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THE Line of Numbers is only the Logarithmes contrived on a Ru∣ler, and the several ranks of figures in the Logarithmes are here express'd by short, and longer, and longest divi∣sions; and they are so contrived in proportion one to another, that as the
Logarithmes by adding together, and substracting one from another pro∣duce the quesita, so here, by turning a pair of Compasses forward or back∣ward, according to due order, from one point to another, doth also bring out the quesita in like manner.
For the length of this Line of Numbers, know, that the longer it is the better it is, and for that purpose it hath been contrived several ways, as first into a Rule of two Foot long, and three Foot long by Mr. Gunter, and I suppose it was therefore called Gunter's Line. Then that Line dou∣bled or laid so together, that you might work either right on, or cross from one to another, by Mr. Windgate afterwards projected in a Circle, by Mr. Oughtred, and also to slide one by another, by the same Author; and last of all projected (and that best of all hitherto, for largeness, and con∣sequently for exactness) into a Ser∣pentine, or winding circular Line, of 5, or 10, or 20 turns, or more or less,
by Mr. Browne, the uses being in all of them in a manner the same, only some with Compasses, as Mr. Gunter's and Mr. Windgate's; and some with flat Compasses, or an opening Index, as Mr. Oughtred's and Mr. Browne's, and one without either as the sliding Rules; but the Rules or Precepts that serve for the use of one, will in∣differently serve for any: But the pro∣jection that I shall chiefly confine my self to, is that of Mr Gunter's; being the most proper for to be inscribed on a Carpenters Rule, for whose sakes I undertake this collection of the most useful, convenient, and proper appli∣cations to their uses in Arithmatick and Geometry. Thus much for defi∣nition of what manner of Lines of Numbers there be, and of what I in∣tend chiefly to handle in this place.
The order of the divisions on this Line of numbers, and commonly on most other, is thus, it begins with 1, and so proceeds with 2, 3, 4, 5, 6, 7, 8, 9; and then 1, 2, 3, 4, 5, 6, 7, 8,
9, 10; whose proper power or order of numeration is thus: The first 1 doth signifie one tenth of any whole number or integer; as one tenth of a Foot, Yard, Ell, Perch, or the like; or the tenth of a penny, shilling, pound, or the like, either in weight, or number, or measure; and so con∣sequently, 2 is 2 tenths; 3, three tenths; and all the small intermediate divisions, are 100 parts of an inte∣ger, or a tenth, of one of the former tenths; so that 1 in the middle, is one whole integer, and 2 onwards two integers, 10 at the end is 10 inte∣gers: Thus the line is in its most proper acception or natural divi∣sion.
But if you are to deal with a greater number then 10, then 1 at the begin∣ning must signifie 1 integer, and in the middle 10 integers, and 10 at the end 100 integers. But if you would have it to a figure more, then the first 1 is ten, the second a hundred, the last 10 a thousand. If you proceed
further, rhen the first 1 is a 100. the middle 1 a 1000. and the 10. at the end is 10000. which is as great a number as you can well discover, on this or most ordinary lines of num∣bers: and so far with convenient care, you may resolve a question very ex∣actly. Now any number being given under 10000. to find the point re∣presenting it on the rule, do thus.
First, Look for the first figure of your number, among the long divisi∣ons, with figures at them, and that leads you to the first figure of your number: then for the second figure,
count so many tenths from that long division onwards, as that second fi∣gure amounteth to; then for the third figure, count from the last tenth, so many centesmes as the third figure contains; and so for the fourth figure, count from the last centesme, so ma∣ny millions, as that fourth figure hath unites, or is in value; and that shall be the point where the number pro∣pounded is on the line of numbers: Take two or three examples.
First, I would find the point upon the line of numbers representing 12. now the first figure of this number is one, therefore I take the middle one for the first figure; then the next figure being 2. I count two tenths from that 1. and that shall be the point representing 12. where usually there is a brass pin with a point in it.
Secondly, To find the point re∣presenting 144. First, as before, I take for 1. the first figure of the number 144 the middle Figure 1, then for the
second Figure (viz. 4.) I count 4. tenths onwards for that: Lastly, for the other 4. I count 4 centesmes fur∣ther, and that is the point for 144.
