(from δένδρον a tree, and μέτρον I measure), an instrument invented by Messrs. Duncombe and Whittel, for which they obtained a patent; and so called from its use in measuring trees. This instrument consists of a semicircle A (fig. 1.), divided into two quadrants, and graduated from the middle: upon the diameter B there hangs a plummet L for fixing the instrument in a vertical position: there is also a chord D parallel to the diameter, and a radius E, passing at right angles through the diameter and chord. From a point on the radius hangs an altimeter C, between the chord and diameter, to which is fixed a small semicircle G, and a screw, to confine it in any position. The altimeter, which is contrived to form the same angle with the radius of the instrument as the tree forms with the horizon, is divided from its centre both ways into forty equal parts: and these parts are again subdivided into halves and quarters. Upon the small semicircle G, on which is accounted the quantity of the angle made by the altimeter and radius, are expressed degrees from 60 to 120, being 30 on each quadrant. The radius is numbered with the same scale of divisions as the altimeter. There is also a nonius to the small semicircle, which shows the quantity of an angle to every five minutes. On the back of the instrument the stock M of the sliding piece is confined to the axis N, which moves concentrically parallel to the elevation index F on the opposite side, to which it is fixed. This index is numbered by a scale of equal divisions with the altimeter and radius: at the end of the index is a nonius, by which the angles of elevation above, or of depression below, the horizon, measured upon the semicircle of the instrument, are determined to every five minutes. There is also a groove in the radius, that slides across the axis by means of a screw I, working between the chord and semicircle of the instrument; and this screw is turned by the key O.
Upon the stock M (fig. 2.) is a sliding piece P, that always acts at right angles with the altimeter, by means of a groove in the latter. To the shank of the sliding piece is affixed a moveable limb Q, which forms the same angle with the altimeter as the bough forms with the body or trunk of the tree. This limb may be of any convenient length, divided into equal parts of the same scale with all the foregoing divisions. At the extremity of the fixed axis, on a centre, an index R, with telescope sights, works horizontally upon the moveable limb of the sliding piece. Upon this horizontal index R may be fixed a small quadrant T, described with any convenient radius from the centre on which the index moves, and divided into 90 degrees, beginning at a right line drawn from the centre at right angles with the fiducial edge of the said index; and upon the extremity of the axis is a nonius, whereby to determine the quantity of an angle upon the quadrant every five minutes. There are also two small circular arches S, S, serving to keep the sights in a parallel position, each containing an equal number of degrees. Upon these arches is measured the angle, subtending a side equal to the difference of the altitudes of the observed objects above the plane of the horizon, and whose base is the nearest distance between the perpendiculars in which these objects are situated. The dendrometer is fitted to a theodolite, and may be used either with or without it as occasion requires.
The principal use of this instrument is for measuring the length and diameter of any tree, perpendicular or oblique to an horizontal plane, or in any situation of the plane on which it rests, or of any figure, whether regular or irregular, and also the length and diameter of the boughs, by mere inspection; and the inventors of it have calculated tables, annexed to their account of the instrument itself, by the help of which the quantity of timber in a tree is obtained without calculation, or the use of the sliding rule. The instrument is rectified by fitting it in a perpendicular position, by means of the plummet, and screwing it to the staff; then the altimeter is placed in the exact position of the tree, whether perpendicular, reclining, or inclining, and forewed fall. If the tree stands on level ground, the horizontal distance from the tree to the axis of the instrument is measured with a tape-line, and the radius is moved with the key till that distance be cut upon it by the infide of the diameter; but if the ground be slanting, the distance from the tree to the instrument is measured, and the elevation index is moved till the point of the tree from which the distance was measured is seen through the sights, and there forewed fall; and the radius is moved backwards or forwards with the key, till this distance is cut upon the elevation index by the perpendicular line of the altimeter; and the horizontal line will be marked upon the radius by the infide of the diameter. In order to obtain the length of the tree, the elevation index is first moved downwards, till the bottom of the tree cut by the horizontal wires is observed through the sights, and the feet and inches marked by the index upon the altimeter below the point of sight or horizontal line are noted down: then the index is moved upwards till the part to which you would measure, cut by the horizontal wires, is seen, and the feet and inches marked on the altimeter above the point of sight are noted: these two quantities added together give the exact length of the tree, which is inserted in a field-book. For the girth of the tree, the circumference in that part where the horizontal distance was taken, is measured with the tape line; and a sixth part of this circumference is added to the distance on the radius, which was before cut by the inside of the diameter, because the tape-line, in taking the distance, cannot be applied to the centre of the body of the tree; then the elevation index is lowered to that part of the tree, of which the diameter is to be taken, and screwed fast. Set the moveable limb of the sliding piece quite straight, and the edge of the horizontal index upon the first division of it. Turn the whole instrument about to the left hand till you see through the sights the left side of the tree cut exactly by the perpendicular wires; then the instrument being fixed, move the sights only upon the sliding piece, till you see the right side of the tree cut also by the perpendicular wires; and you will find the true diameter marked by the horizontal index upon the sliding piece, which is to be entered in a distinct column of the field-book.
