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Morris Kline
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June 15 - June 28, 2019
KOYRE, ALEXANDRE: From the Closed World to the Infinite Universe, Chaps. 1 through 4, The Johns Hopkins Press, Baltimore, 1957.
No nature except an extraordinary one could ever easily formulate a theory. PLATO
The most significant mathematical creation of that century and the one which proved to be most fruitful for the modern development of mathematics and science is the calculus. Like Euclidean geometry, it is a landmark of human thought.
Suppose one knows the acceleration of a moving body at each instant of time. How does one find the speed and distance traveled at any instant? When the acceleration is constant, one can multiply the acceleration by the time of travel and obtain the speed acquired, but this procedure does not yield correct results when the acceleration is variable.
The third major problem was that of finding the maximum and minimum values of a function. When a bullet is fired straight up, one may wish to know how high it will go.
Our great concern in this age for faster transportation and communication is a concern with rate of change.
Yeah buddy... + Einstein's bounds defined essentially by the speed of light.
Circulation of the blood in one’s body means quantity of blood per unit time passing through a specific artery or a collection of arteries, and here, too, it is rate of change which counts. The rate of physiological activity, that is the metabolic rate, measured in terms of the rate of consumption of oxygen per second, is a rate of change. To sum up: the rate of change of one variable with respect to another is a physically useful quantity in many situations.
The rates of change which are of interest to laymen and even to many specialists are average rates.
This book is insulting your stupidity? Or just highlighting its bounds? :-)
If a person traveling in an automobile strikes a tree, it is not his average rate of speed for the time he has traveled from the starting point to the tree that matters. It is his speed at the instant of collision which determines whether or not he will survive the accident. Here we have an instantaneous speed or instantaneous rate of change of distance with respect to time.
Now I got it!
This number is called the limit of the set of average speeds. We should note that the instantaneous speed is not defined as the quotient of distance and time. Rather it is the limit approached by average speeds as the intervals over which these average speeds are computed approach zero.
The process we have just examined, called the method of increments, is basic in the calculus.
To appreciate what the limit process achieves we might consider an analogy. Suppose that a marksman seeks to hit a particular spot on a target. Even if he is a good shot, he is not likely to hit the given spot squarely, but will hit all around it and indeed come close. A bystander observing the location of the hits will readily determine the exact spot at which the marksman is aiming, by noting the concentration of the hits. This process of inferring the precise location at which the marksman is aiming is analogous to determining the instantaneous speed from a knowledge of the average speeds.
...more
More laws are vain where less will serve. ROBERT HOOKE
We shall now present some examples of the usefulness of integration in physical problems. Galileo had found that all objects falling to earth from points near the surface of the earth possess the same acceleration, namely 32 ft/sec2. This acceleration is constant; that is, it is the same at each instant of the fall. Now the acceleration at any one instant is the instantaneous rate of change of velocity with respect to time.
The answer is that Euclidean geometry is adequate only to treat figures bounded by straight line segments and by circles. This limitation is inherent in the subject.
Though it is by no means evident, the calculus proves to be the very mathematical tool which enables us to calculate the lengths of curves, the areas bounded by curves, and the volumes bounded by surfaces.
This is the velocity necessary to escape from our trouble-infested earth.
\o/
The symbol is an abbreviated S and is intended to denote that we are dealing with the limit of a sum. For areas this sum is a sum of rectangles.
The volume of sphere approximated by a sum of cubical volumes.
This is a striking plot.
As a matter of history, Newton solved the very problem we have been considering and proved that the earth attracts a small mass as if the entire mass of the earth were concentrated at its center. In other words, although the earth’s mass is distributed over a large region, it happens to be true that a spherical mass attracting a small mass can be treated as if its mass were concentrated at its center.
Yet they were sure that their ideas were sound because they made sense physically and intuitively, and because the methods gave results which agreed with observations and experiments. Both gave many versions in the attempt to hit upon the precise concepts, but neither was successful.
A more drastic opinion was offered by the mathematician Michel Rolle (1652–1719). He taught that the calculus was a collection of ingenious fallacies.
Voltaire called the calculus “the art of numbering and measuring exactly a Thing whose existence cannot be conceived.”
This history of the development of the calculus is significant because it illustrates the way in which mathematics progresses. Ideas are first grasped intuitively and extensively explored before they become fully clarified and precisely formulated even in the minds of the best mathematicians.
