*THE POPULAR SCIENCE MONTHLY.*

(Chapter VI.), gives this demonstration, and supposes it to be given for the first time.

The Hindoos, however, were not skilled in geometry. One of their authors even chides another for attempting to prove geometrically what can be seen by experience. One of the aphorisms of the present treatise is as follows: "That figure, though rectilinear, of which sides are proposed by some presumptuous person, wherein one side equals or exceeds the sum of the other sides, may be known to be no figure;" and the proof of this is thus given, "Let straight rods, of the length of the proposed sides, be placed on the ground, and the incongruity will be apparent."

The geometry of the circle in "Lílívatí" is the best feature of the book on plane figures. The "rule" of the text is that the ratio of the diameter to the circumference is 39271250 or 3.1416 exactly.

This is given in the text without demonstration, but one of the commentators thus establishes it: the side of the inscribed hexagon is first found to be equal to the radius; the side of the dodecagon is derived from this; "from which, in like manner, may be found the side of a polygon with twenty-four sides; and so on, doubling the number of sides in the polygon until the side be near to the arc. The sum of such sides will be the circumference of the circle, nearly." The side of the polygon of three hundred and eighty-four sides is found, and the ratio given above is deduced.

The explanation of the method of finding the area of the circle is somewhat indirect, and is likewise ingenious. The circle is divided into two semicircles by a diameter: if this diameter is 14, the semi-circumference is equal to 2112391250. Suppose a number of radii drawn, and the semi-circumference developed into a right line; each half of the circle will become a saw-shaped figure (Fig. 1); placing these

Fig. 1. | ||

together, we should have a rectangle, Fig. 2, of equal area with the circle. This, of course, leads to the formula, , *area circle*

To find the surface of the sphere, and its contents, similar methods are employed.

The following sections are concerned with some practical questions, as the determination of the number of boards which can be cut from a prism of wood, the number of measures of grain in a mound, and formulæ for the length of the shadows of gnomons. Sections on the sub-