## Some Practical RULES & EXAMPLES FOR CASK-GAVGING.

THe Corner-stone in the whole Fabrick of Cask-Gauging, as full, was long since laid by Mr. Oughtred, taking a Cask to be the Frustum of a Spheroid, under which capacity they are generally received, though indeed there have been, and daily are found some Cask differing in form, and really are more Parabolical than Spheroidal, I shall therefore lay down a plain Method for the performance of the Work (viz. finding their Content) under these four Considerations:

- As Spheroidal,
- As The Frustum of a Parabolical Spindle,
- As The Frustum of a Parabolical Conoid,
- As The Frustums of two Cones abutting upon one common Base.

[Page 192] These severally, with and without a Table of Area's of Circles.

And forasmuch as the Dimensions must be the first thing known, before the Content can be found, I shall therefore shew the young Tyro, how by some of the Dimensions to find the rest, if any obstruction prohibit the taking of all.

The Boung-diameter, and Head-diameter, and Diagonal, to find the Casks length.

First subduct the semi-difference of Diameters from the Boung-diameter, and Square the Remainder, which Square subduct from the Square of the Diagonal, and the Remainder is the Square of the Casks semi-length.

Example.

Let BD be the Boung-diameter = 29 Inches,

HE be the Head-diameter = 23 Inches,

BE be the Diagonal = 35.3836>Inches,

SD the semi-difference = 3 Inches:

Q. the Length = LT?

The Square of BE = 1252

Square of BD − SD = 676

Sqaure of semi-length = 576 √ 24 [...]

48 = LT

This very Quest. was intended by Mr. Smith, p. 176. but through a Mistake it was left out.

[Page 193] The Boung-diameter, Diagonal, and Length, to find the Head-diameter.

The Rule.

From the quadrupled Square of the Diagonal subduct the Square of the Length, (which done) the Square Root of the Remainder is equal to the Sum of the Boung-diameter and one Head-diameter.

Example.

The Square of Diagonal = 5008

Square of the Length = 2304 [...]

Remainder = 2704 √ = 52

Subduct the Boung-diameter = 29 [...]

Head-diameter = 23

The Head-diameter, Boung diameter, and the Length, to find the Diagonal.

The Rule.

To the Square of the Semi-length add the Square of the Boung diameter, less the Semi-difference of Diameters, and the Square Root of their Sum is equal to the Diagonal.

[Page 194] Example.

Square of Semi-length = 576

Square of 26 = 676 [...]

Square of Diagonal = 1252 √ 35.3836>

A Cask taken as the Frustum of a Spheroid, cut with two Plane Parallels, each Plane bisecting the Axis at right Angles,

- B the Boung-diameter = 29 Inches,
- H E the Head-diameter = 23 Inches,
- L T the Length = 48 Inches:

Q. The Content in Wine Gallons?

The Rule.

To the doubled Square of the Boung-diameter add the Square of the Head-diameter, their Aggregate multiply by the Length, and to the Product add the tenth part of it self, more one third of that tenth part, and from the Sum cut off as many places toward the right hand as were in the Multiplicand.

Example.

The doubled Square of 29 = 1682

The Square of 23 = 529 [...]

The Aggregate = 2211[Page 195] [...] The Answer = 120.2784 Wine Gallons.

Another way.

The (q)square. of the Boung-diameter = 841

The (q)square. of the Head-diameter = 529 [...]

The Sum of their Squares = 1370

The Semi-sum = 685

The Semi-diff. of Squares = 156 [...]

Their Aggregate = 2211

The Length = 48 [...]

16788

8844 [...]

The Product = 106128

The tenth part = 106128

The [...] part or ⅓ of [...] = 35376 [...]

The Answer in Wine Gall. 120.2784

[Page 196] The same Cask being taken as the Frustum of a Parabolical Spindle, the Content may be thus found.

(q)square. of the Boung-diameter = 841

(q)square. of the Head-diameter = 529 [...]

Their Aggregate = 1370

The Semi-sum = 685

The tenth part of the Diff. = 312 [...]

The Length = 48 [...]

The Product = 1001376

Tenth of the Product = 1001376

⅓ of [...]

The Answer in Wine Gall. 113.48928

If taken as the Frustum of a Parabolical Conoid, cut as before mentioned, the Content may be found as in this

Example.

(q)square. of the Boung-diameter = 841

(q)square. of the Head-diameter = 529 [...]

The Sum = 1370[Page 197] [...]

The Product = 98640

Tenth part of the Product = 98640

[...]

The Answer in Wine Gall. 111.7920

If a Cask of the same Dimensions be taken as the middle Frustum of two Cones abutting upon one common Base, cut with two Planes parallel, and each bisecting the Axis at Right Angles, the Content in Wine Gallons may be found as in this

Example.

(q)square. of the Boung-diameter = 841

(q)square. of the Head-diameter = 529 [...]

The Sum = 1370

The Semi-sum = 685 [...]

Semi (q)square. of diff. of Diam. = 18 [...] [Page 198] [...] The Answer in Wine Gall. 110.8128

For finding the Capacity of these, or any other Vessels, it is convenient to have always in readiness a Table of Area's of Circles in Wine and Ale Gallons: I think it unnecessary to swell this intended small Volume with them, there being two lately Printed, exactly Calculated to every tenth part and quarter of an Inch, and also a Table of Area's of Segments of a Circle, by my good Friend Mr. Iohn Smith, in his Book of Gauging, to whom in gratitude I am obliged to render my hearty acknowledgment for many favours and kind assistances in these Studies; yet that you may be able to find any Area of a Circle upon demand, in Wine or Ale Gallons, without a Table, take this

Rule.

