Scientific method

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An 18th-century depiction of early experimentation in the field of chemistry.


The chief characteristic which distinguishes the scientific method from other methods of acquiring knowledge is that scientists seek to let reality speak for itself, supporting a theory when a theory's predictions are confirmed and challenging a theory when its predictions prove false.[4] Although procedures vary from one field of inquiry to another, identifiable features distinguish scientific inquiry from other methods of obtaining knowledge. Scientific researchers propose hypotheses as explanations of phenomena and design experimental studies to test these hypotheses via predictions which can be derived from them. These steps must be repeatable to guard against mistake or confusion in any particular experimenter. Theories that encompass wider domains of inquiry may bind many independently derived hypotheses together in a coherent, supportive structure. Theories, in turn, may help form new hypotheses or place groups of hypotheses into context.



The DNA example below is a synopsis of this method
According to Morris Kline,[8] "Modern science owes its present flourishing state to a new scientific method which was fashioned almost entirely by Galileo Galilei" (1564−1642). Dudley Shapere[9] takes a more measured view of Galileo's contribution.

The scientific method is the process by which science is carried out.[10] Science builds on previous knowledge, and this can lead to improvements and refinements over time.[11] The scientific method can function in the same way,[12][13] meaning that it can become more effective at understanding, and even generating new knowledge.[14][15] For example, the concept of falsification (first proposed in 1934) formalizes the attempt to disprove hypotheses rather than prove them.[16]

This model can be seen to underlay the scientific revolution.[17] One thousand years ago, Alhazen demonstrated the importance of forming questions and subsequently testing them,[18] an approach which was advocated by Galileo in 1638 with the publication of Two New Sciences.[19] The current method is based on a hypothetico-deductive model[20] formulated in the 20th century, although it has undergone significant revision since first proposed (for a more formal discussion, see below).


The overall process involves making conjectures (hypotheses), deriving predictions from them as logical consequences, and then carrying out experiments based on those predictions to determine whether the original conjecture was correct.[21] There are difficulties in a formulaic statement of method, however. Though the scientific method is often presented as a fixed sequence of steps, they are better considered as general principles.[22] Not all steps take place in every scientific inquiry (or to the same degree), and are not always in the same order. As noted by William Whewell (1794–1866), "invention, sagacity, [and] genius"[23] are required at every step.

Formulation of a question










DNA example

  • Question: Previous investigation of DNA had determined its chemical composition (the four nucleotides), the structure of each individual nucleotide, and other properties. It had been identified as the carrier of genetic information by the Avery–MacLeod–McCarty experiment in 1944,[27] but the mechanism of how genetic information was stored in DNA was unclear.
  • Hypothesis: Francis Crick and James D. Watson hypothesized that DNA had a helical structure.[28]
  • Prediction: If DNA had a helical structure, its X-ray diffraction pattern would be X-shaped.[29][30] This prediction was determined using the mathematics of the helix transform, which had been derived by Cochran, Crick and Vand[31] (and independently by Stokes). This prediction was a mathematical construct, completely independent from the biological problem at hand.
  • Analysis: When Watson saw the detailed diffraction pattern, he immediately recognized it as a helix.[32][33] He and Crick then produced their model, using this information along with the previously known information about DNA's composition and about molecular interactions such as hydrogen bonds.[34]


Other components




External review


Data recording and sharing

Scientists typically are careful in recording their data, a requirement promoted by Ludwik Fleck (1896–1961) and others.[38] Though not typically required, they might be requested to supply this data to other scientists who wish to replicate their original results (or parts of their original results), extending to the sharing of any experimental samples that may be difficult to obtain.[39]

Scientific inquiry

Scientific inquiry generally aims to obtain knowledge in the form of testable explanations that can be used to predict the results of future experiments. This allows scientists to gain a better understanding of the topic being studied, and later be able to use that understanding to intervene in its causal mechanisms (such as to cure disease). The better an explanation is at making predictions, the more useful it frequently can be, and the more likely it is to be correct. The most successful explanations, which explain and make accurate predictions in a wide range of circumstances, are called often called scientific theories.

