Budget of Paradoxes/D

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A Budget of Paradoxes by Augustus De Morgan

[ 108 ]


A late discourse ... by Sir Kenelme Digby.... Rendered into English by R. White. London, 1658, 12mo.

On this work see Notes and Queries, 2d series, vii. 231, 299, 445, viii. 190. It contains the celebrated sympathetic powder. I am still in much doubt as to the connection of Digby with this tract.[191] Without entering on the subject here, I observe that in Birch's History of the Royal Society,[192] to which both Digby and White belonged, Digby, though he brought many things before the Society, never mentioned the powder, which is connected only with the names of Evelyn[193] and Sir Gilbert Talbot.[194] The sympathetic powder was that which cured by anointing the weapon with its salve instead of the wound. I have long been convinced that it was efficacious. The directions were to keep the [ 109 ] wound clean and cool, and to take care of diet, rubbing the salve on the knife or sword.[195] If we remember the dreadful notions upon drugs which prevailed, both as to quantity and quality, we shall readily see that any way of not dressing the wound would have been useful. If the physicians had taken the hint, had been careful of diet etc., and had poured the little barrels of medicine down the throat of a practicable doll, they would have had their magical cures as well as the surgeons.[196] Matters are much improved now; the quantity of medicine given, even by orthodox physicians, would have been called infinitesimal by their professional ancestors. Accordingly, the College of Physicians has a right to abandon its motto, which is Ars longa, vita brevis, meaning Practice is long, so life is short.



Examinatio et emendatio Mathematicæ Hodiernæ. By Thomas Hobbes. London, 1666, 4to.

In six dialogues: the sixth contains a quadrature of the circle.[197] But there is another edition of this work, without place or date on the title-page, in which the quadrature is omitted. This seems to be connected with the publication [ 110 ] of another quadrature, without date, but about 1670, as may be judged from its professing to answer a tract of Wallis, printed in 1669.[198] The title is "Quadratura circuli, cubatio sphæræ, duplicatio cubi," 4to.[199] Hobbes, who began in 1655, was very wrong in his quadrature; but, though not a Gregory St. Vincent,[200] he was not the ignoramus in geometry that he is sometimes supposed. His writings, erroneous as they are in many things, contain acute remarks on points of principle. He is wronged by being coupled with Joseph Scaliger, as the two great instances of men of letters who have come into geometry to help the mathematicians out of their difficulty. I have never seen Scaliger's quadrature,[201] except in the answers of Adrianus Romanus,[202] Vieta and Clavius, and in the extracts of Kastner.[203] Scaliger had no right to such strong opponents: Erasmus or Bentley might just as well have tried the problem, and either would have done much better in any twenty minutes of his life.[204]



Scaliger inspired some mathematicians with great respect for his geometrical knowledge. Vieta, the first man of his time, who answered him, had such regard for his opponent [ 111 ] as made him conceal Scaliger's name. Not that he is very respectful in his manner of proceeding: the following dry quiz on his opponent's logic must have been very cutting, being true. "In grammaticis, dare navibus Austros, et dare naves Austris, sunt æque significantia. Sed in Geometricis, aliud est adsumpsisse circulum BCD non esse majorem triginta sex segmentis BCDF, aliud circulo BCD non esse majora triginta sex segmenta BCDF. Illa adsumptiuncula vera est, hæc falsa."[205] Isaac Casaubon,[206] in one of his letters to De Thou,[207] relates that, he and another paying a visit to Vieta, the conversation fell upon Scaliger, of whom the host said that he believed Scaliger was the only man who perfectly understood mathematical writers, especially the Greek ones: and that he thought more of Scaliger when wrong than of many others when right; "pluris se Scaligerum vel errantem facere quam multos κατορθούντας."[208] This must have been before Scaliger's quadrature (1594). There is an old story of some one saying, "Mallem cum Scaligero errare, quam cum Clavio recte sapere."[209] This I cannot help suspecting to have been a version of Vieta's speech with Clavius satirically inserted, on account of the great hostility which Vieta showed towards Clavius in the latter years of his life.

Montucla could not have read with care either Scaliger's quadrature or Clavius's refutation. He gives the first a wrong date: he assures the world that there is no question about Scaliger's quadrature being wrong, in the eyes of geometers at least: and he states that Clavius mortified him [ 112 ] extremely by showing that it made the circle less than its inscribed dodecagon, which is, of course, equivalent to asserting that a straight line is not always the shortest distance between two points. Did Clavius show this? No, it was Scaliger himself who showed it, boasted of it, and declared it to be a "noble paradox" that a theorem false in geometry is true in arithmetic; a thing, he says with great triumph, not noticed by Archimedes himself! He says in so many words that the periphery of the dodecagon is greater than that of the circle; and that the more sides there are to the inscribed figure, the more does it exceed the circle in which it is. And here are the words, on the independent testimonies of Clavius and Kastner:

"Ambitus dodecagoni circulo inscribendi plus potest quam circuli ambitus. Et quanto deinceps plurium laterum fuerit polygonum circulo inscribendum, tanto plus poterit ambitus polygoni quam ambitus circuli."[210]

There is much resemblance between Joseph Scaliger and William Hamilton,[211] in a certain impetuousity of character, and inaptitude to think of quantity. Scaliger maintained that the arc of a circle is less than its chord in arithmetic, though greater in geometry; Hamilton arrived at two quantities which are identical, but the greater the one the less the other. But, on the whole, I liken Hamilton rather to Julius than to Joseph. On this last hero of literature I repeat Thomas Edwards,[212] who says that a man is unlearned who, be his other knowledge what it may, does not [ 113 ] understand the subject he writes about. And now one of many instances in which literature gives to literature character in science. Anthony Teissier,[213] the learned annotator of De Thou's biographies, says of Finæus, "Il se vanta sans raison avoir trouvé la quadrature du cercle; la gloire de cette admirable découverte était réservée à Joseph Scalinger, comme l'a écrit Scévole de St. Marthe."[214]



