VERIFICATION OF THE QUADRATURE OF THE CIRCLE. Let either of the above quadrants be divided into 196 squares, having 14 on each side; then the line which is the side of the whole inscribed square will divide the quadrant into two equal parts, each of which contains 98 squares. And the arc of the circle belonging to the quadrant will divide the outside 98 squares into two parts, which shall be to each other as 4 is to 3. For the arc of the quadrant cuts 21 of the squares, which form a complete arch; and there are 142 squares within those which are cut by the arc and 33 without; then if these 21 squares be divided in the proportion of 4 to 3, there will be 12 squares that will fall within the circle and 9 that will fall without it; then 142 + 12 = 154, the number of squares within the arc, and there are 196 squares in the quadrant. 196 14 = Then 154 is to 196 as 11 is to 14, for Again, of the 154 blocks within the arc, if 98 be deducted, we shall have 56 between the side of the inscribed square and the arc of the circle; and if to the 33 blocks without the arc 9 be added, we shall have 42 blocks without the arc of the circle; then 56 is to 42 as 4 is to 3. Again, if the sides of the squares which form the arch of the quadrant, viz.: 5+2+2+2 = 11 X 2 = 22. be multiplied by 4, we shall have 88 sides for the circumference of the circle, and as each of the quadrants has 14 squares on each side, we shall have 28 sides for the diameter of the circle. = 22 ÷ 7 Then dividing the given circumference by the given diameter, we have 88 ÷ 28 3.142857, or 34, which is the the true ratio of the circumference to the diameter of the given circle. = ΤΟ THE AMERICAN PEOPLE WHOSE LOVE FOR LEARNING AND DEVOTION TO THE TRUTH, ARE ONLY EQUALED BY THE MAGNIFICENT CONTRIBUTIONS WHICH THEY HAVE MADE TO THE CAUSE OF EDUCATION, THIS VOLUME IS RESPECTFULLY INSCRIBED AS A CHEERFUL CONTRIBUTION ΤΟ THE CAUSE WHICH WE ALL ADVOCATE IN COMMON, AND AS A SMALL TESTIMONIAL OF THE ESTEEM IN WHICH THEY ARE HELD BY THE AUTHOR. THE QUADRATURE OF THE CIRCLE, THE SQUARE ROOT OF TWO, AND THE RIGHT-ANGLED TRIANGLE, BY WILLIAM ALEXANDER MYERS, FIRST EDITION. "Where is the wise."-1st Cor., i, 20. "Now the serpent was more subtile than any of the CINCINNATI: WILSTACH, BALDWIN & CO., Nos. 141 AND 143 RACE STREET. 1873. PRINTERS, VERIFICATION OF THE QUADRATURE OF THE CIRCLE. Let either of the above quadrants be divided into 196 squares, having 14 on each side; then the line which is the side of the whole inscribed square will divide the quadrant into two equal parts, each of which contains 98 squares. And the arc of the circle belonging to the quadrant will divide the outside 98 squares into two parts, which shall be to each other as 4 is to 3. For the arc of the quadrant cuts 21 of the squares, which form a complete arch; and there are 142 squares within those which are cut by the arc and 33 without; then if these 21 squares be divided in the proportion of 4 to 3, there will be 12 squares that will fall within the circle and 9 that will fall without it; then 142 +12= 154, the number of squares within the arc, and there are 196 squares in the quadrant. 196 14 = Then 154 is to 196 as 11 is to 14, for Again, of the 154 blocks within the arc, if 98 be deducted, we shall have 56 between the side of the inscribed square and the arc of the circle; and if to the 33 blocks without the arc 9 be added, we shall have 42 blocks without the arc of the circle; then 56 is to 42 as 4 is to 3. Again, if the sides of the squares which form the arch of the quadrant, viz.: 5+2+2+2 = 11 X 2 22. be multiplied by 4, we shall have 88 sides for the circumference of the circle, and as each of the quadrants has 14 squares on each side, we shall have 28 sides for the diam eter of the circle. Then dividing the given circumference by the given diameter, we have 88 ÷ 28 3.142857, = 22 ÷ 7 = or 34, which is the the true ratio of the circumference to the diameter of the given circle. |