Overtone

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https://en.wikipedia.org/wiki/Overtone

Vibrational modes of an ideal string, dividing the string length into integer divisions, producing harmonic partials f, 2f, 3f, 4f, etc. (where f means fundamental frequency).

An overtone is any resonant frequency above the fundamental frequency of a sound. (An overtone may or may not be a harmonic) In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental is the lowest pitch. While the fundamental is usually heard most prominently, overtones are actually present in any pitch except a true sine wave. The relative volume or amplitude of various overtone partials is one of the key identifying features of timbre, or the individual characteristic of a sound.

Using the model of Fourier analysis, the fundamental and the overtones together are called partials. Harmonics, or more precisely, harmonic partials, are partials whose frequencies are numerical integer multiples of the fundamental (including the fundamental, which is 1 times itself). These overlapping terms are variously used when discussing the acoustic behavior of musical instruments. (See etymology below.) The model of Fourier analysis provides for the inclusion of inharmonic partials, which are partials whose frequencies are not whole-number ratios of the fundamental (such as 1.1 or 2.14179).

Duration: 17 seconds.0:17

Main tone (110 Hz) and first 15 overtones (16 harmonic partials) (listen)

Allowed and forbidden standing waves, and thus harmonics

When a resonant system such as a blown pipe or plucked string is excited, a number of overtones may be produced along with the fundamental tone. In simple cases, such as for most musical instruments, the frequencies of these tones are the same as (or close to) the harmonics. Examples of exceptions include the circular drum – a timpani whose first overtone is about 1.6 times its fundamental resonance frequency, gongs and cymbals, and brass instruments. The human vocal tract is able to produce highly variable amplitudes of the overtones, called formants, which define different vowels.

Explanation

Most oscillators, from a plucked guitar string to a flute that is blown, will naturally vibrate at a series of distinct frequencies known as normal modes. The lowest normal mode frequency is known as the fundamental frequency, while the higher frequencies are called overtones. Often, when an oscillator is excited — for example, by plucking a guitar string — it will oscillate at several of its modal frequencies at the same time. So when a note is played, this gives the sensation of hearing other frequencies (overtones) above the lowest frequency (the fundamental).

Timbre is the quality that gives the listener the ability to distinguish between the sound of different instruments. The timbre of an instrument is determined by which overtones it emphasizes. That is to say, the relative volumes of these overtones to each other determines the specific "flavor", "color" or "tone" of sound of that family of instruments. The intensity of each of these overtones is rarely constant for the duration of a note. Over time, different overtones may decay at different rates, causing the relative intensity of each overtone to rise or fall independent of the overall volume of the sound. A carefully trained ear can hear these changes even in a single note. This is why the timbre of a note may be perceived differently when played staccato or legato.

A driven non-linear oscillator, such as the vocal folds, a blown wind instrument, or a bowed violin string (but not a struck guitar string or bell) will oscillate in a periodic, non-sinusoidal manner. This generates the impression of sound at integer multiple frequencies of the fundamental known as harmonics, or more precisely, harmonic partials. For most string instruments and other long and thin instruments such as a bassoon, the first few overtones are quite close to integer multiples of the fundamental frequency, producing an approximation to a harmonic series. Thus, in music, overtones are often called harmonics. Depending upon how the string is plucked or bowed, different overtones can be emphasized.

However, some overtones in some instruments may not be of a close integer multiplication of the fundamental frequency, thus causing a small dissonance. "High quality" instruments are usually built in such a manner that their individual notes do not create disharmonious overtones. In fact, the flared end of a brass instrument is not to make the instrument sound louder, but to correct for tube length “end effects” that would otherwise make the overtones significantly different from integer harmonics. This is illustrated by the following:

Consider a guitar string. Its idealized 1st overtone would be exactly twice its fundamental if its length were shortened by ½, perhaps by lightly pressing a guitar string at the 12th fret; however, if a vibrating string is examined, it will be seen that the string does not vibrate flush to the bridge and nut, but it instead has a small “dead length” of string at each end. This dead length actually varies from string to string, being more pronounced with thicker and/or stiffer strings. This means that halving the physical string length does not halve the actual string vibration length, and, hence, the overtones will not be exact multiples of a fundamental frequency. The effect is so pronounced that properly set up guitars will angle the bridge such that the thinner strings will progressively have a length up to few millimeters shorter than the thicker strings. Not doing so would result in inharmonious chords made up of two or more strings. Similar considerations apply to tube instruments.

Musical usage term

Physical representation of third (O₃) and fifth (O₅) overtones of a cylindrical pipe closed at one end. F is the fundamental frequency; the third overtone is the third harmonic, 3F, and the fifth overtone is the fifth harmonic, 5F for such a pipe, which is a good model for a pan flute.

An overtone is a partial (a "partial wave" or "constituent frequency") that can be either a harmonic partial (a harmonic) other than the fundamental, or an inharmonic partial. A harmonic frequency is an integer multiple of the fundamental frequency. An inharmonic frequency is a non-integer multiple of a fundamental frequency.

An example of harmonic overtones: (absolute harmony)

Frequency	Order	Name 1	Name 2	Name 3
1 · f =   440 Hz	n = 1	fundamental tone	1st harmonic	1st partial
2 · f =   880 Hz	n = 2	2nd overtone	2nd harmonic	2nd partial
3 · f = 1320 Hz	n = 3	3rd overtone	3rd harmonic	3rd partial
4 · f = 1760 Hz	n = 4	4th overtone	4th harmonic	4th partial

Some musical instruments produce overtones that are slightly sharper or flatter than true harmonics. The sharpness or flatness of their overtones is one of the elements that contributes to their sound. Due to phase inconsistencies between the fundamental and the partial harmonic, this also has the effect of making their waveforms not perfectly periodic.

Musical instruments that can create notes of any desired duration and definite pitch have harmonic partials. A tuning fork, provided it is sounded with a mallet (or equivalent) that is reasonably soft, has a tone that consists very nearly of the fundamental, alone; it has a sinusoidal waveform. Nevertheless, music consisting of pure sinusoids was found to be unsatisfactory in the early 20th century.

Etymology

In Hermann von Helmholtz's classic "On The Sensations Of Tone" he used the German "Obertöne" which was a contraction of "Oberpartialtöne", or in English: "upper partial tones". According to Alexander Ellis (in pages 24–25 of his English translation of Helmholtz), the similarity of German "ober" to English "over" caused a Prof. Tyndall to mistranslate Helmholtz' term, thus creating "overtone". Ellis disparages the term "overtone" for its awkward implications. Because "overtone" makes the upper partials seem like such a distinct phenomena, it leads to the mathematical problem where the first overtone is the second partial. Also, unlike discussion of "partials", the word "overtone" has connotations that have led people to wonder about the presence of "undertones" (a term sometimes confused with "difference tones" but also used in speculation about a hypothetical "undertone series").

"Overtones" in choral music

In barbershop music, a style of four-part singing, the word overtone is often used in a related but particular manner. It refers to a psychoacoustic effect in which a listener hears an audible pitch that is higher than, and different from, the fundamentals of the four pitches being sung by the quartet. The barbershop singer's "overtone" is created by the interactions of the upper partial tones in each singer's note (and by sum and difference frequencies created by nonlinear interactions within the ear). Similar effects can be found in other a cappella polyphonic music such as the music of the Republic of Georgia and the Sardinian cantu a tenore. Overtones are naturally highlighted when singing in a particularly resonant space, such as a church; one theory of the development of polyphony in Europe holds that singers of Gregorian chant, originally monophonic, began to hear the overtones of their monophonic song and to imitate these pitches - with the fifth, octave, and major third being the loudest vocal overtones, it is one explanation of the development of the triad and the idea of consonance in music.

