CHAPTER 9

HISTORY OF BABILON MATHEMATICS

A. Development of Mathematics in Babylonian Era

In 626 BC, after Assyrian rule destroyed by the death of the king Asshurbanipal, the
Babylonians rise again under Chaldean dynasty rule and shape civilization New Babylon14.
The history of world civilization records that The Babylonians played a major role in various
fields. In the field of science, the Babylonians have made progress, one of which is in math
field. The Babylonians are considered as nation that has the highest mathematical knowledge.
So that the development of mathematics in Mesopotamia is more known as "Babylonian
Mathematics" because the area Babylonia played a major role as a place for learning.
Babylonian mathematics refers to all mathematics that developed by the Mesopotamians
since Sumerian leadership to the dawn of civilization hellenistic. Babylonian civilization in
Mesopotamia replaced the Sumerian and Akkadian civilizations. In the numeral form used,
the Babylonians inherited the idea of the Sumerians, namely using the system mixed
sexadecimal numeration with base 10 and know place value. Base 10 is used because the
numbers 1 through 59 are formed from the symbols "units" and the "tens" symbol placed into
a single unit. This number system came into use around 2000 BC. But the weakness of the
Babylonian number system has not been recognize the symbol zero16. It was only a few
centuries later, about 200 BC, that the Babylonians had represents zero marked with a space.
The following are 59 Babylonian numeral symbols.

Babylonian mathematician Otto Neugebauer concluded the results of his research that
mathematics Babylon has reached a high level17. Nation The Babylonians had developed
algebra. the math they develop already advanced because they can complete quadratic
equations, third and fourth power equations. And have known the relationship of the sides of
a right triangle since early 1900 BC.

The Babylonians had knowledge of multiplication and division tables. Math

knowledge Babylon is obtained from the discovery of approximately 400 clay slabs
excavated since the 1850s. This clay slab was written while the clay was still wet, then
burned in the furnace or dried in the sun. Some ancient manuscripts related to mathematical
knowledge has been found at Yale, Columbia, and Paris from Babylonian times. In university
Columbia, there is a catalog of processed ancient manuscripts Mesopotamia written by G. A.
Plimpton containing mathematical problems. This catalog is numbered 322 so known as
Plimpton 322.

The manuscript contains a mathematical table from the era between 1900-1600 BC20.
plimpton manuscript 322 in the form of a table consisting of four columns and fifteen rows
containing the corresponding numbers form the number Pythagorean triples. Mostly clay
slabs too contains the topics of fractions, algebra, equations square and cubic, regular number
calculation, inverse multiplication, and twin prime numbers. From the discovery of the clay
plate indicates that at that time the Babylonians had using algebra, but only limited to the
stage theoretical. Then from here is the basis of development next algebra. In solving algebra,
nation The Babylonians used problem solving techniques using idegeometry. This geometric
idea is a problem solving process by manipulating data that actually based on the rules that
have been set. Based on the discovery of several manuscripts mathematics in Babylon,
further inspired Muslim scientists to develop mathematics next. Like Thabit bin Qurrah,
known as the greatest geographer of that time. He was born in Haran, Mesopotamia in 833 C.
 Thabit translated Archimedes' originalstranslated into Arabic manuscripts. Tsabit's
translations were found in Cairo and later spread to western society. In 1929 the book was
translated into German. Besides Archimedes, some of Euclid's works have also been
translated by Thabit, namely On the Promises of Euclid; on the proposition of Euclid, and a
book on the theorem and The question that arises if two straight lines are cut by one line.
There is also Euclid's Elements book which This is the starting point for the development of
the study of geometry among Muslim scientists after being translated by Thabit. With
geometric methods, he was able to solve triple equations problem. Equations geometry
developed by Tsabit gets great concern among Muslim scientists. The experts mathematics
considers the solution made by Tsabit classified as creative, because the books he translated
he can fully master, and be developed by him. In a relatively short time, the method used
developed by the Babylonians then arrived the hands of the Greeks. Aspects of mathematics
The Babylonians who had come to Greece had increased the quality of mathematical work by
not only believing in its physical forms, but also being strengthened by mathematical proofs.

