If we conceive the two surfaces of a crystal to be cut by a plane perpendicular to their common section, the ap parent angle contained by the two lines which form the boundary of the section, when the eye is perpendicular to the section, is evidently the inclination of the planes. But if the cutting plane is not perpendicular to the common section, the apparent angles of the lines which form the boundaries of the section when viewed by an eye perpen dicular to it, is evidently greater or less than the real angle of the crystal, according to the position of the cutting plane. If the observer, however, places himself in such a manner, that the common section of the planes is paral lel to the axis of his eye, then the apparent angle formed by the bounding lines of the section, whatever be the posi tion of the cutting-plane, is the real angle of the crystal. By placing the crystal, therefore, in this position, in the focus of the goniometrical microscope, which shall be hereafter described, and measuring the apparent angle formed by the bounding lines, we obtain, by a very simple process, the inclination of the planes.
This will be understood from Fig. 7. in which ABCDEF is a crystal, ABC a section of it perpendicular to AD, and A b c an oblique section, Now, though BAC is the real angle of the crystal, yet, when the oblique section A b c is viewed by the observer at 0, its bounding lines A b, A c are apparently coincident with the lines AB, AC, whose inclination is the real angle of the planes ; and therefore, if we measure by a proper instrument, the apparent angle contained by the oblique lines A b, A c, we obtain a measure of the real angle of the crystal.
'fhe angles of the crystal may also be advantageously deduced, from the plane angles by which any of the solid angles is contained. 'fhe plane angles are first measured with great accuracy by the goniometrical microscope, or the angular micrometer adapted to a microscope, and the inclination of the planes is deduced from a trigonometrical formula. Whatever be the number of plane angles which contain the solid angle, we can always reduce the solid angle to one which is formed by three plane angles, and determine by the formula the inclination of any two of them. Thus, if the solid angle at A, Fig. 8. is contained by five plane angles, and if it is required to find the inclina tion of the planes ABC, ACD, we first measure the plane angles CAB, CAD, and also the angle contained by the lines AB, AD; so that we have now reduced the solid an gle contained by five plane angles, into one contained by three plane angles, CAB, CAD, BAD.
Legendre, in his Elements of Geometry, has given a very elegant solution of this problem by a plain construc tion ; and it is easy, from his solution, to form an instru ment for sliming the angles of the planes without the trou ble of calculation. Thus let the angles BAC, CAD, DA E, Fig. 9. be made equal to the three plane angles by which the solid angle is contained. Make AB equal to AE, and from the points B, E, let fall the perpendiculars BC, ED on the lines AC, AD, and let them meet at O. From the point C as a centre, with the radius CB, describe the semi circle BFG. From the point 0 draw OF at right angles to CO, and from F, where it meets the semicircle, draw FC. The angle GCF is the inclination of the two planes, CAD, CAB. In order to construct an instrument on this principle, to save the trouble of projection or calculation, we have only to form a graduated circle BHEG, with three moveable radii, AC, AD, AE, and a fixed radius AB. The moveable radii must have vernier scales at their extremi ties, that they may be set so as to contain the three plane angles which form the solid angle. Two moveable arms BG, EO, the former of which is divided into any number of equal parts, turn round the extremities B, E : and, by means of a reflecting mirror on their exterior sides, they can be set in such a position as to be perpendicular to the radii AC, All. When this is clone, the number of equal parts between C and 0, divided by the number between B and C, is the natural co-sine of the angle GCF ; and therefore, by entering a table of sines with this number, the inclination of the two planes will be found.
In order to obtain a more accurate result, however, we must have recourse to a trigonometrical formula. Let A, Fig. 10, be the solid angle, and let it be required to deter mine, by means of the three plane angles, the inclination of the surfaces ACB, ACD. Draw AM, AN in the planes ACB, ACD, and perpendicular to the common section AC ; join BM, DN. Then it is obvious, that the angle MAN is the inclination of the planes required, and that the angle BAD, which is an oblique section of the prism BM, will he equal to MAN when it is reduced to the plane AMNT. By considering that the inclinations of the bound ing lines of the oblique section of the prism, to the bound ing lines of the perpendicular section, are measured by the angles DAN, BAM, the complements of the two given plane angles CAD, CAB, we shall obtain, by spherical trigonometry, the following formula : Hence 2 —= and 9 = 64° 32' 36" the angle of the surfaces of the crystal.