Patent medicine

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Patent_medicine 

A patent medicine, sometimes called a proprietary medicine, is an over-the-counter (nonprescription) medicine or medicinal preparation that is typically protected and advertised by a trademark and trade name (and sometimes a patent) and claimed to be effective against minor disorders and symptoms. Its contents are typically incompletely disclosed. Antiseptics, analgesics, some sedatives, laxatives, and antacids, cold and cough medicines, and various skin preparations are included in the group. The safety and effectiveness of patent medicines and their sale is controlled and regulated by the Food and Drug Administration in the United States and corresponding authorities in other countries.

E. W. Kemble's "Death's Laboratory" on the cover of Collier's (June 3, 1905)

The term is sometimes still used to describe quack remedies of unproven effectiveness and questionable safety sold especially by peddlers in past centuries, who often also called them elixirs, tonics, or liniments. Current examples of quack remedies are sometimes called nostrums or panaceas, but easier to understand terms like scam, cure-all, or pseudoscience are more common.

Patent medicines were one of the first major product categories that the advertising industry promoted; patent medicine promoters pioneered many advertising and sales techniques that were later used for other products. Patent medicine advertising often marketed products as being medical panaceas (or at least a treatment for many diseases) and emphasized exotic ingredients and endorsements from purported experts or celebrities, which may or may not have been true. Patent medicine sales were increasingly constricted in the United States in the early 20th century as the Food and Drug Administration and Federal Trade Commission added ever-increasing regulations to prevent fraud, unintentional poisoning and deceptive advertising. Sellers of liniments, claimed to contain snake oil and falsely promoted as a cure-all, made the snake oil salesman a lasting symbol for a charlatan.

Patent medicines and advertising

Mug-wump, "for all venereal diseases"

The phrase "patent medicine" comes from the late 17th century marketing of medical elixirs, when those who found favour with royalty were issued letters patent authorising the use of the royal endorsement in advertising. Few if any of the nostrums were actually patented; chemical patents did not come into use in the United States until 1925. Furthermore, patenting one of these remedies would have meant publicly disclosing its ingredients, which most promoters sought to avoid.

Advertisement kept these patent medications in the public eye and gave the belief that no disease was beyond the cure of patent medication. “The medicine man’s key task quickly became not production but sales, the job of persuading ailing citizens to buy his particular brand from among the hundreds offered. Whether unscrupulous or self-deluded, nostrum makers set about this task with cleverness and zeal.”

Instead, the compounders of such nostrums used a primitive version of branding to distinguish their products from the crowd of their competitors. Many extant brands from the era live on today in brands such as Luden's cough drops, Lydia E. Pinkham's vegetable compound for women, Fletcher's Castoria and even Angostura bitters, which was once marketed as a stomachic. Though sold at high prices, many of these products were made from cheap ingredients. Their composition was well known within the pharmacy trade, and druggists manufactured and sold (for a slightly lower price) medicines of almost identical composition. To protect profits, the branded medicine advertisements emphasized brand names, and urged the public to "accept no substitutes".

At least in the earliest days, the history of patent medicines is coextensive with scientific medicine. Empirical medicine, and the beginning of the application of the scientific method to medicine, began to yield a few orthodoxly acceptable herbal and mineral drugs for the physician's arsenal. These few remedies, on the other hand, were inadequate to cover the bewildering variety of diseases and symptoms. Beyond these patches of evidence-based application, people used other methods, such as occultism; the "doctrine of signatures" – essentially, the application of sympathetic magic to pharmacology – held that nature had hidden clues to medically effective drugs in their resemblances to the human body and its parts. This led medical men to hope, at least, that, say, walnut shells might be good for skull fractures. Homeopathy, the notion that illness is binary and can be treated by ingredients that cause the same symptoms in healthy people, was another outgrowth of this early era of medicine. Given the state of the pharmacopoeia, and patients' demands for something to take, physicians began making "blunderbuss" concoctions of various drugs, proven and unproven. These concoctions were the ancestors of the several nostrums.

Touting these nostrums was one of the first major projects of the advertising industry. The marketing of nostrums under implausible claims has a long history. In Henry Fielding's Tom Jones (1749), allusion is made to the sale of medical compounds claimed to be universal panaceas:

As to Squire Western, he was seldom out of the sick-room, unless when he was engaged either in the field or over his bottle. Nay, he would sometimes retire hither to take his beer, and it was not without difficulty that he was prevented from forcing Jones to take his beer too: for no quack ever held his nostrum to be a more general panacea than he did this; which, he said, had more virtue in it than was in all the physic in an apothecary's shop.

1914 advertisement implying approval by the U.S. government

Within the English-speaking world, patent medicines are as old as journalism. "Anderson's Pills" were first made in England in the 1630s; the recipe was allegedly learned in Venice by a Scot who claimed to be physician to King Charles I. Daffy's Elixir was invented about 1647 and remained popular in Britain and the USA until the late 19th century. The use of "letters patent" to obtain exclusive marketing rights to certain labelled formulas and their marketing fueled the circulation of early newspapers. The use of invented names began early. In 1726 a patent was also granted to the makers of Dr Bateman's Pectoral Drops; at least on the documents that survive, there was no Dr. Bateman. This was the enterprise of a Benjamin Okell and a group of promoters who owned a warehouse and a print shop to promote the product.

A number of American institutions owe their existence to the patent medicine industry, most notably a number of the older almanacs, which were originally given away as promotional items by patent medicine manufacturers. Perhaps the most successful industry that grew up out of the business of patent medicine advertisements, though, was founded by William H. Gannett in Maine in 1866. There were few circulating newspapers in Maine in that era, so Gannett founded a periodical, Comfort, whose chief purpose was to propose the merits of Oxien, a nostrum made from the fruit of the baobab tree, to the rural folks of Maine. Gannett's newspaper became the first publication of Guy Gannett Communications, which eventually owned four Maine dailies and several television stations. (The family-owned firm is unrelated to the Gannett Corporation that publishes USA Today.) An early pioneer in the use of advertising to promote patent medicine was New York businessman Benjamin Brandreth, whose "Vegetable Universal Pill" eventually became one of the best-selling patent medicines in the United States. “…A congressional committee in 1849 reported that Brandreth was the nation’s largest proprietary advertiser… Between 1862 and 1863 Brandreth’s average annual gross income surpassed $600,000…” For fifty years Brandreth’s name was a household word in the United States. Indeed, the Brandreth pills were so well known they received mention in Herman Melville's classic novel Moby-Dick.

Kickapoo Indian "Sagwa", sold at medicine shows

Another publicity method – undertaken mostly by smaller firms – was the medicine show, a traveling circus of sorts that offered vaudeville-style entertainments on a small scale, and climaxed in a pitch for some sort of cure-all nostrum. "Muscle man" acts were especially popular on these tours, for this enabled the salesman to tout the physical vigour the product supposedly offered. The showmen frequently employed shills, who stepped forward from the crowd to offer "unsolicited" testimonials about the benefits of the medicine. Often, the nostrum was manufactured and bottled in the wagon in which the show travelled. The Kickapoo Indian Medicine Company became one of the largest and most successful medicine show operators. Their shows had an American Indian or Wild West theme, and employed many American Indians as spokespeople such as the Modoc War scout Donald McKay. The "medicine show" lived on in American folklore and Western movies long after they vanished from public life.

Ingredients and their uses

Sick Made Well, Weak Made Strong, Elixir of Life, etc. Typical ad for patent medicine.

Supposed ingredients

Kilmer's Swamp Root

Many promoters desired to lend their preparations a sense of exoticism and mystery. Unlikely ingredients such as the baobab fruit in Oxien were a recurring theme. A famous patent medicine of the period was Dr. Kilmer's Swamp Root; unspecified roots found in swamps had remarkable effects on the kidneys, according to its literature.

