Appendix

I think it right to give the proof that the ratio of the circumference to the diameter is incommensurable. This method of proof was given by Lambert,672 in the Berlin Memoirs for 1761, and has been also given in the notes to Legendre’s673 Geometry, and to the English translation of the same. Though not elementary algebra, it is within the reach of a student of ordinary books.674

Let a continued fraction, such as

a
——
b + c
——
d + e
-
f + etc.,

be abbreviated into a/b+ c/d+ e/f+ etc.: each fraction being understood as falling down to the side of the preceding sign +. In every such fraction we may suppose b, d, f, etc. positive; a, c, e, &c. being as required: and all are supposed integers. If this succession be continued ad infinitum, and if a/b, c/d, e/f, etc. all lie between -1 and +1, exclusive, the limit of the fraction must be incommensurable with unity; that is, cannot be A/B, where A and B are integers.

First, whatever this limit may be, it lies between -1 and +1. This is obviously the case with any fraction p/(q + ω), where ω is between ±1: for, p/q, being < 1, and p and q integer, cannot be brought up to 1, by the value of ω. Hence, if we take any of the fractions a/b, a/b+ c/d, a/b+ c/d+ e/f, etc.

say a/b+ c/d+ e/f+ g/h we have, g/h being between ±1, so is e/f+ g/h, so therefore is c/d+ e/f+ g/h; and so therefore is a/b+ c/d+ e/f+ g/h.

Now, if possible, let a/b+ c/d+ etc. be A/B at the limit; A and B being integers. Let

P = A c/d+ e/f+ etc., Q = P e/f+ g/h+ etc., R = Q g/h + i/k + etc.

P, Q, R, etc. being integer or fractional, as may be. It is easily shown that all must be integer: for

A/B = a/b+ P/A, or, P = aB - bA

P/A = c/d+ Q/P, or, Q = cA - dP

Q/P = e/f+ R/Q, or, R = eP - fQ etc., etc. Now, since a, B, b, A, are integers, so also is P; and thence Q; and thence R, etc. But since A/B, P/A, Q/P, R/Q, etc. are all between -1 and +1, it follows that the unlimited succession of integers P, Q, R, are each less in numerical value than the preceding. Now there can be no such unlimited succession of descending integers: consequently, it is impossible that a/b+ c/d+, etc. can have a commensurable limit.

It easily follows that the continued fraction is incommensurable if a/b, c/d, etc., being at first greater than unity, become and continue less than unity after some one point. Say that i/k, l/m,... are all less than unity. Then the fraction i/k+ l/m+... is incommensurable, as proved: let it be κ. Then g/(h + κ) is incommensurable, say λ; e/(f + λ) is the same, say μ; also c/(d + μ), say ν, and a/(b + ν), say ρ. But ρ is the fraction a/b+ c/d+... itself; which is therefore incommensurable.

Let φz represent

1 + a

z + a²

2z(z+1) + a³

2·3·z(z+1)(z+2) +...

Let z be positive: this series is convergent for all values of a, and approaches without limit to unity as z increases without limit. Change z into z + 1, and form φz - φ(z+1): the following equation will result—

φz - φ(z+1) = a

z(z+1) φ(z+2)

or a = a
z φ(z+1)
φz · z + a
z φ(z+1)
φz · a
z+1 φ(z+2)
φ(z+1)

or a = ψz z + ψ(z+1)

ψz being (a/z)(φ(z+1)/φz); of which observe that it diminishes without limit as z increases without limit. Accordingly, we have

ψz = a
z+ ψ(z+1) = a
z+ a
(z+1)+ ψ(z+2) = a
z+ a
(z+1)+ a
(z+2)+ ψ(z+3)

And, ψ(z + n) diminishing without limit, we have

a
z · φ(z+1)
φz = a
z+ a
(z+1)+ a
(z+2)+ a
(z+3)+...

Let z = ½; and let 4a = -x². Then

a
z φ(z+1) is - x²
2 1 - x²
2·3 + x⁴
2·3·4·5... or - x
2 sin x.

Again

φz is 1 - x²

2 + x⁴

2·3·4 or cos x:

and the continued fraction is

- ¼x²
½+ - ¼x²
(3/2)+ - ¼x²
(5/2)+... or - x
2 x
1+ - x²
3+ - x²
5+...

whence

tan x = x
1+ - x²
3+ - x²
5+ - x²
7+...

Or, as written in the usual way,

tan x = x
——
1 - x²
——
3 - x²
——
5 - x²
——
7 -...

This result may be proved in various ways: it may also be verified by calculation. To do this, remember that if

a₁
b₁+ a₂
b₂+ a₃
b₃+... a_n
b_n = P_n
Q_n ; then

P₁=a₁,	P₂=b₂ P₁,	P₃=b₃ P₂+a₃ P₁,	P₄=b₄ P₃+a₄ P₂, etc.
Q₁=b₁,	Q₂=b₂ Q₁+a₂,	Q₃=b₃ Q₂+a₃ Q₁,	Q₄=b₄ Q₃+a₄ Q₂, etc.

in the case before us we have

a₁=x,	a₂=-x²,	a₃=-x²,	a₄=-x²,	a₅=-x², etc.
b₁=1,	b₂=3,	b₃=5,	b₄=7,	b₅=9, etc.

P₁=x	Q₁=1
P₂=3x	Q₂=3-x²
P₃=15x-x³	Q₃=15-6x²
P₄=105x-10x³	Q₄=105-45x²+x⁴
P₅=945x-105x³+x⁵	Q₅=945-420x²+15x⁴
P₆=10395x-1260x³+21x⁵	Q₆=10395-4725x²+210x⁴-x⁶

We can use this algebraically, or arithmetically. If we divide P_n by Q_n, we shall find a series agreeing with the known series for tan x, as far as n terms. That series is

x + x³
3 + 2x⁵
15 + 17x⁷
315 + 62x⁹
2835 +...

Take P₅, and divide it by Q₅ in the common way, and the first five terms will be as here written. Now take x =.1, which means that the angle is to be one tenth of the actual unit, or, in degrees 5°.729578. We find that when x =.1, P₆ = 1038.24021, Q₆ = 10347.770999; whence P₆ divided by Q₆ gives.1003346711. Now 5°.729578 is 5°43′46½″; and from the old tables of Rheticus675—no modern tables carry the tangents so far—the tangent of this angle is.1003347670.

Now let x = ¼π; in which case tan x = 1. If ¼π be commensurable with the unit, let it be (m/n), m and n being integers: we know that ¼π < 1. We have then

1= (m/n)
1- (m²/n²)
3- (m²/n²)
5-... = m
n- m²
3n- m²
5n- m²
7n-...

Now it is clear that m²/3n, m²/5n, m²/7n, etc. must at last become and continue severally less than unity. The continued fraction is therefore incommensurable, and cannot be unity. Consequently π² cannot be commensurable: that is, π is an incommensurable quantity, and so also is π².

I thought I should end with a grave bit of appendix, deeply mathematical: but paradox follows me wherever I go. The foregoing is—in my own language—from Dr. (now Sir David) Brewster’s676 English edition of Legendre’s Geometry, (Edinburgh, 1824, 8vo.) translated by some one who is not named. I picked up a notion, which others had at Cambridge in 1825, that the translator was the late Mr. Galbraith,677 then known at Edinburgh as a writer and teacher.

But it turns out that it was by a very different person, and one destined to shine in quite another walk; it was a young man named Thomas Carlyle.678 He prefixed, from his own pen, a thoughtful and ingenious essay on Proportion, as good a substitute for the fifth Book of Euclid as could have been given in the space; and quite enough to show that he would have been a distinguished teacher and thinker on first principles. But he left the field immediately.

(The following is the passage referred to at Vol. II, page 54.)

Michael Stifelius679 edited, in 1554, a second edition of the Algebra (Die Coss.), of Christopher Rudolff.680 This is one of the earliest works in which + and - are used.

Stifelius was a queer man. He has introduced into this very work of Rudolff his own interpretation of the number of the Beast. He determined to fix the character of Pope Leo: so he picked the numeral letters from LEODECIMVS, and by taking in X from LEO X. and striking out M as standing for mysterium, he hit the number exactly. This discovery completed his conversion to Luther, and his determination to throw off his monastic vows. Luther dealt with him as straight-forwardly as with Melanchthon about his astrology: he accepted the conclusions, but told him to clear his mind of all the premises about the Beast. Stifelius did not take the advice, and proceeded to settle the end of the world out of the prophet Daniel: he fixed on October, 1533. The parishioners of some cure which he held, having full faith, began to spend their savings in all kinds of good eating and drinking; we may charitably hope this was not the way of preparing for the event which their pastor pointed out. They succeeded in making themselves as fit for Heaven as Lazarus, so far as beggary went: but when the time came, and the world lasted on, they wanted to kill their deceiver, and would have done so but for the interference of Luther.

Notes

1 See Vol. I, page 255, note 6 {584}.

2 “I have no need for this hypothesis.”

3 “Ah, it is a beautiful hypothesis; it explains many things.”

4 “What we know is very slight; what we don’t know is immense.”

5 Brewster relates (Life of Sir Isaac Newton, Vol. II, p. 407) that, a short time before his death, Newton remarked: “I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.”

6 See Vol. I, p. 292, note 1 {632}.

7 “What is all that!”

8 “I have some good news to tell you: at the Bureau of Longitudes they have just received a letter from Germany announcing that M. Bessel has verified by observation your theoretical discoveries on the satellites of Jupiter.”

9 “Man follows only phantoms.”

10 See Vol. I, page 382, note 13 {786}.

11 Dieudonné Thiébault (1733-1807) was a Jesuit in his early life, but he left the order and took up the study of law. In 1765 he went to Prussia and became a favorite of Frederick the Great. He returned to France in 1785 and became head of the Lycée at Versailles.

12 Memories of Twenty Years of Residence in Berlin. There was a second French and an English edition in 1805.

13 Richard Joachim Heinrich von Mollendorff (1724-1816) began his career as a page of Frederick the Great (1740) and became field marshal (1793) and commander of the Prussian army on the Rhine (1794).

14 Hugues Bernard Maret (1763-1839) was not Duc de Bassano in 1807, this title not being conferred upon him until 1809. He was ambassador to England in 1792 and to Naples in 1793. Napoleon made him head of the cabinet and his special confidant. The Bourbons exiled him in 1816.

15 Denis Diderot (1713-1784), whose Lettre sur les aveugles (1749) introduced him to the world as a philosopher, and whose work on the Encyclopédie is so well known.

16 “Sir, (a + bⁿ) / n = x, whence God exists; answer!”

17 This was one James Laurie of Musselburgh.

18 Jelinger Cookson Symons (1809-1860) was an office-holder with a decided leaning towards the improvement of education and social conditions. He wrote A Plea for Schools (1847), The Industrial Capacities of South Wales (1855), and Lunar Motion (1856), to which last work the critic probably refers.

19 “Protimalethes” followed this by another work along the same line the following year, The Independence of the Testimony of St. Matthew and St. John tested and vindicated by the theory of chances.

²⁰ Wilson had already taken up the lance against science in his Strictures on Geology and Astronomy, in reference to a supposed want of harmony between these sciences and some parts of Divine Revelation, Glasgow, 1843. He had also ventured upon poetry in his Pleasures of Piety, Glasgow, 1837.

21 Mrs. Borron was Elizabeth Willesford Mills before her marriage. She made an attempt at literature in her Sibyl’s Leaves, London (printed at Devonport), 1826.

22 See Vol. I, page 386, note 10 {801}.

23 See Vol. I, page 43, notes 7 {32} and 8 {33}.

24 His flying machine, designed in 1843, was one of the earliest attempts at aviation on any extensive scale.

25 Erasmus Darwin (1731-1802) was the grandfather of Charles Darwin. The work here mentioned had great influence, being translated into French, Portuguese, and Italian. Canning parodied it in his Loves of the Triangles.

26 See Vol. I, page 147, note 1 {312}.

27 The notes on this page were written on the day of the funeral of Wilbur Wright, June 1, 1912, the man who realized all of these prophecies, and then died a victim of municipal crime,—of typhoid fever.

28 John Charles, third Earl Spencer (1782-1845), to whose efforts the Reform Bill was greatly indebted for its final success.

29 This was published in London in 1851 instead of 1848.

30 This appeared in 1846.

31 This was done in The Circle Squared, published at Brighton in 1865.

³² It first appeared in 1847, under the title, The Scriptural Calendar and Chronological Reformer, 1848. Including a review of tracts by Dr. Wardlaw and others on the Sabbath question. By W. H. Black. The one above mentioned, for 1849, was printed in 1848, and was also by Black (1808-1872). He was pastor of the Seventh Day Baptists and was interested in archeology and in books. He catalogued the manuscripts of the Ashmolean Museum at Oxford.

33 William Upton, a Trinity College man, Dublin. He also wrote Upton’s Physioglyphics, London, 1844; Pars prima. Geometria vindicata; antiquorumque Problematum, ad hoc tempus desperatorum, Trisectionis Anguli, Circulique Quadraturae, Solutio, per Eucliden effecta, London (printed at Southampton), 1847; The Uptonian Trisection, London, 1866; and The Circle Squared, London, 1872.

34 For example, if θ = 90° we should have 3 cos 30° = 1 + √(4 - sin² 90°), or 3.½ √3 = 1 + √3, or ½ √3 = 1.

35 Nathaniel Wallich (1786-1854) was surgeon at the Danish settlement at Serampore when the East India Company took over the control in 1807. He entered the British medical service and was invalided to England in 1828. His Plantae Asiaticae Rariores (3 vols., London, 1830-1832) was recognized as a standard. He became vice-president of the Linnean Society, F. R. S., and fellow of the Royal Asiatic Society.

36 But if θ = 90° this asserts that or that

cos 30° = (sin 270°. cos 225° + sin² 90°. sin 225°)

√(sin² 270°. cos² 225° + sin⁴ 90° + sin 270°. sin 450°. sin² 90°) ,

½ √3 = -1. (-1 /√2) + 1. (-1/√2)

√(1. ½ + 1 - 1. 1. 1) = 0 / √½,

so that De Morgan must have made some error in copying.

37 John Bonnycastle (died in 1821) was professor of mathematics at Woolwich. His edition of Bossut’s History of Mathematics (1803), and his works on elementary mathematics were well known.

