equation to which there could be no corresponding plane geometrical solution.1 This is a case of reductio ad absurdum, a form of argument of a totally different character. Similarly no number of failures to obtain a general solution of equations of the fifth degree would establish the impossibility of the task, but in an indirect mode, equivalent to a reductio ad absurdum, the impossibility is considered to be proved.

1 Peacock, Algebre, vol. ii. p. 344.

2 Ibid, p. 359. Serret, Algèbre Supérieure, 2nd ed. p. 304.

CHAPTER XIX.

EXPERIMENT.

WE may now consider the great advantages which we enjoy in examining the combinations of phenomena when things are within our reach and capable of being experimented on. We are said to experiment when we bring substances together under various conditions of temperature, pressure, electric disturbance, chemical action, &c., and then record the changes observed. Our object in inductive investigation is to ascertain exactly the group of circumstances or conditions which being present, a certain other group of phenomena will follow. If we denote by A the antecedent group, and by X subsequent phenomena, our object will usually be to discover a law of the form A AX, the meaning of which is that where A is X will happen.

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The circumstances which might be enumerated as present in the simplest experiment are very numerous, in fact almost infinite. Rub two sticks together and consider what 'would be an exhaustive statement of the conditions. There are the form, hardness, organic structure, and all the chemical qualities of the wood; the pressure and velocity of the rubbing; the temperature, pressure, and all the chemical qualities of the surrounding air; the proximity of the earth with its attractive and electric powers; the temperature and other properties of the persons producing motion; the radiation from the sun, and to and from the sky; the electric excitement possibly existing in any overhanging cloud; even the positions of the heavenly bodies must be mentioned. On à priori grounds it is

unsafe to assume that any one of these circumstances is without effect, and it is only by experience that we can single out those precise conditions from which the observed heat of friction proceeds.

The great method of experiment consists in removing, one at a time, each of those conditions which may be imagined to have an influence on the result. Our object in the experiment of rubbing sticks is to discover the exact circumstances under which heat appears. Now the presence of air may be requisite; therefore prepare a vacuum, and rub the sticks in every respect as before, except that it is done in vacuo. If heat still appears we may say that air is not, in the presence of the other circumstances, a requisite condition. The conduction of heat from neighbouring bodies may be a condition. Prevent this by making all the surrounding bodies ice cold, which is what Davy aimed at in rubbing two pieces of ice together. If heat still appears we have eliminated another condition, and so we may go on until it becomes apparent that the expenditure of energy in the friction of two bodies is the sole condition of the production of heat.

The great difficulty of experiment arises from the fact that we must not assume the conditions to be independent. Previous to experiment we have no right to say that the rubbing of two sticks will produce heat in the same way when air is absent as before. We may have heat produced in one way when air is present, and in another when air is absent. The inquiry branches out into two lines, and we ought to try in both cases whether cutting off a supply of heat by conduction prevents its evolution in friction. The same branching out of the inquiry occurs with regard to every circumstance which enters into the experiment.

Regarding only four circumstances, say A, B, C, D, we ought to test not only the combinations ABCD, ABCd, ABCD, AbCD, a BCD, but we ought really to go through the whole of the combinations given in the fifth column of the Logical Alphabet. The effect of the absence of each condition should be tried both in the presence and absence of every other condition, and every selection of those conditions. Perfect and exhaustive experimentation would, in short, consist in examining natural phenomena in all their possible combinations and registering all

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relations between conditions and results which are found capable of existence. It would thus resemble the exclusion of contradictory combinations carried out in the Indirect Method of Inference, except that the exclusion of combinations is grounded not on prior logical premises, but on à posteriori results of actual trial.

The reader will perceive, however, that such exhaustive investigation is practically impossible, because the number of requisite experiments would be immensely great. Four antecedents only would require sixteen experiments; twelve antecedents would require 4096, and the number increases as the powers of two. The result is that the experimenter has to fall back upon his own tact and experience in selecting those experiments which are most likely to yield him significant facts. It is at this point that logical rules and forms begin to fail in giving aid. The logical rule is-Try all possible combinations; but this being impracticable, the experimentalist necessarily abandons strict logical method, and trusts to his own insight. Analogy, as we shall see, gives some assistance, and attention should be concentrated on those kinds of conditions which have been found important in like cases. But we are now entirely in the region of probability, and the experimenter, while he is confidently pursuing what he thinks the right clue, may be overlooking the one condition of importance. It is an impressive lesson, for instance, that Newton pursued all his exquisite researches on the spectrum unsuspicious of the fact that if he reduced the hole in the shutter to a narrow slit, all the mysteries of the bright and dark lines were within his grasp, provided of course that his prisms were sufficiently good to define the rays. In like manner we know not what slight alteration in the most familiar experiments may not open the way to realms of new discovery.

Practical difficulties, also, encumber the progress of the physicist. It is often impossible to alter one condition without altering others at the same time; and thus we may not get the pure effect of the condition in question. Some conditions may be absolutely incapable of alteration; others may be with great difficulty, or only in a certain degree, removable. A very treacherous source of error is the existence of unknown conditions, which of course we

cannot remove except by accident. These difficulties we will shortly consider in succession.

It is beautiful to observe how the alteration of a single circumstance sometimes conclusively explains a phenomenon. An instance is found in Faraday's investigation of the behaviour of Lycopodium spores scattered on a vibrating plate. It was observed that these minute spores collected together at the points of greatest motion, whereas sand and all heavy particles collected at the nodes, where the motion was least. It happily occurred to Faraday to try the experiment in the exhausted receiver of an airpump, and it was then found that the light powder behaved exactly like heavy powder. A conclusive proof was thus obtained that the presence of air was the condition of importance, doubtless because it was thrown into eddies by the motion of the plate, and carried the Lycopodium to the points of greatest agitation. Sand was too heavy to be

carried by the air.

Exclusion of Indifferent Circumstances.

From what has been already said it will be apparent that the detection and exclusion of indifferent circumstances is a work of importance, because it allows the concentration of attention upon circumstances which contain the principal condition. Many beautiful instances may be given where all the most obvious antecedents have been shown to have no part in the production of a phenomenon. A person might suppose that the peculiar colours of motherof-pearl were due to the chemical qualities of the substance. Much trouble might have been spent in following out that notion by comparing the chemical qualities of various iridescent substances. But Brewster accidentally took an impression from a piece of mother-of-pearl in a cement of resin and bees'-wax, and finding the colours repeated upon the surface of the wax, he proceeded to take other impressions in balsam, fusible metal, lead, gum arabic, isinglass, &c., and always found the iridescent colours the same. thus proved that the chemical nature of the substance is a matter of indifference, and that the form of the surface is the real condition of such colours. Nearly the same may

1 Treatise on Optics, by Brewster, Cab. Cyclo. p. 117.

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