Algae

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Algae
A variety of algae growing on the sea bed in shallow waters
Scientific classification
Domain: Eukaryota
Included groups
Archaeplastida Chlorophyta (green algae) Rhodophyta (red algae) Glaucophyta Charophyta Rhizaria, Excavata Chlorarachniophytes Euglenids Chromista, Alveolata Heterokonts Bacillariophyceae (Diatoms) Axodines Bolidomonas Eustigmatophyceae Phaeophyceae (brown algae) Chrysophyceae (golden algae) Raphidophyceae Synurophyceae Xanthophyceae (yellow-green algae) Cryptophyta Dinoflagellata Haptophyta (Cyanobacteria) (blue-green algae)
Excluded groups
(Cyanobacteria) (blue-green algae) Protista

The lineage of algae according to Thomas Cavalier-Smith. The exact number and placement of endosymbiotic events is currently unknown, so this diagram can be taken only as a general guide.^[1][2] It represents the most parsimonious way of explaining the three types of endosymbiotic origins of plastids. These types include the endosymbiotic events of cyanobacteria, red algae and green algae, leading to the hypothesis of the supergroups Archaeplastida, Chromalveolata and Cabozoa respectively. However, the monophyly of Cabozoa has been refuted and the monophylies of Archaeplastida and Chromalveolata are currently strongly challenged.^{[citation needed]} Endosymbiotic events are noted by dotted lines.

Algae (/ˈældʒiː/ or /ˈælɡiː/; singular alga /ˈælɡə/, Latin for "seaweed") are a very large and diverse group of eukaryotic organisms, ranging from unicellular genera such as Chlorella and the diatoms to multicellular forms such as the giant kelp, a large brown alga that may grow up to 50 meters in length. Most are autotrophic and lack many of the distinct cell and tissue types found in land plants such as stomata, xylem and phloem. The largest and most complex marine algae are called seaweeds, while the most complex freshwater forms are the Charophyta, a division of algae that includes Spirogyra and the stoneworts.

There is no generally accepted definition of algae. One definition is that algae "have chlorophyll as their primary photosynthetic pigment and lack a sterile covering of cells around their reproductive cells".^[3] Other authors exclude all prokaryotes^[4] and thus do not consider cyanobacteria (blue-green algae) as algae.^[5]

Algae constitute a polyphyletic group^[4] since they do not include a common ancestor, and although their plastids seem to have a single origin, from cyanobacteria,^[1] they were acquired in different ways. Green algae are examples of algae that have primary chloroplasts derived from endosymbiotic cyanobacteria. Diatoms are examples of algae with secondary chloroplasts derived from an endosymbiotic red alga.^[6]

Algae exhibit a wide range of reproductive strategies, from simple asexual cell division to complex forms of sexual reproduction.^[7]

Algae lack the various structures that characterize land plants, such as the phyllids (leaf-like structures) of bryophytes, rhizoids in nonvascular plants, and the roots, leaves, and other organs that are found in tracheophytes (vascular plants). Most are phototrophic, although some groups^[which?] contain members that are mixotrophic, deriving energy both from photosynthesis and uptake of organic carbon either by osmotrophy, myzotrophy, or phagotrophy. Some unicellular species of green algae, many golden algae, euglenids, dinoflagellates and other algae have become heterotrophs (also called colorless or apochlorotic algae), sometimes parasitic, relying entirely on external energy sources and have limited or no photosynthetic apparatus.^[8][9] Some other heterotrophic organisms, like the apicomplexans, are also derived from cells whose ancestors possessed plastids, but are not traditionally considered as algae. Algae have photosynthetic machinery ultimately derived from cyanobacteria that produce oxygen as a by-product of photosynthesis, unlike other photosynthetic bacteria such as purple and green sulfur bacteria. Fossilized filamentous algae from the Vindhya basin have been dated back to 1.6 to 1.7 billion years ago.^[10]

§Etymology and study

The singular alga is the Latin word for a particular seaweed^[which?] and retains that meaning in English.^[11] The etymology is obscure. Although some speculate that it is related to Latin algēre, "be cold",^[12] there is no known reason to associate seaweed with temperature. A more likely source is alliga, "binding, entwining."^[13]
The Ancient Greek word for seaweed was φῦκος (fūkos or phykos), which could mean either the seaweed (probably red algae) or a red dye derived from it. The Latinization, fūcus, meant primarily the cosmetic rouge. The etymology is uncertain, but a strong candidate has long been some word related to the Biblical פוך (pūk), "paint" (if not that word itself), a cosmetic eye-shadow used by the ancient Egyptians and other inhabitants of the eastern Mediterranean. It could be any color: black, red, green, blue.^[14]

Accordingly the modern study of marine and freshwater, algae is called either phycology or algology, depending on whether the Greek or Latin root is used. The name Fucus appears in a number of taxa.

§Classification

False-color Scanning electron micrograph of the unicellular coccolithophore Gephyrocapsa oceanica

Most algae contain chloroplasts that are similar in structure to cyanobacteria. Chloroplasts contain circular DNA like that in cyanobacteria and presumably represent reduced endosymbiotic cyanobacteria. However, the exact origin of the chloroplasts is different among separate lineages of algae, reflecting their acquisition during different endosymbiotic events. The table below describes the composition of the three major groups of algae. Their lineage relationships are shown in the figure in the upper right. Many of these groups contain some members that are no longer photosynthetic. Some retain plastids, but not chloroplasts, while others have lost plastids entirely.

Phylogeny based on plastid^[15] not nucleocytoplasmic genealogy:

	Cyanobacteria
	Cyanelles Rhodoplasts Rhodophytes Heterokonts Cryptophytes Haptophytes Chloroplasts Euglenophytes Chlorophytes Charophytes Higher plants (Embryophyta) Chlorarachniophytes