38 The bibliographies give Husaín Rifki as the translator, a practical geometry as the work, and 1802 as the date.

39 See Vol. I, page 309, note 2 {670}.

40 Probably in The Improvement of the Mind which Isaac Watts (1674-1748) published in 1741. His Horae Lyricae appeared in 1706, and the Hymns, by which he is still well known, in 1707.

41 Spencer Joshua Alwyne Compton, second Marquis of Northampton (1790-1851), was a poet, a scientist, and a statesman. He was president of the Royal Society from 1838 to 1849.

42 Besides the writings here mentioned Perigal published a work on Geometric Maps (London, 1853), and Graphic Demonstrations of Geometric Problems (1891).

43 See Vol. II, page 5, note 18.

44 James Ferguson (1710-1776) was a portrait painter, an astronomer, and a popular writer and lecturer on various subjects.

45 In the old ballad of King Alfred and the Shepherd, when the latter is tempting the disguised king into his service, he says:
“Of whig and whey we have good store,
And keep good pease-straw fire.”

Whig is then a preparation of milk. But another commonly cited derivation may be suspected from the word whiggamor being used before whig, as applied to the political party; whig may be a contraction. Perhaps both derivations conspired: the word whiggamor, said to be a word of command to the horses, might contract into whig, and the contraction might be welcomed for its own native meaning.—A. De M.

46 This was p. 147 in the first edition.

47 St. Augustine (354-430) was bishop of Hippo. His Confessiones, in 13 books, was written in 397, and his De Civitate Dei in 426.

48 “He was wont to indulge in certain Punic subtleties lest he should weary the reader by much speaking.”

49 John Milner (1751-1826), bishop of Castabala, a well-known antiquarian.

50 It will be said that when the final happiness is spoken of in “sure and certain hope,” it is the Resurrection, generally; but when afterwards application is made to the individual, simple “hope” is all that is predicated which merely means “wish?” I know it: but just before the general declaration, it is declared that it has pleased God of his great mercy to take unto Himself, the soul of our dear brother: and between the “hopes” hearty thanks are given that it has pleased God to deliver our dear brother out of the miseries of this wicked world, with an additional prayer that the number of the elect may shortly be accomplished. All which means, that our dear brother is declared to be taken to God, to be in a place not so miserable as this world—a description which excludes the “wicked place”—and to be of the elect. Yes, but it will be said again! do you not know that when this Liturgy was framed, all who were not in the road to Heaven were excommunicated burial service read over them. Supposing the fact to have been true in old time, which is a very spicy supposition, how does that excuse the present practice? Have you a right always to say what you believe cannot always be true, because you think it was once always true? Yes, but, choose whom you please, you cannot be certain He is not gone to Heaven. True, and choose which Bishop you please, you cannot be demonstratively certain, he is not a concealed unbeliever: may I therefore say of the whole bench, singulatim et seriatim, that they are unbelievers? No! No! The voice of common sense, of which common logic is a part, is slowly opening the eyes of the multitude to the unprincipled reasoning of theologians. Remember 1819. What chance had Parliamentary Reform when the House of Commons thanked the Manchester sabre-men? If you do not reform your Liturgy, it will be reformed for you, and sooner than you think! The dishonest interpretations, by defence of which even the minds of children are corrupted, and which throw their shoots into literature and commerce, will be sent to the place whence they came: and over the door of the established organization for teaching religion will be posted the following notice:
“Shift and Subterfuge, Shuffle and Dodge,
No longer here allowed to lodge!”

All this ought to be written by some one who belongs to the Establishment: in him, it would be quite prudent and proper; in me, it is kind and charitable.—A. De M.

51 But few do have access to it, for the work is not at all common, and this Piscator is rarely mentioned.

52 This derivation has been omitted.—S. E. De M.

53 A blow for a blow. Roland and Oliver were two of the paladins of Charlemagne whose exploits were so alike that each was constantly receiving credit for what the other did. Finally they met and fought for five days on an island in the Rhine, but even at the end of that period it was merely a drawn battle.

54 “In the name of the church.”

55 “From the chair,” officially.

56 Nicholas Patrick Stephen Wiseman (1802-1865), whose elevation to the archbishopric of Westminster and the cardinalate (1850) led to the act prohibiting Roman Catholics from assuming episcopal titles in England, a law that was never enforced.

57 He was born in 1812 and was converted to Catholicism in 1839. He founded the Tablet in London in 1840, removing its office to Dublin in 1849. He became M. P. in 1852, and at the time of his death (1855) he was preparing a memorial to the Pope asking him to annul the proclamation of an Irish bishop prohibiting his priests from taking part in politics.

58 John Guillim (1565-1621) was the first to systematize and illustrate the whole science of heraldry. He published A display of Heraldrie: manifesting a more easie accesse to the knowledge thereof in 1610.

59 “Faith.”

60 “Faithful.”

61 “For the faith vindicated.”

62 The words are of the same root, and hence our word fiddle. Some suppose this root means a rope, which, as that to which you trust, becomes, in one divergence, confidence itself—just as a rock, and other words, come to mean reliance—and in another, a little string.—A. De M.

63 The Greek lexicographer, a Christian, living after 1000 A. D. His lexicon was first printed at Milan in 1499.

64 Skindapsos.

65 This was John William Burgon (1813-1888), Gresham professor of theology (1867) and dean of Chichester. He was an ultra-conservative, opposing the revised version of the New Testament, and saying of the admission of women to the university examinations that it was “a thing inexpedient and immodest.”

66 Ekklesia, or ecclesia.

67 Ennomos ekklesia.

68 “Without doubt I shall perish forever.”

69 “Every man is an animal.” “Sortes is a man.” “Sortes is an animal.”

70 “For a special purpose.”

71 Heraclitus of Ephesus, the weeping philosopher, 6th century B. C.

72 Democritus, the laughing philosopher, founder of the atomistic theory, 5th century B. C.

73 “Ends to which.”

74 “Ends from which.”

75 “In just as many syllables,” “With just as many letters,” “In just as many words.”

76 “I shall make a way,” “I shall find a way.”

77 The notion that the Evil Spirit is a functionary liable to be dismissed for not attending to his duty, is, so far as my reading goes, utterly unknown in theology. My first wrinkle on the subject was the remark of the Somersetshire farmer upon Palmer the poisoner— “Well! if the Devil don’t take he, he didn’t ought to be allowed to be devil no longer.”—A. De M.

William Palmer (1824-1856) was a member of the Royal College of Surgeons and practised medicine at London. He was hanged in 1856 for having poisoned a friend and was also suspected of having poisoned his wife and brother for their insurance money, besides being guilty of numerous other murders. His trial was very much in the public attention at the time.

78 Advantages and dangers.

79 The old priory of St. Mary of Bethlehem in London, was used as an asylum for the insane. The name was corrupted to Bedlam.

80 Referring to the common English pronunciation of St. John, almost Sinjin. John St. John Long (1798-1834), an Irishman by birth, practised medicine in London. He claimed to have found a specific for rheumatism and tuberculosis, but upon the death of one of his patients in 1830 he was tried for manslaughter. He died of tuberculosis four years later, refusing to take his own treatment.

81 William of Occam (d. 1349), so called from his birthplace, Ockham, in Surrey. He was a Franciscan, and lectured on philosophy in the Sorbonne.

82 He signs himself “James Hopkins, schoolmaster,” and this seems to have been his only published effort.

83 Joseph Ady (1770-1852) was a famous swindler. One of his best-known schemes was to send out letters informing the recipients that they would learn something to their advantage on payment of a certain sum. He spent some time in prison.

84 Sir Peter Laurie (c. 1779-1861) was worth referring to, for he was prominent as a magistrate and was honored because of his interest in all social reforms. He made a fortune as a contractor, became sheriff of London in 1823, and was knighted in the following year. He became Lord Mayor of London in 1832.

⁸⁵ See Vol. I, page 321, note 2 {691}. The Astronomy in a nutshell appeared in 1860. The Herald of Astrology was first published in London in 1831, “by Zadkiel the Seer.” It was continued as The Astrological Almanac (London, 1834), as Zadkiel’s Almanac and Herald of Astrology (ibid., 1835, edited by R. J. Morrison, and subsequently by A. J. Pearce), and as Raphael’s Prophetic Almanac (1840-1855).

86 See Vol. I, page 172, note 3 {382}.

87 See Vol. I, page 87, note 4 {133}.

88 Franz Xaver, Freiherr von Zach (1754-1832) was director of the observatory at Seeberge near Gotha. He wrote the Tabulae speciales aberrationis et mutationis (1806-7), Novae et correctae tabulae solis (1792), and L’attraction des montagnes et ses effets sur le fil à plomb (1814).

89 Jean Louis Pons (1761-1831) was connected with the observatory at Marseilles for thirty years (1789-1819). He later became director of the observatory at Marlia, near Lucca, and subsequently filled the same office at Florence. He was an indefatigable searcher for comets, discovering 37 between 1801 and 1827, among them being the one that bears Encke’s name.

90 This hypothesis has now become an established fact.

91 John Chetwode Eustace (c. 1762-1815) was born in Ireland. Although a Roman Catholic priest he lived for a time at Cambridge where he did some tutoring. His Classical Tour appeared in 1813 and went through several editions.

92 “Crimes should be exposed when they are punished, but disgraceful acts should be hidden.”

93 Henri Hureau de Sénarmont (1808-1862) was professor of mineralogy at the Ecole des mines and examiner at the Ecole polytechnique at Paris.

94 Augustin Jean Fresnel (1788-1827), “Ingenieur des ponts et chaussées,” gave the first experimental proofs of the wave theory of light. He studied the questions of interference and polarization, and determined the approximate velocity of light.

95 “As is my custom.”

96 Francis Heywood (1796-1858) made the first English translation of Kant’s Critick of Pure Reason (1838, reprinted in 1848). The Analysis came out, as here stated, in 1844.

97 Louise Renée de Keroualle, Duchess of Portsmouth and Aubigny (1649-1734), was a favorite of Charles II. She used her influence to keep him under the control of Louis XIV.

98 William Chiffinch (c. 1602-1688) was page of the king’s bed-chamber and keeper of the private closet to Charles II. He was one of the king’s intimates and was an unscrupulous henchman.

99 “Well devised.”

100 “John Bellingham Inglis. His Philobiblion “translated from the first edition (of Ricardus d’Aungervile, Bishop of Durham), 1473,” appeared at London in 1832. It was republished in America (Albany, N. Y.) in 1864.

101 “What are you laughing at?”

102 See Vol. I, page 314, note 4 {681}.

103 See Vol. I, page 112, note 7 {211}.

104 Referring to Hamilton’s edition of the Collected Works of Dugald Stewart, 10 volumes, Edinburgh, 1854-58. It is not commonly remembered that Stewart (1753-1828) taught mathematics at the University of Edinburgh before he took up philosophy.

105 This was Hamilton’s edition of the Works of Thomas Reid (2 vols., Edinburgh, 1846-1863). Reid (1710-1796) included mathematics in his work in philosophy at Aberdeen. In 1764 he succeeded Adam Smith at Glasgow.

106 Edward Irving (1792-1834), the famous preacher. At first he assisted Dr. Chalmers at Glasgow, but in 1822 he went to London where he met with great success. A few years later he became mentally unbalanced and was finally expelled from his church (1832) for heresy. He was a great friend of Carlyle.

¹⁰⁷ He also wrote a number of other paradoxes, including An Essay towards a Science of Consciousness (1838), Instinctive Natural Religion (1858), Popular Treatise on the structure, diseases, and treatment of the human teeth (1837), and On Headache (1859).

108 James Smith (1801-1857), known as Shepherd Smith, was a socialist and a mystic, with a philosophy that was wittily described as “Oriental pantheism translated into Scotch.” He was editor of several journals.

109 Joanna Southcott (1750-1814) was known for her rhyming prophecies in which she announced herself as the woman spoken of in Revelations xii. She had at one time as many as 100,000 disciples, and she established a sect that long survived her.

110 Thales, c. 640-548 B. C.

111 Pythagoras, 580-501 B. C.

112 Anaxagoras, 499-428 B. C., the last of the Ionian school, teacher of Euripides and Pericles. Plutarch speaks of him as having squared the circle.

113 Oinopides of Chios, contemporary of Anaxagoras. Proclus tells us that Oinopides was the first to show how to let fall a perpendicular to a line from an external point.

114 Bryson and Antiphon, contemporaries of Socrates, invented the so-called method of exhaustions, one of the forerunners of the calculus.

115 He wrote, c. 440 B. C., the first elementary textbook on mathematics in the Greek language. The “lunes of Hippocrates” are well known in geometry.

116 Jabir ben Aflah. He lived c. 1085, at Seville, and wrote on astronomy and spherical trigonometry. The Gebri filii Affla Hispalensis de astronomia libri IX was published at Nuremberg in 1533.

117 Hieronymus Cardanus, or Girolamo Cardano (1501-1576), the great algebraist. His Artis magnae sive de regulis Algebrae was published at Nuremberg in 1545.

118 Nicolo Tartaglia (c. 1500-1557), the great rival of Cardan.

119 See note 5 {98}, Vol. I, page 69.

120 See note 10 {124}, Vol. I., page 83.

121 See note 9 {123}, Vol. I, page 83.

122 Pierre Hérigone lived in Paris the first half of the 17th century. His Cours mathématique (6 vols., 1634-1644) had some standing but was not at all original.

123 Franciscus van Schooten (died in 1661) was professor of mathematics at Leyden. He edited Descartes’s La Géométrie.

124 Florimond de Beaune (1601-1652) was the first Frenchman to write a commentary on Descartes’s La Géométrie. He did some noteworthy work in the theory of curves.

125 See note 3 {23}, Vol. I, page 41.

126 Olivier de Serres (b. in 1539) was a writer on agriculture. Montucla speaks of him in his Quadrature du cercle (page 227) as having asserted that the circle is twice the inscribed equilateral triangle, although, as De Morgan points out, this did not fairly interpret his position.

