SIGNS AND DEFINITIONS.

THE following signs are occasionally used in this Treatise :—

This (+) is the sign of addition, called plus, and signifies that the num-
bers before and after it are to be added together.

This (-) is the sign of subtraction, called minus, and signifies that the
number following it is to be deducted from the number preceding it.

This (X) is the sign of multiplication, and signifies that the numbers
on each side of it are to be multiplied into each other.

This (÷) is the sign of division, and signifies that the number prece-
ding it is to be divided by the number following it.

=

This (=) is the sign of equality, and signifies that the number or num-
bers before it equal the number or numbers following it. Thus, 8-8 ;
84-12; 9-36; 455 ·9; (6 × 3) ÷ 2-9; and (12-4)
X36-4. When numbers are included between the sign of a pa-
renthesis, as (4+6), it signifies that the numbers thus included are to be
taken together, and their sum or difference is to be added to, or multi-
plied, or divided by the number following or preceding, according as the
sign following or preceding indicates.

This ( : ) and this ( : :) are the signs of proportion. Thus, 3: 6 :: 4 :
that is, as 3 is to 6, so is 4 to 8.

This (9) is the sign of the square root, and signifies that the square
root of the number following is to be extracted. Thus, √9

√25—5.

3; and

A mean proportional between two numbers is the square root of their
product; and a third proportional to two numbers is the quotient obtained
by dividing the square of one of the numbers by the other.

The rectangle of two numbers signifies their product.

The word apothegm is used to signify the distance from the centre of
any figure to the centre of one of its sides; it is likewise used to signify
the perpendicular height of a triangle.

An absciss is the segment, or part of a line. See ¶ 29 and 30.

THE MECHANIC'S ASSISTANT.

DESCRIPTION OF THE SLIDING RULE.

1. THIS useful instrument was first invented by Thomas Everard, an Englishman, in the year 1683, and was made by Isaac Carver, a celebrated mechanic who resided in the vicinity of London. When shut, the rule is one foot in length, but by opening and drawing out the slider, it may be extended to three feet. One side of the rule, together with the under side of the slider, is divided into inches and eighths, or sixteenths of an inch; whilst the edge of the rule is divided into tenths and hundredths of a foot. By this means inches and eighths of an inch may be reduced to tenths and hundredths of a foot, simply by examining the edge of the rule opposite any number of inches and eighths.

On the face of the rule over the slider is a line of numbers called the A line. (This line was invented by Mr. Edmund Gunter, who applied the logarithms of numbers to a rule, by taking the lengths expressed by the figures in those logarithms from a scale of equal parts, and applying them to lines, as laid down on the rule.) On the slider there are two lines exactly like the former, or A line, the upper line being called the B line, and the lower one the C line. Under the slider is a line called the D line, or girt line. And at the right hand, near the end of the rule and opposite the lines which they represent, will be found the letters A, B, C, and D.

The numbers on these four lines increase from left to right; and the manner of reading off on the lines A, B, and C is the If the 1 at the beginning of the line A, be called onetenth, then the next primary division towards the right, marked

same.

2, will represent two-tenths, and the 3 will represent threetenths, the 4 four-tenths, the 5 five-tenths, the 6 six-tenths, the 7 seven-tenths, the 8 eight-tenths, the 9 nine-tenths, and the 1 at the middle of the rule will be ten-tenths, or one; and the following 12 will be twelve-tenths, or one and two-tenths; and the following 2 will count two, the 3 will count three, the 4 four, the 5 five, the 6 six; and so on to the end of the line, the 10 at the end of the line representing ten. Again, if the 1 at the beginning of the lines A, B, or C, represents one, or unity, the 2 following will count two, the 3 three, the 4 four, the 5 five; the 1 at the middle of the line will represent ten, the following 2 will be twenty, the 3 will stand for thirty, the 4 for forty, the 5 for fifty, the 6 for sixty, and so on, the 10 at the right representing one hundred. If we begin again at the left, and call the 1 ten, the following 2 will represent twenty, the 3 will stand for thirty, the 4 for forty, the middle 1 for one hundred, the following 2 for two hundred, the 3 for three hundred, the 4 for four hundred, and the 1 at the end, or the 10, will stand for one thousand.

Thus any value, as, for example, ten, or one hundred, or one thousand, or ten thousand, may be put on the 1 at the beginning of the line, or on the 1 at the middle of the rule; and then the numbers will increase towards the right, (as illustrated above,) but will diminish to the left in the same ratio. If, for example, the 10 at A be called ten thousand, the 9 to the left will represent nine thousand, the 8 will stand for eight thousand, the 7 for seven thousand, the 1 at the middle of the line for one thousand, the 9 for nine hundred, the 8 for eight hundred, and the 1 at the left for one hundred. Or, if we call the 10 at A, one, the 9 to the left will be ninetenths, the 8 will stand for eight-tenths, the 7 for seven-tenths, the 1 at the middle of the line for one-tenth, the 9 to the left for nine-hundredths, the 8 for eight-hundredths, and so on; the 1 at the left representing one-hundredth, or 80.

It is manifest that the value of the subdivisions must depend on that of the primary divisions; that is, the intermediate spaces must have a proportional value. If, for example, the 1 at the left hand be called one unit, and the 2 on the right

two units, then, as there are ten long marks between 1 and 2 to denote the principal subdivisions, each of these marks must be called one-tenth. Thus, if we call the 1 one unit, the first long mark between 1 and 2 will represent one and one-tenth, or 1.1, and the second one and two-tenths, or 1.2, and the third 1.3, the fourth 1.4, the fifth 1.5, or one and a half, and so on to the next primary division, which represents two units. Then between 2 and 3 there are ten marks, or if more than ten there will be ten long marks to denote the principal subdivisions, and these must be reckoned as before, the first mark (or first long mark) after the 2 representing 2.1, the second 2.2, the third 2.3, and so on to 3; and in like manner we continue to reckon till we come to 1 or 10 at the middle of the rule; unless (as is sometimes the case) there should be but five spaces or divisions laid down on the rule between 8 and 9, and 9 and 10, in which case you must count these spaces two-tenths, the first after the 9 representing nine and two-tenths, or 9.2, the second 9.4, the third 9.6, and so on to 1, or 10 at the middle of the line. Then, as there are ten long marks, or ten principal subdivisions between 10 and 20, each of these must be reckoned one unit; the first, therefore, will represent 11, the second 12, the third 13, the fourth 14, and so on. Or, beginning at the left hand at 1, and calling the 1 one-tenth, and the 2 following two-tenths, then, as there are ten principal divisions between 1 and 2, or one-tenth and two-tenths, each of these must be reckoned one-tenth of onetenth, or one-hundredth. Calling the 1 one-tenth, or .1, the first long mark will represent .11, or eleven hundredths, the second .12, the third .13, the fourth .14, and so on to .2. And between .2 and .3, each mark, or (if there be more than ten spaces or divisions) each long mark must be counted as before, one-tenth of one-tenth, or one-hundredth. Between .3 and 4, between .4 and .5, and so on to .8, there are ten marks or divisions between each of the principal divisions; and consequently each of these minor divisions must be valued as before, being called one-hundredth. Between .8 and .9, and .9 and .10, if there be ten spaces, each will be equal to onehundredth; but if there be but five divisions, each space must