# Who discovered the thin lens equation $\frac{1}{u}+\frac{1}{v}=\frac{1}{f}$?

According to Weisstein's webpage it was Halley in 1693 (quoting Steinhaus); but I've also seen it attributed to Cotes, Huygens, even Gauss (eg Britannica). Wikipedia's History of Optics does not give an originator.

A related question would be: Who invented the 2-ray method of geometrical optics of thin lenses, now taught in schools? Gauss? The proof of the lens equation follows by similar triangles from these lens diagrams. The earliest online book I can find that has something similar is Euler's (eg page 36).

Add: I know Euclid and Hero had some geometric optics, but it is not near proving the lens formula. Kepler drew the first geometric optics diagrams of this sort, but they are still not close.

"In the twentieth proposition of Tractatus, Huygens extends the results for thin lenses to non-parallel rays... Huygens had to treat all cases of positive and negative lens sides separately, but the result comes down to the modern formula $$\frac{1}{p'}+\frac{1}{p}=\frac{1}{f}$$" [p.18].
"Barrow, who treated optics as a mathematical playground, did not dwell on the paraxial (Gaussian optics) that is most useful to the theory of optical instruments. He nonetheless gave relations equivalent to the various cases of the modern formula $$\frac{n}{p'}+\frac{1}{p}=\frac{n-1}{R}$$ relating the algebraic distances $$p$$ and $$p'$$ (from the intersection of the axis with the refracting surface) of two conjugate points of a spherical refracting sphere with the index $$n$$ and the radius $$R$$, as well as unnecessarily complicated relations of conjugation for thin lenses. In contrast, Huygens’s early private treatises were aimed at improving the telescopes that he was designing with his brother. In his Dioptrica of 1653, he derived a variant of the now familiar $$\frac{1}{p'}+\frac{1}{p}=\frac{1}{f}\text{ with }\frac{1}{f}=(n-1)\left(\frac{1}{R}-\frac{1}{R'}\right)$$ for a thin lens of index $$n$$, radiuses $$R$$ and $$R'$$, and focal length $$f$$. This formula first appeared in print forty years later: case by case in Molyneux’s treatise, and in algebraic form in a memoir by Edmond Halley also including thick lenses." [pp. 75-6]
"With these powerful tools, Hamilton developed the study of caustics, foci, and various kinds of aberrations. In a memoir read at the British Association meeting of 1833, he briefly indicated how to use the $$T$$ function in the case of optical instruments with axial symmetry... The $$T$$ function thus leads to the basic concepts and relations of paraxial optics, although Carl Friedrich Gauss was first to define them in a general manner in a theory conceived in his youth and published belatedly in 1840. In his demonstrations, Gauss relied on the combination of the linear transformations connecting the parameters (inclination and transverse coordinates in a fixed plane) of incoming and outgoing rays for each refracting surface, as is most commonly done nowadays." [pp. 266-7]