3.2 Attempts to prove the Parallel Postulate
3 min read•july 22, 2024
Throughout history, mathematicians tried to prove Euclid's using his other axioms. Notable attempts came from , , Saccheri, and Legendre. Each approach had unique flaws, often assuming what they aimed to prove.
These failed attempts led to a crucial realization: the Parallel Postulate is independent of Euclid's other axioms. This discovery paved the way for , expanding our understanding of mathematical systems and influencing fields beyond mathematics.
Historical Attempts to Prove the Parallel Postulate
Historical attempts at Euclid's Fifth Postulate
Top images from around the web for Historical attempts at Euclid's Fifth Postulate
Euclid’s Fifth Postulate and Convergence of Non-Parallel Straight Lines View original
Is this image relevant?
Euclid’s Fifth Postulate and Convergence of Non-Parallel Straight Lines View original
Is this image relevant?
Euclid’s Fifth Postulate and Convergence of Non-Parallel Straight Lines View original
Is this image relevant?
Euclid’s Fifth Postulate and Convergence of Non-Parallel Straight Lines View original
Is this image relevant?
Euclid’s Fifth Postulate and Convergence of Non-Parallel Straight Lines View original
Is this image relevant?
1 of 3
Top images from around the web for Historical attempts at Euclid's Fifth Postulate
Euclid’s Fifth Postulate and Convergence of Non-Parallel Straight Lines View original
Is this image relevant?
Euclid’s Fifth Postulate and Convergence of Non-Parallel Straight Lines View original
Is this image relevant?
Euclid’s Fifth Postulate and Convergence of Non-Parallel Straight Lines View original
Is this image relevant?
Euclid’s Fifth Postulate and Convergence of Non-Parallel Straight Lines View original
Is this image relevant?
Euclid’s Fifth Postulate and Convergence of Non-Parallel Straight Lines View original
Is this image relevant?
1 of 3
- Mathematicians throughout history attempted to prove the Parallel Postulate using Euclid's other axioms and postulates
- Parallel Postulate, also known as , states that given a line and a point not on the line, there exists a unique line through the point parallel to the given line
- Notable attempts to prove the Parallel Postulate include:
- Proclus (5th century AD) tried to prove the postulate using the concept of
- Ibn al-Haytham (11th century) used the idea of motion in his attempted proof
- (18th century) explored the consequences of denying the Parallel Postulate
- (18th-19th century) assumed the sum of the angles in a triangle is equal to 180 degrees
- These mathematicians tried to derive the Parallel Postulate from Euclid's other axioms and postulates, which include:
- Two points can be connected by a straight line
- Straight lines can be extended indefinitely
- Circles can be drawn with any center and radius
- All right angles are congruent
Flaws in attempted proofs
- Proclus:
- Assumed the existence of a line equidistant from another line, which is equivalent to assuming the Parallel Postulate
- Equidistant lines imply the existence of parallel lines, making the proof circular
- Ibn al-Haytham:
- Used the concept of motion in his proof, which is not part of Euclid's axioms and postulates
- Motion is an extraneous concept not found in Euclid's static geometry
- Relied on the assumption that the perpendicular distance between two parallel lines is constant, which is equivalent to assuming the Parallel Postulate
- Constant distance between parallel lines is a consequence of the Parallel Postulate, not a justification for it
- Used the concept of motion in his proof, which is not part of Euclid's axioms and postulates
- Legendre:
- Attempted to prove the Parallel Postulate by assuming the sum of the angles in a triangle is equal to 180 degrees
- This assumption is equivalent to the Parallel Postulate itself
- The sum of the angles in a triangle being 180 degrees is a result of the Parallel Postulate, not a substitute for it
- Attempted to prove the Parallel Postulate by assuming the sum of the angles in a triangle is equal to 180 degrees
Unprovability of Parallel Postulate
- Parallel Postulate is logically independent of Euclid's other axioms and postulates
- Cannot be derived from the other axioms and postulates alone
- Requires additional assumptions or a different set of axioms to prove
- Attempts to prove the Parallel Postulate often relied on assumptions that were equivalent to the postulate itself
- These assumptions were disguised forms of the Parallel Postulate
- Proofs were circular, assuming what they set out to prove
- Development of non-Euclidean geometries in the 19th century demonstrated the independence of the Parallel Postulate
- Mathematicians such as , , and created consistent geometric systems that denied the Parallel Postulate while retaining Euclid's other axioms and postulates
- These geometries (hyperbolic and elliptic) showed that the Parallel Postulate is not a necessary truth
Impact on non-Euclidean geometries
- Failed attempts to prove the Parallel Postulate led mathematicians to question the foundations of Euclidean geometry
- Sparked interest in the possibility of alternative geometric systems
- Encouraged exploration of the consequences of denying or modifying the Parallel Postulate
- Realization that the Parallel Postulate is independent of Euclid's other axioms and postulates paved the way for the development of non-Euclidean geometries
- Mathematicians began developing consistent geometric systems without the Parallel Postulate
- Creation of consistent non-Euclidean geometries, such as hyperbolic and , demonstrated that:
- Euclidean geometry is not the only possible geometric system
- Parallel Postulate is not a necessary truth but rather a choice that leads to a specific type of geometry
- Development of non-Euclidean geometries had far-reaching consequences, influencing fields such as:
- Mathematics: expanded the scope and understanding of geometry
- Physics: Einstein's theory of general relativity relied on non-Euclidean geometry (curved spacetime)
- Philosophy: challenged the nature of truth and the role of axioms in mathematics
Key Terms to Review (16)
19th-century mathematics: 19th-century mathematics refers to the period during the 1800s when significant advancements were made in various mathematical fields, particularly in geometry. This era saw the development of Non-Euclidean geometries, a crucial shift from classical Euclidean thought, which influenced concepts of space and the nature of mathematical truth.
