CONTENTS. Addition, subtraction, multiplication, and division of ditto Numerical problem on the reduction of ratios Surds, reduction, addition, subtraction, multiplication, division, Conchoid, cissoid, cycloid, quadratrix, catenary Table of relations, strains, &c. in catenarian curves, applicable Mechanical powers, lever, wheel and axle, pulley, inclined plane, General application of the principles of statics to the equilibrium Illustrations and applications: hydrostatic paradox and bellows, Bramah's press, embankments, strength of pipes of oak or Floating bodies, buoyancy, Farey's self-acting flood gate Specific gravities, Coates's hydrostatic balance, tables of specific gravities and weights of various substances Rules for weights of leaden pipes, and rims of cast-iron fly- Pumps, sucking, lifting, forcing, fire-engine, Clark's quicksilver pump, Archimedes's screw, spiral pump, hydraulic ram, &c. 333 Wind and windmills, Smeaton, Coulomb, &c. Steam and steam-engines; Savery, Newcomen and Cawley, Useful tables and remarks on steam-engines, rail-roads, canals, Active strength, or animal energy, as, of men, horses, &c. Schulze's experiments, Coulomb's, Bevan's, Morisot's, Regnier's, 353 (To be placed at the end.) 1. Isometrical Perspective. A COMMON-PLACE BOOK, &c. &c. CHAPTER I. ARITHMETIC. SECTION I.-Definitions and Notation. ARITHMETIC is the science of numbers. We give the name of number to the assemblage of many units, or of many parts of an assumed unit; unit being the quantity which, among all those of the same kind, forms a whole, which may be regarded as the base or element. Thus, when I speak of one house, one guinea, I speak of units, of which the first is the thing called a house, the second that called a guinea. But when I say four houses, ten guineas, three quarters of a guinea, I speak of numbers, of which the first is the unit house repeated four times; the second is the unit guinea repeated ten times; the third is the fourth part of the unit guinea repeated three times. In every particular classification of numbers, the unit is a measure taken arbitrarily, or established by usage and convention. Numbers formed by the repetition of an unbroken unit are called whole numbers, or integers, as seven miles, thirty shillings: those which are formed by the assemblage of many parts of a unit are called fractional numbers, or simply fractions; as two-thirds of a yard, three-eighths of a mile. When the unit is restricted to a certain thing in particular, as one man, one horse, one pound, the collection of many of those units is called a concrete number, as ten men, twenty horses, fifty pounds. But if the unit does not denote any parti cular thing, and is expressed simply by one, numbers which are constituted of such units are denominated discrete or abstract, as five, ten, thirty. Hence, it is evident that abstract numbers. |