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following year at the station on Leith Hill, near Dorking, rendering the station visible at the distance of 45 miles, though the hill itself was never once seen. The utility of thus employing the sun's reflected rays being established by these results, an instrument was invented by Captain Drummond, Royal Engineers, in lieu of the former temporary expedients, for directing the rays upon the station to be illuminated, the description of which will be found in his Paper on the means of facilitating the observations of distant stations, published in the "Philosophical Transactions for 1826," and from whence the above remarks have been taken. In using this "Heliostat" it is only necessary for the assistant, who is posted as near as possible to the station, to keep the enlightened object in the focus of the telescope, and the mirror is adjusted instrumentally so as to always reflect them upon the station and keep it illuminated. But a contrivance was still wanting to produce a light sufficiently brilliant to answer for distant stations at night. Bengal lights had been used by General Roy, which were succeeded by argand lamps and parabolic reflectors, and these again, by a large planoconvex lens, prepared by MM. Fresnel and Arago, and used by the latter gentleman conjointly with General Colby and Captain Kater, and by the light of which a station, distant 48 miles, was observed. The light invented by Captain Drummond, and described in the volume of the "Philosophical Transactions" alluded to, however, far surpassed all previous contrivances in intensity. A ball of lime, about a quarter of an inch in diameter, placed in the focus of a parabolic reflector, and raised to an intense heat by a stream of oxygen gas directed through a flame of alcohol, produced a light eighty times as intense as that given by an argand burner. A station on the hill in the barony of Ennishowen, of great importance, could not be seen from Devis Mountain, near Belfast, and this instrument was consequently sent there by General Colby; and, in spite of boisterous and hazy weather, the light was brilliantly visible at the distance of 67 miles, and would have been so at a much greater distance. Drummond's light might be also made available in determining the difference of longitudes by signals, which will be explained hereafter *; but difficulties connected

* It is also eminently calculated for those lighthouses where powerful illumination is required. In the "Philosophical Transactions" for 1830 is a paper of Captain Drum

with its management, as well as the cost of the apparatus, have prevented its being brought into use on the Ordnance Survey.

It has been already stated that the sides of the principal triangles should increase as rapidly as possible from the measured base. The accompanying sketch will show how this is to be managed without admitting any ill-conditioned triangles.





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AB is supposed to be the measured base of 3 miles, or any other length, and C and D the nearest trigonometrical points. All the angles being observed, the distances of C and D from the extremities of the base are calculated with the greatest accuracy. In each of the triangles DA C and D BC, then, we have the two sides and the contained angles to find D C, one calculation acting mond's on this subject, containing the results of a course of experiments carried on by order of the Trinity Board. The lime in these experiments was exposed to streams of oxygen and hydrogen gas from separate gasometers, instead of passing the oxygen gas through a flame of alcohol, which was done on the survey for the convenience of carriage, though at an increased expense.

as a check upon the correctness of the other. This line, DC, is again made the base from which the distances of the trigonometrical stations E and F are computed from D and C; and the length of EF is afterwards obtained in the two triangles DEF and FEC. In like manner the relative positions of the points H, G, K, &c., are obtained, and this system should be pursued till the trigonometrical stations arrive at the required distance apart.

On the Ordnance Survey, both of England and Ireland, the largest sized instruments, 3 feet in diameter, were used for fixing the principal stations*. The angles at the vertices of the secondary triangles were observed with the second-class theodolites. The sides of these triangles were, on an average, about 10 or 12 miles long, and the intervals between them were divided into small triangles, with sides of from 1 to 3 miles in length; a smaller theodolite, of 7 inches diameter, being used for measuring the angles. All

points of the secondary or-
der of triangles, which were
fixed upon during the pro-
gress of the principal trian-
gulation, were observed with
the largest instrument; and
a number of the minor sta-
tions, mills, churches, &c., A
were observed with the se-
cond-class theodolites from
different stations: thus the
connexion between the three
classes of triangles was esta-
blished, and the positions of
many of the minor stations
which had been determined
by calculation from a series

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* The large class of theodolites used upon an accurate triangulation require some protection from the weather. Light portable frame-work erections, covered with canvas, or boarding, are used on the Ordnance Survey.-See the article "Observatory Portable" in the Aide Mémoire.

of small triangles were checked by being made the vertices of larger triangles, based upon sides of those of the second order.

