Chapter 6

position; that the thing is really a _pillowry_; and that I am, like Perrette's pot of milk,

"Bien pose sur un coussinet."[383]

Joanna Southcott[384] never had a follower who believed in her with more humble piety than Mr. James Smith believes in himself. After all that has happened to him, he asks me with high confidence to "favor the writer with a proof" that I still continue of opinion that "the best of the argument is in my jokes, and the best of the joke is in his arguments." I will not so favor him. At the very outset I told him in plain English that he has the whiphand of all the reasoners in the world, and in plain French that _il a perdu le droit d'etre frappe de l'evidence_[385]; I might have said _pendu_.[386] To which I now add, in plain Latin, _Sapienti pauca, indocto nihil_.[387] The law of Chancery says that he who will have equity must do equity: the law of reasoning says that he who will have proof must see proof.

The introduction of things quite irrelevant, by way of reproach, is an argument in universal request: and it often happens that the argument so produced really tells against the producer. So common is it that we forget how boyish it is; but we are strikingly reminded when it actually comes from a boy. In a certain police court, certain small boys were arraigned for conspiring to hoot an obnoxious individual on his way from one of their school exhibitions. This proceeding was necessary, because there seemed to be a permanent conspiracy to annoy the gentleman; and the {240} masters did not feel able to interfere in what took place outside the school. So the boys were arraigned; and their friends, as silly in their way as themselves, allowed one of them to make the defence, instead of employing counsel; and did not even give them any useful hints. The defence was as follows; and any one who does not see how richly it sets off the defences of bigger boys in bigger matters has much to learn. The innocent conviction that there was answer in the latter part is delightful. Of course fine and recognizance followed.

A---- said the boys had received great provocation from B----. He was constantly threatening them with a horsewhip which he carried in his hand [the boy did not say what had passed to induce him to take such a weapon], and he had repeatedly insulted the master, which the boys could not stand. B---- had in his own drawing-room told him (A----) that he had drawn his sword against the master and thrown away the scabbard. B---- knew well that if he came to the college he would catch it, and then he went off through a side door--which was no sign of pluck; and then he brought Mrs. B---- with him, thinking that her presence would protect him.

My readers may expect a word on Mr. Thom's sermons, after my account of his queer doings about 666. He is evidently an honest and devout man, much wanting in discrimination. He has a sermon about private _judgment_, in which he halts between the logical and legal meanings of the word. He loathes those who apply their private judgment to the word of God: here he means those who decide what it _ought to be_. He seems in other places aware that the theological phrase means taking right to determine what it _is_. He uses his own private judgment very freely, and is strong in the conclusion that others ought not to use theirs except as he tells them how; he leaves all the rest of mankind free to think with him. In this he is not original: his fame must rest on his senary tripod. {241}

JAMES SMITH ONCE MORE.

Mr. James Smith's procedures are not caricature of reasoning; they are caricature of blundering. The old way of proving that 2 = 1 is solemn earnest compared with his demonstrations. As follows:[388]

Let x = 1 Then x^2 = x And x^2 - 1 = x - 1 Divide both sides by x - 1; then x + 1 = 1; but x = 1, whence 2 = 1.

When a man is regularly snubbed, bullied, blown up, walked into, and put down, there is usually some reaction in his favor, a kind of deostracism, which cannot bear to hear him always called the blunderer. I hope it will be so in this case. There is nothing I more desire than to see _sects_ of paradoxers. There are fully five thousand adults in England who ought to be the followers of some one false quadrature. And I have most hope of 3-1/8, because I think Mr. James Smith better fitted to be the leader of an organized infatuation than any one I know of. He wants no pity, and will get none. He has energy, means, good humor, strong conviction, character, and popularity in his own circle. And, most indispensable point of all, he sticks at nothing;

"In coelum jusseris, ibit."[389]

When my instructor found I did not print an acceptance of what I have quoted, he addressed me as follows (_Corr._, Sept 23):--

"In this life, however, we must do our duty, and, when {242} necessary, use the rod, not in a spirit of revenge, but for the benefit of the culprit and the good of society. Now, Sir, the opportunity has been thrown in your way of slipping out of the pillory without risk of serious injury; but, like an obstinate urchin, you have chosen to quarrel with your opportunity and remain there, and thus you compel me to deal with you as schoolmasters used to do with stupid boys in bygone days--that is to say, you force me to the use of the critic's rod, compel me to put you where little Jack Horner sat, and, as a warning to other naughty boys, to ornament you with a dunce's cap. The task I set you was a very simple one, as I shall make manifest at the proper time."

In one or more places, as well as this, Mr. Smith shows that he does not know the legend of little Jack Horner, whom he imagines to be put in the corner as a bad boy. This is curious; for there had been many allusions to the story in the journal he was writing in, and the Christmas pie had become altered into the Seaforth [pi].

Mr. Smith is satisfied at last that--what between argument and punishment he has convinced me. He says (_Corr._, Jan. 27, 1866): "I tell him without hesitation that he knows the true ratio of diameter to circumference as well as I do, and if he be wise he will admit it." I should hope I do, and better; but there is no occasion to admit what everybody knows.

I have often wished that we could have a slight glimpse of the reception which was given to some of the old cyclometers: but we have nothing, except the grave disapprobation of historians. I am resolved to give the New Zealander a chance of knowing a little more than this about one of them at least; and, by the fortunate entrance into life of the _Correspondent_, I am able to do it. I omit sober mathematical answers, of which there were several. The following letter is grave earnest:

"Sir,--I have watched Mr. James Smith's writings on this subject from the first, and I did hope that, as the more {243} he departs from truth the more easy it must be to refute him, [this by no means always true] some of your correspondents would by this time have done so. I own that I am unable to detect the fallacy of his argument; and I am quite certain that '[Pi]' is wrong, in No. 23, where he declares that Mr. Smith is 'ignorant of the very elements of mathematical truth.' I have observed an immense amount of geometrical reasoning on his part, and I cannot see that it is either fair or honest to deny this, which may be regarded as the 'elements' of mathematical truth. Would it not be better for '[Pi]' to answer Mr. Smith, to refute his arguments, to point out their fallacies, and to save learners from error, than to plunge into gross insult and unmanly abuse? Would it not be well, also, that Professor De Morgan should favour us with a little reasoning?

"I have hitherto seen no attempt to overthrow Mr. Smith's arguments; I trust that this will not continue, since the subject is one of immense importance to science in general, especially to nautical science, and all that thereto belongs.

Yours, etc.,

A CAPTAIN, R.N."

On looking at this homoeopathic treatment of the 3-1/8 quadrature--remember, homoeopathic, _similia similibus_,[390] not infinitesimal--and at the imputation thrown upon it, I asked myself, what _is_ vulgarity? No two agree, except in this, that every one sees vulgarity in what is directed against himself. Mark the world, and see if anything be so common as the description of the other side's remarks as "vulgar attempt at wit." "I suppose you think that very witty:" the answer is "No my friend! your remark shows that you feel it as wit, so that the purpose is answered; I keep my razor for something else than cutting blocks;" I am inclined to think that "out of place" is a necessary attribute of true vulgarity. And further, it is to be noticed that nothing is {244} unproducible--_salvo pudore_[391]--which has classical authority, modern or ancient, in its favor. "He is a vulgar fellow; I asked him what he was upon, and what do you think he answered, My legs!"--"Well, and has he not justification? what do you find in Terence? _Quid agitur? Statur._"[392] I do not even blench from my principle where I find that it brings what is called "taking a sight" within permissible forms of expression: Rabelais not only establishes its antiquity, but makes it English. Our old translation[393] has it thus (book 2. ch. 19):

"Then made the Englishman this sign. His left hand, all open, he lifted up into the air, then instantly shut into his fist the four fingers thereof; and his thumb extended at length he placed upon the tip of his nose. Presently after he lifted up his right hand all open and abased and bent it downwards, putting the thumb thereof in the very place where the little finger of the left hand did close in the fist, and the four right hand fingers he softly moved in the air. Then contrarily he did with the right hand what he had done with the left, and with the left what he had done with the right."

An impressive sight! The making of a fist of the left hand is a great addition of power, and should be followed in modern practice. The gentle sullation of the front fingers, with the clenched fist behind them, says as plainly as possible, Put _suaviter in modo_ in the van, but don't forget to have _fortiter in re_[394] in the rear.

{245}

My Budget was announced (March 23, 1867) for completion on the 30th. Mr. James Smith wrote five letters, one before the completion, four after it; the five contained 68 pages of quarto letter paper. Mr. J. S. had picked up a clerical correspondent, with whom he was in the heat of battle.

"_March 27._--Dear Sir. Very truly yours. Duty; for my own sake; just time left to retrieve my errors; sends copy of letter to clergyman; new proof never before thought of; merest tyro would laugh if I were to stifle it, whether by rhodomontade or silent contempt; keep your temper. I shall be convinced; and if world be right in supposing me incapable of a foul act, I shall proclaim glorious discovery in the _Athenaeum_.

"_April 15._--Sir,... My dear Sir, Your sincere tutelary. Copy of another letter to clergyman; discovery tested by logarithms; reasons such as none but a knave or a sinner can resist. Let me advise you to take counsel before it is too late! Keep your temper. Let not your _pride_ get the better of your discretion! Screw up your courage, my good friend, and _resolve_ to show the world that you are an _honest_ man....

"_April 20._--Sir ... Your very sincere and favorite tutelary. I have long played the _cur_, snapping and snarling...; suddenly lost my power, and became _half-starved_ dog without _spirit_ to bark; try if air cannot restore me; calls himself the _thistle_ in allusion to my other tutelary, the _thorn_; Would I prefer his next work to be, 'A whip for the Mathematical Cur, Prof. De M.' In some previous letter which I have mislaid, he told me his next would be 'a muzzle for the Mathematical Bull dog, Prof. De M.'

"_April 23._--Sir. Very sincerely yours. More letters to clergyman; you may as well knock your head against a stone wall to improve your intellect as attempt to controvert my proofs. [I thought so too; and tried neither]. {246}

"_May 6._--My dear Sir. Very sincerely yours. All to myself, and nothing to note.

"_July 2._--No more in this interval. All that precedes is a desperate attempt to induce me to continue my descriptions: notoriety at any price."

I dare say the matter is finished: the record of so marked an instance of self-delusion will be useful.

I append to the foregoing a letter from Dr. Whewell[395] to Mr. James Smith. The Master of Trinity was conspicuous as a rough customer, an intellectual bully, an overbearing disputant: the character was as well established as that of Sam Johnson. But there was a marked difference. It was said of Johnson that if his pistol missed fire, he would knock you down with the butt end of it: but Whewell, in like case, always acknowledged the miss, and loaded again or not, as the case might be. He reminded me of Dennis Brulgruddery, who says to Dan, Pacify me with a good reason, and you'll find me a dutiful master. I knew him from the time when he was my teacher at Cambridge, more than forty years. As a teacher, he was anything but dictatorial, and he was perfectly accessible to proposal of objections. He came in contact with me in his slashing way twice in our after joint lives, and on both occasions he acknowledged himself overcome, by that change of manner, and apologetic mode of continuance, which I had seen him employ towards others under like circumstances.

I had expressed my wish to have a _thermometer of probability_, with impossibility at one end, as 2 and 2 make 5, and necessity at the other, as 2 and 2 make 4, and a graduated rise of examples between them. Down came a blow: "What! put necessary and contingent propositions together! It's absurd!" I pointed out that the two kinds of necessity are but such extremes of probability as 0 and [infinity] are of number, and illustrated by an urn with 1 white and _n_ black {247} balls, _n_ increasing without limit. It was frankly seen, and the point yielded; a large company was present.

Again, in a large party, after dinner, and politics being the subject, I was proceeding, in discussion with Mr. Whewell, with "I think"...--"Ugh! _you_ think!" was the answer. I repeated my phrase, and gave as a reason the words which Lord Grey[396] had used in the House of Lords the night before (the celebrated advice to the Bishops to set their houses in order). He had not heard of this, and his manner changed in an instant: he was the rational discutient all the rest of the evening, having previously been nothing but a disputant with all the distinctions strongly marked.

I have said that Whewell was gentle with his pupils; it was the same with all who wanted teaching: it was only on an armed enemy that he drew his weapon. The letter which he wrote to Mr. J. Smith is an instance: and as it applies with perfect fidelity to the efforts of unreasoning above described, I give it here. Mr. James Smith is skilfully exposed, and felt it; as is proved by "putting the writer in the stocks."

"The Lodge, Cambridge, September 14th, 1862.

"Sir,--I have received your explanation of your proposition that the circumference of the circle is to its diameter as 25 to 8. I am afraid I shall disappoint you by saying that I see no force in your proof: and I should hope that you will see that there is no force in it if you consider this: In the whole course of the proof, though the word cycle occurs, there is no property of the circle employed. You may do this: you may put the word _hexagon_ or _dodecagon_, or any other word describing a polygon in the place of _Circle_ in your proof, and the proof would be just as good as before. Does not this satisfy you that you cannot have proved a property of that special figure--a circle? {248}

"Or you may do this: calculate the side of a polygon of 24 sides inscribed in a circle. I think you are a Mathematician enough to do this. You will find that if the radius of the circle be one, the side of this polygon is .264 etc. Now, the arc which this side subtends is according to your proposition 3.125/12 = .2604, and therefore the chord is greater than its arc, which you will allow is impossible.

"I shall be glad if these arguments satisfy you, and

"I am, Sir, your obedient Servant,

"W. WHEWELL."

AN M.P.'S ARITHMETIC.

In the debate of May, 1866, on Electoral Qualifications, a question arose about arithmetical capability. Mr. Gladstone asked how many members of the House could divide 1330l. 7s. 6d. by 2l. 13s. 8d. Six hundred and fifty-eight, answered one member; the thing cannot be done, answered another. There is an old paradox to which this relates: it arises out of the ignorance of the distinction between abstract and concrete arithmetic. _Magnitude_ may be divided by _magnitude_; and the answer is number: how often does 12d. contain 4d.; answer three times. _Magnitude_ may be divided by _number_, and the answer is _magnitude_: 12d. is divided in four equal parts, what is each part? Answer three _pence_. The honorable objector, whose name I suppress, trusting that he has mended his ways, gave the following utterance:

"With regard to the division sum, it was quite possible to divide by a sum, but not by money. How could any one divide money by 2l. 16s. 8d.? (Laughter.) The question might be asked, 'How many times 2s. will go into 1l.?' but that was not dividing by money; it was simply dividing 20 by 2. He might be asked, 'How many times will 6s. 8d. go into a pound?' but it was only required to divide 240 by 80. If the right hon. gentleman were to ask the hon. {249} member for Brighton (Professor Fawcett),[397] or any other authority, he would receive the same answer--viz., that it was possible to divide by a sum, but not by money. (Hear.)"

I shall leave all comment for the second edition, if I publish one.[398] I shall be sure to have something to laugh at. Anything said from a respectable quarter, or supposed to be said, is sure to find defenders. Sam Johnson, a sound arithmetician, comparing himself, and what he alone had done in three years, with forty French Academicians and their forty years, said it proved that an Englishman is to a Frenchman as 40 x 40 to 3, or as 1600 to 3. Boswell, who was no great hand at arithmetic, made him say that an Englishman is to a Frenchman as 3 to 1600. When I pointed this out, the supposed Johnson was defended through thick and thin in _Notes and Queries_.

I am now curious to see whether the following will find a palliator. It is from "Tristram Shandy," book V. chapter 3. There are two curious idioms, "for for" and "half in half"; but these have nothing to do with my point:

"A blessing which tied up my father's tongue, and a misfortune which set it loose with a good grace, were pretty equal: sometimes, indeed, the misfortune was the better of the two; for, for instance, where the pleasure of harangue was as _ten_, and the pain of the misfortune but as _five_, my father gained half in half; and consequently was as well again off as if it had never befallen him."

This is a jolly confusion of ideas; and wants nothing but a defender to make it perfect. A person who invests five {250} with a return of ten, and one who loses five with one hand and gains ten with the other, both leave off five richer than they began, no doubt. The first gains "half in half," more properly "half _on_ half," that is, of the return, 10, the second 5 is gain upon the first 5 invested. "Half _in_ half" is a queer way of saying cent. per cent. If the 5l. invested be all the man had in the world, he comes out, after the gain, twice as well off as he began, with reference to his whole fortune. But it is very odd to say that balance of 5l. gain is _twice_ as good as if nothing had befallen, either loss or gain. A mathematician thinks 5 an infinite number of times as great as 0. The whole confusion is not so apparent when money is in question: for money is money whether gained or lost. But though pleasure and pain stand to one another in the same algebraical relation as money gained and lost, yet there is more than algebra can take account of in the difference.

Next, Ri. Milward[399] (Richard, no doubt, but it cannot be proved) who published Selden's[400] Table Talk, which he had collected while serving as amanuensis, makes Selden say, "A subsidy was counted the fifth part of a man's estate; and so fifty subsidies is five and forty times more than a man is worth." For _times_ read _subsidies_, which seems part of the confusion, and there remains the making all the subsidies equal to the first, though the whole of which they are to be the fifths is perpetually diminished.

Thirdly, there is the confusion of the great misomath {251} of our own day, who discovered two quantities which he avers to be identically the same, but the greater the one the less the other. He had a truth in his mind, which his notions of quantity were inadequate to clothe in language. This erroneous phraseology has not found a defender; and I am almost inclined to say, with Falstaff, The poor abuses of the time want countenance.

ERRONEOUS ARITHMETICAL NOTIONS.

"Shallow numerists," as Cocker[401] is made to call them, have long been at work upon the question how to _multiply_ money by money. It is, I have observed, a very common way of amusing the tedium of a sea voyage: I have had more than one bet referred to me. Because an oblong of five inches by four inches contains 5 x 4 or 20 _square_ inches, people say that five inches multiplied by four inches _is_ twenty _square_ inches: and, thinking that they have multiplied length by length, they stare when they are told that money cannot be multiplied by money. One of my betters made it an argument for the thing being impossible, that there is no _square money_: what could I do but suggest that postage-stamps should be made legal tender. Multiplication must be _repetition_: the repeating process must be indicated by _number_ of times. I once had difficulty in persuading another of my betters that if you repeat five shillings as often as there are hairs in a horse's tail, you do not _multiply five shillings by a horsetail_.[402]

I am very sorry to say that these wrong notions have found support--I think they do so no longer--in the University of Cambridge. In 1856 or 1857, an examiner was displaced by a vote of the Senate. The pretext was that he was too severe an examiner: but it was well known that {252} great dissatisfaction had been expressed, far and wide through the Colleges, at an absurd question which he had given. He actually proposed such a fraction as

6s. 3d. --------. 17s. 4d.

As common sense gained a hearing very soon, there is no occasion to say more. In 1858, it was proposed at a college examination, to divide 22557 days, 20 hours, 20 minutes, 48 seconds, by 57 minutes, 12 seconds, and also to explain the fraction

32l. 18s. 8d. -------------. 62l. 12s. 9d.

All paradoxy, in matters of demonstration, arises out of muddle about first principles. Who can say how much of it is to be laid at the door of the University of Cambridge, for not taking care of the elements of arithmetical thought?

ON LITERARY BARGAINS.

The phenomena of the two ends of society, when brought together, give interesting comparisons: I mean the early beginnings of thought and literature, and our own high and finished state, as we think it. There is one very remarkable point. In the early day, the letter was matter of the closest adherence, and implied meanings were not admitted.

The blessing of Isaac meant for Esau, went to false Jacob, in spite of the imposition; and the writer of Genesis seems to intend to give the notion that Isaac had no power to pronounce it null and void. And "Jacob's policy, whereby he became rich"--as the chapter-heading puts it--in speckled and spotted stock, is not considered as a violation of the agreement, which contemplated natural proportions. In {253} the story of Lycurgus the lawgiver is held to have behaved fairly when he bound the Spartans to obey his laws until he returned--intimating a short absence--he intending never to return. And Vishnoo, when he asked the usurper for three steps of territory as a dwarf, and then enlarged himself until he could bring heaven and earth under the bargain, was thought clever, certainly, but quite fair.

There is nothing of this kind recognized in our day: so far good. But there is a bad contrary: the age is apt, in interpretation, to upset the letter in favor of the view--very often the after thought--of one side only. The case of John Palmer,[403] the improver of the mail coach system, is smothered. He was to have an office and a salary, and 2-1/2 per cent for life on the increased _revenue_ of the Post-Office. His rights turned out so large, that Government would not pay them. For misconduct, real or pretended, they turned him out of his _office_: but his bargain as to the percentage had nothing to do with his future conduct; it was payment for his _plan_. I know nothing, except from the debates of 1808 in the two Houses: if any one can redeem the credit of the nation, the field is open. When I was young, the old stagers spoke of this transaction sparingly, and dismissed it speedily.

The government did not choose to remember what private persons must remember, and are made to remember, if needful. When Dr. Lardner[404] made his bargain with the {254} publishers for the _Cabinet Cyclopaedia_ he proposed that he, as editor, should have a certain sum for every hundred sold above a certain number: the publishers, who did not think there was any chance of reaching the turning sale of this stipulation, readily consented. But it turned out that Dr. Lardner saw further than they: the returns under this stipulation gave him a very handsome addition to his other receipts. The publishers stared; but they paid. They had no idea of standing out that the amount was too much for an editor; they knew that, though the editor had a percentage, they had all the rest; and they would not have felt aggrieved if he had received ten times as much. But governments, which cannot be brought to book before a sworn jury, are ruled only by public opinion. John Palmer's day was also the day of Thomas Fyshe Palmer,[405] and the governments, in their prosecutions for sedition, knew that these would have a reflex action upon the minds of all who wrote about public affairs.

DECLARATION OF BELIEF

1864-65.--It often happens that persons combine to maintain and enforce an opinion; but it is, in our state of society, a paradox to unite for the sole purpose of blaming the opposite side. To invite educated men to do this, and above all, men of learning or science, is the next paradoxical thing of all. But this was done by a small combination in 1864. They got together and drew up a _declaration_, to be signed by "students of the natural sciences," who were to express their "sincere regret that researches into {255} scientific truth are perverted by some in our own times into occasion for casting doubt upon the truth and authenticity of the Holy Scriptures." In words of ambiguous sophistry, they proceeded to request, in effect, that people would be pleased to adopt the views of churches as to the _complete_ inspiration of all the canonical books. The great question whether the Word of God is _in_ the Bible, or whether the Word of God is _all_ the Bible, was quietly taken for granted in favor of the second view; to the end that men of science might be induced to blame those who took the first view. The first public attention was drawn to the subject by Sir John Herschel,[406] who in refusing to sign the writ sent to him, administered a rebuke in the _Athenaeum_, which would have opened most eyes to see that the case was hopeless. The words of a man whose _suaviter in modo_ makes his _fortiter in re_[407] cut blocks with a razor are worth preserving:

"I consider the act of calling upon me publicly to avow or disavow, to approve or disapprove, in writing, any religious doctrine or statement, however carefully or cautiously drawn up (in other words, to append my name to a religious manifesto) to be an infringement of that social forbearance which guards the freedom of religious opinion in this country with especial sanctity.... I consider this movement simply mischievous, having a direct tendency (by putting forward a new Shibboleth, a new verbal test of religious partisanship) to add a fresh element of discord to the already too discordant relations of the Christian world.... But no nicety of wording, no artifice of human language, will suffice to discriminate the hundredth part of the shades of meaning in which the most world-wide differences of thought on such subjects may be involved; or prevent the most gentle worded and apparently justifiable expression of regret, so embodied, from grating on the {256} feelings of thousands of estimable and well-intentioned men with all the harshness of controversial hostility."

Other doses were administered by Sir J. Bowring,[408] Sir W. Rowan Hamilton,[409] and myself. The signed declaration was promised for Christmas, 1864: but nothing presentable was then ready; and it was near Midsummer, 1865, before it was published. Persons often incautiously put their names without seeing the _character_ of a document, because they coincide in its _opinions_. In this way, probably, fifteen respectable names were procured before printing; and these, when committed, were hawked as part of an application to "solicit the favor" of other signatures. It is likely enough no one of the fifteen saw that the declaration was, not _maintenance_ of their own opinion, but _regret_ (a civil word for _blame_) that others should _think differently_.

When the list appeared, there were no fewer than 716 names! But analysis showed that this roll was not a specimen of the mature science of the country. The collection was very miscellaneous: 38 were designated as "students of the College of Chemistry," meaning young men who attended lectures in that college. But as all the Royal Society had been applied to, a test results as follows. Of Fellows of the Royal Society, 600 in number, 62 gave their signatures; of writers in the _Philosophical Transactions_, 166 in number, 19 gave their signatures. Roughly speaking, then, only one out of ten could be got to express disapprobation of the free comparison of the results of science with the statements of the canonical books. And I am satisfied that many of these thought they were signing only a declaration of difference of opinion, not of blame for that difference. The number of persons is not small who, when it comes to signing printed documents, would put their names to a declaration that the coffee-pot ought to be taken down-stairs, meaning that the teapot ought to be brought {257} up-stairs. And many of them would defend it. Some would say that the two things are not contradictory; which, with a snort or two of contempt, would be very effective. Others would, in the candid and quiet tone, point out that it is all one, because coffee is usually taken before tea, and it keeps the table clear to send away the coffee-pot before the teapot is brought up.

The original signatures were decently interred in the Bodleian Library: and the advocates of scattering indefinite blame for indefinite sins of opinion among indefinite persons are, I understand, divided in opinion about the time at which the next attempt shall be made upon men of scientific studies: some are for the Greek Calends, and others for the Roman Olympiads. But, with their usual love of indefiniteness, they have determined that the choice shall be argued upon the basis that which comes first cannot be settled, and is of no consequence.

I give the declaration entire, as a curiosity: and parallel with it I give a substitute which was proposed in the _Athenaeum_, as worthy to be signed both by students of theology, and by students of science, especially in past time. When a new attempt is made, it will be worth while to look at both:

_Declaration._ _Proposed Substitute._

We, the undersigned Students We, the undersigned Students of the Natural Sciences, of Theology and of Nature, desire to express our sincere desire to express our sincere regret, that researches into regret, that common notions of scientific truth are perverted religious truth are perverted by some in our own times into by some in our own times into occasion for casting doubt occasion for casting reproach upon the Truth and upon the advocates of Authenticity of the Holy demonstrated or highly Scriptures. probable scientific theories.

{258} We conceive that it is We conceive that it is impossible for the Word of impossible for the Word of God, as written in the book of God, as correctly read in the nature, and God's Word written Book of Nature, and the Word in Holy Scripture, to of God, as truly interpreted contradict one another, out of the Holy Scripture, to however much they may appear contradict one another, to differ. however much they may appear to differ. We are not forgetful that We are not forgetful that Physical Science is not neither theological complete, but is only in a interpretation nor physical condition of progress, and knowledge is yet complete, but that at present our finite that both are in a condition reason enables us only to see of progress; and that at as through a glass darkly, present our finite reason enables us only to see both one and the other as through a glass darkly [the writers of the original declaration have distinctively applied to physical science the phrase by which St. Paul denotes the imperfections of theological vision, which they tacitly assume to be quite perfect], and we confidently believe, and we confidently believe, that a time will come when the that a time will come when the two records will be seen to two records will be seen to agree in every particular. We agree in every particular. We cannot but deplore that cannot but deplore that Natural Science should be Religion should be looked upon looked upon with suspicion by with suspicion by some and many who do not make a study Science by others, of the of it, merely on account of students of either who do not the unadvised manner in which make a study of the {259} some are placing it in other, merely on account of opposition to Holy Writ. the unadvised manner in which some are placing Religion in opposition to Science, and some are placing Science in opposition to Religion. We believe that it is the duty We believe that it is the duty of every Scientific Student to of every theological student investigate nature simply for to investigate the Scripture, the purpose of elucidating and of every scientific truth, student to investigate Nature, simply for the purpose of elucidating truth. and that if he finds that some And if either should find that of his results appear to be in some of his results appear to contradiction to the Written be in contradiction, whether Word, or rather to his own to Scripture or to Nature, or _interpretations_ of it, which rather to his own may be erroneous, he should _interpretation_ of one or the not presumptuously affirm that other, which may be erroneous, his own conclusions must be he should not affirm as with right, and the statements of certainty that his own Scripture wrong; conclusion must be right, and the other interpretation wrong: rather, leave the two side by but should leave the two side side till it shall please God by side for further inquiry to allow us to see the manner into both, until it shall in which they may be please God to allow us to reconciled; arrive at the manner in which they may be reconciled. and, instead of insisting upon In the mean while, instead of the seeming differences insisting, and least of all between Science and the with acrimony or injurious Scriptures, it would be as {260} statements about others, well to rest in faith upon the upon the seeming differences points in which they agree. between Science and the Scriptures, it would be a thousand times better to rest in faith as to our future state, in hope as to our coming knowledge, and in charity as to our present differences.

The distinctness of the fallacies is creditable to the composers, and shows that scientific habits tend to clearness, even to sophistry. Nowhere does it so plainly stand out that the _Written Word_ means the sense in which the accuser takes it, while the sense of the other side is _their interpretation_. The infallible church on one side, arrayed against heretical pravity on the other, is seen in all subjects in which men differ. At school there were various games in which one or another advantage was the right of those who first called for it. In adult argument the same thing is often attempted: we often hear--I cried _Church_ first!

I end with the answer which I myself gave to the application: its revival may possibly save me from a repetition of the like. If there be anything I hate more than another it is the proposal to place any persons, especially those who allow freedom to me, under any abridgment of their liberty to think, to infer, and to publish. If they break the law, take the law; but do not make the law: [Greek: agoraioi agontai enkaleitosan allelois.][410] I would rather be asked to take shares in an argyrosteretic company (with limited liability) for breaking into houses by night on fork and spoon errands. I should put aside this proposal with _nothing but laughter_. It was a joke against Sam Rogers[411] that his appearance was very like that of a corpse. The _John Bull_ {261} newspaper--suppose we now say Theodore Hook[412]--averred that when he hailed a coach one night in St. Paul's Churchyard, the jarvey said, "Ho! ho! my man; I'm not going to be taken in that way: go back to your grave!" This is the answer I shall make for the future to any relics of a former time who shall want to call me off the stand for their own purposes. What obligation have I to admit that they belong to our world?

"SCRIPTURE AND SCIENCE.

"_The Writ De Haeretico Commiserando._[413]

Nov. 14, 1864.

"This document was sent to me four days ago. It 'solicits the favor'--I thought at first it was a grocer's supplication for tea and sugar patronage--of my signature to expression of 'sincere regret' that some persons unnamed--general warrants are illegal--differ from what I am supposed--by persons whom it does not concern--to hold about Scripture and Science in their real or alleged discrepancies.

"No such favor from me: for three reasons. First, I agree with Sir. J. Herschel that the solicitation is an intrusion to be publicly repelled. Secondly, I do _not_ regret that others should differ from me, think what I may: those others are as good as I, and as well able to think, and as much entitled to their conclusions. Thirdly, even if I did regret, I should be ashamed to put my name to bad chemistry made to do duty for good reasoning. The declaration is an awkward attempt to saturate sophism with truism; but the sophism is left largely in excess.

{262}

"I owe the inquisitors a grudge for taking down my conceit of myself. For two months I have crowed in my own mind over my friend Sir J. Herschel, fancying that the promoters instinctively knew better than to bring their fallacies before a writer on logic. Ah! my dear Sir John! thought I, if you had shown yourself to be well up in _Barbara Celarent_,[414] and had ever and anon astonished the natives with the distinction between _simpliciter_ and _secundum quid_, no autograph-hunters would have baited a trap with _non sequitur_[415] to catch your signature. What can I say now? I hide my diminished head, diminished by the horns which I have been compelled to draw in.

"Those who make personal solicitation for support to an opinion about religion are bound to know their men. The king had a right to Brother Neale's money, because Brother Neale offered it. Had he put his hand into purse after purse by way of finding out all who were of Brother Neale's mind, he would have been justly met by a rap on the knuckles whenever he missed his mark.

"The kind of test before me is the utmost our time will allow of that inquisition into opinion which has been the curse of Christianity ever since the State took Providence under its protection. The writ _de haeretico commiserando_ is little more than the smell of the empty cask: and those who issue it may represent the old woman with her

"O suavis anima, quale in te dicam bonum Antehac fuisse; tales cum sint reliquiae."[416]

It is no excuse that the illegitimate bantling is a very little one. Its parents may think themselves hardly treated when they are called lineal successors of Tony Fire-the-faggot: {263} but, degenerate though they be, such is their ancestry. Let every allowance be made for them: but their unholy fire must be trodden out; so long as a spark is left, nothing but fuel is wanted to make a blaze. If this cannot be done, let the flame be confined to theology, though even there it burns with diminished vigor: and let charity, candor, sense, and ridicule, be ready to play upon it whenever there is any chance of its extending to literature and science.

"What would be the consequence if this test-signing absurdity were to grow? Deep would call unto deep; counter-declaration would answer declaration, each stronger than the one before. The moves would go on like the dispute of two German students, of whom each is bound to a sharper retort on a graduated scale, until at last comes _dummer Junge_![417]--and then they must fight. There is a gentleman in the upper fifteen of the signers of the writ--the hawking of whose names appears to me very bad taste--whom I met in cordial cooperation for many a year at a scientific board. All I knew about his religion was that he, as a clergyman, must in some sense or other receive the 39 Articles:--all that he could know about mine was that I was some kind of heretic, or so reputed. If we had come to signing opposite manifestoes, turn-about, we might have found ourselves in the lowest depths of party discussion at our very council-table. I trust the list of subscribers to the declaration, when it comes to be published, will show that the bulk of those who have really added to our knowledge have seen the thing in its true light.

"The promoters--I say nothing about the subscribers--of the movement will, I trust, not feel aggrieved at the course I have taken or the remarks I have made. Walter Scott says that before we judge Napoleon by the temptation to which he yielded, we ought to remember how much he may have resisted: I invite them to apply this rule to myself; they can have no idea of the feeling with which I {264} contemplate all attempts to repress freedom of inquiry, nor of the loathing with which I recoil from the proposal to be art and part. They have asked me to give a public opinion upon a certain point. It is true that they have had the kindness to tender both the opinion they wish me to form, and the shape in which they would have it appear: I will let them draw me out, but I will not let them take me in. If they will put an asterisk to my name, and this letter to the asterisk, they are welcome to my signature. As I do not expect them to relish this proposal, I will not solicit the favor of its adoption. But they have given a right to think, for they have asked me to think; to publish, for they have asked me to allow them to publish; to blame them, for they have asked me to blame their betters. Should they venture to find fault because my direction of disapproval, publicly given, is half a revolution different from theirs, they will be known as having presented a loaded document at the head of a traveler in the highway of discussion, with--Your signature or your silence!"

THE FLY-LEAF PARADOX.

The paradox being the proposition of something which runs counter to what would generally be thought likely, may present itself in many ways. There is a _fly-leaf paradox_, which puzzled me for many years, until I found a probable solution. I frequently saw, in the blank leaves of old books, learned books, Bibles of a time when a Bible was very costly, etc., the name of an owner who, by the handwriting and spelling, must have been an illiterate person or a child, followed by the date of the book itself. Accordingly, this uneducated person or young child seemed to be the first owner, which in many cases was not credible. Looking one day at a Barker's[418] Bible of 1599, I saw an {265} inscription in a child's writing, which certainly belonged to a much later date. It was "Martha Taylor, her book, giuen me by Granny Scott to keep for her sake." With this the usual verses, followed by 1599, the date of the book. But it so chanced that the blank page opposite the title, on which the above was written, was a verso of the last leaf of a prayer book, which had been bound before the Bible; and on the recto of this leaf was a colophon, with the date 1632. It struck me immediately that uneducated persons and children, having seen dates written under names, and not being quite up in chronology, did frequently finish off with the date of the book, which stared them in the face.

Always write in your books. You may be a silly person--for though your reading my book is rather a contrary presumption, yet it is not conclusive--and your observations may be silly or irrelevant, but you cannot tell what use they may be of long after you are gone where Budgeteers cease from troubling.

I picked up the following book, printed by J. Franklin[419] at Boston, during the period in which his younger brother Benjamin was his apprentice. And as Benjamin was apprenticed very early, and is recorded as having learned the mechanical art very rapidly, there is some presumption that part of it may be his work, though he was but thirteen at the time. As this set of editions of Hodder[420] (by {266} Mose[421]) is not mentioned, to my knowledge, I give the title in full:

"Hodder's Arithmetick: or that necessary art made most easy: Being explained in a way familiar to the capacity of any that desire to learn it in a little time. By James Hodder, Writing-master. The Five and twentieth edition, revised, augmented, and above a thousand faults amended, by Henry Mose, late servant and successor to the author. Boston: printed by J. Franklin, for S. Phillips, N. Buttolph, B. Elliot, D. Henchman, G. Phillips, J. Elliot, and E. Negus, booksellers in Boston, and sold at their shops. 1719."

The book is a very small octavo, the type and execution are creditable, the woodcut at the beginning is clumsy. It is a perfect copy, page for page, of the English editions of Mose's Hodder, of which the one called seventeenth is of London, 1690. There is not a syllable to show that the edition above described might not be of Boston in England. Presumptions, but not very strong ones, might be derived from the name of _Franklin_, and from the large number of booksellers who combined in the undertaking. It chanced, however, that a former owner had made the following note in my copy:

"Wednessday, July y^e 14, 1796, att ten in y^e forenoon we sail^d from Boston, came too twice, once in King Rode, and once in y^e Narrows. Sail^d by y^e lighthouse in y^e even^g."

{267}

No ordinary map would decide these points: so I had to apply to my friend Sir Francis Beaufort,[422] and the charts at the Admiralty decided immediately for Massachusetts.

PARADOXES OF ORTHOGRAPHY AND COMPUTATION.

The French are able paradoxers in their spelling of foreign names. The Abbe Sabatier de Castres,[423] in 1772, gives an account of an imaginary dialogue between Swif, Adisson, Otwai, and Bolingbrocke. I had hoped that this was a thing of former days, like the literal roasting of heretics; but the charity which hopeth all things must hope for disappointments. Looking at a recent work on the history of the popes, I found referred to, in the matter of Urban VIII[424] and Galileo, references to the works of two Englishmen, the Rev. Win Worewel and the Rev. Raden Powen. [Wm. Whewell and Baden Powell].[425]

I must not forget the "moderate computation" paradox. This is the way by which large figures are usually obtained. Anything surprisingly great is got by the "lowest computation," anything as surprisingly small by the "utmost computation"; and these are the two great subdivisions of "moderate computation." In this way we learn that 70,000 persons were executed in one reign, and 150,000 persons {268} burned for witchcraft in one century. Sometimes this computation is very close. By a card before me it appears that all the Christians, including those dispersed in heathen countries, those of Great Britain and Ireland excepted, are 198,728,000 people, and pay their clergy 8,852,000l. But 6,400,000 people pay the clergy of the Anglo-Irish Establishment 8,896,000l.; and 14,600,000 of other denominations pay 1,024,000l. When I read moderate computations, I always think of Voltaire and the "memoires du fameux eveque de Chiapa, par lesquels il parait qu'il avait egorge, ou brule, ou noye dix millions d'infideles en Amerique pour les convertir. Je crus que cet eveque exaggerait; mais quand on reduisait ces sacrifices a cinq millions de victimes, cela serait encore admirable."[426]

CENTRIFUGAL FORCE.

My Budget has been arranged by authors. This is the only plan, for much of the remark is personal: the peculiarities of the paradoxer are a large part of the interest of the paradox. As to subject-matter, there are points which stand strongly out; the quadrature of the circle, for instance. But there are others which cannot be drawn out so as to be conspicuous in a review of writers: as one instance, I may take the _centrifugal force_.

When I was about nine years old I was taken to hear a course of lectures, given by an itinerant lecturer in a country town, to get as much as I could of the second half of a good, sound, philosophical omniscience. The first half (and sometimes more) comes by nature. To this end I smelt chemicals, learned that they were different kinds of _gin_, saw young wags try to kiss the girls under the excuse of what was called _laughing gas_--which I was sure {269} was not to blame for more than five per cent of the requisite assurance--and so forth. This was all well so far as it went; but there was also the excessive notion of creative power exhibited in the millions of miles of the solar system, of which power I wondered they did not give a still grander idea by expressing the distances in inches. But even this was nothing to the ingenious contrivance of the centrifugal force. "You have heard what I have said of the wonderful centripetal force, by which Divine Wisdom has retained the planets in their orbits round the Sun. But, ladies and gentlemen, it must be clear to you that if there were no other force in action, this centripetal force would draw our earth and the other planets into the Sun, and universal ruin would ensue. To prevent such a catastrophe, the same wisdom has implanted a centrifugal force of the same amount, and directly opposite," etc. I had never heard of Alfonso X of Castile,[427] but I ventured to think that if Divine Wisdom had just let the planets alone it would come to the same thing, with equal and opposite troubles saved. The paradoxers deal largely in speculation conducted upon the above explanation. They provide external agents for what they call the centrifugal force. Some make the sun's rays keep the planets off, without a thought about what would become of our poor eyes if the _push_ of the light which falls on the earth were a counterpoise to all its gravitation. The true explanation cannot be given here, for want of room.

CAMBRIDGE POETS.

Sometimes a person who has a point to carry will assert a singular fact or prediction for the sake of his point; and {270} this paradox has almost obtained the sole use of the name. Persons who have reputation to care for should beware how they adopt this plan, which now and then eventuates a spanker, as the American editor said. Lord Byron, in "English Bards, etc." (1809), ridiculing Cambridge poetry, wrote as follows:

"But where fair Isis rolls her purer wave, The partial muse delighted loves to lave; On her green banks a greener wreath she wove, To crown the bards that haunt her classic grove; Where Richards[428] wakes a genuine poet's fires, And modern Britons glory in their sires."[429]

There is some account of the Rev. Geo. Richards, Fellow of Oriel and Vicar of Bampton, (M.A. in 1791) in the _Living Authors_ by Watkins[430] and Shoberl[431] (1816). In Rivers's _Living Authors_, of 1798, which is best fitted for citation, as being published before Lord Byron wrote, he is spoken of in high terms. The _Aboriginal Britons_ was an Oxford (special) prize poem, of 1791. Charles Lamb mentions Richards as his school-fellow at Christ's Hospital, "author of the _Aboriginal Britons_, the most spirited of the Oxford Prize Poems: a pale, studious Grecian."

