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{{header
 | title    = [[../|A Budget of Paradoxes]]
 | author   = Augustus De Morgan
 | section  = 1840-1847
 | previous = [[../H|1825-1839]]
 | next     = [[../J|1847-1849]]
 | notes    = {{no scan}}
}}

{{pagenum|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.{{ref|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.

&nbsp;

===SOME PRIMITIVE DARWINISM.===

:Vestiges of the Natural History of Creation.{{ref|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 {{pagenum|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.

&nbsp;

===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 {{pagenum|346}} merely Human Institutions of the kind: A Letter very respectfully addressed to the Rev. James Everett,{{ref|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.{{ref|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.

{{pagenum|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.

&nbsp;

===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.] {{pagenum|348}}

&nbsp;

: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,{{ref|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.

&nbsp;

: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,{{ref|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. {{pagenum|349}} Magnetism and electricity are, in physics, the member of parliament and the cabman—at every man's bidding, as Henry Warburton{{ref|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‘Cook. He makes <math>\pi</math> to be <math>\scriptstyle 2 + 2\sqrt{(8\sqrt{2} - 11)}</math>.

&nbsp;

===THE DUPLICATION PROBLEM.===

:Refutation of a Pamphlet written by the Rev. John Mackey, R.C.P.,{{ref|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.{{ref|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 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 in {{pagenum|350}} a letter from the late Dean Peacock{{ref|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.

&nbsp;

===A NEW VALUE OF <math>\pi</math>.===

: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 <math>\scriptstyle 3\frac{1}{21}</math> diameters.

&nbsp;

===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 {{pagenum|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 {{small-caps|Author}} ''of'' {{small-caps|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 {{small-caps|Globe}}. It is this {{'}}''Crisis''{{'}} that has so generally '{{smaller|ROUSED}}' at this moment the {{'}}''nations throughout the Earth''{{'}} that no equal interest has ever before been excited by {{small-caps|Man}}; unless it be in that caused by the '{{small-caps|Pagan-Temple in Rome}},' which is recorded by the elder Pliny, {{'}}''Nat. Hist.''{{'}} i. 23. iii. 3. {{small-caps|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 {{small-caps|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''{{'}} ({{Polytonic|ἡμέρας ἙΞΗΚΟΝΤΑ}}—{{'}}''threescore or sixty days''{{'}}—{{'}}''I have appointed each'' "{{small-caps|Day}}" ''for a'' "{{small-caps|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 {{small-caps|our blessed Lord}}: viz. '{{small-caps|This Gospel}} ''of the Kingdom shall be preached in'' {{smaller|ALL}} the {{small-caps|Earth}}, ''for a Witness to'' {{small-caps|all Nations; and then}}, ''shall the'' {{small-caps|End come}}.{{'}} The {{'}}''next 18 years''{{'}} must therefore supply the interval of the {{'}}''special Episcopal forerunners''.{{'}}

{{c|(Matt. xxiv. 14.)}} 

"See the '{{small-caps|Jewish Intelligencer}}' of the present month (''April''), p. 153, for the '''Debates in Parliament'',' respecting {{pagenum|352}} the {{small-caps|Bishop of Jerusalem}}, ''viz.'' Dr. Bowring,{{ref|731}} Mr. Hume,{{ref|732}} Sir R. Inglis,{{ref|733}} Sir R. Peel,{{ref|734}} Viscount Palmerston.{{ref|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.

&nbsp;

===THE NUMBER OF THE BEAST.===

:''Wealth'' the name and number of the Beast, 666, in the Book of Revelation. [by John Taylor.{{ref|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 {{Polytonic|εὐπορία}}, 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 {{pagenum|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.

&nbsp;

:A practical Treatise on Eclipses ... with remarks on the anomalies of the present Theory of the Tides. By T. Kerigan,{{ref|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 ''Athenæum'' journal. Mr. Kerigan was a lieutenant in the Navy: he obtained admission to the Royal Society just before the publication of his book.

&nbsp;

:A new theory of Gravitation. By Joseph Denison,{{ref|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."

&nbsp;

===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, {{pagenum|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{{ref|739}} statement of the rule for finding Easter, belonging to distant years, had been made by Sir Harris Nicolas!{{ref|740}} Now as I myself furnished my friend Sir H.&nbsp;N. with Delambre's digest of {{pagenum|355}} Clavius's{{ref|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.&nbsp;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.&nbsp;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.&nbsp;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 {{pagenum|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.

==={{smaller|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 {{pagenum|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{{ref|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 {{pagenum|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''{{ref|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{{ref|744}} and Theodoret.{{ref|745}} "We also send {{pagenum|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{{ref|746}} met with difficulty in 414. S. Leo,{{ref|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{{ref|748}} employed Victorinus{{ref|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 {{pagenum|360}} this cycle bears the name of Dionysius Exiguus,{{ref|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,{{ref|751}} Sacrobosco,{{ref|752}} Cardinal Cusa,{{ref|753}} Regiomontanus,{{ref|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 its ''mean 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 (say ''mean'') 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 correct {{pagenum|361}} as to the ''mean'' moon, it would be occasionally false as to the ''true'' one: 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 defined ''the moon'' as 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{{ref|755}} years. This period, in the old Calendar, was taken to contain exactly 235 ''lunations'', 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 moon ''of this cycle'' gains 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 returned {{pagenum|362}} permanently 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{{ref|756}} submitted to the Roman Catholic Princes and Universities a plan presented to him by the representatives of Aloysius Lilius,{{ref|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: this {{pagenum|363}} was published{{ref|758}} in 1582, with the Pope's bull (dated February 24, 1581) prefixed. A larger work was prepared by Clavius, containing fuller explanation, and entitled ''Romani 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 term ''moon'', 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 moon ''inclusive'', 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 it ''should be more than one day before or after'' the mean full moon.... We have taken pains that in our cycle the new moons should ''follow'' the 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 the {{pagenum|364}} astronomers, 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 the ''fourteenth'' day of the moon. Our legislature lays down the day of the ''full 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 the ''fifteenth'' day, 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 the ''real 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 of ''first visibility'' of 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 the ''fourteenth day'' of the ''calendar'' moon 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 the ''fourteenth day'' happens 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 has {{pagenum|365}} a tendency to make the English explanation often practically accordant with the Calendar. For the fourteenth day of Clavius ''is'' generally 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{{ref|759}} tables.

12. ''Rule for determining Easter Day of the Gregorian Calendar in any year of the new style.'' To the several parts {{pagenum|366}} of 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

<table summary="Dominical letters" title="Dominical letters">
<tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td valign="middle">(7; dominical letter G).</td></tr>
<tr><td>A</td><td>B</td><td>C</td><td>D</td><td>E</td><td>F</td><td>G</td></tr>
</table>

VIII. Divide I. by 19, the remainder (or 19, if no remainder) is the ''golden 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 the ''Calendar'' moon at the beginning of the year. (6).

<table width="100%" summary="Epacts" title="Epacts"><tr><td valign="top" style="width:50%">
''When the Epact is 23, or less.''

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)

</td><td valign="top" style="width:50%">
''When the Epact is greater than 23.''

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.

</td></tr></table>

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, and {{pagenum|367}} the 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.

<table border="1" summary="Easter calculations" title="Easter calculations">
<tr><td>{{small-caps|Given Year}}</td><td align="right">1592</td><td align="right">1637</td><td align="right">1723</td><td align="right">1853</td><td align="right">2018</td><td align="right">4686</td></tr>
<tr><td align="right">I.</td><td align="right">1593</td><td align="right">1638</td><td align="right">1724</td><td align="right">1854</td><td align="right">2019</td><td align="right">4687</td></tr>
<tr><td align="right">II.</td><td align="right">398</td><td align="right">409</td><td align="right">430</td><td align="right">463</td><td align="right">504</td><td align="right">1171</td></tr>
<tr><td align="right">III.</td><td align="right">&mdash;</td><td align="right">0</td><td align="right">1</td><td align="right">2</td><td align="right">4</td><td align="right">30</td></tr>
<tr><td align="right">IV.</td><td align="right">&mdash;</td><td align="right">0</td><td align="right">0</td><td align="right">0</td><td align="right">1</td><td align="right">7</td></tr>
<tr><td align="right">V.</td><td align="right">1991</td><td align="right">2047</td><td align="right">2153</td><td align="right">2315</td><td align="right">2520</td><td align="right">5835</td></tr>
<tr><td align="right">VI.</td><td align="right">3</td><td align="right">3</td><td align="right">4</td><td align="right">5</td><td align="right">0</td><td align="right">4</td></tr>
<tr><td align="right">VII.</td><td align="right">4</td><td align="right">4</td><td align="right">3</td><td align="right">2</td><td align="right">7</td><td align="right">3</td></tr>
<tr><td align="right">VIII.</td><td align="right">16</td><td align="right">4</td><td align="right">14</td><td align="right">11</td><td align="right">5</td><td align="right">13</td></tr>
<tr><td align="right">IX.</td><td align="right">&mdash;</td><td align="right">&mdash;</td><td align="right">0</td><td align="right">0</td><td align="right">0</td><td align="right">1</td></tr>
<tr><td align="right">X.</td><td align="right">0</td><td align="right">0</td><td align="right">0</td><td align="right">1</td><td align="right">1</td><td align="right">10</td></tr>
<tr><td align="right">XI.</td><td align="right">16</td><td align="right">4</td><td align="right">24</td><td align="right">21</td><td align="right">15</td><td align="right">13</td></tr>
<tr><td align="right">XII.</td><td align="right">16</td><td align="right">4</td><td align="right">23</td><td align="right">20</td><td align="right">13</td><td align="right">0 say 30</td></tr>
<tr><td align="right">XIII.</td><td align="right">29</td><td align="right">41</td><td align="right">22</td><td align="right">25</td><td align="right">32</td><td align="right">45</td></tr>
<tr><td align="right">XIV.</td><td align="right">4</td><td align="right">2</td><td align="right">4</td><td align="right">7</td><td align="right">7</td><td align="right">6</td></tr>
<tr><td align="right">XV.</td><td align="right">29</td><td align="right">43</td><td align="right">28</td><td align="right">27</td><td align="right">32</td><td align="right">49</td></tr>
<tr><td align="right">Easter Day</td><td align="right">Mar.29</td><td align="right">Apr.12</td><td align="right">Mar.28</td><td align="right">Mar.27</td><td align="right">Apr.1</td><td align="right">Apr.18</td></tr>
</table>

13. ''Rule for determining Easter Day of the Antegregorian Calendar in any year of the old style.'' To the several parts of the rule are annexed, by way of example, the results for the year 1287. The steps are numbered to correspond with the steps of the Gregorian rule, so that it can be seen what augmentations the latter requires.

I. Set down the given year. (1287).

II. Take the quotient of the given year divided by 4, neglecting the remainder (321).

V. Take 4 more than the sum of I. and II. (1612).

VI. Find the remainder of V. divided by 7. (2).

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

<table summary="Dominical letters" title="Dominical letters">
<tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td valign="middle">(5; dominical letter E).</td></tr>
<tr><td>A</td><td>B</td><td>C</td><td>D</td><td>E</td><td>F</td><td>G</td></tr>
</table>

VIII. Divide one more than the given year by 19, the remainder (or 19 if no remainder) is the golden number. (15).

XII. Divide 3 less than 11 times VIII. by 30; the remainder (or 30 if there be no remainder) is the epact. (12).

{{pagenum|368}}

<table width="100%" summary="Epacts" title="Epacts"><tr><td valign="top" style="width:50%">
''When the Epact is 23, or less.''

XIII. Subtract XII., the epact, from 45. (33).

XIV. Subtract the epact from 27, divide by 7, and keep the remainder, or 7, if there be no remainder, (1).

</td><td valign="top" style="width:50%">
''When the Epact is greater than 23.''

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.

</td></tr></table>

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, and the result is the day of April, on which Easter Sunday (old style) falls. (37; Easter Day is April 6).

These rules completely represent the old and new Calendars, so far as Easter is concerned. For further explanation we must refer to the articles cited at the commencement.

The annexed is the table of new and full moons of the Gregorian Calendar, cleared of the errors made for the purpose of preventing Easter from coinciding with the Jewish Passover.

The second table (page 370) contains ''epacts'', or ages of the moon at the beginning of the year: thus in 1913, the epact is 22, in 1868 it is 6. This table goes from 1850 to 1999: should the New Zealander not have arrived by that time, and should the churches of England and Rome then survive, the epact table may be continued from their liturgy-books. The way of using the table is as follows: Take the epact of the required year, and find it in the first or last column of the first table, in line with it are seen the calendar days of new and full moon. Thus, when the epact is 17, the new and full moons of March fall on the 13th and 28th. The result is, for the most part, correct: but in a minority of cases there is an error of a day. When this happens, the error is almost always a fraction of a day much less than twelve hours. Thus, when the table gives full moon on the 27th, and the real truth is the 28th, we may be sure it is early on the 28th.

