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Answers to Knot 9
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§ 1. The Buckets

Problem. — Lardner states that a solid, immersed in a fluid, displaces an amount equal to itself in bulk. How can this be true of a small bucket floating in a larger one?

Solution. — Lardner means, by “displaces”, “occupies a space which might be filled with water without any change in the surroundings.” If the portion of the floating bucket, which is above the water, could be annihilated1, and the rest of it transformed into water, the surrounding water would not change its position: which agrees with Lardner’s statement.

Five answers have been received, none of which explains the difficulty arising from the well-known fact that a floating body is the same weight as the displaced fluid. Hecla says that “Only that portion of the smaller bucket which descends2 below the original level of the water can be properly said to be immersed, and only an equal bulk of water is displaced.” Hence, according to Hecla, a solid whose weight was equal to that of an equal bulk of water, would not float till the whole of it was below “the original level” of the water: but, as a matter of fact, it would float as soon as it was all under water. Magpie3 says the fallacy is “the assumption that one body can displace another from a place where it isn’t”, and that Lardner’s assertion is incorrect, except when the containing vessel4 “was originally full to the brim”. But the question of floating depends on the present state of things, not on past history. Old King Cole takes the same view as Hecla. Tympanum and Vindex assume that “displaced” means “raised above its original level”, and merely explain how it comes to pass that the water, so raised, is less in bulk than the immersed portion of bucket, and thus land themselves — or rather set themselves floating — in the same boat as Hecla.

I regret that there is no Class List to publish for this Problem.

§ 2. Balbus’s Essay

Problem. — Balbus states that if a certain solid be immersed in a certain vessel of water, the water will rise through a series of distances, two inches, one inch, half an inch, etc., which series has no end. He concludes that the water will rise without limit. Is this true?

Solution. — No. This series can never reach 4 inches, since, however many terms we take, we are always short of 4 inches by an amount equal to the last term taken.

Three answers have been received — but only two seem to me worthy5 of honours.

Tympanum says that the statement about the stick “is merely a blind, to which the old answer may well be applied6, solvitur ambulando, or rather mergendo”. I trust Tympanum will not test this in his own person, by taking the place of the man in Balbus’s Essay! He would infallibly be drowned.

Old King Cole rightly points out that the series, 2, 1, etc., is a decreasing geometrical progression: while Vindex rightly identifies the fallacy as that of “Achilles and the Tortoise”.

Class List.
I.

Old King Cole. Vindex.

§ 3. The Garden

Problem. — An oblong garden, half a yard longer than wide, consists entirely7 of a gravel8 walk, spirally arranged, a yard wide and 3630 yards long. Find the dimensions of the garden.

Answer. — 60, 60½.

Solution. — The number of yards and fractions of a yard traversed in walking along a straight piece of walk, is evidently the same as the number of square yards and fractions of a square yard contained in that piece of walk: and the distance trsversed in passing through a square yard at a corner, is evidently a yard. Hence the area of the garden is 3630 square yards: i.e. if x be the width, x(x+1/2)=3630. Solving this quadratic, we find x=60. Hence the dimentions are 60, 60½.

Twelve answers have been received — seven right and five wrong.

C. G. L., Nabob, Old Crow, and Tympanum assume that the number of yards in the length of the path is equal to the number of square yards in the garden. This is true, but should have been proved. But each is guilty of darker deeds. C. G. L.‘s “working” consists of dividing 3630 by 60. Whence came this divisor, O Segiel? Divination9? Or was it a dream? I fear this solution is worth noting. Old Crow’s is shorter, and so (if possible) worth rather less. He says the answer “is at once seen to be 60 x 60½”! Nabob’s calculation is short, but “as rich as a Nabob” in error. He says that the square root of 3630, multiplied by 2, equals the length plus the breadth. That is 60.25 x 2 = 120½. His first assertion is only true of a square garden. His second is irrelevant10, since 60.25 is not the square root of 3630! Nay11, Bob, this will not do! Tympanum says that, by extracting the square root of 3630, we get 60 yards with a remainder of 30/60, or half a yard, which we add so as to make the oblong 60 x 60½. This is very terrible: but worse remains12 behind. Tympanum proceeds thus: “But why should there be the half-yard at all? Because without it there would be no space at all for flowers. By means of it, we find reserved in the very centre a small plot of ground, two yards long by half a yard wide, the only space not occupied by walk.” But Balbus expressly said that the walk “used up the whole of the area”, O Tympanum! My tympa is exhausted13: my brain is num! I can say no more,

Hecla indulges, again and again, in that most fatal of all habits in computation — the making two mistakes which cancel each other. She takes x as the width of the garden, in yards, and x+½ as its length, and makes her first “coil” the sum of x-½, x-½, x-1, x-1, i.e. 4x-3: but the fourth term should be x-1½, so that her first coil is ½ a yard too long. Her second coil is the sum of x-2½, x-2½, x-3, x-3: here the first term should be x-2 and the last x-3½: these two mistakes cancel and this coil is therefore right. And the same thing is true of every other coil but the last, which needs an extra half-yard to reach the end of the path: and this exactly balances the mistake in the first coil. Thus the sum-total of the coils comes right though the working is all wrong.

Of the seven who are right, Dinah Mite14, Janet, Magpie, and Taffy make the same assumption as C. G. L. and Co. They then solve by a quadratic. Magpie also tries it by arithmetical progression, but fails to notice that the first and last “coils” have special values.

