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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 premisses 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.
Besides the theorems of number, those of geometry also come in as premisses, where the effects take place in space, and involve motion and extension, as in mechanics, optics, acoustics, astronomy.
"'On what special subject of the theorems and topics does your desire for vocality seem to be connected with? I asks. "'I ain't particular, says Andy. 'I am equally good and varicose on all subjects. I can take up the matter of Russian immigration, or the poetry of John W. Keats, or the tariff, or Kabyle literature, or drainage, and make my audience weep, cry, sob and shed tears by turns.
No one disputed the proposition, for no one thoroughly understood it. Alban had not the courage to debate his pet theorems at such a time, and the parson was too intent upon denouncing the national want of seriousness to enter upon such abstruse questions as the banker would willingly have discussed.
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 outcome has been that while they have taken up, as they have understood them, the concepts that rule the civic life of these other, maturer nations, they have apprehended and developed these theorems of civic life in the terms and by the logic enforced in that system of control and surveillance known to them by workday experience, the only empirical terms at hand.
It would be a great facility if we could say the good of society consists of such and such elements; one of these elements requires such conditions, another such others; the government, then, which unites in the greatest degree all these conditions, must be the best. The theory of government would thus be built up from the separate theorems of the elements which compose a good state of society.
The ideal of the school was to fit the children for the struggle into which their lives would thrust them, so that the boy who could spell and read and cipher was supposed to be ready for his life work. Those whose ambition led them into the subtleties of Euclid's problems and theorems were supposed to be in preparation for somewhat higher spheres of life.
After a time it arranges them in groups and learns the first elements of number. But when it has advanced to higher mathematics, the blocks, or figures on the blackboard, become only symbols or means of illustrating the great theorems and propositions of that science.
He refers us to mathematics: the things which make up the world are related to God as the properties of a geometrical figure to its concepts, as theorems to the axiom, as the deduction to the principle, which from eternity contains all that follows from it and retains this even while putting it forth.
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