THE PRINCIPLES AND PRACTICE OF ARITHMETIC, COMPRISING THE NATURE AND USE OF LOGARITHMS, WITH THE GAGERS, AND LAND-SURVEYORS, DESIGNED FOR THE USE OF STUDENTS. BY JOHN HIND, M.A. F.C.P.S. F.R.A.S. LATE FELLOW AND TUTOR OF SIDNEY SUSSEX COLLEGE, EIGHTH EDITION. With a New Appendix of Miscellaneous Questions. CAMBRIDGE: DEIGHTON, BELL AND CO. LONDON: BELL AND DALDY. 1856. 181. c. 30. ADVERTISEMENT. In the present Treatise it has been the Author's endeavour to combine what is necessary of the Philosophy of the Science of Arithmetic with the Practice of the Art of Numbers; and it is here considered sufficient to place before the student an outline of the plan which has been adopted in the arrangement, with a short account of the more important divisions. The first Chapter commences with the elementary Definitions; it then proceeds to the explanation of Notation and Numeration, which are both exemplified in a great variety of instances; and concludes with the consideration of the Fundamental Operations of the Science as applied to pure or abstract numerical magnitudes. In the second Chapter, the Fundamental Operations have been extended to mixed or concrete numerical magnitudes, consisting of various denominations; and some important remarks are introduced. The third Chapter treats of the First Principles of The Rule of Three, sometimes called The Golden Rule; and comprises a collection of examples intended to illustrate its different views. The fourth Chapter contains The Doctrine of Fractions, usually termed Vulgar Fractions; and concludes with many of their applications to practical purposes. The fifth Chapter developes The Theory of Decimals, commonly called Decimal Fractions; and points out most of the uses to which Decimals are peculiarly adapted. In the sixth Chapter are discussed the Doctrines of Ratio and Proportion, from the Principles of which are deduced all the Rules of consequence in the affairs of Commerce; and it concludes with the solution of a few miscellaneous questions, explaining some technical terms. The seventh Chapter contains the Practice of Involution and Evolution, with The Arithmetic of Surds or Irrational Quantities. The object of the eighth Chapter is The Nature and Properties of Logarithms derived from the simplest principles; and the practical advantages afforded by Logarithmic Tables are pointed out in appropriate examples. The ninth Chapter is The Application of Arithmetic to Geometry: and the Calculations of Artificers, Gagers, and Land-Surveyors, are explained and exemplified in it. In this Chapter will also be found an account of the Imperial Weights and Measures, and their origin and relation to each other; as well as of the Calendar adopted in the time of Julius Cæsar, and its subsequent improvement in the time of Pope Gregory the Thirteenth, with the requisite Calculations worked out. The rest is an Appendix, in which the Fundamental Rules have been derived from Elementary Principles, upon the extension of which the present system of Arithmetic is established. Throughout the work, it has been attempted to trace the source of every Rule which is given, and to investigate the reasons upon which it is founded: and by means of particular examples comprising nothing but what is common to every other example of the same kind, to attain in Arithmetic the kind of evidence which is relied upon in Geometry, or in any other demonstrative Science. Single and Double Position are omitted, as most of the examples usually given to illustrate these rules, may be solved by the principles here explained, not to mention that they are merely Algebraical Formula enunciated at length. No notice has been taken of Arithmetical and Geometrical Progression, of Permutations and Combinations, and of Annuities and Reversions, because they depend upon Formulæ expressed by general symbols, which the student would find a difficulty in making use of, without at least a knowledge of the Notation and Fundamental Operations of Algebra; |