As discussed previously in Flat Earth Challenges, checking distance and area measurements is a fairly simple way to show it is impossible to map our Earth to a flat plane.

Just pick 4 cities that are all fairly distant from one another (8000 km or more is perfect). The distances don't even have to be all that accurate, as we will show in our example. If you pick 4 cities that are closer to each other then you need more accurate distances. I don't want to have to care that much about having super accurate distances, I want something that is glaringly obvious.

For my example I've used Sydney, Johannesburg, Buenos Aires, and London -- which I've also used before to show how utterly ridiculous the Gleason map is...

But that's just one Flat Earth map. Let's destroy all Flat maps in one go...

The Method

What I did is pick the cities with the three longest routes and placed those first three cities that distance apart on a flat plane (Sydney, London, Buenos Aires) using a scale of 10 km to 1 pixel.

Then you draw an arc from each of those cities with a radius equal to their distance to the fourth city (Johannesburg in this case). If the Earth is flat, then the distances from Sydney, Buenos Aires, & London to Johannesburg will all cross at just one point. This is simple plane geometry.

But as you can see below, the triangulation doesn't even come close to fitting on a flat plane. They are many thousands of kilometers off - ridiculously far off. This isn't because we don't know the distances, these distances only fit on a Globe.

And this doesn't use any particular map or rely on compass directions, angle measurements, or anything else. Just the distances.

The Usual Suspects

The excuse that we don't really know the distances is just absurd and comes from Flat Earther cognitive dissonance only. Any sailor can tell you that 1° of latitude is always about 111km -- it doesn't change as you go North and it doesn't change even in the Southern Hemisphere. You can drive it -- don't trust your GPS -- get a theodolite and measure the angular change in Polaris yourself. These pathetic appeals to "whaaaaaa, everyone is lying to us" is just utter nonsense -- there is no excuse for such profound ignorance when you can pretty easily verify this for yourself.

See another version using Cities in the US:

So help me reading this picture. I'm trying to fit Chicago at the intersection of the black, red and green lines but apparently I'm really off. pic.twitter.com/zMsDlQOhqd

— Il Lupo Perde Il Pelo Ma Non Il Vizio (@iloveberlu) December 18, 2017

This image of Mt. Rainier taken by Shannon Winslow (author of historical fiction in the tradition of Jane Austen) features a shadow that is cast upwards on the clouds with a clear gap between the mountain top and the shadow.

This places the Sun over South America, near Latitude: 23° 09' South, Longitude: 70° 58' West as shown by Date and Time -- along a heading of about 134 degrees.

And since we're right in line with the shadow this puts us Southeast of Tacoma, possibly near the Orting/Carbonado area. But there are no bodies of water Southeast of Mt Rainier that would explain the upward casted shadow.

Much less the EXTREME grasping at straws this represents from the Flat Earth crowd given where the Sun is overhead...

But it makes perfect sense on a Globe.

Indeed, if we pull back as far as we can in Google Earth (which isn't nearly as distant as the Sun unfortunately) we can see this puts that area of Washington right on the edge where it would need to be when we're over the marked spot on the Globe. Funny how that works.

Let's say we have two people viewing the moon who are 10300km apart along the same line of latitude at the same time. How much of a shift against the more distant background stars should the moon appear to shift?

This approximation assumes that the sublunar point is roughly between the two observers.

Well, first we need to find their actual linear distance (rather than the distance over the curvature of the Earth, which is what you get from Google Earth).

Let's define our variables:
\[ R = 6371393 m \;\; \text{Earth Average Radius} \]
\[ d = 10300000 m \;\; \text{Earth distance along curvature} \]
We can find the angle in radians from the arc length using:
\[ \theta = d/R \]
and we can find the length of a chord using the half-sine rule:
\[ \text{crd} \, \theta = 2 \sin \frac{\theta}{2} \]
Plugging that in we find that the straight-line distance is slightly shorter than around the curvature:
\[ 6371393 \times 2 \sin(\frac{1}{2} \frac{10300000}{6371393}) \approx 9214000m \]
Next we can find the angular size of a shift of some distance \( g \) from a distance \( r \):
\[ \alpha = 2 \arctan (\frac{1}{2} \frac{g}{r}) \]
and :
\[ r = 384400000m \;\; \text{distance to the moon} \]
\[ g = 9214000m \;\; \text{distance between observers} \]
And we find that the expected angular shift is about:
\[ \alpha = 2 \arctan (\frac{1}{2} \frac{9214000m}{384400000m}) \]
\[ \approx 1.373° \nonumber \]
\[ \approx 2.7 \times \text{angular diameter of the moon} \nonumber \]

This seems to be a good match to explain this:

No takers for my little quiz?

Here’s a hint: simultaneous views from Stockholm and Cape Town (very different latitudes, nearly same longitude), both oriented North up đź‘‡.

(Among other things note the curious undulating shape of the path of the moon) pic.twitter.com/gqeOVh3ayv

That's a good first approximation anyway. There would also be some (usually small) refraction effects if the view of the Moon was lower on the horizon for one observer, and, of course, you need to know the current Earth-Moon distance and not just some average value.

Nothing new here, just wanted to capture these proofs into a single location for easy reference. I've tried to arrange them into a form that is easy to understand and follow.

Kepler's First Law: Ellipses

\(\require{cancel}\)Proof that a Newtonian force between two masses will produce an elliptical orbit - this is a very textbook approach using polar coordinates, I'm just capturing it here for reference. There are many other approaches, and this has likely been done millions of times now.

