CHAPTER XXI

PARADOXES AND CURIOSITIES

=2101.= The pseudomath is a person who handles mathematics as a monkey handles the razor. The creature tried to shave himself as he had seen his master do; but, not having any notion of the angle at which the razor was to be held, he cut his own throat. He never tried it a second time, poor animal! but the pseudomath keeps on in his work, proclaims himself clean shaved, and all the rest of the world hairy.

The graphomath is a person who, having no mathematics, attempts to describe a mathematician. Novelists perform in this way: even Walter Scott now and then burns his fingers. His dreaming calculator, Davy Ramsay, swears “by the bones of the immortal Napier.” Scott thought that the philomaths worshipped relics: so they do in one sense.--DE MORGAN, A.

_Budget of Paradoxes (London, 1872), p. 473._

=2102.= Proof requires a person who can give and a person who can receive....

A blind man said, As to the Sun, I’ll take my Bible oath there’s none; For if there had been one to show They would have shown it long ago. How came he such a goose to be? Did he not know he couldn’t see? Not he. --DE MORGAN, A.

_Budget of Paradoxes (London, 1872), p. 262._

=2103.= Mathematical research, with all its wealth of hidden treasure, is all too apt to yield nothing to our research: for it is haunted by certain _ignes fatui_--delusive phantoms, that float before us, and seem so fair, and are _all but_ in our grasp, so nearly that it never seems to need more than _one_ step further, and the prize shall be ours! Alas for him who has been turned aside from real research by one of these spectres--who has found a music in its mocking laughter--and who wastes his life and energy in the desperate chase!--DODGSON, C. L.

_A new Theory of Parallels (London, 1895), Introduction._

=2104.= As lightning clears the air of impalpable vapours, so an incisive paradox frees the human intelligence from the lethargic influence of latent and unsuspected assumptions. Paradox is the slayer of Prejudice.--SYLVESTER, J. J.

_On a Lady’s Fan etc. Collected Mathematical Papers, Vol. 3, p. 36._

=2105.= When a paradoxer parades capital letters and diagrams which are as good as Newton’s to all who know nothing about it, some persons wonder why science does not rise and triturate the whole thing. This is why: all who are fit to read the refutation are satisfied already, and can, if they please, detect the paradoxer for themselves. Those who are not fit to do this would not know the difference between the true answer and the new capitals and diagrams on which the delighted paradoxer would declare that he had crumbled the philosophers, and not they him.

--DE MORGAN, A.

_A Budget of Paradoxes (London, 1872), p. 484._

=2106.= Demonstrative reason never raises the cry of _Church in Danger!_ and it cannot have any Dictionary of heresies except a Budget of Paradoxes. Mistaken claimants are left to Time and his extinguisher, with the approbation of all non-claimants: there is no need of a succession of exposures. Time gets through the job in his own workmanlike manner.--DE MORGAN, A.

_A Budget of Paradoxes (London, 1872), p. 485._

=2107.= 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 until they are over,”--may the Curiosities of Literature outlive the Follies of Science!--DE MORGAN, A.

_A Budget of Paradoxes (London, 1872), p. 71._

=2108.= Montucla says, speaking of France, that he finds three notions prevalent among cyclometers: 1. That there is a large reward offered for success; 2. That the longitude problem depends on that success; 3. That the solution is the great end and object of geometry. The same three notions are equally prevalent among the same class in England. No reward has ever been offered by the government of either country. The longitude problem in no way depends upon perfect solution; existing approximations are sufficient to a point of accuracy far beyond what can be wanted. And geometry, content with what exists, has long passed on to other matters. Sometimes a cyclometer persuades a skipper who has made land in the wrong place that the astronomers are at fault, for using a wrong measure of the circle; and the skipper thinks it a very comfortable solution! And this is the utmost that the problem has to do with longitude.--DE MORGAN, A.

