  Analysis is clearly being understood here in the regressive  sense—as involving the working back from ‘what is  sought’, taken as assumed, to something more fundamental by  means of which it can then be established, through its converse,  synthesis. For example, to demonstrate Pythagoras's  theorem—that the square on the hypotenuse of a right-angled  triangle is equal to the sum of the squares on the other two  sides—we may assume as ‘given’ a right-angled triangle with the three squares drawn on its sides. In investigating the properties of  this complex figure we may draw further (auxiliary) lines between  particular points and find that there are a number of congruent  triangles, from which we can begin to work out the relationship  between the relevant areas. Pythagoras's theorem thus depends  on theorems about congruent triangles, and once these—and  other—theorems have been identified (and themselves proved),  Pythagoras's theorem can be proved. (The theorem is  demonstrated in Proposition 47 of Book I of Euclid's  Elements.)