Translation of Taylor series into LFT expansions
Reinhold Heckmann
1Introduction
In Exact Real Arithmetic,real numbers are represented as potentially in?nite streams of information units,called digits.In this paper,we work in the framework of Linear Frac-tional Transformations(LFT’s,also known as M¨o bius transformations)that provide an elegant approach to real number arithmetic(Gosper1972,Vuillemin1990,Nielsen and Kornerup1995,Potts and Edalat1996,Edalat and Potts1997,Potts1998b).One-dimensional LFT’s are used as digits and to implement basic unary functions,while two-dimensional LFT’s provide binary operations such as addition and multiplication, and can be combined to obtain in?nite expression trees denoting transcendental func-tions.Peter Potts(1998a,1998b)derived these expression trees from continued fraction expansions of the transcendental functions.In contrast,we show how to derive LFT ex-pression trees from power series expansions,which are available for a greater range of functions.
In Section2,we present the LFT approach in some detail.Section3contains the main results of the paper.We?rst derive an LFT expansion from a power series using Horner’s scheme(Section3.1).The results are not very satisfactory.Thus,we show how LFT expansions may be modi?ed using algebraic transformations(Section3.2).A particular such transformation,presented in Section3.3,yields satisfactory results for standard functions,as shown in the?nal examples section4.
2Exact real arithmetic by linear fractional transformations
In this section,we recall the framework of exact real arithmetic via LFT’s(Gosper 1972,Vuillemin1990,Nielsen and Kornerup1995).We do not follow exactly the version used by the group of Edalat and Potts at Imperial College(Potts and Edalat 1996,Edalat and Potts1997,Potts1998a,Potts1998b),but change the base interval from0∞to11.The reasons for this change and its pros and cons were discussed in(Heckmann2000).
2.1LFT’s and matrices
Linear Fractional Transformations(LFT’s)are functions x ax c
More precisely,LFT’s are functions from to where∞is the one-point compacti?cation of the real line,and is plus an additional ‘unde?ned’value.The value∞arises as r0with r0,and on the other hand,the value of the LFT at∞is de?ned as a b.The unde?ned value arises as00,and the value of any LFT at is.
The four parameters of an LFT may be presented as a2-2-matrix M a
b c
d
of integers,
hereafter matrix.Thus,any matrix M a
b c
d
denotes an LFT M with M x ax c
x y
.The contractivity of an
af?ne matrix a
0c
d
is a d,and hence the contractivity of c;a is a.
2.2Representing reals by LFT’s
Using the base interval B11,a sequence A0,A1,...of LFT’s induces a sequence I1,I2,...of intervals by
I n A0A n1B(5) The sequence of intervals is nested if I n I n1for all n 1.The inclusion I n I n1
means A0A n1B A0A n B.If the matrices A0,...,A n1are non-singular,this is equivalent to A n B B.Therefore in the non-singular case,the sequence of intervals is nested iff all LFT’s A n with n1are re?ning.Note that A0need not be re?ning.
A nested sequence of intervals I n c n r n converges to a single point x iff r n0as n∞.Even if the sequence of intervals c n r n is not nested,it may converge to a single point x in the sense that c n x and r n0.But this is not suf?cient to say that the intervals approximate x(consider for instance the case I n2n1n1n3n). Therefore,we require the interval sequence to be nested.In theory,it is suf?cient that the interval sequence is eventually nested,i.e.,I n I n1for almost all n,because we may omit the?rst few intervals and obtain a nested sequence.Yet this procedure is not always practical as we shall see later.
In the sequel,we shall write I n y if the two end points converge to y,I n y if I n y and the intervals are eventually nested,and I n y if I n y and the intervals are nested from the beginning.
A sequence of matrices that generates a nested sequence of intervals converging to x may be considered as a representation of x.Because of the usage of matrix multiplica-tion in(5),we write x∏∞n0A n,using a(formal)in?nite product notation.Many real numbers can be elegantly represented by such in?nite products.
To control the information?ow in computations with reals,it turned out to be useful to convert these representations into a kind of standard form,using a leading exponent matrix(Heckmann2000)or sign matrix followed by a sequence of digit matrices(Potts and Edalat1997,Edalat and Potts1997).For the purposes of the present paper,it is not necessary to delve into the details of these special representations.Instead,we may assume that real numbers are represented by arbitrary in?nite products where all matrices but the?rst one are re?ning(to obtain a nested sequence of intervals)with contractivities bounded by some c1(to ensure the interval sequence shrinks to a single point).
