A philosophical essay on probabilities

A philosophical essay on probabilities

Then the character Δ indicates a finite difference of the primitive function in the case where the index varies by unity; and the n th power of this character placed before the primitive function will indicate the finite n th difference of this function. Applying the same method to some other objects of its knowledge, it has succeeded in referring to general laws observed phenomena and in foreseeing those which given circumstances ought to produce. This is the place to define the word extraordinary. The laws which the different kinds of animals follow in this regard seem to me worthy of the attention of naturalists. BIG EssayWriter. Such is the general law of the probability of results indicated by a great number of observations. The knowledge of the laws of the system of the world acquired in the interval had dissipated the fears begotten by the ignorance of the true relationship of man to the universe; and Halley, a philosophical essay on probabilities, having recognized the identity of this comet with those of the years, andannounced its next return for the end a philosophical essay on probabilities the year or the beginning of the year

TABLE OF CONTENTS.

Theory of probabilities is a branch of mathematics that studies the regularities of random phenomena such as random events, random variables, their properties and operations on them. This study appeared in Middle Ages and its aim was in analysis of games of chance. French scientist Pierre Simon Laplace made great impact in the studying of theory of probabilities. A Philosophical Essay on Probabilities is his book, which is dedicated to this branch of science. Achievements of Laplace and content of his book about the theories of probabilities will be discussed in this outline. Team of smartwriters. org offers custom writing of papers. Today we are talking about the Pierre Simon Laplace.

Birth of great scientist happened in Normandy, which is one of the French regions. His life was surprisingly long because he died at the age of 78 years old. However, he did not wasted his time for nothing. Pierre Simon Laplace was author of books and treatises:. He put forward it knowing the theory of Buffon but not knowing the theory of Kant The planets were born on the border of the nebula by condensation of the chilled vapor in the equatorial plane and because of cooling nebula gradually shrank, spinning faster and faster, a philosophical essay on probabilities.

Centrifugal force a philosophical essay on probabilities equal to the power of gravity and numerous rings were produced, which divided on new rings, with the help of condensing. These rings first created gas planets and after that central clot turned to Sun. In this theory, formation of all bodies in the Solar system, such as Sun, planet and satellites happened at the same time. There are enough information about the figures of planets and tides, theory of gravitation and history of astronomy in this book as about the rings of Saturn and atmospheres of planets. He offers method for calculating the orbits of celestial bodiesreplacing it in with a new way, in his first paper about celestial mechanics, which he did all his life.

This meant that Solar system is apparently stable. Inhe created a dynamic theory of tides, a philosophical essay on probabilities. On 24 JuneLaplace together with the chemist Lavoisier first synthesized water, by combining oxygen and hydrogen. It can be found many latest discoveries of the theory of probabilities, made by other mathematicians in this paper. It covers some of the issues of the game theory, theorem of Bernoulli and its connection with the integral of the normal distribution and so on. As it mentioned earlier in this outline, theory of probabilities became widespread as study in Middle Ages.

It was first try to do mathematical analysis of games of chance such as craps and roulette. Initially, its basic concepts did not have strictly mathematical type, they could be treated as some empirical facts as to the properties of real events, and they were formulated in a visual representation. Jacob Bernoulli introduced an important contribution to the theory of probabilities. However, mainly Pierre Simon Laplace proved first limited theorem. Albert Einstein invented theorem of relativity later. You can read about him and his papers in essay on Albert Einstein.

We know famous equation of Laplace, which concerned to partial derivatives. Theory of probabilities arose as a science from the persuasion, which stated that deterministic laws were the heart of mass random events. This theory studies given patterns. Test is called the implementation of certain set of conditions, which can be played an unlimited number of times. Set of conditions includes random factors. For example, coin flip a philosophical essay on probabilities be as a philosophical essay on probabilities. The result of the test is event. They are divided on:. For example, let us look on examples of events, looking at this occurrence. When we toss the cube, the impossible event of this action will be its fall on the edge, random event will be dropout of any edge, and equally probable event — its fall on the even edge.

Theory of probability learns all these cases. Interesting paper of scientist was published in and served as continuation of previous treatise, which had name Analytical Theory of Probabilities. In the book, which was published ina philosophical essay on probabilities, Laplace examined many effective results of statistics. Therefore, this book revealed scientific part of theory of probabilities. Book, which was published inhad content that is more interesting for us. Let us examine it. We are talking about Philosophical Essay About Probabilities. This book is written about inductive method of thinking, which uses the theory of probabilities during it.

An interesting fact, that Pierre Simon Laplace was mainly physicist and had no relation with philosophy. He was closely linked with physics and astronomy, as it is described in first paragraph. However, his impact in philosophy is huge. How could he make such impact? The reason is in that A philosophical essay on probabilities was the representative of idea of determinism. This idea was central in the philosophy. He considered, that all is predictable in our world. As we already know, he worked hardly in these spheres of science. These branches of science showed that all in our a philosophical essay on probabilities is obeyed to the certain laws.

Another certain question appears about the link of determinism and theory of probabilities. Theoretical opportunity exists in order to count any event, which is based from the previous event. All these principles in the form of theory of probabilities are described in this paper. It is possible to think, that he symbolizes the spirit of a philosophical essay on probabilities in a stronger form. Pierre Simon Laplace can be rightly considered as not only great physicist and astronomer, but also great philosopher. He managed to link popular at that time theory of probabilities with idea of determinism. Laplace was first person, who managed to achieve scientific explanation of human fatalism.

Scientist could not suspect that he did great impact in philosophy of that time. These ideas were marked in his philosophical treatise. If you like this essay, you can place order with your topic of paper on our site. We will be glad to help you. Services of our site mean writing of any your assignment. There are no difficult topics for us. Blog Our Latest News Home Blog. Philosophical Essay On Probabilities as Masterpiece Of Time. Main Achievements of Pierre Simon Laplace Team of smartwriters. Pierre Simon Laplace was author of books and treatises: The Presentation of The World System; About The Cause Of Universal Gravitation And The Secular Inequalities Of The Planets, That Depend on It; Analytical Theory of Probabilities; Celestial Mechanics.

