NUMBER 13

Notes on

The
Perspeculum
of
Galileo
-
1610

In 1610, Galileo electrified the intelligensia of Europe, from England to Bohemia, with his discovery of the modern universe, as announced in his little book Sidereus Nuncius (The Starry Messenger). The book also described, however briefly, the perspeculum – the instrument he used to make his discoveries.

The perspeculum itself – which Galileo called in Italian the occhiale (the recognizably modern term Telescopium didn't appear until April 1611) – is the subject of this study.

Just how many telescopes Galileo himself fashioned is unknown. The usual quantity mentioned in modern discussions is "over a hundred." Demand was high; royalty and ecclesiastical authorities throughout Europe clamored for these marvels after reading Sidereus Nuncius. Of this unknown number, three, complete or in part, seem to have survived. They are Telescope I, Telescope II, and an objective in a fancy commemorative frame, which I refer to as the Medicean lens. All are in the Museo di Storia della Scienza (Museum of the History of Science) in Florence. (The Museum's web site is rearranged on occasion, so the link may stop working at any time – it's good as of August 2004.)

	Modern photographs of "Galileo's telescopes" are of the Florentine specimens (left), or copies of Telescopes I and II in the Science Museum in South Kensington, London (Telescope I, at right). Note the trivial fact that the Florence photograph is sometimes reproduced flipped left-to-right. It appears here in its proper orientation.
	The optical performance of Telescopes I and II, and the Medician lens, have been examined several times during the 20th century. See Abetti, G., L'Universo, 4, Sept. 1923, No. 9 (685-692), and Ronchi, V., L'Universo, 4, Oct. 1923, No. 10 (791-804) (hereinafter referred to as the 1923 papers). Their findings are mentioned in Henry C. King, The History of the Telescope, 1955 (42-43). For a brief modern examination with a digital phaseshift Fizeau interferometer, see Vincenzo Greco, Giuseppe Molesini, and Franco Quercioli, Optical Tests of Galileo's Lenses, Nature, 358.9 (July, 1992):101 (hereinafter the 1992 paper).

Attribution

The attributions of Telescopes I and II to Galileo are not entirely certain. I have yet to find more than a cursory justification for the claim that Galileo himself made these instruments. The 1992 paper says that "the authenticity of the telescope tubes seems certain, but some doubts remain about the lenses," but cites no authority. The Museo di Storia della Scienza website notes that an inventory anno 1702 implies that the eyepiece of Telescope II was, at that time, loose. By the nineteenth century the loose lens had been lost, and replaced by another lens. The 1992 paper says nothing explicitly about that; however, it has interferograms of all three objective lenses, but only one eyepiece (the original one, in Telescope I).

The Medicean lens is more positively attributed to Galileo himself. It is the objective lens of the perspeculum of Sidereus Nuncius fame. Attribution is based on the lens's known travels. As the Museum's web site tells it,

Galileo donated the lens .... to Grand Duke Ferdinand II. At a later date, the lens was accidentally cracked. After Galileo's death (1642) the lens was kept in the Guardaroba [Wardrobe] of Prince (later Cardinal) Leopold de' Medici. When Leopold died (1675), the lens was added to the Medici collection in the Uffizi Gallery .... In 1677, the Medici commissioned Vittorio Crosten to build the ebony frame in which the lens has since been preserved .... The collection remained [in the Uffizi Gallery] until 1793, when it was transferred to the Museo di Fisica e Storia Naturale. In the mid-nineteenth century, the lens was displayed in the Tribuna di Galileo with other Galilean memorabilia.

Unfortunately, even this much is uncertain. Another story claims that Vincenzo Viviani, the ageing Galileo's disciple and amanuensis, gave the broken lens to Prince Leopold de' Medici after Galileo's death, and that Leopold had it mounted in the ornamental frame (Baxandall, D., Trans. Opt. Soc., 24, p. 309, 1923). This doesn't square particularly well with the museum's version.

