IV. PRACTICAL MENSURATION 259. Nature of the Work. In this country the demand for supplementary exercises in practical mensuration is increasing. The work which is required is based not merely upon the demonstrative geometry of the plane and of the simpler solids as set forth in this text, but also upon the actual measurements of lines and upon the trigonometry of the right triangle as it is now taught in connection with algebra in many schools. Material of this kind may safely replace certain of the propositions of plane geometry, such as those which involve inequalities, certain theorems in relation to the circle, and certain parts of Book IV, and it may also take the place of various propositions in Book VI. It is here offered as optional work for the use of those teachers who wish to modify the standard course in demonstrative geometry, as given in this text, by the introduction of a moderate amount of work in advanced mensuration. No attempt has been made to include any exercises on the mensuration of the conic sections, which is more advantageously treated in the calculus or in connection with the propositions of analytic geometry. Furthermore, such work is not so practical for the general student as that which relates to the more common plane and solid figures. As giving proper training in space perception, without involving the logic of demonstration, it is believed that teachers will find this material of great value. It is generally conceded that plane geometry furnishes a sufficient amount of training in deductive reasoning for an initial course, and that the value of solid geometry lies chiefly in its presentation of spatial relations. Such a presentation is made more vital by work of the nature and extent set forth in the following pages. 260. Symbols and Formulas of Plane Geometry. The following symbols and formulas are needed in the exercises: Primes indicate that there are two parts of the same name, such as the bases b and b' of a trapezoid. FORMULAS TO BE MEMORIZED Parallelogram, A=bh Trapezoid, Abh Circumference, C=2πr = πd Area of a circle, A = πr2 = 4 πď2 <A = 60° The result by this formula is approximate. A = 360 πr20 Equilateral triangle, A = 0.4330 s2 <0 = 120° r = 0.5774 s a = 0.2887 s A = 1.7205 s2 r = 0.8506 s a = 0.6882 s a = 0.8660 s r = 1.3065 s SIMPSON'S RULE FOR AREAS To find the area between the x axis and a continuous curve, Simpson's Rule, which is named for its inventor, is often used. In the figure here shown the base y y2 y3 A h Yn B Y1, Y2, Y3,, Yn. The approximate area is then found by the following formula: A = {} h[(Y1+Yn)+2(Y3+Y¿+Yr + · · ·) + 4(Y1⁄2 + Y4 + Yε + · · ·)] This formula may be expressed in words as follows: To find the approximate area between a continuous curve and the x axis, add the extreme ordinates, twice the sum of the other odd ordinates, and four times the sum of the even ordinates, and then multiply the result by one third the common distance between the successive ordinates. The approximation is closer if the curve is undulating (wave-like) as in the figure shown above. The formula and rule need not be memorized. In the work in practical mensuration the symbols (') and (") are used for feet and inches respectively. The application of the rule may be illustrated by the case of a curve which has the ordinates 16.5', 21', 24', 25', 29.5', 33', 30', 28.5', 28.5', 29', 30', 25.5', 21.5', 20', 20.5', and a common distance (h) between the ordinates of 7.68'. We then have y1+Yn = 16.5′ + 20.5′ = 37′. •)=2(24′+29.5′ + 30′ + 28.5' + 30'+21.5') 4(Y2+Y4+ Y6 + · · ·)=4(21′ + 25′+ 33′ + 28.5′+ 29′+25.5′+20) Hence =728'. A = × 7.68 (37 + 327 +728) sq. ft. = 2795.52 sq. ft. 261. Symbols and Formulas of Solid Geometry. In addition to the symbols and formulas given in § 260, the following are needed in the exercises: E = spherical excess of a polygon In the case of cylinders and cones, only cylinders and cones of revolution are considered. It is unnecessary to give formulas for the lateral area of a prism and for the area of the total surface of a cone or a cylinder, as these areas can be found by taking the sum of other known areas. FORMULAS FOR REFERENCE Frustum of a pyramid, V = {h(B+B'+√BB') 12 V={h(B+B'+√√BB') = } πh(r2+r22 +rr') Here S is the area of the zone forming the base of the sector. Spherical segment of one base, V=Th2 (3r-h) Instead of using this formula the student may consider the spherical segment as the difference between a spherical sector and a cone. 262. Trigonometry Presupposed. The exercises assume a knowledge of four functions of an angle and the ability to use these func- a C cos A = Tables of these functions are given on pages 196-203. 263. Trigonometric Formulas. The following formulas are 264. Use of the Tables. In the tables on pages 196-203 the functions are given for every 0.1°, or for every 6'. In the columns of differences the difference for every 1' is given. For example, to find sin 55° 20′, find sin 55° 18′ on page 197, and to it add 3 (for 0.0003) found under 2'; that is, sin 55° 20' 0.8221+ 0.0003 = 0.8224 In finding cos 35° 45', for example, since the cosine decreases as the angle increases, we subtract the difference (indicated in the table as a minus difference). Hence cos 35° 45' 0.8121-0.0005 = 0.8116 = |