Mathematical Instruments
Book I. Additions. Ch. II.

# Of the Construction and Use of the Four-Foot Gauging-Rod.

This Rod, whose Use is to find the Quantities of Liquors contained in any kinds of Vessels, is usually made of Box-Wood, and Consists of four Rules, each a Foot long, and about $$\frac{3}{8}$$ of an Inch square, joined together by three Brass Joints; by which means the Rod is rendered four Foot long, when the four Rules are quite opened, and but one Foot when they are folded together.

On the first Face of this Rod are placed two Diagonal Lines, one for Beer, and the other for Wine; by means of which the Content of any common Vessel in Beer or Wine Gallons may be readily found, in putting the Rod in at the Bung-hole of the Vessel until it meets the Intersection of the Head of the Vessel with the opposite Staves to the Bung-hole. For distinction of this Line, there is writ thereon Beer and Wine Gallons.

On the second Face of this Rod, are, a Line of Inches, and the Gauge Line, which is a Line expressing the Area’s of Circles, whose Diameters are the correspondent Inches in Ale Gallons. At the Beginning of it is writ, Ale Area.

On the third Face are three Scales of Lines; the first, at the end of which is writ Hogshead, is for finding how many Gallons there is in a Hogshead, when it is not full, lying with it’s Axis parallel to the Horizon. The second Line, at the End of which is writ B.L. signifies Butt lying, is for the same Use as that for the Hogshead. The third Line is to find how much Liquor is wanting to fill up a Butt when it is standing. At the End of it is writ B.S. signifying a Butt standing.

Half way the fourth Face of the Gauging-Rod are three Scales of Lines, to find the Wants in a Firkin, Kilderkin, and Barrel, lying with their Axes parallel to the Horizon. They are distinguished by the Letters F.K.B. signifying a Firkin, Kilderkin, and Barrel.

## Construction of the two Diagonal Lines.

These two Diagonal Lines are put upon this Gauging-Rod, in the same manner that our Author, in the last Use of the Line of Solids in the second Book directs, for putting on the Diagonals on his Gauging-Rod, viz. by taking the Diagonal of some Vessel that is similar, or nighly similar to the Vessels, whose Contents in Beer, or Wine Gallons, are afterwards, by means of them, to be found; and then knowing how many Gallons in Beer and Wine the aforesaid Vessel contains, which Gallons must be let against the Inches, or Parts of Inches of their Diagonals Length, on the Diagonal-Face of the Gauging-Rod. Now to find how many Inches, or Parts, the Diagonal of any other similar Vessel must be, when it’s Content in Beer and Wine-Gallons is given; you must say, As the Content of the first Vessel, which is known, is to the Cube of the Length of it’s Diagonal; So is the Content of that other similar Vessel, in Beer or Wine-Gallons, to the Cube of the Length of it's Diagonal: the Cube-Root of which extracted, will give the Length of the Diagonal sought. As for Example, suppose a little Vessel similar, or nighly similar to English Vessels of a usual Form, contains 1 Beer Gallon, or about 1$$\frac{1}{4}$$ Wine Gallon, and the Diagonal is found to be 7.75 Inches; what will be the Diagonal of a similar Vessel, containing 2 Beer Gallons, or 2.8 Wine Gallons? Say, As 1 Gallon is to the Cube of 7.75, which is 465.48437, So is 2 Gallons to the Cube of the Diagonal sought, 930.96875, whose Root will be 9.72 Inches, and so much will be the Length of the Diagonal: therefore set 2 Beer Gallons on the Diagonal Face of the Rod, against 9.72 Inches. In this manner may the Diagonal Face of the Rod be divided from 1 Beer Gallon to 240, and from 1 Wine Gallon to 300, and subdivided in half Gallons, as on the Rod.

