For the purpose of initiating an improvement by petition it is customary to adopt a fixed scale for the measure of the benefits, based upon distance, that will probably be derived from the improvement. Some legislative bodies have enacted definite rules for evaluating “influence” in petitioning. Generally the rule is based upon some mathematical variation. For example that the assessed value or influence of property of uniform width extending back from the roadway shall vary as the square root of the maximum distance back. In the figure on page 313, a lot of one-unit area fronting the street is given a value of 31.62. This is from the mathematical formula _y_² = 1000_x_ where _y_ represents the assessed value or influence in petitioning, and _x_, the distance back, considering the value of _y_ = 100 for _x_ = 10. To draw the curve mark off on a straight line ten equal distances; at the mid-point of these distances or units erect perpendiculars. From the formula calculate values for _y_ as shown in the table; lay these off on the verticals and plot the curve through their extremities. To clarify this some, suppose that upon the center of the first space, there being one unit area or lot here, there is stacked up the value of the assessed benefits 32 (31.62) silver dollars. On the next space, since there are two lots extending back from the street, the stack would contain 45 (44.72) silver dollars--continue this for each space and for the number of lots extending back. A curved line passing through the tops of the stacks representing the assessed values will be the influence curve plotted. [Illustration: +---+------+------+ |_x_| _y_² | _y_ | +---+------+------+ | 1 | 1,000| 31.62| | 2 | 2,000| 44.72| | 3 | 3,000| 54.77| | 4 | 4,000| 63.25| | 5 | 5,000| 70.71| | 6 | 6,000| 77.46| | 7 | 7,000| 83.67| | 8 | 8,000| 89.44| | 9 | 9,000| 94.87| |10 |10,000|100.00| +---+------+------+ Assessment curve.] For the purpose of initiating an improvement the unit in which the prospective benefits are to be measured is usually adopted by the governing or assessing authorities. Dollars will not do because the cost will not be known until after the improvement has been finished. In the case of roads and streets the unit quite generally used is the “front-foot.” The number of front-feet in any paving district will be the same as the number of abutting feet along the street to be improved. A different definition for “front-foot” is given on page 318. The petitioning power or influence of the several properties constituting the whole frontage is proportional to the number of front-feet assigned to each property, and these are assigned according to the adopted rule which is supposed more or less closely to measure the benefits to be derived from the improvement.