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 three times its diameter by a part which is less than $$\scriptstyle{1 \over 7}$$ but more than $$\scriptstyle{10 \over 71}$$ of the diameter." This approximation is exact enough for most purposes.

The Quadrature of the Parabola contains two solutions to the problem—one mechanical, the other geometrical. The method of exhaustion is used in both.

Archimedes studied also the ellipse and accomplished its quadrature, but to the hyperbola he seems to have paid less attention. It is believed that he wrote a book on conic sections.

Of all his discoveries Archimedes prized most highly those in his Sphere and Cylinder. In it are proved the new theorems, that the surface of a sphere is equal to four times a great circle; that the surface of a segment of a sphere is equal to a circle whose radius is the straight line drawn from the vertex of the segment to the circumference of its basal circle; that the volume and the surface of a sphere are $$\scriptstyle{\frac{2}{3}}$$ of the volume and surface, respectively, of the cylinder circumscribed about the sphere. Archimedes desired that the figure to the last proposition be inscribed on his tomb. This was ordered done by Marcellus.

The spiral now called the "spiral of Archimedes," and described in the book On Spirals, was discovered by Archimedes, and not, as some believe, by his friend Conon.[3] His treatise thereon is, perhaps, the most wonderful of all his works. Nowadays, subjects of this kind are made easy by the use of the infinitesimal calculus. In its stead the ancients used the method of exhaustion. Nowhere is the fertility of his genius more grandly displayed than in his masterly use of this method. With Euclid and his predecessors the method of exhaustion was only the means of proving propositions which must have been seen and believed before they were proved. But in the hands of Archimedes it became an instrument of discovery.[9]