Ratios and proportions are symmetric.

A:B = B:A and A:B :: C:D = C:D :: A:B.

But when ratios are converted to quotients or fractions, they are no longer symmetric. There must be a convention as to which is the denominator and which is the numerator.

In an ordinary fraction or quotient or rate the numerator is above the line (the vinculum) or left of the slash, and the denominator is below the line (the vinculum) or right of the slash. Additionally, in an ordinary quotient or rate the denominator is one.

However, there is also what may be called a reciprocal fraction or reciprocal quotient or reciprocal rate that is the opposite: the numerator is *below* the line (the vinculum) or *right* of the slash, and the denominator is *above* the line (the vinculum) or *left* of the slash. Thus there are two numbers (or a single number with an implied number one) but their meaning may change.

For example, uniform motion is defined as

S_{1} : S_{2} :: T_{1} : T_{2}.

But in reality either distance is measured over a given time or time is measured over a given distance.

Thus we may write

ΔS / ΔT = V, average time speed.

But we may equally well write

ΔT / ΔS = W, average space pace.

In general, if we look at motion functionally, we may write

ΔS / ΔT = S(ΔT).

And we may equally well write

ΔT / ΔS = T(ΔS).

These rates use ordinary algebra if their *denominators* are equal:

(ΔS_{1} / ΔT) + (ΔS_{2} / ΔT) = (ΔS_{1} + ΔS_{2}) / ΔT

and

(ΔT_{1} / ΔS) + (ΔT_{2} / ΔS) = (ΔT_{1} + ΔT_{2}) / ΔS

Their difference quotients also use ordinary algebra:

and

For the derivatives take the limits to get

and

Addition of these derivatives is straightforward:

V_{1}(T) + V_{2}(T) = V_{3}(T)

and

W_{1}(S) + W_{2}(S) = W_{3}(S)

But there are two other options. We can have a speed with a given distance:

ΔS / ΔT_{n} = 1 / (ΔT_{n} / ΔS) = 1/W = V^{~}, average space speed,

or a pace with a given time:

ΔT / ΔS_{n} = 1 / (ΔS_{n} / ΔT) = 1/V = W^{~}, average time pace.

These rates use *harmonic algebra* if their *numerators *are equal:

(ΔS / ΔT_{1}) ⊕ (ΔS / ΔT_{2}) = ((ΔT_{1} / ΔS) + (ΔT_{2} / ΔS))^{−1} = ((ΔT_{1} + ΔT_{2}) / ΔS)^{−1} = ΔS / (ΔT_{1} + ΔT_{2})

and

(ΔT / ΔC ⊕ (ΔT / ΔS_{2}) = ((ΔS_{1} / ΔT) + (ΔS_{2} / ΔT))^{−1} = ((ΔS_{1} + ΔS_{2}) / ΔT)^{−1} = ΔT / (ΔS_{1} + ΔS_{2})

Their difference quotients use harmonic algebra:

and

These lead to the reciprocal derivatives

and

Addition of these derivatives uses harmonic addition, that is, taking their reciprocals before and after addition:

and

Averaging these latter two rates uses the harmonic mean.