Thomas Harriot and the rise of atomism

[In 1591 Thomas Harriot prepared a large triangular chart of numbers, with the the explanatory note:]

"There are three speciall groundplats vpon the which may be orderly piled bullets: The triangle: the square: and the oblonge. Concerning pilong there are two questions: one; the nomber of bulletes to be piled being geven with the forme of the gound plat, to know how many must be placed in every rank, with how many rankes in the sayd ground plat.

"The second a pile being made to knowe the nomber of bulletes therein conteyned.

"ffor the aunsweringe of which two questions this table I haue calculated for the purpose."

Obviously, this is a quick reference chart prepared for Ralegh to give information on the ground space required for the storage of cannon balls in connection with the stacking of armaments for his marauding vessels. The chart is ingeniously arranged so that it is possible to read directly the number of cannon balls on the ground or in a pyramid pile with triangular, square, or oblong base. All of this Harriot had worked out by the laws of mathematical progression (not as Miss Rukeyser suggests by experiment), as the rough calculations accompanying the chart make clear. It is interesting to note that on adjacent sheets, Harriot moved, as a mathematician naturally would, into the theory of the sums of the squares, and attempted to determine graphically all the possible configurations that discrete particles could assume -- a study which led him inevitably to the corpuscular or atomic theory of matter originally deriving from Lucretius and Epicurus.

quoted from Thomas Harriot: A Biography by John W. Shirley, Oxford, 1983, page 242.

Robert Kargon on Harriot's correspondence with Kepler

Hariot's theory of matter appears to have been virtually that of Democritus, Hero of Alexandria, and, in a large measure, that of Epicurus and Lucretius. According to Hariot the universe is composed of atoms with void space interposed. The atoms themselves are eternal and continuous. Physical properties result from the magnitude, shape, and motion of these atoms, or corpuscles compounded from them...

Probably the most interesting application of Hariot's atomic theory was in the field of optics. In a letter to Kepler on 2 December 1606 Hariot outlined his views. Why, he asked, when a light ray falls upon the surface of a transparent medium, is it partially reflected and partially refracted? Since by the principle of uniformity, a single point cannot both reflect and transmit light, the answer must lie in the supposition that the ray is resisted by some points and not others.

"A dense diaphanous body, therefore, which to the sense appears to be continuous in all parts, is not actually continuous. But it has corporeal parts which resist the rays, and incorporeal parts vacua which the rays penetrate..."

It was here that Hariot advised Kepler to abstract himself mathematically into an atom in order to enter `Nature's house'. In his reply of 2 August 1607, Kepler declined to follow Harriot, ad atomos et vacua. Kepler preferred to think of the reflection-refraction problem in terms of the union of two opposing qualities--transparence and opacity. Hariot was surprised. `If those assumptions and reasons satisfy you, I am amazed."

The preceding passage is quoted from R. Kargon, Atomism in England from Hariot to Newton, Oxford, 1966, page 18.

Robert Halleux's commentary on constructions in Kepler's Strena

A première vue, ces constructions évoquent l'atomisme et ne sont pas sans analogies, mutatis mutandis, avec les représentations modernes de la structure cristilline. On pense ici aux premières ébauches de la philosophie mécaniste et corpusculaire qui se font jour dans la première moitié du XVIIe siècle. Cecil Schneer pense même que Kepler tente ici une conciliation entre atomisme et pythagorisme.

Colin Hardie remarque à ce propos que le mathématicien anglais Thomas Harriot (1560-1621) étudia dès 1600 le close-packing d'atomes sphériques. On connait peu de chose de Harriot, dont l'oeuvre est en grande partie manuscrite. On sait seulement qu'il fut un des tenants de l'atomisme. Harriot fut en correspondance avec Kepler à propos de questions d'optique. Kepler expliquait l'arc-en-ciel en prenant comme plus petit élément composant une goutte d'eau parfaitement ronde. Dans une lettre de 2 décembre 1606, Harriot invite Kepler à recourir aux atomes pour expliquer la réfraction de la lumière. Dans sa réponse, datée du 2 août 1607, Kepler refuse de le suivre jusqu'aux atomes et au vide. La discussion se poursuit en 1608 et en 1609.

