from each other, and the mean perpendicular depth of the pit 16.8 feet.

Ans. The area of the first section = 6422.19; the area of the second = 5749.26; the area of the third=5113.03; the area of the fourth = 4561.2; the area of the fifth 3955.885; and the content 86584.365 cubic feet 3206.828 cubic yards, the answer required.

DIRECTIONS FOR

MEASURING PONDS AND MILL-DAMS. PONDS and mill-dams are commonly dug by the cubic yard; and assume a variety of shapes.

If the top and bottom of a pond or dam be rectangles, it may be treated as directed in the last Problem, Case I.; if the top and bottom be triangles, trapeziums, or polygons, the content may generally be found by Case II., of the last Problem; but if both these methods fail, you must proceed as directed in Problem III., for drains and canals. In this case, the contents of the pieces at the ends, resembling wedges, must be found by Problem 9, Part IV., Sect. I.

Mr. W. Putsey, Teacher of the Mathematics, at Pickering, informs me that the ponds made upon the wolds in Yorkshire, are generally of a conical shape; hence their contents may be found by Problem 7, Part IV., Sect. I.

Mr. P. who is a practical measurer, has also very kindly communicated the following method of taking the dimensions of a conical pond in which there is water.

EXAMPLE.

Let ABC denote a perpen- A dicular section of a conical pond, whose dimensions are required.

Extend a cord over the pond, with which take the

diameter AB; make a ring fast to the middle of the

cord or diameter, as at D, through which put the end of the plumb-line ADC; let your assistant keep one end of the diameter at B; while you hold the other, and the plumb-line at A; permit the plummet to descend to the bottom of the pond, as at C; then draw back both the cords, and measure CD, which will be the perpendicular depth of the pond.

Note 1. When the top of a pond is not a perfect circle, measure two diameters at right angles to each other; and take half their sum for a mean diameter. 2. Cellars generally form parallelopipedons; and when they are dug by the cubic yard, their contents may be found by Prob. II., Sect. I., Part IV.

MEASURING EMBANKMENTS.

THE most correct method of measuring embankments, is to proceed in the same manner as directed in Problem III., for drains and canals; and if the following observations be well understood, no difficulties will arise in taking the dimensions, and finding the areas of the sections.

E

Let ABCD represent a transverse, perpendicular section of an embankment, made upon level ground; then if DC be parallel to AB, it is evident that the section is a trapezoid, the area of which may be found by Problem 8, Part II.

A

The distance of the parallel sides, or height of the section, may be obtained by placing a staff perpendicular to the horizon, at A or B, and producing DC to E or F; then will AE or BF be the perpendicular distance of the parallel sides AB, DC; and AB, the breadth of the bottom of the embankment, is evidently equal to ED + DC + CF.

Again, let GHKL denote a perpendicular, transverse section of an embankment, made upon uneven ground; then it is evident that the section is a trapezium; and as the diagonals and perpendi

culars cannot be measured, the content must be found in the following manner: Produce LK, the horizontal line of the top, both ways, to M and N, and let fall the perpendiculars MG and NH; then by Problem 8, Part ÌI., half the sum of these parallel sides multiplied by their perpendicular distance MN, will give the area of the trapezoid GHNM, from which subtract the sum of the areas of the two right-angled triangles GML, HNK, and the remainder will be the area of the section GHKL.

Note. Unskilful measurers affect to determine the contents of embankments by finding what they call mean breadths and thicknesses; but no person who has a scientific knowledge of mensuration, will have recourse to such an erroneous process, if the foregoing method can by any means be adopted. See Drains and Canals, Note 1.

DIRECTIONS FOR

MEASURING QUARRIES.

THE baring of quarries is generally done by the cubic yard; and sometimes the stones themselves are got by the same measurement.

Quarries, in general, are very irregular; they may, however, commonly be measured by some of the methods already described for canals, marlpits, ponds,

&c.