Galileo himself gives us few technical details about his telescopes, or their production history. He seems to have kept his glass grinding and polishing methods to himself until nearly 1638, the year he went blind. A few hints about the telescopes can be wrested from his letters and from Sidereus Nuncius.

Sidereus Nuncius

The book is widely available in translation. The English translations are those by Edward Stafford Carlos (The Sidereal Messenger of Galileo Galilei and a Part of the Preface to Kepler's Dioptrics, 1880), Stillman Drake (Discoveries and Opinions of Galileo, 1957), and Albert van Helden (Sidereus Nuncius or The Sidereal Messenger,1989). There are no important differences between these three so far as the telescopes are concerned. Galileo rushed his book into print to publicize his very recent discoveries; consequently, it contains quite a few errors. Drake and van Helden review those, van Helden perhaps more completely. In Sidereus Nuncius, Galileo announced (1) his devotion to Cosimo II de' Medici, Fourth Grand Duke of Tuscany, (2) completion of several telescopes, briefly describing three, (3) a procedure to determine the magnifying power of a telescope, (4) observations of and speculations about the moon's surface topology, and the nature and origin of "earthshine," (5) multiple new stars visible in Orion, Taurus, and Cancer, and (6) the Medicean planets (the four largest Jovian moons) and their orbits .... quite a lot, really, for such a little book.

Sidereus Nuncius appeared in March 1610. Observational astronomy immediately advanced at a pace we moderns sometimes imagine to be unique to our own era. By July 1610 Galileo had noticed a strange lump on either side of Saturn (although he never quite resolved these lumps as rings – a discovery which would wait for Christian Huygens, in 1659). In August 1610 Galileo had observed the phases of Venus, and by mid-1611 had observed sunspots – at first by direct observation (ouch!), later by projection, a technique attributed to one of his students. Galileo was not the only one pursuing these studies. There were others, some claiming – not entirely convincingly – priority for some of these discoveries, notably Simon Marius, Johann Fabricius, Christopher Scheiner, and Thomas Harriot. The Archbishop of Cologne gave Johannes Kepler a telescope in 1610. In his Dioptrice of 1611, Johannes Kepler described (1) spherical aberration, (2) the anatomy of the eye and the formation of an inverted image on the retina, and (3) a telescope using a single positive lens as an eyepiece (although Kepler doesn't seem to have actually made or used such a telescope). And modern observational astronomy was well on its way.

As an aside .... Galileo died in 1642, the year of Newton's birth. Newton leads us into the story of the development of the reflecting telescope .... but we'll leave that for another day.

Galileo's other writings

During this period, 1609 to 1623, there are other extant Galileo writings on astonomical subjects.

January 7 letter. This important letter is dated January 7, 1610. There are two copies, one (at the Vatican) complete with drawings of the moon. However, the addressee is not noted on the copy, and it is not clear that it was ever sent.

There are several interesting lines in this letter [Drake translations].

First is a note that Galileo is about to complete a telescope which would bring him even closer to the moon, as near as two terrestrial radii. Two terrestrial radii would be 8,000 miles. With the moon 239,000 miles distant, that implies 30x for the power of the telescope. Drake notes that he actually wrote "diameters" rather than radii – making the telescope magnification 15x, rather than 30x – but continues, "it is evident that he expected the new instrument to be about thirty-power." Unfortunately Drake doesn't note why this is evident.

Later Galileo writes, "... and only this evening I have seen Jupiter accompanied by three fixed stars ... "

And just before the close of the letter, he writes, "It is good that the convex glass, which is the one far from the eye, should be partly covered and that the opening left should be oval in shape, since thus are objects seen much more distinctly."

January 30 letter. In a letter to Belisario Vinta dated January 30, 1610, Galileo mentions that he and Cosimo had examined the moon, "though imperfectly, I not having had yet an instrument of the excellence that I now have." Vinta was secretary to the Grand Duke.

