OF CONCISE SYSTEM OF IN THEORY AND PRACTICE, FOR THE Use of Schools, Private Students, and Practical Men: COMPREHENDING ALGEBRA, PRACTICAL GEOMETRY, LOGARITHMS, PLANE AND SPHERICAL TRIGONOMETRY, MENSURATION OF SURFACES, SOLIDS, HEIGHTS, AND DISTANCES ; LAND-SURVEYING, GAUGING, MENSURATION OF ARTIFICERS WORKS, &c. WITH CONTAINING THE MORE USEFUL PROPOSITIONS OF GEOMETRY, CONIC SECTIONS, FLUXIONS, AND DEMONSTRATIONS OF THE RULES IN THE BODY OF THE WORK. THE SECOND EDITION, THOROUGHLY REVISED, WITH MANY IMPORTANT ADDITIONS AND IMPROVEMENTS: TANGENTS, AND THE AREAS OF CIRCULAR SEGMENTS. ILLUSTRATED BY UPWARDS OF THREE HUNDRED WOOD-CUTS. BY ALEXANDER INGRAM, Author of Elements of Euclid, Principles of Arithmetic, Editor of an improved Edition of Melrose's Arithmetic, &c. &c. PUBLISHED BY OLIVER & BOYD, EDINBURGH; AND SIMPKIN & MARSHALL, LONDON. 6. The LIMITS of Ratios, FLUXIONS, and FLUENTS, previously forming an Appendix to the Algebra, are now incorporated with the General Appendix, which is so arranged as to exhibit a comprehensive and satisfactory view of the whole theory. And as an introduction to the study of NavigATION and NauTICAL ASTRONOMY, a section on SPHERICAL TRIGONOMETRY, with examples of its application, has been inserted. Hence it will appear that no part of the science really valuable has been omitted. 6. To adapt the work to all the purposes of teaching, due regard has also been paid to the variety of Exercises added to each Problem, which will be found more than double the number contained in the first impression,-an addition of the highest importance in a text-book. 7. Besides many new and useful Tables interspersed throughout the work, there are now added TABLES of the LOGARITHMS of NUMBERS from 1 to 10.000, of LOGARITHMIC SINES and TANGENTS to every Degree and Minute, and of NATURAL SINES and TANGENTS to every Five Minutes of the Quadrant. These have all been carefully stereotyped from new types, and, in order to obtain the utmost possible accuracy, rigidly collated with the Tables of BRIGGS, VLACQ, SHERWIN, GARDINER, CALLET, TAYLOR, HUTTON, BAB ORDA, and also with those of GALBRAITH, which are especially distinguished for accuracy. The Table of the AREAS of CIRCULAR SEGMENTS has likewise been collated with several other more extensive ones. Such is a brief and cursory view of the leading features now introduced into this edition. But, exclusive altogether of the great amount of new matter, and independent of many minor improvements, the whole work has undergone a careful, rigorous, and minute revision ;—what was obscure has been illustrated, and what was defective has been supplied. The errors which had formerly escaped notice have been corrected ; and, with the view of securing perfect accuracy, the Author availed himself of the assistance of an eminent Mathematician in examining every calculation ; and although it would be presumptuous to assert that the work is immaculate, yet the Publishers feel assured that no error of importance will be found. Finally, when the Publishers consider the success attending the work in a less perfect shape, they confidently hope that the variety and importance of the contents of the present edition, as well as the perspicuous and familiar manner in which these are treated, taken along with the numerous and extensive additions and improvements introduced throughout, will give it a still higher claim to public favour, and render it more instrumental in facilitating the acquirement of mathematical knowledge, and in disseminating a taste for that science among all classes of students; and, as an additional recommendation, they may venture to affirm, that while it is in many respects the most complete, it is unquestionably the cheapest work of the kind ever published. EDINBURGH, January 22, 1830. ORIGINAL PREFACE. SEVERAL treatises on Mensuration have made their appearance within the last fifty years. Among these, Dr Hutton's large work has deservedly acquired the highest celebrity. It treats fully both of the theory and practice of the science, and may be consulted with advantage by persons employed in any kind of measurement. But the scientific part of that work can be read by such only as are well acquainted with the higher branches of Mathematics, and hence the student must have frequent recourse to other publications, to enable him to understand it; while the practical part involves such a multiplicity of rules for the same thing, without distinguishing sufficiently the various cases in which they can be applied, that he is liable to be perplexed with their variety; and nothing has been done by later writers to remove the difficulty. A book on Mensuration is therefore still wanted, embracing the whole theory and practice in such a way, that both, though kept separate, may be rendered intelligible to every reader, without the necessity of having recourse to other publications, and arranged in a condensed form, so as to comprise a complete system of the science in a small compass. Such are the objects of the present publication. The practical part of this work consists of plain rules for performing