Friday, November 25, 2005

Charlatan's Who Use Graviton?

Are Gravity people Charlatan's?:)I certainly don't think so.:)

well this is a good perspective with which one could move forward and explain it for us lay people here? :)

Lubos Motl:
The graviton is, on the contrary, an example of a correct derivation from semiclassical gravity - a legitimate approximate unification of GR and QM. Its existence follows from the theories we have, even given some degree of ignorance of quantum gravity at higher energies, and at the semiclassical level, it is absolutely analogous to the photon.

The only difference is the value of the spin, the geometric interpretation of the graviton, and ultraviolet divergences from loops.

I might have had wrong ideas here about what the graviton as a force carrier "proposed?" To exemplified what gravity a further extension of the theory of general relativity? Lubos sets it straight then on such joinings.

This is the crucial difference between the dark energy and modified gravity hypothesis, since, by the former, no observable deviation is predicted at short distances," Dvali says. "Virtual gravitons exploit every possible route between the objects, and the leakage opens up a huge number of multidimensional detours, which bring about a change in the law of gravity."

Dvali adds that the impact of modified gravity is able to be tested by experiments other than the large distance cosmological observations. One example is the Lunar Laser Ranging experiment that monitors the lunar orbit with an extraordinary precision by shooting the lasers to the moon and detecting the reflected beam. The beam is reflected by retro-reflecting mirrors originally placed on the lunar surface by the astronauts of the Apollo 11 mission.

I myself might find it nice to have the origins of how this graviton came about. How one might be mistaken to have seen the bulk as a teaming with them(blackholes?), and such congregations telling, about places stronger then, while others are weaker.

How telling is the photon as it travels through these spaces? What was the initial trigger that set things free as Hawking radiation? Some analogies there to consider as well:)

So it would be nice then if one could find analogies that would sit well and sink deep. You know that the general public likes to think easy, and not finding relevant all the dressings of mathematical explanations. Or do they?

Is it wrong to move so far ahead theoretically to be called a charlatan, by those who recognized the limitiations of experimentally proving it?


  1. Dear Plato,

    There is quite a lot going on in the spacetime fabric, is there not?

    Gauge bosons for electromagnetic forces (photons) and gravity (gravitons), Higgs field bosons for mass, not to mention Einstein's warped spacetime.

    It is beginning to look rather cluttered, like the mechanical ether of Maxwell, is it not?

    On the subject of mathematical physics charlatans, think about the electromagnetism of Maxwell.

    One source is A.F. Chalmers’ article, ‘Maxwell and the Displacement Current’ (Physics Education, vol. 10, 1975, pp. 45-9). Chalmers states that Orwell’s novel 1984 helps to illustrate how the tale was fabricated:

    ‘… history was constantly rewritten in such a way that it invariably appeared consistent with the reigning ideology.’

    Maxwell tried to fix his original calculation deliberately in order to obtain the anticipated value for the speed of light, proven by Part 3 of his paper, On Physical Lines of Force (January 1862), as Chalmers explains:

    ‘Maxwell’s derivation contains an error, due to a faulty application of elasticity theory. If this error is corrected, we find that Maxwell’s model in fact yields a velocity of propagation in the electromagnetic medium which is a factor of root 2 smaller than the velocity of light.’

    It took three years for Maxwell to finally force-fit his ‘displacement current’ theory to take the form which allows it to give the already-known speed of light without the 41% error. Chalmers noted: ‘the change was not explicitly acknowledged by Maxwell.’

    Weber in 1856, not Maxwell in 1865, was the first to notice that, by dimensional analysis (which Maxwell popularised), 1/(square root of product of magnetic force permeability and electric force permittivity) = light speed.

    Maxwell after a lot of failures (like Keplers trial-and-error road to planetary laws) ended up with a cyclical light model in which a changing electric field creates a magnetic field, which creates an electric field, and so on. Sadly, his picture of a light ray in Article 791 of Maxwell's 1873 Treatise on Electricity and Magnetism, showing in-phase electric and magnetic fields at right angles to one another, has been accused of causing confusion and of being incompatible with his light-wave theory (the illustration is still widely used today!).

    Perhaps the solution to unified force theory requires a re-examination of exactly what Maxwell's equations say.

    I'm well aware Maxwell's equations work, and where they work. What I'm saying is that at some stage physicists will have to start asking heretical questions about the model, or lack of a real model, for Maxwell's equations.

