Teaching of elementary mathematics

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1902 Excerpt: ...made a beginning before Euclid's time. Thymaridas of Paros, whose personal history is quite unknown, had already solved some simple equations, and had been the first to use the expressions given or defined (wpiafievoi), and unknown or undefined (aopiaToi),2 and it seems not improbable that the quadratic equation was somewhat' familiar before the Alexandrian school was founded.3 Aristotle, too, had employed letters to indicate unknown quantities in the statement of a problem, although not in an equation.4 2 Cantor, I, p. 148; Gow, p. 97, 107. 8 Cantor, I, p. 301; but see Heath's Diophantos, p. 139. 4 Gow, p. 105. The most notable advance before the Christian era was made by Heron of Alexandria, about 100 B.C. Breaking away from the pure geometry of his predecessors, and not hesitating to speak of the fourth power of lines, he solved the quadratic equation1 and even ran up against imaginary roots.2 This was the turning-point of Greek mathematics, the downfall of their pure geometry, the rise of a new discipline. But it is to Diophantus that we owe the first serious attempt to work out this new science. An Alexandrian, living in the fourth century, probably in the first half, he wrote a work, 'Api6'/MjrtKa, almost entirely devoted to algebra.3 This work is the first one known to have been written upon algebra alone (or chiefly). Diophantus uses only one unknown quantity, 6 apiO/uk or 6 aopurTos api0fjuh, symbolizing it by?' or s'.4 The square he calls Svvafiii, power (its symbol 8), the cube Kiso? («), and he also gives names to the fourth, fifth, and sixth powers. He has symbols for equality and for subtraction, and the modern expression xs--$x2 + 8x--1 he would write in the form Kid0'rjBiefid,1 a form not particularly more difficult than our own. The na...