Newton's grazing-angle ring table follows his fitted rule, which runs 27% low at 85°

In Book II of the Opticks, Newton gives a table of the air thickness at a coloured ring for thirteen viewing angles, and beside it a rule he fitted to the table. We put every row beside exact thin-film optics and beside his rule. From perpendicular light to 60 degrees in the air, his table matches the secant law of modern optics within 0.41 percent.c1 From 65 to 90 degrees every row matches his own 106-means rule within 1 percent, and at 85 degrees his tabled thickness sits 26.7 percent below the secant.c2 Exact theory says the secant holds at every angle below the critical one, so the steep rows follow his rule and not the light. Our reading is that those rows were computed from the rule; the text alone cannot settle it. The same model checks his best-known number: under standard daylight weighted by the eye, the first dark ring falls at 275.7 nm, 3.4 percent below Newton's 1/89000 inch (285.4 nm).c3 All five preregistered predictions held and both controls failed as required.

Newton's ringsDefinition: Newton's rings optics The bright and dark circles seen where a curved lens rests on flat glass, made by light reflected from the two faces of the thin air gap between them, whose thickness grows with distance from the point of contact. Excludes rings made by diffraction at an edge or an aperture. full definition are the bright and dark circles seen where a curved lens rests on flat glass. In Book II, Part I of the Opticks [1], Newton measured them with a lens of known curvature and turned ring diameters into thicknesses of the air gap. In Observation 6 he gives the gap at the first dark ring as 1/89000 inch for light near the perpendicular. In Observation 7 he tilts the view and tabulates the thickness at one ring for thirteen values of the obliquityDefinition: obliquity optics, in Newton's usage The angle between the light's path and the perpendicular to the glass. Newton's table gives it twice, once inside the glass and once in the air gap. Excludes the tilt of the glass itself, which stays fixed. full definition, scaled so that perpendicular light gives 10. He says his values came "partly by measuring" and partly by other means, including two prisms for the steepest angles.

Below the table he states a rule he "seem[s] to gather" from it. We call it the 106-means ruleDefinition: 106-means rule history of optics; Newton's own rule in Opticks, Book II Newton's statement that the thickness goes as the secant of an angle whose sine is the first of 106 arithmetical means between the sines of incidence and refraction, counted from the bigger sine. Excludes the secant law, which uses the air angle alone. full definition: the thickness goes as the secant of an angle whose sine is the first of 106 arithmetical means between the sines of incidence and refraction, counted from the bigger sine. Modern optics gives a simpler answer, the secant lawDefinition: secant law thin-film optics The statement that the air thickness at which a given ring appears grows as one over the cosine of the light's angle in the air gap, scaled here so that perpendicular light gives 10. Excludes any rule built from the angle in the glass. full definition: the thickness at a fixed ring grows as one over the cosine of the angle in the air.

The two agree at shallow angles and part at steep ones. At 90 degrees in the air the secant is infinite, and Newton's last row reads 122 and a half. The question is which of the two his rows follow, and from which angle on.

The literature check for this paper ran three searches on 9 November 2026. Shapiro's study of Newton's theories of coloured bodies and fits [3] works from Newton's manuscripts and the experiments behind Book II, and is the most likely place for an earlier analysis of this table; we found its catalogue and publisher records and did not read it. Morizot [4] traces how Thomas Young turned Newton's ring thicknesses into the first wavelengths; we read its abstract. Lee and Lee [5] give open code for the colours of layered thin films from the same multiple-beam sums we use; we read its abstract. None of the searches returned an analysis of the Observation 7 table against exact theory. That is what the searches returned, and it does not show that no such analysis exists.

The design was written in a dated preregistrationDefinition: preregistration research method (metascience) A dated written plan for a study, fixed and hash-locked before any result is seen: the question, the data, the statistic, the controls, the forecasts and the kill conditions. Excludes a plan revised after looking at the outcome, and a protocol written only for the record after the run. full definition on 28 October 2026. The file states that it was written before the model or the study ran. It was not hash-locked at the time, and its result section was appended to the same file after the run, so the hash given in the reproduction section is of the file as it now stands.

One check came before the file. We had computed the rule by hand at the 90 degree row and found about 122.9 against Newton's 122 and a half. The preregistration says so, and that check is why prediction E exists.

The five predictions, with their bounds:

The file also names two controlsDefinition: control statistics and experimental design A version of the data with the structure under test removed (for example shuffled or pooled), scored the same way as the real data, so the real score has something to beat. Excludes a comparison with a different dataset or with a published number. full definition that had to fail, given in the controls section. If E holds and D fails beyond 60 degrees, the steep rows follow the fitted rule, and C says the true thickness keeps growing as the secant.

The primary text is the fourth edition of the Opticks, 1730, as transcribed by Project Gutenberg (ebook 33504) [1], with the Newton Project transcription as a second copy [2]. Both are public domain. The thickness column, the glass angles and the air angles of Observation 7 were typed by hand from that text into the study script. Table 2 gives every value that entered.

The white light is the CIE D65 daylight table and the eye's weighting is the CIE 1931 2 degree observer, both from the official CIE data tables [6]. The glass is N-BK7, with an index of 1.5175 at 571 nm from the model's own refraction table.