The first step in composing choral music with overtone singing is to discover what the singers can be expected to do successfully without extensive practice. The second step is to find a musical context in which those techniques could be effective, not mere special effects. It was initially hypothesized that beginners would be able to:

• glissando through the partials of a given fundamental, ascending or descending, fast, or slow
• use vowels/text for relative pitch gestures on indeterminate partials specifying the given shape without specifying particular partials
• improvise on partials of the given fundamental, ad lib., freely, or in giving style or manner
• find and sustain a particular partial (requires interval recognition)
• by extension, move to an adjacent partial, above or below, and alternate between the two

Singers should not be asked to change the fundamental pitch while overtone singing and changing partials should always be to an adjacent partial. When a particular partial is to be specified, time should be allowed (a beat or so) for the singers to get the harmonics to "speak" and find the correct one.

String instruments

Main article: String harmonic

Playing a harmonic on a string. Here, "+7" indicates that the string is held down at the position for raising the pitch by 7 half notes, that is, at the seventh fret for a fretted instrument.

String instruments can also produce multiphonic tones when strings are divided in two pieces or the sound is somehow distorted. The sitar has sympathetic strings which help to bring out the overtones while one is playing. The overtones are also highly important in the tanpura, the drone instrument in traditional North and South Indian music, in which loose strings tuned at octaves and fifths are plucked and designed to buzz to create sympathetic resonance and highlight the cascading sound of the overtones.

Western string instruments, such as the violin, may be played close to the bridge (a technique called "sul ponticello" or "am Steg") which causes the note to split into overtones while attaining a distinctive glassy, metallic sound. Various techniques of bow pressure may also be used to bring out the overtones, as well as using string nodes to produce natural harmonics. On violin family instruments, overtones can be played with the bow or by plucking. Scores and parts for Western violin family instruments indicate where the performer is to play harmonics. The most well-known technique on a guitar is playing flageolet tones or using distortion effects. The ancient Chinese instrument the guqin contains a scale based on the knotted positions of overtones. The Vietnamese đàn bầu functions on flageolet tones. Other multiphonic extended techniques used are prepared piano, prepared guitar and 3rd bridge.

Wind instruments

Wind instruments manipulate the overtone series significantly in the normal production of sound, but various playing techniques may be used to produce multiphonics which bring out the overtones of the instrument. On many woodwind instruments, alternate fingerings are used. "Overblowing", or adding intensely exaggerated air pressure, can also cause notes to split into their overtones. In brass instruments, multiphonics may be produced by singing into the instrument while playing a note at the same time, causing the two pitches to interact - if the sung pitch is at specific harmonic intervals with the played pitch, the two sounds will blend and produce additional notes by the phenomenon of sum and difference tones.

Non-western wind instruments also exploit overtones in playing, and some may highlight the overtone sound exceptionally. Instruments like the didgeridoo are highly dependent on the interaction and manipulation of overtones achieved by the performer changing their mouth shape while playing, or singing and playing simultaneously. Likewise, when playing a harmonica or pitch pipe, one may alter the shape of their mouth to amplify specific overtones. Though not a wind instrument, a similar technique is used for playing the jaw harp: the performer amplifies the instrument's overtones by changing the shape, and therefore the resonance, of their vocal tract.

Brass Instruments

Brass instruments originally had no valves, and could only play the notes in the natural overtone, or harmonic series.

Brass instruments still rely heavily on the overtone series to produce notes: the tuba typically has 3-4 valves, the tenor trombone has 7 slide positions, the trumpet has 3 valves, and the French horn typically has 4 valves. Each instrument can play (within their respective ranges) the notes of the overtone series in different keys with each fingering combination (open, 1, 2, 12, 123, etc). The role of each valve or rotor (excluding trombone) is as follows: 1st valve lowers major 2nd, 2nd valve lowers minor 2nd, 3rd valve-lowers minor 3rd, 4th valve-lowers perfect 4th (found on piccolo trumpet, certain euphoniums, and many tubas). The French horn has a trigger key that opens other tubing and is pitched a perfect fourth higher; this allows for greater ease between different registers of the instrument. Valves allow brass instruments to play chromatic notes, as well as notes within the overtone series (open valve = C overtone series, 2nd valve = B overtone series on the C Trumpet) by changing air speed and lip vibrations.

The tuba, trombone, and trumpet play notes within the first few octaves of the overtone series, where the partials are farther apart. The French horn sounds notes in a higher octave of the overtone series, so the partials are closer together and make it more difficult to play the correct pitches and partials.

Overtone singing

Overtone singing is a traditional form of singing in many parts of the Himalayas and Altay; Tibetans, Mongols and Tuvans are known for their overtone singing. In these contexts it is often referred to as throat singing or khoomei, though it should not be confused with Inuit throat singing, which is produced by different means. There is also the possibility to create the overtone out of fundamental tones without any stress on the throat.

Also, the overtone is very important in singing to take care of vocal tract shaping, to improve color, resonance, and text declamation. During practice overtone singing, it helps the singer to remove unnecessary pressure on the muscle, especially around the throat. So if one can "find" a single overtone, then one will know where the sensation needs to be in order to bring out vocal resonance in general, helping to find the resonance in one's own voice on any vowel and in any register.

Overtones in music composition

The primacy of the triad in Western harmony comes from the first four partials of the overtone series. The eighth through fourteenth partials resemble the equal tempered acoustic scale.

When this scale is rendered as a chord, it is called the lydian dominant thirteenth chord. This chord appears throughout Western music, but is notably used as the basis of jazz harmony, features prominently in the music of Franz Liszt, Claude Debussy, Maurice Ravel, and appears as the Mystic chord in the music of Alexander Scriabin.

Because the overtone series rises infinitely from the fundamental with no periodicity, in Western music the equal temperament scale was designed to create synchronicity between different octaves. This was achieved by de-tuning certain intervals, such as the perfect fifth. A true perfect fifth is 702 cents above the fundamental, but equal temperament flattens it by two cents. The difference is only barely perceptible, and allows both for the illusion of the scale being in-tune with itself across multiple octaves, and for tonalities based on all 12 chromatic notes to sound in-tune.

Western classical composers have also made use of the overtone series through orchestration. In his treatise "Principles of Orchestration," Russian composer Nikolai Rimsky-Korsakov says the overtone series "may serve as a guide to the orchestral arrangement of chords". Rimsky-Korsakov then demonstrates how to voice a C major triad according to the overtone series, using partials 1, 2, 3, 4, 5, 6, 8, 10, 12, and 16.

In the 20th century, exposure to non-Western music and further scientific acoustical discoveries led some Western composers to explore alternate tuning systems. Harry Partch for example designed a tuning system that divides the octave into 43 tones, with each tone based on the overtone series. The music of Ben Johnston uses many different tuning systems, including his String Quartet No. 5 which divides the octave into more than 100 tones.

Spectral music is a genre developed by Gérard Grisey and Tristan Murail in the 1970s and 80s, under the auspices of IRCAM. Broadly, spectral music deals with resonance and acoustics as compositional elements. For example, in Grisey's seminal work Partiels, the composer used a sonogram to analyze the true sonic characteristics of the lowest note on a tenor trombone (E2). The analysis revealed which overtones were most prominent from that sound, and Partiels was then composed around the analysis. Another seminal spectral work is Tristan Murail's Gondwana for orchestra. This work begins with a spectral analysis of a bell, and gradually transforms it into the spectral analysis of a brass instrument. Other spectralists and post-spectralists include Jonathan Harvey, Kaija Saariaho, and Georg Friedrich Haas.

