230 SECTION MECHANICS. 



probably Amsler's Planimeter. The principle of this, although really 

 very beautiful, is a little intricate, and has not always been well under- 

 stood. It depends upon this principle that if you take a bar of definite 

 length, and give it a small motion, then you may measure the surface 

 swept over by that bar by simply multiplying the length of the bar 

 into the travel of the middle point resolved at right angles to the bar. 

 I can explain this perhaps more clearly by the aid of a diagram. The 

 way that principle is made use of in Amsler's planimeter is this. He 

 puts a mall roller jon tjhe bar in the direction of its own length, and 



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if the bar moves me wheel simply slides, but if transversely it rolls. If 

 the bar moves in any intermediate direction it both rolls and slides, but 

 the wheel records only that component of the motion which is at right 

 angles to the bar. That of course only applies to small motions of the 

 bar, but as it applies to every small motion it must be true for the sum 

 of them as well as for every part. Supposing you have an irregular 

 curve to measure, you make one end of the bar follow the curve and 

 let the other end reciprocate along a certain curve. Its longitudinal 

 motion will have no record taken of it, but the motion at right angles 

 to the bar will be accurately counted by the little wheel. As it moves 

 in one direction, the wheel will run one way, and as it goes round the 

 other, it will go the other way ; and the difference of those two will be 

 recorded when the bar gets into its original position: and that, read to 

 a proper scale, and multiplied into the length of the bar, will give the 

 difference between the line swept out and the true area all round ; 

 consequently the difference between the readings of the wheel in its 

 first position and when it gets back to its first position will give the 

 area. It is quite immaterial what this curve may be so long as one 

 end of the bar moves backwards and forwards along the same curve, 

 no matter what ; the reading will be exactly the same. Those who 

 have used it will be aware that the wheel is not put at its middle point 

 as I have put it, but it may be proved geometrically that so long as you 

 measure a closed curve with a bar, one end of which is always outside 

 the curve, it is immaterial upon what part of the bar you put the little 

 wheel. It is, of course, material, if you are only going to measure a 

 part of the curve, but not when you measure the whole curve. That, 

 then, contains the whole principle of Amsler's Planimeter, that the 

 surface swept out by the bar is measured by the travel of the middle 



