Quotes about proportion
page 6

David Ben-Gurion photo

“Yet for many of us, anti-Semitic feeling had little to do with our dedication [to Zionism]. I personally never suffered anti-Semitic persecution. Plonsk was remarkably free of it, or at least the Jews felt well protected in the cocoon of their community life. Nevertheless, and I think this very significant, it was Plonsk that sent the highest proportion of Jews to Eretz Israel from any town in Poland of comparable size. We emigrated not for negative reasons of escape but for the positive purpose of rebuilding a homeland, a place where we wouldn't be perpetual strangers and that through our toil would become irrevocably our own. Life in Plonsk was peaceful enough. There were three main communities: Russians, Jews and Poles. Each lived apart from the others. The Russians as the occupiers kept a firm hand on the civil administration. There were no Polish or Jewish officials. Officials or the police almost never interfered in dealings between Jewish and Polish communities. They disliked both equally and took an aloof attitude to the town's day-to-day life. The number of Jews and Poles in the city were roughly equal, about five thousand each. The Jews, however, formed a compact, centralized group occupying the innermost districts whilst the Poles were more scattered, living in outlying areas and shading off into the peasantry. Consequently, when a gang of Jewish boys met a Polish gang the latter would almost inevitably represent a single suburb and thus be poorer in fighting potential than the Jews who even if their numbers were initially fewer could quickly call on reinforcements from the entire quarter. Far from being afraid of them, they were rather afraid of us. In general, however, relations were amicable, though distant.”

David Ben-Gurion (1886–1973) Israeli politician, Zionist leader, prime minister of Israel

Memoirs : David Ben-Gurion (1970), p. 36

Jean-Baptiste Say photo

“But what must be the character of that policy, which aims at national prosperity through the impoverishment of a large proportion of the home producers, with a view to supply foreigners at a cheaper rate, and give them all the benifet of the national privation and self denial?”

Jean-Baptiste Say (1767–1832) French economist and businessman

Source: A Treatise On Political Economy (Fourth Edition) (1832), Book II, On Distribution, Chapter VII, Section IV, p. 340

L. Ron Hubbard photo
Willem de Sitter photo
Karl Schmidt-Rottluff photo

“On occasion I came to exaggerate certain forms, in violation of scientific proportion but in accordance with the balance of their spiritual relationships to each other. I made heads vastly oversized in relation to other parts of the body, because the head is the point of concentration of all the psyche, all expression.”

Karl Schmidt-Rottluff (1884–1976) German artist

In Gerhard Wietek, Schmidt-Rottluff: Graphik, Verlag Karl Thiemig, Munich, 1971, p. 100; as quoted in 'Portfolios', Alexander Dückers; in German Expressionist Prints and Drawings - Essays Vol 1.; published by Museum Associates, Los Angeles County Museum of Art, California & Prestel-Verlag, Germany, 1986, p. 111

Samuel Butler photo

“The turtle obviously had no sense of proportion; it differed so widely from myself that I could not comprehend it; and as this word occurred to me, it occurred also that until my body comprehended its body in a physical material sense, neither would my mind be able to comprehend its mind with any thoroughness. For unity of mind can only be consummated by unity of body; everything, therefore, must be in some respects both knave and fool to all that which has not eaten it, or by which it has not been eaten. As long as the turtle was in the window and I in the street outside, there was no chance of our comprehending one another.
Nevertheless, I knew that I could get it to agree with me if I could so effectually buttonhole and fasten on to it as to eat it. Most men have an easy method with turtle soup, and I had no misgiving but that if I could bring my first premise to bear I should prove the better reasoner. My difficulty lay in this initial process, for I had not with me the argument that would alone compel Mr. Sweeting to think that I ought to be allowed to convert the turtles — I mean I had no money in my pocket. No missionary enterprise can be carried on without any money at all, but even so small a sum as half a crown would, I suppose, have enabled me to bring the turtle partly round, and with many half-crowns I could in time no doubt convert the lot, for the turtle needs must go where the money drives. If, as is alleged, the world stands on a turtle, the turtle stands on money. No money no turtle. As for money, that stands on opinion, credit, trust, faith — things that, though highly material in connection with money, are still of immaterial essence.”

Samuel Butler (1835–1902) novelist

Ramblings In Cheapside (1890)

David Hume photo
Boris Yeltsin photo

“A sense of proportion and humanitarian action are not issues for terrorists. Their aim is that of killing and destroying.”

