John Phin.

The seven follies of science; a popular account of the most famous scientific impossibilities and the attempts which have been made to solve them. To which is added a small budget of interesting paradoxes, illusions, and marvels online

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Online LibraryJohn PhinThe seven follies of science; a popular account of the most famous scientific impossibilities and the attempts which have been made to solve them. To which is added a small budget of interesting paradoxes, illusions, and marvels → online text (page 2 of 12)
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he offered a reward for the detection of any error, and ac-
tually deposited 10,000 francs as earnest of 300,000. But
the courts would not allow any one to recover.


In the last number of the Athenaeum for 1855 a corres-
pondent says " the thing is no longer a problem but an
axiom." He makes the square equal to a circle by making
each side equal to a quarter of the circumference. As De
Morgan says, he does not know that the area of the circle
is greater than that of any other figure of the same cir-

Such ideas are evidently akin to the poetic notion of the
quadrature. Aristophanes, in the "Birds," introduces a
geometer, who announces his intention to make a square
circle. And Pope in the "Dunciad" delivers himself as
follows :

Mad Mathesis alone was unconfined,
Too mad for mere material chains to bind,
Now to pure space lifts her ecstatic stare,
Now, running round the circle, finds it square.

The author's note explains that this "regards the wild
and fruitless attempts of squaring the circle." The poetic
idea seems to be that the geometers try to make a square

As stated by all recognized authorities, the problem is
this : To describe a square which shall be exactly equal in
area to a given circle.

The solution of this problem may be given in two ways:

(1) the arithmetical method, by which the area of a circle
is found and expressed numerically in square measure, and

(2) the geometrical quadrature, by which a square, equal in
area to a given circle, is described by means of rule and
compasses alone.

Of course, if we know the area of the circle, it is
easy to find the side of a square of equal area ; this can be
done by simply extracting the square root of the area, pro-


vided the number is one of which it is possible to extract
the square root. Thus, if we have a circle which contains
100 square feet, a square with sides of 10 feet would be
exactly equal to it. But the ascertaining of the area of the
circle is the very point where the difficulty comes in ; the
dimensions of circles are usually stated in the lengths of
the diameters, and when this is the case, the problem re-
solves itself into another, which is : To find the area of a
circle when the diameter is given.

Now Archimedes proved that the area of any circle is

equal to that of a triangle whose base has the same -
length as the circumference and whose altitude or height
is equal to the radius. Therefore if we can find the length
of the circumference when the diameter is given, we are in
possession of all the points needed to enable us to " square
the circle."

In this form the problem is known to mathematicians as
that of the rectification of the curve.

In a practical form this problem must have presented
itself to intelligent workmen at a very early stage in the
progress of operative mechanics. Architects, builders,
blacksmiths, and the makers of chariot wheels and vessels
of various kinds must have had occasion to compare the
diameters and circumferences of round articles. Thus
in I Kings, vii, 23, it is said of Hiram of Tyre that "he
made a molten sea, ten cubits from the one brim to the
other; it was round all about * * * and a line of
thirty cubits did compass it round about," from which it
has been inferred that among the Jews, at that time, the
accepted ratio was 3 to I, and perhaps, with the crude
measuring instruments of that age, this was as near as could
be expected. And this ratio seems to have been accepted


by the Babylonians, the Chinese, and probably also by the
Greeks, in the earliest times. At the same time we must
not forget that these statements in regard to the ratio
come to us through historians and prophets, and may not
have been the figures used by trained mechanics. An
error of one foot in a hoop made to go round a tub or cis-
tern of seven feet in diameter, would hardly be tolerated
even in an apprentice.

The Egyptians seem to have reached a closer approxima-
tion, for from a calculation in the Rhind papyrus, the ratio of
3. 1 6 to I seems to have been at one time in use. It is prob-
able, however, that in these early times the ratio accepted
by mechanics in general was determined by actual meas-
urement, and this, as we shall see hereafter, is quite
capable of giving results accurate to the second fractional
place, even with very common apparatus.

