Francesco Fontana (1580–1656) from practice to rules of calculation of lens systems

Abstract

In 1646, Francesco Fontana (1580–1656) published his Novae Coelestium Terresriumque Rerum Observationes which includes discussions of optical properties of systems of lenses, e.g., telescope and microscope. Our study of the Novae Coelestium shows that the advance Fontana made in optics could not have been accomplished on the basis of the traditional spectacle optics which was the dominant practice at his time. Though spectacle and telescope making share the same optical elements, improving eyesight and constructing telescope are different practices based on different principles. The production of powerful astronomical telescopes demanded objective lenses with much longer focal length and eyepiece lenses with much shorter focal length than the range of focal length of lenses used for spectacles, respectively. Moreover, higher standard of precision and purity of the glass was required. The transition from the practice by which optical components were chosen from ready-made spectacle lenses to lenses which were produced according to predetermined specifications (e.g., calculation of focal length) was anything but straight forward. We argue that Fontana developed the optical knowledge necessary for improving the performance of optical systems. Essentially, he formulated—based on rich practical experience—a set of rules of calculation by which optical properties of a lens system could be determined and adjusted as required.

Appendix

During the first half of the seventeenth century, attempts were made by men of science, mathematicians, and practitioners to improve the optical performance of the telescope. The production of low powered telescopes for terrestrial applications was relatively simple; such instruments, assembled by lens makers, spread out in Europe since the 1610 s. However, the production of powerful astronomical telescopes demanded objective lenses with much longer focal length and eyepiece lenses with much shorter focal length than the range of focal length of lenses used for spectacles, respectively. Moreover, high standard of precision and purity of the glass was required. The transition from the practice by which optical components were chosen from ready-made spectacle lenses to lenses which were produced according to predetermined specifications (e.g., calculation of focal length) for the use of advanced telescopes was therefore anything but straight forward. The Appendix offers historical data concerning the state of the art of optics in the first half of the seventeenth century. This constitutes the background for Fontana's original endeavor to develop a general mathematical expression for calculating the focal length of a lens in terms of its diameter of curvatures in a rigorous manner for the design and production of advanced telescopes.

• A. Johannes Kepler (1571–1630)

In his Dioptrice (1611), Kepler discussed the effects lenses have on the path of the rays refracted by converging and diverging surfaces. He applied a theory of image formation which he had proposed in his Ad Vitellionem paralipomena quibus astronomiae pars optica traditur (1604),³⁴ to explain the optical properties of lenses and the theory of the telescope.³⁵ In the introduction to Dioptrice, Kepler acknowledged that his “mathematical book is not easy to understand and it requires not only a clever reader, but also a particularly intellectual mind, and an incredible desire to learn the causes of things.”³⁶

In Dioptrice, Kepler shewed that a single spherical surface, either convex or concave, converges or diverges the parallel incidence rays, towards a point on the optical axis at a distance of one and a half diameter of curvature behind the convex surface, and one diameter of curvature behind the reverse concave surface. In the case of bi-convex lenses having similar convexity on both sides of the lens, the parallel incidence rays to the optical axis converge towards a point on the optical axis at a distance of half the diameter of curvature behind the lens’ surfaces.³⁷ These optical phenomena were empirically known by optical practitioners, and were formulated as rules which Kepler sought to theorize. To obtain a better understanding of refraction in combination of two or three lenses, Kepler developed an approximation valid for angles of incidence below 30 degrees by which the angle of refraction was assumed to be 2/3 of the angle of incidence (Zik 2003, 499–502; Malet 2003, 109–111).³⁸ Given this approximation, Kepler geometrically analyzed refraction through a lens having two surfaces of unequal diameter of curvatures.

Kepler explained the magnifying power of a converging lens; he showed that this power can be reduced by the counter-effect of a diverging lens. He acknowledged that the relative degree of convexity and concavity of a meniscus lens' surfaces not only determined the amount of convergence and divergence of the rays respectively, but it also determined the distance between the lenses mounted in the telescope's tube (Malet 2003, 119–127). However, Kepler had no rule to determine the focal distance in terms of the diameter of curvature of the refracting surfaces of a lens. He suggested that the rays will meet the optical axis somewhere between the focal points of the two different radii of curvatures of the lens' surfaces.

