Frames /sing

kvond

Tag Archives: Optics

The Text of van Gutschoven’s Letter to Huygens No. 1148

[Posted here is the full original text of Huygens letter 1147, for which there are some comments here A Method of Grinding Small, Spherical Lenses: Spinoza ]

No. 1147.

[G. van Gutschoven] à [Christiaan Huygens].

[1663]

Apendice au No. 1146.

Instrumentum ad vitra minoris sphaerae terenda.

Laminam eream AB schamno tornatorio affixam excavabis secundum circulum CD cuneo parato in formam circuli aequalis circulo vitri formandi. dice aequalis circulo maximae sphaerulae cuius lens particulam referat. in hoc excabato canali CD, atteres vitra capulae affixa et canali appressa : modo capulam in manu continuo dum vitrum attertur vertas, ut ex omni parte vitrum aequaliter atteratur : consultissimum autem erit laminam horizontalier circumagatur, nam hoc modo non tam facile arena decidet, vltimam autem polituran vitro addes simili lamina sed stannea terra tripolitana inserta : vel lamina simili lignea ex lingo aliquot molliori, quale est salicis, vel populi.

Lentes cavas formabis sphaerula stannea vel plombea EF schamno tornatorio affixa ut vides, polies vero eadem sphaerula vel lignea ut superius dictum terra tripolitana inserta.

Hoc concavo cono schamno tornatorio affixo, scabies vitrorum limbos polies, ne dum ultimam in charta inducimus vitro polituram, particulae tenuiores vitri exfilientes et in poros chartae sese infinuantes vitrum deturpent.

A Method of Grinding Small, Spherical Lenses: Spinoza

Van Gutschoven’s Design for Grinding Small Lenses: Letter No. 1147

We have in a letter written to Christiaan Huygens by G. van Gutshoven, descriptions and diagrams of the essential processes for grinding small spherical lenses, as they were likely shared by most contemporaries of the age. The letter is surely a response to a request from Christiaan who may have been in need of smaller lens grinding techniques, either due to his future interest in compound eye-pieces for telescopes, or in regard to the question of the best lenses for microscopes which would later arise in discussions between himself and Johannes Hudde and Spinoza as well. In any case, van Guschoven an Antwerp mathematician, was Huygens’ initial teacher of the complete essentials of lens grinding in the first place, which he gave to him ten years before in a letter dated Feb. 10, 1653. It was by the aid of these instructions, among others,  that Huygens was able to grind one of the most powerful lenses in Europe, and discover the rings of Saturn in 1656.

This letter is dated only as 1663 by the editors of Huygens’s Oeuvres. 1663 was the year that Spinoza had moved to Voorburg, where the Huygens family kept their Hofwijck country estate. That spring Spinoza rented rooms in the home of master painter Daniel Tydeman, but a five minute walk from the Hofwijck. But Christiaanwas not yet there. He was living in Paris with his father who was attempting to curry the diplomatic favor of King Louis XIV, an effort which would result in Constantijn Sr.’s son becoming the secretary to Louis’ Royal Academy of the Sciences, in 1666.

None of this has occurred yet. Christiaan and Spinoza have not yet met (unless they crossed paths momentarily in the summer of 1663, when a traveling Christiaan took leave of Paris to go to London in the off-season). Huygens would not arrive in Holland and develop his relationship to Spinoza until after May of 1664.

What this letter reveals to us though is the basic mechanism and techniques used in the grinding of small lenses. We know that Spinoza made microscopes (and telescopes) at least since the year of 1661, and in his coming debates over techniques and optics with Christiaan he would champion much smaller, more highly curved lenses for microscopes, against Huygens’ designs of lesser magnification. One would think that from van Gutschoven’s descriptions we can receive a sense of the physical practices that preoccupied Spinoza for many of his daylight hours, specializing at times in these smaller lenses.

It should be noted that the Huygens brothers by this time are among the best lens-grinders in Europe, and Christiaan had already worked on several sophistocated semi-automated designs of grinding machines. These instructions must have been experienced as extremely rudimentary to Christiaan (or perhaps, it is from another date).

The letter has three figures, pictured below. The first of these shows a vertical grinding form that is likely of iron or copper. One can see the core movement of a lathe, as foot petal likely drove the strap that turns the shaft, spinning the form concentrically. For larger lenses the form would be hollow, holding the concavity of a curvature that one would want the glass to have. Here though, the small lens is to be ground in the “canal” near its lip:

“Now in this hollowed out canal C D you will grind glasses affixed to a handle and pressed into the canal, with the handle in the hand continuously; while grinding the glass you would turn it until all parts of the glass are equally ground.”

After this equanimity is roughly achieved, attention is turned to the “laminate” or layered strip A B, which turns so the top of it is horizontal to the turner’s bench. By van Gutschoven’sdirection, the laminate is of a soft wood, polar or willow. (Other techniques of the day call for paper.) The roughness of dimples are by hand ground away, and Tripoli, which is a chalky substance made of the remains of microscopic marine life, is added to the laminate to smooth the way.

After this, there is a third process recommended which can either be done in a concave wooden form G H, it too aided by Tripoli, pictured here:

Or, what seems to be a pillow (plombae), affixed to a lathe shaft EG:

There are several things of interest here. The date of the letter makes this description contemporaneous with Spinoza’s own practices, so one might assume a basic correspondence. The grooved canal method strikes one as similar to those a-centric grinding techniques discs used by diamond polishers which Spinoza may have come in contact with either briefly as a merchant of gems, or simply by growing up in a community where gem polishing. The process remained unchanged for several hundred years as late 19th century illustration below shows:

Like the diamond polisher, it is quite possible that Spinoza’s form was oriented horizontally, and not as van Gutschoven suggests, vertically. This was part of a gradual change in lens grinding techniques, much of it initiated by gem polishing influences. The horizontal mould simply made the glass easier to control, and the variable polishes to be administered more cleanly. For this reason, any polishing with Tripoli also occurred on a horizontal, turning wheel. The grinding forms designs that I have seen that the Huygens were using now all had a horizontally oriented lap. 

The second thing to note that in 1667 and 1668, after Christiaan had come to know Spinoza and become familiar with Spinoza’s techniques, he clearly did not still feel comfortable with the limits of van Gutschoven’s design, whenever he had received it, as he in repeated letters urged his brother about the fineness of Spinoza’s small lens polishing. Spinoza’s technique was not that of van Gutschoven. It is my feeling that he had developed, either though his associations in the community he grew up in, those influenced by the practices of gem polishing, means of polishing that were not common to the rest of Europe. Whether these be methods of grit application, the use of diamond dust, particular designs of a simple but effective lathe, one can only surmise. But it seems that Spinoza’s glasses were of a quality and luminosity that made them distinct.

Here is the Latin Text of van Gutschoven’s letter: The Text of van Gutschoven’s Letter to Huygens No. 1148

Conclusion

Aside from this I would want our investigative imagination to extend itself to the physical understanding of these practices, and the conceptual impression they would leave upon a thinking man who engaged in them repeatedly. This has been a theme of my thinking, that if Spinoza had been a potter we may do well to think about his metaphysics and arguments in terms of the potter’s wheel with which he was familiar. The grinding lathe is not so different from the potter’s wheel, and van Gutshoven’s diagrams give us a visual vocabulary for the kinds of effects and exertions that Spinoza produced in perfecting his craft. What in particular these diagrams allow, apart from the general understanding of the grinding lathe, is the picture of a grooved grinding practice, the canal, which varies from the greater method of placing a glass blank within a concave metal form. If indeed Spinoza used this method for his small objectives for microscopes, we can think along with him in the craft of it, and see him bent over the lip of the spinning canal.

As pictured here before, here is an example of a foot petaled lathe from the year 1647, that used by Hevelius. It may give us a dynamic sense of the physical engagement:

Here is a closer look at the Hevelius Lathe: Spinoza’s Grinding Lathe: An Extended Hypothesis

Simple or Compound: Spinoza’s Microscopes

Smaller Objective Lenses Produce Finer Representations

A very suggestive clue to the kinds of microscopes Spinoza may have produced is Christiaan Huygens’ admission to his brother Constantijn in a May 11 1667 letter that Spinoza was right in one regard, that smaller objective lenses do produce finer images. This has been cited by Wim Klever to be a sign of Huygens making a concession to Spinoza in a fairly substantial question of lensed magnification:

After some disagreement he had to confess in the end that Spinoza was right: “It is true that experience confirms what is said by Spinoza, namely that the small objectives in the microscope represent the objects much finer than the large ones” [OC4, 140, May 11, 1668]

Cambridge Companion to Spinoza, Wim Klever, “Spinoza’s Life and Works” (33)

And this is how I have read the citation as well, not having access to the original context. But some questions arise. Does this admission allow us to conclude that Spinoza was specifically making compounded microscopes, the kind that Huygens favored? Or are “objective” lenses to be understood to be lenses both of single and compound microscopes. What makes this interesting is that if we accept the easiest path, and assume that Huygens is talking about compound microscopes, then there may be some evidence that clouds our understanding of what Huygens would mean.

Edward Ruestow tells us that be believes that Christiaan Huygens in his 1654 beginnings already had experience constructing microscopes using the smallest lenses possible. If so, Spinoza’s claim regarding compound microscopes would not be new to him (or his brother). Ruestow puts the Huygens account in the context of the larger question of the powers of small objective lenses:

It was not obvious in the early seventeenth century that the smaller the lens – or more precisely, the smaller the radius of its surface curvature – the greater its power of magnification, but smaller and more sharply curved lenses were soon being ground as microscope objectives, at first apparently because, with their shorter focal lengths, they allowed the instrument to be brought closer to the object being observed. The curvature of a small cherry ascribed by Peirsec to the objective of Drebbel’s microscope was already a considerable departure from a spectacle lens…

Whatever the intial reason for resorting to smaller objective lenses, however, it was not such as to produce a continuing effort to reduce their size still further. (A lens, after all, could come too close to the object for convenience.) In 1654 a youthful Christiaan Huygens, already making his own first microscopes or preparing to do so, appears to have ordered a lens as sharply curved as a local lens maker could grind it, and it may indeed have been a planoconvex objective lens with which he worked that year whose curvature, with a radius of roughly 8mm, was still to that of Drebbel’s (i.e. to the curvature one might ascribe to a small cherry). Fourteen years later, however, Christiaan was inclined to lenses with a focal distance of roughly an inch, and he pointedly rejected small lenses as objectives – primarily it seems, because of their shallow depth of focus…In 1680 members of the Royal Society were admiring a biconvex lens no more than one-twentieth of an inch in diameter, and Christiaan Huygens, now with a very altered outlook, would write that the perfection of the compound microscope (of two lenses) was to be sought in the smallness of its objective lens. He claimed at the end of his life that the magnification such instruments could achieve was limited only by how small those lenses could be made and used [note: On the other hand, he also recognized that there was an absolute limit for the size of any aperture, beyond which the image become confused.] (13)

[Ruestow footnotes that the 1654 microscope described as constructed by Christiaan above, is thought by J. van Zuylen is rather the Drebbel microscope purchased by Christiaan's father, Constantijn Sr.]

The Microscope in the Dutch Republic, Edward G. Ruestow

Not only is Huygens’s turn around described, no doubt fueled by his own famed success with the single lens, simple microscope, just after Spinoza’s death, but also Ruestow suggests that Huygens indeed already knew what Spinoza’s claimed, that smaller objectives indeed do make larger magnifications, his objection being not that the magnification is inferior, but simply that the depth of field makes observation problematic. It is unclear if Ruestow’s reading of the 1654 for is correct, so we cannot say for certain that Huygens had this experience with smaller objectives, but it is interesting that Ruestow cites the same year as his concession to Spinoza, (1668, “fourteen years later” without direct citation), as the year when Huygens makes clear what his objection to smaller objectives is. This raises the question: Is the “confession” in context part of an admission of the obvious between Christiaan and his brother, something of the order, “As Spinoza says objectives represent objects with greater detail, but the depth of field is awful? (Again, because I do not have the text I cannot check this.) 

Or, does Ruestow make a mistake? Is it not letters written 14 years, but only 11 years later, when Huygens in his debate with Johannes Hudde seems to have readily accepted the possibility of greater magnification, but makes his preference in terms of depth of field. As Marian Fournier sums: 

Hudde discussed the merits of these lense with Huygens [OC5, April 5, 10 and 17 1665: 308-9, 318, 330-1], who declined their use. He particularly deplored their very limited lack of depth of field. He found it inconvenient that with such a small lens one could not see the upper and underside of an object, a hair for instance, at the same time. The compound microscope had, because of the much smaller magnification, greater defintion so that the objects were visible in their entirety and therefore the compound instrument was more expedient in Huygens’ view (579) 

“Huygens’ Design of the Simple Microscope”

It is important that Hudde is not only championing smaller objectives, he is attempting to persuade Huygens that the very small bead-lenses of simple microscopes are best. Hudde had this technique of microscopy from as early as 1663, perhaps as early as 1657, and he taught it to Swammerdam. In the context of these letters, apparently written just as Huygens and Spinoza are getting to know each other in Voorburg, Huygens’ 1668 brotherly admission reads either as a distinct point in regards to compound microscopes, or signifies a larger concession in terms of his debate with Hudde. There are some indications that Hudde and Spinoza would have known each other in 1661, as they both figure as highly influential to Leiden Cartesians in Borch’s Diary. And Spinoza was a maker of microscopes, as Hudde was an enthusiast of the instrument even then. It makes good that there would have been some cross-pollination in the thinking of both instrument maker’s techniques in those days, but of this we cannot be sure. 

Against the notion that Spinoza has argued for simple microscope smaller objectives with Huygens is perhaps the compound microscopes achieved by the Italian Divini. Divini, in following Kepler’s Dioptice, realizes a compound microscope whose ever descreasing size of the objective increases its magnification. I believe that there is good evidence that Spinoza was a close reader of Kepler’s (see my interpretation of Spinoza’s optical letters: Deciphering Spinoza’s Optical Letters ). If Spinoza was making compound lenses, and he had argued with Huygens that the smaller the objective the better, it seems that it would have been the kind of microscope described below, following the reasoning of Kepler, which he would have made. 

First, Silvio Bedini sets out the principle of Divini’s construction: 

Divini was an optical instrument-maker who established himself in Rome in about 1646 and eventually achieved note as a maker of lenses and telescopes. In a work on optics published in Bologna in 1660 by Conte Carlo Antonio Manzini, the author describes a microscope which Divini had constructed in 1648, based on Proposition 37 of the Dioptrice of Johann Kepler. This was a compound instrument which utilized a convex lens for both the eye-piece and as the objective was reduced so were the magnification and the perfection of the instrument increased (386).

Then he typifies a class of microscope of which Divini was known to have constructed with this line of analysis:

One form consisted of a combination of four tubes, made of cardboard covered with paper. Each tube was slightly larger than the previous one, and slid over the former. An external collar at the lower end of each tube served as a stop to the next tube. The ocular lens was enclosed in a metal or wooden diaphragm attached to the uppermost end of the largest tube. The object-lens was likewise enclosed in a wooden or metal cell and attached to the bottom of the lowermost or smallest tube. The rims of the external collars were marked with the digits I, II, and III, in either Roman or Arabic digits, which served as keys to the magnification of the various lengths as noted on each of the tubes. The lowermost of the tubes slid within the metal socket ring of the support and served as an adjustment between the object-lens and the object. The instrument was supported on a tripod made of wood or metal. It consisted of a socket-ring to which three flat feet were attached (384).

 And lastly he presents an example of this type, which he calls Type A:

(Pictured left, a 1668 microscope attributed to Divini):The socket-ring and feet are flat and made of tin, and the cardboard body tubes are covered with grey paper, with the digits 1, 2, and 3 inscribed on the collar tubes. The lowermost tube slides with the socket-ring for adjustment of the distance between the object-lens attached to the nose-piece in a metal cell, and the object. The ocular lens is enclosed in a metal holder at the upper end of the body tube. It consists of two plano-convex lenses with the convex surfaces in contact. The original instrument had a magnification of 41 to 143 diameters. The instrument measured 16 1/2 inches in height when fully extended and the diameter of the largest body tube was 1 1/2 inches. A replica of this instrument, accurate in every detail, was made by John Mayall, Jr., of London in 1888 (385-386).

