The Cosmic Ladder

27 Nov, 2025

The universe is big. Stupidly big. And it doesn’t exactly include a "You Are Here" marker. Everything is far away, everything is zooming, and everything is on fire. Worse still, the whole place is fashioned so that even simple distances between objects are ridiculously hard to determine. You can't go to a star and count your steps, you can't scream at it and measure the echo. You can't even trust your own eyes: the Sun and the Moon look the same size, which is adorable, considering that the Sun could obliterate the Moon like a bowling ball crushing a grain of sand. So as far as figuring things out goes, the universe is about as unhelpful as it gets.

Given all this, our success in piecing together even a small part of it is frankly astonishing. Sure, it took centuries of painstaking observations, brave extrapolations, and flashes of outright genius, but in the end we somehow managed to stack argument atop argument into a wobbly Jenga™ tower stretching from our tiny rock all the way to the stars. In a fit of victorious optimism, we now call this wobbly Jenga™ tower of precariously balanced arguments The Cosmic Ladder. Even though this name is itself a euphemism¹ of cosmic proportions, it provides us with some very neat imagery: that of consecutive rungs. So buckle up for The First Four or so Rungs of the Cosmic Ladder or How Far School Maths Can Be Stretched Before We Have to Invent Something New.²

The First Rung: The Earth

It’s 350 BC. The Earth sits confidently at the centre of creation. The Sun and Moon revolve around it, bringing day and night - and, if you like, the seasons - while the stars hang motionless, awe-inspiring in their indifferent majesty.

Then there’s Aristotle, perched somewhere in Greece, staring at the Moon, drinking in its quiet beauty. Utterly flabbergasted - much like anyone might be tonight. He already knows, from the old Greek teachings,³ that lunar eclipses happen when the Moon passes through Earth’s shadow. And as he lies there, he begins to appreciate it. In fact, he notices something marvellous: even though only a small part of Earth’s shadow ever falls on the Moon, he realises that it is always a fragment of a perfect circle. Not slightly oval. Not vaguely round. Circular. He reasons that a flat disc, the shape most naturally assumed for Earth at the time, would cast an ellipse depending on the viewing angle. Moreover, the only geometric object whose shadow remains circular from every vantage point is a sphere.⁴ And just like that, he deduces that Earth must be a globe. No instruments, no measurements. Just open-minded observation and sharp thinking.

Fast forward about a hundred years. Another Greek philosopher, one Eratosthenes, finally takes the spherical Earth seriously enough to attempt measuring its radius. The setup is gloriously simple: he knows that at noon on the summer solstice, the Sun is directly overhead in Syene (modern Aswan), so vertical sticks cast no shadow. Meanwhile, in Alexandria, vertical sticks do cast shadows. By measuring the angle (about ${7.2}^{\circ}$ ) of the shadow in Alexandria and the distance⁵ (about 788 $km$ ) between the two cities, he calculates the Earth’s circumference using the arc length formula

\frac{{7.1}^{\circ}}{360^{\circ}} ~ \frac{788 km}{Circumference of Earth},

which yields

Circumference of Earth ~ 39400 km .

This corresponds to a value of $R_{E} ~ 6270 km$ for Earth's radius, which, amazingly, falls just short of the modern measurement $R_{E} = 6371 km$ .

Now, let that sink in. Armed only with a bunch of sticks and his wit, Eratosthenes was able to determine Earth’s circumference within a couple of percent of modern measurements. Feeling useless yet? Wait for the next rungs - it gets much worse.

The Second Rung: The Moon

Knowing the shape and size of Earth was a pretty big deal. And it whet our appetite for more. Could we perhaps use a similar kind of stick wizardry to work out the sizes of and distances to other celestial bodies - like, say, the obnoxiously bright and impossible-to-overlook Moon? Astonishingly, the answer is again yes.

