And consequently r=4.3–.036=4.264, nearly. Again let 4.264 and 4.265 be the two assumed numbers ; then First Sup. Second Sup.

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2. Given 9: – 15)* +zva'-90, to find an approximate value of a. - * Here, by a few trials, it will be soon found, that the value of a lies between 10 and 11 ; which let, therefore, be the two assumed numbers, agreeably to the directions given in the rule. - - - - o . . . . . . Then First Sup. Second Sup. 25 . ... (32°–15)* . .84.64 31 622 . . ava. . .36.482 ... 56,622 Results 121,192. 121' 122 . ." 11 . . 121.122

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... And consequently al-11–.482= 10 518 Again, let 10.3 and 19.6 be the two assumed numbers,

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An exponential quantity is that which is to be raised to some unknown power, or which has a variable quantity

for its index; as - - ot - * * : -a”, aw, z*, or a “, &c.- : And an exponential equation is that which is formed between any expression of this kind and some other quan

tity, whose value is known; as * --

a *=b, aro-a, &c.

3. Where it is to be observed, that the first of these equations, when converted into logarithms, is the same as

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is the same as a log. z=log. a.
In the latter of which cases, the value of the unknown

quantity a may be determined, to any degree of exact

ness, by the method of double position, as follows:

- – - • * o * h *. auls. J. B.No-oo o l ** ~, > . -Find, by trial, as in the rule before laid down, twe numbers as near the number sought as possible, and substitute them in the given equation.

a log. a = log. a,

instead of the unknown quantity, noting the results obtained from each.

Then, as the difference of these results is to the difference of the two assumed numbers, so is the difference between the true result, given in the question, and either of the former, to the correction of the number belonging to the result used ; which correction being added to that number, when it is too little, or subtracted from it, when it is too great, will give the root required, nearly.

And, if the number, thus determined, and the nearest of the two former, or any other that appears to be nearer, be taken as the assumed roots and the operation be repeated as before, a new value of the unknown quantity will be obtained still more correct than the first ; and so on, proceeding in this manner, as far as may be thought necessary.


1. Given zo-100, to find an approximate value of w. Here, by the above formula, we have

a log a-log. 100–2, And since x is readily found, by a few trials, to be

nearly in the middle between 3 and 4, but rather nearer the latter than the former, let 3.5 and 3.6 be taken for the two assumed numbers.

Then log. 3.5–.5440680, which, being multiplied by 3.5, gives 1.904238=first result;

And log. 3.6=.5563025, which, being multiplied by 3.6, gives 2.002689 for the second result. Whence

2.002689 . . 3.6 . . 2.002689
.904238 . . 3.5 . . 2.

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For the first correction; which, taken from 3.6 leaves z=3.59727, nearly.

. And as this value is found, by trial, to be rather too small, let 3.597.27 and 3.597.28 be taken as the two assumed numbers.

Then log. 3.597.27=.5559781, which being multiplied by 3.59727, gives 1.9999854= first result.

And log. 3.59728–,5559743, which, being multiplied by 3.51728, gives 1.9999953=second result. Whence 1.9999953 . . 3.59.728 . . 2. 1.99998.54 . . 3. 597.27 . . 1.999995.3

.0000099 : .00001 :: .0000047 : 00000474747 For the second correction ; which, added to 3.597.28, gives r=3.597.28474747, extremely near the truth.

2. Given z*=2000, to find an approximate value of r. Ans. ac-4.82782263

3. Given (6x) =96, to find the approximate value of a. Ans. ace=1.8826432

4. Given z*=123456789, to find the value of 2. Ans, 86400268

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The binomial theorem is a general algebraical expression, or formula, by which any power, or root of a given quantity, consisting of two terms, is expanded into a series; the form of which, as it was first proposed by Newton, being as follows:

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7? Where P is the first term of the binomial, a the second term divided by the first, # the index of the power, or

root, and a, b, c, &c. the terms immediately preceding those in which they are first found, including their signs ... + or —. * Which theorem may be readily applied to any particular case, by substituting the numbers, or letters, in the given example, for P, Q, m, and n, in either of the : above formulae, and then finding the result according to the rule (!).

& (l) This celebrated theorem, which is of the most extensive

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