By Maria Zack, Elaine Landry
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Extra info for Research in History and Philosophy of Mathematics: The CSHPM 2014 Annual Meeting in St. Catharines, Ontario
Sample text
Since the scales are logarithmic, the logarithm of the ratio of a to b is the distance between the points labeled a and the point labeled b on the line of numbers. This can be used to find a fourth proportional (the Golden Rule or Regula Aurea of problem solving techniques), with two simple movements of a pair of dividers. First open the dividers so that its legs extend from point a to point b on the Line of Numbers. Then move the dividers so that one leg points to the point c: The other leg will point to point d whose value as read on the scale is the desired fourth proportional.
2 Line of Superficies The purpose of these lines is to allow the solution of proportional analogies that relate the ratio of the areas of similar figures to the ratio of the lengths of corresponding sides. The lengths of corresponding sides are measured on the Line of Lines, whereas the areas of the similar figures are measured on the Line of Superficies. These lines may also be used to determine the ratio of areas in similar figures whose ratio of corresponding sides are known, or to determine the ratio by which the side of a given figure must be increased or decreased in order to enlarge or reduce the area according to a desired ratio.
Some of the more common included extra lines are described below. S. Silverberg Fig. 6 The Lines of Quadrature are used to determine the length of the side of a square equal in area to a polygon with a side of known length or vice versa a circle equal in area to any given square, or determine the length of the side of a regular polygon of 5 through 10 sides equal in area to a given circle or a given square. The Lines of Quadrature were labeled Q, 5, 6, 7, 8, 9, 10, and S. The numerical labels indicate that the transverse distance between like numbers is the length of a regular polygon with that number of sides.