THE PRACTICAL GAUGER …

THE PRACTICAL GAUGER: BEING A Plain and Easie Method OF GAUGING ALL SORTS OF Brewing Vessels.

Whereunto is added, A SHORT SYNOPSIS OF THE Laws of EXCISE.

By IOHN MAYNE.

London, Printed by W. G. for N. Brooke, at the Angel in Cornhill. 1676.

To the Honourable PEREGRENE BERTIE, ESQUIRE.

May it please Your Honour,

MY present Service is so much the more valuable to me, since I was preferred thereto by Your Recommendation, which I shall endeavour to preserve with un­wearied Industry and constant Fidelity. Your Favour, Sir, is a Blessing my Soul is too narrow to enclose, and me [Page] thinks I can hardly be reconciled to that Divinity which calls Pride a Sin. But if I am guilty, I heartily beg Your par­don, and permission to subscribe,

YOUR HONOURS Most humbly devoted Servant, Iohn Mayne.

To the Worshipful PETER CALVERD, SAMUEL VINCENT, EDMUND BOSTOCK, IOHN PARSONS, And RICHARD BRET, ESQUIRES, FARMERS of His Majesties Revenue of EXCISE within the Kingdom of England, Dominion of Wales, and Town of Berwick upon Tweed; Iohn Mayne

Wisheth Increase of Happiness here, and everlastingly hereafter.

To the Reader.

HAving been commanded by my Masters into the North-west Parts of this Kingdome, I found there many ingenious Men, imployed in their Service, viz. the Revenue of Excise, which were willing to gain some little knowledge in the Art of Gau­ging, but had been frighted therefrom by the seeming difficulty of obtaining their desires. I thought it no disservice to endea­vour to inform these Men, upon whose shoulders the whole Matter of Fact, in this Affair, between the King and the Subject lay. [Page] And accordingly I wrote and read to some of them these short Rules thou hast now before thee, en­deavouring to make the Art plain and easie, even to the mea­nest Capacity, without much ex­pence of Time in reading the Text, or breaking his Teeth with hard Terms of Art: And not ha­ving leisure to write so many Copies, as my Friends in those Parts desi­red, and I could afford them, I am prevailed with to commit it to the Press. Farewell.

I. M.

THE PRACTICAL GAUGER.

IT would be very convenient, that every Gauger were well acquain­ted with the Art of Decimal Arithmetick; but it is necessary that he be so well seen in Vulgar, as to be able to add, subtract, multiply, and divide any whole Numbers.

It is also necessary, that he under­stand the three sorts of Quantity, [Page 2] viz. a Line, a Superficies, and a Solid.

A Line hath length, but no breadth.

A Superficies hath length and breadth, but no depth.

A Solid hath length, breadth, and depth.

Moreover there is no kind of Quantity but is commensurable by some Common Measure thereto as­sign'd, as a Line by a Line of Inches, Feet, Poles, Furlongs, &c. and a Superficies by a Superficies, as the Square Inch, &c. and also a Solid by a Solid: So when it is known how many Inches, Poles, or Furlongs is contained in any Line, the length of that Line is said to be known; and when it is known how many square Inches, square Feet, or square Per­ches are contained within any Super­ficies, the Content or Area of that Superficies is said to be known; and also when it is known how many solid Inches are contained in any [Page 3] Solid, the Content of that Solid is known.

To find the Content of a Back in the form of a Square or Parallelogram.

The Rule is:

Multiply the length by the breadth, and the Product divide by 282 (the number of solid Inches contained in the Ale Gallon) and the Quotient is the Ale Gallons contained upon one Inch in depth of that Back or Tun.

A Square.

[Page 4] [...]

An Oblong or Parallelogram.

[...]

[Page 5]One contains 10 Gallons, and 205/282 parts of a Gallon; the other 21 Gal­lons and 128/282 of a Gallon. To reduce which Fraction into Pints, you need only multiply the Remainder or Nu­merator by 8, and divide by the old Divisor, and the Quote will be the Pints contained in that Fraction.

To find the Area or Content of one Inch of a Back or Tun, in the form of a Triangle, in Ale Gallons:

The Rule is:

Multiply the Line a c ( viz. the Line falling perpendicularly from any Angle on its opposite Base) by half the Line b c d, here the Base, and the Product divide by 282, the Quo­tient will be the Ale Gallons contained in one Inch of depth upon that Tri­angle.

[Page 6] Example.

[figure]

[...]

There is contain'd in the first Inch of this Back or Tun 63 Ale Gallons, 6 Pints, and above an half.

[Page 7]But admitting the Back or Tun be of an irregular Form, divide it into Triangles, and let fall Perpendiculars in each, and find their several Area's, as in the last Example, then add them together, and you have the Content or Area of the whole Figure.

Example.

[figure]

First I find the Triangle abe.

[...]

[Page 8]Now in the other two Triangles, the two Perpendiculars falling upon one common Base, I multiply the Sum of the Perpendiculars by half the Base, and that Product is the Area or Con­tent of both Triangles.

[...]

So doth one Inch of this Back or Tun appear to contain of Ale Mea­sure, [Page 9] 9 Barrels, 3 Firkins, and 3 Gallons.

Here Note, That formerly the Ale Gallon was accompted to con­tain 288 ¼ Cube Inches; but by the care and pains of my good Friend, Mr. Nicholas Gunton, the just quan­tity of the Quart, remaining in the hands of the Chamberlains of His Majesties Exchequer, appears to be 70 ½, as some of late have found to their no small cost. And I am of opinion, that if the Wine Gallon were carefully examined, it would prove to contain less than it is commonly holden to do, by so much as would improve the Revenue of the Crown some thousands per annum.

