cour une plus grande étendue, et que je voudrois faire lire à tout le monde. Je témoigne aussi à My Lady Hervey, l'obligation que je lui ai, de m'avoir fait connoître un auteur qui prouve à chaque mot, que la littérature n'est ennemie que de l'ignorance et des travers, qui mérite d'avoir des Maty pour amis, et qui d'ailleurs honore et fortifie notre langue par l'usage que son esprit en sait faire. Si j'étois plus savant, j'appuyerois sur le mérite des discussions, et sur la justesse des observations. CAYLUS. N° XIV. Geo. Lewis Scott, Esquire,* to EDWARD GIBBON, junior. SUPPOSING you settled in quarters, dear Sir, I obey your commands, and send you my thoughts, relating to the pursuit of your mathematical stu avidity, this little work; and wish that it was more extensive, and read universally. I would also express my thanks to Lady Hervey, for making me acquainted with an author who proves in every page that learning is hostile only to ignorance and prejudice; who deserves to have a Maty for his friend, and who adds honour and strength to our language by the use which he so ably makes of it. Were I more learned I should dwell on the merit of the discussions, and the justness of the observations. * A very able mathematician. dies. You told me, you had read Clairaut's Algebra, and the three first books of l’Hopital's Conic Sections. You did not mention the Elements of Geometry you had perused. Whatever they were, whether Euclid's, or by some other, you will do well, if you have not applied yourself that way for some time past, to go over them again, and render the conclusions familiar to your memory. You may defer, however, a very critical inquiry into the principles and reasoning of geometers, till Dr. Simson's new edition of Euclid (now in the press) appears. I would have you study that book well; in the mean time recapitulate Clairaut and l’Hopital, so far as you have gone, and then go through the remainder of the Marquis's books with care. The fifth book will be an Introduction to the Analyse des Infiniment petits ;" to which I would advise you to proceed, after finishing the Conic Sections. The Infiniment petits may want a comment; Crousaz has written one, but it is a wretched performance: he did not understand the first principles of the science he undertook to illustrate; and his geometry shews, that he did not understand the first principles of geometry. There is a posthumous work of M. Varignon’s, called Eclaircissemens sur l'Analyse des Infiniment petits. Paris, 1725, 4to. This will be often of use to you. However, it must be owned, that the notion of the Infiniment petits, or Infinitesimals, as we call them, is too bold an assumption, and too remote from the principles of the ancients, our masters in geometry; and and has given a handle to an ingenious author (Berkeley, late Bishop of Cloyne) to attack the logic of modern mathematicians. He has been answered by many, but by none so clearly as by Mr. Maclaurin, in his Fluxions, (2 vols. in 4to.) where you will meet with a collection of the most valuable discoveries in the mathematical and physico-mathematical sciences. I recommend this author to you; but whether you ought to read him immediately after M. de l'Hopital, may be a question. I think you may be satisfied at first with reading his introduction, and chap. 1. book I. of the grounds of the Method of Fluxions, and then proceed to chap. 12. of the same book, § 495 to $ 505 inclusive, where he treats of the Method of Infinitesimals, and of the Limits of Ratios. You may then read chap. 1. book II. $ 697 to § 714 inclusive; and this you may do immediately after reading the first section of the Analyse des Infiniment petits : or if you please, you may postpone a critical inquiry into the principles of Infinitesimals and Fluxions, till you have seen the use and application of this doctrine in the drawing of Tangents, and in finding the Maxima and Minima of Geometrical Magnitudes. Annal. des Infin. pet. § 2 and 3. When you have read the beginning of l’Hopital's 4th sect. to sect. 65 inclusive, you may read Maclaurin's chap. 2, 3, and 4; where he fully explains the nature of these higher orders of Fluxions, and applies the notion to geometrical figures. Your principles principles being then firmly established, you may finish M. de l’Hopital. Your next step must be to the inverse method of Fluxions, called by the French calcul intégral. Monsieur de Bougainville has given us a treatise upon this subject, Paris, 1754, 4to. under the title Traité du Calcul intégral pour servir de suite à l'Analyse des Infiniment petits. You should have it; but though he explains the methods hitherto found out for the determination of Fluents from given Fluxions, or in the French'style, pour trouver les intégrales des différences donneés; yet as he has not shewn the use and application of this doctrine, as de l’Hopital did, with respect to that part which he treats of, M. de Bougainville's book is, for that reason, not so well suited to beginners as could be wished. You may therefore take Carré's book in 4to, printed at Paris, 1700, and entitled Méthode pour la Mesure des Surfaces, &c. par l’Application du Calcul intégral. Only I must caution you against depending upon him in his fourth section, where he treats of the centre of oscillation and percussion; he having made several mistakes there, as M. de Mairan has shewn, p. 196. Mem. de l'Acad. Royale des Sciences, édit. Paris, 1735. After Carré; you may read Bougainville. I have recommended French authors to you, because you are a thorough master of that language, and because, by their studying style and clearness. of expression, they seem to me best adapted to beginners. Our authors are often profound and acute, ner. acute, but their laconisms, and neglect of expres. sion, often perplex beginners. I except Mr. Maclaurin, who is very clear; but then he has such a vast variety of matter, that a great part of his book is, on that account, too difficult for a begin I might recommend other authors to you, as a course of elements; for instance, you might read Mr. Thomas Simpson's Geometry, Algebra, Trigonometry, and Fluxions; all which contain a great variety of good things. In his Geometry he departs from Euclid without a sufficient reason. However, you may read him after Dr. Robert Simson's Euclid, or together with it, and take notice of what is new in Thomas Simpson. His Algebra you may, join with Clairaut; and the rather that Clairaut has been sparing of particular problems, and has, besides, omitted several useful applications of Algebra. Simpson's Fluxions may go hand in hand with l’Hopital, Maclaurin, Carré, and Bougainville. If you come to have a competent knowledge of these authors, you will be far advanced, and you may proceed to the works of Newton, Cotes, the Bernoulli's, Dr. Moivre, &c. as your inclination and time will permit. Sir Isaac Newton's treatise of the Quadrature of Curves has been well commented by Mr. Stewart, and is of itself a good institution of Fluxions. Sir Isaac's Algebra is commented in several places by Clairaut, and in more in Maclaurin's Algebra; and New-. ton's famous Principia are explained by the Minims Jacquirs et le Seur, Geneva, 4 vols. 4to. Cotes is explained |