DIVISION being the reverse of multiplication, therefore extend from the divisor to unity, that extent will reach from the dividend to the quotient.

Suppose 64 to be divided by 4, extend from 4 to 1, that ex-. tent will reach from 64 to 16, the quotient.

N. B. This extent in division is to be taken backwards from the dividend to the quotient, but in multiplication it is taken forward from the multiplicand to the product, they being contrary to one another.

PROPORTION, or the RULE OF THREE, being performed by multiplication and division, therefore extend from the first term to the second, that extent will reach from the third term 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 22, that extent will reach from 14 to 44 the same way.

In like manner may any other proportion, of any denomination, be worked, which makes this line of general use, particularly in measuring superficies and solids, which is done by extending from 1 to the breadth, that extent will reach from the length to the superficial content.

Example. Suppose a plank or board 15 inches broad, and 27 feet long, the content of which is required.

Extend from 1 to 1 foot 3 inches, 1.25, that extent will reach from 27 feet to 33.75 feet, the superficial content. Or extend from 12 inches to 15, &c.

The solid content of any bale, box, chest, &c. is found by extending from 1 to the breadth, that extent will reach from the depth to a fourth number, and the extent from 1 to that fourth number, will reach from the length to the solid content.

Example 1st. What is the content of a square pillar, whose length is 21 feet 9 inches, and breadth 1 foot 3 inches?

The extent from 1 to 1.25, will reach from 1.25 to 1.56, the content of 1 foot in length; again, the extent from 1 to 1.56, will reach from the length 21.75 to 33.98 or 34, the solid content in feet. Example 2d. Suppose a square piece of timber, 1.25, feet broad, .56 deep, and 36 long, be given, to find the content.

Extend from 1 to 1.25, that extent will reach from .56 to .7, then extend from 1 to .7, that extent will reach from 36 to 25.2 the solid content. In like manner may the contents of any bales, &c. be found, which, divided by 40, will give the tonnage.

3dly. The line of sines, marked (Sin.) begins at the left hand, and is figured thus: 1, 2, 3, 4, 5, &c. to 10; then 20, 30, 40, &c. to 90, ending at the right hand, where is a brass centre pin, here, and in all lines under it, are called degrees.

4thly. The line of versed sines, marked (V. S.) begins at the right hand, against 90° on the sines, and from thence figured towards the left hand, thus: 10, 20, 30, 40, &c. end at the left

hand-about 169°; each of the subdivisions, from 10 to 30, are 2 degrees, and from thence to 90, it is single degrees, and from thence to the end, each degree is divided into 15 minutes.

5thly. The line of tangents, marked (Tang.) begins at the left hand, as do the sines; from thence it is figured to the right hand, thus: 1, 2, 3, &c. to 10, and so on, 20, 30, 40, and 45, at the right hand, where is a little brass pin, just under and even with 90° iti the sines; from thence back again it is figured 50, 60, 70, 80, &c. to 89, ending at the left hand where it began at 1 degree. The subdivisions of this line are the same as those of the sines.

6thly. The line of the meridional parts, marked (Mer.) begins at the right hand, and is numbered thus: 10, 20, 30, to the left hand, where it ends at 87 degrees. This line, with the line of equal parts, marked (EP) under it, are used together, and only in Mercator's sailing. The uppermost line contains the degree of the meridians, or latitude, in a Mercator's chart; and the lower is the equator, and contains the degrees of longitude.

ON THE

DESCRIPTION AND USE OF THE SECTOR.

