Compass and Straightedge Construction

A “construction” is drawing geometric figures with a high degree of accuracy. The construction performed constitutes both a proof of the existence of a geometric object and the solution of the problem. The ancient Greeks were convinced that all plane figures can be constructed with a compass and a straightedge alone. Their methods of bisecting a line and an angle are ingenious and hard to improve on. They worked with all numbers geometrically. A length was chosen to represent the number 1, and all other numbers were expressed in terms of this length. They solved equations with unknowns by series of geometric constructions. The answers were line segments whose length were the unknown value sought. The Greeks imposed the restrictions of straightedge and compass for the construction of the problems. It is supposed that this tradition was started by Plato Greece’s greatest philosopher. He claimed that more complicated instruments called for manual skill unworthy of a thinker. The Greeks failed to obtain the solution of the famous problems under the restrictions specified not due to the lack of ingenuity of the geometers. The Greeks’ persistent efforts to find compass-and-straightedge ways of trisecting an angle, squaring the circle and duplicating the cube were not futile for almost 2000 years. The Greeks made great math discoveries on the way. The desire to gain full understanding of the theoretical character of the problems inspired many great mathematicians- among them Descartes,Gauss, Poncelet, Lindemann – to mention but a few. The long years of labour on these “impractical”, “worthless” problems indicate the care, patience, persistence and rigour of mathematicians in their attempts to perform the constructions and justify them theoretically. The problems did not exhaust themselves. Even nowadays some authors of the scientific papers issued “solutions” containing some fallacies. The search for the rigorous solution resulted in great discoveries and novel developments in maths.

Vocabulary

accuracy	точность
degree	зд. степень
to constitute	составлять
to convince	убеждать
compass	циркуль
straightedge	линейка
to express	выражать
to impose	налагать (обязательство)
restriction	ограничение
unworthy	недостойный
to specify	устанавливать, определять
ingenuity	изобретательность
to gain = to get
to indicate	указывать на
ingenious	изобретательный

1. Answer the following questions:

1) What inspired great mathematicians like Descartes, Gauss and others?

2) How did Greeks regard numbers?

3) How did they express line segments?

4) What restrictions did the Greeks introduce for the construction of the problems?

5) Why did the Greeks fail to obtain the solution of some famous problems?

2. Find the English equivalents for the Russian words and word combinations:

строить, сооружать; упорство и настойчивость; издавать, выпускать; поиск решения; полное понимание; настойчивые усилия; недостойно мыслителя; в виде длины; отсутствие изобретательности; тщетные попытки; не говоря уже о других; вводить новые понятия; уравнения с неизвестными.

3. Pay attention to the negative prefixes and see that you know the roots of these words.

insoluble	unsolved
unknown	non-existence
unworthy	irrational
unimportant	unskilled
impossibility

4. Translate from Russian into English.

1) Греки не смогли получить решения некоторых известных проблем.

2) Греки прилагали настойчивые усилия, чтобы найти способ квадратирования круга.

3) Долгие годы упорного труда привели к желаемым результатам.

4) Даже сегодня некоторые авторы предлагают решения, которые содержат ошибки (неточности).

5) Древние греки были убеждены, что все фигуры на плоскости можно построить с помощью линейки и циркуля.

Text. 13.

Euclid's Elements

Euclid’s Elements are the first remarkable attempt to build all geometry. Euclid succeeded in basing his development of geometry on a system. A logical self-sufficient system must start somewhere. To be precise about what his abstract terms include Euclid begins his logical system with the first principles of definitions, axioms and postulates. Euclid's definitions were rightly criticized.

"A point is that which has no part" is the definition we are not told what a point is but rather what it is not. After centuries of vain effort it was realized that one must give up definitions of the kind attempted by Euclid. These must be a foundation on which to build, i.e., undefined terms. Euclid's fundamental propositions from which further statements follow logically are divided into "postulates" and "axioms". Modern maths ignores the distinction between these terms. A derivation of a theorem involves a proof. The precise and rigorous sense which the Greeks gave to this word may be understood by studying Euclid's Elements. This sense is not changed because what constituted a proof for Euclid some tacit assumptions that converted some of his proofs into invalid demonstrations. Besides, Euclid's first principles are insufficient for the derivation of all the 465 propositions. The fact of the impossibility of deriving the proposition on parallels from the rest of postulates and axioms was clearly recognized by Euclid himself. Despite some shortcomings and the inadequacy of Euclid's definitions, the Elements are a work of genius. There is no textbook in the history of mankind which retained a position of prominence for as long time as this work of Euclid. Nowadays high-school geometry is based principally on Euclid's accomplishment. Even the emergence of non- Euclidean geometries did not spoil the "image" of Euclid or of his Elements.

Vocabulary

attempt	попытка
to succeed in	добиться успеха
remarkable	замечательный
self-sufficient	самодостаточный
precise	точный
vain effort	тщетные усилия
give up	отказаться от ч.-л.
to ignore	игнорировать
sense	смысл, значение
invalid	необоснованный, недействительный
to recognize	признавать
despite	несмотря на
shortcomings	недостатки
prominence	выдающееся положение

1. Answer the following questions:

1) What is high-school geometry based on?

2) What did Euclid recognize himself as insufficient?

3) Why were Euclid’s definitions criticized?

4) What definitions did Euclid attempt and are they valid nowadays?

5) What does modern maths ignore?

2. Find the Russian word on the right for the English ones.

to involve	предположение
assumption	выводить
invalid	недостатки
derivation	включать в себя
to derive	сохранить положение
shortcomings	предложение
to retain a position	недействительный
proposition	происхождение

3. Find in the text adjectives with the suffix “ive”.

4. Translate from Russian into English.

1) Несмотря на некоторые недостатки, труд Евклида был произведением гения.

2) То, что было доказательством для Евклида, и сейчас остается доказательством и для нас.

3) Смысл его работы не изменился.

4) После многих столетий ученым стало ясно, что нужен фундамент, на котором можно построить термины, не имеющие определения.

5) В истории человечества нет учебника, который бы сохранил на долгие годы такое выдающееся положение.

5. Find Russian equivalents for the English word combinations:

Further statements; are clearly recognized; inadequacy of definitions; exhaustive study; to involve a proof. self-sufficient system; to ignore the distinction; to reveal the shortcomings; to retain a position of prominence; great accomplishment.

Text. 14.