ON PHYSICS, OR NATURAL PHILOSOPHY. No. XXII. (Continued from page 304.) PNEUMATIC AND HYDRAULIC MACHINES. wardly. These valves are kept shut by small coiled springs. The action of the valves is the same as that of the valves in the condenser; and as in the latter there is a limit to the condensation, so there is the same limit in the condensing syringe. This limit depends on the ratio which exists between the two volumes of air included under the piston, when it is at the top or at the bottom of the barrel. If the volume of air, when the piston is at the bottom of the barrel, be oneThe Condenser.-The condenser is an apparatus which is em-sixtieth part of the volume of air when it is at the top of the oyed to condense air or any other gas. As its form differs barrel, we can only condense the air up to 60 atmospheres; ut little from that of the air-pump, with the exception of the valves, it will be sufficient to give here a longitudinal section of this machine, in order to show the action of the valves, which open downwards, whereas in the air-pump they open awards. These valves, of which the one is represented at a ir the bottom of the piston, and the other at o in the bottom of the barrel, fig. 103, are conical, and are kept shut by a Fig. 103: Fig. 105. Fig. 104. e spring. When the piston P is raised, the air is rarefied biow it, the valve o is kept shut by the spring, and the valve 2 opens by the pressure of the atmosphere, which permits the exterior air to enter the barrel. When the piston is lowered, the air which is below it is compressed, the valve a is shut, whilst the valve o is opened and admits the compressed air, which is transmitted to the receiver R. At every stroke of the piston, the mass of air contained in the barrel is passed into the receiver. Yet there is a limit to the tension of the condensed air; for during the condensation, a period will arrive when the elastic force of the air in the barrel, even when the piston is at the bottom, is no longer equal to that of the air in the receiver, and then no more air will pass into the latter, because the valves will remain shut, in consequence of the pressure of the interior air. In the condenser, the tension of the air is measured by means of a small manometer of compressed air communicating with the receiver. In this machine, the receiver must be strongly fastened to the platen, otherwise it would be driven off by the elastic force of the condensed air. For this purpose, the receiver is constructed of a strong glass cylindric vessel open at both ends, and having its edges well ground and well greased. The lower edge rests on the platen A, and the upper edge is shut by a strong glass plate B, perforated at equal distances by four holes, through which pass four iron rods D, fastened to the platen. By means of these rods and the screws E, the glass plate B is firmly fixed to the cylinder, and the whole to the platen. In order to prevent accidents by the breaking of the cylinder, in consequence of the pressure of the condensed air or gas, it is surrounded by an iron grating. This machine has few practical applications; but under the following form it is of very frequent use. for beyond that pressure, the tension of the air in the receiver K would be greater than that of the air in the barrel, and then the bottom valve would not open to give admission to an additional quantity of air. Condensed Air Fountain.-The condensed air fountain is represented in fig. 104. It is composed of a brass cylinder K, furnished at the top with a tube and stop-cock c, upon which the condensing syringe is screwed. A tube н, open at both ends, projects to the bottom of the cylinder, or reservoir K. A quantity of water is put into this reservoir, the stop-cock c is opened, and the condensing syringe A is put in operation. The condensed air enters the reservoir by the tube н, and presses on the upper surface of the water. If then the stop-cock c be shut, and the syringe A be unscrewed, and, instead of it, a tube or ajutage be fastened to the tube и, the water will instantly issue vertically, like a spring or fountain, as soon as the stop-cock c is opened. The apparatus in fig. 104 is also employed in the absorption of gases by water. To effect this, the stop-cock B, by means of the tube D, is made to communicate with the vessel full of gas which is to be absorbed, as, for instance, carbonic acid. The condensing syringe draws the gas from the vessel and condenses it in the reservoir K, where it is absorbed; and the Condensing Syringe. The condensing syringe is a kind of quantity thus absorbed increases, as before observed, in proforcing pump, composed of a single barrel, A, fig. 104, of portion to the degree of condensation to which the gas is subsmall diameter, in which a solid piston (that is, a piston with-jected. By the application of a similar apparatus, aerated or out valves) is made to work by the operation of the hand. gaseous waters are manufactured. The barrel is furnished with a screw by which it can be The Air-gun.