Monthly Archives: August 2000

The begginning of Calculus

Today morning I tried to “discover” the history of calculus and searched the web . I expected to find something starting from Newton and Leibnitz .

But I was wrong . The history or more appropriately the idea dates back to the Greeks ! These dudes were crazy .

 

      After inventing the naural numbers ( 1,2,3 etc ) the greeks thought

of more interesting problems . The Greek philosopher Zeno proposed that

motion was impossible . He said that to go from a place A to B you would

first have to reach the midpoint of AB . Lets call this C . Inorder to

reach C from A you would first have to reach the midpoint of AC etc .

Thus there would be infinite midpoints between A and B and u can never reach

B ! The proposition was crazy but the idea was great . This was the first

time someone had thought of infinities , infinite points between 2 points and

infinitisimal distances .

 

      In 370 BC some Greek Mathematicians thought of a tool called

” The method of Exhaustions”  . The method is : to find the area of a

figure put smaller figures in it to completely fill the given figure.

This will help you find the area .

 

      This method was first used by Arichmedes to find the area of

a parabola. He put a triangle in the parabola . The filled the remaining

area with smalller triangles and thus calculated the area .

 

      So historically integration was invented before differentiation .

 

      In the 16th century Kepler and another dude Cavelieri thought of

any area composed of an infinite no of infinitisimal rectangles .

 

      Fermat invented a method to solve problems related to maxima and

minima of curves . This was the beginning of differentiation .

 

      However the real inventors of Calculus as a subject were

Newton and Leibnitz . They independently combined all the abovementioned

tools/methods to form a subject . They were the first to realize that

differentiation and integration were opposites of each other . They

were the first to form a notation for differentiation and integration .

 

      Newton actually invented calculus while trying to make his

theory of gravitation . He had an intution that the mass of a sphere can

be assumed to be concentrated at the centre to find the gravitational force

outside the sphere  . But he needed a proof . He invented calculus and

got the proof .

 

      There was a bitter rivalary between Newton and Leibnitz over the

question of who invented Calculus . Both parties fought long and hard .

They attacked each other personally . In the end England cut itself

off the mathematics of the continent . The English mathematicians refused

to use the dx/dt notation for differentiation and lazy S notation for

integration because they were invented by leibnitz .

 

      Meanwhile in the continent during the 18th and 19th century began

a age of great mathematical discoveries powered by this great technique

called Calculus .

How we keep balance

 

      Today’s VERITAS is inspired by a Discovery Channel program that I saw on Saturday  .

 

      Have you ever wondered as to how we keep our balance ? How do we always know what position to return to even after making sharp turns . How do we know even while playing games like foot ball , how much to bend so that we dont fall .

 

      3 organs/parts of the body are responsible for helping us maintain this balance :

 

1) The muscles : They signal to the brain how much thay have streched

   and thus knowing from experience how much they have to unststetch .

 

2) The eyes : Gives us our spatial position wrt to the horizion/surroundings

 

3) The inner ear : This organ is the most important organ for balance .

   There are 3 semicircular canals and two other structures ( utricle and

   saccule ) in the inner ear and they help us understand the tilting

   of our head . This system ( called Vestibular system ) consists of

   fluid and sensory hair cells .

 

   As our head moves the fluid moves and makes the hair cells bend 

   and send signals to the brain about the position of the head .

   The signals may be of orientation , rotation , starting/stopping.

 

   The brain then coordinates all these 3 information and knows what

   position we are in . The brain then sends signals to the eyes to

   move appropriately to keep focus.

 

   All this happens very fast . Imagine a footballer who bends down to kick a

   ball keeping his eyes on the ball and then bends sideways to dodge a

   opponent . Talk  about computing speed !!  

 

   As you can see the sense of balance depends on gravity . The normal position

   of the hair cells in the ear is in the direction of gravity . So that

   is the reason that Astronauts in space loose their sense of balance after

   some time in weightlessness . Their brain no longer relies on the ear

    to understand the head’s position. The brain is confused by the arbit

   info given by the ear and does not trust it anymore .

 

   So after the astronauts return to earth it takes their ears some time to

   adjust to gravity and balance themselves  properly .  

Sun Spots and Solar Cycle

 

      Our sun is a raging inferno . Yet it too has a weather . There are places on it which are cooler or warmer than the areas around it .

In 1610 Galileo first observed some dark spots on the Sun . Since then regular observations have been made of these Sunspots .

 

      Sunspots are areas on the SUn that are relatively cooler . The “Sunspot Number” is a count of the number of sunspots seen on a particular day . On July 21 2000 when I am writing this VERITAS the sunspot no is 342 . You can see it at the SOlar Physics site of NASA .

 

      The Sunspot Number gives us an idea of the amount of activity going on in the Sun . The higher the sunspot number the more the SOlar activity .

Scientists have observed that the “Little Ice Age ” that happened  between

1645 to 1715  AD coincided with a lack of sun spots in the sun during that time . The rivers froze and earth was chillier than normal during this time.

Maybe the lack of sun spots ( lower Solar activity ) means ice age on earth .

 

      In 1859 Schwabe noticed a pattern in the sun spot numbers over the years . The number of sun spots goes through a 11 year cycle .

In 1989 there was a maximum and it dropped off to nearly zero in 1995 and the next maximum is this year ( 2000 AD ) .

 

          How are these sunspots formed ? Theorists believe that the formation of sunspots is related to changes in Sun’s Magnetic field at certain places . The sun’s magnetic field gets twisted at some places and sun spots occur . This shear of Magnetic fields causes periodic release of energy . This release of Energy is called SOlar Flares . They dont last long . In a matter of a few minutes a Solar Flare can release energy equivalent to a billion megatons of TNT . Matter is heated to millions of degrees and flung out of the sun . When this matter( particles) strikes the Earth’s Atmosphere they cause sudden weather changes and problems in cummunication systems . People travelling in planes can get as much radiation as a Medical Xray. The Solar flares are also the cause of Aurora Boleris ( Northern lights ) .

 

      Visit the site www.sunspots.com for more info . It is a HOT site !

This site also contains archives of sun spot number averages from 1755 to the present .