800 MOON motions of the planets, and withal so intricate, that little interest attaches to it. The student of Arabian science may find much to interest him in the astronomical speculations of the Arabs, but this people do not seem to have furnished anything in the way of suggestive theory. In the fourth book of De Revolu- twnibus^- where we find the lunar theory of Copernicus, no writer later than Ptolemy is referred to. Moreover, as already intimated, the work of Copernicus in this particu lar direction forms little more than an episode in the his tory of the subject. The working hypothesis of the great founder of modern astronomy was borrowed from the ancients, and was that the celestial motions were all either circular or compounded of circular motions. The hypo thesis of equal circular motions, though accepted by Ptolemy in name, was so strained by him in its applications that little was left of it in the Almagest (the Arabic translation of his Syntaxis). But, by taking the privilege of compound ing circular motions indefinitely in other words, of adding one epicycle to another Copernicus was enabled to repre sent the planetary and lunar inequalities on a uniform system, though his heavens were perhaps worse " scribbled o er " than those of Ptolemy. To one epicycle representing the equation of the centre he added another for the evection, and thus represented the longitude of the moon both at quadratures and oppositions. But the third inequality, "variation," which attains its maxima at the octants and vanishes at all four quarters, was unknown to him. To Tycho Brahe is commonly and justly ascribed the discovery of the variation. Joseph Bertrand of Paris has indeed claimed the discovery for Abu 1-Wefa, an Arabian astro nomer, and has made it appear probable that Abil 1-Wefa really detected inequalities in the moon s motion which we now know to have been the variation. But he has not shown, on the part of the Arabian, any such exact de scription of the phenomena as is necessary to make clear his claim to the discovery. As regards Tycho, although he discovered the fact, he could add nothing in the way of suggestive theory. To the double epicycle of Copernicus he was obliged to add a motion of the centre of the whole lunar orbit round a circle whose circumference passed through the centre of the earth, two revolutions round this circle being made in each lunation. Kepler, by intro ducing a moving ellipse having the earth as its focus, was enabled to make a nearer approach to the truth than any of his predecessors. But the geometrical hypotheses by which he represented the inequalities due to the action of the sun form no greater epoch in the progress of science than do the geometrical constructions of his predecessors. We may therefore dispose of the ancient history of the lunar theory by saying that the only real progress from Hipparchus to Newton consisted in the more exact deter mination of the mean motions of the moon, its perigee and its line of nodes, and ir> the discovery of three new inequalities, the representation of which required geometri cal constructions increasing in complexity with every step. The modern lunar theory commenced with Newton, and consists in determining the motion of the moon deductively from the theory of gravitation. But the great founder of modern mechanics did not employ the method best adapted to lead to the desired result, and hence his efforts to con struct a lunar theory are of more interest as illustrations of his wonderful power and correctness in mathematical reasoning than as germs of new methods of research. He succeeded perfectly in explaining the elliptic motion of two mutually attracting bodies round their common centre of gravity by geometrical constructions. But when the prob- 1 The full title, De Revolntionibus Orbium Coslestium- Libri VI. (small folio, Nuremberg, 1543). lem was one of determining the variations from the elliptic motion which would be produced by a third body, such constructions could lead only to approximate results. The path to modern methods was opened up by the Continental mathematicians, whose great work consisted in reducing the problem to one of pure algebra. The chasm between the laws of motion laid down by Newton and a problem of algebra seems so difficult to bridge over that it is worth while to show in what the real spirit of the modern method consists. We call to mind the statement of Newton s first two laws of motion : that a body uninfluenced by any force moves in a straight line and with uniform velocity for ever, and that the change of motion is proportional to the force impressed upon the body and in the direction of such force. These two laws admit of being expressed in alge braic language thus : let us put m the mass of a material point ; x its distance from any fixed plane whatever ; t the time ; X the sum of the components of all the forces acting upon the point in the direction perpendicular to the fixed plane, it being supposed that each force is resolved into three mutually perpendicular components, one of which is perpendicular to the fixed plane ; then the differential equation expresses Newton s first two laws of motion with a com pleteness and precision which is entirely wanting in all statements in ordinary language. The latter can be no thing more than lame attempts to express the equation in language which may be understood by the non-mathe matical reader, but which bear the same relation to the algebraic equation that a statement of the operations of the Bank of England in the symbolic language of a tribe of savages would bear to the bank statement in pounds, shillings, and pence. By taking two other f >lanes, perpen dicular to each other and to the first plane, we have three equations like the one last written. The law of gravitation and Newton s third law of motion enable us to substitute for X and the other forces the masses and coordinates of the various attracting bodies. Thus the data of the problem are expressed by a triplet of three equations for each attract ing body. The integration of these equations is a problem of pure algebra, which, when solved, leads to expressions that give the position of each body in terms of the time, which is what is wanted. The special form which it is necessary to give the equations has not been radically changed during the century and a half since this method of research was opened out. The end aimed at is the algebraic expression of all the quantities involved in the form of an infinite series of terms, each consisting of a constant coefficient multiplied by the sine or cosine of an angle increasing uniformly with the time. It is indeed a remarkable fact that, notwithstanding the great advances which modern mathematics has made in the discovery of functions more general than the old-fashioned sines and cosines of ele mentary trigonometry, especially of elliptic functions, yet the form of development adopted by the mathematicians of the last century has remained without essential change. It will be instructive to notice the general and simple property of the trigonometric functions to which is due their great advan tage in the problems of celestial mechanics. It may be expressed thus: If we have any number of quantities, each of which is ex pressed in the form of a trigonometric scries in which the angles increase uniformly with the time, then all the powers and 2)roducts of these quantities, and all their differentials and integrals with respect to the time, may be expressed in scries of the same form. This theorem needs only an illustration by an example. Let our quan tities be X and Y, and let us suppose them expressed in the form X = a cos A + b cos B + c cos L + , &c. Y= a sin A + b sin B + c sin G + , &c. , in which we may suppose that the quantities a, b, c, &c., converge
towards zero. In forming their product, the first term will be