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COMPOSITION

Abstract (Précis) Writing

Compass and Straightedge Constructions

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 segment 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 lengths 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 famous problems are insoluble because they involve irrational numbers that cannot be constructed by Euclidean methods.) 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 la‘llacies. The search for the rigorous solution resulted in great discoveries and novel developments in maths. It introduced new geometric concepts (e.g., conic sections), raised a number of important theoretical questions (e.g., to prove the impossibility of the solution) and suggested an entirely new direction for scientific research (e.g., the extension and further generalization of number concept).

«Конструкция» - это рисование геометрических фигур с высокой степенью точности. Выполненная конструкция является как доказательством существования геометрического объекта, так и решением задачи. Древние греки были убеждены, что все фигуры самолета могут быть построены только с помощью компаса и линейки. Их методы деления пополам отрезка и угла гениальны и их трудно улучшить. Они работали со всеми числами геометрически. Длина была выбрана для представления числа 1, а все остальные числа были выражены через эту длину.

Они решали уравнения с неизвестными с помощью ряда геометрических построений. Ответами были отрезки, длина которых была неизвестным искомым значением. Греки наложили ограничения на линейность и компас для построения задач. Предполагается, что эта традиция была начата Платоном, величайшим философом Греции. Он утверждал, что более сложные инструменты требуют ручного труда, недостойного мыслителя. Грекам не удалось найти решение знаменитых задач по указанным ограничениям не из-за отсутствия изобретательности геометров. (Знаменитые проблемы неразрешимы, потому что они включают в себя иррациональные числа, которые не могут быть построены евклидовыми методами.) Постоянные попытки греков найти компас и прямолинейные способы трехугольной обработки угла, возведения в квадрат круга и дублирования куба были бесполезны для почти 2000 лет. По пути греки сделали великие математические открытия. Желание получить полное понимание теоретического характера проблем вдохновило многих великих математиков, в том числе Декарта, Гаусса, Понселе, Линдеманна, - лишь некоторые из них. Долгие годы работы над этими «непрактичными», «бесполезными» проблемами указывают на осторожность, терпение, настойчивость и строгость математиков в их попытках выполнить конструкции и обосновать их теоретически. Проблемы не исчерпали себя. Даже в наши дни некоторые авторы научных работ издают «решения», содержащие некоторые ошибки. Поиск строгого решения привел к большим открытиям и новым разработкам в математике. Он ввел новые геометрические концепции (например, конические сечения), поднял ряд важных теоретических вопросов (например, чтобы доказать невозможность решения) и предложил совершенно новое направление для научных исследований (например, расширение и дальнейшее обобщение концепции числа).

LABORATORY PRACTICE

I. We honour ancient Greece as the cradle of modern science; it was in ancient Greece that the first math, astronomical and physical theories originated and developed. 2. The Greeks’ contributions to philosophy, art, literature and architecture are as significant today as they were in antiquity. 3. Nevertheless, the contribution of the Greeks that determines most the character ofthe present-day civilization was their maths. 4. The Greeks must be credited with the founding of maths as scientific discipline; even among the Greeks themselves maths was set up as the standard for all the sciences. 5. The Greeks were the first people to pursue maths as an art for its own sake. Pure maths emerged when the Greeks began to think of numbers as numbers and of shapes as shapes. 6. The Greeks were the first to formulate the two mental processes vital to all math progress: abstraction and proof. 7. Abstraction is the art of perceiving common qualities in different things and forming a general idea therefrom. 8. Proof is the art of arguing from premises to a conclusion in such a way that no flaws can be picked in any step of the argument. 9. Using the information of the premises the Greeks proved by a reasoning process, known as deduction, the inescapable conclusion. 10. There are two main forms of thinking —- deduction and induction. For the former we are chiefly indebted to the Greeks. They first saw clearly revealed the great power of announcing general axioms or assumptions and deducing from these a useful array of implied propositions. ll. lnductive thinking proceeds in the opposite direction from deduction. Starting from the facts of experience, it leads us to infer general conclusions. lnductive reasoning produces in most cases an uncertain inference. l2. Deductive reasoning is flawless, definite and absolute. Its specific inferences follow, inescapably from the general assumptions. [3. The Greeks converted maths from empirical science into a deductive system of thought. Greece is the mother of logic. A logical deductive system must start somewhere and according to the Greeks‘ criterion it must start with a list of definitions, axioms and postulates. 14. It is always better in pure science —- the Greeks claimed — to assume as little as possible at the start and from a few assumptions to deduce as much as one can. 15. The Greeks created the theory ofthe logical discourse and they embodied it in the first model of material axiomatic system — Euclidean geometry. l6. Euclid was genius for system; his work Elements is a monument of the classical age maths. There were many “elements” before Euclid; there was none after him. 17. Right up to and including the present time, Euclid’s masterpiece serves as the highest standard of logic, rigour and perfect reasoning for all scientific treatments. 18. The Greeks had only one space and only one geometry; these were absolute concepts. [9. For more than twenty centuries mathematicians did not doubt the absolute truth of Euclidean geometry. Euclidean geometry was all ofgeometry; it is no more. 20. The challenging idea of a non—Euclidean geometry originated in the 19th century, simultaneously and independently in different countries. 2]. Lobachevsky —- one of the greatest Russian mathematicians — revolutionized the science of space and objects in space. 22. With the discovery of non-Euclidean geometry, mathematicians realized that there is more than one conceivable space and hence more than one geometry. 23. In the twentieth century geometry lost its former intimate connection with physical space and the study of “abstract spaces” was inaugurated. 24. The creators of non-Euclidean geometry did not think of its practical applications. It was pure science. 25. Hilbert built a model of non-Euclidean geometry, thereby the pure science received its theoretical justification. 26. After the days of Lobachevsky it became the fashion to challenge axioms. 21. There developed the concept offormal axiomatics and postulate sets for a large variety of geometries were investigated. Axiomatics as a science came into being. 28. Whereas the axiomatic method was formerly used for explaining the foundations of maths, nowadays it is a tool for concrete math research. 29. Einstein applied Riemann’s and Minkovsky‘s non—Euclidean geometries in his relativity theory. Thus, pure science obtained its practical justification. 30. Einstein’s geometry is four-dimensional. Space—time is its fourth dimension. Contemporary mathematicians speak of nth dimensional geometries. 3l. Euclidean geometry nowadays is only one applied science furnishing an interpretation of Hilbert’s pure science. There are an infinite number of others besides. 32. Today mathematicians claim that geometry is not a separate math discipline, but a particular point of view — a particular way of looking at a subject.