Boris Yeltsin (1931–2007) 1st President of Russia and Chairman of the Supreme Soviet of the RSFSR

Speech at a summit of the Organisation for Security and Cooperation in Europe in Istanbul, Turkey, as quoted in BBC World Service (19 November 1999)
1990s

C. Wright Mills photo
Richard Cobden photo

“I am not one to advocate the reducing of our navy in any degree below that proportion to the French navy which the exigencies of our service require; and, mind what I say, here is just what the French Government would admit as freely as you would. England has four times, at least, the amount of mercantile tonnage to protect at sea that France has, and that surely gives us a legitimate pretension to have a larger navy than France. Besides, this country is an island; we cannot communicate with any part of the world except by sea. France, on the other hand, has a frontier upon land, by which she can communicate with the whole world. We have, I think, unfortunately for ourselves, about a hundred times the amount of territory beyond the seas to protect, as colonies and dependencies, that France has. France has also twice or three times as large an army as England had. All these things give us a right to have a navy somewhat in the proportion to the French navy which we find to have existed if we look back over the past century. Nobody has disputed it. I would be the last person who would ever advocate any undue change in this proportion. On the contrary—I have said it in the House of Commons, and I repeat it to you—if the French Government showed a sinister design to increase their navy to an equality with ours; then, after every explanation to prevent such an absurd waste, I should vote 100 millions sterling rather than allow that navy to be increased to a level with ours—because I should say that any attempt of that sort without any legitimate grounds, would argue some sinister design upon this country.”

Richard Cobden (1804–1865) English manufacturer and Radical and Liberal statesman

Speech in Rochdale (26 June 1861), quoted in John Bright and J. E. Thorold Rogers (eds.), Speeches on Questions of Public Policy by Richard Cobden, M.P. Volume II (London: T. Fisher Unwin, 1908), pp. 433-4.
1860s

Thomas Jefferson photo
Claude Bernard photo

“Science increases our power in proportion as it lowers our pride.”

Claude Bernard (1813–1878) French physiologist

Bulletin of New York Academy of Medicine, Vol. IV (1928)

Judi Dench photo

“I think you’ve got to have your feet planted firmly on the ground, especially in this business, and you must not believe things that are said or written about you, because everything gets out of proportion one way or the other.”

Judi Dench (1934) English film, stage and television actress

Judi Dench – Grand Dame of Cinema http://www.comingsoon.net/extras/news/12246-judi-dench-grand-dame-of-cinema (December 6, 2005)

Calvin Coolidge photo

“Anybody can reduce taxes, but it is not so easy to stand in the gap and resist the passage of increasing appropriation bills which would make tax reduction impossible. It will be very easy to measure the strength of the attachment to reduced taxation by the power with which increased appropriations are resisted. If at the close of the present session the Congress has kept within the budget which I propose to present, it will then be possible to have a moderate amount of tax reduction and all the tax reform that the Congress may wish for during the next fiscal year. The country is now feeling the direct stimulus which came from the passage of the last revenue bill, and under the assurance of a reasonable system of taxation there is every prospect of an era of prosperity of unprecedented proportions. But it would be idle to expect any such results unless business can continue free from excess profits taxation and be accorded a system of surtaxes at rates which have for their object not the punishment of success or the discouragement of business, but the production of the greatest amount of revenue from large incomes. I am convinced that the larger incomes of the country would actually yield more revenue to the Government if the basis of taxation were scientifically revised downward. Moreover the effect of the present method of this taxation is to increase the cost of interest. on productive enterprise and to increase the burden of rent. It is altogether likely that such reduction would so encourage and stimulate investment that it would firmly establish our country in the economic leadership of the world.”

Calvin Coolidge (1872–1933) American politician, 30th president of the United States (in office from 1923 to 1929)

1920s, Second State of the Union Address (1924)

Ethan Allen photo
Ted Nelson photo
Thomas Little Heath photo
Horace Greeley photo

