Checking Gilbert White’s sums

I’ve spent the last eight days reading Gilbert White’s The Natural History of Selborne (1789), the letters of a Hampshire curate who watched one parish for forty years. White counts and measures, and when he gives a number I’ve tried to check it, either by redoing his arithmetic or against something independent. Here are four of his calculations, roughly from best to worst.

1. Rushlights: right, and conservative

In a letter of 1775 White explains how cottagers made cheap light from peeled rushes dipped in bacon fat, and then he times one:

A good rush, which measured in length two feet four inches and an half, being minuted, burnt only three minutes short of an hour.

He counted “upwards of one thousand six hundred” rushes in a pound. Then, to be safe, he assumes each burns “only half an hour”: 800 hours of light, “a time exceeding thirty-three entire days, for three shillings.” Each rush costs “1/33 of a farthing” before dipping and “1/11 afterwards,” so a family gets “5½ hours of comfortable light for a farthing.” A halfpenny candle lasts about two hours, so the poor who buy candles get “only two hours’ light for their money instead of eleven.”

Every figure checks: 800 hours is 33.3 days, 48 farthings over 1,600 rushes is 1/33, 144 over 1,600 is 1/11, and 5½ hours per farthing is 11 per halfpenny. Because he assumed half the burn time he had measured, the real advantage was probably greater than he claimed.

2. The echo: right, and he had the explanation without seeing it

In 1778 White describes an echo off a stone hop-kiln that would “repeat ten syllables most articulately and distinctly, especially if quick dactyls were chosen”, Tityre, tu patulae recubans, but in “slow, heavy, embarrassed spondees of the same number of syllables”, Monstrum horrendum, informe, ingens, “we could perceive a return but of four or five.”

He measured the distance as 258 yards and compared it with Robert Plot’s rule of 120 feet per syllable, under which ten syllables should need 400 yards. His sums are right: 258 yards is “near 75 feet” a syllable, and 258 to 400 is close to his “five to eight.” He puts the shortfall down to “some latitude… according to time and place.”

The explanation is in his own paragraph. Sound covers the 774 feet out and back in about 1.4 seconds. Any syllable still being spoken when the first one returns is masked by the speaker’s own voice, so an echo gives back about as many syllables as you can say in its round-trip time. Ten quick dactyls in 1.4 seconds is about seven syllables a second, which is brisk but sayable. Four or five slow spondees is about three a second. Plot’s 120 feet per syllable assumes roughly 4.6 a second, an ordinary speaking pace. White’s echo beat the rule because he was chanting fast, and his dactyl-and-spondee comparison measures exactly that.

3. The stilt’s legs: the sum is right, but it’s the wrong sum

In April 1779 five black-winged stilts were shot at Frensham Pond. White had one, “drawn and stuffed with pepper”: 4¼ ounces, with eight inches of thigh and leg. He compares it with a flamingo of “about four pounds” and “about twenty inches” of leg. Four pounds is fifteen times 4¼ ounces, so a flamingo built like a stilt “must have one hundred and twenty inches and a fraction of legs; viz., somewhat more than ten feet; such a monstrous proportion as the world never saw!”

The multiplication is correct (8 × 15.06 = 120.5). The scaling is not. When an animal gets heavier and keeps its shape, its lengths grow with the cube root of its weight, not in proportion to it. Galileo set this out in 1638. The cube root of 15.06 is 2.47, so a stilt enlarged to flamingo weight would stand on about 19.8 inches of leg. White’s flamingo has twenty. By his own figures, the two birds have almost exactly the same proportions.

That also explains his next sentence: “If you should try the experiment in still larger birds the disparity would still increase.” Under linear scaling every larger bird will look short-legged, because the method builds that result in. (His stilt had been gutted, so the living bird was somewhat heavier, which would make it, if anything, a shade less leggy than the flamingo.)

4. The solstice: an observation nobody could make

In one letter White proposes two tall posts, set where the sun goes down at midwinter and at midsummer. It “would soon appear,” he says, “that there is no such thing, strictly speaking, as a solstice,” because the setting point never stands still. Near midsummer the whole disc would set north of the summer post “for about three nights; but on the middle night of the three, sensibly more remote than the former or following.”

The main point is right: the setting sun turns around at the solstice rather than pausing there. To test the detail, I computed the sunset azimuths for Selborne in June 1778 from a modern ephemeris (JPL DE440). The most northerly sunset was on June 21. June 20’s sunset was 4.5 arcseconds less northerly and June 22’s was 43 arcseconds less. The eye’s limit is roughly 60 arcseconds, and the bending of light at the horizon varies from night to night by far more than either figure. So “sensibly” is the one word that fails: nobody standing at a post could have seen the middle night win. White phrases the whole passage as what “would soon appear,” and as far as I can tell he never says he did it. It’s a thought experiment with an observer’s confidence added.

What the four have in common

In all three sums White’s arithmetic was right. Where he went wrong, he chose the wrong thing to calculate (linear instead of cube-root scaling), or he described what a perfect observer would see and called it sensible. He measured carefully, he timed his rush with a watch, and he reported his failed experiments. I think he’d have enjoyed being checked. He did the same to Virgil, by shouting at his bees through a speaking-trumpet to see whether echoes disturbed them. (They didn’t.)

Sources: the Selborne text is the Project Gutenberg edition (Letters XXVI, XXXVIII, XLIV and XLIX to Daines Barrington in the edition I read). The sunset azimuths are my own calculation with Skyfield and DE440. The speed of sound assumes a cool evening (about 1,107 ft/s at 10 °C).