Here is a review of our work this month in the History of Mathematics:
Early tribes used a base 2 to count: 1,2,2-1,2-2,2-3, or a base 3: 1,2,3,3-1,3-2,3-3; then they advance to a number system using their fingers and toes, a base 20 system; remnants of these ancient systems still are echoed in the quatre-vingt and quatre-vingt-dix , four twenties and four twenties plus ten, 80 and 90 in French counting, and the 20 shilling pound in Britain. The abstruse British monetary system, with ha‘pennies, sixpence, et-al, went ½,1,3,6,12,30,240,252, until quite recently.
Babylonian astronomers, astrologers and mathematicians used a base 60, divisible by l,2,3,4,5,6,10,12,15,20,30and 60. It fit well into their division of the year into 360 days, and the circle into 360 degrees, concepts still very much used in timekeeping, astronomy and geometry.
Because of the physiological uniformity, the ten base system appeared throughout many cultures, from Tehran to Hong Kong, and in the abacus. The Hindus developed our 1-9, further augmented by the Ghobar numerals, and decimal system. Florence banned the use of decimal numbers in the 13th century. We studied and drew the number symbols of the Arabic, Chinese, Mayan, Egyptian, Greek, Roman, Bank check numerals.
We studied the ideas of Proof and Abstraction as developed by the Greeks. We explored general premises, axioms, and specific premises, postulates; then induction and deduction, theorems and corollaries. They also readily employed reductio ad absurdum to make a Proof (reduction to the absurd ).
The first early Greek geometer to make a decisive contribution to geometry was Thales, a retired, wealthy trader. His five Propositions are the base for classic mathematics. The class did a careful geometric rendering of Thales’ fourth principle, and were surprised and delighted while discovering its validity. Thales was no less thrilled and was said to have sacrificed a bull to celebrate his discovery.
Angles in a semicircle – Any angle inscribed within a semicircle is a right angle ( 90 degrees ).
Under Thales’ recommendation his student Pythagorus travelled widely among the Mesopotamians, Persians and priests of Zoroaster. Then in 540 B.C. in Crotona, Pythagorus founded his cult. He taught his disciples, along with Math, to worship numbers, to believe in reincarnation, to never eat beans, and to remain anonymous. They believed the universe was made up of whole numbers, with ther even numbers female and the odd numbers male; thus marriage was 5, the sum of 3 and 2.
Although the Chinese and Babylonians had similar concepts around the same time, Pythagorus was the first to prove his most famous theorem of the right triangle, a square + b square = c square. This is still used widely in science, architecture and carpentry. We drew the 3-45 triangle and
Pythagorean Theorem using squares.
Pythagorus also found that musical intervals are governed by ratios of whole numbers. The Pythagorean religion of whole numbers and the music of the spheres was dealt a rude awakening with the discovery pf the Irrational, such as square root of 2.
One of our most comprehensive and time-consuming but absorbing lessons was drawing Pythagorus’ proof , using compass and ruler, of Inscribing a Pentagon in a Circle. Each succeeding step had to be carefully achieved in order to insure that the last step was geometrically correct. It took some redoing but the children persevered and did good work.
The Eleatics, rivals of Pythagorus, were interested in science. Their leader, Zeno presented his famous Paradox, of Achilles never catching the tortoise.
Around 300B.C. in Alexandria, Egypt, the greatest of all Geometers collected the theorems of his
Predecessors, including Democritus, Hippocrates, Archytas, into terse, clear terms, and a single whole. His masterpiece “The Elements”, is considered one of the primary books in human history, like the Bible. “ It’s commanding lucidity and style “ make it one of the most clearly reasoned works ever assembled.
The “Elements” contains 13 books which prove all the human race knew or still knows about lines, points, circles and solids, brilliantly deduced by Euclid , from 5 axioms and 5 postulates. Algebra, infinity, number theory and calculus, all find their foundation in this remarkable work.
Apollonius wrote “ Conics“, which details his work with cones sliced to create circles and ellipses, parabolas and hyperbolas.
These proofs led to the science of ballistics, missiles and rockets, the paths which satellites and moons follow influenced by the gravity of planets and stars.
The famed and influential philosopher was devoted to geometry and insisted that all proofs be done with only straight-edge and compass. Over the gates of his famed Academy he is purported to have inscribed “ Let no man ignorant of geometry enter “. Because of his prestige and high standing in Greek society all mathematicians had to work under the narrow bounds of Platonic discipline, even the modern-thinking Archimedes
Considered by most, along with Newton and Gauss, one of the three greatest mathematicians of all time,
he tackled the task of writing really large numbers, in his treatise “ The Sand Reckoner “. Using the Greek Myriad, or 10,000, he called a myriad of myriads, or 100,000,000, the First Order of Numbers.
