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## Who invented calculus?

One of the physicists in our book Ten Physicists Who Transformed Our Understanding of Reality (follow this link for more information on the book) is, not surprisingly, Isaac Newton. In fact, he is number 1 in the list. One could argue that he practically invented the subject of physics. We decided to call him the ‘father of physics’, with Galileo (whose life preceded Newton’s) being given the title of ‘grandfather’.

Newton was, clearly, a man of genius. But he was also a nasty, vindictive bastard (not to mince my words!). He didn’t really have any close friends in his life; there were plenty of people who admired him and respected him, and of course he had colleagues. But, apart from a niece whom he seemed to dote on in later life, and two men with whom he probably had love affairs, he was not a man who sought company. He was probably autistic, but lived at a time before such conditions were diagnosed or talked about.

Isaac Newton (1643-1727), the ‘father of physics’. He relished in feuding with other scientists

One sort of interaction that he did seem to enjoy with other people though was feuds. In fact, he seemed to thrive on feuding with other scientists. He loved to argue with others, which is not uncommon amongst academics. He had strong opinions which he liked to defend; this is normal. But, Newton took these disputes to an extreme; if he fell out with someone he would do everything he could to destroy that person.

Although I am sure that he had many ‘minor’ arguments, he had three main feuds with fellow scientists. These three men were

• Robert Hooke – curator of experiments at the Royal Society
• Gottfried (von) Leibniz – the German mathematician
• John Flamsteed – the first Astronomer Royal

In each case, he did his level best to destroy the other man. Each of these feuds is discussed in more detail in our book, but in this blogpost I will give a brief summary of his feud with Leibniz.

The feud came about because Newton refused to believe that Leibniz had independently come up with the mathematical idea of calculus. It was a recurring theme throughout Newton’s life that he sincerely believed that he was special. He had deep religious views (some would say extreme religious views). As part of these views, he believed that he had been specially chosen by God to understand things that others would never be able to understand.

Thus, when he heard that Leibniz had developed a mathematics similar to his own ‘theory of fluxions’ (as Newton called it), he naturally assumed that the German had stolen it from him. There then ensued a 30-year dispute between the two men, with Newton very much the aggressor.

Gottfried (von) Leibniz (1646-1716), German mathematician and co-inventor of calculus

It escalated from a dispute to a feud, and culminated in the Royal Society commissioning an ‘official investigation’ to establish propriety for the invention of calculus. When the report came out in 1713 it came out in Newton’s favour. But, by this time Newton was not only President of the Royal Society, but he had secretly authored the entire report. It was anything but impartial. Leibniz died the following year, a broken man from Newton’s relentless attacks.

One should, of course, be able to to admire a person for their work but not admire them in the least for the person that they were. Newton, in my mind, falls very firmly into this category. His contribution to physics is unparalleled, but I don’t think he was the kind of person one would want to know or even come across if one could help it!

Ten Physicists Who Transformed Our Understanding of Reality is available now. Follow this link to order

## When did Newton die?

In the last week I have been putting together the final edits for the book that I have been writing with Brian Clegg – Ten Physicists who transformed our understanding of reality, which will be published later this year. One of the issues which we needed to clarify in this editing process were the dates of Newton’s birth and death. The reason this is an issue is that the calendar system was changed in the period between the 1500s and 1700s, which spans the years that Newton was alive. In 1582 the Gregorian calendar was introduced by Pope Gregory XIII, because the Julian calendar’s system of having a leap year every four years is not exactly correct (I will blog separately about the details of why having a leap year every four years is not correct).

‘Ten Physicists who transformed our understanding of reality” will be out December

Different countries adopted the Gregorian calendar at different times, with Catholic countries adopting it before Protestant ones. Newton was born in England in the mid-1600s, and when he was born England was still using the Julian calendar. Under the Julian calendar, he was born in the early hours of the 25th of December 1642. But, by that time, many European countries were using the Gregorian calendar, and so had someone in e.g. France heard of his birth on that day (imagine radio existed!), their calendar would have said that the date was the 4th of January! But, which year, 1642 or 1643? This is where another subtlety of calendars arrises, because starting the year on the 1st of January is something else that changed during this period.

In England, the year traditionally began on the 25th of March, and so the 4th of January (the one 10 days after the 25th of December) was still in 1642! The 4th of January 1642 actually came after the 25th of December 1642, because the year did not change to 1643 until the 25th of March! (The year starting on the 25th of March is also why the financial year (Tax year) in Britain still starts on the 6th of April, the date to which that the 25th of March was adjusted when the Gregorian calendar was finally adopted in Britain.)

