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| author | nfenwick <nfenwick@pglaf.org> | 2025-02-07 10:50:05 -0800 |
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| committer | nfenwick <nfenwick@pglaf.org> | 2025-02-07 10:50:05 -0800 |
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diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..d7b82bc --- /dev/null +++ b/.gitattributes @@ -0,0 +1,4 @@ +*.txt text eol=lf +*.htm text eol=lf +*.html text eol=lf +*.md text eol=lf diff --git a/LICENSE.txt b/LICENSE.txt new file mode 100644 index 0000000..6312041 --- /dev/null +++ b/LICENSE.txt @@ -0,0 +1,11 @@ +This eBook, including all associated images, markup, improvements, +metadata, and any other content or labor, has been confirmed to be +in the PUBLIC DOMAIN IN THE UNITED STATES. + +Procedures for determining public domain status are described in +the "Copyright How-To" at https://www.gutenberg.org. + +No investigation has been made concerning possible copyrights in +jurisdictions other than the United States. Anyone seeking to utilize +this eBook outside of the United States should confirm copyright +status under the laws that apply to them. diff --git a/README.md b/README.md new file mode 100644 index 0000000..3863391 --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #54976 (https://www.gutenberg.org/ebooks/54976) diff --git a/old/54976-0.txt b/old/54976-0.txt deleted file mode 100644 index 0faa188..0000000 --- a/old/54976-0.txt +++ /dev/null @@ -1,1906 +0,0 @@ -Project Gutenberg's The Molecular Tactics of a Crystal, by Lord Kelvin - -This eBook is for the use of anyone anywhere in the United States and most -other parts of the world at no cost and with almost no restrictions -whatsoever. You may copy it, give it away or re-use it under the terms of -the Project Gutenberg License included with this eBook or online at -www.gutenberg.org. If you are not located in the United States, you'll have -to check the laws of the country where you are located before using this ebook. - -Title: The Molecular Tactics of a Crystal - -Author: Lord Kelvin - -Release Date: June 24, 2017 [EBook #54976] - -Language: English - -Character set encoding: UTF-8 - -*** START OF THIS PROJECT GUTENBERG EBOOK THE MOLECULAR TACTICS OF A CRYSTAL *** - - - - -Produced by Turgut Dincer and The Online Distributed -Proofreading Team at http://www.pgdp.net (This file was -produced from images generously made available by The -Internet Archive/American Libraries.) - - - - - - - - THE MOLECULAR TACTICS OF - A CRYSTAL - - _LORD KELVIN_ - - London - HENRY FROWDE - OXFORD UNIVERSITY PRESS WAREHOUSE - AMEN CORNER, E.C. - -[Illustration] - - New York - MACMILLAN & CO., 66 FIFTH AVENUE - - - - - THE - MOLECULAR TACTICS OF - A CRYSTAL - - BY - - LORD KELVIN, P.R.S. - - PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF GLASGOW - AND FELLOW OF PETERHOUSE, CAMBRIDGE - - - _Being the Second ROBERT BOYLE LECTURE, delivered before - the Oxford University Junior Scientific Club - on Tuesday, May 16, 1893_ - - - WITH TWENTY ILLUSTRATIONS - - - Oxford - AT THE CLARENDON PRESS - 1894 - - Oxford - PRINTED AT THE CLARENDON PRESS - BY HORACE HART, PRINTER TO THE UNIVERSITY - - - - -ON THE MOLECULAR TACTICS OF A CRYSTAL - -By LORD KELVIN, P.R.S. - - -§ 1. My subject this evening is not the physical properties of -crystals, not even their dynamics; it is merely the geometry of the -structure--the arrangement of the molecules in the constitution of a -crystal. Every crystal is a homogeneous assemblage of small bodies -or molecules. The converse proposition is scarcely true, unless in -a very extended sense of the term crystal (§ 20 below). I can best -explain a homogeneous assemblage of molecules by asking you to think -of a homogeneous assemblage of people. To be homogeneous every person -of the assemblage must be equal and similar to every other: they must -be seated in rows or standing in rows in a perfectly similar manner. -Each person, except those on the borders of the assemblage, must have -a neighbour on one side and an equi-distant neighbour on the other: a -neighbour on the left front and an equi-distant neighbour behind on the -right, a neighbour on the right front and an equi-distant neighbour -behind on the left. His two neighbours in front and his two neighbours -behind are members of two rows equal and similar to the rows consisting -of himself and his right-hand and left-hand neighbours, and their -neighbours’ neighbours indefinitely to right and left. In particular -cases the nearest of the front and rear neighbours may be right in -front and right in rear; but we must not confine our attention to the -rectangularly grouped assemblages thus constituted. Now let there be -equal and similar assemblages on floors above and below that which -we have been considering, and let there be any indefinitely great -number of floors at equal distances from one another above and below. -Think of any one person on any intermediate floor and of his nearest -neighbours on the floors above and below. These three persons must be -exactly in one line; this, in virtue of the homogeneousness of the -assemblages on the three floors, will secure that every person on the -intermediate floor is exactly in line with his nearest neighbours above -and below. The same condition of alignment must be fulfilled by every -three consecutive floors, and we thus have a homogeneous assemblage of -people in three dimensions of space. In particular cases every person’s -nearest neighbour in the floor above may be vertically over him, but -we must not confine our attention to assemblages thus rectangularly -grouped in vertical lines. - -§ 2. Consider now any particular person _C_ (Fig. 1) on any -intermediate floor, _D_ and _D′_ his nearest neighbours, _E_ and _E′_ -his next nearest neighbours all on his own floor. His next next nearest -neighbours on that floor will be in the positions _F_ and _F′_ in the -diagram. Thus we see that each person _C_ is surrounded by six persons, -_DD′_, _EE′_ and _FF′_, being his nearest, his next nearest, and his -next next nearest neighbours on his own floor. Excluding for simplicity -the special cases of rectangular grouping, we see that the angles of -the six equal and similar triangles _CDE_, _CEF_, &c., are all acute: -and because the six triangles are equal and similar we see that the -three pairs of mutually remote sides of the hexagon _DEFD′E′F′_ are -equal and parallel. - -[Illustration: FIG. 1] - -§ 3. Let now _A_, _A′_, _A″_, &c., denote places of persons of the -homogeneous assemblage on the floor immediately above, and _B_, _B′_, -_B″_, &c. on the floor immediately below, the floor of _C_. In the -diagram let _a_, _a′_, _a″_ be points in which the floor of _CDE_ is -cut by perpendiculars to it through _A_, _A′_, _A″_ of the floor above, -and _b_, _b′_, _b″_ by perpendiculars from _B_, _B′_, _B″_ of the floor -below. Of all the perpendiculars from the floors immediately above -and below, just two, one from each, cut the area of the parallelogram -_CDEF_: and they cut it in points similarly situated in respect to -the oppositely oriented triangles into which it is divided by either -of its diagonals. Hence if _a_ lies in the triangle _CDE_, the other -five triangles of the hexagon must be cut in the corresponding points, -as shown in the diagram. Thus, if we think only of the floor of _C_ -and of the floor immediately above it, we have points _A_, _A′_, _A″_ -vertically above _a_, _a′_, _a″_. Imagine now a triangular pyramid, -or tetrahedron, standing on the base _CDE_ and having _A_ for vertex: -we see that each of its sides _ACD_, _ADE_, _AEC_, is an acute angled -triangle, because, as we have already seen, _CDE_ is an acute angled -triangle, and because the shortest of the three distances, _CA_, _DA_, -_EA_, is (§ 2) greater than _CE_ (though it may be either greater than -or less than _DE_). Hence the tetrahedron _CDEA_ has all its angles -acute; not only the angles of its triangular faces, but the six angles -between the planes of its four faces. This important theorem regarding -homogeneous assemblages was given by Bravais, to whom we owe the whole -doctrine of homogeneous assemblages in its most perfect simplicity and -complete generality. Similarly we see that we have equal and similar -tetrahedrons on the bases _D′CF_, _E′F′C_; and three other tetrahedrons -below the floor of _C_, having the oppositely oriented triangles -_CD′E′_, &c. for their bases and _B_, _B′_, _B″_ for their vertices. -These three tetrahedrons are equal and heterochirally[1] similar to the -first three. The consideration of these acute angled tetrahedrons, is -of fundamental importance in respect to the engineering of an elastic -solid, or crystal, according to Boscovich. So also is the consideration -of the cluster of thirteen points _C_ and the six neighbours -_DEFD′E′F′_ in the plane of the diagram, and the three neighbours -_AA′A″_ on the floor above, and _BB′B″_ on the floor below. - -§ 4. The case in which each of the four faces of each of the -tetrahedrons of § 3 is an equilateral triangle is particularly -interesting. An assemblage fulfilling this condition may conveniently -be called an ‘equilateral homogeneous assemblage,’ or, for brevity, an -‘equilateral assemblage.’ In an equilateral assemblage _C_’s twelve -neighbours are all equi-distant from it. I hold in my hand a cluster of -thirteen little black balls, made up by taking one of them and placing -the twelve others in contact with it (and therefore packed in the -closest possible order), and fixing them all together by fish-glue. -You see it looks, in size, colour, and shape, quite like a mulberry. -The accompanying diagram shows a stereoscopic view of a similar cluster -of balls painted white for the photograph. - -[Illustration: FIG. 2.] - -§ 5. By adding ball after ball to such a cluster of thirteen, and -always taking care to place each additional ball in some position in -which it is properly in line with others, so as to make the whole -assemblage homogeneous, we can exercise ourselves in a very interesting -manner in the building up of any possible form of crystal of the class -called ‘cubic’ by some writers and ‘octahedral’ by others. You see -before you several examples. I advise any of you who wish to study -crystallography to contract with a wood-turner, or a maker of beads for -furniture tassels or for rosaries, for a thousand wooden balls of about -half an inch diameter each. Holes through them will do no harm and may -even be useful; but make sure that the balls are as nearly equal to one -another, and each as nearly spherical, as possible. - -[Illustration: FIG. 3.] - - -§ 6. You see here before you a large model which I have made to -illustrate a homogeneous assemblage of points, on a plan first given, -I believe, by Mr. William Barlow (_Nature_, December 20 and 27, 1883). -The roof of the model is a lattice-frame (Fig. 3) consisting of two -sets of eight parallel wooden bars crossing one another, and kept -together by pins through the middles of the crossings. As you see, I -can alter it to make parallelograms of all degrees of obliquity till -the bars touch, and again you see I can make them all squares. - - -§ 7. The joint pivots are (for cheapness of construction) of copper -wire, each bent to make a hook below the lattice frame. On these -sixty-four hooks are hung sixty-four fine cords, firmly stretched by -little lead weights. Each of these cords (Fig. 4) bears eight short -perforated wooden cylinders, which may be slipped up and down to any -desired position[2]. They are at present actually placed at distances -consecutively each equal to the distance from joint to joint of the -lattice frame. - -[Illustration: FIG. 4.] - - -§ 8. The roof of the model is hung by four cords, nearly vertical, of -independently variable lengths, passing over hooks from fixed points -above, and kept stretched by weights, each equal to one quarter of -the weight of roof and pendants. You see now by altering the angles -of the lattice work and placing it horizontal or in any inclined -plane, as I am allowed to do readily by the manner in which it is -hung, I have three independent variables, by varying which I can show -you all varieties of homogeneous assemblages, in which three of the -neighbours of every point are at equal distances from it. You see -here, for example, we have the equilateral assemblage. I have adjusted -the lattice roof to the proper angle, and its plane to the proper -inclination to the vertical, to make a wholly equilateral assemblage -of the little cylinders of wood on the vertical cords, a case, as -we have seen, of special importance. If I vary also the distances -between the little pieces of wood on the cords; and the distances -between the joints of the lattice work (variations easily understood, -though not conveniently producible in one model without more of -mechanical construction than would be worth making), I have three -other independent variables. By properly varying these six independent -variables, three angles and three lengths, we may give any assigned -value to each edge of one of the fundamental tetrahedrons of § 3. - - -§ 9. Our assemblage of people would not be homogeneous unless its -members were all equal and similar and in precisely similar attitudes, -and were all looking the same way. You understand what a number of -people seated or standing on a floor or plain and looking the same way -means. But the expression ‘looking’ is not conveniently applicable to -things that have no eyes, and we want a more comprehensive mode of -expression. We have it in the words ‘orientation,’ ‘oriented,’ and -(verb) ‘to orient,’ suggested by an extension of the idea involved in -the word ‘orientation,’ first used to signify positions relatively -to east and west of ancient Greek and Egyptian temples and Christian -churches. But for the orientation of a house or temple we have only -one angle, and that angle is called ‘azimuth’ (the name given to an -angle in a horizontal plane). For orientation in three dimensions of -space we must extend our ideas and consider position with reference to -east and west and up and down. A man lying on his side with his head -to the north and looking east, would not be similarly oriented to a -man standing upright and looking east. To provide for the complete -specification of how a body is oriented in space we must have in the -body a plane of reference, and a line of reference in this plane, -belonging to the body and moving with it. We must also have a fixed -plane and a fixed line of reference in it, relatively to which the -orientation of the moveable body is to be specified; as, for example, -a horizontal plane and the east and west horizontal line in it. The -position of a body is completely specified when the angle between the -plane of reference belonging to it, and the fixed plane is given; and -when the angles between the line of intersection of the two planes and -the lines of reference in them are also given. Thus we see that three -angles are necessary and sufficient to specify the orientation of a -moveable body, and we see how the specification is conveniently given -in terms of three angles. - - -§ 10. To illustrate this take a book lying on the table before you with -its side next the title-page up, and its back to the north. I now lift -the east edge (the top of the book), keeping the bottom edge north and -south on the table till the book is inclined, let us say, 20° to the -table. Next, without altering this angle of 20°, between the side of -the book and the table, I turn the book round a vertical axis, through -45° till the bottom edge lies north-east and south-west. Lastly, -keeping the book in the plane to which it has been thus brought, I turn -it round in this plane through 35°. These three angles of 20°, 45°, -and 35°, specify, with reference to the horizontal plane of the table -and the east and west line in it, the orientation of the book in the -position to which you have seen me bring it, and in which I hold it -before you. - - -§ 11. In Figs. 5 and 6 you see two assemblages, each of twelve equal -and similar molecules in a plane. Fig. 5, in which the molecules are -all same-ways oriented, is one homogeneous assemblage of twenty-four -molecules. Fig. 6, in which in one set of rows the molecules are -alternately oriented two different ways, may either be regarded as -two homogeneous assemblages, each of twelve single molecules; or one -homogeneous assemblage of twelve pairs of those single molecules. - -[Illustration: FIG. 5.] - - -§ 12. I must now call your attention to a purely geometrical -question[3] of vital interest with respect to homogeneous assemblages -in general, and particularly the homogeneous assemblage of molecules -constituting a crystal:--_what can we take as ‘the’ boundary or ‘a’ -boundary enclosing each molecule with whatever portion of space around -it we are at liberty to choose for_ _it, and separating it from -neighbours and their portions of space given to them in homogeneous -fairness?_ - -[Illustration: FIG. 6.] - - -§ 13. If we had only mathematical points to consider we should be at -liberty to choose the simple obvious partitioning by three sets of -parallel planes. Even this may be done in an infinite number of ways, -thus:--Beginning with any point _P_ of the assemblage, choose any other -three points _A_, _B_, _C_, far or near, provided only that they are -not in one plane with _P_, and that there is no other point of the -assemblage in the lines _PA_, _PB_, _PC_, or within the volume of the -parallelepiped of which these lines are conterminous edges, or within -the areas of any of the faces of this parallelepiped. There will be -points of the assemblage at each of the corners of this parallelepiped -and at all the corners of the parallelepipeds equal and similar to -it which we find by drawing sets of equi-distant planes parallel to -its three pairs of faces. (A diagram is unnecessary.) Every point of -the assemblage is thus at the intersection of three planes, which is -also the point of meeting of eight neighbouring parallelepipeds. Shift -now any one of the points of the assemblage to a position within the -volume of any one of the eight parallelepipeds, and give equal parallel -motions to all the other points of the assemblage. Thus we have every -point in a parallelepipedal cell of its own, and all the points of the -assemblage are similarly placed in their cells, which are themselves -equal and similar. - - -§ 14. But now if, instead of a single point for each member of the -assemblage, we have a group of points, or a globe or cube or other -geometrical figure, or an individual of a homogeneous assemblage of -equal, similar, similarly dressed, and similarly oriented ladies, -sitting in rows, or a homogeneous assemblage of trees closely planted -in regular geometrical order on a plane with equal and similar -distributions of molecules, and parallel planes above and below, -we may find that the best conditioned plane-faced parallelepipedal -partitioning which we can choose would cut off portions properly -belonging to one molecule of the assemblage and give them to the cells -of neighbours. To find a cell enclosing all that belongs to each -individual, for example, every part of each lady’s dress, however -complexly it may be folded among portions of the equal and similar -dresses of neighbours; or, every twig, leaf, and rootlet of each one -of the homogeneous assemblage of trees; we must alter the boundary by -give-and-take across the plane faces of the primitive parallelepipedal -cells, so that each cell shall enclose all that belongs to one -molecule, and therefore (because of the homogeneousness of the -partitioning) nothing belonging to any other molecule. The geometrical -problem thus presented, wonderfully complex as it may be in cases -such as some of those which I have suggested, is easily performed for -any possible case if we begin with any particular parallelepipedal -partitioning determined for corresponding points of the assemblage -as explained in § 13, for any homogeneous assemblage of single -points. We may prescribe to ourselves that the corners are to remain -unchanged, but if so they must to begin with either in interfaces of -contact between the individual molecules, or in vacant space among -the molecules. If this condition is fulfilled for one corner it is -fulfilled for all, as the corners are essentially corresponding points -relatively to the assemblage. - - -§ 15. Begin now with any one of the twelve straight lines between -corners which constitute the twelve edges of the parallelepiped, and -alter it arbitrarily to any curved or crooked line between the same -pair of corners, subject only to the conditions (1) that it does not -penetrate the substance of any member of the assemblage, and (2) that -it is not cut by equal and similar parallel curves[4] between other -pairs of corners. - -[Illustration: FIG. 7.] - -Considering now the three fours of parallel edges of the -parallelepiped, let the straight lines of one set of four be altered to -equal and similar parallel curves in the manner which I have described; -and proceed by the same rule for the other two sets of four edges. We -thus have three fours of parallel curved edges instead of the three -fours of parallel straight edges of our primitive parallelepiped with -corners (each a point of intersection of three edges) unchanged. -Take now the quadrilateral of four curves substituted for the four -straight edges of one face of the parallelepiped. We may call this -quadrilateral a curvilineal parallelogram, because it is a circuit -composed of two pairs of equal parallel curves. Draw now a curved -surface (an infinitely thin sheet of perfectly extensible india-rubber -if you please to think of it so) bordered by the four edges of our -curvilineal parallelogram, and so shaped as not to cut any of the -substance of any molecule of the assemblage. Do the same thing with -an exactly similar and parallel sheet relatively to the opposite face -of the parallelepiped; and again the same for each of the two other -pairs of parallel faces. We thus have a curved-faced parallelepiped -enclosing the whole of one molecule and no part of any other; and by -similar procedure we find a similar boundary for every other molecule -of the assemblage. Each wall of each of these cells is common to two -neighbouring molecules, and there is no vacant space anywhere between -them or at corners. Fig. 7 illustrates this kind of partitioning by -showing a plane section parallel to one pair of plane faces of the -primitive parallelepiped, for an ideal case. The plane diagram is in -fact a realization of the two-dimensional problem of partitioning the -pine pattern of a Persian carpet by parallelograms about as nearly -rectilinear as we can make them. In the diagram faint straight lines -are drawn to show the primitive parallelogrammatic partitioning. -It will be seen that of all the crossings (marked with dots in the -diagram) every one is similarly situated to every other in respect to -the homogeneously repeated pattern figures: _A_, _B_, _C_, _D_ are four -of them at the corners of one cell. - - -§ 16. Confining our attention for a short time to the homogeneous -division of a plane, remark that the division into parallelograms -by two sets of crossing parallels is singular in this respect--each -cell is contiguous with three neighbours at every corner. Any -shifting, large or small, of the parallelograms by relative sliding -in one direction or another violates this condition, brings us to -a configuration like that of the faces of regularly hewn stones in -ordinary bonded masonry, and gives a partitioning which fulfils the -condition that at each corner each cell has only two neighbours. Each -cell is now virtually a hexagon, as will be seen by the letters _A_, -_B_, _C_, _D_, _E_, _F_ in the diagram Fig. 8. _A_ and _D_ are to be -reckoned as corners, each with an interior angle of 180°. In this -diagram the continuous heavy lines and the continuous faint lines -crossing them show a primitive parallelogrammatic partition by two sets -of continuous parallel