Chapter III: Part 3

"1853. William Shanks,[136] _Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Tables_, London, 1853. (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing. For instance, one step is the calculation of the reciprocal of 601.5^{601}; and the result is given. The number of pages required to describe these results is 87. Mr. Shanks has also thrown off, as chips or splinters, the values of the base of Napier's logarithms, and of its logarithms of 2, 3, 5, 10, to 137 decimals; and the value of the modulus .4342 ... to 136 decimals: with the 13th, 25th, 37th ... up to the 721st powers of 2. These tremendous stretches of calculation--at least we so call them in our day--are useful in several respects; they prove more than {64} the capacity of this or that computer for labor and accuracy; they show that there is in the community an increase of skill and courage. We say in the community: we fully believe that the unequalled turnip which every now and then appears in the newspapers is a sufficient presumption that the average turnip is growing bigger, and the whole crop heavier. All who know the history of the quadrature are aware that the several increases of numbers of decimals to which [pi] has been carried have been indications of a general increase in the power to calculate, and in courage to face the labor. Here is a comparison of two different times. In the day of Cocker,[137] the pupil was directed to perform a common subtraction with a voice-accompaniment of this kind: '7 from 4 I cannot, but add 10, 7 from 14 remains 7, set down 7 and carry 1; 8 and 1 which I carry is 9, 9 from 2 I cannot, etc.' We have before us the announcement of the following _table_, undated, as open to inspection at the Crystal Palace, Sydenham, in two diagrams of 7 ft. 2 in, by 6 ft. 6 in.: 'The figure 9 involved into the 912th power, and antecedent powers or involutions, containing upwards of 73,000 figures. Also, the proofs of the above, containing upwards of 146,000 figures. By Samuel Fancourt, of Mincing Lane, London, and completed by him in the year 1837, at the age of sixteen. N.B. The whole operation performed by simple arithmetic.' The young operator calculated by successive squaring the 2d, 4th, 8th, etc., powers up to the 512th, with proof by division. But 511 multiplications by 9, in the short (or 10-1) way, would have been much easier. The 2d, 32d, 64th, 128th, 256th, and 512th powers are given at the back of the announcement. The powers of 2 have been calculated for many purposes. In Vol. II of his _Magia Universalis Naturae et Artis_, Herbipoli, 1658, 4to, the Jesuit Gaspar Schott[138] having discovered, on some grounds of theological {65} magic, that the degrees of grace of the Virgin Mary were in number the 256th power of 2, calculated that number. Whether or no his number correctly represented the result he announced, he certainly calculated it rightly, as we find by comparison with Mr. Shanks."

There is a point about Mr. Shanks's 608 figures of the value of [pi] which attracts attention, perhaps without deserving it. It might be expected that, in so many figures, the nine digits and the cipher would occur each about the same number of times; that is, each about 61 times. But the fact stands thus: 3 occurs 68 times; 9 and 2 occur 67 times each; 4 occurs 64 times; 1 and 6 occur 62 times each; 0 occurs 60 times; 8 occurs 58 times; 5 occurs 56 times; and 7 occurs only 44 times. Now, if all the digits were equally likely, and 608 drawings were made, it is 45 to 1 against the number of sevens being as distant from the probable average (say 61) as 44 on one side or 78 on the other. There must be some reason why the number 7 is thus deprived of its fair share in the structure. Here is a field of speculation in which two branches of inquirers might unite. There is but one number which is treated with an unfairness which is incredible as an accident; and that number is the mystic number _seven_! If the cyclometers and the apocalyptics would lay their heads together until they come to a unanimous verdict on this phenomenon, and would publish nothing until they are of one mind, they would earn the gratitude of their race.--I was wrong: it is the Pyramid-speculator who should have been appealed to. A correspondent of my friend Prof. Piazzi Smyth[139] notices that 3 is the number of most frequency, and that 3-1/7 is the nearest approximation to it in simple digits. Professor Smyth himself, whose word on Egypt is paradox of a very high order, backed by a great quantity of useful labor, the results which will be made available by those who do not receive {66} the paradoxes, is inclined to see confirmation for some of his theory in these phenomena.

CURIOUS CALCULATIONS.

These paradoxes of calculation sometimes appear as illustrations of the value of a new method. In 1863, Mr. G. Suffield,[140] M.A., and Mr. J. R. Lunn,[141] M.A., of Clare College and of St. John's College, Cambridge, published the whole quotient of 10000 ... divided by 7699, throughout the whole of one of the recurring periods, having 7698 digits. This was done in illustration of Mr. Suffield's method of _Synthetic division_.

Another instance of computation carried to paradoxical length, in order to illustrate a method, is the solution of x^3 - 2x = 5, the example given of Newton's method, on which all improvements have been tested. In 1831, Fourier's[142] posthumous work on equations showed 33 figures of solution, got with enormous labor. Thinking this a good opportunity to illustrate the superiority of the method of W. G. Horner,[143] not yet known in France, and not much known in {67} England, I proposed to one of my classes, in 1841, to beat Fourier on this point, as a Christmas exercise. I received several answers, agreeing with each other, to 50 places of decimals. In 1848, I repeated the proposal, requesting that 50 places might be exceeded: I obtained answers of 75, 65, 63, 58, 57, and 52 places. But one answer, by Mr. W. Harris Johnston,[144] of Dundalk, and of the Excise Office, went to 101 decimal places. To test the accuracy of this, I requested Mr. Johnston to undertake another equation, connected with the former one in a way which I did not explain. His solution verified the former one, but he was unable to see the connection, even when his result was obtained. My reader may be as much at a loss: the two solutions are:

2.0945514815423265...
9.0544851845767340...

