What is the reasoning behind using "moment" in the "moment of inertia"?

Linear inertia is called mass. Rotational inertia is called moment of inertia.

Moment of inertia is an odd choice for the term for this property. It doesn't seem to "fit" with the style or pattern of a simple term like mass, although it is the rotational equivalent to mass. Other rotational properties keep a pattern,

• like angular velocity,
• angular acceleration,
• angular momentum etc.;
• but there is no "angular mass". It is instead out-of-the-blue called moment of inertia.
• Sure, there is also no "angular force", but at least that is instead given a simple, clean term: torque.

Moment of inertia really seems like the odd one in the group, breaking any chance of a pattern.

Apparently, Euler introduced this term in his work "Theoria motus corporum solidorum seu rigidorum" from 1765. From excerpts I can find from that book (unfortunately I cannot read the Latin of the book itself), I cannot find any reasoning behind using that particular term.

The word moment seems to originate from the Latin word momentum, meaning movement/change/alteration, and thus it can make sense to not purely use the word as "a brief duration" but also in relation to physical motion. But that does not explain much about why moment of inertia suddenly is the choice for the property.

What is the reasoning for using the term moment of inertia as rotational inertia?

• Moment, "momentum", "movimentum", "movere ( verb) " , movement". So " movement of inertia". Does this make any sense?
– user11223
Mar 19, 2020 at 0:02
• I have given a detailed stack discussion of the reasoning here in a physics context with comments on the history if useful. Dec 20, 2020 at 4:19

If Euler introduced the term and did not explain his reasoning we can only speculate as to what he had in mind. Euler himself was followed on many notational and terminological choices simply because he put them together in well structured and comprehensive books.

But Euler likely followed the precedent with the "moment of force". According to Worthington's Dynamics of Rotation (1900), it came from a curious metamorphosis of the meaning of the word "moment" (originally, short duration), and the older meaning was still live at least at the time of Worthington's writing. According to EtymOnline, "moment" is used in old French as "importance" since the 12th century (in English since 1520s, think of "momentous"), such use in Latin must have come even earlier.

In On the Equilibrium of Planes Archimedes referred to the joint "importance" of force and the place where it was applied to maintaining the equilibrium. Commandino's Latin translation of Archimedes in Liber De Centro Gravitatis Solidorum (1565) was:"The center of gravity of each solid figure is that point within it, about which on all sides parts of equal moment stand". Now, one can read "moment" here as "importance", but one can also read it as referring to some specific quantity.

And that is, apparently, how it came to be read. "Moment" in physics now generally refers to quantities that are something times distance, and sums or integrals thereof. Higher moments involve the powers of distance. "Moment" in mathematics has a similar meaning, only applied more abstractly. Here is Worthington:

"The word moment was first used in Mechanics in its now rather old-fashioned sense of 'importance' or 'consequence' and the moment of a force about an axis meant the importance of the force with respect to its power to generate in matter rotation about the axis; and again, the moment of inertia of a body with respect to an axis is a phrase invented to express the importance of the inertia of the body when we endeavour to turn it about the axis. When we say that the moment of a force about an axis varies as the force, and as the distance of its line of action from the axis, we are not so much defining the phrase 'moment of a force' as expressing the result of experiments made with a view to ascertaining the circumstances under which forces are equivalent to each other as regards their turning power. It is important that the student should bear in mind this original meaning of the word, so that such phrases as 'moment of a force' and 'moment of inertia' may at once call up an idea instead of merely a quantity.

But the word 'moment' has also come to be used by analogy in a purely technical sense, in such expressions as the 'moment of a mass about an axis' or 'the moment of an area with respect to a plane', which require definition in each case. In these instances there is not always any corresponding physical idea, and such phrases stand, both historically and scientifically, on a different footing."

• "If Euler introduced the term and did not explain his reasoning we can only speculate as to what he had in mind" Euler does seem to explain it/give his reasoning in ch.3, article 363. I have given a detailed stack discussion here. Dec 20, 2020 at 4:19

For one thing, "inertial mass" could have turned out to be different from "gravitational mass," and observing that they are the same was a big "win" for physics theory.

For another, quoting wikipedia,

Inertia is one of the primary manifestations of mass, which is a quantitative property of physical systems

So inertia , linear or otherwise, is not the same as mass.

• Thank you for your answer. I am aware of this fact, and this does not really have much to do with the odd choice of naming the terms. That inertia and gravitational mass might have been different things would apply to both the idea of mass and moment of inertia. Feb 6, 2020 at 22:18