"Without loss of generality" is a standard in the mathematical lexicon, and I am writing to ask if anyone knows where the expression was popularized. (The idea has been around since antiquity, I'm sure, but the expression itself might not be that old.)

1$\begingroup$ Reflecting on the works listed by mhum and myself, it strikes me that none of them is a translation of any ancient text. This may lend credence to the hypothesis that he expression took off with Euler (who rarely discussed logic) and perhaps Leibniz. But as they say, absence of evidence is not evidence of absence, especially with such tools as Google. (Example.) Isn't there any Greek equivalent, or else medieval tradition Leibniz got it from? $\endgroup$ – Francois Ziegler May 10 '14 at 21:36

3$\begingroup$ @FrancoisZiegler I think some believe Euclid wrote some proofs of cases that were supposed to be representative of something more general, but didn't explicitly spell that out in any way. $\endgroup$ – Mark S. May 11 '14 at 16:50

2$\begingroup$ Without loss of generality we may assume that the term was the first thing communicated between mathematicians. (E.g. "Without loss of generality, you have five sheep and I have three. Then if I steal one sheep, we'll have the same number of sheep.) $\endgroup$ – Asaf Karagila Feb 25 '15 at 0:39
I think one reason JSTOR doesn't have “loss of generality” before 1831 is that fewer scientists wrote in English. But one finds (with minor variants merged, translations *starred, and year first published in [brackets]):
French (1674–1831):
 sauf l'universalité (Leibniz 1674 [1903], 1679 [1761])
 sans faire tort à la généralité (Leibniz 1679 [1761])
 sans affaiblir la généralité (Euler 1757)
 sans nuire à la généralité (Condorcet 1775; Lagrange 1785; Legendre 1798; Abel 1827; Dirichlet 1828; Quetelet 1828)
 sans déroger à la généralité (Lagrange 1783, 1788)
 sans diminuer la généralité (Cousin 1787; Legendre 1798, 1825; Lacroix 1799, 1803; Brisson 1808; Poisson 1823; Abel 1823, 1829, 1829)
 sans rien ôter à la généralité (Lagrange 1788, 1811; Poncelet 1822)
 sans altérer la généralité (Monge 1795; Legendre 1797, 1802; Garnier 1808; Fourier 1829; Liouville 1831)
 sans détruire la généralité (*Euler 1796)
 sans perdre de sa généralité (Lacroix 1797; Legendre 1798; Biot 1829)
 sans restreindre la généralité (Legendre 1798; Lagrange 1815; Poisson 1829, 1831)
 sauf la généralité (Pagani 1826).
Latin (1676–1842):
 salva generalitate (Leibniz 1676 [1858], 1676 [1993]; Euler 1770; Jacobi 1842)
 salva universalitate (Leibniz 1679 [1875]; Christmann 1815)
 non impedit generalitatem (Leibniz 1691 [2009])
