called the Rule of Three from having three
numbers given to find a fourth; but more
properly, the Rule of Proportion, because
by it we find a fourth number proportional
to three given numbers: and because of the
necessary and extensive use of it, it is called
the Golden Rule. But to give a definition
of it, with regard to numbers of particular
and determinate things, it is the rule by
which we find a number of any kind of
things, as money, weight, &c. so propor-
tional to a given number of the same things,
as another number of the same or different
things is to a third number of the last kind
of thing. For the four numbers that are
proportional must either be all applied to
one kind of things; or two of them must
be of one kind, and the remaining two of
another: because there can be no propor-
tion, and consequently no comparison of
quantities of different species; as, for ex-
ample, of three shillings and four days: or
of six men and four yards. All questions
that fall under this rule may be distinguished
into two kinds: the first contains those
wherein it is simply and directly proposed
to find a fourth proportional to three given
numbers taken in a certain order: as if it
were proposed to find a sum of money so
proportioned to one hundred pounds as
sixty-four pounds ten shillings is to eighteen
pounds six shillings and eight-pence, or as
forty pounds eight shillings is to six hundred
weight. The second kind contains all such
questions wherein we are left to discover,
from the nature and circumstances of the
question, that a fourth proportional is
Sought; and consequently, how the state of
the proportion, or comparison of the term,
is to be made; which depends upon a clear
understanding of the nature of the question
and proportion. After the given terms are
duly ordered, what remains to be done is
to find a fourth proportional. But to re-
move all difficulties as much as possible,
the whole solution is reduced to the follow-
ing general rule, which contains what is
necessary for solving such questions wherein
the state of the proportion is given; in order
to which it is necessary to premise these
observations.

1. In all questions that fall under the following rule there is a supposition and a demand; two of the given numbers contain a supposition, upon the conditions whereof a demand is made, to which the other given term belongs; and it is therefore said to raise the question; because the number sought has such a connection with it as one

of these in the supposition has to the other. For example: if three yards of cloth cost 41. 10s. (here is the supposition) what are 7 yards 3 quarters worth? here is the demand or question raised upon 7 yards 3 quarters, and the former supposition.

2. In the question there will sometimes be a superfluous term; that is, a term which, though it makes a circumstance in the question, yet it is not concerned in the proportion, because it is equally so in both the supposition and demand. This superfluous term is always known by being twice mentioned either directly, or by some word that refers to it. Example, if three men spend 20l. in 10 days, how much, at that rate, will they spend in 25 days? Here the three men is a superfluous term, the proportion being among the other three given terms, with the number sought; so that any number of men may be as well supposed as 3.

Rule. 1. The superfluous term (if there is oue) being cast out, state the other three terms thus: of the two terms in the supposition, one is like the thing sought (that is, of the same kind of thing the same way ap plied); set that one in the second or middle place; the other term of the supposition set in the first place, or on the left hand of the middle; and the term that raises the question, or with which the answer is connected, set in the third place, or on the right hand; and thus the extremes are like one another, and the middle term like the thing sought: also the first and second terms contain the supposition, and the third raises the question; so that the third and fourth have the same dependance or connection as the first and second. 2. Make all the three terms simple numbers of the lowest denominations expressed, so that the extremes be of one name. Then, 3. Repeat the questions from the numbers thus stated and reduced (arguing from the supposition to the demand), and observe whether the number sought ought to be greater or lesser than the middle term, which the nature of the question, rightly conceived, will determine; and, accordingly, multiply the middle term by the greater or lesser extreme, and divide the product by the other, the quote is like the middle term, and is the complete answer, if there is no remainder; but if there is, then, 4. Reduce the remainder to the denomination next below that of the middle term, and divide by the same divisor, the quotient is another part of the answer in this new denomination. And if there is here also a remainder, re

latter is numbered from 7 to 36, 4 inches from the other end. We shall point out some of the uses of this rule.

The application of the inches, in measuring lengths, breadths, &c. is obvious. That of the Gunter's line, see under the article GUNTER'S LINE.

