How do we actually know?

How it was done

• c. 240 BC · Eratosthenes, librarian at Alexandria, heard that at noon on the summer solstice the Sun shone straight down a well at Syene (modern Aswan), so a vertical stick there cast no shadow. On the same day at Alexandria, a stick's shadow showed the Sun was 1/50 of a full circle (7.2°) away from overhead. Since the Sun is so far away that its rays arrive parallel, that 7.2° is also the angle between the two cities at the centre of the Earth. The cities were said to be 5,000 stadia apart, so the whole circle is 50 × 5,000 = 250,000 stadia (he later rounded it to 252,000).
• c. 1025 · Al-Bīrūnī needed no second city. From the top of a mountain near Nandana (in today's Pakistan) he measured how far the horizon dipped below the horizontal (about 0.57°). Knowing the mountain's height from triangulation, a single triangle gives the radius. His figure, about 6,340 km, is within 0.5% of the truth.
• 1617 · Willebrord Snell invented triangulation: a chain of triangles between church towers in the Netherlands, of which only one baseline needs to be paced out. All the other sides follow from measured angles. His degree of latitude came out about 3½% too short, but the method became the backbone of all surveying for 350 years.
• 1669–70 · Jean Picard triangulated from Paris to Amiens with telescopic sights and found one degree of latitude to be 57,060 toises (about 111.2 km), a radius of about 6,372 km. Newton used Picard's value to check that the force holding the Moon in orbit is the same gravity that drops an apple.
• 1735–44 · The French geodesic missions to Lapland and Peru found a degree of latitude to be longer near the pole. The Earth is flattened, as Newton had predicted from its spin, so there is no single "radius" any more.

Did it hold up?

Eratosthenes' reasoning is flawless and is still taught unchanged. His inputs were rough: Syene is not exactly on the Tropic of Cancer, it is not due south of Alexandria, and 5,000 stadia was probably a round number from camel caravans or royal pacers. Worst of all, we don't know which stadion he used. With the Egyptian stadion of about 157.5 m his answer is astonishingly close (within 1%). With the Attic stadion of about 185 m it is about 16% too large. Either way, the method was right and the answer was the right size.

Try itEratosthenes with a friend. Find someone 300–1,000 km due north or south of you. On the same day, both of you stand a stick vertically (check it with a plumb line) on level ground and measure the shortest shadow, which comes at local solar noon. Neither of you needs the Sun overhead: the method works with the difference of the two shadow angles. Map distances are fine for the north–south separation.

Try itWatch the Sun set twice. This method needs no friend at all. Lie on a beach facing a flat sea horizon and start a stopwatch as the last sliver of Sun disappears. Then stand up quickly: the Sun is visible again, because your eyes are higher and can see a little further around the curve. Stop the watch when it vanishes a second time. The higher your eyes, the longer the delay. On a smaller Earth the horizon dips faster and the delay is longer. Dennis Rawlins published this in the American Journal of Physics in 1979.

The Earth's axis is tipped by about 23½° relative to its orbit. That tilt causes the seasons, and it sets the latitudes of the tropics and the polar circles. Two things can be measured: the tilt itself (the obliquity), and the slow circling of the axis, like a wobbling spinning top. That circling is called precession, and it means the pole star changes over the centuries.

