Chapter XI: Part 11

[Since the publication of the above, it has been stated that the author is Mr. Oliver Byrne, the author of the _Dual Arithmetic_ mentioned further on: E. B. Revilo seems to be obviously a reversal.]

LOGIC HAS NO PARADOXERS.

Old and new logic contrasted: being an attempt to elucidate, for
ordinary comprehension, how Lord Bacon delivered the human mind from
its 2,000 years' enslavement under Aristotle. By Justin Brenan.[706]
London, 1839, 12mo.

Logic, though the other exact science, has not had the sort of assailants who have clustered about mathematics. There is a sect which disputes the utility of logic, but there are no special points, like the quadrature of the circle, which {331} excite dispute among those who admit other things. The old story about Aristotle having one logic to trammel us, and Bacon another to set us free,--always laughed at by those who really knew either Aristotle or Bacon,--now begins to be understood by a large section of the educated world. The author of this tract connects the old logic with the indecencies of the classical writers, and the new with moral purity: he appeals to women, who, "when they see plainly the demoralizing tendency of syllogistic logic, they will no doubt exert their powerful influence against it, and support the Baconian method." This is the only work against logic which I can introduce, but it is a rare one, I mean in contents. I quote the author's idea of a syllogism:

"The basis of this system is the syllogism. This is a form of couching the substance of your argument or investigation into one short line or sentence--then corroborating or supporting it in another, and drawing your conclusion or proof in a third."

On this definition he gives an example, as follows: "Every sin deserves death," the substance of the "argument or investigation." Then comes, "Every unlawful wish is a sin," which "corroborates or supports" the preceding: and, lastly, "therefore every unlawful wish deserves death," which is the "conclusion or proof." We learn, also, that "sometimes the first is called the premises (_sic_), and sometimes the first premiss"; as also that "the first is sometimes called the proposition, or subject, or affirmative, and the next the predicate, and sometimes the middle term." To which is added, with a mark of exclamation at the end, "but in analyzing the syllogism, there is a middle term, and a predicate too, in each of the lines!" It is clear that Aristotle never enslaved this mind.

I have said that logic has no paradoxers, but I was speaking of old time. This science has slept until our own day: Hamilton[707] says there has been "no progress made in {332} the _general_ development of the syllogism since the time of Aristotle; and in regard to the few _partial_ improvements, the professed historians seem altogether ignorant." But in our time, the paradoxer, the opponent of common opinion, has appeared in this field. I do not refer to Prof. Boole,[708] who is not a _paradoxer_, but a _discoverer_: his system could neither oppose nor support common opinion, for its grounds were not in the conception of any one. I speak especially of two others, who fought like cat and dog: one was dogmatical, the other categorical. The first was Hamilton himself--Sir William Hamilton of Edinburgh, the metaphysician, not Sir William _Rowan_ Hamilton[709] of Dublin, the mathematician, a combination of peculiar genius with unprecedented learning, erudite in all he could want except mathematics, for which he had no turn, and in which he had not even a schoolboy's knowledge, thanks to the Oxford of his younger day. The other was the author of this work, so fully described in Hamilton's writings that there is no occasion to describe him here. I shall try to say a few words in common language about the paradoxers.

Hamilton's great paradox was the _quantification of the predicate_; a fearful phrase, easily explained. We all know that when we say "Men are animals," a form wholly unquantified in phrase, we speak of _all_ men, but not of all animals: it is _some or all_, some may be all for aught the proposition says. This some-may-be-all-for-aught-we-say, or _not-none,_ is the logician's _some_. One would suppose {333} that "all men are some animals," would have been the logical phrase in all time: but the predicate never was quantified. The few who alluded to the possibility of such a thing found reasons for not adopting it over and above the great reason, that Aristotle did not adopt it. For Aristotle never ruled in physics or metaphysics _in the old time_ with near so much of absolute sway as he has ruled in logic _down to our own time_. The logicians knew that in the proposition "all men are animals" the "animal" is not _universal_, but _particular_ yet no one dared to say that _all_ men are _some_ animals, and to invent the phrase, "_some_ animals are _all_ men" until Hamilton leaped the ditch, and not only completed a system of enunciation, but applied it to syllogism.

My own case is as peculiar as his: I have proposed to introduce mathematical _thought_ into logic to an extent which makes the old stagers cry:

"St. Aristotle! what wild notions!
Serve a _ne exeat regno_[710] on him!"

Hard upon twenty years ago, a friend and opponent who stands high in these matters, and who is not nearly such a sectary of Aristotle and establishment as most, wrote to me as follows: "It is said that next to the man who forms the taste of the nation, the greatest genius is the man who corrupts it. I mean therefore no disrespect, but very much the reverse, when I say that I have hitherto always considered you as a great logical heresiarch." Coleridge says he thinks that it was Sir Joshua Reynolds who made the remark: which, to copy a bull I once heard, I cannot deny, because I was not there when he said it. My friend did not call me to repentance and reconciliation with the church: I think he had a guess that I was a reprobate sinner. My offences at that time were but small: I went on spinning syllogism systems, all alien from the common logic, until I had six, the initial letters of which, put together, from the {334} names I gave before I saw what they would make, bar all repentance by the words

RUE NOT!

leaving to the followers of the old school the comfortable option of placing the letters thus:

TRUE? NO!

