ments of science and the arts, you will not likely object to the insertion of the following simple illustrations of several important propositions. I. For sin a. By trigonometrical formulæ, it is shown that sin } a = √✓ (s — b) (s — c) and cosa√ ъс II. For the area of a triangle when the three sides are given. Let a, b, and c, (fig. 5,) be the three sides, and s half their sum, and C D = h. Let A C B, (fig. 6,) be an arc, and let A C 2 a and CB = 2 b; then angle n is measured by b, and m by a. And since the chord of an arc is equal to twice the sine of its half, A C 2 sin a, BC = 2 sin b, and A B = 2 sin (a + b.) Now, radius cos n = A C: A D, or 1: cos b = 2 sin a: A D, therefore, A D = 2 sin a. cos b; and in the same way B D = 2 sin b. cos a; hence by adding A D + B D or A B or 2 sin (a + b) = 2 sin a. cos b+ 2 cos a. sin b; and sin (a + b) = sin a. cos b + cos a. sin b. IV. For the value of the sine and cosine of an arc in terms of an arc. Let sin x A x + B x3 +, &c. neglecting even powers as their co-efficients would be found to be = o. Then = A + B x2 + Cx4, &c.; and as a diminishes sin x x approaches to 1 as its limit, hence A = 1. sin x When x becomes 2 x, observing that A = 1, sin 2 x 2 x + 8 B x3 + 16 C x5 +, &c. But sin 2 x 2 sin x, cos x, and, therefore, cos x = 3.B x23 (5 CB2) x4 +, &c. 166 x6 THE GLASGOW x2 But 1 sin 2x = cos 2x = 1 - x2 - 2 B x — (2 AC + B2) &c. &c. Comparing the corresponding co-efficients of these two values of cos x, and observing that A = 1. To the Editors of the GLASGOW MECHANICS' MAGAZINE. Gentlemen,—As I have not an opportunity of reading your Magazine weekly, it was only two days ago that I first read W. C.'s judicious criticism on my observations on the Crank. It is always pleasing to arrive at truth, and he has lent his aid to its thorough investigation. He has certainly shown that the loss of power is not so great as one-third; but it will be found, from the very nature of the crank, that there is some loss of power occasioned by its use. He has not, however, shown the fallacy in the reasoning. This fallacy lyes in the following argument :-When the power is applied at P, (see fig. 7, No. LXII. Vol. III.) only a part of it proportional to P K is effec tive. This argument is perfectly consistent with mechanical principles in the case of a number of weights sliding down the circumference of a fixed wheel, as, by the principles of the inclined plane, a portion of their weight would be supported by the plane, and another portion, proportional to P K, exerted; but circumstances are entirely changed when the wheel is moveable round its centre. Any weight placed at P, although it could not balance an equal weight at L, would support it in equilibrium at a point in the circumference, diametrically opposite to L, and hence all the weights on one half of the circumference being made to descend through the diameter of the crank, would raise a series of equal weights on the opposite circumference through an equal height; and, therefore, if the diameter of the crank were equal to the length of the stroke of the piston, no power would be lost. But unless the length of the beam be infinite, this diameter will not be equal to the length of the stroke. If, however, it be very long in comparison to it, the difference would be trifling; but then much power would be expended in dragging the beam alternately from rest to a state of motion. Taking the length of the beam, 12 feet, and that of the stroke, 3 feet, the extremity of the beam and that the weight of the water contained in one foot length of it to be one stone. Let the height of the fall be 20 feet, and, for simplicity, let the ratio of the circumference to the diameter of a circle be that of 3 to 1. Then the semi-circumference of the wheel is 30 feet, and if it move as fast as the water, the weight of water in the buckets will be 30 stone; when the wheel is made to perform a half revolution, the water in the buckets should raise 30 stone, placed in the centre of gravity of the opposite semi-circumference, to the height of 20 feet in the course of 1 minute: now, if the double overshot wheel move as fast as the water, its buckets will contain 20 stone weight of water, which, in two-thirds of a minute, would raise 20 stone to the height of 20 feet, or 30 stone in 1 minute. So that the performance of both wheels is the same. ON THE OVERSHOT WHEEL. To the Editors of the GLASGOW MECHANICS' MAGAZINE. GENTLEMEN,-Having had the opinion of some intelligent mechanics that in the overshot wheel onethird of the power is lost, and this appearing on the first consideration to be very probable, from the circumstance of a portion of the water in the buckets near the top and bottom of the wheel appearing to be inefficient, from their apparently unfavourable mechanical advantage, I imagined that the double overshot wheel would remedy the defect.