Chapter 13
Origin of the Irrational

94. The Discovery of Irrational Lines. The Greeks attributed the discovery of the Irrational to the mathematician and philosopher Pythagoras³⁸ (525 b. c.).

If, as is altogether probable,³⁹ the most famous theorem of Pythagoras—that the square on the hypothenuse of a right triangle is equal to the sum of the squares on the other two sides—was suggested to him by the fact that $3^2+4^2=5^2$, in connection with the fact that the triangle whose sides are 3, 4, 5, is right-angled,—for both almost certainly fell within the knowledge of the Egyptians,—he would naturally have sought, after he had succeeded in demonstrating the geometric theorem generally, for number triplets corresponding to the sides of any right triangle as do 3, 4, 5 to the sides of the particular triangle.

The search of course proved fruitless, fruitless even in the case which is geometrically the simplest, that of the isosceles right triangle. To discover that it was necessarily fruitless; in the face of preconceived ideas and the apparent testimony of the senses, to conceive that lines may exist which have no common unit of measure, however small that unit be taken; to demonstrate that the hypothenuse and side of the isosceles right triangle actually are such a pair of lines, was the great achievement of Pythagoras.⁴⁰

95. Consequences of this Discovery in Greek Mathematics. One must know the antecedents and follow the consequences of this discovery to realize its great significance. It was the first recognition of the fundamental difference between the geometric magnitudes and number, which Aristotle formulated brilliantly 200 years later in his famous distinction between the continuous and the discrete, and as such was potent in bringing about that complete banishment of numerical reckoning from geometry which is so characteristic of this department of Greek mathematics in its best, its creative period.

No one before Pythagoras had questioned the possibility of expressing all size relations among lines and surfaces in terms of number,—rational number of course. Indeed, except that it recorded a few facts regarding congruence of figures gathered by observation, the Egyptian geometry was nothing else than a meagre collection of formulas for computing areas. The earliest geometry was metrical.

But to the severely logical Greek no alternative seemed possible, when once it was known that lines exist whose lengths—whatever unit be chosen for measuring them—cannot both be integers, than to have done with number and measurement in geometry altogether. Congruence became not only the final but the sole test of equality. For the study of size relations among unequal magnitudes a pure geometric theory of proportion was created, in which proportion, not ratio, was the primary idea, the method of exhaustions making the theory available for figures bounded by curved lines and surfaces.

The outcome was the system of geometry which Euclid expounds in his Elements and of which Apollonius makes splendid use in his Conics, a system absolutely free from extraneous concepts or methods, yet, within its limits, of great power.

It need hardly be added that it never occurred to the Greeks to meet the difficulty which Pythagoras’ discovery had brought to light by inventing an irrational number, itself incommensurable with rational numbers. For artificial concepts such as that they had neither talent nor liking.

On the other hand, they did develop the theory of irrational magnitudes as a department of their geometry, the irrational line, surface, or solid being one incommensurable with some chosen (rational) line, surface, solid. Such a theory forms the content of the most elaborate book of Euclid’s Elements, the 10th.

96. Approximate Values of Irrationals. In the practical or metrical geometry which grew up after the pure geometry had reached its culmination, and which attained in the works of Hero the Surveyor almost the proportions of our modern elementary mensuration,⁴¹ approximate values of irrational numbers played a very important rôle. Nor do such approximations appear for the first time in Hero. In Archimedes’ “Measure of the Circle” a number of excellent approximations occur, among them the famous approximation $\frac{22}{7}$ for $\pi$, the ratio of the circumference of a circle to its diameter. The approximation $\frac{7}{5}$ for $\sqrt{2}$ is reputed to be as old as Plato.

It is not certain how these approximations were effected.⁴² They involve the use of some method for extracting square roots. The earliest explicit statement of the method in common use to-day for extracting square roots of numbers (whether exactly or approximately) occurs in the commentary of Theon of Alexandria (380 a. d. ) on Ptolemy’s Almagest. Theon, who like Ptolemy employs sexagesimal fractions, thus finds the length of the side of a square containing $4500^{\circ }$ to be $67^{\circ }{1}^{\prime }55^{\prime \prime }$.

97. The Later History of the Irrational is deferred to the chapters which follow (§§ 106, 108, 112, 121, 129).

It will be found that the Indians permitted the simplest forms of irrational numbers, surds, in their algebra, and that they were followed in this by the Arabians and the mathematicians of the Renaissance, but that the general irrational did not make its way into algebra until after Descartes.