The Ann Arbor document. The University of Michigan Library, Rare Books and Special Collection, has the earliest page of Galileo's satellite observations now extant. Drake calls it, not unreasonably, the Ann Arbor document, after the University's home town. The page is a summary of observations Galileo made in mid-January, 1610.

Letters on Sunspots. Galileo's three Letters on Sunspots (dated May, August, and December 1612), published in Rome in 1613, were nominally a conversation with Mark Welser, a merchant of Augsburg and noted scientific dilletante. However, the real correspondent was Father Christoper Scheiner, a Jesuit professor at Ingolstadt. These letters are important both for their information on the earliest discoveries of sunspots, and for alerting just about everyone who could read Italian about Galileo's views on the Copernican system. These views would later land Galileo in difficult straits. The Letters, unfortunately, tell us little about the observation instruments themselves.

Letter to the Grand Duchess Christina. This letter dates from 1615, although it was not published until 1635, and even then was suppressed in Catholic countries. Christina was the mother of the Grand Duke Cosimo II. The letter was an explanation of Galileo's thoughts on the Copernican system and its relation to Scripture. Like the Letters on Sunspots, it is a historically important document, but tells us little about telescopes.

Discourse on Comets and The Assayer. Two lectures, published in 1619 as the Discourse on Comets, were officially authored by Galileo's student Mario Guiducci, but were acknowledged by Guiducci to be principally his old master's ideas. Among other things, the Discourse attacked the cometary theories of Father Horatio Grassi. Grassi responded with a counterattack, The Astronomical and Philosophical Balance. Galileo responded with The Assayer in 1623.

Summary of Galileo's written descriptions

Galileo's early telescopes

Galileo didn't claim the telescope as his invention, though he implied that he was the first to understand it, in the sense of being able to design one, rather than stumble upon one while fooling around with random lenses.

When Galileo first got wind of the new invention, he was a professor of mathematics at the Universiy of the Republic of Venice (now the University of Padua). During a visit to Venice he heard rumors of a device made by a "certain Fleming" (in Sidereus Nuncius, Latin Belga, English Dutchman or Fleming) with peculiar properties. A few days later a letter from "a noble Frenchman at Paris, Jacques Badovere" arrived, confirming the rumors. (Badovere, a former student of Galileo's, had written his letter in reply to a query from a Venetian, Paolo Sarpi, who had heard about the mysterious device late in 1608; Sarpi shared the interesting information with his old friend Galileo). Galileo at this time had never seen one of these Dutch telescopes. He puzzled out the operation of the device from what little was known in those days of the laws of refraction. He returned to Padua, and in a short time – according to one of his letters, less than a day – had made his first telescope. In Sidereus Nuncius, he describes it as a lead tube with a plano-convex object lens and a plano-concave eye lens, with a magnification of 3X. The short construction time means that Galileo made no attempt to grind these lenses – he obviously used spectacle lenses he either bought or happened to have lying around.

Galileo wrote, in Sidereus Nuncius, "Next I constructed another one, more accurate, which represented objects as enlarged more than sixty times." [Drake translation]. From the context he obviously meant that the image was sixty times larger in area – so linear magnification would have been the square root of 60, or slightly less than 8x. This instrument was completed sometime around August 1609. He took it with him to Venice, and spent about a month demonstrating it to various dignitaries. The fact that it gave erect images doubtless made it more impressive during tests on terrestrial objects and ships approaching the port. Galileo made much of the telescope's obvious military applications. He donated it to the Senate of the Republic, then went back to his workshops in Padua, to begin work on a more powerful instrument.

Some secondary sources claim that it was the earlier 3X instrument which Galileo donated to the Senate, but the 8X instrument would be more consistent with his written claims for its performance.

