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 MorganRelated topics
add approach assign beginning continuing deficit different equal far find following four halve last law limit mean near nor now passing quantity result say see series several take term three truth unit valueRelated quotes
As the Bajauris were rebels and at enmity with the people of Islam, and as, by reason of the heathenish and hostile customs prevailing in their midst, the very name of Islam was rooted out from their tribe, they were put to general massacre and their wives and children were made captive. At a guess more than 3000 men went to their death; as the fight did not reach to the eastern side of the fort, a few got away there. The fort taken, we entered and inspected it. On the walls, in houses, streets and alleys, the dead lay, in what numbers! Comers and goers to and fro were passing over the bodies... With mind easy about the important affairs of the Bajaur fort, we marched, on Tuesday the 9th of Muharram, one kuroh (2m) down the dale of Bajaur and ordered that a tower of heads should be set up on the rising ground. On Wednesday the 10th of Muharram, we rode out to visit the Bajaur fort. There was a wine-party in Khawaja Kalan's house, several goat-skins of wine having been brought.
Babur
Ever since [ Copernicus], writers eager to drive the lesson home have urged us [...] to abandon all sentimental egoism, and to see ourselves objectively in the true perspective of time and space. What precisely does this mean? In a full 'main feature' film, recapitulating faithfully the complete history of the universe, the rise of human beings from the first beginnings of man to the achievements of the twentieth century would flash by in a single second. Alternatively, if we decided to examine the universe objectively in the sense of paying equal attention to portions of equal mass, this would result in a lifelong preoccupation with interstellar dust, relieved only at brief intervals by a survey of incandescent masses of hydrogen - not in a thousand million lifetimes would the turn come to give man even a second's notice. It goes without saying that no one - scientists included - looks at the universe in this way, whatever lip-service is given to 'objectivity.
Michael Polanyi
When Mrs. Casaubon was announced he started up as from an electric shock, and felt a tingling at his fingerends. Any one observing him would have seen a change in his complexion, in the adjustment of his facial muscles, in the vividness of his glance, which might have made them imagine that every molecule in his body had passed the message of a magic touch. And so it had. For effective magic is transcendent nature; and who shall measure the subtlety of those touches which convey the quality of soul as well as body, and make a man's passion for one woman differ from his passion for another as joy in the morning light over valley and river and white mountain-top differs from joy among Chinese lanterns and glass panels? Will, too, was made of very impressible stuff. The bow of a violin drawn near him cleverly, would at one stroke change the aspect of the world for him, and his point of view shifted- as easily as his mood. Dorothea's entrance was the freshness of morning.
George Eliot
Chemical signs ought to be letters, for the greater facility of writing, and not to disfigure a printed book ... I shall take therefore for the chemical sign, the initial letter of the Latin name of each elementary substance: but as several have the same initial letter, I shall distinguish them in the following manner:- 1. In the class which I shall call metalloids, I shall employ the initial letter only, even when this letter is common to the metalloid and to some metal. 2. In the class of metals, I shall distinguish those that have the same initials with another metal, or a metalloid, by writing the first two letters of the word. 3. If the first two letters be common to two metals, I shall, in that case, add to the initial letter the first consonant which they have not in common: for example, S = sulphur, Si = silicium, St = stibium (antimony), Sn = stannum (tin), C = carbonicum, Co = colbaltum (colbalt), Cu = cuprum (copper), O = oxygen, Os = osmium, &c.
Jons Jacob Berzelius
Seeing several of these paintings [his series paintings 'Hommage to the square', Josef Albers painted in 1963-64] next to each other makes it obvious that each painting is an instrumentation on its own.
This means that they all are of different palettes, and, therefore, so to speak, of different climates.
Choice of the colors used, as well as their order, is aimed at an interaction – influencing and changing each other forth and back.
Thus, character and feeling alter from painting to painting without any additional 'handwriting' or, so-called, texture.
Though the underlying symmetrical and quasi-concentric order of squares remains the same in all paintings – in proportion and placement – these same squares group or single themselves, connect and separate in many different ways
In consequence, they move forth and back, in and out, and grow up and down and near and far, as well as enlarged and diminished. All this to proclaim color autonomy as a means of plastic organization.
Josef Albers
