(178.) In the verification of a law whose expression is _quantitative_, not only must its generality be established by the trial of it in as various circumstances as possible, but every such trial must be one of precise measurement. And in such cases the means taken for subjecting it to trial ought to be so devised as to repeat and multiply a great number of times any deviation (if any exist); so that, let it be ever so small, it shall at last become sensible.
(179.) For instance, let the law to be verified be, that _the gravity of every material body is in the direct proportion of its mass_, which is only another mode of expressing Galileo’s law above mentioned. The time of falling from any moderate height cannot be measured with precision enough for our purpose: but if it can be repeated a very great multitude of times _without any loss or gain_ in the intervals, and the whole amount of the times of fall so repeated measured by a clock; and if at the same time the resistance of the air can be rendered _exactly alike_ for all the bodies tried, we have here Galileo’s trial in a much more refined state; and it is evident that almost unlimited exactness may be obtained. Now, all this Newton accomplished by the simple and elegant contrivance of enclosing in a hollow pendulum the same weights of a great number of substances the most different that could be found in all respects, as gold, glass, wood, water, wheat, &c.[44], and ascertaining the time required for the pendulum so charged to make a great number of oscillations; in each of which it is clear the weights had to fall, and be raised again successively, without loss of time, through the same _identical_ spaces. Thus any difference, however inconsiderable, that might exist in the time of one such fall and rise would be multiplied and accumulated till they became sensible. And none having been discovered by so delicate a process in any case, the law was considered verified both in respect of generality and exactness. This, however, is nothing to the verifications afforded by astronomical phenomena, where the deviations, if any, accumulate for thousands of years instead of a few hours.
(180.) The surest and best characteristic of a well-founded and extensive induction, however, is when verifications of it spring up, as it were, spontaneously, into notice, from quarters where they might be least expected, or even among instances of that very kind which were at first considered hostile to them. Evidence of this kind is irresistible, and compels assent with a weight which scarcely any other possesses. To give an example: M. Mitscherlich had announced a law to this effect--_that_ the chemical elements of which all bodies consist are susceptible of being classified in distinct groups, which he termed _isomorphous_ groups; and _that_ these groups are so related, that when similar combinations are formed of individuals belonging to two, three, or more of them, such combinations will crystallize in the same geometrical forms. To this curious and important law there appeared a remarkable exception. According to professor Mitscherlich, the arsenic and phosphoric acids _are_ similar combinations coming under the meaning of his law, and their combinations with soda and water, forming the salts known to chemists under the names of arseniate and phosphate of soda, ought, if the law were general, to crystallize in identical shapes. The fact, however, was understood to be otherwise. But lately, Mr. Clarke, a British chemist, having examined the two salts attentively, ascertained the fact that their compositions deviate essentially from that similarity which M. Mitscherlich’s law requires; and that, therefore, the exception in question disappears. This was something: but, pursuing the subject further, the same ingenious enquirer happily succeeded in producing a _new_ phosphate of soda, differing from that generally known in containing a different proportion of water, and agreeing in composition exactly with the arseniate. The crystals of this new salt, when examined, were found by him to be precisely identical in form with those of the arseniate: thus verifying, in a most striking and totally unexpected manner, the law in question, or, as it is called, the law of isomorphism.