As I never heard of Richards as a poet,[432] I conclude that his fame is defunct, except in what may prove to be a very ambiguous kind of immortality, conferred by Lord Byron. The awkwardness of a case which time has broken down {271} is increased by the eulogist himself adding so powerful a name to the list of Cambridge poets, that his college has placed his statue in the library, more conspicuously than that of Newton in the chapel; and this although the greatness of poetic fame had some serious drawbacks in the moral character of some of his writings. And it will be found on inquiry that Byron, to get his instance against Cambridge, had to go back eighteen years, passing over seven intermediate productions, of which he had either never heard, or which he would not cite as waking a genuine poet's fires.

The conclusion seems to be that the _Aboriginal Britons_ is a remarkable youthful production, not equalled by subsequent efforts.

To enhance the position in which the satirist placed himself, two things should be remembered. First, the glowing and justifiable terms in which Byron had spoken,--a hundred and odd lines before he found it convenient to say no Cambridge poet could compare with Richards,--of a Cambridge poet who died only three years before Byron wrote, and produced greatly admired works while actually studying in the University. The fame of Kirke White[433] still lives; and future literary critics may perhaps compare his writings and those of Richards, simply by reason of the curious relation in which they are here placed alongside of each other. And it is much to Byron's credit that, in speaking of the deceased Cambridge poet, he forgot his own argument and its exigencies, and proved himself only a paradoxer _pro re nata_.

Secondly, Byron was very unfortunate in another passage of the same poem:

{272}

"What varied wonders tempt us as they pass! The cow-pox, tractors, galvanism, and gas. In turns appear, to make the vulgar stare, Till the swoln bubble bursts--and all is air!"

Three of the bubbles have burst to mighty ends. The metallic tractors are disused; but the force which, if anything, they put in action, is at this day, under the name of mesmerism, used, prohibited, respected, scorned, assailed, defended, asserted, denied, declared utterly obscure, and universally known. It was hard lines to select for candidates for oblivion not one of whom got in. I shall myself, I am assured, be some day cited for laughing at the great discovery of ----: the blank is left for my reader to fill up in his own way; but I think I shall not be so unlucky in four different ways.

FALSIFIED PREDICTION.

The narration before the fact, as prophecy has been called, sometimes quite as true as the narration after the fact, is very ridiculous when it is wrong. Why, the pre-narrator could not know; the post-narrator might have known. A good collection of unlucky predictions might be made: I hardly know one so fit to go with Byron's as that of the Rev. Daniel Rivers, already quoted, about Johnson's biographers. Peter Pindar[434] may be excused, as personal satire was his object, for addressing Boswell and Mrs. Piozzi[435] as follows:

"Instead of adding splendor to his name, Your books are downright gibbets to his fame; You never with posterity can thrive, 'Tis by the Rambler's death alone you live."

But Rivers, in prose narrative, was not so excusable. He says:

{273}

"As admirers of the learning and moral excellence of their hero, we glow at almost every page with indignation that his weaknesses and his failings should be disclosed to public view.... Johnson, after the luster he had reflected on the name of Thrale ... was to have his memory tortured and abused by her detested itch for scribbling. More injury, we will venture to affirm, has been done to the fame of Johnson by this Lady and her late biographical helpmate, than his most avowed enemies have been able to effect: and if his character becomes unpopular with some of his successors, it is to those gossiping friends he is indebted for the favor."

Poor dear old Sam! the best known dead man alive! clever, good-hearted, logical, ugly bear! Where would he have been if it had not been for Boswell and Thrale, and their imitators? What would biography have been if Boswell had not shown how to write a life?

Rivers is to be commended for not throwing a single Stone at Mrs. Thrale's second marriage. This poor lady begins to receive a little justice. The literary world seems to have found out that a blue-stocking dame who keeps open house for a set among them has a right, if it so please her, to marry again without taking measures to carry on the cake-shop. I was before my age in this respect: as a boy-reader of Boswell, and a few other things that fell in my way, I came to a clearness that the conduct of society towards Mrs. Piozzi was _blackguard_. She wanted nothing but what was in that day a woman's only efficient protection, a male relation with a brace of pistols, and a competent notion of using them.

BYRON AND WORDSWORTH.

Byron's mistake about Hallam in the Pindar story may be worth placing among absurdities. For elucidation, suppose that some poet were now to speak-- {274}

"Of man's first disobedience, and the fruit Eve gave to Adam in his birthday suit--"

and some critic were to call it nonsense, would that critic be laughing at Milton? Payne Knight,[436] in his _Taste_, translated part of Gray's _Bard_ into Greek. Some of his lines are

[Greek: therma d' ho tengon dakrua stonachais] [Greek: oulon melos phoberai] [Greek: eeide phonai.]

Literally thus:

"Wetting warm tears with groans, Continuous chant with fearful Voice he sang."

On which Hallam remarks: "The twelfth line [our first] is nonsense." And so it is, a poet can no more wet his tears with his groans than wet his ale with his whistle. Now this first line is from Pindar, but is only part of the sense; in full it is:

[Greek: therma de tengon dakrua stonachais] [Greek: horthion phonase.]

Pindar's [Greek: tengon] must be Englished by _shedding_, and he stands alone in this use. He says, "shedding warm tears, he cried out loud, with groans." Byron speaks of

"Classic Hallam, much renowned for Greek:"

and represents him as criticising _the Greek_ of all Payne's lines, and not discovering that "the lines" were Pindar's {275} until after publication. Byron was too much of a scholar to make this blunder himself: he either accepted the facts from report, or else took satirical licence. And why not? If you want to laugh at a person, and he will not give occasion, whose fault is it that you are obliged to make it? Hallam did criticise some of Payne Knight's Greek; but with the caution of his character, he remarked that possibly some of these queer phrases might be "critic-traps" justified by some one use of some one author. I remember well having a Latin essay to write at Cambridge, in which I took care to insert a few monstrous and unusual idioms from Cicero: a person with a Nizolius,[437] and without scruples may get scores of them. So when my tutor raised his voice against these oddities, I was up to him, for I came down upon him with Cicero, chapter and verse, and got round him. And so my own solecisms, many of them, passed unchallenged.

Byron had more good in his nature than he was fond of letting out: whether he was a soured misanthrope, or whether his _vein_ lay that way in poetry, and he felt it necessary to fit his demeanor to it, are matters far beyond me. Mr. Crabb Robinson[438] told me the following story more than once. He was at Charles Lamb's chambers in the Temple when Wordsworth came in, with the new _Edinburgh Review_ in his hand, and fume on his countenance. "These reviewers," said he, "put me out of patience! Here is a young man--they say he is a lord--who has written a volume of poetry; and these fellows, just because he is a lord, set upon him, laugh at him, and sneer at his writing. The young man will do something, if he goes on as he has begun. But these reviewers seem to think {276} that nobody may write poetry, unless he lives in a garret." Crabb Robinson told this long after to Lady Byron, who said, "Ah! if Byron had known that, he would never have attacked Wordsworth. He went one day to meet Wordsworth at dinner; when he came home I said, 'Well, how did the young poet get on with the old one?' 'Why, to tell you the truth,' said he, 'I had but one feeling from the beginning of the visit to the end, and that was--_reverence_!'" Lady Byron told my wife that her husband had a very great respect for Wordsworth. I suppose he would have said--as the Archangel said to his Satan--"Our difference is po[li = e]tical."

I suspect that Fielding would, if all were known, be ranked among the unlucky railers at supposed paradox. In his _Miscellanies_ (1742, 8vo) he wrote a satire on the Chrysippus or Guinea, an animal which multiplies itself by division, like the polypus. This he supposes to have been drawn up by Petrus Gualterus, meaning the famous usurer, Peter Walter. He calls it a paper "proper to be read before the R----l Society": and next year, 1743, a quarto reprint was made to resemble a paper in the _Philosophical Transactions_. So far as I can make out, one object is ridicule of what the zoologists said about the polypus: a reprint in the form of the _Transactions_ was certainly satire on the Society, not on Peter Walter and his knack of multiplying guineas.

Old poets have recognized the quadrature of the circle as a well-known difficulty. Dante compares himself, when bewildered, to a geometer who cannot find the principle on which the circle is to be measured:

"Quale e 'l geometra che tutto s' affige Per misurar lo cerchio, e non ritruova, Pensando qual principio ond' egli indige."[439]

{277} And Quarles[440] speaks as follows of the _summum bonum_:

"Or is't a tart idea, to procure An edge, and keep the practic soul in ure, Like that dear chymic dust, or puzzling quadrature?"

The poetic notion of the quadrature must not be forgotten. Aristophanes, in the _Birds_, introduces a geometer who announces his intention to _make a square circle_. Pope, in the _Dunciad_, delivers himself as follows, with a Greek pronunciation rather strange in a translator of Homer. Probably Pope recognized, as a general rule, the very common practice of throwing back the accent in defiance of quantity, seen in o'rator, au'ditor, se'nator, ca'tenary, etc.

"Mad _Mathesis_ alone was unconfined, Too mad for mere material chains to bind,-- Now to pure space lifts her ecstatic stare, Now, running round the circle, finds it square."

The author's note explains that this "regards the wild and fruitless attempts of squaring the circle." The poetic idea seems to be that the geometers try to make a square circle. Disraeli quotes it as "finds _its_ square," but the originals do not support this reading.

DE BECOURT.

I have come in the way of a work, entitled _The Grave of Human Philosophies_ (1827), translated from the French of R. de Becourt[441] by A. Dalmas. It supports, but I suspect not very accurately, the views of the old Hindu books. {278} That the sun is only 450 miles from us, and only 40 miles in diameter, may be passed over; my affair is with the state of mind into which persons of M. Becourt's temperament are brought by a fancy. He fully grants, as certain, four millions of years as the duration of the Hindu race, and 1956 as that of the universe. It must be admitted he is not wholly wrong in saying that our errors about the universe proceed from our ignorance of its origin, antiquity, organization, laws, and final destination. Living in an age of light, he "avails himself of that opportunity" to remove this veil of darkness, etc. The system of the Brahmins is the only true one: he adds that it has never before been attempted, as it could not be obtained except by him. The author requests us first, to lay aside prejudice; next, to read all he says in the order in which he says it: we may then pronounce judgment upon a work which begins by taking the Brahmins for granted. All the paradoxers make the same requests. They do not see that compliance would bring thousands of systems before the world every year: we have scores as it is. How is a poor candid inquirer to choose. Fortunately, the mind has its grand jury as well as its little one: and it will not put a book upon its trial without a _prima facie_ case in its favor. And with most of those who really search for themselves, that case is never made out without evidence of knowledge, standing out clear and strong, in the book to be examined.

BEQUEST OF A QUADRATURE.

There is much private history which will never come to light, _caret quia vate sacro_,[442] because no Budgeteer comes across it. Many years ago a man of business, whose life was passed in banking, amused his leisure with quadrature, was successful of course, and bequeathed the result in a sealed book, which the legatee was enjoined not to sell {279} under a thousand pounds. The true ratio was 3.1416: I have the anecdote from the legatee's executor, who opened the book. That a banker should square the circle is very credible: but how could a City man come by the notion that a thousand pounds could be got for it? A friend of mine, one of the twins of my zodiac, will spend a thousand pounds, if he have not done it already, in black and white cyclometry: but I will answer for it that he, a man of sound business notions, never entertained the idea of [pi] recouping him, as they now say. I speak of individual success: of course if a company were formed, especially if it were of unlimited lie-ability, the shares would be taken. No offence; there is nothing but what a pun will either sanctify, justify, or nullify:

"It comes o'er the soul like the sweet South That breathes upon a bank of _vile hits_."

The shares would be at a premium of 3-1/8 on the day after issue. If they presented me with the number of shares I deserve, for suggestion and advertisement, I should stand up for the Archpriest of St. Vitus[443] and 3-1/5, with a view to a little more gold on the bridge.

I now insert a couple of reviews, one about Cyclopaedias, one about epistolary collections. Should any reader wish for explanation of this insertion, I ask him to reflect a moment, and imagine me set to justify all the additions now before him! In truth these reviews are the repositories of many odds and ends: they were not made to the books; the materials were in my notes, and the books came as to a ready-made clothes shop, and found what would fit them. Many remember Curll's[444] bequest of some very good titles {280} which only wanted treatises written to them. Well! here were some tolerable reviews--as times go--which only wanted books fitted to them. Accordingly, some tags were made to join on the books; and then as the reader sees.

I should find it hard to explain why the insertion is made in this place rather than another. But again, suppose I were put to make such an explanation throughout the volume. The improver who laid out grounds and always studied what he called _unexpectedness_, was asked what name he gave it for those who walked over his grounds a second time. He was silenced; but I have an answer: It is that which is given by the very procedure of taking up my book a second time.

REVIEW OF CYCLOPAEDIAS.

October 19, 1861. _The English Cyclopaedia._ Conducted by Charles Knight.[445] 22 vols.: viz., _Geography_, 4 vols.; _Biography_, 6 vols.; _Natural History_, 4 vols.; _Arts and Sciences_, 8 vols. (Bradbury & Evans.)

_The Encyclopaedia Britannica: a Dictionary of Arts, Sciences, and General Literature._ Eighth Edition. 21 vols. and Index. (Black.)

The two editions above described are completed at the same time: and they stand at the head of the two great branches into which pantological undertakings are divided, as at once the largest and the best of their classes.

When the works are brought together, the first thing that strikes the eye is the syllable of difference in the names. The word _Cyclopaedia_ is a bit of modern purism. Though [Greek: enkuklopaideia][446] is not absolutely Greek of Greece, we learn from both Pliny[447] and Quintilian[448] that the circle {281} of the sciences was so called by the Greeks, and Vitruvius[449] has thence naturalized _encyclium_ in Latin. Nevertheless we admit that the initial _en_ would have euphonized but badly with the word _Penny_: and the _English Cyclopaedia_ is the augmented, revised, and distributed edition of the _Penny Cyclopaedia_. It has indeed been said that Cyclopaedia should mean the education _of_ a circle, just as Cyropaedia is the education _of_ Cyrus. But this is easily upset by Aristotle's word [Greek: kuklophoria],[450] motion _in_ a circle, and by many other cases, for which see the lexicon.

The earliest printed Encyclopaedia of this kind was perhaps the famous "myrrour of the worlde," which Caxton[451] translated from the French and printed in 1480. The original Latin is of the thirteenth century, or earlier. This is a collection of very short treatises. In or shortly after 1496 appeared the _Margarita Philosophica_ of Gregory Reisch,[452] the same we must suppose, who was confessor to the Emperor Maximilian.[453] This is again a collection of treatises, of much more pretension: and the estimation formed of it is proved by the number of editions it went through. In 1531 appeared the little collection of _works_ of Ringelberg,[454] which is truly called an Encyclopaedia by {282} Morhof, though the thumbs and fingers of the two hands will meet over the length of its one volume. There are more small collections; but we pass on to the first work to which the name of _Encyclopaedia_ is given. This is a ponderous _Scientiarum Omnium Encyclopaedia_ of Alsted,[455] in four folio volumes, commonly bound in two: published in 1629 and again in 1649; the true parent of all the Encyclopaedias, or collections of treatises, or works in which that character predominates. The first great _dictionary_ may perhaps be taken to be Hofman's _Lexicon Universale_[456] (1677); but Chambers's[457] (so called) _Dictionary_ (1728) has a better claim. And we support our proposed nomenclature by observing that Alsted accidentally called his work _En_cyclopaedia, and Chambers simply Cyclopaedia.

We shall make one little extract from the _myrrour_, and one from Ringelberg. Caxton's author makes a singular remark for his time; and one well worthy of attention. The grammar rules of a language, he says, must have been invented by foreigners: "And whan any suche tonge was perfytely had and usyd amonge any people, than other people not used to the same tonge caused rulys to be made wherby they myght lerne the same tonge ... and suche rulys be called the gramer of that tonge." Ringelberg says that if the right nostril bleed, the little finger of the right hand should be crooked, and squeezed with great force; and the same for the left.

{283}

We pass on to _the_ Encyclopedie,[458] commenced in 1751; the work which has, in many minds, connected the word _encyclopaedist_ with that of _infidel_. Readers of our day are surprised when they look into this work, and wonder what has become of all the irreligion. The truth is, that the work--though denounced _ab ovo_[459] on account of the character of its supporters--was neither adapted, nor intended, to excite any particular remark on the subject: no work of which D'Alembert[460] was co-editor would have been started on any such plan. For, first, he was a real _sceptic_: that is, doubtful, with a mind not made up. Next, he valued his quiet more than anything; and would as soon have gone to sleep over an hornet's nest as have contemplated a systematic attack upon either religion or government. As to Diderot[461]--of whose varied career of thought it is difficult to fix the character of any one moment, but who is very frequently taken among us for a pure atheist--we will quote one sentence from the article "_Encyclopedie_," which he wrote himself:--"Dans le moral, il n'y a que Dieu qui doit servir de modele a 1'homme; dans les art, que la nature."[462]

A great many readers in our country have but a very hazy idea of the difference between the political Encyclopaedia, as we may call it, and the _Encyclopedie Methodique_,[463] which we always take to be meant--whether rightly or not we cannot tell--when we hear of the "great French Encyclopaedia." This work, which takes much from its {284} predecessor, professing to correct it, was begun in 1792, and finished in 1832. There are 166 volumes of text, and 6439 plates, which are sometimes incorporated with the text, sometimes make about 40 more volumes. This is still the monster production of the kind; though probably the German Cyclopaedia of Ersch and Gruber,[464] which was begun in 1818, and is still in progress, will beat it in size. The great French work is a collection of dictionaries; it consists of Cyclopaedias of all the separate branches of knowledge. It is not a work, but a collection of works, one or another department is to be bought from time to time; but we never heard of a complete set for sale in one lot. As ships grow longer and longer, the question arises what limit there is to the length. One answer is, that it will never do to try such a length that the stern will be rotten before the prow is finished. This wholesome rule has not been attended to in the matter before us; the earlier parts of the great French work were antiquated before the whole were completed: something of the kind will happen to that of Ersch and Gruber.

The production of a great dictionary of either of the kinds is far from an easy task. There is one way of managing the _En_cyclopaedia which has been largely resorted to; indeed, we may say that no such work has been free from it. This plan is to throw all the attention upon the great treatises, and to resort to paste and scissors, or some process of equally easy character, for the smaller articles. However it may be done, it has been the rule that the Encyclopaedia of treatises should have its supplemental Dictionary of a very incomplete character. It is true that the treatises are intended to do a good deal; and that the Index, if it be good, knits the treatises and the dictionary into one whole of reference. Still there are two stools, and between them a great deal will fall to the ground. The dictionary portion of the _Britannica_ is not to be compared with its {285} treatises; the part called Miscellaneous and Lexicographical in the _Metropolitana_[465] is a great failure. The defect is incompleteness. The biographical portion, for example, of the Britannica is very defective: of many names of note in literature and science, which become known to the reader from the treatises, there is no account whatever in the dictionary. So that the reader who has learnt the results of a life in astronomy, for example, must go to some other work to know when that life began and ended. This defect has run through all the editions; it is in the casting of the work. The reader must learn to take the results at their true value, which is not small. He must accustom himself to regard the Britannica as a splendid body of treatises on all that can be called heads of knowledge, both greater and smaller; with help from the accompanying dictionary, but not of the most complete character. Practically, we believe, this defect cannot be avoided: two plans of essentially different structure cannot be associated on the condition of each or either being allowed to abbreviate the other.

The defect of all others which it is most difficult to avoid is inequality of performance. Take any dictionary you please, of any kind which requires the association of a number of contributors, and this defect must result. We do not merely mean that some will do their work better than others; this of course: we mean that there will be structural differences of execution, affecting the relative extent of the different parts of the whole, as well as every other point by which a work can be judged. A wise editor will not attempt any strong measures of correction: he will remember that if some portions be below the rest, which is a disadvantage, it follows that some portions must be above the rest, which is an advantage. The only practical level, if {286} level there must be, is that of mediocrity, if not of absolute worthlessness: any attempt to secure equality of strength will result in equality of weakness. Efficient development may be cut down into meager brevity, and in this way only can apparent equality of plan be secured throughout. It is far preferable to count upon differences of execution, and to proceed upon the acknowledged expectation that the prominent merits of the work will be settled by the accidental character of the contributors; it being held impossible that any editorial efforts can secure a uniform standard of goodness. Wherever the greatest power is found, it should be suffered to produce its natural effect. There are, indeed, critics who think that the merit of a book, like the strength of a chain, is that of its weakest part: but there are others who know that the parallel does not hold, and who will remember that the union of many writers must show exaggeration of the inequalities which almost always exist in the production of one person. The true plan is to foster all the good that can be got, and to give development in the directions in which most resources are found: a Cyclopaedia, like a plant, should grow towards the light.

The _Penny Cyclopaedia_ had its share of this kind of defect or excellence, according to the way in which the measure is taken. The circumstance is not so much noticed as might be expected, and this because many a person is in the habit of using such a dictionary chiefly with relation to one subject, his own; and more still want it for the pure dictionary purpose, which does not go much beyond the meaning of the word. But the person of full and varied reference feels the differences; and criticism makes capital of them. The Useful Knowledge Society was always odious to the organs of religious bigotry; and one of them, adverting to the fact that geography was treated with great ability, and most unusual fullness, in the _Penny Cyclopaedia_, announced it by making it the sole merit of {287} the work that, with sufficient addition, it would make a tolerably good gazetteer.

Some of our readers may still have hanging about them the feelings derived from this old repugnance of a class to all that did not associate direct doctrinal teaching of religion with every attempt to communicate knowledge. I will take one more instance, by way of pointing out the extent to which stupidity can go. If there be an astronomical fact of the telescopic character which, next after Saturn's ring and Jupiter's satellites, was known to all the world, it was the existence of multitudes of double stars, treble stars, etc. A respectable quarterly of the theological cast, which in mercy we refrain from naming, was ignorant of this common knowledge,--imagined that the mention of such systems was a blunder of one of the writers in the _Penny Cyclopaedia_, and lashed the presumed ignorance of the statement in the following words, delivered in April, 1837:

"We have forgotten the name of that Sidrophel who lately discovered that the fixed stars were not single stars, but appear in the heavens like soles at Billingsgate, in pairs; while a second astronomer, under the influence of that competition in trade which the political economists tell us is so advantageous to the public, professes to show us, through his superior telescope, that the apparently single stars are really three. Before such wondrous mandarins of science, how continually must _homunculi_ like ourselves keep in the background, lest we come between the wind and their nobility."

Certainly these little men ought to have kept in the background; but they did not: and the growing reputation of the work which they assailed has chronicled them in literary history; grubs in amber.

This important matter of inequality, which has led us so far, is one to which the _Encyclopaedia_ is as subject as the _Cyclopaedia_; but it is not so easily recognized as a fault. {288} We receive the first book as mainly a collection of treatises: we know their authors, and we treat them as individuals. We see, for instance, the names of two leading writers on Optics, Brewster[466] and Herschel.[467] It would not at all surprise us if either of these writers should be found criticising the other by name, even though the very view opposed should be contained in the same _Encyclopaedia_ with the criticism. And in like manner, we should hold it no wonder if we found some third writer not comparable to either of those we have named. It is not so in the _Cyclopaedia_: here we do not know the author, except by inference from a list of which we never think while consulting the work. We do not dissent from this or that author: we blame the book.

The _Encyclopaedia Britannica_ is an old friend. Though it holds a proud place in our present literature, yet the time was when it stood by itself, more complete and more clear than anything which was to be found elsewhere. There must be studious men alive in plenty who remember when they were studious boys, what a literary luxury it was to pass a few days in the house of a friend who had a copy of this work. The present edition is a worthy successor of those which went before. The last three editions, terminating in 1824, 1842, and 1861, seem to show that a lunar cycle cannot pass without an amended and augmented edition. Detailed criticism is out of the question; but we may notice the effective continuance of the plan of giving general historical dissertations on the progress of knowledge. Of some of these dissertations we have had to take separate notice; and all will be referred to in our ordinary treatment of current literature.[468]

The literary excellence of these two extensive undertakings is of the same high character. To many this will {289} need justification: they will not easily concede to the cheap and recent work a right to stand on the same shelf with the old and tried magazine, newly replenished with the best of everything. Those who are cognizant by use of the kind of material which fills the _Penny Cyclopaedia_ will need no further evidence: to others we shall quote a very remarkable and certainly very complete testimony. The _Cyclopaedia of the Physical Sciences_, published by Dr. Nichol[469] in 1857 (noticed by us, April 4), is one of the most original of our special dictionaries. The following is an extract from the editor's preface:

"When I assented to Mr. Griffin's proposal that I should edit such a Cyclopaedia, I had it in my mind that I might make the _scissors_ eminently effective. Alas! on narrowly examining our best Cyclopaedias, I found that the scissors had become blunted through too frequent and vigorous use. One great exception exists: viz., the _Penny Cyclopaedia_ of Charles Knight.[470] The cheapest and the least pretending, it is really the most philosophical of our _scientific_ dictionaries. It is not made up of a series of treatises, some good and many indifferent, but is a thorough _Dictionary_, well proportioned and generally written by the best men of the time. The more closely it is examined, the more deeply will our obligation be felt to the intelligence and conscientiousness of its projector and editor."

After Dr. Nichol's candid and amusing announcement of his scissorial purpose, it is but fair to state that nothing of the kind was ultimately carried into effect, even upon the work in which he found so much to praise. I quote this testimony because it is of a peculiar kind.

{290}

The success of the _Penny Magazine_ led Mr. Charles Knight in 1832 to propose to the Useful Knowledge Society a Cyclopaedia in weekly penny numbers. These two works stamp the name of the projector on the literature of our day in very legible characters. Eight volumes of 480 pages each were contemplated; and Mr. Long[471] and Mr. Knight were to take the joint management. The plan embraced a popular account of Art and Science, with very brief biographical and geographical information. The early numbers of the work had some of the _Penny Magazine_ character: no one can look at the pictures of the Abbot and Abbess in their robes without seeing this. By the time the second volume was completed, it was clearly seen that the plan was working out its own extension: a great development of design was submitted to, and Mr. Long became sole editor. Contributors could not be found to make articles of the requisite power in the assigned space. One of them told us that when he heard of the eight volumes, happening to want a shelf to be near at hand for containing the work as it went on, he ordered it to be made to hold twenty-five volumes easily. But the inexorable logic of facts beat him after all: for the complete work contained twenty-six volumes and two thick volumes of Supplement.

The penny issue was brought to an end by the state of the law, which required, in 1833, that the first and last page of everything sold separately should contain the name and address of the printer. The penny numbers contained this imprint on the fold of the outer leaf: and _qui tam_[472] informations were laid against the agents in various towns. {291} It became necessary to call in the stock; and the penny issue was abandoned. Monthly parts were substituted, which varied in bulk, as the demands of the plan became more urgent, and in price from one sixpence to three. The second volume of Supplement appeared in 1846, and during the fourteen years of issue no one monthly part was ever behind its time. This result is mainly due to the peculiar qualities of Mr. Long, who unites the talents of the scholar and the editor in a degree which is altogether unusual. If any one should imagine that a mixed mass of contributors is a punctual piece of machinery, let him take to editing upon that hypothesis, and he shall see what he shall see and learn what he shall learn.

The _English_ contains about ten per cent more matter than the _Penny Cyclopaedia_ and its supplements; including the third supplementary volume of 1848, which we now mention for the first time. The literary work of the two editions cost within 500l. and 50,000l.: that of the two editions of the _Britannica_ cost 41,000l. But then it is to be remembered that the _Britannica_ had matter to begin upon, which had been paid for in the former editions. Roughly speaking, it is probable that the authorship of a page of the same size would have cost nearly the same in one as in the other.

The longest articles in the _Penny Cyclopaedia_ were "Rome" in 98 columns and "Yorkshire" in 86 columns. The only article which can be called a treatise is the Astronomer Royal's "Gravitation," founded on the method of Newton in the eleventh section, but carried to a much greater extent. In the _English Cyclopaedia_, the longest article of geography is "Asia," in 45 columns. In natural history the antelopes demand 36 columns. In biography, "Wellington" uses up 42 columns, and his great military opponent 41 columns. In the division of Arts and Sciences, which includes much of a social and commercial character, the length of articles often depends upon the state of the {292} times with regard to the subject. Our readers would not hit the longest article of this department in twenty guesses: it is "Deaf and Dumb" in 60 columns. As other specimens, we may cite Astronomy, 19; Banking, 36; Blind, 24; British Museum, 35; Cotton, 27; Drama, 26; Gravitation, 50; Libraries, 50; Painting, 34; Railways, 18; Sculpture, 36; Steam, etc., 37; Table, 40; Telegraph, 30; Welsh language and literature, 39; Wool, 21. These are the long articles of special subdivisions: the words under which the _En_cyclopaedia gives treatises are not so prominent. As in Algebra, 10; Chemistry, 12; Geometry, 8; Logic, 14; Mathematics, 5; Music, 9. But the difference between the collection of treatises and the dictionary may be illustrated thus: though "Mathematics" have only five columns, "Mathematics, recent terminology of," has eight: and this article we believe to be by Mr. Cayley,[473] who certainly ought to know his subject, being himself a large manufacturer of the new terms which he explains. Again, though "Music" _in genere_, as the schoolmen said, has only nine columns, "Temperament and Tuning," has eight, and "Chord" alone has two. And so on.

In a dictionary of this kind it is difficult to make a total clearance of _personality_: by which we mean that exhibition of peculiar opinion which is offensive to taste when it is shifted from the individual on the corporate book. The treatise of the known author may, as we have said, carry that author's controversies on its own shoulders: and even his crotchets, if we may use such a word. But {293} the dictionary should not put itself into antagonism with general feeling, nor even with the feelings of classes. We refer particularly to the ordinary and editorial teaching of the article. If, indeed, the writer, being at issue with mankind, should confess the difference, and give abstract of his full grounds, the case is altered: the editor then, as it were, admits a correspondent to a statement of his own individual views. The dictionary portion of the Britannica is quite clear of any lapses on this point, so far as we know: the treatises and dissertations rest upon their authors. The Penny Cyclopaedia was all but clear: and great need was there that it should have been so. The Useful Knowledge Society, starting on the principle of perfect neutrality in politics and religion, was obliged to keep strict watch against the entrance of all attempt even to look over the hedge. There were two--we believe only two--instances of what we have called personality. The first was in the article "Bunyan." It is worth while to extract all that is said--in an article of thirty lines--about a writer who is all but universally held to be the greatest master of allegory that ever wrote:

"His works were collected in two volumes, folio, 1736-7: among them 'The Pilgrim's Progress' has attained the greatest notoriety. If a judgment is to be formed of the merits of a book by the number of times it has been reprinted, and the many languages into which it has been translated, no production in English literature is superior to this coarse allegory. On a composition which has been extolled by Dr. Johnson, and which in our own times has received a very high critical opinion in its favor [probably Southey], it is hazardous to venture a disapproval, and we, perhaps, speak the opinion of a small minority when we confess that to us it appears to be mean, jejune and wearisome."

--If the unfortunate critic who thus individualized himself had been a sedulous reader of Bunyan, his power over {294} English would not have been so _jejune_ as to have needed that fearful word. This little bit of criticism excited much amusement at the time of its publication: but it was so thoroughly exceptional and individual that it was seldom or never charged on the book. The second instance occurred in the article "Socinians." It had been arranged that the head-words of Christian sects should be intrusted to members of the sects themselves, on the understanding that the articles should simply set forth the accounts which the sects themselves give of their own doctrines. Thus the article on the Roman Church was written by Dr. Wiseman.[474] But the Unitarians were not allowed to come within the rule: as in other quarters, they were treated as the gypsies of Christianity. Under the head "Socinians"--a name repudiated by themselves--an opponent was allowed not merely to state their alleged doctrines in his own way, but to apply strong terms, such as "audacious unfairness," to some of their doings. The protests which were made against this invasion of the understanding produced, in due time, the article "Unitarians," written by one of that persuasion. We need not say that these errors have been amended in the English Cyclopaedia: and our chief purpose in mentioning them is to remark, that this is all we can find on the points in question against twenty-eight large volumes produced by an editor whose task was monthly, and whose issue was never delayed a single hour. How much was arrested before publication none but himself can say. We have not alluded to one or two remonstrances on questions of absolute fact, which are beside the present purpose.

Both kinds of encyclopaedic works have been fashioned upon predecessors, from the very earliest which had a predecessor to be founded upon; and the undertakings before us will be themselves the ancestors of a line of successors. Those who write in such collections should be {295} careful what they say, for no one can tell how long a mis-statement may live. On this point we will give the history of a pair of epithets. When the historian De Thou[475] died, and left the splendid library which was catalogued by Bouillaud[476] and the brothers Dupuis[477] (Bullialdus and Puteanus), there was a manuscript of De Thou's friend Vieta,[478] the _Harmonicon Coeleste_, of which it is on record, under Bouillaud's hand, that he himself lent it to Cosmo de' Medici,[479] to which must be added that M. Libri[480] found it in the Magliabecchi Library at Florence in our own day. Bouillaud, it seems, entirely forgot what he had done. Something, probably, that Peter Dupuis said to Bouillaud, while they were at work on the catalogue, remained on his memory, and was published by him in 1645, long after; to the effect that Dupuis lent the manuscript to Mersenne,[481] from whom it was procured by some intending plagiarist, who would not give it back. This was repeated by Sherburne,[482] in 1675, who speaks of the work, which "being communicated to Mersennus was, by some perfidious acquaintance of that honest-minded person, surreptitiously taken from him, and irrecoverably lost or suppressed, to the unspeakable detriment of the lettered world." Now let the {296} reader look through the dictionaries of the last century and the present, scientific or general, at the article, "Vieta," and he will be amused with the constant recurrence of "honest-minded" Mersenne, and his "surreptitious" acquaintance. We cannot have seen less than thirty copies of these epithets.

REVIEW OF MACCLESFIELD LETTERS.

October 18, 1862. _Correspondence of Scientific Men of the Seventeenth Century, in the Collection of the Earl of Macclesfield._[483] 2 vols. (Oxford, University Press.)

Though the title-page of this collection bears the date 1841, it is only just completed by the publication of its Table of Contents and Index. Without these, a work of the kind is useless for consultation, and cannot make its way. The reason of the delay will appear: its effect is well known to us. We have found inquirers into the history of science singularly ignorant of things which this collection might have taught them.

In the same year, 1841, the Historical Society of Science, which had but a brief existence, published a collection of letters, eighty-three in number, edited by Mr. Halliwell,[484] of English men of science, which dovetails with the one before us, and is for the most part of a prior date. The two should be bound up together. The smaller collection runs from 1562 to 1682; the larger, from 1606 to past 1700. We shall speak of the two as the Museum collection and the Macclesfield collection. And near them should be placed, in every scientific library, the valuable collection published, by Mr. Edleston,[485] for Trinity College, in 1850.

{297}

The history of these letters runs back to famous John Collins, the attorney-general of the mathematics, as he has been called, who wrote to everybody, heard from everybody, and sent copies of everybody's letter to everybody else. He was in England what Mersenne[486] was in France: as early as 1671, E. Bernard[487] addresses him as "the very Mersennus and intelligence of this age." John Collins[488] was never more than accountant to the Excise Office, to which he was promoted from teaching writing and ciphering, at the Restoration: he died in 1682. We have had a man of the same office in our own day, the late Prof. Schumacher,[489] who made the little Danish Observatory of Altona the junction of all the lines by which astronomical information was conveyed from one country to another. When the collision took place between Denmark and the Duchies, the English Government, moved by the Astronomical Society, instructed its diplomatic agents to represent strongly to the Danish Government, when occasion should arise, the great importance of the Observatory of Altona to the astronomical communications of the whole world. But Schumacher had his own celebrated journal, the _Astronomische Nachrichten_, by which to work out part of his plan; private correspondence was his supplementary assistant. Collins had only correspondence to rely on. Nothing is better known than that it was Collins's collection which furnished the materials put forward by the Committee of the Royal Society in 1712, as a defence of Newton against the partisans of Leibnitz. The noted _Commercium Epistolicum_ is but the abbreviation of a title which runs on with "D. Johannis Collins et aliorum ..."

The whole of this collection passed into the hands of {298} William Jones,[490] the father of the Indian Judge of the same name, who died in 1749. Jones was originally a teacher, but was presented with a valuable sinecure by the interest of George, second Earl of Macclesfield, the mover of the bill for the change of style in Britain, who died President of the Royal Society. This change of style may perhaps be traced to the union of energies which were brought into concert by the accident of a common teacher: Lord Macclesfield and Lord Chesterfield,[491] the mover and the seconder, and Daval,[492] who drew the bill, were pupils of De Moivre.[493] Jones, who was a respectable mathematician though not an inventor, collected the largest mathematical library of his day, and became possessor of the papers of Collins, which contained those of Oughtred[494] and others. Some of these papers passed into the custody of the Royal Society: but the bulk was either bequeathed to, or purchased by, Lord Macclesfield; and thus they found their way to Shirburn Castle, where they still remain.

A little before 1836, this collection attracted the attention of a searching inquirer into points of mathematical history, the late Professor Rigaud,[495] who died in 1839. He examined the whole collection of letters, obtained Lord Macclesfield's consent to their publication, and induced the Oxford Press to bear the expense. It must be particularly remembered that there still remains at Shirburn Castle a {299} valuable mass of non-epistolary manuscripts. So far as we can see, the best chance of a further examination and publication lies in public encouragement of the collection now before us: the Oxford Press might be induced to extend its operations if it were found that the results were really of interest to the literary and scientific world. Rigaud died before the work was completed, and the publication was actually made by one of his sons, S. Jordan Rigaud,[496] who died Bishop of Antigua. But this publication was little noticed, for the reasons given. The completion now published consists of a sufficient table of contents, of the briefest kind, by Professor De Morgan, and an excellent index by the Rev. John Rigaud.[497] The work is now fairly started on its career.

If we were charged to write a volume with the title "Small things in their connection with great," we could not do better than choose the small part of this collection of letters as our basis. The names, as well as the contents, are both great and small: the great names, those which are known to every mathematician who has any infusion of the history of his pursuit, are Briggs,[498] Oughtred, Charles Cavendish,[499] Gascoigne,[500] Seth Ward,[501] Wallis,[502] {300} Hu[y]gens,[503] Collins,[504] William Petty,[505] Hooke,[506] Boyle,[507] Pell,[508] Oldenburg,[509] Brancker,[510] Slusius,[511] Bertit,[512] Bernard,[513] Borelli,[514] Mouton,[515] Pardies,[516] Fermat,[517] Towneley,[518] Auzout,[519] {301} D. Gregory,[520] Halley,[521] Machin,[522] Montmort,[523] Cotes,[524] Jones,[525] Saunderson,[526] Reyneau,[527] Brook Taylor,[528] Maupertuis,[529] Bouguer,[530] La Condamine,[531] Folkes,[532] Macclesfield,[533] {302} Baker,[534] Barrow,[535] Flamsteed,[536] Lord Brounker,[537] J. Gregory,[538] Newton[539] and Keill.[540] To these the Museum collection adds the names of Thomas Digges,[541] Dee,[542] Tycho Brahe,[543] Harriot,[544] Lydyat,[545] Briggs,[546] Warner,[547] Tarporley, Pell,[548] Lilly,[549] Oldenburg,[550] Collins,[551] Morland.[552]

{303}

The first who appears on the scene is the celebrated Oughtred, who is related to have died of joy at the Restoration: but it should be added, by way of excuse, that he was eighty-six years old. He is an animal of extinct race, an Eton mathematician. Few Eton men, even of the minority which knows what a sliding rule is, are aware that the inventor was of their own school and college: but they may be excused, for Dr. Hutton,[553] so far as his Dictionary bears witness, seems not to have known it any more than they. A glance at one of his letters reminds us of a letter from the Astronomer Royal on the discovery of Neptune, which we printed March 20, 1847. Mr. Airy[554] there contends, and proves it both by Leverrier[555] and by Adams,[556] that the limited publication of a private letter is more efficient than the more general publication of a printed memoir. The same may be true of a dead letter, as opposed to a dead book. Our eye was caught by a letter of Oughtred (1629), containing systematic use of contractions for the words _sine_, _cosine_, etc., prefixed to the symbol of the angle. This is so very important a step, simple as it is, that Euler[557] is justly held to have greatly advanced trigonometry by its introduction. Nobody that we know of has noticed that Oughtred was master of the improvement, and willing to have taught it, if people would have learnt. After looking at his dead letter, we naturally turned to his dead book on trigonometry, and there we found the abbreviations _s_, _sco_, _t_, _tco_, _se_, _seco_, regularly established as part of the system of the work. But not one of those who have investigated the contending claims of Euler and Thomas {304} Simpson[558] has chanced to know of Oughtred's "Trigonometrie": and the present revival is due to his letter, not to his book.

A casual reader, turning over the pages, would imagine that almost all the letters had been printed, either in the General Dictionary, or in Birch,[559] etc.: so often does the supplementary remark begin with "this letter has been printed in ----." For ourselves we thought, until we counted, that a large majority of the letters had been given, either in whole or in part. But the positive strikes the mind more forcibly than the negative: we find that all of which any portion has been in type makes up very little more than a quarter; the cases in which the whole letter is given being a minority of this quarter. The person who has been best ransacked is Flamsteed: of 36 letters from him, 34 had been previously given in whole or in part. Of 59 letters to and from Newton, only 17 have been culled.

