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 b, 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 stride on each close to the edge over which the measurement is to be taken."

APPENDIX.

SINCE this Treatise passed from under the hand of its author, some improvements upon the construction and methods of use of old instruments have been made, and various new instruments and methods have been devised. It is proposed in this Appendix (1864) to notice, with brevity, some of the practically more important of these, rather in the way of an index, whereby those who are interested, or engaged in the actual use of mathematical instruments, may be enabled to make further inquiry for themselves, as to those improvements that time has brought about, and to adopt such as they may each deem advantageous.

Theodolites.-Transit theodolites, in which the telescope can be moved round the entire circle in the vertical plane, have to a great extent superseded, and justly, the older forms having a limited vertical arc. They are more expensive, larger, and heavier than the older form, but possess much greater capability and accuracy.

As commonly made, the aperture of the telescope is too small to afford sufficient light, either for very distant stations, or for working very early or very late. The enlargement of the telescope would be a great improvement, as would also be the application of a Newtonian eye-piece to the instrument, by which observation at high angles with the telescope would be rendered much more convenient.

The transit theodolite upon the usual London pattern is by no means free from certain objections, as to the loss of adjustment and accuracy due to wear of the vertical and horizontal centres, and to flexure of the supports of the telescope. These evils have been attempted to be got rid of, and at the same time some subordinate advantages proposed being attained, by different makers. Amongst these partially novel forms, that of Pastorelli, London, in which the horizontal axis of the telescope passes through a single ponderous bent arm springing from the horizontal plates; and that of Yeates, of Dublin, in which it is fixed between the jaws of a U-formed support

of brass, cast in one piece, and with central bearing for the motion in azimuth only are the most noteworthy. The latter we ourselves prefer.

An entirely new form of theodolite has been designed by M. D'Abbadie, and is made in Paris. The telescope is fixed with its line of collimation parallel to the plane of the horizontal circles, which are provided with crossed spirit-levels, and divided in the usual way for the reading of horizontal angles by the aid of a second telescope fixed to the lower plate. Vertical angles are read off by turning the telescope upon its own line of collimation as an axis. The telescope is provided with a vertical circle, &c., whose plane is fixed truly at right angles to the line of collimation. To measure the vertical angle subtended by two objects-one, suppose the lower one, is observed as seen reflected in a totally reflecting prism in the telescope tube, whose plane of reflection is parallel to the line of collima

tion, and the vertical circle is then brought to zero and clamped to the telescope. The latter, with the circle, are now moved or twisted round on the line of collimation as an axis, until the higher of the two objects is seen reflected in the same prism, the precise position being fixed by cross wires in the field, &c., when the angle between the two is at once read off upon the vertical circle, whose centre is in the line of collimation. This instrument is very original in conception, and is said to work well when with perfect accuracy. It presents the advantages of not being easily thrown out of adjustment by carriage or rough usage, is light even when strongly made, and not as costly as the older forms.

Levelling Instruments.-A very excellent levelling telescope for the pocket, to be used without a stand, and intended for ascertaining approximate levels, equal to that of the observer's eye, is made by Elliott, London. The bubble of the spirit-level is by reflection from a diagonal prism seen in the field of the telescope.

A very beautiful little hand instrument, applicable to the same purpose, and also to fixing three points in a right line, or offsets at right angles to it, has been produced by Mr. Adie.

Sextants.-Both nautical and box sextants have been greatly improved in some of the details, and in accuracy, of late years. The patent protracting sextant, which was some years since invented by the late Sir Howard Douglas, and of which a considerable number were constructed by Troughton and Simms, is an extremely convenient little instrument for military sketch survey, or for any reconnaissance surveying. It admits of the observed angle being at once protracted or ruled off upon the paper, without altering the instrument, and laid down, if requisite, to scale, which is transferred from one divided edge of the instrument. This form of pocket sextant is very little known; and, probably from that reason only, has scarcely at all come into use.

German Instruments.-A very cheap form of instrument is much employed in France and Germany for measuring both horizontal and vertical angles, consisting of two cylinders or octagonal hollow prisms of brass, the axes of which coincide. One of the prisms rotates on its axis, the other being relatively fixed, and the conter minous cylindric edges on the exterior of the two cylinders are divided. Both cylinders have four long narrow slits with vertical wires at right angles to each other, through which, without any telescope, objects may be observed. The whole instrument is adapted to a ball and socket joint, at the axis of the relatively fixed cylinder, so that the coincident axes can be placed either vertically or horizontally, or at any intermediate angle. Some of these are provided with two spirit-levels. Instruments for surveying and for levelling, of the character of theodolites and levels, are made in Germany, of some ingenuity as to simplicity, and at very low prices; but they are worthy of little dependence, and in any but the most careful hands, soon go out of order, from want of substantiality. A great number of forms of these were shown in the foreign departments at the Exhibition of 1862.

The legs or other woodwork of German instruments are very com