It is far too much the custom now to rely on the analogy of algebra to justify the introduction of imaginaries into geometry. Analogy, however, is no justification unless we first prove the exact correspondence of the fields of investigation. In analytical geometry the identification of the two fields is permissible, and is easily explained; but in pure geometry any reference to algebra, expressed or implied, is irrelevant and misleading. The elements of pure geometry have no dependence on calculation.

With his unaided faculties, a man is comparatively limited in performance. His vision, hearing and legs operate only over small distances, he cannot lift very heavy weights nor weave much cloth in an hour. But by making use of devices such as the aeroplane, the telephone, and the powered loom, he can exploit natural properties and forces, and thereby attain what is literally a superhuman standard of performance in all kinds of activities.

At the beginning of the seventeenth century π had already been calculated to 35 places of decimals by the Archimedean method of finding in succession the perimeters of inscribed and circumscribed polygons, each having twice as many sides as its predecessor.

But at this time several mathematicians, notably Snell, were seeking shorter methods of approximation, and finally Huygens in thirteen propositions proved or disproved the assumptions they made, and in three more propositions, making use of the known position of the centre of gravity of a parabolic segment and a theorem of his own on the centre of gravity of a segment of a circle, showed how to find a closer upper limit for π from any polygon than had yet been obtained, and by a seventeenth theorem “on centre of gravity”, of which he does not give the proof or even the enunciation, he found also a closer lower limit.