The letters have been modernized in spelling, and, to some extent, in algebraical notation; it also seems that conjectural methods of introducing interpolations into the text have been necessary. For all this we are sorry: the scientific value of the collection is little altered, but its literary value is somewhat lowered. But it could not be helped: the printers could not work from the originals, and Professor Rigaud had to copy everything himself. A fac-simile must have been the work of more time than he had to give: had he attempted it, his death would have cut short the whole undertaking, instead of allowing him to prepare everything but a preface, and to superintend the printing of one of the volumes. We may also add, that we believe we have notices of _all_ the letters in the Macclesfield collection. We judge this because several which are too trivial to print are numbered and described; and those would certainly not have been noticed if _any_ omissions had {305} been made. And we know that every letter was removed from Shirburn Castle to Oxford.

Two persons emerge from oblivion in this series of letters. The first is Michael Dary,[560] an obscure mathematician, who was in correspondence with Newton and other stars. He was a gauger at Bristol, by the interest of Collins; afterwards a candidate for the mathematical school at Christ's Hospital, with a certificate from Newton: he was then a gunner in the Tower, and is lastly described by Wallis as "Mr. Dary, the tobacco-cutter, a knowing man in algebra." In 1674, Dary writes to Newton at Cambridge, as follows:--"Although I sent you three papers yesterday, I cannot refrain from sending you this. I have had fresh thoughts this morning." Two months afterwards poor Newton writes to Collins, "Mr. Dary is very solicitous about mathematics": but in spite of the persecution, he subscribes himself to Dary "your loving friend." Dary's _problem_ is that of finding the rate of interest of an annuity of which the value and term are given. Dary's _theorem_, which he seems to have invented specially for the solution of his problem, though it is of wide range, can be exhibited to mathematical readers even in our columns. In modern language, it is that the limit of [phi]^{_n_}_x_, when _n_ increases without limit, is a solution of [phi]_x_ = _x_. We have mentioned the I. Newton to whom Dary looked up; we add a word about the one on whom he looked down. Dr. John Newton,[561] a sedulous publisher of logarithms, tables of interest, etc., who began his career before Isaac Newton, sometimes puzzles those who do not know him, when described as I. Newton. The scientific world was of opinion that all that was valuable in one of his works was taken from Dary's private communications.

{306}

The second character above alluded to is one who carried mathematical researches a far greater length than Newton himself: the assistance which he rendered in this respect, even to Newton, has never been acknowledged in modern times: though the work before us shows that his contemporaries were fully aware of it, and never thought of concealing it. In his theory of gravitation, in which, so far as he went, we have every reason to believe he was prior to Newton, he did not extend his calculations to the distance of the moon; his views in this matter were purely terrestrial, and led him to charge according to weight. He was John Stiles, the London and Cambridge carrier: his name is a household word in the Macclesfield Letters, and is even enshrined in the depths of Birch's quartos. Dary informs Newton--let us do his memory this justice--that he had paid John Stiles for the carriage. At the time when the railroad to Cambridge was opened, a correspondent recommended the directors, in our columns, to call an engine by the name of John Stiles, and never to let that name go off the road. We do not know whether the advice was followed: if not, we repeat it.

Little points of life and manners come out occasionally. Baker, the author of a work on algebra much esteemed at the time, wrote to Collins that their circumstances are alike, "having a just and equal number of chargeable olive-branches, and being in the same predicament and blessed condemnation with you, not more preaching than unpaid, and preaching the art of contentment to others, am forced to practise it." But the last sentence of his letter runs as follows: "I have sent by the bearer ... twenty shillings, as a token to you; desiring you to accept of it, as a small taste from Yours, Thos. Baker." In our day, men of a station to pay parish taxes do not offer their friends hard money to buy liquor. But Flamsteed[562] writes to Collins as follows: "Last week he sent us down the counterpart, which {307} my father has scaled, and I return up to you by the carrier, with 5l. to be paid to Mr. Leneve for the writing, I have added 2s. 6d. over, which will pay the expenses and serve to drink, with him." This would seem as odd to us as it would have seemed thirty years ago that half-a-crown should pay carriage for a deed from Derby to London, and leave margin for a bottle of wine: in our day, the Post-office and the French treaty would just manage it between them. But Flamsteed does not limit his friend to one bottle; he adds, "If you expend more than the half-crown, I will make it good after Whitsuntide." Collins does not remember exactly where he had met James Gregory, and mentions two equally likely places thus: "Sir, it was once my good hap to meet with you in an alehouse or in Sion College." There is a little proof how universally the dinner-hour was twelve o'clock. Astronomers well know the method of finding time by equal altitudes of the sun before and after noon: Huyghens calls it "le moyen de deux egales hauteurs du soleil devant et apres _diner_."[563]

There is one mention of "Mr. Cocker,[564] our famous English graver and writer, now a schoolmaster at Northampton." This is the true Cocker: his genuine works are specimens of writing, such as engraved copy-books, including some on arithmetic, with copper-plate questions and space for the working; also a book of forms for law-stationers, with specimens of legal handwriting. It is recorded somewhere that Cocker and another, whose name we forget, competed with the Italians in the beauty of their flourishes. This was his real fame: and in these matters he was great. The eighth edition of his book of law forms (1675), published shortly after Cocker's death, has a preface signed "J. H." This was John Hawkins, who became possessed of Cocker's papers--at least he said so--and {308} subsequently forged the famous Arithmetic,[565] a second work on Decimal Arithmetic, and an English dictionary, all attributed to Cocker. The proofs of this are set out in De Morgan's _Arithmetical Books_. Among many other corroborative circumstances, the clumsy forger, after declaring that Cocker to his dying day resisted strong solicitation to publish his Arithmetic, makes him write in the preface _Ille ego qui quondam_[566] of this kind: "I have been instrumental to the benefit of many, by virtue of those useful arts, writing and engraving; and do _now_, with the same _wonted alacrity_, cast this my arithmetical mite into the public treasury." The book itself is not comparable in merit to at least half-a-dozen others. How then comes Cocker to be the impersonation of Arithmetic? Unless some one can show proof, which we have never found, that he was so before 1756, the matter is to be accounted for thus.

Arthur Murphy,[567] the dramatist, was by taste a man of letters, and ended by being the translator of Tacitus; though many do not know that the two are one. His friends had tried to make him a man of business; and no doubt he had been well plied with commercial arithmetic. His first dramatic performance, the farce of "The Apprentice," produced in 1756, is about an idle young man who must needs turn actor. Two of the best known books of the day in arithmetic were those of Cocker and Wingate.[568] Murphy chooses _Wingate_ to be the name of an old merchant who {309} delights in vulgar fractions, and _Cocker_ to be his arithmetical catchword--"You read Shakespeare! get Cocker's Arithmetic! you may buy it for a shilling on any stall; best book that ever was wrote!" and so on. The farce became very popular, and, as we believe, was the means of elevating Cocker to his present pedestal, where Wingate would have been, if his name had had the droller sound of the two to English ears.

A notoriety of an older day turns up, Major-General Lambert.[569] The common story is that he was banished to Guernsey, where he passed thirty years in confinement, rearing and painting flowers. But Baker, in 1678, represents him as a prisoner at Plymouth, sending equations for solution as a challenge: probably his place of confinement was varied, and his occupation also.

[General Lambert was removed to Plymouth, probably about 1668. His daughter captured the son of the Governor of Guernsey, who therefore probably was reckoned an unsafe custodier thenceforward; though he assured the king that he had turned the young couple out of doors, and had never given them a penny. Great importance was attached to Lambert's safe detention: probably the remaining republicans looked upon him as to be their next Cromwell, if such a thing were to be. There were standing orders to shoot him at once on the first appearance of any enemy before the island. See _Notes and Queries_, 3d S. iv. 89.]

Collins informs James Gregory that "some of the Royal Academy wrote over to Mr. Oldenburg, who was desired to impart the same to the Council of the Royal Society, that the French King was willing to allow pensions to one or two learned Englishmen, but they never made any answer {310} to such a proposal." This was written in 1671, and the thing probably happened several years before. Mr. De Morgan communicated the account of the proposal to Lord Macaulay, who replied that he did not think that any Englishman _received_ a literary pension from Louis; but that there is a curious letter, about 1664, from the French Ambassador, in which he says that he has, by his master's orders, been making inquiries as to the state of learning in England, and that he is sorry to find that the best writer is _the infamous Miltonus_. On two such independent testimonies it may be held proved that the French King had attempted to buy a little adherence from English literature and science; and the silent contempt of the Royal Society is an honorable fact in their history.

Another little bit of politics is as follows. Oughtred is informed that "Mr. Foster,[570] our Lecturer on Astronomy at Gresham College, is put out because he will not kneel down at the communion-table. A Scotsman [Mungo Murray], one that is _verbi bis minister_,[571] is now lecturer in Mr. Foster's place." Ward in his work on the Gresham Professors,[572] suppresses the reason, and the suppression lowers the character of his book. Foster was expelled in 1636, and re-elected on a vacancy in 1641, when Puritanism had gained strength.

The correspondence of Newton would require deeper sifting than could be given in such an article as the present. The first of the letters (1669) is curious, as presenting the {311} appearance of forms belonging to the great calculus which, in this paragraph, we ought to call that of fluxions. We find, of the date February 18, 1669-70, what we believe is the earliest manifestation of that morbid part of Newton's temperament which has been so variously represented. He had solved a problem--being that which we have called Dary's--on which he writes as follows: "The solution of the annuity problem, if it will be of any use, you have my leave to insert into the _Philosophical Transactions_, so it be without my name to it. For I see not what there is desirable in public esteem, were I able to acquire and maintain it. It would perhaps increase my acquaintance, the thing which I chiefly study to decline."

Three letters touch upon "the experiment of glass rubbed to cause various motions in bits of paper underneath": they are supplements to the account given by Newton to the Royal Society, and printed by Birch. It was Newton, so far as appears, who added _glass_ to the substances known to be electric. Soon afterwards we come to a little bit of the history of the appointment to the Mint. It has appeared from the researches of late years that Newton was long an aspirant for public employment: the only coolness which is known to have taken place between him and Charles Montague[573] [Halifax] arose out of his imagining that his friend was not in earnest about getting him into the public service. March 14, 1696, Newton writes thus to Halley: "And if the rumour of preferment for me in the Mint should hereafter, upon the death of Mr. Hoar [the comptroller], or any other occasion, be revived, I pray that you would {312} endeavour to obviate it by acquainting your friends that I neither _put in_ for _any_ place in the Mint, nor would meddle with _Mr. Hoar's place_, were it offered to me." This means that Mr. Hoar's place had been suggested, which Newton seems to have declined. Five days afterwards, Montague writes to Newton that he is to have the _Wardenship_. It is fair to Newton to say that in all probability this was not--or only in a smaller degree--a question of personal dignity, or of salary. It must by this time have been clear to him that the minister, though long bound to make him an object of patronage, was actually seeking him for the Mint, because he wanted both Newton's name and his talents for business--which he knew to be great--in the weighty and dangerous operation of restoring the coinage. It may have been, and probably was, the case that Newton had a tolerably accurate notion of what he would have to do, and of what degree of power would be necessary to enable him to do it in his own way.

We have said that the non-epistolary manuscripts are still unexamined. There is a chance that one of them may answer a question of two centuries' standing, which is worth answering, because it has been so often asked. About 1640, Warner,[574] afterwards assisted by Pell,[575] commenced a table of _antilogarithms_, of the kind which Dodson[576] afterwards constructed anew and published. In the Museum collection there is inquiry after inquiry from Charles Cavendish,[577] first, as to when the _Analogics_, as he called them, would be finished; next, when they would be printed. Pell answers, in 1644, that Warner left his papers to a kinsman, who had become bankrupt, and proceeds thus:

"I am not a little afraid that all Mr. Warner's papers, {313} and no small share of my labours therein, are seazed upon, and most unmathematically divided between the sequestrators and creditors, who (not being able to ballance the account where there appeare so many numbers, and much troubled at the sight of so many crosses and circles in the superstitious Algebra and that black art of Geometry) will, no doubt, determine once in their lives to become figure-casters, and so vote them all to be throwen into the fire, if some good body doe not reprieve them for pye-bottoms, for which purposes you know analogicall numbers are incomparably apt, if they be accurately calculated."

Pell afterwards told Wallis[578] that the papers had fallen into the hands of Dr. Busby,[579] and Collins[580] writes that they were left in the hands of Dr. Thorndike,[581] a prebendary of Westminster; whence Rigaud[582] seems to say that Thorndike had left them to Dr. Busby. Birch[583] says that he procured for the Royal Society four boxes from Busby's trustees, containing papers of Warner and Pell: but there is no other tradition of such things in the Society. But in the Birch manuscripts at the British Museum, there turns up, as printed in what we call the Museum collection, a list of Warner's papers, with _Collins's_ receipt to Dr. Thorndike at the bottom, and engagement to restore them on demand. The date is December 14, 1667; Wallis's statement being in 1693. It is possible that Busby may be a mistake altogether: he was very unlikely to have had charge of any mathematical papers: there may have been a confusion between the Prebendary of Westminster and the Head Master of Westminster School. If so, in all probability Thorndike handed {314} the cumbrous lot over to the notorious collector of mathematical papers, blessing himself that he got rid of them in a manner which would insure their return if he were called upon by the owners to restore them. It is much against this hypothesis that Dodson, who certainly recalculated, can say nothing more about Warner than a repetition of Wallis's story: though, had Collins kept the papers, they would probably have been in Jones's possession at the very time when Dodson, who was a friend of Jones and a user of his library, was engaged on his own computations. But even books, and still more manuscripts, are often singularly overlooked; and it remains not very improbable that Warner's table is now at Shirburn Castle, among the unexamined manuscripts.

CYCLOMETRY AND STEEL PENS.

_Redit labor actus in orbem._[584] Among the matters which have come to me since the Budget opened, there is a pamphlet of quadrature of two pages and a half from Professor Recalcati,[585] already mentioned. It ends with "Quelque objection qu'on fasse touchant les raisonnements ci-dessus on tombera toujours dans l'absurde."[586] A civil engineer--so he says--has made the quadrature "no longer a problem, but an axiom." As follows: "Take the quadrant of a circle whose circumference is given, square the quadrant which gives the true square of the circle. Because 30 / 4 = 7.5 x 7.5 = 56.25 = the positive square of a circle whose circumference is 30." Brevity, the soul of wit, is the "wings of mighty-winds" to quadrature, and sends it "flying all abroad." A _surbodhicary_--something like M.A. or LL.D., I understand--at Calcutta, published in 1863 the division of an {315} angle into any odd number of parts, demonstration and all in--when the diagram is omitted--one page, good-sized, well-leaded type, small duodecimo. But in the Preface he acknowledges "sheer inability" to execute his task. Mr. William Dean, of Todmorden, in 1863, announced 3-9/64 as proved both practically and geometrically: he has been already mentioned anonymously. Next I have the tract of Don Juan Larriva, published at Leiria in 1856, and dedicated to Queen Victoria. Mr. W. Peters,[587] already mentioned, who has for some months been circulating diagrams on a card, publishes (August, 1865) _The Circle Squared_. He agrees with the Archpriest of St. Vitus. He hints that a larger publication will depend partly on the support he receives, and partly on the castigation, for which last, of course, he looks to me. Cyclometers have their several styles of wit; so have anticyclometers too, for that matter. Mr. Peters will not allow me any extra-journal being: I am essentially a quotation from the _Athenaeum_; "A. De Morgan" _et praeterea nihil_.[588] If he had to pay for keeping me set up, he would find out his mistake, and would be glad to compound handsomely for a stereotype. Next comes a magnificent sheet of pasteboard, printed on both sides. Having glanced at it and detected quadrature, I began methodically at the beginning--"By Royal Command," with the lion and unicorn, and all that comes between. Mercy on us! thought I to myself: has Her Majesty referred the question to the Judicial Committee of the Privy Council, where all the great difficulties go now-a-days, and is this proclamation the result? On reading further I was relieved by finding that the first side is entirely an advertisement of Joseph Gillott's[589] steel pens, with engraving of his {316} premises, and notice of novel application of his unrivalled machinery. The second side begins with "the circle rectified" by W. E. Walker,[590] who finds [pi] = 3.141594789624155.... This is an off-shoot from an accurate geometrical rectification, on which is to be presumed Mr. Gillott's new machinery is founded. I have no doubt that Mr. Walker's error, which is only in the sixth place of decimals, will not hurt the pens, unless it be by the slightest possible increase of the tendency to open at the points. This arises from Mr. Walker having rectified above proof by .000002136034362....

Lastly, I, even I myself, who have long felt that I was a quadrature below par, have solved the problem by means which, in the present state of the law of libel, I dare not divulge. But the result is permitted; and it goes far to explain all the discordances. The ratio of the circumference to the diameter is not always the same! Not that it varies with the radius; the geometers are right enough on that point: but it varies with the time, in a manner depending upon the difference of the true longitudes of the Sun and Moon. A friend of mine--at least until he misbehaved--insisted on the mean right ascensions: but I served him as Abraham served his guest in Franklin's parable. The true formula is, A and a being the Sun's and Moon's longitudes,

[pi] = 3-13/80 + 3/80 cos(A - a).

Mr. James Smith obtained his quadrature at full moon; the Archpriest of St. Vitus and some others at new moon. Until I can venture to publish the demonstration, I recommend the reader to do as I do, which is to adopt 3.14159..., and to think of the matter only at the two points of the lunar month at which it is correct. The _Nautical Almanac_ will no doubt give these points in a short time: I am in correspondence with the Admiralty, with nothing {317} to get over except what I must call a perverse notion on the part of the Superintendent of the _Almanac_, who suspects one correction depending on the Moon's latitude; and the Astronomer Royal leans towards another depending on the date of the Queen's accession. I have no patience with these men: what can the Moon's node of the Queen's reign possibly have to do with the ratio in question? But this is the way with all the regular men of science; Newton is to them etc. etc. etc. etc.

The following method of finding the circumference of a circle (taken from a paper by Mr. S. Drach[591] in the _Phil. Mag._, Jan. 1863, Suppl.) is as accurate as the use of 3.14159265. From three diameters deduct 8-thousandths and 7-millionths of a diameter; to the result add five per cent. We have then not quite enough; but the shortcoming is at the rate of about an inch and a sixtieth of an inch in 14,000 miles.

JACOB BEHMEN.

Though I have met with nothing but a little tract from the school of Jacob Behmen[592] (or Boehme; I keep to the old English version of his name), yet there has been more, and of a more recent date. I am told of an "Introduction to Theosophy [_Theo_ private, I suppose, as in theological]; or, the Science of the Mystery of Christ," published in 1854, mostly from the writings of William Law[593]: and also of a volume of 688 pages, of the same year, printed for private circulation, containing notes for a biography of William Law. The editor of the first work wishes to grow "a {318} generation of perfect Christians" by founding a Theosophic College, for which he requests the public to raise a hundred thousand pounds. There is a good account of Jacob Behmen in the _Penny Cyclopaedia_. The author mentions inaccurate accounts, one of which he quotes, as follows: "He derived all his mystical and rapturous doctrine from Wood's[594] _Athenae Oxonienses_, Vol. I, p. 610, and _Hist. et Antiq. Acad. Oxon._, Vol. II, p. 308." On which the author remarks that Wood was born after Behmen's death. There must have been a few words which slipped out: what is meant is that Behmen "derived his doctrine from _Robert Fludd_,[595] _for whom see_ Wood's etc. etc." Even this is absurd enough: for Behmen began to publish in 1610, and Fludd in 1616. Fludd was a Rosicrucian, and a mystic of a different type from Behmen. I have some of his works, and could produce out of them paradoxes enough, according to our ways of thinking, to fit out a host. But the Rosicrucian system was a recognized school of its day, and Fludd, a man of great learning, had abettors enough in all which he advanced, and predecessors in most of it.

[A Correspondent has recently sent a short summary of the claims of Jacob Behmen to rank higher than I have placed him. I shall gladly insert this summary in the book I contemplate, as a statement of what is said of Behmen far less liable to suspicion of exaggeration than anything I could write. I shall add a few extracts from Behmen himself, in support of his right to be in my list.]

"_Jacob Behmen._--That Prof. De Morgan classes Jacob Behmen among paradoxers can only be attributed to the fact of his being avowedly unacquainted with the writings {319} of that author. Perhaps you may think a few words from one who knows them well of sufficient interest to the learned Professor, and your readers in general, to be worthy of space in your columns. The metaphysical system of Behmen--the most perfect and only true one--still awaits a qualified commentator. Behmen's countryman, Dionysius Andreas Freher,[596] who spent the greater part of his life in this country, and whose exposition of Behmen exists only in MS., filling many volumes, written in English, with the exception of two, written in German, with numerous beautiful, highly ingenious, and elaborate illustrations,--copies of some of which are in the British Museum, but all the originals of which are in the possession of the gentleman who is the editor of the two works alluded to by Professor De Morgan,--this Freher was the first to philosophically expound Behmen's system, which was afterwards, with the help of these MSS., as it were, popularized by William Law; but both Freher and Law confined themselves chiefly to its theological aspect. In Behmen, however, is to be found, not only the true ground of all theology, but also that of all physical science. He demonstrated with a fullness, accuracy, completeness and certainty that leave nothing to be desired, the innermost ground of Deity and Nature; and, confining myself to the latter, I can from my own knowledge assert, that in Behmen's writings is to be found the true and clear demonstration of every physical fact that has been discovered since his day. Thus, the science of electricity, which was not yet in existence when he wrote, is there anticipated; and not only does Behmen describe all the now known phenomena of that force, but he even gives us the origin, generation and birth of electricity itself. Again, positive evidence can be adduced that Newton derived all his knowledge of gravitation and its {320} laws from Behmen, with whom gravitation or attraction is, and very properly so, as he shows us, the first of the seven properties of Nature. The theory defended by Mr. Grove,[597] at the Nottingham meeting of last year, that all the apparently distinct causes of moral and physical phenomena are but so many manifestations of one central force, and that Continuity is the law of nature, is clearly laid down, and its truth demonstrated, by Behmen, as well as the distinction between spirit and matter, and that the moral and material world is pervaded by a sublime unity. And though all this was not admitted in Behmen's days, because science was not then sufficiently advanced to understand the deep sense of our author, many of his passages, then unintelligible, or apparently absurd, read by the light of the present age, are found to contain the positive enunciation of principles at whose discovery and establishment science has only just arrived by wearisome and painful investigations. Every new scientific discovery goes to prove his profound and intuitive insight into the most secret workings of nature; and if scientific men, instead of sharing the prejudice arising from ignorance of Behmen's system, would place themselves on the vantage ground it affords, they would at once find themselves on an eminence whence they could behold all the arcana of nature. Behmen's system, in fact, shows us the _inside_ of things, while modern physical science is content with looking at the _outside_. Behmen traces back every outward manifestation or development to its one central root,--to that one central energy which, as yet, is only suspected; every link in the chain of his demonstration is perfect, and there is not one link wanting. He carries us from the out-births of the circumference, along the radius to the center, {321} or point, and beyond that even to the zero, demonstrating the constitution of the zero, or nothing, with mathematical precision. C. W. H."

And so Behmen is no subject for the Budget! I waited until I should chance to light on one of his volumes, knowing that any volume would do, and almost any page. My first hap was on the second volume of the edition of 1664 (4to, published by M. Richardson) and opening near the beginning, a turn or two brought me to page 13, where I saw about _sulphur_ and _mercurius_ as follows:

"Thus SUL is the soul, in an herb it is the oil, and in man also, according to the spirit of _this_ world in the third principle, which is continually generated out of the anguish of the will in the mind, and the Brimstone-worm is the Spirit, which hath the fire and _burneth_: PHUR is the sour wheel in itself which causeth that.

"_Mercurius_ comprehendeth all the four forms, even as the life springeth up, and yet hath not its dark beginning in the Center as the PHUR hath, but after the flash of fire, when the sour dark form is terrified, where the hardness is turned into pliant sharpness, and where the second will (_viz._ the will of nature, which is called the Anguish) ariseth, there Mercurius hath its original. For MER is the shivering wheel, very horrible, sharp, venomous, and hostile; which assimulateth it thus in the sourness in the flash of fire, where the sour wrathful life _ariseth_. The syllable CU is the pressing out, of the _Anxious_ will of the mind, from Nature: which is climbing up, and _willeth_ to be out aloft. RI is the comprehension of the flash of fire, which in MER giveth a clear sound and tune. For the flash maketh the tune, and it is the Salt-Spirit which _soundeth_, and its form (or quality) is gritty like sand, and herein arise noises, sounds and voices, and thus CU comprehendeth the flash, and so the pressure is as a _wind_ which thrusteth, and giveth a spirit to the flash, so that it liveth and burneth. Thus the {322} syllable US is called the burning fire, which with the spirit continually driveth itself forth: and the syllable CU presseth continually upon the flash."

Shades of Tauler[598] and Paracelsus,[599] how strangely you do mix! Well may Hallam call Germany the native soil of Mysticism. Had Behmen been the least of a scholar, he would not have divided _sulph-ur_ and _merc-ur-i-us_ as he has done: and the inflexion _us_, that boy of all work, would have been rejected. I think it will be held that a writer from whom hundreds of pages like the above could be brought together, is fit for the Budget. If Sampson Arnold Mackay[600] had tied his etymologies to a mystical Christology, instead of a mystical infidelity, he might have had a school of followers. The nonsense about Newton borrowing gravitation from Behmen passes only with those who know neither what Newton did, nor what was done before him.

The above reminds me of a class of paradoxers whom I wonder that I forgot; they are without exception the greatest bores of all, because they can put the small end of their paradox into any literary conversation whatever. I mean the people who have heard the local pronunciation of celebrated names, and attempt not only to imitate it, but to impose on others their broken German or Arabic, or what not. They also learn the vernacular names of those who are generally spoken of in their Latin forms; at least, they learn a few cases, and hawk them as evidences of erudition. They are miserably mistaken: scholarship, as a rule, {323} always accepts the vernacular form of a name which has vernacular celebrity. Hallam writes Behmen: his index-maker, rather superfluously, gives "_Behmen_ or Boehm." And he retains Melanchthon,[601] the name given by Reuchlin[602] to his little kinsman Schwartzerd, because the world has adopted it: but he will none of Capnio, the name which Reuchlin fitted on to himself, because the world has not adopted it. He calls the old forms pedantry: but he sees that the rejection of well-established results of pedantry would be greater pedantry still. The paradoxers assume the question that it is more _correct_ to sound a man by lame imitation of his own countrymen than as usual in the country in which the sound is to be made. Against them are, first, the world at large; next, an overpowering majority of those who know something about surnames and their history. Some thirty years ago--a fact--there appeared at the police-office a complainant who found his own law. In the course of his argument, he asked, "What does Kitty say?"--"Who's Kitty?" said the magistrate, "your wife, or your nurse?"--"Sir! I mean Kitty, the celebrated lawyer."--"Oh!" said the magistrate, "I suspect you mean Mr. Chitty,[603] the author of the great work on pleading."--"I do sir! But Chitty is an Italian name, and ought to be pronounced _Kitty_." This man was a full-blown flower: but there is many a modest bud; and all ought either to blush when seen or to waste their pronunciation on the desert air.

{324}

A PLEA FOR KING CUSTOM.

I stand up for King Custom, or _Usus_, as Horace called him, with whom is _arbitrium_ the decision, and _jus_ the right, and _norma_ the way of deciding, simply because he has _potestas_ the power. He may admit one and another principle to advise: but Custom is not a constitutional king; he may listen to his cabinet, but he decides for himself: and if the ministry should resign, he blesses his stars and does without them. We have a glorious liberty in England of owning neither dictionary, grammar, nor spelling-book: as many as choose write by either of the three, and decide all disputed points their own way, those following them who please.

Throughout this book I have called people by the names which denote them in their books, or by our vernacular names. This is the intelligible way of proceeding. I might, for instance (Vol. I, p. 44), have spoken of Charles de Bovelles,[604] of Lefevre d'Etaples,[605] of Pelerin,[606] and of Etienne.[607] But I prefer the old plan. Those who like another plan better, are welcome to substitute with a pen, when they know what to write; when they do not, it is clear that they would not have understood me if I had given modern names.

The principal advisers of King Custom are as follows. First, there is Etymology, the _chiffonnier_, or general rag-merchant, who has made such a fortune of late years in his own business that he begins to be considered highly respectable. He gives advice which is more thought of than followed, partly on account of the fearful extremes into which he runs. He lately asked some boys of sixteen, at a matriculation examination in _English_, to what branch of {325} the Indo-Germanic family they felt inclined to refer the Pushto language, and what changes in the force of the letters took place in passing from Greek into Moeso-Gothic. Because all syllables were once words, he is a little inclined to insist that they shall be so still. He would gladly rule English with a Saxon rod, which might be permitted with a certain discretion which he has never attained: and when opposed, he defends himself with analogies of the Aryan family until those who hear him long for the discovery of an Athanasyus. He will transport a word beyond seas--he is recorder of Rhematopolis--on circumstantial evidence which looks like mystery gone mad; but, strange to say, something very often comes to light after sentence is passed which proves the soundness of the conviction.

The next adviser is Logic, a swearing old justice of peace, quorum, and rotulorum, whose excesses brought on such a fit of the gout that for many years he was unable to move. He is now mending, and his friends say he has sown his wild oats. He has some influence with the educated subjects of Custom, and will have more, if he can learn the line at which interference ought to stop: with them he has succeeded in making an affirmative of two negatives; but the vulgar won't never have nothing to say to him. He has always railed at Milton for writing that Eve was the fairest of her daughters; but has never satisfactorily shown what Milton ought to have said instead.

The third adviser has more influence with the mass of the subjects of King Custom than the other two put together; his name is Fiddlefaddle, the toy-shop keeper; and the other two put him forward to do their worst work. In return, he often uses their names without authority. He took Etymology to witness that _means_ to an end must be plural: and he would have any one method to be a _mean_. But Etymology proved him wrong, King Custom referred him to his Catechism, in which is "a means whereby we receive the same," and Analogy--a subordinate of {326} Etymology--asked whether he thought it a great _new_ to hear that he was wrong. It was either this Fiddlefaddle, or Lindley Murray[608] his traveler, who persuaded the Miss Slipslops, of the Ladies Seminary, to put "The Misses Slipslop" over the gate. Sixty years ago, this bagman called at all the girls' schools, and got many of the teachers to insist on the pupils saying "Is it not" and "Can I not" for "Isn't it" and "Can't I": of which it came that the poor girls were dreadfully laughed at by their irreverent brothers when they went home for the holidays. Had this bad adviser not been severely checked, he might by this time have proposed our saying "The Queen's of England son," declaring, in the name of Logic, that the prince was the Queen's son, not England's.

Lastly, there is Typography the metallurgist, an executive officer who is always at work in secret, and whose lawless mode of advising is often done by carrying his notions into effect without leave given. He it is who never ceases suggesting that the same word is not to occur in a second place within sight of the first. When the Authorized Version was first printed, he began this trick at the passage, "Let there be light, and there was light;" he drew a line on the proof under the second _light_, and wrote "_luminosity?_" opposite. He is strongest in the punctuations and other signs; he has a pepper-box full of commas always by his side. He puts everything under marks of quotation which he has ever heard before. An earnest preacher, in a very moving sermon, used the phrase Alas! and alack a day! Typography stuck up the inverted commas because he had read the old Anglo-Indian toast, "A lass and a lac a day!" If any one should have the sense to leave out of his Greek {327} the unmeaning scratches which they call accents, he goes to a lexicon and puts them in. He is powerful in routine; but when two routines interlace or overlap, he frequently takes the wrong one.

Subject to bad advice, and sometimes misled for a season, King Custom goes on his quiet way and is sure to be right at last.

"Treason does never prosper: what's the reason? Why, when it prospers, none dare call it treason."

Language is in constant fermentation, and all that is thrown in, so far as it is not fit to assimilate, is thrown off; and this without any obvious struggle. In the meanwhile every one who has read good authors, from Shakspeare downward, knows what is and what is not English; and knows, also, that our language is not one and indivisible. Two very different turns of phrase may both be equally good, and as good as can be: we may be relieved of the consequences of contempt of one court by _habeas corpus_ issuing out of another.

TEST OF LANGUAGE.

Hallam remarks that the Authorized Version of the Bible is not in the language of the time of James the First: that it is not the English of Raleigh or of Bacon. Here arises the question whether Raleigh and Bacon are the true expositors of the language of their time; and whether they were not rather the incipient promoters of a change which was successfully resisted by--among other things--the Authorized Version of the Testaments. I am not prepared to concede that I should have given to the English which would have been fashioned upon that of Bacon by imitators, such as they usually are, the admiration which is forced from me by Bacon's English from Bacon's pen. On this point we have a notable parallel. Samuel Johnson {328} commands our admiration, at least in his matured style: but we nauseate his followers. It is an opinion of mine that the works of the leading writers of an age are seldom the proper specimens of the language of their day, when that language is in its state of progression. I judge of a language by the colloquial idiom of educated men: that is, I take this to be the best medium between the extreme cases of one who is ignorant of grammar and one who is perched upon a style. Dialogue is what I want to judge by, and plain dialogue: so I choose Robert Recorde[609] and his pupil in the _Castle of Knowledge_, written before 1556. When Dr. Robert gets into his altitudes of instruction, he differs from his own common phraseology as much as probably did Bacon when he wrote morals and philosophy. But every now and then I come to a little plain talk about a common thing, of which I propose to show a specimen. Anything can be made to look old by such changes as _makes_ into _maketh_, with a little old spelling. I shall invert these changes, using the newer form of inflexion, and the modern spelling: with no other variation whatever.

"_Scholar._ Yet the reason of that is easy enough to be conceived, for when the day is at the longest the Sun must needs shine the more time, and so must it needs shine the less time when the day is at the shortest: this reason I have heard many men declare.

_Master._ That may be called a crabbed reason, for it {329} goes backward like a crab. The day makes not the Sun to shine, but the Sun shining makes the day. And so the length of the day makes not the Sun to shine long, neither the shortness of the day causes not [_sic_] the Sun to shine the lesser time, but contrariwise the long shining of the Sun makes the long day, and the short shining of the Sun makes the lesser day: else answer me what makes the days long or short?

_Scholar._ I have heard wise men say that Summer makes the long days, and Winter makes the long nights.

_Master._ They might have said more wisely, that long days make summer and short days make winter.

_Scholar._ Why, all that seems one thing to me.

_Master._ Is it all one to say, God made the earth, and the earth made God? Covetousness overcomes all men, and all men overcome covetousness?

_Scholar._ No, not so; for here the effect is turned to be the cause, and the agent is made the patient.

_Master._ So is it to say Summer makes long days, when you should say: Long days make summer.

_Scholar._ I perceive it now: but I was so blinded with the vulgar error, that if you had demanded of me further what did make the summer, I had been like to have answered that green leaves do make summer; and the sooner by remembrance of an old saying that a year should come in which the summer should not be known but by the green leaves.

_Master._ Yet this saying does not import that green leaves do make summer, but that they betoken summer; so are they the sign and not the cause of summer."

I have taken a whole page of our author, without omission, that the reader may see that I do not pick out sentences convenient for my purpose. I have done nothing but alter the third person of the verb and the spelling: but great is the effect thereof. We say "the Sun shining makes the day"; Recorde, "the Sonne shynynge maketh the daye." {330} These points apart, we see a resemblance between our English and that of three hundred years ago, in the common talk of educated persons, which will allow us to affirm that the language of the authorized Bible must have been very close to that of its time. For I cannot admit that much change can have taken place in fifty years: and the language of the version represents both our common English and that of Recorde with very close approximation. Take sentences from Bacon and Raleigh, and it will be apparent that these writers will be held to differ from all three, Recorde, the version, and ourselves, by differences of the same character. But we speak of Recorde's conversation, and of our own. We conclude that it is the plain and almost colloquial character of the Authorized Version which distinguishes it from the English of Bacon and Raleigh, by approximating it to the common idiom of the time. If any one will cast an eye upon the letters of instruction written by Cecil[610] and the Bishop of London to the translators themselves, or to the general directions sent to them in the King's name, he will find that these plain business compositions differ from the English of Bacon and Raleigh by the same sort of differences which distinguish the version itself.

PRONUNCIATION.

The foreign word, or the word of a district, or class of people, passes into the general vernacular; but it is long before the specially learned will acknowledge the right of those with whom they come in contact to follow general usage. The rule is simple: so long as a word is technical or local, those who know its technical or local pronunciation may reasonably employ it. But when the word has become general, the specialist is not very wise if he refuse to follow {331} the mass, and perfectly foolish if he insist on others following him. There have been a few who demanded that Euler should be pronounced in the German fashion:[611] Euler has long been the property of the world at large; what does it matter how his own countrymen pronounce the letters? Shall we insist on the French pronouncing _Newton_ without that final _tong_ which they never fail to give him? They would be wise enough to laugh at us if we did. We remember that a pedant who was insisting on all the pronunciations being retained, was met by a maxim in contradiction, invented at the moment, and fathered upon Kaen-foo-tzee,[612] an authority which he was challenged to dispute. Whom did you speak of? said the bewildered man of accuracy. Learn your own system, was the answer, before you impose it on others; Confucius says that too.[613]

The old English has _fote_, _fode_, _loke_, _coke_, _roke_, etc., for _foot_, etc. And _above_ rhymes in Chaucer to _remove_. Suspecting that the broader sounds are the older, we may surmise that _remove_ and _food_ have retained their old sounds, and that _cook_, once _coke_, would have rhymed to our _Luke_, the vowel being brought a little nearer, perhaps, to the _o_ in our present _coke_, the fuel, probably so called as used by cooks. If this be so, the Chief Justice _Cook_[614] of our lawyers, and the _Coke_ (pronounced like the fuel) of the greater part of the world, are equally wrong. The lawyer has no right whatever to fasten his pronunciation upon us: even leaving aside the general custom, he cannot prove himself right, and is probably wrong. Those who {332} know the village of Rokeby (pronounced Rookby) despise the world for not knowing how to name Walter Scott's poem: that same world never asked a question about the matter, and the reception of the parody of _Jokeby_, which soon appeared, was a sufficient indication of their notion. Those who would fasten the hodiernal sound upon us may be reminded that the question is, not what they call it now, but what it was called in Cromwell's time. Throw away general usage as a lawgiver, and this is the point which emerges. Probably _R[=u]ke-by_ would be right, with a little turning of the Italian [=u] towards [=o] of modern English.

[Some of the above is from an old review. I do not always notice such insertions: I take nothing but my own writings. A friend once said to me, "Ah! you got that out of the _Athenaeum_!" "Excuse me," said I. "the _Athenaeum_ got that out of me!"]

APOLOGIES TO CLUVIER.

It is part of my function to do justice to any cyclometers whose methods have been wrongly described by any orthodox sneerers (myself included). In this character I must notice _Dethlevus Cluverius_,[615] as the Leipzig Acts call him (probably Dethleu Cluvier), grandson of the celebrated geographer, Philip Cluvier. The grandson was a Fellow of the Royal Society, elected on the same day as Halley,[616] November 30, 1678: I suppose he lived in England. This {333} man is quizzed in the Leipzig Acts for 1686; and, if Montucla insinuate rightly, by Leibnitz, who is further suspected of wanting to embroil Cluvier with his own opponent Nieuwentiit,[617] on the matter of infinitesimals. So far good: I have nothing against Leibnitz, who though he was ironical, told us what he laughed at. But Montucla has behaved very unfairly: he represents Cluvier as placing the essence of his method in the solution of the problem _construere mundum divinae menti analogum_, to construct a world corresponding to the divine mind. Nothing to begin with: no way of proceeding. Now, it ought to have been _ex data linea construere_,[618] etc.: there is a given line, which is something to go on. Further, there is a way of proceeding: it is to find the product of 1, 2, 3, 4, etc. for ever. Moreover, Montucla charges Cluvier with _unsquaring_ the parabola, which Archimedes had squared as tight as a glove. But he never mentions how very nearly Cluvier agrees with the Greek: they only differ by 1 divided by 3n^2, where n is the infinite number of parts of which a parabola is composed. This must have been the conceit that tickled Leibnitz, and made him wish that Cluvier and Nieuwentiit should fight it out. Cluvier, was admitted, on terms of irony, into the Leipzig Acts: he appeared on a more serious footing in London. It is very rare for one cyclometer to refute another: _les corsaires ne se battent pas_.[619] The only instance I recall is that of M. Cluvier, who (_Phil. Trans._, 1686, No. 185) refuted M. Mallemont de Messange,[620] who {334} published at Paris in 1686. He does it in a very serious style, and shows himself a mathematician. And yet in the year in which, in the _Phil. Trans._, he was a geometer, and one who rebukes his squarer for quoting Matthew xi. 25, in that very year he was the visionary who, in the Leipzig Acts, professed to build a world resembling the divine mind by multiplying together 1, 2, 3, 4, etc. up to infinity.

THE RAINBOW PARADOX.

There is a very pretty opening for a paradox which has never found its paradoxer in print. The philosophers teach that the rainbow is not material: it comes from rain-drops, but those rain-drops do not _take_ color. They only _give_ it, as lenses and mirrors; and each one drop gives _all_ the colors, but throws them in different directions. Accordingly, the same drop which furnishes red light to one spectator will furnish violet to another, properly placed. Enter the paradoxer whom I have to invent. The philosopher has gulled you nicely. Look into the water, and you will see the reflected rainbow: take a looking-glass held sideways, and you see another reflection. How could this be, if there were nothing colored to reflect? The paradoxer's facts are true: and what are called the reflected rainbows are _other_ rainbows, caused by those _other_ drops which are placed so as to give the colors to the eye after reflection, at the water or the looking-glass. A few years ago an artist exhibited a picture with a rainbow and its apparent reflection: he simply copied what he had seen. When his picture was examined, some started the idea that there could be no reflection of a rainbow; they were right: they inferred that the artist had made a mistake; they were wrong. When it was explained, some agreed and some dissented. Wanted, {335} immediately, an able paradoxer: testimonials to be forwarded to either end of the rainbow, No. 1. No circle-squarer need apply, His Variegatedness having been pleased to adopt 3.14159... from Noah downwards.

TYCHO BRAHE REVIVED.

The system of Tycho Brahe,[621] with some alteration and addition, has been revived and contended for in our own day by a Dane, W. Zytphen,[622] who has published _The Motion of the Sun in the Universe_, (second edition) Copenhagen, 1865, 8vo, and _Le Mouvement Sideral_, 1865, 8vo. I make an extract.

