The Arc Addendum to the Burning Mirror Solution
”The Math Behind Burning Mirrors” contradicts the widely held view that ”circles make poor approximations for parabolas”. For small angles, the geometry of a circle and a parabola converge to within a few parts per thousand. This is a very good approximation for long focal length mirrors. It also shows that the ancients had the facility to make these mirrors with the pendulum and potter’s wheel construction method.
The huge numbers that emerge from the calculations arise because the curves are getting closer to the perfect parabola. The ideal curve has an infinite concentration factor under the approximation used, where the sun is considered a point source. The theoretical numbers are much too high because the sun is not an exact point, but is actually spread over a few degrees of the sky. The sun’s arc parabola calculation brings the concentration factors down to realistic numbers. The formula for the sun image is independent of dish size and simply proportional to the focal length. It relies on the height and distance to the sun combined with the focal length, the longer the f.p. the larger the sun disk image. This approach actually makes the power levels much easier to calculate.
For a perfect parabola with a focal length of 1m, the real sun image will be 9.2mm, regardless of whether the dish is 5cm or 5m wide. A 2m wide dish with a one-meter focal length delivers a concentration factor of approximately 47MegaWatts per square meter, which is considerable. When the focal length is shortened, the intensity increases. For the same dish with a 0.5m focal length the real sun image is about 4mm wide, which produces an intensity of over 200MegaWatts per square meter. Both of these devices are incredibly powerful even when placed alongside the majority of modern lasers. The weakest one is nearly three times more potent than the solar device used in tests to melt stones and vaporize metals.
Mechanical methods can produce these curves because they are relatively deep and do not entail the precision cutting of the shallower curves. Ancient shield making techniques suffice to make the rough shape, followed by a laborious guided grinding and polishing procedure. This manual aspect can be sped up by using a potter’s wheel in a similar fashion to the long focal length devices. Instead of using a moving grinder on a pendulum, a fixed parabolic shaped grinder would be used. These methods are touched on in the Secrets of the Sun Sects. There is not much debate over ancient abilities to make these shapes since they are widely found in the artifacts from shields to bowls.
Archimedes and Syracuse
The persistent problem has always been how the ancients made long focal length mirrors. This is tightly bound to the famous Archimedes story of burning mirrors at Syracuse. The rationale runs that the ancients could not have burned the roman ships because they could not make or did not have parabolic mirrors with long enough focal lengths. Under the arc approximation, it seems that even if they did have near perfect parabolic mirrors with very long focal lengths it would still be very difficult.
Even if the high quality curves of pendulum method are considered perfect, then using the width of the sun means there will be a large disk of light on the target regardless. If the focal length of the dish is 30m the real sun image will be over 27cm across. If the ship is at 50m the sun disk image will be over 46cm wide. These are huge circles of light and would require similarly huge reflectors to provide enough energy to start a fire.
In ”Secrets of the Sun Sects”, the account of Archimedes Burning Mirrors is concluded with this paragraph.
”This short account neatly summarizes the use of burning mirrors as exceptional weaponry of ancient Greece. It also details the usual counter arguments that make it all seem a ‘bit far-fetched’. The ancient method of manufacture makes not only small short-range mirrors possible, but the technique is scalable for larger, longer focal length reflectors exactly as described. In fact, it appears that the longer range mirrors are easier to produce. A two-meter mirror with a sharp hundred-meter focal point is easier to construct than a fifty-centimeter device with a two-meter focal length. As shown, the power also increases radically as the focal point gets longer, which is counter to many modern methods of build. Whilst this possibility does arise, it is probably a red herring in the search for burning mirrors, misdirection is a useful tool for the concealing historian. The commonly held alternative views are more than likely correct. The Carthagians did have catapults and pitch, which is a much easier combination to fire the approaching boats.”
Under the Fusniak approximation, the statements remain true with the exception of the power increasing as the focal length increases, this only holds true under the point of light approximation. There is however, a small window in which the account could have some validity even under this more accurate calculation. It is linked to the limits of the size and types of devices found in the archaeological record and references to problems the Greeks had extending the range of the burning mirrors.
There is a recently discovered manuscript, which appears to be an Arabic translation of a supposedly lost Greek tract on the theory of conic sections. This is thought to have been written by Archimedes during the 2nd century BCE. This provides some interesting clues.
