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But when great and ingenious artists behold their so inept performances, not undeservedly do they ridicule the blindness of such men; since sane judgment abhors nothing so much as a picture perpetrated with no technical knowledge, although with plenty of care and diligence. Now the sole reason why painters of this sort are not aware of their own error is that they have not learnt Geometry, without which no one can either be or become an absolute artist; but the blame for this should be laid upon their masters, who are themselves ignorant of this art.
The Art of Measurement. 1525.
Loci: Convergence
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An Investigation of Historical Geometric Constructions
Quadrature of the Circle
Hippocrates of Chios (born around 470 B.C.E.), not to be mistaken for Hippocrates of Cos who is the father of Greek medicine, focused much of his studies on solving the three constructions of Greek Antiquity. Although little is known about Hippocrates’ life, it is thought that he traveled from the Island of Chios to Athens to settle a lawsuit in connection with either being deceived by customhouse officers at Byzantium or having his ships looted by pirates. While settling the court issue whose outcome is unknown, Hippocrates attended lectures of Athenian scholars, provoking his interest in solving the three classical problems of Greek Antiquity. Heath notes that Hippocrates had three main contributions to mathematics:
- He was, so far as we know, the first to compile a book of Elements.
- In an attempt to square the circle he showed how to square certain classes of ‘lunes’.
- He was the first to observe that the problem of doubling the cube can be reduced to that of finding two mean proportionals in continued proportion between two given straight lines, with the result that the problem was ever afterwards attacked in the latter form. (Heath, 1963, p. 121)
As described in his second accomplishment, Hippocrates calculated the area of different types of lunes, which are regions between arcs of two circles as shown in figure 12, in his attempt to construct a square with the same area as a circle. Hippocrates calculated the area of three types of lunes, that is, a lune with an outer arc greater than a semicircle, one with an outer arc less than a semicircle, and one with an outer arc equal to a semicircle.
Figure 12: Examples of lunes.
It should be noted that Hippocrates was well aware of properties of circles, in particular that the areas of two circles are proportional to the squares on their diameters. In addition, he knew the Pythagoreans had shown that the sum of the square areas on two legs of a right triangle is equal to the square area on the hypotenuse. With this in mind, we begin interpreting Hippocrates’ method for calculating the area of a lune with an outer arc equal to a semicircle and share how to use Geometer’s Sketchpad to explore this proof.