“Seventeenth Century Italian Compound Microscopes” Silvio A. Bedini

 This 16 1/2 inch compound microscope indeed may not have been the type that Huygens’ comment allows us to conclude that Spinoza built, but it does follow a Keplerian reasoning which employed the plano-convex lenses that Spinoza favored in telescopes, one that imposed the imparitive of smaller and smaller objective lenses. It is more my suspicion that Spinoza had in mind simple microscopes, but we cannot rule out the compound scope, or even that he was thinking about both.

Futher, Spinoza’s favor of spherical lenses and his ideal notion that such spheres provide a peripheral focus of rays (found in letters 39 and 49), seems to be in keeping with the extreme refraction in smaller objectives in microscopes, although he attributes this advantage to telescopes. More than in telescopes, the spherical advantage in conglobed, simple lensed microscopes, would seem to make much less of the prominent question of spherical aberration. But in the case of either compounds or simples, the increase curvature, and minuteness of the object lens would fit more closely with Spinoza’s arguments about magnification, and Descartes’ failure to treat it in terms other than the distance of the crossing of rays.

Huygens’s Criticism of Descartes’ La Dioptrique

In order to understand Spinoza’s dissatisfaction with and objection to Descartes’ La Dioptrique  (found in letters 39 and 40 linked below), one has to understand the opinions of those contemporary to Spinoza. Below I post a selection from Fokko Jan Dijksterhuis comprehensive book on Christian Huygens, who is well-noted for having been Spinoza’s neighbor in Voorburg.  

La Dioptrique was written, Descartes said in the opening discourse, for the benefit of craftsmen who would have to grind and apply his elliptic and hyperbolic lenses. Therefore the mathematical content was kept to a minimum. Apparently this implied that Descartes need not elaborate a theory of the dioptrical properties of lenses. Descartes adopted the term Kepler had coined for the mathematical study of lenses. He had not, however, adopted the spirit of Kepler’s study. Dioptrice and La Dioptrique approached the telescope from opposite directions. Kepler had discussed actual telescopes and drudged on properties of lenses that did not fit mathematics so neatly. Descartes prescribed what the telescope should be according to mathematical theory. The telescope, having been invented and thus far cultivated by experience and fortune, could not reach a state of perfection by explaining its difficulties. Huygens was harsh in his judgement of La Dioptrique. In 1693, he wrote:

“Monsieur Descartes did not know what would be the effect of his hyperbolic telescopes, and assumed incomparably more about it than he should have. He did not understand sufficiently the theory of dioptrics, as his poor build-up demonstration of the telescope reveals” (OC10 402-403)

We can say that Descartes, according the Huygens, had failed to develop a theory of the telescope. He had ignored the questions that really mattered according the Huygens: an exact theory of the dioptrical properties of lenses and their configurations. La Dioptrique glanced over the telescope that existed only in ideal world of mathematics (37).

Lenses and Waves: Christiaan Huygens and the Mathematical Science of Optics in the Seventeenth Century, Fokko Jan Dijksterhuis

Here we get a sense of Spinoza’s complaint that Descartes may have avoided certain questions due to mathematical complexity, and that Descartes is not dealing with actual telescopes. It is significant to follow that indeed the analogical form of arguments in the La Dioptrique was designed to be in favor of the craftsmen and artisans who were meant to grind Descartes’ miraculous hyperbolic lens, while in life (and theory), it was the great tension with this human and technical reality that that produced the Cartesian telescope failure.

A line by line interpretation of Spinoza’s Optical letters: Deciphering Spinoza’s Optical Letters

Jerry Rothman’s website dedicated to Kepler’s Dioptrice: here

 

The Hole at the “Center of Vision”

Spinoza as Seer

In a sense, if we are to understand Spinoza’s optical influences we have to come to at least consider what seeing, or more helpfully, perceiving meant for Spinoza, for behind any optical conceptions Spinoza had lies the very act of actively engaging the world. Much as Descartes worked from definitive values of what clear perception was, wrestling with both empirical experiment and mathematical analysis, so too Spinoza held core positions on what clear perception involved, and these factors into the nature of Spinoza’s break with his precursor. It has become my running thought in this research that if we can generalize, Descartes’ model of clear perception involved the hyperbola’s capacity to refract rays come from a single point on an object, to another co-ordinate point on the surface of the back of the eye, and that importantly this point fell upon the central axis of the hyperbola, a mathematical line which expressed, or was the locus of, the human freedom of Will. This point of focus was – at least in the accounts of vision where Descartesis in praise of the hyperbola and the remarkable representational accuracy of the eye – the fulcrum of a naturalized embrace of narrow focus, frontal clarity.

The hyperbolas central point of focus as a model of clarity

The hyperbola's central point of focus as a model of clarity

This seems to be contrasted in Spinoza with an emphasis upon the multiplicity of visual axes that a spherical lens affords (Spinoza’s optical letters are talked about here: Deciphering Spinoza’s Optical Letters ). Spinoza was privileged enough to be familiar with thinkers who argued that spherical aberrationwas over-emphasized as a problem, and he seemed to hold that there was also a problem of “hyperbolic aberration” (my term), that is the inability of such lenses to focus rays cleanly to any points peripheral to the exact center-line of vision. Whatever one is to make of the impact such aberrationwould have had on telescope construction, it is plain that Spinoza’s view of a model of vision was panoramic, that is, anything that had clarity in the center, was clear due to its place within the context of the clarity of all that surrounded. Instead of a vague and confused border of “confused images” which only “serve” the central crispness (Kepler), because Spinoza felt that we looked with the Mind and not the eyes – something that Descartes also argued but withthe burden of theological-theoretical commitments to a free faculty of Will – Spinoza holds that ideal vision embraces the tableau, the scope of things. The hyperbola’s acute focus, as Spinoza understood it, just did not provide the convincing analogical force of what clarity would mean. I think it safe to say, neither thinker, Spinoza or Descartes, had a sure enough idea of what exact effect spherial aberration had on telescopes nor how refractionproduced its images, and it was their different notions of mental claritywhich governed their arguments for ideal lense shapes, filling in the blanks of what was known.

a modified diagram from Spinozas letter 39 designed to bring out the capacity of spherical lenses to focus peripheral rays

a modified diagram from Spinoza's letter 39 designed to bring out the difference between spherical and hyperbolic lens focus as it pertains to peripheral rays

Within this overview of differences, it is worthwhile to consider my guiding assumption of this research: that Spinoza’s experiences as a lens-grinder and instrument maker (not to mention his social standing having come from an artisan class) decisively gave him a craftsman’s appreciation of perception, one that reflects itself in his metaphysis. To get a firmer grasp on what a “craftsman’s appreciation of perception” is, I turned to Richard Sennett’s book on the subject, The Craftsman. There he writes adroitly on the nature of craftsman perceptions, thinking processes and environments, in particular the relationship to tools and on-site difficulties. This has been of great value. In his sum of craftsman perceptions he turns to “cognitive dissonance’ theory to help explain how the craft perception functions. This strikes me to be of use in pointing out just where Spinoza and Descartes seem to optically diverge. Below he discusses the nature of “focal attention” (he mentions two examples he has discussed previously, the house the philosopher Wittgenstein designed and had built, in which he infamously had the ceiling height of a room changed 3 cm, just as the worksite was being cleaned up; and Gehry’s explorations into the processes of forming titanium, designed for the rippling skin of his Bilbao project).

The capacity to localize names the power to specify where something important is happening…Localizing can result from sensory stimulation, when in a dissection a scalpel unexpected hard matter; at this moment the anatomist’s hand movements become slower and smaller. Localization can also occur when the sensory stimulation is of something missing, absent or ambiguous. An abscess in the body, sending the physical signal of a loss of tension, will localize the hand movement…

In cognitive studies, localizing is sometimes called “focal attention.” Gregory Bateson and Leon Festinger suppose that human beings focus on the difficulties and contradictions they call “cognitive dissonances.” Wittgenstein’s obsession with the precise height of a celing in one room of his house [citing that the philosopher had a ceiling lowered 3 centimeters in a house he had designed just the construction was being completed] derives from what he perceivedj as a cognitive dissonance in his rules of proportion. Localization can also occur when something works successfully. Once Frank Gehry could make titanium quilting work [citing the design of a material of specific relfective and textural capacities], he became more focused on the possibilities of the material. These complicated experiences of cognitive dissonance trace directly, as Festinger has argued, from animal behavior; the behavior consists in an animal’s capacity to attend to “here” or “this.” Parallel processing in the brain activates different neural circuits to establish the attention. In human beings, particularly in people practicing a craft, this animal thinking locates specifically where a material, a practice, or a problem matters.

The capacity to question is no less or more a matter of investigating the locale. Neurologists who follow the cognitive dissonance model believe the brain does something like image in sequence the fact that all the doors in a mental room are locked. There is then no longer doubt, but curiosity remains, the brain asking if different keys have locked them and, if so, why.  Questioning can also occur through operational success…This is explained neurologically as a matter of a new circuit connection being activated between the brains different regions. The newly active pathway makes possible further parallel processing – not instantly, not all at once. “Questioning” means, physiologically, dwelling in an incipient state; the pondering brain is considering its circuit options (278-279).

Gregory Bateson

Gregory Bateson

It is not my intention to claim that Spinoza holds a proto-cognitive dissonance theory, though there are some signficant and suggetive correspondences (a social dimension to agreement, determintative conditioning and holistic forces in judgment). Rather, I would like to put it the other way around, to use Sennett’s point about the nature of focal attention to shed some light upon Spinoza’s difficulty in accepting Descartes notion of an ideal crystal clear center of vision. If we simplify, we could say that Descartes was concerned with identifying and constructing means of “clear and distinct” perceptions or thoughts which would define idealvision (mental and otherwise). His engagement with the hyperbolic lens is at least analogically connected to his engagementwithhyperbolic doubt, each designed to focus the mind on a central clarity. What Sennett’s appeal to the craftsman experience of Cognitive Dissonance does is help expose a rift in the very center of focus which Descartes hopes to at least rhetorically stabilize. Focal Attention may be best understood as an irreconcilable line of fragmenting possibility and dys-clarity, and not the consummate moment-after experience of veritability. Modelsof the mind which have most thoroughly drawn upon the visual metaphor for truth mostly have taken the clarity of a perception as the exemplar of correspondence. I see two dogs, and I know that they are twodogs and this clarity is established against a figure-ground constrast. But a Cognitive Dissonance approach seems to suggest is a much finer grain look at what perception is. That is, when our focal attention is turned from this thing to that thing, this aspect to that aspect, it is not clarity which guides our view, but dys-clarity, a fuzziness of the possible and the incomplete. The eye may apprehend the distinction between a figure and the ground but it does not stop there; it continues to trace the significance of relation of elements that both compose the figure, and distinguish it from its context. The processes that give birth to a single distinction carry on in a relational, distinguishing manner. This destabilization of the center, and its resolution through its coherence with a whole, I believe is expressed in several ways in Spinoza thinking.

The depth of field analogy for knowing and Being

The depth of field analogy for knowing and Being

But first a short defintion of the concept

Cognitive Dissonance: “the uncomfortable tension that comes from holding two conflicting thoughts at the same time, or from engaging in behavior that conflicts with one’s beliefs. More precisely, it is the perception of incompatibility between two cognitions.”

Spinoza and the Trace of Consciousness: the grain of wood

Sennett is concerned with the experiences and perceptions which guide the craftsman through his work, the careful notice of differences in materials, possibilities and designs, how a hand passes over wood grain or the mind might connect one part to another part. It is the waythat the mind glides over difficulties and solutions. Taken in its visual state, it is the way that the eye focuses upon this or that, leading itself across the bed of differences. And it is my intuition that Spinoza’s lived practices with craft that gave him a distinct sense of what it means to perceive and distinguish.

What is necessary is to establish just what it is that lies at the center of focus, if it not a crystalline clarity. And there are two selections of the Ethics which I have in mind in response to the Cognitive Dissonance lead. The first is Spinoza’s maxim concerning what it is that we imagine to be the case. It is important to realize that when Spinoza talks about the imagination, he means a confluence of both sensory experiences, and the beliefs we form about them, so much so the latter cannot be separated out from the former. The ideas we hold – or more properly, the ideational states we are in – determine our imaginary, phenomenological experience of the world.

Spinoza writes in part 3 of the Ethics, proposition 12:

The Mind, as far as it can, strives to imagine those things that increase or aid the Body’s power of acting.

This is the key proposition on which Spinoza will found our imaginary relationship to both to the world, but also to others. Through the mind’s force of positive imagining, it brings coherence to our body’s relationships, and thus improves our ability to act more freely (as our own cause). He continues to explain in the demonstration:

Hence, as long as the mind imagines those things that increase or aid our body’s power of acting, the Body is is affected with modes that increase or aid its power of acting, and consequently the Mind’s power of thinking is increased or aided.

In Spinoza’s view, though our relationship to the world may be imaginary (that is, we may not fully understanding the causes and effects involved), if we imagine a relation which improves our power to act, we will experience Joy (defined as an increase in this power, DOA 2), and thus the Mind will tend to continue to imagine in this fashion. Any imaginary improvement, if it results in Joy, is also an improvment in the power of thinking, and thus there is an imaginary, though non-optimal, path to greater power and freedom.

Hopefully the rough connection to Cognitive Dissonance theory will be seen. There is a tendency in perception and belief which determines the mind to think in a more coherent fashion. When there is dissonance – that is, a disjunction between one’sown ideationaland physical states and the states of the world – the imaginary value is to resolve this. In a sense, the imaginationis guided by the resolution of a center of dissonance, bringing the body into concert with its own powers as far as it understands them. (I leave aside the ladder of rational, causal understanding.) 

For Spinoza there is a cohering balast that centers the processes of imaginary experiences of the world. This is reflected in the most characteristic experiences of consciousness, the passing from one thought to other, as if in a chain. When Spinoza presents his General Definition of the Affects, he radically asserts that our chain of thoughts, most generally, are the result of the Mind affirming one state of the Body or another, such that each affirmation leads either to an increase or decrease of the power to act. These changes are the result of affects which express the adequacy of the ideas which compose our mind:

E3: General Defintion of the Affects: An affect, that is called a Passion of the mind is a confused idea, whereby the Mind affirms concerning its Body, or any part thereof, a force for existence (existendi vis) greater or less than before, and by the presence of which the mind is determined to think of this rather than that.

Exp: I say, first, that an Affect or passion of the mind is a confused idea. For we have shown that the mind is only passive, in so far as it has inadequate or confused ideas. (E3P3)
I say, further, whereby the mind affirms concerning its body or any part thereof a force for existence greater [or less] than before. For all the ideas of bodies, which we possess, denote rather the actual disposition of our own body (E2P16C2) than the nature of an external body. But the idea which constitutes the reality of a passion must denote or express the disposition of the body, or of some part thereof, which is possessed by the body, or some part thereof, because its power of action or force for existence is increased or diminished, helped or hindered.
But it must be noted that, when I say a greater or less force for existence than before, I do not mean that the mind compares the present with the past disposition of the body, but that the idea which constitutes the reality of an affect affirms something of the body, which, in fact, involves more or less of reality than before.

There are two significant aspects of this definition I would like to point out. The first is the ateleological view Spinoza takes toward these kind of ideational affirmations of the body. The mind does not arrive at its present affirmation state through a comparison of a present state with a past one, but rather makes of its present existence a repeated and continual “concrescence” (to borrow wrecklessly from Whitehead’s wordsmithing). Any perceptual grasp of the world, insofar as it involves a shift in degrees power and Joy, can be seen as coming from a comprehensive grasp in a sphere of understanding. To put it another way, if we adopt the cognitive dissonance model of perception and belief holding, the running line of potentiated dissonance which guides and centers our focal points of attention becomes repeatedly resolved in the affirmative embrace of a perception/thought/state of the body, made in context with the whole. Clarityarrives not due to the crispness of an axis of perception, but due to the resolution of that line within the panorama either of the visual tableau, or the ideas we hold. Seeing something clearly, thus, is fundamentally a connective and comprehensive apperception.

