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Transcript
Part I
The Complete English Text of the Optical portions of
Spinoza’s Letters 39 and 40:
Spinoza’s Letter 39
To the most humane and sagacious Jarig Jelles, from B.d.S.
[The Original, written in Dutch, is lost. It may be the text reproduced in the Dutch
edition of the O.P. The Latin is a translation.]
Most humane Sir,
Various obstacles have hindered me from replying any sooner to your letter. I
have looked at and read over what you noted regarding the Dioptica of
Descartes. 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, 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, 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.
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.
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 something
that could be affirmed of no other figure, although the hyperbola and the ellipse
have infinite diameters. 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. 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. If you desire to see the proof of this I am ready to send it
to you whenever you wish.
Voorburg, 3 March 1667
Spinoza’s 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. But they are considered to be so because the object is so far from us that
the aperature of the telescope, in comparison with its distance, can be considered
as no more than a point. Moreover, it is certain that, in order to see an entire
object, we need not only rays coming from a single point but also all the other
rays that come from all the other points. And therefore it is also necessary that,
on passing through the glass, they should come together in as many other foci.
And although the eye is not so exactly constructed that all the rays coming from
different points of an object come together in just so many
foci at the back of the
eye, yet it is certain that the figures that can bring this about
are to be preferred above all others. Now since a definite
segment of a circle can bring it about that all the rays
coming from one point are (using the language of
Mechanics) brought together at another point on its
diameter, it will also bring together all the other rays which
come from other points of the object, at so many other
points. For from any point on an object a line can be drawn
passing through the center of a circle, although for that
purpose the aperature 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.
What I here say of the circle cannot be said of the ellipse or
the hyperbola, and far less of other more complex figures, since from one single
of the object only one line can be drawn passing through both the foci. This what
I intended to say in my first letter regarding this matter.
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. So, after
sending you my cordial greetings, it remains only for me to say that I am, etc.
Voorburg, 25 March 1667
Part II
Deciphering Spinoza’s 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.
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-plano lenses). 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 droplenses 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”.
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 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:
…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 alalogusto our Grass, Shrubs, and Trees; and most of
these incompassing Hills may be covered withsothinavegetable 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
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 overenthused 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
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.
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, my bolding).
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 (my bolding and color emphasis)
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
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
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 convexplano 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, 318319).
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
“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
[Above, Spinoza’s diagram from letter 40, to the leff ot 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.