Antebellum South

From Wikipedia, the free encyclopedia

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

The Antebellum South era (from Latin: ante bellum, lit. 'before the war') was a period in the history of the Southern United States that extended from the conclusion of the War of 1812 to the start of the American Civil War in 1861. This era was marked by the prevalent practice of slavery and the associated societal norms it cultivated. Over the course of this period, Southern leaders underwent a transformation in their perspective on slavery. Initially regarded as an awkward and temporary institution, it gradually evolved into a defended concept, with proponents arguing for its positive merits, while simultaneously vehemently opposing the burgeoning abolitionist movement.

Society was stratified, inegalitarian, and perceived by immigrants as lacking in opportunities. Consequently, the manufacturing base lagged behind that of the non-slave states. Wealth inequality grew as the larger landholders took the greater share of the profits generated by enslaved persons, which also helped to entrench their power as a political class.

As the country expanded westward, slavery's propagation became a major issue in national politics, eventually boiling over into the Civil War. In the years that followed the Civil War, the war was romanticized by historical revisionists to protect three central assertions: that the Confederate cause was heroic, that enslaved people were happy and satisfied, and that slavery was not the primary cause of the war. This phenomenon has continued to the present day to contribute to racism, gender roles, and religious attitudes in the South, and to a lesser extent in the rest of the country.

History

In the 18th century, the Atlantic slave trade brought enslaved Africans to the South during the colonial period as a source of labor for the harvesting of crops. There were almost 700,000 enslaved persons in the U.S. in 1790, which was approximately 18 percent of the total population or roughly one in six people. By the end of the 18th century, slavery was in decline, with states in the North beginning to ban the institution and planters in the South realizing that they had no crops successful enough to make slavery financially viable. This would change with the invention of the cotton gin by Eli Whitney in the mid-1790s, which changed a once-tedious manual cleaning and fiber separation process into a faster, less labor-intensive procedure. Suddenly, cotton could be processed more cheaply and efficiently, resulting in slavery becoming very profitable and a large plantation system developing to support the expanding industry. In the 15 years between the invention of the cotton gin and the passage of the Act Prohibiting Importation of Slaves, the slave trade increased and slavery became more prevalent in the southern United States.

Economic structure

The Antebellum South saw large expansions in agriculture during the early 19th century, spurred on by increased demand for cotton for the new textile factories of the industrial North. In light of the successful cotton industry, anti-industrial and anti-urban attitudes resulting from a belief that agrarian life would continue to be the best way forward, the Southern economy experienced little industrialization or other manufacturing development. The Southern economy was characterized by a low level of capital accumulation (largely slave-labor-based) and a shortage of liquid capital, which led to a South dependent on export trade. This was in contrast to the North and West, which were able to rely on their own domestic markets. Since the Southern domestic market consisted primarily of plantations focused on a few specific crops, Southern states imported sustenance commodities from the West and manufactured goods from England and the North.

The plantation system can be seen as the factory system applied to agriculture, with a concentration of labor under skilled management. But while the industrial manufacturing-based labor economy of the North was driven by growing demand, maintenance of the plantation economic system depended upon slave labor, which was abundant and cheap.

The five major commodities of the Southern agricultural economy were cotton, grain, tobacco, sugar, and rice, with cotton the leading cash crop. These commodities were concentrated in the Deep South (Mississippi, Alabama, and Louisiana).

Inefficiency of slave-based agriculture

See also: Treatment of slaves in the United States

The leading historian of the era was Ulrich Bonnell Phillips, who studied slavery not so much as a political issue between North and South, but as a social and economic system. He focused on the large plantations that dominated the South.

Phillips addressed the unprofitability of slave labor and slavery's ill effects on the Southern economy. An example of pioneering comparative work was A Jamaica Slave Plantation (1914). His methods inspired the "Phillips school" of slavery studies, between 1900 and 1950.

Phillips argued that large-scale plantation slavery was inefficient and not progressive. It had reached its geographical limits by 1860 or so, and therefore eventually had to fade away (as happened in Brazil). In The Decadence of the Plantation System (1910), he argued that slavery was an unprofitable relic that persisted because it produced social status, honor, and political power. "Most farmers in the South had small-to-medium-sized farms with few slaves, but the large plantation owner's wealth, often reflected in the number of slaves they owned, afforded them considerable prestige and political power."

Phillips contended that masters treated enslaved persons relatively well; his views on that issue were later sharply rejected by Kenneth M. Stampp. His conclusions about the economic decline of slavery were challenged in 1958 by Alfred H. Conrad and John R. Meyer in a landmark study published in the Journal of Political Economy. Their arguments were further developed by Robert Fogel and Stanley L. Engerman, who argued in their 1974 book, Time on the Cross, that slavery was both efficient and profitable, as long as the price of cotton was high enough. In turn, Fogel and Engerman came under attack from other historians of slavery.

Effects of economy on social structure

See also: The Impending Crisis of the South

As slavery began to displace indentured servitude as the principal supply of labor in the plantation systems of the South, the economic nature of the institution of slavery aided in the increased inequality of wealth seen in the antebellum South. The demand for slave labor and the U.S. ban on importing more slaves from Africa drove up prices for slaves, making it profitable for smaller farms in older settled areas such as Virginia to sell their slaves further south and west. The actuarial risk, or the potential loss in investment of owning slaves from death, disability, etc. was much greater for small plantation owners. Accentuated by the rise in price of slaves seen just prior to the Civil War, the overall costs associated with owning slaves to the individual plantation owner led to the concentration of slave ownership seen at the eve of the Civil War.

Social structure

Much of the Antebellum South was rural, and in line with the plantation system, largely agricultural. With the exception of New Orleans, Charleston, Richmond, and Louisville the slave states had no large cities, and the urban population of the South could not compare to that of the Northeast, or even that of the agrarian West. This led to a sharp division in class in the southern states, between the landowning "master" class, yeoman farmers, poor whites, and slaves; while in the northern and western states, much of the social spectrum was dominated by a wide range of different laboring classes.

