Setting The Scales   |   My Collection

One vivid memory from high school is my physics teacher, Mr. Upchurch, teaching us how to use slide rules. He did it in such a way that made finding answers to calculations with a couple of sticks fun and easy. First, he showed us how to add with two meter sticks. It was pretty simple really. Put the "end" of the top stick over a number on the bottom stick, move your finger over to another number on the top stick, read the sum of the two numbers under your finger on the bottom stick. WOW, that really worked!

Things got a little fuzzy for me when he made us do the math to convert the linear meter stick scale to a logarithm scale. But when I realized the only thing that changed was the scale on the sticks, I was back on track.

I never did like logarithms, but I got really good at using a slide rule. In fact, for several years after my own physics students started using calculators, I could get the answer to most problems quicker on my slide rule than they could on their calculators.

Mr. Upchurch lived to see calculators make slide rules obsolete and his problem student teaching another generation of physics students, but never dreamed of having his slide rule curriculum on something called the World Wide Web.

Mr. Upchurch, this is for you.

Jim Askew
 

Parts of a slide rule:

  • Body - upper and lower fixed bars
  • Slide - the movable middle bar
  • Cursor - the hairline


To use a slide rule, one must know the following:

  • How to read the scales.
  • How to "set" the slide and cursor for each operation.
  • How to estimate the result and determine the decimal point.
    • Estimation: convert the problem to round numbers that can be easily estimated.
    • Common Sense: for most practical problems, there is usually only one place for the decimal point in which the answer is reasonable.
    • Scientific Notation: carry out the indicated operations then use the laws of exponents to combine the exponents until a single power of 10 is indicated.

A Personal Opinion: I am convinced that the problems many of today's students have with fractions and decimals can be traced directly to the replacement of the slide rule by calculators. Students punch numbers into a calculator and expect it to provide the correct answer. The skills of estimation and carrying decimals are no longer practiced.

Slide Rule Scales:

The number of different scales, their length, and the number of subdivisions are largely determined by the size of the slide rule. Larger slide rules are better than smaller slide rules because of the detail of the scales.
  • C and D scales: used for the common operations of multiplication, division, and proportion.
    • These scales are also used to convert between radians and degrees.
    • Log formula for these scales is d = M Log10 n
    • For this equation, M is the length of the rule (10 inches), and d is the distance between 1 and n on the rule.

    • This produces a graduated scale.

    • Slide two of these scales across each other to begin calculating.

  • CF and DF scales: "folded" C and D scales beginning with pie. The index is near the middle of the scale.
    • Using "folded" scales increases operation speed by allowing the calculation without resetting the slide.
    • The circumference of a circle is found with the DF and D scales.
  • CI and DI scales: just like the C or D scale except it increases from right to left.
  • A and B scales: these scales are shrunk to half the length of the C and D scales and printed twice on the same line.
  • K scale: this scale is shrunk to one third the length of the D scale and printed three times on the same line.
  • L scale: used with the C or D scales to find the mantissa of the common logarithm (base 10) of a number.
    • The logarithm of a number is the exponent to which a given base (10) must be raised to produce the number.
    • A logarithm consists of two parts:
      1. The characteristic is the integer (left of the decimal).
      2. The mantissa is the decimal fraction (right of the decimal).
  • S scale: used to find the approximate value of the sine or cosine of any angle between 5.7 and 90 degrees.
    • Since sin X = cos (90 - X), the same graduations serve for both sines and cosines.
  • T scale: used to find the tangent or cotangent of any angle between 5.7 and 84.3 degrees.
    • Since tan X = cot (90 - X), the same graduations serve for both tangents and cotangents.
  • ST scale: used to find trig functions for small angles, less than 5.7 degrees.

Scale Graduations:

  • Graduations - the subdivisions of a scale
  • Primary graduations - lines on any scale with large numbers above or below them
  • Secondary graduations - ten shorter line divisions between the primary graduations
  • Tertiary graduations - shorter subdivision lines between secondary graduations
  • Left index - the first numbered mark at the left of a scale
  • Right index - the first numbered mark at the right of a scale
  • Special graduations:
    • Pie - found on all basic scales
    • ' - represents pie/4 or 0.7854 - found on A, B, C, and D scales near the right-hand end (just short of 8). Used for finding the area of circles
    • R - represents 57.3 - found on C, D, and CI scales. Used in changing from radians to degrees and conversely

Setting the Scales for Operations:     Practice the setting below with a virtual slide rule.

