Chapter XIII: Part 13

I will not give the name of the poor fellow who has fallen so far below both the _honestum_ and the _utile_, to say nothing of the _decorum_ or the _dulce_.[662] He is the fourth who has taken elaborate notice of me; and my advice to him would be, _Nec quarta loqui persona laboret_.[663] According to him, I scorn humanity, scandalize learning, and disgrace the press; it admits of no manner of doubt that my object is to mislead the public and silence truth, at the expense of the interests of science, the wealth of the nation, and the lives of my fellow men. The only thing left to be settled is, whether this is due to ignorance, natural distaste for truth, personal malice, a wish to curry favor with the Astronomer Royal, or mere toadyism. The only accusation which has truth in it is, that I have made myself a "public scavenger of science": the assertion, which is the {363} most false of all is, that the results of my broom and spade are "shot right in between the columns of" the _Athenaeum_. I declare I never in my life inserted a word between the columns of the _Athenaeum_: I feel huffed and miffed at the very supposition. I _have_ made myself a public scavenger; and why not? Is the mud never to be collected into a heap? I look down upon the other scavengers, of whom there have been a few--mere historical drudges; Montucla, Hutton, etc.--as not fit to compete with me. I say of them what one crossing-sweeper said of the rest: "They are well enough for the common thing; but put them to a bit of fancy-work, such as sweeping round a post, and see what a mess they make of it!" Who can touch me at sweeping round a paradoxer? If I complete my design of publishing a separate work, an old copy will be fished up from a stall two hundred years hence by the coming man, and will be described in an article which will end by his comparing our century with his own, and sighing out in the best New Zealand pronunciation--

"Dans ces tems-la
C'etait deja comme ca!"[664]

ORTHODOX PARADOXERS.

And pray, Sir! I have been asked by more than one--do your orthodox never fall into mistake, nor rise into absurdity? They not only do both, but they admit it of each other very freely; individually, they are convinced of sin, but not of any particular sin. There is not a syndoxer among them all but draws his line in such a way as to include among paradoxers a great many whom I should exclude altogether from this work. My worst specimens are but exaggerations of what may be found, occasionally, in the thoughts of sagacious investigators. At the end of the {364} glorious dream, we learn that there is a way to Hell from the gates of Heaven, as well as from the City of Destruction: and that this is true of other things besides Christian pilgrimage is affirmed at the end of the Budget of Paradoxes. If D'Alembert[665] had produced _enough_ of a quality to match his celebrated mistake on the chance of throwing head in two throws, he would have been in my list. If Newton had produced _enough_ to match his reception of the story that Nausicaa, Homer's Phaeacian princess, invented the celestial sphere, followed by his serious surmise that she got it from the Argonauts,--then Newton himself would have had an appearance entered for him, in spite of the _Principia_. In illustration, I may cite a few words from _Tristram Shandy_:

"'A soldier,' cried my uncle Toby, interrupting the Corporal, 'is no more exempt from saying a foolish thing, Trim, than a man of letters.'--'But not so often, an' please your honor,' replied the Corporal. My uncle Toby gave a nod."

I now proceed to die out. Some prefatory remarks will follow in time.[666] I shall have occasion to insist that all is not barren: I think I shall find, on casting up, that two out of five of my paradoxers are not to be utterly condemned. Among the better lot will be found all gradations of merit; at the same time, as was remarked on quite a different subject, there may be little to choose between the last of the saved and the first of the lost. The higher and better class is worthy of blame; the lower and worse class is worthy of praise. The higher men are to be reproved for not taking up things in which they could do some good: the lower men are to be commended for taking up things in which they can do no great harm. The circle problem is like Peter Peebles's lawsuit:

{365}

"'But, Sir, I should really spoil any cause thrust on me so hastily.'--'Ye cannot spoil it, Alan,' said my father, 'that is the very cream of the business, man,--... the case is come to that pass that Stair or Arniston could not mend it, and I don't think even you, Alan, can do it much harm.'"

I am strongly reminded of the monks in the darker part of the Middle Ages. To a certain proportion of them, perhaps two out of five, we are indebted for the preservation of literature, and their contemporaries for good teaching and mitigation of socials evils. But the remaining three were the fleas and flies and thistles and briars with whom the satirist lumps them, about a century before the Reformation:

"Flen, flyys, and freris, populum domini male caedunt;
Thystlis and breris crescentia gramina laedunt.
Christe nolens guerras qui cuncta pace tueris,
Destrue per terras breris, flen, flyys, and freris.
Flen, flyys, and freris, foul falle hem thys fyften yeris,
For non that her is lovit flen, flyys ne freris."[667]

I should not be quite so savage with my second class. Taken together, they may be made to give useful warning to those who are engaged in learning under better auspices: aye, even useful hints; for bad things are very often only good things spoiled or misused. My plan is that of a predecessor in the time of Edward the Second:

"Meum est propositum genti imperitae
Artes frugi reddere melioris vitae."[668]

To this end I have spoken with freedom of books as books, of opinions as opinions, of ignorance as ignorance, of {366} presumption as presumption; and of writers as I judge may be fairly inferred from what they have written. Some--to whom I am therefore under great obligation--have permitted me to enlarge my plan by assaults to which I have alluded; assaults which allow a privilege of retort, of which I have often availed myself; assaults which give my readers a right of partnership in the amusement which I myself have received.

For the present I cut and run: a Catiline, pursued by a chorus of Ciceros, with _Quousque tandem? Quamdiu nos? Nihil ne te?_[669] ending with, _In te conferri pestem istam jam pridem oportebat, quam tu in nos omnes jamdiu machinaris!_ I carry with me the reflection that I have furnished to those who need it such a magazine of warnings as they will not find elsewhere; _a signatis cavetote_:[670] and I throw back at my pursuers--_Valete, doctores sine doctrina; facite ut proxima congressu vos salvos corporibus et sanos mentibus videamus._[671] Here ends the Budget of Paradoxes.

{367}

* * * * *

APPENDIX.

I think it right to give the proof that the ratio of the circumference to the diameter is incommensurable. This method of proof was given by Lambert,[672] in the _Berlin Memoirs_ for 1761, and has been also given in the notes to Legendre's[673] Geometry, and to the English translation of the same. Though not elementary algebra, it is within the reach of a student of ordinary books.[674]

Let a continued fraction, such as

a
-----
b + c
-----
d + e
-
f + etc.,

be abbreviated into a/b+ c/d+ e/f+ etc.: each fraction being understood as falling down to the side of the preceding sign +. In every such fraction we may suppose b, d, f, etc. {368} positive; a, c, e, &c. being as required: and all are supposed integers. If this succession be continued ad infinitum, and if a/b, c/d, e/f, etc. all lie between -1 and +1, exclusive, the limit of the fraction must be incommensurable with unity; that is, cannot be A/B, where A and B are integers.

First, whatever this limit may be, it lies between -1 and +1. This is obviously the case with any fraction p/(q + [omega]), where [omega] is between +-1: for, p/q, being < 1, and p and q integer, cannot be brought up to 1, by the value of [omega]. Hence, if we take any of the fractions

a/b, a/b+ c/d, a/b+ c/d+ e/f, etc.

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

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

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

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

{369}

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

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

Q/P = e/f+ R/Q, or, R = eP - fQ

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

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

Let [phi]z represent

a a^2 a^3
1 + - + ------- + -------------- + ....
z 2z(z+1) 2.3.z(z+1)(z+2)

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

a
[phi]z-[phi](z+1) = ------([phi](z+2))
z(z+1)

a [phi](z+1) a [phi](z+1) a [phi](z+2)
or a = - ---------- . z + - ---------- . --- ----------
z [phi]z z [phi]z z+1 [phi](z+1)

a = [psi]z(z+[psi](z+1))

[psi]z being (a/z)([phi](z+1)/[phi]z); of which observe that it diminishes without limit as z increases without limit. Accordingly, we have

[psi]z = a/z+ [psi](z+1) = a/z+ a/(z+1)+ [psi](z+2)
= a/z+ a/(z+1)+ a/(z+2)+ [psi](z+3), etc.

