row limits of our Table.-Suppose we would divide 195 by 13. Look for the logarithm of the dividend 195; it is 2.290035; from this subtract the logarithm of the divisor, which is 1.113943; the difference is 1.176092, which we find to be the logarithm of 15, as nearly as can be, so 15 is the quotient, on a division of 195 by 13. And thus might any number, within the limits of our Tables, be divided by any smaller number, merely by subtracting the logarithm of the divisor from that of the dividend, and then finding the difference.

326. It appears, however, in the latter instance, that we do not find, in our small table, a logarithm precisely the same, as the difference; it is 1.176091, instead of 1.176092. And thus it will occasionally be; we shall not find precisely the same figures in the logarithm ; but it directs us to the proper quotient; it answers its purpose. And that is all that we can require. Thus it will be, occasionally, in all Tables of Logarithms. And, although these Tables serve in numerous, nay, in countless instances, to abridge the labour of making extensive calculations, yet have the Tables their limits. The most extensive of them hitherto published, or ever likely to be published, not furnishing those who use them with any, not with every number that they may require, at once, and by a simple, by a mere inspection. With the volumes of these Tables there are always instructions, though not always very clear instructions, as to the manner of using them, and the process, in many instances, is one of no small difficulty, even to those who understand them well, but who are not in the constant practice of using them. It is, even to persons of this description, often a matter of considerable labour; and then, too, there is, on consulting such piles of figures, no small danger of casting the eye on a wrong one. So that, although it may be desirable to know just this much of them, in order

that we may not feel ourselves humbled in the presence of mere calculators, I would by no means. advise any one, whose occupation does not lie in calculating the situations and distances of planets and of stars, to have anything further to do with Tables of Logarithms. We must, however, before we dismiss them, show the manner of extracting the roots of numbers by their aid.

327. The Root of a number, is that number which, involved into itself produces the given number. It is a very simple matter to involve a number; a mere multiplication of the number by itself. But, as we have seen, it is a process of some intricacy, requiring great care and attention, to evolve a number, that number being one of considerable magnitude. It is, as we have seen in paragraph 182, to find, not merely the quotient, but the divisor, also.

328. Now here come Tables of Logarithms to furnish us with these divisors and quotients, almost on a mere inspection. We have just seen-paragraphs 321 and 322-that if we take the logarithm of a number, and double it, that is, multiply it by 2, we have, in this double, the index, or logarithm of the second power of the given number. Thus, the logarithm of 9 is .954243; multiplied by 2 we have 1.908486, which is the logarithm of the square, or second power of 9, Now, if a multiplication by 2 furnish us, as it does, with the logarithm of the square of a number, the division by 2 will reverse the process, and so furnish us with the logarithm, that is to say, with the index or guide to, the square root of that number. And so, of course, it does. Divide the logarithm of 81 by 2, and we have that of 9, which is the square root. Thus it is in small numbers, and thus would it be in numbers of any magnitude. THE DIVISION OF THE LOGARITHM OF ANY NUMBER BY TWO GIVES US THE LOGARITHM OF THE SQUARE ROOT OF THAT NUMBER.

329. And, as the division of the logarithm of a number by 2, gives us the logarithm of the square root of that number; just so will its DIVISION BY THREE GIVE US THE LOGARITHM OF THE CUBE ROOT.

330. All this appears very simple and easy, naturally calling forth the question-"If the roots of numbers can be thus easily evolved, why trouble us with the intricate, and labourious method which you have just been teaching? I answer, in the first place, that I have not written this work, nor this branch of the work, for the purpose of teaching the extraction of the roots of numbers. This I hope it does teach, and the PRINCIPLES, also, on which the processes proceed; and so the work may be useful in that respect. But this has been with me a very subordinate consideration; little more than an adventitious circumstance. The nature of the human mind, its propensities, its capabilities, its powers; these have been, with me, long a favourite object of study and contemplation. And I have written this work chiefly, almost entirely, as I state in the preface, for the purpose of leading that mind, in those who may be pleased to read the work, to a pleasing, an orderly, and a vigorous exercise of its powers: aware, as I am, that when thus trained, the mind may choose its subject; and may exult and revel in its attainments, and in the consciousness of its improved capacity and power.

331. To return, however, to our subject, namely, the uses of logarithmic Tables, and so to conclude the work. It would be a waste of time, indeed, to extract the roots of numbers by the processes before taught, if those roots could be found thus in tables. The roots of small numbers may be thus found, and so might those of large, were there tables of due extent. But these are not to be had; nor are they desirable. Already it makes the head dizzy to look

at a volume of the tables now constructed; and we must have additional volume upon volume; a work of indescribable, and of thankless labour, and of great cost too, ere we should have tables to furnish us thus, at once, with the roots of numbers, even to the amount of that which we have evolved on page 132. With the aid of, perhaps any tables now in print, a number of that magnitude would require, in the extraction of its root, so many references, some statements, and so much vigilance, in order to avoid error; that, as I have before stated, unless done by a person in the constant practice of using the tables, however well he may understand their uses, the difficulty, and labour of the operation are not thereby abated. In all the operations of arithmetic, the regular process, the ordinary rule, founded on the principle of the operation, is the safer course; and the shorter also, unless we be in the habitual practice of some expedient; without which practice, expedients, like mental calculations, will but waste our time, and betray us into uncertainty and error.

TABLES OF SIGNS, TERMS, &c.

USED IN ARITHMETIC.

OF SIGNS.

The mark of ADDITION, called PLUS; which, in the Latin, signifies MORE. When placed between two numbers, it means the first number with the addition of the second: that is to say, it means both numbers taken together. Thus 4+5 are equal to 9; 4 + 5 + 2 are 11.

A short horizontal line is the mark of SUBTRACTION, signifying LESS. It is called MINUS, the Latin word for LESS. Its use and meaning are the reverse of the former sign, thus, 9 — 5 is 4; and 11-2—5 is 4.

× The sign of MULTIPLICATION. It is a small cross, but differs from that for Addition, being two transverse sloping lines; like the cross appropriated to St. Andrew. It has not a name, but it signifies that any numbers, between which it may be placed, are to be multiplied, or regarded as multiplied, into each other, as 8 × 9 are 72. 5 × 3 × 7 are 105. To translate this mark into a word, we say "into."

The sign of DIVISION; without a name: but used for the word "by," meaning, when placed between two numbers, that the larger is to be, or to be considered as being, divided by the smaller. Thus 324 is 8; 847 is 12.

= The sign of EQUALITY, and used instead of the words "equal to. Thus 4+5= 9; 4 + 5 + 2

11; 9.

- 4

=

5;

: Two dots, placed thus, are used as THE SIGN OF PROPORTION; and, placed between two numbers, they invite our attention to the proportion which the first bears to the second. They are to be read thus, 8: 16; that is, 8 is to 16; or 8 is in the proportion to 16.

:: Four dots, or points, are the second mark of comparison, or proportion, between numbers; meaning as,' or "so is." They are not used except to follow the two dots just described in the last definition, and then they are placed and read thus, 8: 16:36. Eight is to 16 as 3 is to 6; or 8 is in the same proportion to 16, as 3 is to 6.