From the Greeks to the Renaissance

MEDIEVAL - p. 1

MEDIEVAL CHRONOLOGY

FROM THE GREEKS TO THE RENAISSANCE

Copyright ©2003 Professor David Singmaster

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NOTE. Transcriptions of Sanskrit and Arabic use a number of characters which are not available in Word. A few of these have close approximations in Word. I use ņ, ş, ţ, to denote the same letters with a dot . under them, and ʼn to denote n with an overdot. I am sure that I avoided inserting some of these diacritical marks when I first started this file and the transcriptions differ considerably. I have not tried to re-examine my sources to determine what the source used. Some of these marks extend beyond the line and I have used wider line spacing where they occur, e.g. in this paragraph.

This includes some chronology of the Greeks, Chinese and Hindus, which is covered more extensively in separate chronologies. This chronology is mainly concerned with the period of the Arabs and medieval Europe and the transmission of ancient knowledge to the Renaissance. I have inserted a few major events in China to show the development there.

My chronologies on Number and on Computing relate to this one.

Some topics whose main development is in this period are trigonometry, induction, calendar reform, time keeping, navigation and discovery and the development of papermaking and printing. I have included relevant earlier and later material on these topics. For a few of these, there are too many entries to cross reference, so I have now made an index here for them.

Dates BC are indicated by - , so 1 BC is -1 and there is no year 0. Intervals BC are indicated with a / , e.g. c-275/-184. Islamic dates are often given as years AH which fall into two years AD – these are indicated with a /, e.g. 927/8.

Number systems. -2400, 499, 522, 595, 662, 820, 876, c980, c1000, 973-1048, 9801037, c1044-1123/4, 1202, early 13C, 1247, 1201-1274, 1318?-1374?, 14C,

Notation, terminology. 1170, early 13C, 1489, 1492, 1525, 1540, 1557, 1540-1603, 1583, 1620, 1626, 1658, 1674,

Abacus, etc. c980, c1000, 1000, 1592, 1617,

House numbering. 1463, c1712.

Double entry book-keeping, banking. 1157, 13C, 1340, 1318?-1374?, 1494,

Compound interest. 1558, 1582,

Negative numbers. c-100, 1202, 1247, c1375, 1484, 1572, 1548-1620, 1629,

Decimal fractions. 263, 952-953, c1000, c1000, c1025, 1172, c1330, ?-1429, 1492, 1585,

Binary system. 1617, 1701,

Number theory. 820, 836-901, c1000, c1010, 1220, c1300, 1510.

Algebra (see also Cubic and higher equations). 820, 820?, c850-930, c1010, c10441123/4, 1145, c1375, 1494, 1557, 1572, 15401603, 1548-1620, 1585, 1591, 1629,

Binomial theorem/Pascal's triangle. c1044-1123/4, 1225, 1201-1274, 1261, 1303, 1356, ?1429, 15481620, 1640?,

Induction. 1550, 1640?, 1838,

Exponents and logarithms. c1010, 1172, c1330, 1323-1382, 1484, 1524, 1614, 1615, 1616, 1617, 1619, 1620?, 1620, 1621, 1624, 1620-1632,

Cubic and higher equations. c1044-1123/4, 1202, 1225, 1328, 1344, 1494, c1506, c1530, 1535, 1539, 1540, 1543, 1545, 1546, 1540-1603,

Euclid's Elements. -300, c390, 754-775, 786-809, 813-833, 830-910, 836-901, 888, c1130, c1175, 12011279, c1270, 1482, 1509, 1533, 1543, 1509-1575, 1562, 1564, 1570. 1537-1612.

Other geometry texts. c1270, 1511, 1509-1575,

Values of π. c-100, c85-c165, 263, c480, 973-1048, 1135-1204, 1202, c1320, ?-1429, 1502, 1551, 1577, 15401603, 1546-1601, 1585, 1593, 1610, 1621, 1667, 1699,

Volume and area of a sphere. c287/212, c-100, 263, c500, 1150,

Trisection. c85-c165,

Parallel postulate. c85-c165, 836-901, c965-1039, c1044-1123/4, 1201-1274, 15371612, See my chronology of Greece for more on this.

