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Carath\'eodory's formalism for classical thermodynamics is a rich alternativeapproach to this theory, although unpopular with students and physicsprofessors. This approach dispenses with the content of thermal machines forthe presentation of the second law of thermodynamics. In this paper, we discussCarath\'eodory's formalism historically, and show how Carath\'eodory's axiom ofthe second law of thermodynamics is derived, didactically, from the Clausiusprinciple and the Kelvin principle. In addition, also providing an objectivecharacter for this paper, in the sense of seeking to popularize the teaching ofCarath\'eodory's formalism in disciplines of classical thermodynamics atundergraduate level, we guide the reader to obtain the entropy and mathematicalcontent of the second law of thermodynamics through this formalism. Finally,considering the wide reviewed literature, the proof we gave for deducing theCarath\'eodory's axiom of the second law of thermodynamics from the Clausiusprinciple is new.

On the Deduction of the Carath\'eodory's Axiom of the Second Law of Thermodynamics from the Clausius and Kelvin Principles