medes, we find a species of knowledge which, we have every reason to believe, did not originate with these celebrated authors; but which, owing to the very imperfect state of scientific history among the ancients, cannot be traced to remoter times, or apportioned among the numerous enquirers to whose industry and genius we actually owe it. Mr. Playfair has, on the whole, accomplished his undertaking with great sucess; so far, at least, as he was able to proceed with it, before his last illness put a stop to his valuable labours. It is known to the reader that the Professor did not live to execute the whole of his plan, and that what is already published of his Dissertation is all that he left fit for the press, or which is to appear as his avowed work. We trust that the interesting portion of the history which yet remains to be performed will be put into the hands of a writer who, imitating the rare qualities of his predecessor, will study above all things the clearness and precision which distinguish his composition; and avoid that pompous declamatory style which, in other contributions to the same Miscellany, so frequently obscures the sense, and mocks the reader with words while he is in search of thoughts. According to the plan sketched out by Mr. Playfair for giving the history of science from the revival of letters to the beginning of the nineteenth century, he confines himself in the First Part of his Dissertation to the period preceding the end of the seventeenth century; or, as he himself expresses it, to that preceding the invention of the fluxionary calculus, and the discovery of the principle of gravitation;—one of the most remarkable epochas, without doubt, in the history of human knowledge. 1 In the Second Part, which was meant to comprehend a review of the progress of science from the period when Newton made known his great discoveries down to the present times, there are no fewer than three sub-division or minor epochs; which, in the view of the Author, were rendered necessary by the rapid advancement of science, and the great variety of subjects to which natural philosophy has extended its enquiries. Were, says he, the history of any particular science to be continued for the whole of the busy interval which this second part embraces, it would leave the other sciences too far behind; and would make it difficult to perceive the mutual action by which they have so much assisted the progress of one another. 66 Considering some sort of sub-division, therefore, as necessary, and observing in the interval which extends from the first of Newton's discoveries to the year 1818, three different conditions of the Physico-Mathematical sciences, all marked and distinguished by great improvements, I have divided the above interval into three corresponding parts. The first of these reaching from the commencement of Newton's discoveries in the year 1663 to a little beyond his death, or to 1730, may be denominated from the men who impressed on it its peculiar character, the period of Newton and Leibnitz. The second which for a similar reason I call that of Euler and D'Alembert, may be regarded as extending from 1730 to 1780; and the third, that of Lagrange and Laplace, from 1780 to 1818. The history of the pure Mathematics comes first in order, which is succeeded by that of Algebra. In the former department, the revival of learning brought to light a treasure in the works of Euclid and Apollonius so rich and complete that very little has been added to it by the successful industry of modern times. The fifteenth century, accordingly was illustrated not so much by the original genius of its authors as by their unwearied zeal in collecting and translating the manuscripts of Grecian geometers, and in thus rendering familiar to their contemporaries the elegant science of the Platonic school. Regiomontanus, the most distinguished author of the age now specified, was particularly successful in this useful toil; and we find that, besides many translations and commentaries for which the scientific world was indebted to him, he added a gift of perhaps more practical value than any which he had derived from the treasures of ancient Greece, the Trigonometry of the Arabians. In the hands of this learned mathematician, the science now named advanced to a great degree of perfection, and approached very near to the condition which it has attained at the present day. He was also the inventor of decimal fractions, or was at least the first to introduce that useful contrivance into arithmetic: thereby giving to numerical computation the utmost degree of simplicity and enlargement which it seems capable of attaining. The sixteenth century passes without any very distinguished name in mathematical science, if we except Mauroclycas of Messina, who is said to have written a treatise on conic sections, as well as to have restored the fifth book of Apollonius on that subject. Like all the mathematicians of his time, however, he was not satisfied with the legitimate province of geometry, nor disposed to confine his researches to the actual phenomena of nature. He laboured to discover from physical facts the future events of moral and civil life: and thus, amidst the clearest proofs of a strong understanding and more than common learning, there remains indubitable evidence that he dealt in astrological prediction. The fame and writings of Cavalieri throw a pleasing lustre over the commencement of the seventeenth century. He was born at Milan in the year 1598; and, having manifested a strong predilection for mathematical studies, he succeeded in distinguishing himself by the invention of a new method for determining the lengths and areas of curves, as also the contents of solids bounded by curved superficies. The principle on which he proceeded, says Mr. P. was that areas may be considered as made up of an infinite number of parallel lines; solids of an infinite number of parallel planes; and even lines themselves, whether curve or straight, of an infinite number of points. The cubature of a solid being thus reduced to the summation of a series of planes, and the quadrature of a curve to the summation of a series of ordinates, each of the investigations was reduced to something more simple. It added to this simplicity not a little, that the sums of series are often more easily found when the number of terms is infinitely great, than when it is finite and actually assigned. At this period the geometrical sciences were advancing so fast in the hands of Kepler, Cavalieri, and Torricelli, that it is not a little difficult to assign to these authors the exact degree of merit to which they were entitled for the several discoveries which were successively brought to light. The quadrature of the cycloid, for example, was