All in all, Kepler's geometrical approximation was sufficient for obtaining an idea of the location of focal points, and understanding of the interaction of system of lenses set in a telescope with human visual perception. Kepler’s Dioptrice was widely read and well appreciated up to the 1670s and even later, but he fell short of finding the specific technical details for calculating the separation of the lenses along the telescope’s tube (Malet 2003, 2005; Zik 2003; Dijksterhuis 2004, 10–13).

• B. Benito Daza de Valdés (1591–c. 1636)

Daza de Valdés was a Dominican friar and an officer of the Spanish Inquisition. He graduated in arts and philosophy at the University of Seville and studied law, which allowed him to practice as a notary of the Holy Office in Seville. Daza de Valdés did not have formal training in optics or mathematics, but in his book, The Use of Eyeglasses (1623), he reported on his systematic approach to solving optical problems, the tools he developed, and how to classify glasses by marking them numerically.³⁹

Spectacle craft masters marked the diameter of curvature of a lens by a number to classify it according to the customer’s age group. In 1623, Daza de Valdés explained that the numbers assigned to eyeglasses are based on portions of the diameter of curvatures of the spheres from which the grinding tools were formed (Daza de Valdés [1623] 2004, 107). He recorded accounts, detailed by craft masters, of practices which had originated in Italy and evolved over a couple of centuries of numbering eyeglass lenses. The earliest written evidence of specifying spectacle lenses for the correction of farsighted (presbyopia) and nearsighted (myopia) visual defects is a letter sent by the Duke of Milan in 1466. He ordered 200 pairs of spectacle lenses specified in accordance with the age of the users (Ilardi 2007, 224–226). In 1585, Tommaso Garzoni (1549–1589), in his popular encyclopedia of occupations practiced in his time, included a chapter on glassmakers. He noted that the curvature of the iron forms used for grinding convex and concave lenses was measured on a scale from 0.5 to 15 punti (points or degree presumably parts of Venetian foot). Garzoni was not an optical practitioner, but his text attests to a common practice applied by spectacle makers for specifying the various lenses in use according to age groups.⁴⁰

The difference between the largest diameter of curvature and the smallest diameter of curvature of the spectacle lenses to which Daza de Valdés referred was measured in length units equal to the local measure of one vara (0.836 m).⁴¹ Then, the measured length was empirically divided into 30 nonlinear parts denoted as degrees (grados),⁴² which corresponded to age 30 to 80 (Daza de Valdés [1623] 2004, 107–118).⁴³ Daza de Valdés reported, inter alia, on the practice of Spanish spectacle makers. Most of the spectacle lenses were made by simply guessing their size and without considering their degrees or the diameter of curvature required for determining the size of the lenses. Daza de Valdés therefore set the solutions required for improving this unprofessional practice. He described a procedure for testing both convex and concave lenses, and how could they be fitted properly to correct individual customer's eyesight. His test chart was made of two circular figures of different size. The lens under test was placed over one of the figures, the concave lens over the larger figure and the convex lens over the smaller, while the observer looked at them from a distance of about 56 cm. The lens placed over the figure is then lifted towards the face with the edge of the lens touching a vertical stick placed on the chart. When the figure, seen through the lens, is equal in size to the other figure seen—not superimposed—either with one or both eyes, the distance of the lens from the chart is marked on the stick and compared to a nonlinear scale of lens numbers, ranging between 2 and 10 degrees, on which the mark provides the number of degrees required for the lens. Apparently, contemporary lens makers did have a numbering unit for specifying lenses and testing method for properly adjusting the lenses. However, the testing method Daza de Valdés described is designed to evaluate the accommodation capacities of the lens on customer's sight, and not to measure the lens’ focal length.⁴⁴ Daza de Valdés may be considered therefore a pioneer in putting the practice of testing visual acuity and prescribing eyeglasses for correcting visual defects in writing (Hofstetter 1988; Bennet 1986, 13‒14).

