20.
This science studies quantities (measurements) 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.
The Greek work on this craft which has been translated
(into Arabic) is the book of Euclid. It is entitled
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)
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
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." 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.
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).
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,
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.
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
Scholars have written many good works on the subject.
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.
Many Greeks wrote works on the subject. The most famous
Muslim author on optics is Ibn al-Haytham. |
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