Budget of Paradoxes/B

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[ 37 ]


Questiones Morales, folio, 1489 [Paris]. By T. Buridan.

This is the title from the Hartwell Catalogue of Law Books. I suppose it is what is elsewhere called the "Commentary on the Ethics of Aristotle," printed in 1489.[13] Buridan[14] (died about 1358) is the creator of the famous ass which, as Burdin's[15] ass, was current in Burgundy, perhaps is, as a vulgar proverb. Spinoza[16] says it was a jenny ass, and that a man would not have been so foolish; but whether the compliment is paid to human or to masculine character does not appear—perhaps to both in one. The story told about the famous paradox is very curious. The Queen of France, Joanna or Jeanne, was in the habit of sewing her lovers up in sacks, and throwing them into the Seine; not for blabbing, but that they might not blab—certainly the safer plan. Buridan was exempted, and, in gratitude, invented the sophism. What it has to do with the matter [ 38 ] has never been explained. Assuredly qui facit per alium facit per se will convict Buridan of prating. The argument is as follows, and is seldom told in full. Buridan was for free-will—that is, will which determines conduct, let motives be ever so evenly balanced. An ass is equally pressed by hunger and by thirst; a bundle of hay is on one side, a pail of water on the other. Surely, you will say, he will not be ass enough to die for want of food or drink; he will then make a choice—that is, will choose between alternatives of equal force. The problem became famous in the schools; some allowed the poor donkey to die of indecision; some denied the possibility of the balance, which was no answer at all.



The following question is more difficult, and involves free-will to all who answer—"Which you please." If the northern hemisphere were land, and all the southern hemisphere water, ought we to call the northern hemisphere an island, or the southern hemisphere a lake? Both the questions would be good exercises for paradoxers who must be kept employed, like Michael Scott's[17] devils. The wizard [ 39 ] knew nothing about squaring the circle, etc., so he set them to make ropes out of sea sand, which puzzled them. Stupid devils; much of our glass is sea sand, and it makes beautiful thread. Had Michael set them to square the circle or to find a perpetual motion, he would have done his work much better. But all this is conjecture: who knows that I have not hit on the very plan he adopted? Perhaps the whole race of paradoxers on hopeless subjects are Michael's subordinates, condemned to transmigration after transmigration, until their task is done.

The above was not a bad guess. A little after the time when the famous Pascal papers[18] were produced, I came into possession of a correspondence which, but for these papers, I should have held too incredible to be put before the world. But when one sheep leaps the ditch, another will follow: so I gave the following account in the Athenæum of October 5, 1867:

"The recorded story is that Michael Scott, being bound by contract to produce perpetual employment for a number of young demons, was worried out of his life in inventing jobs for them, until at last he set them to make ropes out of sea sand, which they never could do. We have obtained a very curious correspondence between the wizard Michael and his demon-slaves; but we do not feel at liberty to say how it came into our hands. We much regret that we did not receive it in time for the British Association. It appears that the story, true as far as it goes, was never finished. The demons easily conquered the rope difficulty, by the simple process of making the sand into glass, and spinning the glass into thread, which they twisted. Michael, thoroughly disconcerted, hit upon the plan of setting some to [ 40 ] square the circle, others to find the perpetual motion, etc. He commanded each of them to transmigrate from one human body into another, until their tasks were done. This explains the whole succession of cyclometers, and all the heroes of the Budget. Some of this correspondence is very recent; it is much blotted, and we are not quite sure of its meaning: it is full of figurative allusions to driving something illegible down a steep into the sea. It looks like a humble petition to be allowed some diversion in the intervals of transmigration; and the answer is—

Rumpat et serpens iter institutum,[19]

—a line of Horace, which the demons interpret as a direction to come athwart the proceedings of the Institute by a sly trick. Until we saw this, we were suspicious of M. Libri,[20] the unvarying blunders of the correspondence look like knowledge. To be always out of the road requires a map: genuine ignorance occasionally lapses into truth. We thought it possible M. Libri might have played the trick to show how easily the French are deceived; but with our present information, our minds are at rest on the subject. We see M. Chasles does not like to avow the real source of information: he will not confess himself a spiritualist."



Philo of Gadara[21] is asserted by Montucla,[22] on the [ 41 ] authority of Eutocius,[23] the commentator on Archimedes, to have squared the circle within the ten-thousandth part of a unit, that is, to four places of decimals. A modern classical dictionary represents it as done by Philo to ten thousand places of decimals. Lacroix comments on Montucla to the effect that myriad (in Greek ten thousand) is here used as we use it, vaguely, for an immense number. On looking into Eutocius, I find that not one definite word is said about the extent to which Philo carried the matter. I give a translation of the passage:

"We ought to know that Apollonius Pergæus, in his Ocytocium [this work is lost], demonstrated the same by other numbers, and came nearer, which seems more accurate, but has nothing to do with Archimedes; for, as before said, he aimed only at going near enough for the wants of life. Neither is Porus of Nicæa fair when he takes Archimedes to task for not giving a line accurately equal to the circumference. He says in his Cerii that his teacher, Philo of Gadara, had given a more accurate approximation (εἰς ἀκριβεστέρους ἀριθμοὺς ἀγάγειν) than that of Archimedes, or than 7 to 22. But all these [the rest as well as Philo] miss the intention. They multiply and divide by tens of thousands, which no one can easily do, unless he be versed in the logistics [fractional computation] of Magnus [now unknown]."

Montucla, or his source, ought not to have made this mistake. He had been at the Greek to correct Philo Gadetanus, as he had often been called, and he had brought away [ 42 ] and quoted ἀπὸ Γάδαρων. Had he read two sentences further, he would have found the mistake.

We here detect a person quite unnoticed hitherto by the moderns, Magnus the arithmetician. The phrase is ironical; it is as if we should say, "To do this a man must be deep in Cocker."[24] Accordingly, Magnus, Baveme,[25] and Cocker, are three personifications of arithmetic; and there may be more.



Aristotle, treating of the category of relation, denies that the quadrature has been found, but appears to assume that it can be done. Boethius,[26] in his comment on the passage, says that it has been done since Aristotle, but that the demonstration is too long for him to give. Those who have no notion of the quadrature question may look at the English Cyclopædia, art. "Quadrature of the Circle."

Tetragonismus. Id est circuli quadratura per Campanum, Archimedem Syracusanum, atque Boetium mathematicæ perspicacissimos adinventa.—At the end, Impressum Venetiis per Ioan. Bapti. Sessa. Anno ab incarnatione Domini, 1503. Die 28 Augusti.

[ 43 ]

This book has never been noticed in the history of the subject, and I cannot find any mention of it. The quadrature of Campanus[27] takes the ratio of Archimedes,[28] 7 to 22 to be absolutely correct; the account given of Archimedes is not a translation of his book; and that of Boetius has more than is in Boethius. This book must stand, with the next, as the earliest in print on the subject, until further showing: Murhard[29] and Kastner[30] have nothing so early. It is edited by Lucas Gauricus,[31] who has given a short preface. Luca Gaurico, Bishop of Civita Ducale, an astrologer of astrologers, published this work at about thirty years of age, and lived to eighty-two. His works are collected in folios, but I do not know whether they contain this production. The poor fellow could never tell his own fortune, because his father neglected to note the hour and minute of his birth. But if there had been anything in astrology, he could have worked back, as Adams[32] and Leverrier[33] did when they caught [ 44 ] Neptune: at sixty he could have examined every minute of his day of birth, by the events of his life, and so would have found the right minute. He could then have gone on, by rules of prophecy. Gauricus was the mathematical teacher of Joseph Scaliger,[34] who did him no credit, as we shall see.



