прочитай текст и выполни задания по тексту Task 4. Read and translate Text A using a dictionary if necessary. 
Text A  
MATHEMATICS DEVELOPMENT 
17th Century Mathematic 
The scientific revolution of the 17th century spurred advances in mathematics 
as well. The founders of modern science – Nicolaus Copernicus, Johannes Kepler, 
Galileo, and Isaac Newton – studied the natural world as mathematicians, and they 
looked for its mathematical laws. Over time, mathematics grew more and more 
abstract as mathematicians sought to establish the foundations of their fields in logic. 
The 17th century opened with the discovery of logarithms by the Scottish 
mathematician John Napier and the Swiss mathematician Justus Byrgius. 
Logarithms enabled mathematicians to extract the roots of numbers and simplified 
27 
many calculations by basing them on addition and subtraction rather than on 
multiplication and division. 
Napier, who was interested in simplification, studied the systems of the Indian 
and Islamic worlds and spent years producing the tables of logarithms that he 
published in 1614. Kepler’s enthusiasm for the tables ensured their rapid spread. 
The 17th century saw the greatest advances in mathematics since the time of 
ancient Greece. The major invention of the century was calculus. Although two great 
thinkers – Sir Isaac Newton of England and Gottfried Wilhelm Leibniz of Germany – have received credit for the invention, they built on the work of others. As Newton 
noted, “If I have seen further, it is by standing on the shoulders of giants.” Major 
advances were also made in numerical calculation and geometry. 
Gottfried Leibniz was born (1st July, 1646) and lived most of his life in 
Germany. His greatest achievement was the invention of integral and differential 
calculus, the system of notation which is still in use today. In England, Isaac Newton 
claimed the distinction and accused 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. 
Differential calculus came out of problems of finding tangents to curves, and 
an account of the method is published in Isaac Barrow’s “Lectiones geometricae” 
(1670). Newton had discovered the method (1665-66) and suggested that Barrow 
include it in his book. 
Leibniz had also discovered the method by 1676, publishing it in 1684. Newton 
did not publish his results until 1687. A bitter dispute arose over the priority for the 
discovery. In fact, it is now known that the two made their discoveries independently 
and that Newton had made it ten years before Leibniz, although Leibniz published 
first. The modern notation of dy/dx and the elongated s for integration are due to 
Leibniz. 
The most important development in geometry during the 17th century was the 
discovery of analytic geometry by Rene Descartes and Pierre de Fermat, working 
in dependently in France. Analytic geometry makes it possible to study geometric 
figures using algebraic equations. By using algebra, Descartes managed to overcome 
the limitations of Euclidean geometry. That resulted in the reversal of the historical 
roles of geometry and algebra. 
The French mathematician Joseph Louis Lagrange observed in the 18th 
century, “As long as algebra and geometry proceeded along separate paths, their 
advance was slow and their applications limited. But when these sciences joined 
company, they drew from each other fresh vitality and thenceforward marched on at a 
rapid pace toward perfection.” 
Descartes’ publications provided the basis for Newton’s mathematical work 
later in the century. Pierre de Fermat, however, regarded his own work on what 
became known as analytic geometry as a reformulation of Appollonius’s treatise on 
28 
conic sections. That treatise had provided the basic work on the geometry of curves 
from ancient times until Descartes. 
18th – 19th Century Mathematics 
During the 18th century, calculus became the cornerstone of mathematical 
analysis on the European continent. Mathematicians applied the discovery to a 
variety of problems in physics, astronomy, and engineering. In the course of doing so, 
they also created new areas of mathematics. 
In France, Joseph Louis Lagrange made substantial contributions in all fields of 
pure mathematics, including differential equations, the calculus of variations, 
probability theory, and the theory of equations. In addition, Lagrange put his 
mathematical skills to work in the solution of practical problems in mechanics and 
astronomy. 
The greatest mathematician of the 18th century, Leonard Euler of Switzerland, 
wrote works that covered the entire fields of pure and applied mathematics. He wrote 
major works on mechanics that preceded Lagrange’s work. He won a number of 
prizes for his work on the orbits of comets and planets, the field known as celestial 
mechanics. But Euler is best known for his works in pure mathematics. In one of his 
works, Introduction to the Analysis of Infinites, published in 1748, he approached 
calculus in terms of functions rather than the geometry of curves. Other works by 
Euler contributed to number theory and differential geometry (the application of 
differential calculus to the study of the properties of curves and curved spaces). 
