In matters of historical research, there is a kind of circumstantial evidence which arises from the combination of known or authenticated facts, with critical argument on the import of doubtful passages of authors, which the reasoner endeavours to interpret, by bringing together, comparing, and making the one illustrate the other; so as to draw from the whole a degree of positive and certain information, which those authenticated facts are not of themselves sufficient to convey, and which those passages, taken separately, are incapable of furnishing. This complex species of evidence, it must be owned, is, with respect to its power of conviction, much inferior to that which arises from the ordinary proofs on which authentic history depends; for example, the testimony of actual witnesses to the facts related; or the positive information of authors, derived from clear and well authenticated records: But, at the same time, as in matters of history we have not always that best kind of evidence on which to found our belief, we are from necessity often compelled to resort to this inferior, circumstantial and analogical species of probation, in order to form to ourselves a rational creed on many matters of doubt, on which the mind is unwilling to rest in absolute uncertainty.

Much light has been thrown upon many of the most important and intricate functions of the animal machine, by accurate inquiries into the processes of nature under extraordinary circumstances. The doctrine of conception, lately advanced, owes its support, in no small degree, to observations made on cases, in which, from adventitious causes, Nature has been led out of her ordinary tract. The following case appears to be of such a description; and, as tending further to elucidate the subject, may probably be not unacceptable to the learned members of this Society.

In drawing up this abstract, I have somewhat enlarged the plan followed in those formerly communicated to the Society. In order to represent more accurately the progress of the seasons, every month is here divided into three parts, and the state of the barometer and thermometer is given for each of these divisions.

Kepler, having discovered the laws that regulate the motion of a planet in its orbit, proposed the following problem, for determining the true place of a planet at any given time: “To draw a straight line DE, from an eccentric point D in the diameter of a semicircle AEB, so that the whole semicircle may be to the sector ADE, in a given ratio.”

In resolving this problem, we are to take the quadrature of the circle for granted: and therefore, if C be the centre of the circle, and if the sector ACM be taken, such, that the whole semicircle is to the sector ACM in the required given ratio, the problem may be otherwise enunciated: “ To draw a straight “ line DE from an eccentric point D, to cut off a sector ADE, that shall be equal to the given sector ACM.”

Sir,

Some time in the month of July 1797, I had a carabine made at Drogheda, of different dimensions, and of a different construction from that of our Light Dragoons : the mode of carrying and of using it is likewise different; and, as I have had considerable experience of this carabine since that period, I take the liberty of recommending it to the notice of his Excellency the Lord Lieutenant, in order that a trial may be made, whether the adoption of such an arm might not be of advantage to the Light Cavalry.

In calculating the effect of the mutual action of two planets upon each other, it has been found necessary to develop the algebraic formula (a2 + b2 — 2ab cos φ)n into a series of this form, A + B cos φ + C cos 2φ + D cos 3φ + &c. Here a and b denote the distances of the planets from the sun; φ denotes the angle of commutation; and the values of n, more immediately the subject of consideration, are —, and —.

The determination of the coefficients A, B, C, &c. in these cases, appears to have been considered as a matter of difficulty by the mathematicians who first applied to the solution of the problem; for they found, that although it was only necessary to compute the first two coefficients A and B, the rest being easily derived from them, yet it did not appear that they could be expressed in finite terms, nor even by means of circular arches, or by logarithms. Recourse was therefore had to other methods, and chiefly to the method of infinite series; but as the series which most readily occurred to them, converged in some cases so slowly as to be in a manner useless, no small degree of analytical address has been found necessary, either to render it more convergent, or to find the sum of a competent number of its terms, with a moderate degree of labour.

It is now generally understood, that by the rectification of a curve line, is meant, not only the method of finding a straight line exactly equal to it, but also the method of expressing it by certain functions of the other lines, whether straight lines or circles, by which the nature of the curve is defined. It is evidently in the latter sense that we must understand the term rectification, when applied to the arches of conic sections, seeing that it has hitherto been found impossible, either to exhibit straight lines equal to them, or to express their relation to their co-ordinates, by algebraic equations, consisting of a finite number of terms.

The zeolite subjected to the following experiments, possesses some of the distinguishing properties common to other stones of the same class, but differs in certain respects from any variety with which I am acquainted. I found it, more than three years ago, in the basaltic rock on which the Castle of Edinburgh is built; and it was inclosed within a mass of prehnite.

Etymological researches have often been branded as trifling, occupying the mind in useless labour, and contributing little to the progress of science, taste, or beneficial knowledge. The justice of this treatment may well be questioned. That this species of study, like many others, has frequently been directed to unimportant objects, cannot be denied; but it by no means follows, that this is the fault of the subject, or that from such researches, however pursued, nothing valuable can be the result. If by etymological investigations, even in their most limited form, essential aids are obtained for facilitating our acquaintance with the writings of antiquity,—if such inquiries have often proved one of the most certain means for attaining accuracy of language and precision of ideas,—if by following them out on an extensive scale, assistance might probably be gained for elucidating some of the obscurities of early history, —and is from this source valuable materials may at all times be drawn for tracing the progress of the intellectual powers, and unfolding the laws of thought through the phenomena of its expressions,—if these, and perhaps other merits, must be allowed to etymological and grammatical disquisitions, they are surely sufficient to rescue from contempt this class of inquiries. If to all this we can add, what I believe will pretty generally be found to hold true, that those who affect most to despite this branch of study, have seldom any thing better to substitute in its place, a sufficient vindication will, I hope, be afforded, for employing a little literary leisure in the culture even of this apparently unpromising field.

In the outset of the Inquiry, it was observed, that the Greek language is a language of regular structure, forming its roots within itself. Following out this idea, I have, in different parts of the investigation, had recourse to the line of cognates and derivatives, as the surest aids for conducting us to the radical sense of a word, where the immediate root appeared to have fallen into difuse, or the signification had diverged considerably from the primitive idea. The grounds upon which I have thus proceeded, may require elucidation.

To the general law, that bodies are expanded by heat, and contracted by cold, water at the point of congelation, and for some degrees of temperature above it, seems to afford a very singular and curious exception.

The circumstances of this remarkable anomaly have been for some time believed to be the following:

When heat is applied to water ice cold, or at a temperature not far distant, it causes a diminution in the bulk of the fluid. The water contracts, and continues to contract, with the augmentation of temperature, till it reaches the 40th or 41st degree. Between this point and the 42d or 43d, it suffers scarcely any perceptible change; but when heated beyond the last-mentioned degree, it begins to expand, and increases in volume with every subsequent rise of temperature.