To prove the existence of real and unequal roots Use of division of roots for facilitating the resolution of an equa tion, when the coefficients are large Method of approximation according to Lagrange Of Proportion and Progression Fundamental principles of proportion and equidifference Of the changes which a proportion may undergo Of progression by differences To determine any term whatever of this progression To determine the sum of the terms Of progression by quotients 222 ib. 226 ib. 229 ib. - 238 239 240 Progressions by quotients, the sum of which has a determinate limit Manner of reducing all the terms of a progression by quotients from the expression of the sum Division of m by m-1, continued to infinity In what cases the quotient of this operation is converging and may be taken for the approximate value of the fraction Of diverging series Theory of exponential quantities and of logarithms m m Remarkable fact, that all numbers may be produced by means of the powers of one ib. 241 243 245 What is meant by the term logarithm, and the base of logarithms 246 Method of calculating a table of logarithms ib. Note-Long's method, and a table of decimal powers The use of logarithms, in finding the numerical value of formulas ib. Application of logarithms to the Rule of Three 255 The logarithms of numbers in progression by quotients form a To compare the values of sums payable at different times ELEMENTS OF ALGEBRA. Preliminary Remarks upon the Transition from Arithmetic to Algebra-Explanation and Use of Algebraic Signs. 1. Ir must have been remarked in the Elementary Treatise of Arithmetic, that there are many questions, the solution of which is composed of two parts; the one having for its object to find to which of the four fundamental rules the determination of the unknown number by means of the numbers given belongs, and the other the application of these rules. The first part, independent of the manner of writing numbers, or of the system of notation, consists, entirely in the development of the consequences which result directly or indirectly from the from the enunciation, or from the manner in which that which is enunciated connects the numbers given with the numbers required, that is to say, from the relations which it establishes between these numbers. If these relations are not complicated, we can for the most part find (by simple reasoning the value of the unknown numbers. In order to this it is necessary to analyze the conditions, which are involved in the relations enunciated, by reducing them to a course of equivalent expressions, of which the last ought to be one of the following; the unknown quantity equal to the sum, or the difference, or the product, or the quotient, of such and such magnitudes. This will be rendered plainer by an example. To divide a given number into two such parts, that the first shall exceed the second by a given difference. In order to this we would observe 1, that, The greater part is equal to the less added to the given excess, and that by consequence, if the less be known, by adding to it this excess we have the greater; 2, that, The greater added to the less forms the number to be divided. Substituting in this last proposition, instead of the words, the greater part, the equivalent expression given above, namely, the less part added to the given excess, we find that, The less part, added to the given excess, added moreover to the less part, forms the number to be divided. But the language may be abridged, thus, Twice the less part, added to the given excess, forms the number to be divided; whence we infer, that, Twice the less part is equal to the number to be divided diminished by the given excess ; and that, Once the less part is equal to half the difference between the number to be divided and the given excess. Or, which is the same thing, The less part is equal to half the number to be divided, diminished by half the given excess. The proposed question then is resolved, since to obtain the parts sought it is sufficient to perform operations purely arithmetical upon the given numbers. If, for example, the number to be divided were 9, and the excess of the greater above the less 5, the less part would be according to the above rule, equal to 2 less, or, or 2; and the greater, being composed of the less plus the excess 5, would be equal to 7. 2. The reasoning, which is so simple in the above problem, but which becomes very complicated in others, consists in general of a certain number of expressions, such as added to, diminished by, is equal to, &c. often repeated. These expressions relate to the operations by which the magnitudes, that enter into the enunciation of the question, are connected among themselves, and it is evident, that the expressions might be abridged by representing each of them by a sign. This is done in the following manner. 19 To denote addition we use the sign +, which signifies plus. For subtraction we use the sign which signifies minus. For multiplication we use the sign ×, which signifies multiplied by. To denote that two quantities are to be divided one by the other, we place the second under the first with a straight line between them; signifies 5 divided by 4. Lastly, to indicate that two quantities are equal, we place between them the sign = which signifies equal. These abbreviations, although very considerable, are still not sufficient, for we are obliged often to repeat the number to be divided, the number given, the less part, the number sought, &c. by which the process is very much retarded. With respect to given quantities, the expedient which first offers itself is, to take for representing them determinate numbers, as in arithmetic, but this not being possible with respect to the unknown quantities, the practice has been to substitute in their stead a conventional sign, which varies as occasion requires. We have agreed to employ the letters of the alphabet, generally using the last; as in arithmetic we put x for the fourth term of a proportion, of which only the three first are known. It is from the use of these several signs that we derive the science of Algebra. I now proceed by means of them to consider the question stated above (1). I shall represent the unknown quantity, or the less number, by the letter x, for example, the number to be divided and the given excess by the two numbers 9 and 5 ; the greater number, which is sought, will be expressed by x + 5, and the sum of the greater and less by x + 5+ x; we have then x + 5 + x = 9; but by writing 2x for twice the quantity x there will result 2x + 5 = 9. This expression shows that 5 must be added to the number 2x to make 9, whence we conclude that By comparing now the import of these abridged expressions, which I have just given by means of the usual signs, with the process of simple reasoning, by which we are lead to the solution, we shall see that the one is only a translation of the other. The number 2, the result of the preceding operations, will answer only for the particular example which is selected, while the course of reasoning considered by itself, by teaching us, that the less part is equal to half the number to be divided, minus half the given excess, renders it evident, that the unknown number is composed of the numbers given, and furnishes a rule by the aid of which we can resolve all the particular cases comprehended in the question. The superiority of this method consists in its having reference to no one number in particular; the numbers given are used throughout without any change in the language by which they are expressed; whereas, by considering the numbers as determinate, we perform upon them, as we proceed, all the operations which are represented, and when we have come to the result there is nothing to show, how the number 2, to which we may arrive by any number of different operations, has been formed from the given numbers 9 and 5. 3. These inconveniences are avoided by using characters to represent the number to be divided and the given excess, that are independent of every particular value, and with which we can therefore perform any calculation. The letters of the alphabet are well adapted to this purpose, and the proposed question by means of them may be enunciated thus, To divide a given number represented by a into two such parts that the greater shall have with respect to the less a given excess represented by b. Denoting always the less by x; The greater will be expressed by x+b; Their sum, or the number to be divided, will be equal to x + x + b, or 2x+b; The first condition of the question then will give 2x + b = a. Now it is manifest that, if it is necessary to add to double of x, or to 2x, the quantity b in order to make the quantity a, it will follow from this, that it is necessary to diminish a by b to obtain 2x, and that consequently 2x ab. We conclude then that half of 2x or x = a 2 b 2 This last result, being translated into ordinary language, by substituting the words and phrases denoted by the letters and signs which it contains, gives the rule found before, according to which, in order to obtain the less of two parts sought we sub |