Natural and Political Observations ... upon the Bills of Mortality. By John Graunt, citizen of London. London, 1662, 4to.[215]

This is a celebrated book, the first great work upon mortality. But the author, going ultra crepidam, has attributed to the motion of the moon in her orbit all the tremors which she gets from a shaky telescope.[216] But there is another paradox about this book: the above absurd opinion is attributed to that excellent mechanist, Sir William Petty, who passed his days among the astronomers. Graunt did not write his own book! Anthony Wood[217] hints that Petty "assisted, or put into a way" his old benefactor: no doubt the two friends talked the matter over many a time. Burnet and Pepys[218] state that Petty wrote the book. It is enough for me that [ 114 ] Graunt, whose honesty was never impeached, uses the plainest incidental professions of authorship throughout; that he was elected into the Royal Society because he was the author; that Petty refers to him as author in scores of places, and published an edition, as editor, after Graunt's death, with Graunt's name of course. The note on Graunt in the Biographia Britannica may be consulted; it seems to me decisive. Mr. C. B. Hodge, an able actuary, has done the best that can be done on the other side in the Assurance Magazine, viii. 234. If I may say what is in my mind, without imputation of disrespect, I suspect some actuaries have a bias: they would rather have Petty the greater for their Coryphæus than Graunt the less.[219]

Pepys is an ordinary gossip: but Burnet's account has an animus which is of a worse kind. He talks of "one Graunt, a Papist, under whose name Sir William Petty[220] published his observations on the bills of mortality." He then gives the cock without a bull story of Graunt being a trustee of the New River Company, and shutting up the cocks and carrying off their keys, just before the fire of London, by which a supply of water was delayed.[221] It was one of the first objections made to Burnet's work, that Graunt was not a trustee at the time; and Maitland, the historian of London, ascertained from the books of the Company that he was not admitted until twenty-three days after the breaking out of the fire. Graunt's first admission [ 115 ] to the Company took place on the very day on which a committee was appointed to inquire into the cause of the fire. So much for Burnet. I incline to the view that Graunt's setting London on fire strongly corroborates his having written on the bills of mortality: every practical man takes stock before he commences a grand operation in business.



De Cometis: or a discourse of the natures and effects of Comets, as they are philosophically, historically, and astrologically considered. With a brief (yet full) account of the III late Comets, or blazing stars, visible to all Europe. And what (in a natural way of judicature) they portend. Together with some observations on the nativity of the Grand Seignior. By John Gadbury, Φιλομαθηματικός. London, 1665, 4to.

Gadbury, though his name descends only in astrology, was a well-informed astronomer.[222] D'Israeli[223] sets down Gadbury, Lilly, Wharton, Booker, etc., as rank rogues: I think him quite wrong. The easy belief in roguery and intentional imposture which prevails in educated society is, to my mind, a greater presumption against the honesty of mankind than all the roguery and imposture itself. Putting aside mere swindling for the sake of gain, and looking at speculation and paradox, I find very little reason to suspect wilful deceit.[224] My opinion of mankind is founded upon the [ 116 ] mournful fact that, so far as I can see, they find within themselves the means of believing in a thousand times as much as there is to believe in, judging by experience. I do not say anything against Isaac D'Israeli for talking his time. We are all in the team, and we all go the road, but we do not all draw.



An essay towards a real character and a philosophical language. By John Wilkins [Dean of Ripon, afterwards Bishop of Chester].[225] London, 1668, folio.

This work is celebrated, but little known. Its object gives it a right to a place among paradoxes. It proposes a language—if that be the proper name—in which things and their relations shall be denoted by signs, not words: so that any person, whatever may be his mother tongue, may read it in his own words. This is an obvious possibility, and, I am afraid, an obvious impracticability. One man may construct such a system—Bishop Wilkins has done it—but where is the man who will learn it? The second tongue makes a language, as the second blow makes a fray. There has been very little curiosity about his performance, the work is scarce; and I do not know where to refer the reader for any account of its details, except, to the partial reprint of Wilkins presently mentioned under 1802, in which there is an unsatisfactory abstract. There is nothing in the Biographia Britannica, except discussion of Anthony Wood's statement that the hint was derived from Dalgarno's book, [ 117 ] De Signis, 1661.[226] Hamilton (Discussions, Art. 5, "Dalgarno") does not say a word on this point, beyond quoting Wood; and Hamilton, though he did now and then write about his countrymen with a rough-nibbed pen, knew perfectly well how to protect their priorities.



Problema Austriacum. Plus ultra Quadratura Circuli. Auctore P. Gregorio a Sancto Vincentio Soc. Jesu., Antwerp, 1647, folio.—Opus Geometricum posthumum ad Mesolabium. By the same. Gandavi [Ghent], 1668, folio.[227]

The first book has more than 1200 pages, on all kinds of geometry. Gregory St. Vincent is the greatest of circle-squarers, and his investigations led him into many truths: he found the property of the area of the hyperbola[228] which led to Napier's logarithms being called hyperbolic. Montucla says of him, with sly truth, that no one has ever squared the circle with so much genius, or, excepting his principal object, with so much success.[229] His reputation, and the many merits of his work, led to a sharp controversy on his quadrature, which ended in its complete exposure by Huyghens and others. He had a small school of followers, who defended him in print.

[ 118 ]



Renati Francisci Slusii Mesolabum. Leodii Eburonum [Liège], 1668, 4to.[230]

The Mesolabum is the solution of the problem of finding two mean proportionals, which Euclid's geometry does not attain. Slusius is a true geometer, and uses the ellipse, etc.: but he is sometimes ranked with the trisecters, for which reason I place him here, with this explanation.