127 Angherà wrote not only the three works here mentioned, but also the Problemi del più alto interesse scientifico, geometricamente risoluti e dimostrati, Naples, 1861. His quadrature was defended by Giovanni Motti in a work entitled Matematica Vera. Falsità del sistema ciclometrico d’Archimede, quadratura del cerchio d’Angherà, ricerca algebraica dei lati di qualunque poligono regolare inscritto in un circolo, Voghera, 1877. The Problemi of 1861 contains Angherà’s portrait, and states that he lived at Malta from 1849 to 1861. It further states that the Malta publications are in part reproduced in this work.

128 This was his friend Paolo Pullicino whose Elogio was pronounced by L. Farrugia at Malta in 1890. He wrote a work La Santa Effegie della Blata Vergine Maria, published at Valetta in 1868.

129 St. Vitus, St. Modestus, and St. Crescentia were all martyred the same day, being torn limb from limb after lions and molten lead had proved of no avail. At least so the story runs.

130 The reference is to Cardinal Wiseman. See Vol. II, page 26, note 56.

131 “Worthy of esteem.”

132 Pedro de Ribadeneira (Ribadeneyra, Rivadeneira), was born at Toledo in 1526 and died in 1611. He held high position in the Jesuit order. The work referred to is the Flos Sanctorum o libro de las vidas de los santos, of which there was an edition at Barcelona in 1643. His life of Loyola (1572) and Historia ecclesiástica del Cisma del reino de Inglaterra were well known.

133 Cæsar Baronius (1538-1607) was made a cardinal in 1595 and became librarian at the Vatican in 1597. The work referred to appeared at Rome in 1589.

134 Mrs. Jameson’s (1794-1860) works were very popular half a century ago, and still have some circulation among art lovers. The first edition of the work mentioned appeared in 1848.

135 The barnyard cock.

136 Shanks did nothing but computing. The title should, of course, read “to 607 Places of Decimals.” He later carried the computation to 707 decimal places. (Proc. Roy. Society, XXI, p. 319.) He also prepared a table of prime numbers up to 60,000. (Proc. Roy. Society, XXII, p. 200.)

137 See Vol. I, page 42, note 4 {24}.

138 See Vol. I, page 64, note 1 {78}.

139 See Vol. I, page 328, note 1 {704}.

140 George Suffield published Synthetic Division in Arithmetic, to which reference is made, in 1863.

141 John Robert Lunn wrote chiefly on Church matters, although he published a work on motion in 1859.

142 Jean Baptiste Joseph, Baron Fourier (1768-1830), sometime professor in the Military School at Paris, and later at the Ecole polytechnique. He is best known by his Théorie analytique de la chaleur (Paris, 1822), in which the Fourier series is used. The work here referred to is the Analyse des équations déterminées (Paris, 1831).

143 William George Horner (1786-1837) acquired a name for himself in mathematics in a curious manner. He was not a university man nor was he a mathematician of any standing. He taught school near Bristol and at Bath, and seems to have stumbled upon his ingenious method for finding the approximate roots of numerical higher equations, including as a special case the extracting of the various roots of numbers. Davies Gilbert presented the method to the Royal Society in 1819, and it was reprinted in the Ladies’ Diary for 1838, and in the Mathematician in 1843. The method was original as far as Horner was concerned, but it is practically identical with the one used by the Chinese algebraist Ch’in Chiu-shang, in his Su-shu Chiu-chang of 1247. But even Ch’in Chiu-shang can hardly be called the discoverer of the method since it is merely the extension of a process for root extracting that appeared in the Chiu-chang Suan-shu of the second century B. C.

144 He afterwards edited Loftus’s Inland Revenue Officers’ Manual (London, 1865). The two equations mentioned were x³ - 2x = 5 and y³ - 90y² + 2500y - 16,000 = 0, in which y = 30 - 10x. Hence each place of y is the complement of the following place of x with respect to 9.

145 Probably the John Power Hicks who wrote a memoir on T. H. Key, London, 1893.

146 Possibly the one who wrote on the quadrature of the circle in 1881.

147 As it is. But what a pity that we have not 12 fingers, with duodecimal fractions instead of decimals! We should then have 0.6 for ½, 0.4 for ⅓, 0.8 for ⅔, 0.3 for ¼, 0.9 for ¾, and 0.16 for ⅛, instead of 0.5, 0.333+, 0.666+, 0.25, 0.75, and 0.125 as we now have with our decimal system. In other words, the most frequently used fractions in business would be much more easily represented on the duodecimal scale than on the decimal scale that we now use.

148 He wrote Hints for an Essay on Anemology and Ombrology (London, 1839-40) and The Music of the Eye (London, 1831).

149 Brigham Young (1801-1877) was born at Whitingham, Vermont, and entered the Mormon church in 1832. In 1840 he was sent as a missionary to England. After the death of Joseph Smith he became president of the Mormons (1847), leading the church to Salt Lake City (1848).

150 Joseph Smith (1805-1844) was also born in Vermont, and was four years the junior of Brigham Young. The Book of Mormon appeared in 1827, and the church was founded in 1830. He was murdered in 1844.

151 Orson Pratt (1811-1881) was one of the twelve apostles of the Mormon Church (1835), and made several missionary journeys to England. He was professor of mathematics in the University of Deseret (the Mormon name for Utah). Besides the paper mentioned Pratt wrote the Divine Authenticity of the Book of Mormon (1849), Cubic and Biquadratic Equations (1866), and a Key to the Universe (1866).

152 “It does not follow.”

153 Dryden (1631-1700) published his Religio Laici in 1682. The use of the word “proportion” in the sense of ratio was common before his time, but he uses it in the sense of having four terms; that is, that price is to price as offence is to offence.

154 Olinthus Gilbert Gregory (1774-1841) succeeded Hutton as professor of mathematics at Woolwich. He was, with De Morgan, much interested in founding the University of London. He wrote on astronomy (1793), mechanics (1806), practical mathematics (1825), and Christian evidences (1811).

155 See Vol. I, page 220, note 6 {482}. The Pensées appeared posthumously in 1670.

156 “The right thing to do is not to wager at all.” “Yes, but you ought to wager; you have started out; and not to wager at all that God exists is to wager that he does not exist.”

157 He lived about 300 A.D., in Africa, and wrote Libri septem adversus Gentes. This was printed at Rome in 1542-3.

158 Pierre Bayle (1647-1706) was professor of philosophy at the Prostestant University at Sedan from 1675 until its dissolution in 1681. He then became professor at Rotterdam (1681-1693). In 1684 he began the publication of his journal of literary criticism Nouvelles de la République des Lettres. He is best known for his erudite Dictionnaire historique et critique (1697).

159 “But Christ himself does not prove what he promises. It is true. For, as I have said, there cannot be any absolute proof of future events. Therefore since it is a condition of future events that they cannot be grasped or comprehended by any efforts of anticipation, is it not more reasonable, out of two alternatives that are uncertain and that are hanging in doubtful expectation, to give credence to the one that gives some hope rather than to the one that offers none at all? For in the former case there is no danger if, as is said to threaten, it becomes empty and void; while in the latter case the danger is greatest, that is, the loss of salvation, if when the time comes it is found that it was not a falsehood.”

160 Gregg wrote several other paradoxes, including the following: The Authentic Report of the extraordinary case of Tresham Dames Gregg... his committal to Bridewell for refusing to give his recognizance (Dublin, 1841), An Appeal to Public Opinion upon a Case of Injury and Wrong... in the case of a question of prerogative that arose between [R. Whately] ... Archbishop of Dublin and the author (London, 1861), The Cosmology of Sir Isaac Newton proved to be in accordance with the Bible (London, 1871), The Steam Locomotive as revealed in the Bible (London 1863) and On the Sacred Law of 1866, conferring perpetual life with immunity from decay and disease. A cento of decisive scriptural oracles strangely discovered (London and Dublin, 1875). These titles will help the reader to understand the man whom De Morgan so pleasantly satirizes.

161 See Vol. I, page 261, note 2 {592}.

162 “They have found it.”

163 The late Greeks used the letters of their alphabet as numerals, adding three early alphabetic characters. The letter χ represented 600, ξ represented 60, and Ϝ stood for 6. This gives 666, the number of the Beast given in the Revelation.

164 “Allowing for necessary exceptions.”

165 Mr. Gregg is not alone in his efforts to use the calculus in original lines, as any one who has read Herbart’s application of the subject to psychology will recall.

166 See Vol. I, page 105, note 4 {188}; page 109, note 1 {197}.

167 The full title shows the plan,—The Decimal System as a whole, in its relation to time, measure, weight, capacity, and money, in unison with each other. But why is this so much worse than the French plan of which we have only the metric system and the decimal division of the angle left?

¹⁶⁸ One of the brothers of Sir Isaac Pitman (1813-1897), the inventor of modern stenography. Of these brothers, Benjamin taught the art in America, Jacob in Australia, and Joseph, Henry, and Frederick in England.

169 For example, The Phonographic Lecturer (London, 1871 etc.), The Phonographic Student (1867, etc.), and The Shorthand Magazine (1866, etc.).

¹⁷⁰ See Vol. II, page 68, note 148.

171 It involves the theory of non-Euclidean geometry, Euclid’s postulate of parallels being used in proving this theorem.

172 Referring to the fact that none of the works of Thales is extant.

173 The author was one B. Bulstrode. Parts 4 and 5 were printed at Calcutta.

174 See Vol. II, page 5, note 18.

175 See Vol. I, page 85, note 2 {129}.

176 Alexander Vasilievich Suvaroff (1729-1800), a Russian general who fought against the Turks, in the Polish wars, and in the early Napoleonic campaigns. When he took Ismail in 1790 he sent this couplet to Empress Catherine.

177 “Newton hath determined rightly,” “Newton hath not determined rightly.”

178 See Vol. I, page 288, note 3 {621}.

179 See Vol. I, page 326, note 1 {700}.

180 “With great honor.”

181 Apparently unknown to biographers. He seems to have written nothing else.

182 Captain Marryat (1792-1848) published Snarley-yow, or the Dog Fiend in 1837.

183 He is not known to biographers, and published nothing else under this name.

184 See Vol. I, page 80, note 5 {119}.

185 He published a Family and Commercial Illustrated Almanack and Year Book... for 1861 (Bath, 1860).

186 Louis Dutens (1730-1812) was born at Tours, but went to England as a young man. He made the first collection of the works of Leibnitz, against the advice of Voltaire, who wrote to him: “Les écrits de Leibnitz sont épars comme les feuilles de la Sybille, et aussi obscurs que les écrits de cette vieille.” The work appeared at Geneva, in six volumes, in 1769.

187 Mungo Park (1771-1806), the first European to explore the Niger (1795-6). His Travels in the Interior of Africa appeared in 1799. He died in Africa.

188 Gerhard Mercator (1512-1594) the well-known map maker of Louvain. The “Mercator’s Projection” was probably made as early as 1550, but the principle of its construction was first set forth by Edward Wright (London, 1599).

189 Quirico Barilli Filopanti wrote a number of works and monographs. He succeeded in getting his Cesare al Rubicone and Degli usi idraulici della Tela in the Memoria letta... all’ Accademia delle Scienze in Bologna (1847, 1866). He also wrote Dio esiste (1881), Dio Liberale (1880), and Sunto della memoria sulle geuranie ossia di alcune singolari relazioni cosmiche della terra e del cielo (1862).

190 The periods of disembodiment may be interesting. They will be seen from the following dates: Descartes (1596-1650), William III (1650-1702); Roger Bacon (1214 to c. 1294), Boccaccio (1313-1375). Charles IX was born in 1550 and died in 1574.

191 His real name was Frederick Parker, and he wrote several works on the Greek language and on religion. Among these were a translation of the New Testament from the Vatican MS. (1864), The Revealed History of Man (1854), An Enquiry respecting the Punctuation of Ancient Greek (1841), and Rules for Ascertaining the sense conveyed in Ancient Greek Manuscripts (1848, the seventh edition appearing in 1862).

¹⁹² See Vol. I, page 352, second note 1 {736}.

The literature on the subject of the Great Pyramid, considered from the standpoint of metrology, is extensive.

193 See Vol. I, page 80, note 5 {119}.

194 Sir Philip Francis (1740-1818) was a Whig politician. The evidence that he was the author of the Letters of Junius (1769-1772) is purely circumstantial. He was clerk in the war office at the time of their publication. In 1774 he was made a member of the Supreme Council of Bengal, and was a vigorous opponent of Warren Hastings, the two fighting a duel in 1780. He entered parliament in 1784 and was among the leaders in the agitation for parliamentary reform.

195 Mrs. Cottle published a number of letters that attracted attention at the time. Among these were letters to the emperor of France and king of Sardinia (1859) relating to the prophecies of the war between France and Austria; to G. C. Lavis and Her Majesty’s Ministers (1859) relating to her claims as a prophetess; and to the “Crowned Heads” at St. James, the King of Prussia, and others (1860), relating to certain passages of Scripture. She also wrote The Lamb’s Book of Life for the New Jerusalem Church and Kingdom, interpreted for all nations (1861).

196 See Vol. I, page 315, note 2 {685}, and Vol. II, page 58, note 109.

197 A Congregational minister, who published a number of sermons, chiefly obituaries, between 1804 and 1851. His Frailty of Human Life, two sermons delivered on the occasion of the death of Princess Charlotte, went through at least three editions.

198 He was secretary of the Congregational Board and editor of the Congregational Year Book (from 1846) and the Congregational Manual.

199 Frederick Denison Maurice (1805-1872) began his preaching as a Unitarian but entered the Established Church in 1831, being ordained in 1834. He was professor of English and History at King’s College, London, from 1840 to 1853. He was one of the founders of Queen’s College for women, and was the first principal of the Working Men’s College, London. The subject referred to by De Morgan is his expression of opinion in his Theological Essays (1853) that future punishment is not eternal. As a result of this expression he lost his professorship at King’s College. In 1866 he was made Knightbridge Professor of Casuistry, Moral Theology, and Moral Philosophy at Cambridge.

200 See Vol. I, page 46, note 1 {42}. Besides the books mentioned in this list he wrote The Ratio between Diameter and Circumference demonstrated by angles, and Euclid’s Theorem, Proposition 32, Book I, proved to be fallacious (Liverpool, 1870). This is the theorem which asserts that the exterior angle of a triangle is equal to the sum of the two opposite interior angles, and that the sum of the interior angles equals two right angles. He also published his Curiosities of Mathematics in 1870, a work containing an extensive correspondence with every one who would pay any attention to him. De Morgan was then too feeble to show any interest in the final effort of the subject of some of his keenest satire.

