The following is based on *Classical Mechanics* by Kibble and Berkshire, 5^{th} ed., Imperial College Press, 2004, with the dual version indented and changes *italicized*.

p.2 The most fundamental assumptions of physics are probably those concerned with the concepts of space and time. We assume that space and time are continuous, that it is meaningful to say that an event occurred at a specific point in space and a specific instant of time, and that there are universal standards of length and time (in the sense that observers in different places and at different times can make meaningful comparisons of their measurements).

The most fundamental assumptions of physics are probably those concerned with the concepts of *length* and *duration*. We assume that *length* and *duration* are continuous, that it is meaningful to say that an event occurred at a specific point in length space and a specific instant of *duration space*, and that there are universal standards of length and *duration* (in the sense that observers in different places and at different times can make meaningful comparisons of their measurements).

In ‘classical’ physics, we assume further that there is a universal time scale (in the sense that two observers who have synchronized their clocks will always agree about the time of any event), that the geometry of space is Euclidean, and that there is no limit in principle to the accuracy with which we can measure all positions and velocities.

In dual ‘classical’ physics, we assume further that there is a universal *length* scale (in the sense that two observers who have *symbasalized* their clocks will always agree about the *base *of any event), that the geometry of *time* is Euclidean, and that there is no limit in principle to the accuracy with which we can measure all *chronations* and *legerities*.

p.3-4 To specify positions and time, each observer may choose a zero of the time scale, an origin in space, and a set of three Cartesian co-ordinate axes. We shall refer to these collectively as a *frame of reference*. The position and time of any event may they be specified with respect to this frame by the three Cartesian co-ordinates *x, y, z* and the time *t*. … The frames used by unaccelerated observers are called *inertial* frames.

p.3-4 To specify *chronations* and *base*, each observer may choose a zero of the *base *scale, an origin in *3D duration*, and a set of three Cartesian co-ordinate axes. We shall refer to these collectively as a *time frame of reference*. The *chronation* and *base *of any event may they be specified with respect to this *time* frame by the three Cartesian co-ordinates *ξ, η, ζ* and the *base **r*. … The *time* frames used by *inexpedienced* observers are called *facile time* frames.

Formally, an inertial frame may be defined to be one with respect to which any isolated body, far removed from all other matter, would move with uniform velocity.

Formally, a *facile* *time* frame may be defined to be one with respect to which any isolated body, far removed from all other *motion*, would move with uniform *legerity*.

p.4-5 It is often convenient to use a notation which does not refer explicitly to a particular set of co-ordinate axes. Instead of using Cartesian co-ordinates *x, y, z* we may specify the position of a point *P* with respect to a given origin *O* by the length and direction of the line *OP*. A quantity which is specified by a magnitude and a direction is called a *vector*; in this case the position vector **r** of *P* with respect to *O*.

p.4-5 It is often convenient to use a notation which does not refer explicitly to a particular set of co-ordinate axes. Instead of using Cartesian co-ordinates *ξ, η, ζ* we may specify the *chronation* of an *event* *Q* with respect to a given origin *O* by the *duration* and direction of the line *OQ*. A quantity which is specified by a magnitude and a direction is called a *vector*; in this case the *chronation* vector **t** of *Q* with respect to *O*.

We shall therefore consider first only small bodies which can be effectively located at a point, so that the position of each, at time t, can be specified by a position vector **r**(*t*).

We shall therefore consider first only small bodies which can be effectively located at an *instant*, so that the *chronation* of each, at *base r*, can be specified by a *chronation* vector **t**(*r*).

p.6 Each body is characterized by a scalar constant, its mass, *m*_{i}. Its momentum **p**_{i} is defined to be mass × velocity:

**p**_{i} = *m*_{i} **v**_{i}.

The equation of motion, which specified how the body will move is Newton’s second law (mass × acceleration = force):

**p**′_{i} = *m*_{i} **a**_{i} = **F**_{i}, (1.1)

where **F**_{i} is the total force acting on the body.

Each body is characterized by a scalar constant, its *vass*, *n*_{i}. Its *fulmentum* **q**_{i} is defined to be *vass* × *legerity*:

**q**_{i} = *n*_{i} **u**_{i}.

The equation of motion, which specified how the body will move is *the dual of* Newton’s second law (*vass* × *expedience* = *rush*):

**q**′_{i} = *n*_{i} **b**_{i} = **Y**_{i}, (1.1)

where **Y**_{i} is the total *rush* acting on the body.

p.7 The two-body forces **F**_{ij} must satisfy Newton’s third law, which asserts that ‘action’ and ‘reaction’ are equal and opposite,

**F**_{ji} = ‒**F**_{ij}. (1.3)

The two-body *rushes* **Y**_{ij} must satisfy *the dual of* Newton’s third law, which asserts that ‘action’ and ‘reaction’ are equal and opposite,

**Y**_{ji} = ‒**Y**_{ij}. (1.3a)

p.11 If we isolate the two bodies from all other matter, and compare their mutually induced accelerations, then according to (1.1) and (1.3),

*m*_{1}**α**_{1} = ‒*m*_{2}**α**_{2}, (1.7)

so that the accelerations are oppositely directed, and inversely proportional to the masses. If we allow two small bodies to collide, then during the collision the effects of more remote bodies are generally negligible in comparison with their effect on each other, and we may treat them approximately as an isolated system.

