l. whythe - boscovich

R. J. Boscovich, S.J., F.R.S. (1711-1787), and the Mathematics of AtomismAuthor(s): L. L. WhyteSource: Notes and Records of the Royal Society of London, Vol. 13, No. 1 (Jun., 1958), pp. 38-48Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/531125 .

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38

R. J. BOSCOVICH, S.J., F.R.S. (I711-I787), AND THE MATHEMATICS OF ATOMISM

By L. L. WHYTE

[Plates 3 and 4]

HIS year is the bicentenary of the publication in 1758 of a pioneer work in the mathematical philosophy of nature: Boscovich's Theoria philo-

sophice naturalis. Though Boscovich has been regarded by the Slavs as one of their greatest scientific minds, worthy to rank beside Copernicus, Lobachew- ski, and Mendeleeff, no adequate biography or general survey of his work is available in any western language. Only in Yugoslavia has a serious attempt been made to preserve his memory and to evaluate his achievements (I). This is regrettable since Boscovich made valuable contributions to many fields of exact science and his atomic theory not only exerted a fertile influence on the development of mathematical physics, but displays features which are of particular interest in relation to the present state of particle theory.

The present note is restricted to a short biographical sketch, a statement of the atomic ideas contained in the Theoria, and an outline of their influence and similarity to methods in use in this century. This unfortunately means that no reference can be made to many other works of great interest.

Roger Joseph Boscovich was a Dalmatian, born in the Free State of

Ragusa (Dubrovnik) on I8 May 1711, who spent the major part of his life in

Italy. At the age of 14 he entered the novitiate of the Society of Jesus in Rome, but his studies in geometry and astronomy raced ahead of his theology, and he was appointed Professor of Mathematics at the Collegium Romanum in 1740, which post he held until I759. After extensive visits to other coun-

tries, including two years, 1756-58, in Vienna where he completed the Theoria, in 1764 he became Professor of Mathematics at the University of Pavia, and in 1770 director of the Brera Observatory at Milan. Three years later the Order of Jesus was suppressed and Boscovich went to Paris at the invitation of Louis XV who offered him asylum in France: 'afin qu'il put se livrer en paix a ses sublimes meditations, et satisfaire son zele ardent pour le

progres des sciences.' He remained in Paris for nine years, holding the position of Directeur de l'optique pour la Marine, an office created for him. His last

years were spent in Italy, and he died in Milan on 13 February I787.



39

This brief summary conceals a life of outstanding interest and variety. For Boscovich was not merely mathematician, astronomer, and physicist- both practical and theoretical-but geodesist, engineer, surveyor, architect, archeologist, diplomatist, and poet. He was consulted by Pope Benedict XIV on the preservation of the dome of St. Peter's, the drainage of the Pontine marshes, and the control of the Tiber, and undertook architectural tasks for the Empress Maria Theresa. He planned observatories, and made a survey and map of the Papal States, including the measurement of an arc of the meridian, and was employed on various diplomatic tasks. Beside all this Boscovich was a devoted Jesuit, and a much travelled man with friends in varied circles in the chief centres of learning.

Throughout his life Boscovich's supreme interest lay in mathematics and the physical sciences. He was primarily a geometer, concerned with the application of geometrical methods to the representation of physical phenomena. From his first study on sun spots written at the age of 25 he published in all over a hundred dissertations, papers, and books, mainly in Latin or Italian, on many branches of mathematics, astronomy, and physics, as well as more practical subjects. Some of his most important papers were on the transit of Mercury and Venus, the Aurora Borealis, the figure of the earth, gravitation, comets, tides, optics, conic sections, spherical trigonometry, the theory of observations, the adjustment of instruments, and the constitution of matter. Boscovich was an enthusiastic follower of Newton, and while holding his chair in Rome became one of the first exponents in Italy of Newtonian ideas. His last publication (1785) was a five volume collection (2) of works on

optics and astronomy, which included a full discussion of the water telescope. Boscovich's writings rapidly gained him a European reputation, and he

received honours from many learned societies, being made a member of the Academie des Sciences of Paris, and the literary Academie des Arcades of Rome. In I760 he spent seven months in England, where he saw much of Nevil Maskelyne and other astronomers, and met many leading personalities. He dined with SamuelJohnson at SirJoshua Reynolds's, conversing in Latin. The Royal Society elected him a Fellow on 15 January 1761, the same day as Reynolds. Two papers by him, on the transit of Venus and on a new micro- meter, were published in the Philosophical Transactions. In I769 the Society commissioned Boscovich to go to California to observe the transit of Venus, but this he did not undertake. Boscovich's experiences in London and Paris are described in his letters to his brother, which unfortunately have not been translated.

