which is certainly false.[34]
{14}
In 1852 I examined a terrific construction, at the request of the late Dr.
Wallich,[35] who was anxious to persuade a poor countryman of his, that
trisection of the angle was waste of time. One of the principles was, that
"magnitude and direction determine each other." The construction was
equivalent to the assertion that, [theta] being any angle, the cosine of
its third part is
sin 3[theta] . cos(5[theta]/2) + sin^2 [theta] sin (5[theta]/2)
divided by the square root of
sin^2 3[theta] . cos^2 (5[theta]/2) + sin^4 [theta] + sin 3[theta] . sin
5[theta] . sin^2 [theta].
This is from my rough notes, and I believe it is correct.[36] It is so
nearly true, unless the angle be very obtuse, that common drawing, applied
to the construction, will not detect the error. There are many formulae of
this kind: and I have several times found a speculator who has discovered
the corresponding construction, has seen the approximate success of his
drawing--often as great as absolute truth could give in graphical
practice,--and has then set about his demonstration, in which he always
succeeds to his own content.
There is a trisection of which I have lost both cutting and reference: I
think it is in the _United Service Journal_. I could not detect any error
in it, though certain there must {15} be one. At least I discovered that
two parts of the diagram were incompatible unless a certain point lay in
line with two others, by which the angle to be trisected--and which was
trisected--was bound to be either 0 deg. or 180 deg..
Aug. 22, 1866. Mr. Upton sticks to his subject. He has just published "The
Uptonian Trisection. Respectfully dedicated to the schoolmasters of the
United Kingdom." It seems to be a new attempt. He takes no notice of the
sentence I have put in italics: nor does he mention my notice of him,
unless he means to include me among those by whom he has been "ridiculed
and sneered at" or "branded as a brainless heretic." I did neither one nor
the other: I thought Mr. Upton a paradoxer to whom it was likely to be
worth while to propound the definite assertion now in italics; and Mr.
Upton does not find it convenient to take issue on the point. He prefers
general assertions about algebra. So long as he cannot meet algebra on the
above question, he may issue as many "respectful challenges" to the
mathematicians as he can find paper to write: he will meet with no
attention.
There is one trisection which is of more importance than that of the angle.
It is easy to get half the paper on which you write for margin; or a
quarter; but very troublesome to get a third. Show us how, easily and
certainly, to fold the paper into three, and you will be a real benefactor
to society.
Early in the century there was a Turkish trisector of the angle, Hussein
Effendi, who published two methods. He was the father of Ameen Bey, who was
well known in England thirty years ago as a most amiable and cultivated
gentleman and an excellent mathematician. He was then a student at
Cambridge; and he died, years ago, in command of the army in Syria. Hussein
Effendi was instructed in mathematics by Ingliz Selim Effendi, who
translated a work {16} of Bonnycastle[37] into Turkish.[38] This Englishman
was Richard Baily, brother of Francis Baily[39] the astronomer, who
emigrated to Turkey in his youth, and adopted the manners of the Turks, but
whether their religion also I never heard, though I should suppose he did.
I now give the letters from the agricultural laborer and his friend,
described on page 12, Vol. I. They are curiosities; and the history of the
quadrature can never be well written without some specimens of this kind:
"Doctor Morgan, Sir. Permit me to address you
"Brute Creation may perhaps enjoy the faculty of beholding visible things
with a more penitrating eye than ourselves. But Spiritual objects are as
far out of their reach as though they had no being
"Nearest therefore to the brute Creation are those men who Suppose
themselves to be so far governed by external objects as to believe nothing
but what they See and feel And Can accomedate to their Shallow
understanding and Imaginations
"My Dear Sir Let us all Consult ourselves by the wise proverb.
"I believe that evry man^s merit & ability aught to be appreciated and
valued In proportion to its worth & utility
"In whatever State or Circumstances they may fortunately or unfortunately
be placed
"And happy it is for evry man to know his worth and place
"When a Gentleman of your Standing in Society Clad with those honors Can
not understand or Solve a problem That is explicitly explained by words and
Letters and {17} mathematically operated by figuers He had best consult the
wise proverd
"Do that which thou Canst understand and Comprehend for thy good.
