Pierre de Fermat

“ But no one before Pascal and Fermat had given the principles and the methods for submitting this subject to calculus, and no one had solved the rather complicated questions of this kind. It is, then, to these two great geometricians that we must refer the first elements of the science of probabilities, the discovery of which can be ranked among the remarkable things which have rendered illustrious the seventeenth century—the century which, has done the greatest honor to the human mind. ”

“ These facts were thought by some to be instances and results of the simplicity and economy of nature; and Fermat, whose researches on maxima and minima are claimed by the Continental mathematicians as the germ of the differential calculus, sought anxiously to trace some similar economy in the 613 more complex case of refraction. He believed that by a metaphysical or cosmological necessity, arising from the simplicity of the universe, light always takes the course which it can traverse in the shortest time. ”

“ A former friend of their father,—the illustrious chemist, Michel Froment,—Bertheroy had now, in his turn, become one of the loftiest glories of France, one to whom chemistry owed much of the extraordinary progress that has made it the mother-science, by which the very face of the earth is being changed. A member of the Institute, laden with offices and honours, he had retained much affection for Pierre, and occasionally visited him in this wise before dinner, by way of relaxation, he would say. ”

“ Hobbs, upon whose account he wrote several Letters to Mersennus, containing many remarks conducing to the Knowledge of the Nature of Reflection and Refraction. But the Person, that did most learnedly and resolutely attack the said Dioptricks, was Monsieur Fermat, {393}writing first about it to Mersennus, who soon communicated his Objections to M. Des-Cartes, who failed not to return his Answer to them. But Fermat replied, and Des-Cartes likewise; and after many reciprocations, in which each party pretended to have the advantage, the matter rested ”

“ Fermat died with the belief that he had found a long-sought-for law of prime numbers in the formula = a prime, but he admitted that he was unable to prove it rigorously. The law is not true, as was pointed out by Euler in the example = 4,294,967,297 = 6,700,417 times 641. The American lightning calculator Zerah Colburn, when a boy, readily found the factors, but was unable to explain the method by which he made his marvellous mental computation. ”

“ On asking if M. Thomas Froment were there, Pierre was led by an old workman to a little shed, where he found the young fellow in the linen jacket of a mechanician, his hands black with filings. He was adjusting some piece of mechanism, and nobody would have suspected him to be a former pupil of the Lycee Condorcet, one of the three clever Froments who had there rendered the name famous. But his only desire had been to act as his father's faithful servant, the arm that forges, the embodiment of the manual toil by which conceptions are realised. ”

“ Later, Scotland brought forth Napier, the inventor of logarithms. The instantaneous appreciation of their value is doubtless the result of superiority in calculation. In Italy, and especially in France, geometry, which for a long time had been an almost stationary science, began to be studied with success. Galileo, Torricelli, Roberval, Fermat, Desargues, Pascal, Descartes, and the English Wallis are the great revolutioners of this science. Theoretical mechanics began to be studied. The foundations were laid by Fermat and Pascal for the theory of numbers and the theory of probability. ”

“ Every number is either a triangular number or the sum of two or three triangular numbers; either a square or the sum of two, three, or four squares; either a pentagonal number or the sum of two, three, four, or five pentagonal numbers; similarly for polygonal numbers in general. The proof of this and other theorems is promised by Fermat in a future work which never appeared. This theorem is also given, with others, in a letter of 1637 (?) ”

“ René François de Sluze (1622–1685) and Johann Hudde (1633–1704) made some improvements on Descartes' and Fermat's methods of drawing tangents, and on the theory of maxima and minima. With Hudde, we find the first use of three variables in analytical geometry. He is the author of an ingenious rule for finding equal roots. We illustrate it by the equation . Taking an arithmetical progression 3, 2, 1, 0, of which the highest term is equal to the degree of the equation, and multiplying each term of the equation respectively by the corresponding term of the progression, we get , or . ”

“ It appears therefore that the honour of having first given a good explanation of the before-quoted passage of Synesius belongs rather to Constantin than to Fermat; but I can readily believe that the latter was not acquainted with the observations made on it by the former. Before I conclude the history of this instrument among the ancients, I shall add two remarks further. It is evidently wrong when one, with Muschenbroek and others, whose opinion I adopted before I engaged in this research, considers Hypatia as the inventress of the hydrometer. ”

“ Peter the Hermit was of small stature, with long beard prematurely whitened by the rigors of his life,—for he was not yet fifty years of age,—with deep and penetrating eyes, fired by the enthusiasm that filled his soul. ”

“ Painters in those days had not the standing in society that they have now, but the voice and manner were unmistakably those of a man of breeding. Even his freedom was not the upstart licence of one trying to assert himself, but the easy expression of a roving imagination.
“I should introduce myself too,” said Logie. “I am Captain James Logie, Lord Balnillo’s brother. But we must rescue your—your—baggage. Where is your postilion?”
Flemington held up the lantern again, and its rays fell upon a man holding the two horses which were standing together under a tree. James went towards them. ”

“ It seemed that he would swoon, and Feversham took him by the arm.
"Is it true?" Trench asked faintly; and before Feversham could answer Abou Fatma went on:—
"Walk forwards very slowly. Before you reach the end of the wall it will be dusk. Draw your cloaks over your heads, wrap these rags about your chains, so that they do not rattle. Then turn and come back, go close to the water beyond the storehouses. I will be there with a man to remove your chains. But keep your faces well covered and do not stop. He will think you slaves." ”

“ Even as Dale turned to face the assembled household, the house-lights, extinguished since the storm, came on in full brilliance—revealing her to them, standing beside Fleming's body with Miss Cornelia's revolver between them.
She shuddered, seeing Fleming's arm flung out awkwardly by his side. No living man could lie in such a posture. ”

