Integral Quotes (15 quotes)

After the discovery of spectral analysis no one trained in physics could doubt the problem of the atom would be solved when physicists had learned to understand the language of spectra. So manifold was the enormous amount of material that has been accumulated in sixty years of spectroscopic research that it seemed at first beyond the possibility of disentanglement. An almost greater enlightenment has resulted from the seven years of Röntgen spectroscopy, inasmuch as it has attacked the problem of the atom at its very root, and illuminates the interior. What we are nowadays hearing of the language of spectra is a true 'music of the spheres' in order and harmony that becomes ever more perfect in spite of the manifold variety. The theory of spectral lines will bear the name of Bohr for all time. But yet another name will be permanently associated with it, that of Planck. All integral laws of spectral lines and of atomic theory spring originally from the quantum theory. It is the mysterious

*organon*on which Nature plays her music of the spectra, and according to the rhythm of which she regulates the structure of the atoms and nuclei.
Beauty, I hear you ask, do not the Graces flee where integrals stretch forth their necks?

But just as much as it is easy to find the differential of a given quantity, so it is difficult to find the integral of a given differential. Moreover, sometimes we cannot say with certainty whether the integral of a given quantity can be found or not.

Good scholars struggle to understand the world in an integral way (pedants bite off tiny bits and worry them to death). These visions of reality ... demand our respect, for they are an intellectual’s only birthright. They are often entirely wrong and always flawed in serious ways, but they must be understood honorably and not subjected to mayhem by the excision of patches.

If texts are unified by a central logic of argument, then their pictorial illustrations are integral to the ensemble, not pretty little trifles included only for aesthetic or commercial value. Primates are visual animals, and (particularly in science) illustration has a language and set of conventions all its own.

In pure mathematics we have a great structure of logically perfect deductions which constitutes an integral part of that great and enduring human heritage which is and should be largely independent of the perhaps temporary existence of any particular geographical location at any particular time. … The enduring value of mathematics, like that of the other sciences and arts, far transcends the daily flux of a changing world. In fact, the apparent stability of mathematics may well be one of the reasons for its attractiveness and for the respect accorded it.

Once when lecturing to a class he [Lord Kelvin] used the word “mathematician,” and then interrupting himself asked his class: “Do you know what a mathematician is?” Stepping to the blackboard he wrote upon it:— [an integral expression equal to the square root of pi]

Then putting his finger on what he had written, he turned to his class and said: “A mathematician is one to whom

Then putting his finger on what he had written, he turned to his class and said: “A mathematician is one to whom

*that*is as obvious as that twice two makes four is to you. Liouville was a mathematician.”
Science is an integral part of culture. It’s not this foreign thing, done by an arcane priesthood. It’s one of the glories of the human intellectual tradition.

The energy of a covalent bond is largely the energy of resonance of two electrons between two atoms. The examination of the form of the resonance integral shows that the resonance energy increases in magnitude with increase in the

*overlapping*of the two atomic orbitals involved in the formation of the bond, the word ‘overlapping” signifying the extent to which regions in space in which the two orbital wave functions have large values coincide... Consequently it is expected that of*two orbitals in an atom the one which can overlap more with an orbital of another atom will form the stronger bond with that atom, and, moreover, the bond formed by a given orbital will tend to lie in that direction in which the orbital is concentrated.*
There are in this world optimists who feel that any symbol that starts off with an integral sign must necessarily denote something that will have every property that they should like an integral to possess. This of course is quite annoying to us rigorous mathematicians; what is even more annoying is that by doing so they often come up with the right answer.

This is one of man's oldest riddles. How can the independence of human volition be harmonized with the fact that we are integral parts of a universe which is subject to the rigid order of nature's laws?

This new integral of Lebesgue is proving itself a wonderful tool. I might compare it with a modern Krupp gun, so easily does it penetrate barriers which were impregnable.

Thought-economy is most highly developed in mathematics, that science which has reached the highest formal development, and on which natural science so frequently calls for assistance. Strange as it may seem, the strength of mathematics lies in the avoidance of all unnecessary thoughts, in the utmost economy of thought-operations. The symbols of order, which we call numbers, form already a system of wonderful simplicity and economy. When in the multiplication of a number with several digits we employ the multiplication table and thus make use of previously accomplished results rather than to repeat them each time, when by the use of tables of logarithms we avoid new numerical calculations by replacing them by others long since performed, when we employ determinants instead of carrying through from the beginning the solution of a system of equations, when we decompose new integral expressions into others that are familiar,—we see in all this but a faint reflection of the intellectual activity of a Lagrange or Cauchy, who with the keen discernment of a military commander marshalls a whole troop of completed operations in the execution of a new one.

We have not been seeing our Spaceship Earth as an integrally-designed machine which to be persistently successful must be comprehended and serviced in total.

[De Morgan relates that some person had made up 800 anagrams on his name, of which he had seen about 650. Commenting on these he says:]

Two of these I have joined in the title-page:

[Ut agendo surgamus arguendo gustamus.]

A few of the others are personal remarks.

Great gun! do us a sum!

is a sneer at my pursuit; but,

Go! great sum! [integral of

Adsum, nugator, suge!

is addressed to a student who continues talking after the lecture has commenced: …

Graduatus sum! nego

applies to one who declined to subscribe for an M.A. degree.

Two of these I have joined in the title-page:

[Ut agendo surgamus arguendo gustamus.]

A few of the others are personal remarks.

Great gun! do us a sum!

is a sneer at my pursuit; but,

Go! great sum! [integral of

*a*to the power*u*to the power*n*with respect to*u*] is more dignified. …Adsum, nugator, suge!

is addressed to a student who continues talking after the lecture has commenced: …

Graduatus sum! nego

applies to one who declined to subscribe for an M.A. degree.