The results are published in the _Mathematician_, Vol. III, p. 290. In 1851, another pupil of mine, Mr. J. Power Hicks,[145] carried the result to 152 decimal places, without knowing what Mr. Johnston had done. The result is in the _English Cyclopaedia_, article INVOLUTION AND EVOLUTION.

I remark that when I write the initial of a Christian name, the most usual name of that initial is understood. I never saw the name of W. G. Horner written at length, until I applied to a relative of his, who told me that he was, as I supposed, Wm. _George_, but that he was named after a relative of that _surname_.

The square root of 2, to 110 decimal places, was given {68} me in 1852 by my pupil, Mr. William Henry Colvill, now (1867) Civil Surgeon at Baghdad. It was

1.4142135623730950488016887242096980785696
7187537694807317667973799073247846210703
885038753432764157273501384623

Mr. James Steel[146] of Birkenhead verified this by actual multiplication, and produced

2 - 2580413 / 10^{117}

as the square.

Calcolo decidozzinale del Barone Silvio Ferrari. Turin, 1854, 4to.

This is a serious proposal to alter our numeral system and to count by twelves. Thus 10 would be twelve, 11 thirteen, etc., two new symbols being invented for ten and eleven. The names of numbers must of course be changed. There are persons who think such changes practicable. I thought this proposal absurd when I first saw it, and I think so still:[147] but the one I shall presently describe beats it so completely in that point, that I have not a smile left for this one.

ON COMETS.

The successful and therefore probably true theory of Comets. London,
1854. (4pp. duodecimo.)

The author is the late Mr. Peter Legh,[148] of Norbury Booths Hall, Knutsford, who published for eight or ten {69} years the _Ombrological Almanac_, a work of asserted discovery in meteorology. The theory of comets is that the joint attraction of the new moon and several planets in the direction of the sun, draws off the gases from the earth, and forms these cometic meteors. But how these meteors come to describe orbits round the sun, and to become capable of having their returns predicted, is not explained.

A NEW PHASE OF MORMONISM.

The Mormon, New York, Saturday, Oct. 27, 1855.

A newspaper headed by a grand picture of starred and striped banners, beehive, and eagle surmounting it. A scroll on each side: on the left, "Mormon creed. Mind your own business. Brigham Young;"[149] on the right, "Given by inspiration of God. Joseph Smith."[150] A leading article on the discoveries of Prof. Orson Pratt[151] says, "Mormonism has long taken the lead in religion: it will soon be in the van both in science and politics." At the beginning of the paper is Professor Pratt's "Law of Planetary Rotation." The cube roots of the densities of the planets are as the square roots of their periods of rotation. The squares of the cube roots of the masses divided by the squares of the diameters are as the periods of rotation. Arithmetical verification attempted, and the whole very modestly stated {70} and commented on. Dated G. S. L. City, Utah Ter., Aug. 1, 1855. If the creed, as above, be correctly given, no wonder the Mormonites are in such bad odor.

MATHEMATICAL ILLUSTRATIONS OF DOCTRINE.

The two estates; or both worlds mathematically considered. London,
1855, small (pp. 16).

The author has published mathematical works with his name. The present tract is intended to illustrate mathematically a point which may be guessed from the title. But the symbols do very little in the way of illustration: thus, x being the _present value_ of the future estate (eternal happiness), and a of all that this world can give, the author impresses it on the mathematician that, x being infinitely greater than a, x + a = x, so that a need not be considered. This will not act much more powerfully on a mathematician by virtue of the symbols than if those same symbols had been dispensed with: even though, as the author adds, "It was this method of neglecting infinitely small quantities that Sir Isaac Newton was indebted to for his greatest discoveries."

There has been a moderate quantity of well-meant attempt to enforce, sometimes motive, sometimes doctrine, by arguments drawn from mathematics, the proponents being persons unskilled in that science for the most part. The ground is very dangerous: for the illustration often turns the other way with greater power, in a manner which requires only a little more knowledge to see. I have, in my life, heard from the pulpit or read, at least a dozen times, that all sin is infinitely great, proved as follows. The greater the being, the greater the sin of any offence against him: therefore the offence committed against an infinite being is infinitely great. Now the mathematician, of which the proposers of this argument are not aware, is perfectly familiar with quantities which increase together, and never cease increasing, but so that one of them remains finite when {71} the other becomes infinite. In fact, the argument is a perfect _non sequitur_.[152] Those who propose it have in their minds, though in a cloudy and indefinite form, the idea of the increase of guilt being _proportionate_ to the increase of greatness in the being offended. But this it would never do to state: for by such statement not only would the argument lose all that it has of the picturesque, but the asserted premise would have no strong air of exact truth. How could any one undertake to appeal to conscience to declare that an offence against a being 4-7/10 times as great as another is exactly, no more and no less, 4-7/10 times as great an offence against the other?

The infinite character of the offence against an infinite being is laid down in Dryden's _Religio Laici_,[153] and is, no doubt, an old argument:

"For, granting we have sinned, and that th' offence
Of man is made against Omnipotence,
Some price that bears proportion must be paid,
And infinite with infinite be weighed.
See then the Deist lost; remorse for vice
Not paid; or, paid, inadequate in price."

