Chios while searching for the ratio was led to the discovery of the exact area of the lune or cresent-shaped figure, which can be demonstrated to be exactly equivalent to a given square. Cardinal Nicholas de Cusa rolled a wheel on a plane, and then he measured the path of one revolution of the wheel and made the ratio to be the 10 = 3.1622776+. He also conceived of the curve made by a point in the rim of the wheel passing through space, called the cycloid, and believed with Charles Bovillus, in the next century, that it was the arc of a circle. Oronce Finée, a Royal professor in 1544 published a quadrature a little more ingenious than Bovillus; Monantheuli, another Royal professor, in 1600, published his ratio. In 1592, Joseph Scaliger published his Nova Cyclometria, giving the ratio, 3.14098+; being shown his error by five geometer,he would not surrender. The only quadrator on record who, it is said, was convinced of his error, and acknowledged the same, was Richard White (Albinus), a Jesuit; his book is called Chrysespis sen Quadratura Circuli. Montucla mentions many others who have spent much time and labor to discover the value of, bringing the history of the subject down to the publication of his work. Montucla says, speaking of France, that he finds three objects prevalent among cyclometers: I. That there was a large reward offered for the solution 2. That the longitude problem depended upon the solution. 3. That the great end and object of geometry depended upon it. Dr. Charles Hutton says, in writing on this subject, that he divides the writers on this problem into two classes: The first, consisting of able geometricians not led away by illusions, are those who seek only for the approximation more and more exact, whose researches have often terminated in discoveries in almost every part of geometry. Second, those who are less acquainted with the principles of geometry and try to solve the problem by analogies and paralogisms. However this may be, the results of many of them greatly differ, and that too among some able geometers. We think the object in view is to find a finite ratio which shall be the true value of π. All desire to find a ratio that shall be finite. The names of many are given, and the results of some are stated. J. E. Montucla's Cyclometers. From Babin's translation of Montucla's "History of the Quadrature," we compile the names of writers on the circle. The list could be extended by including many others who wrote against the quadrateurs. Mr. Babin gives only 14 of the results of Montucla's cyclometers. Archimedes, between 318 and 31 Duchesne, Simon, Bryson, De Bovillus, Charles, Leistner, Longomontanus, Lowenstein, Christian, 3.75 Falcon, Sir James, 1587, 3.1415929+ Hipocrates, of Chios, 3.142 Romanus, Adrianus, (16 decimals) Lambert, Metiús, Peter, Scaliger, Joseph, Wayvel, S. Daniel, Petrus, Cornelius, Van Ceulen, Lud., Anaxagoras, Antiphon, (32 decimals) Mallemant, of Messange, Mathulon, 1728, Meton, of the Meton-cycle, The Academy of Sciences gave notice it would examine no more quadratures, no more trisections, no more duplications, and no more perpetual motions. De Vausenville asked this question: "Would the quadrature be found if means could be devised for determining the center of gravity of a sector of a circle in common parts of the radius and circumference of the same circle?" Dr. Hutton answerd "Yes." The Extension of the Decimals of . The extension of the decimals of the orthodox value of # is credited to several mathematicians as follows: Thomas F. de Lagny, (16-1734) (Radcliffe Library, Manuscript, Oxford,) Clausen and Dase, of Germany, independently, William Rutherford, 1843, William Shanks, 1853, William Shanks, 1873, 707 Decimals. =3.141592 653589 793238 462643 383279 502884 197169 399375 105820 974944 592307 816406 286208 998628 034825 342117 067982 148086 513282 306647 093844 609550 582231 725359 408128 481117 450284 102701 938521 105559 644622 948954 930381 964428 810975 665933 446128 475648 233786 783165 271201 909145 648566 923460 348610 454326 648213 393607 260249 141273 724587 006606 315588 174881 520920 962829 254091 715364 367892 590360 011330 530548 820466 521384 146951 941511 609433 057270 365759 591953 092186 117381 932611 793105 118548 074462 379834 749567 351885 752724 891227 938183 011949 129833 673362 441936 643086 021395 016092 448077 230943 628553 096620 275569 397986 950222 474996 206074 970304 123668 861995 110089 202383 770213 141694 119029 885825 446816 397999 046597 000817 002963 123773 813420 841307 914511 839805 70985± Common logarithm of Naperian logarithm of Square of Square root of Naperian base, Constants. 0.49714987269413385435+ T= 0.318309886183790671537767526745 π 9.869604401089358618834490999876 1.772453850905516027298167483341 2.718281828459045235360287471352 Common logarithm of € = 0.434294481903251827651128918916 Naperian logarithm of e = 1.000000000000000000000000000000 I. 2. 3. 4. Arithmetical Coincidences. In a circle of diameter 4, the perimeter of the circumscribed In a circle of diameter 4, the circumference is equal to the area. 5. In a circle of circumference, the diameter is equal to area of circumscribed square. 6. If diameter of a sphere is 6, the surface is equal to soliidty. 7. 8. 9. If surface of a sphere is 6, the diameter is equal to solidity. If diameter of a sphere is 1, the circumference is equal to surface. If circumference of a sphere is 1, the diameter is equal to surface. 10. The side of the inscribed equilateral triangle is exactly equal to the area of the circumscribed hexagon. S 11. The altitude of the inscribed triangle is exactly equal to the area of the inscribed dodecagon. ly } 12. The side of the inscribed square is exactly equal to the area of the inscribed octagon. 13. Twice the area of the inscribed equilateral trianis exactly equivalent to area of the inscribed hexagon. 14. The side of the inscribed hexagon is exactly equal? to the area of the inscribed square. .75 √.75 .75 √.50 ✓.421875 .50 the diameter of a circle. 3 = convex surface of cylinder. 3 ÷ 4 area of circle. radius. D'Israeli has arranged the "six follies of science," as follows: 1. The quadrature of the circle. 2. The duplication of the cube. 3. The perpetual motion. 4. The philosopher's stone. 6. Astrology. 5. Magic. De Morgan says he ought to have added the trisection of an angle, so to have made the mystic number seven. Another, anonymous author has arranged the 12 follies as follows: MATHEMATICAL. MYSTICAL. 1. The discovery of prime numbers. I. The elixir of life. 2. The duplication of the cube. 2. The philosopher's stone. A bibliography of books on, and inventions for, perpetual motion was published in London, 1861, 8vo., entitled "Perpetuum Mobile : or Search for Self-Motive Power," by Henry Dircks. Gen. T. Perronet Thompson has published in his "Geometry without Axioms," a list of thirty titles on the "Theory of Parallels," the heads of which are in the "Penny-" and "English-Cyclopædias," Art. "Parallels." Lagrange, in one of the later years of his life, imagined that he had overcome the difficulty. He wrote a paper, which he took with him. to the Institute, and began to read it; but in the first paragraph he saw something he had not before observed; he muttered" Ill faut que j'y songe encore," and put the paper in his pocket. Johnnes Von Gumpach has published his work on "The Moon's Rotation on her Axis," London, 1856, and gives 34 titles on the discussion of the subject. He says her rotation is "a bare physical impossibility." Chambers' Encyclopædia, Article, "Quadrature of the Circle," says: "If an equation could be discovered for a + √b, a and b representing irrational quantities, it would be welcomed as the solution of the grand problem." Theodore Faber proposed the following equation as that desideratum: √a+√b = √a+b+ √4ab. The result is however an |
