A visit to Tehran

The taxi service that my host had arranged called the night before to tell me that because of morning traffic, and the distance of the conference to our apartment, he would need to come at 7:30am.  I said OK and hoped that I would be able to sleep and wake up in time, as the night before the whole family was up at 2 am having “breakfast”, and then sound sleep till around noon.  The traffic turned out to be surprisingly light as we drove southwest of Tehran, toward a region that had a number of fruit orchards. 

The conference, 22^nd Iranian Biomedical Engineering meeting, was held at a rather beautiful location, a series of buildings placed in a nicely landscaped campus.  Our building had intricate Islamic style carvings on the ceilings and walls.  The main conference hall was decorated with mosaic of mirrors and tiles, and unexpectedly, two styles of chairs: large, heavily upholstered rows of chairs up front, and smaller, more modestly upholstered chairs in the rows back.  The professors sat up front, the students in the back.

The conference started with speakers who were not scientists, but government officials.  They were there mostly to talk to the cameras (their speech was recorded for the evening news apparently): talk of budgets, lack of budgets, and need for budgets.  One lovely thing was the moderator, who read a short poem before introducing each speaker.  This mixture of “hard” sciences with the “soft” arts, equations and poetry, is one of the pleasures of attending a meeting in Iran.

One of the officials described a new institute, set up about a year ago following a large donation from a wealthy Iranian in Canada.   The institute was named after the donor: Movafaghian Research Center in Neurorehab Technologies.  A young scientist working there, Saeed Behzadipour, later described that the first floor housed a rehab clinic where patients tried out student-designed devices like an exoskeleton robot, and a balancing system.  The second floor housed the students and scientists, and the third floor housed start-up companies that commercialized the devices.  How nice, I thought; finally the wealth and success of a few expatriates coming back to do some good.

My talk was in the afternoon.  Looking at the conference hall, it seemed that most of students were female (later I learned that almost 70% of the undergraduate in Iran were female).  I gave a talk that described my student Yousef Salimpour’s work on non-invasive cortical stimulation in Parkinson’s disease.  My talk was in English, which is a little embarrassing.  I remember my dad coming to one of my talks many years back.  Afterwards he told me that he thought no one understood a word I said.  But after I finished there was a line of students who asked questions.  The last student in line was a young lady who had brought me a gift, a beautiful little book. 

I ended my talk with a poem from Abo Ali Sina, the 12^th century Persian physician.  I had read the poem on the plane ride, having discovered it in my new Iranian passport.  Each page had a drawing of a monument in remembrance of a Persian poet or artist, and a few words from them.  What a nice passport, I thought, focusing on Persian artists and scientists.

The conference had a modest exhibition hall.  There I saw a company that displayed an exoskeleton robot that wrapped around the legs and torso and walked stroke patients, and another company with a robot that moved a patient out of their bed and had them stand up.  Many of the companies displayed hospital instruments, dialysis machines and the like -- all quite proud of the fact that they were able to build the machines despite crippling economic sanctions.  (Perhaps because I was raised in the American west, I have always admired people who aspire to be self-sufficient.)

During the regular sessions I heard a talk by a student , Sahar Jahani, who had done an experiment using functional near infrared spectroscopy, measuring activity in the prefrontal cortex during an attention task.  It was a nicely designed and executed study, showing that activity in the dorsolateral prefrontal cortex increased with task difficulty.  But what was really impressive was the fact that she was using an instrument designed and built by an earlier group of engineering students. 

The conference book listed around 200 abstracts, all in English.  It seemed that despite years of sanctions, science had survived.

IPM

Some of the very best neuroscientists in Iran work at a place called Institute for Research in Fundamental Sciences (IPM), in the School of Cognitive Science.  What stands out is that while almost everywhere else in the world research on non-human primate brain is declining, with laboratories closing due to the high monetary and political costs of neurophysiological research, here is a place where research is expanding.  I saw three laboratories working on the cerebral cortex and the neural basis of vision. 