Remember that Newton's Law is:
\[
\begin{align}
F &= G \frac{Mm}{r^2} \\
&\therefore \nonumber \\
F &= m a
\end{align}
\]

We define a unit vector in the radial direction \( \hat{r} \) along the angle \( \theta \):
\[ \hat{r} = \hat{x} \cos \theta + \hat{y} \sin \theta \]
Therefore \( \hat{r} \) changes as per angle \(\theta\) perpendicular to \( \hat{r} \), giving us the tangential unit vector \( \hat{\theta} \)
\[ \frac{\mathrm{d}\hat{r}}{\mathrm{d}\theta} = \hat{x} ( - \sin \theta ) + \hat{y} \cos \theta = \hat{\theta} \]
And the derivative of \( \hat{\theta} \) is another 90° rotation, giving us \( -\hat{r} \):
\[ \frac{\mathrm{d}\hat{\theta}}{\mathrm{d}\theta} = \hat{x} ( - \cos \theta ) + \hat{y} ( - \sin \theta ) = -\hat{r} \]
Find Acceleration \( \vec{a} \) in polar coordinates, first we find velocity \( \vec{v} \) as the change in radial vector over time:
\[ \vec{v} = \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t} \left( r \hat{r} \right) \]
Applying the product rule and the chain rule we get our expression for how fast you are moving radially (\(\dot{r}\) in \(\hat{r}\)), and how fast you are moving in the tangential direction (\(r \dot{\theta}\) in \(\hat{\theta}\)):
\[
\vec{v} = \frac{\mathrm{d}r}{\mathrm{d}t} \hat{r} + r \frac{\mathrm{d}\theta}{\mathrm{d}t} \frac{\mathrm{d}\hat{r}}{\mathrm{d}\theta}
= \dot{r} \hat{r} + r \dot{\theta} \hat{\theta}
\]
Acceleration is the derivative of velocity:
\[ \vec{a} = \frac{\mathrm{d}\vec{v}}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t} \left(
\dot{r} \hat{r} + r \dot{\theta} \hat{\theta}
\right)
\]
\[
\vec{a} = \color{Red}{\ddot{r} \hat{r}} + \color{ForestGreen}{\dot{r} \dot{\theta} \hat{\theta} + \dot{r} \dot{\theta} \hat{\theta} + r \ddot{\theta} \hat{\theta}} + \color{Red}{r \dot{\theta} (-\hat{r}) \dot{\theta}}
\]
Grouping terms in \( \hat{r} \; \text{and} \; \hat{\theta} \) we get:
\[ \vec{a} = \color{Red}{\hat{r} \left( \ddot{r} - r \hat{\theta}^2 \right)} + \color{ForestGreen}{ \hat{\theta} \left( 2 \dot{r} \dot{\theta} + r \ddot{\theta} \right) } \label{refa} \]
From Newton
\[ \vec{F} = m \vec{a} \]
We only need the radial term from \(\eqref{refa}\) since the Force is only in the radial direction, substitute along with Law of Gravity:
\[ - \frac{GM\cancel{m}}{r^2} \cancel{\hat{r}} = \cancel{m} \cancel{\hat{r}} ( \ddot{r} - r \dot{\theta}^2 ) \]
The radial unit vector \( \hat{r} \) and mass term \( m \) cancel out, giving us this equation that we must show is equal:
\[ - \frac{GM}{r^2} = \ddot{r} - r \dot{\theta}^2 \label{refb} \]
The angular momentum is a constant because there are no torques acting on our system, the only Force is gravity acting radially inwards so we can define \( \dot{\theta} \):
\[ L = m r^2 \dot{\theta} \; \therefore \; \dot{\theta} = \frac{L}{m r^2} \label{eqL} \]
And substitute this into \eqref{refb}:
\[ - \frac{GM}{r^2} \stackrel{?}{=} \ddot{r} - \frac{L^2}{m^2 r^3} \label{refc} \]
Next we need to show that the second time derivative of this equation of an ellipse works in our force equation:
\[ r = \frac{a(1 - \epsilon^2)}{1 + \epsilon \cos \theta} \]
First derivative [Wolfram|Alpha]:
\[ \frac{\mathrm{d}r}{\mathrm{d}t} = \frac{\mathrm{d}r}{\mathrm{d}\theta} \frac{\mathrm{d}\theta}{\mathrm{d}t} = \frac{-a(1-\epsilon^2) ( - \epsilon \sin \theta )}{(1+\epsilon \cos \theta)^2} \dot{\theta} \]
Substitute for \( \dot{\theta} \) and again for \( r \)
\[ \frac{\mathrm{d}r}{\mathrm{d}t} = \frac{\cancel{a(1-\epsilon^2)} ( \epsilon \sin \theta )}{\cancel{(1+\epsilon \cos \theta)^2}} \frac{L \cancel{(1 + \epsilon \cos \theta)^2}}{m a(1-\epsilon^2) \cancel{a(1-\epsilon^2)}} \]
\[ \frac{\mathrm{d}r}{\mathrm{d}t} = \frac{L \epsilon \sin \theta}{m a (1-\epsilon^2)} \]
Second derivative:
\[
\frac{\mathrm{d}^2 r}{\mathrm{d} t^2} = \frac{\mathrm{d}}{\mathrm{d}\theta} \left( \frac{\mathrm{d} r}{\mathrm{d} t} \right) \left( \frac{\mathrm{d}\theta}{\mathrm{d} t} \right) =
\frac{L \epsilon \cos \theta}{m a (1-\epsilon^2)} \frac{L}{m r^2}
\]
Which we can substitute back into our force equation \(\eqref{refc}\), giving us:
\[ - \frac{GM}{\cancel{r^2}} \cancel{\color{Red}{r^2}} \color{Red}{m^2} \stackrel{?}{=} \left( \frac{L^2 \epsilon \cos \theta}{\cancel{m^2} a(1-\epsilon^2) \cancel{r^2}} - \frac{L^2}{\cancel{m^2 r^2} r} \right) \cancel{\color{Red}{{r^2}{m^2}}} \]
Cancel out the mass and radius terms and substitute for \( r \) again:
\[ -GMm^2 \stackrel{?}{=} \frac{L^2 (\epsilon \cos \theta)}{a(1-\epsilon^2)} - \frac{L^2(1+\epsilon \cos \theta)}{a(1-\epsilon^2)} \]
Which will now simplify further and we can solve for \( L \):
\[ L^2 = GMm^2 \; a(1-\epsilon^2) \]
Both sides are constants being equal with no dependence on any term, but defined by the ellipse of semi-major axis \( a \) and eccentricity \( \epsilon \).

