Archimedes的科学遗产：英文全译本与古希腊数学的探索

需积分: 10 179 浏览量更新于2024-07-19 收藏 2.03MB PDF 举报

"The Works of Archimedes - Netz" 《阿基米德著作集》是由Reviel Netz翻译的经典数学著作，旨在将古希腊最伟大的科学家之一阿基米德的工作全面地引入英文世界。这是第一部完全翻译成英文的阿基米德作品集，同时也是首次将一位古代主要数学家的作品附带批判性版面的图解和Eutocius对阿基米德的古代注解翻译成英文。这部作品集特别关注了基于1998年重新发现的阿基米德手稿（阿基米德抄本）的新证据，这一手稿在1915年至1998年间失传。 Reviel Netz在翻译过程中不仅关注文本本身，还深入研究了写作和阅读这些工作时的认知实践。他的目标是恢复这些文本作为交流行为的原始功能。Netz特别强调了阿基米德作品的审美层面，整体上的评论提供了一种创新的方法来研究数学文本。 Netz本人是斯坦福大学古典学的副教授，他的第一部著作《希腊数学中的推理构造：认知历史研究》（1999年）曾获得联合奖项。他的阿基米德著作集第一卷《球体与圆柱体》的翻译工作，在阿基米德抄本重新出现后，为研究提供了理想的基础。这部作品集的重要之处在于它结合了学术严谨性和可读性，使得现代读者能够理解阿基米德的思想和方法，揭示了古代数学的深度和复杂性。通过Netz的工作，我们得以一窥古希腊科学的辉煌，特别是阿基米德在几何学、物理学和工程学领域的贡献。此外，该书的评论部分深入探讨了阿基米德如何通过文字和图示进行思考，以及他的工作如何在当时的社会和文化背景下起到传播知识的作用。这种深入分析为我们理解古代科学家如何构建和传达他们的理论提供了一个独特的视角。《阿基米德著作集》是一部里程碑式的作品，对于数学史、科学史和古典学的研究者，以及对阿基米德感兴趣的广大读者来说，都是一份宝贵的资源。Netz的翻译和评论不仅重振了阿基米德的思想，也为我们理解古代科学思想的形成和发展提供了新的洞察。

4 introduction

A two-part set of comments follows each proposition (or, in some cases,

units of text other than propositions):

The ﬁrst are textual comments. Generally speaking, I follow Heiberg’s

(1910–15) edition, which seems to remain nearly unchanged, for the

books On the Sphere and the Cylinder, even with the new readings of

the Palimpsest. In some cases I deviate from Heiberg’s text, and such

deviations (excepting some trivial cases) are argued for in the textual

comments. In other cases – which are very common – I follow Heiberg’s

text, while doubting Heiberg’s judgment concerning the following ques-

tion. Which parts of the text are genuine and which are interpolated?

Heiberg marked what he considered interpolated, by square brackets

([. . .]). I print Heiberg’s square brackets in my translation, but I ver y

often question them within the textual comments.

The second are general comments. My purpose there is to develop an

interpretation of certain features of Archimedes’ writing. The com-

ments have the character not of a reference work, but of a monograph.

This translation differs from other versions in its close proximity to the

original; it maps, as it were, a space very near the original writing. It is on

this space that I tend to focus in my general comments. Thus I choose

to say relatively little on wider mathematical issues (which could be

equally accessed through a very distant translation), only brieﬂy supply

biographical and bibliographical discussions, and often focus instead

on narrower cognitive or even linguistic issues. I offer three apologies

for this choice. First, such comments on cognitive and linguistic detail

are frequently necessary for understanding the basic meaning of the

text. Second, I believe such details offer, taken as a whole, a central

perspective on Greek mathematical practices in general, as well as on

Archimedes’ individual character as an author. Third and most im-

portant, having now read many comments made in the past by earlier

authors, I can no longer see such comments as “deﬁnitive.” Mine are

“comments,” not “commentary,” and I choose to concentrate on what I

perceive to be of relevance to contemporary scholarship, based on my

own interest and expertise. Other comments, of many different kinds,

will certainly be made by future readers of Archimedes. Readers inter-

ested in more mathematical commentary should use Eutocius as well as

Dijksterhuis (1987), those interested in more biographical and histori-

cal detail on the mathematicians mentioned should use Knorr (1986),

(1989), and those looking for more bibliographic references should

use Knorr (1987) (which remains, sixteen years later nearly complete).

