Conundrums of the geometric and algebraic kind

Conundrums of the geometric and algebraic kind
Located in Berlin, Germany, is a Babylonian tablet with the solution to the diagonal of a 40 by 10 rectangle written on it as 40 + 102/ (2 40). This formula, which claims that the square root of the sum of a2 + b2 may be anticipated as a + b2/2a, is a highly useful approximation approach and can be seen often in later Greek geometric works. These two illustrations of roots show that the Babylonians had the mathematical prowess to analyse geometric forms. They also demonstrate that the Babylonians understood the relationship between the right triangle’s hypotenuse and its two legs almost a thousand years before it was used by the Greeks (now commonly known as the Pythagorean theorem). If you have difficulty dividing, try not to worry too much and visit their website.
Rectangle base and height problems using product and sum are prevalent in Babylonian puzzles. Using this data, the scribe arrived at the conclusion that (b h)2 = (b + h)2 4bh. Equally, we could calculate the sum if we knew both the product and the difference. Now that we have the total and difference, we could also calculate the other side using these steps: 2b = (b + h) + (b h), and 2h = (b + h) (b h). This approach is similar to solving a generic quadratic in one variable. Nonetheless, there are documented instances of Babylonian scribes solving quadratic equations by adjusting the value of a single unknown, much as we do today.
Despite the fact that these Babylonian quadratic methods are sometimes credited as the earliest examples of algebra, there are essential inconsistencies. Although the scribes undoubtedly knew that their solutions could be generalised, they always described them in terms of particular circumstances rather than as the derivation of universal equations and identities, indicating a lack of algebraic symbolism. As a result, they were unable to provide convincing justifications for their answer. Since comparable algorithmic procedures are already commonplace as a result of the widespread availability of computers, the employment of sequential methods rather than equations is less likely to distract from an evaluation of their job.
Rectangles have a base (b), a height (h), and a diagonal (d) that must all meet the relation b2 + h2 = d2. This was known to the aforementioned Babylonian scribes. Despite the fact that there are examples where all three terms are integers, such as (3, 4, 5), it is more common for the third term to be irrational if one randomly chooses the first two (5, 12, 13). (These answers will be referred to as Pythagorean triples in specific instances.)
(The h-column entries were likely part of a cut piece and were thus calculated using the b- and d-column values.) The proper order of the lines is revealed by looking at a column holding the values of d2/h2 (brackets represent missing or illegible numbers): [1 59 0] 15, [1 56 56]. 58 14 50 6 15,…, [1] 23 13 46 40. The angle created by the diagonal and the base shifts from slightly over 45 degrees to just under 60 degrees as the sequence progresses. Aspects of the series reflect the fact that 2d/h = p/q + q/p and 2b/h = p/q q/p for any numbers p and q, indicating that the scribe was familiar with this approach for discovering all such number triples. (As was previously established about the multiplication tables, the numbers p and q in the table are ordinary integers that are part of the standard set of reciprocals.) While experts may disagree on the mechanics of how and why the table was constructed, no one disputes the breadth of understanding it displays.
Algebraic computations for astronomical purposes
It is possible that the Babylonian sexagesimal approach can do computations that are considerably more involved than those in the ancient problem books. But its significance wasn’t completely appreciated until the advent of mathematical astronomy in the Seleucid era. This is because astronomy was developed to predict major events like moon eclipses and changes in the planets’ orbital periods (conjunctions, oppositions, stationary points, and first and last visibility). In order to calculate these coordinates (in latitude and longitude with respect to the Sun’s apparent yearly motion), they came up with an arithmetic progression based on the addition of the required terms. A table outlining future positions was created after all the data was collected, and it would remain in place for as long as the scribe felt it necessary. Despite the mathematical nature of the procedure, the tabulated numbers may be thought of as exhibiting a linear “zigzag” approximation of a true sinusoidal oscillation. Parameters (such as periods, angular range between peak and lowest values, etc.) can only be identified after decades of observations, but a prediction was made using computer algorithms.
Aspects of this system were rapidly disseminated to the Greeks (maybe a century or less). Hipparchus (second century BCE) obtained parameters from the Mesopotamians and employed their sexagesimal way to calculation, even though he preferred the geometric method of his Greek forefathers. The Greeks introduced it to the Middle East, and it was used extensively in mathematical astronomy throughout Europe during the Renaissance and early modern eras. Minutes and seconds are still widely used in the calculation of time and angles.
In fact, it’s possible that the Greeks were exposed to certain features of Old Babylonian mathematics even earlier, in the fifth century BCE, when Greek geometry was still in its infancy. Numerous similarities have been identified by the experts. For more on Greek mathematics, read the section below. Some Babylonian quadratic procedures are similar to the Greeks’ “application of area” approach (although in a geometric, not arithmetic, form). The Babylonian technique of determining square roots was frequently utilised in Greek geometric calculations, suggesting that there may have been some common subtleties in technical language. Although the Greeks were unquestionably influential, when and how the ancient Mesopotamians made substantial contributions to Western mathematics is unknown.