“It is picked out from numbers progressing in continuous proportion. Of continuous progressions, an arithmetical is one which proceeds by equal intervals; a geometrical one which advances by unequal and proportionally increasing or decreasing intervals. Arithmetical progressions: 1, 2, 3, 4, 5, 6, 7, &c.; or 2, 4, 6, 8, 10, 12, 14, 16, &c, Geometrical progressions: 1, 2, 4, 8, 16, 32, 64, &c.; or 243, 81, 27, 9, 3, 1.”

— John Napier

The Construction of the Wonderful Canon of Logarithms (1889)

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John Napier 46

Scottish mathematician 1550–1617

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“Counting is the most simple and primitive of narratives -- 1 2 3 4 5 6 7 8 9 10 -- a tale with a beginning, a middle and an end and a sense of progression -- arriving at a finish of two digits -- a goal attained, a dénouement reached.”

Peter Greenaway (1942) British film director

Fear of Drowning By Numbers

“Suppose we a certain Number of things exposed, different each from other, as a, b, c, d, e, &c.; The question is, how many ways the order of these may be varied? as, for instance, how many changes may be Rung upon a certain Number of Bells; or, how many ways (by way of Anagram) a certain Number of (different) Letters may be differently ordered?
Alt.1,21) If the thing exposed be but One, as a, it is certain, that the order can be but one. That is 1.
2) If Two be exposed, as a, b, it is also manifest, that they may be taken in a double order, as ab, ba, and no more. That is 1 x 2 = 2. Alt.3
3) If Three be exposed; as a, b, c: Then, beginning with a, the other two b, c, may (by art. 2,) be disposed according to Two different orders, as bc, cb; whence arise Two Changes (or varieties of order) beginning with a as abc, acb: And, in like manner it may be shewed, that there be as many beginning with b; because the other two, a, c, may be so varied, as bac, bca. And again as many beginning with c as cab, cba. And therefore, in all, Three times Two. That is 1 x 2, x 3 = 6.
Alt.34) If Four be exposed as a, b, c, d; Then, beginning with a, the other Three may (by art. preceeding) be disposed six several ways. And (by the same reason) as many beginning with b, and as many beginning with c, and as many beginning with d. And therefore, in all, Four times six, or 24. That is, the Number answering to the case next foregoing, so many times taken as is the Number of things here exposed. That is 1 x 2 x 3, x 4 = 6 x 4 = 24.
5) And in like manner it may be shewed, that this Number 24 Multiplied by 5, that is 120 = 24 x 5 = 1 x 2 x 3 x 4 x 5, is the number of alternations (or changes of order) of Five things exposed. (Or, the Number of Changes on Five Bells.) For each of these five being put in the first place, the other four will (by art. preceeding) admit of 24 varieties, that is, in all, five times 24. And in like manner, this Number 120 Multiplied by 6, shews the Number of Alternations of 6 things exposed; and so onward, by continual Multiplication by the conse quent Numbers 7, 8, 9, &c.;
6) That is, how many so ever of Numbers, in their natural Consecution, beginning from 1, being continually Multiplied, give us the Number of Alternations (or Change of order) of which so many things are capable as is the last of the Numbers so Multiplied. As for instance, the Number of Changes in Ringing Five Bells, is 1 x 2 x 3 x 4 x 5 = 120. In Six Bells, 1 x 2 x 3 x 4 x 5 x 6 = 120 x 6 = 720. In Seven Bells, 720 x 7 = 5040. In Eight Bells, 5040 x 8 = 40320, And so onward, as far as we please.”

John Wallis (1616–1703) English mathematician

Source: A Discourse of Combinations, Alterations, and Aliquot Parts (1685), Ch.II Of Alternations, or the different Change of Order, in any Number of Things proposed.

“Ten Practical Experiments
1. Spacing (use gray appearance charcoal)
2. Shape (in varying proportions)
3. Arrangement
4. Rhythm
5. Drawn Line (charcoal and brush)
6. Masses(Chinese and Japanese ink painting)
7. Tone (difference)
8. Color ( suggested by stain glass)
9. Dark and Light(crayon on black paper)
10. Intensities”

Arthur Wesley Dow (1857–1922) painter from the United States

A Course in Fine Arts- Arthur Dow- Bulletin of College of Art of Association of America Vol 1 no 4 September 1918
A Course in Fine Arts

“The Rule of Three, or Golden Rule of Arithmeticall whole Numbers. Be the three termes given 2 3 4. …To finde their fourth proporcionall Terme: that is to say, in such Reason to the third terme 4, as the second terme 3, is to the first terme 2 [Modern notation: \frac{x}{4} = \frac{3}{2}]. …Multiply the second terme 3, by the third terme 4, & giveth the product 12: which dividing by the first terme 2, giveth the Quotient 6: I say that 6 is the fourth proportional terme required.”