the various operations requisite in Trigonometry, Mensuration, Surveying, Gauging, 8c. These rules are illustrated by proper examples, one or more of which is wrought for the assistance of the learner. A demonstration of the rule is sometimes annexed to it in the form of a note, when this can be done in an easy and concise manner. But the more difficult demonstrations are reserved for the Appendix. By pursuing this method, the author has endeavoured to render the book fit for the use of every person who wishes to study Mensuration with facility and success. The treatise on Practical Geometry, which is prefixed to the Trigonometry, will enable the student to draw his figures; while the rules delivered in the following part of the work will direct him how to find their contents, and the lengths of their lines; and a little reflection will qualify him to compare these lengths or contents with one another. In such a state, the work will be found a most useful guide to practical measurers, and well adapted to the use of schools. The rules may be applied directly in aïl ordinary cases. If one shall occur which requires investigation, the method of conducting this process may be learned from the treatise on Algebra, which is prefixed to the work. In the treatise on Algebra, great care has been taken to remove irregularities, and other difficulties, of which beginners usually complain; and the demonstrations of the fundamental rules are generalized, and deduced from one principle intimately connected with the nature of abstract quantity. A short Appendix is annexed to this part, which treats of the management of indeterminate problems, of the relations of variable quantities, and of the limits of ratios, with as much of the practice of Fluxions and Fluents as is requisite in this performance. The Practical Geometry, though short, contains every thing necessary for what follows. Some new methods of operation are introduced, and the lines and angles are generally expressed in numbers. In the Mensuration, the application of the series for finding the circumference of the circle, of which the diameter is unit, has been taken from Euler, and appears to be as simple as it can be made. New rules are given for approximating to the length of an arc of a circle, and to the area of a segment of it, which are both easier and more accurate than those formerly employed by the use of roots. The method of forming the most common solids with pasteboard is introduced, because it renders the reader familiar with their shapes, and illustrates the rules for finding their superficies. Land-surveying, Gauging, &c., are the application of Trigonometry and Mensuration to practical purposes. Great plainness has therefore been studied in explaining them, and the shortest, easiest, and most approved methods of practice have been adopted. The Appendix is appropriated to the demonstration of the rules delivered in the preceding parts of the work. Such of the principles of Geometry and of Conic Sections are introduced as are necessary for enabling the reader to understand the demonstration of the rules, without having recourse to other publications. Here accuracy is rigidly adhered to. Many new demonstrations are given, which are more simple than those that were formerly employed. The theory of Parallel Lines has been rendered as plain and concise as possible. The principles of Conic Sections have been deduced from the ratio of the curve, or its relation to the focus and directrix,-a method which has been generally held by mathematicians to be superior to every other. The leading propositions only are delivered ; but they are so regulated as to introduce principles from which the other properties of these curves may be easily derived. The student who has abundance of time should begin with Algebra, and then read the Appendix to the work and the Practical Geometry together; after which, he should go regularly through the book, in the order in which it is printed. In doing this, he may acquire as much knowledge of Mathematics as will be sufficient for ordinary purposes, and be enabled to prosecute that most extensive science with pleasure and advantage. If his time and other pursuits do not admit of such a regular progress, he may study separately any of the practical branches best adapted to his taste, or the purpose to which he intends to apply them. CONTENTS. Reduction of Fractions......... Addition and Subtraction of Fractions,...... Multiplication and Division of Fractions,.. To add and subtract Surds,...... To multiply and divide Surds............ Involution and Evolution of Surds, ....... To find the Square Root of a Compound Surd, ............. Extermination of Unknown Quantities,...... To resolve a Quadratic Equation............... Solution of Questions,............ Questions producing Simple Equations, Questions producing Quadratic Equations,. Arithmetical Progression,............ Geometrical Progression.................... Interest and Annuities, ................ A General Method of forming Series,.. Reversion of Series, .......... To calculate Logarithms by Series,...... |