However, the challenge of reaching beyond Earth quickly outgrew Eratosthenes’ literal measuring stick. The vast distances involved meant that a more substantial reference was required: if we wanted to make progress, we would need a much bigger stick. In a rather poetic turn of events, the Earth itself became that new stick. In fact, the stick thing wasn’t the only idea we refurbished either: we also turned back to the very same lunar eclipses that allowed us to deduce Earth’s spherical shape in the first place.

But let's take one thing at a time. Some decades after Eratosthenes had figured out the size of Earth, yet another Greek stargazer entered the scene: Aristarchus de Samos. He understood that because the Sun is so far away, its light reaches Earth in nearly parallel rays. As a result, the shadow cast by Earth in the Sun's light remains itself largely Earth-sized - about $2 R_{E}$ across. By measuring how long the Moon spends fully immersed in Earth’s shadow during a total lunar eclipse (about 4 $hours$ ) and comparing that to the time it takes the Moon to complete one orbit around Earth (about 28 $days$ ), he could relate the fraction of orbit spent inside Earth's shadow (about $2 R_{E}$ wide) to the full orbital circumference of $2 π D_{M}$ :

\frac{28 days}{4 hours} = \frac{2 π D_{M}}{2 R_{E}} .

Solving for the Earth-Moon distance $D_{M}$ then gives

D_{M} ~ 61 R_{E} ~ 382470 km .

Having determined the Earth–Moon distance $D_{M}$ , one can go ahead and infer the Moon’s radius $R_{M}$ using essentially the same trick again. This time, the key observation is that the duration of a Moon rise corresponds to the time it takes for Earth’s rotation to carry the line of sight once across the Moon's diameter. This tells us that the Moon-rise interval (about $2 min$ ) has the same fractional relation to Earth’s full rotation period ( $24 hours$ ) as the Moon’s diameter $2 R_{M}$ has to its full orbit $2 π D_{M}$ :

\frac{2 min}{24 hours} = \frac{2 R_{M}}{2 π D_{M}} .

With the Earth-Moon distance $D_{M}$ from before, this yields the Moon radius

R_{M} ~ 0.27 R_{E} ~ 1669 km,

which is, yet again, only a couple percent off the modern measurement $R_{M} = 1737.4 km$ .

It is fair to say that this went outrageously well. Aristarchus thought so too, so he naturally decided to try and conquer the Sun next.

The Third Rung: The Sun

His plan was straightforward enough. During his long hours of staring at the Moon he had realised that at half-moon the Earth, the Moon, and the Sun form a right triangle, with the right angle at the Moon’s position. He knew that if he could somehow measure the angle $θ_{E}$ at the Earth,⁶ he could plug it into a trusty bit of ancient trigonometry,

D_{S} = \frac{D_{M}}{\cos θ_{E}},

to calculate the Sun-Earth distance $D_{S}$ in terms of the now known Moon-Earth distance $D_{M}$ . At least in theory.

In practice, this is where the universe stopped indulging him.

The problem is that the above trigonometric formula is extraordinarily sensitive to small errors $δ θ_{E}$ in the angle $θ_{E}$ at Earth.⁷ Aristarchus, with the prehistoric tools⁸ available to him, did an amazing job: he arrived at $θ_{E} \approx 87^{\circ}$ which is only about ${2.5}^{\circ}$ shy of the true value $θ_{E} \approx {89.5}^{\circ}$ . Unfortunately, this time the tiny ${2.5}^{\circ}$ slip nuked the entire calculation: because $\cos (87^{\circ}) \approx 0.05$ and $\cos ({89.5}^{\circ}) \approx 0.009$ differ by an entire order of magnitude, Aristarchus concluded that the Sun was about twenty times farther away than the Moon, when in reality the Sun-Earth distance $D_{S}$ is over four hundred times the Moon-Earth distance $D_{M}$ .