Galileo's third telescope is not mentioned in Sidereus Nuncius, but is discussed in the January 7 letter. In November 1609 he completed this instrument, of something around 20X. He used it to begin his astronomical studies. The first recorded lunar observation with this instrument is dated December 1, 1609. From the end of November to mid-December, Galileo observed, and sketched, the phases of the moon. Eight of these original sketches have survived. Galileo's telescope showed that the surface of the moon was not smooth, but, contrary to the classical conceit, was lumpy and irregular, much like the earth. He also saw three of the Jovian moons (though did not yet realize that they were moons), and saw that the planets, unlike the stars, could be resolved into distinct disks.

Galileo's January 7 letter also mentioned that he was about to finish a 30X instrument. This he later described in Sidereus Nuncius – "Finally, sparing neither labor nor expense, I succeeded in constructing for myself so excellent an instrument that objects seen by means of it appeared nearly one thousand times larger and over thirty times closer than when regarded with our natural vision" [Drake translation]. This is, in modern terms, a telescope of something a bit over 30X. It is most likely the instrument Galileo used for the bulk of the observations described in such detail in Sidereus Nuncius.

So, Galileo's first telescopes were -

Telescope 1 – a low-powered "proof of concept" instrument
Telescope 2 – an 8x instrument which he donated to the city-state of Venice
Telescope 3 – a 20x instrument with which he started his astronomical observations
Telescope 4 – a 30+x instrument with which he continued his astronomical observations

Can any of these be identified as the instruments now in the museum in Florence?

The Florence Specimens

Telescope I
As described by the Museo di Storia della Scienza, this is a tube made of two hemicylinderical pieces of wood held together by copper wire, the whole covered with paper. Construction is attributed to Galileo, circa 1610.

Telescope II
As described by the Museo di Storia della Scienza, this is a tube built up of strips of wood, covered in red leather (now faded to brown) with gold tooling. Construction is attributed to Galileo, circa late 1609 – early 1610.

Medicean lens

The lens is broken but retains its field stop. The lens and stop are mounted in an inscribed frame. The Museum description seems far more taken with the frame than with the lens.

The optical performance of Telescopes I and II, and the Medician lens, have been examined several times during the 20th century. See Abetti, G., L'Universo, 4, Sept. 1923, No. 9 (685-692), and Ronchi, V., L'Universo, 4, Oct. 1923, No. 10 (791-804) (hereinafter referred to as the 1923 papers). Their findings are mentioned in Henry C. King, The History of the Telescope, 1955 (42-43). For a brief modern examination with a digital phaseshift Fizeau interferometer, see Vincenzo Greco, Giuseppe Molesini, and Franco Quercioli, Optical Tests of Galileo's Lenses, Nature, 358.9 (July, 1992):101 (hereinafter the 1992 paper).

The table below gives the dimensions and optical properties of the three specimens, as given in various modern sources.

Source "Museo" – Museo di Storia della Scienza
Source "Greco" – Greco, Molesini, and Quercioli (Optical Tests of Galileo's Lenses, 1992)
Source "King" – Henry King (The History of the Telescope, 1955). King's numbers are taken from the Abetti and Ronchi papers of 1923.

Linear dimensions converted to millimeters.

	Telescope I			Telescope II			Medicean lens
source	Museo	Greco	King	Museo	Greco	King	Museo	Greco	King
objective diameter	51	51	51	37	37	37	38	58
objective aperture		26	26	15	16	16		38	38
objective focal length	1330	1330	1327	980	980			1710	1690
objective front radius		2700			535			940
objective back radius		950		plano	plano			12000
objective center thickness	2.5	2.5		2.0	2.0			4.0
objective speed		f/51			f/61			f/45
eyepiece diameter	26	26	26	22	22		n/a	n/a	n/a
eyepiece aperture		11	11		16		n/a	n/a	n/a
eyepiece focal length	-94	-94.0		-47.5	-47.5		n/a	n/a	n/a
eyepiece front radius	plano	plano			51.5		n/a	n/a	n/a
eyepiece back radius	48.5	48.5			51.5		n/a	n/a	n/a
eyepiece center thickness	3.0	3.0		1.8	1.8		n/a	n/a	n/a
telescope length	1360			980			n/a	n/a	n/a
telescope magnification	14x		14x	21x		~20x	n/a	n/a	n/a
telescope field of view	15arcmin		15arcmin	15arcmin		15arcmin	n/a	n/a	n/a
telescope angular resolution		see below	15arcsec		see below	10arcsec	n/a	n/a	n/a

