• Galileo Galilei (1564-1642) heard from France, in 1609, that Lippershey in Holland had constructed a sort of "enlarging monocular." This device, made of a single lens of positive power at the first end of a tube and of a negative lens on a sliding tube, was in fact a chance arrangement of eyeglass lenses available on the market which thus may only magnify distant objects by two or three times. It must be considered as a poor half part of our ancient opera glasses, and was totally useless for astronomical observations. Over a few months's time Galileo fully understood its principle and transformed it into a "telescope" by constructing three of them known as telescopes No. 1, 2 and 3.1 In 1610, with telescope No. 3 he discovered Jupiter's satellites, Venus' phases, and the Sun's rotation (observations of Sunspots by the naked eye and natural camera obscuras were reported in China since 28 BC and later in Persia).
From the rustic enlarging monocular, Galileo discovered the basic optical features for obtaining two-lens systems with large magnifications, i.e. a telescope - an invention which he must be credited with -: with a plano-convex lens as his objective and one of various divergent lenses as an eyepiece, he derived afocal systems of large magnifications, i.e. large beam compressions. His second difficult task, and not the least, was to build accurate lenses able to provide such high magnifications (Fig. 1.3).
It is remarkable that all his objective lenses are close to a plano-convex shape which come up from currently available equiconvex lenses that, in order to obtain larger focal lengths, he probably re-figured by himself; this is relevant to his objective lens of telescope No. 3 - the only surviving piece of this telescope - which shows two concentric shapes on the same side: a flat or quasi-flat central zone defining the clear aperture surrounded by a useless convex surface. Galilieo obtained magnifications somewhat higher than 20 requiring deep divergent lenses down to fifi
Fig. 1.3 Galileo's first refracting telescopes. (Up) No. 1: length 980mm, magnification 21, clear aperture ~16mm, f/61. (Down) No. 2: length 1,360 mm, magnification 14, clear aperture ~26 mm, f/51. (Institute and Museum of History of Sciences, Florence)
1 Astronomers generally admit that Giambattista della Porta (1535-1615) built, in Murano, the first "monocular opera glasses" (or low power spyglasses) around 1580 - as he clearly mentioned in private correspondence and in Magia Naturalis - and that Lippershey was one among several Dutch authors who built replicas of an Italian model, about one foot long, dated 1580, and then did not obtained patent letters (cf. Danjon & Couder [1.4], p. 589 to 601). However none of them proposed or elaborated a design with sufficiently large magnification to be called a telescope.
47 mm focal length which he figured by himself because eyeglass makers did not make strong enough negative lenses for correcting such a huge myopia. Although his afocal design gave a view of objects at infinity, he naturally moved the diverging lens towards the objective by a slight amount in order to vision the sky at the eye's punctum proximum distance. His mother found it lucrative to sell lenses to persons who asked for them. Interferometric analyses of some Galileo telescope optics conserved at the Science Museum in Florence show that the emerging wavefronts were "diffraction limited" at a single wavelength; Galileo would not have grasped the nuance of such a compliment, however.
Galileo published his astronomical discoveries with Telescope No. 3 - with a first lens focal length f1 = 1,650 mm, f1/D = ^50 - in Sidereus Nuncius (1610), where he states having used a magnification up to 30 and recognizes that the objective lens could be replaced by a concave mirror.2
• Johannes Kepler (1571-1630) introduced the term "focus" in his work of 1604, Ad Vitellionem Paralipomena. In optics, he was the first to establish the conjugate distance relation for a given focal length. He noticed that the human eye works with an inverted image on the retina. In Dioptrice (1611), Kepler discussed the theory of telescope and enounced the rule giving the magnification as the ratio of the focal lengths of the two lenses. He described a refractor with a positive eyepiece but he never used one. The first positive eyepieces were used by C. Scheiner and later by F. Fontana in 1646; however this is generally known as Kepler's eyepiece.
• Willebrord Snell (1580-1626) discovered the sine law of refraction in 1621 from experiment. He died in 1626 without publishing his discovery. It was first published by Descartes in his Dioptrique (1637) without reference to Snell who communicated it privately to several people including Descartes (cf. Born and Wolf ).
