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which Van t' Hoff and Wislicenus have shown are perfectly capable of explaining the difference in the behaviour of the two acids.

It is obvious that the first formula represents maleic acid, as the proximity of the carboxyl groups -CO-OH facilitates the formation of an anhydride.

In fumaric acid the carboxyl groups are diametrically opposite each other.

Both acids combine with the elements of water, forming inactive malic acid, which can be split up into two optically active isomerides. The addition of the elements of water takes place in each of the two possible ways.

The introduction of the idea of a difference in the arrangement of the atoms in space into the constitutional formulæ of organic compounds has provided a satisfactory explanation for numerous cases of isomerism which could not be formerly accounted for. It has also led to the discovery of numerous relations between the arrangement of the atoms and the properties of compounds. The hypothesis of asymmetrically linked carbon atoms was first propounded in 1874, and it now ranks as one of the most firmly established of the doctrines of chemistry.

§ 59. The Absolute Dimensions of Molecules and Atoms.—The molecules, the constitution of which has been discussed in the preceding paragraphs, are not indefinitely small, although much smaller than any magnitude perceptible to our unaided or even to the aided senses. As to the magnitude of the molecules themselves, it is at present impossible to give any exact determinations; still the limits within which the dimensions must lie can be approximately determined. Such approximations may, as was shown by Sir William Thomson in 1871, be arrived at by the aid of various physical phenomena; his conclusions. have been confirmed and extended by other investigators. From certain optical phenomena, for instance, from the disper

sion accompanying the refraction of light, it may be concluded, with some degree of probability, that the molecules of transparent materials, such as glass, water, and the like, are greater than the ten-thousandth part of a wave-length of light, which latter amounts again to only a few ten-thousandth parts of a millimetre. Similar conclusions may be drawn also from the destruction of colouring matters on solution, and again from the contact electricity of metals, and the heat produced by the attraction of metallic plates oppositely electrified, from the minimum thickness which soap bubbles can attain without bursting, and especially from the properties of gases and the liquids produced by their condensation.

The highly developed kinetic theory of gases shows, for instance, that certain relationships exist between the dimensions of gaseous particles, their velocity, and the path which they traverse before they come in contact with one another. From these relationships approximations may be made as to the weight and the mass of the molecules, and at the same time also of the atoms.

All these investigations have proved with approximate agreement that the diameters of the molecules of different substances are smaller than the ten-millionth part of a millimetre, but at the same time not indefinitely smaller than this.

These approximations agree fairly well with the determination of the weights of the atoms mentioned in § 21, which also show that certain limits exist within which the value of these extremely small quantities must lie.

§ 60. Aggregation of the Molecules. The particles of matter of which we are cognisant by our senses are produced by the heaping together of the molecules. These, according to the foregoing, must exist in enormous numbers, even in the smallest visible and ponderable mass.

The mode of aggregation of these molecules must vary, and these differences will give rise to the different states of matter. In the solid condition the particles are held together in an unalterable position; in the liquid state they are so held that the particles move easily among one another in such a manner that no two particles remain neighbours for any length of time. Between these two conditions, forming as it were the passage

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between the extremes, we have the soft, plastic, and viscous states of matter in which the particles may move with greater or less difficulty, under the influence of the force of gravity or pressure, without destroying the continuity of the whole mass.

In the gaseous state the attraction of the particles for one another ceases, so that these separate particles move away into space unless they are prevented from doing so by impassable boundaries.

§ 61. The Effect of Heat.—In no one of these conditions can we assume that the particles are in a state of absolute rest; we must rather imagine that in each one the particles possess a certain motion, which is perceptible to us as heat, and this movement becomes the more active the greater the amount of heat the bodies take up. The form of this motion is not fully understood; still in the solid state each particle can only move round a certain fixed position of equilibrium, this motion being either vibratory or rotatory.

In liquids the particles must be imagined as moving over one another, so that they leave no spaces between them, whereas in the gaseous or vaporous condition the particles are so far separated from one another that they move into space along rectilinear paths until they come in contact with some hindrance by which they are diverted from their path. A consequence of an accelerated motion of the particles is to be found in the expansion of bodies by heat, because more space is required for these extended movements.

