...proceeds entirely from the undeterminate meaning of words, which is corrected by juster definitions. That the square of the hypothenuse is equal to the squares of the other two sides, cannot be known, let the terms be ever so exactly defined, without a train of reasoning...

...Indian thinkers had framed out the atomic theory, and had, it is said, found the proof of the theorem that the square of the hypothenuse is equal to the squares of the other two sides of a right-angled triangle. In much later times the decimal system was perfected by...

...opinions that there must necessarily be hostile mathematical sects, some affirming and some denying, that the square of the hypothenuse is equal to the squares of the sides. But we do not think either the one analogy or the other of the smallest value. Our way of ascertaining...

...opinions that there must necessarily be hostile mathematical sects, some affirming, and some denying, that the square of the hypothenuse is equal to the squares of the sides. But we do not think either the one analogy or the other of the smallest value. Our way of ascertaining...

...matters of fact and relations of ideas. An example of what he means by the relations of ideas is, ' That the square of the hypothenuse is equal to the squares of two sides,' which Hume says, * is discoverable by the mere operation of thought, without dependence...

...angles of every rectilinear triangle are altogether equal to two right angles. In right-angled triangles the square of the hypothenuse is equal to the squares of the remaining sides. On these truths vast sciences have been reared. What the first principles of Euclid...

...proceeds entirely from the undeterminate meaning of words, which is corrected by juster definitions. That the square of the hypothenuse is equal to the squares of the other two sides, cannot be known, let the terms be ever so exactly defined, without a train of reasoning...

...arithmetic; and in short, every affirmation which is either intuitively or demonstratively certain. That the square of the hypothenuse is equal to the squares of the two sides, is a proposition which expresses a relation between these figures. That three times five...

...them'; and changed his mind only when he came to the proposition, 1,47, 'that in a right angled Triangle the square of the Hypothenuse is equal to the squares of the two other sides'. A story is only a story. But look at Newton's copy 13 of Barrow's 1 655 Latin epitome...

...demonstrative reasoning when establishing arithmetical propositions as well as geometrical propositions such as "the square of the hypothenuse is equal to the squares of the other two sides,"2 whereas we use experimental reasoning to show that fire burns or "that a body at...