History of "without loss of generality"

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 (1792-1824):

• 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 (1809-1830):

• 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).

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. v-x, 1-144

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. 171-259

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. 199-296

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.

"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. 245-258.

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) 439-453.