Chapter IV: Part 4

This gentleman is well known to the readers of the _Athenaeum_, in which, for nearly twenty years, he has inserted, as advertisements, long arguments in favor of Christians keeping the Jewish Sabbath, beginning on Friday Evening. The present letter maintains that, by the force of the definite article, the _days_ of creation may not be consecutive, but may have any time--millions of years--between them. This ingenious way of reconciling the author of Genesis and the indications of geology is worthy to be added to the list, already pretty numerous. Mr. Heinfetter has taken such pains to make himself a public agitator, that {95} I do not feel it to be any invasion of private life if I state that I have heard he is a large corn-dealer. No doubt he is a member of the congregation whose almanac has already been described.

The great Pyramid. Why was it built? And who built it? By John Taylor,
1859,[192] 12mo.

This work is very learned, and may be referred to for the history of previous speculations. It professes to connect the dimensions of the Pyramid with a system of metrology which is supposed to have left strong traces in the systems of modern times; showing the Egyptians to have had good approximate knowledge of the dimensions of the earth, and of the quadrature of the circle. These are points on which coincidence is hard to distinguish from intention. Sir John Herschel[193] noticed this work, and gave several coincidences, in the _Athenaeum_, Nos. 1696 and 1697, April 28 and May 5, 1860: and there are some remarks by Mr. Taylor in No. 1701, June 2, 1860.

Mr. Taylor's most recent publication is--

The battle of the Standards: the ancient, of four thousand years,
against the modern, of the last fifty years--the less perfect of the
two. London, 1864, 12mo.

This is intended as an appendix to the work on the Pyramid. Mr. Taylor distinctly attributes the original system to revelation, of which he says the Great Pyramid is the record. We are advancing, he remarks, towards the end of the Christian dispensation, and he adds that it is satisfactory to see that we retain the standards which were given by unwritten revelation 700 years before Moses. This is lighting the candle at both ends; for myself, I shall not undertake to deny or affirm either what is said about the dark past or what is hinted about the dark future.

{96}

My old friend Mr. Taylor is well known as the author of the argument which has convinced many, even most, that Sir Philip Francis[194] was Junius: pamphlet, 1813; supplement, 1817; second edition "The Identity of Junius with a distinguished living character established," London, 1818, 8vo. He told me that Sir Philip Francis, in a short conversation with him, made only this remark, "You may depend upon it you are quite mistaken:" the phrase appears to me remarkable; it has an air of criticism on the book, free from all personal denial. He also mentioned that a hearer told him that Sir Philip said, speaking of writers on the question,--"Those fellows, for half-a-crown, would prove that Jesus Christ was Junius."

Mr. Taylor implies, I think, that he is the first who started the suggestion that Sir Philip Francis was Junius, which I have no means either of confirming or refuting. If it be so [and I now know that Mr. Taylor himself never heard of any predecessor], the circumstance is very remarkable: it is seldom indeed that the first proposer of any solution of a great and vexed question is the person who so nearly establishes his point in general opinion as Mr. Taylor has done.

As to the Junius question in general, there is a little bit of the philosophy of horse-racing which may be usefully applied. A man who is so confident of his horse that he places him far above any other, may nevertheless, and does, refuse to give odds against all in the field: for many small adverse chances united make a big chance for one or other of the opponents. I suspect Mr. Taylor has made it at least 20 to 1 for Francis against any one competitor who has been named: but what the odds may be against the {97} whole field is more difficult to settle. What if the real Junius should be some person not yet named?

Mr. Jopling, _Leisure Hour_, May 23, 1863, relies on the porphyry coffer of the Great Pyramid, in which he finds "the most ancient and accurate standard of measure in existence."

I am shocked at being obliged to place a thoughtful and learned writer, and an old friend, before such a successor as he here meets with. But chronological arrangement defies all other arrangement.

(I had hoped that the preceding account would have met Mr. Taylor's eye in print: but he died during the last summer. For a man of a very thoughtful and quiet temperament, he had a curious turn for vexed questions. But he reflected very long and very patiently before he published: and all his works are valuable for their accurate learning, whichever side the reader may take.)

MRS. ELIZABETH COTTLE.

1859. _The Cottle Church._--For more than twenty years printed papers have been sent about in the name of Elizabeth Cottle.[195] It is not so remarkable that such papers should be concocted as that they should circulate for such a length of time without attracting public attention. Eighty years ago Mrs. Cottle might have rivalled Lieut. Brothers or Joanna Southcott.[196] Long hence, when the now current volumes of our journals are well-ransacked works of reference, those who look into them will be glad to see this {98} feature of our time: I therefore make a few extracts, faithfully copied as to type. The Italic is from the New Testament; the Roman is the requisite interpretation:

"Robert Cottle '_was numbered_ (5196) _with the transgressors_' at the back of the Church in Norwood Cemetery, May 12, 1858--Isa. liii. 12. The Rev. J. G. Collinson, Minister of St. James's Church, Chapham, the then district church, before All Saints was built, read the funeral service _over the Sepulchre wherein never before man was laid_.

