The modern formulation of proof by induction was not developed until the 17th century, but some later commentators consider it implicit in some of Euclid's proofs, e. g., the proof of the infinitude of primes.

This work was a precursor of a modern formulation in terms of constructive type theory.

The explanation here is reworded in terms of the modern graph theory formulation above.

A practical method for investigating neutrino oscillations was first suggested by Bruno Pontecorvo in 1957 using an analogy with kaon oscillations ; over the subsequent 10 years he developed the mathematical formalism and the modern formulation of vacuum oscillations.

Because it can be difficult to prove " exclusive control ", this element has largely given way in modern cases to a less rigid formulation: that the evidence eliminates, to a sufficient degree, other responsible causes ( including the conduct of the plaintiff and third parties ).

This change in perspective was characteristic of the shift from the Christian mediaval period to the modern period, it had been anticipated in other fields and now Descartes was giving it a formulation in the field of philosophy.

The modern formulation of statistical mechanics is based on the description of the physical system by an ensemble that represents all possible configurations of the system and the probability of realizing each configuration.

In the formulation of Kant, who wrote some of the most influential modern treatises on the subject, the great achievement of reason is that it is able to exercise a kind of universal law-making.

The ideas gained limited exposure until they were independently rediscovered and further developed by Laplace, who first published the modern formulation in his 1812 Théorie analytique des probabilités.

The earliest formulation of the self in modern psychology form the distinction between the self as I, the subjective knower, and the self as Me, the object that is known.

The Declaration of Geneva was intended as a revision of the Oath of Hippocrates to a formulation of that oath's moral truths that could be comprehended and acknowledged in a modern way.

The modern formulation of identity is that of Gottfried Leibniz, who held that x is the same as y if and only if every predicate true of x is true of y as well.

: First formulation of modern politics with the publication of Machiavelli's The Prince.

Indeed, Strauss wrote that Heidegger's thinking must be understood and confronted before any complete formulation of modern political theory is possible.

The modern formulation of Warshall's algorithm as three nested for-loops was first described by Peter Ingerman, also in 1962.

Temple argued against this position in his " On Ancient and Modern Learning " ( where he provided the first English formulation of the commonplace that modern critics see more only because they are dwarves standing on the shoulders of giants ), and Temple's somewhat naive essay prompted a small flurry of responses.

In its investigation of herbal and chemical ingredients, the work of the pharma may be regarded as a precursor of the modern sciences of chemistry and pharmacology, prior to the formulation of the scientific method.

The modern formulation of the antiperspirant was patented by Jules Montenier on January 28, 1941.

The modern formulation of the antiperspirant was patented by Jules Montenier on January 28, 1941.

Grundtvig and his followers are credited with being very influential in the formulation of modern Danish national consciousness.

Nosaltres, els valencians marked the dawn of Valencian nationalism in its modern formulation.

The modern formulation of this is sometimes called the Liouville-Arnold theorem, and the underlying concept of integrability is referred to as Liouville integrability.

The earliest formulation of the self in modern psychology derived from the distinction between the self as I, the subjective knower, and the self as Me, the object that is known.

The formulation goes back to the ancient Greek philosopher Empedocles, and the theologian William Paley set out a variation on this argument in 1802, to refute ( in later pages ) a claim that there had been a wide range of initial creations, with less viable forms eliminated by nature to leave the modern range of species:

In modern mathematics, Euclidean spaces form the prototypes for other, more complicated geometric objects.

In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry.

General relativity, or the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1916 and the current description of gravitation in modern physics.

The tower's smooth aluminum and glass surface and crisp unadorned geometric form demonstrate Pei's adherence to the mainstream of 20th-century modern design.

The modern study of mineralogy was founded on the principles of crystallography ( the origins of geometric crystallography, itself, can be traced back to the mineralogy practiced in the eighteenth and nineteenth centuries ) and to the microscopic study of rock sections with the invention of the microscope in the 17th century.

He is the author of one of the most important treatises on algebra written before modern times, the Treatise on Demonstration of Problems of Algebra, which includes a geometric method for solving cubic equations by intersecting a hyperbola with a circle.

The first modern shorthand systems were geometric.

In modern times, high-speed mirror galvanometers are employed in laser light shows to move the laser beams and produce colorful geometric patterns in fog around the audience.

The geometric patterns, such as modern North African zellige work, made of small tiles each of a single colour but different and regular shapes, are often referred to as " mosaic ", which is not strictly correct.

( Its modern abstractions include deformation quantization and geometric quantization.

In California, he became celebrated for rigorously geometric but airy structures that symbolized a West Coast variation on the mid-century modern residence.

On the other hand, Euclidian optics provided a geometric model that was able to account for this, as well as the length of shadows and reflections in mirrors, because Euclid believed that the visual " rays " could only travel in straight lines ( something which is commonly accepted in modern science ).

What made Cubism progressive and truly modern, according to Metzinger and Gleizes, was its the new geometric armature ; with that it broke free from the immobility of 3-dimensional Euclidean geometry and attained a dynamic representation of the 4-dimensional continuum in which we live, a better representation of reality, of life's experience, something that could be grasped through the senses ( not through the eye ) and expressed onto a canvas.

It took less than two decades to find a modern geometric variational meaning for that operator that links it to the Marr – Hildreth ( zero crossing of the Laplacian ) edge detector.

A later paper of Hilbert ( 1893 ) dealt with the same questions in more constructive and geometric ways, but remained virtually unknown until David Mumford brought these ideas back to life in the 1960s, in a considerably more general and modern form, in his geometric invariant theory.

Subsequent work on algebraic spaces and algebraic stacks by Deligne, Mumford, and Michael Artin, originally in the context of moduli problems, has further enhanced the geometric flexibility of modern algebraic geometry.

This is easily confirmed under modern expansions of Euclidean geometry, and had lasting consequences at its introduction, allowing the construction of almost all the geometric concepts of the time.

In modern geometric terms, the theorem gives necessary and sufficient conditions for the existence of a foliation by maximal integral manifolds each of whose tangent bundles are spanned by a given family of vector fields ( satisfying an integrability condition ) in much the same way as an integral curve may be assigned to a single vector field.

All metric theories satisfy the Einstein equivalence principle, which in modern geometric language states that in a very small region ( too small to exhibit measurable curvature effects ), all the laws of physics known in special relativity are valid in local Lorentz frames.

For the series Stylized Sculpture ( 2007 ), Sugimoto selected distinctive garments by celebrated couturiers from the collection of the Kyoto Costume Institute, shot in chiaroscuro on headless mannequins — from Madeleine Vionnet ’ s precociously modern T-dress and Balenciaga ’ s wasp-waisted billowing ensemble to Yves St Laurent ’ s strict geometric Mondrian shift and Issey Miyake ’ s sail-like slip.

Some modern weavers have begun to favor bold geometric patterns over the traditional Mahi ( Herati ) design and have added colors such as turquoise and purple to the more traditional red, pink, ivory, green, and blue.

Later in the 20th century, the women of Gee's Bend developed a distinctive, bold, and sophisticated quilting style based on traditional African-American quilts with a geometric simplicity that developed separately but was like that of Amish quilts and modern art.