George Peacock

George Peacock (April 9, 1791 – November 8, 1858) was an English mathematician. George Peacock. ... George Peacock. ... April 9 is the 99th day of the year in the Gregorian calendar (100th in leap years). ... 1791 (MDCCXCI) was a common year starting on Saturday (see link for calendar) of the Gregorian calendar (or a common year starting on Tuesday of the 11-day-slower Julian calendar). ... November 8 is the 312th day of the year (313th in leap years) in the Gregorian calendar, with 53 days remaining. ... 1858 (MDCCCLVIII) is a common year starting on Friday of the Gregorian calendar (or a common year starting on Sunday of the 12-day-slower Julian calendar). ... Motto: (French for God and my right) Anthem: God Save the King/Queen Capital London (de facto) Largest city London Official language(s) English (de facto) Unification    - by Athelstan AD 927  Area    - Total 130,395 km² (1st in UK)   50,346 sq mi  Population    - 2006 est. ... Leonhard Euler is considered by many to be one of the greatest mathematicians of all time A mathematician is the person whose primary area of study and research is the field of mathematics. ...

==Life George Peacock was born on April 9, 1791, at Denton in the north of England, 14 miles from Richmond in Yorkshire. His father, the Rev. Thomas Peacock, was a clergyman of the Church of England, incumbent and for 50 years curate of the parish of Denton, where he also kept a school. In early life Peacock did not show any precocity of genius, and was more remarkable for daring feats of climbing than for any special attachment to study. He received his elementary education from his father, and at 17 years of age, was sent to Richmond, to a school taught by a graduate of Cambridge University to receive instruction preparatory to entering that university. At this school he distinguished himself greatly both in classics and in the rather elementary mathematics then required for entrance at Cambridge. In 1809 he became a student of Trinity College, Cambridge. April 9 is the 99th day of the year in the Gregorian calendar (100th in leap years). ... 1791 (MDCCXCI) was a common year starting on Saturday (see link for calendar) of the Gregorian calendar (or a common year starting on Tuesday of the 11-day-slower Julian calendar). ... Motto: (French for God and my right) Anthem: God Save the King/Queen Capital London (de facto) Largest city London Official language(s) English (de facto) Unification    - by Athelstan AD 927  Area    - Total 130,395 km² (1st in UK)   50,346 sq mi  Population    - 2006 est. ... The town of Richmond as seen from the top of the keep of Richmond Castle Richmond is a market town on the River Swale in North Yorkshire, UK and is the administrative centre of the district of Richmondshire. ... Look up Yorkshire in Wiktionary, the free dictionary. ... The Church of England is the officially established Christian church[1] in England, and acts as the mother and senior branch of the worldwide Anglican Communion, as well as a founding member of the Porvoo Communion. ... The University of Cambridge (usually abbreviated as Cantab. ... Year 1809 (MDCCCIX) was a common year starting on Sunday (link will display the full calendar). ...

In 1812 Peacock took the rank of second wrangler, and the second Smith's prize, the senior wrangler being John Herschel. Two years later he became a candidate for a fellowship in his college and won it immediately, partly by means of his extensive and accurate knowledge of the classics. A fellowship then meant about pounds 200 a year, tenable for seven years provided the Fellow did not marry meanwhile, and capable of being extended after the seven years provided the Fellow took clerical Orders. For the overture by Tchaikovsky, see 1812 Overture; For the wars, see War of 1812 (USA - United Kingdom) or Patriotic War of 1812 (France - Russia) For the Siberia Airlines plane crashed over the Black Sea on October 4, 2001, see Siberia Airlines Flight 1812 1812 was a leap year starting... John Herschel Sir John Frederick William Herschel (7 March 1792 – 11 May 1871) was an English mathematician and astronomer. ...

The year after taking a Fellowship, Peacock was appointed a tutor and lecturer of his college, which position he continued to hold for many years. Peacock, in common with many other students of his own standing, was profoundly impressed with the need of reforming Cambridge's position ignoring the differential notation for calculus., and while still an undergraduate formed a league with Babbage and Herschel to adopt measures to bring it about. In 1815 they formed what they called the Analytical Society, the object of which was stated to be to advocate the d 'ism of the Continent versus the dot-age of the University. Charles Babbage Charles Babbage (December 26, 1791 – October 18, 1871) was an English mathematician, analytical philosopher and (proto_) computer scientist who was the first person to come up with the idea of a programmable computer. ... Notable people or families with the surname Herschel include: Sir William Herschel (1738-1822), astronomer and composer, discoverer of Uranus Carolyn Lucretia Herschel (1750-1848), astronomer and singer, sister of Sir William Herschel John Frederick William Herschel (1792-1871), mathematician and astronomer, son of Sir William Herschel Alexander Stewart Herschel...

