In the absence of mathematical means of investigating motion, the ideas of philosopher Aristotle (384-322 B.C.) gained dominance in physics, particularly his "Law of Falling Bodies": an object fell to the earth according to its weight -- the heavier, the faster the fall. Over the centuries, a few sceptics individually pointed out a "paradox" in this notion. Given a 2-unitweight object and a 1-uniweight one, the first should, by this "Law" fall much faster than the second (numerical proportions were ignored). Tied together, the 3-unitweight object should fall FASTER than the 2-unitweight object, but -- at the same time -- SLOWER, since the 1-unitweight object should hold back the 2-unitweight object. But such cavilling was ignored.

Circa 300 B.C., in Alexandria, Egypt, Greek Euclid (450-380 B.C.) published his Elements of Geometry, culminating the work of Thales, Pythagoras, Eudoxus, and others. (Only The Bible has appeared in more copies than The Elements.) Experts attribute the first four Books to the work of Pythagoras. And the Book Five contains the great work of Eudoxus on proportions.

Circa 220 B.C., Archimedes refined the Eudoxian method of exhaustion to approximate the area of a circle, an early estimate of p (ratio of circumerence to diameter of a circle -- a further step toward "the integral calculus", vindicating Pythagorean Geometric-Arithmetic.

The great Alexandrian Period (332 B.C.-420 A.D.) steam cars lurched down the streets and giant robots moved arms under steam propulsion, scaring the faithful. Imperial Roman engineers developed efficient windmills and water mills (For example, in the first century B.C., the Roman engineer Vitrivius described the undershot water wheel, later so useful in medieval and post-medieval Europe, yet rarely used by Romans.) These efficient watermills and windmills could have replaced the slave-power of the time. But, with no theoretical mechanics to show this and give hope of improvements, slavery continued in Europe for hundreds of more years.

415 A.D., in Alexandria, Egypt, Christian mobs assassinated Hypatia, greatest woman mathematician of ancient times. A few days later, these mobs burned the Library of Alexandria (greatest ancient repository of mathematics and science writings) -- destroying all the works of Eudoxus and almost all of those of Archimedes, as well as the works of so many other creators.

In 527 B. C. Justinian, Roman Emperor of the West wing of the Roman Empire, became also Emperor of the East. He decided that the "pagan" learning of the Academy founded in Plato's name and other philsophical schools threatened orthodox Christianity, so, in 529, all these schools were closed, effectively ending speculative research in Europe. The challenge was taken up by Islamic scholars who not only preserved the knowledge of the ancients but advanced it in "forbidden areas".

One of the greatest was the astronomer Mohammed ibn Musa al-Khowarizmi (circa 825 A.S.) whose name inspired the label "algorithm" for a "sure-fire" method for problem-solving. And the title of his book on Arithmetic inspired our word for "algebra". (Evidence recently emerged that pre-Renaissance Islamic -- and perhaps Jewish -- scholars made discoveries later attributed to European scholars in the Renaissance period.)

The most important Islamic advance was to arithmetize motion and challenge the Aristotlean "Law of Falling Bodies". Peculiarly, Islamic arguments for arithmetizing motion derived from analogies in pharmacy and optics. It appeared possible to change magnitudes in medicines and in light intensities, hence, it was extrapolated to motion.

In this "Medieval-Renaissance" period, Arabic artists excelled in the geometric art of figrative adornment. The finest example of this is The Alhambra in Grenada (Spain), a fortess and palace. Mathematicians recognize 128 distinct patterns of "the wallpaper-tile group". All of these appear in the decorations of The Alhambra (as in an Alhambra portal or an Alhambra relief or an Alhambra mosaic).

As you may read in The Medieval Machine: The Industrial Revolution of the Middle Ages, by Jean Gimpel, 1977, in the 12th century (A.D.) began what should be called "The Mechanical Industrial Revolution". Monks -- desiring to raise their food and other needs, yet have time for prayer and meditation -- were given special permission by the Pope to read pagan texts showing how to build watermills and windmills. In this century and the next, these mills spread across Europe. (The Domesday Book of 1086 lists 5624 water-wheel driven mills in England south of Trent, or about one mill for each 400 persons.) Clearly, further extension of these mills would have eliminated much of the slavepower then existing, but "the lesson" was still not obvious enough.

In 1202, Fibonacci (a.k.a. Leonardo of Pisa) published Liber abaci ("book of the abacus"), the first European books to use decimal notation in arithmetic (which Fibonacci learned in his native North Africa). The spread of decimal arithmetic made possible the great "Age of Navigation" and "Age of Trading", which followed -- although historians choose to ignore this.

William of Ockam (1280?-1349?), sketches notion of inertia, a physical concept later credited to Newton. (The charactor of William of Baskerville played by Sean Connery in the film, The Name of the Rose, was modeled on Ockam.) Jean Buridan (1295?-1359), French philosopher, in 1350's developed the notion of impetus, a physical concept close to the concept of inertia.

1320 A.D., Thomas Bradwardine (1290-1349) studied at Merton College Oxford, became chancellor of St. Paul's Cathedral. Bradwardine was a noted mathematician as well as theologian, known as 'the profound doctor'. Bypassing both the Platonic and Aristotlean traditions, Bradwardine studied bodies in uniform motion and ratios of speed in the treatise De proportionibus velocitatum in motibus (1328). Bradwardine was one of the '''Oxford Calculators''', of Merton College, Oxford University, studying mechanics with William Heytesbury, Richard Swineshead, and John Dumbleton. The Oxford Calculators distinguished '''kinematics''' from '''dynamics''', emphasizing kinematics, and investigating '''instantaneous velocity'''. They first formulated the '''mean speed theorem: a body moving with constant velocity travels distance and time equal to an accelerated body whose velocity is half the final speed of the accelerated body'''. They also demonstrated this theorem -- essence of "The Law of Falling Bodies" -- long before Galileo is credited with this. In '''Tractatus de proportionibus''' (1328), Thomas Bradwardine extended Eudoxus' '''theory of proportions''' to anticipate the concept of '''exponential growth''', later developed by the Bernoullis and Euler, with '''compound interest''' as a special case.Bradwardine became Archbishop of Canterbury in 1349 but died, a month later, of the Black Plague. Bradwardine is mentioned (as "Bisshop Bradwardyne") in Chaucer's Canterbury Tales, in "The Nun's Priest's Tale", line 476.

This Mechanical Industrial Revolution (described above) waned with "The Black Death" (bubonic plague), peaking in 1348. The Black Death was caused by the dearth of cats to control the rat population. For centuries, the superstitious had killed cats, alleging them to be the "familiars" of witches. (Cats' eyes  can catch any light in a dark room or cave, lighting up when nothing else does. Also, in cold climes, stroking a cat invokes static electricity, and the sparks scintillate in the dark.) One Pope even advocated exterminating cats. So, by the middle of the 14th century, the dearth of cats allowed the rat population to multiply to epidemic proportions. The rats  carried fleas. And the fleas carried the Plague. In 5 years, the Plague killed 25 million, 1/3 of Europe's Population. In some regions, 50% of the population died in "The Black Death". Many regions lacked blacksmiths, coopers, wheelrights, other craftsmen.

Training had been oral. The Church now allowed more general reading of "pagan" books, initiating The Renaissance. (Tell this story to teen-agers who "want to work with their hands", but ask why they need reading skills!)

Chronology, continued.