In Dialog 4 of his book, Daza de Valdés listed the optical properties of nine Galilean telescopes he seemed to have examined. The optical properties he provided for each telescope were the degrees of the convex and concave lenses and the length of the telescope’s tubes (Daza de Valdés [1623] 2004, 171, 175). Deza's de Valdés assigned greater importance to the convex lens than to the concave lens. When the convex lens is of good quality, the rest does not pose much difficulties, for almost any quality of the concave lens will then be adequate. He emphasized that the length of the telescope tube is determined by the distance in degrees which its convex lens requires. Apart from the lens’ numbers and the overall length of the telescope’s tubes, he did not provide in this list any further details of the used lenses. In fact, he did not show any awareness of the relation between the lens’ numbers and the magnifying power of the telescopes. Daza de Valdés followed thereby the prevailing understanding that the eyepiece did not play a role in the magnification; rather, its purpose was to sharpen the image formed by the objective which meant that the telescope’s two lenses were not considered a single optical system (Daza de Valdés [1623] 2004, espec. 175, 171–178; Zik and Hon 2017b).

Daza de Valdés detailed the method for assessing the magnification of a telescope. One should place an object at a distance where it can be seen with the naked eye. Then, one should move away from it until one can no longer see the object with the naked eye. Next, one should observe the object with a telescope from the distance at which seeing it with the naked eye was not possible. Then, one should measure the distance to the object when one was able to see it, and divide this distance by the longest distance at which one was able to see the object with the telescope. The resulting ratio amounts, on this account, to the magnification of the telescope (Daza de Valdés [1623] 2004, 174).

Daza de Valdés reported on nine telescopes. In the table, we juxtapose the optical properties of the lenses set in the telescopes which Daza de Valdés recorded with the results of optical calculations using the modern OSLO code.⁴⁵

No.	Item	Valdés’ telescopes	OSLO	Δ length
1	Convex	0.25 degree	0.299 diopter
	Concave	3 degrees	3.588 diopters
	Length	4 vara = 3.344 m	3.069 m	0.275 m
	Magnification		12
2	Convex	0.5 degree	0.598 diopter
	Concave	6 degrees	7.177 diopters
	Length	2 vara = 1.672 m	1.535 m	0.137 m
	Magnification		12
3	Convex	1 degree	1.196 diopter
	Concave	8 degrees	9.57 diopters
	Length	1.25 vara = 1.045 m	0.733 m	0.312 m
	Magnification		8
4	Convex	2 degrees	2.392 diopters
	Concave	12 degrees	14.354 diopters
	Length	0.75 vara = 0.627 m	0.351 m	0.276 m
	Magnification		6
5	Convex	3 degrees	3.588 diopters
	Concave	16 degrees	19.14 diopters
	Length	0.33 vara = 0.276 m	0.23 m	0.046 m
	Magnification		5.33
6	Convex	4 degrees	4.785 diopters
	Concave	20 degrees	23.92 diopters
	Length	0.33 vara = 0.276 m	0.17 m	0.106 m
	Magnification		5
7	Convex	8 degrees	9.57 diopters
	Concave	30 degrees	35.89 diopters
	Length	6 fingers = 0.108 m	0.0796 m	0.0284 m
	Magnification		3.75
8	Convex	10 degrees	11.96 diopters
	Concave	40 degrees	47.85 diopters
	Length	6 fingers = 0.108 m	0.0657 m	0.0423 m
	Magnification		4
9	Convex	12 degrees	14.35 diopters
	Concave	70degrees	83.73 diopters
	Length	4 fingers = 0.072 m	0.0607 m	0.0113 m
	Magnification		5.83

The differences between the tubes’ length reported by Daza de Valdés and the results of the computations using OSLO, shown in column “Δ length” of the table, are of significant importance. These differences suggest that Daza de Valdés and the practitioners with whom he probably had consulted were not aware of the fact that the length of a Galilean telescope is determined by the sum of its lenses’ focal lengths.

The limited technological capacities hindered the production of convex lenses of 0.25 degree and 0.5 degree, and concave lenses of 30, 40, and 70 degrees, which were not part of the standard inventory used for spectacle lenses, thus making it difficult to improve the telescopes’ visual performance. Daza de Valdés described qualitatively the optical performance of the telescopes, mostly in reference to terrestrial objects, the moon, but not with respect to the planets (Daza de Valdés [1623] 2004, 172–174).