"_Hewn on the stone_, 'at the mouth of the Sepulchre,' is his name,--Robert Cottle, born at Bristol, June 2, 1774; died at Kirkstall Lodge, Clapham Park, May 6, 1858. _And that day_ (May 12, 1858) _was the preparation_ (day and year for 'the PREPARED place for you'--Cottleites---by the widowed mother of the Father's house, at Kirkstall Lodge--John xiv. 2, 3). _And the Sabbath_ (Christmas Day, Dec. 25, 1859) _drew on_ (for the resurrection of the Christian body on 'the third [Protestant Sun]-day'--1 Cor. xv. 35). _Why seek ye the living_ (God of the New Jerusalem--Heb. xii. 22; Rev. iii. 12) _among the dead_ (men): _he_ (the God of Jesus) _is not here_ (in the grave), _but is risen_ (in the person of the Holy Ghost, from the supper of 'the dead in the second death' of Paganism). _Remember how he spake unto you_ (in the church of the Rev. George Clayton,[197] April 14, 1839). _I will not drink henceforth_ (at this last Cottle supper) _of the fruit of this_ (Trinity) _vine, until that day_ (Christmas Day, 1859), _when I_ (Elizabeth Cottle) _drink it new with you_ (Cottleites) _in my Father's kingdom_--John xv. _If this_ (Trinitarian) _cup may not pass away from me_ (Elizabeth Cottle, April 14, 1839), _except I drink it_ ('new with you Cottleites, in my Father's Kingdom'), _thy will be done_--Matt. xxvi. 29, 42, 64. 'Our Father which art (God) in Heaven,' _hallowed be thy name, thy_ (Cottle) _kingdom_ {99} _come, thy will be done in earth, as it is_ (done) _in_ (the new) _Heaven_ (and new earth of the new name of Cottle--Rev. xxi. 1; iii. 12).

"... Queen Elizabeth, from A.D. 1558 to 1566. _And this_ WORD _yet once more_ (by a second Elizabeth--the WORD of his oath) _signifieth_ (at John Scott's baptism of the Holy Ghost) _the removing of those things_ (those Gods and those doctrines) _that are made_ (according to the Creeds and Commandments of men) _that those things_ (in the moral law of God) _which cannot be shaken_ (as a rule of faith and practice) _may remain, wherefore we receiving_ (from Elizabeth) _a kingdom_ (of God,) _which cannot be moved_ (by Satan) _let us have grace_ (in his Grace of Canterbury) _whereby we may serve God acceptably_ (with the acceptable sacrifice of Elizabeth's body and blood of the communion of the Holy Ghost) _with reverence_ (for truth) _and godly fear_ (of the unpardonable sin of blasphemy against the Holy Ghost) _for our God_ (the Holy Ghost) _is a consuming fire_ (to the nation that will not serve him in the Cottle Church). We cannot defend ourselves against the Almighty, and if He is our defence, no nation can invade us.

"In verse 4 the Church of St. Peter is _in prison between four quaternions of soldiers_--the Holy Alliance of 1815. Rev. vii. i. Elizabeth, _the Angel of the Lord_ Jesus _appears_ to the Jewish and Christian body with _the vision_ of prophecy to the Rev. Geo. Clayton and his clerical brethren, April 8th, 1839. _Rhoda_ was the name of her maid at Putney Terrace who used _to open the door to her Peter_, the Rev. Robert Ashton,[198] the Pastor of 'the little flock' 'of 120 names together, assembled in an upper (school) room' at Putney Chapel, to which little flock she gave the revelation (Acts. i. 13, 15) _of Jesus the same_ King of the Jews _yesterday_ at the prayer meeting, Dec. 31, 1841, _and to-day_, {100} Jan. 1, 1842, _and for ever_. See book of Life, page 24. Matt. xviii. 19, xxi. 13-16. In verse 6 the Italian body of St. Peter _is sleeping_ 'in the second death' _between the two_ Imperial _soldiers_ of France and Austria. The Emperor of France from Jan. 1, to July 11, 1859, causes the Italian _chains of St. Peter to fall off from his_ Imperial _hands_.

"_I say unto thee_, Robert Ashton, _thou art Peter_, a stone, _and upon this rock_, of truth, _will I_ Elizabeth, the angel of Jesus, _build my_ Cottle _Church, and the gates of hell_, the doors of St. Peter, at Rome, shall not prevail against it--Matt. xvi. 18. Rev. iii. 7-12."

This will be enough for the purpose. When any one who pleases can circulate new revelations of this kind, uninterrupted and unattended to, new revelations will cease to be a good investment of excentricity. I take it for granted that the gentlemen whose names are mentioned have nothing to do with the circulars or their doctrines. Any lady who may happen to be intrusted with a revelation may nominate her own pastor, or any other clergyman, one of her apostles; and it is difficult to say to what court the nominees can appeal to get the commission abrogated.

_March 16, 1865._ During the last two years the circulars have continued. It is hinted that funds are low: and two gentlemen who are represented as gone "to Bethlehem asylum in despair" say that Mrs. Cottle "will spend all that she hath, while Her Majesty's Ministers are flourishing on the wages of sin." The following is perhaps one of the most remarkable passages in the whole:

"_Extol and magnify Him_ (Jehovah, the Everlasting God, see the Magnificat and Luke i. 45, 46--68--73--79), _that rideth_ (by rail and steam over land and sea, from his holy habitation at Kirkstall Lodge, Psa. lxxvii. 19, 20), _upon the_ (Cottle) _heavens, as it were_ (Sept. 9, 1864, see pages 21, 170), _upon an_ (exercising, Psa. cxxxi. 1), _horse_-(chair, bought of Mr. John Ward, Leicester-square)." {101}

I have pretty good evidence that there is a clergyman who thinks Mrs. Cottle a very sensible woman.

