Geometry

But, because the origin of arts and sciences is to be considered according to the present revolution of the universe, we must affirm, in conformity with the most general tradition, that geometry was first invented by the Egyptians, deriving its origin from the mensuration of their fields: since this, indeed, was necessary to them, on account of the inundation of the Nile washing away the boundaries of land belonging to each. … When Thales, therefore, first went into Egypt, he transferred this knowledge from thence into Greece: and he invented many things himself, and communicated to his successors the principles of many. … But Euclid was not much junior to these, who collected elements, and constructed many of those things which were invented by Eudoxus; and perfected many which were discovered by Theætetus. Besides, he reduced to invincible demonstrations, such things as were exhibited by others with a weaker arm. But he lived in the times of the first Ptolemy: for Archimedes mentions Euclid, in his first book, and also in others. Besides, they relate that Euclid was asked by Ptolemy, whether there was any shorter way to the attainment of geometry than by his elementary institution, and that he answered, there was no other royal path which led to geometry.

The discipline of Geometry is said to have been first discovered by the Egyptians, because, when the Nile overflowed and covered the possessions of all with silt, the beginning of dividing the land by lines and measures gave the name to the art. Which afterwards, advancing further by the keenness of the wise, measure also the spaces of sea and sky and air. For, provoked by zeal, they began thus after the measuring of the earth to inquire into the spaces of the sky: at what interval the moon was from the earth, the sun itself from the moon, and how great a measure extended to the vertex of the sky, and so they distinguished the intervals of the sky itself and the compass of the orb by a number of stadia, with probable reasoning. But because this discipline began from the measurement of the earth, it also preserved from its beginning the name. For geometry is named from earth and measure. For earth in Greek is called _gē_, measure _metra_. The art of this discipline contains in itself lines, intervals, magnitudes and figures, and in figures dimensions and numbers.

Chapter XI. On the fourfold division of geometry

The division of geometry is fourfold: into the plane, into numerable magnitude, into rational magnitude, and into solid figures. Plane figures are those which are contained by length and breadth, which are, according to Plato, five in number. Numerable magnitude is that which can be divided by the numbers of arithmetic. Rational magnitudes are those whose measure we can know; irrational, those whose quantity of measure is not known.