— But it will be found that the weight of the water in every bucket will support an equal weight placed diametrically opposite, and thus the wheel acts like a number of levers of equal arms, having a common centre of motion, and no power is thus lost. The same conclusion will be arrived at in a more satisfactory manner from the following consideration :-Suppose the water in the canal or lead to move with a velocity of 30 feet per minute, I am, yours, &c. Dollar, 11th April, 1825. A. B. POWER OF BUILDING MATERIALS TO RESIST FROST. A CURIOUS process has been devised by Mr. Brard, for the estimation of the power possessed by building materials of resisting the disintegrating action of frost and weather. He imitated this action by the spontaneous crystallization of a solution of Glauber salts, the crystals producing the same effect upon the particles of the stones submitted to trial, as the formation of ice would have done by the exposure of the same stone to cold after being moistened. The importance of Mr. Brard's process, and its great utility to architects, The stone having been selected, the cubes cut, and the solution of sulphate of soda (saturated at common temperature) made, the solution is to be boiled, and whilst boiling freely the specimens are to be introduced. The stones are to be boiled for half an hour and not longer: M. Vicat having shown, that afterwards the effect surpasses that of frost. The specimens are then to be withdrawn one after the other, and suspended by threads, so that they do not touch each other, or any thing but the thread; a vessel containing some of the clean solution in which they have been boiled, is to be placed beneath each, after which the vessel and its accompanying specimen are not to be separated. If the weather be not too moist or too cold, the specimens will be found covered in 24 hours, with small white saline needles. They are then to be plunged, each into the particular portion of solution beneath it, when the needles fall off, and are again to be suspended in the air. A repetition of this process is to take place each time the needles are well formed. the stone under trial is capable of resisting the action of frost, the salt will remove nothing from it, and neither grains, nor scales, nor fragments, will be found at the bottom of the solution beneath. On the contrary, with a stone which gives way to the weather, it will be seen that even on the If first day the salt will remove particles from it, the cube will lose its angles and edges, and ultimately there will be found at the bottom of the vessel all that has been de'tached during the trial. The trial should be concluded at the end of the fifth day after the salt has first appeared in crystals. The formation of crystals may be facilitated by moistening the stone as soon as they have appeared on any one point; this may be repeated five or six times a day. Great care should be taken that the saturation of the water by the salt be effected at common temperatures. The experiments of M. Vicat and others have demonstrated, that stone, which resists perfectly the action of frost or of cold solution of sulphate of soda, gives way entirely when exposed to the action of a saturated hot solution: the same is the case frequently also, if the trial be continued beyond the fifth day. Mortars and bricks which had withstood ten winters gave way to saturated hot solution; and M. de Thury found, that lias and other stones which had resisted the weather for ages, were disintegrated by the same excessive kind of trial: from which it may be concluded, that stones which can resist these trials would scarcely undergo any change by exposure to weather for any length of time. If it be required to estimate comparatively the power possessed by two or more kinds of stone to resist the action of frost; the portion of matter separated from them, and lying in the solution beneath, is to be collected, washed, dried, and weighed; and the weight indicates the proportion in which the samples tried would suffer by exposure to weather and frost. MR. CLELAND'S PAMPHLET. * THIS Pamphlet, as all Mr. Cleland's generally do, contains much useful information. Besides a history of the Steam Engine-an account of the Meetings in London and Glasgow for erecting Monuments to Mr. Watt-and a history of the application of Steam to the propelling of vessels-it contains many valuable Tables. We extract the following: This The first Steam Engine erected in Glasgow, for spinning cotton, was put up in January, 1792, in Messrs. William Scott & Co.'s (afterwards Tod & Stevenson's) Cotton Mill, near Springfield, nearly opposite what is now the Steam Boat Quay. was seven years after Messrs. Boulton & Watt put up their first Steam Engine for spinning cotton, in Messrs. Robinson's Mill, at Papplewick. Number of Steam Engines in the City of Glasgow and its Suburbs, in April, 1825. Claud Girdwood & Co. 2 engines, 1 of 261...... Engine and Machine Making, Found * Historical Account of the Steam Engine, and its application in propelling Vessels; with an Account of the Proceedings at London, Glasgow, and Greenock, for erecting Monuments to Mr. Watt; also, a detailed Account of the Number and, Power of Steam Engines in Glasgow. Compiled by James Cleland, Superintendent of Public Works for the City, Member of the Chamber of Commerce, &c. &c. Pp. 68. Glasgow; W. R. M'Phun. 1825. |