Amid all the fruitless attempts which have appeared, there was still one avenue to the object of pursuit overlooked, to which the common, and well-known principles of hydrostatics seemed to direct the way; this was the principle, that any body specifically, or bulk for bulk, lighter than common air, will rise and swim in it. Consequently, if I attach a certain quantity of vessels, at equal distances, round the circumference or rim of a wheel, so contrived as that one half of the vessels shall be exhausted on one side of the wheel, and the other half filled with air on the opposite side; in this case the exhausted vessels will attain the highest part of the wheel, and the full ones the lowest. But to render the matter more explicit, I must refer to the prefixed drawing. A, B, C, D, are four vessels, connected to the wheels, E, E, (though only one is shown to prevent confusion) by round pins, a, a, a, a, which project from the vessels on each side, and enter into corresponding holes in the wheels E, E. The wheels, E, E, are caused to revolve, by the space under the vessel, B, being a vacuum, and, therefore, lighter than the same portion of air; a little before the vessel, B, reaches the highest point of the wheels, it begins to close, and opens the opposite vessel, D, in the same manner as the vessel C opens A, because the pressure of the atmosphere on the vessel C, is equal to the pressure on A. Instead of common packing to make the vessels air-tight, mercury is substituted, which has infinitely less friction, and is never out of order; it is represented by the black marks in the drawing. The particles of mercury not being entirely free from friction, a little power is requisite to open and shut the vessels; this is effected by the rods, F, F, connected to the levers, G, G, G, G, by chains; the rods, F, F, give motion to other rods, Í H, by the rollers, b, b, b, b, acting against the collars on the rod, H, H. Again, the levers, G, G, G, G, are successively worked, by sliding over the roller, P. The connecting rods, H, H, are so adjusted as not to draw the vessels out of their upright position, which would let the mercury escape; also, the lower vessels, Á and D, are made rather larger in diameter than B and C, so as the pressure of the atmosphere may' counterpoise the weight of the vessels A, C, and B, D, with their connecting rods, &c. I doubt not in the least, that if a ounces per cubic foot. Halifax. ALPHA. P. S. This machine might be converted into a reciprocating engine, if there would be any advantage in that, by placing the vessels at each end of a beam. SEATON'S FIN-WHEEL CARRIAGES. Sir, I have to tender you my acknowledgements for having, with so much readiness, obtained and published Mr. Seaton's address. I called upon him in consequence, and had a full inspection of the machine, which more than came up to the idea I had formed of it, from the description which appeared in your Magazine. The construction of the wheels is admirable, and the springs at the ends of the shafts, for the protection of the horse's chest, or rather shoulders, though very simple, are extremely valuable. Your's very truly, LITHOGRAPHY. Lithography has been hitherto regarded as an imperfect art, in consequence of the supposed impossibility of retouching drawings on stone after impressions had been taken ON THE SKOTOR, 123 reckon A B in parts of the same scale, we shall have its value in those parts equal to the numerical quotient. But this is done by measuring A B along the limb, C A, or C B; by which we find its value 5.6. To complete proportions by the line of lines is an easy corollary from the properties of similar triangles. For instance, in finding a third proportional, CA: A B:: C a: a b, and if a C, be taken equal to A B, the parallel, ab, is the third proportional. In finding the 4th proportional, we use a similar method to the foreto range our going, taking care terms upon the scale, so that the defective one shall fall upon a parallel distance. To find a mean proportional between the numbers; or, which is the same thing, expressed geometrically, to find the side of a square equal to a given rectangle, B This is the principle of the rule. Now for the practice; because this proportion gives A B ten times larger than it should be; and, besides, 24.64 cannot be reckoned on the primary divisions; we, therefore, diminish the divisor or the dividend, as is most convenient, ten times; by which it is evident we bring out. A B ten times smaller than before. Instead, then, of 24-64, let us reckon 2-464, then constructing the figure as before, with regard only to the altered value of the dividend; we have 44: 2-464: 10: A B 2.464 X 10 which is the same thing 4.4 1 = By Euclid, VI. 13, A S = AB+ BC; and B P is the mean proportional; also S PAS; and SB AB A S, or A SB C. Hence the rule," Extend the sector so that the angle of the lines at the centre, B, is a right angle. Take BS the difference between either of the quantities, and half their sum. Then, having the distance S P sum of quantities between the compasses, place one foot in the point S, and sweep them till the other foot rests on B P. BP measured from the centre is the mean proportional." The lines of the sector form a right angle when it is opened, so that 5, measured from the centre, is a transverse distance, between 3 on one limb and 4 on the other. (To be continued.) DARLEY'S POPULAR GEOMETRY DARLEY'S POPULAR GEOMETRY.