And, [psi](z + n) diminishing without limit, we have

a/z . [phi](z+1)/[phi]z = (a/z+) (a/(z+1)+) (a/(z+2)+) (a/((z+3)+ ...))

Let z = 1/2; and let 4a = -x^2. Then (a/z)[phi](z+1) is -(x^2/2) ( 1 - x^2/(2.3) + x^4/(2.3.4.5...)) or -(x/2) sin x. Again [phi]z is 1 - x^2/2 + x^4/(2.3.4) or cos x: and the continued fraction is

(1/4)x^2/(1/2)+ (1/4)x^2/(3/2)+ (1/4)x^2/(5/2)+ ...
or -x/2 x/1+ -x^2/3+ -x^2/5+
...

{371} whence tan x = x/1+ -x^2/3+ -x^2/5+ -x^2/7+ ...

Or, as written in the usual way,

tan x = x
-------
1 - x^2
-------
3 - x^2
-------
5 - x^2
-------
7 - ...

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

a_1/b_1+ a_2/b_2+ a_3/b_3+ ... a_n/b_n = P_n/Q_n; then

P_1=a_1, P_2=b_2 P_1, P_3=b_3 P_2+a_3 P_1, P_4=b_4 P_3+a_4 P_2, etc.
Q_1=b_1, Q_2=b_2 Q_1+a_2, Q_3=b_3 Q_2+a_3 Q_1, Q_4=b_4 Q_3+a_4 Q_2, etc.

in the case before us we have

a_1=x, a_2=-x^2, a_3=-x^2, a_4=-x^2, a_5=-x^2, etc.
b_1=1, b_2=3, b_3=5, b_4=7, b_5=9, etc.

P_1=x Q_1=1
P_2=3x Q_2=3-x^2
P_3=15x-x^3 Q_3=15-6x^2
P_4=105x-10x^3 Q_4=105-45x^2+x^4
P_5=945x-105x^3+x^5 Q_5=945-420x^2+15x^4
P_6=10395x-1260x^3+21x^5 Q_6=10395-4725x^2+210x^4-x^6

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

x + x^3/3 + 2x^5/15 + 17x^7/315 + 62x^9/2835 + ...

{372} Take P_5, and divide it by Q_5 in the common way, and the first five terms will be as here written. Now take _x_ = .1, which means that the angle is to be one tenth of the actual unit, or, in degrees 5 deg..729578. We find that when x = .1, P_6 = 1038.24021, Q_6 = 10347.770999; whence P_6 divided by Q_6 gives .1003346711. Now 5 deg..729578 is 5 deg.43'46-1/2"; and from the old tables of Rheticus[675]--no modern tables carry the tangents so far--the tangent of this angle is .1003347670.

Now let x = (1/4)[pi]; in which case tan x = 1. If (1/4)[pi] be commensurable with the unit, let it be (m/n), m and n being integers: we know that (1/4)[pi] < 1. We have then

1=(m/n)/1- (m^2/n^2)/3- (m^2/n^2)/5- ... = m/n- m^2/3n- m^2/5n- m^2/7n-
...

Now it is clear that m^2/3n, m^2/5n, m^2/7n, etc. must at last become and continue severally less than unity. The continued fraction is therefore incommensurable, and cannot be unity. Consequently [pi]^2 cannot be commensurable: that is, [pi] is an incommensurable quantity, and so also is [pi]^2.

I thought I should end with a grave bit of appendix, deeply mathematical: but paradox follows me wherever I go. The foregoing is--in my own language--from Dr. (now Sir David) Brewster's[676] English edition of Legendre's Geometry, (Edinburgh, 1824, 8vo.) translated by some one who is not named. I picked up a notion, which others had at Cambridge in 1825, that the translator was the late Mr. Galbraith,[677] then known at Edinburgh as a writer and teacher.

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

* * * * *

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

Michael Stifelius[679] edited, in 1554, a second edition of the Algebra (_Die Coss._), of Christopher Rudolff.[680] This is one of the earliest works in which + and - are used.

Stifelius was a queer man. He has introduced into this very work of Rudolff his own interpretation of the number of the Beast. He determined to fix the character of Pope Leo: so he picked the numeral letters from LEODECIMVS, and by taking in X from LEO X. and striking out M as standing for _mysterium_, he hit the number exactly. This discovery completed his conversion to Luther, and his determination to throw off his monastic vows. Luther dealt with him as straight-forwardly as with Melanchthon about his astrology: he accepted the conclusions, but told him to clear his mind of all the premises about the Beast. Stifelius {374} did not take the advice, and proceeded to settle the end of the world out of the prophet Daniel: he fixed on October, 1533. The parishioners of some cure which he held, having full faith, began to spend their savings in all kinds of good eating and drinking; we may charitably hope this was not the way of preparing for the event which their pastor pointed out. They succeeded in making themselves as fit for Heaven as Lazarus, so far as beggary went: but when the time came, and the world lasted on, they wanted to kill their deceiver, and would have done so but for the interference of Luther. {375}

* * * * *

INDEX.

Pages denoted by numerals of this kind (_287_) refer to biographical notes, chiefly by the editor. Numerals like 426 refer to books discussed by De Morgan, or to leading topics in the text. Numerals like 126 indicate minor references.

Abbott, Justice, I, _181_.
Abernethy, J., II, _219_.
Aboriginal Britons, a poem, II, 270.
Academy of Sciences, French, I, 163.
Adair, J., I, _223_.
Adam, M., I, _65_.
Adams, J. C., I, _43_, 82, 385, 388; II, 131, 135, 140, 303.
Ady, Joseph, II, 42, _42_.
Agnew, H. C., I, 328.
Agricola, J., I, 394.
Agricultural Laborer's letter, II, 16.
Agrippa, H. C., I, _48_, 48.
Ainsworth, W. H., II, _132_.
Airy, I, _85_, 88, 152, 242; II, 85, 140, 150, 303, 347.
Alchemy, I, 125.
Alfonso X (El Sabio), II, _269_.
Alford, H., II, _221_.
Alfred, King, Ballad of, II, 22.
Algebra, I, 121.
Algebraic symbols, I, 121.
Almanac, I, 300; II, 147, 148, 207.
(_See Easter._)
Aloysius Lilius, I, 362.
Alsted, J. H., II, _282_.
Ameen Bey, II, 15.
Amicable Society, I, 347.
Ampere, I, 86.
Amphisbaena serpent, I, 31.
Anagrams, De Morgan, I, 138.
Anaxagoras, II, _59_.
Anghera, II, 60, _60_, 61, 279.
Annuities, Fallacies of, I, 157.
Antichrist, I, 130.
Antimony, I, 125.
Antinewtonism, I, 162.
Antinomians, I, 394.
Antiphon, II, _59_.
Antonie, F., I, _126_, 126.
Apollonius, I, 41, 107.
Apparitions, II, 47.
Arago, I, _243_, 390.
Aratus, II, _167_.
Arbuthnot, I, _133_, 134.
Archer, H., II, 90.
Archimedes, I, 5, 11, 42, 83, 107.
Archytas, I, _53_.
Argoli, I, _104_.
Aristocrat, as a scientist, I, 131.
Aristotle, I, 5, 331.
Arnobius, II, _73_.
Arson, P. J., II, _207_.
Ashton, R., II, _99_.
Astrology, I, 118, 127, 128, 350; II, 43.
Astronomer's Drinking Song, I, 380.
Astronomical Aphorisms, I, 398.
Paradox, I, 394.
Police Report, I, 390.
Society. (_See Royal Astronomical Society._)
Astronomy, Bailly's exaggerated view of, I, 166.
Astunica, Didacas, I, 90.
Athanasian Creed, I, 371.
Atheists, Philosophical, I, 1.
Atoms, II, 191.
{376}
Attraction, I, 136, 151, 155.
Augustine, St., II, _23_.
Aurora borealis, I, 134.
Austen, Jane, I, 191.
Auzout, A., II, _300_.
Aviation, Early ideas of, II, 8.

