Page:A History of Mathematics (1893).djvu/99

 How shall the estates be divided so as to satisfy the will? The celebrated Roman jurist, Salvianus Julianas, decided that the estates shall be divided into seven equal parts, of which the son receives four, the wife two, the daughter one.

We next consider Roman geometry. He who expects to find in Rome a science of geometry, with definitions, axioms, theorems, and proofs arranged in logical order, will be disappointed. The only geometry known was a practical geometry, which, like the old Egyptian, consisted only of empirical rules. This practical geometry was employed in surveying. Treatises thereon have come down to us, compiled by the Roman surveyors, called agrimensores or gromatici. One would naturally expect rules to be clearly formulated. But no; they are left to be abstracted by the reader from a mass of numerical examples. "The total impression is as though the Roman gromatic were thousands of years older than Greek geometry, and as though a deluge were lying between the two." Some of their rules were probably inherited from the Etruscans, but others are identical with those of Heron. Among the latter is that for finding the area of a triangle from its sides and the approximate formula, $$\scriptstyle{\frac{13}{30}a^2}$$, for the area of equilateral triangles (a being one of the sides). But the latter area was also calculated by the formulas $$\scriptstyle{\frac{1}{2}(a^2+a)}$$ and $$\scriptstyle{\frac{1}{2}{a^2}}$$, the first of which was unknown to Heron. Probably the expression $$\scriptstyle{\frac{1}{2}{a^2}}$$ was derived from the Egyptian formula $$\scriptstyle{{{a+b} \over 2} \cdot {{c+d} \over 2}}$$ for the determination of the surface of a quadrilateral. This Egyptian formula was used by the Romans for finding the area, not only of rectangles, but of any quadrilaterals whatever. Indeed, the gromatici considered it even sufficiently accurate to determine the areas of cities, laid out irregularly, simply by measuring their circumferences.[7] Whatever Egyptian geometry the Romans possessed was transplanted across the Mediterranean at the