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History of Mathematics / History of Pyramid (4)
« on: March 15, 2015, 02:42:35 PM »
The End of the Pyramid Era

Pyramids continued to be built throughout the fifth and sixth dynasties, but the general quality and scale of their construction declined over this period, along with the power and wealth of the kings themselves. In the later Old Kingdom pyramids, beginning with that of King Unas (2375-2345 B.C), pyramid builders began to inscribe written accounts of events in the king’s reign on the walls of the burial chamber and the rest of the pyramid’s interior. Known as pyramid texts, these are the earliest significant religious compositions known from ancient Egypt.

The last of the great pyramid builders was Pepy II (2278-2184 B.C.), the second king of the sixth dynasty, who came to power as a young boy and ruled for 94 years. By the time of his rule, Old Kingdom prosperity was dwindling, and the pharaoh had lost some of his quasi-divine status as the power of non-royal administrative officials grew. Pepy II’s pyramid, built at Saqqara and completed some 30 years into his reign, was much shorter (172 feet) than others of the Old Kingdom. With Pepy’s death, the kingdom and strong central government virtually collapsed, and Egypt entered a turbulent phase known as the First Intermediate Period. Later kings, of the 12th dynasty, would return to pyramid building during the so-called Middle Kingdom phase, but it was never on the same scale as the Great Pyramids.

The Pyramids Today

Tomb robbers and other vandals in both ancient and modern times removed most of the bodies and funeral goods from Egypt’s pyramids and plundered their exteriors as well. Stripped of most of their smooth white limestone coverings, the Great Pyramids no longer reach their original heights; Khufu’s, for example, measures only 451 feet high. Nonetheless, millions of people continue to visit the pyramids each year, drawn by their towering grandeur and the enduring allure of Egypt’s rich and glorious past.

History of Mathematics / History of Pyramid (3)
« on: March 15, 2015, 02:41:17 PM »
The Great Pyramids of Giza

No pyramids are more celebrated than the Great Pyramids of Giza, located on a plateau on the west bank of the Nile River, on the outskirts of modern-day Cairo. The oldest and largest of the three pyramids at Giza, known as the Great Pyramid, is the only surviving structure out of the famed seven wonders of the ancient world. It was built for Khufu (Cheops, in Greek), Sneferu’s successor and the second of the eight kings of the fourth dynasty. Though Khufu reigned for 23 years (2589-2566 B.C.), relatively little is known of his reign beyond the grandeur of his pyramid. The sides of the pyramid’s base average 755.75 feet (230 meters), and its original height was 481.4 feet (147 meters), making it the largest pyramid in the world. Three small pyramids built for Khufu’s queens are lined up next to the Great Pyramid, and a tomb was found nearby containing the empty sarcophagus of his mother, Queen Hetepheres. Like other pyramids, Khufu’s is surrounded by rows of mastabas, where relatives or officials of the king were buried to accompany and support him in the afterlife.

The middle pyramid at Giza was built for Khufu’s son Khafre (2558-2532 B.C). A unique feature built inside Khafre’s pyramid complex was the Great Sphinx, a guardian statue carved in limestone with the head of a man and the body of a lion. It was the largest statue in the ancient world, measuring 240 feet long and 66 feet high. In the 18th dynasty (c. 1500 B.C.) the Great Sphinx would come to be worshiped itself, as the image of a local form of the god Horus. The southernmost pyramid at Giza was built for Khafre’s son Menkaure (2532-2503 B.C.). It is the shortest of the three pyramids (218 feet) and is a precursor of the smaller pyramids that would be constructed during the fifth and sixth dynasties.

Approximately 2.3 million blocks of stone (averaging about 2.5 tons each) had to be cut, transported and assembled to build Khufu’s Great Pyramid. The ancient Greek historian Herodotus wrote that it took 20 years to build and required the labor of 100,000 men, but later archaeological evidence suggests that the workforce might actually have been around 20,000. Though some popular versions of history held that the pyramids were built by slaves or foreigners forced into labor, skeletons excavated from the area show that the workers were probably native Egyptian agricultural laborers who worked on the pyramids during the time of year when the Nile River flooded much of the land nearby.

History of Mathematics / History of Pyramid (2)
« on: March 15, 2015, 02:40:17 PM »
The Early Pyramids

From the beginning of the Dynastic Era (2950 B.C.), royal tombs were carved into rock and covered with flat-roofed rectangular structures known as “mastabas,” which were precursors to the pyramids. The oldest known pyramid in Egypt was built around 2630 B.C. at Saqqara, for the third dynasty’s King Djoser. Known as the Step Pyramid, it began as a traditional mastaba but grew into something much more ambitious. As the story goes, the pyramid’s architect was Imhotep, a priest and healer who some 1,400 years later would be deified as the patron saint of scribes and physicians. Over the course of Djoser’s nearly 20-year reign, pyramid builders assembled six stepped layers of stone (as opposed to mud-brick, like most earlier tombs) that eventually reached a height of 204 feet (62 meters); it was the tallest building of its time. The Step Pyramid was surrounded by a complex of courtyards, temples and shrines, where Djoser would enjoy his afterlife.

