ON PLANIMETERS. 



499 



The point T describes during this process the whole boundary of the 

 area, going always in the same sense along it. If this sense be changed, 

 the motion of QT will also be reversed, and consequently the area will 



become negative. 



This shows that the sense of an area is determined by the sense given to 

 its boundary ; and, further, that the area is jjositive if it lies to the left of a 

 person tvho goes round the boundary in its positive sense. If it lies to the 



right it will be negative. 



In the application of the Integral Calculus to the evaluation of areas 



Fig. 3. 



the rule of sign generally adopted is in case of rectangular co-ordinates 

 the reverse to the one here adopted; in using polar co-ordinates, however, 

 it agrees with it. As it would be very inconvenient to use here two rules, 

 a choice had to be made ; and I consider on the whole the rule adopted as 

 the one most generally useful. 



If the closed curve cuts itself, as in fig. 5, take anywhere in the 



plane a line OX, not shown in the figure, then a point T on the curve, 

 and draw the perpendicular TQ to OX. On moving T along the whole 

 curve til it returns to its starting-point the line QT will sweep over a 

 perfectly definite area, and this is taken as the area enclosed by the <^iven 

 curve. To see what it means we may apply again the rule of si^n and 

 see how often and in what sense the line QT passes over any given point. 

 It will be found that every point without the curve is passed over either 



K K 2 