The finding of two mean proportionals is the preliminary to the famous old problem of the duplication of the cube, proposed by Apollo (not Apollonius) himself. D'Israeli speaks of the "six follies of science,"—the quadrature, the duplication, the perpetual motion, the philosopher's stone, magic, and astrology. He might as well have added the trisection, to make the mystic number seven: but had he done so, he would still have been very lenient; only seven follies in all science, from mathematics to chemistry! Science might have said to such a judge—as convicts used to say who got seven years, expecting it for life, "Thank you, my Lord, and may you sit there till they are over,"—may the Curiosities of Literature outlive the Follies of Science!



1668. In this year James Gregory, in his Vera Circuli et Hyperbolæ Quadratura,[231] held himself to have proved that [ 119 ] the geometrical quadrature of the circle is impossible. Few mathematicians read this very abstruse speculation, and opinion is somewhat divided. The regular circle-squarers attempt the arithmetical quadrature, which has long been proved to be impossible. Very few attempt the geometrical quadrature. One of the last is Malacarne, an Italian, who published his Solution Géométrique, at Paris, in 1825. His method would make the circumference less than three times the diameter.



La Géométrie Françoise, ou la Pratique aisée.... La quadracture du cercle. Par le Sieur de Beaulieu, Ingénieur, Géographe du Roi ... Paris, 1676, 8vo. [not Pontault de Beaulieu, the celebrated topographer; he died in 1674].[232]

If this book had been a fair specimen, I might have pointed to it in connection with contemporary English works, and made a scornful comparison. But it is not a fair specimen. Beaulieu was attached to the Royal Household, and throughout the century it may be suspected that the household forced a royal road to geometry. Fifty years before, Beaugrand, the king's secretary, made a fool of himself, and [so?] contrived to pass for a geometer. He had interest enough to get Desargues, the most powerful geometer of his time,[233] the teacher and friend of Pascal, prohibited from [ 120 ] lecturing. See some letters on the History of Perspective, which I wrote in the Athenæum, in October and November, 1861. Montucla, who does not seem to know the true secret of Beaugrand's greatness, describes him as "un certain M. de Beaugrand, mathématicien, fort mal traité par Descartes, et à ce qu'il paroit avec justice."[234]

Beaulieu's quadrature amounts to a geometrical construction[235] which gives \scriptstyle\pi = \sqrt{10}. His depth may be ascertained from the following extracts. First on Copernicus:

"Copernic, Allemand, ne s'est pas moins rendu illustre par ses doctes écrits; et nous pourrions dire de luy, qu'il seroit le seul et unique en la force de ses Problèmes, si sa trop grande présomption ne l'avoit porté à avancer en cette Science une proposition aussi absurde, qu'elle est contre la Foy et raison, en faisant la circonférence d'un Cercle fixe, immobile, et le centre mobile, sur lequel principe Géométrique, il a avancé en son Traitté Astrologique le Soleil fixe, et la Terre mobile."[236]

I digress here to point out that though our quadrators, etc., very often, and our historians sometimes, assert that men of the character of Copernicus, etc., were treated with contempt and abuse until their day of ascendancy came, nothing can be more incorrect. From Tycho Brahé[237] to Beaulieu, there is but one expression of admiration for the genius of Copernicus. There is an exception, which, I [ 121 ] believe, has been quite misunderstood. Maurolycus,[238] in his De Sphæra, written many years before its posthumous publication in 1575, and which it is not certain he would have published, speaking of the safety with which various authors may be read after his cautions, says, "Toleratur et Nicolaus Copernicus qui Solem fixum et Terram in girum circumverti posuit: et scutica potius, aut flagello, quam reprehensione dignus est."[239] Maurolycus was a mild and somewhat contemptuous satirist, when expressing disapproval: as we should now say, he pooh-poohed his opponents; but, unless the above be an instance, he was never savage nor impetuous. I am fully satisfied that the meaning of the sentence is, that Copernicus, who turned the earth like a boy's top, ought rather to have a whip given him wherewith to keep up his plaything than a serious refutation. To speak of tolerating a person as being more worthy of a flogging than an argument, is almost a contradiction.

I will now extract Beaulieu's treatise on algebra, entire.

"L'Algebre est la science curieuse des Sçavans et specialement d'un General d'Armée ou Capitaine, pour promptement ranger une Armée en bataille, et nombre de Mousquetaires et Piquiers qui composent les bataillons d'icelle, outre les figures de l'Arithmetique. Cette science a 5 figures particulieres en cette sorte. P signifie plus au commerce, et à l'Armée Piquiers. M signifie moins, et Mousquetaire en l'Art des bataillons. [It is quite true that P and M were used for plus and minus in a great many old works.] R signifie racine en la mesure du Cube, et en l'Armée rang. Q signifie quaré en l'un et l'autre usage. C signifie cube en la mesure, et Cavallerie en la composition des bataillons et escadrons. Quant à l'operation de cette science, c'est [ 122 ] d'additionner un plus d'avec plus, la somme sera plus, et moins d'avec plus, on soustrait le moindre du plus, et la reste est la somme requise ou nombre trouvé. Je dis seulement cecy en passant pour ceux qui n'en sçavent rien du tout."[240]

This is the algebra of the Royal Household, seventy-three years after the death of Vieta. Quære, is it possible that the fame of Vieta, who himself held very high stations in the household all his life, could have given people the notion that when such an officer chose to declare himself an algebraist, he must be one indeed? This would explain Beaugrand, Beaulieu, and all the beaux. Beaugrand—not only secretary to the king, but "mathematician" to the Duke of Orleans—I wonder what his "fool" could have been like, if indeed he kept the offices separate,—would have been in my list if I had possessed his Geostatique, published about 1638.[241] He makes bodies diminish in weight as they approach the earth, because the effect of a weight on a lever is less as it approaches the fulcrum.