In hoc opere contenta Epitome.... Liber de quadratura Circuli.... Paris, 1503, folio.

The quadrator is Charles Bovillus,[35] who adopted the views of Cardinal Cusa,[36] presently mentioned. Montucla is hard on his compatriot, who, he says, was only saved from the laughter of geometers by his obscurity. Persons must guard against most historians of mathematics in one point: they frequently attribute to his own age the obscurity which a writer has in their own time. This tract was printed by Henry Stephens,[37] at the instigation of Faber Stapulensis,[38] [ 45 ] and is recorded by Dechales,[39] etc. It was also introduced into the Margarita Philosophica of 1815,[40] in the same appendix with the new perspective from Viator. This is not extreme obscurity, by any means. The quadrature deserved it; but that is another point.

It is stated by Montucla that Bovillus makes . But Montucla cites a work of 1507, Introductorium Geometricum, which I have never seen.[41] He finds in it an account which Bovillus gives of the quadrature of the peasant laborer, and describes it as agreeing with his own. But the description makes which it thus appears Bovillus could not distinguish from . It seems also that this about which we shall see so much in the sequel, takes its rise in the thoughtful head of a poor laborer. It does him great honor, being so near the truth, and he having no means of instruction. In our day, when an ignorant person chooses to bring his fancy forward in opposition to demonstration which he will not study, he is deservedly laughed at.

[ 46 ]



Mr. James Smith,[42] of Liverpool—hereinafter notorified—attributes the first announcement of to M. Joseph Lacomme, a French well-sinker, of whom he gives the following account:

"In the year 1836, at which time Lacomme could neither read nor write, he had constructed a circular reservoir and wished to know the quantity of stone that would be required to pave the bottom, and for this purpose called on a professor of mathematics. On putting his question and giving the diameter, he was surprised at getting the following answer from the Professor: Qu'il lui était impossible de le lui dire au juste, attendu que personne n'avait encore pu trouver d'une manière exacte le rapport de la circonférence au diametre.[43] From this he was led to attempt the solution of the problem. His first process was purely mechanical, and he was so far convinced he had made the discovery that he took to educating himself, and became an expert arithmetician, and then found that arithmetical results agreed with his mechanical experiments. He appears to have eked out a bare existence for many years by teaching arithmetic, all the time struggling to get a hearing from some of the learned societies, but without success. In the year 1855 he found his way to Paris, where, as if by accident, he made the acquaintance of a young gentleman, son of M. Winter, a commissioner of police, and taught him his peculiar methods of calculation. The young man was so enchanted that he strongly recommended Lacomme to his father, and [ 47 ] subsequently through M. Winter he obtained an introduction to the President of the Society of Arts and Sciences of Paris. A committee of the society was appointed to examine and report upon his discovery, and the society at its séance of March 17, 1856, awarded a silver medal of the first class to M. Joseph Lacomme for his discovery of the true ratio of diameter to circumference in a circle. He subsequently received three other medals from other societies. While writing this I have his likeness before me, with his medals on his breast, which stands as a frontispiece to a short biography of this extraordinary man, for which I am indebted to the gentleman who did me the honor to publish a French translation of the pamphlet I distributed at the meeting of the British Association for the Advancement of Science, at Oxford, in 1860."—Correspondent, May 3, 1866.

My inquiries show that the story of the medals is not incredible. There are at Paris little private societies which have not so much claim to be exponents of scientific opinion as our own Mechanics' Institutes. Some of them were intended to give a false lustre: as the "Institut Historique," the members of which are "Membre de l'Institut Historique." That M. Lacomme should have got four medals from societies of this class is very possible: that he should have received one from any society at Paris which has the least claim to give one is as yet simply incredible.



Nicolai de Cusa Opera Omnia. Venice, 1514. 3 vols. folio.

The real title is "Hæc accurata recognitio trium voluminum operum clariss. P. Nicolai Cusæ ... proxime sequens pagina monstrat."[44] Cardinal Cusa, who died in 1464, is one of the earliest modern attempters. His quadrature is found in the second volume, and is now quite unreadable.

[ 48 ] In these early days every quadrator found a geometrical opponent, who finished him. Regimontanus[45] did this office for the Cardinal.



De Occulta Philosophia libri III. By Henry Cornelius Agrippa. Lyons, 1550, 8vo.
De incertitudine et vanitate scientiarum. By the same. Cologne, 1531, 8vo.

The first editions of these works were of 1530, as well as I can make out; but the first was in progress in 1510.[46] In the second work Agrippa repents of having wasted time on the magic of the first; but all those who actually deal with demons are destined to eternal fire with Jamnes and Mambres and Simon Magus. This means, as is the fact, that his occult philosophy did not actually enter upon black magic, but confined itself to the power of the stars, of numbers, etc. The fourth book, which appeared after the death of Agrippa, and really concerns dealing with evil spirits, is undoubtedly spurious. It is very difficult to make out what Agrippa really believed on the subject. I have introduced his books as the most marked specimens of treatises on magic, a paradox of our day, though not far from orthodoxy in his; and here I should have ended my notice, if I had not casually found something more interesting to the reader of our day.

[ 49 ]



Walter Scott, it is well known, was curious on all matters connected with magic, and has used them very widely. But it is hardly known how much pains he has taken to be correct, and to give the real thing. The most decided detail of a magical process which is found in his writings is that of Dousterswivel in The Antiquary; and it is obvious, by his accuracy of process, that he does not intend the adept for a mere impostor, but for one who had a lurking belief in the efficacy of his own processes, coupled with intent to make a fraudulent use of them. The materials for the process are taken from Agrippa. I first quote Mr. Dousterswivel:

"... I take a silver plate when she [the moon] is in her fifteenth mansion, which mansion is in de head of Libra, and I engrave upon one side de worts Schedbarschemoth Schartachan [ch should be t]—dat is, de Intelligence of de Intelligence of de moon—and I make his picture like a flying serpent with a turkey-cock's head—vary well—Then upon this side I make de table of de moon, which is a square of nine, multiplied into itself, with eighty-one numbers [nine] on every side and diameter nine...."

In the De Occulta Philosophia, p. 290, we find that the fifteenth mansion of the moon incipit capite Libræ, and is good pro extrahendis thesauris, the object being to discover hidden treasure. In p. 246, we learn that a silver plate must be used with the moon. In p. 248, we have the words which denote the Intelligence, etc. But, owing to the falling of a number into a wrong line, or the misplacement of a line, one or other—which takes place in all the editions I have examined—Scott has, sad to say, got hold of the wrong words; he has written down the demon of the demons of the moon. Instead of the gibberish above, it should have been Malcha betarsisim hed beruah schenhakim. In p. 253, we have the magic square of the moon, with eighty-one numbers, and the symbol for the Intelligence, which Scott likens to a flying [ 50 ] serpent with a turkey-cock's head. He was obliged to say something; but I will stake my character—and so save a woodcut—on the scratches being more like a pair of legs, one shorter than the other, without a body, jumping over a six-barred gate placed side uppermost. Those who thought that Scott forged his own nonsense, will henceforth stand corrected. As to the spirit Peolphan, etc., no doubt Scott got it from the authors he elsewhere mentions, Nicolaus Remigius[47] and Petrus Thyracus; but this last word should be Thyræus.

