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1、第四章科技术语构词中常见的前缀和后缀第四章科技术语构词中常见的前缀和后缀科技英语常见词缀举例如下：科技英语常见词缀举例如下：一、前缀（一、前缀（prefixes）1. a- 非，不非，不asymmetry 非对称性非对称性astray 偏离偏离2. aero- 空气，飞机空气，飞机aerodynamics 空气动力学空气动力学aero-craft carrier 航空母舰航空母舰 3. all- 全全all-weather transportation 全天候运输全天候运输all-round champion 全能冠军全能冠军4. anti- 反、逆、抗、防、耐反、逆、抗、防、耐antifre
2、eze pump 防冻泵防冻泵anti-phase 反相反相5. astro- 天文，宇宙天文，宇宙astronavigation 天文导航天文导航astrovehicle 宇宙飞行器宇宙飞行器6. auto- 自动，自自动，自autocoder 自动编码器自动编码器autochart 自动流程图自动流程图7. bi- 双，重双，重 bipolar relay 双极继电器双极继电器bicarbonate 重碳酸盐重碳酸盐8. bio- 生物，生生物，生bionics 仿生学仿生学biocatalyst 生物催化剂生物催化剂 9. cine- 电影电影cinefilm 电影胶片电影胶片 cine
3、microscopy 电影显微术电影显微术 10. co- 共，同共，同 coaxial cable 同轴电缆同轴电缆cosine 余弦余弦11. contro- 反，逆反，逆controsurge winding 防冲屏蔽绕组防冲屏蔽绕组controflow 逆流逆流12. counter- 逆，反逆，反counteractant 中和剂中和剂counterpart 对方对方13. de- 除，脱除，脱defroster 除霜剂除霜剂derailment 脱轨脱轨14. deci- 十分之一十分之一 ，十分，十分decimal 十进制十进制decibel 分贝分贝15. di- 双，偶，两双
4、，偶，两diode 二极管二极管dibit 两位两位16. du- 二，双二，双duplicate 一式两份一式两份duoplasmatron 双等离子体发射器双等离子体发射器17. equi- 同等，均同等，均equipartition 均分均分equilibrium 均衡均衡18. ferro- 铁，钢铁，钢ferroalloy 铁合金铁合金ferroconcrete 钢筋混泥土钢筋混泥土19. gyro- 陀螺，旋转陀螺，旋转gyroscope 陀螺仪陀螺仪gyrosphere 回转球回转球20. hecto- 百百hectare 公顷公顷hectolitre 百升百升21. hemi-
5、 半半hemisphere 半球半球hemipyramid 半锥体半锥体22. hepta- 七七heptagon 七角形七角形heptad 七价原子七价原子23. hexa- 六六hexagon 六角形六角形hexaploid 六倍体六倍体24. homo- 同，相似同，相似homogeneity 同种同种homoclime 相同气候相同气候25. hydro- 水，氢化水，氢化hydroelectric locomotive 液力传动内燃机车液力传动内燃机车hydrofining 氢化提纯氢化提纯26. hyper- 高，超，重，过高，超，重，过hypertension 高血压高血压hyp
6、ervelocity 超高速超高速27. infra- 下，亚，次下，亚，次infrared rays 红外线红外线infrastructure 地基地基28. inter- 相互，际间相互，际间internet 互联网互联网intercity train 城际列车城际列车29. iso- 等，同等，同isotherm 等温线等温线isotope 同位素同位素30. macro- 大，宏观，常量大，宏观，常量macrocontrol 宏观调控宏观调控microanalysis 常量分析常量分析31. mal- 不，失不，失malnutrition 营养不良营养不良malfunction 失灵失
7、灵32. mega- 兆，百万兆，百万megalopolis 特大城市特大城市megaton 百万吨级百万吨级33. micro- 微观微观microelectronics 微电子学微电子学microfilm 缩微胶卷缩微胶卷34. mono- 单，一单，一monocle 单片眼镜单片眼镜monochord 单弦琴单弦琴35. multi- 多多multiplane 多翼机多翼机multipole 多极多极36. ortho- 正，直正，直orthocenter 垂心垂心orthograph 正投影图正投影图37. over- 过，超过，超overrun 超限超限overexposure 曝光
8、过度曝光过度38. photo- 光电，光敏光电，光敏photosynthesis 光合作用光合作用photoreceptor 感光器感光器39. poly- 多，聚多，聚polymer 聚合物聚合物polyethylene 聚乙烯聚乙烯 40. post- 后后post-acceleration 偏转后加速偏转后加速postmeridian 午后午后41. pseudo- 伪，假伪，假pseudo-program 伪程序伪程序pseudophotoesthesia 光幻觉光幻觉42. quadr- 四，二次四，二次quadruplicate 一式四份一式四份quadraphonic 四声道立
9、体声四声道立体声43. quasi- 准，拟准，拟quasi-high-speed 准高速准高速quasi-ordering 拟序拟序44. semi- 半半 semi-final 半决赛半决赛semitrailer 半挂车半挂车45. stereo- 立体的立体的 stereooptics 立体光学立体光学stereophone 立体声耳机立体声耳机46. sub- 子，亚子，亚 subloop 子回路子回路subsoil 亚土层亚土层47. super- 超超superhighway 超级公路超级公路superconductor 超导体超导体48. tele- 远，电远，电telemetr
10、y 遥测术遥测术telecommunication 电信电信49. thermo- 热热thermostat 恒温器恒温器thermoscreen 隔热屏隔热屏50. trans- 超，越，变换超，越，变换transparency 透明度透明度transpassivation 过钝化过钝化51. tri-，ter- 三三tertiary 第三产业第三产业triplicate 一式三份一式三份52. ultra- 超，过超，过ultrasonic 超声波超声波ultrastructure 超微结构超微结构二、后缀（二、后缀（suffixes）1. -er（or）器，机器，机air-oil boo
11、ster 气气-液增压器液增压器air compressor 空气压缩机空气压缩机2. -ite 石，矿物石，矿物granite 花岗石花岗石sulphite 亚硫酸盐亚硫酸盐3. -meter 表，计表，计speedometer 速度计速度计ohmmeter 电阻表电阻表4. -ity 性，度性，度absolute viscosity 绝对粘度绝对粘度track elasticity 轨道弹性轨道弹性5. -scope 仪，镜仪，镜spectroscope 光谱仪光谱仪telescope 望远镜望远镜6. -graph 仪，机仪，机spectrograph 摄谱仪摄谱仪telegraphy
12、电报机电报机（2）空间顺序（）空间顺序（spatial order）按按事事物物在在空空间间所所处处位位置置的的排排列列情情况况所所进进行行的的一一种种描描述述方方法法，比比如如通通常常是是由由前前到到后后、由由内内而而外外、从从左左至至右右、从从顶顶部部到到底底部部、从从近近处处到到远远处处等等。这这对对仪仪器器设设备备的的描描述述是是很很有有用的。用的。 表示位置的常用语有以下一些：表示位置的常用语有以下一些：A. 名词名词 front, back, side, center, middle, top, bottom, end, edge, interior, inside, exteri
13、or, outside, border, perimeter B. 形容词形容词 front, back, rear, central, superior, upper, inferior, lower, inner, interior, outer, exterior, near, far, distant, parallel, perpendicular, vertical, horizontalC. 副词副词forward, backward, sideways, up, down, upward, clockwise, counter-clockwise D. 介词介词 in fron
14、t of, behind, at the back of, on, above, over, under, below, beneath, underneath, at the top of, along, in the middle of, between, through, among, within, throughout, outside, around, beyond, at the end of, across, from, past, opposite, near, towards, to the right of, from to （3）分门别类法（）分门别类法（classif
15、ication或或division）把一个广泛的题材分成几个类别，然后分类叙述。把一个广泛的题材分成几个类别，然后分类叙述。常用的词汇有：常用的词汇有： A. 形容词形容词 major, main, primary, secondary, former, latter, most important, least importantB. 名词名词 class, kind, type, division, classification, aspect, part, characteristic C. 动词动词 divide into, group into, put into, fall int
16、o, consist of, classify as (into), categorize as (into) A. 用于比较的词语用于比较的词语similarly, likewise, correspondingly, in the same way, to resemble, to equal, both and , either or, neither nor, not only but also , like, identical, equally, similar to, equal to, a similarity B. 用于对照的词语用于对照的词语 to differ from
17、(in), to contrast with, to distinguish between, in contrast, on the other hand, not as as, unlike, while, whereas, although, differently, even if, in contrast to, as opposed to, distinct from, different from, however, but, more (less) than, a contrast, a dissimilarity, a difference 写摘要常用的句式写摘要常用的句式:
18、 :介绍论文内容或作者的观点介绍论文内容或作者的观点The authors present aprinciple that predict作者提出了一种作者提出了一种原理，该原理能预测原理，该原理能预测 The paper presents the data in terms of本文用本文用给出了给出了的数据的数据The brief history of is presented.本文介绍了本文介绍了 的简史。的简史。The authors describe a configuration of作者描述了一种作者描述了一种构造。构造。This paper describes the prin
19、ciples and techniques of 本文描述了本文描述了的原理和技术。的原理和技术。Performance goals and various approaches to are briefly described.简述了性能目标和达到简述了性能目标和达到的各种途径的各种途径。This paper addresses the problem of本文论述本文论述的问题。的问题。The first paper deals with第一篇论文论述了第一篇论文论述了This paper is devoted to assessment of本文专论如何评估本文专论如何评估This pa
20、per is concerned with the analysis of本文对本文对进行了分析。进行了分析。The article looks at 本文考察了本文考察了介绍科技论文研讨的课题介绍科技论文研讨的课题The performance characteristics ofare studied theoretically and experimentally.对对的性能特征进行了理论和实验研究。的性能特征进行了理论和实验研究。The paper examinesand consider本文研究了本文研究了并考虑了并考虑了 The authors consider two speci
21、fic subjects which作者考虑了两个作者考虑了两个的专门课题。的专门课题。 A principle of constructingis considered in this paper本文考虑了建立本文考虑了建立的原则。的原则。This article discusses the reasons for and offers an insight into本文讨论了本文讨论了的原因并提出了对的原因并提出了对的观点。的观点。The influence ofonis investigated.研究了研究了对对所产生的影响。所产生的影响。This paper analyzes some
22、 important characteristics of本文分析了本文分析了 的一些重要特性。的一些重要特性。This process is analyzed using使用使用对这一过程进行了分析。对这一过程进行了分析。介绍论文涉及的范围介绍论文涉及的范围Some of the specific topics discussed are:论述的专门课题包括论述的专门课题包括The article contains some practical recommendations on本文包含了一些有关本文包含了一些有关的实际建议。的实际建议。The scope of the research
23、covers本研究的范围涉及本研究的范围涉及 Subjects covered include涉及的课题包括涉及的课题包括综述与概括对某一领域的研究课题综述与概括对某一领域的研究课题This paper reviews, summarizes the theory fromviewpoint, discusses, and presents 本本文文综综述述了了，从从观观点点概概述述了了这这一一理理论论，并并讨论了讨论了，还提出了还提出了 介绍论文重点介绍论文重点 This report concentrates on 本报告的重点是本报告的重点是Attention is concentrat
24、ed on重点是重点是Particular attention is/was paid to特别重视特别重视 There has been a focusand attention is being paid to一直是重点，而且现在都重视一直是重点，而且现在都重视The greatest emphasis has been on极其重视极其重视The primary emphasis in this paper is on本文的重点是本文的重点是介绍论文的目的介绍论文的目的This paper seeks to justifyin terms of本文追求的目标是从本文追求的目标是从角度来论
25、证角度来论证One of the purposes of this study is本研究的一个目的是本研究的一个目的是The aim of this study is to carry out analysis for本项研究的目的是对本项研究的目的是对进行分析。进行分析。 The primary objective of the study was to determine本研究的主要目标是确定本研究的主要目标是确定The work is aimed at examining本研究的目的是研究本研究的目的是研究介绍成果的获取和开发介绍成果的获取和开发This paper provides
26、the quantitative background to本文为本文为提供了量化基础。提供了量化基础。The computed simulation ofis provided for为为进行了进行了计算机模拟。计算机模拟。The authors have developed two method of作者制定了作者制定了的两种方法。的两种方法。A theoretical model has been developed to predict研制了一种理论模型来预测研制了一种理论模型来预测 A theoretical treatment of a new model ofis given.本
27、文给出了本文给出了的新模型的理论处理方法。的新模型的理论处理方法。Data pertinent toare given.给出了有关给出了有关的资料。的资料。介绍观察和证明介绍观察和证明New evidence indicates that新的证据表明新的证据表明 In this paper, the authors show how文中作者们说明了怎样文中作者们说明了怎样 Studies have shown that 研究表明研究表明It is shown that已证明已证明In this paper it is demonstrated that本文已经证明本文已经证明介绍运算和计量介绍
28、运算和计量As part of, the authors have measured the minimum velocities of作为作为的一部分，作者测量了的一部分，作者测量了的最小速度。的最小速度。The main body orientation is measured by means of主体定位是通过主体定位是通过计量的。计量的。This paper determines the energetically rational range of本文确定了本文确定了的能量合理范围。的能量合理范围。介绍应用与用途介绍应用与用途Two basic technologies are u