The use of the other side is that with which we are now concerned. 1. The breadth of any surface, as board, glass, &c. being given, to find how much in length makes a square foot. Find the number of inches the surface is broad, in the line of board measure, and right against it is the number of inches required. Thus, if the surface were eight inches broad, eighteen inches will be found to make a superficial foot. Or more readily thus: apply the rule to the breadth of the board, or glass, that end, marked 36, being eqnal with the edge, the other edge of the surface will show the inches, and quarters of inches, which go to a square foot. 2. Use of the table at the end of the board measure. If a surface be one inch broad, how many inches long will make a superficial foot? look in the upper row of figures for one inch, and under it in the second row is twelve inches, the answer to the question. 3. Use of the line of timber-measure. This resembles the former; for having learned how much the piece is square, look for that number on the line of the timber-measure; the space thence to the end of the rule is the length which, at that breadth, makes a foot of timber. Thus, if the piece be nine inches square, the length necessary to make a solid foot of timber is 214 inches. If the timber be small, and under nine inches square, seek the square in the upper rank of the table, and immediately under it is the feet and inches that make a solid foot. If the piece be not exactly square, but broader at one end than the other, the method is to add the two together, and take half the sum for the side of the square. For round timber the method is to girt it round with a string, and to allow the fourth part for the side of the square; but this method is erroneous, for hereby you lose nearly one fifth of the true solidity; though this is the method at present practised in buying and selling timber.

RULE, Coggeshall's sliding, is chiefly used for measuring the superficies and solidity of timber, &c. It consists of two rulers, each a foot long, one of which slides in a groove made along the middle of the ⚫ther.

On the sliding side of the rule are four lines of numbers, three whereof are double; that is, are lines to two radiuses; and one, a single broken line of numbers: the three first, marked A, B, C, are figured 1, 2, 3, &c. to 9; then 1, 2, 3, &c. to 10. The single line, called the girt-line, and marked D, whose radius is equal to the two radiuses of any of the other lines, is broke for the easier measurement of timber, and figured 4, 5, 6, 7, 8, 9, 10, 20, 30, &c. From 4 to 5 it is divided into ten parts, and each tenth subdivided into 2, and so on, from 5 to 6, &c. On the backside of the rule are, 1. A line of inch-measure, from 1 to 12; each inch being divided and subdivided. 2. A line of foot measure, consisting of one foot, divided into 100 equal parts, and figured 10, 20, 30, &c. The back part of the sliding piece is divided into inches, halves, &c. and figured from 12 to 24; so that when drawn wholly out, there may be a measure of two feet.

"Use of Coggeshal's Rule for measuring plane superficies.” 1. To measure a square: suppose, for instance, each of the sides 5 feet; set 1 on the line B, to 5 on the line A; then against 5 on the line B is 25 feet, the content of the square on the line A. 2. To measure a long square. Suppose the longest side 18 feet, and the shortest 10; set 1 on the line B, to 10 on the line A; then against 18 feet, on the line B, is 180 feet, the contents on the line A. 3. To measure a rhombus. Suppose the side 12 feet, and the length of a perpendicular let fall from one of the obtuse angles to the opposite side, 9 feet; set 1 on the line B, 12, the length of the side on the line A: then against 9, the length of the perpendicular on the line B, is 108 feet, the content. 4. To measure a triangle. Suppose the base 7 feet, and the length of the perpendicular let fall from the opposite angle to the base 4 feet; set 1 on the line B, to 7 on the line A; then against half the perpendicular, which is 2 on the line B, is 14 on the line A, for the content of the triangle. 5. To find the content of a circle, its diameter being given. Suppose the diameter 3.5 feet; set 11 on the girt line D, to 95 on the line C; then against 3.5 feet on D, is 9.6 on C, which is the content of the circle in feet. 6. To find the content of an oval or ellipsis. Suppose the longest diameter 9 feet, and the shortest 4. Find a mean proportional between the two, by setting the greater 9 on the girt line, to 9 on the liue C; then against the less number 4

on the line is C 6, the mean proportional sought. This done, find the content of a circle, whose diameter is 6 feet; this, when found, by the last article, will be equal to the content of the ellipsis sought.