It should however be stated that the question is not about absolute truth or falsehood. No one denies that anything I call an inference is an inference: they say that my alterations are _extra-logical_; that they are _material_, not _formal_; and that logic is a _formal_ science.

The distinction between material and formal is easily made, where the usual perversions are not required. A _form_ is an empty machine, such as "Every X is Y"; it may be supplied with _matter_, as in "Every _man_ is _animal_." The logicians will not see that their _formal_ proposition, "Every X is Y," is material in three points, the degree of assertion, the quantity of the proposition, and the copula. The purely formal proposition is "There is the probability [alpha] that X stands in the relation L to Y." The time will come when it will be regretted that logic went without paradoxers for two thousand years: and when much that has been said on the distinction of form and matter will breed jokes.

I give one instance of one mood of each of the systems, in the order of the letters first written above.

_Relative._--In this system the formal relation is taken, that is, the copula may be any whatever. As a material instance, in which the _relations_ are those of consanguinity (of men understood), take the following: X is the brother of Y; X is not the uncle of Z; therefore, Z is not the child of Y. The discussion of relation, and of the objections to the extension, is in the _Cambridge Transactions_, Vol. X, Part 2; a crabbed conglomerate.

_Undecided._--In this system one premise, and want of power over another, infer want of power over a conclusion. {335} As "Some men are not capable of tracing consequences; we cannot be sure that there are beings responsible for consequences who are incapable of tracing consequences; therefore, we cannot be sure that all men are responsible for the consequences of their actions."

_Exemplar._--This, long after it suggested itself to me as a means of correcting a defect in Hamilton's system, I saw to be the very system of Aristotle himself, though his followers have drifted into another. It makes its subject and predicate examples, thus: Any one man is an animal; any one animal is a mortal; therefore, any one man is a mortal.

_Numerical._--Suppose 100 Ys to exist: then if 70 Xs be Ys, and 40 Zs be Ys, it follows that 10 Xs (at least) are Zs. Hamilton, whose mind could not generalize on symbols, saw that the word _most_ would come under this system, and admitted, as valid, such a syllogism as "most Ys are Xs; most Ys are Zs; therefore, some Xs are Zs."

_Onymatic._--This is the ordinary system much enlarged in propositional forms. It is fully discussed in my _Syllabus of Logic_.

_Transposed._--In this syllogism the quantity in one premise is transposed into the other. As, some Xs are not Ys; for every X there is a Y which is Z; therefore, some Zs are not Xs.

Sir William Hamilton of Edinburgh was one of the best friends and allies I ever had. When I first began to publish speculation on this subject, he introduced me to the logical world as having plagiarized from him. This drew their attention: a mathematician might have written about logic under forms which had something of mathematical look long enough before the Aristotelians would have troubled themselves with him: as was done by John Bernoulli,[711] {336} James Bernoulli,[712] Lambert,[713] and Gergonne;[714] who, when our discussion began, were not known even to omnilegent Hamilton. He retracted his accusation of _wilful_ theft in a manly way when he found it untenable; but on this point he wavered a little, and was convinced to the last that I had taken his principle unconsciously. He thought I had done the same with Ploucquet[715] and Lambert. It was his pet notion that I did not understand the commonest principles of logic, that I did not always know the difference between the middle term of a syllogism and its conclusion. It went against his grain to imagine that a mathematician could be a logician. So long as he took me to be riding my own hobby, he laughed consumedly: but when he thought he could make out that I was mounted behind Ploucquet or Lambert, the current ran thus: "It would indeed have been little short of a miracle had he, ignorant even of the common principles of logic, been able of himself to rise to generalization so lofty and so accurate as are supposed in the peculiar doctrines of both the rival logicians, Lambert and Ploucquet--how useless soever these may in practice prove to be." All this has been sufficiently discussed elsewhere: "but, masters, remember that I am an ass."

I know that I never saw Lambert's work until after all Hamilton supposed me to have taken was written: he himself, who read almost everything, knew nothing about it until after I did. I cannot prove what I say about my knowledge of Lambert: but the means of doing it may turn up. For, by the casual turning up of an old letter, I _have_ {337} found the means of clearing myself as to Ploucquet. Hamilton assumed that (unconsciously) I took from Ploucquet the notion of a logical notation in which the symbol of the conclusion is seen in the joint symbols of the premises. For example, in my own fashion I write down ( . ) ( . ), two symbols of premises. By these symbols I see that there is a valid conclusion, and that it may be written in symbol by striking out the two middle parentheses, which gives ( . . ) and reading the two negative dots as an affirmative. And so I see in ( . ) ( . ) that ( ) is the conclusion. This, in full, is the perception that "all are either Xs or Ys" and "all are either Ys or Zs" necessitates "some Xs are Zs." Now in Ploucquet's book of 1763, is found, "Deleatur in praemissis medius; id quod restat indicat conclusionem."[716] In the paper in which I explain my symbols--which are altogether different from Ploucquet's--there is found "Erase the symbols of the middle term; the remaining symbols show the inference." There is very great likeness: and I would have excused Hamilton for his notion if he had fairly given reference to the part of the book in which his quotation was found. For I had shown in my _Formal Logic_ what part of Ploucquet's book I had used: and a fair disputant would either have strengthened his point by showing that I had been at his part of the book, or allowed me the advantage of it being apparent that I had not given evidence of having seen that part of the book. My good friend, though an honest man, was sometimes unwilling to allow due advantage to controversial opponents.

