252 PHILOSOPHICAL TRANSACTIONS. [aNNO 1668. 



thor was on his travels in that country. In the preface of the present work he 

 observes, that the defect of algebra in the mensuration of curved jfigures may in 

 some manner be supplied, if out of some essential property of any such figure 

 thence be given a method of changing it into another equal figure, having 

 known properties, and of that into another, and so on, till at last it be changed 

 into some known quantity ; which he says is eff^ected in this work. 



To square a circle organically, or divide an angle in a given ratio, he 

 supposes there is no easier method, than by the common linea quadratrix, the 

 properties whereof are treated at large in Leotaudi Cyclomathia, Lugduni, 

 1663, in 4to. 



He then remarks that all things concerning logarithms, and the composi- 

 tion of ratios, may be performed by help of a curved line, drawn through the 

 tops of a rank of lines in continual proportion, standing as perpendiculars on a 

 right line and at equal distance, being the logistic or logarithmic curve. That 

 however the operations performed thereby are not to be accounted geometrical, 

 because they are not performed by the sole aid of rule and compass. The 

 confirmation of which the author thus demonstrates, that no cubic equation 

 irreducible to a quadratic, can be resolved by the sole aid of rule and compass. 

 For every cubic equation has either only one real root or three real roots 3 

 hence if they could be found by the sole aid of rule and compass, or by the in- 

 tersection of a circle and a right line, then aright line should cut a circle either 

 in one point or three points ; either of which is absurd. And for the like rea- 

 son a cubic equation, having three real roots, can never be reduced to a pure 

 equation which has only one root ; for in these equations, it is impossible, by- 

 aid of any reduction, to change an imaginary root into a real one, and the 

 converse. 



The book itself contains these several heads : — 1. The mensuration of sundry 

 solids, with general methods for that purpose. He here cubes or measures 

 either of the segments of a parabolical conoid cut by a plane parallel to the axis. 

 • — 2. The mensuration or plaining of the surfaces of divers solids and spiral 

 spaces unknown to antiquity, and not treated of by any modern authors, till of 

 very late years ; from whom the author differs in his method : particularly, he 

 finds a circle equal to the surface of a parabolical or hyperbolical conoid, resem- 

 bling a cup or bowl ; viz. when the revolution is about their axes. Prop. 46 and 

 49. Also, the parabolical hour-glass or solid, when the revolution is about a 

 tangent at the vertex, Prop. 52. Also the oblong spheroid. Prop. 47, 48 ; 

 and Prop. 67 , the surface of any segment of a cone. Generally it its shown. 

 Prop. 36, that the surface of every round solid is equal to a rectangle, whose 

 base is the cirGumference of the figure, by the rotation whereof the solid is 