”The manuscript, written around CE 902, is a translation of a Greek manuscript on the code of research of burning mirrors. It outlined an important application of geometry that developed into new concepts on optics by the 10th century. This is probably the oldest copy of the optics manuscript known, though an identical copy made during the 14th Century exists in India. The manuscript references the burning mirrors of the Greeks, who are said to have discovered how to set light to objects thirty cubits away. They wanted to extend this achievement and meet a challenge to set light to objects at a distance of one hundred cubits. In Alexandria, during the 3rd and 2nd centuries BCE, burning mirrors were an important subject of research. Conon of Alexandria, Archimedes, Dosithcus, and Apollonitis are all described as dabbling in the area.”
If one follows the dimensions mentioned in this paper, a few conclusions can be made with the physics. The ”30 cubits” is about 14m, a range that would probably have put Archimedes life in jeopardy as the Romans arrived. However, this would indicate that the sun image would be about 13cm across. In order for a dish to produce fire with that image, it would need over 120 times the area. This results in a very flat dish with a diameter of 150cm, which is just about in line with the shield sizes of the time.
If the mirror range was the slightly safer distance of 30m the dish would need to be about 3m wide. The ”100 cubit” (45m) goal mentioned would require a dish of over 4.5m, which is probably why it remained just an objective. Whilst sun dishes ”twice the height of a man” have been noted in South America, I am not aware of any that large in ancient Greece. These factors leave the solution to the Burning Mirror problem in tact and the Syracuse story a ‘red herring’ in the search for uses of sun dishes in antiquity. The mirrors can retain their Trojan combat use as blinding devices, but are far less likely to be used as solar cannons for Archimedes.
Implications for the Solar Devices
It is worth summarizing the effects the arc approximation has on the extensive range of devices that are described in Secrets of the Sun Sects. The vast majority of the tools have been tested at least on a scale that is practical. There are a few applications that require adjustment for scaling or mechanical reasons.
SOLAR CHAMBERS: About half of the ancient devices are based on the simple premise that dark stones warm up when exposed to sunlight. The ancient solar chambers that utilize this property only require flat reflectors to work and are completely unaffected by the parabolic amendment. This means the simple and inexpensive domestic and industrial cookers remain perfectly viable. Likewise, the water heating, distillation, sterilizing and pumping equipment is still feasible in the ancient world. There is little doubt given all the evidence that the crop drying techniques were applied on a grand scale. All of these items still perform beautifully at both the small and large scale.
PARABOLIC DEVICES: For the parabolic dish devices, the changes only occur on the implementation side, the abilities remain the same. The power is still there at the heart of the burning mirrors, the intense beam is still the most potent entity in antiquity. The relatively shorter focal lengths mean that the use becomes slightly more restricted. In one or two cases, this means the devices have to employ a secondary reflector, which complicates matters slightly. Whilst these improvements have been added to the modern examples in The Sun Devices, when ancient uses were considered simpler was always chosen over complexity. This was primarily because the tool artifacts found are incomplete so less parts, means more likely.
COOKING: The parabolic cooking methods still hold in their entirety, the potency of the devices at this range remain unchanged, fried foods, solar grilled and even boiled remain unchanged. The ancient soldier could still cook his meal with the upturned shield. Any real volume cooking remains within the domain of the solar chambers mentioned above.
OPTICAL: The optical devices are unchanged by the arc addendum, these mirrors will still make excellent components in any reflector telescope ancient or modern. The implications remain that the ancients were scanning the skies with instruments rather than just the naked eye.
METALWORK: On the materials side, most metals could all still be melted, vaporized or worked whilst hot. The smith still had a solar forge in which he could liquefy metals then cast objects along with the ability to bend and fuse others with the intense heat of the beam. He could fuse some metals with a small dish, but not all. There remains the marginally more complicated method, which involves using an iron heated in the larger dish to carry out the same task. Cutting with light is no longer an option without a secondary dish concentrating the first image.
RECYCLING: Recycling methods remain unchanged, though it would not be possible to wander around a dump vaporizing rubbish with ease. This application was aimed primarily at the modern user, but clearly, in the ancient world metal objects would be recycled when damaged.
REFINING: It is noted that the smith and the ancient alchemist were probably the same person in deep antiquity. He still had the power and ability to readily experiment with refining techniques. He was more restricted in where he could practice this art unless furnished with a large flat dish to wander around. Instead of just pointing the dish at a variety of stones, each would have to be placed within the deep dish to find out if there were any useful effects or products to be had. This new material synthesis concept holds for ancient as well as modern. The largest problem here is that the limits on volumes are more restricted since fresnel arrangements of mirrors cannot be used easily.
