See also:

• INFINITE (from Lat. in, not, finis, end or limit; cf. findere, to deave)

INFINITE (from See also:

• LAT

Lat. in, not, finis, end or limit; cf. findere, to deave)  , a See also:

• TERM

term applied in See also:

• COMMON

common usage to anything of vast See also:

• SIZE

size . Strictly, however, the epithet implies the See also:

• ABSENCE (Lat. absentia)

absence of all See also:

• LIMITATION, STATUTES OF

limitation . As such it is used specially in (1) See also:

• THEOLOGY

theology and See also:

• METAPHYSICS
• METAPHYSICS, or METAPHYSIC (from Gr. /sera, after, d v med, things of nature, lots, i.e. the natural universe)

metaphysics, (2) See also:

• MATHEMATICS (Gr. /saBnµar1Kil, Sc. vOcvn or E7rlQTt'µn; from p iN.La, "learning" or "science ")

mathematics . r . Tracing the See also:

• HISTORY

history of the See also:

• WORLD

world to the earliest date for which there is any See also:

• KIND (O. E. ge-cynde, from the same root as is seen in " kin," supra)

kind of See also:

• EVIDENCE (Lat. evidentia, evideri, to appear clearly)

evidence, we are faced with the problem that for everything there is a See also:

• PRIOR (from Lat. prior—former, and hence superior, through O. Fr. priour)
• PRIOR, MATTHEW (1664-1721)

prior something: the mind is unable to conceive an See also:

• ABSOLUTE (Lat. absolvere, to loose, set free)

absolute beginning (" ex nihilo nihil ") . Mundane distances become trivial when compared with the distance from the See also:

• EARTH
• EARTH (a word common to Teutonic languages, cf. Ger. Erde, Dutch aarde, Swed. and Dan. jord; outside Teutonic it appears only in the Gr. 'pq'e, on the ground; it has been connected by some etymologists with the Aryan root ar-, to plough, which is seen in
• EARTH, FIGURE OF
• EARTH, FIGURE OF THE

earth of the See also:

• SUN
• SUN (0. Eng. sonne, Ger. sonne. Fr. soleil, Lat. sol, Gr. ijXior, from which comes helio- in various English compounds)

sun and still more of other heavenly bodies: hence we infer See also:

• INFINITE (from Lat. in, not, finis, end or limit; cf. findere, to deave)

infinite space . Similarly by continual subdivision we reach the See also:

• IDEA (Gr. Ibia, connected with i&eiv, to see; cf. Lat. species from specere, to look at)

idea of the infinitely small . For these inferences there is indeed no actual See also:

• PHYSICAL

physical evidence: infinity is a See also:

• MENTAL

mental concept . As such the term has played an important See also:

• PART

part in the philosophical and theological See also:

• SPECULATION

speculation . In See also:

• EARLY
• EARLY, JUBAL ANDERSON (1816-1894)

early See also:

• GREEK
• GREEK, ETRUSCAN AND

Greek See also:

• PHILOSOPHY (Gr. gthos, fond of, and vo4 (a, wisdom)

philosophy the See also:

• ATTEMPT (Lat. adtemptare, attentare, to try)

attempt to arrive at a physical explanation of existence led the Ionian thinkers to postulate various primal elements (e.g. See also:

• WATER

water, See also:

• FIRE (in O. Eng. f j'r; the word is common to West German languages, cf. Dutch vuur, Ger. Feuer; the pre-Teutonic form is seen in Sanskrit pu, pavaka, and Gr. irup; the ultimate origin is usually taken to be a root meaning to purify, cf. Lat. purus)

fire, See also:

• AIR (from an Indo-European root meaning " breathe," " blow ")
• AIR, or ASBEN

air) or simply the infinite See also:

• ROD (O.E. rodd, probably related to Norw. rudda, stick, rodda, stake)
• ROD, EDOUARD (1857-1910)

rod retpov (see IONIAN SCHOOL) . Both See also:

• PLATO

Plato and See also:

• ARISTOTLE
• ARISTOTLE (384-322 B.C.)