Most experimental results do not result in large changes in human understanding; improvements in theoretical scientific understanding is typically the result of a gradual process that synthesizes the results of different experiments, by various researchers, sometimes across different domains of science.[40] Scientific models vary in the extent to which they have been experimentally tested and for how long, and in their acceptance in the scientific community. In general, explanations become accepted by a scientific community as evidence in favor is presented, and as presumptions that are inconsistent with the evidence are falsified.

Properties of scientific inquiry

Flying gallop falsified; see image below.

Scientific knowledge is closely tied to empirical findings, and typically remains subject to falsification if new experimental observation incompatible with it is found. That is, no theory can ever be considered completely certain, since new evidence falsifying it might be discovered. If such evidence is found, a new theory may be proposed, or (more commonly) it is found that minor modifications to the previous theory are sufficient to explain the new evidence. The strength of a theory can be related to how long it has persisted without falsification of its core principles.

Confirmed theories are also subject to subsumption by more accurate theories. For example, thousands of years of scientific observations of the planets were explained almost perfectly by Newton's laws. However, these laws were then determined to be special cases of a more general theory (relativity), which explained both the (previously unexplained) exceptions to Newton's laws and predicting and explaining other observations such as the deflection of light by gravity. Thus, independent, unconnected, scientific observations can be connected to each other, unified by principles of increasing explanatory power.[41]

Since new theories often should be able to explain more than the previous one, any successor theory capable of subsuming it should meet an even higher standard, explaining both the larger, unified body of observations explained by the previous theory and unifying that with even more observations. In other words, if the theory becomes more accurate with time, it will become increasingly difficult to produce a more successful theory, simply because of the success of what already exists.[41] For example, the theory of evolution explains the diversity of life on Earth, how species adapt to their environments, and many other patterns observed in the natural world;[42][43] its most recent major modification was unification with genetics to form the modern evolutionary synthesis. In subsequent modifications, it has also subsumed aspects of many other fields such as biochemistry and molecular biology.

Beliefs and biases

Scientific methodology typically directs that hypotheses be tested in controlled conditions which can be reproduced by others. The scientific community's pursuit of experimental control and reproducibility can diminish the effects of any one individuals' cognitive biases.




In contrast to the requirement for scientific knowledge to correspond to objective fact, poorly attested beliefs (such as various myths) can be believed and acted upon irrespective of truth,[45] often taking advantage of the narrative fallacy that when narrative is constructed its elements become easier to believe.[46][47] Myths intended to be taken as objectively true must have their elements assumed a priori, while science requires testing and validation a posteriori before ideas are accepted.[48]

Elements of the scientific method


Four essential elements[49][50][51] of the scientific method[52] are iterations,[53][54] recursions,[55] interleavings, or orderings of the following:



A linearized, pragmatic scheme of the four points above is sometimes offered as a guideline for proceeding:[64]

  1. Define a question
  2. Gather information and resources (observe)
  3. Form an explanatory hypothesis
  5. Analyze the data
  7. Publish results
  8. Retest (frequently done by other scientists)


While this schema outlines a typical hypothesis/testing method,[65] it should also be noted that a number of philosophers, historians and sociologists of science (perhaps most notably Paul Feyerabend) claim that such descriptions of scientific method have little relation to the ways that science is actually practiced.

The "operational" paradigm combines the concepts of operational definition, instrumentalism, and utility:

The essential elements of scientific method are operations, observations, models, and a utility function for evaluating models.[66][not in citation given]





Andreas Vesalius, (1546)[67]








DNA icon (25x25).png


Another example: precession of Mercury

Precession of the perihelion (exaggerated)


Hypothesis development

Main article: Hypothesis formation




William Glen observes that




DNA icon (25x25).png

Linus Pauling proposed that DNA might be a triple helix.[71] This hypothesis was also considered by Francis Crick and James D. Watson but discarded. When Watson and Crick learned of Pauling's hypothesis, they understood from existing data that Pauling was wrong[72] and that Pauling would soon admit his difficulties with that structure. So, the race was on to figure out the correct structure (except that Pauling did not realize at the time that he was in a race) ..3. DNA-predictions

Predictions from the hypothesis

Main article: Prediction in science





DNA icon (25x25).png

James D. Watson, Francis Crick, and others hypothesized that DNA had a helical structure. This implied that DNA's X-ray diffraction pattern would be 'x shaped'.[30][73] This prediction followed from the work of Cochran, Crick and Vand[31] (and independently by Stokes). The Cochran-Crick-Vand-Stokes theorem provided a mathematical explanation for the empirical observation that diffraction from helical structures produces x shaped patterns.


