THE NEW ARTIFICIAL GAUGING LINE OR ROD: Together with rules concerning the use thereof: Invented and written by WILLIAM OUGHTRED.  
who In all due and respective observance praesenteth the same To the Right Honourable LL. Sir NICOLAS RAINTON Lord Major of London for this present year, and RALPH FREEMAN Alderman Lord Major elect for the year now ensuing, and To the Worshipful GEORGE ETHRAGE the late Master, and Captain JOHN MILLER the present Master of the Company of Vintner's.  
And to the whole body of that Right Worshipful society.  
London Printed by Aug. Mathewes. 1633.   

Right Honourable, and Right Worshipful.  
I Doubt not, but as every worthy and laudable work is subject to obloquy and ill construction: So this my good intent and useful invention, which I here present unto you Right Honourable Lords, and you Right Worshipful, into whose clientlie I betake both myself and it, shall meet with some overcurious and supercilious censurers, who will misse-judge and misse-say the same: though not for the art (against which I persuade myself they shall find little cause to except) yet in respect of my particular calling: of the height and dignity whereof, such small and low cogitations may seem to be unworthy. But may it please them to consider, that Theologie is 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, the chief and principal Lady and Mistress of all other faculties; unto which all callings in this life, for their just, faithful, and conscionable execution, are to comply, and be accountable. And seeing God in his sacred word biddeth not only cease to do evil, but also learn 
Esa. ●. 16.  to do well, and seek judgement; that Divine may be thought best to perform his duty, that shall not only reprove injustice and wrong dealing, but shall also even in particular actions inform the conscience both what is right, and how to perform it. This I have herein endeavoured to effect. The holy Scripture is frequent in showing the abomination of false and deceitful weights, and in forbidding to 
M●c. 6. 11. ●ev. 19 35. Prov. 16. 11.  do unjustly in line, in weight, or in measure: and telleth us that a true weight, and a just balance are of the Lord, and that all the weights in the bag are his work. If I therefore by the help of God, and the knowledge he hath been pleased to give me, shall exhibit unto this renowned City, a line and rule to measure vessels with according to true art, and shall teach how to reform an error, which hath for some time (through ignorance of better) usurped the place of truth; and that with much more facility, than it is committed: I hope I shall not justly be thought to wander out of the limits of my profession and calling. Now the occasion whereby I invented it was this.  
Many years ago, I devised for my private use an instrument, which I 
The occasion whereby this gauging instrument was invented.  called the Circles of proportion: which is nothing else but the Logarithmes of Numbers, Sins, and Tangents set on circles. And I writ in latin the manifold uses thereof, not only in Arithmetical, Geometrical, and Astronomical calculations, but also in diverse other accounts, practices, and questions, which are occurrent in the civil society and conversation of mankind: among which one was concerning the measuring or Gauging of vessels: wherein taking the form of them to be spharoide, or like a long sphere, with the two ends equally cut off, (as generally both by the ancienter and later writers is supposed, and in the fabric of such vessels is still aimed at by the Makers) I delivered out of the grounds and inventions of that most admirable Artist Archimedes, the true and artificial way of finding their several contents: and applied it unto the use of that my instrument, with as much ease and expeditnesse as I could. Neither did I at all discourse upon the Mechanical waves practised by other men, much less refute any of their errors, but only barely and positively delivered the truth. These my notes, and that my instrument (after many years) at the persuasion of a friend of mine, who proffered to translate them into English, I was content he should set out and make common, myself not having any hand in that publication. When now the translation was ready for the Printer, my friend having perused that which Master Gunter in his second book of the Cross staff Chapt. 4. writeth very obscurely about Gauging of vessels, asked me what he meant by the mean Diameter after the usual manner? I answered him that in the usual manner of art there was no medium or mean, but the mean between the two extreme terms in some kind of Proportion: and that all proportion being either Arithmetical, Geometrical, or Musical, he being a Scholar and an Artist, must needs (as I thought) by the mean understand one of those three: and most likely the mean arithmetical which is the greatest of them. For supposing the two extreme terms to be 9 and 4, between these the mean arithmetical is 61/●, which is half the sum of both: the mean geometrical is 6: and the mean in musical proportion is 5●/13. And yet said I, the mean arithmetical is too little, as in the mensuration of tapering timber I proved: much more either of the two rest. And other mean diameter, which may give the true content of a sphaeroides, art affordeth none. My friend replied, Master Gunter is generally acknowledged to have been a man of skill, and you utterly unknown: and he that shall find your book disagreeing from his documents, will without any more ado reject it; you shall therefore do well to write in a few lines why you differ from Mr. Gunter, and to show that his way by the mean diameter cannot stand with art. Which his counsel I seeing not to be unnecessary, especially in regard of the great pains he had taken in the translation of my notes, took a pen, and presently in the margin of his translation writ that short advertisement touching Mr. Gunters mean diameter.  
