decimals, their error of length could not be made visible to the inhabitants of the twentieth globe below us, unless their microscopes were relatively very much more powerful than ours.

employment. People were told that if they would sell their steam-engines for old iron, they might buy new machines with the money, which would work as long as they held together without costing a farthing for fuel. Certainly, had the scheme been pro- By the geometrical quadrature is meant posed to me, I should have declined to join the determination of a square equal to the until I had derived assurance from seeing the circle, using only Euclid's allowance of donkey who originated it turned into a head-means; that is, using only the straight line over-heels perpetual motion by tying a heavy and circle as in Euclid's first three postuweight to his tail and an exhausted receiver lates. On this matter James Gregory, in to his nose. 1668, published an asserted demonstration of

But he would be a bold man who would be very positive on the point: even though there are trains of reasoning, different from Gregory's which render it in the highest degree improbable, which are in fact all but demonstration themselves, that the geometrical quadrature is impossible.

3. Quadrature of the circle. The arith- the impossibility of the geometrical quadrametical quadrature involves the determination ture. The matter is so difficult, and proofs of the circumference by a definite arithmeti- of a negative so slippery, that mathematical multiplier, which shall be perfectly accu- cians are rather shy of pronouncing positive rate. Lambert proved that the multiplier opinions. Montucla, in the first edition of must be an interminable decimal fraction; the work presently mentioned only ventured and the proof may be found in Legendre's to say that it was very like demonstration. geometry, and in Brewster's translation of In the second edition, after further reflection, that work. The arithmeticians have given he gave his opinion that the point was dem. plenty of approximate multipliers. The last onstrated. I read James Gregory's tract one, and the most accurate of all, was pub- many years ago, and left off with an impres⚫lished a few years ago by Mr. W. Shanks, of sion that probably more attentive consideraHoughton-le-Spring, a calculator to whom tion would compel me to agree with its aumultiplication is no vexation, &c. He pub- thor. lished the requisite multiplier (which mathematicians denote by ) to six hundred and seven decimal places, of which 441 were verified by Dr. Rutherford. To give an idea of the power of this multiplier, we must try to master such a supposition as the following. There are living things on our globe so To say that a given problem cannot be small that, if due proportion were observed, solved, because two thousand years of trial the corpuscles of their blood would be no have not succeeded, is unsafe: far more more than a millionth of an inch in diame- powerful means may be invented. But when ter. Suppose another globe like ours, but the question is to solve a problem with cerso much larger that our great globe itself tain given means and no others, it is not so is but fit to be a corpuscle in the blood of unsafe to affirm that the problem is insoluone of its animalcules: and call this the first ble. By hypothesis, we are to use no means globe above us. Let there be another globe except those which have been used for two so large that this first globe above us is but a corpuscle in the animalcule of that globe; and call this the second globe above us. Go on in this way till we come to the twentieth globe above us. Next, let the minute corpuscle on our globe be another globe like ours, with everything in proportion; and call this the first globe below us. Take a bloodcorpuscle from the animalcule of that globe, and make it the second globe below us: and so on down to the twentieth globe below us. Then if the inhabitants of the twentieth globe above us were to calculate the circumference of their globe from its diameter by the 607

thousand years; it becomes exceedingly probable that all which those means can do has been done, in a question which has been tried by hundreds of men of genius, patience and proved success in other things.

4. Trisection of the Angle.-The question is to cut any given angle into three equal parts, with no more assistance than is conceded in Euclid's first three postulates. It is well known that this problem depends upon representing geometrically the three roots of a cubic equation which has all its roots real: whoever can do either can do the other. Now the geometrical solution, as the

word geometrical is understood, of a cubic equation, has never been attained; and all the à priori considerations which have so much force with those who are used to them are in favor of the solution being impossible. A person used to algebraic geometry cannot conceive how, by intersections of circles and straight lines, a problem should be solved which has three answers, and three only.

The reader may find details on this subject in the articles QUADRATURE and TRISECTION in the Penny Cyclopædia. But further information will be found in Montucla's Histoire des Récherches sur la Quadrature du Cercle, Paris, 1831, 8vo. (second edition). This work contains, besides the vagaries of the insufficiently informed, an account of the attempts of older days, which ended in useful To sum up the whole. The problem of discovery. In later times the whole subject the three bodies has such solution as hun- has lapsed into burlesque; the few who have dreds of other problems have: approximate made rational attempts being lost in the in character, but wanting only pains and pa- crowd who have made absurd misconceptions tience to carry the approximation to any de- of the problem. To square the circle has sired extent. The problem of the perpetual become a byword, though many do not know motion is a physical absurdity. The arith- the problem under a change of terms, say the metical quadrature of the circle has been rectification of the circumference. For exproved impossible in finite terms, but 607 ample, when Mr. Goulburn was a candidate decimal places of the interminable series have for the University of Cambridge in 1831, been found, and 441 of them verified. Of some wags of the opposite faction sent the the geometrical quadrature an asserted proof following to a morning paper, which actually of impossibility exists, which no one who inserted it (May 4) in triumphant answer to has read it ventures to gainsay, but in favor the objection against their candidate's want of which no one speaks very positively. The of Cambridge knowledge:trisection of the angle has no alleged proof of its impossibility. But were this the proper place, an account might be given of those considerations which lead all who have thought much on the subject to feel sure that the difficulty arises from the restrictions placed upon the means of solution amounting to a little too much dictation to the nature of things. For it must be remembered that the problem is not to square the circle, nor to trisect the angle, but to square the circle or trisect the angle without recourse to I need hardly say that mathematicians any means except those afforded by Euclid's know no lunar caustic, except what the first three postulates. This limitation is fre- chemists call nitrate of silver. And so much quently omitted; and persons are led to con- for the impossible problems, which have

clude that mathematicians have never shown how to square a circle, or to trisect an angle than which nothing can be more untrue. I may take occasion to raise a Query in some future communication, whether these difficulties would ever have existed if Euclid's ideas

of solid geometry had been as well arranged as his ideas of plane geometry.

COFFEE-HOUSES, EARLY MENTION OF.-Burton Bays, Anat. Mel., part i. sect. 2., m. 2. s. 2.:

'Tis the summum bonum of our tradesmen their felicity, life, and soul, their chief comfort to be merry together in an alehouse or tavern, as our modern Muscovites do in their medeinns, and Turks in their coffee-houses, which much resemble our taverns."

"We understand that although, owing to circumstances with which the public are not concerned, Mr. Goulburn declined becoming a candidate for University honors, his scientific attainments are far from inconsiderable. He is well known to be the author of an Essay in the Philosophical Transactions on the accurate rectification of a circular arc, and Caustic-a problem likely to become of great an investigation of the equation to the Lunar use in nautical astronomy.'

caught so many ingenious minds, and almost always held them tight. For this reason, I should advise any one not to try them;

Video quod vestigia Intrantium multa, at nulla exeuntium." A. DE MORGAN.

From Household Words.

AGNES LEE.

CHAPTER THE FIRST.

acid. Yet Mrs. Huxley went on in the even tenor of her way, struggling with straitened means, delicate health, recurring washingMRS. WARREN was a charming woman-as days, and her husband's temper. Her ecolike the popular notion of a perfect angel as nomical feebleness, and the difficulties of any body could hope to find, if they took the keeping her weekly bills in a state of longest summer day for the search. She was liquidation, were greatly complicated in conan Irishwoman, the widow of an English sequence of all the poor people in the parish gentleman of large fortune, who had left her coming to her as to a sort of earthly Proviendowed with an ample jointure and a hand-dence, to supply all they lacked in the shape some manor-house in Staffordshire. She was of food, physic, raiment, and good advice. young, bright, fascinating, and thoroughly Strangers said that Mrs. Huxley looked fretgood-natured; she enjoyed nothing so much ful, and that it was a pity a clergyman's wife as making people happy, and would sacrifice should have such unattractive manners; that her own pleasure or convenience even, for an it must be a trial to such a pleasant, genial entire stranger, provided the necessities of man as her husband to have a partner so the case had been brought before her with unlike himself, and all that. The recording sufficient eloquence or emphasis. She did angel might have given a different verdict: every thing in the easiest and most graceful the poor of her parish knew her value. manner, and had the virtue of forgetting all The family at the Rectory consisted of one about it herself, as soon as the occasion had daughter, named Miriam, and an orphan passed away. She was devoted to her friends, niece of Mr. Huxley's whom they had and loved them dearly, so long as they were adopted. Mr. Huxley had made many diffithere to assist themselves; but, if they went culties when this plan was first proposed. He away, she never thought of them till the next objected to the expense, and wished the girl time she saw them, when she was again as to be sent as an articled pupil to some cheap fond of them as ever. With all her gene-school, where she might qualify herself to berosity, however, her tradespeople complained come a nursery governess, or to wait on young that she did not pay her bills; that she did ladies. This he said on the plea that, as they very shabby things, and that she drove would not be able to give her any fortune, it dreadfully hard bargains. A poor woman would be cruel to give her a taste for comforts whom she had employed to do some plain she could not hereafter expect; that it was best work, declared contemptuously that she would sooner work for Jews than for charitable ladies: they screwed down so in the price, and kept folks waiting so long for their money.