Now having shewed thee how to find the superficial Quantity of any Figure enclosed or bounded by Right Lines, if the Dimensions be alike above and below, multiply the Area by the Depth and you have the Con­tent [Page 10] of the whole; or by any part, and you will find thereby the Solidity of that part.

But if the Dimensions above and below be unequal, take your Dimen­sions or cross Diameters at every Foot or half Foot, and so find the Solidity.

Of a Circle.

A Circle is a Figure contained or bounded by one Line, which is called the Circumference, as the Line abgd.

[figure]

[Page 11]Within which Line there is a Point, from whence all Lines drawn from the Circumference are equal: And that Point is called the Center, as the Point c: And through which all Right Lines drawn from one side of the Circumference to the other divide the Circle into two equal parts, and that Line is called the Diameter, as the Line a c b; and the two parts of the Circle, divided by the Diameter, are called Semi [...]circles, as the part a c b g or a c b d.

And although there is not yet found any true proportion between a Square and a Circle, (a Square Inch being our Common Measure) yet is there an Approximation found by Vanculen, which comes almost infinitely near the truth, being as Unity to 3.14159 &c. to the Square of the Semi-diameter. But the old Proportion comes near enough for common practice, viz. As 14 to 11, so is the Square of the Diameter to the Area in Square Inches, &c.

[Page 12] Example.

[...]

Which Area in Square Inches divi­ded by 282 quote the Ale Gallons con­tained in one Inch of depth in that Circle, or by 231 the Wine Gallons.

[Page 13] Example.

[...]

But I would advise my young Gau­ger not to trouble himself with the [Page 14] Inches, his business being to find the Gallons the shortest way he can: In order whereunto, let him observe the following Rule.

Having the Diameter of a Circle in Inches, to find the Area or Content in Ale or Wine Gallons.

The Rule.

Square the Diameter ( viz. multi­ply it by it self) and that Square or Product divide by 359 for Ale, and 294 for Wine, and the Quotes will be the Ale or Wine Gallons respe­ctively, that shall be contain'd in a Circle of that Diameter, and one Inch in depth.

[...]

[Page 15] [...]

If the Tun be in form of a Cy­linder, viz. like the Rolling-stone of a Garden, the Circles above and be­low of equal Diameters, then to find the Content of that Tun, or any part thereof, you need only find the Area of the Circle, and multiply by the whole Depth, or such part as you desire, and the Product will be the Solidity of the whole, or part, re­spectively.

[figure]

[Page 16] Diameter ab or cd = 80 Inches, the Area before found to be 17 gallons 6 pints, and the Depth = 40 Inches: To find the Content.

[...]

The Content of this Tun in Beer Mea­sure.

But if your round Tun have un­equal Dimensions above and below, it is then taken to be the part of a Cone or round Pyramid, having the top cut off as the following Diagram; the whole Cone = abcdefg, the part or Frustum = abcdeg.

[Page 17]

[figure]

Having the two Diameters and Depth of the Tun, to produce the whole Cone the Rule is by Proportion thus:

As the Semidifference of Diameters is to half the Diameter at the Base: So is the height of the Frustum to the Cones whole Axis.

Admit ab 80, cd 140, eg 50, then is the Difference 60, half the Base 70.

[Page 18] Example.

[...]

Having all these Dimensions, the Content of the Tun is not hard to be found.

For if you find the Area of the Base, and multiply that Area by ⅓ of the Altitude, or ⅓ of the Area by the whole Altitude, the Product is the So­lidity of the whole Cone. Then ha­ving found the whole Cone, find also the lesser Cone, and subduct that from the greater, the Remainder will be the Content of the Frustum.

Or if you would find the Content of the Frustum, without producing the whole Cone,

The Rule is:

Multiply the Sum of the two Dia­meters by it self, then multiply the two Diameters by each other, sub­duct the lesser Product from the grea­ter, and the Remainder multiply by the depth, the last Product divide by 1077, and the Quote is the Ale Gal­lons contained in that Tun.

[...]

[Page 20] [...]

The Content appears to be 1727 Ale Gallons, or 47 Barrels, 3 Firkins, 8 Gallons.

If it be desired in Feet or Inches, you may find all the Differences of the Diameters, at every Foot, half Foot, or Inch, by this Proportion:

As the whole Depth, is to the Diffe­rence of Diameters: So is any part of the Depth, to its respective difference of Diameter.

Then by Addition or Subduction, you have the Diameters all the way upward or downward.

[Page 21]Or, if you please, you may take the Diameters actually, in the midst of every Foot or half Foot, and seek its Area in the Table of Area's hereto annexed, which Area multiplied by the Foot or half Foot gives the Soli­dity accordingly, without any sensible errour. This way I would oblige the young Gauger to, if he be not ready at Proportion.

Example.

[figure]

[...]

[Page 22]All these Mean Diameters cut this Tun into so many Cylinders.

[...]

Now having the Area of each Circle, which is an Arithmetical Mean of the half Foot, multiply this Area by 6, and the Product is the Solidity of that half Foot.

[...]

This Tun cast up by the Rule, pag. 19.

[...] [Page 23] [...]

Whereby appears not a Gallon diffe­rence.

If the Conjugate or Cross Diame­ters above and below are not equal, then do Geometricians call that Tun Eliptical; to reduce which to a Circle, there ought to be a Geometrical Mean taken, but the common practice [Page 24] is to add them together, and take the half, which if the Diameters differ not much, the errour is the less.

Coppers are commonly taken at eve­ry 6 Inches, as that Tun in pag. 21.

