# History of “without loss of generality”

"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.)

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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? – Francois Ziegler May 10 '14 at 21:36
@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. – Mark S. May 11 '14 at 16:50
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.) – Asaf Karagila Feb 25 at 0:39

I think one of the reasons JSTOR doesn't have "loss of generality" before 1831 is that fewer scientists wrote in English. But one finds (with minor variants merged, and translations *starred)

The French also often expressed the same idea by simply writing "ce qui est permis".

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 him and anything published by Leibniz (or any precursor or successor).
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In Kant's "Kritik der reinen Vernunft", 1781: "unbeschadet seiner Allgemeinheit" – Moritz Firsching May 8 '14 at 12:00
@TorstenSchoeneberg You are quite right -- I just couldn't resist linking these uproarious letters. – Francois Ziegler May 8 '14 at 14:10
Nice! BTW I'd say "ce qui est permis" and "salva generalitate" have quite different meanings. "Without loss of generality" indicates that a proof to be given for a specific case can be applied mutatis mutandis to all other cases ("If three balls are each colored either red or blue, then there is a monochromatic tuple. Pf: Suppose without loss of generality that the first ball is red..."). Whereas "ce qui est permis" could also indicate "we may assume that we are in this case, because all the other cases are trivial" which, amusingly, is how I see WLOG (incorrectly) used more often than not. – Marcel T. May 9 '14 at 4:44
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. – David Roberts May 9 '14 at 8:10
@DavidRoberts: Even in the same word (piècen, satisfacirt, ...). – Emil Jeřábek May 10 '14 at 11:48

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

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.

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@FrancoisZiegler - and what about in Latin? – David Roberts May 8 '14 at 6:10
This completely explains the Cayley-Hamilton theorem: WLOG, you can replace the eigenvalue by its matrix :p – 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. 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.

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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.

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"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. – mhum May 8 '14 at 0:06
@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. – Sanath K. Devalapurkar May 8 '14 at 0:10
False positive... this Transactions volume dates from 1900. – Francois Ziegler May 8 '14 at 0:18
OTOH, Google has "ohne Beschränkung der Allgemeinheit" in 1829. – Francois Ziegler May 8 '14 at 0:31
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. – Brendan McKay May 8 '14 at 0:43