I gave a one day short-course, summarizing the work of the last couple of decades on how the brain controls movements of the eyes; contributions of the basal ganglia, superior colliculus, and the cerebellum.  We started at 9:30am, with a room that was so full that, to my delight, they kept on bringing in chairs.  The sight of all those eager students, and their wonderful questions that followed every slide, was a source of energy that fed me for the whole day (the lecture ended at 4:30pm).  I then met with individual students and visited labs.  The labs were recording from neurons in the brain with amplifiers and other electronics that were all homemade, another example of survival in the face of sanction imposed scarcity.  When the day had come to an end, I felt that I had experienced one of the best visits that I had had to any scientific institution.

With the students at the short course on Decisions and Actions, held at Institute for Research in Fundamental Sciences (IPM), Center for Cognitive Science.

Tehran

The most unusual feature of the city of Tehran is its sheer number of private banks (that is, not-government owned): driving through the city, it is hard to pass a block of a major street without passing a couple of bank branches.  There are more numerous than corner delis.  On the display windows are rates for certificates of deposit: 20% annual interest.  This implies that inflation must be much higher than that.

I sat and listened to the song of the caged love birds, hanging outside the corner grocer, and watched a  Persian alley cat cross the street.  The best thing about my short visit, however, was the quality of food.  The bread baker a few doors down from the apartment baked a thin flat bread, and sold it hot as it came out of the clay oven for around 15 cents.  The vegetables, particularly the ordinary humble tomatoes, were as red in the inside as the gorgeous color on the outside.  The dairy products, the sheep’s milk cheeses, were spectacular.  The cream pastries that I bought from the confectionery a block away were as good as those that I had had in Paris.  But what made the experience special was that this quality came at a fraction of the price that I had paid back home in America.

Exercise and dopamine

The November sun was shining through the glass roof, making moving shadows that lit up the shallow end of the swimming pool.  I was doing a few laps, thoroughly enjoying my afternoon at the local rec center.  I wondered, why is it that exercise makes me feel good?

Over the last two decades, much of the research on the neural basis of reward has focused on dopamine, a neurotransmitter that is released by a relatively small number of neurons in the midbrain, and sent to pretty much all the rest of the brain and other organs.  When an animal sees something that it likes, for example food, these neurons in the midbrain release dopamine.  The magnitude of the release is related to the subjective value of the rewarding stimulus.  So when you see ice cream, your brain probably releases more dopamine than when you see broccoli. 

However, when you have to work to acquire the rewarding item, then the magnitude of dopamine release becomes smaller as the required effort becomes larger.  That is, if you have to run a mile to get the ice cream, then the broccoli may seem like a better choice.  So it is puzzling that exercise, which basically involves generating a lot of effort, should result in an increase in the production of dopamine.  But there is indeed some evidence for this.

In 1994, Satoshi Hattori, Makoto Naoi, and Hitoo Nishino in Nagoya, Japan trained 6 rats to run on a treadmill.  After training, they measured dopamine levels via micro-dialysis in a region of the basal ganglia (striatum) at 20 minute intervals (this became the baseline measure of dopamine).  They then had the animals run for 20 minutes at a slow, medium, or fast speed, re-measured dopamine levels during the run, and then each 20 minutes after completion of the run (for another 3 hours).  They found that during the medium and fast runs, dopamine levels increased.  Interestingly, dopamine levels remained elevated for about 1.5 hours after completion of the run (in the figure below, the x-axis has bins of 20 minute duration).  As a result, the study demonstrated that dopamine levels increased with physical exercise, and remained elevated beyond completion of the exercise.

But this result was in rats.  Does the same thing happen in humans?  In 2000, Gene-Jack Wang, Nora Volkow and colleagues at State University of New York at Stony Brook asked 12 people who regularly exercised to participate in a PET study, where they used a scanning technique to indirectly measure dopamine levels in a region of the basal ganglia (striatum).  They scanned the brain and measured dopamine levels at baseline, and then they had the volunteers run on a treadmill for 30 minutes.  Following the run, they again scanned the brain.  Surprisingly, they found no significant changes in dopamine levels (as shown in the figure below).  In these people who exercised regularly, the run on the treadmill seemed to have had no particular effects on the dopamine levels.