To find the Content of a Coppers Crown, take this Rule:

Square the Diameter at the Base, to this add 4/3 of the Square of the Crowns Altitude, the Sum multiply by the Altitude, the last Product divide by 718, the Quote is the Ale Gallons contained in that Crown.

To find the Content of the Mash Tun, in Quarters, Bushels, and Gal­lons, there can be no certain Rule in taking the Depth of the Goods, by reason of the difference in the good­ness of Mault, some spending it self much more in the wetting than other; yet that considered, there may be some estimate given thereto, where­by a very considerable fraud may be discovered.

[Page 25]In a square Mash Tun, divide the Product of the Length and Breadth by 227 for indifferent, or by 200 for the finest, and the Product will be the Gallons wetted; and for round Mash Tuns, let your Divisors be 288 or 260, respectively.

For Cask-Gauging, wholly full, I have given many Precepts and Ex­amples in a Treatise called The Mer­chants Companion, Printed about a year since; but that the Reader may not be left here wholly without, take the following Rule, which considers the Cask as the Frustum of a Spheroid, that being the most general form.

To find the Content of a Cask in Ale or Wine Gallons.

The Rule.

To the doubled Square of the Boung Diameter add the Square of the Head Diameter, and that Sum [Page 26] multiply by the Casks Length, the last Product divide by 1077, and the Quote is the Ale Gallons, or by 882 quotes the Wine Gallons contained in that Cask.

Example.

[...] [Page 27] [...] The Content of this Cask is 73 Ale Gallons, 2 Pints, and almost an half.

Or, by the Table of Area's, thus:

To twice the Area of the Boung Circle in Gallons and parts add the [Page 28] Area of the Head Circle, the Sum of these multiply tby ⅓ of the Casks Length, and the Product is the Con­tent in Gallons and parts, as in the former Example.

[...]

Which Fraction being reduced is 2 Pints and better, as was before found.

A TABLE OF Area's of CIRCLES IN ALE GALLONS AND MILESIMAL PARTS [...]
To every Quarter of an Inch, From 1 to 12 Foot of Diameter.

[Page 30]

Circles Area's in Ale Gallons.
Inches. Area. .25 .5 .75
10 0.278 0.293 0.307 0.322
11 0.337 0.352 0.368 0.384
12 0.401 0.418 0.435 0.453
13 0.471 0.489 0.508 0.527
14 0.546 0.565 0.586 0.606
15 0.627 0.648 0.669 0.691
16 0.713 0.731 0.758 0.781
17 0.805 0.824 0.853 0.877
18 0.902 0.923 0.953 0.979
19 1.005 1.027 1.059 1.086
20 1.114 1.142 1.170 1.199
21 1.228 1.258 1.287 1.317
22 1.348 1.379 1.410 1.441
23 1.473 1.505 1.538 1.571
24 1.604 1.638 1.672 1.706
25 1.741 1.776 1.811 1.847
26 1.883 1.919 1.956 1.993
27 2.030 2.068 2.106 2.175
28 2.184 2.223 2.262 2.302
29 2.342 2.383 2.424 2.465

[Page 31]

Circles Area's in Ale Gallons.
Inches. Area. .25 .5 .75
30 2.507 2.549 2.591 2.634
31 2.676 2.720 2.764 2.808
32 2.852 2.896 2.942 2.987
33 3.033 3.079 3.126 3.172
34 3.220 3.267 3.315 3.363
35 3.412 3.461 3.510 3.560
36 3.610 3.660 3.710 3.761
37 3.813 3.864 3.916 3.969
38 4.022 4.075 4.128 4.182
39 4.236 4.291 4.345 4.401
40 4.456 4.512 4.568 4.625
41 4.682 4.739 4.797 4.855
42 4.913 4.972 5.031 5.090
43 5.150 5.210 5.270 5.331
44 5.392 5.453 5.515 5.577
45 5.640 5.703 5.766 5.829
46 5.893 5.957 6.022 6.087
47 6.152 6.218 6.284 6.350
48 6.417 6.484 6.551 6.619
49 6.687 6.755 6.824 6.893

[Page 32]

Circles Area's in Ale Gallons.
Inches. Area. .25 .5 .75
50 6.963 7.033 7.103 7.173
51 7.244 7.315 7.387 7.459
52 7.531 7.603 7.676 7.750
53 7.823 7.897 7.972 8.046
54 8.121 8.197 8.272 8.348
55 8.425 8.502 8.579 8.656
56 8.734 8.812 8.891 8.970
57 9.049 9.128 9.208 9.288
58 9.369 9.450 9.531 9.613
59 9.695 9.777 9.860 9.943
60 10.026 10.110 10.194 10.279
61 10.363 10.448 10.534 10.620
62 10.706 10.792 10.879 10.970
63 11.054 11.142 11.230 11.320
64 11.408 11.497 11.587 11.677
65 11.767 11.868 11.949 12.040
66 12.132 12.224 12.316 12.409
67 12.502 12.596 12.690 12.784
68 12.878 12.973 13.068 13.164
69 13.260 13.356 13 453 13.550

[Page 33]

Circles Area's in Ale Gallons.
Inches. Area. .25 .5 .75
70 13.647 13.745 13.843 13.941
71 14.040 14.139 14.238 14.338
72 14.438 14.539 14.639 14.740
73 14.842 14.944 15.046 15.148
74 15.251 15.354 15.458 15.562
75 15.066 15.771 15.876 15.981
76 16.087 16.193 16.299 16.406
77 16.513 16.620 16.728 16.836
78 16.945 17.053 17.162 17.270
79 17.382 17.492 17.603 17.713
80 17.825 17.936 18.048 18.160
81 18.273 18.386 18.499 18.613
82 18.727 18.841 18.956 19.071
83 19.187 19.302 19.418 19.535
84 19.652 19.769 19.886 20.004
85 20.122 20.241 20.360 20.479
86 20.599 20.718 20.840 20.959
87 21.088 21.202 21.323 21.445
88 21.560 21.691 21.814 21.937
89 22.061 22.185 22.309 22.434