The tendency of Scott's mind towards prophecy is very marked, and it is always fulfilled. Hyder, in his disguise, calls out to Tippoo: "Cursed is the prince who barters justice for lust; he shall die in the gate by the sword of the stranger." Tippoo was killed in a gateway at Seringapatam.[48]



Orontii Finaei ... Quadratura Circuli. Paris, 1544, 4to.

Orontius[49] squared the circle out of all comprehension; but he was killed by a feather from his own wing. His [ 51 ] former pupil, John Buteo,[50] the same who—I believe for the first time—calculated the question of Noah's ark, as to its power to hold all the animals and stores, unsquared him completely. Orontius was the author of very many works, and died in 1555. Among the laudatory verses which, as was usual, precede this work, there is one of a rare character: a congratulatory ode to the wife of the author. The French now call this writer Oronce Finée; but there is much difficulty about delatinization. Is this more correct than Oronce Fine, which the translator of De Thou uses? Or than Horonce Phine, which older writers give? I cannot understand why M. de Viette[51] should be called Viète, because his Latin name is Vieta. It is difficult to restore Buteo; for not only now is butor a blockhead as well as a bird, but we really cannot know what kind of bird Buteo stood for. We may be sure that Madame Fine was Denise Blanche; for Dionysia Candida can mean nothing else. Let her shade rejoice in the fame which Hubertus Sussannæus has given her.

I ought to add that the quadrature of Orontius, and solutions of all the other difficulties, were first published in De Rebus Mathematicis Hactenus Desideratis,[52] of which I have not the date.


[ 52 ]


Nicolai Raymari Ursi Dithmarsi Fundamentum Astronomicum, id est, nova doctrina sinuum et triangulorum.... Strasburg, 1588, 4to.[53]

People choose the name of this astronomer for themselves: I take Ursus, because he was a bear. This book gave the quadrature of Simon Duchesne,[54] or à Quercu, which excited Peter Metius,[55] as presently noticed. It also gave that unintelligible reference to Justus Byrgius which has been used in the discussion about the invention of logarithms.[56]

The real name of Duchesne is Van der Eycke. I have met with a tract in Dutch, Letterkundige Aanteekeningen, upon Van Eycke, Van Ceulen,[57] etc., by J. J. Dodt van Flensburg,[58] which I make out to be since 1841 in date. I should [ 53 ] much like a translation of this tract to be printed, say in the Phil. Mag. Dutch would be clear English if it were properly spelt. For example, learn-master would be seen at once to be teacher; but they will spell it leermeester. Of these they write as van deze; widow they make weduwe. All this is plain to me, who never saw a Dutch dictionary in my life; but many of their misspellings are quite unconquerable.



Jacobus Falco Valentinus, miles Ordinis Montesiani, hanc circuli quadraturam invenit. Antwerp, 1589, 4to.[59]

The attempt is more than commonly worthless; but as Montucla and others have referred to the verses at the end, and as the tract is of the rarest, I will quote them:

Circulus loquitur.
Vocabar ante circulus
Eramque curvus undique
Ut alta solis orbita
Et arcus ille nubium.
Eram figura nobilis
Carensque sola origine
Carensque sola termino.
Modo indecora prodeo
Novisque fœdor angulis.
Nec hoc peregit Archytas[60]
Neque Icari pater neque
Tuus, Iapete, filius.
Quis ergo casus aut Deus
Meam quadravit aream?
Respondet auctor.
Ad alta Turiæ ostia
Lacumque limpidissimum
Sita est beata civitas
[ 54 ]
Parum Saguntus abfuit
Abestque Sucro plusculum.
Hic est poeta quispiam
Libenter astra consulens
Sibique semper arrogans
Negata doctioribus,
Senex ubique cogitans
Sui frequenter immemor
Nec explicare circinum
Nec exarare lineas
Sciens ut ipse prædicat.
Hic ergo bellus artifex
Tuam quadravit aream.[61]

Falco's verses are pretty, if the ˘-mysteries be correct; but of these things I have forgotten—what I knew. [One mistake has been pointed out to me: it is Archȳtas].

As a specimen of the way in which history is written, I copy the account which Montucla—who is accurate when he writes about what he has seen—gives of these verses. He gives the date 1587; he places the verses at the beginning instead of the end; he says the circle thanks its quadrator affectionately; and he says the good and modest chevalier gives all the glory to the patron saint of his order. All of little consequence, as it happens; but writing at second-hand makes as complete mistakes about more important matters.

[ 55 ]



Petri Bungi Bergomatis Numerorum mysteria. Bergomi [Bergamo], 1591, 4to. Second Edition.

The first edition is said to be of 1585;[62] the third, Paris, 1618. Bungus is not for my purpose on his own score, but those who gave the numbers their mysterious characters: he is but a collector. He quotes or uses 402 authors, as we are informed by his list; this just beats Warburton,[63] whom some eulogist or satirist, I forget which, holds up as having used 400 authors in some one work. Bungus goes through 1, 2, 3, etc., and gives the account of everything remarkable in which each number occurs; his accounts not being always mysterious. The numbers which have nothing to say for themselves are omitted: thus there is a gap between 50 and 60. In treating 666, Bungus, a good Catholic, could not compliment the Pope with it, but he fixes it on Martin Luther with a little forcing. If from A to I represent 1-10, from K to S 10-90, and from T to Z 100-500, we see:

M A R T I N   L U T E R A
30 1 80 100 9 40   20 200 100 5 80 1

which gives 666. Again, in Hebrew, Lulter does the same:

ר ת ל ו ל
200 400 30 6 30

And thus two can play at any game. The second is better than the first: to Latinize the surname and not the Christian [ 56 ] name is very unscholarlike. The last number mentioned is a thousand millions; all greater numbers are dismissed in half a page. Then follows an accurate distinction between number and multitude—a thing much wanted both in arithmetic and logic.



What may be the use of such a book as this? The last occasion on which it was used was the following. Fifteen or sixteen years ago the Royal Society determined to restrict the number of yearly admissions to fifteen men of science, and noblemen ad libitum; the men of science being selected and recommended by the Council, with a power, since practically surrendered, to the Society to elect more. This plan appears to me to be directly against the spirit of their charter, the true intent of which is, that all who are fit should be allowed to promote natural knowledge in association, from and after the time at which they are both fit and willing. It is also working more absurdly from year to year; the tariff of fifteen per annum will soon amount to the practical exclusion of many who would be very useful. This begins to be felt already, I suspect. But, as appears above, the body of the Society has the remedy in its own hands. When the alteration was discussed by the Council, my friend the late Mr. Galloway,[64] then one of the body, opposed it strongly, and inquired particularly into the reason why fifteen, of all numbers, was the one to be selected. Was it because fifteen is seven and eight, typifying the Old Testament Sabbath, and the New Testament day of the resurrection following? Was it because Paul strove fifteen days against Peter, proving that he was a doctor both of the Old and New Testament? Was it because the prophet Hosea bought a lady [ 57 ] for fifteen pieces of silver? Was it because, according to Micah, seven shepherds and eight chiefs should waste the Assyrians? Was it because Ecclesiastes commands equal reverence to be given to both Testaments—such was the interpretation—in the words "Give a portion to seven, and also to eight"? Was it because the waters of the Deluge rose fifteen cubits above the mountains?—or because they lasted fifteen decades of days? Was it because Ezekiel's temple had fifteen steps? Was it because Jacob's ladder has been supposed to have had fifteen steps? Was it because fifteen years were added to the life of Hezekiah? Was it because the feast of unleavened bread was on the fifteenth day of the month? Was it because the scene of the Ascension was fifteen stadia from Jerusalem? Was it because the stone-masons and porters employed in Solomon's temple amounted to fifteen myriads? etc. The Council were amused and astounded by the volley of fifteens which was fired at them; they knowing nothing about Bungus, of which Mr. Galloway—who did not, as the French say, indicate his sources—possessed the copy now before me. In giving this anecdote I give a specimen of the book, which is exceedingly rare. Should another edition ever appear, which is not very probable, he would be but a bungling Bungus who should forget the fifteen of the Royal Society.