The ultimate perception is, as Spinoza argues, the perceptual Idea of God, one whose scope and speed of embrace brings clarity to all other affects and imaginations. From the 5th part of the Ethics:

P13: The more an image is joined with other images, the more often it flourishes.

P14 The Mind can bring it about that all the Body’s affection, or images of things, are related to the idea of God.

The second aspect of the General Defintion of the Affects I want to point out is that the chain of thoughts which make our everyday consciousness are not centered upon a Will which controls them, but rather are an expression of the ideas that make up our MInd; thus our ideational states determine the line of imaginary and cognitive processes which include our visual perceptions (clarity of perception cannot be the model of knowing), yet only insofar as these are understood as affirming our physical states. There is no center of vision nor of judgment. Rather there is the confluxof repeated changes in the power to act, something that reveals itself not in binary of Being and Non-Being, but along a gradated spectrum of Being, wherein the power one has is a function of the degree of Being one has.

All this proceeds too fast, for I have not properly connected Spinoza ateleological, affirmational understanding of perceptions and thought-chains to the kinds of curiosity and tensions that arise even the the smallest of conscious distinctions. What a Cognitive Dissonance model of perception and belief provides, I have suggested, is the idea that there is a fissure at the center of the eye’s focus, and that this rift is only closed through the coherent orientation to our experiences at the edge of that rift, in relation to all that lies at the margins. Any philosophical view that in a binary strictly equates focal clarity with Being, and all else with Nothingness or Non-Being, does not fully appreciate the recommendations that a metaphor of visual experience would provide; for at the very center of the eye, if we follow Spinoza’s thinking, lies not the undoubtable truthof one proposition, or the pure assurance of an object seen, but rather the living line of the electric destablized possibility for greater Joy or freedom. Perceptions are a body’s forward lean. In Spinoza’s terms, this line is the shore-point of our realized power to act, and thus occurs along the affects we experience, as they are expressed in both the ideas that make up our mind, and the states our body is in. The very center of focus is our fluxuations in perfection and Joy.

Descartes not Representational Despite His Love of Lens

Now at this point really I would like to take the opportunity to make clear that I have for the sake of contrast been unfair to Descartes, for by and large when he seeks clear thoughts he does not have in mind a clarity which operates independently of other understandings. He, like Spinoza, sees a global and connective sense in truth, one which puts any clear perceptions of the world in the context of the natural dispositions of the Intellect and our soul’s relationship to God. His use of skepticism and doubt is likely at most pedagogical. There has been too much groping at what has become a cadaver of Descartes’ notions of Ideas, without notice of the living relationship such concepts hold in his overall natural science and theological scheme. Nadler, Yolton and Behan (his new piece “Descartes’s Semiotic Realism” forthcoming), all have worked to show that most of our modern conceptions of Descartes’ Representationalism are ill-considered, forwarded by a chain of deformations: first Malebranche, then Reid, and lastly to great effect, Rorty. Much of what we rail against as invidiously “Cartesian” is not really something Descartes would champion. I think the arguments of Nadler et al are very well taken, and expose a tendency of philosophy, for all its sophistication, to organize itself around oppositions simple to grasp. And thus it does us some good to look closer at the forefather of the great Substance divorce between the Mind and the Body.

This is a strange thing to say, considering that much of my contrast between Spinoza’s view of perception and Descartes’ view seems to rely upon representational models of what is known. Spinoza objects to the representational notion of clarity, what he calls “falling into pictures” because he feels that representation simply is inadequate to express what happens when we hold ideas about the world. As I have presented it, Descartes seems too seduced by the visual metaphor of a center of vision becoming clear, a ring of focus, which then can be traced down an ancient heritage of an Ocular philosophy of Presence, where the revealing aletheia of Being stands out from the confusions and negations of Non-Being, playing out the 1s and 2s of dialectical Greek counting. But I would put forward that Descartes is only drawn in this direction against, or at least in tension to, a more comprehensive understanding of perception, one in which the Mind “sees” in a very unrepresentational way, with the “mind’s eye” (a phrase that likely Spinoza takes from Descartes). It is my sense that only Descartes theological commitments to the soul and its freedom of choice expressed through the judgments of the Will which force Descartes away from what he would otherwise be more comfortable with, into an account of vision which emphasize visual clarity along a central axis of focus. It is the need for a localizable edge of judgment, most amenable to an analogy of the otherwise blurred field of view, overdetermined by an essential binary of clear and unclear, which pulls Descartes back into pictures. We see this in the development of his Dioptrics away from the non-representationalistexplanations he begins with.

Descartes’ Blindman

The greatest example of Descartes non-representational concept of mental “seeing” is his analogy of a blindman who sees the world through the use of two sticks, literally feeling the world into accurate appraisal. But first, like Spinoza, Descartes warns us not to fall into pictures. Here he points up the semiotic stimulations of our thought. It is not on the basis of resemblance that we come to know or sense things:

…it is necessary to beware of assuming that in order to sense, the mind needs to perceive certain images transmitted by the objects to the brain, as our philosophers commonly suppose; or, at least, the nature of these images must be conceived quite otherwise than as they do. For, inasmuch as [the philosophers] to not consider anything about these images except that they must resemble the objects they represent, it is impossible for them to show us how they can be formed by these objects, received by the external sense organs, and transmitted by the nerves of the brain…instead we should consider that there are many other things besides pictures which can stimulate our thought, such as for instance, signs and words, which do not in any way resemble the things which they signify (forth discourse, trans. Olscamp)

And then here Descartes draws on the very physical modes of sensing, or seeing through a stick:

It sometimes doubtless happened to you, while walking in the night without a light through places which are a little difficult, that it became necessary to use a stick in order to guide yourself; and you have then been able to notice that you felt, through the medium of the stick, the diverse objects around you, and that you were even able to tell whether they were trees, or stones, or sand…(first discourse)

…just as when the blind man of whom we have spoken above touches some object with his cane, it is certain that the objects do not transmit anything to him except that, by making his cane move in different ways according to their different inherent qualities, they likewise and in some way move the nerves of his hand, and then the places in his brain where the nerves originate. Thus his mind is caused to perceive as many different qualities in these bodies, as there are varieties in the movements that they cause in his brain…(fourth discourse)

Descartes figure 18, Dioptrics

It is Descartes conception of light that the tendency of rays communicate themselves without movement, instanteously across space, just as a blindman’s stick seems to. When rays connect to our eyes, Descartes understands our sensing to be that of connective stimulation. When we see objects, we are seeing like a blindman, with sensations directly transmitted to our nerves. He compares a blindman holding two sticks to the baton centers of vision of each of our eyes, emphasizing that the image itself is not what is directly communicated to the Intellect through the nerves.

 So you must not be surprised that objects can be in their true position, even though the picture they imprint upon the eye is inverted; for this is just like our blind man being able to sense the object B, which is two his right, by means of his left hand, and the object D, which is to his left by means of his right hand at one and the same time. And just as the blind man does not judge that a body is double, although he touches it with two hands, so likewise when both our eyes are disposed in this manner which is required in order to carry our attention toward one and the same location, they need only cause us to see a single object there, even though a picture is formed in each of our eyes (sixth discourse).

The eyes using the two batons of central rays of light

[These citations discussed some here: Descartes and The Blind Man’s Cane ]

It would seem that there is within Descartes thought a primary distinction, as Yolton and Behan argue, between signifying and representing; the stimulations of the senses communicate themselves directly through the nerves in a signifying process not based on essential resemblance. The problem is that such a signifying mode of interpretation does not favorably present itself to the requirements of an Individuated and free action of the Will. Where, and before what would the signification process end…the pineal gland? This puts Descartes in tension with himself, as the analogy of visual clarity, embodied by the pursuit of hyperbolic focus in lenses, pulls him back toward representationalist notions. I don’t at all believe that Descartes holds such a representationalist idea of knowledge, but rather suspect that it is only the independence of the freedom of the will which again and again forces its intrusion, under an auspices of directed and establishing clarity. The resting place of hyperbolic doubt, the cogito, assures a clear focus relation on which all relations can be reconstituted, owing to God. 

Conclusion: Spinoza and Craft

What makes this most interesting is that because Spinoza objects to Descartes at the most radical level of the Will itself, denying the rationality of such a theological vestige (Ethics, 2p48s records the critique of boththe will and representation), he remains unencumbered by the need to take from vision a strict Being/Non-Being binary of optical focus and blurring, center and margin. Instead he draws on, if we can be bold enough to assume it, another luminous analogy, that of Plotinus’s Neo-Platonism, the notion that light radiates in a sphere (put forth by Kepler), and that it expresses itself in gradations of ever-weakening power and cohesion, understood as degrees of Non-Being and power. Spinoza positions himself in the Augustinian, Plotinus line of thought which makes of evil a privation, but he does so at the epistemological, yet by virtue of his parallelism, still bodily level, where the degree of the adequacy of our ideas result in real, affectual experiences of the fluctuations of our power and perfection. Instead of a center of vision which affirms a crisp focus of assured clarity, Spinoza’s center of vision is the breaking wave of the affirmation of our own body’s power, its capacity to act, understood within the context of the full scope of tableau of what is “seen”. As our eyes, fingers, ears, mind flits from thing to thing, we are constantly in states of imagined increases of pleasure and power, owed to the coherence of causes and effects. While central clarity may help incise distinctions of importance, these distinctions only grow meaningful and distinct in the full context of the margins.

It is my sense that Spinoza gained something of this metaphysical insight, in addition to the great variety of sources we might name, from his experiences as a craftsman. His patient polishing of propositions not only reflect in form his careful polishing of lenses, but the content of his thought I believe express the sensitive, non-representational experiences of judgment that come from working with materials, designs and tools in a comprehensive fashion. Spinoza’s refusal to admit Descartes Substance divorce of mind and body perhaps came from his bodily experiences of shaping and sensing glass under tension. While Descartes spent much of his time in mathematics and theory, informing and confirming his hypotheses at times through experiments, he lacked hands-on knowledge of what mechanical construction and application required. In a sense, his vision was mechanical, but his hands were not. One cannot help but realize that Spinoza’s cybernetic turnings of the grinding lathe (either with his off-hand or by foot pedal), communicated a complex of sensations and judgments far too subtle and rapid to place the crown of knowing upon a independent and freely functioning Will. Instead, as the lathe was tensioned in a flux of speeds and grits, and his eyes caught the traces of changes, as his hand holding the torquing glass blank felt the moment to moment consequences of his lathe’s turning – in one great curcuit – he necessarily understood the shore of perceptions within a comprehensive and assembled bodily whole of communications. The coherence that a craftsman brings betweeen his own hands, the limits and possibility of tools, the variations and states of material, amid a continuous, creative line of “dissonance”, a hole in the center of the percieved, non-absolute differentiations of grades, their deviations in form, doubtlessly expressed itself in Spinoza’s own embrace of the union between body and mind, and the careful consideration of the moment to moment changes in the body’s capacity to act.

Spinoza’s Lens-Grinding Equipment

 

Door to the Hofwijck Estate where Spinoza likely strode

Door to the Hofwijck Estate where Spinoza likely strode

 

Spinoza Purchases Lens Grinding Laps

With these I may have ended, in truth, but because for me new dishes for glasses being polished may be fashioned, such is the spirit, your council in this matter I would be eager to hear. I do not see what we may profit in ‘turning’ convex-concave glasses…

…A further reason why convex-concave glasses are less satisfactory, apart from the fact that they require twice the labour and expense, is that the rays, being not all directed to one and the same point, never fall perpendicular on the concave surface. – Letter 36, June 1666

This is what Spinoza writes to mathematician and microscopist Johannes Hudde, part of a correspondence that had begun before the end of the previous year, a correspondence which may have had its impetus in another lettered exchange: the on-going discussions on probability and actuary models between Hudde and Christiaan Huygens of that same year. Spinoza was getting to know his neighbor Huygens, and ends up writing to Hudde, someone he may have known since Rijnsburg and Leiden in the early 60’s.

The value of this letter for those investigating any potential lens-grinding practices Spinoza may have had is that here Spinoza cues that he had his metal shapes or laps made for him, at least at this time in the summer of 1666. The context of these remarks is Spinoza’s argument for the superiority of convex-plano lenses, using the same mathematical analysis of refraction that Hudde uses in his brief “Specilla circularia” (1655). Huygens, the previous summer, had personally calculated to a new degree of precision the phenomena of spherical aberration using convex-plano lenses (something Spinoza may have been privy to), and as Huygens has just left for Paris in April, Spinoza asks Hudde for both practical and theoretical optical advice.

I only wish to present to this context information about the kinds of workmen the Huygens brothers used over the years for their own telescopic lens-grinding projects:

View from the Huygens Estate, the Hofwijck

View from the Huygens Estate, the Hofwijck

 

Christiaan Huygens’ Marbler and Chimney-Sweep

There is no doubt that at first their work consisted solely in grinding and polishing the glass. Even the metal shapes, on which the lenses were ground, were obtained from the outside. Their first ones were of iron (in 1656 Caspar Kalthof supplied one of these, [OC1, 380-81]), later they used copper shapes but for many years they did not make any themselves. In 1662 Christiaan still stated quite emphatically that he had never bothered with making shapes, although he did correct and finish them (OC4 53), and in 1666 in Paris he had a copper shape supplied to him (OC6, 87). But by 1668 we hear that Constantijn makes small shapes (for eyepieces) himself, on a lathe (OC6, 209), and it would appear that later he learned to make larger ones as well, for the instructions for making Telescope-lens (written in 1685 by the two brothers together) contains  detailed instructions about this part of the work (OC21, 251).

“Christiaan Huygens and his instrument makers” (1979), J. H. Leopold

And,

It is not entirely clear if the brothers made their own eyepieces around 1660, but they did not do so later on. Occasionally their correspondence contains references to local craftsman who prepared glass or ground eyepieces; the brothers focused on the delicate work of grinding object lenses.

In 1667 and 1668, Constantijn employed Cornelis Langedelf for polishing glass and grinding eyepieces, and in 1683 this same man delivered the tubes for one of Constantijn’s telescopes. From 1682 the brothers preferred the services of Dirk van der Hoeven, who lived nearby in The Hague. At the same time the brothers also did business with a marbler van der Burgh, who supplied them with grinding laps and glass. The relationships the brothers had with these two craftsmen were not identical. In the case of van der Hoeven – he was often simply referred to as Dirk or the chimney-sweeper – it was only his labour was hired. The brothers supplied the materials and tools, including the grinding laps. Van der Burgh, on the other hand, had a workshop of his own, and the Huygens[es] were were not his only clients. Moreover, one might expect this marbler to have been a more skillful grinder than his chimney-sweeping fellow citizen. So, it was probably was not the routine preparatory work that Constantijn asked of van der Burgh in April 1686, when he sent him two pieces of glass to be flattened 15.

“The Lens Production by Christiaan and Constantijn Huygens” (1999), Anne C. van Helden and Rob H. van Gent

What seems evident from both the Helden and Gent account, and the interpretation of Leopold is that it was not uncommon at all to hire-out for the production of the metal laps in which lenses would be ground. It seems clear from Spinoza’s letter 32 to Oldenburg in November of 1665 that the Huygenses were at least in the possession of a lathe that not only could grind lenses, but also laps or pans, for it is regarding this very (semi-automated) turning of the pans that Spinoza had his greatest doubts:

The said Huygens has been a totally occupied man, and so he is, with polishing glass dioptrics; to that end a workshop he has outfitted, and in it he is able to “turn” pans – as is said, it’s certainly polished – what tho’ thusly he will have accomplished I don’t know, nor, to admit a truth, strongly do I desire to know. For me, as is said, experience has taught that with spherical pans, being polished by a free hand is safer and better than any machine. [See: Spinoza’s Comments on Huygens’s Progress .]