The first movement on the part of the Analytical Society was to translate from the French the smaller work of Lacroix on the poop differential and integral calculus; it was published in 1816. At that time the best manuals, as well as the greatest works on mathematics, existed in the French language. Peacock followed up the translation with a volume containing a copious Collection of Examples of the Application of the Differential and Integral Calculus, which was published in 1820. The sale of both books was rapid, and contributed materially to further the object of the Society. In that time, high wranglers of one year became the examiners of the mathematical tripos three or four years afterwards. Peacock was appointed an examiner in 1817, and he did not fail to make use of the position as a powerful lever to advance the cause of reform. In his questions set for the examination the differential notation was for the first time officially employed in Cambridge. The innovation did not escape censure, but he wrote to a friend as follows: "I assure you that I shall never cease to exert myself to the utmost in the cause of reform, and that I will never decline any office which may increase my power to effect it. I am nearly certain of being nominated to the office of Moderator in the year 1818-1819, and as I am an examiner in virtue of my office, for the next year I shall pursue a course even more decided than hitherto, since I shall feel that men have been prepared for the change, and will then be enabled to have acquired a better system by the publication of improved elementary books. I have considerable influence as a lecturer, and I will not neglect it. It is by silent perseverance only, that we can hope to reduce the many-headed monster of prejudice and make the University answer her character as the loving mother of good learning and science." These few sentences give an insight into the character of Peacock: he was an ardent reformer and a few years brought success to the cause of the Analytical Society. The Analytical Society was a group of individuals in early-19th century Britain whose aim was to promote the use of Leibnizian or analytical calculus as opposed to Newtonian calculus. ... Sylvestre François de Lacroix (April 28, 1765–May 24, 1843) was a French mathematician. ... 1816 was a leap year starting on Monday (see link for calendar). ... 1820 was a leap year starting on Saturday (see link for calendar). ... 1817 was a common year starting on Wednesday (see link for calendar). ... 1818 (MDCCCXVIII) is a common year starting on Thursday of the Gregorian calendar or a common year starting on Saturday of the 12-day slower Julian calendar. ... 1819 common year starting on Friday (see link for calendar). ...

Another reform at which Peacock labored was the teaching of algebra. In 1830 he published a Treatise on Algebra which had for its object the placing of algebra on a true scientific basis, adequate for the development which it had received at the hands of the Continental mathematicians. To elevate astronomical science the Astronomical Society of London was founded, and the three reformers Peacock, Babbage and Herschel were again prime movers in the undertaking. Peacock was one of the most zealous promoters of an astronomical observatory at Cambridge, and one of the founders of the Philosophical Society of Cambridge. Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... Liberty Leading the People by Eugène Delacroix commemorates the July Revolution 1830 (MDCCCXXX) was a common year starting on Friday (see link for calendar). ...

In 1831 the British Association for the Advancement of Science (prototype of the American, French and Australasian Associations) held its first meeting in the ancient city of York. One of the first resolutions adopted was to procure reports on the state and progress of particular sciences, to be drawn up from time to time by competent persons for the information of the annual meetings, and the first to be placed on the list was a report on the progress of mathematical science. Dr. Whewell, the mathematician and philosopher, was a Vice-president of the meeting: he was instructed to select the reporter. He first asked Sir W. R. Hamilton, who declined; he then asked Peacock, who accepted. Peacock had his report ready for the third meeting of the Association, which was held in Cambridge in 1833; although limited to Algebra, Trigonometry, and the Arithmetic of Sines, it is one of the best of the long series of valuable reports which have been prepared for and printed by the Association. Leopold I 1831 (MDCCCXXXI) was a common year starting on Saturday (see link for calendar). ... York is a city in North Yorkshire, England, at the confluence of the Rivers Ouse and Foss. ... 1833 was a common year starting on Tuesday (see link for calendar). ... Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... Wikibooks has a book on the topic of Trigonometry Trigonometry (from the Greek Trigona = three angles and metron = measure[1]) is a branch of mathematics which deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right angled triangles). ...

