Hi I'm Toby and welcome back to another video. Today I'd like to show you through
some of the early works of Isaac Newton. These are very important contributions
to science and mathematics made while Newton was quarantined at home during
the Great Plague of London. This was a plague that went from 1665 to 1666, it
was spread by fleas and killed around a hundred thousand people over the course
of 18 months. Newton had been studying at Cambridge University and had just
obtained his degree in 1665 when the university had to close as a precaution
against the plague. So Newton returned home to Woolsthorpe Manor and over the
next two years there developed his theories on calculus, optics and
gravitation. It is at this home where the now famous apple tree is located. 1666
was Newton's Year of Wonders, he was aged 23 at the time and he later said
that "in those days I was in the prime of my age for invention and minded
mathematics and philosophy more than at any time since". In this video I would
just like to focus on the work Newton did during this time on the topic of
calculus and these pages are printouts from the Cambridge University digital
library so the links for you to see the originals yourself will be in the
description. This first document written by Newton and dated October 1666 has
come to be called the 1666 Tract on Fluxions. It is the first collection of
Newton's ideas on calculus. It is dated in a different ink so this date could
have been added later on and there was quite a rivalry between Newton and
Leibniz over who first described calculus but regardless this document is
a collection of Newton's ideas since 1665 which can also be seen scattered
through his earlier notebooks. It says to resolve
problems by motion these following propositions are sufficient. He then lays
out a bunch of propositions and does some examples. To Newton calculus was all
about motion and he was trying to understand motion because the physics
problems he was also working on required some kind of mathematical base such as
for gravitation. Neither Newton nor Leibniz were the first to completely come
up with calculus they were both building upon the work
done by earlier mathematicians such as Descartes. Let's work through this
demonstration here ourselves to see what was going on. We're going to think about
the motion of a moving particle so at one moment in time we could describe the
position of the particle with an x and a y. These coordinates Newton would later
go on to call fluents as they are flowing quantities. What
we want to know is how we would describe this particle after a small period of
time. So if it has moved to here we would describe its X position as being X plus
some new quantity and likewise in the Y direction. To find how much it has moved
we would find the velocity of the particle and multiply it by the small
change in time. Newton refers to velocities in two
directions as being P and Q and he uses O to denote a very small infinitesimal
change in time, so in the X direction we would write this as P times O. O is not
equal to zero but it is very small so after this change in time our X has gone
to X plus PO and our Y has gone to y plus QO.
Instead of using P and Q Newton would later go on to use X dot and Y dot as
the velocities and he would call these the fluxions but that would have to wait
until a 1671 rewrite of this work. We can see it in Newton's notes here he says "if
the described lines be X and Y in one moment they will be X + PO and Y + QO in
the next. Newton then does a demonstration where the relation between lines X and Y
is X cubed minus A B X plus A cubed minus D Y squared equals zero. He says "I
may substitute X plus PO and Y + QO into the places of X + Y." To expand something
like this he would have used the binomial expansion theorem which he also
had a hand in developing. I will denote these zeros here with a line through
them to make sure that they aren't confused with these O's in here which
are not equal to zero. Newton then writes out the result of
this expansion and he says that since we know X cubed minus ADX plus A cubed
minus D Y squared was zero we can find those terms in our expansion and cancel
them out and then he writes the result of doing that. The next step is to divide
every term by O and were able to do that since O is never actually equal to
zero and this is what we get. Newton then says "also those terms are infinitely
little in which O is, therefore omitting them there rests" and he gives the
solution. So if we cancel all of our terms with an O in it as
being so small that they're not going to contribute then we get the same solution
that Newton gets and it is our time derivative of what we started with.
