In view of what we discussed about the study of Astronomy chronologically, it may be better to study it pedagogically. Of course we will take detours into historical alleys that may interest us. But by and large we will learn astronomy as humans would have in the absence of brilliant minds making extraordinary leaps of intuition and imagination. But before we get involved in astronomy proper there are some linguistic pointers that we need to clear. The need for this special language is quite simple. Words in any language we are acquainted to have special significance. To the extent that many have asserted that physical reality owes its existence to our ability to conceptualise it, i.e., assign words and meanings to it. Renè Descartes felt that the world of ideas and physical realities were forever divorced. Followers of George Berkeley thought that when an object wasn't percieved it didn't exist (viz. the world dissappeared when you close your eyes). He specifically states that he didn't believe this himself and he shouldn't be thought of as having ``fathered'' this concept. This debate has a long tradition and is another one of the many debates between theologists and scientists. You can see that issues like free will are at stake here. However most scientists sidestep the issue altogether by using an entirely different language that has no excess baggage of connotations associated with it. Fortunately for us this is a very rudimentary language and we can hope to pick it up without much effort. But it is important to pick it up before we venture further.
Renè Descartes (b. 1596) is regarded as the originator of modern
analytic science. He was the son of a minor nobility and had been left
enough money to be a philosopher. He left France and went to
Netherlands because he felt a greater freedom working there. In the
preface of his great book, Discourse on Method, he states that
he didn't think himself particularily clever, but wrote the book
anyway because he felt the one thing people shared equal amounts of
was good sense and presented with proper questions could analyse with
equal ability. He felt the reason for people arriving at different
answers after their analyses was only a matter of personal taste and
access to information. He wrote the book in french so as to reach a
largest possible audience. It was the first serious book on
Metaphysics that was not written in Latin. To see how radical this was
remember that this was when Cardinal Richelieu (The Red Eminence) was
in power in France and had declared war on progressive thinking of any
kind. In this extraordinary book, which discussed Optics, Meteorology
and Analytic Geometry, he presented conventions that have been used by
scientists ever since. One is to represent known quantities as 
and unknown quantities as 
. Another was the
use superscript for powers, 
etc. In Analytic Geometry he
initiated a concept that is much more than a useful convention. He
showed that it was very useful to give it some concrete meaning to the
words here and there particularily when describing geometric shapes,
like a line. Choose two straight lines that were at right angles to
each other. They are called axes and the point they meet is called the
origin. By convention the horizontal axis is called x axis and the
vertical y axis. Then along each axes at uniform distances we can
put in tickmarks. This allows us to map the surface we drew the
axes on, with a grid that we can make as fine as we like. Then every
point on that surface can be described by the label of the grid cell
it lies in. These labels 
are called the coordinates
of the cell. You can immidiately see the beauty of this simple
system. Any curve on the surface is immidiately characterised in terms
of the sequence of grid cells it crosses. A curve is the list of
values of y for each x value of the cell the curve crosses. We've
connected algebra and geometry! Any curve on the surface is identical
to the algebraic connection between x and y. The particular
connection between y and x for a particular curve is called a
function. So a circle centered on the origin is the same as the
function 
, where r is the radius of the circle. For
a general curve this is written as y = f(x). This also allows us to
define continuity of a function. If the geometric curve
corresponding to the algebraic function is continuous, so is the
function.
Consider a curved line C, as we learnt in the previous section
this is same as the function y = f(x). Imagine magnifying a little
piece of that curve, and then magnifying a piece of that that
magnified piece, and so forth. As we get smaller and smaller pieces of
the curve we will end up one that looks straighter and
straighter. This is why the Earth looks flat to us, even when we know
it is a sphere (roughly speaking). Because we are looking at such a
small piece of it, that the curvature in it is not discernible to
us. So the curve C can be approximated by many little segments
of straight lines put together end to end. The smaller (and more
numerous) these straight segments the better the approximation (in the
sense of smaller deviations) to the curve. But the function associated
with a straight line is particularily simple, 
. This means we can describe the curve C by
the list of slopes and intercepts of all the little segments we
approximated the curve with. In fact the intercept is redundant,
because note that for each segment, it is given by the end of the last
straight segment! This means to characterise a curve (i.e., a
function), we need the coordinates of one end of the curve 
and the slope of all the little segments making up the
curve. This list of slopes when the segments are infinitsmally long
are called the derivatives of the curve and denoted as
. What does infinitsmal mean? Think of Zeno's
Achilles paradox to understand this. Zeno was a Greek philosopher who
lived around 450 BC who Aristotle named as the inventor of
dialectics. He gave several paradoxes that weren't explained until
modern concepts of continuity and infinity were understood. The
paradox concerns a race between the fleet-footed Achilles and a
slow-moving tortoise. The two start moving at the same moment, but if
the tortoise is initially given a head start and continues to move
ahead, Achilles can run at any speed and will never catch up with
it. Zeno's argument rests on the presumption that Achilles must first
reach the point where the tortoise started, by which time the tortoise
will have moved ahead, even if but a small distance, to another point;
by the time Achilles traverses the distance to this latter point, the
tortoise will have moved ahead to another, and so on. The solution of
course, as Aristotle pointed out, has to do with the ground being a
continuum. We can break up a finite distance into an infinite number
of intervals of infinitsmal length. This doesn't make the distance suddenly
go from finite to infinite. As we said before the smaller the segment
the better the approximation the segments make to the actual
curve. But after a certain point, to all practical purposes there
isn't any discernible difference to the approximation made by the
segments to the curve, by reducing the size of the segments. The slope
of the segments at this point can be called the derivative of the
curve in the location of the segment. Now say the segment is
characterised by the label of its midpoint (x, y). Then there is a
number equal to the derivative that can be related to
each value of x. But recall the definition of a function as a list
of numbers (or y labels) for each x label. Then we can say the
derivative of a function is also a function! And we can draw the
geometric curve corresponding to this function. If this curve,
corresponding to the derivative, is continuous then the function is
called differentiable. Now of course we can do this whole
process for the function corresponding to the derivative of the first
function, ad nauseum.
