Edgar Dehn: Algebraic Charts
- Quadratic Equation General Form with Real Roots
- Cubic Equation Reduced Form with Real Roots
- Cubic Equation Reduced Form with Imaginary Roots
- Quadratic and Cubic Equations With Real or Imaginary Roots
- Cubic Equation General Form with a Real Root
- Biquadratic Equation Reduced Form with a Real Root
Every quadratic equation can be solved on Chart No. 4. If the roots are real, they should be found more accurately on Chart No. 1.
Every cubic equation can be solved on Chart No. 5, with the second term removed also on Chart No. 4. To obtain more accurate roots, Chart No. 2 should be used if all roots are real, otherwise Chart No. 3.
Biquadratic equations with the second term removed can be solved on Chart No. 6. But Charts Nos. 5 and 6 give only real roots; noting the sum and the product of possibly remaining imaginary roots, those may be found on Chart No. 4.
To put equations in a form given on the charts, approximate values of the coefficients will suffice.
If best approximation is wanted, multiply the roots by the greatest number, or divide the roots by the smallest number, still permitting the solution on a chart.
Use the ruler with the frosted side up and erase the lines again with a cleaner.
Chart No. 1: Quadratic Equation
general form with real roots
The roots of quadratic equations are obtained from the chart as the coördinates of points where curves intersect representing coefficients of the equations as functions of the roots.
The procedure of finding the roots of a given equation is as follows:
- Put the equation in a form given on the chart.
- Find the line labeled with the number p in a circle if q is plus, in a hexagon if q is minus, or interpolate such a line.
- Find the curve labeled with the number q in a rectangle, or interpolate such a curve.
- Find the intersection of the line p and the curve q.
- Find the numerical values of the roots as the coördinates of the point of intersection in one such way:
- Place the ruler over the vertical strip with the point, mark the point by tracing a cross along the lines which intersect, and read the roots putting the ruler over the scale of the chart.
- Mark the distances of the point from the nearest vertical and horizontal lines on a piece of paper reading the measure on the scale of the chart.
- Guess the first distance and subtract one root from p with opposite sign to obtain the other.
- Adjust the signs of the roots which are alike if q is plus, else different, the sign of the greater root being opposite to the sign of p.
Multiply or divide the roots, if necessary, to bring the intersection within the chart. To multiply the roots by k, multiply p by k and q by k².
Hold the chart with the corner up, or use Chart No. 4, if the roots are equal or nearly so.
x₁ = 7, x₂ = 5
x₁ = 5, x₂ = −3
Chart No. 2: Cubic Equation
reduced form with real roots
Two roots of cubic equations are obtained from the chart as the coördinates of points where curves intersect representing coefficients of the equations as functions of the roots.
The procedure of finding the roots of a given equation is as follows:
- Put the equation in the form given on the chart. Change the sign of q, if necessary, which changes the sign of every root.
- Find the curve labeled with the number p in a circle, or interpolate such a curve.
- Find the curve labeled with the number q in a rectangle, or interpolate such a curve.
- Find the intersection of the curves p and q.
- Find two roots as the coördinates of the point of intersection in one such way:
- Place the ruler over the vertical strip with the point, mark the point by tracing a cross along the curves which intersect, and read the roots putting the ruler over the scale of the chart.
- Mark the distances of the point from the nearest vertical and horizontal lines on a piece of paper reading the measure on the scale of the chart.
- Estimate approximately the two distances.
- Find the third root as minus the sum of the first two.
Multiply or divide the roots, if necessary, to bring the intersection within the chart. To multiply the roots by k, multiply p by k² and q by k³. Hold the chart with the corner up, or use Chart No. 4, if two roots are equal or nearly so.
y³ − 28y + 48 = 0
y³ − + = 0
y₃ = − (y₁ + y₂ )= −6
Chart No. 3: Cubic Equation
reduced form with imaginary roots
The real roots of cubic equations are found on the chart as numbers associated with diagonal lines obtained in the plane of the coefficients by varying the value of the unknown.
The procedure of finding the roots of a given equation is as follows:
- Put the equation in the form given on the chart. Change the sign of q, if necessary, which changes the sign of every root.
- Find the point (p, q) on the chart.
- Find the diagonal line through (p, q), or interpolate such a line.
- Find the real root as the number associated with the line.
- Find the imaginary roots using the formulas and scales of the chart.