It is, however, a remarkable fact that in the passage from one state of aggregation to another bodies take up the heat which disappears as such, so that it is no longer recognised by the senses or by the thermometer. This so-called latent heat serves, doubtless in a great part, to produce those movements of the particles which are characteristic of the new condition; in part, perhaps, also to overcome the forces of attraction between the particles, assuming such forces to exist.

The expansion exhibited by the majority of substances in melting may also be attributed to the increase of these external movements.

In addition to the motion of the molecules we must also assume that the atoms constituting these molecules are likewise

in a state of motion, and this again would be altered by the application of heat. In monatomic molecules, consisting only of one atom, as, for instance, in the case of the molecule of gaseous mercury, which has been proved to be monatomic by Kundt and Warburg, such atomic movements will not occur.

§ 62. Homogeneous Solid Bodies. When similar molecules collect together to form a solid aggregate, a solid body is produced, which will have a structure determined entirely by the relative position of the particles. In the formless, or amorphous, condition the arrangement of the particles would be similar in each direction throughout the mass of the body, whilst in the case of crystals in certain directions it would be found to be different from others, and these differences are shown not only in the relation of the external boundaries of the crystals by plane surfaces, but also in any piece of crystal taken from any part of the interior. These differences are shown in the solidity, the hardness, the cleavage of the crystals in certain directions, the expansion by heat, the conduction of heat, the velocity and refraction of light, the colour of the same, and in some cases also in certain peculiar electrical phenomena produced by heating or cooling. Such differences can only find their explanation in a different arrangement of the molecules. We may assume that the molecules are brought nearer together in one direction than they are in another; but the reason for such an arrangement of the molecules must be sought for in the molecules themselves; so we must assume that in these, certain directions are different from others, and that the particles arrange themselves near one another, so that the directions or axes are parallel in all or are otherwise regularly arranged.

All possible regularities with regard to the disposition of points in space have been geometrically investigated by Leonard Sohncke, and their relations to the different systems of crystals established. The greater the symmetry of the distribution of such points, the simpler is the crystal system; and in full accord with this it is found that substances of the simplest composition, as, for instance, the elements and the compounds composed of a few atoms, form, as a rule, crystals belonging to the regular and hexagonal systems; whereas molecules composed of many atoms— for instance, the majority of organic compounds-yield aggregates

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which in a few cases crystallise with little or no evidence of symmetry. In amorphous substances the particles must be imagined as arranged irregularly, for this is the only way in which the particles could be arranged so that in any finite mass all sections would be the same. In many of their properties, e.g. behaviour with polarised light, amorphous bodies resemble the substances crystallising in the regular system; but this is not the case for other properties, such as cohesion, hardness, and cleavage.

§ 63. Heterogeneous Solid Molecular Aggregates.—A solid body may also be produced by the grouping together of different kinds of molecules. Many substances crystallise with water of crystallisation; still these compounds would appear rather to be homogeneous aggregates, for every molecule is united with a definite number of molecules of water, and the molecules so produced are regularly grouped into new and larger ones. A few only of the compounds containing water crystallise in the regular system: as, for instance, the alums, the twenty-four molecules of water being so arranged around the salt molecule as to produce an aggregate homogeneous in all directions. The so-called double salts are similarly constituted to the compounds containing water of crystallisation, and these must be reckoned amongst the homogeneous aggregates, and also all other combinations produced in accordance with the laws of stœchiometry.

The mixed crystals of isomorphous bodies in which the constituents occur in varying and changing proportions must, on the other hand, be considered as heterogeneous aggregates. Thus, for example, the so-called vitriols, that is, the sulphates of magnesium, copper, zinc, iron, manganese, nickel, and cobalt, crystallising with water of crystallisation, may be crystallised together in any proportions, which is true also of other isomorphous substances. This crystallisation together takes place only when the compounds are of analogous constitution, and when the isomorphous constituent is able to take up approximately the same space. If this condition is not exactly satisfied, then an angle of the crystal of one substance would be altered to a greater or less extent by the entrance into that crystal of another body. For instance, calcspar (CaCO3) crystallises in rhombohedra, the angle being 105° 5', whereas