April 10, 1997
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Calculus
Sir Isaac Newton
Born: 4 Jan 1643 in Woolsthorpe, Lincolnshire, England
Died: 31 March 1727 in London, England
Newton laid the foundation for differential and integral calculus. His work on optics and gravitation make him the greatest scientist the world has known.
Issac Newton (1642-1727) was like perhaps only Archimedes and Aristotle before him', a person off the scale of normal genius. He was one whose ``shaped the categories of the human intellect''. It is not possible to measure Newton in any ordinary sense.
If he had not invented calculus - as he is ascribed to have done - he would still be one of the great thinkers of all time.
His career included contributions to:
In the opening sections of the Principia Newton had so generalized and clarified Galileo's ideas on motion that ever since we refer to them as ``Newton's laws of motion."
Then Newton went on to combine these laws with Kepler's laws and with Huygens law of centripetal motion to establish the unifying principle in the universe that any two particles attract each other according as the inverse square law of distance.
This had been anticipated by Robert Hooke as well as Edmund Halley. But Hooke's concepts were intuitive. Newton convinced the world by carrying off the mathematics needed for the proof.
In 1693 Newton has a nervous breakdown, after which he substantially retired from research.
He was also Master of the Mint following the publication of the Principia. He took an active interest in his duties and became the scourge of counterfeiters, sending many to the gallows.
In 1703, he was elected president of the Royal Society and assumed the role of patriarch of English science. In 1705 (08?) he was knighted, the first scientist so honored.
Over the years he had furious debates with other scientists, notably Robert Hooke and John Flamsteed.
It is generally agreed that Newton developed calculus before Gottfried Wilhelm Leibnitz seriously pursued mathematics. It is also agreed that Leibnitz developed it independently. Leibnitz published in 1684.
A fracas of priority of discovery developed into a small war. Newton was drawn in; and once his temper was triggered by accusations of dishonesty, his anger was beyond constraint. Leibnitz's conduct though not pleasant, paled beside that of Newton. Said his assistant Whiston:
Newton was of the most fearful, cautious and suspicious temper that I ever knew.
Newton's mathematical works include:
Newton's work on the binomial theorem is nothing short of remarkable. He begins, as did Wallis, by making area computations of the curves , and tabulating the results. He noticed the Pascal triangle and reconstructed the formula
for positive integers n.
Now to get to compute , i.e. n=1/2, he simply applied this relation with n=1/2. This of course generated an infinite series because the terms do not terminate.
Next he generalized to function of the form for any n. This gave him the general binomial theorem - but not a proof.
He was able to determine the power series for by integrating the series for , written according as the binomial series. In modern notation, we have
Now integrate to get the series
With this he was able to compute logarithms of the number , , , to 50 places of accuracy. Then using identities such as
he was able to compute the logarithm of many numbers.
Next he worked out the power series for , and ultimately found the power series for using his method of affected equations. The reason for this apparent reversal of what we would think to be the order of discovery is that
Thus the binomial series and integration term-by-term could be applied.
The confirmations he achieved using his power series method justified in his mind the ultimate correctness of this procedure. But convergence?
Newton was unconcerned with questions of convergence.
Newton developed algorithms for calculating fluxions defined in modern terms as
to solve the problems:
He assumes a form f(x,y) = 0 and produces the differential equation
using the procedure of Hudde. His method builds into it the product rule for derivatives.
He justifies this rule by defining the moment
substituting and resolving the terms àla Fermat. Note the term o is viewed as infinitely small.
At this time infinitesimals have been completely accepted by some while wholy rejected by other. That is, the infinitesimal is a real object, not a potentiality or convenience of expression!!!!
There is, I must emphasize, no theory of any of this infinitesimal analysis. Mathematicians are ``flying about by the seat of their pants", just doing it, and not all worried about the grand Aristotelian/Euclidean plan.
To resolve the ``length of space" question, Newton reverses the procedure if possible. This is an antiderivative approach. Otherwise he resorts to power series.
Example. Consider the equation
is resolved as
Applying the binomial theorem we get for the plus root
Hence one solution is
The other is determined similarly.
Newton discovered a method for finding roots of equations which is still used today.