“VI. We complain that the Confiscation Act which you approved is habitually disregarded by your Generals, and that no word of rebuke for them from you has yet reached the public ear. Fremont's Proclamation and Hunter's Order favoring Emancipation were promptly annulled by you; while Halleck's No. 3, forbidding fugitives from Slavery to Rebels to come within his lines-- an order as unmilitary as inhuman, and which received the hearty approbation of every traitor in America-- with scores of like tendency, have never provoked even your own remonstrance. We complain that the officers of your Armies have habitually repelled rather than invited approach of slaves who would have gladly taken the risks of escaping from their Rebel masters to our camps, bringing intelligence often of inestimable value to the Union cause. We complain that those who have thus escaped to us, avowing a willingness to do for us whatever might be required, have been brutally and madly repulsed, and often surrendered to be scourged, maimed and tortured by the ruffian traitors, who pretend to own them. We complain that a large proportion of our regular Army Officers, with many of the Volunteers, evince far more solicitude to uphold Slavery than to put down the Rebellion. And finally, we complain that you, Mr. President, elected as a Republican, knowing well what an abomination Slavery is, and how emphatically it is the core and essence of this atrocious Rebellion, seem never to interfere with these atrocities, and never give a direction to your Military subordinates, which does not appear to have been conceived in the interest of Slavery rather than of Freedom.”

Horace Greeley (1811–1872) American politician and publisher

1860s, The Prayer of the Twenty Millions (1862)

Margaret Sanger photo
Anthony Burgess photo
William F. Buckley Jr. photo

“Government can't do anything for you except in proportion as it can do something to you.”

William F. Buckley Jr. (1925–2008) American conservative author and commentator

As quoted in "Broken Government: Where the right went wrong," CNN (2006-11-03).

Douglas Adams photo
Theo van Doesburg photo
Gregory Scott Paul photo
Nassim Nicholas Taleb photo
Sheldon L. Glashow photo
Rudolf Clausius photo

“Our culture has confined our imaginations with an uninspiring vision of God. He's been reduced to a manageable deity of consumable proportions.”

The Divine Commodity: Discovering A Faith Beyond Consumer Christianity (2009, Zondervan)

Halldór Laxness photo
Henry James photo
John Ramsay McCulloch photo
Harry Truman photo
Thomas Little Heath photo
Gerhard Richter photo
Thomas Henry Huxley photo

“Even in the important matter of cranial capacity, Men differ more widely from one another than they do from the Apes; while the lowest Apes differ as much, in proportion, from the highest, as the latter does from Man.”

Thomas Henry Huxley (1825–1895) English biologist and comparative anatomist

Source: 1860s, Evidence as to Man's Place in Nature (1863), Ch.2, p. 95

Henry M. Leland photo

“On the train I was going over the problem of Sixes versus Fours and the disturbing periodic vibrations with which the 'six-cylinder manufacturers were contending. I realized the emphasis our competitors were placing on the fact that six smaller cylinders, producing the same maximum power as four larger ones, would result in smaller individual impulses, and consequent smoother action.
I knew that we were having good results with well-balanced four-cylinder motors. I first reasoned that if six light cylinders gave the same maximum power and lighter impulses than the tour, then eight still smaller cylinders would give still lighter impulses than the six cylinders. I also reasoned that, because of the lighter weight, those eight cylinder pistons could be run at higher speeds than either sixes or fours. Furthermore I did not like the six crankshaft. If made small enough to be in proportion with those light pistons, the extra length might introduce those undesirable vibrations; if made heavy enough to avoid; if made heavy enough to avoid these periodic vibrations there was the wight problem contend with.
As I lay awake pondering these factors, the idea came to me that we were having good success with four-cylinder motors; we would surely have equally good results with blocks of lighter four cylinders and pistons. Why not make up those smaller blocks of lighter four cylinders and pistons, and put two of the blocks together at an angle and avoid that troublesome long crankshaft. The more I thought of this idea on that trip, the more convinced I became that it could be worked out.”

Henry M. Leland (1843–1932) American businessman

Source: Master of Precision: Henry M. Leland, 1966, p. 147; Leland talking about his idea for a V8 engine around 1913-14. Partly cited in: Alexander Richard Crabb (1969), Birth of a giant: the men and incidents that gave America the motorcar. p. 315

Jerome Corsi photo

“This was the first time WND had found a major intellectual leader behind the push to integrate North America suggesting that a crisis of 9-11 proportions might be just what was needed to advance the process toward establishing a North American Union and the amero.”

Jerome Corsi (1946) American conservative author

"North American Union leader says merger just crisis away", http://www.worldnetdaily.com/news/article.asp?ARTICLE_ID=53378 WorldNetDaily (2006-12-15)

C. Northcote Parkinson photo

“The time spent on any item of the agenda will be in inverse proportion to the sum involved.”