"I would recommend that Such Gentleman Change his business
"And appropriate his time and attention to a Sunday School to Learn what he
Could and keep the Litle Children form durting their Close
"With Sincere feelings of Gratitude for your weakness and Inability I am
"Sir your Superior in Mathematics ----"
"1849 June th29."
"Dor Morgin Sir
"I wrote and Sent my work to Professor ---- of ---- State of ---- United
States
"I am now in the possession of the facts that he highly approves of my
work. And Says he will Insure me Reward in the States
"I write this that you may understand that I have knowledge of the unfair
way that I am treated In my own nati County
"I am told and have reasons to believe that it is the Clergy that treat me
so unjust.
"I am not Desirous of heaping Disonors upon my own nation. But if I have to
Leave this kingdom without my Just dues. The world Shall know how I am and
have been treated.
"I am Sir Desirous of my
"Just dues ----"
"1849 July 3."
"July 7th, 1849.
"Sir, I have been given to understand that a friend of mine one whom I
shall never be ashamed to acknowledge as {18} such tho' lowly his origine;
nay not only not ashamed but proud of doing so for I am one of those who
esteem and respect a man according to his ability and probity, deeming with
Dr. Watts 'that the mind is the standard of the man,'[40] has laid before
you and asked your opinion of his extraordinary performance, viz. the
quadrature of the circle, he did this with the firmest belief that you
would not only treat the matter in a straightforward manner but with the
conviction that from your known or supposed knowledge of mathematicks would
have given an upright and honorable decision upon the subject; but the
question is have you done so? Could I say yes I would with the greatest of
pleasure and have congratulated you upon your decision whatever it might
have been but I am sorry to say that I cannot your letter is a paltry
evasion, you say 'that it is a great pity that you (Mr. ----) should have
attempted this (the quadrature of the circle) for your mathematical
knowledge is not sufficient to make you know in what the problem consists,'
you don't say in what it does consist _according to your ideas_, oh! no
nothing of the sort, you enter into no disquisition upon the subject in
order to show where you think Mr. ---- is wrong and why you have not is
simply--_because you cannot_--you know that he has done it and what is if I
am not wrongly informed _you have been heard to say so_. He has done what
you nor any other mathematician as those who call themselves such have
done. And what is the reason that you will not candidly acknowledge to him
as you have to others that he has squared the circle shall I tell you? it
is because he has performed the feat to obtain the glory of which
mathematicians have battled from time immemorial that they might encircle
their brows with a wreath of laurels far more glorious than ever conqueror
won it is simply this that it is a poor man a {19} humble artisan who has
gained that victory that you don't like to acknowledge it you don't like to
be beaten and worse to acknowledge that you have miscalculated, you have in
short too small a soul to acknowledge that he is right.
"I was asked my opinion and _I_ gave it unhesitatingly in the affirmative
and I am backed in my opinion not only by Mr. ---- a mathematician and
watchmaker residing in the boro of Southwark but by no less an authority
than the Professor of mathematics of ---- College ---- ---- United States
Mr. ---- and I presume that he at least is your equal as an authority and
Mr. ---- says that the government of the U.S. will recompense M. D. for the
discovery he has made if so what a reflection upon Old england the boasted
land of freedom the nursery of arts and sciences that her sons are obliged
to go to a foreign country to obtain that recompense to which they are
justly entitled
"In conclusion I had to contradict an assertion you made to the effect that
'there is not nor ever was any reward offered by the government of this
country for the discovery of the quadrature of the circle.' I beg to inform
you that there _was_ but that it having been deemed an impossibility the
government has withdrawn it. I do this upon no less an authority than the
Marquis of Northampton.[41]
"I am, sir, yours ----"
"Dr. Morgan."