But to my point. The only work of Ploucquet I ever saw was lent me by my friend Dr. Logan,[717] with whom I have often corresponded on logic, etc. I chanced (in 1865) {338} to turn up the letter which he sent me (Sept. 12, 1847) _with the book_. Part of it runs thus: "I congratulate you on your success in your logical researches [that is, in asking for the book, I had described some results]. Since the reading of your first paper I have been satisfied as to the possibility of inventing a logical notation in which the rationale of the inference is contained in the symbol, though I never attempted to verify it [what I communicated, then, satisfied the writer that I had done and communicated what he, from my previous paper, suspected to be practicable]. I send you Ploucquet's dissertation....'

It now being manifest that I cannot be souring grapes which have been taken from me, I will say what I never said in print before. There is not the slightest merit in making the symbols of the premises yield that of the conclusion by erasure: _the thing must do itself in every system which symbolises quantities_. For in every syllogism (except the inverted _Bramantip_ of the Aristotelians) the conclusion is manifest in this way without symbols. This _Bramantip_ destroys system in the Aristotelian lot: and circumstances which I have pointed out destroy it in Hamilton's own collection. But in that enlargement of the reputed Aristotelian system which I have called _onymatic_, and in that correction of Hamilton's system which I have called _exemplar_, the rule of erasure is universal, and may be seen without symbols.

Our first controversy was in 1846. In 1847, in my _Formal Logic_, I gave him back a little satire for satire, just to show, as I stated, that I could employ ridicule if I pleased. He was so offended with the appendix in which this was contained, that he would not accept the copy of the book I sent him, but returned it. Copies of controversial works, sent from opponent to opponent, are not _presents_, in the usual sense: it was a marked success to make him angry enough to forget this. It had some effect however: during the rest of his life I wished to avoid provocation; for I {339} could not feel sure that excitement might not produce consequences. I allowed his slashing account of me in the _Discussions_ to pass unanswered: and before that, when he proposed to open a controversy in the _Athenaeum_ upon my second Cambridge paper, I merely deferred the dispute until the next edition of my _Formal Logic_. I cannot expect the account in the _Discussions_ to amuse an unconcerned reader as much as it amused myself: but for a cut-and-thrust, might-and-main, tooth-and-nail, hammer-and-tongs assault, I can particularly recommend it. I never knew, until I read it, how much I should enjoy a thundering onslought on myself, done with racy insolence by a master hand, to whom my good genius had whispered _Ita feri ut se sentiat emori_.[718] Since that time I have, as the Irishman said, become "dry moulded for want of a bating." Some of my paradoxers have done their best: but theirs is mere twopenny--"small swipes," as Peter Peebles said. Brandy for heroes! I hope a reviewer or two will have mercy on me, and will give me as good discipline as Strafford would have given Hampden and his set: "much beholden," said he, "should they be to any one that should thoroughly take pains with them in that kind"--meaning _objective_ flagellation. And I shall be the same to any one who will serve me so--but in a literary and periodical sense: my corporeal cuticle is as thin as my neighbors'.

Sir W. H. was suffering under local paralysis before our controversy commenced: and though his mind was quite unaffected, a retort of as downright a character as the attack might have produced serious effect upon a person who had shown himself sensible of ridicule. Had a second attack of his disorder followed an answer from me, I should have been held to have caused it: though, looking at Hamilton's genial love of combat, I strongly suspected that a retort in kind

{340}

"Would cheer his heart, and warm his blood,
And make him fight, and do him good."

But I could not venture to risk it. So all I did, in reply to the article in the _Discussions_, was to write to him the following note: which, as illustrating an etiquette of controversy, I insert.

"I beg to acknowledge and thank you for.... It is necessary that I should say a word on my retention of this work, with reference to your return of the copy of my _Formal Logic_, which I presented to you on its publication: a return made on the ground of your disapproval of the account of our controversy which that work contained. According to my view of the subject, any one whose dealing with the author of a book is specially attacked in it, has a right to expect from the author that part of the book in which the attack is made, together with so much of the remaining part as is fairly context. And I hold that the acceptance by the party assailed of such work or part of a work does not imply any amount of approval of the contents, or of want of disapproval. On this principle (though I am not prepared to add the word _alone_) I forwarded to you the whole of my work on _Formal Logic_ and my second Cambridge Memoir. And on this principle I should have held you wanting in due regard to my literary rights if you had not forwarded to me your asterisked pages, with all else that was necessary to a full understanding of their scope and meaning, so far as the contents of the book would furnish it. For the remaining portion, which it would be a hundred pities to separate from the pages in which I am directly concerned, I am your debtor on another principle; and shall be glad to remain so if you will allow me to make a feint of balancing the account by the offer of two small works on subjects as little connected with our discussion as the _Epistolae Obscurorum Virorum_, or the Lutheran dispute. I trust that by accepting my _Opuscula_ you will enable me to avoid the {341} use of the knife, and leave me to cut you up with the pen as occasion shall serve, I remain, etc. (April 21, 1852)."

I received polite thanks, but not a word about the body of the letter: my argument, I suppose, was admitted.

SOME DOGGEREL AND COUNTER DOGGEREL.