    Best wishes,

  2. Hi Nigel,

    It is beginning to look rather cluttered, like the mechanical ether of Maxwell, is it not?

    To me there is a leavening of sorts in the culminative of vision.

    Geometrically undertanding how we look at Gaussian coordinates, it becomes amazing that such aspects of the field, could have ever been looked at in such a way to point directly to what was gained in understanding the metric? You agree?

    There is a historical passage that was lead through by the geometrics, that if followed revealed the greater understanding of what is unleashed in our views of the spacetime fabric.

    It lead from insights of Gauss and Maxwell, to be incorporated in the course and direction of the keeping things to branes, while the spacetime fabric has been extended to the bulk?

    Riemann curvatures were a positive aspect that grew from this. But there are negative aspects of this thinking that are further revealled cosmologically. If such dynamical processes are understood, what is revealed in such cosmological events?

    That such happenings within the cosmo is a very dynamcial process using such geometries?

    Gravitonic proportions(dimensions) seem very real to me, even on a socialogical scale, while we dealt with the origins of the mathematical explanations and experimentally proven has yet to be done. :)

    Click on Dampt-dwellers name, and follow it to quantum gravity.

    You'll undertand then that such efforts by myself are not the roads that a charlatoan would embrace, although I am indeed biased from consumption of the models.

  3. Dear Plato,

    I don't think the mainstream M-theory of Witten is deliberately wrong, but most charlatans think they are right.

    You can tell Witten is a charlatan because he said:

    ‘String theory has the remarkable property of predicting gravity’. (Edward Witten in the April 1996 issue of Physics Today, repudiated by Roger Penrose on page 896 of his book Road to Reality, 2004: ‘in addition to the dimensionality issue, the string theory approach is (so far, in almost all respects) restricted to being merely a perturbation theory’.)

    Witten is misleading people accidentally by being over-enthusiastic with no testable evidence. String theory does not predict for the strength constant of gravity, G!

    This means that my work on gravitational force mechanism which does predict gravity correctly is suppressed, as is Lunsford's work which does away with the cosmological constant like my model.

    Whereas my work is extremely simple, dealing with mechanisms, Lunsford has gone for the big problem.

    Danny Ross Lunsford’s major paper, published in Int. J. Theor. Phys., v 43 (2004), No. 1, pp.161-177, was submitted to but was removed from by censorship apparently since it investigated a 6-dimensional spacetime which again is not exactly worshipping Witten’s 10/11 dimensional M-theory. It is however on the CERN document server at , and it shows the errors in the historical attempts by Kaluza, Pauli, Klein, Einstein, Mayer, Eddington and Weyl. It proceeds to the correct unification of general relativity and Maxwell’s equations, finding 4-d spacetime inadequate:

    ‘… We see now that we are in trouble in 4-d. The first three [dimensions] will lead to 4th order differential equations in the metric. Even if these may be differentially reduced to match up with gravitation as we know it, we cannot be satisfied with such a process, and in all likelihood there is a large excess of unphysical solutions at hand. … Only first in six dimensions can we form simple rational invariants that lead to a sensible variational principle. The volume factor now has weight 3, so the possible scalars are weight -3, and we have the possibilities [equations]. In contrast to the situation in 4-d, all of these will lead to second order equations for the g, and all are irreducible - no arbitrary factors will appear in the variation principle. We pick the first one. The others are unsuitable … It is remarkable that without ever introducing electrons, we have recovered the essential elements of electrodynamics, justifying Einstein’s famous statement …’

    D.R. Lunsford shows that 6 dimensions in SO(3,3) should replace the Kaluza-Klein 5-dimensional spacetime, unifying GR and electromagnetism:

    ‘One striking feature of these equations ... is the absent gravitational constant - in fact the ratio of scalars in front of the energy tensor plays that role. This explains the odd role of G in general relativity and its scaling behavior. The ratio has conformal weight 1 and so G has a natural dimensionfulness that prevents it from being a proper coupling constant - so this theory explains why ordinary general relativity, even in the linear approximation and the quantum theory built on it, cannot be regularized.’

    It is sad that this work is suppressed without being properly checked.

    After I pushed Lubos Motl, he declared Lunsford was correct in 6-d unification of electromagnetism and general relativity. Motl refers to:

    "Lunsford's ingenious work."