Aristotle devoted much thought to the discussion as to which is most truly real, the finite See also:

• OBJECTS

objects of sense, or the universal idea of each thing laid up in the mind of See also:

• GOD

God; what is the nature of that unity which lies behind the multiplicity and difference of perceived objects ? The same problem, variously expressed, has engaged the See also:

• ATTENTION (from Lat. ad-tendo, await, expect; the condition of being " stretched " or " tense ")

attention of philosophers throughout the ages .

In See also:

• CHRISTIAN
• CHRISTIAN, WILLIAM (1608-1663)

Christian theology God is conceived as infinite in See also:

• POWER [WILLIAM GRATTAN] TYRONE (1797-1841)

power, knowledge and goodness, uncreated and immortal: in some See also:

• ORIENTAL

Oriental systems the end of See also:

• MAN
• MAN, ISLE OF (anc. Mona)

man is absorption into the infinite, his perfection the breaking down of his human limitations . The metaphysical and theological conception is open to the agnostic objection that the finite mind of man is by See also:

• HYPOTHESIS (from Gr. inrorcOivat, to put under; cf. Lat. suppositio, from sub-ponere)

hypothesis unable to cognize or apprehend not only an infinite See also:

• OBJECT

object, but even the very conception of infinity itself; from this stand-point the infinite is regarded as merely a postulate, as it were an unknown quantity (cf . Al -1 in mathematics) . The same difficulty may be expressed in another way if we regard the infinite as unconditioned (cf . See also:

• SIR

Sir See also:

• WILLIAM
• WILLIAM (1143-1214)
• WILLIAM (1227-1256)
• WILLIAM (1J33-1584)
• WILLIAM (A.S. Wilhelm, O. Norse Vilhidlmr; O. H. Ger. Willahelm, Willahalm, M. H. Ger. Willehelm, Willehalm, Mod.Ger. Wilhelm; Du. Willem; O. Fr. Villalme, Mod. Fr. Guillaume; from " will," Goth. vilja, and " helm," Goth. hilms, Old Norse hidlmr, meaning
• WILLIAM (c. 1130-C. 1190)
• WILLIAM, 13TH

William See also:

• HAMILTON
• HAMILTON (GRAND or ASHUANIPI)
• HAMILTON, ALEXANDER (1757-1804)
• HAMILTON, ANTHONY, or ANTOINE (1646-1720)
• HAMILTON, ELIZABETH (1758–1816)
• HAMILTON, EMMA, LADY (c. 1765-1815)
• HAMILTON, JAMES (1769-1831)
• HAMILTON, JAMES HAMILTON, 1ST DUKE OF (1606-1649)
• HAMILTON, JOHN (c. 1511–1571)
• HAMILTON, MARQUESSES AND DUKES OF
• HAMILTON, PATRICK (1504-1528)
• HAMILTON, ROBERT (1743-1829)
• HAMILTON, SIR WILLIAM
• HAMILTON, SIR WILLIAM (1730-1803)
• HAMILTON, SIR WILLIAM ROWAN (1805-1865)
• HAMILTON, THOMAS (1789-1842)
• HAMILTON, WILLIAM (1704-1754)
• HAMILTON, WILLIAM GERARD (1729-1796)

Hamilton's " philosophy of the unconditioned," and See also:

• HERBERT (FAMILY)
• HERBERT, GEORGE (1593-1633)
• HERBERT, HENRY WILLIAM
• HERBERT, SIR THOMAS (1606-1682)

Herbert See also:

• SPENCER
• SPENCER, HERBERT (1820-1903)
• SPENCER, JOHN CHARLES SPENCER, 3RD EARL (1782-1845)
• SPENCER, JOHNPOYNTZ SPENCER
• SPENCER, WILLIAM ROBERT (1769-1834)

Spencer's See also:

• DOCTRINE

doctrine of the infinite " unknowable"); if it is argued that knowledge of a thing arises only from the recognition of its See also:

• DIFFERENCES, CALCULUS OF (Theory of Finite Differences)

differences from other things (i.e. from its limitations), it follows that knowledge of the infinite is impossible, for the infinite is by hypothesis unrelated . With this conception of the infinite as absolutely unconditioned should be compared what may be described roughly as lesser infinities which can be philosophically conceived and mathematically demonstrated . Thus a point, which is by See also:

