Spiral Mechanics. The Mesopotamian Arithmetic and Geometry of antiquity. Discovered in 3200 BC. Demonstrably true for all values of A, B, x and y. First applications were the invention of writing and base 60. Introduced to ancient Greece by Pythagoras, used by Plato and Archimedes, and erased from history by Aristotle and Euclid.
Aryabhata was an Indian mathematician and astronomer from the classical age of Indian mathematics and astronomy. Some of his major works were the Aryabhatiya and Arya-Siddhanta. In the Aryabhatiya, he wrote about topics like astronomy, trigonometry, algebra, and arithmetic. He discovered that the Earth rotates and orbits the sun. He also approximated pi and invented the concept of zero. Aryabhata made many contributions to mathematics including formulas to calculate triangle and circle areas, as well as sums of series.
The document provides an overview of mathematics in ancient Greece, covering important Greek mathematicians and their contributions. It discusses Thales of Miletus, who made early advances in geometry. It also covers Pythagoras and the Pythagorean school, whose discoveries included relationships between musical intervals and ratios. Euclid is discussed for writing the influential Elements, compiling earlier work. Archimedes made advances in areas and used a method of exhaustion to calculate pi. The document provides context on Plato, Aristotle, and others who helped develop mathematics.
Trigonometry developed from studying right triangles in ancient Egypt and Babylon, with early work done by Hipparchus and Ptolemy. It was further advanced by Indian, Islamic, and Chinese mathematicians. Key developments include Madhava's sine table, al-Khwarizmi's sine and cosine tables, and Shen Kuo and Guo Shoujing's work in spherical trigonometry. European mathematicians like Regiomontanus, Rheticus, and Euler established trigonometry as a distinct field and defined functions analytically. Trigonometry is now used in many areas beyond triangle calculations.
History of mathematics - Pedagogy of MathematicsJEMIMASULTANA32
It includes Prehistory: from primitive counting to Numeral systems, Archaic mathematics in Mesopotamia and egypt, Birth of mathematics as a deductive science in Greece: Thales and Pythagoras and Role of Aryabhatta in Indian Mathematics.
This document discusses the history and key concepts of real numbers. It provides background on how real numbers developed from ancient civilizations working with simple fractions to the formal acceptance of irrational numbers. Key figures discussed include Euclid, Hippasus, and developments in ancient Egypt, India, Greece, the Middle Ages, and Alexandria. Fundamental ideas covered include Euclid's lemma, the fundamental theorem of arithmetic, prime factorisation, and the distinction between rational and irrational numbers.
Decoding the ancient_greek_astronomical_calculator_known_as_the_antikythera_m...Sérgio Sacani
The document summarizes research into decoding the ancient Greek astronomical calculator known as the Antikythera Mechanism. Key findings include:
1) Researchers used high-resolution X-ray tomography and digital imaging to further decode inscriptions and reconstruct the device's functions.
2) The mechanism included geared dials to track the Metonic and Callippic lunar cycles and the Saros eclipse cycle, displaying astronomical periods with unexpected sophistication for the time.
3) Glyphs on the Saros dial were correlated with historical eclipse data, identifying the mechanism's method of predicting lunar and solar eclipses based on Babylonian arithmetic cycles.
Pythagoras was a Greek mathematician born in 560 BC on the island of Samos. He traveled to Egypt to study with priests and may have learned geometric principles there. Pythagoras founded a school in Croton, Italy where he and his followers studied mathematics and believed that everything could be explained by numbers. Some of Pythagoras' key contributions included proving the Pythagorean theorem, discovering that the planets were spheres moving in space, and identifying five regular solids.
Earliest methods used to solve quadratic equations were geometric. Babylonian cuneiform tablets from around 1800-1600 BCE contain problems that can be reduced to solving quadratic equations, showing they understood techniques. The Egyptians also solved quadratic equations geometrically in the Middle Kingdom around 2050-1650 BCE. Later mathematicians like Euclid, Brahmagupta, and al-Khwārizmī developed more algebraic methods, with Brahmagupta explicitly describing the quadratic formula around 628 AD. The need for convenience ultimately led to the discovery of the general quadratic formula, first obtained by Simon Stevin in 1594 and published by René Descartes in 1637 in the modern form still used today.
Aryabhata was an Indian mathematician and astronomer from the classical age of Indian mathematics and astronomy. Some of his major works were the Aryabhatiya and Arya-Siddhanta. In the Aryabhatiya, he wrote about topics like astronomy, trigonometry, algebra, and arithmetic. He discovered that the Earth rotates and orbits the sun. He also approximated pi and invented the concept of zero. Aryabhata made many contributions to mathematics including formulas to calculate triangle and circle areas, as well as sums of series.
The document provides an overview of mathematics in ancient Greece, covering important Greek mathematicians and their contributions. It discusses Thales of Miletus, who made early advances in geometry. It also covers Pythagoras and the Pythagorean school, whose discoveries included relationships between musical intervals and ratios. Euclid is discussed for writing the influential Elements, compiling earlier work. Archimedes made advances in areas and used a method of exhaustion to calculate pi. The document provides context on Plato, Aristotle, and others who helped develop mathematics.
