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Whether the admissibility of these postulates rests on intuition, or on proof, may be a matter of dispute; but in either case they are the premises on which the theorems depend; and while these are retained it would make no difference in the certainty of geometrical truths, though every definition in Euclid, and every technical term therein defined, were laid aside.
All that is left to us of musical effort before the seventeenth century, proves to me that early musicians knew melody only; they were ignorant of harmony and its immense resources. Music is at once a science and an art. It is rooted in physics and mathematics, hence it is a science; inspiration makes it an art, unconsciously utilizing the theorems of science.
From whence, we see that all Theorems too abstruse for Vulgar Apprehensions, which Christianity is believ'd to Teach, however Divine Truths, are yet no part of the Doctrine of Salvation.
And then, that I did not fall into the ambition of ordinary sophists, either to write tracts concerning the common theorems, or to exhort men unto virtue and the study of philosophy by public orations; as also that I never by way of ostentation did affect to show myself an active able man, for any kind of bodily exercises.
Had the real or supposed advantages of introducing motion into geometrical conceptions been suggested to Euclid, we may suppose him to have replied that the theorems of space are independent of time; that the idea of motion necessarily implies time, and that, in consequence, to avail ourselves of it would be to introduce an extraneous element into geometry.
Now this is also precisely what we mean, when we say that some tyrants are lawful sovereigns; which, therefore, is not a second proposition inferred from the first, any more than the English translation of Euclid’s Elements is a collection of theorems different from, and consequences of, those contained in the Greek original.
She had reason'd likewise from her own powers laid down theorems drawn consequences, and come to no conclusion. To clear up all, she had twice asked Doctor Slop, 'if poor captain Shandy was ever likely to recover of his wound ? He is recovered, Doctor Slop would say What! quite? Quite: madam But what do you mean by a recovery? Mrs. Wadman would say.
God is, and therefore all properties of him are, just as every property of a circle exists in it as soon as the circle exists. We may if we like, for convenience, throw our theorems into the future, and say, e.g. that if two lines in a circle cut each other, the rectangle under the parts of the one will equal that under the parts of the other.
There are, undoubtedly, the most ample reasons for stating both the principles and the theorems in their general form, and these will be explained presently, so far as explanation is requisite.
The Analysis appeared in 1829, and, whatever its defects of incompleteness and one-sidedness from a philosophical point of view, shows in the highest degree Mill's powers of close, vigorous statement; and lays down with singular clearness the psychological doctrine, which from his point of view supplied the fundamental theorems of knowledge in general.