David Eugene Smith Quotes (1860-1944)

“The excellent work of Tropfke is an example of the tendency to break away from the mere chronological recital of facts.”

“[W]hile we put number into objects, on the other hand we derive our idea of number only from the presence of the world external to the mind. We see a group of people, and we begin by making an abstraction ("people"), and we say, "Here are ten people"—thus calling them all by the one abstract name, even though the individuals be very different. "A careful observation shows us, however, that there are no objects exactly alike; but by a mental operation of which we are quite unconscious, although it holds within itself the entire secret of mathematical abstraction, we take in objects which seem to be alike, rejecting for the time being their differences. Here is to be found the source of calculation." So the idea of number is generated in the mind by the sense perception of a group of things supposed to be alike.”

“Among his [John Wallis'] interesting discoveries was the relation     frac{4}{pi} = frac32cdotfrac34cdotfrac54cdotfrac56cdotfrac76cdotfrac78cdots one of the early values of π involving infinite products.”

“More than any other of his predecessors Plato appreciated the scientific possibilities of geometry... By his teaching he laid the foundation of the science, insisting upon accurate definitions, and logical proof. His opposition to the materialists, who saw in geometry only what was immediately useful to the artisan and the mechanic, is made clear by Plutarch in his Life of Marcellus... "Plato's indignation at it and his invections against it as the mere corruption and annihilation of the one good geometry, which was thus shamefully turning its back upon the unembodied objects of pure intelligence."”

“More than any other of his predecessors Plato appreciated the scientific possibilities of geometry... By his teaching he laid the foundation of the science, insisting upon accurate definitions, and logical proof. His opposition to the materialists, who saw in geometry only what was immediately useful to the artisan and the mechanic, is made clear by Plutarch in his Life of Marcellus... "Plato's indignation at it and his invections against it as the mere corruption and annihilation of the one good geometry, which was thus shamefully turning its back upon the unembodied objects of pure intelligence."”

“William Frend (1757-1841) whose daughter Sophia Elizabeth became De Morgan's wife (1837) ...came under De Morgan's definition of a true paradoxer, carrying on a zealous warfare for what he thought right. As a result of his Address to the Inhabitants of Cambridge 1787 and... he was deprived of his tutorship in 1788. A little later he was banished from Cambridge because of his denunciation of the abuses of the Church and his condemnation of the liturgy. His eccentricity is seen in his declining to use negative quantities in the operations of algebra. He finally became an actuary at London and was prominent in radical associations. He was a mathematician of ability, having been second wrangler and having nearly attained the first place, and he was also an excellent scholar in Latin Greek and Hebrew.”

“In a rough way we may summarize the conclusions of the writers to whom reference has chiefly been made, as follows: 1. Let the child learn to count things, thus getting the notion of number. These things are, for the purpose of counting, considered alike, and they may be single objects or groups. 2. Let him acquire the number series, exercising with it beyond the circle of actually counted things. 3. In the learning of symbols it does not seem to be a matter of moment as to whether these are given with the first presentation of number or not. They must, however, be acquired soon. 4. Unconsciously and gradually the child will acquire the idea (never expressed to him in words) of the one-to-one correspondence of number, name, symbol, and thereafter the pure concept of number will play a small part in his arithmetical calculations. 5. The ratio idea of number should be introduced early, and applied in the work with fractions.”

“[T]here is a possibility that some of the works ascribed to Sened ibn 'Alī are really works of Al-KhowarazmI, whose name immediately precedes his.”

“One of the first important steps to be taken in modern times... was due to Desargues. In a work published in 1639 Desargues set forth the foundation of the theory of four harmonic points, not as done today but based on the fact that the product of the distances of two conjugate points from the center is constant. He also treated the theory of poles and polars, although not using these terms.”

“Aside from Cauchy, the greatest contributory to the theory [of determinants] was Carl Gustav Jacob Jacobi. With him the word "determinant" received its final acceptance. He early used the functional determinant which Sylvester has called the Jacobian, and in his famous memoirs in Crelle's Journal for 1841 he considered these forms as well as that class of alternating functions which Sylvester has called alternants.”

“Grégoire de Saint-Vincent... was a Jesuit, taught mathematics in Rome and Prag (1629-1631), and was afterwards called to Spain by Phillip IV as tutor to his son... He wrote two works on geometry [Principia Matheseos Univerales (1651); Exercitationum Mathematicarum Libri quinque (1657)], giving in one of them the quadrature of the hyperbola referred to its asymptotes, and showing that as the area increased in arithmetic series the abscissas increased in geometric series.”

“Julius Wilhelm Richard Dedekind stands out as one of the most prominent contributors of the 19th century to the theory of algebraic numbers. He wrote various important memoirs on the binomial equation and on the theory of modular and Abelian functions, but is best known for his treatises Was sind und was sollen die Zahlen? (1888) and Stetigkeit und irrationale Zahlen (1872). In the latter work he set forth his idea of the Schnitt (cut) in relation to irrational numbers,—an idea he had in mind as early as 1858.”

“Johannes Tropfke... described the history of those individual parts of mathematics that he believed were most important for mathematics as taught in secondary schools. He intended his history to inform teachers about the origin of special problems, terms, and methods in school mathematics. ...Tropfke's approach to the history of mathematics at this time was new and even now is not yet out of date. The only comparable work is the second volume of D.E Smith's History of Mathematics... which gives far less detailed information.”