Emerson's Doctrine of Fluxions came out in 1743. This has also been always much esteemed in England. It con tains a great number of applications ; but as it places the subject less within the reach of a beginner, Simpson's book is, we believe, more popular.
It is to the celebrated Euler that the calculus is indebt ed for its greatest improvements. Indeed these are far too numerous to find a place in the brief view which our limits allow us to give of the progress of the science ; even the titles of his various memoirs would fill several of our pages: his more remarkable works will be given in the list of books relating to the subject in the conclusion. That branch of the calculus which treats of the higher class of problems, De maximis et minimis, such as the solid of least resistance, the curve of swiftest descent, &c. was first re duced by him to the form of a distinct theory, in his Me thodus inveniendi Lineal Curvas Maximi Minimive propri etatr gaudentes, Sive solutio Problematis Isoperimetrici latissimo senau accepti, (1744.) This theory was improved and new modelled by Lagrange, and denominated the Me thod of Variations. It is a remarkable instance of Euler's candour, that he took up the subject a second time, and laying aside his own theory, treated it according to La Grange's views, employing also the same notation. Euler's writings on the analysis of infinites;and the differential and integral calculus, are a treasure of analytical knowledge, richer than was ever before produced by the labours of an individual.
A discovery made by an Italian mathematician, the Mar quis Fagnano, or Fagnani, has contributed considerably to the improvement of a branch of the fluxional calculus. He found that it is always possible to assign two arcs of an ellipse, reckoned from one extremity of each axis, such, that their difference may be expressed by algebraic quan tities ; and that any hyperbola has a similar property. This curious theorem, w hich has led to some remarkable transfor mations of fluxional formula, appears to have been but lit tle known in Britain, as we do not recollect to have seen it of the mathematical works published in this country until it was also found by our ingenious countryman, Mr Landen, who added to it another remarkable discovery, namely, that any hyperbolic arc may always be rectified by means of two elliptic arcs. This theorem was of great im
portance, because it reduced to elliptic arcs all fluents that had before been expressed by hyperbolic arcs. Legendre followed in the tract of Landen's discovery, and chewed, in the Memoirs of the Academy of Sciences of Paris, for 1786, that the rectification of any ellipse may be reduced to that of two others, which have their eccentricities as small or as great as we please.
It is a circumstance highly honourable to female genius, that we have to mention an excellent treatise on this diffi cult branch of mathematics from the pen of a lady ; we al lude to Analytical Institutions in four books, written origi nally in Italian, by Donna Maria Gaetano. Agnesi, Professor of the Mathematics and Philosophy in the University of Bologna. This work was first published in 1748. A ma thematician of great eminence, Mr Bossut, translated the second volume of it into French, and inserted it in his course of mathematics, as the best treatise he could furnish on the differential and integral calculus. And an English mathe matician, Mr Colson, (who translated Newton's Fluxions), translated it also into English, having studied the Italian lan guage at an advanced period of life, for the express pur pose of making himself master of the work. The histo rian of the mathematics, Montucla, bestows great praise on this extraordinary woman ; and her own countryman, Frisi, who has himself excelled so much both in pure and mixed mathematics, calls her work Opus nitidissimum,ingeniossis simum, et eerie maximum quad adhuc ex fremine alicujus calamo proderat. For an account of this lady, see AGNES1.
The invention of the Arithmetic of Sines, which is clue to Euler, has contributed greatly to the improvement of the calculus, and to its application to the physico-mathemati cal sciences. Indeed, to Euler we are more or less indebt ed for almost every improvement it has received. He first discovered the criterion by which it may be determined whether a fluxional equation admits of an exact integral or not ; but it was also found, about the same time, by two French mathematicians, Fontaine and Clairaut : this was about the year 1739 or 1740.