such a manner that their axes coincided. At A, fig. 168, the focus of one of them, is placed a small iron-wire basket of red-hot coals, and in the focus of the other, at B, is placed an inflammable body, as a piece of amadou or tinder. Then, the rays emitted from the source of heat, at A, are reflected, the first time, from the mirror at whose focus this source is placed; and taking, in consequence of this reflection, a direction parallel to the axis, these rays are reflected the second time from the other mirror and meet at its focus B. This is proved at once by the result; for the piece of tinder placed at this point is set on fire: but either above or below the focus this phenomenon does not take place. This experiment proves that heat and light are reflected according to the same laws. Moreover, if we place at the focus a lighted candle, and at the focus в a piece of ground glass, there will be seen in the latter a luminous focus exactly in the place where the tinder was set on fire. This shows that a luminous focus and a calorific focus are formed at the same point; the reflection takes place, therefore, in both cases, according to the same laws. ice, the surrounding air being at 12° or 15° Centigrade, and in the focus of the other a differential thermometer, this instrument will indicate a reduction of several degrees of temperature. This phenomenon seems, at first sight, to be the effect of the frigorific rays emitted from the ice; but this apparent reflection of cold, as it is called, is explained on the principles, above stated, concerning the equilibrium of temperature which bodies always tend to establish between each other. There is still an exchange of caloric, just as in the experiment of the setting of the tinder on fire, only the parts of the phenomenon are changed; and in this case it is the thermometer which is the warm body. As the rays which it emits are stronger than those emitted by the ice, there is no compensation between the heat which it gives and the heat which it receives; whence its reduction of temperature. It is on the same principle that we explain the cold which is felt when we are near walls in plaster, stone, or marble; and, in general, when we are close to any body whose temperature is lower than our own, Fig. 168: Reflecting Power.-The property which bodies possess of It is owing to the high temperature which can be obtained | by means of the foci of concave mirrors that they have been called burning mirrors. It is related that Archimedes set on fire the vessels of the Romans before Syracuse by the application of such mirrors. M. Buffon constructed burning mirrors of such power as to prove that the fact ascribed to the ingenuity of Archimedes was possible. These mirrors were formed of a great number of pieces of silvered glass about eight inches long and six inches broad. They could be turned independently of each other in any direction, so that the reflected rays could be made to meet in the some point. With 128 pieces of glass and a burning summer's sun, Buffon set on fire a tarred plank of wood at the distance of about 220 feet. Reflection in a Vacuum.-Caloric is reflected in a vacuum as well as in air. This was proved by Sir Humphrey Davy by the following experiment. Under the exhausted receiver of an air-pump, two small mirrors were placed facing each other; in the focus of the one there was a very sensible thermometer, and in the focus of the other a source of electric heat, consisting of platinum wire, which was rendered incandescent by making the current of a voltaic pile pass through it; the thermometer rose immediately, by several degrees, on the application of the current- a phenomenon which was only due to the reflected caloric, for the thermometer would have experienced no elevation of temperature had it not been exactly in the focus of the second mirror. Apparent Reflection of Cold.-If two reflectors are arranged facing each other, as represented in fig. 168, and instead of red-hot coals we place in the focus of one of them a piece of reflecting a greater or less portion of the heat which falls upon them, is called their reflecting power. This power varies considerably in different substances. In order to ascertain experimentally the reflecting power of a variety of substances without constructing a number of reflecting mirrors, Sir John Leslie arranged his experiments in the manner represented in fig. 169. The source of heat is a hollow cube, M, filled with boiling water. On the axis of a spherical reflector, between the focus and the mirror, is fixed a plate, A, formed of the substance whose reflecting power is required. With this arrangement, the rays emitted from the source and reflected the first time, fall on the plate A, and are thence reflected a second time, forming their focus between the plate and the mirror, in a point where the bulb of a thermoscope is placed. Now, the mirror and the thermoscope remaining unchanged, and the water in the cube being kept always at 100° Centigrade, the temperature indicated by the thermoscope varies with the nature of the substance of which the plate A is formed when subjected to the experiment; whence we deduce not the absolute reflecting power of a body, but the ratio of this power to that of another body, assumed as a standard of comparison; that is, in conformity with what has been said on the application of Newton's law, the temperatures indicated by this instrument are proportional to the quantities of heat which it receives. Thus, if a plate of glass causes the differential thermometer to rise 1o, and a plate of lead causes it to rise 6°, we conclude that the heat reflected by lead is six times greater than that reflected by glass; for the quantity