C. Northcote Parkinson (1909–1993) British naval historian

Source: Parkinson's Law: and Other Studies in Administration. (1957), p. 24. : Popularly known as Parkinson's Law of Triviality).

Mahmud of Ghazni photo

“The king, in his zeal to propagate the faith, now marched against the Hindoos of Nagrakote [Nagarkot Kangra], breaking down their idols and razing their temples. The fort, at that time denominated the Fort of Bheem, was closely invested by the Mahomedans, who had first laid waste the country around it with fire and sword.'…'In the year AH 402 (AD 1011), Mahmood resolved on the conquest of Tahnesur [Thanesar (Haryana)], in the kingdom of Hindoostan. It had reached the ears of the king that Tahnesur was held in the same veneration by idolaters, as Mecca by the faithful; that they had there set up a number of idols, the principal of which they called Jugsom, pretending that it had existed ever since the creation. Mahmood having reached Punjab, required, according to the subsisting treaty with Anundpal, that his army should not be molested on its march through his country…'The Raja's brother, with two thousand horse was also sent to meet the army, and to deliver the following message:- "My brother is the subject and tributary of the King, but he begs permission to acquaint his Majesty, that Tahnesur is the principal place of worship of the inhabitants of the country: that if it is required by the religion of Mahmood to subvert the religion of others, he has already acquitted himself of that duty, in the destruction of the temple of Nagrakote. But if he should be pleased to alter his resolution regarding Tahnesur, Anundpal promises that the amount of the revenues of that country shall be annually paid to Mahmood; that a sum shall also be paid to reimburse him for the expense of his expedition, besides which, on his own part he will present him with fifty elephants, and jewels to a considerable amount." Mahmood replied, "The religion of the faithful inculcates the following tenet: That in proportion as the tenets of the prophet are diffused, and his followers exert themselves in the subversion of idolatry, so shall be their reward in heaven; that, therefore, it behoved him, with the assistance of God, to root out the worship of idols from the face of all India. How then should he spare Tahnesur?"… This answer was communicated to the Raja of Dehly, who, resolving to oppose the invaders, sent messengers throughout Hindoostan to acquaint the other rajas that Mahmood, without provocation, was marching with a vast army to destroy Tahnesur, now under his immediate protection. He observed, that if a barrier was not expeditiously raised against this roaring torrent, the country of Hindoostan would be soon overwhelmed, and that it behoved them to unite their forces at Tahnesur, to avert the impending calamity….”

Mahmud of Ghazni (971–1030) Sultan of Ghazni

Tarikh-i-Firishta, translated by John Briggs under the title History of the Rise of the Mahomedan Power in India, first published in 1829, New Delhi Reprint 1981, Vol. I, pp. 27-37.
Quotes from Muslim medieval histories

Sung-Yoon Lee photo
Henry Adams photo
David Hume photo
Stanislav Grof photo
John Dee photo
Henri Matisse photo
Gino Severini photo
Gerhard Richter photo
Enoch Powell photo
Jared Diamond photo
Bram van Velde photo

“Because it's an adventure out of all proportion. You have to devote all your strength to it and it's never enough.”

Bram van Velde (1895–1981) Dutch painter

two short quotes, 24 August 1970; p. 78
1970's, Conversations with Samuel Beckett and Bram van Velde (1970 - 1972)

Adam Smith photo

“The value of money is in proportion to the quantity of the necessaries of life which it will purchase.”

Adam Smith (1723–1790) Scottish moral philosopher and political economist

Source: (1776), Book V, Chapter II, Part II, Article IV.

Mark Steyn photo

“Bisexuality is the proportional representation of sexuality in a world where most of us — straight or gay — operate a first-past-the-post system.”

