Chapter 12

"St. Aristotle! what wild notions!Serve ane exeat regno[710]on him!"

"St. Aristotle! what wild notions!

Serve ane 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 thenames 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 areextra-logical; that they arematerial, notformal; and that logic is aformalscience.

The distinction between material and formal is easily made, where the usual perversions are not required. Aformis an empty machine, such as "Every X is Y"; it may be supplied withmatter, as in "Everymanisanimal." The logicians will not see that theirformalproposition, "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 probabilityαthat 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 therelationsare 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 theCambridge 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.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 wordmostwould 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 mySyllabus 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]James Bernoulli,[712]Lambert,[713]and Gergonne;[714]who, when our discussion began, were not known even to omnilegent Hamilton. He retracted his accusation ofwilfultheft 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, Ihavefound 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 præmissis 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 myFormal Logicwhat 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)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 invertedBramantipof the Aristotelians) the conclusion is manifest in this way without symbols. ThisBramantipdestroys 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 calledonymatic, and in that correction of Hamilton's system which I have calledexemplar, the rule of erasure is universal, and may be seen without symbols.

Our first controversy was in 1846. In 1847, in myFormal 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 notpresents, 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 Icould not feel sure that excitement might not produce consequences. I allowed his slashing account of me in theDiscussionsto pass unanswered: and before that, when he proposed to open a controversy in theAthenæumupon my second Cambridge paper, I merely deferred the dispute until the next edition of myFormal Logic. I cannot expect the account in theDiscussionsto 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 whisperedIta 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"—meaningobjectiveflagellation. 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

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

"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 theDiscussions, 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 myFormal 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 wordalone) I forwarded to you the whole of my work onFormal Logicand 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 theEpistolæ Obscurorum Virorum, or the Lutheran dispute. I trust that by accepting myOpusculayou will enable me to avoid theuse 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 followingjeu d'esprit, orjeu de bêtise,[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 togetherin memoriam[technicam] of the way in which A E I O are used in logic: he added U, Y, for the addition ofmeet, 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 ofthis,these,all,Whilst E denies ofany;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;And this affirms, as that deniesOfmore,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 butsome;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 withoutAn 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?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'shalf-and-half, the gentleman means;Don't you see he talks ofscore?That's the bit of memorandumThat 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 itsemi-definite,But you pay for it all just the same.* * * * * *

DOGGEREL.BY SIR W. HAMILTON.A it affirms ofthis,these,all,Whilst E denies ofany;I it affirms (whilst O denies)Of some (or few, or many).

DOGGEREL.

BY SIR W. HAMILTON.

A it affirms ofthis,these,all,

Whilst E denies ofany;

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.

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.

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;And this affirms, as that deniesOfmore,most(half,plus,some).

Indefinito-definites,

I, UI, YO, last we come;

And this affirms, as that denies

Ofmore,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."

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

Great I takes up butsome;

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 withoutAn Aristotle to match.

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.

"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?

It puzzled me like anything;

In fact, it puzzled me worse:

Isn't schoolman's logic hard enough,

Without being in Sibyl's verse?

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.

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'shalf-and-half, the gentleman means;Don't you see he talks ofscore?That's the bit of memorandumThat we chalk behind the door.

It'shalf-and-half, the gentleman means;

Don't you see he talks ofscore?

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.

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?

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 itsemi-definite,But you pay for it all just the same.* * * * * *

When you call for a pot of half-and-half,

If you're lost to sense of shame,

You may leave itsemi-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."

"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."

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 isdescribedby 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 whichit is as reasonable to speculate as on that of theVestiges. I met long ago with a splendid player on the guitar, who assured me, and was confirmed by his friends, that henever 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—actuallyget our tails again? What if the first man who was detected with such an appendage should be obliged to confess himself the author of theVestiges—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." Theeleven! Not to mention the Iscariot which Le Verrier and Adams calculated into existence, there is more than a septuagint ofnewplanetoids.

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 andmerely 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 allprivateprovision 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 aprivateprovision 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.

