BERNOULLI (James)

, another mathematical branch of the foregoing celebrated family. He was born at Basil in October 1759; being the son of John Bernoulli, and grandson of the first John Bernoulli, before mentioned, and the nephew of Daniel Bernoulli last noticed above. Our author's elder brother John, who still lives at Berlin, is also well known in the republic of science, particularly for his astronomical labours.

The gentleman to whom this article relates, was educated, as most of his relations had been, for the profession of law: but his genius led him very early into the study of mathematics; and at 20 years of age he read public lectures on experimental philosophy in the university of Basil, for his uncle Daniel Bernoulli, whom he hoped to have succeeded as professor. Being disappointed in this view, he resolved to leave his native place, and to seek his fortune elsewhere; hence he accepted the office of secretary to Count Breuner, the emperor's envoy to the republic of Venice; and in this city he remained till the year 1786, when, on the recommendation of his countryman, M. Fuss, he was invited to Petersburgh to succeed M. Lexell in the academy there, where he continued till his death, which happened the 3d of July 1789, at not quite 30 years of age, and when he had been married only two months, to the youngest daughter of John Albert Euler, the son of the so celebrated Leonard Euler.

Impossible or Imaginary BINOMIAL. After this article, in pa. 208, the middle of col. 1, add what here follows.

In the foregoing article are given several rules for the roots of Binomials. Dr. Maskelyne, the Astronomer Royal, has also given a method of finding any power of an Impossible Binomial, by another like Binomial. This rule is given in his Introduction prefixed to Taylor's Tables of Logarithms, pa. 56; and is as follows.

The logarithms of a and b being given, it is required to find the power of the Impossible Binomial a ± √-b2 whose index is m/n, that is, to find (a ± √-b2)m/n by another Impossible Binomial; and thence the value of (a + √-b2)m/n + (a - √-b2)m/n, which is always possible, whether a or b be the greater of the two.

Solution. Put b/a = tang. z. Then , where the first or second of these two last expressions is to be used, according as z is an extreme or mean arc; or rather, because b/a is not only the tangent of z, but also of z + 360°, z + 720°, &c; therefore the factor in the answer will have several values, viz, 2 cos.(m/n)z; 2 cos.(m/n)(z + 360°); 2 cos.(m/n)(z + 720°); &c; the number of which, if m and n be whole numbers, and the fraction m/n be in its least terms, will be equal to the denominator n; otherwise infinite.

By Logarithms. Put log. b + 10-log. a = log. tan. z. Then log. ; where the first or second expression is to be used, according as z is an extreme or mean arc. Moreover by taking successively, l. cos.(m/n)z; l. cos.(m/n)(z + 360°); l.cos.m/n (z + 720°); &c, there will arise several distinct answers to the question, agreeably to the remark above.

BINOMIAL Theorem. Francis Maseres, Esq. (Cursitor Baron of the Exchequer) has communicated | the following observations on the Binomial theorem, and its demonstration; viz, About the year 1666 the celebrated Sir Isaac Newton discovered that, if m were put for any whole number whatsoever, the coefficients of the terms of the mth power of 1 + x would be 1, m/1, m/1.(m - 1)/2, m/1.(m - 1)/2.(m - 2)/3, &c, till we come to the term (m - (m -1))/m, which will be the last term. But how he discovered this proposition, he has not told us, nor has he even attempted to give a demonstration of it. Dr. John Wallis, of Oxford, informs us (in his Algebra, chap. 85, pa. 319) that he had endeavoured to find this manner of generating these coefficients one from another, but without success; and he was greatly delighted with the discovery, when he found that Mr. Newton had made it. But he likewise has omitted to give a demonstration of it, as well as Sir Isaac Newton; and probably he did not know how to demonstrate it.

Sir Isaac Newton, after he had discovered this rule for generating the coefficients of the powers of 1 + x when the indexes of those powers were whole numbers, conjectured that it might possibly be true likewise when they were fractions. He therefore resolved to try whether it was or not, by applying it to such indexes in a few easy instances, and particularly to the indexes 1/2 and 1/3, which, if the rule held good in the case of fractional indexes, would enable him to find serieses equal to the values of ―(1 + x))1/2 and ―(1 + x))1/3, or the square-root and the cube-root of the Binomial quantity 1 + x. And, when he had in this manner obtained a series for ―(1 + x))1/2, which he suspected to be equal to ―(1 + x))1/2, or the square root of 1 + x, he multiplied the said series into itself, and found that the product was 1 + x; and when he had obtained a series for ―(1 + x))1/3 he multiplied the said series twice into itself, and found that the product was 1 + x; and thence he concluded that the former series was really equal to the square-root of 1 + x, and that the latter series was really equal to its cube-root. And from these and a few more such trials, in which he found the rule to answer, he concluded universally that the rule was always true, whether the index m stood for a whole number or a fraction of any kind, as 1/2, 1/3, 2/3, 3/2, 5/9, 9/5, or, in general p/q.