John Luther Adams is known for his extensive use of the overtone series, as well as his tendency to allow musicians to make their own groupings and play at their own pace to alter the sonic experience. For example, his piece Sila: The Breath of the World can be played by 16 to 80 musicians and are separated into their own groups. The piece is set on sixteen "harmonic clouds" that are grounded on the first sixteen overtones of low B-flat. Another example is John Luther Adam's piece Everything That Rises, which grew out of his piece Sila: The Breath of the World. Everything That Rises is a piece for string quartet that has sixteen harmonic clouds that are built off of the fundamental tone (C0)

A Modest Proposal

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A Modest Proposal

Author	Jonathan Swift
Genre	Satirical essay
Publication date	1729
Text	A Modest Proposal at Wikisource

A Modest Proposal for Preventing the Children of Poor People from Being a Burthen to Their Parents or Country, and for Making Them Beneficial to the Publick, commonly referred to as A Modest Proposal, is a Juvenalian satirical essay written and published by Anglo-Irish writer and clergyman Jonathan Swift in 1729. The essay suggests that poor people in Ireland could ease their economic troubles by selling their children as food to the elite. Swift's use of satirical hyperbole was intended to mock hostile attitudes towards the poor and anti-Catholicism among the Protestant Ascendancy as well as the Dublin Castle administration's policies in general. In English writing, the phrase "a modest proposal" is now conventionally an allusion to this style of straight-faced satire.

Synopsis

A painting of Jonathan Swift

Swift's essay is widely held to be one of the greatest examples of sustained irony in the history of English literature. Much of its shock value derives from the fact that the first portion of the essay describes the plight of starving beggars in Ireland, so that the reader is unprepared for the surprise of Swift's solution when he states: "A young healthy child well nursed, is, at a year old, a most delicious nourishing and wholesome food, whether stewed, roasted, baked, or boiled; and I make no doubt that it will equally serve in a fricassee, or a ragout."

Swift goes to great lengths to support his argument, including a list of possible preparation styles for the children, and calculations showing the financial benefits of his suggestion. He uses methods of argument throughout his essay which lampoon the then-influential William Petty and the social engineering popular among followers of Francis Bacon. These lampoons include appealing to the authority of "a very knowing American of my acquaintance in London" and "the famous Psalmanazar, a native of the island Formosa" (who had already confessed to not being from Formosa in 1706).

In the tradition of Roman satire, Swift introduces the reforms he is actually suggesting by paralipsis:

Therefore let no man talk to me of other expedients: Of taxing our absentees at five shillings a pound: Of using neither clothes, nor household furniture, except what is of our own growth and manufacture: Of utterly rejecting the materials and instruments that promote foreign luxury: Of curing the expensiveness of pride, vanity, idleness, and gaming in our women: Of introducing a vein of parsimony, prudence and temperance: Of learning to love our country, wherein we differ even from Laplanders, and the inhabitants of Topinamboo: Of quitting our animosities and factions, nor acting any longer like the Jews, who were murdering one another at the very moment their city was taken: Of being a little cautious not to sell our country and consciences for nothing: Of teaching landlords to have at least one degree of mercy towards their tenants. Lastly, of putting a spirit of honesty, industry, and skill into our shop-keepers, who, if a resolution could now be taken to buy only our native goods, would immediately unite to cheat and exact upon us in the price, the measure, and the goodness, nor could ever yet be brought to make one fair proposal of just dealing, though often and earnestly invited to it. Therefore I repeat, let no man talk to me of these and the like expedients, 'till he hath at least some glympse of hope, that there will ever be some hearty and sincere attempt to put them into practice.

Population solutions

George Wittkowsky argued that Swift's main target in A Modest Proposal was not the conditions in Ireland, but rather the can-do spirit of the times that led people to devise a number of illogical schemes that would purportedly solve social and economic ills. Swift was especially attacking projects that tried to fix population and labour issues with a simple cure-all solution. A memorable example of these sorts of schemes "involved the idea of running the poor through a joint-stock company". In response, Swift's Modest Proposal was "a burlesque of projects concerning the poor" that were in vogue during the early 18th century.

Ian McBride argues that the point of A Modest Proposal was to "find a suitably decisive means of dehumanizing the settlers who had failed so comprehensively to meet their social responsibilities." A Modest Proposal also targets the calculating way people perceived the poor in designing their projects. The pamphlet targets reformers who "regard people as commodities". In the piece, Swift adopts the "technique of a political arithmetician" to show the utter ridiculousness of trying to prove any proposal with dispassionate statistics.

Critics differ about Swift's intentions in using this faux-mathematical philosophy. Edmund Wilson argues that statistically "the logic of the 'Modest proposal' can be compared with defence of crime (arrogated to Marx) in which he argues that crime takes care of the superfluous population". Wittkowsky counters that Swift's satiric use of statistical analysis is an effort to enhance his satire that "springs from a spirit of bitter mockery, not from the delight in calculations for their own sake".

Rhetoric

Author Charles K. Smith argues that Swift's rhetorical style persuades the reader to detest the speaker and pity the Irish. Swift's specific strategy is twofold, using a "trap" to create sympathy for the Irish and a dislike of the narrator who, in the span of one sentence, "details vividly and with rhetorical emphasis the grinding poverty" but feels emotion solely for members of his own class. Swift's use of gripping details of poverty and his narrator's cool approach towards them create "two opposing points of view" that "alienate the reader, perhaps unconsciously, from a narrator who can view with 'melancholy' detachment a subject that Swift has directed us, rhetorically, to see in a much less detached way."

Swift has his proposer further degrade the Irish by using language ordinarily reserved for animals. Lewis argues that the speaker uses "the vocabulary of animal husbandry" to describe the Irish. Once the children have been commodified, Swift's rhetoric can easily turn "people into animals, then meat, and from meat, logically, into tonnage worth a price per pound".

Swift uses the proposer's serious tone to highlight the absurdity of his proposal. In making his argument, the speaker uses the conventional, textbook-approved order of argument from Swift's time (which was derived from the Latin rhetorician Quintilian). The contrast between the "careful control against the almost inconceivable perversion of his scheme" and "the ridiculousness of the proposal" create a situation in which the reader has "to consider just what perverted values and assumptions would allow such a diligent, thoughtful, and conventional man to propose so perverse a plan".

Influences

Scholars have speculated about which earlier works Swift may have had in mind when he wrote A Modest Proposal.

Tertullian's Apology

James William Johnson argues that A Modest Proposal was largely influenced and inspired by Tertullian's Apology: a satirical attack against early Roman persecution of Christianity. Johnson believes that Swift saw major similarities between the two situations. Johnson notes Swift's obvious affinity for Tertullian and the bold stylistic and structural similarities between the works A Modest Proposal and Apology. In structure, Johnson points out the same central theme, that of cannibalism and the eating of babies as well as the same final argument, that "human depravity is such that men will attempt to justify their own cruelty by accusing their victims of being lower than human". Stylistically, Swift and Tertullian share the same command of sarcasm and language. In agreement with Johnson, Donald C. Baker points out the similarity between both authors' tones and use of irony. Baker notes the uncanny way that both authors imply an ironic "justification by ownership" over the subject of sacrificing children—Tertullian while attacking pagan parents, and Swift while attacking the mistreatment of the poor in Ireland.

Defoe's The Generous Projector

It has also been argued that A Modest Proposal was, at least in part, a response to the 1728 essay The Generous Projector or, A Friendly Proposal to Prevent Murder and Other Enormous Abuses, By Erecting an Hospital for Foundlings and Bastard Children by Swift's rival Daniel Defoe.