100,000,000 to the 100,000,000 th power, or a myriad of myriads multiplied a myriad of myriad times, he called Numbers of the First Period. He then multiplied this number by itself a myriad of myriad times, and came up with a number with 80 million billion zeros!
Archimedes discovered how to calculate the volume of a sphere as 2/3 the volume of the smallest possible cylinder which will enclose it. The sphere and cylinder diagram was engraved, at his request , on his tombstone. He was also a physicist and engineer of the highest order. He is most known as the inspired, absent minded discoverer, who ran naked through the streets of Syracuse crying “ Eureka! Eureka! “ when in the bath he had discovered the basic laws of hydraulic engineering. He used these laws, when hired by King Hieron of Syracuse to find out if the court jeweler was substituting silver for gold in his crowns.
Archimedes wrote the proofs of the mathematical laws of the lever still used, and the laws of pulleys and finding the center of gravity that enable to this day every skyscraper to stand and every bridge to span.
The historian Plutarch writes of how Archimedes kept an invading Roman fleet at bay for three years with his catapults and other inventions.
Known as the Father of Algebra, Diophantine equations, indeterminate equations, and what has evolved into Theory of Numbers, Diophantes gave us his famous autobiographic riddle to figure out his age; x=x/6+x/12+x/7+5+x/2+4.
After the Roman onslaught the mathematics of creative inquiry went dark. One brave Greek intellectual, Hypatia, lectured at the University of Alexandria around 400A.D. in mathematics and had a huge following. A sectarian Christian mob dragged her to their church, skinned her with oyster shells and burned her, for the terrible heresy of geometry. Math went dark for centuries.
In 825A.D. al-Khowarizmi wrote the first clear textbook on algebra “al-jabr w’al-muqabala’’ or the art of bringing together unknowns to match a known quantity. The word in the title “al-jabr” bringing together, gave us our word “algebra”.
Leonardo da Pisa, known as Fibonacci, was on of the first mathematicians to openly consider the existence of and importance of negative numbers . Despite Fibonacci’s recognition that an equation might have a negative solution, most mathematicians doubted the validity of negative numbers until the Renaissance.
Friar Luca Pacioli, in “Summa de Arithmetica”, 1494, challenged that no one had solved the cubic equation. This challenge was taken up by Ferro, Univ. of Bologna, who secretly developed x to the third + ax = b. He gave this to his student, Fior, who used it in a famous algebra match with Tartaglia, the Stammerer, who had his own secret weapon for solving cubics,x to third + ax to the second = b. With a crowd gathered in the inn, and a big purse of gold on the line, Tartaglia won. But the most noteworthy antagonist to confront Tartaglia, was Cordano, astrologer for kings, physician, compulsive gambler, teetering on bankruptcy and prison. Cordano cajoled and flattered Tartaglia’s secret solution from him; the Pope gave Cordano a pension. But what incensed Tartaglia was Cordano publishing the Stammerer’s solution in his famous, monumental treatise “ Ars Magna”, 1545. In this work Cardano, ( who credited Tartaglia ) gave the world the solution to cubics and quartics ( solved by his student Ferrari ).
In “Ars Magna” Cardano formally accepted negative numbers and gave the laws which govern them.
Known as perhaps the greatest engraver, etcher and graphic artist of all time, Durer was a wonderful amateur mathematician.
The class drew, using compass and ruler, Durer’s proof for inscribing a regular pentagon in a circle. Using external circles, this solution proved to be much easier than Pythagorus’. We wrote down and explored Durer’s famous Magic Square, wherein numbers in every direction and in many geometric configurations add up to 34. The children tried to invent their own Magic Squares.
We made a times table square where the square numbers make the diagonal. We explored the relationships of the number families 1,2,3,4,5,6,7,8,9,0 for the ones, 9,8,7,6,5,4,3,2,1,0 for the nines; 2,4,6,8,0 for the twos, 4,8,2,6,0 for the fours,6,2,8,4,0, for the sixes, 8,6,4,2,0, for the eights; 3,6,9,2,5,8,1,4,7,0,for the threes, 7,4,1,8,5,2,9,6,3,0 for the sevens; 5,0,5,0,5,0, for the fives, 0,0,0,0,0,0 for the tens.
We discussed Prime numbers, and the use of “complex primes”, a prime times a prime, as a way of protecting computer programs from being hacked and entered. We discussed the enormous paper two feet thick thousands of pages long, which is the largest known single prime number.
We’ve seen how exciting and animated the history of Mathematics is. We’ll continue with Newton, a play about Gauss, Einstein, quantum physics, topology, applied mathematics, etc. when we revisit this exploration.