were neither sparing nor nice in the names they applied to the Gnostic heretics-devils, snakes, hounds, wolves, vipers, and firstborn of Satan. These names of course do not give us much light except upon those who used them; but when less thoroughly aroused with theological passion, and consequently in less bitter mood, they said that the Gnostics derived their teaching from Pythagoras and Plato and Heraclitus and Cleanthes, and from the mystery institutions of Greece, Egypt and the East generally. This was the truth, but instead of being a reproach it was their glory. This meant that the teaching was the best in the religious teaching of the ancient world. Instead of coming into a world of universal darkness with its one divine light of truth, Christianity came from the same source as Gnosticism. In the Epistles of Paul we have echoes of what was taught in Egypt and Greece two or three hundred years before. There is nothing of which we are so sure as the existence of a well-developed and well-defined doctrine in the Hellenistic world of the first centuries before Christ, of the descent of man from the heavenly or archetypal man, and of his return to pristine glory with the experience he has gathered from his contact with, and conquest of, the world of matter and form. This Paul calls the "mystery" of Christ, the mystery hid from ages and generations, but now made manifest.

The story of a Christ who was the Saviour of the world, the divine man who was the representative of a great spiritual process, the mediator between God and men, the ideal man who was overcome in his struggles for human salvation but conquered in being overcome, is the story which the world has repeated to itself over and over again. It is not original with the New Testament, every feature of it was familiar to those who were initiated into the mysteries. This should be enough to show us that we are not in the presence of literal fact. There is no doubt that the crucifixion as Paul conceived it had cosmic significance-it is not merely the death of a martyr. The center and soul of the gnosis of the ancient world was the Cross. The technical phrase for it among the Gnostics is one used by Paul, the "cross the power of God." Wherever the gnosis had established itself the kernel was the cross. It is obvious that in these places it could not mean the death of Jesus for that was a local happening. It meant the great world-passion, the sacrifice of God in the creation, Deity laying down his life in the universe of matter and form. And to Paul the cross was the symbol of this heart-moving conception.

The interpretation of Paul's determination not to know anything among men save Jesus Christ and him crucified that makes him mean to refer to a series of historic facts eviscerates it of all real content. In the creation was the Calvary of Deity. The cross is thus the background plan of the universe. To know the cross from this higher standpoint is to know all there is to know; there is nothing beyond this. The cross was the symbol of a profound mystery which opened up the heart of Deity himself to the gaze of the world. The divine sufferer was God himself, who in creating the universe sacrificed himself for it. The cross, therefore, represents the greatest of all sacrifices, not something that happened once and once for all, but something that is eternal and timeless-the sacrifice of God in and for his own creation that could not be unless he poured his own life into it and restricted himself within the forms of matter. "Confessedly great is the mystery of godliness." Unthinkable in its magnitude is this sacrifice, for it means nothing less than the identification of the infinite with the finite in its lowest forms. Here is the profoundest mystery open to human contemplation to speak of which is possible only in forms of symbol and parable. The literal truth is too vast, too mysterious, too sublime, to be made known to human comprehension. It is the mystery before which angels, we are told, veil their faces; and to gain a single glimpse of it one may well surrender all other knowledge and determine, as Paul did, to know nothing else. Here is the oldest form of the Christian faith. The story of Jesus is the parable of this infinitely larger truth. It is the symbol of the Lamb slain from the foundation of the world, that is, prior to human history, the emblem of divine body and blood voluntarily sacrificed in outward physical nature and entombed in the lower consciousness of man. It was the claim of the second century Gnostics that Christianity was none other than the consummation of the inner doctrine of the mystery institutions of all the nations; and it is this interpretation of the Gospel story which is set forth in the apocryphal Gospels and Acts. The end of them all was the revelation of the mystery of man which is none other than the mystery of Christ. DUNDEE, SCOTLAND. K. C. ANDERSON.

HAMILTON'S HODOGRAPH.

In Mach's Mechanics the space devoted to the hodograph of Sir William Rowan Hamilton is barely a page, half of which is

taken up by the diagram; the diagram, too, is not drawn strictly according to the construction of Hamilton, but according to the usual manner of present-day text-books. This seems to me to be something of an injustice to an exceedingly brilliant piece of work; for from it and Hamilton's theorem of the isochronism of the circular hodograph can be deduced the more important properties of motion in orbits described to a center of force according to the Newtonian law, Lambert's theorem for the area of an elliptic sector in terms of the bounding radii vectores and the chord of the sector, and Euler's corresponding expression for the area of a parabolic sector. The proofs are exceedingly simple and demand very little knowledge of the properties of conics, and no calculus or coordinate geometry at all. For other interesting and elegant applications of the hodograph, reference should be made to Tait's Quaternions and papers in the Proc. R. S. E., and to Tait and Steele's Dynamics of a Particle. But the matter which follows should, I think, be sufficient to corroborate the opinion expressed above.

