74
SPACE AND GEOMETRY
ties of rigid bodies exist only for the smallest time and space elements.
If an oblique-angled triangle having the sides
a, b, and c be displaced in the direction of the side
b, only a and c will, by the principle above stated, describe equivalent parallelograms, which are alike in an equal pair of parallel sides on the same parallels. If a make with b a right angle, and. the triangle be displaced at right angles to c, the distance
c, the side c will describe the square c2, white the two other sides will describe parallelograms the combined areas of which are equal to the area of the
Fig. 11.
square. But the two parallelograms are, by the observation which just precedes, equivalent respectivelj to a2 and b2,—and with this the Pythagorean theorem is reached.
The same result may also be attained (Fig. ri] by first sliding the triangle a distance a at right angles to a, and then a distance b at right angles tcit does not lead in this instance to new points of view. Conservation of volume is a property which rigid and liquid bodies possess in common, and was idealized by the old physics as impenetrability. In the case of rigid bodies, we have the additional attribute that the distances between all the parts are preserved, while in the case of liquids, the proper-that neither the Greek geometrical nor the Hindu arithmetical deductions of the so-called Pythagorean Theorem could avoid the consideration of areas. One essential point on