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can be seen on the surface of the earth. To the product of the hight of the object into the diameter of the earth, add the square of the hight, and extract the square root of the sum. To find the distance of any heavenly body, whose horizontal parallax is known. As radius, to the semi-diameter of the earth; so is the cotangent of the horizontal parallax, to the distance.

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NAVIGATION. Navigation is the art of conducting a ship on the ocean. Plane sailing is the method of calculating the situation and progress of a ship by means of a plane triangle, in which the particulars given or required are, the course, the distance, the difference of latitude, and the departThe Course is the angle between a meridian line passing through the ship, and the direction in which she sails. The Distance is the length of the line which the vessel describes in a given line. The Difference of Latitude is the distance between two parallels of latitude measured on a meridian. Departure is the deviation of a ship east or west from a meridian. The distance, departure, and difference of latitude, are measured in geographical miles or minutes, one of which is equal to the sixtieth part of a degree at the equator. The distance, departure, and difference of latitude may be given in length by the sides of a right angled plane triangle. The traverse tables are tables in which are given the departure and difference of latitude, for every degree in the quadrant, and for every quarter of a degree. The sailing of a ship on a parallel of latitude, is called parallel sailing. Middle latitude sailing is when the ship sails oblique in respect to a meridian. The middle latitude is equal to half the sum of the two extremes of latitude, if they are both north or both south; but to half their difference, if one is north and the other south. Mercator's sailing is when the calculations are made according to Mercator's chart, on which the meridians are drawn parallel to each other, and the degrees of latitude are proportionally enlarged. The solutions in Mercator's sailing are founded on the similarity of two right angled triangles, in one of which the perpendicular sides are the proper difference of latitude and the departure; and in the other, the meridional difference of latitude and the difference of longitude. Traverse sailing is when a ship is continually varying her direction. Resolving a traverse is reducing a compound course to a single one.

If the difference of latitude and difference of longitude be calculated for each part of the compound course; the whole difference of latitude and difference of longitude will be found by addition and subtraction. Oblique sailing is the application of oblique angled trigonometry to the solution of certain problems in navigation. Current sailing is when the progress or direction of a ship is affected by a current. Hadley's quadrant is an instrument by which the altitudes of the heavenly bodies, and their distances from each other, are measured at sea. Its superiority is owing to the fact, that the observations made with it are not materially affected by the motion of the vessel.

SURVEYING. The most common method of surveying a field, is to measure the length of each of the sides, and the angles which they make with the meridian. The essential

parts of a surveyor's compass are a graduated circle, a magnetic needle, and sight holes for taking the direction of any object. The surveyor's chain is four rods long, and is diviIdid into 100 links. There are in common use two methods of finding the contents of a piece of land; one by dividing the plot into triangles, and the other by calculating the de. parture and difference of latitude for each of the sides. Rule for finding the area of a field by departure and difference of latitude. 1. Find the northing or southing, and the easting or westing, for each side of the field, and place them in distinct columns in a table. To these add a column of Meridian Distances, for the distance of one end of each side of the field from a given meridian; a column of multipliers, to contain the pairs of meridian distances, for the two ends of each of the sides; and columns for the north and south areas. 2. Suppose a meridian line to be drawn without the field, at any given distance from the first station; and place the assumed distance at the head of the column of meridian distances. To this add the first departure, if both be east or both west; but subtract, if one be east and the other west; and place the sum or difference in the column of meridian distances, against the first course. To or from the last number, add or subtract the second departure, &c. 3. For the column of multipliers, add together the first and second number in the column of meridian distances; the second and third, the third and fourth, &c., placing the sums opposite the several courses. 4. Multiply each number in the column of

