数学专业英语(2)

1、Mathematical English 2: Geometry and TrigonometryMathematical EnglishDr. Xiaomin ZhangEmail: zhangxiaomin2.2 Geometry and TrigonometryTEXT A Why study geometry?Why do we study geometry? The student beginning the study of this text may well ask, “What is geometry? What can I expect to gain from this

2、study?”Many leading institutions of higher learning have recognized that positive benefits can be gained by all who study this branch of mathematics. This is evident from the fact that they require study of geometry as a prerequisite to matriculation in those schools.Geometry had its origin long ago

3、 in the measurements by the Babylonians and Egyptians of their lands inundated by the floods of the Nile River. The Greek word geometry is derived from geo, meaning “earth”, and metron, meaning “measure”. As early as 2000 B.C. we find the land surveyors of these people re-establishing vanishing land

4、marks and boundaries by utilizing the truths of geometry.Geometry is a science that deals with forms made by lines. A study of geometry is an essential part of the training of the successful engineer, scientist, architect, and draftsman. The carpenter, machinist, stonecutter, artist, and designer al

5、l apply the facts of geometry in their trades. In this course the student will learn a great deal about geometric figures such as lines, angles, triangles, circles, and designs and patterns of many kinds.One of the most important objectives derived from a study of geometry is making the student be m

6、ore critical in his listening, reading, and thinking. In studying geometry he is led away from the practice of blind acceptance of statements and ideas and is taught to think clearly and critically before forming conclusions.There are many other less direct benefits the student of geometry may gain.

7、 Among these one must include training in the exact use of the English language and in the ability to analyze a new situation or problem into its basic parts, and utilizing perseverance, originality, and logical reasoning in solving the problem. An appreciation for the orderliness and beauty of geom

8、etric forms that abound in mans works and of the creations of nature will be a byproduct of the study of geometry. The student should also develop an awareness of the contributions of mathematics and mathematicians to our culture and civilization.TEXT B Some geometrical terms1. Solids and planes. A

9、solid is a three-dimensional figure. Common examples of solid are cube, sphere, cylinder, cone and pyramid.A cube has six faces which are smooth and flat. These faces are called plane surfaces or simply planes. A plane surface has two dimensions, length and width .The surface of a blackboard or a ta

10、bletop is an example of a plane surface.2. Lines and line segments. We are all familiar with lines, but it is difficult to define the term. A line may be represented by the mark made by moving a pencil or pen across a piece of paper. A line may be considered as having only one dimension, length. Alt

11、hough when we draw a line we give it breadth and thickness, we think only of the length of the trace when considering the line. A point has no length, no width, and no thickness, but marks a position. We are familiar with such expressions as pencil point and needle point. We represent a point by a s

12、mall dot and name it by a capital letter printed beside it, as “point A” in Fig. 2-2-1.The line is named by labeling two points on it with capital letters or one small letter near it. The straight line in Fig. 2-2-2 is read “line AB” or “line l”. A straight line extends infinitely far in two directi

13、ons and has no ends. The part of the line between two points on the line is termed a line segment. A line segment is named by the two end points. Thus, in Fig. 2-2-2, we refer to AB as line segment of line l. When no confusion may result, the expression “line segment AB” is often replaced by segment

14、 AB or, simply, line AB.There are three kinds of lines: the straight line, the broken line, and the curved line. A curved line or, simply, curve is line no part of which is straight. A broken line is composed of joined, straight line segments, as ABCDE of Fig. 2-2-3.3. Parts of a circle. A circle is

15、 a closed curve lying in one plane, all points of which are equidistant from a fixed point called the center (Fig. 2-2-4). The symbol for a circle is . In Fig. 2-2-4, O is the center of ABC, or simply of O. A line segment drawn from the center of the circle to a point on the circle is a radius (plur

16、al, radii) of the circle. OA, OB, and OC are radii of O, A diameter of a circle is a line segment through the center of the circle with endpoints on the circle. A diameter is equal to two radii. A chord is any line segment joining two points on the circle. ED is a chord of the circle in Fig. 2-2-4.F

17、rom this definition is should be apparent that a diameter is a chord. Any part of a circle is an arc, such as arc AE. Points A and E divide the circle into minor arc AE and major arc ABE. A diameter divides a circle into two arcs termed semicircles, such as AB. The circumference is the length of a c

18、ircle.SUPPLEMENT A Ruler-and-compass constructionsA number of ancient problems in geometry involve the construction of lengths or angles using only an idealised ruler and compass. The ruler is indeed a straightedge, and may not be marked; the compass may only be set to already constructed distances,

19、 and used to describe circular arcs.Some famous ruler-and-compass problems have been proved impossible, in several cases by the results of Galois theory. In spite of these impossibility proofs, some mathematical amateurs persist in trying to solve these problems. Many of them fail to understand that