Supergroup affiliation	Members	Endosymbiont	Summary
Primoplantae/ Archaeplastida	Chlorophyta Rhodophyta Glaucophyta	Cyanobacteria	These algae have primary chloroplasts, i.e. the chloroplasts are surrounded by two membranes and probably developed through a single endosymbiotic event. The chloroplasts of red algae have chlorophylls a and c (often), and phycobilins, while those of green algae have chloroplasts with chlorophyll a and b without phycobilins. Higher plants are pigmented similarly to green algae and probably developed from them, and thus Chlorophyta is a sister taxon to the plants; sometimes Chlorophyta, Charophyta and land plants are grouped together as Viridiplantae.
Excavata and Rhizaria	Chlorarachniophytes Euglenids	Green algae	These groups have green chloroplasts containing chlorophylls a and b.[16] Their chloroplasts are surrounded by four and three membranes, respectively, and were probably retained from ingested green algae. Chlorarachniophytes, which belong to the phylum Cercozoa, contain a small nucleomorph, which is a relict of the algae's nucleus. Euglenids, which belong to the phylum Euglenozoa, live primarily in freshwater and have chloroplasts with only three membranes. It has been suggested that the endosymbiotic green algae were acquired through myzocytosis rather than phagocytosis.
Chromista and Alveolata	Heterokonts Haptophyta Cryptomonads Dinoflagellates	Red algae	These groups have chloroplasts containing chlorophylls a and c, and phycobilins.The shape varies from plant to plant. they may be of discoid, plate-like, reticulate, cup-shaped, spiral or ribbon shaped. They have one or more pyrenoids to preserve protein and starch. The latter chlorophyll type is not known from any prokaryotes or primary chloroplasts, but genetic similarities with red algae suggest a relationship there.[17] In the first three of these groups (Chromista), the chloroplast has four membranes, retaining a nucleomorph in Cryptomonads, and they likely share a common pigmented ancestor, although other evidence casts doubt on whether the Heterokonts, Haptophyta, and Cryptomonads are in fact more closely related to each other than to other groups.[2][18] The typical dinoflagellate chloroplast has three membranes, but there is considerable diversity in chloroplasts within the group, and it appears there were a number of endosymbiotic events.[1] The Apicomplexa, a group of closely related parasites, also have plastids called apicoplasts. Apicoplasts are not photosynthetic but appear to have a common origin with Dinoflagellate chloroplasts.[1]

Title page of Gmelin's Historia Fucorum, dated 1768

Linnaeus, in Species Plantarum (1753),^[19] the starting point for modern botanical nomenclature, recognized 14 genera of algae, of which only 4 are currently considered among algae.^[20] In Systema Naturae, Linnaeus described the genera Volvox and Corallina, among the animals.

In 1768, Samuel Gottlieb Gmelin (1744–1774) published the Historia Fucorum, the first work dedicated to marine algae and the first book on marine biology to use the then new binomial nomenclature of Linnaeus. It included elaborate illustrations of seaweed and marine algae on folded leaves.^[21][22]

W.H.Harvey (1811—1866) and Lamouroux (1813)^[23] were the first to divide macroscopic algae into four divisions based on their pigmentation. This is the first use of a biochemical criterion in plant systematics. Harvey's four divisions are: red algae (Rhodospermae), brown algae (Melanospermae), green algae (Chlorospermae) and Diatomaceae.^[24][25]

At this time, microscopic algae were discovered and reported by a different group of workers (e.g., O. F. Müller and Ehrenberg) studying the Infusoria (microscopic organisms). Unlike macroalgae, which were clearly viewed as plants, microalgae were frequently considered animals because they are often motile.^[26] Even the non-motile (coccoid) microalgae were sometimes merely seen as stages of the life cycle of plants, macroalgae or animals.^[27][28]

Although used as a taxonomic category in some pre-Darwinian classifications, e.g., Linnaeus (1753), de Jussieu (1789), Horaninow (1843), Agassiz (1859), Wilson & Cassin (1864), in further classifications, the "algae" are seen as an artificial, polyphyletic group.

Throughout 20th century, most classifications treated as divisions or classes of algae the following groups: cyanophytes, rhodophytes, chrysophytes, xanthophytes, bacillariophytes, phaeophytes, pyrrhophytes (cryptophytes and dinophytes), euglenophytes and chlorophytes. Later, many new groups were discovered (e.g., Bolidophyceae), and others were splintered from older groups: charophytes and glaucophytes (from chlorophytes), many heterokontophytes (e.g., haptophytes and synurophytes from chrysophytes, or eustigmatophytes from xanthophytes), and chlorarachniophytes (also from xanthophytes).

With the abandon of plant-animal dichotomous classification, most algae groups (sometimes all) were included in Protista, later also abandoned in favour of Eukaryota. However, as a legacy of the older plant-life scheme, some algae groups treated also as protozoans in the past still have duplicated classifications (see ambiregnal protists).

Some parasitic algae (e.g., the green algae Prototheca and Helicosporidium, parasites of metazoans, or Cephaleuros, parasites of plants) were originally classified as fungi, sporozoans or protistans of incertae sedis,^[29] while others (e.g., the green algae Phyllosiphon and Rhodochytrium, parasites of plants, or the red algae Pterocladiophila and Gelidiocolax mammillatus, parasites of other red algae, or the dinoflagellates Oodinium, parasites of fish) had their relationship with algae conjectured early. In other cases, some groups were originally characterized as parasitic algae (e.g., Chlorochytrium), but later were seen as endophytic algae.^[30] Furthermore, groups like the apicomplexans are also parasites derived from ancestors that possessed plastids, but are not included in any group traditionally seen as algae.

§Relationship to higher plants

The first plants on earth probably evolved from shallow freshwater charophyte algae much like Chara almost 500 million years ago. These probably had an isomorphic alternation of generations and were probably filamentous. Fossils of isolated land plant spores suggest land plants may have been around as long as 475 million years ago.^[31][32]

§Morphology

The kelp forest exhibit at the Monterey Bay Aquarium. A three-dimensional, multicellular thallus

A range of algal morphologies are exhibited, and convergence of features in unrelated groups is common. The only groups to exhibit three-dimensional multicellular thalli are the reds and browns, and some chlorophytes.^[33] Apical growth is constrained to subsets of these groups: the florideophyte reds, various browns, and the charophytes.^[33] The form of charophytes is quite different from those of reds and browns, because they have distinct nodes, separated by internode 'stems'; whorls of branches reminiscent of the horsetails occur at the nodes.^[33] Conceptacles are another polyphyletic trait; they appear in the coralline algae and the Hildenbrandiales, as well as the browns.^[33]

Most of the simpler algae are unicellular flagellates or amoeboids, but colonial and non-motile forms have developed independently among several of the groups. Some of the more common organizational levels, more than one of which may occur in the life cycle of a species, are

• Colonial: small, regular groups of motile cells
• Capsoid: individual non-motile cells embedded in mucilage
• Coccoid: individual non-motile cells with cell walls
• Palmelloid: non-motile cells embedded in mucilage
• Filamentous: a string of non-motile cells connected together, sometimes branching
• Parenchymatous: cells forming a thallus with partial differentiation of tissues

Cyanobacteria Merismopedia

In three lines even higher levels of organization have been reached, with full tissue differentiation. These are the brown algae,^[34]—some of which may reach 50 m in length (kelps)^[35]—the red algae,^[36] and the green algae.^[37] The most complex forms are found among the green algae (see Charales and Charophyta), in a lineage that eventually led to the higher land plants. The point where these non-algal plants begin and algae stop is usually taken to be the presence of reproductive organs with protective cell layers, a characteristic not found in the other alga groups.

