The Seven Follies of Science [2nd ed.]

ct that there is a general impression abroad that the governments of Great Britain and France have offered large rewards for its solution. De Morgan tells of a Jesuit who

problem was worthless, and even if it had be

anded from the Lord Chancellor the sum of one hundred thousand pounds, which he claimed to be the amount of the reward offered and which he desired should be handed over fo

Academy of Sciences to recover a reward to which he felt himself entitled. It ought to be needless to say that there never was a reward offere

ually prevalent among the same class in England. No reward has ever been offered by the government of either country. The longitude problem in no way depends upon perfect solution; existing approximations are sufficient to a point of accuracy far beyond what can be wanted. And geometry, content with what exists, has

ical investigations-the famous computations of Mr. Shanks, to which we shall have occasion to refer hereafter, were submitted to the Royal Society of London and published in their Transactions. Attempts to "square the circle," when made intelligently, were not only commendable but have been productive of the most valuable results. At the same time there is no problem, with the possible exception of that of perpetual motion, that has caused more waste of time and effort on the part

ircle, but both these problems are amongst the very simplest in practical geometry, the solutions being given in the sixth and seventh propositions of the Fourth Book of Euclid. Other definitions have been given, some of them quite absurd. Thus in France, in 1753, M. de Causans, of the Guards, cut a circular piece of t

" He makes the square equal to a circle by making each side equal to a quarter of the circumference. As De Morga

phanes, in the "Birds," introduces a geometer, who announces his intention t

s alone was

re material ch

ace lifts her

und the circle,

fruitless attempts of squaring the circle." The poetic idea

roblem is this: To describe a square which sha

of a circle is found and expressed numerically in square measure, and (2) the geometrical quadrature,

ract 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 difficult

e circumference and whose altitude or height is equal to the radius. Therefore if we can find the length of the circu

wn to mathematicians as that of

olten 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 1, 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 Babylonia

t one time in use. It is probable, however, that in these early times the ratio accepted by mechanics in general was determined by actual measureme

nner; he took the circumference of the circle as intermediate between the perimeters of the inscribed and the circums

, a symbol which was introduced by Euler, between 1737 and 1748, and which is now adopted all over the world. I have, howeve

ble piece of work; the immature condition of arithmetic, at the time, was the only re

y are usually called, nor any system equivalent to our decimal system, was known to these

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 a

CVIII by MDLVII, without using Arabic or common numerals. Professor McArthur, in his article

labor. The notation of the Romans, in particular, could adapt itself so ill to arithmetical operations, that nearly all their calculation

ven by Ptolemy, in the "Great Syntaxis." He made the

to the conclusion that the square root of 10 was the true value of the ratio. He was led to this by calculating the perimeters 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

fractional places, and in 1585 Peter Metius, the father of Adrian, by a lucky step reached the now famous fraction 355?11

This result, which "in his life he found by much labor," was engraved upon his tomb

us in 1705, Abraham Sharp carried it to 72 places; Machin (1706) to 100 places; Rutherford (1841) to 208 plac

hat used on this page, these figures

he value of the ratio of the circumfe

793 23846 26433

09 74944 59230

11 70679 82148

84

ne-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 appreciated by a consider

d seven hundred places the results are simply astounding. Professor De Morgan, in his "Budget of Paradoxes," gives the following illustration

ule 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 twentieth globe above us the 607 deci

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

ly 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 ru

ole 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

nds even more forcibly than either of tho

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 o

d so accurately that the difference in weight between the two plates (the circle and the square)

which would embarrass us in similar calculations on the small scale and confine ourselves to the purely mathematical aspec

onomers 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

monstrated 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 th

ssible. Those who desire to examine the question more fully, will find a very clear discuss

tio we can ascertain the circumference 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,

dividing the circumference into twenty-two parts and setting off seven of

s the diameter of the circle add a fifth of the side of the square; the result will differ from the circumference of the circle by less

g.

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 add

hem CE, equal to the tangent of 30°, and the other AF, equal to three times the radius. If the line FE be then drawn, it will be equal to the semi-circumference of the circle

g.

of the diameter, we can describe a square which shall be e

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

g.

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 t

given by Ahmes in the Rhind papyrus 4000 years ago, is very simple and sufficiently accurate for many pra

and the error does not excee

nd that which was probably the oldest, 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 meta

This is used extensively at the present day by country wheelwrights for measuring tires. It consists of a w

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

er, 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 req

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 fo

d the circumference by working backward the rule announced by Archimedes, viz.: that the area of a circle is equal

been laid upon this problem; that like the hidden treasures of the pirates of old it is protected from the attacks of ordina

nstration, but of practical mechanical measurements. For even when working in wood it is easy to measure to the half or even the one-fourth of the hundredth of an inch, and on a ten-inch circle this will b

e, or even one-thousandth of an inch closer to the standard 7.854. Now if the work be done with anything like the accuracy with which good machinists work,

d to a professor of mathematics for information in regard to the amount of stone required to pave the circular bottom of a well, and was told that it was impossible "to give a correct answer, because the exact ratio of the diameter of a circle to its circumference had never been determined"! This absolutely true but very unpr

athematician, Sir William Rowan Hamilton, tried to convince him of his error, but without success. Professor Whewell's demonstration is so neat and so simple that I make no apology for giving it here. It is in the form of a letter to Mr. Smith: "You may do this: calculate the side of a polygon of 24 sides inscribed in a cir

ously, viz., that of cutting a circle and a square out of the same piece of sheet metal and weighing them, would have done so. And yet by this metho

He announces it as the re-discovery of a long lost secret, which consists in the knowledge of a certain line called "the Nicomedean line." This announc

wizard, Michael Scott, have no

TNO

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

ical illustration, but I have thought it well to call the attention of the