Mark Steyn (1959) Canadian writer

" Sorry, but voters prefer straight choices http://www.telegraph.co.uk/opinion/main.jhtml?xml=/opinion/2006/01/31/do3102.xml", Daily Telegraph, 31 January 2006

Francis Marion Crawford photo
Harry V. Jaffa photo
Thomas Henry Huxley photo

“We know, that, in the individual man, consciousness grows from a dim glimmer to its full light, whether we consider the infant advancing in years, or the adult emerging from slumber and swoon. We know, further, that the lower animals possess, though less developed, that part of the brain which we have every reason to believe to be the organ of consciousness in man; and as, in other cases, function and organ are proportional, so we have a right to conclude it is with the brain; and that the brutes, though they may not possess our intensity of consciousness, and though, from the absence of language, they can have no trains of thoughts, but only trains of feelings, yet have a consciousness which, more or less distinctly, foreshadows our own. I confess that, in view of the struggle for existence which goes on in the animal world, and of the frightful quantity of pain with which it must be accompanied, I should be glad if the probabilities were in favour of Descartes' hypothesis; but, on the other hand, considering the terrible practical consequences to domestic animals which might ensue from any error on our part, it is as well to err on the right side, if we err at all, and deal with them as weaker brethren, who are bound, like the rest of us, to pay their toll for living, and suffer what is needful for the general good.”

Thomas Henry Huxley (1825–1895) English biologist and comparative anatomist

1870s, On the Hypothesis that Animals are Automata, and Its History (1874)

Jerry Coyne photo
Gregory Benford photo
Henrietta Swan Leavitt photo

“The range of H 1255 is only four tenths of a magnitude, and on account of its brightness it is difficult to observe on all plates except those taken with the 1-inch Cooke lens. It seemed necessary, therefore, to take unusual precautions in order to secure accurate observations, and to give each one its full weight. Accordingly, one hundred and thirty six photographs were selected, including nearly all of those taken with the Cooke lens, and also those taken with the 8 inch Bache Telescope on which the variable was certainly faint. Four independent estimates of brightness were made on each plate, and means were taken, thus reducing the probable error one half. The phase was computed for each observation, thus covering all parts of the light curve. …H 1255 and H 1303 differ from the other variables in a marked degree as in each case the duration of the phase of minimum is very long in proportion to the length of the period. This fact led to considerable difficulty in determining their periods as they were apparently at their minimum brightness for some time before and after the actual minima occurred. In H 1255, the change in brightness is obviously continuous throughout the period, although it is much more rapid near minimum than near maximum. This is clearly seen in Plate IV, Figs. 5 and 6.”

Henrietta Swan Leavitt (1868–1921) astronomer

"Ten Variable Stars of the Algol Type" http://books.google.com/books?id=UkdWAAAAYAAJ&pg=PA87 (1908) Annals of the Astronomical Observatory of Harvard College Vol.60. No.5

Adam Smith photo
Isaac Barrow photo
Ossip Zadkine photo
Robert Gascoyne-Cecil, 3rd Marquess of Salisbury photo
Gerhard Richter photo

“I believe I am looking for rightness. My work has so much to do with reality that I wanted to have a corresponding rightness. That excludes painting in imitation. In nature everything is always right: the structure is right, the proportions are good, the colours fit the forms. If you imitate that in painting, it becomes false.”

Gerhard Richter (1932) German visual artist, born 1932

Interview with Anna Tilroe, 1987; as cited on collected quotes on the website of Gerhard Richter: on 'Abstract paintings' https://www.gerhard-richter.com/en/quotes/subjects-2/abstract-paintings-7
1980's

Dana Gioia photo
Thomas Carlyle photo
Charles Lyell photo
El Greco photo

“Anyway, I would not be happy to see a beautiful, well-proportioned woman, no matter from which point of view, however extravagant, not only lose her beauty in order to, I would say, increase in size according to the law of vision, but no longer appear beautiful, and, in fact, become monstrous.”

El Greco (1541–1614) Greek painter, sculptor and architect

Quote from the marginalia, which El Greco inscribed in his copy of Daniele Barbaro's translation of Vitruvius' De architectura; as quoted by Liane Lefaivre and Alexander Tzonis, The Emergence of Modern Architecture: A Documentary History from 1000 to 1810 https://books.google.com/books?id=4xB9k7-Neb8C&pg=PT184; Routledge, New York, 2004) p. 165