Given the results in rats, this result in humans was puzzling, and to my knowledge still remains unresolved.   

Fortunately, over the last decade there have been  advances in our ability to directly record from dopamine neurons in the primate brain.  With these recordings it is possible to see how dopamine responds to both reward and effort.

In 2015, Chiara Varazzani, Sebastian Bouret, and their colleagues in Paris, France, trained thirsty monkeys to squeeze a handle in order to receive juice.  There were 3 juice amounts (small, medium, and large amounts of juice) and 3 amounts of effort (small, medium, and large amounts of force), producing a total of 9 conditions. 

There was a symbol associated with each of the 9 conditions.  For example, when the cue was a long, narrow rectangle, it meant that the monkey would have to squeeze the handle by a small amount to get a small amount of juice.  When the cue was a circle, it meant that the monkey would have to squeeze by a large amount to get the same small amount of juice.  In this way, the symbols defined both the amount of reward and the required effort.  Once the symbol was removed, a “go” cue appeared, instructing the animal to actually produce the force.  If the animal produced the right amount of force, it received the reward.

The authors recorded from 90 dopamine neurons in the substantia nigra, another region in the basal ganglia, and found that when the monkey saw the symbol indicating the reward and effort levels, the dopamine cells responded more with increasing reward.  However, as the required effort increased, the dopamine response became smaller.  Therefore, when the animal received information regarding effort and reward contingencies of the upcoming trial, it produced more dopamine if the trial was to include a large amount of juice, but less dopamine if the trial was to require a large amount of force.  In a sense, dopamine acted like a sum of reward minus effort, signaling the value (or utility) of the upcoming event.  At the time of cue, dopamine levels signaled how much the animal “liked” the following trial: the brain got more dopamine if the cue promised a lot of juice, and required only a small amount of effort.

Interestingly, during the time that the animal actually squeezed the handle and produced the required force, dopamine cells once again responded, but now with increased rates when the required force was larger (in the figure below, the x-axis is time, with each interval 200ms).  That is, the same cells that earlier had reduced their discharge when told that the trial would involve a large amount of effort, now increased their discharge during the actual production of the large effort. 

These results highlight the dual nature of dopamine.  When the brain is deciding between two options that each promise some amount of reward, and require some amount of effort, dopamine response becomes larger with greater promised reward, and becomes smaller with greater required effort.  Maybe this is why we tend to pick the more rewarding, less effortful option.  However, when the brain is sending motor commands to actually perform the option that we selected, dopamine responds more with the increasing effort, and now seems impervious to the promised reward.  Maybe this is why when we exercise, that is, when we spend effort, we tend to feel as if we are rewarded: because during exercise, dopamine makes effort seem like reward.

                                          

References

Satoshi Hattori, Makoto Naoi, and Hitoo Nishino (1994) Striatal dopamine turnover during treadmill running in the rat: relation to the speed of running.  Brain Research Bulletin 35:41-49.

Chiara Varazzani, Aurore San-Galli, Sophie Gilardeau, and Sebastien Bouret (2015) Noradrenaline and dopamine neurons in the reward/effort trade-off: a direct electrophysiological comparison in behaving monkeys.  Journal of Neuroscience 35:7866-7877.

Gene-Jack Wang, Nora D. Volkow, Joanna S. Fowler, et al. (2000) PET studies of the effects of aerobic exercise on human striatal dopamine release.  Journal of Nuclear Medicine 41:1352-1356.

Happiness and the aging brain

In 2008, the Gallop organization randomly called around 355,000 people in the United States, asking them about their state of happiness.  They wanted to quantify how this state changed as people transitioned from youth, to middle age, to old age.  To assess global well-being, they asked the following:

“Please imagine a ladder with steps numbered from 0 at the bottom to 10 at the top.  The top of the ladder represents the best possible life for you, and the bottom of the ladder represents the worst possible life for you.  On which step of the ladder would you say you personally feel you stand at this time?”

