Alois. — To describe a Straight Line by Linh-ioork. 443 



If a fixed point, A, on the circumference of a circle be 

 joined with any other point, P, and a length, AQ, be measured 

 on AP such that the product of the lengths AP and AQ 

 always has the same value, then, as P moves round the 

 circle, Q will move on a straight line. 



Let AB (figs. 1 and 2) be the diameter of the circle 

 through A. Take AC so that the product of AB and AC is 

 equal to the constant value of that of AP and AQ. Then C is 

 .a fixed and determinable point. 



If then CQ and BP be joined, since AP AQ is equal to AB 

 AC, it follows that PBCQ is a cyclic quadrilateral, and there- 

 fore the angle ACQ is equal to the angle APB— that is, is a 

 right angle. Hence Q always lies on the straight line drawn 

 through C at right angles to AC. 



The Peaucellier cell, as the framework is called, consists of 

 seven bars, four of which are of one length, two more are 

 equal to one another, but unequal to the former, while the 

 seventh may be taken arbitrarily. These are jointed together 

 as in fig. 3, the four equal bars forming a rhombus, ABPD, 

 the other two equal ones being attached to this at B and D, 



and to each other at O, while the seventh is attached to the 

 rhombus at A. 



The points C and are fixed points, and the distance 