The 1992 paper informs us that the lens curves are "polished to a good spherical shape," and estimates the glass index of refraction at 1.51 to 1.55. It notes no gross material defects in the glass, such as bubbles, odd colors, or visible striæ. The interferogram of the eyepiece of Telescope I shows a pattern of ring shadows, implying that perhaps the surface shows remnants of a turning process during fabrication. The paper has this to say about the Medician lens -

"The best quality belongs to the single lens, which can be considered as nearly diffraction-limited .... According to the Rayleigh criterion, its resolution at 633 nm is of the order of 3 arcseconds. Of course, the optical performance of the telescopes is degraded for several reasons, mainly chromatic aberration. Computer simulations taking dispersion into account lead to estimates of only 10-20 arcseconds resolution over the visible spectrum."

Are the Florence Specimens Really Galileo's Early Telescopes?

Telescope I remains a mystery. There seems to be nothing to overtly connect it with Galileo. Its general configuration and construction details are of the right type to be one of Galileo's efforts of this period, but there seems to be no written description of Galileo's which matches it. Its provenance as part of the vast Medici collections seems to be the only factor linking it to Galileo himself.

Telescope II seems, at least initially, a bit more promising. The speculative date – late 1609 to early 1610 – implies to me that the Museum is perhaps hopeful that this is the third one Galileo built, the 20X instrument mentioned in his January 7 letter, but not described in Sidereus Nuncius. But Galileo wrote about aperture stops in the same letter – "it is good that the convex glass, which is the one far from the eye [i.e., the objective], should be partly covered and that the opening left should be oval in shape, since thus are objects seen much more distinctly" [Drake translation]. This seems to describe a fairly serious amount of astigmatism in the objective lens of the telescope, and it certainly seems to be his third telescope, the 20X one, which Galileo is here discussing. The interferogram of the objective of Telescope II (in the 1992 paper) does indeed show measurable astigmatism; however, the aperture stop is round, not oval. For that reason I suspect that the Museum's Telescope II is not Galileo's third effort, but was actually constructed a bit later (if it was even made by Galileo at all).

Now for the Medicean lens. This is supposedly the lens with which Galileo discovered the Medicean planets, or the major Jovian moons. As his letter of January 7 makes clear, Galileo had observed at least three of the four large moons with the 20X instrument. But from his glowing description in Sidereus Nuncius, it seems inevitable that the 30+X instrument became Galileo's primary tool as soon as it was completed, sometime not long after January 7. So which telescope is the one which Galileo used to discover the Medicean planets, the 20X or the 30+X? The major clue is again the aperture stop. From the letter, the lens of the 20X telescope should show some appreciable astigmatism, and have an oval stop. The Medicean lens shows negligible astigmatism, and has a perfectly round stop. The reasonable conclusion is that the Medicean lens is the lens from the 30+X telescope. It follows that the focal length of the eyepiece of the 30+X telescope must have been -57 mm (for 30X) to -52 mm (for 33X).

The Lenses

So, who actually made these lenses – Galileo, or some humble spectacle maker of Padua?