201 See Vol. I, page 332, note 4 {709}.

202 See Vol. I, page 101, note 4 {174}.

203 “The circle-squaring disease”; literally, “the circle-measuring disease.”

204 See Vol. II, page 63, note 136.

205 William Rutherford (c. 1798-1871), teacher of mathematics at Woolwich, secretary of the Royal Astronomical Society, editor of The Mathematician, and author of various textbooks. The Extension of π to 440 places, appeared in the Proceedings of the Royal Society in 1853 (p. 274).

206 Charles Knight (1791-1873) was associated with De Morgan for many years. After 1828 he superintended the publications of the Society for the Diffusion of Useful Knowledge, to which De Morgan contributed, and he edited the Penny Cyclopedia (1833-1844) for which De Morgan wrote the articles on mathematics.

207 Sir William Hamilton. See Vol. I, page 112, note 7 {211}.

208 Adam Smith (1723-1790) was not only known for his Wealth of Nations (1776), but for his Theory of Moral Sentiments (1759), published while he was professor of moral philosophy at Glasgow (1752-1764). He was Lord Rector of the university in 1787.

209 See Vol. I, page 332, note 4 {709}.

210 “Whip.”

211 “Terrible lash.”

212 “An accomplished fact [an accomplished fault].”

213 See Extracts from the Diary and Letters of Mrs. Mary Cobb, London, 1805.

214 “Gentle in manner.”

215 “Brave in action.” The motto of Earl Newborough was “Suaviter in modo, fortiter in re.”

216 “Reduction to an absurdity,” a method of proof occasionally used in geometry and in logic.

217 “He has lost the right of being moved (struck) by evidence.”

218 For radix quadratus. The usual root sign is supposed to be derived from r (for radix), and at one time q was commonly used for square, as in Viète’s style of writing Aq for A².

219 The Garde Douloureuse was a castle in the marches of Wales and received its name because of its exposure to attacks by the Welsh.

220 “Out of the fight.”

221 “Hidden.”

222 John Cam Hobhouse (1786-1869), Baron Broughton, was committed to Newgate for two months in 1819 for his anonymous pamphlet, A Trifling Mistake. This was a great advertisement for him, and upon his release he was at once elected to parliament for Westminster. He was a strong supporter of all reform measures, and was Secretary for War in 1832. He was created Baron Broughton de Gyfford in 1851.

223 Thomas Erskine (1750-1823), the famous orator. He became Lord Chancellor in 1806, but sat in the House of Commons most of his life.

224 The above is explained in the MS. by a paragraph referring to some anagrams, in one of which, by help of the orthography suggested, a designation for this cyclometer was obtained from the letters of his name.—S. E. De M.

225 “A personal verb agrees with its subject.”

226 See Vol. I, page 326, note 1 {700}.

227 See Vol. I, page 326, note 2 {701}.

228 Apparently unknown to biographers.

229 The Bibliotheca Mathematica of Ludwig Adolph Sohncke (1807-1853), professor of mathematics at Königsberg and Halle, covered the period from 1830 to 1854, being completed by W. Engelmann. It appeared in 1854.

230 See Vol. I, page 392, note 2 {805}.

231 See Vol. I, page 43, note 7 {32}.

232 See Vol. II, page 91, note 187.

233 Mason made a notable balloon trip from London to Weilburg, in the Duchy of Nassau, in November, 1836, covering 500 miles in 18 hours. He published an account of this trip in 1837, and a work entitled Aeronautica in 1838.

234 William Harrison Ainsworth (1805-1885) the novelist.

235 On this question see Vol. I, page 326, note 2 {701}.

236 Major General Alfred Wilks Drayson, author of various works on geology, astronomy, military surveying, and adventure.

237 Hailes also wrote several other paradoxes on astronomy and circle squaring during the period 1843-1872.

238 See Vol. I, page 43, note 8 {33}.

239 See Vol. I, page 43, note 7 {32}.

240 “Very small errors are not to be condemned.”

241 He seems to have written nothing else.

242 Besides the paradoxes here mentioned by De Morgan he wrote several other works, including the following: Abriss der Babylonisch-Assyrischen Geschichte (Mannheim, 1854), A Popular Inquiry into the Moon’s rotation on her axis (London, 1856), Practical Tables for the reduction of the Mahometan dates to the Christian kalendar (London, 1856), Grundzüge einer neuen Weltlehre (Munich, 1860), and On the historical Antiquity of the People of Egypt (London, 1863).

243 Dircks (1806-1873) was a civil engineer of prominence, and a member of the British Association and the Royal Society of Edinburgh. He wrote (1863) on “Pepper’s Ghost,” an ingenious optical illusion invented by him. There was a second edition of the Perpetuum Mobile in 1870.

244 George Stephenson (1781-1848), the inventor of the first successful steam locomotive. His first engine was tried in 1814.

245 Robert Stephenson (1803-1859), the only son of George. Most of the early improvements in locomotive manufacture were due to him. He was also well known for his construction of great bridges.

246 “In its proper place.”

247 “A fool always finds a bigger fool to admire him.”

248 See Vol. I, page 43, note 7 {32}.

249 See Vol. I, page 43, note 8 {33}.

250 See Vol. I, page 85, note 2 {129}.

251 See Vol. I, page 390, note 1 {390}.

252 From 1823 to 1852 it was edited by I. C. Robertson; from 1852 to 1857 by R. A. Brooman; and from 1857 to 1863 by Brooman and E. J. Reed.

253 Sir James Ivory (1765-1842) was, as a young man, manager of a flax mill in Scotland. In 1804 he was made professor of mathematics at the Royal Military College, then at Marlow and later at Sandhurst. He was deeply interested in mathematical physics, and there is a theorem on the attraction of ellipsoids that bears his name. He was awarded three medals of the Royal Society, and was knighted together with Herschel and Brewster, in 1831.

254 See Vol. I, page 56, note 1 {64}.

255 See Vol. I, page 153, note 5 {338}.

256 See Vol. I, page 309, note 2 {670}.

257 See Vol. I, page 87, note 4 {133}.

258 George Canning (1770-1857), the Tory statesman and friend of Scott, was much interested in founding the Quarterly Review (1808) and was a contributor to its pages.

[259] See Vol. I, page 186, note 14 {418}.

260 See Vol. II, page 141, note 252.

261 De Morgan had a number of excellent articles in this publication.

262 See Vol. I, page 279, note 1 {611}.

263 James Orchard Halliwell (1820-1889), afterwards Halliwell-Phillips, came into prominence as a writer at an early age. When he was seventeen he wrote a series of lives of mathematicians for the Parthenon. His Rara Mathematica appeared when he was but nineteen. He was a great bibliophile and an enthusiastic student of Shakespeare.

264 This was written at the age of twenty-two.

265 The subject of this criticism is of long past date, and as it has only been introduced by the author as an instance of faulty editorship, I have omitted the name of the writer of the libel, and a few lines of further detail.—S. E. De M.

266 “Condemned souls.”

267 The editor of the Mechanics’ Magazine died soon after the above was written.—S. E. De M.

268 Thomas Stephens Davies (1795-1851) was mathematical master at Woolwich and F. R. S. He contributed a series of “Geometrical Notes” to the Mechanics’ Magazine and edited the Mathematician. He also published a number of text-books.

269 See Vol. II, page 66, note 143.

270 The Dictionary of Greek and Roman Biography (1849), edited by Sir William Smith (1813-1893), whose other dictionaries on classical and biblical matters are well known.

271 “O J. S.! This is the worst! the greatest possible injury!”

272 See Vol. I, page 44, note 9 {34} and page 110, note 5 {201}.

273 “If there’s a man whom the judge’s pitiless sentence awaiteth,
His head condemned to penalties and tribulations,
Let neither penitentiaries tire him with laborer’s burdens
Nor let his stiffened hands be harrassed by work in the mines.
He must square the circle! For what else do I care?—all
Known punishments this one task hath surely included.”

274 Houlston was in the customs service. He also published Inklings of Areal Autometry, London, 1874.

275 This is Frederick C. Bakewell. He had already published Natural Evidence of a Future Life (London, 1835), Philosophical Conversations (London, 1833, with other editions), and Electric Science (London, 1853, with other editions).

276 Henry F. A. Pratt had already published A Dissertation on the power of the intercepted pressure of the Atmosphere (London, 1844) and The Genealogy of Creation (1861). Later he published a work On Orbital Motion (1863), and Astronomical Investigations (1865).

277 See Vol. I, page 260, note 1 {591}.

278 Thomas Rawson Birks (1810-1883), a theologian and controversialist, fellow of Trinity College, Cambridge, and (1872) professor of moral philosophy in that university. He wrote Modern Rationalism (1853), The Bible and Modern Thought (1861), The First Principles of Moral Science (1873), and Modern Physical Fatalism and the Doctrine of Evolution (1876), the last being an attack on Herbert Spencer’s First Principles.

279 Pseudonym for William Thorn. In the following year (1863) he published a second work, The Thorn-Tree: being a History of Thorn Worship, a reply to Bishop Colenso’s work entitled The Pentateuch and the Book of Joshua critically examined.

280 Besides The Pestilence (1866) he published The True Church (1851), The Church and her destinies (1855), Religious reformation imperatively demanded (1864), and The Bible plan unfolded (second edition, 1872).

281 See Vol. II, page 97, note 195.

282 Sir George Cornewall Lewis (1806-1863) also wrote an Essay on the Origin and Formation of the Romance Languages (1835), an Essay on the Government of Dependencies (1841), and an Essay on Foreign Jurisdiction and the Extradition of Criminals (1859). He was Chancellor of the Exchequer in 1855 and Home Secretary in 1859.

283 Henry Malden (1800-1876), a classical scholar, fellow of Trinity College, Cambridge, and professor of Greek at University College (1831-1876), then (1831) the University of London. He wrote a History of Rome to 390 B. C. (1830), and On the Origin of Universities and Academical Degrees (1835).

284 Henry Longueville Mansel (1820-1871), theologian and metaphysician, reader in theology at Magdalen College, Oxford (1855), and professor of ecclesiastical history and Dean of St. Paul’s (1866). He wrote on metaphysics, and his Bampton Lectures (1858) were reprinted several times.

285 “Hejus gave freely, gave freely. God is propitious, God is favorable to him who gives freely. God is honored with a banquet of eggs at the cross roads, the god of the world. God, with benignant spirit, desired in sacrifice a goat, a bull to be carried within the precincts of the holy place. God, twice propitiated, blesses the pit of the sacred libation.”

286 Eudoxus of Cnidus (408-355 B. C.) had much to do with the early scientific astronomy of the Greeks. The fifth book of Euclid is generally attributed to him. His astronomical works are known chiefly through the poetical version of Aratus mentioned in note 13, page 167.

287 Simplicius, a native of Cilicia, lived in the 6th century of our era. He was driven from Athens by Justinian and went to Persia (531), but he returned later and had some fame as a teacher.

288 See Vol. I, page 160, note 3 {348}.

289 See Vol. I, page 76, note 3 {112}.

290 “Through right and wrong.”

291 “It is therefore to arrive at this parallelism, or to preserve it, that Copernicus feared to be obliged to have recourse to this equal and opposite movement which destroys the effect which he attributed so freely to the first, of deranging the parallelism.”

292 A contemporary of Plato and a disciple of Aristotle.

293 Meton’s solstice, the beginning of the Metonic cycles, has been placed at 432 B. C. Ptolemy states that he made the length of the year 365¼ + 1/72 days.

294 Aratus lived about 270 B. C., at the court of Antigonus of Macedonia, and probably practiced medicine there. He was the author of two astronomical poems, the Φαινόμενα, apparently based on the lost work of Eudoxus, and the Διοσηεῖα based on Aristotle’s Meteorologica and De Signis Ventorum of Theophrastus.

295 “The nineteen (-year) cycle of the shining sun.”

296 Claudius Salmasius (1588-1653), or Claude Saumaise, was a distinguished classicist, and professor at the University of Leyden. The word ἠλείοιο means Elian, thus making the phrase refer to the brilliant one of Elis.

297 Sir William Brown (1784-1864). In 1800 the family moved to Baltimore, and there the father, Alexander Brown, became prominent in the linen trade. William went to Liverpool where he acquired great wealth as a merchant and banker. He was made a baronet in 1863.

298 Robert Lowe (1811-1892), viscount Sherbrooke, was a fellow of Magdalen College, Oxford (1835). He went to Australia in 1842 and was very successful at the bar. He returned to England in 1850 and became leader writer on the Times. He was many years in parliament, and in 1880 was raised to the peerage.

299 See Vol. I, page 42, note 4 {24}.

300 Francis Walkingame (fl. about 1751-1785), whose Tutor’s Assistant went through many editions from 1751-1854.

301 Davies Gilbert (1767-1839). His family name was Giddy, but he assumed his wife’s name. He sat in parliament from 1806 to 1832. In 1819 he secured the establishment of the Cape of Good Hope observatory. He was Treasurer (1820-1827) and President (1827-1830) of the Royal Society.

302 See Vol. I, page 55, note 2 {63}.

303 Sir Jonathan Frederick Pollock (1783-1870) entered parliament in 1831 and was knighted in 1834.

304 Joseph Hume (1777-1855) entered parliament in 1812 and for thirty years was leader of the Radical party.

305 “What! when I say, ‘Nicole, bring me my slippers,’ is that prose?”

306 Captain Basil Hall (1788-1844), a naval officer, carried on a series of pendulum observations in 1820-1822, while on a cruise of the west coast of North America. The results were published in 1823 in the Philosophical Transactions. He also wrote two popular works on travel that went through numerous editions.

307 Anthony Ashley Cooper (1801-1885), Earl of Shaftesbury. His name is connected with philanthropic work and factory legislation.

308 See Vol. I, page 207, note 12 {469}.

309 See Vol. I, page 80, note 5 {119}.

310 Sir Thomas Maclear (1794-1879), an Irishman by birth, became Astronomer Royal at the Cape of Good Hope in 1833. He was an indefatigable observer. He was knighted in 1860.

