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96
PYTHAGOREAN ARITHMETIC
which give
579
x+y+z+u = ~{z + u)=-{u + y) = -iy + z).
Therefore
5, 7 . 9 , .
x + y + 0 + u = q{x + y) — j(x + z) = ~(x + u).
In this case we take A, the least common multiple of 5. 7, 9,
or 315, and put
x y A z u — L — 315,
x-\-y — — L — 189,
5
X + Z = -L= 180,
7
x + u= ~L = 175,
9
whence
544 -315 229
In order that x may be integral, we have to take 2 L, or 630,
instead of L, or 315, and the solution is x = 229, y = 149,
0 = 131, u = 121.
(y) Area of rectangles in relation to perimeter.
Sluse^in letters to Huygens dated Oct. 4, 1657, and Oct. 25,
1658, alludes to a property of the numbers 16 and 18 of
which he had read somewhere in Plutarch that it was known
to the Pythagoreans, namely that each of these numbers
represents the perimeter as well as the area of a rectangle ;
for 4.4 = 2.4 + 2.4 and 3.6 = 2.3 + 2.6. I have not found the
passage of Plutarch, but the property of 16 is mentioned in the
Theologumena Arithmetices, where it is said that 16 is the only
square the area of which is equal to its perimeter, the peri
meter of smaller squares being greater, and that of all larger
squares being less, than the area. 2 We do not know whether
the Pythagoreans proved that 16 and 18 were the only num
bers having the property in question; but it is likely enough
that they did, for the proof amounts to finding the integral
Œuvres complètes de C. Huygens, pp. 64, 260.
Theol. Ar., pp, 10, 23 (Ast).
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