350 degrees is the optimum temperature of the Maillard reaction. This is the process of browning food to gain flavor without being too hot to instantly burn it.
This has been scientifically understood since the 1920’s when the first ovens with thermostats were developed.
But the basics of this reaction have been understood for centuries. Prior to the invention of the oven thermostat recipes called for an oven to be low, med, or hot but with a cast iron or clay oven that definition varied.
A ‘medium’ temperature in a cast iron oven is around 350 degrees.
In most baked foods, the goal is to achieve a nice browning on the outside and cook things properly inside. To get this most foods like cookies, breads and meats (and toast) rely on the Maillard reaction.
The Maillard reaction occurs at around 300F (there is a range 280-320 ish) and most ovens are imprecise enough that setting it to 350 is more reliable. And there needs to be a buffer because 300 is the target temp of the food to be cooked not the temperature as measured by an external sensor.
If you bake below this temperature, things will still cook but will be left pretty colorless (and lacking in the taste that develops through the browning)
The temperature cannot be too high (for most foods) because big pieces of meat and breads might burn/char on the outside before the inside is safely and tastefully cooked. So 350F is used as a compromise to give the best combination of cooking thoroughly and still having enough browning.
For some breads it would be safer to set a modern oven to around 325. For flat breads that cook quickly, possibly higher. For roasts, it is normal to start at a high temp 450 and then reduce the temps to allow the meats to cook through.
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