Already in ancient Egypt and Babylon it was known how to compute the volume of a truncated quadrangular pyramed, for instance, long before axioms were introduced in Geometry. What Euclid added was an axiomatic approach to an already known subject, besides, possibly, adding a few new theorems of his own.
It is not known how two-dimensional or three-dimensional geometry were discovered. All we have are some legends recorded by the people who lived centuries after these discoveries. It is quite plausible that 2-dimensional and 3-dimensional geometry were discovered simultaneously.
One of the legends says that Thales measured the height of an Egyptian pyramid. It gives no detail about how exactly he did it, but it is quite possible that he had to use some geometry in space. Thales was probably the earliest mathematician on whom we have any record, but his work did not survive and details which are mentioned by later writers are unclear. It wasn't really discovered in a mathematical sense of "exploration of higher dimensions" but more as a result of everyday experiences.
This is not clear from everyday experiences. You don't find perfect triangles in nature. They're a geometrical construct that uses the concept of a line which itself is not "well defined" by modern standards in The Elements. However, we see the world in 3 dimensions. To give "perfect meaning" to shapes, they used the 2D analogs and built what we call "surfaces of revolution" to define some basic shapes they were familiar with mostly owing to their experiences with pottery perhaps :.
In theory you could just rotate any primitive shape and get a corresponding 3D shape. We've lost Euclid's book on Conics so we don't know if they ever rotated those to get some interesting shapes.
The cube was just the most primitive shape. But I'm sure they made some vases in those days so I'd conjecture they did something :.
They studied some "other shapes" like the platonic solids, the various "hedrons" tetra-, dodeca-, icosa So, they just analyzed the shapes and explored the corresponding 2D counterparts i.
They concluded that volumes are proportional to the "cube" of the "variable" i. Finding exact formulae was a separate thing e. Euclid 'perfected' this method of exhaustion as we call it, and published a clean proof in his books. If you read the elements you'll see, that 3D analysis was a natural extension owing to everyday experiences. They proved what they could and what made most sense "by analyzing" shapes or cutting them e.
If angles in trigonometry can be seen as two hands of a clock creating different angles as they divide a circle, the chart of numbers on Plimpton describes angles that are created when lines bisect a square. Calculating angles in this manner allows for exact ratios, instead of the irrational numbers and approximations of Greek geometry. Mansfield had a hunch that these mathematics may have been developed alongside the rise of private property and the need to mark boundaries and borders.
He searched for evidence to prove his hypothesis and he eventually found it. The nearly forgotten tablet Si. Made in Iraq between to B. Alongside notes indicating marshland, a tower, and a threshing floor, a field plan measures the boundaries of the land with extreme precision.
Using the example from tablet Si. He further explains his theory with the support of cultural texts that clue us into ancient practices for surveying land. In his paper, Mansfield includes a portion of a text in which a senior scribe is chastising a junior scribe for making the wrong calculations,. The field pegs you are unable to place; you cannot figure out its shape, so that when wronged men have a quarrel you are not able to bring peace, but you allow brother to attack brother. Humboldt University of Berlin.
The British Museum: Mesopotamia. Image source, Mathieu Ossendrijver. Five Babylonian tablets revealed that these ancient people were using sophisticated geometry. The Babylonians detailed their use of trapezoids to track the planet Jupiter. Related Topics. Astronomy History. Published 21 January Published 14 June Published 10 September
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