[ 123 ]



Remarks upon two late ingenious discourses.... By Dr. Henry More.[242] London, 1676, 8vo.

In 1673 and 1675, Matthew Hale,[243] then Chief Justice, published two tracts, an "Essay touching Gravitation," and "Difficiles Nugæ" on the Torricellian experiment. Here are the answers by the learned and voluminous Henry More. The whole would be useful to any one engaged in research about ante-Newtonian notions of gravitation.


Observations touching the principles of natural motions; and especially touching rarefaction and condensation.... By the author of Difficiles Nugæ. London, 1677, 8vo.

This is another tract of Chief Justice Hale, published the year after his death. The reader will remember that motion, in old philosophy, meant any change from state to state: what we now describe as motion was local motion. This is a very philosophical book, about flux and materia prima, virtus activa and essentialis, and other fundamentals. I think Stephen Hales, the author of the "Vegetable Statics," has the writings of the Chief Justice sometimes attributed to him, which is very puny justice indeed.[244] Matthew Hale died in 1676, and from his devotion to science it probably arose that his famous Pleas of the Crown[245] and other law works did not appear until after his death. One of his [ 124 ] contemporaries was the astronomer Thomas Street, whose Caroline Tables[246] were several times printed: another contemporary was his brother judge, Sir Thomas Street.[247] But of the astronomer absolutely nothing is known: it is very unlikely that he and the judge were the same person, but there is not a bit of positive evidence either for or against, so far as can be ascertained. Halley[248]—no less a person—published two editions of the Caroline Tables, no doubt after the death of the author: strange indeed that neither Halley nor any one else should leave evidence that Street was born or died.

Matthew Hale gave rise to an instance of the lengths a lawyer will go when before a jury who cannot detect him. Sir Samuel Shepherd,[249] the Attorney General, in opening Hone's[250] first trial, calls him "one who was the most learned man that ever adorned the Bench, the most even man that ever blessed domestic life, the most eminent man that ever advanced the progress of science, and one of the [very moderate] best and most purely religious men that ever lived."

[ 125 ]



Basil Valentine his triumphant Chariot of Antimony, with annotations of Theodore Kirkringius, M.D. With the true book of the learned Synesius, a Greek abbot, taken out of the Emperour's library, concerning the Philosopher's Stone. London, 1678, 8vo.[251]

There are said to be three Hamburg editions of the collected works of Valentine, who discovered the common antimony, and is said to have given the name antimoine, in a curious way. Finding that the pigs of his convent throve upon it, he gave it to his brethren, who died of it.[252] The impulse given to chemistry by R. Boyle[253] seems to have brought out a vast number of translations, as in the following tract:



Collectanea Chymica: A collection of ten several treatises in chymistry, concerning the liquor Alkehest, the Mercury of Philosophers, and other curiosities worthy the perusal. Written by Eir. Philaletha,[254] Anonymus, J. B. Van-Helmont,[255] Dr. Fr. [ 126 ] Antonie,[256] Bernhard Earl of Trevisan,[257] Sir Geo. Ripley,[258] Rog. Bacon,[259] Geo. Starkie,[260] Sir Hugh Platt,[261] and the Tomb of Semiramis. See more in the contents. London, 1684, 8vo.

In the advertisements at the ends of these tracts there are upwards of a hundred English tracts, nearly all of the period, and most of them translations. Alchemy looks up since the chemists have found perfectly different substances composed of the same elements and proportions. It is true the chemists cannot yet transmute; but they may in time: they poke about most assiduously. It seems, then, that the conviction that alchemy must be impossible was a delusion: but we do not mention it.

[ 127 ]

The astrologers and the alchemists caught it in company in the following, of which I have an unreferenced note.

"Mendacem et futilem hominem nominare qui volunt, calendariographum dicunt; at qui sceleratum simul ac impostorem, chimicum.[262]

"Crede ratem ventis corpus ne crede chimistis;
Est quævis chimica tutior aura fide."[263]

Among the smaller paradoxes of the day is that of the Times newspaper, which always spells it chymistry: but so, I believe, do Johnson, Walker, and others. The Arabic work is very likely formed from the Greek: but it may be connected either with χημεια or with χυμεια.


Lettre d'un gentil-homme de province à une dame de qualité, sur le sujet de la Comète. Paris, 1681, 4to.

An opponent of astrology, whom I strongly suspect to have been one of the members of the Academy of Sciences under the name of a country gentleman,[264] writes very good sense on the tremors excited by comets.


The Petitioning-Comet: or a brief Chronology of all the famous Comets and their events, that have happened from the birth of Christ to this very day. Together with a modest enquiry into this present comet, London, 1681, 4to.

A satirical tract against the cometic prophecy:

"This present comet (it's true) is of a menacing aspect, but if the new parliament (for whose convention so many good men pray) continue long to sit, I fear not but the star will lose its virulence and malignancy, or at least its portent be averted from this our nation; which being the humble request to God of all good men, makes me thus entitle it, a Petitioning-Comet."

[ 128 ]

The following anecdote is new to me:

"Queen Elizabeth (1558) being then at Richmond, and being disswaded from looking on a comet which did then appear, made answer, jacta est alea, the dice are thrown; thereby intimating that the pre-order'd providence of God was above the influence of any star or comet."

The argument was worth nothing: for the comet might have been on the dice with the event; the astrologers said no more, at least the more rational ones, who were about half of the whole.


An astrological and theological discourse upon this present great conjunction (the like whereof hath not (likely) been in some ages) ushered in by a great comet. London, 1682, 4to. By C. N.[265]

The author foretells the approaching "sabbatical jubilee," but will not fix the date: he recounts the failures of his predecessors.