Native American themes were also useful: natives, imagined to be noble savages, were thought to be in tune with nature, and heirs to a body of traditional lore about herbal remedies and natural cures. One example of this approach from the period was Kickapoo Indian Sagwa, a product of the Kickapoo Indian Medicine Company of Connecticut (completely unrelated to the real Kickapoo Indian tribe of Oklahoma), supposedly based on a Native American recipe. This nostrum was the inspiration for Al Capp's "Kickapoo Joy Juice," featured in the comic strip, "Li'l Abner". Another benefit of claiming traditional native origins was that it was nearly impossible to disprove. A good example of this is the story behind Dr. Morse's Indian Root Pills, which was the mainstay of the Comstock patent medicine business. According to text on a wrapper on every box of pills, Dr. Morse was a trained medical doctor who enriched his education by travelling extensively throughout Asia, Africa, and Europe. He supposedly lived among the American Indians for three years, during which time he discovered the healing properties of various plants and roots that he eventually combined into Dr. Morse's Indian Root Pills. No one knows if Dr. Morse ever actually existed.

Other promoters took an opposite tack from timeless herbal wisdom. Nearly any scientific discovery or exotic locale could inspire a key ingredient or principle in a patent medicine. Consumers were invited to invoke the power of electromagnetism to heal their ailments. In the nineteenth century, electricity and radio were gee-whiz scientific advances that found their way into patent medicine advertising, especially after Luigi Galvani showed that electricity influenced the muscles. Devices meant to electrify the body were sold; nostrums were compounded that purported to attract electrical energy or make the body more conductive. "Violet ray machines" were sold as rejuvenation devices, and balding men could seek solace in an "electric fez" purported to regrow hair. Albert Abrams was a well known practitioner of electrical quackery, claiming the ability to diagnose and treat diseases over long distances by radio. In 1913 the quack John R. Brinkley, calling himself an "Electro Medic Doctor," began injecting men with colored water as a virility cure, claiming it was "electric medicine from Germany." (Brinkley would go on to even greater infamy through transplanting goat testicles into men's scrotums as a virility treatment.)

Towards the end of the period, a number of radioactive medicines, containing uranium or radium, were marketed. Some of these actually contained the ingredients promised, and there were a number of tragedies among their devotees. Most notoriously, steel heir Eben McBurney Byers was a supporter of the popular radium water Radithor, developed by the medical con artist William J. A. Bailey. Byers contracted fatal radium poisoning and had to have his jaw removed in an unsuccessful attempt to save him from bone cancer after drinking nearly 1400 bottles of Bailey's "radium water." Water irradiators were sold that promised to infuse water placed within them with radon, which was thought to be healthy at the time.

Actual ingredients

Contrary to what is often believed, some patent medicines did, in fact, deliver the promised results, albeit with very dangerous ingredients. For example, medicines advertised as "infant soothers" contained opium, then a legal drug. Those advertised as "catarrh snuff" contained cocaine, also legal. While various herbs, touted or alluded to, were talked up in the advertising, their actual effects often came from procaine extracts or grain alcohol. Those containing opiates were at least effective in relieving pain, coughs, and diarrhea, though they could result in addiction. This hazard was sufficiently well known that many were advertised as causing none of the harmful effects of opium (though many of those so advertised actually did contain opium).

Until the twentieth century, alcohol was the most controversial ingredient, for it was widely recognised that the "medicines" could continue to be sold for their alleged curative properties even in prohibition states and counties. Many of the medicines were in fact liqueurs of various sorts, flavoured with herbs said to have medicinal properties. Some examples include:

• Cannabis indica, the low growing variants of cannabis with a high level of THC.
• Peruna was a famous "Prohibition tonic," weighing in at around 18% grain alcohol. A nostrum known as "Jamaican ginger" was ordered to change its formula by Prohibition officials. To fool a chemical test some vendors added a toxic chemical, tricresyl phosphate, an organophosphate compound that produced organophosphate-induced delayed neuropathy, a chronic nerve damage syndrome similar to that caused by certain nerve agents. Unwary imbibers suffered a form of paralysis that came to be known as jake-leg.
• Clark Stanley, the "Rattlesnake King", produced Stanley's snake oil, publicly processing rattlesnakes at the World's Columbian Exposition in Chicago. His liniment, when seized and tested by the federal government in 1917, was found to contain mineral oil, 1% fatty oil, red pepper, turpentine and camphor. This is not too unlike modern capsaicin and camphor liniments.
• The original formulation of Coca-Cola used coca leaves, an indirect source of cocaine, and was marketed as an energy rejuvenator. Unlike most patent medicines of its era, it did not contain alcohol.
• Some herbal preparations included laxatives such as senna or diuretics, to give the compounds some obvious physical effects.

When journalists and physicians began focusing on the narcotic contents of the patent medicines, some of their makers began replacing the opium tincture laudanum with acetanilide, a particularly toxic non-steroidal anti-inflammatory drug with analgesic as well as antipyretic properties that had been introduced into medical practice under the name Antifebrin by A. Cahn and P. Hepp in 1886. But this ingredient change probably killed more of the nostrum's users than the original narcotics did, since acetanilide not only alarmingly caused cyanosis due to methemoglobinemia, but was later discovered to cause liver and kidney damage.

The occasional reports of acetanilide-induced cyanosis prompted the search for less toxic aniline derivatives. Phenacetin was one such derivative; it was eventually withdrawn after it was found to be a carcinogen. After several conflicting results over the ensuing fifty years, it was ultimately established in 1948 that acetanilide was mostly metabolized to paracetamol (known in the United States as USAN: acetaminophen) in the human body, and that it was this metabolite that was responsible for its analgesic and antipyretic properties. Acetanilide is no longer used as a drug in its own right, although the success of its metabolite – paracetamol (acetaminophen) – is well known.

Supposed uses

Bonnore's Electro Magnetic Bathing Fluid was claimed to help many unrelated ailments.

Patent medicines were supposedly able to cure just about everything. Nostrums were openly sold that claimed to cure or prevent venereal diseases, tuberculosis, and cancer. Bonnore's Electro Magnetic Bathing Fluid claimed to cure cholera, neuralgia, epilepsy, scarlet fever, necrosis, mercurial eruptions, paralysis, hip diseases, chronic abscesses, and "female complaints". William Radam's Microbe Killer, a product sold widely on both sides of the Atlantic in the 1890s and early 1900s, had the bold claim 'Cures All Diseases' prominently embossed on the front of the bottle. Ebeneezer Sibly ('Dr Sibly') in late 18th and early 19th century Britain went so far as to advertise that his Solar Tincture was able to "restore life in the event of sudden death", amongst other marvels.

Every manufacturer published long lists of testimonials that described their product curing all sorts of human ailments. Fortunately for both makers and users, the illnesses they claimed were cured were almost invariably self-diagnosed – and the claims of the writers to have been healed of cancer or tuberculosis by the nostrum should be considered in this light.

The end of the patent medicine era

Clark Stanley's Snake Oil Liniment.

Muckraker journalists and other investigators began to publicize instances of death, drug addiction, and other hazards from the compounds. This took no small courage by the publishing industry that circulated these claims, since the typical newspaper of the period relied heavily on the patent medicines. In 1905, Samuel Hopkins Adams published an exposé entitled "The Great American Fraud" in Collier's Weekly that led to the passage of the first Pure Food and Drug Act in 1906. This statute did not ban the alcohol, narcotics, and stimulants in the medicines; it required them to be labeled as such, and curbed some of the more misleading, overstated, or fraudulent claims that appeared on the labels. In 1936 the statute was revised to ban them, and the United States entered a long period of ever more drastic reductions in the medications available unmediated by physicians and prescriptions. Morris Fishbein, editor of the Journal of the American Medical Association, who was active in the first half of the 20th century, based much of his career on exposing quacks and driving them out of business.

In more recent years, also, various herbal concoctions have been marketed as "nutritional supplements". While their advertisements are careful not to cross the line into making explicit medical claims, and often bear a disclaimer that asserts that the products have not been tested and are not intended to diagnose or treat any disease, they are nevertheless marketed as remedies of various sorts. Weight loss "while you sleep" and similar claims are frequently found on these compounds (cf., Calorad, Relacore, etc.). Despite the ban on such claims, salesmen still occasionally (and illegally) make such claims; Jim Bakker, a disgraced televangelist, sells a colloidal silver gel that he claims will cure all venereal diseases and SARS-related coronaviruses. One of the most notorious such elixirs, however, calls itself "Enzyte", widely advertised for "natural male enhancement" – that is, penis enlargement. Despite being a compound of herbs, minerals, and vitamins, Enzyte formerly promoted itself under a fake scientific name Suffragium asotas. Enzyte's makers translate this phrase as "better sex," but it is in fact ungrammatical Latin for "refuge for the dissipated".

