CHAP.I.] GEOMETRICAL SQUARE ROOTS OF NEGATIVE UNITY. 131 minus the square of the number which denotes (comp. 133) the length of the Index of that Right Quotient: as appears from Fig. 41, bis, in which oв is only an ordinate, and not (as before) a radius, of the semicircle ABA'; for we have thus, 149. Thus every Right Radial is, in the present System, one of the Square Roots of Negative Unity; and may therefore be said to be one of the Values of the Symbol √-1; which celebrated symbol has thus a certain degree of vagueness, or at least of indetermination, of meaning in this theory, on account of which we shall not often employ it. For although it thus admits of a perfectly clear and geometrically real Interpretation, as denoting what has been above called a Right Radial Quotient, yet the Plane of that Quotient is arbitrary; and therefore the symbol itself must be considered to have (in the present system) indefinitely many values; or in other words the Equation, has (in the Calculus of Quaternions) indefinitely many Roots,* which are all Geometrical Reals: besides any other roots, of a purely symbolical character, which the same equation may be conceived to possess, and which may be called Geometrical Imaginaries. Conversely, if q be any real quaternion, which It will be subsequently shown, that if x, y, z be any three scalars, of which the sum of the squares is unity, so that x2 + y2+ z2 = 1; and if i, j, k be any three right radials, in three mutually rectangular planes; then the expression, q= ix + jy + kz, denotes another right radial, which satisfies (as such, and by symbolical laws to be assigned) the equation qa =—1; and is therefore one of the geometrically real values of the symbol V-1. + Such imaginaries will be found to offer themselves, in the treatment by Quaternions (or rather by what will be called Biquaternions), of ideal intersections, and of ideal contacts, in geometry; but we confine our attention, for the present, to geometrical reals alone. Compare the Notes to page 90. satisfies the equation q2 = 1, it must be a right radial; for if, = as in Fig. 42, we suppose that ▲ AOB & BOC, с = OA, and this square of q cannot become equal to (1.) If then we meet the equation, Fig. 42. A B where a = OA, and p = OP, as before, we shall know that the locus of the point r is the circumference of a circle, with o for its centre, and with a radius which has the same length as the line OA; while the plane of the circle is perpendicular to that given line. In other words, the locus of P is a great circle, on a sphere of which the centre is the origin; and the given point A, on the same spheric surface, is one of the poles of that circle. (2.) In general, the equation q2=-a2, where a is any (real) scalar, requires that the quaternion q (if real) should be some right quotient (132); the number a denoting the length of the index (133), of that right quotient or quaternion (comp. Art. 148, and Fig. 41, bis). But the plane of q is still entirely arbitrary; and therefore the equation q2 = - a2, like the equation q2=-1, which it includes, must be considered to have (in the present system) indefinitely many geometrically real roots. in which we may suppose that a> 0, expresses that the locus of the point P is a (new) circular circumference, with the line oa for its axis, and with a radius of which the length = =ax the length of oa. 150. It may be added that the index (133), and the axis (128), of a right radial (147), are the same; and that its reciprocal (134), its conjugate (137), and its opposite (143), are all equal to each other. Conversely, if the reciprocal of a given quaternion q be equal to the opposite It being understood, that the axis of a circle is a right line perpendicular to the plane of that circle, and passing through its centre. CHAP. I.] RADIAL QUOTIENTS considered as versors. 133 of that quaternion, then q is a right radial; because its square, q2, is then equal (comp. 136) to the quaternion itself, divided by its opposite; and therefore (by 143) to negative unity. But the conjugate of every radial quotient is equal to the reciprocal of that quotient; because if, in Fig. 36, we conceive that the three lines OA, OB, OB' are equally long, or if, in Fig. 39, we prolong the arc вA, by an equal arc AB', we have the equation, SECTION 8.-On the Versor of a Quaternion, or of a Vector; and on some General Formula of Transformation. 151. When a quaternion q = ß: a is thus a radial quotient (146), or when the lengths of the two lines a and ẞ are equal, the effect of this quaternion q, considered as a FACTOR (103), in the equation qaß, is simply the turning of the multiplicand-line a, in the plane of q (119), and towards the hand determined by the direction of the positive axis Ax.q (129), through the angle denoted by q (130); so as to bring that line a (or a revolving line which had coincided therewith) into a new direction: namely, into that of the product-line ß. And with reference to this conceived operation of turning, we shall now say that every Radial Quotient is a VErsor. 