(Indeed, as mentioned above, Archimedes is not intensively studied.)

Following the translation of Archimedes’ work, I add a translation of

Eutocius’ commentar y to Archimedes. This is a competent commentary

and the only one of its kind to survive from antiquity. Often, it offers a

very useful commentary on the mathematical detail, and in many cases it

has unique historical signiﬁcance for Archimedes and for Greek mathemat-

ics in general. The translation of Eutocius follows the conventions of the

translation of Archimedes, but I do not add comments to his text, instead

supplying, where necessary, fuller footnotes.

preliminary notes: conventions 5

A special feature of this work is a critical edition of the diagrams. Instead of

drawing my own diagrams to ﬁt the text, I produce a reconstruction based on the

independent extant manuscripts, adding a critical apparatus with the variations

between the manuscripts. As I have argued elsewhere (Netz [for thcoming]), I

believe that this reconstruction may represent the diagrams as available in late

antiquity and, possibly, at least in some cases, as produced by Archimedes him-

self. Thus they offer another, vital clue to our main question, how Archimedes

thought and wrote. I shall return below to explain brieﬂy the purpose and

practices of this critical edition.

Before the translation itself I now add a few brief preliminary notes.

2 preliminary notes: conventions

2.1 Some special conventions of Greek mathematics

In the following I note certain practices to be found in Archimedes’ text that a

modern reader might ﬁnd, at ﬁrst, confusing.

1. Greek word order is much freer than English word order and so, selecting

from among the wider set of options, Greek authors can choose one word order

over another to emphasize a certain idea. Thus, for instance, instead of writing

“A is equal to B,” Greek authors might write “to B is equal A.” This would

stress that the main information concerns B, not A – word order would make B,

not A, the focus. (For instance, we may have been told something about B, and

now we are being told the extra property of B, that it is equal to A.) Generally

speaking, such word order cannot be kept in the English, but I try to note it

when it is of special signiﬁcance, usually in a footnote.

2. The summation of objects is often done in Greek through ordinary con-

junction. Thus “the squares AB and EZH” will often stand for what

we may call “the square AB plus the square EZH.” As an extension of

this, the ordinary plural form can serve, as well, to represent summation: “the

squares AB, EZH” (even without the “and” connector!) will then mean

“the square AB plus the square EZH.” In such cases, the sense of the ex-

pression is in itself ambiguous (the following predicate may apply to the sum

of the objects, or it may apply to each individually), but such expressions are

generally speaking easily disambiguated in context. Note also that while such

“implicit” summations are very frequent, summation is often more explicit and

may be represented by connectors such as “together with,” “taken together,”

or simply “with.”

3. The main expression of Greek mathematics is that of proportion:

AsAistoB,soisCtoD.

(A, B, C, and D being some mathematical objects.) This expression is often

represented symbolically, in modern texts, by

A:B::C:D

and I will use such symbolism in my footnotes and commentary. In the main

text I shall translate, of course, the original non-symbolic form. Note especially

that this expression may become even more concise, e.g.:

AsAistoB,CtoD,AsAtoB,CtoD.

6 introduction

And that it may have more complex syntax, especially:

A has to B the same ratio as C has to D, A has toBagreater ratio

than C has to D.

The last example involves an obvious extension of proportion, to ratio-

inequalities, i.e. A:B>C:D. More concisely, this may be expressed by:

AhastoBagreater ratio than C to D.

4. Greek mathematical propositions have, in many cases, the following six

parts:

Enunciation, in which the claim of the proposition is made, in general terms,

without reference to the diagram. It is important to note that, generally

speaking, the enunciation is equivalent to a conditional statement that if x

is the case, then so is y.

