THE SQUARING OF THE CIRCLE 
225 
curvilinear figure to be measured, is particularly characteristic 
of Archimedes. 
We come now to the real rectifications or quadratures of 
circles effected by means of higher curves, the construction 
of which is more ‘ mechanical ’ than that of the circle. Some 
of these carves were applied to solve more than one of the 
three classical problems, and it is not always easy to determine 
for which purpose they were originally destined by their 
inventors, because the accounts of the different authorities 
do not quite agree. Iamblichus, speaking of the quadrature 
of the circle, said that 
‘Archimedes effected it by means of the spiral-shaped curve, 
Nicomedes by means of the curve known by the special name 
quadratrix {rerpctycovi^ova-a), Apollonius by means of a certain 
curve which he himself calls “sister of the cochloid” but 
which is the same as Nicoraedes’s curve, and finally Carpus 
by means of a certain curve which he simply calls (the curve 
arising) “from a double motion”.’ 1 
Pappus says that 
‘ for the squaring of the circle Dinostratus, Nicomedes and 
certain other and later geometers used a certain curve which 
took its name from its property; for those geometers called it 
quadratrix.’ 2 
Lastly, Proclus, speaking of the trisection of any angle, 
says that 
‘ Nicomedes trisected any rectilineal angle by means of the 
concho idal curves, the construction, order and properties of 
which he handed down, being himself the discoverer of their 
peculiar character. Others have done the same thing by 
means of the quadratrices of Hippias and Nicomedes. . . . 
Others again, starting from the spirals of Archimedes, divided 
any given rectilineal angle in any given ratio.’ 3 
All these passages refer to the quadratrix invented by 
Hippias of Elis. The first two seem to imply that it was not 
used by Hippias himself for squaring the circle, but that it 
was Dinostratus (a brother of Menaechmus) and other later 
geometers who first applied it to that purpose; Iamblichus 
and Pappus do not even mention the name of Hippias. We 
might conclude that Hippias originally intended his curve to 
1 Iambi, ap. Simpl. in Categ., p. 192. 19-24 K., 64 b 13-18 Br. 
2 Pappus, iv, pp. 250. 33-252. 3. 3 Proclus on Eucl. I, p. 272. 1-12. 
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