Some of Laplace's results in the theory of probabilities are simplified in S.
Thus the probabilities are delicately balanced.
A volume entitled Opera posthuma (Leiden, 1703) contained his "Dioptrica," in which the ratio between the respective focal lengths of object-glass and eye-glass is given as the measure of magnifying power, together with the shorter essays De vitris figurandis, De corona et parheliis, &c. An early tract De ratiociniis tin ludo aleae, printed in 16J7 with Schooten's Exercitationes mathematicae, is notable as one of the first formal treatises on the theory of probabilities; nor should his investigations of the properties of the cissoid, logarithmic and catenary curves be left unnoticed.
The assumption may be a reasonable one, and if the results agree with probabilities as deduced from the rest of the evidence it is wise to adopt it; if on the other hand the other evidence seems in any serious degree contrary to those results it may be surmised that the assumption is faulty in some particular.
In fact a little consideration of the theory of probabilities will show it to be infinitely probable that such an object should really have some movement of rotation, no matter by what causes the nebula may have originated.
After all, it is only a question of probabilities, and the difficulties of fitting a wife and child into the story seem to be very great, whether we conceive them left behind by Demetrius in Italy, or sent out of the country before him.
These new probabilities open up considerable possibilities in research with regard to the relations of the early Minoans and other Aegeans with Syria and Egypt and the undoubted fact of the resemblances of Minoan on the one hand to Syrian and Egyptian religions and funerary practices, and on the other hand to those of the Etruscans.
Under the general heading "Algebra and Theory of Numbers" occur the subheadings "Elements of Algebra," with the topics rational polynomials, permutations, &c., partitions, probabilities; "Linear Substitutions," with the topics determinants, &c., linear substitutions, general theory of quantics; "Theory of Algebraic Equations," with the topics existence of roots, separation of and approximation to, theory of Galois, &c. "Theory of Numbers," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers.
In the few instances where it is said to have been found wild the probabilities are that the tree was an escape from cultivation.
The probabilities are that in the end the production of a rubber as nearly as possible free from water and impurities and of constant composition will be realized as best meeting the requirements of the modern manufacturer.
The theory of probabilities, which Laplace described as common sense expressed in mathematical language, engaged his attention from its importance in physics and astronomy; and he applied his theory, not only to the ordinary problems of chances, but also to the inquiry into the causes of phenomena, vital statistics and future events.
The device known as the method of least squares, for reducing numerous equations of condition to the number of unknown quantities to be determined, had been adopted as a practically convenient rule by Gauss and Legendre; but Laplace first treated it as a problem in probabilities, and proved by an intricate and difficult course of reasoning that it was also the most advantageous, the mean of the probabilities of error in the determination of the elements being thereby reduced to a minimum.
Laplace published in 1779 the method of generating functions, the foundation of his theory of probabilities, and the first part of his Theorie analytique is devoted to the exposition of its principles, which in their simplest form consist in treating the successive values of any function as the coefficients in the expansion of another function with reference to a different variable.
The balance of probabilities falls, however, in favour of the court hypothesis.
His mathematical writings, which account for some forty entries in the Royal Society's catalogue of scientific papers, cover a wide range of subjects, such" s the theory of probabilities, quadratic forms, theory of integrals, gearings, the construction of geographical maps, &c. He also published a Traite de la theorie des nombres.
In these works he shows how the numbers representing the individual qualities of man are grouped round the numbers referring to the "average man" in a manner exactly corresponding to that in which single results of observation are grouped round the mean result, so that the principles of the theory of probabilities may be applied to statistical researches on the subjects.
It is possible that this is the oldest Coptic version, and this view is supported by the general probabilities of the spread of Christianity in Egypt.
There is no doubt but that in the Maccabean times and onward 218 was the shekel; but the use of the word darkemon by Ezra and Nehemiah, and the probabilities of their case, point to the daragmaneh, 1/60 maneh or shekel of Assyria; and the mention of 1/3 shekel by Nehemiah as poll tax nearly proves that the 129 and not 218 grains is intended, as 218 is not divisible by 3.
Although he strictly limited his prophetic pretensions to the estimate of tendencies and probabilities, his forecasts were none the less in demand.
No doubt, rational evidences had appeared in books of rhetoric, as we see from Plato's Phaedrus, 266-267,where we find proofs,probabilities, refutation and maxim, but mixed up with other evidences.
II, 77 a 31-33); in that for appropriate principles it substitutes the probabilities of authority era g vbo a) which are the opinions of all, or of the majority, or of the wise (Top. i.
Hence it is readily seen that it is among cousin marriages that the greater probabilities exist that two individuals bearing identical characters will meet, than in the population at large.
The entire problem is not without its difficulties still, after all the research lavished upon it, but the probabilities seem to converge upon the conclusion that Paul was never released from his imprisonment, and consequently that he never revisited the East.
The tails when split into two or three, with small strips of narrow tape so as to separate the otherwise dense fur, formerly made very handsome sets of trimmings, ties and muffs, and the probabilities are, as with other fashions, such use will have its period of revival.
In weighing these probabilities he proceeded upon a particular theory which in its leading features he had derived from J.
On the whole the probabilities of the two sites seem to balance, and it is practically impossible without further discoveries to decide between them.
It shows the average number of years which persons of a given age, or, as here, of all ages, will live, on the assumption that they are subject to the calculated probabilities of survival.
The probabilities seem in favour of the Academy cartoon.
De Morgan's other principal mathematical works were The Elements of Algebra (1835), a valuable but somewhat dry elementary treatise; the [[Essay]] on Probabilities (1838), forming the 107th volume of Lardner's Cyclopaedia, which forms a valuable introduction to the subject; and The Elements of Trigonometry and Trigonometrical Analysis, preliminary to the Differential Calculus (1837).
The remainder of the book is concerned for the most part with the probabilities on which human life practically turns, as he and Butler are fond of reminding us.
All our " interpretations of nature " are inadequate; only reasonable probabilities, not final rational certainties.
Probabilities also seem to favour the view that this may, to a certain extent, be true of the four great planets of our system.
This boundless region affords at the best probabilities, ultimately grounded on moral faith, all beyond lies within the veil.
We learn little from Locke as to the rationale of the probabilities on which man thus depends when he deals with the past, Te the distant or the future.
The origin of Ferrara is uncertain, and probabilities are against the supposition that it occupies the site of the ancient Forum Alieni.
But he has been more generally known by his Doctrine of Chances, or Method of Calculating the Probabilities of Events at Play.
His labours in the decline of life were chiefly directed to the doctrine of probabilities in reference to practical purposes, and in particular to economical subjects, as, for example, to inoculation, and to the duration of married life in the two sexes, as well as to the relative proportion of male and female births.
The Memoirs of the Berlin Academy from 1761 to 1784 contain many of his papers, which treat of such subjects as resistance of fluids, magnetism, comets, probabilities, the problem of three bodies, meteorology, &c. In the Acta Helvetica (1752-1760) and in the Nova acta erudita (1763-1769) several of his contributions appear.
Amid all the variation in their details, and the apparent confusion introduced by Napoleon's habit of suggesting alternatives and discussing probabilities, and in spite of the preparations ostensibly made for an expedition to Ireland, which was to have sailed from Brest and to have carried 30,000 troops commanded by Augereau, the real purpose of Napoleon was neither altered nor concealed.
The second part of the Laws of Thought contained a corresponding attempt to discover a general method in probabilities, which should enable us from the given probabilities of any system of events to determine the consequent probability of any other event logically connected with the given events.