Thirdly, To find the point repre∣senting 1728.
First, As before, for 1000. I take the middle 1. on the line.
Secondly, For 7. I reckon seven tenths onward, and that is 700.
Thirdly, For 2. reckon two cen∣tesmes from that 7th. tenth for 20.
And lastly, For 8. you must rea∣sonably estimate that following cen∣tesme, to be divided into 10. parts (if it be not express'd, which in lines of ordinary length cannot be done) and 8. of that supposed 10. is the precise point for 1728. the number propoun∣ded to be found, and the like of any number whatsoever.
But if you were to find a fraction, or broken number, then you must consider, that properly, or absolutely, the line doth express none but deci∣mal fractions: as thus, 1/10 or 1/100 or
1/1000 and more neerer the rule in com∣mon acception cannot express; as one inch, and one tenth, or one hundredth or one thousandth part of an inch, foot, yard, perch, or the like, in weight, number, or time, it being ca∣pable to be applyed to any thing in a decimal way: (but if you would use other fractions, as quarters, half quar∣ters, sixteens, twelves, or the like, you may reasonably read them, or else re∣duce them into decimals, from those fractions, of which more in the fol∣lowing Chapters;) for more plainess sake, take two or three observations:
1. That you may call the 1. at the beginning, either one thousand, one hundred, or one tenth, or one abso∣lutely, that is, one integer, or whole number, or ten integers, or a hundred, or a thousand integers, and the like may you call 1. in the middle, or 10. at the end. 2. That whatsoever value or denomination you put on 1, the same value or denomination all the other figures must have successively, either
increasing forward, or decreasing backwards, and their intermediate di∣visions accordingly, as for example;
If I call 1 at the beginning of the line, one tenth of any integer, then 2 following must be two tenths, 3. three tenths, &c. and 1 in the middle 1 integer, 2 two integers, and 10 at the end must be ten integers.
But if one at the beginning be one integer, then 1 in the middle must be 10 integers, and 10 at the end 100 integers, and all the intermediate fi∣gures 20, 30, 40, 50, 60, 70, 80, 90 integers, and every longest division between the figures, 21, 22, 23, 24, 25, 26, &c. integers, and the shor∣test divisions tenths of those integers, and so in proportion infinitely: as [110 1. 10] [1. 10. 100.] [10. 100-1000.] [100. 1000. 10000.] in all which 4 examples, the first order of Figures, viz. 1/10 1. 10. 100 is repre∣sented by the first 1. on the line of numbers: the second order of Fi∣gures, viz. 1. 10. 100. 1000. is re∣presented
by the middle 1 on the line of numbers: the last order or Place of Figures, viz. 10. 100. 1000. 10000. is represented by the 10. at the end of the line of numbers.
3. That I may be plain (yet fur∣ther) if a number be propounded of 4 Figures, having two cyphers in the middle, as 1005. it is expressed on the line between that prime to which it doth belong, and the next centesme or small division next to it; but if you were to take 5005. where there are not so many divisions, you must ima∣gin them so to be, and reasonably estimate them accordingly. Thus much for numeration on the line, or naming any point found on the Rule, in its proper value and significati∣on.
GEometrical proportion is when divers numbers being compar'd together, differ among themselves, increasing or decreasing, after the rate or reason of these numbers, 2.4 8.16.32. for as 2 is half 4. so is 8 half 16. and as this is continued, so it may be also discontinued, as 3.6.14.28; for though 3 is half 6 and 14 half 28. yet 6 is not half 14, not in proportion to it as 3 is to 6: there is also Arithme∣tical, and Musical proportion; but of that in other more large discourses, being not material to our present purpose (though I may hint it after∣ward.)
To find this by the numbers, ex∣tend the Compasses upon the line of numbers, from one number to ano∣ther, this done, if you apply that ex∣tent (upwards or downwards, as you would either increase or diminish) from either of the numbers propoun∣ded, the moveable point will stay on the 3d proportional number required. Also the same extent applyed the same way from the third, will give you a fourth, and from the fourth a fifth, &c.
