The three right-angled triangles in the first sector are (1) cxi, (2) cxr, and (3) yir: and the two oblique-angled triangles are (1) cyi, and (2) icr.

The two equiangular triangles are yir and car.

They are equiangular, first, because the two angles at i and x are right angles, and the angles at y and c each the complement of r, which is common to both; second, because yi is parallel to c, the angle iyr is equal to the angle xcr (Theor. III. Part. I.); and as the angle at r is common, and the remaining angles right angles, the triangles are therefore equiangular.

The angles of the remaining three triangles are dissimilar.

The two chief lines sought in this sector are the normal ce, representing the height of the first lock, and the arc ei, representing the length of the first level of the canal.

If we suppose the surveying-staff to be divided into two companies, one with the theodolite and levelling-staff, and the other with the chain, and that the station-poles along the canal-line are ranged, those shown on the vertical plane being c, i, n, and b, then the field operations will be performed and the measurements entered in the three field-books that would be open under such an hypothesis, as follows:

At station e the angle yci is measured with the theodolite, the vertical bearing being taken from y to i. (Before removing the instrument the horizontal bearings are also taken; but as they are similar to those of the railway survey, p. 385, the directions in this section are confined to the survey of the vertical plane.) The bearings are next taken at station i; first the angle of depression yic, which gives at the same time icx. The measurement of the angles with the theodolite, and normals with the levelling-staff, now proceed together, cy representing the first one that is found by the latter instrument. When the angle of depression yic is accurately taken and entered in the field-book, the signal is given to those in charge of the levelling-staff, who enter the height cy in their fieldbook, which measurement includes the height of the theodolite above the ground at station i. This latter has therefore to be taken at each angle of depression, and afterwards deducted from the former, to obtain the true normal found. The theodolite is then turned round, and the vertical bearing min taken before the station-pole is replaced. It is next removed to station n, when the levelling-staff is taken to station i.

During the above operations, the chain may be employed in measuring the side lines of the horizontal survey; but when those working the levelling-staff replace station-pole i, and proceed to n, then the diagonal or base line ci may be driven.

This concludes the field operations for the first level ei or sector eri. Those of the other two sectors mrn and oru are performed in a similar manner.

From the two angles thus measured, viz., yci and yic, the other angles of the five triangles may be determined. Thus 180° - (yci + yic) gives cyi and xcr; yic-icx; 90° - icx-cix; 90° - xcr=r; and icx + xcr= = icr.

From the two sides found, viz., ci and cy, and the above angles, the other sides of the triangles may now be determined by the following formulæ :

The two lines yi and ir may also be obtained by formulas for ob lique-angled triangles, Part VIII. Section II., and the secants yr and cr by Theorem VII. Part I., as shown in Example IV. The line ce is now got by subtracting cr from ir; and ey representing the correction for curvature of the horizontal level iy from the arc ei, is obtained by subtracting ir from yr, or the radius of the arc from its

secant.

The arc ei, representing the length of the first level of the canal, remains to be determined.

The difference between radius and secant is found with geometrical accuracy in several ways, but the true length of a curve or of an arc of a circle, as ei, is only approximated. Thus, it is less than its tangent, but greater than its chord, as will be seen from the diagram. Taking the diameter, for example, to be 1, the circumference of a circle has been calculated to be 3.14159; now 3.141 would lie within the circle, while 3.142 would lie without. Between these two lesser and greater differences, another one, viz., 3.141592265358979 lies; still it is upon the inner side of the curve : and in this manner the decimal has been extended to upwards of two hundred places, and might be carried to as many more. But for all practical purposes it may be closed at 3-1416 without, or 3.1415 within, the former being generally adopted. From such data the following rules have been deduced:

:

To find the circumference, multiply the diameter by 3.1416.

To find an arc, multiply 0174533 by radius, and the number of degrees which the arc contains.

Irrigation-works.

Irrigation-works may be considered with a view to their survey under the following three heads :-1. Bottom irrigation; 2. Surface irrigation; 3. The modern system of applying liquid manure to land.

1. Extensive examples of the first kind are to be found in our West India colonies, and many other parts of the world. The works consist principally of parallel canals and ditches, intersecting comparatively level land.

The nature of the survey is therefore similar to that under Example III., the canals and ditches being used for the twofold purpose of surface drainage in the rainy season and bottom irrigation during the intervening periods of drought.

2. Surface irrigation, the second example, is principally practised for the growth of rice and grass. The water is generally conveyed to the fields in both cases by gravitation, either from canals or rivers, naturally or artificially formed, but it is differently applied afterwards, paddy fields consisting of a series of dead levels, while watermeadows lie at various inclinations.