Setting-out, in which the antecedent of the claim is re-stated, in particular

terms referring to the diagram (with the example above, x is re-stated in

particular reference to the diagram).

Deﬁnition of goal, in which the consequent of the claim is re-stated, as

an exhortation addressed by the author to himself: “I say that ...,”“itis

required to prove that ...,”againintheparticular terms of the diagram

(with the same example, we can say that y is re-stated in particular reference

to the diagram).

Construction, in which added mathematical objects (beyond those required

by the setting-out) may be introduced.

Proof , in which the particular claim is proved.

Conclusion, in which the conclusion is reiterated for the general claim from

the enunciation.

Some of these parts will be missing in most Archimedean propositions, but the

scheme remains a useful analytical tool, and I shall use it as such in my com-

mentary. The reader should be prepared in particular for the following difﬁculty.

It is often very difﬁcult to follow the enunciations as they are presented. Since

they do not refer to the particular diagram, they use completely general terms,

and since they aspire to great precision, they may have complex qualiﬁcations

and combinations of terms. I wish to exonerate myself: this is not a problem of

my translation, but of Greek mathematics. Most modern readers ﬁnd that they

can best understand such enunciations by reading, ﬁrst, the setting-out and the

deﬁnition of goal, with the aid of the diagram. Having read this, a better sense

of the dramatis personae is gained, and the enunciation may be deciphered. In

all probability the ancients did the same.

2.2 Special conventions adopted in this translation

1. The main “<...>” policy:

Greek mathematical proofs always refer to concrete objects, realized in the

diagram. Because Greek has a deﬁnite article with a rich morphology, it can

elide the reference to the objects, leaving the deﬁnite article alone. Thus the

Greek may contain expressions such as

“The by the AB, B”

preliminary notes: conventions 7

whose reference is

“The <rectangle contained> by the <lines> AB, B”

(the morphology of the word “the” determines, in the original Greek, the iden-

tity of the elided expressions, given of course the expectations created by the

genre).

In this translation, most such elided expressions are usually added inside

pointed brackets, so as to make it possible for the reader to appreciate the radical

concision of the original formulation, and the concreteness of reference –

while allowing me to represent the considerable variability of elision (very

often, expressions have only partial elision). This variability, of course, will be

seen in the ﬂuctuating positions of pointed brackets:

“The <rectangle contained> by the <lines> AB, B,” as against, e.g.

“The <rectangle> contained by the <lines> AB, B.”

(Notice that I do not at all strive at consistency inside pointed brackets. Inside

pointed brackets I put whatever seems to me, in context, most useful to the

reader; the duties of consistency are limited to the translation proper, outside

pointed brackets.)

The main exception to my general pointed-brackets policy concerns points

and lines. These are so frequently referred to in the text that to insist, always,

upon a strict representation of the original, with expressions such as

“The <point> A,” “The <line> AB”

would be tedious, while serving little purpose. I thus usually write, simply:

A, AB

and, in the less common cases of a non-elliptic form:

“The point A,” “The line AB”

The price paid for this is that (relatively rarely) it is necessary to stress that the

objects in question are points or lines, and while the elliptic Greek expresses

this through the deﬁnite ar ticle, my elliptic “A,” “AB” does not. Hence I need

to introduce, here and there, the expressions:

“The <point> A,” “The <line> AB”

but notice that these stand for precisely the same as

A, AB.

2. The “<=..>” sign is also used, in an obvious way, to mean essentially the

same as the “[Scilicet. . . .]” abbreviation. Most often, the expression following

the “=” will disambiguate pronouns which are ambiguous in the English (but

which, in the Greek, were unambiguous thanks to their morphology).

3. Square brackets in the translation (“[. . .]”) represent the square brackets

in Heiberg’s (1910–15) edition. They signify units of text which according to

Heiberg were interpolated.

