INTRODUCTION. SOME ACCOUNT OF THE HISTORY OF THE MATHEMATICS, AND OF THE PROGRESSIVE DEVELOPEMENT OF THE PRINCIPLES OF THE HE student who desires to acquainc himself with the reason, object, and uses of the several departments into which mathematical study is dividedl, will derive great advantage from becoming acquainted with the origin and progress of the successive inventions by which men have advanced from the power of estimating numbers, to the highest branches of geometrical analysis. Such a history necessarily includes the definitions and reason of every process, and if read together, or when any new branch is entered upon, it will enlarge, correct, and fix the notions of the student better than any abstract definitions. MontuclA's great work has served as the basis of all our English histories of the Science ; but the present sketch is abstracted chiefly from the able Dissertations by the late Professor PLAYFAIR, prefixed to the Supplements of the Encyclopædia Britannica, and partly from the elegant history of the Abbé Bossut. In nothing was the inventive and elegant genius of the Greeks better exemplified than in their GEOMETRY. The elementary truths of that science were connected by Euclid into one great chain, beginning from the axioms, and extending to the properties of the five regular solids ; the whole digested into such admirable order, and explained with such clearness and precision, that no work of superior excellence has appeared. ARCHIMEDES assailed the more difficult problems of geometry, and by means of the method of Exhaustions, demonstrated many curious and important theorems, with regard to the lengths and areas of curves, and the contents of solids. APOLLONIUS treated of the Conic Sections,—the Curves which, after the circle, are the most simple and important in geometry. Another great invention, the Geometrical Analysis, ascribed very generally to the Platonic school, but most successfully cultivated by Apollonius, is one of the most ingenious and beautiful contrivances in the mathematics. It is a method of discovering truth by reasoning concerning things unknown, or propositions merely supposed, as if the one were given, or the others were really true. By this analytical process, therefore, the thing required is discovered, and we are at the same time put in possession of an instrument by which new truths may be found out, and which, when skill in using it has been acquired by practice, may be applied to an unlimited extent. A similar process enables us to discover the demonstrations of propositions, supposed to be true, or, if not true, to discover that they are false. Trigonometry, which had never been known to the Greeks as a separate science, and which took that form in Arabia, advanced, in the hands of REGIOMONTANUS, in the 15th century, to a great degree of perfection, and approached very near to the condition which it has attained at the present day. He also introduced the use of decimal fractions into arithmetic. Cut off in the prime of life ; his untimely death, amidst innumerable projects for the advancement of science, is even at this day a matter of regret. He was buried in the Pantheon at Rome; and the honours paid to him at his death prove that science had now become a distinction which the great were disposed to recognise. Werner, who lived in the end of this century, is the first among the moderns who appears to have been acquainted with the geometrical analysis. He resolved Archimedes's problem of cutting a sphere into two segments, having a given ratio to one another. MAUROLYCUS of Messina flourished in the middle of the sixteenth century, and besides furnishing many valuable translations and commentaries, he wrote a treatise on the conic sections, which is highly esteemed. He endeavoured also to restore the fifth book of the conics of Apollonius, in which that geometer treated of the maxima and minima of the conic sections. In the early part of the seventeenth century, CAVALLERI was pare ticularly distinguished, and made an advance in the higher geome. try, which occupies the middle place between the discoveries of Archimedes and those of Newton. For the purpose of determining the lengths and areas of curves, and the contents of solids contained within curve superficies, the ancients had invented a method, to which the name of Exhuustions has been given. Whenever it is required to measure the space bounded by curve lines, the length of a curve itself, or the solid contained within a curve superficies, the investigation does not fall within the range of elementary geometry. Rectilineal figures are compared by help of the notion of equality which is derived from the coincidence of magnitudes both similar and equal. This principle, which is quite general with respect to recti. lineal figures, must fail, when we would compare curvilineal and rectilineal spaces with one another, and make the latter serve as measures of the former, because no addition or subtraction of rectie lineal figures can ever produce a figure which is curvilineal. It is possible, indeed, to combine curvilineal figures, so as to producs one that is rectilineal ; but this principle is of very limited extent In the difficulty to which geometers were thus reduced, it occurred, that, by inscribing a rectilineal figure within a curve, and circum scribing another round it, two limits could be ovrained, one greater and the other less than the area required. It was also evident, that, by increasing the number, and diminishing the sides of those figures, the two limits might be brought continually nearer to one another, and of course nearer to the curvilinear area, which was always intermediate between them.. In prosecuting this sort of approrimation, a result was at length found out, which must have occasioned no less surprise than delight to the methematician who first encountered it; namely, that, when the series of inscribed figures was continually increased, by multiplying the number of the sides, and diminishing their size, there was an assignable rectilineal area, to which they continually approached, so as to come nearer it than any difference that could be supposed. The same limit would also be observed to belong to the circumscribed figures, and therefore it could be no other than the curvilineal area required. A truth of this sort occurred to ARCHIMEDES, when he found that two-thirds of the rectangle, under the ordinate and abscissa of a parabola, was a limit always greater than the inscribed rectilineal figure, and less than the circumscribed. In some other curves, a similar conclusion was found, and Archimedes contrived to show that it was impossible to suppose that the area of the curve could differ from the said limit, without admitting that the circumscribed figure might become less, or the inscribed figure greater than the curve itself. The method of Exhaustions was the name given to the indirect demonstrations thus formed; and though few things more ingenious than this method have been