[I make a remark on the different colors which the same person gives to one story, according to the bias under which he tells it. My friend Galloway told me how he had quizzed the Council of the Royal Society, to my great amusement. Whenever I am struck by the words of any one, I carry away a vivid recollection of position, gestures, tones, etc. I do not know whether this be common or uncommon. I never recall this joke without seeing before me my friend, leaning against his bookcase, with Bungus open in his hand, and a certain half-depreciatory tone which he often used [ 58 ] when speaking of himself. Long after his death, an F.R.S. who was present at the discussion, told me the story. I did not say I had heard it, but I watched him, with Galloway at the bookcase before me. I wanted to see whether the two would agree as to the fact of an enormous budget of fifteens having been fired at the Council, and they did agree perfectly. But when the paragraph of the Budget appeared in the Athenæum, my friend, who seemed rather to object to the showing-up, assured me that the thing was grossly exaggerated; there was indeed a fifteen or two, but nothing like the number I had given. I had, however, taken sharp note of the previous narration.



I will give another instance. An Indian officer gave me an account of an elephant, as follows. A detachment was on the march, and one of the gun-carriages got a wheel off the track, so that it was also off the ground, and hanging over a precipice. If the bullocks had moved a step, carriages, bullocks, and all must have been precipitated. No one knew what could be done until some one proposed to bring up an elephant, and let him manage it his own way. The elephant took a moment's survey of the fix, put his trunk under the axle of the free wheel, and waited. The surrounders, who saw what he meant, moved the bullocks gently forward, the elephant followed, supporting the axle, until there was ground under the wheel, when he let it quietly down. From all I had heard of the elephant, this was not too much to believe. But when, years afterwards, I reminded my friend of his story, he assured me that I had misunderstood him, that the elephant was directed to put his trunk under the wheel, and saw in a moment why. This is reasonable sagacity, and very likely the correct account; but I am quite sure that, in the fit of elephant-worship under which the story was first told, it was told as I have first stated it.] [ 59 ]



[Jordani Bruni Nolani de Monade, Numero et Figura ... item de Innumerabilibus, Immenso, et Infigurabili ... Frankfort, 1591, 8vo.[65]

I cannot imagine how I came to omit a writer whom I have known so many years, unless the following story will explain it. The officer reproved the boatswain for perpetual swearing; the boatswain answered that he heard the officers swear. "Only in an emergency," said the officer. "That's just it," replied the other; "a boatswain's life is a life of 'mergency." Giordano Bruno was all paradox; and my mind was not alive to his paradoxes, just as my ears might have become dead to the boatswain's oaths. He was, as has been said, a vorticist before Descartes,[66] an optimist before Leibnitz, a Copernican before Galileo. It would be easy to collect a hundred strange opinions of his. He was born about 1550, and was roasted alive at Rome, February 17, 1600, for the maintenance and defence of the holy Church, and the rights and liberties of the same. These last words are from the writ of our own good James I, under which Leggatt[67] was roasted at Smithfield, in March 1612; and if I had a copy of the instrument under which Wightman[68] was roasted at Lichfield, a month afterwards, I daresay I should [ 60 ] find something quite as edifying. I extract an account which I gave of Bruno in the Comp. Alm. for 1855:

"He was first a Dominican priest, then a Calvinist; and was roasted alive at Rome, in 1600, for as many heresies of opinion, religious and philosophical, as ever lit one fire. Some defenders of the papal cause have at least worded their accusations so to be understood as imputing to him villainous actions. But it is positively certain that his death was due to opinions alone, and that retractation, even after sentence, would have saved him. There exists a remarkable letter, written from Rome on the very day of the murder, by Scioppius[69] (the celebrated scholar, a waspish convert from Lutheranism, known by his hatred to Protestants and Jesuits) to Rittershusius,[70] a well-known Lutheran writer on civil and canon law, whose works are in the index of prohibited books. This letter has been reprinted by Libri (vol. iv. p. 407). The writer informs his friend (whom he wished to convince that even a Lutheran would have burnt Bruno) that all Rome would tell him that Bruno died for Lutheranism; but this is because the Italians do not know the difference between one heresy and another, in which simplicity (says the writer) may God preserve them. That is to say, they knew the difference between a live heretic and a roasted one by actual inspection, but had no idea of the difference between a Lutheran and a Calvinist. The countrymen of Boccaccio would have smiled at the idea which the German scholar entertained of them. They said Bruno was burnt for Lutheranism, a name under which they classed all Protestants: and they are better witnesses than Schopp, or Scioppius. He then proceeds to describe to his Protestant friend (to whom he would certainly not have omitted any act which both their churches would have condemned) the mass of opinions with which Bruno was charged; as that there [ 61 ] are innumerable worlds, that souls migrate, that Moses was a magician, that the Scriptures are a dream, that only the Hebrews descended from Adam and Eve, that the devils would be saved, that Christ was a magician and deservedly put to death, etc. In fact, says he, Bruno has advanced all that was ever brought forward by all heathen philosophers, and by all heretics, ancient and modern. A time for retractation was given, both before sentence and after, which should be noted, as well for the wretched palliation which it may afford, as for the additional proof it gives that opinions, and opinions only, brought him to the stake. In this medley of charges the Scriptures are a dream, while Adam, Eve, devils, and salvation are truths, and the Saviour a deceiver. We have examined no work of Bruno except the De Monade, etc., mentioned in the text. A strong though strange theism runs through the whole, and Moses, Christ, the Fathers, etc., are cited in a manner which excites no remark either way. Among the versions of the cause of Bruno's death is atheism: but this word was very often used to denote rejection of revelation, not merely in the common course of dispute, but by such writers, for instance, as Brucker[71] and Morhof.[72] Thus Morhof says of the De Monade, etc., that it exhibits no manifest signs of atheism. What he means by the word is clear enough, when he thus speaks of a work which acknowledges God in hundreds of places, and rejects opinions as blasphemous in several. The work of Bruno in which his astronomical opinions are contained is De Monade, etc. (Frankfort, 1591, 8vo). He is the most thorough-going Copernican possible, and throws out almost every opinion, true or false, which has ever been discussed by astronomers, from the theory of innumerable inhabited worlds and systems to that [ 62 ] of the planetary nature of comets. Libri (vol. iv)[73] has reprinted the most striking part of his expressions of Copernican opinion."