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 topremiumspaid than the family of one who died old. Every one who understands life assurance sees that—bonusapart—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, theAmicable 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.]

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 theMechanic's Magazineinformation 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.

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 squarerootof the distance: and suspects magnetism of doing the difference between this and Newton's law.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. McCook. He makesπto be 2 + 2√(8√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 theCambridge Transactions, are all of high genius. The only account of him which I know of is that which I wrote for theSupplementof thePenny Cyclopædia. 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 ina 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 whosespecialeducation is in advance of hisgeneral; 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 OFπ.

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 thezodiacal 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 ordinaryastronomy, 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 theAuthorofCreationto cause (to Hishuman and reasoningCreatures of this generation, by a 'combined' appearance in HisZodiacalandCometicalsystem) a 'warning Crisis' of universal concernment to this ourGlobe. 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 byMan; 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 thistwofold unionof the 'Zodiacal' and 'Cometical' systems: yet it is nevertheless a most 'Thrilling Warning,' toallthe inhabitants of this precarious and transitoryEarth. 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' (ἡμέρας ἙΞΗΚΟΝΤΑ—'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 ofour blessed Lord: viz. 'This Gospelof the Kingdom shall be preached inALLtheEarth,for a Witness toall Nations; and then,shall theEnd 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,' respectingtheBishop 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 theLondon 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.

Wealththe 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 inεὐπορία, 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 generalattempt 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 theAthenæumjournal. 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.

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,in defiance of the rule, and of several protests. A chronological panic was beginning in December 1844, which was stopped by theTimesnewspaper printing extracts from an article of mine in theCompanion to the Almanacfor 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 theChristian 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 theoldreading-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 ofClavius'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 highestarithmeticalauthority 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, whichwould 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 therules in24 Geo. II cap. 23. (known as the act for thechange of style) contradicted thepreceptgiven 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 ofMarch; and if the Full Moon happens upon a Sunday,Easter Dayis the Sunday after."

But in 1818 and 1845, the full moon fell on a Sunday, and yet the rules gavethat same Sundayfor 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 thePassover, others to keep thefestival of the Resurrection. The first afterwards obtained the name ofQuartadecimans, from their Easter being always kept on thefourteenth dayof 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 noperfectanniversary 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 bedone. 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 anycalendarquestions. 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, beingQuartadecimans. 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 thepassover[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 sendyou 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 festivalat the same time as the Jews, will in future conformto the Romans and to us, and to all who have of old observedour mannerof 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. Butthis 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.

5. The mode of applying astronomy to chronology has always involved these two principles. First, the actual position of the heavenly body is not the object of consideration, but what astronomers call itsmean place, which may be described thus. Let a fictitious sun or moon move in the heavens, in such manner as to revolve among the fixed stars at an average rate, avoiding the alternate accelerations and retardations which take place in every planetary motion. Thus the fictitious (saymean) sun and moon are always very near to the real sun and moon. The ordinary clocks show time by the mean, not the real, sun: and it was always laid down that Easter depends on the opposition (or full moon) of the mean sun and moon, not of the real ones. Thus we see that, were the Calendar ever so correctas to themeanmoon, it would be occasionally false as to thetrueone: if, for instance, the opposition of the mean sun and moon took place at one second before midnight, and that of the real bodies only two seconds afterwards, the calendar day of full moon would be one day before that of the common almanacs. Here is a way in which the discussions of 1818 and 1845 might have arisen: the British legislature has definedthe moonas the regulator of the paschal calendar. But this was only a part of the mistake.

6. Secondly, in the absence of perfectly accurate knowledge of the solar and lunar motion (and for convenience, even if such knowledge existed), cycles are, and always have been taken, which serve to represent those motions nearly. The famous Metonic cycle, which is introduced into ecclesiastical chronology under the name of the cycle of the golden numbers, is a period of 19 Julian[755]years. This period, in the old Calendar, was taken to contain exactly 235lunations, or intervals between new moons, of the mean moon. Now the state of the case is:

19 average Julian years make 6939 days 18 hours.

