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The expressions for elliptic, or hyperbolic areas, and for the logarithm of any number, are exactly of the same form. The terms of this expression are found from the cosines of a series of arcs or sectors, which constitute a geometrical progression. In like manner, the terms of the expres sion for an elliptic or hyperbolic sector, are formed from a series of abscissæ corresponding to elliptic and hy perbolic sectors, each of which is onehalf of that before it. These abscissæ are found by precisely the same formula in the two curves; and in the beginning of the paper, general theorems are investigated, which express the relation between the co-ordinates corresponding to an elliptic and hyperbolic sector, and those which correspond to any multiple of that sector. The theorems are deduced from a single property common to both curves, with out employing any geometrical constructions, and without introducing impossible quantities.

Two other papers, one, on a Petrifaction found near Edinburgh, and another, on the Effects of injecting a Solution of Opium into a Vein, in an anomalous Nervous Affection, were then read, after which the Society adjourned till November.

On the 6th of November, the Society resumed its sittings; and on the 27th, Sir Walter Scott, Bart., was elected President, in the room of Sir

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James Hall, Bart., who had resigned. At this meeting, the following resolution, moved by Dr Hope, and seconded by Sir George Mackenzie, Bart., was unanimously adopted, and ordered to be transmitted to Sir James Hall. "The Royal Society having, in compliance with the wish of Sir James Hall, Bart., refrained from again placing him at their head, beg to avail themselves of this opportunity to offer him their best thanks, both for his long and zealous services as their President, and for the numerous valuable communications with which he has enriched their Transactions, and contributed materially to maintain the repu tation of the Society."

At the meeting of the 4th of December, a notice by Dr Brewster was read, on the Distribution of Heat in the Arctic Regions. The object of this notice, which may be considered a continuation of the Doctor's former paper, on the Mean Temperature of the Earth, was to point out the remark. able agreement between the formula which he had formerly given, and the observations of Captain Parry. He shewed that the Pole of the earth was not the coldest point, but that there were two Poles of maximum cold, situated at a distance from the Pole, and in the meridians passing through North America and Siberia. The agreement of this view of the distribution of heat, with the series of observations reduced by Humboldt, was also pointed out;

and, by giving a position to the two Poles, and assuming their temperature, the author deduced a general formula for all meridians, and shewed, that the differences between the calculated and observed results were far within the limits of the errors of observation. The form of the isothermal lines resembles, generally, that of the isochromatic curves which surround the resultant axes of crystals, with two axes of double refraction. In conclusion, the author pointed out the analogy between the magnetic and isothermal curves in the Polar regions; and, noticing their similarity of position, and conjecturing that these isothermal lines might have a motion of revolution round the Pole of the earth, he shewed how, upon such a supposition, the low temperature of ancient Europe might be explained, and how we might account for the remains of plants, and land and sea animals, being found in climates where they could not now exist.

On the 18th, three papers were read; but we shall only notice Major Rennell's Remarks on the Currents between the Parallels of Cape Finisterre and the Canary Islands. The general result is, that navigators who depart from the parallel of the southern part of the Bay of Biscay, (or say 45°,) and sail in the usual track southward, will be assailed by a S.E. current, and then by an easterly one, till they have passed the parallel of Cape Finisterre, when the current will again turn to the S. of E. and gradually become a S.E. current, till having passed Cape St Vincent, it becomes easterly again. This current is very general across the mouth of the Bay, between Cape St Vincent and Cape Cantin. Beyond this Bay, the current again becomes S. E., and continues as far as the parallel of 25°, and is felt beyond Madeira westward, that is at least 130 leagues from the coast of Africa. The rate of this current varies considerably; that is,

from 12 to 20, or more miles, in 24 hours. Major Rennell thinks 16 miles rather below the mean rate. Hence, a ship sailing in the usual track to Madeira or the Canaries, will be carried to the S. E. at the rate of 16 miles per day, or from 150 to 160 miles in the course of her voyage; and on a S.E. by S. course, from 80 to 90 miles to the E. of her intended port. If we suppose a S.E. course, the error in easting will be no less than 109 miles, a distance which would carry a ship bound for Teneriffe to Allegranza, or Fortaventura, and one making for Allegranza to the Barbary Coast. Vessels bound to the Canaries, or intending to sail between those islands and the mainland of Africa, if without chronometers, as that class of merchant ships generally are, should, therefore, to every day's reckoning, add ten miles for easting. "It is this current," adds Major R." that has furnished the roving Arabs of the desart with their victims from every nation, and the good Mr Willshire with objects of benevo lence."

