Chapter 4 Mirrors, the arc and the corrections
How a sextant turns light into an angle
Two small mirrors fold the sky down onto the horizon. Their geometry doubles every turn of the arm, keeps the two images together while the ship rolls, and lets you read an angle to a fraction of an arcminute.
Every method in the last two chapters comes down to one measurement: the angle between a body in the sky and the horizon, or between the Moon and a star. Measuring it on land, with a plumb line and a steady stand, is not hard. Measuring it on a deck that rolls 10° either way, to a small fraction of a degree, is. The sextant does it with two small mirrors, an arc engraved on brass, and some geometry that is much cleverer than it looks.
Its job is to make two directions coincide in one eyepiece. You look straight ahead at the horizon through a half-silvered glass. A second mirror on a swinging arm catches the Sun or a star and folds its image down into the same view. When the image sits exactly on the horizon, the arm’s position on the arc tells you the angle between them. The measurement is a coincidence, not a pointing, and that difference is why the instrument works at sea.
Two mirrors and a doubled arc
Drag the index arm along the arc. Follow the light: it comes from the Sun to the index mirror at the top, down to the horizon mirror, and into the telescope, joining the direct view of the horizon. Watch the view through the telescope in the corner and bring the Sun’s image down until its lower edge touches the sea.
Notice that the arm turns through only half the angle you measure. Set the arc to 40° and the arm has moved 20°. This is the double-reflection principle, and you can see why with two facts about mirrors. First, a mirror turns a ray by twice the angle it makes with the mirror’s surface. Second, when a ray reflects off two mirrors in turn, the two turns combine into a single rotation of
At zero the two mirrors are parallel, so a ray turned by the first is turned back by the second, and you see straight through: the sky directly ahead lands on the horizon. Turn the index mirror by θ and the sky that reaches your eye comes from 2θ higher. A 60° arc, one sixth of a circle, can therefore measure angles up to 120°, which is where the name comes from. Instrument makers engrave each half-degree of arm motion as a full degree, so you read the angle directly.
The coincidence principle is what makes the instrument usable at sea. When the ship rolls, the horizon and the reflected Sun move together through the telescope, because both views pass through the same glass held in the same hand. The angle between them is a property of the mirrors, not of how steady you are. The older instruments in the last section of this chapter all ask you to aim at two things at once; the reflecting instrument asks you only to keep them touching.
Hadley, Godfrey, Campbell
The first practical reflecting instrument was presented to the Royal Society by John Hadley in a paper read on 13 May 1731.[1] documented Hadley wrote that it was meant for use “where the motion of the objects” makes ordinary instruments unsteady.[2] Its arc was an eighth of a circle, so it measured up to 90°; it was called Hadley’s quadrant, and later the octant. In Philadelphia, the glazier Thomas Godfrey had built a similar instrument around 1730, and his patron James Logan pressed his claim to the Royal Society.[3] Historians generally treat the two as independent inventions, with Hadley first in print. contested
Ninety degrees was enough for altitudes but not for lunar distances, which can be much larger (the Moon as a clock). When the Board of Longitude had Captain John Campbell test Tobias Mayer’s design for a full reflecting circle at sea in the late 1750s, he found it awkward and had the instrument maker John Bird build a 60° arc instead: the sextant.[4] documented Sources date the first one to 1757 or 1759. contested So the sextant grew directly out of the lunar method, and Maskelyne’s almanac of 1767 created the demand for it.[5]
Reading the arc
An arc 20 cm in radius gives about 1.7 mm per degree of reading, far too little to divide into minutes by eye. Two inventions solved that. The vernier, a short auxiliary scale that slides along the main one, lets you read a fraction of a division by seeing which of its lines lines up exactly with a line on the arc. The later micrometer drum turns a fine screw that moves the arm along a toothed rack; one turn of the drum is one degree, and its rim is divided into 60 minutes, with a small vernier beside it for tenths.[6]
Try a few readings in Practice mode. The arc and the drum are most often misread near a whole degree: if the drum shows 58′, the arc index is almost on the next degree mark but has not reached it. Reading to 0.2′ is a property of the scale. It does not mean any sight is that good, as the rest of this chapter shows.
From sextant altitude to observed altitude
What the arc reads is the sextant altitude, Hs: the angle between the visible sea horizon and the edge of the body, as seen from your eye, through the air, with whatever small error your instrument has. What the almanac and the sight-reduction tables need is the observed altitude, Ho: the altitude of the body’s center above the true horizontal, as it would be seen from Earth’s center with no atmosphere. Five corrections connect them.[6][7]
Index error. If the mirrors are not exactly parallel at zero, every reading is off by the same amount. You find it by setting the arm near zero and bringing the true horizon and its reflection into one straight line. An error “on the arc” (a positive reading at true zero) is subtracted; one “off the arc” is added.
