A right angle short by a seventh of a degree

The radius of Venus’s orbit follows from one angle: at greatest elongation the triangle Sun–Venus–Earth has its right angle at Venus, and the sine of the elongation is the orbit’s radius in units of the Earth’s. The same triangle, with the right angle at a different body, was used some eighteen centuries earlier on the Moon, and it asked a far more ambitious question — not the relative sizes of two orbits round the Sun, but how much further away the Sun is than the Moon.

The construction belongs to Aristarchus of Samos, and it is correct. A phase is a matter of geometry, not of shadow: the Moon is exactly half lit when the line from the Moon to the Sun is perpendicular to the line from the Moon to the Earth. At that instant — dichotomy — the Earth, the Moon and the Sun form a right triangle with the right angle at the Moon, and the angle between the Moon and the Sun seen from the Earth, θ, satisfies

$$\frac{d_{\rm Sun}}{d_{\rm Moon}} = \frac{1}{\cos\theta}$$

No distance is measured. One angle gives a ratio of two distances, and Aristarchus measured the angle as 87°, which gives nineteen.

Where the construction fails #

The Sun is about 389 times further away than the Moon, so the angle Aristarchus needed was 89.853° — short of a right angle by 0.147°, about a quarter of the Moon’s own apparent diameter. He measured 87°, short by three degrees, twenty times too much.

That is not an argument that Aristarchus was careless. It is an argument about what the secant does near a right angle. The ratio goes as $1/\cos\theta$, and its logarithmic derivative is $\tan\theta$, which near 90° is the ratio itself per radian. At 87° a tenth of a degree changes the answer by three per cent; at 89° by ten; at the true angle, a tenth of a degree is the difference between 290 and 590. The measurement is exact in principle and asks for a precision of a hundredth of a degree to give the ratio to ten per cent.

No naked-eye instrument of the third century BC could approach that, and the instrument was not really the difficulty. The angle has to be measured at an instant, and the instant is defined by a judgement — the moment the terminator, the boundary between the lit and dark halves of the Moon, is a straight line. The terminator runs across mountains and craters, whose shadows make it ragged, and to the eye it looks straight for hours either side of the true dichotomy. In those hours the Moon moves by several times the 0.147° being sought.

What the eye would have to see #

The judgement can be made quantitative, and the numbers show why no amount of care would have rescued it.

The fraction of the Moon’s disc that is lit, as seen from the Earth, is $(1 + \cos i)/2$, where $i$ is the phase angle at the Moon between the Sun and the Earth. At dichotomy $i = 90°$ and the fraction is one half, and near that point it changes by half the change in $i$, measured in radians. The true half moon and the moment the Moon is at quadrature — 90° from the Sun as seen from the Earth — are separated by the 0.147° being sought. Across that separation the lit fraction changes by 0.13 per cent.

The terminator gives the same verdict in a form that can be compared with the eye’s resolution. The terminator is the Moon’s limb seen at an angle: an ellipse whose semi-minor axis is the Moon’s radius times $\cos i$. At exact dichotomy it is a straight line; 0.147° away from it, the line bulges by the Moon’s radius times the sine of 0.147°. The Moon’s radius on the sky is about 15.5 arcminutes, so the bulge is 2.4 arcseconds — against an unaided eye that resolves about 60. The instant Aristarchus needed was defined by a curvature twenty-five times too small to see, on a boundary broken up by the same mountains that make the lunar limb ragged when it is used to time an eclipse.

What the wrong answer still got right #

The conclusion that survived the error is the one that mattered most. Nineteen is wrong by a factor of twenty, but it is much larger than one, and Aristarchus drew the inference directly. The Sun and the Moon have nearly the same apparent size — the fact that makes total solar eclipses possible — so if the Sun is nineteen times further away it must be nineteen times larger in diameter than the Moon. Combined with an estimate of the Moon’s size from the width of the Earth’s shadow during a lunar eclipse, it made the Sun several times larger than the Earth.

A body several times larger than the Earth going round a smaller one seemed wrong to Aristarchus, and he is reported to have proposed the alternative: the Earth goes round the Sun. The measurement that was twenty times too small was large enough to support a heliocentric argument eighteen centuries before Copernicus, and the proposal was not adopted, partly because it predicted a stellar parallax that nobody could see. The triangle that reaches the stars would not be closed for another two thousand years.

