292 



THE HUMAN MOTOR 



A by a universal joint. The point P is fixed at a point in the 

 plane of the curved figure S, whose area it is desired to ascertain, 

 and the point T is moved over the boundary of the figure. 



During this movement a roller R, fixed to the rod, turns through 

 an arc proportional to the area to be measured. Then S = 

 K X n, K, being an instrumental constant, n the number of 

 revolutions and fractions of revolutions of the roller. A counter 

 on the rod AP gives the number of complete revolutions and a 

 vernier, fixed to the roller, the fractions of a revolution. 



Suppose 2,500 units to be indicated before, and 2,560 after the 

 measurement. Then n = 2,560 2,500 = 60. Again the joint 

 A, being placed at a point of the rod AT, the value of K is found 

 at this point ; it is marked thereon in square millimetres. 



Let K = 9 square millimetres. Then : 



S = Kx=9x60 = 540 square millimetres. 



The correction of the errors, or the compensation, is made 

 by leaving the point P where 

 it was, and bringing the roller 



to a position symmetrically [^ A ^-^ "*%. 



opposite to the first position 

 (fig. 200). In reality, there- 

 fore, two measurements, Sj 

 and S 2 , are made ; the exact 

 value will be : 



c _ 



+ S 



A simple way of finding areas is that of weighing the graphs 



( 80) and comparing them 

 with the weight of a square 

 centimetre of the same paper. 

 Homogeneous paper must be 

 used. The error is about 5%. 



Finally, it can be seen that 

 the area of a graph is equal 

 to the sum, either of a series 

 of small equal rectangles, as 

 already seen, or more exactly 

 of a series of small trapeziums. 



y y 



d d 



Fir. 201 



Area of a graph (quadrature) 



In this case, divide the base 

 or the line of the abscissae Ox 

 into a sufficiently large number 



of equal parts, d (fig. 201). The value of S is" furnished by 



Poncelet's formula: 



