CHAPTER XV. 



TWO OR MORE LINEAR FUNCTIONS MADE SQUARES. 



Diophantus, II, 12, solved +2=D, z+3=D (the first instance of 

 a " double equality ") by resolving the difference of the two linear functions 

 into two factors in a suitable manner; here he took 4 and 1/4. Take the 

 square of half the difference of the two factors and equate it to the smaller 

 expression, whence 225/64 = +2. Or equate the square of half the sum 

 of the factors to the greater expression. To solve without using a double 

 equation, take x = y z 2 and make x+3 = ?/ 2 +l a square, say by equating 

 it to (y 4) 2 , whence ?/ = 15/8. 



Diophantus II, 13 relates to 9 x=D, 21 z=D; while II, 14 relates 

 to x n= D, x m= D. 



Diophantus, III, 5, 6, required three numbers such that their sum is 

 a square and the sum of any pair exceeds the third by a square. Hence 

 the sum of the three squares is a square, as for 4, 9, 36. 



Diophantus, III, 7, 8, required three numbers whose sum and sums by 

 pairs are squares. Let the sum of all three be (z-fl) 2 , the sum of the first 

 and second be a; 2 , the sum of the second and third be (x I) 2 . Then the 

 sum of the first and third is 6x-\-l and equals 121 if x = 20. 



Diophantus III, 9 relates to three numbers in arithmetical progression 

 whose sums by pairs are squares. Since a; 2 , (z+1) 2 , (x 8) 2 are in A.P. 

 if 2 = 31/10, we seek three numbers whose sums by twos are the numbers 

 961, 1681, 2401 just found. 



Diophantus III, 10 relates to three numbers such that the sum of any 

 pair of them added to a given number a gives a square, and such that the 

 sum of the three added to a gives a square. For a = 3, take the sum of the 

 first two to be 2 +4x+l, the sum of the last two to be z 2 +6x+6, and the 

 sum of all three to be x 2 +8x+13. Then the numbers are 2x-\-7, x 2 +2x 6, 

 4z+12. The sum 6#+22 of the first, third, and a, is the square 100 if 

 a: = 13. In III, 11, a is negative. 



Diophantus, III, 18 and IV, 35, noted that his method does not make 

 ax+b and cx+d squares if a : c is not the ratio of two squares. 1 



Diophantus, IV, 14, made x+l, y+l, x+y+1, x y+1 squares. 



Diophantus, IV, 22, found three numbers in G. P., the difference of 

 any two being a square. In V, 1 [2], he found three numbers in G. P. 

 such that each less [plus] the same given number is a square. 



Diophantus, IV, 45, made 8x+4 and 6x+4 squares by subtraction. 



Diophantus, V, 12, 14, treated the problems to divide unity into 2 or 

 3 parts such that, if the same given number is added to each part, the sums 

 will be squares [see Chs. VI, VII]. 



1 Cf. G. H. F. Nesselmann, Algebra der Griechen, 1842, 335-40. Cf. 86 of Ch. XIX. 



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