By ὀνοματοποιια the Greek grammarians understood that principle, or tendency in the growth of language, according to which certain words are formed by an imitation of the sounds which they signify. Thus, ὀγχ, the root of the Greek word ὀγχᾶσθαι, to bray, may be considered to have been formed of a human mimicry of that animal to which human beings of the lowest cerebral capacity are peculiarly compared; and in the same way, laogh, the Gaelic for a calf, seems to contain a sound to which only the throats of Highland calves, Highland chieftains, and Highland crofters are competent. The word onomatopœia, like some other technical terms of the old grammarians, is not particularly happy, for it means only and generally word-making, or rather name-making, and says nothing of the principle by which the special class of words in question is made. Instead of this term, therefore, I should prefer to speak of the imitative or pictorial principle in the formation of human speech; and I should contrast the whole class of words in which the operation of this principle can be traced, with another class, derived from ideas or notions about the thing to be named in the mind of the word-maker. Thus, the modern Greeks call a cock πετεινό, that is, the fowl, or flying animal, from πέτομαι, to fly; and the Latin word, equus, a horse, if it comes, as Professor Müuller says, from the Sanscrit root â'su, swift, will be another word formed on the same principle. The roots of these words I propose to call notional roots, as contrasted with the onomatopoetic class of roots, which I propose to call pictorial roots, or roots formed by phonic imitation.

My attention was called to the subject of Cataract, in consequence of having, about forty years ago, experienced an incipient attack of that complaint, and studied its progress and cure.

While engaged in a game at chess with Sir James Hall, who was a very slow player, I amused myself in the intervals with looking at the streams of light which radiated from the flame of a candle in certain positions of the eyelids. In one of these observations I was surprised by a new phenomenon, of which I did not immediately see the cause. The flame of the candle was surrounded with lines of light, of an imperfectly triangular form, some parts of which were deeply tinged with the prismatic colours. Upon going home from the chess club, this optical figure was seen more distinctly round the moon, and of course it appeared, with more or less brightness, round every source of light.

The affection of Half-vision, or Half-blindness as it has been called, was first distinctly described by Dr Wollaston, in a paper “On Semidecussation of the Optic Nerves,” published in the Philosophical Transactions for 1824. “It is now more than twenty years,” he says, “since I was first affected with this peculiar state of vision, in consequence of violent exercise I had taken for two or three hours before. I suddenly found that I could see but half the face of a man whom I met, and it was the same with every object I looked at. In attempting to read the name Johnson over a door, I saw only son, the commencement of the name being wholly obliterated from my view. In this instance, the loss of sight was towards my left, and was the same, whether I looked with my right eye or my left. This blindness was not so complete as to amount to absolute blackness, but was a shaded darkness, without definite outline. The complaint lasted only about a quarter of an hour.” In 1822, Dr Wollaston had another attack of hemiopsy, with this difference, that the blindness was to the right of the centre of vision, and he has referred to three other cases among his friends; but in these, the affection was accompanied with headache and indigestion.

On a fluid of so much importance as the blood, observations with any pretension to accuracy can hardly be too often made and repeated, more especially when we consider its great instability, its little uniformity, and the differences of opinion entertained by physiologists respecting some of its most remarkable properties.

Such is the persuasion which has influenced me in engaging in the present inquiry, and in submitting its results to the Society.

The following series of theorems is given as an illustration of the modern method of trilinear co-ordinates, having been wrought out after perusal of Mr Ferrar's very lucid and elegant treatise on that subject. The demonstrations present no difficulty, requiring nothing more complicated than the formation of determinants of two and three places. Accordingly, after exhibiting the method of proof in a few instances, I have merely given the enunciations of the remaining propositions. As the series of theorems advances the manipulation becomes, of course, a little more complicated; but the co-ordinates and co-efficients always appear in such symmetry as very greatly abbreviates the task, and guarantees its accuracy. Two, or perhaps three, of these seventy-two theorems are known mathematical truths; but that so many new consecutive propositions should be so easily found, and so easily proved, is a convincing evidence of the simplicity, fertility, and power of this new and beautiful method.

A short paper of mine on Fagnani's theorem, and on Confocal Conic Sections, was inserted in the twenty-third volume of the Transactions of the Royal Society. Some of the conclusions of that paper can, however, be obtained more simply, as I will now proceed to show.

I will, in the first place, resume the problem—

“To find the intersection of a confocal ellipse and hyperbola.”

Since the curves have the same foci, and therefore the same centre, let the distance between the centre and focus be called unity, since it is the same for both curves. Let a, b, be the axes of the ellipse, A, B, those of the hyperbola. Then we have 1 = a2 − b2 = A2 + B2, which equation expresses the condition of confocality.

In the year 1861 I laid before the Royal Society of Edinburgh a theorem concerning the time of descent in a circular arc, by help of which that time can be computed with great ease and rapidity. A concise statement of it is printed in the fourth volume of the Society's Proceedings at page 419.