A judgment of the comet which became first generally visible to us in Dublin, December 13, about 15 minutes before 5 in the evening, A.D. 1680. By a person of quality. Dublin, 1682, 4to.

The author argues against cometic astrology with great ability.


A prophecy on the conjunction of Saturn and Jupiter in this present year 1682. With some prophetical predictions of what is likely to ensue therefrom in the year 1684. By John Case, Student in physic and astrology.[266] London, 1682, 4to.

[ 129 ]

According to this writer, great conjunctions of Jupiter and Saturn occur "in the fiery trigon," about once in 800 years. Of these there are to be seven: six happened in the several times of Enoch, Noah, Moses, Solomon, Christ, Charlemagne. The seventh, which is to happen at "the lamb's marriage with the bride," seems to be that of 1682; but this is only vaguely hinted.


De Quadrature van de Circkel. By Jacob Marcelis. Amsterdam, 1698, 4to.
Ampliatie en demonstratie wegens de Quadrature ... By Jacob Marcelis. Amsterdam, 1699, 4to.
Eenvoudig vertoog briev-wys geschrevem am J. Marcelis ... Amsterdam, 1702, 4to.
De sleutel en openinge van de quadrature ... Amsterdam, 1704, 4to.

Who shall contradict Jacob Marcelis?[267] He says the circumference contains the diameter exactly times


But he does not come very near, as the young arithmetician will find.



Theologiæ Christianæ Principia Mathematica. Auctore Johanne Craig.[268] London, 1699, 4to.

This is a celebrated speculation, and has been reprinted abroad, and seriously answered. Craig is known in the early history of fluxions, and was a good mathematician. [ 130 ] He professed to calculate, on the hypothesis that the suspicions against historical evidence increase with the square of the time, how long it will take the evidence of Christianity to die out. He finds, by formulæ, that had it been oral only, it would have gone out A.D. 800; but, by aid of the written evidence, it will last till A.D. 3150. At this period he places the second coming, which is deferred until the extinction of evidence, on the authority of the question "When the Son of Man cometh, shall he find faith on the earth?" It is a pity that Craig's theory was not adopted: it would have spared a hundred treatises on the end of the world, founded on no better knowledge than his, and many of them falsified by the event. The most recent (October, 1863) is a tract in proof of Louis Napoleon being Antichrist, the Beast, the eighth Head, etc.; and the present dispensation is to close soon after 1864.

In order rightly to judge Craig, who added speculations on the variations of pleasure and pain treated as functions of time, it is necessary to remember that in Newton's day the idea of force, as a quantity to be measured, and as following a law of variation, was very new: so likewise was that of probability, or belief, as an object of measurement.[269] The success of the Principia of Newton put it into many heads to speculate about applying notions of quantity to other things not then brought under measurement. Craig imitated Newton's title, and evidently thought he was making a step in advance: but it is not every one who can plough with Samson's heifer.

It is likely enough that Craig took a hint, directly or indirectly, from Mohammedan writers, who make a reply to the argument that the Koran has not the evidence derived [ 131 ] from miracles. They say that, as evidence of Christian miracles is daily becoming weaker, a time must at last arrive when it will fail of affording assurance that they were miracles at all: whence would arise the necessity of another prophet and other miracles. Lee,[270] the Cambridge Orientalist, from whom the above words are taken, almost certainly never heard of Craig or his theory.


191 ^  There is some substantial basis for De Morgan's doubts as to the connection of that mirandula of his age, Sir Kenelm Digby (1603-1665), with the famous poudre de sympathie. It is true that he was just the one to prepare such a powder. A dilletante in everything,—learning, war, diplomacy, religion, letters, and science—he was the one to exploit a fraud of this nature. He was an astrologer, an alchemist, and a fabricator of tales, and well did Henry Stubbes characterize him as "the very Pliny of our age for lying." He first speaks of the powder in a lecture given at Montpellier in 1658, and in the same year he published the address at Paris under the title: Discours fait en une célèbre assemblée par le chevalier Digby .... touchant la guérison de playes par la poudre de sympathie. The London edition referred to by De Morgan also came out in 1658, and several editions followed it in England, France and Germany. But Nathaniel Highmore in his History of Generation (1651) referred to the concoction as "Talbot's Powder" some years before Digby took it up. The basis seems to have been vitriol, and it was claimed that it would heal a wound by simply being applied to a bandage taken from it.

192 ^  This work by Thomas Birch (1705-1766) came out in 1756-57. Birch was a voluminous writer on English history. He was a friend of Dr. Johnson and of Walpole, and he wrote a life of Robert Boyle.

193 ^  We know so much about John Evelyn (1620-1706) through the diary which he began at the age of eleven, that we forget his works on navigation and architecture.

194 ^  I suppose this was the seventh Earl of Shrewsbury (1553-1616).

195 ^  This is interesting in view of the modern aseptic practice of surgery and the antiseptic treatment of wounds inaugurated by the late Lord Lister.

196 ^  Perhaps De Morgan had not heard the bon mot of Dr. Holmes: "I firmly believe that if the whole materia medica could be sunk to the bottom of the sea, it would be all the better for mankind and all the worse for the fishes."

197 ^  The full title is worth giving, because it shows the mathematical interests of Hobbes, and the nature of the six dialogues: Examinatio et emendatio mathematicae hodiernae qualis explicatur in libris Johannis Wallisii geometriae professoris Saviliani in Academia Oxoniensi: distributa in sex dialogos (1. De mathematicae origine ...; 2. De principiis traditis ab Euclide; 3. De demonstratione operationum arithmeticarum ...; 4. De rationibus; 5. De angula contactus, de sectionibus coni, et arithmetica infinitorum; 6. Dimensio circuli tribus methodis demonstrata ... item cycloidis verae descriptio et proprietates aliquot.) Londini, 1660 (not 1666). For a full discussion of the controversy over the circle, see George Croom Robertson's biography of Hobbes in the eleventh edition of the Encyclopaedia Britannica.