{{pagenum|369}}

<table border="1" summary="True table of moons" title="True table of moons">
<tr><td align="center">&nbsp;</td><td align="center">Jan.</td><td align="center">Feb.</td><td align="center">Mar.</td><td align="center">Apr.</td><td align="center">May </td><td align="center">June</td><td align="center">July</td><td align="center">Aug.</td><td align="center">Sep.</td><td align="center">Oct.</td><td align="center">Nov.</td><td align="center">Dec.</td><td align="center">&nbsp;</td></tr>
<tr><td align="center">1</td><td align="center">29</td><td align="center">27</td><td align="center">29</td><td align="center">27</td><td align="center">27</td><td align="center">25</td><td align="center">25</td><td align="center">23</td><td align="center">22</td><td align="center">21</td><td align="center">20</td><td align="center">19</td><td align="center">1</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">14</td><td align="center">13</td><td align="center">14</td><td align="center">13</td><td align="center">12</td><td align="center">11</td><td align="center">10</td><td align="center">9</td><td align="center">7</td><td align="center">7</td><td align="center">5</td><td align="center">5</td><td align="center"></td></tr>
<tr><td align="center">2</td><td align="center">28</td><td align="center">26</td><td align="center">28</td><td align="center">26</td><td align="center">26</td><td align="center">24</td><td align="center">24</td><td align="center">22</td><td align="center">21</td><td align="center">20</td><td align="center">19</td><td align="center">18</td><td align="center">2</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">13</td><td align="center">12</td><td align="center">13</td><td align="center">12</td><td align="center">11</td><td align="center">10</td><td align="center">9</td><td align="center">8</td><td align="center">6</td><td align="center">6</td><td align="center">4</td><td align="center">4</td><td align="center"></td></tr>
<tr><td align="center">3</td><td align="center">27</td><td align="center">25</td><td align="center">27</td><td align="center">25</td><td align="center">25</td><td align="center">23</td><td align="center">23</td><td align="center">21</td><td align="center">20</td><td align="center">19</td><td align="center">18</td><td align="center">17</td><td align="center">3</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">12</td><td align="center">11</td><td align="center">12</td><td align="center">11</td><td align="center">10</td><td align="center">9</td><td align="center">8</td><td align="center">7</td><td align="center">5</td><td align="center">5</td><td align="center">3</td><td align="center">3</td><td align="center"></td></tr>
<tr><td align="center">4</td><td align="center">26</td><td align="center">24</td><td align="center">26</td><td align="center">24</td><td align="center">24</td><td align="center">22</td><td align="center">22</td><td align="center">20</td><td align="center">19</td><td align="center">18</td><td align="center">17</td><td align="center">16</td><td align="center">4</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">11</td><td align="center">10</td><td align="center">11</td><td align="center">10</td><td align="center">9</td><td align="center">8</td><td align="center">7</td><td align="center">6</td><td align="center">4</td><td align="center">4</td><td align="center">2</td><td align="center">2,31</td><td align="center"></td></tr>
<tr><td align="center">5</td><td align="center">25</td><td align="center">23</td><td align="center">25</td><td align="center">23</td><td align="center">23</td><td align="center">21</td><td align="center">21</td><td align="center">19</td><td align="center">18</td><td align="center">17</td><td align="center">16</td><td align="center">15</td><td align="center">5</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">10</td><td align="center">9</td><td align="center">10</td><td align="center">9</td><td align="center">8</td><td align="center">7</td><td align="center">6</td><td align="center">5</td><td align="center">3</td><td align="center">3</td><td align="center">1</td><td align="center">1,30</td><td align="center"></td></tr>
<tr><td align="center">6</td><td align="center">24</td><td align="center">22</td><td align="center">24</td><td align="center">22</td><td align="center">22</td><td align="center">20</td><td align="center">20</td><td align="center">18</td><td align="center">17</td><td align="center">16</td><td align="center">15</td><td align="center">14</td><td align="center">6</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">9</td><td align="center">8</td><td align="center">9</td><td align="center">8</td><td align="center">7</td><td align="center">6</td><td align="center">5</td><td align="center">4</td><td align="center">2</td><td align="center">2,31</td><td align="center">30</td><td align="center">29</td><td align="center"></td></tr>
<tr><td align="center">7</td><td align="center">23</td><td align="center">21</td><td align="center">23</td><td align="center">21</td><td align="center">21</td><td align="center">19</td><td align="center">19</td><td align="center">17</td><td align="center">16</td><td align="center">15</td><td align="center">14</td><td align="center">13</td><td align="center">7</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">8</td><td align="center">7</td><td align="center">8</td><td align="center">7</td><td align="center">6</td><td align="center">5</td><td align="center">4</td><td align="center">3</td><td align="center">1</td><td align="center">1,30</td><td align="center">29</td><td align="center">28</td><td align="center"></td></tr>
<tr><td align="center">8</td><td align="center">22</td><td align="center">20</td><td align="center">22</td><td align="center">20</td><td align="center">20</td><td align="center">18</td><td align="center">18</td><td align="center">16</td><td align="center">15</td><td align="center">14</td><td align="center">13</td><td align="center">12</td><td align="center">8</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">7</td><td align="center">6</td><td align="center">7</td><td align="center">6</td><td align="center">5</td><td align="center">4</td><td align="center">3</td><td align="center">2,31</td><td align="center">30</td><td align="center">29</td><td align="center">28</td><td align="center">27</td><td align="center"></td></tr>
<tr><td align="center">9</td><td align="center">21</td><td align="center">19</td><td align="center">21</td><td align="center">19</td><td align="center">19</td><td align="center">17</td><td align="center">17</td><td align="center">15</td><td align="center">14</td><td align="center">13</td><td align="center">12</td><td align="center">11</td><td align="center">9</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">6</td><td align="center">5</td><td align="center">6</td><td align="center">5</td><td align="center">4</td><td align="center">3</td><td align="center">2</td><td align="center">1,30</td><td align="center">29</td><td align="center">28</td><td align="center">27</td><td align="center">26</td><td align="center"></td></tr>
<tr><td align="center">10</td><td align="center">20</td><td align="center">18</td><td align="center">20</td><td align="center">18</td><td align="center">18</td><td align="center">16</td><td align="center">16</td><td align="center">14</td><td align="center">13</td><td align="center">12</td><td align="center">11</td><td align="center">10</td><td align="center">10</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">5</td><td align="center">4</td><td align="center">5</td><td align="center">4</td><td align="center">3</td><td align="center">2</td><td align="center">1,31</td><td align="center">29</td><td align="center">28</td><td align="center">27</td><td align="center">26</td><td align="center">25</td><td align="center"></td></tr>
<tr><td align="center">11</td><td align="center">19</td><td align="center">17</td><td align="center">19</td><td align="center">17</td><td align="center">17</td><td align="center">15</td><td align="center">15</td><td align="center">13</td><td align="center">12</td><td align="center">11</td><td align="center">10</td><td align="center">9</td><td align="center">11</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">4</td><td align="center">3</td><td align="center">4</td><td align="center">3</td><td align="center">2</td><td align="center">1,30</td><td align="center">30</td><td align="center">28</td><td align="center">27</td><td align="center">26</td><td align="center">25</td><td align="center">24</td><td align="center"></td></tr>
<tr><td align="center">12</td><td align="center">18</td><td align="center">16</td><td align="center">18</td><td align="center">16</td><td align="center">16</td><td align="center">14</td><td align="center">14</td><td align="center">12</td><td align="center">11</td><td align="center">10</td><td align="center">9</td><td align="center">8</td><td align="center">12</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">3</td><td align="center">2</td><td align="center">3</td><td align="center">2</td><td align="center">1,31</td><td align="center">29</td><td align="center">29</td><td align="center">27</td><td align="center">26</td><td align="center">25</td><td align="center">24</td><td align="center">23</td><td align="center"></td></tr>
<tr><td align="center">13</td><td align="center">17</td><td align="center">15</td><td align="center">17</td><td align="center">15</td><td align="center">15</td><td align="center">13</td><td align="center">13</td><td align="center">11</td><td align="center">10</td><td align="center">9</td><td align="center">8</td><td align="center">7</td><td align="center">13</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">2</td><td align="center">1</td><td align="center">2</td><td align="center">1,30</td><td align="center">30</td><td align="center">28</td><td align="center">28</td><td align="center">26</td><td align="center">25</td><td align="center">24</td><td align="center">23</td><td align="center">22</td><td align="center"></td></tr>
<tr><td align="center">14</td><td align="center">16</td><td align="center">14</td><td align="center">16</td><td align="center">14</td><td align="center">14</td><td align="center">12</td><td align="center">12</td><td align="center">10</td><td align="center">9</td><td align="center">8</td><td align="center">7</td><td align="center">6</td><td align="center">14</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">1,31</td><td align="center">&mdash;</td><td align="center">1,31</td><td align="center">29</td><td align="center">29</td><td align="center">27</td><td align="center">27</td><td align="center">25</td><td align="center">24</td><td align="center">23</td><td align="center">22</td><td align="center">21</td><td align="center"></td></tr>
<tr><td align="center">15</td><td align="center">15</td><td align="center">13</td><td align="center">15</td><td align="center">13</td><td align="center">13</td><td align="center">11</td><td align="center">11</td><td align="center">9</td><td align="center">8</td><td align="center">7</td><td align="center">6</td><td align="center">5</td><td align="center">15</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">30</td><td align="center">28</td><td align="center">30</td><td align="center">28</td><td align="center">28</td><td align="center">26</td><td align="center">26</td><td align="center">24</td><td align="center">23</td><td align="center">22</td><td align="center">21</td><td align="center">20</td><td align="center"></td></tr>
<tr><td align="center">16</td><td align="center">14</td><td align="center">12</td><td align="center">14</td><td align="center">12</td><td align="center">12</td><td align="center">10</td><td align="center">10</td><td align="center">8</td><td align="center">7</td><td align="center">6</td><td align="center">5</td><td align="center">4</td><td align="center">16</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">29</td><td align="center">27</td><td align="center">29</td><td align="center">27</td><td align="center">27</td><td align="center">25</td><td align="center">25</td><td align="center">23</td><td align="center">22</td><td align="center">21</td><td align="center">20</td><td align="center">19</td><td align="center"></td></tr>
<tr><td align="center">17</td><td align="center">13</td><td align="center">11</td><td align="center">13</td><td align="center">11</td><td align="center">11</td><td align="center">9</td><td align="center">9</td><td align="center">7</td><td align="center">6</td><td align="center">5</td><td align="center">4</td><td align="center">3</td><td align="center">17</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">28</td><td align="center">26</td><td align="center">28</td><td align="center">26</td><td align="center">26</td><td align="center">24</td><td align="center">24</td><td align="center">22</td><td align="center">21</td><td align="center">20</td><td align="center">19</td><td align="center">18</td><td align="center"></td></tr>
<tr><td align="center">18</td><td align="center">12</td><td align="center">10</td><td align="center">12</td><td align="center">10</td><td align="center">10</td><td align="center">8</td><td align="center">8</td><td align="center">6</td><td align="center">5</td><td align="center">4</td><td align="center">3</td><td align="center">2</td><td align="center">18</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">27</td><td align="center">25</td><td align="center">27</td><td align="center">25</td><td align="center">25</td><td align="center">23</td><td align="center">23</td><td align="center">21</td><td align="center">20</td><td align="center">19</td><td align="center">18</td><td align="center">17</td><td align="center"></td></tr>
<tr><td align="center">19</td><td align="center">11</td><td align="center">9</td><td align="center">11</td><td align="center">9</td><td align="center">9</td><td align="center">7</td><td align="center">7</td><td align="center">5</td><td align="center">4</td><td align="center">3</td><td align="center">2</td><td align="center">1,31</td><td align="center">19</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">26</td><td align="center">24</td><td align="center">26</td><td align="center">24</td><td align="center">24</td><td align="center">22</td><td align="center">22</td><td align="center">20</td><td align="center">19</td><td align="center">18</td><td align="center">17</td><td align="center">16</td><td align="center"></td></tr>
<tr><td align="center">20</td><td align="center">10</td><td align="center">8</td><td align="center">10</td><td align="center">8</td><td align="center">8</td><td align="center">6</td><td align="center">6</td><td align="center">4</td><td align="center">3</td><td align="center">2</td><td align="center">1,31</td><td align="center">30</td><td align="center">20</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">25</td><td align="center">23</td><td align="center">25</td><td align="center">23</td><td align="center">23</td><td align="center">21</td><td align="center">21</td><td align="center">19</td><td align="center">18</td><td align="center">17</td><td align="center">16</td><td align="center">15</td><td align="center"></td></tr>
<tr><td align="center">21</td><td align="center">9</td><td align="center">7</td><td align="center">9</td><td align="center">7</td><td align="center">7</td><td align="center">5</td><td align="center">5</td><td align="center">3</td><td align="center">2</td><td align="center">1,31</td><td align="center">29</td><td align="center">29</td><td align="center">21</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">24</td><td align="center">22</td><td align="center">24</td><td align="center">22</td><td align="center">22</td><td align="center">20</td><td align="center">20</td><td align="center">18</td><td align="center">17</td><td align="center">16</td><td align="center">15</td><td align="center">14</td><td align="center"></td></tr>
<tr><td align="center">22</td><td align="center">8</td><td align="center">6</td><td align="center">8</td><td align="center">6</td><td align="center">6</td><td align="center">4</td><td align="center">4</td><td align="center">2</td><td align="center">1,30</td><td align="center">30</td><td align="center">28</td><td align="center">28</td><td align="center">22</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">23</td><td align="center">21</td><td align="center">23</td><td align="center">21</td><td align="center">21</td><td align="center">19</td><td align="center">19</td><td align="center">17</td><td align="center">16</td><td align="center">15</td><td align="center">14</td><td align="center">13</td><td align="center"></td></tr>
<tr><td align="center">23</td><td align="center">7</td><td align="center">5</td><td align="center">7</td><td align="center">5</td><td align="center">5</td><td align="center">3</td><td align="center">3</td><td align="center">1,31</td><td align="center">29</td><td align="center">29</td><td align="center">27</td><td align="center">27</td><td align="center">23</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">22</td><td align="center">20</td><td align="center">22</td><td align="center">20</td><td align="center">20</td><td align="center">18</td><td align="center">18</td><td align="center">16</td><td align="center">15</td><td align="center">14</td><td align="center">13</td><td align="center">12</td><td align="center"></td></tr>
<tr><td align="center">24</td><td align="center">6</td><td align="center">5</td><td align="center">6</td><td align="center">5</td><td align="center">4</td><td align="center">3</td><td align="center">2</td><td align="center">1,30</td><td align="center">29</td><td align="center">28</td><td align="center">27</td><td align="center">26</td><td align="center">24</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">21</td><td align="center">19</td><td align="center">21</td><td align="center">19</td><td align="center">19</td><td align="center">17</td><td align="center">17</td><td align="center">15</td><td align="center">14</td><td align="center">13</td><td align="center">12</td><td align="center">11</td><td align="center"></td></tr>
<tr><td align="center">25</td><td align="center">5</td><td align="center">4</td><td align="center">5</td><td align="center">4</td><td align="center">3</td><td align="center">2</td><td align="center">1,31</td><td align="center">29</td><td align="center">28</td><td align="center">27</td><td align="center">26</td><td align="center">25</td><td align="center">25</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">20</td><td align="center">19</td><td align="center">20</td><td align="center">19</td><td align="center">18</td><td align="center">17</td><td align="center">16</td><td align="center">15</td><td align="center">13</td><td align="center">13</td><td align="center">11</td><td align="center">11</td><td align="center"></td></tr>
<tr><td align="center">26</td><td align="center">4</td><td align="center">3</td><td align="center">4</td><td align="center">3</td><td align="center">2</td><td align="center">1,30</td><td align="center">30</td><td align="center">28</td><td align="center">27</td><td align="center">26</td><td align="center">25</td><td align="center">24</td><td align="center">26</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">19</td><td align="center">18</td><td align="center">19</td><td align="center">18</td><td align="center">17</td><td align="center">16</td><td align="center">15</td><td align="center">14</td><td align="center">12</td><td align="center">12</td><td align="center">10</td><td align="center">10</td><td align="center"></td></tr>
<tr><td align="center">27</td><td align="center">3</td><td align="center">2</td><td align="center">3</td><td align="center">2</td><td align="center">1,31</td><td align="center">29</td><td align="center">29</td><td align="center">27</td><td align="center">26</td><td align="center">25</td><td align="center">24</td><td align="center">23</td><td align="center">27</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">18</td><td align="center">17</td><td align="center">18</td><td align="center">17</td><td align="center">16</td><td align="center">15</td><td align="center">14</td><td align="center">13</td><td align="center">11</td><td align="center">11</td><td align="center">9</td><td align="center">9</td><td align="center"></td></tr>
<tr><td align="center">28</td><td align="center">2</td><td align="center">1</td><td align="center">2</td><td align="center">1,30</td><td align="center">30</td><td align="center">28</td><td align="center">28</td><td align="center">26</td><td align="center">25</td><td align="center">24</td><td align="center">23</td><td align="center">22</td><td align="center">28</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">17</td><td align="center">16</td><td align="center">17</td><td align="center">16</td><td align="center">15</td><td align="center">14</td><td align="center">13</td><td align="center">12</td><td align="center">10</td><td align="center">10</td><td align="center">8</td><td align="center">8</td><td align="center"></td></tr>
<tr><td align="center">29</td><td align="center">1,31</td><td align="center">&mdash;</td><td align="center">1,31</td><td align="center">29</td><td align="center">29</td><td align="center">27</td><td align="center">27</td><td align="center">25</td><td align="center">24</td><td align="center">23</td><td align="center">22</td><td align="center">21</td><td align="center">29</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">16</td><td align="center">15</td><td align="center">16</td><td align="center">15</td><td align="center">14</td><td align="center">13</td><td align="center">12</td><td align="center">11</td><td align="center">9</td><td align="center">9</td><td align="center">7</td><td align="center">7</td><td align="center"></td></tr>
<tr><td align="center">30</td><td align="center">30</td><td align="center">28</td><td align="center">30</td><td align="center">28</td><td align="center">28</td><td align="center">26</td><td align="center">26</td><td align="center">24</td><td align="center">23</td><td align="center">22</td><td align="center">21</td><td align="center">20</td><td align="center">30</td></tr>
<tr><td align="center">&nbsp;</td><td align="center">15</td><td align="center">14</td><td align="center">15</td><td align="center">14</td><td align="center">13</td><td align="center">12</td><td align="center">11</td><td align="center">10</td><td align="center">8</td><td align="center">8</td><td align="center">6</td><td align="center">6</td><td align="center"></td></tr>
<tr><td align="center">&nbsp;</td><td align="center">Jan.</td><td align="center">Feb.</td><td align="center">Mar.</td><td align="center">Apr.</td><td align="center">May</td><td align="center">June</td><td align="center">July</td><td align="center">Aug.</td><td align="center">Sep.</td><td align="center">Oct.</td><td align="center">Nov.</td><td align="center">Dec.</td><td align="center">&nbsp;</td></tr>
</table>