Whether anything good came of this Huygens lathe we cannot know. What is significant though in this combination of evidence, is that Spinoza seems to have made use of someone like the marbler Dirk van der Hoeven, at The Hague, just as the Huygens did, but also that Spinoza maintained a priority using the free had to either polish these purchased laps, or to polish lenses in them. That a chimney-sweep and a marbler would both be hired by someone as wealthy as the Huygens family, suggests a rather wide-spread and eclectic economic foundation for the procurement of these services and other related grinding services, something that did not require a specialist.

It is interesting to place Spinoza somewhere between the handyman Chimney-sweep and the savant Christiaan Huygens. Perhaps, if we take a more refined glance back through history, he seems to be between holding the straight-forward lathe experience of marbler-turner van der Hoeven and the specialized knowledge of Christiaan’s brother Constantijn, who spared no expense in carrying out Christiaan’s designs and theories.

As I have mentioned several times on this weblog, the picture is a bit more complicated than that. Christiaan Huygens had to bow to Spinoza’s assertion that the smaller objective lens makes a better microscope, and marveled at the polish that Spinoza was able to achieve in his microscopes, a polish achieved by “by means of the instrument” in a method that Christiaan did not seem to know. The speciality of knowledge did not restrict itself to just microscopes, but to telescope lenses as well. It is reasonable that the Huygenses purchased telescopes, microscopes and lathes from the Spinoza estate upon his death, and there does seem to have been something special about Spinoza’s laps (one’s he likely polished), as Constantijn used one in 1687, ten years after Spinoza’s death:

[I] have ground a glass of 42 feet at one side in the dish of Spinoza’s clear and bright in 1 hour, without once taking it from the dish in order to inspect it, so that I had no scratches on that side ” (Oeuvres completes vol. XI, p. 732, footnote). [cited by Wim Klever]

Spinoza, it would seem, used a man like van der Hoeven, but held at least to some particular degree both theoretical and craft advantages over Christiaan Huygens.

Approaching Huygens

Approaching Huygens

Spinoza as Craftsman: A Closer Look

The Craftsman

An initial synthesis of some of the aspects of my research and a more refined notion of craftsmanship: writing to think through.

This is not the place to write a review of Richard Sennett’s provocative, subtly powerful book The Craftsman. I can only say in that vein, it is a book I highly recommend for those who enjoy looking at recurrent themes in culture and technology, wih an eye to informed thinking about the challenges and opportunities that face us today. The book is chock-full of illuminating analogies and narratives of human progress in problem solving [what he does with chicken recipes is astounding]. Aside from this general appreciation I believe Mr. Sennett’s observations on the significance of the wisdom of the craftsman has great consonance with the same point I am have been attempting to establish: Spinoza’s view of seeing technology as a continuous interface between bodies in assemblage. To put it one way the Craftsman represents the bodily knowledge shore between ideas held and our capacity to act, something I believe is embraced by Spinoza’s parallel grasp of both Mind and Body, thought and extension.

But from this book I would like to concentrate on a few distinctions of analysis Sennett makes to study they way that tools are used, and to assess their epistemic value as a means of intuition and inference. I cite from his chapter “Arousing Tools”.

Mr. Sennett starts with a general appraisal of the place of tools in our process of learning and acting:

Getting better at using tools comes to us, in part, when the tools challenge us, and this challenge often occurs just because the tools are not fit-for-purpose. They may not be good enough or it’s hard to figure out how to use them. The challenge becomes greater when we are obliguedto use these tools to repair or undo mistakes. In bothcreationand repair, the challenge can be met by adapting the form of a tool, or improvising with it as it is, using it in ways it was not meant for. However we come to use it, the very incompleteness of the tool has taught us something (194).

He goes onto make a distinction between a fit-for-purpose tool and an all-purpose tool. The example he uses for each is a screwdriver. A philips-head screwdriver is fit for one purpose, yet a flat-blade driver has a flexibility to it, “it can gouge, lift, and line as well as screw”. It is tempting to apply Spinoza’s definition of power to this distinction, the flat-blade tool actually has a greater number of ways it can “affect or be affected”. This does not make it a better tool at any given instance, for it its capacities must be assessed in assemblage to situations and to among other things, screws, but there does seem to be a fundamental difference to be noted.

The all-purpose tool has something of an openness to it:

…in its sheer variety this all-purpose tool admits all manner of unfathomed possibilities: it, too, can expand our skills if only our imagination rises to the occasion. Without hesitation, the flat-edged screwdriver can be described as sublime – the word sublime standing, as it does in philsophy and the arts, for the potentially strange. In craftwork, that sentiment focuses especially on objects very simple in form that seemingly can do anything.

I’d like to stop here for a moment, for I am attempting to look through Richard Sennett’s lens at the discoveries I have come upon in regard to Spinoza’s technical-optical practices. There is no doubt that Spinoza, at least in life, if not in philosophical theory, embodies the core values of the craftsman. He lived a thrift life upon the intentionally thin thread of the earnings of his lens-grinding and instrument making. He came from the artisan class of Amsterdam Sephardic Jewry, and devoted himself to the perfection of things for their own sake, a mark of craftsmanship. And his hands-on experiences with materials and the lathe gave him a distinctly craftsman approach to problems of light and glass. If we ask, For Spinoza, what tools would strike him as “sublime” in the sense that Sennett defines it? It would seem safe to say that the lathe in its simplicity and multiplicity of uses would be sublime. Nearly every required lens could be ground on his bench provided the right glass, abrasives and forms were available. This spinning concentric action must have been “potentially strange” and a source of many, perhaps unconscious, inferences.

I think though that in my close reading of Spinoza’s two Optical Letters, it would seem that Spinoza also regarded the spherical lens as sublime. In his argument for its superiority its essential multiplicity, as long as one accepted its one inherent drawback of slight spherical aberration, gave it a certain panopticalquality, the way that it addressed rays coming from all points of a field of view. One can say that this understanding came from a failure to properly understand the optics involved, but I suspect that it is rather overestimated how correct the understanding of contemporary optics was. A theorist could hold what we take to be a “true” optical belief, but only due to fact that he also held a great number of beliefs we would regard as false. In a sense, to assess the truth or falsity of a scientific belief, one should perhaps take a more tool-oriented view toward these beliefs, asking how they worked to solve which problems.

Keeping this in mind, I think it safe to say that Descartes’ also found his discovered (imagined) hyperbolic lens “sublime”. For Descartes the simplicity of its form (a hyperbola could be made with two sticks and string in the garden he tells us), the elegance of its line bespoke a remarkable potential for applicability. But there is something to the foundational thinking of Descartes’ hyperbola (shown in Kepler), which also seemed to make it something like a fit-for-purpose tool, a philips-head. The way that Descartes saw it, it matched the human eye, acting like a hand and glove prosthetic extension of the organ out into space. It fit its purpose. Additionally to this, it not only fit the shape and powers of the eye, but it also snugly secured itself within Descartes larger conception of perception and the exercise of the Will. The hyperbola’s narrow aim of focus matched Descartes’ idea of the “clear and distinct” object of perception, crispened in a field of obscured images. This field was a background for the exercise of valuations and free choice. It was this fit-for-purpose aspect of the lens that I believe Spinoza instinctively had some objection to. It could be argued that Spinoza’s embrace of spherical lenses also had a fit-for-purpose quality to it, as indeed it did. The sphere was evocative of the completeness of mental-vision that in the pre-requisite acceptance of inadequate and imaginary ideas, strived for a total awareness of the causal matrices that brought things together. What needs to be pointed out is that Spinoza is not the only holder of such conceptual relations. Rather, they are likely endemic to all descriptive, problem-solving thought. At the time the understanding of refraction and spherical aberration was still quite confused, even in the minds of those that would make the most progress in these areas.

In a certain way, Mr. Sennett wants us to know that the very ambiguity of a tool, the fuzziness of its ability to work, is that which stimulates us to imagine what must be the case that would make the tool work (or require a new tool). He tells a wonderful story about the problems that arose with the invention the surgical scalpel (15th century), born of the introduction of silica into the iron, giving a new and remarkable sharpness. It enabled precise anatomical dissections and discovery, as well as advances in living operations. Remarkably this new knife produced the difficulty of now to cut, what techniques of muscle movements could handle this suddenly fine instrument (less shoulder and a co-operations of figures Sennett tells us). It seems it took a century or more to be able to master these new powers. The tool imposes itself upon the imagination in the context of the resistances of the world.

It is easy to think that Descartes had in mind that his hyperbolic lens would act something like the new scalpel, like his hyperbolic doubt, it would cut through the murky imperfections of the field and clearly select out what is essential. It would carry the eye to distant things, by extending the eye itself. Generally it is assumed that Descartes valuably got it right with the hyperbola, as he identified a solution to spherical aberration (already pointed out by Kepler). And if we are to imagine that he got it wrong, there are likely two ways he was wrong. The first was the widely assumed connection between what turned out to be chromatic aberration (the obscuringcolored ring in telescope observations), and spherical aberration. But, as Graham Burnett has pointed out, he also got wrong his assumption that he could just take a mathematical formula, engineer on paper a machine that could would carry it out, and with a little advice and sweat from a hired craftsman, his paradisaical lens could easily be produced. Again and again he expressed frustration with hired artisans that either they or the materials were failing him.

Four Stages

Sennett gives us a guide to stages of technological development that might help us see Descartes difficulty more clearly, and perhaps gain the value of Spinoza’s craft-sensitive, if metaphysical objections. First he says that there is reformatting. Reformatting is simply the “willingness to see if a tool or practice can be changed in use”. The example offered is the way that Christopher Wren used chiaroscuro techniques of shading in his illustrations of specimens shown in Robert Hooke’s microscope, so to produce plates more vivid than what could actually have been seen in the poor quality glass and light, while still being perceivedas quite accurate. Descartes indeed employed reformatting in both his development of mathematical tools (a forwarding of the Law of Refraction), and his inventive design of an automatic lens-lathe, but lacked any capacity to reformat techniques of placing materials in relation to each other in real machines, assuming that this was the least difficult portion of the process, just getting the matter to do what the mind wished it to.

Secondly, there is adjacency. “Two unlike domains are brought close together; the closer they are, the more stimulating their twined presence”. This seems precisely what Descartes did in terms of his imagination of the eye and his hyperbolic lens. By placing the two hyperbolic forms, organic and glass, together in an assemblage, his inferences to the clarity of thought and vision took wide gallop.

Thirdly, is intuitive leap or surprise. “Although you were preparing for it you didn’t know in advance precisely what you would make of the close comparison. In this third stage you begin to dredging up a tacit knowledge into consciousness to to the comparing – and you are surprised. Surprise is a way of telling yourself that something you know can be other than you assumed” (211). This no doubt lead to the writing of the Dioptrics, and his more than decade-long pursuit of a grinding machine capable of making his very precise lens.

Fourth is Gravity. “The final stage is recognition that a leap does not defy gravity; unresolved problems remain unresolved in the transfer of skills and practices…The recognition that an intuitive leap cannot defy gravity matters more largely because it corrects a frequently held fantasy about technology transfer. This is that importing a procedure will clarify a murky problem; more often, the technical import, like any immigrant, will bring with it its own problems” It is in gravity that Descartes seems to have failed, and is there no better evidence of this than Descartes (feigned?) glee that his craftsman de Beaune had severely cut his hand trying to make the desired lens, proving to Descartes the superiority of his rational vision! Descartes lacked the connection to material practices to adequately transfer his familiar terrain of mathematics and geometry to lathes and glass. His lens was impossible to make.

These four stages help us enframe something of Descartes’ failings and also help suggest to us something of Spinoza’s resistance to the lens and its attendant conceptions. There is one more tidbit that Richard Sennett offers us in this chapter that I want to bring to the issue. He says that in his distinction between a fit-for-purpose tool and an all-purpose tool, as they are designed specifically to either repair something broken or seek to improve it upon repair,

The tool that simply restores is likely to be put mentally in the toolbox of fit-for-purpose-only, whereas the all purpose tool allow us to explore deeper the act of making a repair. The difference matters because it signal two sorts of emotional responses we make to an object that doesn’t work. We can want simply to relieve its frustration and will employ a fit-for-purpose tool to do so. Or we can tolerate the frustration because we are now also curious; the possibilities of making a dynamic repair will stimulate, and the multi-purpose tool will serve as a curiosity’s instrument (200).

It is fair to say that both Spinoza and Descartes, in their philosophies, imagined the possibility of radical improvements in the minds and lives of their readers. Descartes’ radical doubt, and Spinoza’s “clear and distinct” excellerated leap to the Totality of Causes: God, are figured as sublime simplicities. And each thinker has characterized the process as that of the making of tools that could make other tools, like a blacksmith (Spinoza borrowing the image from Descartes). What I would want to ask would be, what is the role of the simplicity of craftsman understanding of the lathe and the lens in Spinoza’s vision of what is clear? For Spinoza I think it likely that he thought of the spherical lens as something like the flat-edged screwdriver, capable of a workman-like enhancement of human bodily powers. This all-purpose property allowed him to remain with this curiosity focused upon a more systematic and comprehensive vision of what made vision clear: the Ideas that we hold. I suspect that it was his hands-on practice of grinding glass to required mathematically describe shapes, and his physical understanding of the fancifulness of Descartes’ hyperbola, that lead him to disparage the enthusiasm for the shape. This did not prevent him from investigating it, but it did keep him in deep suspicion of metaphysical and scientific foundations of its importance.

I think also that this resistance to the hyperbola, and the fundamental acceptance of spherical aberration as a slight and perhaps necessary impairment, draws our attention to Spinoza’s understanding that real change occurs in the world of human imagination and affects, within the realities of our weaknesses. Despite his great concentration upon rationality and abstract reasoning, the model of Euclid’s Geometry, it is within human sociability that the hands touch the matter at hand. The realities of the properties of present conditions are the actual vectors of the power we may have. Any active use of reason does not transcend its earth. As he understood that any change in power and freedom requires a change in the body, and that any ideational move was a physical move, so must he to some degree have gleaned this understanding from the resistances of the cloudy glass and the spinning of the lathe, in the cybernetic feedback of his body’s own actions and experiences. The polishing of a lens was prosthetic, but not prosthetic to the eye, prosthetic and expressive of the body and mind.

I think that that in the gap between the two Substances of Descartes lies Spinoza at his lathe.

A related post writen some time before: The Lathe Mind: What Spinoza Meant by “Individual”

Deciphering Spinoza’s Optical Letters

Line By Line

Below is my reading of Spinoza’s Optical Letters (39 and 40) as best as I have been able to extract interpretations from them. They are letters that are in general ignored, or when brushed over, taken to be evidence for Spinoza’s incompetence in optical matters. It seems that few have thought to examine in detail Spinoza’s point, or the texts he likely had in mind when formulating his opinion and drawing his diagrams. It should be said right from the start that I am at a disadvantage in this, as I have no formal knowledge of optics, either in a contemporary sense, nor in terms of 17th century theory, other than my investigation into Spinoza lens-grinding and its influence upon his metaphysics. In this research, the reading of this letter has proved integral, for it is one of the very few sources of confirmed scientific description offered by Spinoza. That being said, ALL of my facts and inferences need to be checked and double checked, due to my formal lack of familiarity with the subject. It is my hope that the forays in this commentary reading, the citations of likely texts of influence and conceptual conclusions would be the beginning of a much closer look at the matter, very likely resulting in the improvements upon, if not outright disagreement with, what is offered here.

[The Below English selections and links to the Latin text: here ]

Spinoza Answers

“I have looked at and read over what you noted regarding the Dioptica of Descartes.”

Spinoza is responding to a question we do not know, as we have lost Jelles’s letter. We can conclude from several points of correspondence that it is a section of Descartes’Dioptrics that Jelles’ question seems to have focused on, the Seventh Discourse titled “Of The Means of Perfecting Vision”. There, Descartes describes the interactions between light rays, lenses and the eye for purposes of magnification, preparing for the Eighth Discourse where he will present the importance of hyperbolic lenses for telescopes, and also onto the Ninth, “The Description of Telescopes”, where that hyperbola is put to use in a specific proposed construction.