Wealth inequality

The conclusion that, while both the North and the South were characterized by a high degree of inequality during the plantation era, the wealth distribution was much more unequal in the South than in the North arises from studies concerned with the equality of land, slave, and wealth distribution. For example, in certain states and counties, due to the concentration of landholding and slave holding, which were highly correlated, six percent of landowners ended up commanding one-third of the gross income and an even higher portion of the net income. The majority of landowners, who had smaller scale plantations, saw a disproportionately small portion in revenues generated by the slavery-driven plantation system.

Effects of social structure on economy

While the two largest classes in the South included land- and slave-owners and slaves, various strata of social classes existed within and between the two. In examining class relations and the banking system in the South, the economic exploitation of slave labor can be seen to arise from a need to maintain certain conditions for the existence of slavery and from a need for each of the remaining social strata to remain in status quo. In order to meet conditions where slavery may continue to exist, members of the master class (e.g., white, landowning, slave-owning) had to compete with other members of the master class to maximize the surplus labor extracted from slaves. Likewise, in order to remain within the same class, members of the master class (and each subsumed class below) must expand their claim on revenues derived from the slave labor surplus.

Mercantilist underpinnings

Mercantilist ideologies largely explain the rise of the plantation system in the United States. In the 16th and 17th centuries under mercantilism, rulers of nations believed that the accumulation of wealth through a favorable balance of trade was the best way to ensure power. As a result, several European nations began to colonize the Americas to take advantage of rich natural resources and encourage exports.

One example of England utilizing its American colonies for economic gain was tobacco. When tobacco was first discovered as a recreational substance, there was a widespread social backlash in England, spearheaded by King James I himself. By the middle of the 17th century, however, Parliament had realized the revenue potential of tobacco and quickly changed its official moral stance towards its use. As a result, tobacco plantations sprung up across the American South in large numbers to support demand in Europe. By 1670, more than half of all tobacco shipped to England was being re-exported to other countries throughout Europe at a premium. In similar ways, the English were able to profit from other American staple crops, such as cotton, rice, and indigo, which "fueled the expansion of the American plantation colonies, transformed the Atlantic into an English inland sea, and led to the creation of the first British Empire."

Many claim that being a part of the British mercantilist system was in the best economic interest of the American colonies as well, as they would not have been able to survive as independent economic entities. Robert Haywood, in his article "Mercantilism and South Carolina Agriculture, 1700–1763", argues that "it was unthinkable that any trade could prosper in the straight-jacket of regimented and restricted international trade, without the guiding hand of a powerful protecting government."

Adverse economic effects

The plantation system created an environment for the South to experience an economic boom in the 17th, 18th and early 19th centuries. However, reliance on both the plantation system and more widespread slave labor, left the South in a precarious economic situation. This was the subject of the highly influential 1857 book The Impending Crisis of the South: How to Meet It, by Hinton Rowan Helper. Following the end of the Civil War and into Reconstruction era (1865–1877), the South experienced economic devastation. Some states that relied less heavily on the plantation system managed to fare better following its downfall. Ulrich Bonnell Phillips contends that the plantation "sadly restricted the opportunity of such men as were of better industrial quality than was required for the field gangs." Essentially, men who would have been otherwise capable of performing other skilled jobs were nonetheless relegated to field work because of the nature of the system.

A 1984 journal article by Claudia Goldin and Kenneth Sokoloff suggested that the South misallocated labor compared to the North, which more eagerly embraced women and child labor in its factories to push forward industrialization due to their relative value to Northern agriculture being lesser than in Southern agriculture.

While the South still attracted immigrants from Europe, the North attracted far more during the early-to-mid 1800s, such that by the time of the American Civil War, the population of the North far exceeded the non-enslaved population of the South per the 1860 United States census. Colin Woodard argued in his 2011 book American Nations that the South was relatively less successful in attracting immigrants due to the South's reputation as a more stratified society. Striving immigrants who sought economic advancement thus tended to favor the more egalitarian North, compared to the more aristocratic South, where there were fewer perceived opportunities for advancement.

History of trigonometry

From Wikipedia, the free encyclopedia

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

Early study of triangles can be traced to Egyptian mathematics (Rhind Mathematical Papyrus) and Babylonian mathematics during the 2nd millennium BC. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata (sixth century AD), who discovered the sine function, cosine function, and versine function.

During the Middle Ages, the study of trigonometry continued in Islamic mathematics, by mathematicians such as al-Khwarizmi and Abu al-Wafa. The knowledge of trigonometric functions passed to Arabia from the Indian Subcontinent. It became an independent discipline in the Islamic world, where all six trigonometric functions were known. Translations of Arabic and Greek texts led to trigonometry being adopted as a subject in the Latin West beginning in the Renaissance with Regiomontanus.

The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics (Isaac Newton and James Stirling) and reaching its modern form with Leonhard Euler (1748).

Etymology

The term "trigonometry" was derived from Greek τρίγωνον trigōnon, "triangle" and μέτρον metron, "measure".

The modern words "sine" and "cosine" are derived from the Latin word sinus via mistranslation from Arabic (see Sine and cosine § Etymology). Particularly Fibonacci's sinus rectus arcus proved influential in establishing the term.

The word tangent comes from Latin tangens meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin secans "cutting" since the line cuts the circle (see the figure at Pythagorean identities).

The prefix "co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter's Canon triangulorum (1620), which defines the cosinus as an abbreviation for the sinus complementi (sine of the complementary angle) and proceeds to define the cotangens similarly.

The words "minute" and "second" are derived from the Latin phrases partes minutae primae and partes minutae secundae. These roughly translate to "first small parts" and "second small parts".