29、sed to lay down 采用了两种基本技术来建立采用了两种基本技术来建立Such a statistical method has been applied to已将这样一种统计法用于已将这样一种统计法用于介绍评估与比较介绍评估与比较常用词语有常用词语有evaluate, assess, compare, agree with等。等。The present work evaluate the low-temperature properties of本项研究评估了本项研究评估了的低温性质的低温性质The role of has been critically assessed by严格
30、评审了严格评审了的作用。的作用。The calculated results agree with the experimental results.计算结果与实验结果完全一致。计算结果与实验结果完全一致。介绍试验与实验介绍试验与实验常用词语有常用词语有test, experiment等。等。The experiments onhave been carried out using使用使用进行了进行了的实验的实验 。The tests were carried out on试验是在试验是在上进行的。上进行的。介绍论证与依据介绍论证与依据常常用用词词语语有有be based on, base o
31、n, take an reference等。等。The method is based on这种方法的基础是这种方法的基础是Constitutive equation ofis formulated based on的构成式是以的构成式是以公式为基础的。公式为基础的。介绍推荐与建议介绍推荐与建议常用的词语有常用的词语有propose, suggest, recommend等。等。The author proposes an approach to作者建议创立一种方法来作者建议创立一种方法来Suggestions were made for further study of提出了进一步研究提出了
32、进一步研究的建议。的建议。 It is suggested that有人建议有人建议 Recommendations are made regarding提出了有关提出了有关的建议。的建议。介绍结论的语句介绍结论的语句常用词语有常用词语有conclude, arrive at等。等。The paper concludes that 本文的结论是本文的结论是 It was concluded that所作出的结论是所作出的结论是 例如：例如：普通英语词普通英语词 科技英语中的非技术词科技英语中的非技术词 （一般语体）（一般语体） （正式语体）（正式语体） about approximately
33、ask inquire begin commence buy purchase carry bear change transform cheap inexpensive end conclude finish complete 普通英语词普通英语词 科技英语中的非技术科技英语中的非技术词词 （一般语体）（一般语体） （正式语体）（正式语体） get obtain give accord have possess method technique obtainable available avoid circumvent quick rapid say remark similar ident
34、ical touch contact try endeavor 普通英语词普通英语词 科技英语中的非技术词科技英语中的非技术词 （一般语体）（一般语体） （正式语体）（正式语体） use employ或或utilize put place fire flame happy excited careful cautious care caution heart center enough sufficient by which whereby in which wherein in the end eventually （2）一一些些短短语语动动词词往往往往由由正正式式动动词词（通通常常来来源源
35、于于拉拉丁丁语语）替替代代，表表现现出出科科技技英英语语行行文文要要求求精精练练，表表达达上力避冗繁。上力避冗繁。例如：例如：短语动词正式动词短语动词正式动词 turn on complete take in absorb push in insert push down depress put up erect put together aggregate put out extinguish put in add 短语动词正式动词短语动词正式动词 wear away erode take away remove send for summon use up consume；exhaust
36、carry out perform come across encounter make up invent put up with tolerate go with accompany fill up occupy 短语动词正式动词短语动词正式动词 bring out introduce break up rupture look at（over） examine find out discover drive forward propel keep up maintain turn upside down invert set fire（light）to ignite makeweak w
37、eaken takeinto places dismantle turninto liquid liquefy be made up of be composed of 下面是写正文时常用的一些句型：下面是写正文时常用的一些句型：阐明论文目的（宗旨阐明论文目的（宗旨）时常用的句型：时常用的句型： a. 本文的主要目的是本文的主要目的是 The purpose of this paper is The primary goal of this research is The intention of this paper is to survey The overall objective of
38、 this study is In this paper, was provided（given, illustrated） Our goal has been to provide In this paper, we aim at The emphasis of this study lies inb. 该项工作是关于该项工作是关于 The present work is concerned with the processes underlying The above work deals with the mechanism involved in Our work at present
39、 is devoted to the changes occurring in The work we have done bears on the effects produced by The work we are doing is closely related to the deliberations described in Compared with the current research, the previous work was in connection with the investigation made by The work presented in this
40、paper focuses on several aspects of the followingThe authors are now initiating some experimental in investigation to establish Our studies were performed to determine the amount of The problem we have outlined deals largely with the study of c. 该项工作的主要目的是该项工作的主要目的是 The chief aim of the present work
41、 is to investigate the feature of The main purpose of our recent work was to examine the properties of The major object of their further work will be research into the nature of Our work in this direction has been intended to provide some information about The work which is being done now is designe
42、d to find some solution to the problem of Work of this kind has been started in order to assess the role of Previous work in this area was initiated in the hope of obtaining data in We did this work hoping to establish The authors undertook the work with a view to show The authors undertook the work
43、 with a view to show By doing this work we expect demonstrate Doing this joint work, we intend to show The primary objective of our work was to define the reasons for redundant protection systems and give examples of systems used in With research of many years, the authors endeavor is to explain why
44、 in The above-mentioned work has been performed with a view of d. 该项研究的主要目的是该项研究的主要目的是 The chief aim of the present study is to find out The primary object of this fundamental research will be to reveal the cause of The main objective of our investigation has been to obtain some knowledge of The stu
45、dy ofwas made in order to clarify The above study has been carried out to determine Their studies were intended to elucidate Those earlier investigations are in hopes of providing evidence for We performed this present study in the hope that we might bring to light some facts about The authors attem
46、pted their studies in order that they might We undertook the above study to the end that we might They initiated the present study so that they might Carrying out this study we hope to find out Performing these studies the author intended to reveal With recent research, the author intends to outline
47、 the framework ofe. 该项实验主要是为了该项实验主要是为了 We made these experiment in order to show a correlation between The author attempted the set of experiment with a view to demonstrate certain phenomena Experiments onwere made in order to measure the amount of Experiments involvinghave been carried out to deter
48、mine Experiments using this technique are under way to test（check） Experiments onare made to answer the question whether Experiment being made by our research group is aimed at obtaining the result of We have designed such an experiment for the purpose of 指明研究范围时常用的句型：指明研究范围时常用的句型： a. 该问题是关于该问题是关于的研
49、究的研究 This problem is concerned （chiefly）with the study of The problem ofdeals（ largely） with the determination of The problem they have advanced bears on the elucidation of The problem to be discussed in this paper is related closely with The problem described previously was directed to the example
50、The problem under investigation involves some significant agreements and disagreements in b. 这是研究这是研究的问题的问题 This is a problem concerned with the nature of This is a problem relating to the influence of This is a problem which deals with the role of This is a problem which bears on the effect of This
51、 is a problem in connection with This is a problem as mentioned previously inc. 该问题在该问题在范围之内（之外）范围之内（之外） The problem in within the scope of Our problem lies beyond the range of The problem you have just outlined seems to be outside of the province ofd. 该项研究涉及到该项研究涉及到范围范围 Studies of these effects cov
52、er various aspects of Investigation oninvolve also specific situation. Our studies with this technique are confined to one particular aspect More recent studies by this worker were chiefly concerned with the influence of The research on the conventional techniques explains well the phenomenon The ne
53、w findings from the experiment agree to the result obtained ine. 该研究进一步推广到该研究进一步推广到范围范围 The authors have extended their studies to some other related species We hope to expand our investigations to include other materials The research we have done deserves further studies. The investigation carried
54、out by the experts suggests an increase of the application in The approach under study predicates a wider range of application. 阐述理论时的常用句型：阐述理论时的常用句型： a. 该理论是该理论是提出来的提出来的 The theory ofwas created 50 years ago. The alternative theory was constructed in 80s. The theory ofwas developed in the 1999s. Th
55、e theory ofwas elaborated in the early 70s. The theory ofwas formulated as early as 2000. The theory ofwas advanced as long ago as the 1995s. The theory ofwas put forward as far back as 1995. The theory ofwas proposed last year. The theory ofwas suggested the year before last year. b. 该理论的内容（特点）是该理论
56、的内容（特点）是 Underlying this theory is the idea that The underlying concept of this theory is The underlying principle of the theory is as follows This is an alternative theory that This is a similar theory to the effect that This is working theory which is based on the idea that A basic feature of this