The model is a glass, air gap, glass stack, the geometry of Newton's lens on a plate over the small area of one ring. For light of wavelength $\lambda$ crossing a gap of thickness $d$ at angle ${\theta }_a$ in the air, write $c_a$ for the cosine of ${\theta }_a$. The phase delay between successive beams is

Let $r_1$ be the amplitude reflection coefficient from glass into air and $r_2$ that from air into glass, each from the Fresnel equations for one polarisation. Write $C$ for the cosine of $\delta$. Summing every reflected and re-reflected beam gives the Airy reflectanceDefinition: Airy reflectance optics; named after George Airy The share of light power a thin layer reflects when every reflected and re-reflected beam inside it is summed, rather than only the first two. Excludes the two-beam approximation, and absorption in the glass. full definition

computed for s and p polarisationDefinition: s and p polarisation optics The two directions of the light's electric field relative to the plane that holds the ray and the perpendicular to the surface: s across that plane and p within it. Unpolarised light is their average. Excludes circular polarisation, which this study does not model. full definition and averaged. For this stack $r_2=-r_1$. That sign change is the half-wave flipDefinition: half-wave flip optics The change of sign of the reflected wave at one face of the air gap and not the other, which makes the centre of the rings dark in reflected light. Excludes the phase delay from the path across the gap. full definition, and it makes the point of contact dark.

For white light the reflectance is weighted by the CIE D65Definition: CIE D65 colour science (International Commission on Illumination) A standard table of the power in average midday daylight at each wavelength, used as the white light in this study. Excludes the measured spectrum of any one day's sunlight, and Newton's own light. full definition power $S\left(\lambda \right)$ and by the eye's brightness curve $y\left(\lambda \right)$, from 380 to 780 nm in 1 nm steps:

$Y$ is the reflected luminanceDefinition: luminance colour science The brightness of light as the standard eye weighs it, computed by weighting power at each wavelength with the CIE 1931 2 degree observer's brightness curve. Excludes hue and saturation, which the study does not score. full definition as a share of the incoming. The study stepped $d$ from 0 to 2,000 nm in 0.1 nm steps and read off the minima and maxima of $Y$. A dark ring is a minimum of $Y$ and a bright ring a maximum.

With thickness scaled to 10 at perpendicular light, the secant law is

For Newton's rule, let $s_a$ and $s_g$ be the sines of the angles in the air and in the glass that his table lists. The first of 106 arithmetical means between them, counted from the bigger, divides the interval into 107 equal steps, and its value $s_{\varphi }$ is the sine of the angle in Newton's secant:

At 90 degrees in the air, $s_a=1$, and eq. 6 stays finite where eq. 4 does not.

Each row's deviation is the absolute difference between Newton's value and the law, divided by the law's value. Prediction C was scored on the model alone: the first dark ring at 571 nm was found at each air angle and its thickness divided by the thickness at perpendicular light.

No planted data were used. The check with a known answer is prediction B: textbook theory places the bright rings at odd multiples of a quarter wavelength, so a model that missed 1 : 3 : 5 : 7 would be wrong before Newton entered.

Two controls had to fail, and both did.

One error of ours was caught before grading. The first run gave prediction C a ratio of exactly 1 at every angle, because the dark-ring search started at 1 nm, where reflectance is still rising out of the dark centre, and stopped at once. The search was changed to take the first minimum after the first maximum, and the study was run again. The model was not changed.

All five predictions held. Table 1 gives each against its bound.

From perpendicular light to 60 degrees in the air, his table matches the secant law of modern optics within 0.41 percent.c1 From 65 to 90 degrees every row matches his own 106-means rule within 1 percent, and at 85 degrees his tabled thickness sits 26.7 percent below the secant.c2 Figure 2 draws the three curves and Figure 3 the deviations row by row. Table 2 lists them.

The 60 degree row equals the secant to the printed precision and sits 0.99 percent from the rule. From 65 degrees on, the order flips: every row is closer to the rule than to the secant, and the gap to the secant grows with angle, to 9.27 percent at 80 degrees and 26.72 percent at 85.

The model itself keeps the secant. At 571 nm the first dark ring follows the secant of the air angle within 0.0075 percent at every angle from 0 to 85 degrees. The two phase shifts at the faces of the gap cancel at every angle below the critical one, so nothing in the physics bends the curve toward the rule.

Under standard daylight weighted by the eye, the first dark ring falls at 275.7 nm, 3.4 percent below Newton's 1/89000 inch (285.4 nm).c3 The first four bright rings fall at 137.7, 413.8, 692.8 and 973.5 nm, and the first four dark rings at 275.7, 553.2, 832.8 and 1112.7 nm.

The split in Table 2 has a plain reading. Newton measured the rings at shallow angles, fitted a rule to those rows, and filled the steep rows from the rule. He wrote that the values came "partly by measuring" and partly by other means, which allows that reading. It is an inference. A rule fitted to the shallow rows would match them, and a row computed from the rule would match it to within the rounding of the printed fractions. Rows that were measured at steep angles and happened to fall on the rule would look the same in the table. Only his manuscripts could tell the two apart.

The rule itself is a good fit to his shallow data and a wrong law. Up to 60 degrees the rule and the secant differ by under 1 percent. Beyond that the rule bends away, and the error reaches 27 percent at 85 degrees.

The 3.4 percent gap at the first dark ring depends on the light and on what counts as dark in a coloured fringe. Our daylight is a modern standard table, and our dark ring is a luminance minimum. Newton's 285.4 nm corresponds to light centred near 571 nm rather than the 551 nm our weighting gives, a shift that sunlight could plausibly give and an eye judging "the darkest Part" of a fringe. We did not test it. Two related studies of ours, on his thick-plate rings and on his colour limits, also find his 1/89000 inch a few percent off, which is a pattern to test and not yet a result.

Young's wavelengths were taken from Newton's ring thicknesses, as Morizot describes [4]. Our result bears on those perpendicular values only through the 3.4 percent gap.