and being much lighter is more convenient for measurements made at any height from the ground; it is also useful for long offsets; but its liability to twist and kink renders it easily broken, while few persons can anywhere be found to repair it. With care one may be safely used for a long time, but the surveyor should not allow it to leave experienced hands even for a moment. Woven tapes strengthened with cords of catgut or wire are in common use for the same purposes, but are far less accurate, and are unsuited for measuring more than two tapes-length in one continuous distance without appreciable error. Common measuring tapes are altogether untrustworthy; they stretch to an extent visible to the naked eye, and shrink after wetting. 4. Wheel-pedometers record on a dial-face the number of revolutions of, or distance passed over by, a wheel rolling on the surface of the ground; on very smooth and even ground the results are moderately fair, the error being principally due to slipping. 5. Pacing and passometers, Simple pacing is, next to the guessing of a bad estimator, the worst mode of obtaining distances; the inequality of the paces under various conditions combined with the errors in counting render it most inaccurate, while the monotony of perpetual counting makes it a positive torture. A passometer can be used to register the number of paces taken; the paces themselves can be best taken by some one that has been trained to take equal paces, or has been under military training. The pacing adopted should be even, but natural, and not strained with the mistaken object of conforming to any arbitrary length; some paced distance, on the flat, uphill and downhill, should be tested by careful measurement to obtain the ratio for reduction to yards, feet, or miles, under these three conditions of any future pacings; the ratio should again be frequently checked. The passometers that effect the reduction to miles are inconvenient for distances less than a mile, and do not yield very accurate results in any case. Guessing or judging distance may, with continued practice and checking, be carried to an accuracy that is surprising; the main points to be noticed are the rise or fall of the ground and the direction from which the light falls on any object at the distant point. Some persons estimate in yards, others in their own paces, which are more readily available for testing such guesses. 6. Telemetrical observation on graduated staves. A graduated staff is held vertically at the required distant point, and its refracted image is observed in the eye-lens of the telescope, where it appears cut by two horizontal threads or wires; the height of the intercepted portion determines the horizontal distance of the staff from the vertical axis of the instrument. The practical development of this principle was chiefly due to Porro, a Piedmontese military engineer. Telemeters have been used in the United States Coast Survey, in a topographical survey in Schuykill county from 1862 to 1865, and in a geological and mining survey at Cape Breton from 1863 to 1866; the error given by them in a single sight of 660 feet being determined to be only four inches, or 26th, this mode of determining distance compares favourably with chain measurement under ordinary conditions. where the error is aboutth. The theory and principles of direct telemetric observation of this nature are best explained together with the history 000 of its development. The following account of Porro's improvements has hence been extracted from an article by De Sénarmont in the Annales des Mines,' 4th series, vol. xvi. page 383. To investigate the principles involved, First when the telescope and its line of sight is truly horizontal, S d FIGURE 1. DIRECT TELEMETRY. Lets be the intercepted height on the staff, i a x the height of its image, the distance of the staff from the objectglass, the distance of the image from the object d the distance of the axis of the instrument in rear of the object lens. Then by proportion=, and by the general for a a mula for the foci of lenses, 2+1=3, hence a=ƒ{+f; x and if A be the distance of the staff from the axis of the instrument A=a+d=f&+f+d in which f+d is a constant for each instrument, varying from about 12 to 24 inches. Secondly, when the line of sight is inclined at a slight angle to the horizon, the required distance becomes A'=(a+f+d) cos p =ƒ{ cos2 $+(ƒ÷d) cos & =ƒ1⁄2 cos2 +ƒ+d very nearly, as & is small. Thirdly, when the graduated staff is not truly ver tical, Let s be the intercepted height when truly vertical, then s's cos 0, and the previous formula becomes A"=ƒ—, cos2 & cos 0+fƒ+d, hence the resulting error in the distance, due to the inclination of the staff to the vertical, becomes equal to Now, if we take roughly in terms of the circular mea hence 2° may be considered as the extreme limit for the deviation from verticality, which shows the necessity of invariably using the plummet with the staff. Porro's modifications of the simple telescope with a two-wire diaphragm were intended to obtain an increase of field and of power in the telescope, and to make the resulting distances count exactly from the vertical axis of the instrument. In the first place, instead of using a single large object-lens, which would require a focal length of twelve times its diameter to eliminate excessive spherical aberration, he used a compound lens of two separate achromatic object-glasses placed one behind the other, C FIGURE 2. PORRO'S LENSES. thus getting a bright clear object. Secondly, in order to avoid the spherical aberration that might result from observing with one eye-lens only when the threads were wide apart in the field of view, he used three eye-lenses, each placed opposite to a corresponding thread or wire, of which he used three; and besides, placed them farther apart without involving error. In the third place, he introduced between the object-glass and the threads an additional lens whose focus was at the same point as that of the object-glass, thus obtaining as a result that all rays after passing through the additional lens become parallel, while the sizes of all objects subtending the same angle at the point |