-This instrument is a gun in which the exfastened to the vessel in which the air or any gas is to be con-pansive force of condensed air forms a substitute for that of densed. Fig. 104 represents the condensing syringe A C, the gas produced by the ignition of gunpowder. On the with a handle for working it, and screwed to a vessel K, in stock, which is hollow and made of wrought-iron, is screwed which the air is to be condensed. Fig. 105 shows the arrange-a force-pump, by means of which the stock may be filled with ment of the valves, which are constructed so that the side air of 10 or 15 atmospheres of pressure. A projectile being vaive e opens inwardly, and the bottom valve s opens out-placed in the usual manner in the barrel, a valve communi100 VOL. IV. in fig. 106, No. 2. In this figure D is a brass cup or vessel, and and N are two glass globes about four or five inches in diameter; B is the long brass tube connecting the cup D with the lower part of the globe N; A is the tube connecting the upper portions of the two globes м and N. Between these two tubes is seen a third, connecting the lower part of the globe м with the atmosphere above the level of the water in D; but this tube is, in this construction, withdrawn, in order to admit of the pouring of water into the globe M, until it be half-full. This being done, the tube is replaced, and water is poured into the cup or cistern. This water descends by the tube B, into the lower globe, and drives the air out of it; this air is condensed in the upper globe, where it acts upon the water and causes it to spring up, as in the diagram. Abstracting the resistance of the air and friction, the water should rise above the cup to a height equal to the difference of the level of the water in the two globes. The Intermittent Fountain.-The models of the intermittent. fountain exhibited in our lecture-rooms explain in a plausible manner the causes of intermittent springs. The upper part of this apparatus, fig. 107, No. 1, is a close vessel or reservoir, unner vessel a, a middle vessel b, and a lower vessel c. These vessels are connected by three tubes: the first, x, descends from the bottom of the upper vessel, nearly to the bottom of the lower vessel; the second, y, rises from the top of the lower vessel nearly to the top of the middle vessel; and the third, z, rises nearly from the bottom of the middle vessel, and terminates in a jet a little above the upper vessel. The operation is as follows:-Water is put into the vessel b, by means of the stop-cock p, it is then closed; water is also put into the vessel a; the stop-cock r in the tube x is then opened, and the water rushes from the upper vessel into the lower one; in this vessel the water is immediately acted on by the compression of the air which it contains, and is forced up the tube y into the vessel b; here the water is again acted on by the compression of the air which this vessel contains, and is forced up the tube z, through the jet, into the atmosphere, rising to a height above the upper vessel, which, theoretically speaking, is as much above the level of the water in the middle vessel as the level of the water in the upper vessel is above the level of the water in the lower vessel. The reason is that the air which is contained in the lower vessel, and in the middle one, supports a pressure determined by a height of water equal to the difference between the two levels of the water in the upper and lower vessels; the water contained in the middle vessel must therefore rise in the tube x, to the height due to this pressure. For the purpose of lecture-room illustration, the following representation of this fountain will be better understood, as seen z, filled with water up to the level ab. A vertical tube passing into this vessel from below, has its upper orifice open, and raised above the level of the liquid ab, and its lower orifice at c is also open. The bottom-piece of the apparatus is double, and the orifice T allows the water which falls on the first bottom to escape into the second, A B, with less velocity than it falls from the ajutages, c, e, f, d. The flow of water from the upper vessel or reservoir continues until the water, by its accumulation, closes up the orifice, at c, of the vertical tube, and the pressure on ab becomes less than the pressure of the atmosphere. This flow begins again after the discharge of a sufficient quantity of water has taken place at the orifice T; and it continues until it is again interrupted in the same manner as before; and so on. This apparatus is, perhaps, more vividly shown in fig. 107, No. 2, where the upper reservoir for the water is a glass globe or pear-shaped vessel, made air-tight by a ground stopper, and having two or three short capillary tubes, D, through which the water runs. A is a strong glass-tube, open at both ends, which is inserted in the globe c, having one end raised above the water-level in the globe, and the other terminating near the central orifice in the brass basin or stand B, whieh supports the apparatus. Here, the globe c being about twothirds full of water, this liquid issues from the orifice D, the interior pressure on the surface being equal to the sum of that of the atmosphere which is transmitted through the tube A, and that of the column of water above D; whilst the exterior pressure is only that of the atmosphere. These circumstances continue so long as the lower end of the tube is open, that is, so long as the tension of the interior air is equal to that of the atmosphere, for the air is admitted into the globe in proportion as the water runs off. But the apparatus being adjusted so that the orifice in the basin or stand B allows less water to escape than that delivered by the orifices D, the level rises by degrees in the basin, and ultimately covers the lower aperture of the tube. The exterior air no longer obtaining admittance into the globe c, the air within it is rarefied in proportion as the water continues to flow, until the sum of the pressures of the column of water CD and of the tension of the air above it, is equal to the exterior pressure of the atmosphere at D; then the flow of the water is stopt; but the basin continuing to empty itself, the extremity of the tube is again uncovered, and the air entering as before, the flow recommences; and the same process is repeated until there be no water left in the globe c. The Siphon.