_A Budget of Paradoxes (London, 1872), p. 96._

=2109.= Gregory St. Vincent is the greatest of circle-squarers, and his investigations led him into many truths: he found the property of the arc of the hyperbola which led to Napier’s logarithms being called hyperbolic. Montucla says of him, with sly truth, that no one ever squared the circle with so much genius, or, excepting his principal object, with so much success.

--DE MORGAN, A.

_A Budget of Paradoxes (London, 1872), p. 70._

=2110.= When I reached geometry, and became acquainted with the proposition the proof of which has been sought for centuries, I felt irresistibly impelled to try my powers at its discovery. You will consider me foolish if I confess that I am still earnestly of the opinion to have succeeded in my attempt.--BOLZANO, BERNARD.

_Selbstbiographie (Wien, 1875), p. 19._

=2111.= The Theory of Parallels.

It is known that to complete the theory it is only necessary to demonstrate the following proposition, which Euclid assumed as an axiom:

Prop. If the sum of the interior angles ECF and DBC which two straight lines EC and DB make with a third line CP is less than two right angles, the lines, if sufficiently produced, will intersect.

Proof. Construct PCA equal to the supplement PBD of CBD, and ECF, FCG, etc. each equal to ACE, so that ACF = 2.ACE, ACG = 3.ACE, etc. Then however small the angle ACE may be, there exists some number n such that n.ACE = ACH will be equal to or greater than ACP.

Again, take BI, IL, etc. each equal to CB, and draw IK, LM, etc. parallel to BD, then the figures ACBD, DBIK, KILM, etc. are congruent, and ACIK = 2.ABCD, ACLM = 3.ACBD, etc.

Take ACNO = n.ACBD, n having the same value as in the expression ACH = n.ACE, then ACNO is certainly less than ACP, since ACNO must be increased by ONP to be equal to ACP. It follows that ACNO is also less than ACH, and by taking the nth part of each of these, that ACBD is less than ACE.

But if ACE is greater than ACBD, CE and BD must intersect, for otherwise ACE would be a part of ACBD.

_Journal für Mathematik, Bd. 2 (1834), p. 198._

=2112.= Are you sure that it is impossible to trisect the angle by _Euclid_? I have not to lament a single hour thrown away on the attempt, but fancy that it is rather a tact, a feeling, than a proof, which makes us think that the thing cannot be done. But would _Gauss’s_ inscription of the regular polygon of seventeen sides have seemed, a century ago, much less an impossible thing, by line and circle?--HAMILTON, W. R.

_Letter to De Morgan (1852)._

=2113.= One of the most curious of these cases [geometrical paradoxers] was that of a student, I am not sure but a graduate, of the University of Virginia, who claimed that geometers were in error in assuming that a line had no thickness. He published a school geometry based on his views, which received the endorsement of a well-known New York school official and, on the basis of this, was actually endorsed, or came very near being endorsed, as a text-book in the public schools of New York.

--NEWCOMB, SIMON.

_The Reminiscences of an Astronomer (Boston and New York, 1903), p. 388._

=2114.= What distinguishes the straight line and circle more than anything else, and properly separates them for the purpose of elementary geometry? Their self-similarity. Every inch of a straight line coincides with every other inch, and off a circle with every other off the same circle. Where, then, did Euclid fail? In not introducing the third curve, which has the same property--the _screw_. The right line, the circle, the screw--the representations of translation, rotation, and the two combined--ought to have been the instruments of geometry. With a screw we should never have heard of the impossibility of trisecting an angle, squaring the circle, etc.--DE MORGAN, A.

_Quoted in Graves’ Life of Sir W. R. Hamilton, Vol. 3 (New York, 1889), p. 342._

=2115.=

Mad Mathesis alone was unconfined, Too mad for mere material chains to bind, Now to pure space lifts her ecstatic stare, Now, running round the circle, finds it square. --POPE, ALEXANDER.