Daza de Valdés did not refer to his sources. However, his description of the optical effects which can be generated with lenses, concave mirrors, and pinhole projections in the last part of Dialog 4, is similar to what Della Porta provided in Book 17 of Magia Naturalis (Della Porta 1589, 259–280). Daza de Valdés’ account of handling of the telescope as an observing and projecting device resembles the account given by Christopher Scheiner (1575–1650) in his Accuratior Disquisitio (1612) (Reeves and Van Helden 2010, 211–230). Scheiner’s Ocvlvs Hoc Est (1619) could be another source for Daza de Valdés concerning issues, such as the anatomy of the eye, the refraction of light rays inside the eye, the functions of the retina, the correspondence between the eye and the Camera obscura, and contraction and expansion of the pupil in the various functions of the eye. Though Daza de Valdés’ contribution to our knowledge of the Spanish ophthalmic practices of his time is significant, his account of telescopic optics fell short of addressing the principles and specific technical details needed for comprehending the complexities involved in the use of optical instruments.

• C. Antonius Maria Schyrleus de Rheita (1604–1660)

In 1645, Rheita published the book Oculus Enoch et Eliae in which he described how terrestrial and astronomical telescopes should be built. The book includes several chapters in which Rheita formulated rules for lens making and offered instructions how to select suitable glass and materials used in these processes. He devised an apparatus for preparing the metal molds used for grinding and polishing the shape of the lenses, and explained the operative rules for obtaining their final shape on a lathe. He further developed a technique for polishing the lenses without deforming their surfaces (Rheita 1645, 340–349). Rheita provided descriptions of telescopes with two converging lenses; telescopes with three lenses through which a straight image is seen; a telescope with four lenses; and a binocular telescope (Rheita 1645, 352–356).⁴⁶

Rheita presented three tables which reveal the way he characterized the optical properties of the lenses used in his telescopes. To produce accurate tools for grinding and polishing lens' surfaces, it is necessary to know the measure of sagitta of the arcs of the metal molds. For this purpose, Rheita introduced the ‟Tabula longitudinis Sagitta circini angularis” by which one could obtain the sagitta for the arcs corresponding to each lens’ mold according to the length of the telescope tube from ¹/₂ Roman foot to 30 Roman feet (Rheita 1645, 349–350).⁴⁷ In the first table, one column presents the length of the telescope' tube denoted in Roman feet, and the adjacent column presents the measure of the sagitta denoted in ten thousandths parts of a foot corresponding to the lengths of the telescopes’ tubes.⁴⁸ In the second table, the ‟Tabula amplitudinis diametric apertura seu foraminis conuexi obiectiui spharici” one could obtain the diameter of the clear aperture stop suitable for the objective lens of the telescope according to the length of the telescope’s tube from ¹/₂ Roman foot to 100 Roman feet (Rheita 1645, 350–351). The left column of the table presents the length of the telescope's tube denoted in Roman feet, and the right column presents the aperture diameter denoted in ten thousandths parts of a foot corresponding to the length of the telescope’s tube.⁴⁹ In the third table, ‟De tubo astrospico monoculo, duobus conuexis debita proportione elaborando”, one could obtain both the diameter of curvature of the objective lens and that of the eyepiece suitable for each telescope according to the length of the telescope’s tube from 1 Roman foot to 50 Roman feet. The left column of the table presents the length of the telescope's tube denoted in Roman feet, and the right column presents parts of the eyepiece diameter of curvature denoted in one-hundredths parts of a foot corresponding to the length of the telescope’s tube.⁵⁰

In describing the stages of the production of telescope, Rheita delved into the minute technical details of the procedure and the tools involved. However, he did not elaborate how the optical properties of the telescopes he produced were acquired, nor did he mention the magnification of his telescopes. When he addressed the properties of a telescope suitable for observing heavenly bodies, he picked up an instrument with a tube length of 12 Roman feet (Rheita 1645, 340). Thus, according to Rheita’s table, the diameter of curvature of the objective lens of that telescope is 2.68 m, and the diameter of curvature of the eyepiece is 30 hundredth parts of a foot (67.02 mm).⁵¹

Rheita worked empirically.⁵² The list of his telescope lenses showed that a constant ratio of 2.5 hundredths of a foot (2.234 mm) should be added for each foot length of the diameter of curvature of the objective lens to obtain the diameter of curvature of the eyepiece. Thus, the multiplication of the length in feet of the telescope’s tube by 2.5 will result in the number of one-hundredths parts of a foot of the diameter of curvature of the eyepiece.⁵³ The same holds for the table of aperture’s diameter. From the rule he gave for explaining the usage of his table of aperture’s diameters, one can deduce that a constant magnitude of 130 parts of a Roman foot should be added to the diameter of the aperture for each tube length of one foot.⁵⁴ Consequently, each part of a Roman foot equals to 223.4 mm/10000 = 0.02234 mm, and that the aperture of the astronomical telescope of 12 feet he proposed above should be 34.85 mm.⁵⁵

