78 



straight line respectively, formed by the falling of a perpendi- 

 cular upon such diameter (or diameter produced) from the given 

 point; — and; — the sum of such rectangles is equal to the square 

 of the diameter if the perpendicular falls within the circle, and 

 their difference if it falls without it ; or their sum in both 

 cases if one of the segments has its place in the semicircle 

 furthest from the point, and is therefore regarded as minus. 

 Now the demonstration of Peaucillier's first, or positive cell, 

 depends on that case of this proposition in which the per- 

 pendicular from the given (or as I have shown, traversing) 

 point on the diameter does not pass it, and its segment within 

 the circle is part of itself, or positive ; while the demonstration 

 of the negative cell depends on the other case. In investigating 

 the curve produced by fixing the bar c, and moving d, as also 

 in the case of other curves produced by the machine, I met 

 with confirmation of this view. 



If c and d are made unequal, D will describe an arc of a 

 circle the centre of which is in d or d produced ; its radius de- 

 pending on the magnitudes of the various bars of the linkage, 

 and on the distance d, viz.: — 



c(a 2 -b 2 )* 

 c 2 -d 2 

 Accordingly, when a is greater than /;, and c greater than d, the 

 concavity of the arc is towards the figure (positive cell) ; the centre 

 of the circle lying in d produced in the direction of A (fig. VI.) 

 and in the same cell, if c is less than d, the concavity of the arc 

 is from the figure. In the negative cell, c must be less than 

 d to draw an arc with its convexity towards the figure ; and 

 the minus sign appears when the centre of the circle lies in 

 the part of d produced which is away from A. I believe this is 

 the first time in the history of mathematics that a circle has 



* Assuming that the path of the point D in fig. VI. is an arc of a circle, and taking 



any distance e, the dotted line between C and A as a variable, we can obtain expressions 



{or the sine and versed sine of the arc in terms of a, b, c, d, and e ; then, as 



sin 2 -f-versin 2 



: =rad. 



aversin 



Sn the simplest form of which fraction the variable e disappears, the assumption must 



be true, 



