Earliest Uses of Symbols for Constants

gamma for the Euler-Mascheroni constant. The earliest use of the lower case gamma (now more common than c) is apparently unknown.

In vol. VII of "L'Intermediaire des mathematiciens" (1900), G. Vacca (from Turin) raised this very question. His question #1998 reads as follows:

In the German "Encyclopaedie" (1900, vol. II, p. 171) it says that Mascheroni has denoted the Euler's constant 0.577... by gamma. According to my research, this author designated it by letter "A".

In "Synopsis" of Mr. Hagen (1891, vol. I, p. 86), it is said that Euler has introduced this symbol in "Acta Petr." (1769, vol. XIV).

Mr. E. Pascal, in his "Repertorium" (1900, vol. I, p. 478 of the German edition) reproduces that suggestion. But, in the quoted volume, and in many memoirs of Euler, I have found that this author has just used the symbol "C", 'et parfois' "O".

Who is the first mathematician that has introduced the symbol gamma for the Euler's constant?

In 1790, Lorenzo Mascheroni (1750-1800; see the sources page) calculated Euler's constant to 32 decimal places:

A = 0,5772 15664 90153 28606 181 1209 0082 39 ...

As a matter of fact, the first 19 places are the correct digits, though curiously enough not the only ones, as we can notice in the following correct approximation:

C = 0,5772 15664 90153 28606 065 1209 0082 40 ...

(This entry was contributed by Julio González Cabillón, who provided the translation from the French of the quote above.)

e for 2.71828... was first used by Leonhard Euler (1707-1783) in a manuscript, Meditatio in Experimenta explosione tormentorum nuper instituta (Meditation on experiments made recently on the firing of cannon), written at the end of 1727 or the beginning of 1728 (when Euler was just 21 years old). The manuscript was first printed in 1862 in Euler's Opera postuma mathematica et physica, Petropoli, edited by P. H. Fuss and N. Fuss (vol ii, pp. 800-804). The manuscript describes seven experiments performed between August 21 and September 2, 1727:

For the number whose logarithm is unity, let e be written, which is 2,7182817... [sic] whose logarithm according to Vlacq is 0,4342944... [translated from Latin by Florian Cajori].

Euler next used e in a letter addressed to Goldbach on November 25, 1731, writing that e "denotes that number whose hyperbolic logarithm is = 1." He used e again in 1736 in his Mechanica.

Maor writes:

Why did he choose the letter e? There is no general consensus. According to one view, Euler chose it because it is the first letter of the word exponential. More likely, the choice came to him naturally as the first "unused" letter of the alphabet, since the letters a, b, c, and d frequently appear elsewhere in mathematics. It seems unlikely that Euler chose the letter because it is the initial of his own name, as occasionally been suggested: he was an extremely modest man and often delayed publication of his own work so that a colleague or student of his would get due credit.

phi for the golden ratio was chosen by the American mathematician Mark Barr (Johnson, page 149). According to Gardner (1961) and Huntley, the letter was chosen because it is the first letter in the name of Phidias who is believed to have used the golden proportion frequently in his sculpture. However, Schwartzman (page 164) implies the letter stands for Fibonacci.

The Greek letter tau is also used for this constant. Ball and Coxeter (1987, page 57) write, "The symbol [symbol] is appropriate because it is the initial of tomh\ ("section"). (Information for this constant provided by Antreas P. Hatzipolakis.)

i for the imaginary unit was first used by Leonhard Euler (1707-1783) in a memoir presented in 1777 but not published until 1794 in his "Institutionum calculi integralis."

On May 5, 1777, Euler addressed to the 'Academiae' the paper "De Formulis Differentialibus Angularibus maxime irrationalibus quas tamen per logarithmos et arcus circulares integrare licet," which was published posthumously in his "Institutionum calculi integralis," second ed., vol. 4, pp. 183-194, Impensis Academiae Imperialis Scientiarum, Petropoli, 1794.

Quoniam mihi quidem alia adhuc via non patet istud praestandi nisi per imaginaria procedendo, formulam \/-1 littera i in posterum designabo, ita ut sit ii = -1 ideoque 1/i = -i.

Throughout his Introductio, Euler consistently writes "\/-1, denoting by i the "numerus infinite magnus" [namely, an infinitely large number]. Nonetheless, there are very few occasions where Euler chose i with a different meaning. Thus, chapter XXI (volume 2) of Euler's Introductio contains the first appearance of i as quantitas imaginaria:

Cum enim numerorum negativorum Logarithmi sint imaginarii (...) erit log(-n) quantitas imaginaria, quae sit = i.

Please note that, in this fascinating passage about logarithms, Euler does NOT introduce the symbol i such that i^2 = -1. (This entry was contributed by Julio González Cabillón.)

pi for 3.14159... In 1647 William Oughtred (1574-1660) used pi to represent the circumference of a circle. He showed the ratio of the diameter of a circle to its circumference as the fraction lower case delta over pi (Schwartzman, page 165).

In 1697 David Gregory used the fraction lower case pi over rho to represent the ratio of the circumference of a circle to its radius (Schwartzman, page 165).

However, the first person actually to use pi to represent the ratio of the circumference to the diameter (3.14159...) was William Jones (1675-1749) in 1706 in Synopsis palmariorum mathesios. It is believed he used the Greek letter pi because it is the first letter in perimetron (= perimeter). From Cajori (vol. 2, page 9):

The modern notation for 3.14159 .... was introduced in 1706. It was in that year that William Jones made himself noted, without being aware that he was doing anything noteworthy, through his designation of the ratio of the length of the circle to its diameter by the letter pi. He took this step without ostentation. No lengthy introduction prepares the reader for the bringing upon the stage of mathematical history this distinguished visitor from the field of Greek letters. It simply came, unheralded, in the following prosaic statement (p. 263):

"There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as 1 to , &c. = 3.14159, &c. = pi. This series (among others for the same purpose, and drawn form the same Principle) I received from the Excellent Analyst, and my much esteem'd Friend Mr. John Machin; and by means thereof, Van Ceulen's Number, or that in Art. 64.38 may be Examin'd with all desirable Ease and Dispatch."

