The Paper Computer

Turn one number to another

Set the red C index to 3 on the fixed D scale. Without moving it again, 2, 5, and 7 on C point to 6, 15, and 21 on D. Drag the multiplier or use the buttons.

A maker’s manual, read again

The anchor is the scanned instruction manual for the Concise Type E circular slide rule, pages 6 and 7. These are transcriptions of its printed examples, followed by calculations made independently here. The scan was visually checked; the diagrams are newly rendered, not facsimiles.

angle(x) = 2π log10(x). Therefore angle(a) + angle(b), modulo 2π, = angle(mantissa(a × b)). The circle keeps the significand and throws away the power of ten.

The disc can show 1.5 for both 3 × 5 = 15 and 0.3 × 5 = 1.5. You restore the decimal point by estimating first. For 3 × 7, “about 20” makes the printed mantissa 2.1 become 21.

What the bend buys, and what it costs

A straight one-decade scale has ends. Products that run past 10 require changing which index you use. On this circle, logarithms live modulo 1: adding a full decade is one full turn, so the scale joins itself. One setting can expose a complete proportional family.

The longer scale does not automatically mean more accurate arithmetic. Line width, two close radial scales, parallax, centre-hole eccentricity, pin play, card warp, and the reader’s setting can erase the geometric advantage.

The CI chain

The inverse CI scale reverses direction. The Concise manual uses it to chain 3 × 4 × 5 = 60. Independent check: log10(3) + log10(4) + log10(5) = 1.778151, whose mantissa angle is log10(6) = 0.778151. The separate estimate, roughly 3 × 20, restores 60 rather than 6.

Manufacture the physical rule

Error budget, before you measure

For the stated ranges, the largest modelled terms are finding the true lid diameter, handling the thread, and the combined disc reading and pivot error; which one dominates depends on the particular lid, build, and reader. Root-sum-square is roughly 1 to 2.3%, so this experiment uses a ±2.5% acceptance band. This is a conservative component model, not a measured calibration certificate.

1. On both sheets, independently measure the horizontal and vertical 100.0 mm checks. Continue only if each is 99.5 to 100.5 mm. The kit ruler measurement above does not replace these two-axis checks.
2. Measure both diagonals of the 100.0 by 20.0 mm rectangle. Their ideal length is 101.980 mm. Continue only if the two measured diagonals agree within 1 mm. Equal diagonals test skew, not absolute scale.
3. Glue each disc to flat cereal-box card before cutting. Cut the fixed disc at radius 85 mm and rotor at radius 76 mm. The printed circular scale edge, not your cut edge, is the datum.
4. Pierce both printed centre targets and two 12 mm radius washers with the same pin. Stack fixed disc, washer, rotor, then washer and optional cursor. Pin into cork, an eraser, or a small corrugated-card pad.

Sharp pin: an adult should supervise children. Keep a cork, eraser, or card pad over the point and never leave it exposed.

Let a lid supply the answer

Choose a round lid about 60 to 100 mm across. Measure its diameter through the apparent centre in several orientations and average. Wrap non-stretch sewing thread once around the rim without overlap, mark the meeting point, straighten it, and measure the circumference. Use your paper disc in division mode to compute circumference divided by diameter. Repeat three times.

Enter all three raw trials. No result is claimed yet.

A median inside 3.14 ± 0.08 is the stated target, about ±2.5%. Every paper quotient must also agree with its independently calculated C ÷ d within 0.08; otherwise the check fails even when the typed median is 3.14. Outside either band is a useful failure. Recheck that the diameter crosses the true centre, thread does not stretch or overlap, the pin has little lateral play, and both printed scale bars passed. Together, the target and agreement checks test the measuring and calculating chain.