Divide the q. of the Diameter by 294.1 for Wine, and by 359.05 for Ale Gallons, and the Quotient exhibits the Area. Or, saith Mr. Smith, Multiply the q. of the Diameter by .0034 for [Page 199] Wine, and by .0027851 for Ale Gallons, and the Product exhibits the Area in such Gallons.

As in these Examples.

The Diameter of a Circle = 21.7: Q. The Circles Area in Wine Gallons?

[...]

The Diameter of a Circle = 26.8: Q. The Area in Ale Gallons?

[...]

For finding the Capacity of a Cask, taken as Spheroidal, by a Table of Area's of Circles in Gallons.

[Page 200] Example.

A Casks Boung-diameter = 29 Inches, Head-diameter = 23, and the Length = 48 Inches: Q The Content in Wine Gallons?

⅓ of the Area of the Boung ⊙ = .9531

⅓ of the Area of the Head ⊙ = .5996 [...]

Their Sum = 1.5527

Semi-sum = .7763

Semi ∽ = .1768 [...]

Area of Mean Circle = 2.5058

The Length = 48 [...]

The Answer in Wine Gall. 120.2784

Another way.

⅓ of the Area of Boung ⊙ = 1.9062

⅓ of the Area of Head ⊙ = .5996 [...]

The Area of Mean ⊙ = 2.5058

The Length = 48 [...]

The Answer = 120.2784

That is, 120 Gallons, 1 Quart, and ¼ of a Pint, ferè.

[Page 201] To find the solid Content of a Cask, when taken as the middle Frustum of a Parabolical Spindle, &c.

The Dimensions as before.

⅓ of the Area of Boung ⊙ = .9531

⅓ of the Area of the Head ⊙ = .5996 [...]

Their Sum = 1.5527

Semi-sum = .7763

Tenth of the Difference = .03535 [...]

Area of Mean ⊙= 2.36435

The Length = 48 [...]

The Answer in Wine Gall. 113.48880

That is, 113 Gallons, and almost 2 Quarts.

And as the Frustum of a Parabolical Conoid, the Capacity is thus found:

⅓ of the Area of the Boung ⊙ = .9531

⅓ of the Area of the Head ⊙ = .5996 [...]

Their Sum = 1.5527

Semi-sum = .7763 [...]

Area of the Mean ⊙= 2.3290

The Length = 48 [...]

The Answer in Wine Gall. 111.792

[Page 202] If a Cask be taken as the middle Frustum of two Cones, abutting upon one common Base, &c.

The Dimensions as before.

⅓ of the Area of the Boung ⊙ = .9531

⅓ of the Area of the Head ⊙ = .5996 [...]

Their Sum = 1.5527

The Semi-sum = .7763 [...]

[...] of Area of 6 the ∽ of Diam. .0204 [...]

Area of Mean ⊙= 2.3086

The Length = 48 [...]

The Answer in Wine Gall. 110.8128

The Ullage, or Wants in a Cask, may be found under these two Considerations:

- 1. A Cask standing on the Head, with the Diameters parallel to the Horizon.
- 2. A Cask lying with the Axe parallel to the Horizon.

### Prop. I.

In a Cask standing on the Head, with the Diameters parallel to the Horizon, some Liquor remaining, to find how many Wine Gallons it is.

[Page 203]Here are these five things necessary to be known:

- 1. The Diameter at the Boung.
- 2. The Diameter at the Head.
- 3. The Length of the Cask.
- 4. The Depth of the Liquor.
- 5. The Diameter of the Liquors superficies.

Example.

[...]

The Diameter o p is thus found, first find the Axis of the whole Spheroid e f, thus; from the Square of half the Boung-diameter (n h) subduct the Square of half the Diameter at the Head, and extract the Square Root of the Remainder: Then by the Rule of Proportion, say, As that q √, is to n h, the Semi-boung-diameter: So is n i, the Casks Semi-length, to e n half the Axis sought.

[Page 204] [...]Diameter of the Liquors Superficies 27.4

Having found the Diameter of the Liquors Superficies:

Then, to ⅔ of the Area of that Circle = 1.70173

Add ⅓ of Area of the Head Circle = .59953 [...]

The Depth of Liquor = 11.6 [...]

The Answer in Wine Gallons = 26.694616

Which subducted from the whole Content, leaves the Ullage or Wants.

### Prop. II.

A Cask lying with its Axe parallel to the Horizon, and having some Liquor remaining in it, to find the Content of the said Liquor in Gallons.

[...]

Let the Dimensions be as before.

In this Proposition there is five Requisites attending:

- h g the Diameter at the Boung = 29.
- a b the Diameter at the Head = 23.
- i k the Length = 48.
- s g the Depth of Liquor = 11.6.

And the Content of the whole Cask in Gallons.

Then by the help of a Table of Area's of Segments of a Circle, whose Area is Unity, and the Radius divided in the Ratio of 1.0000 Parts, say by the Rule of Proportion: [...] versed Sine or Arrow of Segment.

[Page 206]Then seeking in the Table you will find .4000, and right against it under the Title Area you will find .37353. Then say: whole Content

1.0000. .37353 ∷ 120.2784. 44.9276

the Liquor remaining.

The Inversion of the Question, viz. To find the Liquor wanting.

As 29. 17.4 ∷ 1.0000. .6

Again,

the Ullage.

As 1.0000. .62647 ∷ 120.2784. 75.3509

The Liquor remaining = 44.9276 [...]

Which together make 120.2785 the Casks whole Capacity.