Ancient

Ancient Near East

The ancient Egyptians and Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries. However, as pre-Hellenic societies lacked the concept of an angle measure, they were limited to studying the sides of triangles instead.

The Babylonian astronomers kept detailed records on the rising and setting of stars, the motion of the planets, and the solar and lunar eclipses, all of which required familiarity with angular distances measured on the celestial sphere. Based on one interpretation of the Plimpton 322 cuneiform tablet (c. 1900 BC), some have even asserted that the ancient Babylonians had a table of secants but does not work in this context as without using circles and angles in the situation modern trigonometric notations will not apply. There is, however, much debate as to whether it is a table of Pythagorean triples, a solution of quadratic equations, or a trigonometric table.

The Egyptians, on the other hand, used a primitive form of trigonometry for building pyramids in the 2nd millennium BC. The Rhind Mathematical Papyrus, written by the Egyptian scribe Ahmes (c. 1680–1620 BC), contains the following problem related to trigonometry:

"If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its seked?"

Ahmes' solution to the problem is the ratio of half the side of the base of the pyramid to its height, or the run-to-rise ratio of its face. In other words, the quantity he found for the seked is the cotangent of the angle to the base of the pyramid and its face.

Classical antiquity

The chord of an angle subtends the arc of the angle.

Ancient Greek and Hellenistic mathematicians made use of the chord. Given a circle and an arc on the circle, the chord is the line that subtends the arc. A chord's perpendicular bisector passes through the center of the circle and bisects the angle. One half of the bisected chord is the sine of one half the bisected angle, that is,

${\displaystyle \mathrm{c}\mathrm{h}\mathrm{o}\mathrm{r}\mathrm{d}\theta =2r\sin \frac{\theta }{2},}$

and consequently the sine function is also known as the half-chord. Due to this relationship, a number of trigonometric identities and theorems that are known today were also known to Hellenistic mathematicians, but in their equivalent chord form.

Although there is no trigonometry in the works of Euclid and Archimedes, in the strict sense of the word, there are theorems presented in a geometric way (rather than a trigonometric way) that are equivalent to specific trigonometric laws or formulas. For instance, propositions twelve and thirteen of book two of the Elements are the laws of cosines for obtuse and acute angles, respectively. Theorems on the lengths of chords are applications of the law of sines. And Archimedes' theorem on broken chords is equivalent to formulas for sines of sums and differences of angles. To compensate for the lack of a table of chords, mathematicians of Aristarchus' time would sometimes use the statement that, in modern notation, sin α/sin β < α/β < tan α/tan β whenever 0° < β < α < 90°, now known as Aristarchus's inequality.

The first trigonometric table was apparently compiled by Hipparchus of Nicaea (180 – 125 BC), who is now consequently known as "the father of trigonometry." Hipparchus was the first to tabulate the corresponding values of arc and chord for a series of angles.

Although it is not known when the systematic use of the 360° circle came into mathematics, it is known that the systematic introduction of the 360° circle came a little after Aristarchus of Samos composed On the Sizes and Distances of the Sun and Moon (c. 260 BC), since he measured an angle in terms of a fraction of a quadrant. It seems that the systematic use of the 360° circle is largely due to Hipparchus and his table of chords. Hipparchus may have taken the idea of this division from Hypsicles who had earlier divided the day into 360 parts, a division of the day that may have been suggested by Babylonian astronomy. In ancient astronomy, the zodiac had been divided into twelve "signs" or thirty-six "decans". A seasonal cycle of roughly 360 days could have corresponded to the signs and decans of the zodiac by dividing each sign into thirty parts and each decan into ten parts. It is due to the Babylonian sexagesimal numeral system that each degree is divided into sixty minutes and each minute is divided into sixty seconds.

Menelaus' theorem

Menelaus of Alexandria (c. 100 AD) wrote in three books his Sphaerica. In Book I, he established a basis for spherical triangles analogous to the Euclidean basis for plane triangles. He established a theorem that is without Euclidean analogue, that two spherical triangles are congruent if corresponding angles are equal, but he did not distinguish between congruent and symmetric spherical triangles. Another theorem that he establishes is that the sum of the angles of a spherical triangle is greater than 180°. Book II of Sphaerica applies spherical geometry to astronomy. And Book III contains the "theorem of Menelaus". He further gave his famous "rule of six quantities".

Later, Claudius Ptolemy (c. 90 – c. 168 AD) expanded upon Hipparchus' Chords in a Circle in his Almagest, or the Mathematical Syntaxis. The Almagest is primarily a work on astronomy, and astronomy relies on trigonometry. Ptolemy's table of chords gives the lengths of chords of a circle of diameter 120 as a function of the number of degrees n in the corresponding arc of the circle, for n ranging from 1/2 to 180 by increments of 1/2. The thirteen books of the Almagest are the most influential and significant trigonometric work of all antiquity. A theorem that was central to Ptolemy's calculation of chords was what is still known today as Ptolemy's theorem, that the sum of the products of the opposite sides of a cyclic quadrilateral is equal to the product of the diagonals. A special case of Ptolemy's theorem appeared as proposition 93 in Euclid's Data. Ptolemy's theorem leads to the equivalent of the four sum-and-difference formulas for sine and cosine that are today known as Ptolemy's formulas, although Ptolemy himself used chords instead of sine and cosine. Ptolemy further derived the equivalent of the half-angle formula

${\displaystyle {\sin }^2\left(\frac{x}{2}\right)=\frac{1-\cos (x)}{2}.}$

Ptolemy used these results to create his trigonometric tables, but whether these tables were derived from Hipparchus' work cannot be determined.

Neither the tables of Hipparchus nor those of Ptolemy have survived to the present day, although descriptions by other ancient authors leave little doubt that they once existed.