57、 theory is The fundamental feature of this theory is as follow The theory is characterized by prove to be the outstanding feature of the theory. c. 根据该理论，可得出根据该理论，可得出 According to Bs theory both processes occur simultaneously. As follows from the theory by N, such situation lead to As can be seen fr
58、om the theory advanced bysuch interactions are the basis of In the light of the theory, we have developed a variational method to handle such problems. Based on the theory, the author has obtained sufficient information from recent experiments with d. 该理论认为该理论认为 The kinetic theory holds that This ne
59、w-developed theory maintains that The theory advanced byclaims that The theory ofpredicts that The principal concept of the theory lies in The investigating theorye. 应用该理论（不）能说明应用该理论（不）能说明 We can interpret these findings in terms of the above theory. We cannot explain these phenomena using the above
60、 theory. We can hardly explain these findings applying this theory. These findings can be interpreted in terms of the above theory Such phenomena cannot be explained on the basis of this theory. The theory can apply to other cases of The theory developed byis valid（invalid）for other species. Xs theo
61、ry proved to be true for all steps of The above theory has been verified by many other authors. The very peculiar theory can be applied to a wider range of experiments in 描述实验时常用的句型：描述实验时常用的句型：a. 某项实验（所用设备）包括某项实验（所用设备）包括 The experiment consisted of three steps, which are described in Table 3. The be
62、st equipment which was used consisted of 5 forcemeasuring transducers and 2 recording instruments furnished by Included in the experiment were a high-pressure vessel assembly, 2UA-3 uranium analyzers, and a complete 1005050cm soil bin. All high-hydrogen creep-rupture tests were carried out in specia
63、lly designed high-pressure equipment rated at The winch is composed of the following main parts: a welded frame, a safety-brake clutch, a chain drum and a worn reducing gear.b. 到到，从未有过某方面的实验从未有过某方面的实验 An experiment of the kind has not been made. The kind of experiment you have in mind has never been
64、 carried out until now. The kind of experiment you mentioned would not be undertaken in the near future. So far we have not performed any experiment of this kind. Until now the authors have not attempted any field experiments of Research on the mobility of ions in neutral gases began more than 75 ye
65、ars ago, but very accurate data obtained through actual experiments have not appeared as yet. However, no direct experimental measurements of this have been reported. c. 进行该项实验，以检验进行该项实验，以检验 We（ have） carried out several sets of experiments to test the validity of They undertook many experiments to
66、support the hypothesis which We initiated experiments to prove They have undertaken special experiments to disprove A number of experiments were performed to check Many studies have to be attempted to verify More than one set of experiments must be initiated to substantiated. 某项实验证实（未能证实）某项实验证实（未能证实
67、） Experiments by M and his associates（have）showed Recent experiments in this area（ have）suggested that The present series of experiments（ have） indicated that The experiments reported here（ have）demonstrated that These experiments involvingconfirm our evidence that Our experiments withfailed to prov
68、ide These highly-technical experiments support our assumption that Our initial experiment confirms that the choice of concept is correct. The above-mentioned experiment has proved the newly develop technique is very reliable. 介绍方法时常用的句型：介绍方法时常用的句型： a. 该方法首先由该方法首先由发明（利用）发明（利用） The method ofwas first
69、developed by N. The method ofwas applied by B. The method ofwas first used by A in the early XXth century. The method ofwas first brought into used by C. The method ofwas came into use as long as 2000. The original proposal of this method was first made and published is 2002. The method of making a
70、frequency standard was not invented till the existence of b. 该方法与其他方法有该方法与其他方法有差别差别 The method ofdoes not differ at all the routine one. The method we use differs greatly from the one we used earlier.The method they developed is somewhat different from the conventional one. The new technique is quit
71、e different from the old one. The newly-elaborated technique has something（nothing，little）in common with the one previously used. The method of deriving a program described here differs from previous practice in thatc. 应用该方法可以进行应用该方法可以进行工作工作 The method allows us to demonstrate This technique enables
72、 us to observe The above procedure makes it possible to defect This method is capable of providing By the method we are able to investigate the insight of We have adopted the integrated filtering cell approach to miniaturize the prototype. The method proves to be useful in The interpretation effects
73、 were successfully evaluated by employing AIA.d. 该方法可称为（描述为）该方法可称为（描述为） The method used is known as The technique we applied is referred to as The procedure they followed can be briefly described as The approach adopted extensively is called Since the method was first put in use，we might as well nam
74、e it ase. 用该方法已获得用该方法已获得 We have obtained this information from recent experiments with Detailed information has been acquisited by the authors using X-ray analysis. The researchers have been recorded valuable data using the newly-developed method. The experts in the research center are satisfactory
75、 with the experimental results achieved by the application of The valuable information can not be obtained without using the method of展示结果时的常用句型：展示结果时的常用句型： a. 该项研究说明（指出、揭示）了该项研究说明（指出、揭示）了 The studies we have performed show that these particles are responsible for These pioneer studies that the auth
76、ors attempted have indicated marked variation in We carried out several studies which have demonstrated that the substance acts on The research we have done suggests an increase in The investigation carried out byhas revealed on resistance of Our research has supported the assumption that Their stud
77、ies have confirmed the opinion that This research bore out the idea that Laboratory studies ofdid not furnish any information about Our research carried out to the same effect failed to reveal any variations in Recent research into the nature offails to support the theory that Most recent studies sh
78、owed no changes in Further research reveals no evidence of All our preliminary results throw light on the nature ofb. 根据该项实验结果可得出根据该项实验结果可得出 More recent experiments in this area lead us to conclude that the phenomenon is related to Most recent experiments to the same effect have led the authors to b
79、elieve that the mechanism out of action. Further experiments in this area have enable these workers to suggest that As a result of our experiments, we concluded that From our experiments, the authors come to realize that From these experiments, one can conclude thatc. 该项工作提供了（没有得出）该项工作提供了（没有得出）结果结果
80、This fruitful work gives explanation of Their fundamental work provides some knowledge of The authors pioneer work has contributed our present understanding of This joint work of many years has led to further progress in Our integrated work of the past few years adds to our current concept of The re
81、search work has brought about a discovery of The research by workers result in a solution of Our work on（has）failed to give Their initial work withdid not result in This work, however, did not result in Further work in this field failed to contribute to These findings of the research have led the au
82、thor to a conclusion that d. 该项实验（研究）取得（没有取得）该项实验（研究）取得（没有取得）结果结果 Some result ofare obtained in the experiment with Sufficient result forhas been observed with the new method Further progress can be provided by these experiments No clear advancement has（so far）been seen in No direct outcome was then
83、 reported in No such finding could be available in No further result can be found with that technique. No experimental evidence has come to the studies ofe. 某某数数据据（图图、表表）总总结结（概概括括）了了从从得得到到的的结果结果 Figure 3 shows the results obtained from studies of Table 5 presents the data provided by the experiments
84、 on This table summarized the data collected during the experiment of The result reported above have been shown in Figure 1. Our experimental data are briefly summarized as follows Some of the authors findings are listed in tables. 作出评价时的常用句型：作出评价时的常用句型： The authors work onhas been very successful.