[Page 34]

Circles Area's in Ale Gallons.
Inches. Area. .25 .5 .75
90 22.559 22.685 22.811 22.937
91 23.063 23.190 23.318 23.445
92 23.573 23.701 23.830 23.959
93 24.088 24.217 24.348 24.478
94 24.609 24.740 24.872 25.003
95 25.136 25.268 25.401 25.534
96 25.667 25.801 25.936 26.070
97 26.205 26.340 26.476 26.612
98 26.748 26.885 27.022 27.159
99 27.297 27.435 27.573 27.712
100 27.851 27.990 28.130 28.270
101 28.411 28.552 28.693 28.834
102 28.976 29.118 29.261 29.404
103 29.547 29.691 29.835 29.979
104 30.124 30.269 30.414 30.560
105 30.706 30.852 30.999 31.146
106 31.293 31.441 31.589 31.738
107 31.887 32.036 32.185 32.335
108 32.485 32.636 32.787 32.970
109 33.090 33.242 33.394 33.577

[Page 35]

Circles Area's in Ale Gallons.
Inches. Area. .25 .5 .75
110 33.700 33.853 34.007 34.192
111 34.315 34.470 34.625 34.812
112 34.936 35.093 35.249 35.437
113 35.563 35.721 35.878 36.037
114 36.195 36.354 36.513 36.673
115 36.833 36.993 37.154 37.315
116 37.476 37.638 37.800 37.962
117 38.125 38.288 38.452 38.616
118 38.780 38.944 39.109 39.274
119 39.440 39.606 39.772 39.938
120 40.105 40.273 40.440 40.608
121 40.777 40.945 41.114 41.284
122 41.453 41.623 41.794 41.965
123 42.136 42.307 42.479 42.651
124 42.824 42.997 43.170 43.343
125 43.517 43.692 43.866 44.041
126 44.216 44.392 44.568 44.744
127 44.921 45.098 45.275 45.453
128 45.631 45.809 45.988 46.167
129 46.347 46.527 46.707 46.887

[Page 36]

Circles Area's in Ale Gallons.
Inches. Area. .25 .5 .75
130 47.068 47.249 47.431 47.613
131 47.795 47.978 48.161 48.384
132 48.527 48.712 48.896 49.081
133 49.266 49.451 49.637 49.823
134 50.009 50.196 50.383 50.571
135 50.758 50.946 51.135 51.324
136 51.513 51.703 51.893 52.083
137 52.273 52.464 52.656 52.847
138 53.039 53.232 53.424 53.618
139 53.811 54.005 54.199 54.393
140 54.58 [...] 54.783 54.979 55.174
141 55.371 55.567 55.764 55.961
142 56.159 56.357 56.555 56.754
143 56.953 57.152 57.352 57.592
144 57.752 57.953 58.154 58.395

[Page 37]To make any Number in the prece­dent Table,

The Rule is:

Divide the Square of the Diameter by 359, and the Quote exhibits the Area of that Circle in Ale Gallons.

Example.

[...]

Thus may be made any Number, greater or lesser than the Table doth exhibit, the difference here being not 1/1000 part of a Gallon.

[Page 38]To make a Table of ⅓ ds of Area's of Ci [...]cles in Wine Gallons,

The Rule is:

Multiply ⅓ of the Square of the Diameter by .0034, or multiply the whole Square by .0034 and the Pro­duct divide by 3; and the Product of the former Work, or the Quotient of the latter, is the Circles Area.

Example.

What is the ⅓ d of the Area of that Circle in Wine Gallons whose Dia­meter is 30 Inches?

[...]

A TABLE Of One thirds of AREA's of CIRCLES IN WINE GALLONS: CALCULATED To every quarter of an Inch, From 10 to 60 Inches of Diameter.

[Page 40]

One thirds of Circles Area's
Inches. Area. .25 .5 .75
10 0.1133 0.1190 0.1250 0.1330
11 0.1371 0.1434 0.1499 0.1565
12 0.1632 0.1701 0.1771 0.1842
13 0.1915 0.1990 0.2066 0.2143
14 0.2231 0.2301 0.2383 0.2466
15 0.2550 0.2636 0.2723 0.2811
16 0.2901 0.2993 0.3086 0.3180
17 0.3275 0.3372 0.3471 0.3570
18 0.3672 0.3775 0.3879 0.3984
19 0.4091 0 4200 0.4310 0.4420
20 0.4533 0.4647 0.4763 0.4880
21 0.4998 0.5117 0.5272 0.5362
22 0.5485 0.5609 0.5734 0.5866
23 0.5995 0.6126 0.6259 0.6393
24 0.6528 0.6665 0.6803 0.6943
25 0.7083 0.7226 0.7370 0.7515
26 0.7661 0.7808 0.7969 0.8110
27 0.8262 0.8416 0.8571 0.8727
28 0.8885 0.9044 0.9206 0.9368
29 0.9531 0.9696 0.9863 1.0031

