Theorem, originally proposed by the celebrated John Bernouilli, and which I have exracted from Bossut's Calcul. Integral, tom. i. p. 484. The learned author, from whose invaluable work I have taken it, has not given the investigation; and I therefore solicit the favour of your inserting it, in hopes that some of your correspondents may be able to furnish a solution to it. I am, Sir, your humble Servant, Theorem. GEORGE HARVEY. A C "Let A 4 B be the semi-circumference of a circle, of which AB, the diameter, is equal to the two semi-axes, A, C, BC, of a given elipse. Divide the semicircumference, A 4 B, into a number, 2, 4, 8, 16, &c., of equal parts; draw straight lines from the point C to all the points of division: take the arithmetical mean of all the straight lines drawn to the points of division denoted by odd numbers, and also the arithmetical mean of the sum of all the lines drawn to the points of the even numbers, and B the half of A B; these two arithmetic means shall be the radii of two circles, the circumference of one of which shall be less than that of the elipse, and the circumference of the other greater. Required the demonstration ? V. Burning Gas in the Appenines. Sir Humphry Davy lately discovered, in the Appenines, a jet of gas burning with great brilliancy, and forming a column of flame six feet high. The gas was pure carbureted hydrogen. This is a fact of considerable interest in a geological point of view. The Appenines consist chiefly of limestone, and belong, unless I am misinformed, to the floetz formation. It would be of importance to know, whether any coal exists in the neighbourhood of this flaming jet of gas, or whether it proceeds from a great depth under the surface. This gas is formed in great abundance at the bottom of dirty stagnant water during the summer time. But that fact throws no apparent light on the formation of the gas iu the mountains of the Appenines. VI. Weather in Iceland during the Winter 1813—14. Some of my meteorological friends are anxious to know through the medium of the Annals of Philosophy, whether there was any thing peculiarly severe in the last winter in Iceland. If any of my readers happen to be acquainted with the matter of fact, it would be considered as an obligation if they would communicate what they know. VII. Chromate of Iron. Chromate of iron occurs in the Bare Hills near Baltimore, in North America, in great abundance. These hills appear to be serpentine, so that the situation of this mineral is the same in America as in France. The chromate of iron occurs in these hills in three states: 1. In compact masses of considerable size, mixed with talc, steatite, &c. In this state, which is I presume the same nearly with the French chromate, it has some resemblance to brown blende. 2. In grains from the size of a mustard seed to that of a pepper corn, imbedded in steatite. 3. Crystallized in regular octahedrons. The colour of these crystals is black, inclining to steel Surface smooth. Lustre vitreous. Fracture conchoidal. Opake, except some specimens which transmit a blood red light. Infusible before the blow-pipe, without addition. Magnetic in a considerable degree. Mr. Hayden, from whom this account is taken, believes that many specimens of supposed octahedral iron ore are really chromate of iron.* grey. VIII. Magnetic Pyrites. Stromeyer has lately subjected this mineral to a new analysis, and as he was at uncommon pains to attain the mineral in a state of purity, it is probable that the result which he obtained is a close approximation to the truth. Mr. Hatchett published the first analysis of this mineral, and made us acquainted with its true nature. It was analysed by Proust, and experiments on the artificial combinations of iron and sulphur were made by Vauquelin. The following is the result which they obtained :— If my number for the weight of an atom of iron be correct, and if magnetic pyrites be composed of one atom iron and two atoms sulphur, its constituents should be Iron. 100 + 55.99 sulphur. This circumstance throws some doubt upon the accuracy of Stromeyer's analysis. He informs us, that magnetic pyrites usually contains a mixture of metallic iron and of cubic pyrites. If this be the case, the chance of error in such an analysis was considerably increased. IX. Whitestone. Among some minerals which I lately received from Freyberg, there is a specimen of the true whitestone of Werner. This rock Bruce's American Journal, No. iv. p. 243. I find is an aggregate of three different minerals; namely, felspar, [JAN. quartz, and garnet. The felspar and quartz are both white, and constitute the greatest part of the rock. The garnet is scattered through the rock in small red grains and crystals. From this specimen it appears that our notions respecting this rock in this country have been hitherto incorrect. I have never seen any rock resembling it in Great Britain, except in cabinets; and in no one case was I able to learn whence the specimen had come. too that Klaphroth's analysis, given in a preceding number of the We see Annals, is of no real utility, as the constituents so found must depend upon the accidental proportions of the three minerals constituting whitestone. X. Height of the Peak of Teneriffe. Of all the measurements of this mountain, there seem to be only three upon which we can rely. Borda measured it trigonometrically in 1776; Lemanon barometrically in 1785; and Cordier barometrically in 1803. Their results are as follows:- Borda... 1905 French toises. 