The mariners magazine, or, Sturmy's mathematical and practical arts containing the description and use of the scale of scales, it being a mathematical ruler, that resolves most mathematical conclusions, and likewise the making and use of the crostaff, quadrant, and the quadrat, nocturnals, and other most useful instruments for all artists and navigators : the art of navigation, resolved geometrically, instrumentally, and by calculation, and by that late excellent invention of logarithms, in the three principal kinds of sailing : with new tables of the longitude and latitude of the most eminent places ... : together with a discourse of the practick part of navigation ..., a new way of surveying land ..., the art of gauging all sorts of vessels ..., the art of dialling by a gnomical scale ... : whereunto is annexed, an abridgment of the penalties and forfeitures, by acts of parliaments appointed, relating to the customs and navigation : also a compendium of fortification, both geometrically and instrumentally / by Capt. Samuel Sturmy.

The ART of Gaging of Vessels. CHAP. VIII. The Ʋse of the Line of Numbers, and the Lines on the Gaging Rod or Staff, and the Rules in Arithmetick in Gaging of all sorts of Vessels, (viz.) to Gage a Cube-Vessel, to Mea∣sure any Square-Vessel, and a Cylinder-Vessel; Also, Bar∣rels, Pipes, or Hogsheads; to Measure a Vessel part out, to Measure a Brewers-Tun, or a Mash-Fat, to Measure a Cone-Vessel, to Measure a Rising or Convex Crown; and also a Convex or Falling Crown in a Brewers-Copper; also a Brewers Oval Tun.

PROBL. I. The true Content of a Solid Measure being known, To find the Gage-Point of the same Measure.

THe Gage-Point of a Solid Measure is the Diameter of a Circle whose superficial Content is equal to the solid Content of the same measure; so the solid Content of a Wine-Gallon according to Win∣chester measure, being found to be 231 Cube-Inches: if you conceive a Circle to contain so many Inches, you shall find the Diameter thereof to be 17:15 by this Rule. Example. A Wine-Vessel at London is said being the 66 Inches in length, and 38 Inches in the Diameter, would contain 324 Gallons. If so, by the Line of Numbers we may divide the space be∣tween 324 and 66 into two equal parts, the middle will fall about 146, and that distance will reach from the Diameter 38 unto 17:15 the Gage-point for a Gallon of Wine or Oyl after London measure: the like reason holdeth for the like measure in all places. Thus likewise you may discover the Gage point for Ale-measure, an Ale-Gallon, as hath been of late discovered containing 282 Cubique-Inches; for as 1 is to 1:273, so is 282 to 356, 3 whose square-root is 18:95 the Gage-point for Ale-measure, because of Wast and Soil exceeding that of Wine above two Inches: or you may find it as before by the Content 256, 3 and the length 66, and the Diameter 38, as before. There are several other Rules to find it, but these may satisfy to save Prolixity, Mr. Phillips, and others, have found and proved by Example. That there is 288 ¾ Cubique Inches in an Ale-Gallon, which I believe is the Truth: But that which is received by Autho∣rity, are these sorts of measures, the Wine-measure is 231 Cubique Inches, and for Ale 282 Cubique-Inches or Beer; and for Drie things, as Corn 272 Inches. These Rules are undeceivable with Authority.

Therefore take notice you must be very careful in all your measures of all sorts of Vessels, their length, breadth, and dephth, as also of the Head and Bong; for all small Errours in them may increase too much in the Content: for the mistake of a quarter of an Inch in a large Vessel, may make you misreckon a Gallon in the Content; therefore how to be careful is best known to the Practicioner more than I can declare by many words.

PROBL. II. The Description of the Gaging-Rod, or Staff.

THe most useful Gaging-Rod is 48 Inches or 50 in length, upon one square there is 2 Lines, a Line of Numbers, and a Line of 48 Inches, every Inch divided into 10 parts for the ready measuring of any Vessels, length, breadth, or depth.

But for the measuring of Great-Vessels, there is two Staffs divided into Inches and 10 parts, made to slide.

