in mathematical instruction know full well that a learner may readily yield his assent to every step of an algebraic process, be fully satisfied as to the truth of the result to which it leads, may even clearly see a valuable truth involved in it, and may yet be very far from perceiving how to turn it to account in any case of actual calculation. Indeed, algebraical formulas, transform them as we will, cannot always be made to indicate the best mode of arithmetical arrangement; and yet much, as regards facility of operation, depends upon this arrangement in many parts of practical mathematics, but especially in Trigonometry. In the present volume, therefore, both the theory and the practice of the science have been introduced, every practical formula being illustrated by examples of the numerical calculation, arranged in the proper form. This plan of combining practice with theory, in works like the present, was always adopted by the earlier English writers, and it is to be regretted that recent authors have, in their admiration of foreign methods, departed so widely, in this respect, from the example of their predecessors, dwelling so much as they do upon the symbols, and so little upon the things signified. In addition to the practical illustration of formulas, a distinct part of the work is devoted to the principles of Navigation and Nautical Astronomy, in which will be found a very short and convenient method of clearing the Lunar Distance, for the purpose of ascertaining the Longitude at Sea. This method is probably new, although, as the analytical expression for it occurs during the investigation of the well known formula of Borda, it is equally probable that it has been noticed before. The supplement appended to the treatise is from the pen of my valued and accomplished friend, T. S. Davies, Esq. , Fellow of the Royal Society of Edinburgh, and of the Royal Astronomical Society of London. It will be found to contain several new and interesting researches, which cannot fail to prove acceptable both to the inquiring student and to the more advanced analyst. J. R. YOUNG. January 1, 1833. Kecd.o 12-20-40. rem, 1. ELEMENTS of the INTEGRAL CALCULUS; with its Applications to Geometry, and to the Summation of Infinite Series, &c. 9s. in cloth. This volume forms one of an Analytical Course.~" More elegant Textbooks do not exist in the English Language, and we trust they will speedily be adopted in our Mathematical Seminaries. The existence of such auxiliaries, will, of itself, we hope, prove an inducement to the cultivation of Analytical Science; for, to the want of such Elementary Works, the indifference hitherto manifested in this country on the subject, is, we apprehend, chiefly to be ascribed. Mr. Young has brought the science within the reach of every intelligent student, and, in so doing, has contributed to the advancement of Mathematical Learning in Great Britain.—The Presbyterian Review, Jan. 1832. 2. The ELEMENTS of the DIFFERENTIAL CALCULUS: comprehending the General Theory of Curve Surfaces and of Curves of Double Curvature. 8s. in cloth. 3. An ELEMENTARY TREATISE on ALGEBRA, Theoretical and Practical; with Attempts to simplify some of the more difficult Parts of the Science, particularly the Demonstration of the Binomial Theo ir its most general form ; the Solution of Equations of the higher orders; the Summation of Infinite Series, &c. 8vo. boards, 10s. 6d. 4. An ELEMENTARY TREATISE on the COMPUTATION of LOGARITHMS. Intended as a Supplement to the various books on Algebra. 12mo. 2s. 6d. 5. ELEMENTS of GEOMETRY; containing a New and Universal Treatise on the Doctrine of Proportion, together with Notes, in which are pointed out and corrected several important Errors that have hitherto remained unnoticed in the Writings of Geometers. 8vo. 8s. 6. The ELEMENTS of ANALYTICAL GEOMETRY; comprehending the Doctrine of the Conic Sections, and the general Theory of Curves and Surfaces of the second order, with a variety of local Problems on Lines and Surfaces. Intended for the use of Mathematical Students in schools and universities. 9s. cloth. 7. ELEMENTS of MECHANICS; comprehending the Theory of Equilibrium and of Motion, and the first Principles of PHYSICAL ASTRONOMY, together with a variety of Statical and Dynamical Problems. Illustrated by numerous plates. 10s. 6d. cloth. • If works like the present be introduced generally into our Schools and Colleges, the continent will not long boast of its immense superiority over the country of Newton, in every branch of modern analytical science.”—The Atlas, July 25, 1830. " Mr. Young is already favorably known to the Public by his writings; and the Treatise on Mechanics, which we now propose briefly to notice, will add considerably to his reputation as the author of Elementary Works of Science. To read the works of Laplace, Ivory, Somerville, &c., a knowledge of the methods employed by these writers is previously required, and render such preparatory works as those of Mr. Young's absolutely necessary."-Presbyterian Review, July 1832. CONTENTS. CHAP. I. Explanation to the Trigonometrical Lines. 1. Definition of Plane Trigonometry 2. On the measurement of angular magnitude 3. Connection between circular arcs and angles 4. On complementary and supplementary arcs 7. Of the tangent, cotangent, secant, and cosecant 8. Of the versed sine, coversed sine, and suversed sine 9. Abridged mode of expressing the trigonometrical lines ib. 10. Manner of rendering trigonometrical expressions homogeneous 8 ib. Properties derivable from the definitions 11. Table of the correlative values of the trigonometrical lines CHAP. 11. Formulas and Rules for the solution of Plane Triangles. 13. Particulars to be observed in applying the formulas ib. Expression of the foregoing formulas in words 14. Of the arithmetical complement 17. Expressions for the angles in terms of the sides 18. Given the sines and cosines of two arcs to find the sine and co- 19. In a plane triangle the sum of any two sides is to their difference as the tangent of half the sum of the opposite angles to the tan- 20. Formulas for determining an angle in terms of the sides 22. Solution of plane triangles in general 23. Examples of the solution of oblique-angled triangles 24. When two sides and the included angle are given CHAP. III. Application of Plane Trigonometry to the Mensuration of CHAP. IV. Investigation of Trigonometrical Formulas 26. Formulas for the sum and difference of two unequal arcs markable property, viz. ib. tan. A + tan. B + tan. C = tan. A tan. B tan. C 28. Formulas for multiple arcs 29. Investigation of De Moivre’s Formula 30. General expressions for sin. n A and cos. n A deduced from De ib. Various formulas for double arcs 32. Formulas involving the half sum and half difference of two arcs 33. To express the sine and cosine of a real arc, by means of imagi- ib. To develop sin." x and cos." z in terms of the sine and cosine of |