During his stay in Paris from 1773 to 1782 Boscovich was mainly con-



40

cemed with researches in optics, but found time for an active social life in scientific and literary circles. He made lasting friendships with Clairault and Lalande, became involved in scientific disputes with d'Alembert and Laplace, and met many visiting scientists, amongst them Benjamin Franklin and Joseph Priestley.

Boscovich's greatest achievement is the work in which he gives a com- prehensive statement of his atomic ideas: Theoria philosophiw naturalis, redacta ad unicam legem virium in natura existentium [A theory of natural philosophy, reduced to a single law of the forces existing in nature]. This book was the mature expression of ideas which Boscovich had published in a series of papers from 1745 onwards. The first edition was published in Vienna in 1758, and reprinted in Vienna in 1759, but a later edition issued in Venice in 1763 may be regarded as definitive, as it was printed under Boscovich's personal supervision. These Latin editions are rare. The Venice edition was used in the only translation yet published: the 1922 Latin-English edition (3), prepared by J. M. Child of Manchester University, which contains a short life of Boscovich.

The Theoria is one of the most original and influential works ever published on the mathematical representation of the physical idea of atomism. The bold words of the title: 'reduced to a single law', indicate the author's intention: to reveal the underlying simplicity of natural phenomena, by showing that the apparent multiplicity of physical forces might be merely the result of applying unsuitable mathematics, and that a single law of particle interaction, appropriately chosen, might account for all the properties of matter, microscopic and macroscopic.

More precisely, the Theoria seeks to demonstrate that many of the physical properties known at the time can be explained qualitatively by making the following assumptions:

I. Physical systems are constituted of simple point-formed particles, all identical. 'Matter consists of points which are perfectly simple, indivisible, of no extent, and separated from one another.'

2. Single particles (being all identical) possess no physical parameters, but pairs of particles display relative accelerations (strictly: a tendency to relative acceleration) determined by one basic law.

3. This law of interaction (4) is repulsive at very small distances, alternates in sign frequently as r, the distance between the two points, increases, and asymptotically approaches the Newtonian law of attraction as r tends to infinity. The interaction law is represented by an oscillatory curve giving the relative acceleration of two particles as a function of their distance.



Plate 3

ROGER JOSEPH BOSCOVICH, S.J., F.R.S.

(From a portrait in the possession of the Franciscan monastery, Dubrovnik.)

[Facing page 40



Plate 4

INUNC A

So :: A

Title-page of the Venice edition of Boscovich's Theoria.



41

4. The entire variety of phenomena is a consequence of this one basic law

applied to varied arrangements of the primary particles. A single oscillatory curve is to cover all physics.

On these assumptions regarding the primary particles Boscovich develops a theory of the properties (a) of structures composed of two, three, or four such particles; and (b) of larger systems composed of an indefinite number of

particles, such as solid, liquid, or gaseous bodies. The aim is a universal theory of micro- and macro-physics, in which all properties are ultimately reducible to the interactions of pairs of point particles. It may be desirable to stress the distinction between the phenomenological assumption of central forces in a

macroscopic point mechanics and Boscovich's postulation of identical point particles as the basis of a micro-theory of the structure of matter.

The theory is developed mainly by geometrical methods and is presented in effect as a hypothetico-deductive scheme to be judged by its empirical validity. Though Boscovich discusses certain wider philosophical questions concerning space and time, the mathematics of the theory is based solely on the

changing spatial relations of the primary particles. It is concerned with observable phenomena, not with their causes. No quantitative predictions are made, the aim being to demonstrate that a single oscillatory interaction law of the type proposed can, in principle, account for a wide range of physical properties. Boscovich knew that the numerical data available at the time were

inadequate to permit the exact quantitative form of the interaction law to be fixed; that was a matter for future investigations. Thus the Theoria presented, in the form of a logical and mathematical scheme, a programme Jor point atomism, in which all the primary particles are identical, a single oscillatory law determines their interactions, only relational quantities enter, and the distinction between empty and occupied space disappears.