Briefly About The Theory of Probabilities As it mentioned earlier in this outline, theory of probabilities became widespread as study in Middle Ages. A philosophical essay on probabilities are divided on: reliable always occur in the result of test ; impossible never happen ; equally probable have equal opportunities to occurless likely or more likely; random may or may not occur in the result of test For example, let us look on examples of events, looking at this occurrence. Philosophical Treatise Of Pierre Simon Laplace Interesting paper of scientist was published in and served as continuation of previous treatise, which had name Analytical Theory of Probabilities. Conclusion Pierre Simon Laplace can be rightly considered as not only great physicist and astronomer, a philosophical essay on probabilities, but also great philosopher.

Calculate Your Price. Popular Categories How to Popular topics Services. Recent posts Do My Coursework at SmartWriters. Assignments Services UK are Working for You! Coursework Writing Service: Best Value for Money What Is Friendship Essay: Who Is A Friend?

my friend essay

The order or the degree of this equation is the difference of rank of its two extreme terms. We are able by its use to determine successively the terms of the series and to continue it indefinitely; but for that it is necessary to know a number of terms of the series equal to the degree of the equation. These terms are the arbitrary constants of the expression of the general term of the series or of the integral of the equation of differences. Let us imagine now below the terms of the preceding series a second series of terms arranged horizontally; let us imagine again below the terms of the second series a third horizontal series, and so on to infinity; and let us suppose the terms of all these series connected by a general equation among several consecutive terms, taken as much in the horizontal as in the vertical sense, and the numbers which indicate their rank in the two senses.

This equation is called the equation of partial finite differences by two indices. Let us imagine in the same way below the plan of the preceding series a second plan of similar series, whose terms should be placed respectively below those of the first plan; let us imagine again below this second plan a third plan of similar series, and so on to infinity; let us suppose all the terms of these series connected by an equation among several consecutive terms taken in the sense of length, width, and depth, and the three numbers which indicate their rank in these three senses. This equation I call the equation of partial finite differences by three indices. Finally, considering the matter in an abstract way {34} and independently of the dimensions of space, let us imagine generally a system of magnitudes, which should be functions of a certain number of indices, and let us suppose among these magnitudes, their relative differences to these indices and the indices themselves, as many equations as there are magnitudes; these equations will be partial finite differences by a certain number of indices.

We are able by their use to determine successively these magnitudes. But in the same manner as the equation by a single index requires for it that we know a certain number of terms of the series, so the equation by two indices requires that we know one or several lines of series whose general terms should be expressed each by an arbitrary function of one of the indices. Similarly the equation by three indices requires that we know one or several plans of series, the general terms of which should be expressed each by an arbitrary function of two indices, and so on. In all these cases we shall be able by successive eliminations to determine a certain term of the series.

But all the equations among which we eliminate being comprised in the same system of equations, all the expressions of the successive terms which we obtain by these eliminations ought to be comprised in one general expression, a function of the indices which determine the rank of the term. This expression is the integral of the proposed equation of differences, and the search for it is the object of integral calculus. Taylor is the first who in his work entitled Metodus incrementorum has considered linear equations of finite differences. He gives the manner of integrating those {35} of the first order with a coefficient and a last term, functions of the index.

In truth the relations of the terms of the arithmetical and geometrical progressions which have always been taken into consideration are the simplest cases of linear equations of differences; but they had not been considered from this point of view. It was one of those which, attaching themselves to general theories, lead to these theories and are consequently veritable discoveries. About the same time Moivre was considering under the name of recurring series the equations of finite differences of a certain order having a constant coefficient.

He succeeded in integrating them in a very ingenious manner. As it is always interesting to follow the progress of inventors, I shall expound the method of Moivre by applying it to a recurring series whose relation among three consecutive terms is given. First he considers the relation among the consecutive terms of a geometrical progression or the equation of two terms which expresses it. Referring it to terms less than unity, he multiplies it in this state by a constant factor and subtracts the product from the first equation. Thus he obtains an equation among three consecutive terms of the geometrical progression. Moivre considers next a second progression whose ratio of terms is the same factor which he has just used. He diminishes similarly by unity the index of the terms of the equation of this new progression.

In this condition he multiplies it by the ratio of the terms of the first progression, and he subtracts the product from the equation of the second progression, which gives him among three consecutive terms of this progression a relation entirely {36} similar to that which he has found for the first progression. Then he observes that if one adds term by term the two progressions, the same ratio exists among any three of these consecutive terms. He compares the coefficients of this ratio to those of the relation of the terms of the proposed recurrent series, and he finds for determining the ratios of the two geometrical progressions an equation of the second degree, whose roots are these ratios. Thus Moivre decomposes the recurrent series into two geometrical progressions, each multiplied by an arbitrary constant which he determines by means of the first two terms of the recurrent series.

This ingenious process is in fact the one that d'Alembert has since employed for the integration of linear equations of infinitely small differences with constant coefficients, and Lagrange has transformed into similar equations of finite differences. Finally, I have considered the linear equations of partial finite differences, first under the name of recurro-recurrent series and afterwards under their own name. The most general and simplest manner of integrating all these equations appears to me that which I have based upon the consideration of discriminant functions, the idea of which is here given. If we conceive a function V of a variable t developed according to the powers of this variable, the coefficient of any one of these powers will be a function of the exponent or index of this power, which index I shall call x.

V is what I call the discriminant function of this coefficient or of the function of the index. Now if we multiply the series of the development of V by a function of the same variable, such, for example, {37} as unity plus two times this variable, the product will be a new discriminant function in which the coefficient of the power x of the variable t will be equal to the coefficient of the same power in V plus twice the coefficient of the power less unity. Thus the function of the index x in the product will be equal to the function of the index x in V plus twice the same function in which the index is diminished by unity. This function of the index x is thus a derivative of the function of the same index in the development of V , a function which I shall call the primitive function of the index.

