Augustus De Morgan
Augustus De Morgan (1806-1871)
Born	(1806-06-27)27 June 1806 Madurai, Madras Presidency, British Raj (now India)
Died	18 March 1871(1871-03-18) (aged 64) London, England
Residence	India  England
Nationality	British
Fields	Mathematician and logician
Institutions	University College London University College School
Alma mater	Trinity College University of Cambridge
Academic advisors	John Philips Higman George Peacock William Whewell
Notable students	Edward Routh James Joseph Sylvester Frederick Guthrie William Stanley Jevons Ada Lovelace Francis Guthrie Stephen Joseph Perry
Known for	De Morgan's laws De Morgan algebra Relation algebra Universal algebra
Influences	George Boole
Influenced	Thomas Corwin Mendenhall
Notes He was the father of William De Morgan.

Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous.^[1] The crater De Morgan on the Moon is named after him.

Biography[link]

Childhood[link]

Augustus De Morgan was born in 1806.^[2] His father was Col. Augustus De Morgan, who held various appointments in the service of the East India Company. His mother descended from James Dodson, who computed a table of anti-logarithms, that is, the numbers corresponding to exact logarithms. Augustus De Morgan became blind in one eye a month or two after he was born. The family moved to England when Augustus was seven months old. As his father and grandfather had both been born in India, De Morgan used to say that he was neither English, nor Scottish, nor Irish, but a Briton "unattached", using the technical term applied to an undergraduate of Oxford or Cambridge who is not a member of any one of the Colleges.

When De Morgan was ten years old, his father died. Mrs. De Morgan resided at various places in the southwest of England, and her son received his elementary education at various schools of no great account. His mathematical talents went unnoticed until he was fourteen, when a family-friend discovered him making an elaborate drawing of a figure in Euclid with ruler and compasses. She explained the aim of Euclid to Augustus, and gave him an initiation into demonstration.

He received his secondary education from Mr. Parsons, a Fellow of Oriel College, Oxford, who appreciated classics better than mathematics. His mother was an active and ardent member of the Church of England, and desired that her son should become a clergyman; but by this time De Morgan had begun to show his non-conforming disposition.

University education[link]

In 1823, at the age of sixteen, he entered Trinity College, Cambridge,^[3] where he came under the influence of George Peacock and William Whewell, who became his life-long friends; from the former he derived an interest in the renovation of algebra, and from the latter an interest in the renovation of logic—the two subjects of his future life work. His Cambridge tutor was John Philips Higman.

At college the flute, on which he played exquisitely, was his recreation. He was prominent in the musical clubs. His love of knowledge for its own sake interfered with training for the great mathematical race; as a consequence he came out fourth wrangler. This entitled him to the degree of Bachelor of Arts; but to take the higher degree of Master of Arts and thereby become eligible for a fellowship it was then necessary to pass a theological test. To the signing of any such test De Morgan felt a strong objection, although he had been brought up in the Church of England. In about 1875 theological tests for academic degrees were abolished in the Universities of Oxford and Cambridge.

London University[link]

As no career was open to him at his own university, he decided to go to the Bar, and took up residence in London; but he much preferred teaching mathematics to reading law. About this time the movement for founding London University (now University College London) took shape. The two ancient universities of Oxford and Cambridge were so guarded by theological tests that no Jew or Dissenter outside the Church of England could enter as a student, still less be appointed to any office. A body of liberal-minded men resolved to meet the difficulty by establishing in London a University on the principle of religious neutrality. De Morgan, then 22 years of age, was appointed Professor of Mathematics. His introductory lecture "On the study of mathematics" is a discourse upon mental education of permanent value which has been recently reprinted in the United States.

The London University was a new institution, and the relations of the Council of management, the Senate of professors and the body of students were not well defined. A dispute arose between the professor of anatomy and his students, and in consequence of the action taken by the Council, several professors resigned, headed by De Morgan. Another professor of mathematics was appointed, who then drowned a few years later. De Morgan had shown himself a prince of teachers: he was invited to return to his chair, which thereafter became the continuous centre of his labours for thirty years.

The same body of reformers—headed by Lord Brougham, a Scotsman eminent both in science and politics who had instituted the London University—founded about the same time a Society for the Diffusion of Useful Knowledge. Its object was to spread scientific and other knowledge by means of cheap and clearly written treatises by the best writers of the time. One of its most voluminous and effective writers was De Morgan. He wrote a great work on The Differential and Integral Calculus which was published by the Society; and he wrote one-sixth of the articles in the Penny Cyclopedia, published by the Society, and issued in penny numbers. When De Morgan came to reside in London he found a congenial friend in William Frend, notwithstanding his mathematical heresy about negative quantities. Both were arithmeticians and actuaries, and their religious views were somewhat similar. Frend lived in what was then a suburb of London, in a country-house formerly occupied by Daniel Defoe and Isaac Watts. De Morgan with his flute was a welcome visitor; and in 1837 he married Sophia Elizabeth, one of Frend's daughters.

The London University of which De Morgan was a professor was a different institution from the University of London. The University of London was founded about ten years later by the Government for the purpose of granting degrees after examination, without any qualification as to residence. The London University was affiliated as a teaching college with the University of London, and its name was changed to University College. The University of London was not a success as an examining body; a teaching University was demanded. De Morgan was a highly successful teacher of mathematics. It was his plan to lecture for an hour, and at the close of each lecture to give out a number of problems and examples illustrative of the subject lectured on; his students were required to sit down to them and bring him the results, which he looked over and returned revised before the next lecture. In De Morgan's opinion, a thorough comprehension and mental assimilation of great principles far outweighed in importance any merely analytical dexterity in the application of half-understood principles to particular cases.

During this period, he also promoted the work of the self-taught Indian mathematician Ramchundra, who has been called De Morgan's Ramanujam. He supervised the publication in London of Ramchundra's book on "Maxima and Minima" in 1859. In the introduction to this book, he acknowledged being aware of the Indian tradition of logic, although it is not known whether this had any influence on his own work.

Family[link]

He married in the autumn of 1837 Sophia Elizabeth, eldest daughter of William Frend (social reformer) and his wife, a granddaughter of Archdeacon Francis Blackburne.^[4]

De Morgan had three sons and four daughters, including fairytale author Mary de Morgan. His eldest son was the potter William De Morgan. His second son George acquired great distinction in mathematics both at University College and the University of London. He and another like-minded alumnus conceived the idea of founding a Mathematical Society in London, where mathematical papers would be not only received (as by the Royal Society) but actually read and discussed. The first meeting was held in University College; De Morgan was the first president, his son the first secretary. It was the beginning of the London Mathematical Society.

Retirement and death[link]

Augustus De Morgan

In 1866 the chair of mental philosophy in University College fell vacant. James Martineau, a Unitarian clergyman and professor of mental philosophy, was recommended formally by the Senate to the Council; but in the Council there were some who objected to a Unitarian clergyman, and others who objected to theistic philosophy. A layman of the school of Bain and Spencer was appointed. De Morgan considered that the old standard of religious neutrality had been hauled down, and forthwith resigned. He was now 60 years of age. His pupils secured him a pension of £500 p.a., but misfortunes followed. Two years later his son George — the "younger Bernoulli", as Augustus loved to hear him called, in allusion to the eminent father-and-son mathematicians of that name — died. This blow was followed by the death of a daughter. Five years after his resignation from University College De Morgan died of nervous prostration on 18 March 1871.

