discovery which is to this day known as the Theorem of PYTHAGORAS—­the law that in every right-angled triangle the square on the side opposite the right angle is equal in area to the sum of the squares on the other two sides.[1] The importance of this discovery can hardly be overestimated.  It is of fundamental importance in most branches of geometry, and the basis of the whole of trigonometry—­the special branch of geometry that deals with the practical mensuration of triangles.  EUCLID devoted the whole of the first book of his Elements of Geometry to establishing the truth of this theorem; how PYTHAGORAS demonstrated it we unfortunately do not know.

[1] Fig. 3 affords an interesting practical demonstration of the truth of this theorem.  If the reader will copy this figure, cut out the squares on the two shorter sides of the triangle and divide them along the lines AD, BE, EF, he will find that the five pieces so obtained can be made exactly to fit the square on the longest side as shown by the dotted lines.  The size and shape of the triangle ABC, so long as it has a right angle at C, is immaterial.  The lines AD, BE are obtained by continuing the sides of the square on the side AB, i.e. the side opposite the right angle, and EF is drawn at right angles to BE.

After absorbing what knowledge was to be gained in Egypt, PYTHAGORAS journeyed to Babylon, where he probably came into contact with even greater traditions and more potent influences and sources of knowledge than in Egypt, for there is reason for believing that the ancient Chaldeans were the builders of the Pyramids and in many ways the intellectual superiors of the Egyptians.

At last, after having travelled still further East, probably as far as India, PYTHAGORAS returned to his birthplace to teach the men of his native land the knowledge he had gained.  But CROESUS was tyrant over Samos, and so oppressive was his rule that none had leisure in which to learn.  Not a student came to PYTHAGORAS, until, in despair, so the story runs, he offered to pay an artisan if he would but learn geometry.  The man accepted, and later, when PYTHAGORAS pretended inability any longer to continue the payments, he offered, so fascinating did he find the subject, to pay his teacher instead if the lessons might only be continued.  PYTHAGORAS no doubt was much gratified at this; and the motto he adopted for his great Brotherhood, of which we shall make the acquaintance in a moment, was in all likelihood based on this event.  It ran, “Honour a figure and a step before a figure and a tribolus”; or, as a freer translation renders it:—­

“A figure and a step onward Not a figure and a florin.”

“At all events, as Mr FRANKLAND remarks, “the motto is a lasting witness to a very singular devotion to knowledge for its own sake."[1]

[1] W. B. FRANKLAND, M.A.:  The Story of Euclid (1902), p. 33