Mensuration. Plane Figures and Right-Angled Triangles. Areas of Rectangle, Parallelogram, Triangle, Rhombus, Quadrilateral, Mensuration. Solids. Areas, Volumes and Weights of Prisms (right and oblique). Cylinder, Hollow Cylinder, Pyramid, Cone, Frustum of Pyramid, Frustum of Cone, Guldin's Theorem. Linear Motion. Momentum. Force. Mass. Work. Power. The Specification of Vectors. Triangle and Polygon of Vectors. Funicular Polygon. Graphic conditions of Equilibrium. PRACTICAL MATHEMATICS. CHAPTER I. MEASUREMENT. UNITS OF LENGTH. Measurement.-The measurement of a quantity is known when we have obtained a number which indicates its magnitude. It is necessary, therefore, to select some definite quantity of the same kind, as a unit, and then proceed to find how many times the unit is contained in the quantity to be measured. The number of times that the unit is contained in the given quantity is the numerical value of the quantity. Nearly all the quantities with which the man of science and the practical man are called upon to deal are concerned either directly or indirectly with length, mass, or time. Fundamental and Derived Units.—The primary or fundamental units are three in number, namely: All derived units are either multiples or sub-multiples of these primary units. Units of Length.-In order that length may be measured there must be a unit and a standard. The unit is a certain definite distance with which all other distances can be compared; and a standard is a bar on which the unit is clearly, P.M. A D accurately, and permanently marked. The two units most generally adopted are the yard and the metre. The British System. In this system the unit of length is the yard. It may be defined as the distance between two lines on a particular bronze bar when the bar is at a certain temperature (62° F.). The bar is deposited at the Standards Office of the Board of Trade. The importance of specifying the temperature at which the distance between the two lines on the bar is the exact length, will be evident when the alteration in the size of bodies due to a change of temperature is considered. Sub-Multiples and Multiples of Yard. The yard is for many purposes inconvenient, and smaller units, or as they are called sub-multiples of the unit of length, are often used. To obtain these smaller units, the yard is divided, first into three equal parts each called a foot. The foot is again divided into twelve equal parts called inches; an inch is further subdivided into eight, ten, sixteen (or more) equal parts. Where fractional parts of a foot are required it is often convenient to express feet and the parts of a foot in a better form than as feet and inches. This is effected by dividing the foot into ten equal parts, the fractional parts are then denoted by 1,, etc., or 1, 2, etc. Multiples of the yard are used when comparatively long distances have to be expressed. Thus, 1 mile=1760 yards= 5280 feet. The multiples and sub-multiples of the unit are given in the following Table. It must be at once remarked that in the British System there is no simple relation connecting the unit of length with the units of area, volume, and mass. It is only by a slow and troublesome process of multiplication and division such as reducing feet or miles to inches, ounces to pounds, etc., that we can proceed to find areas, volumes, etc. British Measures of Length. [The unit is divided by 3 and 36, etc.; also multiplied by 2, 5,220, and 1760.] 12 inches=1 foot. 3 feet=1 yard (unit). 2 yards=1 fathom. 5 yards = 1 rod, or pole. 40 poles 1 furlong. 8 furlongs = 1760 yards = 1 mile. or 5280 feet) The above table will serve to show how very inconvenient the British Unit and its sub-multiples and multiples are for most purposes, since so cumbersome a calculation is required to convert the one into the other. Fig. 1 shows one end of a 12 inch steel scale or straight edge. An inch is represented divided into 16 equal parts, and for the convenience of the person using the scale some of the lines are made longer than others. This enables a dimension to be read off much more quickly and accurately than would otherwise be the case. Thus the cross-line at c dividing an inch into two equal parts is seen to be longer than any cross line between a and c, or between c and b. In a similar manner the cross FIG. 1.-Inches divided into 8 and 16 equal parts. A decimetre divided into 10 centimetres and 100 millimetres. (The inches and centimetres are not drawn to true scale, but their comparative dimensions may be seen.) lines indicating the quarter inches are longer than those indicating the eighths, these in turn being longer than the sixteenths. As the number of divisions increases the lines naturally become more crowded together, until, as shown, the distances between the divisions when an inch is divided into 32 equal parts, or those indicating the sixty-fourths become very minute. The French or Metric System. The Metric System is extensively used for all scientific, and in many cases for com |
