from it, equal in length to a times the linear unit, will fulfil the requirements of the single symbol a. But inasmuch as an indefinite number of equal lines may be drawn from any one point, thus far we have no means of determining which of all such lines is intended; hence arises the necessity of some other symbols to indicate direction, or, as they are called, symbols of direction or affection. One or two of the most simple cases of these we proceed to explain, feeling assured that the principle of explanation is so entirely in harmony with the usual meaning of and, that it ought not to be omitted in an elementary treatise; and also because it enables us to shew that an algebraical curve, though apparently discontinuous and confined within certain fixed limits, is not in reality so, but extends to infinity in all directions. Other parts of the theory (some of which are as yet not sufficiently established) we omit, as unsuited to our present object.

162.] Suppose o, fig. 27, to be the point from which lines are to be measured, and oa = a times the linear unit to be drawn from o towards the right hand. Now since, as we said above, any line drawn from o, a times the linear unit in length, will be symbolized by a, it is necessary to fix on some originating direction; suppose this to be oA, and any line measured from o towards A to be affected with the symbol of direction + ; if then, after a line has undergone any operation or a series of operations, it comes into the position oa, it is still to be symbolized by and, if the line be a, by + a. Such an operation we might conceive to be a reciprocating one, the line at one time being in the position oA, and at another in the position o'a', having moved sideways, and assumed all intermediate positions. Or we may conceive that the line oa (see fig. 28) has revolved round the point o, and, having turned in the plane of the paper through 360°, has again come into its original position, and so on continually; and it is manifest that as often as it has revolved through any multiple of 360°, it has assumed its original position oA, and is therefore to be symbolized by + a. So also there are many conceivable ways in which the line may have moved, and that periodically, and at the end of a complete period be in the position oA. But have we any other customary mode of indicating direction, to serve as a guide which of these conceivable operations to take? We have.

Whenever a line equal in length to a is measured from o towards the left, we symbolize it by -a; if therefore either (—) were a symbol for the operation of one oscillation having been performed on the line, i. e. the line having passed into the position o'a' (see fig. 27): or (—) symbolized the line oa (fig. 28), 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 be 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, are what in our ordinary geometrical conceptions we do not use nor contemplate. 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 revolving 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 e.g. 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 + be the symbol of turning through 360°, after one revolution the symbol of affection is + on the back of +, i. e. 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.

In conformity then with the algebraical law of indices is

the symbol of that operation, which, being performed twice, one on the back of the other, 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 (-) +;

$4
+ =

=

=

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

2n+1

2

turning a line through 360°. Similarly again (+) is equivalent to, for it is equivalent to +”+b = +”(+)* = +” —; and this coincides with the ambiguity we have always in the sign of +, for it may, as far as the form + teaches, be either + or -. If therefore the +, whose root has to be extracted, be raised to an even power, its root is to be affected with a positive sign; but if the + be +2n+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

1st, +" = +,

2n+1

2d, + 2 =

163.] So again + symbolizes that operation which, being performed twice, one on the back of the other, is equivalent to +, i.e. to (−), and, being performed four times successively one on the back of the other, 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 = ++*= −(−)*, the line which is affected with it is to be drawn in the direc

4n+1

tion oл3. Similarly + indicates a line drawn in the direcOA3.

4n+3

tion oA2, and + 4 a line drawn in the direction oAg. So also

+means that that line with which it is affected is to be drawn

360°

at an angle of to the originating direction OA.

n

164.] We have inserted the above method of explaining symbols of direction, which are usually termed Impossible 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 0 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

represents a line of length p drawn at an angle to the originating direction. The two methods coincide at those points which will be most useful in the sequel; thus let 0 = O, 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 0 originating line produced backwards; let 0 270°, and (1) be√-1 p, and is in a direction at right angles to and below the originating direction; and if = 360° the line becomes+p, and lies in its original direction.

comes

1, it is to

=

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, Oм = p cos and PM = p sin 0, and as PM is affected with 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 op of the parallelogram, of which oм and MP are the two containing sides.

It is also to be observed, that

cose + √1 sin◊ = e

(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.

165.] 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 a. Similarly if y be 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 not only represents 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 x is affected with ± √√ as long as is less than a, and y is less than + b.

But let a be greater than + a, then we may write (3) in the form

b

y = ± √ = 2 {x2—a2}*;

a

(5)

which equation, short of the symbol, represents an hyperbola whose transverse axis is 2a, and conjugate axis 2b, and whose asymptotes are as drawn in fig. 30; but which hyperbola, when the I is introduced, is in the plane containing the