Projective geometry. 1593-1662, 1640?

Perspective. 1413?, c1425, 1435, 1397-1475, 1478, 1525, 1540, 1600,

Polyhedra, Tessellations, etc. 940-998, 1498, 1525, 1595, 1596, 1619,

Iterative methods. c830, c1225(?), 1247, ?-1429,

Areas. 909-946,

Development of 360 degrees. Mid -5C, c-180, c-180/c-125, c85-c165,

Trigonometry. -300, c-180/c-125, c85-c165, 4C, 499, 505, 724, c830, 858-929, 940998, 973-1048, Late 10C, c1130, 1145, c1175, 1220, 1201-1274, 1321, c1400, 1464, 1533, 15391541, 15401603, 1594, 1595, 1667, 1744, 1748, 1759, c1870,

Prosthaphaeresis. 858-929, 1005, 1514, c1585, 1540-1603, 1546-1601,

Trigonometric tables. c-180/c-125, c85-c165, 499, 820, 858-929, 940-998, c1000, 9731048, ?-1429, 1440, 1551, 1596, 1613, 1620,

Calculus, analysis, series. c287/212, c500, 628, 932, 1150, c1350, 1356, 1323-1382, c1400, 1502, 1592, 1665, 1667, 1699.

Astronomy – see also: Calendars. mid -5C, c-180, c-120?, c85-c165, 531-579, c840, 840, c970, 1005, 9731048, 980-1037, 1120, 1145, 1201-1274, 1292, c1350, c1428, 14011464, 1483, 15391541, 1543, 1551, 1509-1575, 1572, 1575, 1576, 15461601, 1600, 1564-1642, 1609, 16091610, 1616, 1619, 1627, 1632, 1637, 1655, 1655-56, 1659, 1667, 1675, 1838-1840,

Observatories. 813-833, 1005, 1201-1274, c1428, 1440, 1471, 1575, 1546-1601, 1637, 1667, 1675,

Astrolabe. c85-c165, c390, 662, c791, 927/8, c970, 973-1048, 11C, 1145, 1295, 1391, 1484,

Cross-staff, Backstaff, Sextant, Theodolite, etc. 1321, 1484, 1512, 1556, 1551, c1594, 1676, 1731,

Telescope/spectacles/microscope/optics. c965-1039, c1200, 1260, c1280, c1300, 1352, 1352, 1568, 1570, 1571, c1590, 1592, 1608, 1609, 1609-1610, 1660, 1663, 1668, c1673,

Calendars. -3761, -776, -752, mid -5C, -432, -4C, c-275/-194, c180/c125, -153, -46, 0, early 4C, 324, 325, 525, 622, 640, 675-735, 724, 768-814, late 10C, 973-1048, c10441123/4, 1247, 1260, 1292, mid 14C, 1436, 1474, 1514, 1582, 1583, 1600, 1649-1660, 1700, 1752, 1793, 1920C, 1967, 20C,

Sun dials, nocturnals, hourglasses. 606, c840, 909-946,

Hourglasses. 8C, 807, 13C, 1309-1313, 1337-1339,

Clocks, watches and chronometers – see also: Longitude. -1550, c-180, c180/c-125, 622, 724, 802, 807, c840, 850, c980, 1005, 11C, 1088, Early 12C, c1280, 1290, c1300, c1330, 1335, 13351347, mid14C, 1350, 1364, 1365, 1370, 1406, c1430, 1497-1500, 1502, c1525, 1525, 1530, 1581or1583, c1585, 1639, 1641, 1656, 1658 (twice), 1660-1664, 1673, 1675, 1676, 16601700, 1714, 1715, 1721, 1725, 1735, 1739, c1755, 1757, 1757-1759, 1762, 1773, c1790, 1793, 1970s,

Waterclocks. -1550, 724, 802, 11C, 1988, c1280,

Standard time and time zones, etc. 1847, 1869, 1880, 1883, 1883-1884, 1963, 20C,

Magnetic compass. -1C, c1000, 1080, c100, 1116, 12C, Late 12C, c1190, c1200, 1260, 1269, Mid 15C, 1492, 1600.