311 Thomas Romney Robinson (1792-1882), another Irish astronomer of prominence. He was a deputy professor at Trinity College, Dublin, but took charge of the Armagh observatory in 1823 and remained there until his death.

312 Sir James South (1785-1867) was in early life a surgeon, but gave up his practice in 1816 and fitted up a private observatory. He contributed to the science of astronomy, particularly with respect to the study of double stars.

313 Sir John Wrottesley (1798-1867), second Baron Wrottesley. Like Sir James South, he took up the study of astronomy after a professional career,—in his case in law. He built a private observatory in 1829 and made a long series of observations, publishing three star catalogues. He was president of the Astronomical Society from 1841 to 1843, and of the Royal Society from 1854 to 1857.

314 He seems to have written nothing else.

315 See Vol. II, page 68, note 147.

316 “The wills are free, and I wish neither the one nor the other.”

317 “The force of inertia conquered.”

318 Reddie also wrote The Mechanics of the Heavens, referred to later in this work. He must not be confused with Judge James Reddie (1773-1852), of Glasgow, who wrote on international law, although this is done in the printed edition of the British Museum catalogue, for he is mentioned by De Morgan somewhat later as alive in 1862.

319 Henry Dunning Macleod (1821-1902), a lawyer and writer on political economy, was a Scotchman by birth. He wrote on economical questions, and lectured on banking at Cambridge (1877) and at King’s College, London (1878). He was a free lance in his field, and was not considered orthodox by the majority of economists of his time. He was an unsuccessful candidate for the chairs of political economy at Cambridge (1863), Edinburgh (1871), and Oxford (1888).

320 See Vol. I, page 252, note 2 {576}.

321 Francis Henry Laing (1816-1889) was a graduate of Queen’s College, Cambridge, and a clergyman in the Church of England until 1846, when he entered the Church of Rome. He taught in various Jesuit colleges until 1862, when his eccentricity was too marked to warrant the Church in allowing him to continue. He published various controversial writings during his later years. Of course if he had known the works of Wessel, Gaus, Buée, Argand, and others, he would not have made such a sorry exhibition of his ignorance of mathematics.

322 See Vol. I, page 329, note 1 {705}. The book went into a second edition in 1864.

323 Thomas Weddle (1817-1853) was, at the time of publishing this paper, a teacher in a private school. In 1851 he became professor of mathematics at Sandhurst. He contributed several papers to the Cambridge and Dublin Mathematical Journal, chiefly on geometry.

324 See Vol. II, page 109, note 205.

325 See Vol. II, page 66, note 143.

326 See Vol. II, page 151, note 268.

327 George Barrett (1752-1821) worked from 1786 to 1811 on a set of life insurance and annuity tables. He invented a plan known as the “columnar method” for the construction of such tables, and as De Morgan states, this was published by Francis Baily, appearing in the appendix to his work on annuities, in the edition of 1813. Some of his tables were used in Babbage’s Comparative View of the various Institutions for the Assurance of Lives (1826).

328 See Vol. I, page 309, note 2 {670}.

329 This was his Practical short and direct Method of Calculating the Logarithm of any given Number, and the Number corresponding to any given Logarithm (1849).

330 This is William Neile (1637-1670), grandson of Richard Neile (not Neal), Archbishop of York. At the age of 19, in 1657, he gave the first rectification of the semicubical parabola. Although he communicated it to Brouncker, Wren, and others, it was not published until 1639, when it appeared in John Wallis’s De Cycloide.

331 I myself “was a considerable part.”

332 He also wrote A Glance at the Universe (“2d thousand” in 1862), and The Resurrection Body (1869).

333 See Vol. I, page 63, note 1 {74}.

334 As Swift gave it in his Poetry. A Rhapsody, it is as follows:
“So, naturalists observe, a flea
Has smaller fleas that on him prey;
And these have smaller still to bite ’em.
And so proceed ad infinitum.”

335 Perhaps 1,600,000,000 years, if Boltwood’s recent computations based on radium disintegration stand the test. This would mean, according to MacCurdy’s estimate, 60,000,000 years since life first appeared on the earth.

336 De Morgan wrote better than he knew, for this work, the Allgemeine Encyclopädie der Wissenschaften und Künste, begun at Leipsic in 1818, is still (1913) unfinished. Section I, A-G, consists of 99 parts in 56 volumes; Section II, H-N, consists of 43 volumes and is not yet completed; and Section III, O-Z, consists of 25 volumes thus far, with most of the work still to be done. Johann Samuel Ersch (1766-1828), the founder, was head librarian at Halle. Johann Gottfried Gruber (1774-1851), his associate, was professor of philosophy at the same university.

337 William Howitt (1792-1879) was a poet, a spiritualist, and a miscellaneous writer. He and his wife became spiritualists about 1850. He wrote numerous popular works on travel, nature and history.

338 See Vol. II, page 55, note 108.

339 As will be inferred from the text, C. D. was Mrs. De Morgan, and A. B. was De Morgan.

340 Jean Meslier (1678-1733), curé of Estrepigny, in Champagne, was a skeptic, but preached only strict orthodoxy to his people. It was only in his manuscript, Mon Testament, that was published after his death, and that caused a great sensation in France, that his antagonism to Christianity became known.

341 Baron Zach relates that a friend of his, in a writing intended for publication, said Un esprit doit se frotter contre un autre. The censors struck it out. The Austrian police have a keen eye for consequences.—A. De M.

“One mind must rub against another.” On Baron Zach, see Vol. II, page 45, note 4.

342 Referring to the first Lord Eldon (1751-1838), who was Lord Chancellor from 1799 to 1827, with the exception of one year.

343 “Sleeping power.”

344 “Causes sleep.”

345 Richard Hooker (c. 1554-1600), a theologian, “the ablest living advocate of the Church of England as by law established.”

346 See Vol. I, page 76, note 3 {112}.

347 “Other I,”—other self.

348 This “utter rejection” has been repeated (1872) by the same writer.—S. E. De M.

349 Edward Jenner (1749-1823) was a physician and biologist. His first experiments in vaccination were made in 1796, and his discovery was published in 1798.

350 See Vol. II, page 38, note 80.

351 “You will go most safely in the middle (way).”

352 Pierre Joseph Arson was known early in the 19th century for his controversy with Hoëné Wronski the mathematician, whom he attacked in his Document pour l’histoire des grands fourbes qui ont figuré sur la terre (1817-1818).

353 “We enter the course by night and are consumed by fire.”

354 See Vol. I, page 51, note 3 {51}.

355 See Vol. I, page 336, note 8 {713}.

356 See Vol. I, page 137, note 8 {286}.

357 See Vol. I, page 229, note 2 {515}.

358 Richard Cobden (1804-1865), the cotton manufacturer and statesman who was prominent in his advocacy of the repeal of the Corn Laws.

359 James Smith (1775-1839), solicitor to the Board of Ordnance. With his brother Horatio he wrote numerous satires. His Horace in London (1813) imitated the Roman poet. His works were collected and published in 1840.

360 Samuel Butler (1612-1680), the poet and satirist, author of Hudibras (1663-1678).

361 “Is it not fine to be sure of one’s action when entering in a combat with another? There, push me a little in order to see. Nicole. Well! what’s the matter? M. Jourdain. Slowly. Ho there! Ho! gently. Deuce take the rascal! Nicole. You told me to push. M. Jourdain. Yes, but you pushed me en tierce, before you pushed en quarte, and you did not give me time to parry.”

362 John Abernethy (1764-1831), the famous physician and surgeon.

363 See Vol. I, page 102, note 5 {175}.

364 “With what measure ye mete, it shall be measured to you again.”

365 Eusebius of Cæsarea (c. 260-340), leader of the moderate party at the Council of Nicæa, and author of a History of the Christian Church in ten books (c. 324 A. D.).

366 Nathaniel Lardner (1684-1768), a non-conformist minister and one of the first to advocate the scientific study of early Christian literature.

367 Henry Alford (1810-1871) Dean of Canterbury (1857-1871) and editor of the Greek Testament (1849-1861).

368 The work was The Number and Names of the Apocalyptic Beasts: with an explanation and application. Part I. London, 1848, as mentioned below. Thom also wrote The Assurance of Faith, or Calvinism identified with Universalism (London, 1833), and various other religious works.

369 See Vol. I, page 222, note 14 {490}.

370 John Hamilton Thom (1808-1894) was converted to Unitarianism and was long a minister in that church, preaching in the Renshaw Street Chapel from 1831 to 1866. De Morgan refers to the Liverpool Unitarian controversy conducted by James Martineau and Henry Giles in response to a challenge by thirteen Anglican Clergy. In 1839 Thom contributed four lectures and a letter to this controversy. Among his religious works were a Life of Blanco White (1845) and Hymns, Chants, and Anthems (1854).

371 The spelling of these names is occasionally changed to meet the condition that the numerical value of the letters shall be 666, “the number of the beast” of Revelations. The names include Julius Cæsar; Valerius Jovius Diocletianus (249-313), emperor from 287 to 305, persecutor of the Christians; Louis, presumably Louis XIV; Gerbert (940-1003), who reigned as Pope Sylvester II from 999 to 1003, known to mathematicians for his abacus and his interest in geometry, and accused by his opponents as being in league with the devil; Linus, the second Bishop of Rome, the successor of Peter; Camillo Borghese (1552-1621), who reigned as Pope Paul V from 1605 to 1621, and who excommunicated all Venice in 1606 for its claim to try ecclesiastics before lay tribunals, thus taking a position which he was forced to abandon; Luther, Calvin; Laud (see Vol. I, page 145, note 7 {307}); Genseric (c. 406-477), king of the Vandals, who sacked Rome in 455 and persecuted the orthodox Christians in Africa; Boniface III, who was pope for nine months in 606; Beza (see Vol. I, page 66, note 6 {83}); Mohammed; βρασκι, who was Giovanni Angelo Braschi (1717-1799), and who reigned as Pope Pius VI from 1775 to 1799, dying in captivity because he declined to resign his temporal power to Napoleon; Bonaparte; and, under Ιον Παυνε, possibly Pope John XIV, who reigned in 983 and 984 during the absence of Boniface VII in Constantinople.

372 The Greek words and names are also occasionally misspelled so as to fit them to the number 666. They are Λατεινος (Latin), ἡ λατινη βασιλεια (the Latin kingdom), ἐκκλησια ἰταλικα (the Italian Church), εὐανθας (blooming), τειταν (Titan), ἀρνουμε (renounce), λαμπετις (the lustrous), ὁ νικητης (conqueror), κακος ὁδηγος (bad guide), ἀληθης βλαβερος (truthful harmful one), παλαι βασκανος (a slanderer of old), ἀμνος ἀδικος (unmanageable lamb), ἀντεμος (Antemos), γενσηρικος (Genseric), εὐινας (with stout fibers), Βενεδικτος (Benedict), Βονιβαζιος γ. παπα ξ. η. ε. ε. α. (Boniface III, pope 68, bishop of bishops I), οὐλπιος (baneful), διος εἰμι ἡ ἡρας (I, a god, am the), ἡ μισσα ἡ παπικη (the papal brief), λουθερανα (Lutheran), σαξονειος (Saxon), Βεζζα αντιθεος (Beza antigod), ἡ αλαζονεια βιου (the illusion of life), Μαομετις (Mahomet); Μαομετης β. (Mahomet II), θεος εἰμι ἐπι γαιης (I am lord over the earth), ἰαπετος (Iapetos, father of Atlas), παπεισκος (Papeiskos), διοκλασιανος (Diocletian), χεινα (Cheina = Cain? China?), βρασκι (Braschi, as explained in note 10), Ιον Παυνε (Paunian violet, but see note 10), κουποκς (cowpox), Βοννεπαρτη (Bonneparte), Ν. Βονηπαρτε (N. Boneparte), εὐπορια (facility), παραδοσις (surrender), το μεγαθηριον (the megathereum, the beast).

373 James Wapshare, whose Harmony of the Word of God in Spirit and in Truth appeared in 1849.

374 The literature relating to the Swastika is too extended to permit of any adequate summary in these notes.

375 Henry Edward Manning (1808-1892), at first an Anglican clergyman, he became a Roman Catholic priest in 1851, and became Cardinal in 1875. He succeeded Cardinal Wiseman as Archbishop of Westminster in 1865. He wrote a number of religious works.

376 John Bright (1811-1889), Quaker, cotton manufacturer, and statesman. He worked with Cobden for free trade, peace, and reform of the electorate.

377 “The fallacy of many questions.”

378 William Wilberforce (1759-1833), best known for his long fight for the abolition of the slave trade.

379 Richard Martin (1754-1834), high sheriff of County Galway and owner of a large estate in Connemara. Curiously enough, he was known both for his readiness in duelling and for his love for animals. He was known as “Humanity Martin,” and in 1822 secured the passage of an act “to prevent the cruel and improper treatment of cattle.” He was one of the founders (1824) of the Royal Society for the Prevention of Cruelty to Animals. He is usually considered the original of Godfrey O’Malley in Lever’s novel, Charles O’Malley.

380 See Vol. I, page 149, note 1 {323}, also text on same page.

[381] See Vol. I, page 44, note 9 {34}, also text, Vol. I, page 110.

382 “Penitential seat.”

383 “Well placed upon the cushion.”

384 See Vol. II, page 58, note 109.

385 “He has lost the right of being influenced by evidence.”

386 “Hung up.”

387 “A few things to the wise, nothing to the unlettered.”

388 The fallacy results from dividing both members of an equation by 0, x - 1 being the same as 1 - 1, and calling the quotients finite.

389 “If you order him to the sky he will go.”

390 Similia similibus curanter, “Like cures like,” the homeopathic motto.

391 “Without harm to the proprieties.”

392 “What are you doing? I am standing here.”

393 Lors feist l’Anglois tel signe. La main gausche toute ouverte il leva hault en l’aer, puis ferma au poing les quatres doigtz d’icelle et le poulce estendu assit sus la pinne du nez. Soubdain après leva la dextre toute ouverte, et toute ouverte la baissa, joignant la poulce au lieu que fermait le petit doigt de la gausche, et les quatre doigtz d’icelle mouvoit lentement en l’aer. Puis au rebours feit de la dextre ce qu’il avoit faict de la gausche, et de la gausche ce que avoit faict de la dextre.—A. De M.