Trigonometry developed from studying right triangles in ancient Egypt and Babylon, with early work done by Hipparchus and Ptolemy. It was further advanced by Indian, Islamic, and Chinese mathematicians. Key developments include Madhava's sine table, al-Khwarizmi's sine and cosine tables, and Shen Kuo and Guo Shoujing's work in spherical trigonometry. European mathematicians like Regiomontanus, Rheticus, and Euler established trigonometry as a distinct field and defined functions analytically. Trigonometry is now used in many areas beyond triangle calculations.
History of mathematics - Pedagogy of MathematicsJEMIMASULTANA32
It includes Prehistory: from primitive counting to Numeral systems, Archaic mathematics in Mesopotamia and egypt, Birth of mathematics as a deductive science in Greece: Thales and Pythagoras and Role of Aryabhatta in Indian Mathematics.
This document discusses the history and key concepts of real numbers. It provides background on how real numbers developed from ancient civilizations working with simple fractions to the formal acceptance of irrational numbers. Key figures discussed include Euclid, Hippasus, and developments in ancient Egypt, India, Greece, the Middle Ages, and Alexandria. Fundamental ideas covered include Euclid's lemma, the fundamental theorem of arithmetic, prime factorisation, and the distinction between rational and irrational numbers.
Decoding the ancient_greek_astronomical_calculator_known_as_the_antikythera_m...Sérgio Sacani
The document summarizes research into decoding the ancient Greek astronomical calculator known as the Antikythera Mechanism. Key findings include:
1) Researchers used high-resolution X-ray tomography and digital imaging to further decode inscriptions and reconstruct the device's functions.
2) The mechanism included geared dials to track the Metonic and Callippic lunar cycles and the Saros eclipse cycle, displaying astronomical periods with unexpected sophistication for the time.
3) Glyphs on the Saros dial were correlated with historical eclipse data, identifying the mechanism's method of predicting lunar and solar eclipses based on Babylonian arithmetic cycles.
Pythagoras was a Greek mathematician born in 560 BC on the island of Samos. He traveled to Egypt to study with priests and may have learned geometric principles there. Pythagoras founded a school in Croton, Italy where he and his followers studied mathematics and believed that everything could be explained by numbers. Some of Pythagoras' key contributions included proving the Pythagorean theorem, discovering that the planets were spheres moving in space, and identifying five regular solids.
Earliest methods used to solve quadratic equations were geometric. Babylonian cuneiform tablets from around 1800-1600 BCE contain problems that can be reduced to solving quadratic equations, showing they understood techniques. The Egyptians also solved quadratic equations geometrically in the Middle Kingdom around 2050-1650 BCE. Later mathematicians like Euclid, Brahmagupta, and al-Khwārizmī developed more algebraic methods, with Brahmagupta explicitly describing the quadratic formula around 628 AD. The need for convenience ultimately led to the discovery of the general quadratic formula, first obtained by Simon Stevin in 1594 and published by René Descartes in 1637 in the modern form still used today.
Pythagoras and Zeno made early contributions to mathematics and philosophy. Pythagoras is credited with the first proof of the Pythagorean theorem, while Zeno conceived paradoxes to support Parmenides' view that motion is illusory. Archimedes made seminal advances in geometry, measurement of pi, and buoyancy. Euclid's Elements was a principal geometry text for over 2000 years, developing proofs from postulates including the parallel postulate. Later mathematicians like Descartes, Fermat, Pascal, Newton, Euler, Cantor further advanced fields like algebra, calculus, probability, and the theory of infinite sets.
Geometry is a branch of mathematics concerned with shapes, sizes, positions, and properties of space. It arose independently in early cultures and emerged in ancient Greece where Euclid formalized it in his influential Elements text around 300 BC. Elements defined geometry axiomatically and influenced mathematics for centuries. It included proofs of theorems like two triangles being congruent if they share two equal angles and one equal side. Euclid's work defined much of the rules and language of geometry still used today.
Pythagoras was a Greek mathematician born around 570 BCE in Samos, Greece. He founded a school in Croton, Italy where he studied mathematics and developed the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Pythagoras made many contributions to mathematics and music. He discovered that the musical scale is based on string length ratios and ratios of whole numbers.
This document contains a quiz on astronomy, comics, and mathematics. It consists of 10 multiple choice rounds covering topics like supernovae, complex numbers, Batman curve equations, Spiderman enemies, and more. The rules specify point values for correct, passing, and incorrect answers at different rounds.
Euclid was a Greek mathematician born around 300 BC in Alexandria, Egypt. He is best known for his work Elements, which laid out the principles of geometry and remains influential to this day. Little is known about Euclid's life directly, as historical references to him are few. Elements presented geometric concepts and results established by earlier mathematicians in a single, logically coherent framework, and introduced rigorous proof-based reasoning that is still used in mathematics. Euclid is also attributed with several other works dealing with geometry and related topics, though some attributions are uncertain.
Aesthetical Beauty of Mathematics and the Pythagorean Theorem.pdfEmily Smith
The document summarizes the aesthetic beauty of mathematics through the Pythagorean theorem. It discusses how the Pythagoreans first observed a connection between beauty and mathematics. The Pythagorean theorem is given as an example that demonstrates qualities like universality, objectivity, truthfulness, aesthetics, resistance and applicability - qualities it shares with art. The document then provides historical context on Pythagoras and outlines several ways the Pythagorean theorem can be proven, including over 370 proofs collected by mathematician Elisha Scott Loomis.
