Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/30

 so also would be the circle on $$\textstyle \frac{1}{2}AB$$ as diameter. Hippocrates recognized the fact that the meniscus is not quadrable, and he made attempts to find other quadrable lunulae in order to make the quadrature of the circle depend on that of such quadrable lunulae. The question of the existence of various kinds of quadrable lunulae was taken up by Th. Clausen in 1840, who discovered four other quadrable lunulae in addition to the one mentioned above. The question was considered in a general manner by Professor Landau of Göttingen in 1890, who pointed out that two of the four lunulae which Clausen supposed to be new were already known to Hippocrates.

From the time of Plato (429—348 ), who emphasized the distinction between Geometry which deals with incorporeal things or images of pure thought and Mechanics which is concerned with things in the external world, the idea became prevalent that problems such as that with which we are concerned should be solved by Euclidean determination only, equivalent on the practical side to the use of two instruments only, the ruler and the compass.

The first really scientific treatment of the problem was undertaken by the greatest of all the Mathematicians of antiquity, Archimedes (287—212 ). In order to understand the mode in which he actually established his very important approximation to the value of $$\pi$$ it is necessary for us to consider in some detail the Greek method of dealing with problems of limits, which in the hands of Archimedes provided a method of performing genuine integrations, such as his determination of the area of a segment of a parabola, and of a considerable number of areas and volumes.

This method is that known as the method of exhaustions, and rests on a principle stated in the enunciation of Euclid 1, as follows:

"Two unequal magnitudes being set out, if from the greater thert be subtracted a magnitude greater than its half, and from that which