The Operation Rule: The Operation Set:
Multiplication:

Set the index of the C scale over one of the factors on the D scale. Move the cursor over the other factor on the C scale, and read the product under the cursor on the D scale.

To find P = XY

Division: (the inverse of multiplication)

Set the divisor (on the C scale) opposite the number to be divided (on the D scale). Read the result, or quotient, on the D scale under the index of the C scale.

To find Q = X / Y

Continued Products:

Set the C index at X on the D scale. Move the cursor over Y on the C scale. Move the C index under the cursor. Move the cursor over Z on the C scale. Continue moving cursor and C index alternately until all numbers have been set. Read the product under the cursor on the D scale.

To find P = XYZ

Combined Multiplication and Division:

Set the cursor over X on the D scale. Move R on the C scale under the cursor. Set the cursor over Y on the C scale. Move S on the C scale under the cursor. Continue moving the cursor and slide alternately untill all numbers have been set. Read the result on the D scale. If there is one more factor in the numerator than the denominator, the result is under the cursor. If the number of factors in the numerator and denominator are the same, the result is under the C index.

To find Q = XYZ / RST

Proportion:

Set R on the C scale opposite S on the D scale. Under T on the C scale, read X on the D scale.

(Note) Notice that when the C index is opposite 2 on the D scale, the ratio of 1:2 is set for all other opposite graduations.

To find X, R/S = T/X

Radians to Degrees:

When the C index is set over any number of radians on the D scale, under R on the C scale read the corresponding number of degrees on the D scale.

Degrees can be converted to radians with the same set.

To convert radians to degrees

Area of a Circle:

Set the B index to ', (0.7854) on the A scale. Move the cursor over the diameter on the C scale. Read the area under the cursor on the A scale.

When the area is known, the diameter can be found with the same set.

To find circle area

Circumference of a Circle:

Set the cursor over the diameter of the circle on the D scale. The circumference is under the cursor on the DF scale.

To find circumference of circle

Reciprocal of a Number:

When any number is set under the cursor on the C scale, its reciprocal is found under the cursor on the CI scale.

(Note) The reciprocal of a number N is 1/N

To find the reciprocal of N

Square Root and Square:

Set the cursor over any number N on the A scale and read the square root of N under the cursor on the D scale.

(Note #1) To find the square root of a number between 0 and 10, use the left half of the A scale. To find the square root of a number between 10 and 100, use the right half of the A scale.

(Note #2) To find the square of a number, reverse the set.

To find the square root of N

Cube Root and Cube:

Set the cursor over any number N on the K scale and read the cube root of N under the cursor on the D scale.

(Note #1) To find the cube root of a number between 0 and 10, use the left third of the K scale. To find the cube root of a number between 10 and 100, use the middle third of the K scale. To find the cube root of a number between 100 and 1000, use the right third of the K scale.

(Note #2) To find the cube of a number, reverse the set.

To find the cube root of N

Logarithms:

If the L scale is on the slide, set the cursor over the number N on the C scale. Read the mantissa of its logarithm under the cursor on the L scale.

If the L scale is on the body, set the cursor over the number N on the D scale. Read the mantissa of its logarithm under the cursor on the L scale.

(Note #1) If N is greater than or equal to 1, the characteristic is one less than the number of places to the left of the decimal in N.

(Note #2) If N is less than 1, the characteristic is negative. Its numerical value is one more than the number of zeros between the decimal and the first significant figure in N.

To find the mantissa for Log of N

Sine/Cosine of an Angle:

Set the cursor over the angle on the S scale. Read the sine/cosine of the angle under the cursor on the C scale.

(Note #1) The numbers printed at the right of the graduations are read when sines are to be found. (read left to right)

(Note #2) The numbers printed at the left of the graduations are read when cosines are to be found. (read right to left)

(Note #3) If the slide is placed so the C and D scales are exactly together, the sine/cosine can also be read on the D scale, and the mantissa of the logarithm of the sine/cosine (Log sin/cos) may then be read on the L scale.

(Note #4) If either of these trig functions is known for an angle less than 90o, set the cursor over the function value on the C scale and read the angle in degrees on the S scale.

