Strange seas of Thought

Newton's Principia, first published in 1687, established in a manner second to none the power of mathematical reasoning in physical theory. In the succeeding three centuries, developments in this same scientific method have led to changes that have completely altered the face of the earth. Yet, how many can claim to have a direct and detailed appreciation of what it was that Newton actually achieved? Only a few have ventured into Newton's forbidding cathedral of scientific understanding with the persistence and ability to appreciate both its genuine miracles and the sublime elegance of the imposing structures to be found within it.

One of the most eminent of those who have done so is Subrahmanyan Chandrasekhar. Now well into his 85th year, he is the world's most distinguished living theoretical astrophysicist. Among the many prizes he has been awarded was the Nobel prize for physics in 1983. Roughly each decade of his scientific life has been devoted to a separate subject, and at the end of each period he has produced a profoundly authoritative text in that area. The last of these major works concerned the mathematical theory of black holes, an important contribution within the context of general relativity: Einstein's geometrical theory of gravity. Before that time, most of Chandrasekhar's profound contributions had been within the framework of Newton's gravitational theory - a theory that had lasted for more than two and a quarter centuries, until it was finally superseded by Einstein's revolutionary conceptions in 1915. It is striking that in Chandrasekhar's latest major work he has returned not only to Newton's older ideas but to the very form in which they were originally put forward - even to examining the nature of the supreme creative powers that lay behind them.

Not many of our present generation will have even attempted to examine Newton's Principia. Some, such as myself, will have gained much of their information from secondhand sources. Others may have attempted to read Newton directly but been turned back by the difficulty of following his arguments in prose. Yet others will have appealed to accounts that are merely descriptive and lack necessary detail. All will be grateful to Chandrasekhar for providing us with an access to the magnificence of the Principia that is exciting and relatively painless. He acts as a superb guide, pointing out ornaments of particular elegance and subtlety, while all the time keeping us in mind of the grandeur of purpose of the magnificent architecture. Phrases such as "the elegance and simplicity of Newton's demonstrations are startling" abound. They act as signposts, telling us, if we are in danger of becoming weary, to pay close attention and perhaps return with more care to what we may have just read. Such attention is always rewarded, as there are arguments here of great beauty and depth that we might otherwise pass by.

Noteworthy is Newton's "superb theorem" that, under the inverse square law, the force acts as if to the exact centre of a spherically symmetric body, proved by a construction that "must have left its readers in helpless wonder" (J. E. Littlewood, quoted by Chandrasekhar). For myself, I was particularly taken also by the extraordinary simplicity and elegance of Newton's geometrical construction for the formula concerning the centripetal force for a general orbital shape in terms of the circle of curvature at the point in question. Closely related is his beautifully succinct geometrical expression for the relationship between the laws of force, with respect to two alternative centres of attraction, which gives rise to the same orbital shape. Two alternative centres of centripetal force (C.F.) can give rise to the same orbital shape if the force laws are related by (C.F.)R;new=(C.F.)S;old SG3/SP.RP2.

Here, R and S are the alternative centres, and SG is parallel to RP, where PG is the tangent to the orbit at P. This construction leads Newton to a superbly elegant derivation of the fundamental fact that it is with the inverse square law of attraction that elliptical orbits arise, the centre of force being a focus of the ellipse. In an argument appearing for the first time only in the second (1713) edition of the Principia, Newton shows, using but a single line of geometry, that this result for the inverse square law is equivalent to the much more easily derived fact that, with a force proportional to the distance, elliptical orbits still arise, but now with the centre of force being the centre of the ellipse. It is striking that this argument was discovered by Newton during his "London years" when, according to some historians, he had virtually lost interest in science - yet it is clear from this discovery that his talents had been in no way diminished despite his heavy duties at the Royal Mint and other prevailing interests. As Chandrasekhar remarks, Newton continued to voyage "through strange seas of Thought, alone" (Wordsworth).

The relationship between the inverse square law and the "proportional to distance" law that is exhibited by their having orbits of the same shape must have struck Newton particularly, for he noted other instances where this kind of phenomenon arises, pointing out a kind of duality that occurs between certain specific laws of attraction. The significance of this duality appears to have lain dormant for some two centuries, with due attention to its potential arising only in 1911, from K. Bohlin, as has been commented upon by V. I. Arnold (described by Chandrasekhar in chapters 31-36). Arnold also points out another of Newton's mathematical contributions, of great profundity for its time, relating to the "transcendence of Abelian integrals", which Newton must have come across while exploring the integrations required in orbital motion in accordance with Kepler's law of areas. Chandrasekhar says that this is "a striking manifestation of Newton's mathematical insight which enabled him, as in this instance, to surpass the level of scientific understanding of his time by 200 years". (In Chandrasekhar's account of this result, a few misprints and inaccuracies of description arise - although in the text as a whole, I noticed only a very few places where such lapses are to be found.) One of the most daunting aspects of Newton's Principia has always been its geometrical language. Very few modern readers are familiar with the geometrical reasoning Newton used. Historians of science often argue that Newton deliberately clothed his arguments in the "archaic" language of Euclid and Archimedes while himself being fully in possession of the more modern techniques of calculus. Some claim he did so either because this was the language his contemporaries would understand or perhaps because he wished to keep secret the more powerful calculus procedures he had developed earlier. Others suggest he was merely expressing his contempt for the coordinate-based approach to geometry promoted by Descartes.

