exterior to the circle, and touching it in five points; then it is still evident, tnat as the circle is wholly contained within the pentagon, it must be smaller than that which contains it. But if the pentagon be described within the circular, touching it at the five angular points, then a similar train of reasoning will show that the circle is larger than the pentagon which it contains. Extending our researches on a similar principle, we prove clearly that a circle is always smaller than a polygon of any number of sides exterior to it; but larger than any polygon contained within it; and this would be equally true if the polygon contained a million sides, and presented a contour not differing, to the eye, from

that of a circle.

Now the next stage in the reasoning is-that by means of the most simple geometry, the exact periphery or circumference, and the exact area, of any figure bounded by straight lines, may be determined with rigorous accuracy; and if we draw two polygons,-say of one hundred sides one within and the other without a circle, we can ascertain the exact areas of those polygons, and affirm that the area of the circle is greater than a certain amount, and less than another certain amount. These two amounts, if the number of sides in the polygons be so large as we here suppose, may be so very nearly alike that either one will give the area of the circle with great

closeness.

=

177

=

By means more or less similar to these, Archimedes found, that if the diameter be called 7, then the circumference will be nearly 22, and that if the square of the diameter be called 14, then the area of the circle will be equal to about 11: this computation was slightly in error, and gave to the area of the circle, too great a measure by about part of the whole. The Hindoos, who were in early ages famous for their skill in mathematics, found that if the diameter of the circle 1, then the area would be 31% an approximation much closer than the other, for it was not in error so much as one-thousandth part of a square inch when the diameter of the circle was one foot. At a later period, however, a European mathematician named Metius discovered a method which makes an extraordinary approach to accuracy, and is at the same time the most simple and easy to be remembered of all that have been proposed. He found that if the diameter of a circle be considered equal to 113, then the circumference would equal 355, or if we multiply the square of the radius by 355, and divide by 133, the area will be given. Now this method is so very nearly correct, that the area of a circle one foot in diameter is given within the fifty thousandth part of a square inch. The numbers employed are easily remembered, for they consist of duplications of the first three odd numbers in regular order, 113355.

Other mathematicians carried the approximation yet closer. Ludolph Van Ceulen worked it out to 36 places of figures, showing that, if the diameter be 1, the circumference will be

3.14159265358979323846264338327950288;

or, that if the last figure be 8, the result will be a little below the truth, and if 9, a little above it. Mr. Sharp, an English mathematician of the seventeenth century, carried the approximation to 72 places of figures; Mr. John Machin to 100 figures: and, eclipsing all others, M. de Lagny worked it out to 128 places of figures. Of the degree of accuracy obtained in this last-named computation, Montucla says:-" If we suppose a circle, the diameter of which is a thousand million of times greater than the distance between the sun and the earth, the error in the circumference would be a thousand million times less than the thickness of a hair."

But the important part of the matter has yet to be stated. None of these computations are quite correct. They approach within limits quite near enough for all practical purposes; but they all deviate a little from the

truth; and eminent mathematicians have shown that the rigorous solution is impossible; in other words, that there are no numbers or collections of numbers which will give the exact ratio of the circumference or of the area of a circle to its diameter.

Persons but slightly acquainted with mathematics have however, puzzled their brains to effect this object; and many ludicrous instances have been adduced by Montucla and Hutton to illustrate this subject. The Cardinal de Cusa rolled a cylinder over a plane, till the point which was first in contact with the plane, touched it again; and then, by a train of reasoning wholly destitute of geometrical precision, he endeavoured to determine the length of the line thus described. Oliver de Serres weighed a circle, and also a triangle equal to an equilateral triangle inscribed within the circle, and imagined that the former was exactly equal to two of the latter; forgetting that the double of this triangle is equal to the hexagon inscribed within the circle, and therefore smaller than the circle itself. A Frenchman named Mathoulon, who, from being a manufacturer of stuffs at Lyons, commenced geometer, claimed the merit of having solved this problem, and deposited one thousand crowns as a reward to any person who should prove that his solution was not correct. The proof was given by M. Nicole, a Member of the French Academy of Sciences, who gave the reward to the General Hospital at Lyons.

Another Frenchman announced that he had succeeded in the problem, and challenged the world to refute him, depositing ten thousand livres as the stake. He reduced the problem to the mechanical process of dividing a circle into quadrants, and then turning these with the angles outwards, so as to form a square, which he asserted to be equal to the circle. Three persons claimed the reward, and the cause was tried at the Châtelet of Paris; but the judges declared that a person's fortune ought not to be diminished on account of the errors of his judgment, unless they were preju dicial to society, whereupon the King decreed that the proposal should be void; and the Academy of Sciences recommended the man to study the elements of geometry.