intersecting lines. The interrupted crossing -lines (heavy) show, for the same homogeneous distribution of single -points or molecules, the virtually hexagonal partitioning which we get -by shifting the boundary from each portion of one of the light lines to -the heavy line next it between the same continuous parallels. - -[Illustration: FIG. 8.] - -Fig. 8 bis represents a further modification of the boundary by -which the 180° angles _A_, _D_, become angles of less than 180°. The -continuous parallel lines (light) and the short light portions of the -crossing lines show the configuration according to Fig. 8, from which -this diagram is derived. - - -§ 17. In these diagrams (Figs. 8 and 8 bis) the object enclosed -is small enough to be enclosable by a primitive parallelogrammatic -partitioning of two sets of continuous crossing parallel straight -lines, and by the partitioning of ‘bonded’ parallelograms both -represented in Fig. 8, and by the derived hexagonal partitioning -represented in Fig. 8 bis, with faint lines showing the primitive and -the secondary parallelograms. In Fig. 7 the objects enclosed were -too large to be enclosable by any rectilinear parallelogrammatic -or hexagonal partitioning. The two sets of parallel faint lines in -Fig. 7 show a primitive parallelogrammatic partitioning and the -corresponding pairs of parallel curves intersecting at the corners of -these parallelograms, of which _A_,_B_,_C_,_D_ is a specimen, show a -corresponding partitioning by curvilineal parallelograms. Fig. 9 shows -for the same homogeneous distribution of objects a better conditioned -partitioning, by hexagons in each of which one pair of parallel -edges is curved. The sets of intersecting parallel straight lines in -Fig. 9 show the same primitive parallelogrammatic partitioning as -in Fig. 7, and the same slightly shifted to suit points chosen for -well-conditionedness of hexagonal partitioning. - -[Illustration: FIG. 8 bis.] - -[Illustration: FIG. 9.] - - -§ 18. For the division of continuous three-dimensional space[5] into -equal, similar, and similarly oriented cells, quite a corresponding -transformation from partitioning by three sets of continuous mutually -intersecting parallel planes to any possible mode of homogeneous -partitioning, may be investigated by working out the three-dimensional -analogue of §§ 16-17. Thus we find that the most general possible -homogeneous partitioning of space with plane interfaces between the -cells gives us fourteen walls to each cell, of which six are three -pairs of equal and parallel parallelograms, and the other eight are -four pairs of equal and parallel hexagons, each hexagon being bounded -by three pairs of equal and parallel straight lines. This figure, being -bounded by fourteen plane faces, is called a tetrakaidekahedron. It -has thirty-six edges of intersection between faces; and twenty-four -corners, in each of which three faces intersect. A particular case -of it, which I call an orthic tetrakaidekahedron, being that in -which the six parallelograms are equal squares, the eight hexagonal -faces are equal equilateral and equiangular hexagons, and the lines -joining corresponding points in the seven pairs of parallel faces -are perpendicular to the planes of the faces, is represented by -a stereoscopic picture in Fig. 10. The thirty-six edges and the -twenty-four corners, which are easily counted in this diagram, occur -in the same relative order in the most general possible partitioning, -whether by plane-faced tetrakaidekahedrons or by the generalized -tetrakaidekahedron described in § 19. - - -§ 19. The most general homogeneous division of space is not -limited to plane-faced cells; but it still consists essentially of -tetrakaidekahedronal cells, each bounded by three pairs of equal and -parallel quadrilateral faces, and four pairs of equal and parallel -hexagonal faces, neither the quadrilaterals nor the hexagons being -necessarily plane. Each of the thirty-six edges may be straight -or crooked or curved; the pairs of opposite edges, whether of the -quadrilaterals or hexagons, need not be equal and parallel; neither -the four corners of each quadrilateral nor the six corners of each -hexagon need be in one plane. But every pair of corresponding edges of -every pair of parallel corresponding faces, whether quadrilateral or -hexagonal, must be equal and parallel. I have described an interesting -case of partitioning by tetrakaidekahedrons of curved faces with curved -edges in a paper[6] published about seven years ago. In this case each -of the quadrilateral faces is plane. Each hexagonal face is a slightly -curved surface having three rectilineal diagonals through its centre in -one plane. - -[Illustration: FIG. 10.] - -The six sectors of the face between these diagonals lie alternately -on opposite sides of their plane, and are bordered by six arcs -of plane curves lying on three pairs of parallel planes. This -tetrakaidekahedronal partitioning fulfils the condition that the -angles between three planes meeting in an edge are everywhere each -120°; a condition that cannot be fulfilled in any plane-faced -tetrakaidekahedron. Each hexagonal wall is an anticlastic surface of -equal opposite curvatures at every point, being the surfaces of minimum -area bordered by six curved edges. It is shown easily and beautifully, -and with a fair approach to accuracy, by choosing six little circular -arcs of wire, and soldering them together by their ends in proper -planes for the six edges of the hexagon; and dipping it in soap -solution and taking it out. - - -§ 20. Returning now to the tactics of a homogeneous assemblage, remark -that the qualities of the assemblage as a whole depend both upon the -character and orientation of each molecule, and on the character of -the homogeneous assemblage formed by corresponding points of the -molecules. After learning the simple mathematics of crystallography, -with its indicial system[7] for defining the faces and edges of a -crystal according to the Bravais rows and nets and tetrahedrons of -molecules in which we think only of a homogeneous assemblage of points, -we are apt to forget that the true crystalline molecule, whatever its -nature may be, has sides, and that generally two opposite sides of each -molecule may be expected to be very different in quality, and we are -almost surprised when mineralogists tell us that two parallel faces on -two sides of a crystal have very different qualities in many natural -crystals. We might almost as well be surprised to find that an army in -battle array, which is a kind of large-grained crystal, presents very -different appearance to any one looking at it from outside, according -as every man in the ranks with his rifle and bayonet faces to the front -or to the rear or to one flank or to the other. - - -§ 21. Consider, for example, the ideal case of a crystal consisting -of hard equal and similar tetrahedronal solids all same-ways oriented. -A thin plate of crystal cut parallel to any one set of the faces of -the constituent tetrahedrons would have very different properties on -its two sides; as the constituent molecules would all present points -outwards on one side and flat surfaces on the other. We might expect -that the two sides of such a plate of crystal would become oppositely -electrified when rubbed by one and the same rubber; and, remembering -that a piece of glass with part of its surface finely ground but not -polished and other parts polished becomes, when rubbed with white -silk, positively electrified over the polished parts and negatively -electrified over the non-polished parts, we might almost expect that -the side of our supposed crystalline plate towards which flat faces -of the constituent molecules are turned would become positively -electrified, and the opposite side, showing free molecular corners, -would become negatively electrified, when both are rubbed by a rubber -of intermediate electric quality. We might also from elementary -knowledge of the fact of piezo-electricity, that is to say, the -development of opposite electricities on the two sides of a crystal -by pressure, expect that our supposed crystalline plate, if pressed -perpendicularly on its two sides, would become positively electrified -on one of them and negatively on the other. - - -§ 22. Intimately connected with the subject of enclosing cells for -molecules of given shape, assembled homogeneously, is the homogeneous -packing together of equal and similar molecules of any given shape. In -every possible case of any infinitely great number of similar bodies -the solution is a homogeneous assemblage. But it may be a homogeneous -assemblage of single solids all oriented the same way, or it may be -a homogeneous assemblage of clusters of two or more of them placed -together in different orientations. For example, let the given bodies -be halves (oblique or not oblique) of any parallelepiped on the two -sides of a dividing plane through a pair of parallel edges. The two -halves are homochirally[8] similar; and, being equal, we may make a -homogeneous assemblage of them by orienting them all the same way -and placing them properly in rows. But the closest packing of this -assemblage would necessarily leave vacant spaces between the bodies: -and we get in reality the closest possible packing of the given bodies -by taking them in pairs oppositely oriented and placed together to form -parallelepipeds. These clusters may be packed together so as to leave -no unoccupied space. - -Whatever the number of pieces in a cluster in the closest possible -packing of solids may be for any particular shape, we may consider each -cluster as itself a given single body, and thus reduce the problem to -the packing closely together of assemblages of individuals all sameways -oriented; and to this problem therefore it is convenient that we should -now confine our attention. - - -§ 23. To avoid complexities such as those which we find in the familiar -problem of homogeneous packing of forks or spoons or tea-cups or bowls, -of any ordinary shape, we shall suppose the given body to be of such -shape that no two of them similarly oriented can touch one another -in more than one point. Wholly convex bodies essentially fulfil this -condition; but it may also be fulfilled by bodies not wholly convex, as -is illustrated in Fig. 11. - -[Illustration: Fig. 11.] - - -§ 24. To find close and closest packing of any number of our solids -_S_{1}_, _S_{2}_, _S_{3}_ ... of shape fulfilling the condition of § 23 -proceed thus:-- - -(1) Bring _S_{2}_ to touch _S_{1}_ at any chosen point _p_ of its -surface (Fig. 12). - -(2) Bring _S_{3}_ to touch _S_{1}_ and _S_{2}_, at _r_ and _q_ -respectively. - -(3) Bring _S_{4}_ (not shown in the diagram) to touch _S_{1}_, _S_{2}_, -and _S_{3}_. - -(4) Place, any number of the bodies together in three rows continuing -the lines of _S_{1}S_{2}_, _S_{1}S_{3}_, _S_{1}S_{4}_, and in three -sets of equi-distant rows parallel to these. This makes a homogeneous -assemblage. In the assemblage so formed the molecules are necessarily -found to be in three sets of rows parallel respectively to the three -pairs _S_{2}S_{3}_, _S_{3}S_{4}_, _S_{4}S_{2}_. The whole space -occupied by an assemblage of _n_ of our solids thus arranged has -clearly _6n_ times the volume of a tetrahedron of corresponding points -of _S_{1}_, _S_{2}_, _S_{3}_, _S_{4}_. Hence the closest of the -close packings obtained by the operations (1) ... (4) is found if we -perform the operations (1), (2), and (3) as to make the volume of this -tetrahedron least possible. - -[Illustration: FIG. 12] - - -§ 25. It is to be remarked that operations (1) and (2) leave for (3) -no liberty of choice for the place of _S_{4}_, except between two -determinate positions on opposite sides of the group _S_{1}_, _S_{2}_, -_S_{3}_. The volume of the tetrahedron will generally be different for -these two positions of _S_{4}_, and, even if the volume chance to be -equal in any case, we have differently shaped assemblages according as -we choose one or other of the two places for _S_{4}_. - -This will be understood by looking at Fig. 12, showing _S_{1}_ -and neighbours on each side of it in the rows of _S_{1}S_{2}_, -_S_{1}S_{3}_, and in a row parallel to that of _S_{2}S_{3}_. The plane -of the diagram is parallel to the planes of corresponding points of -these seven bodies, and the diagram is a projection of these bodies -by lines parallel to the intersections of the tangent planes through -_p_ and _r_. If the three tangent planes through _p_, _q_, and _r_, -intersected in parallel lines, _q_ would be seen like _p_ and _r_ as a -point of contact between the outlines of two of the bodies; but this is -only a particular case, and in general _q_ must, as indicated in the -diagram, be concealed by one or other of the two bodies of which it is -the point of contact. Now imagining, to fix our ideas and facilitate -brevity of expression, that the planes of corresponding points of the -seven bodies are horizontal, we see clearly that _S_{4}_ may be brought -into proper position to touch _S_{1}_, _S_{2}_, and _S_{3}_ either from -above or from below; and that there is one determinate place for it if -we bring it into position from above, and another determinate place for -it if we bring it from below. - - -§ 26. If we look from above at the solids of which Fig. 12 shows the -outline, we see essentially a hollow leading down to a perforation -between _S_{1}_, _S_{2}_, _S_{3}_, and if we look from below we see a -hollow leading upwards to the same perforation: this for brevity we -shall call the perforation _pqr_. The diagram shows around _S_{1}_ -six hollows leading down to perforations, of which two are similar -to _pqr_, and the other three, of which _p′q′r′_ indicates one, are -similar one to another but are dissimilar to _pqr_. If we bring _S_{4}_ -from above into position to touch _S_{1}_, _S_{2}_, and _S_{3}_, its -place thus found is in the hollow _pqr_, and the places of all the -solids in the layer above that of the diagram are necessarily in the -hollows similar to _pqr_. In this case the solids in the layer below -that of the diagram must lie in the hollows below the perforations -dissimilar to _pqr_, in order to make a single homogeneous assemblage. -In the other case, _S_{4}_ brought up from below finds its place on the -under side of the hollow _pqr_, and all solids of the lower layer find -similar places: while solids in the layer above that of the diagram -find their places in the hollows similar to _p´q´r´_. In the first -case there are no bodies of the upper layer in the hollows above the -perforations _similar_ to _p´q´r´_, and no bodies of the lower layer in -the hollows below the perforations _similar_ to _pqr_. In the second -case there are no bodies of the upper layer in the hollows above the -perforations _similar_ to _pqr_, and none of the under layer in the -hollows below the perforations _similar_ to _p´q´r´_. - - -§ 27. Going back now to operation (1) of § 23, remark that when the -point of contact _p_ is arbitrarily chosen on one of the two bodies -_S_{1}_, the point of contact on the other will be the point on it -corresponding to the point or one of the points of _S_{1}_, where its -tangent plane is parallel to the tangent plane at _p_. If _S_{1}_ -is wholly convex it has only two points at which the tangent planes -are parallel to a given plane, and therefore the operation (1) is -determinate and unambiguous. But if there is any concavity there will -be four or some greater even number of tangent planes parallel to any -one of some planes, while there will be other planes to each of which -only one pair of tangent planes is parallel. Hence, operation (1), -though still determinate, will have a multiplicity of solutions, or -only a single solution, according to the choice made of the position of -_p_. - -Henceforth however, to avoid needless complications of ideas, we shall -suppose our solids to be wholly convex; and of some such unsymmetrical -shape as those indicated in Fig. 12 of § 25, and shown by stereoscopic -photograph in Fig. 13 of § 36. With or without this convenient -limitation, operation (1) has two freedoms, as _p_ may be chosen -freely on the surface of _S_{1}_; and operation (2) has clearly just -one freedom after operation (1) has been performed. Thus, for a solid -of any given shape, we have three disposables, or, as commonly called -in mathematics, three ‘independent variables,’ all free for making a -homogeneous assemblage according to the rule of § 22. - - -§ 28. In the homogeneous assemblage defined in § 24, each solid, -_S_{1}_, is touched at twelve points, being the three points of -contact with _S_{2}_, _S_{3}_, _S_{4}_, and the three 3’s of points -on _S_{1}_ corresponding to the points on _S_{2}_, _S_{3}_, _S_{4}_, -at which these bodies are touched by the others of the quartet. This -statement is somewhat difficult to follow, and we see more clearly -the twelve points of contact by not confining our attention to the -quartet _S_{1}_, _S_{2}_, _S_{3}_, _S_{4}_ (convenient as this is -for some purposes), but completing the assemblage and considering -six neighbours around _S_{1}_ in one plane layer of the solids as -shown in Fig. 12, with their six points _prq″p′r′q″′_ of contact with -_S_{1}_; and the three neighbours of the two adjacent parallel layers -which touch it above and below. This cluster of thirteen, _S_{1}_ -and twelve neighbours, is shown for the case of spherical bodies in -the stereoscopic photograph of § 4 above. We might of course, if we -pleased, have begun with the plane layer of which _S_{1}_, _S_{2}_, -_S_{4}_ are members, or with that of which _S_{1}_, _S_{3}_, _S_{4}_ -are members, or with the plane layer parallel to the fourth side -_S_{2}_ _S_{3}_ _S_{4}_ of the tetrahedron: and thus we have four -different ways of grouping the twelve points of contact on _S_{1}_ into -one set of six and two sets of three. - - -§ 29. In this assemblage we have what I call ‘close order’ or ‘close -packing.’ For closest of close packings the volume of the tetrahedron -(§ 24) of corresponding points of _S_{1}_, _S_{2}_, _S_{3}_, and -_S_{4}_ must be a minimum, and the least of minimums if, as generally -will be the case, there are two more different configurations for -each of which the volume is a minimum. There will in general also be -configurations of minimax volume and of maximum volume, subject to -the condition that each body is touched by twelve similarly oriented -neighbours. - - -§ 30. Pause for a moment to consider the interesting kinematical -and dynamical problems presented by a close homogeneous assemblage -of smooth solid bodies of given convex shape, whether perfectly -frictionless or exerting resistance against mutual sliding according to -the ordinarily stated law of friction between dry hard solid bodies. -First imagine that they are all similarly oriented and each in contact -with twelve neighbours, except outlying individuals (which there must -be at the boundary if the assemblage is finite, and each of which is -touched by some number of neighbours less than twelve). The coherent -assemblage thus defined constitutes a kinematic frame or skeleton -for an elastic solid of very peculiar properties. Instead of the six -freedoms, or disposables, of strain presented by a natural solid it has -only three. Change of shape of the whole can only take place in virtue -of rotation of the constituent parts relatively to any one chosen row -of them, and the plane through it and another chosen row. - - -§ 31. Suppose first the solids to be not only perfectly smooth but -perfectly frictionless. Let the assemblage be subjected to equal -positive or negative pressure inwards all around its boundary. Every -position of minimum, minimax, or maximum volume will be a position of -equilibrium. If the pressure is positive the equilibrium will be stable -if, and unstable unless, the volume is a minimum. If the pressure -is negative the equilibrium will be stable if, and unstable unless, -the volume is a maximum. Configurations of minimax volume will be -essentially unstable. - - -§ 32. Consider now the assemblage of § 31 in a position of stable -equilibrium under the influence of a given constant uniform pressure -inwards all round its boundary. It will have rigidity in simple -proportion to the amount of this pressure. If now by the superposition -of non-uniform pressure at the boundary, for example equal and opposite -pressures on two sides of the assemblage, a finite change of shape is -produced: the whole assemblage essentially swells in bulk. This is the -‘dilatancy’ which Osborne Reynolds has described[9] in an exceedingly -interesting manner with reference to a sack of wheat or sand, or an -india-rubber bag tightly filled with sand or even small shot. Consider, -for example, a sack of wheat filled quite full and standing up open. It -is limp and flexible. Now shake it down well, fill it quite full, shake -again, so as to get as much into it as possible, and tie the mouth very -tightly close. The sack becomes almost as stiff as a log of wood of -the same shape. Open the mouth partially, and it becomes again limp, -especially in the upper parts of the bag. In Reynolds’ observations on -india-rubber bags of small shot his ‘dilatancy’ depends, essentially -and wholly, on breaches of some of the contacts which exist between the -molecules in their configuration of minimum volume: and it is possible -that in all his cases the dilatations which he observed are _chiefly_, -if not wholly, due to such breaches of contact. - -But it is possible, it almost seems probable, that in bags or boxes -of sand or powder, of some kinds of smooth rounded bodies of any -shape, not spherical or ellipsoidal, subjected persistently to unequal -pressures in different directions, and well shaken, stable positions -of equilibrium are found with almost all the particles each touched by -twelve others. - -Here is a curious subject of Natural History through all ages till -1885, when Reynolds brought it into the province of Natural Philosophy -by the following highly interesting statement:--‘A well-marked -phenomenon receives its explanation at once from the existence of -dilatancy in sand. When the falling tide leaves the sand firm, as the -foot falls on it the sand whitens and appears momentarily to dry round -the foot. When this happens the sand is full of water, the surface of -which is kept up to that of the sand by capillary attractions; the -pressure of the foot causing dilatation of the sand more water is -required, which has to be obtained either by depressing the level of -the surface against the capillary attractions, or by drawing water -through the interstices of the surrounding sand. This latter requires -time to accomplish, so that for the moment the capillary forces are -overcome; the surface of the water is lowered below that of the sand, -leaving the latter white or drier until a sufficient supply has been -obtained from below, when the surface rises and wets the sand again. -On raising the foot it is generally seen that the sand under the foot -and around becomes momentarily wet; this is because, on the distorting -forces being removed, the sand again contracts, and the excess of water -finds momentary relief at the surface.’ - -This proves that the sand under the foot, as well as the surface around -it, must be dry for a short time after the foot is pressed upon it, -though we cannot see it whitened, as the foot is not transparent. That -it is so has been verified by Mr. Alex. Galt, Experimental Instructor -in the Physical Laboratory of Glasgow University, by laying a small -square of plate-glass on wet sand on the sea-shore of Helensburgh, and -suddenly pressing on it by a stout stick with nearly all his weight. -He found the sand, both under the glass and around it in contact with -the air, all became white at the same moment. Of all the two hundred -thousand million men, women, and children who, from the beginning -of the world, have ever walked on wet sand, how many, prior to the -British Association Meeting at Aberdeen in 1885, if asked, ‘Is the sand -compressed under your foot?’ would have answered otherwise than ‘Yes!’