4. Two sequences of numbering appear inside standard brackets. The Latin

alphabet sequence “(a) . . . (b) . . .” is used to mark the sequence of constructions:

as each new item is added to the construction of the geometrical conﬁguration

(following the setting-out) I mark this with a letter in the sequence of the Latin

alphabet. Similarly, the Arabic number sequence “(1) . . . (2) . . .” is used

to mark the sequence of assertions made in the course of the proof: as each

new assertion is made (what may be called “a step in the argument”), I mark

this with a number. This is meant for ease of reference: the footnotes and the

8 introduction

commentary refer to constructions and to claims according to their letters or

numbers. Note that this is purely my editorial intervention, and that the

original text had nothing corresponding to such brackets. (The same is true

for punctuation in general, for which see point 6 below.) Also note that these

sequences refer only to construction and proof: enunciation, setting-out, and

deﬁnition of goal are not marked in similar ways.

5. The “/. . ./” symbolism: for ease of reference, I ﬁnd it useful to add in

titles for elements of the text of Archimedes, whether general titles such as

“introduction” or numbers referring to propositions. I suspect Archimedes’

original text had neither, and such titles and numbers are therefore mere aids

for the reader in navigating the text.

6. Ancient Greek texts were written without spacing or punctuation: they

were simply a continuous stream of letters. Thus punctuation as used in moder n

editions reﬂects, at best, the judgments of late antiquity and the middle ages,

more often the judgments of the modern editor. I thus use punctuation freely,

as another editorial tool designed to help the reader, but in general I try to keep

Heiberg’s punctuation, in deference to his superb grasp of the Greek mathe-

matical language, and in order to facilitate simultaneous use of my translation

and Heiberg’s edition.

7. Greek diagrams can be characterized as “qualitative” rather than “quan-

titative.” This is very difﬁcult to deﬁne precisely, and is best understood as

a warning: do not assume that relations of size in the diagram represent

relations of size in the depicted geometrical objects. Thus, two geometri-

cal lines may be assumed equal, while their diagrammatic representation is

of two unequal lines and, even more confusingly, two geometrical lines may

be assumed unequal, while their diag rammatic representation is of two equal

lines. Similar considerations apply to angles etc. What the diagram most clearly

does represent are relations of connection between the geometrical constituents

of the conﬁguration (what might be loosely termed “topological properties”).

Thus, in an extreme case, the diagram may concentrate on representing the

fact that two lines touch at a single point, ignoring another geometrical fact,

that one of the lines is straight while the other is curved. This happens in

a series of propositions from 21 onwards, in which a dodecagon is repre-

sented by twelve arcs; but this is an extreme case, and generally the diag ram

may be relied upon for such basic qualitative distinctions as straight/curved.

See the following note on purpose and practices of the critical edition of

diagrams.

2.3 Purpose and practices of the critical edition of diagrams

The main purpose of the critical edition of diagrams is to reconstruct the

earliest form of diagrams recoverable from the manuscript evidence. It should

be stressed that the diagrams across the manuscript tradition are strikingly

similar to each other, often in quite trivial detail, so that there is hardly a question

that they derive from a common archetype. For most of the text translated here,

diagrams are preserved only for one Byzantine tradition, that of codex A (see

below, note on the text of Archimedes). However, for most of the diagrams from

剩余386页未读，继续阅读

liangalei

粉丝: 0
资源: 2

Archimedes的科学遗产：英文全译本与古希腊数学的探索

PyPI 官网下载 | archimedes-python-client-2.2.3.tar.gz

GVC-Archimedes-开源

Archimedes-开源

Archimedes-July-9:节点简介

archimedes-legacy:良好的“旧”阿基米德软件的稍作清理的版本

archimedes-pelican:结合 Angular + Pelican 用于 Python 驱动的 web 应用程序。 http

Python客户端库archimedes-python-client-2.2.3发布

Archimedes Emulator-开源

Trammel of Archimedes Dynamic Force Analysis：阿基米德trammel全范围运动动态力分析的完整代码。-matlab开发

Archimedes: An architecture open CAD-开源

最新资源