It was this very secrecy, this continual mystery from beginning to end, that favoured deceptions of the grossest and most bungling character, as viewed by the light of modern chemistry. Alchemy no doubt tended to improve Medical science, by the introduction of many new mineral and vegetable preparations, but the healing art treated after the manner of the Hermetic Art, was laid open to every description of quackery. It is not our intention, however, to enlarge on this department, which has steadily advanced at every stage of improvement in chemical science. SQUARING THE CIRCLE. Of Mathematical Problems, the most perplexing to ancient and modern mathematicians, although of late years said to be satisfactorily demonstrated, and no longer desiderata of Geometry, are-- 1. The Quadrature or Squaring of the Circle;--2. The Duplication, or doubling of the Cube;--and 3. The Trisection of the Angle. In his "Popular Astronomy,"[11] Professor Arago, treating on the surface of a circle, observes that,-- It is mathematically equal to the product of the length of the circumference, multiplied by half the radius. To square a circle of a given diameter in mètres, is the same as giving the number of squares, of a mètre in each side, of which the surface is the equivalent. If, the diameter being given, the exact circumference were known by a sort of inspiration, the superficial extent of the circular space would be deducible from the two numbers, by the mere multiplication of the numerical length of the circumference by the fourth of the diameter, or half the radius. But, the circumference being deducible from the diameter only by approximation, the surface alluded to cannot be computed with mathematical rigour; yet the result can be obtained with all desirable precision by the aid of the ratios usually given for such purpose; for instance, the area of the space included within a circle of thirty-eight millions of leagues radius, may be determined within such a degree of precision that the probable error shall not exceed the space of a mite. [11] See Translation, by Admiral W. H. Smith, and Robert Grant, M.A., in 2 vols. 8vo. 1855, Vol. I., page 10. "The sect of squarers then," Arago adds,--"are searching after a solution which is proved to be impossible, and which, moreover, would be of no practical use, even if their foolish hopes were crowned with success." In the "Birds" of Aristophanes, the character is introduced of a geometer, who is going to make a square circle, showing how early this chimerical performance became an object of ridicule.