“On the question as to why the images at the back of the eye become larger or smaller, he takes account of no other cause than the crossing of the rays proceeding from the different points of the object, according as they begin to cross one another nearer to or further from to eye…”

This is the beginning of Spinoza’s attack on Descartes’ rendition of how light refracts through lenses to form images of various sizes at the back of the eye. In the Seventh Discourse Descartes claims to have exhausted all factors that can influence the size of the image, which he numbers at three:

As to the size of images, it is to be noted that this depends solely on three things, namely, on the distance between the object and the place where the rays that it sends from its different points towards the back of the eye intersect; next on the distance between this same place and the base of the eye; and finally, on the refraction of these rays (trans. Olscamp).”

His descriptions that follow are varied. Among his either trite or fanciful augmentations he considers moving the object closer to the eye, then the impossibility of lengthening the eye itself, and lastly musing that if the refraction of the crystaline humor would spread rays more outward, then so too should magnification be achieved. This seems to be the extent to which Descartes will treat the factor of refraction in this discourse (hence perhaps Spinoza’s claim of the repression of a very important factor); but what Spinoza has cast his critical eye upon, I believe, is Descartes characterization of the solution to questions of magnification achieved by fundamentally extending of the distance of the intersection of rays:

There remains but one other means for augmenting the size of images, namely, by causing the rays that come from diverse points of the object to intersect as far as possible from the back of the eye; but this is incomparably the most important and the most significant of all. For it is the only means which can be used for inaccessable objects as well as for accessable ones, and its effect has no limitations; thus we can, by making use of it, increase the size of images indefinitely.

It is good to note that in his description of the strategies of telescope magnification, Descartes is operating under an extended analogy, that the telescope can work like a prosthetic lengthening of the human eye, causing the refraction that would regularly occur at the eye’s surface to happen much farther out, as if the retina were being placed at the end of a very long eye. This is his mechanical concept.

Descartes distance-analysis of magnification (and an assertion of the significance of the hyperbola) is then carried forth in the Ninth Discourse, where again Descartes will treat magnification in terms of the proximity to the eye of the crossing of rays, which here he will call the “burning point” of the lens. The descriptions occur both in the context of solutions to far and near sightedness as well as in proposals to the proper construction of telescopes, and generally follow this idea that one is primarily lengthening the eye. 

“…and so he does not consider the size of the angle which the rays make when they cross one another at the surface of the eye. Although this last cause would be principle (sit praecipua ) to be noted in telescopes…”

What Spinoza is pointing out is that when constructing telescopes, as he understands it, the aim is to increase the magnitude of the angle of rays upon the surface of the eye (the cornea), something not solely achievable merely through the adjustment of the distance of the “burning point” or the crossing of the rays of the lens from the eye. Attention to the angle of intersection is for Spinoza a more accurate discriminator probably because it leads to calculations of refraction which include the angle of incidence upon the lens, giving emphasis upon the varying refractive properties of different shapes and thicknesses of lenses in combination, some of which can increase magnification without lengthening the telescope. Descartes conceived of the objective and eyepiece lenses as mimicking the shape and powers of the eye’s lens(es), just further out in space. Though he states at several points that we do not know the exact shape of the human eye, under this homological view, he still sees a correspondence between his proposed hyperbolic-shaped lenses and those of the eye, likely drawing upon Kepler’s observation that the human crystalline humor was of a hyperbolic shape.

The fuller aspects of the factor of refraction – the third factor listed in Descartes three – are left out in such a distance calculation, Spinoza wants us to see. As mentioned, in the combination of lenses, depending upon their shape and powers, the required lengthening of the telescope can be shortened (Spinoza presents just this sort of argument to Hudde in Letter 36, arguing for the efficacy of convex-planolenses). One can also say that this same emphasis on the powers of refraction was also at play in Spinoza’s debate with Huygens over the kinds of objective lenses which were best for microscopes. Huygens finally had to privately admit in a letter written to his brother a year after these two letters, that Spinoza was right, smaller objective lenses with much greater powers of refraction and requiring much shorter tubes indeed made better microscopes (we do not know if Spinoza had in mind the smallest of lenses, the ground drop-lenses that Hudde, Vossius and van Leeuwenhoek used, but he may have). It should be said that Huygens’ admission goes a long way toward qualifying Spinoza’s optical competence, for Spinoza’s claim could not have simply been a blind assertion for Huygens to have taken it seriously. Descartes to his pardon is writing only three decades after the invention of the telescope, and Spinoza three decades after that. Be that as it may, Descartes’ measure is simply too imprecise a measure in Spinoza’s mind, certainly not a factor significant enough to be called “incomparably the most important and the most significant of all”.

Because Jelles’ question seems to have been about the length of telescopes that would be required to achieve magnification of details of the surface of the moon (the source of this discussed below), it is to some degree fitting for Spinoza to draw his attention away from the analysis of the distance of the “burning point”, toward the more pertinent factor of the angle of rays as they occur at the surface of the eye and calculations of refraction, but it is suspected that he wants to express something beyond Jelles’ question, for focal and telescope length indeed remained a dominant pursuit of most refractive telescope improvements. And Spinoza indeed comes to additional conclusions, aside from Descartes imprecision. Spinoza suspects that Descartes is obscuring an important factor of lens refraction by moving the point of analysis away from the angle of rays at the surface of the eye. This factors is, I believe, the question of the capacity to focus rays coming at angles oblique to the central axis of the lens, (that is, come from parts of an object off-center to the central line of gaze). Spinoza feels that Descartes is hiding a weakness in his much treasured hyperbola.

“…nonetheless, he seems deliberately to have passed over it in silence, because, I imagine, he knew of no other means of gathering rays proceeding in parallel from different points onto as many other points, and therefore he could not determine this angle mathematically.”

Descartes, in Spinoza’s view, wants to talk only of the crossing of rays closer to or farther from the surface of the eye, under a conception of physically lengthening the eye, and not the magnitude of the angle they make at the surface of the eye because he lacks the mathematical capacity to deal with calculations of refraction which involved rays coming obliquely to the lens. For simplicity’s sake, Descartes was only precise when dealing with rays coming parallel to the center axis of the lens, and so are cleanly refracted to a central point of focus, and it is this analysis that grants the hyperbola its essential value. In considering this reason Spinoza likely has in mind Descartes’ admission of the difficulty of calculation when describing the best shapes of lenses for clear vision. As well as the admitted problem of complexity, Descartes also addresses the merely approximate capacties of the hyperbola to focus oblique rays.

[regarding the focusing of rays that come off-center from the main axis]…and second, that through their means the rays which come from other points of the object, such as E, E, enter into the eye in approximately the same manner as F, F [E and F representing extreme ends of an object viewed under lenses which adjust for far and near sightedness]. And note that I say here only, “approximately” not “as much as possible.” For aside from the fact that it would be difficult to determine through Geometry, among an infinity of shapes which can be used for the same purpose, those which are exactly the most suitable, this would be utterly useless; for since the eye itself does not cause all the rays coming from diverse points to converge in exactly as many other diverse points, because of this the lenses would doubtless not be the best suited to render the vision quite distinct, and it is impossible in this matter to choose otherwise than approximately, because the precise shape of the eye cannot be known to us. – Seventh Discourse

This is an important passage for several reasons, but first because it comes the closest to the question of the focus of rays para-axial to the center. Again, one must keep in mind that Descartes is thinking about trying to make lenses of a shape that are exact to the shape (or powers) of the eye. Here he is thinking about ever more exotic geometrical shapes which may achieve this, and insists upon the fruitlessness of such a pursuit; it is significant that in contrast to this, Spinoza imagines rather a very simple solution to the question of aberration: the acceptance of spherical aberration and the embrace of the advantage of spherical omni-axial focus. The quoted passage directly precedes Descartes’ summation of the three factors in magnification, with which I began my citations. And I will return to the latter parts of this passage later when we investigate Spinoza’s critique of the hyperbola and the eye. (Note: Aside from this direct reference to Descartes on the issue of calculation, perhaps Spinoza considers also James Gregory, who had some difficulty calculating paraxial rays for his hyperbolae and parabolae in his Optica Promota, though writing an entire treatise devoted to their value.)

Nonetheless, Spinoza suspects that Descartes has shifted the analysis of magnification not simply because it is not amenable to calculation, but more so because, had Descartes engaged the proper investigation, he would have had to face an essential advantage of spherical lense, lessening to some degree his hyperbolic panacea to the problems of the telescope. Again, we will leave aside for the moment Descartes’ justification of this approximation on the basis of the human eye and Nature.

Soft Focus: Spherical Aberration

“Perhaps he was silent so as not to give any preference to the circle above other figures which he introduced; for there is not doubt that in this matter the circle surpasses all other figures that can be discovered.”

Spinoza goes on to expound for Jelles the virtues of the simple circle, as it expresses itself in spherical lenses. One has to keep in mind that since the publishing of Descartes’ Dioptrics  (1637), there had been a near obsessional pursuit of the grinding of hyperbolic lenses, a lens of such necessary precision that no human hand was able to achieve it. The hyperbolic lens promised – falsely, but for reasons no one would understand until Newton’s discovery of the spectrum character of light in 1672 – a solution to the problem of spherical aberration. Spherical aberration is simply the soft focus of parallel rays that occurs when refracted by a spherical lens. Kepler in his Paralipomena provides a diagram which illustrates this property:

As one can see, rays that are incident to the edges of the lens (α, β) cross higher up from the point of focus, which lies upon the axis (ω). It was thought that this deviation was a severe limitation on the powers of magnification. With the clearing away of the bluish, obscuring ring that haloed all telescopic vision, the hope was for new, immensely powerful telescopes. And it was to this mad chase for the hyperbola that Spinoza was opposed, on several levels, one of which was the idea that spherical lense shapes actually had a theoretical advantage over hyperbolics: the capacity to focus rays along an infinity of axis:

diagram letter 39

“[referring to the above] For because a circle is everywhere the same, it has the same properties everywhere. If, for example, circle ABCD should have the property that all rays coming from direction A and parallel to axis AB are refracted at its surface in such a way that they thereafter all meet at point B; and also all rays coming from point C and parallel to axis CD are refracted at its surface so that they all meet together at point D…,”

This is a very important point in the letter, for I believe it has been misread by some. At the same time that Spinoza seems to be asserting something painfully obvious in terms of the geometry of a circle, he, at first blush, in bringing this geometry to real lenses appears to be making a serious blunder. And, as I hope to show later, beneath both of these facts there is a subtle and deeper phenomenal-epistemic philosophical point being made, one that echoes through to the roots of Cartesian, and perhaps even Western, metaphysics. Let me treat the first two in turns, and then the third in parts.

The first point is obvious. As we can see from the diagram Spinoza provides, each of the refractive relationships of rays parallel to one axis are symmetical to the same relationships of other parallel rays to another axis. The trick comes in Spinoza’s second sentence, where he seems to be asserting an optical property of actual spherical lenses. As one email correspondent to me concluded, (paraphrased) “Spinoza thinks that the focal point of such a lens lies on the diameter, and this only occurs in rare cases.” The index of refraction of glass simply is not 2 in most cases. Spinoza seems to be making an enormous optical blunder in leaving the refractive index of the glass out, opening himself to a modern objection that he simply does not know the significance of the all important Law of Refraction, put forth by Descartes. This is a similiar prima facie reading done by Alan Gabbey in his widely read essay “Spinoza’s natural science and methodology”, found in The Cambridge Companion to Spinoza,

One’s immediate suspicions of error is readily confirmed by a straight forward application of Descartes Law of refraction. For the circle to have to the dioptrical property Spinoza claims, the refractive index of the glass would have to be a function of the angle of incidence, a condition of which there is not the slightest hint in the letter…[he is] apparently unaware of the importance the “[other] figures”…that Descartes had constructed (154).

The problem with these readings, among many, is that Spinoza is not at all asserting that there exists such a lens which would have this refractive property (Gabbey’s concerns about Spinoza’s awareness of the Law of Refraction should be answered by looking his familiarity with Johannes Huddes “Specilla circularia”, in letter 36, which will be taken up later). I have corrected a weakness in the prominent English translation of the text which helps to bring out the distinction I am making. If one looks at the sentence closely, Spinoza is presenting an if-then assertion (he uses the subjective in the intitial clause). IF, and only if, a circular lens can be said to have the focusing property along axis AB, THEN it would have the same property along axis CD. To repeat, he is not asserting such a property in real glass and therefore he remits any refractive index reference because it is not germane to his point; he is only at this point emphasizing the property of an infinity of axes of focus, and he is using a hypothetical sphere for several reasons.

The first reason I suspect is that he is trying to draw out the remarkable resonance of spherical forms, making his diagram evocative of notions of completeness and internal consistency. This is of course not an optical concern, but we have to consider it as an influence. We have a similiar looking diagram presented by Spinoza in the Ethics, showing an argued relationship between Substance and the modes that express it. As Spinoza writes:

diagram from the Ethics 2, prop 8, scholia

The nature of a circle is such that if any number of straight lines intersect within it, the rectangles formed by their segments will be equal to one another; thus, infinite equal rectangles are contained in a circle. Yet none of these rectangles can be said to exist, except in so far as the circle exists; nor can the idea of any of these rectangles be said to exist, except in so far as they are comprehended in the idea of the circle.” E2p8s

There is perhaps much speculation to be made as to Spinoza’s feelings about the the interweave of causes that express themselves in modes and the apparitions of focus generated by hypothetical spherical lenses (are modal expressions seen in some way like a confluence of rays?), but at this point I only want to point out Spinoza’s affinity for the sphere, and thus this one possible reason for using a full sphere to illustrate an optical property of spherical lenses. (Remember, this is just an informal letter written to a friend, and not meant as a treatise.)

The second reason is that Spinoza very likely is thinking of a real sphere, that is, the “aqueous globe” that Kepler used to investigate refraction in his Paralipomena, a work in which he was the first to articulate with mathematical precision the dynamics of spherical aberration (before there was a telescope, in 1604), and also was the first to suggest the hyperbola as the resolving figure for such aberration. Here is Kepler’s diagram of his sphere through which he gazed at various distances, illustrating his Proposition 14: “Problem: In an aqueous globe, to determine the places of intersection of any radiations parallel to an axis”.

Keplers diagram from proposition 14

Kepler's diagram from proposition 14

Thus, Spinoza’s use of a sphere in his diagram has at least two readings that have heretofore not been noticed. The first is that his description is operating at solely the hypothetical level, asserting the abstract properties of spherical symmetry, but secondly, he is referencing, or at least has in mind, a primary historical optical text, in all likelihood the text which spurred Descartes’ enthusiasm for the hyperbola in the first place (likely read by Descartes around 1620). It is precisely in this parallel fashion, between the geometrical and the manifest, that Spinoza seems to work his optical understanding.

The third reason that Spinoza is using a full sphere to illustrate his principle of omni-axial refraction is that Descartes’ treatise deals not only with lenses, but also with the human (and ox) eye. And this eye in diagrams is represented as a sphere. I will return to this point a little later, because as he encounters Descartes, he is making an argument, however loosely, against not only his optics, but his essential concepts of clear perception. By taking up a full sphere in his objection, he also poses a relation to Descartes schemas of the eye.

Aside from Descartes’ pseudo-spherical diagram of the eye, we have to consider as an additional influence Hooke’s spherical depiction of the eye with two pencils of rays focused along different axes, used to illustrate the reception of color (pictured below left). The reason why I mention this diagram is not only because it bears some resemblance to Spinoza’s, but also because Hooke’s extraordinary Micrographia might have been the source of Jelles’ question, as I will soon address, and so may have been a text Spinoza thought of in his answer, though we are not sure if he ever read it, or even looked at it, as it was in published in English. Christiaan Huygens owned a copy of it and it was the subject of a conversation between the two. If Spionoza indeed visited the Hofwijck several times, it is hard to believe that he would not have looked closely at this page of diagrams.

figure 5, Robert Hookes Micrographia

 

 

figure 5, Robert Hooke's Micrographia

“…this is something that could be affirmed of no other figure, although the hyperbola and the ellipse have infinite diameters.”