2022 MOHAMED EL NASCHIE, SCOTT OLSEN - 09 PAGINAS - THE GOLDEN MEAN NUMBER SY...WITO4
This paper explores the golden mean number system and its roots in Plato's philosophy. It discusses how the golden mean number system naturally emerges from Plato's principles of the One and the Indefinite Dyad. The paper shows how quantum parameters like the pre-quantum particle, pre-quantum wave, and Einstein spacetime align with Plato's similes in the Republic. It also reveals an underlying paradigmatic symmetry where any golden power can simultaneously represent geometric, arithmetic, and harmonic means. This symmetry links all aspects of the golden powers in a structure of interdependence.
1. Geometry originated in ancient Egypt and was further developed by ancient Greek mathematicians like Thales, Pythagoras, Plato, Euclid, Archimedes, and others who made discoveries in areas like triangles, circles, volumes, and mechanics.
2. Euclid's Elements collected earlier geometric theorems into a coherent logical system and was used as the standard geometry textbook for over 2000 years.
3. Many of these early geometers were based around the Mediterranean, including Thales in Miletus, Pythagoras on Samos island, Plato in Athens, and Archimedes in Syracuse.
This document provides an overview of Plato's philosophy and its influence on ancient Greek astronomy. It describes Plato's divided line concept of reality and the allegory of the cave, which illustrated his view that true knowledge comes from rational thought rather than the senses. It also discusses how Plato's student Eudoxus attempted to "save the phenomena" of planetary motions by proposing a geometric system of concentric spheres for each planet, influencing later efforts to understand the cosmos through mathematics. While Eudoxus' model did not accurately predict planetary positions, it established a standard for rational explanations of natural phenomena.
The document provides a high-level overview of major milestones in the history of mathematics, including:
1) Early mathematical texts from Babylonian (c. 1900 BC), Egyptian (c. 2000-1800 BC), and Indian (c. 9th century BC) civilizations that approximated values like pi.
2) Key figures like Pythagoras, Euler, and Euclid of Alexandria, considered the "Father of Geometry", who authored the influential Elements textbook.
3) The progression of mathematical study in places like Egypt, India, and Mesopotamia over different historical periods under civilizations like the Sumerians, Greeks, Arabs, and more.
This document provides an overview of the history and development of geometry. It discusses how geometry originated with early peoples discovering principles like the Pythagorean theorem thousands of years before Pythagoras. It then covers the major developments of geometry in ancient cultures like Egypt, Babylon, Greece, China, Islamic caliphates, and the modern era. Key figures discussed include Euclid, who introduced rigorous logic and axioms still used today, and Archimedes, considered one of the greatest mathematicians for his approximations of pi and work on limits.
This document provides an overview of ancient mathematics in Babylon and Egypt. It describes how early mathematics developed out of practical needs in early civilizations along rivers like the Nile, Tigris, Euphrates, Indus, and Huangho. Archaeologists have uncovered hundreds of thousands of clay tablets in Mesopotamia containing early mathematical concepts. These include arithmetic, algebra, geometry, and early use of tables and formulas. Egyptian mathematics is also discussed and sources of early mathematical knowledge from Egypt are described, including papyri, monuments, and other inscriptions.
The document is a student paper on the history of mathematics. It covers the development of mathematics from prehistoric times through modern eras in different regions, including Prehistoric, Babylonian, Egyptian, Greek, Chinese, Indian, Islamic, Medieval European, Renaissance, and Modern mathematics. Key developments discussed include the earliest numerical notations and mathematical objects from prehistoric times, the sexagesimal numeral system of Babylonian mathematics, Egyptian contributions preserved in papyri, Greek advances in logic and deductive reasoning, China's place-value decimal system, and the flowering of mathematics during the Islamic Golden Age.
The document is a student paper on the history of mathematics. It covers the development of mathematics from prehistoric times through modern eras in different regions, including Prehistoric, Babylonian, Egyptian, Greek, Chinese, Indian, Islamic, Medieval European, Renaissance, and Modern mathematics. Some of the key developments highlighted include the earliest numerical notations and arithmetic concepts in prehistoric times, the sexagesimal numeral system of the Babylonians, Egyptian contributions to geometry and fractions, Greek advances in logic and proof-based mathematics, China's place-value decimal system, and the introduction of algebra and Arabic numerals through Islamic mathematics.
This document provides an overview of the history of mathematics from prehistoric times through modern times. It discusses early developments in places like Babylonia, Egypt, Greece, China, and India. Key contributions included early number systems, arithmetic operations, and early geometry concepts in places like ancient Mesopotamia and Egypt. Greek mathematics made large advances through rigorous deductive reasoning and the foundations of logic. Places like China and India also made important contributions, with China developing a very advanced decimal place-value system called rod numerals. The document outlines the major developments in mathematics across different time periods and civilizations.
The document is a student paper on the history of mathematics. It covers the development of mathematics from prehistoric times through modern eras in different regions, including Prehistoric, Babylonian, Egyptian, Greek, Chinese, Indian, Islamic, Medieval European, Renaissance, and Modern mathematics. The paper provides an overview of key mathematical concepts, texts, and figures from each historical period and location.