Let us designate the derivative function by the letter Alembert placed before the primitive function. The derivation indicated by this letter will depend upon the multiplier of V , which we will call T and which we will suppose developed like V by the ratio to the powers of the variable t. If we multiply anew by T the product of V by T , which is equivalent to multiplying V by T² , we shall form a third discriminant function, in which the coefficient of the x th power of t will be a derivative similar to the corresponding coefficient of the preceding product; it may be expressed by the same character δ placed before the preceding derivative, and then this character will be written twice before the primitive function of x.

But in place of writing it thus twice we give it 2 for an exponent. Continuing thus, we see generally that if we multiply V by the n th power of T , we shall have the coefficient of the x th power of t in the product of V by the n th power of T by placing before the primitive function the character δ with n for an exponent. Let us suppose, for example, that T be unity divided {38} by t ; then in the product of V by T the coefficient of the x th power of t will be the coefficient of the power greater by unity in V ; this coefficient in the product of V by the n th power of T will then be the primitive function in which x is augmented by n units.

Let us consider now a new function Z of t , developed like V and T according to the powers of t ; let us designate by the character Δ placed before the primitive function the coefficient of the x th power of t in the product of V by Z ; this coefficient in the product of V by the n th power of Z will be expressed by the character Δ affected by the exponent n and placed before the primitive function of x. It will be then the finite difference of the primitive function of the index x. Then the character Δ indicates a finite difference of the primitive function in the case where the index varies by unity; and the n th power of this character placed before the primitive function will indicate the finite n th difference of this function.

Developing this power in the ratio of the powers of Z , the product of V by the various terms of this development will be the discriminant functions of these same terms in which we substitute in place of the powers of Z the {39} corresponding finite differences of the primitive function of the index. We shall thus obtain the primitive function whose index is augmented by any number n by means of its differences. Supposing that T and Z always have the preceding values, we shall have Z equal to the binomial T - 1; the product of V by the n th power of Z will then be equal to the product of V by the development of the n th power of the binomial T - 1.

Repassing from the discriminant functions to their coefficients as has just been done, we shall have the n th difference of the primitive function expressed by the development of the n th power of the binomial T - 1, in which we substitute for the powers of T this same function whose index is augmented by the exponent of the power, and for the independent term of t , which is unity, the primitive function, which gives this difference by means of the consecutive terms of this function. Placing δ before the primitive function expressing the derivative of this function, which multiplies the x power of t in the product of V by T , and Δ expressing the same derivative in the product of V by Z , we are led {40} by that which precedes to this general result: whatever may be the function of the variable t represented by T and Z , we may, in the development of all the identical equations susceptible of being formed among these functions, substitute the characters δ and Δ in place of T and Z , provided that we write the primitive function of the index in series with the powers and with the products of the powers of the characters, and that we multiply by this function the independent terms of these characters.

We are able by means of this general result to transform any certain power of a difference of the primitive function of the index x , in which x varies by unity, into a series of differences of the same function in which x varies by a certain number of units and reciprocally. The n th power of T is equal to the n th power of this difference. If in this equality we substitute in place of T and Z the characters δ and Δ , and after the development we place at the end of each term the primitive function of the index x , we shall have the n th difference of this function in which x varies by i units expressed by a series of differences of the same function in which x varies by unity.

This series is {41} only a transformation of the difference which it expresses and which is identical with it; but it is in similar transformations that the power of analysis resides. The generality of analysis permits us to suppose in this expression that n is negative. Then the negative powers of δ and Δ indicate the integrals. Indeed the n th difference of the primitive function having for a discriminant function the product of V by the n th power of the binomial one divided by t less unity, the primitive function which is the n th integral of this difference has for a discriminant function that of the same difference multiplied by the n th power taken less than the binomial one divided by t minus one, a power to which the same power of the character Δ corresponds; this power indicates then an integral of the same order, the index x varying by unity; and the negative powers of δ indicate equally the integrals x varying by i units.

We see, thus, in the clearest and simplest manner the rationality of the analysis observed among the positive powers and differences, and among the negative powers and the integrals. If the function indicated by δ placed before the primitive function is zero, we shall have an equation of finite differences, and V will be the discriminant function of its integral. In order to obtain this discriminant function we shall observe that in the product of V by T all the powers of t ought to disappear except the powers inferior to the order of the equation of differences; V is then equal to a fraction whose denominator is T and whose numerator is a polynomial in which the highest power of t is less by unity than the order of the {42} equation of differences.

The arbitrary coefficients of the various powers of t in this polynomial, including the power zero, will be determined by as many values of the primitive function of the index when we make successively x equal to zero, to one, to two, etc. When the equation of differences is given we determine T by putting all its terms in the first member and zero in the second; by substituting in the first member unity in place of the function which has the largest index; the first power of t in place of the primitive function in which this index is diminished by unity; the second power of t for the primitive function where this index is diminished by two units, and so on. The coefficient of the x th power of t in the development of the preceding expression of V will be the primitive function of x or the integral of the equation of finite differences.

Analysis furnishes for this development various means, among which we may choose that one which is most suitable for the question proposed; this is an advantage of this method of integration. Let us conceive now that V be a function of the two variables t and t´ developed according to the powers and products of these variables; the coefficient of any product of the powers x and x´ of t and t´ will be a function of the exponents or indices x and x´ of these powers; this function I shall call the primitive function of which V is the discriminant function. Let us multiply V by a function T of the two variables t and t´ developed like V in ratio of the powers and the products of these variables; the product will be the discriminant function of a derivative of the primitive function; if T , for example, is equal to the {43} variable t plus the variable t´ minus two, this derivative will be the primitive function of which we diminish by unity the index x plus this same primitive function of which we diminish by unity the index x´ less two times the primitive function.

