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Take, for instance, any of the definitions laid down as premisses in Euclid’s Elements; the definition, let us say, of a circle.
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.
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.
Yet there have been and still are geometricians and philosophers, and even some of the most distinguished, who doubt whether the whole universe, or to speak more widely the whole of being, was only created in Euclid’s geometry; they even dare to dream that two parallel lines, which according to Euclid can never meet on earth, may meet somewhere in infinity.
The definition of the circle, therefore, is to one of Euclid’s demonstrations, exactly what, according to Stewart, the axioms are; that is, the demonstration does not depend on it, but yet if we deny it the demonstration fails.
Take, for instance, any of the definitions laid down as premises in Euclid’s Elements; the definition, let us say, of a circle.
At the age of sixteen Pascal had already acquired a scientific reputation. He was still only twelve years of age, but Euclid’s Elements, as soon as put into his hands, were mastered by him without any explanation. By-and-by he began to take an active part in the scientific discussions which took place at his father’s house; and his achievement in Conic Sections has been already narrated.
Even were the subject-matter without meaning, though in truth the style cannot really be abstracted from the sense, still the style would, on that supposition, remain as perfect and original a work as Euclid’s elements or a symphony of Beethoven.
The definition of the circle, therefore, is to one of Euclid’s demonstrations, exactly what, according to Stewart, the axioms are; that is, the demonstration does not depend on it, but yet if we deny it the demonstration fails.
Such, for instance, would be Euclid’s Elements; they relate to truths universal and eternal; they are not mere thoughts, but things: they exist in themselves, not by virtue of our understanding them, not in dependence upon our will, but in what is called the nature of things, or at least on conditions external to us.
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