I find among my miscellaneous papers the following _jeu d'esprit_, or _jeu de betise_,[719] whichever the reader pleases--I care not--intended, before I saw ground for abstaining, to have, as the phrase is, come in somehow. I think I could manage to bring anything into anything: certainly into a Budget of Paradoxes. Sir W. H. rather piqued himself upon some caniculars, or doggerel verses, which he had put together _in memoriam_ [_technicam_] of the way in which A E I O are used in logic: he added U, Y, for the addition of _meet_, etc., to the system. I took the liberty of concocting some counter-doggerel, just to show that a mathematician may have architectonic power as well as a metaphysician.

DOGGEREL.
BY SIR W. HAMILTON.
A it affirms of _this_, _these_, _all_,
Whilst E denies of _any_;
I it affirms (whilst O denies)
Of some (or few, or many).

Thus A affirms, as E denies,
And definitely either;
Thus I affirms, as O denies,
And definitely neither.

A half, left semidefinite,
Is worthy of its score;
U, then, affirms, as Y denies,
This, neither less nor more.

Indefinito-definites,
I, UI, YO, last we come;
{342}
And this affirms, as that denies
Of _more_, _most_ (_half_, _plus_, _some_).

COUNTER DOGGEREL.
BY PROF. DE MORGAN.
(1847.)
Great A affirms of all;
Sir William does so too:
When the subject is "my suspicion,"
And the predicate "must be true."

Great E denies of all;
Sir William of all but one:
When he speaks about this present time,
And of those who in logic have done.

Great I takes up but _some_;
Sir William! my dear soul!
Why then in all your writings,
Does "Great I" fill[720] the whole!

Great O says some are not;
Sir William's readers catch,
That some (modern) Athens is not without
An Aristotle to match.

"A half, left semi-definite,
Is worthy of its score:"
This looked very much like balderdash,
And neither less nor more.

It puzzled me like anything;
In fact, it puzzled me worse:
Isn't schoolman's logic hard enough,
Without being in Sibyl's verse?

{343}
At last, thinks I, 'tis German;
And I'll try it with some beer!
The landlord asked what bothered me so,
And at once he made it clear.

It's _half-and-half_, the gentleman means;
Don't you see he talks of _score_?
That's the bit of memorandum
That we chalk behind the door.

_Semi-definite_'s outlandish;
But I see, in half a squint,
That he speaks of the lubbers who call for a quart,
When they can't manage more than a pint.

Now I'll read it into English,
And then you'll answer me this:
If it isn't good logic all the world round,
I should like to know what is?

When you call for a pot of half-and-half,
If you're lost to sense of shame,
You may leave it _semi-definite_,
But you pay for it all just the same.
* * * * * *

I am unspeakably comforted when I look over the above in remembering that the question is not whether it be Pindaric or Horatian, but whether the copy be as good as the original. And I say it is: and will take no denial.

Long live--long will live--the glad memory of William Hamilton, Good, Learned, Acute, and Disputatious! He fought upon principle: the motto of his book is:

"Truth, like a torch, the more it's shook it shines."

There is something in this; but metaphors, like puddings, quarrels, rivers, and arguments, always have two sides to them. For instance,

"Truth, like a torch, the more it's shook it shines;
But those who want to use it, hold it steady.
They shake the flame who like a glare to gaze at,
They keep it still who want a light to see by."

{344}

ANOTHER THEORY OF PARALLELS.

Theory of Parallels. The proof of Euclid's axiom looked for in the
properties of the Equiangular Spiral. By Lieut-Col. G. Perronet
Thompson.[721] The same, second edition, revised and corrected. The
same, third edition, shortened, and freed from dependence on the theory
of limits. The same, fourth edition, ditto, ditto. All London, 1840,
8vo.

To explain these editions it should be noted that General Thompson rapidly modified his notions, and republished his tracts accordingly.

SOME PRIMITIVE DARWINISM.

Vestiges of the Natural History of Creation.[722] London, 1840, 12mo.

This is the first edition of this celebrated work. Its form is a case of the theory: the book is an undeniable duodecimo, but the size of its paper gives it the look of not the smallest of octavos. Does not this illustrate the law of development, the gradation of families, the transference of species, and so on? If so, I claim the discovery of this esoteric testimony of the book to its own contents; I defy any one to point out the reviewer who has mentioned it. The work itself is described by its author as "the first attempt to connect the natural sciences into a history of creation." The attempt was commenced, and has been carried on, both with marked talent, and will be continued. Great advantage will result: at the worst we are but in the alchemy of some new chemistry, or the astrology of some new astronomy. Perhaps it would be as well not to be too sure on the matter, until we have an antidote to possible consequences as exhibited under another theory, on which {345} it is as reasonable to speculate as on that of the _Vestiges_. I met long ago with a splendid player on the guitar, who assured me, and was confirmed by his friends, that he _never practised_, except in thought, and did not possess an instrument: he kept his fingers acting in his mind, until they got their habits; and thus he learnt the most difficult novelties of execution. Now what if this should be a minor segment of a higher law? What if, by constantly thinking of ourselves as descended from primeval monkeys, we should--if it be true--actually _get our tails again_? What if the first man who was detected with such an appendage should be obliged to confess himself the author of the _Vestiges_--a person yet unknown--who would naturally get the start of his species by having had the earliest habit of thinking on the matter? I confess I never hear a man of note talk fluently about it without a curious glance at his proportions, to see whether there may be ground to conjecture that he may have more of "mortal coil" than others, in anaxyridical concealment. I do not feel sure that even a paternal love for his theory would induce him, in the case I am supposing, to exhibit himself at the British Association,

With a hole behind which his tail peeped through.

