other mathematical lectures, were read by Mr. Ludlam, during many successive years, to the junior students of St. John's; and were received with a degree of approbation adequate to their merit. After the decease of that popular instructer, his lectures were published. They have passed through several editions, and are still used in the university of Cambridge, and also in many mathematical schools in Britain. Mr. Ludlam's Rudiments of Mathematics have been long considered one of the best introductions to Algebra, to Euclid's Elements, and to Plane Trigonometry, in the English language. "In simplicity and perspicuity of style, and in clearness and accuracy of demonstration, (says a teacher of experience), this author is not inferior to his great master, the famous professor Saunderson; and in these respects is not excelled by any writer of the present time. Hence his trigonometry is admirably adapted to the use of students in our colleges and higher schools." In his advertisement to the Elements of Plane Trigonomery Mr. Ludlam informs his readers that "he has selected. such propositions as may probably be wanted in the usual course of academical studies, especially in astronomy and natural philosophy; and that he has taken the liberty of borrowing demonstrations, and even of altering them, when he thought they could be improved." A number of detached articles, and the greater part of the third section, are extracted from Playfair's Trigonometry. The first general division of the third section is taken from Ludlam, Vince, and Bonnycastle; * but mostly from the first. The first and second sections contain the elements of plane trigonometry, the practical and numerical solutions of all the cases of rectilinear triangles, and the mensuration of the heights and distances of objects. These two sections are complete of themselves. * This useful and meritorious writer appears to have followed Legendre's plan, and method of demonstration, and often translates from his Trigonometry without acknowledging the obligation. Legendre's Trigonometry is probably equal, if not superior, to any ele mentary treatise of the same extent. It is uncommonly perspicuous and elegant, and is well adapted to the use of students who are previously acquainted with algebra and geometry. His Elements of Geometry and Trigonometry have not been translated from the original French; but a translation of them would make a valuable addition to our stock of recent mathematical science, which is certainly very defective. Ludlam, who has been our principal guide through plane trigonometry, has not written on spherical; and therefore we have been obliged to follow other authors of experience and ability in the remaining part of this work. Those who have afforded us most assistance are Playfair, Vince, and Bonnycastle, from whose writings the greater part of spherical trigonometry has been extracted. The works of Legendre, Horsley, Lacroix, and others, have furnished some articles of impor tance. It is no small recommendation of the different works which have supplied the materials of this treatise, that the authors of them have been, for many years, public teachers of mathematics in foreign universities. He who writes for the instruction of youth ought to possess the experience and skill, which the art and practice of teaching can alone communicate. Men who have not received the benefit of a liberal education, and are not conversant in the business of tuition, are apt to form plausible theories of the discipline and economy of schools and colleges; but they are generally founded on ideal principles, and therefore cannot be reduced to rational, and successful practice. In the third section of plane trigonometry the changes of the algebraic signs of the lines described in and about a circle, commonly denominated trigonometrical lines, and the principal geometrical properties of those lines, are fully explained and demonstrated. A perfect knowledge of the changes of the signs of those lines, and of their chief properties, is indispensable in the theory of central forces and astronomy, and in many parts of mechanical philosophy, especially in the resolution of physical problems. This is one of the most useful and necessary parts of trigonometry, without the knowledge of which the student can make no proficiency in the higher branches of mathematics. The construction of trigonometrical tables is annexed to plane trigonometry, because it is a direct application of the trigonometrical theorems previously demonstrated. This and the theorems respecting the sines and cosines of arcs," which (says professor Playfair) are the foundation of those applications of trigonometry lately introduced, with so much advantage, into the higher geometry," are extracted from Playfair's Elements of Trigonometry, with a few additions. "Though the solution of triangles forms a distinct and material part of trigonometry, yet a treatise, which should be now b confined to that object, would be justly deemed defective. Many of the most useful formulæ can be easily obtained; and in the present state of science every treatise upon trigonometry should comprise some portion of trigonometrical analysis." British Review for March 1811. The definitions and preliminary principles contained in the introduction to spherical trigonometry are intended to be annexed, with their demonstrations, to the second edition of Playfair's Geometry, under the title of Elements of Spherical Geometry. They properly belong to the elements of geometry, though they are usually prefixed to treatises of spherical trigonometry. With the same kind of impropriety might the common properties of rectilinear triangles be prefixed to plane trigonometry. It may perhaps be said that this treatise of trigonometry. might be taught after any other book of geometry, beside Playfair's, if spherics, or the common properties of the sphere, had been connected with it. But to this objection we may reply, that no other edition of Euclid's Elements of Geometry can continue much longer in use in any places of education where the rudiments of mathematics are systematically and successfully cultivated, because Playfair's contains so