Spinoza here declares the exclusivity of a property that only spheres and their portions possess. It is hard to tell exactly at what level Spinoza is making his objection. Is it entirely at the theoretical level of optics that Spinoza believes hyperbolic lenses to be impaired, such that even if people could manufacture them with ease, they still wouldn’t be desired. If so, he would be guilty of a fairly fundamental blindness to potential advantages in telescope construction that such a lens would grant, rather universally understood. If indeed he was an accomplished builder of telescopes – and we have some evidence that he may have been – this would be a difficult thing to reconcile, forcing us to adopt an estimation of a much more craftsman level understanding of his trade. But it is possible that Spinoza is asserting a combine critique of hyperbolic lenses, one that takes into account the difficulty in making them. There are signs that spherical aberration after Descartes was taken to be a much greater problem than it calculably was, and Spinoza brings out a drawback to hyperbolic focus that adds one more demerit to an already impossible-to-make lens. Thus, as a pragmatic instrument maker he may not be assessing such lenses only in the abstract, but in reality. It may be that Spinoza sees the ideal of the hyperbolic lenses as simply unnecessary, given the serviceability of spheres, and the perceived advantage of oblique focus. This question needs to be answered at the level of optical soundness alone, but such an answer has to take in account the great variety of understandings in Spinoza’s day and age, even among those that supposedly “got it right”. For instance, such an elementary and widely accepted phenomena as “spherical aberration” was neither defined, nor labeled in the same way, by any two thinkers; nor were its empirical effects on lensed vision grasped. We often project our understanding backwards upon those that seem most proximate to our truths. Spinoza’s opinions on aberration seem to reside exactly in that fog of optical understandings that were just beginning to clear.

Man on the Moon

“So the case is as you describe; that is, if no account is taken of anything except the focal lenth of the eye or of the telescope, we should be obliged to manufacture very long telescopes before we could see objects on the moon as distinctly as those on earth.”

Here we possibly get a sense of Jelles’ question. It must have come from a reflection upon Descartes’ comments on crossing of rays at various distances from the eye, posed as a question to whether we might be able to view the Moon with such clarity as we see things here – remember, Descartes’ promised infinite powers of magnification. I mentioned already that Jelles’ question may have come in reference to Hooke’s work. We must first overcome the problem of language of course, for I do know that Jelles read English, though it is possible that he read a personal translation of a passage, as Huygens had translated a passage for Hudde. But given these barriers, I believe there is enough correspondence to make a hypothesis that is not too extravagant: Jelles had recently read a portion of Hooke’s Micrographia. The reason that I suspect this, is that the Micrographia published with extraordinarily vivid plates of magnified insects and materials, concludes with a speculative/visual account of what may be on the moon, seen through his 30-foot telescope (and a suggested 60 ft. telescope), coupled with a close up illustration of a moon’s “Vale” crater, he writes of an earthly lunar realm:

Hookes Vale

…for through these it appears a very spacious Vale, incompassed with a ridge of Hills, not very high in comparison of many other in the Moon, nor yet very steep…and from several appearances of it, seems to be some fruitful place, that is, to have its surface all covered over with some kinds of vegatable substances; for in all portions of the light on it, it seems to give a fainter reflection then the more barren tops of the incompassing Hills, and those a much fainter then divers other cragged, chalky, or rocky Mountains of the Moon. So that I am not unapt to think that the Vale may have Vegetables analogus to our Grass, Shrubs, and Trees; and most of these incompassing Hills may be covered with so thin a vegetable Coat, as we may observe the Hills with us to be, such as the Short Sheep pasture which covers the Hills of Salisbury Plains.

As one can see from this marvelous, evocative passage, the suggestion that the moon’s vales are pastorially covered with rich meadows, calling up even flocks of sheep before the mind, one can easily see that Jelles has something like this in mind when he asks what it would take to see objects on the moon, as we can see objects on the Earth. One might speculate that, having read such a passage, Jelles had a spiritual or theological concern in mind and excitment over the possibility of other people on the moon, but this would be perhaps only wistful supposition on our part. But it is too much to suppose that it was likely Hooke’s description of the moon Jelles was thinking of when he wrote his question to Spinoza, for not only are the details of an Earth-like moon present, but also Hooke’s urging of the reader to use a more power and much longer telescope than he used. Spinoza is responding directly to this aspect of telescope length.

(An alternate thought may be that Jelles had come upon Hevelius’s Selenographia, sive, Lunae descriptio 1647, filled with richly engraved plates of the moon’s surface. It did not have the same fanciful description of moon meadows, and was not circulated with the acclaim of Hooke’s Micrographia, but it did name features of the moon after Earth landmarks, giving it an Alps, a Caucasus and an Island of Sicily.)

If we allow this supposition of a posed question on Jelles’s part, we might be able to construct something of Spinoza’s thinking in his response. It would seem, in our mind’s-eye, that Jelles had read Hooke’s description of the moon and his urge for a longer telescope and set about checking Descartes’ Dioptrics if it were the case that we really would have to build an extraordinarily long telescope to see the details that Hooke invoked (indeed Huygens built a 123 ft. arial telescope; and Hevelius one of 150 ft., pictured below).

Hevelius 150 ft. arial telescope

Hevelius' 150 ft. arial telescope

Following this evolution of the question, it would seem that Jelles came to Descartes’ treatment of magnification in the Seventh (and related) Discourses, one that defined the power of magnification by the all important distance of the crossing of rays from the surface of the eye, treating the telescope as an extended eye. If indeed Jelles was not familiar with optical theory he may have taken this increase of distance for an explanation why telescopes had to be so very long to see the moon with desired detail. It would seem natural for Jelles to pose this question to Spinoza, who not only was regarded as the expert on Descartes in the Collegiant group, but also was a grinder of lenses and a designer of telescopes.

If this hypothetical narrative of the question is correct, Spinoza responded in a slightly misdirected way, taking the opportunity to vent an objection to Descartes thinking which did not have acute bearing upon Jelles’s question. For Descartes’ description of a “burning point” distance and Spinoza’s emphasis on the angle of incidence of rays oblique to the center axis, makes no major difference in the conclusion that Jelles came to, that indeed it would take a very long telescope to do what Jelles imagined, and Spinoza admits as much, above. Yet, when Spinoza qualifies his answer “if no account is taken of anything except the focal lenth of the eye or of the telescope” he is pointing to, one imagines, factors of refraction, for instance in compound telescopes and lenses of different combinations, which do not obviate the contemporary need for very long telescopes, but may affect the length.

Aside from this admission, Spinoza has taken the opportunity to express his displeasure over a perceived Cartesian obscurance, one that has lead to an over-enthused pursuit of an impossible lens, and as we have seen, in this context Spinoza puts forward his own esteem for the spherical lens, and the sphere in general. But this is no triffling matter, for out of Spinoza’s close-cropped critique of Descartes’ Dioptrics run several working metaphors between vision and knowledge, and a history of thinking about the optics of the hyperbola that originates in Kepler (made manifest, I contend, in a full-blown metaphysics in Descartes). Though Spinoza’s objection is small, it touches a fracture in thinking about the Body and Perception, a deep-running crack which might not have direct factual bearing on optical theory, but does have bearing on its founding conceptions. As I have already suggested, we have to keep in mind here that though we are used to thinking of a field of science as a closed set of tested truths oriented to that discipline, at this point in history, just when the (metaphysically) mechanical conception of the world was taking hold, it is not easy, or even advisable, to separate out optical theories from much broader categories of thought, such as metaphysics and the rhetorics of philosophy. For example, how one imagined light to move (was it a firery corpuscula, or like waves in a pond?), refract and focus was in part an expression of one’s overall world picture of how causes and effects related, and of what bodies and motions were composed: and such theories ever involved concepts of perception.

“But as I have said, the chief consideration is the size of the angle made by the rays issuing from different points when they cross one another at the surface of the eye. And this angle also becomes greater or less as the foci of the glasses fitted in the telescope differ to a greater or lesser degree.”

Spinoza reiterates his point that it is the intersecting angles of incidence at the surface of the eye which determined the size of the image seen through a telescope. He finally connects the factor of the angle of incidence and intersection to the foci of lenses themselves. It is tempting to think that Spinoza in his mention of lenses is also thinking of compound forms such as the three-lens eyepiece invented by Rheita in 1645, or as he was already familiar through visits to Christiaan Huygens’s home in 1665, proposed resolutions of spherical aberration by a complex of spherical lenses. Such combinations would be based upon angle of incident calculations.

“If you wish to see the demonstration of this I am ready to send it to you whenever you wish.”

 

Spinoza will send this evidence in his next letter (pictured at bottom).

Letter 40 “…I now proceed to answer your other letter dated 9 March, in which you ask for a further explanation of what I wrote in my previous letter concerning the figure of a circle. This you will easily be able to understand if you will please note that all the rays that are supposed to fall in parallel on the anterior of the glass of the telescope are not really parallel because they all come from one and the same point.”

Jelles has apparently had some difficulty with understanding Spinoza’s explanation. It is interesting because this confusion on Jelles’ part has actually been taken as evidence that Spinoza not only is impaired in his understanding of optics (this may be the case, but Jelles’ confusion, I don’t believe, is worthy of being evidence of it), but that those close to Spinoza around this time became aware that Spinoza’s optical knowledge was superficial at best, something not to be questioned too deeply.

As Michael John Petry writes:

“There is evidence that after 1666 Spinoza’s ideas on theoretical optics were less sought after by his friends and acquaintences…Even JarigJelleswasquiteevidently dissatisfied with the way in which Spinoza explained the apparent anomaly in Descartes’ Dioptrics” (Spinoza’s Algebraic Calculation of the Rainbow & Calculation of Chances, 96)

Petry cites other evidence which needs to be addressed (primarily Huygens’ letters), but a close reading of the nature of Jelles implied question does not seem to support in any way the notion that Spinoza’s optical knowledge had been exposed as a fraud of some sort. Alan Gabbey as well, who maintains serious doubts about Spinoza’s optical proficiency, seems to focus on Spinoza’s need to explain himself to Jelles as a sign that he is somewhat confused:

In his next letter…to Jelles, who has asked for a clarification, Spinoza explained that light rays from a relatively distant object are in fact only approximately parallel, since they arrive as “cones of rays” from different points on the object. Yet he maintained the same property of the cirlce in the case of ray cones, apparently unaware of the importance of the “[other] figures” [the famous "Ovals of Descartes"] (154).

It seems quite clear that Spinoza was aware of the “importance” of these figures, at least he was aware of Hudde’s and Huygens’ attempt to minimize that importance. But Gabbey here seems to suggest that Spinoza is evading a point of confusion by simply changing descriptions, instead of parallel rays of light, Spinoza now uses “cones of rays”. For these reasons of suspicion it is better to go slow here.

The question that Jelles raised apparently has to do with the reading of Spinoza’s circular diagram and its focus of two pencils of light rays, for Spinoza imagines that if Jelles understands these pencils as cones of rays his confusion will be cleared up. To take the simplest tact, it may very well be that Jelles, upon seeing Spinoza’s diagram, turned back to Descartes’ text in order to apply it, and found there a diagram which was quite different. What Jelles may have seen was Descartes’ figure 14 from the Fifth Discourse (pictured below, left), or really any of his diagrams which depict the interaction of rays with the eye:

figure 14 from the Fifth Discourse of the Dioptrics

figure 14 from the Fifth Discourse of the Dioptrics

One can see how in this context Jelles may have been confused by Spinoza’s diagram of the focus of two pencils of rays, and even by the accusation that Descartes is being somehow imprecise, for the illustration seems to depict rays as something like cones of rays, not rays flowing parallel to an axis, as they are in Spinoza’s drawing. Aside from this plain confusion, Jelles’ question may have dealt with some other more detailed aspect, for instance, a question about the importance of a lens’s ability to focus rays oblique to its center. If so, Spinoza would require not only that Jelles understand that rays come in cones, but also have a fuller sense of how those rays refract upon the eye, perhaps provided by the diagram that will follow. In either case, rather than understand Spinoza’s change in descriptive terminology as an attempt to dodge his incomprehension, Spinoza simply appears to be guiding Jelles in the reconsilation of both kinds of diagrams, or preparing ground for a more complete explanation.

Note: Regarding the analytical descriptions of a pencil of parallel of rays or “cones of rays” there is no standing confusion between them. They exhibit two different ways of analyzing the refractive properties of light. But there is more than this, the use of the phrase “cones of rays” by Spinoza gives a clue to what texts he has in mind in his answer. The orgin of this phrase for Spinoza likely comes from Kepler’s Paralipomena  (1604), in a very significant passage. As mentioned, Kepler has already provided a description of the phenomena of spherical aberration (shown in diagrams including the one I first cited here), and forwarded the hyperbola as a figure that would solve this difficulty. Further, he has claimed that the crystalline humor of the human eye has a hyperbolic shape. Here Kepler describes how light, having proceded from each point of an object in a cone of rays (truly radiating in a sphere), intersects the eye’s lens at varying degrees of clarity. The cone that radiates directly along the axis of the lens is the most accurately refracted:

All the lines of the direct cone [a cone whose axis is the same as the axis of the cornea and crystalline] are approximately perpendicular to the crystalline, none of those of the oblique cones are, The direct cone is cut equally by the anterior surface of the crystalline; the oblique cones are are cut very unequally, because where the anterior surface of the crystalline is more inclined [aspherical], it cuts the oblique cone more deeply. The direct cone cuts the hyberbolic surface of the crystalline, or the boss, circularly and equally; the oblique cone cuts its unequally. All the rays of the direct cone are gathered together at one point in the retina, which is the chief thing in the process; the lines of the oblique cones cannot quite be gathered together, because of the causes previously mentioned here, as a result, the picture is more confused. The direct cone aims the middle ray at center of the retina; the oblique cones aim the rays to the side…(Paralipomena 174)

This passage has multiple points of importance, in part because I suspect that it is the orgin passage of Descartes’ enthusiasm for the hyperbola, but also, as I will show later, for a naturalized justification for hyperbolic vision, something which will play to Spinoza’s optical critique. But at this point it is just sufficient to register the citation as a reference point for Spinoza’s phrase. We have already pointed out that Spinoza may have Kepler’s aquaeous globe in mind for his intial diagram, so there is something distinctly Keplerian in Spinoza’s approach.

Another reference point for Spinoza’s phrase is James Gregory’s 1663 Optical Promota, a treatise written without the aid of Descartes’ Dioptrics, but which all the same proposed parabolic and hyperbolic solutions to refraction aberrations and proposed reflective mirror telescopes to avoid the problem altogether. This text we know Spinoza had in his personal library, and he seems to be reasoning from it in part. Gregory regularly uses both “pencils of rays” and “cones of rays” as modes of analysis.

As a point of reference for us, he offers these defintions to begin his work:

6. Parallel rays are those which are always equally distant each to the other amonst themselves.

7. Diverging rays are those which concur in a point when produced in both directions: those rays produced in the opposite direction to the motion from the ray-bearing cone – the apex of the cone is the point of concurrence of the rays.

8. Converging rays are those rays are those which concur in a point in the direction of the motion when produced in both directions; these rays are called a pencil, and the point of concurrence the apex of the pencil…

10. An image before the eye [i.e. a real image], arises from the apices of the light bearing cones from single radiating points of matter brought together in a single surface.

Pencils of parallel rays feature in many of the diagrams, within the understanding that rays proceed as cones. So seems to me that Spinoza is operating with both Kepler and Gregory in mind as he answers Jelles’ question.

“But they are considered to be so because the object is so far from us that the aperture of the telescope, in comparison with its distance, can be considered as no more than a point.”

Spinoza follows Gregory’s Fourth Postulate: “The rays coming from remote visible objects are considered parallel.”

“Moreover, it is certain that, in order to see an entire object, we need not only rays coming from a single point but also all the other rays that come from all the other points.”