## Construction of the Gauge-Line on the second Face of the Rod.

On this Line is set the Gallons, and hundred Parts of Gallons, that any Cylinder, an Inch deep, and any Inches and Parts, from 1 to 46 in Diameter, contains of Ale. As for Example; against 1.9 Inches stands .01 of a Gallon, denoted by a Dot; against 2.63 Inches stands .02 of a Gallon. The Tenths of the Gallons are denoted by the Figures 1, 2, 3, 4, 1&c. as .1 of a Gallon is set against 5.96 Inches; .2 against 8.44 Inches, and 1 Gallon against 18.95 Inches, as per Figure. The Construction of this Line is thus: Because 282 solid Inches make an Ale Gallon, therefore the Diameter of a Cylinder, one Inch deep, whose Content is an Ale Gallon, or 282 solid Inches, will be 18.95 Inches; whence against 18.95 Inches, on the same Face of the Gauging-Rod, set, on the Line drawn to contain the Divisions of the Gauge-Line, 1 Gallon. Now to find the Diameter of a Cylinder one Inch deep, that shall contain the .01 Part of a Gallon, say, As 1 Gallon is to the .01 Part of a Gallon, So is the Square of 18.95 Inches, which is 359, to the Square of the Diameter of the Cylinder, containing the hundredth Part of a Gallon, which will be found by extracting the square Root of that Quantity 1.9 Inch: therefore set the first Dot against 1.9 of an Inch. Again, to find against what Inches, or Parts, .02 of a Gallon must be placed, say, As 1 is to .02, So is 359 to the Square of the Number of Inches, or Parts, whose Root extracted will give 2.63 Inches; against which make a second Dot for .02 of a Gallon. In this manner proceed for all the other Divisions on the Gauge-Line, always making 1 and 359 the two first Terms of the Proportion, and the Gallons or Parts the third; so shall the fourth be the Square of the Inches, or Parts, that the Gallons, or Parts expressed in the third Term, are to be set against. The Reason of the aforesaid Proportion is, that Cylinders, of equal Altitudes, are to each other as their Bases, and Circles as the Squares of their Diameters.

## Construction of the Scales on the third and fourth Faces.

The first Scale of Lines on the third Face, which serves for finding the Gallons wanting in a Hogshead posited with it’s Axis parallel to the Horizon, or lying down, contains the Divisions from 1 Gallon to 54 Gallons, which is the Number of Ale-Gallons a Hogshead contains when full.

The second Scale of Lines, on the same Face, containing the Divisions from 1 Gallon to 108 Gallons, which are the Number of Ale-Gallons contained in a Butt, is for the same Use as the first Scale of Lines when the Butt is lying.

The third Scale, likewise numbered from 1 Gallon to 108, is for finding how many Gallons is wanting in a Butt standing upright.

The three Scales of Lines, on part of the fourth Face, are, as I have already said, for finding the Wants in a Firkin, Kilderkin, and Barrel lying down, in Ale-Gallons. The readiest way to make the Divisions of either of these Scales of Lines for their correspondent Vessels, when lying down, as for a Hogshead, is to pour in first one Gallon of Water, and then put the Rod downright into the Bung-hole to the opposite Staves; then where the Surface of the Water cuts the third Face of the Rod (because the Scale of Lines for the Hogshead is on that Face) make the Division for 1 Gallon; then pour in another Gallon, and where the Surface of the Water cuts the Rod, make the Division for 2 Gallons. Again, pour in another Gallon, and where the Surface of the Water cuts the Rod, make the Division for three Gallons. Proceed thus, by pouring in of one Gallon successively after another, and making of Divisions at every Place in the Face of the Rod, to which the Water arises, until the Hogshead be full, and then the Scale for a Hogshead, on the third Face, will be divided, proceed, in the same manner, in making the Divisions for the other Scales of Lines used in finding the Wants in the several Vessels aforementioned lying down. And taking off the Head of a Butt that is standing, and pouring of Water in the same manner as in the Hogshead, putting the Rod downright into the Butt, and making Divisions on the Rod, as was done for the Hogshead, the Line will be finished, when figured.

Note, The Divisions for Half-Gallons, marked by long Dots on the fourth Face, are made by pouring in of Half-Gallons successively, &c.

## Use.Use of the Diagonal Lines on the Gauging-Rod

### To find the Content of a Vessel in Beer or Wine-Gallons.

Put the brased End of the Gauging-Rod into the Bung-hole of the Cask, with the Diagonal Lines upwards, and thrust the brased End to the meeting of the Head and Staves.

