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wardly. These valves are kept shut by small coiled springs. MW PHYSICS, OR NATURAL PHILOSOPHY.

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 No. XXII.

condensation, so there is the same limit in the condensing

syringe. This limit depends on the ratio which exists between (Continued from page 304.)

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 PNEUMATIC AND HYDRAULIC MACHINES.

air, when the piston is at the bottom of the barrel, be oneYou have condenser.—The condenser is an apparatus which is em- sixtieth part of the volume of air when it is at the top of the proyed to condense air or any other gas. As its form differs barrel, we can only condense the air up to 60 atmospheres; sout little from that of the air-pump, with the exception of the valves, it will be sufficient to give here a longitudinal section on this machine, in order to show the action of the valves,

Fig. 205.

Fig. 104. which open downwards, whereas in the air-pump they open mapywards. These valves, of which the one is represented at a in the bottrm of the piston, and the other at o in the bottom

che barrel, fig. 103, are conical, and are kept shut by a

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:))21 spring. When the piston P is raised, the air is rarefied wiow it, the valve o is kept shut by the spring, and the valve k 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, whils: 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 che receiver. Yet there is a limit to the tension of the condensed air; for during the condensation, a period will arrive for beyond that pressure, the tension of the air in the receiver when the elastic force of the air in the barrel, even when the K would be greater than that of the air in the barrel, and piston is at the bottom, is no longer equal to that of the air in then the bottom valve would not open to give admission to an the receiver, and then no more air will pass into the latter, additional quantity of air. because the valves will remain shut, in consequence of the Condensed Air Fountain.-The condensed air fountain is repressure of the interior air.

presented in fig. 104. It is composed of a brass cylinder K, In the condenser, the tension of the air is measured by means furnished at the top with a tube and stop-cock c, upon which of a small manometer of compressed air communicating with the condensing syringe is screwed. A tube , open at both the receiver. In this machine, the receiver must be strongly ends, projects to the bottom of the cylinder, or reservoir K. fastened to the platen, otherwise it would be driven off by the A quantity of water is put into this reservoir, the stop-cock elastic force of the condensed air. For this purpose, the re- c is opened, and the condensing syringe A is put in

operaceiver is constructed of a strong glass cylindric vessel open at tion. The condensed air enters the reservoir by the tube , both ends, and having its edges well ground and well greased. and presses on the upper surface of the water. If then the The lower edge rests on the platen A, and the upper edge is stop-cock c be shut, and the syringe a be unscrewed, and, shut by a strong glass plate B, perforated at equal distances by instead of it, a tube or ajutage be fastened to the tube 6, the four holes, through which pass four iron rods D, fastened to water will instantly issue vertically, like a spring or fountain, the platen. By means of these rods and the screws E, the as soon as the stop-cock c is opened. glass plate B is firmly fixed to the cylinder, and the whole to The apparatus in fig. 104 is also employed in the absorption the platen. In order to prevent accidents by the breaking of of gases by water. To effect this, the stop-cock B, by means the cylinder, in consequence of the pressure of the condensed of the tube d, is made to communicate with the vessel full of air or gas, it is surrounded by an iron grating. This machine gas which is to be absorbed, as, for instance, carbonic acid. has few practical applications; but under the following form The condensing syringe draws the gas from the vessel and it is of very frequent use.

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, äg. 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 &, 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 a opens inwardly, and the bottom valve s opens out- placed in the usual manner in the barrel, a yalve communi

100

TOL. IV.

cating with the stock and the barrel, is opened by means of a (in fig. 106, No. 2. In this figure D is a brass cup or vessel, and a trigger, and the air escaping from the former with great force, and n are two glass globes about four or five inches in diamethe projectile is discharged from the latter. The valve closing ter; B is the long brass tube connecting the cup D with the immediately that this is done, the air contained in the stock still lower part of the globe N; A is the tube connecting the upper possesses a very considerable elastic force, and several balls portions of the two globes M and N. Between these two tubes can be discharged without the introduction of a fresh quantity is seen a third, connecting the lower part of the globe m with of air.

the atmosphere above the level of the water in d; but this tube The Fountain of Hero.--A variety of hydraulic machines is

, in this construction, withdrawn, in order to admit of the have in modern times been constructed on the principle of pouring of water into the globe m, until it be half-full. This liero's mountain ; such as the Hungarian machine employed being done, the tube is replaced, and water is poured into the for raising water from the mines of Schemnitz, the machine of cup or cistern. This water descends by the tube B, into the Datrouville. the mechanism of Girard's lamps, &c. It is re- lower globe, and drives the air out of it; this air is condensed presentea in ag. 106, No. 1, and is composed of three vessels; an in the upper globe, where it acts upon the water and causes it

to spring up, as in the diagram. * Abstracting the resistance Fig. 106. No.2.

