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Summary

As indicated at the start of the previous chapter, the differential calculus and its complement, the integral calculus, together form the most widely used tool for the analysis of physical systems. The link that connects the two is that they both deal with the effects of vanishingly small changes in related quantities; one seeks to obtain the ratio of two such changes, the other uses such a ratio to calculate the variation in one of the quantities resulting from a change in the other.

Any change in the value of any one property (or variable) of a physical system almost always results in the values of some or all of its other properties being altered; in general, the size of each consequential change depends upon the current values of all of the variables. As a result, during a finite change in any one of the values, that of x say, those associated with all of the other variables are continuously changing, making computation of the final situation difficult, if not impossible. The solution to this difficulty is provided by the integral calculus, which allows only vanishingly small changes, and, after any such change in one variable, brings all the other values ‘up to date’ (by infinitesimal amounts) before allowing any further change.

Summary

All scientists will know the importance of experiment and observation and, equally, be aware that the results of some experiments depend to a degree on chance. For example, in an experiment to measure the heights of a random sample of people, we would not be in the least surprised if all the heights were found to be different; but, if the experiment were repeated often enough, we would expect to find some sort of regularity in the results. Statistical methods are concerned with the analysis of real experimental data of this sort.

In this final chapter we discuss the subject of probability, which is the theoretical basis for most statistical methods. Our development of probability will be with an eye to its eventual applications in statistics, with little emphasis on the axioms and theorems approach favoured by most pure mathematicians.

We first discuss the terminology required, with particular reference to the convenient graphical representation of experimental results as Venn diagrams. The concepts of random variables and distributions of random variables are then introduced. It is here that the connection with statistics is made; we assert that the results of many experiments are random variables and that those results have some sort of regularity, represented by a distribution. Finally, the defining equations for some important distributions, together with some useful quantities that characterise them, are introduced and discussed.

Summary

Since Mathematical Methods for Physics and Engineering by Riley, Hobson and Bence (Cambridge: Cambridge University Press, 1998), hereafter denoted by MMPE, was first published, the range of material it covers has increased with each subsequent edition (2002 and 2006). Most of the additions have been in the form of introductory material covering polynomial equations, partial fractions, binomial expansions, coordinate geometry and a variety of basic methods of proof, though the third edition of MMPE also extended the range, but not the general level, of the areas to which the methods developed in the book could be applied. Recent feedback suggests that still further adjustments would be beneficial. In so far as content is concerned, the inclusion of some additional introductory material such as powers, logarithms, the sinusoidal and exponential functions, inequalities and the handling of physical dimensions, would make the starting level of the book better match that of some of its readers.

To incorporate these changes, and others aimed at increasing the user-friendliness of the text, into the current third edition of MMPE would inevitably produce a text that would be too ponderous for many students, to say nothing of the problems the physical production and transportation of such a large volume would entail.

Summary

In Chapters 3 and 4 we discussed functions f of only one variable x, which were usually written f(x). Certain constants and parameters may also have appeared in the definition of f, e.g. f(x) = ax + 2 contains the constant 2 and the parameter a, but only x was considered as a variable and only the derivatives f (n)(x) = dnf/dxn were defined.

However, we can equally well consider functions that depend on more than one variable, e.g. the function f(x, y) = x 2 + 3xy, which depends on the two variables x and y. For any pair of values x, y, the function f(x, y) has a well-defined value, e.g. f(2, 3) = 22. This notion can clearly be extended to functions dependent on more than two variables. For the n-variable case, we write f(x 1, x 2, …, xn ) for a function that depends on the variables x 1, x 2, …, xn . When n = 2, x 1 and x 2 correspond to the variables x and y used above.

Functions of one variable, like f(x), can be represented by a graph on a plane sheet of paper, and it is apparent that functions of two variables can, with a little effort, be represented by a surface in three-dimensional space.

Summary

In Chapter 9 we discussed the algebra of vectors and in Chapter 10 we considered how to transform one vector into another using a linear operator. In this chapter and the next we discuss the calculus of vectors, i.e. the differentiation and integration both of vectors describing particular bodies, such as the velocity of a particle, and of vector fields, in which a vector is defined as a function of the coordinates throughout some volume (one-, two- or three-dimensional). Since the aim of this chapter is to develop methods for handling multi-dimensional physical situations, we will assume throughout that the functions with which we have to deal have sufficiently amenable mathematical properties, in particular that they are continuous and differentiable.

Differentiation of vectors

Let us consider a vector a that is a function of a scalar variable u. By this we mean that with each value of u we associate a vector a(u). For example, in Cartesian coordinates a(u) = ax (u)i + ay (u)j + az (u)k, where ax (u), ay (u) and az (u) are scalar functions of u and are the components of the vector a(u) in the x-, y- and z-directions respectively. We note that if a(u) is continuous at some point u = u 0 then this implies that each of the Cartesian components ax (u), ay (u) and az (u) is also continuous there.

