Early Beacons

How Lighthouses Work 1 gave us the tools: the inverse square law, the horizon, and the difference between luminous and geographic range. This chapter uses them on the first real lights. The early story is short to tell. Builders learned to raise a fire high (the Pharos of Alexandria), then found that a fire is hungry (the coal beacon on the Isle of May), then that a candle is weak (the candle-lit Eddystone towers), and finally that a tower has to stand against the sea (John Smeaton's stone tower of 1759). Each step has a number attached, and each number can be checked.

The Pharos of Alexandria

The Wikipedia article on the Lighthouse of Alexandria (read directly) says it was built by the Ptolemaic Kingdom: Ptolemy I Soter commissioned it shortly after declaring himself king in 305 BC, and his son Ptolemy II Philadelphus finished it, over twelve years and at a cost of "800 talents of silver". The infobox gives the construction as between 284 and 246 BC, while the article's text puts Ptolemy II's reign at 280 to 247 BC, a small mismatch that shows how loose ancient dates are. Pliny the Elder named Sostratus as the architect, but the article says this conclusion is disputed.

It had three tiers: square at the bottom, octagonal in the middle and cylindrical at the top. The height is not settled. The article's lead says "at least 100 metres", its infobox gives 103 to 118 m, and the medieval Arab writer Al-Idrisi (1154) reckoned about 162 m. I use 103 to 118 m below, and treat it as a range rather than a fact. The light itself is also described two ways: "a furnace at the top", and a mirror at the apex that reflected sunlight by day with a fire lit at night. The article does not reconcile the two, and neither do I.

Its end was gradual. Earthquakes damaged it in 796 and 951, the 956 earthquake brought down the top 20 m, and the 1303 earthquake caused the final major damage. The remnant vanished in 1480 when Qaitbay built a fort on the site with the fallen stone. A medieval legend that a Byzantine agent undermined the foundations is presented by the article as legend only.

This is the geographic range formula from How Lighthouses Work 1. A sailor 5 m above the water could have seen a light at the top of the Pharos from about 25 to 27 nautical miles, if it was bright enough, and the whole disputed range of heights makes only a 7% difference to the horizon (the last line, the square root of 118/103). The height argument matters less to the range than it sounds, because range grows with the square root of height. Whether the fire was bright enough is the other half of the question, and for that we need a fire.

A Fire Is Hungry: The Isle of May Beacon

Open fires on towers, hills and headlands were the standard light for most of history. The best-documented Scottish one is on the Isle of May in the Firth of Forth. The Wikipedia article on the island (read directly) says a coal-fired beacon was established in 1635 (some sources say 1636), burned about 400 tons of coal a year, needed three men to tend it, and was funded by a tonnage fee on passing ships. A search for further sources agreed on the 400 tons, and added that up to 3 tons could be burned in a single night (this is from the summary of a Canmore entry that I did not open, so treat the 3 tons as a search summary). The beacon was replaced by Robert Stevenson's stone lighthouse, which began operating on 1 September 1816. The 150 tons a year that appears in some sources belongs to the later lighthouse, not the beacon.

On average that is 1.1 tons a day, and on the worst night about 250 kg an hour, a heat output of roughly 1.67 MW, the output of several hundred domestic electric fires (assuming about 2 kW each). Almost all of it was wasted as heat and smoke. The point is not the exact figure, which depends on the assumed 24 MJ/kg and the 12-hour night, but the scale: keeping a fire going meant hauling, storing and feeding coal every night of the year, with three people paid by shipping dues. That cost, as much as the poor light, drove the search for better lamps, which is the subject of the next chapter.

A Candle Is Weak: The Eddystone Towers

Eddystone, a reef off Plymouth, shows how the problems stack up. The Wikipedia article on Eddystone Lighthouse (read directly) lists four towers on the rock:

The article's text and its infobox disagree on some details (Rudyerd's shape; Smeaton's height as 18 m in one place and 22 m in another; the year Smeaton's tower was deactivated). I use the figures from the Smeaton's Tower article, which I also read directly: 72 ft (22 m) high, 26 ft (8 m) across at the base and 17 ft (5 m) at the top. Wikipedia lists the Rudyerd candle count as 24 and Smeaton's as 24, and the fourth tower is the first where the figures are in candela (a first-order lens with two six-wick oil burners) rather than candles. In 1882 the Smeaton tower was taken down and its upper part rebuilt on Plymouth Hoe, where it stands as a memorial.

