As an example, let us double the number of decimal places already obtained, which are contained in 3·46410161513. The remainder is 537253550831, the divisor 692820323026, and the process is as in (B). Hence the square root of 12 is, 3·4641016151377545870549; which is true to the last figure, and a little too great; but the substitution of 8 instead of 9 on the right hand would make it too small. EXERCISES. Numbers. | Square roots. ·001728 | ·0415692194 64·34 | 8·02122185 8074 | 89·8554394 10 | 3·16227766 1·57 | 1·2529964086141667788495 SECTION VIII. ON THE PROPORTION OF NUMBERS. 170. When two numbers are named in any problem, it is usually necessary, in some way or other, to compare the two; that is, by considering the two together, to establish some connexion between them, which may be useful in future operations. The first method which suggests itself, and the most simple, is to observe which is the greater, and by how much it differs from the other. The connexion thus established between two numbers may also hold good of two other numbers; for example, 8 differs from 19 by 11, and 100 differs from 111 by the same number. In this point of view, 8 stands to 19 in the same situation in which 100 stands to 111, the first of both couples differing in the same degree from the second. The four numbers thus noticed, viz.: 8, 19, 100, 111, are said to be in _arithmetical[26] proportion_. When four numbers are thus placed, the first and last are called the _extremes_, and the second and third the _means_. It is obvious that 111 + 8 = 100 + 19, that is, the sum of the extremes is equal to the sum of the means. And this is not accidental, arising from the particular numbers we have taken, but must be the case in every arithmetical proportion; for in 111 + 8, by (35), any diminution of 111 will not affect the sum, provided a corresponding increase be given to 8; and, by the definition just given, one mean is as much less than 111 as the other is greater than 8. [26] This is a very incorrect name, since the term ‘arithmetical’ applies equally to every notion in this book. It is necessary, however, that the pupil should use words in the sense in which they will be used in his succeeding studies. 171. A set or series of numbers is said to be in _continued_ arithmetical proportion, or in arithmetical _progression_, when the difference between every two succeeding terms of the series is the same. This is the case in the following series: 1, 2, 3, 4, 5, &c. 3, 6, 9, 12, 15, &c. (1½), 2, (2½), 3, (3½), &c. The difference between two succeeding terms is called the common difference. In the three series just given, the common differences are, 1, 3, and ½.