This shows that a Newtonian force (proportional to the product of the masses and inversely proportional to the square of their distances) acting mutually between two masses will produce an elliptical orbit.

Kepler's Second Law: Equal Areas

Next we want to show that a planet in orbit would sweep out equal areas over equal time.

We can find the angular momentum:
\[ \vec{L} = \vec{r} \times \vec{p} \]
We only need the perpendicular component, so we can drop the vectors:
\[ L = r p_\perp = r (m v_\perp) \]
And \( v_\perp = r \omega \):
\[ L = r m (r \omega) \]
Therefore, as we also found in the First Law \( \eqref{eqL} \):
\[ L = m r^2 \omega \]
and we find that:
\[ r^2 \omega = \frac{L}{m} \label{eqr2} \]
Now we need to show that for the integral:
\[ A = \int \frac{1}{2} r^2 \mathrm{d}\theta \]
that \( r^2 \) is a constant. The derivative with respect to \(\theta\) is:
\[ \frac{\mathrm{d}A}{\mathrm{d}\theta} = \frac{1}{2} r^2 \]
and with respect to time (\(t\)):
\[ \frac{\mathrm{d}A}{\mathrm{d}t} = \frac{\mathrm{d}A}{\mathrm{d}\theta} \frac{\mathrm{d}\theta}{\mathrm{d}r} = \frac{1}{2} r^2 \frac{\mathrm{d}\theta}{\mathrm{d}t} \]
but \( \frac{\mathrm{d}\theta}{\mathrm{d}t} \) is just \( \omega \):
\[ \frac{\mathrm{d}A}{\mathrm{d}t} = \frac{1}{2} r^2 \omega \]
And we can replace \( r^2 \omega \) from \( \eqref{eqr2} \):
\[ \frac{\mathrm{d}A}{\mathrm{d}t} = \frac{1}{2} \frac{L}{m} \]
Which is a constant.

Kepler's Third Law: Harmonies

Finally, we want to show that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

Since we know that centripetal force is \( m \frac{v^2}{R} \) we can consider what Force is needed:
\[ \frac{GMm}{R^2} = m \frac{v^2}{R} \]
For a nearly circular orbit we can substitute \( v^2 = \frac{4 \pi^2 R^2}{T^2} \)
\[ \frac{G M \cancel{m}}{R^2} = \frac{\cancel{m} 4 \pi^2 R \cancel{R}}{\cancel{R} T^2} \]
And we find:
\[ \frac{T^2}{R^3} = \frac{4 \pi^2}{G M} \]
Therefore this ratio depends only on M and our ratio would be constant for masses in orbit around it.

This is something I've been working on for a while and I hope you will find something new in it and you will almost certainly find some place where I've made an error (contact me on twitter @ColdDimSum to report errors). I don't think I've made any grievous errors but possibly have some things out of order, misattributed, or wrong in the fine details or due to my clumsy wording (I'm not a professional scientist nor writer nor science writer, I'm a professional software developer and I have been writing programs for over 35 years). For any errors I apologize -- but please consider these my notes on the subject and always find a good Primary source to substantiate anything specific. Thankfully, scientific theories are valid based solely on the authority of the evidence and not on who thought of them or when.

But I do think I have some insights I can share based on my studies and I hope my errors do not detract from the overall story.

In all likelihood, nothing that we see, feel, taste, or otherwise experience actually exists exactly as it seems to our human senses. There is no color 'pink' in Nature, it is a mixture of a very tiny slice of the electromagnetic spectrum that depends on the peculiarities of human visual senses (pigments in our eyes that stimulate the rods and cones in our retina) and processing in our brain in order to be experienced as 'pink'. Even amongst humans, the experience of 'pink' is not universal, some people lack the pigments necessary to sense the light frequencies necessary. Some humans lack taste buds, some humans lack auditory senses, some lack all visual senses, some lack nociception (a sense of pain) but still have normal somatosensation (touch) -- I cannot say with certainty that there is any sense that is truly universal to all humans, even the way we think and experience 'Self' might well be different in the extremes.

So it likely is with Gravity. In fact, in the Einsteinian view, Gravity is a consequence of the shape of Spacetime itself rather than a proper force, and this view is supported by the fact that, when seemingly being accelerated by Gravity, a body does not measure a proper acceleration, unlike all other sources of acceleration. As shown below, the acceleration drops to zero during the free fall (indicating a near weightless free fall).

Accelerometer recording during iPhone Free Fall

By the end of this article we will be able to estimate the height from which this iPhone was released, noting that it falls for about 12/20ths of a second.

However, this doesn't mean that Gravity doesn't exist. There is some underlying phenomena which we observe, measure, and number and we relate to this experience indirectly. It's unlikely that will or can ever truly know what exists because we are stuck inside of reality. Imagine an AI in a simulated world -- how could it ever know that underlying its experience is trillions of logic operations? It would instead observe that objects do not instantly appear from one place to another, but seem to move at some limiting speed -- only that you also don't seem to be able to make assumptions about 'where' the object is except when you are measuring it. But it could never, through experiment alone, deduce the logic underlying its own experience.