394 Suaviter in modo, fortiter in re, “Gentle in manners, firm in action.”

395 See Vol. I, page 101, note 4 {174}.

396 See Vol. I, page 315, note 3 {686}.

397 Henry Fawcett (1833-1884) became totally blind in 1858, but in spite of this handicap he became professor of political economy at Cambridge and sat in parliament for a number of years. He championed the cause of reform and in particular he was prominent in the protection of the native interests of India. The establishing of the parcels post (1882) took place while he was postmaster general (1880-1884).

398 Of course the whole thing depends upon what definition of division is taken. We can multiply 2 ft. by 3 ft. if we define multiplication so as to allow it, or 2 ft. by 3 lb, getting foot-pounds, as is done in physics.

399 Richard Milward (1609-1680), for so the name is usually given, was rector of Great Braxted (Essex) and canon of Windsor. He was long the amanuensis of John Selden, and the Table Talk was published nine years after Milward’s death, from notes that he left. Some doubt has been cast upon the authenticity of the work owing to many of the opinions that it ascribes to Selden.

400 John Selden (1584-1654) was a jurist, legal antiquary, and Oriental scholar. He sat in the Long Parliament, and while an advocate of reform he was not an extremist. He was sent to the Tower for his support of the resolution against “tonnage and poundage,” in 1629. His History of Tythes (1618) was suppressed at the demand of the bishops. His De Diis Syriis (1617) is still esteemed a classic on Semitic mythology.

401 See Vol. I, page 42, note 4 {24}.

402 See Vol. II, page 249, note 398.

403 John Palmer (1742-1818) was a theatrical manager. In 1782 he set forth a plan for forwarding the mails by stage coaches instead of by postmen. Pitt adopted the plan in 1784. Palmer was made comptroller-general of the post office in 1786 and was dismissed six years later for arbitrarily suspending a deputy. He had been verbally promised 2½% on the increased revenue, but Pitt gave him only a pension of £3000. In 1813 he was awarded £50,000 in addition to his pension.

404 Dionysius Lardner (1793-1859), professor of natural philosophy in London University (now University College). His Cabinet Cyclopædia (1829-1849) contained 133 volumes. De Morgan wrote on probabilities, and Lardner on various branches of mathematics, and there were many other well-known contributors. Lardner is said to have made $200,000 on a lecture tour in America.

405 Thomas Fysche Palmer (1747-1802) joined the Unitarians in 1783, and in 1785 took a charge in Dundee. He was arrested for sedition because of an address that it was falsely alleged that he gave before a society known as the “Friends of Liberty.” As a matter of fact the address was given by an uneducated weaver, and Palmer was merely asked to revise it, declining to do even this. Nevertheless he was sentenced to Botany Bay (1794) for seven years. The trial aroused great indignation.

406 See Vol. I, page 80, note 5 {119}.

407 See Vol. II, page 244, note 394.

408 See Vol. I, page 352, note 1 {731}.

409 See Vol. I, page 332, note 4 {709}.

410 “The lawyers are brought into court; let them accuse each other.”

411 Samuel Rogers (1763-1855), the poet and art connoisseur. He declined the laureateship on the death of Wordsworth (1850). Byron, his pretended friend, wrote a lampoon (1818) ridiculing his cadaverous appearance.

412 Theodore Edward Hook (1788-1841), the well-known wit. He is satirized as Mr. Wagg in Vanity Fair. The John Bull was founded in 1820 and Hook was made editor.

413 “On pitying the heretic.”

414 A term of medieval logic. Barbara: All M is P, all S is M, hence all S is P. Celarent: No M is P, all S is M, hence no S is P.

[415] “Simply,” “According to which,” “It does not follow.”

416 “O sweet soul, what good shall I declare
That heretofore was thine, since such are thy remains!”

417 “Stupid fellow!”

418 Christopher Barker (c. 1529-1599), also called Barkar, was the Queen’s printer. He began to publish books in 1569, but did no actual printing until 1576. In 1575 the Geneva Bible was first printed in England, the work being done for Barker. He published 38 partial or complete editions of the Bible from 1575 to 1588, and 34 were published by his deputies (1588-1599).

419 James Franklin (1697-1735) was born in Boston, Mass., and was sent to London to learn the printer’s trade. He returned in 1717 and started a printing house. Benjamin, his brother, was apprenticed to him but ran away (1723). James published the New England Courant (1721-1727), and Benjamin is said to have begun his literary career by writing for it.

420 James Hodder was a writing master in Tokenhouse Yard, Lothbury, in 1661, and later kept a boarding school in Bromley-by-Bow. His famous arithmetic appeared at London in 1661 and went through many editions. It was the basis of Cocker’s work. (See Vol. I, page 42, note 4 {24}.) It was long thought to have been the first arithmetic published in America, and it was the first English one. There was, however, an arithmetic published much earlier than this, in Mexico, the Sumario compendioso... con algunas reglas tocantes al Aritmética, by “Juan Diaz Freyle,” in 1556.

421 Henry Mose, Hodder’s successor, kept a school in Sherborne Lane, London.

422 Rear Admiral Sir Francis Beaufort (1774-1857), F.R.S., was hydrographer to the Navy from 1829 to 1855. He prepared an atlas that was printed by the Society for the Diffusion of Useful Knowledge.

423 Antoine Sabatier (1742-1817), born at Castres, was known as the Abbé but was really nothing more than a “clerc tonsuré.” He lived at Court and was pensioned to write against the philosophers of the Voltaire group. He posed as the defender of morality, a commodity of which he seems to have possessed not the slightest trace.

424 Maffeo Barberini was pope, as Urban VIII, from 1623 to 1644. It was during his ambitious reign that Galileo was summoned to Rome to make his recantation (1633), the exact nature of which is still a matter of dispute.

425 This Baden Powell (1796-1860) was the Savilian professor of geometry (1827-1860) at Oxford.

426 “Memoirs of the famous bishop of Chiapa, by which it appears that he had butchered or burned or drowned ten million infidels in America in order to convert them. I believe that this bishop exaggerated; but if we should reduce these sacrifices to five million victims, this would still be admirable.”

427 Alfonso X (1221-1284), known as El Sabio (the Wise), was interested in astronomy and caused the Alphonsine Tables to be prepared. These table were used by astronomers for a long time. It is said that when the Ptolemaic system of the universe was explained to him he remarked that if he had been present at the Creation he could have shown how to arrange things in a much simpler fashion.

428 George Richards (c. 1767-1837), fellow of Oriel (1790-1796), Bampton lecturer (1800), Vicar of St. Martin’s-in-the-Fields, Westminster (1824), and a poet of no mean ability.

429 The “Aboriginal Britons,” an excellent poem, by Richards. (Note by Byron.)—A. De M.

430 John Watkins (d. after 1831), a teacher and miscellaneous writer.

431 Frederic Shoberl (1775-1853), a miscellaneous writer.

432 He wrote, besides the Aboriginal Britons, Songs of the Aboriginal Bards (1792), Modern France: a Poem (1793), Odin, a drama (1804), Emma, a drama on the model of the Greek theatre (1804), Poems (2 volumes, 1804), and a Monody on the Death of Lord Nelson (1806).

433 Henry Kirke White (1785-1806), published his first volume of poems at the age of 18. Southey and William Wilberforce became interested in him and procured for him a sizarship at St. John’s College, Cambridge. He at once showed great brilliancy, but he died of tuberculosis at the age of 21.

434 John Wolcot, known as Peter Pindar (1738-1819), was a London physician. He wrote numerous satirical poems. His Bozzy and Piozzi, or the British Biographers, appeared in 1786, and reached the 9th edition in 1788.

435 See Vol. I, page 235, note 8 {532}.

436 Richard Payne Knight (1750-1824) was a collector of bronzes, gems, and coins, many of his pieces being now in the British Museum. He sat in parliament for twenty-six years (1780-1806), but took no active part in legislation. He opposed the acquisition of the Elgin Marbles, holding them to be of little importance. His Analytical Inquiry into the Principles of Taste appeared in 1808.

437 Mario Nizzoli (1498-1566), a well-known student of Cicero, was for a time professor at the University of Parma. His Observationes in M. Tullium Ciceronem appeared at Pratalboino in 1535. It was revised by his nephew under the title Thesaurus Ciceronianus (Venice, 1570).

438 See Vol. I, page 314, note 4 {681}.

439 “Like the geometer, who bends all his powers
To measure the circle, and does not succeed,
Thinking what principle he needs.”

440 Francis Quarles (1592-1644), a religious poet. He wrote paraphrases of the Bible and numerous elegies. In the early days of the revolutionary struggle he sided with the Royalists. One of his most popular works was the Emblems (1635), with illustrations by William Marshall.

441 Regnault de Bécourt wrote La Création du monde, ou Système d’organisation primitive suivi de l’interprétation des principaux phénomènes et accidents que se sont opérés dans la nature depuis l’origine de univers jusqu’à nos jours (1816). This may be the work translated by Dalmas.

442 “Because it lacks a holy prophet.”

443 Angherà. See Vol. II, page 60, note 127.

444 Edmund Curll (1675-1747), a well-known bookseller, publisher, and pamphleteer. He was for a time at “The Peacock without Temple Bar,” and later at “The Dial and Bible against St. Dunstan’s Church.” He was fined repeatedly for publishing immoral works, and once stood in the pillory for it. He is ridiculed in the Dunciad for having been tossed in a blanket by the boys of Westminster School because of an oration that displeased them.

445 See Vol. II, page 109, note 206.

446 Encyclopædia.

447 Author of the Historia Naturalis (77 A.D.)

448 Author of the De Institutione Oratorio Libri XII (c. 91 A.D.)

449 His De Architectures Libri X was not merely a work on architecture and building, but on the education of the architect.

450 Cyclophoria.

451 William Caxton (c. 1422-c.1492), sometime Governor of the Company of Merchant Adventurers in Bruges (between 1449 and 1470). He learned the art of printing either at Bruges or Cologne, and between 1471 and 1477 set up a press at Westminster. Tradition says that the first book printed in England was his Game and Playe of Chesse (1474). The Myrrour of the Worlde and th’ymage of the same appeared in 1480. It contains a brief statement on arithmetic, the first mathematics to appear in print in England.

452 See Vol. I, page 45, note 6 {40}. De Morgan is wrong as to the date of the Margarita Philosophica. The first edition appeared at Freiburg in 1503.

453 Reisch was confessor to Maximilian I (1459-1519), King of the Romans (1486) and Emperor (1493-1519).

454 Joachim Sterck Ringelbergh (c. 1499-c. 1536), teacher of philosophy and mathematics in various cities of France and Germany. His Institutionum astronomicarum libri III appeared at Basel in 1528, his Cosmographia at Paris in 1529, and his Opera at Leyden in 1531.

455 Johannes Heinrich Alsted (1588-1638) was professor of philosophy and theology at his birthplace, Herborn, in Nassau, and later at Weissenberg. He published several works, including the Elementale mathematicum (1611), Systema physicae harmonicae (1612), Methodus admirandorum mathematicorum (1613), Encyclopædia septem tomis distincta (1630), and the work mentioned above.

456 Johann Jakob Hoffmann (1635-1706), professor of Greek and history at his birthplace, Basel. He also wrote the Epitome metrica historiæ universalis civilis et sacræ ab orbe condito (1686).

457 Ephraim Chambers (c. 1680-1740), a crotchety, penurious, but kind-hearted freethinker. His Cyclopædia, or an Universal Dictionary was translated into French and is said to have suggested the great Encyclopédie.

458 Encyclopédie, ou Dictionnaire raisonné des sciences, des arts et des métiers, par un société de gens de lettres. Mis en ordre et publié par M. Diderot, et quant à la partie mathématique, par M. d’Alembert. Paris, 1751-1780, 35 volumes.

459 “From the egg” (state).

460 See Vol. I, page 382, note 12 {785}.

461 See Vol. II, page 4, note 15.

462 “In morals nothing should serve man as a model but God; in the arts, nothing but nature.”

463 Encyclopédie Méthodique, ou par ordre de matières. Paris, 1782-1832, 166½ volumes.

464 See Vol. II, page 193, note 336.

465 Encyclopædia Metropolitana; or, Universal Dictionary of Knowledge. London, 1845, 29 volumes. A second edition came out in 1848-1858 in 40 volumes.

466 See Vol. I, page 137, note 8 {286}.

467 See Vol. I, page 80, note 5 {119}.

468 De Morgan should be alive to satirize some of the statements on the history of mathematics in the eleventh edition.

469 John Pringle Nichol (1804-1859), Regius professor of astronomy at Glasgow and a popular lecturer on the subject. He lectured in the United States in 1848-1849. His Views of the Architecture of the Heavens (1838) was a very popular work, and his Planetary System (1848, 1850) contains the first suggestion for the study of sun spots by the aid of photography.

470 See Vol. II, page 109, note 206.

471 George Long (1800-1879), a native of Poulton, in Lancashire, was called to the University of Virginia when he was only twenty-four years old as professor of ancient languages. He returned to England in 1828 to become professor of Greek at London University. From 1833 to 1849 he edited the twenty-nine volumes of the Penny Cyclopædia. He was an authority on Roman law.

472 A legal phrase, “Qui tam pro domina regina, quam pro se ipso sequitur,”—“Who sues as much on the Queen’s account as on his own.”

473 Arthur Cayley (1821-1895) was a fellow of Trinity College, Cambridge (1842-1846) and was afterwards a lawyer (1849-1863). During his fourteen years at the bar he published some two hundred mathematical papers. In 1863 he became professor of mathematics at Cambridge, and so remained until his death. His collected papers, nine hundred in number, were published by the Cambridge Press in 13 volumes (1889-1898). He contributed extensively to the theory of invariants and covariants. De Morgan’s reference to his coining of new names is justified, although his contemporary, Professor Sylvester, so far surpassed him in this respect as to have been dubbed “the mathematical Adam.”

474 See Vol. II, page 26, note 56.

475 See Vol. I, page 111, note 3 {207}.

476 See Vol. I, page 87, note 6 {135}.

477 Pierre Dupuy (1582-1651) was a friend and relative of De Thou. With the collaboration of his brother and Nicolas Rigault he published the 1620 and 1626 editions of De Thou’s History. He also wrote on law and history. His younger brother, Jacques (died in 1656), edited his works. The two had a valuable collection of books and manuscripts which they bequeathed to the Royal Library at Paris.

