the series 50, 25, 12, &c., and the time required to make these bounds will be the sum of the geometricai

50

25

1 12 2

progression (16), (1 (5)(1) , &c. which sum

12

16

12

,

If this is added to the sum of the for

The value of this expression exceeds a little 141, so that the whole time before the body comes to a state of rest is about 14 seconds, or less than + of a minute.

A question is often proposed in Mathematical geography, which leads to conclusions similar to the above. The question is this: Suppose the earth to be a perfect sphere, entirely covered with water, and that a vessel starting from the equator, sails without deviating from its course or velocity, always in the direction of North-east, when will it reach the north pole of the earth ?

The answer is generally given, that it will never reach the pole. This is wrong, for although the track of the vessel is in the form somewhat of a spiral, which makes an infinite number of turns about the earth before reaching the pole, still the whole time required to perform these turns, or to pass over its whole track, is finite. The technical name of this curve is the Loxodromic

curve.

POSITION.

There is a method by which many questions may be accurately wrought, and by which all others may be

solved approximately. It consists in making one or more assumptions for the answer, and then from the error or errors thus arising, to deduce the true answer, or its approximate value, when only an approximation can be

obtained.

This method is called Position, and sometimes it is called the Rule of False, or the Rule of False Position, or, which is far better, the Rule of Trial and Error. This rule admits of two varieties, Single Position, and

Double Position.

In Single Position, only one assumption is required, while in Double Position two assumptions are necessary. Single Position may be employed in the solution of questions, in which the required number is in any manner increased or decreased by any given part of itself; that is, when it is multiplied or divided by any given

number.

Double Position must be used when the result obtained by increasing or decreasing the required number in any given ratio, is also increased or decreased by some number independent of the required number. Or when any root or power of the required number is either directly or indirectly given in the conditions of the question.

SINGLE POSITION.

From the above definition of Single Position, it follows that if the number assumed is only one-half the true number, the result will be only one-half of the result of the question. If the assumed number is twice as great

as the true number, the result will be twice as great as the result of the question. And in general, the result obtained will be to the result of the question, as the assumed number is to the true number. Hence, we deduce for Single Position this

RULE.

Assume any convenient number, and perform on it the operations required by the question, then, as the result thus obtained, is to the result of the question, so is the assumed number to the true number required.

EXAMPLES.

1. Find a number such that being increased by onehalf, one-third, one-fourth, and one-fifth of itself, the sum shall be 1644.

If we assume 120 for the number, its half will be 60, its third 40, its fourth 30, and its fifth 24. Hence, 120 increased by its half, its third, its fourth, and its fifth, becomes 274 for our result, and

274:1644::120:720=the number sought.

2. A father bequeaths to his three sons $10700, in such a manner that the share of the first being multiplied by 5, that of the second by 6, and that of the third by 7, the products will be equal. What was each one's share?

Since the first multiplied by 5, the second by 6, and the third by 7, give equal products, it follows that their portions are as the reciprocals of these numbers, that is, as,,. These fractions reduced to a common denominator give the following numerators: 42, 35, 30. We will therefore assume that the first had $42, the second

$35 and the third $30. Taking the sum, we find $107

for our result. Hence,

107:10700:: $42: $4200=portion of first son. 107:10700::$35: $3500= do. second

107:10700::$30:$3000= do.

third

66

66

3. A, B, and C, joined their stock and gained $360, of which A received a certain sum, B received 3 times as much as A, and C as much as A and B together. What share of the gain had each?

Suppose A received $10, then as B received 31⁄2 times as much, he must have had $35, and as C had as much as A and B together, his portion must have been $45. Hence, all together received $90, which is our result. Hence, 90: 360:: $10: $40= A's gain.

Consequently, B's gain was $140, and C's gain $180, so that all together gained $360.

We will not extend the method of Single Position any further, since all questions under this rule may be solved by the succeeding rule for Double Position, or indeed without Position, by the method of Analysis.

DOUBLE POSITION.

The method of Double Position requires two assumed numbers, and depends upon the principle that the differences between the true and assumed numbers, are to each other as the differences between the result given in the question and the results arising from the assumed numbers. This principle is rigidly correct for such questions as, when solved by Algebra, would give rise to simple equations; but it is only approximately correct

for all other questions. Admitting the above principle, we readily deduce the following

RULE.

Assume two different numbers, and perform on them separately the operations indicated in the question. Then, as the difference of the results thus obtained, is to the difference of the assumed numbers, so is the difference between the true result and either of the others, to the correction to be applied to the assumed number which gave this result. Add the correction to this number, if the corresponding result was too small; if too large, it must be subtracted.

When the question is of such a nature as to admit only of an approximate solution, we may for a second approximation assume the number already found for the first, and that one of the two first assumptions which was nearer the true answer, or any other number that may appear to be still nearer. In this way, by repeating the operation as often as may be necessary, the true result may be approximated to any assigned degree of accuracy.

The above method of approximation by Double Position is of very little value in the usual questions of Arithmetic, but it becomes of the greatest utility in Algebra, affording, in many cases, a very concise and convenient mode of approximating the roots of equations, and finding the values of unknown quantities in very complex expressions, without making the usual reductions.

EXAMPLES.

1. Required a number, from which if 2 be subtracted,