Summary

The first two chapters of this book review the basic arithmetic, algebra and geometry of which a working knowledge is presumed in the rest of the text; many students will have at least some familiarity with much, if not all, of it. However, the considerable choice now available in what is to be studied for secondary-education examination purposes means that none of it can be taken for granted. The reader may make a preliminary assessment of which areas need further study or revision by first attempting the problems at the ends of the chapters. Unlike the problems associated with all other chapters, those for the first two are divided into named sections and each problem deals almost exclusively with a single topic.

This opening chapter explains the basic definitions and uses associated with some of the most common mathematical procedures and tools; these are the components from which the mathematical methods developed in more advanced texts are built. So as to keep the explanations as free from detailed mathematical working as possible – and, in some cases, because results from later chapters have to be anticipated – some justifications and proofs have been placed in appendices. The reader who chooses to omit them on a first reading should return to them after the appropriate material has been studied.

Summary

In De or. 3 the term oratio is used in at least seven different, if sometimes overlapping senses:

Summary

The purpose of this edition is to furnish advanced students of Latin literature with assistance in reading and interpreting book 3 of Cicero's De oratore. To this end I have sought to provide an accurate and readable text as well as what seems necessary information about its syntax, usage, and style, its historical, literary, and philosophical background, and the subtle and often unexpected progression of its thought and argument.

In attempting to discharge this task I owe much, even when, for reasons of space, the debt is not explicitly acknowledged, to earlier commentaries, including those of Ernesti, Ellendt, Sorof, and Wilkins, to more recent scholarly research, especially that of E. Fantham, to the exemplary translation and guide to De or. by J. May and J. Wisse (= M–W), and, above all, to the first four volumes of the great Kommentar of A. D. Leeman and his associates (= Komm.); the fifth volume, by Wisse, M. Winterbottom, and Fantham, which covers most of book 3, did not, unfortunately, become available until my own commentary was complete, too late to be consulted without further delaying an already much-delayed project. Other works that have been of use are listed in the References, but this does not pretend to be an exhaustive bibliography, and in general citations of secondary material are limited to recent works in English which can in turn direct readers to earlier scholarship.

Summary

CICERO'S PROEM: THE FATES OF THE CHARACTERS IN THE DIALOGUE

The book begins on a sombre note, with Cic. looking past the end of the dialogue to the death of Cra. just a few days later (1n.) and to the calamities which would overtake the other participants, their friends, and even their enemies in the years to come. It is striking that in his initial account of the dialogue's setting (1.24–9) Cic. did not mention any of this, and although some of his readers would have recalled the events without prompting, his cataloguing of them here seems meant to add a special poignancy to some, at least, of Cra.'s speech and especially to the remarks of the soon-to-be renegade Sulp. (147n.).

1 Instituentimihi … renouauit: a peculiarily Ciceronian construction, in which a clause is bracketed by the dat. part. of a verb of ‘thinking’ at or near the start, then a finite verb of ‘getting an idea’ at or near the end (Laughton 1964: 37–8; cf. 13, 1.1, 6, 2.128). Quinte frater: Q. Tullius Cicero, the addressee (1.1, 2.1) and ‘instigator’ (cf. 13 below, 1.4–5, 2.10–11) of De or. He was in Italy during the time of its composition (55), in between serving as legate for Pompey in Sardinia (56), then for Caesar in Gaul (54–52). sermonem … disputationem: when mentioned together, sermo tends to indicate ‘speech in general’, disputatio more specific ‘argumentation’ (22, 107, 211, 2.16, 19–21, ThLL 1440), and Cic.

Summary

In De or. and elsewhere the term loci (= Gk topoi (16n.)) is used of several different types of ‘argument/lines of argument’. The first usage (type a) is from technical rhetoric, and refers to standard, ‘ready-made’ arguments linked to various areas of knowledge, categories of status (70n.), and ‘means of persuasion’ (23n.) which could be found listed in handbooks. Because they are not tied to a particular case (e.g. the Lindbergh kidnapping) but can be applied to any case of a given type (first degree kidnapping) or, when they serve the ‘ethical’ or ‘pathetic’ functions, to more than one type of case (any involving a helpless victim), they are sometimes called loci communes (106n.).

Antonius, whose task is to explain the ‘discovery’ (inuentio) of arguments, has little use for this first type of loci (cf. 2.117, 130, 133), and offers instead something quite different. His loci (type b) are ‘abstract argument patterns which help an orator to devise all his arguments himself’ (M–W 34) whatever the subject matter, issue, or purpose at hand (e.g. the locus ‘from dissimilarity’ (2.169) as the ‘source’ for an argument concerning the inhumanity of Hauptmann, the Lindbergh kidnapper). As Catulus recognizes (2.152), what Antonius expounds is a version of the so-called ‘topical method’ of Aristotle (Top., Rhet.), who likewise distinguishes between ‘ready-made arguments’ and ‘abstract argument patterns’. He calls the former idia, eide, or idiai protaseis (‘specifics’, ‘species’, ‘specific materials’; see Rhet. 1.2.21–2, 4.1–13.7, 2.1.9) if they are tied to the individual genres of oratory, or, if they are ‘common‘ to all of the genres, koina (= communia) eide (Rhet. 2.18.2–19, 27, 22.1–12; cf. 1.3.7–9, 7.1–41, 9.35–41, 14.1–7, 2.1.1–11.7).