However, in France (for example) they switched to starting their year on the 1st of January in 1564 (prior to this France started their year at Easter), so again this hypothetical person in France who heard of Newton’s birth would have said the date was the 4th of January 1643 (for more about this see here).

The first page of the Papal bull announcing the introduction of the Gregorian calendar, which was published on the 24th of February 1582

A similar confusion arises over Newton’s death. Under the Julian calendar, he died on the 20th of March 1726. At the time of his death, England was still using the Julian calendar, and was also still starting its year on the 25th of March (it switched to the Gregorian calendar and to starting its year on the 1st of January in 1752). So, had Newton died just 5 days later his date of death would have been the 25th of March 1727, which to any casual reader would imply he was a year older than he actually was. To someone in France, the date of Newton’s death would have been the 30th of March 1727.

Confused? 😉

## Newton’s 3rd law of motion

In this blog here, I tried to explain Newton’s 1st law of motion, the so-called law of inertia. Then, the following week, in this blog here, I tried to explain Newton’s 2nd law, the one with the equation $F=ma$, which tells us how a body accelerates if there is a (resultant) force applied to it. This week, I will finish up this series by trying to explain Newton’s 3rd law.

His 3rd law is very important. Without it cars wouldn’t work, we wouldn’t be able to walk forwards and rockets wouldn’t work in space. The 3rd law is often stated as “to every action, there is an equal and opposite reaction”. Another way of saying this is that, if you push on something, it will push back. It is the reason you get a recoil from a gun when it fires a bullet. The explosion of the bullet propels the bullet forward, and an equal force propels the gun back into your shoulder.

Due to Newton’s 3rd law, when you fire a gun and the bullet comes hurtling out of the front, an equal and opposite reaction means the gun recoils into your shoulder.

You might be asking yourself “hang on, the gun doesn’t come back at the same speed as the bullet goes out of the front, so how can it be equal and opposite?”. Well, remember it is the force which is equal and opposite. Force is mass multiplied by acceleration (Newton’s 2nd law), and the gun is many times more massive than the bullet. This means that the force applied to the bullet produces much more acceleration than the equal force applied to the gun, so the gun moves back much more slowly than the bullet goes out of the front (thankfully!).

When we walk, we are actually using Newton’s 3rd law to move ourselves forward. We push our foot onto the ground, but also slightly backwards (the difference between walking and just standing). The backwards push of our foot leads to the ground pushing us forwards. Try walking on slippery ice, and you quickly realise that it doesn’t work too well when friction is reduced and your foot slides as you try and push backwards on the ground.

A rocket moving through empty space is an example of Newton’s third law which often confuses people. This is because people don’t see what the rocket is pushing against. The answer is – nothing! But that does not mean there isn’t an equal and opposite reaction. The rocket sends gasses out of the back of the rocket through burning fuel (essentially a controlled explosion), and in pushing the fuel out of the back the equal and opposite reaction to this push is that the rocket is pushed in the opposite direction, forwards. The gasses coming out of the back do not need to be pushing against anything, the mere act of their coming out of the back means the 3rd law reaction is to push the rocket forwards.

When a rocket in empty space moves forwards the gasses coming out of the back do not push against anything, but Newton’s 3rd law means that the gasses being pushed out through the exhaust cause an equal and opposite reaction – the pushing forwards of the rocket.

So, that is it. With these three laws of motion, which Newton outlined in his 1687 masterpiece The Principia (which I blogged about here), the foundations of mechanics had been laid down. These foundations are still in use today, more than 300 years later. Despite Einstein showing that Newton’s laws of motion have their limitations (written in its simple form, Newton’s 2nd law of motion is not entirely correct), we still use them on a daily basis because we so rarely experience the situations where relativistic mechanics is needed.

## Newton’s 2nd law of motion, force and acceleration

Last week, I blogged about Newton’s 1st law of motion, and the concept of inertia. At the end I said that this week I would discuss what happens to an object if a force is applied. Or, to put it more correctly, an “external resultant force” is applied. This is what Newton’s 2nd law of motion is all about – the effect on a body of an applied force.

Newton’s three laws of motion appear in his masterpiece, The Principia, which was published in 1687.