Despite this oopsie, Aristarchus’ attempt wasn’t entirely in vain. Even with the disastrously underestimated Sun-Earth distance, he was able to deduce that the Sun had to be much larger than the Moon - roughly 20 times larger - and also several times larger than the Earth itself.⁹ Guided by the idea that small things revolve around big things - a notion that had neatly aligned with his geocentric worldview when he'd figured out the relative sizes of Earth and Moon before - he boldly proposed the first known heliocentric viewpoint: perhaps it is the Earth that orbits the Sun, rather than the other way around.

Alas, Aristarchus lacked the observational precision to make his case convincing in his time, so today the credit for a workable heliocentric model typically goes to Copernicus, who worked out the maths centuries later.¹⁰ Still, Aristarchus' contributions were enough to earn him a dusty lunar crater for posterity. Oh, the twisted ways of fame.

The Fourth Rung: The Orbits of Planets

Copernicus' heliocentric model assumed circular orbits and constant orbiting speeds — partly because circles were considered the perfect geometric objects and partly because it kept calculations manageable. Within this framework, repetitions of celestial configurations visible from Earth allowed him to tease out the orbital periods of the planets, establishing their order — Mercury, Venus, Earth, Mars, Jupiter, Saturn — and estimating rough relative distances.

Just decades after Copernicus, another giant of cosmology entered the scene: Johannes Kepler. He was not satisfied with orbital periods alone and longed to uncover the sizes of the orbits. Specifically, he was dying to confirm a wild theory he had dreamed up. You see, to Kepler, the spacing of the planets could not be arbitrary — it had to obey some deeper cosmic order, a hidden architecture of the heavens waiting to be revealed. In his Mysterium Cosmographicum (1596), he therefore proposed that the six planets known at the time were arranged in a geometric balance dictated by the five Platonic solids — the octahedron, icosahedron, dodecahedron, tetrahedron, and cube. The idea was to start with the sphere defined by the orbit of the innermost planet, Mercury, and inscribe it within the smallest possible octahedron. Around that octahedron, the smallest possible circumscribed sphere then defines the orbit of the next planet, Venus. Venus' sphere is in turn inscribed in the smallest possible next Platonic solid, the icosahedron, whose circumscribed sphere gives the orbit of Earth. This pattern continues outward, with the dodecahedron placing Mars, the tetrahedron Jupiter, and the cube Saturn, thus creating a nested structure in which orbital spheres and Platonic solids alternate to impose an elegant mathematical harmony on the wilderness of the cosmos. Like any proper nerd would, Kepler admired the idea of a universe whose architecture reflected the perfection of mathematics.

Hungry to prove this theory, Kepler turned to the most precise observational data available: the extensive records of planetary positions collected by giga-geek Tycho Brahe. Historical anecdote has it that Kepler acquired access to Brahe's data under unusual circumstances, having to beg, intern, and strategically outlive his mentor.¹¹ Yet when he finally had the data, the Platonic solid model simply refused to fit. Kepler was, understandably, devastated. Still, there was a sliver of consolation. In benchmarking the observational data against the prevailing Copernican theory, he had noticed that it did not fit either; he might not get his cosmic victory, but perhaps he could at least take the established picture down with him.

The fact that the Copernican theory did not fit the observational data presented a serious theoretical problem: circular orbits and uniform motion could no longer be upheld! Importantly, these things were far from decorative assumptions; they were the very scaffolding that made the system intelligible at all! Once circularity was abandoned, the observational positions of the planets were no longer easy to interpret. Every measurement became relative, dependent on moving reference frames, and the geometry of the solar system seemed impossibly tangled.

Kepler’s genius was to see a way through. He realised that a fixed reference point didn’t need to be absolutely stationary — it only had to return to the same position reliably. By observing Mars at the same point in its orbit year after year, he could anchor Earth’s position relative to it. With only angular measurements and the insistence that Earth’s orbit must form a continuous curve, he adjusted the unknown distances until the geometry finally worked. Circular orbits failed; ellipses survived. Once Earth’s motion was pinned down, the elliptical orbits of all the other planets fell into place as well.