disputed by the last named of the above mathematicians, and Roberval, a French writer of considerable originality and invention; and the question of priority in the invention of the elegant problem alluded to, still remains undetermined. The claims of the two philosophers roused the national feeling of France and Italy; and the zeal of each in pursuing the controversy, has so perplexed the point at issue, that it is now extremely difficult to say on which side the truth is to be found. Torricelli, as the author observes, was a man of a mild, amiable, and candid disposition; Roberval, of a temper irritable, violent, and envious; so that in as far as the testimony of the individuals themselves is concerned, there is no doubt which ought to preponderate. ALGEBRA is the subject of which the history is next brought under review. In regard to the origin of this most ingenious art much obscurity still prevails among the learned; and the most recent inquiries have not contributed much to gratify our curiosity on that interesting head. In the work of Bombelli, an Italian algebraist, there is a notice purporting that he had seen in the Vatican library, a manuscript of a certain Diophantus, a Greek author, which he admired so much that he had formed the design of translating it. He adds that in this manuscript he had found the Indian authors often quoted; from which it appeared, by a very obvious inference, that algebra was known to the Indians before it was known to the Arabians. It is remarked, however, by Dr. Hutton, from whose history of Algebra the fact now stated is derived, that there is nothing of all this to be found in the work of Diophantus, which was published about three years after the time when Bombelli wrote. There is at all events a mystery here which it would be desirable to have cleared up: for, in the first place, it is not easy to conceive how Bombelli could be so far mistaken in regard to a manuscript of which he gives so particular an account; whilst, on the other hand, our later and more perfect acquaintance with the Algebra of the Hindus renders it extremely probable that the Greeks drew that portion of their science as well as several others from the philosophers of India. Whatever doubt there may be on this point, it is certain that the knowledge of Algebra was first communicated to Europe through the medium of Arabian treatises. In the beginning of the thirteenth century, Leonardo, a merchant, belonging to the small state of Pisa, having made frequent voyages to the East in the course of his commercial adventures, returned to his native country, enriched by the traffic and instructed by the science of Arabia and Palestine. The use of the Arabic system of numeral notation had been already conveyed into the Low Countries by Gerbert, a monk, who had likewise learned it from the Moors; and in this way the mathematicians of Italy, France, and Germany, were supplied with the most elegant and ingenious contrivances that mankind have yet discovered for expressing the relations of number and quantity. The improvements which were from time to time introduced into this beautiful art, during the sixteenth century, consisted rather in the use of a less cumbrous apparatus than in any new views respecting principle. In the following age considerable advances were made in the application of algebra to geometry; and no name is more deserving of honour, as an ardent and successful labourer in preparing the instrument for this its highest use, than that of Thomas Herriot, the author of a treatise entitled "Artis Analytica Praxis." Indeed, by a succession of discoveries which have immortalized the genius of that period, the algebraical analysis was brought to a state of perfection little short of that which it has attained at the present moment. It was prepared, says Professor Playfair, for the step which was about to be taken by Descartes, and which forms one of the most important epochas in the history of the mathematical sciences. This was the application of the algebraic analysis, to define the nature and investigate the properties of curve lines, and consequently to represent the nature of variable quantity. It is often said that Descartes is the first who applied algebra to geometry; but this is inaccurate: for such applications had been made before, particularly by Vieta, in his treatise on angular sections. The invention just mentioned is the undisputed property of Descartes, and opened up vast fields of discovery for those who were to come after him. The scientific world received, in the beginning of the seventeenth century, a highly valuable gift from the profound mind of Napier, the inventor of logarithms. The account of the wonderful discovery in the properties of numbers on which this most ingenious contrivance for facilitating calcu lation is founded, is given by Professor Playfair with equal candour and eloquence. The increasing expertness of mathematicians in the use of the algebraical analysis would, in the course of time, have suggested the structure of that piece of numerical mechanism with which the name will be for ever associated and had the invention been delayed to the end of the seventeenth century, it would have come about with so little effort as not to confer on the author more than a trifling portion of the celebrity which it justly procured a hundred years earlier. In another respect also, Napier has been particularly fortunate. : Many inventions have been eclipsed or obscured by new discoveries; or they have been so altered by subsequent improvements, that their original form can hardly be recognised, and, in some instances, has been entirely forgotten. This has almost always happened to the discoveries made at an early period in the progress of science, and before their principles were fully unfolded. It has been quite otherwise with the invention of logarithms, which came out of the hands of the authors so perfect, that it has never received but one material improvement, that which it derived, as has just been said, from the ingenuity of his friend in conjunction with his own. Subsequent improvements in science, instead of offering any thing that could supplant this invention, have only enlarged the circle to which its utility extended. Logarithms have been applied to numberless purposes which were not thought of at the time of their first construction. Even the city of their author did not see the immense fertility of the principle which he had discovered; he calculated his tables merely to facilitate arithmetical, and chiefly trigonometrical computation, and little imagined that he was at the same time constructing a scale whereon to measure the density of the strata of the atmosphere and the heights of mountains; that he was actually com saga |