{{pagenum|370}}

<table border="1" summary="True table of epacts" title="True table of epacts">
<tr><td align="center">&nbsp;</td><td align="center">0</td><td align="center">1</td><td align="center">2</td><td align="center">3</td><td align="center">4</td><td align="center">5</td><td align="center">6</td><td align="center">7</td><td align="center">8</td><td align="center">9</td></tr>
<tr><td align="center">185</td><td align="center">17</td><td align="center">28</td><td align="center">9</td><td align="center">20</td><td align="center">2</td><td align="center">12</td><td align="center">23</td><td align="center">4</td><td align="center">15</td><td align="center">26</td></tr>
<tr><td align="center">186</td><td align="center">7</td><td align="center">18</td><td align="center">30</td><td align="center">11</td><td align="center">22</td><td align="center">3</td><td align="center">14</td><td align="center">25</td><td align="center">6</td><td align="center">17</td></tr>
<tr><td align="center">187</td><td align="center">28</td><td align="center">9</td><td align="center">20</td><td align="center">1</td><td align="center">12</td><td align="center">23</td><td align="center">4</td><td align="center">15</td><td align="center">26</td><td align="center">7</td></tr>
<tr><td align="center">188</td><td align="center">18</td><td align="center">30</td><td align="center">11</td><td align="center">22</td><td align="center">3</td><td align="center">14</td><td align="center">25</td><td align="center">6</td><td align="center">17</td><td align="center">28</td></tr>
<tr><td align="center">189</td><td align="center">9</td><td align="center">21</td><td align="center">1</td><td align="center">12</td><td align="center">23</td><td align="center">4</td><td align="center">15</td><td align="center">26</td><td align="center">7</td><td align="center">18</td></tr>
<tr><td align="center">190</td><td align="center">29</td><td align="center">10</td><td align="center">21</td><td align="center">2</td><td align="center">13</td><td align="center">24</td><td align="center">5</td><td align="center">16</td><td align="center">27</td><td align="center">8</td></tr>
<tr><td align="center">191</td><td align="center">19</td><td align="center">30</td><td align="center">11</td><td align="center">22</td><td align="center">3</td><td align="center">14</td><td align="center">26</td><td align="center">6</td><td align="center">17</td><td align="center">29</td></tr>
<tr><td align="center">192</td><td align="center">10</td><td align="center">21</td><td align="center">2</td><td align="center">13</td><td align="center">24</td><td align="center">5</td><td align="center">16</td><td align="center">27</td><td align="center">8</td><td align="center">19</td></tr>
<tr><td align="center">193</td><td align="center">30</td><td align="center">11</td><td align="center">22</td><td align="center">3</td><td align="center">14</td><td align="center">26</td><td align="center">6</td><td align="center">17</td><td align="center">29</td><td align="center">10</td></tr>
<tr><td align="center">194</td><td align="center">21</td><td align="center">2</td><td align="center">13</td><td align="center">24</td><td align="center">5</td><td align="center">16</td><td align="center">27</td><td align="center">8</td><td align="center">19</td><td align="center">30</td></tr>
<tr><td align="center">195</td><td align="center">11</td><td align="center">22</td><td align="center">3</td><td align="center">14</td><td align="center">26</td><td align="center">6</td><td align="center">17</td><td align="center">29</td><td align="center">10</td><td align="center">21</td></tr>
<tr><td align="center">196</td><td align="center">2</td><td align="center">13</td><td align="center">24</td><td align="center">5</td><td align="center">16</td><td align="center">27</td><td align="center">8</td><td align="center">19</td><td align="center">30</td><td align="center">11</td></tr>
<tr><td align="center">197</td><td align="center">22</td><td align="center">3</td><td align="center">14</td><td align="center">26</td><td align="center">6</td><td align="center">17</td><td align="center">29</td><td align="center">10</td><td align="center">21</td><td align="center">2</td></tr>
<tr><td align="center">198</td><td align="center">13</td><td align="center">24</td><td align="center">5</td><td align="center">16</td><td align="center">27</td><td align="center">8</td><td align="center">19</td><td align="center">30</td><td align="center">11</td><td align="center">22</td></tr>
<tr><td align="center">199</td><td align="center">3</td><td align="center">14</td><td align="center">26</td><td align="center">6</td><td align="center">17</td><td align="center">29</td><td align="center">10</td><td align="center">21</td><td align="center">2</td><td align="center">13</td></tr>
</table>

For example, the year 1867. The epact is 25, and we find in the table:

<table summary="Moons for 1867" title="Moons for 1867">
<tr><td>&nbsp;</td><td align="center">J.</td><td align="center">F.</td><td align="center">M.</td><td align="center">AP.</td><td align="center">M.</td><td align="center">JU.</td><td align="center">JL.</td><td align="center">AU.</td><td align="center">S.</td><td align="center">O.</td><td align="center">N.</td><td align="center">D.</td></tr>
<tr><td>New</td><td align="center">5+</td><td align="center">4</td><td align="center">5+</td><td align="center">4</td><td align="center">3+</td><td align="center">2</td><td align="center">1,31</td><td align="center">29</td><td align="center">28-</td><td align="center">27</td><td align="center">26</td><td align="center">25</td></tr>
<tr><td>Full</td><td align="center">20</td><td align="center">19-</td><td align="center">20</td><td align="center">19-</td><td align="center">18</td><td align="center">17</td><td align="center">16</td><td align="center">15</td><td align="center">13-</td><td align="center">13</td><td align="center">11+</td><td align="center">11</td></tr>
</table>

When the truth is the day after + is written after the date; when the day before, -. Thus, the new moon of March is on the 6th; the full moon of April is on the 18th. {{pagenum|371}}

I now introduce a small paradox of my own; and as I am not able to prove it, I am compelled to declare that any one who shall dissent must be either very foolish or very dishonest, and will make me quite uncomfortable about the state of his soul. This being settled once for all, I proceed to say that the necessity of arriving at the truth about the assertions that the Nicene Council laid down astronomical tests led me to look at Fathers, Church histories, etc. to an extent which I never dreamed of before. One conclusion which I arrived at was, that the Nicene Fathers had a knack of sticking to the question which many later councils could not acquire. In our own day, it is not permitted to Convocation seriously to discuss any one of the points which are bearing so hard upon their resources of defence—the cursing clauses of the Athanasian Creed, for example. And it may be collected that the prohibition arises partly from fear that there is no saying where a beginning, if allowed, would end. There seems to be a suspicion that debate, once let loose, would play up old Trent with the liturgy, and bring the whole book to book. But if any one will examine the real Nicene Creed, without the augmentation, he will admire the way in which the framers stuck to the point, and settled what they had to decide, according to their view of it. With such a presumption of good sense in their favor, it becomes easier to believe in any claim which may be made on their behalf to tact or sagacity in settling any other matter. And I strongly suspect such a claim may be made for them on the Easter question.

