TOL. III.] PHILOSOPHICAL TRANSACTIONS. 817 



that Guldin erred through forgetfulness. In proposition 2Q he shows, that the 

 parabola bears such a proportion to a circle described about the base as a dia- 

 meter, as the axis of the parabola doth to that circumference of a circle, whose 

 radius is equal to the distance of the centre of gravity of the semi-parabola from 

 the axis. 4. Contains divers endeavours and manifold new ways towards the 

 obtaining the quadrature of the circle in 12 propositions. 5. Contains ten pro- 

 positions, from 41 to 51, in the 42d whereof he finds a sphere equal to an 

 hyperbolical ring solid ; whence divers ways are opened towards the attaining the 

 quadrature of the hyperbola : and he finds a sphere equal to a ring made by the 

 rotation of a segment of an hyperbola, and of the segment of a circle thereto 

 annexed, described about the base of the hyperbola as a chord line : Then he 

 absolutely cubes certain hoofs cut out of an hyperbolical cylinder, and thence de- 

 rives other ways towards the obtaining the quadrature of the hyperbola. 6. De- 

 livers 3 theorems, showing the proportion between an hyperbola and a circle : 

 which are conceived to be wholly new. 



But these theorems suppose the quadrature of both figures known, viz. that 

 of a circle, in requiring the length of the circumference of a circle, described by 

 the centre of gravity of an hyperbola ; which centre cannot be found, without 

 giving the quadrature or area of the hyperbola : which has been most happily 

 performed by M. Mercator in his Logarithmotechnia, and further advanced by 

 Dr. Wallis, in N° 38 of these Transactions; and by Mr. Gregory also further 

 promoted and otherwise performed, in his Exercitationes Geometricae, where 

 he shows the same methods and approaches to be likewise applicable to the 

 circle. 



A Continuation of Ohservatioiis on Vegetables. By Dr. E. Tonge, 



N' 44, p. 877.* 



If no rain come to the roots of trees, nor other moisture, they will not grow, 

 but if the points of the roots only be watered, though all the rest remain dry, 

 as it happens naturally in fir trees, they may grow ver}' well. For the points of 

 the roots shoot out yearly a sharp-pointed tender part, somewhat like the sharp 

 bud on the end of a sprig, by which the root not only enlarges itself in the 

 earth, as the branch does in the air, but also receives its nourishment. And 

 that tender part moves towards the best moistened and the tenderest earth ; so 

 that to promote the growth of trees, it is very effectual to loosen the earth of 

 trees about the points of the roots. 



The roots of plum and lime trees inoculated upon, will shoot out their buds. 



♦ See No. 43, p. 304. 