§Physiology

Many algae, particularly members of the Characeae,^[38] have served as model experimental organisms to understand the mechanisms of the water permeability of membranes, osmoregulation, turgor regulation, salt tolerance, cytoplasmic streaming, and the generation of action potentials.

Phytohormones are found not only in higher plants, but in algae too.^[39]

§Symbiotic algae

Some species of algae form symbiotic relationships with other organisms. In these symbioses, the algae supply photosynthates (organic substances) to the host organism providing protection to the algal cells. The host organism derives some or all of its energy requirements from the algae. Examples are as follows.

§Lichens

Rock lichens in Ireland

Lichens are defined by the International Association for Lichenology to be "an association of a fungus and a photosynthetic symbiont resulting in a stable vegetative body having a specific structure."^[40] The fungi, or mycobionts, are mainly from the Ascomycota with a few from the Basidiomycota. They are not found alone in nature but when they began to associate is not known.^[41] One mycobiont associates with the same phycobiont species, rarely two, from the green algae, except that alternatively the mycobiont may associate with a species of cyanobacteria (hence "photobiont" is the more accurate term). A photobiont may be associated with many different mycobionts or may live independently; accordingly, lichens are named and classified as fungal species.^[42] The association is termed a morphogenesis because the lichen has a form and capabilities not possessed by the symbiont species alone (they can be experimentally isolated). It is possible that the photobiont triggers otherwise latent genes in the mycobiont.^[43]

§Coral reefs

Floridian coral reef

Coral reefs are accumulated from the calcareous exoskeletons of marine invertebrates of the order Scleractinia (stony corals). As animals they metabolize sugar and oxygen to obtain energy for their cell-building processes, including secretion of the exoskeleton, with water and carbon dioxide as byproducts. As the reef is the result of a favorable equilibrium between construction by the corals and destruction by marine erosion, the rate at which metabolism can proceed determines the growth or deterioration of the reef.

Dinoflagellates (algal protists) are often endosymbionts in the cells of marine invertebrates, where they accelerate host-cell metabolism by generating immediately available sugar and oxygen through photosynthesis using incident light and the carbon dioxide produced by the host. Stony corals that are reef-building corals (hermatypic corals) require endosymbiotic algae from the genus Symbiodinium to be in a healthy condition.^[44] The loss of Symbiodinium from the host is known as coral bleaching, a condition which leads to the deterioration of a reef.

§Sea sponges

Green algae live close to the surface of some sponges, for example, breadcrumb sponge (Halichondria panicea). The alga is thus protected from predators; the sponge is provided with oxygen and sugars which can account for 50 to 80% of sponge growth in some species.^[45]

§Life-cycle

Rhodophyta, Chlorophyta and Heterokontophyta, the three main algal phyla, have life-cycles which show tremendous variation with considerable complexity. In general there is an asexual phase where the seaweed's cells are diploid, a sexual phase where the cells are haploid followed by fusion of the male and female gametes. Asexual reproduction is advantageous in that it permits efficient population increases, but less variation is possible. Sexual reproduction allows more variation, but is more costly. Often there is no strict alternation between the sporophyte and also because there is often an asexual phase, which could include the fragmentation of the thallus.^[35][46]

For more details on this topic, see Conceptacle.

§Numbers

Algae on coastal rocks at Shihtiping in Taiwan

The Algal Collection of the US National Herbarium (located in the National Museum of Natural History) consists of approximately 320,500 dried specimens, which, although not exhaustive (no exhaustive collection exists), gives an idea of the order of magnitude of the number of algal species (that number remains unknown).^[47] Estimates vary widely. For example, according to one standard textbook,^[48] in the British Isles the UK Biodiversity Steering Group Report estimated there to be 20000 algal species in the UK. Another checklist reports only about 5000 species. Regarding the difference of about 15000 species, the text concludes: "It will require many detailed field surveys before it is possible to provide a reliable estimate of the total number of species ..."

Regional and group estimates have been made as well:

• 5000–5500 species of red algae worldwide
• "some 1300 in Australian Seas"^[49]
• 400 seaweed species for the western coastline of South Africa,^[50] and 212 species from the coast of KwaZulu-Natal.^[51] Some of these are duplicates as the range extends across both coasts, and the total recorded is probably about 500 species. Most of these are listed in List of seaweeds of South Africa. These exclude phytoplankton and crustose corallines.
• 669 marine species from California (US)^[52]
• 642 in the check-list of Britain and Ireland^[53]

and so on, but lacking any scientific basis or reliable sources, these numbers have no more credibility than the British ones mentioned above. Most estimates also omit microscopic algae, such as phytoplankton.

The most recent estimate suggests a total number of 72,500 algal species worldwide.^[54]

§Distribution

The distribution of algal species has been fairly well studied since the founding of phytogeography in the mid-19th century AD.^[55] Algae spread mainly by the dispersal of spores analogously to the dispersal of Plantae by seeds and spores. Spores are everywhere in all parts of the Earth: the waters fresh and marine, the atmosphere, free-floating and in precipitation or mixed with dust, the humus and in other organisms, such as humans. Whether a spore is to grow into an organism depends on the combination of the species and the environmental conditions of where the spore lands.

The spores of fresh-water algae are dispersed mainly by running water and wind, as well as by living carriers.^[56] The bodies of water into which they are transported are chemically selective.^{[clarification needed]} Marine spores are spread by currents. Ocean water is temperature selective, resulting in phytogeographic zones, regions and provinces.^[57]

To some degree the distribution of algae is subject to floristic discontinuities caused by geographical features, such as Antarctica, long distances of ocean or general land masses. It is therefore possible to identify species occurring by locality, such as "Pacific Algae" or "North Sea Algae". When they occur out of their localities, it is usually possible to hypothesize a transport mechanism, such as the hulls of ships. For example, Ulva reticulata and Ulva fasciata travelled from the mainland to Hawaii in this manner.

