DESCRIPTION AND USE

OF

GUNTER'S SCALE.

HILE the Reader is perufing the following, it is proper he fhould have a GUNTER'S SCALE before him.

WH

Gunter's Scale hath set upon it these eight lines following: ift. Sine rhumbs, marked (SR) is a line which contains the logarithms of the natural fine of every point and quarter point of the Mariner's Compafs, figured from the left hand towards the right, with 1, 2, 3, 4, 5, 6, 7, to 8, where is a brass pin, and where it can be done, into halves and quarters.

2d. Tangent rhumbs, marked (TR) alfo correfponds to the logarithm of the tangent of every point of the compafs, and is figured 1, 2, 3, 4, where there is a pin, and from thence towards the left hand with 5, 6, 7.

3d. The line of numbers marked (Num.) contains the logarithms of the numbers, and is figured thus; near the left hand it begins at 1, and towards the right hand is 2, 3, 4, 5, 6, 7, 8, 9; and then I, at which is a brass centre pin, going ftill on 2, 3, 4, 5, 6, 7, 8, 9, and 10 at the end, where there is another brafs pin; (as this line is generally much ufed, it requires a larger defcription.) 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 I may be reckoned i tenth, or 1 hundredth, or I thousandth part, &c. then the next is 2 tenth, or 2 hundredth, or 2 thousandth parts, &c. fo that if the first one be efteemed 1, the middle I 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 firft I is 10, the next 2 is 20, 3 is 30, so on, making the middle I now 100, the next 2 is 200, 3 is 300, 4 is 400, and 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 I is I, and the next 2 is 2, and 10 at the end is now 10. Again, if the first I be counted I hundredth part, the next is 2 hundredth parts, the middle one is now 10 hundredth parts, or 1 tenth part, and the next 2 is 2 tenth parts, and ro 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, muft all the intermediate ftrokes, or fubdivifions, be increased or diminished; that is, if the firft 1 at the left hand be counted I, then 2 (on the right hand of it) is 2, and each fubdivifion between them now is 1 tenth part, and fo all the way to the middle 1, which now is 10, the next. 2 is 20, now the longer ftrokes between 1 and 2 are to be counted from 1, thus; 11, 12,

(where is a brass pin), then 13, 14, 15, fometimes a longer ftroke than the reft, then 16, 17, 18, 19, 20, at the figure 2; and all the shorter Atrokes between them longer, are now each to be counted for I tenth part from the middle one to the next 2, now 20, from whence the longer ftrokes between the figures are units, thus 21, 22, 23, &c. to 3, which now is 30, and the shorter ftrokes each between them, now is the tenth part of an integer; from 3, each short stroke or divifion, is I tenth part of an unit. Again, if I at the left hand be 10, the figures between it and the middle i are common tens; and the fubdivifions between each figure are units; from the middle 1 to 10 at the end; each figure is fo many hundredths; and between these figures each longer divifion is 10; from the middle 1 to 2, each less divifion is 2 units; and, from 2 to the end, each shorter division is 5 units. From this description it will be eafy to find the divifions reprefenting any given number, thus: Suppofe the point reprefenting the number 12 was required: Take the divifion at the figure 1, in the middle, for the firft figure of 12; then, for the fecond figure, count 2 tenths, or longer ftrokes to the right hand, and this laft is the point reprefenting 12, where is the brafs pin.

Again, Suppose the number 22 were required, the first figure being 2, I take the divifion to the figure 2, and for the 2d figure 2, count 2 tenths onwards, and that is the point reprefenting 22.

Again, Suppofe 1728 were required; for the firft figure 1, I take the middle 1, for the second figure 7, count onwards as before, and that is 1700; then for the third 2 count 2 tenths from the last, and it represents 1720; laftly, for the 4th figure 8, estimate 8 parts out of 10 of the next fmaller divifion, or a little less than 10, this point, laft found, reprefents 1728.

Required the point, reprefenting the number 435: from the 4 in the 2d interval count towards 5 on the right, three of the larger divifions, and one of the fmaller, and that will be the divifion expreffing 435, and the like of other numbers, which by a little practice is readily done.

All fractions found in this line muff be decimals; and if they are not, they must be reduced into decimals, which is easily done by extending the compaffes from the denominator to the numerator; that extent laid upon I in the middle will reach to the decimal required.

