N accumulation of quantities

1 2 3 4 5 6 7

n methodology n infinitesimal

n differential calculus n integral calculus

n vast n vital

n tangent n latter

n coordinate nchord

n sake n distinction

1 A line segment joining two points on a curve is a...........................

2 A......................... is a line or surface that touches another.

3 The area of maths used to determine areas, volumes and lengths is called.......................

4 The area of maths relating to changes in variable is called...........................

5 If something is close to zero it is....................

6 You need to eat well for the.......................... of your health.

7There is a..........................amount of knowledge to learn in sciences.

8 There are two theories - one from ancient times and a modern one. The........................,

the modern one, is widely accepted now.

9 She claimed the..........................of having solved the equation.

10 A.........................is a number that identifies a position relative to a straight line.

11..........................is the system of methods followed in an area of study.

12..........................measures areas under a curve, distance travelled, or volume displaced.

13 If something is...........................it is of the utmost importance.

Reading

Gottfried Leibniz*

Gottfried Leibniz was born and lived most of his life in Germany. He made visits to both Paris and London, for the sake of learning and study, but spent the vast majority of his working life as an employee of German royalty, as a philosopher, engineer and mathematician. It is for the latter that he is best remembered. His greatest achievement was as an inventor of calculus, the system of notation which is still in use today. Leibniz is remembered as an inventor, not the inventor of calculus. In England, Isaac Newton claimed the distinction, and was later to accuse Leibniz of plagiarism, that is, stealing somebody else's ideas but stating that they are original. Modern-day historians however, regard Leibniz as having arrived at his conclusions independently of Newton. They point out that there are important differences in the writings of both men. Newton, it must be said, was very protective of his achievements and jealous of others' success. It is important to mention that Leibniz published his writings on calculus three years before Newton published his most important work.

Leibniz was the first to use function to represent geometric concepts. Among other terms, Leibniz used what is now everyday language in mathematics to describe these concepts. Words such as tangent and chord, were first used by Leibniz. He also saw that linear equations in algebra could be arranged into matrices. It was in this significant piece of work on calculus that he introduced mathematics and the world to the word coordinate. He also made important advances in algebra and logic in ways that still today, three hundred years later, have an impact on mathematics.

Leibniz' importance for modern mathematics can be understood through his work. He was especially interested in infinitesimal calculus. This is an area of calculus developed from geometry and algebra. It is divided into two parts. There is differential calculus, which is concerned with measuring rates of change of quantities. And there is integral calculus, which studies the accumulation of quantities. That is, Leibniz was looking at ways of measuring the speed and the distance travelled, for example. Today, calculations of this type are used not only in mathematics but in every branch of science and in many fields which apply a scientific methodology, such as economics and statistics.

Despite the disagreements between Leibniz and Newton, modern mathematicians recognise each of them as being vital to the development of modern mathematics. Newton was certainly the first to apply calculus to the problems of physics. In mathematics itself, it is to Leibniz that we look for our system of writing equations and for the language we use to refer to the concepts. While both reached their understanding without the benefit of reading each other's work, it remains a fact that Leibniz was first to publish.

Pronunciation guide

Gottfried Leibniz

Infinitesimal

Plagiarism

E Comprehension

Read the text and answer the questions in your own words.

1 For what contribution to mathematics is Leibniz best remembered?

2 Who was Leibniz' main rival? About what did they disagree?

3 Which important geometrical terms did Leibniz invent?

4 What other areas of work also use Leibniz' calculus?

5 Who is considered more important for the development of modern mathematics?

Before you read