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A Budget of Paradoxes, Volume II
A Budget of Paradoxes, Volume II

Chapter III

Chapter 3 of A Budget of Paradoxes, Volume II is here in full — 395 sentences across 79 reading sections, with audio for every sentence and word.

Cover illustration for A Budget of Paradoxes, Volume II
Context Illustration Reading 1/79 · 5 lines
English subtitles · tap any word to listen 1 / 5
CELEBRATED APPROXIMATIONS OF [pi].
The following is an extract from the _English Cyclopaedia_, Art.
TABLES: "1853.
William Shanks,[136] _Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Tables_, London, 1853.
(QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.
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LEARNER DICTIONARY
Archivist
/ˈɑːrkɪvɪst/
Editorial verification pending

A keeper or guardian of historical records and memories.

A Budget of Paradoxes, Volume II

Chapter III

Chapter 3 of A Budget of Paradoxes, Volume II is here in full — 395 sentences across 79 reading sections, with audio for every sentence and word.

Reading 1/79 · 5 reading units · Lines 1–5

Line 1
CELEBRATED APPROXIMATIONS OF [pi].
/ˈseləbreɪtɪd əˌprɑːksəˈmeɪʃənz ʌv paɪ/
Line 2
The following is an extract from the _English Cyclopaedia_, Art.
Line 3
TABLES: "1853.
Line 4
William Shanks,[136] _Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Tables_, London, 1853.
Line 5
(QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

Learner Dictionary

These definitions explain words found in the current reading and keep every example grounded in its original sentence.

Oxford-style entry order: headword, part of speech, US IPA, concise definition, contextual example, and audio.

approximations noun US /əˌprɑːksəˈmeɪʃənz/

plural of “approximation” — an approximate calculation of quantity or degree or worth

From this reading “CELEBRATED APPROXIMATIONS OF [pi].”

comprising verb US /kəmˈpraɪzɪŋ/

-ing form of “comprise” — be composed of

From this reading “William Shanks,[136] _Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Tables_, London, 1853.”

Contributions noun US /ˌkɑːntrəˈbjuːʃənz/

plural of “contribution” — the effort contributed by a person in bringing about a result

decimals noun US /ˈdesəməlz/

plural of “decimal” — a proper fraction whose denominator is a power of 10

From this reading “(QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.”

Mathematics noun US /ˌmæθəˈmætɪks/

a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement

From this reading “William Shanks,[136] _Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Tables_, London, 1853.”

Rectification noun US /ˌrektəfəˈkeɪʃən/

(chemistry) the process of refinement or purification of a substance by distillation

From this reading “William Shanks,[136] _Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Tables_, London, 1853.”

Shanks noun US /ʃæŋks/

plural of “shank” — a cut of meat (beef or veal or mutton or lamb) from the upper part of the leg

From this reading “William Shanks,[136] _Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Tables_, London, 1853.”

subordinate adjective US /səˈbɔːrdəneɪt/

lower in rank or importance

From this reading “(QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.”

Word partnerships in these lines

English keeps certain words together, and knowing the pair is worth more than knowing each word alone. Every partnership below was counted across our whole published library, and earns its place by appearing far more often than two unrelated words of the same frequency would.

Shown as this reading writes them, with the form the library counts, what kind of pairing it is, and how often it occurs.

extract from verb + particle

In this reading “The following is an extract from the _English Cyclopaedia_, Art.”

All words in this reading · 51

Every word in these five lines is listed here, and every one of them can be played on its own.

Dictionary details are shown only when verified; these are not licensed Oxford entries.

136

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading William Shanks,[136] _Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Tables_, London, 1853.

1853

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading TABLES: "1853.

527

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

607

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading William Shanks,[136] _Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Tables_, London, 1853.

adeterminerUS /ə/

used before a singular noun to mean one of a group, when it is not a particular one

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

addedverbUS /ˈædəd/

past form of “add” — make an addition (to); join or combine or unite with others; increase the quality, quantity, size or scope of

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

andeterminerUS /æn/

the form of “a” used before a word that begins with a vowel sound

From this reading The following is an extract from the _English Cyclopaedia_, Art.

APPROXIMATIONSnounUS /əˌprɑːksəˈmeɪʃənz/

plural of “approximation” — an approximate calculation of quantity or degree or worth

From this reading CELEBRATED APPROXIMATIONS OF [pi].

