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all the squares eight equal triangles arise, four of which must, consequently, be equal to the moiety of the great quadrate A D. We know that the line A T multiplied by itself is like the area of two triangles, and A K gives the area of two triangles equal to them; the sum of them is therefore four triangles. But the line H T, multiplied by itself, gives likewise the area of four such triangles. We perceive, therefore, that the sum of A T multiplied by itself, added to A H multiplied by itself, is equal to T H multiplied by itself. This is the observation which we were desirous to elucidate. Here is the figure to it:
Quadrangles are of five kinds: firstly, with right (55) angles and equal sides; secondly, with right angles and unequal sides; thirdly, the rhombus, with equal sides and unequal angles; fourthly, the rhomboid, the length of which differs from its breadth, and the angles of which are unequal, only that the two long and the two short sides are respectively of equal length; fifthly, quadrangles with unequal sides and angles.
First kind.—The area of any quadrangle with equal sides and right angles, or with unequal sides and right