[Page 41]

in Wine Gallons.
Inches. Area. .25 .5 .75
30 1.0200 1.0370 1.0543 1.0716
31 1.0891 1.1064 1.1245 1.1425
32 1.1605 1.1787 1.1971 1.2156
33 1.2342 1.2530 1.2719 1.2910
34 1.3101 1.3298 1.3490 1.3686
35 1.3883 1.4082 1.4283 1.4485
36 1.4688 1.4893 1.5099 1.5306
37 1.5513 1.5726 1.5938 1.6184
38 1.6365 1.6581 1.6799 1.7018
39 1.7238 1.7460 1.7683 1.7908
40 1.8133 1.836 [...] 1.8589 1.8819
41 1.9051 1.9285 1.9519 1.9755
42 1.9992 2.0231 [...].0471 2.0713
43 2.0955 2.1199 2.1445 2.1693
44 2.1941 2.2291 2.2443 2.2696
45 2.2950 2.3206 2.3463 2.3722
46 2.3981 2.4243 [...].4506 2.4770
47 2.5035 2.5303 2.5571 2.5840
48 2.6112 2.6385 2.6659 2.6934
49 2.7211 2.7489 2.7736 2.8051

[Page 42]

One thirds of Circles Area's.
Inches. Area. .25 .5 .75
50 2.8333 2.8617 2.8867 2.9190
51 2.9478 2.9768 3.0059 3.0352
52 3.0645 3.0941 3.1237 3.1536
53 3.1835 3 [...]2136 3.2439 3.2743
54 3.3048 3.3355 3.3663 3.3972
55 3.4283 3.4596 3.4910 3.5224
56 3.5541 3.5859 3.6179 3.6500
57 3.6822 3.7146 3.7471 3.7797
58 3.8125 3.8454 3.8786 3.9118
59 3.9451 3.9786 4.0123 4.0461
60 4.0800 4.1141 4.1483 4.1826

To Gauge a Cask by the precedent Table,

The Rule is:

To the Double of the Boung Dia­meter add the Head Diameter, their Sum multiply by the Length, and the Product is the Content.

[Page 43] Example.

A Casks Boung Diameter 29.5 Inches, Head Diameter 23, and the Length 48 Inches, I demand its Content in Wine Gallons?

[...]

Answer 123 ½ Wine Gallons, fer [...].

A TABLE OF EXCISE, At iij. s. iij. d. per Barrel, To every Firkin, From One to Thirty Barrels.

[Page 46]

A Table of Excise, at 3 s. 3 d. per Bar.
Bar­rels.   ¼
l. s. d. l. s. d. q.
1 00 03 03 00 04 00 3
2 00 06 06 00 07 03 3
3 00 09 09 00 10 06 3
4 00 13 00 00 13 09 3
5 00 16 03 00 17 00 3
6 00 19 06 01 00 03 3
7 01 02 09 01 03 06 3
8 01 06 00 01 06 09 3
9 01 09 03 01 10 00 3
10 01 12 06 01 13 03 3
11 01 15 09 01 16 06 3
12 01 19 00 01 19 09 3
13 02 02 03 02 03 00 3
14 02 05 06 02 06 03 3
15 02 03 09 02 09 06 3
16 02 12 00 01 12 09 3
17 02 15 03 02 16 00 3
18 02 18 06 02 19 03 3
19 03 01 09 03 02 06 3
20 03 05 00 03 05 09 3
21 03 08 03 03 09 00 3
22 03 11 06 03 12 03 3
23 03 14 09 03 15 06 3
24 03 18 00 03 18 09 3
25 04 01 03 04 02 00 3
26 04 04 06 04 05 03 3
27 04 07 09 04 08 06 3
28 04 11 00 04 11 09 3
29 04 14 03 04 15 00 3
30 04 17 06 04 18 03 3

[Page 47]

From 1 to 30 Barrels [...]
½ [...]/4
l s. d. q. l. s. d. q.
00 04 10 2 00 05 08 1
00 08 01 2 00 08 11 1
00 11 04 2 00 12 02 1
00 14 07 2 00 15 05 1
00 17 10 2 00 18 08 1
01 01 01 2 01 01 11 1
01 04 04 2 01 05 02 1
01 07 07 2 01 08 05 1
01 10 10 2 01 11 08 1
01 14 01 2 01 14 11 1
01 17 04 2 01 18 02 1
02 00 07 2 02 01 05 1
02 03 10 2 02 04 08 1
02 07 01 2 02 07 11 1
02 10 04 2 02 11 02 1
02 13 07 2 02 14 05 1
02 16 10 2 02 17 08 1
03 00 01 2 03 00 11 1
03 03 04 2 03 04 02 1
03 06 07 2 03 07 05 1
03 09 10 2 03 10 08 1
03 13 01 2 03 13 11 1
03 16 04 2 03 17 02 1
03 19 07 2 04 00 05 1
04 02 10 2 04 03 08 1
04 06 01 2 04 06 11 1
04 09 04 2 04 10 02 1
04 12 07 2 04 13 05 1
04 15 10 2 04 16 08 1
04 19 01 2 04 19 11 1

A TABLE OF THE Content of Cylinders IN ALE GALLONS AND CENTESIMAL PARTS, From twelve to sixty Inches of Diameter, And to eight Inches in Depth.