Yet they not only applied them to physical problems, but used the calculus to evolve new branches of mathematics, differential equations, differential geometry, the calculus of variations, and others. They had the confidence to proceed so far along uncertain ground because their methods yielded correct physical results.
All of these, together with the calculus, constitute a division of mathematics, called analysis, which is considerably more extensive than algebra or geometry. But some glimpse of this most significant mathematical creation of modern times may in itself afford a rich enough reward.
New ideas, always far more effective in determining the course of cultures than wars or political events, began to work their influence, and the growing use of books permitted the leaders to reach large groups of people.
Interesting observation.
The eighteenth century has been called the Age of Reason. It was not the first period of history in which reason played a dominant role, but it was the first in which the intellectual elite emboldened by some successes in physical science dared to apply reason to the reconstruction of an entire civilization. The spirit and outlook of the leaders are indicated by their reference to their own age as the Enlightenment.
COLERUS, EGMONT: From Simple Numbers to the Calculus, Chaps. 24 through 34, Wm. Heinemann Ltd., London, 1954.
KASNER, EDWARD and JAMES R. NEWMAN: Mathematics and the Imagination, Chap. 9, Simon and Schuster, Inc., New York, 1940.
SAWYER, W. W.: What Is Calculus About? Random House, New York, 1961.
SINGH, JAGIT: Great Ideas of Modern Mathematics: Their Nature and Use, Chap. 3, Dover Publications, Inc., New York, 1959.
WIENER, PHILIP P. AND AARON NOLAND: Roots of Scientific Thought, pp. 412–442, Basic Books, Inc., New York, 1957.
WIGHTMAN, WM. P. D.: The Growth of Scientific Ideas, Chap. 9, Yale University Press, New Haven, 1953.
All the effects of nature are only mathematical results of a small number of immutable laws. P. S. LAPLACE
In the seventeenth century one of the most pressing problems of the times was time itself.
The advantage of radians over degrees is simply that it is a more convenient unit. Since an angle of 90° is of the same size as an angle of 1.57 radians, we now have to deal only with 1.57 instead of 90 units.
“There is nothing strange in the circle being the origin of any and every marvel.”
By Galileo’s principle these two motions are independent of each other.
Motion appears in many aspects—but there are two obvious kinds, one which appears in astronomy and another which is the echo of that. As the eyes are made for astronomy so are the ears made for the motion which produces harmony: and thus we have two sister sciences, as the Pythagoreans teach, and we assent. PLATO
In which book this was written?
But by a stroke of good luck mathematics provided the very theorem which gives us remarkable insight into all complex sounds. The stroke of good luck was the mathematician Joseph Fourier (1768–1830).
Fourier was the son of a French tailor. While attending a military school he became intrigued with mathematics.
His chief contribution, a book entitled The Analytical Theory of Heat (1822), is one of the great classics of mathematics.
Fourier’s celebrated theorem says that any periodic function is a sum of simple sine functions of the form D sin 2πft. Moreover, the frequencies of these component functions are all integral multiples of one frequency. To illustrate the significance of this theorem, let us suppose that y is a periodic function of t.
But Fourier’s theorem says that every such function is a sum of simple sine functions of the type illustrated in (6). Each simple sine function corresponds to a simple sound such as is given off by a tuning fork. Hence one arrives at the important conclusion that every musical sound is a sum of simple sounds.
The assertion that every musical sound is no more than a combination of simple sounds is so surprising that, although it is backed by unassailable mathematics, one wishes to see it confirmed by experimental evidence. Such evidence is available.
The most significant revolutions in this world are not political. Political revolutions hardly change the daily life of man or, if they do, exert a short-term effect which may even be reversed by subsequent revolutions. The significant upheavals are caused by new ideas; these far more effectively, powerfully, and lastingly alter the lives of civilized human beings.
This Kline guy is damn entertaining.
The two concepts which have most profoundly revolutionized our intellectual development since the nineteenth century are evolution and non-Euclidean geometry.
Gauss’s scientific interests, like those of Archimedes and Newton, were unbelievably broad. He was, for example, a great inventor.
Though his greatest achievements were in mathematics and he is therefore most often described as a mathematician, it would be more appropriate to call him a student of nature.
