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Updated: May 10, 2025
I therefore began, at the beginning, and before spring I had gone through that old Euclid's geometry, and could demonstrate every proposition like a book. Then in the spring, when I had got through with it, I said to myself one day, 'Ah, do you know now when a thing is proved? And I answered, 'Yes, sir, I do. 'Then you may go back to the law shop; and I went."
Whether Euclid's axioms are true, is a question as to which the pure mathematician is indifferent; and, what is more, it is a question which it is theoretically impossible to answer with certainty in the affirmative. Thus the geometer leaves to the man of science to decide, as best he may, what axioms are most nearly true in the actual world.
Once, made suddenly conscious by defeat of his lack of book-culture, he took up Euclid's geometry, and resolutely studied and re-studied it. Doubtless that helped him in the close logic which often characterized his speeches.
What giant power has cleft the way for these waters Vulcan or Neptune? Or was it laid down in Euclid's adventurous age, when the Titans went into bankruptcy? The train increases its speed to regain the time lost in uncoupling the disabled parlour-car, and this increased speed is chiefly felt at the tail of the great iron dragon. I have to cling tightly to the brass rod in front of the windows.
Further assuming the bullet to have illustrated Euclid's definition of a straight line, such a line, followed back from the yew tree to the spot where the rifle rested, would pass through the hole in the cardboard! In other words, there is only one place from which it is possible to see the flame of the candle through the hole in the cardboard: the place where the rifle rested!
"Bravo!" murmured Mr. Marrier. Edward Henry in the gloom caught Mr. Seven Sachs's unalterable observant smile across the table. "Well, Mr. Machin?" said Carlo Trent. Edward Henry had felt a tremor at the vibrations of Rose Euclid's voice.
Then he takes Euclid's proposition to the effect that the superficial area of globes increases in the proportion of the square of the diameter, whilst the volume increases in the proportion of the cube of the same diameter, and he considers that if one only constructs the globe of thin metal, of sufficient size, and exhausts the air in the manner that he suggests, such a globe will be so far lighter than the surrounding atmosphere that it will not only rise, but will be capable of lifting weights.
'God geometrizes, it used to be said; and it was believed that Euclid's elements literally reproduced his geometrizing. There is an eternal and unchangeable 'reason'; and its voice was supposed to reverberate in Barbara and Celarent.
So inordinate was his desire of making speedy advances in the paths of learning, that within the compass of three months, he diligently perused the writings of Locke and Malebranche, and made himself master of the first six and of the eleventh and twelfth books of Euclid's Elements.
Lastly, we must assume certain other things which are less obvious and cannot be proved but yet have to be accepted; these are called postulates, because they make a demand on the faith of the learner. Euclid's Postulates are of this kind, especially that known as the parallel-postulate.
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