The Satanic doctrine that a church may employ force in aid of its dogma is supposed to be obsolete in England, except as an individual paradox; but this is difficult to settle. Opinions are much divided as to what the Roman Church would do in England, if she could: any one who doubts that she claims the right does not deserve an answer. When the hopes of the Tractarian section of the High Church were in bloom, before the most conspicuous intellects among them had transgressed their ministry, that they might go to their own place, I had the curiosity to see how far it could be ascertained whether they held the only doctrine which makes me the personal enemy of a sect. I found in one of their tracts the assumption of a right to persecute, modified by an asserted conviction that force was not efficient. I cannot now say that this tract was one of the celebrated ninety; and on looking at the collection I find it so poorly furnished with contents, etc., that nothing but searching through three thick volumes would decide. In these volumes I find, augmenting as we go on, declarations about the character and power of "the Church" which have a suspicious appearance. The suspicion is increased by that curious piece of sophistry, No. 87, on religious reserve. The queer paradoxes of that tract leave us in doubt as to everything but this, that the church(man) is not bound to give his whole counsel in all things, and not bound to say what the things are in which he does not give it. It is likely enough that some of the "rights and liberties" are but scantily described. There is now no fear; but the time was when, if not fear, there might be a looking for of fear to come; nobody could then be so [ 63 ] sure as we now are that the lion was only asleep. There was every appearance of a harder fight at hand than was really found needful.

Among other exquisite quirks of interpretation in the No. 87 above mentioned is the following. God himself employs reserve; he is said to be decked with light as with a garment (the old or prayer-book version of Psalm civ. 2). To an ordinary apprehension this would be a strong image of display, manifestation, revelation; but there is something more. "Does not a garment veil in some measure that which it clothes? Is not that very light concealment?"

This No. 87, admitted into a series, fixes upon the managers of the series, who permitted its introduction, a strong presumption of that underhand intent with which they were charged. At the same time it is honorable to our liberty that this series could be published: though its promoters were greatly shocked when the Essayists and Bishop Colenso[74] took a swing on the other side. When No. 90 was under discussion, Dr. Maitland,[75] the librarian at Lambeth, asked Archbishop Howley[76] a question about No. 89. "I did not so much as know there was a No. 89," was the answer. I am almost sure I have seen this in print, and quite sure that Dr. Maitland told it to me. It is creditable that there was so much freedom; but No. 90 was too bad, and was stopped.

The Tractarian mania has now (October 1866) settled down into a chronic vestment disease, complicated with fits of transubstantiation, which has taken the name of [ 64 ] Ritualism. The common sense of our national character will not put up with a continuance of this grotesque folly; millinery in all its branches will at last be advertised only over the proper shops. I am told that the Ritualists give short and practical sermons; if so, they may do good in the end. The English Establishment has always contained those who want an excitement; the New Testament, in its plain meaning, can do little for them. Since the Revolution, Jacobitism, Wesleyanism, Evangelicism, Puseyism,[77] and Ritualism, have come on in turn, and have furnished hot water for those who could not wash without it. If the Ritualists should succeed in substituting short and practical teaching for the high-spiced lectures of the doctrinalists, they will be remembered with praise. John the Baptist would perhaps not have brought all Jerusalem out into the wilderness by his plain and good sermons: it was the camel's hair and the locusts which got him a congregation, and which, perhaps, added force to his precepts. When at school I heard a dialogue, between an usher and the man who cleaned the shoes, about Mr. ——, a minister, a very corporate body with due area of waistcoat. "He is a man of great erudition," said the first. "Ah, yes sir," said Joe; "any one can see that who looks at that silk waistcoat."]



[When I said at the outset that I had only taken books from my own store, I should have added that I did not make any search for information given as part of a work. Had I looked through all my books, I might have made some curious additions. For instance, in Schott's Magia Naturalis[78] [ 65 ] (vol. iii. pp. 756-778) is an account of the quadrature of Gephyrauder, as he is misprinted in Montucla. He was Thomas Gephyrander Salicetus; and he published two editions, in 1608 and 1609.[79] I never even heard of a copy of either. His work is of the extreme of absurdity: he makes a distinction between geometrical and arithmetical fractions, and evolves theorems from it. More curious than his quadrature is his name; what are we to make of it? If a German, he is probably a German form of Bridgeman. and Salicetus refers him to Weiden. But Thomas was hardly a German Christian name of his time; of 526 German philosophers, physicians, lawyers, and theologians who were biographed by Melchior Adam,[80] only two are of this name. Of these one is Thomas Erastus,[81] the physician whose theological writings against the Church as a separate power have given the name of Erastians to those who follow his doctrine, whether they have heard of him or not. Erastus is little known; accordingly, some have supposed that he must be Erastus, the friend of St. Paul and Timothy (Acts xix. 22; 2 Tim. iv. 20; Rom. xvi. 23), but what this gentleman did to earn the character is not hinted at. Few words would have done: Gaius (Rom. xvi. 23) has an immortality which many more noted men have missed, given by John Bunyan, out of seven words of St. Paul. I was once told that the Erastians got their name from Blastus, and I could not solve bl = er: at last I remembered that Blastus was a chamberlain[82] as well as Erastus; hence the association which [ 66 ] caused the mistake. The real heresiarch was a physician who died in 1583; his heresy was promulgated in a work, published immediately after his death by his widow, De Excommunicatione Ecclesiastica. He denied the power of excommunication on the principle above stated; and was answered by Besa.[83] The work was translated by Dr. R. Lee[84] (Edinb. 1844, 8vo). The other is Thomas Grynæus,[85] a theologian, nephew of Simon, who first printed Euclid in Greek; of him Adam says that of works he published none, of learned sons four. If Gephyrander were a Frenchman, his name is not so easily guessed at; but he must have been of La Saussaye. The account given by Schott is taken from a certain Father Philip Colbinus, who wrote against him.

In some manuscripts lately given to the Royal Society, David Gregory,[86] who seems to have seen Gephyrander's work, calls him Salicetus Westphalus, which is probably on the title-page. But the only Weiden I can find is in Bavaria. Murhard has both editions in his Catalogue, but had plainly never seen the books: he gives the author as Thomas Gep. Hyandrus, Salicettus Westphalus. Murhard is a very old referee of mine; but who the non nominandus was to see Montucla's Gephyrander in Murhard's Gep. Hyandrus, both writers being usually accurate?]



A plain discoverie of the whole Revelation of St. John ... whereunto are annexed certain oracles of Sibylla.... Set Foorth by John Napeir L. of Marchiston. London, 1611, 4to.[87]

[ 67 ]

The first edition was Edinburgh, 1593,[88] 4to. Napier[89] always believed that his great mission was to upset the Pope, and that logarithms, and such things, were merely episodes and relaxations. It is a pity that so many books have been written about this matter, while Napier, as good as any, is forgotten and unread. He is one of the first who gave us the six thousand years. "There is a sentence of the house of Elias reserved in all ages, bearing these words: The world shall stand six thousand years, and then it shall be consumed by fire: two thousand yeares voide or without lawe, two thousand yeares under the law, and two thousand yeares shall be the daies of the Messias...."