Whatever else the case may be, we can observe and measure the effects of Gravity and assign these observations to Laws and form Theories that attempt to explain 'why' at the level of experience to which we have access. The AI would not necessarily be wrong to conclude there is a maximum speed within the realm of physics to which it has access, this apparent 'maximum speed' would be emergent from the deeper physics to which it necessarily remains ignorant.

So this question of the reality of Gravity (and is it a fundamental force or an emergent force) is really irrelevant to the question at hand -- this doesn't mean that Gravity doesn't exist -- when you drop an object it clearly accelerates towards the center of the Earth (and we can measure how the motion of atoms are affected by nearby large masses, using a process called Cold Atom Interferometry). So we have direct empirical data for Gravity - it behaves like no other Force having no proper acceleration, cannot be shielded, doesn't have a polarity, does not depend upon the alignment of spin, is proportional only to the mass (not the material or its composition).

It doesn't really matter "what" Gravity is, what matters is what we do know about how it works and what are the unavoidable consequences of that knowledge, in our experience of the world -- our Physics.

So I would like to take a look at the history of this concept, explore how we know what we know, what the consequences are, and maybe get a sense some of what we don't know.

Gravity

Allow me to jump to the end of the story briefly so that we might see where we are headed...

Law of Universal Gravitation

every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them

This is summed up mathematically as \[F = G\frac{mM}{r^2}\] The consequences of this discovery are remarkable and not immediately obvious from such a lowly description. The impact of this simple Law is far reaching, touching on almost everything we experience. Without Gravity we wouldn't have stars, and without stars we wouldn't have planets, and without planets there would be no us to wonder.

And here we have to be careful to understand that this formula applies to a point mass, and while Newton will show that a large rigid body can be effectively treated as a point mass at the Center of Gravity - in the real world rocks and planets are not perfect rigid masses; so the deeper truth is that things get even more complex in the fine details but even that complexity is the product of our humble formula acting over a multitude of little masses.

Now let's glimpse the tiniest bit of the story behind how we got here... Long before Copernicus, an Indian philosopher, Yajnavalkya (circa 9th Century BCE) might be credited as the first person to put down into words that the planets follow the Sun when he wrote:

The sun strings these worlds – the earth, the planets, the atmosphere – to himself on a thread. —Shatapatha Brahmana, 8.7.3.10

Lending weight to this being a heliocentric expression, and not just a poetic one, he also measured the distances between the Sun-Moon and Earth-Moon and recognized the Earth was spherical. [See Shatapatha Brahmana, Full Text] During the same period it was also written:

The Sun never sets nor rises. When people think the sun is setting, it is not so; they are mistaken. It only changes about after reaching the end of the day and makes night below and day to what is on the other side. —Aitareya Brahmana

Hundreds of years later, Greek astronomers, such as Aristarchus of Samos in the 3rd century BCE also proposed heliocentric models but, at that time, the models of Aristotle (384-322 BCE) and Ptolemy (~2nd century BCE) won out because we didn't have the data of sufficient accuracy to distinguish the models. Ptolemy's model is purely empirical based on cycles upon cycles and is accurate enough for basic astronomical observations -- the problem, as we will see, is that it isn't right. You have to keep adding and adding unexplained corrections to the model.

Unfortunately we don't have Aristarchus' full work - but we have mentions of it by Archimedes (287-212 BCE) and perhaps fragments of copies.

But these were all speculative or philosophical models, however genius in their time, lacking sufficient empirical data to judge them on their strengths and weaknesses. We would have to wait about 1500 years before the next leap forward.

Bede(c.672-735) - deduced that the Moon’s gravitational pull on the Earth was responsible for causing our tides and worked out a formula for predicting the tidal times. Nicolaus Copernicus (1473-1543) we flash forward to around 1514 with Copernicus quietly writing a manuscript (De revolutionibus orbium coelestium / On the Revolutions of the Celestial Spheres) that would change the world with his new ideas about planetary motion placing the Sun at the center, at least of his Universe (he didn't have a telescope, so he didn't yet know about other galaxies or moon around other planets).

The Copernican Revolution (or shift away from the Ptolemaic system) was built on the following assumptions (some of which proved to be incorrect, others deeply insightful):

There is no one center of the celestial spheres.

The center of the Earth is not the center of the universe, but is the center of the lunar sphere.

All other spheres revolve about the sun as their midpoint, therefore the sun is the center of the universe.

The ratio of the Earth's distance from the sun to the height of the firmament (outermost celestial sphere containing the stars) is so much smaller than the ratio of the Earth's radius to its distance from the sun that the distance from the Earth to the Sun is imperceptible in comparison with the height of the firmament.

Whatever motion appears in the firmament arises not from any motion of the firmament, but from the Earth's motion. The Earth, together with its circumjacent elements, performs a complete rotation on its fixed poles in a daily motion, while the firmament and highest heaven abide unchanged.

What appear to us as motions of the Sun arise not from its motion but from the motion of the Earth and our sphere, with which we revolve about the Sun like any other planet. The Earth has, then, more than one motion.

The apparent retrograde and direct motion of the planets arises not from their motion but from the Earth's. The motion of the Earth alone, therefore, suffices to explain so many apparent inequalities in the heavens.

This would irrevocably remove the Earth from the center of the Universe, despite it's errors and failings. The idea was too powerful and technology was about to catch up. Ships were sailing to previously unfathomably distant lands powered by TIME -- that is, accurate clocks (and knowledge of the Trade Winds). Sailors had long known how to take the position of stars and figure out their Latitude fairly accurately -- the angle of Polaris over the horizon is good to within a degree (Polaris is 2/3 of a degree off center so it's altitude, or angle of the horizon, varies over each day slightly. But once you got close to home you could narrow it down to a dozen or so miles and easily find your way... if only you could find your Longitude. Longitude is much more difficult to find with the stars alone. You need to know exactly what time it is -- a few seconds off and your position drifts further and further away from true. And sailors needed accurate star maps and computations of the tables that enabled them to use a Sextant to find their position sufficiently. It's a complex bit of spherical trigonometry that most sailors couldn't directly perform (nor most modern people either!) But that's another story (see Marine Chronometer).

