Ruling Ten Thousand Lines Without Drifting

When you rule a diffraction grating, you’re commanding a diamond stylus to scribe thousands of parallel lines at precise intervals—say, 600 lines per millimetre. Each line position is calculated from the last. If your lead screw has a periodic error of even a micron, that error shows up as ghost spectral lines at predictable wavelengths. The errors don’t average out; they accumulate with a signature.

Floating-point arithmetic has the same problem. Add 0.0001 to a running total ten thousand times and you’d expect 1.0, but rounding errors compound. By the time you’ve ruled your ten-thousandth line, you’re microns off course, and your spectrum has ghosts.

William Kahan published a fix in 1965: compensated summation. You track the rounding error separately and fold it back into the next addition. It’s two extra lines per sum, but it keeps your accumulation honest across thousands of iterations.

Here’s the naive approach that drifts:

var sum: Float = 0.0
for _ in 1...10000 {
    sum += 0.0001
}
print("Naive: \(sum)")  // 1.0001483 — drifted by 148 ppm

var kahanSum: Float = 0.0
var c: Float = 0.0
for _ in 1...10000 {
    let y = 0.0001 - c
    let t = kahanSum + y
    c = (t - kahanSum) - y
    kahanSum = t
}
print("Kahan: \(kahanSum)")  // 1.0000000 — held the line

Same drift, same fix, in Perl:

my $sum = 0.0;
$sum += 0.0001 for 1..10000;
printf "Naive: %.7f\n", $sum;

my ($kahan, $c) = (0.0, 0.0);
for (1..10000) {
    my $y = 0.0001 - $c;
    my $t = $kahan + $y;
    $c = ($t - $kahan) - $y;
    $kahan = $t;
}
printf "Kahan: %.7f\n", $kahan;

The error you correct at step 5,000 would otherwise corrupt every line thereafter. Ruling engines compensate mechanically with correction cams and temperature-controlled chambers. Kahan compensates arithmetically by remembering what the last addition lost.