greeks, and at length their languishing state till the destruction of the school of Alexandria. In the second period they are revived and cultivated by the arabs, who carry these sciences with them into some of the countries of Europe. This reaches nearly to the end of the fifteenth century. Some time after this they are diffused, and make a rapid progress among all the nations in Europe of any consequence; and this third period brings us to the discovery of fluxions, where the fourth and last period begins. These four periods will constitute the general divisions of this work. At first view it might seem, that for the sake of perspicuity I should go through each branch of the mathematics successively without interruption: but this method, applied indiscriminately to every part and every age, has some inconveniences. The different branches of the mathematics have been formed and developed by degrees, and frequently one has promoted another. A proposition in mechanics has given birth to a complete theory of geometry; and then it would be impoffible to give an account of the one without explaining the other, and thus being led into details, often prolix, and even foreign to the true and principal object. Besides, a disagreeable void in the general picture, or too striking a disproportion in the parts, would sometimes occur; for all the sciences have not advanced with equal pace, some appearing at times stationary, while others have been making a rapid progress. These observations are more particularly just with regard to the second and fourth ages of mathematics: and frequent instances instances of them will be seen, when we come to the application of fluxions to mechanics and astronomy. The first age is that, in which the thread of each science is most uniform and distinct, so that every part of the mathematics may be kept separate. Of this advantage I have availed myself as much as possible; but in the following periods I have not been able completely to preserve the same order. I must request the reader's assent, therefore, to a plan, which the nature of the subject appeared to me to exact. It is superfluous to make another observation, which will naturally suggest itself. Frequently it will be seen, that the historical documents, necessary to form a complete and uninterrupted narrative, are very rude, or very defective: on the other hand, ornament and fiction are incongruous to the strictness of my subject. In such sterile parts, therefore, I can hope for attention only from those readers, who find treasures even in the ruins of the edifice of science. CHAP. PERIOD THE FIRST. STATE OF THE MATHEMATICS, FROM THEIR ORIGIN TO THE DESTRUCTION OF THE ALEXANDRIAN SCHOOL. CHAP. I. Origin and Progress of Arithmetic. THERE is no idea more simple, or more easy to conceive, than that of number or multitude. As soon as the understanding of a child begins to unfold itself, he can count his fingers, the trees around him, and the other objects that are before his eyes. These operations are performed at first without order, without method, and by the help of memory alone: but means of extending them, and of subjecting them to a kind of regular form, are soon found. Different as the objects to be counted might be, as the same method was always pursued, it was easily perceived, that their nature might be left out of consideration; and to represent them, general symbols were invented, which afterward assumed particular values, adapted to each question, that was to be resolved. Thus for instance little balls were employed, strung strung together like the beads of a rosary, or knots in a cord: each ball denoted a sheep, or a tree; and the whole assemblage of balls, the flock, or the grove. The invention of writing advanced the art of numeration a step farther. On a table covered with dust characters chosen arbitrarily to express numbers were traced, and thus calculations, to a certain extent, were capable of being performed. All nations, if we except the ancient chinese, and an obscure tribe mentioned by Aristotle, distributed numbers into periods, composed each of ten units. This custom can scarcely be attributed to any thing but our habit of counting in childhood by the fingers, which, with some very rare exceptions, are uniformly The ancients equally agreed in representing numbers by the letters of the alphabet; and the dif ferent periods of tens were distinguished by accents put over the numeral letters, as was the practice of the greeks, or by different combinations of the numerals, as was done by the romans. All these methods of notation, particularly that of the romans, were very complex and embarrassing when calcula- tions of any extent were to be performed. Strabo, who lived in the reign of Augustus, says in his Geography, that in his time the invention of arithmetic, as well as that of writing, was ascribed to the phenicians. The establishment of this opinion would find the less difficulty, because the phenicians, being the most ancient mercantile nation, must naturally have improved a science, of which they were making constant use: but the principles of arithmetic were known to the egyptians and chaldeans long be fore fore we hear any thing of the phenicians, who pro bably acquired them from their egyptian neighbours. The mathematics had already taken root in Greece, when Thales appeared: but the impulse he gave them constitutes the era, from which we begin to reckon their real advancement. [A. c. 640.] We know not whether this philosopher made any particular discoveries in arithmetic: his inclination led him principally to the study of geometry, physics, and astronomy. He travelled a long while in Egypt and India. Enriched with the knowledge he had acquired in foreign countries, and which he improved by his own reflections, he returned to Miletus, the place of his birth, and there founded the celebrated ionian school, which divided itself into several branches or sects, embracing every part of philosophy, and spreading themselves through several of the grecian cities. Some time after, Pythagoras of Samos rendered himself illustrious by his vast erudition, and the singularity of his philosophical opinions. Never man more ardently sought glory; never man more deserved it, or raised himself to a higher pitch of reputation. He had all the ambition of a conqueror: full of zeal for extending the empire of the sciences, and not contented with having instructed his countrymen, he went to Italy, and founded a school, which in a short time acquired such celebrity, that he reckoned princes and legislators among his disciples. That almost every part of the mathematics has important obligations to him, will be seen as each comes under consideration. The |