Mandeville's Modest Defence of Publick Stews

Bernard Mandeville's Modest Defence of Publick Stews asked to introduce public and state-controlled bordellos. The 1726 paper acknowledges women's interests and—while not being a completely satirical text—has also been discussed as an inspiration for Jonathan Swift's title. Mandeville had by 1705 already become famous for The Fable of the Bees and deliberations on private vices and public benefits.

John Locke's First Treatise of Government

John Locke commented: "Be it then as Sir Robert says, that Anciently, it was usual for Men to sell and Castrate their Children. Let it be, that they exposed them; Add to it, if you please, for this is still greater Power, that they begat them for their Tables to fat and eat them: If this proves a right to do so, we may, by the same Argument, justifie Adultery, Incest and Sodomy, for there are examples of these too, both Ancient and Modern; Sins, which I suppose, have the Principle Aggravation from this, that they cross the main intention of Nature, which willeth the increase of Mankind, and the continuation of the Species in the highest perfection, and the distinction of Families, with the Security of the Marriage Bed, as necessary thereunto". (First Treatise, sec. 59).

Economic themes

Robert Phiddian's article "Have you eaten yet? The Reader in A Modest Proposal" focuses on two aspects of A Modest Proposal: the voice of Swift and the voice of the Proposer. Phiddian stresses that a reader of the pamphlet must learn to distinguish between the satirical voice of Jonathan Swift and the apparent economic projections of the Proposer. He reminds readers that "there is a gap between the narrator's meaning and the text's, and that a moral-political argument is being carried out by means of parody".

While Swift's proposal is obviously not a serious economic proposal, George Wittkowsky, author of "Swift's Modest Proposal: The Biography of an Early Georgian Pamphlet", argues that to understand the piece fully it is important to understand the economics of Swift's time. Wittowsky argued that an insufficient number of critics have taken the time to focus directly on mercantilism and theories of labour in Georgian era Britain. "If one regards the Modest Proposal simply as a criticism of condition, about all one can say is that conditions were bad and that Swift's irony brilliantly underscored this fact".

"People are the riches of a nation"

At the start of a new industrial age in the 18th century, it was believed that "people are the riches of the nation", and there was a general faith in an economy that paid its workers low wages because high wages meant workers would work less. Furthermore, "in the mercantilist view no child was too young to go into industry". In those times, the "somewhat more humane attitudes of an earlier day had all but disappeared and the laborer had come to be regarded as a commodity".

Louis A. Landa composed a conducive analysis when he noted that it would have been healthier for the Irish economy to more appropriately utilize their human assets by giving the people an opportunity to "become a source of wealth to the nation" or else they "must turn to begging and thievery". This opportunity may have included giving the farmers more coin to work for, diversifying their professions, or even consider enslaving their people to lower coin usage and build up financial stock in Ireland. Landa wrote that, "Swift is maintaining that the maxim—people are the riches of a nation—applies to Ireland only if Ireland is permitted slavery or cannibalism."

Landa presents Swift's A Modest Proposal as a critique of the popular and unjustified maxim of mercantilism in the 18th century that "people are the riches of a nation". Swift presents the dire state of Ireland and shows that mere population itself, in Ireland's case, did not always mean greater wealth and economy. The uncontrolled maxim fails to take into account that a person who does not produce in an economic or political way makes a country poorer, not richer. Swift also recognises the implications of this fact in making mercantilist philosophy a paradox: the wealth of a country is based on the poverty of the majority of its citizens. Landa argued that Swift was putting the onus "on England of vitiating the working of natural economic law in Ireland" by denying Irishmen "the same natural rights common to the rest of mankind."

Public reaction

Allen Bathurst, 1st Earl Bathurst

Swift's essay created a backlash within Georgian society after its publication. The work was aimed at the elite, and they responded in turn. Several prominent members of society wrote to Swift regarding the work. Lord Bathurst's letter (12 February 1729–30) intimated that he certainly understood the message, and interpreted it as a work of comedy:

I did immediately propose it to Lady Bathurst, as your advice, particularly for her last boy, which was born the plumpest, finest thing, that could be seen; but she fell in a passion, and bid me send you word, that she would not follow your direction, but that she would breed him up to be a parson, and he should live upon the fat of the land; or a lawyer, and then, instead of being eat himself, he should devour others. You know women in passion never mind what they say; but, as she is a very reasonable woman, I have almost brought her over now to your opinion; and having convinced her, that as matters stood, we could not possibly maintain all the nine, she does begin to think it reasonable the youngest should raise fortunes for the eldest: and upon that foot a man may perform family duty with more courage and zeal; for, if he should happen to get twins, the selling of one might provide for the other. Or if, by any accident, while his wife lies in with one child, he should get a second upon the body of another woman, he might dispose of the fattest of the two, and that would help to breed up the other. The more I think upon this scheme, the more reasonable it appears to me; and it ought by no means to be confined to Ireland; for, in all probability, we shall, in a very little time, be altogether as poor here as you are there. I believe, indeed, we shall carry it farther, and not confine our luxury only to the eating of children; for I happened to peep the other day into a large assembly [Parliament] not far from Westminster-hall, and I found them roasting a great fat fellow, [ Walpole again ] For my own part, I had not the least inclination to a slice of him; but, if I guessed right, four or five of the company had a devilish mind to be at him. Well, adieu, you begin now to wish I had ended, when I might have done it so conveniently.

Modern usage

A Modest Video Game Proposal is the title of an open letter sent by activist/former attorney Jack Thompson on 10 October 2005.

The 2012 horror film Butcher Boys, written by the original The Texas Chain Saw Massacre scribe Kim Henkel, is said to be an updating of Jonathan Swift's A Modest Proposal. Henkel imagined the descendants of folks who actually took Swift up on his proposal. The film opens with a quote from J. Swift.

The 2023 song "Eat Your Young" written by Irish musician Hozier might be a reference to "A Modest Proposal". It combines themes regarding the anti-war and anti-income-inequality movement, and uses Swift's essay as a framework to compare those modern problems to those same problems during Swift's time.

The July 2023 Channel 4 mockumentary Gregg Wallace: The British Miracle Meat, written by British comedy writer Matt Edmonds, updates A Modest Proposal and presents it in a similar format to Wallace's Inside the Factory, with human meat given as a potential solution to the UK's cost of living crisis. The words "a modest proposal" are used in Wallace's summing up at the end of the programme, and Swift is credited.

Mass

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Mass
A 2 kg (4.4 lb) cast iron weight used for balances

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. The object's mass also determines the strength of its gravitational attraction to other bodies.

The SI base unit of mass is the kilogram (kg). In physics, mass is not the same as weight, even though mass is often determined by measuring the object's weight using a spring scale, rather than balance scale comparing it directly with known masses. An object on the Moon would weigh less than it does on Earth because of the lower gravity, but it would still have the same mass. This is because weight is a force, while mass is the property that (along with gravity) determines the strength of this force.

In the Standard Model of physics, the mass of elementary particles is believed to be a result of their coupling with the Higgs boson in what is known as the Brout–Englert–Higgs mechanism.

Phenomena

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, current experiments have found no difference in results regardless of how it is measured:

• Inertial mass measures an object's resistance to being accelerated by a force (represented by the relationship F = ma).
• Active gravitational mass determines the strength of the gravitational field generated by an object.
• Passive gravitational mass measures the gravitational force exerted on an object in a known gravitational field.

