ON THE QUADRATURE OF THE CIRCLE.

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, 251found 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 252imagines 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.