In 1585 a Spanish friar published a most whimsical work; at least so it appears to others, though we doubt not that he deemed it a work of serious importance. It was a dialogue between himself and a circle, in which the latter thanks him very courteously for having solved its quadrature. Others connected the quadrature of the circle with subjects to which it has no kind of relation. For instance one person discovered a method which he thought would not only quadrate a circle, but would also assist in converting Jews, Pagans, and Mohammedans to Christianity; another fancied he saw a connexion between his method and the vision of Ezekiel; a third thought that the doctrines of Original Sin and of the Trinity, were elucidated by this method of quadrating the

circle.

But it would be useless to follow out the subject further; as most of the instances recorded present similar indications of absence of sound judgment and of all geometrical knowledge. Suffice it to say that the quadrature of the circle, in a geometrical sense, cannot be effected; but that all the useful purposes to which such a problem might be applied, can be attained sufficiently near by the approximative methods before alluded to, of which the most useful is, Diameter: 113:: the Circumference: 355, or the Diameter is to the Circumference, as 113 is to 355; for it matters not whether the second and third of the proportion be transposed.

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We have mentioned that the Grand Master of the Temple, Arnold de Torroge, died on his journey to England, as he travelled in company with the patriarch Heraclius, on a mission to Henry II. He was succeeded by Gerard de Riderfort, in 1185. The patriarch returned in disgust and discontent to the Holy Land, for he was unable to persuade the king to perform the promised penance, ie, to hasten to the succour of the distressed Christians at Jerusalem. Before quitting the king he is said to have used threatenings and warnings, and to have foretold that the monarch would be forsaken of God, since he had at that time forsaken His holy cause. But it appears from authentic records, that this same patriarch was ill qualified to deal out these denunciations on the faults of others, since he was himself living in open contempt of the precepts of the moral law. There is a sad detail of vice and profligacy connected with his name, which we have no intention of laying before our readers, but which we thus allude to, to show what was the state of the Latin church at that time. One of the old chroniclers attributes the loss of Jerusalem to such causes as these, and says, "when Jesus Christ saw the iniquity and wickedness which they committed, in the very place Where He was crucified, He could no longer suffer it."

The period of the truce with Saladin was passed by the Templars and the rest of the Christians in ineffectual endeavours to gain sufficient strength for the coming

VOL. XXI.

contest, and, unfortunately, in stormy discussions respecting the government of their own kingdom. Meanwhile Saladin was carefully consolidating and strengthening his power, exhorting his followers, and raising their enthusiasm to the highest pitch. The first encounter took place near the brook Kishon, where one of Saladin's sons (Malek-el-Afdal, Most excellent Prince), at the head of seven thousand Mussulmen, cut to pieces a body of united Templars and Hospitallers, who had rashly engaged with them. The Grand Master of the Hospital was left dead upon the field, and only the Grand Master of the Temple and two knights broke through the ranks of the Moslems and made their escape. This disastrous battle took place on the 10th of May, 1187.

On the 4th of July of the same year the fate of the holy city was decided at the great battle of Tiberias. The Templars led the van of the Christian army against the hosts of Saladin, which amounted to eighty thousand horse and foot, and appeared over the hilly country like "mountains in movement, or like the agitated waves of the sea." The Templars are also described as "horrible in arms, having their whole bodies cased in triple mail." The latter rushed like lions on the infidels and nothing could withstand their furious charge. Saladin set fire to the grass and shrubs between the two armies, and the wind blew the smoke and flames in the faces of the

659

Acre, and the Pilgrim's Castle, a strong fort of the Templars. These were at length invested, and the Grand Master William de Beaujeu took the command of the garrison. The old and feeble were sent away to the island of Cyprus, then the seat of the Latin kingdom, and none remained in the devoted city of Acre, but such as were prepared to suffer martyrdom rather than yield to the infidels. Military engines of the most formidable construction were set in operation by the besiegers: six hundred instruments of destruction were directed against the fortifications, and the battering machines were of such immense size and weight that a hundred wagons were required to transport the separate timbers of one of them. All the military contrivances which the skill of that age could produce were employed to facilitate the assault. After thirty-three days of constant fighting, the great tower or key of the fortifications was thrown down. At length the double wall was forced, and a body of Mamlooks penetrated to the centre of the city. The Knights drove them back with immense carnage, and precipitated their bodies from the walls. At length the number of the Templars was reduced to three hundred, and these fought their way to the strong Temple, at Acre, and shut themselves up. This little band was at length destroyed beneath the mins of their Temple, which the Sultan had caused to be undermined.

