20. The geometrical sciences.

This science studies quantities (measurements) ⁶⁵³ Quantities (measurements) may be continuous, like lines, planes, and (geometrical) solids, or discontinuous, like numbers. It also studies the essential properties of the quantities (measurements), as, for instance:

The angles of any triangle are equal to two right angles.

Parallel lines do not intersect anywhere, even when they extend to infinity.

The opposite angles formed when two lines intersect are equal to each other.

The result of multiplying the first and the third of four quantities in a proportion is equal to that of multiplying the second and the fourth. ⁶⁵⁴ And so on.

The Greek work on this craft which has been translated (into Arabic) is the book of Euclid. It is entitled Kitab al­usul wa-l-arkan ("Book of Basic Principles and Pillars").⁶⁵⁵ It is the simplest ⁶⁵⁶ book on the subject for students. It was the first Greek work to be translated in Islam in the days of Abu Jafar al-Mansur. The existing recensions differ, de­pending on the respective translators. There are the recensions of }Hunayn b. Ishaq,⁶⁵⁷ Thabit b. Qurrah,⁶⁵⁸ and Yusuf b. al-Hajjaj.⁶⁵⁹

The work contains fifteen books, four on the planes, one on proportions, another one on the relationship of planes to each other, three on numbers, the tenth on rational and irrational (quantities) ⁶⁶⁰ the "roots" - and five on solids.

Many abridgments of Euclid's work have been written. Avicenna, for instance, devoted a special monograph treatment to it in (the section on) the mathematical disciplines in the Shifa'. Ibn as-Salt ⁶⁶¹ made another abridgment in the Kitab al-Iqtisar, and the same was done by others. Many scholars have also written commentaries on it. It is the starting point of the geometrical sciences in general.

It should be known that geometry enlightens the intellect and sets one's mind right. All its proofs are very clear and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this convenient way, the person who knows geometry acquires intelligence. It has been assumed that the following statement was written upon Plato's door: "No one who is not a geometrician may enter our house." ⁶⁶²

Our teachers used to say that one's application to geometry does to the mind what soap does to a garment. It washes off stains and cleanses it of grease and dirt. The reason for this is that geometry is well arranged and orderly, as we have mentioned.

Spherical figures, conic sections, (and mechanics)

A subdivision of this discipline is the geometrical study of spherical figures (spherical trigonometry) and conic sec­tions. There are two Greek works on spherical figures, namely, the works of Theodosius and Menelaus on planes and sections of (spherical figures). ⁶⁶³ In (mathematical) instruction, the book by Theodosius is (studied) before the book by Menelaus, since many of the (latter's) proofs depend on the former. Both works are needed by those who want to study astronomy, because the astronomical proofs depend on (the material contained in) them. All astronomical discussion is concerned with the heavenly spheres and the sections and circles found in connection with them as the result of the various motions, as we shall mention. (Astron­omy,) therefore, depends on knowledge of the laws governing planes and sections of spherical figures.

Conic sections also are a branch of geometry. This discipline is concerned with study of the figures and sections occurring in connection with cones. It proves the properties of cones by means of geometrical proofs based upon elementary geometry. Its usefulness is apparent in practical crafts that have to do with bodies, such as carpentry and architecture. It is also useful for making remarkable statues and rare large objects (effigies, hayakil) ⁶⁶⁴ and for moving loads and transporting large objects (hayakil) with the help of mechanical contrivances, engineering (techniques), pulleys, and similar things.

There exists a book on mechanics that mentions every astonishing, remarkable technique and nice mechanical contrivance. It is often difficult to understand, because the geometrical proofs occurring in it are difficult. People have copies of it. They ascribe it to the Banu Shakir.⁶⁶⁵

Surveying

 Another subdivision of geometry is surveying. This discipline is needed to survey the land. This means that it serves to find the measurements of a given piece of land in terms of spans, cubits, or other (units), or to establish the relationship of one piece of land to another when they are compared in this way. Such surveying is needed to determine the land tax on (wheat) fields, lands, and orchards. It is also needed for dividing enclosures ⁶⁶⁶ and lands among partners or heirs, and similar things.

Scholars have written many good works on the subject.

Optics

 Another subdivision of geometry is optics. This science explains the reasons for errors in visual perception, on the basis of knowledge as to how they occur. Visual perception takes place through a cone formed by rays, the top of which is the point of vision and the base of which is the object seen. Now, errors often occur. Nearby things appear large. Things that are far away appear small. Furthermore, small objects appear large under water or behind transparent bodies. Drops of rain as they fall appear to form a straight line, flame a circle, and so on.

This discipline explains with geometrical proofs the reasons for these things and how they come about. Among many other similar things, optics also explains the difference in the view of the moon at different latitudes. ⁶⁶⁷ Knowledge of the visibility of the new moon and of the occurrence of eclipses is based on that. There are many other such things.

Many Greeks wrote works on the subject. The most famous Muslim author on optics is Ibn al-Haytham.⁶⁶⁸ Others, too, have written works on the subject. It is a branch of the mathematical sciences.