    Best wishes,

    Feynman gravity home page

  4. Note also that Peter Woit has a great deal of work dating back to 1988, as does Tony Smith.

    Tony Smith’s CERN document server paper, EXT-2004-031, uses the Lie algebra E6 to avoid 1-1 boson-fermion supersymmetry:‘As usually formulated string theory works in 26 dimensions, but deals only with bosons … Superstring theory as usually formulated introduces fermions through a 1-1 supersymmetry between fermions and bosons, resulting in a reduction of spacetime dimensions from 26 to 10. The purpose of this paper is to construct … using the structure of E6 to build a string theory without 1-1 supersymmetry that nevertheless describes gravity and the Standard Model…’

    Peter Woit goes in for a completely non-string approach based on building from quantum field theory of spinors,

    Woit has some sensible ideas on how to proceed with the Standard Model: ‘Supersymmetric quantum mechanics, spinors and the standard model’, Nuclear Physics, v. B303 (1988), pp. 329-42; ‘Topological quantum theories and representation theory’, Differential Geometric Methods in Theoretical Physics: Physics and Geometry, Proceedings of NATO Advanced Research Workshop, Ling-Lie Chau and Werner Nahm, Eds., Plenum Press, 1990, pp. 533-45:

    ‘… [it] should be defined over a Euclidean signature four dimensional space since even the simplest free quantum field theory path integral is ill-defined in a Minkowski signature. If one chooses a complex structure at each point in space-time, one picks out a U(2) [is a proper subset of] SO(4) (perhaps better thought of as a U(2) [is a proper subset of] Spin^c (4)) and … it is argued that one can consistently think of this as an internal symmetry. Now recall our construction of the spin representation for Spin(2n) as A *(C^n) applied to a ‘vacuum’ vector.

    ‘Under U(2), the spin representation has the quantum numbers of a standard model generation of leptons… A generation of quarks has the same transformation properties except that one has to take the ‘vacuum’ vector to transform under the U(1) with charge 4/3, which is the charge that makes the overall average U(1) charge of a generation of leptons and quarks to be zero. The above comments are … just meant to indicate how the most basic geometry of spinors and Clifford algebras in low dimensions is rich enough to encompass the standard model and seems to be naturally reflected in the electro-weak symmetry properties of Standard Model particles…

    ‘For the last eighteen years particle theory has been dominated by a single approach to the unification of the Standard Model interactions and quantum gravity. This line of thought has hardened into a new orthodoxy that postulates an unknown fundamental supersymmetric theory involving strings and other degrees of freedom with characteristic scale around the Planck length. …It is a striking fact that there is absolutely no evidence whatsoever for this complex and unattractive conjectural theory. There is not even a serious proposal for what the dynamics of the fundamental ‘M-theory’ is supposed to be or any reason at all to believe that its dynamics would produce a vacuum state with the desired properties. The sole argument generally given to justify this picture of the world is that perturbative string theories have a massless spin two mode and thus could provide an explanation of gravity, if one ever managed to find an underlying theory for which perturbative string theory is the perturbative expansion.’ – Dr P. Woit, Quantum Field Theory and Representation Theory: A Sketch (2002),

    On the topic of gravity and the big bang, Woit won't publish anything under his own name yet, but he has 'allowed' postings saying that the fourth dimension is expanding.

    All this means is that time is increasing, which is true because the universe is aging as it expands:


    However this insight only produces predictions with my model, which uses the expansion rate as a function of time past to give gravitational mechanism;

    We are seeing the past with distance in the big bang! Gravity consists of gauge boson radiation at light speed, coming from the past just like light itself. Hence, there is apparent acceleration (variation in speed from 0 toward c with times past of 15,000,000,000 years toward 0), so F=ma gives outward big bang force. The 3rd law says there’s equal inward (Higgs field/gauge boson) force, which gets shielded by mass, proving gravity to within 1.65%

    Best wishes

  5. Hi Plato,

    I've just seen Penrose's talk to the Perimeter Institute (linked from Woit's blog).

    Penrose says that Witten has 'changed tack' within the 'last year' is now talking about string-twistor theory. This doesn't really impress me, as everything Witten touches seems to be nonsense.

    This seems to be gluon interactions, which seems to be a fiddle, because there are logically 9 gluons (from 3x3 colour interactions), but you can only have 8 to make QCD work.