Thus the point E in the figure is determined from the base BC; and O from both DC and AD, forming a connection between the larger and smaller order of triangles, and constituting a series of checks upon the latter.

The length of the sides of the smallest triangles must depend upon the intended method of filling up the interior. If the contents within the boundaries of parishes, estates, &c., are to be calculated, the distances between these points must be diminished to one or two miles for an inclosed country, and two or three, perhaps, for one more open. If no contents are required, and the object of the triangulation is solely to ensure the accuracy of a topographical survey, the distances may be augmented according to the degree of minutiæ required, and the scale upon which the work is to be laid down.

The direction of one of the sides of the principal triangles must also be determined with regard to the meridian. The methods of ascertaining this angle, termed its azimuth, will be described hereafter.

It is also advisable not merely to measure the angles between the different trigonometrical points, but to observe them all with reference to certain stations previously fixed upon for that purpose.

If for any cause it has been found advisable to commence the triangulation before the base has been measured, the sides of the triangles may be calculated from an assumed base, and corrected afterwards for the difference between this imaginary quantity and the real length of the base line; or, if the length of the base is subsequently found to have been incorrectly ascertained, the triangulation may be corrected in a similar manner. Thus, suppose CB the assumed, and AB the real length of the base—also EB and AE the real distance to the trigonometrical point E, and DB and DC those calculated from the as- A sumed base, then A E evidently = CD. CB On the Continent, the instrument that has been generally used for measuring the angles of the principal and secondary triangles






and EB BD. AB.



is Borda's repeating circle *; but the theodolite is universally preferred in England, and those of the larger description, in their present improved state, are in fact portable Altitude and Azimuth instruments. The theodolite possesses the great advantage of reducing, instrumentally, the angles taken between objects situated in a plane oblique to the horizon to their horizontal values, which reduction, in any instrument measuring the exact angular distance between two objects having different zenith distances, is a matter of calculation depending upon the zenith distances or co-altitudes of the objects observed †. The formula given by Dr. Pearson for this correction when the obliquity is inconsiderable, which must always be the case in angles observed between distant objects on the horizon, is as follows:

A being the angle of position observed, H and h the altitudes of

* For a detailed account of this instrument, which is so seldom met with in England, see pages 89 to 99, "Géodesie, par Francœur;" also page 142, vol. i. “Puissant, Géodesie." There is also a very able paper upon the nature of the repeating circle by Mr. Troughton in the first volume of the Memoirs of the Astronomical Society.

The portability of this instrument is one of its great recommendations; but it seems to be always liable to some constant error, which cannot be removed by any number of repetitions, and the causes of which are still unknown. With all the skill of the most careful and scientific observers, the repeating circle has never been found to give the accurate results expected from it, though in theory the principle of repetition appears calculated to prevent almost the possibility of error.

This will be evident from the figure below, taken from page 220 of Woodhouse's Trigonometry.

Let O be the station of the observer, A and

B the two objects whose altitudes above the horizon are not equal; then the angle subtended by them at O is AOB measured by AB; but if Za, Zb, are each = 90°, then a b, and not AB, measures the angle a Z b, which is the horizontal angle required. The difference, then, between the observed angle AO B and a Zb, is the correction to be applied as the reduction to the horizon. The horizontal distances between these stations of different elevations may be found from having the reciprocal angles of elevation and depres

sion, and the measured or calculated distances,


which being considered as the hypothenuse of the triangle, the distances sought are the bases. From these the horizontal angles may be calculated if required.

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