Designating whatever T may be by the character δ placed before the primitive function, this derivative, the product of V by the n th power of T , will be the discriminant function of the derivative of the primitive function before which one places the n th power of the character δ. Hence result the theorems analogous to those which are relative to functions of a single variable. Suppose the function indicated by the character δ be zero; one will have an equation of partial differences. If, for example, we make as before T equal to the variable t plus the variable t´ - 2, we have zero equal to the primitive function of which we diminish by unity the index x plus the same function of which we diminish by unity the index x´ minus two times the primitive function.

The discriminant function V of the primitive function or of the integral of this equation ought then to be such that its product by T does not include at all the products of t by t´ ; but V may include separately the powers of t and those of t´ , that is to say, an arbitrary function of t and an arbitrary function of t´ ; V is then a fraction whose numerator is the sum of these two arbitrary functions and whose denominator is T. The coefficient of the product of the x th power of t by the x´ power of t´ in the development of this fraction will then be the integral of the preceding equation of partial differences.

This method of integrating this kind of equations seems to me the simplest and the easiest by {44} the employment of the various analytical processes for the development of rational fractions. Considering equations of infinitely small partial differences as equations of finite partial differences in which nothing is neglected, we are able to throw light upon the obscure points of their calculus, which have been the subject of great discussions among geometricians. It is thus that I have demonstrated the possibility of introducing discontinued functions in their integrals, provided that the discontinuity takes place only for the differentials of the order of these equations or of a superior order.

The transcendent results of calculus are, like all the abstractions of the understanding, general signs whose true meaning may be ascertained only by repassing by metaphysical analysis to the elementary ideas which have led to them; this often presents great difficulties, for the human mind tries still less to transport itself into the future than to retire within itself. The comparison of infinitely small differences with finite differences is able similarly to shed great light upon the metaphysics of infinitesimal calculus. It is easily proven that the finite n th difference of a function in which the increase of the variable is E being divided by the n th power of E , the quotient reduced in series by ratio to the powers of the increase E is formed by a first term independent of E.

In the measure that E diminishes, the series approaches more and more this first term from which it can differ only by quantities less than any assignable magnitude. Considering from this point of view the infinitely small differences, we see that the various operations of differential calculus amount to comparing separately in the development of identical expressions the finite terms or those independent of the increments of the variables which are regarded as infinitely small; this is rigorously exact, these increments being indeterminant.

Thus differential calculus has all the exactitude of other algebraic operations. The same exactitude is found in the applications of differential calculus to geometry and mechanics. If we imagine a curve cut by a secant at two adjacent points, naming E the interval of the ordinates of these two points, E will be the increment of the abscissa from the first to the second ordinate. It is easy to see that the corresponding increment of the ordinate will be the product of E by the first ordinate divided by its subsecant; augmenting then in this equation of the curve the first ordinate by this increment, we shall have the equation relative to the second ordinate.

The difference of these two equations will be a third equation which, developed by the ratio of the powers of E and divided by E , will have its first term independent of E , which will be the limit of this development. This term, equal to zero, will give then the limit of the subsecants, a limit which is evidently the subtangent. This singularly happy method of obtaining the subtangent is due to Fermat, who has extended it to {46} transcendent curves. This great geometrician expresses by the character E the increment of the abscissa; and considering only the first power of this increment, he determines exactly as we do by differential calculus the subtangents of the curves, their points of inflection, the maxima and minima of their ordinates, and in general those of rational functions.

We see likewise by his beautiful solution of the problem of the refraction of light inserted in the Collection of the Letters of Descartes that he knows how to extend his methods to irrational functions in freeing them from irrationalities by the elevation of the roots to powers. Fermat should be regarded, then, as the true discoverer of Differential Calculus. Newton has since rendered this calculus more analytical in his Method of Fluxions , and simplified and generalized the processes by his beautiful theorem of the binomial. Finally, about the same time Leibnitz has enriched differential calculus by a notation which, by indicating the passage from the finite to the infinitely small, adds to the advantage of expressing the general results of calculus that of giving the first approximate values of the differences and of the sums of the quantities; this notation is adapted of itself to the calculus of partial differentials.

We are often led to expressions which contain so many terms and factors that the numerical substitutions are impracticable. This takes place in questions of probability when we consider a great number of events. Meanwhile it is necessary to have the numerical value of the formulæ in order to know with what probability the results are indicated, which the events develop by multiplication. It is necessary especially to have the {47} law according to which this probability continually approaches certainty, which it will finally attain if the number of events were infinite. In order to obtain this law I considered that the definite integrals of differentials multiplied by the factors raised to great powers would give by integration the formulæ composed of a great number of terms and factors.

This remark brought me to the idea of transforming into similar integrals the complicated expressions of analysis and the integrals of the equation of differences. I fulfilled this condition by a method which gives at the same time the function comprised under the integral sign and the limits of the integration. It offers this remarkable thing, that the function is the same discriminant function of the expressions and the proposed equations; this attaches this method to the theory of discriminant functions of which it is thus the complement. Further, it would only be a question of reducing the definite integral to a converging series. This I have obtained by a process which makes the series converge with as much more rapidity as the formula which it represents is more complicated, so that it is more exact as it becomes more necessary.

Frequently the series has for a factor the square root of the ratio of the circumference to the diameter; sometimes it depends upon other transcendents whose number is infinite. An important remark which pertains to great generality of analysis, and which permits us to extend this method to formulæ and to equations of difference which the theory of probability presents most frequently, is that the series to which one comes by supposing the limits of the definite integrals to be real and positive {48} take place equally in the case where the equation which determines these limits has only negative or imaginary roots. These passages from the positive to the negative and from the real to the imaginary, of which I first have made use, have led me further to the values of many singular definite integrals, which I have accordingly demonstrated directly.

