THE NEVV ARTIFI­CIAL GAVGING LINE OR ROD: Together with rules concerning the use thereof: Invented and written by WIL­LIAM OVGHTRED.

who In all due and respective observance praesenteth the same To the Right Honourable LL. Sir NICOLAS RAINTON Lord Major of London for this praesent yeare, and RALFE FREEMAN Alderman Lord Major elect for the yeare now ensuing, and To the Worshipfull GEORGE ETHRAGE the late Master, and Captaine IOHN MILLER the praesent Master of the Company of Vinteners.

And to the whole body of that Right Worshipfull societie.

London Printed by Aug. Mathewes. 1633.

Right Honourable, and Right Worshipfull.

I Doubt not, but as every worthy and laudable worke is subject to ob­liquie and ill constru­ction: So this my good intent and usefull invention, which I here present unto you Right Honou­rable Lords, and you Right Worship­full, into whose clientlie I betake both my selfe and it, shall meet with some over-curious and supercilious censurers, who will misse-judge and misse-say the same: though not for the art (against which I perswade my selfe 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 seeme to be unworthy. But may it please them to consider, [Page 2] that Theologie is [...], the chiefe and principall Lady and Mistresse of all o­ther faculties; unto which all callings in this life, for their just, faithfull, and conscionable execution, are to com­ply, and bee accountable. And seeing God in his sacred word biddeth not onely cease to doe evill, but also learne Esa. [...]. 16. to doe well, and seeke judgement; that Divine may bee thought best to per­forme his dutie, that shall not onely reprove injustice and wrong dealing, but shall also even in particular acti­ons informe the conscience both what is right, and how to performe it. This I have herein endeavoured to effect. The holy Scripture is frequent in shewing the abomination of false and deceitful weights, and in forbidding to M [...]c. 6. 11. [...]ev. 19 35. Prov. 16. 11. doe 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 bagge are his worke. If I therefore by the helpe of God, and the knowledge hee hath beene pleased to give me, shall exhi­bite [Page 3] unto this renowned Citie, a line and rule to measure vessels with ac­cording to true art, and shall teach how to reforme an errour, which hath for some time (through igno­rance of better) usurped the place of truth; and that with much more fa­cility, then 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 yeares agoe, I devised for my private use an instrument, which I The occa­sion vvher­by this gauging in­strument was inven­ted. called the Circles of proportion: which is nothing else but the Loga­rithmes of Numbers, Sines, and Tan­gents set on circles. And I writ in la­tine the manifold uses thereof, not onely in Arithmetical, Geometrical, and Astronomical calculations, but al­so in divers other accounts, practises, and questions, which are occurrent in the civill societie and conversation of mankind: among which one was concerning the measuring or Gauging [Page 4] of vessels: wherein taking the forme of them to be spharoide, or like a long sphaere, 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 aymed at by the Makers) I delive­red out of the grounds and inventi­ons of that most admirable Artist Ar­chimedes, the true and artificiall way of finding their severall contents: and applyed 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 Mecha­nicall waves practised by other men, much lesse refute any of their errors, but onely barely and positively deli­vered the truth. These my notes, and that my instrument (after many yeares) at the perswasion of a friend of mine, who profered to translate them into English, I was content hee should set out and make common, my selfe not having any hand in that publication. When now the transla­tion was ready for the Printer, my [Page 5] friend having perused that which Master Gunter in his second booke of the Crosse staffe Chapt. 4. writeth very obscurely about Gauging of vessels, asked mee what he meant by the meane Diameter after the vsuall manner? I answered him that in the vsuall manner of art there was no medium or meane, but the meane be­tweene the two extreme termes in some kind of Proportion: and that all proportion being either Arithme­tical, Geometrical, or Musical, hee be­ing a Scholer and an Artist, must needs (as I thought) by the meane under­stand one of those three: and most likely the meane arithmetical which is the greatest of them. For suppo­sing the two extreme termes to bee 9 and 4, betweene these the meane arithmetical is 61/ [...], which is halfe the summe of both: the meane geome­trical is 6: and the meane in musical proportion is 5 [...]/13. And yet said I, the meane arithmetical is too little, as in the mensuration of tapering timber I proved: much more either of the [Page 6] two rest. And other meane diame­ter, which may give the true content of a sphaeroides, art affordeth none. My friend replyed, Master Gunter is generally acknowledged to have bin a man of skill, and you utterly un­knowne: and hee that shall find your booke disagreeing from his docu­ments, will without any more adoe reject it; you shall therefore doe well to write in a few lines why you differ from Mr. Gunter, and to shew that his way by the meane diameter cannot stand with art. Which his counsell I seeing not to be unnecessary, especially in regard of the great paines hee had taken in the translation of my notes, tooke a pen, and presently in the mar­gine of his translation writ that short advertisement touching Mr. Gunters meane diameter.