I. The first use of the curve is ascribed to Bradley,1 and it is probably his definition that is generally given in elementary textbooks, where its only use is to obtain the acceleration towards the center of a particle moving in a circle with uniform speed.

DEF. 1. If a point be in motion with any velocity in any orbit, and if at any instant a line be drawn from a fixed point representing on some chosen scale the velocity of the point at the instant in magnitude and direction, the locus of its end is the "hodograph" of the motion.

Not even in the particular case of motion in a circle does this, the usual definition, so readily demonstrate the connection between the hodograph and the orbit as the definition originally given by Hamilton in a paper communicated to the Royal Irish Academy in 1846,2 which is as follows:

DEF. 2. "If, in an orbit round a center of force, there be taken on the perpendicular from the center on the tangent at each point a length equal to the velocity at that point of the orbit, the extremities of these lengths will trace out the hodograph."

Note. The use of the word "equal" does not introduce any difficulty, unless we attempt to verify the formulas obtained by reference to the theory of dimensional units. It is to be noted that

1

Sir Robert Ball, Article on "Gravitation," Encyc. Brit. 'Proc. Roy. Ir. Acad., 1847.

the definition of Hamilton is more specialized than the other, referring only to central orbits.

II. Newton showed that for any central force the areas swept out by the radius vector are proportional to the times. The following is a simple proof of this theorem.

If O is the center of force, and P, Q, R points on the orbit separated by very small unit intervals of time, and MQN, PM, RN are drawn perpendicular to OQ, MQ, QN respectively; then MQ, QN measure the velocities transverse to OQ, before and after the position Q.

It follows from Newton's theorem that, if the rate at which the area is swept out is denoted by th, and the perpendicular from O on the tangent at P by p, then pv=h.

Hence, in Fig. 2, ON.OQ= h, and Q is the image of N in a circle whose radius is √ī; also, if P', Q' are near points to P, Q, and the tangents at P, P' meet in T, then Q, Q' are the poles of two lines passing through T, and therefore QQ' is the polar of T. That is, the tangent at Q is the polar of P.

The hodograph is therefore the polar reciprocal of the orbit. It is also evident that OM=h/OP; or, in other words, the product of the length of the perpendicular from the center on the tangent to the hodograph and the length of the corresponding radius vector of the orbit is constant (=h).

Again, since the tangent at Q is the direction of the velocity of Q in the hodograph, this velocity is perpendicular to the radius

'Newton, Principia, Book I, Prop. 1.

vector. The hodograph given by the usual text-book definition is therefore identical with that given by Hamilton's definition, when turned through a right angle.

Further, since OQ, OQ' represent the velocities at P, P' turned through a right angle, it follows that QQ' represents the change of velocity between P and P' when turned through a right angle. Hence the radial acceleration of P is measured by the velocity of Q in the hodograph.

IV. The angle between the normals at QQ' is equal to the angle between the radii OP, OP'.

Hence in Fig. 3 we have

QQ/p=PN/r=2▲ OPP'/r2;

and if the acceleration in the orbit is denoted by f, then

f/p=h/r2.

This formula for the radius of curvature of the hodograph solves immediately the case

fα1/r2.

Fig. 3.

N

For, if fx 1/r2, p is constant, and the hodograph is therefore a circle, and its polar reciprocal is a conic section with the pole as a focus; conversely, if the orbit is conic, with a center of force at a focus, the polar reciprocal is a circle, p is constant, and the law of force is fx 1/r2.

The feet of the perpendiculars from the focus to the tangents to the orbit lie on the auxiliary circle (the tangent at the vertex in the case of the parabola). Hence the hodograph is an inverse of the auxiliary circle or the tangent at the vertex in the case of the parabola. It follows that the focus is within, on, or without the hodograph, according as the orbit is an ellipse, parabola, or hyperbola.

On account of the fact that the hodograph of an orbit described under the Newtonian law, fx 1/r2, is a circle, Hamilton designated the law as the "Law of the Circular Hodograph."

V. If the law is fp/r2, the diameter of the hodograph is easily seen to be 2p/h. Also, the diameter through O of the hodograph corresponds to the diameter aa' of the orbit and is similarly divided at O.