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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

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Spinoza the Merchant: The Canary Islands, Sugar and Diamonds and Leprosy

Spinoza as Merchant: New Reflections on Old Biographical Material

Below I exerpt significant passages from the very valuable translation of “Mercator et Autodictus” written by A. M. Vas Dias and W. G. van der Tak in 1932, filled with primary source material ubiquitously used. These selections are as to the kind of trade that Spinoza’s family business might have engaged in. Nadler follows this text for instance in concluding that the Spinoza name traded in dried fruits, but in looking at the evidence there does seem room for additional inferences, including the trade of Algerian oils and pipes, and the possibility of Canary Island or Brazilian sugar.

Mercator et Autodictus

Found in the notarial archives of Amsterdam:

A deed dated July 15th, 1631, passed before notary Daniel Brendan (register 941) containing the statement of two porters that on May 27th and June 18th of that year, they, acting upon the request of couriers of Amsterdam and sheriff Hendrik Hudde respectively, carried goods to the Weighing-house from a certain warehouse on the Prinsengracht in which miscellaneous merchandise as stored, such as sugar, brazil-wood and candied ginger, of which warehouse Philips Pelt and Michael d’Espinosa kept the keys.

 To nr.2a (Notary archives of Amsterdam register 942):

To this deed we are told that a shipment of fifty small casks of raisins which Michael should have recieved from Malaga [Spain] according the the bill of charge, did not arrive in good order. Besides learning that Michael must have traded in dried citrus, we also hear that he was living on Vloolenburg in 1633. 

Now we give two other deeds in French, passed before the notary Benedict Baddel:

in the first, passed on July 20th, 1651(register 967 folio 304), Michael D’Espinosa grants a power of attorney to Jacob Boeve, merchant in London, to institute legal proceedings to regain possession of goods, belonging to him, that had been seized by the English Admiralty as coming from Portugal. The goods consisted of pipes as small casks of Algerian oil.

The registries of the notary Baddel mention two more deeds in Dutch (registers 964 and 970):

…the first, passed on November 27, 1651, contains a statement by Simon Rodrigues Nunes at the request of Michael D’Espinosa, that at the house of Julian Lanson, also merchant in Amsterdam, he requested payment of the latter’s share in the expenses made in the reclamation of the ship “Prince” that was seized by the English on its way from the Canary Islandsto Amsterdam; the expenses referred to had been advanced by Antonio Fernandes Carvejal in London.

Febuary 15 1655:

…the honourable William van Erpecum, about forty years old, and Jarich Jelles, thiry-five years old, both merchants within this said city, well known to me, said the notary. And the request of Simon Rodrigues Nunes, also a merchant within the same city, they attested…they made the following purchases and received from the requisitionist, namely the said Van Erpecum, five casks of long raisins at twenty-eight guilders a hundred, and the said Jarich Jelles twenty-seven casks also of long raisins at twenty-seven guilders a hundred.

“Jelles dealt in spices and according the the deed [NAA 975] he did buisness with Portuguese Jews from whom he bought raisins. Michael De Spinoza also traded in dried fruits.”

The evidence points toward an interesting matrix of potential trade practices. The record of 1633 does indicate that Michael Spinoza traded in raisins, and in 1655 we have the suggestive record that Jarich Jelles did as well, keeping some continuity across two decades of business. Yethere we have as well a record of trade in Algerian oil and pipes, and an interesting piece of evidence that Michael held the keys to a warehouse that was filled with Brazilian trade items, notably the cash crops sugar and brazil-wood. The authors take this to be proof of Michael’s trustworthiness, but there would seem to be more than this. There is additional evidence that Michael Spinoza traded in sugar, as he attempted to re-acquire the goods of the ship “Prince” seized by the English. This ship came from the Canary Islands which had for the last century or so been dominated by the effective and mostly brutal economies of sugar production. By the 17th century though, sugar production dramatically had dropped due to Brazilian sugar expansion, and its primary export had been replaced by the sweet dessert wine, Malvasia, meant for both American and British consumption. Yet, the Canary Islands were on the trade route to the Americas, so a ship coming from there destined for Amsterdam likely and predominantly carried the Brazilian sugar (or brazil-wood), or even still Canary sugar. Evidence for additional Canary Islands, sugar-trade relations are found in Spanish Inquisitional records:

Gullan-Whur, citing Israel Revah’s “Spinoza et le Dr. Juan de Prado” (1959), writes of this:

An echoing description was given to the Inquisition by Captain Miguel Pérez de Maltranilla, a day after Fray Tomás’s. The captain, visiting a Canary Islands physician convalescing from leprosy in Amsterdam, stumbled upon a discussion group at the sick physician’s home, where he distinguished two men “who had abandoned the Jewish religion” from two other, allegedly practicing, Jews (who should not, of course, have been “under the same roof or come within four cubits”) of Benedictus or De Prado (90).