Size and shape of the earth. c-275/-194, c85-c165, 6C, 622, 724, 820, 825, c830, 9731048, 1323-1382,

Early travel and other contacts – see also: Exploration. -103, 0-200 (twice), 166, 399414, 431, 615, 636, c685, 771, 786-809, 800, 863, 915, c1140, 1245, 1252, late 13C, 1271-1295, 1322-1356, 1415, 1453.

Exploration and discovery – see also: Magnetic compass, Size of earth, Early travel, Longitude and navigation, Maps. c825, c1140, 13C, 1415, 1419, 1405-1433, 1427, 1394-1460, 1456, 1473?, 1482, 1482, 1487, 1488, 1492, 1494, 1497, 1497, 1500, 1507, 1511, 1513, 1514, 15191522, c1594, 1607, 1607, 1616,

Longitude and navigation – see also: Astrolabe; Cross-staff, Back-staff, Sextant, etc. 1295, 1484, c1500, 1514, 1524, 1530, 1545, 1551, 1559, c1594, 1598, 1616, 1622, c1650, 1707, 1714, 1735, 1741, 1761, 1762, 1767, 1773,

Maps. c85-c165, 622, c1140, c1260-1310, 1569, c1594, 1595, c1650.

Date line. 1323-1382, 1519-1522,

Crossbow. 1139,

Gunpowder, cannons, guns, rockets. 668, 9C, c900, 1044, 1118, 1132, 12-13C, 1230or1231, 1192-1280, 13C, 1260, 1280, c1290?, c1300, 1301, 1312, 1324, 1326, 1331, 1332, 1338, 1346, 1346, 1350, 13561377, c1375, 1377, 1380, 1420s, 1453, 1453, 1561, 1613,

Libraries, books, etc. -546/-527, c-315, c-260, -259?, c-150, c-275/-194, -213, -196, 88/-86, 39, 125, 267, 431, 531-579, 642, c685, 813-833, 863, 969, c970, c987, 1005, 10091031, 1258, c1330, 1492, 1608, 1653, 1691, 1701, 1705,

Paper: -196, c105, 751, 794, 874, 952-953, 10C, 1005, 1102, 1150s, 1276, 1348, 1390, 1494, 16C, 1588, early 1700s.

Printing: c750, 770, Late 8C, 868, 10-11C, 1005, 1041-1048, 1083, 14C, 1403, 1423, 14401450, 1460, 1461 (thrice), 1462, c1470, 1473?, 1473, 1475, 1476, 1482, 1500, 1638, 1690.

Printing of mathematical works: 1083, 1478, 1482, 1482, 1489, 1491, 1494, 1500, 1509, 1510, 1511, 1512, 1515, 1519, 1525, 1533, 1537, 1538, 1540, 1544, 1509-1575, 1588.

Chess. c500, c550, c600, 802, c840, c890, c880-946, 1010, 1061, c1100, 1062-1167, c1275 (twice), 1283, 1476,

Cards. 969, 11C, 1120, 1299, 1329, 1367, 1371, 1377, 1379, 1392, 1398, 1423, c1450, 1457, 1617,

Dice. 11C, 1193, 1447, c1564,

Backgammon. 1283,

Probability. 1447, 1494, c1564, 1654, 1657,

Technological innovations.

General: 768-814, 11C.

Agriculture and food technology and use of horse. Stirrup: 4C, c694, 732. Sugar:5C, 642. Heavy Plough: 6C. Horse harness: 8-9C. Crop rotation and introduction of legumes: 768-814. Horseshoes: 9-10C. Rabbits: 1176. Salting herring: 1359.

Use of power. Waterwheels and watermills: 1C, 5C, 768-814, c1000, 1086, Early12C, 1515. Windmills: 644, 768-814, c870, 10C, Early 12C, c1200, 15C. Tidemills:11C.

Machinery. Crank: c825, Early 12C, c1420, 1480. Treadle: Mid 2C, Late 12C, 1480. Compound crank/carpenter's brace: c1420. Grindstone: c825, 1480. Flywheel: late 11C. Automata: Early 12C, 1299, 1464. Speed Governor: Early 12C. Rudder: c1200. Spinning wheel: c1280. Drawloom: 14C. Canal lock: 1396, c1487. Metal screws: 1405. Hooke's Law: 1678.

