points by writing the three first letters of the alphabet at them. Having the base line and two angles of the triangle, the two other sides can be found. At the extremities of these sides take angles to some other distant stations judiciously chosen; then having two angles and one side in each of these new triangles given, the remaining sides can be found; which you sketch in their proper places in your field book; and thus you proceed, adding triangle to triangle, and noting down the angles measured in each, till you extend the connected series of triangles over the entire of the ground to be surveyed. When the boundaries of the survey are curved, take in as much of it as possible by the triangulation, and measure the remaining small segments by erecting offsets from the extreme sides of the triangles to the curved boundaries. Calculate the contents of these small segments by considering all the parts between the offsets as trapezoids; or divide the sum of the offsets by the number of them, accounting that for one of them where the boundary meets the station line, or extreme side of the triangle; then multiply the quotient, which is the mean breadth, by the entire length, and the product will be the area.

In laying off the ordinates, care should be taken to have them equally distant from one another, as otherwise the area found by the above rule would be incorrect.

For another rule for the same purpose, see the author's Treatise on Mensuration for the use of the Irish National Schools.

After having taken the angles of two or three triangles, begin to fill in, that is, lay down the hedges, ditches, buildings, &c. If any of these be close to a station line, take offsets to it from that line, as taught at the commencement of this work. But should these be distant from a station line, run a new line from an ascertained point (by measurement) in the station line, so as to pass close by, or through, the objects to be represented on the map; then take offsets to them on this new line. The direction of this new line is fixed by taking the angle it makes with the station line whose position is given.

In fixing your stations, you must take care to have the triangles as nearly equilateral as the face of the country you are surveying will admit; as in case of a very small error being committed in taking the angles, the small error thus committed will affect the area least, when the triangle is truly equilateral; but will increase the error more and more, as the triangle is removed in its form from equilateral.

In conducting a great trigonometrical survey, its success depends, in a great measure, upon the accuracy with which the base line is measured. Hence it is, that no expense, time, or labour has been spared when such a delicate operation required it. The surveys of France and England furnish proofs of the extraordinary patience, industry, and scientific acquirements of the distinguished individuals who conducted them. But the survey of Ireland, conducted by Col. Colby, assisted by an able corps of Royal Engineers, far excels any thing of the kind ever undertaken: it will remain an everlasting monument, testifying the persevering industry, the scientific talent, and the superior practical the illustrious corps, whose labours in this great

work shall long be claimed and boasted as a national honor. A detailed account of this survey would be a valuable document; which it is hoped shall be soon published.

In conducting large trigonometrical surveys, various instruments have been used to measure a base line with that scrupulous exactitude which such great undertakings required. A chain has been used, composed of long links, and strained by equal weights, in every case acting over pulleys fixed on the tops of pedestals attached to the ground, and made truly vertical on their tops, so that the quantity of its sag or depression is ascertained-cylindrical glass rods have been used, supported at short intervals, to prevent deflection, and secure accuracy of measurement. Various other instruments have been employed, such as, compensating chains, deal rods, &c. But such precautions, however desirable and necessary in national surveys, are too delicate, tedious, and costly to be used in the ordinary measurement of land, which is our main object in the present part of this work.

In a survey of the Phoenix Park by the author's son, under his own immediate superintendence, the foregoing directions and observations have been fully illustrated. The base line, AB, was measured with great care on that smooth and level ground known by the name of "the Fifteen Acres." The first thing done before the base line had been actually measured, was to ascertain the other points most likely to answer as convenient and commanding stations.

Having selected the points A and B for the extremities of the base line, the points next considered favourably situated for new stations may be seen at E, C, and D. Then after having carefully measured AB, the theodolite was placed at A, and the angles EAC, CAB, and BAD were observed, and noted down in a field-book, with the length of AB. This being done, the theodolite was removed to B, from which point the angles DBA and ABC were measured. Then having planted the theodolite at C, the angle ACE was observed.

To ensure correct results, every angle was measured on different parts of the horizontal plate of the instrument, and a mean of all the readings noted in the fieldbook.

Though this part of the Phoenix Park is free from hedges, trees, or other objects that might obstruct the view, yet the triangles here laid down, having reference to other parts of the survey, are far from the form which we would desire, under different circumstances.

The grounds surrounding the Chief Secretary's Lodge and the Military School rendered it necessary, on account of their peculiar position relatively to the base line, to make choice of the stations at the places above described. The rest of the Park is so very close, being full of clumps, bushes, &c., little or no regard has been paid, in the selection of stations, to the form of the triangles, advantage only having been taken of every opening, so as to enable us to measure two angles of every triangle.

Now having given the side AB, and the two angles ABC, BAC, the sides AC and BC were calculated, and thence the area of the triangle. The same data is given in the triangles AEC and ABD, from which the sides of each are calculated respectively, and thence their areas.

See Plate.

It frequently happened during the survey, that the instrument could not be placed at the stations, and therefore it became necessary to reduce the angles to the centre of the stations. We effected this by various methods, according to the circumstances of the case. AB and BC are two stone walls. B The length of AB, and the angle BAC being known, it became necessary to ascertain the angle BCА.

A

C

The distances AC and BC not being easily measured, it therefore became necessary to find the angle ACB; to do which, the instrument was placed at D, and the angle ADC measured, as also the side DC; then the angle DAC was measured, from which the angle DCA was found. Next the side DB was measured, being very short, and the angle BDC; then having the two sides BD, DC, the included angle, BDC, was measured, and the angle DCB found by calculation, which being added to the angle ACD already found, gave the required angle AСВ.

Nothing worth adverting to occurred in the survey, till we arrived at the Phoenix Pillar. The top of the pillar, which we shall suppose at B (see last figure), was one of our stations; a straight pole attached to the top of a large lofty tree at A, was another; and a picket placed on the top of a fence, opposite the Viceregal Lodge, was a third. The distance AB was known from the preceding part of the work, but none of the angles. Here was a case in which we required to measure the three angles of a triangle, without being able to place the theodolite at any of the stations.