"The instruments used for measuring the angles at which the planes of crystals incline to each other, are called Goniometers.

"The mutual inclination of any two planes, as of a and b, fig. 1, is indicated by the angle formed by two lines, e d, ef, drawn upon them from any point, e, on the edge at which they meet, and perpendicular to that edge.

"Now it is known that if two right lines, as gf, dh, fig. 2, cross each other at any point, e, the opposite angles, def, geh, are equal. If, therefore, the lines, gf, dh, are supposed to be very thin and narrow plates, and to be attached together by a pin at e, serving as an axis to permit the point, f, to be brought nearer either to d, or to h, and that the edges, ed, ef, of those plates, are applied to the planes of the crystal, fig. 1, so as to rest upon the lines, ed, ef, it is obvious that the angle, geh, of the moveable plates would be exactly equal to the angle, def, of the crystal.

Fig. 1.

d

Fig. 2.

a

"The common goniometer is a small instrument for measuring this angle, geh, of the moveable plates. It consists of a semicircle, fig. 3, divided into 360 equal parts, or half degrees, and a pair of moveable arms, d h, gf, fig. 4, the semicircle having a pin at i, which fits into a hole in the moveable arms

"The method of using this instrument is to apply the edges, de, ef, of the moveable arms to the two adjacent planes of any crystal, so that they shall actually touch or rest upon those planes in directions perpendicular to their edge. The arm, dh, is then to be laid on the plate, mn, of the semicircle, fig. 3, the hole at e being suffered to drop on the pin at i, and the edge nearest to h of the arm will then indicate on the semicircle, as in fig. 5, the number of degrees which the measured angle contains.

"When this instrument is applied to the planes of a crystal, the points, d and f, fig. 4, should be previously brought sufficiently near together for the edges, de, ef, to form a more acute angle than that about to be measured. The edges being then gently pressed upon the crystal, the points, d and f, will be gradually separated, until the edges coincide so accurately with the planes that no light can be perceived between them.

Fig. 5.

"The common goniometer is, however, incapable of affording very precise results, owing to the occasional imperfection of the planes of crystals, their frequent minuteness, and the difficulty of applying the instrument with the requisite degree of precision.

"The more perfect instrument, and one of the highest value to crystallography, is the reflecting goniometer, invented by Dr. Wollaston, which will give the inclination of planes whose area is less than 100000 of an inch, to less than a minute of a degree.

"This instrument has been less resorted to than might, from its importance to the science, have been expected, owing, perhaps, to an opinion of its use being attended with some difficulty. But the observance of simple rules will render its application easy.

“The principle of the instrument may be thus explained :—

"Let ab, fig. 6, represent a crystal, of which one plane only is visible in the figure, attached to a circle, graduated on its edge, and moveable on its axis at o; and let a and b mark the position of the two planes whose mutual inclination is required.

"And let the lines, oe, og, represent imaginary lines, resting on those planes in directions perpendicular to their common edge, and the dots at i and h, some permanent marks in a line with the center, o.

i.

120

Fig. 6.

e h

"Let the circle be in such a position that the line, o e, would pass through the dot at h, if extended in that direction, as in fig. 6.

Fig. 7.

"If the circle now be turned round with its attached crystal, as in fig. 7, until the imaginary line, og, is brought into the position of the line, o e, in fig. 6, the number 120 will stand opposite the dot at i. This is the number of degrees at which the planes a and b incline to each other. For if the line og be extended in the direction oi, as in fig. 7, it is obvious that the lines, o e, oi, which are perpendicular to the common edge of the planes, a and b, would intercept exactly 120° of the circle.

"Hence an instrument constructed upon the principle of these diagrams is capable of giving with accuracy the mutual inclination of any two planes which reflect objects with sufficient distinctness, if the means can be found for placing

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them successively in the relative positions shown in the two preceding figures.

"This purpose is effected by causing an object, as the line at m, fig. 8, to be reflected successively from the two planes, a and b, at the same angle. It is well known that the images of objects are reflected from bright planes at the same angle as that at which their rays fall on those planes; and that

when the image of an object reflected from a horizontal plane is observed, it appears so much below the reflecting surface as the object itself is above.