I give Napier's parting salute: it is a killing dilemma:

"In summar conclusion, if thou o Rome aledges thyselfe reformed, and to beleeue true Christianisme, then beleeue Saint John the Disciple, whome Christ loued, publikely here in this Reuelation proclaiming thy wracke, but if thou remain Ethnick in thy priuate thoghts, beleeuing[90] the old Oracles of the Sibyls reuerently keeped somtime in thy Capitol: then doth here this Sibyll proclame also thy wracke. Repent therefore alwayes, in this thy latter breath, as thou louest thine Eternall salvation. Amen."

—Strange that Napier should not have seen that this appeal could not succeed, unless the prophecies of the Apocalypse were no true prophecies at all.


13 ^  The work is the Questiones Joannis Buridani super X libros Aristotelis ad Nicomachum, curante Egidio Delfo ... Parisiis, 1489, folio. It also appeared at Paris in editions of 1499, 1513, and 1518, and at Oxford in 1637.

14 ^  Jean Buridan was born at Béthune about 1298, and died at Paris about 1358. He was professor of philosophy at the University of Paris and several times held the office of Rector. As a philosopher he was classed among the nominalists.

15 ^  So in the original.

16 ^  Baruch Spinoza, or Benedict de Spinoza as he later called himself, the pantheistic philosopher, excommunicated from the Jewish faith for heresy, was born at Amsterdam in 1632 and died there in 1677.

17 ^  Michael Scott, or Scot, was born about 1190, probably in Fifeshire, Scotland, and died about 1291. He was one of the best known savants of the court of Emperor Frederick II, and wrote upon astrology, alchemy, and the occult sciences. He was looked upon as a great magician and is mentioned among the wizards in Dante's Inferno.

"That other, round the loins
So slender of his shape, was Michael Scot,
Practised in every slight of magic wile." Inferno, XX.

Boccaccio also speaks of him: "It is not long since there was in this city (Florence) a great master in necromancy, who was called Michele Scotto, because he was a Scot." Decameron, Dec. Giorno.

Scott's mention of him in Canto Second of his Lay of the Last Minstrel, is well known:

"In these fair climes, it was my lot
To meet the wondrous Michael Scott;
A wizard of such dreaded fame,
That when, in Salamanca's cave,
Him listed his magic wand to wave,
The bells would ring in Notre Dame!"

Sir Walter's notes upon him are of interest.

18 ^  These were some of the forgeries which Michel Chasles (1793-1880) was duped into buying. They purported to be a correspondence between Pascal and Newton and to show that the former had anticipated some of the discoveries of the great English physicist and mathematician. That they were forgeries was shown by Sir David Brewster in 1855.

19 ^  "Let the serpent also break from its appointed path."

20 ^  Guglielmo Brutus Icilius Timoleon Libri-Carucci della Sommaja, born at Florence in 1803; died at Fiesole in 1869. His Histoire des Sciences Mathématiques appeared at Paris in 1838, the entire first edition of volume I, save some half dozen that he had carried home, being burned on the day that the printing was completed. He was a great collector of early printed works on mathematics, and was accused of having stolen large numbers of them from other libraries. This accusation took him to London, where he bitterly attacked his accusers. There were two auction sales of his library, and a number of his books found their way into De Morgan's collection.

21 ^  Philo of Gadara lived in the second century B.C. He was a pupil of Sporus, who worked on the problem of the two mean proportionals.

22 ^  In his Histoire des Mathématiques, the first edition of which appeared in 1758. Jean Etienne Montucla was born at Lyons in 1725 and died at Versailles in 1799. He was therefore only thirty-three years old when his great work appeared. The second edition, with additions by D'Alembert, appeared in 1799-1802. He also wrote a work on the quadrature of the circle, Histoire des recherches sur la Quadrature du Cercle, which appeared in 1754.

23 ^  Eutocius of Ascalon was born in 480 A.D. He wrote commentaries on the first four books of the conics of Apollonius of Perga (247-222 B.C.). He also wrote on the Sphere and Cylinder and the Quadrature of the Circle, and on the two books on Equilibrium of Archimedes (287-212 B.C.)

24 ^  Edward Cocker was born in 1631 and died between 1671 and 1677. His famous arithmetic appeared in 1677 and went through many editions. It was written in a style that appealed to teachers, and was so popular that the expression "According to Cocker" became a household phrase. Early in the nineteenth century there was a similar saying in America, "According to Daboll," whose arithmetic had some points of analogy to that of Cocker. Each had a well-known prototype in the ancient saying, "He reckons like Nicomachus of Gerasa."

25 ^  So in the original, for Barrême. François Barrême was to France what Cocker was to England. He was born at Lyons in 1640, and died at Paris in 1703. He published several arithmetics, dedicating them to his patron, Colbert. One of the best known of his works is L'arithmétique, ou le livre facile pour apprendre l'arithmétique soi-mème, 1677. The French word barême or barrême, a ready-reckoner, is derived from his name.

26 ^  Born at Rome, about 480 A.D.; died at Pavia, 524. Gibbon speaks of him as "the last of the Romans whom Cato or Tully could have acknowledged for their countryman." His works on arithmetic, music, and geometry were classics in the medieval schools.

27 ^  Johannes Campanus, of Novarra, was chaplain to Pope Urban IV (1261-1264). He was one of the early medieval translators of Euclid from the Arabic into Latin, and the first printed edition of the Elements (Venice, 1482) was from his translation. In this work he probably depended not a little upon at least two or three earlier scholars. He also wrote De computo ecclesiastico Calendarium, and De quadratura circuli.

28 ^  Archimedes gave , and as the limits of the ratio of the circumference to the diameter of a circle.

29 ^  Friedrich W. A. Murhard was born at Cassel in 1779 and died there in 1853. His Bibliotheca Mathematica, Leipsic, 1797-1805, is ill arranged and inaccurate, but it is still a helpful bibliography. De Morgan speaks somewhere of his indebtedness to it.

30 ^  Abraham Gotthelf Kästner was born at Leipsic in 1719, and died at Göttingen in 1800. He was professor of mathematics and physics at Göttingen. His Geschichte der Mathematik (1796-1800) was a work of considerable merit. In the text of the Budget of Paradoxes the name appears throughout as Kastner instead of Kästner.

31 ^  Lucas Gauricus, or Luca Gaurico, born at Giffoni, near Naples, in 1476; died at Rome in 1558. He was an astrologer and mathematician, and was professor of mathematics at Ferrara in 1531. In 1545 he became bishop of Cività Ducale.

32 ^  John Couch Adams was born at Lidcot, Cornwall, in 1819, and died in 1892. He and Leverrier predicted the discovery of Neptune from the perturbations in Uranus.

33 ^  Urbain-Jean-Joseph Leverrier was born at Saint-Lô, Manche, in 1811, and died at Paris in 1877. It was his data respecting the perturbations of Uranus that were used by Adams and himself in locating Neptune.

34 ^  Joseph-Juste Scaliger, the celebrated philologist, was born at Agen in 1540, and died at Leyden in 1609. His Cyclometrica elementa, to which De Morgan refers, appeared at Leyden in 1594.

35 ^  The title is: In hoc libra contenta.... Introductio i geometriā.... Liber de quadratura circuli. Liber de cubicatione sphere. Perspectiva introductio. Carolus Bovillus, or Charles Bouvelles (Boüelles, Bouilles, Bouvel), was born at Saucourt, Picardy, about 1470, and died at Noyon about 1533. He was canon and professor of theology at Noyon. His Introductio contains considerable work on star polygons, a favorite study in the Middle Ages and early Renaissance. His work Que hoc volumine continētur. Liber de intellectu. Liber de sensu, etc., appeared at Paris in 1509-10.

