knowledge, as well as to assist and promote the progress of real genius and industry, by removing obstacles, and making the way plain and easy. Hence arises the multiplicity of easy introductions, easy grammars, games, &c. which we have in every branch of learning, works which are all useful as far as they go; but it must be remarked, that if they remove difficulties out of the scholar's way, instead of teaching him how to encounter and surmount them, these performances, however they may be patronized and praised, are of but little value. Scientific games, it is allowed, are pleasing and instructive amusements for the nursery; but whatever they may seem to promise, it is making sad game of the sciences, to suppose that these can be acquired by play. I am persuaded that none ever did, or ever will attain to useful or honourable proficiency in any branch of learning, without proportionate labour and application ©. If what has been said be correct, it will not only account for the great number of easy elementary treatises that has appeared, but will shew that an almost endless variety is absolutely necessary to accommodate the various tastes of learners; it will be a sufficient apology for adding one to the number, as well as for the plan on which it is written. In the following work, it is proposed to combine more advantages than are to be met with in any single book on the subject, viz. historical, theoretical, and practical knowledge, and to accompany the whole with explanations so exceedingly simple and easy, that it is presumed to be im "Nothing can be more absurd" (says the author of Hermes) " than the common notion of instruction; as if science were to be poured into the mind like water into a cistern, that passively waits to receive all that comes. The growth of knowledge resembles the growth of fruit; however external causes may in some degree co-operate, it is the internal vigour and virtue of the tree that must ripen the juices to their just maturity." Harris. "To lead a child to suppose, that he is to do nothing which is not conducive to pleasure, is to give him a degree of levity, and a turn for dissipation, which will certainly prevent his improvement, and may perhaps occasion, his ruin.", Knox on Education, p. 19. possible that any person of moderate talents will fail to understand them. It supposes the learner to be, in the proper sense of the word, a beginner, consequently unacquainted with even the rudiments of science; and from common principles known and acknowledged by all, it proceeds by easy and almost imperceptible gradations, to lead him on (with the aid of Simson's Euclid and a Table of . Logarithms, both which it explains) to the attainment of a considerable degree of mathematical knowledge, with scarcely any assistance from a master. The work is divided into ten parts, in which the subjects treated of are-Arithmetic, Algebra, Logarithms, Common Geometry, Trigonometry, and the Conic Sections; each preceded by a popular history of its rise and progressive improvements: to which are added, by way of notes, brief memoirs of the principal authors mentioned in the text; some account of their writings, discoveries, improvements, &c. with a variety of useful information of a miscellaneous nature, respecting the Mathematical Sciences. Part I. begins with an Historical Account of Arithmetic, explaining, to a considerable extent, the nature and construction of numbers, and proceeds by laying down in a plain and simple manner, what are usually called the four fundamental rules: next follow in order, Reduction, the Compound Rules, Proportion Direct, Inverse, and Compound; the Rules of Practice, the theory and practice of Fractional Arithmetic, Vulgar, Decimal, and Duodecimal; Involution, Evolution, and Progression, both Arithmetical and Geometrical; the whole demonstrated, exemplified, and explained: and as simplicity and clearness were always the objects aimed d Plato calls Arithmetic and Geometry "The wings of the mathematician ;" "Arithmetic" (says M. Ozanam) "may be considered as the mathematician's right wing, because without this Geometry would be very imperfect; this justifies the common practice of beginning the Mathematics, with the study of Arithmetic." at, it is hoped no obstacle will be found in the learner's way which may not easily be surmounted. Under these heads, which comprise the whole of Elementary Arithmetic, is given a great number of particular rules and observations, not to be found in any other work, but which are necessary, in order fully to explain the theory, and facilitate the practice of numbers. Besides the examples fully wrought out and explained, several others are introduced under each rule, with their answers only, and a few are given without answers. Part H. contains an Historical Account of Logarithms, the theory and practice of Logarithmical Arithmetic, with numerous examples, problems, and explanations. Part III. contains the History of Algebra, and its fundamental rules; Rules for solving Simple and Quadratic Equations, in which one, two, three, or more unknown quantities are included; and, lastly, a collection of