To Archimedes, however, is generally accorded the
credit of the first attempt to solve the problem in a
scientific manner ; he took the circumference of the circle
as intermediate between the perimeters of the inscribed
and the circumscribed polygons, and reached the conclusion
that the ratio lay between 3^ and 3^, or between 3.1428
and 3.1408.

This ratio, in its more accurate form of 3.141592 . . is
now known by the Greek letter TT (pronounced like the
common word pie), a symbol which was introduced by
Euler, between 1737 and 1748, and which is now adopted
all over the world. I have, however, used the term ratio,
or value of the ratio instead, throughout this chapter, as
probably being more familiar to my readers.

Professor Muir justly says of this achievement of
Archimedes, that it is " a most notable piece of work ; the


immature condition of arithmetic, at the time, was the only
real obstacle preventing the evaluation of the ratio to any
degree of accuracy whatever."

And when we remember that neither the numerals now
in use nor the Arabic numerals, as they are usually called,
nor any system equivalent to our decimal system, was
known to these early mathematicians, such a calculation
as that made by Archimedes was a wonderful feat.

If any of my readers, who are familiar with the Hebrew
or Greek numbers, and the mode of representing them by
letters, will try to do any of those more elaborate sums
which, when worked out by modern methods, are mere
child's play in the hands of any of the bright scholars in
our common schools, they will fully appreciate the diffi-
culties under which Archimedes labored.

Or, if ignorant of Greek and Hebrew, let them try it
with the Roman numerals, and multiply XCVIII by
MDLVII, without using Arabic or common numerals.
Professor McArthur, in his article on " Arithmetic " in the
Encyclopaedia Britannica, makes the following statement
on this point :

" The methods that preceded the adoption of the Arabic
numerals were all comparatively unwieldy, and very simple
processes involved great labor. The notation of the Ro-
mans, in particular, could adapt itself so ill to arithmetical
operations, that nearly all their calculations had to be
made by the abacus. One of the best and most manage-
able of the ancient systems is the Greek, though that, too,
is very clumsy."

After Archimedes, the most notable result was that
given by Ptolemy, in the " Great Syntaxis." He made
the ratio 3.141552, which was a very close approximation.

For several centuries there was little progress towards


a more accurate determination of the ratio. Among the
Hindoos, as early as the sixth century, the now well-known
value, 3.1416, had been obtained by Arya-Bhata, and a
little later another of their mathematicians came to the
conclusion that the square root of 10 was the true value
of the ratio. He was led to this by calculating the perim-
eters of the successive inscribed polygons of 12, 24, 48,
and 96 sides, and finding that the greater the number of
sides the nearer the perimeter of the polygon approached
the square root of 10. He therefore thought that the
perimeter or circumference of the circle itself would be the
square root of exactly 10. It is too great, however, being
3.1622 instead of 3.14159. . . The same idea is attrib-
uted to Bovillus, by Montucla.

By calculating the perimeters of the inscribed and cir-
cumscribed polygons, Vieta (1579) carried his approxima-
tion to ten fractional places, and in 1585 Peter Metius,
the father of Adrian, by a lucky step reached the now
famous fraction -||J, or 3.14159292, which is correct to the
sixth fractional place. The error does not exceed one part
in thirteen millions.

At the beginning of the seventeenth century, Ludolph
Van Ceulen reached 35 places. This result, which " in his
life he found by much labor," was engraved upon his
tombstone in St. Peter's Church, Leyden. The monu-
ment has now unfortunately disappeared.

From this time on, various mathematicians succeeded,
by improved methods, in increasing the approximation.
Thus in 1705, Abraham Sharp carried it to 72 places;
Machin (1706) to 100 places; Rutherford (1841) to 208
places, and Mr. Shanks in 1853, to 607 places. The
same computer in 1873 reached the enormous number of
707 places.


Printed in type of the same size as that used on this
page, these figures would form a line nearly six feet long.