? - -(Contrast with this the case of walking over a bed of wet sea-weed!) - - -§ 33. In the case of globes packed together in closest order (and -therefore also in the case of ellipsoids, if all similarly oriented), -our condition of coherent contact between each molecule and twelve -neighbours implies absolute rigidity of form and constancy of bulk. -Hence our convex solid must be neither ellipsoidal nor spherical -in order that there may be the changes of form and changes of bulk -which we have been considering as dependent on three independent -variables specifying the orientation of each solid relatively to rows -of the assemblage. An interesting dynamical problem is presented by -supposing any mutual forces, such as might be produced by springs, to -act between the solid molecules, and investigating configurations of -equilibrium on the supposition of frictionless contacts. The solution -of it of course is that the potential energy of the springs must be a -minimum or a minimax or a maximum for equilibrium, and a minimum for -stable equilibrium. The solution will be a configuration of minimum or -minimax, or maximum, volume, only in the case of pressure equal in all -directions. - - -§ 34. A purely geometrical question, of no importance in respect to -the molecular tactics of a crystal but of considerable interest in -pure mathematics, is forced on our attention by our having seen (§ 27) -that a homogeneous assemblage of solids of given shape, each touched -by twelve neighbours, has three freedoms which may be conveniently -taken as the three angles specifying the orientation of each molecule -relatively to rows of the assemblage as explained in § 30. - -Consider a solid _S_{1}_ and the twelve neighbours which touch it, -and try if it is possible to cause it to touch more than twelve of -the bodies. Attach ends of three thick flexible wires to any places -on the surface of _S_{1}_; carry the wires through interstices of the -assemblage, and attach their other ends at any three places of _A_, -_B_, _C_, respectively, these being any three of the bodies outside -the cluster of _S_{1}_ and its twelve neighbours. Cut the wires across -at any chosen positions in them; and round off the cut ends, just -leaving contact between the rounded ends, which we shall call _f′f_, -_g′g_, _h′h_. Do homogeneously for every other solid of the assemblage -what we have done for _S_{1}_. Now bend the wires slightly so as to -separate the pairs of points of contact, taking care to keep them -from touching any other bodies which they pass near on their courses -between _S_{1}_ and _A_, _B_, _C_ respectively. After having done this, -thoroughly rigidify all the wires thus altered. We may now, having -three independent variables at our disposal, so change the orientation -of the molecules, relatively to rows of the assemblage, as to bring -_f′f_, _g′g_, and _h′h_ again into contact. We have thus six fresh -points of _S_{1}_; of which three are _f′_, _g′_, _h′_; and the other -three are on the three extensions of _S_{1}_ corresponding to the -single extensions of _A_, _B_, _C_ respectively, which we have been -making. Thus we have a _real_ solution of the interesting geometrical -problem:--It is required so to form a homogeneous assemblage of solids -of any arbitrarily given shape that each solid shall be touched by -eighteen others. This problem is determinate, because the making of -the three contacts _f′f_, _g′g_, _h′h_, uses up the three independent -variables left at our disposal after we have first formed a homogeneous -assemblage with twelve points of contact on each solid. But our manner -of finding a shape for each solid which can allow the solution of the -problem to be real, proves that the solution is essentially imaginary -for every wholly convex shape. - - -§ 35. Pausing for a moment longer to consider afresh the geometrical -problem of putting arbitrarily given equal and similar solids together -to make a homogeneous assemblage of which each member is touched by -eighteen others, we see immediately that it is determinate (whether it -has any real solution or not), because when the shape of each body is -given we have nine disposables for fixing the assemblage: six for the -character of the assemblage of the corresponding points, and three for -the orientation of each molecule relatively to rows of the assemblage -of corresponding points. These nine disposables are determined by the -condition that each body has nine pairs of contacts with others. - -Suppose now a homogeneous assemblage of the given bodies, in open -order with no contacts, to be arbitrarily made according to any nine -arbitrarily chosen values for the six distances between a point of -_S_{1}_ and the corresponding points of its six pairs of nearest and -next nearest neighbours (§ 1 above), and the three angles (§ 9 above) -specifying the orientation of each body relatively to rows of the -assemblage. We may choose in any nine rows through _S_{1}_ any nine -pairs of bodies at equal distances on the two sides of _S_{1}_ far or -near, for the eighteen bodies which are to be in contact with _S_{1}_. -Hence there is an infinite number of solutions of the problem of which -only a finite number can be real. Every solution of the problem of -eighteen contacts is imaginary when the shape is wholly convex. - -[Illustration: FIG. 13.] - - -§ 36. Without for a moment imagining the molecules of matter to be -hard solids of convex shape, we may derive valuable lessons in the -tactics of real crystals by studying the assemblage described in §§ -24 and 25 and represented in Figs. 12 and 13. I must for the present -forego the very attractive subject of the tactics presented by faces -not parallel to one or other of the four faces of the primitive -tetrahedrons which we found in § 24, and ask you only to think of the -two sides of a plate of crystal parallel to any one of them, that is to -say, an assemblage of such layers as those represented geometrically -in Fig. 12 and shown in stereoscopic view in Fig. 13. If, as is the -case with the solids[10] photographed in Fig. 13, the under side of -each solid is nearly plane but slightly convex, and the top is somewhat -sharply curved, we have the kind of difference between the upper and -under of the two parallel sides of the crystal which I have already -described to you in § 21 above. In this case the assemblage is formed -by letting the solids fall down from above and settle in the hollows to -which they come most readily, or which give them the stablest position. -It would, we may suppose, be the hollows _p′ q′ r′_, not _p q r_, (Fig. -12) that would be chosen; and thus, of the two formations described in -§ 25, we should have that in which the hollows above _p′ q′ r′_ are -occupied by the comparatively flat under sides of the molecules of -the layer above, and the hollows below the apertures _p q r_ by the -comparatively sharp tops of the molecules of the layers below. - - -§ 37. For many cases of natural crystals of the wholly asymmetric -character, the true forces between the crystalline molecules will -determine precisely the same tactics of crystallization as would be -determined by the influence of gravity and fluid viscosity in the -settlement from water, of sand composed of uniform molecules of the -wholly unsymmetrical convex shape represented in Figs. 12 and 13. -Thus we can readily believe that a real crystal which is growing -by additions to the face seen in Fig. 12, would give layer after -layer regularly as I have just described. But if by some change of -circumstances the plate, already grown to a thickness of many layers in -this way, should come to have the side facing _from_ us in the diagram -exposed to the mother-liquor, or mother-gas, and begin to grow from -that face, the tactics might probably be that each molecule would find -its resting-place with its most nearly plane side in the wider hollows -under _p′ q′ r′_, instead of with its sharpest corner in the narrower -and steeper hollows under _p q r_, as are the molecules in the layer -below that shown in the diagram in the first formation. The result -would be a compound crystal consisting of two parts, of different -crystalline quality, cohering perfectly together on the two sides of -an interfacial plane. It seems probable that this double structure may -be found in nature, presented by crystals of the wholly unsymmetric -class, though it may not hitherto have been observed or described in -crystallographic treatises. - -[Illustration: FIG. 14.] - - -§ 38. This asymmetric double crystal becomes simply the well-known -symmetrical ‘twin-crystal’[11] in the particular case in which each of -the constituent molecules is symmetrical on the two sides of a plane -through it parallel to the plane of our diagrams, and also on the two -sides of some plane perpendicular to this plane. We see, in fact, that -in this case if we cut in two the double crystal by the plane of Fig. -14, and turn one part ideally through 180° round the intersection of -these two planes, we bring it into perfect coincidence with the other -part. - -This we readily understand by looking at Fig. 14, in which the solid -shown in outline may be either an egg-shaped figure of revolution, or -may be such a figure flattened by compression perpendicular to the -plane of the diagram. The most readily chosen and the most stable -resting-places for the constituents of each successive layer might be -the wider hollows _p′ q′ r′_: and therefore if, from a single layer to -begin with, the assemblage were to grow by layer after layer added to -it on each side, it might probably grow as a twin-crystal. But it might -also be that the presence of a molecule in the wider hollow _p′ q′ r′_ -on one side, might render the occupation of the corresponding hollow on -the other side by another molecule less probable, or even impossible. -Hence, according to the configuration and the molecular forces of the -particular crystalline molecule in natural crystallization, there may -be necessarily, or almost necessarily, the twin, when growth proceeds -simultaneously on the two sides: or the twin growth may be impossible, -because the first occupation of the wider hollows on one side, may -compel the continuity of the crystalline quality throughout, by leaving -only the narrower hollows _p q r_ free for occupation by molecules -attaching themselves on the other side. - - -§ 39. Or the character of the crystalline molecule may be such -that when the assemblage grows by the addition of layer after layer -on one side only, with a not very strongly decided preference to the -wider hollows _p′ q′ r′_, some change of circumstances may cause -the molecules of one layer to place themselves in a hollow _p q r_. -The molecules in the next layer after this would find the hollows -_p′ q′ r′_ occupied on the far side, and would thus have a bias in -favour of the hollows _p q r_. Thus layer after layer might be added, -constituting a twinned portion of the growth, growing, however, -with less strong security for continued homogeneousness than when -the crystal was growing, as at first, by occupation of the wider -hollows _p′ q′ r′_. A slight disturbance might again occur, causing -the molecules of a fresh layer to settle, not in the narrow hollows -_p q r_, but in the wider hollows _p′ q′ r′_, notwithstanding the -nearness of molecules already occupying the wider hollows on the -other side. Disturbances such as these occurring irregularly during -the growth of a crystal, might produce a large number of successive -twinnings at parallel planes with irregular intervals between them, -or a large number of twinnings in planes at equal intervals might be -produced by some regular periodic disturbance occurring for a certain -number of periods, and then ceasing. Whether regular and periodic, or -irregular, the tendency would be that the number of twinnings should -be even, and that after the disturbances cease the crystal should -go on growing in the first manner, because of the permanent bias in -favour of the wider hollows _p′ q′ r′_. These changes of molecular -tactics, which we have been necessarily led to by the consideration -of the fortuitous concourse of molecules, are no doubt exemplified in -a large variety of twinnings and counter-twinnings found in natural -minerals. In the artificial crystallization of chlorate of potash they -are of frequent occurrence, as is proved, not only by the twinnings and -counter-twinnings readily seen in the crystalline forms, but also by -the brilliant iridescence observed in many of the crystals found among -a large multitude, which was investigated scientifically by Sir George -Stokes ten years ago, and described in a communication to the Royal -Society ‘On a remarkable phenomenon of crystalline reflection’ (_Proc. -R.S._, vol. xxxviii, 1885, p. 174). - - -§ 40. A very interesting phenomenon, presented by what was originally -a clear homogeneous crystal of chlorate of potash, and was altered by -heating to about 245°-248° Cent., which I am able to show you through -the kindness of Lord Rayleigh, and of its discoverer, Mr. Madan, -presents another very wonderful case of changing molecular tactics, -most instructive in respect of the molecular constitution of elastic -solids. When I hold this plate before you with the perpendicular to -its plane inclined at 10° or more to your line of vision, you see a -tinsel-like appearance, almost as bright as if it were a plate of -polished silver, on this little area, which is a thin plate of chlorate -of potash cemented for preservation between two pieces of glass; and, -when I hold a light behind, you see that the little plate is almost -perfectly opaque like metal foil. But now when I hold it nearly -perpendicular to your line of vision the tinsel-like appearance is -lost. You can see clearly through the plate, and you also see that very -little light is reflected from it. As a result both of Mr. Madan’s own -investigations, and further observations by himself, Lord Rayleigh came -to the conclusion that the almost total reflection of white light which -you see is due to the reflection of light at many interfacial planes -between successive layers of twinned and counter-twinned crystal of -small irregular thicknesses, and not to any splits or cavities or any -other deviation from homogeneousness than that presented by homogeneous -portions of oppositely twinned-crystals in thorough molecular contact -at the interfaces. - - -§ 41. When the primitive clear crystal was first heated very gradually -by Madan to near its melting-point (359° according to Carnelly), it -remained clear, and only acquired the tinsel appearance after it had -cooled to about 245° or 248°[12]. Rayleigh found that if a crystal -thus altered was again and again heated it always lost the tinsel -appearance, and became perfectly clear at some temperature considerably -below the melting-point, and regained it at about the same temperature -in cooling. It seems, therefore, certain that at temperatures above -248°, and below the melting-point, the molecules had so much of thermal -motions as to keep them hovering about the positions of _p q r_, _p′ q′ -r′_, of our diagrams, but not enough to do away with the rigidity of -the solid; and that when cooled below 248° the molecules were allowed -to settle in one or other of the two configurations, but with little -of bias for one in preference to the other. It is certainly a very -remarkable fact in Natural History, discovered by these observations, -that, when the molecules come together to form a crystal out of the -watery solution, there should be so much more decided a bias in favour -of continued homogeneousness of the assemblage than when, by cooling, -they are allowed to settle from their agitations in a rigid, but nearly -melting, solid. - - -§ 42. But even in crystallization from watery solution of chlorate of -potash the bias in favour of thorough homogeneousness is not in every -contingency decisive. In the first place, beginning, as the formation -seems to begin, from a single molecular plane layer such as that -ideally shown in Fig. 14, it goes on, not to make a homogeneous crystal -on the two sides of this layer, but probably always so as to form a -twin-crystal on its two sides, exactly as described in § 38, and, if -so, certainly for the reason there stated. This is what Madan calls -the ‘inveterate tendency to produce twins (such as would assuredly -drive a Malthus to despair)[13]’; and it is to this that he alludes as -‘the inevitable twin-plate’ in the passage from his paper given in the -foot-note to § 41 above. - - -§ 43. In the second place, I must tell you that many of the crystals -produced from the watery solution by the ordinary process of slow -evaporation and crystallization, show twinnings and counter-twinnings -at irregular intervals in the otherwise homogeneous crystal on either -one or both sides of the main central twin-plane, which henceforth, -for brevity, I shall call (adopting the hypothesis already explained, -which seems to me undoubtedly true) the ‘initial plane.’ Each twinning -is followed, I believe, by a counter-twinning at a very short distance -from it; at all events Lord Rayleigh’s observations[14] prove that the -whole number of twinnings and counter-twinnings in a thin disturbed -stratum of the crystal on one side of the main central twin-plane -is generally, perhaps always, even; so that, except through some -comparatively very small part or parts of the whole thickness, the -crystal on either side of the middle or initial plane is homogeneous. -This is exactly the generally regular growth which I have described -to you (§ 39) as interrupted occasionally or accidentally by some -unexplained disturbing cause, but with an essential bias to the -homogeneous continuance of the more easy or natural one of the two -configurations. - - -§ 44. I have now great pleasure in showing you a most interesting -collection of the iridescent crystals of chlorate of potash, each -carefully mounted for preservation between two glass plates, which have -been kindly lent to us for this evening by Mr. Madan. In March, 1854, -Dr. W. Bird Herapath sent to Prof. Stokes some crystals of chlorate -of potash showing the brilliant and beautiful colours you now see, -and, thirty years later, Prof. E. J. Mills recalled his attention to -the subject by sending him ‘a fine collection of splendidly coloured -crystals of chlorate of potash of considerable size, several of the -plates having an area of a square inch or more, and all of them -being thick enough to handle without difficulty.’ The consequence -was that Stokes made a searching examination into the character of -the phenomenon, and gave the short, but splendidly interesting, -communication to the Royal Society of which I have already told you. -The existence of these beautifully coloured crystals had been well -known to chemical manufacturers for a long time, but it does not appear -that any mention of them was to be found in any scientific journal or -treatise prior to Stokes’ paper of 1885. He found that the colour was -due to twinnings and counter-twinnings in a very thin disturbed stratum -of the crystal showing itself by a very fine line, dark or glistening, -according to the direction of the incident light when a transverse -section of the plate of crystal was examined in a microscope. By -comparison with a spore of lycopodium he estimated that the breadth -of this line, and therefore the thickness of the disturbed stratum of -the crystal, ranged somewhere about the one-thousandth of an inch. He -found that the stratum was visibly thicker in those crystals which -showed red colour than in those which showed blue. He concluded that -‘the seat of the coloration is certainly a thin twinned stratum’ (that -is to say, a homogeneous portion of crystal between a twinning and a -counter-twinning), and found that ‘a single twin-plane does not show -anything of the kind.’ - - -§ 45. A year or two later Lord Rayleigh entered on the subject with -an exhaustive mathematical investigation of the reflection of light at -a twin-plane of a crystal (_Philosophical Magazine_, September, 1888), -by the application of which, in a second paper ‘On the remarkable -phenomenon of Crystalline Reflection described by Prof. Stokes,’ -published in the same number of the _Philosophical Magazine_, he gave -what seems certainly the true explanation of the results of Sir George -Stokes’ experimental analysis of these beautiful phenomena. He came -very decidedly to the conclusion that the selective quality of the -iridescent portion of the crystal, in virtue of which it reflects -almost totally light nearly of one particular wave-length for one -particular direction of incidence (on which the brilliance of the -coloration depends), cannot be due to merely a single twin-stratum, -but that it essentially is due to a considerable number of parallel -twin-strata at nearly equal distances. The light reflected by this -complex stratum is, for any particular direction of incident and -reflected ray, chiefly that of which the wave-length is equal to twice -the length of the period of the twinning and counter-twinning, on a -line drawn through the stratum in the direction of either the incident -or the reflected ray. - - -§ 46. It seems to me probable that each twinning is essentially -followed closely by a counter-twinning. Probably three or four of these -twin-strata might suffice to give colour; but in any of the brilliant -specimens as many as twenty or thirty, or more, might probably be -necessary to give so nearly monochromatic light as was proved by -Stokes’ prismatic analysis of the colours observed in many of his -specimens. The disturbed stratum of about a one-thousandth of an inch -thickness, seen by him in the microscope, amply suffices for the 5, -10, or 100 half wave-lengths required by Rayleigh’s theory to account -for perceptible or brilliant coloration. But what _can_ be the cause -of any approach to regular periodicity in the structure sufficiently -good to give the colours actually observed? Periodical motion of -the mother-liquor relatively to the growing crystal might possibly -account for it. But Lord Rayleigh tells us that he tried rocking the -pan containing the solution without result. Influence of light has -been suggested, and I believe tried, also without result, by several -enquirers. We know, by the beautiful discovery of Edmond Becquerel, -of the prismatic colours photographed on a prepared silver plate by -the solar spectrum, that ‘standing waves’ (that is to say, vibrations -with stationary nodes and stationary places of maximum vibration), -due to co-existence of incident and reflected waves, do produce such -a periodic structure as that which Rayleigh’s theory shows capable -of giving a corresponding tint when illuminated by white light. It -is difficult, therefore, not to think that light may be effective in -producing the periodic structure in the crystallization of chlorate of -potash, to which the iridescence is due. Still, experimental evidence -seems against this tempting theory, and we must perforce be content -with the question unanswered:--What can be the cause of 5, or 10, or -100 pairs of twinning and counter-twinning following one another in -the crystallization with sufficient regularity to give the colour: and -why, if there are twinnings and counter-twinnings, are they not at -irregular intervals, as those produced by Madan’s process, and giving -the observed white tinsel-like appearance with no coloration? - - -§ 47. And now I have sadly taxed your patience: and I fear I have -exhausted it and not exhausted my subject! I feel I have not got -halfway through what I hoped I might be able to put before you this -evening regarding the molecular structure of crystals. I particularly -desired to speak to you of quartz crystal with its ternary symmetry -and its chirality[15]; and to have told you of the etching[16] by -hydrofluoric acid which, as it were, commences to unbuild the crystal -by taking away molecule after molecule, but not in the reverse order of -the primary up-building; and which thus