SQUARING THE CIRCLE.

There must be a singular charm about insoluble problems, since there are never wanting persons who are willing to attack them. I doubt not that at this moment there are persons who are devoting their energies to Squaring the Circle, in the full belief that important advantages would accrue¹ to science—and possibly a considerable pecuniary² profit to themselves—if they could succeed in solving it. Quite recently, applications have been made to the Paris Academy of Sciences, to ascertain³ what was the amount which that body was authorised to pay over to anyone who should square the circle. So seriously, indeed, was the secretary annoyed by applications of this sort, that it was found necessary to announce in the daily journals that not only was the Academy not authorised to pay any sum at all, but that it had determined⁴ never to give the least attention to those who fancied they had mastered the famous problem.

It is a singular circumstance that people have even attacked the problem without knowing exactly what289 its nature is. One ingenious workman, to whom the difficulty had been propounded⁵, actually set to work to invent an arrangement for measuring the circumference⁶ of the circle; and was perfectly⁷ satisfied that he had thus solved a problem which had mastered all the mathematicians⁹ of ancient and modern times. That we may not fall into a similar error, let us clearly understand what it is that is required for the solution of the problem of ‘squaring the circle.’

To begin with, we must note that the term ‘squaring the circle’ is rather a misnomer¹⁰; because the true problem to be solved is the determination of the length of a circle’s circumference when the diameter is known. Of course, the solution of this problem, or, as it is termed, the rectification¹¹ of the circle, involves the solution of the other, or the quadrature, of the circle. But it is well to keep the simpler issue before us.

Many have supposed that there exists some exact relation between the circumference and the diameter of the circle, and that the problem to be solved is the determination of this relation. Suppose, for example, that the approximate relation discovered by Archimedes (who found, that if a circle’s diameter is represented by seven, the circumference may be almost exactly represented by twenty-two) were strictly¹² correct, and that Archimedes had proved it to be so; then, according to this view, he would have solved the great problem; and it is to determine a relation of some such sort that many persons have set themselves.290 Now, undoubtedly¹³, if any relation of this sort could be established, the problem would be solved; but as a matter of fact no such relation exists, and the solution of the problem does not require that there should be any relation of the sort. For example, we do not look on the determination of the diagonal of a square (whose side is known) as an insoluble, or as otherwise than a very simple problem. Yet in this case no exact relation exists. We cannot possibly express both the side and the diagonal of a square in whole numbers, no matter what unit of measurement we adopt: or, to put the matter in another way, we cannot possibly divide both the side and the diagonal into equal parts (which shall be the same along each), no matter how small we take the parts. If we divide the side into 1,000 parts, there will be 1,414 such parts, and a piece over in the diagonal; if we divide the side into 10,000 parts, there will be 14,142, and still a little piece over, in the diagonal; and so on for ever. Similarly, the mere¹⁴ fact that no exact relation exists between the diameter and the circumference of a circle is no bar whatever to the solution of the great problem.

Before leaving this part of the subject, however, I may mention a relation which is very easily remembered, and is very nearly exact—much more so, at any rate, than that of Archimedes. Write down the numbers 113,355, that is, the first three odd numbers each repeated twice over. Then separate the six numbers into two sets of three, thus,—113)?355,291 and proceed with the division thus indicated. The result, 3·1415929?..., expresses the circumference of a circle whose diameter is 1, correctly to the sixth decimal place, the true relation being 3·14159265.

Again, many people imagine that mathematicians are still in a state of uncertainty¹⁵ as to the relation which exists between the circumference and the diameter of the circle. If this were so, scientific societies might well hold out a reward to anyone who could enlighten them; for the determination of this relation (with satisfactory exactitude) may be held to lie at the foundation of the whole of our modern system of mathematics. I need hardly say that no doubt whatever rests on the matter. A hundred different methods are known to mathematicians by which the circumference may be calculated from the diameter with any required degree of exactness. Here is a simple one, for example:—Take any number of the fractions formed by putting one as a numerator over the successive odd numbers. Add together the alternate ones beginning with the first, which, of course, is unity¹⁶. Add together the remainder. Subtract the second sum from the first. The remainder will express the circumference (the diameter being taken as unity) to any required degree of exactness. We have merely to take enough fractions. The process would, of course, be a very laborious¹⁷ one, if great exactness were required, and as a matter of fact mathematicians have made use of much more292 convenient methods for determining the required relation: but the method is strictly exact.