Zero symbol (0). Around 2300-1600 BC, the Babylonians used a symbol to mark a change in placevalue of digits, which was in effect a zero, but this symbol was not used in calculations.

The Chinese, as early as the fourth century BC, represented zero as a blank space on a counting board (Johnson, page 160).

The Encyclopaedia Britannica says, "Hindu literature gives evidence that the zero may have been known before the birth of Christ, but no inscription has been found with such a symbol before the 9th century." Another source says the first zero in the Hindu system was represented by a dot and was found in a text written by Bakhshali, date unknown.

Burton says that about A. D. 150, the Alexandrian astronomer Ptolemy began using the omicron [which looks something like a zero] in the manner of our zero, not only in a medial but also in a terminal position. He says there is no evidence that Ptolemy regarded the symbol as a number by itself that could enter into computations with other numbers. Omicron is the first letter of the Greek word for "nothing." However, Len Berggren says, "Ptolemy probably did not use omicron to denote 0. Papyri from the period when Ptolemy lived show a small 'o' with a bar over it as the symbol for 0, and the small 'o' alone doesn't come in until the Byzantine period. Even in that period Neugebauer considers it unlikely that the small 'o' stood for the Greek word ouden (= nothing). See the discussion in the second edition of Neugebauer's Exact Sciences in Antiquity, esp. pp. 13 - 14."

By the third century A.D., Hindu mathematicians were using a heavy dot to mark its place in calculations. The dot was eventually replaced by an empty circle (Johnson, page 160).

The earliest date the Mayan symbol for zero, which was used only as a placeholder and does not resemble our zero, has been traced to is A. D. 357 (Boyer).

The earliest date Bell has found to be fairly proven for the use of zero was A. D. 505 in a Pancasiddhantika by Varahamihira.

There are claims that the zero symbol appeared in inscriptions in Cambodia and Sumatra as early as 683 (Johnson, page 160).

The oldest surviving use of an empty circle for zero is on a temple inscription in Gvalior, India, and dates from 840 (Johnson, page 160).

The zero symbol first appears in print in the 1200s, according to Burton. Smith says that Ch'in Kiu-shao (or Tsin Kiu tschaou or Ts'in K'ieou-Chao) of China used 0 in 1247 or 1257 in The Nine Sections of Mathematics.

The first European mathematician to advocate the use of zero was Leonardo of Pisa (1180-1250) (or Fibonacci), who used the term zephirum in Liber Abaci.

Rida A. K. Irani, in a paper in Centaurus, vol. 4 (1955), pp. 1-12 gives forms of the Hindu-Arabic characters as they occur in dated Arabic manuscripts, and shows that the medieval Arabs used a small 'o' for zero. This tends to degenerate to a dot in late (e.g. 17th century) manuscripts.

a + bi (notation for complex numbers) was introduced by Leonhard Euler (1707-1783).

Letters of the alphabet for variables. Diophantus (fl. about 250-275) used a Greek letter with an accent to represent an unknown. G. H. F. Nesselmann takes this symbol to be the final sigma and remarks that probably its selection was prompted by the fact that it was the only letter in the Greek alphabet which was not used in writing numbers. However, differing opinions exist (Cajor vol. 1, page 71).

Michael Stifel used A, B, C, D, and F, as well as x for unknowns in 1544 in Arithmetica integra, according to the DSB.

In 1591 Francois Vieta (1540-1603) was the first person to use letters for unknowns and constants in algebraic equations. He used vowels for unknowns and consonants for given numbers (all capital letters) in In artem analyticem isogoge. Vieta wrote:

Quod oopus, ut arte aliqua juventur, symbolo constanti et perpetuo ac bene conspicuo date magnitudines ab incertis quaesititiis distinguantur ut [illegible in Cajori] magnitudines quaesititias elemento A aliave litera volcali, E, I, O, V, Y [illegible in Cajori] elementis B, G, D, aliisve consonis designando. [That this work may be aided by a certain artifice, given magnitudes are to be distinguished from the undetermined unknowns by a constant and very clear symbol, as, for instance, by designating the unknown magnitude by means of a letter A or some other vowel E, I, O, U, or Y, and the given magnitudes by means of letters B, G, and D or other consonants.]

Thomas Harriot (1560-1621) in Artis Analyticae Praxis, ad Aequationes Algebraicas used lower case vowels for unknowns and lower case consonants for known quantities.

In 1637, Rene Descartes (1596-1650) in his La Geometrie used letters toward the end of the alphabet, especially x, for unknowns, and letters near the beginning of the alphabet for given quantities (Cajori vol. 1, page 381). Descartes introduced the equation ax + by = c, which is still used to describe the equation of a line (Johnson, page 145).

Johnson says (on page 145):

The predominant use of the letter x to represent an unknown value came about in an interesting way. During the printing of La Geometrie and its appendix, Discours de La Methode, which introduced coordinate geometry, the printer reached a dilemma. While the text was being typeset, the printer began to run short of the last letters of the alphabet. He asked Descartes if it mattered whether x, y, or z was used in each of the book's many equations. Descartes replied that it made no difference which of the three letters was used to designate an unknown quantity. The printer selected x for most of the unknowns, since the letters y and z are used in the French language more frequently than is x.

Kline says Descartes used letters for positive numbers only and "Not until John Hudde (1633-1704) did so in 1657 was a letter used for positive and negative numbers."