152. A Versor has thus, in general, a plane, an axis, and an angle; namely, those of the Radial (146) to which it corresponds, or is equal: the only difference between them being a difference in the points of viewt from which they are respectively regarded; namely, the radial as the quotient, q, in the * Hence, in the notation of norms (145, (11.)), if Nq = 1, then 9 is a radial; and conversely, the norm of a radial quotient is always equal to positive unity. In a slightly metaphysical mode of expression it may be said, that the radial quotient is the result of an analysis, wherein two radii of one sphere (or circle) are compared, as regards their relative direction; and that the equal versor is the instrument of a corresponding synthesis, wherein one radius is conceived to be generated, by a certain rotation, from the other. satisfies the equation q2 = 1, it must be a right radial; for if, as in Fig. 42, we suppose that ▲ AOB ∞ BOC, we shall have C and this square of q cannot become equal to negative unity, except by oc being = OA, or = OA' in Fig. 41; that is, by the line oв being at right angles to the line oa, and being at the same time equally long, as in O Fig. 40. (1.) If then we meet the equation, A Fig. 42. B where a = OA, and p=OP, as before, we shall know that the locus of the point P is the circumference of a circle, with o for its centre, and with a radius which has the same length as the line oa; while the plane of the circle is perpendicular to that given line. In other words, the locus of P is a great circle, on a sphere of which the centre is the origin; and the given point a, on the same spheric surface, is one of the poles of that circle. (2.) In general, the equation q2=-a2, where a is any (real) scalar, requires that the quaternion q (if real) should be some right quotient (132); the number a denoting the length of the index (133), of that right quotient or quaternion (comp. Art. 148, and Fig. 41, bis). But the plane of q is still entirely arbitrary; and therefore the equation like the equation q2: 11 q2 = — a2, 1, which it includes, must be considered to have (in the present system) indefinitely many geometrically real roots. in which we may suppose that a> 0, expresses that the locus of the point P is a (new) circular circumference, with the line oA for its axis, and with a radius of which the length =ax the length of oa. 150. It may be added that the index (133), and the axis (128), of a right radial (147), are the same; and that its reciprocal (134), its conjugate (137), and its opposite (143), are all equal to each other. Conversely, if the reciprocal of a given quaternion q be equal to the opposite * It being understood, that the axis of a circle is a right line perpendicular to the plane of that circle, and passing through its centre. CHAP. I.] RADIAL QUOTIENTS CONSIDERED As versors. 133 of that quaternion, then q is a right radial; because its square, q', is then equal (comp. 136) to the quaternion itself, divided by its opposite; and therefore (by 143) to negative unity. But the conjugate of every radial quotient is equal to the reciprocal of that quotient; because if, in Fig. 36, we conceive that the three lines OA, OB, OB' are equally long, or if, in Fig. 39, we prolong the arc BA, by an equal arc AB', we have the equation, SECTION 8.-On the Versor of a Quaternion, or of a Vector; and on some General Formula of Transformation. 151. When a quaternion q =ẞ: a is thus a radial quotient (146), or when the lengths of the two lines a and ẞ are equal, the effect of this quaternion q, considered as a FACTOR (103), in the equation qaẞ, is simply the turning of the multiplicand-line a, in the plane of q (119), and towards the hand determined by the direction of the positive axis Ax.q (129), through the angle denoted by q (130); so as to bring that line a (or a revolving line which had coincided therewith) into a new direction: namely, into that of the product-line B. And with reference to this conceived operation of turning, we shall now say that every Radial Quotient is a VERSOR. 152. A Versor has thus, in general, a plane, an axis, and an angle; namely, those of the Radial (146) to which it corresponds, or is equal: the only difference between them being a difference in the points of view† from which they are respectively regarded; namely, the radial as the quotient, q, in the Hence, in the notation of norms (145, (11.)), if Nq= 1, then q is a radial ; and conversely, the norm of a radial quotient is always equal to positive unity. † In a slightly metaphysical mode of expression it may be said, that the radial quotient is the result of an analysis, wherein two radii of one sphere (or circle) are compared, as regards their relative direction; and that the equal versor is the instrument of a corresponding synthesis, wherein one radius is conceived to be generated, by a certain rotation, from the other. |