Abstract: A nomographic apparatus for providing solutions to a mathematical relationship has a panel and several scales along three axes on a surface of the panel, the scales being arranged and calibrated in a nomographic relationship so that values on a straight line intersecting the scales satisfy the mathematical relationship. Guides are adjacent two selected ones of the scales, the two selected scales being separated by at least one other scale. First and second sliders are movable along the guides and strings extend between the sliders in substantially straight lines. The first and second sliders are positionable so that the strings intersect first and second locations on the selected scales, the locations representing selected numerical values of the mathematical relationship. As a result, a third value of the mathematical relationship is readable adjacent the other of the scales.

Abstract: A method for graphically representing binary data in a condensed, machine-readable form, includes forming a pattern of information-carrying frame and synchronization lines defining a geometric reference system which forms boundaries of one or more data fields. The reference system carries machine-readable marks such as a bar code identifying the pattern as a data-field reference system and providing orientation and other information. The reference system also includes synchronization lines forming boundaries of each data field and providing a location reference. Within the reference system the data field has dot locations functioning as data-transmission elements. Each location has or lacks a mark, representing a "1" or "0" binary bit. Each data element location has a known geometric relationship to the synchronization lines of the reference system so that coordinates of each individual mark can be determined precisely.