85、Our work involving studies ofdid not prove to be encouraging The new information has been insufficient to indicate that the change in dramatic. One of the assets of this technique is that it is very simple. The theory byseemed quite competent. We consider the new idea quite likely a suitable one. Th
86、e data available in literature are quite important. The newly-developed method has certain advantage over the existing methods. The author has satisfactorily come to a conclusion that These highly-technical experiments failed to provided any correlation between The theory ofdid not explain how much
87、modifications arose. These findings are not consistent with the current theory. So far there is not enough convincing evidence showing The above hypothesis seems rather interesting. The data obtained appear to be very similar to those reported earlier by There may be a different approach to the abov
88、e findings. Our experimental result does not seem to suggest the following assumptions. 利用上述句式，表示肯定评价的形容词，利用上述句式，表示肯定评价的形容词，如：如： good, satisfactory, competent, convincing, adequate, reliable, correct, valid, helpful, fruitful, successful, stimulating, attractive, interesting, promising, exciting, ca
89、reful, profound, comprehensive, useful, encouraging, extensive, clear, sufficient, etc. 表示否定评价的形容词，如：表示否定评价的形容词，如： deficient, faulty, inadequate, incomplete, unconvincing, unsatisfactory, meaningless, unreliable, incorrect, wrong, invalid, confusing, inexplicit, contradictory, controversial, diverge
90、nt, puzzling, conflicting, ambiguous, insufficient, etc. 表示感谢的句型很多，常见的有以下一些：表示感谢的句型很多，常见的有以下一些： I would like to thank我要感谢我要感谢 The author wishes to express his sincere thanks to 作者愿对作者愿对表示由衷的感谢。表示由衷的感谢。 I would like to acknowledge 我要感谢我要感谢The author wishes to make the acknowledgement of作者愿感谢作者愿感谢 Gra
91、teful acknowledgement is made to 对对深表感谢。深表感谢。 The author is in acknowledgement of 作者要感谢作者要感谢I acknowledge我要感谢我要感谢 I would like to express my gratitude to 我要感谢我要感谢 Special gratitude is owed to 我要特别感谢我要特别感谢 A special gratitude is expressed to 我要特别感谢我要特别感谢It is a pleasure to record my gratitude to 我要感谢
92、我要感谢 I am grateful to 我要感谢我要感谢 I greatly appreciate 我对我对极为感谢。极为感谢。 I would like to express my appreciation of 我要感谢我要感谢 I am greatly indebted to 我对我对极为感谢（深表谢意）极为感谢（深表谢意）。 I am indebtedness to 我对我对表示感谢。表示感谢。 We express our indebtedness to 我们对我们对表示感谢。表示感谢。I wish to thank 我要感谢我要感谢 Thanks must be given t
93、o 我得感谢我得感谢 Thanks must be extended（或或expressed）to 我得感谢我得感谢 Special thanks go to 我要特别感谢我要特别感谢 I offer my thanks to 我要感谢我要感谢A word of thanks to 我要感谢我要感谢 Particular recognition is due to我要特别感谢我要特别感谢We wish to give special thanks to 我们愿特别感谢我们愿特别感谢 His sincere thanks are offered to 作者对作者对表示衷心的感谢。表示衷心的感谢。
94、 We are thankful to 我们要感谢我们要感谢 We wish to record our appreciation to 我们愿感谢我们愿感谢 With a great sense of debt, we thank 我们对我们对深表感谢深表感谢。第四篇数、符号、公式、图形第四篇数、符号、公式、图形的英文表达（简介）的英文表达（简介）第二十九章数的表达第二十九章数的表达 数数的的系系统统：自自然然数数（Natural Numbers）、整整数数（Integers）、有有理理数数（Rational Numbers）、无无理理数数（Irrational Numbers）、实实数数（
95、Real Numbers）、虚虚数数（ Pure Imaginary Numbers） 、 复复 数数 （ Complex Numbers） 一、整数一、整数 35,896,732,546,285 读读作作：thirty-five trillion eight hundred and ninety-six billion seven hundred and thirty-two million five hundred and forty-six thousand two hundred and eighty-five 10亿以上的数，英式英语和美式英语表达不同。亿以上的数，英式英语和美式英语
96、表达不同。 billion：1109（美式）；美式）； 11012（英式）英式） trillion：11012（美式）；美式）； 11018（英式）英式） quadrillion：11015（美式）；美式）； 11024（英式）英式） quintillion： 11018 （美式美式）； 11030 （英式）英式） sextillion：11021（美式）；美式）； 11036（英式）英式） septillion：11024 （美式）；美式）； 11042 （英式）英式） octillion：11027 （美式）；美式）； 11048 （英式）英式） nonillion：11030（美式）；美
97、式）； 11054 英式）英式） decillion： 11033（美式）；美式）； 11060 英式）英式） 注注意意在在上上述述这这样样的的数数词词中中，hundred，thousand，million，billion，trillion等词一般是单数形式。等词一般是单数形式。 例如：例如：two hundred，three million，five trillion，etc. 第第403号房间号房间Room 403 读作：读作：room four oh one 电电话话号号码码：0371-7426329 Telephone Number: 0371-7426329 读读作作：zero th
98、ree seven one seven four two six three two nine 97次列车次列车Train Number 97 读作：读作：train ninety-seven或或the ninety-seventh train 美国西北航空公司美国西北航空公司3644航班航班Flight NW 3644 读作：读作：flight NW three six four four 公元前公元前753年（年（in）the 753 BC 读读作作：（in）the year seven hundred and fifty-three BC 20世纪世纪70年代年代1970s或或1970s
99、 读作：（读作：（in）nineteen seventies 1945年年8月月in August, 1945 读作：读作：in August nineteen forty-five 1997年年7月月1日日on July 1(st), 1997或或on 1(st ) July, 1997 读读作作：on July the first ninety-seven或或on the first of July nineteen ninety-seven 上午九点十分上午九点十分09:10 a.m. 读作：读作：（oh）nine ten a.m. 二、二、小数和分数小数和分数 329,671,239.
100、953579 读读作作：three hundred and twenty-nine million six hundred and seventy-one thousand two hundred and thirty-nine point nine five three five seven nine 2,050.0357 读读作作：two thousand and fifty point zero three five seven. .01 读作：读作：one hundredth（point zero one） .00001 读读作作：one millionth（point zero ze
101、ro zero zero one/ /point four zero one） 9.621 读作：读作：nine point six two one 读读作作：zero point three fives equals zero point five recurring 读作：读作： zero point three seven recurring 读读作作：zero point three seven repetend three, seven 读作：读作： zero point seven repetent seven 读读 作作 ： zero point two four, seven
102、three recurring/ /zero point two four, seven three repetend seven three 读读作作：Zero point four zeros one equals one over one hundred thousand equals one over ten to the fifth power equals ten to the minus fifth powe. 0.0000053= 5.30.000001= 5.310 6 读读作作：Minus zero point five zeros five three equals mi
103、nus five point three times zero point five zeros one equals minus five point three times ten to the minus sixth power. 读读作作：five and five-eighths（five and five over eight或或five and five divided by eight） 0.25= 0.25 1000 = 250 读读作作：Zero point two five equals zero point two five times thousand per mil
104、le equals two hundred and fifty per mille. 读读作作：Three divided by seven equals zero point four two eight five seven one four two eight five seven one, etc. equals zero point four two eight five seven one repetend four two eight five seven one. 读读作作：Zero point zero zero three percent equals zero point
105、 zero zero three over one hundred equals zero point four zeros three.第三十章符号的表达第三十章符号的表达 一、一般数学符号一、一般数学符号 2 + 3 读作：读作：two plus three 10 b 读作：读作：ten minus b x= 3 读作：读作：x equals minus or plus five x y 读作：读作：x times y或或x multiplied by y 9 4 读作：读作：nine divided by four 1 : 2 读作：读作：the ratio of one to two
106、a : b 读作：读作：a is to b 0.3% 读作：读作：zero point three percent 3.8 读作：读作：three point eight per mille a2b2 (ab)(ab) 读读作作：a squared minus b squared equals the quantity a plus b times the quantity a minus b.或或a squared minus b squared equals open parenthesis a plus b close parenthesis times open parenthesis
107、 a minus b close parenthesis. 读读作作：Open parenthesis six plus five and three-fourths minus four point six six multiplied by three close parenthesis divided by three and one seventh 读作：读作：The quantity a plus b to the nth power equals a to the nth power plus n times the quantity a to the n minus one po
108、wer times b plus n times the quantity n minus one divided by two times the quantity a to the n minus two power times b squared down to plus b to the nth power. 读作：读作：k equals eighty-one times two-thirds to the eight power. 读作：读作：x to the third power plus one overdivided by the quantity x squared plu
109、s one to the two-thirds power. 读读作作：The limit as n approaches infinity of the quantity one over n squared times the sum of n positive integers.或或The limit as n approaches infinity of the quantity one over n squared times one plus two plus up to plus n. 读读 作作 ： Fifteen times x squared minus open brac
110、ket minus four times x squared plus six open parenthesis x minus x squared close parenthesis minus three times x close bracket. 读读作作：One-half open brace, a open bracket b plus open parenthesis c minus d close parenthesis close bracket close brace. (ab)(ab) a2b2 读读作作：The product of the sum and differ
111、ence of two terms a and b equals the difference of their squares. 读读作作：c is identical with fivec is equivalent to fivec is identical to five. 读作：读作：c plus d times a is not equal to zero. 读作：读作：G equals m times g, and G is in direct proportion to m. 读读 作作 ： t equals three divide by seven is approxima
112、tely equal to zero point four two nine. 读读作作：Two over four is corresponding to ninety degrees. a 0 读读作作：Ell en x equals the logarithm to the base e of x, for x is greater than zero. f (x) = 1 + ln (x-2) 读读作作：The function f of x equals one plus ell en the quantity x minus two. 六、三角函数、反三角函数、双曲函数和反双曲函数
113、六、三角函数、反三角函数、双曲函数和反双曲函数 sin3 3sin 4sin3 读读作作：The sine of three theta is equivalent to three times the sine of theta minus four times the quantity sine theta cubed. 读读作作：The cosine of t equals minus the square root of one minus the quantity sine t squared. 读作：读作：The tangent of pi over four equals 1.
114、读读作作：The cotangent of t equals x over y equals the cosine of t over the sine of t. 读读作作：The secant of pi over four equals the square root of two. csc2 ctg2 1 读读作作：The cosecant alpha squared minus the cotangent alpha squared equals one. 读读作作：The arcsinethe inverse sine of minus one-half equals minus
115、the quantity pi over six. 读读作作：One-half y equals the arctangent of the square root of the quantity one minus x over one plus x. 读读作作：The arcsecant of x plus the arccosecant of x equals pi over two. 读读作作：The hyperbolic sine of x equals the quantity e to the xth power minus e to the minus xth power ov
116、er two. 读读作作：The hyperbolic tangent of x equals the quantity e to the xth power minus e to the minus xth power over e to the xth power plus e to the minus xth power. 读读作作：The arc-hyperbolic sine of x equals ell en the quantity x plus or minus the square root of x squared plus one. 七、集合七、集合 K = 1, 2,
117、 3, 4, 5 读读作作：K equals the set whose members（elements）are one, two, three, four, five. I1 = 0, 1, 1, 2, , n, n 读读作作：I sub one equals the set whose elements are zero, one, minus one, two, minus two, etc., till n, minus n. S = x | 5 x 7 读读作作：The set S is the set of numbers x is between five and seven.