• Rene Descartes (1595-1650) thoroughly elaborated the general theory of stig-matic curves based on analytic geometry - that he created for this purpose - and simultaneously introduced the standard symbolic writing which we are familiar with. It was known from Diocles (cf. Toomer ) for the paraboloid, and probably by Pappus for the ellipsoid and hyperboloid, that only conicoid mirrors provide a perfect reflected image of an axial source point.
In La Géométrie (1637), Descartes  introduces the complete theory of perfect axial imagery by aspherical surfaces that cancel the spherical aberration. It contains the equations of stigmatic surfaces of mirrors or lenses for a finite or infinite distance conjugate. For mirrors, the meridian sections of stigmatic conicoids appear as degenerated second degree curves. For lenses, the analytic geometry allowed
2 A few years later, N. Peiresc observing the Moon with a Galileo refractor begun drawing a map of it with the help of P. Gassendi and of a distinguished engraver C. Mellan; at mid work he discovered the Moon libration - oscillations of ~ 8° and 6° in longitude and latitude - and then completed his task with three maps done. He was one of the few scientists to defend Galileo against the Vatican Inquisition which condemned all published works on heliocentrism - among them De Revolutionibus Orbium Coelestium by N. Copernicus (1543) - and, in 1600, condemned the Copernician G. Bruno to the stake.
Fig. 1.4 Descartes' ovals: Consider a given source point F in a medium of refractive index unity, and its conjugate G in medium n. A refracting surface - diopter - of stigmatic shape satisfies FC + n CG = constant. The locus of C points is drawn, with a constant tension of the string ECKCG, along the marked straight edge FE as it pivots around F (La Géométrie, 1637 ) (cf. Chap. 9)
him to derive the stigmatic ovoids (cf. Chap. 9), which meridian sections, namely Descartes's ovals, are fourth degree curves.
Using the formalism of Greek geometrical methods, Descartes gave a famous construction of the ovals with the marked straight edge and a string (cf. Arnaudies and Delozoide ) which provides, through a refractive surface, the stigmatism of axial conjugates at finite distance (Fig. 1.4). All possible shapes of stigmatic lenses, designed with one spherical surface which is centered on the object or image, are displayed in La Dioptrique , which is part of Discours de la Methode.
No further advance was made in Descartes' theory of stigmatic surfaces until Petzval (1843) and Seidel (1856) established the complete theory with field aberrations, more than two centuries later.
• Christiaan Huygens (1629-1695), who recognized the importance of atmospheric seeing, built a 5.7 cm aperture refractor of 4 m focal length in 1655 (singlet objective lens at f/70) with which he discovered Titan. Refractors then increased further in size with Hevelius, Cassini, and others. In 1686, Constantin Huygens built several refractors so-called "aerials" - the tubes were open to the air - reaching 22-cm aperture for a focal length of 70 m (objective lens at f/300). Another example is the ~f/500 singlet objective at the Marseille Observatory (Caplan ), apparently used around 1700.
Throughout the period 1609-1740, single lens objectives evolved towards slower f-ratios, which still did not require any asphericity correction, but suffered hugely from chromatic aberrations and mainly from axial chromatism.
The axial chromatism provides a first order variation of the focal length with the wavelength. Further slow down of the f-ratio was not the right way to minimize its angular size: with such huge focal lengths, the human eye could not see any image at all by lack of sensitivity or integration time.
Lead oxide glass, known in Antiquity, was reinvented in English glass factories around 1620. A standard production process was set up in 1675 by Ravenscroft. This material, so-called crystal of England or light flint (LF) glass, offered the white brightness of (quartz-) crystal and was easy to elaborate from closed crucibles. Its refractive index at the yellow helium line was nd = 1.58 instead of 1.52 for the crown (K) or borosilicate (BK) glasses.
