In his 2014 book, Giovanni Ferraro writes at beginning of chapter 1, section 1 on page 7:
Capitolo I
Esempi e metodi dimostrativi
- Introduzione
In The Calculus as Algebraic Analysis, Craig Fraser, riferendosi
all'opera di Eulero e Lagrange, osserva:A theorem is often regarded as demonstrated if verified for several
examples, the assumption being that the reasoning in question could be
adapted to any other example one chose to consider (Fraser [1989, p.
328]).Le parole di Fraser colgono un aspetto poco indagato della matematica
dell'illuminismo.
I am not fluent in Italian but the last sentence seems to indicate that Ferraro endorses Fraser's position as expressed in the passage cited in the original English without Italian translation.
I was rubbing my eyes as I was reading this so I decided to check in Fraser's original, thinking that perhaps the comment is taken out of context. I found the following longer passage on Fraser's page 328 quoted by Ferraro:
The calculus of EULER and LAGRANGE differs from later analysis in its
assumptions about mathematical existence. The relation of this
calculus to geometry or arithmetic is one of correspondence rather
than representation. Its objects are formulas constructed from
variables and constants using elementary and transcendental operations
and the composition of functions. When EULER and LAGRANGE use the term
"continuous" function they are referring to a function given by a
single analytical expression; "continuity" means continuity of
algebraic form. A theorem is often regarded as demonstrated if
verified for several examples, the assumption being that the reasoning
in question could be adapted to any other example one chose to
consider.
Let us examine Fraser's hypothesis that in Euler and Lagrange, allegedly "a theorem is often regarded as demonstrated if verified for several examples."
I don't see Fraser presenting any evidence for this. Now Wallis sometimes used a principle of "induction" in an informal sense that a formula verified for several values of $n$ should be true for all $n$, but for this he was already criticized by his contemporaries, a century before Euler and Lagrange.
Several articles were recently published examining Euler's proof of the infinite product formula for the sine function. The proof may rely on hidden lemmas, but it is a sophisticated argument that is a far cry from anything that could be described as "verification for several examples."
It seems to me that this passage from Fraser is symptomatic of an attitude of general disdain for the great masters of the past. Such an attitude unfortunately is found among a number of received historians. For example, we find the following comment:
Euler's attempts at explaining the foundations of calculus in terms of differentials, which are and are not zero, are dreadfully weak.
(p. 6 in Gray, J. “A short life of Euler.'' BSHM Bull. 23 (2008), no.1, 1–12).
In a similar vein, in a footnote on 18th century notation, Ferraro presents a novel claim that
for 18th-century mathematicians, there was no difference between finite and infinite sums.
(footnote 8, p. 294 in Ferraro, G. “Some aspects of Euler's theory of series: inexplicable functions and the Euler-Maclaurin summation formula.'' Historia Mathematica 25, no. 3, 290–317.)
Far from being a side comment, the claim is emphasized a decade later in the Preface to his 2008 book:
a distinction between finite and infinite sums was lacking, and this gave rise to formal procedures consisting of the infinite extension of finite procedures
(p. viii in Ferraro, G. The rise and development of the theory of series up to the early 1820s. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, New York.)
Grabiner doesn't hesitate to speak about
shaky eighteenth-century arguments
(p. 358 in Grabiner, J. “Is mathematical truth time-dependent?'' Amer. Math. Monthly 81 (1974), 354–365); it is difficult to evaluate her claim since she does not specify the arguments in question.
Instead of viewing Fraser's passage as problematic, Ferraro opens his book with it, which is surely a sign of endorsement. The attitude of disdain toward the masters seems to have permeated the field to such an extent that it has acquired the status of a sine qua non of a true specialist.
In my study of Euler I have seen sophisticated arguments rather than proofs by example, except for isolated instances such as de Moivre's formula. On the other hand Euler's oeuvre is vast.
Question. Can Euler be said to have proved theorems by example in other than a handful of exceptional cases, in any meaningful sense?
Note 1. Some editors requested examples of what I described above as a disdainful attitude toward the masters of the past on the part of some historians. I provided a couple of additional ones. Editors are invited to provide examples they have encountered; I believe they are ubiquitous.
Note 2. We tried to set the record straight on Euler in this recent article and also here.
Best Answer
There's some evidence that precisely the opposite can be said: that Euler is aware of the fallacies of proving theorems by example (of course, this does not necessarily mean he has never used it). One memorable instance is his Exemplum Memorabile Inductionis Fallacis, where he described how he was almost led to conjecture a recursive formula for a particular numerical sequence until he found that they disagreed on the 10th term. (There are other reasons for that formula to have been plausible; that and other topics are discussed in this article.)
(Incidentally the "right" formula is now quite well-known.)