powers, which is also called involution, is therefore nothing more than the multiplication together of equal factors, and is easy enough. But the reverse operationthat is, to find the factor which, involved in this manner, shall produce a given number -is a problem not so readily disposed of. The factor referred to is called the root of the power; so that the reverse problem spoken of is the problem of the extraction of roots. To extract the square root of a given number, is to find a number which, when squared, or raised to the second power, or, which is the same thing, when multiplied by itself, shall reproduce the given number. The rule for this operation is as follows: RULE.-Commencing at the units' figure, cut off two figures, then two more, and so on, thus dividing the number into periods, as they are called. If the number of figures be even, we shall therefore have two figures in every period; but if odd, the first period, on the left, will have but a single figure. Take the square root of this first period, or find the root of the nearest square to it, which nearest square is not, however, to be greater than the number forming the first period. The root thus found will be the leading figure of the root of the proposed number, and the square of this leading figure is to be subtracted from the first period. To the remainder annex the second period: the resulting number will be the first dividend, and the divisor for it is to be found as follows: Put twice the root-figure, just found, in the divisor's place; the leading portion of the complete divisor will thus be obtained. This leading portion, as in common division, is enough to suggest the corresponding quotient-figure, which is now to be found, and annexed to the former root-figure, and also to the leading portion of the divisor employed in finding it. the divisor will then be completed; so that proceeding as in common division, a second remainder will be obtained, to which the next period being united, a second dividend will result. To find the corresponding divisor, put twice the number, of two figures, now in the root's place, in the divisor's place: we shall thus have the leading portion of the complete divisor, by aid of which a third root-figure may be found; which, as before, is to be annexed to the former root-figures, and also to the incomplete divisor, in order to complete it; and the work is to be then carried another step, as in common division; and so on, till all the periods have been brought down, and annexed, step after step, to the successive remainders, thus supplying the successive dividends. If there be a remainder after all the periods have been brought down, the operation may be continued by annexing successive periods of ciphers to the successive remainders, or successive periods. of the decimals in the given number, should any be there: in either case, the root-figures become decimals from the time that ciphers or decimals are annexed. As the root multiplied by itself produces the original number, the decimal places in the root must, of course, always be just half as many as have been employed in determining it. And the root is only to be regarded as an approximation to the truth, whenever a remainder is left at the step where the work terminates. As in division of decimals, the departure from strict accuracy may be made as minute as we please by extending the decimals of the root. Here follow three examples, worked by these precepts: the first is to find the square root of 459684, the second to find the square root of 31640625, and the third to find the square root of 3.65. The symbol for the operation of the square root is √; which is no doubt a degenerated form of r, the initial letter of the word radix, or root: it appears from the foregoing operations that √459684678; √316406255625; √/3·65 = 1·910497, &c. ; the last root being true as far as the sixth decimal place. By the same process it is found that (1.) √473256687.936. (2.) √784-375=28·0067. (4.) √3236068 = •5688645. (3.) √113316625. (5.) √7943=√/794.228.18155425. I here terminate this introductory treatise on the general principles of arithmetical computation. The plan of the present work has precluded the possibility of any extensive application of these principles to the various particular objects usually considered, under special heads, in books entirely devoted to Arithmetic. But I believe rothing has here been omitted which can be considered as essential to the clear understanding of these practical inquiries; for a comprehensive view of which, in moderate compass, the learner is referred to the work mentioned at page 29; and, for further remarks on the philosophy of the subject, to Professor De Morgan's "Elements of Arithmetic." Those who wish to inquire into the early history and progress of Arithmetic, should consult Peacock's elaborate article in the “Encyclopædia Metropolitana,” and De Morgan's "Account of Arithmetical Books;" also, Leslie's "Philosophy of Arithmetic,” and Delambre's "Arithmétique des Grècs," or the Review of it in No. XXXV. of the "Edinburgh Review." To those especially who regard the literature of Arithmetic, this latter work, and the able review of it to which I have alluded, will afford matter of the highest interest; since the Greeks were ignorant of our system of decimal notation, and marked their Lumbers by the letters of their alphabet. The explanation of the process above given, for the extraction of the square root, will be found in the ALGEBRA—the department of science to which the investigation of the principle of it properly belongs; and it is to this subject that the learner must refer for the necessary particulars respecting the cube-root. A TABLE OF THOSE FACTORS OF THE COMPOSITE NUMBERS FROM 75 TO 10000, WHICH FALL WITHIN THE LIMITS OF THE MULTIPLICATION TABLE. PLANE GEOMETRY. THE ELEMENTS OF EUCLID: BOOKS I.-VI. In order to acquire clear conceptions of a point, a line, and a surface, with the deñnitions of which Euclid sets out, it will be best for the learner to consider them, at first, in connection with a solid; that is, with something that has length, breadth, and thickness. All external objects-things that can be seen and felt-are solids; and however small the solid may be, it must have some length, some breadth, and some thickness: using these terms in their ordinary acceptation—that is, in the sense in which they are used in common discourse. The boundaries of a solid are called the surface, or the superficies, of the solid; and, as a boundary, cannot have thickness; because, if it had, it would be a part of the solid, and not a side, or face, or boundary of it: it follows that a surface can have length and breadth only. Again: the surface itself has its boundaries, or limits: these boundaries are called lines: and as a boundary of a surface cannot have either breadth or thickness, since it is no part of the surface, much less a part of the solid, it follows that a line has length only. Lastly, a line has its limits-a beginning and a termination: we speak of these, in common language, as the ends of the line. Euclid calls them points; and it is plain that a point, being no part of the line, cannot have length; and as breadth and thickness are excluded even from the line itself, it follows that a point has no dimensions or magnitude. It merely indicates position; the position, namely, of the commencement or termination of a line. Euclid frequently speaks of taking a point in a line, without meaning an extremity of the line; but we may conceive a line to be crossed, or cut by other lines, and thus to be divided into shorter portions. Each portion has its extremities; so that we may conceive as many points in the line as we please. We cannot represent length only to the eye; it is necessary, therefore, in contemplating the black marks on paper, by which lines are represented, entirely to disregard the breadth of them; and to fix the attention upon the length alone: the eye may see breadth and thickness; the mind takes note of length only. A straight line is that which lies evenly between its extreme points. |