66 : 66 mical notation; while it is to the Arabs that Europeans are indebted for the knowledge and for the name of algebra. The science was first introduced into Europe in the begining of the thirteenth century, by Leonardo Bonacci, a merchant of Pisa, who obtained his knowledge from the Arabs or Moors and a curious illustration of its transmission through an Italian medium, as well as of its original forms, is found in the name of "cossic art," or rule of coss," by which it is often called by old writers. Thus, in 1557, appeared a book by Robert Recorde, a native of Wales, entitled, "The Whetstone of Witte, which is the seconde parte of Arithmeticke, containing the Extraction of Rootes, the Cossike Practice, with the Rules of Equations, and the Workes of Surde Nombres." The name of coss or cossic, is in fact the Italian cosa, "thing," which was used to signify the unknown quantity, for which the letter x is now employed. For a long time no one thought of employing letters to signify quantities, and symbols or marks to express operations; but these were gradually introduced in the shape of abbreviations. "Cosa" signified the first power of the unknown quantity; censa," the second power or square, which in English was corrupted into zenzig, or zensus, and "cubo" or cubus," signified the cube or third power. The celebrated Cardan, who died at Rome, in 1576, made great advances in algebra, and sometimes used the letters of the alphabet to express quantities, but his cotemporary Tartaglia of Brescia, though less famous, was more profound. Both men made important discoveries for finding the roots of cubic equations. The latter used the letter R, as a contraction for radix or root; and the character (√) employed for the purpose at the present day, is, in fact, only a form of the same letter. Other writers on algebra appeared very nearly at the same time indifferent parts of Europe; but the man, who, at that period, contributed most to the improvement of the science, was Francis Vieta, born at Fontenai, in Poitou, in 1540. He it was, who first introduced the general use of the letters of the alphabet to express quantities, but he only employed the capitals, and expressed known quantities by the consonants, and the unknown ones by vowels. He gave new demonstrations of the rule for resolving cubic and even biquadratic equations, and introduced several other improvements in the science; which, after making further strides under Albert Girard, in Flanders, and Thomas Harriot, in England, was carried to a wonderful degree of perfection, by the immortal Des Cartes, who was born in 1596, and died in 1650. The mathematical works of this great philosopher consist chiefly of the application of algebra to geometry, and to the investigation of curves. gebra had now acquired a regular and permanent form, and the writers by whom it was treated, either as a whole or in detached parts, became so numerous, that the mere mention of their names and order of succession, would carry us beyond the limits which we have proposed to ourselves in this paper. Suffice it to say, that among the galaxy of mathematical writers, who flourished during the seventeenth century, and the first half of the eighteenth, we find such stars of the first magnitude as Des Cartes, Kepler, Baron Napier, Cavalieri, Brounker, Fermat, Pascal, Wall Bernouilli, Barrow, Hooke, Huygens, L'Hopital, Leibnitz, and New: followed by a host of other celebrated men, to whom the wonderful Al coveries of the two last named philosophers opened up new fields of science. There is assuredly a fullness of time for every great step of intellectual progress. We sometimes hear that a man has lived before his time, but very seldom that a great scientific discovery has been made before the wants of the age, which it is calculated to meet, have been ripe for its appearance. Almost every important discovery and advance, in science or art, has been preceded by a chain of other discoveries and advances, by which the minds of men have been prepared for it; and these precursor discoveries often set a similar train of ideas in motion in the same direction, n different minds, separated by distance and unconnected by any medium of communication. Thus, fluxions and the differential calculus, the greatest discoveries made in mathematical science in modern times, and both, in a manner, identical in nature, were made almost at the same moment by Newton and Leibnitz, as a result of perfectly independent action in those two mighty intellects. Yet the two discoveries had certain antecedents, which conducted to them by a certain natural course, and without which it is possible that neither of them would have been brought to light. The method of indivisibles of Cavalieri, Wallis's arithmetic of infinites, and Fermat's theory of maxima and minima, were natural precursors of the infinitessimal differences of the new calculus; and Napier's supposed generation of the logarithmic curve by the equable motion of a variable line, naturally set the mind of Newton on the track of his theory, according to which all magnitudes are supposed to be generated by the motion of points, or lines, or surfaces, variable or invariable. Had not the other theories or methods been previously invented, it may be a question whether fluxions or the differential calculus would ever have been thought of; although there can be hardly any question that the pre-existing systems would never have led to these great results had not such minds as those of Newton and Leibnitz been sent by Providence at the proper juncture to take advantage of them. There appears, then, to be no sufficient ground for the controversy between English and continental scholars as to the priority of these two great discoveries. Newton was the first