Dip. From a height above the sea, the visible horizon lies below the true horizontal, because Earth curves away. The almanac’s formula, including normal refraction, is
Refraction. The atmosphere bends light downward as it enters denser air, so every body appears higher than it is. The effect is small overhead and large near the horizon: about 1′ at 45°, 5′ at 10°, and more than 30′ at the horizon itself. Navigators use a formula like Bennett’s,
Semidiameter. You bring the Sun’s or Moon’s lower edge (lower limb) or upper edge (upper limb) to the horizon, not its center, because an edge is much easier to judge. Adding the semidiameter, about 16′ for the Sun, moves the measurement to the center. The Moon’s semidiameter varies from about 14.7′ to 16.8′ with its distance and is enlarged slightly when it is high.
Parallax. The almanac gives positions as seen from Earth’s center. You stand on the surface, 6,371 km away from it, so nearby bodies appear lower than they would from the center. For the Sun the effect is under 0.15′. For the Moon it is enormous, up to 61½′ at the horizon, and falls off as the cosine of the altitude.[9]
Each arcminute of these corrections is a nautical mile of position. Leave out the semidiameter of a Sun sight and your line of position is 16 miles out. That is why they matter, and why, for a Moon sight, they are not optional.
On its side, and on land
The same instrument measures any angle in the plane of its frame. For a lunar distance you hold it tilted, or nearly flat, so that its plane contains both the Moon and the star, then rock it about the line between them until the star’s image swings across the Moon’s bright limb and touches it. The swinging rule is the same; only the reference has changed from the horizon to the Moon’s edge. Surveyors and coastal pilots used sextants flat, to measure the horizontal angle between two landmarks.
On land there is no sea horizon. Explorers and surveyors used an artificial horizon instead: a shallow tray of mercury, or later a dark glass leveled with a spirit level, that reflects the sky. You measure from the body down to its own reflection, which lies as far below the true horizontal as the body is above it, so the reading is twice the altitude, and there is no dip to correct. The price is that you cannot measure altitudes much above 60° with a sextant whose arc stops at 120°.[6]
A moving platform and a tilted arc
The reflecting principle cancels the ship’s motion along the line of sight, but it cannot know which way is up. If you hold the sextant even a few degrees off vertical, you measure the arc from the body down to a point on the horizon that is not directly beneath it, and that arc is longer. The figure shows what you see through the telescope as the instrument tilts.
Navigators handle this by swinging the arc. You rock the instrument gently from side to side about the telescope’s axis, and the body’s image swings like a pendulum bob in a shallow curve. You adjust the drum until the bottom of that swing grazes the horizon. Because the reading changes only as the square of the tilt,
One task, six instruments
The sextant was not the first instrument for measuring altitudes at sea, and the instruments before it were not crude drafts of it. Each one was made for particular bodies, latitudes, conditions and people, and each was used well in its own setting. The figure gives each the same two tasks, a low Polaris and a high noon Sun, and simulates twenty sights by one practiced observer in a moderate sea.
Kamal. A small board on a knotted cord, held at a set distance from the eye so that its lower edge sits on the horizon and its upper edge on a star. Arab and Indian navigators in the Indian Ocean measured star altitudes in a unit called the isba‘, or finger, about 1.6°, and Ibn Mājid’s fifteenth-century writings record them for many stars.[10] reconstructed How old the kamal itself is, and how it relates to other finger-measure methods, is contested; claims of a ninth-century origin rest on thin evidence. contested It suits the low stars of low latitudes, where Polaris stands only a few fingers high, and cannot reach a high Sun at all.
Mariner’s astrolabe. A heavy bronze wheel hung from a ring so that it found its own vertical, with a sighting bar pivoted at the center. It needs no horizon, which is useful at night or in haze, but wind and roll swing it. Iberian pilots used it for the noon Sun from the late fifteenth century; the oldest known example, recovered from the wreck of the Esmeralda off Oman, dates from about 1496–1501.[11] documented
Cross-staff. A staff held at the eye, with a sliding cross-piece (transom) whose ends you set on the horizon and the body. You must look at both at once, and for the Sun you must look into it; small changes in where you hold the staff against your eye bias the result.[12]
Backstaff. John Davis described it in The Seamans Secrets in 1594.[13] documented You stand with your back to the Sun and line up the shadow of one vane with the horizon seen through a slit in another. No glare, and a long arc, so it reads to a few arcminutes; it served English and Dutch ships through the seventeenth century.[12]
Octant and sextant. The reflecting instruments described above. Their scatter at sea is set less by the scale than by the horizon, the light and the observer: a good navigator with a good sextant expects single sights within about a minute of arc, and averages several.[6] illustrative
Columbus, often pictured with a sextant, could not have had one; it was invented more than 250 years after his first voyage. He navigated mainly by dead reckoning, and the latitudes he recorded from quadrant sights in 1492 were badly wrong, a reminder that instruments of that era demanded practice and steady conditions.[14] documented
Resolution is not accuracy
A modern sextant reads to 0.2′, about 370 meters of position. Nobody gets fixes that good from it routinely at sea. The horizon may be hazy or lit by moonlight that hides its true edge; abnormal dip can move it by several arcminutes; the deck is moving; refraction near the horizon is uncertain; a sight timed 4 seconds late moves the line of position by up to a mile. Each sight gives one line of position, and a fix needs two or more, each with its own errors. The next chapter, From a sight to a fix, turns these angles into a position and shows how much the errors matter. And every one of them depended on knowing the time, which is where Longitude is a clock problem began.