The first sentence of this book (1840) is a cast of the log, which shows our rate of progress. "It is familiar knowledge that the earth which we inhabit is a globe of somewhat less than 8,000 miles in diameter, being one of a series of eleven which revolve at different distances around the sun." The _eleven_! Not to mention the Iscariot which Le Verrier and Adams calculated into existence, there is more than a septuagint of _new_ planetoids.

ON RELIGIOUS INSURANCE.

The Constitution and Rules of the Ancient and Universal 'Benefit
Society' established by Jesus Christ, exhibited, and its advantages and
claims maintained, against all Modern and {346} merely Human
Institutions of the kind: A Letter very respectfully addressed to the
Rev. James Everett,[723] and occasioned by certain remarks made by him,
in a speech to the Members of the 'Wesleyan Centenary Institute'
Benefit Society. Dated York, Dec. 7, 1840. By Thomas Smith.[724] 12mo,
(pp. 8.)

The Wesleyan minister addressed had advocated provision against old age, etc.: the writer declares all _private_ provision un-Christian. After decent maintenance and relief of family claims of indigence, he holds that all the rest is to go to the "Benefit Society," of which he draws up the rules, in technical form, with chapters of "Officers," "Contributors" etc., from the Acts of the Apostles, etc., and some of the early Fathers. He holds that a Christian may not "make a _private_ provision against the contingencies of the future": and that the great "Benefit Society" is the divinely-ordained recipient of all the surplus of his income; capital, beyond what is necessary for business, he is to have none. A real good speculator shuts his eyes by instinct, when opening them would not serve the purpose: he has the vizor of the Irish fairy tale, which fell of itself over the eyes of the wearer the moment he turned them upon the enchanted light which would have destroyed him if he had caught sight of it. "Whiles it remained, was it not thine own? and after it was sold, was it (the purchase-money) not in thine own power?" would have been awkward to quote, and accordingly nothing is stated except the well-known result, which is rule 3, cap. 5, "Prevention of Abuses." By putting his principles together, the author can be made, logically, to mean that the successors of the apostles should put to death all contributors who are detected in not paying their full premiums.

{347}

I have known one or two cases in which policy-holders have surrendered their policies through having arrived at a conviction that direct provision is unlawful. So far as I could make it out, these parties did not think it unlawful to lay by out of income, except when this was done in a manner which involved calculation of death-chances. It is singular they did not see that the entrance of chance of death was the entrance of the very principle of the benefit society described in the Acts of the Apostles. The family of the one who died young received more in proportion to _premiums_ paid than the family of one who died old. Every one who understands life assurance sees that--_bonus_ apart--the difference between an assurance office and a savings bank consists in the adoption, _pro tanto_, of the principle of community of goods. In the original constitution of the oldest assurance office, the _Amicable Society_, the plan with which they started was nothing but this: persons of all ages under forty-five paid one common premium, and the proceeds were divided among the representatives of those who died within the year.

THE TWO OLD PARADOXES AGAIN.

[I omitted from its proper place a manuscript quadrature (3.1416 exactly) addressed to an eminent mathematician, dated in 1842 from the debtor's ward of a country gaol. The unfortunate speculator says, "I have labored many years to find the precise ratio." I have heard of several cases in which squaring the circle has produced an inability to square accounts. I remind those who feel a kind of inspiration to employ native genius upon difficulties, without gradual progression from elements, that the call is one which becomes stronger and stronger, and may lead, as it has led, to abandonment of the duties of life, and all the consequences.] {348}

1842. Provisional Prospectus of the Double Acting Rotary Engine
Company. Also Mechanic's Magazine, March 26, 1842.

Perpetual motion by a drum with one vertical half in mercury, the other in a vacuum: the drum, I suppose, working round forever to find an easy position. Steam to be superseded: steam and electricity convulsions of nature never intended by Providence for the use of man. The price of the present engines, as old iron, will buy new engines that will work without fuel and at no expense. Guaranteed by the Count de Predaval,[725] the discoverer. I was to have been a Director, but my name got no further than ink, and not so far as official notification of the honor, partly owing to my having communicated to the _Mechanic's Magazine_ information privately given to me, which gave premature publicity, and knocked up the plan.

An Exposition of the Nature, Force, Action, and other properties of
Gravitation on the Planets. London, 1842, 12mo.

An Investigation of the principles of the Rules for determining the
Measures of the Areas and Circumferences of Circular Plane Surfaces ...
London, 1844, 8vo.

These are anonymous; but the author (whom I believe to be Mr. Denison,[726] presently noted) is described as author of a new system of mathematics, and also of mechanics. He had need have both, for he shows that the line which has a square equal to a given circle, has a cube equal to the sphere on the same diameter: that is, in old mathematics, the diameter is to the circumference as 9 to 16! Again, admitting that the velocities of planets in circular orbits are inversely as the square roots of their distances, that is, admitting Kepler's law, he manages to prove that gravitation is inversely as the square _root_ of the distance: and suspects magnetism of doing the difference between this and Newton's law. {349} Magnetism and electricity are, in physics, the member of parliament and the cabman--at every man's bidding, as Henry Warburton[727] said.

The above is an outrageous quadrature. In the preceding year, 1841, was published what I suppose at first to be a Maori quadrature, by Maccook. But I get it from a cutting out of some French periodical, and I incline to think that it must be by a Mr. M^cCook. He makes [pi] to be 2 + 2[root](8[root]2 - 11).

