A Modern Zoroastrian

by S. Laing

xdy + ydx = 2zdz

we know that the left-hand side is the differential of xy, and therefore that by integrating it we shall get xy; while the right side is the differential of z² which we shall get by integrating it. The relation expressed therefore is that xy = z², or, in other words, that a rectangle whose sides are x and y exactly equals a square whose side is z.

Fig. 1.

Fig. 2.

Fig. 3.

The use of this device in assisting calculation will be apparent if we take the case of an area bounded by a curved line. We cannot directly calculate this area, but we can easily tell that of a rectangle. Now it is evident that if we inscribe rectangles in this area ABC, the more rectangles we inscribe the less will be the error in taking their sum as equal to the curved area. This is apparent if we compare fig. 2 with fig. 3. Suppose we take a point P on the curve, call BN = x and PN = y, and suppose Nn to be dx, the differentially small increment of x, and pq = dy the corresponding small increment of y. The area of the rectangle PqnN = PN × Nn = ydx, and differs from the true curvilinear area PpnN by less than the little rectangle of Pq × pq or of dx.dy. But, as we have seen, if we push our division to the first infinitesimal order, or make Nn and pq differentials of x and y, dx.dy may be neglected—i.e. multiply the number of rectangles indefinitely, and the sum of their areas will differ from the true area inclosed by the curve by an error which is evanescent.

If then x and y are connected by some fixed law, as must be the case if the extremity of y traces out some regular curve, the relation between them may be expressed by an equation, which will remain one however often it may be differentiated or again integrated, and whatever modifications or transformations it may receive by mathematical processes which do not alter the essential equality of the two sides connected by the symbol of equality =. Thus by differentiating and casting off as evanescent all differentials of a lower order than that which we are working with, we may arrive at forms of which we know the integrals, and by integrating get back to the results in ordinary numbers, which we were in search of but could not attain directly.

The same thing will apply if our symbols are more numerous, and if they express relations of motion as well as of space, or, in fact, any relations which are governed by fixed laws expressible by equations. If I have succeeded in conveying to the readers any idea of this celebrated calculus, they will perceive what an analogy it presents to the idea of modern physical and chemical science, that of molecules, atoms, and ether, forming differentials of successive orders of the infinitely small. It is certainly most remarkable that while the former was a purely intellectual idea based on mathematical abstractions, and which was invented and worked as an instrument for solving the most intricate astronomical problems for nearly two centuries, without a suspicion that it represented any objective reality: the latter idea, based on actual experiment, seems to show that differentials and integrals have their real counterpart in nature and represent fundamental facts in the constitution of the universe.

Those who are of a mystic or metaphysical turn of mind may discern in this, arguments for matter and laws of matter being after all only manifestations of one universal, all-pervading mind; but in following such speculations we should be deserting the solid earth for cloudland, and passing the limit of positive knowledge into the region where reflections of our own hopes, fears, religious feelings, and poetical sentiments form and dissolve themselves against the background of the great unknown. For the present, therefore, I confine myself to pointing out how these undoubted truths of mathematical science, which have verified themselves in the practical form of enabling us to predict eclipses and construct nautical almanacs, correspond with and throw light upon the equally certain facts of this succession of infinitely small quantities of successive orders in the constitution of matter.

An attempt has recently been made, based on abstruse mathematical calculations, to carry our knowledge of the constitution of matter one step further back, and identify atoms with ether. This is attempted by the vortex theory of Helmholz, Sir W. Thomson, and Professor Tait. It is singular how some of the ultimate facts discovered by the refinements of science correspond with some of the most trivial amusements. Thus the blowing of soap-bubbles gives the best clue to the movement of waves of light, and through them to the dimensions of molecules and atoms; and the collision of billiard-balls, knocked about at random, to the movements of those minute bodies, and the kinetic theory of gases. In the case of the vortex theory the idea is given by the rings of smoke which certain adroit smokers amuse themselves by puffing into the air. These rings float for a considerable time, retaining their circular form, and showing their elasticity by oscillating about it and returning to it if their form is altered, and by rebounding and vibrating energetically, just as two solid elastic bodies would do, if two rings come into collision. If we try to cut them in two, they recede before the knife, or bend round it, returning, when the external force is removed, to their original form without the loss of a single particle, and preserving their own individuality through every change of form and of velocity. This persistence of form they owe to the fact that their particles are revolving in small circles at right angles to the axis or circumference of the larger circle which forms the ring; motion thus giving them stability, very much as in the familiar instance of the bicycle. They burst at last because they are formed and rotate in the air, which is a resisting medium; but mathematical calculation shows that in a perfect fluid free from all friction these vortex rings would be indivisible and indestructible: in other words, they would be atoms.

The vortex theory assumes, therefore, that the universe consists of one uniform primary substance, a fluid which fills all space, and that what we call matter consists of portions of this fluid which have become animated with vortex motion. The innumerable atoms which form molecules, and through molecules all the diversified forms of matter of the material universe, are therefore simply so many vortex rings, each perfectly limited, distinct, and indestructible, both as to its form, mass, and mode of motion. They cannot change or disappear, nor can they be formed spontaneously. Those of the same kind are constituted after the same fashion, and therefore are endowed with the same properties.