Not long after that translation was in print, word was sent me into the countrey, that an old man (that said he was) the Gauger of London came earnestly inquiring after me, and was very angry; professing hee would vin­dicate [Page 7] Mr. Gunters credit, and speake for him, that could not answer for him­selfe: and that I had taught a false way of Gauging vessels: and that both hee and many others, had bought of my bookes; which much troubled him: and hee was very solicitous how they should come by their money againe: and that hee never saw such a booke 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 booke, and he should have his money againe for it.

At my next comming to London, hee hearing of my being in towne, came to mee, 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. Hee said I understood not what Mr. Gunter meant by the meane diameter [Page 8] after the vsuall manner: For Mr. Gunter (though hee said not so much) did understand the manner vsuall in London; and that hee him selfe follow­ed in gauging: which was to halfe the summe of the diameters at the head and bung, and to adde one quarter of their difference: and that this was a true way, and a better way then mine: and that if hee should use my way, it would require much more paines, and longer time: and that hee had a man, whom hee had taught, which had little or no skill in Arithmetic, and could scarce multiply, yet in his way hee should gauge with any man in England; and that if that his way were not right, hee might doe more wrong in a few yeares then any Aldermans state in London is worth; and that hee was sworne to doe justly. I answered, that the greater tye lay upon him to doe justly, and the more hurt hee might doe by misgauging, the more thankes hee owed mee, for shewing him the right; and with the more gladnesse hee should em­brace it, notwithstanding it might [Page 9] have some more difficultie: for though I should hold my peace (as Mr. Gunter it seemeth did) yet hee were never the more justifiable: and wished him to looke to his con­science: especially seeing now hee might informe himselfe in the truth. Hee said hee found indeed by his expe­rience, that the adding of a quarter of the difference was somewhat too much; and therefore that now hee began to take one fift part thereof: and yet both wayes were better then mine; and hee would not learne of mee to gauge vessels. And so went his way in great choler and displeasure. I did indeed wonder at that his assertion; that whe­ther hee tooke the quarter, or the fift part of the difference, yet both are better then the way of art, which I had delivered: for even in that ex­ample, which in my booke is set downe, 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 hee but made triall, hee should have found the [Page 10] content computed by one quarter of the difference, to have beene gallons 110 and almost an halfe; and by one fift part, to have beene gallons 1051/10; whereas the true measure by art is gal­lons 107▪ and an halfe and somewhat better; which is almost in the very midst betweene both: so that if any man will suppose one of his wayes, bee it the former, or the latter, to bee right: yet the other must of neces­sitie bee further from the truth, then that which I taught; because it fal­ling betweene both, must needs bee neerer to each, then one of them is to the other.

Not long after Mr. Elias Allen (a man well knowne and esteemed by all men of art for his skilfulnesse 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 hee might doe by his manner of Gauging, if it were not true. Which his speech Mr. George Ethrege, then Master of [Page 11] the right worshipfull Company of Vinteners, 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 easie performance; that they might examine and try their vessels themselves: and said it would bee an invention very acceptable, and of great use, and a meanes to prevent much wrong that many times is done. Which inquisition and studie, to find out a true and perfect instrument for measuring of vessels (though at first without any hope of effecting any singular thing therin after such a num­ber of learned artists, and writers in this kind, with the meanest of which I am not to bee compared) I under­tooke, merely our of my love to the society of mankind, and my desire of advancing truth and justice; and at [Page 12] last by the helpe of God, who is righteous, and loveth righteousnesse, have brought to passe: And herein withall humble and affectionate ser­viceablenesse, both unto you my right honourable LL. upon whom the principall care of all such admensura­tions within this most flourishing and illustrious Citie doth rest, and of you right worshipfull the Masters and whole Company of Vintners, and to as many of this most renowned Citie as shall have cause to use the same, I pre­sent this my new gauging instrument. Onely desiring thus much, that you would be pleased to vouchsafe it your acceptance with the same loving and benevolent respect, as I my selfe had unto you, and the truth, in the investi­gation thereof.