of heat emitted by the source is the same, the concave mirror reflects the same portion of it to each, and the difference can only depend on the reflecting power of the plates at A. According to this mode of experimenting, and by representing the reflecting power of brass by 100, when taken as a standard of comparison, Leslie constructed the following table of the relative reflecting powers of different substances: 100 80 M. Melloni has also investigated the reflecting powers of Relative Reflecting Powers. bodies, and it follows from his experiments and those of Leslie, that the reflecting power of metals is much greater than that of other bodies, as shown in the preceding table. M. Melloni has proved, by means of his apparatus, that of all the metals, mercury has the greatest reflecting power. There are causes, however, which modify the reflecting power of the same body. : 70 60 13 Fig. 169. LESSONS IN CHEMISTRY.-No. XXX. As a preliminary to our cupelling operations, we shall require a small piece of lead, say a shot, a piece of wood charcoal, and a blowpipe. First excavate in the smooth surface of the piece of charcoal a slight depression, and into this deposit the lead; which being done, direct upon the lead the continuous outside halo of a blowpipe-flame, as represented by the accompanying sketch, fig. 27. The metal will fuse at first; then, becoming converted into oxide, the surface of the charcoal, to the extent perhaps of a quarter of an inch all round, will assume a yellowish or reddish aspect, due to the presence of oxide of lead. By continuing sufficiently long the blowpipe operation, all the metal admits of being oxidised. Careful attention of the phenomena which present themselves during the operation of fusion, will moreover show that the oxide of lead thus produced is remarkably fusible; will show that the oxide becomes, in point of fact, quite liquid, soaking into the substance of the charcoal, where it may be observed on breaking the latter across. It follows, therefore, that if instead of charcoal we had employed some other material of greater absorbent power, all the oxide might have been removed; filtered away, thus to speak, so soon as formed. We are beginning to see the principle upon which depends the cupelling operation. Let the student now assume a general case. Let it be assumed, I say, that instead of lead alone exposed to an oxidising heat, we had to deal with an alloy of lead and some other metal, the latter being capable of fusion but not oxidation; then it follows theoretically, that separation of the two metals might be effected by taking advantage of the chemical peculiarity just mentioned. In other words, while all the lead should be susceptible of oxidation and absorption, all of the second metal remained behind. Let us now proceed one step further in our theoretical investigation cumstances most favourable to oxidation, and still no oxidation ensue; hence the appellation "noble" metals, by which they are frequently designated. For the present, we will solely direct our attention to silver, that metal having already come under our consideration, whereas gold has not. Supposing, then, the previously-described blowpipe operation to have been prosecuted, not on a piece of mere lead but on a compound or alloy of lead and silver, it should follow theoretically from what we have already stated, that separation of the two might have been effected had our charcoal been sufficiently absorbent of the fused oxide of lead. Practically, charcoal is not sufficiently absorbent, therefore another material possessing the necessary quality has to be found,-that material is bone ash. Exemplification of the Properties f Bone Ash in reference to Cupellation.-Having procured a tobacco-pipe, ram into the bowl some bone earth moistened with water, or still better with a little beer. When rammed full, make a small excavation on the surface, and then place the filled pipe aside in a hot place, say an oven, and there allow it to remain until the contents are quite dry; the apparatus will then be ready for the use to which we shall apply it. operations, the cupels are frequently no less than five, six, or into a furnace in readiness for the cupelling operation. It The accompanying sketch, fig. 30, represents a muffle inserted Fuse together on a charcoal support a small leaden shot with a still smaller bead of silver. The fusion can be readily effected by means of a blowpipe-jet, even though the flame employed be that of an ordinary candle; nevertheless, if the student experience any difficulty, he may employ instead of the candle a spirit lamp. When fused, allow it to cool, and when cold, deposit it in the little cavity already excavated on the surface of the compact bone earth rammed into the tobacco-well as its posterior wall, are perforated with holes round or This muffle is made of refractory clay ware, and its sides, as pipe bowl. These directions having been followed, let the operation of fusion be repeated by directing down upon the elongated, for the purpose of allowing the free passage of the compound bead the outer or oxidising jet of the blowpipe air necessary to cause oxidation. flame. Presently the bead will begin to oxidise; the oxide will fuse into a thin transparent liquid, which immediately on its generation is absorbed into the substance of the bone earth and disappears. In this manner the operation will proceed until every particle of the lead becomes dissipated, and the pure silver remains. There will be no difficulty experienced in determining the period when the total separation of the two metals has been effected. So long as the operation of oxidation proceeds, the compound bead will not only have a dull surface, but