On le voit, il ne s'agit pas dans la Strena ni d'atomes, ni de corpuscules. L'unité élémentaire est la plus petite partie visible, la goutte d'eau. C'est la représentation que Kepler adopte en optique, par exemple dans les Paralipomènes à Vitellion.

Si Kepler est tributaire d'une influence quelconque dans cet atomisme sans atomes, c'est du côté des mathématiques qu'il faut la chercher. Un rapprochement s'impose avec l'arithmo-géométrie des anciens Pythagoriciens, qui est un véritable atomisme mathématique, puisqu'elle représente les nombres par des points groupés de manière à former des figures....

quoted from p24, Introduction by R. Halleux to l'Etrenne, C.N.R.S. 1975.

A passage from Kepler's Strena, written 1609

For in general equal pellets when collected in any vessel, come to a mutual arrangement in two modes according to the two modes of arranging them in a plane.

If equal pellets are loose in the same horizontal plane and you drive them together so tightly that they touch each other, they come together either in a three-cornered or in a four-cornered pattern. In the former case six surround one; in the latter four. Throughout there is the same pattern of contact between all the pellets except the outermost. With a five-sided pattern uniformity cannot be maintained. A six-sided pattern breaks up into three-sided. Thus there are only the two patterns as described.

Now if you proceed to pack the solid bodies as tightly as possible, and set the files that are first arranged on the level on top of others, layer on layer, the pellets will be either squared (A in diagram), or in triangles (B in diagram). If squared, either each single pellet of the upper range will rest on a single pellet of the lower, or, on the other hand, each single pellet of the upper range will settle between every four of the lower. In the former mode any pellet is touched by four neighbours in the same plane, and by one above and one below, and so on throughout, each touched by six others. The arrangement will be cubic, and the pellets, when subjected to pressure, will become cubes. But this will not be the tightest pack. In the second mode not only is every pellet touched by it four neighbours in the same plane, but also by four in the plane above and by four below, and so throughout one will be touched by twelve, and under pressure spherical pellets will become rhomboid. This arrangement will be more comparable to the octahedron and pyramid. This arrangement will be the tightest possible, so that in no other arrangement could more pellets be stuffed into the same container.

En général, des globules égaux, rassemblées dans un récipient quelconque, s'ordonnent de deux manières, selon les deux modes de leur disposition dans un plan quelconque.

En effet, si vous r'eunissez à l'étroit des globules égaux errant sur un même plan horizontal en sorte qu'ils se touchent réciproquement, ils s'unissent ou bien en forme triangulaire ou bien en forme carrée : dans le premier cas, six globules en entourent un, dan l'autre cas, il y en a quatre. La proportion du contact est la même pour tous les globules sauf les derniers. L'égalité ne peut être gardée par la forme pentagonale, l'hexagone se résout en triangles : de telle sorte que les duex ordres précités sont les seuls.

Si on en arrive à la construction solide la plus serrée qui puisse se faire, et qu'on superpose les uns aux autres les rangs d'abord ajustés en plan, ils seront ou bien en carré A ou en triangle B. S'ils sont en carré, ou bien un globe de l'ordre supérieur sera au-dessus d'un globe de l'ordre inférieur, ou, au contraire, chaque globe du rang supériur sera touché par les quatre qui léntourent dans le même plan, par un au-dessus de lui et par un en dessous, et ainsi au total par six autres. L'ordre sera cubique, et après compression on obtiendra des cubes, mais ce ne sera pas la disposition la plus serrée. Dans la seconde manière, chaque globe est en contact, non seulement avec les quatre globes qui l'entourent dans le même plan, mais aussi avec quatre en dessous de lui, avec quatre au-dessus de lui et ainsi au total avec douze : par compression, ces globes donneront des rhombiques. On rapprochera plutôt cet ordre-ci de l'octaèdre et de la pyramide. L'assemblage sera très serré, de sorte qu'ensuite aucunce disposition ne permettra d'entasser un plus grand nombre de globules dans le même récipient.

