Fluxions and Dragons:
Both Newton and Leibniz are credited as the inventors of
Calculus. We know that Newton came first and that Leibniz
published it first. In Newton's biography "Never at Rest"
by Westfall we read that Leibiniz could have had a brief
to Newton's work because Newton's editor, angry at the
master (Newton had a difficult character) could have
shown the piece to Leibnizm without Newton knowing this.
Who knows really? Was a brief peek at Newton's work enough
for Leibniz? This sounds very Mozart's! In any case,
Leibniz's approach is quite different from Newton's. In
style, scope, nomenclature... Quite a different beast.
History is so cumbersome. Most of the fame is attributed to
Newton. However, we all learn Leibniz's approach, at school
and even at college. Maybe because Leibniz was able to
popularize his method across Europe. Maybe because his
approach looks less dirty than Newton's...
When I knew that the calculus I was taught was Leibniz's
I thought: "Weird, but who cares, it must be a notation
thing. The important thing is to be able to calculate
things, and both methods will lead to the same results.
History is for historians, while in math and physics is
the result what we care for." How wrong I was!
I have been studying Newton's "The Method of Fluxions and
Infinite Series", published after Newton's death. My
version includes the annotations by John Colson, the
5th Lucasian (Newton was the 2nd). These annotations are
extremely useful, since Newton's explanations are,
understandably, very short. He did not understand how
dumb the rest of people are.
I prefer to call this work "Fluxions and Dragons" because
the work makes extensive use of infinite series, which
are like dragons to me. And I like that it sounds a bit
like "Dungeons and Dragons". When I explore this work
I really have the feeling of going through tunnels and
dungeons full of very exotic dragons. By the way, dragons
are always treated as monsters that need to be killed.
Please, stop killing dragons, even in fiction!
I am still through the middle of this study and already
found many fascinating things. In the first part you learn
how to deal with series, how to manipulate them in
astonishing ways. When you are used to closed forms,
Leibiniz style, you find very strange such use of series.
A closed form is for example sin(x), when in series form
you would write
x - x^3/3! + x^5/5! - &c
where &c is Newton wrote etc (et cetera, literally
and the rest). How inelegant! An infinite dragon that is
so difficult to manipulate! I was taught to think that
sin(x) was the true form, while the series was just an
expansion to approximate the true function when needed.
But no, the series form is just a definition of the function,
as valid as the closed form. And Newton's work shows how
to think in terms of them. Sure, it is quite a dirty approach
, but it gives you a very different power. For example,
for derivatives (fluxions in Newton's language, recall
that usual names and nomenclature are almost all Leibiniz's)
you can use both methods and obtain the same results,
but while Leibniz gives you the Moses tables of derivation,
Newton gives you a much deeper understanding of how such
miraculous transformations take place, and the rules are
For integration, we still don't know how the Moses-Leibniz
look like. Are they complete? How many commandments of
integration are there? We don't know. However, Newton's
method allows you to perform rapid integration of
functions that with Leibiniz you would not know how to begin.The elegance of closed forms also has a clear downside.
I fell, once again, frustrated by how are we taught in
school and college. I had a few of very good teachers,
good in the sense that they meant well and did a lot of
effort to do their job right. The rest were simply awful.
Especially in college, where I can safely say more than
99% of all where authentic frauds. But even the good ones
were bad in the sense that they really did not know well
what they were teaching.
Because if a teacher would know about calculus, then it
(I use it as gender neutral here) would have talked about
Newton's 1st method all the time. It would also have
talked about Leibniz's method, of course. And don't forget
that Newton came up with a 3rd method of doing calculus!
A purely geometrical method which is the one he used in
his Principia. I know that there would be no time to cover
all in detail, but a good teacher should talk about these
approaches, how different powers are present in each of
them, and invite us to study them at home.
In the last years I am trying to learn from primary
sources (not exclusively, of course) and I have found
an immense stream of amazing things that are nowhere to
be found in textbooks. I am developing a kind of "hate"
to most textbooks which sell you a more "modern" view so
that they are more "accessible" and more refined, since
the most recent in science is the most desirable. This is
so wrong. What about the jewels and wonders that are
only to be found in the primary sources? And even worse:
where is the spirit of the creator? As time passes I find
textbooks more aseptic, sterile and superficial.
Allow me to give an example that I recently discovered.
It is about calculating fluxions. We all know that there
is a "rule" for that when you face a polynomial. And,
since series are concatenated polynomial terms, we should
not bother about other types of functions, since all of
them can be expressed as series. So, if we face a term
A x^n, its fluxion becomes simple A n x^(n-1). You can
explain the rule as "lower the exponent so that it
acts as a multiplying prefactor, and then lower the
degree of the exponent by one". So simple!
But wait, we are implicitly assuming here that we were
performing a derivative with respect to x here, which
is not necessarily the case. Let's follow Newton and
perform the fluxion with respect to time, and later
define time as whatever you want. A notation that is
a Newton original and that we still use it is the dot
notation, which unfortunately cannot be written here in
plain text. It consists in drawing a small dot at the top
of a variable to indicate its time derivative. We can
place more than one dot, indicating further derivatives,
always with respect to time.
In plain text we could write [x·] instead for a first
derivative, (x··) for second derivative, &c. So that
we can write for the fluxion of A x^n
A n x^(n-1) [x·]
So the rule for the fluxion can also be stated as "multiply
by the exponent (n) and then multiply by [x·]/x, which
has the same effect:
(A x^n)(n[x·]/x) = A n x^(n-1) [x·]
So far, nothing very fancy. Now fast your seatbelt because
the wonderful thing is approaching. Let's separate the
two things to be multiplied. On the one hand, we multiply
by [x·]/x. Let's keep this. On the other hand, however,
let's be more flexible in the number to be multiplied.
