About the Work
This text comprises four parts of a larger work by Augustus De Morgan, presented here as excerpts from his Budget of Paradoxes. The material is a collection of essays, reviews, and reflections on a wide range of subjects, unified by De Morgan’s characteristic wit, erudition, and skepticism toward unsound reasoning. The first part opens with a discussion of a religious declaration aimed at scientists in 1864–65, then moves through various topics including orthography, computation, poetry, and pseudoscience. The second part is a review of two major encyclopædias, examining their history, structure, and inherent flaws. The third part reviews a collection of seventeenth-century scientific correspondence, focusing on the lives and work of early mathematicians and natural philosophers. The fourth part addresses theosophy, language, pronunciation, and a curious anecdote about a rainbow paradox. Throughout, De Morgan engages with paradoxers—those who hold and publish views contrary to established knowledge—and defends rigorous, rational inquiry.
Context
The notes indicate that the first part of this text was written in the context of a specific historical event: the “Declaration of Belief” of 1864–65. A small group of people drew up a declaration for “students of the natural sciences” to sign, expressing “sincere regret” that scientific research was being “perverted” into casting doubt on the truth and authenticity of the Holy Scriptures. This declaration took for granted that the Word of God is all the Bible, not merely in the Bible. Sir John Herschel publicly refused to sign, rebuking the promoters in the Athenæum and calling the movement “mischievous” and an infringement of social forbearance guarding religious freedom. Other notable refusals came from Sir J. Bowring and Sir W. Rowan Hamilton. The declaration was promised for Christmas 1864 but was not published until near Midsummer 1865. Of 600 Fellows of the Royal Society, only 62 signed, and of 166 writers in the Philosophical Transactions, only 19 signed. The original signatures were deposited in the Bodleian Library. De Morgan himself refused to sign, giving three reasons: agreement with Herschel, no regret that others differ, and shame at signing “bad chemistry made to do duty for good reasoning.”
The second part is a review dated October 19, 1861, of two completed encyclopædias: the English Cyclopædia, conducted by Charles Knight, and the Encyclopædia Britannica, Eighth Edition. Both were completed simultaneously. The third part is a review dated October 18, 1862, of the Correspondence of Scientific Men of the Seventeenth Century in the Earl of Macclesfield’s collection, published at Oxford in 1841 but only completed by its Table of Contents and Index in 1862. The fourth part contains no explicit date but continues De Morgan’s ongoing commentary on various paradoxes and intellectual follies.
Structure of the Book
The text is divided into four distinct parts, each containing multiple sections or topics. The first part begins with the “Declaration of Belief” controversy, followed by De Morgan’s letter of refusal titled “Scripture and Science.” It then presents “The Fly-Leaf Paradox,” “Paradoxes of Orthography and Computation,” “Centrifugal Force,” “Cambridge Poets,” “Falsified Prediction,” “Byron and Wordsworth,” and “De Bécourt.” The second part is a single extended review of cyclopædias, covering the term itself, early encyclopædias, the French Encyclopédie, the Encyclopédie Méthodique, the dictionary-treatise problem, inequality of performance, and an anecdote about double stars. The third part is a review of the Macclesfield Letters, discussing the collection’s history, the figure of John Collins, the transmission of the papers, and notable correspondents including Oughtred, Dary, and Stiles. The fourth part addresses Jacob Behmen and theosophy, then moves to “A Plea for King Custom,” “Test of Language,” “Pronunciation,” “Apologies to Cluvier,” and “The Rainbow Paradox.”
Detailed Section-by-Section Summary
Part 1
The first part opens with the “Declaration of Belief” of 1864–65. A small combination of persons drew up a declaration for “students of the natural sciences” to sign, expressing “sincere regret” that scientific research is “perverted” into casting doubt on the truth and authenticity of the Holy Scriptures. The declaration took for granted that the Word of God is all the Bible, not merely in the Bible. Sir John Herschel refused to sign, rebuking the promoters in the Athenæum, calling the movement “mischievous” and an infringement of social forbearance guarding religious freedom. Other refusals came from Sir J. Bowring, Sir W. Rowan Hamilton, and De Morgan himself. The declaration was promised for Christmas 1864 but published near Midsummer 1865. Fifteen names were procured before printing; the final list had 716 names. Of 600 Fellows of the Royal Society, only 62 signed; of 166 writers in the Philosophical Transactions, only 19 signed. The original signatures were deposited in the Bodleian Library. De Morgan gives the declaration entire, with a substitute proposed in the Athenæum. He ends with his own answer to the application, refusing to abridge anyone’s liberty to think, infer, and publish, preferring to be asked to take shares in an “argyrosteretic company” (for stealing forks and spoons) than to sign.