_The Dunciad, Bk. 4, lines 31-34._

=2116.=

Or is’t a tart idea, to procure An edge, and keep the practic soul in ure, Like that dear Chymic dust, or puzzling quadrature? --QUARLES, PHILIP.

_Quoted by De Morgan: Budget of Paradoxes (London, 1872), p. 436._

=2117.=

Quale è’l geometra che tutto s’ affige Per misurar lo cerchio, e non ritruova, Pensando qual principio ond’ egli indige. --DANTE.

_Paradise, canto 33, lines 122-125._

[As doth the expert geometer appear Who seeks to square the circle, and whose skill Finds not the law with which his course to steer.[12]]

_Quoted in Frankland’s Story of Euclid (London, 1902), p. 101._

[12] For another rendition of these same lines see 1858.

=2118.=

In _Mathematicks_ he was greater Than _Tycho Brahe_, or _Erra Pater_: For he, by _Geometrick_ scale, Could take the size of _Pots of Ale_; Resolve by Signs and Tangents streight, If _Bread_ or _Butter_ wanted weight; And wisely tell what hour o’ th’ day The Clock doth strike, by _Algebra_. --BUTLER, SAMUEL.

_Hudibras, Part 1, canto 1, lines_ 119-126.

=2119.= I have often been surprised that Mathematics, the quintessence of truth, should have found admirers so few and so languid. Frequent considerations and minute scrutiny have at length unravelled the cause; viz. that though Reason is feasted, Imagination is starved; whilst Reason is luxuriating in its proper Paradise, Imagination is wearily travelling on a dreary desert.--COLERIDGE, SAMUEL.

_A Mathematical Problem._

=2120.= At last we entered the palace, and proceeded into the chamber of presence where I saw the king seated on his throne, attended on each side by persons of prime quality. Before the throne, was a large table filled with globes and spheres, and mathematical instruments of all kinds. His majesty took not the least notice of us, although our entrance was not without sufficient noise, by the concourse of all persons belonging to the court. But he was then deep in a problem, and we attended an hour, before he could solve it. There stood by him, on each side, a young page with flaps in their hands, and when they saw he was at leisure, one of them gently struck his mouth, and the other his right ear; at which he started like one awaked on the sudden, and looking toward me and the company I was in, recollected the occasion of our coming, whereof he had been informed before. He spake some words, whereupon immediately a young man with a flap came to my side, and flapt me gently on the right ear, but I made signs, as well as I could, that I had no occasion for such an instrument; which, as I afterwards found, gave his majesty, and the whole court, a very mean opinion of my understanding. The king, as far as I could conjecture, asked me several questions, and I addressed myself to him in all the languages I had. When it was found, that I could neither understand nor be understood, I was conducted by his order to an apartment in his palace, (this prince being distinguished above all his predecessors, for his hospitality to strangers) where two servants were appointed to attend me. My dinner was brought, and four persons of quality, did me the honour to dine with me. We had two courses of three dishes each. In the first course, there was a shoulder of mutton cut into an equilateral triangle, a piece of beef into a rhomboides, and a pudding into a cycloid. The second course, was, two ducks trussed up in the form of fiddles; sausages and puddings, resembling flutes and haut-boys, and a breast of veal in the shape of a harp. The servants cut our bread into cones, cylinders, parallelograms, and several other mathematical figures.--SWIFT, JONATHAN.

_Gulliver’s Travels; A Voyage to Laputa; Chap. 2._

=2121.= Those to whom the king had entrusted me, observing how ill I was clad, ordered a taylor to come next morning, and take measure for a suit of cloaths. This operator did his office after a different manner, from those of his trade in Europe. He first took my altitude by a quadrant, and then, with rule and compasses, described the dimensions and outlines of my whole body, all which he entered upon paper; and in six days, brought my cloaths very ill made, and quite out of shape, by happening to mistake a figure in the calculation. But my comfort was, that I observed such accidents very frequent, and little regarded.

