infinite 
and then says, "A mediator is required " ; upon which he 
gives the solution that the infinite, both of the world in 
space, and in time and in division, is never before the re- 
gressus, or progressus, but in it. This truth then lies in 
the rightly apprehended conception of the infinite. Thus 
one misunderstands himself if he imagines that he can 
think the infinite, of whatever kind it may be, as some- 
thing objectively present and complete, and independent 
of the regressus. 
Schopenhauer, Will, tr. by Haldane and Kempe, ii. llo. 
If Zero is the sign of a vanished quantity, the Infinite is 
the sign of that Continuity of Existence which has been 
ideally divided into discrete parts in the affixing of limits. 
0. H. Lewes, Probs. of Life and Mind, II. vi. I 5. 
(ct) A large number ; a crowd. 
Their gates are walled vp ; and there are infinite of 
Frier-like companions passing to and fro in the Citie. 
Purchas, Pilgrimage, p. 428. 
Gods defend me, 
What multitudes they are, what infinites! 
Fletcher, Bonduca, iii. 5. 
Arithmetic of infinites, a name given by Dr. Wallis to 
a method invented by him for the summation of infinite 
series. 
infinitely (in'fi-nit-li), adv. Innumerably; im- 
measurably ; incomparably ; in the highest con- 
ceivable degree : often used in hyperbole : as, 
to be infinitely obliged for favors. 
I am a soldier, and a bachelor, lady ; 
And such a wife as you I could love infinitely. 
Fletcher, Rule a Wife, i. 0. 
We know that a good constitution is infinitely better 
than the best despot. Hacaulay, Milton. 
Matter is concluded not to be infinitely divisible. 
A. Daniell, Prin. of Phys., p. 194. 
infiniteness (in'fi-nit-nes), n. The state of be- 
ing infinite ; infinity ; immensity. 
Let us always bear about us such impressions of rever- 
ence, and fear of God, that we may humble ourselves be- 
fore his Almightiness, and express that infinite distance 
between his infiniteneiui and our weaknesses. Jcr. Taylor. 
If we consider the quality of the person appearing, that 
he was no other than the eternal Son of God, how ought 
we to be wrapt with wonder and astonishment at the in- 
finitenets of the divine condescention I 
Abp. Sharp, Works, I. xi. 
infinitesimal (in"fi-ni-tes'i-mal), a. and n. [= 
F. infinitesimal = Sp. Pg. infinitesimal = It. 
infinitesimalc, < NL. infinitesimalis, infinitesi- 
mal, < infinitesimus (fern, infinitesima (> It. in- 
finitesima = Pg. infinitesima = F. infinitesime), 
sc. pars, an infinitesimal), a quasi-ordinal nu- 
meral, a term of an infinite convergent series 
which is preceded by an infinite number of 
terms, hence infinitely small, < infinitum, a 
quasi-cardinal, neut. of L. infinitus, infinite: 
see infinite. For the ordinal termination, cf. 
centesimal, millesimal, etc.] I. a. Infinitely 
or indefinitely small ; less than any assignable 
quantity. 
The distance between them may be either infinite or 
infinitesimal, according to the measure used. 
H. Spencer, Prin. of Psychol. 
Its Ihomeopathy's] leaders have long ceased to insist 
upon infinitesimal dosage as an essential principle of 
treatment. Pop. Sci. Mo., XXII. 537. 
Infinitesimal analysis, in ninth. See analysis, s M. 
Infinitesimal transformation. See transformation. 