The document is a student paper on the history of mathematics. It covers the development of mathematics from prehistoric times through modern eras in different regions, including Prehistoric, Babylonian, Egyptian, Greek, Chinese, Indian, Islamic, Medieval European, Renaissance, and Modern mathematics. The paper provides an overview of key mathematical concepts, texts, and figures from each historical period and location.
The document is a student paper on the history of mathematics. It covers the development of mathematics from prehistoric times through modern eras in different regions, including Prehistoric, Babylonian, Egyptian, Greek, Chinese, Indian, Islamic, Medieval European, Renaissance, and Modern mathematics. The paper provides an overview of key mathematical concepts, discoveries, and texts from each historical period and culture.
This document provides an overview of ancient Egyptian mathematics and its timeline. It discusses the Egyptian numeral system, which was additive, as well as their arithmetic operations of addition, multiplication and division. The Egyptians were able to solve linear equations and used arithmetic and geometric progressions. They could also express fractions as a sum of unit fractions. Overall, the document demonstrates the Egyptians had sophisticated mathematical knowledge and methods as early as 3000 BC.
The power point explains about the life history and contribution of Pythagoras.It also helps us to understand the development of the Pythagoras formula.It also attempts to solve few problems based on Pythagoras.
The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptxMAGOTI ERNEST
Although Artemia has been known to man for centuries, its use as a food for the culture of larval organisms apparently began only in the 1930s, when several investigators found that it made an excellent food for newly hatched fish larvae (Litvinenko et al., 2023). As aquaculture developed in the 1960s and ‘70s, the use of Artemia also became more widespread, due both to its convenience and to its nutritional value for larval organisms (Arenas-Pardo et al., 2024). The fact that Artemia dormant cysts can be stored for long periods in cans, and then used as an off-the-shelf food requiring only 24 h of incubation makes them the most convenient, least labor-intensive, live food available for aquaculture (Sorgeloos & Roubach, 2021). The nutritional value of Artemia, especially for marine organisms, is not constant, but varies both geographically and temporally. During the last decade, however, both the causes of Artemia nutritional variability and methods to improve poorquality Artemia have been identified (Loufi et al., 2024).
Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
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Pythagoras and Zeno made early contributions to mathematics and philosophy. Pythagoras is credited with the first proof of the Pythagorean theorem, while Zeno conceived paradoxes to support Parmenides' view that motion is illusory. Archimedes made seminal advances in geometry, measurement of pi, and buoyancy. Euclid's Elements was a principal geometry text for over 2000 years, developing proofs from postulates including the parallel postulate. Later mathematicians like Descartes, Fermat, Pascal, Newton, Euler, Cantor further advanced fields like algebra, calculus, probability, and the theory of infinite sets.
Geometry is a branch of mathematics concerned with shapes, sizes, positions, and properties of space. It arose independently in early cultures and emerged in ancient Greece where Euclid formalized it in his influential Elements text around 300 BC. Elements defined geometry axiomatically and influenced mathematics for centuries. It included proofs of theorems like two triangles being congruent if they share two equal angles and one equal side. Euclid's work defined much of the rules and language of geometry still used today.
Pythagoras was a Greek mathematician born around 570 BCE in Samos, Greece. He founded a school in Croton, Italy where he studied mathematics and developed the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Pythagoras made many contributions to mathematics and music. He discovered that the musical scale is based on string length ratios and ratios of whole numbers.
This document contains a quiz on astronomy, comics, and mathematics. It consists of 10 multiple choice rounds covering topics like supernovae, complex numbers, Batman curve equations, Spiderman enemies, and more. The rules specify point values for correct, passing, and incorrect answers at different rounds.
Euclid was a Greek mathematician born around 300 BC in Alexandria, Egypt. He is best known for his work Elements, which laid out the principles of geometry and remains influential to this day. Little is known about Euclid's life directly, as historical references to him are few. Elements presented geometric concepts and results established by earlier mathematicians in a single, logically coherent framework, and introduced rigorous proof-based reasoning that is still used in mathematics. Euclid is also attributed with several other works dealing with geometry and related topics, though some attributions are uncertain.
Aesthetical Beauty of Mathematics and the Pythagorean Theorem.pdfEmily Smith
The document summarizes the aesthetic beauty of mathematics through the Pythagorean theorem. It discusses how the Pythagoreans first observed a connection between beauty and mathematics. The Pythagorean theorem is given as an example that demonstrates qualities like universality, objectivity, truthfulness, aesthetics, resistance and applicability - qualities it shares with art. The document then provides historical context on Pythagoras and outlines several ways the Pythagorean theorem can be proven, including over 370 proofs collected by mathematician Elisha Scott Loomis.
2022 MOHAMED EL NASCHIE, SCOTT OLSEN - 09 PAGINAS - THE GOLDEN MEAN NUMBER SY...WITO4
This paper explores the golden mean number system and its roots in Plato's philosophy. It discusses how the golden mean number system naturally emerges from Plato's principles of the One and the Indefinite Dyad. The paper shows how quantum parameters like the pre-quantum particle, pre-quantum wave, and Einstein spacetime align with Plato's similes in the Republic. It also reveals an underlying paradigmatic symmetry where any golden power can simultaneously represent geometric, arithmetic, and harmonic means. This symmetry links all aspects of the golden powers in a structure of interdependence.