2.3Tensors
To represent sums,products,and quotients,two-dimensional LFT’s are employed.They are characterised by8integer parameters,and thus can be represented by2-4-matrices
of integers,called tensors.A tensor T a
b c
d
e
f
g
h
denotes the bistrict function T:
given by T x y axy cx ey g
matrices C n from the tensor,thereby producing a representation∏∞n0C n for the result T x y.
For?xed x,the function T x:y T x y is an LFT with T x ax e
bx f cx g
dx h
.Con-
versely,any matrix M x whose entries linearly depend on a parameter x can be turned into a tensor T such that M x y T x y.The representations of the values of tran-scendental functions will generally involve such parametrised matrices.For instance, we shall later derive the representation
e x x
x1
1
∞
∏
n1
x
x
n1
from the Taylor series e x∑∞n0x n
limsup n∞n 0
∞and1
a n1R limsup
n∞
a n
a n1
converges,its limit is R.For instance,in e x∑∞n01 a n1
n1!
3.1The Horner evaluation of power series
Horner’s rule to evaluate polynomials can easily be adapted to obtain an LFT expansion
from a power series f x ∑∞n 0a n x n
with convergence radius R :
∞
∑n 0
a n x n
a 0
x
∞
∑n 1
a n x n
1
a 0;x
∞
∑n 0
a n
1x
n
(8)
with the af?ne LFT a 0;x
λy a 0
xy .This suggests that f x is given by
f x
∞
∏
n 0
a n ;x
∞
∏
n 0
x 0a n
1
(9)
To verify this guess,we ?rst determine A 0
A n
1
for A i
a i ;x :
a 0;x
a n 1;x
n 1
∑i 0
a i x i ;x n
(10)
This is obvious for n 1.The inductive step follows from ∑n
1i
0a i x i ;
x n
a n ;x
∑n 1i 1a i x
i x n a n ;x n x by (1).From (10),we immediately obtain I n A 0A n 1B s n x n where s n ∑n 1i 0a i x i .
For x R ,we know s n
f x as n ∞,and x n 0holds if and only if x 1.Thus,I n f x holds for x min R 1.For x 1,the radii x n do not tend to 0.Thus,we have already lost part of the convergence area if R 1.
The LFT’s a n ;x are non-singular iff x 0.For x 0,we have f x
a 0and I n I 1a 00.Thus,we always have a nested sequence of intervals converging to the right value in this singular situation.
The LFT’s a n ;x are re?ning iff a n
x 1.Thus in case x 0,the generated sequence of intervals is nested iff x 1a n for all n 1,iff x 1sup n 1a n ,
and it is eventually nested if x 1limsup n ∞a n .
Summary of the properties of the Horner evaluation:For x
R ,we have
I n f x if x
1;I n f x
if x
1limsup n ∞
a n ;
I n
f x
if x
1
sup n
1
a n .
Let us consider some examples:
For f x
1
1x
x n .For x
1,the center points converge to 1
1x .Yet A n is re?ning only if x
0.
Thus,for x0,the sequence of intervals is not nested—not even eventually nested.In fact,for0x1,
1
1x
x n
1x
is not even contained in any of the intervals I n! For f x e x∑∞n01
n!
;x∏∞n0n!x01n!with uncomfortably big entries.The general theory tells us that the end points of I n tend to e x iff x1(since for x1,the radii do not converge to0).Matrix A n is re?ning iff x11
n!will hold for suf?ciently large n.Hence,in contrast to
the?rst example,the function value e x is contained in almost all intervals I n if x 1.
3.2Transformation of the LFT expansion
The LFT expansions derived from Horner’s rule in the previous section are not very good:they are rarely nested,loose out part of the convergence circle,and contain un-comfortably large entries for important functions like e x.There is a general way to transform LFT expansions which may lead to improvements.
Given a sequence A0,A1,A2,...of matrices,the corresponding sequence of intervals I1,I2,...is given by I n P n B(n1)where P n A0A n1is the product of the?rst n matrices.Now,we may choose any sequence U1,U2,...of non-singular matrices and throw in products U i U i of U i and its inverse U i into the original product:
P n A0A n1A0U1U1A1U2U2U n1U n1A n1
With the new names Q n P n U n for n1,B0A0U1,and B i U i A i U i1for i1,we obtain
Q n P n U n B0B1B n1
This means we have de?ned a new sequence B0,B1,...of matrices which denotes the sequence of intervals J n Q n B.