Tycho Brahe (1546-1601) was wealthy and owned an island near Copenhagen where he built an observatory and designed new versions of the sextant and quadrant allowing him to make naked-eye observations accurate to within about 1 arcminute - unparalleled in his day and working without the aid of telescopes he very carefully recorded the position of the stars and planets over time. His work, especially observations of Mars, would prove critical to Kepler in developing his laws of planetary motion (despite Brahe personally being opposed to the heliocentric model). He died from a burst bladder (some suspected poison) before his data could bear fruit, he was, above all, an empiricist and likely would have become convinced by the data. Johannes Kepler (1571-1630) worked briefly with and analyzed Brahe's data (and there are stories within stories behind that, included the Catholic church trying to force Kepler to convert). He first assumed planets went in circles but that would have put Brahe's observations of Mars about 8 arcminutes off, which was too great an error give how methodological and careful Brahe had been, so he started over and formulated his three Laws of planetary motion:

Law of Ellipses: Planets went in ellipses with the Sun near one foci.

Law of Equal Areas: Planets sweep out equal areas over equal times.

Law of Harmonies: The ratio of the squares of the sidereal periods of revolution of the planets (\(T^2\)) to the cubes of their mean distances from the Sun (\(R^3\)) is a constant (\(T^2/R^3\)).

Galileo Galilei (1564-1642) investigating the laws of motion discovered the principle of inertia - which says that an object that is moving in a straight-line will continue to move in a straight-line unless some Force acts upon it. The Aristotelian view of motion up to that time was that some force must be continuously applied to keep an object in motion. Galileo observed that it was forces such as air resistance and friction tend to apply forces to slow objects.

He also discovered Galilean invariance which says that Laws of Physics are the same in all inertial frames of reference -- which has far reaching consequences in itself.

The example Galileo considered was that on a ship travelling perfectly smoothly an observer enclosed below deck wouldn't be able to tell if the ship was moving or stationary.

And finally, Aristotle has proposed that objects fall at a rate proportional to their mass and Galileo showed that this was not correct and that, ignoring air resistance, all objects accelerate at the same rate.

Sir Isaac Newton (1642-1726?) Much of what transpired previously would come to fruition in the mind of Newton and consequently this is our longest entry.

LAW I: Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

LAW II: The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

LAW III: To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

We can see that when Newton pens "If I have seen further it is by standing on ye sholders of Giants" in 1675, he is referring to a vast body of work that preceded him. Years of toil, in the heat or frozen air, carefully measuring out the positions of stars and planets ever more carefully and precisely, all the mechanical work to improve the instruments, and all the intellectual thought he inherited. Even the concept expressed is attributed to Bernard of Chartres (Latin: nanos gigantum humeris insidentes) in the 12th century.

Newton studied what happens when a Force acts upon objects and discovered the principles behind accelerated motion -- that a Force is needed to change the velocity. If you apply the Force in the direction of motion then it will speed up, if it changes direction then the Force must have been at some angle to the direction of motion.

Newton also discovered that the Force can be measured as the product of two effects -- how much the velocity changes over some interval of time (the acceleration) and the inertial coefficient, or mass of the object.

Giving us the famous equation for Newton's Second Law of Motion (note that Newton himself didn't state it this directly) \[\text{Force} = \text{mass} \times \text{acceleration}\] This is a direct consequence of the more general law stated earlier: given G (the gravitational constant), M=Earth's mass, and r=our distance from the center of Earth we find:
\[\begin{align*}F &= m \times \left[\frac{G \times M}{r^2}\right] \\
F &= m \times \left[\frac{(6.67408 \times 10^{-11} \; \mathrm{m^3 \; kg^{-1} \; s^{-2}}) \times (5.972 \times 10^{24} \mathrm{kg})}{(6371393\;\mathrm{m})^2}\right] \\
F &= m \times \left[9.818\;\mathrm{m/s^2}\right]
\end{align*}\] Note: Now the cool thing is that when your Theory gives you a different result (\(9.818\;\mathrm{m/s^2}\)) than you actually measure you have to account for that difference or your Theory is wrong. In this case, we already know that the actual value for 'g' varies by latitude, altitude, Earth's exact density variations, your relative speed, etc -- so we've already accounted for all of those differences in reality. I'm just giving the average value of 'g' in this case, to get the actual value you would have to sum up force vectors between every particle in the Universe -- but that's never required in practice. In practice, \(9.8\;\mathrm{m/s^2}\) gives us about as much accuracy as we're likely to be capable of measuring.

And because this works we can also weigh an object by knowing how much force it takes to accelerate that object by some amount. This is, in fact, how scales work. They measure the force resulting from the acceleration of gravity and divide it by \(9.8\;\mathrm{m/s^2}\), converted to whatever units your scale works in (pounds, kg, grams, micrograms, etc). There are also balance scales which work on the principle of directly comparing the force generated by your test mass against some reference mass until it's balanced.

What we find is that, if you take the same mass to different latitudes on the Earth and use the same scale calibration as your starting latitude, you can actually see the change in weight, as shown here:

It also turned out that these generic and simple empirical laws of Forces and Motions, combined with a simple Law of Gravity (the mutual attraction proportional to mass and inversely proportional to the square of the distances) explained the observed motions of the Planets deduced by Kepler.