The error was not corrected quickly. The eclipse-diagram method attributed to Hipparchus, which combined the sizes of the Earth’s shadow and of the Moon, gave a solar distance of about twelve hundred Earth radii, and that figure — too small by a factor of twenty — held through Ptolemy and the Middle Ages. It was consistent with Aristarchus’s ratio, and the consistency of two methods that shared no measurement was taken, understandably, as confirmation. What they shared was not a measurement but a sensitivity: both methods reduce, at the step that matters, to a small difference between two nearly equal angles.

The distance the method was paired with #

A ratio needs a scale, and the Moon supplied it more easily than the Sun could. The Moon is close enough that its position against the stars differs by about a degree between an observer who sees it overhead and one who sees it on the horizon — a parallax seven times the angle the dichotomy method needed, and large enough to measure with the instruments that failed on 0.147°. Hipparchus obtained a lunar distance of about sixty Earth radii from the parallax and from the geometry of eclipses, and the modern mean is 60.3.

So the two distances in Aristarchus’s ratio were not equally hard. One rested on a parallax of a degree and was right to a few per cent; the other rested on a departure from a right angle of a seventh of a degree and was wrong by a factor of twenty. The solar distance that came out of combining them was the product of a good number and a bad one, and it carried the bad one’s error undiluted into every figure for the size of the solar system for eighteen centuries. The same parallax that measures the Moon is, at the Sun’s distance, nine arcseconds — and it was the size of that angle, not any lack of method, that kept the scale of the solar system out of reach.

The relative scale was a different matter entirely. The law that links period and size gives every planet’s distance in units of the Earth’s from nothing but the periods, and the periods are the most precisely measured quantities in ancient astronomy. By the seventeenth century the shape of the solar system was known to a few parts in a thousand and its size to no better than a factor of two. A single well-measured length anywhere in it — the Earth–Mars distance at one opposition, or the Earth–Venus distance at one transit — would fix every other, which is why the expeditions that eventually measured one were worth mounting across oceans.

The same angle as a clock #

An angle that cannot be measured to a hundredth of a degree can sometimes be turned into a time that can be measured to a few minutes. The dichotomy method has such a form, and it shows both why the idea is good and why the sky defeats it.

If the Sun were infinitely far away, the half moons would fall exactly at quadrature, 90° either side of the Sun, and the Moon would spend exactly half a month getting from first-quarter half moon to last-quarter half moon through full, and half a month back through new. At a finite distance each half moon falls short of quadrature by δ = 90° − θ, on the side towards the Sun. The Moon must therefore gain 180° + 2δ on the Sun through full and only 180° − 2δ through new. At the Moon’s mean rate of gain, 12.19° a day, the two halves of the month differ by 4δ divided by that rate.

The conversion is valuable because it changes what has to be judged. The angle method needed the direction to the Sun and the direction to the Moon at a single instant; the timing method needs two instants, each judged the same way, separated by a fortnight, and it needs them for many months so that errors in judging the terminator average down. The asymmetry it measures is 70 minutes out of a fortnight, which is the same fractional precision as before — but a fortnight is a long time over which to accumulate a precise count of days, and seventy minutes is not a small thing to a water clock.

The noise the Moon supplies #

The method fails for a reason that has nothing to do with the Sun. The Moon does not move at its mean rate.

An orbit with an eccentricity of 0.055 carries the Moon faster than average near perigee and slower near apogee, and the difference between the two halves of any given month is dominated by where perigee happens to fall relative to full moon. The effect is up to a day either way, and it changes from month to month as perigee advances round the orbit every 8.85 years. To extract a 70-minute asymmetry from a 23-hour systematic that varies on a nine-year cycle, the lunar orbit itself would have to be modelled to better than the signal — which, for the ancient and medieval astronomers who had the method, it was not.

There is a quiet lesson in which effect wins. The signal comes from the Sun’s finite distance; the noise comes from the Moon’s own dynamics. An experiment designed around the idealisation of a circular orbit is limited by the departure from that idealisation, not by the instrument, and it is limited at a level twenty times the effect being sought. By the time lunar theory was good to a few minutes of time, in the eighteenth century, the solar distance was being measured by far better means.