"How can one explain Copernically that the velocity of the Moon must be added to the velocity of the Earth on the one place in the Earth's orbit, to learn how far the Moon has advanced from one fixed star to another; but in another place in the orbit these velocities must be subtracted (the movements taking place in opposite directions) to attain the same result? In the Copernican and other systems, it is well known that the Moon, abstracting from the insignificant excentricity of the orbit, always in twenty-four hours performs an equally long distance. Why has Copernicus never been denominated Fundamentus or Fundator? Because he has never convinced anybody so thoroughly that this otherwise so natural epithet has occurred to the mind."

Really the second question is more effective against Newton than against Copernicus; for it upsets gravity: the first is of great depth.

{336}

JAMES SMITH WILL NOT DOWN.

The _Correspondent_ journal makes a little episode in the history of my Budget (born May, 1865, died April, 1866). It consisted entirely of letters written by correspondents. In August, a correspondent who signed "Fair Play"--and who I was afterwards told was a lady--thought it would be a good joke to bring in the Cyclometers. Accordingly a letter was written, complaining that though Mr. Sylvester's[623] demonstration of Newton's theorem--then attracting public attention--was duly lauded, the possibly greater discovery of the quadrature seemed to be blushing unseen, and wasting etc. It went on as follows:

"Prof. De Morgan, who, from his position in the scientific world, might fairly afford to look favourably on less practised efforts than his own, seems to delight in ridiculing the discoverer. Science is, of course, a very respectable person when he comes out and makes himself useful in the world [it must have been a lady; each sex gives science to the other]: but when, like a monk of the Middle Ages, he shuts himself up [it must have been a lady; they always snub the bachelors] in his cloistered cell, repeating his mumpsimus from day to day, and despising the labourers on the outside, we begin to think of Galileo,[624] Jenner,[625] Harvey,[626] and other glorious trios, who have been contemned ..."

The writer then called upon Mr. James Smith[627] to come {337} forward. The irony was not seen; and that day fortnight appeared the first of more than thirty letters from his pen. Mr. Smith was followed by Mr. Reddie,[628] Zadkiel,[629] and others, on their several subjects. To some of the letters I have referred; to others I shall come. The _Correspondent_ was to become a first-class scientific journal; the time had arrived at which truth had an organ: and I received formal notice that I could not stifle it by silence, nor convert it into falsehood by ridicule. When my reader sees my extracts, he will readily believe my declaration that I should have been the last to stifle a publication which was every week what James Mill[630] would call a dose of capital for my Budget. A few anti-paradoxers brought in common sense: but to the mass of the readers of the journal it all seemed to be the difference between Tweedledum and Tweedledee. Some said that the influx of scientific paradoxes killed the journal: but my belief is that they made it last longer than it otherwise would have done. Twenty years ago I recommended the paradoxers to combine and publish their views in a common journal: with a catholic editor, who had no pet theory, but a stern determination not to exclude anything merely for absurdity. I suspect it would answer very well. A strong title, or motto, would be wanted: not so coarse as was roared out in a Cambridge mob when I was an undergraduate--"No King! No Church! No House of Lords! No nothing, blast me!"--but something on that _principle_.

At the end of 1867 I addressed the following letter to the _Athenaeum_:

PSEUDOMATH, PHILOMATH, AND GRAPHOMATH.

_December 31, 1867_

Many thanks for the present of Mr. James Smith's letters {338} of Sept. 28 and of Oct. 10 and 12. He asks where you will be if you read and digest his letters: you probably will be somewhere first. He afterwards asks what the WE of the _Athenaeum_ will be if, finding it impossible to controvert, it should refuse to print. I answer for you, that We-We of the _Athenaeum_, not being Wa-Wa the wild goose, so conspicuous in "Hiawatha," will leave what controverts itself to print itself, if it please.

_Philomath_ is a good old word, easier to write and speak than _mathematician_. It wants the words between which I have placed it. They are not well formed: _pseudomathete_ and _graphomathete_ would be better: but they will do. I give an instance of each.

The _pseudomath_ is a person who handles mathematics as the monkey handled the razor. The creature tried to shave himself as he had seen his master do; but, not having any notion of the angle at which the razor was to be held, he cut his own throat. He never tried a second time, poor animal! but the pseudomath keeps on at his work, proclaims himself clean-shaved, and all the rest of the world hairy. So great is the difference between moral and physical phenomena! Mr. James Smith is, beyond doubt, the great pseudomath of our time. His 3-1/8 is the least of a wonderful chain of discoveries. His books, like Whitbread's barrels, will one day reach from Simpkin & Marshall's to Kew, placed upright, or to Windsor laid length-ways. The Queen will run away on their near approach, as Bishop Hatto did from the rats: but Mr. James Smith will follow her were it to John o' Groats.

The _philomath_, for my present purpose, must be exhibited as giving a lesson to presumption. The following anecdote is found in Thiebault's[631] _Souvenirs de vingt ans de sejours a Berlin_, published in 1804. The book itself got a high character for truth. In 1807 Marshal Mollendorff[632] {339} answered an inquiry of the Duc de Bassano,[633] by saying that it was the most veracious of books, written by the most honest of men. Thiebault does not claim personal knowledge of the anecdote, but he vouches for its being received as true all over the north of Europe.[634]

Diderot[635] paid a visit to Russia at the invitation of Catherine the Second. At that time he was an atheist, or at least talked atheism: it would be easy to prove him either one thing or the other from his writings. His lively sallies on this subject much amused the Empress, and all the younger part of her Court. But some of the older courtiers suggested that it was hardly prudent to allow such unreserved exhibitions. The Empress thought so too, but did not like to muzzle her guest by an express prohibition: so a plot was contrived. The scorner was informed that an eminent mathematician had an algebraical proof of the existence of God, which he would communicate before the whole Court, if agreeable. Diderot gladly consented. The mathematician, who is not named, was Euler.[636] He came to Diderot with the gravest air, and in a tone of perfect conviction said, "_Monsieur!_

(a + b^n)/n = x

_donc Dieu existe; repondez!_"[637] Diderot, to whom algebra was Hebrew, though this is expressed in a very roundabout way by Thiebault--and whom we may suppose to have expected some verbal argument of alleged algebraical closeness, was disconcerted; while peals of laughter sounded on all sides. Next day he asked permission to return to France, which was granted. An algebraist would have {340} turned the tables completely, by saying, "Monsieur! vous savez bien que votre raisonnement demande le developpement de x suivant les puissances entieres de n".[638] Goldsmith could not have seen the anecdote, or he might have been supposed to have drawn from it a hint as to the way in which the Squire demolished poor Moses.

The _graphomath_ is a person who, having no mathematics, attempts to describe a mathematician. Novelists perform in this way: even Walter Scott now and then burns his fingers. His dreaming calculator, Davy Ramsay, swears "by the bones of the immortal Napier." Scott thought that the the philomaths worshiped relics: so they do, in one sense. Look into Hutton's[639] Dictionary for _Napier's Bones_, and you shall learn all about the little knick-knacks by which he did multiplication and division. But never a bone of his own did he contribute; he preferred elephants' tusks. The author of _Headlong Hall_[640] makes a grand error, which is quite high science: he says that Laplace proved the precession of the equinoxes to be a periodical inequality. He should have said the variation of the obliquity. But the finest instance is the following: Mr. Warren,[641] in his well-wrought tale of the martyr-philosopher, was incautious enough to invent the symbols by which his _savant_ satisfied himself Laplace[642] was right on a doubtful point. And this is what he put together--

[sqrt]-3a^2, [rectangle]y^2 / z^2 + 9 - n = 9, n x log e.

Now, to Diderot and the mass of mankind this might be Laplace all over: and, in a forged note of Pascal, would {341} prove him quite up to gravitation. But I know of nothing like it, except in the lately received story of the American orator, who was called on for some Latin, and perorated thus: "Committing the destiny of the country to your hands, Gentlemen, I may without fear declare, in the language of the noble Roman poet,

E pluribus unum, Multum in parvo, Ultima Thule, Sine qua non."[643]

But the American got nearer to Horace than the martyr-philosopher to Laplace. For all the words are in Horace, except _Thule_, which might have been there. But [rectangle] is not a symbol wanted by Laplace; nor can we see how it could have been; in fact, it is not recognized in algebra. As to the junctions, etc., Laplace and Horace are about equally well imitated.

Further thanks for Mr. Smith's letters to you of Oct. 15, 18, 19, 28, and Nov. 4, 15. The last of these letters has two curious discoveries. First, Mr. Smith declares that he has _seen_ the editor of the _Athenaeum_: in several previous letters he mentions a name. If he knew a little of journalism he would be aware that editors are a peculiar race, obtained by natural selection. They are never seen, even by their officials; only heard down a pipe. Secondly, "an ellipse or oval" is composed of four arcs of circles. Mr. Smith has got hold of the construction I was taught, when a boy, for a pretty four-arc oval. But my teachers knew better than to call it an ellipse: Mr. Smith does not; but he produces from it such confirmation of 3-1/8 as would convince any _honest_ editor.

Surely the cyclometer is a Darwinite development of a spider, who is always at circles, and always begins again when his web is brushed away. He informs you that he {342} has been privileged to discover truths unknown to the scientific world. This we know; but he proceeds to show that he is equally fortunate in art. He goes on to say that he will make use of you to bring those truths to light, "just as an artist makes use of a dummy for the purpose of arranging his drapery." The painter's lay-figure is for flowing robes; the hairdresser's dummy is for curly locks. Mr. James Smith should read Sam Weller's pathetic story of the "four wax dummies." As to _his_ use of a dummy, it is quite correct. When I was at University College, I walked one day into a room in which my Latin colleague was examining. One of the questions was, "Give the lives and fates of Sp. Maelius,[644] and Sp. Cassius."[645] Umph! said I, surely all know that Spurius Maelius was whipped for adulterating flour, and that Spurius Cassius was hanged for passing bad money. Now, a robe arranged on a dummy would look just like the toga of Cassius on the gallows. Accordingly, Mr. Smith is right in the drapery-hanger which he has chosen: he has been detected in the attempt to pass bad circles. He complains bitterly that his geometry, instead of being read and understood by you, is handed over to me to be treated after my scurrilous fashion. It is clear enough that he would rather be handled in this way than not handled at all, or why does he go on writing? He must know by this time that it is a part of the institution that his "untruthful and absurd trash" shall be distilled into mine at the rate of about 3-1/8 pages of the first to one column of the second. Your readers will never know how much they gain by the process, until Mr. James Smith publishes it all in a big book, or until they get hold of what he has already published. I have six pounds avoirdupois of pamphlets and letters; and there is more than half a pound of letters {343} written to you in the last two months. Your compositor must feel aggrieved by the rejection of these clearly written documents, without erasures, and on one side only. Your correspondent has all the makings of a good contributor, except the knowledge of his subject and the sense to get it. He is, in fact, only a mask: of whom the fox

"O quanta species, inquit, cerebrum non habet."[646]

I do not despair of Mr. Smith on any question which does not involve that unfortunate two-stick wicket at which he persists in bowling. He has published many papers; he has forwarded them to mathematicians: and he cannot get answers; perhaps not even readers. Does he think that he would get more notice if you were to print him in your journal? Who would study his columns? Not the mathematician, we know; and he knows. Would others? His balls are aimed too wide to be blocked by any one who is near the wicket. He has long ceased to be worth the answer which a new invader may get. Rowan Hamilton,[647] years ago, completely knocked him over; and he has never attempted to point out any error in the short and easy method by which that powerful investigator condescended to show that, be right who may, he must be wrong. There are some persons who feel inclined to think that Mr. Smith should be argued with: let those persons understand that he has been argued with, refuted, and has never attempted to stick a pen into the refutation. He stated that it was a remarkable paradox, easily explicable; and that is all. After this evasion, Mr. James Smith is below the necessity of being told that he is unworthy of answer. His friends complain that I do nothing but _chaff_ him. Absurd! I winnow him; and if nothing but chaff results, whose fault is that? I am usefully employed: for he is the type of a class which ought to be known, and which I have done much to make known.

{344}

Nothing came of this until July 1869, when I received a reprint of the above letter, with a comment, described as Appendix D of a work in course of publication on the geometry of the circle. The _Athenaeum_ journal received the same: but the Editor, in his private capacity, received the whole work, being _The Geometry of the Circle and Mathematics as applied to Geometry by Mathematicians, shown to be a mockery, delusion, and a snare_, Liverpool, 8vo, 1869. Mr. J. S. here appears in deep fight with Professor Whitworth,[648] and Mr. Wilson,[649] the author of the alleged amendment of Euclid. How these accomplished mathematicians could be inveigled into continued discussion is inexplicable. Mr. Whitworth began by complaining of Mr. Smith's attacks upon mathematicians, continued to correspond after he was convinced that J. S. proved an arc and its chord to be equal, and only retreated when J. S. charged him with believing in 3-1/8, and refusing acknowledgment. Mr. Wilson was introduced to J. S. by a volunteer defense of his geometry from the assaults of the _Athenaeum_. This the editor would not publish; so J. S. sent a copy to Mr. Wilson himself. Some correspondence ensued, but Mr. Wilson soon found out his man, and withdrew.

There is a little derision of the _Athenaeum_ and a merited punishment for "that unscrupulous critic and contemptible mathematical twaddler, De Morgan."

MR. REDDIE'S ASTRONOMY.

At p. 183 I mentioned Mr. Reddie,[650] the author of _Vis Inertiae Victa_ and of _Victoria toto coelo_,[651] which last is not {345} an address to the whole heaven, either from a Roman Goddess or a British Queen, whatever a scholar may suppose. Between these Mr. Reddie has published _The Mechanics of the Heavens_, 8vo, 1862: this I never saw until he sent it to me, with an invitation to notice it, he very well knowing that it would catch. His speculations do battle with common notions of mathematics and of mechanics, which, to use a feminine idiom, he blasphemes so you can't think! and I suspect that if you do not blaspheme them too, _you_ can't think. He appeals to the "truly scientific," and would be glad to have readers who have read what he controverts, i.e., Newton's _Principia_: I wish he may get them; I mean I hope he may obtain them. To none but these would an account of his speculations be intelligible: I accordingly disposed of him in a very short paragraph of description. Now many paradoxers desire notice, even though it be disparaging. I have letters from more than one--besides what have been sent to the Editor of the _Athenaeum_--complaining that they are not laughed at; although they deserve it, they tell me, as much as some whom I have inserted. Mr. Reddie informs me that I have not said a single word against his books, though I have given nearly a column to sixteen-string arithmetic, and as much to animalcule universes. What need to say anything to readers of Newton against a book from which I quoted that revolution by gravitation is _demonstrably_ impossible? It would be as useless as evidence against a man who has pleaded guilty. Mr. Reddie derisively thanks me for "small mercies"; he wrote me private letters; he published them, and more, in the _Correspondent_. He gave me, _pro viribus suis_,[652] such a dressing you can't think, both for my Budget non-notice, and for reviews which he assumed me to have written. He outlawed himself by declaring (_Correspondent_, Nov. 11, 1856) that I--in a review--had made a quotation which was "garbled, evidently on purpose {346} to make it appear that" he "was advocating solely a geocentric hypothesis, which is not true." In fact, he did his best to get larger "mercy." And he shall have it; and at a length which shall content him, unless his mecometer be an insatiable apparatus. But I fear that in other respects I shall no more satisfy him than the Irish drummer satisfied the poor culprit when, after several times changing the direction of the stroke at earnest entreaty, he was at last provoked to call out, "Bad cess to ye, ye spalpeen! strike where one will, there's no _plasing_ ye!"

Mr. Reddie attaches much force to Berkeley's[653] old arguments against the doctrine of fluxions, and advances objections to Newton's second section, which he takes to be new. To me they appear "such as have been often made," to copy a description given in a review: though I have no doubt Mr. Reddie got them out of himself. But the whole matter comes to this: Mr. Reddie challenged answer, especially from the British Association, and got none. He presumes that this is because he is right, and cannot be answered: the Association is willing to risk itself upon the counter-notion that he is wrong, and need not be answered; because so wrong that none who could understand an answer would be likely to want one.

Mr. Reddie demands my attention to a point which had already particularly struck me, as giving the means of showing to _all_ readers the kind of confusion into which paradoxers are apt to fall, in spite of the clearest instruction. It is a very honest blunder, and requires notice: it may otherwise mislead some, who may suppose that no one able to read could be mistaken about so simple a matter, {347} let him be ever so wrong about Newton. According to his own mis-statement, in less than five months he made the Astronomer Royal abandon the theory of the solar motion in space. The announcement is made in August, 1865, as follows: the italics are not mine:

"The third (_Victoria ..._), although only published in September, 1863, has already had its triumph. _It is the book that forced the Astronomer Royal of England, after publicly teaching the contrary for years, to come to the conclusion, "strange as it may appear," that "the whole question of solar motion in space is at the present time in doubt and abeyance."_ This admission is made in the Annual Report of the Council of the Royal Astronomical Society, published in the Society's _Monthly Notices_ for February, 1864."

It is added that solar motion is "full of self-contradiction, which "the astronomers" simply overlooked, but which they dare not now deny after being once pointed out."

The following is another of his accounts of the matter, given in the _Correspondent_, No. 18, 1865:

"... You ought, when you came to put me in the 'Budget,' to have been aware of the Report of the Council of the Royal Astronomical Society, where it appears that Professor Airy,[654] with a better appreciation of my demonstrations, had admitted--'strange,' say the Council, 'as it may appear,'--that 'the whole question of solar motion in space [and here Mr. Reddie omits some words] is now in _doubt and abeyance_.' You were culpable as a public teacher of no little pretensions, if you were 'unaware' of this. If aware of it, you ought not to have suppressed such an important testimony to my really having been 'very successful' in drawing the teeth of the pegtops, though you thought them so firmly fixed. And if you still suppress {348} it, in your Appendix, or when you reprint your 'Budget,' you will then be guilty of a _suppressio veri_, also of further injury to me, who have never injured you...."

Mr. Reddie must have been very well satisfied in his own mind before he ventured such a challenge, with an answer from me looming in the distance. The following is the passage of the Report of the Council, etc., from which he quotes:

"And yet, strange to say, notwithstanding the near coincidence of all the results of the before-mentioned independent methods of investigation, the inevitable logical inference deduced by Mr. Airy is, that the whole question of solar motion in space, _so far at least as accounting for the proper motion of the stars is concerned_, [I have put in italics the words omitted by Mr. Reddie] appears to remain at this moment in doubt and abeyance."

Mr. Reddie has forked me, as he thinks, on a dilemma: if unaware, culpable ignorance; if aware, suppressive intention. But the thing is a _trilemma_, and the third horn, on which I elect to be placed, is surmounted by a doubly-stuffed seat. First, Mr. Airy has not changed his opinion about the _fact_ of solar motion in space, but only suspends it as to the sufficiency of present means to give the amount and direction of the motion. Secondly, all that is alluded to in the Astronomical Report was said and printed before the Victoria proclamation appeared. So that the author, instead of drawing the tooth of the Astronomer Royal's pegtop, has burnt his own doll's nose.

William Herschel,[655] and after him about six other astronomers, had aimed at determining, by the proper motions of the stars, the point of the heavens towards which the solar system is moving: their results were tolerably accordant. Mr. Airy, in 1859, proposed an improved method, and, applying it to stars of large proper motion, produced {349} much the same result as Herschel. Mr. E. Dunkin,[656] one of Mr. Airy's staff at Greenwich, applied Mr. Airy's method to a very large number of stars, and produced, again, nearly the same result as before. This paper was read to the Astronomical Society in _March_, 1863, was printed in abstract in the _Notice_ of that month, was printed in full in the volume then current, and was referred to in the Annual Report of the Council in _February_, 1864, under the name of "the Astronomer Royal's elaborate investigation, as exhibited by Mr. Dunkin." Both Mr. Airy and Mr. Dunkin express grave doubts as to the sufficiency of the data: and, regarding the coincidence of all the results as highly curious, feel it necessary to wait for calculations made on better data. The report of the Council states these doubts. Mr. Reddie, who only published in _September_, 1863, happened to see the Report of February, 1864, assumes that the doubts were then first expressed, and declares that his book of September had the triumph of forcing the Astronomer Royal to abandon the _fact_ of motion of the solar system by the February following. Had Mr. Reddie, when he saw that the Council were avowedly describing a memoir presented some time before, taken the precaution to find out _when_ that memoir was presented, he would perhaps have seen that doubts of the results obtained, expressed by one astronomer in March, 1863, and by another in 1859, could not have been due to his publication of September, 1863. And any one else would have learnt that neither astronomer doubts the _solar motion_, though both doubt the sufficiency of present means to determine its _amount_ and _direction_. This is implied in the omitted words, which Mr. Reddie--whose omission would have been dishonest if he had seen their meaning--no doubt took for pleonasm, superfluity, overmuchness. The rashness which pushed him headlong {350} into the quillet that _his_ thunderbolt had stopped the chariot of the Sun and knocked the Greenwich Phaeton off the box, is the same which betrayed him into yet grander error--which deserves the full word, _quidlibet_--about the _Principia_ of Newton. There has been no change of opinion at all. When a person undertakes a long investigation, his opinion is that, at a certain date, there is _prima facie_ ground for thinking a sound result may be obtained. Should it happen that the investigation ends in doubt upon the sufficiency of the grounds, the investigator is not put in the wrong. He knew beforehand that there was an alternative: and he takes the horn of the alternative indicated by his calculations. The two sides of this case present an instructive contrast. Eight astronomers produce nearly the same result, and yet the last two doubt the sufficiency of their means: compare them with the what's-his-name who rushes in where thing-em-bobs fear to tread.

I was not aware, until I had written what precedes, that Mr. Airy had given a sufficient answer on the point. Mr. Reddie says (_Correspondent_, Jan. 20, 1866):

"I claim to have forced Professor Airy to give up the notion of 'solar motion in space' altogether, for he admits it to be 'at present in doubt and abeyance.' I first made that claim in a letter addressed to the Astronomer Royal himself in June, 1864, and in replying, very courteously, to other portions of my letter, he did not gainsay that part of it."

Mr. Reddie is not ready at reading satire, or he never would have so missed the meaning of the courteous reply on one point, and the total silence upon another. Mr. Airy must be one of those peculiar persons who, when they do not think an assertion worth notice, let it alone, without noticing it by a notification of non-notice. He would never commit the bull of "Sir! I will not say a word on that subject." He would put it thus, "Sir! I will only say ten words on that subject,"--and, having thus said them, would {351} proceed to something else. He assumed, as a matter of form, that Mr. Reddie would draw the proper inference from his silence: and this because he did not care whether or no the assumption was correct.

The _Mechanics of the Heavens_, which Mr. Reddie sends to be noticed, shall be noticed, so far as an extract goes:

"My connection with this subject is, indeed, very simply explained. In endeavoring to understand the laws of physical astronomy as generally taught, I happened to entertain some doubt whether gravitating bodies could revolve, and having afterwards imbibed some vague idea that the laws of the universe were chemical and physical rather than mechanical, and somehow connected with electricity and magnetism as opposing correlative forces--most probably suggested to my mind, as to many others, by the transcendent discoveries made in electro-magnetism by Professor Faraday[657]--my former doubts about gravitation were revived, and I was led very naturally to try and discover whether a gravitating body really could revolve; and I became convinced it could not, before I had ever presumed to look into the demonstrations of the _Principia_."

This is enough against the book, without a word from me: I insert it only to show those who know the subject what manner of writer Mr. Reddie is. It is clear that "presumed" is a slip of the pen; it should have been _condescended_.

Mr. Reddie represents me as dreaming over paltry paradoxes. He is right; many of my paradoxes are paltry: he is wrong; I am wide awake to them. A single moth, beetle, or butterfly, may be a paltry thing; but when a cabinet is arranged by genus and species, we then begin to admire the {352} infinite variety of a system constructed on a wonderful sameness of leading characteristics. And why should paradoxes be denied that collective importance, paltry as many of them may individually be, which is accorded to moths, beetles, or butterflies? Mr. Reddie himself sees that "there is a method in" my "mode of dealing with paradoxes." I hope I have atoned for the scantiness of my former article, and put the demonstrated impossibility of gravitation on that level with Hubongramillposanfy arithmetic and inhabited atoms which the demonstrator--not quite without reason--claims for it.

In the Introduction to a collected edition of the three works, Mr. Reddie describes his _Mechanism of the Heavens_, from which I have just quoted, as--

"a public challenge offered to the British Association and the mathematicians at Cambridge, in August, 1862, calling upon them to point to a single demonstration in the _Principia_ or elsewhere, which even attempts to prove that Universal Gravitation is possible, or to show that a gravitating body could possibly revolve about a center of attraction. The challenge was not accepted, and never will be. No such demonstration exists. And the public must judge for themselves as to the character of a so-called "certain science," which thus shrinks from rigid examination, and dares not defend itself when publicly attacked: also of the character of its teachers, who can be content to remain dumb under such circumstances."

ON PARADOXERS IN GENERAL.

The above is the commonplace talk of the class, of which I proceed to speak without more application to this paradoxer than to that. It reminds one of the funny young rascals who used, in times not yet quite forgotten, to abuse the passengers, as long as they could keep up with the {353} stage coach; dropping off at last with "Why don't you get down and thrash us? You're afraid, you're afraid!" They will allow the public to judge for themselves, but with somewhat of the feeling of the worthy uncle in _Tom Jones_, who, though he would let young people choose for themselves, would _have them_ choose wisely. They try to be so awfully moral and so ghastly satirical that they must be answered: and they are best answered in their own division. We have all heard of the way in which sailors cat's-pawed the monkeys: they taunted the dwellers in the trees with stones, and the monkeys taunted them with cocoa-nuts in return. But these were silly dendrobats: had they belonged to the British Association they would have said--No! No! dear friends; it is not in the itinerary: if you want nuts, you must climb, as we do. The public has referred the question to Time: the procedure of this great king I venture to describe, from precedents, by an adaptation of some smart anapaestic tetrameters--your anapaest is the foot for satire to halt on, both in Greek and English--which I read about twenty years ago, and with the point of which I was much tickled. Poetasters were laughed at; but Mr. Slum, whom I employed--Mr. Charles Dickens obliged me with his address--converted the idea into that of a hit at mathematicasters, as easily as he turned the Warren acrostic into Jarley. As he observed, when I settled his little account, it is cheaper than any prose, though the broom was not stolen quite ready made:

_Forty stripes save one for the smaller Paradoxers._

Hark to the wisdom the sages preach Who never have learnt what they try to teach. We are the lights of the age, they say! We are the men, and the thinkers we! So we build up guess-work the livelong day, In a topsy-turvy sort of way, Some with and some wanting _a_ plus b. Let the British Association fuss; What are theirs to the feats to be wrought by us? {354} Shall the earth stand still? Will the round come square? Must Isaac's book be the nest of a mare? Ought the moon to be taught by the laws of space To turn half round without right-about-face? Our whimsey crotchets will manage it all; Deep! Deep! posterity will them call! Though the world, for the present, lets them fall Down! Down! to the twopenny box of the stall!

Thus they--But the marplot Time stands by, With a knowing wink in his funny old eye. He grasps by the top an immense fool's cap, Which he calls a philosophaster-trap: And rightly enough, for while these little men Croak loud as a concert of frogs in a fen, He first singles out one, and then another, Down goes the cap--lo! a moment's pother, A spirit like that which a rushlight utters As just at the last it kicks and gutters: When the cruel smotherer is raised again Only snuff, and but little of that, will remain.

But though _uno avulso_ thus comes every day _Non deficit alter_ is also in play: For the vacant parts are, one and all, Soon taken by puppets just as small; Who chirp, chirp, chirp, with a grasshopper's glee, We're the lamps of the Universe, We! We! We! But Time, whose speech is never long,-- He hasn't time for it--stops the song And says--Lilliput lamps! leave the twopenny boxes, And shine in the Budget of Paradoxes!

When a paradoxer parades capital letters and diagrams which are as good as Newton's to all who know nothing about it, some persons wonder why science does not rise and triturate the whole thing. This is why: all who are fit to read the refutation are satisfied already, and can, if they please, detect the paradoxer for themselves. Those who are not fit to do this would not know the difference between the true answer and the new capitals and diagrams on which the delighted paradoxer would declare {355} that he had crumbled the philosophers, and not they him. Trust him for having the last word: and what matters it whether he crow the unanswerable sooner or later? There are but two courses to take. One is to wait until he has committed himself in something which all can understand, as Mr. Reddie has done in his fancy about the Astronomer Royal's change of opinion: he can then be put in his true place. The other is to construct a Budget of Paradoxes, that the world may see how the thing is always going on, and that the picture I have concocted by cribbing and spoiling a bit of poetry is drawn from life. He who wonders at there being no answer has seen one or two: he does not know that there are always fifty with equal claims, each of whom regards his being ranked with the rest as forty-nine distinct and several slanders upon himself, the great Mully Ully Gue. And the fifty would soon be five hundred if any notice were taken of them. They call mankind to witness that science _will not_ defend itself, though publicly attacked in terms which might sting a pickpocket into standing up for his character: science, in return, allows mankind to witness or not, at pleasure, that it _does not_ defend itself, and yet receives no injury from centuries of assault. Demonstrative reason never raises the cry of _Church in Danger_! and it cannot have any Dictionary of Heresies except a Budget of Paradoxes. Mistaken claimants are left to Time and his extinguisher, with the approbation of all thinking non-claimants: there is no need of a succession of exposures. Time gets through the job in his own workmanlike manner as already described.

On looking back more than twenty years, I find among my cuttings the following passage, relating to a person who had signalized himself by an effort to teach comets to the conductor of the _Nautical Almanac_:

"Our brethren of the literary class have not the least idea of the small amount of appearance of knowledge {356} which sets up the scientific charlatan. Their world is large, and there are many who have that moderate knowledge, and perception of what is knowledge, before which extreme ignorance is detected in its first prank. There is a public of moderate cultivation, for the most part sound in its judgment, always ready in its decisions. Accordingly, all their successful pretenders have _some pretension_. It is not so in science. Those who have a right to judge are fewer and farther between. The consequence is, that many scientific pretenders have _nothing but pretension_."

This is nearly as applicable now as then. It is impossible to make those who have not studied for themselves fully aware of the truth of what I have quoted. The best chance is collection of cases; in fact, a Budget of Paradoxes. Those who have no knowledge of the subject can thus argue from the seen to the unseen. All can feel the impracticability of the Hubongramillposanfy numeration, and the absurdity of the equality of contour of a regular pentagon and hexagon in one and the same circle. Many may accordingly be satisfied, on the assurance of those who have studied, that there is as much of impracticability, or as much of absurdity, in things which are hidden under

"Sines, tangents, secants, radius, cosines Subtangents, segments and all those signs; Enough to prove that he who read 'em Was just as mad as he who made 'em."

Not that I mean to be disrespectful to mathematical terms: they are short and easily explained, and compete favorably with those of most other subjects: for instance, with

"Horse-pleas, traverses, demurrers, Jeofails, imparlances, and errors, Averments, bars, and protestandos, And puis d'arreign continuandos."

{357}

From which it appears that, taking the selections made by satirists for our samples, there are, one with another, four letters more in a law term than in one of mathematics. But pleading has been simplified of late years.

All paradoxers can publish; and any one who likes may read. But this is not enough; they find that they cannot publish, or those who can find they are _not_ read, and they lay their plans athwart the noses of those who, they think, ought to read. To recommend them to be content with publication, like other authors, is an affront: of this I will give the reader an amusing instance. My good nature, of which I keep a stock, though I do not use it all up in this Budget, prompts me to conceal the name.

I received the following letter, accompanied by a prospectus of a work on metaphysics, physics, astronomy, etc. The author is evidently one whom I should delight to honor:

"Sir,--A friend of mine has mentioned your name in terms of panigeric [_sic_], as being of high standing in mathematics, and of greatly original thought. I send you the enclosed without comment; and, assuming that the bent of your mind is in free inquiry, shall feel a pleasure in showing you my portfolio, which, as a mathematician, you will acknowledge to be deeply interesting, even in an educational point of view. The work is complete, and the system so far perfected as to place it above criticism; and, so far as regards astronomy, as will Ptolemy beyond rivalry [_sic_: no doubt some words omitted]. Believe me to be, Sir, with the profoundest respect, etc. The work is the result of thirty-five years' travel and observation, labor, expense, and self-abnegation."

I replied to the effect that my time was fully occupied, and that I was obliged to decline discussion with many persons who have views of their own; that the proper way is to publish, so that those who choose may read when they can find leisure. I added that I should advise a precursor in the shape of a small pamphlet, as two octavo volumes {358} would be too much for most persons. This was sound advice; but it is not the first, second, or third time that it has proved very unpalatable. I received the following answer, to which I take the liberty of prefixing a bit of leonine wisdom:

"Si doceas stultum, laetum non dat tibi vultum; Odit te multum; vellet te scire sepultum.[658]"

"Sir,--I pray you pardon the error I unintentionally have fallen into; deceived by the F.R.S. [I am not F.R.S.] I took you to be a man of science [_omnis homo est animal, Sortes est homo, ergo Sortes est animal_][659] instead of the mere mathematician, or human calculating-machine. Believe me, Sir, you also have mistaken your mission, as I have mine. I wrote to you as I would to any other man well up in mathematics, with the intent to call your attention to a singular fact of omission by Euclid, and other great mathematicians: and, in selecting you, I did you an honor which, from what I have just now heard, was entirely out of place. I think, considering the nature of the work set forth in the prospectus, you are guilty of both folly and presumption, in assuming the character of a patron; for your own sense ought to have assured you that was such my object I should not have sought him in a De Morgan, who exists only by patronage of others. On the other hand, I deem it to be an unpardonable piece of presumption in offering your advice upon a subject the magnitude, importance, and real utility of which you know nothing about: by doing so you have offered me a direct insult. The system is a manual of Philosophy, a one inseparable whole of metaphysics and physic; embracing points the most interesting, laws the most important, {359} doctrines the most essential to advance man in accordance with the spirit of the times. I may not live to see it in print; for, at ----, life at best is uncertain: but, live or die, be assured Sir, it is not my intention to debase the work by seeking patronage, or pandering to the public taste. Your advice was the less needed, seeing I am an old-established ----. I remain, etc.--P.S. You will oblige me by returning the prospectus of my work."

My reader will, I am sure, not take this transition from the "profoundest respect" to the loftiest insolence for an _apocraphical_ correspondence, to use a word I find in the Prospectus: on my honor it is genuine. He will be better employed in discovering whether I exist by patronizing others, or by being patronized by them. I make any one who can find it out a fair offer: I will give him my patronage if I turn out to be Bufo, on condition he gives me his, if I turn out to be Bavius.[660] I need hardly say that I considered the last letter to be one of those to which no answer is so good as no answer.

These letters remind me in one respect of the correspondents of the newspapers. My other party wrote because a friend had pointed me out: but he would not have written if he had known what another friend told him just in time for the second letter. The man who sends his complaint to the newspaper very often says, in effect, "Don't imagine, Sir, that I read your columns; but a friend who sometimes does has told me ..." It is worded thus: "My attention {360} has been directed to an article in your paper of ..." Many thanks to my friend's friends for not mentioning the Budget: had my friend's attention been directed to it I might have lost a striking example of the paradoxer in search of a patron. That my Friend was on this scent in the first letter is revealed in the second. Language was given to man to conceal his thoughts; but it is not every one who can do it.

Among the most valuable information which my readers will get from me is comparison of the reactions of paradoxers, when not admitted to argument, or when laughed at. Of course, they are misrepresented; and at this they are angry, or which is the same thing, take great pains to assure the reader that they are not. So far natural, and so far good; anything short of concession of a case which must be seriously met by counter-reasons is sure to be misrepresentation. My friend Mr. James Smith and my friend Mr. Reddie are both terribly misrepresented: they resent it by some insinuations in which it is not easy to detect whether I am a conscious smotherer of truth, or only muddle-headed and ignorant. [This was written before I received my last communication from Mr. James Smith. He tells me that I am wrong in saying that his work in which I stand in the pillory is all reprint: I have no doubt I confounded some of it with some of the manuscript or slips which I had received from my much not-agreed-with correspondent. He adds that my mistake was intentional, and that my reason is obvious to the reader. This _is_ information, as the sea-serpent said when he read in the newspaper that he had a mane and tusks.]

THE DOUBLE VAHU PROCESS.

My friend Dr. Thorn[661] sees deeper into my mystery. By the way, he still sends an occasional touch at the old {361} subject; and he wants me particularly to tell my readers that the Latin numeral letters, if M be left out, give 666. And so they do: witness DCLXVI. A person who thinks of the origin of symbols will soon see that 666 is our number because we have five fingers on each hand: had we had but four, our mystic number would have been expressed by 555, and would have stood for our present 365. Had n been the number on each hand, the great number would have been

(n + 1) (4n^2 + 2n + 1)

With no finger on each hand, the number would have been 1: with one finger less than none at all on each hand, it would have been 0. But what does this mean? Here is a question for an algebraical paradoxer! So soon as we have found out how many fingers the inhabitants of any one planet have on each hand, we have the means of knowing their number of the Beast, and thence all about them. Very much struck with this hint of discovery, I turned my attention to the means of developing it. The first point was to clear my vision of all the old cataracts. I propose the following experiment, subject of course to the consent of parties. Let Dr. Thorn Double-Vahu Mr. James Smith, and Thau Mr. Reddie: if either be deparadoxed by the treatment, I will consent to undergo it myself. Provided always that the temperature required be not so high as the Doctor hints at: if the Turkish Baths will do for this world, I am content.

The three paradoxers last named and myself have a pentasyllable convention, under which, though we go far beyond civility, we keep within civilization. Though Mr. James Smith pronounced that I must be dishonest if I did not see his argument, which he knew I should not do [to say nothing of recent accusation]; though Dr. Thorn declared me a competitor for fire and brimstone--and my wife, too, which doubles the joke: though Mr. Reddie {362} was certain I had garbled him, evidently on purpose to make falsehood appear truth; yet all three profess respect for me as to everything but power to see truth, or candor to admit it. And on the other hand, though these were the modes of opening communication with me, and though I have no doubt that all three are proper persons of whom to inquire whether I should go up-stairs or down-stairs, etc., yet I am satisfied they are thoroughly respectable men, as to everything but reasoning. And I dare say our several professions are far more true in extent than in many which are made under more parliamentary form. We find excuses for each other: they make allowances for my being hoodwinked by Aristotle, by Newton, by the Devil; and I permit them to feel, for I know they cannot get on without it, that their reasons are such as none but a knave or a sinner can resist. But _they_ are content with cutting a slice each out of my character: neither of them is more than an uncle, a Bone-a-part; I now come to a dreadful nephew, Bone-the-whole.

I will not give the name of the poor fellow who has fallen so far below both the _honestum_ and the _utile_, to say nothing of the _decorum_ or the _dulce_.[662] He is the fourth who has taken elaborate notice of me; and my advice to him would be, _Nec quarta loqui persona laboret_.[663] According to him, I scorn humanity, scandalize learning, and disgrace the press; it admits of no manner of doubt that my object is to mislead the public and silence truth, at the expense of the interests of science, the wealth of the nation, and the lives of my fellow men. The only thing left to be settled is, whether this is due to ignorance, natural distaste for truth, personal malice, a wish to curry favor with the Astronomer Royal, or mere toadyism. The only accusation which has truth in it is, that I have made myself a "public scavenger of science": the assertion, which is the {363} most false of all is, that the results of my broom and spade are "shot right in between the columns of" the _Athenaeum_. I declare I never in my life inserted a word between the columns of the _Athenaeum_: I feel huffed and miffed at the very supposition. I _have_ made myself a public scavenger; and why not? Is the mud never to be collected into a heap? I look down upon the other scavengers, of whom there have been a few--mere historical drudges; Montucla, Hutton, etc.--as not fit to compete with me. I say of them what one crossing-sweeper said of the rest: "They are well enough for the common thing; but put them to a bit of fancy-work, such as sweeping round a post, and see what a mess they make of it!" Who can touch me at sweeping round a paradoxer? If I complete my design of publishing a separate work, an old copy will be fished up from a stall two hundred years hence by the coming man, and will be described in an article which will end by his comparing our century with his own, and sighing out in the best New Zealand pronunciation--

"Dans ces tems-la C'etait deja comme ca!"[664]

ORTHODOX PARADOXERS.

And pray, Sir! I have been asked by more than one--do your orthodox never fall into mistake, nor rise into absurdity? They not only do both, but they admit it of each other very freely; individually, they are convinced of sin, but not of any particular sin. There is not a syndoxer among them all but draws his line in such a way as to include among paradoxers a great many whom I should exclude altogether from this work. My worst specimens are but exaggerations of what may be found, occasionally, in the thoughts of sagacious investigators. At the end of the {364} glorious dream, we learn that there is a way to Hell from the gates of Heaven, as well as from the City of Destruction: and that this is true of other things besides Christian pilgrimage is affirmed at the end of the Budget of Paradoxes. If D'Alembert[665] had produced _enough_ of a quality to match his celebrated mistake on the chance of throwing head in two throws, he would have been in my list. If Newton had produced _enough_ to match his reception of the story that Nausicaa, Homer's Phaeacian princess, invented the celestial sphere, followed by his serious surmise that she got it from the Argonauts,--then Newton himself would have had an appearance entered for him, in spite of the _Principia_. In illustration, I may cite a few words from _Tristram Shandy_:

"'A soldier,' cried my uncle Toby, interrupting the Corporal, 'is no more exempt from saying a foolish thing, Trim, than a man of letters.'--'But not so often, an' please your honor,' replied the Corporal. My uncle Toby gave a nod."