Neither indeed may I omit to re­port the noble favour that the right The tryall made of this gau­ging instru­ment: and the certain­ty of it. worshipfull Company of Vintners did afford, though unknowne, unto mee. For when Mr. Elias Allen had fini­shed up one of those my instruments [Page 13] or gauging rods, and had brought it to their Hall, they presently deputed certaine of their society to see the experience and performance thereof, at the Taverne by Leaden-hall under the signe of the Kings-head: and they tooke the paines to examine the truth of it in many and sundry kinds of wine vessels; where, as I have beene told (for I was not there pre­sent my selfe) 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 markes set thereupon exceeded the same measures by two or three, and sometimes more then foure gal­lons in a vessell. Whereupon they agreed with Mr. Allen for a price, and bespake 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 [Page 14] I demand none) I cannot but ac­knowledge with much thankeful­nesse.

Neither is the facilitie of wor­king The facili­tie of wor­king by this new rod, compared with the trouble­somenesse of the other erroneous way. therewith any whit inferior to the certaintie of it: but even in mea­suring of vessels by this true rule much lesse paines 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 ynches must be added together, and halfe the summe of them taken and kept.

Secondly the diameter at the head is to bee subducted out of the diameter at the bung, and the remaines to bee 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 bee added to the halfe summe kept; to make up a meane diameter.

Lastly there remaineth a proportion to bee wrought by multiplication and division; which is this,

[Page 15] As the quadrate of 17⌊15 (which Mr Gunter calleth his gauge point) is to the quadrate of the meane diameter last found,

So is the length of the vessell measu­red in inches, to the content of the same vessell in wine gallons.

Such a deale of paines, and such a multiplicitie of worke is undergone in their ordinary gauging: and yet when all is done, the capacitie or con­tent of the vessell so found is merely conjectural, sometimes falser and sometimes truer, according as the dia­meters of the vessell are more or lesse unequall.

But this way of art, which I pro­pose, besides that it is constant and uni­versall in all kinds of wine vessells, is most easily performed only with one single addition & multiplication; as shall anon bee declared both by rule, and examples. Soe that now I hope I have made Mr Gauger amends for [Page 16] the too much difficultie and paines hee thought my booke put him to; and that henceforth hee will bee as earnest to vindicate my credit also, as hee hath egerly traduced it before he rightly knew me. For although I cannot so well excuse his rashnesse in impertinent interposing himselfe in questions hee had small skill in: yet I perswade my selfe he had no malice against my person, who was then ut­terly unknowne to him. And for my part (for ought I remember) I had not so much as heard of any such offi­cer about the Citie. But I wholly impute his offence at me partly to the high estimation hee had of Mr. Gun­ter, from whom to dissent he thought to be a hainous matter: and partly to his dissidence and distrust of my rule, how it might hold when it came to tryal. And therefore I doe willingly returne againe into favour with him, and recommend this my invention and my selfe to his future approbati­on and friendship; assuring him that what I have before written was not [Page 17] intented against him: but onely to shew the true occasion how I fell up­on this search and disquisition: which in regard of my profession seemed to bee 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 vessell is in forme of a perfect sphaeroide with the two ends equally cut off. The truth of which ground appeareth both by the generall consent of almost all who have written of gauging: and by the fabric of such kind of vessels; which by the workemen and makers thereof (so farreforth as in practise they are able) are intended to bee such. And now lately by tryal made thereof with the Standard, whereunto it is found to hold greater correspondence in all kind of those vessels, then could bee hoped for, or almost imagined.

Secondly, that a sphaere, or spharoide containeth two third parts of a cylin­der hauing the same length and thick­nesse; and consequently that the so­lid [Page 18] convexitie betweene two cylin­ders, one within the sphaeroide, and the other without touching it, and having the same center and height, is equall to two third parts of the difference of those two cylinders: which hath long since beene taught by the ancient Siracusan Archimedes in his first booke of the Sphaere and Cylinder, & in his booke of Cone-like and sphaere-like magnitudes: & of late by our English Archimedes Mr Hen­ry Briggs, in his Treatice of Arithme­tica logarith mica.