* Mr. Darley is a mathematician after our own hearts. He holds, as all rational scholars do, that geometry is the only foundation of perfect knowledge, and thinks so highly of Euclid as to affirm, that his book of Elements has been of more signal use to the world than all its other uninspired writings put together. He makes confessions at the same time such as you rarely hear from the lips of your professional and profound geometrician, (and Mr. D. is both.) He admits, (in substance or in words,) that the cultivators of this branch of knowledge are accustomed to claim for it a perfection which cannot possibly belong to it; that in their endeavours to exalt it into a perfectly abstract science, they have soared to the clouds and no further; that they have needlessly thrown an air of mystery around a science which treats of the simplest propositions, and is derived from the commonest and humblest sources of information; and that while professing to allow only the most self-evident truths, and the most rigid demonstrations, they have indulged in some as gross absurdities as were ever palmed on human credulity; and that Euclid's Elements, though it does beat all other books hollow, the bible excepted, has, like all other things mundane, its faults and deficiencies, which it is "a shame" to us moderns not to have long ere now rectified and supplied. He presumes, as we have presumed to do in our notices of the geometrical works of M.Dupin and Mr. Lee, vol. vi. pp. 413-435, to make light of the notion, (laid down as a principle by Euclid, and the first with which he sets out!) that "a point is that which has no parts," or in other words, that "a point is a magnitude which has no magni A System of Popular Geometry, containing in a few lessons so much of the Elements of Euclid as is necessary and sufficient for a right understanding of every art and science in its leading truths and general principles. By George Darley, A. B. John Taylor, London, price 4s. 6d. tudé ! He thinks with us, that some of the most standard definitions, (such as that a straight line is “that which lies even, i. e. straight, between its extreme points,") are mere verbal illusions, which teach nothing whatever, but the art of making what is already as simple as can be, somewhat less simple, by means of mere words and phrases; he illustrates, by some striking instances, the absurdity and confusion into which geometers have been universally led in their proofs of indirect propositions, by that subtle sort of trickery which we have styled writing with two meanings to their pens, (p. 437, vol. vi.)-treating things at one time as only supposed, and at another as actual realities,-telling us to do that which cannot be done, instead of telling us only to suppose it done, &c. He shews how that most bothersome of all the propositions in Euclid, nicknamed the pons asinorum, or "asses' bridge," might, but for the sake of a little learned refinement and display, have been rendered so plain and straight forward a thing, that none but the veriest ass could have boggled at it; nay, so mathematically heretical is this Mr. Darley, that he will not take for granted what Euclid has pronounced to be (of all things in the world!) self-evident, namely, "that if a right line intersect two right lines, and make the two internal angles at the same side of the intersecting line together, less than two right angles, these two latter right angles will meet if produced sufficiently." (Euclid's 12th axiom.) But contends, on the contrary, that this is a monstrous and unwarrantable" assumpsion, and that the doctrine of parallels, which Euclid has founded upon it, and on which almost the whole system of geometry, as now studied, depends, must necessarily be so erroneous, that it can scarcely be altered for the worse. 66 To make Euclid's Elements perfect, is still, however, far from being the aim of Mr. Darley's present pub.. lication. He leaves that, he says, "to abler heads and more ambitious hearts." His work professes, in truth, to be no more than" a Compendium |