"Proof by example" is a technique used by Euclid, who often proved results that hold e.g. for n integers in a typical case, say for 3 integers, as well as by Diophantus, who had to choose values for his parameters due to his lack of algebraic notation. I regard both versions as complete proofs.
This is apparently not what Fraser is referring to; Euler did generalize from examples to theorems in his Algebra, where he transferred correct results from "rings of integers" ${\mathbb Z}[i]$ to general quadratic rings without proof; but Euler wrote his algebra when he was old and completely blind, and perhaps it is fair to say that Euler was collecting evidence for his "method" rather than regarding these examples as proofs. I am not aware of a single example where Euler explicitly said that he regarded the verification of examples as a proof, but I only have read his number theoretical work in detail. The idea that Lagrange proved results by examples is ridiculous.