The obscure citation is usually used to draw out the fact that Spinoza may have been still connected to issue of Hebraic religiosity after his ban from the community, and important possibility. But here I want to add it to his possible Canary Islands and sugar associations. In 1659 Spinoza was in the house of a Canary Island physician [Nadler reports elsewise, that the man is merely visiting a physician, perhaps Dr. Reinoso, and not a physician himself, and Klever simply identifies him a “chevalier”; Gullan-Whur reads the man himself as a physician, noting that the Leper’s hospital was by the city wall on Vloyenburg; the argument could procede either way]. Whether Spinoza knew him through his past trade practices, or through the nature of the “discussion group” (or both), or simply through the house owned by a name of the same name, Guerra, is of course undecided.

Apart from general knowledge of Spinoza’s doings and concerns, the reason for looking closely into this is that Spinoza’s personal connection to colonial exploitation may have had a bearing upon either his renunciation of mercantile wealth, or on his personal politics later to be voiced in two treatises. Sugar embodies in many ways the pitfalls of affective indulgence and the systematic, brutal control of others.

The Waking Nightmare of Sugar, “Physician Heal Thyself”

This may be no more than a fancy, but we have a curious record 5 years later in Spinoza’s life of the possible impress of the leperous-physician event. The notable merchant and collegiant Peter Balling’s son has died (whether the death was recent is in question), Spinoza writes to comfort him as to how a father may have somehow presaged his son’s death, in an auditory hallucination. Spinoza tells of a waking dream he had in the Winter of Rijnsburg, where a “scabby Brazilian” stared him hauntingly in the face:

I think I can both illustrate and confirm what I say by another occurrence, which befell me at Rhijnsburg last winter. When one morning, after the day had dawned, I woke up from a very unpleasant dream, the images, which had presented themselves to me in sleep, remained before my eyes just as vividly as though the things had been real, especially the image of a certain black and scabrous Brazilian whom I had never seen before. This image disappeared for the most part when, in order to divert my thoughts, I cast my eyes on a boot, or something else. But, as soon as I lifted my eyes again without fixing my attention on any particular object, the same image of this same Ethiopian appeared with the same vividness again and again, until the head of it gradually vanished (translation modified), – Letter 17, July 20th 1664.

Politically sensitive readers such as Antonio Negri have taken this dream to represent a extraordinary fracture in Spinoza’s System, one that will break off his writing of the Ethics, and a turn toward the requirement of a Theological-Political Treatise. While I will refrain from such a grand, but perhaps attractive interpretation, I will suggest that there are certain correspondences between Spinoza’s dream, his no doubt powerful sympathies for a mourning friend, and our record of the likely spiritual-tinged meetings at the house of a physician from the Canary Islands.

For one, the visiting man [a physician or otherwise] is leprous, as would seem the Brazilian figure is. Sugarcane historically had dominated the slave economies of both Brazil and the Canary Islands, so conflating the two seeming fitting. In addition, more symbolically, the disease of leprosy is the New Testament disease par excellence, symbolizing man’s fallen state. If the man himself was a physician, the image of a physician that cannot heal himself certain recalls the proverb from the fourth chapter of the Gospel of Luke, when Jesus begins his public ministry. It worth quoting in full, since one wonders if the passage was on the minds or on the lips of any of those Christians who may have been in attendance. (Spinoza is in the company of Dr. Reinoso who may be an attending physician):

14 And Jesus returned to Galilee in the power of the Spirit, and news about Him spread through all the surrounding district. 15 And He began teaching in their synagogues and was praised by all. 16 And He came to Nazareth, where He had been brought up; and as was His custom, He entered the synagogue on the Sabbath, and stood up to read. 17 And the book of the prophet Isaiah was handed to Him. And He opened the book and found the place where it was written,

18 “THE SPIRIT OF THE LORD IS UPON ME,
BECAUSE HE ANOINTED ME TO PREACH THE GOSPEL TO THE POOR.
HE HAS SENT ME TO PROCLAIM RELEASE TO THE CAPTIVES,
AND RECOVERY OF SIGHT TO THE BLIND,
TO SET FREE THOSE WHO ARE OPPRESSED,
19 TO PROCLAIM THE FAVORABLE YEAR OF THE LORD.”