 sine detrimento universalitatis (Euler 1744, 1748, 1764)
 non restringitur amplitudo (Euler 1747, 1750, 1764, 1770)
 non limitatur generalitas (Euler 1761)
 sine detrimento amplitudinis (Euler 1764, 1766, 1769)
 sine detrimento generalitatis (Euler 1769; Gauss 1818)
 nihil de universalitate perire (Euler 1782)
 nihil de amplitudine amittere (Euler 1783)
 non restringitur generalitas (Euler 1794).
Pig latin (1695, 1735):
 a generalitate hindert aber nichts (Leibniz 1695 [2004: III, 6])
 pro praesenti negotio general genug (Bernoulli 1735 [1843]).
German (1779–1830):
 ohne der Allgemeinheit Abbruch zu tun (Lagrange 1779; Plücker 1828, 1828, 1829)
 unbeschadet der Allgemeinheit (*de Bicquilley 1788; *Lagrange 1791, 1797; Fischer 1792; Bolzano 1804; Littrow 1823; Möbius 1827; Plücker 1828; Naumann 1830; *Euler 1830)
 ohne dass die Allgemeinheit leidet (*Euler 1791; Unger 1827)
 ohne der Allgemeinheit zu schaden (Lorenz 1792; Umpfenbach 1823; Littrow 1823, 1827; Grunert 1824; von Ettingshausen 1827; Jacobi 1828)
 ohne Allgemeinheit zu verlieren (*Lagrange 1797; *Lacroix 1800, 1822; von Münchow 1826)
 ohne der Allgemeinheit Eintrag zu tun (*Lacroix 1822; *Lagrange 1823; Umpfenbach 1823; Littrow 1823, 1827; Jacobi 1828; *Euler 1829)
 ohne die Allgemeinheit zu schmälern (*Lacroix 1822)
 ohne die Allgemeinheit zu beschränken (Littrow 1827; Fischer 1829)
 ohne Nachtheil für die Allgemeinheit (Gauss 1828)
 ohne Beeinträchtigung der Allgemeinheit (*Euler 1829; Moth 1829).
Italian (17921824):
 senza nulla togliere alla generalità (Paoli 1792, 1799, 1803, 1804; Brunacci 1808; Poletti 1824)
 senza limitare la generalità (Pezzi 1792; Forni 1811; Frullani 1816)
 salva la generalità (Paoli 1799, 1803, 1804; Forni 1811)
 senza alterare la generalità (Brunacci 1804, 1807; Magistrini 1806).
English (18091830):
 without losing its generality (*Legendre 1809; *Lacroix 1816; Wilder 1830; Morton 1830)
 without affecting the generality (*Lacroix 1816; Ryan 1828)
 without diminishing the generality (Morton 1830)
 without detracting from the generality (Morton 1830).
Summing up:
 Clearly Euler had a significant role in popularizing the expression after 1740.
 Yet there remains, for now, the puzzle of the missing link between Euler and anything published by Leibniz (or any precursor or successor of Leibniz).
End Note: The French also often expressed the same idea by simply writing “ce qui est permis”. Such rhetorical turns of phrase are briefly discussed and called “indifferent hypotheses” in François Rostand, Sur la clarté des démonstrations mathématiques (Vrin, Paris, 1962, pp. 79–80).