Mathematical work[link]

De Morgan was a brilliant and witty writer, whether as a controversialist or as a correspondent. In his time there flourished two Sir William Hamiltons who have often been confounded. The one was Sir William Hamilton, 9th Baronet (that is, his title was inherited), a Scotsman, professor of logic and metaphysics at the University of Edinburgh; the other was a knight (that is, won the title), an Irishman, professor at astronomy in the University of Dublin. The baronet contributed to logic, especially the doctrine of the quantification of the predicate; the knight, whose full name was William Rowan Hamilton, contributed to mathematics, especially geometric algebra, and first described the Quaternions. De Morgan was interested in the work of both, and corresponded with both; but the correspondence with the Scotsman ended in a public controversy, whereas that with the Irishman was marked by friendship and terminated only by death. In one of his letters to Rowan, De Morgan says,

"Be it known unto you that I have discovered that you and the other Sir W. H. are reciprocal polars with respect to me (intellectually and morally, for the Scottish baronet is a polar bear, and you, I was going to say, are a polar gentleman). When I send a bit of investigation to Edinburgh, the W. H. of that ilk says I took it from him. When I send you one, you take it from me, generalize it at a glance, bestow it thus generalized upon society at large, and make me the second discoverer of a known theorem."

The correspondence of De Morgan with Hamilton the mathematician extended over twenty-four years; it contains discussions not only of mathematical matters, but also of subjects of general interest. It is marked by geniality on the part of Hamilton and by wit on the part of De Morgan. The following is a specimen: Hamilton wrote,

"My copy of Berkeley's work is not mine; like Berkeley, you know, I am an Irishman."

De Morgan replied,

"Your phrase 'my copy is not mine' is not a bull. It is perfectly good English to use the same word in two different senses in one sentence, particularly when there is usage. Incongruity of language is no bull, for it expresses meaning. But incongruity of ideas (as in the case of the Irishman who was pulling up the rope, and finding it did not finish, cried out that somebody had cut off the other end of it) is the genuine bull."

De Morgan was full of personal peculiarities. On the occasion of the installation of his friend, Lord Brougham, as Rector of the University of Edinburgh, the Senate offered to confer on him the honorary degree of LL. D.; he declined the honour as a misnomer. He once printed his name: Augustus De Morgan, H - O - M - O - P - A - U - C - A - R - U - M - L - I - T - E - R - A - R - U - M (Latin for "man of few letters").^{[citation needed]}

He disliked the provinces outside London, and while his family enjoyed the seaside, and men of science were having a good time at a meeting of the British Association in the country he remained in the hot and dusty libraries of the metropolis. He said that he felt like Socrates, who declared that the farther he was from Athens the farther was he from happiness. He never sought to become a Fellow of the Royal Society, and he never attended a meeting of the Society; he said that he had no ideas or sympathies in common with the physical philosopher. His attitude was possibly due to his physical infirmity, which prevented him from being either an observer or an experimenter. He never voted at an election, and he never visited the House of Commons, or the Tower of London, or Westminster Abbey.

Were the writings of De Morgan published in the form of collected works, they would form a small library, for example his writings for the Useful Knowledge Society. Mainly through the efforts of Peacock and Whewell, a Philosophical Society had been inaugurated at Cambridge; and to its Transactions De Morgan contributed four memoirs on the foundations of algebra, and an equal number on formal logic. The best presentation of his view of algebra is found in a volume, entitled Trigonometry and Double Algebra, published in 1849; and his earlier view of formal logic is found in a volume published in 1847. His most distinctive work is styled a Budget of Paradoxes; it originally appeared as letters in the columns of the Athenæum journal; it was revised and extended by De Morgan in the last years of his life, and was published posthumously by his widow.

George Peacock's theory of algebra was much improved by D. F. Gregory, a younger member of the Cambridge School, who laid stress not on the permanence of equivalent forms, but on the permanence of certain formal laws. This new theory of algebra as the science of symbols and of their laws of combination was carried to its logical issue by De Morgan; and his doctrine on the subject is still followed by English algebraists in general. Thus George Chrystal founds his Textbook of Algebra on De Morgan's theory; although an attentive reader may remark that he practically abandons it when he takes up the subject of infinite series. De Morgan's theory is stated in his volume on Trigonometry and Double Algebra. In the chapter (of the book) headed "On symbolic algebra" he writes:

"In abandoning the meaning of symbols, we also abandon those of the words which describe them. Thus addition is to be, for the present, a sound void of sense. It is a mode of combination represented by Failed to parse (Missing texvc executable; please see math/README to configure.): +

when Failed to parse (Missing texvc executable; please see math/README to configure.): +

receives its meaning, so also will the word addition. It is most important that the student should bear in mind that, with one exception, no word nor sign of arithmetic or algebra has one atom of meaning throughout this chapter, the object of which is symbols, and their laws of combination, giving a symbolic algebra which may hereafter become the grammar of a hundred distinct significant algebras. If any one were to assert that Failed to parse (Missing texvc executable; please see math/README to configure.): +
and Failed to parse (Missing texvc executable; please see math/README to configure.): -
might mean reward and punishment, and Failed to parse (Missing texvc executable; please see math/README to configure.): A

, Failed to parse (Missing texvc executable; please see math/README to configure.): B , Failed to parse (Missing texvc executable; please see math/README to configure.): C , etc., might stand for virtues and vices, the reader might believe him, or contradict him, as he pleases, but not out of this chapter. The one exception above noted, which has some share of meaning, is the sign Failed to parse (Missing texvc executable; please see math/README to configure.): =

placed between two symbols as in Failed to parse (Missing texvc executable; please see math/README to configure.): A = B

. It indicates that the two symbols have the same resulting meaning, by whatever steps attained. That Failed to parse (Missing texvc executable; please see math/README to configure.): A

and Failed to parse (Missing texvc executable; please see math/README to configure.): B

, if quantities, are the same amount of quantity; that if operations, they are of the same effect, etc."

Here, it may be asked, why does the symbol Failed to parse (Missing texvc executable; please see math/README to configure.): =

prove refractory to the symbolic theory? De Morgan admits that there is one exception; but an exception proves the rule, not in the usual but illogical sense of establishing it, but in the old and logical sense of testing its validity. If an exception can be established, the rule must fall, or at least must be modified. Here I am talking not of grammatical rules, but of the rules of science or nature.

De Morgan proceeds to give an inventory of the fundamental symbols of algebra, and also an inventory of the laws of algebra. The symbols are 0, 1, +, −, ×, ÷, ()⁽⁾, and letters; these only, all others are derived. His inventory of the fundamental laws is expressed under fourteen heads, but some of them are merely definitions. The laws proper may be reduced to the following, which, as he admits, are not all independent of one another:

1. Law of signs. + + = +, + − = −, − + = −, − − = +, × × = ×, × ÷ = ÷, ÷ × = ÷, ÷ ÷ = ×.
2. Commutative law. a+b = b+a, ab=ba.
3. Distributive law. a(b+c) = ab+ac.
4. Index laws. a^b×a^c=a^b+c, (a^b)^c=a^bc, (ab)^d= a^d×b^d.
5. a−a=0, a÷a=1.

The last two may be called the rules of reduction. De Morgan professes to give a complete inventory of the laws which the symbols of algebra must obey, for he says, "Any system of symbols which obeys these laws and no others, except they be formed by combination of these laws, and which uses the preceding symbols and no others, except they be new symbols invented in abbreviation of combinations of these symbols, is symbolic algebra." From his point of view, none of the above principles are rules; they are formal laws, that is, arbitrarily chosen relations to which the algebraic symbols must be subject. He does not mention the law, which had already been pointed out by Gregory, namely, Failed to parse (Missing texvc executable; please see math/README to configure.): (a+b)+c = a+(b+c), (ab)c = a(bc)

and to which was afterwards given the name of the law of association. If the commutative law fails, the associative may hold good; but not vice versa. It is an unfortunate thing for the symbolist or formalist that in universal arithmetic Failed to parse (Missing texvc executable; please see math/README to configure.): m^n
is not equal to Failed to parse (Missing texvc executable; please see math/README to configure.): n^m

for then the commutative law would have full scope. Why does he not give it full scope? Because the foundations of algebra are, after all, real not formal, material not symbolic. To the formalists the index operations are exceedingly refractory, in consequence of which some take no account of them, but relegate them to applied mathematics. To give an inventory of the laws which the symbols of algebra must obey is an impossible task, and reminds one not a little of the task of those philosophers who attempt to give an inventory of the a priori knowledge of the mind.