Dryden, in the words "bears proportion" is in verse more accurate than most of the recent repeaters in prose. And this is not the only case of the kind in his argumentative poetry.

My old friend, the late Dr. Olinthus Gregory,[154] who was a sound and learned mathematician, adopted this dangerous kind of illustration in his _Letters on the Christian Religion_. {72} He argued, by parallel, from what he supposed to be the necessarily mysterious nature of the _impossible_ quantity of algebra to the necessarily mysterious nature of certain doctrines of his system of Christianity. But all the difficulty and mystery of the impossible quantity is now cleared away by the advance of algebraical thought: and yet Dr. Gregory's book continues to be sold, and no doubt the illustration is still accepted as appropriate.

The mode of argument used by the author of the tract above named has a striking defect. He talks of reducing this world and the next to "present value," as an actuary does with successive lives or next presentations. Does value make interest? and if not, why? And if it do, then the present value of an eternity is _not_ infinitely great. Who is ignorant that a perpetual annuity at five per cent is worth only twenty years' purchase? This point ought to be discussed by a person who treats heaven as a deferred perpetual annuity. I do not ask him to do so, and would rather he did not; but if he _will_ do it, he must either deal with the question of discount, or be asked the reason why.

When a very young man, I was frequently exhorted to one or another view of religion by pastors and others who thought that a mathematical argument would be irresistible. And I heard the following more than once, and have since seen it in print, I forget where. Since eternal happiness belonged to the particular views in question, a benefit infinitely great, then, even if the probability of their arguments were small, or even infinitely small, yet the product of the chance and benefit, according to the usual rule, might give a result which no one ought in prudence to pass over. They did not see that this applied to all systems as well as their own. I take this argument to be the most perverse of all the perversions I have heard or read on the subject: there is some high authority for it, whom I forget.

The moral of all this is, that such things as the preceding should be kept out of the way of those who are not {73} mathematicians, because they do not understand the argument; and of those who are, because they do.

[The high authority referred to above is Pascal, an early cultivator of mathematical probability, and obviously too much enamoured of his new pursuit. But he conceives himself bound to wager on one side or the other. To the argument (_Pensees_, ch. 7)[155] that "le juste est de ne point parier," he answers, "Oui: mais il faut parier: vous etes embarque; et ne parier point que Dieu est, c'est parier qu'il n'est pas."[156] Leaving Pascal's argument to make its way with a person who, _being a sceptic_, is yet positive that the issue is salvation or perdition, if a God there be,--for the case as put by Pascal requires this,--I shall merely observe that a person who elects to believe in God, as the best chance of gain, is not one who, according to Pascal's creed, or any other worth naming, will really secure that gain. I wonder whether Pascal's curious imagination ever presented to him in sleep his convert, in the future state, shaken out of a red-hot dice-box upon a red-hot hazard-table, as perhaps he might have been, if Dante had been the later of the two. The original idea is due to the elder Arnobius,[157] who, as cited by Bayle,[158] speaks thus:

"Sed et ipse [Christus] quae pollicetur, non probat. Ita est. Nulla enim, ut dixi, futurorum potest existere comprobatio. Cum ergo haec sit conditio futurorum, ut teneri et comprehendi nullius possint anticipationis attactu; nonne {74} purior ratio est, ex duobus incertis, et in ambigua expectatione pendentibus, id potius credere, quod aliquas spes ferat, quam omnino quod nullas? In illo enim periculi nihil est, si quod dicitur imminere, cassum fiat et vacuum: in hoc damnum est maximum, id est salutis amissio, si cum tempus advenerit aperiatur non fuisse mendacium."[159]

Really Arnobius seems to have got as much out of the notion, in the third century, as if he had been fourteen centuries later, with the arithmetic of chances to help him.]

NOVUM ORGANUM MORALIUM.

The Sentinel, vol. ix. no. 27. London, Saturday, May 26, 1855.

This is the first London number of an Irish paper, Protestant in politics. It opens with "Suggestions on the subject of a _Novum Organum Moralium_," which is the application of algebra and the differential calculus to morals, socials, and politics. There is also a leading article on the subject, and some applications in notes to other articles. A separate publication was afterwards made, with the addition of a long Preface; the author being a clergyman who I presume must have been the editor of the _Sentinel_.

Suggestions as to the employment of a _Novum Organum Moralium_. Or,
thoughts on the nature of the Differential Calculus, and on the
application of its principles to metaphysics, with a view to the
attainment of demonstration and certainty in moral, {75} political and
ecclesiastical affairs. By Tresham Dames Gregg,[160] Chaplain of St.
Mary's, within the church of St. Nicholas intra muros, Dublin. London,
1859, 8vo. (pp. xl + 32).