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,

Fig. 106. No.).

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under vessel a, a middle vessel b, and a lower vessel c. These Dessers 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 filied with water up to the level ab. A vertical tube passing lower vessel nearly to the top of the middle vessel ; and the into this vessel from below, has its upper orifice open, and third, z, rises nearly from the bottom of the middle vessel, and raised above the level of the liquid ab, and its lower orifice at terminates in a jet a little above the upper vessel. The opera- c is also open. The bottom-piece of the apparatus is double, tion is as follows: -Water is put into the vessel 6, by means and the orifice T allows the water which falls on the first of the stop-cock p, it is then closed; water is also put into the bottom to escape into the second, A B, with less velocity than vessel a; the stop-cock r in the tube x is then opened, and the it falls from the ajutages, c, e, f, d. The flow of water from water rushes from the upper vessel into the lower one; in this the upper vessel or reservoir continues until the water, by its vessel the water is immediately acted on by the compression of accumulation, closes up the orifice, at c, of the vertical tube, the air which it contains, and is forced up the tube y into the and the pressure on becomes less than the pressure of the vessel b; here the water is again acted on by the compression atmosphere. This flow begins again after the discharge of a of the air which this vessel contains, and is forced up the tube sufficient quantity of water has taken place at the orifice T; %, through the jet, into the atmosphere, rising to a height and it continues until it is again interrupted in the same manabove the upper vessel, which, theoretically speaking, is as ner as before ; and so on. much above the level of the water in the middle vessel as the

This apparatus is, perhaps, more vividly shown in fig. 107, level of the water in the upper vessel is above the level of the No. 2; where the upper reservoir for the water is a glass globe water in the lower vessel. The reason is that the air which is or pear-shaped vessel, made air-tight by a ground stopper, contained in the lower vessel, and in the middle one, supports and having two or three short capillary tubes, D, through a pressure determined by a height of water equal to the differ- which the water runs. A is a strong glass-tube, open at both ence between the two levels of the water in the upper and ends, which is inserted in the globe c, having one end raised lower vessels; the water contained in the middle vessel must above the water-level in the globe, and the other terminating therefore rise in the tube %; to the height due to this pressure. near the central orifice in the brass basin or stand B, which

For the purpose of lecture-room illustration, the following supports the apparatus. Here, the globe c being about twowepresentation of this fountain will be better understood, as seen thirds full of water, this liquid issues from the orifice D, the interior pressure on the surface being equal to the sum of that the velocity of the flow will be greater or less in proportion to of tħe atmusphere which is transmitted through the tube A, the difference of level between the orifice B and the surface of and that of the column of water above D; whilst the exterior the liquid atc. From this, it is evident, that the siphon cannot pressure is only that of the atmosphere. These circumstances act in a vacuum; and that it equally fails, when the height of continue so long as the lower end of the tube is open, that is, the column cd is greater than that of a column of the liquid 80 long as the tension of the interior air is equal to that of the whose weight is equal to the pressure of the atmosphere. atmosphere, for the air is admitted into the globe in proportion as the water runs off. But the apparatus being adjusted

Fig. 109,

Fig. 110. 80 that he orifice in the basin or etand 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;

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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 in variable. This object is attained by the arrangement shown in fig. 110. The siphon is 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 a is lowered, the siphon is lowered with it; hence, the difference between the heights ab and b c remains always the

same.

