The Verification Venue · pointed at a feeling everyone has
The Year That Keeps Getting Shorter
You don't imagine that time speeds up as you age — but your clock never changes. What changes is a fraction. A year is one fifth of a five-year-old's life and one fiftieth of a fifty-year-old's, and that shrinking ratio is recomputable to the decimal.
There are two classic ways to turn that intuition into arithmetic, and they disagree by a lot. Drag your age and a remembered age below; both models recompute, side by side, exactly how much shorter this year feels than that one — and draw the curve of your accumulating felt time.
The age you're living from.
A summer that felt endless, say.
A year now feels, vs. then
— as long
Janet — · Lemlich —
Half your felt life is gone by
age —
Lemlich: felt-time midpoint of a life to your age
Here is the split. Janet's proportional law (Paul Janet, 1877; the version William James quotes in 1890) says the felt length of a real interval is its fraction of the life lived so far — the felt rate goes as 1/age. The cumulative felt time is then S = ln(age), and every doubling of age — 5 → 10 → 20 → 40 → 80 — feels exactly equal, because ln(2a) − ln(a) = ln 2 for any age. By that law a year at 50 feels just 5/50 = 1/10 of a year at 5.
Lemlich's law (Robert Lemlich, 1975) starts from a subtler hypothesis: a real interval is felt against your total accumulated subjective time, not your real age. That single change — dS/dt = k/S — integrates to S = 2√age, a felt rate of 1/√age, and a much gentler verdict: a year at 50 feels √(5/50) ≈ 0.32 — about a third, not a tenth. But it pays for the gentleness with a sharper twist: under Lemlich, half of your subjectively-felt life is already behind you by age 20 (for a life to 80), because the felt midpoint solves √a = √80 / 2.
The check — every number recomputed in front of you
For your current pair of ages, both laws, computed live from the formulas above:
Janet's signature property, the one the verifier nails to ln 2 = 0.693147 — every age-doubling carries the same felt span:
| doubling | Janet S-span = ln(2a)−ln(a) | = ln 2? |
|---|
The curve on the canvas is the exact cumulative-felt-time integral for each law — ln(age) and 2√age — not a stylised animation; the verifier confirms the trapezoid integral of each felt rate lands on its closed form to < 1e−4. Run it yourself: node research/the-year-that-keeps-getting-shorter/verify-the-year-that-keeps-getting-shorter.mjs
What's idealised here, and what's exactly true
Exactly true (it's arithmetic). Given each model's defining law, every number on this page is an exact consequence. Janet's 1/age rate makes age-doublings feel equal (each ln 2) and makes a year at 50 feel 1/10 of a year at 5. Lemlich's dS/dt = k/S integrates uniquely to S = 2√a, a felt rate 1/√a, a year-at-50-vs-5 of √(1/10) ≈ 0.316, and a felt half-life at exactly T/4. None of that is in dispute.
Free choices. The constant k in Lemlich's ODE is set to 2 only to make S = 2√a clean; any positive k gives the same ratios. The check also displays cumulative felt-time values and spans; Lemlich's displayed S values use k = 2. Janet's S = ln(age) needs an arbitrary lower anchor (we plot from age 1, where ln 1 = 0); felt time before the first birthday isn't meaningfully modelled by either curve. The sliders run to age 90 and treat age as a bare number, ignoring that lifespan itself is uncertain.
Where the clean story breaks. These are descriptive curve-fits to recollection polls, not mechanisms. Lemlich (1975) tested his law by asking college students and older adults to rate how fast time passed at various ages, and reported "generally good agreement" — but that is a fit to remembered impressions, not a measured clock. The deeper move, that an interval is judged against accumulated subjective time, is itself a hypothesis Lemlich assumes, not a finding. Modern work credits other forces entirely — novelty and routine (a full, new year lays down more memory and reads as longer in hindsight), attention, and dopaminergic "internal clock" effects. The proportional intuition is a powerful pump, not a settled cause.
So which is right? Neither is a law of nature. Lemlich's square-root curve is usually judged the better fit to how people actually report time speeding up — gentler than Janet's brutal 1/age — but "better fit to a poll" is the whole claim, and the honest verdict is that the felt acceleration is real, multiply-caused, and only approximately captured by either tidy fraction.