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The common tool for solving these three problems is the equilateral triangle Eucl. Finally, the construction of a perpendicular to a line AB from a point C outside of it see Fig. The entrance of Postulate 4. The fourth postulate expresses the homogeneity of the plane, the absence of any privileged direction, and allows one to compare, add and subtract the 34 2 The Elements of Euclid angles around a point. In the situation of Fig.

Let E lie on the line DB, i. Therefore, E and C lie on the same line. If two straight lines cut one another, they make the opposite angles equal to one another, i. By Eucl. By Post. If one side of a triangle is produced at C see Fig. Let E be the midpoint of AC Eucl.

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We produce BE Post. The grey angles at E are equal Eucl. In the geometry on the sphere, which we will discuss in more detail in Section 5. The reason is that the point F , which in our example becomes the South Pole, is no longer certain to remain in the open sector between the produced lines CA and BC. Various theorems of Euclid on the congruence of triangles determined by certain side lengths or angles see Fig.

The ambiguous case ASS last picture is not mentioned by Euclid. For an inequality involving the angles and sides of a triangle Eucl. Congruence theorems for triangles Eucl. This result has been ridiculed as being evident even to an ass. For if one puts the ass at one vertex of the triangle and hay at another, the ass will follow the side that joins the two vertices and will not make the detour through the third vertex digni ipsi, qui cum Asino foenum essent, Heath, , vol. Proclus gave a long logical-philosophical answer. In this case, we write a k b for short.

If a and b were not parallel, they would meet in a point G, see Fig. The entrance of Postulate 5.

Geometry by Its History

This part of geometry is called absolute geometry. Drawing a parallel to a given line through a given point A. One can also use two orthogonal lines Eucl. Proclus made the following statement in his commentary: There exists at most one line through a given point A which is parallel to a given line.

This is a very old theorem, certainly known to Thales.

The remainder of Book I. Book II. This book contains geometrical algebra, i. For instance, the product of two numbers a, b is represented geometrically by the area of a rectangle with sides a and b. We have for example the following relations, Eucl. The two light grey rectangles are the same. For a direct proof of 2.


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With the advance of algebra, the above propositions can all be obtained from Eucl. It allows the quadrature of a rectangle, i. Properties of Circles and Angles The third book is devoted to circles and angles. For instance, Eucl. Angles of a quadrilateral inscribed in a circle Eucl. We consider the triangle ABC in Fig.

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The result is thus a consequence of Eucl. Another proof of Eucl. It is clear from Fig. Euclid did not consider angles greater than 2 ; hence he would not have presented such a proof.

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Being less scrupulous, we see by Eucl. Hence 2. Let E be a point outside a circle and consider a line through E that cuts the circle in two points A and B. Further let T be the point of tangency of a tangent through E see Fig. Exercise 17 on page Corollary Clavius Let A, B, C and D denote four points on a circle. This is clear from Eucl. We can also prove this corollary directly by Eucl. Then Eucl. The particular case of Eucl.

Book IV. This book treats circles, inscribed in or circumscribed to triangles, squares, regular pentagons Eucl. The more elegant proof that we gave in Chap. The book ends with the construction of the regular sided polygon Eucl. Barrow; see Heath, , vol. The theory of proportions. This theory is due to Eudoxus and has been greatly admired.


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  • It concerns ratios of irrational quantities and their properties. One constantly works with inequalities that are multiplied by integers. One thereby squeezes irrational quantities between rational ones, somewhat in the style of Dedekind cuts years later.

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    Book VI. Thales-like theorems. One joins B to E and C to D. Number Theory 43 Proof. Euclid proves this theorem as an application of Eucl. As the points on the angle bisector have the same distance from both sides a consequence of Eucl. The results are not geometrical, but the way of thinking is, at least for Euclid.

    The main tool is the observation, already known from Book V in particular Eucl. This leads to Eucl. Given two numbers not relatively prime, to find their greatest common measure. The Euclidean algorithm.