[_The Cottle Church._ Had I chanced to light upon it at the time of writing, I should certainly have given the following. A printed letter to the _Western Times_, by Mr. Robert Cottle, was accompanied by a manuscript letter from Mrs. Cottle, apparently a circular. The date was Nov^{r}. 1853, and the subject was the procedure against Mr. Maurice[199] at King's College for doubting that God would punish human sins by an existence of torture lasting through years numbered by millions of millions of millions of millions (repeat the word _millions_ without end,) etc. The memory of Mr. Cottle has, I think, a right to the quotation: he seems to have been no participator in the notions of his wife:

"The clergy of the Established Church, taken at the round number of 20,000, may, in their first estate, be likened to 20,000 gold blanks, destined to become sovereigns, in succession,--they are placed between the matrix of the Mint, when, by the pressure of the screw, they receive the impress that fits them to become part of the current coin of the realm. In a way somewhat analogous this great body of the clergy have each passed through the crucibles of Oxford and Cambridge,--have been assayed by the Bishop's chaplain, touching the health of their souls, and the validity of their call by the Divine Spirit, and then the gentle pressure of a prelate's hand upon their heads; and the words--'Receive the Holy Ghost,' have, in a brief space of time, wrought a {102} change in them, much akin to the miracle of transubstantiation--the priests are completed, and they become the current ecclesiastical coin of our country. The whole body of clergy, here spoken of, have undergone the preliminary induction of baptism and confirmation; and all have been duly ordained, _professing_ to hold one faith, and to believe in the selfsame doctrines! In short, to be as identical as the 20,000 sovereigns, if compared one with the other. But mind is not malleable and ductile, like gold; and all the preparations of tests, creeds, and catechisms will not insure uniformity of belief. No stamp of orthodoxy will produce the same impress on the minds of different men. Variety is manifest, and patent, upon everything mental and material. The Almighty has not created, nor man fashioned, two things alike! How futile, then, is the attempt to shape and mould man's apprehension of divine truth by one fallible standard of man's invention! If proof of this be required, an appeal might be made to history and the experience of eighteen hundred years."

This is an argument of force against the reasonableness of expecting tens of thousands of educated readers of the New Testament to find the doctrine above described in it. The lady's argument against the doctrine itself is very striking. Speaking of an outcry on this matter among the Dissenters against one of their body, who was the son of "the White Stone (Rev. ii. 17), or the Roman cement-maker," she says--

"If the doctrine for which they so wickedly fight were true, what would become of the black gentlemen for whose redemption I have been sacrificed from April 8 1839."

There are certainly very curious points about this revelation. There have been many surmises about the final restoration of the infernal spirits, from the earliest ages of Christianity until our own day: a collection of them would be worth making. On reading this in proof, I see a possibility that by "black gentlemen" may be meant the clergy: {103} I suppose my first interpretation must have been suggested by context: I leave the point to the reader's sagacity.]

JAMES SMITH, ARCH-PARADOXER.

The Problem of squaring the circle solved; or, the circumference and
area of the circle discovered. By James Smith.[200] London, 1859, 8vo.

On the relations of a square inscribed in a circle. Read at the British
Association, Sept. 1859, published in the Liverpool Courier, Oct. 8,
1859, and reprinted in broadsheet.

The question: Are there any commensurable relations between a circle
and other Geometrical figures? Answered by a member of the British
Association ... London, 1860, 8vo.--[This has been translated into
French by M. Armand Grange, Bordeaux, 1863, 8vo.]

The Quadrature of the Circle. Correspondence between an eminent
mathematician and James Smith, Esq. (Member of the Mersey Docks and
Harbour Board), London, 1861, 8vo. (pp. 200).

Letter to the ... British Association ... by James Smith, Esq.
Liverpool, 1861, 8vo.

Letter to the ... British Association ... by James Smith, Esq.
Liverpool, 1862, 8vo.--[These letters the author promised to continue.]

A Nut to crack for the readers of Professor De Morgan's 'Budget of
Paradoxes.' By James Smith, Esq. Liverpool, 1863, 8vo.

Paper read at the Liverpool Literary and Philosophical Society,
reported in the Liverpool Daily Courier, Jan. 26, 1864. Reprinted as a
pamphlet.

The Quadrature of the circle, or the true ratio between the diameter
and circumference geometrically and mathematically demonstrated. By
James Smith, Esq. Liverpool, 1865, 8vo.

{104}

[On the relations between the dimensions and distances of the Sun,
Moon, and Earth; a paper read before the Literary and Philosophical
Society of Liverpool, Jan. 25, 1864. By James Smith, Esq.

The British Association in Jeopardy, and Dr. Whewell, the Master of
Trinity, in the stocks without hope of escape. Printed for the authors
(J. S. confessed, and also hidden under _Nauticus_). (No date, 1865).

The British Association in Jeopardy, and Professor De Morgan in the
Pillory without hope of escape. London, 1866, 8vo.]

When my work appeared in numbers, I had not anything like an adequate idea of Mr. James Smith's superiority to the rest of the world in the points in which he is superior. He is beyond a doubt the ablest head at unreasoning, and the greatest hand at writing it, of all who have tried in our day to attach their names to an error. Common cyclometers sink into puny orthodoxy by his side.