How it was done

• c. 230 BC · Eratosthenes (the ratio is usually credited to him; Ptolemy reports it) measured the Sun's noon height at both solstices. The arc between the summer and winter Sun was 11/83 of a full circle, 47°42′, so the tilt is half of that: 23°51′.
• c. 130 BC · Hipparchus compared his position of the star Spica with one Timocharis had made about 160 years earlier. Spica had slid 2° closer to the autumn equinox point. The stars weren't moving: the equinox was, which means the Earth's axis is turning. He put the rate at "at least 1° per century".
• c. 150 · Ptolemy kept both numbers, 23°51′ and 1° per century. As the chart below shows, copying a value is not the same as measuring it.
• c. 900 · Al-Battānī at Raqqa found 23°35′, clearly smaller. Medieval astronomers argued over whether the tilt really changes. A popular theory, "trepidation", had the equinoxes rocking back and forth. It was wrong, but it was an honest attempt to fit measurements that disagreed.
• 1437 · Ulugh Beg at Samarkand built a meridian arc about 40 m in radius into a hillside and measured 23°30′17″, within half an arcminute of the true value for his time.
• 1687 → · Newton explained precession: the Sun and Moon pull on the Earth's equatorial bulge and twist its axis. In the 18th century Lagrange and Laplace showed that the planets slowly tilt the Earth's orbit, which is why the obliquity drifts. Today we know it swings between 22.1° and 24.5° every 41,000 years, one of the rhythms behind the ice ages.

Did it hold up?

Better than anyone at the time could know. Plot the old values against the modern theory and they are not simply wrong: the tilt really was larger. Eratosthenes is about 8′ high for his own century. That is a quarter of the Sun's width, typical for a gnomon whose shadow edge is blurred by the Sun's disc. Ptolemy copied his number 370 years later, by when the true tilt had shrunk further, so Ptolemy's error is bigger. Al-Battānī and Ulugh Beg, with large and carefully built instruments, follow the decline to within an arcminute or less. Hipparchus' precession, "at least 1° per century", was an honest lower bound; the truth is 1.4°.

As the axis circles, the north celestial pole traces a circle 23½° in radius among the stars. Today it points close to Polaris; in 2100 the two will be just 0.46° apart. When the pyramids were built, the pole star was Thuban in Draco. In about 12,000 years it will be Vega. Drag the year and watch.

Try itTwo solstice noons. Fix a vertical stick (a gnomon) on level ground and mark its shortest noon shadow on the June solstice and again on the December solstice. Put a small ball or a disc with a hole on top of the stick: its round shadow lets you find the centre of the Sun's image rather than its fuzzy edge. Half the difference of the two Sun angles is the tilt, and their average gives your latitude. The calculator removes atmospheric refraction, which lifts the low winter Sun by several arcminutes.

Try itPrecession from old star catalogues. Take any star's position measured relative to an equinox long ago, compare it with today's, and divide by the years in between. Hipparchus did exactly that with Spica. You can also check Ptolemy's catalogue: its positions are famously about 1° off, which is part of the long-running argument over whether Ptolemy measured them himself or recycled Hipparchus' list.

An eccentricity of 0 is a circle and an eccentricity of 1 is an open parabola. The planets' orbits are so nearly circular that if you drew Earth's orbit to scale on paper, you couldn't tell it from a circle with a ruler. What gives the ellipse away is speed: by Kepler's second law, a planet sweeps out equal areas in equal times, so it moves faster when it is closer to the Sun.

How it was done

• c. 150 BC · Hipparchus timed the seasons: spring lasted 94½ days but summer only 92½. If the Sun went round a circle at constant speed with the Earth at its centre, the seasons would be equal. He moved the Earth off-centre by 1/24 of the radius, an "eccentric circle". That number is (to first order) twice the eccentricity we use today.
• c. 150 · Ptolemy added the equant, a point off-centre around which a planet's motion looks uniform. It was a clever hack that reproduced most of the speed variation.
• 1543 · Copernicus put the Sun at the centre but kept perfect circles, so he still needed small epicycles.
• 1576–1601 · Tycho Brahe measured planet positions with huge naked-eye instruments to about 1–2 arcminutes. That was ten times better than anyone before him.
• 1609 · Johannes Kepler fought Mars for six years. His best circle-and-equant model missed Tycho's observations by 8 arcminutes, and he refused to blame Tycho. Mars returns to the same point in space every 687 days while the Earth is somewhere else, so he could triangulate Mars from two places on Earth's orbit (try it in the simulation below). He found an ellipse with the Sun at one focus. "These eight minutes alone," he wrote, "led to the reform of all of astronomy."
• 1687 · Isaac Newton derived all three of Kepler's laws from an inverse-square force of gravity. Ellipses stopped being a description and became a consequence.