Indian mathematics

See also: Indian Mathematics and Indian astronomy

Some of the early and very significant developments of trigonometry were in India. Influential works from the 4th–5th century AD, known as the Siddhantas (of which there were five, the most important of which is the Surya Siddhanta) first defined the sine as the modern relationship between half an angle and half a chord, while also defining the cosine, versine, and inverse sine. Soon afterwards, another Indian mathematician and astronomer, Aryabhata (476–550 AD), collected and expanded upon the developments of the Siddhantas in an important work called the Aryabhatiya. The Siddhantas and the Aryabhatiya contain the earliest surviving tables of sine values and versine (1 − cosine) values, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places. They used the words jya for sine, kojya for cosine, utkrama-jya for versine, and otkram jya for inverse sine. The words jya and kojya eventually became sine and cosine respectively after a mistranslation described above.

In the 7th century, Bhaskara I produced a formula for calculating the sine of an acute angle without the use of a table. He also gave the following approximation formula for sin(x), which had a relative error of less than 1.9%:

${\displaystyle \sin x\approx \frac{16x(\pi -x)}{5{\pi }^2-4x(\pi -x)},\left(0\le x\le \pi \right).}$

Later in the 7th century, Brahmagupta redeveloped the formula

${\displaystyle 1-{\sin }^2(x)={\cos }^2(x)={\sin }^2\left(\frac{\pi }{2}-x\right)}$

(also derived earlier, as mentioned above) and the Brahmagupta interpolation formula for computing sine values.

Madhava (c. 1400) made early strides in the analysis of trigonometric functions and their infinite series expansions. He developed the concepts of the power series and Taylor series, and produced the power series expansions of sine, cosine, tangent, and arctangent. Using the Taylor series approximations of sine and cosine, he produced a sine table to 12 decimal places of accuracy and a cosine table to 9 decimal places of accuracy. He also gave the power series of π and the angle, radius, diameter, and circumference of a circle in terms of trigonometric functions. His works were expanded by his followers at the Kerala School up to the 16th century.

No.	Series	Name	Western discoverers of the series and approximate dates of discovery
1	$${\displaystyle \sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+\dots }$$	Madhava's sine series	Isaac Newton (1670) and Wilhelm Leibniz (1676)
2	${\displaystyle \cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+\dots }$	Madhava's cosine series	Isaac Newton (1670) and Wilhelm Leibniz (1676)
3	${\displaystyle \arctan x=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\dots }$	Madhava's arctangent series	James Gregory (1671) and Wilhelm Leibniz (1676)

The Indian text the Yuktibhāṣā contains proof for the expansion of the sine and cosine functions and the derivation and proof of the power series for inverse tangent, discovered by Madhava. The Yuktibhāṣā also contains rules for finding the sines and the cosines of the sum and difference of two angles.

Chinese mathematics

Guo Shoujing (1231–1316)

In China, Aryabhata's table of sines were translated into the Chinese mathematical book of the Kaiyuan Zhanjing, compiled in 718 AD during the Tang dynasty. Although the Chinese excelled in other fields of mathematics such as solid geometry, binomial theorem, and complex algebraic formulas, early forms of trigonometry were not as widely appreciated as in the earlier Greek, Hellenistic, Indian and Islamic worlds. Instead, the early Chinese used an empirical substitute known as chong cha, while practical use of plane trigonometry in using the sine, the tangent, and the secant were known. However, this embryonic state of trigonometry in China slowly began to change and advance during the Song dynasty (960–1279), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendrical science and astronomical calculations. The polymath Chinese scientist, mathematician and official Shen Kuo (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs. Victor J. Katz writes that in Shen's formula "technique of intersecting circles", he created an approximation of the arc s of a circle given the diameter d, sagitta v, and length c of the chord subtending the arc, the length of which he approximated as

${\displaystyle s=c+\frac{2v^2}{d}.}$

Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for spherical trigonometry developed in the 13th century by the mathematician and astronomer Guo Shoujing (1231–1316). As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculations to improve the calendar system and Chinese astronomy. Along with a later 17th-century Chinese illustration of Guo's mathematical proofs, Needham states that:

Guo used a quadrangular spherical pyramid, the basal quadrilateral of which consisted of one equatorial and one ecliptic arc, together with two meridian arcs, one of which passed through the summer solstice point...By such methods he was able to obtain the du lü (degrees of equator corresponding to degrees of ecliptic), the ji cha (values of chords for given ecliptic arcs), and the cha lü (difference between chords of arcs differing by 1 degree).

Despite the achievements of Shen and Guo's work in trigonometry, another substantial work in Chinese trigonometry would not be published again until 1607, with the dual publication of Euclid's Elements by Chinese official and astronomer Xu Guangqi (1562–1633) and the Italian Jesuit Matteo Ricci (1552–1610).

Medieval

Islamic world

Page from The Compendious Book on Calculation by Completion and Balancing by Muhammad ibn Mūsā al-Khwārizmī (c. AD 820)

Previous works from India and Greece were later translated and expanded in the medieval Islamic world by Muslim mathematicians of mostly Persian and Arab descent, who enunciated a large number of theorems which freed the subject of trigonometry from dependence upon the complete quadrilateral, as was the case in Hellenistic mathematics due to the application of Menelaus' theorem. According to E. S. Kennedy, it was after this development in Islamic mathematics that "the first real trigonometry emerged, in the sense that only then did the object of study become the spherical or plane triangle, its sides and angles."

Methods dealing with spherical triangles were also known, particularly the method of Menelaus of Alexandria, who developed "Menelaus' theorem" to deal with spherical problems. However, E. S. Kennedy points out that while it was possible in pre-Islamic mathematics to compute the magnitudes of a spherical figure, in principle, by use of the table of chords and Menelaus' theorem, the application of the theorem to spherical problems was very difficult in practice. In order to observe holy days on the Islamic calendar in which timings were determined by phases of the moon, astronomers initially used Menelaus' method to calculate the place of the moon and stars, though this method proved to be clumsy and difficult. It involved setting up two intersecting right triangles; by applying Menelaus' theorem it was possible to solve one of the six sides, but only if the other five sides were known. To tell the time from the sun's altitude, for instance, repeated applications of Menelaus' theorem were required. For medieval Islamic astronomers, there was an obvious challenge to find a simpler trigonometric method.