The behavior of this singular character induces me to pay him the compliment which Achilles paid Hector, to drag him round the walls again and again. He was treated with unusual notice and in the most gentle manner. The unnamed mathematician, E. M. bestowed a volume of mild correspondence upon him; Rowan Hamilton[201] quietly proved him wrong in a way accessible to an ordinary schoolboy; Whewell,[202] as we shall see, gave him the means of seeing himself wrong, even more easily than by Hamilton's method. Nothing would do; it was small kick and silly fling at all; and he exposed his conceit by alleging that he, James Smith, had placed Whewell in the stocks. He will therefore be universally pronounced a proper object of the severest literary punishment: but the opinion of all who can put two propositions together will be that of the many strokes I have given, the hardest and most telling are my republications of his own attempts to reason.

He will come out of my hands in the position he ought {105} to hold, the Supreme Pontiff of cyclometers, the vicegerent of St. Vitus upon earth, the Mamamouchi of burlesque on inference. I begin with a review of him which appeared in the _Athenaeum_ of May 11, 1861. Mr. Smith says I wrote it: this I neither affirm nor deny; to do either would be a sin against the editorial system elsewhere described. Many persons tell me they know me by my style; let them form a guess: I can only say that many have declared as above while fastening on me something which I had never seen nor heard of.

The Quadrature of the Circle: Correspondence between an Eminent
Mathematician and James Smith, Esq. (Edinburgh, Oliver & Boyd; London,
Simpkin, Marshall & Co.)

"A few weeks ago we were in perpetual motion. We did not then suppose that anything would tempt us on a circle-squaring expedition: but the circumstances of the book above named have a peculiarity which induces us to give it a few words.

"Mr. James Smith, a gentleman residing near Liverpool, was some years ago seized with the _morbus cyclometricus_.[203] The symptoms soon took a defined form: his circumference shrank into exactly 3-1/8 times his diameter, instead of close to 3-16/113, which the mathematician knows to be so near to truth that the error is hardly at the rate of a foot in 2,000 miles. This shrinking of the circumference remained until it became absolutely necessary that it should be examined by the British Association. This body, which as Mr. James Smith found to his sorrow, has some interest in 'jealously guarding the mysteries of their profession,' refused at first to entertain the question. On this Mr. Smith changed his 'tactics' and the name of his paper, and smuggled in the subject under the form of 'The Relations of a Circle inscribed in a Square'! The paper was thus forced upon the Association, for Mr. Smith informs us that he {106} 'gave the Section to understand that he was not the man that would permit even the British Association to trifle with him.' In other words, the Association bore with and were bored with the paper, as the shortest way out of the matter. Mr. Smith also circulated a pamphlet. Some kind-hearted man, who did not know the disorder as well as we do, and who appears in Mr. Smith's handsome octavo as E. M.--the initials of 'eminent mathematician'--wrote to him and offered to show him in a page that he was all wrong. Mr. Smith thereupon opened a correspondence, which is the bulk of the volume. When the correspondence was far advanced, Mr. Smith announced his intention to publish. His benevolent instructor--we mean in intention--protested against the publication, saying 'I do not wish to be gibbeted to the world as having been foolish enough to enter upon what I feel now to have been a ridiculous enterprise.'

"For this Mr. Smith cared nothing: he persisted in the publication, and the book is before us. Mr. Smith has had so much grace as to conceal his kind adviser's name under E. M., that is to say, he has divided the wrong among all who may be suspected of having attempted so hopeless a task as that of putting a little sense into his head. He has violated the decencies of private life. Against the will of the kind-hearted man who undertook his case, he has published letters which were intended for no other purpose than to clear his poor head of a hopeless delusion. He deserves the severest castigation; and he will get it: his abuse of confidence will stick by him all his days. Not that he has done his benefactor--in intention, again--any harm. The patience with which E. M. put the blunders into intelligible form, and the perseverance with which he tried to find a cranny-hole for common reasoning to get in at, are more than respectable: they are admirable. It is, we can assure E. M., a good thing that the nature of the circle-squarer should be so completely exposed as in this volume. The benefit which he intended Mr. James Smith may be {107} conferred upon others. And we should very much like to know his name, and if agreeable to him, to publish it. As to Mr. James Smith, we can only say this: he is not mad. Madmen reason rightly upon wrong premises: Mr. Smith reasons wrongly upon no premises at all.

"E. M. very soon found out that, to all appearance, Mr. Smith got a circle of 3-1/8 times the diameter by making it the supposition to set out with that there was such a circle; and then finding certain consequences which, so it happened, were not inconsistent with the supposition on which they were made. Error is sometimes self-consistent. However, E. M., to be quite sure of his ground, wrote a short letter, stating what he took to be Mr. Smith's hypothesis, containing the following: 'On AC as diameter, describe the circle D, which by hypothesis shall be equal to three and one-eighth times the length of AC.... I beg, before proceeding further, to ask whether I have rightly stated your argument.' To which Mr. Smith replied: 'You have stated my argument with perfect accuracy.' Still E. M. went on, and we could not help, after the above, taking these letters as the initials of Everlasting Mercy. At last, however, when Mr. Smith flatly denied that the area of the circle lies between those of the inscribed and circumscribed polygons, E. M. was fairly beaten, and gave up the task. Mr. Smith was left to write his preface, to talk about the certain victory of truth--which, oddly enough, is the consolation of all hopelessly mistaken men; to compare himself with Galileo; and to expose to the world the perverse behavior of the Astronomer Royal, on whom he wanted to fasten a conversation, and who replied, 'It would be a waste of time, Sir, to listen to anything you could have to say on such a subject.'