118、 读作：读作：four is a member of N 读作：读作：minus one is not a member of N A是是B的子集，或的子集，或A包含于包含于B 读作：读作： A is a subset of set B or A is a part of B. AA = A 读读作作：The set consisting of all element common to A equals A itself. AB= BA 读读作作：The set consisting of all elements common to A and B equals the set consi
119、sting of all elements common to B and A. 若若AB= AB，则则A=B 读读作作：If the set consisting of all elements of A or B or both equals the set consisting of all elements of B or A both, then set A equals set B.第三十一章初等代数公式的表达第三十一章初等代数公式的表达 一、有理数的运算一、有理数的运算 加法的解称为和。加法的解称为和。 The sum is the answer in an addition.
120、加数加数+加数（被加数）加数（被加数）=和（合计）和（合计） Addend+Addend ( Augend )= Sum ( Total ) 减法的解称为差减法的解称为差 The remainder or difference is the answer in subtraction. 被减数被减数-减数减数=差差 Minuend - Subtrahend= Remainder 乘法的解称为积乘法的解称为积 The product is the solution for multiplication. 被乘数被乘数乘数乘数=积积 MultiplicandMultiplier=Product 除
121、法的解称为商除法的解称为商 The quotient is the answer in division. 被除数被除数除数除数=商商 DividendDivisor=Quotient 零除以任何一个不等于零的数，都得零。零除以任何一个不等于零的数，都得零。 Zero divided by any real number which is not zero equals zero. 正数的任何次幂都是正数正数的任何次幂都是正数。 Any power of a positive number is positive. 负数的奇次幂是负数，负数的偶次幂是正数。负数的奇次幂是负数，负数的偶次幂是正数
122、。 The odd power of a negative number is negative and the even power of a negative number is positive. (b m) n=b mn 读读作作：The nth power of b to the mth power equals the m times n power of b. a + b = b + a a b = b a 读读作作：a plus b equals b plus a; a times b equals b times a. a + ( b + c ) = ( a + b ) +
123、c 读读作作：a plus the quantity b plus c equals the quantity a plus b plus c. a ( b + c) = a b + a c 读读作作：a times the quantity b plus c is equal to a times b plus a times c. 二、整式的运算二、整式的运算 在在加加、减减、乘乘、除除、乘乘方方和和开开方方等等运运算算中中由由数数字字和和字字母（或变量母（或变量）表达的式子称为代数式。表达的式子称为代数式。 In the operation of addition, subtractio
124、n, multiplication, division, exponentiation（raising to power）and extraction of roots, the expression formed from any combination of numbers and variables is called an algebraic expression. 分分母母中中不不含含有有字字母母（或或变变量量）的的代代数数式式叫叫整整式式。整整式式中包括单项式和多项式。中包括单项式和多项式。 The algebraic expression without letters or v
125、ariables in denominators is called an integral expression. An integral expression includes monomials and polynomials. 除数字系数外其它均相同的项称为同类项。除数字系数外其它均相同的项称为同类项。 The terms that are identical, except for their numerical coefficients, are called similar terms or like terms. 同底数幂相乘，底数不变，指数相加。同底数幂相乘，底数不变，指数相
126、加。 In multiplication of power with the same base, keep the base and add their exponents. 多多项项式式除除以以单单项项式式，先先把把这这个个多多项项式式的的每每一一项项除除以以这这个个单单项式，再把所得的商相加。项式，再把所得的商相加。 When a polynomial is divided by a monomial, first have each term of the polynomial divided by the monomial, then add their quotients. 把把一
127、一个个多多项项式式化化为为n个个整整式式的的积积的的形形式式，叫叫做做把把这这个个多多项项式因式分解，也叫做把多项式分解因式。式因式分解，也叫做把多项式分解因式。 The process of changing a polynomial into a product of n integral expression is called factoring or factoring a polynomial. 把把多多项项式式各各项项都都有有的的因因式式提提到到括括号号外外的的过过程程称称为为提提取取公因式。公因式。 The process of abstracting the common f
128、actors of a polynomial out of their parentheses is called abstracting common factor. 三、分式和根式三、分式和根式 两两个个整整式式相相除除，且且作作为为除除数数的的整整式式中中含含有有字字母母（或或变变量），就叫做分式。量），就叫做分式。 Quotients of two algebraic expression with denominator containing letters（ or variables） are called fractional expression. 把把分分式式的的分分子子与与
129、分分母母分分解解因因式式，然然后后约约去去分分子子与与分分母母的公因式的过程叫做约分。的公因式的过程叫做约分。 Factor both the numerator and denominator of a fraction and then cancel their common factors. This process is called cancellation. 分分式式中中各各分分母母所所有有因因式式的的最最高高次次幂幂的的积积叫叫做做最最简简公公分分母。母。 The product of all the different prime factors in the denomina
130、tors of the fractions, each taken the greatest number of times it occurs in any of the denominators, is called the least common denominator. 把把n个个异异分分母母的的分分式式化化成成与与原原来来的的分分式式相相等等的的同同分分母母的分式，这个过程叫做通分。的分式，这个过程叫做通分。 The process of reducing fractions that have no common denominator to those that have a
131、common denominator is called the reduction of fractions to a common denominator. 约约去去被被开开方方数数或或式式与与根根指指数数之之间间的的公公约约数数，根根式式的的值不变。值不变。 If we eliminate the common factor between the power of the radicand or expression and the index of the radical and the value of the radical expression remains unchange
132、able. 把分母中的根式化去，叫做分母有理化。把分母中的根式化去，叫做分母有理化。 The process to eliminate the radical in a denominator is called rationalizing the denominator. 被被开开方方数数的的每每一一个个因因式式的的指指数数都都小小于于根根指指数数且且分分母母不不含根式的根式叫做最简根式。含根式的根式叫做最简根式。 The radical expression in which the index of every factor of the radical is less than the
133、 index of the radical and which is free of denominator is called the simplest radical or radical expression. 对于分数指数幂，它的运算仍符合幂运算的指数律。对于分数指数幂，它的运算仍符合幂运算的指数律。 For the power of a fractional exponent, its operation also follows the laws of exponents for power operation. 第三十二章方程和不等式的表达第三十二章方程和不等式的表达 一、方程的
134、表达一、方程的表达 含有未知数的等式称为方程。含有未知数的等式称为方程。 Any statement that two expression containing variables are equal is called an equation. 使使方方程程左左右右两两边边值值相相等等的的未未知知数数的的解解，叫叫做做方方程程的的解解，只有一个未知数的方程的解，叫做根。只有一个未知数的方程的解，叫做根。 The solution of an unknown that make both sides of an equation identical is called the solutio
135、n of the equation. The solution of an equation which has only one unknown is called the root of the equation. 方方程程中中变变量量的的最最高高次次数数为为2次次的的方方程程称称为为二二次次方方程程，也称为一元二次方程。也称为一元二次方程。 An equation whose highest degree of variable is two is called the quadratic equation or the second-degree equation. 由多个变量、多个方
136、程组成的一组方程称为方程组。由多个变量、多个方程组成的一组方程称为方程组。 A collection of two or more equations with two or more variables is called a system of equations. 方方程程的的求求解解因因式式分分解解法法配配方方法法公公式式求解法求解法 根的判别式根的判别式 Solution for Equations; Factoring; Completing the Square; Quadratic Formulas; Discriminant of the Quadratic Form. 加减
137、消元法加减消元法代入消元法代入消元法 Solution by the Addition Method; Substitution Method. 二、不等式的表达二、不等式的表达 用用关关系系符符号号，和和连连接接的的表表达达式式称称为为不不等等式。式。 The expressions which are connected by order symbols and are called inequalities. 不不等等式式中中含含有有一一个个未未知知数数，且且未未知知数数的的次次数数都都是是1的的不不等等式式叫叫做做一一元元一一次次不不等等式式。满满足足不不等等式式条条件件的的未未知知数
138、数的解称为不等式的解，它通常指数轴上一定区间的数。的解称为不等式的解，它通常指数轴上一定区间的数。 An inequality containing one unknown whose degree is 1 is called a first-degree inequality with one variable. The solution for the unknown to satisfy the inequality is called the solution set to the inequality, and it it usually a series of numbers i
139、n the interval of a number-line. 带有绝对值的不等式称为绝对不等式。带有绝对值的不等式称为绝对不等式。 Inequality involving absolute values are called the inequality with absolute values. 不不等等号号两两边边未未知知数数的的次次数数高高于于2次次的的不不等等式式称称为为多多次次不等式。不等式。 The inequality in which the order of the unknown on both sides is greater than 2 is called a
140、polynomial inequality.第三十三章矩阵和行列式的表达第三十三章矩阵和行列式的表达 一、矩阵的表达一、矩阵的表达 矩阵是矩阵是 m n 个数排列成个数排列成 m 行行 n 列的长方形阵列。列的长方形阵列。 A matrix is a rectangular array of entries with dimension mn, where m is the number of rows and n is the number of columns of the entries. 通通常常矩矩阵阵可可以以记记为为Amn=aij， aij称称为为矩矩阵阵的的元元素素，第第一一个个
141、下下标标指指行行号号，第第二二个个下下标标指指列列号号。矩矩阵阵的的元元素素可可以以是是实实数数，或或代代数数式式。 Amn又又称称为为mn阶阶矩矩阵阵，当当m=n时时，称称为为方方阵阵。而而m1阶阶矩矩阵阵称称为为列列矩矩阵阵， 1 n阶阶矩矩阵阵称称为为行矩阵。行矩阵。 In general, a matrix can de written as Amn=aij in which aij is called the element of the matrix. The first subscript indicates the row, and the second subscript i