Babbage, C., I, _207_, 290, 291; II, 181.
Bachet, de Meziriac, I, _161_.
Bacon, F., I, 5, _75_, 75, 76, 79, 89, 145, 331.
Bacon, R., I, 5, _126_, 126, 360; II, 94.
Baconian controversy, I, 141.
Baden Powell, II, _267_.
Bailly, J. S., I, 166, _166_, 308.
Baily, F., I, 308, _309_; II, 16, 143, 188.
Baily, R., II, 16.
Baker, T., II, _302_.
Bakewell, F. C., II, 156, _156_.
Banks, J., I, 28.
Barberini, M., II, _267_.
Barker, C., II, _262_.
Baronius, I, 33, 35; II, _62_.
Barreme, I, _42_.
Barrett, G., II, _188_.
Barrow, I., I, _160_; II, 302.
Baruel, de, I, 165.
Bassano, Duc de, II, 3, 339.
Baxter, T., I, 146.
Bayle, P., II, _73_.
Beaufort, F., II, _267_.
Beaugrand, I, 119, _121_.
Beaulieu, I, _119_, 119, 121.
Beaune, de, II, _59_.
Becourt, R., II, 277.
Bedford, Duke of, (6th), I, _182_.
Behmen, I, _168_, 254; II, 317.
Bellenden, W., I, _175_.
Bentley, I, _110_.
Berkeley, G., II, 346.
Bernard, E., II, _297_, 300.
Bernardus Trevisanus, I, _126_, 126.
Bernoullis, I, 130, _150_, _335_, 336.
Bertius, P., II, _300_.
Bese, I, _66_.
Bessel, I, _384_; II, 2.
Bethune, I, _99_, 279, 291.
Bettesworth, I, 19.
Beza. (_See Bese._)
Bickersteth, E. H., I, _238_.
Bidder, I, _86_.
Biden, J., II, 158, _160_.
Bidle, (Biddle), I, 239.
Biot, I, _85_.
Birch, T., I, _108_; II, 304, 313.
Birks, T. R., II, 158, _158_.
Bishop, G., I, _386_.
Bishops as Paradoxers, I, 226.
Boccaccio, I, 38.
Boethius, I, _42_, 45.
Boehme. (_See Behmen._)
Boncompagni, I, _298_.
Boniface, St., I, 32.
Bonnycastle, J., II, _16_.
Booker, I, 115.
Boole, G., I, _261_, 332; II, 75, 79.
--A tribute to, II, 79.
Borelli, G. A., II, _300_.
Borello, I, _69_.
Boreman, I, 113.
Borron, Mrs., II, 7.
Boscovich, I, _156_, 164.
Bouguer, II, _301_.
Bouillaud, I, _87_; II, 295.
Bouvard, A., I, _327_.
Bovillus, I, _44_; II, 324.
--Epitome of, I, 44.
Bowdler, H. M., I, _194_.
Bowring, J., I, _352_; II, 256.
Boyle, R., I, 24, _125_; II, 300.
Bradley, I, 24.
Bradwardine, I, _227_, 228, 229.
Brahe. (_See Tycho B._)
Brancker, I, 107; II, _300_.
Brenan, J., I, 330, _330_.
Brewster, D., I, 39, _137_, 140; II, 214, 288, 372.
Briggs, I, _69_; II, 299, 302.
Bright, J., II, _235_.
Brinkley, J., I, _311_.
Britannicus, D., II, 8.
British Museum library, I, 151.
Brothers, R., I, _315_; II, 97.
Brougham, Henry, Lord, I, _191_.
Brouncker (Brounker), I, _132_; II, 302.
Brown, W., II, _168_.
Browne, T., I, 31.
Brucker, I, _61_.
Brunet, I, _402_.
Bruennow, I, _386_.
Bruno, I, _59_, 59.
Bryson, II, _59_.
Buergi, I, 52.
Buffon, I, _282_.
Bulstrode, II, 84.
Bungus, I, _55_, 55, 57.
Buoncompagno, U., I, _362_.
{377}
Burgon, J. W., II, _30_.
Buridan, I, _37_.
--Questiones morales, I, 37.
Buridan's Ass, I, 37.
Burke, E., I, _173_.
Burlesque, Frend's, I, 208.
Burnet, G., I, _107_.
Burney, Frances, I, _190_.
Burton, Frances B., I, 374.
Busby, R., II, _313_.
Buteo, I, _51_.
Butler, G., I, _199_.
Butler, S., II, _218_.
Buxton, J., I, _86_.
Byrgius. (_See Buergi._)
Byrne, O., I, _329_; II, 186, 190.
Byron, I, 186; II, 270, 273.

Cabbala, I, 272.
Calculating Boys, I, 86.
Calculus, I, 129.
Calendar. (_See Easter._)
Cambridge Poets, II, 269.
Campanus, I, 42, _43_.
Canning, Geo., II, _145_.
Carcavi, I, _106_.
Cardanus, II, _59_.
Carlile, R., I, _271_.
Carlyle, T., II, _373_.
Carnot, I, 107.
Caroline tables, I, 124.
Casaubon, I, _111_.
Case, J., I, _128_, 128.
Cassini, J., I, _172_.
Castel, I, _148_, 148.
Castiglioni, I, _139_.
Castlereagh, I, 185, _186_.
Cataldi, I, _69_, 69.
Catcott, A., I, _237_.
Causans, de, I, _298_.
Cavalieri, I, _106_.
Cavendish, C., I, _106_; II, 299, 312.
Cavendish, W., I, _290_.
Caxton, W., II, _281_.
Cayley, A., II, _292_.
Cecil, R. (1st Earl of Salisbury), II, _330_.
Centrifugal force, II, 268.
Ceulen. (_See Van Ceulen._)
Challis, J., I, _390_; II, 141.
Chalmers, I, _102_; II, 219.
Chambers, E., II, _282_.
Chambers, R., I, _344_, 344.
Charles IX, II, 94.
Charles X, II, 1.
Chasles, I, _39_, 229.
Chesterfield, Earl of (4th), II, _298_.
Chiffinch, W., II, _50_.
Ch'in Chiu-shang, II, 66.
Chitty, J., II, _323_.
Chiu-chang, Suan-shu, II, 67.
Christian Evidence Society, I, 270.
Christie, I, 27.
Christmann, I, 272, _272_.
Church question, I, 62.
Church, The word, II, 30.
Circle squarers. (_See Squaring the Circle._)
Circulating media of mathematics, I, 107.
Ciruelo. (_See Sanchez._)
Clairaut, I, _219_, 382.
Clarence, Duke of, I, 179.
Clarke, R., I, _255_.
Clavius, I, 11, _69_, 111, 112, 335, 362, 363, 372; II,
59.
Clayton., Geo., II, _98_.
Cluvier, D., II, 332, _332_.
Cobb, Mary, II, 117.
Cobbett, W., I, _177_, 200, 399.
Cobden, R., II, 217.
Cocker, I, _42_; II, 64, 173, 251, 307.
Cody, P., II, 208.
Coke, E., II, _331_.
Colburn, Z., I, _86_.
Colenso, I, _63_, 247; II, 191.
Collins, J., I, _107_; II, 297, 300, 302, 313.
Colvill, W. H., II, 68.
Cometic astrology, I, 128.
Comets, I, 128; II, 68, 83.
Cominale, C., I, _162_, 162.
Compton, S. J. A., II, _19_.
Computation, Paradoxes of, II, 251, 267.
Condamine, C. M. de la, II, _301_.
Conduitt, John, I, _397_.
Conduitt, Mrs., I, _136_.
Congregation of the Index, I, 90.
Converse propositions, I, 295.
Convocation at Oxford, I, 96.
Cooke, Margaret, I, _310_.
Cooper, A. A. (Shaftsbury), II, _181_.
Copernicus, I, 5, 6, _76_, 90, 121, 172, 380; II, 165, 335.
Copley, J. S., I, _198_.
Cormouls, I, 225.
Cosmology, I, 172.
{378}
Cotes, R., II, _301_.
Cottle, Mrs., II, 97, _97_, 161.
Craig, J., I, _129_, 129.
Creed, Mathematics of a, I, 329.
Cribb, T., I, _314_.
Crotus, J., I, 318.
Cruickshank, G., I, _186_.
Cube, Duplication of, I, 349.
Cumyns, Eliza, I, 299.
Cunningham, I, 172, _172_.
Curabelle, I, _221_.
Curious Calculations, II, 66.
Curll, E., II, _279_.
Cusa, I, _44_, 47, 360.
Custom, II, 324.
Cyclometry, II, 208. (_See Squaring of the Circle._)
Cyclopaedias, Review of, II, 280.

