I. Найдите в тексте интернациональные слова, переведите их.
II. Выберите в колонке В эквиваленты к словам колонки А.
| A
| B
| | 1.ordinate
2. perpendicular
3. abscissa
4. smooth curve
5. summarize
6. calculate
| a) непрерывная кривая
b) абсцисса
c) вычислять
d) ордината
e) суммировать
f) перпендикуляр
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III. Заполните пропуски подходящими по смыслу словами.
1. Each equation involving x and y can be pictured as a curve by interpreting x and y as ….
a) relation of points b) coordinates of points
c) mental notions d) symbols
2. Since each of these pairs of coordinates represents a point on the curve we can plot these points and join them by a ….
a) ordinary line b) smooth curve
c) vertical line d) perpendicular
3. The horizontal line is called the ….
a) X-axis b) Y-value
c) Y-axis d) X-value
4. The distance from P to the Y-axis, is called ….
a) origin b) ordinate of P
c) abscissa of P d) X-axis
5. The vertical line is called the ….
a) Y-value b) X-value
c) X-axis d) Y-axis
Text IV
To discuss the equation of a curve Descartes introduced a horizontal line called the X-axis, a point O on the line called the origin, and a vertical line through O called Y-axis. If P is any point on a curve, there are two numbers that describe its position. The first is the distance from O to the foot of the perpendicuar, from P to the X-axis. This number, called X-value, is the abscissa of P. The second number is the distance from P to the Y-axis, called Y-value or ordinate of P. These two numbers are called the coordinates of P and are generally written as P (x, y). The curve itself is then described algebraically by stating some equation which holds for x and y values of points on that curve and only for those points.
The heart of Descartes's and Fermat's idea is the following. To each curve there belongs an equation that uniquely describes the points of that curve and no other points. Conversely, each equation involving x and y can be pictured as a curve by interpreting x and y as coordinates of points.
Thus formally stated: the equation of any curve is an algebraic equality which is satisfied by the coordinates of all points on the curve but not the coordinates of any other point.
Since each of these pairs of coordinates represents a point on the curve we can plot these points and join them by a smooth curve. The more coordinates we calculate, the more points can, be plotted and the more accurately the curve can be drawn.
Beyond the analysis of properties of individual curves, the association of equation and curve makes possible a host of scientific applications of mathematics. Among the practical applications of mathematics we shall merely mention that all the conic sections possess the properties that make these curves effectively employed in lenses, telescopes, microscopes, X-ray machines, radio antennas, searchlights and hundreds of other major devices. When Kepler introduced the conic sections in astronomy they became basic in all astronomical calculations including those of eclipses and paths of comets.
To summarize, it was not so much the use of coordinates that made the work of Descartes and Fermat so important; coordinates were used effectively in antiquity, especially in the geometry of Appolonius, and again in the fourteenth century in a more primitive form in the latitude of forms of Oresme. Descartes saw as the objective of his work the cooperation of algebra and geometry to the end that mathematics might have the best aspects of both branches. In the end, however, it turned out that geometry lost popularity in the partnership. Pure geometry was so overshadowed that it made little progress during the next century and a half, during which time infinitesimal analysis went through a progress of arithmetization that amounted almost to a revolution.
IV. Выберите правильный ответ на вопрос в соответствии с содержанием текста.
1. Who introduced the conic sections in astronomy?
a) Kepler b) Descartes
c) Fermat d) Appolonius
2. What makes it possible the scientific applications of mathematics?
a) the association of equation and curve.
b) the combination of algebra and mathematical theory of probability.
c) the association of line and circle.
d) the combination of physics and mechanics.
3. How is the number of the distance from O to the foot of the perpendicular from P to the X-axis called?
a) ordinate of P b) Y-axis
c) Origin d) Abscissa of P (X- value)
4. What is the heart of Descartes’s and Fermat’s idea?
a) to each curve there belongs an equality that uniquely describes many points on the line.
b) to each curve there belongs an equality that uniquely describes the points on that smooth curve.
5. What shall we receive, if we plot points and join them?
a) smooth curve b) cube
c) rectangle d) triangle
V. Выберите заголовок для данного текста, в соответствии с его содержанием.
a. Coordinate system of Descartes
b. Kepler’s astronomical calculations
c. Horizontal and vertical lines
d. Abscissa and ordinate
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