Mathematicians succeeded in firming the foundations of analysis and 
discovered the existence of additional geometries and algebras and more than one 
kind of infinity. 
The 19th century began with the German mathematician Carl Frederich 
Gauss. He ranks as one of the greatest mathematicians of the world. His book 
Inquiries into Arithmetic published in 1801 marks the beginning of modern era in 
number theory. 
Gauss called mathematics the queen of sciences and number theory the queen 
of mathematics. Almost from the introduction of calculus, efforts had been made to 
supply a rigorous foundation for it. Every mathematician made some effort to 
produce a logical justification for calculus and failed. Although calculus clearly 
worked in solving problems, mathematicians lacked rigorous proof that explained 
why it worked. Finally, in 1821, the French mathematician Augustin Louis Cauchy 
established a rigorous foundation for calculus with his theory of limits, a purely 
arithmetic theory. 
Later, mathematicians found Cauchy’s formulation still too vague because it 
did not provide a logical definition of real number. The necessary precision for 
calculus and mathematical analysis was attained in the 1850s by the German 
mathematician Karl T. W. Weierstrass and his followers. 
Another important advance in analysis came from the French mathematician 
Jean Baptiste Fourier, who studied infinite series in which the terms are 
29 
trigonometric functions. Known today as Fourier series, they are still powerful tools 
in pure and applied mathematics. 
The investigation of Fourier series led another German mathematician, Georg 
Cantor, to the study of infinite sets and to the arithmetic of infinite numbers. Georg 
Cantor began his mathematical investigations in number theory and went on to create 
set theory. In the course of his early studies of Fourier series, he developed a theory 
of irrational numbers. Cantor and another German mathematician, Julius W. R. 
Dedekind, defined the irrational numbers and established their properties. These 
explanations hastened the abandonment of many 19th century mathematical 
principles. When Cantor introduced his theory of sets, it was attacked as a disease 
from which mathematics would soon recover. However, it now forms part of the 
foundations of mathematics. The application of set theory greatly advanced 
mathematics in the 20th century. 
Adopted from Пушкина Е.Н. English for Mathematicians and Information Technologies Learners =  
Английский для студентов, изучающих математику и информационные технологии:  
учебно-методическое пособие [Электронный ресурс] / Е.Н. Пушкина. –  
Нижний Новгород, ННГУ, 2019. – 88 с. 
AFTER TEXT TASKS 
Task 5. Answer the following questions. 
1. What scholars are considered to be the founders of modern science? 
2. Why did mathematics grow more and more abstract? 
3. Who were logarithms discovered by? 
4. What did logarithms enable mathematicians to do? 
5. What was the major invention of the 17th century? 
6. What is the essence of analytic geometry? 
7. Why did a dispute arise between Leibniz and Newton? 
8. What enabled Descartes to overcome the limitations of Euclidean geometry? 
9. Whose publications provided the basis for Newton’s mathematical work later in 
the century? 
10. What did the discovery of calculus lead to? 
11. What was Lagrange’s contribution to pure and applied mathematics? 
12. What did Euler’s works contribute to? 
13. What is the essence of differential geometry? 
14. What event marked the beginning of modern era in number theory? 
15. When was a rigorous foundation for calculus finally supplied? 
16. What is the theoretical and practical value of Fourier series? 
17. What was Georg Cantor’s contribution to mathematical studies? 
18. Who were irrational numbers investigated and defined by? 
19. What was the first reaction to Cantor’s set theory? 
20. Was the attitude to the discovery later changed? 
30 
Task 6. Look through the text to search for unfamiliar words and try to 
understand their general meanings. 
Task 7. Write down the transcription and definitions of unfamiliar words, 
practise reading the words and try to remember them. 
Task 8. Give Russian equivalents to these word combinations.  
1. cornerstone 
2. substantial 
3. major works 
4. to rank as 
5. to lack smth. 