We may then consider these passages as a means of discovery parallel to induction and analogy long employed by geometricians, at first with an extreme reserve, afterwards with entire confidence, since a great number of examples has justified its use. In the mean time it is always necessary to confirm by direct demonstrations the results obtained by these divers means. I have named the ensemble of the preceding methods the Calculus of Discriminant Functions ; this calculus serves as a basis for the work which I have published under the title of the Analytical Theory of Probabilities. It is connected with the simple idea of indicating the repeated multiplications of a quantity by itself or its entire and positive powers by writing toward the top of the letter which expresses it the numbers which mark the degrees of these powers.

This notation, employed by Descartes in his Geometry and generally adopted since the publication of this important work, is a little thing, especially when compared with the theory of curves and variable functions by which this great geometrician has established the foundations of modern calculus. But the language of analysis, most perfect of all, being in itself a powerful instrument of discoveries, its notations, especially when they are necessary and happily conceived, are so many {49} germs of new calculi. This is rendered appreciable by this example. Wallis, who in his work entitled Arithmetica Infinitorum , one of those which have most contributed to the progress of analysis, has interested himself especially in following the thread of induction and analogy, considered that if one divides the exponent of a letter by two, three, etc.

Extending by analogy this result to the case where division is impossible, he considered a quantity raised to a fractional exponent as the root of the degree indicated by the denominator of this fraction—namely, of the quantity raised to a power indicated by the numerator. He observed then that, according to the Cartesian notation, the multiplication of two powers of the same letter amounts to adding their exponents, and that their division amounts to subtracting the exponents of the power of the divisor from that of the power of the dividend, when the second of these exponents is greater than the first. Wallis extended this result to the case where the first exponent is equal to or greater than the second, which makes the difference zero or negative.

He supposed then that a negative exponent indicates unity divided by the quantity raised to the same exponent taken positively. These remarks led him to integrate generally the monomial differentials, whence he inferred the definite integrals of a particular kind of binomial differentials whose exponent is a positive integral {50} number. The observation then of the law of the numbers which express these integrals, a series of interpolations and happy inductions where one perceives the germ of the calculus of definite integrals which has so much exercised geometricians and which is one of the fundaments of my new Theory of Probabilities , gave him the ratio of the area of the circle to the square of its diameter expressed by an infinite product, which, when one stops it, confines this ratio to limits more and more converging; this is one of the most singular results in analysis.

But it is remarkable that Wallis, who had so well considered the fractional exponents of radical powers, should have continued to note these powers as had been done before him. Newton in his Letters to Oldembourg , if I am not mistaken, was the first to employ the notation of these powers by fractional exponents. Comparing by the way of induction, of which Wallis had made such a beautiful use, the exponents of the powers of the binomial with the coefficients of the terms of its development in the case where this exponent is integral and positive, he determined the law of these coefficients and extended it by analogy to fractional and negative powers.

These various results, based upon the notation of Descartes, show his influence on the progress of analysis. It has still the advantage of giving the simplest and fairest idea of logarithms, which are indeed only the exponents of a magnitude whose successive powers, increasing by infinitely small degrees, can represent all numbers. But the most important extension that this notation has received is that of variable exponents, which constitutes exponential calculus, one of the most fruitful {51} branches of modern analysis. Leibnitz was the first to indicate the transcendents by variable exponents, and thereby he has completed the system of elements of which a finite function can be composed; for every finite explicit function of a variable may be reduced in the last analysis to simple magnitudes, combined by the method of addition, subtraction, multiplication, and division and raised to constant or variable powers.

The roots of the equations formed from these elements are the implicit functions of the variable. It is thus that a variable has for a logarithm the exponent of the power which is equal to it in the series of the powers of the number whose hyperbolic logarithm is unity, and the logarithm of a variable of it is an implicit function. Leibnitz thought to give to his differential character the same exponents as to magnitudes; but then in place of indicating the repeated multiplications of the same magnitude these exponents indicate the repeated differentiations of the same function. This new extension of the Cartesian notation led Leibnitz to the analogy of positive powers with the differentials, and the negative powers with the integrals. Lagrange has followed this singular analogy in all its developments; and by series of inductions which may be regarded as one of the most beautiful applications which have ever been made of the method of induction he has arrived at general formulæ which are as curious as useful on the transformations of differences and of integrals the ones into the others when the variables have divers finite increments and when these increments are infinitely small.

But he has not given the demonstrations of it which appear to him difficult. The theory of discriminant {52} functions extends the Cartesian notations to some of its characters; it shows with proof the analogy of the powers and operations indicated by these characters; so that it may still be regarded as the exponential calculus of characters. All that concerns the series and the integration of equations of differences springs from it with an extreme facility. The combinations which games present were the object of the first investigations of probabilities. In an infinite variety of these combinations many of them lend themselves readily to calculus; others require more difficult calculi; and the difficulties increasing in the measure that the combinations become more complicated, the desire to surmount them and curiosity have excited geometricians to perfect more and more this kind of analysis.

It has been seen already that the benefits of a lottery are easily determined by the theory of combinations. But it is more difficult to know in how many draws one can bet one against one, for example that all the numbers will be drawn, n being the number of numbers, r that of the numbers drawn at each draw, and i the unknown number of draws. The expression of the probability of drawing all the {54} numbers depends upon the n th finite difference of the i power of a product of r consecutive numbers. When the number n is considerable the search for the value of i which renders this probability equal to ½ becomes impossible at least unless this difference is converted into a very converging series.

This is easily done by the method here below indicated by the approximations of functions of very large numbers. It is found thus since the lottery is composed of ten thousand numbers, one of which is drawn at each draw, that there is a disadvantage in betting one against one that all the numbers will be drawn in draws and an advantage in making the same bet for draws. In the lottery of France this bet is disadvantageous for 85 draws and advantageous for 86 draws. Let us consider again two players, A and B, playing together at heads and tails in such a manner that at each throw if heads turns up A gives one counter to B, who gives him one if tails turns up; the number of counters of B is limited, while that of A is unlimited, and the game is to end only when B shall have no more counters.