If a quarry be so irregular that none of these methods can be adopted, the general method is to take such dimensions as will give the area of a mean horizontal section; then this area being multiplied by the mean depth of the quarry, the product is taken for the

content.

Note 1. Sometimes a quarry may be most correctly measured by dividing it into several parts; and taking the dimensions of each part separately.

2. When the stones that are left jutting out of the sides of the quarry, are measured with the vacuity, mean dimensions of these stones must be taken, and their contents subtracted, in order to obtain the true content of the va cuity.

DIRECTIONS FOR

MEASURING COAL-HEAPS.

WHEN the stock of a colliery is to be taken, or an exchange of tenants takes place, it becomes necessary to ascertain the quantity of coals which are laid unsold at the pits; and as pit-heaps are generally very irregular, both in extent and thickness, it is no easy task to find their contents with a tolerable degree of accuracy.

The method generally adopted is, to take such dimensions as will give the area of a mean horizontal section of the heap; and to multiply this area by the mean thickness, for the content.

Note 1. The perpendicular height of any point at the extremity of a coalheap, may be obtained in the following manner: Place a spirit or water-level upon the top of the heap, close by the edge; let your assistant hold a pole upon the ground, in a perpendicular direction, at the bottom of the heap; direct the level towards the pole, and note that point in it which is seen through the level; measure the distance between the bottom of the pole and this point: subtract the height of the level from the said distance; and the remainder will be the height of the heap, at the place where the observation is made.

2. When a coal-heap is upon level ground, the height or thickness of any place between the pit and the extremity of the heap, may be found thus: By Note 1, find the height of the heap where it is most elevated; then let your assistant place the pole upon that part of the heap the height of which you wish to obtain ; and without removing the level from the highest point of the heap, direct it towards the pole; measure the distance between the bottom of the pole and the point seen through the level; from this distance take the height of the level; then if the remainder be subtracted from the greatest height of the heap, before found, you will obtain the height required.

3. If the ground upon which a coal-heap is laid be level, and the upper surface of the heap a regular inclined plane, rising gradually from the pit to the utmost extremity of the heap, which is sometimes the case, it is evident that the mean thickness of the heap will be equal to half the sum of the two heights taken at that part of the heap adjoining the pit, and at the utmost extremity or highest point of the heap,

4. When a coal-heap is extremely irregular, it is generally necessary to di· vide it into several parts, and take the dimensions of each part separately. In ' this case, heights must be taken in such places as are most likely to give the mean thickness of each part; and when the ground is not level, a proper allowance must be made for this circumstance.

5. In some parts of England, 5 pecks, Winchester measure, or 2638 cubic inches, make a bushel of coals, and 36 bushels a chaldron; therefore, if the cubic inches in a coal-heap be divided by 2688, or the cubic feet by 1.555, the quotient will be the number of bushels contained in the heap; but as this measure is not general, every person who measures a coal-heap, ought to make himself acquainted with the customary measure of the place.

6. Notwithstanding what has been advanced on the subject of measuring quarries and coal-heaps, a great deal will always depend upon the ingenuity of the measurer; for it is impossible to give directions that will suit every particular case to be met with in the practice of measuring these irregular figures.

Having, in the foregoing pages, treated of the Mensuration of Drains and Canals, it is presumed that a short Account of a few of the principal ones, will not be unacceptable to the young Reader; as it will tend to give him some Idea of the great Improvements that Agriculture and Commerce have received, and are daily receiving by the use of them; and will inform him what stupendous Works have been effected by the Ingenuity, Perseverance, and united Efforts of Men.

A DESCRIPTION

OF SOME OF THE

PRINCIPAL

CANALS

In England, Scotland, France, & China.

CANALS are to be met with in every civilized country; and perhaps it will not be going too far to say, that the internal commerce of no nation has received greater improvements by them, than that of Great Britain.

THE DUKE OF BRIDGEWATER'S CANAL, is a work that begins at Worsley, seven miles from Manchester; where, at the