Remember that this is a derivative with respect to time so it's like we had
differentiated this implicitly. X cubed has become 3X squared times P which is
dx/dt. That is exactly what you would get differentiating this and taking into
account that X and Y are functions of T. This technique was a little
controversial because O is treated both as being not zero and close enough to
zero to cancel out and nowadays we use the notation of limits to make it more
rigorous. We would say that in the limit of O going towards zero this term would
go towards zero. Now this whole demonstration here was actually a
demonstration of proposition seven so why don't we have a look back at
proposition seven to see what it was. Proposition seven starts here and it
talks about "having an equation expressing the relation between two or
more lines X,Y and Z and the relation of their velocities P, Q and R may be found
by putting all the terms on one side of the equation so that they are equal to
nothing and first multiply each term by how many times P over X as X have
dimensions in the terms." Let's work through what he's doing here. We're going
to start with the same equation that we had before and we'll find its derivative
this way which is the same as saying the relation of the velocities. So we times
each term by P over X for how many dimensions of X it has so for example
this first term has three dimensions of X so we will do X cubed times three
times P/X. The second term has one dimension of X.
The third and fourth terms have zero dimensions of X so we times it by
zero times P over X which is the same as multiplying it by zero so I'm not going
to write it down. He says that "secondly, multiply each term by how many times Q
over Y as Y has dimension in it." So we do the same thing now for Y. The first three
terms have zero dimension of Y so they would be multiplied by zero but this
last term has two dimensions of Y so he would times it by two times Q over Y. If
you had a third variable you would keep going.
He says the sum of all of these products shall be equal to nothing, yep so it's
equal to 0, and the equation gives the relation of the velocities. So if we tidy
this equation up this X will cancel with one of the X's up there and we will get
3X squared times P. These X's here will cancel we'll get A B P. One of the Y's
will cancel we'll get minus 2 D Y Q equals to zero and indeed this is the
time derivative of what we started with. It was a quicker method than the one
we just did before but it's the same result. I find that really awesome that
we were able to actually follow along with Newton as he's doing some
derivatives. We can see at the end here that he gives us a general form for
doing this. Later in the document he goes on to apply these theorems to resolving
of problems you can see here he's drawing tangents to crooked lines and
over here, problem five is to find the nature of the crooked line whose area is
expressed by any given equation. That is the nature of the area being
given to find the nature of the crooked line whose area it is. All this talk
of area might remind you of integration and Newton does go on to really talk
about the link between differentiation and integration and finding the area
beneath a curve. He seems aware that these two things are
inverses and that is the fundamental theorem of calculus. This is another
document written by Newton but at a later time and I just wanted to show it
to you because it shows that he does later go on to use the terminology
fluxions where instead of using P and Q for the velocities he uses X dot and Y
dot but the rest of the examples here are the same. Fluxions were kind of
central to the whole Leibniz-Newton controversy and in fact the last
document I would like to show you is a letter in Latin that Newton wrote to
Leibniz via another middle man called Henry Oldenburg and this was dated the
24th of October 1676 and in it Newton explains fluxions but
actually conceals his words in code because he was so suspicious of Leibniz.
Around here he says "I cannot proceed with the explanation of the fluxions now,
I have preferred to conceal it thus" and then he puts this random string of
numbers and letters and it looks a bit like a Bitcoin address but actually it's
an enciphered phrase and this phrase here defines the relationship between the
fluent and fluxion. Now what he did to get this, was he took
the phrase in Latin saying "given an equation that consists of any number of
flowing quantities to find the flexions and vice-versa" and he counted the
frequency of each letter and ordered them alphabetically. If there were three
or more of a letter then he would put a number
in front so it looks like there were six A's and nine N's. Even if Leibniz could
decode this, the phrase doesn't seem to show much, although it must have given
Newton some kind of security over the idea. The fact he could put it down on
paper and make a claim in this fight about his ideas being stolen. This issue
of ownership became so complicated because Newton's work on fluxions and
calculus and many other things wasn't published until a long time after he
wrote it. Even Leibniz's work wasn't published straight away. I thought
this little document here would be a nice insight into the relationship
between the two men. So that's been a look at some of Newton's original work
even as a math student I don't think I had really seen much of his original
thinking so I hope you found this video a little bit interesting. You can go to
the links in the description to the Cambridge University digital library to
see these documents as well as many more about Newton. This video was made
possible by my viewers who support me on patreon so if you'd like to help me
continue making videos like this then please consider supporting me there too.
It's patreon.com/Tibees. Thank you to my current patrons and a
special mention to today's Patron Cats of the Day, Ghandalf the Gray and Beau.