198 ^  This is his Animadversions upon Mr. Hobbes' late book De principiis et ratiocinatione geometrarum, 1666, or his Hobbianae quadraturae circuli, cubationis sphaerae et duplicationis cubi confutatio, also of 1669.

199 ^  This is the work of 1669 referred to above.

200 ^  Gregoire de St. Vincent (1584-1667) published his Opus geometricum quadraturae circuli et sectionum coni at Antwerp in 1647.

201 ^  This appears in J. Scaligeri cyclometrica elementa duo, Lugduni Batav., 1594.

202 ^  Adriaen van Roomen (1561-1615) gave the value of \scriptstyle\pi to sixteen decimal places in his Ideae mathematicae pars prima (1593), and wrote his In Archimedis circuli dimensionem expositio & analysis in 1597.

203 ^  Kästner. See note 30.

204 ^  Bentley (1662-1742) might have done it, for as the head of Trinity College, Cambridge, and a follower of Newton, he knew some mathematics. Erasmus (1466-1536) lived a little too early to attempt it, although his brilliant satire might have been used to good advantage against those who did try.

205 ^  "In grammar, to give the winds to the ships and to give the ships to the winds mean the same thing. But in geometry it is one thing to assume the circle BCD not greater than thirty-six segments BCDF, and another (to assume) the thirty-six segments BCDF not greater than the circle. The one assumption is true, the other false."

206 ^  The Greek scholar (1559-1614) who edited a Greek and Latin edition of Aristotle in 1590.

207 ^  Jacques Auguste de Thou (1553-1617), the historian and statesman.

208 ^  "To value Scaliger higher even when wrong, than the multitude when right."

209 ^  "I would rather err with Scaliger than be right with Clavius."

210 ^  "The perimeter of the dodecagon to be inscribed in a circle is greater than the perimeter of the circle. And the more sides a polygon to be inscribed in a circle successively has, so much the greater will the perimeter of the polygon be than the perimeter of the circle."

211 ^  De Morgan took, perhaps, the more delight in speaking thus of Sir William Hamilton (1788-1856) because of a spirited controversy that they had in 1847 over the theory of logic. Possibly, too, Sir William's low opinion of mathematics had its influence.

212 ^  Edwards (1699-1757) wrote The canons of criticism (1747) in which he gave a scathing burlesque on Warburton's Shakespeare. It went through six editions.

213 ^  Antoine Teissier (born in 1632) published his Eloges des hommes savants, tirés de l'histoire de M. de Thou in 1683.

214 ^  "He boasted without reason of having found the quadrature of the circle. The glory of this admirable discovery was reserved for Joseph Scaliger, as Scévole de St. Marthe has written."

215 ^  Natural and political observations mentioned in the following Index, and made upon the Bills of Mortality.... With reference to the government, religion, trade, growth, ayre, and diseases of the said city. London, 1662, 4to. The book went through several editions.

216 ^  Ne sutor ultra crepidam, "Let the cobbler stick to his last," as we now say.

217 ^  The author (1632-1695) of the Historia et Antiquitates Universitatis Oxoniensis (1674). See note 163.

218 ^  The mathematical guild owes Samuel Pepys (1633-1703) for something besides his famous diary (1659-1669). Not only was he president of the Royal Society (1684), but he was interested in establishing Sir William Boreman's mathematical school at Greenwich.

219 ^  John Graunt (1620-1674) was a draper by trade, and was a member of the Common Council of London until he lost office by turning Romanist. Although a shopkeeper, he was elected to the Royal Society on the special recommendation of Charles II. Petty edited the fifth edition of his work, adding much to its size and value, and this may be the basis of Burnet's account of the authorship.

220 ^  Petty (1623-1687) was a mathematician and economist, and a friend of Pell and Sir Charles Cavendish. His survey of Ireland, made for Cromwell, was one of the first to be made on a large scale in a scientific manner. He was one of the founders of the Royal Society.

221 ^  The story probably arose from Graunt's recent conversion to the Roman Catholic faith.

222 ^  He was born in 1627 and died in 1704. He published a series of ephemerides, beginning in 1659. He was imprisoned in 1679, at the time of the "Popish Plot," and again for treason in 1690. His important astrological works are the Animal Cornatum, or the Horn'd Beast (1654) and The Nativity of the late King Charls (1659).

223 ^  Isaac D'Israeli (1766-1848), in his Curiosities of Literature (1791), speaking of Lilly, says: "I shall observe of this egregious astronomer, that there is in this work, so much artless narrative, and at the same time so much palpable imposture, that it is difficult to know when he is speaking what he really believes to be the truth." He goes on to say that Lilly relates that "those adepts whose characters he has drawn were the lowest miscreants of the town. Most of them had taken the air in the pillory, and others had conjured themselves up to the gallows. This seems a true statement of facts."

224 ^  It is difficult to estimate William Lilly (1602-1681) fairly. His Merlini Anglici ephemeris, issued annually from 1642 to 1681, brought him a great deal of money. Sir George Wharton (1617-1681) also published an almanac annually from 1641 to 1666. He tried to expose John Booker (1603-1677) by a work entitled Mercurio-Coelicio-Mastix; or, an Anti-caveat to all such, as have (heretofore) had the misfortune to be Cheated and Deluded by that Grand and Traiterous Impostor of this Rebellious Age, John Booker, 1644. Booker was "licenser of mathematical [astrological] publications," and as such he had quarrels with Lilly, Wharton, and others.

225 ^  See note 171.

226 ^  This is the Ars Signorum, vulgo character universalis et lingua philosophica, that appeared at London in 1661, 8vo. George Dalgarno anticipated modern methods in the teaching of the deaf and dumb.

227 ^  See note 200.