Did it hold up?

Kepler's laws are still how orbits are computed to first order. Real orbits are slightly perturbed ellipses: Jupiter tugs on Mars, and the Sun tugs on the Moon so strongly that the Moon's eccentricity wobbles between about 0.026 and 0.077. After Newtonian perturbations were accounted for, Mercury's perihelion still crept forward by 43 arcseconds per century. That tiny excess was explained only by Einstein's general relativity in 1915.

Try itHipparchus' calendar trick, for Earth. Look up (or observe) the dates of the equinoxes. Between the March and September equinoxes the Earth covers exactly half of its orbit, as seen from the Sun, but it takes about 7½ days longer than the other half, because it is near aphelion in July and moving slowly. To first order, the difference is (4e/π) × year × sin ν. Here ν = 77° is the angle between perihelion and the March equinox as seen from the Sun.

Try itMeasure the Sun's size in January and July. Punch a pinhole in foil, project the Sun onto white card 2–3 m away and measure the disk's diameter. Never look at the Sun directly. Distance is inversely proportional to apparent size, so the eccentricity is (big − small) ÷ (big + small). Expect a difference of about 3.4%, roughly 0.3 mm at 1 m, so use a long throw and measure carefully. The same works for the Moon: photograph it with the same lens and zoom at a perigee ("supermoon") and an apogee, then count pixels across the disk.

The Sun's radius is the easy part: its angular size is simple to measure (about half a degree). Its distance is the hard part, and it was called "the noblest problem in astronomy". Once you know the distance, the radius is just distance × angle.

How it was done

• c. 270 BC · Aristarchus of Samos noticed that when the Moon is exactly half lit, the angle Sun–Moon–Earth is 90°. Measuring the angle Moon–Earth–Sun at that moment then fixes the shape of the triangle. He estimated 87°, which made the Sun 18 to 20 times farther than the Moon. The Sun looks the same size as the Moon, so it must also be about 19 times larger, and much larger than the Earth. Aristarchus apparently drew the obvious conclusion that the big thing sits in the middle, 1,800 years before Copernicus.
• c. 150 · Ptolemy, using eclipse geometry, put the Sun at about 1,210 Earth radii, about 20 times too close. This was the accepted value until the 1600s.
• 1672 · Giovanni Cassini and Jean Richer observed Mars at the same time from Paris and from Cayenne in French Guiana. The shift of Mars against the stars between the two places (parallax) gave Mars's distance. Kepler's relative scale then gave everything else: a solar parallax of about 9.5″, or about 138 million km. It was the first realistic value, 7% too small.
• 1761, 1769 · Transits of Venus. Edmond Halley had shown that timing Venus crossing the Sun's face from widely separated places on Earth gives the parallax. Expeditions went all over the globe, among them James Cook's first voyage to Tahiti. The "black drop" effect (Venus seemed to stick to the Sun's edge) smeared the timings by tens of seconds. Encke's 1824 reduction gave 8.58″, about 153 million km.
• 1874, 1882 · More transits, with photography. Simon Newcomb's 8.80″ (149.5 million km) became the international standard in 1896.
• 1931 · Asteroid Eros passed close to Earth. A worldwide campaign of 24 observatories measured its parallax on photographic plates, and Harold Spencer Jones obtained 8.790″, about 149.67 million km.
• 1961 · Radar echoes from Venus (Jodrell Bank, MIT, JPL Goldstone) timed the round trip of a radio pulse. The AU jumped from four significant figures to seven: 149,598,500 ± 500 km.

Did it hold up?

Aristarchus' geometry is perfect. His measurement could not have worked. The true angle is 89.85°, so the Sun is 390 times farther than the Moon, and the difference from 90° is a tenth of the Moon's own width. Nobody can judge the exact moment of half moon that precisely, because the terminator looks straight for hours. Try the presets above: an error of 2.8° in the angle becomes a factor of 20 in the answer. It is a classic ill-conditioned measurement. Still, even his underestimate showed that the Sun dwarfs the Earth, and that was the right conclusion.