On the second side is two Lines, the first to Gage by the Head, and the second by the Bong, which added together multiplyed in the length, will give the Contents; As by Example in the following Problem, and Use of a Table of Wine measure.

And the third square is two Diagonal Lines, for the Gage of Wine the first; and for Ale, the second: which shews the Contents to the 1/10 part of a Gallon according to 282 Cubique-Inches in a Beer or Ale-Gallon, the Use in Probl. 7.

On the fourth side is a Line of Segments, or 63 Gallons divided into 1000 parts, as you may have the Use by the following Table in Probl. 8. The making of this Staff is best known to the Instrument-Maker, by reason it must be exactly done; and you may have them of Mr. Philip Standridge in Bristol, and by Mr. Hays, and John Brown in London, Mathematical Instrument-Makers.

PROBL. III. The Description of Symbols of words for Brevity in Arithmetick.

VVHere these following Characters, are placed, you are to Work by these Rules; and that will resolve your Question.

• + Plus or Addition, which is as much as to say add.
• − Minus or Substraction, then you must substract.
• × In or Multiplication, now you are to multiply.
• /2 To Divide by 2 or any other Number under the line.
• = Equal to the thing desired.
• q Square the Number given.
• 2 − q Twice squared, when 2 stands before the Letters.
• C Cube the Numbers.
• Z Sum and Z q Square of the Sum.
• √ q To Extract the square Root.
• Z The Sums of the Squares.
• X Difference, and X. difference of the squares.
• X q Squares of the difference.
• AE The Rectangular or Plain of them, which is the Product of 2 Numbers multiplyed.

PROBL. IV. How to Measure a Cubical Vessel.

SUppose we have a Cubical Vessel to measure, whose sides let be ABCDEF, which let be every way 24 Inches, and I desire to know how many Gallons of Wine or Ale the same will hold.* 1.1

For Beer or Ale by the Line of Numbers.

Extend the Compasses always from the Gage-Point (which for Ale is 16 8/10) unto the side of the Cube 24 Inches, the same extent will reach from the same 24, turning twice over unto 49 Gallons, and better.

For Wine.

Extend the Compasses always from the Gage-Point, which for Wine is 15 2/10 unto the side of the Cube 24 Inches: the same extent will reach from the same 24, turning twice over unto 59 85/200 Gallons, which is almost 60 Gallons of Wine.

The Arithmetical way.

AB,/282 C = Gall. of Ale 49. AB,/231 C = Gallons of Wine 59 ••5/100.

PROBL. V. How to Measure any Square Vessel.

SUppose we have a Square Vessel to Measure, whose side AB let be 72 Inches, and breadth AC 32,* 1.2 and the depth CD 8 Inches.

By the Line of Numbers for Ale.

You must first find a mean proportion between the length AB 72, and the breadth AC 32, by multiplying it together, and taking the Square Root thereof, or taking the middle way between 72, and 32 on the Line of Numbers, and you will find it 48 for the mean.

Now Extending the Compasses from the Gage-point 16 8/10 to the mean Number 48 Inches, the same extent will reach from the depth CD 8 Inches, turning twice over unto 65 36/200 Gallons.

For Wine.

To find how many Wine-Gallons it is, Work by the Gage-point 15 2/20 as you did in the last Rule, and you will find near 79 8/10 Gallons; or you may find a mean proportion between the breadth AC 32, and the depth CD 8: which will be 16 Inches, and so Work according to the former Rule.

How to Work the same without the Gage-Point.

Example for all. Extend the Compasses from the Ale-Gallon 282 unto the length AB 72: the same distance will reach from the breadth AC 32 unto 8: 17/104 Gallons: for an Inch depth, so for 8 Inches you may presently find it to be 65 36/100 Gallons.

For Wine-Gallons.

Take the Numbers 231 the Gage-point, which by the former work you shall find 9 975/1009 Gallons for 1 Inch depth.

The Mathematical way.

ABX: ACX CD / 282 = Ale-Gallons 8 17/100

ABX AC XCD / 231 = Wine-Gallons 9 975/1000

The Browers Coolers are measured all one as this Vessel is.

