George Peacock
Encyclopedia
George Peacock was an English
English people
The English are a nation and ethnic group native to England, who speak English. The English identity is of early mediaeval origin, when they were known in Old English as the Anglecynn. England is now a country of the United Kingdom, and the majority of English people in England are British Citizens...

 mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

.

Life

Peacock was born on 9 April 1791 at Thornton Hall, Denton, near Darlington
Darlington
Darlington is a market town in the Borough of Darlington, part of the ceremonial county of County Durham, England. It lies on the small River Skerne, a tributary of the River Tees, not far from the main river. It is the main population centre in the borough, with a population of 97,838 as of 2001...

, County Durham. His father, the Rev. Thomas Peacock, was a clergyman of the Church of England
Church of England
The Church of England is the officially established Christian church in England and the Mother Church of the worldwide Anglican Communion. The church considers itself within the tradition of Western Christianity and dates its formal establishment principally to the mission to England by St...

, 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
University of Cambridge
The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...

 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
Trinity College, Cambridge
Trinity College is a constituent college of the University of Cambridge. Trinity has more members than any other college in Cambridge or Oxford, with around 700 undergraduates, 430 graduates, and over 170 Fellows...

.

In 1812 Peacock took the rank of Second Wrangler, and the second Smith's prize
Smith's Prize
The Smith's Prize was the name of each of two prizes awarded annually to two research students in theoretical Physics, mathematics and applied mathematics at the University of Cambridge, Cambridge, England.- History :...

, the senior wrangler being John Herschel
John Herschel
Sir John Frederick William Herschel, 1st Baronet KH, FRS ,was an English mathematician, astronomer, chemist, and experimental photographer/inventor, who in some years also did valuable botanical work...

. 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.

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
Charles Babbage
Charles Babbage, FRS was an English mathematician, philosopher, inventor and mechanical engineer who originated the concept of a programmable computer...

 and Herschel
John Herschel
Sir John Frederick William Herschel, 1st Baronet KH, FRS ,was an English mathematician, astronomer, chemist, and experimental photographer/inventor, who in some years also did valuable botanical work...

 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.

The first movement on the part of the Analytical Society
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. The latter system came into being in the 18th century as an invention of Sir Isaac Newton, and was in use...

 was to translate from the French the smaller work of Lacroix
Sylvestre François Lacroix
Sylvestre François Lacroix was a French mathematician.He was born in Paris, and was raised in a poor family who still managed to obtain a good education for their son. He displayed a particular talent for mathematics, calculating the motions of theplanets by the age of 14...

 on the 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.

Another reform at which Peacock labored was the teaching of algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

. 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.

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
York
York is a walled city, situated at the confluence of the Rivers Ouse and Foss in North Yorkshire, England. The city has a rich heritage and has provided the backdrop to major political events throughout much of its two millennia of existence...

. 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
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

, Trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

, 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.

In 1837 Peacock was appointed Lowndean Professor of Astronomy in the University of Cambridge, the chair afterwards occupied by Adams
John Couch Adams
John Couch Adams was a British mathematician and astronomer. Adams was born in Laneast, near Launceston, Cornwall, and died in Cambridge. The Cornish name Couch is pronounced "cooch"....

, the co-discoverer of Neptune
Neptune
Neptune is the eighth and farthest planet from the Sun in the Solar System. Named for the Roman god of the sea, it is the fourth-largest planet by diameter and the third largest by mass. Neptune is 17 times the mass of Earth and is slightly more massive than its near-twin Uranus, which is 15 times...

, and later occupied by Sir Robert Ball
Robert Stawell Ball
Sir Robert Stawell Ball was an Irish astronomer. He worked for Lord Rosse from 1865 to 1867. In 1867 he became Professor of Applied Mathematics at the Royal College of Science in Dublin. In 1874 Ball was appointed Royal Astronomer of Ireland and Andrews Professor of Astronomy in the University...