II. . In math., a fictitious quantity so small 
that by successive additions to itself no sensi- 
ble quantity, such as the unit of quantity, 
could ever be generated. If n is a finite quantity, 
and i an infinitesimal, we always assume a + i = a, a fun- 
damental proposition of the infinitesimal calculus ; but 
whether this is because the infinitesimal is a fictitious 
quantity strictly zero, or because equality is used in a 
generalized sense in which this is true, is a question of 
logic, concerning which mathematicians are not agreed 
Most writers use the method of limits (which see, under 
limit), which avoids this and other difficulties. It is as- 
sumed that all the mathematical operations can be per- 
formed on these quantities. Every power of an infinitesi- 
mal is infinitely smaller than any inferior power of the same 
infinitesimal. (See infinite, 1.) Any infinitesimal may be 
assumed as a base or standard, by comparison with which 
the magnitudes of others are estimated. The base itself 
is said to be of the first order, its square of the second 
order, its cube of the third order, etc. Finite quantities 
are of the zero order of infinitesimals, and infinite quan- 
tities are generally of negative orders. The logarithm of 
any infinitesimal of a finite order is of order zero, although 
it is infinite. In like manner, in every order of infinitesi- 
mals there are quantities infinitely greater and quanti- 
ties infinitely smaller than the power of the base of that 
order. The square, cube, etc., of an infinitesimal of the 
zero order remains of the zero order : yet there is nothing 
peculiar about these infinitesimals ; any one of them might 
have been taken as the base, and then its square would 
have been reckoned as of the second order, while the infi- 
nitesimal In comparison with which it was of the zero 
order would now appear as of the infinite order. 
Infinites are composed of Unites in no other sense than 
as finites are composed of infinitesimals. 
Dr. Clark, Fourth Reply to Leibnitz. 
infinitesimally (in"fi-ni-tes'i-mal-i), adv. By 
infinitesimals ; in infinitely small quantities ; 
to an infinitesimal extent or in an infinitesimal 
degree. 
3084 
Just as he himself forms an infinitesimally small part 
of the universe, so his personal knowledge is utterly in- 
commensurate with the sum total of existence. 
J. Owen, Evenings with Skeptics, II. 18. 
infinitiont (iu-fi-nish'on), n. [= OF. infinicion, 
< L. infinitio(n-), boundlessness, infinity, < in- 
finitus, 'boundless : see infinite.'] Infinitation; 
negation. 
For what Joy is so great but the conceipt 
Of falling to his Infinition 
Of blacke Non-essence will confound it strelght? 
Dames, Wittes Pilgrimage, p. 23. 
infinitival (in-fm-i-ti'val or in-fin'i-ti-val), a. 
[< infinitive + -al.] In gram., of or belonging 
to the infinitive. 
To all verbs, then, from the Anglo-Saxon, to all based 
on the uncorrupted infinitival stems of Latin verbs of 
the first conjugation, and to all substantives, wheuceso- 
ever sprung, we annex -able only. 
F. Hall, -Able and Reliable, p. 47. 
infinitive (in-fin'i-tiv), a. and n. [= F. infini- 
tif= Pr. infinitiu, enfenitiit = Sp. Pg. It. infini- 
tiro = D. infinitivus = G. Dan. Sw. infinitiv, < 
LL. infinitivus, unlimited, indefinite (modus 
infinitivtis or simply infinitivus, the infinitive 
mode), < L. infinitus, unlimited: see infinite."] 
1. a. In gram., unlimited; indefinite: noting 
a certain verb-form sometimes called the in- 
finitive mode. See II. 
II. n. 1. Ingram., a certain verb-form ex- 
pressing the general sense of the verb without 
restriction in regard to person or number, as 
English give, German geben, French donner, 
Latin dare, Greek itd&ixu. In the grammar of Lat- 
in and of the most familiar modern languages, it is used 
as the representative form of the whole verb-system. It 
is by origin simply a verbal noun in an oblique case (often- 
est dative); and hence its tendency to use with a stereo- 
typed prefixed preposition, as to in English, zu (= English 
to) in German, at(= English at) in Scandinavian, de ('of') 
or d (' to 1 ) in French, and so on ; but the preposition is no 
part of the infinitive. In the old grammars, and in many 
recent ones, it is called a mode ; but the term is objec- 
tionable, and is going out of use. Abbreviated inf. 
2f. An endless quantity or number; an infinity. 
Fie, that the spyrit of a single man 
Should contradict innumerable wills ; 
Fie, that infinitiueit of forces can 
Nor may effect what one conceit fulfills. 
G. llarkham, Sir R. Grinuile, p. 69. 
Historical infinitive. See historical, 4. 
infinitively (in-fin'i-tiv-li), adv. In gram., in 
the manner of an infinitive, 
infinite (in-n-ne'to), a. [It., = E. infinite, q. v.] 