“ Standing upon that verandah, with the quiet pool of water in front of him, Feversham had told his story quietly and without exaggeration. He had related how he had fallen in with Abou Fatma at Suakin, how he had planned the recovery of the letters, how the two men had travelled together as far as Obak, and since Abou Fatma dared not go farther, how he himself, driving his grey donkey, had gone on alone to Berber. ”

“ And from it hath been named Loch Guala Umai in the island of Daim, which is in the realm of Ulster; for underneath the lake unto this day is that vat, hidden in a secret place. Also Conor took with him the queen, even Nuagel, the daughter of Erg, and her daughter Ferb, and three times fifty maidens with her. And immediately after this, Ferb and her maidens all died from the sorrow that they felt at the death of the young man Mani; Nuagel also died of grief for the death of her husband and her two sons. ”

“ I left the Scots Brigade and took service with Russia, and I joined Peter Lacy, who was on his way to fight in Poland. Fighting was all I wanted, and God knows I had it. I did not want to be killed, but to kill. Then I grew weary of that, but I still stayed with Lacy, and followed him to fight the Turks. We outlive trouble in time, Flemington; we outlive it, though we cannot outlive memory. We outlast it—that is a better word. ”

“ The hurried words that Archie had spoken about Logie as they left the hovel together had told him little, and that little seemed to him inadequate to explain the tremendous consequences that had followed. What had Logie said or done that had power to turn him out of his way? A man may meet many admirable characters among his enemies without having his efforts paralyzed by the encounter. Flemington was not new to his trade, and had been long enough in the secret service to know its requirements. ”

“ He held it to be dishonourable that any man at all in the world should come with numbers no greater than his own and take the house where he was; therefore he and his warriors chased the Fomorians, and they drove them from the palace. In that hour the hand of Mani was not the hand of a healer, for nine of the men of the Fomorians fell at his first attack. And then to the front of the fight came the pirate of Greater Asia—even Fabric of the venomous tooth ”

“ On the very next evening Chippet and I had our first meal in the house of the Flemings. Because I am a realistic novelist, there is nothing, I suppose, in this modern world of ours that can possibly surprise me. The Flemings' house, Mr. Fleming, the numerous little Flemings, and the great Mrs. Farbman herself, were all exactly as I had expected them. There is something to me strangely depressing about Golders Green, and the Flemings' house would have been depressing anywhere. ”

“ My heart is broken for you, boy!
Then came Fiannamail, the son of Fergus Forderg, to the castle, and three times fifty was the number of the warriors that were with him. And a herald who preceded him gave to Ferb the news of his coming, and to Fiannamail he brought back a sad message to tell him of the deeds which had been done. Fiannamail immediately flew into a rage, and sought eagerly for tidings of where Conor was to be found, and he and Ferb between them made this song:—
VIII
Ferb. ”

“ Homeless women and children had fled down its secret tracks; hunted men had given up their souls under its heights. The rich plainland of Angus had sent its sons to fight for the Prince in the North, and of those who survived to make their way back to their homes, many had been overtaken by the pursuit that had swept down behind them through the hills. No place had a darker record than Glen Esk.
Archie Flemington rode down the Glen with his companion some little way in front of the corporal and the three men who followed them. ”

“ Night fell on the barn at last. Wattie slept sound, with the yellow cur’s muzzle against his shoulder; but he awoke towards midnight, for Archie’s feverish voice was coming from the corner in which he lay. He inclined his ear, [Pg 202] attracted by the recurrent name of Logie which ran through the disconnected babblings, rising again and again like some half-drowned object carried along a swift stream. The darkness made every word seem more distinct.
“Listen to me!” cried Flemington. “Logie! Logie! you do not understand . . . it is safe . . . it is burnt! ”

“ Another apparent evidence of date, which Windisch notes in his preface, is the higher literary quality of the Connaught version. Both versions are apparently, at least, as old as the tenth century, but the Ulster or ballad version and the Egerton one, which in the main agrees with it, tell the story baldly; while the Connaught story-teller seems to have amplified several suggestions in the original tale: in the laments of Nuagel over Gerg, and of Ferb over Mani, also in the dramatic dialogues between Ferb and the two Connaught heroes, Fiannamail and Donnell the Red, much power is shown. ”

“ Charles Prevost answered his doubts and questions with real and kindly anxiety to enlighten him, now explaining curious things, now making instructive remarks which opened new and wide horizons before the boy's virgin mind. But in spite of the Fleming's kindness, which sometimes seemed natural and at other times only courtly manners which had become a second nature, the child's innate sharpness showed him that Prevost always hid him from the gaze of the people ”

“ The first peculiarly mediaeval type that contrived to secure a foothold in eighteenth-century literature was the hermit, a figure which seems to have had a natural attraction, not only for romanticizing poets like Shenstone and Collins, but for the whole generation of verse writers from Parnell to Goldsmith, Percy and Beattie—each of whom composed a "Hermit"—and even for the authors of "Rasselas" and "Tom Jones," in whose fictions he becomes a stock character, as a fountain of wisdom and of moral precepts. ”

“ There was no more about Logie, and Wattie could only make out the word ‘Ardguys,’ which he knew as the name of a place beyond Forfar; and as he had discovered in Brechin that Flemington lived somewhere in those parts, he guessed that his thoughts were roving about his home. His breathing grew less laboured, and the watcher could hear at last that he slept. The moon dropped, and with her going the crevices lost their greyness and the barn grew black. The beggar, who was a healthy sleeper, laid himself down again, and in the middle of his cogitations passed into oblivion. ”