BINOMIAL

, a quantity consisting of two terms or members connected by the sign + or -, viz, plus or minus; as a+b, or 3a-2c, or a2+b, or x2-2√c, &c; denoting the sum or the difference of the two terms. But the difference is also sometimes named a residual, and by Euclid an apotome. The term binomial was first introduced by Robert Recorde; see his algebra, pa. 462.

Binomial Line, or Surd, is that in which at least one of the parts is a surd. Euclid enumerates six kinds of binomial lines or surds, in the 10th book of his Elements, which are exactly similar to the six residuals or apotomes there treated of also, and of which an account is given under the art. Apotome, which see. Those apotomes become binomials by only changing the sign of the latter term from minus to plus, which therefore are as below.

Euclid's 6 Binomial Lines.
First binomial 3+√5,
2d binomial √18+4,
3d binomial √24+√18,
4th binomial 4+√3,
5th binomial √6+2,
6th binomial √6+√2.

To extract the Square Root of a Binomial, as of a+√b, or √c+√b. Various rules have been given for this purpose. The first is that of Lucas De Burgo, in his Summa de Arith. &c, which is this: When one part, as a, is rational, divide it into two parts such, that their product may be equal to 1/4th of the number under the radical b; then shall the sum of the roots of those parts be the root of the binomial sought: or their difference is the root when the quantity is residual. That is, if c+e=a, and ce=(1/4)b; then is the root sought. As if the binomial be 23+√448; then the parts of 23 are 16 and 7, and their product is 112, which is 1/4th of 448; therefore the sum of their roots 4+√7 is the root sought of 23+√448.

De Burgo gives also another rule for the same extractions, which is this: The given binomial being, for example, √c+√b, its root will be ; conseq. 4+√7 is the root sought, as before. Again, if the binomial be √18+√10; here c=18, | and b=10; theref. is the root of √18+√10 sought. And this latter rule has been used by all other authors, down to the present time. To extract the Cubic and other higher Roots of a Binomial. This is useful in resolving cubic and higher equations, and was introduced with the resolution of those equations by Tartalea and Cardan. The rules for such extractions are in great measure tentative; and some of the principal ones are the following.

Tartalea's Rule for the Cube Root of a Binomial p+q. This rule is given in his 9th book of Miscellaneous Questions, quest. 40; and it is made out from either of the terms, p or q, of the binomial, taken fingly, in this manner: Separate either term, as p, into two such parts that the one of them may be a cubic number, and the other part divisible by 3 without a remainder; then the cube root of the said cubic part will be one term of the root, and the other term will be the square root of the quotient arising from dividing the aforesaid third part by the first term just found. So if p be divided into r3+3s, then the root is r+√s/r. For example, to extract the cube root of √108+10. Suppose the part 10 be taken: this separates into the parts 1 and 9, the former of which is a cube, and the latter divisible by 3; that is r3=1, and 3s=9; hence r=1, and s=3; consequently is the cubic root of √108+10 sought. Again, to use the other term √108: this divides into √27+√27, of which the former is a cube, and the latter divisible by 3; that is, the cube root, the same as before.

Bombelli's Rule for the Cubic Root of the Binomial a+√-b. First find ; then, by trials, search out a number c, and a square root √d, such that the sum of their squares ; then shall c+√-d be the cube root of a+√-b sought. For example, to find the cube root of 2+√-121: here ; then taking c=2, and d=1, it is , and , as it ought; therefore 2+√-1 is the cube root of 2+√-121 sought.—Bombelli gave also a rule for the cube root of the binomial a+√b, but it is good for nothing.

Albert Girard's Rule for the Cube Root of a Binomial. This is given in his Invention Nouvelle en l'Algebre, and is explained by him thus: Let 72+√5120 be the given binomial whose cubic root is sought.

 The square of 72 the greater term is 5184 and of the less term is 5120 their difference 64 its cube root 4,
which 4 must be the difference between the squares of the two terms of the root sought; and as the rational part 72 of the given binomial is the greater term, therefore the rational part of the required root
 2+√0 3+√5 4+√12 5+√20
will be the greater part also; consequently the cubic root sought must be one of the binominals here set in the margin, where the difference of the squares of the terms is always 4, as required; and to find out which of them it must be, proceed thus: The first, 2+√0 must be rejected, because one term of it is 0 or nothing; also because 5 exceeds the cube root of 72, or √20 exceeds the cube root of √5120, therefore 5+√20, and all after it must be rejected too; so that the root must be either 3+√5 or 4+√12, if the given quantity has a binomial root: to know which of these is to be taken, it must be considered that the rational term of the root must measure the rational term given; and also the irrational term of the root must measure the irrational term given; then, on examination it is found that both 3 and 4 measure or divide the 72 without a remainder, but that only the √5, and not √12, measures √5120; consequently none but 3+√5 can be the cube root of the given quantity 72+√5120; which is found to answer, by cubing the said root 3+√5.