Not long after that translation was in print, word was sent me into the country, that an old man (that said he was) the Gauger of London came earnestly enquiring after me, and was very angry; professing he would vindicate Mr. Gunter's credit, and speak for him, that could not answer for himself: and that I had taught a false way of Gauging vessels: and that both he and many others, had bought of my books; which much troubled him: and he was very solicitous how they should come by their money again: and that he never saw such a book in his life; for he could not tell what to make of it. The translator told him, that when he, or any other for him, should disprove that way of measuring vessels, which I had delivered; let him bring his book, and he should have his money again for it.  
At my next coming to London, he hearing of my being in town, came to me, and expostulated with me, why I would presume to question any thing that Mr. Gunter writ? I answered him, that it is a fault incident to the students of these arts, to be very incredulous, especially if they see demonstration to the contrary. He said I understood not what Mr. Gunter meant by the mean diameter after the usual manner: For Mr. Gunter (though he said not so much) did understand the manner usual in London; and that he himself followed in gauging: which was to half the sum of the diameters at the head and bung, and to add one quarter of their difference: and that this was a true way, and a better way than mine: and that if he should use my way, it would require much more pains, and longer time: and that he had a man, whom he had taught, which had little or no skill in Arithmetic, and could scarce multiply, yet in his way he should gauge with any man in England; and that if that his way were not right, he might do more wrong in a few years than any Alderman's state in London is worth; and that he was sworn to do justly. I answered, that the greater tie lay upon him to do justly, and the more hurt he might do by misgauging, the more thankes he owed me, for showing him the right; and with the more gladness he should embrace it, notwithstanding it might have some more difficulty: for though I should hold my peace (as Mr. Gunter it seemeth did) yet he were never the more justifiable: and wished him to look to his conscience: especially seeing now he might inform himself in the truth. He said he found indeed by his experience, that the adding of a quarter of the difference was somewhat too much; and therefore that now he began to take one fift part thereof: and yet both ways were better than mine; and he would not learn of me to gauge vessels. And so went his way in great choler and displeasure. I did indeed wonder at that his assertion; that whether he took the quarter, or the fift part of the difference, yet both are better than the way of art, which I had delivered: for even in that example, which in my book is set down, of a vessel having the diameter at the bung 32 inches, and the diameter at the head 18 inches, and the length 40 inches; had he but made trial, he should have found the content computed by one quarter of the difference, to have been gallons 110 and almost an half; and by one fift part, to have been gallons 1051/10; whereas the true measure by art is gallons 107▪ and an half and somewhat better; which is almost in the very midst between both: so that if any man will suppose one of his ways, be it the former, or the latter, to be right: yet the other must of necessity be further from the truth, then that which I taught; because it falling between both, must needs be nearer to each, than one of them is to the other.  
Not long after Mr. Elias Allen (a man well known and esteemed by all men of art for his skilfulness in making instruments in metal) being in the company of some gentlemen of good quality and worth, upon occasion related these former words of the Gauger, what great detriment he might do by his manner of Gauging, if it were not true. Which his speech Mr. George Ethrege, than Master of the right worshipful Company of Vintner's, hearing and observing, said that indeed he doubted not, but there was much wrong done by misgauging wine vessels; and that many times they found it to their cost; but they could not easily remedy it. And therefore invited Mr. Allen to request me to devise a Gauging instrument, which might be according to true art, and of familiar and easy performance; that they might examine and try their vessels themselves: and said it would be an invention very acceptable, and of great use, and a means to prevent much wrong that many times is done. Which inquisition and study, to find out a true and perfect instrument for measuring of vessels (though at first without any hope of effecting any singular thing therein