Of the papers read at the different meetings in January, the only one we shall notice here, is Mr Scoresby's Description of a Magnetimeter for Measuring the Dip of the Needle. This instrument, consisting of a small table of brass, 4 inches square, and 33 inches in height, having a plate of brass attached to it by hinges, and movable by means of a wheel and pinion, through an arch of 250° of a vertical circle. This plate has a small straight groove running from end to end, for the purpose of receiving bars of metal, the polarity of which is to be determined. These bars are readily fixed to the plate, by being slipped through a circular aperture in the end of a spring, which, perforating the movable plate, and acting downward, firmly embraces any substance laid along the groove. The angular position of the movable

plate is marked by a graduated circle screwed upon the side of the table. On the brass table is placed a movable flat plate of brass, divided into rhumbs and degrees, and furnished with a magnetic needle, with an agate cap traversing on a brass or steel point. The needle can be changed according to the nature of the circumstances, a very light, and strongly magnetized one being used in delicate experiments. The compass, or plate, carrying the needle, being movable, its distance from the bar resting on the movable plate, can be varied at pleasure. The centre of the hinges is one tenth of an inch above the level of the table; the magnetized needle stands at the same elevation; and the bars in use being one-fourth of an inch in diameter, are sunk in the groove of the movable plate to such depth, that their axis, or centre, precisely corresponds with the centre of the hinges; hence the middle of the extremity of each bar is at the same ele. vation, and at the same distance from the needle, in every position of the movable limb. To give firmness to the instrument, in making experiments, the table is fixed by the feet to a mass of lead, of seven or eight pounds weight. By means of this plate of lead, which has a screw at each corner, the whole apparatus is readily put into a horizontal position. With this apparatus, Mr Scoresby made a series of experiments, which are fully detailed in the Transactions of the Royal Society of Edinburgh, vol. IX. p. 243, to which we beg leave to refer.

ings, and a description of a 25 feet reflecting telescope, constructed by Mr John Ramage of Aberdeen. Except ing those of the celebrated Sir W. Herschell, this, we believe, is the largest telescope ever constructed, and does honour to Scotland, as well as its ingenious author. The speculum is 25 feet in focal length, and 15 inches in diameter. The method of observing is by the front view; the power is from 50 to 1500; and the mechanism by which the observer and the instrument are moved, is so simple and well con trived, that it can be managed and di. rected to any point of the heavens as readily as a three feet achromatic telescope.

On the 2d of April, there was read to the Society a letter from Sir David Milne, Bart., to Professor Russel, giving an account of the method of fishing for large corals in the Island of Bermuda. They are found round the island in large abundance, in about three feet water at low tide; and are principally attached to the edge of the coral reefs, where the water deepens suddenly to 10 or 18 feet. They are fixed to the reef by a kind of stalk, and seem of all sizes, from two inches to several feet in diameter, growing in clusters like mushrooms. When first taken out of the water, they are of a light brown or snuff colour; but after a few weeks exposure to the sun, they become perfectly white.

A great variety of papers, not calling for particular notice, were read, betwixt this and the 18th of June, At the meeting of the 5th of March, when the Society adjourned its meetthere were laid before the Society drawings till November.

ROYAL INSTITUTE OF FRANCE.

THE Royal Institute of France, as our readers may perhaps know, consists of four branches, called Academies; the Academy of Sciences, the Academy of Inscriptions and Belles Lettres, the Academy of Fine Arts, and the French Academy, each of which strictly confines itself to its own appropriate department of knowledge or of art. Of these different branches, however, by far the most distinguished, is the Academy of Sciences, which reckons among its members, the most eminent scientific characters of the age; and to the proceedings of which alone, the present abstract will, therefore, be entirely restricted. This determination has been rendered imperative on our part, both by the narrow limits to which we are confined, and the comparatively unimportant and uninteresting character of the proceedings of the other three academies, which, from the decided preference given to science, both during the revolution, and subsequently under the imperial government, have lost much of that reputation, (particularly the

tions." The property of which M. Biot speaks, is the power which glass acquires, while in a state of vibration to depolarize the light which traverses it. Having prepared a larger faisceau of polarized light, he received it on a plate of black glass, so placed as to occasion no reflection, and the actual state of which he had examined before interposing it to the path of the light. Some traces were now discovered, of colours corresponding to the tints of the primary rings of Newton's table, and bearing an evident analogy to those presented by longitudinal pieces of glass, which have been powerfully heated, and then rapidly cooled. There was this peculiarity, however, that these traces were most sensible towards the middle of the piece of glass, whether viewed in front, or laterally, and that they rapidly diminished towards the two sides of this middle part, so as to disappear entirely at the extremities. But when the plate of glass was interposed in the path of the ray, so that the ray might pass directly through its thickness, which was

ademy of Inscriptions and Belles about seven millimétres, (.27559 inches,) Lettres,) which they possessed under the ancient regime.

