By Jacques Sesiano

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**Example text**

Example 2. 5] To divide a given number, which is the sum of two squares, into two other squares. 35 Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, & generaliter nullam in inﬁnitum ultra quadratum potestatem in duos (sic) eiusdem nominis fas est dividere. Cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. ) the republication of the 1621 edition of Diophantus, including Fermat’s remarks, by Fermat’s son, Diophanti Alexandrini Arithmeticorum libri sex (.

Figure 6 The Arithmetica is not a work of theory but a collection of problems. Primarily, these problems are indeterminate and quadratic (or reducible to a quadratic problem), with solutions that must be positive and rational (not necessarily integral, as the modern use of the expression “Diophantine equation” would imply). The few glimpses of theory contained in the Arithmetica appear in conditions that may be necessary for obtaining a rational solution; Diophantus mentions such conditions at the outset, immediately following the statement.

3]. The area together with the perimeter of a right-angled triangle is 280 feet; to separate the sides and to ﬁnd the area. I proceed as follows. Always search for the numbers making up (280). Now 280 is made up by twice 140, 4 (times) 70, 5 (times) 56, 7 (times) 40, 8 (times) 35, 10 (times) 28, 14 (times) 20. We have observed that the 8 and 35 will satisfy the given requirement. ]28 Always take two away from the 8, there remains 6 feet. Then the 35 and the 6 together, there results 41 feet. Multiply this by itself, there results 1681 feet.