B. Babylonian Number System

Babylonian mathematics was written using the sexagesimal (base-60) system. From
this derived the use of the numbers 60 seconds for a minute, 60 minutes for an hour, and 360
(60 x 6) degrees for a circle, as well as the use of seconds and minutes on a circular arc
representing fractions of degrees. .No zero was discovered at this time. The Babylonian
advances in mathematics were supported by the fact that 60 had many divisors.

The Babylonians had a true place-value system, in which the numbers written in the
left column represent the larger value, as in the decimal system. However, there is a lack of
decimal comma equivalence, so the place value of a symbol often has to be approximated by
its context. At this time also not found the number zero. For a certain positional system a
number convention is needed that shows the uniqueness of a number. For example the
decimal 12345 means: 1 × 104 + 2 × 103 + 3 × 102 + 4 × 10 + 5.

The Babylonian sexagesimal positional system adheres to the above method of writing,
namely that the far right position is for units up to 59, one side to the left is for 60 × n , where
1 ≤ 𝑛 ≤ 59 and so on. Now we use a notation where the numbers are separated by commas,
for example, 1,57,46,40 represents the sexagesimal number 1 × 603 + 57 × 602 + 46 × 60 +
40, in decimal notation the value is 424000.
Much of our knowledge of mathematics developed in the Mesopotamian region, developed
initially by the Sumerians and later by the Akkadians and others, is relatively recent. This
knowledge is called Babylonian mathematics, as if it came from only one nation. For some
time, it has been known that the vast Babylonian collection of antiquities in British museums,
the Louvre, Yale, and the universities of Pennsylvania consists of many ancient writing
tablets of an unusual type which have not yet been deciphered. Serious research conducted by
Otto Neugebauer, which came to fruition in the 1930s, revealed that all these manuscripts
were mathematical tables and texts, and thus the key to "reading" the contents of the ancient
Babylonian manuscripts.

The Babylonians, freed by their extraordinary numbering system from the tedious
work of computation, became tireless compilers of arithmetical tables, some of which were
extraordinary in complexity and breadth. Many tables give the squares of the numbers 1
through 50 as well as the cubes, square roots, and cube roots of these numbers. A tablet now
in the Berlin Museum gives a list of not only n 2 and n3 for n = 1, 2, …., 10, 20, 30, 40, 50, but
also the sum of n2 + n3.

C. Babylonian Geometry

Geometry was used by the Babylonians from 2000 to 1600 BC. They calculate the
circumference of a circle using three times the diameter, the area of the circle is used one
twelfth of the square of the circumference with = 3,14. The volume of an upright cylinder is
calculated by multiplying the area of the base by the height.

D. Summary

The Babylonians were a tribe that lived in the Mesopotamian region (located between
two major rivers, the Euphrates and the Tigris). The area of Mespotamia is now known as the
country of Iraq.

Babylonian mathematics refers to all of the mathematics used by the Mesopotamians in what
is now Iraq since the beginning of the Hellenistic civilization. Named "Babylonian
Mathematics” because of Babylon's primary role as a place of learning.

Babylonian mathematics was written using the sexagesimal (base-60) system. From
this derived the use of the numbers 60 seconds for a minute, 60 minutes for an hour, and 360
(60 x 6) degrees for a circle, as well as the use of seconds and minutes on a circular arc
representing fractions of degrees. .No zero was discovered at this time. The Babylonian
advances in mathematics were supported by the fact that 60 had many divisors.