They found that youngest people, those in their late teens and early 20s, felt quite upbeat about their life, placing themselves high up on the ladder.  Unfortunately, this sense of well-being dropped as age increased, reaching its lowest point around the age of 50.  But as age increased beyond 50, the sense of well-being increased dramatically, continuing to grow even into the 8th and 9th decade of life. 

Well-being ladder as a function of age in America, as assessed in 2008

The trend was consistent in both men and women.  And so amazingly, people in America felt that they reached their lowest point on the ladder of life around the time they reached midlife.  What is it about aging beyond the teen years that made people feel worse about their sense of well-being, and why did this process reverse in the fifth decade of life?

Stress and worry decline in mid life

Statistical analysis of the data revealed that some obvious things that one might think affects sense of well-being did not alter the age-dependent process.  For example, gender, being unemployed, having a child living at home, and not having a partner had no significant effects on the trends.  That is, regardless of these factors, people simply became less happy as they aged toward midlife, and then something seemed to change, allowing them to regain happiness as they got older.

The authors of the study, writing in the Proceedings of the National Academy of Science, speculated that perhaps this change had something to do with increased wisdom and emotional intelligence of the aged.  They wrote: “older people have an increased ability to self-regulate their emotions and view their situations positively.”

They provided some data to back their speculation.  That same Gallop poll had also asked the respondents to evaluate how they felt yesterday.  They asked about specific affects like worry, stress, and anger: “Did you experience stress during a lot of the day yesterday?  Did you experience worry during a lot of the day yesterday?”

They found that regardless of age, women reported that their yesterday had greater stress and worry than men.  However, there was a dramatic age-dependent effect.  When young people evaluated their yesterday, they reported much more stress and worry than older people.  But for people in their late 40s and early 50s, their yesterday was suddenly much less stressful than for people 10 years younger.  For people in their 60s, their yesterday was even more peaceful.

The emotional response to what could've been

This questions of why people (at least Americans) feel a reduced sense of wellbeing with age, and why this process reverses at midlife, remain unanswered.  One intriguing line of research is that with age, the brain alters how it evaluates lost opportunities.  

Stefanie Brassen and colleagues asked a group of young people (around 25 years old), a group of healthy older people (around 66 years old), and a group of late-life depressed elderly (also around 66 years old), to watch a monitor which showed 8 boxes.  Seven of those boxes contained gold, but one had a devil in it.  Boxes could be opened in sequence, and as long as the box contained gold, they could keep accumulating the reward.  But if the box contained a devil, they would lose everything.  So the participants could decide to stop and collect their gains, or continue.  Importantly, if they decided to stop, the position of the devil was revealed.  This indicated how far they could have safely continued, thereby showing them the missed opportunity.

Young people responded to the missed opportunity by being aggressive on the next trial—the greater the missed opportunity on the current trial, the greater the risk that they took on their next attempt.  Surprisingly, following a missed opportunity the depressed elderly did the same as the young people, taking a bigger risk.  However, healthy elderly did not respond this way.  Following a missed opportunity, they did not increase their risk taking.

To measure the emotional response to the missed opportunities, the authors measured skin conductance and found that this measure was modulated in the depressed elderly but not in the healthy elderly.  While these data were being collected, the authors also measured brain activity using fMRI and found that whereas all groups responded similarly to winning and losing, the main difference was in the response to a missed opportunity.  Both the young and the depressed elderly had a strong response when they observed the missed opportunity.  The healthy elderly, however, only responded to real losses, and not missed opportunities.

The data suggested that healthy aging was associated with a reduced responsiveness to lost opportunities.  When the healthy elderly made their decision, they were happy if they gained something, and did not care so much if that gain was less than optimal.  That is, they did not respond emotionally to the fact that they could have made even a better decision.

On the other hand, both the young folks and the depressed elderly responded emotionally once they found out that they could have made a better decision, despite the fact that the decision that they had made had produced a gain.  That gain, in retrospect, was not good enough.  And in some ways not being as good as it could have been felt like a loss to the young folks and the depressed elderly.