Dr. Wallis's Rule for the Cube Root of a Binomial a±mb or a±m√-b. In these forms the greatest rational part m is extracted out of the radical part, leaving only b the least radical part possible under the radical sign. He then observes that if the given quantity have a binomial root, it must be of this form c±nb, with the same radical b. Then to find the value of c and n, he raises this root to the 3d power, which gives , which must be=a±mb the given quantity; hence putting the rational part of the one quantity equal to that of the other, and also the radical part of the one equal to that of the other, gives . Then assuming several values of n, from the last equation he finds the value of c; hence if these values of c and n, substituted in the first equation, make it just, they are right; but if not, another value of n is assumed, and so on, till the first equation hold true. And it is to be noted that n is always an integer or else the half of an integer. For example, if the cube root of 135 ± √1825 be required, or 135±78√3; here a=135, m=78, and b=3; hence ; then assuming n=1, this last equation becomes , from whence c is found=5; which values of c and n being substituted in the first equation , makes , but ought to be 135, shewing that c is too great, and consequently n taken too little. Let n therefore be assumed=2, so shall , and c=3; and the first equation becomes as it ought, which shews that the true value of n is 2, and that of c is 3; hence then the cube root of 135±78√3 or c±nb is | 3±2√3 or 3±√12. And in like manner is the process instituted when the number in the radical is negative, as the cube root of 81±30√-3, which is (9/2)±(1/2)√-3.

Another rule for extracting the cube root of an imaginary binomial was also given by Demoivre, at the end of Saunderson's Algebra, by means of the trisection of an arc or angle.

Sir I. Newton's Rule for any Root of a Binomial a±b. In his Universal Arith. is given a rule for the square root of a binomial, which is the same as the 2d of Lucas de Burgo, before given; and also a general rule for any root of a binomial, which I have not met with elsewhere; and it is this: Of the given quantity a±b, let a be the greater term, and c the index of the root to be extracted. Seek the least number n whose power nc can be divided by aa-bb without a remainder, and let the quotient be q; Compute in the nearest integer number, which call r; divide aq by its greatest rational divisor, calling the quotient s; and let the nearest integer number above be the root sought, if the root can be extracted. And this rule is demonstrated by s'Gravesande in his commentary on Newton's Arithmetic. And many numeral examples, illustrating this rule, are given in s'Gravesande's Algebra, abovementioned, pa. 160, as also in Newton's Univers. Arith. pa. 53 2d edit. and in Maclaurin's Algebra pa. 118. Other rules may be found in Schooten's Commentary on the Geometry of Descartes, and elsewhere.

Impossible or Imaginary Binomial, is a binomial which has one of its terms an impossible or an imaginary quantity; as a+√-b.

Binomial Curve, is a curve whose ordinate is expressed by a binomial quantity; as the curve whose ordinate is . Stirling, Method. Diff. pa. 58.

Binomial Theorem, is used to denote the celebrated theorem given by Sir I. Newton for raising a binomial to any power, or for extracting any root of it by an approximating infinite series. It was known by Stifelius, and others, about the beginning of the 16th century, how to raise the integral powers, not barely by a continued multiplication of the binomial given, but Stifelius formed also a table of numbers by a continued addition, which shewed by inspection the coefficients of the terms of any power of the binomial, contained within the limits of the table; but still they could not independent of a table, and of any of the lower powers, raise any power of a binomial at once, by determining its terms one from another only, viz, the 2d term from the 1st, the 3d from the 2d, and so on as far as we please, by a general rule; and much less could they extract general algebraic roots in infinite series by any rule whatever.

For although the nature and construction of that table, which is a table of figurate numbers, was so early known, and employed in the raising of powers, and the extracting the roots of pure numbers; yet it was only by raising the numbers one from another by con tinual additions, and then taking them from the table when wanted, till Mr. Briggs first pointed out the way of raising any line in the table by itself, without any of the preceding lines; and thus teaching to raise the terms of any power of a binomial, independent of any of the other powers; and so gave the substance of the binomial theorem in words, wanting only the algebraic notation in symbols; as is shewn at large at pa. 75 of the historical introduction to my Mathematical Tables. Whatever was known however of this matter, related only to pure or integral powers, no one before Newton having thought of extracting roots by infinite series. He happily discovered that, by considering powers and roots in a continued series, roots being as powers having fractional exponents, the same binomial series would equally serve for them all, whether the index should be fractional or integral, or whether the series be finite or infinite. The truth of this method however was long known only by trial in particular cases, and by induction from analogy; nor does it appear that even Newton himself ever attempted any direct proof of it: however, various demonstrations of the theorem have since been given by the more modern mathematicians, some of which are by means of the doctrine of fluxions, and others, more legally, from the pure principles of algebra only: for a full account of which, see pa. 71 &c, of my Mathematical Tracts, vol. 1.