For the boughs: let the distance on the radius be now reduced to its former quantity, and the elevation index moved upwards till the bough is seen through the sights, and screwed fast. Set the moveable part of the sliding piece in a position parallel to the bough, and the edge of the horizontal index on the first division of it. Turn the whole instrument about till you see through the sights the shoot of the bough close to the trunk cut by the perpendicular wires; then move the sights till you see the other end of the bough cut by the said wires, and note the feet and inches marked by the horizontal index on the moveable limb of the sliding piece, which will give the true length of the bough to be inserted in the field-book. And the girth of the bough may be obtained by directing the sights to that part of it whose girth is desired; then by moving the elevation index downwards till you see the under side of the bough cut by the horizontal wires, and there noting the feet and inches marked by the said index on the altimeter; after which, let the elevation index be moved upwards, till the upper side of the bough cut by the horizontal wires is seen; the feet and inches marked upon the altimeter are to be noted as before. The former quantity subtracted from the latter will give the true diameter of the bough, which is entered in the field-book. The true solidity both of the body of the tree and of the boughs may be found from the diameter and lengths in tables calculated for this purpose.
The dendrometer, fitted to a theodolite, may be applied to measuring the heights and distances of objects, accessible or inaccessible, whether situated in planes parallel or oblique to the plane in which the instrument is placed. It may be also used for taking all angles, whether vertical, horizontal, or oblique, in any position of the planes in which they are formed; and thus for facilitating the practical operations of engineering, land-surveying, levelling, mining, &c., and for performing the various cases of plane trigonometry without calculation; of which the inventors have subjoined to their account of this instrument many examples.
VOL. VII. Part I.
an instrument for measuring distances by a single observation, which has been proposed by Mr Fitt of Pendeford, near Wolverhampton, and of which the following is the description in the words of the author.
"The idea of an instrument to measure distances by a single observation, has sometimes been discussed, both in conversation and upon paper; and, though the subject has generally been treated with neglect, and even with a kind of contempt, by found mathematicians, upon an idea of its extravagance and eccentricity, or upon a supposition of its being founded upon false principles, yet I cannot but strongly recommend it to the attention of the ingenious mathematical instrument maker, as an article perhaps capable of being brought to a higher degree of perfection than has generally been supposed.
"The method of determining distances by two observations, from either end of a base line, is well known to every one in the least degree conversant with plane trigonometry: that of determining such distances by one observation has been less explained and understood; and to this I wish to call the attention of the ingenious, whose local circumstances of situation may enable them to investigate and improve the subject.
"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, of the observer being in the centre, the bulk or dimensions of such object must be known, either by measure or estimation, 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 instrument, 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 attendant; 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, I shall just beg leave to illustrate by an example; suppose A, (fig. 3.) the place of the instrument; Dendrome 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, may be, with little trouble, calculated as follows.
