Mathematical Instruments

# Of the Construction and Uses of the Plotting Scale, and an improved Protractor.

The Plotting-Scale is generally made of Box-Wood, and form times of Brass, Ivory, or Silver, exactly a Foot, or half a Foot in Length, about an Inch and a half-broad, and of a convenient Thickness: Those that are but half a Foot long, have that Length given them, that thereby they may be put into Cafes of Instruments.

On one Side of this Scale is placed seven several Scales of Lines, five of which are divided into as many equal Parts as the Length of the Plotting-Scale will permit. The other two are likewise equal Parts, but have two Lines of Chords of different Lengths joined to them. The first of the equal Divisions, on the first Scale of Lines, is subdivided into 10 equal Parts, at the Beginning of which is set the Number 10; signifying, that ten of those Subdivisions make an Inch: that is, in this Case, every of the Divisions on the first Scale, is exactly an Inch; at the End of the first of which, is set 0; at the End of the second 1; at the End of the third 2; and so on to the End of the Scale. The first of the equal Divisions, on the second Scale of Lines, which are lesser than the Divisions on the first Scale, is likewise subdivided into 10 equal Parts, and hath the Number 16 set at the Beginning of it, signifying, that 16 of those Subdivisions make an Inch, or one of the Divisions $$\frac{10}{16}$$ of an Inch; at the End of the first of which is placed 0; at the End of the second 1; at the End of the third 2: and so on to the End of the Scale. The first of the equal Divisions on the third Scale of Lines, which are lesser than the Divisions of the precedent Scale’s, is also subdivided into 10 equal Parts; at the Beginning of which is set the Number 20; signifying, that 20 of those Subdivisions go to make an Inch, or that one of the Divisions is $$\frac{10}{20}$$ or $$\frac{1}{2}$$ of an Inch, which Divisions are marked, 0, 1, 2, 3, and so on to the End of the Scale, Understand the same for the other four Scales, at the Beginnings of which are writ, 24, 32, 40, 48; only the Divisions of the two last Scales of Lines are not continued to the End of the Scale, because of two Lines of Chords of different Lengths, the Beginnings of which are marked by the Letters C, C, signifying Chords. The Construction of which see in the next Chapter.

Note, Each of the aforesaid Scales of Lines are aptly distinguished from one another, by being called Scales of 10, 16, 20, 24, 32, or 48, in an Inch; as the first Scale, is a Scale of 10 in an Inch; the second, 16 in an Inch; the third, 20 in an Inch; the fourth, 24; and so on.

On the back Side of this Scale, is placed a Diagonal Scale; the first of whose Divisions, which is half an Inch, if the Scale is a Foot long, and one fourth, if the Scale is but half a Foot long, is diagonally subdivided into 100 equal Parts. Also at the other End of the Scale is another Diagonal Subdivision of an Inch into 100 equal Parts, if the Scale is a loot long; but if it is half a Foot, the Subdivision is of half an Inch into 100 equal Parts. The Figure of this Diagonal Scale, and what our Author has already said of it, in Use 8, is sufficient to shew it’s Construction and Use.

There is also next to the Diagonal Scale, a Foot divided into 100 equal Parts, if the Scale is a Foot long, every 10 of which are numbered 10, 20, 30, &c. There is likewise next to that the Divisions of Inches, numbered, 1, 2, 3, &c. each of which is subdivided into ten equal Parts.

## Use of the Plotting-Scale.

This Scale’s principal Use is to lay down Chains and Links taken in surveying Land.

### Use I.Any Distance being measured by your Chain, to lay it down upon Paper.

Suppose, that measuring along a Hedge, or the Distance between any two Marks, or Places, with your Chain, you find the Length thereof to contain 6 Chains, 50 Links. Now to take this Distance from your Scale, and lay it down upon Paper, do thus:

First draw the Line AB, then place one Foot of your Compasses upon your Scale at the Figure 6, for the 6 Chains, and extend the other Foot to 5 of the Subdivisions (which represents the 50 Links), then set this Distance upon the Line drawn from A to B, and the Line AB will contain 6 Chains, 50 Links, if you take the Distance from the Scale of 10 in an Inch.

But if you would have the Line shorter, and yet to contain 6 Chains 50 Links, then take your Distance from a smaller Scale, as of 16, 20, 24, &c. in an Inch, and then the 6 Chains, co Links, will end at C: if taken from the Scale of 16 in an Inch; or at D, if taken from the Scale of 20 in an Inch, &c. either of which Lines will contain 6 Chains, 50 Links, and be proportional one to another, as the Scales from which they were taken. And in this manner any Number of Chains and Links may be taken from any of the Scales.

### Use II.Of the Construction and Use of an improved Protractor

This Protractor is made of Brass, as the others commonly are, and has likewise it’s Semicircular Limb divided into 180 Degrees; there is an Index adjusted in the Center of this Protractor, by means of which, an Angle of any Number of Degrees and Minutes, may be protracted: there is a Circle cut out in the Piece, whose Edge, next to the Limb, serves for the Diameter of the Semicircle, the Center of this Circle is in the Center of the Limb, and it is cut sloping, so that it makes the Frustum of a Cone, the greatest Base being underneath. In this Circle is adjusted a Ring, to which the Ring of the End of the Index is riveted; by which means the Index will move freely about the Limb. There is a little Steel Point fixed to the Ring, adjusted in the aforesaid Circle, the End of which terminates in the Center of the Circle; the End of this Point must be laid to the angular Point to be protracted.

The Index consists of two Pieces, one End of that which comes out beyond the Limb of the Protractor is cut slopewise, so as exactly to fit the Edge of the Limb of the Protractor, which is likewise doped underneath, and is fastened to the other Piece; by which means the Index is kept down close to the Limb.

The Divisions on both Edges of that Part of the Index beyond the Limb, are 60 equal Parts of the Portions of Circles (passing thro’ the Center of the Protractor, and two Points assumed in the outward Edge of the Limb of that Piece of the Index nighest the Center), intercepted by two other right Lines drawn from the Center; so that they each make, with Lines drawn to the assumed Points from the Center, Angles of one Degree.

To lay off any Number of Degrees and Minutes by this Protractor, you must move the Index, so that one of the Lines drawn upon the Limb, from one of the aforementioned Points, may be upon the Number of Degrees sought; and then pricking off as many of the equal Parts on the proper Edge of the Index, as there are Minutes given, and drawing a Line from the Center, to that Point so pricked off, you will have an Angle, with the Diameter of the Protractor, of the proposed Number of Degrees or Minutes. The reason of this Contrivance is from Prop. 27. Lib. 3. Eucl. where it is proved that Angles insisting upon the same Arcs, in equal Circles, or in the same Circle (for it is the same thing), are equal.