--SWIFT, JONATHAN.

_Gulliver’s Travels; A Voyage to Laputa, Chap. 2._

=2122.= The knowledge I had in mathematics, gave me great assistance in acquiring their phraseology, which depended much upon that science, and music; and in the latter I was not unskilled. Their ideas are perpetually conversant in lines and figures. If they would, for example, praise the beauty of a woman, or any other animal, they describe it by rhombs, circles, parallelograms, ellipses, and other geometrical terms, or by words of art drawn from music, needless here to repeat. I observed in the king’s kitchen all sorts of mathematical and musical instruments, after the figures of which, they cut up the joints that were served to his majesty’s table.--SWIFT, JONATHAN.

_Gulliver’s Travels; A Voyage to Laputa, Chap. 2._

=2123.= I was at the mathematical school, where the master taught his pupils, after a method, scarce imaginable to us in Europe. The propositions, and demonstrations, were fairly written on a thin wafer, with ink composed of a cephalic tincture. This, the student was to swallow upon a fasting stomach, and for three days following, eat nothing but bread and water. As the wafer digested, the tincture mounted to his brain, bearing the proposition along with it. But the success has not hitherto been answerable, partly by some error in the _quantum_ or composition, and partly by the perverseness of lads; to whom this bolus is so nauseous, that they generally steal aside, and discharge it upwards, before it can operate; neither have they been yet persuaded to use so long an abstinence as the prescription requires.--SWIFT, JONATHAN.

_Gulliver’s Travels; A Voyage to Laputa, Chap. 5._

=2124.= It is worth observing that some of those who disparage some branch of study in which they are deficient, will often affect more contempt for it than they really feel. And not unfrequently they will take pains to have it thought that they are themselves well versed in it, or that they easily might be, if they thought it worth while;--in short, that it is not from hanging too high that the grapes are called sour.

Thus, Swift, in the person of Gulliver, represents himself, while deriding the extravagant passion for Mathematics among the Laputians, as being a good mathematician. Yet he betrays his utter ignorance, by speaking “of a pudding in the _form of a cycloid_:” evidently taking the cycloid for a _figure_, instead of a _line_. This may help to explain the difficulty he is said to have had in obtaining his Degree.--WHATELY, R.

_Annotations to Bacon’s Essays, Essay L._

=2125.= It is natural to think that an abstract science cannot be of much importance in the affairs of human life, because it has omitted from its consideration everything of real interest. It will be remembered that Swift, in his description of Gulliver’s voyage to Laputa, is of two minds on this point. He describes the mathematicians of that country as silly and useless dreamers, whose attention has to be awakened by flappers. Also, the mathematical tailor measures his height by a quadrant, and deduces his other dimensions by a rule and compasses, producing a suit of very ill-fitting clothes. On the other hand, the mathematicians of Laputa, by their marvellous invention of the magnetic island floating in the air, ruled the country and maintained their ascendency over their subjects. Swift, indeed, lived at a time peculiarly unsuited for gibes at contemporary mathematicians. Newton’s _Principia_ had just been written, one of the great forces which have transformed the modern world. Swift might just as well have laughed at an earthquake.

--WHITEHEAD, A. N.

_An Introduction to Mathematics (New York, 1911), p. 10._

=2126.= [Illustration: A geometrical drawing including square and four triangles to demonstrate a graphical proof of the theorem of Pythagoras as described in the poem.]

Here I am as you may see a² + b² - ab When two Triangles on me stand Square of hypothen^e is plann’d But if I stand on them instead, The squares of both the sides are read. --AIRY, G. B.

_Quoted in Graves’ Life of Sir W. R. Hamilton, Vol. 3 (New York, 1889), p. 502._

=2127.= π = 3.141 592 653 589 793 238 462 643 383 279 ...