Out of the consideration of these two principles, or grounds, I framed my rule, which I set downe in my booke of the Circles of proportion, Chapt. 9. in these words. Measure The rule of Gauging framed out of those tvvo grounds. the two diameters of the vessell, in yn­ches, 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 adde together two third parts, of the greater circle, and one [Page 19] third part of the lesse; Lastly, multi­ply the aggregate by the length: so shall you have the content of the vessell, either in cubic ynches, 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 yn­ches, the one to bee 272¼ the other 231) I taught how to reduce into gallons and hundreth parts, by divi­ding the whole number of ynches contained in the vessell by either of those two numbers respectively. And this is that very rule, the worke 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 apply­ed to this my new gauging line or rod.

It consisteth of two rulers of brasse The gau­ging rod described: and of the severall parts there­of. about 32 ynches of length, which al­so are halfe an ynch broad, and a quar­ter of an ynch thick: that being set together they may make halfe an ynch square. [Page 20] At one end of both those rulers are two little sockets of brasse fastned on strongly: by which the rulers are held together, and made to move one upon another, and to bee drawne 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 serue them fast.

These two little sockets encom­passe not both rulers quite round: but have each of them a slit about a quar­ter of an ynch 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 principall 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 thirtie: of which eve­ry particular division standeth for one hundreth part of a wine gallon: so that they are in all a wine gallons [Page 21] and 30 hundred parts. And for the more perspicuous distinguishing of them they are figured by tens 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 betweene e­very one of those figures ten divi­sions, or ten hundreth parts.

At the other end of both the ru­lers is a hooke framed into right or square angles to be scrued on: the one with a Scrue-pin, the other with a matrix or Scrue-plate. And above these hookes the edge of both ru­lers is divided into ynches, beginning exactly equall with the ends of the hookes: and figured with 5, 10, 15, 20, 25, 27, 30, &c. And each ynch is subdivided into ten equall parts. Which division of ynches, together with the hookes, serve to take the length of all vessels; that the ends of the barrell boards overreaching the bottoms, may not bee an impediment to the measuring thereof.

Vpon the other edge of each ruler [Page 22] 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 tens, as 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300: and I call it the line of denarie 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 fiftie.

Thus having shewed the descrip­tion The use of the gauging rod taught in foure rules. of the severall 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 downe these fower rules,

Rule I. How to take the length of a Vessell.

HAving first sitted the two hookes upon the ends of the two rulers, so that the line of ynches may be uppermost, draw out the ru­lers in length, and apply them to the vessell long wise, so that the two hookes may embrace the two bot­toms thereof: at that length scrue them fast together with the long Scrue-pin in the socket: then take off the hooke, which hath the Scrue-pin: and looke how many ynches the soc­ket of the ruler which hath the hooke on doth cut in the ruler with­out the hooke: for that shall be the length of the vessell from out side to out side in ynches: wherefore if out of that measure you take the thicknes of the two bottoms (which may bee easily estimated by a spigot hole) you shall have the true length of the ves­sell within.

But if the length of the vessell be [Page 24] lesse then 27 ynches, you cannot use both hookes, as before was done. Wherefore apply the ruler which hath the matrix or Scrue-plate to the vessell, setting that hooke against one of the bottoms: and where the di­visions of ynches in that ruler begin­neth, stick up a knife: then remove that same hooke to the other bot­tome; and the ynches, which the knife cutteth, shall give the length of the vessell from out side to outside.

Rule II. How to take the Diameters of a Vessell at the Bunge, and at the Head. and how to adde them toge­ther, that you may finde the summe of them.

THe Diameters are not measured by ynches, as the length is, but by Wine gallons, and hundreth parts: in this maner. Take off the hook which hath the scrue-pinne, and dipp that end of the Ruler into the Vessell at the bung; and looke how many parts of the divisions of Gallons the inside [Page 25] of the bung-hole doth cut, the same shall bee the Diameter at the bung. In like maner apply the same end of the Ruler to the head of the Vessell without; and looke how many parts of the divisions of Gallons it is broad over the midd'st or center thereof; the same shall be the Diameter at the head.

But if the Diameter at the bung be greater then that Ruler is long: draw out the rulers, and scrue them at their whole length: and so measure there­with the Diameters, as was before shewed.

