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Proportion; A Substitute for the Fifth Book of Euclid

F. S. Macaulay, G. A. Gibson

1900
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Mathematical Gazette
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... e argument, and asserts that "The conclusion of Wantzel, that the roots cannot be indicated in algebraic language, is equivalent to saying that there are no roots " ! At the beginning of his preface Mr. M'Ginnis tells us that his book appears "at the request of many able mathematicians, teachers, and scholars throughout the United States," and follows this by a list of " a few " of them, which includes, with others, two professors of mathematics (one of whom professes " Languages " as well), a President of a College, a Principal of a High School, and a State Superintendent. Assuming that they have not been unkind enough to play a practical joke, it is difficult to form a high opinion of their intellectual capacity. ; 8vo., pp. 27.) Prof. Gibson's pamphlet has received the formal approval of the Edinburgh Mathematical Society, and is printed in the Proceedings of the Society for the current year. We welcome it as a genuine and not unsuccessful attempt to provide a satisfactory substitute for the Fifth Book of Euclid. By a fortunate coincidence, it appears at the same time as a more elaborate attempt of the same nature by Prof. M. J. M. Hill, F.R.S. ("Euclid, Books V. and VI.," Cambridge University Press.) The coincidence will have happy results if it leads to a general discussion and to some practical improvements in the teaching of proportion in elementary geometry. That such improvements are much needed is very clearly shown by Prof. Hill, both in the book referred to and in the School World for September and October, 1899. We hope that Prof. Hill's work will be reviewed later in these columns, and we refer to it at present solely for the sake of comparison. In criticising Prof. Gibson's pamphlet we are not questioning its present opportuneness and value, but merely giving expression to personal views and predilections on a debateable question of method and procedure. Prof. Gibson advocates an entire departure from Euclid's method by recommending two fundamental alterations: first, that ratio should be defined as a number from the outset, and second, that the consideration of the ratio of like commensurable magnitudes should be separated from and precede the consideration of the ratio of like incommensurable magnitudes. In ? 3 it is apparently implied, though not formally stated, that the symbol -A represents m times the 1th part of the magnitude A, when m and n are positive integers. In ? 6 the following definition is given:-" If A and B be two like magnitudes having a common measure M, so that A = m M, B= n M, and therefore A = -B, the ratio of A to B is defined to be the fraction

doi:10.2307/3603369
fatcat:c7qp7vetnrhy3eqatj3jivf7r4