The largest circle we have much to do with in scientific questions is the earth’s equator. As a matter of curiosity, we may inquire what the circumference of the earth’s orbit is; but as we are far from being sure of the exact length of the radius¹⁸ of that orbit (that is, of the earth’s distance from the sun), it is clear that we do not need a very exact relation between the circumference and the diameter in dealing¹⁹ with that enormous circle. Confining ourselves, therefore, to the circle of the earth’s equator, let us see what exactness we seem to require. We will suppose for a moment that it is possible to measure round the earth’s equator without losing count of a single yard, and that we want to gather from our estimate what the diameter of this great circle may be. This seems, indeed, the only use to which, in this case, we can put our knowledge of the relation we are dealing with. We have then a circle some twenty-five thousand miles round, and each mile contains one thousand seven hundred and sixty yards: or in all there are some forty-four million yards in the circumference, and therefore (roughly) some fourteen million yards in the diameter of this great circle. Hence, if our relation is correct within a fourteen-millionth part of the diameter, or a forty-four millionth part of the circumference, we are safe from any error exceeding a yard. All we want, then, is that the number expressing the circumference (the diameter293 being unity) should be true to the eighth decimal place, as quoted above (p. 291, l. 5).

But as I have said, mathematicians have not been content with a computation of this sort. They have calculated the number not to the eighth, but to the six hundred and twentieth decimal place. Now, if we remember that each new decimal makes the result ten times more exact, we shall begin to see what a waste of time there has been in this tremendous calculation. We all remember the story of the horse which had twenty-four nails in its shoes, and was valued at the sum obtained by adding together a farthing for the first nail, a halfpenny for the next, a penny for the next, and so on, doubling twenty-four times. The result was counted by thousands of pounds. The old miser²⁰ who paid at a similar rate for a grave eighteen feet deep (doubling for each foot), killed himself when he heard the total. But now consider the effect of multiplying by ten, six hundred and twenty times. A fraction, with that enormous number for denominator, and unity for numerator, expresses the minuteness of the error which would result if the ‘long value’ of the circumference were made use of. Let an illustration show the force of this:—

It has been estimated that light, which could eight times circle the earth in a second, takes 50,000 years in reaching us from the faintest stars seen in Lord Rosse’s giant reflector. Suppose we knew the exact length of the tremendous line which extends from the294 earth to such a star, and wanted, for some inconceivable purpose, to know the length of the circumference of a circle, of which that line was the radius. The value deduced from the above-mentioned calculation of the relation between the circumference and the diameter would differ from the truth by a length which would be imperceptible under the most powerful microscope ever yet constructed. Nay²¹, the radius we have conceived, enormous as it is, might be increased a million-fold, or a million times a million-fold, with the same result. And the area of the circle formed with this increased radius would be determinable with so much accuracy, that the error, if presented in the form of a minute square, would be utterly²² imperceptible under a microscope a million times more powerful than the best ever yet constructed by man.

Not only has the length of the circumference been calculated once in this unnecessarily exact manner, but a second calculator has gone over the work independently. The two results are of course identical figure for figure.

It will be asked then, what is the problem about which so great a work has been made? The problem is, in fact, utterly insignificant²³; its only interest lies in the fact that it is insoluble—a property which it shares along with many other problems, as the trisection of an angle, the duplication of a cube, and so on.

The problem is simply this: Having given the295 diameter of a circle, to determine, by a geometrical construction, in which only straight lines and circles shall be made use of, the side of a square, equal in area to the circle. As I have said, the problem is solved, if, by a construction of the kind described, we can determine the length of the circumference; because then the rectangle under half this length and the radius is equal in area to the circle, and it is a simple problem to describe a square equal to a given rectangle.

To illustrate²⁴ the kind of construction required, I give an approximate solution which is remarkably²⁵ simple, and, so far as I am aware, not generally known. Describe a square about the given circle, touching²⁶ it at the ends of two diameters, AOB, COB, at right angles to each other, and join CA; let COAE be one of the quarters of the circumscribing²⁷ square, and from E draw EG, cutting off from AO a fourth part AG of its length, and from AC the portion AH. Then three sides of the circumscribing square together with AH are very nearly equal to the circumference of the circle. The difference is so small, that in a circle two feet in diameter, it would be less than the two-hundredth part of an inch. If this construction were exact, the great problem would have been solved.