I collect from many little indications, both before and after the Council, that the division of the Christian world into Judaical and Gentile, though not giving rise to a sectarian distinction expressed by names, was of far greater force and meaning than historians prominently admit. I took ''note'' of many indications of this, but not ''notes'', as it was not to my purpose. If it were so, we must admire the discretion of the Council. The Easter question was the {{pagenum|372}} fighting ground of the struggle: the Eastern or Judaical Christians, with some varieties of usage and meaning, would have the Passover itself to be the great feast, but taken in a Christian sense; the Western or Gentile Christians, would have the commemoration of the Resurrection, connected with the Passover only by chronology. To shift the Passover in time, under its name, ''Pascha'', without allusion to any of the force of the change, was gently cutting away the ground from under the feet of the Conservatives. And it was done in a very quiet way: no allusion to the precise character of the change; no hint that the question was about two different festivals: "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." The Judaizers meant to be keeping the Passover ''as'' a Christian feast: they are gently assumed to be keeping, ''not'' the Passover, ''but'' a Christian feast; and a doctrinal decision is quietly, but efficiently, announced under the form of a chronological ordinance. Had the Council issued theses of doctrine, and excommunicated all dissentients, the rupture of the East and West would have taken place earlier by centuries than it did. The only place in which I ever saw any part of my paradox advanced, was in an article in the ''Examiner'' newspaper, towards the end of 1866, after the above was written.

A story about Christopher Clavius, the workman of the new Calendar. I chanced to pick up "Albertus Pighius Campensis de æquinoctiorum solsticiorumque inventione... Ejusdem de ratione Paschalis celebrationis, De que Restitutione ecclesiastici Kalendarii," Paris, 1520, folio.{{ref|760}} On the title-page were decayed words followed by "..''hristophor.. C..ii'', 1556 (or 8)," the last blank not entirely erased by time, but showing the lower halves of an ''l'' and of an ''a'', and {{pagenum|373}} rather too much room for a ''v''. It looked very like ''E Libris Christophori Clavii'' 1556. By the courtesy of some members of the Jesuit body in London, I procured a tracing of the signature of Clavius from Rome, and the shapes of the letters, and the modes of junction and disjunction, put the matter beyond question. Even the extra space was explained; he wrote himself Cla''u''ius. Now in 1556, Clavius was nineteen years old: it thus appears probable that the framer of the Gregorian Calendar was selected, not merely as a learned astronomer, but as one who had attended to the calendar, and to works on its reformation, from early youth. When on the subject I found reason to think that Clavius had really read this work, and taken from it a phrase or two and a notion or two. Observe the advantage of writing the baptismal name at full length.

&nbsp;

===A COUPLE OF MINOR PARADOXES.===

:The discovery of a general resolution of all superior finite equations, of every numerical both algebraick and transcendent form. By A.&nbsp;P. Vogel,{{ref|761}} mathematician at Leipzick. Leipzick and London, 1845, 8vo.

This work is written in the English of a German who has not mastered the idiom: but it is always intelligible. It professes to solve equations of every degree "in a more extent sense, and till to every degree of exactness." The general solution of equations of ''all'' degrees is a vexed question, which cannot have the mysterious interest of the circle problem, and is of a comparatively modern date.{{ref|762}} Mr. Vogel {{pagenum|374}} announces a forthcoming treatise in which are resolved the "last impossibilities of pure mathematics."

&nbsp;

:Elective Polarity the Universal Agent. By Frances Barbara Burton, authoress of 'Astronomy familiarized,' 'Physical Astronomy,' &c. London, 1845, 8vo.{{ref|763}}

The title gives a notion of the theory. The first sentence states, that 12,500 years ago α Lyræ was the pole-star, and attributes the immense magnitude of the now fossil animals to a star of such "polaric intensity as Vega pouring its magnetic streams through our planet." Miss Burton was a lady of property, and of very respectable acquirements, especially in Hebrew; she was eccentric in all things.

1867.—Miss Burton is revived by the writer of a book on meteorology which makes use of the planets: she is one of his leading minds.{{ref|764}}

&nbsp;

===SPECULATIVE THOUGHT IN ENGLAND.===

In the year 1845 the old ''Mathematical Society'' was merged in the Astronomical Society. The circle-squarers, etc., thrive more in England than in any other country: there are most weeds where there is the largest crop. Speculation, though not encouraged by our Government so much as by those of the Continent, has had, not indeed such forcing, but much wider diffusion: few tanks, but many rivulets. On this point I quote from the preface to the reprint of the work of Ramchundra,{{ref|765}} which I superintended for the late Court of Directors of the East India Company.

{{pagenum|375}}

"That sound judgment which gives men well to know what is best for them, as well as that faculty of invention which leads to development of resources and to the increase of wealth and comfort, are both materially advanced, perhaps cannot rapidly be advanced without, a great taste for pure speculation among the general mass of the people, down to the lowest of those who can read and write. England is a marked example. Many persons will be surprised at this assertion. They imagine that our country is the great instance of the refusal of all ''unpractical'' knowledge in favor of what is ''useful''. I affirm, on the contrary, that there is no country in Europe in which there has been so wide a diffusion of speculation, theory, or what other unpractical word the reader pleases. In our country, the scientific ''society'' is always formed and maintained by the people; in every other, the scientific ''academy''—most aptly named—has been the creation of the government, of which it has never ceased to be the nursling. In all the parts of England in which manufacturing pursuits have given the artisan some command of time, the cultivation of mathematics and other speculative studies has been, as is well known, a very frequent occupation. In no other country has the weaver at his loom bent over the ''Principia'' of Newton; in no other country has the man of weekly wages maintained his own scientific periodical. With us, since the beginning of the last century, scores upon scores—perhaps hundreds, for I am far from knowing all—of annuals have run, some their ten years, some their half-century, some their century and a half, containing questions to be answered, from which many of our examiners in the universities have culled materials for the academical contests. And these questions have always been answered, and in cases without number by the lower order of purchasers, the mechanics, the weavers, and the printers' workmen. I cannot here digress to point out the manner in which the concentration of manufactures, and the general diffusion of education, have affected the {{pagenum|376}} state of things; I speak of the time during which the present system took its rise, and of the circumstances under which many of its most effective promoters were trained. In all this there is nothing which stands out, like the state-nourished academy, with its few great names and brilliant single achievements. This country has differed from all others in the wide diffusion of the disposition to speculate, which disposition has found its place among the ordinary habits of life, moderate in its action, healthy in its amount."

&nbsp;

===THE OLD MATHEMATICAL SOCIETY.===

Among the most remarkable proofs of the diffusion of speculation was the Mathematical Society, which flourished from 1717 to 1845. Its habitat was Spitalfields, and I think most of its existence was passed in Crispin Street. It was originally a plain society, belonging to the studious artisan. The members met for discussion once a week; and I believe I am correct in saying that each man had his pipe, his pot, and his problem. One of their old rules was that, "If any member shall so far forget himself and the respect due to the Society as in the warmth of debate to threaten or offer personal violence to any other member, he shall be liable to immediate expulsion, or to pay such fine as the majority of the members present shall decide." But their great rule, printed large on the back of the title page of their last book of regulations, was "By the constitution of the Society, it is the duty of every member, if he be asked any mathematical or philosophical question by another member, to instruct him in the plainest and easiest manner he is able." We shall presently see that, in old time, the rule had a more homely form.

I have been told that De Moivre{{ref|766}} was a member of this {{pagenum|377}} Society. This I cannot verify: circumstances render it unlikely; even though the French refugees clustered in Spitalfields; many of them were of the Society, which there is some reason to think was founded by them. But Dolland,{{ref|767}} Thomas Simpson,{{ref|768}} Saunderson,{{ref|769}} Crossley,{{ref|770}} and others of known name, were certainly members. The Society gradually declined, and in 1845 was reduced to nineteen members. An arrangement was made by which sixteen of these members, who where not already in the Astronomical Society became Fellows without contribution, all the books and other property of the old Society being transferred to the new one. I was one of the committee which made the preliminary inquiries, and the reason of the decline was soon manifest. The only question which could arise was whether the members of the society of working men—for this repute still continued—were of that class of educated men who could associate with the Fellows of the Astronomical Society on terms agreeable to all parties. We found that the artisan element had been extinct for many years; there was not a man but might, as to education, manners, and position, have become a Fellow in the usual way. The fact was that life in Spitalfields had become harder: and the weaver could {{pagenum|378}} only live from hand to mouth, and not up to the brain. The material of the old Society no longer existed.

In 1798, experimental lectures were given, a small charge for admission being taken at the door: by this hangs a tale—and a song. Many years ago, I found among papers of a deceased friend, who certainly never had anything to do with the Society, and who passed all his life far from London, a song, headed "Song sung by the Mathematical Society in London, at a dinner given Mr. Fletcher,{{ref|771}} a solicitor, who had defended the Society gratis." Mr. Williams,{{ref|772}} the Assistant Secretary of the Astronomical Society, formerly Secretary of the Mathematical Society, remembered that the Society had had a solicitor named Fletcher among the members. Some years elapsed before it struck me that my old friend Benjamin Gompertz,{{ref|773}} who had long been a member, might have some recollection of the matter. The following is an extract of a letter from him (July 9, 1861):

"As to the Mathematical Society, of which I was a member when only 18 years of age, [Mr. G. was born in 1779], having been, contrary to the rules, elected under the age of 21. How I came to be a member of that Society—and continued so until it joined the Astronomical Society, and was then the President—was: I happened to pass a bookseller's small shop, of second-hand books, kept by a poor taylor, but a good mathematician, John Griffiths. I was very pleased to meet a mathematician, and I asked him if he would give me some lessons; and his reply was that I was more capable to teach him, but he belonged to a society of mathematicians, and he would introduce me. I accepted the offer, and I was elected, and had many scholars then to teach, as {{pagenum|379}} one of the rules was, if a member asked for information, and applied to any one who could give it, he was obliged to give it, or fine one penny. Though I might say much with respect to the Society which would be interesting, I will for the present reply only to your question. I well knew Mr. Fletcher, who was a very clever and very scientific person. He did, as solicitor, defend an action brought by an informer against the Society—I think for 5,000''l.''—for giving lectures to the public in philosophical subjects [i.e., for unlicensed public exhibition with money taken at the doors]. I think the price for admission was one shilling, and we used to have, if I rightly recollect, from two to three hundred visitors. Mr. Fletcher was successful in his defence, and we got out of our trouble. There was a collection made to reward his services, but he did not accept of any reward: and I think we gave him a dinner, as you state, and enjoyed ourselves; no doubt with astronomical songs and other songs; but my recollection does not enable me to say if the astronomical song was a drinking song. I think the anxiety caused by that action was the cause of some of the members' death. [They had, no doubt, broken the law in ignorance; and by the sum named, the informer must have been present, and sued for a penalty on every shilling he could prove to have been taken]."

I by no means guarantee that the whole song I proceed to give is what was sung at the dinner: I suspect, by the completeness of the chain, that augmentations have been made. My deceased friend was just the man to add some verses, or the addition may have been made before it came into his hands, or since his decease, for the scraps containing the verses passed through several hands before they came into mine. We may, however, be pretty sure that the original is substantially contained in what is given, and that the character is therefore preserved. I have had myself to repair damages every now and then, in the way of conjectural restoration of defects caused by ill-usage. {{pagenum|380}}

&nbsp;

===THE ASTRONOMER'S DRINKING SONG.===

:"Whoe'er would search the starry sky,
::Its secrets to divine, sir,
:Should take his glass—I mean, should try
::A glass or two of wine, sir!
:True virtue lies in golden mean,
::And man must wet his clay, sir;
:Join these two maxims, and 'tis seen
::He should drink his bottle a day, sir!

:"Old Archimedes, reverend sage!
::By trump of fame renowned, sir,
:Deep problems solved in every page,
::And the sphere's curved surface found,{{ref|774}} sir:
:Himself he would have far outshone,
::And borne a wider sway, sir,
:Had he our modern secret known,
::And drank a bottle a day, sir!

:"When Ptolemy,{{ref|775}} now long ago,
::Believed the earth stood still, sir,
:He never would have blundered so,
::Had he but drunk his fill, sir:
:He'd then have felt{{ref|776}} it circulate,
::And would have learnt to say, sir,
:The true way to investigate
::Is to drink your bottle a day, sir!

:"Copernicus,{{ref|777}} that learned wight,
::The glory of his nation,
:With draughts of wine refreshed his sight,
::And saw the earth's rotation;
:{{pagenum|381}}
:Each planet then its orb described,
::The moon got under way, sir;
:These truths from nature he imbibed
::For he drank his bottle a day, sir!

:"The noble{{ref|778}} Tycho placed the stars,
::Each in its due location;
:He lost his nose{{ref|779}} by spite of Mars,
::But that was no privation:
:Had he but lost his mouth, I grant
::He would have felt dismay, sir,
:Bless you! ''he'' knew what he should want
::To drink his bottle a day, sir!

