The Birth Of Calculus (1986)

Summary

This 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

• 📚 Calculus is a fundamental tool in modern mathematics, with key contributions from Isaac Newton and Gottfried Wilhelm Leibniz.
• 🕰️ Newton first discovered calculus around 1665-1666, while Leibniz made his independent discovery about 10 years later.
• 📖 Both Newton and Leibniz's original writings on their discoveries are preserved, providing insight into their mathematical processes.
• 🎓 Newton's early work in calculus was influenced by his studies at Trinity College, Cambridge, and his intense work period from 1665.
• 📐 Newton's initial approach to calculus involved improving methods for finding tangents to polynomial curves, using circles and the method of normals.
• 🔍 Newton realized the inverse relationship between tangent problems and area problems, leading to his development of methods for calculating areas under curves.
• 🛣️ Newton's work extended to 'mechanical curves,' such as the cycloid, which are defined by motion rather than polynomial equations.
• 🔄 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.
• 📈 Leibniz's calculus was influenced by the study of area and tangency problems, and his approach was more geometric, using infinite sums of ordinates.
• 📝 Leibniz introduced the integral sign (∫) and the differential symbol (d), laying the foundation for differentiation and integration in calculus.
• 🔧 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

00:00

📚 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.

05:01

🔍 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.

10:03

📈 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.

15:03

📘 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.

20:04

🌟 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.

Keywords

💡Calculus

Calculus is a branch of mathematics that deals with the study of continuous change, typically expressed in terms of derivatives and integrals. In the video, calculus is presented as a fundamental tool of modern mathematics, invented by Isaac Newton and Gottfried Wilhelm Leibniz. The script discusses the development of calculus as it relates to the geometric properties of figures, such as finding tangents and areas, and the automation of these processes through systematic rules.

💡Isaac Newton

Isaac Newton is a pivotal figure in the history of calculus, having made significant contributions to its development. The script describes Newton's early work in Cambridge, where he developed methods for finding tangents to curves and areas under them, which laid the groundwork for his calculus. Newton's approach to calculus was deeply connected to the concept of motion and change over time, as illustrated by his work with mechanical curves and the development of his fluxions.

💡Gottfried Wilhelm Leibniz

Gottfried Wilhelm Leibniz is credited alongside Newton as the co-inventor of calculus. The script highlights Leibniz's independent discovery of calculus about a decade after Newton. Leibniz's approach to calculus was more geometrical, focusing on the concept of differentials and the summation of infinitesimals to find areas and tangents. His introduction of the integral symbol and the development of rules for differentiation and integration are significant contributions discussed in the video.

💡Tangents

In the context of the video, tangents refer to the lines that touch a curve at a single point, and their study is central to the development of calculus. The script describes the historical methods of finding tangents, such as using circles with centers on the x-axis, and how Newton improved upon these methods to develop a universal theorem for tangents to 'crooked lines,' which is a key aspect of his calculus.

💡Areas

The concept of areas is integral to calculus, particularly in the study of integrals. The script discusses how Newton and Leibniz approached the calculation of areas under curves, with Newton using the relationship between tangents and areas and Leibniz considering areas as summations of infinitesimally small ordinates. The automation of area calculation through systematic rules is a significant theme in the development of calculus.

💡Fluxions

Fluxions is the term used by Newton to describe the instantaneous rates of change of variables, which is a fundamental concept in calculus. The script mentions that Newton's calculus was initially referred to in terms of fluxions, and his method for finding tangents and areas was based on the idea of motion and change over time, as seen in his work with curves defined by motion.

💡Differentials

Differentials, in Leibniz's calculus, represent infinite linear points of curves, which are considered as being composed of infinite-sided polygons. The script explains that Leibniz's approach to calculus was based on a geometric analogy, where differentials allowed for the calculation of areas and tangents through summation and difference operations, respectively.

💡Mechanical Curves

Mechanical curves, as discussed in the script, are curves defined by motion rather than by polynomial equations. Newton's work with mechanical curves, such as the cycloid, contributed to his perception of a curve as a result of movement and helped him develop a new method for finding tangents based on the ratio of velocities of two points moving along the curve.

💡Integral Sign

The integral sign, introduced by Leibniz, is a fundamental symbol in calculus used to denote the process of integration. The script describes the moment when Leibniz first used this symbol, which has remained unchanged to the present day, as a significant development in the notation and formalization of calculus.

💡Derivative

Although not explicitly mentioned in the transcript, the concept of the derivative is closely related to the process of finding tangents and is a core aspect of calculus. Derivatives represent the rate at which a quantity changes with respect to another quantity, and in the script, Newton's work on tangents to curves and Leibniz's work on differentials both lead to the development of differentiation rules.

💡Automating Reasoning Processes

Leibniz's work on calculus was driven by his ambition to mechanize all reasoning processes, including mathematical proofs. The script mentions his invention of a calculating machine and his lifelong pursuit to formalize the rules of logic, which influenced his approach to calculus and the invention of a powerful and flexible notation for his calculus.

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

you

you