Notes on
Bos, HJ ( nd) 'Differentials, HigherOrder
Differentials and the Derivative in the
Leibnizian Calculus' ,
www.tau.ac.il/~corry/teaching/toldot/download/Bos1974.pdf
Dave Harris
The story so far, as far as I understand it.
There was a problem in calculating changes in
direction or slopes of lines in graphs. The
problems seemed to lie in the observation that the
line was a continuous variable, an infinite series
of points. This meant that strictly
speaking, areas and the like could not be
calculated with precision, because there was
always an area extending beyond the artificial
point to which we had given a number, and units
were fundamentally arbitrary. That was OK if we
stuck to the (rather mysterious) geometric
dimensions (the normal 3?) . However, a further
reservation meant that this would be confined only
to solid objects and their geometries—beyond that,
we had no agreed dimensions and therefore no
agreed quantities.This prevented any further
abstraction, so that points on the graph for
Descartes remained as single concrete
calculations, as it were, not more abstract
functions but concrete relations between discrete
variables.
Apparently, Descartes had attempted to tackle this
by providing a series of algebraic equations to
explain the slope of a curve in terms of variable
quantities of X and Y. This included working with
ratios between lines (line segments) which would
be precise. If we let algebraic terms stand for
those line segments, we can multiply the terms. We
do not need to use real numbers. We can substitute
real numbers for solutions, but that involves an
arbitrary operation of choosing units again.
Leibniz is going to generalize to take ratios even
further away from real numbers and geometric
dimensions, and make them into functions: he was
one of those who coined the term along with
Bernouille and Euler (incidentally, they had to
agree on terms  another was the 'integral' which
Bernouille preferred to Leibniz's term 'sum').
Originally, Euler's functions related to single
variables, later to many, another move from
geometric space. He is going to establish a ratio
between changes in x and changes in y, dx and dy.
Changes are to be expressed in an abstract way not
arithmetic, as differentials. He still has
problems though with the notion of the infinitely
small, and will lead to a replacement for
Leibniz's system  a further abstraction from the
specific calculations of differentials based in
infinitessimals  the derivative.
It seems that this dilemma shows a move from
geometry to analysis is how Bos describes it, or a
move towards the concept of a mathematical
function, where one variable maps on to another.
Mathematicians became more interested in these
generally, and stopped worrying about real world
equivalents (scholasticism  but what machinic
possibles unfolded!). Leibniz began his inquiries
with the geometry of graphs and the function
emerged. Terms become fully abstract
symbols. Leibniz had already investigated the
issue of differentials to explain sequences—the
differences between one number and another, one
solution and another. Again there are
regular mathematical formulae to explain how these
differentials can add up, and you don't have to be
much of a philosopher to realise that infinitely
small differences lie at one end and infinitely
large sums of these differences at the
other. The work was done originally on
number sequences so Leibniz had to extrapolate to
changes in coordinates on graphs  he saw that
differentials could be used to explain the use of
tangents, and sums the 'quadrature' (area under
the curve) .This helps us explain the slope of a
tangent as changes on the X and Y axes of a graph
without actually putting any values or numbers
(real numbers to be precise) on them.
What about curves representing a relation between
the Y axis and the X axis? Common sense
suggests that we can pretend these curves are
straight lines, by drawing a tangent to the curve
as a hypotenuse of a triangle (the other two sides
being distances on X and Y). A simple
tangent on a sharp curve (NB tangents
always connect two points on a curve at
infinitely small distances from each other)
will obviously include bits outside the curve,
however. To try and pin down that area we
can draw another tangent that turns that area into
a triangle, then subtract it from the first
total. Again there will be surpluses, so yet
another triangle can manage those—and so on to
infinity. Apparently, Leibniz said that what
we were doing is drawing an infinitely sided
polygon from an infinite number of tangents
touching the curve, together with the familiar
three sided square, so to speak, provide by lines
on X and Y axes connecting to the tangent.
We then have to remember we are not assigning real
values to the areas of the triangles, because the
hypotenuses are infinite, but comparing their
differences as we move round the curve: we have to
do this with continual variables. It is the best
we can do to make the infinite finite  we can at
least generate finite sequences (based on the
earlier application to numbers?). So the variable
becomes understood as an infinite sequence of
differentials, a series or progression (sometimes
Leibniz uses terms that imply progression or
growth). As we approximate more and more closely
the shape of the curve, the differences (residues)
get smaller and smaller.
So we now have two key operations.