:"Cold water makes no lucky hits;
::On mysteries the head runs:
:Small drink let Kepler{{ref|780}} time his wits
::On the regular polyhedrons:
:He took to wine, and it changed the chime,
::His genius swept away, sir,
:Through area varying{{ref|781}} as the time
::At the rate of a bottle a day, sir!

:"Poor Galileo,{{ref|782}} forced to rat
::Before the Inquisition,
:''E pur si muove''{{ref|783}} was the pat
::He gave them in addition:
:{{pagenum|382}}
:He meant, whate'er you think you prove,
::The earth must go its way, sirs;
:Spite of your teeth I'll make it move,
::For I'll drink my bottle a day, sirs!

:"Great Newton, who was never beat
::Whatever fools may think, sir;
:Though sometimes he forgot to eat,
::He never forgot to drink, sir:
:Descartes{{ref|784}} took nought but lemonade,
::To conquer him was play, sir;
:The first advance that Newton made
::Was to drink his bottle a day, sir!

:"D'Alembert,{{ref|785}} Euler,{{ref|786}} and Clairaut,{{ref|787}}
::Though they increased our store, sir,
:Much further had been seen to go
::Had they tippled a little more, sir!
:Lagrange{{ref|788}} gets mellow with Laplace,{{ref|789}}
::And both are wont to say, sir,
:The ''philosophe'' who's not an ass
::Will drink his bottle a day, sir!

:"Astronomers! what can avail
::Those who calumniate us;
:Experiment can never fail
::With such an apparatus:
:Let him who'd have his merits known
::Remember what I say, sir;
:Fair science shines on him alone
::Who drinks his bottle a day, sir!

:{{pagenum|383}}
:"How light we reck of those who mock
::By this we'll make to appear, sir,
:We'll dine by the sidereal{{ref|790}} clock
::For one more bottle a year, sir:
:But choose which pendulum you will,
::You'll never make your way, sir,
:Unless you drink—and drink your fill,—
::At least a bottle a day, sir!"

Old times are changed, old manners gone!

There is a new Mathematical Society,{{ref|791}} and I am, at this present writing (1866), its first President. We are very high in the newest developments, and bid fair to take a place among the scientific establishments. Benjamin Gompertz, who was President of the old Society when it expired, was the link between the old and new body: he was a member of ''ours'' at his death. But not a drop of liquor is seen at our meetings, except a decanter of water: all our heavy is a fermentation of symbols; and we do not draw it mild. There is no penny fine for reticence or occult science; and as to a song! not the ghost of a chance.

&nbsp;

1826. The time may have come when the original documents connected with the discovery of Neptune may be worth revising. The following are extracts from the ''Athenæum'' of October 3 and October 17:

&nbsp;

===LE VERRIER'S{{ref|792}} PLANET.===

We have received, at the last moment before making up for press, the following letter from Sir John Herschel,{{ref|793}} {{pagenum|384}} in reference to the matter referred to in the communication from Mr. Hind{{ref|794}} given below:

:::::"Collingwood, Oct. 1.

"In my address to the British Association assembled at Southampton, on the occasion of my resigning the chair to Sir R. Murchison,{{ref|795}} I stated, among the remarkable astronomical events of the last twelvemonth, that it had added a new planet to our list,—adding, 'it has done more,—it has given us the probable prospect of the discovery of another. We see it as Columbus saw America from the shores of Spain. Its movements have been felt, trembling along the far-reaching line of our analysis, with a certainty hardly inferior to that of ocular demonstration.'—These expressions are not reported in any of the papers which profess to give an account of the proceedings, but I appeal to all present whether they were not used.

"Give me leave to state my reasons for this confidence; and, in so doing, to call attention to some facts which deserve to be put on record in the history of this noble discovery. On July 12, 1842, the late illustrious astronomer, Bessel,{{ref|796}} honored me with a visit at my present residence. On the evening of that day, conversing on the great work of the planetary reductions undertaken by the Astronomer Royal{{ref|797}}—then in progress, and since published,{{ref|798}}—M. Bessel remarked that the motions of Uranus, as he had satisfied {{pagenum|385}} himself by careful examination of the recorded observations, could not be accounted for by the perturbations of the known planets; and that the deviations far exceeded any possible limits of error of observation. In reply to the question, Whether the deviations in question might not be due to the action of an unknown planet?—he stated that he considered it highly probable that such was the case,—being systematic, and such as might be produced by an exterior planet. I then inquired whether he had attempted, from the indications afforded by these perturbations, to discover the position of the unknown body,—in order that 'a hue and cry' might be raised for it. From his reply, the words of which I do not call to mind, I collected that he had not then gone into that inquiry; but proposed to do so, having now completed certain works which had occupied too much of his time. And, accordingly, in a letter which I received from him after his return to Königsberg, dated November 14, 1842, he says,—'In reference to our conversation at Collingwood, I ''announce'' to you (''melde'' ich Ihnen) that Uranus is not forgotten.' Doubtless, therefore, among his papers will be found some researches on the subject.

"The remarkable calculations of M. Le Verrier—which have pointed out, as now appears, nearly the true situation of the new planet, by resolving the inverse problem of the perturbations—if uncorroborated by repetition of the numerical calculations by another hand, or by independent investigation from another quarter, would hardly justify so strong an assurance as that conveyed by my expressions above alluded to. But it was known to me, at that time, (I will take the liberty to cite the Astronomer Royal as my authority) that a similar investigation had been independently entered into, and a conclusion as to the situation of the new planet very nearly coincident with M. Le Verrier's arrived at (in entire ignorance of his conclusions), by a young Cambridge mathematician, Mr. Adams;{{ref|799}}—who will, I hope, {{pagenum|386}} pardon this mention of his name (the matter being one of great historical moment),—and who will, doubtless, in his own good time and manner, place his calculations before the public.

:::::"J. F. W. HERSCHEL."

===''Discovery of Le Verrier's Planet.''===

Mr. Hind announces to the ''Times'' that he has received a letter from Dr. Brünnow, of the Royal Observatory at Berlin, giving the very important information that Le Verrier's planet was found by M. Galle, on the night of September 23. "In announcing this grand discovery," he says, "I think it better to copy Dr. Brünnow's{{ref|800}} letter."

&nbsp;

:::::"Berlin, Sept. 25.

"My dear Sir—M. Le Verrier's planet was discovered here the 23d of September, by M. Galle.{{ref|801}} It is a star of the 8th magnitude, but with a diameter of two or three seconds. Here are its places:

<table class="nobctr" summary="Locations of Neptune" title="Locations of Neptune">
<tr><td>&nbsp;</td><td>h.</td><td>m.</td><td>s.</td><td>&nbsp;</td><td align="center" colspan="4">R. A.</td><td>&nbsp;</td><td align="center" colspan="4">Declination.</td></tr>
<tr><td>Sept. 23,</td><td align="right">12</td><td align="right">0</td><td align="right">14.6 M.T.</td><td>&nbsp;</td><td align="right">328°</td><td align="right">19'</td><td align="right">16.0"</td><td>&nbsp;</td><td align="right">-13°</td><td align="right">24'</td><td align="right">8.2"</td></tr>
<tr><td>Sept. 24,</td><td align="right">8</td><td align="right">54</td><td align="right">40.9 M.T.</td><td>&nbsp;</td><td align="right">328°</td><td align="right">18'</td><td align="right">14.3"</td><td>&nbsp;</td><td align="right">-13°</td><td align="right">24'</td><td align="right">29.7"</td></tr>
</table>

The planet is now retrograde, its motion amounting daily to four seconds of time.

:"Yours most respectfully, {{small-caps|Brünnow}}."

"This discovery," Mr. Hind says, "may be justly considered one of the greatest triumphs of theoretical astronomy;" and he adds, in a postscript, that the planet was observed at Mr. Bishop's{{ref|802}} Observatory, in the Regent's Park, {{pagenum|387}} on Wednesday night, notwithstanding the moonlight and hazy sky. "It appears bright," he says, "and with a power of 320 I can see the disc. The following position is the result of instrumental comparisons with 33 Aquarii:

Sept. 30, at 8h. 16m. 21s. Greenwich mean time—

<table class="nob" summary="Location of Neptune" title="Location of Neptune">
<tr><td class="spac">Right ascension of planet</td><td>21h.</td><td>52m.</td><td>47.15s.</td></tr>
<tr><td class="spac">South declination</td><td>13°</td><td>27'</td><td>20"."</td></tr>
</table>

&nbsp;

===THE NEW PLANET.===

:::::"Cambridge Observatory, Oct. 15.

"The allusion made by Sir John Herschel, in his letter contained in the ''Athenæum'' of October 3, to the theoretical researches of Mr. Adams, respecting the newly-discovered planet, has induced me to request that you would make the following communication public. It is right that I should first say that I have Mr. Adams's permission to make the statements that follow, so far as they relate to his labors. I do not propose to enter into a detail of the steps by which Mr. Adams was led, by his spontaneous and independent researches, to a conclusion that a planet must exist more distant than Uranus. The matter is of too great historical moment not to receive a more formal record than it would be proper to give here. My immediate object is to show, while the attention of the scientific public is more particularly directed to the subject, that, with respect to this remarkable discovery, English astronomers may lay claim to some merit.

"Mr. Adams formed the resolution of trying, by calculation, to account for the anomalies in the motion of Uranus on the hypothesis of a more distant planet, when he was an undergraduate in this university, and when his exertions for the academical distinction, which he obtained in January 1843, left him no time for pursuing the research. In the course of that year, he arrived at an approximation to the position of the supposed planet; which, however, he did not consider to be worthy of confidence, on account of his not {{pagenum|388}} having employed a sufficient number of observations of Uranus. Accordingly, he requested my intervention to obtain for him the early Greenwich observations, then in course of reduction;—which the Astronomer Royal immediately supplied, in the kindest possible manner. This was in February, 1844. In September, 1845, Mr. Adams communicated to me values which he had obtained for the heliocentric longitude, excentricity of orbit, longitude of perihelion, and mass, of an assumed exterior planet,—deduced entirely from unaccounted-for perturbations of Uranus. The same results, somewhat corrected, he communicated, in October, to the Astronomer Royal. M. Le Verrier, in an investigation which was published in June of 1846, assigned very nearly the same heliocentric longitude for the probable position of the planet as Mr. Adams had arrived at, but gave no results respecting its mass and the form of its orbit. The coincidence as to position from two entirely independent investigations naturally inspired confidence; and the Astronomer Royal shortly after suggested the employing of the Northumberland telescope of this observatory in a systematic search after the hypothetical planet; recommending, at the same time, a definite plan of operations. I undertook to make the search,—and commenced observing on July 29. The observations were directed, in the first instance, to the part of the heavens which theory had pointed out as the most probable place of the planet; in selecting which I was guided by a paper drawn up for me by Mr. Adams. Not having hour xxi. of the Berlin star-maps—of the publication of which I was not aware—I had to proceed on the principle of comparison of observations made at intervals. On July 30, I went over a zone 9' broad, in such a manner as to include all stars to the eleventh magnitude. On August 4, I took a broader zone and recorded a place of the planet. My next observations were on August 12; when I met with a star of the eighth magnitude in the zone which I had gone over on July 30,—and which did not then {{pagenum|389}} contain this star. Of course, this was the planet;—the place of which was, thus, recorded a second time in four days of observing. A comparison of the observations of July 30 and August 12 would, according to the principle of search which I employed, have shown me the planet. I did not make the comparison till after the detection of it at Berlin—partly because I had an impression that a much more extensive search was required to give any probability of discovery—and partly from the press of other occupation. The planet, however, was ''secured'', and two positions of it recorded six weeks earlier here than in any other observatory,—and in a systematic search expressly undertaken for that purpose. I give now the positions of the planet on August 4 and August 12.

{| class="nob" {{ts|bracetable}} summary="Locations of Neptune" title="Locations of Neptune"
|-
|Greenwich mean time.
|-
|rowspan="2" valign="middle"|Aug. 4, 13h. 36m. 25s. ...
|rowspan="2" valign="middle" {{ts|brace}}}|{{brace3|100%|l}}
|R.A.
|21h.
|58m.
|14.70s
|-
|N.P.D.||102°||57'||32.2"
|-
|rowspan="2" valign="middle"|Aug. 12, 13h. 3m. 26s. ...
|rowspan="2" valign="middle" {{ts|vmi|brace}}|{{brace3|100%|l}}
|R.A.||21h.||57m.||26.13s.
|-
|N.P.D.||103°||2'||&nbsp;0.2"
|}

"From these places compared with recent observations Mr. Adams has obtained the following results:

<table class="nob" summary="Orbital elements of Neptune" title="Orbital elements of Neptune">
<tr><td>Distance of the planet from the sun ...</td><td align="right">30.05</td></tr>
<tr><td>Inclination of the orbit ...</td><td align="right">1° 45'</td></tr>
<tr><td>Longitude of the descending node ...</td><td align="right">309° 43'</td></tr>
<tr><td>Heliocentric longitude, Aug. 4 ...</td><td align="right">326° 39'</td></tr>
</table>

"The present distance from the sun is, therefore, thirty times the earth's mean distance;—which is somewhat less than the theory had indicated. The other elements of the orbit cannot be approximated to till the observations shall have been continued for a longer period.

