Ground truth · the price of light

The Evening You Could Afford

For nearly all of human history, the light you could buy was not enough to read by, and the light that was enough to read by, you could not buy. Put a real source over a real page below and watch the illuminance fall off as the inverse square says it must. Then watch what an hour of work bought, from a Babylonian sesame-oil lamp to a lamp you can order tonight. Every number here is recomputed from a 1996 paper's own printed inputs, checked against measurements made after it was written, and carried thirty-four years past the point where it stops.

I. One light, one page

The paragraph in the panel is real. It is the appendix of the paper this whole layer rests on, describing the experiment that produced its oldest number. Choose a light, move it up and down, and see whether you could read it.

Two things fall out of that panel that no amount of prose delivers. The first is how steeply light dies. Bring a candle close enough to make the middle of the page bright and the corner goes dark, because the corner is further away and the light arrives at a slant. There is a best height, and it is not a matter of taste: the illuminance at a point r away from directly beneath the source is largest when the source sits at h = r/√2, and the ratio of centre to corner at that height is exactly 3√3 ≈ 5.196, whatever the lamp. That is why people held the book up to the candle rather than setting the candle over the book.

The second is the sheer scale of the gap. A tallow candle at a normal reading distance puts about lux on the page. A modern lighting standard asks for lux for reading. Full moonlight at a temperate latitude is around lux. The candle sits almost exactly at the geometric mean of the two: about times moonlight, and about times short of the standard. Between the light you can read by and the light you can only see shapes by, a candle is the halfway house, and every reader before the gas mantle lived in it.

II. What an hour of work bought

In 1996 William Nordhaus published a price series for light running from the fires of Peking man to a compact fluorescent bulb, measured not in money but in labour: how many hours of work it took to buy a thousand lumen-hours. Money across four thousand years is close to meaningless. An hour of work is not.

The last row is not in the paper. Nordhaus stops at a first-generation compact fluorescent bulb in 1992, and the row after it is computed here by his own method, from three public files committed alongside this page: the average hourly earnings of all private employees, the average retail price of a kilowatt-hour, and the certified efficacy of every integrated-LED general service lamp legally sold in the United States. That last file holds lamps. Their median efficacy in the general-service band is lumens per watt; the frontier lamp used for the row above is a purchasable 850 lumen, 5.0 watt lamp at 170.

III. One per cent of a working year

A price series is abstract until you spend something on it. So here is one fixed yardstick, applied identically to every era: take the earnings from two thousand hours of work, spend one per cent of them on light at that era's frontier price, and ask what you get. The wage cancels out entirely, which is the point. One per cent of two thousand hours of work is twenty hours of work, whoever you are and whenever you live.

The turn happens with gas. Before it, one per cent of a working year buys about an hour of one weak flame per night, and light is something you ration by the hour. After it, the same one per cent buys more light than a household can use, and the question stops being what you can afford and becomes what you want lit. Nothing in the domestic history of Europe is a sharper line than that one, and it does not appear in any price index.

IV. What it cost to read

Put the two halves together. Section I says how much light a page needs. Section II says what light cost. Multiply.

To hold the far corner of a single page at the lux a modern standard asks for, using the best achievable geometry, takes tallow candles. Not a metaphor: thirty-odd actual candles, burning at once, over one book.

Spermaceti candles, the good ones, were brighter per candle and far dearer per lumen, so they cost more, not less: hours of work per hour of reading. And the comparison is deliberately generous to the past in one way and harsh in another. Generous, because it prices only fuel and ignores the candles themselves, the holders, the snuffing, the smoke, and the very real chance of setting the house on fire. Harsh, because nobody in 1800 was trying to hit a twenty-first-century office standard: they read by one candle and ruined their eyes, which is a choice the arithmetic above explains rather than judges.

V. The record that could not see it

Here is why Nordhaus wrote the paper. Official price statistics tracked the price of a candle, and then the price of a gallon of kerosene, and then the price of a kilowatt-hour, as three separate goods, each with its own index, none of them linked to the others by what they were for. Measured that way, the price of light in the United States rose. Measured as the price of the thing people actually wanted, it collapsed.