many corrections, additions, and improvements, and is so far superior to all former editions of the celebrated work of the Greek geometer. Indeed Playfair's Geometry is now generally used as a textbook in our colleges and higher schools; and the teachers uniformly acknowledge its utility and excellence. It can be read by students in much less time, and with far less difficulty and labour, than Simson's or any other edition of Euclid's Elements of Geometry. Some teachers would perhaps be satisfied with the elementary propositions of trigonometry, and would be willing to dispense with their application to the numerical solution of the cases of triangles, and to the mensuration of heights and distances, &c. But this confined plan does not accord with the import and object of trigonometry, which is the art of finding the dimensions of the sides and angles of triangles; or, agreeably to professor Playfair's definition, " is the application of arithmetic to geometry, or, more precisely, of number to express the relations of the sides and angles of triangles to one another." Hence the nature and design of this work comprehend a theoretical and practical treatise of plane and spherical trigonometry, which shall be adapted to numerical computation; and, as far as this science is concerned, shall serve as an introduction to mechanical philosophy and the higher branches of mathematics. Besides, when theory and practice are thus united, the work may probably be useful to some young men who have acquired a competent knowledge of the elements of algebra and geometry, but cannot obtain the assistance of a preceptor. Rules, without suitable examples to illustrate them, are of little use to youth, because they do not make a lasting impression upon their memory, and consequently are soon forgotten. Hence the elementary propositions in this work are illustrated by a variety of numerical examples, in the solutions of all the cases of plane and spherical triangles, and in the mensuration of the heights and distances of terrestrial objects. By the solution of these numerical problems, the learner will acquire a facility in the use of the tables of logarithms, and in trigonometrical calculations, which will promote his progress in the study of navigation, physical astronomy, and other branches of mixed mathematics. In some articles, algebra has been applied to facilitate and shorten the demonstrations, which would have been long and difficult by pure geometry. The sciences of algebra and geometry afford mutual assistance in mathematical investigations, and should be applied jointly when they cannot easily effect the object separately, or when one conduces essentially to the aid of the other. In favour of this practice we have the authority of the ablest mathematicians during the last fifty years; and algebra is constantly applied to geometry, by the most eminent writers of the present time, whenever it can be introduced with advantage into a mathematical process. As this work contains many properties of plane and spherical triangles, which are useful, and even necessary, in certain parts of mathematics, but are not requisite in the solution of the common cases of triangles, it is proper to inform the reader what articles may be omitted entirely, or deferred till he has made some proficiency in other branches of mathematics. The articles which may be wholly omitted, or postponed to a future time, are specified in page iv. It was the intention of the publisher to subjoin the stereographic projection of the sphere; but he found that this could not be done without increasing the size and price of the book too much. The mere principles of projection are short; but without their practical application they would be of little benefit to the learner. It was therefore judged most expedient to reject this part of the original plan, and to refer the reader to the best works on the different projections of the sphere. See Horsley's Mathematics, vol. iii, Webber's Mathematics, vol. ii. I do not refer to Emerson's Treatise on the Projection of the Sphere, because it has been remarked by Dr. Horsley and other mathematicians, that " Emerson's works are composed with great depth of knowledge, and upon sound principles; but without elegance, without order, and without perspicuity." But writings of this character do not afford models fit for the study and imitation of youth. Of all the recent English writers, Dr. Horsley, bishop of Rochester, appears to have treated of the different projections of the sphere most fully and elaborately. His book is composed with the precision and prolixity of the ancient geometers, whose works he professes to admire. "We conceive (says he) that we have treated this subject of projection in a manner which goes to the bottom of it, brings many curious properties to light, not observed by former writers, and clears up many difficulties."* The learner will not be duly qualified to study this treatise with advantage, if he do not possess a competent knowledge of vulgar and decimal fractions, of the elements of algebra and geometry, and of practical geometry. Furnished with those preparatory qualifications, he will be able to read the whole book without much difficulty. Most of the demonstrations in this work are simple and plain. A few in spherical trigonometry are, from the nature of the propositions, more artificial and difficult; but they are so diffuse that a reader, who is acquainted with the elements of algebra and geometry, will easily understand them. In no book of equal extent will a learner meet so few impediments to retard his progress. The publisher does not presume to claim this book as author, but only as editor and proprietor. He has prefixed his name in the title page, because, without a name, it could not be distinguished from other works of the same kind, and consequently could not be obtained by persons residing at a distance from the places where it is published and sold. Philadelphia, August 1811. F. N. * A republication of this part of the bishop's book would probably be an acceptable present to the professors of mathematics in our colleges. It appears to be studied in the university of Oxford. |