[Page 50]

Content of Cylinders
Diam. DEPTH.
Area. 2 3 4
12 0.40 0.80 1.20 1.60
13 0.47 0.94 1.41 1.88
14 0.55 1.10 1.65 2.20
15 0.63 1.26 1.89 2.52
16 0.71 1.42 2.13 2.84
17 0.80 1.60 2.40 3.20
18 0.90 1.80 2.70 3.60
19 1.00 2.00 3.00 4.00
20 1.11 2.22 3.33 4.44
21 1.23 2.46 3.69 4.92
22 1.35 2.70 4.05 5.40
23 1.47 2.94 4.41 5.88
24 1.60 3.20 4.80 6.40
25 1 [...]74 3.48 5.22 6.96
26 1.88 3.76 5.64 7.72
27 2.03 4.06 6.09 8.12
28 2.18 4.36 6.54 8.72
29 2.34 4.68 7.02 9.36
30 2.51 5.02 7.53 10.04
31 2.68 5.36 8.04 10.72

[Page 51]

in Gallons and hundred parts.
DEPTH.
5 6 7 8
2.00 2.40 2.80 3.20
2.35 2.62 3.29 3.76
2.75 3.30 3.85 4.40
3.15 3.78 4.41 5.04
3.55 4.26 4.97 5.68
4.00 4.80 5.60 6.40
4.50 5.40 6.30 7.20
5.00 6.00 7.00 8.00
5.55 6.66 7.77 8.88
6.15 7.38 8.61 9.84
6.75 8.10 9.45 10.80
7.35 8.82 10.29 11.76
8.00 9.60 11.20 12.80
8.70 10.44 12.18 13.92
9.40 11.28 13.16 15.04
10.15 12.18 14.21 16.24
10.90 13.08 15.26 17.44
11.70 14.04 16.38 18.72
12.55 15.06 17.57 20.08
13.40 16.08 18.76 21.44

[Page 52]

Content of Cylinders
Diam. DEPTH.
Area. 2 3 4
32 2.85 5.70 8.55 11.40
33 3.03 6.06 9.09 12.12
34 3.22 6.44 9.66 12.88
35 3.41 6.82 10.23 13.64
36 3.61 7.22 10.83 14.44
37 3.81 7.62 11.43 15.24
38 4.02 8.04 12.06 16.08
39 4.24 8.48 12.72 16.96
40 4.46 8.92 13.38 17.84
41 4.68 9.36 14.04 18.72
42 4.91 9.82 14.73 19.64
43 5.15 10.30 15.45 20.60
44 5.39 10.78 16.17 21.56
45 5 [...]64 11.28 16.92 22.56
46 5.89 11.78 17.67 23.56
47 6.15 12.30 18.45 24.60
48 6.4 [...] 12.84 19.26 25.68
49 6.69 13.38 20.07 26.76
50 6.96 13.92 20.88 27.84
51 7.24 14.48 21.72 28.96

[Page 53]

in Gallons and hundred parts.
DEPTH.
5 6 7 8
14.25 17.10 19.95 22.80
15.15 18.18 21.21 24.24
16.10 19.32 22.54 25.76
17.05 20.46 23.87 27.28
18.05 21.66 25.27 28.88
19.05 22.86 26.67 30.48
20.10 24.12 28.14 32.16
21.20 25.44 29.68 33.92
22.30 26.76 31.22 35.68
23.40 28.08 32.76 37.44
24.55 29.46 34.37 39.28
25.75 30.90 36.05 41.20
26.95 32.34 37.73 43.12
28.20 33.84 39.48 45.12
29.45 35.34 41.23 47.12
30.75 36.90 43.05 49.20
32.10 38.52 44.94 51.36
33.45 40.14 46.83 53.52
34.80 41.76 48.72 55.68
36.20 43.44 50.68 57.92

[Page 54]

Content of Cylinders
Diam. DEPTH.
Area. 2 3 4
52 7.53 15.06 22.59 30.12
53 7.82 15.64 23.46 31.28
54 8.12 16.24 24.36 32.48
55 8.42 16.84 25.26 33.68
56 8.73 17.46 26.19 34.92
57 9.05 18.10 27.15 36.20
58 9.37 18.74 28.11 37.48
59 9.69 19.38 29.07 38.76
60 10.03 20.06 30.09 40.12

[Page 55]

in Gallons and hundred parts.
DEPTH.
5 6 7 8
37.65 45.18 52.71 60.24
39.10 46.92 54.74 62.56
40.60 48.72 56.84 64.96
42.10 50.52 58.94 67.36
43.65 52.38 61.11 69.84
45.25 54.30 63.35 72.40
46.84 56.22 65.59 74.96
48.45 58.14 67.83 77.52
50.15 60.18 70.21 80.24

The Use of the precedent Table.

Having taken the Diameter of the Vessel, and also the Depth of the Li­quor, seek the Diameter in the first Co­lumn of the Table, and casting your Eye upwards, you find over all the other Columns (except that for 1 Inch of Depth, over which stands the Title Area) one of the Digits, amongst which you shall find your Depth of Liquor, if it exceed not 8 Inches; then under that Figure, and against your Diameter you shall find your number of Gallons and parts.

Example.

To find the Content of a Cylinder, whose Diameter is 38, and Depth 4 Inches.

I first seek the Diam. 38, and against it in the Column under 4 I find 16.08, which is 16 Gallons and 8/100 parts.

[...]To reduce which Fraction into Pints, multiply it by 8, and cut off two places in the Product, the figures to the left hand are Pints, and those cut off parts of a Pint, as in the Margin.

A TABLE OF AREA's of SEGMENTS OF A CIRCLE, Whose whole Area is 2, and the Radius divided into 100 p ts, Calculated To the 1/10000 part of a Square Inch.