Knowledge of glass and glass-working dates to remote antiquity, but until the modern era glass was opaque and useless for lenses. The few magnifiers known from the ancient era were fabricated from rock crystal, a clear, colorless form of quartz. These magnifiers were used for reading, and for generating heat from sunlight – to start fires or cauterize wounds. The first recognizable eyeglasses appeared in the late 13th century, in Italy. By the 15th century, negative (concave) lenses were being used to correct myopia; and, in Venice, the first clear glass was made, and exported in large quantities. It can't have been long before someone noticed that clear glass would make a dandy eyeglass lens, more easily obtained in useful sizes and more easily worked than rock crystal. The next notable optical advance in eyeglasses came in 1780, with Ben Franklin's invention of bifocals. By 1827, George Airey (later the seventh Astronomer Royal, and a Big Name in physics, astronomy, and optics) was experimenting with cylindrical lenses to correct astigmatism. About 1804, William Wollaston (another Big Name, mainly in chemistry, physics, optics, and physiology) noticed that meniscus lenses would be superior to simple lenses for eyeglass use.

So, in the period of concern here, the late 16th century, spectacles were being made of simple glass lenses. There may have been some rock crystal lenses being made, but in most situations there would be little advantage. The refractive index of clear quartz – 1.55 – is just about identical to that of the Venetian glass of that era, so there are no obvious optical advantages to using quartz lenses. And quartz is a relatively hard material, difficult to grind to shape. Glass spectacle lenses were plano-convex or plano-concave, or sometimes perhaps double convex or double concave – but all were so-called "simple" lenses. Meniscus lenses, bifocals, and more complex figures to correct astigmatism, had yet to appear.

Exactly who made some of these spectacle lenses into the first device which we would recognize as a telescope remains obscure, but it had happened by 1608 at the latest, when Hans Lippershey of Zeeland applied for a patent. These earliest telescopes would all have been of low power – 2X to 3X, like Galileo's first effort.

Incidentally, Galileo professed a preference for Florintine glass over Venetian, claiming one to be clearer than the other. However, this was when he was trying to relocate from the university at Padua to the court at Florence, and he could conceivably have been trying to flatter his would-be patrons in Florence.

Aberrations of the Human Eye and Their Corrections

The eye's loss of close-focus ability, normal with increasing age, is presbyopia. Reading glasses compensate for it. Modern reading glasses are commonly available in a range of 1.25 to 2.75 diopter. Not quite as common, but not rare either, are 1 through 4 diopter glasses. 4.5 and 5.0 dipoter glasses are available, but are intended to compensate for macular degeneration rather than normal presbyopia. Diopter is the reciprocal of the focal length in meters. So, for example, a 1.0 diopter lens has a focal length of 1 meter, or 1000 mm; 4.0 diopter, 250 mm; 2.5 diopter, 400 mm. Note that stronger lenses have higher diopter numbers but shorter focal lengths. A 1.0 diopter reading glass isn't of much use. Anyone whose eyesight can be corrected by such a weak lens really doesn't need it, and is unlikely to bother with spectacles at all. Making the safe assumption that the human eye hasn't evolved much in the last four centuries, and the more tenuous assumption that spectacle makers of the early 17th century would only bother to make lenses which were likely to sell, we can guess that the weakest positive spectacle lens commercially available in Galileo's day would have been perhaps a 2.0 diopter lens, with a focal length of 500 mm.

The other extreme, myopia, is (in simple cases) corrected with negative lenses. "Moderate" myopia can be corrected with lenses in the -3.0 to -6.0 diopter range; "severe" myopia requires lenses up to -9.0 diopter. Approximately 90% of modern sufferers need corrections of less than -6.0 diopters. If we make the wild guess that -6.0 diopters was, for economic reasons, the practical limit for early 17th century commerical lens manufacture, we find -167 mm to be the shortest focal length readily available.

Eyeglass Lenses Used as Telescope Lenses

For a telescope operating in its "far field" (always the case for astronomical telescopes), the magnification of the telescope is the focal length of the objective divided by the focal length of the eyepiece – a simple relationship which Galileo must have found after minimal experimentation. A 2.0 diopter objective and a -6.0 diopter eyepiece would make a telescope of almost exactly 3x. From contemporary descriptions, this seems to have been the approximate magnifying power of the Flemish or Dutch telescopes which were appearing in Europe about 1610, and the power Galileo ascribes to his first telescope. I postulate that 3x was the maximum power possible with off-the-shelf eyeglass lenses circa 1610.