THIS instrument consists of two legs or rulers, representing the radius of a circle, moveable round a joint in the centre; on each face are drawn several lines or scales from the centre to almost the end of the legs, and are drawn on both legs, that every scale may have its fellow, and are called sectoral lines. There are other lines drawn parallel to the edges of the legs, and must be used with the sector quite open, the use of which is explained in the description of the Gunter scale. On one face are two lines of chords to 60 degrees, marked Cho. or C., two scales of equal parts to 10, marked Lin. or L., two lines of secants to 75 degrees, marked Sec. or S., two lines of polygons marked pol. Upon the other face the sectoral lines are two scales of sines to 90 degrees, marked Sin. or S., two lines of tangents to 45 degrees, marked, Tan. or T., two lines of upper tangents to supply the defect of the former, extending from 45 degrees to 75 degrees, and marked t. ; several pair of sectoral lines are numbered from the centre, and so arranged as to make equal angles at the centre; therefore at whatever distance the sector is opened, the angles will always correspond; that is, the distance or radius from 60 to 60 on the line of chords, are equal to 10 and 10 on the line of lines, 45 and 45 on the line of tangents, and 90 and 90 on the line of sines.

The lines of chords, sines, &c. are constructed as those on the Gunter scale, making 60 on the line of chords the radius of the

circle.

D

DIVISION being the reverse of multiplication, therefore extend from the divisor to unity, that extent will reach from the dividend to the quotient.

Suppose 64 to be divided by 4, extend from 4 to 1, that extent will reach from 64 to 16, the quotient.

N. B. This extent in division is to be taken backwards from the dividend to the quotient, but in multiplication it is taken forward from the multiplicand to the product, they being contrary to one another. PROPORTION, Or the RULE OF THREE, being performed by multiplication and division, therefore extend from the first term to the second, that extent will reach from the third term 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 22, that extent will reach from 14 to 44 the same way.

In like manner may any other proportion, of any denomination, be worked, which makes this line of general use, particularly in measuring superficies and solids, which is done by extending from 1 to the breadth, that extent will reach from the length to the superficial content.

Example. Suppose a plank or board 15 inches broad, and 27 feet long, the content of which is required.

Extend from 1 to 1 foot 3 inches, 1.25, that extent will reach from 27 feet to 33.75 feet, the superficial content. Or extend from 12 inches to 15, &c.

The solid content of any bale, box, chest, &c. is found by extending from 1 to the breadth, that extent will reach from the depth to a fourth number, and the extent from 1 to that fourth number, will reach from the length to the solid content.

Example 1st. What is the content of a square pillar, whose length is 21 feet 9 inches, and breadth 1 foot 3 inches?

The extent from 1 to 1.25, will reach from 1.25 to 1.56, the content of 1 foot in length; again, the extent from 1 to 1.56, will reach from the length 21.75 to 33.98 or 34, the solid content in feet. feet Example 2d. Suppose a square piece of timber, 1.25, broad, .56 deep, and 36 long, be given, to find the content. Extend from 1 to 1.25, that extent will reach from .56 to .7, then extend from 1 to .7, that extent will reach from 36 to 25.2 the contents of any bales, the solid content. In like manner may &c. be found, which, divided by 40, will give the tonnage.

3dly. The line of sines, marked (Sin.) begins at the left hand, and is figured thus: 1, 2, 3, 4, 5, &c. to 10; then 20, 30, 40, &c. to 90, ending at the right hand, where is a brass centre pin, here, and in all lines under it, are called degrees.

4thly. The line of versed sines, marked (V. S.) begins at the right hand, against 90° on the sines, and from thence figured towards the left hand, thus: 10, 20, 30, 40, &c. end at the left

1

hand-about 169°; each of the subdivisions, from 10 to 30, are 2 degrees, and from thence to 90, it is single degrees, and from thence to the end, each degree is divided into 15 minutes.

5thly. The line of tangents, marked (Tang.) begins at the left hand, as do the sines; from thence it is figured to the right hand, thus: 1, 2, 3, &c. to 10, and so on, 20, 30, 40, and 45, at the right hand, where is a little brass pin, just under and even with 90° iti the sines; from thence back again it is figured 50, 60, 70, 80, &c. to 89, ending at the left hand where it began at 1 degree. The subdivisions of this line are the same as those of the sines.