Vitruvius photo
George Gershwin photo
Simon Kuznets photo
Charles A. Beard photo

“I present, for what it is worth, and may prove to be worth, the following bill of axioms or aphorisms on public administration, as fitting this important occasion.
# The continuous and fairly efficient discharge of certain functions by government, central and local, is a necessary condition for the existence of any great society.
# As a society becomes more complicated, as its division of labor ramifies more widely, as its commerce extends, as technology takes the place of handicrafts and local self-sufficiency, the functions of government increase in number and in their vital relationships to the fortunes of society and individuals.
# Any government in such a complicated society, consequently any such society itself, is strong in proportion to its capacity to administer the functions that are brought into being.
# Legislation respecting these functions, difficult as it is, is relatively easy as compared with the enforcement of legislation, that is, the effective discharge of these functions in their most minute ramifications and for the public welfare.
# When a form of government, such as ours, provides for legal changes, by the process of discussion and open decision, to fit social changes, then effective and wise administration becomes the central prerequisite for the perdurance of government and society — to use a metaphor, becomes a foundation of government as a going concern.
# Unless the members of an administrative system are drawn from various classes and regions, unless careers are open in it to talents, unless the way is prepared by an appropriate scheme of general education, unless public officials are subjected to internal and external criticism of a constructive nature, then the public personnel will become a bureaucracy dangerous to society and to popular government.
# Unless, as David Lilienthal has recently pointed out in an address on the Tennessee Valley Authority, an administrative system is so constructed and operated as to keep alive local and individual responsibilities, it is likely to destroy the basic well-springs of activity, hope, and enthusiasm necessary to popular government and to the following of a democratic civilization.”

Charles A. Beard (1874–1948) American historian

Administration, A Foundation of Government (1940)

“Suppose then I want to give myself a little training in the art of reasoning; suppose I want to get out of the region of conjecture and probability, free myself from the difficult task of weighing evidence, and putting instances together to arrive at general propositions, and simply desire to know how to deal with my general propositions when I get them, and how to deduce right inferences from them; it is clear that I shall obtain this sort of discipline best in those departments of thought in which the first principles are unquestionably true. For in all 59 our thinking, if we come to erroneous conclusions, we come to them either by accepting false premises to start with—in which case our reasoning, however good, will not save us from error; or by reasoning badly, in which case the data we start from may be perfectly sound, and yet our conclusions may be false. But in the mathematical or pure sciences,—geometry, arithmetic, algebra, trigonometry, the calculus of variations or of curves,—we know at least that there is not, and cannot be, error in our first principles, and we may therefore fasten our whole attention upon the processes. As mere exercises in logic, therefore, these sciences, based as they all are on primary truths relating to space and number, have always been supposed to furnish the most exact discipline. When Plato wrote over the portal of his school. “Let no one ignorant of geometry enter here,” he did not mean that questions relating to lines and surfaces would be discussed by his disciples. On the contrary, the topics to which he directed their attention were some of the deepest problems,—social, political, moral,—on which the mind could exercise itself. Plato and his followers tried to think out together conclusions respecting the being, the duty, and the destiny of man, and the relation in which he stood to the gods and to the unseen world. What had geometry to do with these things? Simply this: That a man whose mind has not undergone a rigorous training in systematic thinking, and in the art of drawing legitimate inferences from premises, was unfitted to enter on the discussion of these high topics; and that the sort of logical discipline which he needed was most likely to be obtained from geometry—the only mathematical science which in Plato’s time had been formulated and reduced to a system. And we in this country [England] have long acted on the same principle. Our future lawyers, clergy, and statesmen are expected at the University to learn a good deal about curves, and angles, and numbers and proportions; not because these subjects have the smallest relation to the needs of their lives, but because in the very act of learning them they are likely to acquire that habit of steadfast and accurate thinking, which is indispensable to success in all the pursuits of life.”

Joshua Girling Fitch (1824–1903) British educationalist

Source: Lectures on Teaching, (1906), pp. 291-292

Paul Signac photo
Ernest Hemingway photo

“Fear of death increases in exact proportion to increase in wealth: Hemingstein's Law on the Dynamics of Dying.”

Ernest Hemingway (1899–1961) American author and journalist

Pt. 2, Ch. 7
Papa Hemingway (1966)