Another example: general relativity



Main article: Experiment





DNA icon (25x25).png


Evaluation and improvement




DNA icon (25x25).png

After considerable fruitless experimentation, being discouraged by their superior from continuing, and numerous false starts,[78][79][80] Watson and Crick were able to infer the essential structure of DNA by concrete modeling of the physical shapes of the nucleotides which comprise it.[34][81] They were guided by the bond lengths which had been deduced by Linus Pauling and by Rosalind Franklin's X-ray diffraction images. ..DNA Example




Models of scientific inquiry

Classical model

The classical model of scientific inquiry derives from Aristotle,[83] who distinguished the forms of approximate and exact reasoning, set out the threefold scheme of abductive, deductive, and inductive inference, and also treated the compound forms such as reasoning by analogy.

Pragmatic model

In 1877,[11] Charles Sanders Peirce (/ˈpɜrs/ like "purse"; 1839–1914) characterized inquiry in general not as the pursuit of truth per se but as the struggle to move from irritating, inhibitory doubts born of surprises, disagreements, and the like, and to reach a secure belief, belief being that on which one is prepared to act. He framed scientific inquiry as part of a broader spectrum and as spurred, like inquiry generally, by actual doubt, not mere verbal or hyperbolic doubt, which he held to be fruitless.[84] He outlined four methods of settling opinion, ordered from least to most successful:






2. Deduction. Two stages:


3. Induction. The long-run validity of the rule of induction is deducible from the principle (presuppositional to reasoning in general[88]) that the real is only the object of the final opinion to which adequate investigation would lead;[96] anything to which no such process would ever lead would not be real. Induction involving ongoing tests or observations follows a method which, sufficiently persisted in, will diminish its error below any predesignate degree. Three stages:

ii. Probation: direct Inductive Argumentation. Crude (the enumeration of instances) or Gradual (new estimate of proportion of truth in the hypothesis after each test). Gradual Induction is Qualitative or Quantitative; if Qualitative, then dependent on weightings of qualities or characters;[97] if Quantitative, then dependent on measurements, or on statistics, or on countings.

Communication and community


Peer review evaluation


Documentation and replication

Main article: Reproducibility

Sometimes experimenters may make systematic errors during their experiments, unconsciously veer from scientific method (Pathological science) for various reasons, or, in rare cases, deliberately report false results. Consequently, it is a common practice for other scientists to attempt to repeat the experiments in order to duplicate the results, thus further validating the hypothesis.



Data sharing

When additional information is needed before a study can be reproduced, the author of the study is expected to provide it promptly. If the author refuses to share data, appeals can be made to the journal editors who published the study or to the institution which funded the research.



Dimensions of practice

Further information: Rhetoric of science

The primary constraints on contemporary science are:

  • Publication, i.e. Peer review
  • Resources (mostly funding)



Philosophy and sociology of science



Norwood Russell Hanson, Imre Lakatos and Thomas Kuhn have done extensive work on the "theory laden" character of observation. Hanson (1958) first coined the term for the idea that all observation is dependent on the conceptual framework of the observer, using the concept of gestalt to show how preconceptions can affect both observation and description.[100] He opens Chapter 1 with a discussion of the Golgi bodies and their initial rejection as an artefact of staining technique, and a discussion of Brahe and Kepler observing the dawn and seeing a "different" sun rise despite the same physiological phenomenon. Kuhn[101] and Feyerabend[102] acknowledge the pioneering significance of his work.




Role of chance in discovery





The development of the scientific method is inseparable from the history of science itself. Ancient Egyptian documents describe empirical methods in astronomy,[109] mathematics,[110] and medicine.[111] In the 7th century BCE, Daniel, a Jewish captive of the Babylonian king Nebuchadnezzar, conducted a scientific experiment complete with a hypothesis, a control group, a treatment group, and a conclusion. The control group partook of the king's delicacies and wine, whereas Daniel's test group limited themselves to vegetables and water.[112] At the end of the test, Daniel's hypothesis was proven true.