In music, perpetual, as a canon whose end leads 
back to the beginning. 
infinitude (iu-fin'i-tud), n. [= F. infinitude, < 
ML. as if * infinitude, < L. infinitus, infinite: 
see infinite."] 1. The state or quality of being 
the greatest possible, or inconceivably great: 
as, the infinitude of power or grace. 
And thou the third subsistence of Divine Infinitude, il- 
lumining Spirit^ the joy and solace of created things. 
Milton, Reformation in Eng., ii. 
Such wide and undetermined prospects are as pleasing 
to the fancy as speculations of eternity or infinitude are 
to the understanding. Spectator, No. 412. 
2. Infinite extension; infinity; innumerable or 
immeasurable quantity: as, an infinitude of 
space or of stars. 
We see all the good sense of the age cut out, and minced 
into almost an infinitude of distinctions. 
Addition, Spectator. 
The infinitude of the universe, in which our system 
dwindles to a grain of sand. 
Sumner, Speech, Cambridge, Aug. 27, 1846. 
infinituple (in-fiu'i-tu-pl), a. [< infinite + 
-uple, as in duple, quadruple, etc.] Multiplied 
an infinite number of times. Wollaston. [Rare.] 
infinity (in-fin'i-ti), n. [= F. infinite = Pr. 
infinitat, enfenitat = Sp. infinidad = Pg. in- 
finidade = It. infinita, < L. infinita(t-)s, bound- 
lessness, endlessness, < infinitus, boundless, 
endless: see infinite."} 1. The condition of be- 
ing infinite or the greatest possible; immea- 
surableness; innumerability ; perfection: as, 
the infinity of God; infinity of duration. 
One whose eternity passeth al time, and whose infinity 
passeth all nombre, that is almightye. 
Sir T. More, Works, p. 636. 
If we dare not trust God with the circumstance of the 
event, and stay his leisure, ... we disrepute the infinity 
of his wisdom. Jer. Taylor, Works (ed. 1836), I. 35. 
This endless addition or addibility . . . of numbers . . . 
is that . . . which gives us the clearest and most distinct 
idea of infinity. Locke, Human Understanding, II. xvi. 8. 
2. Unlimited extension; figuratively, exhaust- 
less quantity or number: as, inconceivable in- 
finity; an infinity of details. 
Here has been that infinity of strangers ! 
B. Jonson, Devil is an Ass, v. 1. 
Adorn'd with an infinity of statues, pictures, stately 
altars, and innumerable reliques. 
Evelyn, Diary. Nov. 19, 1644. 
infirmary 
In an infinity of things thus relative, a mind which sees 
not infinitely can see nothing fully. 
Shafteslmri/, Moralists, ii. 4, quoted in Fowler, p. 111. 
3. In geom., the part of space at an infinite dis- 
tance from the part chiefly considered. Owing 
to the mechanical properties of the straight line, the most 
important mode of geometrical transformation is that 
which transforms every straight line into a straight line, 
its position only being changed. But this transforms the 
part of space at infinity into a plane, just as in a perspec- 
tive view of an unbounded plane the infinitely distant parts 
are compressed into a line. Hence, mathematicians gen- 
erally speak of the plane at infinity, or the line at infinity 
in a plane. In analytical geometry the plane at infinity is 
best considered as two coincident planes, fastened together 
at an imaginary nodal circle, and constituting a degenerate 
sphere, called the sphere at infinity. (See absolute, n., 2.) 
In the theory of functions, the most important mode of 
transformation of the plane is one which preserves the 
magnitudes of all angles, and this transforms the infinite- 
ly distant parts of the plane into a point ; hence, in that 
branch of mathematics, the point at infinity is spoken of. 
infirm (in-ferm'), a. [< ME. infirm, < OF. in- 
firm, enferm, anferm, emfarm, F. infirme = Pr. 
eferm, enferm = Sp. Pg. enfcrmo == It. infermo, 
infirmo, < L. infirmus, not strong or firm, < in- 
priv. + firmus, strong: see firm."] 1. Not firm, 
stable, or strong ; lacking- stability or solidity ; 
faltering; feeble: as, an infirm support; an in- 
firm judgment. 