Mapping is possible for select species only: "there are many valid examples of confined distribution patterns."^[58] For example, Clathromorphum is an arctic genus and is not mapped far south of there.^[59] On the other hand, scientists regard the overall data as insufficient due to the "difficulties of undertaking such studies."^[60]

§Ecology

Phytoplankton, Lake Chuzenji

Algae are prominent in bodies of water, common in terrestrial environments and are found in unusual environments, such as on snow and on ice. Seaweeds grow mostly in shallow marine waters, under 100 metres (330 ft); however some have been recorded to a depth of 360 metres (1,180 ft).^[61]

The various sorts of algae play significant roles in aquatic ecology. Microscopic forms that live suspended in the water column (phytoplankton) provide the food base for most marine food chains. In very high densities (algal blooms) these algae may discolor the water and outcompete, poison, or asphyxiate other life forms.

Algae are variously sensitive to different factors, which has made them useful as biological indicators in the Ballantine Scale and its modification.

On the basis of their habitat, algae can be categorized as: aquatic (planktonic, benthic, marine, freshwater), terrestrial, aerial (subareial),^[62] lithophytic, halophytic (or euryhaline), psammon, thermophilic, cryophilic, epibiont (epiphytic, epizoic), endosymbiont (endophytic, endozoic), parasitic, calcifilic or lichenic (phycobiont).^[63]

§Cultural associations

In Classical Chinese, the word 藻 is used both for "algae" and (in the modest tradition of the imperial scholars) for "literary talent". The third island in Kunming Lake beside the Summer Palace in Beijing is known as the Zaojian Tang Dao which thus simultaneously means "Island of the Algae-Viewing Hall" and "Island of the Hall for Reflecting on Literary Talent".

§Uses

Harvesting algae

§Agar

Agar, a gelatinous substance derived from red algae, has a number of commercial uses.^[64] It is a good medium on which to grow bacteria and fungi as most microorganisms cannot digest agar.

§Alginates

Alginic acid, or alginate, is extracted from brown algae. Its uses range from gelling agents in food, to medical dressings. Alginic acid also has been used in the field of biotechnology as a biocompatible medium for cell encapsulation and cell immobilization. Molecular cuisine is also a user of the substance for its gelling properties, by which it becomes a delivery vehicle for flavours.

Between 100,000 and 170,000 wet tons of Macrocystis are harvested annually in New Mexico for alginate extraction and abalone feed.^[65][66]

§Energy source

To be competitive and independent from fluctuating support from (local) policy on the long run, biofuels should equal or beat the cost level of fossil fuels. Here, algae based fuels hold great promise,^[67][68] directly related to the potential to produce more biomass per unit area in a year than any other form of biomass. The break-even point for algae-based biofuels is estimated to occur by 2025.^[69]

§Fertilizer

Seaweed-fertilized gardens on Inisheer

For centuries seaweed has been used as a fertilizer; George Owen of Henllys writing in the 16th century referring to drift weed in South Wales:^[70]

This kind of ore they often gather and lay on great heapes, where it heteth and rotteth, and will have a strong and loathsome smell; when being so rotten they cast on the land, as they do their muck, and thereof springeth good corn, especially barley ... After spring-tydes or great rigs of the sea, they fetch it in sacks on horse backes, and carie the same three, four, or five miles, and cast it on the lande, which doth very much better the ground for corn and grass.

Today, algae are used by humans in many ways; for example, as fertilizers, soil conditioners and livestock feed.^[71]
Aquatic and microscopic species are cultured in clear tanks or ponds and are either harvested or used to treat effluents pumped through the ponds. Algaculture on a large scale is an important type of aquaculture in some places. Maerl is commonly used as a soil conditioner.

§Nutrition

Dulse, a type of food

Naturally growing seaweeds are an important source of food, especially in Asia. They provide many vitamins including: A, B₁, B₂, B₆, niacin and C, and are rich in iodine, potassium, iron, magnesium and calcium.^[72] In addition commercially cultivated microalgae, including both algae and cyanobacteria, are marketed as nutritional supplements, such as Spirulina,^[73] Chlorella and the Vitamin-C supplement, Dunaliella, high in beta-carotene.

Algae are national foods of many nations: China consumes more than 70 species, including fat choy, a cyanobacterium considered a vegetable; Japan, over 20 species;^[74] Ireland, dulse; Chile, cochayuyo.^[75] Laver is used to make "laver bread" in Wales where it is known as bara lawr; in Korea, gim; in Japan, nori and aonori. It is also used along the west coast of North America from California to British Columbia, in Hawaii and by the Māori of New Zealand. Sea lettuce and badderlocks are a salad ingredient in Scotland, Ireland, Greenland and Iceland.

The oils from some algae have high levels of unsaturated fatty acids. For example, Parietochloris incisa is very high in arachidonic acid, where it reaches up to 47% of the triglyceride pool.^[76] Some varieties of algae favored by vegetarianism and veganism contain the long-chain, essential omega-3 fatty acids, Docosahexaenoic acid (DHA) and Eicosapentaenoic acid (EPA). Fish oil contains the omega-3 fatty acids, but the original source is algae (microalgae in particular), which are eaten by marine life such as copepods and are passed up the food chain.^[77] Algae has emerged in recent years as a popular source of omega-3 fatty acids for vegetarians who cannot get long-chain EPA and DHA from other vegetarian sources such as flaxseed oil, which only contains the short-chain Alpha-Linolenic acid (ALA).

§Pollution control

• Sewage can be treated with algae, reducing the usage of large amounts of toxic chemicals that would otherwise be needed.
• Algae can be used to capture fertilizers in runoff from farms. When subsequently harvested, the enriched algae itself can be used as fertilizer.
• Aquariums and ponds can be filtered using algae, which absorb nutrients from the water in a device called an algae scrubber, also known as an algae turf scrubber (A T S) .^{[78][79][80][81]}

Agricultural Research Service scientists found that 60–90% of nitrogen runoff and 70–100% of phosphorus runoff can be captured from manure effluents using a horizontal algae scrubber, also called an algal turf scrubber (ATS).
Scientists developed the ATS, which are shallow, 100-foot raceways of nylon netting where algae colonies can form, and studied its efficacy for three years. They found that algae can readily be used to reduce the nutrient runoff from agricultural fields and increase the quality of water flowing into rivers, streams, and oceans. The enriched algae itself also can be used as a fertilizer. Researchers collected and dried the nutrient-rich algae from the ATS and studied its potential as an organic fertilizer. They found that cucumber and corn seedlings grew just as well using ATS organic fertilizer as they did with commercial fertilizers.^[82] Algae scrubbers, using bubbling upflow or vertical waterfall versions, are now also being used to filter aquariums and ponds.