Surviving consumer products from the patent medicine era

A horse drawn Bromo Seltzer wagon.

A number of brands of consumer products that date from the patent medicine era are still on the market and available today. Their ingredients may have changed from the original formulas; the claims made for the benefits they offer have typically been seriously revised, but in general at least some of them, like Bayer Aspirin, have genuine medical uses. These brands include:

Lydia Pinkham's Herb Medicine (circa 1875) remains on the market today.

 

Main article: List of patent medicines

• Absorbine Jr.
• Anacin/Anadin
• Andrews Liver Salts
• Aspro aspirin tablets
• Bayer Aspirin
• BC Powder
• Bromo-Seltzer
• Carter's Little Liver Pills (currently sold as Carter's Little Pills)
• Chlorodyne
• Doan's Pills
• Fletcher's Castoria
• Geritol
• Goody's Powder
• Dr. J.H. McLean's Volcanic Oil
• Lobeila Cough Syrup
• Lorman’s Indian Oil
• Luden's Throat Drops
• Lydia E. Pinkham's Vegetable Compound
• Minard's Liniment
• Phillips' Milk of Magnesia
• Smith Brothers Throat Drops
• Vicks VapoRub

A number of patent medicines are produced in China. Among the best known of these is Shou Wu Chih, a black, alcoholic liquid that the makers claimed turned gray hair black.

Products no longer sold under medicinal claims

Some consumer products were once marketed as patent medicines, but have been repurposed and are no longer sold for medicinal purposes. Their original ingredients may have been changed to remove drugs, as was done with Coca-Cola. The compound may also simply be used in a different capacity, as in the case of Angostura Bitters, now associated chiefly with cocktails.

• 7-Up
• Angostura Bitters
• Bovril
• Buckfast Tonic Wine
• Coca-Cola
• Dr Pepper
• Fernet Branca
• Graham crackers
• Grape-Nuts
• Hires Root Beer
• Moxie brand soda
• Pepsi Cola
• Tonic water

Terraforming of Mars

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Terraforming_of_Mars 

Artist's conception of the process of terraforming Mars

The terraforming of Mars or the terraformation of Mars is a hypothetical procedure that would consist of a planetary engineering project or concurrent projects, with the goal of transforming Mars from a planet hostile to terrestrial life to one that can sustainably host humans and other lifeforms free of protection or mediation. The process would presumably involve the rehabilitation of the planet's extant climate, atmosphere, and surface through a variety of resource-intensive initiatives, and the installation of a novel ecological system or systems.

Justifications for choosing Mars over other potential terraforming targets include the presence of water and a geological history that suggests it once harbored a dense atmosphere similar to Earth's. Hazards and difficulties include low gravity, low light levels relative to Earth's, and the lack of a magnetic field.

Disagreement exists about whether current technology could render the planet habitable. Other objections include ethical concerns about terraforming and the considerable cost that such an undertaking would involve. Reasons for terraforming the planet include allaying concerns about resource use and depletion on Earth and arguments that the altering and subsequent or concurrent settlement of other planets decreases the odds of humanity's extinction.

Motivation and side effects

See also: Ethics of terraforming

 

Illustration of plants growing in an imaginary Mars base.

Future population growth, demand for resources, and an alternate solution to the Doomsday argument may require human colonization of bodies other than Earth, such as Mars, the Moon, and other objects. Space colonization would facilitate harvesting the Solar System's energy and material resources.

In many aspects, Mars is the most Earth-like of all the other planets in the Solar System. It is thought that Mars had a more Earth-like environment early in its geological history, with a thicker atmosphere and abundant water that was lost over the course of hundreds of millions of years through atmospheric escape. Given the foundations of similarity and proximity, Mars would make one of the most plausible terraforming targets in the Solar System.

Side effects of terraforming include the potential displacement or destruction of indigenous life, even if microbial, if such life exists.

Challenges and limitations

See also: Colonization of Mars

 

This diagram shows the change in the atmosphere escaping from Mars if it was close to the average temperature on Earth. Mars is thought to have been warm in the past (due to evidence of liquid water on the surface) and terraforming would make it warm again. At these temperatures oxygen and nitrogen would escape into space much faster than they do today.

The Martian environment presents several terraforming challenges to overcome and the extent of terraforming may be limited by certain key environmental factors. Here is a list of some of the ways in which Mars differs from Earth, which terraforming seeks to address:

• Reduced light levels (about 60% of Earth) 
• Low surface gravity (38% of Earth's)
• Unbreatheable atmosphere
• Atmospheric pressure (about 1% of Earth's; well below the Armstrong limit)
• Ionizing solar and cosmic radiation at the surface 
• Average temperature −63 °C (210 K; −81 °F) compared to Earth average of 14 °C (287 K; 57 °F))
• Molecular instability - bonds between atoms break down in critical molecules such as organic compounds
• Global dust storms
• No natural food source
• Toxic soil
• No global magnetic field to shield against the solar wind

Countering the effects of space weather

See also: Health threat from cosmic rays

Mars doesn't have an intrinsic global magnetic field, but the solar wind directly interacts with the atmosphere of Mars, leading to the formation of a magnetosphere from magnetic field tubes. This poses challenges for mitigating solar radiation and retaining an atmosphere.

The lack of a magnetic field, its relatively small mass, and its atmospheric photochemistry, all would have contributed to the evaporation and loss of its surface liquid water over time. Solar wind–induced ejection of Martian atmospheric atoms has been detected by Mars-orbiting probes, indicating that the solar wind has stripped the Martian atmosphere over time. For comparison, while Venus has a dense atmosphere, it has only traces of water vapor (20 ppm) as it lacks a large, dipole induced, magnetic field. Earth's ozone layer provides additional protection. Ultraviolet light is blocked before it can dissociate water into hydrogen and oxygen.

Low gravity and pressure

The surface gravity on Mars is 38% of that on Earth. It is not known if this is enough to prevent the health problems associated with weightlessness.

Mars's CO
₂ atmosphere has about 1% the pressure of the Earth's at sea level. It is estimated that there is sufficient CO
₂ ice in the regolith and the south polar cap to form a 30 to 60 kilopascals [kPa] (4.4 to 8.7 psi) atmosphere if it is released by planetary warming. The reappearance of liquid water on the Martian surface would add to the warming effects and atmospheric density, but the lower gravity of Mars requires 2.6 times Earth's column airmass to obtain the optimum 100 kPa (15 psi) pressure at the surface. Additional volatiles to increase the atmosphere's density must be supplied from an external source, such as redirecting several massive asteroids (40-400 billion tonnes total) containing ammonia (NH
₃) as a source of nitrogen.

Breathing on Mars

Current conditions in the Martian atmosphere, at less than 1 kPa (0.15 psi) of atmospheric pressure, are significantly below the Armstrong limit of 6 kPa (0.87 psi) where very low pressure causes exposed bodily liquids such as saliva, tears, and the liquids wetting the alveoli within the lungs to boil away. Without a pressure suit, no amount of breathable oxygen delivered by any means will sustain oxygen-breathing life for more than a few minutes. In the NASA technical report Rapid (Explosive) Decompression Emergencies in Pressure-Suited Subjects, after exposure to pressure below the Armstrong limit, a survivor reported that his "last conscious memory was of the water on his tongue beginning to boil". In these conditions humans die within minutes unless a pressure suit provides life support.

If Mars' atmospheric pressure could rise above 19 kPa (2.8 psi), then a pressure suit would not be required. Visitors would only need to wear a mask that supplied 100% oxygen under positive pressure. A further increase to 24 kPa (3.5 psi) of atmospheric pressure would allow a simple mask supplying pure oxygen. This might look similar to mountain climbers who venture into pressures below 37 kPa (5.4 psi), also called the death zone, where an insufficient amount of bottled oxygen has often resulted in hypoxia with fatalities. However, if the increase in atmospheric pressure was achieved by increasing CO₂ (or other toxic gas) the mask would have to ensure the external atmosphere did not enter the breathing apparatus. CO₂ concentrations as low as 1% cause drowsiness in humans. Concentrations of 7% to 10% may cause suffocation, even in the presence of sufficient oxygen. (See Carbon dioxide toxicity.)