Boscovich was able to show that many of the physical properties known at his time concerning collisions, cohesion, sound, rigid and flexible rods, crystals, fluids, changes of state, etc., as well as gravitation, could be covered

qualitatively by a law of the type proposed. He ascribed chemical differences to the contrasted particle structures of the different chemical substances.

Boscovich says of his theory: 'it so happens that the principal points of all the most distinguished theories of the day, interlocking and as it were cemented together in a truly marvellous way, are combined in it.' But his intention had not been to make a synthesis of contemporary methods. He had set himself a specific task: to eliminate the discontinuity involved in the impact of finite bodies by treating collisions as the expression of continuous interactions between particles treated as points. Thus he found himself



42

combining the Newtonian concept of vis or force, attractive and repulsive, with the Leibnizian concept of monads or point centres of action, but thereby transforming them.

Greek material atomism, with its finite extended units of matter, had

already been revived by many thinkers, such as Magnenus, Gassendi, Boyle, and Newton. Though Boscovich was a keen disciple of Newton, he emanci-

pated himself from this traditional conception and adopted instead what has been called 'Pythagorean' or geometrical atomism (5), the view that the units of existence are indivisible points, or numbers of such points arranged in

geometrical patterns. He replaced Newton's 'solid, massy, hard, impenetrable' particles of various sizes and shapes by a single class of interacting physical points (6), and called this his 'new world'.

It is difficult for us two centuries later to appreciate the daring originality of the substitution of point centres of interaction for finite pieces of matter, a

metaphysical emancipation opening new realms to mathematical physics, and

profoundly influencing the subsequent development of physical theory. Boscovich's transformation of the basis of physical science is one of the

greatest steps ever taken to liberate thought from the immediate impressions of the senses (7). By discarding the redundant concept of 'material stuff'

occupying space, the fundamental concepts of physics were at once simplified and rendered more powerful. The dualism of discrete units of matter separated by empty space was replaced by the single and more fertile conception of the

spatial relations of discrete points, and thereby many pseudo-problems eliminated (e.g. those concerned with 'absolute', i.e. non-relational space) and new

possibilities opened up (e.g. the penetration of bodies by high speed particles, and existence of matter of very high density).

As Faraday (8) emphasized, the superiority of point particles over any other atomistic model lies in the fact that they assume the minimum possible, and throw the main responsibility on to the laws of interaction required by the observations. This is appropriate, since the apparent properties of particles are evidenced only in their interactions. But current particle theories still ascribe contrasted properties to different classes of particles and it is too early to guess what kind of fundamental model may be employed in some future unified theory. Thus no final conclusions can yet be drawn regarding the Boscovichian conception of a system of identical points in three-dimensional

space. This remains an ideal, which at present appears too simple to cover the facts.

Though references are frequently made to Boscovich's 'atoms', this is

misleading. He was not concerned with chemical atoms, which were only



43

discovered subsequently, but with hypothetical primary constituents of matter which he called puncta or particula. These are the permanent units whose stable or changing arrangements constitute physical systems. Since Boscovich assumes that chemical properties are the result of different groupings of his point particles, had the concept of a chemical atom existed in his day he would have inferred that under appropriate conditions all atoms would be unstable and mutually transformable. This is a natural consequence of his assumption that all physical systems represent different arrangements of one type of fundamental particle. In current particle physics there is no equivalent to Boscovich's single class of permanent particles, but nucleons, as the relatively permanent constituents of nuclei, come nearest to it.

When the Theoria was published the feature which evoked most interest was not the interaction law but the underlying assumption that physical phenomena express the interplay of point-formed entities. This view, though alien to the naive commonsense conception of matter, is so simple and powerful, and lends itself so readily to mathematical treatment, that it is not surprising that it was increasingly adopted during the nineteenth century until it found its clearest expression in classical electron theory. Even where the fundamental constituents of matter were assumed to be of finite size, they were usually treated mathematically as if they were point centres. (The vortex atom is a contrary example.) Only with the advent of wave mechanics were point particles replaced by a more powerful conception.

The concept of point particles probably occurred to others independently, but Boscovich was the first to base a systematic theory on this model of the primary constituents of matter. Moreover there can be no doubt of the extensive influence of the Theoria, which was known to most leading mathematical physicists from the end of the eighteenth to the opening of this century. Here two issues must be distinguished. Boscovich's point particles had a negligible effect on the development of chemical atomic theory; it was in the fundamental mathematical physics of the atomic constitution of matter that they proved fertile.