How far does a candle-lit tower shine?

The Wikipedia article on the candela says a common candle emits roughly 1 cd, and I use 1 cd per candle as an assumption. The luminous-range function is the one from How Lighthouses Work 1, repeated here so the snippet runs on its own.

The columns are the tower, the number of candles, the nominal range in nautical miles (luminous range at 10 nm visibility), and the geographic range for an eye 5 m up, using only the tower's height (the rock adds some more). The result is clear and a little cruel: 24 candles give a nominal range of about 3.5 nm, while the horizon would allow over 14 nm. The light, not the tower, was the limit, and 60 candles reach only 4.7 nm, not 2.5 times as far, because of the inverse square law. Treat this as an order-of-magnitude picture: a lantern's candles are spread around the compass, not all aimed at one sailor, and the real range depends on the glass, the weather and the watcher. Even so, the gap between 3.5 and 14 nm explains the next three chapters: more light on the same tower.

Smeaton's Tower: A Stone Oak Tree

The first two towers were largely of wood, and neither survived the sea or the fire. John Smeaton's third tower (work began in 1756; the foundation stone, weighing 2¼ tons, was carried out to the reef on 12 June 1756, and the light was first lit on 16 October 1759) was the first to be thought through as engineering. The Smeaton's Tower article (read directly) says it was built of granite blocks joined with dovetail joints and marble dowels, used hydraulic lime (a Roman-era cement that sets underwater, which Smeaton rediscovered), and was modelled on an oak tree, wide at the base and narrowing upwards, for strength. The cost was £40,000. The article on hydraulic lime (also read directly) says hydraulic limes set by reacting with water, so they harden even when submerged, unlike air lime, which hardens by absorbing carbon dioxide. It credits Smeaton with pioneering hydraulic lime in the 18th century.

Why the base is wider

A tower of stone has to carry its own weight. The lower courses carry everything above them, so the stress there is greatest. A taper puts more stone where the load is greatest and saves it where it is not. The code below models Smeaton's tower as a solid truncated cone with the sourced dimensions (22 m high, 8 m across at the base, 5 m at the top) and an assumed granite density of 2,650 kg per cubic metre. The real tower is not solid (it has rooms and a stair), so these figures overestimate the weight and the stress. They show the pattern, not the real loads.

The columns are the height above the base (m), the diameter there (m), the weight of everything above (meganewtons) and the average compressive stress (kilopascals). The load climbs from 1.08 MN at 20 m to 19.32 MN at the base, and the stress from 49 kPa to 384 kPa, so the lowest stones bear nearly eight times what the highest do. (A kilopascal is a thousand newtons per square metre. Granite is generally far stronger than this in compression (general knowledge, not checked here), so the stone itself is not near failing under its own weight. The sea is the threat, as How Lighthouses Work 6 will show.) The solid model gives about 1,969 tonnes. The real tower, which is hollow in part, must weigh less; I have not verified its true mass.

Does the wind tip it over?

A second check is whether a gust could overturn the tower. A force at half the height (a crude assumption) tries to tip the tower about the edge of its base, while the weight acts through the centre, a base radius away. The ratio of the two moments is a safety factor. The wind pressures here are assumed round numbers and the 0.6 is an assumed drag factor for a round shape.

The columns are the wind pressure (Pa), the force (kN), the overturning moment (MN·m), the restoring moment (MN·m) and the safety factor. Even at 6,000 Pa, a violent gust, the weight resists with a factor of 13.6. Wind alone is not what kills a squat stone tower; the sea is, which is why the shape and joints matter more than the weight. Waves hit a reef tower far harder than air, and also pull at individual stones, and that is where dovetails come in.

What a dovetail does

A dovetail is a joint cut so that the two pieces cannot be pulled apart in the direction of the load. In a course of masonry, a wave can push one stone out of its place, held back only by friction. If the stones are interlocked, the wave must move the whole ring at once. A toy model shows the idea: friction on each stone is the friction coefficient times the weight on it, and a ring of n interlocked stones resists n times as much. Every number below is illustrative, and this is a model of the principle, not a calculation of Smeaton's joints.

One block resists 35 kN by friction alone, and a locked ring of 12 resists 424 kN, twelve times as much. The marble dowels and the hydraulic lime add to this, but the numbers say nothing about the real strength of Smeaton's joints, which I did not find a source for. What the model does show is why the idea works: it makes a hundred small stones act as one big one.