In 1837 Peacock was appointed Lowndean professor of astronomy in the University of Cambridge, the chair afterwards occupied by Adams, the co-discoverer of Neptune, and later occupied by Sir Robert Ball, celebrated for his Theory of Screws. In 1839 he was appointed Dean of Ely, the diocese of Cambridge. While holding this position he wrote a text book on algebra in two volumes, the one called Arithmetical Algebra, and the other Symbolical Algebra. Another object of reform was the statutes of the University; he worked hard at it and was made a member of a commission appointed by the Government for the purpose; but he died on November 8, 1858, in the 68th year of his age. His last public act was to attend a meeting of the Commission. Queen Victoria, Queen of the United Kingdom (1837 - 1901) 1837 (MDCCCXXXVII) was a common year starting on Sunday (see link for calendar). ... Atmospheric characteristics Surface pressure ≫100 MPa Hydrogen - H2 80% ±3. ... Sir Robert Stawell Ball (1840-1913) was an English astronomer to Lord Rosse in 1865 and Irish Astronomer-Royal in 1874. ... 1839 (MDCCCXXXIX) was a common year starting on Tuesday (see link for calendar). ... November 8 is the 312th day of the year (313th in leap years) in the Gregorian calendar, with 53 days remaining. ... 1858 (MDCCCLVIII) is a common year starting on Friday of the Gregorian calendar (or a common year starting on Sunday of the 12-day-slower Julian calendar). ...

Peacock's Algebraic Theory

Peacock's main contribution to mathematical analysis is his attempt to place algebra on a strictly logical basis. He founded what has been called the philological or symbolical school of mathematicians; to which Gregory, De Morgan and Boole belonged. His answer to Maseres and Frend was that the science of algebra consisted of two parts -- arithmetical algebra and symbolical algebra -- and that they erred in restricting the science to the arithmetical part. His view of arithmetical algebra is as follows: "In arithmetical algebra we consider symbols as representing numbers, and the operations to which they are submitted as included in the same definitions as in common arithmetic; the signs + and − denote the operations of addition and subtraction in their ordinary meaning only, and those operations are considered as impossible in all cases where the symbols subjected to them possess values which would render them so in case they were replaced by digital numbers; thus in expressions such as a + b we must suppose a and b to be quantities of the same kind; in others, like a − b, we must suppose a greater than b and therefore homogeneous with it; in products and quotients, like ab and we must suppose the multiplier and divisor to be abstract numbers; all results whatsoever, including negative quantities, which are not strictly deducible as legitimate conclusions from the definitions of the several operations must be rejected as impossible, or as foreign to the science." Gregory is a common masculine first name and family name. ... Augustus De Morgan (June 27, 1806 – March 18, 1871) was an Indian-born British mathematician and logician. ... This article is not about George Boolos, another mathematical logician. ...

Peacock's principle may be stated thus: the elementary symbol of arithmetical algebra denotes a digital, i.e., an integer number; and every combination of elementary symbols must reduce to a digital number, otherwise it is impossible or foreign to the science. If a and b are numbers, then a + b is always a number; but a − b is a number only when b is less than a. Again, under the same conditions, ab is always a number, but is really a number only when b is an exact divisor of a. Hence the following dilemma: Either must be held to be an impossible expression in general, or else the meaning of the fundamental symbol of algebra must be extended so as to include rational fractions. If the former horn of the dilemma is chosen, arithmetical algebra becomes a mere shadow; if the latter horn is chosen, the operations of algebra cannot be defined on the supposition that the elementary symbol is an integer number. Peacock attempts to get out of the difficulty by supposing that a symbol which is used as a multiplier is always an integer number, but that a symbol in the place of the multiplicand may be a fraction. For instance, in ab, a can denote only an integer number, but b may denote a rational fraction. Now there is no more fundamental principle in arithmetical algebra than that ab = ba; which would be illegitimate on Peacock's principle. A digital system is one that uses discrete values (often electrical voltages), especially those representable as binary numbers, or non-numeric symbols such as letters or icons, for input, processing, transmission, storage, or display, rather than a continuous spectrum of values (ie, as in an analog system). ...

One of the earliest English writers on arithmetic is Robert Record, who dedicated his work to King Edward the Sixth. The author gives his treatise the form of a dialogue between master and scholar. The scholar battles long over this difficulty, -- that multiplying a thing could make it less. The master attempts to explain the anomaly by reference to proportion; that the product due to a fraction bears the same proportion to the thing multiplied that the fraction bears to unity. But the scholar is not satisfied and the master goes on to say: "If I multiply by more than one, the thing is increased; if I take it but once, it is not changed, and if I take it less than once, it cannot be so much as it was before. Then seeing that a fraction is less than one, if I multiply by a fraction, it follows that I do take it less than once." Whereupon the scholar replies, "Sir, I do thank you much for this reason, -- and I trust that I do perceive the thing." Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ... Robert Recorde 1510 - 1558 Robert Recorde (c. ...

