having been turned through 180°, either would account for the negative sign of affection, and (-) would be the symbol of the operation; but under the first hypothesis, the line at one stage of the process will be half on the positive side of the origin and half on the negative; if therefore the operation is continuous, which it is, in passing from + to there should be some symbol to indicate that particular stage; it does not however appear that we have any symbol of the kind; and such a motion, and a line in such a state, we do not use nor contemplate in our ordinary geometrical conceptions. Let us therefore consider whether we have not symbols to indicate a line in any intermediate position between OA and OA1, conceiving the line to pass from the one position to the other by means of a revolution through 180°.

As we said before, whenever the line is measured from o in the direction oA, it is to be affected with a sign. Taking therefore o as the origin of line, and oa as the direction line from which symbols and operations of affection are to be originated, whenever a line, as, for example, oA, has turned an integral number of times through 360°, it is to be affected with the sign with which it started. If therefore it was affected with the sign at first, indicating that it started from oA, and if + is the symbol of turning through 360°, after one revolution the symbol of affection is on the back of +, that is, according to the index law, +2; similarly after two revolutions, +3; and after (n-1) revolutions, +". Supposing therefore that the line which is of the length a, when along the originating direction oA, is unaffected with any sign: + a means that the line has turned through 360°, and has come again into the position whence it started; and so +"a means that a line of length a has revolved n times from the direction of origination, and is in the position oA; whence it appears, in accordance with the arithmetical meaning and law of +, that is, for symbolical purposes of direction, equivalent to +", n being a whole number.

is

In conformity then with the algebraical law of indices the symbol of that operation, which, being performed twice successively, brings the symbol into the value +; that is, if + signifies turning the line through 360°, (+) indicates turning it through 180°; but - symbolizes this operation,

+ = and (-)2 = +;

or the operation symbolized by (-) performed twice successively is equivalent to the operation signified by +, and means

2n+1

turning a line through 360°. Similarly again (+) is equivalent to; for it is equivalent to + "+ = + "(+)* = +”—; and this coincides with the usual ambiguity in the sign of +, for it may as far as the form + teaches, be either + or If therefore the +, whose root is to be extracted, is raised to an even power, its root is to be affected with a positive sign; but if the+is+2+1, then the square root is +"+, which is equivalent to, and the root must be affected with the negative sign. Hence also it is plain that √(−a) × √(−a), which equals a2, can only be a; because the +, with which a2 is affected under the radical, is of only the first power. Therefore we have shewn that in symbolical geometry

190.] So again + symbolizes that operation which, being performed twice successively, is equivalent to +, that is, to (−); and, being performed four times successively, is equivalent to +;

+ * = (−)$;

and therefore, as - indicates that a line is to be turned through 180°, so (-) means that a line is to be turned through 90°. Whenever then a line is affected with (-), which is equivalent to, as its symbol of direction, that line is to be drawn at right angles to the original direction of origination, viz. in the direction OA2, see fig. 28; and whenever the symbol of direction is ++, which = +*+* = − (−)3, the line which is affected with it is to be drawn in the direction oA3. Similarly + 4 in

1

4n+3

4n+1

dicates a line drawn in the direction OA2, and + 4 a line drawn in the direction o Ag. So also means that the line 360° with which it is affected is to be drawn at an angle of to the originating direction oa.

n

191.] We have inserted the above method of explaining symbols of direction, which are usually termed Impossible and

Imaginary and passed over in silence, because it is clearer to the perception than another method which has received copious elucidation from Dr. Peacock and Mr. Warren: that viz. in which cos + √1 sin is considered as the symbol, and whereby, when it is affixed to a line, say p, the direction is indicated in which the line is to be drawn; thus

p (cos + √1 sin 0)

(1)

represents a line of length p drawn at an angle 0 to the originating direction. The two methods coincide at those points which will be most useful in the sequel; thus let = 0, then the line represented by (1) is p, and coincident with the zero operation, that is, with the line of origination; let = 90°, the line becomes p-1, and is at right angles to the originating direction; let = 180°, and the line is p, that is, is the originating line produced backwards; let 0 270°, and (1) becomes√1p, and is in a direction at right angles to and below the originating direction; and if comes+p, and lies in its original direction.