In the early 9th century AD, Muhammad ibn Mūsā al-Khwārizmī produced accurate sine and cosine tables. He was also a pioneer in spherical trigonometry. In 830 AD, Habash al-Hasib al-Marwazi discovered the tangent and the cotangent and produced the first table of these trigonometric functions. Muhammad ibn Jābir al-Harrānī al-Battānī (Albatenius) (853–929 AD) discovered the secant and the cosecant, and produced the first table of cosecants for each degree from 1° to 90°.

By the 10th century AD, in the work of Abū al-Wafā' al-Būzjānī, all six trigonometric functions were used. Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of tangent values. He also developed the following trigonometric formula:

${\displaystyle \sin (2x)=2\sin (x)\cos (x)}$ (a special case of Ptolemy's angle-addition formula; see above)

In his original text, Abū al-Wafā' states: "If we want that, we multiply the given sine by the cosine minutes, and the result is half the sine of the double". Abū al-Wafā also established the angle addition and difference identities presented with complete proofs:

${\displaystyle \sin (\alpha \pm \beta )=\sqrt{{\sin }^2\alpha -(\sin \alpha \sin \beta {)}^2}\pm \sqrt{{\sin }^2\beta -(\sin \alpha \sin \beta {)}^2}}$

${\displaystyle \sin (\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta }$

For the second one, the text states: "We multiply the sine of each of the two arcs by the cosine of the other minutes. If we want the sine of the sum, we add the products, if we want the sine of the difference, we take their difference".

He also discovered the law of sines for spherical trigonometry:

${\displaystyle \frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c}.}$

Also in the late 10th and early 11th centuries AD, the Egyptian astronomer Ibn Yunus performed many careful trigonometric calculations and demonstrated the following trigonometric identity:

${\displaystyle \cos a\cos b=\frac{\cos (a+b)+\cos (a-b)}{2}}$

Al-Jayyani (989–1079) of al-Andalus wrote The book of unknown arcs of a sphere, which is considered "the first treatise on spherical trigonometry". It "contains formulae for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle." This treatise later had a "strong influence on European mathematics", and his "definition of ratios as numbers" and "method of solving a spherical triangle when all sides are unknown" are likely to have influenced Regiomontanus.

The method of triangulation was first developed by Muslim mathematicians, who applied it to practical uses such as surveying and Islamic geography, as described by Abu Rayhan Biruni in the early 11th century. Biruni himself introduced triangulation techniques to measure the size of the Earth and the distances between various places. In the late 11th century, Omar Khayyám (1048–1131) solved cubic equations using approximate numerical solutions found by interpolation in trigonometric tables. In the 13th century, Naṣīr al-Dīn al-Ṭūsī was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form. He listed the six distinct cases of a right-angled triangle in spherical trigonometry, and in his Book on the Complete Quadrilateral, he stated the law of sines for plane and spherical triangles, discovered the law of tangents for spherical triangles, and provided proofs for both these laws. Nasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right.

The law of cosines, in geometric form, can be found as propositions II.12–13 in Euclid's Elements (c. 300 BC), but was not used for the solution of triangles per se. Medieval Islamic mathematicians developed a method for finding the third side of an arbitrary triangle given two sides and the included angle based on the same concept but more similar to the modern formulation of the law of cosines. A sketch of the method can be found in Naṣīr al-Dīn al-Ṭūsī's Book on the Complete Quadrilateral (c. 1250), and the same method is described in more detail in Jamshīd al-Kāshī's Key of Arithmetic (1427). Al-Kāshī also computed the sine of 1° accurate to 8 sexagesimal digits, and constructed the most accurate trigonometric tables to date, accurate to four sexagesimal places (equivalent to 8 decimal places) for each 1° of arc. Al-Kāshī presumably worked on Ulugh Beg's even more comprehensive trigonometric tables, with five-place (sexagesimal) entries for each minute of arc.

European renaissance

In 1342, Levi ben Gershon, known as Gersonides, wrote On Sines, Chords and Arcs, in particular proving the sine law for plane triangles and giving five-figure sine tables.

A simplified trigonometric table, the "toleta de marteloio", was used by sailors in the Mediterranean Sea during the 14th-15th Centuries to calculate navigation courses. It is described by Ramon Llull of Majorca in 1295, and laid out in the 1436 atlas of Venetian captain Andrea Bianco.

Regiomontanus was perhaps the first mathematician in Europe to treat trigonometry as a distinct mathematical discipline, in his De triangulis omnimodis written in 1464, as well as his later Tabulae directionum which included the tangent function, unnamed.

Modern

The Opus palatinum de triangulis of Georg Joachim Rheticus, a student of Copernicus, was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596.

In the 17th century, Isaac Newton and James Stirling developed the general Newton–Stirling interpolation formula for trigonometric functions.

In the 18th century, Leonhard Euler's Introduction in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, deriving their infinite series and presenting "Euler's formula" e^ix = cos x + i sin x. Euler used the near-modern abbreviations sin., cos., tang., cot., sec., and cosec. Prior to this, Roger Cotes had computed the derivative of sine in his Harmonia Mensurarum (1722). Also in the 18th century, Brook Taylor defined the general Taylor series and gave the series expansions and approximations for all six trigonometric functions. The works of James Gregory in the 17th century and Colin Maclaurin in the 18th century were also very influential in the development of trigonometric series.

