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 pronounce 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 title 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
circumference 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 circumference 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 cock 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
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.