TANGENT TO HARMONIC CURVE.

43

Now the parabola being a central projection of a circle, the points T, A, M, A' are projected into t, a, m and a point at an infinite distance, say a'. Since a' is at an infinite distance ta': ma' = 1. But since the four points form a harmonic range, ta: am ta: ma' = 1, or ta= am, as we before proved by another method. This determines the tangent to a parabola at any point p.

a

m

To determine the tangent to the harmonic curve, we must remember that it is formed by unrolling an ellipse from a cylinder. Let ac be the ellipse, ab its orthogonal projection, the circular section of the cylinder by a plane

perpendicular to the axis, pm perpendicular to that plane, pt and mt tangents to the ellipse and circle respectively, at tangent to both of them. A plane touching the cylinder along the line pm will clearly cut the planes of the ellipse and circle in tangents to them at p and m, which must meet at t on at the line of intersection of those planes. The second figure represents the circle ab in the plane of the paper, and the third figure the result of unwrapping one-half the ellipse. Now

mp : bc (fig. 1)= lk: bc = al : ab (fig. 1 or 2). Therefore (since mp = bn, fig. 3)

bn: bcal ab (fig. 2),

and consequently

2qn: bc=2lm: ab=lm : om = rm : tm = al : tm.

That is al mt=2nq: bc, but pm: albc ab, there

fore

pm: mt = 2nq; ab (of fig. 2) =πnq : ab (of fig. 3), ab (fig. 3) = π,ab (fig. 2).

since

Thus

pm: mt=π.ng: ab.

Hence the inclination of the tangent is greatest when nq is greatest, or when n is the centre of bc. The point of greatest inclination is called a point of inflexion, because the curve stops bending upwards and begins to bend downwards.

EXACT DEFINITION OF TANGENT.

We have regarded the tangent at a point a of a curve as the final position of a line cutting the curve in two points p, q, when the line is made to move so that p, q coalesce at a. This method indeed will always find the tangent when there is one. But we have seen that when the curve has a sharp point at a there is properly no tangent at the point a.

In the case of a sharp point in the curve, we may draw a line ab through it, and then turn this line round until b moves up and coalesces with a. The final position at of this line may be called the tangent up to a. Similarly if we draw a line ac cutting the curve on the other side, and turn this round until c moves up to a, the final position at' of this

line may be called the tangent on from a. So that we have a tangent up to a and a tangent on from a, but no tangent at a, properly so called.

The final position of ab when b has moved up to a, is however not so well defined in this case as when there is no sharp point. For then if we turn the line a little too far round a, it will cut the curve on the other side. But when there is a sharp point, there are intermediate

NEWTON'S CRITERION.

45

positions between at and at', such as pq, in which the line does not cut the curve on either side of a.

To improve this definition, we observe that the true tangent at has the property that if we turn it ever so little in the direction ab it will cut the curve

between a and b. Hence, when such a line ab has been drawn, it is always possible to find a point x on the curve such that ax shall lie between at and ab; or (which is the same thing) so that

ax shall make with at an angle less than that which ab makes. This is very obvious in the case of the circle, for the angle bat ba'a, and cat xa'a; so that we have only to draw through a' a line a'x making with a'a a less angle than a'b makes.

=

=

This rule may be stated as follows: If at is the tangent at the point a, it is possible to find a point x on the curve near to a so that the angle xat shall be less than any proposed angle, however small. For let tab be the proposed angle; however small it is, the line ab must cut the curve if at is a tangent, and then we have only to take a point between a and b.

The proof fails, however, when the curve is wavy. In the figure we can take a point

x between a and b so that ax does not lie between at and ab.

This

only means that we have begun too

far away from a. If we take b somewhere between a and the nearest point of inflexion c, the proof of the rule will hold good.

So guarded, the rule amounts to Newton's criterion for a tangent. Even this criterion, however, is baffled by some curves which can be conceived. Suppose two circles

to touch one another at the point a, and between these circles let us draw a wavy curve, like the harmonic curve, except that the waves become smaller and smaller as they approach a; and let us suppose the shape of the waves, that is, the ratio of their height

t

to their length, to be kept always the same. Then it will be impossible to take b so near to a that there shall be no point of inflexion between them. Also it is clear in this case that there is no real tangent at a; for however near we get to a, the direction of the curve sways from side to side through the same range.

If, however, the waves are so drawn that the ratio of their height to their length becomes smaller and smaller as they approach a, so that they get more and more flat without any limit, then although the proof of the rule fails as before, there is a real tangent at a, namely, the common tangent at to the two circles.

a

t

In both these cases our criterion for a tangent is satisfied; that is to say, there is a line at such that by taking a point x on the curve near enough to a, the angle cat can be made less than any proposed angle. Yet in one of these cases this line at is a tangent, and in the other it is not. We must therefore find a better criterion, which will distinguish between these cases.

For

The tangent to a circle has the following property. If we take any two points p and q between a and b, the chord pq makes with the tangent at a an angle less than aob. the angle between pq and at is equal to aom, where om is perpendicular to pq. Let pq be called a chord inside ab, even if p is at a or q is at b. Then we can

find a point b such that every chord inside ab makes with at an angle less than a proposed angle, however small. For we have only to draw the angle aob a little less than the proposed angle.

Now the second of our exceptional curves, that which really has a tangent, has also this property, that we can find a point b so near to a that every chord inside ab shall make with at an angle less than a proposed angle,

DEFINITION OF TANGENT.

47

however small. For since the waves get flatter and flatter without limit, the tangents at the successive points of inflexion make with at angles which decrease without limit. We have then only to find a point of inflexion whose tangent makes with at an angle less than the proposed angle, and take b at this point or between it and a.

But the first curve has not this property, for the inclinations of the tangents at the points of inflexion are always the same, and any one of these counts as a chord inside ab.

We shall now therefore make this definition :

When there is a line at through a point a of a curve having the property that, any angle being proposed, however small, it is always possible to find a point b so near to a on one side that every chord inside ab makes with at an angle less than the proposed angle; then this line at is called the tangent of the curve up to the point a on that side.

When there are tangents up to the point a on both sides, and these two are in one straight line, that straight line is called the tangent at a. In this case the curve is said to be elementally straight in the neighbourhood of a. It has the property that the more it is magnified, the straighter it looks.

Going back to our first and simpler definition of a tangent, as the final position of a line pq which is made to move so that p and q coalesce at a, we see that not only does it always find the tangent when there is one, but that when there is not, the final position of pq will not be determinate, but will depend upon the way in which and q are made to coalesce at a. When therefore this method gives us a determinate line, we may be sure that that line is really a tangent.

p

The problem which we have now to consider is the following:-Suppose that we know the position of a point