1. Geometry originated in ancient Egypt and was further developed by ancient Greek mathematicians like Thales, Pythagoras, Plato, Euclid, Archimedes, and others who made discoveries in areas like triangles, circles, volumes, and mechanics.
2. Euclid's Elements collected earlier geometric theorems into a coherent logical system and was used as the standard geometry textbook for over 2000 years.
3. Many of these early geometers were based around the Mediterranean, including Thales in Miletus, Pythagoras on Samos island, Plato in Athens, and Archimedes in Syracuse.
This document provides an overview of Plato's philosophy and its influence on ancient Greek astronomy. It describes Plato's divided line concept of reality and the allegory of the cave, which illustrated his view that true knowledge comes from rational thought rather than the senses. It also discusses how Plato's student Eudoxus attempted to "save the phenomena" of planetary motions by proposing a geometric system of concentric spheres for each planet, influencing later efforts to understand the cosmos through mathematics. While Eudoxus' model did not accurately predict planetary positions, it established a standard for rational explanations of natural phenomena.
The document provides a high-level overview of major milestones in the history of mathematics, including:
1) Early mathematical texts from Babylonian (c. 1900 BC), Egyptian (c. 2000-1800 BC), and Indian (c. 9th century BC) civilizations that approximated values like pi.
2) Key figures like Pythagoras, Euler, and Euclid of Alexandria, considered the "Father of Geometry", who authored the influential Elements textbook.
3) The progression of mathematical study in places like Egypt, India, and Mesopotamia over different historical periods under civilizations like the Sumerians, Greeks, Arabs, and more.
This document provides an overview of the history and development of geometry. It discusses how geometry originated with early peoples discovering principles like the Pythagorean theorem thousands of years before Pythagoras. It then covers the major developments of geometry in ancient cultures like Egypt, Babylon, Greece, China, Islamic caliphates, and the modern era. Key figures discussed include Euclid, who introduced rigorous logic and axioms still used today, and Archimedes, considered one of the greatest mathematicians for his approximations of pi and work on limits.
This document provides an overview of ancient mathematics in Babylon and Egypt. It describes how early mathematics developed out of practical needs in early civilizations along rivers like the Nile, Tigris, Euphrates, Indus, and Huangho. Archaeologists have uncovered hundreds of thousands of clay tablets in Mesopotamia containing early mathematical concepts. These include arithmetic, algebra, geometry, and early use of tables and formulas. Egyptian mathematics is also discussed and sources of early mathematical knowledge from Egypt are described, including papyri, monuments, and other inscriptions.
The document is a student paper on the history of mathematics. It covers the development of mathematics from prehistoric times through modern eras in different regions, including Prehistoric, Babylonian, Egyptian, Greek, Chinese, Indian, Islamic, Medieval European, Renaissance, and Modern mathematics. Key developments discussed include the earliest numerical notations and mathematical objects from prehistoric times, the sexagesimal numeral system of Babylonian mathematics, Egyptian contributions preserved in papyri, Greek advances in logic and deductive reasoning, China's place-value decimal system, and the flowering of mathematics during the Islamic Golden Age.
The document is a student paper on the history of mathematics. It covers the development of mathematics from prehistoric times through modern eras in different regions, including Prehistoric, Babylonian, Egyptian, Greek, Chinese, Indian, Islamic, Medieval European, Renaissance, and Modern mathematics. Some of the key developments highlighted include the earliest numerical notations and arithmetic concepts in prehistoric times, the sexagesimal numeral system of the Babylonians, Egyptian contributions to geometry and fractions, Greek advances in logic and proof-based mathematics, China's place-value decimal system, and the introduction of algebra and Arabic numerals through Islamic mathematics.
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The document is a student paper on the history of mathematics. It covers the development of mathematics from prehistoric times through modern eras in different regions, including Prehistoric, Babylonian, Egyptian, Greek, Chinese, Indian, Islamic, Medieval European, Renaissance, and Modern mathematics. The paper provides an overview of key mathematical concepts, texts, and figures from each historical period and location.
The document is a student paper on the history of mathematics. It covers the development of mathematics from prehistoric times through modern eras in different regions, including Prehistoric, Babylonian, Egyptian, Greek, Chinese, Indian, Islamic, Medieval European, Renaissance, and Modern mathematics. The paper provides an overview of key mathematical concepts, texts, and figures from each historical period and location.
The document is a student paper on the history of mathematics. It covers the development of mathematics from prehistoric times through modern eras in different regions, including Prehistoric, Babylonian, Egyptian, Greek, Chinese, Indian, Islamic, Medieval European, Renaissance, and Modern mathematics. The paper provides an overview of key mathematical concepts, discoveries, and texts from each historical period and culture.
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The power point explains about the life history and contribution of Pythagoras.It also helps us to understand the development of the Pythagoras formula.It also attempts to solve few problems based on Pythagoras.