In 2021 however, the NASA spaceship Perseverance was able to make oxygen on Mars. The process is complex and takes a lot of time to produce a small amount of oxygen.

Advantages

See also: Atmosphere of Mars

 

Hypothetical terraformed Mars

According to scientists, Mars exists on the outer edge of the habitable zone, a region of the Solar System where liquid water on the surface may be supported if concentrated greenhouse gases could increase the atmospheric pressure. The lack of both a magnetic field and geologic activity on Mars may be a result of its relatively small size, which allowed the interior to cool more quickly than Earth's, although the details of such a process are still not well understood.

There are strong indications that Mars once had an atmosphere as thick as Earth's during an earlier stage in its development, and that its pressure supported abundant liquid water at the surface. Although water appears to have once been present on the Martian surface, ground ice currently exists from mid-latitudes to the poles. The soil and atmosphere of Mars contain many of the main elements crucial to life, including sulfur, nitrogen, hydrogen, oxygen, phosphorus and carbon.

Any climate change induced in the near term is likely to be driven by greenhouse warming produced by an increase in atmospheric carbon dioxide (CO
₂) and a consequent increase in atmospheric water vapor. These two gases are the only likely sources of greenhouse warming that are available in large quantities in Mars' environment. Large amounts of water ice exist below the Martian surface, as well as on the surface at the poles, where it is mixed with dry ice, frozen CO₂. Significant amounts of water are located at the south pole of Mars, which, if melted, would correspond to a planetwide ocean 5–11 meters deep. Frozen carbon dioxide (CO₂) at the poles sublimes into the atmosphere during the Martian summers, and small amounts of water residue are left behind, which fast winds sweep off the poles at speeds approaching 400 km/h (250 mph). This seasonal occurrence transports large amounts of dust and water ice into the atmosphere, forming Earth-like ice clouds.

Most of the oxygen in the Martian atmosphere is present as carbon dioxide (CO₂), the main atmospheric component. Molecular oxygen (O₂) only exists in trace amounts. Large amounts of oxygen can be also found in metal oxides on the Martian surface, and in the soil, in the form of per-nitrates. An analysis of soil samples taken by the Phoenix lander indicated the presence of perchlorate, which has been used to liberate oxygen in chemical oxygen generators. Electrolysis could be employed to separate water on Mars into oxygen and hydrogen if sufficient liquid water and electricity were available. However, if vented into the atmosphere it would escape into space.

Proposed methods and strategies

Comparison of dry atmosphere

Atmospheric property	Mars	Earth
Pressure	0.61 kPa (0.088 psi)	101.3 kPa (14.69 psi)
Carbon dioxide (CO2)	96.0%	0.04%
Argon (Ar)	2.1%	0.93%
Nitrogen (N2)	1.9%	78.08%
Oxygen (O2)	0.145%	20.94%

Terraforming Mars would entail three major interlaced changes: building up the magnetosphere, building up the atmosphere, and raising the temperature. The atmosphere of Mars is relatively thin and has a very low surface pressure. Because its atmosphere consists mainly of CO₂, a known greenhouse gas, once Mars begins to heat, the CO₂ may help to keep thermal energy near the surface. Moreover, as it heats, more CO₂ should enter the atmosphere from the frozen reserves on the poles, enhancing the greenhouse effect. This means that the two processes of building the atmosphere and heating it would augment each other, favoring terraforming. However, it would be difficult to keep the atmosphere together because of the lack of a protective global magnetic field against erosion by the solar wind.

Importing ammonia

One method of augmenting the Martian atmosphere is to introduce ammonia (NH₃). Large amounts of ammonia are likely to exist in frozen form on minor planets orbiting in the outer Solar System. It might be possible to redirect the orbits of these or smaller ammonia-rich objects so that they collide with Mars, thereby transferring the ammonia into the Martian atmosphere. Ammonia is not stable in the Martian atmosphere, however. It breaks down into (diatomic) nitrogen and hydrogen after a few hours. Thus, though ammonia is a powerful greenhouse gas, it is unlikely to generate much planetary warming. Presumably, the nitrogen gas would eventually be depleted by the same processes that stripped Mars of much of its original atmosphere, but these processes are thought to have required hundreds of millions of years. Being much lighter, the hydrogen would be removed much more quickly. Carbon dioxide is 2.5 times the density of ammonia, and nitrogen gas, which Mars barely holds on to, is more than 1.5 times the density, so any imported ammonia that did not break down would also be lost quickly into space.

Importing hydrocarbons

Another way to create a Martian atmosphere would be to import methane (CH₄) or other hydrocarbons, which are common in Titan's atmosphere and on its surface; the methane could be vented into the atmosphere where it would act to compound the greenhouse effect. However, like ammonia (NH₃), methane (CH₄) is a relatively light gas. It is in fact even less dense than ammonia and so would similarly be lost into space if it was introduced, and at a faster rate than ammonia. Even if a method could be found to prevent it escaping into space, methane can exist in the Martian atmosphere for only a limited period before it is destroyed. Estimates of its lifetime range from 0.6–4 years.

Use of fluorine compounds

Especially powerful greenhouse gases, such as sulfur hexafluoride, chlorofluorocarbons (CFCs), or perfluorocarbons (PFCs), have been suggested both as a means of initially warming Mars and of maintaining long-term climate stability. These gases are proposed for introduction because they generate a greenhouse effect thousands of times stronger than that of CO₂. Fluorine-based compounds such as sulphur hexafluoride and perfluorocarbons are preferable to chlorine-based ones as the latter destroys ozone. It has been estimated that approximately 0.3 microbars of CFCs would need to be introduced into Mars' atmosphere in order to sublimate the south polar CO₂ glaciers. This is equivalent to a mass of approximately 39 million tonnes, that is, about three times the amount of CFCs manufactured on Earth from 1972 to 1992 (when CFC production was banned by international treaty). Maintaining the temperature would require continual production of such compounds as they are destroyed due to photolysis. It has been estimated that introducing 170 kilotons of optimal greenhouse compounds (CF₃CF₂CF₃, CF₃SCF₂CF₃, SF₆, SF₅CF₃, SF₄(CF₃)₂) annually would be sufficient to maintain a 70-K greenhouse effect given a terraformed atmosphere with earth-like pressure and composition.

Typical proposals envision producing the gases on Mars using locally extracted materials, nuclear power, and a significant industrial effort. The potential for mining fluorine-containing minerals to obtain the raw material necessary for the production of CFCs and PFCs is supported by mineralogical surveys of Mars that estimate the elemental presence of fluorine in the bulk composition of Mars at 32 ppm by mass (as compared to 19.4 ppm for the Earth).

Alternatively, CFCs might be introduced by sending rockets with payloads of compressed CFCs on collision courses with Mars. When the rockets crashed into the surface they would release their payloads into the atmosphere. A steady barrage of these "CFC rockets" would need to be sustained for a little over a decade while Mars changed chemically and became warmer.

Use of orbital mirrors

Mirrors made of thin aluminized PET film could be placed in orbit around Mars to increase the total insolation it receives. This would direct the sunlight onto the surface and could increase Mars's surface temperature directly. The 125 km radius mirror could be positioned as a statite, using its effectiveness as a solar sail to orbit in a stationary position relative to Mars, near the poles, to sublimate the CO
₂ ice sheet and contribute to the warming greenhouse effect. However, certain problems have been found with this. The main concern is the difficulty of launching large mirrors off of earth.

Albedo reduction

Reducing the albedo of the Martian surface would also make more efficient use of incoming sunlight in terms of heat absorption. This could be done by spreading dark dust from Mars's moons, Phobos and Deimos, which are among the blackest bodies in the Solar System; or by introducing dark extremophile microbial life forms such as lichens, algae and bacteria. The ground would then absorb more sunlight, warming the atmosphere. However, Mars is already the second darkest planet in the solar system, absorbing over 70% of incoming sunlight so the scope for darkening it further is small.