Rheita’s book provides ample details—some of them original—of technical issues related to the way of constructing terrestrial and celestial telescopes. Yet, with regard to the telescope mounted with three lenses, the telescope mounted with four lenses (which he considered as his secretum),⁵⁶ and the binocular telescope, Rheita gave only a general description of the lens setup. Rheita was a skillful optician who contributed to the development of practical rules during the 1650s. His apparatus for improving the production of metal molds on a lathe, and the process of polishing the lenses, were indeed original. However, with regard to the optical data of telescopes mounted with two lenses, the details Rheita provided are surprisingly similar to the format of the list provided by Benito Daza de Valdés in 1623 (see Appendix B, above). Like Benito Daza de Valdés, Rheita specified only the length of the telescope’s tube and the diameter of curvature of the configuring mold of the telescope’s eyepieces. He did not refer to the magnification of the telescopes. Moreover, the short telescopes in Rheita's table whose tube length varied between one foot (22.34 cm) and nine feet (2.01 m) were not feasible.⁵⁷

A survey of old telescopes mounted with two convex lenses manufactured in Italy before 1650 confirms that the strongest eyepieces then used did not have focal lengths shorter than 50 mm.⁵⁸ The same holds with respect to the diameters of the objective lenses of telescopes longer than 18 feet listed in Rheita's table, which had lenses larger in diameter than 50 mm.⁵⁹ It stands to reason that the largest diameter of the objective lenses then used did not exceed 50 mm.⁶⁰ However, the 25 telescopes with two lenses listed in Rheita's table magnified no more than 40 times. These instruments were not particularly impressive in comparison with the Galilean telescopes of good quality which could have been acquired in the markets.⁶¹

In addition, Rheita produced telescopes with three and four convex lenses.⁶² To enlarge the field of view of his telescopes, Rheita mounted a small convex objective lens in a narrow tube in front of the instrument. The other three convex lenses composing the eyepiece were mounted in the far end of a larger in diameter main tube (Rheita 1645, 352). To be sure, the field of view of this optical layout was enlarged, but the cumulative thickness of the four lenses reduced the throughput of this optical system. Moreover, the small diameter of the objective lens affects the telescope’s capacity for light gathering, reducing the contrast and degrading the quality of the image. Given the technological limitations at the time, and the absence of objective lenses of substantially larger apertures, these optical devices were useless for astronomical observations.

The optical properties of Rheita's telescopes were determined according to the data provided by the ready-made tables he drew. It transpires that a coefficient of a constant ratio between the diameter of curvature of the configuring tool’s objective lens and that of the eyepiece lens was added to each foot of length of the telescopes. The same holds for the table denoting the diameter of clear apertures of the telescopes, and the table denoting the sagitta's of the configuring tools. The magnitudes of the coefficients were calculated on the basis of cumulative experience originated in various optician guilds which resulted in the requisite measures.⁶³ Yet, without any standardization, Rheita's explanation of the rules for applying the table's data was in effect an ideal extrapolation which had nothing to do with real optics. Rheita did not trace the path of light rays through the telescope, nor did he calculate the focal length of the lenses he used. He rehearsed optical data which were empirically compiled and improved through decades of practice and were known among the guilds’ masters of craft. However, due to his experience and technical expertise, Rheita was able to modify the optical properties of his lenses empirically and produce terrestrial telescopes of better quality than the instruments available at the time in the markets.

