DENDROMETER, an instrument for measuring distances by a single observation, which has been proposed by Mr Pitt 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 sound 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 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, 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 , 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 , or , then, inches = distance AB.
"Or, if the angle A be , then inches.
"Hence it appears, that an error of of a minute in the angle A, would cause an error of 9 inches and a half in the distance AB, or about 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 admeasurement), 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 theo-
dolite, 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 sine 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 = , then, per first method, yards = the distance BO, or AO. Or, by the second method,
To the log. of the side AO = 229 yards = 2.3601840
Or,
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 some 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.
Irometer
Denb. "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.
3. 5. "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 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 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.