De Morgan then presents “Scripture and Science” (Nov. 14, 1864), his letter refusing signature for three reasons: agreement with Herschel, no regret that others differ, and shame at signing “bad chemistry made to do duty for good reasoning.” He calls the declaration an attempt to “saturate sophism with truism.” He recounts a story of Sir John Herschel and the distinction between simpliciter and secundum quid. He warns against test-signing leading to counter-declarations and party discord. He quotes Walter Scott on Napoleon, inviting promoters to remember how much he may have resisted.
Next comes “The Fly-Leaf Paradox”: De Morgan puzzled over illiterate or child owners’ names in old books followed by the book’s date. He found a solution in a Barker’s Bible of 1599, where the inscription was on a verso of a leaf from a prayer book dated 1632, bound before the Bible. Uneducated persons finished with the date of the book staring at them. He advises always writing in books. He describes a copy of Hodder’s Arithmetick (1719, Boston, printed by J. Franklin, likely with young Benjamin as apprentice), a perfect copy of English editions, with a former owner’s note dated July 14, 1796, about sailing from Boston; Sir Francis Beaufort’s charts at the Admiralty decided for Massachusetts.
“Paradoxes of Orthography and Computation” notes French misspellings (Swif, Adisson, Otwai, Bolingbrocke; “Win Worewel” and “Raden Powen” for Whewell and Powell). The “moderate computation” paradox yields huge or tiny figures; De Morgan cites 70,000 executed in one reign, 150,000 burned for witchcraft, and a card showing 198,728,000 Christians paying clergy 8,852,000l., while 6,400,000 pay the Anglo-Irish Establishment 8,896,000l. He quotes Voltaire on the Bishop of Chiapa.
“Centrifugal Force” recounts De Morgan’s childhood lecture where the lecturer explained that centrifugal force balances centripetal force to keep planets from falling into the Sun; De Morgan thought leaving planets alone would save equal and opposite troubles. Paradoxers provide external agents for centrifugal force, e.g., sun’s rays.
“Cambridge Poets” discusses Lord Byron’s English Bards (1809) praising the Rev. Geo. Richards, Fellow of Oriel, for the Aboriginal Britons (Oxford prize poem, 1791), while ignoring Cambridge poets. Byron’s college placed his statue in the library, more conspicuously than Newton’s in the chapel. Byron also mocked cow-pox, tractors, galvanism, and gas; three of these bubbles have burst to mighty ends (mesmerism survives).
“Falsified Prediction” quotes the Rev. Daniel Rivers on Johnson’s biographers (Boswell and Mrs. Piozzi), predicting injury to Johnson’s fame; De Morgan defends Boswell and Thrale, calling conduct toward Mrs. Piozzi “blackguard.”
“Byron and Wordsworth” recounts Payne Knight’s Greek translation of Gray’s Bard, which Hallam called nonsense; the first line was from Pindar. Byron represented Hallam as not discovering this. De Morgan tells of Crabb Robinson’s story: Wordsworth came to Charles Lamb’s chambers fuming at reviewers who sneered at a young lord’s poetry; Lady Byron later said Byron had “reverence” for Wordsworth. De Morgan suspects Fielding’s Chrysippus (1743) satirized the Royal Society, not just Peter Walter. Dante and Quarles mention the quadrature of the circle; Aristophanes’ Birds has a geometer making a square circle; Pope’s Dunciad speaks of “Mad Mathesis” running round the circle and finding it square.
“De Bécourt” describes The Grave of Human Philosophies (1827), translated from R. de Bécourt by A. Dalmas, supporting Hindu views: sun 450 miles away, 40 miles in diameter, four million years for .
Part 3
The third part reviews the Correspondence of Scientific Men of the Seventeenth Century in the Earl of Macclesfield’s collection (2 vols., Oxford, 1841), only completed by its Table of Contents and Index in 1862. The delay rendered the work useless for consultation; inquirers into scientific history remained ignorant of its contents.
The 1841 Historical Society of Science published a smaller collection of eighty-three letters edited by Mr. Halliwell, running 1562–1682, which dovetails with the Macclesfield collection (1606–past 1700). De Morgan calls these the Museum and Macclesfield collections, recommending they be bound together, alongside Mr. Edleston’s 1850 Trinity College collection.