The Academy of Sciences commenced its sittings for the year 1820, on the 3d of January, when M. Boué read a Memoir, entitled, "A Geological Essay on Scotland," which he has since published, in an enlarged form, thereby superseding the necessity of any particular notice in this place. On the 17th, M. Biot read a paper, entitled, "On a New Physical Property which Plates of Glass acquire, when they perform Longitudinal Vibra

a slight change was with difficulty perceived, in the languid reflection which took place on the blackened glass, prepared to absorb the polarized ray; but if one-half of the plate of glass, taken by the middle, was rubbed with a moistened cloth, so as to excite longitudinal vibrations, while the other half was placed in the path of the faisceau of polarized light, at each time a sound was produced, a brilliant flash of white light shone on the surface of the absorbent glass, which proves that a change had taken place in the direc

tion of the polarization; and the more the sound (its tone remaining the same) became full and intense, the brighter was the light thus perceived, while the moment it ceased, the absorbent glass resumed its original state, and the polarization its primitive direction. If, instead of transmitting the polarized faisceau across the thickness of the plate, which was seven millimétres, it was directed across its breadth, which was 30, (1.18110 inches,) immediately delicate lines of colour, analogous to the first order of rings, appeared in the direction of the length of the plate, modifying the primitive coloured striæ, and presenting, not only the bluish white of the first order, but descending even to the orange. M. Biot concluded the detail of these curious experiments, by some remarks on the ratio of the distribution of the light elicited by these vibratory motions in the particles of the plates of glass, and by a comparison of the results with some previous experiments by MM. Savart and Chladny.

At the same sitting, M. Dumeril, in the name of a committee, made a report of a case in practical medicine, which had been communicated to the Academy by M. Chomel. A young woman, after having experienced several attacks of hysteria, was seized with a periodical cough, which commenced every day about two o'clock, and continued with extraordinary violence for seven or eight hours. This periodical access of coughing had been transformed into real attacks of hysteria, and this conversion, provoked, or produced, by the administration of extract of belladona, (deadly nightshade,) was, in this particular case, the more fortunate, as the attacks of hysteria yielded easily to the administration of quinquina (Peruvian bark).

The proceedings of the Academy, in February, do not call for any par

ticular notice. At the sitting of the 6th of March, M. Cauchy made a report on a Note of M. Lepely, relative to the summation of descending progressions. This Note contains the curious proposition, which, however, may be demonstrated by a simple comparison of triangles, that if, in a trapezium having three sides equal to one another, and smaller than the fourth, the two sides not parallel be produced till they meet, the lengths comprised under these two sides, between the point where they meet, and the base of the trapezium, will be numerically equal to the sum of the descending geometrical progression, which shall have for its ratio, the relation between the two parallel sides; and for its term, the smallest of these sides.

On the 13th, M. de Humboldt read a most ingenious memoir, "On the Nocturnal Increase in the Intensity of Sound." This phenomenon has been remarked from the remotest antiquity. Aristotle speaks of it in his Problems, (Sect. XI. quæst. 5 and 23,) and Plutarch in his Dialogues, (Sympos. Lib. VIII. cap. 3. vol. II. p. 720, Franck. Edit. 1620,) and numerous attempts have at different times been made, to give a satisfactory solution of a fact, which every one must have observed. Prior to that proposed in the memoir of this celebrated traveller, we are not aware of any that deserve the name of scientific, or serve, even in an imperfect degree, to explain and connect the numerous facts which have been observed and recorded. In this paper, the ingenious author considers only the increase of intensity in a tranquil state of the atmosphere, there being no difficulty as to that produced by the wind during the night, and which is modified by the relation subsisting between the direction of the wind and that of the sonorous ray. Under the same zone, for example, between the tropics, the nocturnal increase of the

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