So it is possible that in youth, a decision that results in a positive outcome, but represents a lost opportunity (because it could have been better), produces an emotional, stressful response. But when that same decision is made after midlife, the brain is less sensitive to the fact that the results could have been better, and more concerned with the fact that the decision produced a gain.  Curiously, this is true for elderly who are healthy, but not elderly who are depressed.  

In his poem, "The Bridge", Henry Wadsworth Longfellow writes:

For my heart was hot and restless,
      And my life was full of care,
And the burden laid upon me
      Seemed greater than I could bear.

But now it has fallen from me,
      It is buried in the sea;
And only the sorrow of others
      Throws its shadow over me.

Yet whenever I cross the river
      On its bridge with wooden piers,
Like the odor of brine from the ocean
      Comes the thought of other years.

And I think how many thousands
      Of care-encumbered men,
Each bearing his burden of sorrow,
      Have crossed the bridge since then.

I see the long procession
      Still passing to and fro,
The young heart hot and restless,
      And the old subdued and slow!

And forever and forever,
      As long as the river flows,
As long as the heart has passions,
      As long as life has woes;

The moon and its broken reflection
      And its shadows shall appear,
As the symbol of love in heaven,
      And its wavering image here.

Sources
Andrew Steptoe, Angus Deaton, Arthur A. Stone (2015) Psychological wellbeing, health and ageing.  Lancet 385: 640–648.

Stone, A. A., Schwartz, J. E., Broderick, J. E., & Deaton, A. (2010). A snapshot of the age distribution of psychological well-being in the United States.Proceedings of the National Academy of Sciences, 107:9985-9990.

Stefanie Brassen, Matthias Gamer, Jan Peters, Sebastian Gluth, and Christian Büchel (2012) Don’t look back in anger! Responsiveness to missed changes in successful and unsuccessful aging.  Science 336:612-614.

The problem of planets: from Aristotle to Newton

The ancient Greeks, along with a number of other civilizations, noticed five “wandering stars” that over many nights appeared to travel against the background of fixed stars in the sky. These “stars” went along the same path as the Sun and the moon, but in the opposite direction. They named the wandering stars Hermes, Aphrodite, Ares, Zeus, and Cronos.  The Romans translated these names into Mercury, Venus, Mars, Jupiter, and Saturn.  The names of the wandering stars, along with the Sun and the moon, became the names of the 7 days of the week. Saturday, Sunday, and Monday are associated with Saturn, the Sun, and the moon. Tuesday is thought to be associated with the Germanic god Tyr (Mars), Wednesday with Wotan (Mercury), Thursday with Thor (Jupiter), and Friday with Frigga (Venus). 

The wandering stars, of course, were no stars at all, but planets.  It took about two thousand years to understand why the planets appeared to wander. The story begins around the time of Aristotle, and ends with Newton. Along the way, humans learned how to use mathematics to represent observations in nature, and this led to the birth of science. In a recent book titled “To Explain the World”, Stephen Weinberg, a physicist and Noble Laureate, tells this story. Here, I simplify his eloquent and thorough text, and highlight the key ideas.

Aristotle and Ptolemy

Anaxagoras, an Ionian Greek born around 500 BC, reasoned that the earth is spherical because when the Sun placed the earth’s shadow on the moon, one could see the round outline of the earth. Aristotle repeated this idea in his book “On the Heavens”, writing: “In eclipses the outline is always curved, and, since it is the interposition of the Earth that makes the eclipse, the form of the line will be caused by the form of the Earth’s surface, which is therefore spherical.” But he also argued that the earth must be stationary and not moving, because if it were moving a rock thrown upward would not fall straight down, but to one side. He wrote: “heavy bodies forcibly thrown quite straight upward return to the point from which they started, even if they are thrown to an unlimited distance.” 

Given that the earth is not moving, how does one explain the fixed and the wandering stars (the planets)? Aristotle, citing an earlier work by Eudoxus of Cnidus, suggested that the fixed stars are carried around the earth on a sphere that revolves once a day from east to west, while the sun and moon and planets are carried around the earth on separate (and transparent) spheres. Now there were lots of problems with this scheme. For example, because the planets were thought to shine with their own light, and the spheres were always the same distance from the earth, the brightness of the planets should not change, which disagreed with observations.