E. Exercise

1. Explain the history of algebra from Babylonian times to the present!

2. Name and explain the mathematicians from Babylonian times and their discoveries!

3. Name the most famous mathematical discoveries of the Babylonian era!

4. What is the Babylonian mathematical number system?

5. Mention the proof of mathematical heritage in Babylonian times!

 CHAPTER 10
EUCLID’S ELEMENTS CONCEPTS AND DEVELOPMENTS
A. Geometri Euclid
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.
The Elements book has been the standard handbook for more than 2000 years and is the most
successful textbook ever compiled by man. As a training tool for the logic of the human
mind, The Elements is far more influential than all of Aristotle's treatises on logic. The
Element begins with definitions, postulates, propositions, theorems, before closing with
proofs using the definitions and postulates already mentioned. This book was published in
Greece in 1482, was translated into Latin and Arabic, and became a textbook on geometry
and logic in the early 1700s.
Euclid set out 5 postulates which later became the subject of discussion. In order to avoid
misinterpretation, the fifth postulate is also presented in English. This is intentional, due to
the emergence of non-Euclidian geometry (which was pioneered by Gauss), which begins by
assuming the fifth postulate is totally wrong. Meanwhile, Euchild gives five postulates
(axioms):
1.    Apa-apa dua titik boleh dihubungkan dengan satu garis lurus.
2.     Apa-apa tembereng garis lurus boleh dipanjangkan di dalam satu garis lurus.
3.    Satu bulatan boleh dilukis dengan menggunakan satu garis lurus sebagai jejaridan satu
lagi titik hujung sebagai pusat.
4.    Semua sudut serenjang adalah kongruen.
5.    Postulat selari. Jika dua garis bersilangan dengan yang ketiga dalam satu cara yang jumlah
sudut dalaman adalah kurang daripada satu lagi, maka dua garis ini mesti bersilangan di atas
satu sama lain sekiranya dipanjangkan secukupnya.
These axioms use the following concepts: points, segments of straight lines and lines, parts of
a line, circles with radii and center, congruent angles, congruent, interior and congruent
angles, sum. The following verbs appear: connect, extend, paint, cross. This circle is
described using postulate 3 is very unique. Postulates 3 and 5 should only be used for straight
geometries; in three dimensions, postulate 3 defines a sphere.
One proof from Euclid's book "Elements" that if given a line segment, an equilateral triangle
includes the segment as one of the three sides. The proof is by the built method: An
equilateral triangle E is created by painting the circles and centered on the points and , and by
taking a cross of the circle as the vertex of the third angle for the triangle. Postulate 5 leads to
the same geometry as the following statement, known as Playfair's Axiom, to which only the
concept may be held within it: "Continuing a point which does not lie on a straight line, only
one line that may be drawn will not meet. given line."
Postulates 1, 2, 3, and 5 affirm that the existence and uniqueness of geometrical diagrams,
and this assertion is a natural construction, we are not told that there is a certain matter of
existence, but the rules are given to create by nothing more than one compass and one
unmarked straight edge. In this respect, Euclid's geometry is more concrete than most modern
axiomatic systems such as set theory, which is in the habit of asserting the existence of
objects without saying how to construct them, or asserting the existence of objects that should
not be constructed in the space of the theory concerned.
In fact, constructs of lines on paper and so on are object models that are better defined in a
formal system, than just examples of the object concerned. For example, a Euclidean straight
line has no width, but any true line will have width. Elements also includes five “regular
notations”, namely:
1.    Perkara yang sama dengan benda yang sama tetapi juga setara antara satu sama lain.
2.    Jika setara ditambahkan kepada persamaan, maka jumlah keseluruhan juga adalah setara.
3.    Jika setara ditolak daripada persamaan, maka akibatnya juga adalah setara.
4.    Perkara yang bertembung di antara satu sama lain juga setara antara satu sama lain “that
coincide with one another equal one another.”
5.    Jumlah keseluruhan juga lebih besar daripada bagian berkenaan.
Euclid also used other properties related to magnitude. 1 is the only part of the basic logic
that Euclid expounds clearly and clearly. 2 and 3 are "arithmetic" principles; note that the
meanings of "add" and "reject" in the context of this original geometry have been given the
same as taken. 1 to 4 literally have similarities, which may also be taken as part of the logical
basis or as a necessary equivalence relationship, such as "conversation," a very precise
definition. 5 is a rheological principle. "Whole", "part", and "tray" require precise definitions.
B. Struktur Geometri Euclid
   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.
Theorem 1. Exterior angle theorem. Triangular exterior corners will be larger than any
secluded interior corners.

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 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.
• 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.

Theorem 3. The sum of two angles of a triangle is less than180O.

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.