Let these two numbers 2 and 4 be propounded to find a third proporti∣onal to them, (that is, to find a num∣ber that shall bear the same propor∣tion to 4. that 2 doth bear to 4.) and then to that 3d, a fourth, fifth, & sixth, &c.) Extend the Compasses upon the first part of the line of numbers, from 2 to 4; this done, if you apply the same extent upwards from 4, the moveable point will fall upon 8, the third pro∣portional required, and then from 8
it will reach to 16. the fourth pro∣portional, & from 16 to 32 the fifth, and from 32 to 64 the sixth propor∣tional: but if you will continue the progression further, then remove the Compasses to 64 in the former part of the line, and the moveable point will stay upon 128 the seventh pro∣portional, and from 128 to 256 the eighth, and from 256 to 512 the ninth, &c. Contrarily to this, if you would diminish, as from 4 to 2, extend the Compasses from 4 to 2, and the moveable point will fall on 1, and from 1 to 5/10 or 5 of ten, which is one half (by the second Problem of the first Chapter) and from 5. to 25. or 2/2 and so forward.
But generally in this, and most o∣ther work make use of the small divi∣sions in the middle of the line, that you may the better estimate the fra∣ctions of the numbers you make use on; for observe, look how much you miss in setting the Compasses to the first and second term, so much on
more will you erre in the fourth; therefore the middle part will be most useful; as for example, as 8 to 11, so is 12, to 16.50. or 5. if you do imagin one integer to be divided but into 10 parts, as they are on the line on a two foot Rule; but on the other end you cannot so well express a small fraction as there you may.
Extend the Compasses from 1 to the Multiplicator, and the same ex∣tent applyed the same way from the Multiplicand, will cause the movea∣ble point to fall on the product.
Let 6 be given to be multiplyed by 5, here if you extend the Compasses from 1 to 5, the same extent will reach from 6 to 30: which number 30 though it be numbred but with 3, yet your reason may regulate you, to call it 30. and not 3; for look what pro∣portion
the first number bears to 1, the same must the other number (or Multiplycand) bear to the Product, which in this place cannot be 3, but 30.
Let 125 be given to be Multiplyed by 144, extend the Compasses from 1 to 125, and the moveable point will fall on 18000. now read to this num∣ber 18000 (so much and no more) you must consider, that as in 125 there is two figures more than in 1. so there must be two figures more in the Product than in the Multiplicand; and as for the order of reading the numbers, you may consider well the first Problem of the first Chapter.
3. As 1. to 25, so 30 to 750 as 1. to 8, so 6 to 48. as 1 to 9, so 9 to 81. as 1 to 12, so 20 to 240. One help more I shall add as to the right com∣putation of the last figure in 4 fi∣gures (for more cannot be well ex∣prest, (on ordinary lines, as that on
a two foot Rule is) but for the true number of figures in the Product; note, that for the most part there is as many as there is in the Multiply∣cator and Multiplycand put together, when the lesser of them doth exceed so many of the first figures of the Product, but if the least of them do not exceed so many of the first fi∣gures of the Product, then it shall have one less then the Multiplycator and Multiplycand put together, as 92 and 68 Multiplyed makes 6256, 4 figures; and 12 Multiplyed by 16, makes but 192, 3 figures, for the reason above-said: now for right na∣ming the last figure, write them down; as thus, 75 by 63. now you Multi∣ply 5 by 3. that is 15. for which you by vulgar Arithmetick, set down 5. and carry 1. therefore 5 is the last fi∣gure in the Product, and it is 4725.
Extend the Compasses from the
Divisor to 1, and the same extent will reach from the Dividend, to the Quo∣tient, or extend the compasses from the Divisor to the Dividend the same extent shall reach the same way from 1 to the quotient; as for example;
Let 750 be a number given to be divided by 25 (the Divisor) I extend the Compasses downward, from 25 to 1, then applying of that extent the same way from 750, the other point of the Compasses will fall up∣on 30, the Quotient sought; or you may say, as 25 is to 750, so is 1 to 30.