If algae or other green life were established, it would also contribute a small amount of oxygen to the atmosphere, though not enough to allow humans to breathe. The conversion process to produce oxygen is highly reliant upon water, without which the CO₂ is mostly converted to carbohydrates. In addition, because on Mars atmospheric oxygen is lost into space (unlike Earth where there is an Oxygen cycle), this would represent a permanent loss from the planet. For both of these reasons it would be necessary to cultivate such life inside a closed system. This would decrease the albedo of the closed system (assuming the growth had a lower albedo than the Martian soil), but would not affect the albedo of the planet as a whole.

On April 26, 2012, scientists reported that lichen survived and showed remarkable results on the adaptation capacity of photosynthetic activity within the simulation time of 34 days under Martian conditions in the Mars Simulation Laboratory (MSL) maintained by the German Aerospace Center (DLR).

One final issue with albedo reduction is the common Martian dust storms. These cover the entire planet for weeks, and not only increase the albedo, but block sunlight from reaching the surface. This has been observed to cause a surface temperature drop which the planet takes months to recover from. Once the dust settles it then covers whatever it lands on, effectively erasing the albedo reduction material from the view of the Sun.

Funded research: ecopoiesis

The Mars Ecopoiesis Test Bed showing its transparent dome to allow for solar heat and photosynthesis, and the cork-screw system to collect and seal Martian soil together with oxygen-producing Earth organisms. Total length is about 7 centimetres (2.8 in).

Since 2014, the NASA Institute for Advanced Concepts (NIAC) program and Techshot Inc are working together to develop sealed biodomes that would employ colonies of oxygen-producing cyanobacteria and algae for the production of molecular oxygen (O₂) on Martian soil. But first they need to test if it works on a small scale on Mars. The proposal is called Mars Ecopoiesis Test Bed. Eugene Boland is the Chief Scientist at Techshot, a company located in Greenville, Indiana. They intend to send small canisters of extremophile photosynthetic algae and cyanobacteria aboard a future rover mission. The rover would cork-screw the 7 cm (2.8 in) canisters into selected sites likely to experience transients of liquid water, drawing some Martian soil and then release oxygen-producing microorganisms to grow within the sealed soil. The hardware would use Martian subsurface ice as its phase changes into liquid water. The system would then look for oxygen given off as metabolic byproduct and report results to a Mars-orbiting relay satellite.

If this experiment works on Mars, they will propose to build several large and sealed structures called biodomes, to produce and harvest oxygen for a future human mission to Mars life support systems. Being able to create oxygen there would provide considerable cost-savings to NASA and allow for longer human visits to Mars than would be possible if astronauts have to transport their own heavy oxygen tanks. This biological process, called ecopoiesis, would be isolated, in contained areas, and is not meant as a type of global planetary engineering for terraforming of Mars's atmosphere, but NASA states that "This will be the first major leap from laboratory studies into the implementation of experimental (as opposed to analytical) planetary in situ research of greatest interest to planetary biology, ecopoiesis, and terraforming."

Research at the University of Arkansas presented in June 2015 suggested that some methanogens could survive in Mars's low pressure. Rebecca Mickol found that in her laboratory, four species of methanogens survived low-pressure conditions that were similar to a subsurface liquid aquifer on Mars. The four species that she tested were Methanothermobacter wolfeii, Methanosarcina barkeri, Methanobacterium formicicum, and Methanococcus maripaludis. Methanogens do not require oxygen or organic nutrients, are non-photosynthetic, use hydrogen as their energy source and carbon dioxide (CO₂) as their carbon source, so they could exist in subsurface environments on Mars.

Protecting the atmosphere

Escaping atmosphere on Mars (carbon, oxygen, and hydrogen) by MAVEN in UV

One key aspect of terraforming Mars is to protect the atmosphere (both present and future-built) from being lost into space. Some scientists hypothesize that creating a planet-wide artificial magnetosphere would be helpful in resolving this issue. According to two NIFS Japanese scientists, it is feasible to do that with current technology by building a system of refrigerated latitudinal superconducting rings, each carrying a sufficient amount of direct current.

In the same report, it is claimed that the economic impact of the system can be minimized by using it also as a planetary energy transfer and storage system (SMES).

Magnetic shield at L₁ orbit

Magnetic shield on L1 orbit around Mars

During the Planetary Science Vision 2050 Workshop in late February 2017, NASA scientist Jim Green proposed a concept of placing a magnetic dipole field between the planet and the Sun to protect it from high-energy solar particles. It would be located at the Mars Lagrange orbit L₁ at about 320 R_♂, creating a partial and distant artificial magnetosphere. The field would need to be "Earth comparable" and sustain 50 μT as measured at 1 Earth-radius. The paper abstract cites that this could be achieved by a magnet with a strength of 1–2 teslas (10,000–20,000 gauss). If constructed, the shield may allow the planet to partially restore its atmosphere.

Plasma torus along the orbit of Phobos

A plasma torus along the orbit of Phobos by ionizing and accelerating particles from the moon may be sufficient to create a magnetic field strong enough to protect a terraformed Mars.

Thermodynamics of terraforming

The overall energy required to sublimate the CO₂ from the south polar ice cap was modeled by Zubrin and McKay in 1993. If using orbital mirrors, an estimated 120 MW-years of electrical energy would be required in order to produce mirrors large enough to vaporize the ice caps. This is considered the most effective method, though the least practical. If using powerful halocarbon greenhouse gases, an order of 1,000 MW-years of electrical energy would be required to accomplish this heating. However, if all of this CO₂ were put into the atmosphere, it would only double the current atmospheric pressure from 6 mbar to 12 mbar, amounting to about 1.2% of Earth's mean sea level pressure. The amount of warming that could be produced today by putting even 100 mbar of CO₂ into the atmosphere is small, roughly of order 10 K. Additionally, once in the atmosphere, it likely would be removed quickly, either by diffusion into the subsurface and adsorption or by re-condensing onto the polar caps.

The surface or atmospheric temperature required to allow liquid water to exist has not been determined, and liquid water conceivably could exist when atmospheric temperatures are as low as 245 K (−28 °C; −19 °F). However, a warming of 10 K is much less than thought necessary in order to produce liquid water.

Fermat's principle

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Fermat%27s_principle 

 

Fig. 1: Fermat's principle in the case of refraction of light at a flat surface between (say) air and water. Given an object-point A in the air, and an observation point B in the water, the refraction point P is that which minimizes the time taken by the light to travel the path APB. If we seek the required value of x, we find that the angles α and β satisfy Snell's law.

Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the path that can be traveled in the least time. In order to be true in all cases, this statement must be weakened by replacing the "least" time with a time that is "stationary" with respect to variations of the path — so that a deviation in the path causes, at most, a second-order change in the traversal time. To put it loosely, a ray path is surrounded by close paths that can be traversed in very close times. It can be shown that this technical definition corresponds to more intuitive notions of a ray, such as a line of sight or the path of a narrow beam.

First proposed by the French mathematician Pierre de Fermat in 1662, as a means of explaining the ordinary law of refraction of light (Fig. 1), Fermat's principle was initially controversial because it seemed to ascribe knowledge and intent to nature. Not until the 19th century was it understood that nature's ability to test alternative paths is merely a fundamental property of waves. If points A and B are given, a wavefront expanding from A sweeps all possible ray paths radiating from A, whether they pass through B or not. If the wavefront reaches point B, it sweeps not only the ray path(s) from A to B, but also an infinitude of nearby paths with the same endpoints. Fermat's principle describes any ray that happens to reach point B; there is no implication that the ray "knew" the quickest path or "intended" to take that path.

Fig. 2: Two points P and P′ on a path from A to B. For the purposes of Fermat's principle, the propagation time from P to P′ is taken as for a point-source at P, not (e.g.) for an arbitrary wavefront W passing through P. The surface Σ  (with unit normal n̂ at P′) is the locus of points that a disturbance at P can reach in the same time that it takes to reach P′; in other words, Σ is the secondary wavefront with radius PP′. (The medium is not assumed to be homogeneous or isotropic.)

