have puzzled both themselves and their readers, is here rendered plain, easy, and intelligible. In Fellowship, several new rules are given. Exchange is likewise treated of in a different manner to what it usually has been, and several useful tables are introduced, which have not hitherto been inserted in books of arithmetic. These tables have, in this edition, been carefully compared with a correct set of tables in the library of Hans Sloane, esq. and likewise with the tables published by the principal writers on exchange, as Kruse, Corbaua, Dubost, &c. Those who wish for farther information on the subject of exchange than is contained in the following treatise, may consult the works above mentioned, or the Universal Cambist, by Dr. Kelly. The nature of Ratios, and Proportions, so far as they relate to commensurable quantities is considered. These subjects are of the highest importance. The learned Whiston, in his Tacquet's Euclid, says, “Si proportionis “ doctrinam e Mathesi abstuleris, nihil fere praeclarum “aut egregium relinques.” ** The second part of the work concludes with some general observations on Numbers odd and even; Square and Cube Numbers, &c. These will serve to raise the curiosity of the learner, and give impulse to his farther enquiries. The Bills of Parcels, Promissory Notes, &c., which in the former editions followed Duodecimals, and concluded Part I., are now classed nearly in the same order at the end of the book, forming Part III. In this edition, the Rules for Annuities at Simple Interest have been omitted, being of no use, except as arithmetical exercises, and the principal cases of Annuities on Lives are introduced in lieu of them. All the rules” and examples have undergone a thorough revisal, and many new ones have been added, in consequence of the distinguished approbation which this work has met with from several of the most respectable and intelligent tutors in the kingdom. No. 1, York-Buildings, New Road, St. Mary-le-bone, London, March, 1822. * Algebraical demonstrations of the rules have been purposely omitted; because to a young student who is learning the elements of the Science, they are perfectly unintelligible. The truth of arithmetical operations should be explained, by the teacher, from the nature of the process; for though the theory and practice of Science ought to go hand in hand, yet every experienced teacher will allow, that, in common arithmetic, the practice must in a great measure precede the theory. The young student, who has made himself perfectly master of the practical parts of arithmetic, and has acquired some knowledge of algebra, will derive considerable advantage from the perusal of the Appendiz to the Complete Practical Arithmetician, annexed to the Key to that work. This Appendix contains a Synopsis of Logarithmical Arithmetic, together with general demonstrations of all the principal rules in the Complete Practical Arithmetician. CONTENTS. * - - PAGE DEFINITIONS • . . . . . . . . . 1 Notation . . . . . . . . . . . . ... 4 Subtraction . . . . . . . Multiplication. . . . . . 8 Division 13 TABLES of English Coin, Weights and Measures • - 18 Com Pou N D Addition . . . . . . 25 Subtraction . . . . 31 Multiplication .. 36 Division. . . . . . . . 39 Reduction • . . . . . . . . . . . . . 42 DirecT Proportion . . . . . . . . 48 lx verse Proportion........ 56 CoM Pound Proportion . . . . 59 vulgaR FRACTIONs, Definitions . . . . . . . . . . . . . . 64 Reduction of Fractions . . . . 65 A DD11 Iow of ditto . . . . . . . . 78 Su BTRAcrion of ditto . . . . . . 79 Multiplication of ditto . . 81 Division of ditto . . . . . . . . 82 Rule of Three Direct in ditto 83 Definitions - - - - - - - - - - - - - - 89 Addition of Decimals . . . . 90 Definitions . . . . . . . . . . . ... 104 Decimals. • . . . . . . . . . ... 104 Decimals. . . . . . . . . . . . . . D1 v is ros of ditto - - - - - - - - INTERest SIMPLE . . . . . . . . CoMM ission - - - - - - - - - - - Discount . . . . . . . . . . . . . . Do to a LE FEL Lows HIP. . . . 156 Arbitration of Exchange- - - - 193 Evolution. Definitions • - - - 201 Square-root - - - - - - - - - - - - 202 Cube-root - - - - - - - - - - - *-* * 209 183 184 • - - - - - - - |