• DEFINITION (Lat. definitio, from de-finire, to set limits to, describe)

definition infinitely small, is as compared with a See also:

• LINE

line a unit: the line is infinite, made up of an infinite number of points, any pair of which have an infinite number of points between them . The line itself, again, in relation to the See also:

• PLANE

plane is a unit, while the plane is infinite, i.e. made up of an infinite number of lines; hence the plane is described as doubly infinite in relation to the point, and a solid as trebly infinite . This is See also:

• SPINOZA, BARUCH (1632-1677)

Spinoza's theory of the " infinitely infinite," the limiting notion of infinity being of a numerical, quantitative aeries, each term of which is a qualitative determination itself quantitatively little, e.g. a line which is quantitatively unlimited (i.e. in length) is qualitatively limited when regarded as an infinitely small unit of a plane . A similar relation exists in thought between the various grades of See also:

• SPECIES

species and genera; the highest genus is the " infinitely infinite," each subordinated genus being infinite in relation to the particulars which it denotes, and finite when regarded as a unit in a higher genus . 2 . In mathematics, the term " infinite " denotes the result of increasing a variable without limit; similarly, the term " infinitesimal," meaning indefinitely small, denotes the result of diminishing the value of a variable without limit, with the See also:

• RESERVATION (Lat. reservare, to keep back)

reservation that it never becomes actually zero .

The application of these conceptions distinguishes See also:

• ANCIENT
• ANCIENT (also spelt ANTIENT; derived, through the Fr. ancien, old, from the late Lat. antianum, from ante, before)

ancient from See also:

• MODERN

modern mathematics . See also:

• ANALYTICAL

Analytical investigations revealed the existence of See also:

• SERIES (a Latin word from serere, to join)

series or sequences which had no limit to the number of terms, as for example the fraction 1/(1—x) which on See also:

• DIVISION (from Lat. dividere, to break up into parts, separate)

division gives the series. r-1-x+x2+ . . . . ; the discussion of these so-called infinite sequences is given in the articles SERIES and See also:

• FUNCTION

FUNCTION . The doctrine of geometrical continuity (q.v.) and the application of See also:

• ALGEBRA (from the Arab. al-jebr wa'l-mugabala, transposition and removal [of terms of an equation], the name of a treatise by Mahommed ben Musa al-Khwarizmi)

algebra to See also:

• GEOMETRY

geometry, See also:

• DEVELOPED

developed in the 16th and 17th centuries mainly by See also:

• KEPLER, JOHANN (1571-1630)

Kepler and See also:

• DESCARTES, RENE (1596-1650)

Descartes, led to the See also:

• DISCOVERY

discovery of many properties which gave to the notion of infinity, as a localized space conception, a predominant importance . A line became continuous, returning into itself by way of infinity; two parallel lines intersect in a point at infinity; all circles pass through two fixed points at infinity (the circular points); two See also:

• SPHERES, MUSIC OF THE

spheres intersect in a fixed circle at infinity; an asymptote became a tangent at infinity; the foci of a conic became the intersections of the tangents from the circular points at infinity; the centre of a conic the See also:

• POLE (FAMILY)
• POLE, REGINALD (1500-1558)
• POLE, RICHARD DE LA (d. 1525)
• POLE, WILLIAM (1814—1900)

pole of the line at infinity, &c . In analytical geometry the line at infinity plays an important part in trilinear co-ordinates . These subjects are treated in GEOMETRY . A notion related to that of infinitesimals is presented in the Greek " method of exhaustion "; the more perfect conception, however, only See also:

• DATES

dates from the r7th See also:

• CENTURY (from Lat. centuria, a division of a hundred men)

century, when it led to the infinitesimal calculus . A See also:

• CURVE (Lat. curvus, bent)

curve came to be treated as a sequence of infinitesimal straight lines; a tangent as the See also:

• EXTENSION (Lat. ex, out ; tendere, to stretch)

extension of an infinitesimal chord; a See also:

• SURFACE

surface or See also:

• AREA

area as a sequence of infinitesimally narrow strips, and a solid as a collection of infinitesimally small cubes (see INFINITESIMAL CALCULUS) .