An elementary course of practical mathematics. Key, Part 21852 |
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altitude Answers arc bc arc whose tangent Area of base Author's Complete Treatise axis Bisect breadth calculating centre chord circumf circumference cone consequently convex surface Corresponding arc cube cylinder decagon Demonstration described diagonal diagram Diam diameter divided EDINBURGH ellipse equal Euclid Euclid's Elements EXERCISES IN CHAPTER expression feet find the area fore frustum given half arc Hence heptagon hexagon inches infinite number inscribed length lines multiplied NE-Ne Nearest on Table number of degrees number of sides octagon parallel parallelogram parallelopiped perp PRACTICAL GEOMETRY prism prismoid PROBLEM VIII PROPOSITION pyramid radius ratio rectangle rectilineal reverse formula right angles Rule sector similar sine of half slant height solid sphere square tabular segment trapezoid triangles true series Undecagon versin vertex wedge
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Page 132 - Multiply the square root of half the sum of the squares of the two axes by *, and the product will be nearly = the circumference. Ex. Taking the same example as before, we hare /24' + IT \/ = — X 3,14159 = 66,6433= the circumference nearly.
Page 162 - The surface of a sphere is equal to the convex surface of the circumscribing cylinder ; and the solidity of the sphere is two thirds the solidity of the circumscribing cylinder.
Page 115 - ... string may be used for the other half, commencing the curve at F or B, as the case may be. This is commonly called "a gardener's oval," because gardeners make use of it for forming ornamental beds for flowers, or in making curves for walks, etc., etc. This method of forming the curve, is based on the well-known property of the ellipse that the sum of any two lines drawn from the foci to their circumference is the same.
Page 161 - Shew that the convex surface of a spherical segment is equal to the area of a circle whose radius is the distance from the pole to the circumference of its base.
Page 151 - DF). Therefore the surface of the frustum is equal to this circle. Lemmas. " 1. Cones having equal height have the same ratio as their bases ; and those having equal bases...
Page 168 - ... about one of the short sides. On the convex surface of the cone, from the vertex to the circumference of the base, innumerable straight lines may be drawn, which in the right cone are all equal to each other. Every cone may be considered as a pyramid of an infinite number of sides. Since, then, the pyramid is the third part of a prism of the same base and altitude, the cone will be the third part of a cylinder of the same base and altitude. When a cone is intersected by a plane we obtain, 1,...
Page 116 - The circumferences of two circles are to each other as their radii, and their areas are to each other as the squares of their radii. Let R and R' be the radii of the circles, C and C" their circumferences, S and S' their areas. Inscribe in the two circles similar regular polygons ; let P and P...
Page 137 - I of the last chapter. For the square, whose area is required, being divided into two right-angled triangles by its diagonal, we know that in either of these, the square of the diagonal is equal to twice the square of the side, since the two sides are equal. /. required square = side2 = £ diag2.
Page 168 - ... indefinitely great, or when the base becomes a continued closed curve, as a circle, an ellipse, &c. ; or, the center of gravity of a cone, right or oblique, and on any base, is one fourth the distance from the center of gravity of the base to the vertex. Ex. 7. To find the center of gravity of a frustum of a cone or pyramid cut off by a plane parallel to the base. Let a be the length of the line drawn from the vertex of the cone, when complete, to the center of gravity of the base, a' that portion...
Page 149 - The area of a regular inscribed polygon, and that of a similar circumscribed polygon being given, to find the areas of the regular inscribed and circumscribed polygons having double the number of sides. Let AB be the side of the given inscribed, and EF that of the given circumscribed polygon.