We have undertaken this article, because we have reason to know, that
there are still some persons who, from ignorance of the real question,
are turning their attention to this useless and exploded problem, and
deceiving themselves and their neighbours into the belief that they have
succeeded in doing that which has been repeatedly shown to be
impossible. By the _rectification_ of a circle is meant the finding of a
straight line which shall be equal in length to the circumference of a
given circle, or the two ends of which when bent round it should exactly
meet; by the _quadrature_ of a circle is meant the finding a square
which shall be equal in surface to a given circle, that is, to the whole
space contained inside its circumference. There is no difficulty in
doing these problems in a manner more than sufficiently correct for all
useful purposes. If, for example, we take a circle as large as the
earth’s orbit, and an atom as small as the smallest insect which a
microscope ever showed in a drop of water, nothing would be more easy
than to find the circumference of this circle so nearly, that the error
committed should be less than the length of that atom. The problem which
has puzzled so many generations, is the finding what is called a
_geometrical_ quadrature of the circle. What this is we proceed to
explain. It is the object of geometry to find out, by reasoning, all
truths which relate to the various figures which a draughtsman can
construct. Not that these figures are precisely the objects of
geometrical reasoning; for a geometrical line has no breadth or
thickness, but only length, while the line we draw with a pencil has a
small degree both of breadth and thickness. It is agreed by geometers to
take as little for granted as possible, and to make all their
propositions arise out of the smallest number of simple truths. It is
also agreed to imagine the existence of as few figures as possible. It
was therefore the practice of the ancient geometers to assume no
problems except the following:--1. A straight line can be drawn from one
point to another. 2. A straight line, when finished, can afterwards be
made longer. 3. A circle can be drawn with any point as a centre, and
any line as a radius. All lines, except the straight line and circle,
and afterwards the conic sections, were called _mechanical_, as
distinguished from geometrical lines; and if any problem arose, the
first attempt was always to solve it geometrically, and only when that
failed, were mechanical means resorted to, or were other curves
constructed, the construction of which _once granted_, solved the
problem. The names geometrical and mechanical, as applied to distinguish
one sort of solution from another, may be improper; but that is not the
question. When a man asserts that he has found a _geometrical_
quadrature of the circle, he either does or does not use the word in the
sense of the ancient geometers. If he does, and his solution is correct,
he has certainly solved the problem; but that no one has yet done this
is universally admitted. If he does _not_ use the word geometrical in
the ancient signification, his solution has nothing to do with the
problem which has hitherto remained unsolved. Many ways have been
discovered of finding the area of a circle, which take something more
for granted than the use of the ruler and compasses only, and any
person, with a reasonable knowledge of mathematics, might add a dozen to
the number in a couple of hours. So much for the geometrical solution of
the problem.
It was proved long before the Christian era, that the circumferences of
two different circles are to one another as the radii, that is, whatever
number of times one circumference contains its radius, the other
circumference contains its radius as many times, or whatever fraction
one radius is of its circumference, the same fraction is the other
radius of its circumference. From this it follows, that if any number of
circles were taken, having for radii a foot, a yard, a mile, &c.,
whatever number of feet and parts of feet would go round the first, the
same number of miles and similar parts of miles would go round the
third, and so on. Hence it became a question of importance to discover
what was the number of units and parts of units contained in the
circumference of a circle whose radius was the unit. Again, it was
proved that the number of square feet and parts of square feet in a
circle of one foot radius, was the same as the number of square miles
and similar parts of square miles contained in the circle of one mile
radius. Archimedes showed that a circle of one foot in radius contained
nearly 3 square feet and ⅐ of a square foot, which does not differ from
the truth by so much as ⅕ of a square inch, and gives the circle too
great by about its three-thousandth part. An ancient measure of the
Hindoos makes it 3 square feet and 177 parts out of 1250 of a square
foot. This is much nearer to the truth, differing from it by about
one-thousandth part of a square inch, but still a little too much.
Metius, who flourished in the beginning of the seventeenth century,
found that if a square foot be divided into 113 parts, the circle of one
foot radius contains about 355 of these parts; a result of surprising
accuracy when the simplicity of the numbers is considered; it is too
great by about the fifty-thousandth part of a square inch. These numbers
may be very easily recollected, since, when put together, they give the
first three odd numbers, each repeated twice; thus, 113355.
The following simple rules will enable every one of our readers to find
the circumference of a circle. If any one of them would go direct round
the world, he would by means of them, if the earth were a perfect
sphere, be able to tell the length of his journey within less than four
yards. From them the word _inch_ may be taken out, and any other unit
substituted.
To find the length of the circumference of a circle, multiply the number
of _inches_ in the diameter (or twice the radius) by 355, and divide the
product by 113. The result is the number of _inches_ in the
circumference.
To find the area, or surface of a circle, multiply the number of
_inches_ in the radius by itself, and that product by 355; divide the
result by 113, the quotient of which is the number of square _inches_ in
the area.
We might go on to describe still more accurate methods; it will be
sufficient to say, that the latest of them gives the area of a circle
true to 127 decimal places, as it is called; that is, if the radius be
1000, &c. feet, the ciphers being 127 in number, the area of the circle
will be obtained without an error of a square foot.