The parallax methods held up but were limited by the instruments: blurry planet disks, the black drop, atmospheric shimmer. Radar replaced them all at once.

Try itThe Sun's angular size, with a pinhole. In a long, dark corridor, project the Sun through a 1–2 mm hole in foil onto card. The image diameter divided by the hole-to-card distance is the Sun's angular diameter (about 1 cm per metre). Multiply by the distance to get the radius. The Moon is almost exactly the same angular size, which is why total eclipses happen at all.

Try itThrough the speed of light. In 1729 James Bradley discovered that stars shift back and forth by about 20½ arcseconds over a year. This is aberration, the way rain seems to slant when you run. The angle is Earth's orbital speed divided by the speed of light. Speed × one year gives the circumference of the orbit, so this route to the AU needs no parallax at all. Today the same speed can be measured with a spectrograph: a star near the ecliptic shows a Doppler shift of ±29.8 km/s over the year. Some well-equipped amateur spectroscopists can resolve that.

Light covers the length of a football pitch in a third of a microsecond. No human reaction is fast enough to notice that, so the first measurement had to borrow a clock from the sky and a distance from the Solar System. The speed of light and the astronomical unit are tied together: Rømer needed the AU to turn a light-time into a speed, and three centuries later radar needed the speed to turn a light-time into the AU.

How it was done

• 1638 · Galileo proposed two people with shuttered lanterns on hills: the second uncovers his lantern the moment he sees the first. The experiment was tried over about a mile and showed no delay. Light needs about 10 microseconds for that, and human reaction takes 0.2 s.
• 1676 · Ole Rømer, a Danish astronomer working at the Paris Observatory, timed eclipses of Jupiter's moon Io, which disappears into Jupiter's shadow every 42½ hours. The eclipses ran early while the Earth approached Jupiter and late while it receded. He predicted that the eclipse of 9 November 1676 would come about 10 minutes late, and it did. Light, he concluded, takes about 22 minutes to cross the Earth's orbit. Huygens (1690) combined that with Cassini's AU and got roughly 212,000 km/s.
• 1729 · James Bradley's aberration of starlight (Chapter 4) gave the ratio of light's speed to the Earth's orbital speed: light takes about 8 min 12 s to reach us from the Sun.
• 1849 · Hippolyte Fizeau made the first measurement entirely on Earth. A beam passed between the teeth of a spinning wheel with 720 teeth, travelled 8.6 km from Suresnes to a mirror on Montmartre, and came back. At 12.6 revolutions per second, the returning light hit a tooth and the beam went dark. That gives 313,000 km/s.
• 1862 · Léon Foucault replaced the wheel with a fast rotating mirror and fitted the light path inside a room: 298,000 km/s. In 1850 he had already shown that light is slower in water, which settled an old argument about the nature of light in favour of waves.
• 1879–1935 · Albert Michelson refined the rotating mirror for half a century: between Mount Wilson and Mount San Antonio, 35 km apart (299,796 km/s in 1926), and finally in a 1.6 km evacuated pipe.
• 1950–1972 · Microwaves and lasers. Louis Essen measured a microwave cavity's resonance (299,792.5 km/s). In 1972 Kenneth Evenson's group at NBS measured a laser's frequency and its wavelength separately and multiplied them: 299,792,456.2 ± 1.1 m/s. The remaining uncertainty came from the definition of the metre itself.
• 1983 · The metre was redefined as the distance light travels in 1/299,792,458 of a second. The speed of light became exact, and it can no longer be measured in SI units. An experiment today measures your ruler, not light.

Did it hold up?