§Bioremediation

The algae Stichococcus bacillaris, has been seen to colonize silicone resins used at archaeological sites; biodeterriorating the synthetic substance.^[83]

§Pigments

The natural pigments (carotenoids and chlorophylls) produced by algae can be used as an alternative to chemical dyes and coloring agents.^[84] Presence of some individual alga pigments, together with specific pigment concentrations ratios, are taxon-specific: analysis of their concentrations with various analytical methods, particularly HPLC, can therefore offer deep insight into the taxonomic composition and relative abundance of natural alga populations in sea water samples.^[85][86]

Carl Friedrich Gauss

From Wikipedia, the free encyclopedia

Carl Friedrich Gauss
Carl Friedrich Gauß (1777–1855), painted by Christian Albrecht Jensen
Born	Johann Carl Friedrich Gauss (1777-04-30)30 April 1777 Brunswick, Duchy of Brunswick-Wolfenbüttel, Holy Roman Empire
Died	23 February 1855(1855-02-23) (aged 77) Göttingen, Kingdom of Hanover
Residence	Kingdom of Hanover
Nationality	German
Fields	Mathematics and physics
Institutions	University of Göttingen
Alma mater	University of Helmstedt
Doctoral advisor	Johann Friedrich Pfaff
Other academic advisors	Johann Christian Martin Bartels
Doctoral students	Christoph Gudermann Christian Ludwig Gerling Richard Dedekind Johann Listing Bernhard Riemann Christian Peters Moritz Cantor
Other notable students	Johann Encke Peter Gustav Lejeune Dirichlet Gotthold Eisenstein Carl Wolfgang Benjamin Goldschmidt Gustav Kirchhoff Ernst Kummer August Ferdinand Möbius L. C. Schnürlein Julius Weisbach
Known for	See full list
Influenced	Sophie Germain Ferdinand Minding
Notable awards	Lalande Prize (1810) Copley Medal (1838)
Signature

Johann Carl Friedrich Gauss (/ɡaʊs/; German: Gauß, pronounced [ɡaʊs] ( listen); Latin: Carolus Fridericus Gauss) (30 April 1777 – 23 February 1855) was a German mathematician who contributed significantly to many fields, including number theory, algebra, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy, matrix theory, and optics.

Sometimes referred to as the Princeps mathematicorum^[1] (Latin, "the Prince of Mathematicians" or "the foremost of mathematicians") and "greatest mathematician since antiquity", Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians.^[2]

§Early years

Statue of Gauss at his birthplace, Brunswick

Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Braunschweig), in the Duchy of Brunswick-Wolfenbüttel (now part of Lower Saxony, Germany), as the son of poor working-class parents.^[3] His mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension, which itself occurs 40 days after Easter. Gauss would later solve this puzzle about his birthdate in the context of finding the date of Easter, deriving methods to compute the date in both past and future years.^[4] He was christened and confirmed in a church near the school he attended as a child.^[5]

Gauss was a child prodigy. There are many anecdotes about his precocity while a toddler, and he made his first ground-breaking mathematical discoveries while still a teenager. He completed Disquisitiones Arithmeticae, his magnum opus, in 1798 at the age of 21, though it was not published until 1801. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day.

Gauss's intellectual abilities attracted the attention of the Duke of Brunswick,^[2] who sent him to the Collegium Carolinum (now Braunschweig University of Technology), which he attended from 1792 to 1795, and to the University of Göttingen from 1795 to 1798. While at university, Gauss independently rediscovered several important theorems;^[6] his breakthrough occurred in 1796 when he showed that any regular polygon with a number of sides which is a Fermat prime (and, consequently, those polygons with any number of sides which is the product of distinct Fermat primes and a power of 2) can be constructed by compass and straightedge. This was a major discovery in an important field of mathematics; construction problems had occupied mathematicians since the days of the Ancient Greeks, and the discovery ultimately led Gauss to choose mathematics instead of philology as a career. Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone. The stonemason declined, stating that the difficult construction would essentially look like a circle.^[7]

The year 1796 was most productive for both Gauss and number theory. He discovered a construction of the heptadecagon on 30 March.^[8] He further advanced modular arithmetic, greatly simplifying manipulations in number theory.^{[citation needed]} On 8 April he became the first to prove the quadratic reciprocity law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic. The prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers.

Gauss also discovered that every positive integer is representable as a sum of at most three triangular numbers on 10 July and then jotted down in his diary the note: "ΕΥΡΗΚΑ! num = Δ + Δ + Δ". On October 1 he published a result on the number of solutions of polynomials with coefficients in finite fields, which 150 years later led to the Weil conjectures.

§Middle years

In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. Mathematicians including Jean le Rond d'Alembert had produced false proofs before him, and Gauss's dissertation contains a critique of d'Alembert's work. Ironically, by today's standard, Gauss's own attempt is not acceptable, owing to implicit use of the Jordan curve theorem. However, he subsequently produced three other proofs, the last one in 1849 being generally rigorous. His attempts clarified the concept of complex numbers considerably along the way.

Gauss also made important contributions to number theory with his 1801 book Disquisitiones Arithmeticae (Latin, Arithmetical Investigations), which, among other things, introduced the symbol ≡ for congruence and used it in a clean presentation of modular arithmetic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon (17-sided polygon) can be constructed with straightedge and compass.

Title page of Gauss's Disquisitiones Arithmeticae

In that same year, Italian astronomer Giuseppe Piazzi discovered the dwarf planet Ceres. Piazzi could only track Ceres for somewhat more than a month, following it for three degrees across the night sky. Then it disappeared temporarily behind the glare of the Sun. Several months later, when Ceres should have reappeared, Piazzi could not locate it: the mathematical tools of the time were not able to extrapolate a position from such a scant amount of data—three degrees represent less than 1% of the total orbit.

Gauss, who was 24 at the time, heard about the problem and tackled it. After three months of intense work, he predicted a position for Ceres in December 1801—just about a year after its first sighting—and this turned out to be accurate within a half-degree when it was rediscovered by Franz Xaver von Zach on 31 December at Gotha, and one day later by Heinrich Olbers in Bremen.

Gauss's method involved determining a conic section in space, given one focus (the Sun) and the conic's intersection with three given lines (lines of sight from the Earth, which is itself moving on an ellipse, to the planet) and given the time it takes the planet to traverse the arcs determined by these lines (from which the lengths of the arcs can be calculated by Kepler's Second Law). This problem leads to an equation of the eighth degree, of which one solution, the Earth's orbit, is known. The solution sought is then separated from the remaining six based on physical conditions. In this work Gauss used comprehensive approximation methods which he created for that purpose.^[9]

One such method was the fast Fourier transform. While this method is traditionally attributed to a 1965 paper by J. W. Cooley and J. W. Tukey, Gauss developed it as a trigonometric interpolation method. His paper, Theoria Interpolationis Methodo Nova Tractata, was only published posthumously in Volume 3 of his collected works. This paper predates the first presentation by Joseph Fourier on the subject in 1807.^[10]

Zach noted that "without the intelligent work and calculations of Doctor Gauss we might not have found Ceres again". Though Gauss had up to that point been financially supported by his stipend from the Duke, he doubted the security of this arrangement, and also did not believe pure mathematics to be important enough to deserve support. Thus he sought a position in astronomy, and in 1807 was appointed Professor of Astronomy and Director of the astronomical observatory in Göttingen, a post he held for the remainder of his life.