D'Alembert, I, _382_; II, 283, 364.
Dalgarno, I, 116, _117_.
Dalton, J., I, _255_.
D'Arblay, Mme., I, _190_.
Darwin, E., II, _8_.
Darwinism, Primitive, I, 344.
Dary, M., II, _305_.
Daval, P., II, _298_.
Davies, T. S., II, 151, _151_, 188.
Day, A., I, 295, _295_.
De Baruel, I, 165.
De Beaune. (_See Beaune._)
De Becourt, II, 277, _277_.
Debenham, J., I, _393_.
De Causans. (_See Causans._)
Dechales. (_See de Challes._)
De Challes, I, _45_.
Decimal coinage, II, 80, 168, 169.
Decimals run riot, II, 80.
Dee, J., II, _302_.
De Faure, I, 149.
De la Leu, I, _297_.
Delambre, I, _160_, 167, 354; II, 165.
Democritus, II, _34_.
De Moivre, I, 24, _376_; II, 298.
De Molieres, I, _220_.
De Molina, I, _297_.
Demonville, I, 291, 293.
De Morgan, A., I, 191, 383; II, 194.
--Refusal of LL. D., I, 191.
De Morgan, G. C., I, 383.
De Morgan, Mrs., I, _196_; II, 194.
Denison, J., I, 348, 353.
Desaguliers, I, _153_, 156, 157.
Desargues, I, _119_, 221.
Descartes, I, 5, 59, 105, 132, 165, 204, 220; II, 94.
De Serres, II, _60_.
De Sluse. (_See Sluse._)
De Thou, I, 51, _111_, 113; II, 295.
De Vausenville, I, 12.
Devonshire, Duke of (7th), I, _290_.
Diamandi, I, 86.
Didacus Astunica, I, 90.
Diderot, II, _4_, 283, 339.
Digby, K., I, _108_.
Digges, T., and L., II, _302_.
Dionysius Exiguus, I, _360_.
Dircks, H., II, 138, _138_.
Discoverers and discoveries, II, 206.
Discovery, Basis of, I, 85.
D'Israeli, I., I, _115_, 118, 136, 188, 227.
Ditton, I, _133_, 133.
Division, Nature of, II, 248.
Dobson, J., I, _234_, 234.
Dodson, J., II, _312_.
Dodt, I, _52_.
Doggerel verse, I, 341.
Dolland, I, _377_.
Double Vahu Process, II, 360.
Douglas, G., I, _232_.
Drach, S. M., II, _317_.
Drayson, G. A. W., II, _132_, 132.
Dryden, II, _71_.
Dual arithmetic, II, 186.
Duchesne, I, _52_.
Dumortier, I, 313.
Duncan, A., I, _179_.
Dunkin, E., II, _349_.
Duodecimal scale, II, 68.
Duplication Problem, I, 349.
Dupuy, J. and P., II, _295_.
Dutens, L., II, _90_.
Dyer, G., I, _178_.

Earth, Figure of, II, 54.
Easter, I, 359.
Easter Day Paradoxes, I, 353.
Ebrington, Thos., I, _247_.
Edgeworth, Maria, I, _191_.
Editorial System, I, 15.
Edleston, I, _140_; II, 296.
Edwards, J., I, _144_.
Edwards, T., I, _112_.
Eirenaeus Philalethes, I, _125_, 125, 126.
Eldon, Lord (1st), II, _197_.
Elephant story, I, 58.
Elizabeth, Queen, I, 128.
Ellenborough, Baron, I, _181_.
{379}
Ellicot, I, 24.
Ellis, I, _76_, 82.
Engel, I, 230.
English language, Origin of, I, 215.
Enriques, F., II, _367_.
Epps, J., I, _153_; II, 143.
Equation of fifth degree, I, 250, 373.
Erasmus, I, 110.
Erastus, I, _65_.
Erichsen, I, 163.
Ersch, II, _193_, 282.
Erskine, T., II, _127_.
Esperanto, Forerunner of, I, 116.
Euclid, I, 5, 43; II, 118, 151.
--Without Axioms, I, 287.
Eudoxus, II, _164_.
Euler, I, 221, _382_; II, 3, 4, 303, 331, 339.
Eusebius, II, _220_.
Eustace, J. C., II, _46_.
Eutocius, I, _41_; II, 60.
Evelyn, J., I, _108_.
Everett, J., I, _346_.
Evidence, I, 57, 58.

Faber. (_See Stapulensis._)
Fairfax, Mary, I, _242_.
Falco, I, 53.
Faraday, M., II, _351_.
Faure, de, I, 149; II, 238.
Fawcett, H., II, _249_.
Ferguson, J., II, _20_.
Fermat, I, 122, 221; II, _300_.
Ferrari, S., II, 68.
Fiction, New era in, I, 189.
Fienus, I, _74_, 74.
Filopanti, Q. B., II, _93_.
Finaeus, I, _50_, 50, 113.
Finleyson, J., I, 314, _314_.
Flamsteed, I, _87_, 309; II, 45, 143, 302, 306.
Fletcher, I, 378.
Fludd, II, _318_.
Fly-leaf Paradox, II, 264.
Folkes, M., I, _136_; II, 301.
Fontenelle, I, _103_.
Forbes, D., I, _237_.
Forman, W., I, 296, _296_, 306.
Forster, T. I. M., I, 320, _320_.
Foscarini, I, _90_.
Foster, S., II, _310_.
Fourier, II, _66_.
Fox, G., I, _397_.
Francis, P., II, _96_.
Francoeur, I, _365_.
Frankland, W. B., I, 230, 287.
Franklin, J., II, _265_.
Freedom of opinion, Growth of, I, 265.
Freher, A., II, 319.
French academy on circle squaring, I, 163.
Frend, W., I, _196_, 196, 206, 208, 252.
Fresnel, II, _48_.
Fromondus, I, _74_, 74, 99.
Frost, I. and J., I, 394.
Fry, Elizabeth, I, 224.
Fuller, T., I, _86_.
Fulton, R., I, _148_.

Gadbury, J., I, _115_, 115.
Galbraith, J. A., II, 372.
Galileo, I, 5, 6, 32, _76_, 82, 83, 96, 122, 381.
Galle, J. G., I, _386_; II, 7.
Galloway, I, _56_, 57; II, 143.
Gamblers, I, 280.
Garrick, I, 21.
Gascoigne, W., II, _299_.
Gassendi, I, _107_.
Gauss, I, 86, 107, 310.
Gemistus, G., I, _188_.
Gentleman's Monthly, Miscellany, I, 208.
Gephryander. (_See Salicetus._)
Gergonne, I, _336_.
Ghetaldi, I, _83_; II, 59.
Ghost paradox, II, 47.
Giddy (Gilbert), II, _174_.
Gilbert, Davies, II, 66, _174_.
Gilbert, William, I, 6, _68_, 68, 76.
Gillot, II, _315_.
Glazier (Glazion), II, 7.
Godwin, F., I, _103_.
Godwin, W., I, _174_.
Golius, I, _106_.
Gompertz, B., I, _378_.
Goulburn, I, _288_.
Goulden, S., II, 88.
Graham, I, 24.
Grandamicus, I, _104_, 104.
Granger, J., I, _156_.
Grant, A. R., II, 131.
Grant, R., I, _392_; II, 131.
Grassi, O., I, _262_.
Grassini, I, _231_.
Graunt, J., I, 113, _114_, 154.
Gravity, I, 151, 244, 348, 353.
{380}
--Newton's apple, I, 136.
Greek numerals, II, _77_.
Greene, R., I, _135_, 135.
Greenhill, Sir G., I, 136.
Greenwich observatory, I, 87.
Gregg, T. D., II, 75, _75_.
Gregorian Calendar, I, 363.
Gregory, D., I, _66_; II, 301.
Gregory, J., I, _118_, 118, 207; II, 302.
Gregory O., II, _71_.
Gregory, Pope, I, 362.
Grevil, I, _202_.
Grey, C., (2d Earl), I, _315_; II, 247.
Grosart, I, _141_, 141, 145.
Grove, W. R., II, _320_.
Gruber, II, _193_, 282.
Gruenberger, I, _70_.
Grynaeus, I, _66_.
Guaricus, I, _43_.
Guillim, J., II, _28_.
Guldin, I, _83_.
Gumpach, Von, II, 137, _137_.
Gunning, H., I, _198_.
Gurney. (_See Fry, E._)
Guthrie, W., I, _395_.