The two Diameters being found out must bee added together, thus: Write downe the Diameter at the bung twice, and under it the Diame­ter at the head onely once, setting like places one under another in the same rowes: that is all the unites in one rowe; 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 be­tweene the unites and the parts, then [Page 26] adde them all three together into one summe 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 onely 96 hundreth parts of a Gallon, viz. 0⌊96. you shall write downe [...] the Diameter at the bung twice, & the dia­meter at the head once and adde them altoge­ther, as you see done in the side: and the summe of the diameters shall bee found 3⌊12, that is gallons 3 and 12 hundreth parts. And thus shall you doe in all Vessels.

Rule III. How to multiply the summe of the Diameters of a Vessell by the length.

THe length of a Vessell being mea­sured in ynches and tenth parts, and the Diameters being found out in Gallons and hundreth parts, and also added together, as hath beene decla­red: [Page 27] The summe of the Diameters is to bee reconed in the line of digit numbers, which is noted with the figures 1, 2, 3, 4, 5, &c. and the length, in the other line of denarie numbers, which is noted with the figures 10, 20, 30, 40, 50, &c. In both which lines of numbers you are to consider, that in the spaces betweene 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 betweene both▪ and in the line of digit num­bers, they signifie so many hundreth parts of a gallon: But in the line of denarie 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 smal­nesse have but 50, yet they are under­stood to bee 100, by imagining every one to containe two.

[Page 28] The nature and valuation of both the lines of numbers, the digit and the denarie, 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 Vessell reconed in the line of denary numbers: there scrue the Rulers fast▪ then in the line of digit numbers recon the summe of the Diameters found out, that at the Bung being doubled, as was taught before: and looke what space and division it poynteth out in the line of denary numbers: the same being rea­sonably aestimated, shall bee the pro­duct sought for.

By aestimating reasonably I under­stand that you consider whether the figure of that space is to bee taken for tenns or for hundreds: which is done very easily: for no man is so simple, as to mistake a Vessell 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 bee given. That the [Page 29] product is to bee aestimated according to that former valuation of the line of denarie numbers: unlesse the Vessell be so small, that the summe 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 summe of the diameters falleth not betweene the unite place in the line of digit numbers, and the socket: divide the same summe by 2, and then multiply the one halfe by the length; so shall you have halfe the product sought for: which being doubled will give you the whole.

Rule IIII. How to gauge or measure the content of any Vessell in Wine gallons.

TAke the length of the Vessell pro­posed in ynch-measure, according to the Rule I, then take the two di­ameters in gallon measure, and adde them together, that at the bung being doubled, according to the Rule II. [Page 30] Lastly multiply the summe 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 Vessell.

The performance and practice of this Worke by Examples.

Example I.

ANd first I will take the Exam­ple Diverse ex­amples shewing the use of this Instru­ment, and the practice of the for­mer rules. in my booke of the Circles of Proportion, Chap 9, of a Ves­sell supposed to bee in length 40 yn­ches; and the diameter at the bung 32 ynches; and the diameter at the head 18 ynches: which two diame­ters being taken with my gauging rod in gallon measure, would have beene at the bung gallons 1 and 16 hundreth parts, and at the head almost 37 hundreth parts of a gallon. Set downe the diameter at the bung twice, and the diameter at the head once, & adde them: the summe of the [Page 31] diameters shall be gallons 2 & almost [...]9 hundreth parts, by [...] Rule II. which being multiplyed by 40, the product shal be gallons 107, and better then an halfe; which is the true content sought for.

Example II.

Suppose a small Ves­sell; whose length is ynches 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?

Adde the two dia­meters [...] together, that at the bung being dou­bled, the summe is gal­lons 1 and 6 hundreth parts, by Rule II. Multiply this summe by the length 45⌊3: and the product shall bee 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 ynches 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?

Adde the two diameters together, that at the bung being doubled: the summe is onely 96 hun­dreth [...] parts of a gallon, by Rule II. Multiply this summe 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 halfe. which is the true content. For the first figure 2 signifyeth not 200 (though it be so marked on the line of denarie numbers) but only 20; as both plaine reason, & also the short rule, at the end of Rule III, wil shew.

Exam. IIII.