Alumnus Etonae attempts to prove what C. G. L. assumes by a particular instance, taking a garden 6 by 5½. He ought to have proved it generally: what is true of one number is not always true of others. Old King Cole solves it by an arithmetical progression. It is right, but too lengthy15 to be worth as much as a quadratic.

Vindex proves it very neatly16, by pointing out that a yard of walk measured along the middle represents a square yard of garden, “whether we consider the straight stretches of walk or the square yards at the angles, in which the middle line goes half a yard in one direction and then turns a right angle and goes half a yard in another direction.”

Class List.
I.

Vindex.

II.

Alumnus Etonae. Old King Cole.

III.

Dinah Mite. Magpie. Janet. Taffy.


点击收听单词发音收听单词发音  

1 annihilated b75d9b14a67fe1d776c0039490aade89     
v.(彻底)消灭( annihilate的过去式和过去分词 );使无效;废止;彻底击溃
参考例句:
  • Our soldiers annihilated a force of three hundred enemy troops. 我军战士消灭了300名敌军。 来自《现代汉英综合大词典》
  • We annihilated the enemy. 我们歼灭了敌人。 来自《简明英汉词典》
2 descends e9fd61c3161a390a0db3b45b3a992bee     
v.下来( descend的第三人称单数 );下去;下降;下斜
参考例句:
  • This festival descends from a religious rite. 这个节日起源于宗教仪式。 来自《简明英汉词典》
  • The path descends steeply to the village. 小路陡直而下直到村子。 来自《简明英汉词典》
3 magpie oAqxF     
n.喜欢收藏物品的人,喜鹊,饶舌者
参考例句:
  • Now and then a magpie would call.不时有喜鹊的叫声。
  • This young man is really a magpie.这个年轻人真是饶舌。
4 vessel 4L1zi     
n.船舶;容器,器皿;管,导管,血管
参考例句:
  • The vessel is fully loaded with cargo for Shanghai.这艘船满载货物驶往上海。
  • You should put the water into a vessel.你应该把水装入容器中。
5 worthy vftwB     
adj.(of)值得的,配得上的;有价值的
参考例句:
  • I did not esteem him to be worthy of trust.我认为他不值得信赖。
  • There occurred nothing that was worthy to be mentioned.没有值得一提的事发生。
6 applied Tz2zXA     
adj.应用的;v.应用,适用
参考例句:
  • She plans to take a course in applied linguistics.她打算学习应用语言学课程。
  • This cream is best applied to the face at night.这种乳霜最好晚上擦脸用。
7 entirely entirely     
ad.全部地,完整地;完全地,彻底地
参考例句:
  • The fire was entirely caused by their neglect of duty. 那场火灾完全是由于他们失职而引起的。
  • His life was entirely given up to the educational work. 他的一生统统献给了教育工作。
8 gravel s6hyT     
n.砂跞;砂砾层;结石
参考例句:
  • We bought six bags of gravel for the garden path.我们购买了六袋碎石用来铺花园的小路。
  • More gravel is needed to fill the hollow in the drive.需要更多的砾石来填平车道上的坑洼。
9 divination LPJzf     
n.占卜,预测
参考例句:
  • Divination is made up of a little error and superstition,plus a lot of fraud.占卜是由一些谬误和迷信构成,再加上大量的欺骗。
  • Katherine McCormack goes beyond horoscopes and provides a quick guide to other forms of divination.凯瑟琳·麦考马克超越了占星并给其它形式的预言提供了快速的指导。
10 irrelevant ZkGy6     
adj.不恰当的,无关系的,不相干的
参考例句:
  • That is completely irrelevant to the subject under discussion.这跟讨论的主题完全不相关。
  • A question about arithmetic is irrelevant in a music lesson.在音乐课上,一个数学的问题是风马牛不相及的。
11 nay unjzAQ     
adv.不;n.反对票,投反对票者
参考例句:
  • He was grateful for and proud of his son's remarkable,nay,unique performance.他为儿子出色的,不,应该是独一无二的表演心怀感激和骄傲。
  • Long essays,nay,whole books have been written on this.许多长篇大论的文章,不,应该说是整部整部的书都是关于这件事的。
12 remains 1kMzTy     
n.剩余物,残留物;遗体,遗迹
参考例句:
  • He ate the remains of food hungrily.他狼吞虎咽地吃剩余的食物。
  • The remains of the meal were fed to the dog.残羹剩饭喂狗了。
13 exhausted 7taz4r     
adj.极其疲惫的,精疲力尽的
参考例句:
  • It was a long haul home and we arrived exhausted.搬运回家的这段路程特别长,到家时我们已筋疲力尽。
  • Jenny was exhausted by the hustle of city life.珍妮被城市生活的忙乱弄得筋疲力尽。
14 mite 4Epxw     
n.极小的东西;小铜币
参考例句:
  • The poor mite was so ill.可怜的孩子病得这么重。
  • He is a mite taller than I.他比我高一点点。
15 lengthy f36yA     
adj.漫长的,冗长的
参考例句:
  • We devoted a lengthy and full discussion to this topic.我们对这个题目进行了长时间的充分讨论。
  • The professor wrote a lengthy book on Napoleon.教授写了一部有关拿破仑的巨著。
16 neatly ynZzBp     
adv.整洁地,干净地,灵巧地,熟练地
参考例句:
  • Sailors know how to wind up a long rope neatly.水手们知道怎样把一条大绳利落地缠好。
  • The child's dress is neatly gathered at the neck.那孩子的衣服在领口处打着整齐的皱褶。


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