PHILOSOPHICAL TRANSACTIONS. 547 



place and room in the world, in and about which they ought to move as long 

 as the world will last. — But their motion, he thinks, cannot be explained by any 

 straight line, for if so, the comet would run from one vortex into another, and 

 so could never transcend 6 signs or 180 degrees, which is against experience, 

 and for that reason he supposes it must be some crooked or bended line, as a 

 circular, elliptical, or some other of that kind. — As to the place of the comets, 

 he thinks it cannot be beneath the moon : For 1 . The parallax proves the con- 

 trary. 2. If the comet should be lower than the moon, it would be also some- 

 times eclipsed by the shadow of the earth, when by experience the comets have 

 at that time a curled circular light in the form of a rose. And 3, the tail be- 

 fore and after the conjunction with the sun would be the shortest of all, against 

 the experience of our sight. Next, as the comets for these reasons are above 

 the moon, so he thinks they cannot have any place among the planets, for their 

 orb requires a room far larger, to avoid the intersection which continually would 

 happen with the planetary orbs. From hence he concludes that there is no 

 place more fit for them than beyond Saturn, according to Descartes' opinion. 

 And thus having adopted the solid orbs and vortices of Descartes, to avoid the 

 intersections of such orbits, he is obliged to find a place in the immense space 

 between Saturn and the fixed stars, where they perform all the motions about an 

 imaginary centre in that space, without ever coming below the orbit of Saturn. 



The Quadrature of the Circle ; from the Leipzic Acts, No, 2. By M. Leibnitu 



Philos. Collect. N°7, P-204. 



Geometricians have always endeavoured to find out the proportions between 

 straight and curve lines. And even to this day, though they have made use of 

 algebraical helps, they do not find the matter so easy to be accomplished by the 

 methods hitherto published. Archimedes was the first, for what we know, who 

 found out the ratio between a cone, a sphere, and a cylinder, of the same al- 

 titude and base, viz. that of the numbers 1, 2, 3, so that the cylinder is triple 

 the cone and -| the sphere ; for which cause he commanded that a sphere and a 

 cylinder should be carved on his tombstone. The same Archimedes found out the 

 quadrature of the parabola. Of late there is invented a method of measuring in- 

 numerable curve-lined figures, especially when the ordinates are in any proportion 

 multiplicate or submultiplicate, direct or reciprocal, of the intercepted abscissas. 



For the figure ABC A (fig. 2, pi. 15,) will be to the circumscribed rectangle 

 ABCD, as an unit is to a number expressing the multiplication of the ratio, 

 increased by an unit. For instance, the intercepted lines A B or DC in a para- 

 bola, being in proportion as the simple natural numbers, viz. as 1, 2, 3, &c. 

 And the ordinates BC being as the squares of those numbers, viz. 1, 4, g, &c. 

 or in duplicate the proportion of them, and the number expressing that ratio 



4 A 2 