478 See Vol. I, page 51, note 3 {51}.

479 It was Cosmo de’ Medici (1590-1621) who was the patron of Galileo.

480 See Vol. I, page 40, note 4 {20}.

481 See Vol. I, page 106, note 4 {188}.

482 Sir Edward Sherburne (1618-1702), a scholar of considerable reputation. The reference by De Morgan is to The Sphere of Marcus Manilius, in the appendix to which is a Catalogue of Astronomers, ancient and modern.

483 George Parker, second Earl of Macclesfield (1697-1764). He erected an observatory at Shirburn Castle, Oxfordshire, in 1739, and fitted it out with the best equipment then available. He was President of the Royal Society in 1752.

484 See Vol. II, page 148, note 263.

485 See Vol. I, page 140, note 7 {296}.

486 See Vol. I, page 106, note 4 {188}.

487 Edward Bernard (1638-1696), although Savilian professor of astronomy at Oxford, was chiefly interested in archeology.

488 See Vol. I, page 107, note 1 {190}.

489 See Vol. I, page 107, note 1 {190}.

490 See Vol. I, page 135, note 3 {281}.

491 Philip Dormer Stanhope, fourth Earl of Chesterfield (1694-1773), well known for the letters written to his son which were published posthumously (1774).

492 Peter Daval (died in 1763), Vice-President of the Royal Society, and an astronomer of some ability.

493 See Vol. I, page 376, note 1 {766}.

494 William Oughtred (c. 1573-1660), a fellow of King’s College, Cambridge, and afterwards vicar of Aldbury, Surrey, wrote the best-known arithmetic and trigonometry of his time. His Arithmeticæ in Numero & Speciebus Institutio... quasi Clavis Mathematicæ est (1631) went through many editions and appeared in English as The Key to the Mathematicks new forged and filed in 1647.

495 See Vol. I, page 140, note 5 {294}.

496 Stephen Jordan Rigaud (1816-1859) was senior assistant master of Westminster School (1846) and head master of Queen Elizabeth’s School at Ipswich (1850). He was made Bishop of Antigua in 1858 and died of yellow fever the following year.

497 He also wrote a memoir of his father, privately printed at Oxford in 1883.

498 See Vol. I, page 69, note 3 {96}.

499 See Vol. I, page 106, note 4 {188}.

500 William Gascoigne was born at Middleton before 1612 and was killed in the battle of Marston Moor in 1644. He was an astronomer and invented the micrometer with movable threads (before 1639).

501 Seth Ward (1617-1689) was deprived of his fellowship at Cambridge for refusing to sign the covenant. He became professor of astronomy at Oxford (1649), Bishop of Exeter (1662), Bishop of Salisbury (1667), and Chancellor of the Garter (1671). He is best known for his solution of Kepler’s problem to approximate a planet’s orbit, which appeared in his Astronomia geometrica in 1656.

502 See Vol. I, page 110, note 2 {198}.

503 See Vol. I, page 100, note 2 {172}.

504 See Vol. I, page 107, note 1 {190}.

505 See Vol. I page 114, note 6 {220}.

506 See Vol. I, page 77, note 4 {118}.

507 See Vol. I, page 125, note 3 {253}.

508 See Vol. I, page 105, note 2 {186}.

509 Heinrich Oldenburgh (1626-1678) was consul in England for the City of Bremen, his birthplace, and afterwards became a private teacher in London. He became secretary of the Royal Society and contributed on physics and astronomy to the Philosophical Transactions.

510 Thomas Brancker, or Branker (1636-1676) wrote the Doctrinæ sphæricæ adumbratio et usus globorum artificialium (1662) and translated the algebra of Rhonius with the help of Pell. The latter work appeared under the title of An Introduction to Algebra (1668), and is noteworthy as having brought before English mathematicians the symbol ÷ for division. The symbol never had any standing on the Continent for this purpose, but thereafter became so popular in England that it is still used in all the English-speaking world.

511 See Vol. I, page 118, note 1 {230}.

512 Pierre Bertius (1565-1629) was a native of Flanders and was educated at London and Leyden. He became a professor at Leyden, and later held the chair of mathematics at the Collège de France. He wrote chiefly on geography.

513 See Vol. II, page 297, note 487.

514 Giovanni Alphonso Borelli (1608-1679) was professor of mathematics at Messina (1646-1656) and at Pisa (1656-1657), after which he taught in Rome at the Convent of St. Panteleon. He wrote several works on geometry, astronomy, and physics.

515 See Vol. I, page 172, note 2 {381}.

516 Ignace Gaston Pardies (c. 1636-1673), a Jesuit, professor of ancient languages and later of mathematics and physics at the Collège of Pau, and afterwards professor of rhetoric at the Collège Louis-le-Grand at Paris. He wrote on geometry, astronomy and physics.

517 Pierre Fermat was born in 1608 (or possibly in 1595) near Toulouse, and died in 1665. Although connected with the parliament of Toulouse, his significant work was in mathematics. He was one of the world’s geniuses in the theory of numbers, and was one of the founders of the theory of probabilities and of analytic geometry. After his death his son published his edition of Diophantus (1670) and his Varia opera mathematica (1679).

518 This may be Christopher Townley (1603-1674) the antiquary, or his nephew, Richard, who improved the micrometer already invented by Gascoigne.

519 Adrien Auzout a native of Rouen, who died at Rome in 1691. He invented a screw micrometer with movable threads (1666) and made many improvements in astronomical instruments.

520 See Vol. I, page 66, note 9 {86}.

521 See Vol. I, page 124, note 7 {248}.

522 John Machin (d. 1751) was professor of astronomy at Gresham College (1713-1751) and secretary of the Royal Society. He translated Newton’s Principia into English. His computation of π to 100 places is given in William Jones’s Synopsis palmariorum matheseos (1706).

523 Pierre Rémond de Montmort (1678-1719) was canon of Notre Dame until his marriage. He was a gentleman of leisure and devoted himself to the study of mathematics, especially of probabilities.

524 Roger Cotes (1682-1716), first Plumian professor of astronomy and physics at Cambridge, and editor of the second edition of Newton’s Principia. His posthumous Harmonia Mensurarum (1722) contains “Cotes’s Theorem” on the binomial equation. Newton said of him, “If Mr. Cotes had lived we had known something.”

525 See Vol. I, page 135, note 3 {281}.

526 See Vol. I, page 377, note 4 {769}.

527 Charles Réné Reyneau (1656-1728) was professor of mathematics at Angers. His Analyse démontrée, ou Manière de resoudre les problèmes de mathématiques (1708) was a successful attempt to popularize the theories of men like Descartes, Newton, Leibnitz, and the Bernoullis.

528 Brook Taylor (1685-1731), secretary of the Royal Society, and student of mathematics and physics. His Methodus incrementorum directa et inversa (1715) was the first treatise on the calculus of finite differences. It contained the well-known theorem that bears his name.

529 Pierre Louis Moreau de Maupertuis (1698-1759) was sent with Clairaut (1735) to measure an arc of a meridian in Lapland. He was head of the physics department in the Berlin Academy from 1745 until 1753. He wrote Sur la figure de la terre (1738) and on geography and astronomy.

530 Pierre Bouguer (1698-1758) was professor of hydrography at Paris, and was one of those sent by the Academy of Sciences to measure an arc of a meridian in Peru (1735). The object of this and the work of Maupertuis was to determine the shape of the earth and see if Newton’s theory was supported.

531 Charles Marie de la Condamine (1701-1774) was a member of the Paris Academy of Sciences and was sent with Bouguer to Peru, for the purpose mentioned in the preceding note. He wrote on the figure of the earth, but was not a scientist of high rank.

532 See Vol. I, page 136, note 5 {283}.

533 See Vol. II, page 296, note 483.

534 Thomas Baker (c. 1625-1689) gave a geometric solution of the biquadratic in his Geometrical Key, or Gate of Equations unlocked (1684).

535 See Vol. I, page 160, note 5 {350}.

536 See Vol. I, page 87, note 4 {133}.

537 See Vol. I, page 132, note 2 {272}.

538 See Vol. I, page 118, second note 1 {231}.

539 The name of Newton is so well known that no note seems necessary. He was born at Woolsthorpe, Lincolnshire, in 1642, and died at Kensington in 1727.

540 John Keill (1671-1721), professor of astronomy at Oxford from 1710, is said to have been the first to teach the Newtonian physics by direct experiment, the apparatus being invented by him for the purpose. He wrote on astronomy and physics. His Epistola de legibus virium centripetarum, in the Philosophical Transactions for 1708, accused Leibnitz of having obtained his ideas of the calculus from Newton, thus starting the priority controversy.

541 Thomas Digges (d. in 1595) wrote An Arithmeticall Militare Treatise, named Stratioticos (1579), and completed A geometrical practise, named Pantometria (1571) that had been begun by his father, Leonard Digges.

542 John Dee (1527-1608), the most famous astrologer of his day, and something of a mathematician, wrote a preface to Billingsley’s translation of Euclid into English (1570).

543 See Vol. I, page 76, note 3 {112}.

544 Thomas Harriot (1560-1621) was tutor in mathematics to Sir Walter Raleigh, who sent him to survey Virginia (1585). He was one of the best English algebraists of his time, but his Artis Analyticæ Praxis ad Aequationes Algebraicas resolvendas (1631) did not appear until ten years after his death.

545 Thomas Lydiat (1572-1626), rector of Alkerton, devoted his life chiefly to the study of chronology, writing upon the subject and taking issue with Scaliger (1601).

546 See Vol. I, page 69, note 3 {96}.

547 Walter Warner edited Harriot’s Artis Analyticae Praxis (1631). Tarporley is not known in mathematics.

548 See Vol. I, page 105, note 2 {186}.

549 See Vol. I, page 115, note 3 {224}.

550 See Vol. II, page 300, note 509.

551 See Vol. I, page 107, note 1 {190}.

552 Sir Samuel Morland (1625-1695) was a diplomat and inventor. For some years he was assistant to John Pell, then ambassador to Switzerland. He wrote on arithmetical instruments invented by him (1673), on hydrostatics (1697) and on church history (1658).

553 See Vol. I, page 153, note 4 {337}.

554 See Vol. I, page 85, note 2 {129}.

555 See Vol. I, page 43, note 8 {33}.

556 See Vol. I, page 43, note 7 {32}.

557 See Vol. I, page 382, note 13 {786}. The history of the subject may be followed in Braunmühl’s Geschichte der Trigonometrie.

558 See Vol. I, page 377, note 3 {768}.

559 See Vol. I, page 108, note 2 {192}.

560 Michael Dary wrote Dary’s Miscellanies (1669), Gauging epitomised (1669), and The general Doctrine of Equation (1664).

561 John Newton (1622-1678), canon of Hereford (1673), educational reformer, and writer on elementary mathematics and astronomy.

562 See Vol. I, page 87, note 4 {133}.

563 “The average of the two equal altitudes of the sun before and after dinner.”

564 See Vol. I, page 42, note 4 {24}.

565 London, 1678. It went though many editions.

566 “This I who once...”

567 Arthur Murphy (1727-1805) worked in a banking house until 1754. He then went on the stage and met with some success at Covent Garden. His first comedy, The Apprentice (1756) was so successful that he left the stage and took to play writing. His translation of Tacitus appeared in 1793, in four volumes.

568 Edmund Wingate (1596-1656) went to Paris in 1624 as tutor to Princess Henrietta Maria and remained there several years. He wrote L’usage de la règle de proportion (Paris, 1624, with an English translation in 1626), Arithmétique Logarithmétique (Paris, 1626, with an English translation in 1635), and Of Natural and Artificial Arithmetick (London, 1630, revised in 1650-1652), part I of which was one of the most popular textbooks ever produced in England.

569 John Lambert (1619-1694) was Major-General during the Revolution and helped to draw up the request for Cromwell to assume the protectorate. He was imprisoned in the Tower by the Rump Parliament. He was confined in Guernsey until the clandestine marriage of his daughter Mary to Charles Hatton, son of the governor, after which he was removed (1667) to St. Nicholas in Plymouth Sound.

570 Samuel Foster (d. in 1652) was made professor of astronomy at Gresham College in March, 1636, but resigned in November of that year, being succeeded by Mungo Murray. Murray vacated his chair by marriage in 1641 and Foster succeeded him. He wrote on dialling and made a number of improvements in geometric instruments.

571 “Twice of the word a minister,” that is, twice a minister of the Gospel.

572 This is The Lives of the Professors of Gresham College to which is prefixed the Life of the Founder, Sir Thomas Gresham, London, 1740. It was written by John Ward (c. 1679-1758), professor of rhetoric (1720) at Gresham College and vice-president (1752) of the Royal Society.

573 Charles Montagu (1661-1715), first Earl of Halifax, was Lord of the Treasury in 1692, and was created Baron Halifax in 1700 and Viscount Sunbury and Earl of Halifax in 1714. He introduced the bill establishing the Bank of England, the bill becoming a law in 1694. He had troubles of his own, without considering Newton, for he was impeached in 1701, and was the subject of a damaging resolution of censure as auditor of the exchequer in 1703. Although nothing came of either of these attacks, he was out of office during much of Queen Anne’s reign.

574 See Vol. II, page 302, note 547.

575 See Vol. I, page 105, note 2 {186}.

576 James Dodson (d. 1757) was master of the Royal Mathematical School, Christ’s Hospital. He was De Morgan’s great-grandfather. The Anti-Logarithmic Canon was published in 1742.

577 See Vol. I, page 106, note 4 {188}.

578 See Vol. I, page 110, note 2 {198}.

579 Richard Busby, (1606-1695), master of Westminster School (1640) had among his pupils Dryden and Locke.

580 See Vol. I, page 107, note 1 {190}.

581 Herbert Thorndike (1598-1672), fellow of Trinity College, Cambridge (1620-1646), and Prebend of Westminster (1661), was a well-known theological writer of the time.

582 See Vol. I, page 140, note 5 {294}.

583 See Vol. I, page 108, note 2 {192}.

584 “Labor performed returns in a circle.”

585 See Vol. II, page 208.

586 “Whatever objections one may make to the above arguments, one always falls into an absurdity.”

