But since it is necessary to add 140 to triple of x to make 230, it follows, that by taking 140 from 230 we have exactly the triple of x, or or whence it follows that 3x230 140, 3 x = 90, 9 x = = 30. By adding to 30 the excess 40 of the middle part above the least, we have 70 for the middle part. By adding to 70 the excess 60 of the greatest above the middle part, we have 130 for the greatest. 6. If the known numbers were different from those which I have used in the enunciation, we should still resolve the question by following the course pursued in the preceding article, but we should be obliged to repeat all the reasonings and all the operations, by which we have arrived at the number 30, because there is nothing to show how this number is composed of 230, 40, and 60. To render the solution independent of the particular values of numbers, and to show how the value of the unknown quantity is fixed by means of the known quantities, I will enunciate the problem thus ; To divide a given number a into three such parts, that the excess of the middle one above the least shall be a given number b, and the excess of the greater above the middle one shall be a given number c. Designating as above by x the unknown quantity and making use of the common signs and the symbols a, b, c, which represent the known quantities in the question, the reasoning already given will be repeated. The least part, the middle part = x+b, the greatest =x+b+c. and the sum of these three makes the number to be divided; hence, x + x + b + x + b + c = ɑ. This expression, which is so simple, may be still further abridged; for since it appears that x enters three times into the number to be divided and b twice, instead of x + x + x, I shall write 3 x, and instead of +b+b, I shall write + 2b, and it will become 3x+2b+c=a. From this last expression it is evident, that it is necessary to add to triple the number represented by x, double the number represented by b, and also the number c, in order to make the number a; it follows then, that if from the number a we take double the number b and also the number c, we shall have exactly the triple of x, or that 3 x α 26 -C. Now being one third of three times x, we thence conclude It should be carefully observed, that having assigned no particular value to the numbers represented by a, b, c, the result to which we have come is equally indeterminate as to the value of x; it shews merely what operations it is necessary to perform upon these numbers, when a value is assigned to them, in order thence to deduce the value of the unknown quantity. be reduced to common language by writing, instead of the letters, the numbers which they represent, and instead of the signs, the kind of operation which they indicate; it will then become, as follows; From the number to be divided, subtract double the excess of the middle part above the least, and also the excess of the greatest above the middle part, and take a third of the remainder. If we apply this rule, we shall determine, by the simple oper ations of arithmetic, the least part. The number to be divided being for example 230, one excess 40, and the other 60, if we subtract as in the preceding article twice 40, or 80, and CO from 230, there will remain 90, of which the third part is 30, as we have found already. If the number to be divided were 520, one excess 50, and the other 120, we should subtract twice 50, or 100, and 120 from 520, and there would remain 300, a third of which or 100 would be the smallest part. The others are found by adding 50 to 100, which makes 150, and 120 more to this, which makes 270, so that the parts sought would be 100, 150, 270, and their sum would be 520, as the question requires. It is because the results in algebra are for the most part only an indication of the operations to be performed upon numbers in order to find others, that they are called in general formulas. This question, although more complicated than that of article 1, may still be resolved by ordinary language, as may be seen in the following table, where against each step is placed a translation of it into algebraic characters. PROBLEM. To divide a number into three such parts, that the excess of the middle one above the least shall be a given number, and the excess of the greatest above the middle one shall be another given number. By common language. SOLUTION. By algebraic characters. The middle part will be the Let the number to be divided be denoted by the excess of the middle part the excess of the greatest least, plus the excess of the The middle part will be x + b. mean above the least. The greatest part will be the middle one, plus the excess of the greatest above the middle The greatest will be a+b+c. one. The three parts will together form the number proposed. Whence the least part, plus the least part, plus the excess of the middle one above the least, plus also the least part, Whence plus the excess of the middle >x+x+b+x+b+c=a. a. b. C. x. Whence three times the least' part, plus twice the excess of least, plus also the excess of 3 x + 2 b + c = a. Whence three times the least Whence in fine, the least part will be equal to a third of what remains after deducting from the number to be divided twice the excess of the middle part above the least, and also the excess of the greatest above the middle one. 3x=α- 2b- -C. x= a-2b C 3 7. The signs mentioned in article 2 are not the only ones used in algebra. New considerations will give rise to others, as we proceed. It must have been observed in article 2, that the multiplication of x by 2, and in articles 5 and 6 that of x by 3 and that of b by 2, is denoted by merely writing the figures before the letters ≈ and b without any sign between them, and I shall express it in this manner hereafter; so that a number placed before a letter is to be considered as multiplied by the number represented by that letter, 5x, 5 a, &c. signify five times X, five times a, &c. 3x or 37, &c. signifies of x or three times a divided by 4, &c. 4 x In general, multiplication will be denoted by writing the factors in order one after the other without any sign between them, whenever it can be done without confusion. Thus the expressions ax, bc, &c. are equivalent to a X x, bxc, &c., but we cannot omit the sign when numbers are concerned, for then 3 × 5, the value of which is 15, becomes 35. In this case we often substitute a point in the place of the usual sign, thus, 3.5. Equations. 8. If the solution of the problems in articles 3 and 6 be examined with attention, it will be found to consist of two parts entirely distinct from each other. In the first place, we express by means of algebraic characters the relations established by the nature of the question between the known and unknown quantities, from which we infer the equality of two quantities among themselves; for instance, in article 3 the quantities 2 x + b and and in article 6 the quantities 3x+2b+c and a. We afterwards deduce from this equality a series of conse quences, which terminate in showing the unknown quantity a to be equal to a number of known quantities connected together by operations, that are familiar to us; this is the second part of the solution. These two parts are found in almost every problem which belongs to algebra. It is not easy, however, at present to give a rule adapted to the first part, which has for its object to reduce the conditions of the question to algebraic expressions. To be able to do this well, it is necessary to become familiar with the characters used in algebra, and to acquire a habit of analyzing a problem in all its circumstances, whether expressed or implied. But when we have once formed the two numbers, which the question supposes equal, there are regular steps for deducing from this expression the value of the unknown quantity, which is the object of the second part of the solution. Before treating of these I shall explain the use of some terms which occur in algebra. An equation is an expression of the equality of two quantities. The quantities which are on one side of the sign = taken. together are called a member; an equation has two members. That which is on the left is called the first member, and the other the second. In the equation 2 x + b = a, 2 x + b is the first member, and a is the second member. The quantities, which compose a member, when they are separated by the sign + or, are called terms. Thus, the first member of the equation 2x + b = a contains two terms, namely, 2x and + b. |