The ancient Greek philosopher Thales in the 6th century BCE refused to accept supernatural, religious or mythological explanations for natural phenomena, proclaiming that every event had a natural cause. The development of deductive reasoning by Plato was an important step towards the scientific method. Empiricism seems to have been formalized by Aristotle, who believed that universal truths could be reached via induction.

For the beginnings of scientific method: Karl Popper writes of Parmenides (fl. 5th century BCE): "So what was really new in Parmenides was his axiomatic-deductive method, which Leucippus and Democritus turned into a hypothetical-deductive method, and thus made part of scientific methodology."[113]

According to David Lindberg, Aristotle (4th century BCE) wrote about the scientific method even if he and his followers did not actually follow what he said. Lindberg also notes that Ptolemy (2nd century CE) and Ibn al-Haytham (11th century CE) are among the early examples of people who carried out scientific experiments. [114] Also, John Losee writes that "the Physics and the Metaphysics contain discussions of certain aspects of scientific method", of which, he says "Aristotle viewed scientific inquiry as a progression from observations to general principles and back to observations."[115]

However, in order for true scientific method to develop, Aristotle could not be taken at face value. Errors in his "On the Heavens" and "Physics" had to be realized and corrected. Moreover, the pagan view common in the world during that era followed two concepts that prevented them from progressing toward a functional scientific method:

  1. Organismic view of nature – nature and created objects are divine or are themselves without beginning or end
  2. Circular reasoning as opposed to linear reasoning.[discuss]

According to Haffner, cultures that were thus debilitated included Chinese, Hindu, Meso-American, Egyptian, Babylonian, Greek and Arabic.[116]

Early Christian leaders such as Clement of Alexandria (150–215) and Basil of Caesarea (330–379) encouraged future generations to view the Greek wisdom as "handmaidens to theology" and science was considered a means to more accurate understanding of the Bible and of God.[117] Augustine of Hippo (354–430) who contributed great philosophical wealth to the Latin Middle Ages, advocated the study of science and was wary of philosophies that disagreed with the Bible, such as astrology and the Greek belief that the world had no beginning.[117] This Christian accommodation with Greek science "laid a foundation for the later widespread, intensive study of natural philosophy during the Late Middle Ages."[117] However, the division of Latin-speaking Western Europe from the Greek-speaking East,[117] followed by barbarian invasions, the Plague of Justinian, and the Islamic invasion,[118] resulted in the West largely losing access to Greek wisdom.

By the 8th century Islam had overrun the Christian lands[119] of Syria, Iraq, Iran and Egypt[120] This swift occupation further severed Western Europe from many of the great works of Aristotle, Plato, Euclid and others, many of which were housed in the great library of Alexandria. Having come upon such a wealth of knowledge, the Arabs, who viewed non-Arab languages as inferior, even as a source of pollution,[121] employed conquered Christians and Jews to translate these works from the native Greek and Syriac into Arabic[122]

Thus equipped, Arab philosopher Alhazen performed optical and physiological experiments, reported in his manifold works, the most famous being Book of Optics (1021).[123] He was thus a forerunner of scientific method, having understood that a controlled environment involving experimentation and measurement is required in order to draw educated conclusions. Other Arab polymaths of the same era produced copious works on mathematics, philosophy, astronomy and alchemy. Most stuck closely to Aristotle, being hesitant to admit that some of Aristotle's thinking was errant,[124] while others strongly criticized him.

During these years, occasionally a paraphrased translation from the Arabic, which itself had been translated from Greek and Syriac, might make its way to the West for scholarly study. It was not until 1204, during which the Latins conquered and took Constantinople from the Byzantines in the name of the fourth Crusade, that a renewed scholarly interest in the original Greek manuscripts began to grow. Due to the new easier access to the libraries of Constantinople by Western scholars, a certain revival in the study and analysis of the original Greek texts by Western scholars began.[125] From that point a functional scientific method that would launch modern science was on the horizon.