20 And He closed the book, gave it back to the attendant and sat down; and the eyes of all in the synagogue were fixed on Him. 21 And He began to say to them, “Today this Scripture has been fulfilled in your hearing. 22 And all were speaking well of Him, and wondering at the gracious words which were falling from His lips; and they were saying, “Is this not Joseph’s son?” 23 And He said to them, “No doubt you will quote this proverb to Me, ‘Physician, heal yourself! Whatever we heard was done at Capernaum, do here in your hometown as well.'”

24 And He said, “Truly I say to you, no prophet is welcome in his hometown. 25 But I say to you in truth, there were many widows in Israel in the days of Elijah, when the sky was shut up for three years and six months, when a great famine came over all the land; 26 and yet Elijah was sent to none of them, but only to Zarephath, in the land of Sidon, to a woman who was a widow. 27 And there were many lepers in Israel in the time of Elisha the prophet; and none of them was cleansed, but only Naaman the Syrian.”

28 And all the people in the synagogue were filled with rage as they heard these things; 29 and they got up and drove Him out of the city, and led Him to the brow of the hill on which their city had been built, in order to throw Him down the cliff.

Let me diverge from my main point for a moment, as the writing seems to be heading this way. This passage from Luke contains several elements consonant with Spinoza’s situation, which in what may only be a marvelous coincidence flow together in figuative ways. Here is marked out the beginning of Jesus’ ministry, and the violent rejection of him by his own home town. Clearly, this too is the place where Spinoza found himself in 1659, upon reflection. We also have the healing of lepers, and for Spinoza, a physician who is struck with leprosy. And lastly, there is the “Old” Testament imperative to release the captives, a command that Jesus saw himself fulfilling. To return to our main point, is it too much to assume that Spinoza’s relating of this dream is something more than simply comforting and interpreting the apparition experience of a bereaved father (I long have accepted this general reading)? Is it beyond likelihood to expect that Balling and even Jelles were at this or other meetings at the house of this diseased man [physician]? What stories of disease, both in politic and in body, were being told about the Spanish owned island? Was the Canary Inquisitional burning of a London Crypto-Jew in effigy the year before talked of? The subject of slavery? The recent seizure of Dutch Brazil?

We cannot tell if any of these connections were consciously made by Spinoza, either in the reading of his own dream, or in decision to relay it to Peter Balling as a comfort, but there is an outline to be traced between Spinoza’s possible association with the sugar trade, the events in the home of the Canary Islands physician, and the haunting figure of a scabrous Brazilian. To my ear, there is oddity to Spinoza’s insistence that the Brazilian floating before him “he had never seen before”. I would think that there would have been few chances to have seen such a figure. Is this assertion simply to enforce the purely imaginary characteristic of the apparition, whose “cause was quite different”? Perhaps. Or, this was a denial of a sorts, as the story is most curious, too much so to dismiss it as only an example of the kinds of tricks the mind can play upon us. He knows him to be a Brazilian, despite later calling him, perhaps generically, an Ethiopian. Dare we risk a dream-interpretation in the conflation of the diseased sugar slave, and the diseased Canary doctor (and the ostracized Jesus), a participating “in the ideal essence” Despite Spinoza’s disavowal? Perhaps that is all we have left. But I am tempted to imagine that indeed Balling had been at the “group discussion” and at some level Spinoza’s family had participated in the sugar trade either of Brazil or the Canary Islands, two “facts” that worked themselves into Spinoza’s dream and perhaps his sense of personal mission.

Spinoza and Diamonds

A last bit of evidence and conclusion taken from Vaz Dias’ report concerns the diamond and jewel trade, and the possibility that Spinoza had turned his mercantile business to include this source of weath. In the record of Spinoza’s arrest of the brother Alveres, a trader in jewels, and the holding of a bill from the diamond family Duartes, the authors write:

The Alveres brothers, alias Nunes, dealt in jewels and came from Paris around 1641 to settle on Uilenburg in Amsterdam in the house called “de Vergulde Valck”. Their business was of doubtful solidity, also in other respects. Through Gabriel Alveres they were related to the Duartes, to which family also belonged Francisca Duarte, known as French Nightingale, who was aquainted with with Pieter Cornelisz. Hooft and is considered a member of the so-called “Muider-circle”…The circumstance that not only the Alveres brother but also the witnesses Manuel Duarte and Manual Levy traded in jewels, leads us to wonder if Spinoza was also involved in this trade.