2. Let 1728 be given to be divi∣ded by 12, say as 12 is to 1, so is 1728 to 144. Extend the Compasses from 12 to 1, and the same extent shall reach the same way from 1728 to 144. or as before, as 12 to 1728. so 1 to 144.
3. If the number be a decimal fraction, then you work as if it were an absolute whole number; but if it be a whole number, joyned to a de∣cimal fraction, it is wrought here as
properly as a whole number, example, •• would divide 111.4 by 1.728. ex∣tend the Compasses from 1. 728. to 1, the same extent applyed from 111 4. shall reach to 64. 5. so again 56. 4. being to be divided by 8. 75. the Quotient will be found to be 6. 45.
Now to know of how many Fi∣gures any Quotient ought to cosist, it is necessary to write the Dividend down, and the Divisor under it, and see how often it may be written un∣der it; for so many figures must there be in the Quotient, as in Dividing this number 12231, by 27. according to the rules of Division. 27. may be written 3 times under the Dividend: therefore there must be 3 Figures in the Quotient; for if you extend the Compasses from 27 to 1, it will reach from 12231, to 453. the Quotient sought for.
Note, That in this, and also in all other questions, it is best to order it so, as that the Compasses may be at
the closest extent; for you may take a close extent more easily and exactly, than you can a large extent, as by ex∣perience you will find.
Extend the Compasses from the first number to the second; that done, the same extent applyed the same way from the third, will reach to the fourth proportional number required.
If the Circumference of a Circle, whose diameter is 7, be 22, what cir∣cumference shall a Circle have, whose diameter is 14? Extend the Com∣passes upward from 7 in the first part, to 14 in the second, and that extent ayplyed the same way, shall reach from 22 to 44. the fourth proportio∣nal required; for so much shall the circumference of a Circle be, whose diameter is 14, — and the contrary if the circumference were given.
Again, A second Example, if 8. foot of Timber be worth 10 shillings, how much is 12 foot worth? extend the Compasses from 8 to 10, (either in the first part or second) the same extent applied the same way from 12, shall reach to 15. which is the answer to the question; for so many shilings is 12 foot worth.
Extend the Compasses from the first of the numbers given, to the second of the same denomination, if that distance be applyed from the third number backwards, it shall reach to the fourth number sought.
If 60 pence be 5 shillings, how much is 30 pence? facit 2.5. two shillings, five tenths of a shilling, that is, being reduced, 2 shillings 6 pence.
Again, If 60 men can raise a Brest∣work of a certain length and breadth in 48 hours, how long will it be ere
40 men can raise such another? Ex∣tend the Compasses from 60. to 40. numbers of like denomination, viz. of Men; this done, that extent apply∣ed the contrary way from 48. will reach to 72, the fourth number you look for. Therefore I conclude that 40 men will perform as much in 72 hours as 60 men will do in 48 hours.
Note, That this back Rule of 3, may for the most part be wrought by the direct Rule of 3.
If you do but duely consider the order of the Question, for you must needs grant that fewer men must have longer time, and the contrary there∣fore the answer must be in proportion to the question, which might have been wrought thus as well. The Ex∣tent from 40 to 60 will reach the same way from 48 to 72 in direct proportion: Or contrarily, as 60 to 40, so is 72 to 48. which you see is but by Turning the question to its di∣rect opperation, according to the true reason of the question.
Thus you have the way for the di∣rect and reverse rule of 3: for the double rule of 3, and Compound rule of 3, this is the rule for it.
Alwayes in the double rule of 3, 5 terms are propounded, and a sixth is required, 3 of which terms are of supposition, and two of demand; now the difficulty is in placing them, which is best done thus; as in this Example.
If 5 Scholars spend 20 l. in 3 months, How many pounds will serve 9 Scholars for 6 months?
Note here the terms of suppositi∣on are the first three, viz. 5, 20, and 3, and the terms of demand are 9 and 6.