If we apply a force to an object it will change its velocity, which means it will accelerate. As I have mentioned before, in physics acceleration has a more precise meaning than it does in everyday life. It doesn’t just mean an object is changing its speed, it can also be keeping a constant speed but changing its direction, such as an object moving at a constant speed in a circle. But, whether an object is changing its speed or changing its direction, it has to accelerate to do this, and so a force needs to be applied.

## The most important equation in physics

The relationship between force and acceleration is given by Newton’s 2nd law of motion, which states that

$\boxed{ F = m a }$

where $F$ is the force, $a$ is the acceleration, and $m$ is the mass of the body. From this equation, along with Newton’s 3rd law (which I will discuss next week), nearly all of mechanics can be derived. For example, this equation tells us that for the Moon to orbit the Earth, it must have a force acting upon it. That force is gravity, and Newton was also the first person to produce an equation to describe gravity. For example, in this blog I showed how we can derive the acceleration felt by a body travelling in a circle, which we call the centripetal acceleration.

Using calculus, this equation also allows us to derive the three equations of motion, equations like $v = u + at$ and $s=ut + \frac{1}{2}at^{2}$, as I did in this blog. It tells us that it is more difficult to accelerate a more massive object than it is a less massive one, which is why you need a more powerful engine in a large truck than you do in e.g. a small car. Along with Newton’s 3rd law, it explains why a bullet comes out at such a high speed from the nozzle of a rifle, but why the recoil of the gun moves much more slowly. As I said, the most important equation in physics.

Next week I will discuss the last of Newton’s laws of motion, his 3rd law.

## Inertia and Newton’s 1st law

This weekend I was helping my youngest daughter revise for her Christmas physics exam. She tells me she enjoys physics, but I’m not sure whether she’s just saying this to please me! Her brother has just started his physics degree this last September, and her elder sister seriously thought of doing physics for A-level before deciding against it; but I suspect my youngest is more on the languages and creative arts side than a scientist. We shall see, she is only 13.

The material we were going over was the basics of motion, or mechanics. She has been learning about forces, pressure, velocity and resistances to motion (friction and air resistance). I asked her if she had been learning Newton’s three laws of motion, and from her answer I wasn’t sure whether she had or not!

I distinctly remember my own first encounter with Newton’s three laws of motion. I was about my daughter’s present age, and due to a Horizon programme about cosmology and particle physics (that I mention in this blog), I had already decided I wanted to study astronomy and physics. It therefore came as a bit of a shock to me when we were presented with Newton’s laws of motion, and I found I could not remember them.

Despite seeming to have a very good ability to remember poems and song lyrics, I am terrible at remembering ‘facts’, and our physics teacher presented Newton’s three laws to us as a series of facts. After several days of trying and failing to remember them in the words he had used (or which the text book had used), I was beginning to have serious doubts that I could go into physics at all.

And then I had an epiphany. I realised that if I understood Newton’s laws of motion, I did not need to remember them. If I could understand them, I could just state them in my own words; and within less than an hour I felt I had understood them thoroughly (although I’d like to think I have a deeper understanding of them now than I did at 13!). I often say to my students that the only things they need to remember in physics are the things they do not understand. That, certainly, has been my own experience.

## Inertia

The concept of inertia is fundamental to our ideas of motion, and yet it is not the easiest concept to understand or explain. But, I will have a go! Galileo was the first person to think of the concept of what we now call ‘inertia’. He realised that a stationary object wants to stay stationary, and you have to do something to it (push it, pull it, or drop it) to get it moving.

Galileo was the first person to outline the concept of inertia.

He also realised that objects which are moving want to carry on moving. This is not obvious, as we all know if we give an object a push it may start to move but will slow down and stop. Galileo realised that objects which were moving stopped because of resistive forces like friction or air resistance, and in the absence of these an object would carry on moving. This is, essentially, the idea of inertia. The tendency a body has to remain at rest or to carry on moving.

## Newton’s 1st law of motion

Newton’s 1st law of motion is essentially a statement of the concept of inertia, and is sometimes called the ‘law of inertia’. If someone asks me to state Newton’s 1st law the wording I use will probably change slightly each time, but the key idea I make sure I try to get across is the concept of inertia. So, a way to state Newton’s 1st law is

A body will maintain a constant velocity unless a force acts upon it

This is, more or less, the most succinct way I can express his 1st law. But, by stating it so succinctly, there are hidden complications. The first is to realise that the term ‘velocity’ has a very precise meaning in physics. I was trying to explain this to my daughter over the weekend (except we were doing it in Welsh, so using the Welsh word ‘buanedd’).

