Discussion of facts, ideas and concepts

Task 13. Answer the following questions.

1. What position did geometry hold at the beginning of the 17th century?

2. What kind of investigations took place within the frames of geometry and algebra and what did they aim at?

3. How did the experiments in mathematics held in the ancient classics differ from those of the 17th century?

4. What was the profound impact of Descartes⁹ mathematics rooted in and what did Descartes try to demonstrate?

5. Did he ever graph an equation?

6. What were the steps Descartes followed in representing curves algebraically?

7. Why is it difficult for a person well schooled in modern math­ematics to enter into and appreciate the linguistic issues in­volved in 17th century mathematics?

8. How did we come to trust symbolic manipulations?

9. What does the author think one must be able to do to rein­force his trust in symbolic manipulations?

10. When was the progress in mathematics made and under what achievements?

11. What student could miss the main point of Leibniz’s symbolic achievements?

12. What issue did Descartes’ curve-drawing devices rise poign­antly?

13. How did the general orientation change during the seven­teenth century?

14. A wide variety of motions and devices used transformed clas­sical problems into locus problems, didn’t they?

15. What is “linkage” and what curves can be constructed with it?

16. Where has the finding of equations that model motion always been a fundamental concern?

Task 14. In the text you can find dither explicit or implicit in­dication of time when some events, facts or ideas important for tracing the historic developments in science occur. The time indi­cations given in the list below follow the order they appear in the text. Tell what events, facts, ideas or technologies are associated with this time. Arrange them according to the chronological order: the beginning of the seventeenth century; throughout the scientific revolution; ancient classics; the seventeenth century; today; modem mathematics; seventeenth century mathematics and science; more than a century after Descartes; throughout the seventeenth century; 200 В. C.; (Leibniz) his day; modem mathematicians; recently; since the eighteenth century; during the seventeenth century; dur­ing the Renaissance; during much of the seventeenth century; mod­em computer technology; advent of dynamic geometry software; re­cently; seventeenth century mechanical geometry.

Task 15. Read the following sentences paying special attention to the italicized words.

1. Geometry, however, held a preeminent position as an older and far more trusted form of mathematics.

2. Throughout the scientific revolution geometry continued to be thought of as the primary and most reliable form of math­ematics.

3. We have all been thoroughly trained in algebra and calculus and have come to rely on this language and grammar as a dominant form of mathematical representation.

4. We inherently trust that these symbolic manipulations will give results that are compatible with geometry; a trust that did not fully emerge in mathematics until the early works of Euler.

5. Such trust became possible because of an extensive set of rep­resentational experiments conducted throughout the seven­teenth century which tested the ability of symbolic algebraic language to represent geometry faithfully.

6. Because of our cultural trust in reliability of symbolic lan­guages applied to geometry, many have learned comparatively little about geometry in its own right.

7. Such progress was possible only after mathematicians had achieved a basic faith in the ability of algebraic language to represent and model geometry accurately.

As you have noticed all the italicized words have a likeness in denotation and can be identified in the context of this article as synonyms that is having the same or very nearly the same essential meaning. They can be assembled in one group with common de­notation which in case of verbs “rely⁵ and “trust” means “to have or place full confidence”; and in case of nouns “trust”, “faith”, “re­liance” denotes “the feelings that a person or thing will not fail in loyalty, duty or service.” Besides being synonyms the words com­ing into this group are distinguished from one another by an added implication or application so they may be, and usually are inter- changable within limits. Now read the discriminating articles for the words earlier grouped as synonyms.