A method to trisect a series of angles having relation to each other; also another to trisect any given angle. By James Sabben. 1848 (two quarto pages).
"The consequence of years of intense thought": very likely, and very sad.
1848. The following was sent to me in manuscript. I give the whole of it:
"_Quadrature of the Circle_.--A quadrant is a curvilinear angle traversing round and at an equal distance from a given point, called a center, no two points in the curve being at the same angle, but irreptitiously graduating from 90 to 60. It is therefore a mean angle of 90 and 60, which is 75, because it is more than 60, and less than 90, approximately from 60 to 90, and from 90 to 60, with equal generation in each irreptitious approximation, therefore meeting in 75, and which is the mean angle of the quadrant.
"Or suppose a line drawn from a given point at 90, and from the same point at 60. Let each of these lines revolve on this point toward each other at an equal ratio. They will become one line at 75, and bisect the curve, which is one-sixth of the entire circle. The result, taking 16 as a diameter, gives an area of 201.072400, and a circumference of 50.2681.
"The original conception, its natural harmony, and the result, to my own mind is a demonstrative truth, which I presume it right to make known, though perhaps at the hazard of unpleasant if not uncourteous remarks."
I have added punctuation: the handwriting and spelling {11} are those of an educated person; the word _irreptitious_ is indubitable. The whole is a natural curiosity.
The quadrature and exact area of the circle demonstrated. By Wm. Peters. 8vo. _n. d._ (circa 1848).[29]
Suggestions as to the necessity for a revolution in philosophy; and prospectus for the establishment of a new quarterly, to be called the _Physical Philosopher and Heterodox Review_. By Q. E. D. 8vo. 1848.
These works are by one author, who also published, as appears by advertisement,
"Newton rescued from the precipitancy of his followers through a century and a half,"[30] and "Dangers along a coast by correcting (as it is called) a ship's reckoning by bearings of the land at night fall, or in a fog, nearly out of print. Subscriptions are requested for a new edition."
The area of a circle is made four-fifths of the circumscribed square: proved on an assumption which it is purposed to explain in a longer essay.[31] The author, as Q. E. D., was in controversy with the _Athenaeum_ journal, and criticised a correspondent, D., who wrote against a certain class of discoverers. He believed the common theories of hydrostatics to be wrong, and one of his questions was:
"Have you ever taken into account anent gravity and gravitation the fact that a five grain cube of cork will of itself half sink in the water, whilst it will take 20 grains of brass, which will sink of itself, to pull under the other half? Fit this if you can, friend D., to your notions of gravity and specific gravity, as applied to the construction of a universal law of gravitation."
This the _Athenaeum_ published--but without some Italics, for which the editor was sharply reproved, as a sufficient {12} specimen of the _quod erat_ D. _monstrandum_: on which the author remarks--"D,--Wherefore the e caret? is it D apostrophe? D', D'M, D'Mo, D'Monstrandum; we cannot find the _wit_ of it." This I conjecture to contain an illusion to the name of the supposed author; but whether De Mocritus, De Mosthenes, or De Moivre was intended, I am not willing to decide.
The Scriptural Calendar and Chronological Reformer, for the statute year 1849. Including a review of recent publications on the Sabbath question. London, 1849, 12mo.[32]
This is the almanac of a sect of Christians who keep the Jewish Sabbath, having a chapel at Mill Yard, Goodman's Fields. They wrote controversial works, and perhaps do so still; but I never chanced to see one.
Geometry _versus_ Algebra; or the trisection of an angle geometrically solved. By W. Upton, B.A.[33] Bath (circa 1849). 8vo.
The author published two tracts under this title, containing different alleged proofs: but neither gives any notice of the change. Both contain the same preface, complaining of the British Association for refusing to examine the production. I suppose that the author, finding his first proof wrong, invented the second, of which the Association never had the offer; and, feeling sure that they would have equally refused to examine the second, thought it justifiable to {13} present that second as the one which they had refused. Mr. Upton has discovered that the common way of finding the circumference is wrong, would set it right if he had leisure, and, in the mean time, has solved the problem of the duplication of the cube.
_The trisector of an angle, if he demand attention from any mathematician, is bound to produce, from his construction, an expression for the sine or cosine of the third part of any angle, in terms of the sine or cosine of the angle itself, obtained by help of no higher than the square root._ The mathematician knows that such a thing cannot be; but the trisector virtually says it can be, and is bound to produce it, to save time. This is the misfortune of most of the solvers of the celebrated problems, that they have not knowledge enough to present those consequences of their results by which they can be easily judged. Sometimes they have the knowledge and quibble out of the use of it. In many cases a person makes an honest beginning and presents what he is sure is a solution. By conference with others he at last feels uneasy, fears the light, and puts self-love in the way of it. Dishonesty sometimes follows. The speculators are, as a class, very apt to imagine that the mathematicians are in fraudulent confederacy against them: I ought rather to say that each one of them consents to the mode in which the rest are treated, and fancies conspiracy against himself. The mania of conspiracy is a very curious subject. I do not mean these remarks to apply to the author before me.
One of Mr. Upton's trisections, if true, would prove the truth of the following equation:
3 cos ([theta] / 3) = 1 + [root](4 - sin^2[theta])
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."