3 1 4 1 5 9 Now I, even I, would celebrate 2 6 5 3 5 In rhymes inapt, the great 8 9 7 9 Immortal Syracusan, rivaled nevermore, 3 2 3 8 4 Who in his wondrous lore, 6 2 6 Passed on before, 4 3 3 8 3 2 7 9 Left men his guidance how to circles mensurate. --ORR, A. C.

_Literary Digest, Vol. 32 (1906), p. 84._

=2128.= I take from a biographical dictionary the first five names of poets, with their ages at death. They are

Aagard, died at 48. Abeille, “ “ 76. Abulola, “ “ 84. Abunowas, “ “ 48. Accords, “ “ 45.

These five ages have the following characters in common:--

1. The difference of the two digits composing the number divided by _three_, leaves a remainder of _one_.

2. The first digit raised to the power indicated by the second, and then divided by _three_, leaves a remainder of _one_.

3. The sum of the prime factors of each age, including _one_ as a prime factor, is divisible by _three_.--PEIRCE, C. S.

_A Theory of Probable Inference; Studies in Logic (Boston, 1883), p. 163._

=2129.= In view of the fact that the offered prize [for the solution of the problem of Fermat’s Greater Theorem] is about $25,000 and that lack of marginal space in his copy of Diophantus was the reason given by Fermat for not communicating his proof, one might be tempted to wish that one could send credit for a dime back through the ages to Fermat and thus secure this coveted prize, if it actually existed. This might, however, result more seriously than one would at first suppose; for if Fermat had bought on credit a dime’s worth of paper even during the year of his death, 1665, and if this bill had been drawing compound interest at the rate of six per cent, since that time, the bill would now amount to more than seven times as much as the prize.

--MILLER, G. A.

_Some Thoughts on Modern Mathematical Research; Science, Vol. 35 (1912), p. 881._

=2130.= _If the Indians hadn’t spent the $24._ In 1626 Peter Minuit, first governor of New Netherland, purchased Manhattan Island from the Indians for about $24. The rate of interest on money is higher in new countries, and gradually decreases as wealth accumulates. Within the present generation the legal rate in the state has fallen from 7% to 6%. Assume for simplicity a uniform rate of 7% from 1626 to the present, and suppose that the Indians had put their $24 at interest at that rate (banking facilities in New York being always taken for granted!) and had added the interest to the principal yearly. What would be the amount now, after 280 years? 24 × (1.07)^{280} = more than 4,042,000,000.

The latest tax assessment available at the time of writing gives the realty for the borough of Manhattan as $3,820,754.181. This is estimated to be 78% of the actual value, making the actual value a little more than $4,898,400,000.

The amount of the Indians’ money would therefore be more than the present assessed valuation but less than the actual valuation.

--WHITE, W. F.

_A Scrap-book of Elementary Mathematics (Chicago, 1908), pp. 47-48._

=2131.= See Mystery to Mathematics fly!--POPE, ALEXANDER.

_The Dunciad, Bk. 4, line 647._

=2132.= The Pythagoreans and Platonists were carried further by this love of simplicity. Pythagoras, by his skill in mathematics, discovered that there can be no more than five regular solid figures, terminated by plane surfaces which are all similar and equal; to wit, the tetrahedron, the cube, the octahedron, the dodecahedron, and the eicosihedron. As nature works in the most simple and regular way, he thought that all elementary bodies must have one or other of those regular figures; and that the discovery of the properties and relations of the regular solids must be a key to open the mysteries of nature.

This notion of the Pythagoreans and Platonists has undoubtedly great beauty and simplicity. Accordingly it prevailed, at least to the time of Euclid. He was a Platonic philosopher, and is said to have wrote all the books of his Elements, in order to discover the properties and relations of the five regular solids. The ancient tradition of the intention of Euclid in writing his elements, is countenanced by the work itself. For the last book of the elements treats of the regular solids, and all the preceding are subservient to the last.--REID, THOMAS.