PROBL. VI. How to Measure a Cylinder Vessel.

SUppose the Diameter of the Head AB be 24 Inches, and the length thereof AC be 30 Inches, To find the contents in Ale-Gallons.* 1.3 Extend the Compasses always from the Gage-point, which for Ale is 18 95/100 Inches unto the diameter 24 Inches; the same distance will reach from the length 30 Inches turned twice over unto 48 13/190 Gal∣lons.

For Wine.

Extend the Compasses from the Gage-point 17 15/100 unto the diameter 24; the same distance will reach from 30 turned twice to 58 7/10 Gallons.

The Arithmetical way.

For Ale. ABqXAC / 359 = Ale-Gallons (viz.) 48 13/190

For Wine. ABqXAC / 294 = Wine-Gallons, (viz.) 58 75/100

PROBL. VII. How to Measure a Globe-Vessel.

SUppose the diameter or height of the Globe be AB 24 Inches: Then to know the Contents in Ale or Wine, it is thus.* 1.4

For Ale.

Extend the Compasses from the Gage-point, which is 23 21/100 unto the diameter AB 24 Inches; the same distance will reach from the same 24 turned twice over unto 25 2/3 Gallons of Ale.

For Wine.

Extend the Compasses from the Gage-point 21 unto the diameter 24 turned twice over, as before, you shall have 31 ⅓ Gallons.

The Arithmetical way.* 1.5

For Ale AB.C / 540 = Gallons, (viz.) 25 60/100

For Wine. AB.C / 440 = Gallons, (viz.) 31 40/190

PROBL. VIII. How to Measure a Barrel, Pipe, Butt, Punching, Hogshead, or small Cask.

SUppose you have a Cask to measure, whose length is AB 27 Inches, and depth at the Bong CD 23 Inches, and breadth at the Head EF 20 Inches.

You are to find a mean-diameter between the Head and the Bong by these Rules.

Take the difference between 23 and 20, which is 3: which being multiplyed always by 7, the Product here is 21, and divided by 10, the Quotient will be 2 1/10 which added to the lesser diameter 20, you have 22 1/19 for the mean-diameter.

Another way to find the mean diameter is thus. A Vessel having 20 Inches diame∣ter at the Head, 23 Inches 〈◊〉〈◊〉 the Bong, I would know the mean-diameter: 20 and 23 makes 43, the half is 21: 50 the lesser taken out of the greater, the difference is 3, which reduced into 10 is 300; then divide by 45, the Product is 6/10 added to 21 50/100 makes 22 1/10 Inches the mean-diameter required.

Then for Ale.

Extend the Compasses from the Gage-point always 18 95/100 unto the Mean-diameter 22 1/10, the same will reach from the length 27 Inches turned twice over, to 36 7/10 Gallons.

For Wine.

Extend the Compasses always from the Gage-point 15 15/200 unto 22 1/10, the same will reach from 27 to 44 and 8/10 Gallons, as before.

The Arithmetical way.

* 1.6For Ale. CD 2 q +: EFqXAB / 1077 = Gallons (viz.) 36 55/10••.

For Wine. CD 2 q +: EFqXAB / 880 = Gallons 44 78/100.

* 1.7There is another way to Work this Vessel or Question, by the Mean-diameter which was before found to be 22 1/10 Inches; and that is after the Cylinder-Vessel, which may be resolved by the Line of Numbers, as before, and by

Arithmetick thus.

* 1.8For Ale. MDqXAB / 359 = Gallons, (viz) 36 73/100

For Wine. MDqXAB / 294 = Gallons, (viz.) 44 85/100 fere.

By the Diagonal Line on the Rod or Staff.

* 1.9Take the measure with your Rod from the Bong hole at C to the lower part of the Head at F, as the Line FC, which in the Example is near 25 4/10 Inches: so if you would know how much Ale the ••ask will hold, you shall find the Bong Hole to cut in the Diagonal Line 36 7/10 Gall.* 1.10 And for Wine it will cut 44 ¾ Gall, the Contents required.

A Table for the Gaging of Vessels.