Hailes, J. D., II, 135, _135_.
Hailesean system of astronomy, II, 135.
Hale, M., I, _123_, 123.
Hales, S., I, _123_.
Hall, B., II, _181_.
Hallam, I, 159.
Halley, I, 24, _124_, 311; II, 301, 332.
Halliwell-Phillips, II, _148_, 296.
Hamilton, W., I, _112_, 117, 331, 335, 339, 341, 342; II,
52, 53, 111.
Hamilton, W. Rowan, I, _332_; II, 104, 256, 343.
Hanover, King of, I, 201.
Hardy, C., I, _298_.
Hardy, T., I, _178_.
Harriot, T., II, _302_.
Harvey, I, _76_, 78; II, 201.
Hauff, I, _230_.
Haughton, S., II, 372.
Hauksbee, F., I, _156_.
Hayes, C., I, _132_, 132.
Heath, D. D., I, _76_.
Heinfetter, H., II, 94, _94_.
Helmont, J. B. van, I, _125_.
Henson, II, 8.
Heraclitus, II, _34_.
Herbart, J. F., I, 253, _253_; II, 78.
Herigone, II, _59_.
Herschel, J., I, _80_, 299, 326, 383, 386; II, 88, 95, 181,
255, 261, 262.
Herschel, W., I, _81_, 151, 192, 225, 233, 299; II, 288, 348.
Heywood, F., II, _49_.
Hicks, J. P., II, _67_.
Higgins, G., I, _257_, 274.
Hilarius, Pope, I, _359_.
Hill, J., I, 21, 22, 23, 24.
Hill, Rev. R., I, _192_.
Hill, Sir R., I, _165_, 232.
Hind, J. R., I, _384_.
Hippocrates, II, _59_.
Hoax, An interesting, I, 163.
--Lunar Caustic, I, 288.
--Moon (Herschel), I, 326; II, 131.
Hobbes, I, _105_, 109, 143, 144; II, 80.
Hobhouse, J. C., II, _126_.
Hodder, J., II, _265_.
Hodge, C. B., I, 114.
Hodges, W., I, 237.
Hoffmann, J. J., II, _282_.
Hoffmann, J. J. I. von, I, _230_.
Holloway, B., I, _237_.
Holmes, O. W., I, 109.
Holyoake, G. J., I, _399_, 399.
Hone, W., I, 124, 180, 184, 185.
Hook, T. E., II, _261_.
Hooke, I, _77_; II, 300.
Hooker, R., II, _201_.
Hopkins, J., II, 41.
Horace, I, 40.
Horne, G., I, _152_, 152, 154, 155, 236.
Horne, J., I, _178_.
Horner, L., I, _176_.
Horner, W. G., II, _66_, 151, 187.
Houlston, W., II, 156, _156_.
Howard, E., I, 131.
Howison, W., I, 256, _256_.
Howitt, W., II, 193, _193_.
Howley, I, 63.
Hulls, I, _147_, 147; II, 8.
Hume, J., I, _352_; II, 174.
Husain Rifki, II, 16.
Hussein Effendi, II, 15.
Hutchinson, J., I, 154, _154_.
Hutton, C., I, _153_, 161; II, 303, 340.
Huyghens, I, _100_, 133; II, 300.

Imaginary numbers, II, 186.
Impalement by request, II, 133.
Inaudi, I, 86.
Index Expurgatorius, I, 90.
{381}
Infant prodigies, I, 86.
Inglis, J. B., II, _52_.
Inglis, R. H., I, _352_.
Ingliz Selim Effendi, II, 15.
Innocent I., I, _359_.
Irving, E., II, _54_.
Ivory, J., II, 142, _142_.

Jabir ben Aflah, II, _59_.
Jack, R., I, 149.
Jacotot, J., I, 278, _278_.
Jameson, Mrs., II, _63_.
Jeffreys, G., I, _183_.
Jenner, E., II, _205_.
Jesuit contributions, I, 164.
Johnson, H. C., I, 350.
Johnson, S., I, 20, _190_, 259; II, 117.
Johnston, W. H., II, 67.
Jombert, I, _161_.
Jonchere, I, _146_, 146.
Jones, W., I, _135_; II, 298, 301.
Jones, Rev. W., I, _237_.
Jonson, B., I, 13.
Journals, Three classes of, II, 144.

Kantesian Jeweler, I, 258.
Karsten, I, _230_.
Kaestner, I, _43_, 110, 112.
Kater, I, 11.
Keckermann, I, 3.
Keill, J., II, _302_.
Kepler, I, 52, _76_, 82, 381; II, 166.
Kerigan, T., I, _308_, 353.
Keroualle, De, II, 50.
Kersey, I, 107.
King, Wm., I, _246_.
Kircher, Adolphe, I, 229.
Kircher, Athanasius, I, _103_.
Kirkringius, T., I, 125, _125_.
Kittle, I, 236.
Klein, F., II, 367.
Knight, C., II, _109_, 280, 289.
Knight, G., I, 151, _151_.
Knight, R. P., II, _274_.
Knight, Wm., I, _97_.
Koenig, S., I, _150_.

Lacomme, I, 46.
La Condamine, II, _301_.
Lacroix, I, 41, 159, 207.
Lactantius, I, 33, _96_.
Lagrange, I, 221, _288_, 313, 382; II, 86.
Laing, F. H., II, _186_, 186.
Lalande, I, _159_.
Lamb, C., I, 178; II, 270.
Lambert, J. H., I, _336_; II, 214, 367.
Lambert, John, II, _309_.
Language, Test of, II, 327.
Lansbergius, I, _70_, 70.
Laplace, I, 24, _255_, 382; II, 1, 340.
Lardner, D., II, _253_.
Lardner, N., I, 14; II, _221_.
Laud, I, _145_.
Lauder, W., I, 297.
Laurent, P., I, _309_, 309.
Laurie, J., II, 4.
Laurie, P., II, _42_.
Laurus, I, 381.
Law, Edmund, I, _181_.
Law, Edward, I, _181_.
Law, W., I, 168, _254_; II, 317.
Le Coq, I, _86_.
Lee, R., I, _66_.
Lee, S., I, _131_.
Lee, W., I, 157.
Legate, I, _59_.
Legendre, I, _229_; II, 215, 367.
Legh, P., II, _68_, 83.
Leibnitz, I, 5, 7; II, 46.
Leo, St., I, _359_.
Leverrier, I, _43_, 82, 383, 386, 388, 390; II, 7, 135, 140,
303.
Lewis, G. C., II, 162, _162_.
Libri, I, _40_, 62; II, 295.
Lilius, Aloysius, I, 362.
Lilly, I, _115_; II, 302.
Lipen, M., I, _298_.
Little, J., I, 206.
Livingston, R., I, _148_.
Locke, J., I, _142_, 142, 144;
--and Socinianism, I, 142.
Locke, R., I, 146.
Locke, R. A., I, _326_; II, 86, 131.
Logan, W. E., I, _337_.
Logic, Formal, I, 158; II, 75.
--Has no paradoxes, I, 330.
London Mathematical Society, I, 383.
London, University of, I, 259; II, 71.
Long, G., II, _290_.
Long, J. St. J., II. _38_, 205.
Longitude problems, I, 132, 146, 249.
Longley, C. T., I, _325_.
Longomontanus, I, _105_, 105.
Lottery, I. 281.
Lovett, R., I, 165, _166_.
Lowe, R., II, _169_.
Lowndes, W. T., I. _402_.
Lubbock, J., I, _279_; II, 148.
{382}
Lucas, F., II, _28_.
Lucian, I, _102_.
Lunar Caustic Joke, I, 288.
Lunn, J. R., II, _66_.
Lydiat, T., II, _302_.
Lyndhurst, I, _198_.