We want to apply this technique to the af?ne matrices A n a n;x coming from Horner’s rule.Since we want to preserve af?nity here,we choose af?ne matrices U n c n;r n (with r n0for non-singularity).Then
A n U n1a n;x c n1;r n1a n c n1x;r n1x
Therefore
B0A0U1a0c1x;r1x
while multiplication with U n from the left yields
B n c n
r n
a n c n1x;r n1x a n
c n c n1x
r n
x
The matrices Q n are given by
Q n P n U n s n;x n c n;r n s n c n x n;r n x n
so that the denoted intervals are J n s n c n x n r n x n.Note that for x0,we have J n a00a0f0.
3.3The standard choice for c n and r n
There are two possibilities to reduce the term a n c n c n1x in B n to just one summand, namely c n0and c n a n.In the?rst case,J n becomes s n r n x n,while in the second, we have J n s n a n x n r n x n s n1r n x n.This shows that the two possibilities are actually not that different.We prefer the second possibility because it appears to be one step ahead of the?rst.
The choice c n a n yields
B0r1x
a0a1x
1
and B n
r n1x
a n1x
r n
for n1
The resulting intervals were already determined to be J n s n1r n x n.Thus,J n
f x if x R and r n x n0.The matrices B n are re?nin
g for x r n
a n1
.The resulting intervals are J n s n1ka n x n.If x R,then ∑∞n0a n x n converges,which implies a n x n0.Thus J n f x holds for all x R.
The matrices B n are re?ning for x ka n
k1
q n.Thus the interval se-quence is eventually nested if x k
k1R.The condition for
a nested sequence is similar,with liminf replaced by inf.
J n f x for x R;
J n f x for x k
k1
R if q n converges);
J n f x for x k
k1increases with k,yet the simplest possibility is k1with k
2
.
Values of x near R should be avoided anyway since the contractivity of B n is x q n which converges to x R if q n is convergent;in this case,we have poor contractivity for x R,and the restriction to x1
4Case studies
Throughout this section,we use the name q for inf n1q n which is important for deter-mining the region where a nested sequence of intervals is obtained.
4.1Exponential function
Recall e x∑∞n01a
n1
1n!
n1
1
2
.
The contractivity is not affected by varying k.Even with k1,representation(11) yields an eventually nested convergent sequence of intervals for all x;the condition x1is only needed to ensure that the sequence is nested from the beginning.
As already mentioned,a representation such as(11)is open to two different interpre-tations.For rational arguments x,it is(equivalent to)an in?nite product of integer
matrices,e.g.,e1
02
1
∏∞n1101n1.In the rational case,it is not absolutely nec-
essary that all matrices but the?rst one are re?ning.Consider for instance x2,where
the?rst three matrices2
03
1
,2
2
2
,and2
2
3
are not re?ning,while the next matrix
2 02
4
and all subsequent ones are re?ning.We may multiply the?rst three matrices
into one which gives e24
019
3
∏∞n1202n3which generates a nested sequence of
intervals.
For general(real)arguments however,representation(11)should be turned into the in?nite tensor expression e x T0x T1x T2x with
T01
1
1
1
and T n
1
1
n1
In this case,the above trick to obtain a nested sequence of intervals even for x1 cannot be applied because there is no way to multiply the?rst few tensors into one(at least in the standard LFT approach).If you try to multiply the corresponding matrices, whose entries depend linearly on x,the entries of the product matrix will be non-linear polynomials in x.
4.2Cosine and hyperbolic cosine
The power series of the cosine function is cos x∑∞n01n
2,and
q n2n12n2,whence q12,which for the standard case gives a bound of 6for a nested sequence.Hence with a bit of scaling
cos x
u
2u
2
∞
∏
n1
u
u
2n12n1
nested for u x2 6.
The function can be extended to the whole real line by cos2x2cos2x 1.
The power series for cosh is similar to that for cos:cosh x∑∞n01
2n1!
.Here,the exponents of x are odd.They can be expressed in terms of x2if one factor of x is taken out:sin x 2n1!
x2n.We get a01,a11
x
u
3u
3
∞
∏
n1
u
u
2n12n3
nested for u x210.
For extension to the whole real line,the formula sin2x2sin x cos x may be used which unfortunately involves cosine as well.Again,sinh is analogous;just replace u by u.
4.4Inverse of tangent
The power series for arctan x is∑∞n01n x2n1
x
∑∞n01n3,q n2n3
x
u
3u
3
∞
∏
n1
2n1u
2n1u
2n3
for u x21 n
x n.Thus q n n1
An arbitrary positive argument of ln can be moved towards1by the equations ln2x
ln x ln2and ln1
ln11
2
.
