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 provedb.

b Peacock, Algebra,' vol. ii. p. 359. Serret, Algèbra Supérieure,' 2nd ed. p. 289.

CHAPTER XIX.

EXPERIMENT.

We now come to consider the great facilities which we enjoy for examining the possible combinations of properties and phenomena when objects are within our reach and capable of manipulation. We are said to experiment, when we bring substances together under various conditions of temperature, pressure, electric disturbance, molecular attraction, &c., and then record the changes observed.

If we denote by A a certain group of antecedent conditions, and by X a certain series of subsequent phenomena, our object will usually be to ascertain a law of the form A = AX, the meaning of which is that where A is X will happen, and we may sometimes rise to the still simpler and higher law A = X, meaning that where A is, and only where A is, X will happen (see vol. i. pp. 146, 149.)

The great object of the art of experiment is to ascertain exactly those circumstances or conditions which are requisite for the happening of any event X. Now the circumstances which might be enumerated as present in the very 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 properties; the temperature and other powers 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. Now on à priori grounds it is unsafe to assume that any one of these circumstances is without effect, and it is only on the results of experience that we can finally 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 practically 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 an independence to exist among the conditions. Thus 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. Now 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, aBCD,

but we ought really to go through the whole of the combinations given in the fifth column of the Logical Abecedarium. The effect of the absence of each condition should be tried both in the presence and absence of every other condition, and every variety of selection of those conditions. Perfect and exhaustive experimentation would, in short, consist in examining natural phenomena in all their possible combinations and registering all relations between conditions and results which are found capable of existence. Experimentation would thus resemble the exclusion of contradictory combinations carried out in the Indirect Method of Inference (chapter vi. vol. i. p. 95), except that the exclusion of any combination is grounded not on prior logical premises, but on à posteriori results of actual trial.

The reader will readily perceive, however, that such exhaustive investigation is practically impossible, because the number of requisite experiments would be immensely great. Four circumstances only would require sixteen experiments; twelve circumstances 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 variations 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 com

binations; but this being impracticable, the experimentalist necessarily abandons strict logical method, and trusts to his own insight. Analogy, as we shall afterwards see, gives some assistance, and attention will probably 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 entirely overlooking the one condition whose importance has been hitherto unsuspected. 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 a similar manner we know not what slight alteration in the most familiar experiments may not open the way to realms of new discovery.

Many additional practical difficulties 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 unsuspected conditions, which we of course cannot remove except by accident. These difficulties we will shortly consider in succession.

It is often beautiful to observe how the alteration of a single circumstance conclusively explains a phenomenon. An excellent 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