Calculating the differences between the triangles
that we draw, the differentials, which are
infinitely small, represented as a triangle or
d. Then we have to add these differences up
to infinity, the infinitely large, to get the area
of the quadrature, using sigma, represented as the
elongated S. The change in terminology
represents a change from concrete operations of
calculating difference and adding them, to
abstract operators which can range to infinity in
both directions. As hinted above, Leibniz's
original term of summing was replaced after
conversation with these friends to that of
integration. With this abstract operation,
we also move to the idea of a function, applicable
to any range of real values, and applying equally
well in reverse, so there is no longer a
difference between dependent and independent
variables. In practice, the extremes of the
infinite were largely ignored, but an additional
problem arose in calculating 'higher order'
(secondary) differentials ('variables
ranging over an ordered sequence'), (17)
Now a really clever bit. Ratios again. Draw a
tangent and extend it down to the x axis. Make it
a triangle with the 'subtangent' (the length on
the x axis from the point at which the tangent
crosses the x axis to the point at which the y
coordinate drops from the tangent to the x axis.
The 3 sides of this triangle are known as alpha
(subtangent) Greek y (height from x axis to
tangent) and Greek r, length of tangent ( see diag
below, p. 18). The ratio between these 3 will be
the same as the ratio of the sides of the
differential triangle. I think this only holds if
we want to extend a regular curve to
infinity. In effect, the original
triangle gets infinitely repeated.
This is
obviously a version of the triangle that
Deleuze (The Fold) and Delanda get
so keen on:
Here, the ratio stays the same as we reduce
the dimensions of the triangle from E to e and
C to c etc. When we locate the line ec exactly
on A, the actual values of the lengths of the
lines are 0  but the ratio remains, fully
free of the real world. I am not sure if
the values really are at 0, though, not at the
infinitely small  perhaps it is the same
thing for all intents and purposes? We can see
the same development in Bos's diagram of
course just project the triangle further and
further to the right, into the infinitely
small. Incidentally, all equations could be
said to remain as a relation when the actual
values are zero? Maybe this is the point 
the triumph of the relation. So. we
began with a measurement problem solved by
turning to differentials and ratios. We saw
that sequences of differentials can extend
to the infinite, as infinitessimals. And
this is where we have ended  'pure' ratio.
Back to the Le9obniz version, I suppose this
does help us see the shape of curves
especially since it is about the relation
between changes. and their
analogous(?) relation to actual lengths.
Another thought strikes me  if the ratios
change will this help us explain changes in
the slopes of the curve? If Y increases, will
this not affect the slope?
I think this is what is at stake in needing to
develop the higher order differentials and
sums. The values of dx and dy can and often do
change, and, guess what, we can assess that as
a differential too  ddx and ddy. And do the
same to get dddx etc Same with sums. Again,
differentials will tend to the infinitely
small, and sums to the infintely large.
Then we get on to the relations between
differentials and sums [dear God, but
apparently we must!]. It is easy to see that
the sum of all the differentials of y is y
itself  $dy=y. But it is also true that d$y
=y ( hard to put this in normal English. If I
have glossed Bos correctly, (19) it
means that the differential/differences
between the terms in the sum of all the values
of y is y itself. Bos says we can see this
better if we think of the differences between
the items in the sum as prdued by infinitely
small quadratures in the first place  but I
still don't really get it. I think the
argument is that as we continue to add
infinitely small quadratures we get close to
the actual variable, and when we get to adding
quadratures of 0 size, we are completing this
process. I will just have to accept for
now that d and $ are reciprocals of each
other]. Horrible but perfectly logical
complications ensue,such as that the
differential and the integral themselves enjoy
a 'pure' relation, as inverses of of each
other.
I am also glimpsing a 'practical' implication,
if we accept that the practical problem was
measuring the area under complex curves. It
will be the sum of all the areas of
quadratures produced by adding tangents to the
slope of the curve. And (maybe) we can see
that sum as the reciprocal of the sum of all
the differences between points on the curve?
Back to the abstract maths. There is now an
operation called integration which 'assigns to
an infintely small variable its integral',
accepting by definition that 'the differential
of the integral equals its original quantity'
(20) (as in $dy=y above?). This is
where Leibniz and Bernouille agreed to use
the latter's term integral rather than sum (
21)This is a further abstraction from
the process of adding the incremental
quadratures. We can reinterpret this adding to
mean not just that in any particular case, an
overall quadrature is made up of the sum of
all the little quadratures, but that the
overall area of the quadrature can be rendered
as the differential of the little quadratures
that make it up. Remember that
calculating the area of the overall quadrature
is our practical aim. I don't know if we are
ever going to be able to substitute real
values for the differentials? We can quantify
relations though.