We ask in how many throws one should bet one to one that the game will end. The expression of the probability that the game will end in an i number of throws is given by a series which comprises a great number of terms and factors if the number of counters of B is considerable; the search for the value of the unknown i which renders this series ½ would then be impossible if we did not reduce the same to a very convergent series. In applying to it the method of which we have just spoken, we find a very simple expression for the unknown from which it results that if, {55} for example, B has a hundred counters, it is a bet of a little less than one against one that the game will end in throws, and a bet of a little more than one against one that it will end in throws.

These two examples added to those we have already given are sufficient to shows how the problems of games have contributed to the perfection of analysis. Inequalities of this kind have upon the results of the calculation of probabilities a sensible influence which deserves particular attention. Let us take the game of heads and tails, and let us suppose that it is equally easy to throw the one or the other side of the coin. Then the probability of throwing heads at the first throw is ½ and that of throwing it twice in succession is ¼. But if there exist in the coin an inequality which causes one of the faces to appear rather than the other without knowing which side is favored by this inequality, the probability of throwing heads at the first throw will always ½; because of our ignorance of which face is favored by the inequality the probability of the simple event is increased if this inequality is favorable to it, just so much is it diminished if the inequality is contrary to it.

But in this same ignorance the probability of throwing heads twice in succession is increased. Indeed this probability is that of throwing heads at the first throw multiplied by the probability {57} that having thrown it at the first throw it will be thrown at the second; but its happening at the first throw is a reason for belief that the inequality of the coin favors it; the unknown inequality increases, then, the probability of throwing heads at the second throw; it consequently increases the product of these two probabilities. In order to submit this matter to calculus let us suppose that this inequality increases by a twentieth the probability of the simple event which it favors. We find thus generally that the constant and unknown causes which favor simple events which are judged equally possible always increase the probability of the repetition of the same simple event.

In an even number of throws heads and tails ought {58} both to happen either an even number of times or odd number of times. The probability of each of these cases is ½ if the possibilities of the two faces are equal; but if there is between them an unknown inequality, this inequality is always favorable to the first case. Two players whose skill is supposed to be equal play on the conditions that at each throw that one who loses gives a counter to his adversary, and that the game continues until one of the players has no more counters.

The calculation of the probabilities shows us that for the equality of the play the throws of the players ought to be an inverse ratio to their counters. But if there is between the players a small unknown inequality, it favors that one of the players who has the smallest number of counters. His probability of winning the game increases if the players agree to double or triple their counters; and it will be ½ or the same as the probability of the other player in the case where the number of their counters should become infinite, preserving always the same ratio. One may correct the influence of these unknown inequalities by submitting them themselves to the chances of hazard. Thus at the play of heads and tails, if one has a second coin which is thrown each time with the first and one agrees to name constantly heads the face turned up by the second coin, the probability of throwing heads twice in succession with the first coin will approach much nearer ¼ than in the case of a single coin.

In this last case the difference is the square of the small increment of possibility that the unknown inequality gives to the face of the first coin which it favors; in the other case this difference is the {59} quadruple product of this square by the corresponding square relative to the second coin. Let there be thrown into an urn a hundred numbers from 1 to in the order of numeration, and after having shaken the urn in order to mix the numbers one is drawn; it is clear that if the mixing has been well done the probabilities of the drawing of the numbers will be the same.

But if we fear that there is among them small differences dependent upon the order according to which the numbers have been thrown into the urn, we shall diminish considerably these differences by throwing into a second urn the numbers according to the order of their drawing from the first urn, and by shaking then this second urn in order to mix the numbers. A third urn, a fourth urn, etc. Amid the variable and unknown causes which we comprehend under the name of chance , and which render uncertain and irregular the march of events, we see appearing, in the measure that they multiply, a striking regularity which seems to hold to a design and which has been considered as a proof of Providence. But in reflecting upon this we soon recognize that this regularity is only the development of the respective possibilities of simple events which ought to present themselves more often when they are more probable.

Let us imagine, for example, an urn which contains white balls and black balls; and let us suppose that each time a ball is drawn it is put back into the urn before proceeding to a new draw. The ratio of the number of the white balls drawn to the number of black balls drawn will be most often very irregular in the first drawings; but the variable causes of this irregularity produce effects alternately favorable and unfavorable to the regular march of events which destroy each other {61} mutually in the totality of a great number of draws, allowing us to perceive more and more the ratio of white balls to the black balls contained in the urn, or the respective possibilities of drawing a white ball or black ball at each draw.

From this results the following theorem. The probability that the ratio of the number of white balls drawn to the total number of balls drawn does not deviate beyond a given interval from the ratio of the number of white balls to the total number of balls contained in the urn, approaches indefinitely to certainty by the indefinite multiplication of events, however small this interval. This theorem indicated by common sense was difficult to demonstrate by analysis. Accordingly the illustrious geometrician Jacques Bernoulli, who first has occupied himself with it, attaches great importance to the demonstrations he has given.

The calculus of discriminant functions applied to this matter not only demonstrates with facility this theorem, but still more it gives the probability that the ratio of the events observed deviates only in certain limits from the true ratio of their respective possibilities. One may draw from the preceding theorem this consequence which ought to be regarded as a general law, namely, that the ratios of the acts of nature are very nearly constant when these acts are considered in great number. Thus in spite of the variety of years the sum of the productions during a considerable number of years is sensibly the same; so that man by useful foresight is able to provide against the irregularity of the seasons by spreading out equally over all the {62} seasons the goods which nature distributes in an unequal manner.

I do not except from the above law results due to moral causes. The ratio of annual births to the population, and that of marriages to births, show only small variations; at Paris the number of annual births is almost the same, and I have heard it said at the post-office in ordinary seasons the number of letters thrown aside on account of defective addresses changes little each year; this has likewise been observed at London. It follows again from this theorem that in a series of events indefinitely prolonged the action of regular and constant causes ought to prevail in the long run over that of irregular causes. It is this which renders the gains of the lotteries just as certain as the products of agriculture; the chances which they reserve assure them a benefit in the totality of a great number of throws.

