A Budget of Paradoxes Volume II Part 5

(P. 592.) "Joanna Southcott[109] is not very gallantly treated by the gentlemen of the Press, who, we believe, without knowing anything about her, merely pick up their idea of her character from the rabble. We once entertained the same rabble idea of her; but having read her works--for we really have read them--we now regard her with great respect. However, there is a great abundance of chaff and straw to her grain; but the grain is good, and as we do not eat either the chaff or straw if we can avoid it, nor even the raw grain, but thrash it and winnow it, and grind it and bake it, we find it, after undergoing this process, not only very palatable, but a special dainty of its kind. But the husk is an insurmountable obstacle to those learned and educated gentlemen who judge of books entirely by the style and the grammar, or those who eat grain as it grows, like the cattle.

Such men would reject all prological revelation; for there never was and probably never will be a revelation by voice and vision communicated in cla.s.sical manner. It would be an invasion of the rights and prerogatives of Humanity, and as contrary to the Divine and Established order of mundane government, as a field of quartern loaves or hot French rolls."

Mr. Smith's book is spiritualism from beginning to end; and my anonymous gainsayer, honest of course, is either ignorant of the work he thinks he has read, or has a most remarkable development of the organ of imperception.]

A CONDENSED HISTORY OF MATHEMATICS.

I cut the following from a Sunday paper in 1849:

"X. Y.--The Chaldeans began the mathematics, in which the Egyptians excelled. Then crossing the sea, by means {59} of Thales,[110] the Milesian, they came into Greece, where they were improved very much by Pythagoras,[111] Anaxagoras,[112] and Anopides[113] of Chios. These were followed by Briso,[114] Antipho, [two circle-squarers; where is Euclid?]

and Hippocrates,[115] but the excellence of the algebraic art was begun by Geber,[116] an Arabian astronomer, and was carried on by Carda.n.u.s,[117]

Tartaglia,[118] Clavius,[119], Stevinus,[120] Ghetaldus,[121]

Herig_e_nius,[122] Fran. Van Schooten [meaning Francis Van Schooten[123]], Florida de Beau_m_e,[124] etc."

Bryso was a mistaken man. Antipho had the disadvantage of being in advance of his age. He had the notion of which the modern geometry has made so much, that of {60} a circle being the polygon of an infinitely great number of sides. He could make no use of it, but the notion itself made him a sophist in the eyes of Aristotle, Eutocius,[125] etc. Geber, an Arab astronomer, and a reputed conjurer in Europe, seems to have given his name to unintelligible language in the word _gibberish_. At one time _algebra_ was traced to him; but very absurdly, though I have heard it suggested that _algebra_ and _gibberish_ must have had one inventor.

Any person who meddles with the circle may find himself the crane who was netted among the geese: as Antipho for one, and Olivier de Serres[126] for another. This last gentleman ascertained, by weighing, that the area of the circle is very nearly that of the square on the side of the inscribed equilateral triangle: which it is, as near as 3.162 ... to 3.141.... He did not pretend to more than approximation; but Montucla and others misunderstood him, and, still worse, misunderstood their own misunderstanding, and made him say the circle was exactly double of the equilateral triangle. He was let out of limbo by Lacroix, in a note to his edition of Montucla's _History of Quadrature_.

ST. VITUS, PATRON OF CYCLOMETERS.

Quadratura del cerchio, trisezione dell' angulo, et duplicazione del cubo, problemi geometricamente risolute e dimostrate dal Reverendo Arciprete di San Vito D. Domenico Anghera,[127] Malta, 1854, 8vo.

{61}

Equazioni geometriche, estratte dalla lettera del Rev. Arciprete ... al Professore Pullicino[128] sulla quadratura del cerchio. Milan, 1855 or 1856, 8vo.

Il Mediterraneo gazetta di Malta, 26 Decembre 1855, No. 909: also 911, 912, 913, 914, 936, 939.

The Malta Times, Tuesday, 9th June 1857.

Misura esatta del cerchio, dal Rev. D. Anghera. Malta, 1857, 12mo.

Quadrature of the circle ... by the Rev. D. Anghera, Archpriest of St.

Vito. Malta, 1858, 12mo.

I have looked for St. Vitus in catalogues of saints, but never found his legend, though he figures as a day-mark in the oldest almanacs. He must be properly accredited, since he was an archpriest. And I p.r.o.nounce and ordain, by right accruing from the trouble I have taken in this subject, that he, St. Vitus, who leads his votaries a never-ending and unmeaning dance, shall henceforth be held and taken to be the patron saint of the circle-squarer. His day is the 15th of June, which is also that of St.

Modestus,[129] with whom the said circle-squarer often has nothing to do.

And he must not put himself under the first saint with a slantendicular reference to the other, as is much to be feared was done by the Cardinal who came to govern England with a t.i.tle containing St. Pudentiana,[130] who shares a day with _St. Dunstan_. The Archpriest of St. Vitus will have it that the square inscribed in a semicircle is half of the semicircle, or the circ.u.mference 3-1/5 diameters. He is active and able, with {62} nothing wrong about him except his paradoxes. In the second tract named he has given the testimonials of crowned heads and ministers, etc. as follows.

Louis-Napoleon gives thanks. The minister at Turin refers it to the Academy of Sciences, and hopes so much labor will be judged _degna di pregio_.[131]

The Vice-Chancellor of Oxford--a blunt Englishman--begs to say that the University has never proposed the problem, as some affirm. The Prince Regent of Baden has received the work with lively interest. The Academy of Vienna is not in a position to enter into the question. The Academy of Turin offers the most _distinct_ thanks. The Academy della Crusca attends only to literature, but gives thanks. The Queen of Spain has received the work with the highest appreciation. The University of Salamanca gives infinite thanks, and feels true satisfaction in having the book. Lord Palmerston gives thanks, by the hand of "William San." The Viceroy of Egypt, not being yet up in Italian, will spend his first moments of leisure in studying the book, when it shall have been translated into French: in the mean time he congratulates the author upon his victory over a problem so long held insoluble. All this is seriously published as a rate in aid of demonstration. If these royal compliments cannot make the circ.u.mference of a circle about 2 per cent. larger than geometry will have it --which is all that is wanted--no wonder that thrones are shaky.

I am informed that the legend of St. Vitus is given by Ribadeneira[132] in his lives of Saints, and that Baronius,[133] in {63} his _Martyrologium Romanum_, refers to several authors who have written concerning him. There is an account in Mrs. Jameson's[134] _History of Sacred and Legendary Art_ (ed. of 1863, p. 544). But it seems that St. Vitus is the patron saint of _all_ dances; so that I was not so far wrong in making him the protector of the cyclometers. Why he is represented with a c.o.c.k is a disputed point, which is now made clear: next after _gallus gallinaceus_[135] himself, there is no crower like the circle-squarer.

CELEBRATED APPROXIMATIONS OF [pi].

The following is an extract from the _English Cyclopaedia_, Art. TABLES:

"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 c.o.c.ker,[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 grat.i.tude 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 quant.i.ty 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 ill.u.s.trations 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 ill.u.s.tration of Mr. Suffield's method of _Synthetic division_.

Another instance of computation carried to paradoxical length, in order to ill.u.s.trate 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 ill.u.s.trate 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 cla.s.ses, 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.)