228 ^  If the hyperbola is referred to the asymptotes as axes, the area between two ordinates (x = a, x = b) is the difference of the logarithms of a and b to the base e. E.g., in the case of the hyperbola xy = 1, the area between x = a and x = 1 is \log{a}.

229 ^  "On ne peut lui refuser la justice de remarquer que personne avant lui ne s'est porté dans cette recherche avec autant de génie, & même, si nous en exceptons son objet principal, avec autant de succès." Quadrature du Cercle, p. 66.

230 ^  The title proceeds: Seu duae mediae proportionales inter extremas datas per circulum et per infinitas hyperbolas, vel ellipses et per quamlibet exhibitae.... René Francois, Baron de Sluse (1622-1685) was canon and chancellor of Liège, and a member of the Royal Society. He also published a work on tangents (1672). The word mesolabium is from the Greek μεσολάβιον or μεσόλαβον, an instrument invented by Eratosthenes for finding two mean proportionals.

231 ^  The full title has some interest: Vera circuli et hyperbolae quadratura cui accedit geometriae pars universalis inserviens quantitatum curvarum transmutationi et mensurae. Authore Jacobo Gregorio Abredonensi Scoto ... Patavii, 1667. That is, James Gregory (1638-1675) of Aberdeen (he was really born near but not in the city), a good Scot, was publishing his work down in Padua. The reason was that he had been studying in Italy, and that this was a product of his youth. He had already (1663) published his Optica promota, and it is not remarkable that his brilliancy brought him a wide circle of friends on the continent and the offer of a pension from Louis XIV. He became professor of mathematics at St Andrews and later at Edinburgh, and invented the first successful reflecting telescope. The distinctive feature of his Vera quadratura is his use of an infinite converging series, a plan that Archimedes used with the parabola.

232 ^  Jean de Beaulieu wrote several works on mathematics, including La lumière de l'arithmétique (n.d.), La lumière des mathématiques (1673), Nouvelle invention d'arithmétique (1677), and some mathematical tables.

233 ^  A just estimate. There were several works published by Gérard Desargues (1593-1661), of which the greatest was the Brouillon Proiect (Paris, 1639). There is an excellent edition of the Œuvres de Desargues by M. Poudra, Paris, 1864.

234 ^  "A certain M. de Beaugrand, a mathematician, very badly treated by Descartes, and, as it appears, rightly so."

235 ^  This is a very old approximation for \pi. One of the latest pretended geometric proofs resulting in this value appeared in New York in 1910, entitled Quadrimetry (privately printed).

236 ^  "Copernicus, a German, made himself no less illustrious by his learned writings; and we might say of him that he stood alone and unique in the strength of his problems, if his excessive presumption had not led him to set forth in this science a proposition so absurd that it is contrary to faith and reason, namely that the circumference of a circle is fixed and immovable while the center is movable: on which geometrical principle he has declared in his astrological treatise that the sun is fixed and the earth is in motion."

237 ^  So in the original.

238 ^  Franciscus Maurolycus (1494-1575) was really the best mathematician produced by Sicily for a long period. He made Latin translations of Theodosius, Menelaus, Euclid, Apollonius, and Archimedes, and wrote on cosmography and other mathematical subjects.

239 ^  "Nicolaus Copernicus is also tolerated who asserted that the sun is fixed and that the earth whirls about it; and he rather deserves a whip or a lash than a reproof."

240 ^  "Algebra is the curious science of scholars, and particularly for a general of an army, or a captain, in order quickly to draw up an army in battle array and to number the musketeers and pikemen who compose it, without the figures of arithmetic. This science has five special figures of this kind: P means plus in commerce and pikemen in the army; M means minus, and musketeer in the art of war;... R signifies root in the measurement of a cube, and rank in the army; Q means square (French quarè, as then spelled) in both cases; C means cube in mensuration, and cavalry in arranging batallions and squadrons. As for the operations of this science, they are as follows: to add a plus and a plus, the sum will be plus; to add minus with plus, take the less from the greater and the remainder will be the sum required or the number to be found. I say this only in passing, for the benefit of those who are wholly ignorant of it."

241 ^  He refers to the Joannis de Beaugrand ... Geostatice, seu de vario pondere gravium secundum varia a terrae (centro) intervalla dissertatio mathematica, Paris, 1636. Pascal relates that de Beaugrand sent all of Roberval's theorems on the cycloid and Fermat's on maxima and minima to Galileo in 1638, pretending that they were his own.

242 ^  More (1614-1687) was a theologian, a fellow of Christ College, Cambridge, and a Christian Platonist.

243 ^  Matthew Hale (1609-1676) the famous jurist, wrote a number of tracts on scientific, moral, and religious subjects. These were collected and published in 1805.

244 ^  They might have been attributed to many a worse man than Dr. Hales (1677-1761), who was a member of the Royal Society and of the Paris Academy, and whose scheme for the ventilation of prisons reduced the mortality at the Savoy prison from one hundred to only four a year. The book to which reference is made is Vegetable Staticks or an Account of some statical experiments on the sap in Vegetables, 1727.

245 ^  Pleas of the Crown; or a Methodical Summary of the Principal Matters relating to the subject, 1678.

246 ^  Thomae Streete Astronomia Carolina, a new theory of the celestial motions, 1661. It also appeared at Nuremberg in 1705, and at London in 1710 and 1716 (Halley's editions). He wrote other works on astronomy.

247 ^  This was the Sir Thomas Street (1626-1696) who passed sentence of death on a Roman Catholic priest for saying mass. The priest was reprieved by the king, but in the light of the present day one would think the justice more in need of pardon. He took part in the trial of the Rye House Conspirators in 1683.

248 ^  Edmund Halley (1656-1742), who succeeded Wallis (1703) as Savilian professor of mathematics at Oxford, and Flamsteed (1720) as head of the Greenwich observatory. It is of interest to note that he was instrumental in getting Newton's Principia printed.