In surveys of this kind, field operations are simple, but multitudinous in character, the levels required for distributing the water being exceedingly numerous; and as they are represented upon the plan, they furnish a corresponding amount of office work.

3. The third practice of applying water to land is by forcing it through pipes, either by gravitation, or by steam, or other power. It may first be thrown, by means of pumping apparatus, into tanks or cisterns, situated on elevated ground, and from thence be distributed throughout the fields by gravitation; or it may be forced directly to the land. The pipes and hydrants are ramified throughout the fields in such a manner that each of the latter has a certain area of land assigned to it, so that the liquid may be showered equally over it by means of a hose of the proper length screwed on to the hydrant. When one area is finished, the liquid is turned off, the hose screwed on to another hydrant, and thus the work proceeds until the several hydrantal areas are gone over.

It does not belong to this work to describe the details of liquid manuring land on this plan. Enough has been said to show the student who may not be acquainted with the practice the nature of the levels that have to be taken, and the survey required in laying down the pipes and hydrants.

The vertical survey closely resembles that of water-works for supplying towns, castles, farm-homesteads, and fields, with water.

It embraces two levels, a lower one at the river or fountain-head, or liquid manure tank, and a higher one at the reservoir, or its equivalent, the height to which the liquid has to be forced. The work of taking the levels is therefore similar to that in canal surveying.

With regard to the horizontal survey, a hose screwed on to a hydrant or mouth of a stopcock would distribute water over a circular area of land. The areas, however, into which fields must be subdivided are necessarily square areas; consequently each square area has to be inscribed, as it were, within the circular area which the length of the hose and jet will cover. As gases are very liable to be disengaged from drainage water and liquid manure, and to collect in the bends of pipes, where they pass over elevated ground, hydrants should be placed at such, so as to draw off the gases which there collect.

Warping land.

When flowing water holding organic and inorganic matter in suspension is turned into an area of land inclosed by embankments, and then allowed to stagnate, such matter is deposited, and termed "warp," and the process "warping."

The annual overflowing of the Nile in Egypt furnishes a practical example on a large scale. Warping has also been extensively practised in this country, the warp being chiefly obtained from tidal rivers, some of which have been termed "muddy to excess," a depth of from six to twelve inches of warp being left by them upon the land in a single summer season.

The survey is similar in character to that for embanking land when the work is not attended with any difficulty. The height and breadth of the embankments at the base are generally shown by figures upon the plan, in the same manner as the area of land inclosed for warping. As the depth of the warp will be directly as that of the water, the depths of the surface of the land at the lowest and highest places below the top of the embankment may have to be taken and shown upon the plan, as different depths of the ocean are directed to be represented in Example III. Such bottom levels are easily found by setting station-poles at the places, and raising flags or marks upon them until such marks appear in a line or in the same plane with the top of the embankment levels.

SECTION IV.

THE METHOD OF REDUCING LOCAL OR CUSTOMARY MEASURES TO

STATUTE MEASURE, AND VICE VERSA; ALSO, THE METHOD OF REDUCING SCOTCH AND IRISH MEASURES TO STATUTE MEASURE, AND VICE VERSA.

It has been already observed that formerly, by custom, the perch varied in different parts of England, and with it consequently the acre also varied in proportion.

In Devonshire and part of Somersetshire 15, in Cornwall 18, in Lancashire 21, and in Cheshire and Staffordshire 24 feet were accounted a perch.

In the common field-lands of Wiltshire, and in some other counties, there was a customary measure of a different nature, viz., of 120 instead of 160 statute perches to an acre; consequently 30 perches of statute measure made 1 rood of customary, or 3 statute roods made 1 customary acre, or 30 statute perches made 1 rood, and 4 such roods made 1 acre, customary measure.

In some places, an acre of this measure was called a day-work, or a day's work of land.

We may also observe that the customary measures of Scotland and Ireland differed very greatly from the English statute measure.

PROBLEM I.

To reduce customary measure to statute measure, or statute measure to customary measure.

GENERAL RULES.

RULE I. To reduce customary measure to statute measure.

Multiply the number of perches, customary measure, by the square feet in a square perch, customary measure; divide the product by the square feet in a square perch, statute measure, and the quotient will be the answer in square perches, which reduce to roods and acres by dividing by 40 and by 4 in the usual manner.

RULE II. As the square yards in an acre, statute measure, are to the square yards in an acre, customary measure; so are any number of acres and decimals, customary measure, to their equivalent in acres and decimals, statute measure. Then reduce the decimals to roods and perches by multiplying by 4 and by 40 in the usual

manner.