The Bonne . . . may . . . nat by the infirme lyht of his 
beemesbrekyn or percenthe inward entrailes of the erthe. 
Chaucer, Boethius, v. meter 2. 
Macb. I'll go no more : 
I am afraid to think what I have done; 
Look on 't again I dare not. 
Lady M. Infirm of purpose ! 
Give me the daggers. Shah., Macbeth, ii. 2. 
Infirm the stalks, unsolid are the leaves. 
Dryden, tr. of Ovid's Metamorph., xv. 307. 
He who fixes on false principles treads on infirm ground. 
South, Sermons. 
Specifically 2. Not sound in health; impaired 
in health or vitality ; enfeebled ; weak : as, in- 
firm in body or constitution. 
Here I stand, your slave, 
A poor, infirm, weak, and despis'd old man. 
Shak., Lear, iii. 2. 
The unhealthful east, 
That breathes the spleen, and searches ev'ry bone 
Of the infirm, is wholesome air to thee. 
Cooper, Task, iv. 365. 
3. Voidable ; obnoxious to legal objection that 
may destroy apparent efficacy. =gyn. i. Vacillat- 
ing, wavering, shaky. 2. Enfeebled, debilitated, sickly, 
decrepit shackly. 
infirmt (in-ferm'), v. t. [< F. infirmer = Pr. 
enfermar, enfirmar = Sp. Pg. enfermar, infirmar 
= It. infirmare, < L. infirmare, make infirm, 
weaken, ML. also be infirm or sick, < infirmus, 
infirm : see infirm, a."] 1. To weaken; enfeeble. 
If they be strong, you do what you can to infirm their 
strength. J. Bradford, Letters (Parker Soc. , 1853), II. 120. 
2. To make less firm or certain ; render doubt- 
ful, questionable, or dubious. 
Some contrary spirits will object this as a sufficient rea- 
son to infirm ail those points. Raleigh, Essays. 
This is not infirmed because we read how God doth seem 
in some things to alter his will, before not determined, but 
dependant upon man's behaviour. 
J. Bradford, Letters (Parker Soc., 1853), II. 180. 
Socrates, . . . professing to affirm nothing, but to infirm. 
that which was affirmed by another, hath exactly expressed 
all the forms of objection, fallacy, and redargution. 
Bacon, Advancement of Learning, ii. 224. 
infirmarert (in-fer'ma-rer),w. [< ML. infirmari- 
s (see infirmarian) -f E. -er 1 .] Same as infir- 
marian. I. Campbell, St. Giles Leet., 1st ser., 
p. 78. 
infirmarian (in-fer-ma'ri-an), . [< ML. infir- 
marius (as defined) (see infirmary) + -an.] An 
officer in a monastery who has charge of the 
quarters for the sick. 
Antony de Madrid . . . had to nurse St Stanislaus in 
his last illness, as infirmarian of Sant' Andrea. 
Life of St. Kotka, p. 83. 
The Community [Freres des Ecoles Chr^tiennesl, which 
during the whole time of the war had sent five hundred in- 
firmariang into the battle fields. 
Quarterly Rev., CLXII. 354. 
infirmary (in-fer'ma-ri), .; pi. infirmaries 
(-riz). [Formerly enfenuerie, by apheresis/er- 
mary,fermery.fermry,firmary, etc. (seeferme- 
ry); < OF. enfermerie (alsofermerie), F. infirme- 
rie = Pr. effermeria, efermaria = Sp. enfermeria 
= Pg. enfermaria = It. infermeria, < ML. infir- 
iarti, au infirmary or hospital (cf. infirmarivs, 
m., one in charge of the sick), < L. infirmus, in- 
firm: see infirm, a."] A place for the treatment 
of the infirm, or persons suffering from disease 
or injury. 
Here, in the farthest Nook of the Meadow, is a little Ban- 
glutting House ; there I sup sometimes in Summer, and I 
make Use of it, as an Infirmary, if any of my Family be 
taken ill with any infectious Disease. 
A". Bailey, tr. of Colloquies of Erasmus, I. 200. 