2
A better way to obtain ln2is at follows:Recall artanh x1
,and therefore,
1x
artanh13
ln2ln113
3
.
9
References
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Gosper,W.(1972):Continued fraction arithmetic.In:Technical Report HAKMEM Item101B, MIT Arti?cial Intelligence Memo239,MIT.
Heckmann,R.(2000):How many argument digits are needed to produce n result digits?In:Real Number Computation,RealComp’98,Indianapolis,Indiana.Elsevier Science B.V.,Amsterdam, URL:http://www.elsevier.nl/locate/entcs/volume24.html(Electronic Notes in Theoretical Com-puter Science,vol.24)
Nielsen,A.,Kornerup,P.(1995):MSB-?rst digit serial arithmetic.J.of https://www.doczj.com/doc/662422732.html,p.Scien. 1(7):523–543
Potts,P.J.(1998a):Ef?cient on-line computation of real functions using exact?oating point. Draft report,Imperial College,London.
URL:http://www.purple?https://www.doczj.com/doc/662422732.html,/?potts/pub/phd/ef?cient.ps.gz
Potts,P.J.(1998b):Exact Real Arithmetic using M¨o bius Transformations.PhD thesis,Imperial College,London.URL:http://www.purple?https://www.doczj.com/doc/662422732.html,/?potts/thesis.ps.gz
Potts,P.J.,Edalat,A.(1996):Exact real arithmetic based on linear fractional transformations. Draft report,Imperial College,London.
URL:http://www.purple?https://www.doczj.com/doc/662422732.html,/?potts/pub/phd/stanford.ps.gz
Potts,P.J.,Edalat,A.(1997):Exact real computer arithmetic.Draft report,Imperial College, London.URL:http://www.purple?https://www.doczj.com/doc/662422732.html,/?potts/pub/phd/normal.ps.gz
Vuillemin,J.E.(1990):Exact real computer arithmetic with continued fractions.IEEE Transac-tions on Computers39(8):1087–1105
Unit 9 How to Grow Old Bertrand A. Russell 1. In spite of the title, this article will really be on how not to grow old, which, at my time of life, is a much more important subject. My first advice would be, to choose your ancestors carefully. Although both my parents died young, I have done well in this respect as regards my other ancestors. My maternal grandfather, it is true, was cut off in the flower of his youth at the age of sixty-seven, but my other three grandparents all lived to be over eighty. Of remoter ancestors I can only discover one who did not live to a great age, and he died of a disease which is now rare, namely, having his head cut off. A great-grandmother of mine, who was a friend of Gibbon, lived to the age of ninety-two, and to her last day remained a terror to all her descendants. My maternal grandmother, after having nine children who survived, one who died in infancy, and many miscarriages, as soon as she became a widow devoted herself to women’s higher education. She was one of the founders of Girton College, and worked hard at opening the medical profession to women. She used to relate how she met in Italy an elderly gentleman who was looking very sad. She inquired the cause of his melancholy and he said that he had just parted fro m his two grandchildren. “Good gracious,” she exclaimed, “I have seventy-two grandchildren, and if I were sad each time I parted from one of them, I should have a dismal existence!” “Madre snaturale,” he replied. But speaking as one of the seventy-two, I prefer her recipe. After the age of eighty she found she had some difficulty in getting to sleep, so she habitually spent the hours from midnight to 3 a.m. in reading popular science. I do not believe that she ever had time to notice that she was growing old. This, I think, is the proper recipe for remaining young. If you have wide and keen interests and activities in which you can still be effective, you will have no reason to think about the merely statistical fact of the number of years you have already lived, still less of the probable brevity of your future. 2. As regards health, I have nothing useful to say since I have little experience of illness. I eat and drink whatever I like, and sleep when I cannot keep awake. I never do anything whatever on the ground that it is good for health, though in actual fact the things I like doing are mostly wholesome. 3. Psychologically there are two dangers to be guarded against in old age. One of these is undue absorption in the past. It does not do to live in memories, in regrets for the good old days, or in sadness about friends who are dead. One’s thoughts must be directed to
