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LII. Notes on a Geometrical Construction for rectifying 

 any Arc of a Circle. By F. A. LlNDEMANN *. 



NOTES have been published recently by M. de Pulligny 

 and byR. E. Baynes giving geometrical constructions 

 for the ratio tt or some simple function of it. All of these are 

 based upon some numerical coincidence which enables it or the 

 function in question to be represented very closely by a ratio 

 of fairly small whole numbers such as 355/113. The following 

 construction may perhaps be of interest as it allows any arc 

 of a circle to be rectified, and as it is based upon no such 

 numerical coincidence but represents an extremely rapidly 

 converging series. In principle, an extraordinary degree of 

 accuracy is obtainable in a very short time ; in practice, 

 it need scarcely be said, it is of no more value for this 

 purpose than any of the constructions whose accuracy can 

 only be verified a posteriori. 



Let AB be the arc whose length is to be determined. 



Draw AT the tangent to AB at the point A. 



Continue OB to D and draw BO parallel to OA. 



Bisect < DBC by line BE and < BAT by line AE which 

 cuts BF at E. 



Draw EG parallel to OA. 



Bisect < FEG by line EH and < EAT by line AH which 

 cuts EH at H. 



This process may be repeated as often as desire;!. In the 

 present instance, for the sake of clearness in the diagram, no 



* Communicated by the Author. 