I have a personal interest in this system, as will appear from the following extract from the newspaper:

"We were subsequently referred to De Morgan's _Formal Logic_ and
Boole's _Laws of Thought_[161] both very elaborate works, and greatly
in the direction taken by ourselves. That the writers amazingly surpass
us in learning we most willingly admit, but we venture to pronounce of
both their learned treatises, that they deal with the subject in a mode
that is scholastic to an excess.... That their works have been for a
considerable space of time before the world and effected nothing, would
argue that they have overlooked the vital nature of the theme.... On
the whole, the writings of De Morgan and Boole go to the full
justification of our principle without in any wise so trenching upon
our ground as to render us open to reproach in claiming our Calculus as
a great discovery.... But we renounce any paltry jealousy as to a
matter so vast. If De Morgan and Boole have had a priority in the case,
to them we cheerfully shall resign the glory and honor. If such be the
truth, they have neither done justice to the discovery, nor to
themselves [quite true]. They have, under the circumstances, acted like
'the foolish man, who roasteth not that which he taketh {76} in
hunting.... It will be sufficient for us, however, to be the Columbus
of these great Americi, and popularize what they found, _if_ they found
it. We, as from the mountain top, will then become _their_ trumpeters,
and cry glory to De Morgan and glory to Boole, under Him who is the
source of all glory, the only good and wise, to Whom be glory for ever!
_If_ they be our predecessors in this matter, they have, under Him,
taken moral questions out of the category of probabilities, and
rendered them perfectly certain. In that case, let their books be read
by those who may doubt the principles this day laid before the world as
a great discovery, by our newspaper. Our cry shall be [Greek:
eurekasi]![162] Let us hope that they will join us, and henceforth keep
their light [_sic_] from under their bushel."

For myself, and for my old friend Mr. Boole, who I am sure would join me, I disclaim both priority, simultaneity, and posteriority, and request that nothing may be trumpeted from the mountain top except our abjuration of all community of thought or operation with this _Novum Organum_.

To such community we can make no more claim than Americus could make to being the forerunner of Columbus who popularized his discoveries. We do not wish for any [Greek: eurekasi] and not even for [Greek: heurekasi]. For self and Boole, I point out what would have convinced either of us that this house is divided against itself.

[Alpha] being an apostolic element, [delta] the doctrinal element, and [Chi] the body of the faithful, the church is [Alpha] [delta] [Chi], we are told. Also, that if [Alpha] become negative, or the Apostolicity become Diabolicity [my words]; or if [delta] become negative, and doctrine become heresy; or if [Chi] become negative, that is, if the faithful become unfaithful; the church becomes negative, "the very opposite to what it ought to be." For self and Boole, I admit this. But--which is not noticed--if [Alpha] and [delta] should _both_ become negative, diabolical origin {77} and heretical doctrine, then the church, [Alpha] [delta] [Chi], is still positive, what it ought to be, unless [Chi] be also negative, or the people unfaithful to it, in which case it is a bad church. Now, self and Boole--though I admit I have not asked my partner--are of opinion that a diabolical church with false doctrine does harm when the people are faithful, and can do good only when the people are unfaithful. We may be wrong, but this is what we _do_ think. Accordingly, we have caught nothing, and can therefore roast nothing of our own: I content myself with roasting a joint of Mr. Gregg's larder.

These mathematical vagaries have uses which will justify a large amount of quotation: and in a score of years this may perhaps be the only attainable record. I therefore proceed.

After observing that by this calculus juries (heaven help them! say I) can calculate damages "almost to a nicety," and further that it is made abundantly evident that c e x is "the general expression for an individual," it is noted that the number of the Beast is not given in the Revelation in words at length, but as [Greek: chxw'].[163] On this the following remark is made:

"Can it be possible that we have in this case a specimen given to us of the arithmetic of heaven, and an expression revealed, which indicates by its function of addibility, the name of the church in question, and of each member of it; and by its function of multiplicability the doctrine, the mission, and the members of the great Synagogue of Apostacy? We merely propound these questions;--we do not pretend to solve them."

After a translation in blank verse--a very pretty one--of the 18th Psalm, the author proceeds as follows, to render it into differential calculus:

{78}

"And the whole tells us just this, that David did what he could. He augmented those elements of his constitution which were (_exceptis excipiendis_)[164] subject to himself, and the Almighty then augmented his personal qualities, and his vocational _status_. Otherwise, to throw the matter into the expression of our notation, the variable e was augmented, and c x rose proportionally. The law of the variation, according to our theory, would be thus expressed. The resultant was David the king c e x [c = r?] (who had been David the shepherd boy), and from the conditions of the theorem we have

du/de = ce(dx/de) + ex(dc/de)x + cx

which, in the terms of ordinary language, just means, the increase of David's educational excellence or qualities--his piety, his prayerfulness, his humility, obedience, etc.--was so great, that when multiplied by his original talent and position, it produced a product so great as to be equal in its amount to royalty, honor, wealth, and power, etc.: in short, to all the attributes of majesty."[165]

The "solution of the family problem" is of high interest. It is to determine the effect on the family in general from a change [of conduct] in one of them. The person chosen is one of the maid-servants.

"Let c e x be the father; c_1e_1x_1 the mother, etc. The family then consists of the maid's master, her mistress, her young master, her young mistress, and fellow servant. Now the master's calling (or c) is to exercise his share of control over this servant, and mind the rest of his business: call this remainder a, and let his calling generally, or all his affairs, be to his maid-servant as m : y, i.e., y = (mz/c); ... {79} and this expression will represent his relation to the servant. Consequently,

c e x = (a + mz/c)e x; otherwise (a + mz/c)e x

is the expression for the father when viewed as the girl's master."

I have no objection to repeat so far; but I will not give the formula for the maid's relation to her young master; for I am not quite sure that all young masters are to be trusted with it. Suffice it that the son will be affected directly as his influence over her, and inversely as his vocational power: if then he should have some influence and no vocational power, the effect on him would be infinite. This is dismal to think of. Further, the formula brings out that if one servant improve, the other must deteriorate, and _vice versa_. This is not the experience of most families: and the author remarks as follows:

"That is, we should venture to say, a very beautiful result, and we may say it yielded us no little astonishment. What our calculation might lead to we never dreamt of; that it should educe a conclusion so recondite that our unassisted power never could have attained to, and which, if we could have conjectured it, would have been at best the most distant probability, that conclusion being itself, as it would appear, the quintessence of truth, afforded us a measure of satisfaction that was not slight."

