The Birth Of Calculus (1986)
Summary
TLDRThis script explores the invention of calculus by Isaac Newton and Gottfried Wilhelm Leibniz, highlighting their independent discoveries and the evolution of their mathematical methods. It delves into Newton's early work on tangents and areas, his development of a universal method for finding tangents, and Leibniz's geometric approach to areas and his introduction of integral and differential calculus symbols. The narrative underscores the significance of their work in advancing mathematical power and the automation of reasoning processes.
Takeaways
- π Calculus is a fundamental tool in modern mathematics, with key contributions from Isaac Newton and Gottfried Wilhelm Leibniz.
- π°οΈ Newton first discovered calculus around 1665-1666, while Leibniz made his independent discovery about 10 years later.
- π Both Newton and Leibniz's original writings on their discoveries are preserved, providing insight into their mathematical processes.
- π Newton's early work in calculus was influenced by his studies at Trinity College, Cambridge, and his intense work period from 1665.
- π Newton's initial approach to calculus involved improving methods for finding tangents to polynomial curves, using circles and the method of normals.
- π Newton realized the inverse relationship between tangent problems and area problems, leading to his development of methods for calculating areas under curves.
- π£οΈ Newton's work extended to 'mechanical curves,' such as the cycloid, which are defined by motion rather than polynomial equations.
- π Newton's key insight was the ratio of velocities of two points moving along the x and y axes, leading to a new method for finding tangents to any curve.
- π Leibniz's calculus was influenced by the study of area and tangency problems, and his approach was more geometric, using infinite sums of ordinates.
- π Leibniz introduced the integral sign (β«) and the differential symbol (d), laying the foundation for differentiation and integration in calculus.
- π§ Both Newton and Leibniz aimed to automate and formalize mathematical reasoning, with Leibniz focusing on creating a universal algorithm for his calculus.
Q & A
Who are the two men credited with inventing calculus?
-Isaac Newton and Gottfried Wilhelm Leibniz are the two men credited with inventing calculus.
When did Newton first discover his calculus?
-Newton first discovered his calculus in 1665 or 1666.
What was the method used by Newton to find the tangent to a polynomial curve?
-Newton used the method of finding a circle with a center on the x-axis that just touches the curve at point P, where the line from the center of the circle to P is called a normal, and the line at right angles to this normal through P is the tangent.
What was the significance of Newton's work with mechanical curves?
-Newton's work with mechanical curves, such as the cycloid, led him to a new way of looking at all curves by perceiving them as generated by movement, which was a fundamental perception of the problem and led to a new method for finding tangents.
What was the key insight that led Newton to a new method for finding tangents?
-Newton's key insight was the concept of the ratio of the velocities of two points moving along the curve, which he used to determine the direction of the tangent.
What was the significance of Leibniz's introduction of the integral sign?
-The introduction of the integral sign by Leibniz was significant as it marked the first occurrence of a symbol that remains unchanged to the present day, symbolizing the sum of all the ordinates in his method for finding areas.
What was Leibniz's approach to finding areas under curves?
-Leibniz's approach to finding areas under curves involved considering the area as made up of all the ordinates taken infinitely close together, which gave the area directly.
How did Leibniz's calculus differ from Newton's in terms of notation and conceptualization?
-Leibniz's calculus used differentials and infinite linear points of curves, focusing on a bold geometrical analogy, while Newton's calculus was more focused on the idea of motion and change in time, using the language of fluxions.
What was the impact of the calculus as described by Newton and Leibniz on the field of mathematics?
-The calculus as described by Newton and Leibniz had a profound impact on mathematics, arguably providing the greatest increase in its power since the time of the Greeks.
What was the motivation behind Leibniz's development of his calculus?
-Leibniz's motivation behind the development of his calculus grew out of his study of contemporary mathematical problems, particularly area and tangency problems, and his desire to systematize mathematical reasoning.
Outlines
π Invention of Calculus by Newton and Leibniz
This paragraph discusses the invention of calculus, a fundamental tool in modern mathematics, attributed to Isaac Newton and Gottfried Wilhelm Leibniz. Newton's discovery dates back to 1665-1666, while Leibniz made his independent discovery a decade later. The original writings of both are preserved in libraries at Cambridge and Hannover, offering insights into their mathematical processes. The narrative begins with Newton's time at Trinity College, Cambridge, where he developed foundational ideas on calculus, optics, and gravitation. Newton's work on finding tangents to polynomial curves and his universal theorem for tangents are highlighted, showcasing his progression in mathematical thought.