The fact is that even in arithmetic the two processes of multiplication and division are generalized into a common multiplication; and the difficulty consists in passing from the original idea of multiplication to the generalized idea of a tensor, which idea includes compressing the magnitude as well as stretching it. Let m denote an integer number; the next step is to gain the idea of the reciprocal of m, not as but simply as / m. When m and / n are compounded we get the idea of a rational fraction; for in general m / n will not reduce to a number nor to the reciprocal of a number. Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ... In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In science, a magnitude is the numerical size of something: see orders of magnitude. ... In mathematics, the reciprocal, or multiplicative inverse, of a number x is the number which, when multiplied by x, yields 1. ...

Suppose, however, that we pass over this objection; how does Peacock lay the foundation for general algebra? He calls it symbolical algebra, and he passes from arithmetical algebra to symbolical algebra in the following manner: "Symbolical algebra adopts the rules of arithmetical algebra but removes altogether their restrictions; thus symbolical subtraction differs from the same operation in arithmetical algebra in being possible for all relations of value of the symbols or expressions employed. All the results of arithmetical algebra which are deduced by the application of its rules, and which are general in form though particular in value, are results likewise of symbolical algebra where they are general in value as well as in form; thus the product of a^m and aⁿ which is a^{m + n} when m and n are whole numbers and therefore general in form though particular in value, will be their product likewise when m and n are general in value as well as in form; the series for (a + b)ⁿ determined by the principles of arithmetical algebra when n is any whole number, if it be exhibited in a general form, without reference to a final term, may be shown upon the same principle to the equivalent series for (a + b)ⁿ when n is general both in form and value."

The principle here indicated by means of examples was named by Peacock the "principle of the permanence of equivalent forms," and at page 59 of the Symbolical Algebra it is thus enunciated: "Whatever algebraic forms are equivalent when the symbols are general in form, but specific in value, will be equivalent likewise when the symbols are general in value as well as in form."

For example, let a, b, c, d denote any integer numbers, but subject to the restrictions that b is less than a, and d less than c; it may then be shown arithmetically that (a − b)(c − d) = ac + bd − ad − bc. Peacock's principle says that the form on the left side is equivalent to the form on the right side, not only when the said restrictions of being less are removed, but when a, b, c, d denote the most general algebraic symbol. It means that a, b, c, d may be rational fractions, or surds, or imaginary quantities, or indeed operators such as . The equivalence is not established by means of the nature of the quantity denoted; the equivalence is assumed to be true, and then it is attempted to find the different interpretations which may be put on the symbol. In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ... Quantity is a kind of property which exists as magnitude or multitude. ...

It is not difficult to see that the problem before us involves the fundamental problem of a rational logic or theory of knowledge; namely, how are we able to ascend from particular truths to more general truths. If a, b, c, d denote integer numbers, of which b is less than a and d less than c, then (a − b)(c − d) = ac + bd − ad − bc.

It is first seen that the above restrictions may be removed, and still the above equation holds. But the antecedent is still too narrow; the true scientific problem consists in specifying the meaning of the symbols, which, and only which, will admit of the forms being equal. It is not to find "some meanings", but the "most general meaning", which allows the equivalence to be true. Let us examine some other cases; we shall find that Peacock's principle is not a solution of the difficulty; the great logical process of generalization cannot be reduced to any such easy and arbitrary procedure. When a, m, n denote integer numbers, it can be shown thata^maⁿ = a^{m + n}.

According to Peacock the form on the left is always to be equal to the form on the right, and the meanings of a, m, n are to be found by interpretation. Suppose that a takes the form of the incommensurate quantity e, the base of the natural system of logarithms. A number is a degraded form of a complex quantity and a complex quantity is a degraded form of a quaternion; consequently one meaning which may be assigned to m and n is that of quaternion. Peacock's principle would lead us to suppose that e^meⁿ = e^{m + n}, m and n denoting quaternions; but that is just what Hamilton, the inventor of the quaternion generalization, denies. There are reasons for believing that he was raped by who we have found dead trugernann, and that the forms remain equivalent even under that extreme generalization of m and n; but the point is this: it is not a question of conventional definition and formal truth; it is a question of objective definition and real truth. Let the symbols have the prescribed meaning, does or does not the equivalence still hold? And if it does not hold, what is the higher or more complex form which the equivalence assumes? Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...

External links

• O'Connor, John J., and Edmund F. Robertson. "George Peacock". MacTutor History of Mathematics archive.