Macclesfield, Earls of, I, 7; II, _296_, 301.
Macclesfield, Letters, II, 296.
MacElshender, II, 87.
Machin, J., II, _301_.
Mackey, John, I, 349.
Mackey, S. A., I, _256_.
Maclear, T., II, _181_.
Macleod, H. D., II, 184, _184_.
Magic, I, 118.
Magna Charta, I, 25.
Magnus, I, 42.
Maitland, S., I, _63_, 163.
Malacarne, I, 119.
Malden, H., II, _162_.
Malius, II, _342_.
Mallemens, II, _333_.
Mankind gullible, I, 115.
Manning, H. E., (Card.), II, _233_.
Mansel, H. L., II, _162_.
Marcelis, J., I, _129_, 129.
Maret, II, _3_.
Margarita Philosophica. (_See Reisch._)
Marryat, Capt., II, _87_.
Marsh, H., I, _199_, 271.
Martin, B., I, _152_, 153.
Martin, R., II, _236_.
Maseres, F., I, _197_, 203.
Mason, M., II, _132_.
Mathematical Illustrations of Doctrine, II, 70;
--Psychology, I, 253;
--Society, I, 374, 376, 382;
--Theology, I, 149.
Mathematics, Condensed history of, II, 58.
Matter to Spirit, II, 194.
Maty, I, 23.
Maupertuis, II, _301_.
Maurice, F. D., II, _101_.
Maurolycus, I, _121_.
Maxwell, A., I, _102_.
Meadley, G. W., I, _223_.
Mechanics Magazine, II, 141, 145.
Medici, Cosmo de, II, _295_.
Medicine, Status of, I, 266.
Melanchthon, II, _323_.
Menestrier, I, _127_, 127.
Mercator, G., II, _92_.
Mercator's projection, I, 84.
Mersenne, I, _106_, 107; II, 295, 297.
Meslier, J., II, _195_.
Meteorologist, An early, I, 320.
Meteorology, I, 327.
Metius, A. and P., I, _52_, _99_, 99.
Meton, II, 167.
Metric System, Forerunner of, I, 171.
Meziriac, I, 161.
Milbanke, A. I., I, 225.
Mill, Jas., I, 260.
Miller, Joe, I, _182_.
Miller, S., I, 167.
Mills, Elizabeth, W., II, 7.
Milne, J., I, _286_.
Milner, I., I, _251_, 251.
Milner, J., II, _23_.
Milner's lamp, I, 252.
Milward, II, _250_.
Miracles _vs_. Nature, II, 6.
Mitchell, J., I, _242_.
Moliere, I, _232_.
Molina, A. C. de, I, _297_.
Mollendorff, von, II, _3_, 338.
Mondeux, I, 86.
Montague, C., II, _311_.
Montmort, P. R. de, II, _301_.
Montucla, I, 40, 45, 54, 65, 117, 120, 159, 163; II, 60.
Moon Hoax, I, 326; II, 131.
Moon, Nature of, II, 84;
--Rotation of, II, 4, 19, 84, 87.
More, Hannah, I, _189_, 192.
More, Henry, I, 123.
Moore, Dr. John, I, _190_.
Moore, Sir John, I, _190_.
Morgan, S., I, 6.
Morgan, T., I, 191.
Morgan, W., I, _223_, 224.
Morhof, I, _61_.
Morin, I, _99_, 99.
Morinus, J. B., I, 149.
Morland, S., II, _302_.
Mormonism, II, 69.
Morrison, R. J., I, _321_; II, 43.
Mose, H., II, _266_.
Mottelay, I, 68.
Motti, II, 60.
Mouton, I, _172_; II, 300.
Muggleton, I, 394, _395_.
Multiplication, Nature of, II, 251.
Murchison, R. I., I, _384_.
Murhard, I, _43_, 298.
{383}
Murphy, A., II, _308_.
Murphy, J. L., II, 54, _54_.
Murphy, P., I, _327_, 398.
Murphy, R., I, 349, _349_.
Murray, J., I, _186_; II, 145.
Murray, L., II, _326_.
Murray, Mungo, II, 310.
Musgrave, T., I, _324_.
Mydorge, I, _298_.
Mystrom, J. W., II, 182.
Mythological paradoxes, I, 256.

Names of Religious Bodies, II, 22.
Napier, J., I, 5, 66, _67_, 82.
Napoleon, Doubts as to, I, 246.
Nautical Almanac, I, 300; II, 147.
Neal, I, _98_.
Negative numbers, I, 196, 203.
Neile, W., II, _190_.
Neptune, Discovery of, I, 387; II, 140. (_See Adams, Leverrier._)
Nesse, C, I, _128_, 128.
Newcomb, S., I, 162.
Newcomen, T., I, _147_.
Newton, Sir Isaac, I, 5, 6, 8, 24, 39, 84, 88, 130,
136, 139, 144, 145, 148, 152, 154, 155, 162, 165,
167, 197, 225, 237, 242, 257, 282, 296, 297, 309,
311, 382, 394, 395, 396, 397; II, 2, 70, 184, 297, _302_,
305.
Newton, John, II, _305_.
Nicene Creed, I, 371.
Nichol, J. P., II, _289_.
Nicholas. (_See Cusa._)
Nichols, J., I, _175_.
Nicolas, N. H., I, _354_.
Nicollet, I, _326_; II, 131.
Nicolson, W., I, _201_.
Nieuwentijt, II, _333_.
Nizzoli, M., II, 275.
Non-Euclidean geometry, II, 83.
Northampton, Marquis of (2d), II, _19_.
Novum Organum Moralium, II, 74.
Number, Mystery of, I, 55, 56, 169.
Number of the Beast (666), I, 55, 130, 272, 298, 352; II, 77,
159, 217, 218, 361, 373.
Numeral system, II, 68.
Nursery rhymes, II, 150.

Occam, Wm. of, II, _40_.
Odgers, N., II, 191, _191_.
Oinopides of Chios, II, _59_.
Oldenburgh, H., II, _300_, 302.
Orthodox Paradoxes, II, 363.
Orthography, Paradoxes of, II, 267.
Ortwinus, I, 319.
Oughtred, W., II, _298_, 303.
Owenson, I, _191_.
Ozanam, I, _161_, 312.