Divide the roots, if necessary, to bring the point (p, q) within the chart. To divide the roots by k, divide p by k² and q by k³.
y³ + 24y − 160 = 0
y₁ = 4
y³ + − = 0
y₃ = −2 − 6i
Chart No. 4: Quadratic and Cubic Equations
with real or imaginary roots
The real roots of quadratic and cubic equations are found on the chart as numbers associated with points on the curves. These numbers are slopes of the curves, except for sign, and the curves are envelopes of lines obtained in the plane of the coefficients by varying the value of the unknown.
To quadratic equations belongs the curve on the right of the q-axis, to cubic equations the curve on the left.
The procedure of finding the roots of a given equation is as follows:
- Put the equation in the form given on the chart.
- Find the point (p, q) on the chart.
- Mark the lines through (p, q) which touch the curve.
- Find the real roots as numbers of the contact points.
- Find the imaginary roots using the formulas and scales of the chart. In the formula for quadratic equations q' denotes the ordinate up to the curve for a given p.
While the point (p, q) is readily located also outside the chart, divide the roots if necessary to bring the contact points within the chart. To divide the roots by k, divide p by k and q by k² in a quadratic equation, and divide p by k² and q by k³ in a cubic equation.
If the point (p, q) is on the curve, the equation has a double root.
If the point (p, q) does not give all real roots which there are, use the point (−p, q) for the quadratic and the point (p, −q) for the cubic equation. The changed sign changes the signs of the roots.
x² + 3 x − 10 = 0
x = 2, −5
x² − 10x + 34 = 0
x = 5±3i
y³ + y − 10 = 0
y = 2, −1±2i
Chart No. 5: Cubic Equation
general form with a real root
The real roots of cubic equations are found on the chart as numbers associated with points on the curves. These numbers are the negative slopes of the curves, and the curves are envelopes of lines obtained in the (q, r)-plane for the parameter p by varying the value of x.
Through points with the same value go slope lines giving the value. Slope lines are not drawn in the region of cusps but have to be marked there so that steeper lines serve steeper parts of the curves.
The chart gives only positive roots. Negative roots are found by changing the signs of p and r, which changes the signs of the roots.
The procedure of finding the real roots of a given equation is as follows:
- Put the equation in the form given on the chart.
- Placing the ruler over the scale of the chart, note the relative position of the point (q, r) in a horizontal strip.
- Mark the point (q, r) on the chart by putting the ruler over the horizontal strip containing the point.
- Find the curve labeled with the number p in a circle, or interpolate such a curve.
- Mark the lines through the point (q, r) which touch the curve p.
- Find the positive roots as numbers associated with the contact points.
- Find the negative roots in a like manner, using the point (q, −r) and the curve −p.
The point (q, r) on the curve p gives in general a double root, but it gives a triple root on the cusps. The locus of cusps is the envelope of the slope lines, and their numbers are noted on its scale.
While the point (q, r) is readily located also outside the chart, divide the roots if necessary to bring the contact points within the chart. To divide the roots by k, divide p by k, q by k² and r by k³.
x₃ = 6 − (x₁+x₂) = 6
Chart No. 6: Biquadratic Equation
reduced form with a real root
The real roots of biquadratic equations are found on the chart as numbers associated with points on the curves. The negative squares of these numbers are the slopes of the curves, and the curves are envelopes of lines obtained in the (p, r)—plane for the parameter q by varying the value of y.
Through points with the same numerical value go slope lines giving the value. Slope lines are not drawn in the region of cusps but have to be marked there so that steeper lines serve steeper parts of the curves.
The curves consist of two branches each. Those to the right of the central parabola q = 0 carry negative numbers, those to the left positive ones, and the points of the central parabola are both positive and negative.
The procedure of finding the real roots of a given equation is as follows:
- Put the equation in the form given on the chart. Change the sign of q, if necessary, which changes the sign of every root.
- Placing the ruler over the scale of the chart, note the relative position of the point (p, r) in a horizontal strip.
- Mark the point (p, r) on the chart by putting the ruler over the horizontal strip containing the point.
- Find the curve labeled with the number q in a rectangle, or interpolate such a curve.
- Mark the lines through the point (p, r) which touch the curve q.
- Find the roots as numbers associated with the contact points.
The point (p, r) on the curve q gives in general a double root; but two double roots differing by sign on the central parabola, and on the cusps a triple root. The locus of cusps is the envelope of the slope lines, and their numbers are noted on its scale.
While the point (p, r) is readily located also outside the chart, divide the roots if necessary to bring the contact points within the chart. To divide the roots by k, divide p by k², q by k³ and r by k⁴.
y₄ = − (y₁ + y₂ + y₃) = −4