Geometry was developed by the ancient Greeks in the language of *length*, but it is an abstraction that may be applied to anything that conforms to its definitions and axioms. Here we apply it to *duration*. We will use Brossard’s “Metric Postulates for Space Geometry” [*American Mathematical Monthly*, Vol. 74, No. 7, Aug.-Sep., 1967, pp. 777-788], which generalizes the plane metric geometry of Birkhoff to three dimensions. First, Brossard:

Primitive notions. *Points* are abstract undefined objects. Primitive terms are: point, distance, line, ray or half-line, half-bundle of rays, and angular measure. The set of all points will be denoted by the letter *S* and some subsets of *S* are called *lines*. The plane as well as the three-dimensional space shall not be taken as primitive terms but will be constructed.

Axioms on points, lines, and distance. The axioms on the points and on the lines are:

E_{1}. *There exist at least two points in S.*

E_{2}. *A line contains at least two points.*

E_{3}. *Through two distinct points there is one and only one line.*

E_{4}. *There exist points not all on the same line.*

A set of points is said to be *collinear* if this set is a subset of a line. Two sets are *collinear* if the union of these sets is collinear. The axioms on distance are:

D_{1}. *If A and B are points, then d(AB) is a nonnegative real number.
*

D_{2}. *For points A and B, d(AB) = 0 if and only if A = B.*

D_{3}. *If A and B are points, then d(AB) = d(BA).*

For distinct points *A, B, C; B* is *between* *A* and *C* if and only if *d*(*AB*) + *d*(*BC*) = *d*(*AC*). If *B* is between *A* and *C* we write [*A, B, C*]. If *A* ≠ *B*, the class of points *P* between *A* and *B* is the *interval AB*. If *A* ≠ *O*, the class of points *P* ≠ *O* such that *d*(*AP*) = | *d*(*OA*) − *d*(*OP*) | is the *ray OA* (or *half-line OA*). *O* is the *end-point* of the ray *OA*. We have furthermore the two following axioms concerning the distance.

D_{4}. *Three distinct points belong to a line if and only if one of them is between the other two.*

D_{5}. *On each ray OA and for each positive real number β there exists a point B such that d(OB) = β.*

The axioms on angles, half-bundles, and the continuity axiom. An *angle* is an ordered couple of rays with the same end-point. The rays are the *sides* of the angle and the common end-point is the *vertex* of the angle. An angle is *flat* in case the sides are distinct and belong to a line. We shall introduce the measure ∠*rs* of an angle *rs* with the axioms A_{1}, A_{2}, A_{3}. The measure of an angle is intuitively conceived as the number obtained with the use of a protractor. If *A* ≠ *O* and *B* ≠ *O*, we write ∠*AOB* for ∠(ray *OA*)(ray *OB*).

A_{1}. *If rs is an angle, then ∠rs is a real number such that 0 ≤ ∠rs ≤ 180.*

A_{2}. *For an angle rs, ∠rs = 0 if and only if r = s*.

A_{3}. *If rs is an angle, then ∠rs = ∠sr.*

A *half-bundle ab*, where *a* and *b* are rays with the same end-point not belonging to a line, is the class of rays x such that x ≠a and *∠bx *= | *∠ab *− *∠ax |.* The common end-point of the rays is the *vertex* of the half-bundle. For the half-bundle *ab* we use the notation *HB*_{ab}. A ray *b* is between the rays *a* and *c* if and only if *a, b, c* have a common end-point, no two of them belong to a line, and *∠ab *+ *∠bc *= *∠ac*; if *b* is between *a* and *c* we write [*a, b, c*]. With those notations we have that x ∈ *HB*_{ab} if and only if [*a, b, x*] or [*a, x, b*] or *x* = *b*.

The axioms on the half-bundles are:

A_{4}. *Three distinct rays a, b, c belong to a half-bundle if and only if one of them is between the other two.*

A_{5}. *If β is a real number between 0 and 180, then there exists on each half-bundle HB _{oa} one ray b such that ∠ob = β.*

We shall now add the so-called continuity axiom.

Continuity Axiom. If [*a, b, c*] and if *A, C* are points on rays *a* and *c* respectively, then there exists a point *B* on ray *b* such that [*A, B, C*]. Conversely, if *a* and *c* are noncollinear rays with the same end-point *O*, *A* ∈ *a*, *C* ∈ *c*, and if *B* is such that [*A, B, C*] then the ray *OB = b* is between the rays *a* and *c* (i.e. [*a, b, c*]).

