Editing Mass (section)

== Pre-Newtonian concepts ==
=== Weight as an amount ===
{{main|Weight}}
[[File:El pesado del corazón en el Papiro de Hunefer.jpg|right|thumb|Depiction of early [[balance scales]] in the [[Papyrus of Hunefer]] (dated to the [[Nineteenth dynasty of Egypt|19th dynasty]], {{circa|1285 BCE}}). The scene shows [[Anubis]] weighing the heart of Hunefer.]]

The concept of [[wikt:amount|amount]] is very old and [[Prehistoric numerals|predates recorded history]]. The concept of "weight" would incorporate "amount" and acquire a double meaning that was not clearly recognized as such.<ref name=":0" />

{{Blockquote|text=What we now know as mass was until the time of Newton called “weight.” ... A goldsmith believed that an ounce of gold was a quantity of gold. ... But the ancients believed that a beam balance also measured “heaviness” which they recognized through their muscular senses. ... Mass and its associated downward force were believed to be the same thing.|author=K. M. Browne|title=The pre-Newtonian meaning of the word “weight”}}

Humans, at some early era, realized that the weight of a collection of similar objects was [[Proportionality (mathematics)|directly proportional]] to the number of objects in the collection:
: <math>W_n \propto n,</math>
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]]:
: <math>\frac{W_n}{n} = \frac{W_m}{m}</math>, or equivalently <math>\frac{W_n}{W_m} = \frac{n}{m}.</math>

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.

Consequently, historical weight standards were often defined in terms of amounts.  The Romans, for example, used the [[carob]] seed ([[Carat (unit)|carat]] or [[siliqua]]) as a measurement standard.  If an object's weight was equivalent to [http://std.dkuug.dk/JTC1/SC2/WG2/docs/n3138.pdf 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 [[Ancient Roman units of measurement|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:
: <math>\frac{\mathrm{ounce}}{\mathrm{pound}} = \frac{W_{144}}{W_{1728}} = \frac{144}{1728} = \frac{1}{12}.</math>

=== Planetary motion ===
{{see also|Kepler's laws of planetary motion}}
In 1600 AD, [[Johannes Kepler]] sought employment with [[Tycho Brahe]], who had some of the most precise astronomical data available.  Using Brahe's precise observations of the planet Mars, Kepler spent the next five years developing his own method for characterizing planetary motion. In 1609, Johannes Kepler published his three laws of planetary motion, explaining how the planets orbit the Sun. In Kepler's final planetary model, he described planetary orbits as following elliptical paths with the Sun at a focal point of the [[ellipse]]. Kepler discovered that the [[square (algebra)|square]] of the [[orbital period]] of each planet is directly [[Proportionality (mathematics)|proportional]] to the [[cube (arithmetic)|cube]] of the [[semi-major axis]] of its orbit, or equivalently, that the [[ratio]] of these two values is constant for all planets in the [[Solar System]].<ref group=note>This constant ratio was later shown to be a direct measure of the Sun's active gravitational mass; it has units of distance cubed per time squared, and is known as the [[standard gravitational parameter]]:
: <math qid=Q140028>\mu=4\pi^2\frac{\text{distance}^3}{\text{time}^2}\propto\text{gravitational mass}</math></ref>

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.

=== Galilean free fall ===
[[File:Galileo.arp.300pix.jpg|left|upright|thumb|Galileo Galilei (1636)]]
[[File:Falling ball.jpg|thumb|right|upright|Distance traveled by a freely falling ball is proportional to the square of the elapsed time.]]
Sometime prior to 1638, Galileo turned his attention to the phenomenon of objects in free fall, attempting to characterize these motions. Galileo was not the first to investigate Earth's gravitational field, nor was he the first to accurately describe its fundamental characteristics.  However, Galileo's reliance on scientific experimentation to establish physical principles would have a profound effect on future generations of scientists. It is unclear if these were just hypothetical experiments used to illustrate a concept, or if they were real experiments performed by Galileo,<ref>{{cite journal |last=Drake |first=S. |date=1979 |title=Galileo's Discovery of the Law of Free Fall |journal=Scientific American |volume=228 |issue=5 |pages=84–92 |bibcode=1973SciAm.228e..84D |doi=10.1038/scientificamerican0573-84}}</ref> but the results obtained from these experiments were both realistic and compelling.  A biography by Galileo's pupil [[Vincenzo Viviani]] stated that Galileo had dropped [[ball]]s of the same material, but different masses, from the [[Leaning Tower of Pisa]] to demonstrate that their time of descent was independent of their mass.<ref group="note">At the time when Viviani asserts that the experiment took place, Galileo had not yet formulated the final version of his law of free fall. He had, however, formulated an earlier version that predicted that bodies ''of the same material'' falling through the same medium would fall at the same speed. See {{cite book |last=Drake |first=S. |date=1978 |title=Galileo at Work |pages=[https://archive.org/details/galileoatwork00stil/page/19 19–20] |publisher=University of Chicago Press |isbn=978-0-226-16226-3 |url=https://archive.org/details/galileoatwork00stil/page/19 }}</ref> In support of this conclusion, Galileo had advanced the following theoretical argument: He asked if two bodies of different masses and different rates of fall are tied by a string, does the combined system fall faster because it is now more massive, or does the lighter body in its slower fall hold back the heavier body?  The only convincing resolution to this question is that all bodies must fall at the same rate.<ref>{{cite book |last=Galileo |first=G. |date=1632 |title=Dialogue Concerning the Two Chief World Systems|title-link=Dialogue Concerning the Two Chief World Systems }}</ref>