Spinoza may be still addressing the nature of Jelles’ request for clarification. He follows the reasoning of Gregory’s Tenth defintion (above). Whether the rays be treated as parallel pencils, or cones does not make a strict difference to Spinoza’s point, though understanding that they are coming to the lense as cones does something to express their spherical nature (one must recall that Kepler asserted that light radiates as a sphere as it can, and even that Hooke proposed that it moves in waves; Spinoza’s attachment to the sphere may be in regards to this). It is the lens’ capacity to gather together these rays come from diverse points of the object, and not just rays parallel to its central axis, that Spinoza emphasizes. In other words, though considered no more than a point, it is a point that must gather rays from a variety of angles.

“And therefore it is also necessary that, on passing through the glass, they should come together in as many other foci.”

It should be noted that Spinoza is talking about glass lenses here, and not the eye’s lens. Spinoza has taken his ideal model of a spherical refraction from the first letter, and has applied it to actual lenses (there is no requirement to the index of refraction of the glass). As Spinoza envisions it, because a glass has to focus rays coming obliquely, the foci along those alternate axes are significant factors in clarity.

Seeing More, or Seeing Narrowly

“And although the eye is not so exactly constructed that all the rays coming from different points of an object come together in just so many foci at the back of the eye, yet it is certain that the figures that can bring this about are to be preferred above all others.”

This is the big sentence, the one that opens up the place from which Spinoza is coming from. What does Spinoza mean “the eye is not so exactly constructed”? How odd. Descartes’ comments on optics indeed are often made in the service of correcting far- and near-sightedness, so there is context for a notion of the “inexactness” of the eye, and for his own uses Descartes picks up on the notion that the eye is limited or flawed: …”in as much as Nature has not given us the means…”, “I still have to warn you as to the faults of the eye”. But this is not what Spinoza has in mind. What I believe Spinoza is thinking about is the hidden heritage behind a naturalizing justification of hyperbolic vision itself. This is not strictly an optical point, as we have come to understand optical theory, but an analogical point. And this distinction organizes itself around the failure that a hyperbolic lens to handle rays oblique to its axis, with clarity, and whether this failure is something to be concerned with.

Keplers drawing the hyperbolic crystalline humor, 167

Kepler's drawing the hyperbolic crystalline humor, 167

Kepler begins the justification. The passage continues on from the conclusion of the one cited above, which ended with an explanation of why the image of the eye is blurred at its borders,

All the rays of the direct cone are gathered together at one point in the retina, which is the chief thing in the process; the lines of the oblique cones cannot quite be gathered together, because of the causes previously mentioned here, as a result, the picture is more confused. The direct cone aims the middle ray at center of the retina; the oblique cones aim the rays to the side…

so the sides of the retina use their measure of sense not for its own sake, but whatever they can do they carry over to the perfection of the direct vision. That is we see an object perfectly when at last we perceive it with all the surroundings of the hemisphere. On this account, oblique vision is least satisfying to the soul, but only invites one to turn the eyes thither so that they may be seen directly (174).

This is a striking passage in that we know the history of the hyperbolic lens, and Descartes’ fascination with it. Due to the hyperbolically shaped crystalline humor (as Kepler reasons it), the image at the border, projected at the edges of the retina, is said to be more confused due to the inability of the lens to focus oblique rays. This is what Spinoza has in mind when he says that the eye is not so exactly constructed. But there is more to this passage. Not only is the image more confused, but Kepler goes so far was to qualify this confused quality as an explanation for why the soul is dissatisfied with oblique vision. At the margins of blurred vision, according to Kepler, the sides of the retina do not “sense” for their own sake, but for the sake of central axis perfection, in effect serving the center. Kepler has provided the hyperbola as the solution for spherical aberration, but has also couched that shape within a larger context of human perception and the nature of what experience satisfies the soul or not.

This theme of the hyperbola’s justifcation through Nature continues. I will leap forward to Gregory’s Optica Promota, a writer who, as I have said, had no access to Descartes’ treatise but did read Kepler closely. At the end of a thorough and brilliant work on the value of hyperbolic and parabolic forms for use in telescopes, Gregory as well evokes Kepler’s notion of the weakness of the hyperbola, along with its naturalization. This is how he ends his Optica :

But against hyperbolic lenses, it is only objected that nothing will be able to be most clearly seen, except a visible point arising on the axis of the instrument. But this weakness [ infirmitas ] (if it would be worthwhile to call it that) is sufficiently manifested in the eye itself, though not to be impuning Nature, for whom nothing is in vain, but how much all things most appropriately she carries out [ peragit]. Nevertheless, withconicallenses and mirrors not granted, it shall be rather with spherical portions used in place of spheriods and paraboloids in catoptrics; as with hyperboloids in dioptrics, in which portions of spheres are less appropriate.

With these we go to the stars – His itur ad astra

Just as Kepler justifies hyperbolic vision by appeal to the eye’s own weakness, redeemed by the roles of the retina and the satisfactions of the soul, so here too Nature herself is the justification of central axis priority. This is a curious naturalization, given that so much of optics addresses the failings or the limitations of Nature. Such a self-contradiction deserves attention, especially with a focus upon the foundations of valuations that make one adjustment to Nature desired, and another not. But here I would like to continue the line of justifications of the hyperbola through the construction of the eye that Spinoza likely has in mind.

Descartes, if you recall from a passage cited above, also justifies the shape of the hyperbolic lens through appeal to the shape of the human eye. After he admits that the foci of rays that come obliquely to the axis of the hyperbola can only approximate a point of focus,

…for since the eye itself does not cause all the rays coming from diverse points to converge in exactly as many other diverse points, because of this the lenses would doubtless not be the best suited to render the vision quite distinct, and it is impossible in this matter to choose otherwise than approximately, because the precise shape of the eye cannot be known to us…

Descartes has not strictly forwarded Kepler’s claim that the crystalline humor has a hyperbolic shape, perhaps because his own anatomical investigations caused him to doubt the accuracy of this, but he maintains Kepler’s reasoning to some degree. While Descartes has long let go of any notion that spherical lenses may be preferred due to their omni-axial focus, he shrugs off the necessity for anything more than approximate foci along these oblique axes. The reason he provides for this is unclear. Either it is proposed that because the eye does not focus oblique rays, the benefits of any lens that does so would simply be lost – yet, if this were the reason, it would not result in the conclusion that such shapes are not best for precise vision, for they would be no worse than his hyperbola; or, he means to say that hyperbolic lenses are simply preferred because their weaknesses are natural weaknesses of the eye, with Nature not to be improved upon. This is emphasized in conclusion of the passage:

…Moreover we will always have to take care, when we thus place some body before our eyes, that we imitate Nature as much as possible, in all things that we see she has observed in constructing them; and that we lose none of the advantages that she has given us, unless it be to gain another more important one. – Seventh Discourse

There is additional evidence for the naturalized justification of the hyperbolic “weakness” (notice the question of valuation in the phrase “important one”). Firstly, when he proposes his notion that the telescope is simply an extension of the eye, Descartes imagines that all the refraction would occur in one lens, thus, “…there will be no more refraction at the entrance of that eye” (120). In this analogical conception of the extended length of the eye Descartes imagines his hyperbola as supplimenting and even supplanting the eye’s refractions. Secondly, when Descartes addresses the possibility that seeing at the borders may be an improvement of vision, he denies this, by virtue of how Nature has endowed our sight. Seeing more is not seeing better.

There is only one other condition which is desirable on the part of the exterior organs, which is that they cause us to perceive as many objects as possible at the same time. And it is to be noted that this condition is not in any way requisite for the improvement for seeing better, but only for the convenience of seeing more; and it should be noted that it is impossible to see more than one object distinctly at the same time, so that this convenience, of seeing many others confusedly, at the same time, is principally useful only in order to ascertain toward what direction we must subsequently turn our eyes in order to look at the one among them which we will wish to consider better. And for this, Nature has so provided that it is impossible for art to add anything to it. - Seventh Discourse

What Kepler has stated as simply the role of the borders of the retina to serve the perfection of the center, Descartes has made an occasion to assert the virtue of the human Will (a cornerstone of his metaphysics, and a cornerstone which Spinoza rejects, which makes the two philosophers quite opposed in their philosophy of ideal perception). For Kepler the edges serve the center, as is shown in the satisfactions of the soul. For Descartes the width of blurred vision becomes only a field upon which the Will manifests itself in making judgements of good and bad. Not only is the hyperbola’s condensed vision naturalized, it is key to how the Individual Will functions. Nature herself has foreclosed the possibility of improving the capacity to see more in a better way. Spinoza’s philsophy of mind’s-eye perception is based on the principle that one sees clearly as one sees more – more at once. (It is interesting that immediately following this assertion Descartes uses the examples of sailors and hunters who are able to improve on Nature’s provisions, but only in the direction of further sharpening their eyes to a more narrow focus. Descartes valuation is both implicit and naturalized.)

It suffices to say that in this long digression what Spinoza means by “the eye is not so exactly constructed” is that the non-spherical shapes of the eye (and our tendencies of vision that come from it) provides a focus that is not optimal. Spinoza here likely conflates his metaphysics and his optics, as perhaps does Descartes. His critique, right down to the root of centralized conceptions of a naturalization of hyperbolic vision, opens to Post-modern and Post-structuralist critiques of marginalization and philosophies of Presence, locating his objection not in the glorification of the human eye, but in the understanding of its limitations. Descartes’ philosophy of “clear and distinct” and its parasitic conceptions of Human Will are cut at in a very essential way. But the question remains, is there an optical advantage to spherical lenses, as they exhibit the flexibility of omni-axial foci? The obvious objection to hyperbolics is that they proved impossible to grind, either by hand, or in the kinds of automated machines that Descartes proposed. As a practiced lens-grinder Spinoza better than most would surely know this. But aside from this serious detraction Spinoza finds one more, and it is one that Kepler, Descartes and Gregory all admit, as they justify it not in optical terms, but in terms of naturalized conceptions of the eye and perception. Perhaps we can assume that Spinoza, out of his love for the sphere, coupled with the Keplerian sense of the spherical radiation of light, the practical considerations of lens grinding, and a epistemological conception of Comprensive Vision, saw in the admitted weakness of the hyperbola (and the eye) something that outweighed the moderate weakness of spherical aberration. In a sense, Spinoza may have seen spherical aberration in terms of his acceptance that almost all of our ideas are Inadequate Ideas. [More of this line of thought written about here: A Diversity of Sight: Descartes vs. Spinoza ]

“Now since a definite segment of a circle can bring it about that all the rays coming from one point are (using the language of Mechanics) brought together at another point on its diameter, it will also bring together all the other rays which come from other points of the object, at so many other points.”

A modified version of the letter 39 diagram, showing what Spinoza believed to be the failings of the hyperbola

A modified version of the letter 39 diagram, showing what Spinoza believed to be the failings of the hyperbola

Spinoza repeats his insistence upon the virtues of spherical lenses. As the modified diagram here shows, the capacity to refract rays along an infinity of axes is in Spinoza’s mind an ideal which hyperbolic forms cannot achieve. He does not accept the notion that an assumed narrow focus of human vision, nor the supposed shape of the crystalline humor (Kepler) determines that “hyperbolic abberation” is negligable to what should be most esteemed. This insistance upon the importance of the sphere calls to mind James Gregory’s description of refraction on sphere of the “densest medium” presented in his first proposition of the Optica:

If truly, everything is examined carefully, then it will seem – on account of the aforementioned reasons -that all the rays, either parallel or non-parallel, which are incident on the circular surface of the densest medium for refraction, are concurrent in the centre of the circle. Now we ask: how does this come about? The answer is: – Well, however a line is drawn incident on the circle, (provided they are co-planar) an axis can be drawn parallel to it and without doubt the circle can be considered a kind of ellipse so that any diameter can be called the axis, from which it appears that the special line sought is the axis of a conic section. - Optica Promota

figures 1 and 2 from the Optima Promota

figures 1 and 2 from the Optima Promota

One feels that there seems something of this ideal conception of the densest medium floating behind Spinoza’s conception of the spherical lens. Material glass somehow manifests for Spinoza, in its particularities of modal expression, these geometric powers of unified focus, and peripheral focus is a part of what Spinoza conceives of as ideal clarity.

“the language of mechanics”

But there is another very important clue in this section of the letter: the phrase “using the language of Mechanics”); for now I believe we get direct reference to Johannes Hudde’s optical treatise “Specilla circularia” (1655), an essential text for understanding Spinoza’s approach to spherical aberration.

Rienk Vermij and Eisso Atzema provided a most valuable, but perhaps sometimes overlooked insight into the 17th century reaction to Descartes resolution to spherical aberration in their article “Specilla circularia: an Unknown Work by Johannes Hudde”. They present Hudde’s small tract (it is not quite nine typed journal pages) which offers a mathematical treatment of the problem of spherical aberration. Interestingly, as it was published anonymously, Hudde’s teacher at Leiden, Van Schooten, actually thought that the work belonged to his star student Christiaan Huygens. Presumably this was because of the closeness it bore to Huygens’ 1653 calculations of aberration, and he wrote him to say as much, and he likely sent him a copy of it as Christiaan requested. Hudde’s approach is a kind of applied mathematics to problems he considered to be pragmatic mechanical issues. In a sense he simply took spherical aberration to be a fact of life when using lenses, and thought it best to precisely measure the phenomena so as to work with it effectively. The hyperbolic quest was likely in his mind a kind of abstract unicorn chasing. He wanted a mechanical solution which he could treat mathematically, hence his ultimate distinction between a “mathematical point” of focus and a “mechanical point”. As Vermij and Atzema write describing this distinction and its use in analysis:

At the basis of Hudde’s solution to the problem is his distinction between mathematical exactness and mechanical exactness. Whereas the first is exactness according the laws of mathematics, the second is exactness as far as can be verified by practical means. After having made this distinction, Hudde claims that parallel incident rays that are refracted in a sphere unite into a mechanically exact point (“puntum mechanicum”). In order to substantiate his claim Huddethen proceeds to the explicit determination of the position of a number of rays after refraction.

Restricting his investigation to the plane, Huddeconsiderstherefraction of seven parallel rays by explicitly computingthepoint of intersection of these rays with the diameter of the circle parallel to the incident rays for given indices of refraction. The closer these rays get to the diameter, the closer these points get to one another until they finally merge into one point. Today, we would call this point the focal point of the circle; Hudde does not use this term.

Returning to spheres, Hudde erects a plane perpendicular to the diameter introduced above and considers the disc illuminated by the rays close to this diameter. He refers to this disc as the “focal plane”. On the basis of the same rays he used earlier, Hudde concludes that the radius of the focal plane is very small compared to the distance of the rays to the diameter. Therefore this disc could be considered as one, mechanically exact point. In other words, parallel rays refracted in a sphere unite into one point (111-112).

From this description one can immediately see a conceptual influence upon Spinoza’s initial diagram of spherical foci, and far from it being the case that Spinoza knew nothing about spherical aberration and the Law of refraction, instead, it would seem that he was working within Hudde’s understanding of a point of focus as “mechanical”. We know that Spinoza had read and reasoned with Hudde’s tract, as he writes to Hudde about its calculations, and proposes his own argument for the superiority of the convex-plano lens. And the reference to “the language of mechanics” seems surely derived straight from Hudde’s thinking. What these considerations suggest is that Spinoza’s objection to the hyperbola to some degree came from his agreement with Hudde that spherical aberration was not a profound problem. As it turns out, given the diameters of telescope apertures that were being used, this was in fact generally correct. Spinoza joined Hudde in thinking that the approximation of the point of focus was the working point of mechanical operations, and the aim of shrinking it down to a mathematical exactness was not worth pursuing (perhaps with some homology in thought to Descartes’ own dismissal of the approximations of focus of rays oblique to the axis of the hyperbola).