In the 19th century, Joseph Fourier discovered the Fourier series during his attempts to find solutions to the heat equation, paving the way for Fourier and harmonic analysis.

Ability

From Wikipedia, the free encyclopedia

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

Abilities are powers an agent has to perform various actions. They include common abilities, like walking, and rare abilities, like performing a double backflip. Abilities are intelligent powers: they are guided by the person's intention and executing them successfully results in an action, which is not true for all types of powers. They are closely related to but not identical with various other concepts, such as disposition, know-how, aptitude, talent, potential, and skill.

Theories of ability aim to articulate the nature of abilities. Traditionally, the conditional analysis has been the most popular approach. According to it, having an ability means one would perform the action in question if one tried to do so. On this view, Michael Phelps has the ability to swim 200 meters in under 2 minutes because he would do so if he tried to. This approach has been criticized in various ways. Some counterexamples involve cases in which the agent is physically able to do something but unable to try, due to a strong aversion. In order to avoid these and other counterexamples, various alternative approaches have been suggested. Modal theories of ability, for example, focus on what is possible for the agent to do. Other suggestions include defining abilities in terms of dispositions and potentials.

An important distinction among abilities is between general abilities and specific abilities. General abilities are abilities possessed by an agent independent of their situation while specific abilities concern what an agent can do in a specific situation. So while an expert piano player always has the general ability to play various piano pieces, they lack the corresponding specific ability in a situation where no piano is present. Another distinction concerns the question of whether successfully performing an action by accident counts as having the corresponding ability. In this sense, an amateur hacker may have the effective ability to hack his boss's email account, because they may be lucky and guess the password correctly, but not the corresponding transparent ability, since they are unable to reliably do so.

The concept of abilities and how they are to be understood is relevant for various related fields. Free will, for example, is often understood as the ability to do otherwise. The debate between compatibilism and incompatibilism concerns the question whether this ability can exist in a world governed by deterministic laws of nature. Autonomy is a closely related concept, which can be defined as the ability of individual or collective agents to govern themselves. Whether an agent has the ability to perform a certain action is important for whether they have a moral obligation to perform this action. If they possess it, they may be morally responsible for performing it or for failing to do so. Like in the free will debate, it is also relevant whether they had the ability to do otherwise. A prominent theory of concepts and concept possession understands these terms in relation to abilities. According to it, it is required that the agent possess both the ability to discriminate between positive and negative cases and the ability to draw inferences to related concepts.

Definition and semantic field

Abilities are powers an agent has to perform various actions. Some abilities are very common among human agents, like the ability to walk or to speak. Other abilities are only possessed by a few, such as the ability to perform a double backflip or to prove Gödel's incompleteness theorem. While all abilities are powers, the converse is not true, i.e. there are some powers that are not abilities. This is the case, for example, for powers that are not possessed by agents, like the power of salt to dissolve in water. But some powers possessed by agents do not constitute abilities either. For example, the power to understand French is not an ability in this sense since it does not involve an action, in contrast to the ability to speak French. This distinction depends on the difference between actions and non-actions. Actions are usually defined as events that an agent performs for a purpose and that are guided by the person's intention, in contrast to mere behavior, like involuntary reflexes. In this sense, abilities can be seen as intelligent powers.

Various terms within the semantic field of the term "ability" are sometimes used as synonyms but have slightly different connotations. Dispositions, for example, are often equated with powers and differ from abilities in the sense that they are not necessarily linked to agents and actions. Abilities are closely related to know-how, as a form of practical knowledge on how to accomplish something. But it has been argued that these two terms may not be identical since know-how belongs more to the side of knowledge of how to do something and less to the power to actually do it. The terms "aptitude" and "talent" usually refer to outstanding inborn abilities. They are often used to express that a certain set of abilities can be acquired when properly used or trained. Abilities acquired through learning are frequently referred to as skills. The term "disability" is usually used for a long-term absence of a general human ability that significantly impairs what activities one can engage in and how one can interact with the world. In this sense, not any lack of an ability constitutes a disability. The more direct antonym of "ability" is "inability" instead.

Theories of ability

Various theories of the essential features of abilities have been proposed. The conditional analysis is the traditionally dominant approach. It defines abilities in terms of what one would do if one had the volition to do so. For modal theories of ability, by contrast, having an ability means that the agent has the possibility to execute the corresponding action. Other approaches include defining abilities in terms of dispositions and potentials. While all the concepts used in these different approaches are closely related, they have slightly different connotations, which often become relevant for avoiding various counterexamples.

Conditional analysis

The conditional analysis of ability is the traditionally dominant approach. It is often traced back to David Hume and defines abilities in terms of what one would do if one wanted to, tried to or had the volition to do so. It is articulated in the form of a conditional expression, for example, as "S has the ability to A iff S would A if S tried to A". On this view, Michael Phelps has the ability to swim 200 meters in under 2 minutes because he would do so if he tried to. The average person, on the other hand, lacks this ability because they would fail if they tried. Similar versions talk of having a volition instead of trying. This view can distinguish between the ability to do something and the possibility that one does something: only having the ability implies that the agent can make something happen according to their will. This definition of ability is closely related to Hume's definition of liberty as "a power of acting or not acting, according to the determinations of the will". But it is often argued that this is different from having a free will in the sense of the capacity of choosing between different courses of action.

This approach has been criticized in various ways, often by citing alleged counterexamples. Some of these counterexamples focus on cases where an ability is actually absent even though it would be present according to the conditional analysis. This is the case, for example, if someone is physically able to perform a certain action but, maybe due to a strong aversion, cannot form the volition to perform this action. So according to the conditional analysis, a person with arachnophobia has the ability to touch a trapped spider because they would do so if they tried. But all things considered, they do not have this ability since their arachnophobia makes it impossible for them to try. Another example involves a woman attacked on a dark street who would have screamed if she had tried to but was too paralyzed by fear to try it. One way to avoid this objection is to distinguish between psychological and non-psychological requirements of abilities. The conditional analysis can then be used as a partial analysis applied only to the non-psychological requirements.