This theorem was first discovered by Sir I. Newton in 1669, and sent in a letter of June 13, 1676, to Mr. Oldenburgh, Secretary to the Royal Society, to be by him communicated to Mr. Leibnitz; and it was in this form: : where p + pq signifies the quantity whose root, or power, or root of any power, is to be found; p being the first term of that quantity; q the quotient of all the rest of the terms divided by that first term; and m/n the numeral index of the power or root of the quantity p+pq, whether it be integral or fractional, positive or negative; and lastly a, b, c, d, &c, are assumed to denote the several terms in their order as they are found, viz, a=the first term p(m/n), b=the 2d term (m/n)aq, c=the 3d term (m-n)/2n bq, and so on. As Newton's general notation of indices was not commonly known, he takes this occasion to explain it; and then he gives many examples of the application of this theorem, one of which is the following.

Ex. 1. To find the value of , that is, to extract the square root of c2+x2 in an infinite series. Here , &c; and therefore the root sought is . |

A variety of other examples are also given in the same place, by which it is shewn that the theorem is of universal application to all sorts of quantities whatever.

This theorem is sometimes represented in other forms, as ; which comes to the same thing. Or also thus .

In another letter to Mr. Oldenburgh, of Oct. 24, 1676, Newton explains the train of reasoning by which he obtained the said theorem, as follows: “In the beginning of my mathematical studies, when I was perusing the works of the celebrated Dr. Wallis (see his Arith. of Insinites, prop. 118, and 121, also his Algebra chap. 82), and considering the series by the interpolation of which he exhibits the area of the circle and hyperbola; for instance, in this series of curves, whose common base or axis is x, and the ordinates respectively ; I perceived that if the areas of the alternate curves, which are x, , &c; could be interpolated, we should obtain the areas of the intermediate ones; the first of which, or , is the area of the circle: now in order to this, it appeared that in all the series the first term was x; that the 2d terms (0/3)x3, (1/3)x3, (2/3)x3, (3/3)x3, &c, were in arithmetical progression; and consequently that the first two terms of all the series to be interpolated would be , &c.

“Now for the interpolation of the rest, I considered that the denominators 1, 3, 5, 7, &c, were in arithmetical progression; and that therefore only the numeral coefficients of the numerators were to be investigated. But these in the alternate areas, which are given, were the same with the figures of which the several powers of 11 consist, viz, of 110, 111, 112, 113 &c; that is, the first 1,

the second 1, 1
the third 1, 2, 1
the fourth 1, 3, 3, 1
the fifth 1, 4, 6, 4, 1
&c.

“I enquired therefore how, in these series, the rest of the terms may be derived from the first two being given; and I found that by putting m for the 2d figure or term, the rest would be produced by the continued multiplication of the terms of this series, .

“For instance, if the 2d term m=4; then shall , or 6, be the 3d term; and , or 4, the 4th term; and , or 1, the 5th term; and , or 0, the 6th; which shew<*> that in this case the series terminates.

“This rule therefore I applied to the series to be interpolated. And since, in the series for the circle, the 2d term was (1/2)x3/3 I put m=1/2, which produced the terms ; and so on ad infinitum. And hence I found that the required area of the circular segment is .

“And in the same manner might be produced the interpolated areas of the other curves: as also the area of the hyperbola and the other alternates in this series . And in the same way also may other series be interpolated, and that too if they should be taken at the distance of two or more terms.

“This was the way then in which I first entered upon these speculations; which I should not have remembered, but that in turning over my papers a few weeks since, I chanced to cast my eyes on those relating to this matter.

“Having proceeded so far, I considered that the terms , &c, that is, 1 , &c, might be interpolated in the same manner as the areas generated by them: and for this, nothing more was required but to omit the denominators 1, 3, 5, 7, &c, in the terms expressing the areas; that is, the coefficients of the terms of the quantity to be interpolated, , will be produced by the continued multiplication of the terms of this series .

“Thus, for example, there would be found |

“Thus then I discovered a general method of reducing radical quantities into infinite series, by the theorem which I sent in the beginning of the former letter, before I knew the same by the extraction of roots.

“But having discovered that way, this other could not long remain unknown: for, to prove the truth of those operations, I multiplied &c, by itself, and the product is 1 - x2, all the rest of the terms vanishing after these, in infinitum. In like manner, &c, twice multiplied by itself, produced 1 - x2. But as this was a certain proof of those conclusions, so I was naturally led to try conversely whether these series, which were thus known to be the roots of the quantity 1-x2, could not be extracted out of it after the manner of arithmetic; and upon trial I found it to succeed. The process for the square root is here set down

“These methods being found, I laid aside the other way by interpolation of series, and used these operations only as a more genuine foundation. Neither was I ignorant of the reduction by division, which is so much easier.” See Collins's Commercium Epistolicum.

And this is all the account that Newton gives of the invention of this theorem, which is engraved on his monument in Westminster Abbey, as one of his greatest discoveries.

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

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