Glass: 1180, 1277.

Atmospheric pressure: 1654,

Translation of texts. 399-414, 531-579, 662, c685, 771, 775?, 786-809, c791, 813-833, 820, 830-910, 836-901, 840, c870, c890, 940-998, c965-1039, 1085, 1120, c1130, 1145, c1150, c1160, c1175, 1238, 1201-1274, c1270, 1342, 1476, 1498, 1515, 1517, 1534, 1543, 15091575, 1570, 1585, 1588, 1616, 1632, 1636,

Use of vernacular languages. 405, c715, 813, 863, c890, 1328, c1375, 1494, 1509, 1511, 1534, 16C, 1509-1575,

Use of Hebrew, Jewish mathematicians. c791, 1120, 1092-1167, 1135-1204, 1321, 1475, 1498, 1517. (Just added, there are probably more entries, e.g. Bonfils?.)

Academies and journals. 529, 786-809, 813-833, 820, 836-901, c860, c870, 1005, 1582, 1603, 1625, c1645, 17C, 1652, 1660-1662, 1665, 1666, 1682,

Universities. -863, c-390, c-1C, 431, 5-12C, 529, 531-579, 651, 7C, 725, c750, 786809, 813-833, 825, 859, 969, c970, c1050, 11C, 1065, 1088, 1096, 1109, 1137, 11331144, 1167, 1096-1270, 1200, 1209, 1214, Early 13C, 1215, 1221 (twice), 1224, Firsthalfofthe13C, 1227/8, 1231, 1253, 1284, 1287, 1290, 1303, 1308, Early 14C, 1331, 1343, 1348 (twice), 1364, 1365, 1367, 1386, 1388, 1389, 1392, 1409, 1411, 1419, 1425, 1426, 1446, 1450, 1460, 1467, 1472, 1477 (twice), 1490, 1491, 1492, 1498, 1502, c1525, 1526, 1529or1530, 1544, 1558, 1559, 1559, 1575, 1581, 1582, 1583, 1585, 1590, 1597, 1614 (twice), 1625, 1632 (twice), 1636(thrice), 1646, 1647, 1665, 1668, 1678, 1693, 1737,

Religious developments. (I haven't included all details of conquest and reconquest nor all religious aspects of the calendar.) 64, 303, 312, 313, c320, early 4C, 324, 391, 431, early6C, c520, 531579, 597, 622, 632, 638, 640, 642, 656, 661, 717-718, 725, 8C, 842, 863, 929, 989, 1054, 1095, 1099, 1187, 1202, 1204, 1245, c1266, 1290, 1291, 1302, 1309, 1367, 1378, 1415, 1469, 1477, 1492, 1517, 1521, 1529, 1530, 1534, 1534, 1540, 1545-1563, 1555, 1568, 1572, 1598, 1600, 1616, 1618, 1621, 1625, 1632, 1648, 1685,

Medical developments. 860-932, 1096, 1096-1270, 1137, 1221, 1616, 1660,

Social developments. (I have just added this and may not have entered all dates of relevance.) 768-814, 1290, 1346-1350, c1360, 1568, 1581, 1621, 1678,

Women in Mathematics and Science. 415,

Warning notes. Many dates are speculative. Generally they indicate just when the person flourished. Older Chinese items are extremely uncertain – it is often argued that the content is much older than the writing, so the Chou Pei Suan Ching may be dated back to 800!!. Writers on Hindu mathematics often assert its beginnings may go back several million (!!) years. Even Greek authors may have dates which are uncertain by 200 years. Islamic dates may be off due to conversion between AH and AD – the Islamic year is about 11 days shorter than the solar year – see 640.

Spellings can vary greatly, especially for names which are transliterated. D. E. Smith

gives a useful discussion of transliteration and pronunciation at the beginning of his History of Mathematics.

Notes:

1 year= 365.242216 or 365.242199 or 365.242197 or 365.24219870 days.

1 lunar month= 29.530598 days.

1 year= 12.368267 lunar months.