After Djoser, the stepped pyramid became the norm for royal burials, although none of those planned by his dynastic successors were completed (probably due to their relatively short reigns). The earliest tomb constructed as a “true” (smooth-sided, not stepped) pyramid was the Red Pyramid at Dahshur, one of three burial structures built for the first king of the fourth dynasty, Sneferu (2613-2589 B.C.) It was named for the color of the limestone blocks used to construct the pyramid’s core.

History of Mathematics / History of Pyramid (1)
« on: March 15, 2015, 02:39:25 PM »
Built during a time when Egypt was one of the richest and most powerful civilizations in the world, the pyramids—especially the Great Pyramids of Giza—are some of the most magnificent man-made structures in history. Their massive scale reflects the unique role that the pharaoh, or king, played in ancient Egyptian society. Though pyramids were built from the beginning of the Old Kingdom to the close of the Ptolemaic period in the fourth century A.D., the peak of pyramid building began with the late third dynasty and continued until roughly the sixth (c. 2325 B.C.). More than 4,000 years later, the Egyptian pyramids still retain much of their majesty, providing a glimpse into the country’s rich and glorious past.

The Pharaoh in Egyptian Society
During the third and fourth dynasties of the Old Kingdom, Egypt enjoyed tremendous economic prosperity and stability. Kings held a unique position in Egyptian society. Somewhere in between human and divine, they were believed to have been chosen by the gods to serve as mediators between them and the people on earth. Because of this, it was in everyone’s interest to keep the king’s majesty intact even after his death, when he was believed to become Osiris, god of the dead. The new pharaoh, in turn, became Horus, the falcon-god who served as protector of the sun-god, Ra.
Ancient Egyptians believed that when the king died, part of his spirit (known as “ka”) remained with his body. To properly care for his spirit, the corpse was mummified, and everything the king would need in the afterlife was buried with him, including gold vessels, food, furniture and other offerings. The pyramids became the focus of a cult of the dead king that was supposed to continue well after his death. Their riches would provide not only for him, but also for the relatives, officials and priests who were buried near him.

History of Mathematics / A Brief History of π
« on: March 15, 2015, 02:31:25 PM »
The ancient Babylonians calculated the area of a circle by taking 3 times the square of its radius, which gave a value of pi = 3. One Babylonian tablet (ca. 1900–1680 BC) indicates a value of 3.125 for pi, which is a closer approximation.

The Rhind Papyrus (ca.1650 BC) gives us insight into the mathematics of ancient Egypt. The Egyptians calculated the area of a circle by a formula that gave the approximate value of 3.1605 for pi.

The first calculation of pi was done by Archimedes of Syracuse (287–212 BC), one of the greatest mathematicians of the ancient world. Archimedes approximated the area of a circle by using the Pythagorean Theorem to find the areas of two regular polygons: the polygon inscribed within the circle and the polygon within which the circle was circumscribed. Since the actual area of the circle lies between the areas of the inscribed and circumscribed polygons, the areas of the polygons gave upper and lower bounds for the area of the circle. Archimedes knew that he had not found the value of pi but only an approximation within those limits. In this way, Archimedes showed that pi is between 3 1/7 and 3 10/71.

A similar approach was used by Zu Chongzhi (429–501), a brilliant Chinese mathematician and astronomer. Zu Chongzhi would not have been familiar with Archimedes’ method—but because his book has been lost, little is known of his work. He calculated the value of the ratio of the circumference of a circle to its diameter to be 355/113. To compute this accuracy for pi, he must have started with an inscribed regular 24,576-gon and performed lengthy calculations involving hundreds of square roots carried out to 9 decimal places.

Mathematicians began using the Greek letter π in the 1700s. Introduced by William Jones in 1706, use of the symbol was popularized by Leonhard Euler, who adopted it in 1737.


History of Mathematics / Brief History of Algebra
« on: March 10, 2015, 01:38:24 PM »
As a branch of mathematics, algebra emerged at the end of 16th century, with the work of François Viète. Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations and is not, nowadays, considered as belonging to algebra.

This article describes the history of the theory of equations, called here "algebra", from the origins to the emergence of algebra as a separate area of mathematics.

To read more:

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