That the writings of Mr. Boole and myself "go to the full justification of" this "principle," is only true in the sense in which the Scotch use, or did use, the word _justification_.

A TRIBUTE TO BOOLE.

[The last number of this Budget had stood in type for months, waiting until there should be a little cessation of correspondence more connected with the things of the day. {80} I had quite forgotten what it was to contain; and little thought, when I read the proof, that my allusions to my friend Mr. Boole, then in life and health, would not be printed till many weeks after his death. Had I remembered what my last number contained, I should have added my expression of regret and admiration to the numerous obituary testimonials, which this great loss to science has called forth.

The system of logic alluded to in the last number of this series is but one of many proofs of genius and patience combined. I might legitimately have entered it among my _paradoxes_, or things counter to general opinion: but it is a paradox which, like that of Copernicus, excited admiration from its first appearance. That the symbolic processes of algebra, invented as tools of numerical calculation, should be competent to express every act of thought, and to furnish the grammar and dictionary of an all-containing system of logic, would not have been believed until it was proved. When Hobbes,[166] in the time of the Commonwealth, published his _Computation or Logique_, he had a remote glimpse of some of the points which are placed in the light of day by Mr. Boole. The unity of the forms of thought in all the applications of reason, however remotely separated, will one day be matter of notoriety and common wonder: and Boole's name will be remembered in connection with one of the most important steps towards the attainment of this knowledge.]

DECIMALS RUN RIOT.

The Decimal System as a whole. By Dover Statter.[167] London and
Liverpool, 1856, 8vo.

{81}

The proposition is to make everything decimal. The day, now 24 hours, is to be made 10 hours. The year is to have ten months, Unusber, Duober, etc. Fortunately there are ten commandments, so there will be neither addition to, nor deduction from, the moral law. But the twelve apostles! Even rejecting Judas, there is a whole apostle of difficulty. These points the author does not touch.

ON PHONETIC SPELLING.

The first book of Phonetic Reading. London, Fred. Pitman,[168] Phonetic
Depot, 20, Paternoster Row, 1856, 12mo.

The Phonetic Journal. Devoted to the propagation of phonetic reading,
phonetic longhand, phonetic shorthand, and phonetic printing. No. 46.
Saturday, 15 November 1856. Vol. 15.

I write the titles of a couple out of several tracts which I have by me. But the number of publications issued by the promoters of this spirited attempt is very large indeed.[169] The attempt itself has had no success with the mass of the public. This I do not regret. Had the world found that the change was useful, I should have gone contentedly with the stream; but not without regretting our old language. I admit the difficulties which our unpronounceable spelling puts in the way of learning to read: and I have no doubt that, as affirmed, it is easier to teach children phonetically, and afterwards to introduce them to our common system, than to proceed in the usual way. But by the usual way I mean proceeding by letters from the very beginning. If, which I am sure is a better plan, children be taught at the commencement very much by _complete words_, as if they were learning Chinese, and be gradually accustomed to {82} resolve the known words into letters, a fraction, perhaps a considerable one, of the advantage of the phonetic system is destroyed. It must be remembered that a phonetic system can only be an approximation. The differences of pronunciation existing among educated persons are so great, that, on the phonetic system, different persons ought to spell differently.

But the phonetic party have produced something which will immortalize their plan: I mean their _shorthand_, which has had a fraction of the success it deserves. All who know anything of shorthand must see that nothing but a phonetic system can be worthy of the name: and the system promulgated is skilfully done. Were I a young man I should apply myself to it systematically. I believe this is the only system in which books were ever published. I wish some one would contribute to a public journal a brief account of the dates and circumstances of the phonetic movement, not forgetting a list of the books published in shorthand.

A child beginning to read by himself may owe terrible dreams and waking images of horror to our spelling, as I did when six years old. In one of the common poetry-books there is an admonition against confining little birds in cages, and the child is asked what if a great giant, amazingly strong, were to take you away, shut you up,

And feed you with vic-tu-als you ne-ver could bear.

The book was hyphened for the beginner's use; and I had not the least idea that _vic-tu-als_ were _vittles_: by the sound of the word I judged they must be of iron; and it entered into my soul.

The worst of the phonetic shorthand book is that they nowhere, so far as I have seen, give _all_ the symbols, in every stage of advancement, together, in one or following pages. It is symbols and talk, more symbols and more talk, etc. A universal view of the signs ought to begin the works. {83}

A HANDFUL OF LITTLE PARADOXERS.

Ombrological Almanac. Seventeenth year. An essay on Anemology and
Ombrology. By Peter Legh,[170] Esq. London, 1856, 12mo.

Mr. Legh, already mentioned, was an intelligent country gentleman, and a legitimate speculator. But the clue was not reserved for him.

The proof that the three angles of a triangle are equal to two right
angles looked for in the inflation of the circle. By Gen. Perronet
Thompson. London, 1856, 8vo. (pp. 4.)

Another attempt, the third, at this old difficulty, which cannot be put into few words of explanation.[171]

Comets considered as volcanoes, and the cause of their velocity and
other phenomena thereby explained. London (_circa_ 1856), 8vo.

