It's been said that in the past, when only the nobility were able to eat an optimal amount of food, that a fat man or woman was considered attractive because it was a way of advertising their wealth. Is there any truth behind this?
There isn't much truth behind it.
Across Europe through the high middle ages (and into the late period), the standard of beauty was fairly consistent: the ideal men and women were willowy and thin, with fair hair and skin and red lips and cheeks. Women were to have a high forehead, curved eyebrows, light eyes, small breasts, and a posture that pushed the stomach forward. In an Old French reworking of a tale from Ovid's Metamorphoses, the beautiful Philomena is described as having small, high breasts, thin thighs, and narrow hips. Art of the period tends to reflect this, with humans, saints, and angels represented with long, thin figures. There was more leeway for men - a beautiful youth could ripen into a handsome man, whose good features, strength, and nobility of manner could excuse excess weight; for women, the period of maidenhood was essentially it - but despite the pre-industrial society, the beauty standard inclined toward thinness.
On the other hand, during the 19th century, while the details of the fashionable figure changed several times, a soft roundness of the face, body, and arms was ideal. (Examples: "Lady Hannah Chetwynd", William Owen, ca. 1815; "Feodora, Princess of Hohenlohe-Langenburg with her daughter Princess Adelaide", Sir George Hayter, 1840; "The Bride", Abraham Solomon, 1856; Actress and socialite Lillie Langtry photographed by William Downey, 1885) Certainly a small waist was prized, and the word "slender" was used frequently in descriptions of fictional beauties, but the ideals encompassed both smallness and softness. Corsetry of the period works on the principle of redistributing fat and flesh, so this is not as much of a contradiction as it first appears, but the matter is still complicated enough that it can't be distilled down to either heaviness or thinness being more attractive. It was possible to be too far along the spectrum in either direction. But it is fairly clear that the muscular, thin arms and thigh gaps considered beautiful today would not have been thought so then.
Sources:
For Appearance' Sake: The Historical Encyclopedia of Good Looks, Beauty, and Grooming, by Victoria Sherrow (well-sourced)
Bodytalk: When Women Speak in Old French Literature, by E. Jane Burns
Women and Gender in Medieval Europe: An Encyclopedia, "Beauty", by Kim Phillips (which also has a good list of sources for learning more)
Medieval Maidens: Young Women and Gender in England, C.1270-c.1540, by Kim Phillips
A follow-up question is: did this transition of beauty standards, if it really existed, occurred during industrialization? And why did it occur?
Art does a good job at representing beauty ideals. What's more difficult is to establish causation--be it class or other contextual factors like industrialization--behind changes in beauty ideals. It has been speculated that the rise in butter consumption in the early modern period, for example, produced an ideal of a larger body type. But it is exceedingly difficult to prove this, and the reality is always far more complicated than such a simplistic hypothesis would suggest; for example, in certain periods fatness is linked to lower class identity and so can not be a means of advertising wealth.
Art does indeed give very explicit statements of beauty ideals in the form of canons of proportions. For example, Albrecht Dürer, who researched and wrote on the canon of human proportions, employed such a canon in his Adam and Eve. Within this one image there are different standards for men and women; both are based on texts and images from classical antiquity.
However, difficulties arise from the fact that there was never one universal abstract ideal of beauty, but rather always a multiplicity of gender-, class-, and context-specific ideals. In much of Western art, the female nude is much more allegorical than we think, and much less of a model to be emulated in real life. When people mention the "heavier" body ideal of the early modern period they are usually thinking of Rubens's nudes. But Rubens was painting variations on classical ideals, embodied in classical statuary, while working in a mode that emphasized abundance and fertility as allegorical concepts. These are allegorical bodies; in contrast, his female portraiture emphasizes the slender, tightly corseted lines of prevailing courtly fashions and embody a very different beauty ideal. His portrait of his wife was not a boudoir shot but an allegory.
The final complication is that art is like fashion: it is nearly impossible to attribute causes to stylistic changes for either. Both represent fairly arbitrary beauty ideals that change simply for the sake of change; see Barthes The Fashion System for an account of this.
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