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Although Artemia has been known to man for centuries, its use as a food for the culture of larval organisms apparently began only in the 1930s, when several investigators found that it made an excellent food for newly hatched fish larvae (Litvinenko et al., 2023). As aquaculture developed in the 1960s and ‘70s, the use of Artemia also became more widespread, due both to its convenience and to its nutritional value for larval organisms (Arenas-Pardo et al., 2024). The fact that Artemia dormant cysts can be stored for long periods in cans, and then used as an off-the-shelf food requiring only 24 h of incubation makes them the most convenient, least labor-intensive, live food available for aquaculture (Sorgeloos & Roubach, 2021). The nutritional value of Artemia, especially for marine organisms, is not constant, but varies both geographically and temporally. During the last decade, however, both the causes of Artemia nutritional variability and methods to improve poorquality Artemia have been identified (Loufi et al., 2024).
Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
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light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
What is greenhouse gasses and how many gasses are there to affect the Earth.moosaasad1975
What are greenhouse gasses how they affect the earth and its environment what is the future of the environment and earth how the weather and the climate effects.
4. - 3 -
Table of Contents
Spiral Mechanics 1
Introduction 4
The Discovery of Heaven 11
The Trojan from Babylon
Egalitarian versus Elitist
Aristotle’s Stratagem: the Final Solution
The Elements of Euclid: the Implementation
The Mesopotamian Identity and the Sexagesimal System
Conclusion
Bibliography
Endnotes
16
30
39
49
54
66
69
70
5. - 4 -
Spiral Mechanics
There are only two mistakes one can make along the road to truth; not going all the way, and not
starting.
– Shakyamuni Buddha
The right to search for the truth implies also a duty; one must not conceal any part of what one has
recognized to be the truth.
– Albert Einstein
Introduction
The most widespread principle among scholars of classical antiquity - the dogma of the time of
Pythagoras and Plato – was the "The Truth" had been fully discovered in earlier times and since
then had only been reformulated by more recent writers. The most widely held belief because it
was not based on faith but on evidence. A dogma because the evidence was mathematical from
start to finish and therefore irrefutable. Everything that is comes from one ultimate truth and all
that is can be traced back to this truth.
The distinction between the writers who reformulated the truth and the writers who distorted
the truth was also easily made. The reformers revealed their source and honored the memory of
the Ancients by generously sharing and embracing the wisdom. The perverts obscured their
existence and selfishly used all the benefits of their own conversion to the exclusion of others. Of
course "everything for me and nothing for the other" is only the despicable maxim of the false
prophets and the corrupt rulers of humanity.
Three thousand years before the birth of Archimedes, the Sumerians discovered the
Archimedean point ("Punctum Archimedis") of arithmetic and geometry. The ideal of "moving
away" from the study object so that it can be viewed in relation to all others. The hypothetical
vantage point from which an observer can observe the subject of research with a view of the
totality. At the beginning of the second half of the fourth millennium BC, this truth found its first
applications in southern Mesopotamia.
6. - 5 -
The Sumerians invented writing and arithmetic. Both applications are more practical and
timeless monuments of this Truth than the enormous monolithic works of stone from before and
after that were erected for her honor. From the very beginning the Sumerians used a
sexagesimal number system for computation. In the following millennium this sexagesimal
number system was expanded with a positional-place value number system. A sexagesimal
positional numerical system four thousand years before Simon Stevin in 1585 published "De
Thiende" and introduced Europe to the Hindu-Arabic numerals of our modern decimal place
value number system. The classical sciences of arithmetic, geometry, astronomy and music and
the classical arts such as literature and speech are all directly derived from this truth. The
Sumerians believed with appropriate justification they had discovered the Gate of Heaven,
because the truth is precisely expressed in a demonstrable number theory of everything. A
perfect theory which exactly expresses (all) the facts. At the end of the third millennium BC the
Sumerian civilization in the south came to an end. The knowledge that had given rise to the
power of the ancient cities of Ur and Larsa was transferred north where a new city had been
founded Bābilim. Babylon meaning ‘The Gate of God.’
Written in cuneiform 𒆍𒀭𒊏𒆠 (KA2.DINGIR.RAki/bāb ili/, “gate of god”).
Pythagoras studied with the priests in Egypt, when in 525 BC Egypt was conquered by the
Persian king Cambyses II (son of Cyrus II) and was taken to Babylon. In Babylon, Pythagoras
learned the truth of the ancients and was taught all its components: arithmetic, geometry,
astronomy, and music. Upon his return from Babylon, Pythagoras founded a commune called
the "mathematikoi" in the Greek colonies of southern Italy. In this commune women were
admitted on equal terms and slavery and private property were prohibited. The essence of
Pythagoras' philosophy is equality thinking or egalitarianism.
Bertrand Russell at the beginning of The History of Western Philosophy says about Pythagoras:
“It is only in quite recent times that it has been possible to say clearly where Pythagoras was
wrong. I do not know of any other man who has been as influential as he was in the sphere of
thought. I say this because what appears as Platonism is, when analysed, found to be in essence
Pythagoreanism. The whole conception of an eternal world, revealed to the intellect but not to the
senses, is derived from him. But for him, Christians would not have thought of Christ as the Word;
but for him, theologians would not have sought logical proofs of God and immortality. But in him
all this is still implicit. How it became explicit will appear.”1
The Pythagorean philosophy is entirely based on the mathematical Identity of the ancient
Mesopotamians. The mathematical truth which Pythagoras introduced in the Hellenic world
was not his famous theorem. No, A2 + B2 = C2 is only the equation that remains when the
Mesopotamian Identity is limited to right angled triangles.