One point, however, must be noted²⁸; the circle is of all curved lines the easiest to draw by mechanical means. But there are others which can be so drawn²⁹. And if such curves as these be admitted as available,296 the problem of the quadrature of the circle can be readily solved. There is a curve, for instance, invented by Dinostratus, which can readily be described mechanically, and has been called the quadratrix of Dinostratus, because it has the property of thus solving the problem we are dealing with.

As such curves can be described with quite as much accuracy as the circle—for, be it remembered, an absolutely perfect circle has never yet been drawn—we see that it is only the limitations which geometers have themselves invented that give this problem its difficulty. Its solution has, as I have said, no value; and no mathematician⁸ would ever think of wasting a moment over the problem—for this reason, simply, that it has long since been demonstrated to be insoluble by simple geometrical methods. So that, when a man says he has squared the circle (and many will say so, if one will only give them a hearing), he shows that either he wholly misunderstands the nature of the problem, or that his ignorance of mathematics has led him to mistake a faulty for a true solution.

(From Chambers’s Journal, January 16, 1869.)

1 accrue   

v.(利息等)增大，增多

参考例句：

• Ability to think will accrue to you from good habits of study.思考能力将因良好的学习习惯而自然增强。
• Money deposited in banks will accrue to us with interest.钱存在银行，利息自生。

2 pecuniary   

adj.金钱的；金钱上的

参考例句：

• She denies obtaining a pecuniary advantage by deception.她否认通过欺骗手段获得经济利益。
• She is so independent that she refused all pecuniary aid.她很独立，所以拒绝一切金钱上的资助。

3 ascertain   

vt.发现，确定，查明，弄清

参考例句：

• It's difficult to ascertain the coal deposits.煤储量很难探明。
• We must ascertain the responsibility in light of different situtations.我们必须根据不同情况判定责任。

4 determined   

adj.坚定的；有决心的

参考例句：

• I have determined on going to Tibet after graduation.我已决定毕业后去西藏。
• He determined to view the rooms behind the office.他决定查看一下办公室后面的房间。

5 propounded   

v.提出（问题、计划等）供考虑[讨论]，提议( propound的过去式和过去分词 )

参考例句：

• the theory of natural selection, first propounded by Charles Darwin 查尔斯∙达尔文首先提出的物竞天择理论
• Indeed it was first propounded by the ubiquitous Thomas Young. 实际上，它是由尽人皆知的杨氏首先提出来的。 来自辞典例句

6 circumference   

n.圆周，周长，圆周线

参考例句：

• It's a mile round the circumference of the field.运动场周长一英里。
• The diameter and the circumference of a circle correlate.圆的直径与圆周有相互关系。

7 perfectly   

adv.完美地，无可非议地，彻底地

参考例句：

• The witnesses were each perfectly certain of what they said.证人们个个对自己所说的话十分肯定。
• Everything that we're doing is all perfectly above board.我们做的每件事情都是光明正大的。

8 mathematician   

n.数学家

参考例句：

• The man with his back to the camera is a mathematician.背对着照相机的人是位数学家。
• The mathematician analyzed his figures again.这位数学家再次分析研究了他的这些数字。

9 mathematicians   

数学家( mathematician的名词复数 )

参考例句：

• Do you suppose our mathematicians are unequal to that? 你以为我们的数学家做不到这一点吗？ 来自英汉文学
• Mathematicians can solve problems with two variables. 数学家们可以用两个变数来解决问题。 来自哲学部分

10 misnomer   

n.误称

参考例句：

• Herbal"tea"is something of a misnomer because these drinks contain no tea at all.花草“茶”是一个误称，因为这类饮料里面根本不含茶。
• Actually," Underground "is a misnomer,because more than half the shops are above ground.实际上，“ 地下 ” 这个名称用之不当，因为半数以上的店铺是在地面上的。

11 rectification   

n. 改正, 改订, 矫正

参考例句：

• The process of producing a shift of the average value is called rectification. 产生平均值移动的过程叫做整流。
• This effect, in analogy to its radiofrequency counterpart, is known as optical rectification. 同它的射频对应物相仿，这种现象称为光学整流。