THE DUPLICATION PROBLEM.

Refutation of a Pamphlet written by the Rev. John Mackey, R.C.P.,[728]
entitled "A method of making a cube double of a cube, founded on the
principles of elementary geometry," wherein his principles are proved
erroneous, and the required solution not yet obtained. By Robert
Murphy.[729] Mallow, 1824, 12mo.

This refutation was the production of an Irish boy of eighteen years old, self-educated in mathematics, the son of a shoemaker at Mallow. He died in 1843, leaving a name which is well known among mathematicians. His works on the theory of equations and on electricity, and his papers in the _Cambridge Transactions_, are all of high genius. The only account of him which I know of is that which I wrote for the _Supplement_ of the _Penny Cyclopaedia_. He was thrown by his talents into a good income at Cambridge, with no social training except penury, and very little intellectual training except mathematics. He fell into dissipation, and his scientific career was almost arrested: but he had great good in him, to my knowledge. A sentence in {350} a letter from the late Dean Peacock[730] to me--giving some advice about the means of serving Murphy--sets out the old case: "Murphy is a man whose _special_ education is in advance of his _general_; and such men are almost always difficult subjects to manage." This article having been omitted in its proper place, I put it at 1843, the date of Murphy's death.

A NEW VALUE OF [pi].

The Invisible Universe disclosed; or, the real Plan and Government of
the Universe. By Henry Coleman Johnson, Esq. London, 1843, 8vo.

The book opens abruptly with:

"First demonstration. Concerning the centre: showing that, because the centre is an innermost point at an equal distance between two extreme points of a right line, and from every two relative and opposite intermediate points, it is composed of the two extreme internal points of each half of the line; each extreme internal point attracting towards itself all parts of that half to which it belongs...."

Of course the circle is squared: and the circumference is 3-1/21 diameters.

SOME MODERN ASTROLOGY.

Combination of the Zodiacal and Cometical Systems. Printed for the
London Society, Exeter Hall. Price Sixpence. (n. d. 1843.)

What this London Society was, or the "combination," did not appear. There was a remarkable comet in 1843, the tail of which was at first confounded with what is called the _zodiacal light_. This nicely-printed little tract, evidently got up with less care for expense than is usual in such works, brings together all the announcements of the astronomers, and adds a short head and tail piece, which I shall quote entire. As the announcements are very ordinary {351} astronomy, the reader will be able to detect, if detection be possible, what is the meaning and force of the "Combination of the Zodiacal and Cometical Systems":

"_Premonition._ It has pleased the AUTHOR _of_ CREATION to cause (to His _human and reasoning_ Creatures of this generation, by a '_combined_' appearance in His _Zodiacal_ and _Cometical_ system) a '_warning Crisis_' of universal concernment to this our GLOBE. It is this '_Crisis_' that has so generally 'ROUSED' at this moment the '_nations throughout the Earth_' that no equal interest has ever before been excited by MAN; unless it be in that caused by the 'PAGAN-TEMPLE IN ROME,' which is recorded by the elder Pliny, '_Nat. Hist._' i. 23. iii. 3. HARDOUIN."

After the accounts given by the unperceiving astronomers, comes what follows:

"Such has been (_hitherto_) the only object discerned by the '_Wise of this World_,' in this _twofold union_ of the '_Zodiacal_' and '_Cometical_' systems: yet it is nevertheless a most '_Thrilling Warning_,' to _all_ the inhabitants of this precarious and transitory EARTH. We have no authorized intimation or reasonable prospective contemplation, of '_current time_' beyond a year 1860, of the present century; or rather, except '_the interval which may now remain from the present year 1843, to a year 1860_' ([Greek: hemeras HEXEKONTA]--'_threescore or sixty days_'--'_I have appointed each_ "DAY" _for a_ "YEAR,"' _Ezek._ iv. 6): and we know, from our '_common experience_,' how speedily such a measure of time will pass away.

"No words can be '_more explicit_' than these of OUR BLESSED LORD: viz. 'THIS GOSPEL _of the Kingdom shall be preached in_ ALL the EARTH, _for a Witness to_ ALL NATIONS; AND THEN, _shall the_ END COME.' The '_next 18 years_' must therefore supply the interval of the '_special Episcopal forerunners_.'

(Matt. xxiv. 14.)

"See the 'JEWISH INTELLIGENCER' of the present month (_April_), p. 153, for the '_Debates in Parliament_,' respecting {352} the BISHOP OF JERUSALEM, _viz._ Dr. Bowring,[731] Mr. Hume,[732] Sir R. Inglis,[733] Sir R. Peel,[734] Viscount Palmerston.[735]"

I have quoted this at length, to show the awful threats which were published at a time of some little excitement about the phenomenon, under the name of the _London Society_. The assumption of a corporate appearance is a very unfair trick: and there are junctures at which harm might be done by it.

THE NUMBER OF THE BEAST.

_Wealth_ the name and number of the Beast, 666, in the Book of
Revelation. [by John Taylor.[736]] London, 1844, 8vo.