    Also, in a neutron or proton you are going to have two identical quarks qhich differ only in colour charge. The whole concept of colour charge originates from the exclusion principle of Pauli, and I think it is a but like epicycles.

    The first thing to do is understand the causal mechanism behind the exclusion principle, the spin thing, which is a magnetism force effect. The electrons and quarks are small normal dipole magnets and align in pairs, cancelling each other's magnetism out, just as when you have a pile of magnets. They don't align themselves naturally into one super magnet, but into pairs pointing opposite ways, because the entropy increases that way. Electrons in an atom feel each other's magnetism as well as the electric force. In fact the polar (radial) magnetic field from the electron core won't be shielded by the polarised vacuum, so it will produce greater magnetic force effects than the electric field from the core which is reduced by a factor of 137 by the polarised vacuum shield.

  6. sorry for not being more responsive, as I work through the nights, and sleep is short, through the days.

    I get back to this when I can.

    IN my post above the entries, I relate logic to the math. It is by following such logic and looking for methods to discern natures underlying reality, that I am saying, math is created?

    If one form of math can be dismissed, can another math be dismissed so easily?

    Peter, Clifford and Lubos all have respective maths to form the basis of their views.

    I do not believe in such conspiracies that would limited intelligent math of nature from being allowed to be expressed from each other?

    I do believe they are starting form a consensus, that Lee Smolin would have like seen, where each of them is taming the idealizations as I see them around string/M theory to a respectable level. This is the responsibility maybe Peter was looking for?

    I do not know how many of my posts he has deleted. He has that right.

    It does not offend me to much as it is his blog to do what he wishes. I do find though, that if I inject a counter view to his fervor pitched that arises from various commentors then such a view as mine would be quickly dismissed for creating a disharmony? Imagine:)

    I would like to think it is to keep the topic in line and show that such comments are dealing with the thoughts he is sharing by blogging. If I fall off the rail then most certainly and I respect this, for my limitations on such developing skills of the math issues, has a long way to go. Before I could ever say that I could fit into such a field of endeavors that these gentlemen work in.

  7. Dear Plato,

    I'm sorry but you have access to, do you not? Other's don't.

    Views and opinions are of little value to Peter or science.

    What counts for science are defendable facts, of which there are none in M-theory. Am I wrong? :)

    Best wishes,

  8. I'm sorry but you have access to, do you not?

    No I do not.

    Views and opinions are of little value to Peter or science.

    Is this not what we are doing here?

    While mathematical forays are being developed into insights of science, how could you not form opinions around one's respective views? :)

    What counts for science are defendable facts

    Yes, of course.

    of which there are none in M-theory.

    If we were to limit and curtail any adventours into the theoretcial world, how could you have ever included, "Gravity derived from GR?"

    By simple logic, this cannot be done, if it works according to the views shared bythose who hold mathematical perspective that would disc mathematics of one form or another? You cannot escape the logic?:)

    Thining of Smolins position, working from special relativity, as I understand it. This might be far from comprehensible features that simple folk like me could espouse?

    John Baez:Gravitational Radiation is to gravity what light is to electromagnetism. It is produced when massive bodies accelerate. You can accelerate any body so as to produce such radiation, but due to the feeble strength of gravity, it is entirely undetectable except when produced by intense astrophysical sources such as supernovae, collisions of black holes, etc. These are quite far from us, typically, but they are so intense that they dwarf all possible laboratory sources of such radiation.

    Gravitational waves have a polarization pattern that causes objects to expand in one direction, while contracting in the perpendicular direction. That is, they have spin two. This is because gravity waves are fluctuations in the tensorial metric of space-time.

    Yet the safe distances measured would have included views on the calorimetric and glast determinations, as viable views of interactive features. I understand this, and the depth of these perceptions? How far can they go back?

    Yes let's keep it fun?:)

    So the work of the Wheeler's and protege's like Kip thornes would have been stopped? The ideas around LIGO built up, saved, until new holographical discs could be manufactured to increase calculations from enormous stats collected? A large interferometer?

    How could we if we followed the thinking of those who want to limit theoretical advancement if we are to ascertain at this time gravity waves do not exist? You see?