5$\begingroup$ In Kant's "Kritik der reinen Vernunft", 1781: "unbeschadet seiner Allgemeinheit" $\endgroup$ – Moritz Firsching May 8 '14 at 12:00

2$\begingroup$ @Torsten Schoeneberg: The book is long, but it is easy to do a full text search. The quote is on the beginning of page 714 in the first edition. II. Teil, 1. Hauptstück, 1. Abschnitt $\endgroup$ – Moritz Firsching May 8 '14 at 19:47

2$\begingroup$ I can't read the German, Latin or French in the linked Bernoulli letter (yes, in one letter, oft in the same sentence!), but the sentiment comes across. $\endgroup$ – David Roberts May 9 '14 at 8:10

2$\begingroup$ @MarcelT. I'm curious, what would you use then instead of WLOG for that purpose? $\endgroup$ – David FernandezBreton May 25 '14 at 18:43

5$\begingroup$ @DavidFernandezBreton WMATWAITCBATOCAT $\endgroup$ – David Steinberg Feb 25 '15 at 5:33
These are the earliest citations I could find for the phrase "loss of generality" in JSTOR. Note how they all slightly differ from the strict "without loss of generality" form. Also note how they're all from William R. Hamilton.
... and many of the new partial differential coefficients vanish, without producing, by this simplification, any real loss of generality
 Third Supplement to an Essay on the Theory of Systems of Rays, William R. Hamilton, The Transactions of the Royal Irish Academy, Vol. 17, (1831), pp. vx, 1144
Mr. Jerrard has therefore accomplished a very remarkable simplification of this general problem, since he has reduced it to the problem of discovering two real functions of two arbitrary real quantities, by showing that, without any real loss of generality, it is permitted to suppose ...
 On the Argument of Abel, Respecting the Impossibility of Expressing a Root of Any General Equation above the Fourth Degree, by Any Finite Combination of Radicals and Rational Functions, William R. Hamilton, The Transactions of the Royal Irish Academy, Vol. 18, (1839), pp. 171259
And if we farther simplify the formulae by supposing $ a = 1, b=0, c=0, d=0$, which will be found in the applications to involve no essential loss of generality ...
 Researches Respecting Quaternions. First Series, William Rowan Hamilton, The Transactions of the Royal Irish Academy, Vol. 21, (1846), pp. 199296
EDITED TO ADD:
Here are the dates and authors for the first 20 instances of "loss of generality" in the above JSTOR search:
1831 Hamilton
1839 Hamilton
1846 Hamilton
1848 Stokes
1854 Cayley
1855 Cayley
1856 Thomson
1857 Cayley
1860 Donkin
1862 Cayley
1863 Schlafli, as communicated (translated?) by Cayley
1864 Cayley
1866 Sylvester
1867 Cayley
1867 Cayley
1868 Cayley
1870 Strutt
1871 Russell
1873 Williamson
1874 Cayley
While the three earliest citations are due to Hamilton, fully half of the first twenty instances in the JSTOR database are due to Cayley. Of course, the JSTOR database is not comprehensive; in particular, it does not include The Transactions of the Cambridge Philosophical Society which contains the earlier Stokes citation that Brendan McKay found.

1$\begingroup$ @FrancoisZiegler  and what about in Latin? $\endgroup$ – David Roberts May 8 '14 at 6:10

$\begingroup$ This completely explains the CayleyHamilton theorem: WLOG, you can replace the eigenvalue by its matrix :p $\endgroup$ – Vidit Nanda May 8 '14 at 12:17
"But in dealing with given quantics, we may without loss of generality consider the covariant as a function of the like form with the quantic,..." — Arthur Cayley, An Introductory Memoir upon Quantics, Philosophical Transactions of the Royal Society of London, Vol. 144, (1854), pp. 245258.
The equivalent phrase "without losing generality" appears earlier in Stokes, On the steady motion of incompressible fluids, Transactions of the Cambridge Philosophical Society, 7 (1842) 439453.
The expression seems to have been used by authors in Cambridge in $1838$. The oldest reference available online which I can find containing this is Transactions of the Cambridge Philosophical Society, by Cambridge Philosophical Society, $1900$, on page $270$, in the following papers  On the Simplest Algebraic Minimal Curves, and the derived Real Minimal Surfaces, Herbert Richmond; On certain Quintic Surfaces which admit of Integrals of the First Kind of Differentials, Arthur Berry; Diophantine Inequalities, G. B. Mathews, and almost all the other papers there.

3$\begingroup$ "Come into existence" may be a bit strong. Perhaps more conservatively, we may say that this data point indicates the earliest appearance in Google's corpus. In particular, its usage in that volume by multiple authors seems to argue against the proposition that this was the first time this phrase was ever used. $\endgroup$ – mhum May 8 '14 at 0:06

$\begingroup$ @mhum I agree with you that "Come into existence" may be a bit strong. However, the above is the earliest reference I can find where this term is used collectively by many authors, implying that it was used before $1804$. Perhaps the term was introduced in Cambridge, since if this is indeed the earliest reference, then since all the authors seem to be familiar with the term. $\endgroup$ – user62675 May 8 '14 at 0:10

$\begingroup$ False positive... this Transactions volume dates from 1900. $\endgroup$ – Francois Ziegler May 8 '14 at 0:18

11$\begingroup$ OTOH, Google has "ohne Beschränkung der Allgemeinheit" in 1829. $\endgroup$ – Francois Ziegler May 8 '14 at 0:31

7$\begingroup$ I don't see your evidence for 1838. Ngrams prove nothing as they are based on the same erroneous publication years. Also, the German example is welcome. $\endgroup$ – Brendan McKay May 8 '14 at 0:43