• Chester Moor Hall invented in 1728 the achromatic objectives - corrected from axial chromatism - by combining two lenses together: a negative flint lens and a positive crown lens. First, experimenting with flint and crown prisms, he carefully measured both their mean deviation angles and color dispersion angles. Then he determined the ratio of the prism angles of a matching prism pair that minimizes the resulting color dispersion which, thus, provided an achromatic deviation. Next, considering a lens pair, Hall stated that if at any given axial height the local prism angle ratio is the same, the chromatism correction will be achieved. Denoting K1 and K2 the optical power (cf. Sect. 1.4.3) of each lens in glass of respective dispersive power Sn1/(n1-1) and Sn2/(n2-1), this means that Hall discovered the achromatism condition K1 8n1/(n1 -1)+ K2 Sn2/(n2 -1) = 0. After designing a lens-pair, in 1733, Hall sub-contracted the optical figuring of the two lenses which, when assembled as a 3.5-cm aperture telescope, revealed results in accordance to his theory.
His results were well understood by Peter Dollond (renowned instrument and lens maker; his son John later succeeded in obtaining a Dollond patent for doublet lens achromats which was Hall's results! ), and proved that the dispersive power of a glass Sn/ (nd -1) completely differs between a flint and a crown. This brings to evidence Newton's error who, by supposing that the dispersive power was linearly the same for all glasses, hastily concluded that achromatization was impossible. Essays by L. Euler in 1742 and by S. Klingenstierna some years later confirmed this error. In establishing his theory of primary chromatism correction, Hall made possible the major advance in the development of refractors.
• Alexis Clairaut (1713-1765) elaborated the theory of achromatic doublet lenses in the period 1756-1762. He more accurately repeated the refractive index measures of crown and flint by Hall and P. Dollond, and concluded that a doublet-lens never could be exactly matched for obtaining achromatism because chromatic residuals will remain (these residuals were later called secondary spectrum). In a first memoir to the Royal Academy of Sciences , Clairaut discussed achromats with a crown first element. In this case, considering that the chromatic aberration of the crown positive lens must be set exactly opposite to that of the flint negative lens, he showed that continuous pairings are possible (with more or less spherical aberration). The assembled lenses provide a net positive power with the same focal length at two different wavelengths.
In the second memoir he investigated various shape achromats and discovered the second solution class with a negative flint lens as the first element. Investigating the two classes, and by varying the mean curvature of the lenses - cambrure in French - he derived relationships for achromats with nulled spherical aberration. Denoting c1; c2 the surface curvatures of the first lens and c3, c4 those of the second lens, Clairaut introduced an equal curvature for both internal surfaces, c2 = c3. Among the infinite number of solutions, this particular solution is known as Clairaut's equal curvature condition of minimizing the number of surfacing tools (which later allowed cementing the lenses for a higher throughput). It falls that this particular solution c2 = c3 is not far from the other particular solution with four differing curvatures which, in addition to the spherical aberration correction, gives the correction of the (off-axis) coma.
In the third memoir of 1762 Clairaut investigated the field imagery and noticed that the focused images asymmetrically aberrated (coma and astigmatism) and did not remain in a plane (field curvature). In a figure, he displays an off-axis blur image that he derived from trigonometrical ray trace. Finally, he derived the two simultaneous algebraic equations for non-cemented achromatic objectives corrected from both spherical aberration and coma. This is the Clairaut aplanatism conditions in the third-order aberration theory which was usually solved algebraically by Clairaut and soon after by J. D'Alembert. An equivalent graphical solution was found much later by A.E. Conrady: considering a (c2, c3) Cartesian plane and a positive crown lens as the first element, Clairaut's conditions are represented by a two-branch hyperbola (c3 - b)2/B2 - (c2 + a)2/A2 = 1 for zero spherical aberration and a straight line for zero coma. From the two solutions corresponding to the two intersection points, only the solution with curvatures c3 = 0.987c2 both negative is useful for aplanatic objectives; the second and freakish solution is with two meniscus lenses of curvatures c3 = 2.520 c2 both positive for a Schott glass BK7-F2 objective (Fig. 1.5-Left).
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