旅游与文化I Part I 1．charming autumn scenery in a most fresh air and clear weather 秋高气爽，秋色宜人 2．the 15th General Assembly Session of the World Tourism Organization 世界旅游组织第15届全体大会3．to travel ten thousand li and read ten thousand books 读万卷书，行万里路 4．enriching themselves mentally and physically 承天地之灵气，接山水之精华 5．tourist arrival 旅游人数 6．foreign currency receipts 外汇收入 7．outbound tourists 出境旅游人数 8．unique, rich and varied tourism resources 得天独厚的旅游资源 9．World Cultural and Natural Heritages sites 世界文化遗产地和世界自然遗产地 10．t o add radiance and charm to each other 交相辉映 11．a thriving modern metropolis 繁华的现代化大都市 12．a patchwork of cottages 村舍星罗棋布 13．t o exist side by side 鳞次栉比 14．I nternational Architecture Exhibition 万国建筑博览会 15．c lock towers and turrets , marble pillars 钟塔、角楼和大理石柱 16．e ach representing a distinctively individual appearance 风格迥异，各领风骚 17．t he rainy season 梅雨季节 18．t o linger longer 留连忘返 19．e xcellence, elegance and the best quality 卓越超群，富丽堂皇，一流质量 20．e mbroidery, inlaid lacquer 刺绣，金漆镶嵌 21．g old and silver jewelleries, water-color woodblock prints 金银首饰，木刻水印 22．c arvings in jade, ivory, bamboo and woven bamboo baskets 玉雕、牙雕，竹雕，竹编筐篮 23．b ird cages, lanterns 鸟笼灯笼 24．d ouble-sided embroidery and sandal wood fans from Suzhou 双面绣和苏州的檀香扇 25．t erracotta teapots from Yixing, and clay figures from Wuxi 宜兴的陶制茶壶和无锡的泥人 26．t he Peach Blossom Fair 桃花节 27．t he Daci Temple Fair 大慈寺庙会 28．t he Chengdu Tourism Festival 成都旅游节 29．a place blessed with favorite climate, fertile land, rich resources and outstanding talents 物华天宝，人杰地灵30．s uperb artistic style of aiming at catching the sprit of the landscape 写意山水 31．a rtistic gems 艺术瑰宝 32．U NESCO Heritage Committee 联合国教科文组织遗产委员会 33．t he list of World cultural heritage 世界文化遗产名录 34．b ronzeware 青铜器 35．b amboo, wood and lacquer ware 竹木漆器 36．i nscribed bones and tortoise shells 甲骨 37．s eals 玺印 38．a rchaeology 39．r estoration room 文物修复馆 旅游与文化II Part II景点描述常用语 Match work: 雄伟壮丽imposing 灯火辉煌glittering
Lesson 1 I’m not feeling well. Let’s talk （M=Mom, T= Tom） M: What,'s the matter, Tom T: I'm not feeling well, Mom M: Do you have a cold T: Yes, I think so. Could you give me some water, please M: Here you are. T: Thank you, Mom. M: Tom, you must go and see a doctor. T: OK, Mom. M: It's cold outside. You must wear your coat. T: OK, Mom. Could you pass me my coat,please M: Here you are. T: Thank you, Mom M: Tell me your teacher's number. I'll call him and tell him you are sick. T: OK. Here it is. 译文 (M=妈妈,T=汤姆) 妈妈:怎么了,汤姆 汤姆:我感觉不舒服,妈妈。 妈妈:你感冒了吗 汤姆:是的,我想是这样的。您能给我一些水吗 妈妈:给你。 汤姆:谢谢您,妈妈。 妈妈:汤姆,你必须去看医生. 汤姆:好的,妈妈。 妈妈:外面很冷。你必须穿你的外套。 汤姆:好的,妈妈。您能把我的外套递给我吗 妈妈:给你。 汤姆:谢谢您,妈妈。 妈妈:告诉我你老师的电话号码。我将给他打电话告诉他你生病了。
Unit 1 A CLASS ACT Florence Cartlidge 1. Growing up in bomb-blitzed Manchester during the Second World War meant times were tough, money was short, anxiety was rife and the pawnshop was a familiar destination for many families, including mine. 2. Yet I could not have asked for more enterprising and optimistic parents. They held our family together with hard work, dignity and bucketloads of cheer. My sturdy and ingenious father could turn his hand to almost anything and was never short of carpentry and handyman work. He even participated in the odd bout of backstreet boxing to make ends meet. For her part, our mum was thrifty and meticulously clean, and her five children were always sent to school well fed, very clean, and attired spotlessly, despite the hard conditions. 3. The trouble was, although my clothes were ironed to a knife-edge, and shoes polished to a gleam, not every item was standard school uniform issue. While Mum had scrimped and saved to obtain most of the gear, I still didn’t have the pres cribed blue blazer and hatband. 