7. - 6 -
The Discovery of Heaven
The Mesopotamian Identity is in modern notation:
𝐴 𝑥
· 𝐵 𝑦
= (
𝐴 𝑥+𝐵 𝑦
2
)
2
− (
𝐵 𝑦 −𝐴 𝑥
2
)
2
the product van unequal factors (Ax · By) = a difference of
squares (Ax < By).
----------------------------------------------------------------------------------------------------------------------
𝐴1
· 𝐴1
= 𝐶2
− 𝐵2
the product of equal factors (A · A) = a difference of squares.
𝐴2
+ 𝐵2
= 𝐶2
The product of Ax and By = is equal to = the square of the half-sum of Ax and By
Minus
the square of the half-difference of Ax and By.
Discrete multitude = difference of two continuous magnitudes.
A rectangle = a gnomon, the difference of two squares.
Above the line is a mathematical Identity. It is always demonstrably true for every value of A, B, x
and y.
---------------------------------------------------------------------------------------------------------------------
Below the line is a mathematical equation. It is always demonstrably true for specific values of
A, B and C.
The Identity explains the dogma about the truth in antiquity as well as the emphasis of the
Pythagoreans on the monad (singularity), dyad (duality), and triad (trinity). It is completely
pointless to philosophically challenge the truthfulness of the Identity. It is demonstrably true for
all variables. This mathematical identity cannot be improved upon. Nobody in antiquity
therefore took a different starting point. Pythagoras, Plato, Archimedes all used the science of
the Identity of arithmetic and geometry. With the notable exception of one person, no one
thought it worthwhile to argue about the Identity or mathematical proof.
8. - 7 -
The difference between the Identity and the Pythagorean Theorem can be illustrated with a
simple example. Fill the Identity with values A = 1, B = 9, x = 1 and y = 1.
𝐴 𝑥
· 𝐵 𝑦
= (
𝐴 𝑥
+ 𝐵 𝑦
2
)
2
− (
𝐵 𝑦
− 𝐴 𝑥
2
)
2
11
· 91
= (
11
+ 91
2
)
2
− (
91
− 11
2
)
2
11
· 91
= (
10
2
)
2
− (
8
2
)
2
11
· 91
= (5)2
− (4)2
-------------------------------------------------------------------------------------------------------------------
32
= 52
− 42
𝐴2
= 𝐶2
− 𝐵2
The values 1.9.1.1 for the Identity look overly simple and insignificant. For the comparison and
the history of mathematics, however, these specific values are fundamental. With these values,
the Identity provides the smallest possible integers that satisfy the Pythagorean theorem. After
all, the 1 by 9 rectangle can also be represented as a 3 by 3 square.
At the same time, filling the Identity with A and B as equal values - that is, making the unequal
factors equal - naturally does not make it counter-factual. The Identity is of course always
correct.
31
· 31
= (
31
+ 31
2
)
2
− (
31
− 31
2
)
2
31
· 31
= (
6
2
)
2
− (
0
2
)
2
31
· 31
= (3)2
− (0)2
An example without a value of 1. Fill the Identity with values A = 7, B = 9, x = 2 and y = 3.
9. - 8 -
𝐴 𝑥
· 𝐵 𝑦
= (
𝐴 𝑥
+ 𝐵 𝑦
2
)
2
− (
𝐵 𝑦
− 𝐴 𝑥
2
)
2
72
· 93
= (
72
+ 93
2
)
2
− (
93
− 72
2
)
2
49 · 729 = (
49 + 729
2
)
2
− (
729 − 49
2
)
2
49 · 729 = (
778
2
)
2
− (
680
2
)
2
49 · 729 = (389)2
− (340)2
49 · 729 = 151.321 − 115.600
49 · 729 = 35.721
A simple example with A = 1 and B = 7 to further illustrate the limitation of the Identity to the
Pythagorean theorem and give a two-dimensional representation of a prime number:
𝐴 · 𝐵 = (
𝐴+𝐵
2
)
2
− (
𝐵−𝐴
2
)
2
1 · 7 = (
1+7
2
)
2
− (
7−1
2
)
2
1 · 7 = (
8
2
)
2
− (
6
2
)
2
1 · 7 = (4)2
− (3)2
Below the geometric figure of a rectangle with a width of 1 and a length of 7, the geometric
representation of the prime number 7 based on its two divisors.
↑
Y
5
4
3
2
1 ⑦
1 2 3 4 5 6 7 X →
10. - 9 -
The difference of the two squares (of 16 and 9) is another two-dimensional representation of the
prime number 7.
The figure that is the difference of two squares is called a gnomon.
MINUS
=
↑
Y
5 42
4
3
2
1
1 2 3 4 5 6 7 X →
1 + 7
2
2
↑
Y
5
4 32
3
2
1
1 2 3 4 5 6 7 X →
7 − 1
2
2
↑
Y 42
-32
=
5
4 ⑦
3
2
1
1 2 3 4 5 6 7 X →
11. - 10 -
𝐴 𝑥
· 𝐵 𝑦
= (
𝐴 𝑥
+ 𝐵 𝑦
2
)
2
− (
𝐵 𝑦
− 𝐴 𝑥
2
)
2
1 · 7 = (
1+7
2
)
2
− (
7−1
2
)
2
Of course, the two squares can - trivially - be laid down as two (of the three) squares on the side
of one rectangular triangle. The larger square on the diagonal and the smaller square as one of
the legs of the triangle. The length of the remaining leg of the triangle will then exactly
correspond to the square root of the surface of the gnomon.
