9. From the same principle it follows, that A B B λA-B. For, leta, and consequently A=BXQ: we shall then have λ λ=λ B+λQ; whence &q=2A-2 B. So that the logarithm of the quotient is equal to the logarithm of the dividend minus that of the divisor; or the logarithm of a fraction is equal to the logarithm of the numerator made less by that of the denominator. = 1 10. Farther, 2 s−" — — n λ s. For A-" = therefore A" 2 A" = 21 — 2 A" — 0 — nλA: which is no other than -λ A= p -λ A. Ρ 12. Suppose there be two systems of logarithms whose roots or bases are r and s. Let any number N have p for its logarithm in the first system, and q for its logarithm in the second : P we shall have N=r1 and N=s; which gives r"=s2, and s = r2. Therefore, taking the logarithms for the system r, we shall have 2s=22r; or, if in the system r we have 2 r=1, then 2, s= 1 px. Thus, knowing the logarithm p of any λε number N, for the system whose base is r, we may obtain the logarithm q of the same number for the system s, by multiplying p by a fraction whose numerator is unity and denominator the logarithm of s taken in the system r. 13. In the system of logarithms first constructed by Baron Napier, the great inventor, r = 2.718281828459, &c. and the exponents are usually denominated Napierian or Hyperbolic logarithms; the latter name being given because of the relation between these logarithms and the lines and asymptotic spaces in the equilateral hyperbola: so that in this system n is always the hyperbolic logarithm of (2-71828, &c.)". But in the system constructed by Mr. Briggs (corresponding with the spaces in a hyperbola whose asymptotes make an angle of 25° 44' 25" 28"), called common or Briggean logarithms, r = 10; so that the common logarithm of any number is the index of that power of 10 which is equal to the said number. -69897 3.955351 Thus, if 50 10 = and 9023 = 10 ; then is 1.69897 the common logarithm of 50, and 3.955351 the common logarithm of 9023. 14. The rules for the management and application of logarithms being given in the best collections of logarithmic tables, are here omitted. The tables published in England by Dr. Hutton, those published in France by Callet, and those recently published by Professor Babbage, may be recommended as the most correct and best fitted for scientific use. Mr. Galbraith's tables are correct and valuable. If the reader wish for neatly arranged tables in small compass, for the practical purposes of a man of business, I would recommend those of Mr. Woollgar, given in the Mechanics' Magazine, those recently published by Mr. J. R. Young, and those given by Mr. Carr, in his valuable "Synopsis of Practical Philosophy." SECTION XIII.-Computation of Formulæ. Since the comprehension, and the numerical computation of formulæ expressed algebraically, are of the utmost consequence to practical men, enabling them to avail themselves advantageously of the theoretical results of men of science, as well as to express in scientific language the results of their own experimental or other researches; it has appeared expedient to present brief treatises of Arithmetic and Algebra. The thorough understanding of these two initiatory departments of science will serve essentially in the application of all that follows in the present volume; and that application may probably be facilitated by a few examples, as below :— Ex. 1. Let a=5, b=12, c=13, and s=a+b+c; then what is the numerical value of the expression, √ [ž s (ž s − a) († s · b) (1⁄2 s - c)], which denotes the area of the triangle whose sides are 5, 12, and 13? Here s a+b+c=5+12+13=30; s=15; sc As a 15-5-10; s-b-15-12-3; Consequently, by substituting the numerical values of the several quantities between the parentheses for them, we shall have = = (15 x 10 x 3 x 2)=√900=30, the value required. Ex. 2. Suppose g= 32, t 6: required the value of gt, an expression denoting the space in feet which a heavy body would fall vertically from quiescence in six seconds, in the latitude of London. Here gt-16 X6-96×6=579 feet. Ex. 3. Given D=6, d=4, h = 12, я = 3.141593; required the value of xh (D2+D d+d), a theorem for the solid content of a conic frustrum whose diameters of the two ends are D, d, and height h. Here d3=36, d d=6×4=24, d2 = 16, 12 = '2618 nearly. Hence h (D2+D d+d2)=2618 (36+24+16) 12 =2618 X 76 × 12=3.141593 × 76=238.761068. 2 Ex. 4. Let a=1, h=25, g=193 inches: what is the value of 2 agh? This being the expression for the cubic inches of water discharged in a second, from an orifice whose area is a, and depth below the upper surface of water in the vessel or reservoir, h, both in inches. Here 2 a ✔g h=2 √(25 × 193)=10✓✓193 = 10 × 13·89244 =138.9244 cubic inches. Ex. 5. Suppose the velocity of the wind to be known in miles per hour; required short approximative expressions for the yards per minute, and for the feet per second. First 176060-88-29-30 nearly. Also 5280÷(60 x 60)=5280-88-44-14 nearly. If, therefore, n denote the number of miles per hour: 30 n will express the yards per minute; and 14 n, the feet per second. These are approximative results to render them correct, where complete accuracy is required, subtract from each result its 45th part, or the fifth part of its ninth part. Thus suppose the wind blows at the rate of 20 miles per hour: 600 yards per minute, or more cor Then 30 n=30 x 20 = rectly 600400-600-13-5863 yards. Also 11⁄2 n=30 feet per second. or, correctly, 30-38-303-29 feet. 45 Conversely, of the feet per second will indicate the miles per hour, correct within the 45th part, which is to be added to obtain the true result. Ex. 6. To find a theorem by means of which it may be ascertained when a general law exists, and what that law is. Suppose, for example, it were required to determine the law which prevailed between the resistances of bodies moving in the air and other resisting media, and the velocities with which they move. Let v, v, denote any two velocities, and R,r the corresponding resistances experienced by a body moving with those velocities: we wish to ascertain what power of v it is to which R is proportional. Let x denote the index or exponent of the power; then will v*: v :: R: r, if a law subsist. Div.the consequents by the antecedents, we have 1: V ::1: R This, expressed logarithmically, log. R or = log. vlog. v Hence, the quotient of the differences of the logs. of the resistances, divided by the difference of the corresponding velocities, will express the exponent a required. This theorem is of very frequent application in reference to the motion of cannon balls, of barges on canals, of carriages on rail roads, &c. and may indeed be applied to the planetary motions. CHAPTER III. PRINCIPLES OF GEOMETRY. Definitions. 1. GEOMETRY is a department of science, by means of which we demonstrate the properties, affections, and measures of all sorts of magnitude. 2. Magnitude is a continued quantity, or any thing that is extended; as a line, surface, or solid. 3. A point is that which has no parts: i. e. neither length, breadth, nor thickness. 4. A line is a length without breadth or thickness. Cor. The extremes of a line are points. 5. A right line is that which lies evenly, or in the same direction, between two points. A curve line continually changes its direction. Cor. Hence there can only be one species of right lines, but there is infinite variety in the species of curves. 6. An angle is the inclination of two lines to one another, meeting in a point, called the angular point. When it is formed by two right lines, it is a plane angle, as a; if by curve lines, it is a curvilinear angle. 7. Aright angle is that which is made by one right line A B falling upon another c D, and making the angles on each side equal, A B C A B D; so that A B does not incline more to one side than another: A B is called a perpendicular. All other angles are called oblique angles. 8. An obtuse angle is greater than a right angle, as R. C B D 9. An acute angle is less than a right angle, as s. 10. Contiguous angles are those made by one line falling upon another, and joining to one another, as R, S. |