Fig. 1.5 Doublet-lens objectives achromatized for an object at infinity in the spectral range [Xc = 486; Xf = 656 nm], the blue and red hydrogen lines, and Xd = 587nm, the yellow helium line. Optimizations with Kerber's condition of focus defined from ray height at y/3/2 on the first surface, the entrance pupil. Effective focal length f = 1. Focal-ratio f/16. The curvatures are exaggerated on the drawings. Left: Clairaut's algebraic conditions, like the later ones by A.E. Conrady, produce a graph of this sort which shows Sphe 3 = 0 as a hyperbola and Coma 3 = 0 as a straight line, for thin air-spaced doublets (cf. for instance Szulc ). The intersection points give two aplanats for crown first; with BK7-F2 from Schott, these are with c% = -2.813, c3 = -2.777 and c2 = 2.631, c3 = 6.636. There are also two other aplanat solution classes for flint first. Right: Clairaut-Mossotti aplanat also called Cemented aplanat. Glass materials developed later, such as Schott SF5 and others, avoid half of the reflected light by cementing the two elements, c2 = c3; however this is only possible for small aperture objectives. In addition they provide a reduced secondary spectrum. With BK7-SF5 objectives and a field diameter of 1 degree, both aplanat solution classes provide the same resolution of 2arsec; this becomes two times better with CaF2-KZFSN4
A similar representation with other two solutions can be obtained with a negative flint lens as the first element.
A detailed historical account on Clairaut's optics work and the latter geometric representation of his algebraic conditions for aplanatic objectives is given by J.A. Church .3
• Joseph Fraunhofer (1787-1826) investigated the diffraction of light by gratings, published his theory of diffraction in 1823, and laid down the basis of spectroscopy. In 1752, the first spectral lines were observed by T. Melvill in the spectra of flames into which metals or salts have been introduced. Each chemical element is associated with a set of spectral lines. Fraunhofer studied the absorption lines of the Sun's spectra, originally observed by W. Wollaston (1802), and determined the wavelengths of the brightest lines of hydrogen, helium, oxygen, sodium, magnesium, iron, and calcium. The yellow helium d-line at 587.561 nm, and the blue H and red Ha hydrogen lines, i.e. the F- and C-lines at 486.132 and 656.272 nm, have been mainly used to characterize the refractive index nd and the reciprocal dispersive power vd = (nd -1)/(nF-nC) - sometimes called V-number - of optical materials. Fraunhofer's works on spectral lines made possible a major advance in the achrom-atization accuracy of doublet lenses.
Then without apparently using the algebraic results of Clairaut which includes the exact correction of coma - at least in theory - as reformulated by d'Alembert in 1764 and 1767, Fraunhofer designed achromatic doublet lenses by successive iterations of trigonometric ray traces. This allowed him to build excellent objectives with a reduced coma although this latter aberration was not exactly nulled.4
• Ottaviano Mossotti (1791-1863), professor of geodesics at the University of Pisa, elaborated the theory of cemented aplanatic doublet lenses in the period 1853-1859. Let us denote n = nd the refractive index of a glass, v = vd its reciprocal dispersive power and K the power of a lens of this glass, so that two lenses of different glass can be characterized by (Ki, vi) and (K2, v2). In Gaussian optics, the Hall achromatic condition K1/v1 + K2/v2 = 0, entails that the resulting focal length is exactly the
3 Clairaut's optical results continue to be misinterpreted in modern literature. For example, H.C. King's The History of the Telescope  is very wide off the mark on p. 157, where we read: "Clairaut managed to reduce astigmatism and [field] curvature to within reasonable limits, but he could do nothing for coma and considered it an irremediable evil of two-lens combinations..." which is totally false since he established the algebraic conditions for thin-lens achromatic aplanats; furthermore Clairaut and D'Alembert realized correctly that astigmatism and field curvature are not too serious in the narrow fields of view of telescopes.
4 Concerning some astronomical results obtained with Fraunhofer objectives, we must mention the Königsberg heliometer as the famous instrument with which Bessel discovered and measured the first stellar parallax, that of 61 Cygni. Referring to this instrument after publishing his aberration theory (1856), Seidel noticed that the coma was corrected [partly] and wrote that "this objective perfectly satisfies the Fraunhofer condition" but there was no such existing Fraunhofer condition since he used iterations of trigonometric ray traces.
This inappropriate claim by Seidel has often been repeated, thus introducing much confusion in the past and present literature. This condition is in fact the Seidel sum Cn = 0 (implicitly included in Claurault's algebraic formulation) and could have only been approximatively satisfied in Fraunhofer objectives.
same for the wavelengths XF and XC, whilst the effective focal length slightly differs since it is defined for the wavelength Xd.