This issue remained unresolved until 650 years later, with Claudius Ptolemy, who in AD 150, working in Alexandria, Egypt, wrote Almagest. Ptolemy gave up on the notion that earth was the center of rotation for the planets, and instead suggested that each planet had a center of rotation that itself went around the earth. For the nearby planets of Venus and Mercury, he proposed that the centers of rotation were always along a line between the earth and the sun, and went around the earth in exactly one year. For Mars, Jupiter, and Saturn, the centers of rotation were beyond the sun.  

Ptolemy's planetary model

Ptolemy wrote: “I know that I am mortal and the creature of a day; but when I search out the massed wheeling circles of the stars, my feet no longer touch the earth, but, side by side with Zeus himself, I take my fill of ambrosia, the food of the gods.”

Copernicus and Tycho Brahe

For centuries the idea that the earth was stationary remained, so that even in the middle ages, scholars like Jean Buridan would reject the idea that the earth could be rotating, not realizing that if earth rotated, then its rotation would give everything, including an arrow that was shot straight up, an impetus. Like all good mentors, Buridan had a student who thought independently. His name was Nicole Oresme. Oresme studied with his mentor Buridan in Paris in 1340s.  In his book “On the Heavens and the Earth”, Oresme rejected Aristotle’s arguments for a stationary earth, stating that when an archer shoots an arrow vertically, the earth’s rotation carries the arrow with it (along with the archer). Therefore this observation is not a demonstration of an immovable earth, but also consistent with a rotating earth.  Aristotle’s argument on a stationary earth took its first major blow.

The idea that the earth might be rotating took center stage with Nicolaus Copernicus, who in 1510 wrote a short, anonymous book titled “Little Commentary”. The book was not published until after the author’s death, but in it he put forth a new theory. He began by asserting that there is no center for the orbits of the celestial bodies: the moon goes around the earth, but all other heavenly bodies go around a point near the sun. He further asserted that the night sky has fixed stars that are much farther away than the sun, and appear to move around the earth only because the earth is rotating on its axis and about the sun.

Tycho Brahe was impressed with the simplicity of Copernicus’ theory, but pointed out a huge problem:  if the earth is moving, what is moving it? After all, earth was made of rocks and dirt, materials that would make something the size of earth weigh an enormous amount. In contrast, ever since Aristotle it was thought that the heavenly bodies were nothing like earth, made of some kind of substance that gave them a natural tendency to undergo rapid circular motion. The problem was, if earth was moving around the sun, what was pushing it, and what was keeping it there in its orbit?

In an ironic twist, to explain motion of the earth it was the Copernican astronomers who called on divine intervention. In a letter to Brahe, Copernican Christoph Rothmann wrote: “These things that vulgar sorts see as absurd at first glance are not easily charged with absurdity, for in fact divine Sapience and Majesty are far greater than they understand.”

Being unimpressed with divine intervention, in 1588 Tycho Brahe pointed out that if one took Ptolemy’s theory and put the moving center of all the planets (except earth) on the sun, and have the sun go around the stationary earth, then much of the observed data would fit just as well as Copernicus’ theory. This “Tychonic” system kept the advantage of a stationary earth, and was mathematically identical to the model of Copernicus.

Tycho Brahe's planetary model

In January of 1610, Galileo used his newly built telescope to look at Jupiter, and saw that “three little stars were positioned near him, small but very bright.” The next night he noticed that the little stars seemed to have moved, and eventually he concluded that the little stars were actually satellites of Jupiter, its moons. This observation was critical, as it was the first discovery of heavenly objects that circled something other than earth. They were a miniature example of what Copernicus had proposed. But Tycho Brahe’s model remained a viable alternative, because the fundamental question for a sun-centric theory remained that if the earth is moving, what could be so powerful as to move it?  

Newton and calculus

In 1665, Issac Newton asked a simple question: how does one compute speed of some object if the distance traveled as a function of time is not constant (or uniform). Suppose x(t) represents position as a function time t. Newton argued that in order to calculate speed, we need to think of an infinitesimally small period of time, which he called o.  Speed becomes:

For example, suppose that

For o an infinitesimal period of time, we can ignore terms that include squared and cubic powers of o. This means that:

Newton called this the "fluxion" of x(t). We now call it the derivative of x(t).  