	Head.	Bong.		Head.	Bong.
D The Diameter in Inches.	G pts.	G pts.		G pts.	G pts.
01	0,001	0.002	31	1,089	2,178
02	0,004	0.009	32	1,160	2.321
03	0,010	0.020	33	1,234	2.468
04	0,018	0 036	34	1,310	2.620
05	0,028	0.056	35	1,388	2.776
06	0,041	0.081	36	1,469	2.938
07	0,056	0.111	37	1,551	3.102
08	0,072	0.145	38	1,636	3.272
09	0,092	0.183	39	1,724	3.448
10	0,113	0.226	40	1,813	3.625
11	0,137	0.274	41	1,904	3.809
12	0,163	0.326	42	2,000	4.000
13	0,192	0.383	43	2,096	4.191
14	0,222	0.444	44	2,194	4.388
15	0,255	0.510	45	2,296	4.588
16	0,290	0.580	46	2,398	4.796
••7	0,328	0.557	47	2,504	5.007
18	0,367	0.734	48	2,611	5.222
19	0,409	0.818	49	2,721	5.442
20	0,453	0.906	50	2,833	5.665
21	0,500	1.000	51	2,948	5.895
22	0,548	1.097	52	3,065	6.129
23	0,600	1.199	53	3,184	6.367
24	0,653	1.305	54	3,305	6.609
25	0,708	1.416	55	3,428	6.856
26	0,766	1.532	56	3,554	7.108
27	0,826	1.692	57	3,682	7.364
28	0,888	1.777	58	3,813	7.625
29	0,953	1.906	59	3,945	7.890
30	1,020	2.040	60	4,080	8.160

* 1.11

The Use of the two Lines upon the Rod marked Head ••nd Bong; and of this Table for Wine-measure.

The Use of this Table is the Root of the usual Making and Use of the lines on the Rod or Staff only. In the Table you have the perfect Number, but you must number upon the Staff, for 10 account 100, and every small Di∣vision is 10; and you must estimate the parts of these small Divisions: then is the Work all one as with this Table, (viz.)

You must measure the Diameter first at the Head, and find the Num∣ber in the Table, or Staff belonging to it; then measure the Diameter at the Bong, and likewise in the Table, or on the Staff, find the Number belong∣ing to that; then add those two toge∣ther, and multiply the Sum thereof by the Inches of the Vessels length, mea∣sured from Head to Head in the In∣side.

The Table and Staff shews for 20 Inches at the Head.	0,453
For 20 Inches at the Bong.	1,199
These 2 added together, make	1652
	27
Which being Multiplyed by	11564
27, the length,	3304
Makes	44604

According to this Operation, it should be 44 Gallons 604/1000 parts, which difference is of no Moment in these Conclusions.

The Table of Segments.

Gals	Parts.	Gal.	Parts.	Gal.	Parts.
63	10000	42	6288	21	3712
½	9705		6223	½	3647
62	9530	41	6158	20	3582
½	9390		6094	½	3517
61	9280	40	6040	19	3452
½	9170		5976	½	3387
60	9065	39	5913	18	3321
½	8962		5850	½	3255
59	8862	38	5787	17	3189
½	8765		5724	½	3123
58	8661	37	5662	16	3056
½	8580		5600	½	2986
57	8491	36	5535	15	2918
½	8404		5476	½	2847
56	8319	35	5415	14	2775
½	8236		5354	½	2703
55	81••4	34	5294	13	2630
½	8072		5234	½	2556
54	7990	33	5174	12	2481
½	7909		5115	½	2405
53	7829	32	5057	11	2328
½	7758		5000	½	2250
52	7672	31	4943	10	2171
½	7595		4885	½	2091
51	7519	30	4826	9	2010
½	7444		4766	50	1928
50	7370	29	4706	8	1846
½	7297		4646	50	1764
40	7225	28	4585	7	1681
½	7153		4542	50	1509
48	7082	27	4462	6	1420
½	7012		4400	50	1329
47	6944	26	4338	5	1235
½	6877		4276	50	1138
46	6811	25	4213	4	1038
½	6679		4150	50	1935
45	6679	24	4087	3	1035
½	6613		4024	50	830
44	6548	23	3960	2	720
½	6483		3906	50	602
43	6418	22	3842	1	470
½	6353		3777	50	295

PROBL. VIII. By the Line of Segments on the Rod or Staff, and also by a Table, How to find the Quantity of Liquor in a Cask that is part full.