As a matter of interest I give here the value of the
ratio of the circumference to the diameter, to 127 places :

3.14159 26535 89793 23846 26433 83279 50288 41971
69399 375 10 58209 74944 59230 78164 06286 20899
86280 34825 34211 7067982148 08651 32723 06647
0938446 +

The degree of accuracy which may be attained by using
a ratio carried to only ten fractional places, far exceeds
anything that can be required in even the finest work, and
indeed it is beyond anything attainable by means of our
present tools and instruments. For example : If the
length of a curve of 100 feet radius were determined by
a value of ten fractional places, the result would not err
by the one-millionth part of an inch, a quantity which is
quite invisible under the best microscopes of the present
day. This shows us that in any calculations relating to
the dimensions of the earth, such as longitude, etc., we
have at our command, in the 127 places of figures
given above, an exactness which for all practical purposes
may be regarded as absolute. This will be best appre-
ciated by a consideration of the fact that if the earth were
a perfect sphere and if we knew its exact diameter, we
could calculate so exactly the length of an iron hoop which
would go round it, that the difference produced by a
change of temperature equal to the millionth of a millionth
part of a degree Fahrenheit, would far exceed the error
arising from the difference between the true ratio and the
result thus reached.

Such minute quantities are far beyond the powers of
conception of even the most thoroughly trained human


mind, but when we come to use six and seven hundred
places the results are simply astounding. Professor
De Morgan, in his " Budget of Paradoxes," gives the fol-
lowing illustration of the extreme accuracy which might be
attained by the use of 607 fractional places, the highest
number which had been reached when he wrote :

" Say that the blood-globule of one of our animalcules
is a millionth of an inch in diameter. 1 Fashion in thought
a globe like our own, but so much larger that our globe is
but a blood-globule in one of its animalcules ; never mind the
microscope which shows the creature being rather a bulky
instrument. Call this the first globule above us. Let the
first globe above us be but a blood-globule, as to size, in the
animalcule of a still larger globe, which call the second
globe above us. Go on in this way to the twentieth globe
above us. Now, go down just as far on the other side.
Let the blood-globule with which we started be a globe
peopled with animals like ours, but rather smaller, and
call this the first globe below us. This is a fine stretch of
progression both ways. Now, give the giant of the twen-
tieth globe above us the 607 decimal places, and, when he
has measured the diameter of his globe with accuracy
worthy of his size, let him calculate the circumference of
his equator from the 607 places. Bring the little phil-
osopher from the twentieth globe below us with his very
best microscope, and set him to see the small error which

1 What follows is an exceedingly forcible illustration of an important
mathematical truth, but at the same time it may be worth noting that
the size of the blood-globules or corpuscles has no relation to the
size of the animal from which they are taken. The blood corpuscle
of the tiny mouse is larger than that of the huge ox. The smallest
blood corpuscle known is that of a species of small deer, and the
largest is that of a lizard-like reptile found in our southern waters
the amphiuma.

These facts do not at all affect the force or value of De Morgan's
mathematical illustration, but I have thought it well to call the atten-
tion of the reader to this point, lest he should receive an erroneous
physiological idea.


the giant must make. He will not succeed, unless his
microscopes be much better for his size than ours are for

It would of course be impossible for any human mind to
grasp the range of such an illustration as that just given.
At the same time these illustrations do serve in some
measure to give us an impression, if not an idea, of the
vastness on the one hand and the minuteness on the other
of the measurements with which we are dealing. I there-
fore offer no apology for giving another example of the
nearness to absolute accuracy with which the circle has
been " squared/'

It is common knowledge that light travels with a ve-
locity of about 185,000 miles per second. In other words,
light would go completely round the earth in a little more
than one-eighth of a second, or, as Herschel puts it, in less
time than it would take a swift runner to make a single
stride. Taking this distance of 185,000 miles per second
as our unit of measurement, let us apply it as follows :

It is generally believed that our solar system is but an
individual unit in a stellar system which may include hun-
dreds of thousands of suns like our own, with all their
attendant planets and moons. This stellar system again
may be to some higher system what our solar system is to
our own stellar system, and there may be several such
gradations of systems, all going to form one complete whole
which, for want of a better name, I shall call a universe.
Now this universe, complete in itself, may be finite and
separated from all other systems of a similar kind by an
empty space, across which even gravitation cannot exert its
influence. Let us suppose that the imaginary boundary of
this great universe is a perfect circle, the extent of which


is such that light, traveling at the rate we have named
(185,000 miles per second), would take millions of millions
of years to pass across it, and let us further suppose that
we know the diameter of this mighty space with perfect
accuracy ; then, using Mr. Shanks' 707 places of decimal
fractions, we could calculate the circumference to such a
degree of accuracy that the error would not be visible under
any microscope now made.

