Compared with the Euler beam theory, the Timoshenko beam theory has the advantages in solution accuracy and in applicable aspect ratios. However, it received some criticisms in two aspects: firstly, it is energy inconsistent, and secondly, it cannot be reduced to the Euler beam theory in a mechanics way. From the reduced standpoint of the higher-order displacement mode, the two lower-order beam theories endowed with a linear axial displacement are systematically studied. The present work shows that the shear free Euler beam theory is different from the first-order shear deformable beam theory because they are basded on different conditions, and hence they cannot be mutually transformed. Thus, it is uncecessary to reduce the Timoshenko beam theory, which poses as an improved first-order shear deformable beam theory, to the Euler beam theory. The lower-order beam theory is derived on the assumption of the higher-order displacement mode, and the physical essence is revealed that, irrespective of the displacement mode, the Timoshenko beam theory is in essence such a lower-order beam theory.