SUppose you would know the Quantity of Liquor in a Cask whose depth at the Bong is 23 Inches, as before, and let the Liquor be in height 16 Inches, and the whole Cask to hold 44 85/190 Gallons.* 1.12

By the Line of Numbers on the Staff, the proportion will be, as the whole depth 23 Inch∣es is to the depth in Liquor 16 Inches, so is 1000 to 692 parts.

Which being sought for in the Segment-line on the Staff, you shall have in the Line by it 46 15/1••0 Gallons.

Now if you extend the Compasses from 63 to 46 75/100 Gallons; the same distance will reach once from 44 85/100, the Contents of the whole Cask to 33 6/10, that is, 33 6/10 Gall. of Wine in the Cask.* 1.13

Then by this Rule always as 63 Gallons is to 46 75/100 Gallons, so is 44 85/100 Gallons to 33 60/100 Gallons of Wine in the Cask.* 1.14

By the same Rule Work for Beer or Ale.

To Work this Arithmetically is somewhat tedious; wherefore I have here Calculated a Table whereby you may perform it very easy by help of the Rule of Three.

Example.* 1.15

In the last Vessel whose depth at the Bong is 23 Inches, and depth in Liquor 16 Inches

The first Rule of Proportion.

As 23 is to 16, so is 10000 to 69 21 parts, which sought for in the nearest Number in the table of Segments you shall have against it 46 75/100 Gallons nearest.

Then again.

As the whole Radius 63 Gallons is to 46 75/100, so is the Gage of your Vessel 44 85/100 Gallons to 33 54/100 Gallons near, as before.* 1.16

After this manner of Working you shall have for 7 Inches depth of Liquor 11 22/120 gallons.

And so by these Rules you may work for any other Cask.

PROBL. IX. How to Measure a Brewers Tun, or a Mash-Fat.

LEt the Tun be ACDE, whose Diameter in the bottom let be ED 98 Inches, and the Diameter at the top AC let be 90 Inches, add both the Diameters toge∣ther, you have 188 Inches; then take the half thereof, and it is 94 Inches, this is the Mean-diameter FG; then get the height of the Tun, which let be AB 40 Inches. Now to know how many Barrels of Ale or Beer it will hold according to 36 Gallons to the Barrel, you shall Work thus.

By the Line of Numbers.

Extend the Compasses always from 113 7/10, which is the Gage-point for a Barrel unto the Mean-diameter 94, the same distance will reach from the height 40 Inches turned twice over unto 27 ½ of a Barrel.

The Arithmetical way by the Mean-diameter.

FGqXAB / 359 = Gallons 984 5/10.

Which being divided by 36, you have 27 Barrels 121 ½ Gallons.

Or thus for Barrels.

FGqXAB / 12924 = Barrels 27 35/••90••, near as before.

This Arithmetical way by the Mean-diameter is not absolute true, yet near enough for Brewers Tuns, by reason there is difference of Diameters between the bottom and the top; yet it is seldom above 7 or 8 Inches: But to have an Exact way which also serveth for Coopers or any, take this way for Working this Tun for an Example.

The truest Arithmetical way.

EDq + AC q ; + EDX. ACZ, XAB / 1077 = 985 Gall. 115/1990

Divide 985 1/10 Gallons by 36 Gallons in a Barrel, and the Quotient 27 Barrels, and 13 Gallons remains: so the one will hold 27 Barrels 13 Gallons 115/1000 parts.

Or thus for Barrels.

EDq + AC q, + EDXAC. ZXAB / 38772 = 27 Barrels 378/1000.

PROBL. X. How to Measure a Cone-Vessel, such as is a Spire of a Steeple, or the like, by having the Height and the Diameter at the Base.