36 ^  Nicolaus Cusanus, Nicolaus Chrypffs or Krebs, was born at Kues on the Mosel in 1401, and died at Todi, Umbria, August 11, 1464. He held positions of honor in the church, including the bishopric of Brescia. He was made a cardinal in 1448. He wrote several works on mathematics, his Opuscula varia appearing about 1490, probably at Strasburg, but published without date or place. His Opera appeared at Paris in 1511 and again in 1514, and at Basel in 1565.

37 ^  Henry Stephens (born at Paris about 1528, died at Lyons in 1598) was one of the most successful printers of his day. He was known as Typographus Parisiensis, and to his press we owe some of the best works of the period.

38 ^  Jacobus Faber Stapulensis (Jacques le Fèvre d'Estaples) was born at Estaples, near Amiens, in 1455, and died at Nérac in 1536. He was a priest, vicar of the bishop of Meaux, lecturer on philosophy at the Collège Lemoine in Paris, and tutor to Charles, son of Francois I. He wrote on philosophy, theology, and mathematics.

39 ^  Claude-François Milliet de Challes was born at Chambéry in 1621, and died at Turin in 1678. He edited Euclidis Elementorum libri octo in 1660, and published a Cursus seu mundus mathematicus, which included a short history of mathematics, in 1674. He also wrote on mathematical geography.

40 ^  This date should be 1503, if he refers to the first edition. It is well known that this is the first encyclopedia worthy the name to appear in print. It was written by Gregorius Reisch (born at Balingen, and died at Freiburg in 1487), prior of the cloister at Freiburg and confessor to Maximilian I. The first edition appeared at Freiburg in 1503, and it passed through many editions in the sixteenth and seventeenth centuries. The title of the 1504 edition reads: Aepitoma omnis phylosophiae. alias Margarita phylosophica tractans de omni genere scibili: Cum additionibus: Quae in alijs non habentur.

41 ^  This is the Introductio in arithmeticam Divi S. Boetii.... Epitome rerum geometricarum ex geometrica introductio C. Bovilli. De quadratura circuli demonstratio ex Campano, that appeared without date about 1507.

42 ^  Born at Liverpool in 1805, and died there about 1872. He was a merchant, and in 1865 he published, at Liverpool, a work entitled The Quadrature of the Circle, or the True Ratio between the Diameter and Circumference geometrically and mathematically demonstrated. In this he gives the ratio as exactly <math\scriptstyle>\scriptstyle 3 \tfrac{1}{8}</math>.

43 ^  "That it would be impossible to tell him exactly, since no one had yet been able to find precisely the ratio of the circumference to the diameter."

44 ^  This is the Paris edition: "Parisiis: ex officina Ascensiana anno Christi ... MDXIIII," as appears by the colophon of the second volume to which De Morgan refers.

45 ^  Regiomontanus, or Johann Müller of Königsberg (Regiomontanus), was born at Königsberg in Franconia, June 5, 1436, and died at Rome July 6, 1476. He studied at Vienna under the great astronomer Peuerbach, and was his most famous pupil. He wrote numerous works, chiefly on astronomy. He is also known by the names Ioannes de Monte Regio, de Regiomonte, Ioannes Germanus de Regiomonte, etc.

46 ^  Henry Cornelius Agrippa was born at Cologne in 1486 and died either at Lyons in 1534 or at Grenoble in 1535. He was professor of theology at Cologne and also at Turin. After the publication of his De Occulta Philosophia he was imprisoned for sorcery. Both works appeared at Antwerp in 1530, and each passed through a large number of editions. A French translation appeared in Paris in 1582, and an English one in London in 1651.

47 ^  Nicolaus Remegius was born in Lorraine in 1554, and died at Nancy in 1600. He was a jurist and historian, and held the office of procurator general to the Duke of Lorraine.

48 ^  This was at the storming of the city by the British on May 4, 1799. From his having been born in India, all this appealed strongly to the interests of De Morgan.

49 ^  Orontius Finaeus, or Oronce Finé, was born at Briançon in 1494 and died at Paris, October 6, 1555. He was imprisoned by François I for refusing to recognize the concordat (1517). He was made professor of mathematics in the Collège Royal (later called the Collège de France) in 1532. He wrote extensively on astronomy and geometry, but was by no means a great scholar. He was a pretentious man, and his works went through several editions. His Protomathesis appeared at Paris in 1530-32. The work referred to by De Morgan is the Quadratura circuli tandem inventa & clarissime demonstrata ... Lutetiae Parisiorum, 1544, fol. In the 1556 edition of his De rebus mathematicis, hactenus desideratis, Libri IIII, published at Paris, the subtitle is: Quibus inter cætera, Circuli quadratura Centum modis, & suprà, per eundem Orontium recenter excogitatis, demonstratus, so that he kept up his efforts until his death.

50 ^  Johannes Buteo (Boteo, Butéon, Bateon) was born in Dauphiné c. 1485-1489, and died in a cloister in 1560 or 1564. Some writers give Charpey as the place and 1492 as the date of his birth, and state that he died at Canar in 1572. He belonged to the order of St. Anthony, and wrote chiefly on geometry, exposing the pretenses of Finaeus. His Opera geometrica appeared at Lyons in 1554, and his Logistica and De quadratura circuli libri duo at Lyons in 1559.

51 ^  This is the great French algebraist, François Viète (Vieta), who was born at Fontenay-le-Comte in 1540, and died at Paris, December 13, 1603. His well-known Isagoge in artem analyticam appeared at Tours in 1591. His Opera mathematica was edited by Van Schooten in 1646.

52 ^  This is the De Rebus mathematicis hactenus desideratis, Libri IIII, that appeared in Paris in 1556. For the title page see Smith, D. E., Rara Arithmetica, Boston, 1908, p. 280.

53 ^  The title is correct except for a colon after Astronomicum. Nicolaus Raimarus Ursus was born in Henstede or Hattstede, in Dithmarschen, and died at Prague in 1599 or 1600. He was a pupil of Tycho Brahe. He also wrote De astronomis hypothesibus (1597) and Arithmetica analytica vulgo Cosa oder Algebra (1601).

54 ^  Born at Dôle, Franche-Comté, about 1550, died in Holland about 1600. The work to which reference is made is the Quadrature du cercle, ou manière de trouver un quarré égal au cercle donné, which appeared at Delft in 1584. Duchesne had the courage of his convictions, not only on circle-squaring but on religion as well, for he was obliged to leave France because of his conversion to Calvinism. De Morgan's statement that his real name is Van der Eycke is curious, since he was French born. The Dutch may have translated his name when he became professor at Delft, but we might equally well say, that his real name was Quercetanus or à Quercu.

55 ^  This was the father of Adriaan Metius (1571-1635). He was a mathematician and military engineer, and suggested the ratio for , a ratio afterwards published by his son. The ratio, then new to Europe, had long been known and used in China, having been found by Tsu Ch'ung-chih (428-499 A.D.).

