# What was the first physical theory to be mathematically expressed?

Newton's theory of gravitation is expressed as a mathematical relationship between distances and masses. It can be deduced from Kepler's equations (which are also mathematical).

Are there even older physical theories formulated by equations?

• I would say Archimedes' Law of the Lever is a good candidate, but I don't know to what extent it was expressed mathematically.
– d_e
Commented Apr 3, 2022 at 11:08
• Perhaps not the oldest, but interesting: Ossendrijver, doi.org/10.1126/science.aad8085 (pdf), "Ancient Babylonian astronomers calculated Jupiter’s position from the area under a time-velocity graph." ~350-50 BCE. Commented Apr 6, 2022 at 18:23

The first physical theory which was mathematically expressed is the Pythagorean theory of music. According to the legend, Pythagoras himself discovered the relation between musical intervals and the ratios of lengths of strings. (Based on this discovery, Pythagoras jumped to the conclusion that "Numbers rule the world", so mathematical physics was born). This seems to be the earliest quantitative discovery of physics. Based on this, a sophisticated theory of music was developed, one can read about it for example in the book:

B. L. van der Waerden, Science awakening. Mathemarics of ancient Egypt, Babylon and Greece, Appendix: Pythagorean teaching on harmony. (Was earlier published as a separate paper in Math. Ann., 118 (1941)).

Remark. This competes with Babylonian mathematical astronomy, but one can argue whether Babylonian astronomy belongs to physics or not. Anyway, this discovery is more ancient than Greek astronomy.

Remark 2. Since Pythagoreans was a secret society, the exact details of this discovery and its date are not known. But surely this predates the first well-documented discoveries in physics which are due to Archimedes (floating bodies, the rues of the lever etc.)

• Thank you for your reply Now that you mention Pythagoras, I wonder if the Pythagorean theorem (which was known long before him) cannot be considered as a result of physics.
– user16484
Commented Apr 3, 2022 at 13:12
• I don't think so. But we really do not know much about Pythagoreans (not speaking of Pythagoras himself), only secondary sources written hundreds years later. Commented Apr 3, 2022 at 16:54
• The earliest primary sources which reached us are Euclid on mathematics and Aristotle on physics. Commented Apr 3, 2022 at 16:56
• Babylonians, Jews, Indians and Chinese had very sophisticated mathematical theories related to astronomy. Commented Apr 4, 2022 at 9:03
• @markvs: 1. As I said, it is disputable whether ancient astronomy was a part of physics. 2. About Babylonians, I mentioned. About Chinese and Indians, I strongly doubt that they had any sophisticated theories of astronomy as early as 6 bc (the time of Pythagoras). Commented Apr 4, 2022 at 13:25

The Almagest, written in the second century contains a mathematical description of the Universe (and, in particular, of the solar system), which is expressed mathematically. Its contents were almost certainly already known to (and copied from) Hipparchus of Nicaea, who lived in he second century BC. In fact, the kind of research that led to this began already at the time of Plato (fifth century BC).

• It worth noting, however, that this ingenious description of the solar system was not an attempt the express any specific physical theory, rather its aim was "to save the phenomena"
– d_e
Commented Apr 3, 2022 at 11:03
• One can argue whether Ptolemy's astronomy belongs to physics. Ptolemy himself did not think so. Commented Apr 3, 2022 at 12:46