**Analysis** is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384–322 B.C.), though *analysis* as a formal concept is a relatively recent development.

The word comes from the Ancient Greek ἀνάλυσις (*analysis*, "a breaking up", from *ana-* "up, throughout" and *lysis* "a loosening").

As a formal concept, the method has variously been ascribed to Alhazen,René Descartes (*Discourse on the Method*), and Galileo Galilei. It has also been ascribed to Isaac Newton, in the form of a practical method of physical discovery (which he did not name).

The field of chemistry uses analysis in at least three ways: to identify the components of a particular chemical compound (qualitative analysis), to identify the proportions of components in a mixture (quantitative analysis), and to break down chemical processes and examine chemical reactions between elements of matter. For an example of its use, analysis of the concentration of elements is important in managing a nuclear reactor, so nuclear scientists will analyze neutron activation to develop discrete measurements within vast samples. A matrix can have a considerable effect on the way a chemical analysis is conducted and the quality of its results. Analysis can be done manually or with a device. Chemical analysis is an important element of national security among the major world powers with materials measurement and signature intelligence (MASINT) capabilities.

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**Mathematical analysis** is a branch of mathematics that studies continuous change and
includes the theories of differentiation, integration, measure, limits, infinite series, and analytic functions.

These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).

Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. The explicit use of infinitesimals appears in Archimedes' *The Method of Mechanical Theorems*, a work rediscovered in the 20th century. In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle.Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere in the 5th century. The Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem in the 12th century.

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**Philosophical analysis** (from Greek: Φιλοσοφική ανάλυση) is a general term for techniques typically used by philosophers in the analytic tradition that involve "breaking down" (i.e. analyzing) philosophical issues. Arguably the most prominent of these techniques is the analysis of concepts (known as *conceptual analysis*). This article will examine the major philosophical techniques associated with the notion of analysis, as well as examine the controversies surrounding it.

While analysis is characteristic of the analytic tradition in philosophy, what is to be analyzed (the *analysandum*) often varies. Some philosophers focus on analyzing linguistic phenomena, such as sentences, while others focus on psychological phenomena, such as sense data. However, arguably the most prominent analyses are of concepts or propositions, which is known as *conceptual analysis* (Foley 1996).

Conceptual analysis consists primarily in breaking down or analyzing concepts into their constituent parts in order to gain knowledge or a better understanding of a particular philosophical issue in which the concept is involved (Beaney 2003). For example, the problem of free will in philosophy involves various key concepts, including the concepts of freedom, moral responsibility, determinism, ability, etc. The method of conceptual analysis tends to approach such a problem by breaking down the key concepts pertaining to the problem and seeing how they interact. Thus, in the long-standing debate on whether free will is compatible with the doctrine of determinism, several philosophers have proposed analyses of the relevant concepts to argue for either compatibilism or incompatibilism.

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