How the ratio was finally measured #

Every successful measurement of the solar distance used a baseline rather than a right angle. The relative scale of the solar system was known to three figures from Kepler’s third law and the synodic periods; what was missing was a single length in metres anywhere in it, and a single length fixes everything.

The first good one came from Mars at opposition in 1672: simultaneous observations from Paris and French Guiana gave its parallax against the stars and hence its distance, and with it the astronomical unit, to within about ten per cent. The transits of Venus in 1761 and 1769, observed from stations spread across the Earth, improved on that — the timing of Venus crossing the Sun’s disc from widely separated places depends on the solar parallax, and the method turns an angle of nine arcseconds into differences of minutes, exactly the trick that the half-month timing attempted. It worked where the lunar version did not because Venus’s orbit was known well enough that its irregularities were smaller than the signal. And every distance measured beyond the solar system inherited the scale it set.

Today the astronomical unit is defined as an exact number of metres, and the distances within the solar system are measured by radar and by spacecraft ranging to metres. The ratio of the Sun’s distance to the Moon’s is 389.2 on average and varies between about 360 and 420 as both orbits carry their bodies nearer and further. With the Moon near perigee the dichotomy angle is about 89.86°; near apogee, about 89.84°. The difference between those two is itself as large as the precision the method would have needed.

What the construction assumes #

The triangle assumes the Moon is a sphere lit by a point source at infinity, which is close enough; and that half-lit means the terminator is a great circle perpendicular to the sunlight, which it is. It assumes the observer is at the Earth’s centre, and an observer on the surface is displaced by up to a degree in the Moon’s apparent position — lunar parallax — which is seven times the angle being sought and has to be removed. And it assumes the Moon is exactly half lit when the terminator looks straight, when in fact the eye’s judgement is biased by the rough surface and by the brightness gradient across the disc, so that the moment of apparent dichotomy is offset from the true one by an amount that depends on the observer.

None of those is fatal alone. Together they set a floor well above 0.147°, and the lesson of the figures is that the floor was never going to be reachable with this construction, whatever the instrument.

A large quantity read from a small departure from symmetry #

A ratio computed from the secant of an angle near 90° is a ratio computed from the difference between that angle and 90°. That difference, not the angle, is the measured quantity, and its relative precision is what sets the answer’s precision. The construction is exact and the sensitivity is fatal — a combination that recurs whenever a large quantity is obtained from a small departure from a symmetric configuration.

Greatest elongation escapes the problem because Venus’s elongation is 46°, where the sine changes gently: a tenth of a degree moves the radius by less than a fifth of a per cent. Mercury’s elongation varies because its orbit is eccentric, which the method records as a spread rather than hiding as a bias. Parallax is the inverse case — a tiny angle measured directly — and its difficulty is honest rather than hidden.

All three constructions are about the same thing, which is what the sky offers an observer who is moving. A planet’s retrograde loop is the difference of two position vectors seen from one of them, and its width is set by their ratio; greatest elongation is the extreme of that difference for an inner planet; dichotomy is the configuration in which the difference between the Moon’s direction and the Sun’s is as close to perpendicular as it gets. In each case the ratio of two distances is hidden in an angle, and in each case what decides whether it can be read is not the geometry but how steeply the angle depends on the ratio at the configuration where it is measured. The geometry is the same; only the derivative differs. Aristarchus’s method hides it: the angle he read was 87°, a perfectly comfortable number, and the quantity that mattered was the three degrees he could not see.

Still open: what the planets’ meetings measure #

The half-month asymmetry is a beat between two motions — the Moon’s gain on the Sun and the Sun’s finite distance — read as a timing difference. Two planets produce a cleaner beat, because both of their motions are nearly uniform and both periods are long. When Jupiter and Saturn meet, their meetings do not recur at the same place but march round the sky, and the pattern they trace turns slowly; the rate at which it turns is a statement about how nearly their two periods fit a ratio of small whole numbers, which is also the rate at which the two planets perturb each other most strongly.