To reach higher powers, Galileo must have realized that he needed objectives of longer focal lengths, eyepices of shorter focal lengths, or both. Either would do, though both present practical difficulties. A longer objective focal length results in a longer and clumsier tube assembly, more difficult to mount and keep steady. A shorter eyepiece focal length requires a more deeply curved lens, more difficult and more time-consuming to grind. Examination of the lens focal length data for the surviving specimens shows that Gallileo pursued both approaches. The focal lengths of the three surviving objectives – 1330, 980, and 1710 – or 0.75, 1.02, and .58 diopter – would render them uselessly weak as spectacle lenses. The eyepiece focal lengths – 94, 47.5, and the hypothetical 57 mm – or 10.6, 21, and 17.5 diopter – must have been difficult to grind, and are far too strong to compensate correctly for the usual range of myopia. (The 47.5 mm eyepiece, the one in Telescope II, is not the original 17th century one, but the replacement must have a focal length fairly close to that of the original lens, or it wouldn't come to a proper focus in the old tube.)

Conclusions

Telescopes Described in Galileo's Writings
my designation	description	use	fate
Telescope 1	~3x	proof of concept	lost
Telescope 2	~8x	donated to Venice	lost
Telescope 3	~20x	observations of three Jovian moons	lost
Telescope 4	~30x	observations of four Jovian moons	object lens preserved

Surviving Telescopes Attributed to Galileo
my designation	description	significance
Telescope I	14x	not one of Galileo's earliest telescopes
Telescope II	21x	not one of Galileo's earliest telescopes
Medicean lens	--	the object lens of Telescope 4

Galileo's January 1 letter describes his then-best telescope as a 20x with an elliptical stop on the objective. This description doesn't fit either Telescope II or the Medecean Lens. This is the basis for my conclusion that Telescope 3 is not Telescope II, and has in fact been lost; and for my conclusion that the Medicean Lens is the object lens from Telescope 4.

Refs

The Sidereal Messenger of Galileo Galilei and a Part of the Preface to Kepler's Dioptrics containing the original account of Galileo's astronomical discoveries. A translation with introduction and notes by Edward Stafford Carlos (London 1880; reprinted, London: Dawsons of Pall Mall, 1960).

Abetti, G., L'Universo, 4, Sept. 1923, No. 9 (685-692)

Ronchi, V., L'Universo, 4, Oct. 1923, No. 10 (791-804)

Vincenzo Greco, Giuseppe Molesini, and Franco Quercioli, "Optical Tests of Galileo's Lenses", Nature, 358.9 (July, 1992):101

Albert van Helden, "The Invention of the Telescope", American Philosophical Society, Transactions 67, part 4 (1977): 5-16, 24, et seq

Stillman Drake, "Galileo's First Telescopic Observations", Journal for the History of Astronomy 7 (1976): 153-68

Edward Rosen, "Galileo on the Distance between the Earth and the Moon", Isis 43 (1952): 344-48

Jean Meeus, "Galileo's First Records of Jupiter's Satellites", Sky & Telescope 24 (1962): 137-39

Baxandall, D., Trans. Opt. Soc., 24, p. 309, 1923

Harold I. Brown, "Galileo on the Telescope and the Eye," Journal for the History of Ideas 46 (1985): 487-501

Vincent Ilardi, "Eyeglasses and Concave Lenses in Fifteenth-Century Florence and Milan: New Documents", Renaissance Quarterly 29 (1976): 341-60

Edward Rosen, "The Invention of Eyeglasses", Journal for the History of Medicine and Allied Sciences 11 (1956): 13-46, 183-218

Jane F. Koretz & George H. Handelman, "How the Human Eye Focuses", Scientific American 259, no. 1 (July 1988), 92-99