in its usual acceptation, is the name of an instrument for measuring trees, of which the reader will find a description in the Encyclopedia Britannica. The same name has been lately given, by William Pitt, Esq; of Pendeford, near Wolverhampton, to an instrument proposed by him for measuring distances by one observation.

The idea of such an instrument is not new. It has been frequently discussed, both in conversation and upon paper; but has been generally treated by sound mathematicians with contempt, on the supposition of its being founded on false principles. Of all this our author is fully aware; but he, notwithstanding, strongly recommends it to the attention of the ingenious mathematical instrument-maker.

To determine distances by one observation, two methods may be proposed, founded on different principles; the one on the supposition of the observer being in the centre, and the object in the circumference, of a circle; the other, on the contrary supposition, of the observer being in the circumference, and the object in the centre.

To determine the distance of any object on the first supposition, the bulk or dimensions of such object must be known, either by measure or elimination, and the angle formed by lines drawn to its extremities being taken by an accurate instrument, the distance is easily calculated; and such calculations may be facilitated by tables or theorems adapted to that purpose. For this method our present instruments, with a nonius, and the whole very accurately divided, are sufficient; the only improvement wanting seems to be the application of a micrometer to such instruments, to enable the observer to read his angle with more minute accuracy, by ascertaining, not only the degrees and parts of a degree, but also the minutes and parts of a minute.

As in this method the bulk of inaccessible objects can only be estimated, the error in distance will be exactly in the proportion of the error in such estimation; little dependence can therefore be placed on distances thus ascertained. For the purposes of surveying, indeed, a staff of known length may be held by an assistant; and the angle from the eye of the observer to its two ends being measured by an accurate instrument, with a micrometer fitted to ascertain minutes and parts of a minute, distances may be thus determined with great accuracy; the application of a micrometer to the theodolite, if it could be depended upon, for thus determining the minute parts of a degree, in small angles, is very much a desideratum with the practical surveyor.

This method of measuring distances, though plain and simple enough, our author illustrates by an example: Suppose A, fig. 1. (See Plate XXI.) the place of the instrument; BC the assistant's staff, with a perpendicular pin at D, to enable the assistant to hold it in its right position; now, if the angle BAC could, by the help of a micrometer, be ascertained to parts of a minute, the distance from A to B, or to C, might be easily calculated by the rules of plane Trigonometry; for which see that article in the Encyclopedia.

But this method of ascertaining distances cannot be applied to inaccessible objects, and it is moreover subject to the inconvenience of an assistant being obliged to go to the object whose distance is required (an inconvenience almost equal to the trouble of actual measurement); therefore the perfection of the second method proposed, if attainable, is principally to be desired; namely, that of conceiving the observation made on the circumference of a circle, whose centre is in the object whose distance is to be ascertained; and, none of our instruments now in use being adapted to this mode of observation, a new construction of a mathematical instrument is therefore proposed, the name intended for which is the dendrometer.

Our author admits, that this name is not now used for the first time, though he thinks that the principle has never been applied in practice, for the familiar purpose of ascertaining terrestrial distances, in surveying, or otherwise, though the same principle has been to generally and successfully applied in determining the distance of the heavenly bodies by means of their parallax.

The following principles of construction are proposed, which may perhaps be otherwise varied and improved. O, fig. 2. the object of whose distance is required; ABCDE the instrument in plan; BC a telescope, placed exactly parallel to the side AE; CE an arch of a circle, whose centre is at A, accurately divided from E in degrees, &c.; AD an index, moveable on the centre A, with a nonius scale at the end D, graduated to apply to the divisions of the arch; also with a telescope, to enable the observer to discriminate the object, or any particular part or side thereof, more accurately. The whole should be mounted on three legs—in the manner of a plain table or theodolite, and furnished with spirit-tubes to adjust it to an horizontal position. The instrument being placed in such position, the telescope BC must be brought upon the object O, or rather upon some particular point or side thereof; when, being there fastened, the index AD must be moved till its telescope exactly strikes the same point of the object; then the divisions on the arch ED mark out the angle DAE, which will be exactly equal to the angle BOA, as is demonstrated in the XV. and XXIX. propositions of Euclid, Book I.; and the side BA, as well as the angles ABO, and BAO, being already known, the distance BO or AO may be easily determined.

As the perfection of this instrument depends altogether upon its accuracy in taking small angles, so that accuracy must depend, not only upon the instrument's being properly fitted with a micrometer, but also in some measure upon the length of the line BA in the figure. That line, therefore, might be extended, by the instrument being constructed to fold or slide out to a greater length when in use; upon which principle, connected with the application of a micrometer, an accurate and useful instrument might certainly be constructed. To adjust such instrument for use, let a staff be held up at a distance, in the manner of fig. 1. exactly equal in length to the distance of the two teleopes, and the index AD being brought exactly upon the side AE, if the two teleopes accurately strike either end of the staff, the instrument is properly adjusted.

The construction of a similar instrument, on the principles of Hadley's quadrant, for naval observations, would also doubtless be an acceptable object in navigation, by enabling the mariner to ascertain the distances of ships, capes, and other objects, at a single observation; and that, perhaps, with greater accuracy than can be done by any method now in use. For this purpose, the following construction is proposed: ABCDE, fig. 3; the instrument in plane; O the object whose distance is required; at A, at C, at E, and at 3, are to be fixed speculums, properly framed and fitted, that at 3 having only its lower part quicksilvered, the upper part being left transparent to view the object; the speculum at A being fixed obliquely, so that a line A 1, drawn perpendicular to its surface, may bisect the angle BAC in equal parts; that at C being perpendicular to the line C 2; those at E and 3 being perpendicular to the index E 3, and that at E being furnished with a sight; the arch DC to be divided from D in the manner of Hadley's quadrant; the movement of the index to be measured as before by a micrometer; and as the length of the line AE would tend to the perfection of the instrument, it may be constructed to fold up in the middle, on the line C 2, into less compass when not in use. The instrument may be adjusted for use by holding up a staff at a distance, as before proposed, whose length is exactly equal to the line AE.

To make an observation by this instrument, it being previously properly adjusted, the eye is to be applied at the sight in the speculum E, and the face turned towards the object; when the object being received on the speculum A, is reflected into that at C, and again into that at E, and that at 3 on the index; the index being then moved till the reflected object in the speculum at 3 exactly coincides with the real object in the transparent part of the glass, the divisions on the arch D 3, subdivided by the micrometer, will determine the angle DE 3 = the angle AOE; from which the distance O may be determined as before.

DENOMINATOR OF A RATIO is the quotient arising from the division of the antecedent by the consequent. Thus, 6 is the denominator of the ratio 30 to 5, because 30 divided by 5 gives 6. It is otherwise called the exponent of the ratio.

DEPRESSION OF A STAR, OR OF THE SUN, is its distance below the horizon; and is measured by an arc of a vertical circle, intercepted between the horizon and the place of the star.

Depression of the Visible Horizon, or Dip of the Horizon, denotes its sinking or dipping below the true horizontal plane, by the observer's eye being raised above the surface of the sea; in consequence of which, the observed altitude of an object is by too much too great.