in use.

larly and easily, by turning a milled nut fitted for that purpose. The compass is fixed to the upper horizontal plate; its ring is divided into 360 degrees, and the bottom of its box is divided into four parts or quadrants, each of which is subdivided to every ten. The magnetic needle is supported, in the middle of the box, upon a steel pin finely pointed; and there is a wire trigger for throwing off the needle when not The horizontal limb consists of two plates, one moveable on the other. The outermost edge of the upper plate is marked so as to serve for an index to the degrees on the lower. The upper plate, together with the compass, the vertical arc, and telescope, is easily turned round by a pinion fixed to a screw. The horizontal plate is divided into half degrees, and numbered from the right hand towards the left. The divisions are subdivided to every minute of a degree. On the upper plate, are a few divisions similar to those on the vertical arc, giving the hundredth parts for measuring the diameter of trees, &c.

The whole instrument fits on the conical ferrule of a strong, brass-headed staff, which has three substantial wooden legs. The top or head of the staff, consists of two brass plates parallel to each other. Four screws pass through the upper plate, and rest on the lower plate, by the action of which, the horizontal limb may be set truly level. For this purpose, a strong pin is fixed to the outside of the plate, and connected with a ball that fits into a socket in the lower plate.

The axis of the pin and ball, are so framed as to be perpendicular to the plate, and, consequently. to the horizontal limb.

Three adjustments must be made, before the instrument is applied to the mensuration of angles. In the first place, care must be taken that the line of collimation (that is, the line of vision passing through the cross hairs) be exactly in the centre of the cylindric rings round the telescope; in the next place, that the level be parallel to this line; and, lastly, that the horizontal limb be so set, that when the vertical arc is at Zero, and the upper part moved round, the bubble of the level will remain in the middle of the open space.

When these adjustments are made, and the instrument is to be applied to practice, the lower plate of the horizontal limb, being supposed to remain unmoved and parallel to the horizon, the telescope is to be directed successively to the different objects, whose angular positions are to be determined by means of two pinions, of which, one turns the upper part of the instrument in a horizontal plane, and the other turns the arc in a vertical plane.

Then the angle which a line passing through the axis of the telescope and any object makes with the horizon, will be indicated by the arc of the vertical circle between 0° and the index engraved on the scale fixed to the upper plate of the horizontal limb of the instrument. the horizontal angle, contained by two vertical planes, conceived to pass through any two objects and the centre of the instrument, will be shown by the arc of the lower plate of the horizontal

Also,

limb over which, the index, engraved on the upper plate, has passed by the direction of the telescope being changed from the one object to the other.

Example. Having measured AE (in the following figure,) a distance of two hundred feet in a direct horizontal line, from the bottom of a tower, the angle, BCD, contained by the horizontal line, CD; and a line drawn from C to the top of the tower, having been measured by a quadrant, or by a theodolite placed at C, was found to be 47° 30'. The centre C of the instrument, was five feet above the line AE, at its extremity E. From these premises, it is required to determine AB, the height of the tower.

In the right angled triangle, CBD is given the side CD 200 feet, and the angle C-47°30'. Then by the rules of plane trigonometry, the radius is to the tangent BCD, as DC is to DB. By employing the logarithmic tables, and proceeding as is taught by plane trigonometry, DB will be found to be equal to 218-3 feet. To which, if DA=CE=5 feet the height of the instrument, be added AB, the height of the tower will be found equal to 223-3 feet.

MENSURATION OF PLANE FIGURES.

Problem.

To find the area of a parallelogram, whether it be a square, a rectangle, a rhombus, or a rhomboid.

Rule 1. Multiply the length by the perpendicular breadth, and the product will be the

area.

2. As the radius is to the sine of any angle of the parallelogram, so is the product of the sides including the angle, to the area of the parallelogram.

Problem.

Any two sides of a right angled triangle being given, to find the remaining side.

Rule 1. When the sides about the right angle are given, to find the hypothenuse. Add together the squares of the sides about the right angle, and the square root of the sum will be the hypothenuse.

2. When the hypothenuse, and one of the sides about the right angle, are given, to find the other side.

1

From the square of the hypothenuse, subtract the square of the given side; and the square root of the remainder will be the other side.

Problem.

To find the area of a triangle. Multiply any one of its sides by the perpendicular let fall upon it from the opposite angle, and half the product will be the area.

Problem.

To find the area of a circle.

Rule 1. Multiply half the circumference by half the diameter, and the product will be the

area.

Rule 2. Multiply the square of the diameter by 7854, and the product will be the area.

Problem.

To find the area of an ellipse.

Multiply the product of the two axes, by the number 7854. for the area of the ellipse.

The foregoing problems seem sufficient to give some idea of the mensuration of plane figures.

LAND SURVEYING.

The instruments most commonly employed in land surveying, are the chain, the plane-table, and cross.

As a statute acre of land is 160 square poles, the measuring chain is made four poles, or sixtysix feet in length, in order that ten square chains, or 100,000 square links, may be equal to an acre. Hence, each link is 7-92 inches in length.

The plane-table is used for drawing plans of fields, and taking such angles as are necessary for calculating their areas. It is of a rectangular form, and is surrounded by a moveable frame, by means of which a sheet of paper may be fixed to its surface. It is furnished with an index, by which a line may be drawn on the paper in the direction of any object in the field; and with scales of equal parts, by which such lines may be made proportional to the distances of the objects from the plane-table when measured