The mass of an object determines its acceleration in the presence of an applied force. The inertia and the inertial mass describe this property of physical bodies at the qualitative and quantitative level respectively. According to Newton's second law of motion, if a body of fixed mass m is subjected to a single force F, its acceleration a is given by F/m. A body's mass also determines the degree to which it generates and is affected by a gravitational field. If a first body of mass m_A is placed at a distance r (center of mass to center of mass) from a second body of mass m_B, each body is subject to an attractive force F_g = Gm_Am_B/r², where G = 6.67×10⁻¹¹ N⋅kg⁻²⋅m² is the "universal gravitational constant". This is sometimes referred to as gravitational mass. Repeated experiments since the 17th century have demonstrated that inertial and gravitational mass are identical; since 1915, this observation has been incorporated a priori in the equivalence principle of general relativity.

Units of mass

Further information: Orders of magnitude (mass)

The kilogram is one of the seven SI base units.

The International System of Units (SI) unit of mass is the kilogram (kg). The kilogram is 1000 grams (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.

However, the mass of the IPK and its national copies have been found to drift over time. The re-definition of the kilogram and several other units came into effect on 20 May 2019, following a final vote by the CGPM in November 2018. The new definition uses only invariant quantities of nature: the speed of light, the caesium hyperfine frequency, the Planck constant and the elementary charge.

Non-SI units accepted for use with SI units include:

• the tonne (t) (or "metric ton"), equal to 1000 kg
• the electronvolt (eV), a unit of energy, used to express mass in units of eV/c² through mass–energy equivalence
• the dalton (Da), equal to 1/12 of the mass of a free carbon-12 atom, approximately 1.66×10⁻²⁷ kg.

Outside the SI system, other units of mass include:

• the slug (sl), an Imperial unit of mass (about 14.6 kg)
• the pound (lb), a unit of mass (about 0.45 kg), which is used alongside the similarly named pound (force) (about 4.5 N), a unit of force
• the Planck mass (about 2.18×10⁻⁸ kg), a quantity derived from fundamental constants
• the solar mass (M_☉), defined as the mass of the Sun, primarily used in astronomy to compare large masses such as stars or galaxies (≈ 1.99×10³⁰ kg)
• the mass of a particle, as identified with its inverse Compton wavelength (1 cm⁻¹ ≘ 3.52×10⁻⁴¹ kg)
• the mass of a star or black hole, as identified with its Schwarzschild radius (1 cm ≘ 6.73×10²⁴ kg).

Definitions

In physical science, one may distinguish conceptually between at least seven different aspects of mass, or seven physical notions that involve the concept of mass. Every experiment to date has shown these seven values to be proportional, and in some cases equal, and this proportionality gives rise to the abstract concept of mass. There are a number of ways mass can be measured or operationally defined:

• Inertial mass is a measure of an object's resistance to acceleration when a force is applied. It is determined by applying a force to an object and measuring the acceleration that results from that force. An object with small inertial mass will accelerate more than an object with large inertial mass when acted upon by the same force. One says the body of greater mass has greater inertia.
• Active gravitational mass is a measure of the strength of an object's gravitational flux (gravitational flux is equal to the surface integral of gravitational field over an enclosing surface). Gravitational field can be measured by allowing a small "test object" to fall freely and measuring its free-fall acceleration. For example, an object in free-fall near the Moon is subject to a smaller gravitational field, and hence accelerates more slowly, than the same object would if it were in free-fall near the Earth. The gravitational field near the Moon is weaker because the Moon has less active gravitational mass.
• Passive gravitational mass is a measure of the strength of an object's interaction with a gravitational field. Passive gravitational mass is determined by dividing an object's weight by its free-fall acceleration. Two objects within the same gravitational field will experience the same acceleration; however, the object with a smaller passive gravitational mass will experience a smaller force (less weight) than the object with a larger passive gravitational mass.
• According to relativity, mass is nothing else than the rest energy of a system of particles, meaning the energy of that system in a reference frame where it has zero momentum. Mass can be converted into other forms of energy according to the principle of mass–energy equivalence. This equivalence is exemplified in a large number of physical processes including pair production, beta decay and nuclear fusion. Pair production and nuclear fusion are processes in which measurable amounts of mass are converted to kinetic energy or vice versa.
• Curvature of spacetime is a relativistic manifestation of the existence of mass. Such curvature is extremely weak and difficult to measure. For this reason, curvature was not discovered until after it was predicted by Einstein's theory of general relativity. Extremely precise atomic clocks on the surface of the Earth, for example, are found to measure less time (run slower) when compared to similar clocks in space. This difference in elapsed time is a form of curvature called gravitational time dilation. Other forms of curvature have been measured using the Gravity Probe B satellite.
• Quantum mass manifests itself as a difference between an object's quantum frequency and its wave number. The quantum mass of a particle is proportional to the inverse Compton wavelength and can be determined through various forms of spectroscopy. In relativistic quantum mechanics, mass is one of the irreducible representation labels of the Poincaré group.

Weight vs. mass

Main article: Mass versus weight

Mass and weight of a given object on Earth and Mars. Weight varies due to different amount of gravitational acceleration whereas mass stays the same.

In everyday usage, mass and "weight" are often used interchangeably. For instance, a person's weight may be stated as 75 kg. 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.

The force known as "weight" is proportional to mass and acceleration in all situations where the mass is accelerated away from free fall. For example, when a body is at rest in a gravitational field (rather than in free fall), it must be accelerated by a force from a scale or the surface of a planetary body such as the Earth or the Moon. This force keeps the object from going into free fall. Weight is the opposing force in such circumstances and is thus determined by the acceleration of free fall. On the surface of the Earth, for example, an object with a mass of 50 kilograms weighs 491 newtons, which means that 491 newtons is being applied to keep the object from going into free fall. By contrast, on the surface of the Moon, the same object still has a mass of 50 kilograms but weighs only 81.5 newtons, because only 81.5 newtons is required to keep this object from going into a free fall on the moon. Restated in mathematical terms, on the surface of the Earth, the weight W of an object is related to its mass m by W = mg, where g = 9.80665 m/s² is the acceleration due to Earth's gravitational field, (expressed as the acceleration experienced by a free-falling object).

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).

Inertial vs. gravitational mass

See also: Eötvös experiment

Although inertial mass, passive gravitational mass and active gravitational mass are conceptually distinct, no experiment has ever unambiguously demonstrated any difference between them. In classical mechanics, Newton's third law implies that active and passive gravitational mass must always be identical (or at least proportional), but the classical theory offers no compelling reason why the gravitational mass has to equal the inertial mass. That it does is merely an empirical fact.

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.

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:

${\displaystyle F=Mg.}$

Given this force, the acceleration of the object can be determined by Newton's second law:

${\displaystyle F=ma.}$

Putting these together, the gravitational acceleration is given by:

${\displaystyle a=\frac{M}{m}g.}$

This says that the ratio of gravitational to inertial mass of any object is equal to some constant K if and only if all objects fall at the same rate in a given gravitational field. This phenomenon is referred to as the "universality of free-fall". In addition, the constant K can be taken as 1 by defining our units appropriately.

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, using the torsion balance pendulum, in 1889. As of 2008, 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.

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.

A stronger version of the equivalence principle, known as the Einstein equivalence principle or the strong equivalence principle, lies at the heart of the general theory of relativity. Einstein's equivalence principle states that within sufficiently small regions of spacetime, it is impossible to distinguish between a uniform acceleration and a uniform gravitational field. Thus, the theory postulates that the force acting on a massive object caused by a gravitational field is a result of the object's tendency to move in a straight line (in other words its inertia) and should therefore be a function of its inertial mass and the strength of the gravitational field.