Summary

[1] Instituenti mihi, Quinte frater, eum sermonem referre et mandare huic tertio libro quem post Antoni disputationem Crassus habuisset, acerba sane recordatio ueterem animi curam molestiamque renouauit. nam illud immortalitate dignum ingenium, illa humanitas, illa uirtus L. Crassi morte exstincta subita est uix diebus decem post eum diem qui hoc et superiore libro continetur. nam ut Romam rediit extremo ludorum scaenicorum die, uehementer commotus oratione ea quae ferebatur habita esse in contione a Philippo, quem dixisse constabat uidendum sibi esse aliud consilium; illo senatu se rem publicam gerere non posse, mane Idibus Septembribus et ille et senatus frequens uocatu Drusi in curiam uenit. ibi cum Drusus multa de Philippo questus esset, rettulit ad senatum de illo ipso, quod in eum ordinem consul tam grauiter in contione esset inuectus. hic, ut saepe inter homines sapientissimos constare uidi, quamquam hoc Crasso, cum aliquid accuratius dixisset, semper fere contigisset, ut numquam dixisse melius putaretur, tamen omnium consensu sic esse tum iudicatum ceteros a Crasso semper omnes, illo autem die etiam ipsum a se superatum. deplorauit enim casum atque orbitatem senatus, cuius ordinis a consule, qui quasi parens bonus aut tutor fidelis esse deberet, tamquam ab aliquo nefario praedone diriperetur patrimonium dignitatis; neque uero esse mirandum, si, cum suis consiliis rem publicam profligasset, consilium senatus a re publica repudiaret. hic cum homini et uehementi et diserto et in primis forti ad resistendum Philippo quasi quasdam uerborum faces admouisset, non tulit ille et grauiter exarsit pignoribusque ablatis Crassum instituit coercere.

Summary

CRASSUS' EDICT CONCERNING THE ‘LATIN RHETORS’ (93N.)

Suet. DGR 25.2 (text of Kaster) Cn. Domitius Ahenobarbus et L. Licinius

Crassus censores ita edixerunt:

renuntiatum est nobis esse homines qui nouum genus disciplinae instituerunt, ad quos iuuentus in ludum conueniat; eos sibi nomen imposuisse Latinos rhetoras, ibi homines adulescentulos dies totos desidere. maiores nostri quae liberos suos discere et quos in ludos itare uellent instituerunt: haec noua, quae praeter consuetudinem ac morem maiorum fiunt, neque placent neque recta uidentur. quapropter et iis qui eos ludos habent et iis qui eo uenire consuerunt uisum est faciundum ut ostenderemus nostram sententiam: nobis non placere.

Summary

CICERO AND DE ORATORE

Circumstances of composition

Between 63, the year of his consulate and his widely supported and acclaimed suppression of Catiline's conspiracy, and 55, when he wrote De or., Cicero experienced, among other reversals, estrangement from the most powerful men in Rome, Pompey, Crassus, and Caesar, whom he declined to abet in their ‘triumvirate’ aimed at dominating the state, and increasing resentment in other quarters, fomented by his arch enemy P. Clodius, for what had come to be regarded as the unlawful execution of some of Catiline's followers. The estrangement and the resentment culminated in his exile from Italy for over a year (58–57) and, despite a triumphant return (Sept. 57) suggesting better times ahead, his hopes of renewed prominence were cut off by warnings from the ‘triumvirs’, who reaffirmed their alliance in the infamous conference at Luca (May 56), and by his disgust and disillusionment with the senate in general, which he came to regard as no less harmful to the state than the ‘triumvirate’ itself.

In these circumstances Cic. turned to the solacia (cf. 14) furnished by literary composition, first (57–55) with poems about Marius (= frr. 15–19 FLP; cf. 8n.) and about his own exile and return (= fr. 14 FLP), later with the philosophical treatises Rep. (54–51) and Leg. (begun in 52). In between – figuratively, perhaps (cf. 27, 56nn.), as well as actually – the poetry and the philosophy came De or., which he completed after considerable care and effort in Nov. of 55.

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4 - Integral calculus

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15 - Elementary probability

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Preface

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7 - Partial differentiation

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11 - Vector calculus

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Index

1 - Arithmetic and geometry

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Abbreviations

SIGLA

Appendix 2 - oratio

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Preface

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References

Indexes

Contents

Commentary

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Appendix 3 - loci, loci communes

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M. TVLLI CICERONIS DE ORATORE LIBER III

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Appendix 1 - Supplementary texts

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Introduction

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