# POSITION

, or *Site,* or *Situation,* in Physics, is an
affection of place, expressing the manner of a body's
being in it.

, in Architecture, denotes the situation of a building, with respect to the points of the horizon. The best it is thought is when the four sides point directly to the four winds.

Position, in Astronomy, relates to the sphere. The position of the sphere is either right, parallel, or oblique, whence arise the inequality of days, the difference of seasons, &c.

*Circles of* Position, are circles passing through the
common intersections of the horizon and meridian, and
through any degree of the ecliptic, or the centre of any
star, or other point in the heavens; used for finding
out the position or situation of any star. These are
usually counted six in number, cutting the equator into
twelve equal parts, which the astrologers call the celestial
houses.

, in Arithmetic, called also False Position, or Supposition, or Rule of False, is a rule so called, because it consists in calculating by false numbers supposed or taken at random, according to the process described in any question or problem proposed, as if they were the true numbers, and then from the results, compared with that given in the question, the true numbers are found. It is sometimes also called Trialand-Error, because it proceeds by trials of false numbers, and thence finds out the true ones by a comparison of the errors.

Position is either Single or Double.

*Single* Position is when only one supposition is employed
in the calculation. And

*Double* Position is that in which two suppositions
are employed.

To the rule of Position properly belong such questions as cannot be resolved from a direct process by any of the other usual rules in arithmetic, and in which the required numbers do not ascend above the first power: such, for example, as most of the questions usually brought to exercise the reduction of simple equations in algebra. But it will not bring out true answers when the numbers sought ascend above the first power; for then the results are not proportional to the Positions,| or supposed numbers, as in the single rule; nor yet the errors to the difference of the true number and each Position, as in the double rule. Yet in all such cases, it is a very good approximation, and in exponential equations, as well as in many other things, it succeeds better than perhaps any other method whatever.

Those questions, in which the results are proportional to their suppositions, belong to Single Position: such are those which require the multiplication or division of the number sought by any n<*>mber; or in which it is to be increased or diminished by itself any number of times, or by any part or parts of it. But those in which the results are not proportional to their positions, belong to the double rule: such are those, in which the numbers sought, or their multiples or parts, are increased or diminished by some given absolute number, which is no known part of the number sought.

To work by *the Single Rule of* Position. Suppose,
take, or assume any number at pleasure, for the number
sought, and proceed with it as if it were the true number,
that is, perform the fame operations with it as, in
the question, are described to be performed with the
number required: then if the result of those operations
be the same with that mentioned or given in the question,
the supposed number is the same as the true one
that was required; but if it be not, make this proportion,
viz, as your result is to that in the question, so is
your supposed false number, to the true one required.

*Example.* Suppose that a person, after spending 1/3
and 1/4 of his money, has yet remaining 60l.; what ium
had he at first?

Suppose he had at first 120l.

Now 1/3 of 120 is | 40 |

and 1/4 of it is | 30 |

their sum is | 70 |

which taken from | 120 |

leaves remaining | 50, instead of 60. |

Therefore as the sum at first.

Proof. | 1/3 of 144 is | 48 |

1/4 of it is | 36 | |

their sum | 84 | |

taken from | 144 | |

leaves just | 60 as per quest. |

*To work by the Double Rule of*Position.

In this rule, make two different suppositions, or assumptions, and work or perform the operations with each, described in the question, exactly as in the single rule: and if neither of the supposed numbers solve the question, that is, produce a result agreeing with that in the question; then observe the errors, or how much each of the false results differs from the true one, and also whether they are too great or too little; marking them with + when too great, and with - when too little. Next multiply, crosswise, each position by the error of the other; and if the errors be of the same affection, that is both +, or both -, subtract the one product from the other, as also the one error from the other, and divide the former of these two remainders by the latter, for the answer, or number sought. But if the errors be unlike, that is, the one +, and the other -, add the two products together, and also the two errors together, and divide the former sum by the latter, for the answer.

And in this rule it is particularly useful to remember this part of the rule, viz. to subtract when the errors are alike, both + or both -, but to add when unlike, or the one + and the other -.

*Example.* A son asking his father how old he was,
received this answer: Your age is now 1/4 of mine; but 5
years ago your age was only 1/5 of mine at that time.
What then were their ages?

First, suppose the son 15; then the father's; also, 5 years ago the son was 10, and the father's must be 55, but ought to be 10 X 5 or 50, therefore the error is 5-. Again, suppose the son 22; then is the father's; also 5 years ago the son was 17, and the father's then 83, but ought to be 17 X 5, or 85, therefore the error is 2 +.

And the errors, being unlike, must be added, theirsum being 7.

Then 15 | 22 |

2 | 5 |

30 | 110 |

30 | |

7) 140 (20 the son's age, and consequently 80 the father's. |

This rule of Position, or trial-and-error, is a good general way of approximating to the roots of the higher equations, to which it may be applied even before the equation is reduced to a final or simple state, by which it often saves much trouble in such reductions. It is also eminently useful in resolving exponential equations, and equations involving arcs, or sines, &c, or logarithms, and in short in any equations that are very intricate and difficult. And even in the extraction of the higher roots of common numbers, it may be very usefully applied. As for instance, to extract the 3d or cubic root of the number 20.—Here it is evident that the root is greater than 2 and less than 3; making these two numbers therefore the suppositions, the process-will be thus:

1st sup. 2^{3} | = | 8 | 2d sup. 3^{3} | = | 27 | |||

given number | 20 | given number | 20 | |||||

1st error | 12 | - | 2d error | 7 | + | |||

3 | 2 | |||||||

12 | 36 | } | add | 14 | ||||

7 | 14 | |||||||

19 ) | 50 | ( | 2.63 the first approximation. |

Again, as it thus appears the cube root of 20 is near 2.6 or 2.7, make supposition of these two, and repeat the proeess with them, thus:

Ist sup. 2.6^{3} | = | 17.576 | 2d sup. 2.7^{3}= | 19.683 | ||

given number | 20. | given number | 20. | |||

Ist error | 2.424 | - | 2d error | 0.317 | - | |

2.7 | 2.6 | |||||

16968 | 1902 | |||||

4848 | 634 | |||||

2.424 | 6.5448 | } | subtr. | .8242 | ||

.317 | .8242 | |||||

2.107 | ) | 5.7206 | (2.714 root sought. |

The rule of Position passed from the Moors into Europe, by Spain and Italy, along with their algebra, or method of equations, which was probably derived from the former.

Position, in Geometry, respects the situation, bearing, or direction of one thing, with regard to another. And Euclid says, “Points, lines, and angles, which have and keep always one and the same place and situation, are said to be given by Position or situation.” Data, def. 4.

POSITIVE *Quantities,* in Algebra, such as are of a
real, affirmative, or additive nature; and which either
have, or are supposed to have, the affirmative or positive
sign + before them; as *a* or + *a,* or *bc,* &c. It is
used in contradistinction from negative quantities, which
are defective or subductive ones, and marked by the
sign -; as - *a,* or - *ab.*