Origin

Main article: Mass generation mechanism

In theoretical physics, a mass generation mechanism is a theory which attempts to explain the origin of mass from the most fundamental laws of physics. To date, a number of different models have been proposed which advocate different views of the origin of mass. The problem is complicated by the fact that the notion of mass is strongly related to the gravitational interaction but a theory of the latter has not been yet reconciled with the currently popular model of particle physics, known as the Standard Model.

Pre-Newtonian concepts

Weight as an amount

Main article: Weight

Depiction of early balance scales in the Papyrus of Hunefer (dated to the 19th dynasty, c. 1285 BCE). The scene shows Anubis weighing the heart of Hunefer.

The concept of amount is very old and predates recorded history. The concept of "weight" would incorporate "amount" and acquire a double meaning that was not clearly recognized as such.

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.

— K. M. Browne, The pre-Newtonian meaning of the word “weight”

Humans, at some early era, realized that the weight of a collection of similar objects was directly proportional to the number of objects in the collection:

${\displaystyle W_n\propto n,}$

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:

${\displaystyle \frac{W_n}{n}=\frac{W_m}{m}}$, or equivalently ${\displaystyle \frac{W_n}{W_m}=\frac{n}{m}.}$

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 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:

${\displaystyle \frac{\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{e}}{\mathrm{p}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}=\frac{W_{144}}{W_{1728}}=\frac{144}{1728}=\frac{1}{12}.}$

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 of the orbital period of each planet is directly proportional to the 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.

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

Galileo Galilei (1636)

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, 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 balls 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. 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.

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. 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 angles 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.

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:

${\displaystyle \text{Distance}\propto {\text{Time}}^2}$

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." 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.

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.

— Isaac Newton, Mathematical principles of natural philosophy, Definition I.

Newtonian mass

Earth's Moon		Mass of Earth
Semi-major axis	Sidereal orbital period
0.002 569 AU	0.074 802 sidereal year	${\displaystyle 1.2{\pi }^2\cdot {10}^{-5}\frac{{\text{AU}}^3}{{\text{y}}^2}=3.986\cdot {10}^{14}\frac{{\text{m}}^3}{{\text{s}}^2}}$
Earth's gravity	Earth's radius
9.806 65 m/s2	6 375 km

Isaac Newton, 1689

Robert Hooke had published his concept of gravitational forces in 1674, stating that all celestial bodies have an attraction or gravitating power towards their own centers, and also attract all the other celestial bodies that are within the sphere of their activity. He further stated that gravitational attraction increases by how much nearer the body wrought upon is to its own center. In correspondence with Isaac Newton from 1679 and 1680, Hooke conjectured that gravitational forces might decrease according to the double of the distance between the two bodies. Hooke urged Newton, who was a pioneer in the development of calculus, to work through the mathematical details of Keplerian orbits to determine if Hooke's hypothesis was correct. Newton's own investigations verified that Hooke was correct, but due to personal differences between the two men, Newton chose not to reveal this to Hooke. Isaac Newton kept quiet about his discoveries until 1684, at which time he told a friend, Edmond Halley, that he had solved the problem of gravitational orbits, but had misplaced the solution in his office. After being encouraged by Halley, Newton decided to develop his ideas about gravity and publish all of his findings. In November 1684, Isaac Newton sent a document to Edmund Halley, now lost but presumed to have been titled De motu corporum in gyrum (Latin for "On the motion of bodies in an orbit"). Halley presented Newton's findings to the Royal Society of London, with a promise that a fuller presentation would follow. Newton later recorded his ideas in a three-book set, entitled Philosophiæ Naturalis Principia Mathematica (English: Mathematical Principles of Natural Philosophy). The first was received by the Royal Society on 28 April 1685–86; the second on 2 March 1686–87; and the third on 6 April 1686–87. The Royal Society published Newton's entire collection at their own expense in May 1686–87.

Isaac Newton had bridged the gap between Kepler's gravitational mass and Galileo's gravitational acceleration, resulting in the discovery of the following relationship which governed both of these:

${\displaystyle \mathbf{g}=-\mu \frac{\hat{\mathbf{R}}}{|\mathbf{R}{|}^2}}$

where g is the apparent acceleration of a body as it passes through a region of space where gravitational fields exist, μ is the gravitational mass (standard gravitational parameter) of the body causing gravitational fields, and R is the radial coordinate (the distance between the centers of the two bodies).

By finding the exact relationship between a body's gravitational mass and its gravitational field, Newton provided a second method for measuring gravitational mass. The mass of the Earth can be determined using Kepler's method (from the orbit of Earth's Moon), or it can be determined by measuring the gravitational acceleration on the Earth's surface, and multiplying that by the square of the Earth's radius. The mass of the Earth is approximately three-millionths of the mass of the Sun. To date, no other accurate method for measuring gravitational mass has been discovered.

Newton's cannonball

A cannon on top of a very high mountain shoots a cannonball horizontally. If the speed is low, the cannonball quickly falls back to Earth (A, B). At intermediate speeds, it will revolve around Earth along an elliptical orbit (C, D). Beyond the escape velocity, it will leave the Earth without returning (E).

Main article: Newton's cannonball

Newton's cannonball was a thought experiment used to bridge the gap between Galileo's gravitational acceleration and Kepler's elliptical orbits. It appeared in Newton's 1728 book A Treatise of the System of the World. According to Galileo's concept of gravitation, a dropped stone falls with constant acceleration down towards the Earth. However, Newton explains that when a stone is thrown horizontally (meaning sideways or perpendicular to Earth's gravity) it follows a curved path. "For a stone projected is by the pressure of its own weight forced out of the rectilinear path, which by the projection alone it should have pursued, and made to describe a curve line in the air; and through that crooked way is at last brought down to the ground. And the greater the velocity is with which it is projected, the farther it goes before it falls to the Earth." Newton further reasons that if an object were "projected in an horizontal direction from the top of a high mountain" with sufficient velocity, "it would reach at last quite beyond the circumference of the Earth, and return to the mountain from which it was projected."

Universal gravitational mass

An apple experiences gravitational fields directed towards every part of the Earth; however, the sum total of these many fields produces a single gravitational field directed towards the Earth's center.

In contrast to earlier theories (e.g. celestial spheres) which stated that the heavens were made of entirely different material, Newton's theory of mass was groundbreaking partly because it introduced universal gravitational mass: every object has gravitational mass, and therefore, every object generates a gravitational field. Newton further assumed that the strength of each object's gravitational field would decrease according to the square of the distance to that object. If a large collection of small objects were formed into a giant spherical body such as the Earth or Sun, Newton calculated the collection would create a gravitational field proportional to the total mass of the body, and inversely proportional to the square of the distance to the body's center.

For example, according to Newton's theory of universal gravitation, each carob seed produces a gravitational field. Therefore, if one were to gather an immense number of carob seeds and form them into an enormous sphere, then the gravitational field of the sphere would be proportional to the number of carob seeds in the sphere. Hence, it should be theoretically possible to determine the exact number of carob seeds that would be required to produce a gravitational field similar to that of the Earth or Sun. In fact, by unit conversion it is a simple matter of abstraction to realize that any traditional mass unit can theoretically be used to measure gravitational mass.

Vertical section drawing of Cavendish's torsion balance instrument including the building in which it was housed. The large balls were hung from a frame so they could be rotated into position next to the small balls by a pulley from outside. Figure 1 of Cavendish's paper.

