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 23 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. Again, of the 154 blocks within the arc, if 98 be deducted, we shall have 56 between the side of the inseribed 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. Then dividing the given circumference by the given diameter, we have 88 28 = 22 ÷ 7 = 3.142857, or 3, which is the true ratio of the circumference to the diameter of the given circle. ག ཡསལ་ THE QUADRATURE OF THE CIRCLE, THE SQUARE ROOT OF TWO, AND THE RIGHT-ANGLED TRIANGLE, BY WILLIAM ALEXANDER MYERS, SECOND EDITION. "Where is the wise."-1st Cor., i, 20. "Now the serpent was more subtile than any of the CINCINNATI: WILSTACH, BALDWIN & CO., PRINTERS, Nos. 141 AND 143 RACE STREET. 1874. AUTHORS MADE USE OF IN THE PRESENT VOLUME. Should the student desire more general information upon the subjects treated of in the present volume, he is referred to the following works which have been freely used by the author wherever they have been found to be of service to his cause. They will be found to be among the best of their kind "Montuclas' History of Mathematics;" "Hutton's Recreations;” “DeMorgan on the Law of Probabilities;" "Elements of Euclid," by Todhunter; "Elements of Euclid," by Thompson; "Davies' Le Gendre;" "Robinson's Geometry;" Chauvenet's Geometry;" ""Loomis' Geometry and Trigonometry;" “Bullfinch's Beauties of Mythology;" "Minifee's Draughting and Architecture;" "Home and School Journal;" "Chambers' Encyclopædia," and the "Douay Bible.” Entered according to Act of Congress, in the Office of the Librarian at Washington, D. C., BY WILLIAM ALEXANDER MYERS, March 31st, 1873. All Rights Reserved. TO 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 TO THE CAUSE WHICH WE ALL ADVOCATE IN COMMON, AND AS A SMALL TESTIMONIAL OF THE ESTEEM IN WHICH THEY ARE HELD BY THE AUTHOR. |