For the purpose of comparing traversal times, the time from one point to the next nominated point is taken as if the first point were a point-source. Without this condition, the traversal time would be ambiguous; for example, if the propagation time from P to P′ were reckoned from an arbitrary wavefront W containing P  (Fig. 2), that time could be made arbitrarily small by suitably angling the wavefront.

Treating a point on the path as a source is the minimum requirement of Huygens' principle, and is part of the explanation of Fermat's principle. But it can also be shown that the geometric construction by which Huygens tried to apply his own principle (as distinct from the principle itself) is simply an invocation of Fermat's principle. Hence all the conclusions that Huygens drew from that construction — including, without limitation, the laws of rectilinear propagation of light, ordinary reflection, ordinary refraction, and the extraordinary refraction of "Iceland crystal" (calcite) — are also consequences of Fermat's principle.

Derivation

Sufficient conditions

Let us suppose that:

1. A disturbance propagates sequentially through a medium (a vacuum or some material, not necessarily homogeneous or isotropic), without action at a distance;
2. During propagation, the influence of the disturbance at any intermediate point P upon surrounding points has a non-zero angular spread (as if P were a source), so that a disturbance originating at any point A arrives at any other point B via an infinitude of paths, by which B receives an infinitude of delayed versions of the disturbance at A; and
3. These delayed versions of the disturbance will reinforce each other at B if they are synchronized within some tolerance.

Then the various propagation paths from A to B will help each other if their traversal times agree within the said tolerance. For a small tolerance (in the limiting case), the permissible range of variations of the path is maximized if the path is such that its traversal time is stationary with respect to the variations, so that a variation of the path causes at most a second-order change in the traversal time.

The most obvious example of a stationarity in traversal time is a (local or global) minimum — that is, a path of least time, as in the "strong" form of Fermat's principle. But that condition is not essential to the argument.

Having established that a path of stationary traversal time is reinforced by a maximally wide corridor of neighboring paths, we still need to explain how this reinforcement corresponds to intuitive notions of a ray. But, for brevity in the explanations, let us first define a ray path as a path of stationary traversal time.

A ray as a signal path (line of sight)

If the corridor of paths reinforcing a ray path from A to B is substantially obstructed, this will significantly alter the disturbance reaching B from A — unlike a similar-sized obstruction outside any such corridor, blocking paths that do not reinforce each other. The former obstruction will significantly disrupt the signal reaching B from A, while the latter will not; thus the ray path marks a signal path. If the signal is visible light, the former obstruction will significantly affect the appearance of an object at A as seen by an observer at B, while the latter will not; so the ray path marks a line of sight.

In optical experiments, a line of sight is routinely assumed to be a ray path.

A ray as an energy path (beam)

Fig. 3: An experiment demonstrating refraction (and partial reflection) of rays — approximated by, or contained in, narrow beams

If the corridor of paths reinforcing a ray path from A to B is substantially obstructed, this will significantly affect the energy reaching B from A — unlike a similar-sized obstruction outside any such corridor. Thus the ray path marks an energy path — as does a beam.

Suppose that a wavefront expanding from point A passes point P, which lies on a ray path from point A to point B. By definition, all points on the wavefront have the same propagation time from A. Now let the wavefront be blocked except for a window, centered on P, and small enough to lie within the corridor of paths that reinforce the ray path from A to B. Then all points on the unobstructed portion of the wavefront will have, nearly enough, equal propagation times to B, but not to points in other directions, so that B will be in the direction of peak intensity of the beam admitted through the window. So the ray path marks the beam. And in optical experiments, a beam is routinely considered as a collection of rays or (if it is narrow) as an approximation to a ray (Fig. 3).

Analogies

According to the "strong" form of Fermat's principle, the problem of finding the path of a light ray from point A in a medium of faster propagation, to point B in a medium of slower propagation (Fig. 1), is analogous to the problem faced by a lifeguard in deciding where to enter the water in order to reach a drowning swimmer as soon as possible, given that the lifeguard can run faster than (s)he can swim. But that analogy falls short of explaining the behavior of the light, because the lifeguard can think about the problem (even if only for an instant) whereas the light presumably cannot. The discovery that ants are capable of similar calculations does not bridge the gap between the animate and the inanimate.

In contrast, the above assumptions (1) to (3) hold for any wavelike disturbance and explain Fermat's principle in purely mechanistic terms, without any imputation of knowledge or purpose.

The principle applies to waves in general, including (e.g.) sound waves in fluids and elastic waves in solids. In a modified form, it even works for matter waves: in quantum mechanics, the classical path of a particle is obtainable by applying Fermat's principle to the associated wave — except that, because the frequency may vary with the path, the stationarity is in the phase shift (or number of cycles) and not necessarily in the time.

Fermat's principle is most familiar, however, in the case of visible light: it is the link between geometrical optics, which describes certain optical phenomena in terms of rays, and the wave theory of light, which explains the same phenomena on the hypothesis that light consists of waves.

Equivalence to Huygens' construction

Fig. 4: Two iterations of Huygens' construction. In the first iteration, the later wavefront W′ is derived from the earlier wavefront W by taking the envelope of all the secondary wavefronts (gray arcs) expanding in a given time from all the points (e.g., P) on W. The arrows show the ray directions.

In this article we distinguish between Huygens' principle, which states that every point crossed by a traveling wave becomes the source of a secondary wave, and Huygens' construction, which is described below.

Let the surface W be a wavefront at time t, and let the surface W′ be the same wavefront at the later time t + Δt (Fig. 4). Let P be a general point on W. Then, according to Huygens' construction,

1. W′ is the envelope (common tangent surface), on the forward side of W, of all the secondary wavefronts each of which would expand in time Δt from a point on W, and
2. if the secondary wavefront expanding from point P in time Δt touches the surface W′ at point P′, then P and P′ lie on a ray.

The construction may be repeated in order to find successive positions of the primary wavefront, and successive points on the ray.

The ray direction given by this construction is the radial direction of the secondary wavefront, and may differ from the normal of the secondary wavefront (cf. Fig. 2), and therefore from the normal of the primary wavefront at the point of tangency. Hence the ray velocity, in magnitude and direction, is the radial velocity of an infinitesimal secondary wavefront, and is generally a function of location and direction.

Now let Q be a point on W close to P, and let Q′ be a point on W′ close to P′. Then, by the construction,

1.   the time taken for a secondary wavefront from P to reach Q′ has at most a second-order dependence on the displacement P′Q′, and
2. the time taken for a secondary wavefront to reach P′ from Q has at most a second-order dependence on the displacement PQ.

By (i), the ray path is a path of stationary traversal time from P to W′; and by (ii), it is a path of stationary traversal time from a point on W to P′.

So Huygens' construction implicitly defines a ray path as a path of stationary traversal time between successive positions of a wavefront, the time being reckoned from a point-source on the earlier wavefront. This conclusion remains valid if the secondary wavefronts are reflected or refracted by surfaces of discontinuity in the properties of the medium, provided that the comparison is restricted to the affect paths and the affected portions of the wavefronts.

Fermat's principle, however, is conventionally expressed in point-to-point terms, not wavefront-to-wavefront terms. Accordingly, let us modify the example by supposing that the wavefront which becomes surface W at time t, and which becomes surface W′ at the later time t + Δt, is emitted from point A at time 0. Let P be a point on W (as before), and B a point on W′. And let A, W, W′, and B be given, so that the problem is to find P.

If P satisfies Huygens' construction, so that the secondary wavefront from P is tangential to W′ at B, then PB is a path of stationary traversal time from W to B. Adding the fixed time from A to W, we find that APB is the path of stationary traversal time from A to B (possibly with a restricted domain of comparison, as noted above), in accordance with Fermat's principle. The argument works just as well in the converse direction, provided that W′ has a well-defined tangent plane at B. Thus Huygens' construction and Fermat's principle are geometrically equivalent.

Through this equivalence, Fermat's principle sustains Huygens' construction and thence all the conclusions that Huygens was able to draw from that construction. In short, "The laws of geometrical optics may be derived from Fermat's principle". With the exception of the Fermat-Huygens principle itself, these laws are special cases in the sense that they depend on further assumptions about the media. Two of them are mentioned under the next heading.