Fig. 111.

and when in action, the bend is uppermost. In order to make The Interinittent Siphon.-The intermittent siphon, as its use of this instrument, it is first inverted, and the shorter name indicates, is one in which the flow is not continuous, branch being kept closed, it is filled to the top of the longer This siphon is arranged in a vessel so that the shorter branch branch with

the liquid to be drawn off. This branch is then is open near the bottom, while the greater branch passes through closed, and the instrument being restored to its right position, it and opens below it. The vessel being supplied with a conthe shorter branch is inserted in the vessel containing the stant flow of water, the level rises by degrees, both in the liquid, and the longer in the vessel to be filled; both ends vessel and in the shorter branch, up to the top or bend of the being then opened, the liquid will flow from the one vessel siphon. The siphon is then filled in consequence of the presinto the other until the level of the liquid be the same in both. sure of the liquid, and the flow takes place as shown in fig. 111. 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, aud with the mouth to draw out the air contained in the tube at the orifice B 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 ON , is that of the pressure of the atmosphere, minus that of a But as the discharge of the siphon is so adjusted that it is column of water whose height is CD. Also, the force which greater than that of the tube which supplies the vessel, the presses on the liquid at B and urges it in the direction BDC is level sinks in the vessel, and the shorter branch of the siphon the pressure of the atmosphere, minus that of the weight of a soon ceases to be immersed in the liquid; the siphon is then column of water whose height is A B. Now the latter column emptied and the flow is interrupted. The vessel, however being greater than the former, it follows that the effective force continuing to be supplied by the constant source, the level again which sets at B is less than that which acts at c. The flow rises, and the same series of operations is periodically then takes place in consequence of the difference of these renewed. forces. Consequently, according to the theorem of Torricelli, 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, firs& 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 miles X

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Fig. 7. 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.

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LESSONS IN CHEMISTRY -No. XXI. . IN SMICII as the metal silver is one that admits of being obtained bodily, evidently, from all its solutions with remarkable facility, by For example, let us suppose A and B to be two test-glasses, of first precipitating it as a chloride, and then decomposing that which a contains an unknown quantity of silver, and B a known chloride by either of the means already described, the student may quantity of chlorine, in any convenient form of combination (for perhaps imagine that, for this very reason, the discovery and the pure chlorine is a gas). This known amount of chlorine, we will quantitative estimation of silver are patters of peculiar certainty. furthermore assume to be 36 grains in weight. Suppose, now, I. is true that this discovery and quantitative estimation are matters the contents of B to be added to the contents of A, and that exactly of great ease and certainty, but not for the reason adverted to the whole of B is required to precipitate the whole of A; no more, Young chemists are in the habit of committing the great error of no less. Then does it not follow, as clearly as the simplest demonassuming that chemical estimation of any given substance neces. stration in geo etry, that the quantity of silver in a must be equal sarily involves bodily presence of that substance. The idea is to 108 grains ? natural, but it is not always correct; on the contrary, the actual Nothing, then, can be easier than the theory of this operation; bodily presence of a substance is seldom effected during the course but, as usual, certain difficulties present themselves in practice, of analysis. Chemistry, in this respect, furnishes an exception to and have to be provided for. In the first place, the solution of the usual rules of evidence: collateral and indirect, being frequently chlorine compound must be absolutely pure ; secondly, means of greater value than immediate and direct evidence. The chemical must be taken to prevent evaporation of the standard solution, relations of the metal silver furnish a remarkable illustration of otherwise its strength would be continually increasing. These - this proposition. We can easily get the metal out of any solution; matters, however, exclusively refer to quantitative chemistry, hence nevertheless, in practice it is found more correct to estimate the I need not further advert to them here. amount of silver by collecting and weighing the amount of chloride Mercury-As it is my object, in the present course of lessons, to generated. So much more correct is the latter process, that it is treat of chemical substances according to the groups in which they adopted as the means of testing the purity of silver in the French present themselves to a practical operator, I cannot do better thane mint. We English do not adopt that process in our mint, because follow the investigation of silver by that of the only other metal it occupies more time; but as to its superior correctness there can which affords a chloride absolutely insoluble in water. I have alnot be two npinions,

ready 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

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 a wodilt of silver and chlorine. Now this is the case, and the pro- wbite 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 Having reference to their chlorides, therefore, it is evident that now treat of its practice, and I sball describe the laws of definite silver, mercury, and lead arrange themselves in one analytical proportionality, and expatiate on the beauties and the probabilities group. Mercury differs from the greater number of metals we of the atomic theory. Meantime remember, if you please--that have already considered, in forming two classes of combinations-the fact of chloride of silver being fixed and invariable as to proto combinations and per combinations. Thus we have protoxide composition, however prepared, furnishes one of the many proofs and peroxide of mercury; protochloride and perchloride of mer. that the composition of chemical compounds generally is fixed and cury; protobromide and perbromide of mercury, and so on. These invariable, and supplies one of the strongest arguments in favour of various salts I shall not treat of in detail, but I shall merely group the atomic theory.