Then with Chalk make a Mark on the middle of the Bung-hole of the Vessel, and also on the Diagonal Lines of the Rod, right against, or over one another, when the brased End is thrust home to the Head and Staves.

Then turn the Gauging-Rod to the other End of the Vessel, and thrust the brased End home to the End as before.

And see if the Mark made on the Gauging-Rod come even with the Mark made on the Bung-Hole, when the Rod was thrust to the other End; which if it be, the Mark made on the Diagonal Lines, will, on the same Lines, shew the whole Content of the Cask in Beer or Wine-Gallons.

But if the Mark first made on the Bung-hole be not right against that made on the Rod, when put the other way; then right against the Mark made on the Bung-hole, make another on the Diagonal Lines: then the Division on the Diagonal Line, between the two Chalks, will shew the Vessel&rsquo;s whole Content in Beer or Wine-Gallons. As for Example; if the Diagonal Line of a Vessel be 28 Inches 4 Tenths, it&rsquo;s Content in Beer-Gallons will be near 51, and in Wine-Gallons 62.

But if a Vessel be open, as a Half-Barrel, Tun, or Copper, and the Measure from the middle on one Side, to the Head and Staves, be 38 Inches, the Diagonal Line gives 122 Beer-Gallons; half of which, viz. 61, is the Content of the open Half-Tub.

But if you have a large Vessel, as a Tun, or Copper, and the Diagonal Line, taken by a long Rule, prove 70 Inches; then the Content of that Vessel may be found thus:

Every Inch, at the Beginning-End of the Diagonal Line, call 10 Inches, then 10 Inches becomes 100 Inches.

And every Tenth of a Gallon call 100 Gallons; and every whole Gallon, with a Figure, call 1000 Gallons. Example, at 44.8 Inches, on the Diagonal Beer-Line, is 200 Gallons; so also at 4 Inches 48 Parts, now called 44 Inches 8 Tenths, is just two Tenths of a Gallon, now called 200 Gallons.

Also if the Diagonal Line be 76 Inches and 7 Tenths, a close Cask, of so great a Diagonal, will hold 1000 Beer-Gallons: but an open Cask but half so much, viz. 500 Beer-Gallons.

For reducing of Wine-Gallons to Beer-Gallons, or, vice versa, by Inspection, this may be done.

Thus 30 Wine-Gallons, is 24$$\frac{1}{2}$$ Beer-Gallons, &c.

## Use of the Gauge-Line.

### Use I.To find the Content of any Cylindrical Vessel in Ale-Gallons.

Seek the Diameter of the Vessel in the Inches, and just against it, on the Gauge-Line, is the Quantity of Ale-Gallons contained in one Inch deep: then this multiplied by the Length of the Cylinder, will give it’s Content in Ale-Gallons. For Example; suppose the Length of the Vessel be 32.06, and the Diameter of it's Base 25 Inches, what is the Content in Ale-Gallons? Right against 25 Inches, on the Gauge-Line, is 1 Gallon, and .745 of a Gallon; which multiplied by 32.06, the Length, gives 55.9447 Gallons for the Content of the Vessel.

### Use II.The Bung-Diameter of a Hogshead is 25 Inches, the Head-Diameter 22 Inches, and the Length 32.06 Inches; to find the Quantity of Ale-Gallons contained in it.

Seek 25, the Bung-Diameter, on the Line of Inches, and right against it, on the Gauge-Line, you will find 1.745; take $$\frac{1}{3}$$ of it, which is .580, and set it down twice. Seek 22 Inches, the Head Diameter, and against it you will find, on the Gauge-Line, 1.356; $$\frac{1}{3}$$ of which added to twice .580, gives 1.6096; which multiplied by the Length 32.06, the Product will be 51.603776, the Content in Ale-Gallons. This Operation supposes, that the aforesaid Hogshead is in the Figure of the middle Frustum of a Spheroid.

The Use of the Lines on the two other Faces of the Rod, is very easy; for you need but put it downright into the Bung-hole (if the Vessel you desire to know the Quantity of Ale-Gallons contained therein be lying), to the opposite Staves; and then where the Surface of the Liquor cuts any one of the Lines appropriated for that Vessel, will be the Number of Gallons contained in that Vessel.