Simon Stevin (1548–1620) Flemish scientist, mathematician and military engineer

Disme: the Art of Tenths, Or, Decimall Arithmetike (1608)

“First, the eight rules of tolerance: 1. Stay out of the way. 2. Don't pick a fight. 3. If challenged, walk away. 4. Avoid eye contact. 5. If your enemy is diurnal, learn to be nocturnal. 6. Vice versa. 7. Possess nothing the other wants. 8. Draw the line at hurting kids: I will fight to the death.
After tolerance is achieved comes play. Here are the nine rules of play: 1. Know when to quit. 2. Learn how to handicap. 3. Learn what frightens the other (cat claws). 4. Don't let it get to you. It's just a game; you mustn't take it seriously (cats have trouble with this one—escalation is always a risk in cat games). 5. Don't eat your playmate. 6. Pay attention to the signals on the other side—for instance, “enough” and “quit.” 7. Don't suddenly change the rules. 8. Don't be a sore loser. 9. Remember: It's only a game.
And if play succeeds, we can move on to the eight rules of friendship: 1. Learn the rules of your opposite number. 2. Recognize that danger is no longer relevant. 3. Take your time. 4. One step at a time. 5. Apologize often by learning the other's words, gestures, sounds, or postures for “I'm sorry.””

Jeffrey Moussaieff Masson (1941) American writer and activist

6. Acknowledge mistakes. 7. Make the offer of friendship more than once. 8. Express curiosity about what the other is like.
Source: Raising the Peaceable Kingdom (2005), Ch. 5

“[T]he astronomical data give the number N of nebulae counted out to a given inferred "distance" d, and in order to determine the curvature… we must express N, or equivalently V, to which it is assumed proportional, in terms of d. …from the second of formulae (3) and… (4)… to the approximation here adopted, 5)V = \frac{4}{3} \pi d^2 (1 + \frac{3}{10} K d^2 + …);…plotting N against… d and comparing… with the formula (5), it should be possible operationally to determine the "curvature" K.”

Howard P. Robertson (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

“The basic unit of writing practice is the timed exercise.
1. Keep your hand moving.
2. Don't cross out.
3. Don't worry about spelling, punctuation, grammar.
4. Lose control.
5. Don't think. Don't get logical.
6. Go for the jugular.
[…] That is the discipline: to continue to sit.”

Natalie Goldberg book Writing Down the Bones

Essay, "First thoughts". p.8, 9
Writing Down the Bones (1986)

“Take a unit, halve it, halve the result, and so on continually. This gives—1 1⁄2 1⁄4 1⁄8 1⁄16 1⁄32 1⁄64 1⁄128 &c.;Add these together, beginning from the first, namely, add the first two, the first three, the first four, &c;… We see then a continual approach to 2, which is not reached, nor ever will be, for the deficit from 2 is always equal to the last term added.
…We say that—1, 1 + 1⁄2, 1 + 1⁄2 + 1⁄4, 1 + 1⁄2 + 1⁄4 + 1⁄8, &c.; &c.;is a series of quantities which continually approximate to the limit 2. Now the truth is, these several quantities are fixed, and do not approximate to 2. …it is we ourselves who approximate to 2, by passing from one to another. Similarly when we say, "let x be a quantity which continually approximates to the limit 2," we mean, let us assign different values to x, each nearer to 2 than the preceding, and following such a law that we shall, by continuing our steps sufficiently far, actually find a value for x which shall be as near to 2 as we please.”

Augustus De Morgan (1806–1871) British mathematician, philosopher and university teacher (1806-1871)

The Differential and Integral Calculus (1836)

“1. Zero is a number.
2. The successor of any number is another number.
3. There are no two numbers with the same successor.
4. Zero is not the successor of a number.
5. Every property of zero, which belongs to the successor of every number with this property, belongs to all numbers.”

Giuseppe Peano (1858–1932) Italian mathematician

As expressed in "The Mathematical Philosophy of Giuseppe Peano" by Hubert C. Kennedy, in Philosophy of Science Vol. 30, No. 3 (July 1963)
Peano axioms

“The ambiguity of progress becomes evident. Without doubt, it offers new possibilities for good, but it also opens up appalling possibilities for evil — possibilities that formerly did not exist. We have all witnessed the way in which progress, in the wrong hands, can become and has indeed become a terrifying progress in evil. If technical progress is not matched by corresponding progress in man's ethical formation, in man's inner growth (cf. Eph 3:16; 2 Cor 4:16), then it is not progress at all, but a threat for man and for the world.”

Pope Benedict XVI (1927) 265th Pope of the Catholic Church

In Encyclical Letter Spe Salvi http://www.vatican.va/holy_father/benedict_xvi/encyclicals/documents/hf_ben-xvi_enc_20071130_spe-salvi_en.html (30 November 2007)
2007

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John Napier 46

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