Artauxiliary verbUS /ɑːrt/

an old form of “are”, used with “thou”

From this reading The following is an extract from the _English Cyclopaedia_, Art.

asconjunctionUS /æz/

at the same time that; because; in the way that

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

becauseconjunctionUS /bɪˈkɔːz/

for the reason that

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

beingauxiliary verbUS /ˈbiːɪŋ/

the -ing form of “be”

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

calculationnounUS /ˌkælkjəˈleɪʃən/

the procedure of calculating; determining something by mathematical or logical methods

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

CELEBRATEDadjectiveUS /ˈseləbreɪtɪd/

widely known and esteemed

From this reading CELEBRATED APPROXIMATIONS OF [pi].

chieflyadverbUS /ˈtʃiːfli/

for the most part

From this reading William Shanks,[136] _Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Tables_, London, 1853.

CirclenounUS /ˈsɜːrkəl/

ellipse in which the two axes are of equal length; a plane curve generated by one point moving at a constant distance from a fixed point

From this reading William Shanks,[136] _Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Tables_, London, 1853.

comprisingverbUS /kəmˈpraɪzɪŋ/

-ing form of “comprise” — be composed of

From this reading William Shanks,[136] _Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Tables_, London, 1853.

ContributionsnounUS /ˌkɑːntrəˈbjuːʃənz/

plural of “contribution” — the effort contributed by a person in bringing about a result

From this reading William Shanks,[136] _Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Tables_, London, 1853.

Cyclopaedianoun

a reference work (often in several volumes) containing articles on various topics (often arranged in alphabetical order) dealing with the entire range of human knowledge or with some particular specialty

From this reading The following is an extract from the _English Cyclopaedia_, Art.

decimalsnounUS /ˈdesəməlz/

plural of “decimal” — a proper fraction whose denominator is a power of 10

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

duringprepositionUS /ˈdʊrɪŋ/

all through a period, or at some point within it

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

EnglishnounUS /ˈɪŋɡlɪʃ/

an Indo-European language belonging to the West Germanic branch; the official language of Britain and the United States and most of the commonwealth countries

From this reading The following is an extract from the _English Cyclopaedia_, Art.

enormousadjectiveUS /ɪˈnɔːrməs/

extraordinarily large in size or extent or amount or power or degree

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

extractverbUS /ˈekstrækt/

remove, usually with some force or effort; also used in an abstract sense

From this reading The following is an extract from the _English Cyclopaedia_, Art.

faradverbUS /fɑːr/

to a considerable degree; very much

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

followingadjectiveUS /ˈfɑːloʊɪŋ/

about to be mentioned or specified

From this reading The following is an extract from the _English Cyclopaedia_, Art.

fromprepositionUS /frʌm/

showing where something starts, or who sent it

From this reading The following is an extract from the _English Cyclopaedia_, Art.

HereadverbUS /hiːr/

in or at this place; where the speaker or writer is

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

isauxiliary verbUS /ɪz/

the form of “be” used with he, she, it, or a singular noun

From this reading The following is an extract from the _English Cyclopaedia_, Art.

itpronounUS /ɪt/

a thing, animal, or idea already mentioned

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

LondonnounUS /ˈlʌndən/

the capital and largest city of England; located on the Thames in southeastern England; financial and industrial and cultural center

From this reading William Shanks,[136] _Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Tables_, London, 1853.

MathematicsnounUS /ˌmæθəˈmætɪks/

a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement

From this reading William Shanks,[136] _Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Tables_, London, 1853.

OFprepositionUS /ʌv/

belonging to, or connected with, somebody or something

From this reading CELEBRATED APPROXIMATIONS OF [pi].

onlyadverbUS /ˈoʊnli/

and nothing more

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

pinounUS /paɪ/

the ratio of the circumference to the diameter of a circle; approximately equal to 3.14159265358979323846...

From this reading CELEBRATED APPROXIMATIONS OF [pi].

PlacesnounUS /ˈpleɪsəz/

plural of “place” — a point located with respect to surface features of some region

From this reading William Shanks,[136] _Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Tables_, London, 1853.

printingnounUS /ˈprɪntɪŋ/

text handwritten in the style of printed matter

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

QUADRATUREnoun

the construction of a square having the same area as some other figure

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

RectificationnounUS /ˌrektəfəˈkeɪʃən/

(chemistry) the process of refinement or purification of a substance by distillation

From this reading William Shanks,[136] _Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Tables_, London, 1853.

remaindernounUS /rɪˈmeɪndər/

something left after other parts have been taken away

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

resultsnounUS /rɪˈzʌlts/

plural of “result” — a phenomenon that follows and is caused by some previous phenomenon

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

ShanksnounUS /ʃæŋks/

plural of “shank” — a cut of meat (beef or veal or mutton or lamb) from the upper part of the leg

