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BABYLONIAN NUMERALS ''Main article:'' Babylonian Numerals The Babylonian system of mathematics was Sexagesimal (base-60) Numeral System . From this we derive the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle. The Babylonians were able to make great advances in mathematics for two reasons. Firstly, the number 60 is a Highly Composite Number , having divisors 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30, facilitating calculations with Fractions . Additionally, unlike the Egyptians and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values (much as in our base ten system: 734 = 7×100 + 3×10 + 4×1). The Babylonians may have been familiar with the general rules for measuring the areas. They measured the circumference of a circle as three times the radius and the area as one-twelfth the square of the circumference, which would be correct if Π is estimated as 3. The volume of a cylinder was taken as the product of the base and the height, however, the volume of the frustum of a cone or a square pyramid was incorrectly taken as the product of the height and half the sum of the bases. The Pythagorean theorem was also known to the Babylonians. Also, there was a recent discovery in which a tablet used Π as 3 and 1/8. The Babylonians are also know for the Babylonian mile, which was a measure of distance equal to about seven miles today. This measurement for distances eventually was converted to a time-mile used for measuring the travel of the Sun, therefore, representing time. (Eves, Chapter 2) OLD BABYLONIAN MATHEMATICS (2000-1600 BC) The Old Babylonian period is the period to which most of the clay tablets on Babylonian mathematics belong, which is why the mathematics of Mesopotamia is commonly known as Babylonian mathematics. The Babylonians from this era divided the day into 24 hours, each hour into 60 minutes, each minute into 60 seconds. This form of counting has survived for 4000 years. To write 5h 25' 30", i.e. 5 hours, 25 minutes, 30 seconds, is just to write the sexagesimal fraction, 5 25/60 30/3600. We adopt the notation 5; 25, 30 for this sexagesimal number, for more details regarding this notation see our article on Babylonian numerals. As a base 10 fraction the sexagesimal number 5; 25, 30 is 5 4/10 2/100 5/1000 which is written as 5.425 in decimal notation. Perhaps the most amazing aspect of the Babylonian's calculating skills was their construction of tables to aid calculation. Two tablets found at Senkerah on the Euphrates in 1854 date from 2000 BC . They give squares of the numbers up to 59 and cubes of the numbers up to 32. The table gives 82 = 1,4 which stands for 82 = 1, 4 = 1 60 + 4 = 64 and so on up to 592 = 58, 1 (= 58 X 60 + 1 = 3481). The Babylonians used the formula ab = + b)2 - a2 - b2 /2 to make multiplication easier. Even better is their formula ab = + b)2 - (a - b)2 /4 which shows that a table of squares is all that is necessary to multiply numbers, simply taking the difference of the two squares that were looked up in the table then taking a quarter of the answer. Division is a harder process. The Babylonians did not have an algorithm for long division. Instead they based their method on the fact that a/b = a (1/b) so all that was necessary was a table of reciprocals. We still have their reciprocal tables going up to the reciprocals of numbers up to several billion. Of course these tables are written in their numerals, but using the sexagesimal notation we introduced above, the beginning of one of their tables would look like: 2 0; 30 3 0; 20 4 0; 15 5 0; 12 6 0; 10 8 0; 7, 30 9 0; 6, 40 10 0; 6 12 0; 5 15 0; 4 16 0; 3, 45 18 0; 3, 20 20 0; 3 24 0; 2, 30 25 0; 2, 24 27 0; 2, 13, 20 Now the table had gaps in it since 1/7, 1/11, 1/13, etc. are not finite base 60 fractions. This did not mean that the Babylonians could not compute 1/13, say. They would write 1/13 = 7/91 = 7 (1/91) = (approx) 7 (1/90) and these values, for example 1/90, were given in their tables. In fact there are fascinating glimpses of the Babylonians coming to terms with the fact that division by 7 would lead to an infinite sexagesimal fraction. A scribe would give a number close to 1/7 and then write statements such as: ''... an approximation is given since 7 does not divide.'' (G G Joseph) Babylonian mathematics went far beyond arithmetical calculations. In our article on Pythagoras's theorem in Babylonian mathematics we examine some of their geometrical ideas and also some basic ideas in number theory. In this article we now examine some algebra which the Babylonians developed, particularly problems which led to equations and their solution. We noted above that the Babylonians were famed as constructors of tables. Now these could be used to solve equations. For example they constructed tables for n3 + n2 then, with the aid of these tables, certain cubic equations could be solved. For example, consider the equation ax3 + bx2 = c Let us stress at once that we are using modern notation and nothing like a symbolic representation existed in Babylonian times. Nevertheless the Babylonians could handle numerical examples of such equations by using rules which indicate that they did have the concept of a typical problem of a given type and a typical method to solve it. For example in the above case they would (in our notation) multiply the equation by a2 and divide it by b3 to get (ax/b)3 + (ax/b)2 = ca2/b3 Putting y = ax/b this gives the equation y3 + y2 = ca2/b3 which could now be solved by looking up the n3 + n2 table for the value of n satisfying n3 + n2 = ca2/b3. When a solution was found for y then x was found by x = by/a. We stress again that all this was done without algebraic notation and showed a remarkable depth of understanding. Again a table would have been looked up to solve the linear equation ax = b. They would consult the 1/n table to find 1/a and then multiply the sexagesimal number given in the table by b. An example of a problem of this type is the following. Suppose, writes a scribe, 2/3 of 2/3 of a certain quantity of barley is taken, 100 units of barley are added and the original quantity recovered. The problem posed by the scribe is to find the quantity of barley. The solution given by the scribe is to compute 0; 40 times 0; 40 to get 0; 26, 40. Subtract this from 1; 00 to get 0; 33, 20. Look up the reciprocal of 0; 33, 