"The part taken by Mr. Adams in the theoretical search after this planet will, perhaps, be considered to justify the suggesting of a name. With his consent, I mention ''Oceanus'' as one which may possibly receive the votes of astronomers.—I {{pagenum|390}} have authority to state that Mr. Adams's investigations will in a short time, be published in detail.

:::::"{{small-caps|J. Challis}}."{{ref|803}}

&nbsp;

===ASTRONOMICAL POLICE REPORT.===

"An ill-looking kind of a body, who declined to give any name, was brought before the Academy of Sciences, charged with having assaulted a gentleman of the name of Uranus in the public highway. The prosecutor was a youngish looking person, wrapped up in two or three great coats; and looked chillier than anything imaginable, except the prisoner,—whose teeth absolutely shook, all the time.

Policeman Le Verrier{{ref|804}} stated that he saw the prosecutor walking along the pavement,—and sometimes turning sideways, and sometimes running up to the railings and jerking about in a strange way. Calculated that somebody must be pulling his coat, or otherwise assaulting him. It was so dark that he could not see; but thought, if he watched the direction in which the next odd move was made, he might find out something. When the time came, he set Brünnow, a constable in another division of the same force, to watch where he told him; and Brünnow caught the prisoner lurking about in the very spot,—trying to look as if he was minding his own business. Had suspected for a long time that somebody was lurking about in the neighborhood. Brünnow was then called, and deposed to his catching the prisoner as described.

''M. Arago.''—Was the prosecutor sober?

''Le Verrier.''—Lord, yes, your worship; no man who had a drop in him ever looks so cold as he did.

''M. Arago.''—Did you see the assault?

''Le Verrier.''—I can't say I did; but I told Brünnow exactly how he'd be crouched down;—just as he was.

{{pagenum|391}}

''M. Arago (to Brünnow).''—Did ''you'' see the assault?

''Brünnow.''—No, your worship; but I caught the prisoner.

''M. Arago.''—How did you know there was any assault at all?

''Le Verrier.''—I reckoned it couldn't be otherwise, when I saw the prosecutor making those odd turns on the pavement.

''M. Arago.''—You reckon and you calculate! Why, you'll tell me, next, that you policemen may sit at home and find out all that's going on in the streets by arithmetic. Did you ever bring a case of this kind before me till now?

''Le Verrier.''—Why, you see, your worship, the police are growing cleverer and cleverer every day. We can't help it:—it grows upon us.

''M. Arago.''—You're getting too clever for me. What does the prosecutor know about the matter?

The prosecutor said, all he knew was that he was pulled behind by somebody several times. On being further examined, he said that he had seen the prisoner often, but did not know his name, nor how he got his living; but had understood he was called Neptune. He himself had paid rates and taxes a good many years now. Had a family of six,—two of whom got their own living.

The prisoner being called on for his defence, said that it was a quarrel. He had pushed the prosecutor—and the prosecutor had pushed him. They had known each other a long time, and were always quarreling;—he did not know why. It was their nature, he supposed. He further said, that the prosecutor had given a false account of himself;—that he went about under different names. Sometimes he was called Uranus, sometimes Herschel, and sometimes Georgium Sidus; and he had no character for regularity in the neighborhood. Indeed, he was sometimes not to be seen for a long time at once.

The prosecutor, on being asked, admitted, after a little hesitation, that he had pushed and pulled the prisoner too. {{pagenum|392}} In the altercation which followed, it was found very difficult to make out which began:—and the worthy magistrate seemed to think they must have begun together.

''M. Arago.''—Prisoner, have you any family?

The prisoner declined answering that question at present. He said he thought the police might as well reckon it out whether he had or not.

''M. Arago'' said he didn't much differ from that opinion.—He then addressed both prosecutor and prisoner; and told them that if they couldn't settle their differences without quarreling in the streets, he should certainly commit them both next time. In the meantime, he called upon both to enter into their own recognizances; and directed the police to have an eye upon both,—observing that the prisoner would be likely to want it a long time, and the prosecutor would be not a hair the worse for it."

&nbsp;

This quib was written by a person who was among the astronomers: and it illustrates the fact that Le Verrier had sole possession of the field until Mr. Challis's letter appeared. Sir John Herschel's previous communication should have paved the way: but the wonder of the discovery drove it out of many heads. There is an excellent account of the whole matter in Professor Grant's{{ref|805}} ''History of Physical Astronomy''. The squib scandalized some grave people, who wrote severe admonitions to the editor. There are formalists who spend much time in writing propriety to journals, to which they serve as foolometers. In a letter to the ''Athenæum'', speaking of the way in which people hawk fine terms for common things, I said that these people ought to have a new translation of the Bible, which should contain the verse "gentleman and lady, created He them." The editor was handsomely fired and brimstoned!

{{pagenum|393}}

&nbsp;

===A NEW THEORY OF TIDES.===

:A new theory of the tides: in which the errors of the usual theory are demonstrated; and proof shewn that the full moon is not the cause of a concomitant spring tide, but actually the cause of the neaps.... By Comm{{sup|r}}. Debenham,{{ref|806}} R.N. London, 1846, 8vo.

The author replied to a criticism in the ''Athenæum'', and I remember how, in a very few words, he showed that he had read nothing on the subject. The reviewer spoke of the forces of the planets (i.e., the Sun and Moon) on the ocean, on which the author remarks, "But N.B. the Sun is no planet, Mr. Critic." Had he read any of the actual investigations on the usual theory, he would have known that to this day the sun and moon continue to be called ''planets''—though the phrase is disappearing—in speaking of the tides; the sense, of course, being the old one, wandering bodies.

A large class of the paradoxers, when they meet with something which taken in their sense is absurd, do not take the trouble to find out the intended meaning, but walk off with the words laden with their own first construction. Such men are hardly fit to walk the streets without an interpreter. I was startled for a moment, at the time when a recent happy—and more recently happier—marriage occupied the public thoughts, by seeing in a haberdasher's window, in staring large letters, an unpunctuated sentence which read itself to me as "Princess Alexandra! collar and cuff!" It immediately occurred to me that had I been any one of some scores of my paradoxers, I should, no doubt, have proceeded to raise the mob against the unscrupulous person who dared to hint to a young bride such maleficent—or at least immellificent—conduct towards her new lord. But, as it was, certain material contexts in the shop window suggested a less {{pagenum|394}} savage explanation. A paradoxer should not stop at reading the advertisements of Newton or Laplace; he should learn to look at the stock of goods.

I think I must have an eye for double readings, when presented: though I never guess riddles. On the day on which I first walked into the ''Panizzi'' reading room{{ref|807}}—as it ought to be called—at the Museum, I began my circuit of the wall-shelves at the ladies' end: and perfectly coincided in the propriety of the Bibles and theological works being placed there. But the very first book I looked on the back of had, in flaming gold letters, the following inscription—"Blast the Antinomians!"{{ref|808}} If a line had been drawn below the first word, Dr. Blast's history of the Antinomians would not have been so fearfully misinterpreted. It seems that neither the binder nor the arranger of the room had caught my reading. The book was removed before the catalogue of books of reference was printed.

&nbsp;

===AN ASTRONOMICAL PARADOXER.===

:Two systems of astronomy: first, the Newtonian system, showing the rise and progress thereof, with a short historical account; the general theory with a variety of remarks thereon: second, the system in accordance with the Holy Scriptures, showing the rise and progress from Enoch, the seventh from Adam, the prophets, Moses, and others, in the first Testament; our Lord Jesus Christ, and his apostles, in the new or second Testament; Reeve and Muggleton, in the third and last Testament; with a variety of remarks thereon. By Isaac Frost.{{ref|809}} London, 1846, 4to.

{{pagenum|395}}

A very handsomely printed volume, with beautiful plates. Many readers who have heard of Muggletonians have never had any distinct idea of Lodowick Muggleton,{{ref|810}} the inspired tailor, (1608-1698) who about 1650 received his commission from heaven, wrote a Testament, founded a sect, and descended to posterity. Of Reeve{{ref|811}} less is usually said; according to Mr. Frost, he and Muggleton are the two "witnesses." I shall content myself with one specimen of Mr. Frost's science:

"I was once invited to hear read over 'Guthrie{{ref|812}} on Astronomy,' and when the reading was concluded I was asked my opinion thereon; when I said, 'Doctor, it appears to me that Sir I. Newton has only given two proofs in support of his theory of the earth revolving round the sun: all the rest is assertion without any proofs.'—'What are they?' inquired the Doctor.—'Well,' I said, 'they are, first, the power of {{pagenum|396}} attraction to keep the earth to the sun; the second is the power of repulsion, by virtue of the centrifugal motion of the earth: all the rest appears to me assertion without proof.' The Doctor considered a short time and then said, 'It certainly did appear so.' I said, 'Sir Isaac has certainly obtained the credit of completing the system, but really he has only half done his work.'—'How is that,' inquired my friend the Doctor. My reply was this: 'You will observe his system shows the earth traverses round the sun on an inclined plane; the consequence is, there are four powers required to make his system complete:

:1st. The power of ''attraction''.
:2ndly. The power of ''repulsion''.
:3rdly. The power of ''ascending'' the inclined plane.
:4thly. The power of ''descending'' the inclined plane.

You will thus easily see the ''four'' powers required, and Newton has only accounted for ''two''; the work is therefore only half done.' Upon due reflection the Doctor said, 'It certainly was necessary to have these ''four'' points cleared up before the system could be said to be complete.'"

&nbsp;

I have no doubt that Mr. Frost, and many others on my list, have really encountered doctors who could be puzzled by such stuff as this, or nearly as bad, among the votaries of existing systems, and have been encouraged thereby to print their objections. But justice requires me to say that from the words "power of repulsion by virtue of the centrifugal motion of the earth," Mr. Frost may be suspected of having something more like a notion of the much-mistaken term "centrifugal force" than many paradoxers of greater fame. The Muggletonian sect is not altogether friendless: over and above this handsome volume, the works of Reeve and Muggleton were printed, in 1832, in three quarto volumes. See ''Notes and Queries, 1st Series'', v, 80; 3d Series, iii, 303. {{pagenum|397}}

[The system laid down by Mr. Frost, though intended to be substantially that of Lodowick Muggleton, is not so vagarious. It is worthy of note how very different have been the fates of two contemporary paradoxers, Muggleton and George Fox.{{ref|813}} They were friends and associates,{{ref|814}} and commenced their careers about the same time, 1647-1650. The followers of Fox have made their sect an institution, and deserve to be called the pioneers of philanthropy. But though there must still be Muggletonians, since expensive books are published by men who take the name, no sect of that name is known to the world. Nevertheless, Fox and Muggleton are men of one type, developed by the same circumstances: it is for those who investigate such men to point out why their teachings have had fates so different. Macaulay says it was because Fox found followers of more sense than himself. True enough: but why did Fox find such followers and not Muggleton? The two were equally crazy, to all appearance: and the difference required must be sought in the doctrines themselves.

Fox was not a ''rational'' man: but the success of his sect and doctrines entitles him to a letter of alteration of the phrase which I am surprised has not become current. When Conduitt,{{ref|815}} the husband of Newton's half-niece, wrote a circular to Newton's friends, just after his death, inviting them to bear their parts in a proper biography, he said, "As Sir I. Newton was a ''national'' man, I think every one ought to contribute to a work intended to do him justice." Here is the very phrase which is often wanted to signify that {{pagenum|398}} celebrity which puts its mark, good or bad, on the national history, in a manner which cannot be asserted of many notorious or famous historical characters. Thus George Fox and Newton are both ''national'' men. Dr. Roget's{{ref|816}} ''Thesaurus'' gives more than fifty synonyms—''colleagues'' would be the better word—of "''celebrated''," any one of which might be applied, either in prose or poetry, to Newton or to his works, no one of which comes near to the meaning which Conduitt's adjective immediately suggests.

The truth is, that we are too ''monarchical'' to be ''national''. We have the Queen's army, the Queen's navy, the Queen's highway, the Queen's English, etc.; nothing is national except the ''debt''. That this remark is not new is an addition to its force; it has hardly been repeated since it was first made. It is some excuse that ''nation'' is not vernacular English: the ''country'' is our word, and ''country man'' is appropriated.]

&nbsp;

:Astronomical Aphorisms, or Theory of Nature; founded on the immutable basis of Meteoric Action. By P. Murphy,{{ref|817}} Esq. London, 1847, 12mo.

This is by the framer of the Weather Almanac, who appeals to that work as corroborative of his theory of planetary temperature, years after all the world knew by experience that this meteorological theory was just as good as the others.

{{pagenum|399}}

&nbsp;

:The conspiracy of the Bullionists as it affects the present system of the money laws. By Caleb Quotem. Birmingham, 1847, 8vo. (pp. 16).

