Excerpt from The Contents of the Fifth and Sixth Books of Euclid: With a Note on Irrational Numbers The argument is made to relate to a thing which is completely defined, viz., the relative multiple scale of two magnitudes; whilst in Euclid's argument it is not made clear what a ratio is; and the lack of information on this point is a serious obstacle to the learner. The determination of the stage at which the idea of ratio should be introduced into the argument is one of great difficulty. Complex as the idea is, it is ...
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Excerpt from The Contents of the Fifth and Sixth Books of Euclid: With a Note on Irrational Numbers The argument is made to relate to a thing which is completely defined, viz., the relative multiple scale of two magnitudes; whilst in Euclid's argument it is not made clear what a ratio is; and the lack of information on this point is a serious obstacle to the learner. The determination of the stage at which the idea of ratio should be introduced into the argument is one of great difficulty. Complex as the idea is, it is formed by every one at an early age. As soon as a child can recognise an object from a drawing of it, he has formed the idea of similar figures, and therefore he is able to see that the ratio of two of the dimensions of the object is the same as that of the corresponding dimensions of the drawing. When however the use of the idea, and its introduction into Algebra and Geometry are under consideration, its complexity becomes apparent. There is no necessity, arising out of the nature of the subject, for introducing the idea of ratio into the statement of any of the Propositions in the Fifth Book, with the possible exception of the 8th, 10th and 13th, which deal with unequal ratios and are not required for the Sixth Book. It is not until the subject of Compounding Ratios is reached that the introduction of the idea becomes desirable. I believe that it will be ultimately recognised that it is best to postpone its introduction until the stage just mentioned has been reached. At the same time I have not ventured to do this in this book, as I desire to conform to established practice, so far as is possible. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at ... This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
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