Description of a Reflective Goniometer

XVI. Description of a reflective Goniometer. By William Hyde Wollaston, M. D. Sec. R. S. Read June 8, 1809. From the advances that have been made of late years in crystallography, a very large proportion of mineral substances may now be recognised, if we can ascertain the angular dimensions of their external forms, or the relative position of those surfaces that are exposed by fracture. But though the modifications of tetrahedrons, of cubes, and of those other regular solids, to which the adventitious aid of geometry could be correctly applied, have been determined with the utmost precision, yet it has been often a subject of regret, that our instruments for measuring the angles of crystals are not possessed of equal accuracy, and that in applying the goniometer to small crystals, where the radius in contact with the surface is necessarily very short, the measures, even when taken with a steady hand, will often deviate too much from the truth to aid us in determining the species to which a substance belongs. A means of remedying this defect has lately occurred to me, by which in most cases the inclination of surfaces may be measured as exactly as is wanted for common purposes, and when the surfaces are sufficiently smooth to reflect a distinct image of distant objects, the position of faces only $\frac{1}{5}$ of an inch in breadth may be determined with as much precision as those of any larger crystals. For this purpose, the ray of light reflected from the surface is employed as radius, instead of the surface itself, and accordingly for a radius of $\frac{1}{50}$ of an inch, we may substitute either the distance of the eye from the crystal, which would naturally be about twelve or fifteen inches; or for greater accuracy we may, by a second mode, substitute the distance of objects seen at a hundred or more yards from us. The instrument which I use, consists of a circle graduated on its edge, and mounted on a horizontal axle, supported by an upright pillar (Plate XI). This axle being perforated, admits the passage of a smaller axle through it, to which any crystal of moderate size may be attached by a piece of wax, with its edge, or intersection of the surfaces, horizontal and parallel to the axis of motion. This position of the crystal is first adjusted, so that by turning the smaller axle, each of the two surfaces, whose inclination is to be measured, will reflect the same light to the eye. The circle is then set to zero, or $180^\circ$, by an index attached to the pillar that supports it. The small axle is then turned till the further surface reflects the light of a candle, or other definite object to the eye; and, lastly, (the eye being kept steadily in the same place) the circle is turned by its larger axle, till the second surface reflects the same light. This second surface is thus ascertained to be in the same position as the former surface had been. The angle through which the circle has moved, is in fact the supplement to the inclination of the surfaces; but as the graduations on its margin are numbered accordingly in an inverted order, the angle is correctly shewn by the index, without need of any computation. It may here be observed, that it is by no means necessary to have a clean uniform fracture for this application of the instrument to the structure of laminated substances; for since all those small portions of a shattered surface, that are parallel to one another (though not in the same plane), glisten at once with the same light, the angle of an irregular fracture may be determined nearly as well, as when the reflecting fragments are actually in the same plane. In this method of taking the measure of an angle, when the eye and candle are only ten or twelve inches distant, a small error may arise from parallax, if the intersection of the planes or edge of the crystal be not accurately in a line with the axis of motion;* but such an error may be rendered insensible, even in that mode of using the instrument, by due care in placing the crystal; and when the surfaces are sufficiently smooth to reflect a distinct image of objects, all error from the same source may be entirely obviated by another method of using it. For this purpose, if the eye be brought within about an inch of the reflecting surface, the reflected image of some distant chimney may be seen inverted beneath its true place, and by --- *I cannot omit mentioning, that Mr. Sowerby had thought of employing reflection for this purpose, nearly at the same time as myself; but did not succeed to his satisfaction, in consequence of an attempt to fix the position of the eye. For when the line of sight is determined by a point connected with the apparatus, the radius employed is thereby limited to the extent of the instrument, and the error from parallax is manifestly increased. turning the small axle may be brought to correspond apparently with the bottom of the house (or with some other distant horizontal line.) In this position the surface accurately bisects the angle, which the height of that house subtends at the eye (or rather at the reflecting surface); then, by turning the whole circle and crystal together, the other surface, however small, may be brought exactly into the same position; and the angle of the surfaces may thus be measured, with a degree of precision which has not hitherto been expected in goniometry. The accuracy, indeed, of this instrument is such, that a circle of moderate dimensions, with a vernier adapted to it, will probably afford corrections to many former observations. I have already remarked one instance of a mistake that prevails respecting the common carbonate of lime, and I am induced to mention it, because this substance is very likely to be employed as a test of the correctness of such a goniometer, by any one who is not convinced of its accuracy from a distinct conception of the principles of its construction. The inclination of the surfaces of a primitive crystal of carbonate of lime is stated, with great appearance of precision, to be $104^\circ 28' 40''$: a result deduced from the supposed position of its axis at an angle of $45^\circ$ with each of the surfaces, and from other seducing circumstances of apparent harmony by simple ratios. But however strong the presumption might be that this angle, which by measurement approaches to $45^\circ$, is actually so, it must nevertheless be in fact about $45^\circ 20'$; for I find the inclination of the surfaces to each other is very nearly, if not accurately $105^\circ$, as it was formerly determined to be by Huygens;* and since the measure of the superficial angle given by Sir Isaac Newton† corresponds with this determination of Huygens, his evidence may be considered as a further confirmation of the same result; for it may be presumed, that he would not adopt the measures of others, without a careful examination. IN THE ANNEXED PLATE, ab. Is the principal circle of the goniometer graduated on its edge. cc. The axle of the circle. d. A milled head by which the circle is turned. ee. The small axle for turning the crystal, without moving the circle. f. A milled head on the small axle. g. A brass plate supported by the pillar, and graduated as a vernier to every five minutes. h. The extremity of a small spring, by which the circle is stopped at $180^\circ$, without the trouble of reading off. ii and kk. Are two centers of motion, the one horizontal, the other vertical for adjusting the position of a crystal: one turned by the handle l, the other by the milled head m. The crystal being attached to a screw-head at the point n (in the center of all the motions), with one of its surfaces as * Huygenii Opera Reliqua, Tom. I. p. 73 — Tract. de Lumine. † Newton's Optics, 8vo. p. 329. Qu. 25, concerning Iceland Crystal. nearly parallel as may be to the milled head $m$, is next rendered truly parallel to the axis by turning the handle $l$ till the reflected image of a horizontal line is seen to be horizontal. By means of the milled head $f$, the second surface is then brought into the position of the first, and if the reflected image from this surface is found not to be horizontal, it is rendered so by turning the milled head $m$, and since this motion is parallel to the first surface, it does not derange the preceding adjustment.