249 ^  Shepherd (born in 1760) was one of the most famous lawyers of his day. He was knighted in 1814 and became Attorney General in 1817.

250 ^  This was William Hone (1780-1842), a book publisher, who wrote satires against the government, and who was tried three times because of his parodies on the catechism, creed, and litany (illustrated by Cruikshank). He was acquitted on all of the charges.

251 ^  Valentinus was a Benedictine monk and was still living at Erfurt in 1413. His Currus triumphalis antimonii appeared in 1624. Synesius was Bishop of Ptolemaide, who died about 430. His works were printed at Paris in 1605. Theodor Kirckring (1640-1693) was a fellow-student of Spinoza's. Besides the commentary on Valentine he left several works on anatomy. His commentary appeared at Amsterdam in 1671. There were several editions of the Chariot.

252 ^  The chief difficulty with this curious "monk-bane" etymology is its absurdity. The real origin of the word has given etymologists a good deal of trouble.

253 ^  Robert Boyle (1627-1691), son of "the Great Earl" (of Cork). Perhaps his best-known discovery is the law concerning the volume of gases.

254 ^  The real name of Eirenaeus Philalethes (born in 1622) is unknown. It may have been Childe. He claimed to have discovered the philosopher's stone in 1645. His tract in this work is The Secret of the Immortal Liquor Alkahest or Ignis-Aqua. See note 260, infra.

255 ^  Johann Baptist van Helmont, Herr von Merode, Royenborg etc. (1577-1644). His chemical discoveries appeared in his Ortus medicinae (1648), which went through many editions.

256 ^  De Morgan should have written up Francis Anthony (1550-1623), whose Panacea aurea sive tractatus duo de auro potabili (Hamburg, 1619) described a panacea that he gave for every ill. He was repeatedly imprisoned for practicing medicine without a license from the Royal College of Physicians.

257 ^  Bernardus Trevisanus (1406-1490), who traveled even through Barbary, Egypt, Palestine, and Persia in search of the philosopher's stone. He wrote several works on alchemy,—De Chemica (1567), De Chemico Miraculo (1583), Traité de la nature de l'oeuf des philosophes (1659), etc., all published long after his death.

258 ^  George Ripley (1415-1490) was an Augustinian monk, later a chamberlain of Innocent VIII, and still later a Carmelite monk. His Liber de mercuris philosophico and other tracts first appeared in Opuscula quaedam chymica (Frankfort, 1614).

259 ^  Besides the Opus majus, and other of the better known works of this celebrated Franciscan (1214-1294), there are numerous tracts on alchemy that appeared in the Thesaurus chymicus (Frankfort, 1603).

260 ^  George Starkey (1606-1665 or 1666) has special interest for American readers. He seems to have been born in the Bermudas and to have obtained the bachelor's degree in England. He then went to America and in 1646 obtained the master's degree at Harvard, apparently under the name of Stirk. He met Eirenaeus Philalethes (see note 254 above) in America and learned alchemy from him. Returning to England, he sold quack medicines there, and died in 1666 from the plague after dissecting a patient who had died of the disease. Among his works was the Liquor Alcahest, or a Discourse of that Immortal Dissolvent of Paracelsus and Helmont, which appeared (1675) some nine years after his death.

261 ^  Platt (1552-1611) was the son of a London brewer. Although he left a manuscript on alchemy, and wrote a book entitled Delights for Ladies to adorne their Persons (1607), he was knighted for some serious work on the chemistry of agriculture, fertilizing, brewing, and the preserving of foods, published in The Jewell House of Art and Nature (1594).

262 ^  "Those who wish to call a man a liar and deceiver speak of him a writer of almanacs; but those who (would call him) a scoundrel and an imposter (speak of him as) a chemist."

263 ^  "Trust your barque to the winds but not your body to a chemist; any breeze is safer than the faith of a chemist."

264 ^  Probably the Jesuit, Père Claude François Menestrier (1631-1705), a well known historian.

265 ^  The author was Christopher Nesse (1621-1705), a belligerent Calvinist, who wrote many controversial works and succeeded in getting excommunicated four times. One of his most virulent works was A Protestant Antidote against the Poison of Popery.

266 ^  John Case (c. 1660-1700) was a famous astrologer and physician. He succeeded to Lilly's practice in London. In a darkened room, wherein he kept an array of mystical apparatus, he pretended to show the credulous the ghosts of their departed relatives. Besides his astrological works he wrote one serious treatise, the Compendium Anatomicum nova methodo institutum (1695), in which he defends Harvey's theories of embryology.

267 ^  Marcelis (1636-after 1714) was a soap maker of Amsterdam. It is to be hoped that he made better soap than values of \pi.

268 ^  John Craig (died in 1731) was a Scotchman, but most of his life was spent at Cambridge reading and writing on mathematics. He endeavored to introduce the Leibnitz differential calculus into England. His mathematical works include the Methodus Figurarum ... Quadraturas determinandi (1685), Tractatus ... de Figurarum Curvilinearum Quadraturis et locis Geometricis (1693), and De Calculo Fluentium libri duo (1718).

269 ^  As is well known, this subject owes much to the Bernoullis. Craig's works on the calculus brought him into controversy with them. He also wrote on other subjects in which they were interested, as in his memoir On the Curve of the quickest descent (1700), On the Solid of least resistance (1700), and the Solution of Bernoulli's problem on Curves (1704).

270 ^  This is Samuel Lee (1783-1852), the young prodigy in languages. He was apprenticed to a carpenter at twelve and learned Greek while working at the trade. Before he was twenty-five he knew Hebrew, Chaldee, Syriac, Samaritan, Persian, and Hindustani. He later became Regius professor of Hebrew at Cambridge.