A later experiment was described in Galileo's ''Two New Sciences'' published in 1638.  One of Galileo's fictional characters, Salviati, describes an experiment using a bronze ball and a wooden ramp.  The wooden ramp was "12 cubits long, half a cubit wide and three finger-breadths thick" with a straight, smooth, polished [[Groove (engineering)|groove]].  The groove was lined with "[[parchment]], also smooth and polished as possible".  And into this groove was placed "a hard, smooth and very round bronze ball".  The ramp was inclined at various [[angle]]s to slow the acceleration enough so that the elapsed time could be measured.  The ball was allowed to roll a known distance down the ramp, and the time taken for the ball to move the known distance was measured.  The time was measured using a water clock described as follows:
:a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length; the water thus collected was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results.<ref>{{cite book |last1=Galileo |first1=G. |date=1638 |title=Discorsi e Dimostrazioni Matematiche, Intorno à Due Nuove Scienze |volume=213 |publisher=[[House of Elzevir|Louis Elsevier]]}}, translated in {{cite book |date=1954 |editor1-last=Crew |editor1-first=H. |editor2-last=de Salvio |editor2-first=A. |title=Mathematical Discourses and Demonstrations, Relating to Two New Sciences |url=http://oll.libertyfund.org/index.php?option=com_staticxt&staticfile=show.php%3Ftitle=753&Itemid=99999999 |publisher=[[Dover Publications]] |isbn=978-1-275-10057-2 |access-date=11 April 2012 |archive-date=1 October 2013 |archive-url=https://web.archive.org/web/20131001015122/http://oll.libertyfund.org/index.php?option=com_staticxt&staticfile=show.php%3Ftitle=753&Itemid=99999999 |url-status=dead }} and also available in {{cite book |editor1-last=Hawking |editor1-first=S. |date=2002 |title=On the Shoulders of Giants |pages=[https://archive.org/details/isbn_9780762413485/page/534 534–535] |publisher=[[Running Press]] |isbn=978-0-7624-1348-5 |url=https://archive.org/details/isbn_9780762413485/page/534 }}</ref>

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:
: <math>{\text{Distance}} \propto {\text{Time}^2}</math>

Galileo had shown that objects in free fall under the influence of the Earth's gravitational field have a constant acceleration, and Galileo's contemporary, Johannes Kepler, had shown that the planets follow elliptical paths under the influence of the Sun's gravitational mass.  However, Galileo's free fall motions and Kepler's planetary motions remained distinct during Galileo's lifetime.

=== Mass as distinct from weight ===
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."<ref name=":0">{{Cite journal |last=Browne |first=K. M. |year=2018 |title=The pre-Newtonian meaning of the word "weight"; a comment on "Kepler and the origins of pre-Newtonian mass" [Am. J. Phys. 85, 115–123 (2017)] |journal=American Journal of Physics |volume=86 |issue=6 |pages=471–74|doi=10.1119/1.5027490 |bibcode=2018AmJPh..86..471B |s2cid=125953814 |doi-access=free }}</ref> 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.<ref>{{Cite book |last=Newton |first=I. |url=https://books.google.com/books?id=Tm0FAAAAQAAJ&pg=PP13 |title=The mathematical principles of natural philosophy |publisher=Printed for Benjamin Motte |year=1729 |pages=1–2 |translator-last=Motte |translator-first=A. |orig-date=1686}}</ref>{{Blockquote|text=The quantity of matter is the measure of the same, arising from its density and bulk conjunctly. ... It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body; for it is proportional to the weight.|author=Isaac Newton|title=Mathematical principles of natural philosophy|source=Definition I.}}