The theorem in question was arrived at by the comparison of two formulæ, the one being the common series and the other an expression given in the “Edinburgh Philosophical Magazine” for November 1828, by a writer under the signature J. W. L. Each of these series is reached by a long train of transformations, developments, and integrations, which require great familiarity with the most advanced branches of the infinitesimal calculus; yet the theorem which results from their comparison has an aspect of extreme simplicity, and seems as if surely it might be attained to by a much shorter and less rugged road. For that reason I did not, at the time, give an account of the manner in which it was arrived at, intending to seek out a better proof. On comparing it with what is known in the theory of elliptic functions, its resemblance to the beautiful theorem of Halle became obvious; but then the coefficients in Halle's formulæ are necessarily less than unit, whereas for this theorem they are required to be greater than unit.

39. In the first part of this paper, read to the Royal Society of Edinburgh in April 1862, and published in their Transactions, I explained the principles of a method devised by me in 1850 for ascertaining the absolute conducting power of substances capable of being formed into long bars; and I also stated the general results of experiments made in 1851 on the Conductivity for heat of wrought Iron.

40. I explained in Art. 14 of that paper, that the publication of the results had been for ten years withheld, partly in consequence of the state of my health which completely interrupted the experiments, but still more from the defective graduation of some of the thermometers used, which made it necessary to submit the instruments to a careful scrutiny, and to repeat with the duly corrected numbers the whole of the elaborate projections of the curves and calculations from them, on which the accuracy of the final results of course depends.

Though it is generally admitted that the cuticle performs an important part in retarding and regulating evaporation from the surface of the body, yet I am not aware of any inquiry hitherto made to determine the fact with exactness.

On account of the importance of the subject, I have been induced to engage in it. The experiments instituted for the purpose have all been of a very simple kind and easily made. They were on the similar parts of dead animals, detached immediately or very soon after the animals had been killed. From one specimen in each instance, the cuticle with the cutis, or the cuticle alone, was removed; whilst from the other these parts of the integuments were left entire. Each was carefully weighed, and then suspended, exposed to the air, side by side. Day after day, with occasional interruptions, or hour after hour, the weighing was repeated, and the result in the loss sustained was noted down. The experiments were made, when not otherwise mentioned, as just described, in a room in which, except in the height of summer, there was commonly a fire by day, its temperature during the day and night varying from about 50° of Fahr. to 55° and 58°. The animals affording the subjects of the trials were the trout, frog, toad, hare, rabbit, pig, thrush, common fowl, blue tit. Even at the risk of tediousness, I shall give the results of the weighing in some detail,—exactness in such trials being the first thing necessary.

During the summer vacation, Mr Henry Perigal of London proposed to me the following problem:—

“To determine the proportions of an epicycloid of which the loops touch each other.”

The solution of this problem contains some points of interest to the general analyst, and exhibits relations between certain trigonometric formulæ and algebraic equations. I, therefore, offer an outline of it to the attention of the Royal Society.

The problem which bears the name of the Italian geometer Malfatti, by whom it was first proposed and solved, has long attracted the attention and exercised the ingenuity of mathematicians, and has been made the subject of many careful and elaborate researches.

The great attention which has been bestowed upon this problem has arisen partly from its intrinsic difficulty, but chiefly on account of the extreme simplicity of the solution finally obtained by Malfatti, which seemed to open new views of geometrical research, and gave reason to hope that simple solutions might in like manner be found of many other geometrical problems usually accounted very difficult or insoluble.

It has always appeared to me that the difficulties which present themselves in investigations concerning the Frequency of Error, and the deduction of the most probable result from a large number of observations by the Method of Least Squares (which is an immediate consequence of the ordinary “Law of Error”), are difficulties of reasoning, or logic, rather than of analysis. Hence I conceive that the elaborate analytical investigations of Laplace, Poisson, and others, do not in anywise present the question in its intrinsic simplicity. They seem to me to be necessitated by the unnatural point of view from which their authors have contemplated the question. It is, undoubtedly, a difficult one; but this is a strong reason for abstaining from the use of unnecessarily elaborate analysis, which, however beautiful in itself, does harm when it masks the real nature of the difficulty it is employed to overcome. I believe that, so far at least as mathematics is concerned, the subject ought to be found extremely simple, if we only approach it in a natural manner.