reveals differences of tactics -in the alternate faces of the six-sided pyramid which terminates at -either end, sometimes at both ends, the six-sided prism constituting -generally the main bulk of the crystal. I must confine myself to giving -you a geometrical symbol for the ternary symmetry of the prism and its -terminal pyramid. - -[Illustration: FIG. 15.] - - -§ 48. Make an equilateral equiangular hexagonal prism, with -its diagonal from edge to edge ninety-five hundredths[17] of its -length. Place a number of these close together, so as to make up -a hexagonal plane layer with its sides perpendicular to the sides -of the constituent hexagonal prisms: see Fig. 15 and imagine the -semicircles replaced by their diameters. You see in each side of the -hexagonal assemblage, edges of the constituent prisms, and you see -at each corner of the assemblage a face (not an edge) of _one_ of -the constituent prisms. Build up a hexagonal prismatic assemblage -by placing layer after layer over it with the constituent prisms of -each layer vertically over those in the layer below; and finish the -assemblage with a six-sided pyramid by building upon the upper end -of the prism, layer after layer of diminishing hexagonal groups, -each less by one circumferential row than the layer below it. You -thus have a crystal of precisely the shape of a symmetrical specimen -of rock crystal, with the faces of its terminal pyramid inclined at -38° 13′ to the faces of the prism from which they spring. But the -assemblage thus constituted has ‘senary’ (or six-rayed symmetry). To -reduce this to ternary symmetry, cut a groove through the middle of -each alternate face of the prismatic molecule, making this groove -in the first place parallel to the edges: and add a corresponding -projection, or fillet, to the middles of the other three faces, so -that two of the cylinders similarly oriented would fit together, with -the projecting fillet on one side of one of them entering the groove -in the anti-corresponding side of the other. The prismatic portion -of the assemblage thus formed shows (see Fig. 15), on its alternate -edges, faces of molecules with projections and faces of molecules with -grooves; and shows only orientational differences between alternate -faces, whether of the pyramid or of the prism. Having gone only so far -from ‘senary’ symmetry, we have exactly the triple, or three-pair, -anti-symmetry required for the piezo-electricity of quartz investigated -so admirably by the brothers Curie[18], who found that a thin plate of -quartz crystal cut from any position perpendicular to a pair of faces -of a symmetrical crystal, becomes positively electrified on one side -and negatively on the other when pulled in a direction perpendicular -to those faces. But this assemblage has not the chiral piezo-electric -quality discovered theoretically by Voigt[19], and experimentally in -quartz and in tourmaline by himself and Riecke[20], nor the well-known -optic chirality of quartz. - -[Illustration: FIG. 16.] - -[Illustration: FIG. 17.] - - -§ 49. Change now the directions of the grooves and fillets to either of -the oblique configurations shown in Fig. 16, which I call right-handed, -because the directions of the projections are tangential to the threads -of a three-thread right-handed screw, and Fig. 17 (left-handed). The -prisms with their grooves and fillets will still all fit together if -they are all right-handed, or all left-handed. - -[Illustration: FIG. 18.] - -Fig. 18 shows the upper side of a hexagonal layer of an assemblage -thus composed of the right-handed molecule of Fig. 16. Fig. 15 -unchanged, still represents a horizontal section through the centres -of the molecules. A prism built up of such layers, and finished at -each end with a pyramid according to the rule of § 48, has all the -qualities of ternary chiral symmetry required for the piezo-electricity -of quartz; for the orientational differences of the alternate pairs -of prismatic faces; for the absolute difference between the alternate -pairs of faces of each pyramid which are shown in the etching by -hydrofluoric acid; for the merely orientational difference between -the parallel faces of the two pyramids; and for the well-known -chiro-optic[21] property of quartz. Look at two contiguous faces _A_, -_B_ of our geometrical model quartz crystal now before you, with its -axis vertical. You will see a difference between them: turn it upside -down; _B_ will be undistinguishable from what _A_ was, and _A_ will be -undistinguishable from what _B_ was. Look at the two terminal pyramids, -and you will find that the face above _A_ and the face below _B_ are -identical in quality, and that they differ from the face above _B_ -and below _A_. This model is composed of the right-handed constituent -molecules shown in Fig. 16. It is so placed before you that the edge of -the prismatic part of the assemblage nearest to you shows you filleted -faces of the prismatic molecules. You see two pyramidal faces; the one -to your right hand, over _B_, presents complicated projections and -hollows at the corners of the constituent molecules; and the pyramidal -face next your left hand, over _A_, presents their unmodified corners. -But it will be the face next your left hand which will present the -complex bristling corners, and the face next your right hand that -will present the simple corners, if, for the model before you, you -substitute a model composed of left-handed molecules such as those -shown in Fig. 17. - - -§ 50. To give all the qualities of symmetry and anti-symmetry of -the pyro-electric and piezo-electric properties of tourmaline -investigated theoretically by Voigt[22], and experimentally by himself -and Friecke[23], make a hollow in one terminal face of each of our -constituent prisms, and a corresponding projection in its other -terminal face. - - -§ 51. Coming back to quartz, we can now understand perfectly the -two kinds of macling which are well known to mineralogists as being -found in many natural specimens of the crystal, and which I call -respectively the orientational macling, and the chiral macling. -In the orientational macling all the crystalline molecules are -right-handed, or all left-handed; but through all of some part of the -crystal, each of our component hexagonal prisms is turned round its -axis through 60° from the position it would have if the structure -were homogeneous throughout. In each of the two parts the structure -is homogeneous, and possesses all the electric and optic properties -which any homogeneous portion of quartz crystal presents, and the -facial properties of natural uncut crystal, shown in the etching by -hydrofluoric acid; but there is a discontinuity at the interface, not -generally plane, between the two parts, which in our geometrical model -would be shown by non-fittings between the molecules on the two sides -of the interface, while all the contiguous molecules in one part, and -all the contiguous molecules in the other part, fit into one another -perfectly. In chiral macling, which is continually found in amethystine -quartz, and sometimes in ordinary clear quartz crystals, some parts -are composed of right-handed molecules, and others of left-handed -molecules. It is not known whether, in this chiral macling, there is or -there is not also the orientational macling on the two sides of each -interface; but we may say probably _not_; because we know that the -orientational macling occurs in nature without any chiral macling, and -because there does not seem reason to expect that chiral macling would -imply orientational macling on the two sides of the same interface. -I would like to have spoken to you more of this most interesting -subject; and to have pointed out to you that some of the simplest and -most natural suppositions we can make as to the chemical forces (or -electrical forces, which probably means the same thing) concerned in -a single chemical molecule of quartz, _SiO_{2}_, and acting between -it and similar neighbouring molecules, would lead essentially to -these molecules coming together in triplets, each necessarily either -right-handed or left-handed, but with as much probability of one -configuration as of the other: and to have shown you that these -triplets of silica 3(_SiO_{2}_) can form a crystalline molecule with -all the properties of ternary chiral symmetry, typified by our grooved -hexagonal prisms, and can build up a quartz crystal by the fortuitous -concourse of atoms. I should like also to have suggested and explained -the possibility that a right-handed crystalline molecule thus formed -may, in natural circumstances of high temperature, or even of great -pressure, become changed into a left-handed crystal, or _vice-versa_. -My watch, however, warns me that I must not enter on this subject. - -[Illustration: FIG. 19.] - - -§ 52. Coming back to mere molecular tactics of crystals, remark that -our assemblage of rounded, thoroughly scalene, tetrahedrons, shown -in the stereoscopic picture (§ 36, Fig. 13 above), essentially has -chirality because each constituent tetrahedron, if wholly scalene, has -chirality[24]. I should like to have explained to you how a single or -double homogeneous assemblage of points has essentially no chirality, -and how three assemblages of single points, or a single assemblage of -triplets of points, can have chirality, though a single triplet of -points cannot have chirality. I should like indeed to have brought -somewhat thoroughly before you the geometrical theory of chirality; -and in illustration to have explained the conditions under which four -points, or two lines, or a line and two points, or a combination of -point, line and plane, can have chirality: and how a homogeneous -assemblage of non-chiral objects can have chirality; but in pity I -forbear, and I thank you for the extreme patience with which you have -listened to me. - - - - -FOOTNOTES: - -[1] See foot-note on § 22 below. - -[2] The holes in the cylinders are bored obliquely, as shown in Fig. -4, which causes them to remain at any desired position on the cord and -allows them to be freed to move up and down by slackening the cord for -a moment. - -[3] ‘On the Homogeneous Division of Space,’ by Lord Kelvin, _Royal -Society Proceedings_, vol. lv, Jan. 18, 1894. - -[4] Similar curves are said to be parallel when the tangents to them at -corresponding points are parallel. - -[5] See foot-note to § 12 above. - -[6] ‘On the Division of Space with Minimum Partitional Area,’ -_Philosophical Magazine_, vol. xxiv, 1887, p. 502, and _Acta -Mathematica_ of the same year. - -[7] A. Levy, _Edinburgh Philosophical Journal_, April, 1822; -Whewell, _Phil. Trans. Royal Society_, 1825; Miller, _Treatise on -Crystallography_. - -[8] I call any geometrical figure, or group of points, _chiral_, and -say that it has chirality, if its image in a plane mirror, ideally -realized, cannot be brought to coincide with itself. Two equal and -similar right hands are homochirally similar. Equal and similar right -and left hands are heterochirally similar or ‘allochirally’ similar -(but heterochirally is better). These are also called ‘enantiomorphs,’ -after a usage introduced, I believe, by German writers. Any chiral -object and its image in a plane mirror are heterochirally similar. - -[9] _Philosophical Magazine_, vol. xx, 1885, second half year, p. 469, -and _British Association Report_, 1885, Aberdeen, p. 896. - -[10] The solids of the photograph are castings in fine plaster of Paris -from a scalene tetrahedron of paraffin wax, with its corners and edges -rounded, used as a pattern. - -[11] ‘A twin-crystal is composed of two crystals joined together in -such a manner that one would come into the position of the other by -revolving through two right angles round an axis which is perpendicular -to a plane which either is, or may be, a face of either crystal. -The axis will be called the twin-axis, and the plane to which it is -perpendicular the twin-plane.’ Miller’s _Treatise on Crystallography_, -p. 103. In the text the word ‘twin-plane,’ quoted from the writings -of Stokes and Rayleigh, is used to signify the plane common to the -two crystals in each of the cases referred to: and not the plane -perpendicular to this plane, in which one part of the crystal must be -rotated to bring it into coincidence with the other, and which is the -twin-plane as defined by Miller. - -[12] ‘A clear transparent crystal of potassium chlorate, from which -the inevitable twin-plate had been ground away so as to reduce it to -a single crystal film about 1 mm. in thickness, was placed between -pieces of mica and laid on a thick iron plate. About 3 cm. from it -was laid a small bit of potassium chlorate, and the heat of a Bunsen -burner was applied below this latter, so as to obtain an indication -when the temperature of the plate was approaching the fusing-point of -the substance (359° _C_ according to Prof. Carnelly). The crystal plate -was carefully watched during the heating, but no depreciation took -place, and no visible alteration was observed, up to the point at which -the small sentinel crystal immediately over the burner began to fuse. -The lamp was now withdrawn, and when the temperature had sunk a few -degrees a remarkable change spread quickly and quietly over the crystal -plate, causing it to reflect light almost as brilliantly as if a film -of silver had been deposited upon it. No further alteration occurred -during the cooling; and the plate, after being ground and polished on -both sides, was mounted with Canada balsam between glass plates for -examination. Many crystals have been similarly treated with precisely -similar results; and the temperature at which the change takes place, -has been determined to lie between 245° and 248°, by heating the -plates upon a bath of melted tin in which a thermometer was immersed. -With single crystal plates no decrepitation has ever been observed, -while with the ordinary twinned-plates it always occurs more or less -violently, each fragment showing the brilliant reflective power above -noticed.’--_Nature_, May 20, 1886. - -[13] _Nature_, May 20, 1886. - -[14] _Philosophical Magazine_, 1888, second half year, p. 260. - -[15] See foot-note to § 22 above. - -[16] Widmanstätten, 1807. Leydolt (1855, Wien. Akad. Ber. 15, 59, T. -9, 10. Baumhauer, Pogg. Ann. 138, 563 (1869); 140, 271; 142, 324; 145, -460; 150, 619.) For an account of these investigations, see Mallard, -_Traité de Crystallographie_ (Paris, 1884), Tome II, chapitre xvi. - -[17] More exactly .9525, being 3/4 × cot 38° 13′; see p. 53. - -[18] J. and P. Curie and C. Friedel, _Comptes Rendus_, 1882, 1883, -1886, 1892. - -[19] Allgemeine Theorie der piëzo- und pyroelectrischen Erscheinungen -an Krystallen. W. Voigt, Königl. Gesellschaft der Wissenschaften zu -Göttingen, August 2, 1890. - -[20] Wiedemann, _Annalen_, 1892, xlv, p. 923. - -[21] Generally miscalled ‘rotational.’ - -[22] See foot-note (2) to p. 54 above. - -[23] See foot-note (3) to p. 54 above. - -[24] See foot-note to § 22 above. - - - THE END - - Oxford - PRINTED AT THE CLARENDON PRESS - BY HORACE HART, PRINTER TO THE UNIVERSITY - - - - - -End of Project Gutenberg's The Molecular Tactics of a Crystal, by Lord Kelvin - -*** END OF THIS PROJECT GUTENBERG EBOOK THE MOLECULAR TACTICS OF A CRYSTAL *** - -***** This file should be named 54976-0.txt or 54976-0.zip ***** -This and all associated files of various formats will be found in: - http://www.gutenberg.org/5/4/9/7/54976/ - -Produced by Turgut Dincer and The Online Distributed -Proofreading Team at http://www.pgdp.net (This file was -produced from images generously made available by The -Internet Archive/American Libraries.) - -Updated editions will replace the previous one--the old editions will -be renamed. - -Creating the works from print editions not protected by U.S. copyright -law means that no one owns a United States copyright in these works, -so the Foundation (and you!) can copy and distribute it in the United -States without permission and without paying copyright -royalties. 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You may copy it, give it away or re-use it under the terms of -the Project Gutenberg License included with this eBook or online at -www.gutenberg.org. If you are not located in the United States, you'll have -to check the laws of the country where you are located before using this ebook. - -Title: The Molecular Tactics of a Crystal - -Author: Lord Kelvin - -Release Date: June 24, 2017 [EBook #54976] - -Language: English - -Character set encoding: UTF-8 - -*** START OF THIS PROJECT GUTENBERG EBOOK THE MOLECULAR TACTICS OF A CRYSTAL *** - - - - -Produced by Turgut Dincer and The Online Distributed -Proofreading Team at http://www.pgdp.net (This file was -produced from images generously made available by The -Internet Archive/American Libraries.) - - - - - - -</pre> - -<p><span class="pagenum"><a name="Page_1" id="Page_1">1</a></span></p> - -<p class="center padt1">THE MOLECULAR TACTICS OF<br /> -A CRYSTAL</p> - -<p class="center"><i>LORD KELVIN</i></p> - -<hr /> - -<p class="center">London<br /> -HENRY FROWDE<br /> -<span class="smcap">Oxford University Press Warehouse<br /> -Amen Corner, E.C.</span> -</p> - -<div class="figcenter"> -<img src="images/i002.jpg" width="100" height="96" alt="" /> -</div> - -<p class="center">New York<br /> -MACMILLAN & CO., 66 FIFTH AVENUE</p> - -<hr /> - -<h1> -THE<br /> -MOLECULAR TACTICS OF<br /> -A CRYSTAL</h1> - -<p class="center"><small><small><small>BY</small></small></small><br /> -<br /> -LORD KELVIN, P.R.S.<br /> -<small><small><small> -PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF GLASGOW<br /> -AND FELLOW OF PETERHOUSE, CAMBRIDGE</small></small></small><br /> -<br /> -<br /> -<small><i>Being the Second <span class="smcap">Robert Boyle Lecture</span>, delivered before<br /> -the Oxford University Junior Scientific Club<br /> -on Tuesday, May 16, 1893</i>/<br /> -<br /> -<br /> -WITH TWENTY ILLUSTRATIONS</small><br /> -<br /> -<br /> -Oxford<br /> -AT THE CLARENDON PRESS<br /> -1894<br /> -</p> - -<hr /> - -<p class="center"><small>Oxford<br /> -PRINTED AT THE CLARENDON PRESS<br /> -<small>BY HORACE HART, PRINTER TO THE UNIVERSITY</small></small> -</p> -<hr /> - -<h2><small>ON THE</small><br /> -MOLECULAR TACTICS OF A CRYSTAL<br /> -<small><span class="smcap">By LORD KELVIN, P.R.S.</span></small></h2> - -<p class="section">§ 1. My subject this evening is not the physical -properties of crystals, not even their dynamics; it is -merely the geometry of the structure—the arrangement -of the molecules in the constitution of a crystal. -Every crystal is a homogeneous assemblage of small -bodies or molecules. The converse proposition is -scarcely true, unless in a very extended sense of the -term crystal (<a href="#s20">§ 20</a> below). I can best explain a homogeneous -assemblage of molecules by asking you to -think of a homogeneous assemblage of people. To be -homogeneous every person of the assemblage must -be equal and similar to every other: they must be -seated in rows or standing in rows in a perfectly similar -manner. Each person, except those on the borders of -the assemblage, must have a neighbour on one side -and an equi-distant neighbour on the other: a neighbour -on the left front and an equi-distant neighbour -behind on the right, a neighbour on the right front -and an equi-distant neighbour behind on the left. His -two neighbours in front and his two neighbours behind -are members of two rows equal and similar to the rows<span class="pagenum"><a name="Page_6" id="Page_6">6</a></span> -consisting of himself and his right-hand and left-hand -neighbours, and their neighbours’ neighbours indefinitely -to right and left. In particular cases the nearest -of the front and rear neighbours may be right in front -and right in rear; but we must not confine our attention -to the rectangularly grouped assemblages thus constituted. -Now let there be equal and similar assemblages -on floors above and below that which we have -been considering, and let there be any indefinitely -great number of floors at equal distances from one -another above and below. Think of any one person -on any intermediate floor and of his nearest neighbours -on the floors above and below. These three persons -must be exactly in one line; this, in virtue of the -homogeneousness of the assemblages on the three -floors, will secure that every person on the intermediate -floor is exactly in line with his nearest neighbours -above and below. The same condition of alignment -must be fulfilled by every three consecutive floors, and -we thus have a homogeneous assemblage of people in -three dimensions of space. In particular cases every -person’s nearest neighbour in the floor above may be -vertically over him, but we must not confine our -attention to assemblages thus rectangularly grouped -in vertical lines.</p> - -<p class="section">§ 2. Consider now any particular person <i>C</i> (Fig. 1) -on any intermediate floor, <i>D</i> and <i>D′</i> his nearest neighbours, -<i>E</i> and <i>E′</i> his next nearest neighbours all on his -own floor. His next next nearest neighbours on that -floor will be in the positions <i>F</i> and <i>F′</i> in the diagram. -Thus we see that each person <i>C</i> is surrounded by six -persons, <i>DD′</i>, <i>EE′</i> and <i>FF′</i>, being his nearest, his next -nearest, and his next next nearest neighbours on his<span class="pagenum"><a name="Page_7" id="Page_7">7</a></span> -own floor. Excluding for simplicity the special cases -of rectangular grouping, we see that the angles of the -six equal and similar triangles <i>CDE</i>, <i>CEF</i>, &c., are all -acute: and because the six -triangles are equal and similar -we see that the three pairs -of mutually remote sides of -the hexagon <i>DEFD′E′F′</i> are -equal and parallel.</p> - -<div class="figright"> -<img src="images/i007.jpg" width="200" height="216" alt="" /> -<div class="caption"><span class="smcap">Fig.</span> 1</div> -</div> - -<p class="section"><a name="s3" id="s3">§ 3.</a> Let now <i>A</i>, <i>A′</i>, <i>A″</i>, &c., -denote places of persons of -the homogeneous assemblage -on the floor immediately -above, and <i>B</i>, <i>B′</i>, <i>B″</i>, &c. on the floor immediately -below, the floor of <i>C</i>. In the diagram let <i>a</i>, <i>a′</i>, <i>a″</i> be -points in which the floor of <i>CDE</i> is cut by perpendiculars -to it through <i>A</i>, <i>A′</i>, <i>A″</i> of the floor above, and -<i>b</i>, <i>b′</i>, <i>b″</i> by perpendiculars from <i>B</i>, <i>B′</i>, <i>B″</i> of the floor -below. Of all the perpendiculars from the floors immediately -above and below, just two, one from each, -cut the area of the parallelogram <i>CDEF</i>: and they -cut it in points similarly situated in respect to the -oppositely oriented triangles into which it is divided -by either of its diagonals. Hence if <i>a</i> lies in the -triangle <i>CDE</i>, the other five triangles of the hexagon -must be cut in the corresponding points, as shown in -the diagram. Thus, if we think only of the floor of -<i>C</i> and of the floor immediately above it, we have points -<i>A</i>, <i>A′</i>, <i>A″</i> vertically above <i>a</i>, <i>a′</i>, <i>a″</i>. Imagine now a -triangular pyramid, or tetrahedron, standing on the -base <i>CDE</i> and having <i>A</i> for vertex: we see that each -of its sides <i>ACD</i>, <i>ADE</i>, <i>AEC</i>, is an acute angled -triangle, because, as we have already seen, <i>CDE</i> is an<span class="pagenum"><a name="Page_8" id="Page_8">8</a></span> -acute angled triangle, and because the shortest of the -three distances, <i>CA</i>, <i>DA</i>, <i>EA</i>, is (§ 2) greater than -<i>CE</i> (though it may be either greater than or less than -<i>DE</i>). Hence the tetrahedron <i>CDEA</i> has all its angles -acute; not only the angles of its triangular faces, but -the six angles between the planes of its four faces. -This important theorem regarding homogeneous assemblages -was given by Bravais, to whom we owe -the whole doctrine of homogeneous assemblages in -its most perfect simplicity and complete generality. -Similarly we see that we have equal and similar tetrahedrons -on the bases <i>D′CF</i>, <i>E′F′C</i>; and three other -tetrahedrons below the floor of <i>C</i>, having the oppositely -oriented triangles <i>CD′E′</i>, &c. for their bases and <i>B</i>, <i>B′</i>, <i>B″</i> -for their vertices. These three tetrahedrons are equal -and heterochirally<a name="FNanchor_1_1" id="FNanchor_1_1"></a><a href="#Footnote_1_1" class="fnanchor">1</a> similar to the first three. The consideration -of these acute angled tetrahedrons, is of -fundamental importance in respect to the engineering -of an elastic solid, or crystal, according to Boscovich. -So also is the consideration of the cluster of thirteen -points <i>C</i> and the six neighbours <i>DEFD′E′F′</i> in the -plane of the diagram, and the three neighbours <i>AA′A″</i> -on the floor above, and <i>BB′B″</i> on the floor below.