atement Mr. James Smith precedes by saying that we
have treated as true what we well knew to be false: and he follows by
saying that we have not read his work, or we should have known the above
facts to be true. Mr. Smith's pretext is as follows. His correspondent
E. M. says, 'My letters were not intended for publication, and I protest
against their being published,' and he subjoins 'Therefore I must desire
that my name may not be used.' The obvious meaning is that E. M. protested
against the publication altogether, but, judging that Mr. Smith was {114}
determined to publish, desired that his name should not be used. That he
afterwards corrected the proofs merely means that he thought it wiser to
let them pass under his own eyes than to leave them entirely to Mr. Smith.

"We have received from Sir W. Rowan Hamilton[209] a proof that the
circumference is more than 3-1/8 diameters, requiring nothing but a
knowledge of four books of Euclid. We give it in brief as an exercise for
our juvenile readers to fill up. It reminds us of the old days when real
geometers used to think it worth while seriously to demolish pretenders.
Mr. Smith's fame is now assured: Sir W. R. Hamilton's brief and easy
exposure will procure him notice in connection with this celebrated
problem.

"It is to be shown that the perimeter of a regular polygon of 20 sides is
greater than 3-1/8 diameters of the circle, and still more, of course, is
the circumference of the circle greater than 3-1/8 diameters.

"1. It follows from the 4th Book of Euclid, that the rectangle under the
side of a regular decagon inscribed in a circle, and that side increased by
the radius, is equal to the square of the radius. But the product 791 (791
+ 1280) is less than 1280 x 1280; if then the radius be 1280 the side of
the decagon is greater than 791.

"2. When a diameter bisects a chord, the square of the chord is equal to
the rectangle under the doubles of the segments of the diameter. But the
product 125 (4 x 1280 - 125) is less than 791 x 791. If then the bisected
chord be a side of the decagon, and if the radius be still 1280, the double
of the lesser segment exceeds 125.

"3. The rectangle under this doubled segment and the radius is equal to the
square of the side of an inscribed regular polygon of 20 sides. But the
product 125 x 1280 is equal to 400 x 400; therefore, the side of the
last-mentioned polygon is greater than 400, if the radius be still 1280. In
other wo