History of Mathematics Ancient Mathematics Some thoughts on dating: • • History does not consist mainly of dates. It is important to have a sense of how events fit together. However the “landmarks” should not overshadow the “landscape.” The AD/BC convention proclaims the central importance of an event that which is actually only believed to be central by a small proportion of mankind. Ancient Ages • • • • Old Stone Age: 5,000,000 BC 10,000 BC Middle Stone Age: 10,000 BC 7,000 BC New Stone Age: 7,000 BC 3,500 BC Bronze Age: 3500 BC 1400 BC Some Early Dates: Stone Age • • • • • • • • 2,400,000 BC 1,500,000 BC 750,000 BC Hominoids in Africa manufacture crude stone tools Homo Erectus (stone tools) Ancient hearths found in caves near Marseilles indicate that Homo Erectus uses fires 350,000 BC Homo Sapiens moved to caves 100,000 BC Earliest known ornament-an amulet made from mammoth’s tooth by a Neanderthal. (Hungrary) 79,000 BC Lamps fueled with animal fat, uses grass for wick. 30,000 BC Tally Sticks; Beads, Bracelets and Pendants worn. Fire ceramics found in Czechoslovakia (not used yet in making bowls) 25,000 BC Artifacts with primitive geometrical designs; Venus figurines; Music (cave paintings, footprints of dancers, carved bonesinstruments?) • • • • • • • • • 20,000 BC Bow & Arrow (Spain), Boomerangs, Sewing Needle (France), Tailored Clothes (Russia), Spear-thrower 13,000 BC Calendars, Maps (Ukraine), Ropes 10,000 BC Dogs (Mesopotamia), Goats & Sheep (Iran, Afghanistan) 9,000 BC Mayan astronomical inscriptions 8,000 BC Potatoes & beans (Peru), Rice (Indo China), Pumpkin (Latin America), Wheat & Barley (Israel) 7,000 BC Clay tokens record number of animals, measure grain in Mesopotamia 6,000 BC Corn (Mexico) 5,000 BC Earliest cities in Mesopotamia (carbon dating), Llama & Alpacas (Peru) 4,241 BC First Recorded Date (Mesopotamia) Bronze Age: 3500 BC 1400 BC • 3500 BC • • • 3300 BC 2600 BC 3000 BC • • 2700 BC 2400 BC Bronze (Egyptians & Babylonians), Potters Wheel, Papyrus (Egyptians), Wheeled Vehicles, Wine (Turkestan), Beer (Mesopotamia), Cattle (Thailand) Egypt united Great Pyramid at Giza Eclipse Predicted; 365 Day Calendar, Sailing ships (Egypt) Silk worms in China Chinese introduce method of taking observations of the sky, still in use today The Ancient World Ishango Bone • • Every culture in the world creates some mathematics, though it may be on a basic level. The most ancient evidence of mathematical activity comes from a wolf’s bone on which 55 notches are carved, grouped in sets of 5. It was found in Czech Republic and believed to be about 35,000 years old. The Ishango bone (from a baboon) dates to around 18,000 BC, and is particularly intriguing since the notches are grouped into sets of 11, 13, 17, and 19, suggesting an interest in prime numbers. Civilization • • • (from civilis, “city-dweller” in Latin) At this time, mankind obtained food by hunting animals and gathering plants. About 10,000 years ago an unknown group of people invented agriculture. This forced people to establish permanent settlements so the plants could be cared for until harvest: the first cities. Since cities were originally farming communities, they were established in regions that provided the 2 fundamental needs of agriculture: fertile soil and a reliable water supply. River flood-plains provided both, and most ancient civilizations developed around them. Ancient Egypt • • • • Civilization developed around the Nile river, whose annual flooding deposited silt that fertilized the fields of Egyptian farmers. As early as 4000 BC, they may have noticed that it took 365 days from one Nile to the next. The Egyptian calendar divided that year into 12 months of 360 days, with 5 extra days celebrated as the birthdays of the main gods of the Egyptian pantheon (holidays or “holy days”). Egypt is divided into two main parts, Lower Egypt and Upper Egypt. • • Lower Egypt consists of the marshy areas where the Nile empties into the Mediterranean, known as the Delta (because of it’s resemblance to the Greek letter, pointing southwards); the term would later be applied to any river’s outlet into a sea or lake. According to tradition, Upper and Lower Egypt were united by Narmer (Menes in Greek) around 3100 BC. • • • • The kings of Egypt were known as pharaohs (“Great House”). They were revered as gods though this did not spare them from being criticized, plotted against, and deposed. The best known feature of Egyptian civilization are the pyramids. Great Pyramid of Giza, finished around 2500 BC is the oldest and is the only one of the Seven Wonders of the World still standing! By 2700 BC, a form of writing had been invented. Since many examples were found adorning the walls of Egyptian temples, it was erroneously believed that the writings were religious in nature. Hence this form of writing became known as hieroglyphic (“sacred writing” in Greek). By 2600 BC, a cursive form of hieroglyphics, called hieratic, was developed ; it was suitable for writing on soft materials such as papyrus, cloth, and leather. Egyptian Mathematics • • • Mathematics is oldest and most continuous pursued of exact sciences. Aristotle in Metaphysics said math began with Egyptian priests . Egypt mathematics was mostly utilitarian but developed into • Algebra: from techniques of calculation • Theoretical Geometry: land measurement All civilizations have developed special symbols for numbers. The use of a stroke to represent “one” is universal. Napoleon’s invasion of Egypt • • • • • • Disastrous military campaign in 1798. 