4. Because of the war, rationing was in place and most schools had relaxed their attitude towards proper uniforms, knowing how hard it was to obtain clothes. Nevertheless, the girls’ school I attended made it strict policy that each of its students was properly attired, and the deputy headmistress who ran the daily assembly made it her mission to teach me a lesson. 5. Despite my attempts at explaining why I couldn’t comply, and despite the fact that I was making slow progress towards the full uniform, every day I would be pulled out of line and made to stand on the stage as a shining example of what not to wear to school. 6. Every day I would battle back tears as I stood in front of my peers, embarrassed and, most often, alone. My punishment also extended to being barred from the gym team or to not taking part in the weekly ballroom dancing classes, which I adored. I desperately
The Chaser John Collier Alan Auste n, as n ervous as a kitte n, went up certa in dark and creaky stairs in the n eighborhood of Pell Street , and peered about for a long time on the dim landing before he found the n ame he wan ted writte n obscurely on one of the doors. He pushed ope n this door, as he had bee n told to do, and found himself in a tiny room, which contained no furn iture but a pla in kitche n table, a rock in g-chair, and an ordinary chair. On one of the dirty buff-colored walls were a couple of shelves, containing in all perhaps a doze n bottles and jars. An old man sat in the rock in g-chair, read ing a n ewspaper. Ala n, without a word, handed him the card he had been given. 人Sit down, Mr. Austen, said the old man very politely. 人I am glad to make your acqua intance. 人Is it true, asked Alan, 人that you have a certain mixture that has ! er ! quite extraordinary effects? 人My dear sir, replied the old man, 人my stock in trade is not very large ! I don …t deal in laxatives and teething mixtures ! but such as it is, it is varied. I think nothing I sell has effects which could be precisely described as ordin ary. 人Well, the fact is ! began Alan. 人Here, for example, interrupted the old man, reaching for a bottle from the shelf. 人Here is a liquid as colorless as water, almost tasteless, quite imperceptible in coffee, wine, or any other beverage. It is also quite imperceptible to any known method of autopsy. 人Do you mean it is a poison? cried Alan, very much horrified. 人Call it a glove-cleaner if you like, said the old man indifferently. 人Maybe it will clean gloves. I have never tried. One might call it a life-cleaner. Lives need cleaning sometimes. 人I want nothing of that sort, said Alan. 人Probably it is just as well, said the old man. 人Do you know the price of this? For one teaspoonful, which is sufficient, I ask five thousa nd dollars. Never less. Not a penny less. 人I hope all your mixtures are not as expensive, said Alan apprehe nsively. 人Oh dear, no, said the old man. 人It would be no good charg ing that sort of price for a love poti on, for example. Young people who n eed a love poti on very seldom have five thousa nd dollars. Otherwise they would not n eed a love poti on. 人I am glad to hear that, said Alan. 人I look at it like this, said the old man. 人Please a customer with one article, and he will come back whe n he n eeds another. Even if it is more costly. He will save up for it, if n ecessary. 人So, said Alan, 人you really do sell love potions? 人If I did not sell love potions, said the old man, reaching for another bottle, 人I should not have mentioned the other matter to you. It is only whe n one is in a positi on to oblige that one can afford to be so con fide ntial. 人And these potions, said Alan. 人They are not just ! just ! er ! 人Oh, no, said the old man. 人Their effects are permanent, and exte nd far bey ond casual impulse. But they in clude it. Boun tifully, in siste ntly. Everlast in gly. 人Dear me! said Alan, attempting a look of scientific detachme nt. "How very in teresti ng! 人But consider the spiritual side, said the old man.
