548 PHILOSOPHICAL TRANSACTIONS. [aNNO 1682. 



being 2, the figure ABC A will be to the rectangle ABCD, as 1 to 2 + 1, or 

 as 1 to 3, or the figure shall be a third part of the rectangle. If AB or CD 

 remaining as the natural numbers 1, 2, 3, and BC, BC be made as the cubes, 

 or 1, 8, 27, &c. as in the cubical parobala, the ratio of the ordinates being tri- 

 plicate of the ratio of the intercepted parts, the figure will be to the rectangle, 

 as 1 to 3 + 1, that is to 4, or a fourth part. But if DC, DC be as squares, and 

 BC, BC as cubes, that is the ratio of BC, BC, be a the ratio of DC, DC; 

 the figure, a semi-cubical parabola, ABC A, to the rectangle ABCD, will be as 

 1 to 4 + 1 , or shall equal f of the rectangle. In reciprocals, to the number ex- 

 pressing the multiplication of the ratio, the sign — or minus must be prefixed. 



But a circle could never be brought within the compass of these rules ; and 

 though it has been laboured for by geometricians in all times, there could never 

 yet be found a number by which the ratio of the circle, to the circumscribed 

 square, could be expressed. 



Nor could the ratio of the circumference to the diameter be ever yet found, 

 which is quadruple of the ratio of the circle to the square. Archimedes indeed, 

 inscribing and circumscribing polygons to a circle, it being larger than the in- 

 scribed and less than the circumscribed, shows a way of giving the limits within 

 which the circle falls, or of giving approaches, by which he shows that the ratio 

 of the circumference to the diameter is more than as 3 to 1, or than as 21 to 7, 

 and less than 22 to 7^ which method others have since prosecuted, viz. Ptolemy, 

 Vieta, Metius, and most of all Ludolphus van Ceulen, who has shown the cir- 

 cumference to be, as 3.14159265358979323846 &c. 

 to the diameter 1.00000000000000000000. 



But these sorts of approaches, though they are useful in practical geometry, 

 give no satisfaction to the mind, which covets the very truth, unless the pro- 

 gression of such numbers could be carried on in infinitum. Many indeed have 

 professed to have found the true quadrature, as Cardinal Cusanus, Oron. Fineus, 

 Jos. Scaliger, Tho. Gepherander, Tho. Hobbs, but all of them falsely, being dis- 

 proved by the calculations of Archimedes, or by those of Ludolphus van Ceulen. 



But because I perceive there are many who do not well understand what it is 

 that is desired in this matter; they are to know that the quadrature, or the turn- 

 ing of the circle into an equal square, or any other right-lined figure, which de- 

 pends on the ratio of the circle to the square of its diameter, or of the circum- 

 ference to its diameter, may be understood to be fourfold, viz. either by calcu- 

 lation, or by linear construction : and each of them again may be either perfectly 

 exact, or else almost, or pretty near. The quadrature by accurate calculation, 

 I call the analytical. That which is done by accurate construction, I call the 

 geometrical. That which is done by calculation pretty near, I call the approach. 

 That which is by construction pretty near, I call the mechanical. The approaches 