There is no doubt that the Duarte family was immensely wealthy due to trade in diamonds, and art, in fact Constantijn Huygens’ Sr. and Christiaan visited their home rather regularly for their mutual love of music. I do not know the precise relationship between the young Manuel Duarte (23) and the famed Diego, but it is perhaps significant that Spinoza’s Latin teacher, the book and art seller Franciscus Van den Enden, had strong art trade relationships to Antwerp through his brother’s shop there, where the Portuguese marano Diego Duartewasone of the most prominent dealers in both art and diamonds. The likely nexus for these two strands is Diego’s son Gaspar Duarte, the high-profileartcollector and diamond merchant of Amsterdam. How Spinoza came to hold a 500 guilder bill from a Duarte jewel-trader, likely as some form of payment, is obsured, but these circles are tighter than one might assume at first glance. At least circumstantially, Van den Enden’s brother must have known Diego Duarte as Franciscus likely knew Gaspar Duarte in Amsterdam, and Van den Enden’s student ends up holding the bill of jewel merchant Manuel Duarte.

Addendum, August 10: There is one final hint that maybe gem-dealing had been in the family business for some time. In December 1650 Michael Spinoza was appointed administrator of the Synagogue Pawnshop-Loan Office (Gullan-Whur adds that there was a note in the Book of Agreements: “That it may be to his benefit!”). Because gems must have formed some substantial part of the deposited, some aspect of gem dealing and associations with other gem dealers, would seem a natural conclusion. The early date of this appointment would place such dealings as lasting to the Spinoza family. 

Addendum, August 27: Spinoza Sr. had a record of dealings with a substanial diamond dealer as early as 1641, Lopo Ramires. As Jonathan Israel describes:

Lopo Ramires (David Curiel), a leading Dutch Sephardi merchant of the first half of the century, regularly remitted sugar, diamonds, dye-woods and spices from Portugal to his brother who lived during the Truce years at Florence and who shipped Italian silks and red coral to Lisbon. The red coral was for re-export on the Portuguese East India galleons to Goa where it was exchanged for diamonds, both Lopo and his brother being major diamond dealers as well as general merchants (Empires and Entrepots, 423).

Addendum, September 3: Spinoza’s possible sugar relations are slightly made more likely in that in 1659 that he was noted to be frequently in the company of the tobacco merchant Pacheco. Tobacco and sugar as commodities went hand and hand.

[Additional discussion of related ideas on Spinoza’s dream here: Spinoza and the Caliban Question ]

[Speculation as to the diamond trade and Spinoza’s lens polishing: Spinoza and the Caliban Question ; A Possible Influence of Diamond Polishing on Assited Lens-grinding ]

An origin of Spinoza’s “cones of rays” explanation, Letter 40

[addendum: in addition to these thoughts, the influence of a more recent source, James Gregory’s Optica Promota (1663) has to be considered]

Kepler and How Spinoza Viewed the Eye and Light

As a point of reference it is important to locate the origin of Spinoza’s phrase “cones of rays” found in his letter 40, since implicit in this phrase is likely the conception of light and refraction which would help us make sense of his objection to Descartes. This phrase has a history of what seems a bit of interpretive confusion, for instance, that expressed by Alan Gabbey in his Cambridge Companion to Spinoza article, “Spinoza’s Natural Science and Methodology”. Here professor Gabbey quotes the phrase as if it embodies the locus of Spinoza’s befuddlement:

Spinoza explained that light rays from a relatively distant object are in fact only approximently parallel, since they arrive as “cones of rays” from different points on the object. Yet he maintains the same property of the circle in the case of ray cones, apparently unaware of the importance of the “[other] figures” [the famous “Ovals of Descartes”] that Descartes had constructed in Book 2 of La Géométrie to provide a general solution to the problem of spherical aberration [Ep 40].

I have already pointed out that Spinoza indeed was not “unaware” of the “importance” of Descartes’ figures (since he was intimate with the debate over that importance), and that part of Gabbey’s difficulty may stem from a weakness in translation, or not taking into account Spinoza’s familiarity with Hudde’s Specilla circularia: here. Spinoza, all the same, is constructing an argument that seems to shift parameters. In Letter 39 he speaks of the capacity of spherical lenses to focus parallel rays to an (approximate) point of focus opposite, along an infinity of axes, and now he tells Jelles that this capacity is to be understood not for parallel rays, but for “cones of rays”, which is more accurate to what is actually occurring. Where does Spinoza get his conception of “cones of rays”?

