is a brass centre pin, going still on 2, 3, 4, 5, 6, 7, 8, 9, and 10 at the end, where there is another centre pin; the first one may be counted for 1, or 10, or 100, or 1000; and then the next 2 is accordingly 2, or 20, or 200, or 2000, &c.

Again; the first 1 may be reckoned 1 tenth, or 1 hun dredth, or 1 thousandth part, &c. then the next 2 is 2 tenths, or 2 hundredths, or 2 thousandth parts, &c. so that if the first 1 be esteemed 1, the middle 1 is then 10, and 2 to its right is 20; 3 is 30; 4 is 40; and 10 at the end is 100.

Again; if the first one be 10, the next 2 is 20; 3 is 30, and so on, making the middle 1 now 100; the next 2 is 200; 3 is 300; 4 is 400, & nd 10 at the end is now 1000. In like manner, if the first 1 be esteemed 1 tenth part, the next 2 is 2 tenth parts; and the middle 1 is 1, and the next 2 is 2, and 10 at the end is now 10..>

Again; if the first 1 be counted 1 hundreth part, the next 2 is 2 hundredth parts; the middle 1 is now 10 hundredth parts, or one tenth part; and the next 2 is 2 tenth parts; and 10 at the end is now but one whole number, or integer.

As the figures are increased or diminished in their value, so in like manner must all the intermediate strokes or subdivisions be increased or diminished; that is, if the first 1 be counted 1, then 2 on the right of it is 2, and each subdivision between them now is one tenth part, and

next 2 is 20; now the longer strokes between one and 2 are to be counted thus, 11, 12, where is a brass pin; then 13, 14, 15; sometimes a longer stroke than the rest, 16, 17, 18, 19, 20, at the figure 2; and all the shorter strokes between the longer, are now each to be counted two tenth parts from the middle 1 to the next 2, now 20 from whence the longer strokes between the figures are units, thus, 21, 22, 23, &c. to 3, which now is 30, and the shorter strokes each between them, now is one tenth part of an integer; from 3, each short stroke or division, is one tenth part of an unit.

Again ; if 1 at the left hand be 10, the figures between it and the middle 1 are common tens; and the subdivisions between each figure are units: from the middle 1 to 10 at the end, each figure is so many hundredths; and between these figures, each longer division is 10; from the middle 1. to 2, each less division is 2 units; and from 2 to the end, each shorter division is 5 units.

From this description, it will be easy to find the divisi ons representing any given number, thus: suppose the point representing 12 is required? Take the division at the figure 1, in the middle for the first figure of 12; then for the second figure, count 2 tenths, or longer strokes, to the right hand, and this last is the point representing 12, where is a brass pin.

Again; suppose the number 22 is required? The fire figure being 2, take the division to the figure 2, and for the second figure 2, count 2 tenths onwards, and that is the point representing 22.

Again; suppose 1728 is required? For the figure 1, take the middle 1; for the second figure 7, count onwards as before, and that is 1700; then for the third figure 2, count 2 tenths from the last, and it represents 1720; lastly, for the fourth figure 8, estimate 8 parts out of 10 of the next smaller division, this point represents 1728.

Required the point representing the number 455 ? From 4 in the second interval, count towards 5 on the right hand, three of the large divisions, and one of the smaller, and that will be the division expressing 435, and the like of other numbers.

All fractions in this line must be decimals; and if they be not, they must be reduced into decimals, which is easily done by extending the compasses from the denominator to the numerator; that extent will reach from 1 in the middle to the decimal required.

EXAMPLE.

Required the decimal fraction equal to 2?

Extend from 4 to 3; that extent will reach from 1. in the middle to 75,.75, the decimal required, towards the left hand; and so of any other vulgar fraction.

MULTIPLICATION,

Is performed on this line, by extending from 1 to the multiplier; that extent will reach from the multiplicand

Suppose it is required to multiply 16 by 4?

Extend from 1 to 4; that extent will reach from 16 to 64, the product.

DIVISION,

Being the reverse of Multiplication; therefore extend from the divisor to 1; that extent will reach from the dividend to the quotient.

Required to divide 64 by 4?

Extend from 4 to 1; that extent will reach from 64 to 16, the quotient.

PROPORTION, OR THE RULE OF THREE,

TON,

Being performed by Multiplication and Division, there-fore extend from the first term to the third; that extent will reach from the second to the fourth.

EXAMPLE.

If the diameter of a circle be 7 inches, and the circumference 22; what is the circumference of another circle, the diameter of which is 14 inches?

Extend from 7 to 14; that extent will reach from 22 to 44, the circumference required.

In like manner, may any other proportion be worked.

To find the superficial content of a board, plank, &c.

Extend from 1 to the breadth; that extent will reach from the length to the superficial content.

EXAMPLE.

Suppose a board or plank 15 inches broad, and 27 feet long; required the content?

Extend from 1 to 1 foot 3 inches; that extent will reach from 27 feet to 33.75 feet, the superficial content.

Or, Extend from 12 inches to 15 inches, &c.

The solid content is found by extending from 1 to the breadth; that extent will reach from 1 to a 4th number; and from 1 to that 4th number, will reach from the length to the solid content.

EXAMPLE I.

What is the content of a square pillar, 21 feet 9 inches long, 1 foot 3 inches broad on each side?

The extent from 1 to 1.25, will reach from 1.25 to