Whether Junius or the Beast be the more difficult to identify, must be referred to Mr. Taylor, the only person who has attempted both. His cogent argument on the political secret is not unworthily matched in his treatment of the theological riddle. He sees the solution in [Greek: euporia], which occurs in the Acts of the Apostles as the word for wealth in one of its most disgusting forms, and makes 666 in the most straightforward way. This explanation has as good a chance as any other. The work contains a general {353} attempt at explanation of the Apocalypse, and some history of opinion on the subject. It has not the prolixity which is so common a fault of apocalyptic commentators.

A practical Treatise on Eclipses ... with remarks on the anomalies of
the present Theory of the Tides. By T. Kerigan,[737] F.R.S. 1844, 8vo.

Containing also a refutation of the theory of the tides, and afterwards increased by a supplement, "Additional facts and arguments against the theory of the tides," in answer to a short notice in the _Athenaeum_ journal. Mr. Kerigan was a lieutenant in the Navy: he obtained admission to the Royal Society just before the publication of his book.

A new theory of Gravitation. By Joseph Denison,[738] Esq. London, 1844,
12mo.

Commentaries on the Principia. By the author of 'A new theory of
Gravitation.' London, 1846, 8vo.

Honor to the speculator who can be put in his proper place by one sentence, be that place where it may.

"But we have shown that the velocities are inversely as the square roots of the mean distances from the sun; wherefore, by equality of ratios, the forces of the sun's gravitation upon them are also inversely as the square roots of their distances from the sun."

EASTER DAY PARADOXERS.

In the years 1818 and 1845 the full moon fell on Easter Day, having been particularly directed to fall before it in the act for the change of style and in the English missals and prayer-books of all time: perhaps it would be more correct to say that Easter Day was directed to fall after the full moon; "but the principle is the same." No explanation was given in 1818, but Easter was kept by the tables, {354} in defiance of the rule, and of several protests. A chronological panic was beginning in December 1844, which was stopped by the _Times_ newspaper printing extracts from an article of mine in the _Companion to the Almanac_ for 1845, which had then just appeared. No one had guessed the true reason, which is that the thing called the moon in the Gregorian Calendar is not the moon of the heavens, but a fictitious imitation put wrong on purpose, as will presently appear, partly to keep Easter out of the way of the Jews' Passover, partly for convenience of calculation. The apparent error happens but rarely; and all the work will perhaps have to be gone over next time. I now give two bits of paradox.

Some theologians were angry at this explanation. A review called the _Christian Observer_ (of which Christianity I do not know) got up a crushing article against me. I did not look at it, feeling sure that an article on such a subject which appeared on January 1, 1845, against a publication made in December 1844, must be a second-hand job. But some years afterwards (Sept. 10, 1850), the reviews, etc. having been just placed at the disposal of readers in the _old_ reading-room of the Museum, I made a tour of inspection, came upon my critic on his perch, and took a look at him. I was very glad to remember this, for, though expecting only second-hand, yet even of this there is good and bad; and I expected to find some hints in the good second-hand of a respectable clerical publication. I read on, therefore, attentively, but not long: I soon came to the information that some additions to Delambre's[739] statement of the rule for finding Easter, belonging to distant years, had been made by Sir Harris Nicolas![740] Now as I myself furnished my friend Sir H. N. with Delambre's digest of {355} Clavius's[741] rule, which I translated out of algebra into common language for the purpose, I was pretty sure this was the ignorant reading of a person to whom Sir H. N. was the highest _arithmetical_ authority on the subject. A person pretending to chronology, without being able to distinguish the historical points--so clearly as they stand out--in which Sir H. N. speaks with authority, from the arithmetical points of pure reckoning on which he does not pretend to do more than directly repeat others, must be as fit to talk about the construction of Easter Tables as the Spanish are to talk French. I need hardly say that the additions for distant years are as much from Clavius as the rest: my reviewer was not deep enough in his subject to know that Clavius made and published, from his rules, the full table up to A.D. 5000, for all the movable feasts of every year! I gave only a glance at the rest: I found I was either knave or fool, with a leaning to the second opinion; and I came away satisfied that my critic was either ignoramus or novice, with a leaning to the first. I afterwards found an ambiguity of expression in Sir H. N.'s account--whether his or mine I could not tell--which might mislead a novice or content an ignoramus, but would have been properly read or further inquired into by a competent person.

The second case is this. Shortly after the publication of my article, a gentleman called at my house, and, finding I was not at home, sent up his card--with a stylish west-end club on it--to my wife, begging for a few words on pressing business. With many well-expressed apologies, he stated that he had been alarmed by hearing that Prof. De M. had an intention of altering Easter next year. Mrs. De M. kept her countenance, and assured him that I had no such intention, and further, that she greatly doubted my having the power to do it. Was she quite sure? his authority was very good: fresh assurances given. He was greatly relieved, for he had some horses training for after Easter, which {356} would not be ready to run if it were altered the wrong way. A doubt comes over him: would Mrs. De M., in the event of her being mistaken, give him the very earliest information? Promise given; profusion of thanks; more apologies; and departure.

Now, candid reader!--or uncandid either!--which most deserves to be laughed at? A public instructor, who undertakes to settle for the world whether a reader of Clavius, the constructor of the Gregorian Calendar, is fool or knave, upon information derived from a compiler--in this matter--of his own day; or a gentleman of horse and dog associations, who, misapprehending something which he heard about a current topic, infers that the reader of Clavius had the ear of the Government on a proposed alteration. I suppose the querist had heard some one say, perhaps, that the day ought to be set right, and some one else remark that I might be consulted, as the only person who had discussed the matter from the original source of the Calendar.