I believe it is found in Kepler’s Paralipomena to  Witelo (1604), a work I am beginning to suspect holds some of Spinoza’s resistance to Descartes. Descartes called Kepler his “first teacher” in optics, so when there is a divergence between the two, Kepler and Descartes, one may perhaps look to Kepler as a source for other resistance to Descartes’ conclusions. (It is a mistake to assume that solely in terms of temporal advancement, all of Descartes deviations from Kepler are corrections, for in some ways Kepler held views antecedent to our better conceptions on the nature of light.) In letters 39 and 40 Spinoza is critiquing Descartes explanation of how image size is produced in telescopes, and he finds in Descartes’ explanation some delinquencies which give undue favor to the hyperbola. Where Spinoza likely draws his conception of “cones of rays” is where Kepler is discussing the manner in which images are formed in the human eye:

Now in order to approach closer to the way this picturing happens, and to prepare myself gradually for the demonstration, I say that this picturing consists of as many pairs of cones as there are points in an object seen, the pairs always being on the same base, the breadth of the crystalline humor, or making use of a small part of it, so that one of the cones is set up with its vertex at the point seen and its base at the crystalline (though it is altered somewhat by refraction in entering the cornea), the other, with the base at the crystalline, common with the former, the vertex at some point of the picture, reaches to the surface of the retina, this too undergoing refraction in departing from the crystalline. And all the outside cones come together at the opening of the uvea [pupil], at which space the intersection of the cones takes place, and right becomes left..

…[now speaking of a single cone of those cones of rays] Thus those rays which previously were spreading out in their progress through the air, are gathered together now that they have encountered in to the cornea, so much so that any great circle described by those rays upon the cornea, which in their decent touch the edges of the opening is wider than the circle of the opening of the uvea; however, these rays, all the way to the opening of the uvea, are so strongly gathered together through such a small depth of the aqueous humor, that now the edges of that opening are trimmed of by the extremes, and by that decent they have made illuminous a portion on the surface of the crystalline humor, if indeed they all have first arisen at a point at a certain and proportionate distance (which is pecular to each eye, and not just the same for all), they fall approximately perpendicularly, because of the similar convexity of the cornea and the crystalline humor. (trans. Donahoe, 170)  

Included in this reference is also the obvious fact that for an object to be seen, light from all its points must be gathered. It is part of Kepler’s picture:

Spinoza writes: “…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.”

Because Spinoza is arguing that the hyperbolic lens – designed to receive rays solely parallel to its one axis – is insufficient for the variety of angles at which light arrives, the question of parallel or coned rays does not seem germane to his argument. His emphasis in the original description seems meant to be in terms of axes, assuming a “mechanical point” of focus definition. (Whether it is ultimately germane to contemporary telescope construction is another question.)

It must be noted, though here is both a most significant implication of the cone of light having a spherical (wave?)front, something ungrasped by Descartes but captured later by Huygens, in the text that follows as Kepler closely describes this action of cones of rays in the eye, he emphasizes the “hyperbolic posterior surface of the crystalline” (171), possibly disturbing the cohesion of Spinoza’s purely spherical ideal of light refraction. If indeed Spinoza is taking Kepler’s description as his source, this gives us to consider how Spinoza might mean the inexactness of the construction of the eye (letter 40). In what way can the eye be considered imperfect, and is there a Kepler source for this notion?

Spinoza writes: “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.”

There is an antecedent to this in Keplers’ description of the action of rays as they come from cones at angles oblique to the axis of the cornea:

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 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

…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). 

Here Kepler seems to be making the exact same point as Spinoza, with an additional hint towards the necessity of the oblique in Spinoza’s concern. The construction of the eye, in so far as its lenses are aspherical, it is retarded its capacity to handle the focus of cones of rays oblique to its single axis. This first calls our attention to the limits of human vision (in individuals and in plan), and then suggests that Spinoza’s point is one of practical application in terms of lenses: that in aiding human vision and constructing telescopes, the symmetry of spherical lenses is preferred for magnification, handling a greater variety of angles of incidence through its infinity of axes.

This does not of course establish the veracity of Spinoza’s argument, but in locating a likely origin for Spinoza’s conception, we at least place Spinoza’s argument within the context of a larger view, to be weighed with all other anti-hyperbolic (Cartesian) positions  of his day (Hudde, Huygens). As I have said, it is my sense that Spinoza derives more than this from Kepler’s account of light. More posts to follow.

