inches; required the side of the greatest square prism that can be cut from it. Ans. 5.6946 inches. (See 65, problem 19; and ¶ 19, example 13.) 113. The area of a right-angled triangle is 25,600 rods, and the sides are in geometrical proportion; now, if 1.27202066 represent the shortest side, I require the sides of the triangle. Ans. 200.6263; 255.2008; and 324.6206 rods. 114. The parallel sides of a trapezoid are 16 and 24 rods, and its area 5 acres; required the length of a line parallel to the parallel sides, that will divide it into two equal parts. Ans. 20.39608 rods. 115. The major axis of the earth's orbit is 191 millions of miles, and its minor axis is 190.976323 miles; required the sun's distance from the centre of the ellipse; that is, the eccentricity of the earth's orbit. Ans. 14 million of miles. 116. The axes of the earth's orbit being the same as stated in the former exercise; what is the parameter of its orbit? Ans. 190,952,879 miles. 117. Should a fortunate miner find a sphere of pure gold 4 inches in diameter, in the placers of California, I demand its true value. Ans. $6683.268. 118. A slide on the side of a mountain in Switzerland, is three miles in length, and its perpendicular height above the lake in which it terminates is one and a half mile; what will be the momentum of a body weighing 500 pounds at the foot of the slide? Ans. 355,977.6 lbs. 119. A bomb requires two pounds of loose powder to fill the internal concavity; it is of cast-iron, and its weight 30 pounds; required the thickness of the shell. Ans. 0.9853 inch. 120. What weight will a balloon, filled with hydrogen gas, sustain in the atmosphere near the earth, its form being that of a parabolic spindle 50 feet long, and its diameter 20 feet. Ans. 608.214 lbs. 121. A cast-iron shell, whose external diameter is 30 inches, will just float in pure water; required its thickness. Ans. 0.70799 inch. 122. If steam be heated to the temperature of 358.88 degrees, what force will it exert on the end of a piston 10 inches in diameter ? Ans. 11,466.84 lbs. 123. If a horse-power be equal to 400 pounds, to how many horse-power will a steam-engine be equal, the diameter of the piston being 30 inches, and the temperature of the steam 418.46 degrees? Ans. 516.0078. 124. A careless baker baked a hemispherical loaf of bread till it was half crust; the crust was of equal thickness throughout the whole loaf, the diameter of its base was 10 inches, and its height, of course, 5 inches; required the thickness of the Ans. Q.66984 inch. crust. 125. The equatorial diameter of the earth is 7,925 miles, and the ratio of the centrifugal force at the equator to the force of equatorial gravity, is as 1 to 289, or; I demand the polar diameter of the earth. Ans. 7,898.772 miles. 126. Required the diameter of a globe from which an octahedron may be cut, whose side is 12 inches. Ans. 16.9705 inches. 127. A tinker is required to make a tin can that will hold 9 pounds of powder well shaken; its form is required to be that of a conic frustum, its top and bottom diameters being 5 and 7 inches; what must be its altitude? Ans. 8.0733.inches. 128. If the equatorial diameter of the earth be 7,925 miles, what will be its diameter between the 44th degrees of north and south latitude, the ratio of the centrifugal force to the force of gravity in latitude 44° being as 1 to 400 ? Ans. 7,913.52 miles. 129. A spile-hammer, whose weight is 250 pounds, falls 16 feet; required its force, or momentum, Ans. 8,000 pounds. 130. If a charge of 6 pounds of powder is sufficient to im pel a ball over a range of 3,600 feet, what charge will be required to give the ball a range of 4,500 feet? Ans. 7.5 pounds. 131. The resistance of the atmosphere to a 12-pound ball, moving with a velocity of 25 feet per second, is half an ounce avoirdupois; and the resistance for velocities less than 1,100 feet, being nearly proportional to the squares of the velocities, it is required to find the resistance opposed to an 18-pound ball moving with a velocity of 1,000 feet per second. Ans. 62.876 pounds. 132. The breadth of a ditch in front of a tower is 48 feet; and from the outer edge of the ditch, the angle of elevation of the top of the tower is 53° 20′; what is the height of the tower? Ans. 64.47 feet. 133. Required the height of a tower, a horizontal base being measured of 245 feet, and the angle of elevation being 35 degrees 24 minutes. Ans. 174.11 feet. 134. If three trees be so planted that the angles of the triangle, at the corners of which they stand, are to each other as the numbers 1, 2, and 4, and that a line of 100 yards will just go round them; required their distances from each other. Ans. 19.8; 35.69; and 44.5 yards, nearly. One-seventh, two-sevenths, and four-sevenths of 180 degrees, will be the angles, viz. 25° 42'5 and 51° 25' and 102° 51'; and their natural sines are .43388 and .78183 and .97493, the sum of which is 2.19064. Then, as 2.19064 100: 0.43388 to the side of the triangle opposite the least angle. 