after such a number of learned artists, and writers in this kind, with the meanest of which I am not to be compared) I undertook, merely our of my love to the society of mankind, and my desire of advancing truth and justice; and at last by the help of God, who is righteous, and loveth righteousness, have brought to pass: And herein withal humble and affectionate serviceableness, both unto you my right honourable LL. upon whom the principal care of all such admensurations within this most flourishing and illustrious City doth rest, and of you right worshipful the Masters and whole Company of Vintners, and to as many of this most renowned City as shall have cause to use the same, I present this my new gauging instrument. Only desiring thus much, that you would be pleased to vouchsafe it your acceptance with the same loving and benevolent respect, as I myself had unto you, and the truth, in the investigation thereof.  
Neither indeed may I omit to report the noble favour that the right 
The trial made of this gauging instrument: and the certainty of it.  worshipful Company of Vintners did afford, though unknown, unto me. For when Mr. Elias Allen had finished up one of those my instruments or gauging rods, and had brought it to their Hall, they presently deputed certain of their society to see the experience and performance thereof, at the Tavern by Leaden-hall under the sign of the Kings-head: and they took the pains to examine the truth of it in many and sundry kinds of wine vessels; where, as I have been told (for I was not there present myself) beyond all expectation they found such an exact agreement with the measure of water they filled in by gallons, after the sealed Standard for wine measure, that in most the difference was scarce sensible; whereas the marks set thereupon exceeded the same measures by two or three, and sometimes more than four gallons in a vessel. Whereupon they agreed with Mr. Allen for a price, and bespoke of him threescore of the same my rods or instruments. Which so noble and courteous respect unto me, to decree the examination of that my invention (though other benefit or recompense for the same I demand none) I cannot but acknowledge with much thankfulness.  
Neither is the facility of working 
The facility of working by this new rod, compared with the troublesomeness of the other erroneous way.  therewith any whit inferior to the certainty of it: but even in measuring of vessels by this true rule much less pains is taken, then that other erroneous and inartificial way doth necessarily require.  
For therein first the two diameters at the bung, and at the head measured in inches must be added together, and half the sum of them taken and kept.  
Secondly the diameter at the head is to be subducted out of the diameter at the bung, and the remains to be divided by 4, or else by 5, to find out the quarter, or fift part of their difference.  
Thirdly that quarter of fift part of the difference must be added to the half sum kept; to make up a mean diameter.  
Lastly there remaineth a proportion to be wrought by multiplication and division; which is this,  
As the quadrate of 17 ⌊ 15 (which Mr Gunter calleth his gauge point) is to the quadrate of the mean diameter last found,  
So is the length of the vessel measured in inches, to the content of the same vessel in wine gallons.  
Such a deal of pains, and such a multiplicity of work is undergone in their ordinary gauging: and yet when all is done, the capacity or content of the vessel so found is merely conjectural, sometimes falser and sometimes truer, according as the diameters of the vessel are more or less unequal.  
But this way of art, which I propose, besides that it is constant and universal in all kinds of wine vessels, is most easily performed only with one single addition & multiplication; as shall anon be declared both by rule, and examples. So that now I hope I have made Mr Gauger amends for the too much difficulty and pains he thought my book put him to; and that henceforth he will be as earnest to vindicate my credit also, as he hath eagerly traduced it before he rightly knew me. For although I cannot so well excuse his rashness in impertinent interposing himself in questions he had small skill in: yet I persuade myself he had no malice against my person, who was then utterly unknown to him. And for my part (for aught I remember) I had not so much as heard of any such officer about the City. But I wholly impute his offence at me partly to the high estimation he had of Mr. Gunter, from whom to descent he thought to be a heinous matter: and partly to his dissidence and distrust of my rule, how it might hold when it came to trial. And therefore I do willingly return again into favour with him, and recommend this my invention and myself to his future approbation and friendship; assuring him that what I have before written was not intented against him: but only to show the true occasion how I fell upon this search and disquisition: which in regard of my profession seemed to be very necessary.  
The principles whereon the way 