, celebrated for his Theory of Screws. In 1839 he was appointed Dean of Ely
Dean of Ely
The position of Dean of Ely Cathedral, in East Anglia, England, was created at the time of the Dissolution of the Monasteries. The first Dean of Ely had been the last Benedictine prior of Ely.-List of Deans:*1541-1557 Robert Steward or Welles...

, 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 in Ely on 8 November 1858, in the 68th year of his age. His last public act was to attend a meeting of the Commission. He is buried in Ely cemetery.

He was elected a Fellow of the Royal Society in January 1818.

Politically he was a Whig.

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
Duncan Farquharson Gregory
Duncan Farquharson Gregory , a Scottish mathematician, was the youngest son of James Gregory and Isabella Macleod .-Education:...

, De Morgan
Augustus De Morgan
Augustus De Morgan was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. The crater De Morgan on the Moon is named after him....

 and Boole
George Boole
George Boole was an English mathematician and philosopher.As the inventor of Boolean logic—the basis of modern digital computer logic—Boole is regarded in hindsight as a founder of the field of computer science. Boole said,...

 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 we must suppose and to be quantities of the same kind; in others, like , we must suppose greater than and therefore homogeneous with it; in products and quotients, like 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."

Peacock's principle may be stated thus: the elementary symbol of arithmetical algebra denotes a digital
Digital
A digital system is a data technology that uses discrete values. By contrast, non-digital systems use a continuous range of values to represent information...

, 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 and are numbers, then is always a number; but is a number only when is less than . Again, under the same conditions, is always a number, but is really a number only when is an exact divisor of . 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 , can denote only an integer number, but may denote a rational fraction. Now there is no more fundamental principle in arithmetical algebra than that ; which would be illegitimate on Peacock's principle.

One of the earliest English writers on arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

 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."

The fact is that even in arithmetic the two processes of multiplication
Multiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....

 and division
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...

 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
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

, which idea includes compressing the magnitude
Magnitude (mathematics)
The magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....

 as well as stretching it. Let denote an integer number; the next step is to gain the idea of the reciprocal
Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...

 of , not as but simply as . When and are compounded we get the idea of a rational fraction; for in general will not reduce to a number nor to the reciprocal of a number.

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 and which is when and are whole numbers and therefore general in form though particular in value, will be their product likewise when and are general in value as well as in form; the series for determined by the principles of arithmetical algebra when 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 when 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 , , , denote any integer numbers, but subject to the restrictions that is less than , and less than ; it may then be shown arithmetically that . 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 , , , denote the most general algebraic symbol. It means that , , , may be rational fractions, or surds, or imaginary quantities, or indeed operators such as . The equivalence
Equivalence
Equivalence or equivalent may refer to:*In chemistry:**Equivalent **Equivalence point**Equivalent weight*In computing:**Turing equivalence *In ethics:**Moral equivalence*In history:...

 is not established by means of the nature of the quantity
Quantity
Quantity is a property that can exist as a magnitude or multitude. Quantities can be compared in terms of "more" or "less" or "equal", or by assigning a numerical value in terms of a unit of measurement. Quantity is among the basic classes of things along with quality, substance, change, and relation...

 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.

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 , , , denote integer numbers, of which is less than and less than , then .

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 , , denote integer numbers, it can be shown that.

According to Peacock the form on the left is always to be equal to the form on the right, and the meanings of , , are to be found by interpretation. Suppose that takes the form of the incommensurate quantity , the base of the natural system of logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

s. A number is a degraded form of a complex quantity and a complex quantity is a degraded form of a quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...

; consequently one meaning which may be assigned to and is that of quaternion. Peacock's principle would lead us to suppose that , and denoting quaternions; but that is just what Hamilton, the inventor of the quaternion generalization, denies. There are reasons for believing that he was mistaken, and that the forms remain equivalent even under that extreme generalization of and ; 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?

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