or, 4; and if 9 = 134, then 9 + 4 4 11 7; If 8711+ 4, then 8 +7 4 11, and 8 if a + b c, then a = c b; and if a cb, then c = b + a. Whence a term, or any number of terms, may be transposed from one side of an equation to the other, without destroying the equality, by changing the sign of each term so transposed. Again, if 9 + 4 = 13, then 9 + 4 — 13 = 0; and if 9 +4 13 = 0, 9 + 4 = 13, or generally, if a + b = c, then a + b - C = 0; and if a + b c = 0, then a + b = c; if a − c + de, then Hence, if all the terms on one side of an equation be transposed with their signs changed, the resulting expression will be equal to 0. Hence, if a term of an equation be multiplied by any number, the multiplier may be omitted, if all the other terms of the equation be divided by that number; and, conversely, if a term of an equation be divided by any number, the divisor may be omitted, if all the other terms of the equation be multiplied by it. If 42 = 16, then 4 = √16; or if 42 = 12 +4, then 4 = √12 + 4; or if a2 = b, then a = √; or if a2 = b + c d, then a= √b+c= d; if √ a = b, then a = b2; if √ a = b Na= a=b_c. c, then Hence, if a power of any required quantity be given, the quantity itself will be obtained by extracting the corresponding root of the given quantity; and if a root of any quantity be given, that quantity will be obtained by the corresponding power of the given quantity. 9. Given x2 + a = b -c to find x. Answer, x√6 C -a. b2 + c2 a2 10. Given 2 b c x = b2 + c2-a2 to find x. Ans. = 2 bx LOGARITHMS are a series of numbers contrived to alleviate the labour attending many troublesome calculations; the operation of multiplication being effected by the addition, and that of division by the subtraction of logarithms. The raising of powers is effected by multiplying the logarithms by the index of the power; and the extraction of roots, by dividing the logarithm by the index of the root. Logarithms may be considered as the exponents of the powers of a certain number, called the root or radix of the system; or they may be considered as an arithmetical series, indicating the places of the numbers in a corresponding geometrical one. The common system of logarithms, or that which is most used, and most useful in common calculations, has 10 for its root; and the logarithm of any number in that system, is that power of 10 which is equal to the given number. 1; Thus 101 is 10, 10 is 100, 103 is 1000; the log. of 10 therefore is that of 100 is 2; and that of 1000 is 3, &c. 1 1 Now 103, 10, 10, 1, 10' 100 &c. form a continually decreasing geometrical series; their logarithms therefore form a continually decreasing arithmetical series. Nos 1000. 100. 10 Logs. 3. 2 1 1 1. 01 &c. Hence the log. of 1 is 0, and the logs of all numbers less than 1 have their indices negative; the log of every number between 1 and 10 is between 0 and 1; the log of every number between 10 and 100 is between 1 and 2; the log of every number between 100 and 1000 is between 2 and 3, &c. The numbers 1, 2, 3, &c., called the indices of the logarithms, are therefore always less by one than the digits contained in the integral part of the number whose logarithm is taken. From what has been said, it is evident that the logs of 10 and all its integral powers will be whole numbers; but to determine from these what the logarithms of other numbers are, is a task attended with a good deal of trouble; and as our object is rather to point out the principal properties of logarithms, and to explain their practical uses, than to give a view of the many refined artifices which have been employed in computing them, we shall merely shew by one example how the logarithm of any number may be determined: and that example may suffice to convince us how much we are indebted to the industry and the skill of those persons, whose labours have furnished us with the extensive tables which we now possess. EXAMPLE. Let it be required to find the logarithm of 9. Here the proposed number lies between 1 and 10. First, then, the log. of 10 is 1, and the log. of 1 is 0; therefore 1+0 ÷ 2 = = 5 is the arithmetical mean, and √10 × 1= √10=3.1622777 the geometrical mean; hence the log. of 3.1622777 is 5. Secondly, the log. of 10 is 1, and the log. of 3.1622777 is 5; therefore 152=75 is the arithmetical mean, and √10 × 3·1622777 = 5.6234132 is the geometrical mean; hence the log. of 5.6234132 is 75. Thirdly, the log. of 10 is 1, and the log. of 5'6234132 is 75; therefore 1 + 75÷2=875 is the arithmetical mean, and √10 × 5′6234132 = 74989422 the geometrical mean; hence the log. of 7.4989422 is '875. Fourthly, the log. of 10 is 1, and the log. of 7·4989422 is 875; therefore 1875 ÷ 2 = 9375 is the arithmetical mean, and √10 × 7·4989422 = 8·6596431 the geometrical mean ; hence the log. of 8.6596431 is '9375. Fifthly, the log. of 10 is 1, and the log. of 86596431 is 9375; therefore 1 ·9375 ÷ 2 = 96875 is the arithmetical mean, and √10 × 8.6596431 = 9.3057204 the geometrical mean; hence the log. of 9.3057204 is '96875. Sixthly, the log. of 8.6596431 is 9375, and the log. of 9.3057204 is 96875; therefore 9375 + ·96875 ÷ 2 = 953125 is the arithmetical mean, and √86596431 × 9.3057204 = 8.9768713 the geometrical mean; hence the log. of 8.9768713 is '953125. And proceeding in this manner, after 25 extractions, it will be found that the logarithm of 8.9999998 is 9542425; which may be taken for the logarithm of 9, as it differs so little from it, that it is sufficiently exact for all practical purposes. And thus were the logarithms of almost all the prime numbers at first computed. Having found in this manner the logarithms of all prime numbers, which are those that have no divisor; as 2, 3, 5, 7, 11, &c. the logarithms of all other numbers may easily be found. For, as 3 x 5 15, the logarithms of 3 and 5 added together will be the logarithm of 15, and as 32 = 9, twice the logarithm of 3 will be the logarithm of 9, &c. And thus may an idea be formed of the method of constructing a table of logarithms. The logarithms constructed to the radix 10 possess one very important practical advantage; which is, that all numbers consisting of the same figures, whether they be integral, fractional, or mixed, have the decimal parts of their logarithms the same. For, the log. of a × 10" is equal to n added to the log. of a; and the log. of a ÷ 10" is equal to the log. of a diminished by n. 5. Find a fourth proportional to 567, 913, and 485. As to perform the required operation without the aid of logarithms, we must divide the product of 913 and 485, by 567; therefore, to perform it logarithmically, from the sum of the logarithms of 913 and 485, we must deduct the logarithm of 567 for the logarithm of the result. Thus 567 log. 2.753583 913 log. 2.960471 485 log. 2.685742 5.646213 Required fourth proportional..781 log. 2.892633 6. Extract the 6th root of 148 148 log. 2170262 62.170262 Required root..23 log. 361710 GEOMETRY. DEFINITIONS. 1. GEOMETRY is the science by which we determine the relations between quantities which have extension. 2. A Point is that which has position, but no magnitude. 3. A Line is length only. 4. A Surface, or Superficies, is a figure which has length and breadth only. 5. A Solid is a figure which has length, breadth, and thickness. 6. A Right Line, or Straight Line, is one which does not change its direction between its extremities. 7. A Crooked Line is one which changes its direction at intervals. 8. A Curve Line is one which continually changes its direction. |