It follows hence, that whosoever taketh for a principle, that a side of a square is a mere line without latitude, and that the root of a square is such a line (as Dr. Wallis continually does) demonstrates nothing. But if a line be divided into what number of equal parts soever, so the line have breadth allowed it (as all lines must, if they be drawn), and the length be to the breadth as number to an unit; the side and the roof will be all of one length. PROP. II. [Illustration] Any number given is produced by the greatest root multiplied into itself, and into the remaining fraction. Let the number given be two hundred squares, the greatest root is 14(4)/(14) squares. I say that 200 is equal to the product of 14 into itself, together with 14 multiplied into (4)/(14). For 14 multiplied into itself makes 196. And 14 into (4)/(14) makes (56)/(14) which is equal to 4. And 4 added to 196 maketh 200; as was to be proved. Or take any other number 8, the greatest root is 2; which multiplied into itself is 4, and the remainder (2)/(4) multiplied into 2, is 4, and both together 8. PROP. III. But the same square calculated geometrically by the like parts, consisteth (by Euclid II. 4) of the same numeral great square 196, and of the two rectangles under the greatest side 14, and the remainder of the side, or (which is all one) of one rectangle under the greatest side, and double the remainder of the side; and further of the square of the less segment; which altogether make 200, and moreover (1)/(49) of those 200 squares, as by the operation itself appeareth thus: The side of the greater segment is 14(4)/(14) 14(4)/(14) Which multiplied into itself makes 200. The product of 14, the greatest segment, into the two fractions (4)/(14), that is, into (4)/(14) (or into twice (2)/(14)) is (56)/(14) (that is 4); and that 4 added to 196 makes 200. Lastly, the product of (2)/(14) into (2)/(14) or (1)/(7) into (1)/(7) is (1)/(49). And so the same square calculated by roots is less by (1)/(49) of one of those two hundred squares, than by the true and geometrical calculation; as was to be demonstrated. CONSEQUENT. It is hence manifest, that whosoever calculates the length of an arc or other line by the extraction of roots, must necessarily make it shorter than the truth, unless the square have a true root. ------- _The Radius of a Circle is a Mean Proportion between the Arc of a Quadrant and two-fifths of the same._