Chandrasekhar's position on this is completely different and much more persuasive. He argues convincingly that Newton's geometrical arguments are themselves more powerful and elegant than the alternative methods of calculus that would be used by virtually all scientists of today. In fact, Chandrasekhar points out in several places that Newton does not shy away from using calculus-based arguments when they are appropriate, but in most cases Newton's geometrical methods are not only more concise and elegant, they reveal deeper principles than would become evident by the use of those formal methods of calculus that nowadays would seem more direct.

This interplay between Newton's geometrical procedures and the more modern approach to the issues with which Newton was concerned - notably celestial mechanics - provides one of the primary themes of Chandrasekhar's book. His initial approach to the reading of the Principia was to examine each of the propositions and its corollaries in turn and then without looking at Newton's argument to supply his own proof using all the standard procedures of modern analysis with which he is so excellently equipped. Only after having devised an appropriate argument to his own satisfaction did Chandrasekhar allow himself to examine Newton's own reasoning. In almost all cases, he found to his astonishment that Newton's "archaic" methods were not only shorter and more elegant but more revealing of the deeper issues (as Chandrasekhar has personally informed me).

It has often been argued that the geometrical methods used in the Principia held back the development of mathematical science in England while the "more powerful" analytical approach to the calculus was followed in the work of Leibniz, Euler, Lagrange and others. Undoubtedly it is true that Leibniz's notation, as developed further by Euler, led to the powerful algebraic and analytical techniques of modern-day calculus - techniques that are the common tools of modern mathematical scientific method. Newton's influence on the way in which calculus is used, at least in formal manipulations, is indeed less than that of Leibniz and Euler.

This may be partly due to the fact that Newton did not publish his approach to the calculus until after Leibniz's version had appeared; but also Leibniz developed his tools in a way that allowed formal manipulation to be carried out with greater ease. One advantage of a good notation, such as that of Leibniz, is that calculations can then be performed without the necessity of a continuing understanding of what the symbols actually mean. This can be valuable in freeing the mind from continual reference to the deeper issues involved in what one is doing, but there is a loss, too, in that one may lose sight of the very principles on which the validity of these manipulations is based. This type of dichotomy, between unthinking computation and the alternative of a continual re-examination of basic principles, is especially evident in scientific research today. The development of modern high-speed computers has enhanced the power of the manipulative approach, to the extent that the importance of the underlying principles is often obscured. Yet a need for continual reference to underlying principles can itself be stultifying. Newton's geometrical methods might well, in the hands of lesser individuals, have led to little more than a scratching of the surface of the problems of planetary motion. But Newton's supreme combination of geometrical mastery with his profound understanding of the power of perturbational methods, joined with his calculational and experimental abilities and his deep physical insight, enabled him to achieve what no one else could have been able to do using such geometrical methods alone.

Since Newton's day, the history of physical science has followed the analytical route rather than Newton's geometrical one, or at least it has largely done so far. Yet, as Chandrasekhar makes abundantly clear, there are many insights still to be gained by re-examining the very different type of geometrical development implicit in Newton's own procedures. And, it may be recalled, Einstein's general relativity is a profoundly geometrical theory!

Those who are familiar with Chandrasekhar's work, in which his exceptional facility with complicated equations is a characteristic feature, may find it surprising that he has devoted so many of his latest years to exploring the delights of Newton's geometry. This surprise is removed, once one appreciates that it is in his artistry with equations rather than with brute-force calculations, that his abilities, in this regard, lie. Moreover, his artistry is not confined merely to equations, as his writings on other topics (for example, Truth and Beauty: Aesthetics and Motivations in Science, 1987) make abundantly clear. It is one of his major tasks, in writing on Newton's Principia, to bring forth the profound beauty that is to be found in Newton's arguments, and in this he succeeds extraordinarily well.

I have benefited greatly from Chandrasekhar's account, which has brought home much more of the excitement and depth of Newton's great work than I had been aware of before. It is comprehensive on matters in the Principia that relate to Newton's universal gravitation and on some gems peripheral to the main arguments, though parts of the Principia are not addressed, such as motion in a resisting medium and most of the matters relating to observation. It is a very individual account, in which Chandrasekhar brings understanding, expertise, and sensitivity to bear on the problems of revealing Newton to the "common reader" (Dr Johnson's phrase). The common reader must be prepared to work hard, however, though the rewards are great for the one who does so.

Scant attention is paid, in Chandrasekhar's account, to the writings of others concerning Newton or the Principia. I can certainly forgive him for this, if forgiveness is needed, as I forgive him also for what may seem to be downgradings of other scientists coming pre- or post-Newton, who are traditionally awarded credit for understandings for which Newton himself can be argued to lay primary claim.

Overall, there is no question that Chandrasekhar has performed a uniquely valuable service. I know I shall often return to his book and continue to take advantage of his insights into Newton's supreme work for years to come.

Sir Roger Penrose is Rouse Ball professor of mathematics, University of Oxford.