of mathematics and physics for the Supplement to the Encyclopædia Britannica; and the composition of this paper, as his nephew informs us, interrupted the execution of the second edition of his Illustrations. This dissertation, the first part of which Mr Playfair did not live to finish, is marked with the able but now faultering hand of its distinguished author. In the 70th year of his age, and haunted with the image of his unfinished work, the drudgery of compilation must have been irksome to a mind less anxious than his, and conscious that it was exhausting its powers on an arena where no laurels could be gained. The history of the mathematical and physical sciences had been already exhausted by the voluminous and profound labours of Montucla. The Abbé Bossut had gleaned its choicest flowers, and prepared them in nervous and simple rhetoric for the taste of less laborious students;-and that branch of the history of physics, for which Mr Playfair's talents were peculiarly adapted, namely, the history of astronomical discovery, had been illustrated by the brief yet powerful narrative of Laplace, by the rich and discursive eloquence of Bailly, and by the varied and searching erudition of Delambre. A field thus pre-occupied, and on which learning and genius had cast their richest offerings, was not likely to be chosen by Mr Playfair for the display of his own powers. He undertook the task to which he had been urged; and that part of it which Providence allowed him to execute, he executed with his usual judgment and discrimination.

Some time after his return from the Continent, Mr. Playfair read to the Royal Society a paper on Volcanoes, which excited great interest, but which is neither mentioned in his life, nor printed along with the rest of his works. On the 3d. December, 1818, he likewise communicated to the same body his Description of the Slide of Alpnach, which has been published as an appendix to his life. These two communications were an earnest of the stores of valuable information which he had accumulated during his travels, and which he meant to give to the world in a series of detached papers. His health, however, had been for some time on the decline; and in the winter of 1818-1819, his labours were often interrupted by a severe attack of a disease in the bladder, which, at his advanced period of life, it was not easy to subdue. An interval of health soothed for awhile the anxieties of his friends; but it was only a deceitful precursor of the fatal attack which carried him off, on the 19th July, 1819, in the 72d year of his

age. The various public bodies with which he was connected followed his remains to the grave, and testified the respect which they cherished for that rare union of worth and talents which marked the character of this much esteemed and deepJy lamented philosopher.

ART. LII.-A New Demonstration of the Binomial Theorem. By WARREN COLBURN.

THE celebrated Binomial Theorem, first invented by Sir Isaac Newton, has been demonstrated by means of the differential Calculus. But in this form it is without the reach of a large portion of students in Algebra; and this mode of demonstration is less satisfactory, because it only shows the truth of it without showing the reason. It has also been demonstrated algebraically on the principle of permutations and combinations. This mode of proof seems to show the reason of the formation, but the reason appears very far fetched, and on the whole rather unsatisfactory. Besides, it combines so many principles, and requires them all to be kept so distinctly in view, that it is very difficult for learners to comprehend it.

The following demonstration, it is believed, obviates some of these difficulties. It is derived directly from the actual formation of the powers by multiplication. It is also more easily comprehended, because it does not require all the parts to be kept in the mind at the same time.

The manner of demonstrating the rule for finding the coefficients only is considered as new. The rule itself is the same that has always been given. The rules for finding the sums of the several kinds of series are not new, but the mode of demonstrating them is believed to be new.

Let a few of the powers of a + be found and their formation attended to.

(a + x) = a*+4a3x+6a2x2+4ax3+x* 4

The law of the formation of the literal part is sufficiently manifest.

In each power there is one term more than the number denoting the power to which it is raised. The first power consists of two terms, the second power of three terms, the third power of four terms, &c.

In every power a is found in every term except the last, and x is found in every term except the first. The exponent of a in the first term is the same as the exponent of the power to which the binomial is raised, and it diminishes by one in each succeeding term.

The exponent of x in the second term is 1, and it increases by one in each succeeding term, until in the last term it is the same as that of a in the first term.

The law of the formation of the coefficients is not so simple, though it is not less remarkable.

The coefficients of the first power, viz. a+ x, are 1, 1;

those of the second power are 1, 2, 1. These are formed from the first as follows. When a is multiplied by a, it produces a, and no other term being produced like it, there is nothing added to it, and it remains with the same coefficient as the a in the multiplicand. In multiplying x by a and afterward a by x, two similar terms are produced, having the coefficients of the a and x in the multiplicand, viz. 1 and 1; and the addition of these forms the 2. The other 1 is produced like the first.

The coefficients of the third, power are 1, 3, 3, 1. The Is are produced from the second power, as those of the second power are produced from the first. In multiplying 2 ax by a, the term produced is 2 a2 x, having the coefficient of the second term of the multiplicand; and in multiplying a2 by x, the term produced is a x, similar to the last, and having the coefficient 1 of the first term of the multiplicand. The addition of the coefficients of these two terms produces the 3 before a x. That is, the coefficient of the second term of the third power is formed by adding together the coefficients of the first and second terms of the second power. In the same manner it may be shown, that the coefficient 3 of the third term of the third power is formed by adding together the coefficients of the second and third terms of the second power.

The following law will be found on examination to be general.

The coefficient of the first term of every power is 1. The coefficient of the second term of every power is formed by adding together the coefficients of the first and second terms of the preceding power. The coefficient of the third term of every power is formed by adding together the coefficients of the second and third terms of the preceding power.

The coefficient of the fourth term of every power is found by adding together the coefficients of the third and fourth terms of the preceding power. And so of the rest.

This law, though perhaps sufficiently evident by inspection, may be easily demonstrated.

Suppose the above law to hold true as far as some power which we may designate by n. The literal part of the nth power will be formed thus.

a”, an―1x, an—2x2, an−3 x3

a xn-1, x2.

We cannot write all the terms without assigning a particular value to n. We can write a few of the first and last. The points between show that the number of terms is indeterminate; there may or may not be more than are written.

VOL. II.-No. 5.

power. which will produce the next higher power, or the (n + 1)th of the third, &c. and let the whole be multiplied by a + x, Suppose that A is the coefficient of the second term, B that

a2+1 + A a” x + B an−1 x2 + C a2 - 2 x 3 + . . . . . . F a2x2-1 + a x2 a2 x + A an−1 x2 + В a2-2 x3 + С an−3 x2 +

a”+1+(1+A)a”x+(A+B)a”−1x2+(B+C)an−2x +(C+)a

Fax2 + xn+1

•·(+F)a2x2−1+(F+1)ax2+x”+1.

a2 + A an−1 x + B a2¬3 x2 + C a2¬3 x3 +........

Faxn−1 + xn

1

x + 1

61