• D. Bonaventura Cavalieri (1598–1647)

In 1647, 1 year after the publication of Fontana's Novae Coelestium (1646), Cavalieri published his Exercitationes Geometricae Sex. In a chapter titled De perspicillorum focis,⁶⁴ he summarized the principles of refraction according to Kepler's treatment of refraction in lenses in Dioptrice. In the chapter titled Regula generalis ad inueniendos lentium seu perspicillorum focos,⁶⁵ Cavalieri analyzed geometrically the path of single rays as they pass through each one of the 26 lenses he studied: plano-convex, plano-concave, bi-convex, bi-concave, as well as converging and diverging meniscus lenses.⁶⁶ Cavalieri realized that a parallel incidence ray, issuing from left to the right side of a lens, is converging towards a focal point on the optical axis, and when issuing from the very same focal point back along the same path, the ray is emerging parallel from the other side of the lens. If a ray is issuing from a point on the optical axis located between the focal point and the lens, it continues to converge or diverge after passing the lens, but less than before. Then, if a ray is issuing from a point on the optical axis further away beyond the focal point, it is converging towards a point on the optical axis beyond the opposite focal point. Cavalieri recognized that a ray issuing from twice the distance of the focal point is converging towards a point located at the same distance on the other side of the lens. Like Kepler, Cavalieri acknowledged that the magnifying power of a converging lens can be reduced by the counter-effect of a diverging lens. And that the relative degree of convexity and concavity of the lens's surfaces not only determined the amount of convergence or divergence of the rays, but also determined the distance of the lens’s focal point.

Cavalieri addressed two equal and one smaller intersecting circles, producing the shape of a bi-convex lens, bisected by a common chord drawn between the points of intersection. The shape of the convex lens was identified as one in which the convexity of the two surfaces is facing opposite directions, while the intersection of the smaller circle produced a lens having the shape of a “banana” in which the convexity of its two surfaces is facing the same direction. Cavalieri calculated the sagitta of each arc of the bisected lens, and the sagittas’ ratio to the diameters of the drawn circles.⁶⁷ In reference to each one of the lenses’s shapes, he formulated the following rules. In the case of a convex lens with two opposite convexities: ‟as the ratio of the sum of both diameter of curvatures of the lens' surfaces is to one of them, so is the ratio of the other diameter of curvature to the distance of the focal point.” And, in the case of a lens with two surfaces facing the same direction, he formulated the following rule: ‟as the ratio of the difference of both diameter of curvatures of the lens' surfaces is to one of them, so is the ratio of the other diameter of curvature to the distance of the focal point,”⁶⁸ The mathematical formulation of Cavalieri's first rule is (d₁ + d₂)/d₁ = d₂/f where d₁ and d₂ are the first and second diameter of curvatures of the convex lens’s surfaces, and f is the focal length. This rule is applicable to bi-convex lenses in which the two converging surfaces facing opposite to each other (Cavalieri 1647, 462, 490).⁶⁹ A numerical example will clarify the rule: in the case of a bi-convex lens with two different converging surfaces in which d₁ = 0.5 m and d₂ = 1 m, the focal length is, (0.5 + 1)/0.5 = 1/f thus, f = 0.33 m. Changing the order of the lens’s surfaces will result in the same focal distance, (1 + 0.5)/1 = 0.5/f, and thus, f = 0.33 m. The mathematical formulation of Cavalieri's second rule is (d₁– d₂)/ d₁ = d₂/f, where d₁ and d₂ are the first and second diameters of curvatures of the “banana” lens’s surfaces, and f is the focal distance. Again, a numerical examples will clarify the rule: in the case of a lens with two different converging surfaces in which d₁ = 0.5 m and d₂ = 1 m, the focal length is (0.5–1)/0.5 = 1/f thus, f = – 1 m, that is, this setup results in a diverging lens. Changing the order of the lens’s surfaces will result in the same focal distance but of a converging lens, (1–0.5)/1 = 0.5/f, and thus,  f = 1 m.

However, despite Cavalieri’s characterization of the aforementioned rules as a ‟general rule applies to find the focus of all the lenses mentioned above,”⁷⁰ the optical calculations showed that the rules could be considered a specific case at best. With the absence of the concept of sign convention in geometrical optics,⁷¹ Cavalieri's rules were short of formulating a general expression suitable for calculating the focal length of any lens shape in terms of its diameter of curvatures (Bennet 1986, 3).

Cavalieri's geometrical analysis and his rules enabled one to find the focal distance of a given converging lenses, or to find by projection the focal distance of diverging lenses of the same diameter of curvatures, that was the extent of generalization Cavalieri accomplished. A general formulation that could predetermine the properties of optical elements or lens systems according to specific optical requirements (e.g., focal length or magnification) was still wanting.