"Having thus disposed of Mr. James Smith, we proceed to a few remarks on the subject: it is one which a journal would never originate, but which is rendered necessary from time to time by the attempts of the autopseustic to become {108} heteropseustic. To the mathematician we have nothing to say: the question is, what kind of assurance can be given to the world at large that the wicked mathematicians are not acting in concert to keep down their superior, Mr. James Smith, the current Galileo of the quadrature of the circle.

"Let us first observe that this question does not stand alone: independently of the millions of similar problems which exist in higher mathematics, the finding of the diagonal of a square has just the same difficulty, namely, the entrance of a pair of lines of which one cannot be definitely expressed by means of the other. We will show the reader who is up to the multiplication-table how he may go on, on, on, ever nearer, never there, in finding the diagonal of a square from the side.

"Write down the following rows of figures, and more, if you like, in the way described:

1 2 5 12 29 70 169 408 985
1 3 7 17 41 99 239 577 1393

After the second, each number is made up of double the last increased by the last but one: thus, 5 is 1 more than twice 2, 12 is 2 more than twice 5, 239 is 41 more than twice 99. Now, take out two adjacent numbers from the upper line, and the one below the first from the lower: as

70 169
99.

Multiply together 99 and 169, giving 16,731. If, then, you will say that 70 diagonals are exactly equal to 99 sides, you are in error about the diagonal, but an error the amount of which is not so great as the 16,731st part of the diagonal. Similarly, to say that five diagonals make exactly seven sides does not involve an error of the 84th part of the diagonal.

"Now, why has not the question of _crossing the square_ been as celebrated as that of _squaring the circle_? Merely because Euclid demonstrated the impossibility of the first {109} question, while that of the second was not demonstrated, completely, until the last century.

"The mathematicians have many methods, totally different from each other, of arriving at one and the same result, their celebrated approximation to the circumference of the circle. An intrepid calculator has, in our own time, carried his approximation to what they call 607 decimal places: this has been done by Mr. Shanks,[204] of Houghton-le-Spring, and Dr. Rutherford[205] has verified 441 of these places. But though 607 looks large, the general public will form but a hazy notion of the extent of accuracy acquired. We have seen, in Charles Knight's[206] _English Cyclopaedia_, an account of the matter which may illustrate the unimaginable, though rationally conceivable, extent of accuracy obtained.

"Say that the blood-globule of one of our animalcules is a millionth of an inch in diameter. Fashion in thought a globe like our own, but so much larger that our globe is but a blood-globule in one of its animalcules: never mind the microscope which shows the creature being rather a bulky instrument. Call this the first globe _above_ us. Let the first globe above us be but a blood-globule, as to size, in the animalcule of a still larger globe, which call the second globe above us. Go on in this way to the twentieth globe above us. Now go down just as far on the other side. Let the blood-globule with which we started be a globe peopled with animals like ours, but rather smaller: {110} and call this the first globe below us. Take a blood-globule out of this globe, people it, and call it the second globe below us: and so on to the twentieth globe below us. This is a fine stretch of progression both ways. Now give the giant of the twentieth globe _above_ us the 607 decimal places, and, when he has measured the diameter of his globe with accuracy worthy of his size, let him calculate the circumference of his equator from the 607 places. Bring the little philosopher from the twentieth globe _below_ us with his very best microscope, and set him to see the small error which the giant must make. He will not succeed, unless his microscopes be much better for his size than ours are for ours.

"Now it must be remembered by any one who would laugh at the closeness of the approximation, that the mathematician generally goes _nearer_; in fact his theorems have usually no error at all. The very person who is bewildered by the preceding description may easily forget that if there were _no error at all_, the Lilliputian of the millionth globe below us could not find a flaw in the Brobdingnagian of the millionth globe above. The three angles of a triangle, of perfect accuracy of form, are _absolutely_ equal to two right angles; no stretch of progression will detect _any_ error.

"Now think of Mr. Lacomme's mathematical adviser (_ante_, Vol. I, p. 46) making a difficulty of advising a stonemason about the quantity of pavement in a circular floor!

"We will now, for our non-calculating reader, put the matter in another way. We see that a circle-squarer can advance, with the utmost confidence, the assertion that when the diameter is 1,000, the circumference is accurately 3,125: the mathematician declaring that it is a trifle more than 3,141-1/2. If the squarer be right, the mathematician has erred by about a 200th part of the whole: or has not kept his accounts right by about 10s. in every 100l. Of course, if he set out with such an error he will accumulate blunder upon blunder. Now, if there be a process in which {111} close knowledge of the circle is requisite, it is in the prediction of the moon's place--say, as to the time of passing the meridian at Greenwich--on a given day. We cannot give the least idea of the complication of details: but common sense will tell us that if a mathematician cannot find his way round the circle without a relative error four times as big as a stockbroker's commission, he must needs be dreadfully out in his attempt to predict the time of passage of the moon. Now, what is the fact? His error is less than a second of time, and the moon takes 27 days odd to revolve. That is to say, setting out with 10s. in 100l. of error in his circumference, he gets within the fifth part of a farthing in 100l. in predicting the moon's transit. Now we cannot think that the respect in which mathematical science is held is great enough--though we find it not small--to make this go down. That respect is founded upon a notion that right ends are got by right means: it will hardly be credited that the truth can be got to farthings out of data which are wrong by shillings. Even the celebrated Hamilton[207] of Edinburgh, who held that in mathematics there was no way of going wrong, was fully impressed with the belief that this was because error was avoided from the beginning. He never went so far as to say that a mathematician who begins wrong must end right somehow.