Two principles or grounds of this gauging instrument.  that I teach is grounded, are these two. First that a wine vessel is in form of a perfect sphaeroide with the two ends equally cut off. The truth of which ground appeareth both by the general consent of almost all who have written of gauging: and by the fabric of such kind of vessels; which by the workmen and makers thereof (so farforth as in practice they are able) are intended to be such. And now lately by trial made thereof with the Standard, whereunto it is found to hold greater correspondence in all kind of those vessels, than could be hoped for, or almost imagined.  
Secondly, that a sphere, or spharoide containeth two third parts of a cylinder having the same length and thickness; and consequently that the solid convexity between two cylinders, one within the sphaeroide, and the other without touching it, and having the same centre and height, is equal to two third parts of the difference of those two cylinders: which hath long since been taught by the ancient Syracusan Archimedes in his first book of the Sphere and Cylinder, & in his book of Cone-like and sphere-like magnitudes: & of late by our English Archimedes Mr Henry Briggs, in his Treatise of Arithmitica logarith mica.  
Out of the consideration of these two principles, or grounds, I framed my rule, which I set down in my book of the Circles of proportion, Chapt. 9 in these words. Measure 
The rule of Gauging framed out of those two grounds.  the two diameters of the vessel, in inches, or else in tenth parts of a foot, the one at the bung hole, the other at the head, and also the length within. And by the diameters found, find out the circles; then add together two third parts, of the greater circle, and one third part of the less; Lastly, multiply the aggregate by the length: so shall you have the content of the vessel, either in cubic inches, or cubic tenth parts of a foot. Which afterward in the end of that Chapter (having first inquired the quantity of a gallon both Ale, and Wine measure in solid inches, the one to be 272¼ the other 231) I taught how to reduce into gallons and hundreth parts, by dividing the whole number of inches contained in the vessel by either of those two numbers respectively. And this is that very rule, the work and practice of which I have by art, with an invention not yet thought upon by any other, and with divisions calculated for that purpose, sitted and applied to this my new gauging line or rod.  
It consisteth of two rulers of brass 
The gauging rod described: and of the several parts thereof.  about 32 inches of length, which also are half an inch broad, and a quarter of an inch thick: that being set together they may make half an inch square. At one end of both those rulers are two little sockets of brass fastened on strongly: by which the rulers are held together, and made to move one upon another, and to be drawn out unto any length, as occasion shall require: and when you have them at the just length, there is upon one of the sockets a long Scrue-pin to serve them fast.  
These two little sockets encompass not both rulers quite round: but have each of them a slit about a quarter of an inch board: that they may not hinder the sight of the divisions, which are upon the broad or flat sides of the rulers. Which divisions are indeed of most principal use: being properly the gauge-divisions: and are now first of all invented, and accommodated to the mensuration of vessels.  
The gauge-divisions are in all two hundred and thirty: of which every particular division standeth for one hundreth part of a wine gallon: so that they are in all a wine gallons and 30 hundred parts. And for the more perspicuous distinguishing of them they are figured by ten thus, 0 ⌊ 10, 0 ⌊ 20, 0 ⌊ 30, 0 ⌊ 40, 0 ⌊ 50, 0 ⌊ 60, 0 ⌊ 70, 0 ⌊ 80, 0 ⌊ 90, 1 ⌊ 00, 1 ⌊ 10, 1 ⌊ 20, 1 ⌊ 30, 1 ⌊ 40, 1 ⌊ 50, 1 ⌊ 60, 1 ⌊ 70, 1 ⌊ 80, 1 ⌊ 90, 2 ⌊ 00, 2 ⌊ 10, 2 ⌊ 20, 2 ⌊ 30: there being between every one of those figures ten divisions, or ten hundreth parts.  
At the other end of both the rulers is a hook framed into right or square angles to be screwed on: the one with a Scrue-pin, the other with a matrix or Scrue-plate. And above these hooks the edge of both rulers is divided into inches, beginning exactly equal with the ends of the hooks: and figured with 5, 10, 15, 20, 25, 27, 30, etc. And each inch is subdivided into ten equal parts. Which division of inches, together with the hooks, serve to take the length of all vessels; that the ends of the barrel board's overreaching the bottoms, may not be an impediment to the measuring thereof.  
Upon the other edge of each ruler is set the the line of Numbers: which is the very same in both: though some what differently figured. For that on the ruler next the socket with the long Scrue-pin, is noted with the figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, which I therefore call the line of digit numbers: and that on the other ruler is figured with ten, as 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300: and I call it the line of denary numbers. Wherein you are to remember that from 1 to 5, and from 10 to 50, and also above 100, every space is subdivided into one hundred parts, but above 5 unto 10, and above 50 unto 100, because the spaces are too small to receive an hundred parts, they are subdivided but into fifty.  
Thus having showed the description 
The use of the gauging rod taught in four rules.  of the several parts of the gauging rod: I now come to the use thereof in measuring any kind of vessel by the wine gallon. For which I will set down these four rules,  