Rømer's reasoning was right and was soon accepted, though Cassini himself resisted it. His number was about 30% low: the true crossing time is 16 min 38 s, not 22 min, and Cassini's AU was 7% too small. Fizeau was 5% high and Foucault 0.6% low. Then came a cautionary tale. Measurements in the 1930s and 1940s, including Michelson's last one, clustered about 17 km/s below the true value, with error bars that excluded it. Raymond Birge's influential 1941 review adopted 299,776 km/s. In 1950 microwave methods jumped straight to the modern value. Historians of science use this as the textbook example of experimenters unconsciously steering toward an accepted number.

Try itFizeau's toothed wheel. Below is Fizeau's apparatus: 720 teeth and a mirror 8,633 m away. Speed the wheel up until the returning light goes dark for the first time. At that speed, the wheel turns by half a tooth gap in the time light takes for the round trip.

Try itA microwave oven and a bar of chocolate. Take out the turntable (or put an upside-down plate over it so nothing rotates). Lay a flat bar of chocolate, a slice of cheese or a layer of marshmallows inside and heat for 15–25 seconds until a few spots just begin to melt. The waves inside form a standing pattern, and the hot spots are half a wavelength apart. Speed = 2 × spot spacing × frequency, and the frequency (usually 2.45 GHz) is printed on the oven's label.

Try itRømer's way. With a small telescope, time several eclipses of Io, from the moment it fades into Jupiter's shadow or reappears from it. Work out the average interval from eclipses close together, then predict an eclipse months later. The miss, divided into the change in the Earth–Jupiter distance, is the light-time per AU.

How it was done

• c. 270 BC · Aristarchus watched lunar eclipses. The Moon is crossing the Earth's own shadow, and the curved edge of that shadow on the Moon is a picture of the Earth's size at the Moon's distance. He judged the shadow to be about 2 Moon-widths across. Because the Sun is far away and looks as big as the Moon, the shadow narrows by exactly one Moon diameter over the Earth–Moon distance. So Earth diameter = shadow + 1 Moon. With the true shadow ratio (about 2.6 to 2.7), the Moon comes out at 0.27 Earth diameters. That is right.
• c. 150 BC · Hipparchus used the solar eclipse of 190 BC. It was total at the Hellespont, while at Alexandria only 4/5 of the Sun was covered, so the Moon's position against the Sun differed between the two places by a fifth of the Sun's width. That is parallax again. He bracketed the distance at roughly 62–73 Earth radii, against a true range of 56–64.
• c. 150 · Ptolemy settled on 59 Earth radii at full and new Moon, within a few percent of the truth. Strangely, his model of the Moon's motion made the Moon's apparent size change by nearly a factor of two each month, which anyone could see was false.
• 1751–53 · Lacaille and Lalande observed the Moon at the same time from the Cape of Good Hope and Berlin, nearly on the same meridian and 90° of latitude apart, and turned its parallax into a modern-quality distance.
• 1946 · Project Diana (US Army) bounced radar off the Moon for the first time, and the round-trip time gave the distance directly.
• 1969 → · Lunar laser ranging. Apollo 11 left a retroreflector. Observatories fire laser pulses at it and catch a handful of returning photons per second, timing the 2.5-second round trip to picoseconds.

Did it hold up?

The eclipse method is one of the great successes of ancient science. It needs no instruments and is almost free of error amplification: an eclipse photograph today gives the Moon's radius to a few percent. The distance follows from the Moon's angular size: the Moon spans about half a degree, so it is about 110 diameters away. Hipparchus' and Ptolemy's distances were good to a few percent, far better than anything they achieved for the Sun.

Try itMeasure the Earth's shadow on the Moon. Photograph a partial lunar eclipse (any phone on a tripod, through binoculars or a small telescope). Click three points along the edge of the dark shadow below and the page fits a circle through them, exactly as you would on your own photo. The shadow's radius divided by the Moon's radius is the ratio Aristarchus needed. The edge is fuzzy because of the penumbra and the Earth's atmosphere, so aim for the middle of the gradient.

