Leibnitz, also, in a paper entitled De proportione Circuli ad quadratum circumscriptum in Numeris rationalibus, published in the Leipsic Acts, 1682, gave several curious numerical series, of a different nature from the former; among which may be reckoned the following:

The sum of an infinite number of terms of which

Also, if the odd terms only of this series be taken, as

the sum of the terms, when they are continued to infinity, will be.

2

And the sum of the even terms of the same series, as

1

taken as in the former instance, will be 4.

Series, translated and published by Colson, in 1736; and his Analysis by Equations of an Infinite Number of Terms, and his Quadratures, both of which were translated and published by Stewart, in 1745. Also the well known treatise of James Gregory, entitled Vera Circuli et Hyperbola quadratura, printed at Padua in 1667, and reprinted the following year at Venice; and the Logarithmotechnia of Mercator, London, 1668;

Lastly, the sum of an infinite number of terms of the same series, omitting every three terms from the first to the fifth, from the fifth to the ninth, &c.; as

will be equal to the area of the circle, of which the inscribed square is; or that of the circumscribed

1

square 2

But if we begin from the second term of the same series, and omit the three intermediate terms between that and the sixth, &c.; as

The sum of an infinite number of these terms will be equal to the area of the space included between the curve and the asymptotes of an equilateral hyperbola, or hyp. log. 2.

He also gave, in 1683, in the same work, the summation of several other series, of a more difficult kind, which have the numerators of the different terms either unity, or the natural, or triangular numbers, &c. and their denominators, certain numbers in geometrical progression, the signs being alternately + and −; as

The sum of an infinite number of terms of which

are, respectively,

20 400
21' 441'

8000

and

9261

Leibnitz, indeed, barely announced these series in the publication above mentioned; but the truth of them was soon after demonstrated, in different ways, by James and John Bernoulli; the former of whom also gave, in a small tract entitled Positiones Arith. de Serieb. Infinitis, &c. 1692 and 1697, various other series, of the form,

the sums of an infinite number of terms of which

are respectively equal to 2, 4, and 8.

He here likewise shows that the sum of the reciprocals of the series of triangular numbers

And that the sum of the reciprocals of the following polygonal numbers,

since been greatly extended, and rendered more general, was a considerable step for that time.

One of the means used by the two illustrious brothers above mentioned, in the resolution of problems of this kind, was to convert the proposed series into several others whose summation is known; but that which they more frequently employed, consists in subtracting from an assumed series the same series deprived of some of its leading terms; in which case, the sum of the resulting series will be determined.

Thus, supposing the following series of the reciprocals of the natural numbers,

without regarding whether the sum be finite or infinite (s).

Then, if from this sum, or the value of s, there be taken the same series, deprived of its first term, we shall have,

(s) The sum of this series, although its terms continually decrease, can be readily shown to be infinite, or greater than any assignable number; as will be done in the next article, which treats of the theory of logarithms; under which head the demonstration properly falls.

Where it appears that this last series, continued to infinity, is equal to 1.

In like manner, if the two first terms of the

same series be omitted, and the rest be subtracted

Which, as before observed, is the sum of this last series, continued ad infinitum.

Also, if the first term of this last series be omitted, and the rest subtracted from it, we shall have

Where the law of the terms, both in the numnerator and denominator, is sufficiently obvious.

The same results may also be readily obtained by the method made use of by de Moivre (Miscel. Analyt. lib. vi, cap. 3); which consists in multiplying an assumed series by some binomial or trinomial expression, and then equating that ex