Considering the Seidel aberration theory (1856) and doublets that satisfy Clairaut's condition of equal internal curvatures c3 = c2, in 1857 Mossotti [110-112] derived particular glass pairs n1, v1 and n2, v2 which provide aplanatic cemented objectives: the Clairaut-Mossotti aplanats corrected from axial chromatism, spherical aberration, and coma.
This led him to iteratively solve fifth degree equations - called Mossotti's equations - with respect to the power ratio K2/K1 of the two lenses. These equations always have three real roots of which only the root K2 /K1 ^-1 when n2 ^ n1 is of practical interest (cf. the grid curves by Chretien  in his Sect. 442). A solution can be derived for any case whether crown or flint comes first provided appropriate glass pairs which in fact were not really existing at this period. These results led H. Harting  to develop tables, which have been reproduced many times, giving the three computed curvatures of cemented aplanatic achromats in function of the glass pairing entries, i.e. indexes and dispersive powers. After 1900, similar theoretical results of this problem were also obtained by E. Abbe and co-workers. This boosted the elaboration of such new glass types also allowing lowered secondary spectrums, notably by O. Schott.
• Alfred Kerber (^1840-date of death unknown) improved the performances of doublet lenses in 1886. He showed  that, after the above aberration corrections, the chromatic variations of spherical aberration - or sphero-chromatism - at the extreme wavelengths Xc and XF now remain dominating at the image residuals because the two least confusion images are not located at the same axial position for these aberration variations (cf. Chretien  in Sect. 367). Kerber concluded that these residuals can be set to a minimal value. Since for first-order spherical aberration the axial location of the least confusion circle is defined by a zonal ray height hK equal to V3/2 = 0.866 times that of the clear aperture hmax, Kerber's condition of achromatization states that:
^ The axial location of the C-red and F-blue images of least confusion must be set in coincidence. This is achieved for the zonal ray height ratio hK/hmax = %/3/2 of the clear aperture height. Then the F and C blur images have the same size and their extremal radii are algebraically balanced for rays at half- andfull-aperture heights.
Hence, the Gaussian first-order achromatism condition K1/v1 + K2/v2 = 0 is now replaced by the non-Gaussian Kerber condition K1 / v1+K2/v2 = K28z, where K = K1 + K2 and the small axial shift 8z is set for the coincidence of C and F blur images. This focus is defined by zonal rays of optical height hK at the entrance pupil, i.e. at first surface of the objective. With automatized raytrace optimizations, the introduction of Kerber's condition provides an extremely robust operand for finding solutions. Another use of Kerber's condition is with the refractive correctors of Schmidt systems (cf. Sect. 4.2.1).
Around 1920, E. Turriere and H. Chretien  introduced the general name Clairaut-Mossotti doublet to refer to the aplanatic cemented doublet lens. Today's optical glasses allow a much better correction of the secondary spectrum (Fig. 1.5). An historical introduction and analysis for various f-ratio Clairaut-Mossotti aplanats is given by Szulc  with a non-Gaussian condition for achromatism.5
These advances, mainly due to Hall, Dollond, Ramsden, Euler, Clairaut, Fraunhofer, Mossotti, Kerber and optical glass factories led by Guinand and successors Mantois and Parra, opened the way to the construction of large discs for achromatic refractors that culminates just before 1900 with 0.8-1 m aperture telescopes in Potsdam, Paris, Mount Hamilton (Lick), and - the largest - Williams Bay (Yerkes), the two later at f/19. With these large telescopes, the residual spherical aberration - partly due to the stress residuals from the casting that led to a center-edge variation of the refractive index - was annulled by slightly aspherizing one surface. However, materials of high purity and homogeneity are difficult to obtain for large lenses and, furthermore the residual secondary spectrum residual is inevitable. This severely limited the spectral domain of astronomical observations so that, even for small size, a telescope objective could not be designed for both photographic and visual work.
Added to these problems is the fact that the telescope length was very long ^20-times the clear aperture diameter. Attempts at introducing a telephoto effect, by using a faster f-ratio objective and a negative doublet-lens focal expander of the Barlow type (1834) - in a kind of dialytic system -, showed that, compared to the single objective case with the same focal length, the image quality cannot be improved even for smaller fields of view.
The conclusion is that the refractor concept had become stagnant soon after 1900. All these problems were next solved with reflectors which significantly progressed since 1800.
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