Adrien-Marie Legendre: Adrien-Marie Legendre was a French mathematician known for his work in number theory, statistics, and geometry. He is particularly famous for his attempts to prove the Parallel Postulate, which became a significant milestone in the study of non-Euclidean geometry and led to the development of alternative geometrical frameworks.
Axiomatic Systems: An axiomatic system is a structured framework consisting of a set of axioms or postulates, which are accepted as true without proof, and rules for deriving theorems from these axioms. This foundational structure allows for the development of mathematical theories and logical reasoning, especially in geometry where different systems can lead to distinct geometrical properties, including Non-Euclidean geometries that challenge traditional Euclidean views, particularly regarding concepts like parallel lines.
Bernhard Riemann: Bernhard Riemann was a German mathematician known for his groundbreaking contributions to analysis, differential geometry, and number theory, particularly through the introduction of Riemannian geometry. His work has had profound implications in understanding complex concepts of space, geometry, and their relationship to physical reality.
Elliptic Geometry: Elliptic geometry is a type of non-Euclidean geometry where the parallel postulate does not hold, and there are no parallel lines—any two lines will eventually intersect. This geometry describes a curved surface, like that of a sphere, where the usual rules of Euclidean geometry are altered, impacting our understanding of concepts such as distance and angle.
Equidistant Lines: Equidistant lines are lines that maintain a constant distance from each other at all points along their length. This concept is significant in geometry as it relates to the behavior of parallel lines and the understanding of space in both Euclidean and non-Euclidean contexts. When discussing attempts to prove the Parallel Postulate, the notion of equidistant lines is crucial in understanding why certain geometries behave differently, particularly when exploring the properties of hyperbolic and elliptic spaces.
Euclid's Fifth Postulate: Euclid's Fifth Postulate, also known as the Parallel Postulate, states that if a line segment intersects two straight lines and makes the interior angles on one side less than two right angles, then those two lines, if extended indefinitely, will meet on that side. This postulate is crucial in Euclidean Geometry, setting it apart from non-Euclidean geometries and leading to various equivalents and attempts at proof.
Giovanni Saccheri: Giovanni Saccheri was an Italian priest and mathematician, best known for his work in non-Euclidean geometry, particularly his attempts to prove the parallel postulate in the 17th century. His notable work, 'Euclides ab Omni Naevo Vindicatus', laid foundational ideas that eventually contributed to the development of hyperbolic geometry and showcased early explorations of space beyond traditional Euclidean concepts.
Hyperbolic geometry: Hyperbolic geometry is a type of non-Euclidean geometry characterized by a space where the parallel postulate does not hold, meaning that through a point not on a line, there are infinitely many lines that do not intersect the original line. This concept fundamentally alters the understanding of shapes, angles, and distances, reshaping perspectives on space, time, and even the fabric of the universe.
Ibn al-haytham: Ibn al-Haytham, also known as Alhazen, was a pioneering Arab mathematician, astronomer, and physicist in the 10th and 11th centuries, credited with significant advancements in optics and the scientific method. His work laid the groundwork for later developments in both mathematics and natural sciences, including the exploration of geometrical optics and the nature of light.
János Bolyai: János Bolyai was a Hungarian mathematician known for his groundbreaking work in the development of non-Euclidean geometry. His contributions challenged the long-held beliefs about the nature of space and parallel lines, significantly influencing the mathematical landscape and paving the way for further exploration of geometrical concepts beyond Euclidean principles.
Logical Independence: Logical independence refers to the property of a statement or proposition where its truth value is not determined by other statements within a logical system. In relation to geometry, particularly in attempts to prove the Parallel Postulate, this concept highlights that certain geometric principles can stand alone, meaning that they cannot be proven true or false based on existing axioms or theorems. This becomes crucial when discussing the limitations of traditional Euclidean geometry and the exploration of non-Euclidean systems.
Nikolai Lobachevsky: Nikolai Lobachevsky was a Russian mathematician known for developing hyperbolic geometry, a groundbreaking concept that deviated from Euclidean principles. His work laid the foundation for non-Euclidean geometry, significantly influencing mathematical thought and our understanding of space.
Non-Euclidean Geometries: Non-Euclidean geometries are systems of geometry that reject the traditional Euclidean postulates, particularly the parallel postulate, which states that through any point not on a given line, there is exactly one line parallel to the given line. These geometries arise in various forms, such as hyperbolic and elliptic geometries, and challenge our understanding of space and angles by allowing for multiple parallels or no parallels at all, leading to different rules for shapes and figures.
Parallel Postulate: The Parallel Postulate is a foundational statement in Euclidean geometry which asserts that if a line is drawn parallel to one side of a triangle, it will not intersect the other two sides. This postulate underpins many concepts in geometry, influencing our understanding of space, the development of non-Euclidean geometries, and the philosophical discussions surrounding the nature of mathematical truth.
Proclus: Proclus was a prominent Greek philosopher of the 5th century CE, known for his work in Neoplatonism and his contributions to the understanding of geometry, especially in relation to the parallel postulate. His writings explored various mathematical concepts and attempted to provide a foundation for geometric truths, which included significant discussions about the nature of parallel lines and their properties.