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