"There is always a difficulty about the mode in which the thinking man of common life is to deal with subjects he has not studied to a professional extent. He must form opinions on matters theological, political, legal, medical, and social. If he can make up his mind to choose a guide, there is, of course, no perplexity: but on all the subjects mentioned the direction-posts point different ways. Now why should he not form his opinion upon an abstract mathematical question? Why not conclude that, as to the circle, it is possible Mr. James Smith may be the man, just {112} as Adam Smith[208] was the man of things then to come, or Luther, or Galileo? It is true that there is an unanimity among mathematicians which prevails in no other class: but this makes the chance of their all being wrong only different in degree. And more than this, is it not generally thought among us that priests and physicians were never so much wrong as when there was most appearance of unanimity among them? To the preceding questions we see no answer except this, that the individual inquirer may as rationally decide a mathematical question for himself as a theological or a medical question, so soon as he can put himself into a position in mathematics, level with that in which he stands in theology or medicine. The every-day thought and reading of common life have a certain resemblance to the thought and reading demanded by the learned faculties. The research, the balance of evidence, the estimation of probabilities, which are used in a question of medicine, are closely akin in character, however different the matter of application, to those which serve a merchant to draw his conclusions about the markets. But the mathematicians have methods of their own, to which nothing in common life bears close analogy, as to the nature of the results or the character of the conclusions. The logic of mathematics is certainly that of common life: but the data are of a different species; they do not admit of doubt. An expert arithmetician, such as is Mr. J. Smith, may fancy that calculation, merely as such, is mathematics: but the value of his book, and in this point of view it is not small, is the full manner in which it shows that a practised arithmetician, venturing into the field of mathematical demonstration, may show himself utterly destitute of all that distinguishes the reasoning geometrical investigator from the calculator.

{113}

"And further, it should be remembered that in mathematics the power of verifying results far exceeds that which is found in anything else: and also the variety of distinct methods by which they can be attained. It follows from all this that a person who desires to be as near the truth as he can will not judge the results of mathematical demonstration to be open to his criticism, in the same degree as results of other kinds. Should he feel compelled to decide, there is no harm done: his circle may be 3-1/8 times its diameter, if it please him. But we must warn him that, in order to get this circle, he must, as Mr. James Smith has done, _make it at home_: the laws of space and thought beg leave respectfully to decline the order."

I will insert now at length, from the _Athenaeum_ of June 8, 1861, the easy refutation given by my deceased friend, with the remarks which precede.

"Mr. James Smith, of whose performance in the way of squaring the circle we spoke some weeks ago in terms short of entire acquiescence, has advertised himself in our columns, as our readers will have seen. He has also forwarded his letter to the Liverpool _Albion_, with an additional statement, which he did not make in _our_ journal. He denies that he has violated the decencies of private life, since his correspondent revised the proofs of his own letters, and his 'protest had respect only to making his name public.' This statement Mr. James Smith precedes by saying that we have treated as true what we well knew to be false: and he follows by saying that we have not read his work, or we should have known the above facts to be true. Mr. Smith's pretext is as follows. His correspondent E. M. says, 'My letters were not intended for publication, and I protest against their being published,' and he subjoins 'Therefore I must desire that my name may not be used.' The obvious meaning is that E. M. protested against the publication altogether, but, judging that Mr. Smith was {114} determined to publish, desired that his name should not be used. That he afterwards corrected the proofs merely means that he thought it wiser to let them pass under his own eyes than to leave them entirely to Mr. Smith.

"We have received from Sir W. Rowan Hamilton[209] a proof that the circumference is more than 3-1/8 diameters, requiring nothing but a knowledge of four books of Euclid. We give it in brief as an exercise for our juvenile readers to fill up. It reminds us of the old days when real geometers used to think it worth while seriously to demolish pretenders. Mr. Smith's fame is now assured: Sir W. R. Hamilton's brief and easy exposure will procure him notice in connection with this celebrated problem.

"It is to be shown that the perimeter of a regular polygon of 20 sides is greater than 3-1/8 diameters of the circle, and still more, of course, is the circumference of the circle greater than 3-1/8 diameters.

"1. It follows from the 4th Book of Euclid, that the rectangle under the side of a regular decagon inscribed in a circle, and that side increased by the radius, is equal to the square of the radius. But the product 791 (791 + 1280) is less than 1280 x 1280; if then the radius be 1280 the side of the decagon is greater than 791.

"2. When a diameter bisects a chord, the square of the chord is equal to the rectangle under the doubles of the segments of the diameter. But the product 125 (4 x 1280 - 125) is less than 791 x 791. If then the bisected chord be a side of the decagon, and if the radius be still 1280, the double of the lesser segment exceeds 125.

"3. The rectangle under this doubled segment and the radius is equal to the square of the side of an inscribed regular polygon of 20 sides. But the product 125 x 1280 is equal to 400 x 400; therefore, the side of the last-mentioned polygon is greater than 400, if the radius be still 1280. In other words, if the radius be represented by the new {115} member 16, and therefore the diameter by 32, this side is greater than 5, and the perimeter exceeds 100. So that, finally, if the diameter be 8, the perimeter of the inscribed regular polygon of 20 sides, and still more the circumference of the circle, is greater than 25: that is, the circumference is more than 3-1/8 diameters."

The last work in the list was thus noticed in the _Athenaeum_, May 27, 1865.