Across a finite range no doubt we could
actually measure changes in x and y etc? .
With unconfined ranges it is difficult because
we would have to keep adding areas to the
infinitely large. We have avoided this again
(made it less likely is how Bos puts it) by
thinking of quantified relations not actual
values We can manipulate these quantities in
relations. We can preserve the original
dimensions to which they refer ( so on a
conventional graph we are measuring
differences in lengths of a line etc).
Then some more stuff which I have to take on
trust. We assume the polygons ( little
quadratures) are regular and this somehow
leads us to assume that the measures of
relations are also regular  kind of nested
is how I see it, so differentials ( and sums)
stretch away to infinity compared to actual
numbers, and higher oirded differentials (and
sums) do the same for the differentials. I
think this is an assumption that helps us
manage concepts of the infinte using the same
principles An arcane dispute reported on 23
led to Leibniz proving that higher
differentials were infinitely small compared
to differentials but were still quantities.
Then  'first order differentials involve a
fundamental indeterminacy' (24) because the
formal definition of a differential as ,say a
line segment of a certain length applies to
segments of many lengths ( so the definition
is too pure as it were) , and there are also
segments of other lines, related to x which
would also agree. This was not notiiced
originally, and makes no difference for
practical calculations in normal geometry (
who cares exactly how small a segment is at
the infinitely small level, and who cares is
there are other possible mathematical lines).
A bigger problem arises though.
What sort of polygon do we draw to represent
the quadratures? Again I am forced to gloss,
but the issue seems to be that the ratios will
vary according to how we draw the polygon 
equal x sides? Equal y sides? Equal sides? ( I
must confess I don't see how the x or y sides
could be equal, but there  ah, these are
projected sides. But if we project we
are surely making the x sides artificially
equal and I can see how this will screw up the
ratios, so why do it? Perhaps we have to if we
cannot directly measure? ).
We also need to make assumptions about the
changes in the dimensions of the differential
triangle above when we get to infinite levels
( thought so). We have to assume one or all
three of the sides remains constant. These can
only be reasonable suppositions or 'choices'
,and we might wish instead to suppose rates of
change ( eg by working out the combinations of
varying each side in turn to give 18
possibilities, apparently, and assuming
uniform change) . Guess what  there are
really infinite choices and thus an infinite
range for the changes in the sides. We have to
pin it down somehow or the differentials (and
sums) will be indeterminate again, capable of
occupying any position on the range, from 0
(no alteration in change) upwards. We can only
deal with this if we consider the area of each
additional little quadrature (which will vary
if the sides of the triangle vary) ,and the
overall quadrature as the sum of them. It will
be dead handy if one of the sides only varies.
At the end of the day, we just have to assume
there is some regularity in the little
quadratures and just exclude 'anomalous
progressions'. Again this is OK for practical
purposes if the anomalies are infinitely
small. Apparently we will still have problems
with 'singularities', (the points at which the
curve changes direction?). In effect, by
assuming constant change in one variable, we
are suggesting that this can be an independent
variable ( a common procedure in stats etc) .
Again this is OK with most actual
calculations, but there are still problems in
theory (one of them turns on whether 0 is a
point on a journey to infinity  28).
Another development is required to fully deal
with indeterminacy without making assumptions
about rates of change in variables. We
have to calculate the slope of curves by
calculating differential equations, the rates
of change of the variables. Differential
equations proceed by following certain rules (
god knows where these come from  well from
Leibniz it seems) p.29, which hold whatever
concrete values apply to the progression of
the variables ( although it is still common in
practice to hold one variable constant
etc) It cut out the need for long
calculations with actual curves etc. I am lost
again, but what we have to do is take the
(conventional, Cartesian? ) formula for a
curve (the example is ay=x sqd which gives you
a parabola, where a is any number?) and apply
the operator d to both sides ( and how do we
derive that exactly? I am still stick with a
need to give a concrete value to d!). We can
get closer and closer to the shape by
repeatedly differentiating as in higherorder
differentiations, which get closer and closer
to infinity etc. We still need to specify if
the differential equation applies to x or y,
though, which still involves a choice of
holding one constant, at least at one stage.
It follows we can predict the shape of a curve
if we enter values in the differential
equations.
Oh good  there are formulas for calculating
derivatives after all ( but I don't understand
them), varying according to whether x or y is
held constant  eg y(x=sq rt ay) apparently
gives you the derivatives of x. Remember a=a
line segment or quantity of length.
Derivative is the calculation of the rate of
change. I need to keep reminding myself. I
also need a break  off to paint windowcills

