“The concept of rotation led to geometrical exponential magnitudes, to the analysis of angles and of trigonometric functions, etc. I was delighted how thorough the analysis thus formed and extended, not only the often very complex and unsymmetric formulae which are fundamental in tidal theory, but also the technique of development parallels the concept.”

— Hermann Grassmann

Ausdehnungslehre (1844)

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Hermann Grassmann10

German polymath, linguist and mathematician 1809–1877

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“The concept of 'measurement' becomes so fuzzy on reflection that it is quite surprising to have it appearing in physical theory at the most fundamental level… does not any analysis of measurement require concepts more fundamental than measurement? And should not the fundamental theory be about these more fundamental concepts?”

John S. Bell (1928–1990) Northern Irish physicist

"Quantum Mechanics for Cosmologists" (1981); published in Quantum Gravity (1981) edited by Christopher Isham, Roger Penrose and Dennis William Sciama, p. 611 - 637

“A work on tidal theory… led me to Lagrange's Mécanique analytique and thereby I returned to those ideas of analysis. All the developments in that work were transformed through the principles of the new analysis in such a simple way that the calculations often came out more than ten times shorter than in Lagrange's work.”

Hermann Grassmann (1809–1877) German polymath, linguist and mathematician

the "ideas of analysis" to which he returned, are those quoted above.
Ausdehnungslehre (1844)

“The concept of centroid as sum led me to examine Möbius' Barycentrische Calcul, a work of which until then I knew only the title; and I was not little pleased to find here the same concept of the summation of points to which I had been led in the course of the development. This was the first, and… the only point of contact which my new system of analysis had with the one that was already known.”

Hermann Grassmann (1809–1877) German polymath, linguist and mathematician

Ausdehnungslehre (1844)

“An engineering science aims to organize the design principles used in engineering practice into a discipline and thus to exhibit the similarities between different areas of engineering practice and to emphasize the power of fundamental concepts. In short, an engineering science is predominated by theoretical analysis and very often uses the tool of advanced mathematics.”

Qian Xuesen (1911–2009) Chinese rocket scientist

Source: Engineering cybernetics, (1954), p. vii

“While I was pursuing the concept of geometrical product, as this idea was established by my father… I concluded that not only rectangles, but also parallelograms, may be viewed as products of two adjacent sides, provided that the sides are viewed not merely as lengths, but rather as directed magnitudes. When I joined this concept of geometrical product with the previously established idea of geometrical sum the most striking harmony resulted. Thus when I multiplied the sum of two vectors by a third coplaner vector, the result coincided (and must always coincide) with the result obtained by multiplying separately each of the two original vectors by the third… and adding together (with due attention to positive and negative values) the two products. [Thus A(B + C) = AB + AC. ]
From this harmony I came to see a whole new area of analysis was opening up which could lead to important results.”

Hermann Grassmann (1809–1877) German polymath, linguist and mathematician

Ausdehnungslehre (1844)

“The second period, which commenced in the middle of the seventeenth century, and lasted for about a century, was characterized by the application of the powerful analytical methods provided by the new Analysis to the determination of analytical expressions for the number π in the form of convergent series, products, and continued fractions. The older geometrical forms of investigation gave way to analytical processes in which the functional relationship as applied to the trigonometrical functions became prominent. The new methods of systematic representation gave rise to a race of calculators of π, who, in their consciousness of the vastly enhance means of calculation placed in their hands by the new Analysis, proceeded to apply the formulae to obtain numerical approximations to π to ever larger numbers of places of decimals, although their efforts were quite useless for the purpose of throwing light upon the true nature of that number. At the end of this period no knowledge had been obtained as regards the number π of the kind likely to throw light upon the possibility or impossibility of the old historical problem of the ideal construction; it was not even definitely known whether the number is rational or irrational. However, one great discovery, destined to furnish the clue to the solution of the problem, was made at this time; that of the relation between the two numbers π and e, as a particular case of those exponential expressions for the trigonometrical functions which form one of the most fundamentally important of the analytical weapons forged during this period.”

E. W. Hobson (1856–1933) British mathematician

Source: Squaring the Circle (1913), pp. 11-12

“Because intersectionality is a concept (a description of the experience of multiple oppressions, without explaining their causes) rather than a theory (which does attempt to explain the root causes of oppressions), it can be applied alongside different theories of oppression--theories informed by Marxism or postmodernism, but also separatism, etc. Because Marxism and postmodernism are often antithetical, their specific uses of the concept of intersectionality can be very different and in very different and contrary ways.”

Sharon Smith (writer) (1956) American historian

A Marxist Case For Intersectionality (2017)

“As I was reading the extract from your paper in the geometric sum and difference… I was struck by the marvelous similarity between your results and those discoveries which I made even as early as 1832…
I conceived the first idea of the geometric sum and difference of two or more lines and also of the geometric product of two or three lines in that year (1832). This idea is in all ways identical to that presented in your paper. But since I was for a long time occupied with entirely different pursuits, I could not develop this idea. It was only in 1839 that I was led back to that idea and pursued this geometrical analysis up to the point where it ought to be applicable to all mechanics. It was possible for me to apply this method of analysis to the theory of tides, and in this I was astounded by the simplicity of the calculations resulting from this method.”

Hermann Grassmann (1809–1877) German polymath, linguist and mathematician

Letter to Saint-Venant (1845) as quoted by Michael J. Crowe, A History of Vector Analysis: The Evolution of the Idea of a Vectorial System (1967)

“The ‘theory of control systems’ in engineering is now a well-developed subject, making use of some remarkably powerful concepts and methods of analysis, especially in relation to problems of stabilization and the prevention of unwanted oscillations.”

Arnold Tustin (1899–1994) British engineer

Source: The Mechanism of Economic Systems (1953), p. v

“The particular conception of ideology operates primarily with a psychology of interests, while the total conception uses a more formal functional analysis, without any reference to motivations, confining itself to an objective description of the structural differences in minds operating in different social settings. The former assumes that this or that interest is the cause of a given lie or deception. The latter presupposes simply that there is a correspondence between a given social situation and a given perspective, point of view, or apperception mass. In this case, while an analysis of constellations of interests may often be necessary it is not to establish causal connections but to characterize the total situation. Thus interest psychology tends to be displaced by an analysis of the correspondence between the situation to be known and the forms of knowledge.”

Karl Mannheim (1893–1947) Hungarian sociologist

Ideology and Utopia (1929)

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