DR. WALLIS, DE MOTU, _Cap._ v. _Prop._ 1. If there be understood an infinite row of quantities beginning with 0 or (1)/(0), and increasing continually according to the natural order of numbers, 0, 1, 2, 3, &c. or according to the order of their squares, as, 0, 1, 4, 9, &c. or according to the order of their cubes, as, 0, 1, 8, 27, &c. whereof the last is given; the proportion of the whole, shall be to a row of as many, that are equal to the last, in the first case, as 1 to 2; in the second case, as 1 to 3; in the third case, as 1 to 4, &c. This proposition is the ground of all his doctrine concerning the centres of gravity of all figures. Wherein may it please you to consider: First, whether there can be understood an infinite row of quantities, whereof the last can be given. Secondly, whether a finite quantity can be divided into an infinite number of lesser quantities, or a finite quantity can consist of an infinite number of parts, which he buildeth on as received from Cavallieri. Thirdly, whether (which in consequence he maintaineth) there be any quantity greater than infinite. Fourthly, whether there be, as he saith, any finite magnitude of which there is no centre of gravity. Fifthly, whether there be any number infinite. For it is one thing to say, that a quantity may be divided perpetually without end, and another thing to say, that a quantity may be divided into an infinite number of parts. Sixthly, if all this be false, whether that whole book of _Arithmetica Infinitorum_, and that definition which he buildeth on, and supposeth to be the doctrine of Cavallieri, be of any use for the confirming or confuting of any propounded doctrine. Humbly praying you would be pleased to declare herein your judgment, the examination thereof being so easy, that there needs no skill either in geometry, or in the Latin tongue, or in the art of logic, but only of the common understanding of mankind to guide your judgment by. THOMAS HOBBES, ROSET. _Prop._ v. _To find a straight line equal to two-fifths of the arc of a quadrant._ I describe a square A B C D, and in it a quadrant D A C. Suppose D T be two-fifths of D C, then will the quadrantal arc T V be two-fifths of the arc C A. Again let D R be a mean proportional between D C and D T; then will the quadrantal arc R S be a mean proportional between the arc C A and the arc T V. Suppose further a right line were given equal to the arc C A, and a quadrantal arc therewith described; then will D C, C A, the arc on C A be continually proportional. Set these proportionals in order by themselves. D C, C A, arc on C A∺ D R, R S, arc on R S∺ D T, T V, arc on T V∺