Try itTime a radio echo. Radio amateurs do "moonbounce" (EME): they transmit on 144 MHz or higher toward the Moon and hear their own signal come back about 2.5 seconds later. Distance = speed of light × delay ÷ 2. If you know a ham with an EME station, the echo delay alone gives the distance to better than 0.1%.

For Mars the history comes in two steps. First the shape of the Solar System was found, with every distance relative to the Earth–Sun distance, and that was surprisingly easy. Then the scale came, which is the AU problem from Chapter 4. Mars's radius is its angular size times its distance.

How it was done

• 1543 · Copernicus showed that in a Sun-centred system each planet's distance follows from simple timing. At opposition Mars is opposite the Sun in our sky. About 106 days later Mars is at quadrature: 90° from the Sun as seen from Earth. In those 106 days Earth and Mars have drifted apart by an angle θ around the Sun, known from their periods, and the right triangle Sun–Earth–Mars gives Mars's distance = 1 / cos θ AU. His value for Mars was 1.52 AU. Ptolemy's system could not give distances at all.
• 1619 · Kepler's third law: period² ∝ distance³. Mars's 1.88-year period gives 1.524 AU directly.
• 1672 · Cassini & Richer measured Mars's parallax from Paris and Cayenne at a close opposition, which put the first kilometres on the whole relative map.
• 1600s–1800s · Telescopic micrometers measured Mars's disk, which is between 4″ and 25″ across, a coin seen from a kilometre or more. Results scattered by ±10%. William Herschel (1784) even "measured" a polar flattening of 1/16, five times too large, because the bright polar caps fooled the eye.
• 1960s–70s · Radar and spacecraft. Goldstone radar ranged Mars in the 1960s. Mariner 9 mapped it from orbit, and the Viking landers (1976–82) were radio-ranged to about 7 m. That was precise enough to measure the relativistic Shapiro delay of radio waves passing the Sun.

Did it hold up?

Copernicus' relative distances were already good to about a percent, and Kepler's to a fraction of a percent. Their geometry was sound. The circular-orbit assumption is the main error: Mars's distance from the Sun varies by ±9%, so the quadrature trick gives a different answer at every opposition. Average several. The absolute scale inherited every error of the AU (7% for Cassini). Telescopic radii were limited by atmospheric blur, which is as large as Mars's whole disk on a bad night.

Try itMars's radius from a photo. At opposition, image Mars through a telescope with a camera, then image a double star of known separation (or use the plate scale, 206,265 × pixel size ÷ focal length, in arcseconds per pixel). That gives Mars's diameter in arcseconds. Multiply by its distance at that date: the Copernicus/Kepler AU value above, or a free ephemeris. The next opposition is on 19 February 2027, with Mars about 101 million km away and 13.8″ across.

Gravity between everyday objects is absurdly weak: two 1 kg balls 10 cm apart pull on each other with about the weight of a grain of dust. Astronomy measures GM (the product) with breathtaking precision from orbital periods, but to split it into G and the Earth's mass you must weigh something in the lab. That is why G is the least precisely known of all the fundamental constants.

How it was done

• 1687 · Newton guessed that the Earth's mean density was 5–6 times that of water (which is right), but thought measuring the pull between two spheres in a lab was hopeless.
• 1774 · Nevil Maskelyne measured how much a plumb line was pulled sideways by the very regular Scottish mountain Schiehallion: about 11.6″. Charles Hutton surveyed the mountain's volume (and invented contour lines to do it) and deduced an Earth density of 4.5 g/cm³. The method was right, but the rock density was guessed.
• 1798 · Henry Cavendish used John Michell's torsion balance: a 1.8 m wooden beam with two small lead balls, hung from a thin wire in a sealed shed and watched by telescope from outside, with two 158 kg lead spheres swung in beside the small balls. The beam turned by a fraction of a degree and oscillated with a period of minutes. From the twist, the period and the geometry he found the Earth's density to be 5.448 g/cm³. He quoted it as "weighing the world"; G as a named constant came later.
• 1895 · C. V. Boys shrank the balance and used fused-quartz fibres, getting 6.658 × 10⁻¹¹.
• 1930 → · Heyl, Luther & Towler, Gundlach & Merkowitz, Quinn, Rosi, Li… Time-of-swing methods, angular-acceleration feedback, beam balances and atom interferometers have all been used. Each experiment claims a precision of tens of parts per million, yet they disagree with each other by several hundred.