Rule I. How to take the length of a Vessel.  
HAving first sitted the two hooks upon the ends of the two rulers, so that the line of inches may be uppermost, draw out the rulers in length, and apply them to the vessel long wise, so that the two hooks may embrace the two bottoms thereof: at that length screw them fast together with the long Scrue-pin in the socket: then take off the hook, which hath the Scrue-pin: and look how many inches the socket of the ruler which hath the hook on doth cut in the ruler without the hook: for that shall be the length of the vessel from out side to out side in inches: wherefore if out of that measure you take the thickness of the two bottoms (which may be easily estimated by a spigot hole) you shall have the true length of the vessel within.  
But if the length of the vessel be less than 27 inches, you cannot use both hooks, as before was done. Wherefore apply the ruler which hath the matrix or Scrue-plate to the vessel, setting that hook against one of the bottoms: and where the divisions of inches in that ruler beginneth, stick up a knife: then remove that same hook to the other bottom; and the inches, which the knife cutteth, shall give the length of the vessel from out side to outside.   

Rule II. How to take the Diameters of a Vessel at the Bunge, and at the Head. and how to add them together, that you may find the sum of them.  
THe Diameters are not measured by inches, as the length is, but by Wine gallons, and hundreth parts: in this manner. Take off the hook which hath the scrue-pinne, and dipp that end of the Ruler into the Vessel at the bung; and look how many parts of the divisions of Gallons the inside of the bunghole doth cut, the same shall be the Diameter at the bung. In like manner apply the same end of the Ruler to the head of the Vessel without; and look how many parts of the divisions of Gallons it is broad over the midst or centre thereof; the same shall be the Diameter at the head.  
But if the Diameter at the bung be greater than that Ruler is long: draw out the rulers, and screw them at their whole length: and so measure therewith the Diameters, as was before showed.  
The two Diameters being found out must be added together, thus: Writ down the Diameter at the bung twice, and under it the Diameter at the head only once, setting like places one under another in the same rows: that is all the unites in one row; all the tenth parts (which are they next the units) in another row; and all the hundreth parts in the last row: and draw a long line between the unites and the parts, then add them all three together into one sum by the way of ordinary Addition. As in example; Suppose a Vessel that hath the diameter at the bung gallon 1 & 8 hundreth parts, viz. 1 ⌊ 08; and the diameter at the head only 96 hundreth parts of a Gallon, viz. 0 ⌊ 96. you shall write down the Diameter at the bung twice, & the diameter at the head once and add them altogether, as you see done in the side: and the sum of the diameters shall be found 3 ⌊ 12, that is gallons 3 and 12 hundreth parts. And thus shall you do in all Vessels.   