6. vague 
7. precision 
8. to attain 
9. abandonment 
10. to advance 
11. simplification 
12. to approach 
13. to elongate 
14. pace 
15. tangents to curves 
Task 9. Find the English equivalents to the following words and word 
combinations. 
1. первенство 
2. сделать открытие 
3. извлекать корни 
4. упростить 
5. плагиат 
6. опубликовать 
7. интегральные и дифференциальные исчисления 
8. система обозначений 
9. претендовать (на что-л.) 
10. совершенство 
11. искривленное пространство 
12. теория чисел 
13. достичь 
14. ускорять 
15. несмотря на то, что 
31 
Task 10. Match the mathematicians with their contributions to science. 
№ Mathematician  
Contribution 
1 Georg Cantor  
a 
infinite 
series in which the terms are 
trigonometric functions 
2 Julius W. R. Dedekind b analytic geometry 
c 
3 Augustin Louis 
Cauchy  
pure mathematics, including differential 
equations, the calculus of variations, 
probability theory, and the theory of equations 
4 Jean Baptiste Fourier  
d a theory of irrational numbers 
e 
5 Augustin Louis 
Cauchy  
tables of logarithms 
6 Joseph Louis Lagrange  f 
g 
7 Rene Descartes and 
Pierre de Fermat 
integral and differential calculus 
irrational numbers and their properties 
8 Gottfried Wilhelm 
Leibniz 
h theory of limits, a purely arithmetic theory 
9 John Napier 
i 
j 
laws of motion 
10 Isaac Newton 
foundation for calculus with his theory of 
limits, a purely arithmetic theory 
Task 11. Mark true (T) or false (F) sentences. 
1. The 17th century opened with the discovery of logarithms by the Swiss 
mathematician John Napier and the Scottish mathematician Justus Byrgius. 
2. Kepler’s enthusiasm for the tables of logarithms ensured their rapid spread. 
3. Gottfried Wilhelm Leibniz of Germany noted, “If I have seen further, it is by 
standing on the shoulders of giants.” 
4. The greatest achievement of Carl Frederich Gauss was the invention of integral 
and differential calculus. 
5. Leonard Euler of Switzerland wrote works that covered the entire fields of pure 
and applied mathematics. 
6. In the work “Introduction to the Analysis of Infinites” Euler approached calculus 
in terms of functions rather than the geometry of curves.  
7. During the 18th century, calculus became the cornerstone of mathematical analysis 
on the European continent. 
8. The 19th century began with the mathematician Jean Baptiste Fourier. His book 
“Inquiries into Arithmetic” marks the beginning of modern era in number theory. 
9. The necessary precision for calculus and mathematical analysis was attained in the 
1850s by the German mathematician Karl T. W. Weierstrass. 
10. The application of set theory greatly advanced mathematics in the 20th century. 
32 
Task 12. Complete the sentences below with the words and phrases from the 
box. 
a. Rene Descartes and Pierre de Fermat 
b. the discovery of calculus 
c. Kepler 
d. preceded Lagrange’s work 
e. Newton and Leibniz 
f. the scientific revolution of the 17th century 
g. the tables of logarithms 
h. physics, astronomy, and engineering 
i. celestial mechanics 
j. Fourier series 
k. Karl T. W. Weierstrass 
l. mechanics and astronomy 
m. number theory and differential 
geometry 
n. Carl Frederich Gauss 
o. Cantor and Dedekind 
1. The Scottish mathematician Napier spent years producing … . 
2. The rapid spread of the tables of logarithms was ensured by … . 
3. The development of analytic geometry was beneficial for both … . 
4. The invention of calculus is connected with the names of … . 
5. A bitter dispute arose over the priority for … . 
6. Advances in mathematics were facilitated by … . 
7. Euler’s major works on mechanics … . 
8. Mathematicians applied the discovery of calculus to … . 
9. Lagrange managed to solve some practical problems in … . 
10. Euler’s works contributed to … . 
11. Euler won a number of prizes for his work on … . 
12. Mathematics was called the queen of sciences by … . 
13. Cantor’s study of infinite sets became possible due to the study of … . 
14. The properties of irrational numbers were established by … . 
15. A precise foundation for calculus was supplied by … .