</p> - -<p class="section">§ 4. The case in which each of the four faces of each -of the tetrahedrons of <a href="#s3">§ 3</a> is an equilateral triangle is -particularly interesting. An assemblage fulfilling this -condition may conveniently be called an ‘equilateral -homogeneous assemblage,’ or, for brevity, an ‘equilateral -assemblage.’ In an equilateral assemblage <i>C</i>’s -twelve neighbours are all equi-distant from it. I hold -in my hand a cluster of thirteen little black balls, made -up by taking one of them and placing the twelve others -<span class="pagenum"><a name="Page_9" id="Page_9">9</a></span> -in contact with it (and therefore packed in the closest -possible order), and fixing them all together by fish-glue. -You see it looks, in size, colour, and shape, -quite like a mulberry. The accompanying diagram -shows a stereoscopic view of a similar cluster of balls -painted white for the photograph.</p> - -<div class="figcenter"> -<img src="images/i009.jpg" width="400" height="238" alt="" /> -<div class="caption"><span class="smcap">Fig. 2.</span></div> -</div> - -<p class="section">§ 5. By adding ball after ball to such a cluster of -thirteen, and always taking care to place each additional -ball in some position in which it is properly in -line with others, so as to make the whole assemblage -homogeneous, we can exercise ourselves in a very -interesting manner in the building up of any possible -form of crystal of the class called ‘cubic’ by some -writers and ‘octahedral’ by others. You see before -you several examples. I advise any of you who wish -to study crystallography to contract with a wood-turner, -or a maker of beads for furniture tassels or for rosaries, -for a thousand wooden balls of about half an inch<span class="pagenum"><a name="Page_10" id="Page_10">10</a></span> -diameter each. Holes through them will do no harm -and may even be useful; but make sure that the balls -are as nearly equal to one another, and each as nearly -spherical, as possible.</p> - -<div class="figcenter"> -<img src="images/i010.jpg" width="300" height="173" alt="" /> -<div class="caption"><span class="smcap">Fig. 3.</span></div> -</div> - -<p class="section">§ 6. You see here before you a large model which -I have made to illustrate a homogeneous assemblage -of points, on a plan first given, I believe, by Mr. William -Barlow (<cite>Nature</cite>, December 20 and 27, 1883). The -roof of the model is a lattice-frame (Fig. 3) consisting -of two sets of eight parallel wooden bars crossing one -another, and kept together by pins through the middles -of the crossings. As you see, I can alter it to make -parallelograms of all degrees of obliquity till the bars -touch, and again you see I can make them all -squares.</p> - -<p class="section">§ 7. The joint pivots are (for cheapness of construction) -of copper wire, each bent to make a hook -below the lattice frame. On these sixty-four hooks -are hung sixty-four fine cords, firmly stretched by -little lead weights. Each of these cords (Fig. 4) bears -eight short perforated wooden cylinders, which may<span class="pagenum"><a name="Page_11" id="Page_11">11</a></span> -be slipped up and down to any desired position<a name="FNanchor_2_2" id="FNanchor_2_2"></a><a href="#Footnote_2_2" class="fnanchor">2</a>. They -are at present actually placed at distances consecutively -each equal to the distance from joint -to joint of the lattice frame.</p> - -<div class="figright"><a name="f4" id="f4"> -<img src="images/i011.jpg" width="125" height="360" alt="" /></a> -<div class="caption"><span class="smcap">Fig. 4.</span></div> -</div> - -<p class="section">§ 8. The roof of the model is hung -by four cords, nearly vertical, of independently -variable lengths, passing -over hooks from fixed points above, -and kept stretched by weights, each -equal to one quarter of the weight of -roof and pendants. You see now -by altering the angles of the lattice -work and placing it horizontal or in -any inclined plane, as I am allowed to -do readily by the manner in which it -is hung, I have three independent -variables, by varying which I can show -you all varieties of homogeneous assemblages, -in which three of the -neighbours of every point are at equal -distances from it. You see here, for -example, we have the equilateral assemblage. I have -adjusted the lattice roof to the proper angle, and -its plane to the proper inclination to the vertical, -to make a wholly equilateral assemblage of the little -cylinders of wood on the vertical cords, a case, as we -have seen, of special importance. If I vary also the -distances between the little pieces of wood on the cords; -and the distances between the joints of the lattice -<span class="pagenum"><a name="Page_12" id="Page_12">12</a></span>work (variations easily understood, though not conveniently -producible in one model without more of -mechanical construction than would be worth making), -I have three other independent variables. By properly -varying these six independent variables, three angles -and three lengths, we may give any assigned value to -each edge of one of the fundamental tetrahedrons of <a href="#s3">§ 3.</a></p> - -<p class="section">§ 9. Our assemblage of people would not be homogeneous -unless its members were all equal and similar -and in precisely similar attitudes, and were all looking -the same way. You understand what a number of people -seated or standing on a floor or plain and looking the -same way means. But the expression ‘looking’ is not -conveniently applicable to things that have no eyes, and -we want a more comprehensive mode of expression. -We have it in the words ‘orientation,’ ‘oriented,’ and -(verb) ‘to orient,’ suggested by an extension of the idea -involved in the word ‘orientation,’ first used to signify -positions relatively to east and west of ancient Greek -and Egyptian temples and Christian churches. But -for the orientation of a house or temple we have only -one angle, and that angle is called ‘azimuth’ (the name -given to an angle in a horizontal plane). For orientation -in three dimensions of space we must extend our -ideas and consider position with reference to east and -west and up and down. A man lying on his side with -his head to the north and looking east, would not -be similarly oriented to a man standing upright and -looking east. To provide for the complete specification -of how a body is oriented in space we must have in the -body a plane of reference, and a line of reference in this -plane, belonging to the body and moving with it. We -must also have a fixed plane and a fixed line of reference<span class="pagenum"><a name="Page_13" id="Page_13">13</a></span> -in it, relatively to which the orientation of the moveable -body is to be specified; as, for example, a horizontal plane -and the east and west horizontal line in it. The position -of a body is completely specified when the angle between -the plane of reference belonging to it, and the fixed -plane is given; and when the angles between the line -of intersection of the two planes and the lines of reference -in them are also given. Thus we see that three -angles are necessary and sufficient to specify the -orientation of a moveable body, and we see how the -specification is conveniently given in terms of three -angles.</p> - -<p class="section">§ 10. To illustrate this take a book lying on the table -before you with its side next the title-page up, and its -back to the north. I now lift the east edge (the top of -the book), keeping the bottom edge north and south on -the table till the book is inclined, let us say, 20° to the -table. Next, without altering this angle of 20°, between -the side of the book and the table, I turn the book -round a vertical axis, through 45° till the bottom edge -lies north-east and south-west. Lastly, keeping the -book in the plane to which it has been thus brought, I -turn it round in this plane through 35°. These three -angles of 20°, 45°, and 35°, specify, with reference to the -horizontal plane of the table and the east and west line -in it, the orientation of the book in the position to -which you have seen me bring it, and in which I hold -it before you.</p> - -<p class="section">§ 11. In Figs. 5 and 6 you see two assemblages, each -of twelve equal and similar molecules in a plane. -Fig. 5, in which the molecules are all same-ways -oriented, is one homogeneous assemblage of twenty-four -molecules. <a href="#f6">Fig. 6</a>, in which in one set of rows the<span class="pagenum"><a name="Page_14" id="Page_14">14</a></span> -molecules are alternately oriented two different -ways, may either be regarded as two homogeneous -assemblages, each of twelve single molecules; or one -homogeneous assemblage of twelve pairs of those -single molecules.</p> - -<div class="figcenter"> -<img src="images/i014.jpg" width="400" height="358" alt="" /> -<div class="caption"><span class="smcap">Fig. 5.</span></div> -</div> - -<p class="section"><a name="s12" id="s12">§ 12.</a> I must now call your attention to a purely -geometrical question<a name="FNanchor_3_3" id="FNanchor_3_3"></a><a href="#Footnote_3_3" class="fnanchor">3</a> of vital interest with respect to -homogeneous assemblages in general, and particularly -the homogeneous assemblage of molecules constituting -a crystal:—<em>what can we take as ‘the’ boundary or -‘a’ boundary enclosing each molecule with whatever -portion of space around it we are at liberty to choose for</em> -<span class="pagenum"><a name="Page_15" id="Page_15">15</a></span><em>it, and separating it from neighbours and their portions -of space given to them in homogeneous fairness?</em></p> - -<div class="figcenter"><a name="f6" id="f6"> -<img src="images/i015.jpg" width="400" height="366" alt="" /></a> -<div class="caption"><span class="smcap">Fig. 6.</span></div> -</div> - -<p class="section"><a name="s13" id="s13">§ 13.</a> If we had only mathematical points to consider -we should be at liberty to choose the simple obvious -partitioning by three sets of parallel planes. Even this -may be done in an infinite number of ways, thus:—Beginning -with any point <i>P</i> of the assemblage, choose -any other three points <i>A</i>, <i>B</i>, <i>C</i>, far or near, provided -only that they are not in one plane with <i>P</i>, and that -there is no other point of the assemblage in the lines -<i>PA</i>, <i>PB</i>, <i>PC</i>, or within the volume of the parallelepiped -of which these lines are conterminous edges, or within -the areas of any of the faces of this parallelepiped. -There will be points of the assemblage at each of the<span class="pagenum"><a name="Page_16" id="Page_16">16</a></span> -corners of this parallelepiped and at all the corners of the -parallelepipeds equal and similar to it which we find -by drawing sets of equi-distant planes parallel to its -three pairs of faces. (A diagram is unnecessary.) Every -point of the assemblage is thus at the intersection of -three planes, which is also the point of meeting of eight -neighbouring parallelepipeds. Shift now any one of -the points of the assemblage to a position within the -volume of any one of the eight parallelepipeds, and -give equal parallel motions to all the other points of -the assemblage. Thus we have every point in a -parallelepipedal cell of its own, and all the points of -the assemblage are similarly placed in their cells, which -are themselves equal and similar.</p> - -<p class="section">§ 14. But now if, instead of a single point for each -member of the assemblage, we have a group of points, -or a globe or cube or other geometrical figure, or an -individual of a homogeneous assemblage of equal, -similar, similarly dressed, and similarly oriented ladies, -sitting in rows, or a homogeneous assemblage of trees -closely planted in regular geometrical order on a plane -with equal and similar distributions of molecules, and -parallel planes above and below, we may find that -the best conditioned plane-faced parallelepipedal partitioning -which we can choose would cut off portions -properly belonging to one molecule of the assemblage -and give them to the cells of neighbours. To find -a cell enclosing all that belongs to each individual, for -example, every part of each lady’s dress, however -complexly it may be folded among portions of the -equal and similar dresses of neighbours; or, every -twig, leaf, and rootlet of each one of the homogeneous -assemblage of trees; we must alter the boundary by<span class="pagenum"><a name="Page_17" id="Page_17">17</a></span> -give-and-take across the plane faces of the primitive -parallelepipedal cells, so that each cell shall enclose all -that belongs to one molecule, and therefore (because -of the homogeneousness of the partitioning) nothing -belonging to any other molecule. The geometrical -problem thus presented, wonderfully complex as it -may be in cases such as some of those which I have -suggested, is easily performed for any possible case -if we begin with any particular parallelepipedal partitioning -determined for corresponding points of the -assemblage as explained in <a href="#s13">§ 13</a>, for any homogeneous -assemblage of single points. We may prescribe to -ourselves that the corners are to remain unchanged, -but if so they must to begin with either in interfaces -of contact between the individual molecules, or in -vacant space among the molecules. If this condition -is fulfilled for one corner it is fulfilled for all, as the -corners are essentially corresponding points relatively -to the assemblage.</p> - -<p class="section">§ 15. Begin now with any one of the twelve straight -lines between corners which constitute the twelve -edges of the parallelepiped, and alter it arbitrarily -to any curved or crooked line between the same pair -of corners, subject only to the conditions (1) that it does -not penetrate the substance of any member of the -assemblage, and (2) that it is not cut by equal and -similar parallel curves<a name="FNanchor_4_4" id="FNanchor_4_4"></a><a href="#Footnote_4_4" class="fnanchor">4</a> between other pairs of corners.</p> - -<div class="figcenter"><a name="f7" id="f7"> -<img src="images/i018.jpg" width="200" height="265" alt="" /></a> -<div class="caption"><span class="smcap">Fig. 7.</span></div> -</div> - -<p>Considering now the three fours of parallel edges -of the parallelepiped, let the straight lines of one set of -four be altered to equal and similar parallel curves in -the manner which I have described; and proceed by -<span class="pagenum"><a name="Page_18" id="Page_18">18</a></span>the same rule for the other two sets of four edges. -We thus have three fours of parallel curved edges -instead of the three fours of parallel straight edges -of our primitive parallelepiped with corners (each -a point of intersection of three edges) unchanged. -Take now the quadrilateral of four curves substituted -for the four straight edges of one face of the parallelepiped. -We may call this quadrilateral a curvilineal -parallelogram, because it is a circuit composed of two -pairs of equal parallel curves. Draw now a curved -surface (an infinitely thin sheet of perfectly extensible -india-rubber if you please to think of it so) bordered by -the four edges of our curvilineal parallelogram, and so -shaped as not to cut any of the substance of any -molecule of the assemblage. Do the same thing with -an exactly similar and parallel sheet relatively to the -opposite face of the parallelepiped; and again the same<span class="pagenum"><a name="Page_19" id="Page_19">19</a></span> -for each of the two other pairs of parallel faces. We -thus have a curved-faced parallelepiped enclosing the -whole of one molecule and no part of any other; and -by similar procedure we find a similar boundary for every -other molecule of the assemblage. Each wall of each of -these cells is common to two neighbouring molecules, -and there is no vacant space anywhere between them -or at corners. <a href="#f7">Fig. 7</a> illustrates this kind of partitioning -by showing a plane section parallel to one pair of plane -faces of the primitive parallelepiped, for an ideal case. -The plane diagram is in fact a realization of the two-dimensional -problem of partitioning the pine pattern -of a Persian carpet by parallelograms about as nearly -rectilinear as we can make them. In the diagram faint -straight lines are drawn to show the primitive parallelogrammatic -partitioning. It will be seen that of all -the crossings (marked with dots in the diagram) every -one is similarly situated to every other in respect to the -homogeneously repeated pattern figures: <i>A</i>, <i>B</i>, <i>C</i>, <i>D</i> are -four of them at the corners of one cell.</p> - -<p class="section"><a name="s16" id="s16">§ 16.</a> Confining our attention for a short time to -the homogeneous division of a plane, remark that the -division into parallelograms by two sets of crossing -parallels is singular in this respect—each cell is contiguous -with three neighbours at every corner. Any -shifting, large or small, of the parallelograms by relative -sliding in one direction or another violates this condition, -brings us to a configuration like that of the -faces of regularly hewn stones in ordinary bonded -masonry, and gives a partitioning which fulfils the -condition that at each corner each cell has only two -neighbours. Each cell is now virtually a hexagon, as -will be seen by the letters <i>A</i>, <i>B</i>, <i>C</i>, <i>D</i>, <i>E</i>, <i>F</i> in the diagram<span class="pagenum"><a name="Page_20" id="Page_20">20</a></span> -Fig. 8. <i>A</i> and <i>D</i> are to be reckoned as corners, each -with an interior angle of 180°. In this diagram the -continuous heavy lines and the continuous faint lines -crossing them show a primitive parallelogrammatic partition -by two sets of continuous parallel intersecting -lines. The interrupted crossing lines (heavy) show, -for the same homogeneous distribution of single points -or molecules, the virtually hexagonal partitioning which -we get by shifting the boundary from each portion of -one of the light lines to the heavy line next it between -the same continuous parallels.</p> - -<div class="figcenter "> -<img src="images/i020.jpg" width="400" height="210" alt="" /> -<div class="caption"><span class="smcap">Fig. 8.</span></div> -</div> - -<div class="figcenter padt1"> -<img src="images/i021.jpg" width="400" height="210" alt="" /> -<div class="caption"><span class="smcap">Fig. 8</span> bis.</div> -</div> - -<p>Fig. 8 bis represents a further modification of the -boundary by which the 180° angles <i>A</i>, <i>D</i>, become -angles of less than 180°. The continuous parallel lines -(light) and the short light portions of the crossing lines -show the configuration according to Fig. 8, from which -this diagram is derived.</p> - -<p class="section">§ 17. In these diagrams (Figs. 8 and 8 bis) the object -enclosed is small enough to be enclosable by a primitive -parallelogrammatic partitioning of two sets of continuous -crossing parallel straight lines, and by the partitioning<span class="pagenum"><a name="Page_21" id="Page_21">21</a></span> -of ‘bonded’ parallelograms both represented in Fig. 8, -and by the derived hexagonal partitioning represented -in Fig. 8 bis, with faint lines showing the primitive and -the secondary parallelograms. In <a href="#f7">Fig. 7</a> the objects enclosed -were too large to be enclosable by any rectilinear -parallelogrammatic or hexagonal partitioning. The two -sets of parallel faint lines in <a href="#f7">Fig. 7</a> show a primitive -parallelogrammatic partitioning and the corresponding -pairs of parallel curves intersecting at the corners of -these parallelograms, of which <i>A</i>,<i>B</i>,<i>C</i>,<i>D</i> is a specimen, -show a corresponding partitioning by curvilineal parallelograms. -Fig. 9 shows for the same homogeneous -distribution of objects a better conditioned partitioning, -by hexagons in each of which one pair of parallel edges -is curved. The sets of intersecting parallel straight -lines in Fig. 9 show the same primitive parallelogrammatic -partitioning as in <a href="#f7">Fig. 7</a>, and the same slightly -shifted to suit points chosen for well-conditionedness of -hexagonal partitioning.</p> - -<div class="figcenter"> -<img src="images/i022.jpg" width="200" height="245" alt="" /> -<div class="caption"><span class="smcap">Fig. 9.</span></div> -</div> - -<p class="section">§ 18. For the division of continuous three-dimensional -space<a name="FNanchor_5_5" id="FNanchor_5_5"></a><a href="#Footnote_5_5" class="fnanchor">5</a> into equal, similar, and similarly oriented cells, -<span class="pagenum"><a name="Page_22" id="Page_22">22</a></span>quite a corresponding transformation from partitioning -by three sets of continuous mutually intersecting parallel -planes to any possible mode of homogeneous partitioning, -may be investigated by working out the three-dimensional -analogue of §§ <a href="#s16">16</a>-17. Thus we find that the -most general possible homogeneous partitioning of space -with plane interfaces between the cells gives us fourteen -walls to each cell, of which six are three pairs of equal -and parallel parallelograms, and the other eight are four -pairs of equal and parallel hexagons, each hexagon -being bounded by three pairs of equal and parallel -straight lines. This figure, being bounded by fourteen -plane faces, is called a tetrakaidekahedron. It has thirty-six -edges of intersection between faces; and twenty-four -corners, in each of which three faces intersect. A particular -case of it, which I call an orthic tetrakaidekahedron, -being that in which the six parallelograms are<span class="pagenum"><a name="Page_23" id="Page_23">23</a></span> -equal squares, the eight hexagonal faces are equal -equilateral and equiangular hexagons, and the lines -joining corresponding points in the seven pairs of -parallel faces are perpendicular to the planes of the -faces, is represented by a stereoscopic picture in <a href="#f10">Fig. 10</a>. -The thirty-six edges and the twenty-four corners, which -are easily counted in this diagram, occur in the same -relative order in the most general possible partitioning, -whether by plane-faced tetrakaidekahedrons or by the -generalized tetrakaidekahedron described in § 19.</p> - -<p class="section">§ 19. The most general homogeneous division of -space is not limited to plane-faced cells; but it still -consists essentially of tetrakaidekahedronal cells, each -bounded by three pairs of equal and parallel quadrilateral -faces, and four pairs of equal and parallel -hexagonal faces, neither the quadrilaterals nor the -hexagons being necessarily plane. Each of the thirty-six -edges may be straight or crooked or curved; the -pairs of opposite edges, whether of the quadrilaterals -or hexagons, need not be equal and parallel; neither -the four corners of each quadrilateral nor the six -corners of each hexagon need be in one plane. But -every pair of corresponding edges of every pair of -parallel corresponding faces, whether quadrilateral or -hexagonal, must be equal and parallel. I have described -an interesting case of partitioning by tetrakaidekahedrons -of curved faces with curved edges in a -paper<a name="FNanchor_6_6" id="FNanchor_6_6"></a><a href="#Footnote_6_6" class="fnanchor">6</a> published about seven years ago. In this case -each of the quadrilateral faces is plane. Each hexagonal -face is a slightly curved surface having three -rectilineal diagonals through its centre in one plane.