Then next for the right placing them observe which of the terms of Supposition is of the same denomina∣tion with the term required, as here the 20 l. is of the same denomina∣tion with the how many pounds re∣quired. Set that always in the second place, and the two terms of suppo∣sition one above another in the first
place, and the terms of demand one above another in the last place thus,
5-20-9
3—pounds—6
Then the work is performed by two single rules of 3 direct, thus:
Extend the Compasses from 5 to 20. the same extent applyed the same way from 9 shall reach to 369 a 4th. this is the first operation: Then as 3 to 36: the 4th. so is 6 to 72 the number of pounds required. By the line of numbers the double rule is wrought as soon as the compound: therefore I shall wave it now:
Four Questions and their Answers, to shew the various forms of working on the line of numbers.
1. If 12 men taise a frame in ten days, in how many days might 8. men raise the like frame? Reason tells me, that fewer men must needs have longer time; therefore the work is thus, as 12 is to 8, so is 10, to 15, by the last Rule, or 8 to 10, so is 12 to 15.
2. If 60 yards of stuff, at 3 quar∣ters of a yard broad, would hang about a certain room; how many yards of stuff of half a yard broad, will serve to hang about the same room? the work is thus, as 510th to 710th ½ so is 60, to 90, or as 75 to 5, so is 60 to 90, wrought backwards.
3. If to make a foot superficial, 12 inches in breadth, do require 12 in∣ches in length, the breadth being 16 inches, how many inches in length must I have to make a foot superfi∣cial? the work is thus, as 16 is to 12, so is 12 to 9, the number of inches to make a foot.
4. If to make a foot solid, a base of 144 inches require 12 inches in height, a base being given of 216 in∣ches, how much in height makes a foot solid? the work then is, as 216 is to 144, so is 12 to 8. or otherwise thus, as 12 is to 216, so is 144 to 8. the height sought.
This Problem concerns questions of proportions between lines and su∣perficies: now if the denomination of the first and second terms be of lines, then extend the Compasses from the first term to the second, (of the same kind or denomination,) this done, that extent applyed twice, the same way from the third term, the move∣able point will stay upon the fourth term required.
If the Content of a Circle, whose diameter is 14, be 154, what will the content of a Circle be, whose diame∣ter is 28?
Here 14 and 28 having the same denomination, viz. of lines, I extend the Compasses from 14 to 28, then applying that extent the same way, from 154 twice, the moveable point will fall on 616 the fourth proporti∣onal sought, that is, first from 154 to 308, and from 308 to 616. But
But if the first denomination be o•• superficial content, then extend th•••• Compasses unto the half of the di∣stance, between the first and second o•• the same denomination; so the sam•• extent will reach from the third t•• the fourth example.
If the content of a Circle, being 154, have a diameter that is 14, wha•• shall the diameter of a Circle be whose content is 616? Divide the di∣stance betwixt 154 and 616 into •••• equal parts, then set one foot in 14 the other shall reach to 28, the dia∣meter required.
The like is for Squares; for if •••• square whose side is 40 foot, contai•••• 1600 foot: how much shall a squar•• contain, whose side is 60 foot? Tak•• the distance from 40 to 60, and appl•• it twice from 1600, and the move a••∣ble point will stay on 3600, the con∣tent sought for.
The use of this Problem, consist∣eth in questions of proportion, be∣tween lines and solids, wherein if the first and second terms have denomi∣nation of lines, then extend the Com∣passes from the first to the second, that extent applyed three times from the third, will cause the moveable point to stay on the fourth proportional re∣quired.
If an Iron Bullet, whose Diameter is 4 inches, shall weigh 9 pounds, what shall another Iron Bullet weigh, whose Diameter is 8 inches? Extend the Compasses from 4 to 8, that ex∣tent applyed the same way 3 times form 9, the moveable point will fall at last on 72, the fourth proportional and weight required, that is, in short thus, as 4 to 8 so 9 to 18, so 18 to 36, so 36 to 72.
But if the two given terms be
weight, or contents of solids, and (the Diameter or side of a square, or) a line is sought for, then divide the space between the two given terms of the same denomination into three parts, and that distance shall reach from the third to the fourth propor∣tional.
Divide the space between 9 and 72 in three parts, that third part shall reach from 8 to 4 (or from 4 to 8, as the question was propounded, either augmenting or diminishing.)