Grosseteste (1175–1253), an English statesman, scientist and Christian theologian, was "the principal figure" in bringing about "a more adequate method of scientific inquiry" by which "medieval scientists were able eventually to outstrip their ancient European and Muslim teachers" (Dales 1973:62). ... His thinking influenced Roger Bacon, who spread Grosseteste's ideas from Oxford to the University of Paris during a visit there in the 1240s. From the prestigious universities in Oxford and Paris, the new experimental science spread rapidly throughout the medieval universities: "And so it went to Galileo, William Gilbert, Francis Bacon, William Harvey, Descartes, Robert Hooke, Newton, Leibniz, and the world of the seventeenth century" (Crombie 1962:15). So it went to us as well.| Hugh G. Gauch, 2003.[126]

Roger Bacon (c. 1214 – 1294) is sometimes credited as one of the earliest European advocates of the modern scientific method inspired by the works of Aristotle.[127]

Roger Bacon (1214–1294), an English thinker and experimenter, is recognized by many to be the father of modern scientific method. His view that mathematics was essential to a correct understanding of natural philosophy was considered to be 400 years ahead of its time.[128] He was viewed as "a lone genius proclaiming the truth about time," having correctly calculated the calendar[128] His work in optics provided the platform on which Newton, Descartes, Huygens and others later transformed the science of light. Bacon's groundbreaking advances were due largely to his discovery that experimental science must be based on mathematics. (186–187) His works Opus Majus and De Speculis Comburentibus contain many "carefully drawn diagrams showing Bacon's meticulous investigations into the behavior of light."[128] He gives detailed descriptions of systematic studies using prisms and measurements by which he shows how a rainbow functions.[128]

Others who advanced scientific method during this era included Albertus Magnus (c. 1193 – 1280), Theodoric of Freiberg, (c. 1250 – c. 1310), William of Ockham (c. 1285 – c. 1350), and Jean Buridan (c. 1300 – c. 1358). These were not only scientists but leaders of the church – Christian archbishops, friars and priests.

By the late 15th century, the physician-scholar Niccolò Leoniceno was finding errors in Pliny's Natural History. As a physician, Leoniceno was concerned about these botanical errors propagating to the materia medica on which medicines were based.[129] To counter this, a botanical garden was established at Orto botanico di Padova, University of Padua (in use for teaching by 1546), in order that medical students might have empirical access to the plants of a pharmacopia. The philosopher and physician Francisco Sanches was led by his medical training at Rome, 1571–73, and by the philosophical skepticism recently placed in the European mainstream by the publication of Sextus Empiricus' "Outlines of Pyrrhonism", to search for a true method of knowing (modus sciendi), as nothing clear can be known by the methods of Aristotle and his followers[130] – for example, syllogism fails upon circular reasoning. Following the physician Galen's method of medicine, Sanches lists the methods of judgement and experience, which are faulty in the wrong hands,[131] and we are left with the bleak statement That Nothing is Known (1581). This challenge was taken up by René Descartes in the next generation (1637), but at the least, Sanches warns us that we ought to refrain from the methods, summaries, and commentaries on Aristotle, if we seek scientific knowledge. In this, he is echoed by Francis Bacon, also influenced by skepticism; Sanches cites the humanist Juan Luis Vives who sought a better educational system, as well as a statement of human rights as a pathway for improvement of the lot of the poor.

The modern scientific method crystallized no later than in the 17th and 18th centuries. In his work Novum Organum (1620) – a reference to Aristotle's OrganonFrancis Bacon outlined a new system of logic to improve upon the old philosophical process of syllogism.[132] Then, in 1637, René Descartes established the framework for scientific method's guiding principles in his treatise, Discourse on Method. The writings of Alhazen, Bacon and Descartes are considered critical in the historical development of the modern scientific method, as are those of John Stuart Mill.[133]


