...right-angled triangles, which are similar to the first triangle, and to one another. In every right-angled triangle, the square of the hypothenuse is equal to the sum of the squares of the other two sides; and, in general, any figure described on the hypothenuse is equal to...

...that the three angles of a triangle are equal to two right angles ; and also, that in any right-angled triangle the square of the hypothenuse is equal to the sum of the squares of the other two sides. In like manner did this ingenious gentleman demonstrate, that it was...

...the matter beyond all dispute by an illustration. In the usual mode of proving that in a rectangular triangle, the square of the hypothenuse is equal to the sum of the squares of the other two sides, the square formed upon the h\ pothenuse is divided by lines into two...

...hecatomb, upon bis discovering what is called the Pythagoric theorem, namely, that in a right-angled triangle, the square of the hypothenuse is equal to the sum of the squares of the sides ? As for the Essenes it is not easy to reconcile their not using animal sacrifices...

...the third side may be found, without the aid of the trigonometrical tables, by the proposition, that the square of the hypothenuse is equal to the sum of the squares of the two perpendicular sides (Euc. 47. I.) If the legs be given, extracting the square root...

...be found by the Square Root, without finding the Angles ; according to the following PROPOSITION ; G In every Right Angled Triangle, the Square of the Hypothenuse is equal to the Sum of the Squares of the two Legs. Hence, The Square of the given Leg being subtracted from the Square of the...

...is equal to the rectangle under their sum and difference. «. ED THEOREM XXXIV. IN any Right-angled Triangle, the square of the Hypothenuse, is equal to the Sum of the Squares of the other two Sides. Let ABC be a right-angled triangle, having the right angle o ; then...

...others ; for the three figures will be proportional to the squares of their homologous sides ; now the square of the hypothenuse is equal to the sum of the squares of the two other sides ; therefore, &c. THEOREM. 223. The parts of two chords which cut each...

...others ; for the three figures will be proportional to the squares of their homologous sides ; now the square of the hypothenuse is equal to the sum of the squares of the two other sides ; therefore, &c. V 223. The parts of two chords which cut each other...

...propositions, that similar triangles have their homologous sides proportional, and that in a right-angled triangle the square of the hypothenuse is equal to the sum of the squares of the two other sides, are the basis of the application of algebra to geometry. But there...