π Newton's Advancements in Calculus and Mechanical Curves
The second paragraph delves into Newton's further exploration of calculus, particularly his work on mechanical curves defined by motion. It describes Newton's method for calculating tangents to curves like the cycloid, which is the path traced by a point on a rolling circle. The paragraph also touches on the historical context of instantaneous direction of motion and how Newton's approach differed from previous mathematicians. His development of a new method for finding tangents by considering the ratio of velocities of two points moving along the curve is detailed, marking a significant step in the evolution of calculus.
π Leibniz's Development of Calculus and His Notation
This paragraph focuses on Leibniz's development of calculus, influenced by his studies of area and tangency problems. It explains Leibniz's geometric approach to understanding areas as composed of lines and his innovative use of the 'omnia' notation to represent infinite summations. The paragraph also describes how Leibniz introduced the integral symbol and established rules for its use, reflecting his systematic approach to mathematical reasoning. The transition from geometric to algebraic notation in Leibniz's work is highlighted, emphasizing the creation of a universal and automated process for calculus.
π Leibniz's Calculus Notation and Its Evolution
The fourth paragraph continues the discussion on Leibniz's calculus, detailing the evolution of his notation. It describes how Leibniz introduced the differential symbol 'D' for differences and how he sought to create a comprehensive set of rules for its application. The paragraph illustrates Leibniz's realization that areas were summations and tangents were differences, leading to the establishment of the foundational symbols and rules of calculus. The narrative also touches on Leibniz's pursuit of a universal automatic process for calculus, showcasing his dedication to formalizing mathematical reasoning.
π The Impact and Comparison of Newton and Leibniz's Calculus
The final paragraph reflects on the impact of Newton and Leibniz's calculus and compares their approaches. It emphasizes the similarities in their focus on geometric properties and the automation of their findings, as well as the differences in their methodologies and notations. Newton's use of motion and fluxions is contrasted with Leibniz's geometric analogy and differentials. The paragraph concludes by highlighting the immense power of calculus and its significance in advancing mathematics since the time of the Greeks.
Mindmap
Keywords
π‘Calculus
π‘Isaac Newton
π‘Gottfried Wilhelm Leibniz
π‘Tangents
π‘Areas
π‘Fluxions
π‘Differentials
π‘Mechanical Curves
π‘Integral Sign
π‘Derivative
π‘Automating Reasoning Processes
Highlights
Calculus is identified as a fundamental tool of modern mathematics, with Isaac Newton and Gottfried Wilhelm Leibniz credited as its inventors.
Newton's initial discovery of calculus in 1665-1666, predating Leibniz's independent discovery by about a decade.
Original writings of both Newton and Leibniz are preserved, offering insight into their mathematical discovery processes.
Newton's development of calculus was intertwined with his work in optics and gravitation during his time at Trinity College, Cambridge.
The method of finding tangents to a curve using circles with centers on the x-axis, a technique improved upon by Newton.
Newton's realization that tangent problems and area problems are inverse to one another, leading to his universal theorem for tangents.
Newton's waste book entries from May 1665, showcasing his mastery of techniques for finding normals and tangents.
Introduction of the concept of 'squaring crooked lines' as a method for calculating areas under curves by Newton.
Newton's shift to mechanical curves and his exploration of the cycloid, a curve traced by a point on a rolling circle.
Newton's innovative approach to finding tangents to all curves using the ratio of velocities of two points moving along the curve.
Leibniz's introduction of the integral sign and the development of rules for its use, marking a significant step in calculus.
Leibniz's method of considering areas as made up of lines, influenced by the work of Cavalieri and Pascal.
Leibniz's realization that areas are summations and tangents are differences, leading to the foundation of differential and integral calculus.
The introduction of the differential symbol 'D' by Leibniz, representing differences and complementing the integral symbol.
Leibniz's pursuit to formalize mathematical reasoning and automate the process of calculus through the invention of a logical machine.
The comparison of Newton's and Leibniz's approaches to calculus, highlighting their similarities and divergences in methodology and notation.
The impact of calculus on mathematics, described as the greatest increase in mathematical power since the time of the Greeks.
