33 3 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 " Lilivati " 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 f-f-f-^, 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 de- 

 rived 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 21^--~-. 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 to- 



FlG. 1. 

 22 



Fig. 2. 



m 



zz 



gether, we should have a rectangle, Fig. 2, of equal area with the circle. 

 This, of course, leads to the formula, ixr 2, area circle = 2-rrr.r ix.r 3 . 



2 



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 for- 

 mulas for the length of the shadows of gnomons. Sections on the sub- 