Special cases

Isotropic media: Rays normal to wavefronts

In an isotropic medium, because the propagation speed is independent of direction, the secondary wavefronts that expand from points on a primary wavefront in a given infinitesimal time are spherical, so that their radii are normal to their common tangent surface at the points of tangency. But their radii mark the ray directions, and their common tangent surface is a general wavefront. Thus the rays are normal (orthogonal) to the wavefronts.

Because much of the teaching of optics concentrates on isotropic media, treating anisotropic media as an optional topic, the assumption that the rays are normal to the wavefronts can become so pervasive that even Fermat's principle is explained under that assumption, although in fact Fermat's principle is more general.

Homogeneous media: Rectilinear propagation

In a homogeneous medium (also called a uniform medium), all the secondary wavefronts that expand from a given primary wavefront W in a given time Δt are congruent and similarly oriented, so that their envelope W′ may be considered as the envelope of a single secondary wavefront which preserves its orientation while its center (source) moves over W. If P is its center while P′ is its point of tangency with W′, then P′ moves parallel to P, so that the plane tangential to W′ at P′ is parallel to the plane tangential to W at P. Let another (congruent and similarly orientated) secondary wavefront be centered on P′, moving with P, and let it meet its envelope W″ at point P″. Then, by the same reasoning, the plane tangential to W″ at P″ is parallel to the other two planes. Hence, due to the congruence and similar orientations, the ray directions PP′ and P′P″ are the same (but not necessarily normal to the wavefronts, since the secondary wavefronts are not necessarily spherical). This construction can be repeated any number of times, giving a straight ray of any length. Thus a homogeneous medium admits rectilinear rays.

Modern version

Formulation in terms of refractive index

Let a path Γ extend from point A to point B. Let s be the arc length measured along the path from A, and let t be the time taken to traverse that arc length at the ray speed ${\displaystyle v_{\mathrm {r} }}$ (that is, at the radial speed of the local secondary wavefront, for each location and direction on the path). Then the traversal time of the entire path Γ is

$${\displaystyle T=\int _{A}^{B}\!dt=\int _{A}^{B}{\frac {ds}{\,v_{\mathrm {r} }\,}}}$$

(1)

(where A and B simply denote the endpoints and are not to be construed as values of t or s). The condition for Γ to be a ray path is that the first-order change in T due to a change in Γ is zero; that is,

$${\displaystyle \delta T=\,\delta \int _{A}^{B}{\frac {ds}{\,v_{\mathrm {r} }\,}}\,=\,0\,.}$$

Now let us define the optical length of a given path (optical path length, OPL) as the distance traversed by a ray in a homogeneous isotropic reference medium (e.g., a vacuum) in the same time that it takes to traverse the given path at the local ray velocity.^[24] Then, if c denotes the propagation speed in the reference medium (e.g., the speed of light in a vacuum), the optical length of a path traversed in time dt is dS = c dt, and the optical length of a path traversed in time T is S = cT.  So, multiplying equation (1) through by c, we obtain

$${\displaystyle S\,=\int _{A}^{B}\!dS\,=\int _{A}^{B}{\frac {\,c\,}{v_{\mathrm {r} }}}\,ds=\int _{A}^{B}\!n_{\mathrm {r} }\,ds\,,}$$

where ${\displaystyle \,n_{\mathrm {r} }=c/v_{\mathrm {r} }\,}$ is the ray index — that is, the refractive index calculated on the ray velocity instead of the usual phase velocity (wave-normal velocity). For an infinitesimal path, we have  ${\displaystyle dS=n_{\mathrm {r} \,}ds\,,\,}$ indicating that the optical length is the physical length multiplied by the ray index: the OPL is a notional geometric quantity, from which time has been factored out. In terms of OPL, the condition for Γ to be a ray path (Fermat's principle) becomes

$${\displaystyle \delta S=\delta \int _{A}^{B}\!n_{\mathrm {r} }\,ds=0.}$$

(2)

This has the form of Maupertuis's principle in classical mechanics (for a single particle), with the ray index in optics taking the role of momentum or velocity in mechanics.

In an isotropic medium, for which the ray velocity is also the phase velocity, we may substitute the usual refractive index n for n_r. 

Relation to Hamilton's principle

If x,y,z are Cartesian coordinates and an overdot denotes differentiation with respect to s , Fermat's principle (2) may be written

$${\displaystyle {\begin{aligned}\delta S&=\,\delta \int _{A}^{B}\!n_{\mathrm {r} }\,{\sqrt {dx^{2}+dy^{2}+dz^{2}}}\\&=\,\delta \int _{A}^{B}\!n_{\mathrm {r} }\,{\sqrt {{\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2}}}~ds\,=\,0\,.\end{aligned}}}$$

In the case of an isotropic medium, we may replace n_r with the normal refractive index  n(x,y,z), which is simply a scalar field. If we then define the optical Lagrangian as

$${\displaystyle L(x,y,z,{\dot {x}},{\dot {y}},{\dot {z}})\,=\,n(x,y,z)\,{\sqrt {{\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2}}}\,,}$$

Fermat's principle becomes

$${\displaystyle \delta S=\,\delta \int _{A}^{B}\!L(x,y,z,{\dot {x}},{\dot {y}},{\dot {z}})\,ds\,=\,0\,.}$$

If the direction of propagation is always such that we can use z instead of s as the parameter of the path (and the overdot to denote differentiation w.r.t. z instead of s), the optical Lagrangian can instead be written

$${\displaystyle L{\big (}x(z),y(z),{\dot {x}}(z),{\dot {y}}(z),z{\big )}=n(x,y,z)\,{\sqrt {1+{\dot {x}}^{2}+{\dot {y}}^{2}}}\,,}$$

so that Fermat's principle becomes

$${\displaystyle \delta S=\,\delta \int _{A}^{B}\!L{\big (}x(z),y(z),{\dot {x}}(z),{\dot {y}}(z),z{\big )}\,dz\,=\,0.}$$

This has the form of Hamilton's principle in classical mechanics, except that the time dimension is missing: the third spatial coordinate in optics takes the role of time in mechanics. The optical Lagrangian is the function which, when integrated w.r.t. the parameter of the path, yields the OPL; it is the foundation of Lagrangian and Hamiltonian optics.

History

Fermat vs. the Cartesians

Pierre de Fermat (1607 –1665)

If a ray follows a straight line, it obviously takes the path of least length. Hero of Alexandria, in his Catoptrics (1st century CE), showed that the ordinary law of reflection off a plane surface follows from the premise that the total length of the ray path is a minimum. In 1657, Pierre de Fermat received from Marin Cureau de la Chambre a copy of newly published treatise, in which La Chambre noted Hero's principle and complained that it did not work for refraction.

Fermat replied that refraction might be brought into the same framework by supposing that light took the path of least resistance, and that different media offered different resistances. His eventual solution, described in a letter to La Chambre dated 1 January 1662, construed "resistance" as inversely proportional to speed, so that light took the path of least time. That premise yielded the ordinary law of refraction, provided that light traveled more slowly in the optically denser medium.

Fermat's solution was a landmark in that it unified the then-known laws of geometrical optics under a variational principle or action principle, setting the precedent for the principle of least action in classical mechanics and the corresponding principles in other fields (see History of variational principles in physics). It was the more notable because it used the method of adequality, which may be understood in retrospect as finding the point where the slope of an infinitesimally short chord is zero, without the intermediate step of finding a general expression for the slope (the derivative).

It was also immediately controversial. The ordinary law of refraction was at that time attributed to René Descartes (d. 1650), who had tried to explain it by supposing that light was a force that propagated instantaneously, or that light was analogous to a tennis ball that traveled faster in the denser medium, either premise being inconsistent with Fermat's.  Descartes' most prominent defender, Claude Clerselier, criticized Fermat for apparently ascribing knowledge and intent to nature, and for failing to explain why nature should prefer to economize on time rather than distance. Clerselier wrote in part:

1. The principle that you take as the basis of your demonstration, namely that nature always acts in the shortest and simplest ways, is merely a moral principle and not a physical one; it is not, and cannot be, the cause of any effect in nature.... For otherwise we would attribute knowledge to nature; but here, by "nature", we understand only this order and this law established in the world as it is, which acts without foresight, without choice, and by a necessary determination.

