westminster Abbey, another on the roof of a neighbouring house, but at a much lower level, and the third in the garden of the same house. At the end of twelve months he found that the gauge on the roof of the Abbey had received 12·099 inches of rain, the gauge on the neighbouring house-top 18·139 inches, and the gauge on the ground 22·608 inches.

This paradoxical result required, of course, to be confirmed by other observers, and in other localities; and the similar results obtained by Dobson, Dalton, Howard, and especially by Arago at the Paris Observatory, and by Phillips at York, have amply supplied all that was wanting in this respect.

In a successful haul of the towing net, off the Island of Sardinia, I obtained a very beautiful Firola, which gave me a good opportunity of testing the truth of my former conclusions with regard to the economy of the Heteropoda in general, and of Firola in particular; and I have great pleasure in submitting the facts arrived at, with the accompanying enlarged figure of the visceral nucleus and the neighbouring parts, to the Royal Society of Edinburgh, as an appendix to my paper “On the Anatomy and Classification of the Heteropoda,” already brought under its consideration.

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.

In the years 1823 and 1826 I communicated to this Society two papers “On the Existence of Two New Fluids in the Cavities of Precious Stones and other Minerals.” These two fluids were generally found together in the same cavity, though sometimes the cavities were occupied only by one of them. They were perfectly transparent and immiscible. The denser of the two occupied the angles of the cavities, or the necks, or narrow passages, or canals which united two or more larger cavities; while the rarer fluid floated, as it were, on the other in deep cavities, or filled the body of shallower ones, with the exception of a circular vacuity, which diminished and disappeared with the slightest increase of temperature, or enlarged itself and disappeared in consequence of the fluid being converted into vapour.

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.

The disintegration of solid bodies by means of active or feeble solvents, or by those invisible processes which go on during long periods of time, has, so far as I know, been studied neither by the chemist nor the natural philosopher. In 1837 I submitted to this Society a paper “On the Optical Figures produced by the Disintegrated Surfaces of Crystals,” containing experiments which I believe have not been repeated, and results which no person has attempted to explain. Since that paper was published, my attention was called to the structure and properties of decomposed glass, in consequence of having had occasion to study the action of its coloured films, in absorbing definite parts of the spectrum.

The laws of the polarisation of light when reflected from the surfaces of solids and fluids, and when refracted and transmitted by translucent and transparent bodies, have been successfully investigated; but no experiments, I believe, have been made on the polarisation of light by rough or unpolished surfaces, such as ground glass, painted surfaces, pounded glass, snow, white powders, and solids and fluids reflecting white light from their interior. When studying the polarisation of the atmosphere, and anxious to discover the cause of its partial polarisation, and of the three neutral points or spots, in which there is no polarisation, I investigated the action of rough surfaces upon light, under the conviction that the sky or atmosphere was a rough surface like any aggregation of white or coloured particles. Had the atmosphere been specular like water or any body with a polished surface, the image of the sun would have been seen in it by reflexion, but being composed of aerial and aqueous molecules, it must reflect the sun's rays like pounded glass, or any white or coloured powders.

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.

The object of this paper will be, to give a connected view of some theorems of importance, which are often found in books rather obscurely demonstrated, and in some cases are inaccurately given, or are liable to exceptions which are not mentioned.

Before entering on the main subject of this paper, the composition of rainwater, I shall give a brief account of the rain that has fallen in the Lake-District during the year just past; and this I shall do chiefly by means of tables. The quantity of rain registered has been so greatly in excess of any former year, that of itself it is deserving of record, and the more so, comparing the weather which has prevailed here with the weather in the south, for the most part, happily for the harvest, as remarkable for an opposite state, an excess of dryness.

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.

During the last half century, observations on the polarisation of the atmosphere were made by several eminent observers—by Arago, Delezenne, Babinet, and Zantedeschi; but no result of special importance was obtained till Arago made the great discovery that there existed in the atmosphere a point, or spot, in which there is no polarisation. At sunrise or sunset he found that this neutral point was 20° or 30° above the point opposite to the sun, or what we may call the Antisolar point. The name of Arago's neutral point has been given to this spot without polarisation. It is best seen after sunset. At St Andrews it is above the horizon all the day, between the middle of November and the end of January.

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.

While engaged last autumn in examining with a hand-lens the contents of a phial into which I had transferred some of the refuse of the dredging-boats employed in the oyster fishery on the coast of South Devon, my attention was attracted by a minute organism which adhered to a fragment of one of the larger Sertularidans. Under this low power it resembled somewhat a Campanularia, with the polype expanded; but, on being removed with a portion of the substance to which it was attached, and placed in a glass trough under the compound microscope, I found that it had closed up, and now resembled in form a cup surmounted by a pyramidal lid, and supported on the summit of a long jointed stem (Plate XIII., fig. 3).