235 average lunations make 6939 days 16 hours 31 minutes.

So that successive cycles of golden numbers, supposing the first to start right, amount to making the new moons fall too late, gradually, so that the mean moonof this cyclegains 1 hour 29 minutes in 19 years upon the mean moon of the heavens, or about a day in 300 years. When the Calendar was reformed, the calendar new moons were four days in advance of the mean moon of the heavens: so that, for instance, calendar full moon on the 18th usually meant real full moon on the 14th.

7. If the difference above had not existed, the moon of the heavens (the mean moon at least), would have returnedpermanently to the same days of the month in 19 years; with an occasional slip arising from the unequal distribution of the leap years, of which a period contains sometimes five and sometimes four. As a general rule, the days of new and full moon in any one year would have been also the days of new and full moon of a year having 19 more units in its date. Again, if there had been no leap years, the days of the month would have returned to the same days of the week every seven years. The introduction of occasional 29ths of February disturbs this, and makes the permanent return of month days to week days occur only after 28 years. If all had been true, the lapse of 28 times 19, or 532 years, would have restored the year in every point: that is, A.D. 1, for instance, and A.D. 533, would have had the same almanac in every matter relating to week days, month days, sun, and moon (mean sun and moon at least). And on the supposition of its truth, the old system of Dionysius was framed. Its errors, are, first, that the moments of mean new moon advance too much by 1 h. 29 m. in 19 average Julian years; secondly, that the average Julian year of 365¼ days is too long by 11 m. 10 s.

8. The Council of Trent, moved by the representations made on the state of the Calendar, referred the consideration of it to the Pope. In 1577, Gregory XIII[756]submitted to the Roman Catholic Princes and Universities a plan presented to him by the representatives of Aloysius Lilius,[757]then deceased. This plan being approved of, the Pope nominated a commission to consider its details, the working member of which was the Jesuit Clavius. A short work was prepared by Clavius, descriptive of the new Calendar: thiswas published[758]in 1582, with the Pope's bull (dated February 24, 1581) prefixed. A larger work was prepared by Clavius, containing fuller explanation, and entitledRomani Calendarii a Gregorio XIII. Pontifice Maximo restituti Explicatio. This was published at Rome in 1603, and again in the collection of the works of Clavius in 1612.

9. The following extracts from Clavius settle the question of the meaning of the termmoon, as used in the Calendar:

"Who, except a few who think they are very sharp-sighted in this matter, is so blind as not to see that the 14th of the moon and the full moon are not the same things in the Church of God?... Although the Church, in finding the new moon, and from it the 14th day,uses neither the true nor the mean motion of the moon, but measures only according to the order of a cycle, it is nevertheless undeniable that the mean full moons found from astronomical tables are of the greatest use in determining the cycle which is to be preferred ... the new moons of which cycle, in order to the due celebration of Easter, should be so arranged that the 14th days of those moons, reckoning from the day of new mooninclusive, should not fall two or more days before the mean full moon, but only one day, or else on the very day itself, or not long after. And even thus far the Church need not take very great pains ... for it is sufficient that all should reckon by the 14th day of the moon in the cycle, even though sometimes itshould be more than one day before or afterthe mean full moon.... We have taken pains that in our cycle the new moons shouldfollowthe real new moons, so that the 14th of the moon should fall either the day before the mean full moon, or on that day, or not long after; and this was done on purpose, for if the new moon of the cycle fell on the same day as the mean new moon of theastronomers, it might chance that we should celebrate Easter on the same day as the Jews or the Quartadeciman heretics, which would be absurd, or else before them, which would be still more absurd."