Problems, teaching the application of Simple and Quadratic Equations, in a great variety of ways; the whole accompanied with notes and easy explanations as above. This completes the first volume. The second volume (Part IV.) begins with Literal Algebra, in which the Problems are analytically investigated, and likewise demonstrated by the method of Synthesis. General conclusions are applied to particular examples, and the methods of converting numeral Problems into general ones; deducing Theorems, Rules, and Corollaries; registering the steps of operations, &c. are laid down and applied in a variety of cases. The doctrine of Ratios, Proportion, Progression, Variable and Dependant Quantities, Interest, Discount, Permutations, Combinations, the Properties of Numbers, &c. are Algebraically investigated, with numerous examples. Part V. explains the nature and theory of Equations in general, their Composition, Depression, Transformation, and Resolution, according to the methods of Newton, Cardan, Euler, Simson, Des Cartes, and others. Various methods of Approximation as laid down by Simson, Raphson, Hutton, Bernoulli, &c. the Solution of Exponental Equations, and Problems for exercise. Part VI. explains the nature and method of resolving indeterminate Problems, both simple and Diophantine. Part VII. shews how to convert Fractions and Binomial Surds into Infinite Series, by Sir Isaac Newton's Binomial Theorem, and otherwise: how to sum, interpolate, and revert a given Series; to which is added, the Algebraic investigation of Logarithms, with Rules for constructing entire tables of those numbers, both common and hyperbolical. Part VIII. treats of Geometry, viz. its history and use, and describes the nature, construction, and use of Mathematical Instruments, to prepare the learner for the practical application of Geometry: this is followed by an easy logical Introduction to the study of Euclid's Elements, considered as a system of demonstration, with Observations on the Definitions, Postulates, and Axioms, and the most remarkable propositions in the first six books, as they stand in Dr. Simson's Translation; with Corollaries, explanations of the difficulties that occur, &c. partly original, and partly selected from Clavius, Barrow, Saville, Austen, Ludlam, Ingram, and Playfair: to which is subjoined, an Appendix, containing some useful propositions, not in Euclid; and an easy system of Practical Geometry and Mensuration, for the purpose of applying Euclid's theory to practice. Part IX. contains the theory and practice of Trigonometry, the investigation of Formula for the Sines, Tangents, Secants, &c. both natural and artificial; with the description, construction, and use of instruments employed in Altimetry, Surveying, Geography, Navigation, &c.; and, lastly, the Mensuration of inaccessible heights and distances. In Part X. is given the History of the Conic Sections, with the principal and most useful properties of those celebrated curves, deduced by an easy and natural method, accompanied with numerous. references to Euclid, for the convenience of the learner. Such is the plan of the work; with respect to its execution, the Author submits with becoming diffidence to the judgment of the public: he is aware of many imperfections, and is too well acquainted with himself, not to suspect that some errors may have escaped him, of which he is unconscious; but he trusts that none will be found of sufficient importance to mislead the student, or materially impede his progress. If any plea for indulgence could be urged or admitted, it might be truly said, that few books have been composed under more unfavourable circumstances than this; but waving every claim of the kind, the Author only requests it will be remembered, first, that his work is intended for beginners; this will account for some apparent prolixity, especially in the explanations, and for the manner in which some of the rules and operations are accounted for, being rather popular than scientific. Secondly, that as he was not within fifty miles of the press, typographical errors are unavoidable; but from the kind attention of two learned friends at Oxford, and the care taken by the Printers, their number is comparatively small. Thirdly, part of the manuscript was at the press and inaccessible to the Author, while he had the other part under correction, which will explain the cause of two or three unnecessary repetitions, should they be discovered. It ought to be acknowledged, that in the prosecution of the subjects here treated of, occasional assistance has been derived from the writings of approved authors and commentators; and, in some instances, their methods and observations have been extended, abridged, or otherwise altered, to suit the plan of the Author: this is allowable in works of an elementary nature, and is not without its advantage, both to the subject and to the reader... Two copies of the greater part of Vol. I. having been more than twelve months in boards, were, by way of experiment, e Veneror inventa Sapientiæ, inventoresque adire tanquam multorum hæreditatem, juvat. Mihi ista acquisita, mihi laborata sunt. Seneca. |