I now proceed to die out. Some prefatory remarks will follow in time.[666] I shall have occasion to insist that all is not barren: I think I shall find, on casting up, that two out of five of my paradoxers are not to be utterly condemned. Among the better lot will be found all gradations of merit; at the same time, as was remarked on quite a different subject, there may be little to choose between the last of the saved and the first of the lost. The higher and better class is worthy of blame; the lower and worse class is worthy of praise. The higher men are to be reproved for not taking up things in which they could do some good: the lower men are to be commended for taking up things in which they can do no great harm. The circle problem is like Peter Peebles's lawsuit:

{365}

"'But, Sir, I should really spoil any cause thrust on me so hastily.'--'Ye cannot spoil it, Alan,' said my father, 'that is the very cream of the business, man,--... the case is come to that pass that Stair or Arniston could not mend it, and I don't think even you, Alan, can do it much harm.'"

I am strongly reminded of the monks in the darker part of the Middle Ages. To a certain proportion of them, perhaps two out of five, we are indebted for the preservation of literature, and their contemporaries for good teaching and mitigation of socials evils. But the remaining three were the fleas and flies and thistles and briars with whom the satirist lumps them, about a century before the Reformation:

"Flen, flyys, and freris, populum domini male caedunt; Thystlis and breris crescentia gramina laedunt. Christe nolens guerras qui cuncta pace tueris, Destrue per terras breris, flen, flyys, and freris. Flen, flyys, and freris, foul falle hem thys fyften yeris, For non that her is lovit flen, flyys ne freris."[667]

I should not be quite so savage with my second class. Taken together, they may be made to give useful warning to those who are engaged in learning under better auspices: aye, even useful hints; for bad things are very often only good things spoiled or misused. My plan is that of a predecessor in the time of Edward the Second:

"Meum est propositum genti imperitae Artes frugi reddere melioris vitae."[668]

To this end I have spoken with freedom of books as books, of opinions as opinions, of ignorance as ignorance, of {366} presumption as presumption; and of writers as I judge may be fairly inferred from what they have written. Some--to whom I am therefore under great obligation--have permitted me to enlarge my plan by assaults to which I have alluded; assaults which allow a privilege of retort, of which I have often availed myself; assaults which give my readers a right of partnership in the amusement which I myself have received.

For the present I cut and run: a Catiline, pursued by a chorus of Ciceros, with _Quousque tandem? Quamdiu nos? Nihil ne te?_[669] ending with, _In te conferri pestem istam jam pridem oportebat, quam tu in nos omnes jamdiu machinaris!_ I carry with me the reflection that I have furnished to those who need it such a magazine of warnings as they will not find elsewhere; _a signatis cavetote_:[670] and I throw back at my pursuers--_Valete, doctores sine doctrina; facite ut proxima congressu vos salvos corporibus et sanos mentibus videamus._[671] Here ends the Budget of Paradoxes.

{367}

* * * * *

APPENDIX.

I think it right to give the proof that the ratio of the circumference to the diameter is incommensurable. This method of proof was given by Lambert,[672] in the _Berlin Memoirs_ for 1761, and has been also given in the notes to Legendre's[673] Geometry, and to the English translation of the same. Though not elementary algebra, it is within the reach of a student of ordinary books.[674]

Let a continued fraction, such as

a ----- b + c ----- d + e - f + etc.,

be abbreviated into a/b+ c/d+ e/f+ etc.: each fraction being understood as falling down to the side of the preceding sign +. In every such fraction we may suppose b, d, f, etc. {368} positive; a, c, e, &c. being as required: and all are supposed integers. If this succession be continued ad infinitum, and if a/b, c/d, e/f, etc. all lie between -1 and +1, exclusive, the limit of the fraction must be incommensurable with unity; that is, cannot be A/B, where A and B are integers.

First, whatever this limit may be, it lies between -1 and +1. This is obviously the case with any fraction p/(q + [omega]), where [omega] is between +-1: for, p/q, being < 1, and p and q integer, cannot be brought up to 1, by the value of [omega]. Hence, if we take any of the fractions

a/b, a/b+ c/d, a/b+ c/d+ e/f, etc.

say a/b+ c/d+ e/f+ g/h we have, g/h being between +-1, so is e/f+ g/h, so therefore is c/d+ e/f+ g/h; and so therefore is a/b+ c/d+ e/f+ g/h.

Now, if possible, let a/b+ c/d+ etc. be A/B at the limit; A and B being integers. Let

P = A c/d+ e/f+ etc., Q = P e/f+ g/h+ etc., R = Q g/h + i/k + etc.

P, Q, R, etc. being integer or fractional, as may be. It is easily shown that all must be integer: for

{369}

A/B = a/b+ P/A, or, P = aB - bA

P/A = c/d+ Q/P, or, Q = cA - dP

Q/P = e/f+ R/Q, or, R = eP - fQ

etc., etc. Now, since a, B, b, A, are integers, so also is P; and thence Q; and thence R, etc. But since A/B, P/A, Q/P, R/Q, etc. are all between -1 and +1, it follows that the unlimited succession of integers P, Q, R, are each less in numerical value than the preceding. Now there can be no such _unlimited_ succession of _descending_ integers: consequently, it is impossible that a/b+ c/d+, etc. can have a commensurable limit.

It easily follows that the continued fraction is incommensurable if a/b, c/d, etc., being at first greater than unity, become and continue less than unity after some one point. Say that i/k, l/m,... are all less than unity. Then the fraction i/k+ l/m+ ... is incommensurable, as proved: let it be [kappa]. Then g/(h + [kappa]) is incommensurable, say [lambda]; e/(f + [lambda]) is the same, say [mu]; also c/(d + [mu]), say [nu], and a/(b + [nu]), say [rho]. But [rho] is the fraction a/b+ c/d+ ... itself; which is therefore incommensurable.

Let [phi]z represent

a a^2 a^3 1 + - + ------- + -------------- + .... z 2z(z+1) 2.3.z(z+1)(z+2)

{370} Let z be positive: this series is convergent for all values of a, and approaches without limit to unity as z increases without limit. Change z into z + 1, and form [phi]z - [phi](z+1): the following equation will result--

a [phi]z-[phi](z+1) = ------([phi](z+2)) z(z+1)

a [phi](z+1) a [phi](z+1) a [phi](z+2) or a = - ---------- . z + - ---------- . --- ---------- z [phi]z z [phi]z z+1 [phi](z+1)

a = [psi]z(z+[psi](z+1))

[psi]z being (a/z)([phi](z+1)/[phi]z); of which observe that it diminishes without limit as z increases without limit. Accordingly, we have

[psi]z = a/z+ [psi](z+1) = a/z+ a/(z+1)+ [psi](z+2) = a/z+ a/(z+1)+ a/(z+2)+ [psi](z+3), etc.

And, [psi](z + n) diminishing without limit, we have

a/z . [phi](z+1)/[phi]z = (a/z+) (a/(z+1)+) (a/(z+2)+) (a/((z+3)+ ...))

Let z = 1/2; and let 4a = -x^2. Then (a/z)[phi](z+1) is -(x^2/2) ( 1 - x^2/(2.3) + x^4/(2.3.4.5...)) or -(x/2) sin x. Again [phi]z is 1 - x^2/2 + x^4/(2.3.4) or cos x: and the continued fraction is

(1/4)x^2/(1/2)+ (1/4)x^2/(3/2)+ (1/4)x^2/(5/2)+ ... or -x/2 x/1+ -x^2/3+ -x^2/5+ ...

{371} whence tan x = x/1+ -x^2/3+ -x^2/5+ -x^2/7+ ...

Or, as written in the usual way,

tan x = x ------- 1 - x^2 ------- 3 - x^2 ------- 5 - x^2 ------- 7 - ...

This result may be proved in various ways: it may also be verified by calculation. To do this, remember that if

a_1/b_1+ a_2/b_2+ a_3/b_3+ ... a_n/b_n = P_n/Q_n; then

P_1=a_1, P_2=b_2 P_1, P_3=b_3 P_2+a_3 P_1, P_4=b_4 P_3+a_4 P_2, etc. Q_1=b_1, Q_2=b_2 Q_1+a_2, Q_3=b_3 Q_2+a_3 Q_1, Q_4=b_4 Q_3+a_4 Q_2, etc.

in the case before us we have

a_1=x, a_2=-x^2, a_3=-x^2, a_4=-x^2, a_5=-x^2, etc. b_1=1, b_2=3, b_3=5, b_4=7, b_5=9, etc.

P_1=x Q_1=1 P_2=3x Q_2=3-x^2 P_3=15x-x^3 Q_3=15-6x^2 P_4=105x-10x^3 Q_4=105-45x^2+x^4 P_5=945x-105x^3+x^5 Q_5=945-420x^2+15x^4 P_6=10395x-1260x^3+21x^5 Q_6=10395-4725x^2+210x^4-x^6

We can use this algebraically, or arithmetically. If we divide P_n by Q_n, we shall find a series agreeing with the known series for tan x, _as far as_ n _terms_. That series is

x + x^3/3 + 2x^5/15 + 17x^7/315 + 62x^9/2835 + ...

{372} Take P_5, and divide it by Q_5 in the common way, and the first five terms will be as here written. Now take _x_ = .1, which means that the angle is to be one tenth of the actual unit, or, in degrees 5 deg..729578. We find that when x = .1, P_6 = 1038.24021, Q_6 = 10347.770999; whence P_6 divided by Q_6 gives .1003346711. Now 5 deg..729578 is 5 deg.43'46-1/2"; and from the old tables of Rheticus[675]--no modern tables carry the tangents so far--the tangent of this angle is .1003347670.

Now let x = (1/4)[pi]; in which case tan x = 1. If (1/4)[pi] be commensurable with the unit, let it be (m/n), m and n being integers: we know that (1/4)[pi] < 1. We have then

1=(m/n)/1- (m^2/n^2)/3- (m^2/n^2)/5- ... = m/n- m^2/3n- m^2/5n- m^2/7n- ...

Now it is clear that m^2/3n, m^2/5n, m^2/7n, etc. must at last become and continue severally less than unity. The continued fraction is therefore incommensurable, and cannot be unity. Consequently [pi]^2 cannot be commensurable: that is, [pi] is an incommensurable quantity, and so also is [pi]^2.

I thought I should end with a grave bit of appendix, deeply mathematical: but paradox follows me wherever I go. The foregoing is--in my own language--from Dr. (now Sir David) Brewster's[676] English edition of Legendre's Geometry, (Edinburgh, 1824, 8vo.) translated by some one who is not named. I picked up a notion, which others had at Cambridge in 1825, that the translator was the late Mr. Galbraith,[677] then known at Edinburgh as a writer and teacher.

{373} But it turns out that it was by a very different person, and one destined to shine in quite another walk; it was a young man named Thomas Carlyle.[678] He prefixed, from his own pen, a thoughtful and ingenious essay on Proportion, as good a substitute for the fifth Book of Euclid as could have been given in the space; and quite enough to show that he would have been a distinguished teacher and thinker on first principles. But he left the field immediately.

* * * * *

(The following is the passage referred to at Vol. II, page 54.)

Michael Stifelius[679] edited, in 1554, a second edition of the Algebra (_Die Coss._), of Christopher Rudolff.[680] This is one of the earliest works in which + and - are used.

Stifelius was a queer man. He has introduced into this very work of Rudolff his own interpretation of the number of the Beast. He determined to fix the character of Pope Leo: so he picked the numeral letters from LEODECIMVS, and by taking in X from LEO X. and striking out M as standing for _mysterium_, he hit the number exactly. This discovery completed his conversion to Luther, and his determination to throw off his monastic vows. Luther dealt with him as straight-forwardly as with Melanchthon about his astrology: he accepted the conclusions, but told him to clear his mind of all the premises about the Beast. Stifelius {374} did not take the advice, and proceeded to settle the end of the world out of the prophet Daniel: he fixed on October, 1533. The parishioners of some cure which he held, having full faith, began to spend their savings in all kinds of good eating and drinking; we may charitably hope this was not the way of preparing for the event which their pastor pointed out. They succeeded in making themselves as fit for Heaven as Lazarus, so far as beggary went: but when the time came, and the world lasted on, they wanted to kill their deceiver, and would have done so but for the interference of Luther. {375}

* * * * *

INDEX.

Pages denoted by numerals of this kind (_287_) refer to biographical notes, chiefly by the editor. Numerals like 426 refer to books discussed by De Morgan, or to leading topics in the text. Numerals like 126 indicate minor references.

Abbott, Justice, I, _181_. Abernethy, J., II, _219_. Aboriginal Britons, a poem, II, 270. Academy of Sciences, French, I, 163. Adair, J., I, _223_. Adam, M., I, _65_. Adams, J. C., I, _43_, 82, 385, 388; II, 131, 135, 140, 303. Ady, Joseph, II, 42, _42_. Agnew, H. C., I, 328. Agricola, J., I, 394. Agricultural Laborer's letter, II, 16. Agrippa, H. C., I, _48_, 48. Ainsworth, W. H., II, _132_. Airy, I, _85_, 88, 152, 242; II, 85, 140, 150, 303, 347. Alchemy, I, 125. Alfonso X (El Sabio), II, _269_. Alford, H., II, _221_. Alfred, King, Ballad of, II, 22. Algebra, I, 121. Algebraic symbols, I, 121. Almanac, I, 300; II, 147, 148, 207. (_See Easter._) Aloysius Lilius, I, 362. Alsted, J. H., II, _282_. Ameen Bey, II, 15. Amicable Society, I, 347. Ampere, I, 86. Amphisbaena serpent, I, 31. Anagrams, De Morgan, I, 138. Anaxagoras, II, _59_. Anghera, II, 60, _60_, 61, 279. Annuities, Fallacies of, I, 157. Antichrist, I, 130. Antimony, I, 125. Antinewtonism, I, 162. Antinomians, I, 394. Antiphon, II, _59_. Antonie, F., I, _126_, 126. Apollonius, I, 41, 107. Apparitions, II, 47. Arago, I, _243_, 390. Aratus, II, _167_. Arbuthnot, I, _133_, 134. Archer, H., II, 90. Archimedes, I, 5, 11, 42, 83, 107. Archytas, I, _53_. Argoli, I, _104_. Aristocrat, as a scientist, I, 131. Aristotle, I, 5, 331. Arnobius, II, _73_. Arson, P. J., II, _207_. Ashton, R., II, _99_. Astrology, I, 118, 127, 128, 350; II, 43. Astronomer's Drinking Song, I, 380. Astronomical Aphorisms, I, 398. Paradox, I, 394. Police Report, I, 390. Society. (_See Royal Astronomical Society._) Astronomy, Bailly's exaggerated view of, I, 166. Astunica, Didacas, I, 90. Athanasian Creed, I, 371. Atheists, Philosophical, I, 1. Atoms, II, 191. {376} Attraction, I, 136, 151, 155. Augustine, St., II, _23_. Aurora borealis, I, 134. Austen, Jane, I, 191. Auzout, A., II, _300_. Aviation, Early ideas of, II, 8.

Babbage, C., I, _207_, 290, 291; II, 181. Bachet, de Meziriac, I, _161_. Bacon, F., I, 5, _75_, 75, 76, 79, 89, 145, 331. Bacon, R., I, 5, _126_, 126, 360; II, 94. Baconian controversy, I, 141. Baden Powell, II, _267_. Bailly, J. S., I, 166, _166_, 308. Baily, F., I, 308, _309_; II, 16, 143, 188. Baily, R., II, 16. Baker, T., II, _302_. Bakewell, F. C., II, 156, _156_. Banks, J., I, 28. Barberini, M., II, _267_. Barker, C., II, _262_. Baronius, I, 33, 35; II, _62_. Barreme, I, _42_. Barrett, G., II, _188_. Barrow, I., I, _160_; II, 302. Baruel, de, I, 165. Bassano, Duc de, II, 3, 339. Baxter, T., I, 146. Bayle, P., II, _73_. Beaufort, F., II, _267_. Beaugrand, I, 119, _121_. Beaulieu, I, _119_, 119, 121. Beaune, de, II, _59_. Becourt, R., II, 277. Bedford, Duke of, (6th), I, _182_. Behmen, I, _168_, 254; II, 317. Bellenden, W., I, _175_. Bentley, I, _110_. Berkeley, G., II, 346. Bernard, E., II, _297_, 300. Bernardus Trevisanus, I, _126_, 126. Bernoullis, I, 130, _150_, _335_, 336. Bertius, P., II, _300_. Bese, I, _66_. Bessel, I, _384_; II, 2. Bethune, I, _99_, 279, 291. Bettesworth, I, 19. Beza. (_See Bese._) Bickersteth, E. H., I, _238_. Bidder, I, _86_. Biden, J., II, 158, _160_. Bidle, (Biddle), I, 239. Biot, I, _85_. Birch, T., I, _108_; II, 304, 313. Birks, T. R., II, 158, _158_. Bishop, G., I, _386_. Bishops as Paradoxers, I, 226. Boccaccio, I, 38. Boethius, I, _42_, 45. Boehme. (_See Behmen._) Boncompagni, I, _298_. Boniface, St., I, 32. Bonnycastle, J., II, _16_. Booker, I, 115. Boole, G., I, _261_, 332; II, 75, 79. --A tribute to, II, 79. Borelli, G. A., II, _300_. Borello, I, _69_. Boreman, I, 113. Borron, Mrs., II, 7. Boscovich, I, _156_, 164. Bouguer, II, _301_. Bouillaud, I, _87_; II, 295. Bouvard, A., I, _327_. Bovillus, I, _44_; II, 324. --Epitome of, I, 44. Bowdler, H. M., I, _194_. Bowring, J., I, _352_; II, 256. Boyle, R., I, 24, _125_; II, 300. Bradley, I, 24. Bradwardine, I, _227_, 228, 229. Brahe. (_See Tycho B._) Brancker, I, 107; II, _300_. Brenan, J., I, 330, _330_. Brewster, D., I, 39, _137_, 140; II, 214, 288, 372. Briggs, I, _69_; II, 299, 302. Bright, J., II, _235_. Brinkley, J., I, _311_. Britannicus, D., II, 8. British Museum library, I, 151. Brothers, R., I, _315_; II, 97. Brougham, Henry, Lord, I, _191_. Brouncker (Brounker), I, _132_; II, 302. Brown, W., II, _168_. Browne, T., I, 31. Brucker, I, _61_. Brunet, I, _402_. Bruennow, I, _386_. Bruno, I, _59_, 59. Bryson, II, _59_. Buergi, I, 52. Buffon, I, _282_. Bulstrode, II, 84. Bungus, I, _55_, 55, 57. Buoncompagno, U., I, _362_. {377} Burgon, J. W., II, _30_. Buridan, I, _37_. --Questiones morales, I, 37. Buridan's Ass, I, 37. Burke, E., I, _173_. Burlesque, Frend's, I, 208. Burnet, G., I, _107_. Burney, Frances, I, _190_. Burton, Frances B., I, 374. Busby, R., II, _313_. Buteo, I, _51_. Butler, G., I, _199_. Butler, S., II, _218_. Buxton, J., I, _86_. Byrgius. (_See Buergi._) Byrne, O., I, _329_; II, 186, 190. Byron, I, 186; II, 270, 273.

Cabbala, I, 272. Calculating Boys, I, 86. Calculus, I, 129. Calendar. (_See Easter._) Cambridge Poets, II, 269. Campanus, I, 42, _43_. Canning, Geo., II, _145_. Carcavi, I, _106_. Cardanus, II, _59_. Carlile, R., I, _271_. Carlyle, T., II, _373_. Carnot, I, 107. Caroline tables, I, 124. Casaubon, I, _111_. Case, J., I, _128_, 128. Cassini, J., I, _172_. Castel, I, _148_, 148. Castiglioni, I, _139_. Castlereagh, I, 185, _186_. Cataldi, I, _69_, 69. Catcott, A., I, _237_. Causans, de, I, _298_. Cavalieri, I, _106_. Cavendish, C., I, _106_; II, 299, 312. Cavendish, W., I, _290_. Caxton, W., II, _281_. Cayley, A., II, _292_. Cecil, R. (1st Earl of Salisbury), II, _330_. Centrifugal force, II, 268. Ceulen. (_See Van Ceulen._) Challis, J., I, _390_; II, 141. Chalmers, I, _102_; II, 219. Chambers, E., II, _282_. Chambers, R., I, _344_, 344. Charles IX, II, 94. Charles X, II, 1. Chasles, I, _39_, 229. Chesterfield, Earl of (4th), II, _298_. Chiffinch, W., II, _50_. Ch'in Chiu-shang, II, 66. Chitty, J., II, _323_. Chiu-chang, Suan-shu, II, 67. Christian Evidence Society, I, 270. Christie, I, 27. Christmann, I, 272, _272_. Church question, I, 62. Church, The word, II, 30. Circle squarers. (_See Squaring the Circle._) Circulating media of mathematics, I, 107. Ciruelo. (_See Sanchez._) Clairaut, I, _219_, 382. Clarence, Duke of, I, 179. Clarke, R., I, _255_. Clavius, I, 11, _69_, 111, 112, 335, 362, 363, 372; II, 59. Clayton., Geo., II, _98_. Cluvier, D., II, 332, _332_. Cobb, Mary, II, 117. Cobbett, W., I, _177_, 200, 399. Cobden, R., II, 217. Cocker, I, _42_; II, 64, 173, 251, 307. Cody, P., II, 208. Coke, E., II, _331_. Colburn, Z., I, _86_. Colenso, I, _63_, 247; II, 191. Collins, J., I, _107_; II, 297, 300, 302, 313. Colvill, W. H., II, 68. Cometic astrology, I, 128. Comets, I, 128; II, 68, 83. Cominale, C., I, _162_, 162. Compton, S. J. A., II, _19_. Computation, Paradoxes of, II, 251, 267. Condamine, C. M. de la, II, _301_. Conduitt, John, I, _397_. Conduitt, Mrs., I, _136_. Congregation of the Index, I, 90. Converse propositions, I, 295. Convocation at Oxford, I, 96. Cooke, Margaret, I, _310_. Cooper, A. A. (Shaftsbury), II, _181_. Copernicus, I, 5, 6, _76_, 90, 121, 172, 380; II, 165, 335. Copley, J. S., I, _198_. Cormouls, I, 225. Cosmology, I, 172. {378} Cotes, R., II, _301_. Cottle, Mrs., II, 97, _97_, 161. Craig, J., I, _129_, 129. Creed, Mathematics of a, I, 329. Cribb, T., I, _314_. Crotus, J., I, 318. Cruickshank, G., I, _186_. Cube, Duplication of, I, 349. Cumyns, Eliza, I, 299. Cunningham, I, 172, _172_. Curabelle, I, _221_. Curious Calculations, II, 66. Curll, E., II, _279_. Cusa, I, _44_, 47, 360. Custom, II, 324. Cyclometry, II, 208. (_See Squaring of the Circle._) Cyclopaedias, Review of, II, 280.

D'Alembert, I, _382_; II, 283, 364. Dalgarno, I, 116, _117_. Dalton, J., I, _255_. D'Arblay, Mme., I, _190_. Darwin, E., II, _8_. Darwinism, Primitive, I, 344. Dary, M., II, _305_. Daval, P., II, _298_. Davies, T. S., II, 151, _151_, 188. Day, A., I, 295, _295_. De Baruel, I, 165. De Beaune. (_See Beaune._) De Becourt, II, 277, _277_. Debenham, J., I, _393_. De Causans. (_See Causans._) Dechales. (_See de Challes._) De Challes, I, _45_. Decimal coinage, II, 80, 168, 169. Decimals run riot, II, 80. Dee, J., II, _302_. De Faure, I, 149. De la Leu, I, _297_. Delambre, I, _160_, 167, 354; II, 165. Democritus, II, _34_. De Moivre, I, 24, _376_; II, 298. De Molieres, I, _220_. De Molina, I, _297_. Demonville, I, 291, 293. De Morgan, A., I, 191, 383; II, 194. --Refusal of LL. D., I, 191. De Morgan, G. C., I, 383. De Morgan, Mrs., I, _196_; II, 194. Denison, J., I, 348, 353. Desaguliers, I, _153_, 156, 157. Desargues, I, _119_, 221. Descartes, I, 5, 59, 105, 132, 165, 204, 220; II, 94. De Serres, II, _60_. De Sluse. (_See Sluse._) De Thou, I, 51, _111_, 113; II, 295. De Vausenville, I, 12. Devonshire, Duke of (7th), I, _290_. Diamandi, I, 86. Didacus Astunica, I, 90. Diderot, II, _4_, 283, 339. Digby, K., I, _108_. Digges, T., and L., II, _302_. Dionysius Exiguus, I, _360_. Dircks, H., II, 138, _138_. Discoverers and discoveries, II, 206. Discovery, Basis of, I, 85. D'Israeli, I., I, _115_, 118, 136, 188, 227. Ditton, I, _133_, 133. Division, Nature of, II, 248. Dobson, J., I, _234_, 234. Dodson, J., II, _312_. Dodt, I, _52_. Doggerel verse, I, 341. Dolland, I, _377_. Double Vahu Process, II, 360. Douglas, G., I, _232_. Drach, S. M., II, _317_. Drayson, G. A. W., II, _132_, 132. Dryden, II, _71_. Dual arithmetic, II, 186. Duchesne, I, _52_. Dumortier, I, 313. Duncan, A., I, _179_. Dunkin, E., II, _349_. Duodecimal scale, II, 68. Duplication Problem, I, 349. Dupuy, J. and P., II, _295_. Dutens, L., II, _90_. Dyer, G., I, _178_.

Earth, Figure of, II, 54. Easter, I, 359. Easter Day Paradoxes, I, 353. Ebrington, Thos., I, _247_. Edgeworth, Maria, I, _191_. Editorial System, I, 15. Edleston, I, _140_; II, 296. Edwards, J., I, _144_. Edwards, T., I, _112_. Eirenaeus Philalethes, I, _125_, 125, 126. Eldon, Lord (1st), II, _197_. Elephant story, I, 58. Elizabeth, Queen, I, 128. Ellenborough, Baron, I, _181_. {379} Ellicot, I, 24. Ellis, I, _76_, 82. Engel, I, 230. English language, Origin of, I, 215. Enriques, F., II, _367_. Epps, J., I, _153_; II, 143. Equation of fifth degree, I, 250, 373. Erasmus, I, 110. Erastus, I, _65_. Erichsen, I, 163. Ersch, II, _193_, 282. Erskine, T., II, _127_. Esperanto, Forerunner of, I, 116. Euclid, I, 5, 43; II, 118, 151. --Without Axioms, I, 287. Eudoxus, II, _164_. Euler, I, 221, _382_; II, 3, 4, 303, 331, 339. Eusebius, II, _220_. Eustace, J. C., II, _46_. Eutocius, I, _41_; II, 60. Evelyn, J., I, _108_. Everett, J., I, _346_. Evidence, I, 57, 58.

Faber. (_See Stapulensis._) Fairfax, Mary, I, _242_. Falco, I, 53. Faraday, M., II, _351_. Faure, de, I, 149; II, 238. Fawcett, H., II, _249_. Ferguson, J., II, _20_. Fermat, I, 122, 221; II, _300_. Ferrari, S., II, 68. Fiction, New era in, I, 189. Fienus, I, _74_, 74. Filopanti, Q. B., II, _93_. Finaeus, I, _50_, 50, 113. Finleyson, J., I, 314, _314_. Flamsteed, I, _87_, 309; II, 45, 143, 302, 306. Fletcher, I, 378. Fludd, II, _318_. Fly-leaf Paradox, II, 264. Folkes, M., I, _136_; II, 301. Fontenelle, I, _103_. Forbes, D., I, _237_. Forman, W., I, 296, _296_, 306. Forster, T. I. M., I, 320, _320_. Foscarini, I, _90_. Foster, S., II, _310_. Fourier, II, _66_. Fox, G., I, _397_. Francis, P., II, _96_. Francoeur, I, _365_. Frankland, W. B., I, 230, 287. Franklin, J., II, _265_. Freedom of opinion, Growth of, I, 265. Freher, A., II, 319. French academy on circle squaring, I, 163. Frend, W., I, _196_, 196, 206, 208, 252. Fresnel, II, _48_. Fromondus, I, _74_, 74, 99. Frost, I. and J., I, 394. Fry, Elizabeth, I, 224. Fuller, T., I, _86_. Fulton, R., I, _148_.

Gadbury, J., I, _115_, 115. Galbraith, J. A., II, 372. Galileo, I, 5, 6, 32, _76_, 82, 83, 96, 122, 381. Galle, J. G., I, _386_; II, 7. Galloway, I, _56_, 57; II, 143. Gamblers, I, 280. Garrick, I, 21. Gascoigne, W., II, _299_. Gassendi, I, _107_. Gauss, I, 86, 107, 310. Gemistus, G., I, _188_. Gentleman's Monthly, Miscellany, I, 208. Gephryander. (_See Salicetus._) Gergonne, I, _336_. Ghetaldi, I, _83_; II, 59. Ghost paradox, II, 47. Giddy (Gilbert), II, _174_. Gilbert, Davies, II, 66, _174_. Gilbert, William, I, 6, _68_, 68, 76. Gillot, II, _315_. Glazier (Glazion), II, 7. Godwin, F., I, _103_. Godwin, W., I, _174_. Golius, I, _106_. Gompertz, B., I, _378_. Goulburn, I, _288_. Goulden, S., II, 88. Graham, I, 24. Grandamicus, I, _104_, 104. Granger, J., I, _156_. Grant, A. R., II, 131. Grant, R., I, _392_; II, 131. Grassi, O., I, _262_. Grassini, I, _231_. Graunt, J., I, 113, _114_, 154. Gravity, I, 151, 244, 348, 353. {380} --Newton's apple, I, 136. Greek numerals, II, _77_. Greene, R., I, _135_, 135. Greenhill, Sir G., I, 136. Greenwich observatory, I, 87. Gregg, T. D., II, 75, _75_. Gregorian Calendar, I, 363. Gregory, D., I, _66_; II, 301. Gregory, J., I, _118_, 118, 207; II, 302. Gregory O., II, _71_. Gregory, Pope, I, 362. Grevil, I, _202_. Grey, C., (2d Earl), I, _315_; II, 247. Grosart, I, _141_, 141, 145. Grove, W. R., II, _320_. Gruber, II, _193_, 282. Gruenberger, I, _70_. Grynaeus, I, _66_. Guaricus, I, _43_. Guillim, J., II, _28_. Guldin, I, _83_. Gumpach, Von, II, 137, _137_. Gunning, H., I, _198_. Gurney. (_See Fry, E._) Guthrie, W., I, _395_.

Hailes, J. D., II, 135, _135_. Hailesean system of astronomy, II, 135. Hale, M., I, _123_, 123. Hales, S., I, _123_. Hall, B., II, _181_. Hallam, I, 159. Halley, I, 24, _124_, 311; II, 301, 332. Halliwell-Phillips, II, _148_, 296. Hamilton, W., I, _112_, 117, 331, 335, 339, 341, 342; II, 52, 53, 111. Hamilton, W. Rowan, I, _332_; II, 104, 256, 343. Hanover, King of, I, 201. Hardy, C., I, _298_. Hardy, T., I, _178_. Harriot, T., II, _302_. Harvey, I, _76_, 78; II, 201. Hauff, I, _230_. Haughton, S., II, 372. Hauksbee, F., I, _156_. Hayes, C., I, _132_, 132. Heath, D. D., I, _76_. Heinfetter, H., II, 94, _94_. Helmont, J. B. van, I, _125_. Henson, II, 8. Heraclitus, II, _34_. Herbart, J. F., I, 253, _253_; II, 78. Herigone, II, _59_. Herschel, J., I, _80_, 299, 326, 383, 386; II, 88, 95, 181, 255, 261, 262. Herschel, W., I, _81_, 151, 192, 225, 233, 299; II, 288, 348. Heywood, F., II, _49_. Hicks, J. P., II, _67_. Higgins, G., I, _257_, 274. Hilarius, Pope, I, _359_. Hill, J., I, 21, 22, 23, 24. Hill, Rev. R., I, _192_. Hill, Sir R., I, _165_, 232. Hind, J. R., I, _384_. Hippocrates, II, _59_. Hoax, An interesting, I, 163. --Lunar Caustic, I, 288. --Moon (Herschel), I, 326; II, 131. Hobbes, I, _105_, 109, 143, 144; II, 80. Hobhouse, J. C., II, _126_. Hodder, J., II, _265_. Hodge, C. B., I, 114. Hodges, W., I, 237. Hoffmann, J. J., II, _282_. Hoffmann, J. J. I. von, I, _230_. Holloway, B., I, _237_. Holmes, O. W., I, 109. Holyoake, G. J., I, _399_, 399. Hone, W., I, 124, 180, 184, 185. Hook, T. E., II, _261_. Hooke, I, _77_; II, 300. Hooker, R., II, _201_. Hopkins, J., II, 41. Horace, I, 40. Horne, G., I, _152_, 152, 154, 155, 236. Horne, J., I, _178_. Horner, L., I, _176_. Horner, W. G., II, _66_, 151, 187. Houlston, W., II, 156, _156_. Howard, E., I, 131. Howison, W., I, 256, _256_. Howitt, W., II, 193, _193_. Howley, I, 63. Hulls, I, _147_, 147; II, 8. Hume, J., I, _352_; II, 174. Husain Rifki, II, 16. Hussein Effendi, II, 15. Hutchinson, J., I, 154, _154_. Hutton, C., I, _153_, 161; II, 303, 340. Huyghens, I, _100_, 133; II, 300.

Imaginary numbers, II, 186. Impalement by request, II, 133. Inaudi, I, 86. Index Expurgatorius, I, 90. {381} Infant prodigies, I, 86. Inglis, J. B., II, _52_. Inglis, R. H., I, _352_. Ingliz Selim Effendi, II, 15. Innocent I., I, _359_. Irving, E., II, _54_. Ivory, J., II, 142, _142_.

Jabir ben Aflah, II, _59_. Jack, R., I, 149. Jacotot, J., I, 278, _278_. Jameson, Mrs., II, _63_. Jeffreys, G., I, _183_. Jenner, E., II, _205_. Jesuit contributions, I, 164. Johnson, H. C., I, 350. Johnson, S., I, 20, _190_, 259; II, 117. Johnston, W. H., II, 67. Jombert, I, _161_. Jonchere, I, _146_, 146. Jones, W., I, _135_; II, 298, 301. Jones, Rev. W., I, _237_. Jonson, B., I, 13. Journals, Three classes of, II, 144.

Kantesian Jeweler, I, 258. Karsten, I, _230_. Kaestner, I, _43_, 110, 112. Kater, I, 11. Keckermann, I, 3. Keill, J., II, _302_. Kepler, I, 52, _76_, 82, 381; II, 166. Kerigan, T., I, _308_, 353. Keroualle, De, II, 50. Kersey, I, 107. King, Wm., I, _246_. Kircher, Adolphe, I, 229. Kircher, Athanasius, I, _103_. Kirkringius, T., I, 125, _125_. Kittle, I, 236. Klein, F., II, 367. Knight, C., II, _109_, 280, 289. Knight, G., I, 151, _151_. Knight, R. P., II, _274_. Knight, Wm., I, _97_. Koenig, S., I, _150_.

Lacomme, I, 46. La Condamine, II, _301_. Lacroix, I, 41, 159, 207. Lactantius, I, 33, _96_. Lagrange, I, 221, _288_, 313, 382; II, 86. Laing, F. H., II, _186_, 186. Lalande, I, _159_. Lamb, C., I, 178; II, 270. Lambert, J. H., I, _336_; II, 214, 367. Lambert, John, II, _309_. Language, Test of, II, 327. Lansbergius, I, _70_, 70. Laplace, I, 24, _255_, 382; II, 1, 340. Lardner, D., II, _253_. Lardner, N., I, 14; II, _221_. Laud, I, _145_. Lauder, W., I, 297. Laurent, P., I, _309_, 309. Laurie, J., II, 4. Laurie, P., II, _42_. Laurus, I, 381. Law, Edmund, I, _181_. Law, Edward, I, _181_. Law, W., I, 168, _254_; II, 317. Le Coq, I, _86_. Lee, R., I, _66_. Lee, S., I, _131_. Lee, W., I, 157. Legate, I, _59_. Legendre, I, _229_; II, 215, 367. Legh, P., II, _68_, 83. Leibnitz, I, 5, 7; II, 46. Leo, St., I, _359_. Leverrier, I, _43_, 82, 383, 386, 388, 390; II, 7, 135, 140, 303. Lewis, G. C., II, 162, _162_. Libri, I, _40_, 62; II, 295. Lilius, Aloysius, I, 362. Lilly, I, _115_; II, 302. Lipen, M., I, _298_. Little, J., I, 206. Livingston, R., I, _148_. Locke, J., I, _142_, 142, 144; --and Socinianism, I, 142. Locke, R., I, 146. Locke, R. A., I, _326_; II, 86, 131. Logan, W. E., I, _337_. Logic, Formal, I, 158; II, 75. --Has no paradoxes, I, 330. London Mathematical Society, I, 383. London, University of, I, 259; II, 71. Long, G., II, _290_. Long, J. St. J., II. _38_, 205. Longitude problems, I, 132, 146, 249. Longley, C. T., I, _325_. Longomontanus, I, _105_, 105. Lottery, I. 281. Lovett, R., I, 165, _166_. Lowe, R., II, _169_. Lowndes, W. T., I. _402_. Lubbock, J., I, _279_; II, 148. {382} Lucas, F., II, _28_. Lucian, I, _102_. Lunar Caustic Joke, I, 288. Lunn, J. R., II, _66_. Lydiat, T., II, _302_. Lyndhurst, I, _198_.

Macclesfield, Earls of, I, 7; II, _296_, 301. Macclesfield, Letters, II, 296. MacElshender, II, 87. Machin, J., II, _301_. Mackey, John, I, 349. Mackey, S. A., I, _256_. Maclear, T., II, _181_. Macleod, H. D., II, 184, _184_. Magic, I, 118. Magna Charta, I, 25. Magnus, I, 42. Maitland, S., I, _63_, 163. Malacarne, I, 119. Malden, H., II, _162_. Malius, II, _342_. Mallemens, II, _333_. Mankind gullible, I, 115. Manning, H. E., (Card.), II, _233_. Mansel, H. L., II, _162_. Marcelis, J., I, _129_, 129. Maret, II, _3_. Margarita Philosophica. (_See Reisch._) Marryat, Capt., II, _87_. Marsh, H., I, _199_, 271. Martin, B., I, _152_, 153. Martin, R., II, _236_. Maseres, F., I, _197_, 203. Mason, M., II, _132_. Mathematical Illustrations of Doctrine, II, 70; --Psychology, I, 253; --Society, I, 374, 376, 382; --Theology, I, 149. Mathematics, Condensed history of, II, 58. Matter to Spirit, II, 194. Maty, I, 23. Maupertuis, II, _301_. Maurice, F. D., II, _101_. Maurolycus, I, _121_. Maxwell, A., I, _102_. Meadley, G. W., I, _223_. Mechanics Magazine, II, 141, 145. Medici, Cosmo de, II, _295_. Medicine, Status of, I, 266. Melanchthon, II, _323_. Menestrier, I, _127_, 127. Mercator, G., II, _92_. Mercator's projection, I, 84. Mersenne, I, _106_, 107; II, 295, 297. Meslier, J., II, _195_. Meteorologist, An early, I, 320. Meteorology, I, 327. Metius, A. and P., I, _52_, _99_, 99. Meton, II, 167. Metric System, Forerunner of, I, 171. Meziriac, I, 161. Milbanke, A. I., I, 225. Mill, Jas., I, 260. Miller, Joe, I, _182_. Miller, S., I, 167. Mills, Elizabeth, W., II, 7. Milne, J., I, _286_. Milner, I., I, _251_, 251. Milner, J., II, _23_. Milner's lamp, I, 252. Milward, II, _250_. Miracles _vs_. Nature, II, 6. Mitchell, J., I, _242_. Moliere, I, _232_. Molina, A. C. de, I, _297_. Mollendorff, von, II, _3_, 338. Mondeux, I, 86. Montague, C., II, _311_. Montmort, P. R. de, II, _301_. Montucla, I, 40, 45, 54, 65, 117, 120, 159, 163; II, 60. Moon Hoax, I, 326; II, 131. Moon, Nature of, II, 84; --Rotation of, II, 4, 19, 84, 87. More, Hannah, I, _189_, 192. More, Henry, I, 123. Moore, Dr. John, I, _190_. Moore, Sir John, I, _190_. Morgan, S., I, 6. Morgan, T., I, 191. Morgan, W., I, _223_, 224. Morhof, I, _61_. Morin, I, _99_, 99. Morinus, J. B., I, 149. Morland, S., II, _302_. Mormonism, II, 69. Morrison, R. J., I, _321_; II, 43. Mose, H., II, _266_. Mottelay, I, 68. Motti, II, 60. Mouton, I, _172_; II, 300. Muggleton, I, 394, _395_. Multiplication, Nature of, II, 251. Murchison, R. I., I, _384_. Murhard, I, _43_, 298. {383} Murphy, A., II, _308_. Murphy, J. L., II, 54, _54_. Murphy, P., I, _327_, 398. Murphy, R., I, 349, _349_. Murray, J., I, _186_; II, 145. Murray, L., II, _326_. Murray, Mungo, II, 310. Musgrave, T., I, _324_. Mydorge, I, _298_. Mystrom, J. W., II, 182. Mythological paradoxes, I, 256.

Names of Religious Bodies, II, 22. Napier, J., I, 5, 66, _67_, 82. Napoleon, Doubts as to, I, 246. Nautical Almanac, I, 300; II, 147. Neal, I, _98_. Negative numbers, I, 196, 203. Neile, W., II, _190_. Neptune, Discovery of, I, 387; II, 140. (_See Adams, Leverrier._) Nesse, C, I, _128_, 128. Newcomb, S., I, 162. Newcomen, T., I, _147_. Newton, Sir Isaac, I, 5, 6, 8, 24, 39, 84, 88, 130, 136, 139, 144, 145, 148, 152, 154, 155, 162, 165, 167, 197, 225, 237, 242, 257, 282, 296, 297, 309, 311, 382, 394, 395, 396, 397; II, 2, 70, 184, 297, _302_, 305. Newton, John, II, _305_. Nicene Creed, I, 371. Nichol, J. P., II, _289_. Nicholas. (_See Cusa._) Nichols, J., I, _175_. Nicolas, N. H., I, _354_. Nicollet, I, _326_; II, 131. Nicolson, W., I, _201_. Nieuwentijt, II, _333_. Nizzoli, M., II, 275. Non-Euclidean geometry, II, 83. Northampton, Marquis of (2d), II, _19_. Novum Organum Moralium, II, 74. Number, Mystery of, I, 55, 56, 169. Number of the Beast (666), I, 55, 130, 272, 298, 352; II, 77, 159, 217, 218, 361, 373. Numeral system, II, 68. Nursery rhymes, II, 150.

