What each measurable number is a function of

One of the figures on what a measurement can fix: The other half of every model: which of its parameters a measurement can recover, which combinations no measurement of that kind can see, and how much a second number of a different shape is worth.

Four essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

How many parameters a curve is worth

The singular values of the design for two, three and four parameters fitted to the same curve. Each is what is left of the measurement after the directions above it have taken their share; the fourth is a fortieth of a per cent of the first.

Every parameter’s uncertainty against where the temperature window starts. The monomer fraction runs off the top as the cold end is given up, and takes everything else with it.

Uncertainty against the number of points, with the square-root law drawn from the smallest run. Sixteen times the data is a little under four times the precision, and the shortfall is in one direction.

Ten directions no frequency can see

The map from methane’s fifty-five independent force constants to its Cartesian Hessian, as a spectrum of directions: forty-five the frequencies respond to and ten they do not. The flat direction is not a direction.

Two ways of being second order

Every pair of parameters that would have produced one observed number, for three observed numbers, with the true system marked. Two of the curves cross at a definite angle and fix a point between them. The third lies almost along the first, and adding it to the first fixes nothing that was not already fixed.

The four measurable numbers against the four parameters, as exponents. The two widths are first order in their own hopping and read zero everywhere else. The two second-order rows both read 2 in the coupling and differ in the separation — one inverse square, one inverse first order — and that single difference is the whole of what follows.

The excess gap divided by the shape departure, against the separation, at three couplings. Three straight lines through the origin, and the slopes barely differ: the quotient of two second-order numbers is first order in the separation and contains no coupling at all.

The product a curve measures

The four directions, best fixed first, each written as the product it is, with the uncertainty a one per cent curve leaves along it.