the three sides. CASE I. Given two angles and a side, to So is the sine of the angle opposite the required side, CASE II. Given two sides and an opposite angle, to find the remaining side and the other two angles. One of the required angles is found, by beginning with a side, and, according to theorem I. stating the proportion. As the side opposite the given angle, To the sine of that angle; So is the side opposte the required angle, CASE III. Given two sides, and the included angle, to find the remaining side, and the other two angles. In this case the angles are found by theorem II. The required side may be found by theorem I. CASE IV. Given the three sides, to find the angles. The solution may be made by dividing the given triangle into two right angled triangles. MENSURATION OF PLANE SURFACES. The area of a parallelogram may be found by multiplying the length by the hight or breadth. The area of a square is found by multiplying one of the sides into itself. When the dimensions are given in feet and inches, the multiplication may be conveniently performed by the arithmetical rule of duodecimals. To find the area of a triangle. a triangle. Multiply one side by half the perpendicular from the opposite angle; or, when the three sides are given, from half their sum, subtract each side severally, multiply together half the sum and the three remainders, and extract the square root of the product. To find the area of a trapezoid. Multiply half the sum of the parallel sides into their perpendicular distance. To find the area of a trapezium Divide the whole figure into triangles, by drawing diagonals, and find the sum of the areas of those triangles. When a trapezium can be inscribed in a circle, the area may be found by either of the following rules. 1. Multiply together any two adjacent sides, and also the two other sides; then multiply half the sum of these products by the sine of the angle included by either of the sides multiplied together. Or, 2. From half the sum of all the sides, subtract each side severally, multiply together the four remainders, and extract the square root of the product. To find the area of a regular polygon. Multiply one of its sides into half its perpendicular distance from the center, and this product into the number of sides. MENSURATION OF THE CIRCLE. To find the circumference of a circle from its diameter. Multiply the diameter by 3.14159, or, multiply the diameter by 22 and divide the product by 7. To find the diameter of a circle from its circumference. Divide the circumference by 3.14159. Or, multiply the circumference by 7 and divide the product by 22. To find the length of an arc of a circle. As 360°, to the number of degrees in the arc; so is the circumference of the circle, to the length of the arc. To find the area of a circle. Multiply the square of the diameter by the decimals .7854. Or, multiply half the diameter into half the circumference. Or, multiply the whole diameter into the whole circumference, and take of the product. To find the area of a sector of a circle. Mul. tiply the radius into half the length of the arc. Or, as 360° to the number of degrees in the arc; so is the area of the circle, to the area of the sector. To find the area of the segment of a circle. Find the area of the sector which has the same arc, and also the area of the triangle formed by the chord of the segment, and the radii of the sector. Then, if the segment be less than a semicircle, subtract the area of the triangle from the area of the sector. But if the seg ment be greater than a semicircle, add the area of the triangle to the area of the sector. To find the area of a circular zone. From the area of the whole circle subtract the two segments on the sides of the zone. To find the area of a lune or crescent. Find the difference of the two segments which are between the arcs of the crescent and its chord. To find the area of a ring, included between the peripheries of two concentric circles. Find the difference of the areas of the two circles. Or, multiply the product of the sum and difference of the two diameters by .7854. MENSURATION OF SOLIDS. To find the solidity of a prism. Multiply the area of the base by the hight. To find the lateral surface of a right prism. Multiply the length into the perimeter of the base. To find the solidity of a pyramid. Multiply the area of the base into of the hight. To find the lateral surface of a regular pyramid. Multiply half the slant hight into the perimeter of the base. To find the solidity of the frustrum of a pyramid. Add together the areas of the two ends, and the square root of the product of these areas; and multiply the sum by of the perpendicular hight of the solid. To find the lateral surface of a frustrum of a regular pyramid. Multiply half the slant hight by the sum of the perimeters of the two ends. To find the solidity of a wedge. Add the length of the edge to twice the length of the base, and multiply the sum by of the product of the hight of the wedge and the breadth of the base. To find the solidity of a rectangular prismoid. To the areas of the two ends, add four times the area of a parallel section equally distant from the ends, and multiply the sum by of the hight. A solid is said to be regular, when all its solid angles are equal, and all its sides are equal and regular polygons. The following are of this description: 1. the tetraedron, whose sides are four triangles; 2. the hexaedron, or cube, whose sides are six squares; 3. the octaedron, whose sides are eight triangles; 4. the dodecaedron, whose sides are twelve pentagons; 5. the icosaedron, whose sides are twenty triangles. To find the surface of a regular solid. Multiply the area of one of the sides by the number of sides. Or, multiply the square of one of the edges, by the surface of a similar