"If, therefore, the planes a and b, fig. 8, are successively brought into such positions as will cause the reflection of the line at m, from each plane, to appear to coincide with another line at n, both planes will be successively placed in the relative positions of the corresponding planes in figs.

Fig. 8.

6 and 7. To bring the planes of any crystal successively into these relative positions, the following directions will be found useful.

"The instrument, as shown in the sketch, fig. 9, should be first placed on a pyramidal stand, and the stand on a small steady table, about six or ten or twelve feet from a flat window. The graduated circular plate should stand perpendicularly from the window, the pin x being horizontal, not in the direction of the axis, as it is usually figured, but with the slit end nearest to the eye.

"Place the crystal which is to be measured on the table, resting on one of the two planes whose inclination is required, and with the edge at which those planes meet, nearest and parallel to the window.

Fig. 9.

Fig. 10.

d

"Attach a portion of wax, about the size of d, to one side of a small brass plate, e, fig. 10; lay the plate on the table with the edge, f, parallel to the window, the side to which the wax is attached being uppermost, and press the end of the wax against the crystal until it adheres; then lift the plate with its attached crystal, and place it in the slit of the pin, x, with that side uppermost which rested on the table.

"Bring the eye now so near the crystal, as, without perceiving the crystal itself, to permit the images of objects reflected from its planes to be distinctly observed, and raise or lower that end of the pin, x, which has the small circular plate on it, until one of the horizontal upper bars of the window is seen reflected from the upper or first plane of the crystal, corresponding with the plane a, fig. 6, and until the image of the bar appears to touch some line below the window, as the edge of the skirting-board where it joins the floor. “Turn the pin, x, on its own axis also, if necessary, until the reflected image of the bar of the window coincides accurately with the observed line below the window.

"Turn now the small circular handle, a, on its axis, until the same bar of the window appears reflected from the second plane of the crystal corresponding with plane 6, figs. 6 and 7, and until it appears to touch the line

below; and having, in adjusting the first plane, turned the pin, x, on its axis, to bring the reflected image of the bar of the window to coincide accurately with the line below, now move the lower end of the pin laterally, either towards or from the instrument, in order to make the image of the same bar, reflected from the second plane, coincide with the same line below,

Having ascertained by repeatedly looking at, and adjusting both planes, that the image of the horizontal bar, reflected successively from each plane, coincides with the observed lower line, the crystal may be considered ready for measurement.

"Let the 180° on the graduated circle be now brought opposite the 0 of the vernier at c, by turning the handle, b; and while the circle is retained accurately in this position, bring the reflected image of the bar from the first plane, to coincide with the line below, by turning the small circular handle, a. Now turn the graduated circle, by means of the handle, b, until the image of the bar, reflected from the second plane, is also observed to coincide with the same line below. In this state of the instrument the vernier at c will indicate the degrees and minutes at which the two planes are inclined to each other.

"The accuracy of the measurements taken with this instrument will depend upon the precision with which the image of the bar, reflected successively from both planes, is made to appear to coincide with the same line below; and also upon the 0, or the 180°, on the graduated circle, being made to stand precisely even with the lower line of the vernier, when the first plane of the crystal is adjusted for measurement. A wire being placed horizontally between two upper bars of the window, and a black line of the same thickness being drawn parallel to it below the window, will contribute to the exactness of the measurement, by being used instead of the bar of the window and any other line.

"Persons beginning to use this instrument are recommended to apply it first to the measurement of fragments at least as large as that represented in fig. 10, and of some substance whose planes are bright. Crystals of carbonate of lime will supply good fragments for this purpose, if they are merely broken by a slight blow of a small hammer.

"For accurate measurement, however, the fragments ought not, when the planes are bright, to exceed the size of that shown in fig. 9, and they ought to be so placed on the instrument, that a line passing through its axis should also pass through the center of the small minute fragment which is to be measured. This position on the instrument ought also to be attended to when the fragments of crystal are large. In which case the common edge of the two planes, whose inclination is required, should be brought very nearly to coincide with the axis of the goniometer; and it is frequently useful to blacken the whole of the planes to be measured, except a narrow stripe on each close to the edge over which the measurement is to be taken."

INDEX.