630 PHILOSOPHICAL TRANSACTIONS. [ANNOI67I. 



into -f is tV- His calculation therefore should have been this : The greater 

 segment is, not 14^^, but 14; which multiplied into itself makes, not 200, 

 but 196 : The rectangle of the greater segment 14, into the lesser -^, is 4 : 

 And this taken a second time is another 4 : The lesser segment, not -^v^ but 

 -Y^, or f , multiplied into itself, is not ^, but -^ : All which added together 

 make not 100^, but 196 + 4 + 4 + ^V =^ 204vV> which is just the same with 

 14_iy multiplied into itself. So that had he known how to multiply a number 

 into a number, especially when incumbered with fractions, which it is manifest 

 he does not, he would have found no disagreement between the arithmetical 

 calculation, and what he calls the geometrical. But I am ashamed (for him) 

 that so great a pretender to such high things in geometry, should be so 

 miserably ignorant of the common operations of practical arithmetic. 



His repeated quadrature he now expresses thus3 the radius of a circle is a 

 mean proportional between the arc of a quadrant and two fifths of the same. 

 But instead of two fifths, he might as well have said the half, or tenth, or 

 hundredth part, &c. ; or (taking T in DC produced beyond C,) the double, 

 decuple, centuple, &c. or what you please : For his demonstration would have 

 proved it, which is this. Describe a square ABCD, and in it a quadrant DC A. 

 In the side DC (continued if need be,) take TD two fifths of DC, (or its half, 

 double, hundredth part, or what you please;) and between DC and DT a mean 

 proportional DR; and describe the quadrantal arcs RS, TV. I say the arcRS 

 is equal to the straight line DC. For seeing the proportion of DC to DT is 

 duplicate of the proportion of DC to DR, it will be also duplicate of the pro- 

 portion of the arc CA to the arc RS, and likewise duplicate of the proportion 

 of the arc RS to the arc TV. Suppose some other arc, less or greater than the 

 arc RS, to be equal to DC, as for example rs; then the proportion of the arc 

 rs to the straight line DT will be duplicate of the proportion of RS to TV, or 

 DR to DT, which is absurd ; because Dr is by construction greater or less than 

 DR. Therefore the arc RS is equal to the side DC; which was to be demon- 

 strated. W^hich demonstration therefore proving indifferently every proportion, 

 does not indeed prove any. In brief: The force of his demonstration is but 

 this; DT being to DC as 1 to 5, or in any other proportion, and DR a mean 

 proportional between them; RS will be so between TV and CA; and therefore 

 rs (greater or less than RS,) will not be a mean proportional between TV and 

 CA: which is true; but why it may not be equal to DC, we have nothing but 

 his word for it; there being nothing to show, that DC is equal to such a mean 

 proportional. Again, though rs be not a mean proportional between TV and 

 CA, yet it may be between tv and CA, which serves his demonstration as well; 

 which is indifferent to any three continual proportionals, as was shown before. 



