This document provides an overview of Euclid's Elements and developments in geometry from Euclid. It discusses Euclid's geometric structure and postulates, including his parallel postulate. It also examines attempts to prove or replace the parallel postulate, such as Playfair's postulate and the works of Proclus and Saccheri. Figures were important in developing Euclid's geometry and understanding problems like the parallel postulate.

1. EUCLID'S ELEMENTS
CONCEPTS AND
DEVELOPMENTS
History of Mathematics
By
Miftahul Jannah (4193311004)
MESP 2019.

2.  Geometri Euclid
 Euclid's geometric structure
 Substitute Euclid's Parallel Postulates
 Euclid's and Playfair's Equivalence Postulates
 The Role of Euclid's Parallel Postulates
 Figures in the Development of Euclid Geometry

3. Euclid can be called a major mathematician. He is known for his legacy in the form of the
mathematical work embodied in the monumental The Elements. The ideas poured into the book
made Euclid considered the mathematics teacher of all time and the greatest Greek
mathematician. Many of the theorems he describes are the work of earlier thinkers, including
Thales, Hippocrates, and Pythagoras in The Elements. In general, however, Euchild is credited
with having arranged these theorems logically, in order to be able to show (undeniably, not
always with the rigorous proofs modern mathematics demands) that it is sufficient to follow five
simple axioms.
euclidean geometry

4. Euclid's geometric structure
The assumptions or postulates that exist for Euclid's plane geometry are:
1. Something will be equal to something or something equal will be equal to each other.
2. If the similarity is added to the similarity, then the sum will be the same.
3. If the similarity is subtracted from the similarity, the difference will be the same.
4. The whole will be greater than the part.
5. Geometric shapes can be moved without changing their size or shape.
6. Each corner has a bisector.
7. Each segment has a midpoint.
8. Two points are only on the only line.
9. Any segment can be expanded by a segment equal to the given segment.
10. A circle can be drawn with any given center and radius.
11. All right angles are equal.
From the postulates above, a number of basic theorems can be deduced, including:
1. Opposite angles are equal.
2. Congruence properties of triangles (si-sd-si, sd-si-sd, si-si-si).
3. The equation of the basic angles of an isosceles triangle and its conversions.
4. The existence of a line that is perpendicular to the line at the point of the line.
5. The existence of a line that is perpendicular to the line passing through the external point.
6. Prove that an angle is equal to the angle with the vertex and the side given previously.
7. Formation of congruent triangles with triangles with the same side on the known sides of the triangle.
We will now prove the exterior angle theorem, as a way of progressing further.

5. Proof. Let ABC be an arbitrary triangle and let D be an extension of through C. First we will show that the exterior
angle is greater than . Let E be the midpoint of AC, and let BE be the extension of its length through E to F. Then
AE=EC=BE=EF and (opposite angles are equal). So (si-su-si), and (due to congruent triangles). Since (the whole
angle is always greater than its part), it follows that .
To show that , extend through C to H, which forms . Then show that , using the procedure of the first part of
the proof: let M be the midpoint of , extend the length of , and so on. To complete the proof, note that these are
opposite angles so they are the same measure. The statement depends on the diagram. Now it's easy to prove some
pretty important results.
Theorem 1. Exterior angle theorem. Triangular exterior
corners will be larger than any secluded interior
corners.

6. Theorem 2. If two lines are divided by a transverse line so that they form pairs of interior angles that are opposite,
then the lines are parallel.
Proof. Recall that two lines in the same plane are said to be parallel if they are indefinite (intersect). Suppose the
transverse line bisects the line l, m at points A, B so as to form a pair of opposite interior angles, 1 and 2, which are
equal in size, and let the line l and line m be not parallel. Then the line l and line m will meet at point C which forms
ABC. C lies on one side of AB or on the other. For the other case, the exterior angle ABC is equal to the remote
interior angle. (e.g., if C is on the same side AB as 2 then the exterior angle 1 is equal to the remote interior angle 2 ).
This contradicts the previous theorem. Therefore, line l and line m are parallel.