π= 3.14159 26535.

The circumference of the earth is 40,010 km = 24,861 mi.

The length of a degree is 111.14 km or 69.06 mi.

-3761Date of creation used as the starting point of the Jewish calendar.

-2400Positional notation in Babylonia, base 60. It is conjectured that this evolved from a system of counting on the joints of the four fingers of one hand – this makes 12 – and then keeping track of the 12s with the five fingers of the other hand, making 60 in all. This also gave a count of 12 hours in the day and 12 in the night.

-1550?Earliest waterclocks (clepsydra) invented in Egypt by Amenemhet.

-863In 570, Merlin of Caledonia mentions a university at Stamford, Lincolnshire, founded by Prince Bladud in -863. I suspect this is entirely legendary.

776First Olympiad, used as the beginning of the Greek calendar developed by Eratosthenes, c-240. The Athenian Festival Calendar had months of alternately 29 and 30 days, with occasional intercalary months, but this was not done systematically and the months were often out of phase with the seasons.

-752Legendary date of foundation of Rome and beginning of the Roman calendar, legendarily established by Romulus. However, Roman scholars already disagreed over this date, variously putting it at -751 or -754, and I have seen both -752 and 753 and even -735 (probably a misprint for -753). Consequently numerical years were never widely used by the Romans. The calendar had a year of only 304 days in ten months, based on the growing season, with March as the first month. The rest of the days were not counted – or else the months drifted through the year?? Romulus's successor, the semi-legendary King Numa Pompilius (c-650 or c-700 or reigned -715/-673) added January at the beginning and February at the end (or both at the beginning??), extending the year to 354 days with months alternating between 30 and 29 days. An extra day was added, because odd numbers were luckier, giving a year of 355 days. To correct for the obvious inaccuracy, Numa decreed an intercalary month of Mercedonius to be added between 23 & 24 February of every third year (or an intercalary month of 22 days every second year or of alternately 22 and 23 days every second year – ah, another source says it was 27 or 28 days every other year, but then the last 5 days of February were dropped, though sometimes the extra month was omitted when things got out of phase), making a year of 377 or 378 days. The last of the methods gives 1465 days in 4 years, making a year of 366.25 days, which was later recognised as too long and supposed to be corrected every 24 years. As can be seen, there is considerable uncertainty – one source says no one really knows how many days were in this month. Just to confuse matters, the name Mercedonius only occurs in Plutarch's Lives, which were written in Greek – the Romans apparently never had a definite name for it beyond 'mensis intercalaris'. In 452 or -153, February was shifted from 12th to 2nd month, so the beginning of the year was 1 Jan. The intercalations were done so poorly that the calendar was out by 117 days in -190 and 75 days in -168. Apparently Roman officials often added months in order to extend their time in office. Caesar reformed the calendar in -46 (qv).

-546/-527Tyranny of Pisistratus at Athens. He is said to be the first to found a public library, though this is doubtful – see c-260.

Mid -5CGreeks determine the year is 365+ days long.

Mid -5CBabylonians are asserted to have divided the zodiac into 360 degrees with 60 minutes to the degree.

-432Meton (and Euctemon?) discovers the 19 year cycle of 235 months. This is the basis of the Greek astronomical calendar which started on 27 Jun -432, but doesn't seem to have been used for any ordinary purposes.

c-390The Pythagorean, Archytas of Tarentum (c-430/c-370), establishes the Quadrivium of Music, Astronomy, Arithmetic and Geometry which remains the basis of university education until the Renaissance, even to the 16C.

-4CCallippus adjusts the Metonic cycle (see -432) by omitting the last day of every fourth cycle. My source claims this makes the year equal to 365 1/4 days, but I can't make this work out.

336/323Reign of Alexander.

c315Ptolemy I Soter founds Museum (c= a research institute) at Alexandria. This includes the Library, but the history of these is given in my Greek Chronology and is too long to carry over to here.

300Euclid: Elements of Geometry. This includes a form of the law of cosines, but it is not clearly stated until Vieta (1593).

c-260Ptolemy II Philadelphus founds first public library in Athens, within a gymnasium called the Ptolemeion.