142、ndicates the column. The entries of a matrix may be real numbers or algebraic expressions. Amn is also called a mn matrix. If m = n, the matrix is called a square matrix. m1 matrix is called a column matrix. 1n matrix is called a row matrix. 如如果果两两个个矩矩阵阵Amn=aij和和Bmn=bij，则则AmnBmn，这这意意味味着着当当也也只只有有当当两两
143、个个矩矩阵阵的的大大小小和和所所有有对对应应元元素素都都应相等时，两个矩阵才相等。应相等时，两个矩阵才相等。 If two matrices Amn=aij and Bmn=bij, then Amn= Bmn, this means that two matrices are equal if and only if they are of the same dimension and all corresponding entries are equal. 如如果果两两个个矩矩阵阵具具有有相相同同的的行行与与列列时时，才才可可以以进进行行相相加加。两矩阵相加就是将两矩阵的对应元素相加。两矩
144、阵相加就是将两矩阵的对应元素相加。 If two matrices have the same dimension, they are said to be conformable for addition. The matrices are added by adding corresponding entries of the given matrices. 矩阵与实数相乘，就是矩阵中的每一元素与实数相乘。矩阵与实数相乘，就是矩阵中的每一元素与实数相乘。 This operation of multiplying each entry in a matrix by a number is
145、called scalar multiplication. 常用矩阵来求线性代数方程。常用矩阵来求线性代数方程。 Matrices are often used to solve for linear algebraic equations. 如如果果矩矩阵阵Amn通通过过有有限限的的初初等等变变换换能能变变成成Bmn，则则称称A与与B为等价矩阵，记为为等价矩阵，记为A B或或B A。 If Amn can be changed into Bmn by a finite number of elementary operation, we call A and B equivalent mat
146、rices, written A B or B A. 二、行列式的表达二、行列式的表达 行行和和列列都都相相等等的的矩矩阵阵称称为为方方阵阵，由由n阶阶方方阵阵形形成成的的n2个个数数可可以以确确定定一一个个实实数数，每每一一个个有有实实数数项项的的Ann都都具具有有一一个个唯唯一一的的实实数数，这这个个实实数数被被称称为为行行列列式式，记记为为(A)或或|A| 。 (Ann)也称为也称为n阶行列式。阶行列式。 A matrix with the same number of rows and column is called a square matrix. n2 numbers in a
147、nth-order matrix may determine a real number. With every square matrix Ann with real number entries, there is associated a unique real number called its determinant, denoted by (A) or|A|. Also, (Ann) is called a determinant of order n. 行行列列式式的的横横排排称称为为行行，竖竖排排称称为为列列，Aij称称为为行行列列式式的的元元素素。 Aij的的第第一一个个下下
148、角角标标代代表表行行，第第二二个个下下角角标标代代表表列列，表明该元素在行列式中所处的位置。表明该元素在行列式中所处的位置。 The horizontal entries of a determinant is called rows and the vertical ones columns. Aij is called the members of the determinant. The first subscript of Aij represents the row and the second the column, which indicate the position of t
149、he members in the determinant. aij的的子子式式是是通通过过消消去去 (Ann)中中的的第第i行行和和第第j列列而而获获得得的的Mij行列式。行列式。 The minor of aij is the determinant Mij obtained by deleting the ith row and the jth column of (Ann). 求求解解行行列列式式的的值值时时，用用子子式式求求解解有有时时可可简简化化计计算算，但但要要考考虑虑子子式式所所带带的的正正负负号号，带带有有正正负负号号的的子子式式称称为为元元素素的的余子式。余子式。 In o
150、btaining the value of a determinant, it is possible to simplify the operation by using minors, but we must consider the positive and negative signs affixed to the minors. The minor with the proper sign affixed is called the cofactor of the corresponding element. 运运用用行行列列式式的的性性质质，可可以以简简化化求求解解行行列列式式的的
151、过过程程，特特别别是对于那些高阶行列式。是对于那些高阶行列式。 Because of the properties of determinant, a process of simplification can be developed to help evaluate determinant, especially those of large order. 如如果果行行列列式式某某一一行行或或列列的的每每一一个个元元素素乘乘以以一一个个常常数数，则则行列式的值乘以常数。行列式的值乘以常数。 If each entry in one row or one column is multipli
152、ed by a constant, the value of the determinant is multiplied by that constant. 如如果果某某一一行行（列列）的的所所有有元元素素的的倍倍数数分分别别加加上上相相应应的的另一行（列）的所有元素，则行列式的值不变。另一行（列）的所有元素，则行列式的值不变。 If a multiple of the entries of one row/column is added to corresponding entries of another row/column, respectively, the value of the
153、 determinant remains the same.第三十四章三角形图形的表达第三十四章三角形图形的表达 一、线和角一、线和角 几几何何上上的的点点是是空空间间里里的的一一个个确确定定位位置置。它它没没有有尺尺寸寸，也不可见，但它是几何位置的唯一表达。也不可见，但它是几何位置的唯一表达。 A geometric point is an element of geometry which has a position but no extension. It has no size, nor ca it been seen. It is a single expression of a
154、geometric position. 几几何何直直线线是是共共线线点点的的集集合合。几几何何线线中中的的直直线线是是由由无无数数个个点点集集合合而而成成，它它没没有有端端点点，可可延延两两个个方方向向无无限限延延伸伸。直线的任意部分没有宽度，但有长度。直线的任意部分没有宽度，但有长度。 A geometric straight line is a set of collinear points. A geometric line has an infinite number of points without endpoints and may be extended indefinitel
155、y in both directions. Any part of a line has no width or thickness but length. 射线和线段都是直线的一部分。射线和线段都是直线的一部分。 A ray and a line segment are both a part of a straight line. 由由不不是是共共线线点点的的几几何何组组成成的的线线称称为为曲曲线线。由由n根根不不同同线段端点相互连接组成的线称为折线。线段端点相互连接组成的线称为折线。 A line composed of sets of uncommon points is called
156、 a curved line. A line composed of n different segments connected with each other is called a broken line. 一一根根直直线线在在平平面面中中的的位位置置有有垂垂直直、水水平平和和倾倾斜斜放放置置三三种方式。种方式。 One straight line in a plane may be in vertical, horizontal and slant or oblique position. 两两根根直直线线在在平平面面中中的的相相互互位位置置有有三三种种：相相交交、平平行行和和垂直。垂
157、直。 there are three positions of each other for two lines in a plane: intersecting, parallel and perpendicular. 由由公公共共端端点点的的两两条条射射线线组组成成的的图图形形叫叫做做角角。这这个个公公共共的端点叫做的端点叫做顶点顶点，这两条射线叫做角的，这两条射线叫做角的边边。 The figure by two rays drawing from a common endpoint is called an angle. This common endpoint is called t
158、he vertex of the angle and the two rays the sides of the angle. 角角还还可可以以看看成成是是一一条条射射线线绕绕着着它它的的端端点点从从一一个个位位置置旋旋转到另一个位置所形成的图形。转到另一个位置所形成的图形。 An angle may also be regarded as a figure formed by a ray revolving about a fixed point from one position to another. 直直角角锐锐角角 钝钝角角 平平角角周周角角优优角角补补角角余余角角对对顶顶角角同同位
159、位角角内内错错角角同旁内角同旁内角 right angle; acute angle; obtuse angle; straight angle; circumferentical angle; reflex angle; supplementary angle; complementary angle; vertical angle; corresponding angle; alternate interior angle; same-side interior angle 二、三角形二、三角形 由由不不在在同同一一条条直直线线上上的的三三条条线线段段首首尾尾顺顺次次相相接接组组成成的的图
160、形叫做三角形。图形叫做三角形。 A triangle is a figure with three segments of a straight line connecting in sequence with each other. 在在三三角角形形中中，组组成成三三角角形形的的线线段段叫叫做做三三角角形形的的边边。三三角角形形相相邻邻两两边边的的公公共共端端点点叫叫做做三三角角形形的的顶顶点点。三三角角形形相相邻邻两边组成的角叫做三角形的两边组成的角叫做三角形的内角内角，简称三角形的，简称三角形的角角。 In the triangle, the segments forming a tri
161、angle are called the sides of the triangle. A common endpoint between adjacent sides of a triangle is called a vertex of the triangle. Angles formed by two adjacent sides of a triangle are called the interior angle of a triangle, simplified as the angles of a triangle. 三三角角形形一一个个角角的的平平分分线线与与这这个个角角的的
162、对对边边相相交交，这这个个角的顶点和交点之间的线段叫做三角形的角的顶点和交点之间的线段叫做三角形的角平分线角平分线。 The bisector of an angle in a triangle intersects the opposite side of the angle and the segment between the vertex of the triangle and the intersecting point is called the angular bisector of the triangle. 连接一个顶点和它对边中点的线叫做三角形的连接一个顶点和它对边中点的线
163、叫做三角形的中线中线。 A line connecting one vertex with its opposite mid-point is called the central line of a triangle. 从从三三角角形形一一个个顶顶点点向向它它的的对对边边作作垂垂线线，顶顶点点和和垂垂足足之之间的线段叫做三角形的间的线段叫做三角形的高高。 Draw a vertical line from a vertex of a triangle to its opposite side, called base, and the segment between the vertex a
164、nd the base is called the altitude of the triangle. 不不等等边边三三角角形形等等腰腰三三角角形形等等边边三三角角形形锐锐角角三三角角形形钝钝角角三三角角形形直直角角三三角角形形斜斜三三角角形形 腰腰顶角顶角底边底边底角底角 nonequilateral triangle; isosceles triangle; equilateral triangle; acute triangle; obtuse triangle; right triangle; slant triangle; waist; vertex angle; base side