Still this is only an approximation, and however nearly the
circumference of a circle has been obtained, it has never been obtained
exactly. Numberless attempts have been made to find the exact ratio of
the circumference to the diameter, but without success; the reason
being, as was afterwards proved, that the thing is impossible. We can
now demonstrate that the ratio of the circumference to the diameter
cannot be _accurately_ expressed in numbers; all we can do with numbers
is, to express it as nearly as we please. Thus, using decimals, we may
say that the circumference of a circle whose diameter is 1 is greater
than 3 and less than 4; greater than 3·1 and less than 3·2; greater than
3·14 and less than 3·15, and so on, assigning nearer and nearer
fractions, between which it must lie, but never coming to an exact
result. Almost every projector who imagines he can solve this problem,
is sure to produce some number or fraction as the _exact_ ratio of the
circumference to the diameter; and it is observable that the less his
knowledge of geometry, the more easily does he overcome the difficulty,
and the more obstinately does he believe himself in the right. Some have
been found hardy enough to deny the common propositions of geometry, in
order to establish their own conclusion on this point. Others, totally
ignorant of geometry, hearing that a circle could not be _exactly_
measured, have imagined that the word _exact_ was used in the sense in
which a carpenter would take it, who, very properly for his purpose,
considers two rods to be of exactly the same length, when they do not
differ to the naked eye. These usually cut a circle out in wood, measure
it with a bit of string, pronounce their result to be perfectly
accurate, and are very much surprised that an ungrateful world does not
perceive their claim to one of the first places in the ranks of science.
We shall give some anecdotes connected with this subject, principally
extracted from Montucla’s History of the Mathematics.
In 1585, a Spanish friar published his quadrature of the circle. His
preface is a dialogue between himself and the circle, who thanks him in
most affectionate terms for having solved the problem. The circle,
however, did not in this case attend to the maxim, “Know thyself,” any
more than some of its squares have since done, for the pretended
quadrature was utterly wrong. The author of it was a modest man, and
ascribed all the honour to the Virgin Mary. Another Knight of the Round
Shield found out by his method that the first book of Euclid was all a
mistake. About the same time a merchant of Rochelle discovered not only
the quadrature of the circle, but with it, and depending upon it, a
method of converting Jews, Pagans, and Mahometans to Christianity. In
1671, an anonymous writer published a treatise with the following title:
‘Demonstration of the Divine Theorem of the quadrature of the circle, of
the bisection of the angle, and of the perpetual motion, and the
connexion of this theorem with the Vision of Ezekiel and the Revelation
of St. John.’ A certain Cluver found out that this problem depended upon
another, which he expressed thus: “Construere mundum divinæ menti
analogum.” The literal translation of this (the _sense_ is unknown) is,
“To build a world resembling the divine mind.” But the most singular
person was one Richard White, an English Jesuit, who, having once
undertaken to square the circle, was afterwards convinced by argument
that he was in the wrong which never happened to any other of this class
of speculators, except perhaps to one Mathulon, a Frenchman of Lyons.
This man offered to give a thousand crowns to any one who would detect
an error in his solution. It was done to his satisfaction, but he
refused to pay the money, and a court of justice decided that it should
be given to the poor. So late as 1750 an Englishman, Henry Sullamar,
found out the area of the circle by means of the number 666 mentioned in
the Revelations. But in 1753, a Captain in the French Guards, did more
for in squaring the circle, which he did with a piece of turf, he hit
upon what he thought was a most obvious connexion between this and the
doctrines of original sin and the Trinity. He offered to bet three
hundred thousand francs that he was right, and actually deposited ten
thousand of them. A young lady, and several other persons easily won the
wager, and brought actions for the money; but the courts declared that
the bet was void.
Such are a few of the most remarkable aberrations of the human mind on
this problem. They show, in the most convincing manner, that presumption
is rarely confined to one subject in the same mind, and that a man who,
without studying a science, conceives himself to be more knowing than
those who have passed their lives in the pursuit of it, must previously
have brought himself to believe that he is almost a God, and is but one
step removed from taking the government of the universe out of the hands
of its Creator, and arranging it according to his own improved notions.
With regard to a geometrical solution, and the possibility or
impossibility of it, we shall now say a few words. We have already
observed that an arithmetical solution is certainly impossible, that is,
there is no number or fraction which _exactly_ represents the
circumference of the circle where the radius is a unit. In 1668, James
Gregory, a well-known name in geometry, asserted that a _geometrical_
quadrature was impossible, that is, no use of the ruler and compasses
could give a square of exactly the same dimensions as a given circle. Of
this he published his demonstration, which was attacked by Huyghens,
another geometer of the same time. The dispute has interested
mathematicians so little for the last century and a half, that few of
them seem to have cared which was right. The _historians_ of mathematics
have, of course, been obliged to give an opinion, and yet Montucla and
Dr. Hutton both forbear to decide the question, each being apparently
somewhat inclined to believe that J. Gregory was right. The
demonstration of the latter appears to us to render it extremely
probable that the geometrical quadrature is impossible; but we will not
venture a positive opinion where such respectable authorities have
declined to give one. But we would recommend any one who imagines he can
give this solution, to learn geometry, to examine the demonstration of
J. Gregory, which he will find in the library of the British Museum, and
find out the error; and it deserves some attention, since neither
Montucla nor Hutton, both very well informed mathematicians, would
positively say it was false.
We would not have entered upon this subject at such length, if it were
not that there appear, from time to time, pretended solutions of this
problem. To any one who is ignorant of geometry, we would recommend to
be sure of two things before he undertakes it; first, that he has an
imagination which will set common sense at defiance, for without this he
will never out-herod Herod so far as to produce any thing worthy of
notice, after the instances which we have mentioned; secondly, that he
has his own good opinion to a very great degree, for otherwise his peace
of mind will be disturbed, either by the neglect or ridicule which it
will be his fate to meet with. To one who understands geometry, and who
imagines himself to be the person destined by Providence to work this
wonder, we have not a word to say; if the study of Euclid has not been
sufficient to teach him more sense, or at least to induce him to wait
until he knows more, we should almost rival him in absurdity, if we
thought him a proper subject for the language of common sense.
---------------------