Two figures are reciprocal when the properties of the first relative to the second are the same as those of the second relative to the first. Several kinds of reciprocity are known to mathematicians, and the theories of Inverse Figures and of Polar Reciprocals have been developed at great length, and have led to remarkable results. I propose to investigate a different kind of geometrical reciprocity, which is also capable of considerable development, and can be applied to the solution of mechanical problems.

Of all the branches of interesting and curious learning, there is none which has been so systematically neglected in this country as mythology—a subject closely connected both with theology and philosophy, and on which those grand intellectual pioneers and architects, the Germans, have expended such a vast amount of profitable and unprofitable labour. The consequence of this neglect has been, that of the few British books we have on the subject, the most noticeable are not free from the dear seduction of favourite ideas, which possess the minds of the writers as by a juggling witchcraft, and prevent them from looking on a rich and various subject with that many-sided sympathy and catholic receptiveness which it requires. In fact, some of our most recent writers on this subject have not advanced a single step, in respect of scientific method, beyond Jacob Bryant, unquestionably the most learned and original speculator on mythology of the last century; but whose great work, nevertheless, can only be compared to a grand chase in the dark, with a few bright flashes of discovery, and happy gleams of suggestion by the way. For these reasons, and to make a necessary protest against some ingenious aberrations of Max Müller, Gladstone, Inman, and Cox in the method of mythological interpretation, I have undertaken to read the present paper; which, if it possess only the negative virtue of warning people to be sober-minded and cautious when entering on a path of so slippery inquiry, cannot be deemed impertinent at the present moment.

The discovery of those numbers which shall, either truly or approximately, represent the ratio of two magnitudes, necessarily attracted the attention of the earliest cultivators of exact science. The definition of the equality of ratios given in Euclid's compilation clearly exposes the nature of the process used in his time. This process consisted in repeating each of the two magnitudes until some multiple of the one agreed perfectly or nearly with a multiple of the other; the numbers of the repetitions, taken in inverse order, represented the ratio. Thus, if the proposed magnitudes were two straight lines, Euclid would have opened two pairs of compasses, one to each distance, and, beginning at some point in an indefinite straight line, he would step the two distances along, bringing up that which lagged behind, until he obtained an exact or a close coincidence.

I was originally attracted to the study of Quaternions by Sir W. R Hamilton's ingeniously devised and most valuable operator

to which he called special attention (Lectures on Quaternions, § 620) on account of its promise of usefulness in physical applications. But I soon found that in order that its full power may be applied, in general investigations, it is necessary that we should have processes of definite integration, of the kinds required in physics, applicable to quaternion symbols and not merely to scalar variables. I often consulted Hamilton about this want, and he promised to endeavour to supply it at some future time. I fancy that shortly before his death he must have in some way supplied it, though he certainly did not print, nor does he appear even to have written, anything on the subject. In one of the last letters I received from him, he said that he intended to conclude the final chapter of his Elements, which is devoted to physical applications, by some sections on the use of the operator mentioned above. That chapter remains unfinished, and as Hamilton rarely wrote down the steps of even a complex train of mathematical reasoning until he had mentally completed it, it is to be feared that this portion of his investigations is entirely lost. So far as the analytical aspect of Quaternions is concerned, this loss is very serious indeed, for there can be little doubt that Hamilton's solution would have been of immense value from the purely mathematical point of view.

In a paper communicated to the Royal Irish Academy in 1841, I gave an account of a large number of experiments on the heat disengaged when acids and bases, taken in the state of dilute solution, enter into combination, and when bases, insoluble in water, are dissolved in dilute acids. The following general conclusions or laws were deduced from those experiments:—

Law 1.—The heat developed in the union of acids and bases is determined by the base and not by the acid, the same base producing, when combined with an equivalent of different acids, nearly the same quantity of heat; but different bases, different quantities.

Law 2.—When a neutral is converted into an acid salt, by combining with one or more atoms of acid, no change of temperature occurs.

Law 3.—When a neutral is converted into a basic salt, by combining with an additional proportion of base, the combination is accompanied with the evolution of heat.

From all the facts which the study of the Hydroida has made apparent, we may regard it as certain that however long zooidal multiplication may continue, this is not sufficient for the perpetuation of the species, but that a period must at last come in the life of the hydroid when by an act of true sexual reproduction, new individuals are produced for the indefinite extension of the species through time.

The innervation of the vascular system is a subject which has engrossed the attention of physiologists ever since the days of Galen. Yet, notwithstanding the number of distinguished observers who have contributed to our knowledge of this difficult topic, there are still many points of the greatest importance which are enveloped in the deepest obscurity, and not a few regarding which opinions are much at variance.

No subject of modern scientific inquiry is more important than the series of deposits in which geology comes in contact with the period of human history. This must be my apology for some of the seemingly trivial details contained in the following paper. When these observations were begun, nothing could be further from my thoughts than any reference to the antiquity of man. But I shall perhaps best introduce the subject by simply narrating the way in which I was led forward step by step, till the whole inquiry assumed the form in which it is here presented.

It is familiarly known as one of the commonest experiments in optics that when a beam of polarised light is passed through a thin film of mica or selenite, and subsequently analysed either by reflection or by double refraction, two colours are seen complementary to one another, and alternating with one another at each 90° of a revolution of the analysing plate or prism.

The combined researches of Anderson and Williams on the basic compounds contained in coal tar have led to the discovery of two well-defined series of organic bases, called respectively the Pyridine and Chinoline series, the members of both of which possess the properties of nitrile bases. The isomerism between the pyridine and the aniline series of bases excited considerable interest at the time of its discovery. The subsequent researches of Williams on the products of the distillation of chinchonine led to the discovery of bases having the same composition as the members of the pyridine and chinoline series of coal tar. When first discovered they were supposed to be identical. Since that time, however, a careful examination and comparison of the tar series of bases with the chinchonine series has led Mr Williams to the interesting discovery that the two lutidines, as also the two chinolines, are in reality not identical, but isomeric. This introduces a greater complexity into the study of the constitution of these compounds.

On the 3d November 1869, a huge Finner whale was stranded on the beach at Gosford Bay, Longniddry, Firth of Forth.

Most of the large Fin whales which have been examined by British and Continental anatomists have been found floating dead on the surface of the sea, and have then been towed ashore by their captors. But, from the account which was given in the Edinburgh daily newspapers, it would appear that, for some days previously, this animal had been recognised by the fishermen, swimming to and fro in the Firth. On the morning of the 3d it was seen from the shore, blowing with great violence from its nostrils, flapping its huge tail, and obviously struggling to disengage itself from the rocks and shoals, amidst which an unusually high tide had permitted it to wander. Shots were fired at it, and, from the wounds produced, blood poured forth which tinged the surrounding waves. As the tide receded, the animal was fairly stranded; and, after some vigorous but ineffectual attempts to disengage itself from its position, it slowly died. The animal lay some yards above low-water mark, so that for several hours each day it could be examined, and photographs taken from various points of view.