Tuesday, February 6, 2018
One explanation why the mathematics found on Egyptian papyri and Babylonian cuneiform tablets is that there were cultural exchanges taking place at this time. Petrie points out that scribes also used the slanting side of a triangle to calculate its area. And that Pythagorean triples are found in the papyrus mathematics. We also find unit fractions and simple sums of series both cultures. Petrie cites an algorithm used to find the sum a geometric series.
Problems 53-60 of the Rhind Mathematical Papyrus are very similar in character to those found in Babylonian mathematics. Babylonian mathematics seems to be slightly more advanced at this time.
Monday, February 5, 2018
Professor Wildberger at the UNSW seems to have anticipated some of the posts that I would have liked to make.
Sunday, February 4, 2018
The Old Babylonians did partitioned triangle problems. An example is YBC 8633.
The cuneiform tablet is somewhat damaged but according to Neugebauer the data for the triangle in the upper left corner can be read as follows.
The calculation of the areas is somewhat off since the base of the inner triangle is 20/60=1/3 and the side, taken to be 1+1/3, is used as the altitude of the triangle. The calculations are as follows with a decimal translation which multiplies the sexagesimal fractions by 60 for convenience.
The tablet would make more sense if 1;12 was used instead of 1;20 for the inner side and 1;11 was used as the altitude.
Perhaps the length of the inner side was misread and the problem then recalculated incorrectly. The Pythagorean theorem would have allowed the scribe to solve for the altitude correctly. The writing is rather cramped and a copyist may have had the same problem working with damaged tablets.
This problem shows how the Old Babylonians might have used steps and a common distance of a line from a point to record changes in direction.
Friday, February 2, 2018
An example of Pythagorean triples from 1800 BC Babylon can be found on the Plimpton 322 cuneiform clay tablet.
Sexagesimal notation is used on the tablet with multiples of ten written sideways and the numbers 1-9 written vertically as counts. The last column on the right is a line count and we see 1=|, 2=||, and 3=|||. Four is written in two rows with two impressions on each row, etc. 10=<, 20=<<, and 30=<<<. Forty is written in two rows with three impressions diagonally on one row and another on the second diagonal row. Zeroes are represented as empty spaces as they would be in a place notation. Two of the Pythagorean triples are in the 2nd and 3rd columns. The 2nd column is one side of the triangle and the 3rd column contains the diagonal. Neugebauer gives a translation and interpretation of the tablet as triangles The 1st column appears to be the square of the ratio of the two sides. We're missing a lot of context associated with the tablet but Robson argues the likelihood that the tablet was made by a bureaucrat favors the interpretation that the tablet represents a list of solutions to a set of similar math problems.
There are some discrepancies in the first column for lines 1, 4, and 13.
On the 1st and 13th lines Neugebauer appears to have misread spaces. The 8th line may contain a calculation error or perhaps transcription errors by a copyist.
Except for the 8th row the numbers in the 1st column are the squares of the ratio of the two sides and are accurate to 15 decimal places.
We can label the sides of the triangle as shown in the figure below.
The sexagesimals on the clay tablet can be transcribed into decimal notation as follows using the values from Wikipedia.
We can compute the missing side using the Pythagorean theorem and we find that the square of the ratio of the two sides matches the sexagesimal values exactly.
We can compare the angles of the triangles along a unit circle and the last column above shows that the deviation from 45° ranges from approximately 0-15°.
Could the list be the ratios of the sides of a set of standard triangles associated with slopes? The Babylonians were more likely to think of angles in terms of triangles and would measure a slope as a displacement x per unit change in height like we find with the Egyptian seked. The angles above range between 45° and 60°.
Supplemental (Feb 3): A skeptic on Plimpton 322. The loss of context for the tablet makes it more difficult to interpret. It may be incomplete since column could be missing on the left. Could it have been forgery? Could it be carbon dated if it contained straw or biological material? Can we rule out a practical joker? Another way we can supply context is to compare the tablet with other mathematical tablets to see if it fits in with them. What would a Babylonian administrator need such a tablet for?
Sunday, January 21, 2018
In the last post we derived formulas for rotations from trigonometric formulas but we may have gotten a little ahead of ourselves. So we need to ask if Pythagoras could have derived the same formulas using geometry. Let's suppose we have two right triangles with rational coordinates (x,y) and (u,v) respectively. To add the rotations we can construct the second triangle, ΔOBE, on the hypotenuse of the first triangle, ΔOAP.
So we need to determine the coordinates of the point B on the unit circle. The major difficulty is to find the sides of triangle ΔEFB. To do so we construct a copy of ΔEFB at the origin, namely, triangle ΔOGD. Line BE is perpendicular to line OA so to find the direction of BE from the origin we need to construct a copy of triangle ΔOAP based on the vertical axis, namely, ΔOCQ. We can then mark off point D a distance v away from the origin. Using the proportions for the similar triangles in triangles ΔOAP and ΔOCQ we can deduce the formulas for OH, EH, GO, and OR. The new coordinates are just sums and differences of these lengths.
So one would not need trigonometry to deduce the formulas for the addition of the two triangles and Pythagoras should have been able to to this. The remaining question is whether he was motivated to do so or not.
Saturday, January 20, 2018
One can reconcile the two sets of triangles by allowing λ to have rational values which are half integers in this case. We can still index the points with integers by using μ=2λ.
Both sets of points will then map onto the same curve.
We can also map all these points onto a unit circle by letting x=a/c and y=b/c which are rational numbers p/q with p and q integers.
The series of points with rational coordinates on the unit circle are not unique. Starting with point on the circle with rational coordinates near the horizontal axis we can use the trigonometric relations for the sum of two angles to step along the circle and get another point with rational coordinates since the rational numbers are closed under the arithmetic operations of addition, subtraction, multiplication and division (excluding zero).
For Pythagoras the proportions of the sides of the triangles and the calculation of lengths and areas would have been his main concern. In ancient Egypt angles were determined by right triangles and was expressed as a seked. The best the Pythagoreans appear to have done in terms of angles were the standard angles of an arc, 30°, 45°, 60° and 90°, and their coordinates.
Friday, January 19, 2018
In his book on Diophantus' work Arithmetic or Numbers Heath states that Pythagoras gives a family of solutions which provide integer values for the sides of a right triangle. How could Pythagoras have accomplished this? The Pythagorean Theorem states that the sum of the squares of the sides are equal to the square of the hypotenuse or a2+b2=c2. We can subtract the square of one side from the square of the hypotenuse and then solve for this side using the other side and an auxiliary variable x.
We get a rational expression for side a and can factor the numerator. Side a is presumed to be an integer so lets assume that b-x is divisible by 2x with the quotient being equal to another auxiliary variable λ. We can then express all the sides of the triangle in terms of the variables x and λ. Noting that all three sides have a common factor, x, we can reduce the expressions for the sides by ignoring the larger similar triangles.
This is the family of solutions attributed to Pythagoras. Evaluating the formulas for integer values of λ does indeed give integer solutions. Note that b is always an odd number. We can find another set of solutions for which b is an even number by manipulating the formulas a little.
We can compare the two sets of numbers by plotting them.
Multiplying all three sides of these triangles by a common integer factor gives more integer solutions. We can also multiply by a rational number to rescale and find smaller similar triangles.
That Pythagoras was able to arrive at the first set of solutions indicates that he had a fairly good grasp of algebra.