A Conflation of Spinoza Diagrams

How Spinoza Thought of the Eye, the Lens and The Modes

Perhaps this is an irresponsible and trite comparison, but sometimes the mind indeed works visually, even in authors as exacting and deductive as Spinoza attempts to be. It is striking that Spinoza uses two very similar diagrams to illustrate on the hand, the powers of spherical lenses to most ideally focus rays across an infinity of axes, (the manifestation of which is subject to the properties of real lenses):

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Letter 39 to Jelles, March 3rd 1667
Benedicti de Spinoza opera quotquot reperta sunt quotquot reperta sunt By Benedictus de Spinoza, Baruch Spinoza, Johannes van Vloten, Jan Pieter Nicolaas Land

Depicted above are the hypothetical intersection of rays, in two sets taken to be parallel, as they arrive at the surface of a spherical lens. Such rays are taken to be then focused at the back of the circumfrance, as the would be at the back of the eye, or as part of the refractions of a lens.

 

In this diagram, Spinoza illustrates how each contingently expressive mode – what is usually taken to have come into existence and then will pass away – are implied by, that is caused by as immanent to, the Idea of an infinity of points that make up a circle. In this way, the rectangles that are immanent to a circle’s circumfence are by analogy seen to be dependent upon that circle. The rectangles come and go, the circle remains eternal. As explained in Ethics IIp8s:

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.

There is the simple coincidence of using a circle to diagram both physical effects, and metaphysical effects (which for Spinoza are of course commensurate). But if one allows a conflation, one that may have occurred within Spinoza’s thinking, in the first we have the effects what occur within the eye, as it interacts with events outside of it, and in the second, we have the effects (modes) as expressed immanent to the circle that contains them.

Because Adequate Ideas are understood by Spinoza to be Ideas uncaused by something external to them, I don’t think it is too big of a leap to understand that when Spinoza is diagramming the effects of light with the eye (and for a lens, post-angle of incidence), he is thinkingof the second diagram. It is perhaps for this reason that Spinoza is not obsessed with the crystality of vision that occupied Descartes in his quest for the hyperbolic lens. The sharpness of an image is but a part played in an assemblage of knowledge. However clearly one’s eye, or lenses work, this simply is not clear thinking. Of course Descartes understood this as well, but there is something to how Descartes and Spinoza each responded to spherical aberration which reveals a difference of emphasis in the very project of mental and physical liberation. I believe in this co-incidence of diagrams, a profound conflation is being accomplished in Spinoza’s process of thinking.

I see hear as well an interesting graphic subsumption of the scattering of rays that occur with spherical aberration, as in being focused they tend about a “mechanical point” [Johannes Hudde]. Much as rays are never entirely focused to a mathematical point (even with real, hyperbolic lenses), so too we never possess wholly adequate ideas. The focus rays as seen in the first diagram (again, if we allow an analogical thought), appear to enact indices found in the second diagram. Is Spinoza at some level conceiving of rays of focus as being parallel to the adequacy of ideas? And is Spinoza’s theoretical acceptance of spherical aberration [a la Hudde] a product of his acceptance of the fundamentally inadequate nature of ideas we hold? Is his mechanical project of lens focusing analogous to a mechanical – that is, pragmatic, rational and crafted – construction of human freedom? These are large and obscured questions.

This certainly does not make up an argument either for Spinoza’s position, or for an interpretation of Spinoza’s position. It is really more an intuition into the kinds of thought processes Spinoza may have been engaged in, in part elicited by the diagrams he used to make things clear. Meant is a direction for future analysis.

The Buttle Principle

A Beetle in the gears of knowing and the notion of the Press of the Mind

Wittgenstein has a beautiful and striking analogy which he folds into his (No) Private Language argument. He compares any fact checking one would do in using a so called private language, to attempting to check for an error by buying several copies of the same edition of a paper. Such a process is cursed by its reflexivity. This analogy is specific to the example of an imaginary table of terms who’s check is only in the imagination:

If the mental image [recollection] of the time-table [for the departure of trains] could not itself be tested for correctness, how could it confirm the correctness of the first memory? (As if someone were to buy several copies of the morning paper to assure himself that what is said were true) (PI, section 265).