587 See Vol. II. page 11, note 29. The Circle Squared; and the solution of the problem adapted to explain the difference between square and superficial measurement appeared at Brighton in 1865.

588 “And beyond that nothing.”

589 Gillott (1759-1873) was the pioneer maker of steel pens by machinery, reducing the price from 1s. each to 4d. a gross. He was a great collector of paintings and old violins.

590 William Edward Walker wrote five works on circle squaring (1853, 1854, 1857, 1862, 1864), mostly and perhaps all published at Birmingham.

591 Solomon M. Drach wrote An easy Rule for formulizing all Epicyclical Curves (London, 1849), On the Circle area and Heptagon-chord (London, 1864), An easy general Rule for filling up all Magic Squares (London, 1873), and Hebrew Almanack-Signs (London, 1877), besides numerous articles in journals.

592 See Vol. I, page 168, note 3 {371}.

593 See Vol. I, page 254, note 2 {580}.

594 See Vol. I, page 98, note 6 {163}.

595 Robert Fludd or Flud (1574-1637) was a physician with a large London practice. He denied the diurnal rotation of the earth, and was attacked by Kepler and Mersenne, and accused of magic by Gassendi. His Apologia Compendiania, Fraternitatem de Rosea Cruce suspicionis... maculis aspersam, veritatis quasi Fluctibus abluens (Leyden, 1616) is one of a large number of works of the mystic type.

596 Consult To the Christianity of the Age. Notes... comprising an elucidation of the scope and contents of the writings... of Dionysius Andreas Freher (1854).

597 Sir William Robert Grove (1811-1896), although called to the bar (1835) and to the bench (1853), is best known for his work as a physicist. He was professor of experimental philosophy (1840-1847) at the London Institution, and invented a battery (1839) known by his name. His Correlation of Physical Forces (1846) went through six editions and was translated into French.

598 Johann Tauler (c. 1300-1361), a Dominican monk of Strassburg, a mystic, closely in touch with the Gottesfreunde of Basel. His Sermons first appeared in print at Leipsic in 1498.

599 Paracelsus (c. 1490-1541). His real name was Theophrastes Bombast von Hohenheim, and he took the name by which he is generally known because he held himself superior to Celsus. He was a famous physician and pharmacist, but was also a mystic and neo-Platonist. He lectured in German on medicine at Basel, but lost his position through the opposition of the orthodox physicians and apothecaries.

600 See Vol. I, page 256, note 2 {588}.

601 Philip Schwarzerd (1497-1560) was professor of Greek at Wittenberg. He helped Luther with his translation of the Bible.

602 Johann Reuchlin (1455-1522), the first great German humanist, was very influential in establishing the study of Greek and Hebrew in Germany. His lectures were mostly delivered privately in Heidelberg and Stuttgart. Unlike Melanchthon, he remained in the Catholic Church.

603 Joseph Chitty (1776-1841) published his Precedents of Pleading in 1808 and his Reports of Cases on Practice and Pleading in 1820-23 (2 volumes).

604 See Vol. I, page 44, note 1 {35}.

605 See Vol. I, page 44, note 4 {38}.

606 Jean Pèlerin, also known as Viator, who wrote on perspective. His work appeared in 1505, with editions in 1509 and 1521.

607 Henry Stephens. See Vol. I, page 44, note 3 {37}.

608 The well-known grammarian (1745-1826). He was born at Swatara, in Pennsylvania, and practised law in New York until 1784, after which he resided in England. His grammar (1795) went through 50 editions, and the abridgment (1818) through 120 editions. Murray’s friend Dalton, the chemist, said that “of all the contrivances invented by human ingenuity for puzzling the brains of the young, Lindley Murray’s grammar was the worst.”

609 Robert Recorde (c. 1510-1558) read and probably taught mathematics and medicine at Cambridge up to 1545. After that he taught mathematics at Oxford and practised medicine in London. His Grounde of Artes, published about 1540, was the first arithmetic published in English that had any influence. It went through many editions. The Castle of Knowledge appeared in 1551. It was a textbook on astronomy and the first to set forth the Copernican theory in England. Like Recorde’s other works it was written on the catechism plan. His Whetstone of Witte... containying thextraction of Rootes: The Cosike practise, with the rule of Equation: and the woorkes of Surde Nombres appeared in 1557, and it is in this work that the modern sign of equality first appears in print. The word “Cosike” is an adjective that was used for a long time in Germany as equivalent to algebraic, being derived from the Italian cosa, which stood for the unknown quantity.

610 Robert Cecil (c. 1563-1612), first Earl of Salisbury, Secretary of State under Elizabeth (1596-1603) and under James I (1603-1612).

611 In America the German pronunciation is at present universal among mathematicians, as in the case of most other German names. This is due, no doubt, to the great influence that Germany has had on American education in the last fifty years.

612 The latest transliteration is substantially K’ung-fu-tzǔ.

613 The tendency seems to be, however, to adopt the forms used of individuals or places as rapidly as the mass of people comes to be prepared for it. Thus the spelling Leipzig, instead of Leipsic, is coming to be very common in America.

614 Sir Edward Coke (1552-1634), the celebrated jurist.

615 Dethlef Cluvier or Clüver (d. 1708 at Hamburg) was a nephew, not a grandson, of Philippe Cluvier, or Philipp Clüver (1580-c. 1623). Dethlef traveled in France and Italy and then taught mathematics in London. He wrote on astronomy and philosophy and also published in the Acta Eruditorum (1686) his Schediasma geometricum de nova infinitorum scientia. Quadratura circuli infinitis modis demonstrata, and his Monitum ad geometras (1687). Philippe was geographer of the Academy of Leyden. His Introductionis in universam geographiam tam veterem quam novam libri sex appeared at Leyden in 1624, about the time of his death.

616 See Vol. I, page 124, note 7 {248}.

617 Bernard Nieuwentijt (1654-1718), a physician and burgomaster at Purmerend. His Considerationes circa Analyseos ad quantitates infinite parvas applicatæ Principia et Calculi Differentialis usum (Amsterdam, 1694) was attacked by Leibnitz. He replied in his Considerationes secundæ (1694), and also wrote the Analysis Infinitorum, seu Curvilineorum Proprietates ex Polygonorum Natura deductæ (1695). His most famous work was on the existence of God, Het Regt Gebruik der Werelt Beschouwingen (1718).

618 “From a given line to construct” etc.

619 “Pirates do not fight one another.”

620 Claude Mallemens (Mallement) de Messanges (1653-1723) was professor of philosophy at the Collège du Plessis, in Paris, for 34 years. The work to which De Morgan refers is probably the Fameux Problème de la quadrature du cercle, résolu géometriquement par le cercle et a ligne droite that appeared in 1683.

621 On Tycho Brahe see Vol. I, page 76, note 3 {112}.

622 Wilhelm Frederik von Zytphen also published the Tidens Ström, a chronological table, in 1840. The work to which De Morgan refers, the Solens Bevægelse i Verdensrummet, appeared first in 1861. De Morgan seems to have missed his Nogl Ord om Cirkelens Quadratur which appeared in 1865, at Copenhagen.

623 James Joseph Sylvester (1814-1897), professor of natural philosophy at University College, London (1837-1841), professor of mathematics at the University of Virginia (1841-1845), actuary in London (1845-1855), professor of mathematics at Woolwich (1877-1884) and at Johns Hopkins University, Baltimore (1877-1884), and Savilian professor of geometry at Oxford (1884-1894).

624 See Vol. I, page 76, note 3 {112}.

625 See Vol. II, page 205, note 349.

626 See Vol. I, page 76, note 3 {112}.

627 See Vol. I, page 46, note 1 {42}.

628 See Vol. II, page 183, note 318.

629 See Vol. I, page 321, note 2 {691}.

630 James Mill, born 1773, died 1836.

631 See Vol. II, page 3, note 11.

632 See Vol. II, page 3, note 13.

633 See Vol. II, page 3, note 14.

634 This anecdote is printed at page 4 (Vol. II); but as it is used in illustration here, and is given more in detail, I have not omitted it.—S.E. De M.

635 See Vol. II, page 4, note 15.

636 See Vol. I, page 382, note 13 {786}.

637 “Monsieur, (a + bⁿ)/n = x, whence God exists; answer that!”

638 “Monsieur, you know very well that your argument requires the development of x according to integral powers of n.”

639 See Vol. I, page 153, note 4 {337}.

640 Thomas Love Peacock (1785-1866) an English novelist and poet.

641 Perhaps Dr. Samuel Warren (1807-1877), the author of Ten Thousand a Year (serially in Blackwood’s in 1839; London, 1841).

642 See Vol. I, page 255, note 6 {584}.

643 “From many, one; much in little; Ultima Thule (the most remote region); without which not.”

644 Spurius Mælius (fl. 440 B. C.), who distributed corn freely among the poor in the famine of 440 B. C. and was assassinated by the patricians.

645 Spurius Cassius Viscellinus, Roman consul in 502, 493, and 486 B. C. Put to death in 485.

646 “O what a fine bearing, he said, that has no brain.”

647 Sir William Rowan Hamilton. See Vol. I, page 332, note 4 {709}.

648 William Allen Whitworth, the author of the well-known Choice and Chance (Cambridge, 1867), and other works.

649 James Maurice Wilson, whose Elementary Geometry appeared in 1868 and went through several editions.

650 See Vol. II, page 183, note 315.

651 “Force of inertia conquered,” and “Victory in the whole heavens.”

652 “With all his might.”

653 George Berkeley (1685-1753), Bishop of Cloyne, the idealistic philosopher and author of the Principles of Human Knowledge (1710), The Analyst, or a Discourse addressed to an Infidel Mathematician (1734), and A Defense of Freethinking in Mathematics (1735). He asserted that space involves the idea of movement without the sensation of resistance. Space sensation less than the “minima sensibilia” is, therefore, impossible. From this he argues that infinitesimals are impossible concepts.

654 See Vol. I, page 85, note 2 {129}.

655 See Vol. I, page 81, note 6 {120}.

656 Edwin Dunkin revised Lardner’s Handbook of Astronomy (1869) and Milner’s The Heavens and the Earth (1873) and wrote The Midnight Sky (1869).

657 Michael Faraday (1791-1867) the celebrated physicist and chemist. He was an assistant to Sir Humphrey Davy (1813) and became professor of chemistry at the Royal Institution, London, in 1827.

658 “If you teach a fool he shows no joyous countenance; he cordially hates you; he wishes you buried.”

659 “Every man is an animal, Sortes is a man, therefore Sortes is an animal.”

660 “May some choice patron bless each grey goose quill;
May every Bavius have his Bufo still.”—Pope, Prologue to the Satires.

Bavius has become proverbial as a bad poet from the lines in Vergil’s Eclogues (III, 90):

“Qui Bavium non odit, amet tua carmina, Maevi,
Atque idem jungat vulpes, et mulgeat hircos.”

“He who does not hate Bavius shall love thy verses, O Maevius; and the same shall yoke foxes and shall milk he-goats.”

Bavius and Maevius were the worst of Latin poets, condemned by Horace as well as Vergil.

661 See Vol. II, page 158, note 279.

662 “Honest,” “useful,” “handsome,” “sweet.”

663 “Let not the fourth man attempt to speak.”

664 “In those old times,—ah
’Twas just like this, ah!”

665 See Vol. I, page 382, note 12 {785}.

666 These remarks were never written.—S. E. De M.

667 “Fleas, flies, and friars, are masters who sadly the people abuse,
And thistles and briars are sure growing grains to abuse.
O Christ, who hatest strife and slayst all things in peace,
Destroy where’er are rife, briars, friars, flies and fleas.
Fleas, flies, and friars foul fall them these fifteen years
For none that there is loveth fleas, flies, nor freres.”

668 “It is my plan to restore to an unskilled race the worthy arts of a better life.”

669 The first sentences of the first oration of Cicero against Catiline: “Quo usque tandem abutere, Catilina, patientia nostra?” (How long, O Catiline, will you abuse our patience?) “Quamdiu etiam furor iste tuus nos eludet?” (How long will this your madness baffle us?) “Nihilne te nocturnum praesidium Palati,... nihil horum ora voltusque moverunt?” (Does the night watch of the Palatium,... do the faces and expressions of all these men fail to move you?) “In te conferri...” (This plague should have been inflicted upon you long ago, which you have plotted against us so long.)

670 “Beware of the things that are marked.”

671 “Farewell, ye teachers without learning! See to it that at our next meeting we may find you strong in body and sound in mind.”

672 See Vol. I, page 336, note 8 {713}.

673 See Vol. I, page 229, note 2 {515}.

674 This proof, although capable of improvement, is left as in the original. Those who may be interested in the mathematics of the question, may consult F. Enriques, Fragen der Elementargeometrie (German by Fleischer), Leipsic, 1907, Part II, p. 267; F. Rudio, Archimedes, Huygens, Lambert, Legendre. Vier Abhandlungen über die Kreismessung, Leipsic, 1892; F. Klein, Famous Problems of Elementary Geometry (English by Beman and Smith), Boston, 1895; J. W. A. Young, Monographs on Modern Mathematics, New York, 1911, Chap. IX (by the editor of the present edition of De Morgan.)

675 See Vol. I, page 69, note 2 {95}.

676 See Vol. I, page 137, note 8 {286}.

677 Joseph Allen Galbraith who, with Samuel Haughton, wrote the Galbraith and Haughton’s Scientific Manuals. (Euclid, 1856; Algebra, 1860; Trigonometry, 1854; Optics, 1854, and others.)

678 This note on Carlyle (1795-1881) is interesting. The translation of Legendre appeared in the same year (1824) as his translation of Goethe’s Wilhelm Meister.

679 Michael Stifel (1487-1567), also known as Stiefel, Styfel, and Stifelius, was an Augustine monk but became a convert to Lutheranism. He was professor of mathematics at Jena (1559-1567). His edition of the Coss appeared at Königsberg in 1553, the first edition having been published in 1525. The + and - signs first appeared in print in Widman’s arithmetic of 1489, but for purposes of algebra this book was one of the first to make them known.

680 Christoff Rudolff was born about 1500 and died between 1540 and 1552. Die Coss appeared in 1525 and his arithmetic in 1526.

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