Newton was considering this question because he wanted to ask about the acceleration that a body would experience as it travels in constant speed about a circle. At any time t, the velocity of this body is a vector tangent to the circle, with amplitude v.  

Suppose that the circle is radius r. After an infinitesimal time o, the body will have traveled by a distance vo, and angle q about the circle. At this new location the speed would still be v, but the velocity vector will have rotated by an angle q. We now have two isosceles triangles that are scaled versions of each other. Therefore, the ratio of the short side to the long side of the two triangles is equal: 

We can re-write the above equation as follows:

Eq. (1)

The term on the left of the above equation is a derivative. It represents the length of the acceleration vector that the body experiences (pointing to the center of the circle) as it rotates with constant speed around the circle.  

Newton realized that this acceleration toward the center is due to a force that is pulling the body toward the center of the circle (otherwise, it would fly off in a straight line, tangent to the circle). That force, he assumed, is proportional to square of the velocity v, divided by radius r.  

Next, Newton considered Kepler’s observation (his third law) that the square of the period of a planet in its orbits is proportional to the cube of the radius of its orbit. The period of a body moving with speed v around a circle of radius r is the circumference  2pr divided by speed v.  And so Kepler’s third law says that 

We can re-write the above equation as follows: 

Eq. (2)

If we now compare Eq. (1) with Eq. (2), we see that the acceleration that was keeping the body moving in circular motion, is also proportional to the reciprocal of squared r. This means that the force that is pulling the body toward the center is proportional to the inverse of the squared distance of the body from the center. This is the inverse square law of gravity.

But the incredible discovery was still one step away. Newton now asked whether the acceleration of the moon in its orbit around the earth is the same acceleration that a body undergoes when it is falling here on earth. To calculate moon’s acceleration, he estimated the distance of the moon to the center of the earth to be around 60 times the radius of earth, or around 314 million meters. Next, he computed the speed of the moon by dividing the circumference of one orbit around the earth by its period of travel (27.3 days, or 2.36 million seconds):

He then used Eq. (1) to compute the acceleration of the moon toward the earth:

This is the moon’s acceleration toward earth. It is quite small, but Newton understood that the acceleration is small because the moon is very far away from earth. An object on the surface of earth accelerates faster because it is at a distance of one radius of earth away from the earth’s center. The moon is 60 times farther. Therefore, using Eq. (2), he argued that the moon’s acceleration should be 1/60^2 that of an object on the surface of the earth.  

Multiplying moon’s acceleration by 60^2 we find the result that a body on the surface of earth should accelerate at around 8 m/s^2.  (The actual value is 9.8 m/s^2. The greatest source of error in Newton’s calculations was that the distance of moon from earth, which he underestimated by around 15%). He then writes:  

“I began to think of gravity extending to the orb of the moon and (having found out how to estimate the force with which [a] globe revolving within a sphere presses the surface of the sphere) from Kepler’s rule of the periodical times of the plants being in sesquilterate proportion of their distances from the center of the orbs, I deduced that the forces which keep the planets in their orbs must [be] reciprocally as the squares of their distances from the centers about which they revolved and thereby compared the moon in her orb with the force of gravity at the surface of the earth and found them answer pretty nearly.”

So what Newton had done was to show that the motion of the moon around the earth described an acceleration toward earth that was due to a force quite identical to the force that acts on an apple on the surface of the earth. The only reason that the moon accelerates much slower toward earth is because the moon is much farther, and therefore the force that it feels from earth is much weaker.  

The acceleration of the apple, the moon, and the planets around the sun, are all governed by the same rules: force grows weaker as the squared distance of one body from another.

Sources:

Dennis Danielson and Christopher M. Graney (2014) The case against Copernicus.  Scientific American, January 2014, pp. 74-77.

Stephen Weinberg (2015) To Explain the World: The discovery of modern science. HarperCollins.