Trigonometry and double algebra[link]

De Morgan's work entitled Trigonometry and Double Algebra consists of two parts; the former of which is a treatise on Trigonometry, and the latter a treatise on generalized algebra which he calls Double Algebra.The first stage in the development of algebra is arithmetic, where numbers only appear and symbols of operations such as Failed to parse (Missing texvc executable; please see math/README to configure.): + , Failed to parse (Missing texvc executable; please see math/README to configure.): \times , etc. The next stage is universal arithmetic, where letters appear instead of numbers, so as to denote numbers universally, and the processes are conducted without knowing the values of the symbols. Let Failed to parse (Missing texvc executable; please see math/README to configure.): a

and Failed to parse (Missing texvc executable; please see math/README to configure.): b
denote any numbers; then such an expression as Failed to parse (Missing texvc executable; please see math/README to configure.): a-b
may be impossible; so that in universal arithmetic there is always a proviso, provided the operation is possible. The third stage is single algebra, where the symbol may denote a quantity forwards or a quantity backwards, and is adequately represented by segments on a straight line passing through an origin. Negative quantities are then no longer impossible; they are represented by the backward segment. But an impossibility still remains in the latter part of such an expression as Failed to parse (Missing texvc executable; please see math/README to configure.): a+b\sqrt{-1}
which arises in the solution of the quadratic equation. The fourth stage is double algebra; the algebraic symbol denotes in general a segment of a line in a given plane; it is a double symbol because it involves two specifications, namely, length and direction; and Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{-1}
is interpreted as denoting a quadrant. The expression Failed to parse (Missing texvc executable; please see math/README to configure.): a+b\sqrt{-1}
then represents a line in the plane having an abscissa Failed to parse (Missing texvc executable; please see math/README to configure.): a
and an ordinate Failed to parse (Missing texvc executable; please see math/README to configure.): b

. Argand and Warren carried double algebra so far; but they were unable to interpret on this theory such an expression as Failed to parse (Missing texvc executable; please see math/README to configure.): e^{a\sqrt{-1}} . De Morgan attempted it by reducing such an expression to the form Failed to parse (Missing texvc executable; please see math/README to configure.): b+q\sqrt{-1} , and he considered that he had shown that it could be always so reduced. The remarkable fact is that this double algebra satisfies all the fundamental laws above enumerated, and as every apparently impossible combination of symbols has been interpreted it looks like the complete form of algebra. In chapter 6 he introduced hyperbolic functions and discussed the connection of common and hyperbolic trigonometry.

If the above theory is true, the next stage of development ought to be triple algebra and if Failed to parse (Missing texvc executable; please see math/README to configure.): a+b\sqrt{-1}

truly represents a line in a given plane, it ought to be possible to find a third term which added to the above would represent a line in space. Argand and some others guessed that it was Failed to parse (Missing texvc executable; please see math/README to configure.): a + b\sqrt{-1} + c\sqrt{-1}\,^{\sqrt{-1}}
although this contradicts the truth established by Euler that Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{-1}\,^{\sqrt{-1}}=e^{-\frac{1}{2} \pi}

. De Morgan and many others worked hard at the problem, but nothing came of it until the problem was taken up by Hamilton. We now see the reason clearly: the symbol of double algebra denotes not a length and a direction; but a multiplier and an angle. In it the angles are confined to one plane; hence the next stage will be a quadruple algebra, when the axis of the plane is made variable. And this gives the answer to the first question; double algebra is nothing but analytical plane trigonometry, and this is why it has been found to be the natural analysis for alternating currents. But De Morgan never got this far; he died with the belief "that double algebra must remain as the full development of the conceptions of arithmetic, so far as those symbols are concerned which arithmetic immediately suggests."

When the study of mathematics revived at the University of Cambridge, so did the study of logic. The moving spirit was Whewell, the Master of Trinity College, whose principal writings were a History of the Inductive Sciences, and Philosophy of the Inductive Sciences. Doubtless De Morgan was influenced in his logical investigations by Whewell; but other influential contemporaries were Sir W. Hamilton of Edinburgh, and Professor Boole of Cork. De Morgan's work on Formal Logic, published in 1847, is principally remarkable for his development of the numerically definite syllogism. The followers of Aristotle say that from two particular propositions such as Some M's are A's , and Some M's are B's nothing follows of necessity about the relation of the A's and B's. But they go further and say in order that any relation about the A's and B's may follow of necessity, the middle term must be taken universally in one of the premises. De Morgan pointed out that from Most M's are A's and Most M's are B's it follows of necessity that some A's are B's and he formulated the numerically definite syllogism which puts this principle in exact quantitative form. Suppose that the number of the M's is Failed to parse (Missing texvc executable; please see math/README to configure.): m , of the M's that are A's is Failed to parse (Missing texvc executable; please see math/README to configure.): a , and of the M's that are B's is Failed to parse (Missing texvc executable; please see math/README to configure.): b

then there are at least Failed to parse (Missing texvc executable; please see math/README to configure.): (a+b-m)

A's that are B's. Suppose that the number of souls on board a steamer was 1000, that 500 were in the saloon, and 700 were lost; it follows of necessity, that at least 700+500-1000, that is, 200, saloon passengers were lost. This single principle suffices to prove the validity of all the Aristotelian moods; it is therefore a fundamental principle in necessary reasoning.

Here then De Morgan had made a great advance by introducing quantification of the terms. At that time Sir W. Hamilton was teaching at Edinburgh a doctrine of the quantification of the predicate, and a correspondence sprang up. However, De Morgan soon perceived that Hamilton's quantification was of a different character; that it meant for example, substituting the two forms The whole of A is the whole of B, and The whole of A is a part of B for the Aristotelian form All A's are B's. Hamilton thought that he had placed the keystone in the Aristotelian arch, as he phrased it; although it must have been a curious arch which could stand 2000 years without a keystone. As a consequence he had no room for De Morgan's innovations. He accused De Morgan of plagiarism, and the controversy raged for years in the columns of the Athenæum, and in the publications of the two writers.

The memoirs on logic which De Morgan contributed to the Transactions of the Cambridge Philosophical Society subsequent to the publication of his book on Formal Logic are by far the most important contributions which he made to the science, especially his fourth memoir, in which he begins work in the broad field of the logic of relatives. This is the true field for the logician of the twentieth century, in which work of the greatest importance is to be done towards improving language and facilitating thinking processes which occur all the time in practical life. Identity and difference are the two relations which have been considered by the logician; but there are many others equally deserving of study, such as equality, equivalence, consanguinity, affinity, etc.

In the introduction to the Budget of Paradoxes De Morgan explains what he means by the word.

"A great many individuals, ever since the rise of the mathematical method, have, each for himself, attacked its direct and indirect consequences. I shall call each of these persons a paradoxer, and his system a paradox. I use the word in the old sense: a paradox is something which is apart from general opinion, either in subject matter, method, or conclusion. Many of the things brought forward would now be called crotchets, which is the nearest word we have to old paradox. But there is this difference, that by calling a thing a crotchet we mean to speak lightly of it; which was not the necessary sense of paradox. Thus in the 16th century many spoke of the earth's motion as the paradox of Copernicus and held the ingenuity of that theory in very high esteem, and some I think who even inclined towards it. In the seventeenth century the deprivation of meaning took place, in England at least."

How can the sound paradoxer be distinguished from the false paradoxer? De Morgan supplies the following test:

"The manner in which a paradoxer will show himself, as to sense or nonsense, will not depend upon what he maintains, but upon whether he has or has not made a sufficient knowledge of what has been done by others, especially as to the mode of doing it, a preliminary to inventing knowledge for himself... New knowledge, when to any purpose, must come by contemplation of old knowledge, in every matter which concerns thought; mechanical contrivance sometimes, not very often, escapes this rule. All the men who are now called discoverers, in every matter ruled by thought, have been men versed in the minds of their predecessors and learned in what had been before them. There is not one exception."