The title explains the book better than the book explains the title.

1856. A stranger applied to me to know what the ideas of a friend of his were worth upon the magnitude of the earth. The matter being one involving points of antiquity, I mentioned various persons whose speculations he seemed to have ignored; among others, Thales. The reply was, "I am instructed by the author to inform you that he is perfectly acquainted with the works of Thales, Euclid, Archimedes, ..." I had some thought of asking whether he had used the Elzevir edition of Thales,[172] which is known to be very incomplete, or that of Professor Niemand with the lections, Nirgend, 1824, 2 vols. folio; just to see whether the {84} last would not have been the very edition he had read. But I refrained, in mercy.

The moon is the image of the Earth, and is not a solid body. By T^{he}
Longitude.[173] (Private Circulation.) In five parts. London, 1856,
1857, 1857; Calcutta, 1858, 1858, 8vo.

The earth is "brought to a focus"; it describes a "looped orbit round the sun." The eclipse of the sun is thus explained: "At the time of eclipses, the image is more or less so directly before or behind the earth that, in the case of new moon, bright rays of the sun fall and bear upon the spot where the figure of the earth is brought to a focus, that is, bear upon the image of the earth, when a darkness beyond is produced reaching to the earth, and the sun becomes more or less eclipsed." How the earth is "brought to a focus" we do not find stated. Writers of this kind always have the argument that some things which have been ridiculed at first have been finally established. Those who put into the lottery had the same kind of argument; but were always answered by being reminded how many blanks there were to one prize. I am loath to pronounce against anything: but it does force itself upon me that the author of these tracts has drawn a blank.

LUNAR MOTION AGAIN.

_Times_, April 6 or 7, 1856. The moon has no rotary motion.

A letter from Mr. Jellinger Symons,[174] inspector of schools, which commenced a controversy of many letters and pamphlets. This dispute comes on at intervals, and will continue to do so. It sometimes arises from inability to understand the character of simple rotation, geometrically; sometimes from not understanding the mechanical doctrine of rotation.

{85}

Lunar Motion. The whole argument stated, and illustrated by diagrams;
with letters from the Astronomer Royal. By Jellinger C. Symons. London,
1856, 8vo.

The Astronomer Royal endeavored to disentangle Mr. J. C. Symons, but failed. Mr. Airy[175] can correct the error of a ship's compasses, because he can put her head which way he pleases: but this he cannot do with a speculator.

Mr. Symons, in this tract, insinuated that the rotation of the moon is one of the silver shrines of the craftsmen. To see a thing so clearly as to be satisfied that all who say they do not see it are telling wilful falsehood, is the nature of man. Many of all sects find much comfort in it, when they think of the others; many unbelievers solace themselves with it against believers; priests of old time founded the right of persecution upon it, and of our time, in some cases, the right of slander: many of the paradoxers make it an argument against students of science. But I must say for men of science, for the whole body, that they are fully persuaded of the honesty of the paradoxers. The simple truth is, that all those I have mentioned, believers, unbelievers, priests, paradoxers, are not so sure they are right in their points of difference that they can safely allow themselves to be persuaded of the honesty of opponents. Those who know demonstration are differently situated. I suspect a train might be laid for the formation of a better habit in this way. We know that Suvaroff[176] taught his Russians at Ismail not to fear the Turks by accustoming them to charge bundles of faggots dressed in turbans, etc.

At which your wise men sneered in phrases witty,
He made no answer--but he took the city!

Would it not be a good thing to exercise boys, in pairs, in the following dialogue:--Sir, you are quite wrong!--Sir, {86} I am sure you honestly think so! This was suggested by what used to take place at Cambridge in my day. By statute, every B.A. was obliged to perform a certain number of disputations, and the _father_ of the college had to affirm that it had been done. Some were performed in earnest: the rest were huddled over as follows. Two candidates occupied the places of the respondent and the opponent: _Recte statuit Newtonus_, said the respondent: _Recte non statuit Newtonus_,[177] said the opponent. This was repeated the requisite number of times, and counted for as many _acts_ and _opponencies_. The parties then changed places, and each unsaid what he had said on the other side of the house: I remember thinking that it was capital drill for the House of Commons, if any of us should ever get there. The process was repeated with every pair of candidates.

The real disputations were very severe exercises. I was badgered for two hours with arguments given and answered in Latin,--or what we called Latin--against Newton's first section, Lagrange's[178] derived functions, and Locke[179] on innate principles. And though I _took off_ everything, and was pronounced by the moderator to have disputed _magno honore_,[180] I never had such a strain of thought in my life. For the inferior opponents were made as sharp as their betters by their tutors, who kept lists of queer objections, drawn from all quarters. The opponents used to meet the day before to compare their arguments, that the same might not come twice over. But, after I left Cambridge, it became the fashion to invite the respondent to be present, who therefore learnt all that was to be brought against him. This made the whole thing a farce: and the disputations were abolished.

{87}

The Doctrine of the Moon's Rotation, considered in a letter to the
Astronomical Censor of the _Athenaeum_. By Jones L. MacElshender.[181]
Edinburgh, 1856, 8vo.

This is an appeal to those cultivated persons who will read it "to overrule the _dicta_ of judges who would sacrifice truth and justice to professional rule, or personal pique, pride, or prejudice"; meaning, the great mass of those who have studied the subject. But how? Suppose the "cultivated persons" were to side with the author, would those who have conclusions to draw and applications to make consent to be wrong because the "general body of intelligent men," who make no special study of the subject, are against them? They would do no such thing: they would request the general body of intelligent men to find their own astronomy, and welcome. But the truth is, that this intelligent body knows better: and no persons know better that they know better than the speculators themselves.