For a physicist, velocity is not the same as speed, even though we may use them interchangeably in everyday language. Speed (‘cyflymder’ in Welsh 😛 ) is a measurement of how quickly an object is moving, but velocity also includes the object’s direction of motion. A stationary object has zero velocity.

Therefore, a more long-winded (but no less correct) way to state Newton’s 1st law is

A body will remain at rest, or carry on moving with a constant speed in a straight line, unless acted upon by a force

The last thing I should mention in relation to the wording of Newton’s 1st law is that, strictly speaking, I should say ‘an external resultant force’, because an object can maintain a constant velocity with more than one force act upon it, as long as those forces are equal in size and opposite in direction. A good every-day example of this is driving a car at a constant speed in a straight line. The car does not continue to do this if we take our foot off of the accelerator, because air resistance and friction will slow the car down. When it is moving at a constant velocity the resistive forces are balanced by the force the engine is transferring to the tyres on the road. An ice puck, once pushed, will take a long time to slow down. This is because the friction on ice is much less than on a rougher surface, so the puck comes closer to acting like Newton’s 1st law says objects should act.

I don’t think my daughter has learnt about Newton’s 2nd law yet, which I often tell my students contains the most important equation in physics. His second law tells us what happens to an object if there is a (resultant external) force acting up on it. I will blog about that next week.

## Derivation of Newton’s equations of motion

Anyone who has studied mechanics / dynamics will have come across Newton’s equations of motion (not to be confused with his laws of motion). The ones I get my students to use are

1. $v = u + at \; \; \text{ (Equ. 1)}$
2. $s = ut + \frac{1}{2} at^{2} \; \; \text{ (Equ. 2)}$
3. $v^{2} = u^{2} + 2 as \; \; \text{ (Equ. 3)}$

where $u$ is the initial velocity, $v$ is the velocity at time $t$, $s$ is the displacement and $a$ is the acceleration. Note, these equations are only true for constant acceleration, but that actually covers quite a lot of situations. They can all be derived from the definition of acceleration.

Newton’s equations of motion can be derived from his 2nd law of motion.

## Derivation of Equation 1

We start off with out definition of acceleration, which is the rate of change of velocity. Writing that mathematically,

$a = \frac{dv}{dt}$

This is an example of a first order differential equation. To solve it we integrate. So we have

$a \int dt = \int dv$

When we integrate without limits, we have to include a constant term, so we can write

$at = v + C$

where $C$ is our constant. To determine the value of the constant we need to put in some conditions, such as (but not necessarily) initial conditions. When $t=0$ we have defined that $v=u$, so we can write

$0 = u + C \rightarrow C=-u \rightarrow \boxed{v = u + at \; \; \text{ (Equ. 1)} }$

## Derivation of Equation 2

To derive equation two, which we notice involves the displacement (the vector equivalent of distance), we do the following

$v = \frac{ds}{dt} = u + at \rightarrow ds = \int u dt + \int at dt$
$s = ut + \frac{at^{2}}{2} + C$

When $t=0 \text{ then } s=0$ so we can write

$0 = 0 + 0 +C \rightarrow C=0 \rightarrow \boxed{s = ut + \frac{1}{2}at^{2} \; \; \text{ (Equ. 2)} }$

## Derivation of Equation 3

To derive equation 3 we use the trick of writing the acceleration $a$ in terms of the velocity $v$ and the displacement $s$. To do this we write

$a = \frac{dv}{dt} = \frac{dv}{ds} \cdot \frac{ds}{dt} = \frac{dv}{ds} \cdot v = v \frac{dv}{ds}$

So, writing

$v \frac{dv}{ds} = a \rightarrow \int v dv = \int a ds \rightarrow \frac{v^{2}}{2} = as + C$
Again, we can work out $C$ by remembering that $s=0 \text{ when } t=0$ so

$\frac{u^{2}}{2} = 0 + C \rightarrow C = \frac{u^{2}}{2}$

and so

$\frac{ v^{2} }{2} = as + \frac{ u^{2} }{2} \rightarrow \boxed{ v^{2} = u^{2} + 2 as \; \; \text{ (Equ. 3)} }$

## The 10 best physicists – no. 3 – Galileo Galilei

At number 3 in The Guardian’s list of the ten best physicists is Galileo Galilei.

