ndoubtedly one of the reasons why this problem has received so much attention from those whose minds certainly have no special leaning towards mathematics, lies in the fact 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 came all the way from South America, bringing with him a quadrature of the circle and a newspaper cutting announcing that a reward was ready for the discovery in England. As a matter of fact his method of solving the problem was worthless, and even if it had been valuable, there would have been no reward.

Another case was that of an agricultural laborer who spent his hard-earned savings on a journey to London, carrying with him an alleged solution of the problem, and who demanded 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 forthwith. When he failed to get the money he and his friends were highly indignant and insisted that the influence of the clergy had deprived the poor man of his just deserts!

And it is related that in the year 1788, one of these deluded individuals, a M. de Vausenville, actually brought an action against the French 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 offered for the solution of this or any other of the problems which are discussed in this volume. Upon this point De Morgan has the following remarks:

In the year 1775 the Royal Academy of Sciences of Paris passed a resolution not to entertain communications which claimed to give solutions of any of the following problems: The duplication of the cube, the trisection of an angle, the quadrature of a circle, or any machine announced as showing perpetual motion. And we have heard that the Royal Society of London passed similar resolutions, but of course in the case of neither society did these resolutions exclude legitimate mathematical 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 of those who have attempted its solution, and who have in almost all cases been ignorant both of the nature of the problem and of the results which have been already attained. From Archimedes down to the present time some of the ablest mathematicians have occupied themselves with the quadrature, or, as it is called in common language, "the squaring of the circle"; but these men are not to be placed in the same class with those to whom the term "circle-squarers" is generally applied.

As already noted, the great difficulty with most circle-squarers is that they are ignorant both of the nature of the problem to be solved and of the results which have been already attained. Sometimes we see it explained as the drawing of a square inside a circle and at other times as the drawing of a square around a circle, 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 turf, squared it, and from the result deduced original sin and the Trinity. He found out that the circle was equal to the square in which it is inscribed, and he offered a reward for the detection of any error, and actually 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 AthenÆum for 1855 a correspondent 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 circuit.

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:

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 circle.

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, provided 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 resolves 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 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 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 cistern of seven feet in diameter, would hardly be tolerated even in an apprentice.

The Egyptians seem to have reached a closer approximation, for from a calculation in the Rhind papyrus, the ratio of 3.16 to 1 seems to have been at one time in use. It is probable, however, that in these early times the ratio accepted by mechanics in general was determined by actual measurement, 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 31/7 and 310/71, or between 3.1428 and 3.1408.

This ratio, in its more accurate form of 3.141592.. is now known by the Greek letter p (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 difficulties 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 EncyclopÆdia Britannica, makes the following statement on this point:

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 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 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 attributed to Bovillus, by Montucla.

By calculating the perimeters of the inscribed and circumscribed polygons, Vieta (1579) carried his approximation to ten fractional places, and in 1585 Peter Metius, the father of Adrian, by a lucky step reached the now famous fraction 355/113, 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 monument 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:

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 appreciated 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 following 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:

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 therefore 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 velocity 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 hundreds 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 mathematical aspect of the case; but the stretch of imagination required is not greater than that demanded by many illustrations of the kind.

So much, then, for what is claimed by the mathematicians; and the certainty that their results are correct, as far as they go, is shown by the predictions made by astronomers 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 determination 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 31/7 to 1. Using this ratio 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, 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 circumference 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. 1, 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 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 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 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 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 circumference 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 undetermined, 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 1 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 diameters 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 circle upon which the cylinder stood.

Since the straight diameter is easily measured with great accuracy, when he had the area he could readily have found the circumference by working backward the rule announced by Archimedes, viz.: that the area of a circle is equal to that of a triangle whose base has the same length as the circumference and whose altitude is equal to the radius.

One would almost fancy that amongst circle-squarers there prevails an idea that some kind of ban or magical prohibition has been laid upon this problem; that like the hidden treasures of the pirates of old it is protected from the attacks of ordinary mortals by some spirit or demoniac influence, which paralyses the mind of the would-be solver and frustrates his efforts.

It is only on such an hypothesis that we can account for the wild attempts of so many men, and the persistence with which they cling to obviously erroneous results in the face not only of mathematical demonstration, 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 bring the circumference to 3.1416 inches, which is a corroboration of the orthodox ratio (3.14159) sufficient to show that any value which is greater than 3.142 or less than 3.141 cannot possibly be correct.

And in regard to the area the proof is quite as simple. It is easy to cut out of sheet metal a circle 10 inches in diameter, and a square of 7.85 on the side, 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, it will be found that the circle and the square will exactly balance each other in weight, thus proving in another way the correctness of the accepted ratio.

But although even as early as before the end of the eighteenth century, the value of the ratio had been accurately determined to 152 places of decimals, the nineteenth century abounded in circle-squarers who brought forward the most absurd arguments in favor of other values. In 1836, a French well-sinker named Lacomme, applied 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 unpractical statement by the professor, set the well-sinker to thinking; he studied mathematics after a fashion, and announced that he had discovered that the circumference was exactly 31/8 times the length of the diameter! For this discovery (?) he was honored by several medals of the first class, bestowed by Parisian societies.

Even as late as the year 1860, a Mr. James Smith of Liverpool, took up this ratio 31/8 to 1, and published several books and pamphlets in which he tried to argue for its accuracy. He even sought to bring it before the British Association for the Advancement of Science. Professors De Morgan and Whewell, and even the famous mathematician, 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 circle. I think you are mathematician enough to do this. You will find that if the radius of the circle be one, the side of the polygon is .264, etc. Now the arc which this side subtends is, according to your proposition, 3.125/12 = .2604, and, therefore, the chord is greater than its arc, which, you will allow, is impossible."

This must seem, even to a school-boy, to be unanswerable, but it did not faze Mr. Smith, and I doubt if even the method which I have suggested previously, 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 method even a common pair of grocer's scales will show to any common-sense person the error of Mr. Smith's value and the correctness of the accepted ratio.

Even a still later instance is found in a writer who, in 1892, contended in the New York "Tribune" for 3.2 instead of 3.1416, as the value of the ratio. 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 announcement gave rise to considerable discussion, and even towards the dawn of the twentieth century 3.2 had its advocates as against the accepted ratio 3.1416.

Verily the slaves of the mighty wizard, Michael Scott, have not yet ceased from their labors!