Editing Volume (section)

=== Ancient history ===
[[File:Pompeji_6_Hohlmaße_aus_Glas.jpg|thumb|6&nbsp;volumetric measures from the ''mens ponderia'' in [[Pompeii]], an ancient municipal institution for the control of weights and measures]]

The precision of volume measurements in the ancient period usually ranges between {{Cvt|10–50|mL|USoz impoz|sigfig=1}}.<ref name="Imhausen-2016" />{{Rp|page=8}} The earliest evidence of volume calculation came from [[ancient Egypt]] and [[Mesopotamia]] as mathematical problems, approximating volume of simple shapes such as [[cuboid]]s, [[cylinder]]s, [[frustum]] and [[cone]]s. These math problems have been written in the [[Moscow Mathematical Papyrus]] (c. 1820 BCE).<ref name="Treese-2018" />{{Rp|page=403}} In the [[Reisner Papyrus]], ancient Egyptians have written concrete units of volume for grain and liquids, as well as a table of length, width, depth, and volume for blocks of material.<ref name="Imhausen-2016">{{Cite book |last=Imhausen |first=Annette |url= |title=Mathematics in Ancient Egypt: A Contextual History |date=2016 |publisher=[[Princeton University Press]] |isbn=978-1-4008-7430-9 |location= |oclc=934433864}}</ref>{{Rp|page=116}} The Egyptians use their units of length (the [[cubit]], [[Palm (unit)|palm]], [[Digit (unit)|digit]]) to devise their units of volume, such as the volume cubit<ref name="Imhausen-2016" />{{Rp|page=117}} or deny<ref name="Treese-2018" />{{Rp|page=396}} (1 cubit × 1 cubit × 1 cubit), volume palm (1 cubit × 1 cubit × 1 palm), and volume digit (1 cubit × 1 cubit × 1 digit).<ref name="Imhausen-2016" />{{Rp|page=117}}

The last three books of [[Euclid's Elements|Euclid's ''Elements'']], written in around 300 BCE, detailed the exact formulas for calculating the volume of [[parallelepiped]]s, cones, [[pyramid]]s, cylinders, and [[sphere]]s. The formula were determined by prior mathematicians by using a primitive form of [[Integral|integration]], by breaking the shapes into smaller and simpler pieces.<ref name="Treese-2018">{{Cite book |last=Treese |first=Steven A. |title=History and Measurement of the Base and Derived Units |date=2018 |publisher=[[Springer Science+Business Media]] |isbn=978-3-319-77577-7 |location=Cham, Switzerland |lccn=2018940415 |oclc=1036766223}}</ref>{{Rp|page=403}} A century later, [[Archimedes]] ({{Circa|287 – 212 BCE}}) devised approximate volume formula of several shapes using the [[method of exhaustion]] approach, meaning to derive solutions from previous known formulas from similar shapes. Primitive integration of shapes was also discovered independently by [[Liu Hui]] in the 3rd century CE, [[Zu Chongzhi]] in the 5th century CE, the [[Middle East]] and [[India]].<ref name="Treese-2018" />{{Rp|page=404}}

Archimedes also devised a way to calculate the volume of an irregular object, by submerging it underwater and measure the difference between the initial and final water volume. The water volume difference is the volume of the object.<ref name="Treese-2018" />{{Rp|page=404}} Though highly popularized, Archimedes probably does not submerge the golden crown to find its volume, and thus its density and purity, due to the extreme precision involved.<ref>{{cite web |last=Rorres |first=Chris |title=The Golden Crown |url=http://www.math.nyu.edu/~crorres/Archimedes/Crown/CrownIntro.html |url-status=live |archive-url=https://web.archive.org/web/20090311051318/http://www.math.nyu.edu/~crorres/Archimedes/Crown/CrownIntro.html |archive-date=11 March 2009 |access-date=24 March 2009 |publisher=[[Drexel University]]}}</ref> Instead, he likely have devised a primitive form of a [[Hydrostatic Balance|hydrostatic balance]]. Here, the crown and a chunk of pure gold with a similar weight are put on both ends of a [[weighing scale]] submerged underwater, which will tip accordingly due to the [[Archimedes' principle]].<ref>{{Cite journal |last=Graf |first=E. H. |date=2004 |title=Just what did Archimedes say about buoyancy? |url=https://aapt.scitation.org/doi/10.1119/1.1737965 |journal=The Physics Teacher |volume=42 |issue=5 |pages=296–299 |bibcode=2004PhTea..42..296G |doi=10.1119/1.1737965 |access-date=2022-08-07 |archive-date=2021-04-14 |archive-url=https://web.archive.org/web/20210414102422/https://aapt.scitation.org/doi/10.1119/1.1737965 |url-status=live }}</ref>