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I am searching for Newton's original formulation of the Law of Cooling. In his article (p.140-143), Newton said:

  • "For the heat which the hot iron communicates in a given time to cold bodies which are near it, that is, the heat which the iron loses in a given time, is proportional to the whole heat of the iron."

  • "For thus equal parts of the air are warmed in equal times and carry away a heat proportional to the heat of the iron."

However, he does not really state that the change in temperature is proportional to the difference in temperatures between the body and the environment. Therefore I wonder, where/how did Newton originally formulate his law of cooling or do the quotes above imply the law and I am just quibbling?

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    $\begingroup$ Given Newton's terminology, "the heat which the iron loses in a given time" is the rate of change of the temperature, and "the whole heat of the iron" is the temperature difference between the iron and the ambient. He adds "if equal time of cooling be taken, the degrees of heat will be in geometrical proportion", which confirms that $\dot{T}\sim-(T-T_a)$ is what he meant, see Besson, History of the Cooling Law. Distinction between heat and temperature was made only later $\endgroup$ – Conifold Mar 22 at 7:42
  • $\begingroup$ I know that the terms heat and temperature were used synonymously at Newton's time, but it still just confuses me. Besson puts it like this: "the excess of the degrees of the heat … were in geometrical progression when the times are in an arithmetical progression (by 'degree of heat' Newton meant what we now call 'temperature', so that 'excess of the degrees of the heat' means ‘temperature difference')." This explanation cleared it up for me so thanks for pointing it out! $\endgroup$ – David A. Mar 22 at 9:37
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This explanation helped me understand Newton's language and answered my question:

"the excess of the degrees of the heat … were in geometrical progression when the times are in an arithmetical progression (by 'degree of heat' Newton meant what we now call 'temperature', so that 'excess of the degrees of the heat' means ‘temperature difference')."

Besson's article

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