Nordhaus's own summary of that gap: the traditional price has risen by a factor of between nine hundred and sixteen hundred relative to the true price, an average divergence of 3.6 per cent a year. His conclusion follows immediately and is the reason the paper is still cited: if light is at all representative of the goods that changed in kind rather than in quantity, then measured real wage growth over two centuries, a factor of about thirteen, is badly understated. His own thought experiment puts the true factor between forty and a hundred and ninety, and he is careful to call that speculative.

VI. What reproduces, and what does not

None of the above is worth anything if the table underneath it is wrong. So the table was rebuilt. Every figure below was transcribed by hand from images of the printed pages, because the scan's own text layer mangles digits, and then recomputed.

The price chain reproduces completely

The labour price of light is the price of light divided by the wage, and that is the whole of the paper's central table. A printed figure is an interval, not a number: 0.124 means somewhere in [0.1235, 0.1245). Dividing the intervals gives every labour price consistent with what is on the page, and the honest question is whether the printed answer lies inside. It does, for of the rows that have both inputs printed. The two efficiency columns of the efficiency table are the same quantity in different clothes and agree across all rows to within .

Three constants the paper never states, recovered

• Thirteen lumens per candlepower. The 1855 experiment table prices its light twice, per candle-hour and per thousand lumen-hours, and the ratio of the two is the conversion he was working in. Across all rows it is , against the 4π = 12.566 that an isotropic point source would give. He states thirteen in a footnote and uses it consistently.
• A two-hundred-hour Babylonian month. The oldest price in the series comes from a lamp of unverified age, sesame oil at a tenth of a shekel per litre, and a wage of one shekel a month. The month is never given in hours. Solve for it and the printed 41.50 needs hours, the appendix's rounded 42 needs . Two hundred, on the nose. Elsewhere in the same table an American farm worker's month is 250 hours.
• The price of electricity, in every electric row. Each of those rows is an electricity price divided by an efficacy. Invert them and the prices come back out. For 1883 the answer is cents per kilowatt-hour, and the paper's own prose, quoting Edison's 1883 price list, independently says approximately twenty-four cents per kilowatt-hour at the dawn of the electric age.

Four places where the paper does not agree with itself

All four are small, all four are in the descriptive layer rather than the price chain, and none of them moves the result. They are listed because a reproduction that reports only its successes is not a reproduction.

The engineering, against measurements made after 1996

The half of the paper that could age is the physics, and some of the measurements that test it did not exist when it was written. The result is more interesting than a list of errors.

VII. A lamp of unverified age

The oldest number in a series that is still cited thirty years later, the price of light in Babylon in 1750 BC, was produced like this. An economist at Yale bought a terra-cotta lamp from a shop in Minneapolis. It was sold to him as Roman and he could not check that, and he says so in print. He filled it with a supermarket brand of cold-pressed sesame oil, pulled the wick out of a modern candle, lit it, and left it burning for seventeen hours with a Minolta illuminance meter beside it. It gave 0.17 foot-candles. He estimated the lit zone at ten square feet, called it 28.6 lumen-hours from a quarter of a cup, and turned that into a price.

It is easy to read that as a story about how thin the foundations are, and it partly is. But run the numbers and something else shows up. The lamp gave about 1.7 lumens. Mills's measured kerosene lanterns, a technology three thousand years later, give 8 to 82. Nordhaus's efficacy for a candle, 0.1009 lumens per watt, sits inside the band you get from NIST's measured candle heat-release of 77 watts once you correct for the anisotropy that Mills measured, and inside the range of the eight real wick lamps Mills tested. The naive isotropic assumption would have given 0.163 and been half again too high. He did not make it. The fuel energy his candle arithmetic implies, MJ/kg, is within one per cent of the net heat of combustion NIST measured for paraffin wax nine years later.

The estimates the paper calls extremely rough really are rough, and this page says by how much. The ones it does not flag hold up against instruments that were not pointed at them for another decade. That is what a good measurement looks like from the inside: honest about which parts are guesses, and right about the rest.