SUppose the Diameter at the Base AB be 98 Inches, and the height DC 490 Inches.

Then by the Line of Numbers for Barrels of Ale or Beer.

Extend the Compasses from the Gage-point 169 9/10 unto the Diameter AB 98, the same will reach from the height of the Cone DC 490,* 1.17 turned twice over unto 121 4/10 Barrels ferè.

This is the best Proportion to Work for great Cones to have it in Barrels, but small Cones have it in Gallons.

Then thus Work.

Extend the Compasses from the Gage-point 32 82/100 unto the Diameter of the Base 98, the same will reach from the height of the Cone 490 twice turned unto 4369 547/1009 Gall.

The Arithmetical way. For Gallons.

ABq × DC / 1077 = 4369 547/1000 Gallons.

Which being divided by 36, you have 121 34/100 Barrels.

Or thus for Barrels.

ABq, XDC / 38772 = 121 347/1000 Barrels.

An Example.

The Brewers Tun before measured may be measured, after this manner by Cones, by this Example in this Figure I have proportioned the same Tun in this Cone, as you may prove thus by, the Rule of Proportion to find the Diameter on the top EF 90, it was before.

Work thus.

As CD 490 is to AB 98, so is CG 450 to EF 90 Inches; and so back again, to find the height of the greater Cone, say, as the difference of the Diameters 8 Inches is to the height of the Tun 40 Inches: so is the Diameter of the bottom AB 98 Inches to the greater height DC 490 Inches, from whence substract 40, there remains the height of the lesser Cone GC 450 Inches.

Now Working as before, for the Contents of each Cone.

The greater Cone will be found to be	4369 Gall.	547 Parts.
And the lesser Cone to contain	3384 Gall.	432 Parts.
Which substracted from the greater Cone, there remains 985 Gallons 115/1000 Parts.	985 Gallons	115/1000

For the Brewers Tun, as before found in the 9 PROBL. which is 27 Barrels 13 Gallons. 1/10

PROBL. XI. How to Measure a Segment or portion of a Globe or Sphere, which serves for a Convex Signet or Rising, or Falling Crown in a Brewers Copper.

ADmit you have the Diameter of the Crown AB 80 Inches, and the height there∣of CD 6 Inches.

A Convex Rising Crown.

* 1.18The Falling Crown is nothing but this Figure, the upper part turned down.

Note that a Crown is seldom less then 2 Inches, nor above 12 Inches; for in Bristol in all their Crowns belonging to the Brewers Coppers, the least that was found, was 1 6/1•• Inch, and the greatest height or depth 11 2/10 Inches.

By the Line of Numbers.

Extend the Compasses from the Gage-point 18 95/100 unto the Diameter AB 80, the same distance will reach from half the height CD 6, which is 3 being turned twice unto 53 Gallons ½ ferè.

The Arithmetical way for Ale or Beer-Gallons.

ABq. X ½ CD / 359 = 53 48/100 Gallons.

PROBL. XII. How to Reduce Ale-measure into Wine; And likewise to Reduce Wine-Gallons into Ale.

For Example.

THere is a Vessel that holds 60 Gallons of Ale; the Question is how many Gallons of Wine it will hold.

The Proportion of the backward Rule of 3.

As 282 Ale is to 231 Wine ∷ so 60 Ale to	73; 19/77 Wine-Gallons.
Or thus. As 94 is to 77 ∷ so is 60 Ale to

The reason is thus, 231 Ale-Gallons is 282 Wine-Gallons, or 77 Ale-Gallons is 94 Wine-Gallons.

Or, as 282 to 231, so is 116. 4 to 95. 30, and as 231:282, so is 95:30 to 116 ••/10, or extend the Compasses from the Ale-gallon 282 to the Wine-gallon 231, the same distance will reach from 60 to 73 2/10 Gallons, or from 77 to 94, or from 94 to 77.

PROBL. XIII. How to Measure a Brewers Oval Tun.

LEt the length in the bottom be AB 120 Inches, and the breadth EF 90; let the length at the top be CD 112 Inches, and the breadth 84; also the depth 40 Inches CA.