《论语十则》——《中国文化经典研读》（整理） 1 子曰:“君子食无求饱，居无求安，敏于事而慎于言，就(1)有道(2)而正(3)焉，可谓好学也已。” 【注释】 (1)就:靠近、看齐。 (2)有道:指有道德的人。 (3)正:匡正、端正。 【译文】 孔子说:“君子，饮食不求饱足，居住不要求舒适，对工作勤劳敏捷，说话却小心谨慎，到有道的人那里去匡正自己，这样可以说是好学了。” 2 2?4 子曰:“吾十有(1)五而志于学，三十而立(2)，四十而不惑(3)，五十而知天命(4)，六十而耳顺(5)，七十而从心所欲不逾矩(6)。” 【注释】 (1)有:同“又”。 (2)立:站得住的意思。 (3)不惑:掌握了知识，不被外界事物所迷惑。 (4)天命:指不能为人力所支配的事情。 (5)耳顺:对此有多种解释。一般而言，指对那些于己不利的意见也能正确对待。 (6)从心所欲不逾矩:从，遵从的意思;逾，越过;矩，规矩。 【译文】 孔子说:“我十五岁立志于学习;三十岁能够自立;四十岁能不被外界事物所迷惑;五十岁懂得了天命;六十岁能正确对待各种言论，不觉得不顺;七十岁能随心所
欲而不越出规矩。” 3 子曰:“由(1)，诲女(2)，知之乎,知之为知之，不知为不知，是知也。” 【注释】 (1)由:姓仲名由，字子路。生于公元前542年，孔子的学生，长期追随孔子。 (2)女:同汝，你。 【译文】 孔子说:“由，我教给你怎样做的话，你明白了吗,知道的就是知道，不知道就是不知道，这就是智慧啊～” 4.颜渊、季路侍(1)。子曰:“盍(2)各言尔志。”子路曰:“原车马，衣轻裘，与朋友共，敝之而无憾。”颜渊曰:“愿无伐(3)善，无施劳(4)。”子路曰:“愿闻子之志。”子曰:“老者安之，朋友信之，少者怀之(5)。” 【注释】 (1)侍:服侍，站在旁边陪着尊贵者叫侍。 (2)盍:何不。 (3)伐:夸耀。 (4)施劳:施，表白。劳，功劳。 (5)少者怀之:让少者得到关怀。 【译文】 颜渊、子路两人侍立在孔子身边。孔子说:“你们何不各自说说自己的志向,”子路说:“愿意拿出自己的车马、衣服、皮袍，同我的朋友共同使用，用坏了也不抱怨。”颜渊说:“我愿意不夸耀自己的长处，不表白自己的功劳。”子路向孔子说:“愿意听听您的志向。”孔子说:“(我的志向是)让年老的安心，让朋友们信任我，让年轻的子弟们得到关怀。” 5 子曰:“知之者不如好之者，好之者不如乐之者。” 【译文】
Unit 3 Out of Step Bill Bryson 1 After living in England for 20 years, my wife and I decided to move back to the United States. We wanted to live in a town small enough that we could walk to the business district, and settled on Hanover, N.H., a typical New England town—pleasant, sedate and compact. It has a broad central green surrounded by the venerable buildings of Dartmouth College, an old-fashioned Main Street and leafy residential neighborhoods. 2 It is, in short, an agreeable, easy place to go about one’ s business on foot, and ye far as I can tell, virtually no one does. 3 Nearly every day, I walk to the post office or library or bookstore, and sometimes, if I am feeling particu larly debonair, I stop at Rosey Jekes Caféa cappuccino. Occasionally, in the evenings, my wife and I stroll up to the Nugget Theatre for a movie or to Murphy’ s on the Green for a beer, I wouldn’ t dream of going to any of these places by car. People ha ve gotten used to my eccentric behavior, but in the early days acquaintances would often pull up to the curb and ask if I wanted a ride. 4“ I’ m going your way,” they would insist when I politely declined. no bother.”
Unit 7 The Chaser John Henry Collier 1 Alan Austen, as nervous as a kitten, went up certain dark and creaky stairs in the neighborhood of Pell Street, and peered about for a long time on the dim hallway before he found the name he wanted written obscurely on one of the doors. 2 He pushed open this door, as he had been told to do, and found himself in a tiny room, which contained no furniture but a plain kitchen table, a rocking-chair, and an ordinary chair. On one of the dirty buff-coloured walls were a couple of shelves, containing in all perhaps a dozen bottles and jars. 3 An old man sat in the rocking-chair, reading a newspaper. Alan, without a word, handed him the card he had been given. “Sit down, Mr. Austen,” said the old man very politely. “I am glad to make your acquaintance.” 4 “Is it true,” asked Alan, “that you have a certain mixture that has … er … quite extraordinary effects?” 5 “My dear sir,” replied the old man, “my sto ck in trade is not very large — I don’t deal in laxatives and teething mixtures —but such as it is, it is varied. I think nothing I sell has effects which could be precisely described as ordinary.” 6 “Well, the fact is …” began Alan. 7 “Here, for example,” interrupted the old man, reaching for a bottle from the shelf. “Here is a liquid as colourless as water, almost tasteless, quite imperceptible in coffee, wine, or any other beverage. It is also quite imperceptible to any known method of autopsy.” 8 “Do you mean it is a poison?” cried Alan, very much horrified. 9 “Call it a glove-cleaner if you like,” said the old man indifferently. “Maybe it will clean gloves. I have never tried. One might call it a life-cleaner. Lives need cleaning sometimes.” 10 “I want nothing of that sort,” said Alan. 11 “Probably it is just as well,” said the old man. “Do you know the price of this? For one teaspoonful, which is sufficient, I ask five thousand dollars. Never less. Not a penny less.” 12 “I hope all your mixtures are not as expensive,” said Alan apprehensively.