Rule III. How to multiply the sum of the Diameters of a Vessel by the length.  
THe length of a Vessel being measured in inches and tenth parts, and the Diameters being found out in Gallons and hundreth parts, and also added together, as hath been declared: The sum of the Diameters is to be reckoned in the line of digit numbers, which is noted with the figures 1, 2, 3, 4, 5, etc. and the length, in the other line of denary numbers, which is noted with the figures 10, 20, 30, 40, 50, etc. In both which lines of numbers you are to consider, that in the spaces between the figures every tenth division is distinguished with a long line: and every single division with a short line: and every fift with a line of a length between both▪ and in the line of digit numbers, they signify so many hundreth parts of a gallon: But in the line of denary numbers (as also in the line of digit numbers beyond 10) every tenth division is for unites; and every single division is for tenth parts. and beyond 100 every tenth division is for ten; and every single division for one unite. Where note, that in those spaces, which by reason of their smallness have but 50, yet they are understood to be 100, by imagining every one to contain two.  
The nature and valuation of both the lines of numbers, the digit and the denary, being understood, multiplication is thus to be wrought. Set the unite place of the line of digit numbers marked with 1, to the place of the length of the Vessel reckoned in the line of denary numbers: there screw the Rulers fast▪ then in the line of digit numbers recon the sum of the Diameters found out, that at the Bung being doubled, as was taught before: and look what space and division it pointeth out in the line of denary numbers: the same being reasonably aestimated, shall be the product sought for.  
By aestimating reasonably I understand that you consider whether the figure of that space is to be taken for tenns or for hundreds: which is done very easily: for no man is so simple, as to mistake a Vessel of ten or twenty Gallons, for one or two hundred; or of one or two hundred gallons, for one of ten or twenty. And yet this short rule may be given. That the product is to be aestimated according to that former valuation of the line of denary numbers: unless the Vessel be so small, that the sum of the diameters amount not to one gallon: for then the figures 100 & 200, are to be taken for 10 and twenty.  
And if at any time it chance, that the sum of the diameters falleth not between the unite place in the line of digit numbers, and the socket: divide the same sum by 2, and then multiply the one half by the length; so shall you have half the product sought for: which being doubled will give you the whole.   