* 1.19A Brewers Oval Tun.

Now to Work this, you must find a Mean-proportion between the length in the bot∣tom 120, and the breadth 90 Inches.

The Arithmetical way.

AB × EF ♈ q = 103 − 92. That is, Multiply the two Numbers together, and of the Product thereof extract the Square Root▪ so shall you have the Mean-propor∣tional Number.

[illustration] geometrical diagrams

By this Rule you will find a Mean-proportion between the 120 at the bottom, and the breadth 90, to be 103 and 92.

And likewise for the top between 112 and 84, will be 97; then as before, you shall find the Sum to be 200 − 92; the half thereof is the Mean-diameter 100; ••6/10•• Inches; so shall you Work all one, as you did in the Round Tun.

How to get the Mean-diameter by the Line of Numbers.

Let the Numbers given be 120, and 90. Extend the Compasses from 90 to 120: Divide that in half, the same distance will reach from 90 to 103 92/100 almost 104 the Mean Number required; and so likewise between the Number 112 and 84, you will find it 97; then as before, you shall find the Sum 200:92 and ½ 100 46/100 Inches.

The Arithmetical way, as before, is thus.

M. Diam: XCA / 12924 = 31 3/36 Barrels of Ale or Beer.

And in Gall. M. Diam. XAC = 1119 219/1000 Gallons of Beer or Ale.

By the Line of Numbers for Barrels.

Extend the Compasses always from the Gage-Point 113 7/10 to the Mean-diameter 100 46/100: the same will reach from the height 40 turned twice over, to the Quantity of Barrels of Ale or Beer 31 2/10.

PROBL. XIV. How to Gage a Vessel by Oughtred's Gage-Rule.

THis is an Instrument by taking the length in Inches and 10 parts, and is as Exact as any way Instrumental extant; both the Diameters at the Head and Bong, with a Line called Oughtred's Gage-Line.

The Use is thus.

Take the Diameter at the Bong with those Divisions before said from that end where the Divisions begin to be numbred, and set that down twice: and on the Diameter of the inside the Head in this manner,* 1.20 and then add them together, as here you see the length in Inches. Suppose to be (30. 82) then say, as 1, is to 1:77, so is 30:82 to 54 55/100 of a Gallon, being a little more then ½ a Gallon, or 54 Gallons ½ the Content of a High-Country Hogshead; and so you may do by any other great or small sort of Cask.

The end of the 8 Chapter of Gaging Vessels.

Notes

• * 1.1

  See Figure 102 in the follow∣ing Scheme. ☞
• * 1.2

  See Figure 103 ☞
• * 1.3

  See Figure 104 ☜
• * 1.4

  See Figure 105 ☜
• * 1.5

  〈 math 〉〈 math 〉
• * 1.6

  M stands for Mean, D. for Diameter, MD for mean-Diameter.
• * 1.7

  See Figure 106 ☞
• * 1.8

  These sort of Cask Gaged 5 several ways.
• * 1.9

  To know what it shall hold in all.
• * 1.10

  Extend the Compasses from 231 to 282, the same ex∣tent will reach from the Con∣tent in Wine-Gallons 44 8/10 to 36 7/10 the Contents in all Gallons.
• * 1.11

  By the same Rule you may know what an Ale-Cask will hold in VVine.
• * 1.12

  See Figure 107 ☜
• * 1.13

  33 − 6/10
• * 1.14

  This table is made to ½ or 50/100 parts of a Gallon: by same Rules you may make him Quarts or Pints.
• * 1.15

  The top or Bong is up∣permost in the Column to the left hand 63 Gal. the bottom is ½ to the right hand Colum.
• * 1.16

  23 − 7:10000 to 3049

  3043 answers 15:76/100

  63; 15.76 so 44 85/290 = 11 22/100 Gall.
• * 1.17

  See Figure 109 ☜
• * 1.18

  See Figure 110 ☞
• * 1.19

  See Figure 111 ☞
• * 1.20

  0	63
  0	63
  0	51
  1	77