Consequently, historical weight standards were often defined in terms of amounts. The Romans, for example, used the carob seed ( carat or siliqua) as a measurement standard. If an object's weight was equivalent to 1728 carob seeds, then the object was said to weigh one Roman pound. If, on the other hand, the object's weight was equivalent to 144 carob seeds then the object was said to weigh one Roman ounce (uncia). The Roman pound and ounce were both defined in terms of different sized collections of the same common mass standard, the carob seed. The ratio of a Roman ounce (144 carob seeds) to a Roman pound (1728 carob seeds) was: where W is the weight of the collection of similar objects and n is the number of objects in the collection. Proportionality, by definition, implies that two values have a constant ratio: Galilean free fall Galileo Galilei (1636) Distance traveled by a freely falling ball is proportional to the square of the elapsed time.

For other situations, such as when objects are subjected to mechanical accelerations from forces other than the resistance of a planetary surface, the weight force is proportional to the mass of an object multiplied by the total acceleration away from free fall, which is called the proper acceleration. Through such mechanisms, objects in elevators, vehicles, centrifuges, and the like, may experience weight forces many times those caused by resistance to the effects of gravity on objects, resulting from planetary surfaces. In such cases, the generalized equation for weight W of an object is related to its mass m by the equation W = – ma, where a is the proper acceleration of the object caused by all influences other than gravity. (Again, if gravity is the only influence, such as occurs when an object falls freely, its weight will be zero).

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According to K. M. Browne: "Kepler formed a [distinct] concept of mass ('amount of matter' ( copia materiae)), but called it 'weight' as did everyone at that time." [9] Finally, in 1686, Newton gave this distinct concept its own name. In the first paragraph of Principia, Newton defined quantity of matter as “density and bulk conjunctly”, and mass as quantity of matter. [13] The International System of Units (SI) unit of mass is the kilogram (kg). The kilogram is 1000grams (g), and was first defined in 1795 as the mass of one cubic decimetre of water at the melting point of ice. However, because precise measurement of a cubic decimetre of water at the specified temperature and pressure was difficult, in 1889 the kilogram was redefined as the mass of a metal object, and thus became independent of the metre and the properties of water, this being a copper prototype of the grave in 1793, the platinum Kilogramme des Archives in 1799, and the platinum-iridium International Prototype of the Kilogram (IPK) in 1889.

Albert Einstein developed his general theory of relativity starting with the assumption that the inertial and passive gravitational masses are the same. This is known as the equivalence principle. Active gravitational mass determines the strength of the gravitational field generated by an object. Galileo found that for an object in free fall, the distance that the object has fallen is always proportional to the square of the elapsed time: Inertial mass measures an object's resistance to being accelerated by a force (represented by the relationship F = ma). An early use of this relationship is a balance scale, which balances the force of one object's weight against the force of another object's weight. The two sides of a balance scale are close enough that the objects experience similar gravitational fields. Hence, if they have similar masses then their weights will also be similar. This allows the scale, by comparing weights, to also compare masses.

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On 25 August 1609, Galileo Galilei demonstrated his first telescope to a group of Venetian merchants, and in early January 1610, Galileo observed four dim objects near Jupiter, which he mistook for stars. However, after a few days of observation, Galileo realized that these "stars" were in fact orbiting Jupiter. These four objects (later named the Galilean moons in honor of their discoverer) were the first celestial bodies observed to orbit something other than the Earth or Sun. Galileo continued to observe these moons over the next eighteen months, and by the middle of 1611, he had obtained remarkably accurate estimates for their periods. The universality of free-fall only applies to systems in which gravity is the only acting force. All other forces, especially friction and air resistance, must be absent or at least negligible. For example, if a hammer and a feather are dropped from the same height through the air on Earth, the feather will take much longer to reach the ground; the feather is not really in free-fall because the force of air resistance upwards against the feather is comparable to the downward force of gravity. On the other hand, if the experiment is performed in a vacuum, in which there is no air resistance, the hammer and the feather should hit the ground at exactly the same time (assuming the acceleration of both objects towards each other, and of the ground towards both objects, for its own part, is negligible). This can easily be done in a high school laboratory by dropping the objects in transparent tubes that have the air removed with a vacuum pump. It is even more dramatic when done in an environment that naturally has a vacuum, as David Scott did on the surface of the Moon during Apollo 15. the electronvolt (eV), a unit of energy, used to express mass in units of eV/ c 2 through mass–energy equivalence

Isaac Newton, Mathematical principles of natural philosophy, Definition I. Newtonian mass Earth's Moon In everyday usage, mass and " weight" are often used interchangeably. For instance, a person's weight may be stated as 75kg. In a constant gravitational field, the weight of an object is proportional to its mass, and it is unproblematic to use the same unit for both concepts. But because of slight differences in the strength of the Earth's gravitational field at different places, the distinction becomes important for measurements with a precision better than a few percent, and for places far from the surface of the Earth, such as in space or on other planets. Conceptually, "mass" (measured in kilograms) refers to an intrinsic property of an object, whereas "weight" (measured in newtons) measures an object's resistance to deviating from its current course of free fall, which can be influenced by the nearby gravitational field. No matter how strong the gravitational field, objects in free fall are weightless, though they still have mass. [6] Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a body, until the discovery of the atom and particle physics. It was found that different atoms and different elementary particles, theoretically with the same amount of matter, have nonetheless different masses. Mass in modern physics has multiple definitions which are conceptually distinct, but physically equivalent. Mass can be experimentally defined as a measure of the body's inertia, meaning the resistance to acceleration (change of velocity) when a net force is applied. [1] The object's mass also determines the strength of its gravitational attraction to other bodies.

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There are several distinct phenomena that can be used to measure mass. Although some theorists have speculated that some of these phenomena could be independent of each other, [2] current experiments have found no difference in results regardless of how it is measured: The particular equivalence often referred to as the "Galilean equivalence principle" or the " weak equivalence principle" has the most important consequence for freely falling objects. Suppose an object has inertial and gravitational masses m and M, respectively. If the only force acting on the object comes from a gravitational field g, the force on the object is: W n n = W m m {\displaystyle {\frac {W_{n}}{n}}={\frac {W_{m}}{m}}} , or equivalently W n W m = n m . {\displaystyle {\frac {W_{n}}{W_{m}}}={\frac {n}{m}}.} The first experiments demonstrating the universality of free-fall were—according to scientific 'folklore'—conducted by Galileo obtained by dropping objects from the Leaning Tower of Pisa. This is most likely apocryphal: he is more likely to have performed his experiments with balls rolling down nearly frictionless inclined planes to slow the motion and increase the timing accuracy. Increasingly precise experiments have been performed, such as those performed by Loránd Eötvös, [7] using the torsion balance pendulum, in 1889. As of 2008 [update], no deviation from universality, and thus from Galilean equivalence, has ever been found, at least to the precision 10 −6. More precise experimental efforts are still being carried out. [8] Astronaut David Scott performs the feather and hammer drop experiment on the Moon.