The letters trace to John Collins, “attorney-general of the mathematics,” who corresponded with everyone and circulated copies. He was England’s Mersenne; E. Bernard addressed him in 1671 as “the very Mersennus and intelligence of this age.” Collins was accountant to the Excise Office, promoted from teaching writing at the Restoration, dying in 1682. De Morgan compares him to Prof. Schumacher of Altona Observatory, whose Danish observatory served as an astronomical communications junction; the English Government, moved by the Astronomical Society, instructed diplomats to stress Altona’s importance during the Denmark-Duchies collision. Collins had only correspondence, unlike Schumacher’s Astronomische Nachrichten. Collins’s collection furnished materials for the Royal Society Committee’s 1712 defence of Newton against Leibnitz’s partisans—the Commercium Epistolicum.
The collection passed to William Jones (father of the Indian Judge), who died 1749. Jones, originally a teacher, received a sinecure through George, second Earl of Macclesfield, who moved the change-of-style bill and died President of the Royal Society. Lord Macclesfield, Lord Chesterfield (seconder), and Daval (who drew the bill) were pupils of De Moivre. Jones collected the largest mathematical library of his day, possessing Collins’s papers including Oughtred’s. Some went to the Royal Society; the bulk went to Lord Macclesfield, reaching Shirburn Castle.
Before 1836, Professor Rigaud (died 1839) examined the letters, obtained consent, and induced Oxford Press to publish. Non-epistolary manuscripts remain at Shirburn. Rigaud died; his son S. Jordan Rigaud (later Bishop of Antigua) completed publication, but it was little noticed. The completion includes a table of contents by Professor De Morgan and an index by Rev. John Rigaud.
De Morgan lists great names in the collection: Briggs, Oughtred, Charles Cavendish, Gascoigne, Seth Ward, Wallis, Huygens, Collins, William Petty, Hooke, Boyle, Pell, Oldenburg, Brancker, Slusius, Bertit, Bernard, Borelli, Mouton, Pardies, Fermat, Towneley, Auzout, D. Gregory, Halley, Machin, Montmort, Cotes, Jones, Saunderson, Reyneau, Brook Taylor, Maupertuis, Bouguer, La Condamine, Folkes, Macclesfield, Baker, Barrow, Flamsteed, Lord Brounker, J. Gregory, Newton, Keill. The Museum collection adds Thomas Digges, Dee, Tycho Brahe, Harriot, Lydyat, Briggs, Warner, Tarporley, Pell, Lilly, Oldenburg, Collins, Morland.
Oughtred first appears, dying of joy at the Restoration at eighty-six. He was an Eton mathematician, inventor of the sliding rule—unknown to Dr. Hutton’s Dictionary. A 1629 letter shows Oughtred’s systematic contractions for sine, cosine, etc., prefixed to angle symbols—a step Euler is credited with advancing trigonometry. Oughtred’s dead book on trigonometry established abbreviations s, sco, t, tco, se, seco; but investigators of Euler and Thomas Simpson’s claims missed Oughtred’s “Trigonometrie.”
Only slightly more than a quarter of letters had been previously printed in whole or part. Flamsteed was best ransacked: 34 of 36 letters given. Of 59 Newton letters, only 17 culled. Spelling and algebraical notation were modernized; conjectural interpolations were necessary. Professor Rigaud copied everything himself; a fac-simile would have exceeded his time.
Two persons emerge from the correspondence. Michael Dary, obscure mathematician corresponding with Newton: gauger at Bristol (through Collins), candidate for Christ’s Hospital mathematical school (with Newton’s certificate), gunner in the Tower, finally “tobacco-cutter, a knowing man in algebra” per Wallis. In 1674 Dary writes Newton: “Although I sent you three papers yesterday, I cannot refrain from sending you this. I have had fresh thoughts this morning.” Newton writes Collins, “Mr. Dary is very solicitous about mathematics,” yet subscribes “your loving friend.” Dary’s problem: finding interest rate of an annuity given value and term. Dary’s theorem: the limit of φnx as n increases without limit is a solution of φx = x. Dr. John Newton, sedulous publisher of logarithms, began before Isaac Newton, sometimes confused with him; the scientific world held his work’s value came from Dary’s private communications.
John Stiles, the London and Cambridge carrier, carried mathematical researches further than Newton; in gravitation theory he was prior to Newton but did not extend calculations to the moon’s distance, charging according to weight. [The text cuts off here.]