Now think back to the connection of a function y = F(x) and its
derivative, . Let us write the derivative as a
function, y = f(x), it is clear f(x) in fact refers to dF(x)/dx
(which is used interchangebly with dy/dx). The connection between
the function f(x) and its ``parent'' function F(x) is called an
integration and is denoted as 
. So you can
think of an integral as an anti-derivative. Note that if we added
any constant c to the function F(x) we'd still have the same
derivative f(x), because the slope of a straight line parallel to
the x axis (which is the curve corresponding to the function y =
const.) is zero. This means that if we know only the derivative f(x) we can
find the integral F(x) upto an arbitrary constant. This corresponds
to what we said in the previous section about needing one of the end
points as well as the slope of a curve to characterise it. This is why
this integral is a called an indefinite integral. If however we are
interested in a particular function (and not upto an arbitrary
constant), we need to evaluate the definite integral,
where a and b are the x labels for the end points of the curve
F(x) we are interested in. The method of solving these was found by
Riemann.
Bernhard Riemann was born in 1826 in Hanover to a family of a Lutheran pastor, and lived an exemplary life that would be a joy to any Gingrichite. Working on an pittance he got from the charity of his students he produced an extraordinary body of work that has been profoundly influential on Mathematical Physics. He finally got a permanant post in 1859 at the age of 33, married in 1862 at the age of 36 and died in 1866 at the age 40. So of a total of 40 years he lived 7 in some degree of security and comfort. In exchange we got off him
Here we are interested in the Riemannian integral. He said, take the
function f(x) and break up the interval [a,b] in x into many
subintervals of size 
centered on the list 
. Incidentally the square brackets in the interval [a,b]
means that both a and b are included in the interval. This is
called a closed interval. Then we have the list of values,
, of the function f(x) evaluated at each 
. The definite
integral is given by
when the interval is infinitismal and N infinite, again
in the sense we used to define derivatives. But 
is the
area of the rectangle of width and height . But this
rectangle is roughly the area between the curve f(x) and the x
axis in the interval of width centered about . So you
can see the definite integral is equal to the area between the curve
f(x) and the x axis.
The traditional way of looking at vectors has been to think of an
object that possesses direction and magnitude. However we will find it
more useful to think of them in a different way. They are essentially
equivalent but the second way allows us manipulate them more
easily. Think of three numbers, 
, denoted by the symbol
. This is a vector (in 3 dimensions, in two dimensions there
would be two numbers instead of three), provided it obeys some
rules. For two vectors 
and 
these rules are:
, implies, 
, 
& 
.
, where, 
.
, where,
. Note that s is a single
number and hence not a vector. It is said to be a scalar, hence scalar
multiplication. This is also known as dot product.
, where,
. Obviously this is called a vector product because the result is
a vector. It is also known as a cross product.
.
. Find the grid cell with that label. Now join
this cell to the origin. You can now visualise the vector as the line
joining the origin and cell labeled by 
and in the direction
going from the origin to the cell. It is clear how the addition works
out. Another useful concept is the realization that if we take the two
vectors 
and 
, all two dimensional
vectors can be written as the combination 
. Note that in the former sense of vectors 
is the unit
vector along x axis and 
the unit vector along y
axis. They are at right angles to each other as 
. They are said to be orthogonal to each other.
Once again consider a curve in the two dimensional surface. But this
time think of the curve as the path of a body moving on the
surface. This is called the trajectory of the body. Provided the
body can't leap off the surface the path will be continuous (in the
sense we spoke of before). Say at some time 
the body is at the
position . After time 
, at 
, say it is at
. The line connecting and
clearly has some magnitude and a direction. So the displacement of the
body is a vector, 
. This is also obvious from the fact that
is a set of two numbers, our other definition of
a vector. Now imagine another two dimensional surface, one axis of
which time and the other x, with the origin at 
. Then
the path of the body in the x-y plane corresponds to a particular
curve in the t-x plane. This curve on the t-x plane will also
be continuous. So we can evaluate the derivative dx/dt of this curve
as we learnt before. This derivative is clearly the rate of change of
the x coordinate of the body with time. Similarily we can find the
rate of change of the y coordinate of the body. Now return to our
original two dimensional surface. We have then at each point of the
body's trajectory, two numbers 
, i.e., a vector (where we
have used the names 
and 
to stand for the derivatives
dx/dt and dy/dt respectively. This vector is called the velocity
of the body. The individual numbers and are called the
x and y components of the velocity, respectively. Now as we did
before, say we treat these velocity labels as coordinates in another
two dimensional surface (called the velocity surface). Then the body
traces a path in this velocity surface as well. And we can again find
the rate of change of the velocities with respect to time. We
will get again two numbers 
. This vector, written
as 
is called the acceleration vector.