From this it appears that Clavius continued the Calendar of his predecessors in the choice of thefourteenthday of the moon. Our legislature lays down the day of thefull moon: and this mistake appears to be rather English than Protestant; for it occurs in missals published in the reign of Queen Mary. The calendar lunation being 29½ days, the middle day is thefifteenthday, and this is and was reckoned as the day of the full moon. There is every right to presume that the original passover was a feast of thereal full moon: but it is most probable that the moons were then reckoned, not from the astronomical conjunction with the sun, which nobody sees except at an eclipse, but from the day offirst visibilityof the new moon. In fine climates this would be the day or two days after conjunction; and the fourteenth day from that of first visibility inclusive, would very often be the day of full moon. The following is then the proper correction of the precept in the Act of Parliament:

Easter Day, on which the rest depend, is always the First Sunday after thefourteenth dayof thecalendarmoon which happens upon or next after the Twenty-first day of March,according to the rules laid down for the construction of the Calendar; and if thefourteenth dayhappens upon a Sunday, Easter Day is the Sunday after.

10. Further, it appears that Clavius valued the celebration of the festival after the Jews, etc., more than astronomical correctness. He gives comparison tables which would startle a believer in the astronomical intention of his Calendar: they are to show that a calendar in which the moon is always made a day older than by him,represents the heavens better than he has done, or meant to do. But it must be observed that this diminution of the real moon's age hasa tendency to make the English explanation often practically accordant with the Calendar. For the fourteenth day of Claviusisgenerally the fifteenth day of the mean moon of the heavens, and therefore most often that of the real moon. But for this, 1818 and 1845 would not have been the only instances of our day in which the English precept would have contradicted the Calendar.

11. In the construction of the Calendar, Clavius adopted the ancient cycle of 532 years, but, we may say, without ever allowing it to run out. At certain periods, a shift is made from one part of the cycle into another. This is done whenever what should be Julian leap year is made a common year, as in 1700, 1800, 1900, 2100, etc. It is also done at certain times to correct the error of 1 h. 19 m., before referred to, in each cycle of golden numbers: Clavius, to meet his view of the amount of that error, put forward the moon's age a day 8 times in 2,500 years. As we cannot enter at full length into the explanation, we must content ourselves with giving a set of rules, independent of tables, by which the reader may find Easter for himself in any year, either by the old Calendar or the new. Any one who has much occasion to find Easters and movable feasts should procure Francœur's[759]tables.

12.Rule for determining Easter Day of the Gregorian Calendar in any year of the new style.To the several partsof the rule are annexed, by way of example, the results for the year 1849.

I. Add 1 to the given year. (1850).

II. Take the quotient of the given year divided by 4, neglecting the remainder. (462).

III. Take 16 from the centurial figures of the given year, if it can be done, and take the remainder. (2).

IV. Take the quotient of III. divided by 4, neglecting the remainder. (0).

V. From the sum of I, II, and IV., subtract III. (2310).

VI. Find the remainder of V. divided by 7. (0).

VII. Subtract VI. from 7; this is the number of the dominical letter

VIII. Divide I. by 19, the remainder (or 19, if no remainder) is thegolden number. (7).

IX. From the centurial figures of the year subtract 17, divide by 25, and keep the quotient. (0).

X. Subtract IX. and 15 from the centurial figures, divide by 3, and keep the quotient. (1).

XI. To VIII. add ten times the next less number, divide by 30, and keep the remainder. (7).

XII. To XI. add X. and IV., and take away III., throwing out thirties, if any. If this give 24, change it into 25. If 25, change it into 26, whenever the golden number is greater than 11. If 0, change it into 30. Thus we have the epact, or age of theCalendarmoon at the beginning of the year. (6).

XIII. Subtract XII., the epact, from 45. (39).

XIV. Subtract the epact from 27, divide by 7, and keep the remainder, or 7, if there be no remainder. (7)

XIII. Subtract XII., the epact, from 75.

XIV. Subtract the epact from 57, divide by 7, and keep the remainder, or 7, if there be no remainder.

XV. To XIII. add VII., the dominical number, (and 7 besides, if XIV. be greater than VII.,) and subtract XIV., the result is the day of March, or if more than 31, subtract 31, andthe result is the day of April, on which Easter Sunday falls. (39; Easter Day is April 8).

In the following examples, the several results leading to the final conclusion are tabulated.

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