  Carath\'eodory's formalism for classical thermodynamics is a rich alternative
approach to this theory, although unpopular with students and physics
professors. This approach dispenses with the content of thermal machines for
the presentation of the second law of thermodynamics. In this paper, we discuss
Carath\'eodory's formalism historically, and show how Carath\'eodory's axiom of
the second law of thermodynamics is derived, didactically, from the Clausius
principle and the Kelvin principle. In addition, also providing an objective
character for this paper, in the sense of seeking to popularize the teaching of
Carath\'eodory's formalism in disciplines of classical thermodynamics at
undergraduate level, we guide the reader to obtain the entropy and mathematical
content of the second law of thermodynamics through this formalism. Finally,
considering the wide reviewed literature, the proof we gave for deducing the
Carath\'eodory's axiom of the second law of thermodynamics from the Clausius
principle is new.
