1、数学专业英语2Mathematical 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 study?”Many leading institutions of higher
2、 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 in the measurements by the Babylonians an
3、d 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 landmarks and boundaries by utilizing the trut
4、hs 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 all apply the facts of geometry in their tra
5、des. 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 more critical in his listening, reading, an
6、d 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. Among these one must include training in
7、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 geometric forms that abound in mans works and
8、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 solid is a three-dimensional figure. Commo
9、n 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 tabletop is an example of a plane surface.2.
10、 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. Although when we draw a line we give it bread
11、th 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 small dot and name it by a capital letter p
12、rinted 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 directions and has no ends. The part of the line
13、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 AB or, simply, line AB.There are three ki
14、nds 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 a closed curve lying in one plane, all po
15、ints 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 (plural, radii) of the circle. OA, OB, and OC a
16、re 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.From this definition is should be apparent
17、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 circle.SUPPLEMENT A Ruler-and-compass const
18、ructionsA 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, and used to describe circular arcs.Some f
19、amous 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 many of these problems are trivially solu
20、ble 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 false ruler-and-compass proofs, as well as oth
21、er 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. Squaring the circle has been proved impossible,
22、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 requiring solution by ruler and compasses alon
23、e, 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 side. This is impossible because the cube r
24、oot 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 requires taking the cube root of an arbitrar
25、y 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 to take a square root with just a ruler, so
26、 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 and On the Sphere and the Cylinder Archime
27、des (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, Gauss and Euler.He was an aristocrat, th
28、e 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 Greeks and Arabs into the Middle Ages, and, fo
29、rtunately, 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 value of ) by summing the areas of inscribed
30、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 Leibniz by almost 2000 years! He found the
31、 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. With calculus it is an easy problem; witho
32、ut 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 it 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 the same volume as an equal w