165、; base angle 任一直角三角形两直角边的平方和等于斜边的平方。任一直角三角形两直角边的平方和等于斜边的平方。 In a triangle, the sum of the squares of the two legs equal to the square of the hypotenuse.第三十五章四边形图形的表达第三十五章四边形图形的表达 一、基本概念一、基本概念 由由不不在在同同一一直直线线上上的的四四条条线线段段首首尾尾顺顺次次相相接接组组成成的的图图形形叫叫做做四四边边形形。组组成成四四边边形形的的各各条条线线段段叫叫做做四四边边形形的的边边，每相邻两条边的公共端点叫做四
166、边形的，每相邻两条边的公共端点叫做四边形的顶点顶点。 The graph with four segments not on the same line connected with one after another is called quadrilateral. Each of the four segment forming a quadrilateral is called its side and the common endpoint of every two adjacent sides is called a vertex of the quadrilateral. 把把四四
167、边边形形的的任任何何一一边边向向两两方方延延长长，如如果果其其它它各各边边都都在在延延长长线线所所得得直直线线的的同同一一旁旁，这这样样的的四四边边形形叫叫做做凸凸四四边边形形。 Extend any side of a quadrilateral to both direction and if all the sides are on the same side of the extended line, the resulting quadrilateral is called a convex quadrilateral. 在在四四边边形形中中，连连接接不不相相邻邻两两个个顶顶点点的的
168、线线段段叫叫做做四四边边形的形的对角线对角线。 In a quadrilateral, the line segment that joint two vertices not adjacent to each other is called a diagonal of the quadrilateral. 四四边边形形相相邻邻两两边边所所组组成成的的角角叫叫做做四四边边形形的的内内角角，简简称称四边形的四边形的角角。 In a quadrilateral, the angles composed of two adjacent sides are called the interior an
169、gles of the quadrilateral, simplified as the angles of a quadrilateral. 在在平平面面内内，由由一一些些线线段段首首尾尾相相接接组组成成的的图图形形叫叫做做多多边边形形，多多边边形形的的边边、顶顶点点、内内角角、外外角角、对对角角线线的的意意义义和四边形相同，多边形有几条边就是几边形。和四边形相同，多边形有几条边就是几边形。 In a plane, the figure composed of several line segments jointed at every endpoint in succession is c
170、alled a polygon, whose sides, vertices, interior angles, exterior angles and diagonals are identical in meaning to those of a quadrilateral. The number of sides of a polygon determines what polygon it is. 二、平行四边形二、平行四边形 两组对边分别平行的四边形叫做两组对边分别平行的四边形叫做平行四边形平行四边形。 The quadrilateral with its opposite side
171、s parallel is called parallelogram. 两组对角分别相等的四边形是平行四边形。两组对角分别相等的四边形是平行四边形。 The quadrilateral in which two pairs of opposite angles are equal, respectively, is a parallelogram. 一组对边平行且相等的四边形是平行四边形。一组对边平行且相等的四边形是平行四边形。 The quadrilateral in which a pair of opposite sides are both parallel and equal is
172、a parallelogram. 三、矩形、菱形、正方形、矩形、菱形、正方形 矩形、菱形、正方形都是特殊的平行四边形。矩形、菱形、正方形都是特殊的平行四边形。 Rectangles, lozenges or rhombuses and squares are all special parallelograms. 矩形的四个角都是直角。矩形的四个角都是直角。 The four angles of a rectangle are all right angles. 菱菱形形的的对对角角线线互互相相垂垂直直，并并且且每每一一条条对对角角线线平平分分一一组组对角。对角。 The diagonals
173、of a lozenge are perpendicular to each other, each diagonal bisecting a pair of opposite angles. 有有一一组组邻邻边边相相等等并并且且有有一一个个角角是是直直角角的的平平行行四四边边形形叫叫做做正正方方形形。正正方方形形具具有有矩矩形形的的性性质质，同同时时又又具具有有菱菱形形的的性质。性质。 The parallelogram with a pair of adjacent sides equal and one angle a right angle is called a square. A
174、square has the qualities of both a rectangle and a lozenge. 四、梯形四、梯形 一一组组对对边边平平行行而而另另一一组组对对边边不不平平行行的的四四边边形形叫叫做做梯梯形形。平平行行的的两两边边叫叫做做梯梯形形的的底底（通通常常把把较较短短的的底底叫叫做做上上底底，较较长长的的底底叫叫做做下下底底），不不平平行行的的两两边边叫叫做做梯梯形形的的腰腰，两两底的距离叫做底的距离叫做梯形的高梯形的高。 The quadrilateral in which one pair of opposite sides are parallel and
175、the other are not parallel is called a trapezoid. The parallel sides are the bases of the trapezoid（usually the short one is called the upper base and the long one the bottom base）; the two sides which are not parallel are called the waist of the trapezoid and the perpendicular distance between the
176、bases is called its altitude. 一一腰腰垂垂直直于于底底的的梯梯形形叫叫做做直直角角梯梯形形，两两腰腰相相等等的的梯梯形叫做等腰梯形。形叫做等腰梯形。 The trapezoid with one waist perpendicular to a base is called a right-angled trapezoid; the trapezoid with two waists equal is called an isosceles trapezoid. 在同一底上的两个角相等的梯形是等腰梯形。在同一底上的两个角相等的梯形是等腰梯形。 The trapez
177、oid with two angles on the same base is called an isosceles trapezoid. 对角线相等的梯形是等腰梯形。对角线相等的梯形是等腰梯形。 The trapezoid with its diagonals equal is called an isosceles trapezoid.第三十六章圆图形的表达第三十六章圆图形的表达 一、基本概念一、基本概念 在在一一个个平平面面内内，线线段段OA绕绕它它的的固固定定端端点点O旋旋转转一一周周，另另一一个个端端点点A随随之之旋旋转转所所形形成成的的图图形形叫叫做做圆圆，固固定定的的端端点点O
178、叫做叫做圆心圆心，线段，线段OA叫做叫做半径半径。 In a plane, the graph in which line segment OA revolves about its fixed endpoint O to 360 is called a circle. The fixed endpoint O is called the center and the segment OA is called the radius. 到定点的距离等于定长的点都在圆上。到定点的距离等于定长的点都在圆上。 All points at a given distance are on the circ
179、le. 连连接接圆圆上上任任意意两两点点的的线线段段叫叫做做弦弦，经经过过圆圆心心的的弦弦叫叫做做直径直径，直径等于半径的两倍。，直径等于半径的两倍。 The segment joining any two points on a circle is called a chord of the circle. The chord passing through the center of the circle is called the diameter, which is twice the radius. 圆上任意两点间的部分叫做圆上任意两点间的部分叫做圆弧圆弧，简称，简称弧弧。 The
180、curve bounded by two given points of a circle is called an arc of circle, simplified as an arc. 圆圆的的任任意意一一条条直直径径的的两两个个端端点点分分圆圆成成两两条条弧弧，每每一一条条弧弧都都叫叫做做半半圆圆。大大于于半半圆圆的的弧弧叫叫做做优优弧弧；小小于于半半圆圆的的弧叫做弧叫做劣弧劣弧。由弦及其所对的弧组成的图形叫做。由弦及其所对的弧组成的图形叫做弓形弓形。 A diameter at any two endpoints divides a circle into two arcs, eit
181、her of which is called a half-circle. The arc greater than a half-circle is called a superior arc and that less than a half-circle is called an inferior arc. The figure consisting of a chord and its opposite arc is called a segment of a circle. 圆心相同，半径不相等的两个圆叫做圆心相同，半径不相等的两个圆叫做同心圆同心圆。 The two circles
182、 having a common center and unequal radii are called concentric circles. 能够重合的两个圆叫做能够重合的两个圆叫做等圆等圆。 Two circles which can be coincident with each other are called equal circles. 经经过过三三角角形形三三个个顶顶点点可可以以作作一一个个圆圆。经经过过三三角角形形各各顶顶点点的的圆圆叫叫做做三三角角形形的的外外接接圆圆，外外接接圆圆的的圆圆心心叫叫做做三三角角形的形的外心外心，这个三角形叫做这个圆的，这个三角形叫做这个圆的内
183、接三角形内接三角形。 A circle can be made by passing through three vertices of a triangle. The circle passing through three vertices of a triangle is called a circumcircle of the triangle whose center is called the circumcenter of the triangle and this triangle is called an inscribed triangle. 顶点在圆心的角称为顶点在圆心的
184、角称为圆心角圆心角。 The angle with its vertex at the center of a circle is called a central angle. 从圆心到弦的距离叫做从圆心到弦的距离叫做弦心矩弦心矩。 The distance from the center of a circle to the chord is called a chord-center moment. 顶点在圆上，并且两边都和圆相交的角叫做顶点在圆上，并且两边都和圆相交的角叫做圆周角圆周角。 The angle whose vertex is on a circle with both s
185、ides intersecting the circle is called a circumferential angle. 二、直线和圆及圆和圆的位置关系二、直线和圆及圆和圆的位置关系 直直线线和和圆圆有有两两个个公公共共点点时时，叫叫做做直直线线和和圆圆相相交交，这这时直线叫做圆的时直线叫做圆的割线割线。 If there two common points between a line and a circle, the line and the circle are said to be intersected where the line is called a secant of
186、 the circle. 直直线线和和圆圆有有唯唯一一公公共共点点时时，叫叫做做直直线线和和圆圆相相切切，这这时直线叫做圆的时直线叫做圆的切线切线，唯一公共点叫做，唯一公共点叫做切点切点。 If there is only common point between a line and a circle, the line is called tangent to the circle where the line is called a tangent line of the circle and the common point is called a tangent point. 直线和
187、圆没有公共点时，叫做直线和圆直线和圆没有公共点时，叫做直线和圆相离相离。 If there is no common point between a line and a circle, the line and the circle are separated. 圆的切线垂直于经过切点的半径。圆的切线垂直于经过切点的半径。 A tangent line of a circle is perpendicular to the radius through the tangent point. 与与三三角角形形各各边边都都相相切切的的圆圆叫叫做做三三角角形形的的内内切切圆圆，内内切切圆圆的的圆圆
188、心心叫叫做做三三角角形形的的内内心心，这这个个三三角角形形叫叫做做圆圆的的外外切三角形切三角形。 The circle tangent to all the sides of a triangle is called the inscribed circle of a triangle whose center is called the incenter of the triangle, and the triangle is called the circumscribed triangle of the circle. 和和多多边边形形的的各各边边都都相相切切的的圆圆叫叫做做多多边边形形