To find sine/cosine of angle X

Tangent/Cotangent of angle:

Set the cursor over the angle on the T scale and read:

  • Tangents of angles from 5.7o to 45o under the cursor on the C scale. Read left to right.
  • Tangents of angles from 45o to 84.3o under the cursor on the CI scale. Read right to left.
  • Cotangents of angles from 45o to 84.3o under the cursor on the C scale.
  • Cotangents of angles from 5.7o to 45o under the cursor on the CI scale.

(Note #1) If the tangent/cotangent is read on the C scale, the decimal point is at the left of the first digit read.

(Note #2) If the tangent/cotangent is read on the CI scale, the decimal point is at the right of the first digit read.

(Note #3) If either of these trig functions is known for an angle less than 90o, set the cursor over the function value on the C or CI scale and read the angle in degrees on the T scale.

To find tangent/cotangent of angle X

5.7o to 45o

45o to 84.3o

Trig Functions for Angles Less Than 5.7o:

Set the cursor over the angle on the ST scale. Read the sine and tangent under the cursor on the C scale. Read the cotangent on the CI scale.

(Note #1) The sine and tangent of angles less than 5.7o are very nearly equal.

(Note #2) Sines and tangents of angles on the ST scale have one zero.

To find sin or tan of an angle

To find cot of an angle

 

My Collection:

The last slide rule I bought as a calculating device was a yellow aluminum Pickett, model N1010-ES TRIG. My first calculator was a TI-30 with a red LED display powered by a 9 volt battery. Both of these are now sitting on a shelf in my office.

Although collecting slide rules is not really a passion, I have kept my eyes open through the last thirty years and now have examples of most of the slide rules that were used in an educational setting.

Pickett

K & E
  • K&E #4081-5 Log Log Duplex Decitrig (20 inch scale)
  • K&E #4081-3 Log Log Duplex Decitrig (10 inch scale)
  • K&E #N4081-3 Log Log Duplex Decitrig (10 inch scale)
  • K&E #4092 with patent dates June 5 '00 and Dec. 22 '08 (10 inch scale)
  • (2) K&E #4092-3 with patent dates June 5 '00 Dec. 22 '08 April 1 '24 (10 inch scale)
  • K&E #N4053-3 Polyphase (10 inch scale)
  • K&E K-12 Prep #68 1892 (10 inch scale, white plastic with green print)
  • K&E Doric model 1992 (5 inch scale, white plastic)
Post Sun
  • Post #1446 (10 inch scale)
  • Post #1447 (10 inch scale)
  • Post #1452W (10 inch scale)
Dietzgen
  • Dietzgen #1732 Maniphase Multiplex with Pats. 2,170,144 & 2,258,722 (10 inch scale)
  • Dietzgen #1768-P (10 inch scale, white plastic with green slide)
  • Dietzgen Phillips #1759 B (10 inch scale)
            Others
  • Charvoz-Roos #Sr-109, 1946 (10 inch scale)
  • ACU-MATH #500 (10 inch scale, white plastic)
  • ACU-MATH #1200 (5 inch scale, white plastic)
  • Aristo #89 (5 inch scale, white plastic)
  • C-THRU S888 (10 inch scale, white plastic)
      This is the student model I still have in quantity to reward students if they master the basic slide rule operations.
  • Disney Mickey Mouse - Walt Disney World
  • Empire Pedigree (10 inch scale, white plastic)
  • Engineering Instruments #250BT (10 inch scale, wood/white plastic)
  • Sterling #587 (5 inch scale, white plastic)
  • Sterling Precision (10 inch scale, white plastic)

I have also found the following slide rule books:

  • The Standard Manual of the Slide Rule, Its History, Principle, and Operation
    J. E. Thompson
    Second Edition, 1952
  • The Log Log Duplex Slide Rule Self Teaching Manual
    Keuffel & Esser Co.
    1932
  • The Log Log Duplex Decitrig Slide Rule Manual #4081
    Keuffel & Esser Co.
    1943
  • How to use Trig Slide Rules
    Maurice L. Hartung
    1953
  • How to use Log Log Slide Rules
    Maurice L. Hartung
    1953
  • A Manual for the Slide Rule
    Paul E. Machovina
    1950
 

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