1902 1920 Mean 1909 This gives us the height of the peak above the level of the sea, 12,207 English feet. XI. Dip of the Magnetic Needle, and Intensity of the Magnetic Forces in the North Atlantic in 1799. 42.25 229 Good. Good. Humboldt's Personal Narrative, ii. 115. XII. Postman of the Province of Jaen de Bracamoros. In order to render the communication between the coasts of the Pacific Ocean, and the province of Jaen de Bracamoros, situated on the east of the Andes, more easy, the postman of Peru descends swimming for two days, first the river Guancabamba, or Chamaya, and afterwards that of Amazons, from Pomahuaca and Ingatambo to Tomependa. He wraps the few letters of which he is the bearer every month, sometimes in a handkerchief, sometimes in a kind of drawers called guayuco, which he winds as a turban round his head. This turban contains also the great knife with which every Indian is armed, rather to cut his way through the forests, than as a weapon of defence. The Chamaya river is not navigable, on account of a great number of small cascades. Its fall, according to Humboldt, from the ford of Pucara to its mouth in the river of the Amazons, below the village of Choras, a distance of only 18 leagues, is 1778 English feet. The postman in order to fatigue himself less in descending the river, supports himself on a log of hombax or ochroma, trees of very light wood. When a ledge of rocks intersects the bed of the river, he lands above the cascade, crosses the forest, and re-enters the water when he sees no further danger. He has no need of taking provisions with him, for he finds a welcome in a great number of huts, surrounded with plantain trees, and situated along the banks of the river, between las Huertas de Pucara, Cavico, Sonanga, and Tomependa. These rivers are not infested with crocodiles; the savage hordes therefore almost all travel like the Peruvian postman.-Humboldt's Researches, p. 254. XIII. Floetz Trap Rocks. It appears from the observations of Professor Jameson that there exist three floetz trap formations: 1. That in old red sandstone. 2. That in the independent coal. 3. The newest floetz trap, which is considered as newer than the other two. It constitutes the floetz trap formation described by Werner. The floetz trap mountains of Scotland belong chiefly to the two first formations. XIV. Granite in Scotland. Two granite mountains have been lately discovered in Roxburghshire, immediately on the borders of England. They appear to be situated in the midst of the transition rocks of that country. XV. Hoffman. Hoffman, well known as the author of the best German work on mineralogy, is just dead. Unfortunately he has left his work unfinished. XVI. Steffens. Dr. Steffens, one of the most eloquent teachers of natural history on the Continent, is a professor at Breslaw, the capital of Silesia. When Silesia was invaded by the French armies, he called together his class, pointed out to them in glowing colours the disgrace of quietly witnessing the pollution of their country by an enemy, and the honour to be obtained by joining in the glorious REMARKS. Eleventh Month.-4. Misty morning: cloudy: shower at night. 5. Overcast, a. m. wet afternoon and evening. 6. The drops of dew were frozen, but so clear as not to appear like hoar frost: a fine day, with large Cirri, and afterwards Cirrostratus and Cumulus: a streaked orange sky at sun-set, 7. Hoar frost, followed by cloudy sky: some rain after dark. 8. Clear morning: hoar frost: about four p. m. the sky became very dark, and there was a storm of rain, attended by a single loud explosion of electricity. It is remarkable that on the same day of the month last year, at the same hour, and in the same direction, (south-west,) we had a similar storm. 9. Cirrus, with the compound modifications: several shooting stars this evening: windy afterwards. 10. A steady breeze from the N.W.: Cirrostratus appeared over the Thames at sun-rise: ice a quarter of an inch thick, and permanent: a brilliant twilight; first lemon colour, surmounted by purple; then, by the blending of the two, a rich deep orange. 11. Hoar frost. I observe this often continues an hour longer on some tufts of saxifraga cœspitosa than any where else in my garden: the plant grows on close spongy masses of fibre, which are bad conductors of heat. 12. It was storm last night; and is said to have thundered and lightened about three a. m, Wet this morning; but the day fine: twilight orange, delicately varied with dusky horizontal striæ. 13. Cloudy and raining at eight a. m., but fine afterwards, with a smart breeze. 14. Cloudy. 15. Cirrostratus, a. m. : then Cirrus nimbiformis, depending and curling beneath: a turbid sky ensued, and wind in the night. 16. A wet stormy morning. 17. Much wind by night. 18. Small rain, a. m.: wet and stormy at night. 20. Hoar frost: a little rain at intervals. 21. Overcast. 22, 23. Hoar frost: misty. 24. The same: near sun-set a hollow wind from S. W., and rain in the night. 25. Wet day. 26. Some rain, a. m. 27. Clear morning, with evaporation: but before noon cloudiness was coming on from the W., and it rained steadily from sun-set till late at night. 28. Wet again in the night. 30. Large elevated Cirri, with Cirrostratus. Twelfth Month.-1. Brisk wind at N.: in the night a slight sprinkling of snow. 3. Wet morning. In the period last reported, the amounts of evaporation were put, by an oversight, in the column headed" Rain," and vice versa; which the reader is requested to correct. TOTTENHAM, Twelfth Month, 1814. L. HOWARD, |