"Use of Coggeshal's Rule in measuring timber." 1°. To measure timber the usual way. Take the length in feet, half feet, and, if required, quarters; then measure half way back again; then girt the tree with a small cord or line; double this line twice very evenly, and measure this fourth part of the girt or perimeter in inches, halves, and quarters. The dimensions thus taken, the timber is to be measured as if square, and the fourth of the girt taken for the side of the square, thus; set 12 on the girt line D, to the length in feet on the line C; then against the side of the square, on the girt-line D, taken in inches, you have, on the line C, the content of the tree in feet. For an instance: suppose the girt of a tree, in the middle, be 60 inches, and the length 30 feet, to find the content, set 12 on the girt-line D, and 30 feet on the line C; then against 15, one fourth of 60, on the girt-line D, is 46.8 feet, the content on the line C. If the length should be 9 inches, and the quarter of the girt 35 inches; here, as the length is beneath a foot, measure it on the line of foot-measure, and see what decimal part of a foot it makes, which you will find .75. Set 12, therefore, on the girt line, to 75 on the first radius of the line C, and against 35 on the girt-line is 64 feet on C, for the content. 2o. To measure round timber the true way. The former method, though that generally in usé is not quite just. To measure timber accurately, instead of the point 12 on the girt-line, use another, viz. 10.635; at which there should be placed a centre-pin. This 10,635 is the side of a square equal to a circle, whose diameter is 12 inches. For an instance: suppose the length 15 feet, and of the girt 42 inches, set the point 10.635 to 15, the length; then against 42 on the girt-line is 233 feet for the content sought; whereas by the common way, there arises only 184 feet. In effect, the common measure is only to the true measure, as 11 to 14. 3°. To measure a cube. Suppose the sides to be 6 feet each; set 12 on the girt-line D, to 6 on C; then against 72 inches (the inches 6 feet) on the girt-line, is 216 feet on C, which is the content required. 4°. To measure unequally-squared timber; that is, where the breadth and depth are not equal. Measure the length

of the piece, and the depth (at the end) is inches: then find a mean proportional between the breadth and depth of the piece. This mean proportional is the side of a square, equal to the end of the piece; which found, the piece may be measured as square timber. For an instance: let the length of the piece of timber be 13 feet, the breadth 23 inches, and the depth 15 inches; set 23 on the girt line D, to 23 on C; then against 13 on C is 17.35 on the girt-line D, for the mean proportional. Again, setting 12 on the girt-line D, to 13 feet, the length of the line C; against 17.35 on the girt-line is 27 feet, the content. 5. To measure taper timber. The length being measured in feet, note one-third of it; which is found thus: set 3 on the line A, to the length on the line B; then against 1 on A is the third part on B: then, if the solid be round, measure the diameter at each end in inches, and subtract the less diameter from the greater; add half the difference to the less diameter; the sum is the diameter in the middle of the piece. Then set 13.54 on the girt to the length of the line C, and against the diameter in the middle on the girt-line is a fourth number on the line C. Again, set 13.54 on the girt-line to the third part of the length on the line C; then against half the difference on the girt-line is another fourth number on the line C; these two fourth numbers, added together, give the content. For an instance: let the length be 27 feet (one third whereof is 9) the greater diameter 22 inches, and the lesser 18; the sum of the two will be 40, their difference 4, and half the difference 2, which, added to the less diameter, gives 20 inches for the diameter in the middle of the piece. Now set 13.54 on the girt-line to 27 on the line C, and against 20 on D is 58.9 feet. Again, set 13.44 of the girt-line to 9 on the line C; and against g on the girt-line (represented by 20) is .196 parts; therefore, by adding 58.9 feet to .196 feet, the sum is 59.096 feet, the content.

If the timber be square, and have the same dimensions; that is, the length 27 feet, the side of the greater end 22 inches, and that of the lesser 18 inches; to find the content, set 12 on the girt-line to 27, the length on the line C, and against 20 inches, the side of the mean square on the girt-line is 75.4 feet. Again, set 12 on the girt-line to 9 feet, one third of the length, on the line C, and against 2 inches, half the difference of the sides of the squares of the ends on the girt-line, is .25 parts of a foot;

RUM, a species of vinous spirit, distilled which is reeved the fall of the tackle, or the from sugar canes.