"I remember that just before the American Association met at Indianapolis in 1890, the local newspapers heralded a great discovery which was to be laid before the assembled savants -- a young man living somewhere in the country had squared the circle. While the meeting was in progress I observed a young man going about with a roll of paper in his hand. He spoke to me and complained that the paper containing his discovery had not been received. I asked him whether his object in presenting the paper was not to get it read, printed and published so that everyone might inform himself of the result; to all of which he assented readily. But, said I, many men have worked at this question, and their results have been tested fully, and they are printed for the benefit of anyone who can read; have you informed yourself of their results? To this there was no assent, but the sickly smile of the false paradoxer"

The Budget consists of a review of a large collection of paradoxical books which De Morgan had accumulated in his own library, partly by purchase at bookstands, partly from books sent to him for review, partly from books sent to him by the authors. He gives the following classification: squarers of the circle, trisectors of the angle, duplicators of the cube, constructors of perpetual motion, subverters of gravitation, stagnators of the earth, builders of the universe. You will still find specimens of all these classes in the New World and in the new century. De Morgan gives his personal knowledge of paradoxers.

"I suspect that I know more of the English class than any man in Britain. I never kept any reckoning: but I know that one year with another? -- and less of late years than in earlier time? -- I have talked to more than five in each year, giving more than a hundred and fifty specimens. Of this I am sure, that it is my own fault if they have not been a thousand. Nobody knows how they swarm, except those to whom they naturally resort. They are in all ranks and occupations, of all ages and characters. They are very earnest people, and their purpose is bona fide, the dissemination of their paradoxes. A great many -- the mass, indeed -- are illiterate, and a great many waste their means, and are in or approaching penury. These discoverers despise one another."

A paradoxer to whom De Morgan paid the compliment which Achilles paid Hector—to drag him round the walls again and again—was James Smith, a successful merchant of Liverpool. He found Failed to parse (Missing texvc executable; please see math/README to configure.): \pi = 3 \frac{1}{8} . His mode of reasoning was a curious caricature of the reductio ad absurdum of Euclid. He said let Failed to parse (Missing texvc executable; please see math/README to configure.): \pi = 3 \frac{1}{8} , and then showed that on that supposition, every other value of Failed to parse (Missing texvc executable; please see math/README to configure.): \pi

must be absurd; consequently Failed to parse (Missing texvc executable; please see math/README to configure.): \pi = 3\frac{1}{8}
is the true value. The following is a specimen of De Morgan's dragging round the walls of Troy:

"Mr. Smith continues to write me long letters, to which he hints that I am to answer. In his last of 31 closely written sides of note paper, he informs me, with reference to my obstinate silence, that though I think myself and am thought by others to be a mathematical Goliath, I have resolved to play the mathematical snail, and keep within my shell. A mathematical snail! This cannot be the thing so called which regulates the striking of a clock; for it would mean that I am to make Mr. Smith sound the true time of day, which I would by no means undertake upon a clock that gains 19 seconds odd in every hour by false quadrative value of Failed to parse (Missing texvc executable; please see math/README to configure.): \pi

. But he ventures to tell me that pebbles from the sling of simple truth and common sense will ultimately crack my shell, and put me hors de combat. The confusion of images is amusing: Goliath turning himself into a snail to avoid Failed to parse (Missing texvc executable; please see math/README to configure.): \pi = 3\frac{1}{8}

and James Smith, Esq., of the Mersey Dock Board: and put hors de combat by pebbles from a sling. If Goliath had crept into a snail shell, David would have cracked the Philistine with his foot. There is something like modesty in the implication that the crack-shell pebble has not yet taken effect; it might have been thought that the slinger would by this time have been singing -- And thrice [and one-eighth] I routed all my foes, And thrice [and one-eighth] I slew the slain."

In the region of pure mathematics De Morgan could detect easily the false from the true paradox; but he was not so proficient in the field of physics. His father-in-law was a paradoxer, and his wife a paradoxer; and in the opinion of the physical philosophers De Morgan himself scarcely escaped. His wife wrote a book describing the phenomena of spiritualism, table-rapping, table-turning, etc.; and De Morgan wrote a preface in which he said that he knew some of the asserted facts, believed others on testimony, but did not pretend to know whether they were caused by spirits, or had some unknown and unimagined origin. From this alternative he left out ordinary material causes. Faraday delivered a lecture on Spiritualism, in which he laid it down that in the investigation we ought to set out with the idea of what is physically possible, or impossible; De Morgan did not believe this.

Relations[link]

De Morgan discovered relation algebra in his Syllabus of a Proposed System of Logic (1966: 208-46), first published in 1860. This algebra was extended by Charles Sanders Peirce (who admired De Morgan and met him shortly before his death), and re-exposited and further extended in vol. 3 of Ernst Schröder's Vorlesungen über die Algebra der Logik. Relation algebra proved critical to the Principia Mathematica of Bertrand Russell and Alfred North Whitehead. In turn, this algebra became the subject of much further work, starting in 1940, by Alfred Tarski and his colleagues and students at the University of California.

Spiritualism[link]

De Morgan later in his life became interested in the phenomena of Spiritualism. In 1849 he had investigated clairvoyance and was impressed by the subject. He later carried out paranormal investigations in his own home with the medium Maria Hayden. The result of these investigations was later published by his wife Sophia. De Morgan believed that his career as a scientist might have been affected if he had revealed his interest in the study of spiritualism so he helped to publish the book anonymously.^[5] The book was published in 1863 titled From Matter to Spirit: The Result of Ten Years Experience in Spirit Manifestations.

According to (Oppenheim, 1988) De Morgan's wife Sophia was a convinced spiritualist but De Morgan shared a third way position on spiritualist phenomena which Oppenheim defined as a "wait-and-see position", he was neither a believer or a skeptic, instead his viewpoint was that the methodology of the physical sciences does not automatically exclude psychic phenomena and that such phenomena may be explainable in time by the possible existence of natural forces which as yet physicists had not identified.^[6]

In the preface of From Matter to Spirit (1963) De Morgan stated:

Thinking it very likely that the universe may contain a few agencies - say half a million - about which no man knows anything, I can not but suspect that a small proportion of these agencies - say five thousand - may be severally competent to the production of all the [spiritualist] phenomena, or may be quite up to the task among them. The physical explanations which I have seen are easy, but miserably insufficient: the spiritualist hypothesis is sufficient, but ponderously difficult. Time and thought will decide, the second asking the first for more results of trial.

John Beloff in Parapsychology: A Concise History (1997) wrote that De Morgan was the first notable scientist in Britain to take an interest in the study of spiritualism and his studies had influenced the decision of William Crookes to also study spiritualism. De Morgan was also an atheist and because of this had debarred him from a position at Oxford and Cambridge.^[7]

Legacy[link]

Beyond his great mathematical legacy, the headquarters of the London Mathematical Society is called De Morgan House and the student society of the Mathematics Department of University College London is called the August De Morgan Society.

Selected writings[link]

• 1836. An Explanation of the Gnomonic Projection of the Sphere. London: Baldwin.
• 1837. Elements of Trigonometry, and Trigonometrical Analysis. London: Taylor & Walton.
• 1837. The Elements of Algebra. London: Taylor & Walton.
• 1838. An Essay on Probabilities. London: Longman, Orme, Brown, Green & Longmans.
• 1840. The Elements of Arithmetic. London: Taylor & Walton.
• 1840. First Notions of Logic, Preparatory to the Study of Geometry. London: Taylor & Walton.
• 1842. The Differential and Integral Calculus. London: Baldwin.
• 1845. The Globes, Celestial and Terrestrial. London: Malby & Co.
• 1847. Formal Logic or The Calculus of Inference. London: Taylor & Walton.
• 1849. Trigonometry and Double Algebra. London: Taylor, Walton & Malbery.
• 1860. Syllabus of a Proposed System of Logic. London: Walton & Malbery.
• 1872. A Budget of Paradoxes London: Longmans, Green.

