Near Addition from Amplitude of Accommodation
Near Addition from Amplitude of Accommodation
The working distance in dioptres, less the share of the amplitude the patient can sustain. Published guidance says a half or two thirds — so both answers are printed, with the gap between them.
Amplitude ⇄ reading addition
Tentative addAmplitude 2.50 D, working distance 40 cm, half the amplitude sustained, age 52
Formula, and the fraction two sources disagree about
- the demand term
- 100 ÷ the working distance in centimetres, which is the reciprocal of the distance in metres and is the definition of the dioptre. 40 cm is 2.50 D, 33 cm is 3.03 D, 25 cm is 4.00 D. The term is a reciprocal, so the demand rises steeply as the task comes closer and a reported working distance that is 5 cm out matters far more at 25 cm than at 70
- the reserve, quoted
- the first source read states it twice: “As a rule of thumb, the patient should be able to maintain up to half or two thirds of their amplitude of accommodation”, and “the reading addition equals the working distance in dioptres minus two thirds or a half of the amplitude of accommodation in dioptres”. The second source read gives only the half: “usable accommodation = ½ × Acc”, with worked examples at age 40 (5.00 D minimum amplitude, 2.50 D usable) and age 48 (3.00 D, 1.50 D). Neither fraction is wrong; the half is the more conservative and gives the larger addition
- what the two conventions cost
- the difference between them is exactly one sixth of the amplitude, because ⅔ − ½ = ⅙. On a 2.50 D amplitude that is 0.42 D — not quite two lens steps. On a 6.00 D amplitude it is 1.00 D. The disagreement therefore matters most in the patients whose accommodation is largest, which is the ones least likely to need an addition at all
- Hofstetter’s three regressions
- read as plain text from a journal article’s methods: minimum expected amplitude = 15.0 − 0.25 × age, average = 18.5 − 0.30 × age, maximum = 25.0 − 0.40 × age, all in dioptres with age in years. The article attributes them to Hofstetter’s 1947 note and to Borish’s Clinical Refraction. They are PREDICTIONS for comparison with a measurement, and this page does not substitute one for the other
- where the regressions stop
- each goes negative at a finite age — the minimum at 60, the average at 61.7, the maximum at 62.5 — and a negative amplitude is not a measurement. The rows above disappear rather than printing one, which is also the clearest statement of what a linear regression on age can and cannot be used for
- why this is tentative
- the figure is a starting lens for a near refraction, refined against the patient’s own range of clear vision at the addition and against the task. The source read notes the method becomes less reliable above about 55, because age-related miosis deepens the depth of focus and the patient manages on less addition than the arithmetic predicts
- the vertex caveat
- the working distance is measured from the spectacle plane, and the addition is a spectacle-plane power. Transferred to a contact lens it needs vertex compensation, which at the small powers an addition usually takes is below the 0.25 D step and at high distance powers is not
Worked example
Amplitude 2.50 D, working distance 40 cm, half the amplitude sustained, age 52
Demand = 100 ÷ 40 = 2.50 D
Half the amplitude = 0.5 × 2.50 = 1.25 D, which is what the patient is expected to sustain
Addition = 2.50 − 1.25 = +1.25 D, and 1.25 is already a dispensable 0.25 D step
The other convention, on the same numbers. Two thirds of 2.50 is 1.67, so the addition becomes 2.50 − 1.67 = +0.83 D — nearest step +0.75. The two published fractions therefore differ by 0.42 D here, which is one sixth of the amplitude and about two lens steps
Against Hofstetter at 52. Minimum expected = 15.0 − 0.25 × 52 = 2.00 D; average = 18.5 − 0.30 × 52 = 2.90 D; maximum = 25.0 − 0.40 × 52 = 4.20 D. The measured 2.50 D sits between the minimum and the average, which is unremarkable
The reserve is what makes this work. Spending the whole amplitude on the task leaves nothing to hold focus with, so an addition computed as demand minus the FULL amplitude — 2.50 − 2.50 = 0.00 D here — predicts no addition is needed for a patient who in practice cannot read for more than a minute. That is the error the fraction exists to prevent
Three working distances. At 25 cm the demand is 4.00 D and the addition (half convention) is 2.75 D; at 33 cm, 3.03 D and 1.78 D; at 70 cm, 1.43 D and 0.18 D. The demand is a reciprocal, so the same amplitude gives very different answers and the working distance has to be the patient's real one
And where a row disappears. Set the age to 62 and the two Hofstetter rows vanish — 15.0 − 0.25 × 62 is −0.50 and 18.5 − 0.30 × 62 is −0.10, and a negative amplitude is not a measurement. The addition itself is unaffected, because the age never entered it
The two published reserve conventions compared
| Amplitude | Demand at 40 cm | Half convention | Two-thirds convention | Difference (amplitude ÷ 6) |
|---|---|---|---|---|
| 1.00 D | 2.50 D | +2.00 D | +1.83 D | 0.17 D |
| 2.00 D | 2.50 D | +1.50 D | +1.17 D | 0.33 D |
| 2.50 D | 2.50 D | +1.25 D | +0.83 D | 0.42 D |
| 3.00 D | 2.50 D | +1.00 D | +0.50 D | 0.50 D |
| 4.00 D | 2.50 D | +0.50 D | −0.17 D | 0.67 D |
| 5.00 D | 2.50 D | 0.00 D | −0.83 D | 0.83 D |
| 6.00 D | 2.50 D | −0.50 D | −1.50 D | 1.00 D |
Hofstetter’s expected amplitudes, and where they stop
| Age | Minimum (15.0 − 0.25 × age) | Average (18.5 − 0.30 × age) | Maximum (25.0 − 0.40 × age) |
|---|---|---|---|
| 20 | 10.00 D | 12.50 D | 17.00 D |
| 40 | 5.00 D | 6.50 D | 9.00 D |
| 48 | 3.00 D | 4.10 D | 5.80 D |
| 52 | 2.00 D | 2.90 D | 4.20 D |
| 55 | 1.25 D | 2.00 D | 3.00 D |
| 60 | Not defined | 0.50 D | 1.00 D |
| 62 | Not defined | Not defined | 0.20 D |
| 63 | Not defined | Not defined | Not defined |
Why the addition is not just the working distance in dioptres
A presbyope holding a book at 40 cm needs 2.50 D of vergence to focus it, and the naive answer is to prescribe whatever they cannot supply themselves. That answer is wrong, and the reason is the whole content of this page: accommodation cannot be held at its maximum. A patient who spends their entire amplitude on the task has nothing left to maintain focus with, and will manage a line or two before the print blurs. The published rule of thumb therefore subtracts only the share of the amplitude the patient can sustain, and keeps the rest in reserve.