Rule FOUR How to gauge or measure the content of any Vessel in Wine gallons.  
TAke the length of the Vessel proposed in ynch-measure, according to the Rule I, then take the two diameters in gallon measure, and add them together, that at the bung being doubled, according to the Rule II. Lastly multiply the sum of the diameters by the length, according to Rule III: and the product being reasonably aestimated, shall give the number of Wine gallons contained in that Vessel.   

The performance and practice of this Work by Examples.  

Example I.  
ANd first I will take the Example 
Divers examples showing the use of this Instrument, and the practice of the former rules.  in my book of the Circles of Proportion, Chap 9, of a Vessel supposed to be in length 40 inches; and the diameter at the bung 32 inches; and the diameter at the head 18 inches: which two diameters being taken with my gauging rod in gallon measure, would have been at the bung gallons 1 and 16 hundreth parts, and at the head almost 37 hundreth parts of a gallon. Set down the diameter at the bung twice, and the diameter at the head once, & add them: the sum of the diameters shall be gallons 2 & almost ●9 hundreth parts, by Rule II. which being multiplied by 40, the product shall be gallons 107, and better than an half; which is the true content sought for.   

Example II.  
Suppose a small Vessel; whose length is inches 45 and 3 tenth parts, viz. 45 ⌊ 3; and the diameter at the bung 38 hundreth parts of a gallon; and the diameter at the head 30 hundreth parts. what is the content?  
Add the two diameters together, that at the bung being doubled, the sum is gallons 1 and 6 hundreth parts, by Rule II. Multiply this sum by the length 45 ⌊ 3: and the product shall be 48 ⌊ 12▪ by Rule III, that is gallons 48 and almost 12 hundreth parts: which is the true content sought for.   

Exam. III.  
Suppose another small vessel, whose length is inches 30 and 7 tenth parts, viz. 30 ⌊ 7. and the diameter at the bung 34 hundreth parts of a gallon: and the diameter at the head 28 hundreth parts, what is the content?  
Add the two diameters together, that at the bung being doubled: the sum is only 96 hundreth parts of a gallon, by Rule II. Multiply this sum by the length 30 ⌊ 7: and the product shall be 29 ⌊ 47, by Rule III, that is gallons 29, and 47 hundreth parts, which is almost an half. which is the true content. For the first figure 2 signifieth not 200 (though it be so marked on the line of denary numbers) but only 20; as both plain reason, & also the short rule, at the end of Rule III, will show.   

Exam. FOUR  
Suppose a great vessel: whose length is inches 70 and an half, viz. 70 ⌊ 5; and the diameter at the bung gallons 2 and 3 hundreth parts, viz, 2 ⌊ 03; and the diameter at the head gallons 1 and 10 hundreth parts, viz. 1 ⌊ 10. what is the content?  
Add the two diameters together, that at the bung being doubled; the sum is gallons 5 and 16 hundreth parts, by Rule II. Multiply this sum by the length 70 ⌊ 5: but because the unit● place of the line of digit numbers, being set to 70 ⌊ 5 in the line of denary numbers, the sum of the diameters 5 ⌊ 16 will overreach beyond the socket: therefore to help this, take half the sum of the diameters, viz. gallons 2 ⌊ 58: and multiply that half sum by the length 70 ⌊ 5: and the product shall be 181 ⌊ 8●, by Rule III, that is gallons 181 and almost 9 tenth parts: which is half the true content. And being doubled shall give gallons 363 ⌊ 78 for the whole content of that great vessel.  
The Examples following are of some vessels measured in the presence of them, which were deputed by the Company of Vintner's to see the trial of this Gauging line or rod.   

Exam. V.  
A Canary pipe whose length was inches 48 and an half, viz. 48 ⌊ 5; and the diameter at the bung 93 hundreth parts of a gallon, viz. 0 ⌊ 93: and the diameter at the head 54 hundreth parts, viz. 0 ⌊ 54. what is the content?  
Add the two diameters together, that at the bung being doubled, the sum is gallons 2 and 4 tenth parts, by Rule II. Multiply this sum by the length 48 ⌊ 5: and the product shall be 116 ⌊ 4 by Rule III, that is gallons 116 & 4 tenth parts: which is the content sought for.   

Examp. VI.  
A Graves-hogshead whose length was inches 31 and 2 tenth parts, viz. 31 ⌊ 2: And the diameter at the bung 73 hundreth part● of a gallon, viz. 0 ⌊ 73; and the diameter at the head 57 hundreth parts, viz. 0 ⌊ ●7. What is the content?  
Add the two diameters together, that at the bung being doubled, the Sum is gallon ● and 3 hundreth parts, by Rule II. Multiply this sum by the length 31 ⌊ 2: and the product shall be 63 ⌊ 34 by Rule III, that is gallons 63 and 34 hundreth parts: which is the content sought for.   

