PART 2A: Minkowski's words are in boldface.
"Before we enter into these questions, let us first make an important observation. When we individualize space and time in any manner, then a straight line parallel to the t-axis corresponds as a world-line to a substance-point at rest, a straight line inclined to the t-axis corresponds to a uniformly moving substance-point, and a world-line curved at will corresponds to a not-uniformly moving substance-point.
"If we consider the world-line passing through any world-point x, y, z, t, and if we there find it parallel to any radius vector OA′ of the above mentioned hyperboloid sheet, we may introduce OA′ as the new time-axis, and in the new conception of space and time thus obtained substance appears at rest at the world-point in question."
The reader might wish to consult the non-technical Relativity Tutorial by Ned Wright of UCLA to see a variety of space-time diagrams illustrating these points.
"Let us now introduce this fundamental axiom:
is always positive or, what amounts to the same thing, every velocity v is always less than c.
"According to this, c would exist as upper limit for all substance velocities and in this fact would lie the deeper significance of the magnitude c."
Consider the case with only an x space-axis, as before. Clearly, if we can write ct² - x² = 1, with both x and ct real, we must have ct > x. This must also be true of small changes in ct and x. Therefore velocity divided by c, the ratio of a small change in x to a small change in ct, must be less than 1. Therefore the velocity itself must be less than c.
"In this other form the axiom has in it something which at first sight is unsatisfactory."
The idea of a maximum velocity which nothing can exceed is philosophically disturbing, rather like the idea of an edge to the universe. If one reaches the edge of the world and takes another step, what happens? If one reaches the natural speed-limit and tries to accelerate, what happens?
"But we must consider that now a modified mechanics will supersede the old --- one into which will enter the square root of the above combination of differentials of the second degree, so that cases involving velocities exceeding that of light will only play some such part as figures with imaginary coordinates play in geometry."
One of the main themes of Eighteenth- and Nineteenth-Century mathematics is the gradual (and reluctant) inclusion of counterintuitive objects -- negative numbers, imaginary numbers, transfinite numbers -- into the world of acceptable theory. Although visual imagery came to be associated with most of these objects, everyone knew that such "geometrical interpretations" were secondary. By associating a class of forbidden and un-visualisable velocities with the imaginaries, Minkowski avoids having to deal with them. Later in the talk, he will expand the role of imaginary numbers to the very centre of relativity.
"The impulse and actual motive for the assumption of the groupG(c) originated through the fact that the differential equation for the transmission of light-waves in empty space is actually characterized by the group G(c). (What is practically an application of this fact is to be found as early as 1887 in a contribution by W. Voigt in Nachrichten der K. Gesselschaft der Wissenschaften zu Göttingen, mathematisch-physikalische Klasse, page 41.)"
Woldemar Voigt, studying the propagation of waves in a moving medium, discovered the Lorentz transformation in 1887! Lorentz himself remarked [Theory of Electrons, Article 169, Footnote 1]: "In a paper ... which to my regret has escaped my notice all these years, Voigt has applied to equations of the form (6) (§ 3 of this book) a transformation equivalent to the formulæ (287) and (288). The idea of the transformations used above (and in § 44) might therefore have been borrowed from Voigt and the proof that it does not alter the form of the equations for the free ether is contained in his paper." Voigt, still alive and active in 1908, is one of Minkowski's senior colleagues at Göttingen.
By coincidence, a paper about Voigt and the Lorentz transformation has appeared on arXiv this very week: On the Origin of the Lorentz Transformation by Wolfgang Engelhardt [2013/03].
"On the other hand the concept of rigid bodies has a meaning only in a mechanics with the group G(∞). If we have an optics with G(c) and if on the other hand rigid bodies existed, it is easy to perceive that by the two hyperboloid sheets belonging to G(c) and to G(∞) a definite t-direction would be determined, and this would have the further result that we must be able to detect by means of suitable rigid optical instruments in the laboratory, a change in the phenomena at different orientations with reference to the direction of the earth's motion. All attempts, however, at this detection, especially a famous interference experiment of Michelson, had a negative result.
"To find an explanation for this, H. A. Lorentz constructed a hypothesis the value of which depends on the invariance of optics for the group G(c). According to Lorentz, every body in motion suffers a contraction in the direction of motion, and for the velocity v this contraction is in the ratio
TIME AND SPACE, by Hermann Minkowski