Transcripts
the calculus one of the most basic and
fundamental tools of modern mathematics
two men can rightly claim to have
invented it Isaac Newton and Gottfried
Wilhelm Leibniz
Nutan actually discovered his calculus
first in 1665 or 1666 Leibniz made his
own independent discovery of it some 10
years later however neither man saw fit
to publish what they'd found for some
years after that what's really
fascinating is that the original
writings recording the discoveries of
both of these men are preserved in the
university library in Cambridge we have
the notebooks that Newton kept between
1665 and 1667 and in Hannover Leibniz
his notes from 1676 are preserved as
well they provide a fascinating glimpse
into the process of mathematical
discovery that both of these men used
and is really exciting to be able to
study them we start our story with
Newton Newton was a student at Trinity
College Cambridge and in January 1665 he
took his degree and became Bachelor of
Arts they then followed two years of
intense work in which many of Newton's
basic ideas on the calculus as well as
optics and gravitation were form we
should restrict ourselves to his
mathematics in May 1665 Newton was
working in Cambridge he was rapidly
mastering and improving on the methods
of Descartes and hooda for finding
tangents the contemporary way of finding
a tangent to a polynomial curve that is
a curve with a polynomial equation was
as follows to find the tangent to this
curve at the point P look at circles
with centers on the x axis passing
through P most circles will cross the
curve at P and re cross it at another
point but one circle will just touch the
curve at P the line from the center of
this circle to P is called a normal and
the line at right angles to this normal
through P is the tangent to both the
circle and the curve hooda who is a
smart mathematician had developed a
cunning way of finding the center
of this circle which used the following
trick invented by firm are in general a
circle cuts the curve in two places
suppose this distance is o now find an
expression for the distance D of the
center of the circle from some
convenient reference point in terms of
Oh finally assume that o actually has a
value of zero the procedure gives a
value for D and so the centre of the
circle and the normal CP can be found
this method was reliable in practice but
it could be complicated to apply this is
what you can call his waste book which
he kept entries on a vast number of
different topics and these are the
mathematical pages which have been taken
out and rebound here on the 20th of May
1665 he made a note which makes it clear
that he had mastered these techniques
for finding normals and tangents and
this very page he writes that he has a
universal theorem for tangents to
crooked lines now Newton was well aware
that tangent problems and area problems
were inverse to one another so every
time he solved the tangent problem he'd
solved the corresponding area problem
and he wrote that up as such here in
this little book he presents a method
whereby to square those crooked lines
which may be squared squared means area
it was the standard terminology of the
time and here he starts writing down the
results 3x squared equals a Y the
parabola has square or area X cubed over
a 4x cubed equals a squared Y has square
or area X to the fourth over a squared
and so on down the page given the
equation of a curve Newton starts by
writing out tables of values for the
area under the curve so by summer 1665
Newton has lasted the techniques of
Descartes and CUDA for finding tangents
to curves
he's also used the inverse relationship
between tangents and areas to write down
the areas under lots of curves and he
finishes by writing down a result which
summarizes the pattern that he is
noticed if ax to the N equals B Y to the
n then n XY over n plus M is the area
under the curve described by Y in the
autumn of 1665 Newton returned to
calculating tangents calculating
tangents is generally Newton's main aim
but now he had switched his attention to
mechanical curves mechanical curves a
curve defined by motion rather than by
polynomial equations the most famous of
these is probably the cycloid a cycloid
is the path traced out by a point on a
circumference of a rolling circle a
kangan to this curve can be thought of
as the instantaneous direction of motion
of a point as it traces out the curve
for the cycloid this direction of motion
can be worked out as follows at this
instant the point on the circumference
of the circle is moving with equal
speeds in the direction the circle is
rolling and along a tangent to the
circle combining these two speeds using
the parallelogram rule gives this
direction of the tangent this idea of
instantaneous direction of motion was
not new Kepler Galileo Torricelli
and robber valve had all exploited it
but none had ever really understood it
Newton dived in copying much of what had
been done before and making the same
mistakes following the traditional
method of the time a point on an
Archimedean spiral would appear to have
velocities in these two directions so
combining the two gives the tangent for
the ellipse the length a plus the length
B is a constant so at any instant the
speed with which a is increasing must
equal the speed with which B is
decreasing so using the parallelogram
law the diagonal gives the direction of
the tangent these sort of constructions
do indeed give tangents
but for completely wrong reasons as was
shown when applied to the Quadra tricks
the Quadra tricks is formed by tracing
the path of the point of intersection of
a horizontal line moving downwards with
uniform velocity and aligned rotating
with constant velocity about the origin
the method used for the spiral and
ellipse says that the tangent at this
point should be a combination of speeds
in these two directions it clearly
didn't work
several mathematicians including
Descartes and Robert Valle attempted to
modify the method but none seemed to
work really satisfactorily
however when Newton had perfected his
