Here it is necessary to mention, that the line BC or AC could not be accurately measured, the former running along a ditch full of water, and the latter through one or two thick bushes. In this case we placed the instrument at D, from which we had a distinct view of the three stations, A, B, C; and having taken the angle DBC, and measured the sides BD, DC, the angles DBC, DCB, became known. We next measured the angle ADC, and the sides AD, DC, from which the angle ACD, and the side AC, were found. Lastly, we observed the angle ADB, which, with the sides AD, DB, already measured, enabled us to find the angles DBA, DAB. Hence, by adding the angles DCB, DCA, we found the angle BCA; also, by adding the angles DAC, DAB, the angle BAC was discovered. These angles mght have been otherwise discovered, but the method here described was considered the best at the time, under all the circumstances of the case. As this problem of reducing angles to the centre of the station is one of frequent occurrence in delicate surveys, it may not be employing the young engineer's time unnecessarily to dwell on the subject at some length. In large surveys, it will often be found more convenient to select the more notable spires, towers, or other conspicuous objects, interspersed over the face of the country, than to plant moveable signals at each point of observation. An instance of this nature occurred in the case last described, in which it was impossible to place the instrument immediately under the points taken as stations. In such cases, the instrument is placed as near the station as is considered judicious, and the error occasioned by the displacement is calculated in the following manner, which perhaps is preferable to the method employed above. If it were required to find the angle ABC, subtended by two objects, A and C, which are the tops of two spires. Before we commence the discussion of this subject, we beg to state that the observer may be considered in three E A C Ο D different positions with respect to the centre, viz., "he is either in a line with the centre and one of the objects, or a line drawn from the centre through his situation would, if produced, pass between the objects; or a line drawn from the centre to the place of the observer, when produced, would pass without the objects." In the first place, let us suppose the observer at D, in a line between the objects B and C; B being the centre of the station, and the angle required being ABC. Then, it is evident that the angle ABC= > CDA DAB. In the second place, let us suppose the observer at O, and B, as before, the centre of the station; and ABC the angle required. In this case it is evident that >AOC+>OAC+ OCA = > ABC + > OAB + > OAC+ > OCA + OCB, each of the sums being equal to two right angles: consequently the > AOC=>ABC+>OAB +>OCB. From this it appears that the angle AOC is greater than the angle ABC by the sum of the B angles OAB and OCB: therefore > ABC=>AOC -(OAB+>OCB.) If we suppose the centre of the station to be at O, and the observer at B, then the angles OAB and OCB must be added to the angle ABC to obtain the angle AOC. In the third place, if we suppose the observer at E, and A to be the centre of the station, and CAB the angle required. Then, EBO+OBC CEB+>ECA+>ACO+>OCB+> =>CAB+>ACO+>OCB+> ABE+EBO+>OBC, each of these sums being equal to two right angles. Consequently the angles CAB and ABE, together, are equal to the sum of the angles CEB and ECA. Therefore > CAB=>CEB +>ECA-> ABE The foregoing observations are illustrated by the following examples, taken from Keith's Trigonometry: Let A and B represent the vanes on two steeples; E the situation of the theodolite upon the steeple A, and O its situation upon the steeple B. Then, suppose AE= 12 feet > CEB= 74° 32′ > CEA=139° 39′ BO-105 feet > AOC= 49° 27′ > COB=137° 55′ It is required to find the angles CAB and ABC, the distance AB being 5000 feet. SOLUTION. The > CEB=>CAB nearly, and AOC=> ABC nearly; with these angles and AB, find AC and BC=4581.8 and 5811.6. Then AC: sin. CEA :: AE : sin. > ECA=5′ 50′′ AB: sin. BEA :: AE : sin. > ABE=7' 29" Hence > CAB=74°30′ 21′′. BC: sin. > COB :: BO : sin. > OCB=4' 10" Hence ABC=49° 21′ 54′′. With the corrected angles CAB, ABC, and the distance AB, the sides AC and BC may be determined. In the prosecution of this survey, every day's work was calculated in the evening, and laid down, as in Plate 2. When time permitted, the area of each triangle was also found, and written in each triangle respectively. In finding the area of the triangles, after having multiplied two sides and the natural sine of their contained angle together, it is not necessary to divide by 2, as the one division of the sum of all the products by 2 will answer, by which the repeated division by 2 will be avoided. The area of each triangle may be given in links, and the aggregate of all the triangles reduced to acres, roods, and perches; by which the time taken to reduce each triangle to these denominations may be saved. The plan of this survey encloses a small space at the principal entrance from the city, not included in the Park. This space is therefore surveyed separately, and its contents deducted from that of the triangulated plan, a thing necessary to be done in all similar cases. When any part of a survey lies external to the bounds of the triangulation, the contents of the external part must be added to that enclosed by the triangulation. In the course of this survey we took care to measure our chain frequently, which we found never to contract; but on the contrary, often found considerable expansion to have taken place, which we always took care to correct. The several enclosures in the Park we surveyed on trigonometrical principles, taking the form of all the buildings by means of very short offsets, drawn to the various angles and corners. That part of the Park bounded by the public road leading to Chapelizod, inclines at a very considerable angle towards its boundary, and is besides impervious, being thickly planted with white-thorns and brambles; it therefore became necessary to survey this part by taking stations on the bank of the river, which runs at a small distance outside the boundary. The portion outside the boundary was then surveyed separately, and deducted from this part. These external stations are not shewn on the plan, as in laying it down, the displaced angles were reduced to others terminating at the boundary. To prove the accuracy of the work in its progress, several lines of verification were measured and compared with the computed lengths, and in general they were found to agree as nearly as could reasonably be expected. The sides of the triangles which are very oblique were sometimes deduced from other data, founded on more regular triangles. The best and most expeditious method of surveying is, doubtless, to cover the ground with a series of triangles, as in the plate to which this survey has reference; taking care to take as many angles from the |