Another form of criticism involves cases where the ability is present even though it would be absent according to the conditional analysis. This argument can be centered on the idea that having an ability does not ensure that each and every execution of it is successful. For example, even a good golfer may miss an easy putt on one occasion. That does not mean that they lack the ability to make this putt but this is what the conditional analysis suggests since they tried it and failed. One reply to this problem is to ascribe to the golfer the general ability, as discussed below, but deny them the specific ability in this particular instance.

Modal approach

Modal theories of ability focus not on what the agent would do under certain circumstances but on what is possible for the agent to do. This possibility is often understood in terms of possible worlds. On this view, an agent has the ability to perform a certain action if there is a complete and consistent way how the world could have been, in which the agent performs the corresponding action. This approach easily captures the idea that an agent can possess an ability without executing it. In this case, the agent does not perform the corresponding action in the actual world but there is a possible world where they perform it.

The problem with the approach described so far is that when the term "possible" is understood in the widest sense, many actions are possible even though the agent actually lacks the ability to perform them. For example, not knowing the combination of the safe, the agent lacks the ability to open the safe. But dialing the right combination is possible, i.e. there is a possible world in which, through a lucky guess, the agent succeeds at opening the safe. Because of such cases, it is necessary to add further conditions to the analysis above. These conditions play the role of restricting which possible worlds are relevant for evaluating ability-claims. Closely related to this is the converse problem concerning lucky performances in the actual world. This problem concerns the fact that an agent may successfully perform an action without possessing the corresponding ability. So a beginner at golf may hit the ball in an uncontrolled manner and through sheer luck achieve a hole-in-one. But the modal approach seems to suggest that such a beginner still has the corresponding ability since what is actual is also possible.

A series of arguments against this approach is due to Anthony Kenny, who holds that various inferences drawn in modal logic are invalid for ability ascriptions. These failures indicate that the modal approach fails to capture the logic of ability ascriptions.

It has also been argued that, strictly speaking, the conditional analysis is not different from the modal approach since it is just one special case of it. This is true if conditional expressions themselves are understood in terms of possible worlds, as suggested, for example, by David Kellogg Lewis and Robert Stalnaker. In this case, many of the arguments directed against the modal approach may equally apply to the conditional analysis.

Other approaches

The dispositional approach defines abilities in terms of dispositions. According to one version, "S has the ability to A in circumstances C iff she has the disposition to A when, in circumstances C, she tries to A". This view is closely related to the conditional analysis but differs from it because the manifestation of dispositions can be prevented through the presence of so-called masks and finks. In these cases, the disposition is still present even though the corresponding conditional is false. Another approach sees abilities as a form of potential to do something. This is different from a disposition since a disposition concerns the relation between a stimulus and a manifestation that follows when the stimulus is present. A potential, on the other hand, is characterized only by its manifestation. In the case of abilities, the manifestation concerns an action.

Types

Whether it is correct to ascribe a certain ability to an agent often depends on which type of ability is meant. General abilities concern what agents can do independent of their current situation, in contrast to specific abilities. To possess an effective ability, it is sufficient if the agent can succeed through a lucky accident, which is not the case for transparent abilities.

General and specific

An important distinction among abilities is between general and specific abilities, sometimes also referred to as global and local abilities. General abilities concern what agents can do generally, i.e. independent of the situation they find themselves in. But abilities often depend for their execution on various conditions that have to be fulfilled in the given circumstances. In this sense, the term "specific ability" is used to describe whether an agent has an ability in a specific situation. So while an expert piano player always has the general ability to play various piano pieces, they lack the corresponding specific ability if they are chained to a wall, if no piano is present or if they are heavily drugged. In such cases, some of the necessary conditions for using the ability are not met. While this example illustrates a case of a general ability without a specific ability, the converse is also possible. Even though most people lack the general ability to jump 2 meters high, they may possess the specific ability to do so when they find themselves on a trampoline. The reason that they lack this general ability is that they would fail to execute it in most circumstances. It would be necessary to succeed in a suitable proportion of the relevant cases for having the general ability as well, as would be the case for a high jump athlete in this example.

It seems that the two terms are interdefinable but there is disagreement as to which one is the more basic term. So a specific ability may be defined as a general ability together with an opportunity. Having a general ability, on the other hand, can be seen as having a specific ability in various relevant situations. A similar distinction can be drawn not just for the term "ability" but also for the wider term "disposition". The distinction between general and specific abilities is not always drawn explicitly in the academic literature. While discussions often focus more on the general sense, sometimes the specific sense is intended. This distinction is relevant for various philosophical issues, specifically for the ability to do otherwise in the free will debate. If this ability is understood as a general ability, it seems to be compatible with determinism. But this seems not to be the case if a specific ability is meant.

Effective and transparent

Another distinction sometimes found in the literature concerns the question of whether successfully performing an action by accident counts as having the corresponding ability. For example, a student in the first grade is able, in a weaker sense, to recite the first 10 digits of Pi insofar as they are able to utter any permutation of the numerals from 0 to 9. But they are not able to do so in a stronger sense since they have not memorized the exact order. The weaker sense is sometimes termed effective abilities, in contrast to transparent abilities corresponding to the stronger sense. Usually, ability ascriptions have the stronger sense in mind, but this is not always the case. For example, the sentence "Usain Bolt can run 100 meters in 9.58 seconds" is usually not taken to mean that Bolt can, at will, arrive at the goal at exactly 9.58 seconds, no more and no less. Instead, he can do something that amounts to this in a weaker sense.