D'autre part, si les dispositions constuites en plan sont triangulaires, alors, dans la construction solide, ou bien chaque globe de la couche supérieure se superpose à un globe de la couche inférieure dans une adaptation de nouveau lâche, ou bien chaque globe de la couche supérieure se trouve entre trois de l'ordre inférieur. De la première manière, chaque globe est touché par six qui l'entourent dans le même plan, par un au-dessus et par un en dessous, c'est-à-dire au total par huit autres. Cet order sera assimilé au prisme, et par compression, on obtiendra à la place des globules, des colonnes de six côtés quadrilatères avec deux bases hexagonales. De la deuxième manière, on obtiendra la même chose que plus haut dans la deuxième possibilité de l'ordre carré.

Mathematical Problems

Lecture delivered before the International Congress of Mathematicians at Paris in 1900

18. Building up of Space from Congruent Polyhedra.

If we enquire for those groups of motions in the plane for which a fundamental region exists, we obtain various answers, according as the plane considered is Riemann's (elliptic), Euclid's, or Lobachevsky's (hyperbolic) ....

Now, while the results and methods of proof applicable to elliptic and hyperbolic space hold directly for n-dimensional space also, the generalization of the theorem for euclidean space seems to offer decided difficulties. The investigation of the following question is therefore desirable: Is there in n-dimensional euclidean space also only a finite number of essentially different kinds of groups of motions with a fundamental region?

A fundamental region of each group of motions, together with the congruent regions arising from the group, evidently fills up space completely. The question arises : Whether polyhedra also exist which do not appear as fundamental regions of groups of motions, by means of which nevertheless by a suitable juxtaposition of congruent copies a complete filling up of all space is possible. I point out the following question, related to the preceding one, and important to number theory and perhaps sometimes useful to physics and chemistry: How can one arrange most densely in space an infinite number of equal solids of given form, e.g., spheres with given radii or regular tetrahedra with given edges (or in prescribed position), that is, how can one so fit them together that the ratio of the filled to the unfilled sapce may be as great as possible?

Kantor's commentary on Hilbert's 18th problem.

Problem 18: "To rebuild space with congruent polyhedra." Hilbert's problem is divided into three distinct parts.

A: "In Euclidean space of n dimensions, show that there are only finitely many different kinds of groups of displacements with a (compact) fundamental domain." In other words, one looks for discrete subgroups with compact quotient of the group E(n) of isometries of Rn. The result was proved by Bieberbach in 1910. The classification of these groups is important in crystallography and it generalizes to questions about lattices in Lie groups.

B: "Tiling of space by a single polyhedron which is not a fundamental domain as in A."

J. Milnor's remarks in Gen. Ref. still apply: This is a very lively topic today. Important developments have been the theory of Penrose tilings and the closely related physical problem of anomalous crystal structure..., as well as Thurston's theory of self-similar fractal tilings.

C: "Packing of spheres. How should spheres of the same radius be arranged in space (of any dimensionality) so as to achieve the greatest density of packing?"

Hilbert's text gives the impression that he did not anticipate the success and the developments this problem would have. The hexagonal packing in the plane is the densest (proof by Thue in 1892, completed by Fejes in 1940). In space, the problem is still not solved. There is very recent progress by Hales. For spheres whose centers lie on a lattice, the problem is solved in up to eight dimensions. The subject has various ramifications: application to the geometry of numbers, deep relations between coding theory and sphere-packing theory, the very rich geometry of the densest known lattices.

L. Fejes Tóth

Thus it seems that the problem can be reduced to the determination of the minimum of a function of a finite number of variables, providing a programme realizable in principle. In view of the intricacy of this function we are far from attempting to determine the exact minimum. But, mindful of the rapid development of our computers, it is imaginable that the minimum may be approximated with great exactitude.