Galileo is often thought of as the founder of experimental science, although there were others before him. But there is no doubting his huge contribution in our early understanding of what we now call physics. He made important observations of the Solar System, including discovering four moons orbiting Jupiter, and showing that Venus must be orbiting the Sun. He showed that a pendulum keeps the same period irrespective of amplitude, and that the period depends on the length of the pendulum. He introduced the concept of inertia, the tendency of bodies to remain at rest or to keep moving.

## Galileo’s brief biography

Galileo was born in 1564 in Pisa, Italy. He was the first of six children; his father Vincenzo Galilei was a famous musician. Galileo initially studied for a medical degree at the University of Pisa. But, in 1581, whilst doing his medical studies, he noticed that a swinging chandelier’s period was not dependent on how large the swings were but rather on the length of the pendulum. This changed the course of his life, he persuaded his father to let him abandon his medical studies and instead he switched to studying mathematics and “natural philosophy” (as physics was commonly known at the time).

In 1589 he was appointed Chair of mathematics at the University of Pisa. In 1592 he moved to the University of Padua, where he taught mathematics, geometry and astronomy. He stayed in this position until 1610. In 1609 Galileo built a telescope and was soon making important discoveries, including proof that Venus orbited the Sun. He published this and other findings, bringing him into conflict with the Catholic Church.

In 1616 Galileo was summoned to Rome and told to stop promoting the idea of a Heliocentric Universe, the idea that the Sun and not the Earth was at the centre of everything. Initially Galileo obeyed, but by 1623 he revived his project of working on a book to argue in favour of the Heliocentric model. This was finally published in 1632, under the title “Dialogue Concerning the Two Chief World Systems”. This book some him placed under house arrest by the Church, under which he lived for the rest of his life. In In 1638 he published “The Two Sciences”, which summarised much of his life’s work. It is for the work in this book that Galileo is often referred to as the “father of Physics”. He died in 1642, and is buried in the Sante Croce in Florence.

## Galileo’s contributions to Physics

Galileo’s title as “the father of Physics” comes about because of his laying the foundations of much of mechanics, which Isaac Newton later formalised in a more mathematical framework. Whilst he was still training for a Medical degree, he noticed in 1581, that the period of oscillation of a chandelier did not depend on how large the swings were, but only on the length of the pendulum. The story is that this observation was made in Pisa Cathedral, where a chandelier was being blown by the wind to swing with different amplitudes. Galileo used his pulse to measure the periods and saw that they were all the same, irrespective of the amplitude of the swing.

After building his own telescope in 1609, based on the design of one built in The Netherlands about which he heard, Galileo made some crucial observations which helped show that the Earth and the other planets orbit the Sun, contrary to the popular view of the time. Key to showing the veracity of the Heliocentric model were Galileo’s discovery of four moons orbiting Jupiter, and his observations of Venus.

Once Galileo had showed that Jupiter had moons orbiting it, it strengthened the argument that not everything went around the Earth. However, Jupiter’s moons still allowed for the planets, the Moon and the Sun to be orbiting the Earth (the Geocentric model), but with Jupiter having its family of moons. However, when Galileo turned his telescope on Venus he was able to see that the planet exhibited all phases, from crescent to full, just like our Moon. Not only this, but the size of Venus when full was smaller than when it was crescent. It is impossible to explain these observations if both the Sun and Venus are orbiting the Earth. If one has a model where Venus is orbiting the Sun and both orbit the Earth it is possible to explain the observations, but the size difference between the crescent and full phases of Venus argues against this interpretation. In a model where Venus orbits the Sun inside of Earth’s orbit about the Sun the observations are explained perfectly in the most simple fashion.

Galileo argued against Aristotle’s view that heavier objects fell more rapidly towards the ground. It is not clear whether he actually conducted the experiment, but he argued that two objects of different weights would fall at the same rate. He introduced the concept of inertia, which is the tendency a body has to either stay at rest or keep moving once it is set in motion. Galileo’s Principle of Inertia stated: “A body moving on a level surface will continue in the same direction at constant speed unless disturbed”. Newton’s first law of motion is essentially a statement of the concept of inertia.

Galileo also made many other contributions to Physics including attempts to measure the speed of light using lanterns, and developing thermometers and better compasses. He was one of the first scientists to realise that much of physics can be explained mathematically.

Personally I think there is little doubt that this pioneer of experimental physics deserves to be in this list of the ten best physicists. What do you think?