To give a better chance of the explanation being at once produced, next time the real full moon and Easter Day shall fall together, I insert here a summary which was printed in the Irish Prayer-book of the Ecclesiastical Society. If the amusement given by paradoxers should prevent a useless discussion some years hence, I and the paradoxers shall have done a little good between us--at any rate, I have done my best to keep the heavy weight afloat by tying bladders to it. I think the next occurrence will be in 1875.

EASTER DAY.

In the years 1818 and 1845, Easter Day, as given by the _rules in_ 24 Geo. II cap. 23. (known as the act for the _change of style_) contradicted the _precept_ given in the preliminary explanations. The precept is as follows:

"_Easter Day_, on which the rest" of the moveable feasts "depend, is always the First Sunday after the Full Moon, which happens upon or next after the Twenty-first Day of {357} _March_; and if the Full Moon happens upon a Sunday, _Easter Day_ is the Sunday after."

But in 1818 and 1845, the full moon fell on a Sunday, and yet the rules gave _that same Sunday_ for Easter Day. Much discussion was produced by this circumstance in 1818: but a repetition of it in 1845 was nearly altogether prevented by a timely[742] reference to the intention of those who conducted the Gregorian reformation of the Calendar. Nevertheless, seeing that the apparent error of the Calendar is due to the precept in the Act of Parliament, which is both erroneous and insufficient, and that the difficulty will recur so often as Easter Day falls on the day of full moon, it may be advisable to select from the two articles cited in the note such of their conclusions and rules, without proof or controversy, as will enable the reader to understand the main points of the Easter question, and, should he desire it, to calculate for himself the Easter of the old or new style, for any given year.

1. In the very earliest age of Christianity, a controversy arose as to the mode of keeping Easter, some desiring to perpetuate the _Passover_, others to keep the _festival of the Resurrection_. The first afterwards obtained the name of _Quartadecimans_, from their Easter being always kept on the _fourteenth day_ of the moon (Exod. xii. 18, Levit. xxiii. 5.). But though it is unquestionable that a Judaizing party existed, it is also likely that many dissented on chronological grounds. It is clear that no _perfect_ anniversary can take place, except when the fourteenth of the moon, and with it the passover, falls on a Friday. Suppose, for instance, it falls on a Tuesday: one of three things must be {358} done. Either (which seems never to have been proposed) the crucifixion and resurrection must be celebrated on Tuesday and Sunday, with a wrong interval; or the former on Tuesday, the latter on Thursday, abandoning the first day of the week; or the former on Friday, and the latter on Sunday, abandoning the paschal commemoration of the crucifixion.

The last mode has been, as every one knows, finally adopted. The disputes of the first three centuries did not turn on any _calendar_ questions. The Easter question was merely the symbol of the struggle between what we may call the Jewish and Gentile sects of Christians: and it nearly divided the Christian world, the Easterns, for the most part, being _Quartadecimans_. It is very important to note that there is no recorded dispute about a method of predicting the new moon, that is, no general dispute leading to formation of sects: there may have been difficulties, and discussions about them. The Metonic cycle, presently mentioned, must have been used by many, perhaps most, churches.

2. The question came before the Nicene Council (A.D. 325) not as an astronomical, but as a doctrinal, question: it was, in fact, this, Shall the _passover_[743] be treated as a part of Christianity? The Council resolved this question in the negative, and the only information on its premises and conclusion, or either, which comes from itself, is contained in the following sentence of the synodical epistle, which epistle is preserved by Socrates[744] and Theodoret.[745] "We also send {359} you the good news concerning the unanimous consent of all in reference to the celebration of the most solemn feast of Easter, for this difference also has been made up by the assistance of your prayers: so that all the brethren in the East, who formerly celebrated this festival _at the same time as the Jews_, will in future conform _to the Romans and to us_, and to all who have of old observed _our manner_ of celebrating Easter." This is all that can be found on the subject: none of the stories about the Council ordaining the astronomical mode of finding Easter, and introducing the Metonic cycle into ecclesiastical reckoning, have any contemporary evidence: the canons which purport to be those of the Nicene Council do not contain a word about Easter; and this is evidence, whether we suppose those canons to be genuine or spurious.

3. The astronomical dispute about a lunar cycle for the prediction of Easter either commenced, or became prominent, by the extinction of greater ones, soon after the time of the Nicene Council. Pope Innocent I[746] met with difficulty in 414. S. Leo,[747] in 454, ordained that Easter of 455 should be April 24; which is right. It is useless to record details of these disputes in a summary: the result was, that in the year 463, Pope Hilarius[748] employed Victorinus[749] of Aquitaine to correct the Calendar, and Victorinus formed a rule which lasted until the sixteenth century. He combined the Metonic cycle and the solar cycle presently described. But {360} this cycle bears the name of Dionysius Exiguus,[750] a Scythian settled at Rome, about A.D. 530, who adapted it to his new yearly reckoning, when he abandoned the era of Diocletian as a commencement, and constructed that which is now in common use.

4. With Dionysius, if not before, terminated all difference as to the mode of keeping Easter which is of historical note: the increasing defects of the Easter Cycle produced in time the remonstrance of persons versed in astronomy, among whom may be mentioned Roger Bacon,[751] Sacrobosco,[752] Cardinal Cusa,[753] Regiomontanus,[754] etc. From the middle of the sixth to that of the sixteenth century, one rule was observed.