What follows is the above abstraction with terms that suggest *duration* instead of *length*.

Primitive notions. *Instants* are abstract undefined objects. Primitive terms are: instant, distance, timeline, ray or half-line, half-bundle of rays, and turn measure. The set of all instants will be denoted by the letter *S* and some subsets of *S* are called *timelines*. The timeplane as well as three-dimensional time shall not be taken as primitive terms but will be constructed.

Axioms on instants, timelines, and distance. The axioms on the instants and timelines are:

E_{1}. *There exist at least two instants in S.*

E_{2}. *A timeline contains at least two instants.*

E_{3}. *Through two distinct instants there is one and only one timeline.*

E_{4}. *There exist instants not all on the same timeline.*

A set of instants is said to be *collinear* if this set is a subset of a timeline. Two sets are *collinear* if the union of these sets is collinear. The axioms on distance are:

D_{1}. *If A and B are instants, then d(AB) is a nonnegative real number.
*

D_{2}. *For instants A and B, d(AB) = 0 if and only if A = B.*

D_{3}. *If A and B are instants, then d(AB) = d(BA).*

For distinct instants *A, B, C; B* is *between* *A* and *C* if and only if *d*(*AB*) + *d*(*BC*) = *d*(*AC*). If *B* is between *A* and *C* we write [*A, B, C*]. If *A* ≠ *B*, the class of instants *P* between *A* and *B* is the *interval AB*. If *A* ≠ *O*, the class of instants *P* ≠ *O* such that *d*(*AP*) = | *d*(*OA*) − *d*(*OP*) | is the *ray OA* (or *half-line OA*). *O* is the *end-instant *of the ray *OA*. We have furthermore the two following axioms concerning the distance.

D_{4}. *Three distinct instants belong to a timeline if and only if one of them is between the other two.*

D_{5}. *On each ray OA and for each positive real number β there exists an instant B such that d(OB) = β.*

The axioms on turns, half-bundles, and the continuity axiom. A *turn* is an ordered couple of rays with the same end-instant. The rays are the *sides* of the turn and the common end-instant is the *vertex* of the turn. A turn is *flat* in case the sides are distinct and belong to a timeline. We shall introduce the measure ∠*rs* of a turn *rs* with the axioms A_{1}, A_{2}, A_{3}. The measure of a turn is intuitively conceived as the number obtained with the use of a dial. If *A* ≠ *O* and *B* ≠ *O*, we write ∠*AOB* for ∠(ray *OA*)(ray *OB*).

A_{1}. *If rs is a turn, then ∠rs is a real number such that 0 ≤ ∠rs ≤ 180.*

A_{2}. *For a turn rs, ∠rs = 0 if and only if r = s*.

A_{3}. *If rs is a turn, then ∠rs = ∠sr.*

A *half-bundle ab*, where *a* and *b* are rays with the same end-instant not belonging to a timeline, is the class of rays x such that x ≠a and *∠bx *= | *∠ab *− *∠ax |.* The common end-instant of the rays is the *vertex* of the half-bundle. For the half-bundle *ab* we use the notation *HB*_{ab}. A ray *b* is between the rays *a* and *c* if and only if *a, b, c* have a common end-instant , no two of them belong to a timeline, and *∠ab *+ *∠bc *= *∠ac*; if *b* is between *a* and *c* we write [*a, b, c*]. With those notations we have that x ∈ *HB*_{ab} if and only if [*a, b, x*] or [*a, x, b*] or *x* = *b*.

The axioms on the half-bundles are:

A_{4}. *Three distinct rays a, b, c belong to a half-bundle if and only if one of them is between the other two.*

A_{5}. *If β is a real number between 0 and 180, then there exists on each half-bundle HB _{oa} one ray b such that ∠ob = β.*

We shall now add the so-called continuity axiom.

Continuity Axiom. If [*a, b, c*] and if *A, C* are instants on rays *a* and *c* respectively, then there exists an instant *B* on ray *b* such that [*A, B, C*]. Conversely, if *a* and *c* are noncollinear rays with the same end-instant *O*, *A* ∈ *a*, *C* ∈ *c*, and if *B* is such that [*A, B, C*] then the ray *OB = b* is between the rays *a* and *c* (i.e. [*a, b, c*]).