devised, and though nothing could be more conclusive than the demonstrations resulting from it, yet it laboured under two very considerable defects. In the first place, the process by which the demonstration was obtained, was long and difficult; and, in the second place, it was indirect, giving no insight into the principle on which the investigation was founded. A more compendious, and more analytical method, was therefore particularly required. CAVALLERI, born at Milan in the year 1598, is the person by whom this great improvement was made. The principle on which he proceeded 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. The rule for summing an infinite series of terms in arithmetical progression had been long known, and the application of it to find the area of a triangle, according to the method of indivisibles, was quan a matter of no difficulty. The next step was, supposing a series of lines in arithmetical progression, and squares to be described on each of them, to find what ratio the sum of all these squares bears to the greatest square, taken as often as there are terms in the progression. Cavalleri showed, that when the number of terms is infinitely great, the first of these sums is just one-third of the second; which led to the cubature of many solids. He then sought for the sum of the cubes of the same lines, and found it to be one-fourth of the greatest, taken as often as there are terms; and, continuing this investigation, he was able to assign the sum of the nth powers of a series in arithmetical progression, supposing always the difference of the terms to be infinitely sniall, and their number to be infinitely great. He thus gave, over geometrical problems of the higher class, the same power which the integral calculus, or the inverse method of fluxions does, in the case when the exponent of the variable tity is an integer. The method of indivisibles, however, was not without difficulties, and could not but be liable to objection, with those accustomed to the rigorous exactness of the ancient geometry. In strictness, lines, however multiplied, can never make an area, or any thing but a line ; nor can areas, however they may be added together, compose a solid, or any thing but an area. This is certainly true, and yet the conclusions of Cavalleri, deduced on a contrary supposition, are true also. It was the doctrine of infinitely small quantities carried to the extreme, and gave at once the result of an infinite series of successive approximations. Nothing, perhaps, more ingenious, and certainly nothing more happy, ever was contrived, than to arrive at the conclusion of all these approximations, without going through the approximations themselves. The Cycloid afforded a number of problems, well calculated to exercise the proficients in the geometry of indivisibles, or of infinites. It is the curve described by a point in the circumference of a circle, while the circle itself rolls in a straight line along a plane. It is not quite certain when this curve, so remarkable for its curious properties, and for the place which it occupies in the history of geometry, first drew the attention of mathematicians. In the year 1639, Galileo informed his friend Torricelli, that, forty years before that time, he had thought of this curve, on account of its shape, and the graceful form it would give to arches in architecture. The same philosopher had endeavoured to find the area of the cycloid ; but though he was one of those who first introduced the consideration of infinites into geometry, he was not expert enough in the use of that doctrine, to be able to resolve this problem. It is still more extraordinary, that the same problem proved too difficult for Caval. leri, though he certainly was in complete possession of the principles by which it was to be resolved. It is, however, not easy to determine whether it be to Torricelli, the scholar of Cavalleri, and his successor in genius and talents, or to Roberval, a French mathematician of the same period, and a man also of great originality and invention, that science is indebted for the first quadrature of the cycloid, or the proof that its area is three times that of its generating circle. Both these mathematicians laid claim to it. The French and Italians each took the part of their own countryman; and in their zeal have so perplexed the question, that it is hard to say on which side the truth is to be found. Torricelli, however, 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. They had both the skill and talent which fitted them for this, or even for more difficult researches. The properties of this curve, its tangents, its length, its curvature, &c. exercised the ingenuity, not only of the geometers just mentioned, but of Wren, Wallis, Huygens, and, even after the invention of the integral calculus, of Newton, Leibnitz, and Bernoulli. mat, who, in his inventive resources, as well as in the correctness of his mathematical taste, yielded to none of his contemporaries, applied at this period infinitely small quantities to determine the warima and minima of the ordinates of curves, as also their tangents. As early as the beginning of the thirteenth century, Leonardo, a merchant of Pisa, having made frequent vists to the East, in the course of commercial adventure, returned to Italy enriched by the traffic, and instructed by the science of those countries. He brought with him the knowledge of ALGEBRA; in 1202. But though Algebra was brought into Europe from Arabia, it is by no means certain that this last is its native country. There is, indeed, reason to think that its invention must be sought for much farther to the East, and probably not nearer than Indostan, Though in all this the moderns received none of their information from the Greeks, yet a work in the Greek language, treating of arithmetical questions, in a manner that may be accounted algebraic, was written by Diophantus of Alexandria, who had composed thirteen books of Arithmetical Questions, and is supposed to have flourished about 150 years after the Christian era. The investigations donot extend beyond quadratic equations; they are, however, extremely ingenious, and prove the author to have been a man of talent, though the instruments he worked with were weak and imperfect. As to the general doctrine of equations, it appears that Cardan was acquainted both with the negative and positive roots, the former of which be called by the name of false roots. He also knew that the number of positive, or, as he called them, true roots, is equal to the number of the changes of the signs of the terms; and that the co-efficient of the second term is the difference between the sum of the true and the false roots. He also had perceived the difficulty of that case of cubic equations, which cannot be reduced to his own rule. He was not able to overcome the difficulty, but showed how, in all cases, an approximation to the roots might be obtained. The properties of algebraic equations were discovered, however, |