It was not difficult for Mrs. Warren to be an angel she had no domestic discipline to test her virtues too severely, nor to ruffle the bird of paradise beauty of her wings. Husbands are daily stumbling-blocks in the path of female perfection; they have the faculty of taking the shine out of the most dazzling appearances. It is easier to be an angel than to be an average good woman under domestic difficulties.

to accustom her betimes to the hardships of her lot. Mrs. Huxley did not often contradict her husband; but, on this occasion, she exerted her powers of speech; she was a mother, and acted as she would have wished another to act by her own Miriam. Mr. Huxley graciously allowed himself to be persuaded, and Agnes Lee, the child of his favorite sister, was adopted into the Rectory nursery on a perfect equality with her cousin. It somehow got to be reported abroad, that Mrs. Huxley had greatly opposed her husband's generosity, and had wished the little orphan to be sent to the workhouse.

The two children grew up together, and Mrs. Huxley was the wife of the hard- were as fond of each other as sisters usually working clergyman in whose parish Mrs. are; but Agnes Lee had the strongest will Warren's manor-house was situated. She and the most energy. So it was she who had a cross husband, who did not adore settled the plays and polity of doll-land, and her, but who (chiefly from the force of habit) who took the lead in all matters of "books, found fault with every thing she did; nothing and work, and needle-play." Agnes was but the purest gold could have stood the twelve, and Miriam fourteen, when the fasciconstant outpouring of so much sulphuric

nating Mrs: Warren came to live at the Warren, saw Miriam at her request and gave Great House. little hope that she would ever be anything

She took up the Rectory people most but a life-long invalid. She was ordered to warmly, and threw herself with enthusiasm keep as much as possible in a recumbent pointo all manner of benevolent schemes for sition. Mrs. Warren was on the point of the benefit of the parish. To the two girls departing for London. Nothing could exceed she seemed like a good fairy. She had them her sympathy and generosity. At first she constantly to her beautiful house, she gave declared she would postpone her journey, to them lessons in singing, and taught them to assist Mrs. Huxley to nurse her sweet dance; her French maid manufactured their Miriam; but she easily gave up that idea bonnets and dresses; she lavished gifts upon when Mrs. Huxley declared, rather dryly, them, she made pets of them, and was never "that there was not the least occasion; for, weary of inventing schemes for giving them as the case was likely to be tedious, it was pleasure. It was delightful to see their en- better to begin as they could go on." Mrs. joyment and to receive their gratitude, and Warren, however, loaded Miriam with presshe never suspected the delicate unobtrusive ents. She made Miriam promise to write to care with which poor cold, stiff, Mrs. Huxley her all she read and thought; and for this contrived that the two girls should never purpose, she gave her a supply of fairy-like fall too heavily upon the hands of their beau- paper and a gold pen. Miriam, on her side, tiful patroness. She also tried to inspire promised to write twice a-week at least, and them with a portion of her own reserve; but to tell Mrs. Warren everything that could that was not so easy. Miriam―a mild, shy, amuse her. Mrs. Warren gave orders to her undemonstrative girl-felt an admiration of gardener to supply the Rectory with fruit, Mrs. Warren that approached to idolatry. It flowers, and vegetables; but either Mrs. took the place of a first love. Mrs. Warren Warren's directions were not clear, or the liked the excitement of being loved with gardener did not choose to act upon them. enthusiasm; but she never calculated the He charged for everything that he sent down responsibility it brought along with it, and and gave as his reason that his mistress paid omitted nothing that could stimulate Miri-him no wages in her absence, but let him am's passionate attachment. Agnes was less pick up what he could. impressionable. She had a aprecocious After Mrs. Warren's departure, she wrote amount of common sense, and Mrs. Warren's for a month; after that, her letters ceased. fascinations, did not take too much hold upon Newspapers supplied their place; and, it her. The Rector was almost as much be- appeared from the notices of fashionable witched as his daughter by the fair widow. life, that Mrs. Warren had taken her place She talked gaily to him, and obliged him to amongst the gayest. At last the newspapers rub up his ancient gallantry, which had fallen ceased; the last that came contained the an into rusty disuse. She dressed all the chil-nouncement that Mrs. Warren had left town dren of his school in green gowns and red for Paris. After this, no more news reached ribbons. She subscribed a painted window the Rectory. The Manor House remained to the church. She talked over two refrac-shut up, and the lodge-keeper said "that tory churchwardens, who had been the tor- the Missis was spending the winters at tment of his life: above all, she admired his Bath." sermons; and, as she was in correspondence with a lord bishop, he had sanguine hopes that her admiration might lead to something better. Mrs. Huxley was the only person who refused to be charmed. She did not contradict the raptures expressed by her husband and daughter, but she heard them in silence.

