furing of Angles according to the common Divifions of the Mariner's Compafs. If the Radius A C be divided into 100 or 1000, &c.. equal Parts, and the Lengths of the feveral Sines, Tangents, and Secants, correfponding to the feveral Arches of the Quadrant, be measured thereby, and thefe Numbers be fet down in a Table, cach in its proper Column, you will by this Means have a Triangular Canon of Numbers, by which the feveral Cafes in Trigonometry may he refolved. The Right Lines graduated as above, being placed feverally upon a Ruler, forin the Inftrument called the Plane Scale; by which the Lines and Angles of all Triangles may be measured. All Right Lines (as the Sides of Plane Triangles, &c. when they are confidered fimply as fuch, without having any Relation to a Circle) are measured by Scales of equal Parts; one of which is fubdivided equally into 10, and this ferves as a common Divifion to all the reft. In moft Scales an Inch is taken for a common Measure, to determine their Largenefs and Number of Parts: what an Inch is divided into, is generally fet at the End of the Scale, as in the Scales A, B, and C, the Numbers 10, 20, 30, fhew that fo many Parts of the Scales A, B, C, are contained in an Inch. By any Scale of equal Parts divided as above, any Number lefs than 100 may be readily taken; but if the Number thould confift of Three Places of Figures, the Value of the Third Figure can only be guessed at: wherefore, in thefe Cafes, it is better to ufe fuch a Scale as D, called a Diagonal Scale, by which any Number of Three Figures may be exactly found.

Having prepared a Ruler of convenient Breadth for your Scale, (which may be an Inch more or less) Firft, near the Edges thereof, draw Two Right Lines af, cg, parallel to each other; then divide one of thefe Lines as af, into equal Parts, according to the Large ness you intend your Scale; and through each of these Divisions draw perpendicular Right Lines as far as the Line cg; next divide the Breadth into 10 equal Parts; and through each of thefe Divifions draw Right Lines parallel to the former af and c g; again, divide the Length, a, b, c, d, each into 10 equal Parts; and from the Point d to the first Divifion in the Line A B, draw a Right Line; then, parallel to that Line, draw Right Lines through all the other, Divifions, and the Scale is done.

in

Befides the Lines already mentioned, there is another on the Plane Scale marked ML, which is joined to a Line of Chords; and fhews how inany Miles Eafting or Wefting make a Degree of Longitude every Latitude; these feveral Lines are generally put on one Side of a Ruler 2 Feet long; and on the other Side are laid down a Scale of the Logarithms of the Sines, Tangents, and Numbers, which is commonly called Gunter's Scale; and as it is of general Ufe, it requires a particular Description.

THE

DESCRIPTION and

O F

USE

GUNTER's SCA L E.

G

UNTER's Scale, hath fet upon it these Eight Lines following.

ift. Sine Rhumbs (marked S R) is a Line which contains the Logarithm of the Sine of every Degree, 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 Brafs Pin, and where it can, it is divided into Halves and Quarters.

2d. Tangent Rhumbs (marked T R) alfo correfponds to the Logarithm of the Tangent of every Degree of the said Compass, and is figured 1, 2, 3, 4, at the Centre, where is a Pin, and from thence towards the Left Hand with 5, 6, 7, it is alfo divided, where it can, into Halves and Quarters.

3d. The Line of Numbers (marked Num.) contains the Logarithm of the Numbers, and is figured thus; near the Left Hand End it begins at 1, and towards the Right Hand is 2, 3, 4, 5, 6, 7, 8, 9 ; then I is the Middle, at which is a Brafs Centre Pin, going ftill on 2, 3, 4, 5, 6, 7, 8, 9, and 10 at the End, where is another Centre Pin: (As this Line is generally ufed, it requires a larger Description) The first I 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 for 1 Tenth, or 1 Hundreth, or I Thousandth Part, &c. then the next is 2 Tenth, or 2 Hundredth, or 2 Thousandth Parts, &c. fo 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 1 is 10, the next 2 is 20, 3 is 30, and 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 I be efteemed 1 Tenth Part, the next 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 I be counted 1 Hundredth Part, the next is 2 Hundredth Parts, the Middle I is now 10 Hundredth Parts or 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 increafed or diminished in their Value, fo in like Manner muft all the intermediate Strokes or Subdivifions be increafed or decreased; that is, if the Firft 1 (at the Left Hand) be counted 1, then 2 (on the Right Hand of it) is 2; and each Subdivifion 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 Strokes between 1 and 2 are to be counted from I thus: 11, 12, (where is a Brafs Pin) then 13, 14, 15, (fometimes a longer Stroke than the reft) then 16, 17, 18, 19, and 20 at the Figure 2; and all the shorter Strokes between thofe longer are now each to be counted for a Tenth Part, 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 between them, each now is a Tenth Part of an Integer; from 3, each fhort Stroke, or little Divifion, is 5 Tenth Parts of an Unit.