2. This same principle would make nature irresolute... For I ask you... when a ray of light must pass from a point in a rare medium to a point in a dense one, is there not reason for nature to hesitate if, by your principle, it must choose the straight line as soon as the bent one, since if the latter proves shorter in time, the former is shorter and simpler in length? Who will decide and who will pronounce? 

Fermat, being unaware of the mechanistic foundations of his own principle, was not well placed to defend it, except as a purely geometric and kinematic proposition.  The wave theory of light, first proposed by Robert Hooke in the year of Fermat's death, and rapidly improved by Ignace-Gaston Pardies and (especially) Christiaan Huygens, contained the necessary foundations; but the recognition of this fact was surprisingly slow.

Huygens's oversight

Christiaan Huygens (1629–1695)

Huygens repeatedly referred to the envelope of his secondary wavefronts as the termination of the movement, meaning that the later wavefront was the outer boundary that the disturbance could reach in a given time, which was therefore the minimum time in which each point on the later wavefront could be reached. But he did not argue that the direction of minimum time was that from the secondary source to the point of tangency; instead, he deduced the ray direction from the extent of the common tangent surface corresponding to a given extent of the initial wavefront. His only endorsement of Fermat's principle was limited in scope: having derived the law of ordinary refraction, for which the rays are normal to the wavefronts, Huygens gave a geometric proof that a ray refracted according to this law takes the path of least time. He would hardly have thought this necessary if he had known that the principle of least time followed directly from the same common-tangent construction by which he had deduced not only the law of ordinary refraction, but also the laws of rectilinear propagation and ordinary reflection (which were also known to follow from Fermat's principle), and a previously unknown law of extraordinary refraction — the last by means of secondary wavefronts that were spheroidal rather than spherical, with the result that the rays were generally oblique to the wavefronts. It was as if Huygens had not noticed that his construction implied Fermat's principle, and even as if he thought he had found an exception to that principle. Manuscript evidence cited by Alan E. Shapiro tends to confirm that Huygens believed the principle of least time to be invalid "in double refraction, where the rays are not normal to the wave fronts".

Shapiro further reports that the only three authorities who accepted "Huygens' principle" in the 17th and 18th centuries, namely Philippe de La Hire, Denis Papin, and Gottfried Wilhelm Leibniz, did so because it accounted for the extraordinary refraction of "Iceland crystal" (calcite) in the same manner as the previously known laws of geometrical optics. But, for the time being, the corresponding extension of Fermat's principle went unnoticed.

Laplace, Young, Fresnel, and Lorentz

Pierre-Simon Laplace (1749–1827)

On 30 January 1809, Pierre-Simon Laplace, reporting on the work of his protégé Étienne-Louis Malus, claimed that the extraordinary refraction of calcite could be explained under the corpuscular theory of light with the aid of Maupertuis's principle of least action: that the integral of speed with respect to distance was a minimum. The corpuscular speed that satisfied this principle was proportional to the reciprocal of the ray speed given by the radius of Huygens' spheroid. Laplace continued:

According to Huygens, the velocity of the extraordinary ray, in the crystal, is simply expressed by the radius of the spheroid; consequently his hypothesis does not agree with the principle of the least action: but it is remarkable that it agrees with the principle of Fermat, which is, that light passes, from a given point without the crystal, to a given point within it, in the least possible time; for it is easy to see that this principle coincides with that of the least action, if we invert the expression of the velocity.

Thomas Young (1773–1829)

Laplace's report was the subject of a wide-ranging rebuttal by Thomas Young, who wrote in part:

The principle of Fermat, although it was assumed by that mathematician on hypothetical, or even imaginary grounds, is in fact a fundamental law with respect to undulatory motion, and is explicitly [sic] the basis of every determination in the Huygenian theory...  Mr. Laplace seems to be unacquainted with this most essential principle of one of the two theories which he compares; for he says, that "it is remarkable," that the Huygenian law of extraordinary refraction agrees with the principle of Fermat; which he would scarcely have observed, if he had been aware that the law was an immediate consequence of the principle.

In fact Laplace was aware that Fermat's principle follows from Huygens' construction in the case of refraction from an isotropic medium to an anisotropic one; a geometric proof was contained in the long version of Laplace's report, printed in 1810.

Young's claim was more general than Laplace's, and likewise upheld Fermat's principle even in the case of extraordinary refraction, in which the rays are generally not perpendicular to the wavefronts. Unfortunately, however, the omitted middle sentence of the quoted paragraph by Young began "The motion of every undulation must necessarily be in a direction perpendicular to its surface..." (emphasis added), and was therefore bound to sow confusion rather than clarity.

Augustin-Jean Fresnel (1788–1827)

No such confusion subsists in Augustin-Jean Fresnel's "Second Memoir" on double refraction (Fresnel, 1827), which addresses Fermat's principle in several places (without naming Fermat), proceeding from the special case in which rays are normal to wavefronts, to the general case in which rays are paths of least time or stationary time. (In the following summary, page numbers refer to Alfred W. Hobson's translation.)

• For refraction of a plane wave at parallel incidence on one face of an anisotropic crystalline wedge (pp. 291–2), in order to find the "first ray arrived" at an observation point beyond the other face of the wedge, it suffices to treat the rays outside the crystal as normal to the wavefronts, and within the crystal to consider only the parallel wavefronts (whatever the ray direction). So in this case, Fresnel does not attempt to trace the complete ray path.
• Next, Fresnel considers a ray refracted from a point-source M inside a crystal, through a point A on the surface, to an observation point B outside (pp. 294–6). The surface passing through B and given by the "locus of the disturbances which arrive first" is, according to Huygens' construction, normal to "the ray AB of swiftest arrival". But this construction requires knowledge of the "surface of the wave" (that is, the secondary wavefront) within the crystal.
• Then he considers a plane wavefront propagating in a medium with non-spherical secondary wavefronts, oriented so that the ray path given by Huygens' construction — from the source of the secondary wavefront to its point of tangency with the subsequent primary wavefront — is not normal to the primary wavefronts (p. 296). He shows that this path is nevertheless "the path of quickest arrival of the disturbance" from the earlier primary wavefront to the point of tangency.
• In a later heading (p. 305) he declares that "The construction of Huygens, which determines the path of swiftest arrival," is applicable to secondary wavefronts of any shape. He then notes that when we apply Huygens' construction to refraction into a crystal with a two-sheeted secondary wavefront, and draw the lines from the two points of tangency to the center of the secondary wavefront, "we shall have the directions of the two paths of swiftest arrival, and consequently of the ordinary and of the extraordinary ray."
• Under the heading "Definition of the word Ray" (p. 309), he concludes that this term must be applied to the line which joins the center of the secondary wave to a point on its surface, whatever the inclination of this line to the surface.
• As a "new consideration" (pp. 310–11), he notes that if a plane wavefront is passed through a small hole centered on point E, then the direction ED of maximum intensity of the resulting beam will be that in which the secondary wave starting from E will "arrive there the first", and the secondary wavefronts from opposite sides of the hole (equidistant from E) will "arrive at D in the same time" as each other. This direction is not assumed to be normal to any wavefront.

Thus Fresnel showed, even for anisotropic media, that the ray path given by Huygens' construction is the path of least time between successive positions of a plane or diverging wavefront, that the ray velocities are the radii of the secondary "wave surface" after unit time, and that a stationary traversal time accounts for the direction of maximum intensity of a beam. However, establishing the general equivalence between Huygens' construction and Fermat's principle would have required further consideration of Fermat's principle in point-to-point terms.

Hendrik Lorentz, in a paper written in 1886 and republished in 1907, deduced the principle of least time in point-to-point form from Huygens' construction. But the essence of his argument was somewhat obscured by an apparent dependence on aether and aether drag.

Lorentz's work was cited in 1959 by Adriaan J. de Witte, who then offered his own argument, which "although in essence the same, is believed to be more cogent and more general." De Witte's treatment is more original than that description might suggest, although limited to two dimensions; it uses calculus of variations to show that Huygens' construction and Fermat's principle lead to the same differential equation for the ray path, and that in the case of Fermat's principle, the converse holds. De Witte also noted that "The matter seems to have escaped treatment in textbooks."