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 manuscripts directly or from the translations made by his countrymen, Gerard of Cremona and Plato of Tivoli. Leonardo's Geometry contains an elegant geometrical demonstration of Heron's formula for the area of a triangle, as a function of its three sides. Leonardo treats the rich material before him with skill and Euclidean rigour.

Of still greater interest than the preceding works are those containing Fibonacci's original investigations. We must here preface that after the publication of the Liber Abaci, Leonardo was presented by the astronomer Dominicus to Emperor Frederick II. of Hohenstaufen. On that occasion, John of Palermo, an imperial notary, proposed several problems, which Leonardo solved promptly. The first problem was to find a number x, such that $$\scriptstyle{x^2+5}$$ and $$\scriptstyle{x^2-5}$$ are each square numbers. The answer is $$\scriptstyle{x=3\frac{5}{12}}$$; for $$\scriptstyle{(3\frac{5}{12})^2+5=(4\frac{1}{12})^2,~(2\frac{5}{12})^2-5=(2\frac{7}{12})^2}$$. His masterly solution of this is given in his liber quadratorum, a copy of which work was sent by him to Frederick II. The problem was not original with John of Palermo, since the Arabs had already solved similar ones. Some parts of Leonardo's solution may have been borrowed from the Arabs, but the method which he employed of building squares by the summation of odd numbers is original with him.

The second problem proposed to Leonardo at the famous scientific tournament which accompanied the presentation of this celebrated algebraist to that great patron of learning, Emperor Frederick II., was the solving of the equation $$\scriptstyle{x^3+2x^2+10x=20}$$. As yet cubic equations had not been solved algebraically. Instead of brooding stubbornly over this knotty problem, and after many failures still entertaining new hopes of success, he changed his method of inquiry and showed by clear and rigorous demonstration that the roots of this equation could not be represented by the Euclidean irrational quantities, or, in other words, that they could not be