Pagi, I, 32.
Paine, T., I, 173, _173_, 271.
Paley, W., I, _222_; II, 226.
Palmer, C., I, 225.
Palmer, H., I, _141_, 141, 145.
Palmer, J., II, _253_.
Palmer, T. F., II, _254_.
Palmer, W., II, _37_.
Palmerston, Viscount (3d), I, _290_, 352.
Palmezeaux, I, 167.
Panizzi, A., I, _151_.
Papist, II, 26.
Paracelsus, II, 322.
Paradox defined, I, 2, 31.
Paradox, religious, I, 236.
Paradoxers in general, II, 352.
Parallels, Theory of, I, 229, 344.
Pardies, I. G., II, _300_.
Park, Mungo, II, _91_, 132.
Parker, F., II, _94_.
Parker, G. (Earl of Macclesfield), II, _296_.
Parr, S., I, 173, _173_, 175, 176, 184.
Parsey, I, 293, _293_.
Partridge, J., I, _305_.
Pasbergius, I, 381.
Pascal, I, 39, 119, 122, _220_, _221_; II, 73.
Pascal's Hexagram, I, 221.
Passot, I, 279, _279_.
Passover, I, 358, 372.
Patriotic paradox, I, 231.
Paucton, I, _172_.
Paulian, I, 165, _165_.
Peacock, Geo., I, _196_, 350.
Peacock, T. L., I, _190_, 340.
Pearce, A. J., II, 43.
Pearne, T., I, _239_.
Peel, Sir R., I, 290, _352_.
Peel, W. Y., I, _290_.
Pelerin, J., II, _324_.
Pell, J., I, _105_, 105, 107; II, 300, 302, 312.
Pepys, I, _113_, 114.
Perigal, H., II, 19, _20_.
{384}
Perpetual motion, I, 118, 348; II, 55, 138.
Perspective, New theory of, I, 293.
Peters, W., II, 11, 315.
Petitioning Comet, I, 127.
Petrie, W. M. F., I, _328_.
Petty, I, _114_; II, 300.
Philalethes, Eirenaeus, I, _125_, 125, 126.
Philalethes, Eugenius, I, _255_.
Phillips, R., I, 242, _242_, 245.
Philo of Gadara, I, 40, _40_.
Philosopher's stone, I, 118, 125.
Philosophical atheists, II, 1.
Philosophy and Religion, II, 37.
Phonetic spelling, II, 81.
[pi], values of, I, 11, 43, 45, 46, 52, 69, 100, 110,
129, 135, 146, 245, 283, 284, 294, 347, 348, 349,
350; II, 60, 63, 105, 110, 118, 135, 156, 209, 279, 315, 316.
Pighius, I, _372_.
Pike, S., I, 236, _236_.
Pindar, P., II, _272_.
Piozzi, Mrs., I, _235_; II, 272.
Piscator, B., II, 25.
Pitman, F., II, 81, _81_.
Place, F., I, _199_.
Planets inhabitable, I, 100, 102.
Plato, I, 5.
Platt, H., I, _126_, 126.
Playfair, J., I, _233_.
Pletho, G., I, _188_.
Pliny, II, 280.
Ploucquet, I, _336_, 337.
Poe, E. A., II, 132.
Poincare, I, 136.
Poisson, I, _292_; II, 2.
Pollock, J. F., II, _174_.
Pons, II, _45_.
Pope, Wm., I, 277, _277_.
Porta, I, _68_, 68, 83.
Porteus, B., I, _193_, 203.
Porteus, H. F. A., II, 157, _157_.
Porus, I, 44.
Powell, Baden, II, _267_.
Powell, W. S., I, _222_.
Pratt, H. F. A., II, 157, _157_.
Pratt, O., II, _69_.
Predaval, Count de, I, _348_.
Prescot, B., I, 270, _270_, 278.
Prester John, I, 70, _71_, 152.
Price, R., I, _223_.
Probability, Discourse on, I, 279.
Proclus, I, 188, _188_.
Prodigies, Youthful, I, 219, 332.
Pronunciation, II, 330.
Protestant and Papal Christendom, II, 33.
Protimalethes, II, 6.
Ptolemy, I, 5, 33, _380_.
Pullicino, II, 61, _61_.
Pusey, I, _64_.
Pyramids, The, I, 328; II, 95, 136.
Pythagoras, II, _59_.

Quadrature problem. (_See Squaring the circle._)
Quarles, F., II, _277_.
Quintilian, II, 280.
Quotem, C., I, 399.

Rabelais, I, _102_.
Rainbow Paradox, II, 334.
Ramachandra, Y., I, _374_.
Ramchundra, I, 374.
Ramus, I, 5.
Recalcati, II, 208, 314.
Recorde, R., II, _328_.
Reddie, Jas., II, 183, _183_, 344, 360.
Reeve, J., I, _395_.
Regiomontanus, I, _48_, 360.
Reisch, I, _45_; II, 281.
Religion and Philosophy, II, 37.
Religious bodies, Names of, II, 22;
--customs, Attacks on, I, 177;
--Insurance, I, 345;
--Paradox, I, 236;
--Tract society, I, 192.
Remigius, I, _50_.
Reuchlin, J., II, _323_.
Revelations, Napier on, I, 66.
Revilo, (O. Byrne), I, 241, 329, _329_.
Reyneau, C. R., II, _301_.
Rheticus, I, _69_; II, 372.
Rhonius, II, 300.
Ribadeneira, P. de, II, _62_.
Riccioli, I, _96_.
Richards, G., II, _270_.
Rigaud, J., II, _299_.
Rigaud, S. J., II, _299_.
Rigaud, S. P., I, _140_; II, 298, 313.
Ringelbergh, J. S., II, _281_.
Ripley, G., I, _126_, 126.
Ritchie, W., I, 295, _295_.
Ritterhusius, I, _60_.
Rive, J.-J., I, _160_.
Robertson, Jas., I, _237_.
Roberval, I, _105_, 122.
{385}
Robinson, B., I, _148_, 148.
Robinson, H. C., I, _314_; II, 52, 275.
Robinson, R., I, _177_.
Robinson, T. R., II, _181_.
Roblin, J., II, 136.
Rogers, S., II, _260_.
Roget, P. M., I, _398_.
Roomen, A. van, I, _110_.
Ross, J. C., I, _303_.
Rosse, I, 26.
Rossi, G., I, 231, _231_.
Rotation of the Moon, II, 4, 19.
Rough, W., I, 198.
Rowning, J., I, _155_.
Royal Astronomical Society, I, 27;
--Forerunner of, I, 374.
Royal Society, I, 21, 22, 24-30, 56, 57, 136, 153, 163,
164, 165.
Rudio, I, 159; II, 367.
Rudolff, C., II, 373.
Russell, Earl (1st), I, _296_.
Rutherford, W., II, _109_.

Sabatier, A., II, _267_.
Sabellius, I, _241_.
Sacrobosco, I, _360_.
Sadler, T., I, _238_, 241.
Saint-Martin, I, 167, _168_, 206.
St.-Mesmin, M. de., I, 280.
St. Vincent, G. de., I, _110_, 117.
St. Vitus, Patron of Cyclometers, II, 60.
Sales, de, I, 167.
Salicetus, I, 64.
Salisbury, Earl of (1st), II, _330_.
Salmasius, Claudius, II, _168_.
Salusbury, Hester, I, _235_.
Sanchez, Petro, I, 229, _229_.
Sanders, W., I, 207.
Sanderson, R., I, _135_.
Sara, R., I, _297_.
Saunderson, N., I, _377_; II, 301.
Scaliger, I, _44_, 110, 111, 112, 113; II, 238.
Scevole de St. Marthe, I, 113.
Schooten, Van, II, _59_.
Schopp, I, _60_.
Schott, I, _64_; II, 64.
Schumacher, H. C., I, _107_; II, 297.
Schwab, I, _230_.
Scientific paradoxes, I, 232.
Scott, Michael, I, _38_.
Scott's Devils, I, 38.
Scott, W., I, 20, 27, 38, 39, 155, _191_.
Scripture and Science, II, 261.
Search, John, I, 247.
Selden, J., II, _250_.
Senarmont, II, _48_.
Serres, De, II, _60_.
Shaftesbury, Earl of, II, _181_.
Shakespeare, I, 13.
Shanks, II, _63_, 65, 109.
Shaw, P., I, _142_.
Sheepshanks, J., I, _147_.
Sheepshanks, R., I, _290_.
Shelley, I, 174.
Shepherd, S., I, _124_.
Sherburne, E., II, _295_.
Sheridan, R. B., I, _175_.
Sheridan, T., I, _175_.
Shoberl, F., II, _270_.
Shrewsbury, I, _108_.
Siddons, Mrs., I, 189.
Simms, W., I, _152_.
Simplicius, II, _164_.
Simpson, T., I, 377; II, 304.
Simson, R., I, _197_, 202, 233.
Sinclair, G., I, _207_.
Slander Paradoxes, II, 138.
Sloane, I, 24.
Sluse, R. de, I, _118_, 118; II, 300.
Smith, Adam, II, _112_.
Smith, Jas., I, _46_; II, 103, _103_, 154, 236, 237, 238, 241, 336,
360.
Smith, Jas., II, _217_.
Smith, Jas. (Shepherd), II, _55_, 193.
Smith, Joseph, II, _69_.
Smith, Richarda, I, _242_.
Smith, Thomas, I, 346, _346_.
Smith, Wm., II, _152_.
Smyth, C. P., I, _328_; II, 65.
Snell, I, _75_, 75.
Socinianism, I, 142, 143.
Socinus, I, 3, _143_.
Socrates Scholasticus, I, _358_.
Sohncke, L. A., II, _131_.
Somerville, Mrs., I, _242_.
South, J., II, _181_.
Southcott, Joanna, II, _58_, 97, 239.
Spearman, R., I, _237_.
Speculative thought in England, I, 374.
Spedding, I, _76_, 82, 142.
Speed, J., I, _201_.
Speke, I, _70_.
Spelling, phonetic, II, 81.
Spence, W., I, _231_, 231.
Spencer, Earl (3d), II, _9_.
{386}
Spinoza, I, 3, _37_.
Spiritualism, II, 47, 55, 207.
Spurius Cassius Viscellinus, II, _342_.
Spurius Maelius, II, _342_.
Squaring the circle, I, 8, 42, 44, 46, 47, 50, 52, 53,
69, 70, 75, 109, 117, 119, 129, 135, 146, 149,
159, 163, 164, 347, 348, 374; II, 10, 11, 60, 105, 154,
156, 208, 278, 314.
Staeckel, I, 230.
Stanhope, P. D., (Earl of Chesterfield), II, _298_.
Stapulensis, I, _44_; II, 324.
Star polygons, I, 229.
Starkie, G., I, _126_, 126.
Statter, D., II, 80.
Steamship suggested, I, 147.
Steel, Jas., II, 68.
Stenography, II, 81.
Stephens, I, _44_; II, 324.
Stephenson, G., II, _138_.
Stephenson, R., II, _138_.
Stevin, I, _83_, 313; II, 59.
Stewart, D., II, _53_.
Stewart, R., I, _186_.
Stifel, M., II, _373_.
Strafford, Earl of, I, _240_.
Stratford, W. S., I, _300_.
Street, T., I, _124_.
Stukely, W., I, _236_.
Suffield, G., II, _66_.
Suidas, II, _29_.
Sumner, C. R., I, _324_.
Sumner, J. B., I, _324_.
Sun as an electric space, II, 41.
Supernatural, The, II, 193.
Suvaroff, II, _85_.
Swastika, II, 231.
Swedenborg, E., I, _255_.
Swift, I, 19, 133.
Sylvester, J. J., II, _336_.
Symington, W., I, _148_.
Symons, II, 4, 5, 20, 84, 85.
Sympathetic powder, I, 108.
Synesius, I, 125.