38000 soldiers sailed from Toulon in 328 ships to try to seize Egypt, but 12 month campaign against English Admiral Nelson stopped them. Napoleon had 167 scholars to make comprehensive inquiry into life of Egypt in ancient and modern times (including Fourier, Monge). Two particular areas of study resulted in significant and long-lasting discoveries or achievements. The savants traveled with the army and the geography was charted and a map of Egypt was drawn (completed in 1806) that remained classified until the end of Napoleon's reign. The great monuments were examined and the science of Egyptology was founded. One of the most important discoveries in that field was the Rosetta Stone. Mathematics on Expedition • • Monge (1746-1818) is considered the father of differential geometry because of his work Application de l'analyse à la géométrie where he introduced the concept of lines of curvature of a surface in 3-space. Fourier (1768-1830) studied the mathematical theory of heat conduction- established the partial differential equation governing heat diffusion and solved it by using infinite series of trigonometric functions. Napoleon Monge Fourier Napoleon Description de l'Egypte, first appeared in 1809; work of 160 scholars on Napolean’s expedition. Etat Moderne,Plate 87, Views of Qait Bey Fortress and the Diamant Rock, drawn c.1798, published in the Panckoucke edition of 1821. Rosetta Stone • • • • The Rosetta Stone is a block of basalt with engravings made on its polished surface. It was named after the village where it was found in 1799, Rashid (known as Rosetta to Europeans) located a few miles from the sea in the western delta of the Nile. It measures 3'9" (114 cm) in height, 2'4-1/2" (72cm) in width and 11" (28cm) in thickness. It weighs just under a ton (762kg). It is somewhat damaged, missing a large part of the upper left-hand corner, and a smaller part of its lower right corner. The chiseled inscriptions are in two languages, Greek and Egyptian, but three scripts. The first of the Egyptian scripts is Hieroglyphs, used 3,000 years ago at the time of the First Dynasty. The second was later determined to be Demotic, a cursive language that evolved from Hieroglyphs and dating from 643 B.C. Napoleon order rubbings for Europeans and 4 casts for Oxford, Cambridge, Edinburgh, Dublin. Hieroglyphic Hieratic Ancient Greek • • • • Jean-Francois Champollion, 1790-1832, born in Lot, last of 7 children, raised in humble circumstances; Met Fourier at 11; Egyptology became his life. He spoke several languages by age of 16; by age of 20, he could was fluent in Latin, Greek, Hebrew, Amharic, Sanskrit, Avestan, Pahlavi, Arabic, Syriac, Chaldean, Persian and Ge'ez in addition to his native French . At age 17 became faculty at University of Grenoble; Father of Modern Egyptology. Finished complete reading of upper panel of Rosetta Stone in 1822. Grave of Champollion (Paris) The names of Cleopatra and Ptolemy were the first words deciphered by Champollion. Champollion's notes of his study of the cartouche of Cleopatra, inscribed on an obelisk found at Philae by Belzoni. • By analyzing the texts of the Rosetta Stone and comparing them with those on the obelisk of Philae, Champollion had the brilliant intuition that the names of the pharaohs in cartouches were in hieroglyphs with a phonetic value. • It was therefore possible to establish an equivalence between hieroglyphic and alphabetic signs. The Rhind Papyrus Rhind (Ahmes) Papyrus: 2000-1800 BC • The papyrus was bought by Rhind in Luxor in 1858 and willed to the British Museum. • Hieratic script from 1650BC by scribe Ahmes • 18ft long and 13in high (center missing) • US Egyptologist Edwin Smith uncovered missing section and sent it in 1906 to NY Historical Society and then in 1922 to British Museum. • • The Rhind Papyrus The text contains eighty-four problems concerned with numerical operations, practical problem-solving, and geometrical shapes. It claims to be a ``thorough study of all things, insight into all that exists, knowledge of all obscure secrets." In fact, it is somewhat less. • It is not a theoretical treatise, but a list of practical problems encountered in administrative and building works. It is a collection of exercises, substantially rhetorical in form, designed primarily for students of mathematics. Included are exercises in • • • • • • • fractions notation arithmetic algebra geometry mensuration The practical mathematical tools for construction. Early Egyptian Multiplication • Egyptian mathematics was recorded and taught by means of problems that were intended as examples to be imitated. Unclear what the Egyptian mathematicians developed their science beyond what was needed for everyday work (unlike the Greeks). It seems remarkably uniform throughout its long history. It was at all stages, built around the operation of addition – so primitive. What do you think that the Egyptians were preoccupied with? Basic operation was adding and doubling. • • – – Multiply: doubling one number and then add the appropriate duplications to form the product. Division: divisor repeatedly doubled to give dividend.