课文翻译 Unit 1 Text A young man finds that strolling along the streets without an obvious purpose can lead to trouble with the law. One misunderstanding leads to another until eventually he must appear in court for trial…… 一个青年发现，在大街上毫无明显目的地游逛会招致警方的责罚。误会一个接一个发生，最终他只得出庭受审…… A Brush with the Law 与警察的一场小冲突 I have only once been in trouble with the law. 我平生只有一次跟警方发生纠葛。 The whole process of being arrested and taken to court was a rather unpleasant experience at the time, but it makes a good story now. 被捕和出庭的整个过程在当时是一件非常不愉快的事，但现在倒成了一篇很好的故事。 What makes it rather disturbing was the arbitrary circumstances both of my arrest and my subsequent fate in court. 这次经历令人可恼之处在于围绕着我的被捕以及随后庭上审讯而出现的种种武断专横的情况。It happened in February about twelve years ago. 事情发生在大约12年前，其时正是2月。 I had left school a couple of months before that and was not due to go to
Unit 1 1 A CLASS ACT 2 3 Florence Cartlidge 4 5 1. Growing up in bomb-blitzed Manchester during the Second World War 6 meant times were tough, money was short, anxiety was rife and the pawnshop was a familiar destination for many families, including mine. 7 8 9 2. Yet I could not have asked for more enterprising and optimistic 10 parents. They held our family together with hard work, dignity and 11 bucketloads of cheer. My sturdy and ingenious father could turn his hand 12 to almost anything and was never short of carpentry and handyman work. 13 He even participated in the odd bout of backstreet boxing to make ends 14 meet. For her part, our mum was thrifty and meticulously clean, and her 15 five children were always sent to school well fed, very clean, and attired 16 spotlessly, despite the hard conditions. 17 18 3. The trouble was, although my clothes were ironed to a knife-edge, 19 and shoes polished to a gleam, not every item was standard school uniform 20 issue. While Mum had scrimped and saved to obtain most of the gear, I 21 still didn’t have the pres cribed blue blazer and hatband. 22 23 4. Because of the war, rationing was in place and most schools had 24 relaxed their attitude towards proper uniforms, knowing how hard it was
Unit 7 The Chaser Key to the Exercises Text comprehension I. Decide which of the following is likely to happen after the story. C II. Judge, according to the text, whether the following statements are true or false. 1. F (Refer to Paragraph 5. The old man says that his stock in trade is not very large, but it is varied and has extraordinary effects.) 2. F (Refer to Paragraphs 11 and 1 3. The price of a glove-cleaner, as he calls it, is very high, five thousand dollars for a teaspoonful, but the love potion is very cheap.) 3. F (Refer to Paragraph 19. The old man claims that the effects of love potions are permanent.) 4. T (Refer to Paragraphs 24 and 28. Austen says that Diana is fond of parties and, although she is everything to him already, she does not care about his love at all. That is why he decides to go to the old man for the love potion and whenever the old man mentions the magic of his potion, he can't help "crying." From that, we can see the man loves the girl very much.) 5. F (The old man sells the love potions almost for nothing because by doing so his customers will come back for a much dearer commodity, the glove-cleaner, to help them out. It is the "death potion" that the old man makes most of his profits from, and intends to sell to his customers.) III. Answer the following questions. 1. What the old man means is that a young man who falls in love one-sidedly is seldom rich enough to win a girl's heart. His words imply that money is one of the crucial factors for love. If a man is not rich, he can rarely expect to be loved by a girl. 2. Refer to Paragraphs 19 to 37. The love potion has powerful, everlasting effects. To begin with, it may produce sexual desire in the person who takes it. And on the spiritual side, it can replace indifference with devotion and scorn with adoration. It will make a gay girl want nothing but solitude and her lover's company. She will feel jealous of him when her lover is with other girls; she will want to be everything to him. She will be only interested in her lover and take every concern of him. Even if he slips a bit, she will forgive him though terribly hurt. In a word, she will fall in love with him if she drinks the love potion. 3. Refer to Paragraphs 39 to 43. It is an irony, by which the author seems to imply that love is far from being precious or desirable. It is easy for a man to fall in love, yet it is hard