-The siphon is a bent tube having two unequal branches used for drawing off liquids, as in fig. 108, No. 1; Fig. 108. No. 1. Fig. 108. No. 2. and when in action, the bend is uppermost. In order to make use of this instrument, it is first inverted, and the shorter branch being kept closed, it is filled to the top of the longer branch with the liquid to be drawn off. This branch is then closed, and the instrument being restored to its right position, the shorter branch is inserted in the vessel containing the liquid, and the longer in the vessel to be filled; both ends being then opened, the liquid will flow from the one vessel into the other until the level of the liquid be the same in both. Another mode of putting the instrument in operation, is to insert the shorter branch into the liquid at c, as in fig. 108, No. 2, and with the mouth to draw out the air contained in the tube at the orifice в of the longer branch. This being done a vacuum is formed, and the liquid in the vessel c rises in the tube by the pressure of the atmosphere, fills it, and continues to flow as before. When the liquid is unfit to touch the mouth, a siphon is used, to which is soldered a second tube M, as in fig. 109, parallel to the longer branch. The air is withdrawn from the siphon by the orifice o of this additional tube, the orifice P of the siphon being kept shut only until the liquid reaches it; otherwise the liquid might rise to the mouth in the additional tube. In order to understand how the flow of the liquid takes place, let it be observed that the force which presses on the liquid at c, in fig. 108, No. 2, and draws it in the direction CB, is that of the pressure of the atmosphere, minus that of a column of water whose height is CD. Also, the force which presses on the liquid at B and urges it in the direction BDC is the pressure of the atmosphere, minus that of the weight of a column of water whose height is A B. Now the latter column being greater than the former, it follows that the effective force which sets at B is less than that which acts at c. The flow then takes place in consequence of the difference of these forces. Consequently, according to the theorem of Torricelli, Siphon with Constant Flow.-In order that the flow of the siphon may be always the same, it is evident from the preceding observations, that the difference between the levels of the liquid in its two branches must be invariable. This object is attained by the arrangement shown in fig. 110. The siphonis preserved in equilibrium by a float a and a weight p, in such a manner that in proportion as the level of the liquid in the vessel H is lowered, the siphon is lowered with it; hence, the difference between the heights a b and b c remains always the same. The Intermittent Siphon.-The intermittent siphon, as its name indicates, is one in which the flow is not continuous, This siphon is arranged in a vessel so that the shorter branch is open near the bottom, while the greater branch passes through it and opens below it. The vessel being supplied with a constant flow of water, the level rises by degrees, both in the vessel and in the shorter branch. up to the top or bend of the siphon. The siphon is then filled in consequence of the pressure of the liquid, and the flow takes place as shown in fig. 111. But as the discharge of the siphon is so adjusted that it is greater than that of the tube which supplies the vessel, the level sinks in the vessel, and the shorter branch of the siphon soon ceases to be immersed in the liquid; the siphon is then emptied and the flow is interrupted. The vessel, however continuing to be supplied by the constant source, the level again rises, and the same series of operations is periodically renewed. In the large water-works, constructed for the supply of towns, apparatus with intermittent flows are often employed to open | of the case, perhaps the student will have already anticipated the or shut the stop-cocks of main-pipes at certain fixed periods. statement that the troublesome process of collecting, washing, For this purpose, vessels supplied by a small but constant run fusing, and weighing the chloride, may be altogether dispensed of water, empty themselves at intervals, and becoming some-with, simply by preparing a chlorine solution of known invariable times heavy and sometimes light, they act, by the aid of counter- and definite strength, weighing a vessel full of this solution, addweights, first in one direction and then in another, on the ing portions of this solution, drop by drop, to the silver solution, keys of the stop-cocks, and produce the effect required. The until no more chloride of silver is deposited; and finally estitheory of the intermittent siphon gives the explanation of mating the amount of loss in weight of the standard test solution natural intermittent fountains which are to be found in various thus employed. places of the globe. Some of these fountains yield a supply of water during several days or months, then they stop during a longer or shorter interval, and after this they begin again to