Properties of such a period if arranged circularly. Cyclical arrange 721. If the n terms, therefore, of a period be arranged circularly, and if we mark in succession every 7th term, (r being prime to n), we shall pass round the circle r times, having marked every term of the period once, and once only, before we return to the first term: thus, if n = 7 and r = 2, and if we arrange the terms of the period ment of the where 5 circuits are made before we return to 1: and similarly in other cases: and it is obvious that the same periods will be found, and in the same order, if we replace a, in every successive term of the period (1), by a2 in one case, and by a in the other, depressing the indices in every case to the residual which arises from dividing them by 7. 722. If we restrict our attention to the imaginary roots of "- 1 = 0, where n is a prime number, or, in other words, to the roots of the equation we shall find that they are capable of an arrangement, by which the terms of the period which they form, may always recur in the same order, upon the replacement of one base by another; this may be effected by arranging the terms of the period a, a3, a3, ... a”-1 them less than n, it follows that s - t is less than n: but we have shewn above, that the least power of a', which is equal to 1, is a"". It may be observed, that the proposition is true of all powers of a base whose indices are prime to n, whether n be a prime number or not. in such an order that their indices may be the residuals of the successive powers of a primitive root of n (Art. 531), which comprehend every number from 1 to n 1 inclusive: thus, if a be 7, of which 3 is a primitive root, the residuals of (1), 0, be distributed in the order of these * These residuals are successively formed by multiplying the preceding residual by the primitive root, or 3: thus, 3 being the first residual, the second is 2, which is the remainder from dividing 3 x 3 by 7: the third is 3 x 2 or 6: the fourth is 4, the remainder from dividing 3 x 6 by 7: the fifth is 5, the remainder from dividing 3 x 4 by 7: the sixth is 1, the remainder from dividing 3 × 5 by 7: they afterwards recur in the same order for ever. If we take the second primitive root of 7, which is 5, we shall get the series of residuals 5, 4, 6, 2, 3, 1, the five first terms of one series being the five first terms of the other series in a reverse order. If n = 13, there are 4 primitive roots corresponding, which give the following periods of residuals : these periods form pairs, in which the first 11 terms follow severally an inverse order: and it may be observed, that the terms of the second pair are the 5th terms of those of the first. The student will find the theory of such primitive roots discussed at considerable length in Art. 531, and those which precede and follow it. Cyclical periods: their great import ance. lar arrangement with those of the period (2), each term being one place in advance: if we replace a by a2 or a2, we get α (4), where each term is two places in advance, when compared with the same period (2): if we replace a by a3 or ao, we get the period where each term is three places in advance: and similar results will be observed to follow, whatever be the term in the series by which a is replaced. 723. Such periods of the imaginary roots of 1, which correspond to prime values of the index, may be properly called cyclical periods, inasmuch as the various derivative periods which result from changes of their bases, will perpetually follow in the same order round the circumference of a circle, where no regard is paid to the initial term: they will be found hereafter to be connected with the most important analytical theories*, and therefore deserve the most careful attention of the student. They form the basis of Gauss' well-known researches respecting the geometrical division of the circle, and of Lagrange's theory of the solution of the equation a 1 for all values of n. See the Disquisitiones Arithmetice of Gauss, Sectio 7ma, and the Résolution des Equations Numériques of Lagrange, Note xiv. CHAPTER XXV. ON THE GENERAL PRINCIPLES OF THE INTERPRETATION THE SIGNS OF AFFECTION WHICH ARE OF the roots 724. WE have explained in a former Chapter (XXIII.) the The use of introduction and use of the roots of 1, as the recipients of the of 1 as the signs of affection, which the application of the general principle recipients of signs of of the "permanence of equivalent forms" (Art. 631) renders ne- affection in Symbocessary in algebraical operations, and more particularly in those lical Algeof Evolution: we have shewn that 1 x rand - 1 x r may be conve- bra. niently used as equivalent to +r and r respectively, where 1 and – 1, which are also the square roots of 1, may be considered as the recipients of the signs + and -: the extraction of the square roots of expressions, such as and r2, or of their equivalents 1 × r2 and - 1 x r2, leads to results which are correctly symbolized by 1×r and √√-1×r: the consideration of higher roots conducts us in a similar manner to expressions such as √1xr and -1xr, or to their equivalents (1) × r and (− 1)ar, where the signs (1) and (−1) are used to designate such affections or qualities of their common subject r, as can be shewn to be consistent with their symbolical properties. lation of perties. 725. We have subsequently investigated (Chapter XXIV.) Recapituthe more important of these properties, with a view to the dis- their symcovery of the conditions which must be made the basis of their bolical prointerpretation: we have shewn, when n is a prime number, that (1) has necessarily n symbolical equivalents or values *, provided it has one such value which is different from 1: and we have further demonstrated that these equivalent values or roots may be considered as the successive powers of a common The term value, when thus used, has no reference to magnitude, but to symbolical form only, or to the quality of magnitude which its symbolical conditions are competent to express all the values of (1)r are equal in magmitude, but different in affection or quality. Research of the condi tions of base, which is one of them, forming periods of n terms, which recur in the same order for ever. 726. If r, therefore, be a specific magnitude to which the their inter- sign (1) is attached, or into which it is multiplied, forming pretation when ap plied to specific magnitudes. The suc cessive terms of a period possess the same sym bolical relation to (1)☆ the expression (1) r; and if a be a base of (1) (Art. 710), or any one of its roots which is different from 1, then (1)r may equally express any one term of the period which are all of them symbolically different from each other. 727. It will be observed that the successive terms of this period bear the same symbolical relation to each other, the ratios being identical with each other, and with a: and it will follow, each other: therefore, that whatever be the affection or quality which a is and must represent therefore quantities of the same magnitude. capable of symbolizing, when applied to the specific magnitude r, it must equally symbolize the affection which connects ar with ar, a3r with aar, and so on, throughout the period, until we reach its last term a"r, which is identical with r. We may conclude, therefore, that the quantities designated by the terms of the period ar, a3r, a3r ... a′′r kind and of must be of the same kind*, and of the same magnitude+; but the same inasmuch as the successive terms of this period are symbolically different from each other, they are also different in their affections but if we enter upon a second period, the same terms will re-appear, and in the same order: and so on from period to period, however far the series of them may be extended. 728. The relative position of equal lines in Geometry, makinterpreta- ing equal angles with each other, and which are the quotients Their geo metrical tion. Quantities may be of the same kind though different in quality or affection: thus all straight lines are quantities of the same kind, though they may differ in relative or in absolute position. For, if not, let the magnitude of the quantity expressed by ar, considered without regard to its affection, be r (1+c): then since a2r bears the same relation to ar that ar bears to r, it will follow that the magnitude expressed by a3r must be r(1+c)2: similarly, the magnitude of a3r will be r(1+c)3, that of a1r, r(1 + c) ... and that of a"r, ... r ( 1 + c)" : but a❞r=r, and therefore r = r (1+c)" : or in other words, c = 0. |