Again, if one at the left Hand be 10, the Figures between it and the Middle one are common Tens; and the Subdivifions, between each Figure, are Units: from the Middle 1 to 10 at the End, each Figure is fo many Hundreds; and between thefe Figures each longer Divifion is 10; from the Middle 1 to 2, each lefs Division is 2 Units; and from 2 to the End, each shorter Stroke is 5 Units.

From this Description it will be eafy to find the Divifions reprefenting a given Number, thus: Suppofe the Point representing the Number 12 was 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 Laft is the Point representing 12.

Again, fuppofe the Number 22 was required, the First Figure being 2, I take the Divifion to the Figure 2 for it, and for the Second Figure 2 count 2 Tenths onwards, and that is the Point reprefenting 22.

Again, fuppofe 1728 was required; for the Firft Figure 1, I take the Middle 1, for the Second Figure 7, count onward as before, and that is 1700; then for the 2, the Third Figure, count 2 Tenths from the Laft, and it represents 1720; laftly, for the Fourth Figure 8, eftimate 8 Tenths of the next fmall Divifion, or a little less than 10, this Point, laft found, reprefents 1728.

Required the Point representing the Number 435? From the 4 in the Second Interval, count, towards 5 on the Right, three of the larger Divifions and one of the smaller, 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, muft 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 1 in the Middle will reach to the Decimal required.

Example, required the Decimal Fraction equal to ? Extend from 4 to 3, that Extent will reach from I in 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 one to 4, that Extent will reach from 16 to 64, the Product required.

Divifion being the Reverse of Multiplication, therefore extend from the Divifor to 1, that Extent will reach from the Dividend to the Quotient.

Suppofe 64 is to be divided by 4, extend from 4 to 1, that Extent will reach from 64 to 16, the Quotient. The fame with any other Numbers

Proportion, or the Rule of Three, being performed by Multiplication and Divifion, therefore extend from the First Term to the Second; that Extent will reach from the Third 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 Circumference required.

In like Manner may any other Proportion, of any Denomination, be worked, which makes this Line of general Ufe, particularly in meafuring of Superfices and Solids, which is done by extending from I to the Breadth, that Extent will reach from the Length to the Superficial Content.

Example, fuppofe 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, that Extent will reach from 27 Feet to 33,75 Feet the Superficial Content; or extend from 12 Inches to 15 Inches, that Extent will reach from 27 Feet to 33,75 Feet.

The Solid Content of any Bale, Box, Cheft, &c. is found by extending from r 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 ft. 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 33,8, the Solid Content in Feet.

Example 2d. Suppofe a Square Piece of Timber 1,25 Feet broad, ,56 deep, and 36 Feet long, be given, to find the Content: Extend from 1 to 1,25, that Extent will reach from ,56, to,7; then extend from I 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 Tonage.

38. The Line of Sines (marked Sines) begins at the Left Hand, and is Figured thus: 1, 2, 3, &c. to 10; then 20, 30, 40, &c. to 90, ending at the Right Hand, where is a Brafs Centre Pin: Thefe Figures never change their Value or Denomination, being here (and in all Lines under it) called Degrees.

4th. The Line of verfed Sines (marked V. S.) begins at the Right Hand, against 90 in the Sines, and from thence Figured towards the Left Hand, thus: 10, 20, 30, 40, &c. ending at the Left Hand End about 169 Degrees; each of the Subdivifions, from 10 to 30, are 2 Degrees; from thence to 90, (it is fingle Degrees) and from thence to the End, each Degree is divided into 15 Minutes.

5th. The Line of Tangents (marked Tan.) begins at the Left Hand as the Sines do; from thence it is Figured to the Right Hand, thus: 1, 2, 3, &c. to 10, and fo on 20, 30, 40, and 45 at the Right Hand, where is a little Brass Centre Pin, juft under and even with go in 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 i Degree. The Subdivifions of this Line are the fame as thofe of the Sines.

6th. The Line of 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 E. P.) under it, are ufed together, and only in Mercator's Sailing. The uppermoft Line contains the Degrees of the Meridian or Latitude, in a Mercator's Chart; and the lower is the Equator, and contains the Degrees of Longitude.