6thly. The line of the meridional parts, marked (Mer.) begins at the right hand, and is numbered thus: 10, 20, 30, to the left hand, where it ends at 87 degrees. This line, with the line of equal parts, marked (EP) under it, are used together, and only in Mercator's sailing. The uppermost line contains the degree of the meridians, or latitude, in a Mercator's chart; and the lower is the equator, and contains the degrees of longitude.

ON THE

DESCRIPTION AND USE OF THE SECTOR.

THIS instrument consists of two legs or rulers, representing the radius of a circle, moveable round a joint in the centre; on each face are drawn several lines or scales from the centre to almost the end of the legs, and are drawn on both legs, that every scale may have its fellow, and are called sectoral lines. There are other lines drawn parallel to the edges of the legs, and must be used with the sector quite open, the use of which is explained in the description of the Gunter scale. On one face are two lines of chords to 60 degrees, marked Cho. or C., two scales of equal parts to 10, marked Lin. or L., two lines of secants to 75 degrees, marked Sec. or S., two lines of polygons marked pol. Upon the other face the sectoral lines are two scales of sines to 90 degrees, marked Sin. or S., two lines of tangents to 45 degrees, marked, Tan. or T., two lines of upper tangents to supply the defect of the former, extending from 45 degrees to 75 degrees, and marked t. ; several pair of sectoral lines are numbered from the centre, and so arranged as to make equal angles at the centre; therefore at whatever distance the sector is opened, the angles will always correspond; that is, the distance or radius from 60 to 60 on the line of chords, are equal to 10 and 10 on the line of lines, 45 and 45 on the line of tangents, and 90 and 90 on the line of sines.

The lines of chords, sines, &c. are constructed as those on the Gunter scale, making 60 on the line of chords the radius of the circle.

D

The sectoral lines are like so many similar triangles, namely, that their corresponding sides are proportional, thus: let AC, AE, represent in plate 1. fig. 1. a pair of sectoral lines, forming the angle CAE, divide each leg into any number of equal parts (say 10) draw lines to any of the corresponding numbers, and each will be a similar triangle to CAE, and if the lines AC, AE, should represent the line of chords, sines, or tangents, and CE the radius, and D on the chord, sine, or tangent, any proposed number, then the transverse measure BD will be the chord, sine, or tangent of that number.

In describing the use of the sector, the term lateral distance is the distance on one leg, only taken from the centre to any part of a sectoral line; and the transverse distance is that taken between any two corresponding divisions on a scale of the same name. All are measured on the lines of each scale that are nearest each other.

The Line of Lines, or Proportional Scale.

The line of lines is used to divide a given line into any number of equal parts: suppose for example 8 deg., take the length of the line given in the compasses, and make it a transverse distance from 8 to 8, then will the transverse distance from 1 to be one of the equal parts, or of the whole; from 2 to 2 will be the 2d, &c. ; but if the line to be divided be too long for the legs of the sector, make any division so that it may be applied to the sector, multiplying each transverse distance by the same number you divided by.

To find a fourth proportional to any 3 given lines or numbers, as suppose 6, 2, and 4, take the lateral distance of 2 in your compasses, and make it the transverse distance at 6, then the transverse distance of 4 will give the lateral distance of 1 and . Or if a ship sailed 64 miles in 8 hours, how many miles did she sail in 5 hours at the same rate of sailing? Make the lateral distance of 64 the transverse distance at 8 and 8, then the transverse distance of 5 and 5 will give the lateral distance of 40, the fourth proportional. Having a chart constructed upon a scale of 5 miles to an inch, the sector is adjusted to a corresponding scale, by making the transverse distance from 5 to 5 equal to one inch. And to reduce a chart of 6 inches to a degree, to one of 4 inches to a degree, make the transverse distance of 6, 6, equal to the lateral distance of 4, then any distance from the chart set off laterally the corresponding transverse distance will be the distance required. And if you have a chart of 3 inches to a mile, to enlarge to 5 inches to a mile, make the transverse distance of 3, 3, equal to the lateral distance of 5, and proceed as before. A third proportional is found to two numbers; thus having 6 and 4 given to find a third proportional, make the transverse distance at 4 and 4, the lateral distance