-259?Ptolemy II Philadelphus (reigned -282/-247) founds Library at Alexandria with Demetrios as Librarian, according to some sources – cf c-315. A source says Ptolemy II acquired Aristotle's library, but I believe it was at Pergamum, cf -88/-86 below.

c-250Callimachus of Cyrene compiles first library catalogue, for the Library at Alexandria, comprising 120 rolls.

c287/212Archimedes. Volume and surface area of the sphere and primitive integration. Estimates π by use of a 96-gon as 223/71 < π < 22/7. His Cattle Problem leads to x24729494y2=1, the first example of Pell's equation.

c275/194Eratosthenes: director of the library at Alexandria from c245, his 'sieve' and measurement of the earth. He gets a circumference of 28,727 miles. He suggests the calendar adopted by Julius Caesar in -46. He also initiates the first classical calendar with years numbered, based on the four-yearly Olympiads which began in -776, denoted the first year of the first Olympiad. This system persisted well into Byzantine times.

213Burning of books under Qin Shi Huangdi.

-196Library founded at Pergamum by Eumenes. In order to discommode these rivals, Egypt prohibited the export of papyrus (the origin of 'paper'), so Pergamum developed and perfected the treating of skins for writing – the word 'parchment' derives from Pergamum.

c180Hypsicles divides day into 360 parts. (I have seen an assertion that the Babylonians divided the zodiac into 360 degrees, c450.)

c180/c125Hipparchus. He begins trigonometry, using Hypsicles' 360 parts to measure angles. He clearly gives sin (a + b) and seems to know sin 2a, sin a/2 and sin2 a + cos2a. Greek trigonometry used chords instead of sines. He may have made a table of chords. He takes year as 365.247222 days (or 365.246667 = 365 74/300 days). He introduces hours of equal length in the day and the night, but these are only used in scientific work.

-153From this time, Roman years start on 1 Jan, while Greek years continue to start in midsummer. (This may have started earlier, possibly -452, cf -752.)

c120?Zhou Bi Suan Jing = ChouPei Suan Ching (The Arithmetical Classic of the Gnomon and the Circular Paths (of Heaven). Variously claimed to date from 11C or 4C. It has star positions from between 575 and 450. Compiled into a book in its present form sometime between 100 and +100. Has the theorem of Pythagoras and applies it.

-103Chinese ships reach east coast of India.

c100Jiu Zhang Suan Shu = Chiu Chang Suan Ching (Nine Chapters on the Mathematical Art. Possibly as late as +50. See also Liu Hui (263) for more history. 246 solved problems. Finds square and cube roots. Lots of surveying, area and volume problems. Solves simultaneous linear equations, both determinate and indeterminate. Earliest known use of negatives. "The method of positive and negative states: for subtracting – same signs take away, different signs add together, positive from nothing makes negative, negative from nothing makes positive; for addition – different signs take away, same signs add together, positive and nothing is positive, negative and nothing makes negative." One problem involves finding a numerical solution of a quadratic by a method which Needham and Wang identify as Horner's method. Says area of a circle is the radius times the circumference/2. The volume of the sphere is found as 9/16 the volume of the circumscribed cube, i.e. V = 9/2 r3, a traditional ratio based on weighing, corresponding to π= 27/8.

-88/-86Revolt against Romans in Asia Minor, lead by Mithradates. Sulla defeats Mithradates and sacks Athens on 1 Mar 86. Aristotle's notebooks are taken from Pergamum as booty by Sulla. Antikythera mechanism possibly taken as booty.

-1CChinese geomancers use a magnetite spoon to indicate north, but the application to navigation is not made until c1190, cf 1269.

c-1CMarseilles was a Greek colony and had a Greek university around -1C, 'the last refuge of Greek teaching in the West'.

