“Of the contemporaries of Newton one of the most prominent was John Wallis. …Wallis was a voluminous writer, and not only are his writings erudite, but they show a genius in mathematics… He was one of the first to recognize the significance of the generalization of exponents to include negative and fractional as well as positive and integral numbers. He recognized also the importance of Cavalieri's method of indivisibles, and employed it in the quadrature of such curves as y=xn, y=x1/n, and y=x0 + x1 + x2 +… He failed in his attempts at the approximate quadrature of the circle by means of series because he was not in possession of the general form of the binomial theorem. He reached the result, however, by another method. He also obtained the equivalent of ds = \! dx \sqrt{1+(\frac{dy}{dx})^2} for the length of an element of a curve, thus connecting the problem of rectification with that of quadrature.”

History of Mathematics (1923) Vol.1

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David Eugene Smith33

American mathematician 1860–1944

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“Wallis was in sympathy with Greek mathematics and astronomy, editing parts of the works of Archimedes, Eutocius, Ptolemy, and Aristarchus; but at the same time he recognized the fact that the analytic method was to replace the synthetic, as when he defined a conic as a curve of the second degree instead of as a section of a cone, and treated it by the aid of coordinates.”

David Eugene Smith (1860–1944) American mathematician

History of Mathematics (1923) Vol.1

“Though Wallis was advanced for his times and accepted negative numbers, he thought they were larger than infinity but not less than zero. In his Arithmetica Infinitorum (1655), he argued that since the ratio a/0, when a is positive, is infinite, then, when the denominator is changed to a negative number, as in a/b with b negative, the ratio must be greater than infinity.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 253.

“Rita: Have they sacked y'?
Frank: Not quite.
Rita: Well, why y' – packing your books away.
Frank: Australia. [After a pause] Some weeks ago – made rather a night of it.
Rita: Did y' bugger the bursar?
Frank: Metaphorically.”

Willy Russell book Educating Rita

Page 50.
Educating Rita (1980)

“Mice: What is the best early training for a writer?

Y. C.: An unhappy childhood.”

Ernest Hemingway (1899–1961) American author and journalist

Source: Ernest Hemingway on Writing

“The discovery of Hippocrates amounted to the discovery of the fact that from the relation
(1)\frac{a}{x} = \frac{x}{y} = \frac{y}{b}it follows that(\frac{a}{x})^3 = [\frac{a}{x} \cdot \frac{x}{y} \cdot \frac{y}{b} =] \frac{a}{b}and if a = 2b, [then (\frac{a}{x})^3 = 2, and]a^3 = 2x^3.The equations (1) are equivalent [by reducing to common denominators or cross multiplication] to the three equations
(2)x^2 = ay, y^2 = bx, xy = ab[or equivalently…y = \frac{x^2}{a}, x = \frac{y^2}{b}, y = \frac{ab}{x} ]Doubling the Cube
the 2 solutions of Menaechmusand the solutions of Menaechmus described by Eutocius amount to the determination of a point as the intersection of the curves represented in a rectangular system of Cartesian coordinates by any two of the equations (2).
Let AO, BO be straight lines placed so as to form a right angle at O, and of length a, b respectively. Produce BO to x and AO to y.
The first solution now consists in drawing a parabola, with vertex O and axis Ox, such that its parameter is equal to BO or b, and a hyperbola with Ox, Oy as asymptotes such that the rectangle under the distances of any point on the curve from Ox, Oy respectively is equal to the rectangle under AO, BO i. e. to ab. If P be the point of intersection of the parabola and hyperbola, and PN, PM be drawn perpendicular to Ox, Oy, i. e. if PN, PM be denoted by y, x, the coordinates of the point P, we shall have

\begin{cases}y^2 = b. ON = b. PM = bx\\ and\\ xy = PM. PN = ab\end{cases}whence\frac{a}{x} = \frac{x}{y} = \frac{y}{b}.
In the second solution of Menaechmus we are to draw the parabola described in the first solution and also the parabola whose vertex is O, axis Oy and parameter equal to a.”

Thomas Little Heath (1861–1940) British civil servant and academic

The point P where the two parabolas intersect is given by<center><math>\begin{cases}y^2 = bx\\x^2 = ay\end{cases}</math></center>whence, as before,<center><math>\frac{a}{x} = \frac{x}{y} = \frac{y}{b}.</math></center>
Apollonius of Perga (1896)

“Neither the good nor the true is self-realizing, so it is not generally a sufficient explanation of why people believe that X that X is true, or of why people do Y that Y is good.”

Raymond Geuss book Philosophy and Real Politics

Philosophy and Real Politics (2008).
Philosophy and Real Politics (2008)

“[B]y banning psychedelic research we have not only given up the study of an interesting drug or group of substances, but also abandoned one of the most promising approaches to the understanding of the human mind and consciousness.”

Stanislav Grof (1931) Czech pychiatrist

LSD psychotherapy (1980), MAPS 2001 edition, Epilogue, p. 300.

“This method of mine takes its beginnings where Cavalieri ends his Method of indivisibles.”

John Wallis (1616–1703) English mathematician

...for as his was the Geometry of indivisibles, so I have chosen to call my method the Arithmetic of infinitesimals.
Arithmetica Infinitorum (1656)

“The purpose of the mathematical theory of statistics is to deal with the relationship between 2 or more variable quantities without assuming that one is a single-valued mathematical function of the rest. The statistician does not think a certain x will produce a single-valued y; not a causative relation but a correlation. The relationship between x and y will be somewhere within a zone and we have to work out the probability that the point (x,y) will lie in different parts of that zone. The physicist is limited and shrinks the zone into a line. Our treatment will fit all the vagueness of biology, sociology, etc. A very wide science.”

Karl Pearson (1857–1936) English mathematician and biometrician

As quoted by E.S. Pearson, Karl Pearson: An Appreciation of Some Aspects of his Life and Work (1938) and cited in Bernard J. Norton, "Karl Pearson and Statistics: The Social Origins of Scientific Innovation" in Social Studies of Science, Vol. 8, No. 1, Theme Issue: Sociology of Mathematics (Feb.,1978), pp. 3-34.

“In the course of the fifteenth century, the sexagesimal division of the radius, in terms of which cords and goniometrical line-segments were expressed, was generally superseded, though not immediately replaced, by a decimal system of positional notation. Instead, mathematicians sought to avoid fractions by taking the Radius equal to a number of units of length of the form {\displaystyle 10^{n}} {\displaystyle 10^{n}}…The first to apply this method was the German astronomer Regiomontanus… the second half of the sixteenth and the first decades of the seventeenth century… observed of a gradual development of this method of Regiomontanus into a complete system of decimal positional fractions. Yet none of the steps taken by… writers is comparable in importance and scope with the progress achieved by Stevin in his De Thiende.”

Eduard Jan Dijksterhuis (1892–1965) Dutch historian

Source: Simon Stevin: Science in the Netherlands around 1600, 1970, p. 17-18

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David Eugene Smith33

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