The earliest evidence of volume calculation emerged in ancient Egypt and Mesopotamia as mathematical problems, approximating the volume of simple shapes such as cuboids, cylinders, frustums, and cones. These mathematical problems were written in the Moscow Mathematical Papyrus around 1820 BCE, where ancient Egyptians devised units of volume based on their units of length. They used the cubit, palm, and digit to create the volume cubit, volume palm, and volume digit, which were calculated as 1 cubit times 1 cubit times 1 cubit, 1 cubit times 1 cubit times 1 palm, and 1 cubit times 1 cubit times 1 digit respectively. The precision of volume measurements in the ancient period usually ranged between 1 and 2 percent, a remarkable feat for a time when standardized containers were not yet available. In ancient times, volume was measured using similar-shaped natural containers, and later on, standardized containers were used to ensure consistency in trade and construction. The Egyptians also wrote concrete units of volume for grain and liquids in the Reisner Papyrus, along with a table of length, width, depth, and volume for blocks of material, demonstrating a sophisticated understanding of three-dimensional space.
Archimedes Principle
Archimedes devised a way to calculate the volume of an irregular object by submerging it underwater and measuring the difference between the initial and final water volume, a method that is now known as Archimedes' principle. Although this story is highly popularized, Archimedes probably did not submerge the golden crown to find its volume and thus its density and purity due to the extreme precision involved. Instead, he likely devised a primitive form of a hydrostatic balance, where the crown and a chunk of pure gold with a similar weight were put on both ends of a weighing scale submerged underwater, which would tip accordingly due to the principle. The last three books of Euclid's Elements, written in around 300 BCE, detailed the exact formulas for calculating the volume of parallelepipeds, cones, pyramids, cylinders, and spheres. The formulas were determined by prior mathematicians by using a primitive form of integration, by breaking the shapes into smaller and simpler pieces. A century later, Archimedes devised approximate volume formulas 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, showing a global effort to understand three-dimensional space.Standardizing the Gallon
In the Middle Ages, many units for measuring volume were made, such as the sester, amber, coomb, and seam. The sheer quantity of such units motivated British kings to standardize them, culminated in the Assize of Bread and Ale statute in 1258 by Henry III of England. The statute standardized weight, length and volume as well as introduced the peny, ounce, pound, gallon and bushel. In 1618, the London Pharmacopoeia adopted the Roman gallon or congius as a basic unit of volume and gave a conversion table to the apothecaries' units of weight. Around this time, volume measurements are becoming more precise and the uncertainty is narrowed to between 0.1 and 0.2 percent. The metric system was formally defined in French law on the 7th of April 1795 using six units, three of which are related to volume: the stère for volume of firewood, the litre for volumes of liquid, and the gramme for mass, defined as the mass of one cubic centimetre of water at the temperature of melting ice. Thirty years later in 1824, the imperial gallon was defined to be the volume occupied by ten pounds of water at 62 degrees Fahrenheit, and this definition was further refined until the United Kingdom's Weights and Measures Act 1985, which makes 1 imperial gallon precisely equal to 4.54609 litres with no use of water.