Chapter XXVIII: Book III: , Chapter 10. On the inventors of geometry and its name

1. The science of geometry is said to have been discovered first by the Egyptians, because when the Nile overflowed and all their lands were overspread with mud, its origin in the division of the land by lines and measurements gave the name to the art. And later, being carried further by the keenness of the philosophers, it measured the spaces of the sea, the heavens, and the air.

2. For, having their attention aroused, students began to search into the spaces of the heavens, after measuring the earth; how far the moon was from the earth, the sun itself from the moon, and how great a measure extended to the summit of the sky; and thus they laid off in numbers of stades with probable reason the very distances of the sky and the circuit of the earth.

3. But since this science arose from the measuring of the earth, it took its name also from its beginning. For _geometria_ is so named from the earth and measuring. For the earth is called γῆ in Greek, and measuring, μέτρον. The art[253] of this science embraces lines, intervals, magnitudes, and figures, and in figures, dimensions and numbers.

[253] _Hujus ars disciplinae_. _Ars_ may be equal to ‘hand-book’
here.

Chapter 11. On the four-fold division of geometry.

1. The four-fold division of geometry is into plane figures, numerical magnitude, rational magnitude, and solid figures.

2. Plane figures are those which are contained by length and breadth. Numerical magnitude is that which can be divided by the numbers of arithmetic.

3. Rational magnitudes are those whose measures we can know, and irrational, those the amount of whose measurement is not known.

4. Solid figures are those that are contained by length, breadth, and thickness, which are five in number, according to Plato.

Chapter 12. On the figures of geometry.

1. The first of the figures on a plane surface is the circle, a figure that is plane, and has a circumference, in the middle of which is a point upon which everything converges (_cuncta convergunt_) which geometers call the center, and the Latins call the point of the circle.

2. A quadrilateral figure is one on a plane surface, and it is contained by four straight lines....

3. A sphere is a figure of rounded form equal in all its parts.

A cube is a solid figure which is contained by length, breadth, and thickness.

5. A cone (_conon_) is a solid figure which narrows from a broad base like the right-angled triangle.

6. A pyramid is a solid figure which narrows to a point from a broad base like fire. For fire in Greek is called πῦρ.

7. Just as all number is contained within ten so the outline of every figure is contained within the circle.

Chapter 13. On the first principles of geometry.

1. ... A point is that which has no part. A line is length without breadth. A straight line is one which lies evenly in respect to its points. A superficies is that which has length and breadth alone.

Chapter 14. On the numbers of geometry.

1. You search into the numbers of geometry as follows: the extremes being multiplied, amount to as much as the means multiplied; as for example, VI and XII being multiplied, make LXXII; the means VIII and IX being multiplied, amount to the same.

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