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Why can a school microscope show you a cheek cell but not a virus? Better lenses do not help past a certain point. The reason was worked out in 1873 by Ernst Abbe, a physicist in Jena who turned microscope making from guesswork into calculation. This post covers his life from reliable biographies, explains numerical aperture, and works through the numbers that set the limit.
Who was Ernst Abbe?
The Neue Deutsche Biographie records that Abbe was born in Eisenach on 23 January 1840 and died in Jena on 14 January 1905. His father worked in a spinning mill. The MacTutor biography from the University of St Andrews describes the 14- to 16-hour working days his father endured and how the mill owner supported Ernst's schooling. He studied at Jena and Göttingen, where his teachers included Wilhelm Weber and Bernhard Riemann, took his doctorate in 1861 and qualified to teach at Jena in 1863.
In the 1860s Carl Zeiss, who ran a small workshop in Jena making optical instruments, asked Abbe for help. The NDB says Zeiss wanted to move instrument making away from "Pröbeln", the old trial-and-error method of testing and correcting. Zeiss made Abbe a partner in 1875.
What did Abbe show in 1873?
His paper Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung (Contributions to the theory of the microscope and of microscopic perception) appeared in December 1873. MacTutor gives the famous result as a resolution limit d = λ/(2nA). In modern notation this is d = λ/(2n sin α), where λ is the wavelength, n is the refractive index of the medium between the specimen and the lens, and α is half the angle of the cone of light the objective accepts.
The product n sin α is called the numerical aperture (NA), so d = λ/(2 NA). The idea behind it is diffraction. Fine detail in a specimen spreads light into a pattern, and the finer the detail, the wider the spread. If the objective cannot collect enough of that spread light, the detail is simply absent from the image, however much you magnify.
The Nobel Prize in Chemistry 2014 popular information says that Abbe's 1873 equation led scientists to believe that optical microscopes could never resolve things smaller than roughly half the wavelength of light, about 0.2 micrometres. Cells and some organelles were visible; single proteins were not.
Worked example: how fine can a microscope see?
Use green light, λ = 550 nm. The objective and oil values below are sample numbers of a realistic size.
Low-power objective, NA = 0.25. d = 550 / (2 × 0.25) = 1100 nm, about 1.1 µm.
High-power dry objective, NA = 0.95. In air n = 1, so sin α = 0.95 and α ≈ 71.8°: a very wide cone. d = 550 / 1.90 ≈ 289 nm.
Oil between lens and slide, NA = 1.40. With oil of refractive index n = 1.515, sin α = 1.40/1.515 = 0.924, so α ≈ 67.5°. d = 550 / 2.80 ≈ 196 nm, close to the 0.2 µm figure.
Bluer light, λ = 450 nm, NA = 1.40. d = 450 / 2.80 ≈ 161 nm.
Notice that in air NA can never reach 1, because sin α is at most 1. Filling the gap with a liquid of higher refractive index is the only way past it, which is why high-power objectives are used with oil.
Common mistakes with resolution questions
Confusing NA with magnification. A 100× objective with a small NA shows less detail than a 60× objective with a larger one.
Forgetting the medium. NA includes n. The same lens cone gives a larger NA in oil than in air.
Using the full cone angle. α is half the angle of the cone, measured from the axis.
Thinking smaller d is worse. d is the smallest separation you can still tell apart. Smaller is better.
The JEE (Main) 2026 syllabus lists the microscope and its magnifying power, and single-slit diffraction, but not resolving power by name. Check the current year's syllabus. For the magnifying power side, read our guide to microscopes and telescopes in optics problems.
What else did Abbe build?
The NDB credits Abbe with founding the new microscopy, both in theory and in a long list of instruments, measuring devices and illumination systems. In 1882 the chemist Otto Schott moved to Jena, and in 1884 the glassworks Schott und Genossen was founded with Abbe's help, making new kinds of optical glass with properties that could be calculated. Abbe's "apochromat" objectives, with much better colour correction, were first announced in 1886.
Carl Zeiss died in 1888, and by 1889 Abbe was the sole owner of the firm. In 1891 he set up the Carl Zeiss Foundation and gave it his fortune. Its statute included an eight-hour working day and pensions for workers, the NDB records, which reflects the mill worker's son he had been.
The limit he described was not the end of the story. Electron beams, with far shorter wavelengths, gave the first electron microscope built by Ernst Ruska. The Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, so you can revise diffraction before you tackle resolution.
What can a student take from how Abbe worked?
Ask what limits the result. Abbe stopped polishing lenses blindly and asked what physics sets the ceiling.
Separate two ideas that look the same. Magnification and resolution feel alike, but only one adds information.
Put numbers on it. A single formula told makers exactly which change would help.
Practice routine: find d for NA = 0.65 with red light of 650 nm, find the NA needed to resolve 250 nm in green light, and explain in one sentence why NA in air is always below 1.
Next step: look at the objectives on your school microscope, note the NA printed on each, and work out d for green light.