</p> -<p><span class="pagenum"><a name="Page_24" id="Page_24">24</a></span></p> - -<div class="figcenter"><a name="f10" id="f10"> -<img src="images/i024.jpg" width="500" height="261" alt="" /></a> -<div class="caption"><span class="smcap">Fig. 10.</span></div> -</div> - -<p><span class="pagenum"><a name="Page_25" id="Page_25">25</a></span></p> - -<p>The six sectors of the face between these diagonals -lie alternately on opposite sides of their plane, and are -bordered by six arcs of plane curves lying on three -pairs of parallel planes. This tetrakaidekahedronal -partitioning fulfils the condition that the angles between -three planes meeting in an edge are everywhere each -120°; a condition that cannot be fulfilled in any plane-faced -tetrakaidekahedron. Each hexagonal wall is an -anticlastic surface of equal opposite curvatures at every -point, being the surfaces of minimum area bordered -by six curved edges. It is shown easily and beautifully, -and with a fair approach to accuracy, by choosing -six little circular arcs of wire, and soldering them -together by their ends in proper planes for the six -edges of the hexagon; and dipping it in soap solution -and taking it out.</p> - -<p class="section"><a name="s20" id="s20">§ 20.</a> Returning now to the tactics of a homogeneous -assemblage, remark that the qualities of the assemblage -as a whole depend both upon the character and orientation -of each molecule, and on the character of the -homogeneous assemblage formed by corresponding -points of the molecules. After learning the simple -mathematics of crystallography, with its indicial system<a name="FNanchor_7_7" id="FNanchor_7_7"></a><a href="#Footnote_7_7" class="fnanchor">7</a> -for defining the faces and edges of a crystal according -to the Bravais rows and nets and tetrahedrons of -molecules in which we think only of a homogeneous -assemblage of points, we are apt to forget that the true -crystalline molecule, whatever its nature may be, has -sides, and that generally two opposite sides of each -molecule may be expected to be very different in -quality, and we are almost surprised when mineralogists -<span class="pagenum"><a name="Page_26" id="Page_26">26</a></span>tell us that two parallel faces on two sides of a crystal -have very different qualities in many natural crystals. -We might almost as well be surprised to find that an -army in battle array, which is a kind of large-grained -crystal, presents very different appearance to any one -looking at it from outside, according as every man in -the ranks with his rifle and bayonet faces to the front -or to the rear or to one flank or to the other.</p> - -<p class="section"><a name="s21" id="s21">§ 21.</a> Consider, for example, the ideal case of a crystal -consisting of hard equal and similar tetrahedronal solids -all sameways oriented. A thin plate of crystal cut -parallel to any one set of the faces of the constituent -tetrahedrons would have very different properties on -its two sides; as the constituent molecules would all -present points outwards on one side and flat surfaces -on the other. We might expect that the two sides of -such a plate of crystal would become oppositely electrified -when rubbed by one and the same rubber; and, -remembering that a piece of glass with part of its -surface finely ground but not polished and other parts -polished becomes, when rubbed with white silk, positively -electrified over the polished parts and negatively -electrified over the non-polished parts, we might almost -expect that the side of our supposed crystalline plate -towards which flat faces of the constituent molecules -are turned would become positively electrified, and -the opposite side, showing free molecular corners, -would become negatively electrified, when both are -rubbed by a rubber of intermediate electric quality. -We might also from elementary knowledge of the fact -of piezo-electricity, that is to say, the development of -opposite electricities on the two sides of a crystal by -pressure, expect that our supposed crystalline plate, if<span class="pagenum"><a name="Page_27" id="Page_27">27</a></span> -pressed perpendicularly on its two sides, would become -positively electrified on one of them and negatively on -the other.</p> - -<p class="section"><a name="s22" id="s22">§ 22.</a> Intimately connected with the subject of enclosing -cells for molecules of given shape, assembled -homogeneously, is the homogeneous packing together -of equal and similar molecules of any given shape. -In every possible case of any infinitely great number -of similar bodies the solution is a homogeneous assemblage. -But it may be a homogeneous assemblage -of single solids all oriented the same way, or it -may be a homogeneous assemblage of clusters of -two or more of them placed together in different -orientations. For example, let the given bodies be -halves (oblique or not oblique) of any parallelepiped -on the two sides of a dividing plane through a pair -of parallel edges. The two halves are homochirally<a name="FNanchor_8_8" id="FNanchor_8_8"></a><a href="#Footnote_8_8" class="fnanchor">8</a> -similar; and, being equal, we may make a homogeneous -assemblage of them by orienting them all the same -way and placing them properly in rows. But the -closest packing of this assemblage would necessarily -leave vacant spaces between the bodies: and we get -in reality the closest possible packing of the given -bodies by taking them in pairs oppositely oriented -and placed together to form parallelepipeds. These -clusters may be packed together so as to leave no -unoccupied space.</p> -<p><span class="pagenum"><a name="Page_28" id="Page_28">28</a></span></p> -<p>Whatever the number of pieces in a cluster in the -closest possible packing of solids may be for any -particular shape, we may consider each cluster as -itself a given single body, and thus reduce the problem -to the packing closely together of assemblages -of individuals all sameways oriented; and to this problem -therefore it is convenient that we should now -confine our attention.</p> - -<div class="figleft100"> -<img src="images/i028.jpg" width="100" height="252" alt="" /> -<div class="caption">Fig. 11.</div> -</div> - -<p class="section"><a name="s23" id="s23">§ 23.</a> To avoid complexities such as those which we -find in the familiar problem of homogeneous -packing of forks or spoons or tea-cups or -bowls, of any ordinary shape, we shall -suppose the given body to be of such -shape that no two of them similarly oriented -can touch one another in more than one -point. Wholly convex bodies essentially -fulfil this condition; but it may also be -fulfilled by bodies not wholly convex, as is -illustrated in Fig. 11.</p> - -<p class="section"><a name="s24" id="s24">§ 24.</a> To find close and closest packing -of any number of our solids <i>S<sub>1</sub></i>, <i>S<sub>2</sub></i>, <i>S<sub>3</sub></i> ... of -shape fulfilling the condition of <a href="#s23">§ 23</a> -proceed thus:—</p> - -<p>(1) Bring <i>S<sub>2</sub></i> to touch <i>S<sub>1</sub></i> at any chosen point <i>p</i> of its -surface (Fig. 12).</p> - -<p>(2) Bring <i>S<sub>3</sub></i> to touch <i>S<sub>1</sub></i> and <i>S<sub>2</sub></i>, at <i>r</i> and <i>q</i> respectively.</p> - -<p>(3) Bring <i>S<sub>4</sub></i> (not shown in the diagram) to touch <i>S<sub>1</sub></i>, -<i>S<sub>2</sub></i>, and <i>S<sub>3</sub></i>.</p> - -<p>(4) Place, any number of the bodies together in three -rows continuing the lines of <i>S<sub>1</sub>S<sub>2</sub></i>, <i>S<sub>1</sub>S<sub>3</sub></i>, <i>S<sub>1</sub>S<sub>4</sub></i>, and in -three sets of equi-distant rows parallel to these. This -makes a homogeneous assemblage. In the assemblage<span class="pagenum"><a name="Page_29" id="Page_29">29</a></span> -so formed the molecules are necessarily found to be in -three sets of rows parallel respectively to the three pairs -<i>S<sub>2</sub>S<sub>3</sub></i>, <i>S<sub>3</sub>S<sub>4</sub></i>, <i>S<sub>4</sub>S<sub>2</sub></i>. The whole space occupied by an -assemblage of <i>n</i> of our solids thus arranged has clearly -<i>6n</i> times the volume of a tetrahedron of corresponding -points of <i>S<sub>1</sub></i>, <i>S<sub>2</sub></i>, <i>S<sub>3</sub></i>, <i>S<sub>4</sub></i>. Hence the closest of the close -packings obtained by the operations (1) ... (4) is found if -we perform the operations (1), (2), and (3) as to make -the volume of this tetrahedron least possible.</p> - -<div class="figcenter"><a name="f12" id="f12"> -<img src="images/i029.jpg" width="200" height="150" alt="" /></a> -<div class="caption"><span class="smcap">Fig. 12</span></div> -</div> - -<p class="section"><a name="s25" id="s25">§ 25.</a> It is to be remarked that operations (1) and -(2) leave for (3) no liberty of choice for the place of -<i>S<sub>4</sub></i>, except between two determinate positions on opposite -sides of the group <i>S<sub>1</sub></i>, <i>S<sub>2</sub></i>, <i>S<sub>3</sub></i>. The volume of -the tetrahedron will generally be different for these -two positions of <i>S<sub>4</sub></i>, and, even if the volume chance -to be equal in any case, we have differently shaped -assemblages according as we choose one or other of -the two places for <i>S<sub>4</sub></i>.</p> - -<p>This will be understood by looking at Fig. 12, -showing <i>S<sub>1</sub></i> and neighbours on each side of it in the -rows of <i>S<sub>1</sub>S<sub>2</sub></i>, <i>S<sub>1</sub>S<sub>3</sub></i>, and in a row parallel to that of -<i>S<sub>2</sub>S<sub>3</sub></i>. The plane of the diagram is parallel to the planes -of corresponding points of these seven bodies, and the<span class="pagenum"><a name="Page_30" id="Page_30">30</a></span> -diagram is a projection of these bodies by lines parallel -to the intersections of the tangent planes through -<i>p</i> and <i>r</i>. If the three tangent planes through <i>p</i>, <i>q</i>, -and <i>r</i>, intersected in parallel lines, <i>q</i> would be seen like -<i>p</i> and <i>r</i> as a point of contact between the outlines of -two of the bodies; but this is only a particular case, -and in general <i>q</i> must, as indicated in the diagram, be -concealed by one or other of the two bodies of which -it is the point of contact. Now imagining, to fix our -ideas and facilitate brevity of expression, that the -planes of corresponding points of the seven bodies are -horizontal, we see clearly that <i>S<sub>4</sub></i> may be brought into -proper position to touch <i>S<sub>1</sub></i>, <i>S<sub>2</sub></i>, and <i>S<sub>3</sub></i> either from above -or from below; and that there is one determinate place -for it if we bring it into position from above, and -another determinate place for it if we bring it from -below.</p> - -<p class="section">§ 26. If we look from above at the solids of which -Fig. 12 shows the outline, we see essentially a hollow -leading down to a perforation between <i>S<sub>1</sub></i>, <i>S<sub>2</sub></i>, <i>S<sub>3</sub></i>, and -if we look from below we see a hollow leading upwards -to the same perforation: this for brevity we shall call -the perforation <i>pqr</i>. The diagram shows around <i>S<sub>1</sub></i> -six hollows leading down to perforations, of which two -are similar to <i>pqr</i>, and the other three, of which <i>p′q′r′</i> -indicates one, are similar one to another but are dissimilar -to <i>pqr</i>. If we bring <i>S<sub>4</sub></i> from above into position -to touch <i>S<sub>1</sub></i>, <i>S<sub>2</sub></i>, and <i>S<sub>3</sub></i>, its place thus found is in the -hollow <i>pqr</i>, and the places of all the solids in the layer -above that of the diagram are necessarily in the hollows -similar to <i>pqr</i>. In this case the solids in the layer -below that of the diagram must lie in the hollows -below the perforations dissimilar to <i>pqr</i>, in order to<span class="pagenum"><a name="Page_31" id="Page_31">31</a></span> -make a single homogeneous assemblage. In the other -case, <i>S<sub>4</sub></i> brought up from below finds its place on the -under side of the hollow <i>pqr</i>, and all solids of the -lower layer find similar places: while solids in the -layer above that of the diagram find their places in the -hollows similar to <i>p′q′r′</i>. In the first case there are -no bodies of the upper layer in the hollows above the -perforations <em>similar</em> to <i>p′q′r′</i>, and no bodies of the -lower layer in the hollows below the perforations -<em>similar</em> to <i>pqr</i>. In the second case there are no bodies -of the upper layer in the hollows above the perforations -<em>similar</em> to <i>pqr</i>, and none of the under layer in the -hollows below the perforations <em>similar</em> to <i>p′q′r′</i>.</p> - -<p class="section">§ 27. Going back now to operation (1) of <a href="#s23">§ 23</a>, remark -that when the point of contact <i>p</i> is arbitrarily chosen on -one of the two bodies <i>S<sub>1</sub></i>, the point of contact on the -other will be the point on it corresponding to the -point or one of the points of <i>S<sub>1</sub></i>, where its tangent -plane is parallel to the tangent plane at <i>p</i>. If <i>S<sub>1</sub></i> is -wholly convex it has only two points at which the -tangent planes are parallel to a given plane, and -therefore the operation (1) is determinate and unambiguous. -But if there is any concavity there will be -four or some greater even number of tangent planes -parallel to any one of some planes, while there will -be other planes to each of which only one pair of -tangent planes is parallel. Hence, operation (1), though -still determinate, will have a multiplicity of solutions, -or only a single solution, according to the choice made -of the position of <i>p</i>.</p> - -<p>Henceforth however, to avoid needless complications -of ideas, we shall suppose our solids to be wholly -convex; and of some such unsymmetrical shape as those<span class="pagenum"><a name="Page_32" id="Page_32">32</a></span> -indicated in <a href="#f12">Fig. 12</a> of § 25, and shown by stereoscopic -photograph in <a href="#f13">Fig. 13</a> of § 36. With or without this -convenient limitation, operation (1) has two freedoms, -as <i>p</i> may be chosen freely on the surface of <i>S<sub>1</sub></i>; and -operation (2) has clearly just one freedom after operation -(1) has been performed. Thus, for a solid of any given -shape, we have three disposables, or, as commonly -called in mathematics, three ‘independent variables,’ all -free for making a homogeneous assemblage according -to the rule of <a href="#s22">§ 22.</a></p> - -<p class="section">§ 28. In the homogeneous assemblage defined in -§ 24, each solid, <i>S<sub>1</sub></i>, is touched at twelve points, being -the three points of contact with <i>S<sub>2</sub></i>, <i>S<sub>3</sub></i>, <i>S<sub>4</sub></i>, and the -three 3’s of points on <i>S<sub>1</sub></i> corresponding to the points -on <i>S<sub>2</sub></i>, <i>S<sub>3</sub></i>, <i>S<sub>4</sub></i>, at which these bodies are touched by -the others of the quartet. This statement is somewhat -difficult to follow, and we see more clearly the -twelve points of contact by not confining our attention -to the quartet <i>S<sub>1</sub></i>, <i>S<sub>2</sub></i>, <i>S<sub>3</sub></i>, <i>S<sub>4</sub></i> (convenient as this is for -some purposes), but completing the assemblage and -considering six neighbours around <i>S<sub>1</sub></i> in one plane -layer of the solids as shown in <a href="#f12">Fig. 12</a>, with their -six points <i>prq″p′r′q″′</i> of contact with <i>S<sub>1</sub></i>; and the three -neighbours of the two adjacent parallel layers which -touch it above and below. This cluster of thirteen, -<i>S<sub>1</sub></i> and twelve neighbours, is shown for the case of -spherical bodies in the stereoscopic photograph of -§ 4 above. We might of course, if we pleased, have -begun with the plane layer of which <i>S<sub>1</sub></i>, <i>S<sub>2</sub></i>, <i>S<sub>4</sub></i> are -members, or with that of which <i>S<sub>1</sub></i>, <i>S<sub>3</sub></i>, <i>S<sub>4</sub></i> are members, -or with the plane layer parallel to the fourth -side <i>S<sub>2</sub></i> <i>S<sub>3</sub></i> <i>S<sub>4</sub></i> of the tetrahedron: and thus we have -four different ways of grouping the twelve points of<span class="pagenum"><a name="Page_33" id="Page_33">33</a></span> -contact on <i>S<sub>1</sub></i> into one set of six and two sets of -three.</p> - -<p class="section">§ 29. In this assemblage we have what I call ‘close -order’ or ‘close packing.’ For closest of close packings -the volume of the tetrahedron (§ 24) of corresponding -points of <i>S<sub>1</sub></i>, <i>S<sub>2</sub></i>, <i>S<sub>3</sub></i>, and <i>S<sub>4</sub></i> must be a minimum, -and the least of minimums if, as generally will be the -case, there are two more different configurations for -each of which the volume is a minimum. There will -in general also be configurations of minimax volume -and of maximum volume, subject to the condition that -each body is touched by twelve similarly oriented -neighbours.</p> - -<p class="section"><a name="s30" id="s30">§ 30.</a> Pause for a moment to consider the interesting -kinematical and dynamical problems presented by a -close homogeneous assemblage of smooth solid bodies -of given convex shape, whether perfectly frictionless -or exerting resistance against mutual sliding according -to the ordinarily stated law of friction between dry -hard solid bodies. First imagine that they are all -similarly oriented and each in contact with twelve -neighbours, except outlying individuals (which there -must be at the boundary if the assemblage is finite, -and each of which is touched by some number of -neighbours less than twelve). The coherent assemblage -thus defined constitutes a kinematic frame or -skeleton for an elastic solid of very peculiar properties. -Instead of the six freedoms, or disposables, of strain -presented by a natural solid it has only three. Change -of shape of the whole can only take place in virtue of -rotation of the constituent parts relatively to any one -chosen row of them, and the plane through it and -another chosen row.</p> - -<p><span class="pagenum"><a name="Page_34" id="Page_34">34</a></span></p> - -<p class="section"><a name="s31" id="s31">§ 31.</a> Suppose first the solids to be not only perfectly -smooth but perfectly frictionless. Let the assemblage -be subjected to equal positive or negative pressure -inwards all around its boundary. Every position of -minimum, minimax, or maximum volume will be a -position of equilibrium. If the pressure is positive the -equilibrium will be stable if, and unstable unless, the -volume is a minimum. If the pressure is negative -the equilibrium will be stable if, and unstable unless, -the volume is a maximum. Configurations of minimax -volume will be essentially unstable.</p> - -<p class="section">§ 32. Consider now the assemblage of <a href="#s31">§ 31</a> in a -position of stable equilibrium under the influence of -a given constant uniform pressure inwards all round -its boundary. It will have rigidity in simple proportion -to the amount of this pressure. If now by the superposition -of non-uniform pressure at the boundary, for -example equal and opposite pressures on two sides -of the assemblage, a finite change of shape is produced: -the whole assemblage essentially swells in -bulk. This is the ‘dilatancy’ which Osborne Reynolds -has described<a name="FNanchor_9_9" id="FNanchor_9_9"></a><a href="#Footnote_9_9" class="fnanchor">9</a> in an exceedingly interesting manner -with reference to a sack of wheat or sand, or an india-rubber -bag tightly filled with sand or even small shot. -Consider, for example, a sack of wheat filled quite -full and standing up open. It is limp and flexible. -Now shake it down well, fill it quite full, shake again, -so as to get as much into it as possible, and tie the -mouth very tightly close. The sack becomes almost -as stiff as a log of wood of the same shape. Open the -mouth partially, and it becomes again limp, especially -<span class="pagenum"><a name="Page_35" id="Page_35">35</a></span>in the upper parts of the bag. In Reynolds’ observations -on india-rubber bags of small shot his ‘dilatancy’ -depends, essentially and wholly, on breaches of -some of the contacts which exist between the molecules -in their configuration of minimum volume: and it is -possible that in all his cases the dilatations which he -observed are <em>chiefly</em>, if not wholly, due to such breaches -of contact.</p> - -<p>But it is possible, it almost seems probable, that in -bags or boxes of sand or powder, of some kinds of -smooth rounded bodies of any shape, not spherical -or ellipsoidal, subjected persistently to unequal pressures -in different directions, and well shaken, stable -positions of equilibrium are found with almost all the -particles each touched by twelve others.</p> - -<p>Here is a curious subject of Natural History -through all ages till 1885, when Reynolds brought it -into the province of Natural Philosophy by the following -highly interesting statement:—‘A well-marked -phenomenon receives its explanation at once from -the existence of dilatancy in sand. When the falling -tide leaves the sand firm, as the foot falls on it -the sand whitens and appears momentarily to dry -round the foot. When this happens the sand is full -of water, the surface of which is kept up to that of -the sand by capillary attractions; the pressure of the -foot causing dilatation of the sand more water is required, -which has to be obtained either by depressing -the level of the surface against the capillary attractions, -or by drawing water through the interstices of the -surrounding sand. This latter requires time to accomplish, -so that for the moment the capillary forces are -overcome; the surface of the water is lowered below<span class="pagenum"><a name="Page_36" id="Page_36">36</a></span> -that of the sand, leaving the latter white or drier until -a sufficient supply has been obtained from below, when -the surface rises and wets the sand again. On raising -the foot it is generally seen that the sand under the foot -and around becomes momentarily wet; this is because, -on the distorting forces being removed, the sand again -contracts, and the excess of water finds momentary -relief at the surface.’</p> - -<p>This proves that the sand under the foot, as well -as the surface around it, must be dry for a short time -after the foot is pressed upon it, though we cannot see -it whitened, as the foot is not transparent. That it is -so has been verified by Mr. Alex. Galt, Experimental -Instructor in the Physical Laboratory of Glasgow -University, by laying a small square of plate-glass on -wet sand on the sea-shore of Helensburgh, and suddenly -pressing on it by a stout stick with nearly all -his weight. He found the sand, both under the glass -and around it in contact with the air, all became white -at the same moment. Of all the two hundred thousand -million men, women, and children who, from the beginning -of the world, have ever walked on wet sand, -how many, prior to the British Association Meeting at -Aberdeen in 1885, if asked, ‘Is the sand compressed -under your foot?’ would have answered otherwise -than ‘Yes!’?</p> - -<p>(Contrast with this the case of walking over a bed -of wet sea-weed!)</p> - -<p class="section">§ 33. In the case of globes packed together in closest -order (and therefore also in the case of ellipsoids, if all -similarly oriented), our condition of coherent contact -between each molecule and twelve neighbours implies -absolute rigidity of form and constancy of bulk.<span class="pagenum"><a name="Page_37" id="Page_37">37</a></span> -Hence our convex solid must be neither ellipsoidal -nor spherical in order that there may be the changes -of form and changes of bulk which we have been -considering as dependent on three independent variables -specifying the orientation of each solid relatively -to rows of the assemblage. An interesting dynamical -problem is presented by supposing any mutual forces, -such as might be produced by springs, to act between -the solid molecules, and investigating configurations -of equilibrium on the supposition of frictionless contacts. -The solution of it of course is that the potential -energy of the springs must be a minimum or a -minimax or a maximum for equilibrium, and a minimum -for stable equilibrium. The solution will be a -configuration of minimum or minimax, or maximum, -volume, only in the case of pressure equal in all -directions.