(Part 2)

1. According to the passage, the earliest drawings that indicate knowledge of mathematics were found in

a) northeastern Congo.

b) ancient India and Pakistan.

c) South Africa.

d) England and Scotland.

2. Megalithic monuments found in different places incorporate in their design

a) small and precise subdivisions.

b) pictorially represented geometric spatial designs.

c) geometric ideas.

d) the circumference of the circle.

3. What mathematical instruments were not discovered in the Indus Valley Civilization of North India and Pakistan?

a) a shell instrument used to measure 8–12 whole sections of the horizon and sky.

b) Pythagorean theorem.

c) a shell instrument that served as a compass to measure angles on plane surfaces or in horizon.

d) an accurate decimal ruler.

4. The word circumference means

a) "the line bounding a circle or any rounded or elliptical area".

b) "a line bounding any area, like triangle, square or polygon".

c) "a line connecting two points on the plane".

d) "the line of the horizon".

5. Which of the following would be an appropriate title for this passage?

a) History of Megalithic Monuments.

b) Early Mathematics.

c) History of Indus Valley Civilization of North India and Pakistan.

d) Mathematics in Early England and Scotland.

6. The word combination concentric circles in the text is closest in meaning to

a) circles having a centre in common.

b) circles having different centres.

c) circles intersecting each other.

d) circles with the same radius.

7. The word value in the text is closest in meaning to

a) worth.

b) amount.

c) effectiveness.

d) price.

8. According to the text, when considering herds of animals, early hunters had the concepts of

a) one, two, and many as well as the idea of none or zero.

b) decimal fractions.

c) the golden ratio.

d) the circumference of the circle and a value of π.

9. What can you state as the main idea of the text?

a) Ancient people utilized the principal planar and spatial geometric ideas.

b) Archeological evidence has led to the conclusion that many basic concepts of mathematics have been discovered and utilized by early civilizations.

c) The Ishango Bone, found in the area of the headwaters of the Nile River (northeastern Congo), dates as early as 20,000 BC is the only known demonstration of mathematical knowledge of early civilizations.

d) Mathematics in ancient India which developed a system of uniform weights and measures, is the earliest in the world.

10. The word combination right angle in the text is translated into Ukrainian as

a) гострий кут.

b) правильний кут.

c) правий кут.

d) прямий кут.

7. Answer the following questions:

1. Where and when was the earliest mathematics in India found? 2. What is the interpretation of the Ishango Bone? 3. What is considered to be the first true evidence of mathematical activities in ancient China? 3. What do the ancient Megalithic monuments incorporate? 4. What mathematical concepts did the Indus Valley Civilization utilize? 5. Which of mathematical instruments discovered in ancient India are still in use nowadays?

8. Translate these sentences into English:

1. Індійські та арабські вчені зробили значний внесок у розвиток математики в IX-V століттях до нашої ери. 2. В багатьох стародавніх текстах розглядається теорема Піфагора, яка вважається найпоширенішим математичним відкриттям після елементарної арифметики і геометрії. 3. Культура Стародавньої Греції, Єгипту, Месопотамії та міста Сиракузи збагатила знання людей в галузі математики. 4. В стародавньому світі бурхливий розвиток математики змінювався століттями застою. 5. Першим справжнім свідченням розвитку математики в Китаї була нумерація костей для передбачення майбутнього. 6. В стародавній Індії та Пакистані застосовували поняття пропорції, десяткових дробів та геометричні фігури, такі як прямокутний паралелепіпед, конус, циліндр, круг і трикутник. 7. Артефакти, знайдені в стародавньому Єгипті свідчать про те, що дана цивілізація не лише володіла геометричними просторовими уявленнями, а й уміла вимірювати час, куруючись рухом зірок. 8. Цивілізація стародавньої Індії володіла знаннями про відношення довжини кола до його діаметра, а отже про величину числа π. 9. Задовго до перших письмових пам’яток було знайдено наскельні малюнки, які свідчать про знання основ математики та вміння доісторичних людей вимірювати час. 10. Індійський рукопис ще не розшифровано, але археологічні знахідки свідчать про досить високий рівень знань з математики у стародавньому світі.

Unit 1.2

Read and translate the text using the dictionary:

(Part 1)

Egyptian mathematics refers to mathematics written in the Egyptian language. From the Hellenistic period, Greek replaced Egyptian as the written language of Egyptian scholars, and from this point Egyptian mathematics merged with Greek and Babylonian mathematics to give rise to Hellenistic mathematics. Mathematical study in Egypt later continued under the Islamic Caliphate as part of Islamic mathematics, when Arabic became the written language of Egyptian scholars.

Using a kind of reed – papyrus – Egyptians made papers. About 1650 B.C. in Ahmes Papyrus we can see how to calculate the fraction and the superficial measure of farmland. Ancient Egyptians said that the area of a circle is repeatedly taken as equal to that of the square of 8/9 of the diameter. They also extracted the volume of a right cylinder and the area of a triangle as well as arithmetic and geometric series, but they handled only a simple first order linear equation.

The oldest mathematical text discovered so far is the Moscow papyrus, which is an Egyptian Middle Kingdom papyrus dated circa 2000 B.C.-1800 B.C. Like many ancient mathematical texts, it consists of what are today called "word problems" or "story problems", which were apparently intended as entertainment. One problem is considered to be of particular importance because it gives a method for finding the volume of a frustum: "If you are told: A truncated pyramid of 6 for the vertical height by 4 on the base, by 2 on the top. You are to square this 4, result 16. You are to double 4, result 8. You are to square 2, result 4. You are to add the 16, the 8, and the 4, result 28. You are to take one third of 6, result 2. You are to take 28 twice, result 56. See, it is 56. You will find it right."

The R-hind papyrus (circa 1650 B.C.) is another major Egyptian mathematical text, an instruction manual in arithmetic and geometry. In addition to giving area formulas and methods for multiplication, division and working with unit fractions, it also contains evidence of other mathematical knowledge, including composite and prime numbers; arithmetic, geometric and harmonic means; and simplistic understandings of both the Sieve of Eratosthenes and perfect number theory (namely, that of the number 6).

Also, three geometric elements contained in the R-hind papyrus suggest the simplest of underpinnings to analytical geometry: (1) first and foremost, how to obtain an approximation of π accurate to within less than one percent; (2) second, an ancient attempt at squaring the circle; and (3) third, the earliest known use of a kind of cotangent.

Finally, the Berlin papyrus (circa 1300 BC) shows that ancient Egyptians could solve a second-order algebraic equation.