3. Read and translate the text:

Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content. The method consists of assuming a small set of intuitively appealing axioms, and then proving many other propositions (theorems) from those axioms. These axioms invoke the following concepts: point, straight line segment and line, side of a line, circle with radius and center, right angle, congruence, inner and right angles, sum. The following verbs appear: join, extend, draw, and intersect.

Although many of Euclid's results had been stated by earlier Greek mathematicians, Euclid was the first to show how these propositions could be fit together into a comprehensive deductive and logical system. The Elements begin with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. The Elements goes on to the solid geometry of three dimensions, and Euclidean geometry was subsequently extended to any finite number of dimensions. Much of the Elements states results of what is now called number theory, proved using geometrical methods. For over two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute sense. It also is no longer taken for granted that Euclidean geometry describes physical space. An implication of Einstein's theory of general relativity is that Euclidean geometry is a good approximation to the properties of physical space only if the gravitational field is not too strong.

Euclid believed that his axioms were self-evident statements about physical reality. This led to deep philosophical difficulties in reconciling the status of knowledge from observation as opposed to knowledge gained by the action of thought and reasoning. Apollonius and other Ancient Greek geometers made an extensive study of the conic sections — curves created by intersecting a cone and a plane. The ones are the ellipse, the parabola and the hyperbola, distinguished by having zero, one, or two intersections with infinity. This turned out to facilitate the work of Galileo, Kepler and Newton in the 17th Century, as these curves accurately modeled the movement of bodies under the influence of gravity.

In the 19th century, it was realized that Euclid's ten axioms and common notions do not prove all of theorems stated in the Elements. For example, Euclid assumed implicitly that any line contains at least two points, but this assumption cannot be proved from the other axioms, and therefore needs to be an axiom itself.

4. Answer the following questions:

1. What kind of system is Euclidean geometry?

2. Why has it been one of the most influential books in history?

3. What do the Euclidean axioms invoke?

4. What do the Elements begin with?

5. Was Euclidean geometry always in use?

6. What did Apollonius and other Ancient Greek geometers make?

7. What works of famous scientists has Euclidean geometry facilitated?

8. How were Euclid's ten axioms presented in the 19^th century?