Did it hold up?

Cavendish's density converts to G ≈ 6.75 × 10⁻¹¹, about 1% high. That is remarkable for 1798, and it stood for nearly a century. The torsion balance is still the workhorse. What did not hold up is the modern hope of nailing G down: the chart shows two centuries of improvement, then a floor at about 10⁻⁴ to 10⁻⁵, where careful experiments scatter beyond their own error bars. Nobody knows why. Unrecognised systematic errors are suspected, not new physics.

Try itA Cavendish balance. Below is a simulated tabletop torsion balance of the kind sold to university labs (or built at home with fishing line, a mirror, lead fishing weights and patience). Two 15 g balls sit on a 10 cm beam hanging from a fine fibre. A laser reflects off a mirror on the beam onto a wall 5 m away. Swing the two 1.5 kg masses to the other side and the beam starts to oscillate around a new rest angle. Read off the period and the shift of the rest position from the trace, then compute G.

Try itThen weigh the Earth. Time a pendulum (a nut on a 1 m string, ten swings) to get g. Because g = GM/R², your G from the balance and your Earth radius from Chapter 1 give the Earth's mass and its density. This is what Cavendish actually reported. Every number in this calculator can come from your own measurements.

Anything that orbits weighs whatever it orbits. Newton turned Kepler's third law into a scale: GM = 4π² a³ / T², where a is the orbit's size and T its period. Every orbit gives the product GM of the central body. You need the G from the previous chapter only to convert that into kilograms. And when you compare two bodies, G cancels. If the distances are measured as angles, even the size of the AU cancels.

How it was done

• 1610 · Galileo discovered four moons circling Jupiter: a miniature solar system that everyone with a telescope could watch.
• 1621 · Kepler noted, using Simon Marius' timings, that Jupiter's moons obey his third law too: the cube of the distance over the square of the period is the same for all four. That constant is Jupiter's "weight", in different units from the Sun's.
• 1687, 1726 · Newton, in the Principia, compared the moons of Jupiter, Saturn's Titan, the Moon and the planets themselves. In the third edition he gave the masses of Jupiter, Saturn and the Earth as fractions of the Sun's (table below). It was the first time anyone had weighed a planet.
• 1798 · Cavendish's torsion balance (Chapter 8) supplied G, and so the first masses in actual kilograms.
• 1973 → · Spacecraft. Pioneer 10's flyby of Jupiter was tracked by the Doppler shift of its radio signal. Galileo and Juno orbited the planet. Each spacecraft is a test particle, and the bending of its path gives Jupiter's GM to better than one part in ten million.

Did it hold up?

Newton's Jupiter is within 2% of the truth. His Earth is almost twice too heavy. The difference is instructive. Jupiter's mass ratio needs only angles and times: how far its moons swing from the planet, how far Jupiter is from the Sun in AU, and the periods. The Earth's ratio needs the Earth's size in AU, which is the solar parallax. Newton assumed 10½″, putting the Sun about 16% too close, and masses scale with distance cubed. His method was right; the AU was not.

Try itWatch Jupiter's moons. Steady binoculars on a tripod show all four Galilean moons as points in a line beside Jupiter. Photograph them each clear night for two or three weeks with a phone held to the eyepiece or a camera on a tripod, and measure each moon's distance from Jupiter. Each moon swings back and forth in a sine curve. Its greatest distance is the orbit's radius as an angle, and the time between repeats is its period. The simulated observing log below shows a month of them.

Try itWeigh the Sun. Any planet works: its distance from the Sun (from the AU chapter) and its year. Every planet should give the same mass. That is Kepler's third law, now with a number attached.