C. Pengganti Postulat Sejajar Euclid

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. This means that if the first statement is
considered a postulate (along with all of Euclid's postulates except the parallel postulates),
then the second statement can be deduced as a theorem; and its conversion, if the second
statement is considered a postulate (along with all of Euclid's postulates except parallel
postulates), then the first statement can be deduced as a theorem. So logically, it doesn't
matter which two statements are to be assumed as postulates and which are to be deduced as
a theorem.
D. Ekivalensi Postulat Euclid dan Playfair
Akan dibuktikan ekivalensi postulat Euclid dan postulat Playfair. Pertama, dengan
mengasumsikan postulat sejajar Euclid, maka akan dideduksi postulat Playfair.
Diketahui garis l dan titik P tidak pada l (gambar 2.5), maka akan ditunjukkan bahwa hanya
ada satu garis melalui P yang tidak pada l. diketahui bahwa ada garis melalui P yang sejajar
dengan l, dan diketahui juga bagaimana cara menggambarnya (akibat 3,teorema 2). Dari P,
dihilangkan garis tegak lurus pada l dengan kaki Q dan pada P garis tegak m yang tegak lurus
pada . Maka garis m sejajar garis l.
Kemudian misalkan garis n sebarang garis melalui P yang berbeda dengan garis m. maka
akan ditunjukkan bahwa garis n bertemu dengan garis l. Misalkan
∠1, ∠2 menunjukkan sudut dimana garis n bertemu dengan  . Maka ∠1 bukan merupakan
sudut siku-siku untuk sebaliknya garis n  dan  garis  m  berimpit, berlawanan dengan asumsi.
Jadi ∠1 atau ∠2 adalah  sudut  lancip, misalnya  ∠1 yang merupakan sudut lancip.

Ringkasannya, garis l dan garis n dibagi oleh garis transversal sehingga membentuk sudut

lancip ∠1 dan sudut siku – siku, yang merupakan sudut interior pada sisi yang sama dari garis
transversal tersebut. Karena jumlah sudut tersebut kurang dari 180 O, postulat sejajar Euclid
dapat diaplikasikan dan disimpulkan bahwa garis n bertemu dengan garis l. Jadi
garis m hanya satu – satunya garis yang melalui P yang sejajar dengan garis l dan
dideduksikan bahwa postulat Playfair dari postulat sejajar Euclid.
Sekarang dengan mengasumsikan postulat Playfair, akan dideduksi postulat sejajar Euclid.

Misalkan garis m dibagi oleh garis transversal dititik Q, P yang membentuk ∠1 dan ∠2,

pasangan sudut interior pada satu sisi garis transversal yang memiliki jumlah sudut kurang
dari 180o ( gambar 2.6 ), adalah :
(1)     ∠1 +  ∠2 = 180O
Misalkan    ∠3  menunjukkan  tambahan  ∠1  yang  terletak  pada  sisi berlawanan  dari ∠1
dan ∠2 ( gambar 2.6 ), maka :
  (2)      ∠1 + ∠3 = 180O
Dari hubungan (1), (2) maka :
(3)     ∠2 < ∠3
Pada titik P, bentuk ∠QPR yang sama dengan dan yang interior dalam
berseberangan dengan ∠3. Maka ∠2 < ∠PQR, sehingga  berbeda dari garis m. menurut
teorema 2,  sejajar dengan l. Karenanya menurut postulat Playfair, m tidak sejajar dengan l.
Oleh karena itu, garis m dan l bertemu.
Seandainya  garis-garis  tersebut  bertemu  di  sisi  berlawanan  dari    dari  ∠1  dan ∠2,
katakanlah di titik E maka ∠2 merupakan sudut eksterior ΔPQE, karenanya ∠2  > ∠3 ,
berlawanan dengan (3). Akibatnya, pengandaian tadi salah, jadi
garis m dan  l  bertemu  pada  sisi  garis  transversal   yang  memuat  ∠1  dan  ∠2.  Jadi, postu
lat sejajar Euclid mengikuti postulat Playfair dan akibatnya dua postulat tersebut menjadi
ekivalen.

