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that diameter of the circle of which d is one-half. For (fig. II.) 

 if ABC be a circle of which AB is a diameter, produced to 

 E, and ED perdendicular to that diameter ; if AD be joined by 

 a line A CD, the angle A CB being in a semicircle is a right 

 angle ; and the angle BAC is common to the two triangles, 

 AB C, ADE ; therefore 



AE : AD :: AC : AB : and AEAB=AD'AC ; 

 so that the locus of D, when the above rectangle is a 

 constant quantity, is in a straight line perpendicular to the 

 diameter AC; and equally so whether D be within or without the 

 circle. Bisect CD in G ; and draw any line GF perpendicular 

 to CD ; and join DF and CF. Then the difference of the 

 squares of AG and GD is equal to the difference of the squares 

 of AF and FC (or FD)\ and is also equal to the rectangle under 

 their sum AD and difference A C ; which has been shown to 

 be constant. But AF and FD are equivalent to the a and b of 

 Peaucellier's linkage ; which accordingly succeeds in drawing a 

 mathematically correct line in DE. Peaucellier's other cell 

 (fig. III.) differs from this in that the four equal bars (3) have 

 the pair of bars (a) inside them ; when, to produce the straight 

 line, the fixed bar or distance d must be linked to the pair a. 

 If otherwise, and the bar c in either linkage were the fixed one, 

 the traversing point D would describe a very curious curve ; 

 the law of which I shall presently state. 



Peaucellier calls the original cell — that with the smaller rhom- 

 bus — his positive cell ; and that with the rhombus outside the 

 pair of bars, negative. It was some time before I recognized 

 the appositeness of these terms ; preferring the more obvious 

 ones of " external" and " internal." But I am convinced, by the 

 consideration of the geometrical proposition on which the proof 

 is based, that Peaucellier's names are the most logical. The 

 proposition, stated to cover all the cases, is : — If straight lines 

 be drawn from any point to the extremities of any diameter of 

 a circle in the same plane as the point, the rectangle under each 

 of such lines and its segment (or segment produced) intercepted 

 in the circle is equal to the rectangle under the diameter of the 

 circle and its segment towards that extremity met by each 