But suppose the general body were to combine, in opposition to those who have studied. Of course all my list must be admitted to their trial; and then arises the question whether both sides are to be heard. If so, the general body of the intelligent must hear all the established side have to say: that is, they must become just as much of students as the inculpated orthodox themselves. And will they not then get into _professional rule_, pique, pride, and prejudice, as the others did? But if, which I suspect, they are intended to judge as they are, they will be in a rare difficulty. All the paradoxers are of like pretensions: they cannot, as a class, be right, for each one contradicts a great many of the rest. There will be the puzzle which silenced the crew of the cutter in Marryat's novel of the Dog Fiend.[182] "A tog is a tog," said Jansen.--"Yes," replied another, "we all know a dog is a dog; but the question is--Is _this_ dog {88} a dog?" And this question would arise upon every dog of them all.

ZETETIC ASTRONOMY.

Zetetic Astronomy: Earth not a globe. 1857 (Broadsheet).

Though only a traveling lecturer's advertisement, there are so many arguments and quotations that it is a little pamphlet. The lecturer gained great praise from provincial newspapers for his ingenuity in proving that the earth is a flat, surrounded by ice. Some of the journals rather incline to the view: but the _Leicester Advertiser_ thinks that the statements "would seem very seriously to invalidate some of the most important conclusions of modern astronomy," while the _Norfolk Herald_ is clear that "there must be a great error on one side or the other." This broadsheet is printed at Aylesbury in 1857, and the lecturer calls himself _Parallax_: but at Trowbridge, in 1849, he was S. Goulden.[183] In this last advertisement is the following announcement: "A paper on the above subjects was read before the Council and Members of the Royal Astronomical Society, Somerset House, Strand, London (Sir John F. W. Herschel,[184] President), Friday, Dec. 8, 1848." No account of such a paper appears in the _Notice_ for that month: I suspect that the above is Mr. S. Goulden's way of representing the following occurrence: Dec. 8, 1848, the Secretary of the Astronomical Society (De Morgan by name) said, at the close of the proceedings,--"Now, gentlemen, if you will promise not to tell the Council, I will read something for your amusement": and he then read a few of the arguments which had been transmitted by the lecturer. The fact is worth noting that from 1849 to 1857, arguments on the roundness or flatness of the earth did itinerate. I have {89} no doubt they did much good: for very few persons have any distinct idea of the evidence for the rotundity of the earth. The _Blackburn Standard_ and _Preston Guardian_ (Dec. 12 and 16, 1849) unite in stating that the lecturer ran away from his second lecture at Burnley, having been rather too hard pressed at the end of his first lecture to explain why the large hull of a ship disappeared before the sails. The persons present and waiting for the second lecture assuaged their disappointment by concluding that the lecturer had slipped off the icy edge of his flat disk, and that he would not be seen again till he peeped up on the opposite side.

But, strange as it may appear, the opposer of the earth's roundness has more of a case--or less of a want of case--than the arithmetical squarer of the circle. The evidence that the earth is round is but cumulative and circumstantial: scores of phenomena ask, separately and independently, what other explanation can be imagined except the sphericity of the earth. The evidence for the earth's figure is tremendously powerful of its kind; but the proof that the circumference is 3.14159265... times the diameter is of a higher kind, being absolute mathematical demonstration.

The Zetetic system still lives in lectures and books; as it ought to do, for there is no way of teaching a truth comparable to opposition. The last I heard of it was in lectures at Plymouth, in October, 1864. Since this time a prospectus has been issued of a work entitled "The Earth not a Globe"; but whether it has been published I do not know. The contents are as follows:

"The Earth a Plane--How circumnavigated.--How time is lost or gained.--Why a ship's hull disappears (when outward bound) before the mast head.--Why the Polar Star sets when we proceed Southward, etc.--Why a pendulum vibrates with less velocity at the Equator than {90} at the Pole.--The allowance for rotundity _supposed_ to be made by surveyors, not made in practice.--Measurement of Arcs of the Meridian unsatisfactory.--Degrees of Longitude North and South of the Equator considered.--Eclipses and Earth's form considered.--The Earth no motion on axis or in orbit.--How the Sun moves above the Earth's surface concentric with the North Pole.--Cause of Day and Night, Winter and Summer; the long alternation of light and darkness at the Pole.--Cause of the Sun rising and setting.--Distance of the Sun from London, 4,028 miles--How measured.--_Challenge to Mathematicians._--Cause of Tides.--Moon self-luminous, NOT a reflector.--Cause of Solar and Lunar eclipses.--Stars _not worlds_; their distance.--Earth, the _only material_ world; its true position in the universe; its condition and ultimate destruction by fire (2 Peter iii.), etc."

I wish there were geoplatylogical lectures in every town; in England (_platylogical_, in composition, need not mean _babbling_). The late Mr. Henry Archer[185] would, if alive, be very much obliged to me for recording his vehement denial of the roundness of the earth: he was excited if he heard any one call it a globe. I cannot produce his proof from the Pyramids, and from some caves in Arabia. He had other curious notions, of course: I should no more believe that a flat earth was a man's only paradox, than I should that Dutens,[186] the editor of Leibnitz, was eccentric only in supplying a tooth which he had lost by one which he found in an Italian tomb, and fully believed that it had once belonged to Scipio Africanus, whose family vault was discovered, it is supposed, in 1780. Mr. Archer is of note as {91} the suggester of the perforated border of the postage-stamps, and, I think, of the way of doing it; for this he got 4000l. reward. He was a civil engineer.