    So you grow with it? The WEbbers paving the way for introspective views about how we should test? Do we say that such advancement stop here while the roads to develping perspective brought the protege's out in force, to develope LIGO and LIsa, Gravity Probe "B" accordingly, from the shoulders of giants? :)


  9. Dear Plato,

    Thanks for replying this time.

    You say you personally want science to be the intellectual adventure playground, because that is fun. This is what Witten and others go along with.

    It drags physics down into the arts, where beauty is subjective and a whole group can believe something which turns out false.

    The crime is that the reality is censored out. I'm interested in facts, not speculations unless they are testable. String theory is a religion. I'm Catholic and have had as much religion as I need to last me for the rest of my life.

    In turning top-level physics into a religion, you corrupt the last stronghold of enlightenment. M-theory, you know deep down, is a crime. It leads to paranoia at top levels, suppressing 'alternatives'.

    I don't think dr peter woit, who deletes many other comments too, is blame for sticking to science in his own blog space. This is not an analogy to suppressing Lunsford, Smith, etc.

    Best wishes,

  10. Plato,

    I can't make sense of your comment:

    "If we were to limit and curtail any adventours into the theoretcial world, how could you have ever included, "Gravity derived from GR?"

    "By simple logic, this cannot be done, if it works according to the views shared bythose who hold mathematical perspective that would disc mathematics of one form or another? You cannot escape the logic?:)"

    It predicts gravity within 1.65%.

  11. Dear Plato,

    General relativity is not a string theory. Einstein built on facts. The contraction of volume and radius which introduces the special features of general relativity over Newtonian gravity, can be obtained several ways without speculation.

    Einstein was solving REAL WORLD problems of energy conservation in gravitational fields which newton had ignored. Newton also made errors in sound "theory", see How about understanding explosions before dealing with the big bang and gravity? Or is real physics too arcane?

    Best wishes,

  12. Another long night and very tired,

    If we were to limit and curtail any adventours into the theoretcial world, how could you have ever included, "Gravity derived from GR?"

    I thought mentioning webber and ligo, contractive and expansive generations would have been inevitable? Do gravity waves exist? What does the aluminum bar mean for detection?

    You say you personally want science to be the intellectual adventure playground, because that is fun. This is what Witten and others go along with.

    I hope you followed the link and found my explanation for what a older gentleman can do now having some time to devote to understanding the issues in relation to mathematics.

    What mathematics is Witten using and can you explain to me, the different forms of math that are being used in string/M theory?

    My next posting to blog will be based on the "foundations of mathematics." If I were to say that the basis of the math is logic, then what is it I am saying?

    In this capacity, it is always necessary for me to continue to probe the information in respect of the views these scientists have. So I know where they have left their tracks through their generalizations. These tracks are the basis of the maths used and how this language defines what they are doing.

    I tend to skip a lot of work by digging for these generalizations. I then look further for comparative views, that might help me to comprehend these issues better.

    In my world, art plays a decisive role. If these artists understand these issues well, then they can paint the limitations of where the math ends? Where the abstract can become visual and can be reproduced. You have to be artistic and creative in this sense to get to the edge of thoughts in these areas.

    I mentioned "cubist art" and one can see this effect when using blog search above for this information of this site.As well as Penrose's use of Escher. Or Salvadore Dali's foray into dimensional valuations of what God or religion might mean in a hyperdimensional sense?

    I regards to the geometry that is unfolding or the gravitational collapse. I mentioned Heisenberg for a reason in relation to plato.

    Isomorphic geometrical functions on a cosmological scale and orbitals? I have a post on this as well somewhere.

    I am not in a position to judge any who work on string/M theory becuase I do not have the capabilites from a math perspective, so I can not rule string/M theory out.

    I cannot judge Lundsford or Peter woit, for again, I do not have capacity to rule out the positions they have.

    The tone that is now being set is much more respectful one even though the positions of Peter and Lubos are 180 degrees.

    I had to know why this was so? So now the understanding of the foundation of math begins to play an important role. As to the basis and foundation, each mathematican might hold in regards to how they are percieving reality.

    What does mathematics mean to them? This might have some basis? I have to understand how in todays world different mathematcians view themselves, in relation to historical developement, that I am going through.

    In regards to sound, as an analogy.

    Bubble nucleation is an attempt to get a hold of the dynamics of that gravitational collapse. What does geometrodynamics mean here in this site of Plato's? What experiments do I talk about in relation?