References and notes[link]

1. ^ De Morgan, (1838) Induction (mathematics), The Penny Cyclopedia.
2. ^ The year of his birth may be found by solving a conundrum proposed by himself, "I was Failed to parse (Missing texvc executable; please see math/README to configure.): x years of age in the year Failed to parse (Missing texvc executable; please see math/README to configure.): x^2 " (He was 43 in 1849). The problem is indeterminate, but it is made strictly determinate by the century of its utterance and the limit to a man's life. Those born in 1722 (1764-42), 1892 (1936-44) and 1980 (2025-45) are similarly privileged.
3. ^ Venn, J.; Venn, J. A., eds. (1922–1958). "De Morgan, Augustus". Alumni Cantabrigienses (10 vols) (online ed.). Cambridge University Press. 
4. ^  "Frend, William". Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900. 
5. ^ Geoffrey K. Nelson Spiritualism and Society 1969, p. 90
6. ^ Janet Oppenheim The Other World: Spiritualism and Psychical Research in England, 1850-1914 1988 p. 335
7. ^ John Beloff Parapsychology: A Concise History 1997, pp. 46-47

Further reading[link]

• De Morgan, A., 1966. Logic: On the Syllogism and Other Logical Writings. Heath, P., ed. Routledge. A useful collection of De Morgan's most important writings on logic.
• De Morgan, Sophia Elizabeth (1882). Memoir of Augustus De Morgan. London: Longmans, Green and Company. http://books.google.com/?id=YiGJGOtbMHcC&printsec=frontcover&dq=Augustus+De+Morgan. 
• Grattan-Guinness, Ivor (2000). The Search for Mathematical Roots 1870-1940. Princeton University Press. 
• Macfarlane, Alexander (1916). Lectures on Ten British Mathematicians of the Nineteenth Century. New York: John Wiley and Sons. http://www.gutenberg.net/etext06/tbmms10p.pdf. 
• Merrill, Daniel D. (1990). Augustus De Morgan and the Logic of Relations. Dordrecht: Kluwer Academic Publishers. 

External links[link]

Wikisource has original works written by or about: Augustus de Morgan

Wikiquote has a collection of quotations related to: Augustus De Morgan

Wikimedia Commons has media related to: Augustus De Morgan

• O'Connor, John J.; Robertson, Edmund F., "Augustus De Morgan", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/De_Morgan.html .
• Papers of Augustus De Morgan held by Senate House Library, University of London
• Library of Augustus De Morgan
• Archival material relating to Augustus De Morgan listed at the UK National Register of Archives
• Portraits of Augustus De Morgan at the National Portrait Gallery, London

This article's introduction may be too long for its overall length. Please help by moving some material from it into the body of the article. For more information please read the layout guide and Wikipedia's lead section guidelines. (May 2012)

Augustus
1st Emperor of the Roman Empire
The statue known as the Augustus of Prima Porta, 1st century
Reign	16 January 27 BC – 19 August AD 14
Full name	Birth to adoption: Gaius Octavius Thurinus Adoption to accession: Gaius Julius Caesar As Emperor: Gaius Julius Caesar Augustus
Born	23 September 63 BC
Birthplace	Rome, Roman Republic
Died	19 August AD 14 (aged 75)
Place of death	Nola, Italia, Roman Empire
Buried	Mausoleum of Augustus, Rome
Predecessor	None (Empire founded)
Successor	Tiberius
Consort to	Clodia Pulchra Scribonia Livia Drusilla
Offspring	Julia the Elder Gaius Caesar (adoptive) Lucius Caesar (adoptive) Agrippa Postumus (adoptive) Tiberius (adoptive)
Father	Gaius Octavius
Mother	Atia Balba Caesonia

These articles cover Ancient Rome and the fall of the Republic

Mark Antony, Cleopatra VII, Assassination of Julius Caesar, Pompey, Theatre of Pompey, Cicero, First Triumvirate, Roman forum, Comitium, Rostra, Curia Julia, Curia Hostilia

Augustus ( /ɔːˈɡʌstəs/;^[1] Classical Latin: [awˈɡʊstʊs]; Latin: Imperator Gaius Julius Caesar Augustus;^{[note 1]} 23 September 63 BC – 19 August AD 14) is considered the first emperor of the Roman Empire, which he ruled alone from 27 BC until his death in 14 AD.^{[note 2]} Born Gaius Octavius Thurinus, he was adopted posthumously by his great-uncle Gaius Julius Caesar in 44 BC via his last will and testament, and between then and 27 BC was officially named Gaius Julius Caesar. In 27 BC the Senate awarded him the honorific Augustus ("the revered one"), and thus consequently he was Gaius Julius Caesar Augustus.^{[note 3]}

The young Octavius came into his inheritance after Caesar's assassination in 44 BC. In 43 BC, Octavian joined forces with Mark Antony and Marcus Aemilius Lepidus in a military dictatorship known as the Second Triumvirate. As a triumvir, Octavian ruled Rome and many of its provinces.^{[note 4]} The triumvirate was eventually torn apart under the competing ambitions of its rulers: Lepidus was driven into exile, and Antony committed suicide following his defeat at the Battle of Actium by Octavian's fleet under the command of Agrippa in 31 BC.

After the demise of the Second Triumvirate, Octavian restored the outward facade of the Roman Republic, with governmental power vested in the Roman Senate, but in practice retained his autocratic power. It took several years to determine the exact framework by which a formally republican state could be led by a sole ruler; the result became known as the Roman Empire. The emperorship was never an office like the Roman dictatorship which Caesar and Sulla had held before him; indeed, he declined it when the Roman populace "entreated him to take on the dictatorship".^[2]

He was not known as what we would call "Emperor" in his lifetime; "Imperator" was (to begin with) simply a title given to an unusually successful general. But Augustus was given, among other titles, that of "princeps", "the first (of citizens)". Originally it had no royal connotations. By law, Augustus held a collection of powers granted to him for life by the Senate, including those of tribune of the plebs and censor. He was consul until 23 BC.^[3]

His substantive power stemmed from financial success and resources gained in conquest, the building of patronage relationships throughout the Empire, the loyalty of many military soldiers and veterans, the authority of the many honors granted by the Senate,^[4] and the respect of the people. Augustus' control over the majority of Rome's legions established an armed threat that could be used against the Senate, allowing him to coerce the Senate's decisions. With his ability to eliminate senatorial opposition by means of arms, the Senate became docile towards him. His rule through patronage, military power, and accumulation of the offices of the defunct Republic became the model for all later imperial governments.

The reign of Augustus initiated an era of relative peace known as the Pax Romana, or Roman peace. Despite continuous wars on the frontiers, and one year-long civil war over the imperial succession, the Mediterranean world remained at peace for more than two centuries. Augustus enlarged the empire dramatically, annexing Egypt, Dalmatia, Pannonia, and Raetia, expanded possessions in Africa, and completed the conquest of Hispania. Beyond the frontiers, he secured the empire with client states, and made peace with Parthia through diplomacy.

He reformed the Roman system of taxation, developed networks of roads with an official courier system, established a standing army, established the Praetorian Guard, and created official police and fire-fighting services for Rome. Much of the city was rebuilt under Augustus; and he wrote a record of his own accomplishments, known as the Res Gestae Divi Augusti, which has survived. Upon his death in AD 14, Augustus was declared a god by the Senate—to be worshipped by the Romans.^[5] His names Augustus and Caesar were adopted by every subsequent emperor; and the eighth month of the Roman calendar, previously named Sextilis, was renamed Augustus (August in English) in his honour. He was succeeded by his adopted son (also stepson and former son-in-law), Tiberius.