At first Miriam wrote in all the enthusiasm and good faith of youthful adoration. Mrs. Warren had begged she would not count with her letter for letter, but have trust in her unalterable attachment, &c., &c.; and Miriam went on writing, long after all answers had ceased. Everything earthly has its limit; and when reciprocity is all on one When Miriam was sixteen, she fell into side, the term is reached rather earlier than delicate health; a slight accident developed it might otherwise have been. Poor Miriam a spinal affection. A London physician, who lay on her couch, and went through all thỏ with his wife was on a short visit to Mrs. | heart-sickening process of disenchantment

about the friendship which she had made the hands will be charming. This furniture will light of her life. She rejoiced moodily in look to more advantage there than it does her physical sufferings, and hoped that she here: and when I have seen you comfortably should soon die, as she could not endure such settled, I shall leave you, to seek my formisery long. The young believe in the eter-tune." nity of all they feel.

She was roused from this sorrow of sentiment by a real affliction. Scarlet fever broke out in the parish. Mr. Huxley caught it, and died, after a fortnight's illness. A life insurance for a thousand pounds, and a few hundreds painfully saved and laid by in the Bank of England, was all the provision that remained to his family.

"My dear, you are so rash, and you talk so fast, I don't hear one word you say," said Mrs. Huxley, querulously.

"I was talking aunt, about a cottage I had seen this morning," said Agnes, gently. "I thought it would just suit us."'

"I am sure I should not like it. It will have stone floors, which will not do for Miriam. You talk so wildly of going to seek your fortune. I am sure I don't know what is to become of us. You are so sanguine: no good ever comes of it. You were all so set up with Mrs. Warren, and you see what

A fortnight after the funeral, Mrs. Huxley and Agnes were sitting sadly before the fire, which had burned low, on a dull, chill November evening. Miriam lay on her couch, and could scarcely be discerned in the deep-came of it." ening shadow. The dusk was gathering "Well, aunt, my belief is, that Mrs. thick, the curtains were not drawn; both Warren would be as good as ever, if she only without and within, the world looked equally saw us; but she cannot recollect people out desolate to these three women. The silence of sight."y was broken only by the sighs of poor Mrs. Huxley; the dull firelight showed her widow's cap, and the glaze of tears upon her pale clay-like cheeks. At length Agnes roused herself. She had taken the lead in the house since the family troubles, and now moved briskly about the room, endeavoring to impart something like comfort. She replenished the fire, trimmed the lamp; and made the old servant bring in tea.

"Now listen, dear aunt; for I have schemed a scheme, which only needs your approval."

"That will go a very little way towards doing good," sighed Mrs. Huxley.

"She loves flattery, and she likes fresh people," said Miriam, bitterly.

Agnes went to the piano, and began to play some old hymn tunes very softly. Jasottr "Agnes, my dear, I cannot bear music. Do come back and sit still," said her aunt.

The next morning Agnes persuaded her aunt to go with her to the Green, to look at the cottage; and, after some objections, Mrs. Huxley agreed that it might be made to do.

Whilst making arrangements for the removal, Agnes thought seriously how she was to obtain a situation of some kind, and anxiously examined what she was qualified to undertake. She knew that she had only herself to depend upon. A few days afterwards the postman brought a letter with a fereign postmark. It was Mrs. Warren's handwriting. Agnes bounded with it into the parlor, exclaiming, "See! who was right about Mrs. Warren? It is for you."

Miriam turned aside her head. Mrs. Huxley put on her spectacles; and, after turning the letter over half-a-dozen times, Oh, it will go further than you think! opened it. A bank-note for twenty pounds said Agnes, cheerfully. "I was up at the fell out. The letter was written in the kindGreen this morning, and I heard that Sam est tone. She had just seen the mention of Blacksmith is going to leave his cottage for Mr. Huxley's death, and wrote on the spur another that is nearer to his smithy. It of, the moment. She was full of self-reproach struck me that the one he is leaving would for her neglect: begged them to believe she just suit you, and Miriam, and old Mary. loved them as much as ever; spoke of Miriam There is a garden; and the cottage in your with great kindness, but without any spe