Occam, Wm. of, II, _40_. Odgers, N., II, 191, _191_. Oinopides of Chios, II, _59_. Oldenburgh, H., II, _300_, 302. Orthodox Paradoxes, II, 363. Orthography, Paradoxes of, II, 267. Ortwinus, I, 319. Oughtred, W., II, _298_, 303. Owenson, I, _191_. Ozanam, I, _161_, 312.

Pagi, I, 32. Paine, T., I, 173, _173_, 271. Paley, W., I, _222_; II, 226. Palmer, C., I, 225. Palmer, H., I, _141_, 141, 145. Palmer, J., II, _253_. Palmer, T. F., II, _254_. Palmer, W., II, _37_. Palmerston, Viscount (3d), I, _290_, 352. Palmezeaux, I, 167. Panizzi, A., I, _151_. Papist, II, 26. Paracelsus, II, 322. Paradox defined, I, 2, 31. Paradox, religious, I, 236. Paradoxers in general, II, 352. Parallels, Theory of, I, 229, 344. Pardies, I. G., II, _300_. Park, Mungo, II, _91_, 132. Parker, F., II, _94_. Parker, G. (Earl of Macclesfield), II, _296_. Parr, S., I, 173, _173_, 175, 176, 184. Parsey, I, 293, _293_. Partridge, J., I, _305_. Pasbergius, I, 381. Pascal, I, 39, 119, 122, _220_, _221_; II, 73. Pascal's Hexagram, I, 221. Passot, I, 279, _279_. Passover, I, 358, 372. Patriotic paradox, I, 231. Paucton, I, _172_. Paulian, I, 165, _165_. Peacock, Geo., I, _196_, 350. Peacock, T. L., I, _190_, 340. Pearce, A. J., II, 43. Pearne, T., I, _239_. Peel, Sir R., I, 290, _352_. Peel, W. Y., I, _290_. Pelerin, J., II, _324_. Pell, J., I, _105_, 105, 107; II, 300, 302, 312. Pepys, I, _113_, 114. Perigal, H., II, 19, _20_. {384} Perpetual motion, I, 118, 348; II, 55, 138. Perspective, New theory of, I, 293. Peters, W., II, 11, 315. Petitioning Comet, I, 127. Petrie, W. M. F., I, _328_. Petty, I, _114_; II, 300. Philalethes, Eirenaeus, I, _125_, 125, 126. Philalethes, Eugenius, I, _255_. Phillips, R., I, 242, _242_, 245. Philo of Gadara, I, 40, _40_. Philosopher's stone, I, 118, 125. Philosophical atheists, II, 1. Philosophy and Religion, II, 37. Phonetic spelling, II, 81. [pi], values of, I, 11, 43, 45, 46, 52, 69, 100, 110, 129, 135, 146, 245, 283, 284, 294, 347, 348, 349, 350; II, 60, 63, 105, 110, 118, 135, 156, 209, 279, 315, 316. Pighius, I, _372_. Pike, S., I, 236, _236_. Pindar, P., II, _272_. Piozzi, Mrs., I, _235_; II, 272. Piscator, B., II, 25. Pitman, F., II, 81, _81_. Place, F., I, _199_. Planets inhabitable, I, 100, 102. Plato, I, 5. Platt, H., I, _126_, 126. Playfair, J., I, _233_. Pletho, G., I, _188_. Pliny, II, 280. Ploucquet, I, _336_, 337. Poe, E. A., II, 132. Poincare, I, 136. Poisson, I, _292_; II, 2. Pollock, J. F., II, _174_. Pons, II, _45_. Pope, Wm., I, 277, _277_. Porta, I, _68_, 68, 83. Porteus, B., I, _193_, 203. Porteus, H. F. A., II, 157, _157_. Porus, I, 44. Powell, Baden, II, _267_. Powell, W. S., I, _222_. Pratt, H. F. A., II, 157, _157_. Pratt, O., II, _69_. Predaval, Count de, I, _348_. Prescot, B., I, 270, _270_, 278. Prester John, I, 70, _71_, 152. Price, R., I, _223_. Probability, Discourse on, I, 279. Proclus, I, 188, _188_. Prodigies, Youthful, I, 219, 332. Pronunciation, II, 330. Protestant and Papal Christendom, II, 33. Protimalethes, II, 6. Ptolemy, I, 5, 33, _380_. Pullicino, II, 61, _61_. Pusey, I, _64_. Pyramids, The, I, 328; II, 95, 136. Pythagoras, II, _59_.

Quadrature problem. (_See Squaring the circle._) Quarles, F., II, _277_. Quintilian, II, 280. Quotem, C., I, 399.

Rabelais, I, _102_. Rainbow Paradox, II, 334. Ramachandra, Y., I, _374_. Ramchundra, I, 374. Ramus, I, 5. Recalcati, II, 208, 314. Recorde, R., II, _328_. Reddie, Jas., II, 183, _183_, 344, 360. Reeve, J., I, _395_. Regiomontanus, I, _48_, 360. Reisch, I, _45_; II, 281. Religion and Philosophy, II, 37. Religious bodies, Names of, II, 22; --customs, Attacks on, I, 177; --Insurance, I, 345; --Paradox, I, 236; --Tract society, I, 192. Remigius, I, _50_. Reuchlin, J., II, _323_. Revelations, Napier on, I, 66. Revilo, (O. Byrne), I, 241, 329, _329_. Reyneau, C. R., II, _301_. Rheticus, I, _69_; II, 372. Rhonius, II, 300. Ribadeneira, P. de, II, _62_. Riccioli, I, _96_. Richards, G., II, _270_. Rigaud, J., II, _299_. Rigaud, S. J., II, _299_. Rigaud, S. P., I, _140_; II, 298, 313. Ringelbergh, J. S., II, _281_. Ripley, G., I, _126_, 126. Ritchie, W., I, 295, _295_. Ritterhusius, I, _60_. Rive, J.-J., I, _160_. Robertson, Jas., I, _237_. Roberval, I, _105_, 122. {385} Robinson, B., I, _148_, 148. Robinson, H. C., I, _314_; II, 52, 275. Robinson, R., I, _177_. Robinson, T. R., II, _181_. Roblin, J., II, 136. Rogers, S., II, _260_. Roget, P. M., I, _398_. Roomen, A. van, I, _110_. Ross, J. C., I, _303_. Rosse, I, 26. Rossi, G., I, 231, _231_. Rotation of the Moon, II, 4, 19. Rough, W., I, 198. Rowning, J., I, _155_. Royal Astronomical Society, I, 27; --Forerunner of, I, 374. Royal Society, I, 21, 22, 24-30, 56, 57, 136, 153, 163, 164, 165. Rudio, I, 159; II, 367. Rudolff, C., II, 373. Russell, Earl (1st), I, _296_. Rutherford, W., II, _109_.

Sabatier, A., II, _267_. Sabellius, I, _241_. Sacrobosco, I, _360_. Sadler, T., I, _238_, 241. Saint-Martin, I, 167, _168_, 206. St.-Mesmin, M. de., I, 280. St. Vincent, G. de., I, _110_, 117. St. Vitus, Patron of Cyclometers, II, 60. Sales, de, I, 167. Salicetus, I, 64. Salisbury, Earl of (1st), II, _330_. Salmasius, Claudius, II, _168_. Salusbury, Hester, I, _235_. Sanchez, Petro, I, 229, _229_. Sanders, W., I, 207. Sanderson, R., I, _135_. Sara, R., I, _297_. Saunderson, N., I, _377_; II, 301. Scaliger, I, _44_, 110, 111, 112, 113; II, 238. Scevole de St. Marthe, I, 113. Schooten, Van, II, _59_. Schopp, I, _60_. Schott, I, _64_; II, 64. Schumacher, H. C., I, _107_; II, 297. Schwab, I, _230_. Scientific paradoxes, I, 232. Scott, Michael, I, _38_. Scott's Devils, I, 38. Scott, W., I, 20, 27, 38, 39, 155, _191_. Scripture and Science, II, 261. Search, John, I, 247. Selden, J., II, _250_. Senarmont, II, _48_. Serres, De, II, _60_. Shaftesbury, Earl of, II, _181_. Shakespeare, I, 13. Shanks, II, _63_, 65, 109. Shaw, P., I, _142_. Sheepshanks, J., I, _147_. Sheepshanks, R., I, _290_. Shelley, I, 174. Shepherd, S., I, _124_. Sherburne, E., II, _295_. Sheridan, R. B., I, _175_. Sheridan, T., I, _175_. Shoberl, F., II, _270_. Shrewsbury, I, _108_. Siddons, Mrs., I, 189. Simms, W., I, _152_. Simplicius, II, _164_. Simpson, T., I, 377; II, 304. Simson, R., I, _197_, 202, 233. Sinclair, G., I, _207_. Slander Paradoxes, II, 138. Sloane, I, 24. Sluse, R. de, I, _118_, 118; II, 300. Smith, Adam, II, _112_. Smith, Jas., I, _46_; II, 103, _103_, 154, 236, 237, 238, 241, 336, 360. Smith, Jas., II, _217_. Smith, Jas. (Shepherd), II, _55_, 193. Smith, Joseph, II, _69_. Smith, Richarda, I, _242_. Smith, Thomas, I, 346, _346_. Smith, Wm., II, _152_. Smyth, C. P., I, _328_; II, 65. Snell, I, _75_, 75. Socinianism, I, 142, 143. Socinus, I, 3, _143_. Socrates Scholasticus, I, _358_. Sohncke, L. A., II, _131_. Somerville, Mrs., I, _242_. South, J., II, _181_. Southcott, Joanna, II, _58_, 97, 239. Spearman, R., I, _237_. Speculative thought in England, I, 374. Spedding, I, _76_, 82, 142. Speed, J., I, _201_. Speke, I, _70_. Spelling, phonetic, II, 81. Spence, W., I, _231_, 231. Spencer, Earl (3d), II, _9_. {386} Spinoza, I, 3, _37_. Spiritualism, II, 47, 55, 207. Spurius Cassius Viscellinus, II, _342_. Spurius Maelius, II, _342_. Squaring the circle, I, 8, 42, 44, 46, 47, 50, 52, 53, 69, 70, 75, 109, 117, 119, 129, 135, 146, 149, 159, 163, 164, 347, 348, 374; II, 10, 11, 60, 105, 154, 156, 208, 278, 314. Staeckel, I, 230. Stanhope, P. D., (Earl of Chesterfield), II, _298_. Stapulensis, I, _44_; II, 324. Star polygons, I, 229. Starkie, G., I, _126_, 126. Statter, D., II, 80. Steamship suggested, I, 147. Steel, Jas., II, 68. Stenography, II, 81. Stephens, I, _44_; II, 324. Stephenson, G., II, _138_. Stephenson, R., II, _138_. Stevin, I, _83_, 313; II, 59. Stewart, D., II, _53_. Stewart, R., I, _186_. Stifel, M., II, _373_. Strafford, Earl of, I, _240_. Stratford, W. S., I, _300_. Street, T., I, _124_. Stukely, W., I, _236_. Suffield, G., II, _66_. Suidas, II, _29_. Sumner, C. R., I, _324_. Sumner, J. B., I, _324_. Sun as an electric space, II, 41. Supernatural, The, II, 193. Suvaroff, II, _85_. Swastika, II, 231. Swedenborg, E., I, _255_. Swift, I, 19, 133. Sylvester, J. J., II, _336_. Symington, W., I, _148_. Symons, II, 4, 5, 20, 84, 85. Sympathetic powder, I, 108. Synesius, I, 125.

Talbot, G., I, _22_, _108_. Talbot's powder, I, 108. Tartaglia, II, 59. Tasse, I, _106_. Tate, J., I, _199_. Tauler, J., II, _322_. Taylor, Brook, II, _301_. Taylor, John, I, _352_; II, 95. Taylor, Robt., I, _270_. Taylor, T., I, 188, _188_. Teissier, I, _113_. Temple, H. J., I, 290. Tenterden, Chief Justice, I, _181_. Thales, II, _59_, 83. Theism independent of Revelation, I, 399. Thelwall, J., I, _178_. Theodoretus, I, _358_. Theological Paradoxes, I, 316. Theology, Mathematical, I, 129, 149. Theophrastus, II, _167_. Thiebault, II, _3_, 338. Thom, D., II, _226_, 240. Thom, J. H., II, _226_. Thompson, P., I, _7_. Thompson, T. P., I, _252_, 287, 344; II, 83, 185. Thomson, Dr., I, 21. Thomson, W., I, _325_. Thorn, W., II, 158, _158_, 360. Thorndike, H., II, _313_. Thrale, Mrs., I, _235_. Thurlow, Baron, I, _222_. Thyraeus, I, 50. Tides, New theory of, I, 393. Tombstones of mathematicians, I, 106. Tonal System, II, 182. Tooke, H., I, _178_. Torriano, E., I, 250. Towneley, II, 300. Townley, C., II, _300_. Trisection problem, I, 118; II, 10, 12, 13, 15. Troughton, I, _152_. Turnor, E., I, _137_. Tycho Brahe, I, 5, _76_, 381; II, 302, 335.

Upton, W., II, 12, _12_, 15. Ursus, I, _52_.

Valentine, B., I, _125_, 125. Van Ceulen, I, 52, 70, 100. Van de Weyer, I, _313_. Van Etten, I, _161_. Van Helmont, I, _125_, 125. Van Roomen, I, _110_. Van Schooten, II, _59_. Vaughan, T., I, _255_. Victorinus, I, _359_. Viete, I, _51_; II, 210, 295. Virgil, St., I, 32, _33_, 34, 99. {387} Virginia, University of, I, 233. Viscellinus, II, _342_. Vitruvius, II, _281_. Vivian, T., I, 172, _172_. Vogel, A. F., I, 373. Voltaire, I, 103, 165, 166, 167, 168, 248; II, 268. Von Gumpach, II, 137, _137_. Von Hutten, I, 318. Von Wolzogen. (_See Wolzogen._) Vyse, R. W. H., I, _328_.

Walker, W. E., II, _316_. Walkingame, F., II, _173_. Wallich, N., II, _14_. Wallis, J., I, 107, 109, 110; II, 299, 313. Walpole, I, _23_, 131. Walsh, John, I, 260, _260_; II, 157. Wapshare, J., II, _230_. Warburton, H., I, _349_. Warburton, Wm., I, _55_, 112; II, 174. Ward, S., II, _299_. Waring, E., I, _203_, 222. Warner, W., II, _302_, 312. Warren, S., II, _340_. Watkins, J., II, _270_. Watson, Bp., I, _223_. Watt, R., I, _102_, 402. Watts, I., II, 18. Weddle, T., II, _187_. Wentworth, Thos., I, _240_. Wharton, I, 115. Whately, R., I, 246, _246_, 324. Whately's Paradox, I, 246. Whewell, I, _101_, 101, 273, 314, 380; II, 104, 246, 247. Whigs, II, 22. Whiston, J., I, _147_. Whiston, W., I, 133, _133_, 146, 156, 311. White, H. K., II, _271_. White, J. B., I, 248. White, R., I, 11. Whitford, I, 105. Whitworth, W. A., II, _344_. Whizgig, On the, I, 254. Wightman, I, 59. Wilberforce, W., II, _236_. Wilkins, J., I, 96, _100_, 116, 226. Williams, J. B., I, _378_. Williams, T., I, 171, _171_. Wilson, Sir J., I, _221_. Wilson, J. M., II, _344_. Wilson, R., II, 7, _7_. Wilson's Theorem, I, _222_. Wingate, E., II, _308_. Winter, I, 46. Wirgman, T., I, 258, _258_. Wiseman, N. P. S., II, _26_, 61, 294. Wolcot, J. (Peter Pindar), II, _272_. Wollstonecraft, I, 173, _173_. Wolzogen, I, _106_. Wood, A., I, _98_. Wood, John, I, _233_. Wood, Wm., I, 246, _246_. Woodley, W., I, 307, _307_. Wordsworth, II, 273. Wright, E., I, 84. Wright, T., I, 151, _151_, 152, 153. Wright, W., II, 9. Wronski, I, 249, _250_. Wrottesley, J. (Baron), II, _181_.

Young, B., II, _69_. Young, J. W. A., II, 367. Young, T., I, 24, 30, _250_. Youthful Prodigies, I, 219. Yvon, I, _297_.

Zach, von, II, _45_, 196. Zachary, Pope, I, 32, 34. Zadkiel, I, _321_; II, 43. Zetetic Astronomy, II, 88. Zodiac, II, 136. Zytphen, II, _335_.

* * * * *

Notes

Transcriber's note: References to Notes in Volume I are shown as in the printed book, with the resequenced footnote numbers in the Project Gutenberg Edition (EText-No. 23100) added thus {123}.

[1] See Vol. I, page 255, note 6 {584}.

[2] "I have no need for this hypothesis."

[3] "Ah, it is a beautiful hypothesis; it explains many things."

[4] "What we know is very slight; what we don't know is immense."

[5] Brewster relates (_Life of Sir Isaac Newton_, Vol. II, p. 407) that, a short time before his death, Newton remarked: "I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me."

[6] See Vol. I, p. 292, note 1 {632}.

[7] "What is all that!"

[8] "I have some good news to tell you: at the Bureau of Longitudes they have just received a letter from Germany announcing that M. Bessel has verified by observation your theoretical discoveries on the satellites of Jupiter."

[9] "Man follows only phantoms."

[10] See Vol. I, page 382, note 13 {786}.

[11] Dieudonne Thiebault (1733-1807) was a Jesuit in his early life, but he left the order and took up the study of law. In 1765 he went to Prussia and became a favorite of Frederick the Great. He returned to France in 1785 and became head of the Lycee at Versailles.

[12] _Memories of Twenty Years of Residence in Berlin._ There was a second French and an English edition in 1805.

[13] Richard Joachim Heinrich von Mollendorff (1724-1816) began his career as a page of Frederick the Great (1740) and became field marshal (1793) and commander of the Prussian army on the Rhine (1794).

[14] Hugues Bernard Maret (1763-1839) was not Duc de Bassano in 1807, this title not being conferred upon him until 1809. He was ambassador to England in 1792 and to Naples in 1793. Napoleon made him head of the cabinet and his special confidant. The Bourbons exiled him in 1816.

[15] Denis Diderot (1713-1784), whose _Lettre sur les aveugles_ (1749) introduced him to the world as a philosopher, and whose work on the _Encyclopedie_ is so well known.

[16] "Sir, (a + b^{n}) / n = x, whence God exists; answer!"

[17] This was one James Laurie of Musselburgh.

[18] Jelinger Cookson Symons (1809-1860) was an office-holder with a decided leaning towards the improvement of education and social conditions. He wrote _A Plea for Schools_ (1847), _The Industrial Capacities of South Wales_ (1855), and _Lunar Motion_ (1856), to which last work the critic probably refers.

[19] "Protimalethes" followed this by another work along the same line the following year, _The Independence of the Testimony of St. Matthew and St. John tested and vindicated by the theory of chances_.

[20] Wilson had already taken up the lance against science in his _Strictures on Geology and Astronomy, in reference to a supposed want of harmony between these sciences and some parts of Divine Revelation_, Glasgow, 1843. He had also ventured upon poetry in his _Pleasures of Piety_, Glasgow, 1837.

[21] Mrs. Borron was Elizabeth Willesford Mills before her marriage. She made an attempt at literature in her _Sibyl's Leaves_, London (printed at Devonport), 1826.

[22] See Vol. I, page 386, note 10 {801}.

[23] See Vol. I, page 43, notes 7 {32} and 8 {33}.

[24] His flying machine, designed in 1843, was one of the earliest attempts at aviation on any extensive scale.

[25] Erasmus Darwin (1731-1802) was the grandfather of Charles Darwin. The work here mentioned had great influence, being translated into French, Portuguese, and Italian. Canning parodied it in his _Loves of the Triangles_.

[26] See Vol. I, page 147, note 1 {312}.

[27] The notes on this page were written on the day of the funeral of Wilbur Wright, June 1, 1912, the man who realized all of these prophecies, and then died a victim of municipal crime,--of typhoid fever.

[28] John Charles, third Earl Spencer (1782-1845), to whose efforts the Reform Bill was greatly indebted for its final success.

[29] This was published in London in 1851 instead of 1848.

[30] This appeared in 1846.

[31] This was done in _The Circle Squared_, published at Brighton in 1865.

[32] It first appeared in 1847, under the title, _The Scriptural Calendar and Chronological Reformer, 1848. Including a review of tracts by Dr. Wardlaw and others on the Sabbath question. By W. H. Black._ The one above mentioned, for 1849, was printed in 1848, and was also by Black (1808-1872). He was pastor of the Seventh Day Baptists and was interested in archeology and in books. He catalogued the manuscripts of the Ashmolean Museum at Oxford.

[33] William Upton, a Trinity College man, Dublin. He also wrote _Upton's Physioglyphics_, London, 1844; _Pars prima. Geometria vindicata; antiquorumque Problematum, ad hoc tempus desperatorum, Trisectionis Anguli, Circulique Quadraturae, Solutio, per Eucliden effecta, London_ (printed at Southampton), 1847; _The Uptonian Trisection_, London, 1866; and _The Circle Squared_, London, 1872.

[34] For example, if [theta] = 90 deg. we should have 3 cos 30 deg. = 1 + [root](4 - sin^2 90 deg.), or 3. 1/2 [root]3 = 1 + [root]3, or 1/2 [root]3 = 1.

[35] Nathaniel Wallich (1786-1854) was surgeon at the Danish settlement at Serampore when the East India Company took over the control in 1807. He entered the British medical service and was invalided to England in 1828. His _Plantae Asiaticae Rariores_ (3 vols., London, 1830-1832) was recognized as a standard. He became vice-president of the Linnean Society, F. R. S., and fellow of the Royal Asiatic Society.

[36] But if [theta] = 90 deg. this asserts that

cos 30 deg. = (sin 270 deg. . cos 225 deg. + sin^2 90 deg. . sin 225 deg.) / [root](sin^2 270 deg. . cos^2 225 deg. + sin^{4} 90 deg. + sin 270 deg. . sin 450 deg. . sin^2 90 deg.),

or that

1/2 [root]3 = (-1 . (-1 /[root]2) + 1 . (-1/[root]2) / [root]1 . 1/2 + 1 - 1 . 1 . 1) = 0 / [root](1/2),

so that De Morgan must have made some error in copying.

[37] John Bonnycastle (died in 1821) was professor of mathematics at Woolwich. His edition of Bossut's _History of Mathematics_ (1803), and his works on elementary mathematics were well known.

[38] The bibliographies give Husain Rifki as the translator, a practical geometry as the work, and 1802 as the date.

[39] See Vol. I, page 309, note 2 {670}.

[40] Probably in _The Improvement of the Mind_ which Isaac Watts (1674-1748) published in 1741. His _Horae Lyricae_ appeared in 1706, and the _Hymns_, by which he is still well known, in 1707.

[41] Spencer Joshua Alwyne Compton, second Marquis of Northampton (1790-1851), was a poet, a scientist, and a statesman. He was president of the Royal Society from 1838 to 1849.

[42] Besides the writings here mentioned Perigal published a work on _Geometric Maps_ (London, 1853), and _Graphic Demonstrations of Geometric Problems_ (1891).

[43] See Vol. II, page 5, note 18.

[44] James Ferguson (1710-1776) was a portrait painter, an astronomer, and a popular writer and lecturer on various subjects.

[45] In the old ballad of King Alfred and the Shepherd, when the latter is tempting the disguised king into his service, he says:

"Of whig and whey we have good store, And keep good pease-straw fire."

_Whig_ is then a preparation of milk. But another commonly cited derivation may be suspected from the word _whiggamor_ being used before _whig_, as applied to the political party; _whig_ may be a contraction. Perhaps both derivations conspired: the word _whiggamor_, said to be a word of command to the horses, might contract into _whig_, and the contraction might be welcomed for its own native meaning.--A. De M.

[46] This was p. 147 in the first edition.

[47] St. Augustine (354-430) was bishop of Hippo. His _Confessiones_, in 13 books, was written in 397, and his _De Civitate Dei_ in 426.

[48] "He was wont to indulge in certain Punic subtleties lest he should weary the reader by much speaking."

[49] John Milner (1751-1826), bishop of Castabala, a well-known antiquarian.

[50] It will be said that when the final happiness is spoken of in "sure and certain hope," it is _the_ Resurrection, generally; but when afterwards application is made to the individual, simple "hope" is all that is predicated which merely means "wish?" I know it: but just before the general declaration, it is declared that it _has_ pleased God of his great mercy to _take unto Himself_, the soul of our dear brother: and between the "hopes" hearty thanks are given that it _has_ pleased God to deliver our dear brother out of the miseries of this wicked world, with an additional prayer that the number of the elect may shortly be accomplished. All which means, that our dear brother is declared to be taken to God, to be in a place not so miserable as this world--a description which excludes the "wicked place"--and to be of the elect. Yes, but it will be said again! do you not know that when this Liturgy was framed, all who were not in the road to Heaven were excommunicated burial service read over them. Supposing the fact to have been true in old time, which is a very spicy supposition, how does that excuse the present practice? Have you a right _always_ to say what you believe _cannot always_ be true, because you think it was once _always_ true? Yes, but, choose whom you please, you cannot be _certain_ He is _not_ gone to Heaven. True, and choose which Bishop you please, you cannot be demonstratively _certain_, he is _not_ a concealed unbeliever: may I therefore say of the whole bench, _singulatim et seriatim_, that they _are_ unbelievers? No! No! The voice of common sense, of which common logic is a part, is slowly opening the eyes of the multitude to the unprincipled reasoning of theologians. Remember 1819. What chance had Parliamentary Reform when the House of Commons thanked the Manchester sabre-men? If you do not reform your Liturgy, it will be reformed for you, and sooner than you think! The dishonest interpretations, by defence of which even the minds of children are corrupted, and which throw their shoots into literature and commerce, will be sent to the place whence they came: and over the door of the established organization for teaching religion will be posted the following notice:

"Shift and Subterfuge, Shuffle and Dodge, No longer here allowed to lodge!"

All this ought to be written by some one who belongs to the Establishment: in him, it would be quite prudent and proper; in me, it is kind and charitable.--A. De M.

[51] But few do have access to it, for the work is not at all common, and this Piscator is rarely mentioned.

[52] This derivation has been omitted.--S. E. De M.

[53] A blow for a blow. Roland and Oliver were two of the paladins of Charlemagne whose exploits were so alike that each was constantly receiving credit for what the other did. Finally they met and fought for five days on an island in the Rhine, but even at the end of that period it was merely a drawn battle.

[54] "In the name of the church."

[55] "From the chair," officially.

[56] Nicholas Patrick Stephen Wiseman (1802-1865), whose elevation to the archbishopric of Westminster and the cardinalate (1850) led to the act prohibiting Roman Catholics from assuming episcopal titles in England, a law that was never enforced.

[57] He was born in 1812 and was converted to Catholicism in 1839. He founded the _Tablet_ in London in 1840, removing its office to Dublin in 1849. He became M. P. in 1852, and at the time of his death (1855) he was preparing a memorial to the Pope asking him to annul the proclamation of an Irish bishop prohibiting his priests from taking part in politics.

[58] John Guillim (1565-1621) was the first to systematize and illustrate the whole science of heraldry. He published _A display of Heraldrie: manifesting a more easie accesse to the knowledge thereof_ in 1610.

[59] "Faith."

[60] "Faithful."

[61] "For the faith vindicated."

[62] The words are of the same root, and hence our word _fiddle_. Some suppose this root means a _rope_, which, as that to which you trust, becomes, in one divergence, confidence itself--just as a _rock_, and other words, come to mean reliance--and in another, a little string.--A. De M.

[63] The Greek lexicographer, a Christian, living after 1000 A. D. His lexicon was first printed at Milan in 1499.

[64] _Skindapsos._

[65] This was John William Burgon (1813-1888), Gresham professor of theology (1867) and dean of Chichester. He was an ultra-conservative, opposing the revised version of the New Testament, and saying of the admission of women to the university examinations that it was "a thing inexpedient and immodest."

[66] _Ekklesia_, or _ecclesia_.

[67] _Ennomos ekklesia._

[68] "Without doubt I shall perish forever."

[69] "Every man is an animal." "Sortes is a man." "Sortes is an animal."

[70] "For a special purpose."

[71] Heraclitus of Ephesus, the weeping philosopher, 6th century B. C.

[72] Democritus, the laughing philosopher, founder of the atomistic theory, 5th century B. C.

[73] "Ends to which."

[74] "Ends from which."

[75] "In just as many syllables," "With just as many letters," "In just as many words."

[76] "I shall make a way," "I shall find a way."

[77] The notion that the Evil Spirit is a functionary liable to be dismissed for not attending to his duty, is, so far as my reading goes, utterly unknown in theology. My first wrinkle on the subject was the remark of the Somersetshire farmer upon Palmer the poisoner-- "Well! if the Devil don't take he, he didn't ought to be allowed to be devil no longer."--A. De M.

William Palmer (1824-1856) was a member of the Royal College of Surgeons and practised medicine at London. He was hanged in 1856 for having poisoned a friend and was also suspected of having poisoned his wife and brother for their insurance money, besides being guilty of numerous other murders. His trial was very much in the public attention at the time.

[78] Advantages and dangers.

[79] The old priory of St. Mary of Bethlehem in London, was used as an asylum for the insane. The name was corrupted to Bedlam.

[80] Referring to the common English pronunciation of St. John, almost Sinjin. John St. John Long (1798-1834), an Irishman by birth, practised medicine in London. He claimed to have found a specific for rheumatism and tuberculosis, but upon the death of one of his patients in 1830 he was tried for manslaughter. He died of tuberculosis four years later, refusing to take his own treatment.

[81] William of Occam (d. 1349), so called from his birthplace, Ockham, in Surrey. He was a Franciscan, and lectured on philosophy in the Sorbonne.

[82] He signs himself "James Hopkins, schoolmaster," and this seems to have been his only published effort.

[83] Joseph Ady (1770-1852) was a famous swindler. One of his best-known schemes was to send out letters informing the recipients that they would learn something to their advantage on payment of a certain sum. He spent some time in prison.

[84] Sir Peter Laurie (c. 1779-1861) was worth referring to, for he was prominent as a magistrate and was honored because of his interest in all social reforms. He made a fortune as a contractor, became sheriff of London in 1823, and was knighted in the following year. He became Lord Mayor of London in 1832.

[85] See Vol. I, page 321, note 2 {691}. The _Astronomy in a nutshell_ appeared in 1860. _The Herald of Astrology_ was first published in London in 1831, "by Zadkiel the Seer." It was continued as _The Astrological Almanac_ (London, 1834), as _Zadkiel's Almanac and Herald of Astrology_ (_ibid._, 1835, edited by R. J. Morrison, and subsequently by A. J. Pearce), and as _Raphael's Prophetic Almanac_ (1840-1855).

[86] See Vol. I, page 172, note 3 {382}.

[87] See Vol. I, page 87, note 4 {133}.

[88] Franz Xaver, Freiherr von Zach (1754-1832) was director of the observatory at Seeberge near Gotha. He wrote the _Tabulae speciales aberrationis et mutationis_ (1806-7), _Novae et correctae tabulae solis_ (1792), and _L'attraction des montagnes et ses effets sur le fil a plomb_ (1814).

[89] Jean Louis Pons (1761-1831) was connected with the observatory at Marseilles for thirty years (1789-1819). He later became director of the observatory at Marlia, near Lucca, and subsequently filled the same office at Florence. He was an indefatigable searcher for comets, discovering 37 between 1801 and 1827, among them being the one that bears Encke's name.

[90] This hypothesis has now become an established fact.

[91] John Chetwode Eustace (c. 1762-1815) was born in Ireland. Although a Roman Catholic priest he lived for a time at Cambridge where he did some tutoring. His _Classical Tour_ appeared in 1813 and went through several editions.

[92] "Crimes should be exposed when they are punished, but disgraceful acts should be hidden."

[93] Henri Hureau de Senarmont (1808-1862) was professor of mineralogy at the _Ecole des mines_ and examiner at the _Ecole polytechnique_ at Paris.

[94] Augustin Jean Fresnel (1788-1827), "Ingenieur des ponts et chaussees," gave the first experimental proofs of the wave theory of light. He studied the questions of interference and polarization, and determined the approximate velocity of light.

[95] "As is my custom."

[96] Francis Heywood (1796-1858) made the first English translation of Kant's _Critick of Pure Reason_ (1838, reprinted in 1848). The _Analysis_ came out, as here stated, in 1844.

[97] Louise Renee de Keroualle, Duchess of Portsmouth and Aubigny (1649-1734), was a favorite of Charles II. She used her influence to keep him under the control of Louis XIV.

[98] William Chiffinch (c. 1602-1688) was page of the king's bed-chamber and keeper of the private closet to Charles II. He was one of the king's intimates and was an unscrupulous henchman.

[99] "Well devised."

[100] "John Bellingham Inglis. His _Philobiblion_ "translated from the first edition (of Ricardus d'Aungervile, Bishop of Durham), 1473," appeared at London in 1832. It was republished in America (Albany, N. Y.) in 1864.

[101] "What are you laughing at?"

[102] See Vol. I, page 314, note 4 {681}.

[103] See Vol. I, page 112, note 7 {211}.

[104] Referring to Hamilton's edition of the _Collected Works of Dugald Stewart_, 10 volumes, Edinburgh, 1854-58. It is not commonly remembered that Stewart (1753-1828) taught mathematics at the University of Edinburgh before he took up philosophy.

[105] This was Hamilton's edition of the _Works of Thomas Reid_ (2 vols., Edinburgh, 1846-1863). Reid (1710-1796) included mathematics in his work in philosophy at Aberdeen. In 1764 he succeeded Adam Smith at Glasgow.

[106] Edward Irving (1792-1834), the famous preacher. At first he assisted Dr. Chalmers at Glasgow, but in 1822 he went to London where he met with great success. A few years later he became mentally unbalanced and was finally expelled from his church (1832) for heresy. He was a great friend of Carlyle.

[107] He also wrote a number of other paradoxes, including _An Essay towards a Science of Consciousness_ (1838), _Instinctive Natural Religion_ (1858), _Popular Treatise on the structure, diseases, and treatment of the human teeth_ (1837), and _On Headache_ (1859).

[108] James Smith (1801-1857), known as Shepherd Smith, was a socialist and a mystic, with a philosophy that was wittily described as "Oriental pantheism translated into Scotch." He was editor of several journals.

[109] Joanna Southcott (1750-1814) was known for her rhyming prophecies in which she announced herself as the woman spoken of in Revelations xii. She had at one time as many as 100,000 disciples, and she established a sect that long survived her.

[110] Thales, c. 640-548 B. C.

[111] Pythagoras, 580-501 B. C.

[112] Anaxagoras, 499-428 B. C., the last of the Ionian school, teacher of Euripides and Pericles. Plutarch speaks of him as having squared the circle.

[113] Oinopides of Chios, contemporary of Anaxagoras. Proclus tells us that Oinopides was the first to show how to let fall a perpendicular to a line from an external point.

[114] Bryson and Antiphon, contemporaries of Socrates, invented the so-called method of exhaustions, one of the forerunners of the calculus.

[115] He wrote, c. 440 B. C., the first elementary textbook on mathematics in the Greek language. The "lunes of Hippocrates" are well known in geometry.

[116] Jabir ben Aflah. He lived c. 1085, at Seville, and wrote on astronomy and spherical trigonometry. The _Gebri filii Affla Hispalensis de astronomia libri_ IX was published at Nuremberg in 1533.

[117] Hieronymus Cardanus, or Girolamo Cardano (1501-1576), the great algebraist. His _Artis magnae sive de regulis Algebrae_ was published at Nuremberg in 1545.

[118] Nicolo Tartaglia (c. 1500-1557), the great rival of Cardan.

[119] See note 5 {98}, Vol. I, page 69.

[120] See note 10 {124}, Vol. I., page 83.

[121] See note 9 {123}, Vol. I, page 83.

[122] Pierre Herigone lived in Paris the first half of the 17th century. His _Cours mathematique_ (6 vols., 1634-1644) had some standing but was not at all original.

[123] Franciscus van Schooten (died in 1661) was professor of mathematics at Leyden. He edited Descartes's _La Geometrie_.

[124] Florimond de Beaune (1601-1652) was the first Frenchman to write a commentary on Descartes's _La Geometrie_. He did some noteworthy work in the theory of curves.

[125] See note 3 {23}, Vol. I, page 41.

[126] Olivier de Serres (b. in 1539) was a writer on agriculture. Montucla speaks of him in his _Quadrature du cercle_ (page 227) as having asserted that the circle is twice the inscribed equilateral triangle, although, as De Morgan points out, this did not fairly interpret his position.

[127] Anghera wrote not only the three works here mentioned, but also the _Problemi del piu alto interesse scientifico, geometricamente risoluti e dimostrati_, Naples, 1861. His quadrature was defended by Giovanni Motti in a work entitled _Matematica Vera. Falsita del sistema ciclometrico d'Archimede, quadratura del cerchio d'Anghera, ricerca algebraica dei lati di qualunque poligono regolare inscritto in un circolo_, Voghera, 1877. The _Problemi_ of 1861 contains Anghera's portrait, and states that he lived at Malta from 1849 to 1861. It further states that the Malta publications are in part reproduced in this work.

[128] This was his friend Paolo Pullicino whose _Elogio_ was pronounced by L. Farrugia at Malta in 1890. He wrote a work _La Santa Effegie della Blata Vergine Maria_, published at Valetta in 1868.

[129] St. Vitus, St. Modestus, and St. Crescentia were all martyred the same day, being torn limb from limb after lions and molten lead had proved of no avail. At least so the story runs.

[130] The reference is to Cardinal Wiseman. See Vol. II, page 26, note 56.

[131] "Worthy of esteem."

[132] Pedro de Ribadeneira (Ribadeneyra, Rivadeneira), was born at Toledo in 1526 and died in 1611. He held high position in the Jesuit order. The work referred to is the _Flos Sanctorum o libro de las vidas de los santos_, of which there was an edition at Barcelona in 1643. His life of Loyola (1572) and _Historia ecclesiastica del Cisma del reino de Inglaterra_ were well known.

[133] Caesar Baronius (1538-1607) was made a cardinal in 1595 and became librarian at the Vatican in 1597. The work referred to appeared at Rome in 1589.

[134] Mrs. Jameson's (1794-1860) works were very popular half a century ago, and still have some circulation among art lovers. The first edition of the work mentioned appeared in 1848.

[135] The barnyard cock.

[136] Shanks did nothing but computing. The title should, of course, read "to 607 Places of _Decimals_." He later carried the computation to 707 decimal places. (_Proc. Roy. Society_, XXI, p. 319.) He also prepared a table of prime numbers up to 60,000. (_Proc. Roy. Society_, XXII, p. 200.)

[137] See Vol. I, page 42, note 4 {24}.

[138] See Vol. I, page 64, note 1 {78}.

[139] See Vol. I, page 328, note 1 {704}.

[140] George Suffield published _Synthetic Division in Arithmetic_, to which reference is made, in 1863.

[141] John Robert Lunn wrote chiefly on Church matters, although he published a work on motion in 1859.

[142] Jean Baptiste Joseph, Baron Fourier (1768-1830), sometime professor in the Military School at Paris, and later at the _Ecole polytechnique_. He is best known by his _Theorie analytique de la chaleur_ (Paris, 1822), in which the Fourier series is used. The work here referred to is the _Analyse des equations determinees_ (Paris, 1831).

[143] William George Horner (1786-1837) acquired a name for himself in mathematics in a curious manner. He was not a university man nor was he a mathematician of any standing. He taught school near Bristol and at Bath, and seems to have stumbled upon his ingenious method for finding the approximate roots of numerical higher equations, including as a special case the extracting of the various roots of numbers. Davies Gilbert presented the method to the Royal Society in 1819, and it was reprinted in the _Ladies' Diary_ for 1838, and in the _Mathematician_ in 1843. The method was original as far as Horner was concerned, but it is practically identical with the one used by the Chinese algebraist Ch'in Chiu-shang, in his _Su-shu Chiu-chang_ of 1247. But even Ch'in Chiu-shang can hardly be called the discoverer of the method since it is merely the extension of a process for root extracting that appeared in the _Chiu-chang Suan-shu_ of the second century B. C.

[144] He afterwards edited Loftus's _Inland Revenue Officers' Manual_ (London, 1865). The two equations mentioned were x^3 - 2x = 5 and y^3 - 90y^2 + 2500y - 16,000 = 0, in which y = 30 - 10x. Hence each place of y is the complement of the following place of x with respect to 9.

[145] Probably the John Power Hicks who wrote a memoir on T. H. Key, London, 1893.

[146] Possibly the one who wrote on the quadrature of the circle in 1881.

[147] As it is. But what a pity that we have not 12 fingers, with duodecimal fractions instead of decimals! We should then have 0.6 for 1/2, 0.4 for 1/3, 0.8 for 2/3, 0.3 for 1/4, 0.9 for 3/4, and 0.16 for 1/8, instead of 0.5, 0.333+, 0.666+, 0.25, 0.75, and 0.125 as we now have with our decimal system. In other words, the most frequently used fractions in business would be much more easily represented on the duodecimal scale than on the decimal scale that we now use.

[148] He wrote Hints for an _Essay on Anemology and Ombrology_ (London, 1839-40) and _The Music of the Eye_ (London, 1831).

[149] Brigham Young (1801-1877) was born at Whitingham, Vermont, and entered the Mormon church in 1832. In 1840 he was sent as a missionary to England. After the death of Joseph Smith he became president of the Mormons (1847), leading the church to Salt Lake City (1848).