7. • Effect 1. Two lines perpendicular to the same line must be parallel.
• Effect 2. There is only one line that is perpendicular to the line through the external point.
• Effect 3. (Existence of parallel lines). If the point P is not on the line l, then there will be at least one line through
P that is parallel to l.
Proof. From P remove the perpendicular to the line l which has the leg at Q, and at P draw the line m
perpendicular to PQ. Then line m is parallel to line l according to the effect 1

8. Theorem 3. The sum of two angles of a triangle is less than 180O.
Bukti. Misalkan ∆ABC merupakan sebarang segitiga. Akan ditunjukkan
bahwa ∠A + ∠B < 180O. Perluas CB melalui B hingga ke D. maka ∠ABD
merupakan sudut eksterior ∆ABC. Dengan menggunakan teorema 1, ∠ABD >∠A,
tetapi ∠ABD = 180O - ∠B.dengan mensubstitusikan untuk ∠ABD pada relasi
pertama, maka : 180O - ∠B > ∠A, atau 180O > ∠A + ∠B. Jadi, ∠A + ∠B < 180O,
dan teorema tersebut terbukti.

9. Euclid's parallel postulate is usually replaced by the following statement:
"There is only one parallel line on the line that passes through the point not on the line."
This statement is called the Playfair postulate. This postulate can be related to Euclid's parallel postulate
because actually these two statements are not the same. The previous statement is a statement about parallel
lines, and the second statement is about meeting lines. In fact the two statements play the same role in the
logical development of geometry. It is said that this statement is logically equivalent.
Substitute Euclid's Parallel Postulates

10. We will prove the equivalence of Euclid's postulates and Playfair's postulates. First, by assuming
Euclid's parallel postulate, Playfair's postulate will be deduced.
Euclid's and Playfair's Equivalence Postulates
Now assuming Playfair's postulate, we will deduce Euclid's parallel postulate.

11. Assuming Euclid's parallel postulates, the following important results can be justified:
1. If two parallel lines are divided by a transverse line, any pair of opposite interior interior angles formed
will be equal.
2. The sum of the angles of any triangle is 180°.
3. The opposite sides of a parallelogram are equal.
4. Parallel lines are always equidistant.
5. The existence of a quadrilateral and a square.
6. Area theory using square units.
7. The theory of equal triangles, which includes the existence of shapes of any size equal to the known
shapes.
The Role of Euclid's Parallel Postulates

12. Figures in the Development of Euclid Geometry
 Bukti Proclus tentang Postulat Sejajar Euclid
Prolus (410-485) provides a "proof" of Euclid's parallel postulate which we summarize as follows:
We assume Euclid's postulates are not parallel postulates. Suppose P is a point not on line l (figure 2.7).
The Prolus argument includes 3 assumptions:
a. If two lines intersect, the distance on a line from one point to another will increase erratically,
because the point is receding (shrinking) endlessly.
b. The shortest segment connecting external points on a line is a perpendicular segment.
c. the distance between two parallel lines is finite.

13.  Percobaan Saccheri untuk Mempertahankan Postulat Euclid
Girolamo Saccheri (1667-1733) undertook an in-depth study of geometry in a book entitled Euclides Vindicatus,
published in the year of his death. He approached the problem of proving Euclid's parallel postulate in a radically
new way. The procedure is equivalent to assuming that Euclid's parallel postulates are wrong, and finding
contradictions by logical reasoning. This will validate the parallel postulate using the principle of the indirect
method.
Saccheri meant the study of quadrilaterals having sides that are the same length and perpendicular to the third
side. Without assuming any parallel postulates, he undertook an in-depth study of the quadrilateral which is now
known as the Saccheri quadrilateral. Let ABCD be a Saccheri quadrilateral with AD = BC and right angles at A, B
(figure 2.10).

14. Thank you!!