_The root 10 is a number of those squares, whereof the whole containeth 100, whereof one square is an unity; therefore the root 10, is 10 squares: Therefore the root of 100 squares is 10 squares, and not the side of any square; because the side of a square is not a superficies, but a line. For as the root of 100 unities is 10 unities, or of 100 soldiers 10 soldiers: so the root of 100 squares is 10 of those squares. Therefore the theorem is false; and more false, when the root is augmented by multiplying it by other greater numbers._ Hence it followeth, that no proposition can either be demonstrated or confuted from this false theorem. Upon which, and upon the numeration of infinites, is grounded all the geometry which Dr. Wallis hath hitherto published. And your said servant humbly prayeth to have your judgment hereupon: and that if you find it to be false, you will be pleased to correct the same: and not to suffer so necessary a science as geometry to be stifled, to save the credit of a professor. TO THE RIGHT HONOURABLE AND OTHERS, THE LEARNED MEMBERS OF THE ROYAL SOCIETY, FOR THE ADVANCEMENT OF THE SCIENCES. --- Your most humble servant Thomas Hobbes presenteth, that the quantity of a line calculated by extraction of roots is not to be truly found. And further presenteth to you the invention of a straight line equal to the arc of a circle. A square root is a number which multiplied into itself produced a number. DEFINITION. And the number so produced is called a square number. For example: Because 10 multiplied by 10 makes 100; the root is 10, and the square number 100. CONSEQUENT. In the natural row of numbers, as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, &c. every one is the square of some number in the same row. But square numbers (beginning at 1) intermit first two numbers, then four, then six, &c. So that none of the intermitted numbers is a square number, nor has any square root. PROP. I. A square root (speaking of quantity) is not a line, such as Euclid defines, without latitude, but a rectangle. [Illustration] Suppose A B C D be the square, and A B, B C, C D, D A, be the sides, and every side divided into 10 equal parts, and lines drawn through the opposite points of division; there will then be made 100 lesser squares, which taken altogether are equal to the square A B C D. Therefore the whole square is 100, whereof one square is an unit; therefore 10 units, which is the root, is ten of the lesser squares, and consequently has latitude; and therefore it cannot be the side of a square, which, according to Euclid, is a line without latitude. CONSEQUENT.