Measuring gravitational mass in terms of traditional mass units is simple in principle, but extremely difficult in practice. According to Newton's theory, all objects produce gravitational fields and it is theoretically possible to collect an immense number of small objects and form them into an enormous gravitating sphere. However, from a practical standpoint, the gravitational fields of small objects are extremely weak and difficult to measure. Newton's books on universal gravitation were published in the 1680s, but the first successful measurement of the Earth's mass in terms of traditional mass units, the Cavendish experiment, did not occur until 1797, over a hundred years later. Henry Cavendish found that the Earth's density was 5.448 ± 0.033 times that of water. As of 2009, the Earth's mass in kilograms is only known to around five digits of accuracy, whereas its gravitational mass is known to over nine significant figures.

Given two objects A and B, of masses M_A and M_B, separated by a displacement R_AB, Newton's law of gravitation states that each object exerts a gravitational force on the other, of magnitude

${\displaystyle {\mathbf{F}}_{\text{AB}}=-G{M}_{\text{A}}{M}_{\text{B}}\frac{{\hat{\mathbf{R}}}_{\text{AB}}}{|{\mathbf{R}}_{\text{AB}}{|}^2}}$,

where G is the universal gravitational constant. The above statement may be reformulated in the following way: if g is the magnitude at a given location in a gravitational field, then the gravitational force on an object with gravitational mass M is

${\displaystyle F=Mg}$.

This is the basis by which masses are determined by weighing. In simple spring scales, for example, the force F is proportional to the displacement of the spring beneath the weighing pan, as per Hooke's law, and the scales are calibrated to take g into account, allowing the mass M to be read off. Assuming the gravitational field is equivalent on both sides of the balance, a balance measures relative weight, giving the relative gravitation mass of each object.

Inertial mass

Mass was traditionally believed to be a measure of the quantity of matter in a physical body, equal to the "amount of matter" in an object. For example, Barre´ de Saint-Venant argued in 1851 that every object contains a number of "points" (basically, interchangeable elementary particles), and that mass is proportional to the number of points the object contains. (In practice, this "amount of matter" definition is adequate for most of classical mechanics, and sometimes remains in use in basic education, if the priority is to teach the difference between mass from weight.) This traditional "amount of matter" belief was contradicted by the fact that different atoms (and, later, different elementary particles) can have different masses, and was further contradicted by Einstein's theory of relativity (1905), which showed that the measurable mass of an object increases when energy is added to it (for example, by increasing its temperature or forcing it near an object that electrically repels it.) This motivates a search for a different definition of mass that is more accurate than the traditional definition of "the amount of matter in an object".

Massmeter, a device for measuring the inertial mass of an astronaut in weightlessness. The mass is calculated via the oscillation period for a spring with the astronaut attached (Tsiolkovsky State Museum of the History of Cosmonautics).

Inertial mass is the mass of an object measured by its resistance to acceleration. This definition has been championed by Ernst Mach and has since been developed into the notion of operationalism by Percy W. Bridgman. The simple classical mechanics definition of mass differs slightly from the definition in the theory of special relativity, but the essential meaning is the same.

In classical mechanics, according to Newton's second law, we say that a body has a mass m if, at any instant of time, it obeys the equation of motion

${\displaystyle \mathbf{F}=m\mathbf{a},}$

where F is the resultant force acting on the body and a is the acceleration of the body's centre of mass. For the moment, we will put aside the question of what "force acting on the body" actually means.

This equation illustrates how mass relates to the inertia of a body. Consider two objects with different masses. If we apply an identical force to each, the object with a bigger mass will experience a smaller acceleration, and the object with a smaller mass will experience a bigger acceleration. We might say that the larger mass exerts a greater "resistance" to changing its state of motion in response to the force.

However, this notion of applying "identical" forces to different objects brings us back to the fact that we have not really defined what a force is. We can sidestep this difficulty with the help of Newton's third law, which states that if one object exerts a force on a second object, it will experience an equal and opposite force. To be precise, suppose we have two objects of constant inertial masses m₁ and m₂. We isolate the two objects from all other physical influences, so that the only forces present are the force exerted on m₁ by m₂, which we denote F₁₂, and the force exerted on m₂ by m₁, which we denote F₂₁. Newton's second law states that

${\displaystyle \begin{array}{rl}{\mathbf{F}}_{\mathbf{12}}& =m_1{\mathbf{a}}_1,\\ {\mathbf{F}}_{\mathbf{21}}& =m_2{\mathbf{a}}_2,\end{array}}$

where a₁ and a₂ are the accelerations of m₁ and m₂, respectively. Suppose that these accelerations are non-zero, so that the forces between the two objects are non-zero. This occurs, for example, if the two objects are in the process of colliding with one another. Newton's third law then states that

${\displaystyle {\mathbf{F}}_{12}=-{\mathbf{F}}_{21};}$

and thus

${\displaystyle m_1=m_2\frac{|{\mathbf{a}}_2|}{|{\mathbf{a}}_1|}.}$

If |a₁| is non-zero, the fraction is well-defined, which allows us to measure the inertial mass of m₁. In this case, m₂ is our "reference" object, and we can define its mass m as (say) 1 kilogram. Then we can measure the mass of any other object in the universe by colliding it with the reference object and measuring the accelerations.

Additionally, mass relates a body's momentum p to its linear velocity v:

${\displaystyle \mathbf{p}=m\mathbf{v}}$,

and the body's kinetic energy K to its velocity:

${\displaystyle K={\displaystyle \frac{1}{2}}m|\mathbf{v}{|}^2}$.

The primary difficulty with Mach's definition of mass is that it fails to take into account the potential energy (or binding energy) needed to bring two masses sufficiently close to one another to perform the measurement of mass. This is most vividly demonstrated by comparing the mass of the proton in the nucleus of deuterium, to the mass of the proton in free space (which is greater by about 0.239%—this is due to the binding energy of deuterium). Thus, for example, if the reference weight m₂ is taken to be the mass of the neutron in free space, and the relative accelerations for the proton and neutron in deuterium are computed, then the above formula over-estimates the mass m₁ (by 0.239%) for the proton in deuterium. At best, Mach's formula can only be used to obtain ratios of masses, that is, as m₁ / m₂ = |a₂| / |a₁|. An additional difficulty was pointed out by Henri Poincaré, which is that the measurement of instantaneous acceleration is impossible: unlike the measurement of time or distance, there is no way to measure acceleration with a single measurement; one must make multiple measurements (of position, time, etc.) and perform a computation to obtain the acceleration. Poincaré termed this to be an "insurmountable flaw" in the Mach definition of mass.

Atomic masses

Main article: Dalton (unit)

Typically, the mass of objects is measured in terms of the kilogram, which since 2019 is defined in terms of fundamental constants of nature. The mass of an atom or other particle can be compared more precisely and more conveniently to that of another atom, and thus scientists developed the dalton (also known as the unified atomic mass unit). By definition, 1 Da (one dalton) is exactly one-twelfth of the mass of a carbon-12 atom, and thus, a carbon-12 atom has a mass of exactly 12 Da.

In relativity

Special relativity

Main article: Mass in special relativity

In some frameworks of special relativity, physicists have used different definitions of the term. In these frameworks, two kinds of mass are defined: rest mass (invariant mass), and relativistic mass (which increases with velocity). Rest mass is the Newtonian mass as measured by an observer moving along with the object. Relativistic mass is the total quantity of energy in a body or system divided by c². The two are related by the following equation:

${\displaystyle m_{\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}}=\gamma (m_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}})}$

where ${\displaystyle \gamma }$ is the Lorentz factor:

${\displaystyle \gamma =\frac{1}{\sqrt{1-v^2/c^2}}}$

The invariant mass of systems is the same for observers in all inertial frames, while the relativistic mass depends on the observer's frame of reference. In order to formulate the equations of physics such that mass values do not change between observers, it is convenient to use rest mass. The rest mass of a body is also related to its energy E and the magnitude of its momentum p by the relativistic energy-momentum equation:

${\displaystyle (m_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}})c^2=\sqrt{E_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}^2-(|\mathbf{p}|c{)}^2}.}$

So long as the system is closed with respect to mass and energy, both kinds of mass are conserved in any given frame of reference. The conservation of mass holds even as some types of particles are converted to others. Matter particles (such as atoms) may be converted to non-matter particles (such as photons of light), but this does not affect the total amount of mass or energy. Although things like heat may not be matter, all types of energy still continue to exhibit mass. Thus, mass and energy do not change into one another in relativity; rather, both are names for the same thing, and neither mass nor energy appear without the other.

