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Updated: May 10, 2025
It is not twenty years since English mathematicians could deplore the fact that, despite certain rather obvious defects of the work of Euclid, no better textbook than this was available. Euclid's work, of course, gives expression to much knowledge that did not originate with him.
With algebraic formulae and Euclid's propositions her fine memory saved her. But with quick intuition she threw herself frankly upon the boy's generosity, and in the evenings together they, with Margaret's assistance, wrestled with the bewildering intricacies of arithmetical problems.
In the first place, he was a boy just fresh from the rougher associations of school life; and, secondly, his inquiring mind was intently occupied in endeavouring to solve a series of mathematical problems that set all Euclid's laws at defiance, as the train whizzed on its way with a `piff-paff! pant-pant! of the great Juggernaut engine, the carriages rattling and jolting as they were dragged along at the tail of the mighty steam demon, swaying to and fro with a rhythmical movement of the wheels, in measured cadence of spondees and dactyls, as if singing to themselves the song of "the Iron Road."
For this purpose the Father worketh hitherto, and Christ works, that man's will may yield and bow itself wholly and happily and lovingly to the great infinite will of the Father in heaven. Brethren! that is the perfection of a man's nature, when his will fits on to God's like one of Euclid's triangles superimposed upon another, and line for line coincides.
Euclid's axioms are useful because they are self-evident; and so long as people make mistakes in geometry, it will be necessary to expose their blundering by bringing out the contradictions involved. As Hobbes observed, people would dispute even geometrical axioms if they had an interest in doing so; and, certainly, they are ready to dispute the plainest doctrines about money.
No abstract expression such as Euclid's Elements, Newton's Principia, or Peano's Formulaire, no matter how rigorous and complete, is a work of art. We admire the mathematician's formula for its simplicity and adequacy; we take delight in its clarity and scope, in the ease with which it enables the mind to master a thousand more special truths, but we do not find it beautiful.
It is curious to observe the triumph of slight incidents over the mind: What incredible weight they have in forming and governing our opinions, both of men and things that trifles, light as air, shall waft a belief into the soul, and plant it so immoveably within it that Euclid's demonstrations, could they be brought to batter it in breach, should not all have power to overthrow it.
Kant, rightly perceiving that Euclid's propositions could not be deduced from Euclid's axioms without the help of the figures, invented a theory of knowledge to account for this fact; and it accounted so successfully that, when the fact is shown to be a mere defect in Euclid, and not a result of the nature of geometrical reasoning, Kant's theory also has to be abandoned.
He proposed to take some leading proposition of Euclid's, and show by construction that its truth was known to us, to demonstrate, for example, that the angles at the base of an isosceles triangle are equal, and that if the equal sides be produced the angles on the other side of the base are equal also, or that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the two other sides.
It is enough to remark that for quite a long time there had been an awareness of the fact that the consistency of Euclid's definitions and proofs fails as soon as one has no longer to do with finite geometrical entities, but with figures which extend into infinity, as for instance when the properties of parallel straight lines come into question.
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