E. Peran Postulat Sejajar Euclid

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.
Euclid's parallel postulates are the source for many very important results. Without these
postulates (or their equivalents), we would not have the long-known broad theory, the
similarity theory, and the famous Pythagorean theory.
The way in which Euclid organizes his theorems implies that in fact Euclid was not
completely satisfied with his parallel postulates. Euclid stated this at the beginning of his
work but he didn't use it until he was unable to make progress without the postulate.
Presumably, Euclid had the intuition that the parallel postulate did not have the intuitive or
simplistic qualities of the other postulates. This feeling was carried out by deep geometricians
for the 20th century. Experts try to deduce parallel postulates from other postulates, or
replace these postulates with postulates that seem more certain.
 F. Tokoh-Tokoh Dalam Perkembangan 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). we form the line m through P parallel to the line l in the usual way. Let be
perpendicular to l at Q, and let m be perpendicular to at P. Now, suppose there is another line
n through P that is parallel to l, then n forms an acute angle with line PQ, which lies, say, on
the right side of . The portion of n to the right of point P is entirely contained in the region
bounded by the lines l, m and . Now suppose X is any point in m that is to the right of point
P, let's say it's perpendicular to l in Y and let the line meet n at Z. Then > . Suppose X goes
backwards on the m line, then it increases erratically, because it is at least as large as the
segment of X that is perpendicular to n.
So it also increases erratically. But the distance between two parallel lines must be limited.
Therefore, it would be a contradiction and a false supposition. So, m is the only line through
P that is parallel to line l. Therefore, Playfair's postulate holds, and is also equivalent to
Euclid's parallel postulate.
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.
(a) and (b) can be justified without the aid of Euclid's parallel postulates. So the core of the
problem of proof is assumption (c). Proclus assumes (c) as an additional postulate. Let us call
this the hidden Proclus assumption postulate. Then it can be stated: Proclus' postulate is
equivalent to Proclus' parallel postulate. Euclid's parallel postulate implies that the distance
 between parallel lines is always constant, and finite. The conversion, through Proclus'
argument can be stated that Proclus' postulate implies Euclid's parallel postulate. Thus,
Proclus replaced the parallel postulate with the equivalent postulate, and did not establish the
validity of the parallel postulate.

 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).

If Euclid's parallel postulate is assumed, then the right angle hypothesis will occur (because
the parallel postulate implies that the sum of the angles of any quadrilateral is 360°).
Saccheri's basic argument is as follows:
“Show that the obtuse angle hypothesis and the acute angle hypothesis both carry a state of
contradiction. This will form the right angle hypothesis which is equivalent to Euclid's
parallel postulate."
Saccheri proved, using a series of well-founded theorems, that the obtuse angle hypothesis
would result in a contradiction.
He considered the implications of the acute angle hypothesis. Among them are a number of
uncommon theorems, two of which we state as follows:
• The sum of the angles of any triangle is less than 180°.
• If l and m are two lines in the plane, then one of the following properties is satisfied:
a. l and m intersect, in the case where the two lines diverge from the point of intersection.
b. l and m do not intersect but have the same perpendicular where the two lines diverge in
both directions from the same perpendicular.
l and m do not intersect and do not have the same perpendicular line, where the two lines
converge in one direction of the step, and diverge in the other direction.
Saccheri did not view it as a contradiction, although he thought it should be considered a
contradiction and it is even known today that Saccheri's hypothetical theory of acute angles is
free of contradictions like Euclid's geometry. Saccheri proved that ∠C = ∠D and then
considered three possibilities related to angles C and D:
1. Hypothesis about right angles (∠C = ∠D = 90°)
1. Hypothesis about obtuse angles (∠C = ∠D > 90°)

2. Hypothesis about acute angle (∠C = ∠D < 90°)

G. Summary
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.
Meanwhile, Euchild gives five postulates (axioms):
1.    Any two points can be connected by a straight line.
2.     Any straight line section may be extended in a straight line.
3.    A circle can be drawn using a straight line as the radius and one endpoint as the
center.
4.    All congruent angles are congruent.
5.    The parallel postulate. If two lines intersect with the third in a way where the sum of
the interior angles is less than one, then these two lines must intersect on top of each other
if they are sufficiently extended.

H. Exercise
1. from the prolus argument that includes 3 assumptions. How did Proclus prove Euclid's
parallel postulate?
2. about the Sacceri experiment which defended Euclid's postulates. what is the
argument presented and how to prove it?
3. state euclid's postulates of mathematics!
4. How do you prove Euclid's third postulate?
5. Mention the figures who discuss Euclid's development of mathematics (at least 3)!