Relativity may be derived as an algebraic relation among differentials. Consider motion in the *x* spatial dimension, with a differential displacement, *dx*, differential velocity displacement, *dv*, and arc (elapsed) time *t*:

*dx*² = (*dx*/d*t*)²d*t*² = *dv*²d*t*² = *d*(*vt*)².

Let there be a constant, *c*:

*dx*² = *d*(*vt*)² = *d*(*cvt*)²/*c*² = *d*(*ct*)² (*v*/c)² = *d*(*ct*)² (1 – (1 – *v*²/*c*²)).

Let *γ*² = 1/(1 – *v*²/*c*²). Then

*dx*² = *d*(*vt*)² = *d*(*ct*)² (1 – 1/*γ*²) = *d*(*ct*)² – *d*(*ct*)²/*γ*² = *d*(*ct*)² – *d*(*ct*/*γ*)² = *d*(*ct*)² – *d*(*cτ*)²,

where *τ* = *t*/*γ*. This may be rearranged as

*d*(*cτ*)² = *d*(*ct*)² – *dx*²,

which equals the Lorentz invariant, *d*(*cτ*)² = *ds*².

Alternately, consider motion in the *t* temporal dimension, with a differential displacement, *dt*, differential velocity displacement, *dv*, and arc length *x*:

*dt*² = (*dt*/d*x*)²d*x*² = *d*(1/*v*)²d*x*² = *d*(*x*/*v*)².

Let there be a constant, *c*:

*dt*² = *d*(*x*/*v*)² = *d*(*x*/*cv*)² *c*² = *d*(*x*/*c*)² (c/*v*)² = *d*(*x*/*c*)² (1 – (1 – *c*²/*v*²)).

Let *λ*² = 1/(1 – *c*²/*v*²). Then

*dt*² = *d*(*x*/*v*)² = *d*(*x*/*c*)² (1 – 1/*λ*²) = *d*(*x*/*c*)² – *d**x*²/(*λc*)² = *d*(*x*/*c*)² – *d*(*x*/*λc*)² = *d*(*x*/*c*)² – *d*(*σ*/*c*)²,

where *σ* = *x*/*λ*. This may be rearranged as

*d*(*σ*/*c*)² = *d*(*x*/*c*)² – *dt*²,

which equals the Lorentz invariant, *d*(*σ*/*c*)² = *dw*².

Equivalently, consider motion in the *ξ* temporal dimension, with a differential displacement, *dξ*, differential *celerity* displacement, *du*, and elapsed space *x*:

*dξ*² = (*dξ*/d*x*)²d*x*² = *du*²d*x*² = *d*(*ux*)².

Let there be a constant, *k* = 1/*c*:

*dξ*² = *d*(*ux*)² = *d*(*kux*)²/*k*² = *d*(*kx*)² (*u*/*k*)² = *d*(*kx*)² (1 – (1 – *u*²/*k*²)).

Let *λ*² = 1/(1 – *u*²/*k*²). Then

*dξ*² = *d*(*ux*)² = *d*(*kx*)² (1 – 1/*λ*²) = *d*(*kx*)² – *d*(*kx*)²/*λ*² = *d*(*kx*)² – *d*(*kx*/*λ*)² = *d*(*kx*)² – *d*(*kσ*)²,

where *σ* = *x*/*λ*. This may be rearranged as

*d*(*kσ*)² = *d*(*kx*)² – *dξ*²,

which equals the Lorentz invariant, *d*(*kσ*)² = *dw*².