Boscovich's ideas evoked a vigorous response in Britain. Joseph Priestley was much interested and wrote to Boscovich in 1778 supporting a plan for a French translation of the Theoria. Priestley adopted Boscovich's view of the penetrability of material bodies, and suggested that a small body moving with sufficient velocity might pass through another body without disordering its constituent particles (9). This was a direct echo of an argument developed by Boscovich in the Theoria (Io). Indeed our present view of a solid body as an open lattice of points is the same as Boscovich's.



44

In I785 Robison, Professor of Physics in Edinburgh, said: 'If we shall ever

acquire the knowledge of a true theory, it will resemble Mr Boscovich's in

many of its features.' The Encyclopaedia Britannica, in the supplementary volumes issued in I801, devoted 14 pages to an analysis of Boscovich's theory. Faraday (8) wrote in I844: 'the safest course appears to be to assume as little as possible, and in that respect the atoms of Boscovich appear to me to have a

great advantage over the more usual notion.' Clerk Maxwell ( I) said the same: 'The best thing we can do is to get rid of the rigid nucleus and substitute for it an atom of Boscovich', and described Boscovich's ideas in some detail in his Encyclopaedia Britannica article Atom (I875). Kelvin cited Boscovich in

many papers, spoke of Boscovich's 'great book,' called Navier's and Clausius' work on the kinetic theory of gases 'all developments of Boscovich's theory', and after some marked oscillations of attitude said in Igo5: 'my present assumption is Boscovichianism pure and simple' (12). Finally, J. J. Thomson

(13) showed in I907 that under Boscovich's oscillatory field only certain central orbits, separated by finite intervals, are stable, thus preparing the way for Bohr's decisive step.

It may appear surprising that Boscovich's ideas, in spite of their apparently speculative mathematical character, found such a sustained response in this

country. But we must note first, that Boscovich based his work on that of Newton, and second, that his emphasis was essentially empirical. The law of interaction was presented not as an arbitrary mathematical form to be imposed on phenomena, but as a formula whose constants were to be determined by subsequent research. Mathematics and observation were well balanced in Boscovich's outlook.

A similar influence could be traced (14) on the European continent, from Clairault, Lalande, d'Alembert, and others who knew Boscovich personally, through Laplace (who praises Boscovich's theory of observations), Poisson, Gay-Lussac, Ampere, and Cauchy to Fechner,Weber, and Lorentz. Saint- Venant, in 1878, called Boscovich 'the most consistent Newtonian of any', but he might more appropriately be regarded as the father of twentieth

century atomism, in which hard finite atoms and particles have been increas-

ingly replaced by point-centres of interaction, though the finite region of

space supposed to be occupied by electrons and other particles is still an embarrassment.

Yet after the first World War awareness of Boscovich's contribution

began to fade. There is irony in this, because his methods are in several

respects more characteristic of the twentieth than of the eighteenth or the nineteenth century, and can only now be properly understood and valued.



45

The striking advantages of his point particles have drawn attention away from other aspects of his theory. It is a remarkable fact that those features (I5) of his method which distinguish it from its predecessors reappear in theories of the twentieth century. In attempting to develop a systematic theory of point particles Boscovich was forced to anticipate several characteristics of the physics of our own time:

I. Unlike the various classical theories of the nineteenth century, which were scale-free, employing no natural lengths, Boscovich's theory was scale- fixed, like quantum theory (I6). In order to permit the existence of a series of distances at which two particles would be in equilibrium-Boscovich called these the 'limit points of cohesion'-he introduced into his interaction law fundamental parameters of length. Their actual values were not known, but he knew that such parameters were necessary and he even discussed how many independent fundamental lengths might be required. Boscovich substituted for the finite size of the traditional material atoms point particles obeying an interaction law which contains natural lengths. This method, which makes the basic laws and not finite pieces of matter responsible for the definiteness of scale of atomic and molecular systems, is used in quantum mechanics.

2. Since all phenomena are regarded as reducible to changes in the relative distances of point particles, the theory is completely relational. There is no place in it for any quantities other than those which express the changing spatial relations of particles. Bertrand Russell (17) noted that '[Leibniz's] relational theory of space should have led him ... to a theory of unextended centres of force ... The true Leibnizian Dynamics is not his own, but that of Boscovich'.

