is less than the arc, augmented by itself often enough, will not exceed the length of the arc? </s>
<s id="id.1.2.2.01.04">But if it will exceed it, why is it said by Aristotle that the arc and the chord do not have a ratio? </s>
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<s id="id.1.2.2.02.00"/>
<s id="id.1.2.2.02.01">Yet there have not been lacking those who try to save Aristotle, by saying, Aristotle only wanted the following, namely, that the curved and the straight are not comparable to each other. </s>
<s id="id.1.2.2.02.02">But these people {1} are by far more ignorant than Aristotle in geometry, since, while they try to show that he himself did not err, they attribute to him an error which is far more serious than that from which they try to cleanse him. </s>
<s id="id.1.2.2.02.03">And first of all, where in geometry have they found mention made of ratio or lack of ratio of the curved and the straight, when a ratio is not found except where there is greater and lesser, that is where there is quantity? </s>
<s id="id.1.2.2.02.04">But who has ever proclaimed the curved and the straight to be quantities? </s>
<s id="id.1.2.2.02.05"> But what greater foolishness could Aristotle ever have imagined, than to say that the curved and the straight are without ratio to each other or comparable?</s>
<s id="id.1.2.2.02.06">For this would be as if one were to say that the triangle and the square are not comparable since the triangle has only three angles, whereas the square has four. </s>
<s id="id.1.2.2.02.07">But what is the point of this when Aristotle did not want what they themselves want?</s>
<s id="id.1.2.2.02.08">For he says the following words in Phys., Book VII, text #24 [248b4-6]: If the straight and the curved are comparable, it turns out that a straight line is equal to a circle; but these are not comparable. {1} </s>
<s id="id.1.2.2.02.09">These are his words. </s>
<s id="id.1.2.2.02.10">But in order that I convince them so that they will never be able to escape, I will argue in the following way. </s>
<s id="id.1.2.2.02.11">Surely they will not deny that a plane surface has a ratio to some part of itself: if this is so, I already have achieved my intent. </s>
<s id="id.1.2.2.02.12">For a circle, inscribed in a square, is a certain part of that square; hence the square has some ratio to the circle: but the square is to the inscribed circle as the perimeter of the square is to the circumference of the circle: that is why the perimeter of the square, which consists of straight lines, has a ratio to the curved circumference of the circle. </s>
<s id="id.1.2.2.02.13">But why do I go further? </s>
<s id="id.1.2.2.02.14">Aristotle casually says: "A straight line equal to the circumference of a circle is not given": that this is false is demonstrated by the divine Archimedes in his treatise On Spirals, proposition [n¼ XX], where a straight line is found equal to the circumference of a circle around the spiral of first revolution. {1} </s>
<s id="id.1.2.2.02.15"> And do not say: This has escaped Aristotle's attention, because Archimedes is much more recent than Aristotle.</s>
<s id="id.1.2.2.02.16">For, if the demonstration of finding a straight line equal to a curved one has escaped Aristotle's attention, the demonstration proving that a straight line equal to a curved one is not given has also escaped his attention </s>