The discovery of Ceres led Gauss to his work on a theory of the motion of planetoids disturbed by large planets, eventually published in 1809 as Theoria motus corporum coelestium in sectionibus conicis solem ambientum (Theory of motion of the celestial bodies moving in conic sections around the Sun). In the process, he so streamlined the cumbersome mathematics of 18th century orbital prediction that his work remains a cornerstone of astronomical computation.^{[citation needed]} It introduced the Gaussian gravitational constant, and contained an influential treatment of the method of least squares, a procedure used in all sciences to this day to minimize the impact of measurement error. Gauss proved the method under the assumption of normally distributed errors (see Gauss–Markov theorem; see also Gaussian). The method had been described earlier by Adrien-Marie Legendre in 1805, but Gauss claimed that he had been using it since 1795.^{[citation needed]}

Gauss's portrait published in Astronomische Nachrichten 1828

In 1818 Gauss, putting his calculation skills to practical use, carried out a geodesic survey of the Kingdom of Hanover, linking up with previous Danish surveys. To aid the survey, Gauss invented the heliotrope, an instrument that uses a mirror to reflect sunlight over great distances, to measure positions.

Gauss also claimed to have discovered the possibility of non-Euclidean geometries but never published it. This discovery was a major paradigm shift in mathematics, as it freed mathematicians from the mistaken belief that Euclid's axioms were the only way to make geometry consistent and non-contradictory. Research on these geometries led to, among other things, Einstein's theory of general relativity, which describes the universe as non-Euclidean. His friend Farkas Wolfgang Bolyai with whom Gauss had sworn "brotherhood and the banner of truth" as a student, had tried in vain for many years to prove the parallel postulate from Euclid's other axioms of geometry. Bolyai's son, János Bolyai, discovered non-Euclidean geometry in 1829; his work was published in 1832. After seeing it, Gauss wrote to Farkas Bolyai: "To praise it would amount to praising myself. For the entire content of the work ... coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years."

Four Gaussian distributions in statistics

This unproved statement put a strain on his relationship with János Bolyai (who thought that Gauss was "stealing" his idea), but it is now generally taken at face value.^[11] Letters from Gauss years before 1829 reveal him obscurely discussing the problem of parallel lines. Waldo Dunnington, a biographer of Gauss, argues in Gauss, Titan of Science that Gauss was in fact in full possession of non-Euclidean geometry long before it was published by János Bolyai, but that he refused to publish any of it because of his fear of controversy.^[12][13]

The geodetic survey of Hanover, which required Gauss to spend summers traveling on horseback for a decade,^[14] fueled Gauss's interest in differential geometry, a field of mathematics dealing with curves and surfaces. Among other things he came up with the notion of Gaussian curvature. This led in 1828 to an important theorem, the Theorema Egregium (remarkable theorem), establishing an important property of the notion of curvature. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring angles and distances on the surface. That is, curvature does not depend on how the surface might be embedded in 3-dimensional space or 2-dimensional space.

In 1821, he was made a foreign member of the Royal Swedish Academy of Sciences.

§Later years and death

Daguerreotype of Gauss on his deathbed, 1855.

Grave of Gauss at Albanifriedhof in Göttingen, Germany.

In 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism (including finding a representation for the unit of magnetism in terms of mass, charge, and time) and the discovery of Kirchhoff's circuit laws in electricity. It was during this time that he formulated his namesake law. They constructed the first electromechanical telegraph in 1833, which connected the observatory with the institute for physics in Göttingen. Gauss ordered a magnetic observatory to be built in the garden of the observatory, and with Weber founded the "Magnetischer Verein" (magnetic club in German), which supported measurements of Earth's magnetic field in many regions of the world. He developed a method of measuring the horizontal intensity of the magnetic field which was in use well into the second half of the 20th century, and worked out the mathematical theory for separating the inner and outer (magnetospheric) sources of Earth's magnetic field.

In 1840, Gauss published his influential Dioptrische Untersuchungen,^[15] in which he gave the first systematic analysis on the formation of images under a paraxial approximation (Gaussian optics).^[16] Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points^[17] and he derived the Gaussian lens formula.^[18]

In 1854, Gauss selected the topic for Bernhard Riemann's Habilitationvortrag, Über die Hypothesen, welche der Geometrie zu Grunde liegen.^[19] On the way home from Riemann's lecture, Weber reported that Gauss was full of praise and excitement.^[20]

Gauss died in Göttingen, in the Kingdom of Hanover (now part of Lower Saxony, Germany) on 23 February 1855^[3] and is interred in the Albanifriedhof cemetery there. Two individuals gave eulogies at his funeral: Gauss's son-in-law Heinrich Ewald and Wolfgang Sartorius von Waltershausen, who was Gauss's close friend and biographer. His brain was preserved and was studied by Rudolf Wagner who found its mass to be 1,492 grams (slightly above average) and the cerebral area equal to 219,588 square millimeters^[21] (340.362 square inches). Highly developed convolutions were also found, which in the early 20th century was suggested as the explanation of his genius.^[22]

§Religious views

Gauss biographer G. Waldo Dunnington describes Gauss's religious views in these terms:

For him science was the means of exposing the immortal nucleus of the human soul. In the days of his full strength it furnished him recreation and, by the prospects which it opened up to him, gave consolation. Toward the end of his life it brought him confidence. Gauss' God was not a cold and distant figment of metaphysics, nor a distorted caricature of embittered theology. To man is not vouchsafed that fullness of knowledge which would warrant his arrogantly holding that his blurred vision is the full light and that there can be none other which might report truth as does his. For Gauss, not he who mumbles his creed, but he who lives it, is accepted. He believed that a life worthily spent here on earth is the best, the only, preparation for heaven. Religion is not a question of literature, but of life. God's revelation is continuous, not contained in tablets of stone or sacred parchment. A book is inspired when it inspires. The unshakeable idea of personal continuance after death, the firm belief in a last regulator of things, in an eternal, just, omniscient, omnipotent God, formed the basis of his religious life, which harmonized completely with his scientific research.^[23]

