# infinitesimal

## Definitions

• WordNet 3.6
• adj infinitesimal infinitely or immeasurably small "two minute whiplike threads of protoplasm","reduced to a microscopic scale"
• n infinitesimal (mathematics) a variable that has zero as its limit
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Webster's Revised Unabridged Dictionary
• n Infinitesimal (Math) An infinitely small quantity; that which is less than any assignable quantity.
• a Infinitesimal Infinitely or indefinitely small; less than any assignable quantity or value; very small.
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Century Dictionary and Cyclopedia
• infinitesimal Infinitely or indefinitely small; less than any assignable quantity.
• n infinitesimal In mathematics, a fictitious quantity so small that by successive additions to itself no sensible quantity, such as the unit of quantity, could ever be generated. If a is a finite quantity, and i an infinitesimal, we always assume a + i = a, a fundamental 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 assumed that all the mathematical operations can be performed on these quantities. Every power of an infinitesimal 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 quantities 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 infinitesimals there are quantities infinitely greater and quantities 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 infinitesimal in comparison with which it was of the zero order would now appear as of the infinite order.
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Chambers's Twentieth Century Dictionary
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## Etymology

Webster's Revised Unabridged Dictionary
Cf. F. infinitésimal, fr. infinitésime, infinitely small, fr. L. infinitus,. See Infinite (a.)

## Usage

### In literature:

Deeper in the pass, where the flood would be confined, their chance for escape would be infinitesimal.
"Rimrock Trail" by J. Allan Dunn
He felt awed; the trivial things which made up so much of his life seemed infinitesimal now, in the face of this mysterious wonder.
"Duncan Polite" by Marian Keith
It does make one feel so infinitesimal, and so immense.
"Daisy's Aunt" by E. F. (Edward Frederic) Benson
Again he saw a supposedly invincible fleet utterly destroyed, saw comrades whiffed out of existence in infinitesimal seconds.
"When the Sleepers Woke" by Arthur Leo Zagat
Life itself becomes infinitesimal.
"The Twins of Suffering Creek" by Ridgwell Cullum
There is an infinitesimal downward sagging, as with incredible deliberation it moves on with its cargo of rock and sand.
"A Mountain Boyhood" by Joe Mills
What infinitesimal changes in the direction of the gentle curvature of the rounded lines!
"The Continental Monthly, Vol. 4, No. 2, August, 1863" by Various
There was the slight, almost infinitesimal tremor of their vibration.
"Astounding Stories of Super-Science, May, 1930" by Various
An infinitesimal beam of daylight had lit up the black horizon of his threatened future.
"The Law-Breakers" by Ridgwell Cullum
An infinitesimal number of women had a bit of School suffrage.
"The History of Woman Suffrage, Volume VI" by Various
In an ideal intelligence, self-consciousness would be infinite, sensation infinitesimal.
"The Religious Sentiment" by Daniel G. Brinton
So much the more infinitesimal were the chances of his being recognised or even suspected for what he was.
"" by E.W. Hornung
What infinitesimal sum had the fox cost the country for every man who rushed after him?
"Marion Fay" by Anthony Trollope
Yet this infinitesimal bit of life has an instinct, the instinct to save itself.
"Prize Orations of the Intercollegiate Peace Association" by Intercollegiate Peace Association
I have known a home to be ruined by just such infinitesimal annoyances.
"The Heart of the New Thought" by Ella Wheeler Wilcox
The things of men were matters of infinitesimal consequence in nature.
"The Bondboy" by George W. (George Washington) Ogden
Even the infinitesimal amount of heat radiated by their half-frozen human bodies caused the ice-folk discomfort.
"Black Amazon of Mars" by Leigh Brackett
Their bites were infinitesimal, but each one smarted like a prick with a hot needle.
"The Test Colony" by Winston Marks
The number of artists who either think for themselves, or express with new force the ideas of others, is infinitesimal.
"Michelangelo" by Romain Rolland
Yet the chances of the prisoner's life being eventually spared were infinitesimal.
"'Tween Snow and Fire" by Bertram Mitford
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### In poetry:

Beware the Sissy Boy my child,
Not because he's very wild;
The Sissy Boy is never that,
Although he'll run if you say "Scat!"
The Sissy Boy's infinitesimal,
He is not worth a duodecimal.
"The Sissy Boy" by Edwin Carty Ranck
"But if I broke asunder all such pettifogging bounds,
And forged a party's Will for (say) Five Hundred Thousand Pounds,
With such an irresistible temptation to a haul,
Of course the sin must be infinitesimally small.
"Mister William" by William Schwenck Gilbert

### In news:

Every incident in her life as a teenager and young adult, parsed infinitesimally.
Quantum entanglements unify the infinitesimal aspects of virtual particles into momentary states between existence and nonexistence, while precise cosmological constants undergird the multitudinous dimensions of space-time.
The little guy transcends his infinitesimal existence in A Bug's Life.
When these impulses interact with Emmi- dent 's own toothpaste, they cause millions of infinitesimal "nano-bubbles" to form and then collapse.
Frankie Rose is a pop sophisticate with a knack for reading the infinitesimal trendlines coursing through Hiplandia.
On Monday, in this column, I ridiculed soccer's attempt to make vital decisions based on supposedly accurate measurement of infinitesimally small distances.
If you stop and think about it, the odds are infinitesimally small that any of us ever turned out to be the precise genetic beings we are.
We got an e-mail, with the subject line "With Infinitesimal Talent," divulging dirty secrets.
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### In science:

For an irreversible infinitesimal change the work is not equal to -pdV.
The Energy-Entropy Principle
Being measure of unpredictability, the latter is the average measure how fast a trajectory deviates under infinitesimally small perturbation of the initial conditions.
Fluctuations and Long-Term Stability: from Coherence to Chaos
TC for an infinitesimal field dH and the H-dependence of M at T=TC.
Ferromagnet in a continuously tuneable random field
Motivation: We propose a new distance metric for DNA sequences, which can be defined on any evolutionary Markov model with infinitesimal generator matrix Q.
A new distance between DNA sequences
The Poynting vector is thus zero and therefore the momentum of the infinitesimal object is now zero.
On the concept of relativistic mass
Or, is there any sensible infinitesimal? It seems no.
How does certainty enter into the mind?
In this process infinitesimally clo se init ial states for finite time can give different terminat ing states.
The Universal Arrow of Time III-IV:(Part III) Nonquantum gravitation theory (Part IV) Quantum gravitation theory
It has finite value in respect to infinitesimally small perturbations.
The Universal Arrow of Time III-IV:(Part III) Nonquantum gravitation theory (Part IV) Quantum gravitation theory
It means that the observer/environment even infinitesimally weakly interact ing with the white hole can affect considerably its evolut ion for finite time.
The Universal Arrow of Time III-IV:(Part III) Nonquantum gravitation theory (Part IV) Quantum gravitation theory
In this case the time wormho le, like a white ho le, appears unstable even with respect to infinitesimally weak perturbations from gravitat ion of travelling object.
The Universal Arrow of Time III-IV:(Part III) Nonquantum gravitation theory (Part IV) Quantum gravitation theory
In general relativity theory, unlike the classical mechanics, two different states for finite time can give infinitesimally close states.
The Universal Arrow of Time III-IV:(Part III) Nonquantum gravitation theory (Part IV) Quantum gravitation theory
This can be seen at the level of the components undergoing infinitesimal conformational changes to traverse a continuous energy landscape, or even at the global system level, where diverse closely spaced states in genotype-space are accessible via environmental fluctuations.
Field-control, phase-transitions, and life's emergence
We also imagine a gentle H-field gradient --stemming from (inhomogeneous) magnetic rocks (Mitra-Delmotte and Mitra 2010a)--that provides both detailed-balance-breaking non-equilibrium as well as asymmetry, to a diffusing magnetic dipole undergoing infinitesimal spin-alignment changes.
Field-control, phase-transitions, and life's emergence
AHO) [1 – 3] where it has been conclusively demonstrated that the naïve perturbation theory (NPT) diverges even for infinitesimal value of the coupli ng strength, presumably due to the eventual dominance of the perturbation correction over the un -perturbed contribution for large amplitude of oscillation.
A new general approximation scheme(NGAS) in quantum theory:application to the anharmonic- and double well oscillators
Lorentz transformat ion, so m term in (7.5) is the infinitesimal generator of a dilatation.
Nonlinear connections and 1+3 splittings of spacetime
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