From this reading William Shanks,[136] _Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Tables_, London, 1853.

stepsnounUS /steps/

a flight of stairs or a flight of steps

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

subordinateadjectiveUS /səˈbɔːrdəneɪt/

lower in rank or importance

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

tablenounUS /ˈteɪbəl/

a set of data arranged in rows and columns

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

TABLESnounUS /ˈteɪbəlz/

plural of “table” — a set of data arranged in rows and columns

From this reading TABLES: "1853.

tabulates

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

ThedeterminerUS /ðə/

used before a noun that is already known, already mentioned, or the only one of its kind

From this reading The following is an extract from the _English Cyclopaedia_, Art.

thisdeterminerUS /ðɪs/

used for a person or thing that is near, or that has just been mentioned

From this reading (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing.

toparticleUS /tuː/

used before the base form of a verb to make the infinitive

From this reading William Shanks,[136] _Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Tables_, London, 1853.

William

Dictionary entry pending editorial verification. Tap Listen to hear the word.

From this reading William Shanks,[136] _Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Tables_, London, 1853.

Chapter 3 of A Budget of Paradoxes, Volume II by Augustus De Morgan holds 395 sentences, grouped into 79 reading sections of at most 5 sentences. The text is the source edition, unabridged, and the chapter records which sentences cover which part of the work so that nothing is dropped. Every word can be tapped for its pronunciation and dictionary entry.

What does chapter 3 of A Budget of Paradoxes, Volume II contain?

Chapter 3 holds 395 sentences of A Budget of Paradoxes, Volume II, unabridged, divided into 79 reading sections.

This page carries chapter 3 of A Budget of Paradoxes, Volume II by Augustus De Morgan exactly as it stands in the source edition, without abridgement. It runs to 395 sentences, grouped into 79 sections of at most 5 sentences so that a reader can take the chapter in steady, finishable pieces. The chapter also stores a coverage record mapping every one of its sentences to a part of the work, which is how the edition demonstrates that no line was dropped.

How do I study chapter 3 of A Budget of Paradoxes, Volume II?

Work through the 79 sections in order, replaying any sentence and tapping unfamiliar words for their entries.

Each of the 79 sections in this chapter is short enough to read once, hear once, and then re-read with the vocabulary in view. Tapping a word plays it alone and opens its dictionary entry with IPA, part of speech and meaning, so the reader is never forced to guess a pronunciation from spelling. Working section by section keeps a long chapter from becoming an undifferentiated wall of text, which is the usual reason a reader abandons a work of this period.

Can chapter 3 be read aloud to me?

Yes — every sentence and every individual word in chapter 3 can be played aloud in United States English.

Audio covers the whole chapter: any sentence can be played, and any single word within it can be played on its own. The voice uses United States English. For a work of this period that matters, because spelling is an unreliable guide to pronunciation for vocabulary that has fallen out of daily use, and a learner who guesses wrong tends to keep the error. Hearing the word in its own sentence, and then alone, settles it.

Questions about reading A Budget of Paradoxes, Volume II

Common questions about reading A Budget of Paradoxes, Volume II by Augustus De Morgan on this page, its source edition and its audio.

Is this the complete text of A Budget of Paradoxes, Volume II?

Yes. All 7048 sentences of Project Gutenberg eBook #26408 are kept, with a record of which sentences cover which part.

Which edition of A Budget of Paradoxes, Volume II is used?

Project Gutenberg eBook #26408, unabridged. The Project Gutenberg header, footer and licence are removed.

How long does A Budget of Paradoxes, Volume II take to read?

About 587 minutes of reading across 1410 sections of 5 sentences each, at an unhurried pace.

How to read A Budget of Paradoxes, Volume II section by section

How to work through A Budget of Paradoxes, Volume II section by section, using the audio and the tap-to-define vocabulary.

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  3. Listen and tap

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Vocabulary met early in A Budget of Paradoxes, Volume II

Words and structures a reader meets early in A Budget of Paradoxes, Volume II, each tappable for pronunciation and meaning.

Typographical
A word that appears in the opening chapter of A Budget of Paradoxes, Volume II; the reader can tap it to hear it and see its dictionary entry.
Errors
A word that appears in the opening chapter of A Budget of Paradoxes, Volume II; the reader can tap it to hear it and see its dictionary entry.
Corrected
A word that appears in the opening chapter of A Budget of Paradoxes, Volume II; the reader can tap it to hear it and see its dictionary entry.
Listed
A word that appears in the opening chapter of A Budget of Paradoxes, Volume II; the reader can tap it to hear it and see its dictionary entry.