A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

I. 9. a — x = m(b — x).
I. 10. x + b = m{a — x).
I. 11. x + b = m(x — a).
DETERMINATE EQUATIONS 485
1.39. (a + x)b+(b + x)a = 2(a + b)x, \
or (a + b) x + (b + x)a = 2 (a + a?) b, L (a > />)
or
(a + b)x + (a + x)b = 2 (b + x)a.)
Diophantus states this problem in this form, ' Given
two numbers (a, b), to find a third number (x) such that
the numbers
(a + x)b, (b + x)a, (a + b)x
are in arithmetical progression.'
The result is of course different according to the order
of magnitude of the three expressions. If a > b (5 and 3
are the numbers in Diophantus), then (a-\-x)b < (b + x)a;
there are consequently three alternatives, since (a + x)b
must be either the least or the middle, and (b + x) a either
the middle or the greatest of the three products. We may
have
(a + x) b < (a + b) x < (b + x)a,
or (a + b) x < {a + x) b < (b + x) a,
or (a + x) b < (b + x) a < (a + b) x,
and the corresponding equations are as set out above.
486 DIOPHANTUS OF ALEXANDRIA
I. 12. x 1 + x 2
= y1 + y2
= a )
x 1
= my 2 ,y 1
==nx 2
(oB 1
>x 2 ,y 1 >y 2 ).
I. 1 3. x x
+ x 2
= y 1
+ y2
— z t
+ z 2
— a'
X l
= #2
1
»
Vl = nZ 2 1 -1 = 2^2
J
I. 15. flj + a = m(2/ — a), y -\-b — n(x—b).
>)
y (x 1
>x 2
,
y1 >y 2
, z 1
>z 2 ).
[Diophantus puts y = £ + a, where £ is his unknown.]
r
I. 16. y + z = a, z + x=zb, x + y = c. [Dioph. puts £=x + y + z.]
I. 17. y + s + w = a, z + iv + x = b,w + x + y = c,x + y + z = d.
[x + y + z + w = g.]
I. 18. y + z — x = a, z + x — y = b, x + y — z = c.
[Dioph. puts
2 £ = x + y + z.]
I. 19. 2/ + + W — # = a, z + w + a? — y = b, w + x + y — z=c,
[2£ = x + y + z + w.]
I. 20. a? + 2/ r£ «, = a, x + y = mz, y + z = nx.
I. 2 1 . x
x + y + z — w = d.
= y + — z, y = z H—ic, = a + - ty (where «>w>2),
with necessary condition.
11.18*. x-Q^x + a) + (£* + *) = 2/- (^ + 6)+ ^ + a)
[Solution wanting.]
= ~C>* + C )
+ G;2/ + O
5
*;
+
2/ + = a.
(ii)
Determinate systems of equations of the first degree.
I. 1. x + y = a, x — y = b.
(I. 2. x + y = a, x = my,
ll. 4. x — y = a, x = my.
I. 3. x + y =
11.
a, x = my + b.
I. 5. x + y = a, — x + - y — b, subject to necessary condition.
/
j
I. 6. x + y — a, — x y = b,
V lib lb
(iii)
Determinate systems of equations reducible to the
first degree.
I. 26. ax — a 2 , bx = oc.
I. 29. x + y = a, x 2 —
2
y = b. [Dioph. puts 2£ = x — y.']
{I. 31. a? = my, x 2 2
+ y = w(a? + 2/).
I. 32. a? = m?/, x 2 2
+ y = n(x — y).
I. 33. a? == m?/, x — 2 2
y = n(x + y).
I. 34. a; = 7>t2/, x 2 —y = 2 n(x — y).
I. 34. Cor. 1. a? = m^/, 033/ = n(x + y).
Cor. 2. a? = m^/, «?/ = n{x~y).