A calculation that leaves the screen

The Straight Line That Adds Squares

Put one straight edge through two lengths. The third crossing predicts a rectangle’s diagonal. Print the chart, make it answer with the computer off, then see the exact boundary of what three fixed parallel scales can calculate.

Make 60, 80, 100 line up

Move either input. The bright index is the only operation.

The right scale reads c = 100.0 mm. 60² + 80² − 100² = 0.

The published construction, reproduced first

On numbered page 2 of Maurice d’Ocagne’s 1925 Esquisse d’ensemble de la nomographie, a functional scale is written x = μ f(z), with the modulus μ expressed in a metric unit such as the millimetre. On numbered page 31, for f₁ + f₂ + f₃ = 0, outer moduli μ₁ and μ₂ force the middle modulus

μ₃ = μ₁μ₂ / (μ₁ + μ₂)

and its carrier divides the outer interval in the ratio μ₁:μ₂. Marshall printed the same modulus formula in 1921. These are reproduced results, not claims of novelty.

An asymmetric check keeps that rule from passing by symmetry: with outer moduli 30 and 20 mm/unit across 100 mm, the middle carrier is 60 mm from the first support, leaving 40 mm, and its modulus is 12 mm/unit. A midpoint rule would instead read 3.4167 for the test 1.2 + 2.3 = 3.5000, an error of 0.0833.

For this chart the outer modulus is chosen as M = 130/19600 = mm per mm². Equal outer moduli put the middle carrier halfway and give M/2. That midpoint is an instantiation, not a test of the general placement rule. The choice of 130 mm makes 140² occupy exactly 130.000 mm.

z (mm)z² (mm²)M z² (mm)

Euclid I.47 supplies the relation in modern notation: a² + b² = c². The table and plotted coordinates are computed, not hand-positioned.

Error budget, before you measure

No reader trial supports a universal ranking or millimetre envelope for this paper instrument. Input placement, output interpolation, index curvature and height, local print distortion, tick width, eyesight and whether the object is rectangular can all matter. Their effects depend on the chosen alignment; do not report a fixed accuracy from this page.

A single affine deformation of the whole sheet, including uniform scaling, unequal axis scaling or shear, preserves collinearity when the printed marks themselves are used. It can make the millimetre labels physically false or cause clipping, and the four dimensional checks below do not detect every shear. Non-affine warp, blur, curl and a bent index do not cancel. Use an independent ruler if physical millimetres matter, commit the chart reading before the direct measurement, and publish all signed residuals.

Print both sheets at 100% / Actual size. Disable Fit, Shrink and headers. Choose A4 or US Letter in the dialog. Essential marks sit within the common 210 × 279.4 mm safe box and are anchored from the top-left, so the geometry does not move sideways between papers. These four repeated axis-length checks catch gross scale errors; they are not four independent measurements and cannot certify right angles, shear or absolute calibration.

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

Cut, crease, stiffen
  1. Reject clipping. On sheet 2, cut one 190 × 8 mm strip on the solid outline. The second is a spare.
  2. Fold lengthwise on the dashed centreline, press the crease, then reopen it. The crease is the fine index.
  3. Tape the strip to the spare or a folded scrap, without bowing it. Keep tape off the centreline.
  4. Lay it across the chart. Do not pin it. Look directly down to avoid parallax.

Safety: cut away from your body on a protected surface. Scissors are sufficient.

Let a rectangle answer

Choose a rigid rectangular face with adjacent sides at least 50 mm and a diagonal no longer than 140 mm. Avoid a bevel, flexible cover, case or rounded corner. Measure the same geometric endpoints consistently. The chart reading and direct measurement must each lie from 70.7 to 140 mm; equality between those two entries alone is not evidence for the rule, because a reader can type the same value twice. The independently calculated prediction is the check.

  1. Use the printed measuring edge to record a and b.
  2. Place the index through a on the left and b on the middle. Commit the c reading before measuring the diagonal.
  3. Measure corner to corner with the printed edge, or transfer it with taut non-stretch thread.
  4. Repeat with a second rectangle, or rotate the first and swap a and b.

Waiting. Blank inputs, sides below 50 mm, and readings outside 70.7–140 mm are refused rather than averaged away.

Exactly what this architecture can say

The class is deliberately narrow: three distinct fixed parallel support lines x₁ < x₂ < x₃; each value chooses one marked point Pᵢ(zᵢ) = (xᵢ,yᵢ(zᵢ)); and a triple is valid exactly when one straight index meets those three points.

01

Expand collinearity

(x₃−x₂)y₁(z₁) − (x₃−x₁)y₂(z₂) + (x₂−x₁)y₃(z₃) = 0

Each coefficient is fixed and nonzero. Absorb coefficients, signs and ordinate offsets into three one-variable functions. Every chart in the declared class therefore expresses a weighted additive-separable relation F₁(z₁)+F₂(z₂)=F₃(z₃).

02

Run it backwards

Given any such relation, choose the three ordinates so the determinant is that relation. Its valid triples are collinear. This proves both directions inside the declared class.

03

Recover the published modulus and placement

Under the normalized orientation y₁=y₀+M₁f₁, y₂=y₀−M₂f₂, y₃=y₀−M₃f₃, coefficient matching requires

(x₃−x₂)M₁ = (x₃−x₁)M₂ = (x₂−x₁)M₃

Hence (x₂−x₁)/(x₃−x₂)=M₁/M₃ and M₂=M₁M₃/(M₁+M₃). Equal outer moduli put the middle exactly halfway. Reversing a scale, shifting ordinates, permuting variables, or applying an incidence-preserving transformation changes the drawing, so “forced” here means this normalized parallel layout.

04

Instantiate the paper

x = (20,105,190) mm
yₐ = 139 + Ma²
yᵦ = 139 − (M/2)b²
y꜀ = 139 − Mc²

Because the carriers are equally spaced, the determinant is proportional to yₐ−2yᵦ+y꜀ = M(a²+b²−c²). Collinearity is therefore equivalent to the Pythagorean relation.

A relation that refuses these three rails

Consider h(u,v)=u+uv+v². If it were locally writable as f₁(u)+f₂(v)=f₃(h) with C² functions and nonzero f₃′, mixed differentiation would force hᵤᵥ/(hᵤhᵥ) to depend only on h.

point A

(u,v)=(11/6, 1/2)
h =
ratio =

point B

(u,v)=(1,1)
h =
ratio =

Same output, unequal ratio. So no chart in this smooth, nondegenerate three-fixed-parallel-straight-scale class represents that relation on a domain containing both points. This says nothing against nonsmooth, multivalued, degenerate, piecewise, curved-scale, nonparallel or multi-index charts.

It also is not a global impossibility claim. D’Ocagne’s numbered pages 15–16 state that every three-variable equation has a general planar representation by three concurrent line systems. This example needs a different chart architecture.

The check, shown

Primary sources: Maurice d’Ocagne, Esquisse d’ensemble de la nomographie (1925), numbered pp. 2, 15–16, 26–31; William Crosby Marshall, Graphical Methods (1921), numbered pp. 173–175; Euclid, Elements I.47, Heath edition; ISO document identifying A4 as 210 × 297 mm; NIST SP 1247, US Letter 215.9 × 279.4 mm.

What remains empirical: no prototype reader study establishes tick readability, placement error, local printer distortion or a fixed accuracy envelope. Publish the actual signed residuals rather than inferring population performance.