The Lorentz transformation is derived merely from the properties of space and time when spaceis empty and Galileo relativity. Additional postulates about the speed of light, reciprocity, and other ones are not necessary. Straight world lines are bijectively mapped onto straight world lines. This known fact is exploited in an illustrative manner. This is extremely useful for teaching special relativity, in particular, at an elementary level. Moreover, the approach described here, (i), provides an example of strict physical thinking, (ii), corrects a widespread erroneous belief, see over-next paragraph, and, (iii), presents an elementary introduction to the largely unknown hyperbolic rotations (the common rotations are circular). The transformation to be found is represented as a kind of rotation times a Lorentzian ‘scale factor’. This crucially simplifies the calculations and is much easier to grasp than a rather abstract ansatz with unknown coefficients. The rotation is proven to be hyperbolic rather than circular. After that, the scale factor turns out to equal unity in a most direct manner. The reciprocity property of the transformation is obtained as a by-product. Not special relativity makes an additional assumption for justifying the appearance of a seemingly additional natural constant, the speed of light in vacuum c, but classical mechanics does whence c disappears. Two common basic assumptions of classical mechanics lead not to the Galileo but to the Lorentz transformation. The existence of a maximum speed of bodies is shown to be a purely kinematic effect, too. Einstein’s second postulate is obtained as a by-product.