ARTICLE II. Observations on the Measuring of the Angles of Crystals. WHEN I Composed, 20 years ago, my Traité de Mineralogie, my collection, which was not very far from its commencement, was affected by the rarity of regular and well-defined crystals among us. It was almost solely with these feeble means that I undertook to apply my theory to all the varieties hitherto described, adding those that were new to myself. It is well known that the study of such bodies requires a copious collection in order to be able to find crystals free from those accidental circumstances which alter the level of the surface and occasion perceptible differences between their inclinations and those derived from invariable laws of structure. These accidental deviations occasioned some of the inaccuracies into which I fell, notwithstanding all my care, and which I should have avoided had ĺ been possessed of different crystals of the same variety to verify my observations. Other inaccuracies were occasioned by imperfections of which I was aware, without being able to extricate myself from the uncertainties to which they gave rise. In such cases I took care to mention that I did not guarantee the accuracy of the measurements.+ Such is the fate of works which constitute the foundation of a great system, especially those which result from a multitude of delicate researches. Some of them indeed exhibit the requisite degree of accuracy, but others still leave uncertainties to be cleared up, by the investigation of objects which admit more decisive conclusions. The great increase of my collection since the publication of my treatise, has put it in my power to correct many of my old determinations. Some of these corrections have been consigned in my Tableau Comparatif; and since the publication of that book, I have continued to occupy myself with the same subject, proposing to insert the new results which I have obtained in the second edition of my Traité de Mineralogie, which I am preparing for the press. I had no other instrument for the determination of the angles but the goniometer invented by M. Carangeot, by means of which one can scarcely hope to come nearer the truth than * Translated from the Journal de Physique, lxxxvii. 233. (October, 1818.) + When treating of the crystals of oxide of tin (Traité de Mineralogie, iv. 153), I employed considerations derived from the law of symmetry, which led me to infer a difference between the primitive form of this mineral and the cube, from which it does not deviate far. But the only crystals which I had (they were macles) did not enable me to verify my notions. I pointed out the difference in my Tableau Comparatif, pp. 284 and 285. within half a degree, or the third of a degree when the crystal measured possesses every desirable perfection. But the method which I had adopted, and which I shall immediately explain, seemed to put it in my power to dispense with a greater degree of precision, by giving me a means of knowing from theory the term at which I ought to stop, amid the various results which I obtained sometimes on one side, and sometimes on another. As the sciences advance, those who cultivate them invent new methods of determining with more precision the quantities which serve as data for the solution of problems. The repeating circle of Borda furnished one of these methods to astronomy and geodesy. Malus employed it to measure, by means of the angles of incidence and reflexion of light, the angles of different natural bodies, which he wished to employ for the development of his beautiful theory of double refraction. Dr. Wollaston, to whom the sciences lie under so many obligations, has contrived another very ingenious instrument, founded on the same principle expressly, for the use of crystallography. The smallness of the size, far from being a reason for excluding crystals, is, in his instrument, rather a motive of preference. This is a prerogative which this distinguished philosopher enjoys to be able to employ the method furnished by physics and chemistry to determine at one time the angles, at another the constituents of a substance, almost too small to be perceived, and which seems to borrow from the extreme dexterity of the hand who performs the experiment what it wants in bulk and weight. Mr. Phillips, who has successfully practised the art of handling this instrument, has published in the Transactions of the Geological Society of London, the results of his measurement of the angles of a variety of crystals; and without comparing them with those which I have obtained, it is merely necessary to consider the way in which the instrument is constructed and graduated to be entitled to conclude that the ordinary goniometer is unable to contend with it, and that we have no reason to hesitate about the choice whenever we wish to obtain the requisite precision in the measurement of the angles of crystals. The results of Mr. Phillips, who had no knowledge of most of the rectifications which I have made of my old measurements, point out very sensible differences with several of those, which seem to complete the proof of the pre-eminence of the reflecting goniometer. And the kind of disgrace into which they have a tendency to bring the one of which I made use may even be a reason for doubting if my theory be as well proved as I believed it to be, and whether it ought not even to be rejected, as not being able to exhibit in its applications that accuracy which constitutes the essence of every theory. I propose, therefore, to show, that my theory, in the state into which it has been brought by the new attempts which I have made to complete it, cannot leave any doubt respecting the accuracy of the results deduced from it; that the determinations of the primitive forms on which I have fixed, lead, with regard to the secondary forms, to the true laws of decrement on which these forms depend; and that the measurements made by the reflecting goniometer itself confirm the existence of those laws. I add that the application of the theory to the mineralogical method has likewise all the accuracy necessary to make the forms of the molecules contribute to the distinction of the species. Finally, without excluding in certain particular cases, the use of measures obtained by the reflecting goniometer, I am convinced that those obtained by the common goniometer, which have the advantage of being at once direct and rapid, are sufficient, either to determine a new variety, or to ascertain to which of the varieties already classed in the method the crystal under examination belongs, though seen only for the first time. I shall give three examples in support of what I have just said. The first two, namely, quartz, and oxide of tin, have been chosen from those of which the determinations have been published either in my Treatise, or my Tableau Comparatif. With the last, sulphate of lead, I have occupied myself more recently. I shall compare the results obtained by the two goniometers; and I shall draw from the comparison consequences which appear to me to guarantee the truth of all that I have advanced. Quartz. Though the crystals of quartz be subject to several anomalies which occasion slight variations in the position of their faces, especially of those which are parallel to the axis; yet it is not difficult to find, among the great variety deposited in collections, some possessed of all the requisite regularity for mechanical measurement. Such in particular are those called hyacinths of compostella, many of which are isolated and complete, and all the faces of which are smooth and perfectly level. With respect to these crystals then, I was in a favourable situation to bring the ratio of their dimensions to a simple limit, capable of leading to results sensibly the same with those of the crystallation. I took for a datum the inclination of one of the faces of the pyramid, such as P (fig. 1, Plate XCII.) on the adjacent face r. I found that it was between 141° and 142°. I supposed it 1410. On this hypothesis, if from the centre c of the base of the pyramid, of which c s is the axis, we draw the perpendicular cr to one of the sides, and then the line r s, we shall have cr s = 51° 45′ and cr: cs :: sin. 38° 15′: sin. 51° 45'. To have the ratio crcs expressed in radical quantities, I take the logarithms of the squares of these two sines, and seeking in the table of natural numbers, those to which they correspond, I obtain crcs: √3833 : √ 6167, or nearly :: √ 38: √ 62 |