Edmond Halley travelled from London to Cambridge in August of 1684 and asked Newton what kind of orbit a planet would follow if it were subject to a attractive force towards the Sun that were inversely proportional to the square of the distance between to the two bodies. Newton had already shown it would be an ellipse but couldn't find his notes and ended up redoing his analysis. He sent them in a short treatise entitled, On the motion of bodies in an orbit.

Halley would then go on to push Newton to publish his Principia.

But first Newton would need to show that you could treat large, mostly spherical masses as one unit with a point mass at the center. Without this it becomes computationally infeasible to consider anything more than a trivial point mass.

And this is indeed one of the most astonishing products of Principia and begins in Section XII where he devises a clever proof by considering a series of 'evanescent orbs' and showing how the forces would sum; for example, Prop. LXXI. Theor. XXXI:

Iisdem positis, dico quod corpusculum extra SphĂ¦ricam superficiem constitutum attrahitur ad centrum SphĂ¦rĂ¦, vi reciproce proportionali quadrato distantiĂ¦ suĂ¦ ab eodem centro. The same things supposed as above, I say that a corpuscle placed without the sphĂ¦rical superficies is attracted towards the centre of the sphere with a force reciprocally proportional to the square of its distance from that centre

Newton avoids directly using his Calculus in Principia feeling that it has not yet found a sound footing, but his arguments (such as dividing a spherical mass into a series of evanescent shells) clearly hint that his insights spring from it.

Newton also showed that the Gravitational explanation worked for bodies here on Earth, the planets of the Solar system, and the moons of Earth, Jupiter and Saturn.

Newton discovered the equivalence of inertial mass and gravitational mass.

Newton estimated the ratio of the masses relative to the Sun for several planets, getting Jupiter correct to within about 2%.

Newton showed that the center of the solar system is not the Earth or Sun, but as a center of gravity near the Sun that constantly changes position as the planets orbit. The center between two masses is called a barycenter.

This, combined with the rotation of the Earth, explained the Tides whereas Bede had only calculated the cycles.

Later, it would be noted, through careful observation of the orbits of the moons of Jupiter, that they were about 8 minutes ahead of the Newtonian calculated schedule when Jupiter was closest to the Earth and about 8 minutes late when Jupiter was furthest from the Earth. This would turn out to be a consequence of the speed of light rather than an error in the calculations.

Calculus

Kepler's earlier work left many questions of analysis open, new approaches to mathematics would be required to solve the unanswered questions -- that is, to know what the consequences of the observations are. These questions and more drove Newton to invent a new calculus - the "ultimate ratio of evanescent quantities".

For the layperson, suffice it to say that the area under the curve of well-behaved, continuous functions can usually be well-approximated by breaking it up into rectangles and adding up the area. And that if you do that very thoughtfully you'll see that patterns form that allow you find the function to calculate that area directly instead of approximating it.

Here is an animation that explains this concept of breaking things up into smaller rectangles to approximate the area, as you carry this out to infinitely many rectangles you approach the correct answer:

For example, the integral (notated as \(\int\)) of \(x^2\) is found to be \(\frac{x^3}{3}\), this means that the area under the parabola \(x^2\), let's say taking x from 0 to 3 is \(\frac{x^3}{3}\) which is 9. This becomes an incredibly powerful tool of analysis and you can verify it for trivial cases like \(x = x\) and see that it simplifies to the same as the area of a right triangle with equal length sides.
\[\int x\; \mathrm{d}x = \frac{x^2}{2}\] so the area of x=1 is 1/2, area of x=2 is 4/2, etc. It is easy to see by inspection that this is true.

Likewise, we can ask about how much a function changes instantaneously at some point (known as the tangent) by finding the derivative of that function. This is approximated similarly but by taking the "RISE OVER RUN" over increasingly tiny intervals.

The derivative, notated by Newton as \(\dot{f}\) (fluxion) or Leibniz as (\(\dfrac{\mathrm{d}}{\mathrm{d}x}\)) or sometimes as \(x'\), of \(\frac{x^2}{2} = x\) -- the opposite of the Integral. So there is a deep relationship between the tangent and the area of functions.

And with that tiny bit of calculus we can already begin to understand accelerated motion much more clearly than we could without it.

If you have a function \(f(t)\) that gives the position of an object at each point in time (t) then the derivative of that function \(f'(t)\) will be the velocity (or speed, which is the magnitude of velocity), and the derivative of speed is the acceleration.

In the "simple" case of a rigid body accelerated by gravity, the acceleration is a constant \(g = 9.8\;\mathrm{m/s^2}\) (technically, as we get further from Earth gravity becomes weaker but for human-scale events the difference is miniscule and we can ignore it for now), so we have:

\(f(t) = gt²/2\) -- this gives us the expected displacement of our object after time (t), in seconds
\(f'(t) = gt\) -- this is the velocity, or the rate of change in our position (the magnitude of which is speed)
\(f''(t) = g\) -- this is our acceleration, \(9.8\;\mathrm{m/s^2}\)

These give us the Equations for a falling body. The derivative of position is velocity, and the derivative of velocity is acceleration. We can do the same in 3-dimensional space using vectors, but the result is, in essence, the same (just more complex to understand).

Newton also didn't try to explain why gravity works the way it does, but rather just explained what was directly observed about it:

I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses; for whatever is not deduced from the phenomena is to be called a hypothesis, and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy.

With all this as the background, Newton observed that if the same Gravity we observe here on Earth was supplying a centripetal force to the Moon and planets, then Kepler's Laws fall out of this Law of Universal Gravity.