Boscovich develops various consequences of this relational method, such as the fact that an expansion or contraction of the entire universe, accompanied by corresponding changes in the laws, would not be observable. He gives (I8) the earliest analysis of space and time measurements in which the spatial and temporal parameters of any system are necessarily affected in some minute degree by its state of motion relatively to all the other particles in the universe (more than a century before Mach). Rods and clocks must therefore be regarded as only approximately constant. In Boscovich's words, as translated by Child: 'in my theory of points at a distance from one another, all the points of a ten-foot rod, while they are being transferred, really change the distance continually.' And again: 'However, there is always some slight change, owing to the fact that the forces which connect the points of matter will be changed to some slight extent, if its position is altered with respect to all the rest of the Universe.' Though Silberstein (I9) has called attention to



46

Boscovich's 'remarkably clear and radical ideas regarding the relativity of space, time, and motion', they have been neglected in most histories of relativistic ideas. (Boscovich did not, of course, introduce the velocity of light as an invariant.)

3. We now come to a feature of Boscovich's method which is of exceptional interest today: its kinematic character. It is the only kinematic particle theory yet proposed.

Here certain terms must be defined. Theories using three primary (i.e. irreducible) dimensional magnitudes (L, T, M) are called mechanical. In such theories mass properties are not reducible to anything else. Theories using only two primary dimensional magnitudes (L, T) are called kinematic, and must derive mass properties from some prior kinematic (space-time) or structural principle. All recent atomic theories are mechanical. Einstein's General Theory of Relativity is a macroscopic kinematic theory, deriving mass properties from space-time geometry. The attempt to substitute a kinematic for a mechanical foundation is also evident in parts of Eddington's and Milne's theories.

Though Boscovich could not express his intention unambiguously, his theory is kinematic, not mechanical (20). This has not yet been sufficiently appreciated. Though Boscovich was a Newtonian, he went beyond Newton and eliminated from his method Newtonian mass as a primary dimensional variable. He was able to do this because he postulated that all the primary particles were identical and thus could obtain the equivalent of mass ratios by counting the number of particles in two systems. Boscovich's vires are not Newtonian forces, but pure kinematic tendencies or accelerations, and his massa are not Newtonian masses, but pure numbers representing the numbers of primary particles in any system.

Clerk Maxwell, in his Atom article did not make this clear, and translated Boscovich's vis as 'force'. Indeed Boscovich's particles were universally interpreted as 'point centres of force' until his translator, J. M. Child, drew attention to this mistake. He says in his Introduction that 'the Theory of Boscovich differs from that of Newton in being purely kinematical', and again, 'Clerk Maxwell, in ascribing mass to a Boscovichian point of matter, seems to have been obsessed by a prejudice'.

As a kinematic thinker, treating mass, density, and force as secondary concepts derivable from structural and kinematic principles, Boscovich is closer to Einstein than to Newton. Of course Boscovich, writing in Latin in 1758 and lacking the equipment of dimensional theory, did not make clear the changed status of'mass' in his theory. Yet he came very close to stating it.



47

(i) 'The mass is equal to the number of points', i.e. the dimensionless ratio of two masses is derivable from geometrical structure, by counting the particles in each. (ii) 'I have come to the conclusion that the idea of mass is not strictly definite and distinct, but that it is vague, arbitrary, and confused.' Perhaps Boscovich is here trying to say that Newtonian mass is not what it claims to be, an independent magnitude, but a function of the spatial structure of a system (in Boscovich's atomic theory).

Particle theory is today challenged by the obscure mass spectrum of the elementary particles, and here something may perhaps be learnt from Boscovich as regards the greater power of kinematic as against mechanical particle theories. Mechanical theories which treat mass as an irreducible magnitude may for that reason be incapable of interpreting the mass ratios. To derive the mass ratios it may be necessary to go behind the concept of mass to prior kinematic principles, just as Einstein did in the macroscopic field. Thus Boscovich's programme for a kinematic atomic theory based on a single class of particles may contain a hint regarding the appropriate form of the future unified theory of particles.

This brief note has been concerned only with Boscovich's atomic ideas, and no reference made to his achievements in pure mathematics, astronomy, optics, and other branches of exact science. It is much to be hoped that a general survey of his work will be undertaken by someone competent in the many fields involved, or alternatively that a selection of the studies already made in the Serbo-Croat language be translated.