Thus fell the last stronghold of the Christians in Palestine, and with it every reasonable hope of recovering possession of the holy city. No sooner was this the case, than the remaining Templars scattered in different countries soon found their order falling into suspicion and neglect. They began to be considered as a useless body, whose large possessions in Europe might be much better, or at least more agreeably employed. A rancorous spirit was awakened among the clergy, who considered the exemptions and privileges of this order as so much detracted from their just rights. So high did this spirit rise among bishops, priests, and the whole body of the clergy, that the Roman pontiff had to express his grief and astonishment at the conduct of the ecclesiastics, and to threaten them with severe measures. And not only were the Templars obnoxious to the Church; they were also unpopular with the European sovereigns and nobles; in fact, they experienced such neglects, unkind suspicions and misrepresentataions, as commonly fall to the lot of those who cease, from whatever cause, to form the stepping-stones to the promotion of others. To join the Templars, at one period, was to rise to honour and renown in this world, and to obtain (as it was falsely believed) a passport to heaven: to succour them in adverse circumstances did not hold out any prospect of worldly advantage, and therefore the claims of mercy and forbearance were soon forgotten. But the Templars themselves were not free from blame: they interfered in the quarrels of different princes; became secular instead of religious warriors; and thus lost the popularity which attached to them as a holy order of men.

At length their ruin was resolved on by a needy and avaricious monarch, Philip the Fair, who longed to possess himself of a portion of their riches. He found an able accomplice in Pope Clement V., a man whose character is painted by Romish historians in the darkest colours. They prevailed on the Grand Master of the Temple to visit France in 1306, with a small retinue, under the pretence of consulting him on measures relative to the recovery of the Holy Land; while secret agents were at work circulating dark rumours respecting the order, and at length the Templars were openly charged with abominable crimes. Upon this, Philip arrested the members of the order and seized on their real and personal property. The accusations were both absurd and horrible; of the former description were those which charged them With worshipping an idol covered with an old skin havng the appearance of a piece of polished oil-clotn, with

carbuncles for eyes, bright as the brightness of heaven; also with burning the bodies of deceased brethren and making the ashes into a powder which they administered to the younger brethren in their food, &c., &c.

The tortures practised on this devoted order, in the hope of getting them in the midst of their agonies to confess to these unjust accusations, were of the most revolting description, and ended in death or madness of numbers; but for a long time they were to no purpose. At length forged letters were produced as if from their superior, the Grand Master, exhorting them to do as their persecutors wished, and under this decception several gave way, and made pretended confessions, whose very improbability destroyed the effect intended to be produced by them. And now a similar scene was to be enacted in England. The accusations were opened against them at the episcopal hall, London, after the most industrious endeavours on the part of the Pope's inquisitors, to substantiate some charge, and after many months' imprisonment of the Templars. These venerable men, after all the hardships and dangers of their past lives, were now degraded and tortured by those who had extolled them to the skies; they had to answer to the most absurd allegations, and to choose suffering of the most dreadful kind, or make a false confession. Nothing could be extorted from them, and after another long season of imprisonment they were called before a second council in 1310. They had been subjected to three years of imprisonment and varied sufferings; torturė had been skilfully applied by French monks, but, as it had been enacted, "without the mutilation or disabling of any limb, and without a violent effusion of blood;' (truly, "the tender mercies of the wicked are cruel;") the Templars knew that by pronouncing a few words of confession they should immediately receive absolution from the Church, and be restored to pardon, peace, and liberty, yet not one of them would depart from the truth, and at the second council they made, instead of a confession, a noble declaration of faith and of their innocency, most unpalatable to the assembled ecclesiastics. Fresh atrocities were then resorted to, and the prisons of the Tower, Aldgate, Ludgate, Newgate, Bishopsgate, and Cripplegate, were the scenes of renewed tortures. At length the constancy of two poor serving brethren was overcome, and their confessions were triumphantly published. An avowal of guilt was also wrung from the chaplain of the order. But the progress made in this way was so slow, that the accusers changed their ground, and endeavoured to convince the Templars of their guilt, in having at former periods received absolution at the hands of the Master, he being a layman. The Templars did not attempt to defend themselves from this charge, and were willing publicly to confess their error in that respect. A sort of compromise was therefore entered into; the Templars made their confession in Latin or Norman French; they were pardoned, and the impression produced on the public mind by their. having made a confession of guilt was sufficient for the purpose of the ecclesiastics.

Their order was now suppressed by the Pope, and their treasures appropriated by the pontiff and the European sovereigns. Their lands were subsequently given to the Knights Hospitallers, but not till enormous claims on the part of the French king and others had been satisfied. The stipend allowed to the surviving Templars was fourpence a day, and this was so irregularly paid that some of them were near dying of hunger, until charity was in some measure awakened on their behalf. Some of them joined the world once more, and presumed to marry, which gave great offence to the Pope, who ordered them to be separated from their wives. It is remarked by historians respecting those who were foremost in the horrid cruelties practised on the Knights. Templars, that every one came to a miserable and untimely end.