A 1 by 7 rectangle with area ⑦ can be represented by a square with sides of √7.
1 · 7 = ( 4 )2
− ( 3)2
-------------------------------------------------------
7
2
= (4)2
− (3)2
(𝐵)2
= (𝐶)2
− (𝐴)2
↑
Y
5
4
3
2
1 ⑦
1 2 3 4 5 6 7 X →
↑
Y 42
-32
=
5
4 ⑦
3
2
1
1 2 3 4 5 6 7 X →
4
3 ⑯
3 ⑨ 4
√7 ⑦
√7
↑
Y 42
-32
=
5
4 ⑦
3
2
1
1 2 3 4 5 6 7 X →
12. - 11 -
Everything from One
Mathematics concerns provable truth and accordingly the Mesopotamian Identity leaves nothing
to the imagination. The Identity gives a complete and consistent axiomatization of mathematics
and two-dimensional space. Whomever can grasp the arithmetic of the Table of one, can grasp
the mathematical configuration of the Universe.
Beginning with the arithmetic of “The table of 1,” the Mesopotamian Identity delivers:
1. The tree of all Pythagorean triples (ABC), with 1 · n2 = Primitive Pythagorean Triple,
see Table I.A .
2. The tree of all quadratic equations, with roots 1 and n, see Table I.B .
Even though the Identity is neutral as regards the number system and is, of course, as true in
decimal as it is in binary, the Mesopotamian Identity grants additional advantages to the
sexagesimal number system (base 60). When A and B are a pair of reciprocal numbers, that is
A=
1
𝐵
and B=
1
𝐴
, the left side of the Identity -A times B- gives the multiplicative identities:
3. A ·
1
𝐴
= 1 and/ or B ·
1
𝐵
= 1, thus one has an identity within the Identity, see Table I.C.
We usually start learning multiplication when we are in 2nd or 3rd grade. Table I.A and I.B cover
the large majority of high school mathematics. Table I.C covers the arithmetic of time, how two
times half an hour makes one hour, four times fifteen minutes makes one hour, eight times 7
minutes and 30 seconds makes one hour, and the square of one minute makes one hour.
Table I.A. The Mesopotamian Identity and the Theorem of Pythagoras.
The Mesopotamian Identity is unparalleled as regards a recipe for Pythagorean triples. Line 7 of
Table I.A features the geometry of ‘1 · 7’ from the previous pages. From A= 1, B= {1, 2, 3, ..},
x= 1and y= 1, i.e. the Multiplication Table of One, the Mesopotamian Identity generates the tree
of (all primitive) Pythagorean triples.
1 · (number)2 = a primitive Pythagorean triple.
1 · (odd number)2 = 1 · 32 = 1 · 9 = ((1+9)/2)2 – ((9-1)/2)2 = 52-42
1 · (even number)2 = 1 · 42 = 1 · 16 = ((1+16)/2)2 – ((16-1)/2)2 = 8.52 – 7.52
14. - 13 -
The even is that which can be divided into two equal parts without a unit intervening in the middle;
and the odd is that which cannot be divided into two equal parts because of the aforesaid
intervention of a unit.
Now this is the definition after the ordinary conception; by the Pythagorean doctrine, however,
the even number is that which admits of division into the greatest and the smallest parts at the
same operation, greatest in size and smallest in quantity, in accordance with the natural
contrariety2 of these two genera; and the odd is that which does not allow this to be done to it, but
is divided into two unequal parts. [Please note definition ‘B’ of Aristotle’s Stratagem!]
In still another way, by the ancient definition, the even is that which can be divided alike into
two equal and two unequal parts, except that the dyad, which is its elementary form, admits but
one division, that into equal parts; and in any division whatsoever it brings to light only one species
of number, however it may be divided, independent of the other. The odd is a number which in any
division whatsoever, which necessarily is a division into unequal parts, shows both the two
species of number together, never without intermixture one with another, but always in one
another’s company.
By definition in terms of each other, the odd is that which differs by a unit from the even in either
direction, that is, toward the greater or the less, and the even is that which differs by a unit in either
direction from the odd, that is, is greater by a unit or less by a unit. [..]”3
The Pythagorean definition of an odd number is the exact opposite of the definition of an odd
number given by Euclid in the Elements:
VII.7 An odd number is that which is not divisible into two equal parts, or that which differs by a
unit from an even number.
Suffices to say here the Pythagorean definition is logically coherent and the Euclidean is not.
Table I.B. The Mesopotamian Identity and Quadratic equations versus the ABC-formula.
From the Identity quadratic equations are as easily constructed as they are solved. The degree
of an equation is given by the sum of x and y, thus x= 1 and y= 1 make a quadratic equation.
17. - 16 -
1
Bertrand Russell, The History of Western Philosophy , p. 37.
2
The natural contrariety of magnitude and quantity:
1, 2, 3, 4, 5, .., ∞
1, ½, ⅓, ¼, ⅕, .., ∞
3
Nicomachus of Gerasa, Introduction to Arithmetic, vertaling van Martin Luther D’Ooge, London:
MacMillan and Company, 1926. pp. 182-191.