A calculator that leaves the screen

The Carry You Can See

Two paper instruments compute the same product. Napier's diagonals ask you to carry; the derived Genaille paths make that carry a line you can follow with a pencil.

Try it in ten seconds

One to ten decimal digits, multiplied directly by one digit from 1 to 9.

574 × 8 = 4592

The default deliberately crosses two nonzero internal carry states, 3 and 5; multiplier 1, zero, and a one-digit multiplicand would make the comparison vacuous. For a separate small physical check, arrange 8 rows of 7 coins, buttons or beans. Follow either sheet for 7 × 8, then count each object once. The required agreement is exactly 56 objects, discrepancy 0.

The published anchor, before the print button

Napier's 1617 construction supplies the complete digit products below. Each cell is tens above units, separated here by a slash. The generation rule is P(d,m)=d×m; q=⌊P/10⌋; r=P−10q. Checksum: the sum of all 90 products is 2025.

m\d

Napier's own chapter III example gives 1615 × 365 = 589475: the rods supply 4845, 9690 and 8075, shifted and added as 484500 + 96900 + 8075. The independent reconstruction and Napier's printed result agree exactly.

A contemporary 1885 notice specifies eleven square rulers in a box 0.01 m thick, 0.11 m wide and 0.18 m long, for a multiplicand of at most ten digits times one digit. This edition's 10 mm strip width is an inference from eleven square rulers occupying 110 mm, not a separately printed ruler measurement. A surviving museum boxed object is catalogued separately as 15 × 120 × 180 mm overall. These figures describe different scopes and are not averaged.

d'Ocagne's worked reading is reproduced here: 42457 × 8 = 339656. Read from the units: 6, then 5, 6, 9, 3, 3.

The triangles do hide the carry. Here it is.

On a digit strip d, multiplier row m, and incoming carry c, every corridor is exactly one transition:

T=m·d+c   output u=T mod 10   next carry c′=⌊T/10⌋

It never escapes its row's tracks. Since 0≤c≤m−1 and 0≤d≤9, then 0≤m·d+c≤10m−1, so 0≤c′≤m−1. After tracing k digits from right to left, the live cards above show the stronger invariant:

m·(N mod 10ᵏ) = Σ uⱼ10ʲ + cₖ10ᵏ

The path does not eliminate carrying. It removes mental carry arithmetic from multiplication by one digit by turning every possible carry into geometry. For a multi-digit multiplier you still need shifted partial products and addition.

Error budget, before measurement: human miscount, wrong strip order, or a lost path at a poorly aligned cut are plausible operation failures, but this edition has no user-trial data that ranks them. Printer scale and anisotropy affect the historical 10 × 180 mm format and edge alignment, but uniform scaling does not change the arithmetic. This edition declares a size-check tolerance of ±0.5 mm on the horizontal 150 mm ruler and the vertical 100 mm check; the historical sources do not supply that tolerance. There is no statistical error bar on an integer product.

Print at 100% / actual size, never Fit. Both A4 and US Letter fit. Measure the sheet's shared 150 mm horizontal ruler and its printed 100 mm vertical bar. Enter the shared ruler first:

Not calibrated. This page cannot vouch for a sheet it has not seen measured.

This page reports the estimated horizontal error but does not silently resize the edition. Correct the print setting and reprint. The orthogonal vertical bar exposes differential feed error: record its measured length minus 100.0 mm. The declared size check passes only when the 150 mm ruler is within ±0.5 mm and the vertical 100 mm bar is within ±0.5 mm; that does not certify cutting, registration, parallelism or endpoint legibility.

  1. Cut the eleven strips on each sheet.
  2. Tape their top edges to scrap paper and align every row boundary; at multiplier 9 the carry-track pitch is only 2 mm.
  3. Put the index at left, then digit strips in written order.
  4. Start the Genaille trace at carry 0 on the rightmost digit and move left. Record each output digit, then read the terminal carry printed on the index; prepend it unless it is zero.
  5. For Napier's sheet, add along the diagonals from right to left, carrying tens into the next diagonal.
  6. Close the computer. Build and recount the 8 by 7 array.

Sources and limits

This is a derived flat paper edition of the arithmetic transition graph, not a facsimile of every face, colour, typeface or triangle of the Belin object. The object-array check validates one small multiplication and your operation, not the historical attribution or all 450 reachable Genaille transitions.