method some months later he returned to
this problem and worked out what the
correct construction should be this work
with mechanical curves seems to a given
Newton a new way of looking at all
curves this is how Newton now perceived
of a curve simultaneously two points
move along in the X direction and along
the Y Direction the distance moved along
the y axis at any time is related to the
distance moved along the x axis by some
relationship which may be a polynomial
equation but could also be some sort of
mechanical link so by interconnecting
these two movements a curve would be
drawn but what Newton was interested in
was working out the ratio of the
velocities of these two points he knew
what the curve was however it was
defined so he knew how any distance
along one axis was related to a distance
along the other axis but Newton's
concept of the way this curve was
generated was by movement and what
Newton wanted to know was how the
velocities of the two points were
related this was a fundamental
perception of the problem and on
November the 3rd
tene 1665 it led Newton to give a new
method for finding tangents he starts by
going back to curves he knows and
showing how to find the ratio of the
velocity Q of Y to the velocity P of X
basically he lets an infinitely small
amount of time elapse in which the point
moves from X Y to X plus little o y plus
little o Q over P he writes what is x
and y in one moment will be X plus
little o and why this little o Q over P
in the next so X plus little o why was
that low Q over P is a point on the
curve that means he can replace X by X
plus little o Y by Y was little o Q over
P in the equation of the curve and then
let little o take the value zero a
perfectly systematic method unlock
dissimilar from what we do today nuking
Caesars on the idea that the ratio of Q
over P that is the ratio of the
velocities will give in the direction of
the tangent he then writes this very
important page in which he claims that
the method is completely general to draw
tangency says the crooked lines however
they may be related to straight ones now
he's completely certain that his method
will give him the tangents at all curves
and all points and he says hitherto may
be reduced the manner of drawing
tangents to mechanical lines see folio
50 folio 50 was his earlier and
incorrect method for drawing tangents to
mechanical lines so now he has a method
for finding tangents to all curves in
particular you can find the tangent to
the Quadra tricks the first time this
has been done in complete generality so
this page marks an important step in the
development of the calculus not only is
it completely general but when it's
applied to curves given by polynomial
equations it
how's Newton to use the rules he had
before for finding tangents but without
the need for who does complicated
calculations it's still mathematically
imprecise though not only is there the
question of relating geometrical
constructions for tangents the
instantaneous velocities there's the
business of relating velocities to
movements in infinitely small amounts of
time through the winter of 1665 Newton
Ponder's the concept of velocities then
in May of 1666 he starts to write up his
results here he says instead of the
ordinary method it would be convenient
and perhaps more natural to use this
namely define the motion of any line or
quantity and then in this little tract
of October 1666
he pulls all his results together not
only are the proofs or demonstrations
more explicit but the whole thing is
more coherent and by putting it all down
in one place he may have intended to
that other people see it he still
doesn't give his velocities any special
name they are what he will later call
fluxions but that has to wait for yet
another rewrite but one of 1671 but this
tract of October 1666 contains Newton's
first presentation of the basic ideas of
the calculus
our story of lightness begins in London
in 1673 in January of that year he
presented to the Royal Society a
calculating machine he had invented
incorporating several novel features he
was elected a fellow of the Royal
Society on the strength of this
invention all his life libel its work to
mechanize all reasoning processes he
wanted to formalize the rules of logic
so that any logical argument or
mathematical proof could be produced by
machine Leibniz saw the calculating
machine he took to the Royal Society as
just the first stage in the development
of such a logical machine and all his
life he worked to improve his
calculating machines this is the sixth
begun in 1690 it was not completed until
after his death some 30 years later
these ideas of live Nets are important
since they do much to explain his way of
working and particular care he went to
to invent a powerful and flexible
notation for his calculus Leibniz his
invention of his calculus grew out of
his study of contemporary mathematical
problems in particular area and tangency
problems so how were they studied at
that time it was under the guidance of
Christian Huygens in Paris that liveness
was to learn his mathematics at that
time it was quite usual to think of an
area as somehow made up of lines this
was a tradition going back to Cavalieri
and more recently defended by pascal so
the computing area you considered all
the Ordnance the notation deriving again
from Cavalieri
was on L from the latin omnia meaning
all l standing for the ordinance why is
this reasonable if we want to compute
this area we would probably pick
ordinates a fixed amount Delta apart we
could then approximate the area by
rectangles now each of these rectangles
has an area of ordinate Li times Delta
so the area we seek is approximately
the sum of all these products this sum
can be re-written as the sum of the
Allies times Delta we'd finish our
calculation by letting Delta get smaller
and smaller
giving better and better approximations
of the area what live Nets believed was
that the area was made up of all the
ordinates taken infinitely close
together when you did this he argued the