Early life[link]

While his paternal family was from the town of Velletri, about 40 kilometres (25 mi) from Rome, Augustus was born in the city of Rome on 23 September 63 BC. He was born at Ox Head, a small property on the Palatine Hill, very close to the Roman Forum. He was given the name Gaius Octavius Thurinus, his cognomen possibly commemorating his father's victory at Thurii over a rebellious band of slaves.^[6][7]

Due to the crowded nature of Rome at the time, Octavius was taken to his father's home village at Velletri to be raised. Octavius only mentions his father's equestrian family briefly in his memoirs. His paternal great-grandfather was a military tribune in Sicily during the Second Punic War. His grandfather had served in several local political offices. His father, also named Gaius Octavius, had been governor of Macedonia.^{[note 5][8]} His mother Atia was the niece of Julius Caesar.

In 59 BC, when he was four years old, his father died.^[9] His mother married a former governor of Syria, Lucius Marcius Philippus.^[10] Philippus claimed descent from Alexander the Great, and was elected consul in 56 BC. Philippus never had much of an interest in young Octavius. Because of this, Octavius was raised by his grandmother (and Julius Caesar's sister), Julia Caesaris.

In 52 or 51 BC, Julia Caesaris died. Octavius delivered the funeral oration for his grandmother.^[11] From this point, his mother and stepfather took a more active role in raising him. He donned the toga virilis four years later,^[12] and was elected to the College of Pontiffs in 47 BC.^[13][14] The following year he was put in charge of the Greek games that were staged in honor of the Temple of Venus Genetrix, built by Julius Caesar.^[14] According to Nicolaus of Damascus, Octavius wished to join Caesar's staff for his campaign in Africa but gave way when his mother protested.^[15] In 46 BC, she consented for him to join Caesar in Hispania, where he planned to fight the forces of Pompey, Caesar's late enemy, but Octavius fell ill and was unable to travel.

When he had recovered, he sailed to the front, but was shipwrecked; after coming ashore with a handful of companions, he crossed hostile territory to Caesar's camp, which impressed his great-uncle considerably.^[12] Velleius Paterculus reports that Caesar afterwards allowed the young man to share his carriage.^[16] When back in Rome, Caesar deposited a new will with the Vestal Virgins, naming Octavius as the prime beneficiary.^[17]

Rise to power[link]

Heir to Caesar[link]

The Death of Caesar, by Jean-Léon Gérôme (1867). On 15 March 44 BC, Octavius's adoptive father Julius Caesar was assassinated by a conspiracy led by Marcus Junius Brutus and Gaius Cassius Longinus.

At the time Caesar was killed on the Ides of March (the 15th) 44 BC, Octavius was studying and undergoing military training in Apollonia, Illyria. Rejecting the advice of some army officers to take refuge with the troops in Macedonia, he sailed to Italia to ascertain if he had any potential political fortunes or security.^[18] After landing at Lupiae near Brundisium, he learned the contents of Caesar's will, and only then did he decide to become Caesar's political heir as well as heir to two-thirds of his estate.^[14][18][19]

Having no living legitimate children,^[20] Caesar had adopted his great-nephew Octavius as his son and main heir.^[21] Upon his adoption, Octavius assumed his great-uncle's name, Gaius Julius Caesar. Although Romans who had been adopted into a new family usually retained their old nomen in cognomen form (e.g. Octavianus for one who had been an Octavius, Aemilianus for one who had been an Aemilius, etc.) there is no evidence that he ever bore the name Octavianus, as it would have made his modest origins too obvious.^[22][23][24]

However, despite the fact that he never officially bore the name Octavianus, to save confusing the dead dictator with his heir, historians often refer to the new Caesar—between his adoption and his assumption, in 27 BC, of the name Augustus—as Octavian.^[25] Mark Antony later charged that Octavian had earned his adoption by Caesar through sexual favours, though Suetonius, in his work Lives of the Twelve Caesars, describes Antony's accusation as political slander.^[26]

To make a successful entry into the upper echelons of the Roman political hierarchy, Octavian could not rely on his limited funds.^[27] After a warm welcome by Caesar's soldiers at Brundisium,^[28] Octavian demanded a portion of the funds that were allotted by Caesar for the intended war against Parthia in the Middle East.^[27] This amounted to 700 million sesterces stored at Brundisium, the staging ground in Italy for military operations in the east.^[29]

A later senatorial investigation into the disappearance of the public funds made no action against Octavian, since he subsequently used that money to raise troops against the Senate's arch enemy, Mark Antony.^[28] Octavian made another bold move in 44 BC when without official permission he appropriated the annual tribute that had been sent from Rome's Near Eastern province to Italy.^[23][30]

Octavian began to bolster his personal forces with Caesar's veteran legionaries and with troops designated for the Parthian war, gathering support by emphasizing his status as heir to Caesar.^[18][31] On his march to Rome through Italy, Octavian's presence and newly acquired funds attracted many, winning over Caesar's former veterans stationed in Campania.^[23] By June he had gathered an army of 3,000 loyal veterans, paying each a salary of 500 denarii.^[32][33][34]

A statue of Augustus as a younger Octavian, dated ca. 30 BC.

Arriving in Rome on 6 May 44 BC,^[23] Octavian found the consul Mark Antony, Caesar's former colleague, in an uneasy truce with the dictator's assassins; they had been granted a general amnesty on 17 March, yet Antony succeeded in driving most of them out of Rome.^[23] This was due to his "inflammatory" eulogy given at Caesar's funeral, mounting public opinion against the assassins.^[23]

Although Mark Antony was amassing political support, Octavian still had opportunity to rival him as the leading member of the faction supporting Caesar. Mark Antony had lost the support of many Romans and supporters of Caesar when he at first opposed the motion to elevate Caesar to divine status.^[35] Octavian failed to persuade Antony to relinquish Caesar's money to him. However, during the summer he managed to win support from Caesarian sympathizers, who saw the younger heir as the lesser evil and hoped to manipulate him, or to bear with him during their efforts to get rid of Antonius.^[36]

In September, the Optimate orator Marcus Tullius Cicero began to attack Antony in a series of speeches portraying Antony as the greatest threat to the order of the Senate.^[37][38] With opinion in Rome turning against him and his year of consular power nearing its end, Antony attempted to pass laws which would lend him control over Cisalpine Gaul, which had been assigned as part of his province, from Decimus Junius Brutus Albinus, one of Caesar's assassins.^[39][40]

Octavian meanwhile built up a private army in Italy by recruiting Caesarian veterans, and on 28 November won over two of Antony's legions with the enticing offer of monetary gain.^[41][42][43] In the face of Octavian's large and capable force, Antony saw the danger of staying in Rome, and to the relief of the Senate he fled to Cisalpine Gaul, which was to be handed to him on 1 January.^[43]

First conflict with Antony[link]

Bust of Augustus in Musei Capitolini, Rome

After Decimus Brutus refused to give up Cisalpine Gaul, Antony besieged him at Mutina.^[44] The resolutions passed by the Senate to stop the violence were rejected by Antony, as the Senate had no army of its own to challenge him; this provided an opportunity for Octavian, who was already known to have armed forces.^[42] Cicero also defended Octavian against Antony's taunts about Octavian's lack of noble lineage; he stated "we have no more brilliant example of traditional piety among our youth."^[45]

This was in part a rebuttal to Antony's opinion of Octavian, as Cicero quoted Antony saying to Octavian, "You, boy, owe everything to your name."^[46][47] In this unlikely alliance orchestrated by the arch anti-Caesarian senator Cicero, the Senate inducted Octavian as senator on 1 January 43 BC, yet he was also given the power to vote alongside the former consuls.^[42][43] In addition, Octavian was granted imperium (commanding power), which made his command of troops legal, sending him to relieve the siege along with Hirtius and Pansa (the consuls for 43 BC).^[42][48] In April 43 BC, Antony's forces were defeated at the battles of Forum Gallorum and Mutina, forcing Antony to retreat to Transalpine Gaul. However, both consuls were killed, leaving Octavian in sole command of their armies.^[49][50]