"Mr. James Smith appears to be tired of waiting for his place in the Budget of Paradoxes, and accordingly publishes a long letter to Professor De Morgan, with various prefaces and postscripts. The letter opens by a hint that the Budget appears at very long intervals, and 'apparently without any sufficient reason for it.' As Mr. Smith hints that he should like to see Mr. De Morgan, whom he calls an 'elephant of mathematics,' 'pumping his brains' 'behind the scenes'--an odd thing for an elephant to do, and an odd place to do it in--to get an answer, we think he may mean to hint that the Budget is delayed until the pump has worked successfully. Mr. Smith is informed that we have had the whole manuscript of the Budget, excepting only a final summing-up, in our hands since October, 1863. [This does not refer to the Supplement.] There has been no delay: we knew from the beginning that a series of historical articles would be frequently interrupted by the things of the day. Mr. James Smith lets out that he has never been able to get a private line from Mr. De Morgan in answer to his communications: we should have guessed it. He says, 'The Professor is an old bird and not to be easily caught, and by no efforts of mine have I been able, up to the present moment, either to induce or twit him into a discussion....' Mr. Smith curtails the proverb: old birds are not to be caught with _chaff_, nor with _twit_, which seems to be Mr. Smith's word for his own chaff, and, so long as the first letter is sounded, a very proper word. Why does he not try a little grain of sense? Mr. Smith evidently {116} thinks that, in his character as an elephant, the Professor has not pumped up brain enough to furnish forth a bird. In serious earnest, Mr. Smith needs no answer. In one thing he excites our curiosity: what is meant by demonstrating 'geometrically _and_ mathematically?'"

I now proceed to my original treatment of the case.

Mr. James Smith will, I have no doubt, be the most uneclipsed circle-squarer of our day. He will not owe this distinction to his being an influential and respected member of the commercial world of Liverpool, even though the power of publishing which his means give him should induce him to issue a whole library upon one paradox. Neither will he owe it to the pains taken with him by a mathematician who corresponded with him until the joint letters filled an octavo volume. Neither will he owe it to the notice taken of him by Sir William Hamilton, of Dublin, who refuted him in a manner intelligible to an ordinary student of Euclid, which refutation he calls a remarkable paradox easily explainable, but without explaining it. What he will owe it to I proceed to show.

Until the publication of the _Nut to Crack_ Mr. James Smith stood among circle-squarers in general. I might have treated him with ridicule, as I have done others: and he says that he does not doubt he shall come in for his share at the tail end of my Budget. But I can make a better job of him than so, as Locke would have phrased it: he is such a very striking example of something I have said on the use of logic that I prefer to make an example of his writings. On one point indeed he well deserves the _scutica_,[210] if not the _horribile flagellum_.[211] He tells me that he will bring his solution to me in such a form as shall compel me to admit it as _un fait accompli_ [_une faute accomplie?_][212] {117} or leave myself open to the humiliating charge of mathematical ignorance and folly. He has also honored me with some private letters. In the first of these he gives me a "piece of information," after which he cannot imagine that I, "as an honest mathematician," can possibly have the slightest hesitation in admitting his solution. There is a tolerable reservoir of modest assurance in a man who writes to a perfect stranger with what he takes for an argument, and gives an oblique threat of imputation of dishonesty in case the argument be not admitted without hesitation; not to speak of the minor charges of ignorance and folly. All this is blind self-confidence, without mixture of malicious meaning; and I rather like it: it makes me understand how Sam Johnson came to say of his old friend Mrs. Cobb,[213]--"I love Moll Cobb for her impudence." I have now done with my friend's _suaviter in modo_,[214] and proceed to his _fortiter in re_[215]: I shall show that he _has_ convicted himself of ignorance and folly, with an honesty and candor worthy of a better value of [pi].

Mr. Smith's method of proving that every circle is 3-1/8 diameters is to assume that it is so,--"if you dislike the term datum, then, by hypothesis, let 8 circumferences be exactly equal to 25 diameters,"--and then to show that every other supposition is thereby made absurd. The right to this assumption is enforced in the "Nut" by the following analogy:

"I think you (!) will not dare (!) to dispute my right to this hypothesis, when I can prove by means of it that every other value of [pi] will lead to the grossest absurdities; unless indeed, you are prepared to dispute the right of Euclid to adopt a false line hypothetically for the purpose {118} of a '_reductio ad absurdum_'[216] demonstration, in pure geometry."

Euclid assumes what he wants to _disprove_, and shows that his _assumption_ leads to absurdity, and so _upsets itself_. Mr. Smith assumes what he wants to _prove_, and shows that _his_ assumption makes _other propositions_ lead to absurdity. This is enough for all who can reason. Mr. James Smith cannot be argued with; he has the whip-hand of all the thinkers in the world. Montucla would have said of Mr. Smith what he said of the gentleman who squared his circle by giving 50 and 49 the same square root, _Il a perdu le droit d'etre frappe de l'evidence_.[217]

It is Mr. Smith's habit, when he finds a conclusion agreeing with its own assumption, to regard that agreement as proof of the assumption. The following is the "piece of information" which will settle me, if I be honest. Assuming [pi] to be 3-1/8, he finds out by working instance after instance that the mean proportional between one-fifth of the area and one-fifth of eight is the radius. That is,

if [pi] = 25/8, sqrt(([pi]r^2)/5 . 8/5) = r.

