ON PLAXIMETERS. 497 



the same reason the work of Maxwell, Lord Kelvin, Boys, Abdank-Aba- 

 kanowicz, and Hele-Shaw will remain practically unnoticed. 



With regard to the integrating apparatus in particular, there is the well- 

 known paper on ' Mechanical Integrators,' by Professor Hele-8haw, in the 

 ' Proceedings ' of the Institute of Civil Engineers for 1885, in which, besides, 

 many descriptions with figures are given of instruments here treated of. 



In this Report there is first given the geometrical theory of generating 

 areas, together with simple descriptions of planimeters based on it. Then 

 follows a historical sketch up to the invention of Amsler's planimeter. 

 Next, this instrument is considered, its errors are discussed, together 

 with those more modern planimeters which have been constructed with a 

 view to avoid these errors. Lastly, some planimeters are described which 

 have recently been introduced. 



The object of a planimeter is to measure an area ; it has, therefore, to 

 solve a geometrical problem by mechanical means. 



To give at once an idea how this is possible, consider a very simple case. 



If a line AB (fig. 1) of finite length I moves parallel to itself to CD, 

 where AB is perpendicular to AC, then it will sweep over the area of a 

 rectangle which will have the value Iw if 

 tv^A.C. Let the line be replaced by a yig. 1. 



material rod QT, and let a wheel W be 

 mounted on it. On placing this apparatus 

 on the paper above the line AB, the wheel 

 resting on the paper, and moving this rod 

 along to CD, it will describe the same 

 area. At the same time the wheel will ' 

 turn and the arc of its circumference, which 

 comes in contact with the paper, will have 

 the length iv. If the circumference is 



graduated and a fixed index is provided, say, at the highest point, the 

 length of this arc can at once be read off. 



This arc, as read off at the index, may be called conveniently the ' roll ' 

 of the wheel (Macfarlane Gray). 



This instrument may be considered as a simple planimetei-, which, how- 

 ever, measures only the areas of rectangles with fixed altitude, and is, there- 

 fore, practically of no use. Nevertheless it will serve to elucidate a great 

 many properties common to nearly all planimeters. 



First we have a geometrical generation of an area by aid of a moving 

 line, and secondly the ' recording apparatus ' represented by the wheel, with 

 its graduation and index. It is advisable always to keep these two ideas 

 quite separate. The one is geometrical, the other kinematic. The former 

 of these will first of all engage our attention. 



Geometricai. Generation of Areas. 



Our instrument teaches us, if the 'rod' QT is moved one way, the 

 ' roll ' of the wheel will increase, whilst it will decrease when moved in the 

 opposite sense. Hence we must consider the 'sense ' in which the motion 

 takes place ; we shall call the one motion positive, the other negative. At 

 the same time we shall call the area generated positive or negative. It 

 will be seen that in this case alone will the area always be measured by the 

 'roll.' 



If the rod QT is turned round so that Q is above B, and T above A 

 1894. K K 



