Hirsch's Geometry: Or, A Sequel to Euclid

Black, Young and Young, 1827 - 264 σελίδες

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Σελίδα 124 - IF a straight line be bisected, and produced to any point: the rectangle contained by the whole line thus produced, and the part of it produced, together with the square...
Σελίδα 34 - ... are proportionals. Let the two triangles ABC, DEF have one angle in the one equal to one angle in the other, viz. the angle BAC to the angle EDF, and the sides about...
Σελίδα 213 - Of all the lines that can be drawn from a given point to a given line, the perpendicular is the shortest.
Σελίδα 14 - ... its appearance to an inhabitant of Jupiter, s its appearance to an inhabitant of Saturn, u to an inhabitant of Uranus, and N to an inhabitant of Neptune. The light and heat which it would supply to each of these planets would be in the exact proportion of the apparent surface of the solar disk, and since the areas of circles are as the squares of their diameters, it would follow that the solar light and heat at Jupiter is 25 times, at Saturn 81 times, at Uranus 324 times, and at Neptune 784 times...
Σελίδα 97 - Find the diameter of a circle whose area is equal to the sum of the areas of two circles whose diameters are 12 in.
Σελίδα 96 - Find the length of the radius of a circle whose area is equal to the sum of the areas of four circles of 10 in., 15 in., 18 in., and 24 in.
Σελίδα 125 - DEF have equal bases and unequal altitudes. ... A ABC > A DEF. (Why ?) .•. A ABC is a maximum. QED Ex. 1015. To divide a given line into two parts so that the rectangle contained by the segments is a maximum. Ex. 1016. In the hypotenuse of a right triangle to find a point so that the sum of the squares of perpendiculars drawn from the point upon the arms is a minimum.
Σελίδα 96 - No portion of the column capital should be considered effective which lies outside of the largest 90° cone that can be included within the outlines of the column capital. When a square, or other symmetrically shaped capital is used, c is 'the diameter of a circle whose area is equal to the area of the base of the largest 90° pyramid which can be included within the outlines of the column capital.
Σελίδα 5 - To a given straight line apply a triangle which shall be equal to a given parallelogram and have one of its angles equal to a given rectilineal angle. 106. Transform a given rectilineal figure into a triangle whose vertex shall be in a given angle of the figure, and whose base shall be in one of the sides. 107. Divide a triangle by two straight lines into three parts which when properly arranged shall form a parallelogram whose angles are of a given magnitude. 108. Shew that a scalene triangle cannot...
Σελίδα 213 - ... deduced the following rule for finding the area of any quadrilateral figure : — " Multiply the sum of the perpendiculars drawn from opposite angles of the figure upon the diagonal joining the other two angles, and take half the product." 15. In Euclid, n. 3, where must be the point of division of the line, so that the rectangle contained by the two parts may be a maximum ? Exemplify in the case where the line is 12 inches long. 16. How may the demonstration of Euclid n. 4, be legitimately shortened...

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