Part 4
The fourth part opens with De Morgan discussing Jacob Behmen (Böhme). He notes he has only encountered a small tract from Behmen’s school, though he is told of an 1854 “Introduction to Theosophy” drawn mostly from William Law’s writings, and a 688-page volume of the same year for private circulation containing notes for a Law biography. The first work’s editor proposes founding a Theosophic College requiring £100,000 from the public.
De Morgan quotes a Penny Cyclopædia account mentioning inaccurate claims that Behmen derived his doctrine from Wood’s Athenæ Oxonienses and Hist. et Antiq. Acad. Oxon., which the author refutes by noting Wood was born after Behmen’s death. The corrected claim—that Behmen derived from Robert Fludd—is also absurd, since Behmen began publishing in 1610 and Fludd in 1616. Fludd was a Rosicrucian mystic of different type.
A correspondent (“C. W. H.”) defends Behmen, claiming his metaphysical system is “the most perfect and only true one,” that Dionysius Andreas Freher first expounded it philosophically (his MSS in the British Museum), and that William Law popularized it. The correspondent asserts Behmen anticipated electricity, that Newton derived gravitation from Behmen, and that Behmen demonstrated all physical science.
De Morgan then quotes Behmen’s own words on sulphur and mercurius from the 1664 edition, page 13, showing Behmen’s etymological divisions (SUL, PHUR, MER, CU, RI, US). He calls this “nonsense” and says the claim about Newton borrowing gravitation “passes only with those who know neither what Newton did, nor what was done before him.”
The passage then shifts to “A Plea for King Custom,” defending vernacular usage over pedantic pronunciation. De Morgan discusses Etymology, Logic, Fiddlefaddle, and Typography as advisers to Custom. He gives the anecdote of a man insisting “Chitty” be pronounced “Kitty.”
In “Test of Language,” De Morgan argues the Authorized Version represents common colloquial English, using Robert Recorde’s Castle of Knowledge dialogue as evidence, modernizing spelling and verb forms to show resemblance.
“Pronunciation” discusses foreign words entering vernacular, citing Euler, Newton, Rokeby, and Chief Justice Cook.
“Apologies to Cluvier” defends Dethlevus Cluverius against Montucla’s misrepresentation, noting Cluvier’s method involved multiplying 1, 2, 3, 4, etc. to infinity, and that he refuted Mallemont de Messange in the Phil. Trans., 1686.
“The Rainbow Paradox” describes a potential paradox about reflected rainbows, where an artist’s picture showing a rainbow’s reflection was doubted, though the reflection was real.
Main Arguments
De Morgan’s central argument throughout the text is a defense of rational inquiry and a critique of paradoxers—those who hold and publish views contrary to established knowledge. In the first part, he argues against the Declaration of Belief, refusing to sign because he agrees with Herschel, has no regret that others differ, and is ashamed at signing “bad chemistry made to do duty for good reasoning.” He warns that test-signing leads to counter-declarations and party discord.
In the second part, De Morgan argues that encyclopædias have inherent structural flaws. The dictionary-treatise problem—the attempt to combine comprehensive treatises with brief dictionary entries—is practically unavoidable and leads to incompleteness, particularly in biographical portions. He argues that inequality of performance among contributors is inevitable and that attempts to secure equality of strength result in equality of weakness. He famously states, “A Cyclopædia, like a plant, should grow towards the light.”
In the third part, De Morgan argues for the importance of the Macclesfield and Museum collections of scientific correspondence, noting that the delay in completing the Macclesfield work rendered it useless for consultation. He highlights the role of John Collins as a central figure in seventeenth-century scientific communication and emphasizes the value of the letters for understanding the development of mathematics and natural philosophy.
In the fourth part, De Morgan refutes claims that Behmen derived his doctrine from Wood or Fludd (showing the chronology makes this impossible) and dismisses the assertion that Newton derived gravitation from Behmen, saying this “passes only with those who know neither what Newton did, nor what was done before him.” He defends vernacular usage over pedantic pronunciation in “A Plea for King Custom.”
Key Concepts
Paradoxers: De Morgan’s term for those who hold and publish views contrary to established knowledge. Throughout the text, he examines various paradoxes—from the Declaration of Belief to Behmen’s theosophy—and subjects them to rational scrutiny.
The Declaration of Belief: A document drawn up in 1864–65 for students of the natural sciences to sign, expressing regret that scientific research casts doubt on the Holy Scriptures. De Morgan analyzes its reception and the controversy it generated.
The Dictionary-Treatise Problem: The structural flaw in encyclopædias that attempt to combine comprehensive treatises with brief dictionary entries, leading to incompleteness and inequality of performance.
