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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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