This pamphlet is one of a class of which I know very little, in which the effects of the laws relating to this or that political bone of contention are imputed to deliberate conspiracy of one class to rob another of what the one knew ought to belong to the other. The success of such writers in believing what they have a bias to believe, would, if they knew themselves, make them think it equally likely that the inculpated classes might really believe what it is ''their'' interest to believe. The idea of a ''guilty'' understanding existing among fundholders, or landholders, or any holders, all the country over, and never detected except by bouncing pamphleteers, is a theory which should have been left for Cobbett{{ref|818}} to propose, and for Apella to believe.{{ref|819}}

[''August'', 1866. A pamphlet shows how to pay the National Debt. Advance paper to railways, etc., receivable in payment of taxes. The railways pay interest and principal in money, with which you pay your national debt, and redeem your notes. Twenty-five years of interest redeems the notes, and then the principal pays the debt. Notes to be kept up to value by penalties.]

&nbsp;

===THEISM INDEPENDENT OF REVELATION.===

:The Reasoner. No. 45. Edited by G.J. Holyoake.{{ref|820}} Price ''2d.'' Is there sufficient proof of the existence of God? 8vo. 1847.

This acorn of the holy oak was forwarded to me with a manuscript note, signed by the editor, on the part of the {{pagenum|400}} "London Society of Theological Utilitarians," who say, "they trust you may be induced to give this momentous subject your consideration." The supposition that a middle-aged person, known as a student of thought on more subjects than one, had that particular subject yet to begin, is a specimen of what I will call the ''assumption-trick'' of controversy, a habit which pervades all sides of all subjects. The tract is a proof of the good policy of letting opinions find their level, without any assistance from the Court of Queen's Bench. Twenty years earlier the thesis would have been positive, "There is sufficient proof of the non-existence of God," and bitter in its tone. As it stands, we have a moderate and respectful treatment—wrong only in making the opponent argue absurdly, as usually happens when one side invents the other—of a question in which a great many Christians have agreed with the atheist: that question being—Can the existence of God be proved independently of revelation? Many very religious persons answer this question in the negative, as well as Mr. Holyoake. And, this point being settled, all who agree in the negative separate into those who can endure scepticism, and those who cannot: the second class find their way to Christianity. This very number of ''The Reasoner'' announces the secession of one of its correspondents, and his adoption of the Christian faith. This would not have happened twenty years before: nor, had it happened, would it have been respectfully announced.

There are people who are very unfortunate in the expression of their meaning. Mr. Holyoake, in the name of the "London Society" etc., forwarded a pamphlet on the existence of God, and said that the Society trusted I "may be induced to give" the subject my "consideration." How could I know the Society was one person, who supposed I had arrived at a conclusion and wanted a "''guiding word''"? But so it seems it was: Mr. Holyoake, in the ''English {{pagenum|401}} Leader'' of October 15, 1864, and in a private letter to me, writes as follows:

"The gentleman who was the author of the argument, and who asked me to send it to Mr. De Morgan, never assumed that that gentleman had 'that particular subject to begin'—on the contrary, he supposed that one whom we all knew to be eminent as a thinker ''had'' come to a conclusion upon it, and would perhaps vouchsafe a guiding word to one who was, as yet, seeking the solution of the Great Problem of Theology. I told my friend that 'Mr. De Morgan was doubtless preoccupied, and that he must be content to wait. On some day of courtesy and leisure he might have the kindness to write.' Nor was I wrong—the answer appears in your pages at the lapse of seventeen years."

I suppose Mr. Holyoake's way of putting his request was the ''stylus curiæ'' of the Society. A worthy Quaker who was sued for debt in the King's Bench was horrified to find himself charged in the declaration with detaining his creditor's money by force and arms, contrary to the peace of our Lord the King, etc. It's only the ''stylus curiæ'', said a friend: I don't know ''curiæ'', said the Quaker, but he shouldn't style us peace-breakers.

The notion that the ''non''-existence of God can be ''proved'', has died out under the light of discussion: had the only lights shone from the pulpit and the prison, so great a step would never have been made. The question now is as above. The dictum that Christianity is "part and parcel of the law of the land" is also abrogated: at the same time, and the coincidence is not an accident, it is becoming somewhat nearer the truth that the law of the land is part and parcel of Christianity. It must also be noticed that ''Christianity'' was part and parcel of the articles of ''war''; and so was ''duelling''. Any officer speaking against religion was to be cashiered; and any officer receiving an affront without, in the last resort, attempting to kill his opponent, was also to be cashiered. Though somewhat of a book-hunter, I {{pagenum|402}} have never been able to ascertain the date of the collected remonstrances of the prelates in the House of Lords against this overt inculcation of murder, under the soft name of ''satisfaction'': it is neither in Watt,{{ref|821}} nor in Lowndes,{{ref|822}} nor in any edition of Brunet;{{ref|823}} and there is no copy in the British Museum. Was the collected edition really published?

[The publication of the above in the ''Athenæum'' has not produced reference to a single copy. The collected edition seems to be doubted. I have even met one or two persons who doubt the fact of the Bishops having remonstrated at all: but their doubt was founded on an absurd supposition, namely, that it was ''no business of theirs''; that it was not the business of the prelates of the church in union with the state to remonstrate against the Crown commanding murder! Some say that the edition was published, but under an irrelevant title, which prevented people from knowing what it was about. Such things have happened: for example, arranged extracts from Wellington's general orders, which would have attracted attention, fell dead under the title of "Principles of War." It is surmised that the book I am looking for also contains the protests of the Reverend bench against other things besides the Thou-shalt-do-murder of the Articles (of war), and is called "First Elements of Religion" or some similar title. Time clears up all things.]

{{rule}}

===Notes===

721 {{note|721}} See note [[../G#endnote_576|576]]. There was also a ''Theory of Parallels'' that differed from these, London, 1853, second edition 1856, third edition 1856.

722 {{note|722}} The work was written by Robert Chambers (1802-1871), the Edinburgh publisher, a friend of Scott and of many of his contemporaries in the literary field. He published the ''Vestiges of the Natural History of Creation'' in 1844, not 1840.

723 {{note|723}} Everett (1784-1872) was at that time a good Wesleyan, but was expelled from the ministry in 1849 for having written ''Wesleyan Takings'' and as under suspicion for having started the ''Fly Sheets'' in 1845. In 1857 he established the United Methodist Free Church.

724 {{note|724}} Smith was a Primitive Methodist preacher. He also wrote an ''Earnest Address to the Methodists'' (1841) and ''The Wealth Question'' (1840?).

725 {{note|725}} He wrote the ''Nouveau traité de Balistique'', Paris, 1837.

726 {{note|726}} Joseph Denison, known to fame only through De Morgan. See also [[#THE NUMBER OF THE BEAST.|THE NUMBER OF THE BEAST.]]

727 {{note|727}} The radical (1784?-1858), advocate of the founding of London university (1826), of medical reform (1827-1834), and of the repeal of the duties on newspapers and corn, and an ardent champion of penny postage.

728 {{note|728}} I. e., Roman Catholic Priest.

729 {{note|729}} Murphy (1806-1843) showed extraordinary powers in mathematics even before the age of thirteen. He became a fellow of Caius College, Cambridge, in 1829, dean in 1831, and examiner in mathematics in London University in 1838.

730 {{note|730}} See note [[../F#endnote_442|442]].

731 {{note|731}} Sir John Bowring (1792-1872), the linguist, writer, and traveler, member of many learned societies and a writer of high reputation in his time. His works were not, however, of genuine merit.

732 {{note|732}} Joseph Hume (1777-1855) served as a surgeon with the British army in India early in the nineteenth century. He returned to England in 1808 and entered parliament as a radical in 1812. He was much interested in all reform movements.

733 {{note|733}} Sir Robert Harry Inglis (1786-1855), a strong Tory, known for his numerous addresses in the House of Commons rather than for any real ability.

734 {{note|734}} Sir Robert Peel (1788-1850) began his parliamentary career in 1809 and was twice prime minister. He was prominent in most of the great reforms of his time.

735 {{note|735}} See note [[../H#endnote_627|627]].

736 {{note|736}} John Taylor (1781-1864) was a publisher, and published several pamphlets opposed to Peel's currency measures. De Morgan refers to his work on the Junius question. This was done early in his career, and resulted in ''A Discovery of the author of the Letters of Junius'' (1813), and ''The Identity of Junius with a distinguished living character established'' (1816), this being Sir Philip Francis.

737 {{note|737}} See note [[../H#endnote_665|665]].

738 {{note|738}} See [[#THE TWO OLD PARADOXES AGAIN.|THE TWO OLD PARADOXES AGAIN.]]

739 {{note|739}} See note [[../F#endnote_348|348]].

740 {{note|740}} Sir Nicholas Harris Nicolas (1799-1848) was a reformer in various lines,&mdash;the Record Commission, the Society of Antiquaries, and the British Museum,&mdash;and his work was not without good results.

741 {{note|741}} See note [[../C#endnote_98|98]].

742 {{note|742}} In the ''Companion to the Almanac'' for 1845 is a paper by Prof. De Morgan, "On the Ecclesiastical Calendar," the statements of which, so far as concerns the Gregorian Calendar, are taken direct from the work of Clavius, the principal agent in the arrangement of the reformed reckoning. This was followed, in the ''Companion to the Almanac'' for 1846, by a second paper, by the same author, headed "On the Earliest Printed Almanacs," much of which is written in direct supplement to the former article.&mdash;S.&nbsp;E. De Morgan.

743 {{note|743}} It may be necessary to remind some English readers that in Latin and its derived European languages, what we call Easter is called the passover (''pascha''). The Quartadecimans had the ''name'' on their side: a possession which often is, in this world, nine points of the law.&mdash;A. De M.

744 {{note|744}} Socrates Scholasticus was born at Constantinople c. 379, and died after 439. His ''Historia Ecclesiastica'' (in Greek) covers the period from Constantine the Great to about 439, and includes the Council of Nicæa. The work was printed in Paris 1544.

745 {{note|745}} Theodoretus or Theodoritus was born at Antioch and died about 457. He was one of the greatest divines of the fifth century, a man of learning, piety, and judicial mind, and a champion of freedom of opinion in all religious matters.

746 {{note|746}} He died in 417. He was a man of great energy and of high attainments.

747 {{note|747}} He died in 461, having reigned as pope for twenty-one years. It was he who induced Attila to spare Rome in 452.

748 {{note|748}} He succeeded Leo as pope in 461, and reigned for seven years.

749 {{note|749}} Victorinus or Victorius Marianus seems to have been born at Limoges. He was a mathematician and astronomer, and the cycle mentioned by De Morgan is one of 532 years, a combination of the Metonic cycle of 19 years with the solar cycle of 28 years. His canon was published at Antwerp in 1633 or 1634, ''De doctrina temporum sive commentarius in Victorii Aquitani et aliorum canones paschales''.

750 {{note|750}} He went to Rome about 497, and died there in 540. He wrote his ''Liber de paschate'' in 525, and it was in this work that the Christian era was first used for calendar purposes.

751 {{note|751}} See note [[../D#endnote_259|259]].

752 {{note|752}} Johannes de Sacrobosco (Holy wood), or John of Holywood. The name was often written, without regard to its etymology, Sacrobusto. He was educated at Oxford and taught in Paris until his death (1256). He did much to make the Hindu-Arabic numerals known to European scholars.

753 {{note|753}} See note [[../B#endnote_36|36]].

754 {{note|754}} See note [[../B#endnote_45|45]].

755 {{note|755}} The Julian year is a year of the Julian Calendar, in which there is leap year every fourth year. Its average length is therefore 365 days and a quarter.&mdash;A. De M.

756 {{note|756}} Ugo Buoncompagno (1502-1585) was elected pope in 1572.

757 {{note|757}} He was a Calabrian, and as early as 1552 was professor of medicine at Perugia. In 1576 his manuscript on the reform of the calendar was presented to the Roman Curia by his brother, Antonius. The manuscript was not printed and it has not been preserved.

758 {{note|758}} The title of this work, which is the authority on all points of the new Calendar, is ''Kalendarium Gregorianum Perpetuum. Cum Privilegio Summi Pontificis Et Aliorum Principum. Romæ, Ex Officina Dominici Basæ. MDLXXXII. Cum Licentia Superiorum'' (quarto, pp. 60).&mdash;A. De M.

759 {{note|759}} ''Manuels-Roret. Théorie du Calendrier et collection de tous les Calendriers des Années passées et futures''.... Par L.&nbsp;B. Franc&oelig;ur,... Paris, à la librairie encyclopédique de Roret, rue Hautefeuille, 10 bis. 1842. (12mo.) In this valuable manual, the 35 possible almanacs are given at length, with such preliminary tables as will enable any one to find, by mere inspection, which almanac he is to choose for any year, whether of old or new style. [1866. I may now refer to my own ''Book of Almanacs'', for the same purpose].&mdash;A. De M.

