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The savage who executes unerringly the exact throw which brings down his game, or his enemy, in the manner most suited to his purpose, under the operation of all the conditions necessarily involved, the weight and form of the weapon, the direction and distance of the object, the action of the wind, &c., owes this power to a long series of previous experiments, the results of which he certainly never framed into any verbal theorems or rules.
The given fact is a mark that a certain relation subsists between the quantities of some of the elements concerned; while the fact sought presupposes a certain relation between the quantities of some other elements: now, if these last quantities are dependent in some known manner upon the former, or vice versa, we can argue from the numerical relation between the one set of quantities, to determine that which subsists between the other set; the theorems of the calculus affording the intermediate links.
I know directly, and of my own knowledge, that I was vexed yesterday, or that I am hungry to-day. Examples of truths which we know only by way of inference, are occurrences which took place while we were absent, the events recorded in history, or the theorems of mathematics.
It is to Fourier, also, that we owe certain theorems, by means of which we may ascend from the differential equations to the integrals, and push the solutions in the majority of cases to the final numerical applications. The first memoir of Fourier on the theory of heat dates from the year 1807.
It only remains to add, that his mind will presently arrive at a fit condition for that most valuable of all exercises for the reflective faculties the making of original demonstrations. Such theorems as those appended to the successive books of the Messrs.
Our minds have reacted so violently towards provable logical theorems and demonstrable mechanical or chemical facts that we have become incapable of metaphysical truth, and try to cast out incredible and silly lies by credible and clever ones, calling in Satan to cast out Satan, and getting more into his clutches than ever in the process.
Not that the Supreme Being is the sole universal lawgiver in the sense of a creator of law, whose will alone determines the boundaries of right and wrong. God is the creator of the beings who are the subjects of law. He is the author of law the one lawgiver in the same sense that he, who first discovered a plain figure, may be said to be the author of all theorems, which may be predicated of it.
And for this opinion he gives, as far as we have been able to discover, no reason whatever, except that everybody who vigorously and freely exercises his understanding on Euclid's Theorems assents to them. "The activity of private judgment," he truly observes, "and the unity and strength of conviction in mathematics vary directly as each other."
The answer to this question is the binomial theorem. And of this nature are all the theorems of the science of number. They assert the identity of the result of different modes of formation. They affirm that some mode of formation from x, and some mode of formation from a certain function of x, produce the same number. Such, as above described, is the aim and end of the calculus.
His intellect, burning with a cold, white light,illumined every point, but betrayed no heat of passion. He applied the rules as impartially as if they were theorems of algebra. Time after time the Rooseveltians protested against the holders of contested seats to vote, but he was unmoved because the rule prescribed that the person had a right to vote.
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