20 in a table to get 1;48. Multiply 1;48 by 1,40 to get the answer 3,0. It is not that easy to understand these calculations by the scribe unless we translate them into modern algebraic notation. We have to solve 2/3 X (2/3)x + 100 = x which is, as the scribe knew, equivalent to solving (1 - 4/9)x = 100. This is why the scribe computed 2/3 2/3 subtracted the answer from 1 to get (1 - 4/9), then looked up 1/(1 - 4/9) and so x was found from 1/(1 - 4/9) multiplied by 100 giving 180 (which is 1; 48 times 1, 40 to get 3, 0 in sexagesimal). To solve a quadratic equation the Babylonians essentially used the standard formula. They considered two types of quadratic equation, namely x2 + bx = c and x2 - bx = c where here b, c were positive but not necessarily integers. The form that their solutions took was, respectively x = √ + c - (b/2) and x = √ + c + (b/2). Notice that in each case this is the positive root from the two roots of the quadratic and the one which will make sense in solving "real" problems. For example problems which led the Babylonians to equations of this type often concerned the area of a rectangle. For example if the area is given and the amount by which the length exceeds the breadth is given, then the breadth satisfies a quadratic equation and then they would apply the first version of the formula above. A problem on a tablet from Old Babylonian times states that the area of a rectangle is 1, 0 and its length exceeds its breadth by 7. The equation x2 + 7x = 1, 0 is, of course, not given by the scribe who finds the answer as follows. Compute half of 7, namely 3; 30, square it to get 12; 15. To this the scribe adds 1, 0 to get 1, 12; 15. Take its square root (from a table of squares) to get 8; 30. From this subtract 3; 30 to give the answer 5 for the breadth of the triangle. Notice that the scribe has effectively solved an equation of the type x2 + bx = c by using x = √ + c - (b/2). Berriman gives 13 typical examples of problems leading to quadratic equations taken from Old Babylonian tablets. If problems involving the area of rectangles lead to quadratic equations, then problems involving the volume of rectangular excavation (a "cellar") lead to cubic equations. The clay tablet BM 85200+ containing 36 problems of this type, is the earliest known attempt to set up and solve cubic equations. Of course the Babylonians did not reach a general formula for solving cubics. This would not be found for well over three thousand years. CHALDEAN MATHEMATICS (626-539 BC) The Chaldea n period is often known as the Neo-Babylonian period due to the second flowering of Babylon as a capital city and center of study. This period provides the second source of Babylonian mathematics, though somewhat more vague than the Old Babylonian mathematics. Since the rediscovery of the Babylonian civilization, it has become apparent that Greek and Hellenistic astronomers, and in particular Hipparchus , borrowed a lot from the Chaldean s. , specifically the collection of texts nowadays called "System B" (sometimes attributed to Kidinnu ). Apparently Hipparchus only confirmed the validity of the periods he learned from the Chaldeans by his newer observations. It is clear that Hipparchus (and Ptolemy after him) had an essentially complete list of eclipse observations covering many centuries. Most likely these had been compiled from the "diary" tablets: these are clay tablets recording all relevant observations that the Chaldeans routinely made. Preserved examples date from 747 BC . This raw material by itself must have been hard to use, and no doubt the Chaldeans themselves compiled extracts of e.g., all observed eclipses (some tablets with a list of all eclipses in a period of time covering a Saros have been found). This allowed them to recognise periodic recurrences of events. Among others they used in System B (cf. ''Almagest'' IV.2):
The Babylonians expressed all periods in synodic Month s, probably because they used a Lunisolar Calendar . Various relations with yearly phenomena led to different values for the length of the year. Similarly various relations between the periods of the Planet s were known. The relations that Ptolemy attributes to Hipparchus in ''Almagest'' IX.3 had all already been used in predictions found on Babylonian clay tablets. All this knowledge was transferred to the introduced his 76-year cycle, which improved upon the 19-year Metonic Cycle , about that time. He had the first year of his first cycle start at the summer solstice of 28 June 330 BC ( Julian Proleptic date), but later he seems to have counted lunar months from the first month after Alexander's decisive battle at Gaugamela in fall 331 BC . So Callippus may have obtained his data from Babylonian sources and his calendar may have been anticipated by Kidinnu. Also it is known that the Babylonian priest known as Berossus wrote around 281 BC a book in Greek on the (rather mythological) history of Babylonia, the '' Babyloniaca '', for the new ruler Antiochus I ; it is said that later he founded a school of Astrology on the Greek island of Kos . Another candidate for teaching the Greeks about Babylonian Astronomy / Astrology was Sudines who was at the court of Attalus I Soter late in the 3rd Century BC . In any case, the translation of the astronomical records required profound knowledge of the Cuneiform Script , the language, and the procedures, so it seems likely that it was done by some unidentified Chaldeans. Now, the Babylonians dated their observations in their lunisolar calendar, in which months and years have varying lengths (29 or 30 days; 12 or 13 months respectively). At the time they did not use a regular calendar (such as based on the Metonic Cycle like they did later), but started a new month based on observations of the New Moon . This made it very tedious to compute the time interval between events. What Hipparchus may have done is transform these records to the ) the courses of both stars (=Sun and Moon) for 600 years were prophecied by Hipparchus, ...". This seems to imply that Hipparchus predicted eclipses for a period of 600 years, but considering the enormous amount of computation required, this is very unlikely. Rather, Hipparchus would have made a list of all eclipses from Nabonasser's time to his own. Other traces of Babylonian practice in Hipparchus' work are:
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