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The grounds, then, of every rule of art, are to be found in the theorems of science. An art, or a body of art, consists of the rules, together with as much of the speculative propositions as comprises the justification of those rules.
On Astronomical Observations; 5. Mathematical Theorems; 6. On the Arabic Language and its Properties; 7. On the Origin of the Soul and Resurrection of the Body; 8. Demonstration of Collateral Lines on the Sphere; 9. An Abridgment of Euclid; 10. On Finity and Infinity; 11. On Physics and Metaphysics; 12. An Encyclopædia of Human Knowledge, in 20 vols., etc., etc.
I do not say whether they were right or wrong in applying your general arguments to their own case. It is not easy, indeed, to make a monopoly of theorems and corollaries.
Here, as elsewhere, we build up our own ideals out of the fragments to be found in the world; and in the end it is hard to say whether the result is a creation or a discovery. It is very desirable, in instruction, not merely to persuade the student of the accuracy of important theorems, but to persuade him in the way which itself has, of all possible ways, the most beauty.
From one slope to the other of the theorems, it grew to a heavy mass; and the mass became a mighty projectile which, flung backwards and retracing its course, split the darkness and spread it into one vast sheet of light. D'Alembert's precept is good and very good, provided you do not abuse it. Too much precipitation in turning over the intractable page might expose you to many a disappointment.
But if the symbols represent any other things than mere numbers, let us say even straight or curve lines, we have then to apply theorems of geometry not true of all lines without exception, and to select those which are true of the lines we are reasoning about. And how can we do this unless we keep completely in mind what particular lines these are?
Of their many discoveries which have been useful for the development of human life, I will cite a few examples. On reviewing these, people will admit that honours ought of necessity to be bestowed upon them. First of all, among the many very useful theorems of Plato, I will cite one as demonstrated by him.
No less a person, my lord, than the great Professor Hoskins van Huysman!" "What!" exclaimed Leighton, with a laugh that was almost boyish for such a serious and learned young man. "The Huysman: the Professor's most doughty antagonist in the arena of symbols and theorems? Oh, now that is good!"
Who has absolute knowledge of a machine, the student who analyses it in mechanical theorems, or the engineer who has lived in comradeship with it, even to sharing the physical sensation of its laboured or easy working, who feels the play of its inner muscles, its likes and dislikes, who notes its movements and the task before it, as the machine itself would do were it conscious, for whom it has become an extension of his own body, a new sensori-motor organ, a group of prearranged gestures and automatic habits?
Thales the Milesian was the author of several of the geometrical theorems and demonstrations now included in the Elements of Euclid. The celebrated fifth proposition of the first book, so famous among all the modern nations of Europe as the great stumbling block in the way of beginners in the study of geometry, was his.
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