56 ^  This was Jost Bürgi, or Justus Byrgius, the Swiss mathematician of whom Kepler wrote in 1627: "Apices logistici Justo Byrgio multis annis ante editionem Neperianam viam præiverunt ad hos ipsissimos logarithmos." He constructed a table of antilogarithms (Arithmetische und geometrische Progress-Tabulen), but it was not published until after Napier's work appeared.

57 ^  Ludolphus Van Ceulen, born at Hildesheim, and died at Leyden in 1610. It was he who first carried the computation of to 35 decimal places.

58 ^  Jens Jenssen Dodt, van Flensburg, a Dutch historian, who died in 1847.

59 ^  I do not know this edition. There was one "Antverpiae apud Petrum Bellerum sub scuto Burgundiae," 4to, in 1591.

60 ^  Archytas of Tarentum (430-365 B.C.) who wrote on proportions, irrationals, and the duplication of the cube.

61 ^   

The Circle Speaks.
"At first a circle I was called,
And was a curve around about
Like lofty orbit of the sun
Or rainbow arch among the clouds.
A noble figure then was I—
And lacking nothing but a start,
And lacking nothing but an end.
But now unlovely do I seem
Polluted by some angles new.
This thing Archytas hath not done
Nor noble sire of Icarus
Nor son of thine, Iapetus.
What accident or god can then
Have quadrated mine area?"
The Author Replies.
"By deepest mouth of Turia
And lake of limpid clearness, lies
A happy state not far removed
From old Saguntus; farther yet
A little way from Sucro town.
In this place doth a poet dwell,
Who oft the stars will closely scan,
And always for himself doth claim
What is denied to wiser men;—
An old man musing here and there
And oft forgetful of himself,
Not knowing how to rightly place
The compasses, nor draw a line,
As he doth of himself relate.
This craftsman fine, in sooth it is
Hath quadrated thine area."

62 ^  Pietro Bongo, or Petrus Bungus, was born at Bergamo, and died there in 1601. His work on the Mystery of Numbers is one of the most exhaustive and erudite ones of the mystic writers. The first edition appeared at Bergamo in 1583-84; the second, at Bergamo in 1584-85; the third, at Venice in 1585; the fourth, at Bergamo in 1590; and the fifth, which De Morgan calls the second, in 1591. Other editions, before the Paris edition to which he refers, appeared in 1599 and 1614; and the colophon of the Paris edition is dated 1617. See the editor's Rara Arithmetica, pp. 380-383.

63 ^  William Warburton (1698-1779), Bishop of Gloucester, whose works got him into numerous literary quarrels, being the subject of frequent satire.

64 ^  Thomas Galloway (1796-1851), who was professor of mathematics at Sandhurst for a time, and was later the actuary of the Amicable Life Assurance Company of London. In the latter capacity he naturally came to be associated with De Morgan.

65 ^  Giordano Bruno was born near Naples about 1550. He left the Dominican order to take up Calvinism, and among his publications was L'expulsion de la bête triomphante. He taught philosophy at Paris and Wittenberg, and some of his works were published in England in 1583-86. Whether or not he was roasted alive "for the maintenance and defence of the holy Church," as De Morgan states, depends upon one's religious point of view. At any rate, he was roasted as a heretic.

66 ^  Referring to part of his Discours de la méthode, Leyden, 1637.

67 ^  Bartholomew Legate, who was born in Essex about 1575. He denied the divinity of Christ and was the last heretic burned at Smithfield.

68 ^  Edward Wightman, born probably in Staffordshire. He was anti-Trinitarian, and claimed to be the Messiah. He was the last man burned for heresy in England.

69 ^  Gaspar Schopp, born at Neumarck in 1576, died at Padua in 1649; grammarian, philologist, and satirist.

70 ^  Konrad Ritterhusius, born at Brunswick in 1560; died at Altdorf in 1613. He was a jurist of some power.

71 ^  Johann Jakob Brucker, born at Augsburg in 1696, died there in 1770. He wrote on the history of philosophy (1731-36, and 1742-44).

72 ^  Daniel Georg Morhof, born at Wismar in 1639, died at Lübeck in 1691. He was rector of the University of Kiel, and professor of eloquence, poetry, and history.

73 ^  In the Histoire des Sciences Mathématiques, vol. IV, note X, pp. 416-435 of the 1841 edition.

74 ^  Colenso (1814-1883), missionary bishop of Natal, was one of the leaders of his day in the field of higher biblical criticism. De Morgan must have admired his mathematical works, which were not without merit.

75 ^  Samuel Roffey Maitland, born at London in 1792; died at Gloucester in 1866. He was an excellent linguist and a critical student of the Bible. He became librarian at Lambeth in 1838.

76 ^  Archbishop Howley (1766-1848) was a thorough Tory. He was one of the opponents of the Roman Catholic Relief bill, the Reform bill, and the Jewish Civil Disabilities Relief bill.

77 ^  We have, in America at least, almost forgotten the great stir made by Edward B. Pusey (1800-1882) in the great Oxford movement in the middle of the nineteenth century. He was professor of Hebrew at Oxford, and canon of Christ Church.

78 ^  That is, his Magia universalis naturae et artis sive recondita naturalium et artificialium rerum scientia, Würzburg, 1657, 4to, with editions at Bamberg in 1671, and at Frankfort in 1677. Gaspard Schott (Königshofen 1608, Würzburg 1666) was a physicist and mathematician, devoting most of his attention to the curiosities of his sciences. His type of mind must have appealed to De Morgan.

79 ^  Salicetti Quadratura circuli nova, perspicua, expedita, veraque tum naturalis, tum geometrica, etc., 1608.—Consideratio nova in opusculum Archimedis de circuli dimensione, etc., 1609.

80 ^  Melchior Adam, who died at Heidelberg in 1622, wrote a collection of biographies which was published at Heidelberg and Frankfort from 1615 to 1620.

81 ^  Born at Baden in 1524; died at Basel in 1583. The Erastians were related to the Zwinglians, and opposed all power of excommunication and the infliction of penalties by a church.

82 ^  See Acts xii. 20.

83 ^  Theodore de Bèse, a French theologian; born at Vezelay, in Burgundy, in 1519; died at Geneva, in 1605.

84 ^  Dr. Robert Lee (1804-1868) had some celebrity in De Morgan's time through his attempt to introduce music and written prayers into the service of the Scotch Presbyterian church.

85 ^  Born at Veringen, Hohenzollern, in 1512; died at Röteln in 1564.

86 ^  Born at Kinnairdie, Bannfshire, in 1661; died at London in 1708. His Astronomiae Physicae et Geometriae Elementa, Oxford, 1702, was an influential work.

87 ^  The title was carelessly copied by De Morgan, not an unusual thing in his case. The original reads: A Plaine Discovery, of the whole Revelation of S. Iohn: set downe in two treatises ... set foorth by John Napier L. of Marchiston ... whereunto are annexed, certaine Oracles of Sibylla ... London ... 1611.

88 ^  I have not seen the first edition, but it seems to have appeared in Edinburgh, in 1593, with a second edition there in 1594. The 1611 edition was the third.

89 ^  It seems rather certain that Napier felt his theological work of greater importance than that in logarithms. He was born at Merchiston, near (now a part of) Edinburgh, in 1550, and died there in 1617, three years after the appearance of his Mirifici logarithmorum canonis descriptio.

90 ^  Followed, in the third edition, from which he quotes, by a comma.