An illustration which may impress some minds even
more forcibly than either of those which we have just
given, is as follows :

Let us suppose that in some titanic iron-works a steel
armor-plate had been forged, perfectly circular in shape
and having a diameter of exactly 185,000,000 miles, or
very nearly that of the orbit of the earth, and a thickness
of 8000 miles, or about that of the diameter of the earth.
Let us further assume that, owing to the attraction of some
immense stellar body, this huge mass has what we would
call a weight corresponding to that which a plate of the
same material would have at the surface of the earth, and
let it be required to calculate the length of the side of a
square plate of the same material and thickness and which
shall be exactly equal to the circular plate.

Using the 707 places of figures of Mr. Shanks, the length
of the required side could be calculated so accurately that
the difference in weight between the two plates (the circle
and the square) would not be sufficient to turn the scale of
the most delicate chemical balance ever constructed.

Of course in assuming the necessary conditions, we are
obliged to leave out of consideration all those more refined
details which would embarrass us in similar calculations on
the small scale and confine ourselves to the purely mathe-


matical aspect of the case ; but the stretch of imagination
required is not greater than that demanded by many illus-
trations of the kind.

So much, then, for what is claimed by the mathemati-
cians ; and the certainty that their results are correct, as far
as they go, is shown by the predictions made by astrono-
mers in regard to the moon's place in the heavens at any
given time. The error is less than a second of time in
twenty-seven days, and upon this the sailor depends for a
knowledge of his position upon the trackless deep. This
is a practical test upon which merchants are willing to
stake, and do stake, billions of dollars every day.

It is now well established that, like the diagonal and
side of a square, the diameter and circumference of any
circle are incommensurable quantities. But, as De Morgan
says, " most of the quadrators are not aware that it has been
fully demonstrated that no two numbers whatsoever can
represent the ratio of the diameter to the circumference,
with perfect accuracy. When, therefore, we are told that
either 8 to 25 or 64 to 201 is the true ratio, we know that
it is no such thing, without the necessity of examination.
The point that is left open, as not fully demonstrated to
be impossible, is the geometrical quadrature, the determina-
tion of the circumference by the straight line and circle,
used as in Euclid."

But since De Morgan wrote, it has been shown that a
Euclidean construction is actually impossible. Those who
desire to examine the question more fully, will find a very
clear discussion of the subject in Klein's "Famous Problems
in Elementary Geometry." (Boston, Ginn & Co.)

There are various geometrical constructions which give
approximate results that are sufficiently accurate for most


practical purposes. One of the oldest of these makes the
ratio 3y to I. Using this ratio we can ascertain the cir-
cumference of a circle of which the diameter is given by
the following method : Divide the diameter into 7 equal
parts by the usual method. Then, having drawn a straight
line, set off on it three times the diameter and one of the
sevenths ; the result will give the circumference with an
error of less than the one twenty-five-hundredth part or
one twenty-fifth of one per cent.

If the circumference had been given, the diameter might
have been found by dividing the circumference into twenty-
two parts and setting off seven of them. This would give
the diameter. A more accurate method is as follows :

Given a circle, of which it is desired to find the length
of the circumference : Inscribe in the given circle a square,
and to three times the diameter of the circle add a fifth of
the side of the square ; the result will differ from the circum-

t I

G F.

Fig. i.

ference of the circle by less than one-seventeen-thousandth
part of it. Another method which gives a result accurate
to the one-seventeen-thousandth part is as follows :

Let AD, Fig. i, be the diameter of the circle, C the
center, and CB the radius perpendicular to AD. Continue
AD and make DE equal to the radius ; then draw BE, and
in AE, continued, make EF equal to it ; if to this line EF,


its fifth part FG be added, the whole line AG will be equal
to the circumference described with the radius CA, within
one-seventeen-thousandth part.