[150] Joseph Smith (1805-1844) was also born in Vermont, and was four years the junior of Brigham Young. The _Book of Mormon_ appeared in 1827, and the church was founded in 1830. He was murdered in 1844.

[151] Orson Pratt (1811-1881) was one of the twelve apostles of the Mormon Church (1835), and made several missionary journeys to England. He was professor of mathematics in the University of Deseret (the Mormon name for Utah). Besides the paper mentioned Pratt wrote the _Divine Authenticity of the Book of Mormon_ (1849), _Cubic and Biquadratic Equations_ (1866), and a _Key to the Universe_ (1866).

[152] "It does not follow."

[153] Dryden (1631-1700) published his _Religio Laici_ in 1682. The use of the word "proportion" in the sense of ratio was common before his time, but he uses it in the sense of having four terms; that is, that price is to price as offence is to offence.

[154] Olinthus Gilbert Gregory (1774-1841) succeeded Hutton as professor of mathematics at Woolwich. He was, with De Morgan, much interested in founding the University of London. He wrote on astronomy (1793), mechanics (1806), practical mathematics (1825), and Christian evidences (1811).

[155] See Vol. I, page 220, note 6 {482}. The _Pensees_ appeared posthumously in 1670.

[156] "The right thing to do is not to wager at all." "Yes, but you ought to wager; you have started out; and not to wager at all that God exists is to wager that he does not exist."

[157] He lived about 300 A.D., in Africa, and wrote _Libri septem adversus Gentes_. This was printed at Rome in 1542-3.

[158] Pierre Bayle (1647-1706) was professor of philosophy at the Prostestant University at Sedan from 1675 until its dissolution in 1681. He then became professor at Rotterdam (1681-1693). In 1684 he began the publication of his journal of literary criticism _Nouvelles de la Republique des Lettres_. He is best known for his erudite _Dictionnaire historique et critique_ (1697).

[159] "But Christ himself does not prove what he promises. It is true. For, as I have said, there cannot be any absolute proof of future events. Therefore since it is a condition of future events that they cannot be grasped or comprehended by any efforts of anticipation, is it not more reasonable, out of two alternatives that are uncertain and that are hanging in doubtful expectation, to give credence to the one that gives some hope rather than to the one that offers none at all? For in the former case there is no danger if, as is said to threaten, it becomes empty and void; while in the latter case the danger is greatest, that is, the loss of salvation, if when the time comes it is found that it was not a falsehood."

[160] Gregg wrote several other paradoxes, including the following: _The Authentic Report of the extraordinary case of Tresham Dames Gregg ... his committal to Bridewell for refusing to give his recognizance_ (Dublin, 1841), _An Appeal to Public Opinion upon a Case of Injury and Wrong ... in the case of a question of prerogative that arose between_ [R. Whately] _... Archbishop of Dublin and the author_ (London, 1861), _The Cosmology of Sir Isaac Newton proved to be in accordance with the Bible_ (London, 1871), _The Steam Locomotive as revealed in the Bible_ (London 1863) and _On the Sacred Law of 1866, conferring perpetual life with immunity from decay and disease. A cento of decisive scriptural oracles strangely discovered_ (London and Dublin, 1875). These titles will help the reader to understand the man whom De Morgan so pleasantly satirizes.

[161] See Vol. I, page 261, note 2 {592}.

[162] "They have found it."

[163] The late Greeks used the letters of their alphabet as numerals, adding three early alphabetic characters. The letter [chi] represented 600, [xi] represented 60, and [digamma] stood for 6. This gives 666, the number of the Beast given in the Revelation.

[164] "Allowing for necessary exceptions."

[165] Mr. Gregg is not alone in his efforts to use the calculus in original lines, as any one who has read Herbart's application of the subject to psychology will recall.

[166] See Vol. I, page 105, note 4 {188}; page 109, note 1 {197}.

[167] The full title shows the plan,--_The Decimal System as a whole, in its relation to time, measure, weight, capacity, and money, in unison with each other._ But why is this so much worse than the French plan of which we have only the metric system and the decimal division of the angle left?

[168] One of the brothers of Sir Isaac Pitman (1813-1897), the inventor of modern stenography. Of these brothers, Benjamin taught the art in America, Jacob in Australia, and Joseph, Henry, and Frederick in England.

[169] For example, _The Phonographic Lecturer_ (London, 1871 etc.), _The Phonographic Student_ (1867, etc.), and _The Shorthand Magazine_ (1866, etc.).

[170] See Vol. II, page 68, note 148.

[171] It involves the theory of non-Euclidean geometry, Euclid's postulate of parallels being used in proving this theorem.

[172] Referring to the fact that none of the works of Thales is extant.

[173] The author was one B. Bulstrode. Parts 4 and 5 were printed at Calcutta.

[174] See Vol. II, page 5, note 18.

[175] See Vol. I, page 85, note 2 {129}.

[176] Alexander Vasilievich Suvaroff (1729-1800), a Russian general who fought against the Turks, in the Polish wars, and in the early Napoleonic campaigns. When he took Ismail in 1790 he sent this couplet to Empress Catherine.

[177] "Newton hath determined rightly," "Newton hath not determined rightly."

[178] See Vol. I, page 288, note 3 {621}.

[179] See Vol. I, page 326, note 1 {700}.

[180] "With great honor."

[181] Apparently unknown to biographers. He seems to have written nothing else.

[182] Captain Marryat (1792-1848) published _Snarley-yow, or the Dog Fiend_ in 1837.

[183] He is not known to biographers, and published nothing else under this name.

[184] See Vol. I, page 80, note 5 {119}.

[185] He published a _Family and Commercial Illustrated Almanack and Year Book ... for 1861_ (Bath, 1860).

[186] Louis Dutens (1730-1812) was born at Tours, but went to England as a young man. He made the first collection of the works of Leibnitz, against the advice of Voltaire, who wrote to him: "Les ecrits de Leibnitz sont epars comme les feuilles de la Sybille, et aussi obscurs que les ecrits de cette vieille." The work appeared at Geneva, in six volumes, in 1769.

[187] Mungo Park (1771-1806), the first European to explore the Niger (1795-6). His _Travels in the Interior of Africa_ appeared in 1799. He died in Africa.

[188] Gerhard Mercator (1512-1594) the well-known map maker of Louvain. The "Mercator's Projection" was probably made as early as 1550, but the principle of its construction was first set forth by Edward Wright (London, 1599).

[189] Quirico Barilli Filopanti wrote a number of works and monographs. He succeeded in getting his _Cesare al Rubicone_ and _Degli_ _usi idraulici della Tela_ in the _Memoria letta ... all' Accademia delle Scienze in Bologna_ (1847, 1866). He also wrote _Dio esiste_ (1881), _Dio Liberale_ (1880), and _Sunto della memoria sulle geuranie ossia di alcune singolari relazioni cosmiche della terra e del cielo_ (1862).

[190] The periods of disembodiment may be interesting. They will be seen from the following dates: Descartes (1596-1650), William III (1650-1702); Roger Bacon (1214 to c. 1294), Boccaccio (1313-1375). Charles IX was born in 1550 and died in 1574.

[191] His real name was Frederick Parker, and he wrote several works on the Greek language and on religion. Among these were a translation of the New Testament from the Vatican MS. (1864), _The Revealed History of Man_ (1854), _An Enquiry respecting the Punctuation of Ancient Greek_ (1841), and _Rules for Ascertaining the sense conveyed in Ancient Greek Manuscripts_ (1848, the seventh edition appearing in 1862).

[192] See Vol. I, page 352, second note 1 {736}.

The literature on the subject of the Great Pyramid, considered from the standpoint of metrology, is extensive.

[193] See Vol. I, page 80, note 5 {119}.

[194] Sir Philip Francis (1740-1818) was a Whig politician. The evidence that he was the author of the _Letters of Junius_ (1769-1772) is purely circumstantial. He was clerk in the war office at the time of their publication. In 1774 he was made a member of the Supreme Council of Bengal, and was a vigorous opponent of Warren Hastings, the two fighting a duel in 1780. He entered parliament in 1784 and was among the leaders in the agitation for parliamentary reform.

[195] Mrs. Cottle published a number of letters that attracted attention at the time. Among these were letters to the emperor of France and king of Sardinia (1859) relating to the prophecies of the war between France and Austria; to G. C. Lavis and Her Majesty's Ministers (1859) relating to her claims as a prophetess; and to the "Crowned Heads" at St. James, the King of Prussia, and others (1860), relating to certain passages of Scripture. She also wrote _The Lamb's Book of Life for the New Jerusalem Church and Kingdom, interpreted for all nations_ (1861).

[196] See Vol. I, page 315, note 2 {685}, and Vol. II, page 58, note 109.

[197] A Congregational minister, who published a number of sermons, chiefly obituaries, between 1804 and 1851. His _Frailty of Human Life_, two sermons delivered on the occasion of the death of Princess Charlotte, went through at least three editions.

[198] He was secretary of the Congregational Board and editor of the _Congregational Year Book_ (from 1846) and the _Congregational Manual_.

[199] Frederick Denison Maurice (1805-1872) began his preaching as a Unitarian but entered the Established Church in 1831, being ordained in 1834. He was professor of English and History at King's College, London, from 1840 to 1853. He was one of the founders of Queen's College for women, and was the first principal of the Working Men's College, London. The subject referred to by De Morgan is his expression of opinion in his _Theological Essays_ (1853) that future punishment is not eternal. As a result of this expression he lost his professorship at King's College. In 1866 he was made Knightbridge Professor of Casuistry, Moral Theology, and Moral Philosophy at Cambridge.

[200] See Vol. I, page 46, note 1 {42}. Besides the books mentioned in this list he wrote _The Ratio between Diameter and Circumference demonstrated by angles, and Euclid's Theorem, Proposition 32, Book I, proved to be fallacious_ (Liverpool, 1870). This is the theorem which asserts that the exterior angle of a triangle is equal to the sum of the two opposite interior angles, and that the sum of the interior angles equals two right angles. He also published his _Curiosities of Mathematics_ in 1870, a work containing an extensive correspondence with every one who would pay any attention to him. De Morgan was then too feeble to show any interest in the final effort of the subject of some of his keenest satire.

[201] See Vol. I, page 332, note 4 {709}.

[202] See Vol. I, page 101, note 4 {174}.

[203] "The circle-squaring disease"; literally, "the circle-measuring disease."

[204] See Vol. II, page 63, note 136.

[205] William Rutherford (c. 1798-1871), teacher of mathematics at Woolwich, secretary of the Royal Astronomical Society, editor of _The Mathematician_, and author of various textbooks. _The Extension of [pi] to 440 places_, appeared in the _Proceedings_ of the Royal Society in 1853 (p. 274).

[206] Charles Knight (1791-1873) was associated with De Morgan for many years. After 1828 he superintended the publications of the Society for the Diffusion of Useful Knowledge, to which De Morgan contributed, and he edited the _Penny Cyclopedia_ (1833-1844) for which De Morgan wrote the articles on mathematics.

[207] Sir William Hamilton. See Vol. I, page 112, note 7 {211}.

[208] Adam Smith (1723-1790) was not only known for his _Wealth of Nations_ (1776), but for his _Theory of Moral Sentiments_ (1759), published while he was professor of moral philosophy at Glasgow (1752-1764). He was Lord Rector of the university in 1787.

[209] See Vol. I, page 332, note 4 {709}.

[210] "Whip."

[211] "Terrible lash."

[212] "An accomplished fact [an accomplished fault]."

[213] See _Extracts from the Diary and Letters of Mrs. Mary Cobb_, London, 1805.

[214] "Gentle in manner."

[215] "Brave in action." The motto of Earl Newborough was "Suaviter in modo, fortiter in re."

[216] "Reduction to an absurdity," a method of proof occasionally used in geometry and in logic.

[217] "He has lost the right of being moved (struck) by evidence."

[218] For _radix quadratus_. The usual root sign is supposed to be derived from _r_ (for radix), and at one time _q_ was commonly used for square, as in Viete's style of writing Aq for A^2.

[219] The Garde Douloureuse was a castle in the marches of Wales and received its name because of its exposure to attacks by the Welsh.

[220] "Out of the fight."

[221] "Hidden."

[222] John Cam Hobhouse (1786-1869), Baron Broughton, was committed to Newgate for two months in 1819 for his anonymous pamphlet, _A Trifling Mistake_. This was a great advertisement for him, and upon his release he was at once elected to parliament for Westminster. He was a strong supporter of all reform measures, and was Secretary for War in 1832. He was created Baron Broughton de Gyfford in 1851.

[223] Thomas Erskine (1750-1823), the famous orator. He became Lord Chancellor in 1806, but sat in the House of Commons most of his life.

[224] The above is explained in the MS. by a paragraph referring to some anagrams, in one of which, by help of the orthography suggested, a designation for this cyclometer was obtained from the letters of his name.--S. E. De M.

[225] "A personal verb agrees with its subject."

[226] See Vol. I, page 326, note 1 {700}.

[227] See Vol. I, page 326, note 2 {701}.

[228] Apparently unknown to biographers.

[229] The _Bibliotheca Mathematica_ of Ludwig Adolph Sohncke (1807-1853), professor of mathematics at Koenigsberg and Halle, covered the period from 1830 to 1854, being completed by W. Engelmann. It appeared in 1854.

[230] See Vol. I, page 392, note 2 {805}.

[231] See Vol. I, page 43, note 7 {32}.

[232] See Vol. II, page 91, note 187.

[233] Mason made a notable balloon trip from London to Weilburg, in the Duchy of Nassau, in November, 1836, covering 500 miles in 18 hours. He published an account of this trip in 1837, and a work entitled _Aeronautica_ in 1838.

[234] William Harrison Ainsworth (1805-1885) the novelist.

[235] On this question see Vol. I, page 326, note 2 {701}.

[236] Major General Alfred Wilks Drayson, author of various works on geology, astronomy, military surveying, and adventure.

[237] Hailes also wrote several other paradoxes on astronomy and circle squaring during the period 1843-1872.

[238] See Vol. I, page 43, note 8 {33}.

[239] See Vol. I, page 43, note 7 {32}.

[240] "Very small errors are not to be condemned."

[241] He seems to have written nothing else.

[242] Besides the paradoxes here mentioned by De Morgan he wrote several other works, including the following: _Abriss der Babylonisch-Assyrischen Geschichte_ (Mannheim, 1854), _A Popular Inquiry into the Moon's rotation on her axis_ (London, 1856), _Practical Tables for the reduction of the Mahometan dates to the Christian kalendar_ (London, 1856), _Grundzuege einer neuen Weltlehre_ (Munich, 1860), and _On the historical Antiquity of the People of Egypt_ (London, 1863).

[243] Dircks (1806-1873) was a civil engineer of prominence, and a member of the British Association and the Royal Society of Edinburgh. He wrote (1863) on "Pepper's Ghost," an ingenious optical illusion invented by him. There was a second edition of the _Perpetuum Mobile_ in 1870.

[244] George Stephenson (1781-1848), the inventor of the first successful steam locomotive. His first engine was tried in 1814.

[245] Robert Stephenson (1803-1859), the only son of George. Most of the early improvements in locomotive manufacture were due to him. He was also well known for his construction of great bridges.

[246] "In its proper place."

[247] "A fool always finds a bigger fool to admire him."

[248] See Vol. I, page 43, note 7 {32}.

[249] See Vol. I, page 43, note 8 {33}.

[250] See Vol. I, page 85, note 2 {129}.

[251] See Vol. I, page 390, note 1 {390}.

[252] From 1823 to 1852 it was edited by I. C. Robertson; from 1852 to 1857 by R. A. Brooman; and from 1857 to 1863 by Brooman and E. J. Reed.

[253] Sir James Ivory (1765-1842) was, as a young man, manager of a flax mill in Scotland. In 1804 he was made professor of mathematics at the Royal Military College, then at Marlow and later at Sandhurst. He was deeply interested in mathematical physics, and there is a theorem on the attraction of ellipsoids that bears his name. He was awarded three medals of the Royal Society, and was knighted together with Herschel and Brewster, in 1831.

[254] See Vol. I, page 56, note 1 {64}.

[255] See Vol. I, page 153, note 5 {338}.

[256] See Vol. I, page 309, note 2 {670}.

[257] See Vol. I, page 87, note 4 {133}.

[258] George Canning (1770-1857), the Tory statesman and friend of Scott, was much interested in founding the _Quarterly Review_ (1808) and was a contributor to its pages.

[259] See Vol. I, page 186, note 14 {418}.

[260] See Vol. II, page 141, note 252.

[261] De Morgan had a number of excellent articles in this publication.

[262] See Vol. I, page 279, note 1 {611}.

[263] James Orchard Halliwell (1820-1889), afterwards Halliwell-Phillips, came into prominence as a writer at an early age. When he was seventeen he wrote a series of lives of mathematicians for the _Parthenon_. His _Rara Mathematica_ appeared when he was but nineteen. He was a great bibliophile and an enthusiastic student of Shakespeare.

[264] This was written at the age of twenty-two.

[265] The subject of this criticism is of long past date, and as it has only been introduced by the author as an instance of faulty editorship, I have omitted the name of the writer of the libel, and a few lines of further detail.--S. E. De M.

[266] "Condemned souls."

[267] The editor of the _Mechanics' Magazine_ died soon after the above was written.--S. E. De M.

[268] Thomas Stephens Davies (1795-1851) was mathematical master at Woolwich and F. R. S. He contributed a series of "Geometrical Notes" to the _Mechanics' Magazine_ and edited the _Mathematician_. He also published a number of text-books.

[269] See Vol. II, page 66, note 143.

[270] The _Dictionary of Greek and Roman Biography_ (1849), edited by Sir William Smith (1813-1893), whose other dictionaries on classical and biblical matters are well known.

[271] "O J. S.! This is the worst! the greatest possible injury!"

[272] See Vol. I, page 44, note 9 {34} and page 110, note 5 {201}.

[273]

"If there's a man whom the judge's pitiless sentence awaiteth, His head condemned to penalties and tribulations, Let neither penitentiaries tire him with laborer's burdens Nor let his stiffened hands be harrassed by work in the mines. He must square the circle! For what else do I care?--all Known punishments this one task hath surely included."

[274] Houlston was in the customs service. He also published _Inklings of Areal Autometry_, London, 1874.

[275] This is Frederick C. Bakewell. He had already published _Natural Evidence of a Future Life_ (London, 1835), _Philosophical Conversations_ (London, 1833, with other editions), and _Electric Science_ (London, 1853, with other editions).

[276] Henry F. A. Pratt had already published _A Dissertation on the power of the intercepted pressure of the Atmosphere_ (London, 1844) and _The Genealogy of Creation_ (1861). Later he published a work _On Orbital Motion_ (1863), and _Astronomical Investigations_ (1865).

[277] See Vol. I, page 260, note 1 {591}.

[278] Thomas Rawson Birks (1810-1883), a theologian and controversialist, fellow of Trinity College, Cambridge, and (1872) professor of moral philosophy in that university. He wrote _Modern Rationalism_ (1853), _The Bible and Modern Thought_ (1861), _The First Principles of Moral Science_ (1873), and _Modern Physical Fatalism and the Doctrine of Evolution_ (1876), the last being an attack on Herbert Spencer's _First Principles_.

[279] Pseudonym for William Thorn. In the following year (1863) he published a second work, _The Thorn-Tree: being a History of Thorn Worship_, a reply to Bishop Colenso's work entitled _The Pentateuch and the Book of Joshua critically examined_.

[280] Besides _The Pestilence_ (1866) he published _The True Church_ (1851), _The Church and her destinies_ (1855), _Religious reformation imperatively demanded_ (1864), and _The Bible plan unfolded_ (second edition, 1872).

[281] See Vol. II, page 97, note 195.

[282] Sir George Cornewall Lewis (1806-1863) also wrote an _Essay on the Origin and Formation of the Romance Languages_ (1835), an _Essay on the Government of Dependencies_ (1841), and an _Essay on Foreign Jurisdiction and the Extradition of Criminals_ (1859). He was Chancellor of the Exchequer in 1855 and Home Secretary in 1859.

[283] Henry Malden (1800-1876), a classical scholar, fellow of Trinity College, Cambridge, and professor of Greek at University College (1831-1876), then (1831) the University of London. He wrote a _History of Rome to 390 B. C._ (1830), and _On the Origin of Universities and Academical Degrees_ (1835).

[284] Henry Longueville Mansel (1820-1871), theologian and metaphysician, reader in theology at Magdalen College, Oxford (1855), and professor of ecclesiastical history and Dean of St. Paul's (1866). He wrote on metaphysics, and his Bampton Lectures (1858) were reprinted several times.

[285] "Hejus gave freely, gave freely. God is propitious, God is favorable to him who gives freely. God is honored with a banquet of eggs at the cross roads, the god of the world. God, with benignant spirit, desired in sacrifice a goat, a bull to be carried within the precincts of the holy place. God, twice propitiated, blesses the pit of the sacred libation."

[286] Eudoxus of Cnidus (408-355 B. C.) had much to do with the early scientific astronomy of the Greeks. The fifth book of Euclid is generally attributed to him. His astronomical works are known chiefly through the poetical version of Aratus mentioned in note 13, page 167.

[287] Simplicius, a native of Cilicia, lived in the 6th century of our era. He was driven from Athens by Justinian and went to Persia (531), but he returned later and had some fame as a teacher.

[288] See Vol. I, page 160, note 3 {348}.

[289] See Vol. I, page 76, note 3 {112}.

[290] "Through right and wrong."

[291] "It is therefore to arrive at this parallelism, or to preserve it, that Copernicus feared to be obliged to have recourse to this equal and opposite movement which destroys the effect which he attributed so freely to the first, of deranging the parallelism."

[292] A contemporary of Plato and a disciple of Aristotle.

[293] Meton's solstice, the beginning of the Metonic cycles, has been placed at 432 B. C. Ptolemy states that he made the length of the year 365-1/4 + 1/72 days.

[294] Aratus lived about 270 B. C., at the court of Antigonus of Macedonia, and probably practiced medicine there. He was the author of two astronomical poems, the [Greek: Phainomena], apparently based on the lost work of Eudoxus, and the [Greek: Dioseeia] based on Aristotle's _Meteorologica_ and _De Signis Ventorum_ of Theophrastus.

[295] "The nineteen (-year) cycle of the shining sun."

[296] Claudius Salmasius (1588-1653), or Claude Saumaise, was a distinguished classicist, and professor at the University of Leyden. The word [Greek: eleioio] means Elian, thus making the phrase refer to the brilliant one of Elis.

[297] Sir William Brown (1784-1864). In 1800 the family moved to Baltimore, and there the father, Alexander Brown, became prominent in the linen trade. William went to Liverpool where he acquired great wealth as a merchant and banker. He was made a baronet in 1863.

[298] Robert Lowe (1811-1892), viscount Sherbrooke, was a fellow of Magdalen College, Oxford (1835). He went to Australia in 1842 and was very successful at the bar. He returned to England in 1850 and became leader writer on the _Times_. He was many years in parliament, and in 1880 was raised to the peerage.

[299] See Vol. I, page 42, note 4 {24}.

[300] Francis Walkingame (fl. about 1751-1785), whose _Tutor's Assistant_ went through many editions from 1751-1854.

[301] Davies Gilbert (1767-1839). His family name was Giddy, but he assumed his wife's name. He sat in parliament from 1806 to 1832. In 1819 he secured the establishment of the Cape of Good Hope observatory. He was Treasurer (1820-1827) and President (1827-1830) of the Royal Society.

[302] See Vol. I, page 55, note 2 {63}.

[303] Sir Jonathan Frederick Pollock (1783-1870) entered parliament in 1831 and was knighted in 1834.

[304] Joseph Hume (1777-1855) entered parliament in 1812 and for thirty years was leader of the Radical party.

[305] "What! when I say, 'Nicole, bring me my slippers,' is that prose?"

[306] Captain Basil Hall (1788-1844), a naval officer, carried on a series of pendulum observations in 1820-1822, while on a cruise of the west coast of North America. The results were published in 1823 in the _Philosophical Transactions_. He also wrote two popular works on travel that went through numerous editions.

[307] Anthony Ashley Cooper (1801-1885), Earl of Shaftesbury. His name is connected with philanthropic work and factory legislation.

[308] See Vol. I, page 207, note 12 {469}.

[309] See Vol. I, page 80, note 5 {119}.

[310] Sir Thomas Maclear (1794-1879), an Irishman by birth, became Astronomer Royal at the Cape of Good Hope in 1833. He was an indefatigable observer. He was knighted in 1860.

[311] Thomas Romney Robinson (1792-1882), another Irish astronomer of prominence. He was a deputy professor at Trinity College, Dublin, but took charge of the Armagh observatory in 1823 and remained there until his death.

[312] Sir James South (1785-1867) was in early life a surgeon, but gave up his practice in 1816 and fitted up a private observatory. He contributed to the science of astronomy, particularly with respect to the study of double stars.

[313] Sir John Wrottesley (1798-1867), second Baron Wrottesley. Like Sir James South, he took up the study of astronomy after a professional career,--in his case in law. He built a private observatory in 1829 and made a long series of observations, publishing three star catalogues. He was president of the Astronomical Society from 1841 to 1843, and of the Royal Society from 1854 to 1857.

[314] He seems to have written nothing else.

[315] See Vol. II, page 68, note 147.

[316] "The wills are free, and I wish neither the one nor the other."

[317] "The force of inertia conquered."

[318] Reddie also wrote _The Mechanics of the Heavens_, referred to later in this work. He must not be confused with Judge James Reddie (1773-1852), of Glasgow, who wrote on international law, although this is done in the printed edition of the British Museum catalogue, for he is mentioned by De Morgan somewhat later as alive in 1862.

[319] Henry Dunning Macleod (1821-1902), a lawyer and writer on political economy, was a Scotchman by birth. He wrote on economical questions, and lectured on banking at Cambridge (1877) and at King's College, London (1878). He was a free lance in his field, and was not considered orthodox by the majority of economists of his time. He was an unsuccessful candidate for the chairs of political economy at Cambridge (1863), Edinburgh (1871), and Oxford (1888).

[320] See Vol. I, page 252, note 2 {576}.

[321] Francis Henry Laing (1816-1889) was a graduate of Queen's College, Cambridge, and a clergyman in the Church of England until 1846, when he entered the Church of Rome. He taught in various Jesuit colleges until 1862, when his eccentricity was too marked to warrant the Church in allowing him to continue. He published various controversial writings during his later years. Of course if he had known the works of Wessel, Gaus, Buee, Argand, and others, he would not have made such a sorry exhibition of his ignorance of mathematics.

[322] See Vol. I, page 329, note 1 {705}. The book went into a second edition in 1864.

[323] Thomas Weddle (1817-1853) was, at the time of publishing this paper, a teacher in a private school. In 1851 he became professor of mathematics at Sandhurst. He contributed several papers to the _Cambridge and Dublin Mathematical Journal_, chiefly on geometry.

[324] See Vol. II, page 109, note 205.

[325] See Vol. II, page 66, note 143.

[326] See Vol. II, page 151, note 268.

[327] George Barrett (1752-1821) worked from 1786 to 1811 on a set of life insurance and annuity tables. He invented a plan known as the "columnar method" for the construction of such tables, and as De Morgan states, this was published by Francis Baily, appearing in the appendix to his work on annuities, in the edition of 1813. Some of his tables were used in Babbage's _Comparative View of the various Institutions for the Assurance of Lives_ (1826).

[328] See Vol. I, page 309, note 2 {670}.

[329] This was his _Practical short and direct Method of Calculating the Logarithm of any given Number, and the Number corresponding to any given Logarithm_ (1849).

[330] This is William Neile (1637-1670), grandson of Richard Neile (not Neal), Archbishop of York. At the age of 19, in 1657, he gave the first rectification of the semicubical parabola. Although he communicated it to Brouncker, Wren, and others, it was not published until 1639, when it appeared in John Wallis's _De Cycloide_.

[331] I myself "was a considerable part."

[332] He also wrote _A Glance at the Universe_ ("2d thousand" in 1862), and _The Resurrection Body_ (1869).

[333] See Vol. I, page 63, note 1 {74}.

[334] As Swift gave it in his _Poetry. A Rhapsody_, it is as follows:

"So, naturalists observe, a flea Has smaller fleas that on him prey; And these have smaller still to bite 'em. And so proceed _ad infinitum_."

[335] Perhaps 1,600,000,000 years, if Boltwood's recent computations based on radium disintegration stand the test. This would mean, according to MacCurdy's estimate, 60,000,000 years since life first appeared on the earth.

[336] De Morgan wrote better than he knew, for this work, the _Allgemeine Encyclopaedie der Wissenschaften und Kuenste_, begun at Leipsic in 1818, is still (1913) unfinished. Section I, A-G, consists of 99 parts in 56 volumes; Section II, H-N, consists of 43 volumes and is not yet completed; and Section III, O-Z, consists of 25 volumes thus far, with most of the work still to be done. Johann Samuel Ersch (1766-1828), the founder, was head librarian at Halle. Johann Gottfried Gruber (1774-1851), his associate, was professor of philosophy at the same university.

[337] William Howitt (1792-1879) was a poet, a spiritualist, and a miscellaneous writer. He and his wife became spiritualists about 1850. He wrote numerous popular works on travel, nature and history.

[338] See Vol. II, page 55, note 108.

[339] As will be inferred from the text, C. D. was Mrs. De Morgan, and A. B. was De Morgan.

[340] Jean Meslier (1678-1733), cure of Estrepigny, in Champagne, was a skeptic, but preached only strict orthodoxy to his people. It was only in his manuscript, _Mon Testament_, that was published after his death, and that caused a great sensation in France, that his antagonism to Christianity became known.

[341] Baron Zach relates that a friend of his, in a writing intended for publication, said _Un esprit doit se frotter contre un autre_. The censors struck it out. The Austrian police have a keen eye for consequences.--A. De M.

"One mind must rub against another." On Baron Zach, see Vol. II, page 45, note 4.

[342] Referring to the first Lord Eldon (1751-1838), who was Lord Chancellor from 1799 to 1827, with the exception of one year.

[343] "Sleeping power."

[344] "Causes sleep."

[345] Richard Hooker (c. 1554-1600), a theologian, "the ablest living advocate of the Church of England as by law established."

[346] See Vol. I, page 76, note 3 {112}.

[347] "Other I,"--other self.

[348] This "utter rejection" has been repeated (1872) by the same writer.--S. E. De M.

[349] Edward Jenner (1749-1823) was a physician and biologist. His first experiments in vaccination were made in 1796, and his discovery was published in 1798.

[350] See Vol. II, page 38, note 80.

[351] "You will go most safely in the middle (way)."

[352] Pierre Joseph Arson was known early in the 19th century for his controversy with Hoene Wronski the mathematician, whom he attacked in his _Document pour l'histoire des grands fourbes qui ont figure sur la terre_ (1817-1818).

[353] "We enter the course by night and are consumed by fire."

[354] See Vol. I, page 51, note 3 {51}.

[355] See Vol. I, page 336, note 8 {713}.

[356] See Vol. I, page 137, note 8 {286}.

[357] See Vol. I, page 229, note 2 {515}.

[358] Richard Cobden (1804-1865), the cotton manufacturer and statesman who was prominent in his advocacy of the repeal of the Corn Laws.

[359] James Smith (1775-1839), solicitor to the Board of Ordnance. With his brother Horatio he wrote numerous satires. His _Horace in London_ (1813) imitated the Roman poet. His works were collected and published in 1840.

[360] Samuel Butler (1612-1680), the poet and satirist, author of _Hudibras_ (1663-1678).

[361] "Is it not fine to be sure of one's action when entering in a combat with another? There, push me a little in order to see. NICOLE. Well! what's the matter? M. JOURDAIN. Slowly. Ho there! Ho! gently. Deuce take the rascal! NICOLE. You told me to push. M. JOURDAIN. Yes, but you pushed me _en tierce_, before you pushed _en quarte_, and you did not give me time to parry."

[362] John Abernethy (1764-1831), the famous physician and surgeon.

[363] See Vol. I, page 102, note 5 {175}.

[364] "With what measure ye mete, it shall be measured to you again."

[365] Eusebius of Caesarea (c. 260-340), leader of the moderate party at the Council of Nicaea, and author of a _History of the Christian Church_ in ten books (c. 324 A. D.).

[366] Nathaniel Lardner (1684-1768), a non-conformist minister and one of the first to advocate the scientific study of early Christian literature.

[367] Henry Alford (1810-1871) Dean of Canterbury (1857-1871) and editor of the Greek Testament (1849-1861).

[368] The work was _The Number and Names of the Apocalyptic Beasts: with an explanation and application. Part I._ London, 1848, as mentioned below. Thom also wrote _The Assurance of Faith, or Calvinism identified with Universalism_ (London, 1833), and various other religious works.

[369] See Vol. I, page 222, note 14 {490}.

[370] John Hamilton Thom (1808-1894) was converted to Unitarianism and was long a minister in that church, preaching in the Renshaw Street Chapel from 1831 to 1866. De Morgan refers to the Liverpool Unitarian controversy conducted by James Martineau and Henry Giles in response to a challenge by thirteen Anglican Clergy. In 1839 Thom contributed four lectures and a letter to this controversy. Among his religious works were a _Life of Blanco White_ (1845) and _Hymns, Chants, and Anthems_ (1854).

[371] The spelling of these names is occasionally changed to meet the condition that the numerical value of the letters shall be 666, "the number of the beast" of Revelations. The names include Julius Caesar; Valerius Jovius Diocletianus (249-313), emperor from 287 to 305, persecutor of the Christians; Louis, presumably Louis XIV; Gerbert (940-1003), who reigned as Pope Sylvester II from 999 to 1003, known to mathematicians for his abacus and his interest in geometry, and accused by his opponents as being in league with the devil; Linus, the second Bishop of Rome, the successor of Peter; Camillo Borghese (1552-1621), who reigned as Pope Paul V from 1605 to 1621, and who excommunicated all Venice in 1606 for its claim to try ecclesiastics before lay tribunals, thus taking a position which he was forced to abandon; Luther, Calvin; Laud (see Vol. I, page 145, note 7 {307}); Genseric (c. 406-477), king of the Vandals, who sacked Rome in 455 and persecuted the orthodox Christians in Africa; Boniface III, who was pope for nine months in 606; Beza (see Vol. I, page 66, note 6 {83}); Mohammed; [Greek: braski], who was Giovanni Angelo Braschi (1717-1799), and who reigned as Pope Pius VI from 1775 to 1799, dying in captivity because he declined to resign his temporal power to Napoleon; Bonaparte; and, under [Greek: Ion Paune], possibly Pope John XIV, who reigned in 983 and 984 during the absence of Boniface VII in Constantinople.

[372] The Greek words and names are also occasionally misspelled so as to fit them to the number 666. They are [Greek: Lateinos] (Latin), [Greek: he latine basileia] (the Latin kingdom), [Greek: ekklesia italika] (the Italian Church), [Greek: euanthas] (blooming), [Greek: teitan] (Titan), [Greek: arnoume] (renounce), [Greek: lampetis] (the lustrous), [Greek: ho niketes] (conqueror), [Greek: kakos hodegos] (bad guide), [Greek: alethes blaberos] (truthful harmful one), [Greek: palai baskanos] (a slanderer of old), [Greek: amnos adikos] (unmanageable lamb), [Greek: antemos] (Antemos), [Greek: genserikos] (Genseric), [Greek: euinas] (with stout fibers), [Greek: Benediktos] (Benedict), [Greek: Bonibazios g. papa x. e. e. e. a.] (Boniface III, pope 68, bishop of bishops I), [Greek: oulpios] (baneful), [Greek: dios eimi he heras] (I, a god, am the), [Greek: he missa he papike] (the papal brief), [Greek: loutherana] (Lutheran), [Greek: saxoneios] (Saxon), [Greek: Bezza antitheos] (Beza antigod), [Greek: he alazoneia biou] (the illusion of life), [Greek: Maometis] (Mahomet); [Greek: Maometes b.] (Mahomet II), [Greek: theos eimi epi gaies] (I am lord over the earth), [Greek: iapetos] (Iapetos, father of Atlas), [Greek: papeiskos] (Papeiskos), [Greek: dioklasianos] (Diocletian), [Greek: cheina] (Cheina = Cain? China?), [Greek: braski] (Braschi, as explained in note 10), [Greek: Ion Paune] (Paunian violet, but see note 10), [Greek: koupoks] (cowpox), [Greek: Bonneparte] (Bonneparte), [Greek: N. Boneparte] (N. Boneparte), [Greek: euporia] (facility), [Greek: paradosis] (surrender), [Greek: to megatherion] (the megathereum, the beast).

[373] James Wapshare, whose _Harmony of the Word of God in Spirit and in Truth_ appeared in 1849.

[374] The literature relating to the _Swastika_ is too extended to permit of any adequate summary in these notes.

[375] Henry Edward Manning (1808-1892), at first an Anglican clergyman, he became a Roman Catholic priest in 1851, and became Cardinal in 1875. He succeeded Cardinal Wiseman as Archbishop of Westminster in 1865. He wrote a number of religious works.

[376] John Bright (1811-1889), Quaker, cotton manufacturer, and statesman. He worked with Cobden for free trade, peace, and reform of the electorate.

[377] "The fallacy of many questions."

[378] William Wilberforce (1759-1833), best known for his long fight for the abolition of the slave trade.

[379] Richard Martin (1754-1834), high sheriff of County Galway and owner of a large estate in Connemara. Curiously enough, he was known both for his readiness in duelling and for his love for animals. He was known as "Humanity Martin," and in 1822 secured the passage of an act "to prevent the cruel and improper treatment of cattle." He was one of the founders (1824) of the Royal Society for the Prevention of Cruelty to Animals. He is usually considered the original of Godfrey O'Malley in Lever's novel, _Charles O'Malley_.

[380] See Vol. I, page 149, note 1 {323}, also text on same page.

[381] See Vol. I, page 44, note 9 {34}, also text, Vol. I, page 110.

[382] "Penitential seat."

[383] "Well placed upon the cushion."

[384] See Vol. II, page 58, note 109.

[385] "He has lost the right of being influenced by evidence."

[386] "Hung up."

[387] "A few things to the wise, nothing to the unlettered."

[388] The fallacy results from dividing both members of an equation by 0, x - 1 being the same as 1 - 1, and calling the quotients finite.

[389] "If you order him to the sky he will go."

[390] _Similia similibus curanter_, "Like cures like," the homeopathic motto.

[391] "Without harm to the proprieties."

[392] "What are you doing? I am standing here."

[393] Lors feist l'Anglois tel signe. La main gausche toute ouverte il leva hault en l'aer, puis ferma au poing les quatres doigtz d'icelle et le poulce estendu assit sus la pinne du nez. Soubdain apres leva la dextre toute ouverte, et toute ouverte la baissa, joignant la poulce au lieu que fermait le petit doigt de la gausche, et les quatre doigtz d'icelle mouvoit lentement en l'aer. Puis au rebours feit de la dextre ce qu'il avoit faict de la gausche, et de la gausche ce que avoit faict de la dextre.--A. De M.

[394] _Suaviter in modo, fortiter in re_, "Gentle in manners, firm in action."

[395] See Vol. I, page 101, note 4 {174}.

[396] See Vol. I, page 315, note 3 {686}.

[397] Henry Fawcett (1833-1884) became totally blind in 1858, but in spite of this handicap he became professor of political economy at Cambridge and sat in parliament for a number of years. He championed the cause of reform and in particular he was prominent in the protection of the native interests of India. The establishing of the parcels post (1882) took place while he was postmaster general (1880-1884).

[398] Of course the whole thing depends upon what definition of division is taken. We can multiply 2 ft. by 3 ft. if we define multiplication so as to allow it, or 2 ft. by 3 lb, getting foot-pounds, as is done in physics.

[399] Richard Milward (1609-1680), for so the name is usually given, was rector of Great Braxted (Essex) and canon of Windsor. He was long the amanuensis of John Selden, and the _Table Talk_ was published nine years after Milward's death, from notes that he left. Some doubt has been cast upon the authenticity of the work owing to many of the opinions that it ascribes to Selden.

[400] John Selden (1584-1654) was a jurist, legal antiquary, and Oriental scholar. He sat in the Long Parliament, and while an advocate of reform he was not an extremist. He was sent to the Tower for his support of the resolution against "tonnage and poundage," in 1629. His _History of Tythes_ (1618) was suppressed at the demand of the bishops. His _De Diis Syriis_ (1617) is still esteemed a classic on Semitic mythology.

[401] See Vol. I, page 42, note 4 {24}.

[402] See Vol. II, page 249, note 398.

[403] John Palmer (1742-1818) was a theatrical manager. In 1782 he set forth a plan for forwarding the mails by stage coaches instead of by postmen. Pitt adopted the plan in 1784. Palmer was made comptroller-general of the post office in 1786 and was dismissed six years later for arbitrarily suspending a deputy. He had been verbally promised 2-1/2% on the increased revenue, but Pitt gave him only a pension of L3000. In 1813 he was awarded L50,000 in addition to his pension.

[404] Dionysius Lardner (1793-1859), professor of natural philosophy in London University (now University College). His _Cabinet Cyclopaedia_ (1829-1849) contained 133 volumes. De Morgan wrote on probabilities, and Lardner on various branches of mathematics, and there were many other well-known contributors. Lardner is said to have made $200,000 on a lecture tour in America.

[405] Thomas Fysche Palmer (1747-1802) joined the Unitarians in 1783, and in 1785 took a charge in Dundee. He was arrested for sedition because of an address that it was falsely alleged that he gave before a society known as the "Friends of Liberty." As a matter of fact the address was given by an uneducated weaver, and Palmer was merely asked to revise it, declining to do even this. Nevertheless he was sentenced to Botany Bay (1794) for seven years. The trial aroused great indignation.

[406] See Vol. I, page 80, note 5 {119}.

[407] See Vol. II, page 244, note 394.

[408] See Vol. I, page 352, note 1 {731}.

[409] See Vol. I, page 332, note 4 {709}.

[410] "The lawyers are brought into court; let them accuse each other."