135. In order to find the height of a conic pyramid, a base line was measured from its base of 130 feet, and the angles of elevation of the top of the pyramid, measured at the extremities of the base, were 31° and 46°; required its slant, and its perpendicular height. Ans. 258.69, and 186.08 feet. The angle of elevation at the base of the pyramid, viz. 46°, subtracted from 180° leaves 134°, the angle between the base line and slant height; consequently, 180-(134+31,) or 15 degrees, equals the angle opposite the base. Hence, as .258819 130 :: 0.515038 to the slant height. And, as 1258.69.719340 to the altitude of the pyramid. 136. Required the height of an inaccessible tower on the opposite side of a river, the length of the horizontal base being 170 feet, and the angles of elevation at its extremities 320 and 58°. Ans. 174.27 feet. UNGULAS. Ungulas are portions cut from pyramids, prismoids, cylinders, and cones, by plane sections not parallel to the base.Ungulas cut off from pyramids, or from triangular or rectangular prisms, or from prismoids, are wedges, or the frustums of wedges, and their solidities may be found accordingly. See 43 and 44. The content of a cylindric ungula, cut off by a plane perpendicular to the base, may be found by multiplying the area of the end by its length. The end will of course be the segment of a circle, the area of which may be found by the rules for circular segments. See 25. To find the solidity of a CYLINDRIC UNGULA cut off by a plane inclined to the base :— Multiply the area of the base of the ungula by the difference between the diameter of the cylinder and twice the versed sine of the base of the ungula, and subtract this product from onesixth of the cube of the chord of the base of the segment, if said base be less than a semicircle, but add this product to onesixth of the cube of said chord when the base is greater than a semicircle; multiply the remainder in the former, or the sum in the latter case, by the length of the ungula, and divide the product by twice the versed sine of the segment's base. 137. Find the solidities of the two wedges into which a frustum of a rectangular pyramid is divided by a plane passing through two of the shorter opposite edges of its ends, the length and breadth of its base being 45 and 30, those of its top 36 and 24, and its height 40. Ans. 25,200 and 18,720. 138. The length of a cylindric ungula is 10 feet, the diameter of the cylinder 18 inches, and the section is 6 inches distant from the axis and perpendicular to the base; what is the solidity of the two ungulas ? Ans. 1.9359; and 17.6715 feet. 139. Find the solidity of a cylindric ungula cut off by a plane inclined to the base, the diameter of the cylinder being 25, the length of the ungula 60, and the versed sine of its base 5. Ans. 1709.92. 140. A cylindric vessel 10 inches in diameter, and partly filled with wine, is inclined till the horizontal surface of the fluid leaves 8 inches of the diameter of the bottom dry, and meets the side of the vessel 24 inches from the bottom; required the number of cubic inches of wine in the vessel. Ans. 109.434 inches. 141. Suppose that the fluid in the same vessel leaves only 2 inches of the bottom diameter dry, and that it rises to the same height as before; what is the quantity of wine? Ans. 734.2169 inches. CONIC UNGULAS are elliptic, parabolic, or hyperbolic, according as the plane, which cuts off the ungula, is an ellipse, or portion of an ellipse, or a parabola, or an hyperbola. (See paragraphs 28, 29, and 30.) To find the solidity of a CONIC UNGULA, cut off by a plane passing through the opposite edges of the ends of the conic frustum: Extract the square root of the product of the two diameters of the conic frustum; multiply this root by the less diameter, and subtract the product from the square of the greater diameter; divide the remainder by the difference of the diameters, and multiply the quotient by .2618, and multiply this product by the product of the greater diameter into the altitude of the frustum, and the result will be the solidity of the greater ungula, which, subtracted from the solidity of the frustum, will give the content of the less ungula. 142. Find the solidity of the greater ungula of a conic frus |