flow. Others stop and resume their flow several times in an hour. These phenomena are explained by supposing the existence of subterranean cavities which are filled more or less slowly with water from springs, and which empty themselves again by fissures which exist under ground, in the same manner -as the intermittent siphon. Fig. 7. A LESSONS IN CHEMISTRY -No. XXI. INASMUCH as the metal silver is one that admits of being obtained bodily, evidently, from all its solutions with remarkable facility, by first precipitating it as a chloride, and then decomposing that chloride by either of the means already described, the student may perhaps imagine that, for this very reason, the discovery and the quantitative estimation of silver are matters of peculiar certainty. E is true that this discovery and quantitative estimation are matters of great ease and certainty, but not for the reason adverted to. Young chemists are in the habit of committing the great error of assuming that chemical estimation of any given substance necessarily involves bodily presence of that substance. The idea is natural, but it is not always correct; on the contrary, the actual bodily presence of a substance is seldom effected during the course of analysis. Chemistry, in this respect, furnishes an exception to the usual rules of evidence: collateral and indirect, being frequently of greater value than immediate and direct evidence. The chemical relations of the metal silver furnish a remarkable illustration of this proposition. We can easily get the metal out of any solution; nevertheless, in practice it is found more correct to estimate the amount of silver by collecting and weighing the amount of chloride generated. So much more correct is the latter process, that it is adopted as the means of testing the purity of silver in the French mint. We English do not adopt that process in our mint, because it occupies more time; but as to its superior correctness there cannot be two opinions. For example, let us suppose A and B to be two test-glasses, of which a contains an unknown quantity of silver, and в a known quantity of chlorine, in any convenient form of combination (for pure chlorine is a gas). This known amount of chlorine, we will furthermore assume to be 36 grains in weight. Suppose, now, the contents of B to be added to the contents of A, and that exactly the whole of B is required to precipitate the whole of A; no more, no less. Then does it not follow, as clearly as the simplest demonstration in geo etry, that the quantity of silver in a must be equal to 108 grains? Nothing, then, can be easier than the theory of this operation; but, as usual, certain difficulties present themselves in practice, and have to be provided for. In the first place, the solution of chlorine compound must be absolutely pure; secondly, means must be taken to prevent evaporation of the standard solution, otherwise its strength would be continually increasing. These matters, however, exclusively refer to quantitative chemistry, hence I need not further advert to them here. Mercury-As it is my object, in the present course of lessons, to treat of chemical substances according to the groups in which they present themselves to a practical operator, I cannot do better than follow the investigation of silver by that of the only other metal which affords a chloride absolutely insoluble in water. I have already mentioned (Lesson xix.), that any dilute metallic solution In order, however, that the amount of chloride generated should which yields a white precipitate on the addition of hydrochloric be a faithful index of the quantity of silver present, one postulate acid or a chloride-I might have also said a solution of chlorineis necessary. It will have occurred to the student, no doubt; for must either be mercury or silver. I lay stress on the word dilute, I have taken several opportunities of expatiating upon parallel because strong lead solutions produce a similar effect, as we shall cases. The composition of chloride of silver must be fixed and un-recognise when discussing that metal. If you have any doubt, varying; a given weight of it must always contain the same relative therefore, dilute the suspected solution and apply heat. If the amount of silver and chlorine. Now this is the case, and the pro-white precipitate disappears, the metal under examination will be position holds good for all chemical compounds whatsoever. Per- lead; if it remain, the metal will be mercury or silver. haps hereafter I shall treat of the philosophy of chemistry, as I now treat of its practice, and I shall describe the laws of definite proportionality, and expatiate on the beauties and the probabilities of the atomic theory. Meantime remember, if you please-that the fact of chloride of silver being fixed and invariable as to composition, however prepared, furnishes one of the many proofs that the composition of chemical compounds generally is fixed and invariable, and supplies one of the strongest arguments in favour of the atomic theory. Well, to proceed. The composition of chloride of silver, dried and fused, to drive off the last remnant of moisture, is, as near as our most delicate balances can inform us, as follows:-Every 144 parts by weight are made up of 108 parts by weight of silver, and 36 parts by weight of chlorine; whence it follows, that, having generated a certain given weight of chloride of silver, absolutely pure and dry, we may ascertain the quantity of silver present in it by a