There are a few problems if we attempt to take this analogy as a knockdown argument for why one cannot have a recursive sense of rule-following and justification. Wittgenstein wants us to know that “justification consists in appealing to something independent”. You might have the feeling that you have remembered a train-time table right, but you cannot justify this feeling unless you appeal to some other, independent criteria (in which for him independent does not consist in another moment of recollection or thought process). Put another way, one can believe that one is following a rule, but one doesn’t know if one actually is until one is checked by an independent process. What seems to be missing from this appeal to outside criteria is that our memories, and our use of them, are not at all like a bunch of copies from the very same press (ones in which, if their are errors, they will simply be reproduced endlessly as the same). If there is a “press” of the mind, it is much more like one which is in print all the time, and one can watch the results of taking one “edition” as correct, and make provisional adjustments if a set of beliefs fail. That is, if one follows only one’s memory, and one misses the train, one might question if there were a better way of finding out when the train would be there. But checking by glancing again at the physical time-table may help with one’s accuracy, but not categorically so. For instance, how one read that time-table anew might not jibe with one’s strong recollection. One might make sense of this by reasoning that one saw it wrong this time, or that the time-table must be out of date, or even some rather odd conspiracies of the world, and one might choose to simply trust one’s memory all the same. There is no easy, conceptual way out to what is “independent”.

[One could say that checking the truth of a report by looking at the multiple products of a process, only occurs when the truth is of an nature where doubt is necessarily cast, where it is not readily believable.]

This is related to what can be called The Buttle Principle [I give immense credit to my wife here for pointing out the concept, and naming it]. Terry Gilliam’s movie Brazil  opens with the bureaucratically automated typing out of the name of a man to be arrested. A beetle body lands on the typewriting mechanism and changes the printing process of a name of a terrorist from “Tuttle” to “Buttle”; in totalitarian justice the wrong man, an innocent shoe repairman, is arrested, tortured and executed. It becomes the “accident” which drives plot of the entire wistful and humorous critique of modern society. But now, given the metaphor of a printing press and editions of knowledge text, just where would the change from T to B lie? For instance, if you did as Wittgenstein parodies, after seeing in the small print of a newspaper that you, Buttle, were wanted for murder, it might do very well to check several copies of the newspaper to make sure it is so. Did a beetle simply fall into the mechanism at just that one moment of pressing the very copy one has in one’s hands? Suddenly (as is often the case with many of Wittgenstein’s otherwise convincing analogies) what sounds so ridiculous at first, when examined closely in real-world possibilities, is less so. 

One might ask, would the (mis)typing of the name “Buttle” in the movie Brazil  be part of the same press of an edition of knowledge? More exactly, are the given processes by which the name “Tuttle” had been inscribed in the system (an officer’s report, an original secretary’s typing), and the one where the name “Buttle ” is inscribed, to be understood as distinct or homogeneous? And in coordintion, would recalling again and again a train time-table in your mind really be simply running off more copies of the same edition of a newspaper? Would there be any sense of checking one aspect against another (what if you recall now that the time you thought that the train arrived was actually the date of your anniversary)? How much would all this self-referential conception of knowing be approaching what Wittgenstein called “a wheel that can be turned though nothing else moves with it” (section 271)? And when such a wheel turns, what is turning with it?

What is of interest here may be that the mechanism of inscription from the film is indeed tracking an alphabetized, rule-following procedure when it is “interrupted” by the fallen beetle. The name Buttle appears in a long string of T’s. To quote from the screenplay. :

The TECHNICIAN gets up and balances a chair on top of his
desk. He climbs up onto it attempting to swat the BEETLE
still buzzing about the room just out of reach. Beneath
him an automatic type-writing machine rattles away
compiling a typed list of names under the heading
“Information Retrieval, Subjects For Detention &
Interview”. The machine is being fed from a spool of paper
which is being rhythmically chopped by an automatic
guillotine which neatly leaves each name on a separate
sheet, with the title above each name, each sheet
following its predecessor into a holding basket. In CLOSE-
UP we see the names on the sheets of paper building up in
the holding basket: TONSTED, Simon … TOPPER, Martin F.
… TROLLOPE, Benjamin G. … TURB, William K. … TURNER,
John D. … Every name begins with T.

INTERVIEWER
Do you think that the government is
winning the battle against
terrorists?

HELPMANN
On yes. Our morale is much higher
than theirs, we’re fielding all their
strokes, running a lot of them out,
and pretty consistently knocking them
for six. I’d say they’re nearly out
of the game.

The TECHNICIAN is tottering on one leg on the chair on the
desk as he strains to swat the BEETLE. Swish, swash, oops,
WHAP! Gottcha!!

INTERVIEWER
But the bombing campaign is now in
its thirteenth year …

HELPMANN
Beginner’s luck.

The BEETLE’s career comes to a halt … squashed flat on
the brilliantly clean ceiling … or has it? As the
TECHNICIAN clambers down from the rickety heights, the
BEETLE’s carcass comes unstuck from the ceiling and drops
silently into the typewriting machine which hiccoughs,
hesitates and then types the letter “B” and hesitates and
then continues so that the next name is BUTTLE, Archibald.