"Suppose the length of the staff BC be 100 inches, or other parts; divide the number 343,500 by the minutes contained in the angle A, the quotient will be the distance AB, or AC, in the same parts.
"The number 343,500 becomes the dividend in this case, because the arch of a circle subtending an angle of 3435 minutes, or 57° 15', is equal in length to the radius, and the object staff BC is supposed divided into 100 equal parts.
"Thus, suppose the angle A be 1°, or 60', then, 60)343300(=5725 inches = distance AB.
"Or, if the angle A be 60'×15, then 60×1)343500(=5715.5 inches.
"Hence it appears, that an error of 1/15 of a minute in the angle A, would cause an error of 9 inches and a half in the distance AB, or about 1/80 part of the whole; the accuracy, therefore, of thus taking distances, depends upon the accuracy wherewith angles can be ascertained; and the error in distance will bear the same proportion to the actual distance, as the error in taking the angle does to the actual angle.
"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.
"This name is not now used for the first time, it was applied in the same way by a gentleman who had, as I have been informed, turned his thoughts to this particular subject; but I do not find that he ever brought his instrument into use, or explained its principles; nor do I understand that this principle has ever been applied in practice, for the familiar purpose of ascertaining terrestrial distances in surveying, or otherwise; though the same principle has been so 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. 4. the object whose distance is required; ABCDE the instrument in plano; 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, the 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 15th and 29th propositions of Euclid, Book I.; and the side BA being already known, the distance BO, or AO, may be easily determined in two different ways; viz. first, by supposing the triangle BOA an isosceles triangle; then multiply the side BA by 3435, as before, and divide the product by the minutes contained in the angle DAE=the angle BOA; the quotient will be the distance BO=AO, very nearly; or, secondly, by supposing the triangle ABO right-angled at B, then, as the sine of the angle found DAE=BOA is to the side known BA, so is the radius to the side AO, or so is the fine of the angle BAO to the side BO. To illustrate this by an example, suppose the side BA=1 yard, the angle found DAE=BOA=9° 15', then, per first method, 15)3435(=229 yards = the distance BO, or AO. Or, by the second method,
As the fine of the angle found 9° 15'= 7.6398160 Is to the side BA=1 yard = 0.0000000 So is radius 90° 0'= 10.0000000
To the log. of the side AO=229 yards = 2.3601840
Or,
As the fine of the angle found 9° 15'= 7.6398160 Is to the side BA=1 yard = 0.0000000 So is the fine of the angle BAO=89° 45'=9.9999959
To the log. of the side BO = 229 yards = 2.3601799
"As the perfection of this instrument depends totally upon its accuracy in taking small angles, which accuracy must depend, for its minute divisions, upon its being fitted with a micrometer; and as the writer of this cannot doubt that the particular mode of doing this must be familiar to the intelligent instrument-maker, he cannot but strongly recommend it to the attention of the ingenious of that profession, as an object which, when perfected, would be a real and considerable improvement in their art, and an useful instrument to the practical surveyor. Its accuracy would also, in force measure, depend upon the length of the line BA in the figure; that line might therefore 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 telescopes, and the index AD being brought exactly upon the side AE, if the two telescopes accurately strike either end of the staff, the instrument is properly adjusted." CYDER PRESS. PLATE CLXVIII.
Fig. 1. DENDROMETER. Fig. 3. A C D B C Fig. 4. Fig. 5. DRAW BRIDGE. Q P N M H E L D S A B E. Mitchell fecit. "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. 5. the instrument in plano; 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 in the middle, on the line C 2, into leaf 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 toward 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.
"It is very probable that this arrangement may be improved, by those who are familiar with the best construction of Hadley's quadrant; which the writer of this professes himself not to be, farther than its general principle. He has not the least doubt that useful practical instruments may be constructed on the principles here described; and, upon this idea, cannot but recommend the subject to the attention of those concerned in the manufacture of similar instruments." Repertory of Arts, vol. i.