</p> - -<p class="section">§ 34. A purely geometrical question, of no importance -in respect to the molecular tactics of a crystal but of -considerable interest in pure mathematics, is forced -on our attention by our having seen (§ 27) that a -homogeneous assemblage of solids of given shape, each -touched by twelve neighbours, has three freedoms -which may be conveniently taken as the three angles -specifying the orientation of each molecule relatively -to rows of the assemblage as explained in <a href="#s30">§ 30.</a></p> - -<p>Consider a solid <i>S<sub>1</sub></i> and the twelve neighbours which -touch it, and try if it is possible to cause it to touch -more than twelve of the bodies. Attach ends of three -thick flexible wires to any places on the surface of <i>S<sub>1</sub></i>; -carry the wires through interstices of the assemblage, -and attach their other ends at any three places of <i>A</i>, <i>B</i>, <i>C</i>, -respectively, these being any three of the bodies outside<span class="pagenum"><a name="Page_38" id="Page_38">38</a></span> -the cluster of <i>S<sub>1</sub></i> and its twelve neighbours. Cut the -wires across at any chosen positions in them; and -round off the cut ends, just leaving contact between the -rounded ends, which we shall call <i>f′f</i>, <i>g′g</i>, <i>h′h</i>. Do -homogeneously for every other solid of the assemblage -what we have done for <i>S<sub>1</sub></i>. Now bend the wires -slightly so as to separate the pairs of points of contact, -taking care to keep them from touching any other -bodies which they pass near on their courses between -<i>S<sub>1</sub></i> and <i>A</i>, <i>B</i>, <i>C</i> respectively. After having done this, -thoroughly rigidify all the wires thus altered. We -may now, having three independent variables at our -disposal, so change the orientation of the molecules, -relatively to rows of the assemblage, as to bring <i>f′f</i>, <i>g′g</i>, -and <i>h′h</i> again into contact. We have thus six fresh -points of <i>S<sub>1</sub></i>; of which three are <i>f′</i>, <i>g′</i>, <i>h′</i>; and the other -three are on the three extensions of <i>S<sub>1</sub></i> corresponding -to the single extensions of <i>A</i>, <i>B</i>, <i>C</i> respectively, which -we have been making. Thus we have a <em>real</em> solution -of the interesting geometrical problem:—It is required -so to form a homogeneous assemblage of solids of any -arbitrarily given shape that each solid shall be touched -by eighteen others. This problem is determinate, -because the making of the three contacts <i>f′f</i>, <i>g′g</i>, <i>h′h</i>, -uses up the three independent variables left at our -disposal after we have first formed a homogeneous -assemblage with twelve points of contact on each solid. -But our manner of finding a shape for each solid which -can allow the solution of the problem to be real, proves -that the solution is essentially imaginary for every -wholly convex shape.</p> - -<p class="section">§ 35. Pausing for a moment longer to consider afresh -the geometrical problem of putting arbitrarily given<span class="pagenum"><a name="Page_39" id="Page_39">39</a></span> -equal and similar solids together to make a homogeneous -assemblage of which each member is touched -by eighteen others, we see immediately that it is -determinate (whether it has any real solution or not), -because when the shape of each body is given we -have nine disposables for fixing the assemblage: six -for the character of the assemblage of the corresponding -points, and three for the orientation of each molecule -relatively to rows of the assemblage of corresponding -points. These nine disposables are determined by the -condition that each body has nine pairs of contacts -with others.</p> - -<p>Suppose now a homogeneous assemblage of the -given bodies, in open order with no contacts, to be -arbitrarily made according to any nine arbitrarily -chosen values for the six distances between a point -of <i>S<sub>1</sub></i> and the corresponding points of its six pairs of -nearest and next nearest neighbours (§ 1 above), and -the three angles (§ 9 above) specifying the orientation -of each body relatively to rows of the assemblage. -We may choose in any nine rows through <i>S<sub>1</sub></i> any nine -pairs of bodies at equal distances on the two sides of -<i>S<sub>1</sub></i> far or near, for the eighteen bodies which are to be -in contact with <i>S<sub>1</sub></i>. Hence there is an infinite number -of solutions of the problem of which only a finite -number can be real. Every solution of the problem -of eighteen contacts is imaginary when the shape is -wholly convex.</p> - -<div class="figcenter"><a name="f13" id="f13"> -<img src="images/i040.jpg" width="400" height="238" alt="" /></a> -<div class="caption"><span class="smcap">Fig. 13.</span></div> -</div> - -<p class="section">§ 36. Without for a moment imagining the molecules -of matter to be hard solids of convex shape, we may -derive valuable lessons in the tactics of real crystals by -studying the assemblage described in §§ <a href="#s24">24</a> and <a href="#s25">25</a> and -represented in Figs. 12 and 13. I must for the present<span class="pagenum"><a name="Page_40" id="Page_40">40</a></span> -forego the very attractive subject of the tactics presented -by faces not parallel to one or other of the four faces of -the primitive tetrahedrons which we found in <a href="#s24">§ 24</a>, and -ask you only to think of the two sides of a plate of crystal -parallel to any one of them, that is to say, an assemblage -of such layers as those represented geometrically in -Fig. 12 and shown in stereoscopic view in Fig. 13. If, -as is the case with the solids<a name="FNanchor_10_10" id="FNanchor_10_10"></a><a href="#Footnote_10_10" class="fnanchor">10</a> photographed in <a href="#f13">Fig. 13</a>, -the under side of each solid is nearly plane but slightly -convex, and the top is somewhat sharply curved, we have -the kind of difference between the upper and under of -the two parallel sides of the crystal which I have already -described to you in <a href="#s21">§ 21</a> above. In this case the assemblage -is formed by letting the solids fall down -from above and settle in the hollows to which they -come most readily, or which give them the stablest -position. It would, we may suppose, be the hollows -<span class="pagenum"><a name="Page_41" id="Page_41">41</a></span><i>p′ q′ r′</i>, not <i>p q r</i>, (Fig. 12) that would be chosen; and thus, -of the two formations described in <a href="#s25">§ 25</a>, we should have -that in which the hollows above <i>p′ q′ r′</i> are occupied by -the comparatively flat under sides of the molecules of -the layer above, and the hollows below the apertures -<i>p q r</i> by the comparatively sharp tops of the molecules -of the layers below.</p> - -<p class="section">§ 37. For many cases of natural crystals of the wholly -asymmetric character, the true forces between the -crystalline molecules will determine precisely the same -tactics of crystallization as would be determined by -the influence of gravity and fluid viscosity in the settlement -from water, of sand composed of uniform molecules -of the wholly unsymmetrical convex shape represented -in Figs. <a href="#f12">12</a> and <a href="#f13">13</a>. Thus we can readily believe that -a real crystal which is growing by additions to the -face seen in <a href="#f12">Fig. 12</a>, would give layer after layer -regularly as I have just described. But if by some -change of circumstances the plate, already grown to -a thickness of many layers in this way, should come -to have the side facing <em>from</em> us in the diagram exposed -to the mother-liquor, or mother-gas, and begin to grow -from that face, the tactics might probably be that each -molecule would find its resting-place with its most -nearly plane side in the wider hollows under <i>p′ q′ r′</i>, -instead of with its sharpest corner in the narrower and -steeper hollows under <i>p q r</i>, as are the molecules in -the layer below that shown in the diagram in the first -formation. The result would be a compound crystal -consisting of two parts, of different crystalline quality, -cohering perfectly together on the two sides of an -interfacial plane. It seems probable that this double -structure may be found in nature, presented by crystals<span class="pagenum"><a name="Page_42" id="Page_42">42</a></span> -of the wholly unsymmetric class, though it may not -hitherto have been observed or described in crystallographic -treatises.</p> - -<div class="figcenter"><a name="f14" id="f14"> -<img src="images/i042.jpg" width="250" height="196" alt="" /></a> -<div class="caption"><span class="smcap">Fig. 14.</span></div> -</div> - -<p class="section"><a name="s38" id="s38">§ 38.</a> This asymmetric double crystal becomes simply -the well-known symmetrical ‘twin-crystal’<a name="FNanchor_11_11" id="FNanchor_11_11"></a><a href="#Footnote_11_11" class="fnanchor">11</a> in the particular -case in which each of the constituent molecules -is symmetrical on the two sides of a plane through it -parallel to the plane of our diagrams, and also on the two -sides of some plane perpendicular to this plane. We -see, in fact, that in this case if we cut in two the double -crystal by the plane of Fig. 14, and turn one part ideally -through 180° round the intersection of these two planes, -we bring it into perfect coincidence with the other part.</p> -<p><span class="pagenum"><a name="Page_43" id="Page_43">43</a></span></p> -<p>This we readily understand by looking at Fig. 14, in -which the solid shown in outline may be either an -egg-shaped figure of revolution, or may be such a figure -flattened by compression perpendicular to the plane -of the diagram. The most readily chosen and the -most stable resting-places for the constituents of each -successive layer might be the wider hollows <i>p′ q′ r′</i>: -and therefore if, from a single layer to begin with, -the assemblage were to grow by layer after layer -added to it on each side, it might probably grow -as a twin-crystal. But it might also be that the -presence of a molecule in the wider hollow <i>p′ q′ r′</i> on -one side, might render the occupation of the corresponding -hollow on the other side by another molecule -less probable, or even impossible. Hence, according -to the configuration and the molecular forces of the -particular crystalline molecule in natural crystallization, -there may be necessarily, or almost necessarily, the -twin, when growth proceeds simultaneously on the two -sides: or the twin growth may be impossible, because -the first occupation of the wider hollows on one side, -may compel the continuity of the crystalline quality -throughout, by leaving only the narrower hollows <i>p q r</i> -free for occupation by molecules attaching themselves -on the other side.</p> - -<p class="section">§ 39. Or the character of the crystalline molecule -may be such that when the assemblage grows by -the addition of layer after layer on one side only, -with a not very strongly decided preference to the -wider hollows <i>p′ q′ r′</i>, some change of circumstances -may cause the molecules of one layer to place -themselves in a hollow <i>p q r</i>. The molecules in the -next layer after this would find the hollows <i>p′ q′ r′</i><span class="pagenum"><a name="Page_44" id="Page_44">44</a></span> -occupied on the far side, and would thus have a bias -in favour of the hollows <i>p q r</i>. Thus layer after layer -might be added, constituting a twinned portion of -the growth, growing, however, with less strong security -for continued homogeneousness than when the -crystal was growing, as at first, by occupation of the -wider hollows <i>p′ q′ r′</i>. A slight disturbance might -again occur, causing the molecules of a fresh layer -to settle, not in the narrow hollows <i>p q r</i>, but in the -wider hollows <i>p′ q′ r′</i>, notwithstanding the nearness of -molecules already occupying the wider hollows on -the other side. Disturbances such as these occurring -irregularly during the growth of a crystal, might produce -a large number of successive twinnings at -parallel planes with irregular intervals between them, -or a large number of twinnings in planes at equal -intervals might be produced by some regular periodic -disturbance occurring for a certain number of periods, -and then ceasing. Whether regular and periodic, or -irregular, the tendency would be that the number of -twinnings should be even, and that after the disturbances -cease the crystal should go on growing in -the first manner, because of the permanent bias in -favour of the wider hollows <i>p′ q′ r′</i>. These changes -of molecular tactics, which we have been necessarily -led to by the consideration of the fortuitous concourse -of molecules, are no doubt exemplified in a large -variety of twinnings and counter-twinnings found in -natural minerals. In the artificial crystallization of -chlorate of potash they are of frequent occurrence, -as is proved, not only by the twinnings and counter-twinnings -readily seen in the crystalline forms, but -also by the brilliant iridescence observed in many of<span class="pagenum"><a name="Page_45" id="Page_45">45</a></span> -the crystals found among a large multitude, which -was investigated scientifically by Sir George Stokes -ten years ago, and described in a communication to -the Royal Society ‘On a remarkable phenomenon of -crystalline reflection’ (<i>Proc. R.S.</i>, vol. xxxviii, 1885, -p. 174).</p> - -<p class="section">§ 40. A very interesting phenomenon, presented by -what was originally a clear homogeneous crystal of -chlorate of potash, and was altered by heating to about -245°-248° Cent., which I am able to show you through -the kindness of Lord Rayleigh, and of its discoverer, -Mr. Madan, presents another very wonderful case of -changing molecular tactics, most instructive in respect -of the molecular constitution of elastic solids. When -I hold this plate before you with the perpendicular -to its plane inclined at 10° or more to your line of -vision, you see a tinsel-like appearance, almost as -bright as if it were a plate of polished silver, on this -little area, which is a thin plate of chlorate of potash -cemented for preservation between two pieces of -glass; and, when I hold a light behind, you see that -the little plate is almost perfectly opaque like metal -foil. But now when I hold it nearly perpendicular -to your line of vision the tinsel-like appearance is -lost. You can see clearly through the plate, and you -also see that very little light is reflected from it. As -a result both of Mr. Madan’s own investigations, and -further observations by himself, Lord Rayleigh came -to the conclusion that the almost total reflection of -white light which you see is due to the reflection of -light at many interfacial planes between successive -layers of twinned and counter-twinned crystal of small -irregular thicknesses, and not to any splits or cavities<span class="pagenum"><a name="Page_46" id="Page_46">46</a></span> -or any other deviation from homogeneousness than -that presented by homogeneous portions of oppositely -twinned-crystals in thorough molecular contact at the -interfaces.</p> - -<p class="section">§ 41. When the primitive clear crystal was first heated -very gradually by Madan to near its melting-point (359° -according to Carnelly), it remained clear, and only acquired -the tinsel appearance after it had cooled to about -245° or 248°<a name="FNanchor_12_12" id="FNanchor_12_12"></a><a href="#Footnote_12_12" class="fnanchor">12</a>. Rayleigh found that if a crystal thus -altered was again and again heated it always lost the -tinsel appearance, and became perfectly clear at some -temperature considerably below the melting-point, and -regained it at about the same temperature in cooling. -It seems, therefore, certain that at temperatures above -248°, and below the melting-point, the molecules had -so much of thermal motions as to keep them hovering -<span class="pagenum"><a name="Page_47" id="Page_47">47</a></span>about the positions of <i>p q r</i>, <i>p′ q′ r′</i>, of our diagrams, -but not enough to do away with the rigidity of the -solid; and that when cooled below 248° the molecules -were allowed to settle in one or other of the two -configurations, but with little of bias for one in preference -to the other. It is certainly a very remarkable -fact in Natural History, discovered by these observations, -that, when the molecules come together to -form a crystal out of the watery solution, there should -be so much more decided a bias in favour of continued -homogeneousness of the assemblage than when, by -cooling, they are allowed to settle from their agitations -in a rigid, but nearly melting, solid.</p> - -<p class="section">§ 42. But even in crystallization from watery solution -of chlorate of potash the bias in favour of thorough -homogeneousness is not in every contingency decisive. -In the first place, beginning, as the formation seems to -begin, from a single molecular plane layer such as that -ideally shown in <a href="#f14">Fig. 14</a>, it goes on, not to make a homogeneous -crystal on the two sides of this layer, but -probably always so as to form a twin-crystal on its two -sides, exactly as described in <a href="#s38">§ 38</a>, and, if so, certainly -for the reason there stated. This is what Madan calls -the ‘inveterate tendency to produce twins (such as -would assuredly drive a Malthus to despair)<a name="FNanchor_13_13" id="FNanchor_13_13"></a><a href="#Footnote_13_13" class="fnanchor">13</a>’; and it is -to this that he alludes as ‘the inevitable twin-plate’ in -the passage from his paper given in the foot-note to -§ 41 above.</p> - -<p class="section">§ 43. In the second place, I must tell you that many -of the crystals produced from the watery solution by -the ordinary process of slow evaporation and crystallization, -show twinnings and counter-twinnings at -<span class="pagenum"><a name="Page_48" id="Page_48">48</a></span>irregular intervals in the otherwise homogeneous -crystal on either one or both sides of the main -central twin-plane, which henceforth, for brevity, I -shall call (adopting the hypothesis already explained, -which seems to me undoubtedly true) the ‘initial -plane.’ Each twinning is followed, I believe, by a -counter-twinning at a very short distance from it; at -all events Lord Rayleigh’s observations<a name="FNanchor_14_14" id="FNanchor_14_14"></a><a href="#Footnote_14_14" class="fnanchor">14</a> prove that -the whole number of twinnings and counter-twinnings -in a thin disturbed stratum of the crystal on one side -of the main central twin-plane is generally, perhaps -always, even; so that, except through some comparatively -very small part or parts of the whole -thickness, the crystal on either side of the middle or -initial plane is homogeneous. This is exactly the -generally regular growth which I have described to -you (§ 39) as interrupted occasionally or accidentally -by some unexplained disturbing cause, but with an -essential bias to the homogeneous continuance of the -more easy or natural one of the two configurations.</p> - -<p class="section">§ 44. I have now great pleasure in showing you -a most interesting collection of the iridescent crystals -of chlorate of potash, each carefully mounted for -preservation between two glass plates, which have -been kindly lent to us for this evening by Mr. Madan. -In March, 1854, Dr. W. Bird Herapath sent to Prof. -Stokes some crystals of chlorate of potash showing -the brilliant and beautiful colours you now see, and, -thirty years later, Prof. E. J. Mills recalled his attention -to the subject by sending him ‘a fine collection of -splendidly coloured crystals of chlorate of potash of -considerable size, several of the plates having an area -<span class="pagenum"><a name="Page_49" id="Page_49">49</a></span>of a square inch or more, and all of them being thick -enough to handle without difficulty.’ The consequence -was that Stokes made a searching examination into -the character of the phenomenon, and gave the short, -but splendidly interesting, communication to the Royal -Society of which I have already told you. The -existence of these beautifully coloured crystals had -been well known to chemical manufacturers for a long -time, but it does not appear that any mention of them -was to be found in any scientific journal or treatise -prior to Stokes’ paper of 1885. He found that the -colour was due to twinnings and counter-twinnings -in a very thin disturbed stratum of the crystal showing -itself by a very fine line, dark or glistening, according -to the direction of the incident light when a transverse -section of the plate of crystal was examined in -a microscope. By comparison with a spore of lycopodium -he estimated that the breadth of this line, and -therefore the thickness of the disturbed stratum of the -crystal, ranged somewhere about the one-thousandth -of an inch. He found that the stratum was visibly -thicker in those crystals which showed red colour -than in those which showed blue. He concluded that -‘the seat of the coloration is certainly a thin twinned -stratum’ (that is to say, a homogeneous portion of -crystal between a twinning and a counter-twinning), -and found that ‘a single twin-plane does not show -anything of the kind.’</p> - -<p class="section">§ 45. A year or two later Lord Rayleigh entered on -the subject with an exhaustive mathematical investigation -of the reflection of light at a twin-plane of a crystal -(<cite>Philosophical Magazine</cite>, September, 1888), by the application -of which, in a second paper ‘On the remarkable<span class="pagenum"><a name="Page_50" id="Page_50">50</a></span> -phenomenon of Crystalline Reflection described by -Prof. Stokes,’ published in the same number of the -<cite>Philosophical Magazine</cite>, he gave what seems certainly -the true explanation of the results of Sir George Stokes’ -experimental analysis of these beautiful phenomena. -He came very decidedly to the conclusion that the -selective quality of the iridescent portion of the crystal, -in virtue of which it reflects almost totally light -nearly of one particular wave-length for one particular -direction of incidence (on which the brilliance of the -coloration depends), cannot be due to merely a single -twin-stratum, but that it essentially is due to a considerable -number of parallel twin-strata at nearly equal -distances. The light reflected by this complex stratum -is, for any particular direction of incident and reflected -ray, chiefly that of which the wave-length is equal to -twice the length of the period of the twinning and -counter-twinning, on a line drawn through the stratum -in the direction of either the incident or the reflected -ray.