Example. Required the decimal fraction equal to 2, extend from 4 to 3, that extent will reach from I on the middle to 75, towards the left hand; the like may be obferved 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 to the product.

Suppofe, for example, it was required to find the product of 16 multiplied by 4, extend from 1 to 4, that extent will reach from 16 to 64, the product required.

DIVISION

DIVISION being the reverfe of Multiplication, therefore extend from the divifor to unity, that extent will reach from the dividend to the quotient.

Suppofe 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 Divifion 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 Divifion, therefore extend from the firft term to the fecond, 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 fame way.

In like manner may any other proportion, of any denomination, be worked, which makes this line of general ufe, particularly in measuring Superfices and Solids, which is done by extending from I to the breadth, that extent will reach from the length to the fuperficial content.

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

Extend from I to I foot 3 inches, 1.25, that extent will reach from 27 feet to 33,75 feet, the fuperficial content. Or extend from 12 inches to 15, &c.

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

Example Ift. What is the content of a fquare 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 1,50, will reach from the length 21,75 to 33,98 or 34, the folid content in feet. Example 2d. Suppofe a fquare 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 folid 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 fines, 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 verfed fines, marked (V.S.) begins at the right hand, against 90° on the fines, and from thence figured towards the left hand, thus: 10, 20, 30, 40, &c. ending at the left hand-about

169°; each of the fubdivifions, from 10 to 30, are 2 degrees, and from thence to 90, it is fingle 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 fines; 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 brafs pin, juft under and even with 90° in the fines; 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 fubdivifions of this line are the same as thofe of the fines.

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 failing. The uppermoft 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.

HIS inftrument confifts of two legs or rulers, representing the Trad instrumente, moveable round a joint in the centre, on

each face are drawn feveral lines or fcales 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 fectoral 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 fcale. On one face are two lines of chords to 60 degrees, marked Cho. or C. two fcales of equal parts to 10, marked Lin. or L. two lines of fecants to 75 degrees, marked Sec. or S. two lines of poligons marked pol. Upon the other face the sectoral lines are two scales of fines to 90 degrees, marked Sin. or S. two lines of tangents to 45 degrees, marked Tan. or T. two lines of upper tangents to fupply the defect of the former, extending from 45 degrees to 75 degrees, and marked t. feveral pair of fectoral lines are numbered from the centre, and fo arranged as to make equal angles at the centre; therefore, at whatever diftance the fector is opened, the angles will always correfpond; 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 fines.

The lines of chords, fines, &c. are conftructed as those on the Gunter fcale, making 60 on the line of chords the radius of the circle.

C

The

PROBLEM IX.

The Angles and Hypothenufe of a Right-angled Triangle given, to find either of the Legs.

Given the hypothenufe 250 leagues, the angle oppofite the base 54° 30', confequently the other angle 35° 30'; the base and perpendicular are required.

90.

€35.00

250

Draw the line CB, and at C make an angle equal to 35° 30' by drawing the line CA, A4.30 take 250 from any convenient scale of equal parts, and fet it off from C to A, from A let fall the perpendicular AB, to cut the line CB, and it i done; for AB measured on the fame fcale gives 145, and CB 203.6 leagues.

36.30.

145

204.6. NOTE. The two acute angles of a right-angled triangle make 90 degrees.

PROBLEM X.

The Angles and one Leg of a Right-angled Triangle being given, to find the Hypothenufe and the other Leg.

The angle ACB 33° 15', the leg AC 285 miles, to find the hypothenfe and the other leg AB.

B

90. C33.45 B56.45

0341

Draw the bafe AC, lay off on it 285 from your fcale of equal parts, from A to C; on A erect the perpendicular AB: with the chord of 60° fweep the arch AD, and on it fet off 331, from your line of chords from A to D, through D and C, draw the right line BC, then BC will measure 341 nearly, and BA 187 nearly, on the fame fcale of equal parts that AC was taken from.'

PROBLEM XI.

33.45

The Hypothenuse and one Leg given, to find the Angles and the

other Leg.

The leg AB 350, the hypothenufe 600 given, to find the angles, and leg BC.

per

;

Draw the bafe CB, on B erect the pendicular AB, on which fet off 350 from B to A, on the point A with an opening of 600. Draw an arch to cut the line BC, in the point C draw AC, and it is done for the angle ACB will measure 35° 41′ on the line of chords, and BC will measure 487 nearly, on the fame fcale of equal parts before used.