If we isolate the two *events* from all other *activities*, and compare their mutually induced *expediences*, then according to (1.1a) and (1.3a),

*n*_{1}**β**_{1} = ‒*n*_{2}**β**_{2}, (1.7a)

so that the *expediences* are oppositely directed, and inversely proportional to the *vasses*. If we allow two small *events* to *coincide*, then during the *coinciding* the effects of more remote *activities* are generally negligible in comparison with their effect on each other, and we may treat them approximately as an isolated system.

The mass ratio can then be determined from measurements of their velocities before and after the collision, by using (1.7) or its immediate consequence, the law of *conservation of momentum*,

*m*_{1}**v**_{1} + *m*_{2}**v**_{2} = constant, (1.8)

The *vass* ratio can then be determined from measurements of their legerities before and after the *coincidence*, by using (1.7a) or its immediate consequence, the law of *conservation of fulmentum*,

*n*_{1}**u**_{1} + *n*_{2}**u**_{2} = constant. (1.8a)

If we wish to separate the definition of mass from the physical content of equation (1.7), we may adopt as a fundamental axiom the following:

⇒ In an isolated two-body system, the accelerations always satisfy the relation **a**_{1} = ‒*k*_{21}**a**_{2}, where the scalar *k*_{21} is, for two given bodies, a constant independent of their positions, velocities and internal states.

If we wish to separate the definition of *vass* from the physical content of equation (1.7), we may adopt as a fundamental axiom the following:

⇒ In an isolated two-body system, the *expediences* always satisfy the relation **b**_{1} = ‒*k*_{21}**b**_{2}, where the scalar *k*_{21} is, for two given bodies, a constant independent of their *chronations*, *legerities* and internal states.

If we choose the first body to be a standard body, and conventionally assign it a unit mass (say *m*_{1} = 1 kg), the we may *define* the mass of the second to be *k*_{21} in units of this standard mass (here *m*_{2} = *k*_{21} kg).

If we choose the first body to be a standard body, and conventionally assign it a unit *vass* (say *n*_{1} = 1 kg^{–1}), the we may *define* the *vass* of the second to be *k*_{21} in units of this standard *vass* (here *n* = *k*_{21} kg^{–1}).

Note that for consistency, we must have *k*_{12} = 1/*k*_{21}. We must also assume of course that if we compare the masses of three bodies in this way, we obtain consistent results:

For any three bodies, the constants *k*_{ij} satisfy *k*_{31} = *k*_{32}*k*_{21}.

It then follows that for *any* two bodies, *k*_{32} is the mass ratio: *k*_{32} = *m*_{3}/*m*_{2}.

Note that for consistency, we must have *k*_{12} = 1/*k*_{21}. We must also assume of course that if we compare the *vasses* of three bodies in this way, we obtain consistent results:

For any three bodies, the constants *k*_{ij} satisfy *k*_{31} = *k*_{32}*k*_{21}.

It then follows that for *any* two bodies, *k*_{32} is the *vass* ratio: *k*_{32} = *n*_{3}/*n*_{2}.

To complete the list of fundamental axioms, we need one which deals with systems containing more than two bodies, analogous to the law of composition of forces, (1.2). This may be stated as follows: (p.12)

⇒The acceleration induced in one body by another is some definite function of their positions, velocities and internal structure, and is unaffected by the presence of other bodies. In a many-body system, the acceleration of any given body is equal to the sum of the accelerations induced in it by each of the other bodies individually.

To complete the list of fundamental axioms, we need one which deals with systems containing more than two bodies, analogous to the law of composition of *rushes*, (1.2a). This may be stated as follows:

⇒The *expedience* induced in one body by another is some definite function of their *chronations*, *legerities* and internal structure, and is unaffected by the presence of other bodies. In a many-body system, the *expedience* of any given body is equal to the sum of the *expediences* induced in it by each of the other bodies individually.

These laws, which appear in a rather unfamiliar form, are actually completely equivalent to Newton’s laws, as stated in the previous section. In view of the apparently fundamental role played by the concept of force in Newtonian mechanics, it is remarkable that we have been able to reformulate the basic laws without mentioning this concept. It can of course be introduced, by defining it through Newton’s second law, (1.1). The utility of the definition arises from the fact that forces satisfy Newton’s third law, (1.3), while accelerations satisfy only the more complicated law, (1.7). Since the mutually induced accelerations of two given bodies are always proportional, they are essentially determined by a single function, and it is useful to introduce the more systematic concept of force, for which this becomes obvious.

These laws, which appear in a rather unfamiliar form, are actually completely equivalent to *the dual of* Newton’s laws, as stated in the previous section. In view of the apparently fundamental role played by the concept of *rush* in *the dual of* Newtonian mechanics, it is remarkable that we have been able to reformulate the basic laws without mentioning this concept. It can of course be introduced, by defining it through *the dual of* Newton’s second law, (1.1a). The utility of the definition arises from the fact that forces satisfy *the dual of* Newton’s third law, (1.3a), while *expediences* satisfy only the more complicated law, (1.7a). Since the mutually induced *expediences* of two given bodies are always proportional, they are essentially determined by a single function, and it is useful to introduce the more systematic concept of *rush*, for which this becomes obvious.