“It is a defect in the existing histories that, while they state generally the contents of, and the main propositions proved in, the great treatises of Archimedes and Apollonius, they make little attempt to describe the procedure by which the results are obtained. I have therefore taken pains, in the most significant cases, to show the course of the argument in sufficient detail to enable a competent mathematician to grasp the method used and to apply it, if he will, to other similar investigations.”

A History of Greek Mathematics (1921) Vol. 1. From Thales to Euclid

Adopted from Wikiquote. Last update June 3, 2021. History

Help us to complete the source, original and additional information

Do you have more details about the quote "It is a defect in the existing histories that, while they state generally the contents of, and the main propositions pr…" by Thomas Little Heath?

Thomas Little Heath46

British civil servant and academic 1861–1940

Related quotes

“Most programming languages are decidedly inferior to mathematical notation and are little used as tools of thought in ways that would be considered significant by, say, an applied mathematician.”

Kenneth E. Iverson (1920–2004) Canadian computer scientist

§0
Notation as a Tool of Thought (1979)

“Another feature of Alexandrian algebra is the absence of any explicit deductive structure. The various types of numbers… were not defined. Nor was there any axiomatic basis on which a deductive structure could be erected. The work of Heron, Nichomachus, and Diophantus, and of Archimedes as far as his arithmetic is concerned, reads like the procedural texts of the Egyptians and Babylonians… The deductive, orderly proof of Euclid and Apollonius, and of Archimedes' geometry is gone. The problems are inductive in spirit, in that they show methods for concrete problems that presumably apply to general classes whose extent is not specified. In view of the fact that as a consequence of the work of the classical Greeks mathematical results were supposed to be derived deductively from an explicit axiomatic basis, the emergence of an independent arithmetic and algebra with no logical structure of its own raised what became one of the great problems of the history of mathematics. This approach to arithmetic and algebra is the clearest indication of the Egyptian and Babylonian influences… Though the Alexandrian Greek algebraists did not seem to be concerned about this deficiency… it did trouble deeply the European mathematicians.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p.144

“The numberless treatises of antiquities, philosophy, mathematics, natural and other history … written originally in, or translated to our tongue are sufficient to lead us a great way into any science our curiosity shall prompt us to. The greatest difficulty we struggled with, was the want of a good art of reasoning, which we had not, that I know of, till that defect was supplied by Locke, whose Essay on Human Understanding makes large amends for the want of all others in that kind.”

Mary Astell (1666–1731) English feminist writer

An Essay in Defence of the Female Sex. P. 54 https://archive.org/details/essayindefenceof00aste

“"Groping" and "muddling through" is usually described as a solution by trial and error…. a series of trials, each of which attempts to correct the error committed by the preceding and, on the whole, the errors diminished as we proceed and the successive trials come closer and closer to the desired final result…. we may wish a better characterization…"successive trials" or "successive corrections" or "successive approximations."… You use successive approximations when… looking for a word in the dictionary… A mathematician may apply the term… to a highly sophisticated procedure… to treat some very advanced problem… that he cannot treat otherwise. The term even applies to science as a whole; the scientific theories which succeed each other, each claiming a better explanation… may appear as successive approximations to the truth.
Therefore, the teacher should not discourage his students from using trial and error—on the contrary, he should encourage the intelligent use of the fundamental method of successive approximations. Yet he should convincingly show that for… many… situations, straightforward algebra is more efficient than successive approximations.”

George Pólya (1887–1985) Hungarian mathematician

George Pólya, Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving (1962)

“The most economical way of obtaining good results is to apply the great, fundamental principles of art; and depend on them for beauty, rather than upon the use of either applied ornament or more expensive materials… much better results are likely to accrue from truth than falsehood, and from architectural [rather] than archaeological methods.”

Ernest Flagg (1857–1947) American architect

Introduction
Small Houses: Their Economic Design and Construction (1922)

“Taken altogether, Washington as a city is most unsatisfactory, and falls more grievously short of the thing attempted than any other of the great undertakings of which I have seen anything in the United States.”

Anthony Trollope (1815–1882) English novelist (1815-1882)

Source: North America (1862), Ch. 21

“"I exist" does not follow from "there is a thought now." The fact that a thought occurs at a given moment does not entail that any other thought has occurred at any other moment, still less that there has occurred a series of thoughts sufficient to constitute a single self. As Hume conclusively showed, no one event intrinsically points to any other. We infer the existence of events which we are not actually observing, with the help of general principle. But these principles must be obtained inductively. By mere deduction from what is immediately given we cannot advance a single step beyond. And, consequently, any attempt to base a deductive system on propositions which describe what is immediately given is bound to be a failure.”

Alfred Jules Ayer book Language, Truth, and Logic

Source: Language, Truth, and Logic (1936), p. 47.

“If a mathematician wishes to disparage the work of one of his colleagues, say, A, the most effective method he finds for doing this is to ask where the results can be applied. The hard pressed man, with his back against the wall, finally unearths the researches of another mathematician B as the locus of the application of his own results. If next B is plagued with a similar question, he will refer to another mathematician C. After a few steps of this kind we find ourselves referred back to the researches of A, and in this way the chain closes.”

Alfred Tarski (1901–1983) Polish-American logician

Source: The Semantic Conception of Truth (1952), p. 41.

“The methods of mathematics are the main topic of the course, not a long list of finished mathematical results with such highly polished proofs that the poor student can only marvel at the results, with no hope of understanding how mathematics is actually created by practicing mathematicians.”

Richard Hamming (1915–1998) American mathematician and information theorist

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

“The Arithmetica… is deficient, sometimes pardonably, sometimes without excuse, in generalization. The book of Porismata, to which Diophantus sometimes refers, seems on the other hand to have been entirely devoted to the discussion of general properties of numbers. It is three times expressly quoted in the Arithmetica… Of all these propositions he says… 'we find it in the Porisms'; but he cites also a great many similar propositions without expressly referring to the Porisms. These latter citations fall into two classes, the first of which contains mere identities, such as the algebraical equivalents of the theorems in Euclid II…. The other class contains general propositions concerning the resolution of numbers into the sum of two, three or four squares…. It will be seen that all these propositions are of the general form which ought to have been but is not adopted in the Arithmetica. We are therefore led to the conclusion that the Porismata, like the pamphlet on Polygonal Numbers, was a synthetic and not an analytic treatise. It is open, however, to anyone to maintain the contrary, since no proof of any porism is now extant.”

James Gow (scholar) (1854–1923) scholar

p, 125
A Short History of Greek Mathematics (1884)

Help us to complete the source, original and additional information

Thomas Little Heath46

Related quotes

Related topics