Suppose a great ves­sell: whose length is ynches 70 and an halfe, viz. 70⌊5; and the diame­ter [Page 33] 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?

Adde the two diameters together, that at the bung being doubled; the summe is gallons 5 and [...] 16 hundreth parts, by Rule II. Multiply this summe 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 denarie num­bers, the summe of the diameters 5⌊16 will over-reach beyond the socket: therefore to helpe this, take halfe the summe of the diameters, viz. gallons 2⌊58: and multiply that halfe summe by the length 70⌊5: and the product shall bee 181⌊8 [...], by Rule III, that is gallons 181 and almost 9 tenth parts: which is halfe the true con­tent. And being doubled shall give gallons 363⌊78 for the whole content of that great vessell.

[Page 34] The Examples following are of some vessels measured in the presence of them, which were deputed by the Company of Vinteners to see the try­all of this Gauging line or rod.

Exam. V.

A Canarie pipe whose length was ynches 48 and an halfe, 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?

Adde the two diameters together, that at the bung being [...] doubled, the summe is gallons 2 and 4 tenth parts, by Rule II. Mul­tiply this summe by the length 48⌊5: and the product shall bee 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 ynches 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 [Page 35] at the head 57 hundreth parts, viz. 0⌊ [...]7. What is the content?

Adde the two dia­meters [...] together, that at the bung being doubled, the Summe is gallon [...] and 3 hun­dreth parts, by Rule II. Multiply this summe by the length 31⌊2: and the product shall bee 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 ynches 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?

Adde the two diame­ters [...] together, that at the bung being doub­led, the summe is gal­lons 1 and 77 hun­dreth [Page 36] parts, by Rule II. Multiply this summe by the length 30⌊82: and the product shall bee 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 beere ves­sells. vessels by the wine gallon: yet it may also fitly be applyed to the mea­suring of the same vessels by the ale or beere gallon. And for this pur­pose I have on the Ruler figured with digit numbers set two little lines or markes, 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.

Seeke out the content of the vessell proposed in wine gallons, upon the ruler figured with denarie numbers, as hath before beene taught: and thereto set the marke (w): and so shall the other marke (a) point to the number of beere gallons contained in the same vessell.

[Page 37] I will shew the practice of this Rule by the vessells measured in the three last examples.

First The Canarie pipe, whose length was ynches 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 beere gallons will it containe?

Set the marke (w) unto the wine measure 116⌊4 reconed upon the ru­ler figured with denarie numbers, scruing it there fast, and the marke (a) shall in the same ruler point out 98⌊76 that is gallons 98 and 76 hun­dreth parts, the content thereof in beere measure.

Againe, The Graves hogshead, whose length was ynches 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 bee gallons 63⌊ [...]4: how many beere gallons will it containe?

Set the marke (w) unto the wine [Page 38] measure 63⌊34 reconed upon the ru­ler figured with denarie numbers, scruing it there fast: and the marke (a) shall in the same Ruler point out 53⌊74, that is gallons 53, and 74 hundreth parts, the content thereof in beere measure.

Lastly the High countrey hogges-head, whose length was ynches 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 bee gallons 54⌊55: how many beere gal­lons will it containe?

Set the marke (w) unto the wine measure 54⌊ [...]5 reconed upon the ru­ler figured with denarie numbers, scruing it there fast: and the marke (a) shall in the same ruler point out 46⌊28, that is gallons 46, and [...]8 hun­dreth parts, the content thereof in beere measure.

[Page 39] 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 plaine and easie that I doubt not but every meane capacitie will bee 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 worke: only the maner of computing the Gauge-divisions I have concealed: both because that speculation is impertinent to the managing and hand-working there­with: and also that because unto men of art by comparing the rule with the performance, it will not bee difficult to find out the reason: but especially because I intend 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 bee serviceable to this most illustrious Citie, as a meanes of kee­ping truth and equitie in that kind, and acceptable to you Right Honou­rable [Page 40] LL, and to you Right Worship­full to whom I present it, and con­ducing unto the glory of Al­mighty God, the author of every good abilitie, it hath obtayned the desired END.

PErlegi hoc opus Mathematicum, cui titulus est, The new artifici­all Gauging Line or Rod, in quo nihil reperio quod non cum utilitate publica imprimatur, modò intra tres menses proximè sequentes Typis man­detur.

Guil. Bray.

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