diftant from the greater end is the centre of the fegment? Anf. 30 feet. 64. A barrel is filled with pure fpirits, and weighs, when full, 66 lb. How many gallons does it contain, allowing 6 lb. for the weight of the barrel? Anf. 8.29 gallons. 65. A column of the atmosphere, whose base is a square inch, weighs 15 lb.; and supposing the atmosphere to prefs equally in all directions,-required the preffure upon a middlefized man, whose surface may be reckoned 16 fquare feet. Anf. 34560 lb. 66. Suppofe the atmosphere, in a mean ftate, balance mercury in the barometer 291 inches high, required the height to which water may be raised by means of a pump, the state of the atmosphere being the fame. Anf. 3414 feet. 67. Suppofe the earth's mean distance from the fun is 82 millions of miles, and goes round him in 365 days 5 hours 49 minutes, at what rate does it travel per hour? Anf. 58776 miles. 68. Light paffes from the fun to the earth in 8 minutes 15 feconds of time, and the velocity of the earth in its orbit is 58776 miles per hour; required the proportion they bear to each other. Anf. The velocity of light is to that of the earth as 993934 is to 979 6. 69. In what time would the earth fall to the fun at the rate of 58776 miles per hour? Anf. 58 days 3 hours 7 minutes 38 feconds. 70. The paving of a fquare infcribed in a femicircle, whose fide coincides with the diameter, and whofe oppofite angles are in the circumference, at 9d. per square foot, coft 331. 158. required the diameter of the circle. Anf. 22.3606 yards. 71. A triangle, whofe three fides are 800, 640, and 360 feet, is infcribed in a circle; it is required to find the diameter of the circle. Anf. 820.211 feet. 72. The 72. The three fides of a triangular pyramid are 312, 360 and 96, and altitude 100 feet; it is required to find the folidity of a cone circumfcribed about the pyramid, and whose altitude is equal to that of the pyramid. Anf 3981978 cubic feet. 73. Required the dimenfions of a cone, whose area of the bafe, curve fuperficies, and folid content, are in geometrical progreffion, and the area of the base equal to the rectangle of the bafe's diameter and axis. Anf. The diameter of the cone's base is 24.7036, and axis 19.4022. 74. The area of an equilateral triangle being 720, required the fide. Anf. 40.7776. 75. Suppose I have a circular inclosure of an acre of ground, how long ought a cord be, that, faftened in the circumference of the inclosure as centre, will strike an arch that wil. divide the faid inclosure into two equal parts? Anf. 45.47898 yards76. A refervoir is fupplied from a pipe of 6 inches bore. How many pipes of 3 inches bore will be fufficient to discharge the fame quantity? Anf. 4 pipes. ALGEBRA. APPENDIX.. ALGEBRA. ALGEBRA LGEBRA is a general method of computation, by which many useful problems in geometry and arithmetic are folved, which, without its aid, would be impoffible. The principles on which the rules are founded are fimilar to thofe in common arithmetic. Certain fymbols and characters are admitted into this fcience, to give it that extent and excellence which it poffeffes above all other methods of computation. Through all the fteps of an algebraic operation, thefe fymbols may be fo conducted as to be preferved diftin&ly in view, with their relations and affections to each other, and at last to produce a canon, or general rule, by which not only the question propofed is folved, but every other queftion of the like conditions. Whereas, in the courfe of an arithmetical operation, the original numbers dif appear. NOTATION. 1.-Algebraic figns only affect thofe fymbols to which they are prefixed. 2. Quantity is that which is made up of parts, or is capable of being encreafed by addition or diminished by fubtraction. Hence a quantity may be introduced into an algebraic compu tation two different ways, either as a decrement, or as an increment, that is, as a negative quantity, or as a positive one. SIGNS. 3. + (plus) fignifies Addition, or that the quantity to which it is prefixed is pofitive *. (minus) fignifies Subtraction, or that the quantity to which it is prefixed is negative. × fignifies Multiplication †. = Equal to, or the sign of equality. (the radical fign) denotes the fquare root of the quanEity annexed. 4. A quantity may be represented by any fymbol or character. It is, however, a pretty general custom to use the first letters of the alphabet, a, b, c, &c. for known quantities, and the laft letters, x, y, z, for unknown ones. In the following com pendium we will follow the general method. To examplify thefe figns, let us fuppofe a=3, b=8, c=12, d=10, e=4, m=6, p=1, and s=5. 5.-Then the fum of a and b is reprefented thus, a+b=11. The difference of d and p, d-p=9. The *When no fign is prefixed to a quantity, is understood. † When no fign is placed between two quantities, x is underflood. The product of a multiplied by c*, The quotient of d divided by s, The fquare root of ac, dxc or de=36. d dsor-=2 Vac-6.. 6. Plus and minus are always opposed to each other. Thus, fignifies lofs; if + fignifies flock, iffignifies gain, negative one. And fo on. 7.—When letters are placed together without any fign between them, they denote the rectangle or product of the quantities they reprefent. Thus, the product of a into b is ab=24. A number prefixed to any quantity is understood to multiply the quantity, and is called its co-efficient. Thus, 6a=18.. 8.-Divifion is often reprefented by placing the dividend in the form of a numerator, and the divifor in that of a denomi 9.-The continual multiplication of quantities, by others of the like kind and dimenfion, is called powers of that quantity, and are commonly expreffed by fmall numbers placed at the corners of the letters, called indices, or exponents. Thus, the fquare of a is a2, the cube a3, the fourth power a*, &c. The exponent of the original quantity, or root, is unity, and is feldom or never expreffed. 10.-Quantities of the like dimenfions, which, by their fuc-. ceffive multiplication, produce any given quantity, are called roots of the given quantity. Thus, a is the iquare root a2. When no fign is marked between two or more quantifies, it denotes their product. |