0

=

= 360°, the line be

In accordance then with the interpretation of √1 which such a symbol as (1) thus used involves, it will be observed that (1) correctly represents two sides of a rectangle; that is, fig. 29, if op = p and Poм= 0, 0м = p cos 0 and PM = p sin 0, and as PM is affected with 1, it is to be measured in a direction PM, which is perpendicular to oм; which lines therefore cannot be added, or subtracted, as they are not in the same line, but we may by an extension of interpretation suppose (1) to represent the diagonal or of the parallelogram, of which OM and MP are the two containing sides.

It is also to be observed, that

cos +1 sin 0 = e√-1,

(2)

and that therefore e-1 may be used as a symbol of direction; wherein expresses the angle of inclination to the originating line of the line which the symbol affects.

192.] To apply these principles to the delineation of plane curves from their equations, suppose y = f(x) to be the equation to the curve; since x and y have already preoccupied the two directions at right angles to each other in the plane of the paper, which is, and conveniently so, called the plane of reference, we must seek for some other course by which a line,

which has been measured in the positive direction, may be made to turn through 180° into the negative. Such we shall have if it is made to revolve in a plane to which the other axis is perpendicular; as, for instance, let a revolve in a plane at right angles to the axis of y, then, whenever a is affected with ±(−), it is to be measured in a plane passing through the axis of y, and perpendicular to the axis of x. Similarly if y is affected with (-), it is to be drawn in the plane passing through the axis of x, and perpendicular to the axis of y. Thus it appears that an equation between x and y may represent not only a curve in the plane of the paper, but also curves in the planes at right angles to it, passing through the axes of x and y.

Let us consider the following examples:

The equation to the ellipse, referred to its centre as origin, and principal axes as coordinate axes, is

and therefore neither y nor is affected with ±√, as long

which equation, short of the symbol √—, represents an hyperbola whose transverse axis is 2 a, and conjugate axis 2b, and whose asymptotes are as drawn in fig. 30; but which hyperbola, when the 1 is introduced, is in the plane containing the line a'oa, and perpendicular to the plane of the paper, and is delineated by the dotted line; also as the equations to the asymptotes are

y = +

X,

a

(6)

they lie in the same plane as the curves, and are represented by the lines oL and OL'.

Similarly, when y is greater than + b, we have

1

x = ± √=I ¦ (y2 — b2) 3,

+

b

(7)

which represents an hyperbola in the plane passing through the line BOB' and perpendicular to the plane of the paper, and which is delineated by the dotted lines of the figure, viz. SBS' and TB'T'; and the equations to its asymptotes are

Thus the general equation to the ellipse not only represents the ellipse in the plane of the paper, but also two hyperbole in planes containing the coordinate axes and perpendicular to the plane of reference.

Similarly the equation

x2 + y2 = a2,

in addition to the circle in the plane of xy, expresses also two rectangular hyperbolæ in planes perpendicular to it, and containing the axes of x and y.

Again, consider the equation to the parabola, viz.

and therefore the curve is in the plane of the paper for all positive values of a; but let a be negative, and we have

which expresses another equal parabola, but turned in the opposite direction and in a plane perpendicular to that of the paper, as is indicated by the dotted curve of fig. 31.

Many other examples of the same kind will occur in the sequel. The explanation of such impossible symbols is necessary to a due adjustment of geometrical interpretation to the law of continuity; for no algebraical formula can, so far as is known, give points of discontinuity, neither therefore ought the geometrical representative to exhibit such; but it does so, unless we interpret those quantities which are affected with ± √− 1.

The preceding Section is but a mere sketch of a method of extensive application, and of only one part of it, viz. of that which relates to √; in solving cubic equations, and in tracing the curves which they represent, we shall meet with such sym