46Julius Caesar, advised by Sosigenes of Alexandria, based on Eratosthenes' suggestion, reforms the calendar to a year of 365.25 days (off by 11 min 14 sec per year). The year 46 (= Roman year 708) has 445 days and is known as the 'last year of confusion'. This was done by adding 23 days to February (i.e. intercalating 'Mercedonius') and two extra months (Undecimber & Duodecimber, having 33 and 34 days) between November and December. (The confusion was partly due to Caesar – he had been Pontifex Maximus, and hence in charge of the calendar, since 52 and he had only inserted one intercalary month which should have been done every two years.) The vernal equinox is intended to be on 25 March. The beginning of the year is moved from 1March to 1January. March, April, May, June are renamed to the Latin forms from which the English words are derived (other sources says these names go back to Romulus or Numa, -8C ??). 8 months are lengthened and one shortened to shift from the 355 day lunar calendar to the 365 day solar calendar. Basically the odd months have 31 days and the even months have 30 days, but February has 29 days in normal years and 30 days in leap years. Leap year day is between 25 & 26 February – so there were two 25ths. Since the 25th was sexto calendas Martias (the sixth day before 1 Mar), a leap year was known as a bissextile year. However, other sources say the repeated day was 23 Feb or 24 Feb. Some sources say Caesar lengthened Quintilis and named it for himself, but Quintilis already had 31 days and was later named Julius by the Senate in -45 (or -46), or after his death in 44. The priests misinterpreted the rules for leap years – because they counted to four inclusively and hence inserted leap years every third year! This was sorted out by Augustus who found out that -8 started three days late. He cancelled the next three leap years, so the next leap year was in +8, so the Julian system ran consistently from +5. He may or may not have adjusted the lengths of some months. Whoever did it, a day was removed from February to lengthen August, so the bissextile day became the 24th of Feb. The seven day week was taken over from the Jews in the 4C, qv. A source says that an eight-day week was used before then. Another source says the weeks alternated between 7 and 8 days. From about 1300 until 1752, the beginning of the year had shifted to 26 (or 25??)March, at least in England, for reasons unknown to me.

44Death of Julius Caesar.

-39Gaius Asinius Pollo founds the first public library in Rome.

-8Augustus orders a census. Most probable year of birth of Christ. Augustus observes that this year starts 3 days late and cancels the next three leap years. Cf -46, +8. At some time, Augustus changed the name of Sextilius to Augustus – the motion of the Senate is preserved. Further, a day is added to Augustus by changing the number of days in later months and stealing a day from February.

0This year does not exist! See 525. This led to the 2000th anniversary of Virgil being celebrated a year early!

14Death of Augustus.

1CWaterwheels in Europe, Egypt, Asia Minor, China, but they are not extensively developed until about 1000. However, cf 5C.

64Buddhism introduced to China.

100Nicomachus of Gerasa.

0-200Silk Road carries Chinese Silk to Roman Syria.

0-200Roman trading post in India.

c105Cai Lun = Ts'ai Lun invents papermaking in China, using mulberry bark with other fibres.

100-200Roman Empire at height.

125Theon of Smyrna.

c125Hadrian establishes a library at Athens.

c85-c165Ptolemy: Almagest uses Hypsicles' and Hipparchus' 360 degrees for a circumference, taking the radius as 60. Introduces minutes and seconds, establishing the use of decimals for whole numbers and sexagesimals for fractions. Includes a table of chords with 30' intervals (equivalent to angles at 15' intervals), accurate to five decimals. He describes the astrolabe and its construction. He estimates π=3;8,30=317/120=3.14166. He knows sin (a + b) (equivalent to Ptolemy's theorem on cyclic quadrilaterals), sin a/2 and the law of sines. Establishes the Ptolemaic System of astronomy. He extends and amends Hipparchus' star catalogue to 1028 stars. He asserts that angle trisection cannot be done 'by geometrical methods'.

His Geography introduces latitude and longitude and assumes the earth's circumference is 18,000 miles, thereby encouraging Columbus's 1492 voyage. He draws his 0o of longitude through the Canaries – the furthest west land known. He gives latitude and longitude of some 8000 places. In other works, he explains and uses orthographic and stereographic projections. There is a traditional map of the world ascribed to Ptolemy, though the oldest extant versions are 12C or 13C – a version was produced by Maximos Planudes, c1300, based on Ptolemy's data. He is often considered the first to try to derive Euclid's Fifth Postulate from the other postulates, but see Aristotle (-384/-322). He nearly discovers the law of refraction.