Relation to other concepts

The concept of abilities is relevant for various other concepts and debates. Disagreements in these fields often depend on how abilities are to be understood. In the free will debate, for example, a central question is whether free will, when understood as the ability to do otherwise, can exist in a world governed by deterministic laws of nature. Free will is closely related to autonomy, which concerns the agent's ability to govern oneself. Another issue concerns whether someone has the moral obligation to perform a certain action and is responsible for succeeding or failing to do so. This issue depends, among other things, on whether the agent has the ability to perform the action in question and on whether they could have done otherwise. The ability-theory of concepts and concept possession defines them in terms of two abilities: the ability to discriminate between positive and negative cases and the ability to draw inferences to related concepts.

Free will

The topic of abilities plays an important role in the free will debate. The free will debate often centers around the question of whether the existence of free will is compatible with determinism, so-called compatibilism, or not, so-called incompatibilism. Free will is frequently defined as the ability to do otherwise while determinism can be defined as the view that the past together with the laws of nature determine everything happening in the present and the future. The conflict arises since, if everything is already fixed by the past, there seems to be no sense in which anyone could act differently than they do, i.e. that there is no place for free will. Such a result might have serious consequences since, according to some theories, people would not be morally responsible for what they do in such a case.

Having an explicit theory of what constitutes an ability is central for deciding whether determinism and free will are compatible. Different theories of ability may lead to different answers to this question. It has been argued that, according to a dispositionalist theory of ability, compatibilism is true since determinism does not exclude unmanifested dispositions. Another argument for compatibilism is due to Susan Wolf, who argues that having the type of ability relevant for moral responsibility is compatible with physical determinism since the ability to perform an action does not imply that this action is physically possible. Peter van Inwagen and others have presented arguments for incompatibilism based on the fact that the laws of nature impose limits on our abilities. These limits are so strict in the case of determinism that the only abilities possessed by anyone are the ones that are actually executed, i.e. there are no abilities to do otherwise than one actually does.

Autonomy

Autonomy is usually defined as the ability to govern oneself. It can be ascribed both to individual agents, like human persons, and to collective agents, like nations. Autonomy is absent when there is no intelligent force governing the entity's behavior at all, as in the case of a simple rock, or when this force does not belong to the governed entity, as when one nation has been invaded by another and now lacks the ability to govern itself. Autonomy is often understood in combination with a rational component, e.g. as the agent's ability to appreciate what reasons they have and to follow the strongest reason. Robert Audi, for example, characterizes autonomy as the self-governing power to bring reasons to bear in directing one's conduct and influencing one's propositional attitudes. Autonomy may also encompass the ability to question one's beliefs and desires and to change them if necessary. Some authors include the condition that decisions involved in self-governing are not determined by forces outside oneself in any way, i.e. that they are a pure expression of one's own will that is not controlled by someone else. In the Kantian tradition, autonomy is often equated with self-legislation, which may be interpreted as laying down laws or principles that are to be followed. This involves the idea that one's ability of self-governance is not just exercised on a case-by-case basis but that one takes up long-term commitments to more general principles governing many different situations.

Obligation and responsibility

The issue of abilities is closely related to the concepts of responsibility and obligation. On the side of obligation, the principle that "ought implies can" is often cited in the ethical literature. Its original formulation is attributed to Immanuel Kant. It states that an agent is only morally obligated to perform a certain action if they are able to perform this action. As a consequence of this principle, one is not justified to blame an agent for something that was out of their control. According to this principle, for example, a person sitting on the shore has no moral obligation to jump into the water to save a child drowning nearby, and should not be blamed for failing to do so, if they are unable to do so due to Paraplegia.

The problem of moral responsibility is closely related to obligation. One difference is that "obligation" tends to be understood more in a forward-looking sense in contrast to backward-looking responsibility. But these are not the only connotations of these terms. A common view concerning moral responsibility is that the ability to control one's behavior is necessary if one is to be responsible for it. This is often connected to the thesis that alternative courses of action were available to the agent, i.e. that the agent had the ability to do otherwise. But some authors, often from the incompatibilist tradition, contend that what matters for responsibility is to act as one chooses, even if no ability to do otherwise was present.

One difficulty for these principles is that our ability to do something at a certain time often depends on having done something else earlier. So a person is usually able to attend a meeting 5 minutes from now if they are currently only a few meters away from the planned location but not if they are hundreds of kilometers away. This seems to lead to the counter-intuitive consequence that people who failed to take their flight due to negligence are not morally responsible for their failure because they currently lack the corresponding ability. One way to respond to this type of example is to allow that the person is not to be blamed for their behavior 5 minutes before the meeting but hold instead that they are to be blamed for their earlier behavior that caused them to miss the flight.

Concepts and concept possession

Concepts are the basic constituents of thoughts, beliefs and propositions. As such, they play a central role for most forms of cognition. A person can only entertain a proposition if they possess the concepts involved in this proposition. For example, the proposition "wombats are animals" involves the concepts "wombat" and "animal". Someone who does not possess the concept "wombat" may still be able to read the sentence but cannot entertain the corresponding proposition. There are various theories concerning how concepts and concept possession are to be understood. One prominent suggestion sees concepts as cognitive abilities of agents. Proponents of this view often identify two central aspects that characterize concept possession: the ability to discriminate between positive and negative cases and the ability to draw inferences from this concept to related concepts. So, on the one hand, a person possessing the concept "wombat" should be able to distinguish wombats from non-wombats (like trees, DVD-players or cats). On the other hand, this person should be able to point out what follows from the fact that something is a wombat, e.g. that it is an animal, that it has short legs or that it has a slow metabolism. It is usually taken that these abilities have to be possessed to a significant degree but that perfection is not necessary. So even some people who are not aware of their slow metabolism may count as possessing the concept "wombat". Opponents of the ability-theory of concepts have argued that the abilities to discriminate and to infer are circular since they already presuppose concept possession instead of explaining it. They tend to defend alternative accounts of concepts, for example, as mental representations or as abstract objects.