[Page 58]

Area's of Segments.
V Area V Area V Area V Area
1 .0017 99 .9983 26 .2066 74 .7934
2 .0048 98 .9952 27 .2178 73 .7822
3 .0087 97 .9913 28 .2292 72 .7708
4 .0134 96 .986 [...] 29 .2407 71 .7593
5 .0187 95 .9813 30 .2523 70 .7477
6 .0245 94 .9755 31 .2640 69 .7360
7 .0308 93 .9692 32 .2759 68 .7241
8 .0375 92 .9625 33 .2878 67 .7122
9 .0446 91 .9554 34 .2998 66 .7002
10 .0520 90 .9480 35 .3119 65 .6881
11 .0598 89 .9402 36 .3241 64 .6759
12 .0680 88 .9320 37 .3364 63 .6636
13 .0764 87 .9236 38 .3487 62 .6513
14 .0851 86 .9149 39 .3611 61 .6389
15 .0941 85 .9059 40 .3735 60 .6265
16 .1033 84 .8967 41 .3860 59 .6140
17 .1127 83 .8873 42 .3986 58 .6014
18 .1224 82 .8776 43 .4112 57 .5888
19 .1323 81 .8677 44 .4238 56 .5762
20 .1424 80 .8576 45 .4364 55 .5636
21 .1527 79 .8473 46 .4491 54 .5509
22 .1631 78 .8369 47 .4618 53 .5382
23 .1737 77 .8263 48 .4745 52 .5255
24 .1845 76 .8155 49 .4873 51 .5127
25 .1955 75 .8045 50 .5000 50 .5000

[Page 59]The Use of the precedent Table is very considerable in Geometry, but my present intention is to apply it to Cask-Gauging, viz. To find the vacant Frustums in a Cask partly full, lying with its Axis parallel to the Horizon, the Cask being taken as the Frustum of a Spheroid cut with two Planes parallel, bisecting the Axis at Right Angles.

And here it is requisite, the Boung and Head Diameters, Casks Length, the whole Content, and dry or wet Inches be known. Then, if the Que­stion be what is wanting, or what is remaining in the Cask? divide either the dry or wet Inches by the Boung Diameter, and the Quote seek in the Table, under V or Versed Sine, against it stands a number, which multiplied by the Content exhibits the Vacuity, if your Dividend were the dry, or the remaining Liquor, if it were the wet Inches.

FINIS.
A SHORT SYNOPSIS OR …

A SHORT SYNOPSIS OR INDEX OF THE LAWS OF EXCISE.

LONDON: Printed by William Godbid, 1676.

The Use of the precedent Table is very considerable in Geometry, but my present intention is to apply it to Cask-Gauging, viz. To find the vacant Fru­stums in a Cask partly full, lying with its Axis parallel to the Horizon, the Cask being taken as the Frustum of a Spheroid cut with two Planes parallel, bisecting the Axis at Right Angles.

And here it is requisite, the Boung and Head Diameters, Casks Length, the whole Content, and dry or wet Inches be known. Then, if the Question be what is wanting, or what is remain­ing in the Cask? divide either the dry or wet Inches by the Boung Diameter, and the Quote seek in the Table, under V or Versed Sine, against it stands a number, which multiplied by the Con­tent exhibits the Vacuity, if your Di­vidend were the dry, or the remaining Liquor, if it were the wet Inches.

[Page 60] Some Examples of the Use of this Table of Area's of Segments, in finding the Vacuity of Cask.

QUEST. I.

What is the Vllage of a Cask, whose Boung Diameter is 28 Inches, Con­tent 60 Gallons, and dry Inches 7?

According to the precedent Rule, I divide 7 by 28, which I do by ad­ding two Cyphers, thus: [...]

Then seeking 25 in the Table in the Column under V, in the next Column against it I find .1955, which number I multiply by the whole Content, and cutting off four places toward the right hand of the Product, it exhibits the [Page 61] Ullage or Wants in that Cask in Gal­lons and parts.

Example.

[...]

So is the Ullage or Wants 11 Gallons and almost ¾ of a Gallon.

QUEST. II.

What quantity of Liquor is there re­maining in this Cask?

Divide the wet Inches by the Boung-Diameter, after this manner: [...]

Against which number (75) I find [Page 62] in the Table .8045, which I multiply by the whole Content as before, and the Product gives the quantity of re­maining Liquor.

Example.

[...]

Now if after Division there happen a Remainer, and that be above half the Divisor, I take the next bigger number; or if it be less than half the Divisor, I take the next lesser number, as in the following Examples.

QUEST. III.

If in the fore-mentioned Cask there be 9 Inches of the Boung Diameter dry, what is the Wants?

[...]

Here the Remainer being under half the Divisor I take 32, [...]

QUEST. IV.

There being 19 wet Inches, what is the remaining Gallons?

[...]

[Page 64]Here the Remainer being above half the Divisor, I take the next bigger number, viz. 68, [...]

THE DESCRIPTION AND …

THE DESCRIPTION AND USE Of the GAUGING-RULE.

THE DESCRIPTION & USE OF THE GAUGING-RULE.

THis Rule is commonly four Foot in length, and is made to double in four joints, for convenient portage: It hath also four Sides, on which are drawn several Lines, viz.

1. There is two Lines called Dia­gonals, the one for Wine, the other for Beer or Ale-measure. These are so commonly known, that I suppose there are few Officers but are well ac­quainted with them; however, lest any should be ignorant, take this following advice.

[Page 68]Put the end which is cut slope-ways in at the Boung-hole, and let i [...] touch the bottom of the Head, the Number that appears at the Boung is the Num­ber of Ale or Wine Gallons respe­ctively.

As for Example.

a Cask.

Put your Rule down at the Boung-hole e to the bottom of the Head c, if 60 appear at the Boung on the Dia­gonal for Wine, then is the Content 60 Wine Gallons, and almost 49 Ale Gallons, for the Sub-divisions between the Numbers from 10 upwards signifie each one Gallon.

This way will give a very good esti­mate of the Content of all Cask, in [...]he form of the London Beer Barrel, [Page 69] or the French wine Hogshead. These Lines being together serve also very well for a Table of Reduction of Wine into Ale measure, and the Converse, by inspection only.

