CHAPTER XXXI. ON THE REPRESENTATION OF STRAIGHT LINES BOTH IN POSITION 823. THE investigations, in the last Chapter, have enabled Restateus not merely to prove the necessary existence of roots of the ment of the equation x" − 1 = 0, results obtained in the last Chapter respecting of 1. which are different from 1, {a proposition which we had previously the roots assumed (Art. 708)}, but likewise to determine, in all cases, their complete analytical values: we have there shewn that those roots are expressed by the n values (and there are no more, Art. 811) of the formula corresponding to the successive values of r between 0 and n- 1 inclusive. 824. In the application of the successive terms of the period The base a 1, a, a2,... a"-1, corresponding to the least angle of transfer, Art. 729, is formed by the n roots of 1, as signs of affection to denote the successive positions of the n radii which divide the circumference determi-' of a circle into n equal parts (Art. 728 and 729) it was shewn nate. that if a was the appropriate sign used to denote the least of the successive angles of transfer, then the other terms of the period, in their order, would correspond to the other angles of transfer in their order: but even when the roots of 1 were explicitly given, as in the case of its cubie, biquadratic and quinary roots, we were unable to connect a specific root with a specific angle of transfer, inasmuch as there existed no manifest symbolical connection between this angle and the analytical form of the root which exclusively corresponded to it: this uncertainty however will no longer be found to prevail in the analytical form of the roots of 1, which Demoivre's formula enables us to assign to them: for, if we make a2 2π 1 a" (cos +/-1 sin) = cos +/- sin, n COS 4π n 2π 4.π 6 п where the successive angles of transfer follow the order of the successive powers of a: but if we had n n n as the base of the period, we should have found where the order of the powers of the base of the period does not follow the order of the angles, and where we must pass twice round the circumference of the circle before we return to the primitive line: in a similar manner, if the base of the period had been assumed to be we should have returned to the primitive line after. r. transits round the circumference, and not before. It appears, therefore, that the base a, which corresponds to the least angle of transfer is determinate, and is in all cases that root of 1, which is expressed by the formula where is the angle of transfer, and n is the denomination 2π n of the root. consideration of the 825. It follows, therefore, in conformity with the conclu- Resumed sions established in Art. 728, that if AB or P be the primitive line, and if AB, AB„ AB„‚.......AB- be drawn from the centre interpretaof the circle to a series of points dividing the circumference into in Art. 728. 29... n-1 2π n n equal parts, and making angles equal to with each other, then the several radii thus drawn will be represented both in magnitude and in position, in their order, by and that if the formation of these analytical values, according to the same law, be continued, the same series of values will be reproduced in the same order: for 2π P{cos 2 (" + 1) = + √1 sin 2 ("+1)*) -p (cos + √1 sin2), n n } = n tions given The same interpreta and, consequently, the n roots of 1, connected as signs of affection with a symbol p which denotes a line, will represent the same line both in magnitude and position in n different positions, 2π making angles equal to with each other, and in no more; the successive lines, in their order, being severally represented by the successive values of 21T P(cos + √-1 sin 2*) n where r has successively every value in the series 0, 1, 2, ... (n − 1). 826. And, generally, if p be the length of a line, making tions an angle 0, with the primitive line, then gene ralized. (cos +√1 sin 8) p will, in conformity with the preceding theory, express it both in magnitude and in position: for if 0= of r and n are whole numbers, then 2rT where the values n + √ √ = 1 sin 2) p, + − or the equivalent expression (cos + √1 sin 0) p will represent the magnitude and position of a line, equal in 2rT magnitude to p, which makes an angle or with the pri mitive line and if no finite integral values of r and n can be 2rT found, which make absolutely equal to 0, yet we can always n determine, by the theory of converging fractions or otherwise, 2rT such values of them as will make the value of approximate to as near as we chuse: it will follow, therefore, under such circumstances, that will represent a line in magnitude and position (Art. 820) as near as we chuse to that which is assumed to be expressed by and if the approximation be indefinitely continued, we may consider the line which is represented by one of these formulæ as geometrically coincident (Art. 169) with that which is assumed to be represented by the other. converging to the value of, the last of them being equal to it: 1 136 13. 14. 1, which 2rT 98π = n 1333 = 13o. 13'.998, which differs from 0 by less than th part of a minute: the last, 500 which gives the accurate value of 0, is the line expressed in magnitude and position by whose sign of affection is the 397th term of the period formed by the 10800 values of the 10800 th root of 1. 827. Inasmuch, therefore, as a line p, making an angle 0 Examples with a primitive line AB, is expressed, both in magnitude and of the interposition with respect to it, by (cos + √1 sin ◊) p, pretations of cos 0 + √1 sin 0 for given values of 0. |