discoverer in point of time, having laid down the principles of his fluxions in a treatise written before 1669, but not published till many years after; while Leibnitz, in the interval, made the same discovery, wholly independent of that of Newton, to which it obtained priority of publication. It is also to be observed that the notation and method of Leibnitz were the best, and have now been universally adopted, so that Newton's fluxions have given way to the continental calculus of differentials even in England. When Columbus discovered America he only prepared the way for a host of explorers, who made fresh discoveries every day; yet, all these later discoveries only served to enhance and give effect to the great discovery of all. In the same way, in the march of mathematical discovery, the new fields opened by Newton and Leibnitz have been explored in every direction, and with extraordinary diligence and utility by a multitude of learned men; but there would, nevertheless, seem to have been a grand halt in the progress of science when those two illustrious philosophers had performed their work. They taught other minds to follow them in their upward flight; but no mind has yet been able to soar higher. This consideration should teach us to respect the learning of past times, and to draw a wide distinction between the general diffusion of knowledge and its original creation or discovery. A thousand facts in science, of which Newton was ignorant, may be known to any ordinary schoolboy; but yet no one would place the schoolboy on a par with Newton. If we are proud of our knowledge, the thought of the little trouble by which we have obtained it, and the little we have done to improve it, might well teach us some humility. We should recollect that thousands of years ago poetry and eloquence, painting, sculpture, and architecture, attained a perfection that has never since been excelled; and mathematical science would also appear to have reached something like its culminating point a long time ago. WORK WHILE IT IS CALLED TO-DAY. Listless falling in idleness down; Men in the silent market-place grouping "No man hath hired us"-pale hands twining, "The red dawn has pass'd, the noon is shining, Then a voice pealed down from the heights of Heaven, I gave you a land as a garden of Eden, Where you and your sons should till and toil; I set your throne by the glorious waters, Where ocean flung round you her mighty bands, "Power I gave to the hands of your leaders, your children grew as the stately cedars That shadowed the streams of Paradise. Has the wild boar trampled my vineyard down?' "True," they answered, faint in replying "Our vines are rent by the wild boar's tusks, The corn on our golden slopes is lying, But our children feed on the remnant husks. Our strong men lavish their blood for others; Our prophets and wise men are heard no more; "From wooded valleys and mountain gorges, Yet, the chant we hear of the new evangels, Then the Lord came down from the heights of Heaven, Came down that garden fair to view, Where the weary men waited from morn till even For some one to give them work to do. "Ye have sinned," he said, and the angel lustre Darkened slowly as summer clouds may, "Weeds are growing where fruit should cluster, Yet, ye stand idle all the day. Have ye trod in the furrows, and worked as truly As men who knew they should reap as they sow? Have ye flung in the seed and watched it duly Day and night lest the tares should grow? To strengthen the fruit its branches bear? "Have ye striven in earnest, working solely To guard my flock in their native fold? And whatever is right shall my angels give ye, "Who knoweth the times of the new dispensations? The last may be first! Shall our country's glory SPERANZA. I HAD minutely inspected every preparation, and I saw that she felt a kind hand had been there. I proposed to her to remain upstairs for the evening; she gladly assented, and I already saw a shade of disappointment in Marguerite's face. It was no wonder; there was an indescribable charm about this lady. It did not consist only in her beauty, though that every moent grew upon my perception, and before I left the room I thought I had ever seen such loveliness before, and tried to imagine what its lustrous majesty must have been before a profaning finger had been impressed upon it. Every movement was the perfection of grace and refinement, every word was melodious. She said she preferred being unattended by Louise, and would make all her own arrangements. "Oh! Miss Armytage," cried Marguerite, 66 may I stay; I can unpack Mrs. Ross's trunks very well, and help her to dress?" The child's face was turned from her, she slightly shook her head, and I said: “No, dear; Mrs. Ross is tired, and better alone; besides, you surely would not leave your papa; you shall bring up some tea if you wish." This contented her, and I drew her away. We joined Mr. Lydyard. I need have given myself very little uneasiness as to what I should say, for Marguerite enjoyed an entire monopoly of the discourse. How beautiful she thought Mrs. Ross, how delightful that she had come, how sad she looked, how much Maud would like her, how much she wished Maud would come home. Had Mrs. Ross any children? 66 "She has had, but years ago," said Mr. Lydyard, with wonderful firmness. And they are dead, too," said Mag, "and her husband—we must not talk about them though, unless she does to us." "I wonder if she will ever tell us anything about them? I thought she had children though, papa, by the way she kissed me." He turned to the window, and remarked that the days were sensibly shorter. Then I sent Marguerite, according to promise, to take Mrs. Ross her tea; she lingered upstairs a long while, leaving her father and r VOL. V. NEW SERIES. |