François Viète photo

“On symbolic use of equalities and proportions. Chapter II.
The analytical method accepts as proven the most famous [ as known from Euclid ] symbolic use of equalities and proportions that are found in items such as:
1. The whole is equal to the sum of its parts.
2. Quantities being equal to the same quantity have equality between themselves. [a = c & b = c => a = b]
3. If equal quantities are added to equal quantities the resulting sums are equal.
4. If equals are subtracted from equal quantities the remains are equal.
5. If equal equal amounts are multiplied by equal amounts the products are equal.
6. If equal amounts are divided by equal amounts, the quotients are equal.
7. If the quantities are in direct proportion so also are they are in inverse and alternate proportion. [a:b::c:d=>b:a::d:c & a:c::b:d]
8. If the quantities in the same proportion are added likewise to amounts in the same proportion, the sums are in proportion. [a:b::c:d => (a+c):(b+d)::c:d]
9. If the quantities in the same proportion are subtracted likewise from amounts in the same proportion, the differences are in proportion. [a:b::c:d => (a-c):(b-d)::c:d]
10. If proportional quantities are multiplied by proportional quantities the products are in proportion. [a:b::c:d & e:f::g:h => ae:bf::cg:dh]
11. If proportional quantities are divided by proportional quantities the quotients are in proportion. [a:b::c:d & e:f::g:h => a/e:b/f::c/g:d/h]
12. A common multiplier or divisor does not change an equality nor a proportion. [a:b::ka:kb & a:b::(a/k):(b/k)]
13. The product of different parts of the same number is equal to the product of the sum of these parts by the same number. [ka + kb = k(a+b)]
14. The result of successive multiplications or divisions of a magnitude by several others is the same regardless of the sequential order of quantities multiplied times or divided into that magnitude.
But the masterful symbolic use of equalities and proportions which the analyst may apply any time is the following:
15. If we have three or four magnitudes and the product of the extremes is equal to the product means, they are in proportion. [ad=bc => a:b::c:d OR ac=b2 => a:b::b:c]
And conversely
10. If we have three or four magnitudes and the first is to the second as the second or the third is to the last, the product of the extremes is equal to that of means. [a:b::c:d => ad=bc OR a:b::b:c => ac=b2]
We can call a proportion the establishment of an equality [equation] and an equality [equation] the resolution of a proportion.”

François Viète (1540–1603) French mathematician

From Frédéric Louis Ritter's French Tr. Introduction à l'art Analytique (1868) utilizing Google translate with reference to English translation in Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (1968) Appendix
In artem analyticem Isagoge (1591)

Barbara Hepworth photo
François de La Rochefoucauld photo
Paul R. Halmos photo
Ignatius Sancho photo
Gardiner Spring photo
Samuel Beckett photo
Thomas Little Heath photo
Samuel Johnson photo

“Why, Sir, it is difficult to settle the proportion of iniquity between them.”

Samuel Johnson (1709–1784) English writer

Feb. 15, 1766, p. 145
Said of Rousseau and Voltaire
Life of Samuel Johnson (1791), Vol II

Charles Lyell photo

“But exactly in proportion as the completeness of the record and our knowledge of it are overrated, in that same degree are many progressionists unconscious of the goal towards which they are drifting.”

Charles Lyell (1797–1875) British lawyer and geologist

Source: The Geological Evidences of the Antiquity of Man (1863), Ch.20, p. 406
Context: No one can believe in transmutation who is not profoundly convinced that all we know in paleontology is as nothing compared to what we have yet to learn, and they who regard the record as so fragmentary, and our acquaintance with the fragments which are extant as so rudimentary, are apt to be astounded at the confidence placed by the progressionists in data which must be defective in the extreme. But exactly in proportion as the completeness of the record and our knowledge of it are overrated, in that same degree are many progressionists unconscious of the goal towards which they are drifting. Their faith in the fullness of the annals leads them to regard all breaks in the series of organic existence, or in the sequence of the fossiliferous rocks, as proofs of original chasms and leaps in the course of nature, signs of the intermittent action of the creational force, or of catastrophes which devastated the habitable surface; and they are therefore fearless of discovering any continuity of plan (except that which must have existed in the Divine mind) which would imply a material connection between the outgoing organisms and the incoming ones.

Ulysses S. Grant photo

“I repeat that the adoption of the fifteenth amendment to the Constitution completes the greatest civil change and constitutes the most important event that has occurred since the nation came into life. The change will be beneficial in proportion to the heed that is given to the urgent recommendations of Washington.”

Ulysses S. Grant (1822–1885) 18th President of the United States

1870s, Message to the Senate and House of Representatives (1870)
Context: In his first annual message to Congress the same views are forcibly presented, and are again urged in his eighth message. I repeat that the adoption of the fifteenth amendment to the Constitution completes the greatest civil change and constitutes the most important event that has occurred since the nation came into life. The change will be beneficial in proportion to the heed that is given to the urgent recommendations of Washington. If these recommendations were important then, with a population of but a few millions, how much more important now, with a population of 40,000,000, and increasing in a rapid ratio.