452 Mr. T. Muir on the first Extension of the term Area. 



right and the other on the left, he uses a narrow border of sha- 

 ding lines drawn at right angles to the circuit, and always on 

 one and the same side throughout the course (see figs. 2, 3) . 



The next diagram being taken to illustrate the application of 

 the first convention may be passed over, as both conventions are 

 used in the consideration of the third diagram, to which we shall 

 now refer, a, do, en } m i, k (fig. 2) are successive positions 



Fig. 2. 



which the describing line occupies in the course of its parallel 

 motion, a being the point where it comes into existence, and /c 

 the point at which it ceases to exist. With this explanation and 

 the guidance of the conventions, any reader may easily verify the 

 result obtained for the area, which is " + ghikg -f kXvk* + ceSqdc 

 -\- abeqfnopa, so that the part eqSe is reckoned twice, and 

 —ImpXl—fcgf/c^ — becb" Moreover, as it is here observed that 

 some portions of area external to the perimeter are swept in by 

 the moving line only to be swept out by it later in its course, 

 there arises the theorem — " The area which a straight line vary- 

 ing in length from zero to zero describes in moving parallel to 

 itself is equal to that of the figure whose perimeter is described 

 by the extremities of the straight line." 



In the next place is considered the case where the describing 

 straight line during its motion never vanishes (and therefore 

 never changes sign), and finally returns to the same position 



* Misprints occur here in the original through a confusion of the Greek 

 k and the Italic k. 