After the discovery of this rule by Sir Isaac Newton, and the publication of it by Dr. Wallis, in his Algebra, chap. 85, in the year 1685, (which l believe was the first time it was published to the world at large, though it was inserted in Sir Isaac Newton's first letter to Mr. Oldenburgh, the secretary to the Royal Society, dated June 13, 1676, and the said letter was shewn to Mr. Leibnitz, and probably to some other of the learned mathematicians of that time) it remained for some years without a demonstration, either in the case of integral powers or of roots. At last however it was demon- strated in the case of integral powers by means of the properties of the figurate numbers, by that learned, sagacious, and accurate mathematician Mr. James Bernoulli, in the 3d chapter of the 2d part of his excellent treatise De Arte Conjectandi, or, On the Art of forming reasonable Conjectures concerning Events that depend on Chance; which appears to me to be by much the best written treatise on the doctrine of Chances that has yet been published, though Mr. Demoivre's book on the same subject may have carried the doctrine something further. This treatise of Mr. James Bernoulli's was not published till the year 1713, which was some years after his death, which happened in August 1705; but there is reason to think that it was composed in the latter years of the preceding century, about the years 1696, 1697, 1698, 1699, and 1700, and even that some parts of it, or some of the propositions inserted in it, had been found out by the author in the years 1689, 1690, 1691, and 1692. For the first part of his very curious tract, intitled, Positiones Arithmeticæ de Seriebus Infinitis was published at Basil or Basle in Switzerland (which was his native place, and in which he was at that time professor of mathematics) in the year 1689; and the second part of the said Positiones (in the 19th Position of which those properties of the figurate numbers from which the Binomial theorem may be deduced, are set down) was published at the same place in the year 1692. But the demonstrations of those properties of the figurate numbers, and of the Binomial theorem, which depends upon them, were never as I believe communicated to the public till the year 1713, when the author's posthumous treatise De Arte Conjectandi made its appearance. These demonstrations are founded on clear and simple principles, and afford as much satisfaction as can well be expected on the subject. But the full display and explanation of these principles, and the deduction of the said properties of the figurate numbers, and ultimately of the Binomial theorem, from them, is a matter of considerable length. It will not therefore be amiss to give a shorter proof of the truth of this important theorem, that shall not require a previous knowledge of the properties of the figurate numbers, but yet shall be equally conclusive with that which is derived from those properties. Now this may be done in the manner following.

Let us suppose that the coefficients of the terms of the first six powers of the Binomial quantity 1 + x have been found, upon trial, to be such as would be produced by the general expressions 1, m/1, m/1.(m - 1)/2, m/1.(m - 1)/2.(m - 2)/3, &c, by substituting in them first 1, then 2, then 3, then 4, then 5, and lastly 6, instead of m. This may easily be tried by raising the said first six powers of 1 + x by repeated multiplications by 1 + x in the common way, and afterwards finding the terms of the same powers by means of the said general expressions above; which will be found to produce the very same terms as arose from the multiplications. After these trials we shall be sure that those general expressions are the true values of the coefficients of the powers of 1 + x at least in the said first six powers. And it will therefore only | remain to be proved that, since the rule is true in the said first six powers, it will also be true in the next following, or the 7th power, and consequently in the 8th, 9th and 10th powers, and in all higher powers whatsoever.

Now, if the coefficients of the 1st, 2d, 3d, 4th, and other following terms of (―(1 + x)))m be denoted by the letters a, b, c, d, &c, respectively, it is evident from the nature of multiplication, that the coefficients of the 1st, 2d, 3d, 4th, and other following terms of the next higher power of 1 + x, to wit, (―(1 + x)))m + 1 will be equal to a, a + b, b + c, c + d, &c, respectively, or to the sums of every two contiguous coefficients of the terms of the preceding series which is = (―(1 + x)))m. This will appear from the operation of multiplication, which is as follows.

a + bx + cx2 + dx3 + ex4 + &c
1 + x
a + bx + cx2 + dx3 + ex4 + &c
  + ax + bx2 + cx3 + dx4 + &c.
Therefore, if (―(1 + x)))m is equal to the series , then (―(1 + x)))m + 1 will be equal to the series .

Now let n be = m + 1. We shall then have to prove that, if the coefficients a, b, c, d, &c, be respectively equal to 1, m/1, (m/1).((m - 1)/2), (m/1).((m - 1)/2).((m - 2)/3), &c, the coefficients a, a + b, b + c, &c, will be respectively equal to 1, n/1, (n/1).((n - 1)/2), (n/1).((n - 1)/2).((n - 2)/3), &c.

In order to prove this, there is nothing more to do than to collect together every two terms of the former of these two series, and then substitute into these sums, n instead of m + 1, when there will immediately come out the terms of the latter series as above, viz, . Q. E. D.

Binomial Theorem, Improvement of. Mr Bonnycastle, of the Royal Mil. Acad. has lately discovered the following ingenious improvement of this theorem, which is now published for the first time.

This celebrated theorem has been given under various forms, since the time of its first invention; but the following property of it is conceived to be new, and capable of an application of which the original series is not susceptible.

The Newtonian theorem, in one of its most commodious forms, is &c; and the new theorem here alluded to, is &c. Of which the investigation is as follows: &c. Then by connecting the several powers of p with all the like powers of n, the latter series will become ; which by abbreviation, &c, becomes . In which last series, the literal parts of the coefficients of the 3d, 4th, 5th, &c terms, are the square, cube, biquadrate, &c, of the coefficient of the 2d term, as will appear either from the actual involution of &c, or by comparing its several powers with the multinomial theorem of Demoivre.

From hence it follows that, | &c. And if &c be put = s, we shall have &c, as was to be shewn.

By a similar mode of deduction, it may also be proved that ; where in this case .

In each of which formulæ, the index n, may be considered either as a whole number, a fraction, a surd, a given or an unknown quantity, as the circumstance may require.

For the application of these theorems, see LOGARITHMS, and Exponential Equations, following.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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* BERNOULLI (James)