Newton didn't know the gravitational constant \((G)\), nor the mass of the Earth or the Sun \((M)\), so he worked in terms of ratios \[
\frac{C}{M}=\frac{c}{m}=\frac{k}{4 \pi^2} \\
\begin{align*}
& \text{where,} \\
& k = \text{universal factor of proportionality} \\
& C = \text{constant for the Sun} \\
& c = \text{constant for planet} \\
& M = \text{mass of Sun} \\
& m = \text{mass of planet}
\end{align*}
\] The \({4 \pi^2}\) arises because when we consider the centripetal acceleration required to keep something in orbit we are considering circular motion around a circumference, where acceleration is \(v^2/r\) and \(v = 2 \pi r/T\)

Let's consider the following orbit using Kepler's laws

\[
\begin{align}
\text{From Kepler's 3rd Law of Harmonies}& \\
\\
\frac{r^3}{T^2} = C \;\;\text{or}\;\; \frac{r}{T} &= \frac{C}{r^2} \;\;\text{or}\;\; T = C r^{3/2} \\
\\
\text{From Kepler's 2nd Law of Equal Areas}& \\
\\
a = \frac{4 \pi^2 r}{T^2} &= \frac{4 \pi^2 C}{r^2} \;\;\; \text{centripetal acceleration} \\
\\
\text{From Newton's 2nd Law of Force}& \\
\\
f &= \text{mass} \cdot \text{acceleration} \\
\\
f &= m \cdot \frac{4 \pi^2 C}{r^2} \;\;\; \text{sun's force on planet} \\
\\
f' &= M \cdot \frac{4 \pi^2 c}{r^2} \;\;\; \text{planet's force on sun} \\
\\
\text{From Newton's 3rd Law of Equal and Opposite Reaction}& \\
\\
f &= f' \\
\\
m\frac{4\pi^2C}{r^2} &=M\frac{4\pi^2c}{r^2} \\
\\
mC &= Mc \\
\\
\frac{C}{M} &= \frac{c}{m} \;\;\; \text{giving us our ratios} \\
\\
\text{Also, we can see that}& \\
\\
4 \pi^2 C &= kM \\
4 \pi^2 c &= km \\
\\
\text{Therefore,}&
\\
f \cdot f' &= m \frac{4 \pi^2 C}{r^2} \cdot M \frac{4 \pi^2 c}{r^2} \\
\\
&= m \frac{km}{r^2} \cdot M \frac{kM}{r^2} \\
\\
&= k^2 \frac{m^2 M^2}{r^4} \\
\\
f &= k \frac{m M}{r^2} \;\;\; \text{Newton's Law of Gravitation}
\end{align}
\]

The planets no longer needed to be "moved" around by Angles but rather they moved inertially forward with a gravitational acceleration towards other masses. Empirical observations of motions here on Earth finally fit the motions observed in the Heavens without appealing to unseen forces.

Astronomy exploded once the telescope (credited to Hans Lippershey) was adapted to this purpose in the early 1600's, allowing ever more accurate observations to be made, with Kepler, Galileo, and Newton all playing roles in improving it. Galileo was merely the first to adapt Lippershey's "Dutch Perspective Glass" to the purpose of astronomy and consequently discovered the four largest moons of Jupiter (Io, Ganymede, Callisto and Europa).

Henry Cavendish (1731-1810) about 100 years after Newton, Henry Cavendish carried out an experiment to measure the density of the Earth (originally conceived by John Michell) by directly measuring the attraction between masses suspended very carefully by a wire.

This gave him an average density for the Earth of about \(5.448 \; \mathrm{g \cdot cm^3}\) which works out to \(G \approx 6.74 \times 10^{-11} \; \mathrm{m^{3} \; kg^{-1} \; s^{-2}}\) -- very close to the value we get today.

Now, according to Newtonian physics, the orbits of the planets should not be perfect ellipses because all the planets also pull on each other just a little bit (but not zero). But when astronomers took Saturn, Jupiter, and Uranus into account (Alexis Bouvard published predictive tables in 1821) and calculated out the resulting orbits, substantial deviations began to be noticed. John Couch Adams and Urbain Le Verrier both performed calculations that predicted the discovery and location of a as yet unseen planet on the basis of this anomaly in the orbit of Uranus -- that planet would be Neptune - located to within about 1 degree of its actual position, solely on the perturbation in the orbit of Uranus.

Around 1859 these ever improving observations would finally call Newton into question when it was noticed that the Perihelion of Mercury (where Mercury is closest to the Sun) was precessing (moving its position around the Sun) in a way that Newtonian gravity did not account for. It would have to wait until 1905 for a young patent clerk to unravel that mystery. But that's a story for another time...

With the improvement in telescopes astronomers began to be able to resolve stars that are very close to each other, such as Kruger 60 in Cepheus with an orbital period of 44.5 years, confirming that these same laws of motion applied.

These laws continued to be successful in explaining other phenomena such as great Globular Clusters and Galaxies and even giant clusters of Galaxies.

The story of Gravity continues on, hundreds of books could be written about it. One of the threads you should follow on your own is to look at how Gravity, along with the Standard Model of physics explains the abundance of the elements, a phenomena known as Nucleosynthesis. Without the energy potential from Gravity stars could not fuse the simpler elements such as Hydrogen into the heavier elements. It also requires nova, Super nova, and neutron star collisions to produce the abundance of elements we measure in the Universe.

So we've learned that the same laws of motion observed here on Earth also explain the observed motions of the planets and we've directly measured gravity, many times, using many different methods.

One unavoidable consequence of Gravity is that any large body will pull itself into something roughly spherical until it reaches hydrostatic equilibrium -- and the larger the body, the greater the forces acting on the whole to achieve this. No body with the acceleration we experience on Earth could remain a large flat disc. So it is with our Moon, the other planets in our Solar System, our own Sun and to the limit of our ability to measure them, the distant stars and their Exoplanets.

