Luigi Cremona

Graphical Statics. Two Treatises on the Graphical Calculus and Reciprocal Figures in Graphical Statics. Luigi Cremona Large Type edition

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1890 edition. Excerpt: ...crooked line AabcdeC, we draw IV parallel to ca s cc' ? dV fljr gcr r till each intersects the side Aa, eB ? Cd and now the whole polygon Aabcde C012345 is reduced to the equivalent quadrilateral ABCD. 103. This is the easiest and most convenient way of finding the areas of figures, the perimeters of which take the most different forms. With a little practice we learn to perform the reduction quite mechanically, and without paying any attention to the actual form of the proposed circuit. This construction moreover permits us to take account of signs, so that in dealing with areas of different signs, the result gives the actual sign belonging to their sum without further a trouble. Take, for example, the self-cutting circuit (Fig. 90) ABC01234, which represents the cross section of an embankment and excavation in earthwork. Fig. 90. Draw 11' parallel to 20, 22' ? 31', 33' ? 42', 42? ? A3', until they meet the side CO, then the given polygon is reduced to the equivalent quadrilateral ABCD, which therefore represents the difference between the areas ABCI4 of the embankment and 70123 of the excavation, which have necessarily different signs. The circuit ABCD has the same sign as the circuit ABCIi, or as the circuit 70123, according as the embankment or the excavation is the larger. 104. Circular Figures. A sector of a circle (Fig. 9i) OAB is equivalent to a triangle OAC, with its vertex at the centre O and its base a portion AC of the tangent equal to the arc AB. In order to obtain approximately the length of the arc AB measured along the tangent, we take an arc a, so small that it can without any sensible error be replaced by its chord a; we then apply the chord a to the given arc AB starting from its extremity B, and...