    I would let the Peter Woit, the Lubos Motl's defend themselves, and stop the pursuate for anti- evangelistic denunciations of string/M theory. There are people who have to explain why this is so, and give very good reasons. If this cannot be done in relation to the maths used, and why these maths are wrong, then the positon from which they are arguing, holds no ground.

    The steps in math are, and have been progressive?

  13. Dear Plato,

    Become an artist if you want art, become a priest if you want M-theory, but please don't insult your readers by telling them that a testable quantum field theory heuristic mechanism based entirely on facts, and which accurately other things like gravity, is a mere opinion.

    If speculations are allowed in science, the speculations of the handful of people in control of the journals will dominate.

    This is why M-theory, the insult to the layman, based on horseshit mathematics (with no connection to the real world, no mechanism, no predictions that are tested) is more important than the tested mechanism, more important than Lunsford. You do not praise 6 dimensions because you can't understand it, you say, although the steps are simple to follow.

    But you don't ignore M-theory, because that is orthodoxy today?

    "If you have got anything new, in substance or in method, and want to propagate it rapidly, you need not expect anything but hindrance from the old practitioner - even though he sat at the feet of Faraday... beetles could do that ... he is very disinclined to disturb his ancient prejudices. But only give him plenty of rope, and when the new views have become fashionably current, he may find it worth his while to adopt them, though, perhaps, in a somewhat sneaking manner, not unmixed with bluster, and make believe he knew all about it when he was a little boy!" - self-taught mathematician Oliver Heaviside (left school at 13), "Electromagnetic Theory Vol. 1", p337, 1893.

    Was t'Hooft around in 1893 when Heaviside wrote this? Did he sit at the feet of Faraday? ;)

    Best wishes,

  14. The gauge boson radiation causing gravity and electromagnetism is DISPLACEMENT CURRENT. Catt shows that Maxwell got his interpretation of this ‘displacement current’ wrong, by ignoring the time it takes light speed electricity to flow along the capacitor plates. His co-authors Drs. Walton and Davidson mathematically worked out how the transmission line theory of Heaviside can be applied to explain the charging curve of a capacitor, which is compared to reality and is a correct prediction. Catt's error follows from Heaviside’s false idea that the light speed electricity Poynting-Heaviside vector is the same as light, with the two conductors guiding the light which travels in the insulator between them. This is false, as we know electricity originates as electrons in conductors and such like, although it is true that the measured speed is that in the insulator not the wires. What is going on is plain from quantum electrodynamics, gauge bosons/photons are being exchanged via the insulator between the two conductors. This is why parallel wires carrying currents attract/repel. In addition, the radio transmitter and receiver aerial form a capacitor with air as the dielectric. The radio waves are displacement current energy, detectable just when the varying current varies the electric field across the transmitter aerial. In the same way, the displacement current flows in the capacitor only while the field in the capacitor plate is varying, due to its charging up or discharging. Maxwell's error was fiddling a theory to fit Weber's 1856 observation that 1/(root of product of permittivity and permeability) = c. This fiddle is like the application by Rayleigh of a wave equation to sound without understanding the pressure and force mechanisms involved in particulate (molecular) sound waves. Planck showed the resolution to the problem with the wave model of light by the quantum theory, while Bohr had shown that Maxwell's light theory was incompatible with the atom. Nobody corrected Maxwell's false theory, however. In reality, ‘displacement current’ is the gauge boson, causing electromagnetic and gravitational forces, and all radio and light waves. Emitted by due to the centripetal acceleration of continuous, uniformly spinning charges (fundamental particles) with no oscillation, it is undetectable radiation, but still carries pressure and force (pressure times area), causing fundamental forces.

  15. ‘Oh, my dear Kepler, how I wish that we could have one hearty laugh together! Here at Padua is the principal professor of philosophy [Professor Cremonini] whom I have repeatedly and urgently requested to look at the moon and planets through my glass, which he pertinaciously refuses to do. Why are you not here? What shouts of laughter we should have at this glorious folly! And to hear the professor of philosophy at Pisa [Professor Giulio Libri] labouring before the Grand Duke with logical arguments, as if with magical incantations, to charm the new planets out of the sky.’

    – Letter from Galileo to Kepler, 1610 (Sir Oliver Lodge, Pioneers of Science, London, 1913, Chapter 4).