Thus favorable and numerous chances being constantly attached to the observation of the eternal principles of reason, of justice, and of humanity which establish and maintain societies, there is a great advantage in conforming to these principles and of grave inconvenience in departing from them. If one consult histories and his own experience, one will see all the facts come to the aid of this result of calculus. Consider the happy effects of institutions founded upon reason and the natural rights of man among the peoples who have known how to establish and preserve them. Consider again the advantages which good faith has procured for the governments who have made it the basis of their conduct and how they have been indemnified for the sacrifices which a scrupulous exactitude in keeping {63} their engagements has cost them.

What immense credit at home! What preponderance abroad! On the contrary, look into what an abyss of misfortunes nations have often been precipitated by the ambition and the perfidy of their chiefs. Every time that a great power intoxicated by the love of conquest aspires to universal domination the sentiment of independence produces among the menaced nations a coalition of which it becomes almost always the victim. Similarly in the midst of the variable causes which extend or restrain the divers states, the natural limits acting as constant causes ought to end by prevailing. It is important then to the stability as well as to the happiness of empires not to extend them beyond those limits into which they are led again without cessation by the action of the causes; just as the waters of the seas raised by violent tempests fall again into their basins by the force of gravity.

It is again a result of the calculus of probabilities confirmed by numerous and melancholy experiences. History treated from the point of view of the influence of constant causes would unite to the interest of curiosity that of offering to man most useful lessons. Sometimes we attribute the inevitable results of these causes to the accidental circumstances which have produced their action. It is, for example, against the nature of things that one people should ever be governed by another when a vast sea or a great distance separates them.

It may be affirmed that in the long run this constant cause, joining itself without ceasing to the variable causes which act in the same way and which the course of time develops, will end by finding them sufficiently {64} strong to give to a subjugated people its natural independence or to unite it to a powerful state which may be contiguous. In a great number of cases, and these are the most important of the analysis of hazards, the possibilities of simple events are unknown and we are forced to search in past events for the indices which can guide us in our conjectures about the causes upon which they depend.

In applying the analysis of discriminant functions to the principle elucidated above on the probability of the causes drawn from the events observed, we are led to the following theorem. When a simple event or one composed of several simple events, as, for instance, in a game, has been repeated a great number of times the possibilities of the simple events which render most probable that which has been observed are those that observation indicates with the greatest probability; in the measure that the observed event is repeated this probability increases and would end by amounting to certainty if the numbers of repetitions should become infinite.

There are two kinds of approximations: the one is relative to the limits taken on all sides of the possibilities which give to the past the greatest probability; the other approximation is related to the probability that these possibilities fall within these limits. The repetition of the compound event increases more and more this probability, the limits remaining the same; it reduces more and more the interval of these limits, the probability remaining the same; in infinity this interval becomes zero and the probability changes to certainty. If we apply this theorem to the ratio of the births of {65} boys to that of girls observed in the different countries of Europe, we find that this ratio, which is everywhere about equal to that of 22 to 21, indicates with an extreme probability a greater facility in the birth of boys.

Considering further that it is the same at Naples and at St. Petersburg, we shall see that in this regard the influence of climate is without effect. We might then suspect, contrary to the common belief, that this predominance of masculine births exists even in the Orient. I have consequently invited the French scholars sent to Egypt to occupy themselves with this interesting question; but the difficulty in obtaining exact information about the births has not permitted them to solve it. Happily, M. de Humboldt has not neglected this matter among the innumerable new things which he has observed and collected in America with so much sagacity, constancy, and courage.

He has found in the tropics the same ratio of the births as we observe in Paris; this ought to make us regard the greater number of masculine births as a general law of the human race. The laws which the different kinds of animals follow in this regard seem to me worthy of the attention of naturalists. The fact that the ratio of births of boys to that of girls differs very little from unity even in the great number of the births observed in a place would offer in this regard a result contrary to the general law, without which we should be right in concluding that this law did not exist.

In order to arrive at this result it is necessary to employ great numbers and to be sure that it is indicated by great probability. Buffon cites, for example, in his Political Arithmetic several communities {66} of Bourgogne where the births of girls have surpassed those of boys. Among these communities that of Carcelle-le-Grignon presents in births during five years girls and boys. The registers of births, which are kept with care in order to assure the condition of the citizens, may serve in determining the population of a great empire without recurring to the enumeration of its inhabitants—a laborious operation and one difficult to make with exactitude.

But for this it is necessary to know the ratio of the population to the annual births. The most precise means of obtaining it consists, first, in choosing in the empire districts distributed in an almost equal manner over its whole surface, so as to render the general result independent of local circumstances; second, in enumerating with care for a given epoch the inhabitants of several communities in each of these districts; third, by determining from the statement of the births during several years which precede and follow this epoch the mean number corresponding to the annual births. This number, divided by that of the inhabitants, will give the ratio of the annual births to the population in a manner more and more accurate as the enumeration becomes more considerable.

The {67} government, convinced of the utility of a similar enumeration, has decided at my request to order its execution. In thirty districts spread out equally over the whole of France, communities have been chosen which would be able to furnish the most exact information. Their enumerations have given individuals as the total number of their inhabitants on the 23d of September, The statement of the births in these communities during the years , , and have given:. Multiplying the number of annual births in France by this ratio, we shall have the population of this kingdom. But what is the probability that the population thus determined will not deviate from the true population beyond a given limit?