189、的的内内切切圆圆。这个多边形叫做圆的这个多边形叫做圆的外切多边形外切多边形。 The circle tangent to the sides of a polygon is called the inscribed circle of a polygon. This polygon is called the circumscribed polygon of the circle. 两两个个圆圆没没有有公公共共点点，并并且且每每个个圆圆上上的的点点都都在在另另一一个个圆的外部时，叫做这两个圆外离。圆的外部时，叫做这两个圆外离。 If there is no common point betwe
190、en two circles and the points of one circle is outside the other, these two circles are said to be separated. 两两个个圆圆有有唯唯一一的的公公共共点点，并并且且除除了了这这个个公公共共点点以以外外，每每个个圆圆上上的的点点都都在在另另一一个个圆圆外外部部时时，叫叫做做这这两两个个圆圆外外切切，这个唯一的公共点叫做切点。这个唯一的公共点叫做切点。 If two circles have a common point, and besides the common point, the p
191、oints of one circle are outside the other, we say that these two circles are circumscribed and the only common point is called a tangent point. 两个圆有两个公共点时，叫做这两个圆相交。两个圆有两个公共点时，叫做这两个圆相交。 If two circles have two common points, we say the two circles are intersected. 两两个个圆圆有有唯唯一一的的公公共共点点，并并且且除除了了这这个个公公共
192、共点点以以外外，一一个个圆圆上上的的点点都都在在另另一一个个圆圆的的内内部部时时，叫叫做做这这两两个个圆圆内内切，这个唯一的公共点叫做切点。切，这个唯一的公共点叫做切点。 Two circles have only one common point, and besides this common point, the points of a circle are all inside the other circle, we say that these two circles are inscribed and this common point is called the tangent
193、 point. 两两个个圆圆没没有有公公共共点点，并并且且一一个个圆圆上上的的点点都都在在另另一一个个圆圆的的内内部部时时叫叫做做这这两两个个圆圆内内含含。两两圆圆同同心心是是两两圆圆内内含含的的一种特例。一种特例。 Two circles have no common point and the points of one circle are inside the other circle, we say that these two circles are inclusive. It is a special case of the inclusive for two circles t
194、o have a common center. 和两个圆都相切的直线，叫做两圆的和两个圆都相切的直线，叫做两圆的公切线公切线。 Lines tangent to two circles are called the common tangent lines of the two circles. 外公切线外公切线内公切线内公切线 exterior common tangent line; interior common tangent line 一一条条弧弧和和经经过过这这条条弧弧的的两两个个端端点点的的两两条条半半径径所所组组成的图形叫做成的图形叫做扇形扇形。 A portion of a
195、circle bounded by two radii of the circle and one of the arcs which they intersect is called a sector of a circle. 弓形面积弓形面积 area of segment of a circle第三十七章空间图形的表达第三十七章空间图形的表达 一、多面体一、多面体 有有两两个个面面互互相相平平行行，其其余余各各面面都都是是四四边边形形，并并且且每每相相邻邻两两个个四四边边形形的的公公共共边边都都互互相相平平行行，由由这这些些面面所所围围成成的的几何图形叫做棱形。几何图形叫做棱形。 A p
196、rism is a polyhedron with two congruent parallel faces and whose other faces are quadrilateral with the common sides of every two adjacent quadrilateral parallel to each other. 棱棱柱柱的的底底面面棱棱柱柱的的侧侧面面棱棱柱柱的的侧侧棱棱棱棱柱柱的的顶顶点点棱棱柱柱的的对对角角线线棱棱柱柱的的高高斜斜棱棱柱柱直棱柱直棱柱正棱柱正棱柱三棱柱三棱柱四棱柱四棱柱五棱柱五棱柱 bases of a prism; lateral
197、faces of a prism; lateral edges of a prism; vertices of a prism; diagonal of a prism; ; altitude of a prism; slant prism; right prism; regular prism; triangle prism; quadrangular prism; pentagonal prism 平平行行六六面面体体直直平平行行六六面面体体长长方方体体正正方方体体 parallel hexahedron; right parallel hexahedron; rectangular so
198、lid; tetragonal solid 有有一一个个面面是是多多边边形形，其其余余各各面面是是有有一一个个公公共共顶顶点点的的三三角形，由这些面所组成的图形叫做角形，由这些面所组成的图形叫做棱锥棱锥。 A polyhedron with one face a polygon and the other faces triangles containing a common vertex is called a pyramid. 如如果果一一个个棱棱锥锥的的底底面面是是正正多多边边形形，并并且且顶顶点点在在底底面面的的射影是底面中心，这样的棱锥叫做射影是底面中心，这样的棱锥叫做正棱锥正棱锥。
199、 The pyramid with a regular polygon for base and with the foot of its altitude at the center of the base is called a regular pyramid. 用用一一个个平平行行于于棱棱锥锥底底面面的的平平面面去去截截棱棱锥锥，底底面面和和截截面面面之间的部分叫做面之间的部分叫做棱台棱台。 Using a plane parallel to the base of a pyramid to cut the pyramid, we get a section of a pyramid b
200、etween the base and the plane which is called a frustum of pyramid. 由由若若干干个个多多边边形形所所围围成成的的几几何何体体叫叫做做多多面面体体。棱棱柱柱、棱台、棱锥等都是棱台、棱锥等都是多面体多面体。 The geometric solid bounded by some polygons is called polyhedron. The prism, frustum and pyramid are all polygons. 二、旋转体二、旋转体 分分别别以以一一矩矩形形的的一一边边、直直角角三三角角形形一一直直角角边边
201、、直直角角梯梯形形垂垂直直于于底底边边的的腰腰所所在在的的直直线线为为旋旋转转轴轴，其其余余各各边边旋旋转转而而形形成成的的曲曲面面所所围围成成的的几几何何体体分分别别叫叫做做圆圆柱柱、圆圆锥锥、圆台圆台。 The geometric solid generated by revolving a plane about a line （the axis of revolving）is called a solid of rotation. The solid of rotation revolving about a side of a rectangle is called a circul
202、ar cylinder, that revolving about a leg of a right triangle is called a circular cone and that revolving about the side perpendicular to the base of a right-angled trapezoid is called a circular-truncated cone. 把把圆圆柱柱、圆圆锥锥、圆圆台台的的侧侧面面沿沿着着它它们们的的一一条条母母线线剪剪开后展开在平面上，展开图的面积就是它们的侧面积。开后展开在平面上，展开图的面积就是它们的侧面积
203、。 Extend the lateral surfaces of a circular cylinder, cone and truncated cone on a plane along one of their generatrices and the areas of the expanded graphs are their lateral areas. 半半圆圆以以它它的的直直径径为为旋旋转转轴轴，旋旋转转所所成成的的曲曲面面叫叫做做球球面面。球球面面所所围围成成的的几几何何体体叫叫做做球球体体，简简称称球球。半半圆圆的的圆圆心心叫叫做做球球心心。连连接接球球心心和和球球面面上上任任
204、意意一一点点的的线线段段叫叫做做球球的的半半径径。连连接接球球面面上上任任意意两两点点并并且且经经过过球球心心的的线线段段叫叫做做球球的的直径直径。 The curved surface formed with a half-circle revolving about its diameter which is used as an axis of revolution is called a sphere. The geometric solid surround by the sphere is called a spheroid, simplified as a sphere. The
205、 center of the half-circle is the center of spheroid. The center and any point on the spherical surface is called the radius of spheroid. The segment joining two points on the spherical surface through the center is called the diameter of sphere. 球球面面被被平平面面所所截截得得的的一一部部分分叫叫做做球球冠冠，截截得得得得圆圆叫叫做做球球冠冠的的底底
206、，垂垂直直于于截截面面的的直直径径被被截截得得的的一一段段叫叫做做球球冠冠的高的高。 The segment of a sphere intersected by a plane is called a spherical crown, the intersected circle is called the base of spherical crown and the segment of the diameter perpendicular to the section is called the altitude of crown. 一一条条平平面面曲曲线线（包包括括直直线线）绕绕它它
207、所所在在的的平平面面内内的的一一条条定定直直线线旋旋转转所所形形成成的的球球面面叫叫做做旋旋转转面面。这这条条定定直直线线叫叫做做旋旋转转轴轴。无无论论旋旋转转到到什什么么位位置置，这这条条曲曲线线都都叫叫做做旋旋转转面面的的母线母线。 The curved surface with one plane curve（including a straight line）revolving about a fixed line in a plane is called the surface of rotation. This fixed line is called the axis of ro
208、tation. No matter where the axis goes, this resulting curve is always called a generatrix of the surface of rotation. 圆柱面圆柱面圆锥面圆锥面环面环面环体环体 surface of cylinder; surface of circular cone; ring surface; ring solid 三、多面体和旋转体的体积三、多面体和旋转体的体积 几何体占有空间部分的大小叫做它的体积。几何体占有空间部分的大小叫做它的体积。 The volume refers to the
209、size in which a geometric solid takes up the space. 要要度度量量一一个个几几何何体体的的体体积积，首首先先要要选选取取一一个个单单位位体体积积作作为为标标准准，然然后后求求出出几几何何体体的的体体积积是是单单位位体体积积的的多多少少倍倍，这这个个倍倍数数就就是是这这个个几几何何体体的的体体积积数数值值。通通常常取取棱棱长长等等于于单单位位长长度度（如如1cm, , 1m等等）的的正正方方体体的的体体积积作作为为体体积积单单位。位。 To measure the volume of a geometric solid, firstly, cho
210、ose a unit volume as a standard, then find how many times the volume of the solid is taken by unit volume, and the times is the quantity of the volume of this geometric solid. Usually choose as the unit volume the volume of a tetragonal solid with the length of edge to the unit length, such as 1 cm
211、and 1 m. 设设长长方方体体的的长长、宽宽、高高分分别别为为a、b、c，长长方方体体的的体体积等于它的长、宽、高的积。积等于它的长、宽、高的积。 Let the length, width and altitude of a rectangular solid be a, b and c, respectively, and then the volume of the rectangular solid equals the product of its length, width and altitude. 正方体的体积等于它的棱长的立方。正方体的体积等于它的棱长的立方。 The volume of a tetragonal solid is equal to the cube of the length of the edge. 柱柱体体（棱棱柱柱、圆圆柱柱）的的体体积积等等于于它它的的底底面面积积与与高高的的乘乘积。积。 The volume of a cylindrical body（a prism, a circular cylinder）is equal to the product of its base area and altitude.
