we have 1 1−1+1 =1=1+1-1-1+1+1, &c., which series contains twice the series found, (Art. 178), 1-1+1-1+1, &c. Now, as we have found this to be equal to 1, it is not extraordinary that we should find, or 1, for the value of that which we have just determined. 1 By making a=1, we shall have =1=1+1~1 128 ·}+&+8—3}2, &c. If a=1, we shall have 1 1 &c. And if we take the four leading terms of this series, we have 1, which is only less than 2. Let us suppose again a=3, and we shall have ==1+3—2—1i+&+, &c. this series is therefore equal to the preceding one, and by subtracting one from the other, we obtain 1-2-3 +, &c., which is necessarily =0. 299 7 183. The method which has been here explained, serves to resolve, generally, all fractions into infinite series; which is often found, as has been observed by EULER in his Algebra, to be of the greatest utility; it is also remarkable, that an infinite series, though it never ceases, may have a determinate value. It should likewise be observed, that from this branch of Mathematics, inventions of the utmost importance have been derived, on which account, the subject deserves to be studied with the greatest attention. 6 1 1 or into 4 5 Ex. 13. It is required to convert to an infinite series. 6 6 Ex. 14. It is required to convert +. + + +&c. 5 25 125 625' 1 1 1 1 +. + ++, &c. 5 52 53 54 CHAPTER III. ON SIMPLE EQUATIONS, INVOLVING ONLY ONE UNKNOWN QUANTITY, 184. In addition to what has been already said, (Art. 34), it may be here observed, that the expression, in algebraic symbols, of two equivalent phrases contained in the enunciation of a question. is called an equation, which, as has been remarked by GARNIER, differs from an equality, in this, that the first comprehends an unknown quantity combined with certain known quantities; whereas the second takes place but between quantities d that are known. Thus, the expression a=+ 2 2' (Art. 102), according to the above remark, is called an equality; because the quantities a, s, and d, are supposed to be known. And the expression x+x—d=s, (Art. 103), is called an equation, because the unknown quantity x, is combined with the given quantities d and s. Also, x-a=0 is an equation which asserts that x-a is equal to nothing, and therefore, that the positive part of the expression is equal to the negative part. 185. A simple equation is that, which contains only the first power of the unknown quantity, or the unknown quantity merely in its simplest form, after the terms of the equation have been properly arranged: Thus, x+a=b; ax+bx=c; or +=d, &c. 4 where denotes the unknown quantity, and the other letters, or numbers, the known quantities. § I. REDUCTION OF SIMPLE EQUATIONS. 186. Any quantity may be transposed from one side of an equation to the other, by changing its sign. Because, in this transposition, the same quantity is merely added to or subtracted from each side of the equation; and, (Art. 48, 49,) if equals be added to or subtracted from equal quantities, the sums or remainders will be equal. Thus, if x+5=12; by subtracting 5 from each side, we shall have x+5-5-12-5; but 5-5=0, and 12-5=7; hence x=7. Also, if x+a-b-2x; by subtracting a from each side, we shall have x+a-a-b--2x—a ; and by adding 2x to each side, we shall have x+a-a+2x=b-2x-a+2x; but a-a=0, and -2x+2x=0; therefore x+2x=b-a, or 3x=b-a. Again, if ax-c=d, and c be added to each side, ax-c+c=d+c, or ax=d+c. Also, if 5x-7=2x+12; by subtracting 2x from each side, we shall have 5x-7-2x-2x+12-2x, or 3x-7=12; subtracting 7, or, which is the same thing, adding +7 to each side of this last equation, and we shall have 3x-7+7=12+7; but 7-7=0,... 3x=19. Finally, if x-a+b=c-2x+d; then, by sub: tracting b from each side, we shall have x=a+bbc 2x+d—b ; |