The Commercium Epistolicum: The 1712 Royal Society Committee’s defence of Newton against Leibnitz’s partisans, which was furnished by materials from John Collins’s collection of correspondence.
Theosophy: De Morgan discusses Jacob Behmen’s mystical system, including claims that it anticipated electricity and that Newton derived gravitation from Behmen—claims De Morgan dismisses as nonsense.
King Custom: De Morgan’s personification of vernacular usage, which he defends against pedantic pronunciation and etymological purism.
The Rainbow Paradox: A potential paradox about reflected rainbows, where an artist’s picture showing a rainbow’s reflection was doubted, though the reflection was real.
Themes
The Conflict Between Science and Religion: The first part directly addresses this theme through the Declaration of Belief controversy. De Morgan refuses to sign, arguing that the declaration is an attempt to “saturate sophism with truism” and that it infringes on liberty to think, infer, and publish. He prefers to be asked to take shares in an “argyrosteretic company” (for stealing forks and spoons) than to sign.
The Nature of Scientific Progress: De Morgan repeatedly emphasizes the importance of understanding the history of science. In the third part, he highlights the value of the Macclesfield and Museum collections for understanding seventeenth-century mathematics and natural philosophy. He notes that Oughtred’s systematic contractions for trigonometric functions were a step Euler is credited with advancing, showing that credit is often misattributed.
The Persistence of Pseudoscience: Throughout the text, De Morgan examines various forms of unsound reasoning—from the Declaration of Belief to Behmen’s theosophy to the centrifugal force paradox. He consistently subjects these to rational scrutiny, exposing their logical flaws.
The Role of Language and Usage: In the fourth part, De Morgan defends vernacular usage over pedantic pronunciation, arguing that the Authorized Version represents common colloquial English. He uses Robert Recorde’s Castle of Knowledge dialogue as evidence, modernizing spelling and verb forms to show resemblance.
The Value of Correspondence and Networks: In the third part, De Morgan emphasizes the importance of John Collins as a central figure in seventeenth-century scientific communication, comparing him to Prof. Schumacher of Altona Observatory. He shows how Collins’s collection furnished materials for the Royal Society Committee’s 1712 defence of Newton.
The Inevitability of Inequality: In the second part, De Morgan argues that inequality of performance in encyclopædias is unavoidable and that attempts to secure equality of strength result in equality of weakness. This theme extends to his broader view of human endeavor.
The Author’s Conclusions
De Morgan concludes that the Declaration of Belief is a misguided attempt to impose religious orthodoxy on scientific inquiry. He refuses to sign, preferring to maintain his liberty to think, infer, and publish. He warns that such declarations lead to counter-declarations and party discord.
On encyclopædias, De Morgan concludes that the dictionary-treatise problem is practically unavoidable and that inequality of performance is inevitable. He argues that critics who think a book’s merit is that of its weakest part are mistaken; the union of many writers exaggerates inequalities. He advises that “A Cyclopædia, like a plant, should grow towards the light.”
On the Macclesfield Letters, De Morgan concludes that the delay in completing the work rendered it useless for consultation, and he emphasizes the importance of the collections for understanding scientific history. He highlights the roles of John Collins, Michael Dary, and John Stiles, showing that obscure figures often made significant contributions.
On Behmen, De Morgan concludes that the claims about his system being “the most perfect and only true one” and that Newton derived gravitation from Behmen are nonsense, passing “only with those who know neither what Newton did, nor what was done before him.” He defends vernacular usage over pedantic pronunciation, arguing for the authority of common usage.
Significance
This text is significant as a window into the intellectual concerns of a prominent Victorian mathematician and logician. De Morgan’s Budget of Paradoxes is a landmark work in the study of pseudoscience and unsound reasoning, and these excerpts demonstrate his characteristic approach: combining rigorous logical analysis with wit, historical erudition, and a defense of rational inquiry. The text is also valuable for its insights into the history of science—particularly the seventeenth-century correspondence collected by John Collins and the development of trigonometry by Oughtred. De Morgan’s discussion of encyclopædias provides a critical perspective on the genre that remains relevant today. His defense of vernacular usage and his critique of pedantic pronunciation reflect broader Victorian debates about language and education. Finally, his treatment of the Declaration of Belief controversy illuminates the tensions between science and religion in Victorian England, a theme that continues to resonate. The text stands as a testament to De Morgan’s belief in the importance of clear thinking, historical accuracy, and intellectual freedom.