After heaping many more rewards on Decimus Brutus than Octavian for defeating Antony, the Senate attempted to give command of the consular legions to Decimus Brutus, yet Octavian decided not to cooperate.^[51] Instead, Octavian stayed in the Po Valley and refused to aid any further offensive against Antony.^[52] In July, an embassy of centurions sent by Octavian entered Rome and demanded that he receive the consulship left vacant by Hirtius and Pansa.^[53]

Octavian also demanded that the decree declaring Antony a public enemy should be rescinded.^[52] When this was refused, he marched on the city with eight legions.^[52] He encountered no military opposition in Rome, and on 19 August 43 BC was elected consul with his relative Quintus Pedius as co-consul.^[54][55] Meanwhile, Antony formed an alliance with Marcus Aemilius Lepidus, another leading Caesarian.^[56]

Second Triumvirate[link]

Proscriptions[link]

Roman aureus bearing the portraits of Mark Antony (left) and Octavian (right), issued in 41 BC to celebrate the establishment of the Second Triumvirate by Octavian, Antony and Marcus Lepidus in 43 BC. Both sides bear the inscription "III VIR R P C", meaning "One of Three Men for the Regulation of the Republic".^[57]

In a meeting near Bologna in October 43 BC, Octavian, Antony, and Lepidus formed a junta called the Second Triumvirate.^[58] This explicit arrogation of special powers lasting five years was then supported by law passed by the plebs, unlike the unofficial First Triumvirate formed by Gnaeus Pompey Magnus, Julius Caesar and Marcus Licinius Crassus.^[58][59] The triumvirs then set in motion proscriptions in which allegedly 300 senators and 2,000 equites were branded as outlaws and deprived of their property and, for those who failed to escape, their lives.^[60]

The estimation that 300 senators were proscribed was presented by Appian, although his earlier contemporary Livy asserted that only 130 senators had been proscribed.^[61] This decree issued by the triumvirate was motivated in part by a need to raise money to pay their troops' salaries for the upcoming conflict against Caesar's assassins, Marcus Junius Brutus and Gaius Cassius Longinus.^[62] Rewards for their arrest gave incentive for Romans to capture those proscribed, while the assets and properties of those arrested were seized by the triumvirs.^[60]

Contemporary Roman historians provide conflicting reports as to which triumvir was more responsible for the proscriptions and killing. However, the sources agree that enacting the proscriptions was a means by all three factions to eliminate political enemies.^[63] Marcus Velleius Paterculus asserted that Octavian tried to avoid proscribing officials whereas Lepidus and Antony were to blame for initiating them.^[64] Cassius Dio defended Augustus as trying to spare as many as possible, whereas Antony and Lepidus, being older and involved in politics longer, had many more enemies to deal with.^[64]

This claim was rejected by Appian, who maintained that Octavian shared an equal interest with Lepidus and Antony in eradicating his enemies.^[65] Suetonius presents the case that Octavian, although reluctant at first to proscribe officials, nonetheless pursued his enemies with more rigor than the other triumvirs.^[63] Plutarch describes the proscriptions as a ruthless and cutthroat swapping of friends and family between Antony, Lepidus, and Octavian. For example, Octavian allowed the proscription of his ally Cicero, Antony the proscription of his maternal uncle Lucius Julius Caesar IV, and Lepidus his brother Paulus.^[64]

A denarius minted c. 18 BC. Obverse: CAESAR AVGVSTVS; reverse: DIVVSIVLIV[S] (DIVINE JULIUS).

Battle of Philippi and division of territory[link]

On 1 January 42 BC, the Senate posthumously recognized Julius Caesar as a divinity of the Roman state, Divus Iulius. Octavian was able to further his cause by emphasizing the fact that he was Divi filius, "Son of God".^[66] Antony and Octavian then sent 28 legions by sea to face the armies of Brutus and Cassius, who had built their base of power in Greece.^[67] After two battles at Philippi in Macedonia in October 42, the Caesarian army was victorious and Brutus and Cassius committed suicide. Mark Antony would later use the examples of these battles as a means to belittle Octavian, as both battles were decisively won with the use of Antony's forces.^[68] In addition to claiming responsibility for both victories, Antony also branded Octavian as a coward for handing over his direct military control to Marcus Vipsanius Agrippa instead.^[68]

After Philippi, a new territorial arrangement was made among the members of the Second Triumvirate. While Antony placed Gaul, the provinces of Hispania, and Italia in the hands of Octavian, Antony traveled east to Egypt where he allied himself with Queen Cleopatra VII, the former lover of Julius Caesar and mother of Caesar's infant son, Caesarion. Lepidus was left with the province of Africa, stymied by Antony who conceded Hispania to Octavian instead.^[69]

Octavian was left to decide where in Italy to settle the tens of thousands of veterans of the Macedonian campaign whom the triumvirs had promised to discharge. The tens of thousands who had fought on the republican side with Brutus and Cassius, who could easily ally with a political opponent of Octavian if not appeased, also required land.^[69] There was no more government-controlled land to allot as settlements for their soldiers, so Octavian had to choose one of two options: alienating many Roman citizens by confiscating their land, or alienating many Roman soldiers who could mount a considerable opposition against him in the Roman heartland; Octavian chose the former.^[70] There were as many as eighteen Roman towns affected by the new settlements, with entire populations driven out or at least given partial evictions.^[71]

Rebellion and marriage alliances[link]

Widespread dissatisfaction with Octavian over his soldiers' settlements encouraged many to rally at the side of Lucius Antonius, who was brother of Mark Antony and supported by a majority in the Senate.^[71] Meanwhile, Octavian asked for a divorce from Clodia Pulchra, the daughter of Fulvia and her first husband Publius Clodius Pulcher. Claiming that his marriage with Clodia had never been consummated, he returned her to her mother, Mark Antony's wife. Fulvia decided to take action. Together with Lucius Antonius she raised an army in Italy to fight for Antony's rights against Octavian. However, Lucius and Fulvia took a political and martial gamble in opposing Octavian, since the Roman army still depended on the triumvirs for their salaries.^[71] Lucius and his allies ended up in a defensive siege at Perusia (modern Perugia), where Octavian forced them into surrender in early 40 BC.^[71]

Lucius and his army were spared due to his kinship with Antony, the strongman of the East, while Fulvia was exiled to Sicyon.^[72] However, Octavian showed no mercy for the mass of allies loyal to Lucius; on 15 March, the anniversary of Julius Caesar's assassination, he had 300 Roman senators and equestrians executed for allying with Lucius.^[73] Perusia was also pillaged and burned as a warning for others.^[72] This bloody event sullied Octavian's reputation and was criticized by many, such as the Augustan poet Sextus Propertius.^[73]

Sextus Pompeius, son of the First Triumvir Pompey and still a renegade general following Julius Caesar's victory over his father, was established in Sicily and Sardinia as part of an agreement reached with the Second Triumvirate in 39 BC.^[74] Both Antony and Octavian were vying for an alliance with Pompeius, who was ironically a member of the republican party, not the Caesarian faction.^[73] Octavian succeeded in a temporary alliance when in 40 BC he married Scribonia, a daughter of Lucius Scribonius Libo who was a follower of Pompeius as well as his father-in-law.^[73] Scribonia conceived Octavian's only natural child, Julia, who was born the same day that he divorced Scribonia to marry Livia Drusilla, little more than a year after his marriage.^[73]

While in Egypt, Antony had been engaged in an affair with Cleopatra and had fathered three children with her.^[75] Aware of his deteriorating relationship with Octavian, Antony left Cleopatra; he sailed to Italy in 40 BC with a large force to oppose Octavian, laying siege to Brundisium. However, this new conflict proved untenable for both Octavian and Antony. Their centurions, who had become important figures politically, refused to fight due to their Caesarian cause, while the legions under their command followed suit.^[76][77] Meanwhile in Sicyon, Antony's wife Fulvia died of a sudden illness while Antony was en route to meet her. Fulvia's death and the mutiny of their centurions allowed the two remaining triumvirs to effect a reconciliation.^[76][77]

In the autumn of 40, Octavian and Antony approved the Treaty of Brundisium, by which Lepidus would remain in Africa, Antony in the East, Octavian in the West. The Italian peninsula was left open to all for the recruitment of soldiers, but in reality, this provision was useless for Antony in the East.^[76] To further cement relations of alliance with Mark Antony, Octavian gave his sister, Octavia Minor, in marriage to Antony in late 40 BC.^[76] During their marriage, Octavia gave birth to two daughters (known as Antonia Major and Antonia Minor).