20、 many of these problems are trivially soluble provided that other geometric transformations are allowed: for example, squaring the circle is possible using geometric constructions, but not possible using ruler and compasses alone. Mathematician Underwood Dudley has made a sideline of collecting fals

21、e ruler-and-compass proofs, as well as other work by mathematical cranks, and has collected them into several books.Squaring the circle The most famous of these problems, “squaring the circle”, involves constructing a square with the same area as a given circle using only ruler and compasses. Squari

22、ng the circle has been proved impossible, as it involves generating a transcendental ratio, namely 1:.Only algebraic ratios can be constructed with ruler and compasses alone. The phrase “squaring the circle” is often used to mean “doing the impossible” for this reason. Without the constraint of requ

23、iring solution by ruler and compasses alone, the problem is easily soluble by a wide variety of geometric and algebraic means, and has been solved many times in antiquity.Doubling the cube Using only ruler and compasses, construct the side of a cube that has twice the volume of a cube with a given s

24、ide. This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication, division, and taking square roots.Angle trisection Using only ruler and compasses, construct an angle that is one-third of a given arbitrary angle. This re

25、quires taking the cube root of an arbitrary complex number with absolute value 1 and is likewise impossible.Constructing with only ruler or only compassIt is possible, as shown by Georg Mohr, to construct anything with just a compass that can be constructed with ruler and compass. It is impossible t

26、o take a square root with just a ruler, so some things cannot be constructed with a ruler that can be constructed with a compass; but given a circle and its center, they can be constructed.Problem How can you determine the midpoint of any given line segment with only compass?SUPPLEMENT B Archimedes

27、and On the Sphere and the Cylinder Archimedes (287 BC-212 BC) was an Ancient mathematician, astronomer, philosopher, physicist and engineer born in the Greek seaport colony of Syracuse. He is considered by some math historians to be one of historys greatest mathematicians, along with possibly Newton

28、, Gauss and Euler.He was an aristocrat, the son of an astronomer, but little is known of his early life except that he studied under followers of Euclid in Alexandria, Egypt before returning to his native Syracuse, then an independent Greek city-state. Several of his books were preserved by the Gree

29、ks and Arabs into the Middle Ages, and, fortunately, the Roman historian Plutarch described a few episodes from his life. In many areas of mathematics as well as in hydrostatics and statics, his work and results were not surpassed for over 1500 years! He approximated the area of circles (and the val

30、ue of ) by summing the areas of inscribed and circumscribed rectangles, and generalized this method of exhaustion, by taking smaller and smaller rectangular areas and summing them, to find the areas and even volumes of several other shapes. This anticipated the results of the calculus of Newton and

31、Leibniz by almost 2000 years! He found the area and tangents to the curve traced by a point moving with uniform speed along a straight line which is revolving with uniform angular speed about a fixed point. This curve, described by r = a in polar coordinates, is now called the spiral of Archimedes.

32、With calculus it is an easy problem; without calculus it is very difficult. The king of Syracuse once asked Archimedes to find a way of determining if one of his crowns was pure gold without destroying the crown in the process. The crown weighed the correct amount but that was not a guarantee that i

33、t was pure gold. The story is told that as Archimedes lowered himself into a bath he noticed that some of the water was displaced by his body and flowed over the edge of the tub. This was just the insight he needed to realize that the crown should not only weigh the right amount but should displace

34、the same volume as an equal weight of pure gold. He was so excited by this idea that he reportedly ran naked through the streets shouting Eureka (I have found it). Give me a place to stand and I will move the earth was his boast when he discovered the laws of levers and pulleys. Since it was impossi

35、ble to challenge that statement directly, he was asked to move a ship which had required a large group of laborers to put into position. Archimedes did so easily by using a compound pulley system. During the war between Rome and Carthage, the Roman fleet decided to attack Syracuse, but Archimedes ha

36、d been at work devising a few surprises. There were catapults with adjustable ranges which could throw objects which weighted over 500 pounds. The ships which survived the catapults were met with poles which reached over the city walls and dropped heavy stones onto the ships. Large grappling hooks a

37、ttached to levers lifted the ships out of the water and then dropped them. During another failed assault, it is said that Archimedes had the soldiers of Syracuse use specially shaped and shined shields to focus the sunlight onto the sails to set them afire. This was more than the terrified sailors c

38、ould stand, and the fleet withdrew. Unfortunately, the city began celebrating a bit early, and Marcellus captured Syracuse by attacking from the landward side during the celebration. Archimedes, who was then, as fate would have it, intent upon working out some problem by a diagram, and having fixed

39、his mind alike and his eyes upon the subject of his speculation, he never noticed the incursion of the Romans, nor that the city was taken. In this transport of study and contemplation, a soldier, unexpectedly coming upon him, commanded him to follow to Marcellus, which he declined to do before he had worked out his problem to a demonstration; the soldier, enraged, drew his sword and ran him through. (Plutarch) Archimedes requested that his tombstone be decorated with a sphere contained in the smallest possible cylinder and inscribed with t