Resolving this problem and applying to its solution the preceding data, I have found that, the number of annual births in France being supposed to be , which brings the population to inhabitants, it is a bet of almost against 1 that the error of this result is not half a million. The ratio of the births of boys to that of girls which the preceding statement offers is that of 22 to 21; and the marriages are to the births as 3 is to 4. At Paris the baptisms of children of both sexes vary a little from the ratio of 22 to Since , the {68} epoch in which one has commenced to distinguish the sexes upon the birth-registers, up to the end of , there have been baptized in this capital boys and girls. The ratio of the two numbers is almost that of 25 to 24; it appears then at Paris that a particular cause approximates an equality of baptisms of the two sexes.

If we apply to this matter the calculus of probabilities, we find that it is a bet of to 1 in favor of the existence of this cause, which is sufficient to authorize the investigation. Upon reflection it has appeared to me that the difference observed holds to this, that the parents in the country and the provinces, finding some advantage in keeping the boys at home, have sent to the Hospital for Foundlings in Paris fewer of them relative to the number of girls according to the ratio of births of the two sexes. This is proved by the statement of the registers of this hospital.

From the beginning of to the end of there were entered boys and girls. This confirms the existence of the assigned cause, namely, that the ratio of births of boys to those of girls is at Paris that of 22 to 21, no attention having been paid to foundlings. The preceding results suppose that we may compare the births to the drawings of balls from an urn which contains an infinite number of white balls and black balls so mixed that at each draw the chances of drawing ought to be the same for each ball; but it is possible that the variations of the same seasons in different years may have some influence upon the annual ratio {69} of the births of boys to those of girls. The Bureau of Longitudes of France publishes each year in its annual the tables of the annual movement of the population of the kingdom.

Applying to this deviation the analysis of probabilities in the hypothesis of the comparison of births to the drawings of balls from an urn, we find that it would be scarcely probable. It appears, then, to indicate that this hypothesis, although closely approximated, is not rigorously exact. In the number of births which we have just stated there are of natural children boys and girls. This result is in the same sense as that of the births of foundlings; and it seems to prove that in the class of natural children the births of the two sexes approach more nearly equality than in the class of legitimate children. The difference of the climates from the north to the south of France does not appear to influence appreciably the ratio of the births of boys and girls.

The constancy of the superiority of the births of boys over girls at Paris and at London since they have been {70} observed has appeared to some scholars to be a proof of Providence, without which they have thought that the irregular causes which disturb without ceasing the course of events ought several times to have rendered the annual births of girls superior to those of boys. But this proof is a new example of the abuse which has been so often made of final causes which always disappear on a searching examination of the questions when we have the necessary data to solve them. The constancy in question is a result of regular causes which give the superiority to the births of boys and which extend it to the anomalies due to hazard when the number of annual births is considerable.

The investigation of the probability that this constancy will maintain itself for a long time belongs to that branch of the analysis of hazards which passes from past events to the probability of future events; and taking as a basis the births observed from to , it is a bet of almost 4 against 1 that at Paris the annual births of boys will constantly surpass for a century the births of girls; there is then no reason to be astonished that this has taken place for a half-century. Let us take another example of the development of constant ratios which events present in the measure that they are multiplied. Let us imagine a series of urns arranged circularly, and each containing a very great number of white balls and black balls; the ratio of white balls to the black in the urns being originally very different and such, for example, that one of these urns contains only white balls, while another contains only black balls.

If one draws a ball from the first urn in order to put it into the second, and, after having {71} shaken the second urn in order to mix well the new ball with the others, one draws a ball to put it into the third urn, and so on to the last urn, from which is drawn a ball to put into the first, and if this series is recommenced continually, the analysis of probability shows us that the ratios of the white balls to the black in these urns will end by being the same and equal to the ratio of the sum of all the white balls to the sum of all the black balls contained in the urns. Thus by this regular mode of change the primitive irregularity of these ratios disappears eventually in order to make room for the most simple order.

In this type of essay you required you make some arguments and support them. The subject of probabilities is mostly expressed in mathematical concepts, like formulas and numbers, which are not possible to use in an essay. You main task is to find a way to discuss this topic from philosophical point of view in ordinary non-mathematical language. So what kind of philosophical arguments about probability you could come up with? You could consider using this argument — does theory of probabilities belong to our everyday life or is it just a scientific concept that has no use for ordinary people? Ask yourself and the readers if probabilities really exist out there in the world in the form of frequencies or propensities? If it is an objective feature of reality or just a subjective believe that exists only in our heads?

Finding a way to support your arguments can be harder than discovering the arguments themselves, but nonetheless it is possible and you have to do it. So how could you support your argument that probabilities it is not just abstract subjective scientific concept but has objective implementation in everyday lives of ordinary people? Consider stating that probability plays crucial role in our reasoning — the way we choose and make decisions every day. We do it unconsciously yes, but it does not change the fact that we use probabilities. It is not an abstract concept, in fact, it is logic. Whenever we find uncertainty — we will find probability that means almost everywhere in human life.

To make it sound more convincing make a real life example, like one from a court room. A fate of man accused of murder depends on the juries who consider all probabilities of the evidence from both sides being true or false. We calculate many things in our heads every day not being aware of using the theory of probabilities. Writing a philosophical essay can be a challenging task especially when the topic is the theory of probabilities. Before beginning to compose, learn the structure of this type of paper, choose your arguments and only then create an outline.

If your life is too busy right now for managing such task you can trust our professional writers to do it for you. If you still hesitate whether you should use such kind of service, please read this article , which will give you 8 reasons to order your essay online. We hope this information will help you to get an excellent mark on your philosophical essay on probabilities. BIG EssayWriter. com Order Now Login Toggle navigation College Essay Essay Editing UK Essay Custom Essay Custom Writing FAQ Prices About Us Contact Blog. Writing A Philosophical Essay On Probabilities. Writing a philosophical essay A philosophical essay as any other types of essay requires specific form and structure. Philosophical essay structure: Introduction. In this very first paragraph you have to present the problem you are going to discuss in the essay.

It has to include a thesis statement and a roadmap of your future essay — briefly tell the reader what you are going to talk about; Description. In this next paragraph you must describe the matter at hands, in our case it is the theory of probabilities.