Apart from his correspondence, there are not many known details about Gauss' personal creed. Many biographers of Gauss disagree with his religious stance, with Bühler and others who considered him a deist with very unorthodox impressions,^[24][25][26] while Dunnington (though admits while Gauss didn't believe literally in all Christian dogmas and it's unknown if he believe in most doctrinal and confessional questions) points out that he was, at least, a nominal Lutheran.^[27]

In connection to this, there's a record of a conversation between Rudolf Wagner and Gauss, in which they discussed William Whewell's book Of the Plurality of Worlds. In this work, Whewell had discarded the possibility of existing life in other planets, on the basis of theological arguments, but this was a position with which both Wagner and Gauss disagreed. Later Wagner explained that he did not fully believe in the Bible, though he confessed that he "envied" those who were able to easily believe.^[24][28] This later led them to discuss the topic of faith, and in some other religious remarks, Gauss said that he had been more influenced by theologians like Paul Gerhardt, than by Moses;^[29] Other of his religious influences included Wilhelm Braubach, Johann Peter Süssmilch, and the New Testament.^[30]

Dunnington further elaborates on Gauss's religious views by writing:

Gauss' religious consciousness was based on an insatiable thirst for truth and a deep feeling of justice extending to intellectual as well as material goods. He conceived spiritual life in the whole universe as a great system of law penetrated by eternal truth, and from this source he gained the firm confidence that death does not end all.^[31]

Gauss declared he firmly believed in the afterlife, and saw spirituality as something essentially important for human beings.^[32] He was quoted stating: "The world would be nonsense, the whole creation an absurdity without immortality,"^[33] and for this statement he was severely criticized by the atheist Eugen Dühring who judged him as a narrow superstitious man.^[34]

Though he was not a church-goer,^[35] Gauss strongly upheld religious tolerance, believing "that one is not justified in disturbing another's religious belief, in which they find consolation for earthly sorrows in time of trouble."^[2] When his son Eugene announced that he wanted to become a Christian missionary, Gauss approved him saying that regardless of the problems within religious organizations, missionary work was "a highly honorable" task.^[36]

§Family

Gauss's daughter Therese (1816—1864)

Gauss's personal life was overshadowed by the early death of his first wife, Johanna Osthoff, in 1809, soon followed by the death of one child, Louis. Gauss plunged into a depression from which he never fully recovered. He married again, to Johanna's best friend named Friederica Wilhelmine Waldeck but commonly known as Minna. When his second wife died in 1831 after a long illness,^[37] one of his daughters, Therese, took over the household and cared for Gauss until the end of his life. His mother lived in his house from 1817 until her death in 1839.^[2]

Gauss had six children. With Johanna (1780–1809), his children were Joseph (1806–1873), Wilhelmina (1808–1846) and Louis (1809–1810). With Minna Waldeck he also had three children: Eugene (1811–1896), Wilhelm (1813–1879) and Therese (1816–1864). Eugene shared a good measure of Gauss's talent in languages and computation.^[38] Therese kept house for Gauss until his death, after which she married.

Gauss eventually had conflicts with his sons. He did not want any of his sons to enter mathematics or science for "fear of lowering the family name", as he believed none of them would surpass his own achievements.^[38] Gauss wanted Eugene to become a lawyer, but Eugene wanted to study languages. They had an argument over a party Eugene held, which Gauss refused to pay for. The son left in anger and, in about 1832, emigrated to the United States, where he was quite successful. While working for the American Fur Company in the Midwest, he learned the Sioux language. Later he moved to Missouri and became a successful businessman. Wilhelm also moved to America in 1837, and settled in Missouri, starting as a farmer and later becoming wealthy in the shoe business in St. Louis. It took many years for Eugene's success to counteract his reputation among Gauss's friends and colleagues.

§Personality

Carl Gauss was an ardent perfectionist and a hard worker. He was never a prolific writer, refusing to publish work which he did not consider complete and above criticism. This was in keeping with his personal motto pauca sed matura ("few, but ripe"). His personal diaries indicate that he had made several important mathematical discoveries years or decades before his contemporaries published them. Mathematical historian Eric Temple Bell said that, had Gauss published all of his discoveries in a timely manner, he would have advanced mathematics by fifty years.^[39]
Though he did take in a few students, Gauss was known to dislike teaching. It is said that he attended only a single scientific conference, which was in Berlin in 1828. However, several of his students became influential mathematicians, among them Richard Dedekind, Bernhard Riemann, and Friedrich Bessel. Before she died, Sophie Germain was recommended by Gauss to receive her honorary degree.

Gauss usually declined to present the intuition behind his often very elegant proofs—he preferred them to appear "out of thin air" and erased all traces of how he discovered them.^{[citation needed]} This is justified, if unsatisfactorily, by Gauss in his "Disquisitiones Arithmeticae", where he states that all analysis (i.e., the paths one travelled to reach the solution of a problem) must be suppressed for sake of brevity.

Gauss supported the monarchy and opposed Napoleon, whom he saw as an outgrowth of revolution.

§Anecdotes

There are several stories of his early genius. According to one, his gifts became very apparent at the age of three when he corrected, mentally and without fault in his calculations, an error his father had made on paper while calculating finances.

Another story has it that in primary school after the young Gauss misbehaved, his teacher, J.G. Büttner, gave him a task: add a list of integers in arithmetic progression; as the story is most often told, these were the numbers from 1 to 100. The young Gauss reputedly produced the correct answer within seconds, to the astonishment of his teacher and his assistant Martin Bartels.

Gauss's presumed method was to realize that pairwise addition of terms from opposite ends of the list yielded identical intermediate sums: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on, for a total sum of 50 × 101 = 5050. However, the details of the story are at best uncertain (see^[40] for discussion of the original Wolfgang Sartorius von Waltershausen source and the changes in other versions); some authors, such as Joseph Rotman in his book A first course in Abstract Algebra, question whether it ever happened.

According to Isaac Asimov, Gauss was once interrupted in the middle of a problem and told that his wife was dying. He is purported to have said, "Tell her to wait a moment till I'm done."^[41] This anecdote is briefly discussed in G. Waldo Dunnington's Gauss, Titan of Science where it is suggested that it is an apocryphal story.

He referred to mathematics as "the queen of sciences"^[42] and supposedly once espoused a belief in the necessity of immediately understanding Euler's identity as a benchmark pursuant to becoming a first-class mathematician.^[43]

§Commemorations

German 10-Deutsche Mark Banknote (1993; discontinued) featuring Gauss

Gauss (aged about 26) on East German stamp produced in 1977. Next to him: heptadecagon, compass and straightedge.