Footnotes:

Us calling the cosmic ladder The Cosmic Ladder instead of The Cosmic Wobbly Jenga™ Tower gives me about the same vibes as Gollum calling the stairs of Cirith Ungol The Stairs of Cirith Ungol instead of The Vertical Death Trap of Treason.↩
This is more than a quip: look closely and you'll see that much of the cosmic ladder is indeed cobbled together from frighteningly elementary maths - essentially today's school maths!↩
The first Greek thinker to propose that the Moon shines by reflected sunlight and that eclipses occur when one celestial body enters another’s shadow was a bloke called Anaxagoras (c. 500–428 BC).↩
Mathematically speaking, in dimensions $d$ greater than two, the only convex shape whose every projection is a $(d - 1)$ -dimensional disc $𝔻^{d - 1}$ is a $(d - 1)$ -dimensional sphere $𝕊^{d - 1}$ .↩
Figuring out the distance between Alexandria and Syene was no small feat. I like to imagine that the tedious task fell to one of his students, labouring over it in what would today be recognised as a graduate project. More likely, Eratosthenes relied on reports of established caravan routes, which indicated that the cities were separated by about $5000 stadia$ , roughly $788 km$ using the commonly cited length of $1 stadion ~ 157.5 m$ .↩
Or, equivalently, the angle $θ_{S}$ at the Sun - we are dealing with a right triangle so that $θ_{E} = 90^{\circ} - θ_{S}$ . I use $θ_{E}$ here because that is the one Aristarchus actually measured.↩
If we estimate the error $δ D_{S}$ induced by a small angular error $δ θ_{E}$ via standard error propagation, $δ D_{S} \approx | \partial D_{S} / \partial θ_{E} | δ θ_{E}$ , we obtain $δ D_{S} \approx | D_{S} \tan θ_{E} | δ θ_{E}$ . Thus, everything hinges on the size of $\tan θ_{E}$ at the angle $θ_{E}$ from which we deviate. Because the Sun lies at an enormous distance, the Earth-Sun line is almost parallel to the Sun-Moon line; equivalently, the Earth-Moon line is almost perpendicular to the Sun-Moon line. As a result, the angle $θ_{E}$ at Earth sits uncomfortably close to $90^{\circ}$ where $\tan θ_{E}$ blows up. This is why even tiny angular slips $δ θ_{E}$ produce absurdly large errors $δ D_{S}$ in the Sun-Earth distance $D_{S}$ .↩
It’s worth pausing to reiterate just how primitive his toolkit actually was: no lenses, no telescopes, no precision timing, just varying numbers of sticks, a good portion of hope and a solid pair of eye balls. In light of that, his achievements are borderline impossible.↩
Here, he exploited the ludicrous coincidence that the Sun and the Moon have the same apparent size in our sky. This means that $R_{M} / D_{M} \approx R_{S} / D_{S}$ , i.e. that the Moon’s radius-to-distance ratio must be roughly equal to the Sun’s radius-to-distance ratio. Rearranging gives the handy estimate $R_{S} \approx (R_{M} D_{S}) / D_{M}$ .↩
Copernicus formulated his heliocentric model roughly between 1514 and 1530, but it was only published as part of De revolutionibus orbium coelestium in 1543, the year he died.↩
Tycho Brahe was notoriously protective of his decades of painstaking observations. He feared that if others got access to his data, they might take the credit for discoveries he had meticulously recorded. Kepler, eager to test his Platonic solid theory, initially had only limited access, even after becoming Brahe’s assistant in Prague. Only after Tycho’s untimely death in 1601 did Kepler inherit the full set of observational records, along with Brahe’s position as Imperial Mathematician. (Huh.) With this treasure trove of precise planetary positions at last in his hands, he was able to unlock the geometry of Earth’s orbit — and eventually the ellipses of all the planets. One might say Kepler’s impatience with cosmic mysteries, combined with good timing, worked very nicely in our favour.↩