12 strictly   

adv.严厉地，严格地；严密地

参考例句：

• His doctor is dieting him strictly.他的医生严格规定他的饮食。
• The guests were seated strictly in order of precedence.客人严格按照地位高低就座。

13 undoubtedly   

adv.确实地，无疑地

参考例句：

• It is undoubtedly she who has said that.这话明明是她说的。
• He is undoubtedly the pride of China.毫无疑问他是中国的骄傲。

14 mere   

adj.纯粹的；仅仅，只不过

参考例句：

• That is a mere repetition of what you said before.那不过是重复了你以前讲的话。
• It's a mere waste of time waiting any longer.再等下去纯粹是浪费时间。

15 uncertainty   

n.易变，靠不住，不确知，不确定的事物

参考例句：

• Her comments will add to the uncertainty of the situation.她的批评将会使局势更加不稳定。
• After six weeks of uncertainty,the strain was beginning to take its toll.6个星期的忐忑不安后，压力开始产生影响了。

16 unity   

n.团结，联合，统一；和睦，协调

参考例句：

• When we speak of unity,we do not mean unprincipled peace.所谓团结，并非一团和气。
• We must strengthen our unity in the face of powerful enemies.大敌当前，我们必须加强团结。

17 laborious   

adj.吃力的，努力的，不流畅

参考例句：

• They had the laborious task of cutting down the huge tree.他们接受了伐大树的艰苦工作。
• Ants and bees are laborious insects.蚂蚁与蜜蜂是勤劳的昆虫。

18 radius   

n.半径，半径范围；有效航程，范围，界限

参考例句：

• He has visited every shop within a radius of two miles.周围两英里以内的店铺他都去过。
• We are measuring the radius of the circle.我们正在测量圆的半径。

19 dealing   

n.经商方法，待人态度

参考例句：

• This store has an excellent reputation for fair dealing.该商店因买卖公道而享有极高的声誉。
• His fair dealing earned our confidence.他的诚实的行为获得我们的信任。

20 miser   

n.守财奴，吝啬鬼 (adj.miserly)

参考例句：

• The miser doesn't like to part with his money.守财奴舍不得花他的钱。
• The demon of greed ruined the miser's happiness.贪得无厌的恶习毁掉了那个守财奴的幸福。

21 nay   

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。

22 utterly   

adv.完全地，绝对地

参考例句：

• Utterly devoted to the people,he gave his life in saving his patients.他忠于人民，把毕生精力用于挽救患者的生命。
• I was utterly ravished by the way she smiled.她的微笑使我完全陶醉了。

23 insignificant   

adj.无关紧要的，可忽略的，无意义的

参考例句：

• In winter the effect was found to be insignificant.在冬季，这种作用是不明显的。
• This problem was insignificant compared to others she faced.这一问题与她面临的其他问题比较起来算不得什么。

24 illustrate   

v.举例说明，阐明；图解，加插图

参考例句：

• The company's bank statements illustrate its success.这家公司的银行报表说明了它的成功。
• This diagram will illustrate what I mean.这个图表可说明我的意思。

25 remarkably   

ad.不同寻常地，相当地

参考例句：

• I thought she was remarkably restrained in the circumstances. 我认为她在那种情况下非常克制。
• He made a remarkably swift recovery. 他康复得相当快。

26 touching   

adj.动人的，使人感伤的

参考例句：

• It was a touching sight.这是一幅动人的景象。
• His letter was touching.他的信很感人。

27 circumscribing   

v.在…周围划线( circumscribe的现在分词 )；划定…范围；限制；限定

参考例句：

• There are laws circumscribing the right of individual citizens to cause bodily harm to others. 有多项法律限制公民的权利，使其不得对他人进行人身伤害。 来自柯林斯例句

28 noted   

adj.著名的，知名的

参考例句：

• The local hotel is noted for its good table.当地的那家酒店以餐食精美而著称。
• Jim is noted for arriving late for work.吉姆上班迟到出了名。

29 drawn   

v.拖，拉，拔出；adj.憔悴的，紧张的

参考例句：

• All the characters in the story are drawn from life.故事中的所有人物都取材于生活。
• Her gaze was drawn irresistibly to the scene outside.她的目光禁不住被外面的风景所吸引。