The P scale is an inverted scale reading from right to left, hence the graduations are in red. The P scale is related to the D scale such that for a number "x" on the D scale, immediately below it on the P scale we have . This of course is only valid for (i.e. ), as we cannot have the square root of a negative number.

(Note we must have i.e. for to have a real value.)

Example 1:

1. Set the hair line over 0.6 on the D scale.
2. Under the hair line read off 0.8 on the P scale as the answer.

Example 2:

1. Set the hair line over 0.8 on the D scale.
2. Under the hair line read off 0.6 on the P scale as the answer.

Note: If then , thus to find we could either find x on the D scale and read off the P scale, or find x on the P scale and read off the D scale.

1. =
2. =
3. =
4. 

From the relationship we can express:

Example: sin 60° = 0.866 then cos 60° = 0.5

1. Set the hair line over 0.866 (i.e. sin 60°) on the D scale.
2. Under the hair line read off 0.5 (i.e. cos 60°) on the P scale as the answer.

1. if sin 35°48’ = 0.585, then cos 35°48’ =
2. if sin 90° = 1.000, then cos 90° =
3. if cos 70° = 0.342, then sin 70° =
4. if cos 81°42’ = 0.1445, then sin 81°42 =

For sines of large angles (i.e. in the region 80 to 90) working from the S scale is very inaccurate, as you can see from a glance at this region S scale.

Take for example sin 84°, the best we could estimate using the S and D scales would be 0.994. It would be impossible to make any more accurate estimation if the questing was 84°20’. A better method is as follows:

Example: sin 84°6’ = 0.9947

1. Set the hair line over 84°6’ (i.e. in the red graduations) on the S scale. (84°6’ in red graduation is the same as 5°54’ in black).
2. Under the hair line read off 0.09947 on the P scale as the answer.

The same situation arises for cosines of small angels. Therefore, using the fact cos 5°54’ = sin 84°6’ we have:

Example: sin 5°54’ = 0.9947

1. Set the hair line over 5°54’ (in black) on the S scale.
2. Under the hair line read off 0.09947 on the P scale as the answer.

Note:

1. For angles in red on the S scale, the P scale gives us the sine.
2. For the angle in black on the S scale, the P scale gives us the cosine.

1. sin 61° =
2. sin 78°30’ =
3. 83°24’ =
4. cos 27°48’ =
5. cos 14°6’ =
6. cos 47°48’ =

The square root of numbers a little less than 1, 100, etc, can be obtained using the D and P scales to a greater degree of accuracy than in the conventional way, with the D (or C) and A (or B) scales.

Example 1: (Fig 13.4)

Express

Note: We must have the form to use the P scale. Thus we subtract 0.911 from 1 to obtain 0.089, and express 0.081 = (0.298)² using the A and D scales.

Once we subtract 0.911 from 1, to obtain 0.089 the procedure is as follows:

1. Set the hair line over 0.089 (i.e. at 8.9) on the A scale.
2. Under the hair line read off 0.9545 on the P scale as the answer.

Example 2:

Express

1. Set the hair line over 0.0245 (i.e. at 2.45) on the A scale.
2. Under the hair line read off 0.9877 on the P scale as the answer.

Note: For we would express it as:

and obtain as in example 2

i.e. = 10x0.9877

therefore = 9.877

1. =
2. =
3. =
4. =
5. =
6. 

This is the form often encountered when using Pythagoras’ Theorem to find the third side of a right triangle. We note that:

Thus, if we calculate using the C and D scales and transfer the result onto the P scale, on the D scale we have . Then we could easily multiply by x to obtain (i.e. .

This answer would be read off the D scale, thus to obtain we would read the answer off the A scale.

Example:

Express

(evaluate in any of the usual ways.)

1. Set the hair line over 0.842 on the P scale. (The is on the D scale under the hair line.)
2. Place the right index of the C scale under the hair line.
3. Reset the hair line over 4.3 on the C scale.
4. Under the hair line read off 2.32 on the D scale as the answer.

Note: If instead of we required on the A scale as the answer.

1. =
2. =
3. =
4. =
5. =
6. 

1. To calculate the ordinates of an ellipse . Transpose the equation to and work from the P to D scale as in 13.6.
2. The following tables give a few other uses of the P scale.

1. =
2. =
3. =
4. =
5. =
6. =
7. =
8. =
9. =
10. =
11. =
12.