Among the curves worked on by Newton were the Cartesian ovals, the Cissoid, the Conchoid, the Cycloid, the Epicycloid, the Epitrochoid, the Hypocycloid, the Hypotrochoid, the Kappa curve and the Serpentine. Newton gave a classification of cubic curves.
Newton gives methods of finding extrema problems normals, tangents and areas.
The concept of limit appears in the Principia as the ``ultimate ratio of evanescent quantities'' which is similar to our own notion of limit of a difference quotient. He goes to some effort to assuage the great bulk of mathematicians still wedded to Greek geometry and thought.
By studying the finest work of the time Newton was led to important new syntheses. To develop them fully he acquired a mastery of analytical techniques unsurpassed in his time. Thus he was able to derive simple and general methods compared with the laborious work of his contemporaries. Newton thought analytically in the modern sense. This was an enormous advantage.
Gottfried Wilhelm von Leibniz
Born: 1 July 1646 in Leipzig, Saxony
Died: 14 Nov 1716 in Hannover, Hanover\
Leibniz developed the present day notation for the differential and integral calculus. He never thought of the derivative as a limit.
Gottfried Wilhelm Leibnitz (1646-1716) did not pursue mathematics seriously until 1672 when he studied with Huygens in Paris.
As a diplomat he made two trips to London, in 1673 and 1676, where it is possible he had access to Newton's manuscript.
Only ten years later he began to publish short pieces on calculus.
Leibnitz's earlier career had been devoted to philosophy and received a doctorate in 1667. His original idea was to work out an algebra of human thought, an attempt to symbolize thought and to work out a combinatorial calculus.
Leibniz founded the Berlin Academy in 1700 and was its first president. He became more and more a recluse in his later years.
Leibnitz' Mathematics
His first investigations were with the harmonic triangles H.
From this he noticed that
This means that sums along diagonals of H are sums of differences. So for example
Also,
Multiplying by 3 we sum the pyramidal numbers
The importance of these ideas rested with their applications of summing differences in geometry. That is, he sees the possibility
where
Leibnitz interpreted the term as area
(i.e. ). This gives in principle his fundamental theorem.
By 1673 he was still struggling to develop a good notation for his calculus and his first calculations were clumsy. On 21 November 1675 he wrote a manuscript using the notation for the first time. The symbol was an elongated S, which of course stood for sum.
In the same manuscript the product rule for differentiation is given. The quotient rule first appeared two years later, in July 1677. Leibnitz was very conscious of notation. He recognizes two separate branches.
Leibnitz' clarity of differencing was applied to the difference triangle, which is the one we use today. From it he derives the sum, product and quotient rules, at first erroneously. It is
and not
as he originally thought.
In 1684 he gives the power rules for powers and roots. The chain rule is transparent from his notation
In 1684 he solves a problem posed by Debeaune to Descartes in 1639, that being to find a curve whose subtangent is a constant:
Leibnitz takes dx=1 and gets ; that is, the ordinates are proportional to their increments. So the curve is logarithmic (``exponential'' in modern terms).
In 1695, he computes the differential of where y and x are variables. With Jacques Bernoulli's suggestion he solves this by taking the logarithm of both sides.
Hence
Leibnitz develops a fundamental theorem: One can find a curve z such that dz/dx = y. It is given by
By 1690 Leibnitz has discovered most ideas in current calculus text books.
Leibnitz was more interested in solving differential equations than finding areas. Among them he derives and solves the familiar differential equation for the sine function. He developed the separation of variables method.
Among the curves worked on by Leibniz were the Astroid, the Catenary, the Cycloid, the Epicycloid, the Epitrochoid, the Hypocycloid, the Hypotrochoid, the semi cubical parabola and the Tractrix.
SUMMARY
Our modern calculus resembles that of Leibnitz far more than Newton. Possibly because of Newton's reluctance to publish Leibnitz's version became better known on the continent. Leibnitz's calculus was somewhat easier to comprehend and apply. This cost English mathematics almost a century of isolation from the continent and the resulting progress implied.
First Calculus Texts:
L'Hospital, Analyse des Infiniment Petits four l'intelligence des lignes courbes, 1696 He makes fundamental statements in the beginning of his text that make clear that he assumes infinitesimals are real objects, though arbitrarily small.
Humphrey Ditton (1675-1715) An Institution of Fluxions, 1706
Charles Hayes (1678-1760) A Treatise on Fluxions, 1706.