William John Macquorn Rankine FRSE FRS (/ˈræŋkɪn/; 5 July 1820 – 24 December 1872) was a Scottish mathematician and physicist. He was a founding contributor, with Rudolf Clausius and William Thomson (Lord Kelvin), to the science of thermodynamics, particularly focusing on its First Law. He developed the Rankine scale, a Fahrenheit-based equivalent to the Celsius-based Kelvin scale of temperature.

Rankine developed a complete theory of the steam engine and indeed of all heat engines. His manuals of engineering science and practice were used for many decades after their publication in the 1850s and 1860s. He published several hundred papers and notes on science and engineering topics, from 1840 onwards, and his interests were extremely varied, including, in his youth, botany, music theory and number theory, and, in his mature years, most major branches of science, mathematics and engineering.

He was also a singer, pianist and cellist as well as a rifleman.

Rankine was born in Edinburgh to Lt David Rankin (sic), a civil engineer from a military background, who later worked on the Edinburgh and Dalkeith Railway (locally known as the Innocent Railway). His mother was Barbara Grahame, of a prominent legal and banking family.

His father moved around Scotland on various projects and the family moved with him. William was initially educated at home, due to his poor health, but he later attended Ayr Academy (1828–29) and then the High School of Glasgow (1830). Around 1830 the family moved to Edinburgh when the father got a post as Manager of the Edinburgh to Dalkeith Railway. The family then lived at 2 Arniston Place.

In 1834 he was sent to the Scottish Naval and Military Academy on Lothian Road in Edinburgh with the mathematician George Lee. By that year William was already highly proficient in mathematics and received, as a gift from his uncle, Isaac Newton's Principia (1687) in the original Latin.

In 1836, Rankine began to study a spectrum of scientific topics at the University of Edinburgh, including natural history under Robert Jameson and natural philosophy under James David Forbes. Under Forbes he was awarded prizes for essays on methods of physical inquiry and on the undulatory (or wave) theory of light. During vacations, he assisted his father who, from 1830, was manager and, later, effective treasurer and engineer of the Edinburgh and Dalkeith Railway which brought coal into the growing city. He left the University of Edinburgh in 1838 without a degree (which was not then unusual) and, perhaps because of straitened family finances, became an apprentice to Sir John Benjamin Macneill, who was at the time surveyor to the Irish Railway Commission. During his pupilage he developed a technique, later known as Rankine's method, for laying out railway curves, fully exploiting the theodolite and making a substantial improvement in accuracy and productivity over existing methods. In fact, the technique was simultaneously in use by other engineers – and in the 1860s there was a minor dispute about Rankine's priority.

The year 1842 also marked Rankine's first attempt to reduce the phenomena of heat to a mathematical form but he was frustrated by his lack of experimental data. At the time of Queen Victoria's visit to Scotland, later that year, he organised a large bonfire situated on Arthur's Seat, constructed with radiating air passages under the fuel. The bonfire served as a beacon to initiate a chain of other bonfires across Scotland.