fumes of oxide will be seen to arise bodily in vapour. Immediately, however, that the last particle of lead has become removed, then the remaining bead will become clean-looking, white, and resplendent, an appearance technically known as the "brightening;" it must now be removed from the source of heat gradually (for reasons hereafter to be described), and allowed to cool. The process of cupellation, by means of which the separation of lead from silver and gold is effected both on the large and the small scale, is an obvious application of the principles just explained. I will describe the process in detail as followed by assayers, and in our English Mint, as well as the Prussian and several foreign Mints,-not the French, however, the authorities of which preferring the moist process of analysis as being more correct, although far less expeditious. The term "cupellation" originates in the circumstance that the bone earth employed in the separative process, instead of being rammed into a containing vessel, the representative of our tobacco-pipe-bowl, is fashioned into a sort of thick-sided cup, or crucible of the shape represented in fig.28, and technically denominated a cupel. Fig. 28. In practice the size of these cupels varies, the usual dimensions ranging from half an inch to an inch and a half in diameter, and the respective thickness or depth of each being somewhat less. I allude, it need scarcely be indicated, to the mint or assaying cupels. For the purpose of large metallurgic should be remarked, however, that for the sake of greater clearness of illustration, the muffle is not inserted quite so far as it would be in practice. Assuming the muffle to be in its place, and the fire lighted, the various stages of the assaying operation are as follow: the cupels being placed mouth downwards are gradually thrust into the muffle, and the muffle-door closed. The fire is now urged until the cupels are brought to a bright red heat. The muffle-door is now opened, the button of alloy dropped in, and the muffle-door closed once more, until the alloy becomes completely fused. From time to time the door is opened, or partially opened, for the purpose of watching the progress of the operation and of admitting a flow of air. Presently the lead becomes oxidised, the oxide fuses, part of it volatilises and passes away in vapour, whilst another, and by far the larger portion, is absorbed by the porous bone earth, which acts just like a sponge. Not the least difficulty can arise as to knowing when the operation has come to a conclusion, the evidences of this being most striking. So long as the change of oxidation goes on, the muffle is pervaded with a peculiar smoke, by observing the colour of which the metals contained in the alloy may be frequently determined. For example, pure lead tinges the cupel, straw yellow verging towards lemon colour; bismuth, straw yellow passing into orange; copper, a grey dirty red or brown according to the proportion in which it is present; iron yields black scoriæ; tin a grey slag; zinc leaves a yellowish hue upon the cupel and generates a very luminous flame; antimony furnishes yellow scoria, and causes the surface of the alloy to assume iridescent hues. So soon as the process of oxidation is completed, all smoke and vapours disappear, as well as the phenomena of iridescence. The silver acquires a peculiar spinning motion, and emits a sort of flash constituting an appearance termed "the brightening." The operation may now be regarded as at an end, but the purified silver must not be immediately removed from the muffle, inasmuch as it is subject whilst in the fused condition to throw off portions of its substance in all directions, constituting what is denominated in practice the phenomena of "spitting." For the purpose of guarding against this spitting, which, if it take place, causes a portion of the silver to be lost, the cupel must not be suddenly withdrawn altogether from the muffle, but removed nearer and nearer to the mouth of the latter by degrees, the fire during the withdrawal being gradually damped. MATHEMATICAL ILLUSTRATIONS.-No. VI. ARITHMETICAL LOGARITHMS. mantissa prefix the index in the manner described in the preceding Lessons, and you will have the required logarithm. Example, required the logarithm of the number 4. Here, looking for 40 in the first column of the table, you find in the same horizontal line, in the adjoining column on the right, and under 0 at " top, the mantissa 6021; to this mantissa prefix 0, which is the index for units, and you have 0.6021 for the logarithm of the number 4. If the logarithm of the number 40 were required, the mantissa would be the same; but the index would be 1, and the logarithm 1·6021. If the logarithm of 400 were required, the mantissa would still be the same; but the index would be 2, and the logarithm 2 6021; and so on. If the logarithm of a number be required which consists of two figures only, as of all numbers between 10 and 99, seek for that number in the first column of the table; and when you have found it, the mantissa of its logarithm you will find in the same horizontal line in the adjoining column on the right, under the figure marked 0 at the top. To this mantissa prefix the index as before, and you will have the complete logarithm. Thus; required the logarithm of the number 78. Here, looking for 78 in the first column of the table, you find in the same horizontal line, in the adjoining column on the right, and under 0 at the top, the mantissa 8921; to this mantissa prefix 1, which is the index for tens, or for a number consisting of two integer figures, and you have 1.8921 for the logarithm of the number 78. If the logarithm of the number 7-8 were required, the