them into proto and per salts. Well, to proceed. The composition of chloride of silver, dried Now the best solutions on which to display the peculiarities of and fused, to drive off the last remnant of moisture, is, as near as these classes will be protonitrate, protochloride, and perchloride of our most delicate balances can inform us, as follows :-Every 144 mercury. I may also as well say at once, that the insoluble chloparts by weight are made up of 108 parts by weight of silver, and ride of mercury, of which I have spoken, is the protochloride com36 parts by weight of chlorine; whence it follows, that, having monly known as “calomel,” under which name you will do well generated a certain given weight of chloride of silver, absolutely to procure some. As regards the protonitrate, you will have to pure and dry, we may ascertain the quantity of silver present in it make it, directions for effecting which will be presently given. The by a rule of simple proportion. If we choose to extract the silver, perchloride is corrosive sublimate of the shops, otherwise known as we can easily do so; but the resulting indication will not be so * oxymuriate of mercury,” xix, p. 292: of this you win require a exact, for the very simple reason, that the manipulative processes few grains. It is a violent poison, as you have been already ininvolved in effecting reduction, however carefully exercised, must formed ; and its antidote, as you already know, is white of egg. be necessarily attended with some slight loss.

Protonitrate of mercury is made by adding aquafortis, mixed Proceeding with the mental investigation of the circumstances with about an equal bulk of water, to quicksilver, and applying

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heat. The quicksilver should be more in quantity. than the nitric same rule, there is no objection to saying that for alle acid epiployed can dissolve.

ten hours We will now proceed with a comparative testing of a protosalt

for this means nothing but there is

five hours and a persalt of mercury, using protonitrate and protochloride as two miles x the number which results frou seeing bosan our specimen of the former, perchloride as our specimen of the five hours can be taken in ten hours. It follows toutor that latter. For this purpose, begin with pouring into one glass (a wine-glass are reducible to pairs by running some two

or more of them into

heterogeneous quantities enter equations by pairs; or at ail events do) a few drops of protonitrate, then fill the glass with water. one by the operation of addition or subtraction. There cannot Repeat the operation, using a few drops of the perchloride solution be the slightest idea of questioning this, or any of the legitimate in another glass. Let us now assume the nature of the metal to be results of what has been called the principle of homogeneity. totally unknown, and begin to examine it analytically. The stu- But the application in this instance was not legitimate, or at dent knows by this time that the first witnesses to be brought into all events not legitimately conducted. There was on the face of court are hydrosulphuric acid solution, hydrosulphate of ammonia, it an unjustifiable operation, consisting in substituting for the and ferrocyanide of potassium. Let both protosalt and persalt be Leslie brought this into full light,

by pointing

out that if the

angles the numbers which expressed their ratios. Professor tested with the first of these tests, and we shall get a result, it may same reasoning were applied to the case where two sides ja and be, of a very peculiar character. If the amounts of the two solutions ) were given and the

angle between them P, it would produce be duly apportioned, the precipitate will be black; nevertheless, if the statement that the remaining side c=g: (a, b, P); in which, the admixture be effected within certain limits, a white precipitate on substituting for a, b,c, the numbers which express their ratios, may ensue; or, finally, we may bave a white changing to black, or there would be the same argument for inferring that c would be black changing to white. This variable indication is characteristie the same whatever was the angle, which is notoriously untrue. of mercury; we will, however, let it pass, and will assume the And this brought out the avowal, that his opponents in the case of precipitate to have been black from the beginning. This being the the angles intended to substitute the ratios, and in the case of the case, what does our operation teach us ? That the solution under sides, not; a mode of arguing.comparable only to the ingenuity consideration contains a metada calcigenous metal ---Dot zinc, or 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 iron, or manganese, or uranium, or nickel; not arsenic or antimony, make him win. cadmium or persalt of tin,

When pushed on this point, they replied, that their reason for Test next with ferrocyanide of potassium (prussiate of potash), substituting the ratios in the case of the angles and not of the We now get a white precipitate, hence the metal in question is not sides, was because the right angle was the natural unit of copper, molybdenum, uranium, or titanium. Thus far qur process angles."* But the fact of a right angle (or more properly four of testing has been equally applied to both protosalt and persalt. right angles, or a turning from the place started from till arriving