Once we account for Relativity, there is simply no experimental, empirical observations that show these Laws are wrong. Period. There is room to doubt that the shape of space 'causes' Gravity (the Theory of Gravity) -- but there is simply zero room to doubt that there is a phenomena of mutual attraction of matter that is proportional to the mass and inversely proportional to the square of the distance -- the empirical Law of Gravity.

So, dear Flat Earther, when you say "Gravity is a lie" -- we simply don't believe you in the slightest, and with good cause. Your claim is vapid and puerile and made without the slightest bit of supporting evidence. Pardon me if I'm utterly unimpressed.

iPhone Free Fall

Now we easily can solve our iPhone free fall question -- the answer is given above as the first equation of motion for a falling body noting that the displacement over some time (t) is the one-half the acceleration times the time squared , where our time is 12/20ths of a second. \[f(t) = gt^2/2\]\[\left( 9.8\;\mathrm{m/s^2} \right) \left(12\;\mathrm{s}/20\right)^2/2 \approx 1.76 \mathrm{m}\] [wolfram|alpha]

The phone was 2.38 meters off the floor and the top of the pillow was at approximately 0.60 meters high (including the bed), giving us a fall distance of approximately 1.78 meters (2.38-0.60).

Do your own experiment and repeat the experiment several times because your accuracy will vary slightly but you should get to within about 2% for a fall that is a couple of meters high (but don't break your phone).

====== footnote: why do I link to Wikipedia? Because each source has dozens or hundreds of further sources - the entry on Newton has 162 citations. The reader is expected to be able to identify Primary sources and use them appropriately.

I love doing these because I found an error in the 'Blue Marble' Wikipedia entry, someone was careless with their timezone conversions and flagged this image as taken at 05:39 UTC which is 6 minutes after launch, clearly wrong. Also why I always check facts against primary sources.

The 'Blue Marble' frame (aka AS17-148-22727) was taken 5 hours 6 minutes later (probably by Jack Schmitt).

So that puts us at 5:39am EST (1039 UTC). See the mix up?

There is also an amazing site called Apollo In Real-Time where you can follow along the whole long, view the photos from around that time in the mission, listen to the mission control recordings, and so forth.

So here we are 7th Dec 1972, 10:49 UTC, about 29,000 km (18,000 miles) over about 30°S 31°E [found here]. That's just an incredible level accuracy.

I have some new toys to share in this installment of 'Yes, the Earth really is Curved even though you can see one distant mountain peak from another', so this should be fun!

The EXIF metadata shows this image was taken by a Panasonic DMC-FZ72 with a focal length of 215mm -- given the 5.62 crop factor of the 1/2.3" sensor in this camera, this gives you a 35mm equivalent focal length of 1200mm giving a full frame view of 1.644° wide and 1.215° high.
Since 1521 x 1014 isn't the exact same aspect ratio as the size of image this camera shoots it's likely that some small amount of cropping has taken place.

To double check this we can draw lines from our viewpoint to each peak and we find that, at the distance of Ferrand (392.48 km), our lines are about 1.7 km apart. This gives us an angle of approximately 0.24817° -- if we then apply that over the full 1521 pixels of the image we get an estimate of 1.63° wide -- very close to the 1.644° we expect so let's just use 1.644° / 1.215° as our field of view.
Now we're in a very good position to size the mountain and compare it to our photo.
First, we need the Law of Perspective, which says that the angular size (Î±) of an object whose height perpendicular to our line of sight (g) at distance (r) is:
Î± = 2*arctan(g/2/r)
Excellent - we just need to know the height and the distance.

Technically, we should find the straight-line distance but at 440 km it doesn't change the answer much. If you want to find the straight-line distance from a curved distance (o) along the surface of the Earth you could use this formula (where R is Earth's radius of curvature, for 42°N latitude R ~ 6369km): g = R*2*sin((o/R)/2) = 6369*2*sin((443/6369)/2) = 442.9 km So let's just keep it simple since this would be about 1/1000th of a degree.

Pic Gaspard (3880m) @ 443 km = 2*arctan(3.880/2/443) = 0.5018° = 461 pixels
Grand Ferrand (2758m) @ 392.48 km = 2*arctan(2.758/2/392.48) = 0.4026° = 375 pixels

Just for scale, one-half a degree is the apparent size of the Moon, so we're talking about a pinkie-at-arms-length size -- but that that is the size those mountains should appear at the distances they are from this observer.
So we can mark those out on the image now as the full height of the mountain at that distance.

As expected, the "sea-level" bottom for the more distant mountain (Gaspard) is lower than the closer mountain (Ferrand) and since it's a larger mountain the total apparent size is larger even though it's slightly further away.

So please do not try to tell me we're seeing the "whole mountain" here.

Now, Walter Bislin has an amazing calculator that will let you place objects in the world and includes a very advanced refraction module (which I left at defaults except to set the 'surveying standard' 7/6th refraction, about 14%. This is very modest amount of refraction and since the photographer tells you that this is very rare to be able to see this, the actual refraction is likely greater than shown here.

Here is Ferrand and Gaspard on a Globe model vs a Flat Earth model. Notice how the Globe model correctly shows just the distant peaks with Ferrand peak above Gaspard even though Ferrand is shorter. When you put these two mountains on a flat plane then Gaspard clearly stands taller.

There is no contest here, the Globe best fits the observations and really ONLY fits the observations. If we could see the entire mountain, even at this great distance with perspective, they would stand vastly larger than seen here.

Another fun tool that comes from the Beyond Horizons page is a 'panorama' rendering tool that will label the mountain peaks. I've used PeakFinder in the past but this one seems to be more full featured. Here is our approximate view:

So this view comports with the Globe model extremely well and utterly destroys the Flat Earth model. The problem is that Flat Earthers do not how to do basic estimations like how big something would be in an image (if you could see the whole thing).