It appears to be the Theory of Games and Economic Behavior (1944) by John von Neumann & Oskar Morgenstern. I have the 1953 edition which is counted as "3d", but by reading the included introductions to the 2nd and to the 3d editions, it appears that nothing of substance has been changed in chapter 3, where I locate the issue of equivalence up to linear transformations.
In sub-chapter 3.4 the authors discuss in general the issue of equivalence under transformations for any system of quantities, mainly from physics. They end this sub-chapter by writing (bold my emphasis)
"One may take the attitude that the only "natural" datum in this domain (utility) is the relation "greater," i.e. the concept of
preference. In this case utilities are numerical up to a monotone
transformation. This is, indeed, the generally accepted standpoint in
economic literature, best expressed in the technique of indifference
curves. To narrow the system of transformations it would be
necessary to discover further "natural" operations or relations in the
domain of utility. Thus it was pointed out by Pareto (in V. Pareto, Manuel d'Economie Politique, Paris, 1907, p. 264) that an equality
relation for utility differences would suffice; in our terminology it
would reduce the transformation system to the linear
transformations. However, since it does not seem that this relation is
really a "natural" one—i.e. one which can be interpreted by
reproducible observations—the suggestion does not achieve the
In sub-chapter 3.5 they start by writing
"The failure of one particular device need not exclude the possibility of achieving the same end by another device. Our
contention is that the domain of utility contains a "natural"
operation which narrows the system of transformations to precisely
the same extent as the other device would have done(*). This is the
combination of two utilities with two given alternative probabilities
$a, 1 — a, (0 < a < 1)$ as described in 3.3.2. The process is so
similar to the formation of centers of gravity mentioned in
3.4.3. that it may be advantageous to use the same terminology."
(*) Note: The "failure of one particular device" refers to the suggestion of Pareto mentioned in the previous passage. So what they are saying here, is that, apart from ordinal preference, there is another "natural operation" related to utility (but not the one suggested by Pareto), that will nevertheless achieve the same purpose as Pareto's suggestion, i.e. to narrow the admissible transformations to linear ones only.
And which is this "natural operation"?
"...The combination of two utilities with two given alternative probabilities $a, 1 — a, (0 < a < 1)$"
which ushers us into the world of expected utility.
The details come next, together with Appendix "The Axiomatic Treatment of Utility" at the end of the book.
PS: By rejecting Pareto's suggestion, the authors make clear that expected utility is not cardinal utility, since what they reject is the existence of "an equality relation between utility differences". This rejection preserves fundamentally the ordinal nature of expected utility.