The two series being thus arranged, the terms in the arithmetical series, are called the logarithms of the corresponding terms in the geometrical series; that is, O is the logarithm of 1, 2 of 4, 3 of 8, and so on. Hence it appears that the logarithms of the terms of the geometrical series have the two following properties. 1. The sum of the logarithms of any two numbers or terms in the geometrical series is equal to the logarithm of that number or term, of the series which is equal to their product. Example. Let the terms of the geometrical series be 4 and 32, then, the terms of the arithmetical series corresponding to them (that is, their logarithms) are 2 and 5. Now, the product of the numbers in the geometrical series is 128, and the sum of their logarithms is 7; and is appears by the table that the logarithm of the former, 128, is 7, the latter number. In like manner, if the numbers or terms of the geometrical series be 16 and 64, the logarithms of which are 4 and 6, we find from the table, that 10 4+6, is the logarithm of 16 x 64; that is, 1024. 2. The difference of the logarithms of any two numbers or terms of the geometrical series, is equal to the logarithm of that term of the series which is equal to the quotient arising from the division of the one number by the other. Example. - Take the terms 128 and 32, the the logarithms of which are 7 and 5. The greater of those numbers, 128, divided by the less, 32, is equal to 4; and the difference of their logarithms is 2. By inspecting the two series, this last number, 2, is found to be the logarithm of the former, 4. In like manner, if the terms of the geometrical series be 1024 and 16, the logarithms of which are 10 and 4, we find that 1024-16=64; and that 10-4-6. But it appears, from the table, that the latter number, 6, is the logarithm of the former, 64. By these two properties of logarithms, may be found, with facility, the product or the quotient of any two terms of a geometrical series, to which there is adapted an arithmetical series; so that each number has its logarithm opposite to it, as in the preceding short table. For, it is evident, that to multiply two numbers, it is necessary only to add their logarithms; and opposite to that logarithm, which is their sum, the required product will be found. Example. Multiply 16 by 128. To the logarithm of 16, which is 4, add the logarithm of 128, which is 7, the sum of those two logarithms, namely 11, the table shows to be the logarithm of 2048; which is the product of 16 multiplied by 128, the product required. To divide any number in the table by any other number, the logarithm of the divisor must be subtracted from the logarithm of the dividend; the remainder being found among the logarithms, opposite to it will appear the quotient sought. Example. Divide 2,048 by 128. From the logarithm of 2048, that is 11, subtract the lo garithm of 128, which is 7; opposite to the remainder, 4, stands 16, which is the quotient sought. In order to form the logarithms of the intervening numbers to those of the table above, a great number of geometrical and arithmetical means were conceived to be interposed between each two adjoining terms of the geometrical and arithmetical series; then, like as the terms of the arithmetical series, are the logarithms of the corresponding terms of the geometrical series, the interpolated terms of the former will also be the logarithms of the corresponding interpo. lated terms of the latter. Upon these principles, tables are formed which, exhibiting the logarithms of all numbers within certain limits, may be applied to simplify calculations. QUESTIONS. What are fluxions? Who claimed the invention of fluxions? What is a constant quantity? What is a variable quantity? What are logarithms? By whom invented? What is the first property of the logarithms of the terms of the geometrical series? What is their second property? How are these properties applied to the multiplication of two numbers? How are they applied to the division of numbers? What are the examples given? CHAP. XV. ALGEBRA. ALGEBRA is a general method of reasoning concerning the relations which magnitudes of every kind bear to each other, in respect of quantity. Or, algebra is a method of computation by symbols, which have been invented for expressing the quantities that are the objects of this science; and also, their mutual relation and dependance. It is sometimes called universal arithmetic; because its first principles and operations are similar to those of common arithmetic. The symbols which it employs to denote magnitudes are, however, more general and more extensive in their application than those employed in common arithmetic. Hence, and from the great facility with which the various relations of magnitudes to one another may be expressed by means of a few signs or characters, the application of algebra to the resolution of problems is more extensive than that of simple arithmetic. Algebra represents quantities of all kinds, by the letters of the alphabet; and the operations to be performed with them are denoted by certain characters, instead of being expressed by words at length. The term algebra is derived from the Arabians, who assert that the science was invented by Mahomet Ben Musa, who flourished about the ninth or tenth century. But algebra seems not to have been entirely unknown to the ancient mathematicians. Algebra is either numeral, or literal, Algebra numeral, or vulgar, is that which is chiefly concerned in the resolution of arithmetical questions. In this, the quantity sought is represented by some letter or character; but all the given quantitities are expressed by numbers. Algebra literal, or specious, or the new alge bra, is that in which all the quantities known, or unknown, are expressed by their species, or letters of the alphabet. Specious algebra is not, like the numeral, confined to certain kinds of problems; but serves, universally, for the investigation or invention of theorems, as well as the solution and demonstration of all kinds of problems, both arithmetical and geometrical. The letters used in algebra, severally, represent either lines or numbers, as the problem is either arithmetical or geometrical; and, together, they express planes, solids, and powers more or less high, as the letters are in greater or less number. For instance, if there be two letters, a, b, they represent a rectangle, whose two sides are expressed, one by the letter a, and the other by the letter b; so that by their mutual multiplication, they produce the plane a b. Where the same letter is repeated twice, as, a, a, they denote a square. Three letters, as, a, b, c, represent a solid, whose three dimensions are expressed by the three letters a, b, c; the length by a, the breadth by b, and the depth by c; so that, by their mutual multiplication, they produce the solid, a b c. As the multiplication of dimensions is expressed by the multiplication of letters, and as the number of these may be so great as to become incommodious, the method pursued is, only to write down the root; and, on its right, to insert the index of the power; that is, the number of letters of which the quantity to be expressed consists; as, a3, a1, and so on; the |