rule of simple proportion. If we choose to extract the silver, we can easily do so; but the resulting indication will not be so exact, for the very simple reason, that the manipulative processes involved in effecting reduction, however carefully exercised, must be necessarily attended with some slight loss. Proceeding with the mental investigation of the circumstances Having reference to their chlorides, therefore, it is evident that silver, mercury, and lead arrange themselves in one analytical group. Mercury differs from the greater number of metals we have already considered, in forming two classes of combinationsproto combinations and per combinations. Thus we have protoxide and peroxide of mercury; protochloride and perchloride of mercury; protobromide and perbromide of mercury, and so on. These various salts I shall not treat of in detail, but I shall merely group them into proto and per salts. Now the best solutions on which to display the peculiarities of these classes will be protonitrate, protochloride, and perchloride of mercury. I may also as well say at once, that the insoluble chloride of mercury, of which I have spoken, is the protochloride commonly known as "calomel," under which name you will do well to procure some. As regards the protonitrate, you will have to make it, directions for effecting which will be presently given. The perchloride is corrosive sublimate of the shops, otherwise known as "oxymuriate of mercury," xix. p. 292: of this you win require a few grains. It is a violent poison, as you have been already informed; and its antidote, as you already know, is white of egg. Protonitrate of mercury is made by adding aquafortis, mixed with about an equal bulk of water, to quicksilver, and applying heat. The quicksilver should be more in quantity than the nitric | same rule, there is no objection to saying that four mões acid employed can dissolve. We will now proceed with a comparative testing of a protosalt and a persalt of mercury, using protonitrate and protochloride as our specimen of the former, perchloride as our specimen of the latter. For this purpose, begin with pouring into one glass (a wine-glass will do) a few drops of protonitrate, then fill the glass with water. Repeat the operation, using a few drops of the perchloride solution in another glass. Let us now assume the nature of the metal to be totally unknown, and begin to examine it analytically. The student knows by this time that the first witnesses to be brought into court are hydrosulphuric acid solution, hydrosulphate of ammonia, and ferrocyanide of potassium. Let both protosalt and persalt be tested with the first of these tests, and we shall get a result, it may be, of a very peculiar character. If the amounts of the two solutions be duly apportioned, the precipitate will be black; nevertheless, if the admixture be effected within certain limits, a white precipitate may ensue; or, finally, we may have a white changing to black, or black changing to white. This variable indication is characteristic of mercury; we will, however, let it pass, and will assume the precipitate to have been black from the beginning. This being the case, what does our operation teach us? That the solution under consideration contains a metal-a calcigenous metal-not zinc, or iron, or manganese, or uranium, or nickel; not arsenic or antimony, cadmium or persalt of tin. Test next with ferrocyanide of potassium (prussiate of potash). We now get a white precipitate, hence the metal in question is not copper, molybdenum, uranium, or titanium. Thus far qur process of testing has been equally applied to both protosalt and persalt. Let us now try what a solution containing chlorine will effect -a solution of common salt, for example. Adding a little of this to the perchloride of mercury, we get no precipitate, hence the information conveyed by our test is very little, and that little negative; but adding a portion of the same test-solution to the protochloride, we get a white precipitate. Now this white precipitate may be indicative of silver, mercury, or perhaps of lead. If it represent the latter, it will dissolve when boiled in contact with water. Remove therefore a little to a test-tube; half fill the testtube with water, and boil. The white substance does not dissolve; it cannot therefore represent lead. But does it stand for silver? If it be chloride of silver, it will readily dissolve in hartshorn. It does not, but turns black. Now this characteristic is indicative of mercury-nothing but mercury. If therefore the unknown metal had come before us as a protosalt, we should have already made it out. Directing our attention now to the other solution, place a few drops of it upon a piece of gold (say a coin), and bring into contact with the liquid and the coin at once a piece of clean iron, say a key; after the lapse of a few seconds a white metallic stain, growing resplendent when dried and rubbed, will appear on the coin. This result is indicative of mercury; and here let us take our leave of the metal mercury for the present. Let us see. LESSONS IN GEOMETRY.-No. XXVIII. LECTURES ON EUCLID. PROPOSITION XXVIII.