multipliers into its corresponding northing or southing, and place the product in the column of north or south areas. The difference between the sum of the north areas, and the sum of the south areas, will be twice the area of the field. To survey a field from two stations. Find the distance of the two stations, and their bearings from each other; then take the bearings of the several corners of the field from each of the stations. To survey a field by measuring from one station. Take the bearings of the several corners of the field, and measure the distance of each from the given station. To survey a field by the chain alone. Measure the sides of the field, and the diagonals by which it is divided into triangles. To survey an irregular boundary by means of offsets. Run a straight line in any convenient direction, and measure the perpendicular distance of each angular point of the bounda ry from this line. To measure the distance between any two points on the surface of the earth, by means of a series of triangles extending from one to the other. Measure a side of one of the triangles for a base line, take the bearing of this or some other side, and measure the angles in each of the triangles. To lay out a given number of acres in the form of a square. Reduce the number of acres to square rods or chains, and extract the square root. This will give one side of the required field. To lay out a field in the form of a parallelogram, when one side and the contents are given. `Divide the number of square rods or chains by the length of the given side. The quotient will be a side perpendicular to the given side. To lay out a piece of land in the form of a parallelogram, the length of which shall be to the breadth in a given ratio. As the length of the parallelogram to its breadth; so is the area, to the area of a square of the same breadth. The area of a parallelogram being given, to lay it out in such a form, that the length shall exceed the breadth by a given difference. To the area of the parallelogram, add one fourth of the square of the difference between the length and the breadth, and from the square root of the sum subtract half the difference of the sides; the remainder will be the breadth of the parallelogram. To lay out a triangle whose areas and angles are given. Calculate the area of the supposed triangle which has the same angles. Then as the area of the assumed triangle, to the area of that which is required; so is the square of any side of the former, to the

square of the corresponding side of the .latter. To divide the area of a triangle into parts having given ratios to each other, by lines drawn from one of the angles to the opposite base. Divide the base in the same proportion as the parts required. To divide an irregular piece of land into any two given parts. Run a line at a venture, near to the true division line required, and find the area of one of the parts. If this be too large or too small, add or subtract, by the preceding articles, a triangle, a trapezoid, or a trapezium, as the case may require.

LEVELLING. The true level is á curve which either coincides with, or is parallel to the surface of water at rest. The apparent level is a straight line which is a tangent to the true level, at the point where the observation is made. To find the difference in the hight of two places by levelling rods. Set up levelling rods perpendicular to the horizon, and at equal distances from the spirit level; observe the points where the line of level strikes the rods before and behind, and measure the hight of their points above the ground; level in the same manner from the second station to the third, from the third to the fourth, &c. The difference between the sum of the hights at the back stations, and at the forward stations, will be the difference between the hight of the first station and the last. To find the dif ference between the true and apparent level, for any given distance. Add one fourth of the square of the earth's diameter, and the square of the distance, extract the square root of the sum, and subtract the semi-diameter of the earth. To find the difference in the height of two places whose distance is known. From the angle of elevation or depression, calculate how far one of the places is above or below the apparent level of the other; and then make allowance for the difference between the apparent and true level.

THE MAGNETIC NEEDLE. The declination of the needle is the angle which it makes with a north and south line. In some places it is 20 or 30 degrees; in others, little or nothing. The declination is not only different in different places, but different in the same place at different times. To draw a true meridian line: take the direction of the pole star when it is farthest west, and also when it is farthest east; and bisect the angle made by these two directions.

CONIC SECTIONS.

A cone is a solid figure formed by the revolution of a right angled triangle about one of its sides. The three conic sections are the parabola, the ellipse, and the hyperbola.

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Properties of the parabola. 1. The latus rectum is equal to four times the distance from the focus to the vertex. The tangent at any point of the curve, bisects the angle formed at that point, by the perpendicular to the directrix, and the line drawn to the focus. 3. If a tangent to any point of the curve cut the axis produced, the points of contact and intersection will be equally distant from the focus. Also, the subtangent is bisected by the vertex, or the subtangent is double the corresponding abscissa. 4. The square of any ordinate to the axis is equal to the rectangle of the corresponding abscissa and the latus rectum. 5. The subnormal is equal to half the latus rectum. 6. The square of an ordinate to any diameter, is equal to four times the rectangle of the corresponding abscissa, and the distance from the vertex of that diameter to the focus. 7. Every diameter bisects all lines in the parabola, drawn parallel to the tangent at its vertex, and terminated both ways by the curve; or every diameter bisects its double ordinates. 8. The parameter to any diameter is equal to four times the distance from the vertex of that diameter to the focus. COR. The square of the ordinate is equal to the parameter into the abscissa. 9. If a diameter be produced to meet any tangent to the parabola, without the curve, the parts of these diameters, between the curve and the tangent, will be as the squares of the intercepted parts of the tangent. 10. The tangents drawn from the two extremities of any double ordinate, intersect the diameter to which that double ordinate belongs, in the same point. (a) If from any point in the curve, there be drawn a tangent, and also a line to meet the curve in some other place; and if any diameter, intercepted by this line, be produced to meet the tangent; then will the curve divide the diameter in the same ratio in which the diameter divides the line. 11. The vertical tangent intersects any other tangent, in the point where a perpendicu lar from the focus upon that tangent intersects it. 12. The normal is a mean proportional between the semi-parameters

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