By Florian Cajori

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**Extra resources for A History of Elementary Mathematics**

**Example text**

2 n2 + 2 n -f 1 = the ^ Ascribed to the Pythagoreans is matical discoveries of antiquity The discovery is t^JJ^ one of the "greatest mathe that of Irrational Quantities. usually supposed to have gro\vn out of the If each of the equal 1 study of the isosceles right triangle. legs is taken as unity, then the hypotenuse, being equal cannot be V2, exactly represented by any number what ever. TTe may imagine that other numbers, say 7 or i, were taken to represent the legs in these and all other cases experi to ; mented upon, no number could be found to exactly measure the After repeated failures, doubtless, length of the hypotenuse.

As there cannot be this problem happy thought which could in the of was nothing shape any geometrical figure thinking, it suggest to the eye the existence of irrationals, their discovery The must have resulted from unaided abstract thought. Pythagoreans saw in irrationals a symbol of the unspeakable. The one who first divulged their theory is said to have suf fered shipwreck in consequence, "for the unspeakable and invisible should always be kept secrt^if___ --- 1 ALLMJLS-, p. 42, thinks it more likely that the discovery to cut a line in extreme and mean ratio.

L, p. 78. der Antiken last the solu Multiply und Modernen Alge Leipzig, 1878, p. 269. " l Let a and d be the first term and thus Perhaps the difference in the required arithmetical progression, then by If ; 38J, 291, 20, 10 5^- ? e. Assuming the sum is 60 x If 60, last but = 100. - = = (a (a 2d)] 3d) 4- (a 4cZ), whence d the difference d is 5-J- times the last term. term = 1, he gets his first progression. The should be 100; hence multiply by If, for We have here a method of solution which appears again later among the Hindus, Arabs, and modern the method of false position.