One of the grandest steps which has ever been made in Dynamical Science is contained in two papers, “On a General Method in Dynamics” contributed to the Philosophical Transactions for 1834 and 1835 by Sir W. R. Hamilton. It is there shown that the complete solution of any kinetical problem, involving the action of a given conservative system of forces, and constraint depending upon the reaction of smooth guiding curves or surfaces, also given, is reducible to the determination of a single quantity called the Characteristic Function of the motion. This quantity is to be found from a partial differential equation of the first order, and second degree; and it has been shown that, from any complete integral of this equation, all the circumstances of the motion may be deduced by differentiation. So far as I can discover, this method has not been applied to inverse problems, of the nature of the Brachistochrone for instance, where the object aimed at is essentially the determination of the constraint requisite to produce a given result. It is easy to see, however, that a large class of such questions may be treated successfully by a process perfectly analogous to that of Hamilton; though the characteristic function in such cases is not the same function (of the quantities determining the motion) as that of the Method of Varying Action.

Coal deposits of Tertiary age have now been found more or less throughout the two great islands (north and south) of New Zealand. They are best known, and they have been chiefly worked, however, in the provinces of Otago, Nelson, Canterbury, and Auckland. Their apparent more meagre development in other provinces is probably simply due to the circumstance that the latter have not, as yet, been so thoroughly explored or so extensively colonised and peopled as the others. The explorations, however, of provincial and geological surveyors, of prospecting goldminers and sheep-owners, and of other pioneers of civilisation, are daily adding to the number of the known coal-fields of New Zealand; and it is probably not going too far to assert, in general terms, that the whole area of its two great islands is studded over with coal-basins of various extent, depth, age, and quality of coal.

Deviations from the usually described arrangements of the parts, of which the human body is composed, have from time to time attracted the attention of the anthropotomist. In many anatomical text-books, as well as in sundry memoirs specially devoted to the subject, numerous examples of such variations have now been recorded. To the scientific anatomist these have always had a certain value, but of late years this department of anatomical inquiry, more especially in connection with variations in the muscular system, has had additional importance and interest attached to it, on account of the attention which has been directed to the correspondence, or want of correspondence, in the muscular arrangements in man and the other mammalia, more particularly the apes.

A brief account of the weather of this period as regards temperature was read before the Royal Society last year. It was drawn up at the request of Professor Balfour, to accompany his paper “On the Remarkable State of Vegetation in the Edinburgh Botanic Garden in December 1863.”

From the 26th of October to the end of December the weather was in every way remarkable. Though frost occurred in the end of October and beginning of November it was not severe, and the temperature continued on the whole seasonable till the 12th of November. From this date till the end of the month it ranged unprecedently high, being 9° above the average temperature of the season. It then fell for the next ten days, but on no occasion below the average; and again rose considerably above the average during the week ending with the 18th of December. Under this genial weather vegetation in the open air advanced rapidly to a state of forwardness not usually seen till the month of March. In December 245 plants were in flower in the Gardens in the open air, and of these 35 were spring flowers. The frost which had occurred was insufficient to damage, to any material extent, 210 autumn-flowering plants; and the high temperature of November, which was as high as what ordinarily occurs in the beginning of May, brought the spring flowers prematurely into bloom, so that there was to be seen the rare spectacle of sweet peas and hepaticas flowering together.

The etymology of the names of places in a country is either a very important element in fixing the ethnology of its inhabitants, or it is a snare and a delusion, just according as the subject is treated. When such names are analysed according to fixed laws, based upon sound philological principles, and a comprehensive observation of facts, they afford results both important and trustworthy; but if treated empirically, and based upon resemblance of sounds alone, they become a mere field for wild conjectures and fanciful etymologies, leading to no certain results. The latter is the ordinary process to which they are subjected. The natural tendency of the human mind is to a mere phonetic etymology of names, both of persons and of places. It is this tendency which has given rise to what may be called punning etymologies, in which the King of Scotland plays so facetious a part, when the first Guthrie had that name fixed upon by the king, from his proposing when asked, how many fish should be prepared, to gut three; and when Rosemarkie received its name because the king, on asking what land he neared, was answered, Ross mark ye. This illustrates the natural tendency to suggest a mere phonetic etymology, in which the sounds of the name of the place appear to resemble the sounds in certain words of a certain language, the language from which the etymology is derived being selected upon no sound philological grounds, but from arbitrary considerations merely.

In examining the colours produced by thin laminæ of the crystalline lens of fishes, I observed a series of rectilineal serrated fringes perpendicular to the direction of the fibres, and produced by inclining the laminæ in a plane cutting these fibres at right angles. I was thus led to imitate these fringes or bands by combining grooves or striæ cut upon glass or steel surfaces, or grooves taken from these surfaces upon isinglass or gums.

In the preceding paper I have described the bands produced by gratings or grooved surfaces with 500 divisions in an inch, when the two grooved surfaces are in contact, and the grooves in the one slightly inclined to those in the other.

The following results were obtained with two gratings, one of wrhich had 2000 and the other 1000 divisions in an inch.

1. When the surfaces are in perfect contact, and the grooves parallel, very irregular bands are seen on the united surfaces, either with a lens or by ordinary vision, and are parallel to the grooves. They are seen only on the 2d, 4th, 6th, &c., spectra on each side of the luminous bar or disc.