F. J. Dijksterhuis summarizes the import of Hudde’s tract, in the context of Descartes’ findings in this way:

The main goal of Specilla circularia was to demonstrate that there was no point in striving after the manufacture of Descartes’ asphericallenses. In practice one legitimately makes do with spherical lenses, because spherical aberrations are sufficiently small. (Lenses and Waves. Diss. 72)

Spinoza has a connection to the other main attempt to resolve the difficulty of aberration from focus using only spherical lenses, that which was conducted by Christiaan Huygens. Spinoza in the summer of 1665 seemed to have visited Huygens’ nearby estate several times, just as Huygens was working on developing a theory of spherical aberration and devising a strategy for counteracting it which did not include hyperbolas. In that summer as Spinoza got to know Huygens, he was busy calculating the the precise measure of the phenomena. In 1653 he had already made calculations on the effects in a convex-plano lens, an effort he now renewed under a new idea: that the combination of defects in glasses may cancel them out, as he wrote:

Until this day it is believed that spherical surfaces are…less apt for this use [of making telescopes]. Nobody has suspected that the defects of convex lenses can be corrected by means of concave lenses. (OC13-1, 318-319).

What followed was a mathematical finding which not only gave Huygens the least aberrant proportions of a convex-plano lens, but also the confirmation of its proper orientation. In addition he found the same for convex-convex lenses. In August of that summer Huygens wrote in celebration:

In the optimal lens the radius of the convex objective side is to the radius of the convex interior side as 1 to 6. EUPHKA. 6 Aug. 1665.

During this time the secretary of the Royal Society was writing Spinoza, trying to get updates on the much anticipated work of Spinoza’s illustrious neighbor (he was about to become the founding Secretary for the Académie Royale des Sciences for Louis XIV. Spinoza writes to Oldenburg:

When I asked Huygens about his Dioptricsandabout another treatise dealing with Parhelia he replied that he was still seeking the answer to a problem in Dioptrics, and that soon as he found the solution he would set that book to print together with his treatise on Parhelia. However for my part I believe he is more concerned withhisjourneytoto France (he is getting ready to to live in France as soon as his father has returned) than with anything else. The problem which he says he is trying to solve in the Dioptrics is as follows: It is possible to arrange the lenses in telescopes in such a way that the deficiency in the one will correct the deficiency of the other and thus bring it about that all parallel rays passing through the objective rays will reach the eye as if they converged on a mathematical point. As yet this seems to me impossible. Further, throughout his Dioptrics, as I have both seen and gathered from him (unless I am mistaken), he treats only spherical figures.

This letter is dated October 7, 1665, two months after Huygens had scribed his Eureka optimalization of the lens shape. Significantly, Huygen found that lenses of this optimal shape actually were not the best for his project of combining lens weaknesses (302-303), rather lenses with greater “weaknesses” were better combined. Several facts can be gleaned from Spinoza’s letter, and perhaps a few others guessed at. Spinoza had both looked at and discussed with Huygens his contemporary work. So the sometimes guarded Huygens was not shy about the details of his project with Spinoza. It may well have been Huygens’ treatment of the convex-plano lens here that caused Spinoza to write to Hudde less than a year later with his own calculations in argument for the superiority of the convex-plano lens, using Hudde’s own Specilla as a model. (Hudde seemed quite interested in Spinoza’s proofs of the unity of God, and the correspondence seems to have begun as early as late 1665.) What cannot be lost is that with a joint awareness of both Hudde’s and Huygens’ attempts to resolve spherical aberration, Spinoza was in a very tight loop of contemporary optical solutions to the problem. Not only is his scientific comprehension trusted by both Huygens and Oldenburg at this point, but perhaps also by Hudde.

What is striking though is Spinoza’s pessimism toward Huygens’ project. Given Spinoza’s optical embrace of spherical lenses (in the letters 39 and 40 we are studying), what would lead Spinoza to such a view he qualifies as “As yet this seems to me impossible.” Is this due to a familiarity with Huygens’ mathematics, and thus comes from his own notable objections? Has Huygens actually shared the frustrations of his experiments? Or is he doubtful because Spinoza has only a vague notion of what Huygens is doing? He seems to deny the very possibility of achieving a mathematical point of focus, though his mind remains tentatively open. His added on thought, Further, throughout his Dioptrics, as I have both seen and gathered from him (unless I am mistaken), he treats only spherical figures” , is also curious. He seems privy to the central idea that Huygens is using spherical lenses to achieve this – what other figure would it be? – but it is possible that Spinoza here qualifies his doubt as a general doubt about sphericals which he only believes Huygens is using in his calculations, showing only a cursory knowledge. Perhaps it is only an addendum of information for Oldenburg.

Huygens indeed would soon find such a solution to aberration writing,

“With concave and convex spherical lenses, to make telescopes that are better than the one made according to what we know now, and that emulate the perfection of those that are made withellipticor hyperbolic lenses” (OC13, 318-319).

I am unsure if he had come to this solution before he left for Paris in mid 1666, or if he would even have shared this discovery with Spinoza, but he also came to the same pessimistic conclusion as Spinoza held, at least for Keplerian telescopes, for his design only worked for those of the Gallelian designs which had fairly low powers of magnification. By combining convex lenses the aberration was only increased. This would be the case until February of 1669, when Huygens finally came up with right combinations of lenses.

“For from any point on an object a line can be drawn passing through the center of a circle, although for that purpose the aperture of the telescope must be made much smaller that it would otherwise be made if there were no need of more than one focus, as you may easily see.”

Again Spinoza returns to his initial point, now putting it in context of real telescopes. Such telescopes required the stopping down of the aperture, something that reduced the impact of spherical aberration; but restricting the aperture reduced the amount of light entering the tube, hence making the image less distinct. I am unsure what Spinoza refers to in “as you may easily see”, for neither of his diagrams seem to distinctly address this aspect. Perhaps Spinoza has in mind two diagrams of the eye that Descartes provides, contrasting the angles of rays entering the eye with a narrow and a wide pupil aperture. Was this a diagram which Jelles had mentioned in his response (below, left)?

Descartes diagram 17 of the eye, Sixth Discourse

 

 

Descartes' diagram 17 of the eye, Sixth Discourse

“What I here say of the circle cannot be said of the ellipse or the hyperbola, and far less of other more complex figures, since from one single point of the object only one line can be drawn passing through both the foci. This what I intended to say in my first letter regarding this matter.”

I am unsure what Spinoza means by “both the foci”, but it appears that he asserts again that because there is only one axis of either hyperbolics or ellipse available to any rays of light arriving for refraction, and that spherical lenses, again, have the advantage that rays come from any particular point of an object then can be focused to a single “mechanical point” along an available axis. Under Spinoza’s conception, this is an advantage that cannot be ignored.

Below I post Spinoza’s last diagram to which he refers with his final remarks. I place it beside Descartes diagram to which it most likely refers. This may be the most telling aspect of Spinoza’s letter, for we have to identify just what Spinoza is making clear as distinct from what Descartes was asserting.

Descartes’ diagram is a variation of as similar diagram which illustrated his prototype idea of forming a single lens made of an objective lens and a tube of water which was imagined to be placed directly upon the eye, making a long prosthetic lens, physically extending the eye. In this version he proposes that because such a watery tube is difficult to use, the tube may be filled with one large glass lens, with surfaces A and B acting as the anterior and posterior surfaces. And yet again acknowledging that the making of such a lens is unlikely, the same diagram is meant to serve as a model of an elementary telescope:

…because there would again be some inconvenience…we will be able to leave the whole inside of this tube empty, and merely place, at its two ends, two lenses which have the same effect as I have just said that the two surfaces GHI and KLMshouldcause. And on this alone is founded the entire invention of these telescopes composed of two lenses placed in the two ends of the tube, which gave me occasion to write this Treatise. – Eighth Discourse

Spinoza’s diagram from Letter 40

 

 

 

Descartes diagram 30, Seventh Discourse

“From the attached diagram you will be able to see the proof that the angle formed at the surface of the eye by rays coming from different points becomes greater or less according to the difference of the foci is greater or less.”

There are several ways to look at Spinoza’s diagram, but it is best to take note of where it diverges from Descartes’ (for Jelles would have had the latter to compare it to). The virtual image of the arrow appearing to be much closer to the eye is eliminated, presumably because the appearance of magnification is not in Spinoza’s point. The refraction of the centerpoint of the arrow remains, and is put in relation to refractions of rays coming from the extreme ends of the arrow. The refractions within the eye have been completely collapsed into an odd, artfully drawn eye, (the touch of lid and lashes actually seem to speak to Colerus’ claim that Spinoza was quite a draftsman, drawing life-like portraits of himself and visitors). Behind this collapse of the eye perhaps we could conclude either a lack of effort to portray his version of refractions into the mechanisms of the eye, or even a failure of understanding, but since this is just a letter to a friend, it probably marks Spinoza’s urge just to get a single optical point of across, and he took more pleasure in drawing an eye than he did tracing out his lines of focus. An additional piece of curiousness, which may be a sign of a very casual approach is that the last arrow in the succession, which to my eye appears to be one supposed to be in the imagination of the mind, Spinoza fails to properly reverse again so that it faces the same direction as the “real” one, although perhaps this is an indication that Spinoza thought of the image as somehow arrived within the nervous system at a point, on its way to be inverted by the imagination (though in the Ethics he scoffs at Descartes’ pituitary concept of projective perception). There is of course the possiblity that I am misreading the diagram, and the the final arrow somehow represents the image as it lies on the retina at the back of the eye. At any rate, it is a confusing addition and one wonders if it is just a part of Spinoza’s musings.

As best I can read, below is an altered version of the diagram designed to emphasize the differences between Descartes’ drawing the Spinoza’s:

The first thing to be addressed, which is not labeled here, is what C is. There is the possibility that it is a crude approximation of the crystalline humor, acknowledged as a refractive surface. If so, the upper arc of the eye and the figure C would form some kind of compound refractive mechanism approximate to what Descartes shows in his eye, here compressed and only signified. But I strongly suspect that C is the pupil of the eye, as the aperture of the telescope has been recently has been referred to in terms of its effect on the requirements of refraction, and in Descartes text there is a definite relationship between the telescope aperture and the pupil of the eye (it has also been proposed to me that C is the eyepiece of the telescope).

The primary difference though is the additional emphasis on the cones of rays that come from either end of the object to be seen (here shaded light blue and magenta). This really seems the entire point of Spinoza’s assertion, that spherical lenses are needed for the non-aberrant focus of oblique cones for a object to be seen clearly. In addition to this, the angle that these rays make at the surface of the eye (indicated) points to Spinoza’s original objection to Descartes incomplete description of what is the most significant factor the construction of a telescope.

What remains is to fully assess this conception of refraction that Spinoza holds. While it is made in the context of historic discussions of the blurred nature of the borders of an image’s perception, it is also true that such an oblique focusing must occur, however slightly, at any point exactly off from the center axis of a hyperbolic lens. It may well be that Spinoza is balancing this aberration of focus in hyperbolic lenses with the found-to-be overstated aberration of spherical focus. Given his comprehensive conception of clear mental vision -seeing more is seeing better – and its attendant critique of the Cartesian Will, given his love for the sphere, perhaps aided by a spherical conception of the propagation light come from Kepler, with Spinoza being much sensitized to the absolute impracticality of ground hyperbolic glasses through his own experiences of glass grinding, it may have been quite natural for Spinoza to hold this optical opinion…though it is beyond my understanding to say definitively so. 

“So, after sending you my cordial greetings, it remains only for me to say that I am, etc.”

This is a curious ending for such a wonderful letter. Perhaps we can assume that once again the editors of his Opera suppressed important personal details.

 

These English selections and links to the Latin text: here

The Optica Promota and Spinoza’s concept of focus

Spinoza: Against Hyperbolic Vision

Optica Promota, figures 1 and 2

Optica Promota, figures 1 and 2, refraction in the densest medium

There are several points of correspondence between Spinoza’s objections to Descartes’ treatment of the magnification of images in the L’Dioptrique in his letters 39 and 40, and the analyses found in James Gregory’s Optica Promota, but perhaps the most significant touching point occurs in Gregory’s conclusion, where he brings up the lone failings of parabolic mirrors and hyperbolic lenses, which Spinoza is sensitive to. Here you have a presentation of the weakness of the hyperbola as an ideal of lenses, and yet the appeal to the human eye as somehow exemplary and natural, which I have pointed out in regards to Kepler’s Paralipomena, [thought about here: A Diversity of Sight: Descartes vs. Spinoza and more broadly here Some Observations on Spinoza’s Sight ]. The entirety of Spinoza’s referential borrowings from Gregory must be fleshed out, (they seem proliferate through these two letters) but for the moment I present this conclusion alone, which marks out the limitations of hyperbolic lenses, apart from their difficulty of manufacture:

But against hyperbolic lenses, it is only objected that nothing will be able to be most clearly seen, except a visible point arising on the axis of the instrument. But this weakness [ infirmitas ] (if it would be worthwhile to call it that) is sufficiently manifested in the eye itself, though not to be impuning Nature, for whom nothing is in vain, but how much all things most appropriately she carries out [ peragit ]. Nevertheless, with conical lenses and mirrors not granted, it shall be rather with spherical portions used in place of spheriods and paraboloids in catoptrics; as with hyperboloids in dioptrics, in which portions of spheres are less appropriate.

With these we go to the stars – His itur ad astra

One can see how Spinoza would have sympathy to the notion that Nature does not err, but the subtle question of the “infirmitas” of human, hyperbolic vision is one that he would pause at. Insofar as human vision is used as a model for mental vision, any weakness “natural” to human beings is not redeemed by that natural state. All states a human being finds herself in are “natural”, but there is a degree of perfection which works as a fulcrum point for change and improvement in Spinoza’s thinking. The question remains whether Spinoza’s appraisal of spherical lenses as ideal is a pragmatic solution to the problem of finding the best lenses possible in the real world, or is a mistaken extension in analogy from his concept of ideal mental vision back down to the question of lens shapes, over-valuing the capacity of spherical lenses to handle otherwise would be paraxial rays.  

Spinoza’s Picture of the Omni-axial Spherical Lens

As Spinoza writes in his letters 39 and 40, following Gregory’s warning about hyperbolics:

Letter 39 – …Perhaps he was silent so as not to give any preference to the circle above other figures which he introduced; for there is not doubt that in this matter the circle surpasses all other figures that can be discovered. If, for example, circle ABCD should have the property that all rays coming from direction A and parallel to axis AB are refracted at its surface in such a way that they thereafter all meet at point B; and also all rays coming from point C and parallel to axis CD are refracted at its surface so that they all meet together at point D, this is something that could be affirmed of no other figure, although the hyperbola and the ellipse have infinite diameters. 

diagram letter 39

And then..

Letter 40 – …Moreover, it is certain that, in order to see an entire object, we need not only rays coming from a single point but also all the other rays that come from all the other points. And therefore it is also necessary that, on passing through the glass, they should come together in as many other foci. And although the eye is not so exactly constructed that all the rays coming from different points of an object come together in just so many foci at the back of the eye, yet it is certain that the figures that can bring this about are to be preferred above all others. Now since a definite segment of a circle can bring it about that all the rays coming from one point are (using the language of Mechanics) brought together at another point on its diameter, it will also bring together all the other rays which come from other points of the object, at so many other points. For from any point on an object a line can be drawn passing through the center of a circle…What I here say of the circle cannot be said of the ellipse or the hyperbola, and far less of other more complex figures, since from one single of the object only one line can be drawn passing through both the foci. This what I intended to say in my first letter regarding this matter.

As said, much more than this needs to be assessed regarding these two optical letters, but in this vein we identify a prominent theme of naturalized, hyperbolic vision and its critique, stretching from Kepler, to Descartes, and alternately Gregory to Spinoza.

The Optica Promota, by James Gregory (1663)

A valuable Spinoza resource: we have an on-line photocopy edition of the 1663 work Optica Promota by James Gregory, an edition found in Spinoza’s library of books. One can see the imprint that Spinoza likely looked at. This copy is found here.

Not only this, Ian Bruce has supplied us with a marvelous English translation and Latin transcript, with explanatory notes, found here in PDF parts.

This is a signficant text, as I believe that Spinoza is arguing with Gregory’s propositions and diagrams in mind in his March 1667 letters 39 and 40 to Jelles, as he makes his objections to Descartes, and perhaps also in his June 1666 letter 36 to Hudde, showing that he had studied this volume at least by this point in time.

Follow

Get every new post delivered to your Inbox.

Join 56 other followers