War with Pompeius[link]

A denarius of Sextus Pompeius, minted for his victory over Octavian's fleet. On the obverse the Pharus of Messina, who defeated Octavian. On the reverse, the monster Scylla.

Sextus Pompeius threatened Octavian in Italy by denying to the peninsula shipments of grain through the Mediterranean; Pompeius' own son was put in charge as naval commander in the effort to cause widespread famine in Italy.^[77] Pompeius' control over the sea prompted him to take on the name Neptuni filius, "son of Neptune".^[78] A temporary peace agreement was reached in 39 BC with the treaty of Misenum; the blockade on Italy was lifted once Octavian granted Pompeius Sardinia, Corsica, Sicily, and the Peloponnese, and ensured him a future position as consul for 35 BC.^[77][78]

The territorial agreement amongst the triumvirs and Sextus Pompeius began to crumble once Octavian divorced Scribonia and married Livia on 17 January 38 BC.^[79] One of Pompeius' naval commanders betrayed him and handed over Corsica and Sardinia to Octavian. However Octavian needed Antony's additional support to attack Pompeius, so an agreement was reached with the Second Triumvirate's extension for another five-year period beginning in 37 BC.^[80][81]

In supporting Octavian, Antony expected to gain support for his own campaign against Parthia, desiring to avenge Rome's defeat at Carrhae in 53 BC.^[81] In an agreement reached at Tarentum, Antony provided 120 ships for Octavian to use against Pompeius, while Octavian was to send 20,000 legionaries to Antony for use against Parthia.^[82] However, Octavian sent only a tenth the number of those promised, which was viewed by Antony as an intentional provocation.^[82]

Octavian and Lepidus launched a joint operation against Sextus in Sicily in 36 BC.^[83] Despite setbacks for Octavian, the naval fleet of Sextus Pompeius was almost entirely destroyed on 3 September by general Agrippa at the naval battle of Naulochus.^[84] Sextus fled with his remaining forces to the east, where he was captured and executed in Miletus by one of Antony's generals the following year.^[84] Both Lepidus and Octavian gathered the surrendered troops of Pompeius, yet Lepidus felt empowered enough to claim Sicily for himself, ordering Octavian to leave.^[84] However, Lepidus' troops deserted him and defected to Octavian since they were weary of fighting and found Octavian's promises of money to be enticing.^[84]

Lepidus surrendered to Octavian and was permitted to retain the office of pontifex maximus (head of the college of priests), but was ejected from the Triumvirate, his public career at an end, and was effectively exiled to a villa at Cape Circei in Italy.^[84][85] The Roman dominions were now divided between Octavian in the West and Antony in the East. To maintain peace and stability in his portion of the Empire, Octavian ensured Rome's citizens of their rights to property. This time he settled his discharged soldiers outside of Italy while returning 30,000 slaves to former Roman owners that had previously fled to Pompeius to join his army and navy.^[86] To ensure his own safety and that of Livia and Octavia once he returned to Rome, Octavian had the Senate grant him, his wife, and his sister tribunal immunity, or sacrosanctitas.^[87]

War with Antony[link]

Anthony and Cleopatra, by Lawrence Alma-Tadema.

Meanwhile, Antony's campaign against Parthia turned disastrous, tarnishing his image as a leader, and the mere 2,000 legionaries sent by Octavian to Antony were hardly enough to replenish his forces.^[88] On the other hand, Cleopatra could restore his army to full strength, and since he was already engaged in a romantic affair with her, he decided to send Octavia back to Rome.^[89] Octavian used this to spread propaganda implying that Antony was becoming less than Roman because he rejected a legitimate Roman spouse for an "Oriental paramour".^[90] In 36 BC, Octavian used a political ploy to make himself look less autocratic and Antony more the villain by proclaiming that the civil wars were coming to an end, and that he would step down as triumvir if only Antony would do the same; Antony refused.^[91]

After Roman troops captured the Kingdom of Armenia in 34 BC, Antony made his son Alexander Helios the ruler of Armenia; he also awarded the title "Queen of Kings" to Cleopatra, acts which Octavian used to convince the Roman Senate that Antony had ambitions to diminish the preeminence of Rome.^[90] When Octavian became consul once again on 1 January 33 BC, he opened the following session in the Senate with a vehement attack on Antony's grants of titles and territories to his relatives and to his queen.^[92]

Defecting consuls and senators rushed over to the side of Antony in disbelief of the propaganda (which turned out to be true), yet so did able ministers desert Antony for Octavian in the autumn of 32 BC.^[93] These defectors, Munatius Plancus and Marcus Titius, gave Octavian the information he needed to confirm with the Senate all the accusations he made against Antony.^[94]

By storming the sanctuary of the Vestal Virgins, Octavian forced their chief priestess to hand over Antony's secret will, which would have given away Roman-conquered territories as kingdoms for his sons to rule, alongside plans to build a tomb in Alexandria for him and his queen to reside upon their deaths.^[95][96] In late 32 BC, the Senate officially revoked Antony's powers as consul and declared war on Cleopatra's regime in Egypt.^[97][98]

The Battle of Actium, by Lorenzo Castro, painted 1672, National Maritime Museum, London.

In early 31 BC, while Antony and Cleopatra were temporarily stationed in Greece, Octavian gained a preliminary victory when the navy under the command of Agrippa successfully ferried troops across the Adriatic Sea.^[99] While Agrippa cut off Antony and Cleopatra's main force from their supply routes at sea, Octavian landed on the mainland opposite the island of Corcyra (modern Corfu) and marched south.^[99] Trapped on land and sea, deserters of Antony's army fled to Octavian's side daily while Octavian's forces were comfortable enough to make preparations.^[99]

In a desperate attempt to break free of the naval blockade, Antony's fleet sailed through the bay of Actium on the western coast of Greece. It was there that Antony's fleet faced the much larger fleet of smaller, more maneuverable ships under commanders Agrippa and Gaius Sosius in the battle of Actium on 2 September 31 BC.^[100] Antony and his remaining forces were only spared due to a last-ditch effort by Cleopatra's fleet that had been waiting nearby.^[101]

Octavian pursued them, and after another defeat in Alexandria on 1 August 30 BC, Antony and Cleopatra committed suicide; Antony fell on his own sword and into Cleopatra's arms, while she let a venomous snake bite her.^[102] Having exploited his position as Caesar's heir to further his own political career, Octavian was only too well aware of the dangers in allowing another to do so and, reportedly commenting that "two Caesars are one too many", he ordered Caesarion—Julius Caesar's son by Cleopatra—to be killed, whilst sparing Cleopatra's children by Antony, with the exception of Antony's older son.^[103][104]

Octavian had previously shown little mercy to military combatants and acted in ways that had proven unpopular with the Roman people, yet he was given credit for pardoning many of his opponents after the Battle of Actium.^[105]

Octavian becomes Augustus[link]

Aureus of Octavian, circa 30 BC, British Museum

After Actium and the defeat of Antony and Cleopatra, Octavian was in a position to rule the entire Republic under an unofficial principate,^[106] but would have to achieve this through incremental power gains, courting the Senate and the people, while upholding the republican traditions of Rome, to appear that he was not aspiring to dictatorship or monarchy.^[107][108] Marching into Rome, Octavian and Marcus Agrippa were elected as dual consuls by the Senate.^[109]