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.

The observations which have been made to determine the magnitude and figure of the earth, have not hitherto led to results completely satisfactory. They have indeed demonsrated the compression or oblateness of the terrestrial spheroid, but they have left an uncertainty as to the quantity of that compression, extending from about the one hundred and seventieth, to the three hundred and thirtieth part of the radius of the equator. Between these two quantities, the former of which is nearly double of the latter, most of the results are placed, but in such a manner that those best entitled to credit are much nearer to the least extreme than to the greatest.

A Comparison of some Observations on the Diurnal Variations of the Barometer, made in Peyrouse's Voyage round the world, with those made at Calcutta by Dr Balfour was read by Professor Playfair.

An Account of an Aurora Borealis, observed in day-light at Aberfoyle in Perthshire, on the 10th February 1799, by Patrick Graham, D. D. minister of Aberfoyle, was communicated by the Reverend Dr Finlayson.

“On the 10th of February 1799, about half-an-hour past 3 o'clock P. M., the sun being then a full hour above the horizon, and shining with an obscure lustre through a leaden-coloured atmosphere, I observed,” says Dr Graham, “the rare phenomenon of an aurora borealis by day-light. The weather, for several days before, had been intensely cold; and during the two preceding days, much snow had fallen. On this day a thaw had come on, and the temperature of the air was mild. The general aspect of the sky was serene. Some dark clouds hung on the horizon between S. W. and W. I was intensely observing a large halo about the fun, of about 20 degrees in semidiameter : It exhibited the prismatic colours, though obscurely, except in one quarter, where it coincided with the skirt of a dark cloud on the horizon, almost directly west. In that portion of the halo, the colours of the iris were very distinctly exhibited.

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.

Sir,

I Have examined my memorandums, concerning the observations I made on the condensed air in the air vault of the Devon Iron Works, near Alloa; and, according to your request, I now transmit you an account of them; and also of an experiment I made, when a partner and manager of these works, in order to increase the produce of blast furnaces.

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.

An account of two intersecting rainbows, seen at Dunglass in East Lothian in July last, was communicated by Professor Playfair.

“At Dunglass, where I happened to be in the beginning of July last, our attention was called one evening, a little before sunset, to a very large and beautiful rainbow, formed on a cloud which hung over the sea, and from which a shower was falling at a considerable distance to the S. E. The sun was about 2° high, so that the arch was not much less than a semicircle, with its highest point elevated about 40°. At the point where the northern extremity of this arch touched the horizon, another arch seemed also to spring from the sea, diverging from the former at an angle of 3° or 4°, on the side toward the sun.

The experiments described in this paper were suggested to me many years ago, when employed in studying the Geological System of the late Dr Hutton, by the following plausible objection, to which it seems liable.

Granite, porphyry, and basaltes, are supposed by Dr Hutton to have flowed in a state of perfect fusion into their present position; but their internal structure, being universally rough and stony, appears to contradict this hypothesis; for the result of the fusion of earthy substances, hitherto observed in our experiments, either is glass, or possesses, in some degree, the vitreous character.

Sir George Mackenzie, Bart. read a paper containing an account of experiments which he had made on the Combustion of the Diamond.

What follows is a short abstract of this paper; the paper itself having been published in Nicholson's Journal for July 1800.

On the 5th of August last, I announced to the Society that I had discovered soda in several varieties of the whinstone of Scotland, and also in lava from Mount Ætna, but did not describe the various experiments to which these substances had been subjected in my examination of them. In the following paper, therefore, I have the honour of laying an account of these experiments before the Society.

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.”

The Reverend Dr William Richardson, late F. T. C. D., having sent to Dr Hope a collection of specimens from the northern coast of Antrim, with a catalogue, and observations, the specimens were exhibited, and the observations were read in the Royal Society, March 1803.

1. I Divide cubic equations into two varieties or species: the one, comprehending all cubic equations with three real roots; the other, all those with only one real root.

2. Let φ denote any angle whatever, and let τ = tan φ, the radius being unity: let also : then from the doctrine of angular sections we have

which being reduced to the form of an equation, is

Z3-3τZ2-3Z+τ=0.

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.