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: