This continues the series of posts, see here.

Let’s begin with Galileo’s figure for uniform motion and uniform accelerated motion:

Let the horizontal uniform motion be situated in a 2D x, y coordinate system:

The dependent uniform acceleration moves in an additional dimension, z, and so has 3D coordinates.

If the independent uniform motion is measured by time, then time has 2D coordinates. The coordinates are proportional to one another, and so may just as well be replaced by a scalar of the signed magnitude. The same can be done if the uniform motion has 3D coordinates. The scalar time is proportional to the common measure of time, with the appropriate rate of motion, it is the same as scalar time.

One can say the same if the independent uniform motion is measured by length, then length has 2D (or 3D) coordinates. This is nothing new, but because of the uniform motion, the coordinate lengths are proportional to one another, and so may just as well be replaced by a single scalar of the signed magnitude. In this way, length becomes a scalar, called here the stance.

This continues posts about Galileo such as here. Look again at this figure from Galileo:

The horizontal line represents the independent variable, which is the horizontal component of a projectile with an initial speed that falls with uniform acceleration. The independent variable need not be horizontal, so let us represent it vertically with the vertical component of motion:

The first two lines represent the independent uniform motion but are measured differently, (1) by stance and (2) by time. The second and third lines show the correspondence between the independent variable (time) and the dependent variable (length). The first and fourth lines have the same relation: stance is the independent variable and duration is the dependent variable. Lines 1 and 2 are proportional, as are lines 3 and 4. As the units change, the functional relationship does not: the dependent variable is proportional to a square of the independent variable.

Time is the measure of duration of a uniform motion that is the reference motion, which means it is the independent variable. Similarly, stance is the measure of length of a uniform motion that is the reference motion, in which case it is the independent variable. The dependent variable for stance is duration in three dimensions, whereas for time the dependent variable is length in three dimensions.

A kinematic frame of reference is a mathematical method to determine the position of points in abstract 3D space and scalar time. An inertial frame of reference is a physical method to measure the position of bodies in physical 3D space and scalar time. The latter is often envisioned as three mutually-perpendicular rigid rods attached at a common point, or a lattice of such rigid rods. In addition, there is envisioned a clock at every node of the lattice, which are all synchronized, which requires a method to synchronize them. The common point is called the origin point.

Such a frame of reference assigns coordinates in 3D space and 1D time to every event.

A kinematic timeframe of reference is a mathematical method to determine the position of points in abstract scalar space and 3D time. An inertial timeframe of reference is a physical method to measure the position of bodies in physical 3D space and scalar time. The latter may be envisioned as three mutually-perpendicular rigid monorails attached at a common point, or a lattice of such rigid monorails, with a regular succession of small monorail vehicles traveling at constant velocity. In addition, there is envisioned a metreloge at every node of the lattice, which are all synstancized, which requires a method to synstancize them. The start of monorails leaving the common point is the origin event.

Such a timeframe of reference assigns coordinates in 1D space and 3D time to every event.

This post relates to others such as this.

Consider Galileo’s figure (see his Dialogues Concerning Two New Sciences, tr. Crew & De Salvio p.249 Fig. 108 or Drake’s translation p.221) below with horizontal and vertical rulers added :

A projectile moves with uniform velocity horizontally to the left and begins to descend at point b. Galileo used the sequence a-b-c-d-e to represent time and the sequence b-o-g-l-n to represent the height of the projectile above the Earth. The sequence b-i-f-h represents the parabolic path of the falling projectile.

Any uniform motion can serve as a reference motion. There are two uses of a reference variable: (i) as a parametric variable, or (ii) as a measurement variable. A parametric variable is an independent variable that provides ordered input for any dependent variable. A measurement variable is a variable that is dependent on the independent variable being measured. In the figure above the parametric variable is the time (duration) of the uniform motion on the horizontal axis, and the measurement variable is the height (length) of the uniform acceleration on the vertical axis.

Combine this with the two measures of motion, length and duration, and there are four possible cases: (1) independent duration variable with dependent length variable; (2) independent length variable with dependent duration variable; (3) independent length variable with dependent length variable; and (4) independent duration variable with dependent duration variable.

The figure above is an example of case (1). Its complement is case (2). Cases (3) and (4) include only one measure, length or duration, and so cannot express a rate of motion. Galileo expresses case (1) as a proportion between ratios of the variables at different times: s₁ : s₂ :: t₁² : t₂², which avoids combining different units in a single ratio, consistent with Eudoxian proportionality.

Consider case (2) in which the independent variable is length. This variable is a stanceline for locating other motions, which is like a timeline except that it expresses an independent length as the order parameter. The dependent reference variable in this case is duration, which measures any independent variable, in this case projectile height. This could be expressed as a proportion between ratios of the variables at different times: t₁ : t₂ :: s₁² : s₂², avoiding different units in a single ratio.

Case (1) enables multiple length variables dependent on one independent variable, the timeline. Case (2) enables multiple duration variables dependent on one independent variable, the stanceline. Rates of motion in case (1) are in units of the independent timeline, which is duration. Rates of motion in case (2) are in units of the independent stanceline, which is length.

Let’s start with one-dimensional, i.e., scalar, functions, f, g, h, and k. Say there is the following functional relation:

s = f(t) = f(h(s)) ≡ g(s) = t,

t = g(s) = g(k(t)) ≡ f(t) = s,

in which s and t are parameters with different units. By implication the functions are either f or its inverse:

s = f(t) = f(f^-1(s)) = t,

t = f^-1(s) = f^-1(f(t)) = s.

Function f takes t-units into s-units, and function f^-1 takes s-units into t-units. The vector versions are as follows:

s = f(t) = f(f^-1(s)) = t,

t = f^-1(s) = f^-1(f(t)) = s.

Motion space is an ordered pair of vectors s and t: (s, t), resulting in their direct sum vector space. Addition is conducted by components: (r, w) + (s, t) = (r + s, w + t). Scalar multiplication is also by component: (a, b) (s, t) = (as, bt). To multiply a scalar and only one component requires the other component to be unity. Thus additive unity is (0, 0) and multiplicative unity is (1, 1).

There are two ways to mask an ordered pair of vectors: left mask (s, t)^⌊ = (s, t) and right mask (s, t)^⌋ = (s, t), where s = |s| and t = |t|. What was described here as expansion and contraction may now be shown more clearly as masking and unmasking. A parametric length vector function is converted to a parametric duration vector function as follows:

r(t) = masked r(t) ↑ unmasked r(t) ↔ (inverted) = unmasked t(r) ↓ masked t(r) = t(r).

r(t) = [r(t), θ(t), φ(t)] ↑ [r´(t´, χ´, ψ´), θ´(t´, χ´, ψ´), φ´(t´, χ´, ψ´)] ↔ (t´(r´, θ´, φ´), χ´(r´, θ´, φ´), ψ´(r´, θ´, φ´)) ↓ [t(r), χ(r), ψ(r)] = t(r).