From 1989 through 2001, Gauss's portrait, a normal distribution curve and some prominent Göttingen buildings were featured on the German ten-mark banknote. The reverse featured the approach for Hanover. Germany has also issued three postage stamps honoring Gauss. One (no. 725) appeared in 1955 on the hundredth anniversary of his death; two others, nos. 1246 and 1811, in 1977, the 200th anniversary of his birth.

Daniel Kehlmann's 2005 novel Die Vermessung der Welt, translated into English as Measuring the World (2006), explores Gauss's life and work through a lens of historical fiction, contrasting them with those of the German explorer Alexander von Humboldt. A film version directed by Detlev Buck was released in 2012.^[44]

In 2007 a bust of Gauss was placed in the Walhalla temple.^[45]

Things named in honor of Gauss include:

• The Gauss Prize, one of the highest honors in mathematics
• Gauss's Law and Gauss's law for magnetism, two of Maxwell's four equations.
• Degaussing, the process of eliminating a magnetic field
• The CGS unit for magnetic field was named gauss in his honour
• The crater Gauss on the Moon^[46]
• Asteroid 1001 Gaussia
• The ship Gauss, used in the Gauss expedition to the Antarctic
• Gaussberg, an extinct volcano discovered by the above-mentioned expedition
• Gauss Tower, an observation tower in Dransfeld, Germany
• In Canadian junior high schools, an annual national mathematics competition (Gauss Mathematics Competition) administered by the Centre for Education in Mathematics and Computing is named in honour of Gauss
• In University of California, Santa Cruz, in Crown College, a dormitory building is named after him
• The Gauss Haus, an NMR center at the University of Utah
• The Carl-Friedrich-Gauß School for Mathematics, Computer Science, Business Administration, Economics, and Social Sciences of Braunschweig University of Technology
• The Gauss Building at the University of Idaho (College of Engineering)
• The Carl-Friedrich-Gauss Gymnasium (a school for grades 5-13) in Worms, Germany

In 1929 the Polish mathematician Marian Rejewski, who would solve the German Enigma cipher machine in December 1932, began studying actuarial statistics at Göttingen. At the request of his Poznań University professor, Zdzisław Krygowski, on arriving at Göttingen Rejewski laid flowers on Gauss's grave.^[47]

§Writings

• 1799: Doctoral dissertation on the fundamental theorem of algebra, with the title: Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse ("New proof of the theorem that every integral algebraic function of one variable can be resolved into real factors (i.e., polynomials) of the first or second degree")
• 1801: Disquisitiones Arithmeticae (Latin). A German translation by H. Maser "Untersuchungen über höhere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition)". New York: Chelsea. 1965. ISBN 0-8284-0191-8. , pp. 1–453. English translation by Arthur A. Clarke "Disquisitiones Arithemeticae (Second, corrected edition)". New York: Springer. 1986. ISBN 0-387-96254-9. .
• 1808: "Theorematis arithmetici demonstratio nova". Göttingen: Commentationes Societatis Regiae Scientiarum Gottingensis. 16. . German translation by H. Maser "Untersuchungen über höhere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition)". New York: Chelsea. 1965. ISBN 0-8284-0191-8. , pp. 457–462 [Introduces Gauss's lemma, uses it in the third proof of quadratic reciprocity]
• 1809: Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium (Theorie der Bewegung der Himmelskörper, die die Sonne in Kegelschnitten umkreisen), Theory of the Motion of Heavenly Bodies Moving about the Sun in Conic Sections (English translation by C. H. Davis), reprinted 1963, Dover, New York.
• 1811: "Summatio serierun quarundam singularium". Göttingen: Commentationes Societatis Regiae Scientiarum Gottingensis. . German translation by H. Maser "Untersuchungen über höhere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition)". New York: Chelsea. 1965. ISBN 0-8284-0191-8. , pp. 463–495 [Determination of the sign of the quadratic Gauss sum, uses this to give the fourth proof of quadratic reciprocity]
• 1812: Disquisitiones Generales Circa Seriem Infinitam
• 1818: "Theorematis fundamentallis in doctrina de residuis quadraticis demonstrationes et amplicationes novae". Göttingen: Commentationes Societatis Regiae Scientiarum Gottingensis. . German translation by H. Maser "Untersuchungen über höhere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition)". New York: Chelsea. 1965. ISBN 0-8284-0191-8. , pp. 496–510 [Fifth and sixth proofs of quadratic reciprocity]
• 1821, 1823 and 1826: Theoria combinationis observationum erroribus minimis obnoxiae. Drei Abhandlungen betreffend die Wahrscheinlichkeitsrechnung als Grundlage des Gauß'schen Fehlerfortpflanzungsgesetzes. (Three essays concerning the calculation of probabilities as the basis of the Gaussian law of error propagation) English translation by G. W. Stewart, 1987, Society for Industrial Mathematics.
• 1827: Disquisitiones generales circa superficies curvas, Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores. Volume VI, pp. 99–146. "General Investigations of Curved Surfaces" (published 1965) Raven Press, New York, translated by A.M.Hiltebeitel and J.C.Morehead.
• 1828: "Theoria residuorum biquadraticorum, Commentatio prima". Göttingen: Commentationes Societatis Regiae Scientiarum Gottingensis. 6. . German translation by H. Maser "Untersuchungen über höhere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition)". New York: Chelsea. 1965. ISBN 0-8284-0191-8. , pp. 511–533 [Elementary facts about biquadratic residues, proves one of the supplements of the law of biquadratic reciprocity (the biquadratic character of 2)]
• 1832: "Theoria residuorum biquadraticorum, Commentatio secunda". Göttingen: Commentationes Societatis Regiae Scientiarum Gottingensis. 7. . German translation by H. Maser "Untersuchungen über höhere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition)". New York: Chelsea. 1965. ISBN 0-8284-0191-8. , pp. 534–586 [Introduces the Gaussian integers, states (without proof) the law of biquadratic reciprocity, proves the supplementary law for 1 + i]
• 1843/44: Untersuchungen über Gegenstände der Höheren Geodäsie. Erste Abhandlung, Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen. Zweiter Band, pp. 3–46
• 1846/47: Untersuchungen über Gegenstände der Höheren Geodäsie. Zweite Abhandlung, Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen. Dritter Band, pp. 3–44
• Mathematisches Tagebuch 1796–1814, Ostwaldts Klassiker, Harri Deutsch Verlag 2005, mit Anmerkungen von Neumamn, ISBN 978-3-8171-3402-1 (English translation with annotations by Jeremy Gray: Expositiones Math. 1984)