which are in continual proportion of the semi-diameter of the arc. And D C, D R, D T are in a continual proportion by construction, and therefore also C A, R S, T V, and arc on C A, arc on R S, arc on T V, in continual proportion. Therefore as D C to R S, so is R S to the arc on T V. And D C, R S, the arc on T V will be continually proportional. And because D C, C A, the arc on C A are also continually proportional, and have the first antecedent D C common; the proportion of the arc on C A to the arc on T V is (by Eucl. xiv. 28) duplicate of the proportion of C A to R S, and the arc on R S a mean proportional between the arc on C A and the arc on T V. Now if D C be greater than R S, also R S must be greater than the arc on T V; and the arc C A greater than the arc on R S. Therefore seeing D C, C A, arc on C A, are continually proportional; the arc on T V, the arc on R S, the arc on C A cannot be continually proportional, which is contrary to what has been demonstrated. Therefore D C is not greater than R S. Suppose, then, R S to be greater than D C, then will the arc on R S be a mean proportional between the arc on T V, and a greater arc than that on C A; and so the inconvenience returneth. Therefore the semidiameter D C is equal to the arc R S, and D R equal to T V, that is to say to two-fifths of the arc C A, which was to be demonstrated. Nor needeth there much geometry for examining of this demonstration. Therefore I submit them both to your censure, as also the whole _Rosetum_, a copy whereof I have caused to be delivered to the secretary of your society. [Illustration] TO THE RIGHT HONOURABLE AND OTHERS, THE LEARNED MEMBERS OF THE ROYAL SOCIETY, FOR THE ADVANCEMENT OF THE SCIENCES. ------- Presenteth to your consideration, your most humble servant Thomas Hobbes, a confutation of a theorem which hath a long time passed for truth; to the great hinderance of Geometry, and also of Natural Philosophy, which thereon dependeth. THE THEOREM. _The four sides of a square being divided into any number of equal parts, for example into 10; and straight lines drawn through the opposite points, which will divide the square into 100 lesser squares; the received opinion, and which Dr. Wallis commonly useth, is, that the root of those 100, namely 10, is the side of the whole square._ THE CONFUTATION.