The following construction gives even still closer results :
Given the semi-circle ABC, Fig 2 ; from the extremities
A and C of its diameter raise two perpendiculars, one of
them CE, equal to the tangent of 30, and the other AF,
equal to three times the radius. If the line FE be then

Fig. 2.

drawn, it will be equal to the semi-circumference of the
circle, within one-hundred-thousandth part nearly. This is
an error of one-thousandth of one per cent, an accuracy
far greater than any mechanic can attain with the tools
now in use.

When we have the length of the circumference and the
length of the diameter, we can describe a square which


shall be equal to the area of the circle. The following is
the method :

Draw a line ACB, Fig. 3, equal to half the circumference
and half the diameter together. Bisect this line in O, and
with O as a center and AO as radius, describe the semi-
circle ADB. Erect a perpendicular CD, at C, cutting the
arc in D ; CD is the side of the required square which can


Fig. 3.

then be constructed in the usual manner. The explanation
of this is that CD is a mean proportional between AC
and CB.

De Morgan says : "The following method of finding the
circumference of a circle (taken from a paper by Mr. S.
Drach in the * Philosophical Magazine,' January, 1863,
Suppl.), is as accurate as the use of eight fractional places:
From three diameters deduct eight-thousandths and seven-
millionths of a diameter ; to the result, add five per cent.
We have then not quite enough ; but the shortcoming is
at the rate of about an inch and a sixtieth of an inch in
14,000 miles."

For obtaining the side of a square which shall be equal
in area to a given circle, the empirical method, given by
Ahmes in the Rhind papyrus 4000 years ago, is very


simple and sufficiently accurate for many practical purposes.
The rule is : Cut off one-ninth of the diameter and construct
a square upon the remainder.

This makes the ratio 3.16.. and the error does not exceed
one-third of one per cent.

There are various mechanical methods of measuring and
comparing the diameter and the circumference of a circle,
and some of them give tolerably accurate results. The
most obvious device and that which was probably the old-
est, is the use of a cord or ribbon for the curved surface
and the usual measuring rule for the diameter. With an
accurately divided rule and a thin metallic ribbon which
does not stretch, it is possible to determine the ratio to the
second fractional place, and with a little care and skill the
third place may be determined quite closely.

An improvement which was no doubt introduced at a
very early day is the measuring wheel or circumferentor.
This is used extensively at the present day by country
wheelwrights for measuring tires. It consists of a wheel
fixed in a frame so that it may be rolled along or over any
surface of which the measurement is desired.

This may of course be used for measuring the circumfer-
ence of any circle and comparing it with the diameter.
De Morgan gives the following instance of its use : A
squarer, having read that the circular ratio was undeter-
mined, advertised in a country paper as follows: "I thought
it very strange that so many great scholars in all ages
should have failed in finding the true ratio and have been
determined to try myself." He kept his method secret,
expecting "to secure the benefit of the discovery," but it
leaked out that he did it by rolling a twelve-inch disk along
a straight rail, and his ratio was 64 to 201 or 3.140625


exactly. As De Morgan says, this is a very creditable piece
of work ; it is not wrong by i in 3000.

Skilful machinists are able to measure to the one-five-
thousandth of an inch ; this, on a two-inch cylinder, would
give the ratio correct to five places, provided we could
measure the curved line as accurately as we can the straight
diameter, but it is difficult to do this by the usual methods.
Perhaps the most accurate plan would be to use a fine wire
and wrap it round the cylinder a number of times, after
which its length could be measured. The result would
of course require correction for the angle which the wire
would necessarily make if the ends did not meet squarely
and also for the diameter of the wire. Very accurate results
have been obtained by this method in measuring the diam-
eters of small rods.

A somewhat original way of finding the area of a circle
was adopted by one squarer. He took a carefully turned
metal cylinder and having measured its length with great
accuracy he adopted the Archimedean method of finding
its cubical contents, that is to say, he immersed it in water
and found out how much it displaced. He then had all
the data required to enable him to calculate the area of the

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Online LibraryJohn PhinThe seven follies of science; a popular account of the most famous scientific impossibilities and the attempts which have been made to solve them. To which is added a small budget of interesting paradoxes, illusions, and marvels → online text (page 2 of 12)