[411] Samuel Rogers (1763-1855), the poet and art connoisseur. He declined the laureateship on the death of Wordsworth (1850). Byron, his pretended friend, wrote a lampoon (1818) ridiculing his cadaverous appearance.

[412] Theodore Edward Hook (1788-1841), the well-known wit. He is satirized as Mr. Wagg in _Vanity Fair_. The _John Bull_ was founded in 1820 and Hook was made editor.

[413] "On pitying the heretic."

[414] A term of medieval logic. Barbara: All M is P, all S is M, hence all S is P. Celarent: No M is P, all S is M, hence no S is P.

[415] "Simply," "According to which," "It does not follow."

[416]

"O sweet soul, what good shall I declare That heretofore was thine, since such are thy remains!"

[417] "Stupid fellow!"

[418] Christopher Barker (c. 1529-1599), also called Barkar, was the Queen's printer. He began to publish books in 1569, but did no actual printing until 1576. In 1575 the Geneva Bible was first printed in England, the work being done for Barker. He published 38 partial or complete editions of the Bible from 1575 to 1588, and 34 were published by his deputies (1588-1599).

[419] James Franklin (1697-1735) was born in Boston, Mass., and was sent to London to learn the printer's trade. He returned in 1717 and started a printing house. Benjamin, his brother, was apprenticed to him but ran away (1723). James published the _New England Courant_ (1721-1727), and Benjamin is said to have begun his literary career by writing for it.

[420] James Hodder was a writing master in Tokenhouse Yard, Lothbury, in 1661, and later kept a boarding school in Bromley-by-Bow. His famous arithmetic appeared at London in 1661 and went through many editions. It was the basis of Cocker's work. (See Vol. I, page 42, note 4 {24}.) It was long thought to have been the first arithmetic published in America, and it was the first English one. There was, however, an arithmetic published much earlier than this, in Mexico, the _Sumario compendioso ... con algunas reglas tocantes al Aritmetica_, by "Juan Diaz Freyle," in 1556.

[421] Henry Mose, Hodder's successor, kept a school in Sherborne Lane, London.

[422] Rear Admiral Sir Francis Beaufort (1774-1857), F.R.S., was hydrographer to the Navy from 1829 to 1855. He prepared an atlas that was printed by the Society for the Diffusion of Useful Knowledge.

[423] Antoine Sabatier (1742-1817), born at Castres, was known as the Abbe but was really nothing more than a "clerc tonsure." He lived at Court and was pensioned to write against the philosophers of the Voltaire group. He posed as the defender of morality, a commodity of which he seems to have possessed not the slightest trace.

[424] Maffeo Barberini was pope, as Urban VIII, from 1623 to 1644. It was during his ambitious reign that Galileo was summoned to Rome to make his recantation (1633), the exact nature of which is still a matter of dispute.

[425] This Baden Powell (1796-1860) was the Savilian professor of geometry (1827-1860) at Oxford.

[426] "Memoirs of the famous bishop of Chiapa, by which it appears that he had butchered or burned or drowned ten million infidels in America in order to convert them. I believe that this bishop exaggerated; but if we should reduce these sacrifices to five million victims, this would still be admirable."

[427] Alfonso X (1221-1284), known as El Sabio (the Wise), was interested in astronomy and caused the Alphonsine Tables to be prepared. These table were used by astronomers for a long time. It is said that when the Ptolemaic system of the universe was explained to him he remarked that if he had been present at the Creation he could have shown how to arrange things in a much simpler fashion.

[428] George Richards (c. 1767-1837), fellow of Oriel (1790-1796), Bampton lecturer (1800), Vicar of St. Martin's-in-the-Fields, Westminster (1824), and a poet of no mean ability.

[429] The "Aboriginal Britons," an excellent poem, by Richards. (Note by Byron.)--A. De M.

[430] John Watkins (d. after 1831), a teacher and miscellaneous writer.

[431] Frederic Shoberl (1775-1853), a miscellaneous writer.

[432] He wrote, besides the _Aboriginal Britons_, _Songs of the Aboriginal Bards_ (1792), _Modern France: a Poem_ (1793), _Odin, a drama_ (1804), _Emma, a drama on the model of the Greek theatre_ (1804), _Poems_ (2 volumes, 1804), and a _Monody on the Death of Lord Nelson_ (1806).

[433] Henry Kirke White (1785-1806), published his first volume of poems at the age of 18. Southey and William Wilberforce became interested in him and procured for him a sizarship at St. John's College, Cambridge. He at once showed great brilliancy, but he died of tuberculosis at the age of 21.

[434] John Wolcot, known as Peter Pindar (1738-1819), was a London physician. He wrote numerous satirical poems. His _Bozzy and Piozzi, or the British Biographers_, appeared in 1786, and reached the 9th edition in 1788.

[435] See Vol. I, page 235, note 8 {532}.

[436] Richard Payne Knight (1750-1824) was a collector of bronzes, gems, and coins, many of his pieces being now in the British Museum. He sat in parliament for twenty-six years (1780-1806), but took no active part in legislation. He opposed the acquisition of the Elgin Marbles, holding them to be of little importance. His _Analytical Inquiry into the Principles of Taste_ appeared in 1808.

[437] Mario Nizzoli (1498-1566), a well-known student of Cicero, was for a time professor at the University of Parma. His _Observationes in M. Tullium Ciceronem_ appeared at Pratalboino in 1535. It was revised by his nephew under the title _Thesaurus Ciceronianus_ (Venice, 1570).

[438] See Vol. I, page 314, note 4 {681}.

[439]

"Like the geometer, who bends all his powers To measure the circle, and does not succeed, Thinking what principle he needs."

[440] Francis Quarles (1592-1644), a religious poet. He wrote paraphrases of the Bible and numerous elegies. In the early days of the revolutionary struggle he sided with the Royalists. One of his most popular works was the _Emblems_ (1635), with illustrations by William Marshall.

[441] Regnault de Becourt wrote _La Creation du monde, ou Systeme d'organisation primitive suivi de l'interpretation des principaux phenomenes et accidents que se sont operes dans la nature depuis l'origine de univers jusqu'a nos jours_ (1816). This may be the work translated by Dalmas.

[442] "Because it lacks a holy prophet."

[443] Anghera. See Vol. II, page 60, note 127.

[444] Edmund Curll (1675-1747), a well-known bookseller, publisher, and pamphleteer. He was for a time at "The Peacock without Temple Bar," and later at "The Dial and Bible against St. Dunstan's Church." He was fined repeatedly for publishing immoral works, and once stood in the pillory for it. He is ridiculed in the _Dunciad_ for having been tossed in a blanket by the boys of Westminster School because of an oration that displeased them.

[445] See Vol. II, page 109, note 206.

[446] Encyclopaedia.

[447] Author of the _Historia Naturalis_ (77 A.D.)

[448] Author of the _De Institutione Oratorio Libri_ XII (c. 91 A.D.)

[449] His _De Architectures Libri_ X was not merely a work on architecture and building, but on the education of the architect.

[450] Cyclophoria.

[451] William Caxton (c. 1422-c.1492), sometime Governor of the Company of Merchant Adventurers in Bruges (between 1449 and 1470). He learned the art of printing either at Bruges or Cologne, and between 1471 and 1477 set up a press at Westminster. Tradition says that the first book printed in England was his _Game and Playe of Chesse_ (1474). The _Myrrour of the Worlde and th'ymage of the same_ appeared in 1480. It contains a brief statement on arithmetic, the first mathematics to appear in print in England.

[452] See Vol. I, page 45, note 6 {40}. De Morgan is wrong as to the date of the _Margarita Philosophica_. The first edition appeared at Freiburg in 1503.

[453] Reisch was confessor to Maximilian I (1459-1519), King of the Romans (1486) and Emperor (1493-1519).

[454] Joachim Sterck Ringelbergh (c. 1499-c. 1536), teacher of philosophy and mathematics in various cities of France and Germany. His _Institutionum astronomicarum libri III_ appeared at Basel in 1528, his _Cosmographia_ at Paris in 1529, and his _Opera_ at Leyden in 1531.

[455] Johannes Heinrich Alsted (1588-1638) was professor of philosophy and theology at his birthplace, Herborn, in Nassau, and later at Weissenberg. He published several works, including the _Elementale mathematicum_ (1611), _Systema physicae harmonicae_ (1612), _Methodus admirandorum mathematicorum_ (1613), _Encyclopaedia septem tomis distincta_ (1630), and the work mentioned above.

[456] Johann Jakob Hoffmann (1635-1706), professor of Greek and history at his birthplace, Basel. He also wrote the _Epitome metrica historiae universalis civilis et sacrae ab orbe condito_ (1686).

[457] Ephraim Chambers (c. 1680-1740), a crotchety, penurious, but kind-hearted freethinker. His _Cyclopaedia, or an Universal Dictionary_ was translated into French and is said to have suggested the great _Encyclopedie_.

[458] _Encyclopedie, ou Dictionnaire raisonne des sciences, des arts et des metiers, par un societe de gens de lettres. Mis en ordre et publie par M. Diderot, et quant a la partie mathematique, par M. d'Alembert._ Paris, 1751-1780, 35 volumes.

[459] "From the egg" (state).

[460] See Vol. I, page 382, note 12 {785}.

[461] See Vol. II, page 4, note 15.

[462] "In morals nothing should serve man as a model but God; in the arts, nothing but nature."

[463] _Encyclopedie Methodique, ou par ordre de matieres._ Paris, 1782-1832, 166-1/2 volumes.

[464] See Vol. II, page 193, note 336.

[465] _Encyclopaedia Metropolitana; or, Universal Dictionary of Knowledge._ London, 1845, 29 volumes. A second edition came out in 1848-1858 in 40 volumes.

[466] See Vol. I, page 137, note 8 {286}.

[467] See Vol. I, page 80, note 5 {119}.

[468] De Morgan should be alive to satirize some of the statements on the history of mathematics in the eleventh edition.

[469] John Pringle Nichol (1804-1859), Regius professor of astronomy at Glasgow and a popular lecturer on the subject. He lectured in the United States in 1848-1849. His _Views of the Architecture of the Heavens_ (1838) was a very popular work, and his _Planetary System_ (1848, 1850) contains the first suggestion for the study of sun spots by the aid of photography.

[470] See Vol. II, page 109, note 206.

[471] George Long (1800-1879), a native of Poulton, in Lancashire, was called to the University of Virginia when he was only twenty-four years old as professor of ancient languages. He returned to England in 1828 to become professor of Greek at London University. From 1833 to 1849 he edited the twenty-nine volumes of the _Penny Cyclopaedia_. He was an authority on Roman law.

[472] A legal phrase, "Qui tam pro domina regina, quam pro se ipso sequitur,"--"Who sues as much on the Queen's account as on his own."

[473] Arthur Cayley (1821-1895) was a fellow of Trinity College, Cambridge (1842-1846) and was afterwards a lawyer (1849-1863). During his fourteen years at the bar he published some two hundred mathematical papers. In 1863 he became professor of mathematics at Cambridge, and so remained until his death. His collected papers, nine hundred in number, were published by the Cambridge Press in 13 volumes (1889-1898). He contributed extensively to the theory of invariants and covariants. De Morgan's reference to his coining of new names is justified, although his contemporary, Professor Sylvester, so far surpassed him in this respect as to have been dubbed "the mathematical Adam."

[474] See Vol. II, page 26, note 56.

[475] See Vol. I, page 111, note 3 {207}.

[476] See Vol. I, page 87, note 6 {135}.

[477] Pierre Dupuy (1582-1651) was a friend and relative of De Thou. With the collaboration of his brother and Nicolas Rigault he published the 1620 and 1626 editions of De Thou's History. He also wrote on law and history. His younger brother, Jacques (died in 1656), edited his works. The two had a valuable collection of books and manuscripts which they bequeathed to the Royal Library at Paris.

[478] See Vol. I, page 51, note 3 {51}.

[479] It was Cosmo de' Medici (1590-1621) who was the patron of Galileo.

[480] See Vol. I, page 40, note 4 {20}.

[481] See Vol. I, page 106, note 4 {188}.

[482] Sir Edward Sherburne (1618-1702), a scholar of considerable reputation. The reference by De Morgan is to _The Sphere of Marcus Manilius_, in the appendix to which is a _Catalogue of Astronomers, ancient and modern_.

[483] George Parker, second Earl of Macclesfield (1697-1764). He erected an observatory at Shirburn Castle, Oxfordshire, in 1739, and fitted it out with the best equipment then available. He was President of the Royal Society in 1752.

[484] See Vol. II, page 148, note 263.

[485] See Vol. I, page 140, note 7 {296}.

[486] See Vol. I, page 106, note 4 {188}.

[487] Edward Bernard (1638-1696), although Savilian professor of astronomy at Oxford, was chiefly interested in archeology.

[488] See Vol. I, page 107, note 1 {190}.

[489] See Vol. I, page 107, note 1 {190}.

[490] See Vol. I, page 135, note 3 {281}.

[491] Philip Dormer Stanhope, fourth Earl of Chesterfield (1694-1773), well known for the letters written to his son which were published posthumously (1774).

[492] Peter Daval (died in 1763), Vice-President of the Royal Society, and an astronomer of some ability.

[493] See Vol. I, page 376, note 1 {766}.

[494] William Oughtred (c. 1573-1660), a fellow of King's College, Cambridge, and afterwards vicar of Aldbury, Surrey, wrote the best-known arithmetic and trigonometry of his time. His _Arithmeticae in Numero & Speciebus Institutio ... quasi Clavis Mathematicae est_ (1631) went through many editions and appeared in English as _The Key to the Mathematicks new forged and filed_ in 1647.

[495] See Vol. I, page 140, note 5 {294}.

[496] Stephen Jordan Rigaud (1816-1859) was senior assistant master of Westminster School (1846) and head master of Queen Elizabeth's School at Ipswich (1850). He was made Bishop of Antigua in 1858 and died of yellow fever the following year.

[497] He also wrote a memoir of his father, privately printed at Oxford in 1883.

[498] See Vol. I, page 69, note 3 {96}.

[499] See Vol. I, page 106, note 4 {188}.

[500] William Gascoigne was born at Middleton before 1612 and was killed in the battle of Marston Moor in 1644. He was an astronomer and invented the micrometer with movable threads (before 1639).

[501] Seth Ward (1617-1689) was deprived of his fellowship at Cambridge for refusing to sign the covenant. He became professor of astronomy at Oxford (1649), Bishop of Exeter (1662), Bishop of Salisbury (1667), and Chancellor of the Garter (1671). He is best known for his solution of Kepler's problem to approximate a planet's orbit, which appeared in his _Astronomia geometrica_ in 1656.

[502] See Vol. I, page 110, note 2 {198}.

[503] See Vol. I, page 100, note 2 {172}.

[504] See Vol. I, page 107, note 1 {190}.

[505] See Vol. I page 114, note 6 {220}.

[506] See Vol. I, page 77, note 4 {118}.

[507] See Vol. I, page 125, note 3 {253}.

[508] See Vol. I, page 105, note 2 {186}.

[509] Heinrich Oldenburgh (1626-1678) was consul in England for the City of Bremen, his birthplace, and afterwards became a private teacher in London. He became secretary of the Royal Society and contributed on physics and astronomy to the _Philosophical Transactions_.

[510] Thomas Brancker, or Branker (1636-1676) wrote the _Doctrinae sphaericae adumbratio et usus globorum artificialium_ (1662) and translated the algebra of Rhonius with the help of Pell. The latter work appeared under the title of _An Introduction to Algebra_ (1668), and is noteworthy as having brought before English mathematicians the symbol / for division. The symbol never had any standing on the Continent for this purpose, but thereafter became so popular in England that it is still used in all the English-speaking world.

[511] See Vol. I, page 118, note 1 {230}.

[512] Pierre Bertius (1565-1629) was a native of Flanders and was educated at London and Leyden. He became a professor at Leyden, and later held the chair of mathematics at the College de France. He wrote chiefly on geography.

[513] See Vol. II, page 297, note 487.

[514] Giovanni Alphonso Borelli (1608-1679) was professor of mathematics at Messina (1646-1656) and at Pisa (1656-1657), after which he taught in Rome at the Convent of St. Panteleon. He wrote several works on geometry, astronomy, and physics.

[515] See Vol. I, page 172, note 2 {381}.

[516] Ignace Gaston Pardies (c. 1636-1673), a Jesuit, professor of ancient languages and later of mathematics and physics at the College of Pau, and afterwards professor of rhetoric at the College Louis-le-Grand at Paris. He wrote on geometry, astronomy and physics.

[517] Pierre Fermat was born in 1608 (or possibly in 1595) near Toulouse, and died in 1665. Although connected with the parliament of Toulouse, his significant work was in mathematics. He was one of the world's geniuses in the theory of numbers, and was one of the founders of the theory of probabilities and of analytic geometry. After his death his son published his edition of Diophantus (1670) and his _Varia opera mathematica_ (1679).

[518] This may be Christopher Townley (1603-1674) the antiquary, or his nephew, Richard, who improved the micrometer already invented by Gascoigne.

[519] Adrien Auzout a native of Rouen, who died at Rome in 1691. He invented a screw micrometer with movable threads (1666) and made many improvements in astronomical instruments.

[520] See Vol. I, page 66, note 9 {86}.

[521] See Vol. I, page 124, note 7 {248}.

[522] John Machin (d. 1751) was professor of astronomy at Gresham College (1713-1751) and secretary of the Royal Society. He translated Newton's _Principia_ into English. His computation of [pi] to 100 places is given in William Jones's _Synopsis palmariorum matheseos_ (1706).

[523] Pierre Remond de Montmort (1678-1719) was canon of Notre Dame until his marriage. He was a gentleman of leisure and devoted himself to the study of mathematics, especially of probabilities.

[524] Roger Cotes (1682-1716), first Plumian professor of astronomy and physics at Cambridge, and editor of the second edition of Newton's _Principia_. His posthumous _Harmonia Mensurarum_ (1722) contains "Cotes's Theorem" on the binomial equation. Newton said of him, "If Mr. Cotes had lived we had known something."

[525] See Vol. I, page 135, note 3 {281}.

[526] See Vol. I, page 377, note 4 {769}.

[527] Charles Rene Reyneau (1656-1728) was professor of mathematics at Angers. His _Analyse demontree, ou Maniere de resoudre les problemes de mathematiques_ (1708) was a successful attempt to popularize the theories of men like Descartes, Newton, Leibnitz, and the Bernoullis.

[528] Brook Taylor (1685-1731), secretary of the Royal Society, and student of mathematics and physics. His _Methodus incrementorum directa et inversa_ (1715) was the first treatise on the calculus of finite differences. It contained the well-known theorem that bears his name.

[529] Pierre Louis Moreau de Maupertuis (1698-1759) was sent with Clairaut (1735) to measure an arc of a meridian in Lapland. He was head of the physics department in the Berlin Academy from 1745 until 1753. He wrote _Sur la figure de la terre_ (1738) and on geography and astronomy.

[530] Pierre Bouguer (1698-1758) was professor of hydrography at Paris, and was one of those sent by the Academy of Sciences to measure an arc of a meridian in Peru (1735). The object of this and the work of Maupertuis was to determine the shape of the earth and see if Newton's theory was supported.

[531] Charles Marie de la Condamine (1701-1774) was a member of the Paris Academy of Sciences and was sent with Bouguer to Peru, for the purpose mentioned in the preceding note. He wrote on the figure of the earth, but was not a scientist of high rank.

[532] See Vol. I, page 136, note 5 {283}.

[533] See Vol. II, page 296, note 483.

[534] Thomas Baker (c. 1625-1689) gave a geometric solution of the biquadratic in his _Geometrical Key, or Gate of Equations unlocked_ (1684).

[535] See Vol. I, page 160, note 5 {350}.

[536] See Vol. I, page 87, note 4 {133}.

[537] See Vol. I, page 132, note 2 {272}.

[538] See Vol. I, page 118, second note 1 {231}.

[539] The name of Newton is so well known that no note seems necessary. He was born at Woolsthorpe, Lincolnshire, in 1642, and died at Kensington in 1727.

[540] John Keill (1671-1721), professor of astronomy at Oxford from 1710, is said to have been the first to teach the Newtonian physics by direct experiment, the apparatus being invented by him for the purpose. He wrote on astronomy and physics. His _Epistola de legibus virium centripetarum_, in the Philosophical Transactions for 1708, accused Leibnitz of having obtained his ideas of the calculus from Newton, thus starting the priority controversy.

[541] Thomas Digges (d. in 1595) wrote _An Arithmeticall Militare Treatise, named Stratioticos_ (1579), and completed _A geometrical practise, named Pantometria_ (1571) that had been begun by his father, Leonard Digges.

[542] John Dee (1527-1608), the most famous astrologer of his day, and something of a mathematician, wrote a preface to Billingsley's translation of Euclid into English (1570).

[543] See Vol. I, page 76, note 3 {112}.

[544] Thomas Harriot (1560-1621) was tutor in mathematics to Sir Walter Raleigh, who sent him to survey Virginia (1585). He was one of the best English algebraists of his time, but his _Artis Analyticae Praxis ad Aequationes Algebraicas resolvendas_ (1631) did not appear until ten years after his death.

[545] Thomas Lydiat (1572-1626), rector of Alkerton, devoted his life chiefly to the study of chronology, writing upon the subject and taking issue with Scaliger (1601).

[546] See Vol. I, page 69, note 3 {96}.

[547] Walter Warner edited Harriot's _Artis Analyticae Praxis_ (1631). Tarporley is not known in mathematics.

[548] See Vol. I, page 105, note 2 {186}.

[549] See Vol. I, page 115, note 3 {224}.

[550] See Vol. II, page 300, note 509.

[551] See Vol. I, page 107, note 1 {190}.

[552] Sir Samuel Morland (1625-1695) was a diplomat and inventor. For some years he was assistant to John Pell, then ambassador to Switzerland. He wrote on arithmetical instruments invented by him (1673), on hydrostatics (1697) and on church history (1658).

[553] See Vol. I, page 153, note 4 {337}.

[554] See Vol. I, page 85, note 2 {129}.

[555] See Vol. I, page 43, note 8 {33}.

[556] See Vol. I, page 43, note 7 {32}.

[557] See Vol. I, page 382, note 13 {786}. The history of the subject may be followed in Braunmuehl's _Geschichte der Trigonometrie_.

[558] See Vol. I, page 377, note 3 {768}.

[559] See Vol. I, page 108, note 2 {192}.

[560] Michael Dary wrote _Dary's Miscellanies_ (1669), _Gauging epitomised_ (1669), and _The general Doctrine of Equation_ (1664).

[561] John Newton (1622-1678), canon of Hereford (1673), educational reformer, and writer on elementary mathematics and astronomy.

[562] See Vol. I, page 87, note 4 {133}.

[563] "The average of the two equal altitudes of the sun before and after dinner."

[564] See Vol. I, page 42, note 4 {24}.

[565] London, 1678. It went though many editions.

[566] "This I who once ..."

[567] Arthur Murphy (1727-1805) worked in a banking house until 1754. He then went on the stage and met with some success at Covent Garden. His first comedy, _The Apprentice_ (1756) was so successful that he left the stage and took to play writing. His translation of Tacitus appeared in 1793, in four volumes.

[568] Edmund Wingate (1596-1656) went to Paris in 1624 as tutor to Princess Henrietta Maria and remained there several years. He wrote _L'usage de la regle de proportion_ (Paris, 1624, with an English translation in 1626), _Arithmetique Logarithmetique_ (Paris, 1626, with an English translation in 1635), and _Of Natural and Artificial Arithmetick_ (London, 1630, revised in 1650-1652), part I of which was one of the most popular textbooks ever produced in England.

[569] John Lambert (1619-1694) was Major-General during the Revolution and helped to draw up the request for Cromwell to assume the protectorate. He was imprisoned in the Tower by the Rump Parliament. He was confined in Guernsey until the clandestine marriage of his daughter Mary to Charles Hatton, son of the governor, after which he was removed (1667) to St. Nicholas in Plymouth Sound.

[570] Samuel Foster (d. in 1652) was made professor of astronomy at Gresham College in March, 1636, but resigned in November of that year, being succeeded by Mungo Murray. Murray vacated his chair by marriage in 1641 and Foster succeeded him. He wrote on dialling and made a number of improvements in geometric instruments.

[571] "Twice of the word a minister," that is, twice a minister of the Gospel.

[572] This is _The Lives of the Professors of Gresham College to which is prefixed the Life of the Founder, Sir Thomas Gresham_, London, 1740. It was written by John Ward (c. 1679-1758), professor of rhetoric (1720) at Gresham College and vice-president (1752) of the Royal Society.

[573] Charles Montagu (1661-1715), first Earl of Halifax, was Lord of the Treasury in 1692, and was created Baron Halifax in 1700 and Viscount Sunbury and Earl of Halifax in 1714. He introduced the bill establishing the Bank of England, the bill becoming a law in 1694. He had troubles of his own, without considering Newton, for he was impeached in 1701, and was the subject of a damaging resolution of censure as auditor of the exchequer in 1703. Although nothing came of either of these attacks, he was out of office during much of Queen Anne's reign.

[574] See Vol. II, page 302, note 547.

[575] See Vol. I, page 105, note 2 {186}.

[576] James Dodson (d. 1757) was master of the Royal Mathematical School, Christ's Hospital. He was De Morgan's great-grandfather. The _Anti-Logarithmic Canon_ was published in 1742.

[577] See Vol. I, page 106, note 4 {188}.

[578] See Vol. I, page 110, note 2 {198}.

[579] Richard Busby, (1606-1695), master of Westminster School (1640) had among his pupils Dryden and Locke.

[580] See Vol. I, page 107, note 1 {190}.

[581] Herbert Thorndike (1598-1672), fellow of Trinity College, Cambridge (1620-1646), and Prebend of Westminster (1661), was a well-known theological writer of the time.

[582] See Vol. I, page 140, note 5 {294}.

[583] See Vol. I, page 108, note 2 {192}.

[584] "Labor performed returns in a circle."

[585] See Vol. II, page 208.

[586] "Whatever objections one may make to the above arguments, one always falls into an absurdity."

[587] See Vol. II. page 11, note 29. _The Circle Squared; and the solution of the problem adapted to explain the difference between square and superficial measurement_ appeared at Brighton in 1865.

[588] "And beyond that nothing."

[589] Gillott (1759-1873) was the pioneer maker of steel pens by machinery, reducing the price from 1s. each to 4d. a gross. He was a great collector of paintings and old violins.

[590] William Edward Walker wrote five works on circle squaring (1853, 1854, 1857, 1862, 1864), mostly and perhaps all published at Birmingham.

[591] Solomon M. Drach wrote _An easy Rule for formulizing all Epicyclical Curves_ (London, 1849), _On the Circle area and Heptagon-chord_ (London, 1864), _An easy general Rule for filling up all Magic Squares_ (London, 1873), and _Hebrew Almanack-Signs_ (London, 1877), besides numerous articles in journals.

[592] See Vol. I, page 168, note 3 {371}.

[593] See Vol. I, page 254, note 2 {580}.

[594] See Vol. I, page 98, note 6 {163}.

[595] Robert Fludd or Flud (1574-1637) was a physician with a large London practice. He denied the diurnal rotation of the earth, and was attacked by Kepler and Mersenne, and accused of magic by Gassendi. His _Apologia Compendiania, Fraternitatem de Rosea Cruce suspicionis ... maculis aspersam, veritatis quasi Fluctibus abluens_ (Leyden, 1616) is one of a large number of works of the mystic type.

[596] Consult _To the Christianity of the Age. Notes ... comprising an elucidation of the scope and contents of the writings ... of Dionysius Andreas Freher_ (1854).

[597] Sir William Robert Grove (1811-1896), although called to the bar (1835) and to the bench (1853), is best known for his work as a physicist. He was professor of experimental philosophy (1840-1847) at the London Institution, and invented a battery (1839) known by his name. His _Correlation of Physical Forces_ (1846) went through six editions and was translated into French.

[598] Johann Tauler (c. 1300-1361), a Dominican monk of Strassburg, a mystic, closely in touch with the Gottesfreunde of Basel. His _Sermons_ first appeared in print at Leipsic in 1498.

[599] Paracelsus (c. 1490-1541). His real name was Theophrastes Bombast von Hohenheim, and he took the name by which he is generally known because he held himself superior to Celsus. He was a famous physician and pharmacist, but was also a mystic and neo-Platonist. He lectured in German on medicine at Basel, but lost his position through the opposition of the orthodox physicians and apothecaries.

[600] See Vol. I, page 256, note 2 {588}.

[601] Philip Schwarzerd (1497-1560) was professor of Greek at Wittenberg. He helped Luther with his translation of the Bible.

[602] Johann Reuchlin (1455-1522), the first great German humanist, was very influential in establishing the study of Greek and Hebrew in Germany. His lectures were mostly delivered privately in Heidelberg and Stuttgart. Unlike Melanchthon, he remained in the Catholic Church.

[603] Joseph Chitty (1776-1841) published his _Precedents of Pleading_ in 1808 and his _Reports of Cases on Practice and Pleading_ in 1820-23 (2 volumes).

[604] See Vol. I, page 44, note 1 {35}.

[605] See Vol. I, page 44, note 4 {38}.

[606] Jean Pelerin, also known as Viator, who wrote on perspective. His work appeared in 1505, with editions in 1509 and 1521.

[607] Henry Stephens. See Vol. I, page 44, note 3 {37}.

[608] The well-known grammarian (1745-1826). He was born at Swatara, in Pennsylvania, and practised law in New York until 1784, after which he resided in England. His grammar (1795) went through 50 editions, and the abridgment (1818) through 120 editions. Murray's friend Dalton, the chemist, said that "of all the contrivances invented by human ingenuity for puzzling the brains of the young, Lindley Murray's grammar was the worst."

[609] Robert Recorde (c. 1510-1558) read and probably taught mathematics and medicine at Cambridge up to 1545. After that he taught mathematics at Oxford and practised medicine in London. His _Grounde of Artes_, published about 1540, was the first arithmetic published in English that had any influence. It went through many editions. The _Castle of Knowledge_ appeared in 1551. It was a textbook on astronomy and the first to set forth the Copernican theory in England. Like Recorde's other works it was written on the catechism plan. His _Whetstone of Witte ... containying thextraction of Rootes: The Cosike practise, with the rule of Equation: and the woorkes of Surde Nombres_ appeared in 1557, and it is in this work that the modern sign of equality first appears in print. The word "Cosike" is an adjective that was used for a long time in Germany as equivalent to algebraic, being derived from the Italian _cosa_, which stood for the unknown quantity.

[610] Robert Cecil (c. 1563-1612), first Earl of Salisbury, Secretary of State under Elizabeth (1596-1603) and under James I (1603-1612).

[611] In America the German pronunciation is at present universal among mathematicians, as in the case of most other German names. This is due, no doubt, to the great influence that Germany has had on American education in the last fifty years.

[612] The latest transliteration is substantially K'ung-fu-tz[vu].

[613] The tendency seems to be, however, to adopt the forms used of individuals or places as rapidly as the mass of people comes to be prepared for it. Thus the spelling Leipzig, instead of Leipsic, is coming to be very common in America.

[614] Sir Edward Coke (1552-1634), the celebrated jurist.

[615] Dethlef Cluvier or Cluever (d. 1708 at Hamburg) was a nephew, not a grandson, of Philippe Cluvier, or Philipp Cluever (1580-c. 1623). Dethlef traveled in France and Italy and then taught mathematics in London. He wrote on astronomy and philosophy and also published in the _Acta Eruditorum_ (1686) his _Schediasma geometricum de nova infinitorum scientia_. _Quadratura circuli infinitis modis demonstrata_, and his _Monitum ad geometras_ (1687). Philippe was geographer of the Academy of Leyden. His _Introductionis in universam geographiam tam veterem quam novam libri sex_ appeared at Leyden in 1624, about the time of his death.

[616] See Vol. I, page 124, note 7 {248}.

[617] Bernard Nieuwentijt (1654-1718), a physician and burgomaster at Purmerend. His _Considerationes circa Analyseos ad quantitates infinite parvas applicatae Principia et Calculi Differentialis usum_ (Amsterdam, 1694) was attacked by Leibnitz. He replied in his _Considerationes secundae_ (1694), and also wrote the _Analysis Infinitorum, seu Curvilineorum Proprietates ex Polygonorum Natura deductae_ (1695). His most famous work was on the existence of God, _Het Regt Gebruik der Werelt Beschouwingen_ (1718).

[618] "From a given line to construct" etc.

[619] "Pirates do not fight one another."

[620] Claude Mallemens (Mallement) de Messanges (1653-1723) was professor of philosophy at the College du Plessis, in Paris, for 34 years. The work to which De Morgan refers is probably the _Fameux Probleme de la quadrature du cercle, resolu geometriquement par le cercle et a ligne droite_ that appeared in 1683.

[621] On Tycho Brahe see Vol. I, page 76, note 3 {112}.

[622] Wilhelm Frederik von Zytphen also published the _Tidens Stroem_, a chronological table, in 1840. The work to which De Morgan refers, the _Solens Bevaegelse i Verdensrummet_, appeared first in 1861. De Morgan seems to have missed his _Nogl Ord om Cirkelens Quadratur_ which appeared in 1865, at Copenhagen.

[623] James Joseph Sylvester (1814-1897), professor of natural philosophy at University College, London (1837-1841), professor of mathematics at the University of Virginia (1841-1845), actuary in London (1845-1855), professor of mathematics at Woolwich (1877-1884) and at Johns Hopkins University, Baltimore (1877-1884), and Savilian professor of geometry at Oxford (1884-1894).

[624] See Vol. I, page 76, note 3 {112}.

[625] See Vol. II, page 205, note 349.

[626] See Vol. I, page 76, note 3 {112}.

[627] See Vol. I, page 46, note 1 {42}.

[628] See Vol. II, page 183, note 318.

[629] See Vol. I, page 321, note 2 {691}.

[630] James Mill, born 1773, died 1836.

[631] See Vol. II, page 3, note 11.

[632] See Vol. II, page 3, note 13.

[633] See Vol. II, page 3, note 14.

[634] This anecdote is printed at page 4 (Vol. II); but as it is used in illustration here, and is given more in detail, I have not omitted it.--S.E. De M.

[635] See Vol. II, page 4, note 15.

[636] See Vol. I, page 382, note 13 {786}.

[637] "Monsieur, (a + b^{n})/n = x, whence God exists; answer that!"

[638] "Monsieur, you know very well that your argument requires the development of x according to integral powers of n."

[639] See Vol. I, page 153, note 4 {337}.

[640] Thomas Love Peacock (1785-1866) an English novelist and poet.

[641] Perhaps Dr. Samuel Warren (1807-1877), the author of _Ten Thousand a Year_ (serially in Blackwood's in 1839; London, 1841).

[642] See Vol. I, page 255, note 6 {584}.

[643] "From many, one; much in little; Ultima Thule (the most remote region); without which not."

[644] Spurius Maelius (fl. 440 B. C.), who distributed corn freely among the poor in the famine of 440 B. C. and was assassinated by the patricians.

[645] Spurius Cassius Viscellinus, Roman consul in 502, 493, and 486 B. C. Put to death in 485.

[646] "O what a fine bearing, he said, that has no brain."

[647] Sir William Rowan Hamilton. See Vol. I, page 332, note 4 {709}.

[648] William Allen Whitworth, the author of the well-known _Choice and Chance_ (Cambridge, 1867), and other works.

[649] James Maurice Wilson, whose _Elementary Geometry_ appeared in 1868 and went through several editions.

[650] See Vol. II, page 183, note 315.

[651] "Force of inertia conquered," and "Victory in the whole heavens."

[652] "With all his might."

[653] George Berkeley (1685-1753), Bishop of Cloyne, the idealistic philosopher and author of the _Principles of Human Knowledge_ (1710), _The Analyst, or a Discourse addressed to an Infidel Mathematician_ (1734), and _A Defense of Freethinking in Mathematics_ (1735). He asserted that space involves the idea of movement without the sensation of resistance. Space sensation less than the "minima sensibilia" is, therefore, impossible. From this he argues that infinitesimals are impossible concepts.

[654] See Vol. I, page 85, note 2 {129}.

[655] See Vol. I, page 81, note 6 {120}.

[656] Edwin Dunkin revised Lardner's _Handbook of Astronomy_ (1869) and Milner's _The Heavens and the Earth_ (1873) and wrote _The Midnight Sky_ (1869).

[657] Michael Faraday (1791-1867) the celebrated physicist and chemist. He was an assistant to Sir Humphrey Davy (1813) and became professor of chemistry at the Royal Institution, London, in 1827.

[658] "If you teach a fool he shows no joyous countenance; he cordially hates you; he wishes you buried."

[659] "Every man is an animal, Sortes is a man, therefore Sortes is an animal."

[660]

"May some choice patron bless each grey goose quill; May every Bavius have his Bufo still."--POPE, _Prologue to the Satires._

Bavius has become proverbial as a bad poet from the lines in Vergil's _Eclogues_ (III, 90):

"Qui Bavium non odit, amet tua carmina, Maevi, Atque idem jungat vulpes, et mulgeat hircos."

"He who does not hate Bavius shall love thy verses, O Maevius; and the same shall yoke foxes and shall milk he-goats."

Bavius and Maevius were the worst of Latin poets, condemned by Horace as well as Vergil.

[661] See Vol. II, page 158, note 279.

[662] "Honest," "useful," "handsome," "sweet."

[663] "Let not the fourth man attempt to speak."

[664]

"In those old times,--ah 'Twas just like this, ah!"

[665] See Vol. I, page 382, note 12 {785}.

[666] These remarks were never written.--S. E. De M.

[667]

"Fleas, flies, and friars, are masters who sadly the people abuse, And thistles and briars are sure growing grains to abuse. O Christ, who hatest strife and slayst all things in peace, Destroy where'er are rife, briars, friars, flies and fleas. Fleas, flies, and friars foul fall them these fifteen years For none that there is loveth fleas, flies, nor freres."

[668] "It is my plan to restore to an unskilled race the worthy arts of a better life."

[669] The first sentences of the first oration of Cicero against Catiline: "Quo usque tandem abutere, Catilina, patientia nostra?" (How long, O Catiline, will you abuse our patience?) "Quamdiu etiam furor iste tuus nos eludet?" (How long will this your madness baffle us?) "Nihilne te nocturnum praesidium Palati, ... nihil horum ora voltusque moverunt?" (Does the night watch of the Palatium, ... do the faces and expressions of all these men fail to move you?) "In te conferri ..." (This plague should have been inflicted upon you long ago, which you have plotted against us so long.)

[670] "Beware of the things that are marked."

[671] "Farewell, ye teachers without learning! See to it that at our next meeting we may find you strong in body and sound in mind."

[672] See Vol. I, page 336, note 8 {713}.

[673] See Vol. I, page 229, note 2 {515}.

[674] This proof, although capable of improvement, is left as in the original. Those who may be interested in the mathematics of the question, may consult F. Enriques, _Fragen der Elementargeometrie_ (German by Fleischer), Leipsic, 1907, Part II, p. 267; F. Rudio, _Archimedes_, _Huygens_, _Lambert_, _Legendre_. _Vier Abhandlungen ueber die Kreismessung_, Leipsic, 1892; F. Klein, _Famous Problems of Elementary Geometry_ (English by Beman and Smith), Boston, 1895; J. W. A. Young, _Monographs on Modern Mathematics_, New York, 1911, Chap. IX (by the editor of the present edition of De Morgan.)

[675] See Vol. I, page 69, note 2 {95}.

[676] See Vol. I, page 137, note 8 {286}.

[677] Joseph Allen Galbraith who, with Samuel Haughton, wrote the Galbraith and Haughton's _Scientific Manuals_. (Euclid, 1856; Algebra, 1860; Trigonometry, 1854; Optics, 1854, and others.)

[678] This note on Carlyle (1795-1881) is interesting. The translation of Legendre appeared in the same year (1824) as his translation of Goethe's _Wilhelm Meister_.

[679] Michael Stifel (1487-1567), also known as Stiefel, Styfel, and Stifelius, was an Augustine monk but became a convert to Lutheranism. He was professor of mathematics at Jena (1559-1567). His edition of the _Coss_ appeared at Koenigsberg in 1553, the first edition having been published in 1525. The + and - signs first appeared in print in Widman's arithmetic of 1489, but for purposes of algebra this book was one of the first to make them known.

[680] Christoff Rudolff was born about 1500 and died between 1540 and 1552. _Die Coss_ appeared in 1525 and his arithmetic in 1526.

* * * * *

Corrections made to printed original.

Page 9, "long-fostered prejudice": 'perjudice' in original.

Page 73, "Pensees, ch. 7": 'Pansees' in original.

Page 127, "and pulled out a plum": 'und' in original.

Page 147, "did not come forward": 'forword' in original.

Page 172, "come into general circulation": 'circulalation' in original.

Ibid., "the more difficult fractions which we have got": 'he have got' in original.

Page 192, "it has been stated": 'started' in original.

Page 216, "the obsolete word tetch of the same meaning": 'meaing' in original.

Page 228, "[Greek: dioklasianos]": 'dioklalasianos' in original.

Page 233. After `Henry E. Manning' were printed two paragraphs `Shilling versus Franc.' and `Teutonic Long Hundred 120 versus 100 or the Decimal question.' These appear to have been set in error, there is no applicable context.

Page 316, "in a manner depending upon the difference": 'maner' in original.

Page 322, "neither what Newton did, nor what was done before him.": 'not' (for 'nor') in original.

Page 344, "Victoria toto coelo": 'tolo' in original.

Page 368, "cannot be brought up to 1": 'up to +-' in original.

Page 371, "Q_2=b_2 Q_1+a_2": 'Q_2=b_2 Q_1-a_2' in original.

Note 50, "all who were not in the road to Heaven were excommunicated": 'excomunicated' in original.

Note 372, "[Greek: he alazoneia biou]": 'alaxoneia' in original.

Ibid., "Iapetos": 'Ispetos' in original.

Ibid., "Papeiskos": 'Paspeisoks' in original.

Ibid., "[Greek: dioklasianos]": 'dioklalasianos' in original.