One cannot read the Theoria without recognizing a universal spirit, a mind of exceptional imaginative power whose way of thinking is closely akin to our own two centuries later. Whatever lies ahead in new developments of particle theory, Boscovich's ideal: to trace everything to the differences of arrangement and relative motion of identical point particles, without any arbitrary quantitative or qualitative distinctions, still sets a standard for research.

NOTES

(i) The most extensive studies of Boscovich's life and work are in Serbo-Croat in various

publications of the Academie des Sciences et des Arts des Slaves du Sud, Zagreb, particularly Rad jug. Akad. Znan. Umj. 87-90 (1887-88), 230-234 (1926-29), and

recently. See also H. V. Gill, Roger Boscovich (Dublin, 1941), a sketch of Boscovich's

anticipations of modern theories; D. Nedelkovitch, La philosophie naturelle et relativiste de R. J. Boscovich (Paris, 1922); L. Cermelj, 'R. J. Boscovich als relativist', Arch. Gesch. Math. II (I928-29), 424; M. Oster, Roger Joseph Boscovich als Naturphilosoph (Inaug. diss., Bonn, I909). For bibliographies of Boscovich's writings see the Latin-English



48

edition of the Theoria, or C. Sommervogel, Bibliotheque de la Compagnie deJdsus (Paris, 1932), vol. I, which is more complete. I wish to thank Dr Z. Markovic of Zagreb for much help in connexion with my studies on Boscovich. Dr Markovic is now editing a series of books, published by the Zagreb Academy, entitled Rudder Boscovic, grada knjiga [Collection of material on life and work]. Professor E. N. da C. Andrade, F.R.S., also made helpful suggestions.

(2) Opera pertinentia ad opticam et astronomiam (Bassani, I785). (3) Open Court Publishing Co., Chicago and London. An interesting critical commentary

on this edition, with information on Boscovich's life, is given in Dr V. VariCak's 60-page review, an English translation of which appears in the Zagrab Academy's Bulletin des travaux de la classe des sciences mathematiques et naturelles [otherwise Izvje3sa o raspravama], 20 (1924), 45. Mr Child has been of great assistance to me in preparing material of Boscovich (see note I6).

(4) See S. Chapman, Nature, Lond. I46 (I940), 607, on the similarity of Boscovich's expression to the complex law of force currently assumed for molecular fields.

(5) On Boscovich and 'Pythagorean atomism' see A. Lalande, Lectures sur la philosophie des sciences (Paris, I893), p. 261.

(6) There is no evidence that Newton considered the possibility that all the small solid

particles were identical, i.e. of the same size and density. Thus Boscovich here went

beyond Newton in two respects: by assuming (i) that all the fundamental particles were identical; and (ii) that they could be treated as points. Boscovich was the first scientist to postulate the identity of all the particles constituting matter.

(7) As Nietzsche recognized. Beyond good and evil (London, I907), §I2.

(8) M. Faraday, Phil. Mag. 24 (1844), 136.

(9) J. Priestley, History and present state of discoveries relating to vision, light, and colours (London, 1772), period VI, lection I, chapter III, particularly pp. 390-I; Disquisitions relating to matter and spirit (London, 1777), p. I9.

(Io) Theoria, 5366, and elsewhere.

(II) J. Clerk Maxwell, Nature, Lond. I6 (1877), 246.

(I2) Lord Kelvin, Phil. Mag. 10 (1905), 695.

(I3)J. J. Thomson, Corpuscular theory of matter(London, 1907), p. I60.

(14) See Fechner, Atomlehre (1864), p. 229; F. A. Lange, Geschichte des Materialismus (Iserlohn, 1873-5), vol. 2, pp. 192-3; F. Moigno, Cosmos, 2 (1852), 373.

(I5) An analysis of these features was given by the author in Nature, Lond. 179 (1957), 284. See also Ann. Sci. 10 (I954), 20.

(16) MrJ. M. Child, Boscovich's translator, informs me that two aspects of the Theoria which struck him especially were 'the explanations of cohesion and of Bohr's atom theory'. (Under Boscovich's law of interaction only certain central orbits are stable. SeeWhyte, loc. cit. note i5.)

(I7) Critical exposition of the philosophy of Leibniz (Cambridge, 1900), p. 9I.

(I8) E.g. in supplements I and II to the Theoria.

(19) L. Silberstein, Theory of relativity (2nd ed., London, 1924), p. 38. (20) On this, see Whyte, loc. cit. note I5.



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