Both rest and relativistic mass can be expressed as an energy by applying the well-known relationship E = mc², yielding rest energy and "relativistic energy" (total system energy) respectively:

${\displaystyle E_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}=(m_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}})c^2}$

${\displaystyle E_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=(m_{\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}})c^2}$

The "relativistic" mass and energy concepts are related to their "rest" counterparts, but they do not have the same value as their rest counterparts in systems where there is a net momentum. Because the relativistic mass is proportional to the energy, it has gradually fallen into disuse among physicists. There is disagreement over whether the concept remains useful pedagogically.

In bound systems, the binding energy must often be subtracted from the mass of the unbound system, because binding energy commonly leaves the system at the time it is bound. The mass of the system changes in this process merely because the system was not closed during the binding process, so the energy escaped. For example, the binding energy of atomic nuclei is often lost in the form of gamma rays when the nuclei are formed, leaving nuclides which have less mass than the free particles (nucleons) of which they are composed.

Mass–energy equivalence also holds in macroscopic systems. For example, if one takes exactly one kilogram of ice, and applies heat, the mass of the resulting melt-water will be more than a kilogram: it will include the mass from the thermal energy (latent heat) used to melt the ice; this follows from the conservation of energy. This number is small but not negligible: about 3.7 nanograms. It is given by the latent heat of melting ice (334 kJ/kg) divided by the speed of light squared (c² ≈ 9×10¹⁶ m²/s²).

General relativity

Main article: Mass in general relativity

In general relativity, the equivalence principle is the equivalence of gravitational and inertial mass. At the core of this assertion is Albert Einstein's idea that the gravitational force as experienced locally while standing on a massive body (such as the Earth) is the same as the pseudo-force experienced by an observer in a non-inertial (i.e. accelerated) frame of reference.

However, it turns out that it is impossible to find an objective general definition for the concept of invariant mass in general relativity. At the core of the problem is the non-linearity of the Einstein field equations, making it impossible to write the gravitational field energy as part of the stress–energy tensor in a way that is invariant for all observers. For a given observer, this can be achieved by the stress–energy–momentum pseudotensor.

In quantum physics

In classical mechanics, the inert mass of a particle appears in the Euler–Lagrange equation as a parameter m:

${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\mathrm{\partial }L}{\mathrm{\partial }{\dot{x}}_i}\right)=m{\ddot{x}}_i.}$

After quantization, replacing the position vector x with a wave function, the parameter m appears in the kinetic energy operator:

${\displaystyle i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathrm{\Psi }(\mathbf{r},t)=\left(-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+V(\mathbf{r})\right)\mathrm{\Psi }(\mathbf{r},t).}$

In the ostensibly covariant (relativistically invariant) Dirac equation, and in natural units, this becomes:

${\displaystyle (-i{\gamma }^{\mu }{\mathrm{\partial }}_{\mu }+m)\psi =0}$

where the "mass" parameter m is now simply a constant associated with the quantum described by the wave function ψ.

In the Standard Model of particle physics as developed in the 1960s, this term arises from the coupling of the field ψ to an additional field Φ, the Higgs field. In the case of fermions, the Higgs mechanism results in the replacement of the term mψ in the Lagrangian with ${\displaystyle G_{\psi }\overline{\psi }\varphi \psi }$. This shifts the explanandum of the value for the mass of each elementary particle to the value of the unknown coupling constant G_ψ.

Tachyonic particles and imaginary (complex) mass

Main articles: Tachyonic field and Tachyon § Mass

A tachyonic field, or simply tachyon, is a quantum field with an imaginary mass. Although tachyons (particles that move faster than light) are a purely hypothetical concept not generally believed to exist, fields with imaginary mass have come to play an important role in modern physics and are discussed in popular books on physics. Under no circumstances do any excitations ever propagate faster than light in such theories—the presence or absence of a tachyonic mass has no effect whatsoever on the maximum velocity of signals (there is no violation of causality). While the field may have imaginary mass, any physical particles do not; the "imaginary mass" shows that the system becomes unstable, and sheds the instability by undergoing a type of phase transition called tachyon condensation (closely related to second order phase transitions) that results in symmetry breaking in current models of particle physics.

The term "tachyon" was coined by Gerald Feinberg in a 1967 paper, but it was soon realized that Feinberg's model in fact did not allow for superluminal speeds. Instead, the imaginary mass creates an instability in the configuration:- any configuration in which one or more field excitations are tachyonic will spontaneously decay, and the resulting configuration contains no physical tachyons. This process is known as tachyon condensation. Well known examples include the condensation of the Higgs boson in particle physics, and ferromagnetism in condensed matter physics.

Although the notion of a tachyonic imaginary mass might seem troubling because there is no classical interpretation of an imaginary mass, the mass is not quantized. Rather, the scalar field is; even for tachyonic quantum fields, the field operators at spacelike separated points still commute (or anticommute), thus preserving causality. Therefore, information still does not propagate faster than light, and solutions grow exponentially, but not superluminally (there is no violation of causality). Tachyon condensation drives a physical system that has reached a local limit and might naively be expected to produce physical tachyons, to an alternate stable state where no physical tachyons exist. Once the tachyonic field reaches the minimum of the potential, its quanta are not tachyons any more but rather are ordinary particles with a positive mass-squared.

This is a special case of the general rule, where unstable massive particles are formally described as having a complex mass, with the real part being their mass in the usual sense, and the imaginary part being the decay rate in natural units. However, in quantum field theory, a particle (a "one-particle state") is roughly defined as a state which is constant over time; i.e., an eigenvalue of the Hamiltonian. An unstable particle is a state which is only approximately constant over time; If it exists long enough to be measured, it can be formally described as having a complex mass, with the real part of the mass greater than its imaginary part. If both parts are of the same magnitude, this is interpreted as a resonance appearing in a scattering process rather than a particle, as it is considered not to exist long enough to be measured independently of the scattering process. In the case of a tachyon, the real part of the mass is zero, and hence no concept of a particle can be attributed to it.

In a Lorentz invariant theory, the same formulas that apply to ordinary slower-than-light particles (sometimes called "bradyons" in discussions of tachyons) must also apply to tachyons. In particular the energy–momentum relation:

${\displaystyle E^2=p^2{c}^2+m^2{c}^4}$

(where p is the relativistic momentum of the bradyon and m is its rest mass) should still apply, along with the formula for the total energy of a particle:

${\displaystyle E=\frac{m{c}^2}{\sqrt{1-\frac{v^2}{c^2}}}.}$

This equation shows that the total energy of a particle (bradyon or tachyon) contains a contribution from its rest mass (the "rest mass–energy") and a contribution from its motion, the kinetic energy. When v is larger than c, the denominator in the equation for the energy is "imaginary", as the value under the radical is negative. Because the total energy must be real, the numerator must also be imaginary: i.e. the rest mass m must be imaginary, as a pure imaginary number divided by another pure imaginary number is a real number.