-THEOREM. (Continued from page 313.) IN the analytical proof reference is made to what is denominated the principle of homogeneity; a principle in itself irrefragable, but, like all others, capable of being ill applied. Wherever quantities are to be equal, it is necessary that they be homogeneons, or of the same kind; for equality is nothing but the capability of coincidence, and things heterogeneous cannot coincide. A mile of length, or two or three or four miles, can by no possibility be equal to an hour of time; the assertion would be ipso facto foolish and unmeaning. But there is no objection to saying that four miles ten hours because the first of these two miles five hours expressions means only the number of times that the quantity two miles can be taken in the quantity four miles, which is the number two; and five hours may be taken the same number of times in ten hours. The things finally declared equal are not heterogeneous, for they are both of them numbers. And by the miles X ten hours, for this means nothing but tuuk low bouten v five hours two miles the number which results from seeing how stan five hours can be taken in ten hours. It follows therefore that heterogeneous quantities enter equations by pairs; or at all events are reducible to pairs by running some two or more of them into one by the operation of addition or subtraction. There cannot be the slightest idea of questioning this, or any of the legitimate results of what has been called the principle of homogeneity. But the application in this instance was not legitimate, or at all events not legitimately conducted. There was on the face of it an unjustifiable operation, consisting in substituting for the angles the numbers which expressed their ratios. Professor Leslie brought this into full light, by pointing out that if the same reasoning were applied to the case where two sides (a and b) were given and the angle between them P, it would produce the statement that the remaining side c=: (a, b, P); in which, on substituting for a, b, c, the numbers which express their ratios, there would be the same argument for inferring that c would be the same whatever was the angle, which is notoriously untrue. And this brought out the avowal, that his opponents in the case of the angles intended to substitute the ratios, and in the case of the sides, not; a mode of arguing.comparable only to the ingenuity he has the power of projecting or not, as shall be required to of the artist, who in playing at "odd or even," holds a ball which make him win. When pushed on this point, they replied, that their reason for substituting the ratios in the case of the angles and not of the sides, was because the right angle was the natural unit of angles. But the fact of a right angle (or more properly four right angles, or a turning from the place started from till arriving at it again) being a convenient object of reference for the comparison of angles in general, is devoid of any proved connexion with the propriety of substituting the ratios in one case, and not substitnting them in the other. When pressed, however, they produced a reason. They said it was because the angle is a portion of a finite whole, the straight line a portion of an infinite whole; so that every given angle is a finite quantity, while every given straight line is a quantity infinitely small, and only the ratios of given straight lines can enter into our calculations with given angles." And this was repeated as a very subtle and very just metaphysical idea; and at the same time strictly analytical." On which all that can be done, is to remark on the absence of any reasonable or demonstrated connexion (even supposing the facts indisputable, which might be questioned), between the facts alleged and the consequences assigned to them. But a circumstance which appears to have escaped Professor Leslie is, that his opponents, till his counter case appeared, had been at the expense of an unnecessary wrong. There was not the slightest necessity for substituting numbers, to produce their argument; for p was just as heterogeneous and intractable when A, B, C were angles, as after numbers had been substituted. It is difficult therefore to surmise any reason for the substitution unless they had a foresight of Leslie's reply. And when that came, they should have said that they would eject p the side, as incapable of homogeneity, but for P the angle they would substitute and then it would be a number, which need not be P R ejected, because c, a, and b may by possibility compound a number. This would at least have held together; but it would have sunk under the unreasonableness of the substitution demanded in one case with intention to refuse it in the other. And this leads to the substantial inference from the whole of the somewhat perplexed controversy which took place at the time; which is, that the original mistake consisted in confounding two sets of things essentially distinct: the quantities, the fixation of which causes another quantity to be necessarily fixed, or what Euclid, in his Book of Data, calls given, and the quantities which * "L'angle est une quantité que je mesure toujours par son rapport avec r'angle droit, car l'angle droit est l'unité naturelle des angles. Dans cette notion très simple, un angle est toujours un nombre. Il n'en est pas de même des lignes: une ligne ne peut entrer dans le calcul, dans une équation quelconque, qu'avec une autre ligne qui sera prise pour unite, ou qui aura un rapport connu avec la ligne unité."-Letter of M. Legendre. Leslie's Rudiments of Plane Geometry. Fourth Edition. Notes and Illustrations, page 296. +Paper of M. le Baron Maurice, in the Bibliothèque Universelle de Genève, Oct. 1819; as given in Dr. Brewster's Edition of Legendre's Geometry, Notes, p. 235. Note by M. Legendre, Ibid. ? R standing for a right angle |