solid whose edges are 1. To find the solidity of a regular solid. Multiply the surface by of the perpendicular distance from the center to one of the sides. Or, multiply the cube of one of the edges, by the solidity of a similar solid whose edges are one. THE CYLINDER, cone, and sphere. To find the convex surface of a right cylinder. Multiply the length into the circumference of the base. To find the solidity of a cylinder. Multiply the area of the base by the hight. To find the convex surface of a right cone. Multiply half the slant hight by the circumference of the base. To find the solidity of a cone. Multiply the area of the base into of the hight. To find the convex surface of a right cone. Multiply half the slant hight, by the sum of the peripheries of the two ends. To find the solidity of a frustrum of a cone. Add together the areas of the two ends, and the square root of the product of these areas; and multiply the same by of the perpendicular hight. To find the surface of a sphere. Multiply the diameter by the circumference. To find the solidity of a sphere. Multiply the cube of the diameter by .5236. Or, multiply the square of the diameter by of the circumference. Or, multiply the surface by of the diameter. To find the convex surface of a segment or zone of a sphere. Multiply the hight of the segment or zone into the circumference of the sphere. To find the solidity of a spherical sector. Multiply the spherical surface by of the radius of the sphere. To find the solidity of a spherical segment. Multiply half the hight of the segment into the area of the base, and the cube of the hight into .5236; and add the two products. To find the solidity of a spherical zone or frustrum. From the solidity of the whole sphere, subtract the two segments on the sides of the zone. Or, add together the squares of the radii of the two ends, and of the square of their distance; and multiply the sum by three times this diameter, and the product by .5236. ISOPERIMETRY. 1. An isosceles triangle has a greater area than any scalene triangle, of equal base and perimeter. 2. A triangle in which two given sides make a right angle, has a greater area than any triangle in which the same sides make an oblique angle. 3. If all the sides except one of a polygon be given, the area will be the greatest, when the given sides are so disposed, that the figure may be inscribed in a semicircle, of which the undetermined side is the diameter. 4. A polygon inscribed in a circle has a greater area, than any polygon of equal perimeter, and the same number of sides, which cannot be inscribed in a circle. 5. When a polygon has a greater area than any other, of the same number of sides, and of equal perimeters, the sides are equal. 6. A regular polygon has a greater area than any other polygon of equal perimeter, and of the same number of sides. 7. If a polygon be described about a circle, the area of the two figures are as their perimeters. 8. A circle has a greater area than any polygon of equal perimeter. 9. A right prism, whose bases are regular polygons, has a less surface than any other right prism of the same solidity, the same altitude, and the same number of sides. 10. A right cylinder has a less surface than any right prism of the same altitude and solidity. 11. A cube has a less surface than any other right parallelopiped of the same solidity. 12. A cube has a greater solidity than any other right parallelopiped, the sum of whose length, breadth, and depth, is equal to the sum of the corresponding dimensions of the cube. 13. If a prism be described about a cylinder, the capacities of the two solids are as their surfaces. 14. A right cylinder whose hight is equal to the diameter of its base, has a greater solidity than any other right cylinder of equal surface. 15. If a sphere be circumscribed by a solid bounded by plane surfaces, the capacities of the two solids are as their surfaces. 16. A sphere has a greater solidity than any regular poly. edron of equal surface. HIGHTS AND DISTANCES. To find the perpendicular hight of an accessible object standing on a horizontal plane. Meas. ure from the object to a convenient station, and then take the angle of elevation subtended by the object. To find the hight of an accessible object standing on an inclined plane. Measure the distance from the object to a convenient station, and take the angles which this base makes with lines drawn from its two ends to the top of the object. To find the hight of an inaccessible object above a horizontal plane. Take two stations in a vertical plane passing through the top of the ob ject, measure the distance from one station to the other, and the angle of elevation at each. To find the hight of any object by observation at two stations. Measure the base line between the two stations, the angles between this base and lines drawn from each of the stations to each end of the object, and the angle subtended by the object, at one of the stations. To find the distance of an inaccessble object. Measure a base line between two stations, and the angles between this and lines drawn from each of the stations to the object. To find the distance between two objects, when the passage from one to the other, in a straight line, is obstructed. Measure the right lines from one station to each of the objects, and the angle included between these lines. To find the distance between two inaccessible objects. Measure a base line between two stations, and the angles between this base and lines drawn from each of the stations to each of the objects. To find the diameter of the earth, from the known hight of a distant mountain, whose summit is just visible in the horizon. From the square of the distance divided by the hight, subtract the hight. To find the greatest distance at which a given object |