sum was of all the ordinates and that
gave you the area directly so to live
nets to find an area is to find on L of
a figure a highly geometric procedure
but he wanted to systematize
mathematical reasoning to see how he
proceeded were in the fortunate position
of being able to go to the Landis
bibliothèque Hannover but tens of
thousands of pages of his writings are
collected with the help of the staff
here who are engaged in the lengthy
business of publishing them we are able
to pick out just a few pages we need
here for example is a crucial one dated
the 26th of October 1675 libraries wants
to find the area under this curve so
that you can see what's going on we've
enlarged it for you and turned it round
like this Leibniz wants to find the blue
area you notice that it was the area of
the whole rectangle - the yellow area
and wrote down his formula this is the
blue area that's his sign for equals
this is the area of the whole rectangle
and that's the area of the yellow bit
live let's then apply this result to a
over X and obtained a result connecting
logarithms with the areas of hyperbolic
sectors the result was not new but live
Nets may well have been surprised by how
easily his new methods obtained it now
on the next page written only three days
later live Nets is in
Wester gating rules for omnia he writes
that omnia y el over a cannot be said to
be equal to
omnia y x omnia el nor is it y x omnia
el then you decide that writing omnia
gets in the way on the next side he says
it will be useful to write this symbol
for omnia and this for omnia el that is
the sum of all the owl's the live needs
this was just the long script s for
summer or some but of course we
recognize it immediately this is the
first occurrence of the integral sign
Leibniz ever keen for the most
appropriate symbol introduced on the
29th of October 1675 a sign that remains
unchanged to the present day he then
proceeded to find some rules for his new
symbol here he writes Omni R X is x
squared over two here Omni are x squared
is X cubed over three and here that
omnia a over B times L is equal to a
over B times omnia L whenever a over B
is a constant with the introduction with
new symbol live Nets is of course
dealing with problems of area but areas
and tangents as leiden its new were
related problems if the sum of the
ordinance made up an area the difference
between two ordinates represented the
increase in the curve over the interval
and when the ordinates moved infinitely
close together the tangent was produced
so enliven its mind was the realization
that areas were summations and tangents
differences and here we see Leibniz
saying just that he writes given L
related to X we have to find omnia L
what can now be done from the contrary
calculus everything
t is if omnia L equals y over a we may
set L equals y a over D consequently
just as omnia increases dimensions so
does D diminish them omnia signifies
some D difference here live Nets
introduced the de symbol D for
difference as he said but he wrote it
underneath because just as omnia
increases dimension D goes up from lines
to areas so his opposite operation must
reduce dimension but live Nets didn't
stick to this notation for very long in
another note this one written 12 days
later the D moves upstairs as he says in
a margin DX is the same as x over D that
is the difference of two neighboring X's
this is a very exciting moment for the
first time the two basic symbols of the
calculus exist and live mates is looking
for rules for their use here he writes d
of x times y is equal to d of XY minus x
dy which we can immediately rewrite as X
dy plus y DX equals D of XY so here in
the space of three weeks in autumn 1675
we see the foundations of the live Nets
in calculus laid down once the
discoveries were made it's interesting
to see how live knits proceeded with
them here's a document written in
mid-july 1677 which makes it clear that
live knits is looking for a way of
casting in his discoveries in a way that
makes them amenable to a universal
automatic process of reasoning he writes
but to explain my ideas neatly and
succinctly I'm obliged to introduce some
new characters and to give them a new
algorithm that is special rules for
their addition subtraction
multiplication division power
routes and equations so he's cast this
discovery in the form of rules for D
here is the roof multiplication give a
product and here give a quotient so to
sum up it's interesting to compare our
two central characters as they stood in
the late 16 70s
they're great calculus or should I say
calculate in the plural as yet
unpublished when we make such a
comparison several interesting points of
agreement emerge as well as several
interesting divergences similarities
first above all both calculus are about
geometric properties of figures areas
and tangents and both men went a long
way to automating their findings and
subjecting the calculus that they
discovered to rules but there are also
several interesting divergences Newton
spoke of fluxions
infinitesimal increases in a variable
where the means he used to find his
fluxions and from the first he was
always attracted to the idea of motion
of change in time as a way of expressing
mathematical ideas and although his
thoughts on that topic grew more
profound as the years went by he was
always wedded to the language of motion
live Nix talked of differentials of
infinite linear points of curves being
made up of infinite sided polygons no
motion here rather a bold geometrical
analogy which whatever else yielded
valid rules for what came to be called
differentiation and integration indeed
the rules which Leibniz found are more
basic to his way of thinking
mathematically than were the equivalent
rules found by Newton it's hard to
overestimate the power of the calculus
as Newton and Leibniz described it
indeed it can be argued that when they
came to publish their findings in the
late sixteen eighty s mathematics
received the greatest increase in its
power since the time of the Greeks
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