(_August 28, 1865._) The _Zetetic Astronomy_ has come into my hands. When, in 1851, I went to see the Great Exhibition, I heard an organ played by a performer who seemed very desirous to exhibit one particular stop. "What do you think of that stop?" I was asked.--"That depends on the name of it," said I.--"Oh! what can the name have to do with the sound? 'that which we call a rose,' etc."--"The name has everything to do with it: if it be a flute-stop, I think it very harsh; but if it be a railway-whistle-stop, I think it very sweet." So as to this book: if it be childish, it is clever; if it be mannish, it is unusually foolish. The flat earth, floating tremulously on the sea; the sun moving always over the flat, giving day when near enough, and night when too far off; the self-luminous moon, with a semi-transparent invisible moon, created to give her an eclipse now and then; the new law of perspective, by which the vanishing of the hull before the masts, usually thought to prove the earth globular, really proves it flat;--all these and other things are well fitted to form exercises for a person who is learning the elements of astronomy. The manner in which the sun dips into the sea, especially in tropical climates, upsets the whole. Mungo Park,[187] I think, gives an African hypothesis which explains phenomena better than this. The sun dips into the western ocean, and the people there cut him in pieces, fry him in a pan, and then join him together again, take him round the underway, and set him up in the east. I hope this book will be read, and that many will be puzzled by it: for there are many whose notions of astronomy deserve no better fate. There is no subject on which there is so little {92} accurate conception as that of the motions of the heavenly bodies. The author, though confident in the extreme, neither impeaches the honesty of those whose opinions he assails, nor allots them any future inconvenience: in these points he is worthy to live on a globe, and to revolve in twenty-four hours.

(_October, 1866._) A follower appears, in a work dedicated to the preceding author: it is _Theoretical Astronomy examined and exposed by Common Sense_. The author has 128 well-stuffed octavo pages. I hope he will not be the last. He prints the newspaper accounts of his work: the _Church Times_ says--not seeing how the satire might be retorted--"We never began to despair of Scripture until we discovered that 'Common Sense' had taken up the cudgels in its defence." This paper considers our author as the type of a _Protestant_. The author himself, who gives a summary of his arguments in verse, has one couplet which is worth quoting:

"How is't that sailors, bound to sea, with _a 'globe'_ would never start,
But in its place will always take _Mercator's_[188] LEVEL _chart_!"

To which I answer:

Why, really Mr. Common Sense, you've never got so far
As to think Mercator's planisphere shows countries as they are;
It won't do to measure distances; it points out how to steer,
But this distortion's not for you; another is, I fear.
The earth must be a cylinder, if seaman's charts be true,
Or else the boundaries, right and left, are one as well as two;
They contradict the notion that we dwell upon a plain,
For straight away, without a turn, will bring you home again.
There are various plane projections; and each one has its use:
I wish a milder word would rhyme--but really you're a goose!

The great wish of persons who expose themselves as above, is to be argued with, and to be treated as reputable {93} and refutable opponents. "Common Sense" reminds us that no amount of "blatant ridicule" will turn right into wrong. He is perfectly correct: but then no amount of bad argument will turn wrong into right. These two things balance; and we are just where we were: but you should answer our arguments, for whom, I ask? Would reason convince this kind of reasoner? The issue is a short and a clear one. If these parties be what I contend they are, then ridicule is made for them: if not, for what or for whom? If they be right, they are only passing through the appointed trial of all good things. Appeal is made to the future: and my Budget is intended to show samples of the long line of heroes who have fallen without victory, each of whom had his day of confidence and his prophecy of success. Let the future decide: they say roundly that the earth is flat; I say flatly that it is round.

The paradoxers all want reason, and not ridicule: they are all accessible, and would yield to conviction. Well then, let them reason with one another! They divide into squads, each with a subject, and as many different opinions as persons in each squad. If they be really what they say they are, the true man of each set can put down all the rest, and can come crowned with glory and girdled with scalps, to the attack on the orthodox misbelievers. But they know, to a man, that the rest are not fit to be reasoned with: they pay the regulars the compliment of believing that the only chance lies with them. They think in their hearts, each one for himself, that ridicule is of fit appliance to the rest.

Miranda. A book divided into three parts, entitled Souls, Numbers,
Stars, on the Neo-Christian Religion ... Vol. i. London, 1858, 1859,
1860. 8vo.

The name of the author is Filopanti.[189] He announces himself as the 49th and last Emanuel: his immediate {94} predecessors were Emanuel Washington, Emanuel Newton, and Emanuel Galileo. He is to collect nations into one family. He knows the transmigrations of the whole human race. Thus Descartes became William III of England: Roger Bacon became Boccaccio. But Charles IX,[190] in retribution for the massacre of St. Bartholomew, was hanged in London under the name of Barthelemy for the murder of Collard: and many of the Protestants whom he killed as King of France were shouting at his death before the Old Bailey.

THE SABBATH--THE GREAT PYRAMID

A Letter to the members of the Anglo-Biblical Institute, dated Sept. 7,
1858, and signed 'Herman Heinfetter.'[191] (Broadsheet.)