This "remarkable general principle" may fail to establish Mr. Smith's quadrature, even in an honest mind, if that mind should happen to know that, a and b being any two numbers whatever, we need only assume--

[pi] = a^2/b, to get at sqrt(([pi]r^2)/a . b/a) = r.

We naturally ask what sort of glimmer can Mr. Smith have of the subject which he professes to treat? On this point he has given satisfactory information. I had mentioned the old problem of finding two mean proportionals, {119} as a preliminary to the duplication of the cube. On this mention Mr. Smith writes as follows. I put a few words in capitals; and I write rq[218] for the sign of the square root, which embarrasses small type:

"This establishes the following _infallible_ rule, for finding two mean proportionals OF EQUAL VALUE, and is more than a preliminary, to the famous old problem of 'Squaring the circle.' Let any finite number, say 20, and its fourth part = (1/4)(20) = 5, be given numbers. Then rq(20 x 5) = rq 100 = 10, is their mean proportional. Let this be a given mean proportional TO FIND ANOTHER MEAN PROPORTIONAL OF EQUAL VALUE. Then

20 x [pi]/4 = 20 x 3.125/4 = 20 x .78125 = 15.625

will be the first number; as

25 : 16 :: rq 20 : rq 8.192: and (rq 8.192)^2 x [pi]/4 = 8.192 x .78125 =
6.4

will be the second number; therefore rq(15.625 x 6.4) = rq 100 = 10, is the required mean proportional.... Now, my good Sir, however competent you may be to prove every man a fool [not _every_ man, Mr. Smith! only _some_; pray learn logical quantification] who now thinks, or in times gone by has thought, the 'Squaring of the Circle' _a possibility_; I doubt, and, on the evidence afforded by your Budget, I cannot help doubting, whether you were ever before competent to find two mean proportionals _by my unique method_."--(_Nut_, pp. 47, 48.) [That I never was, I solemnly declare!]

All readers can be made to see the following exposure. When 5 and 20 are given, x is a mean proportional when in 5, x, 20, 5 is to x as x to 20. And x must be 10. But x and y are two mean proportionals when in 5, x, y, 20, x {120} is a mean proportional between 5 and y, and y is a mean proportional between x and 20. And these means are x = 5 [cuberoot]4, y = 5 [cuberoot]16. But Mr. Smith finds _one_ mean, finds it _again_ in a roundabout way, and produces 10 and 10 as the two (equal!) means, in solution of the "famous old problem." This is enough: if more were wanted, there is more where this came from. Let it not be forgotten that Mr. Smith has found a translator abroad, two, perhaps three, followers at home, and--most surprising of all--a real mathematician to try to set him right. And this mathematician did not discover the character of the subsoil of the land he was trying to cultivate until a goodly octavo volume of letters had passed and repassed. I have noticed, in more quarters than one, an apparent want of perception of the _full_ amount of Mr. Smith's ignorance: persons who have not been in contact with the non-geometrical circle-squarers have a kind of doubt as to whether anybody can carry things so far. But I am an "old bird" as Mr. Smith himself calls me; a Simorg, an "all-knowing Bird of Ages" in matters of cyclometry.

The curious phenomena of thought here exhibited illustrate, as above said, a remark I have long ago made on the effect of proper study of logic. Most persons reason well enough on matter to which they are accustomed, and in terms with which they are familiar. But in unaccustomed matter, and with use of strange terms, few except those who are practised in the abstractions of pure logic can be tolerably sure to keep their feet. And one of the reasons is easily stated: terms which are not quite familiar partake of the vagueness of the X and Y on which the student of logic learns to see the formal force of a proposition independently of its material elements.

I make the following quotation from my fourth paper on logic in the _Cambridge Transactions_:

"The uncultivated reason proceeds by a process almost entirely material. Though the necessary law of thought {121} must determine the conclusion of the ploughboy as much as that of Aristotle himself, the ploughboy's conclusion will only be tolerably sure when the matter of it is such as comes within his usual cognizance. He knows that geese being all birds does not make all birds geese, but mainly because there are ducks, chickens, partridges, etc. A beginner in geometry, when asked what follows from 'Every A is B,' answers 'Every B is A.' That is, the necessary laws of thought, except in minds which have examined their tools, are not very sure to work correct conclusions except upon familiar matter.... As the cultivation of the individual increases, the laws of thought which are of most usual application are applied to familiar matter with tolerable safety. But difficulty and risk of error make a new appearance with a new subject; and this, in most cases, until new subjects are familiar things, unusual matter common, untried nomenclature habitual; that is, until it is a habit to be occupied upon a novelty. It is observed that many persons reason well in some things and badly in others; and this is attributed to the consequence of employing the mind too much upon one or another subject. But those who know the truth of the preceding remarks will not have far to seek for what is often, perhaps most often, the true reason.... I maintain that logic tends to make the power of reason over the unusual and unfamiliar more nearly equal to the power over the usual and familiar than it would otherwise be. The second is increased; but the first is almost created."

Mr. James Smith, by bringing ignorance, folly, dishonesty into contact with my name, in the way of conditional insinuation, has done me a good turn: he has given me right to a freedom of personal remark which I might have declined to take in the case of a person who is useful and respected in matters which he understands.

Tit for tat is logic all the world over. By the way, what has become of the rest of the maxim: we never hear it {122} now. When I was a boy, in some parts of the country at least, it ran thus:

"Tit for tat;
Butter for fat:
If you kill my dog,
I'll kill your cat."