2. On another Side or Face there is put a Line of Inches, from 1 to 48 Inches, and each decimally divided; and also upon the same Side you have Oughtred's Gauge-Line, it being a Line of One thirds of Area's of Circles, in Wine Gallons, by which you may Gauge a Cask after this manner: Put your Rule down at the Boung per­pendicularly, observing what Number appears j [...]t even with the inside of the Cask, admit it be 7, set that down twice, then take the Diameter at the Head, and let that shew you 6 upon the same Line, set that down to the former, add these three Numbers to­gether, and multiply the Sum by the Casks Length, here 30, then cut off one place from the Product toward the right hand, and the Figures toward the left hand are your Number of [Page 70] Wine Gallons contained in the Cask.

Example.

[...]

Note, If your Diameter fall a­mongst the Divisions, between the Numbers, you must cut off two places from the Product.

Example.

[...]

Content as before 60.00 Wine Gall.

[Page 71]3. On a third Face of this Rule (which meets the precedent Line of One thirds of Area's upon one Angle or Edge of the Rule) is put a Line of equal parts, numbred from 1 to 96, and is divided into halves: This Line considered together with that before mentioned, do make a Table of Area's of Circles in Ale Gallons, so that if you find your Diameter in this Line, turn up the other Face, and against your Diameter you shall have the Area of your Circle in Ale measure.

As for Example:

The Diameter of a Circle is 19 In­ches, the Area of that Circle upon the other Edge in Oughtrea's Line is a little above one Gallon.

Again, the Diameter being 30 In­ches, the Area is 2.5 Gallons; and if the Diameter be 67 Inches, it holds 12.5 Gallons upon one Inch of depth.

The Use of these Lines, thus toge­ther, [Page 72] is the same with that of the Table of Area's page 29, &c.

4. On the fourth Side of this Rule is drawn a Line of Numbers, vulgarly called Gunters's Line, which Line with a pair of Compasses is of ex­cellent use upon sundry occasions, it being a Line of Logari [...]hms, and by it is performed Multiplic [...]ion, Divi­sion, Extraction of the Square and Cube Roots, and many other Calcu­lations Arithmetical: Of this Line a­lone, are two or three Books of like magnitude with this already printed, to which for those things I r [...]fer you, and sh [...]ll here only apply it to Cask-Gauging; of which take the following Instructions:

  • First learn to find any Number upon the Line; from 1 to 10 you have the Figures Arithmetically placed, and the subdivisions are tenths; but from 10 to 20 the divisions signifie each an additional Unite, and so to 100.
  • At 17.2 you have a small brass Pin, whereon to set the Foot of your Com­passes, [Page 73] and is called the Gauge-point for Wine Gallons, almost at 19 is the Gauge-point for Ale Gallons, the first hath w g, and the other a g placed over it, by which they are easily known.
  • To Gauge a Cask by this Line, you must first find the Diameter at the Head and Boung, and also the Casks Length, by the Line of Inches: These being had, find your mean Diameter, by ad­ding double the Boung Diameter to once the Diameter at the Head, and divide their Sum by 3, the Quote take for your Mean: Then with your Com­passes set one Foot in the Gauge-point, and extend the other to the mean Dia­meter upon your Line of Numbers, so keeping your points at that distance, set one Foot at the number expressing the Casks Length, and from thence double the distance of the Feet of the Compasses exhibits the Content in Ale or Wine Gallons respectively.

As for Example:

A Cask, Boung Diameter 27, Head [Page 74] 24, Length 30 Inches, if the Con­tent be required in Ale or Wine Gal­lons, I find the mean Diameter accor­ding to the Rule thus: [...]

The mean Diameter being 26, I take my Rule and Compasses, if the Question be Ale Gallons, and set one Foot in the Gauge-point for Ale, and the other I extend to 26, then I take off the Compasses so extended, and setting one Foot at 30, the Length, giving the Compasses one turn upon the other Foot, whereby to take the dou­ble distance, and the point toucheth at 57, which is the Content of that Cask in Ale Gallons.

[Page 77]If the Question be Wine Gallons, I take the distance from the Gauge-point for Wine to 26, the mean Dia­meter, and the Compasses applied to the Casks Length, and turned as before, exhibits 69, the Content of the Cask in Wine Gallons.

This is a very quick and easie way of Gauging Cask, and is also an ap­proximation near enough the truth for common practice.

Having the mean Diameter of any Conical Tun, and the Depth of Li­quor, the Quantity is found after the same manner.

Example.

The mean Diameter of a Tun 28, Depth of Liquor 29, Quantity of Ale Gallons will be found 63 ½.

The common way of finding the mean Diameter of a Conical Tun, is by adding the Diameters above and below together, and take the half. This I allow, as an easie and practical way [Page 76] for young Gaugers; but note, that the greater the difference of the Dia­meters are, the greater is your errour, but in Diameters that differ not much, it doth very well.

There is also another Line, that runs parallel with this Line of Numbers, and is called a Line of Segments, but I like not the Hypothesis upon which it is framed, and the way of finding the Wants of a Cask (lying with its Axis parallel to the Horizon, being partly empty of Liquor) made so plain and easie, by the precedent Ex­ampl [...]s upon the Table of Area's of Segments, I had thought to have left it out, but lest any having a Rule and not Table by him, should have such occasion, let him take one

Example.

A Casks Boung 24 Inches, wet 18 Inches, Content 50 Ale Gallons, what is the Ullage or Wants in this Cask?

As 24 on the Line of Numbers, is to [Page 77] Radius on the Segments: So is 6 on the Numbers, to 17.8 on the Segments.

Then, As Vnity, to 50 on the Num­bers: So 17.8 on the Numbers, to 8.9 Ale Gallons, the Wants required.

FINIS.

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