Example VII.  
A High-countrey hogshead whose length was inches 30, and 82 hundreth parts, viz. 30 ⌊ 82: And the diameter at the bung 63 hundreth parts of a gallon: viz. 0 ⌊ 63 and the diameter at the head 51 hundreth parts, viz. 0 ⌊ 51. What is the content?  
Add the two diameters together, that at the bung being doubled, the sum is gallons 1 and 77 hundreth parts, by Rule II. Multiply this sum by the length 30 ⌊ 82: and the product shall be 54 ⌊ 55, by Rule III, that is gallons 54 and 55 hundreth parts: which is the true content sought for.  
Although this Gauging instrument is properly framed for measuring 
The use of this Instrument in gauging beer vessels.  vessels by the wine gallon: yet it may also fitly be applied to the measuring of the same vessels by the ale or beer gallon. And for this purpose I have on the Ruler figured with digit numbers set two little lines or marks, the one at 272¼, noted with the letter (w); and the other at 231, noted with the letter (a). The use of which I deliver in this Rule following.  
Seek out the content of the vessel proposed in wine gallons, upon the ruler figured with denary numbers, as hath before been taught: and thereto set the mark (w): and so shall the other mark (a) point to the number of beer gallons contained in the same vessel.  
I will show the practice of this Rule by the vessels measured in the three last examples.  
First The Canary pipe, whose length was inches 48 ⌊ 5: and the diameter at the bung gallon 0 ⌊ 93: and the diameter at the head gallon 0 ⌊ 54: the measure whereof was by Example V found to be gallons 116 ⌊ 4: how many beer gallons will it contain?  
Set the mark (w) unto the wine measure 116 ⌊ 4 reckoned upon the ruler figured with denary numbers, serving it there fast, and the mark (a) shall in the same ruler point out 98 ⌊ 76 that is gallons 98 and 76 hundreth parts, the content thereof in beer measure.  
Again, The Graves hogshead, whose length was inches 31 ⌊ 2: and the diameter at the bung gallon 0 ⌊ 73: and the diameter at the head gallon 0 ⌊ 57: the measure whereof was by Example VI found to be gallons 63 ⌊ ●4: how many beer gallons will it contain?  
Set the mark (w) unto the wine measure 63 ⌊ 34 reckoned upon the ruler figured with denary numbers, serving it there fast: and the mark (a) shall in the same Ruler point out 53 ⌊ 74, that is gallons 53, and 74 hundreth parts, the content thereof in beer measure.  
Lastly the High country hogshead, whose length was inches 30 ⌊ 82: and the diameter at the bung gallon 0 ⌊ 63: and the diameter at the head gallon 0 ⌊ 51: the measure whereof was by Example VII found to be gallons 54 ⌊ 55: how many beer gallons will it contain?  
Set the mark (w) unto the wine measure 54 ⌊ ●5 reckoned upon the ruler figured with denary numbers, serving it there fast: and the mark (a) shall in the same ruler point out 46 ⌊ 28, that is gallons 46, and ●8 hundreth parts, the content thereof in beer measure.  
And now I have finished what I determined to write concerning the use of this my new Gauging line or rod: and have made it so plain and easy that I doubt not but every mean capacity will be able with a little care to apprehend the meaning and practice: I have also delivered the Rule which I follow, and whereupon I ground this work: only the manner of computing the Gauge-divisions I have concealed: both because that speculation is impertinent to the managing and handworking therewith: and also that because unto men of art by comparing the rule with the performance, it will not be difficult to find out the reason: but especially because I intent and wish the benefit of making and fabricating this Instrument, unto Mr. Elias Allen, who gave the occasion of it, and at whose request I invented it. And if it shall be serviceable to this most illustrious City, as a means of keeping truth and equity in that kind, and acceptable to you Right Honourable LL, and to you Right Worshipful to whom I present it, and conducing unto the glory of Almighty God, the author of every good ability, it hath obtained the desired END.     

PErlegi hoc opus Mathematicum, cui titulus est, The new artificial Gauging Line or Rod, in quo nihil reperio quod non cum utilitate publica imprimatur, modò intra tres menses proximè sequentes Typis mandetur.  
Ex aedibus Lambethanis, Octob. 10. 1633. Guil. Bray.   

THese Instruments are made in brass by Elias Allen over against St. Clement's Church without Temple-bar: where also those who are desirous may be instructed in the practical use thereof: and such as shall have occasion may have vessels gauged.