mantissa would be the same, but the index would be 0, and the logarithm 0-8921. If the logarithm of the number 78 were required, the mantissa would still be the same; but the index would be I, and the logarithm I-8921; and so on. If the logarithm of a number be required which consists of three figures, as of all numbers between 100 and 999, seek for the first two figures of the number as in the preceding case, that is, in the first column of the table; and when these are found, you will then find the mantissa of its logarithm in the the right, under the third figure of the number at the top. To same horizontal line in one of the ten adjoining columns on this prefix the proper index, and you will have the logarithm required. Thus, let the logarithm of 476 be required. Here, looking for 47 in the first column of the table, you find in one of the ten adjoining columns on the right, and under 6 at the top, the mantissa 6776; to this prefix 2, which is the index for hundreds or for a number consisting of three integer figures, and you have 2 6776 for the logarithm of the number 476. If the logarithms of the numbers 47.6, 4.76, 476, or 0476 were required, the operation for finding the mantissa of each would be the same, and they would be, on the prin 'es now fully explained to our students, 1.6776, 0.6776, TABLES OF LOGARITHMS AND ANTILOGARITHMS. THE following is one of the tables promised in our last Lesson; it will be found very useful, not only to our students who are endeavouring to make themselves acquainted with logarithms, but also to persons who are desirous of abridging calculations of any description, especially those connected with the Mathematical and Philosophical Sciences. The first table, called the Table of Logarithms, contains the logarithms, or rather the mantissa of the logarithms, of all numbers from 1 to 10,000, according to the common system, of which the base is 10. The decimal part of a logarithm is called its mantissa, and the integral part is called its index or characteristic. Thus in the logarithms 0.477121, 1041393, and 3005609, the decimal parts 477121, 041393, and 005609, are the manti776, and 76776 respectively. and the integral parts 0, 1, and 3 are the indices or cheras teristics. The mantissæ of the logarithms in the first table extend only to four decimal places; but these are reckoned sufficient for ordinary purposes. If, however, a greater degree of accuracy be required than can be obtained from this table, recourse must be had to more extensive tables; of these the best are Hutton's or Babbage's Tables of Logarithms. Let us now proceed to explain our own tables contained in the two following pages. In the first vertical column of the table are contained the first two figures of any given number, whose logarithm is required, within the range above mentioned; and this column is headed, First Two Figures. In the next ten vertical columns is contained the third figure of any such number; these ten columns are headed, Third Figure. In the next nine vertical columns is contained the fourth figure of any such number; and these nine columns are headed, Fourth Figure. If the logarithms of a number be required, which consists of one figure only, as of the nine digits, seek for that figure with a cipher annexed to it in the first column of the table; and when it is found, then you will find the mantissa of its logarithm in the same horizontal line in the adjoining column on the right, under the figure marked 0 at the top. To this If the logarithm of a number be required which consists of four figures, as of all numbers be.ween 1000 and 9999, seek for the mantissa corresponding to the first three figures, as in the preceding case, and in the same horizontal line in one of the fourth figure at the top, a number which is to be added to the nine columns, headed Fourth Figure, you will find, under the mantissa, in order to make it the complete mantissa required; to this prefix the index as before, and you will have the logarithm sought. For example, let it be required to find the logarithm of the number 5768. Here, looking for the mantissa of the first three figures 576, as in the preceding case, you find 7604; and in the same horizontal line with it, under the fourth figure 8, you find the number 6, which is to be added to 7604; this being done, you have 7610 for the complete mantissa; prefixing the index 3, according to previous directions, you have 3.7610 for the complete logarithm required. If the logarithms of 57680, 576 8, 5.768 or 005768 were required, the operation for finding the mantissa would still be the same; but the indices, according to the previous rules, would be different, the logarithms being respectively 4.7610, 2·7610, 0·7610, and 3-7610. Having thus explained the method of finding the logarithms of numbers from the table, we ought now to show how to perform arithmetical calculations by their means; but we delay doing so till our next Lesson, when we shall also give and explain our Table of Antilogarithms. 47 6812 6902 6532 6542 6551 6561 6571 6580 6590 6599 6609 6618 22222 6 4 2 4 5 50 51 53 54 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 1 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 1 2 52 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 1 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 1 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396 22222 2 55 56 67 58 59 60 62 63 64 7782 7789 7796 7803 7810 7818 7825 7832 61 7853 7860 7868 7875 7882 7889 7896 7903 7910 7924 7931 7938 7945 7952 7959 7966 7973 7980 7987 1 1 7993 8000 8007 8014 8021 8028 8035 8041 8048 8055 1 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 1 1 6995 6666 a 88∞∞∞ 88777 77766 87777 79466 7 69966 |