_Essays on the Powers of the Human Mind (Edinburgh, 1812), Vol. 2, p. 400._

=2133.= In the Timæus [of Plato] it is asserted that the particles of the various elements have the forms of these [the regular] solids. Fire has the Pyramid; Earth has the Cube; Water the Octahedron; Air the Icosahedron; and the Dodecahedron is the plan of the Universe itself. It was natural that when Plato had learnt that other mathematical properties had a bearing upon the constitution of the Universe, he should suppose that the singular property of space, which the existence of this limited and varied class of solids implied, should have some corresponding property in the Universe, which exists in space.

--WHEWELL, W.

_History of the Inductive Sciences, 3rd Edition, Additions to Bk. 2._

=2134.= The orbit of the earth is a circle: round the sphere to which this circle belongs, describe a dodecahedron; the sphere including this will give the orbit of Mars. Round Mars describe a tetrahedron; the circle including this will be the orbit of Jupiter. Describe a cube round Jupiter’s orbit; the circle including this will be the orbit of Saturn. Now inscribe in the earth’s orbit an icosahedron; the circle inscribed in it will be the orbit of Venus. Inscribe an octahedron in the orbit of Venus; the circle inscribed in it will be Mercury’s orbit. This is the reason of the number of the planets.--KEPLER.

_Mysterium Cosmographicum [Whewell]._

=2135.= It will not be thought surprising that Plato expected that Astronomy, when further advanced, would be able to render an account of many things for which she has not accounted even to this day. Thus, in the passage in the seventh Book of the _Republic_, he says that the philosopher requires a reason for the proportion of the day to the month, and the month to the year, deeper and more substantial than mere observation can give. Yet Astronomy has not yet shown us any reason why the proportion of the times of the earth’s rotation on its axis, the moon’s revolution round the earth, and the earth’s revolution round the sun, might not have been made by the Creator quite different from what they are. But in asking Mathematical Astronomy for reasons which she cannot give, Plato was only doing what a great astronomical discoverer, Kepler, did at a later period. One of the questions which Kepler especially wished to have answered was, why there are five planets, and why at such particular distances from the sun? And it is still more curious that he thought he had found the reason of these things, in the relation of those five regular solids which Plato was desirous of introducing into the philosophy of the universe.... Kepler regards the law which thus determines the number and magnitude of the planetary orbits by means of the five regular solids as a discovery no less remarkable and certain than the Three Laws which give his name its imperishable place in the history of astronomy.--WHEWELL, W.

_History of the Inductive Sciences, 3rd Edition, Additions to Bk. 3._

=2136.= Pythagorean philosophers ... maintained that of two combatants, he would conquer, the sum of the numbers expressed by the characters of whose names exceeded the sum of those expressed by the other. It was upon this principle that they explained the relative prowess and fate of the heroes in Homer, Πατροκλος, Ἑκτορ and Αχιλλευς, the sum of the numbers in whose names are 861, 1225, and 1276 respectively.--PEACOCK, GEORGE.

_Encyclopedia of Pure Mathematics (London, 1847); Article “Arithmetic,” sect. 38._

=2137.= Round numbers are always false.--JOHNSON, SAMUEL.

_Johnsoniana; Apothegms, Sentiment, etc._

=2138.= Numero deus impare gaudet [God in number odd rejoices.]

--VIRGIL.

_Eclogue, 8, 77._

=2139.= Why is it that we entertain the belief that for every purpose odd numbers are the most effectual?--PLINY.

_Natural History, Bk. 28, chap. 5._

=2140.=

“Then here goes another,” says he, “to make sure, Fore there’s luck in odd numbers,” says Rory O’Moore. --LOVER, S.

_Rory O’Moore._

=2141.= This is the third time; I hope, good luck lies in odd numbers.... They say, there is divinity in odd numbers, either in nativity, chance, or death.--SHAKESPEARE.

_The Merry Wives of Windsor, Act 5,