STONEWORK: The closely linked areas of stone working and ceramics are the most affected techniques. All of the methods tested still work in exactly the same way provided the object is small. Ceramic pots or stones can have a glaze applied quickly and neatly within the confines of the dish. It is only the larger objects that become more awkward to work. Huge rocks can still be shattered by heat, simply by pointing the beam from a shield-sized dish toward the desired fracture point and pouring water on afterward.
When a huge object is to be glazed where it is standing, there are issues with respect to the dimensions of the dish relative to the focal length. These can be overcome in two ways. The first involves using a larger dish than originally envisioned. The second involves the primitive or double dish Cassegrain set ups described. The advantage of this more complicated arrangement is that it can do the fine work at high powers and apply the finishes at slightly lower power without restrictions. The primitive version involves using a flat panel on the object to reflect light onto a short f.p. dish. The result is an off center beam with lower power but flexibility in use. This is the same method as Lindroth proved with a small 30cm dish and a 2mm beam. It cuts via vaporization. Back from its maximum power, it can also glaze stones with ceramic paints or the natural mica within the stone.
The last is a true Cassegrain device akin to the designs of the modern patented solar cutters/polishers. The first dish points directly at the sun and directs the light to an inline dish. This second mirror reflects the beam back down through a hole in the center of the first. Both dishes are effectively concentrating the light to very high powers, which will cut through just about anything. Obviously, the power can be reduced for other tasks by using the device at a short distance from the true f.p. The issue with this device is that whilst it is relatively simple looking, the geometry is not. Dishes have been found with the necessary hole, dimensions and curves though the tripod for the second dish has not. There are some objects from antiquity that could do the task, but it is much preferable to find them all in one piece or at least in the vicinity. This more complicated set up and geometry may help explain why the stone cutting technique was so easily lost.
GEMS: Gem processing remains an easy and lucrative sideline for the solar artisan. The approximation does not alter the speed, ease or new/old techniques in this arena. The natural gems are simply placed in the beam for partial or complete transforms.
Experimental Confirmation and Failings
The details of the calculation have been checked by much better physicists and engineers than myself and none of them spotted the arc amendment. It seems to have been forgotten behind the headline that spherical reflectors actually make excellent long focal length mirrors and the astounding effects produced in test. To be fair, the guys were more intrigued by the effects and the possibilities raised by cheap long focal length mirrors. The tests were carried out with mirrors with focal lengths of a few meters at most. Invariably those built with the ancient method were small with high fp to dish ratios. This was because aim was to test the theory that the angle of pendulum dictates the accuracy of the curve to a parabola. Deep dishes can be made with standard mechanical methods now and in antiquity, there was no need to test these. All results seemed to fall into line with the predictions of the calculation, given the quality of the devices.
The test for the burning mirrors of Syracuse simply followed on from making small mirrors that could start fires easily. Under the point of light approximation, there was no reason to think that at greater distances the beam would be less intense. Under the arc approximation, the dish has to increase in a proportion equal to the increase in size of the focal point to maintain the potency. As the calculations above show, this does still permit a mirror just at the limit of the technology to burn a ship, but it was right at the limit of ancient mirror construction techniques as well.
The results were materially quite amazing, vaporized rather than melted metals, glass rather than fractured stones, all aspects at achieved at very high temperatures. The problem in experiment seemed to be keeping the power down rather than not enough. The prospect of even higher powers at lower angles seemed to be confirmed by scaled down versions. What we were in fact doing was wandering between the improvements in power caused by more accurate parabola construction and extremely high energy inputs from the larger cruder devices.
The sun’s arc approximation adds limits to the upper power of these devices; it does not change what has been done or what is possible. The ancient method of constructing relatively flat curves still allows for a greater degree of flexibility. The accuracy of the curves to the ideal parabola is still more than adequate for the purposes of burning mirrors. The wide range of applications was still available to the ancient artisans and scientists. The devices are still incredibly useful today.
The new calculation method removes the tendency of the point source calculation to increase to infinity in a neat and elegant way. It puts more realistic power figures on the devices that can be confirmed across all sizes and focal lengths. There is still a matrix of mirror sizes and focal length that need constructing and testing, hopefully, some will try. The new approximation suggested has greatly simplified the method of calculating the potency of these devices, for which the author is grateful.