Editing History of geometry (section)

==Early geometry==
The earliest recorded beginnings of geometry can be traced to early peoples, such as the [[Indus Valley civilization|ancient Indus Valley]] (see [[Indian mathematics#Prehistory|Harappan mathematics]]) and ancient [[Babylonia]] (see [[Babylonian mathematics]]) from around 3000&nbsp;BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in [[surveying]], [[construction]], [[astronomy]], and various crafts. Among these were some surprisingly sophisticated principles, and a modern mathematician might be hard put to derive some of them without the use of [[calculus]] and algebra. For example, both the [[Egyptians]] and the [[Babylon]]ians were aware of versions of the [[Pythagorean theorem]] about 1500 years before [[Pythagoras]] and the Indian [[Sulba Sutras]] around 800&nbsp;BC contained the first statements of the theorem; the Egyptians had a correct formula for the volume of a [[frustum]] of a square pyramid.

===Egyptian geometry===
{{main|Egyptian geometry}}

The ancient Egyptians knew that they could approximate the area of a circle as follows:<ref name="School Mathematics">Ray C. Jurgensen, Alfred J. Donnelly, and Mary P. Dolciani. Editorial Advisors Andrew M. Gleason, Albert E. Meder, Jr. ''Modern School Mathematics: Geometry'' (Student's Edition). Houghton Mifflin Company, Boston, 1972, p.&nbsp;52. {{ISBN|0-395-13102-2}}. Teachers Edition {{ISBN|0-395-13103-0}}.</ref>

::::Area of Circle ≈ [ (Diameter) x 8/9 ]<sup>2</sup>.

Problem 50 of the [[Ahmes]] papyrus uses these methods to calculate the area of a circle, according to a rule that the area is equal to the square of 8/9 of the circle's diameter. This assumes that [[pi|{{pi}}]] is 4&times;(8/9)<sup>2</sup> (or 3.160493...), with an error of slightly over 0.63 percent. This value was slightly less accurate than the calculations of the [[Babylonia]]ns (25/8 = 3.125, within 0.53 percent), but was not otherwise surpassed until [[Archimedes]]' approximation of 211875/67441 = 3.14163, which had an error of just over 1 in 10,000.

Ahmes knew of the modern 22/7 as an approximation for {{pi}}, and used it to split a hekat, hekat x 22/x x 7/22 = hekat;{{cn|date=July 2018}} however, Ahmes continued to use the traditional 256/81 value for {{pi}} for computing his hekat volume found in a cylinder.

Problem 48 involved using a square with side 9 units. This square was cut into a 3x3 grid. The diagonal of the corner squares were used to make an irregular octagon with an area of 63 units. This gave a second value for {{pi}} of 3.111...

The two problems together indicate a range of values for {{pi}} between 3.11 and 3.16.

Problem 14 in the [[Moscow Mathematical Papyrus]] gives the only ancient example finding the volume of a [[frustum]] of a pyramid, describing the correct formula:
:<math>V = \frac{1}{3} h(a^2 + ab + b^2)</math>
where ''a'' and ''b'' are the base and top side lengths of the truncated pyramid and ''h'' is the height.

===Babylonian geometry===
{{main|Babylonian mathematics}}

The Babylonians may have known the general rules for measuring areas and volumes. They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if ''[[Pi|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 known 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.<ref>Eves, Chapter 2.</ref> There have been recent discoveries showing that ancient Babylonians may have discovered astronomical geometry nearly 1400 years before Europeans did.<ref>{{cite news| url = https://www.washingtonpost.com/news/speaking-of-science/wp/2016/01/28/clay-tablets-reveal-babylonians-invented-astronomical-geometry-1400-years-before-europeans/| title = Clay tablets reveal Babylonians discovered astronomical geometry 1,400 years before Europeans - The Washington Post| newspaper = [[The Washington Post]]}}</ref>

===Vedic India geometry===

[[Image:Rigveda MS2097.jpg|thumb|right|''[[Rigveda]]'' manuscript in [[Devanagari]]]]
The Indian [[Vedic period]] had a tradition of geometry, mostly expressed in the construction of elaborate altars.
Early Indian texts (1st millennium BC) on this topic include the ''[[Satapatha Brahmana]]'' and the ''[[Shulba Sutras|Śulba Sūtras]]''.<ref>A. Seidenberg, 1978. The origin of mathematics. Archive for the history of Exact Sciences, vol 18.</ref><ref>{{Harv|Staal|1999}}</ref><ref>Most mathematical problems considered in the ''Śulba Sūtras'' spring from "a single theological requirement,"  that of constructing fire altars which have different shapes but occupy the same area. The altars were required to be constructed of five layers of burnt brick, with the further condition that each layer consist of 200 bricks and that no two adjacent layers have congruent arrangements of bricks. {{Harv|Hayashi|2003|p=118}}</ref>

The ''Śulba Sūtras'' has been described as "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians."{{Sfn|Hayashi|2005|p=363}} They make use of [[Pythagorean triples]],{{Sfn|Knudsen|2018|p=87}}<ref>Pythagorean triples are triples of integers <math> (a,b,c) </math> with the property: <math>a^2+b^2=c^2</math>. Thus, <math>3^2+4^2=5^2</math>, <math>8^2+15^2=17^2</math>, <math>12^2+35^2=37^2</math> etc.</ref> which are particular cases of [[Diophantine equations]].<ref name="cooke198">{{Harv|Cooke|2005|p=198}}: "The arithmetic content of the ''Śulva Sūtras'' consists of rules for finding Pythagorean triples such as (3, 4, 5), (5, 12, 13), (8, 15, 17), and (12, 35, 37). It is not certain what practical use these arithmetic rules had. The best conjecture is that they were part of religious ritual. A Hindu home was required to have three fires burning at three different altars. The three altars were to be of different shapes, but all three were to have the same area. These conditions led to certain "Diophantine" problems, a particular case of which is the generation of Pythagorean triples, so as to make one square integer equal to the sum of two others."</ref>

According to mathematician S. G. Dani, the Babylonian cuneiform tablet [[Plimpton 322]] written c. 1850 BC<ref>Mathematics Department, University of British Columbia, [http://www.math.ubc.ca/~cass/courses/m446-03/pl322/pl322.html ''The Babylonian tabled Plimpton 322''].</ref> "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple,<ref>Three positive integers <math>(a, b, c) </math> form a ''primitive'' Pythagorean triple if <math> c^2=a^2+b^2</math> and if the highest common factor of <math> a, b, c </math> is 1. In the particular Plimpton322 example, this means that <math> 13500^2+ 12709^2= 18541^2 </math> and that the three numbers do not have any common factors. However some scholars have disputed the Pythagorean interpretation of this tablet; see Plimpton 322 for details.</ref> indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia in 1850&nbsp;BC.{{Sfn|Dani|2003|p=223}} "Since these tablets predate the Sulbasutras period by several centuries, taking into account the contextual appearance of some of the triples, it is reasonable to expect that similar understanding would have been there in India."{{Sfn|Dani|2003|p=223}} Dani goes on to say:{{Sfn|Dani|2003|p=223–224}}

<blockquote> As the main objective of the ''Sulvasutras'' was to describe the constructions of altars and the geometric principles involved in them, the subject of Pythagorean triples, even if it had been well understood may still not have featured in the ''Sulvasutras''. The occurrence of the triples in the ''Sulvasutras'' is comparable to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and would not correspond directly to the overall knowledge on the topic at that time. Since, unfortunately, no other contemporaneous sources have been found it may never be possible to settle this issue satisfactorily.</blockquote>