A History of Mathematics From Mesopotamia to Modernity

Understanding the‘ScientificRevolution’ 159

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Fig. 6Archimedes’ idea: ABC is the triangle whose base is the circumference and whose height is the radius. BC’ is the perimeter of
the polygon, and its area equal that of triangle ABC’. If the area of the circle is greater than ABC, then it is greater than that of some
ABC’, since they approach ABC as near as you like; but then it is greater than the area of some polygon, which is absurd.

SIMP. I am quite aware that a squared number is one which results from the multiplication of
another number by itself; thus 4, 9, etc., are squared numbers which come from multiplying 2, 3,
etc. by themselves.
SALV. Very well; and you also know that just as the products are called squares so the factors
are called sides or roots; while on the other hand those numbers which do not consist of two
equal factors are not squares. Therefore if I assert that all numbers, including both squares and
non-squares, are more than the squares alone, I shall speak the truth, shall I not?
SIMP. Precisely so.
SALV. But if I inquire how many roots there are, it cannot be denied that there are as many
as there are numbers, because every number is a root of some square. This being granted we must
say that there are as many squares as there are numbers because they are just as numerous as
their roots, and all the numbers are roots...[Salviati develops his point, and shows that as the
proportion of squares gets smaller the more numbers we consider.]
SAGR. What then must one conclude under these circumstances?
SALV. So far as I can see we can only infer that the totality of all numbers is infinite, that the
number of squares is infinite, and that the number of their roots is infinite; neither is the number of
squares less than the totality of all numbers, nor the latter greater than the former; and finally the
attributes “equal”, “greater”, and “less”, are not applicable to infinite, but only to finite, quantities.

Solutions to exercises

1. The natural way to proceed would be as follows. Given a circle C, letabe its diameter. Construct a
   line B whose length is 3^17 times A—this is straightforward, Euclid quite early gives a method for
   dividing a line into (for example) seven equal parts. This we take to be equal to the circumference
   of C, and now the area of C is equal to that of a right angled triangle whose height is A and
   whose base is B. Finally, Euclid II.14 gives a method of constructing a square whose area is
   equal to the triangle (you construct a ‘mean proportional’ between A and B/2, a line whose
   length is

√

1

2 AB).