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A + b) (a + b) = a2 + ab + ab + b2 ,= a f 2ab ' b' . a-b D Binomial formulae (a b)' - a' (a - b)' - a' (a b) (a - b) The formula for (a - b)' i uperflu u. tri tly peaking. L b 1. ). By appl ing these formulae the quare of an algebrai um can be ritten down and a um an be factorized . In both direction one obtain aid for mental calculations. Examples: 1. ,'. 2. n'. 1. Ss)'. 4. 1601 - 560b + 49b l - 64e l = (4a - 7b)' 5. 5-2 Illustration of (a - b)1 - a l 64e' = (40 - 7b 36 = 159964. 41616. 360. - 8c) (4a - 7b - 8e).

In Greek mathematics the fourth proportional was constructed geometrically, by the method of geometrical algebra. But the computational treatment of proportions and the calculus of the 'rule of three' were not developed in Europe until the 15th-17th century, in particular, in con- 40 1. Fundamental operations on rational numbers nection with commercial calculations. Such problems form one of the main constituents of the widely used arithmetic books and the principal teaching topic of the arithmeticians and 'cossists'.

How many meters of wire of the same material. but of diameter d1 = 6 mm have the mass = 90 kg? - Since the wires consist of the same material, their masses are in the ratio of their volumes. Hence By the product equation the solution is: m, : ml = (dU4) ni, : (di/4) n/" m,: m, = dIll : dl/,. I _ m,d~/1 el = m, , - mld~ , in which corresponding quantities (II and I,; d, and d2 : m, and m2) must be measured in the same units. In this numerical example one calculates the Icngth of 436 m. 24:x = 1200:1000 Example 4: A service station has enough fuel for 24 days if the daily 24' 1000 sale is 1000 gallons.

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Actes Du Congres International Des Mathematiciens 1970 by Anon.

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