Beginning in the 1930s, Karl Popper argued that there is no such thing as inductive reasoning.[134] All inferences ever made, including in science, are purely[135] deductive according to this view. Accordingly, he claimed that the empirical character of science has nothing to do with induction – but with the deductive property of falsifiability that scientific hypotheses have. Contrasting his views with inductivism and positivism, he even denied the existence of the scientific method: "(1) There is no method of discovering a scientific theory (2) There is no method for ascertaining the truth of a scientific hypothesis, i.e., no method of verification; (3) There is no method for ascertaining whether a hypothesis is 'probable', or probably true".[136] Instead, he held that there is only one universal method, a method not particular to science: The negative method of criticism, or colloquially termed trial and error. It covers not only all products of the human mind, including science, mathematics, philosophy, art and so on, but also the evolution of life. Following Peirce and others, Popper argued that science is fallible and has no authority.[136] In contrast to empiricist-inductivist views, he welcomed metaphysics and philosophical discussion and even gave qualified support to myths[137] and pseudosciences.[138] Popper's view has become known as critical rationalism.

Although science in a broad sense existed before the modern era, and in many historical civilizations (as described above), modern science is so distinct in its approach and successful in its results that it now defines what science is in the strictest sense of the term.[139]

Relationship with mathematics


Mathematical work and scientific work can inspire each other.[141] For example, the technical concept of time arose in science, and timelessness was a hallmark of a mathematical topic. But today, the Poincaré conjecture has been proven using time as a mathematical concept in which objects can flow (see Ricci flow).


George Pólya's work on problem solving,[142] the construction of mathematical proofs, and heuristic[143][144] show that the mathematical method and the scientific method differ in detail, while nevertheless resembling each other in using iterative or recursive steps.

Mathematical method Scientific method
1 Understanding Characterization from experience and observation
2 Analysis Hypothesis: a proposed explanation
3 Synthesis Deduction: prediction from the hypothesis
4 Review/Extend Test and experiment

In Pólya's view, understanding involves restating unfamiliar definitions in your own words, resorting to geometrical figures, and questioning what we know and do not know already; analysis, which Pólya takes from Pappus,[145] involves free and heuristic construction of plausible arguments, working backward from the goal, and devising a plan for constructing the proof; synthesis is the strict Euclidean exposition of step-by-step details[146] of the proof; review involves reconsidering and re-examining the result and the path taken to it.




See also

Problems and issues

History, philosophy, sociology


  1. ^ a b Goldhaber & Nieto 2010, p. 940
  3. ^ From the Oxford English Dictionary definition for "scientific".
  12. ^ Gauch 2003, p. 1: This is the principle of noncontradiction.


  15. ^ a b Taleb contributes a brief description of anti-fragility,
  20. ^ Godfrey-Smith 2003 p. 236.
  22. ^ Gauch 2003, p. 3
  23. ^ History of Inductive Science (1837), and in Philosophy of Inductive Science (1840)
  25. ^ This phrasing is attributed to Marshall Nirenberg.
  27. ^ a b McCarty1985
  35. ^ Galileo Galilei (1638) Two new sciences
  36. ^ Reconstruction of Galileo Galilei's experiment - the inclined plane
  38. ^ Fleck 1979, pp. xxvii–xxviii
  39. ^ "NIH Data Sharing Policy."
  41. ^ a b Brody 1993, pp. 44–45
  46. ^ Imre Lakatos (1976), Proofs and Refutations. Taleb 2007, p. 72 lists ways to avoid narrative fallacy and confirmation bias.
  50. ^ Jevons 1874, pp. 265–6.
  52. ^ Galileo 1638, pp. v–xii,1–300
  65. ^ See, e.g., Gauch 2003, esp. chapters 5–8
  66. ^ Cartwright, Nancy (1983), How the Laws of Physics Lie. Oxford: Oxford University Press. ISBN 0-19-824704-4
  69. ^ McElheny 2004 p.34
  70. ^ Glen 1994, pp. 37–38.
  74. ^ McElheny 2004 p.68: Nature April 25, 1953.
  77. ^ al-Battani, De Motu Stellarum translation from Arabic to Latin in 1116, as cited by "Battani, al-" (c. 858 – 929) Encyclopaedia Britannica, 15th. ed. Al-Battani is known for his accurate observations at al-Raqqah in Syria, beginning in 877. His work includes measurement of the annual precession of the equinoxes.
  85. ^ But see Scientific method and religion.
  87. ^ " ... in order to learn, one must desire to learn ..." – Peirce (1899), "F.R.L." [First Rule of Logic], Collected Papers v. 1, paragraphs 135-40, Eprint at the Wayback Machine (archived January 6, 2012)