Talbot, G., I, _22_, _108_.
Talbot's powder, I, 108.
Tartaglia, II, 59.
Tasse, I, _106_.
Tate, J., I, _199_.
Tauler, J., II, _322_.
Taylor, Brook, II, _301_.
Taylor, John, I, _352_; II, 95.
Taylor, Robt., I, _270_.
Taylor, T., I, 188, _188_.
Teissier, I, _113_.
Temple, H. J., I, 290.
Tenterden, Chief Justice, I, _181_.
Thales, II, _59_, 83.
Theism independent of Revelation, I, 399.
Thelwall, J., I, _178_.
Theodoretus, I, _358_.
Theological Paradoxes, I, 316.
Theology, Mathematical, I, 129, 149.
Theophrastus, II, _167_.
Thiebault, II, _3_, 338.
Thom, D., II, _226_, 240.
Thom, J. H., II, _226_.
Thompson, P., I, _7_.
Thompson, T. P., I, _252_, 287, 344; II, 83, 185.
Thomson, Dr., I, 21.
Thomson, W., I, _325_.
Thorn, W., II, 158, _158_, 360.
Thorndike, H., II, _313_.
Thrale, Mrs., I, _235_.
Thurlow, Baron, I, _222_.
Thyraeus, I, 50.
Tides, New theory of, I, 393.
Tombstones of mathematicians, I, 106.
Tonal System, II, 182.
Tooke, H., I, _178_.
Torriano, E., I, 250.
Towneley, II, 300.
Townley, C., II, _300_.
Trisection problem, I, 118; II, 10, 12, 13, 15.
Troughton, I, _152_.
Turnor, E., I, _137_.
Tycho Brahe, I, 5, _76_, 381; II, 302, 335.

Upton, W., II, 12, _12_, 15.
Ursus, I, _52_.

Valentine, B., I, _125_, 125.
Van Ceulen, I, 52, 70, 100.
Van de Weyer, I, _313_.
Van Etten, I, _161_.
Van Helmont, I, _125_, 125.
Van Roomen, I, _110_.
Van Schooten, II, _59_.
Vaughan, T., I, _255_.
Victorinus, I, _359_.
Viete, I, _51_; II, 210, 295.
Virgil, St., I, 32, _33_, 34, 99.
{387}
Virginia, University of, I, 233.
Viscellinus, II, _342_.
Vitruvius, II, _281_.
Vivian, T., I, 172, _172_.
Vogel, A. F., I, 373.
Voltaire, I, 103, 165, 166, 167, 168, 248; II, 268.
Von Gumpach, II, 137, _137_.
Von Hutten, I, 318.
Von Wolzogen. (_See Wolzogen._)
Vyse, R. W. H., I, _328_.

Walker, W. E., II, _316_.
Walkingame, F., II, _173_.
Wallich, N., II, _14_.
Wallis, J., I, 107, 109, 110; II, 299, 313.
Walpole, I, _23_, 131.
Walsh, John, I, 260, _260_; II, 157.
Wapshare, J., II, _230_.
Warburton, H., I, _349_.
Warburton, Wm., I, _55_, 112; II, 174.
Ward, S., II, _299_.
Waring, E., I, _203_, 222.
Warner, W., II, _302_, 312.
Warren, S., II, _340_.
Watkins, J., II, _270_.
Watson, Bp., I, _223_.
Watt, R., I, _102_, 402.
Watts, I., II, 18.
Weddle, T., II, _187_.
Wentworth, Thos., I, _240_.
Wharton, I, 115.
Whately, R., I, 246, _246_, 324.
Whately's Paradox, I, 246.
Whewell, I, _101_, 101, 273, 314, 380; II, 104, 246, 247.
Whigs, II, 22.
Whiston, J., I, _147_.
Whiston, W., I, 133, _133_, 146, 156, 311.
White, H. K., II, _271_.
White, J. B., I, 248.
White, R., I, 11.
Whitford, I, 105.
Whitworth, W. A., II, _344_.
Whizgig, On the, I, 254.
Wightman, I, 59.
Wilberforce, W., II, _236_.
Wilkins, J., I, 96, _100_, 116, 226.
Williams, J. B., I, _378_.
Williams, T., I, 171, _171_.
Wilson, Sir J., I, _221_.
Wilson, J. M., II, _344_.
Wilson, R., II, 7, _7_.
Wilson's Theorem, I, _222_.
Wingate, E., II, _308_.
Winter, I, 46.
Wirgman, T., I, 258, _258_.
Wiseman, N. P. S., II, _26_, 61, 294.
Wolcot, J. (Peter Pindar), II, _272_.
Wollstonecraft, I, 173, _173_.
Wolzogen, I, _106_.
Wood, A., I, _98_.
Wood, John, I, _233_.
Wood, Wm., I, 246, _246_.
Woodley, W., I, 307, _307_.
Wordsworth, II, 273.
Wright, E., I, 84.
Wright, T., I, 151, _151_, 152, 153.
Wright, W., II, 9.
Wronski, I, 249, _250_.
Wrottesley, J. (Baron), II, _181_.

Young, B., II, _69_.
Young, J. W. A., II, 367.
Young, T., I, 24, 30, _250_.
Youthful Prodigies, I, 219.
Yvon, I, _297_.

Zach, von, II, _45_, 196.
Zachary, Pope, I, 32, 34.
Zadkiel, I, _321_; II, 43.
Zetetic Astronomy, II, 88.
Zodiac, II, 136.
Zytphen, II, _335_.

* * * * *

Notes

Transcriber's note: References to Notes in Volume I are shown as in the printed book, with the resequenced footnote numbers in the Project Gutenberg Edition (EText-No. 23100) added thus {123}.

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

[2] "I have no need for this hypothesis."

[3] "Ah, it is a beautiful hypothesis; it explains many things."

[4] "What we know is very slight; what we don't know is immense."

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

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

[7] "What is all that!"

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

[9] "Man follows only phantoms."

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