The TECHNICIAN fails to notice this and the machine
continues smoothly TUTWOOD, Thomas T. … TUZCZLOW,
Peter…

What I suggest is that there indeed is a component of justification (though justification is not reduced to it) which indeed is like checking several copies of the same edition, a self-editing proof, whereby one could internally look at the inscription stages of “Tuttle” and the context of one copy of “Buttle” and say, there is an error there.  I believe that this is the case because even in intersubjective conditions the appeal to something “independent” only ends up being the causal nature of the world. That is, a group of people sharing criteria still have no “independent” appeal for their means of justification, other than the causal results of following them. (What is the ultimately “independent” criteria which is available to the totalitarian system of justice, concerning Mr. Buttle’s innocence?) As a consequence, part of the mechanism for justification is also the internal sense of cohesion betweencriteria events, the special rational character with which beliefs stick together and support each other. Buttle just should not be in the T’s.

In a certain sense, me checking whether I remembered a train schedule right by turning over my own recollections, is like the totalitarian beareucracy in Brazil checking over whether they arrested the right man. If indeed they did pay attention to the internal discrepancy of texts, the Tuttle to Buttle shift, and be self-critical to it, they might have an additional explanation for the constitutive pleas of innocence by Mr. Buttle. The wheel that so turns is always connected to the world, and it is experienced as having its turnings caused by events in the world. The recursivity of an internal cohesion, though not sufficient for intersubjective justification, plays as a grounding for its possibility. The “independence” is always relative to a dependence, which in the end is causal. And coherent self-reference is always open to self- (and therefore other) critique.

This calls to mind another analogy of the printing press, one used by Spinoza to explain Descartes “proof” of God.

For example, if someone were to ask through what cause a certain determinate body is set in motion, we could answer that it is determined to such motion by another body, and this again by another, and so on to infinity. We could reply in this way, I say, because the question is only about motion, and by continuing to posit another body we assign sufficient and eternal cause to this motion. But if I see a book containing excellent thoughts and beautifully written in the hands of a common man and I ask him whence he has such a book, and he replies that he has copied it from another book belonging to another common man who could also write beautifully, and so on to infinity, he does not satisfy me. For I am asking him not only about the form and the arrangement of the letters, which which alone his answer is concerned, but also about the thoughts and meaning expressed in their arrangement, and this he does not answer by his progression to infinity.

(Letter to Jelles (40), March 25 1667

I hope that you notice the comparison in printing press analogies. We have in Wittgenstein the “absurd” notion that if we only referred to our own sense impressions and our beliefs about them, we would be like someone who is looking again and again at multiple copies of the same edition of a newspaper. And we have in Spinoza the notion that if we simply refer to the recursivity of actions of the proliferation of copies of a book (rule-followings?), we really have gotten nowhere in answering the larger question for an “independent” (conceptually distinct) cause of their production. The causation, either in the case of a self-referential series of experiences which attest to facts of the world, or a proliferation of rule-following expressions taken as shared criteria which produces formal justification, is “the world” experienced as causing both our experiences and our beliefs, and the experiences and beliefs of others. And writing, as an inscription, is understood to be an affective process. That is, both our experiences and our beliefs cause and condition the inscription process itself. Part of having beliefs is understanding that self-regulation and critique takes in account The Buttle Principle. That is, our experiences of a fact may indeed be the result of non-intentional error (the “beetle” in the system). As such, their cause can lie within physical causation, and ignored. All the same, the The Buttle Principle also allows that errors can be re-inscribed back into the intentionality of the system (Buttle must be guilty if the system finds him so, the train must be late since my memory never fails me). In this way cohesion can, as an autonomic sense of “right”, overide any Intersubjective Critique or Reality Principle that might serve as a correction. This is part of the ballast that subjectivity provides to social forms of knowing.

For Spinoza, if one could encapsulate, this causation ultimately resides in an immanentive expression of a totality which is taken to be vastly causal, which from our perspective is bootstrapped largely through affective (Joy rather than Sadness) and imaginary (picturing what makes us more powerful) means. For Wittgenstein it is much more a case of an immanence of organization which bubbles up, games stacked on games, as criteria become shared and communicated, part of this dependent/independent differential which helps create the “public” nature of language. In the middle, I believe, the two meet.

It should also be of a happy note that the beetle of Terry Gilliam’s film conflates the Ungeziefer of Kafka’s Gregor’s subjectivity, and Wittgenstein’s own Beetle in a box. One could say, two isometric reflections of the same phenomena.