How large that share is depends on which source you read, and the two read for this page do not agree. One states it as a range in its own words — the patient “should be able to maintain up to half or two thirds of their amplitude of accommodation”, and the reading addition is “the working distance in dioptres minus two thirds or a half of the amplitude”. The other gives only the half, as “usable accommodation = ½ × Acc”, and works two examples that way. Both are printed above because neither is wrong. The gap between them is exactly one sixth of the amplitude — ⅔ less ½ — which is 0.42 D on a 2.50 D amplitude and a full dioptre on a 6.00 D one, so the sources disagree most about the patients who need an addition least.
The demand term is a reciprocal and behaves like one. 40 cm is 2.50 D, 33 cm is 3.03 D, 25 cm is 4.00 D and 70 cm is 1.43 D, so a working distance guessed 5 cm short costs 0.33 D at 25 cm and 0.10 D at 70. The distance has to be the patient’s actual one, measured from the spectacle plane, and a patient with a short reach or a specific task — a musician’s stand, a dashboard, a sewing machine — is not the 40 cm the textbook default assumes. The amplitude has to be measured rather than predicted, too, and measured amplitudes depend on method: push-up over-reads, because the target’s angular size grows as it approaches and the blur is noticed late.
Hofstetter’s three regressions are printed above for comparison, not for substitution: minimum 15.0 − 0.25 × age, average 18.5 − 0.30 × age, maximum 25.0 − 0.40 × age. They are worth having because a measured amplitude well below the minimum expected for an age is itself a finding. They are also a clean illustration of a linear fit outliving its data — each line crosses zero at a finite age, the minimum at 60 and the maximum at 62.5, and this page prints nothing beyond that rather than a negative amplitude. Two last caveats. The method is less reliable above about 55, where age-related miosis deepens the depth of focus and patients manage on less addition than the arithmetic gives. And the output is a spectacle-plane power: moved to a contact lens it needs vertex compensation, which is negligible for an addition on its own and is not once it rides on a high distance power. The figure is a tentative lens, refined against the patient’s range of clear vision.
Frequently asked questions
How do I calculate a near addition from the amplitude of accommodation?
Take the working distance in dioptres — 100 divided by the distance in centimetres — and subtract the share of the amplitude the patient can sustain. With a 2.50 D amplitude at 40 cm and the half convention: 2.50 − 1.25 = +1.25 D.
Is half or two thirds of the amplitude held in reserve?
Published guidance gives both, and this page prints both answers. One source states that the patient should be able to maintain “up to half or two thirds” of their amplitude, so the addition is the demand minus a half or minus two thirds; another gives usable accommodation as half the amplitude. The two differ by exactly one sixth of the amplitude, and the half convention is the more conservative because it gives the larger addition.
What is Hofstetter’s formula for amplitude of accommodation?
Three linear regressions on age, in dioptres: minimum expected 15.0 − 0.25 × age, average 18.5 − 0.30 × age, maximum 25.0 − 0.40 × age. At 40 the minimum is 5.00 D and at 48 it is 3.00 D. Each line crosses zero at a finite age — 60, 61.7 and 62.5 respectively — beyond which it predicts a negative amplitude and means nothing.
Why can’t the patient use their whole amplitude for reading?
Because accommodation at its maximum cannot be held. A patient who needs their full amplitude to focus the task will clear it momentarily and lose it, which is why an addition computed as demand minus the full amplitude predicts that a struggling reader needs nothing. Subtracting only half or two thirds leaves the margin that makes sustained near work possible.
What does a negative addition mean?
That on the convention chosen the amplitude already covers the demand at that distance, with the figure’s magnitude as the margin. This page reports it as computed rather than clamping it to zero, because the size of the margin is the informative part — a margin of 0.12 D and one of 1.50 D are different situations. It is not a prescription for minus power at near.
Related calculators
References
- Patel T. Optometric examination of the older patient — part 7. Optician CET series, opticianonline.net. Accessed 9 October 2026.
- Open Exam Prep. Section 2.3: Usable Accommodation and Range of Vision, National Opticianry Competency Examination study guide. open-exam-prep.com. Accessed 9 October 2026.
- Yun J-H, Hwang H-Y, Kim SW, Kim H-M, Son J-S. Comparative study of the maximum accommodative amplitude in 20’s and 40’s myopia. J Korean Oph Opt Soc. 2012;17(3):273-8. (Reproduces Hofstetter’s three age-amplitude formulas as plain text, attributing them to Hofstetter HW, The Pennsylvania Optometrist 1947 and to Borish’s Clinical Refraction, 2nd ed.)
Not medical advice. For healthcare professionals and education. Reference intervals vary by laboratory and assay — always use your own laboratory's. Never base a dose or a treatment decision on this page alone. Full disclaimer at calcengines.com/disclaimer/