Louis Benjamin Franc&oelig;ur (1773-1849), after holding positions in the Ecole polytechnique (1804) and the Lycée Charlemagne (1805), became professor of higher algebra in the University of Paris (1809). His ''Cours complet des mathématiques pures'' was well received, and he also wrote on mechanics, astronomy, and geodesy.

760 {{note|760}} Albertus Pighius, or Albert Pigghe, was born at Kempen c. 1490 and died at Utrecht in 1542. He was a mathematician and a firm defender of the faith, asserting the supremacy of the Pope and attacking both Luther and Calvin. He spent some time in Rome. His greatest work was his ''Hierarchiæ ecclesiasticæ assertio'' (1538).

761 {{note|761}} This was A. F. Vogel. The work was his translation from the German edition which appeared at Leipsic the same year, ''Entdeckung einer numerischen General-Auflösung aller höheren endlichen Gleichungen von jeder beliebigen algebraischen und transcendenten Form''.

762 {{note|762}} The latest edition of Burnside and Panton's ''Theory of Equations'' has this brief summary of the present status of the problem: "Demonstrations have been given by Abel and Wantzel (see Serret's ''Cours d'Algèbre Supérieure'', Art. 516) of the impossibility of resolving algebraically equations unrestricted in form, of a degree higher than the fourth. A transcendental solution, however, of the quintic has been given by M. Hermite, in a form involving elliptic integrals."

763 {{note|763}} There was a second edition of this work in 1846. The author's ''Astronomy Simplified'' was published in 1838, and the ''Thoughts on Physical Astronomy'' in 1840, with a second edition in 1842.

764 {{note|764}} This was ''The Science of the Weather, by several authors... edited by B.'', Glasgow, 1867.

765 {{note|765}} This was Y. Ramachandra, son of Sundara L&#x101;la. He was a teacher of science in Delhi College, and the work to which De Morgan refers is ''A Treatise on problems of Maxima and Minima solved by Algebra'', which appeared at Calcutta in 1850. De Morgan's edition was published at London nine years later.

766 {{note|766}} Abraham de Moivre (1667-1754), French refugee in London, poor, studying under difficulties, was a man with tastes in some respects like those of De Morgan. For one thing, he was a lover of books, and he had a good deal of interest in the theory of probabilities to which De Morgan also gave much thought. His introduction of imaginary quantities into trigonometry was an event of importance in the history of mathematics, and the theorem that bears his name, <math>\scriptstyle (\cos\phi + i\sin\phi)^n = \cos n\phi + i\sin n\phi</math>, is one of the most important ones in all analysis.

767 {{note|767}} John Dolland (1706-1761), the silk weaver who became the greatest maker of optical instruments in his time.

768 {{note|768}} Thomas Simpson (1710-1761), also a weaver, taking his leisure from his loom at Spitalfields to teach mathematics. His ''New Treatise on Fluxions'' (1737) was written only two years after he began working in London, and six years later he was appointed professor of mathematics at Woolwich. He wrote many works on mathematics and Simpson's Formulas for computing trigonometric tables are still given in the text-books.

769 {{note|769}} Nicholas Saunderson (1682-1739), the blind mathematician. He lost his eyesight through smallpox when only a year old. At the age of 25 he began lecturing at Cambridge on the principles of the Newtonian philosophy. His ''Algebra'', in two large volumes, was long the standard treatise on the subject.

770 {{note|770}} He was not in the class with the others mentioned.

771 {{note|771}} Not known in the literature of mathematics.

772 {{note|772}} Probably J. Butler Williams whose ''Practical Geodesy'' appeared in 1842, with a third edition in 1855.

773 {{note|773}} Benjamin Gompertz (1779-1865) was debarred as a Jew from a university education. He studied mathematics privately and became president of the Mathematical Society. De Morgan knew him professionally through the fact that he was prominent in actuarial work.

774 {{note|774}} Referring to the contributions of Archimedes (287-212 B.C.) to the mensuration of the sphere.

775 {{note|775}} The famous Alexandrian astronomer (c. 87-c. 165 A.D.), author of the ''Almagest'', a treatise founded on the works of Hipparchus.

776 {{note|776}} Dr. Whewell, when I communicated this song to him, started the opinion, which I had before him, that this was a very good idea, of which too little was made.&mdash;A. De M.

777 {{note|777}} See note [[../C#endnote_117|117]].

778 {{note|778}} The common epithet of rank: ''nobilis Tycho'', as he was a nobleman. The writer had been at history.&mdash;A. De M.

See note [[../C#endnote_117|117]].

779 {{note|779}} He lost it in a duel, with Manderupius Pasbergius. A contemporary, T.&nbsp;B. Laurus, insinuates that they fought to settle which was the best mathematician! This seems odd, but it must be remembered they fought in the dark, "''in tenebris densis''"; and it is a nice problem to shave off a nose in the dark, without any other harm.&mdash;A. De M.

Was this T. B. Laurus Joannes Baptista Laurus or Giovanni Battista Lauro (1581-1621), the poet and writer?

780 {{note|780}} See note [[../C#endnote_117|117]].

781 {{note|781}} Referring to Kepler's celebrated law of planetary motion. He had previously wasted his time on analogies between the planetary orbits and the polyhedrons.&mdash;A. De M.

782 {{note|782}} See note [[../C#endnote_117|117]].

783 {{note|783}} "It does move though."

784 {{note|784}} As great a lie as ever was told: but in 1800 a compliment to Newton without a fling at Descartes would have been held a lopsided structure.&mdash;A. De M.

785 {{note|785}} Jean-le-Rond D'Alembert (1717-1783), the foundling who was left on the steps of Jean-le-Rond in Paris, and who became one of the greatest mathematical physicists and astronomers of his century.

786 {{note|786}} Leonhard Euler (1707-1783), friend of the Bernoullis, the greatest of Swiss mathematicians, prominent in the theory of numbers, and known for discoveries in all lines of mathematics as then studied.

787 {{note|787}} See notes See note [[../G#endnote_478|478]], [[../G#endnote_479|479]].

788 {{note|788}} See note [[../H#endnote_621|621]].

789 {{note|789}} See note [[../G#endnote_584|584]].

790 {{note|790}} The ''siderial'' day is about four minutes short of the solar; there are 366 sidereal days in the year.&mdash;A. De M.

791 {{note|791}} The founding of the London Mathematical Society is discussed by Mrs. De Morgan in her ''Memoir'' (p. 281). The idea came from a conversation between her brilliant son, George Campbell De Morgan, and his friend Arthur Cowper Ranyard in 1864. The meeting of organization was held on Nov. 7, 1864, with Professor De Morgan in the chair, and the first regular meeting on January 16, 1865.

792 {{note|792}} See note [[../B#endnote_33|33]].

793 {{note|793}} See note [[../C#endnote_119|119]].

794 {{note|794}} John Russell Hind (b. 1823), the astronomer. Between 1847 and 1854 he discovered ten planetoids.

795 {{note|795}} Sir Roderick Impey Murchison (1792-1871), the great geologist. He was knighted in 1846 and devoted the latter part of his life to the work of the Royal Geographical Society and to the geology of Scotland.

796 {{note|796}} Friedrich Wilhelm Bessel (1784-1846), the astronomer and physicist. He was professor of astronomy at Königsberg.

797 {{note|797}} This was the ''Reduction of the Observations of Planets made ... from 1750 to 1830: computed ... under the superintendence of George Biddell Airy'' (1848). See note 129, page 85.

798 {{note|798}} The expense of this magnificent work was defrayed by Government grants, obtained, at the instance of the British Association, in 1833&mdash;A. De M.

799 {{note|799}} See note [[../B#endnote_32|32]].

800 {{note|800}} Franz Friedrich Ernst Brünnow (1821-1891) was at that time or shortly before director of the observatory at Dusseldorf. He then went to Berlin and thence (1854) to Ann Arbor, Michigan. He then went to Dublin and finally became Royal Astronomer of Ireland.

801 {{note|801}} Johann Gottfried Galle (1812-1910), at that time connected with the Berlin observatory, and later professor of astronomy at Breslau.

802 {{note|802}} George Bishop (1785-1861), in whose observatory in Regent's Park important observations were made by Dawes, Hind, and Marth.

803 {{note|803}} James Challis (1803-1882), director of the Cambridge observatory, and successor of Airy as Plumian professor of astronomy.

804 {{note|804}} On Leverrier and Arago see note [[../B#endnote_33|33]] and note [[../G#endnote_561|561]].

805 {{note|805}} Robert Grant's (1814-1892) ''History of Physical Astronomy from the Earliest Ages to the Middle of the Nineteenth Century'' appeared in 1852. He was professor of astronomy and director of the observatory at Glasgow.

806 {{note|806}} John Debenham was more interested in religion than in astronomy. He wrote ''The Strait Gate; or, the true scripture doctrine of salvation clearly explained'', London, 1843, and ''Tractatus de magis et Bethlehemæ stella et Christi in deserto tentatione'', privately printed at London in 1845.

807 {{note|807}} More properly the Sydney Smirke reading room, since it was built from his designs.

808 {{note|808}} The Antinomians were followers of Johannes Agricola (1494-1566). They believed that Christians as such were released from all obligations to the Old Testament. Some went so far as to assert that, since all Christians were sanctified, they could not lose this sanctity even though they disobeyed God. The sect was prominent in England in the seventeenth century, and was transferred to New England. Here it suffered a check in the condemnation of Mrs. Ann Hutchinson (1636) by the Newton Synod.

809 {{note|809}} Aside from this work and his publications on Reeve and Muggleton he wrote nothing. With Joseph Frost he published ''A list'' ''of Books and general index to J. Reeve and L. Muggleton's works'' (1846), ''Divine Songs of the Muggletonians'' (1829), and the work mentioned on page 396. ''The works of J. Reeve and L. Muggleton'' (1832).

810 {{note|810}} About 1650 he and his cousin John Reeve (1608-1658) began to have visions. As part of their creed they taught that astronomy was opposed by the Bible. They asserted that the sun moves about the earth, and Reeve figured out that heaven was exactly six miles away. Both Muggleton and Reeve were imprisoned for their unitarian views. Muggleton wrote a ''Transcendant Spirituall Treatise'' (1652). I have before me ''A true Interpretation of All the Chief Texts ... of the whole Book of the Revelation of St. John.... By Lodowick Muggleton, one of the two last Commissioned Witnesses & Prophets of the onely high, immortal, glorious God, Christ Jesus'' (1665), in which the interpretation of the "number of the beast" occupies four pages without arriving anywhere.

811 {{note|811}} In 1652 he was, in a vision, named as the Lord's "last messenger," with Muggleton as his "mouth," and died six years later, probably of nervous tension resulting from his divine "illumination." He was the more spiritual of the two.

812 {{note|812}} William Guthrie (1708-1770) was a historian and political writer. His ''History of England'' (1744-1751) was the first attempt to base history on parliamentary records. He also wrote a ''General History of Scotland'' in 10 volumes (1767). The work to which Frost refers is the ''Geographical, Historical, and Commercial Grammar'' (1770) which contained an astronomical part by J. Ferguson. By 1827 it had passed through 24 editions.

813 {{note|813}} George Fox (1624-1691), founder of the Society of Friends; a mystic and a disciple of Boehme. He was eight times imprisoned for heresy.

814 {{note|814}} If they were friends they were literary antagonists, for Muggleton wrote against Fox ''The Neck of the Quakers Broken'' (1663), and Fox replied in 1667. Muggleton also wrote ''A Looking Glass for George Fox''.

815 {{note|815}} John Conduitt (1688-1737), who married (1717) Newton's half niece, Mrs. Katherine Barton. See note  [[../E#endnote_284|284]].

816 {{note|816}} Probably Peter Mark Roget's (1779-1869) ''Thesaurus of English Words'' (1852) is not much used at present, but it went through 28 editions in his lifetime. Few who use the valuable work are aware that Roget was a professor of physiology at the Royal Institution (London), that he achieved his title of F. R. S. because of his work in perfecting the slide rule, and that he followed Sir John Herschel as secretary of the Royal Society.

817 {{note|817}} See note [[../H#endnote_703|703]]. This work went into a second edition in the year of its first publication.

818 {{note|818}} See note [[../F#endnote_398|398]].

819 {{note|819}} See note [[../G#endnote_528|528]].

820 {{note|820}} George Jacob Holyoake (1817-1906) entered into a controversial life at an early age. In 1841 he was imprisoned for six months for blasphemy. He founded and edited ''The Reasoner'' (Vols. 1-26, 1846-1861). In his later life he did much to promote cooperation among the working class.

821 {{note|821}} See note [[../C#endnote_176|176]].

822 {{note|822}} William Thomas Lowndes (1798-1843), whose ''Bibliographer's Manual of English Literature'', 4 vols., London, 1834 (also 1857-1864, and 1869) is a classic in its line.

823 {{note|823}} Jacques Charles Brunet (1780-1867), the author of the great French bibliography, the ''Manuel du Libraire'' (1810).