Let us now try what a solution containing chlorine will effect at it again) being a convenient object of reference for the compari- solution of common salt, for example. Adding a little son of angles in general, is devoid of any proved connexion with of this to the perchloride of mercury, we get no precipitate, the propriety of substituting the ratios in one case, and not subhence the information conveyed by our test is very little, and that stitnting them in the other. little negative; but adding a portion of the same test-solution to

When pressed, however, they produced a reason. They said

it was because the protochloride, we get a white precipitate. Now this white pre straight line a portion of an infinite whole; so that every given

the angle is a portion of a finite whole, the cipitate may be indicative of silver, mercury, or perhaps of lead. If angle is a finite quantity, while every given straight line is a it represent the latter, it will dissolve when boiled in contact with quantity infinitely small, and only the ratios of given straight water. Remove therefore a little to a test-tube ; half fill the test- lines can enter into our calculations with given angles.” And cube with water, and boil. The white substance does not dissolve; this was repeated as "a very subtle and very just metaphysical it cannot therefore represent lead. But does it stand for silver idea ; and at the same time strictly, analytical."I On which all Let us see.

If it be chloride of silver, it will readily dissolve in that can be done, is to remark on the absence of any reasonable hartshorn. It does not, but turns black. Now this characteristic or demonstrated connexion (even supposing the facts indisputable, is indicative of mercury-nothing but mercury. If therefore the which might be questioned), between the facts alleged and the unknown metal had come before us as a protosalt, we should have consequences assigned to them. already made it out. Directing our attention now to the other Leslie is, that his opponents, till his counter case appeared, had

But a circumstance which appears to have escaped Professor solution, place a few drops of it upon a piece of gold (say a coin), been at the expense of an unnecessary

wrong. There was not and bring into contact with the liquid and the coin at once a piece the slightest necessity for substituting numbers, to produce their of clean iron, say a key; after the lapse of a few seconds a white argument; for p was just as heterogeneous and intractable when metallic stain, growing resplendent when dried and rubbed, will A, B, C were angles, as after numbers had been substituted. It appear on the coin. This result is indicative of mercury; and here is difficult therefore to surmise any reason for the substitution let us take our leave of the metal mercury for the present. 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 substi

tute, f and then it would be a number, which need not be LESSONS IN GEOMETRY.-No, XXVIII.

ejected, because c, d, and b may by possibility compound a num

ber. This would at least have held together; but it would have LECTURES ON EUCLID.

sunk under the unreasonableness of the substitution demanded

in one case with intention to refuse it in the other. PROPOSITION XXVIII.---THEOREN,

And this leads to the substantial inference from the whole of

the somewhat perplexed controversy which took place at the time; (Continued frour page 313.)

which is, that the original mistake consisted in confounding two In the analytical proof reference is made to what is denominated sets of things essentially distinct: the quantities, the fixation of the principle of homogeneity; a principle in itself irrefragable, which causes another quantity to be necessarily fixed, or what but, like all others, capable of being ill applied. Wherever Euclid, in his Book of Data, calls given, and the quantities which quantities are to be equal, it is necessary that they be homogeneons, or of the same kind; for equality is nothing but the coincide. Alcidence, and things heterogeneous cannot

“L'angle est une quantité que je mesure toujours par son rapport avec by no possibility be equal to an huur of time; the assertion would notion très simple, un angle est toujours un nombre. Il n'en est pas de :

o ême des lignes : une ligne ne peut entrer dans le calcul, dans

quelconque, qu'avec une autre ligne qui sera prise pour unite, ou qui aura to saying that

four miles_ten hours because the first of these un rapport .connu avec la ligne unité." --Letter of M. Legendre. Leslie's uwo miles five hours

Řndiments of Plane Geometry. Fourth Edition. Notes and Illustrations, expressions menus only the number of times that the quantity page 296. two miles can be taken in the quantity four miles, which is the Geneve, Oct. 1819; as given in Dr. Brewster's Edition of Legendre's

+ Paper of M. le Baron Maurice, in the Bibliothèque Universelle de number two; and five hours may be taken the same number of Geometry, Notes, p. 235. times in ten liours. The things finally declared equal are not # Note by M. Legendre, Ibid. · heterogencous, for they are both of the numbers. And by the B standing for a right angle

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