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% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
% (dwilkins@maths.tcd.ie)
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% Trinity College, 2000.
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\centerline{\Largebf ELEMENTARY PROOF, THAT EIGHT}
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\centerline{\Largebf PERIMETERS, OF THE REGULAR INSCRIBED}
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\centerline{\Largebf POLYGON OF TWENTY SIDES, EXCEED}
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\centerline{\Largebf TWENTY-FIVE DIAMETERS OF THE CIRCLE}
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\centerline{\Largebf By}
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\centerline{\Largebf William Rowan Hamilton}
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\centerline{\largerm (Philosophical Magazine, 23 (1862), pp.\ 267--269.)}
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\centerline{\largerm Edited by David R. Wilkins}
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\centerline{\largerm 2000}
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\noindent
{\largeit Elementary Proof, that Eight Perimeters, of the Regular
inscribed Polygon of Twenty Sides, exceed Twenty-five Diameters
of the Circle. By\/} {\largerm Professor Sir}
{\largesc William Rowan Hamilton}, {\largeit LL.D.,
\&c.}\footnote*{Communicated by the Author.}
\bigbreak
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\centerline{[{\it The London, Edinburgh and Dublin Philosophical
Magazine and Journal of Science},}
\centerline{4th series, vol.~xxiii (1862), pp. 267--269.]}
\bigskip
It was proved by Archimedes that 71 perimeters, of a regular
polygon of 96 sides inscribed in a circle, exceed 223 diameters;
whence follows easily the well-known theorem, that eight
circumferences of a circle exceed twenty-five diameters, or that
$8 \pi > 25$. Yet the following elementary proof, that eight
perimeters of the regular inscribed polygon of {\it twenty
sides\/} are greater than twenty-five diameters, has not perhaps
hitherto appeared in any scientific\footnote\dag{A sketch of the
proof was published, at the request of a friend, in an eminent
literary journal last summer, but in a connexion not likely to
attract the attention of mathematical readers in general. At all
events, it pretends to no merit but that of brevity, and the
simplicity of the principles on which it rests.}
work or periodical; and if a page of the Philosophical Magazine
can be spared for its insertion, some readers may find it
interesting from its extreme simplicity. In fact, for completely
understanding it, no preparation is required beyond the four
first Books of Euclid, and the few first Rules of Arithmetic,
together with some rudimentary knowledge of the connexion between
arithmetic and geometry.
\bigbreak
1.
It follows from the Fourth Book of Euclid's `Elements,' that the
rectangle under the side of the regular decagon inscribed in a
circle, and the same side increased by the radius, is equal to
the square of the radius. But the product of the two numbers,
791 and 2071, whereof the latter is equal to the former increased
by 1280, is less than the square of 1280 (because 1638161 is less
than 1638400). If then the radius be divided into 1280 equal
parts, the side of the inscribed decagon must be greater than a
line which consists of 791 such parts; or briefly, if the radius
be equal to 1280, the side of the decagon exceeds 791.
\bigbreak
2.
When a diameter of a circle bisects a chord, the square of the
chord is equal, by the Third Book, to the rectangle under the
doubled segments of that diameter. But the product of the two
numbers, 125 and 4995, which together make up 5120, or the double
of the double of 1280, is less than the square of 791 (because
624375 is less than 625681). If then the radius be still
represented by 1280, and therefore the doubled diameter by 5120,
and if the bisected chord be a side of the regular decagon, and
therefore greater (by what has been just proved) than 791, the
lesser segment of the diameter is greater than the line
represented by 125.
\bigbreak
3.
The rectangle under this doubled segment and the radius, is equal
to the square of the side of the regular inscribed polygon of
twenty sides. But the product of 125 and 1280 is equal to the
square of 400; and if the radius be still 1280, it has been
proved that the doubled segment exceeds 125; with this
representation of the radius, the side of the inscribed polygon
of twenty sides exceeds therefore the line represented by 400;
and the perimeter of that polygon is consequently greater than
8000.
\bigbreak
4.
Dividing then the numbers 1280 and 8000 by their greatest common
measure 320, we find that if the radius be now represented by the
number~4, or the diameter by 8, the perimeter of the polygon will
be greater than the line represented by 25; or in other words,
that {\it eight perimeters of the regular inscribed polygon of
twenty sides\/} (and by a still stronger reason, {\it eight
circumferences of the circle\/} itself) {\it exceed twenty five
diameters}.
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Observatory, March~7, 1862.
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