</p> - -<p class="section">§ 46. It seems to me probable that each twinning is -essentially followed closely by a counter-twinning. -Probably three or four of these twin-strata might suffice -to give colour; but in any of the brilliant specimens as -many as twenty or thirty, or more, might probably be -necessary to give so nearly monochromatic light as was -proved by Stokes’ prismatic analysis of the colours -observed in many of his specimens. The disturbed -stratum of about a one-thousandth of an inch thickness, -seen by him in the microscope, amply suffices for the -5, 10, or 100 half wave-lengths required by Rayleigh’s -theory to account for perceptible or brilliant coloration. -But what <em>can</em> be the cause of any approach to regular<span class="pagenum"><a name="Page_51" id="Page_51">51</a></span> -periodicity in the structure sufficiently good to give the -colours actually observed? Periodical motion of the -mother-liquor relatively to the growing crystal might -possibly account for it. But Lord Rayleigh tells us -that he tried rocking the pan containing the solution -without result. Influence of light has been suggested, -and I believe tried, also without result, by several -enquirers. We know, by the beautiful discovery of -Edmond Becquerel, of the prismatic colours photographed -on a prepared silver plate by the solar spectrum, -that ‘standing waves’ (that is to say, vibrations -with stationary nodes and stationary places of maximum -vibration), due to co-existence of incident and reflected -waves, do produce such a periodic structure as that -which Rayleigh’s theory shows capable of giving a -corresponding tint when illuminated by white light. -It is difficult, therefore, not to think that light may -be effective in producing the periodic structure in -the crystallization of chlorate of potash, to which the -iridescence is due. Still, experimental evidence seems -against this tempting theory, and we must perforce be -content with the question unanswered:—What can -be the cause of 5, or 10, or 100 pairs of twinning and -counter-twinning following one another in the crystallization -with sufficient regularity to give the colour: -and why, if there are twinnings and counter-twinnings, -are they not at irregular intervals, as those produced -by Madan’s process, and giving the observed white -tinsel-like appearance with no coloration?</p> - -<p class="section">§ 47. And now I have sadly taxed your patience: -and I fear I have exhausted it and not exhausted my -subject! I feel I have not got halfway through what -I hoped I might be able to put before you this evening<span class="pagenum"><a name="Page_52" id="Page_52">52</a></span> -regarding the molecular structure of crystals. I particularly -desired to speak to you of quartz crystal -with its ternary symmetry and its chirality<a name="FNanchor_15_15" id="FNanchor_15_15"></a><a href="#Footnote_15_15" class="fnanchor">15</a>; and to -have told you of the etching<a name="FNanchor_16_16" id="FNanchor_16_16"></a><a href="#Footnote_16_16" class="fnanchor">16</a> by hydrofluoric acid -which, as it were, commences to unbuild the crystal -by taking away molecule after molecule, but not in the -reverse order of the primary up-building; and which -thus reveals differences of tactics in the alternate faces -of the six-sided pyramid which terminates at either -end, sometimes at both ends, the six-sided prism -constituting generally the main bulk of the crystal. -I must confine myself to giving you a geometrical -symbol for the ternary symmetry of the prism and -its terminal pyramid.</p> - -<div class="figcenter"><a name="f15" id="f15"> -<img src="images/i053.jpg" width="250" height="229" alt="" /></a> -<div class="caption"><span class="smcap">Fig. 15.</span></div> -</div> - -<p class="section"><a name="s48" id="s48">§ 48.</a> Make an equilateral equiangular hexagonal -prism, with its diagonal from edge to edge ninety-five -hundredths<a name="FNanchor_17_17" id="FNanchor_17_17"></a><a href="#Footnote_17_17" class="fnanchor">17</a> of its length. Place a number of these -close together, so as to make up a hexagonal plane layer -with its sides perpendicular to the sides of the constituent -hexagonal prisms: see Fig. 15 and imagine the -semicircles replaced by their diameters. You see in -each side of the hexagonal assemblage, edges of the -constituent prisms, and you see at each corner of -the assemblage a face (not an edge) of <em>one</em> of the -constituent prisms. Build up a hexagonal prismatic -assemblage by placing layer after layer over it with -the constituent prisms of each layer vertically over -those in the layer below; and finish the assemblage -<span class="pagenum"><a name="Page_53" id="Page_53">53</a></span>with a six-sided pyramid by building upon the upper -end of the prism, layer after layer of diminishing -hexagonal groups, each less by one circumferential -row than the layer below it. You thus have a crystal -of precisely the shape of a symmetrical specimen of -rock crystal, with the faces of its terminal pyramid -inclined at 38° 13′ to the faces of the prism from which -they spring. But the assemblage thus constituted has -‘senary’ (or six-rayed symmetry). To reduce this to -ternary symmetry, cut a groove through the middle -of each alternate face of the prismatic molecule, making -this groove in the first place parallel to the edges: and -add a corresponding projection, or fillet, to the middles -of the other three faces, so that two of the cylinders -similarly oriented would fit together, with the projecting -fillet on one side of one of them entering the -groove in the anti-corresponding side of the other. -The prismatic portion of the assemblage thus formed -shows (see Fig. 15), on its alternate edges, faces of<span class="pagenum"><a name="Page_54" id="Page_54">54</a></span> -molecules with projections and faces of molecules -with grooves; and shows only orientational differences -between alternate faces, whether of the pyramid or -of the prism. Having gone only so far from ‘senary’ -symmetry, we have exactly the triple, or three-pair, -anti-symmetry required for the piezo-electricity of -quartz investigated so admirably by the brothers -Curie<a name="FNanchor_18_18" id="FNanchor_18_18"></a><a href="#Footnote_18_18" class="fnanchor">18</a>, who found that a thin plate of quartz crystal -<span class="figleft200"><a name="f16" id="f16"> -<img src="images/i054.jpg" width="200" height="228" alt="" /></a> -<span class="caption"><span class="smcap">Fig. 16. Fig. 17.</span></span> -</span> -cut from any position perpendicular to a pair of faces -of a symmetrical crystal, becomes positively electrified -on one side and negatively on the other when pulled -in a direction perpendicular to those faces. But this -assemblage has not the chiral piezo-electric quality -discovered theoretically by -Voigt<a name="FNanchor_19_19" id="FNanchor_19_19"></a><a href="#Footnote_19_19" class="fnanchor">19</a>, and experimentally -in quartz and in tourmaline -by himself and Riecke<a name="FNanchor_20_20" id="FNanchor_20_20"></a><a href="#Footnote_20_20" class="fnanchor">20</a>, nor -the well-known optic chirality -of quartz.</p> - -<p class="section">§ 49. Change now the directions -of the grooves and -fillets to either of the oblique -configurations shown in Fig. -16, which I call right-handed, -because the directions of the -projections are tangential to the threads of a three-thread -right-handed screw, and Fig. 17 (left-handed). The -prisms with their grooves and fillets will still all fit together -if they are all right-handed, or all left-handed.</p> - -<p><span class="pagenum"><a name="Page_55" id="Page_55">55</a></span></p> - -<div class="figcenter"> -<img src="images/i055.jpg" width="250" height="224" alt="" /> -<div class="caption"><span class="smcap">Fig. 18.</span></div> -</div> - -<p>Fig. 18 shows the upper side of a hexagonal layer of an -assemblage thus composed of the right-handed molecule -of <a href="#f16">Fig. 16</a>. <a href="#f15">Fig. 15</a> unchanged, still represents -a horizontal section through the centres of the molecules. -A prism built up of such layers, and finished -at each end with a pyramid according to the rule -of <a href="#s48">§ 48</a>, has all the qualities of ternary chiral symmetry -required for the piezo-electricity of quartz; for -the orientational differences of the alternate pairs of -prismatic faces; for the absolute difference between -the alternate pairs of faces of each pyramid which -are shown in the etching by hydrofluoric acid; for -the merely orientational difference between the parallel -faces of the two pyramids; and for the well-known -chiro-optic<a name="FNanchor_21_21" id="FNanchor_21_21"></a><a href="#Footnote_21_21" class="fnanchor">21</a> property of quartz. Look at two contiguous -faces <i>A</i>, <i>B</i> of our geometrical model quartz -crystal now before you, with its axis vertical. You -will see a difference between them: turn it upside -<span class="pagenum"><a name="Page_56" id="Page_56">56</a></span>down; <i>B</i> will be undistinguishable from what <i>A</i> was, -and <i>A</i> will be undistinguishable from what <i>B</i> was. -Look at the two terminal pyramids, and you will -find that the face above <i>A</i> and the face below <i>B</i> are -identical in quality, and that they differ from the face -above <i>B</i> and below <i>A</i>. This model is composed of the -right-handed constituent molecules shown in <a href="#f16">Fig. 16</a>. -It is so placed before you that the edge of the prismatic -part of the assemblage nearest to you shows -you filleted faces of the prismatic molecules. You -see two pyramidal faces; the one to your right hand, -over <i>B</i>, presents complicated projections and hollows -at the corners of the constituent molecules; and the -pyramidal face next your left hand, over <i>A</i>, presents -their unmodified corners. But it will be the face next -your left hand which will present the complex bristling -corners, and the face next your right hand that will -present the simple corners, if, for the model before -you, you substitute a model composed of left-handed -molecules such as those shown in <a href="#f16">Fig. 17</a>.</p> - -<p class="section">§ 50. To give all the qualities of symmetry and -anti-symmetry of the pyro-electric and piezo-electric -properties of tourmaline investigated theoretically by -Voigt<a name="FNanchor_22_22" id="FNanchor_22_22"></a><a href="#Footnote_22_22" class="fnanchor">22</a>, and experimentally by himself and Friecke<a name="FNanchor_23_23" id="FNanchor_23_23"></a><a href="#Footnote_23_23" class="fnanchor">23</a>, -make a hollow in one terminal face of each of our -constituent prisms, and a corresponding projection in -its other terminal face.</p> - -<p class="section">§ 51. Coming back to quartz, we can now understand -perfectly the two kinds of macling which are well -known to mineralogists as being found in many natural -specimens of the crystal, and which I call respectively -<span class="pagenum"><a name="Page_57" id="Page_57">57</a></span>the orientational macling, and the chiral macling. In -the orientational macling all the crystalline molecules -are right-handed, or all left-handed; but through all -of some part of the crystal, each of our component -hexagonal prisms is turned round its axis through 60° -from the position it would have if the structure were -homogeneous throughout. In each of the two parts -the structure is homogeneous, and possesses all the -electric and optic properties which any homogeneous -portion of quartz crystal presents, and the facial properties -of natural uncut crystal, shown in the etching -by hydrofluoric acid; but there is a discontinuity at -the interface, not generally plane, between the two -parts, which in our geometrical model would be shown -by non-fittings between the molecules on the two sides -of the interface, while all the contiguous molecules in -one part, and all the contiguous molecules in the other -part, fit into one another perfectly. In chiral macling, -which is continually found in amethystine quartz, and -sometimes in ordinary clear quartz crystals, some parts -are composed of right-handed molecules, and others of -left-handed molecules. It is not known whether, in -this chiral macling, there is or there is not also the -orientational macling on the two sides of each interface; -but we may say probably <em>not</em>; because we know that -the orientational macling occurs in nature without any -chiral macling, and because there does not seem reason -to expect that chiral macling would imply orientational -macling on the two sides of the same interface. I would -like to have spoken to you more of this most interesting -subject; and to have pointed out to you that some -of the simplest and most natural suppositions we can -make as to the chemical forces (or electrical forces,<span class="pagenum"><a name="Page_58" id="Page_58">58</a></span> -which probably means the same thing) concerned in -a single chemical molecule of quartz, <i>SiO<sub>2</sub></i>, and acting -<span class="figleft50"> -<img src="images/i058.jpg" width="50" height="99" alt="" /> -<span class="caption"><span class="smcap">Fig. 19.</span></span> -</span> -between it and similar neighbouring molecules, -would lead essentially to these molecules coming -together in triplets, each necessarily either right-handed -or left-handed, but with as much probability -of one configuration as of the other: and -to have shown you that these triplets of silica -3(<i>SiO<sub>2</sub></i>) can form a crystalline molecule with -all the properties of ternary chiral symmetry, typified -by our grooved hexagonal prisms, and can build -up a quartz crystal by the fortuitous concourse of -atoms. I should like also to have suggested and explained -the possibility that a right-handed crystalline -molecule thus formed may, in natural circumstances -of high temperature, or even of great pressure, become -changed into a left-handed crystal, or <em>vice-versa</em>. My -watch, however, warns me that I must not enter on -this subject.</p> - -<p class="section">§ 52. Coming back to mere molecular tactics of -crystals, remark that our assemblage of rounded, -thoroughly scalene, tetrahedrons, shown in the stereoscopic -picture (§ 36, <a href="#f13">Fig. 13</a> above), essentially has -chirality because each constituent tetrahedron, if wholly -scalene, has chirality<a name="FNanchor_24_24" id="FNanchor_24_24"></a><a href="#Footnote_24_24" class="fnanchor">24</a>. I should like to have explained -to you how a single or double homogeneous assemblage -of points has essentially no chirality, and how three -assemblages of single points, or a single assemblage -of triplets of points, can have chirality, though a single -triplet of points cannot have chirality. I should like -indeed to have brought somewhat thoroughly before -you the geometrical theory of chirality; and in -<span class="pagenum"><a name="Page_59" id="Page_59">59</a></span>illustration to have explained the conditions under -which four points, or two lines, or a line and two -points, or a combination of point, line and plane, can -have chirality: and how a homogeneous assemblage -of non-chiral objects can have chirality; but in pity -I forbear, and I thank you for the extreme patience -with which you have listened to me.</p> - -<h2>FOOTNOTES:</h2> - -<div class="footnote"> - -<p><a name="Footnote_1_1" id="Footnote_1_1"></a><a href="#FNanchor_1_1"><span class="label">1</span></a> See <a href="#Footnote_8_8">foot-note</a> on § 22 below.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_2_2" id="Footnote_2_2"></a><a href="#FNanchor_2_2"><span class="label">2</span></a> The holes in the cylinders are bored obliquely, as shown in <a href="#f4">Fig. 4</a>, -which causes them to remain at any desired position on the cord and -allows them to be freed to move up and down by slackening the cord for -a moment.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_3_3" id="Footnote_3_3"></a><a href="#FNanchor_3_3"><span class="label">3</span></a> ‘On the Homogeneous Division of Space,’ by Lord Kelvin, <cite>Royal -Society Proceedings</cite>, vol. lv, Jan. 18, 1894.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_4_4" id="Footnote_4_4"></a><a href="#FNanchor_4_4"><span class="label">4</span></a> Similar curves are said to be parallel when the tangents to them at -corresponding points are parallel.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_5_5" id="Footnote_5_5"></a><a href="#FNanchor_5_5"><span class="label">5</span></a> See <a href="#Footnote_3_3">foot-note</a> to § 12 above.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_6_6" id="Footnote_6_6"></a><a href="#FNanchor_6_6"><span class="label">6</span></a> ‘On the Division of Space with Minimum Partitional Area,’ <cite>Philosophical -Magazine</cite>, vol. xxiv, 1887, p. 502, and <cite>Acta Mathematica</cite> of the same year.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_7_7" id="Footnote_7_7"></a><a href="#FNanchor_7_7"><span class="label">7</span></a> A. Levy, <cite>Edinburgh Philosophical Journal</cite>, April, 1822; Whewell, <cite>Phil. -Trans. Royal Society</cite>, 1825; Miller, <cite>Treatise on Crystallography</cite>.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_8_8" id="Footnote_8_8"></a><a href="#FNanchor_8_8"><span class="label">8</span></a> I call any geometrical figure, or group of points, <em>chiral</em>, and say that -it has chirality, if its image in a plane mirror, ideally realized, cannot be -brought to coincide with itself. Two equal and similar right hands are -homochirally similar. Equal and similar right and left hands are heterochirally -similar or ‘allochirally’ similar (but heterochirally is better). -These are also called ‘enantiomorphs,’ after a usage introduced, I believe, -by German writers. Any chiral object and its image in a plane mirror are -heterochirally similar.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_9_9" id="Footnote_9_9"></a><a href="#FNanchor_9_9"><span class="label">9</span></a> <cite>Philosophical Magazine</cite>, vol. xx, 1885, second half year, p. 469, and -<cite>British Association Report</cite>, 1885, Aberdeen, p. 896.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_10_10" id="Footnote_10_10"></a><a href="#FNanchor_10_10"><span class="label">10</span></a> The solids of the photograph are castings in fine plaster of Paris from -a scalene tetrahedron of paraffin wax, with its corners and edges rounded, -used as a pattern.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_11_11" id="Footnote_11_11"></a><a href="#FNanchor_11_11"><span class="label">11</span></a> ‘A twin-crystal is composed of two crystals joined together in such -a manner that one would come into the position of the other by revolving -through two right angles round an axis which is perpendicular to a plane -which either is, or may be, a face of either crystal. The axis will be called -the twin-axis, and the plane to which it is perpendicular the twin-plane.’ -Miller’s <cite>Treatise on Crystallography</cite>, p. 103. In the text the word ‘twin-plane,’ -quoted from the writings of Stokes and Rayleigh, is used to signify -the plane common to the two crystals in each of the cases referred to: and -not the plane perpendicular to this plane, in which one part of the crystal -must be rotated to bring it into coincidence with the other, and which is -the twin-plane as defined by Miller.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_12_12" id="Footnote_12_12"></a><a href="#FNanchor_12_12"><span class="label">12</span></a> ‘A clear transparent crystal of potassium chlorate, from which the -inevitable twin-plate had been ground away so as to reduce it to a single -crystal film about 1 mm. in thickness, was placed between pieces of mica -and laid on a thick iron plate. About 3 cm. from it was laid a small bit of -potassium chlorate, and the heat of a Bunsen burner was applied below -this latter, so as to obtain an indication when the temperature of the plate -was approaching the fusing-point of the substance (359° <i>C</i> according to -Prof. Carnelly). The crystal plate was carefully watched during the -heating, but no depreciation took place, and no visible alteration was observed, -up to the point at which the small sentinel crystal immediately over -the burner began to fuse. The lamp was now withdrawn, and when the -temperature had sunk a few degrees a remarkable change spread quickly -and quietly over the crystal plate, causing it to reflect light almost as -brilliantly as if a film of silver had been deposited upon it. No further -alteration occurred during the cooling; and the plate, after being ground -and polished on both sides, was mounted with Canada balsam between -glass plates for examination. Many crystals have been similarly treated -with precisely similar results; and the temperature at which the change -takes place, has been determined to lie between 245° and 248°, by heating -the plates upon a bath of melted tin in which a thermometer was immersed. -With single crystal plates no decrepitation has ever been observed, while -with the ordinary twinned-plates it always occurs more or less violently, -each fragment showing the brilliant reflective power above noticed.’—<cite>Nature</cite>, -May 20, 1886.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_13_13" id="Footnote_13_13"></a><a href="#FNanchor_13_13"><span class="label">13</span></a> <cite>Nature</cite>, May 20, 1886.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_14_14" id="Footnote_14_14"></a><a href="#FNanchor_14_14"><span class="label">14</span></a> <cite>Philosophical Magazine</cite>, 1888, second half year, p. 260.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_15_15" id="Footnote_15_15"></a><a href="#FNanchor_15_15"><span class="label">15</span></a> See <a href="#Footnote_8_8">foot-note</a> to § 22 above.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_16_16" id="Footnote_16_16"></a><a href="#FNanchor_16_16"><span class="label">16</span></a> Widmanstätten, 1807. Leydolt (1855, Wien. Akad. Ber. 15, 59, T. -9, 10. Baumhauer, Pogg. Ann. 138, 563 (1869); 140, 271; 142, 324; 145, -460; 150, 619.) For an account of these investigations, see Mallard, <cite lang="fr" xml:lang="fr">Traité -de Crystallographie</cite> (Paris, 1884), Tome II, chapitre xvi.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_17_17" id="Footnote_17_17"></a><a href="#FNanchor_17_17"><span class="label">17</span></a> More exactly .9525, being 3/4 × cot 38° 13′; see p. 53.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_18_18" id="Footnote_18_18"></a><a href="#FNanchor_18_18"><span class="label">18</span></a> J. and P. Curie and C. Friedel, <cite lang="fr" xml:lang="fr">Comptes Rendus</cite>, 1882, 1883, 1886, 1892.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_19_19" id="Footnote_19_19"></a><a href="#FNanchor_19_19"><span class="label">19</span></a> <span lang="de" xml:lang="de">Allgemeine Theorie der piëzo- und pyroelectrischen Erscheinungen an -Krystallen. W. Voigt, Königl. Gesellschaft der Wissenschaften zu Göttingen</span>, -August 2, 1890.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_20_20" id="Footnote_20_20"></a><a href="#FNanchor_20_20"><span class="label">20</span></a> Wiedemann, <cite lang="de" xml:lang="de">Annalen</cite>, 1892, xlv, p. 923.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_21_21" id="Footnote_21_21"></a><a href="#FNanchor_21_21"><span class="label">21</span></a> Generally miscalled ‘rotational.’</p></div> - -<div class="footnote"> - -<p><a name="Footnote_22_22" id="Footnote_22_22"></a><a href="#FNanchor_22_22"><span class="label">22</span></a> See <a href="#Footnote_19_19">foot-note (2)</a> to p. 54 above.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_23_23" id="Footnote_23_23"></a><a href="#FNanchor_23_23"><span class="label">23</span></a> See <a href="#Footnote_20_20">foot-note (3)</a> to p. 54 above.</p></div> - -<div class="footnote"> - -<p><a name="Footnote_24_24" id="Footnote_24_24"></a><a href="#FNanchor_24_24"><span class="label">24</span></a> See <a href="#Footnote_8_8">foot-note</a> to § 22 above.</p></div> - -<p class="center">THE END</p> - -<hr /> - -<p class="center"><small>Oxford<br /> -PRINTED AT THE CLARENDON PRESS<br /> -<small>BY HORACE HART, PRINTER TO THE UNIVERSITY</small></small> -</p> - - - - - - - - -<pre> - - - - - -End of Project Gutenberg's The Molecular Tactics of a Crystal, by Lord Kelvin - -*** END OF THIS PROJECT GUTENBERG EBOOK THE MOLECULAR TACTICS OF A CRYSTAL *** - -***** This file should be named 54976-h.htm or 54976-h.zip ***** -This and all associated files of various formats will be found in: - http://www.gutenberg.org/5/4/9/7/54976/ - -Produced by Turgut Dincer and The Online Distributed -Proofreading Team at http://www.pgdp.net (This file was -produced from images generously made available by The -Internet Archive/American Libraries.) - -Updated editions will replace the previous one--the old editions will -be renamed. - -Creating the works from print editions not protected by U.S. copyright -law means that no one owns a United States copyright in these works, -so the Foundation (and you!) can copy and distribute it in the United -States without permission and without paying copyright -royalties. 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