What if, instead of n, the degree of the term, the exponent,
we multiply by another number? Sounds weird, doesn't it?
Consider this function:
We want the fluxion (the time derivative) of it. If we
follow the rules learned at school, we do
which by the way needs the chain rule, gues who invented
that! Then we can group terms like
The same result is obtained if you follow the rule of
multiplying each term by [x·]/x or [y·]/y and also
multiplying by the corresponding exponent of the term.
Rewrite the previous function so as to have all
x exponents explicitly written:
x^3-ax^2+ayx^1-y^3x^0 = 0
Then we assume the progression of exponents as 3,2,1,0,...
and then multiply each term by [x·] and also by its
member of the progression. The Moses rule for a
derivative is equivalent to assign the 3 to x^3, the 2 to
the x^2 term, and so on.
Of course, you also need to do the same for y. First,
we write the funcion so that it looks like a progression
of y powers:
-y^3+0y^2+axy^1+(x^3-ax^2)y^0 = 0
and then assume a progression 3,2,1,0,... and assign
3 to the y^3 term, the 2 to the y^2 term, and so on.
The last step is just to add both expressions and they
will be equal to 0.
The jewel comes now: what if we follow this exact same
process but instead of assuming the progression 3,2,1,0,...
we assume another one? For example, let's assume the
progression 4,3,2,1,... and see what we get.
4x^3[x·]/x-3ax^2[x·]/x+2ayx^1[x·]/x-1y^3x^0[x·]/x = 0
Notice how we always multiply by [x·]/x and how we
multiply each term by a number according to our
progression. We can simplify to
4x^2[x·]-3ax[x·]+2ay[x·]-y^3[x·]/x = 0
Now it is time to process the y's. Should we assume the
progression 4,3,2,1,... here? Not necessarily! Let's go
really wild and assume 2,1,0,-1,-2,... So that we get
which can be polished to
Now we must collect the two expressions and add them,
which, after some arrangements, give
This is a very different result! You can ask, what if we
would assume 2,1,0,... for x and 4,3,2,... for y? Then
we would obtain
What about the general case, where we assume a progression
m+3,m+2,m+1,... for x and n+3,n+2,n+1,... for y? Notice
how we recover the "classical" result for n=m=0. But in
general we obtain the following ratio of fluxions:
[y·]/[x·] = A/B
A = ((m+3)x^2-(m+2)ax+(m+1)ay-my^3/x)
B = ((n+3)y^2-(n+1)ax-nx^3/y+nax^2/y)
We see how we can obtain as many Fluxional Equations as
we please, by choosing m and n! Of course you may wonder,
but not all of them can be correct! Wrong! All of them
are correct! As John Colson says, and I am doing an
extensive use of his annotations, "this variety of Solutions
will beget no ambiguity in the Conclusion, as possibly
might have been suspected."
I will give no proof here of this. Do it yourself if you
want, or check it with some numerical cases. The basic
procedure is just to solve x or y from the initial
equation and then to substitute in the Fluxional Equation.
You will see how the result will not depend on either m
Now, you may think this is interesting but not useful,
because with our college method we arrived so fast to a
valid solution. Why should we interested in other
solutions? Well, because PERHAPS in some cases the
Fluxional Equation obtained for m=n=0 is not the most
elegant or simple one! You don't believe it? Let's
develop the following example:
2y^3+x^2y-2cyz+3yz^2-z^3 = 0
where there are three flowing quantities. If we proceed
with the progression of the indices, the exponents
2xy[x·]+(6y^2+x^2-2cz+3z^2)[y·]+(6yz-3z^2-2cy)[z·] = 0
while if we choose the progression as to produce "the
greatest destruction of terms", which means 2,1,0,... for
x and for y and 3,2,1,... for x, we arrive to
By just observing the most suitable progression to
obtain the fluxional equation we have been able to
have two terms less!
Of course you can say this is not a big deal, since from
the first and longer expression you could recognize that
the string -2cz+3z^2 is already present in the original
equation and it is equal to (z^3-2y^3-x^2y)/y and
substitute. But this way is less elegant, less deep and
not always evident as the equation becomes more complicated.
Another example: take the equation
3x^3+xy-2xy^2 = 0
Notice how the usual method yields
Now practice yourself the other method with
a generic m for x and n for y and see how
Now, this is subjective, but having this I would choose
m=-1 and n=0 so that we obtain
It takes a little practice to choose the right indices
before applying the fluxion rule, but the results are
clearly worth the effort.
Again, you can see from the original equation that
9x^2+y-2y^2=6x^2+3x^2+y-2y^2=6x^2, but if you downplay
the importance of this method only because you can do
it with a less elegant method, then you are already
sucking the life out of the spirit of calculus and you
may be ready for modern textbooks production.
Newton's method deeply relates series, progressions of
indices and fluxions. I find this astonishing.
And Newton's text is full of
ultra-clever tricks like this that I am not aware of
them being present elsewhere. Many times I have performed
time (or total) derivatives of many expressions, and
probably I have carried too many terms simply because
none of my teachers knew that derivation had some
degrees of freedom that you can choose in advance so
as to optimize the simplicity of the result. This is what
distinguishes the good from the master. And, as I am
neither good nor master, I at least want to learn from
Today we have talked about Fluxions. Another day we will
talk more about Dragons. But before I have to keep