The formula \Eq{(6.2)} for the composition of velocities in the same straight \index{Addition of velocities}% \index{Composition of velocities}% line may be written \[ \tanh^{-1} v/c = \tanh^{-1} v'/c - \tanh^{-1} u/c. \Tag{(7.3)} \] The quantity $\tanh^{-1} v/c$ has been called by Robb the \emph{rapidity} corresponding \index{Rapidity}% to the velocity~$v$. Thus \Eq{(7.3)}~shows that relative rapidities in the same direction compound according to the simple addition-law. Since $\tanh^{-1} 1 = \infty$ , the velocity of light corresponds to infinite rapidity. We cannot reach infinite rapidity by adding any finite number of finite rapidities; therefore we cannot reach the velocity of light by compounding any finite number of relative velocities less than that of light. \PageSep{23} There is an essential discontinuity between speeds greater than and less than that of light which is illustrated by the following example. If two points \index{Light!velocity of}% move in the same direction with velocities \[ v_{1} = c + \epsilon,\quad v_{2} = c - \epsilon \] respectively, their relative velocity is by~\Eq{(6.2)} \[ \frac{v_{1} - v_{2}}{1 - v_{1}v_{2}/c^{2}} = \frac{2\epsilon}{1 - (c^{2} - \epsilon^{2})/c^{2}} = \frac{2c^{2}}{\epsilon}, \] which tends to infinity as $\epsilon$~is made infinitely small! If the fundamental velocity is exactly $300,000$~km.\ per~sec., and two points move in the same direction with speeds of $300,001$ and $299,999$~km.\ per~sec., the speed of one relative to the other is $180,000,000,000$ km.\ per~sec. The barrier at $300,000$~km.\ per~sec.\ is not to be crossed by approaching it. A particle which is aiming to reach a speed of $300,001$~km.\ per~sec.\ might naturally hope to attain its object by continually increasing its speed; but when it has reached $299,999$~km.\ per~sec., and takes stock of the position, it sees its goal very much farther off than when it started. A particle of matter is a structure whose linear extension is timelike. We might perhaps imagine an analogous structure ranged along a spacelike track. That would be an attempt to picture a particle travelling with a velocity greater than that of light; but since the structure would differ fundamentally from matter as known to us, there seems no reason to think that it would be recognised by us as a particle of matter, even if its existence were possible. For a suitably chosen observer a spacelike track can lie wholly in an instantaneous space. The structure would exist along a line in space at one moment; at preceding and succeeding moments it would be non-existent. Such instantaneous intrusions must profoundly modify the continuity of evolution from past to future. In default of any evidence of the existence of these spacelike particles we shall assume that they are impossible structures. \Section{8.}{Immediate consciousness of time} \index{Time!immediate consciousness of}%