...acrimony. No mathematician ever impeached the sceptic who should deny that in a right angled triangle the square of the hypothenuse is equal to the squares of the other two sides; he is so thoroughly convinced of the truth of this proposition, that incredulity provokes a smile,...

...are already developed, must compassionate us as much when we cry out with rapture at discovering that the square of the hypothenuse is equal to the squares of the two sides, as we should look with pity upon an emmet, triumphantly exclaiming, that the straw on which...

...affirmation that there really was nothing for the comparing mind to perceive ? The proposition, that the square of the hypothenuse is equal to the squares of the other two sides, is a proposition which does not surely express a relation between the figure of a right-angled triangle...

...single right-angled triangle, it would be quite as " absolutely necessary," and eternally true, that the square of the hypothenuse is equal to the squares of the other two sides, as if the whole Universe had consisted of nothing else but right-angled triangles. But Relative Necessity...

...power of at once enabling Zerah Colburn to perform his mental calculations, Pythagoras to prove that the square of the hypothenuse is equal to the squares of the other two sides of a right-angled triangle, and Laplace to write " La Mecanique Celeste." Observation has corrected...

...evening star was the same : he was the first also to demonstrate that in every rightangled triangle, the square of the hypothenuse is equal to the squares of the other two sides. It is said that Pythagoras was so transported with the discovery of this famous theorem, that, supposing...

...corresponding sides, the areas of the circles will also be proportional to the squares of their radii. Thus, as the square of the hypothenuse is equal to the squares of the two sides, a circle, of which this hypothenuse is the radius, will be equal to two circlea having for...

...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...

...when applied to the ordinary exercises of the understanding. One mathematical student believes that the square of the hypothenuse is equal to the squares of the sides inclosing the right angle, because he has read Euclid's forty-seventh proposition. Another, wholly...

...the two following; the sides of similar triangles are proportionals; and in right-angled triangles the square of the hypothenuse is equal to the squares of the two sides. And I am not afraid to suppose several unknown quantities, that I may reduce the proposed...