  96. ^ Peirce (1878), "The Probability of Induction", Popular Science Monthly, v. 12, pp. 705–18, see 718 Google Books; 718 via Internet Archive. Reprinted often, including (Collected Papers v. 2, paragraphs 669-93), (The Essential Peirce v. 1, pp. 155–69).
  105. ^
    • Intellectual Impostures, Economist Books, 2003
  109. ^ The ancient Egyptians observed that heliacal rising of a certain star, Sothis (Greek for Sopdet (Egyptian), known to the West as Sirius), marked the annual flooding of the Nile river. See Neugebauer, Otto (1969) [1957], The Exact Sciences in Antiquity (2 ed.), Dover Publications, ISBN 978-0-486-22332-2 , p.82, and also the 1911 Britannica, "Egypt".
  110. ^ The Rhind papyrus lists practical examples in arithmetic and geometry – 1911 Britannica, "Egypt".
  111. ^ The Ebers papyrus lists some of the 'mysteries of the physician', as cited in the 1911 Britannica, "Egypt"
  112. ^ Holy Bible, Book of Daniel 1:8–16
  113. ^ Popper, Karl (1998). The world of Parmenides: essays on the Presocratic enlightenment. Routledge. p. 91. ISBN 0415173019. 
  114. ^ Lindberg, David (2007). The beginnings of western science: the European scientific tradition in philosophical, religious, and institutional context, Prehistory to A.D. 1450. The University of Chicago Press. p. 362. ISBN 0226482057. 
  115. ^ Losee, John (2001). A Historical Introduction to the Philosophy of Science. Oxford University Press. pp. 4–5. ISBN 0198700555. 
  116. ^ Haffner, Paul. “Mystery of the Church”. Gracewing, Herefordshire, UK, 2007, p. 263.
  117. ^ a b c d Grant, Edward. “The Foundations of Modern Science in the Middle Ages”. Cambridge University Press, UK, pp. 4–5.
  118. ^ Hodges, Richard and David Whitehouse. “Mohammed, Charlemagne and the Origins of Europe”. Cornell University Press, Ithaca, NY, 1983, p. 76.
  119. ^ Lewis, Bernard. “Muslim Discovery of Europe”. W. W. Norton and Company Ltd., New York, NY, 2001, p. 74.
  120. ^ Dowley, Tim, Ed. "The Baker Atlas of Christian History." 2001, p. 89
  121. ^ Lewis, Muslim Discovery, p. 72.
  122. ^ Meri, Josef W. and Jere L. Bacharach. “Medieval Islamic Civilization”. Vol. 1 Index A – K. 2006, p. 304.
  124. ^ Jaki, Stanley. “Science and Creation: From Eternal Cycles to an Oscillating Universe”. Science History Publications, NY, 1974, p. 206.
  125. ^ Mathpages. “Translating Aristotle”. Accessed July 2013.
  126. ^ Gauch 2003, pp. 52–53
  127. ^ George Sampson (1970). The concise Cambridge history of English literature. Cambridge University Press. p.174. ISBN 0-521-09581-6
  128. ^ a b c d Clegg, Brian. “The First Scientist: A Life of Roger Bacon”. Carroll and Graf Publishers, NY, 2003, p. 2.
  131. ^ Sanches, Limbrick & Thomson 1988, p. 278.
  134. ^ Logik der Forschung, new appendices *XVII*XIX (not yet available in the English edition Logic of scientific discovery)
  135. ^ Logic of Scientific discovery, p. 20
  136. ^ a b Karl Popper: On the non-existence of scientific method. Realism and the Aim of Science (1983)
  137. ^ Karl Popper: Science: Conjectures and Refutations. Conjectures and Refuations, section VII
  138. ^ Karl Popper: On knowledge. In search of a better world, section II
  142. ^ Pólya 1957 2nd ed.
  145. ^ Pólya 1957, p. 142
  146. ^ Pólya 1957, p. 144
  147. ^ Mackay 1991 p.100


Further reading

External links