Also if a cube whose side is 6 inches contain 216 inches, how many inches shall a cube contain, whose side is 12 inches; Extend the Compasses from 6 to 12, that extent measured from 216 in the first part of the line of numbers three times, shall at last fall upon 1728, in the second part of the line of numbers; for note, if you had begun on the second part, you would at three times turning, have fallen be∣yond the end of the line, and the con∣trary as above, holds here in squares also.
This may be done without the help of the line of numbers: nevertheless, because it serves to find the next fol∣lowing, I shall here insert it, though I thought to pass both this and the next over in silence, yet to set forth the excellency of number, I have set them down; and the Rule is this, Add half the difference of the given terms to the lesser of them, and that aggre∣gate (or sum) is the Arithmetical mean required.
Let 20 and 40 be the terms given, now if you substract one out of the o∣ther, their difference is 60, whose half difference 30, added to 20, the lesser term makes 50, and that is the Arith∣metical mean sought.
Multiply the difference to the
terms by the lesser term, and add likewise the sa••••e terms together; this done, if you divide that Pro∣duct by the sum of the terms, and to the Quotient add the lesser term, that last sum is the Music•••• mean required: or shorter thus,
Multiply the terms one by another, and divide the Product by their sum, and the Quotient doubled is the Mu∣sical mean required.
The numbers given being 8 and 12 multiplyed together, make 96, that divided by 20, the sum of 8 and 12, the Quotient is 4 80, which doubled is 9-6 10 s, the Musical mean requi∣red. This may be done by the line of numbers; otherwise thus, find the Arithmetical mean between 8 and 12, and then the analogy or agreement is thus,
As the Arithmetical mean found is to the greater term, so is the lesser term to the Musical mean required.
Divide the space on the line of numbers, between the two extreme numbers, into two equal parts; and the point will stay at the mean pro∣portional required. So the extreme numbers being 8 and 32, the middle point between them will be found to be 16.
Divide the space between the two extreme numbers into 3 equal parts, and the two middle points, dividing the space shall shew the two mean proportionals. As for example, let 8 and 27, be two extremes, the two means will be found to be 12, and 18, which are the two means sought for.
The Square root of every number
is always the mean proportional be∣tween 1, and that number for which you would find a square root; but yet with this general caution, if the fi∣gures of the number be even, that is 2, 4, 6, 8, 10, &c. then you must look for the unit (or one) at the beginning of the line, and the number in the se∣cond part, and the root in the first part, or rather reckon 10 at the end to be the unit, and then both root and square will fall backwards to∣ward the middle in the second length or part of the line: but if they be odd, then the middle one will be most convenient to be counted the unity, and both root and square will be found from thence forwards to∣ward 10. so that according to this rule, the square of 9, will be found to be 3, the square of 64, will be found to be 8, the square of 144, to be 12. the square of 1444, to be 38. the square of 57600; to be 240. the square of 972196, will be found to be 986. and so for any other number.
Now to know of how many figures any root ought to consist, put a prick under the first figure, the third, the fifth and the seventh, if there be so many; and look how many pricks, so many Figures there must be in the Root.
The Cubique root is always by the first of two mean proportionals be∣tween 1 and the Number given, and therefore to be found by dividing the space between them into three equal Parts: So by this means the root of 1728 will be found to be 12, the root of 17280 is neer 26, the root of 172800 is almost 56, although the point on the Rule representing all the square numbers is in one place, yet by altering the unit it produceth various points and numbers, for their respe∣ctive proper roots. The Rule of find
which is in this manner: You must set (or suppose pricks to be set) pricks under the first figure to the left hand, the fourth figure, the seventh and the tenth; now if by this means the last prick to the left hand shall fall on the last figure, as it doth in 1728, then the unit will be best placed at 1 in the middle of the Line, and the Root, the Square and Cube will all fall forward toward the end of the Line.
But if it fall on the last but 1, as it doth in 17280 then the unit may be placed at 1 in the beginning of the Line, and the Cube in the second length, or else the unit may be pla∣ced at 10 in the end of the Line, and the Cube in the first part of the Line, you may help your self, as in the first Problem of the 2 Chapter.) But if the last prick fall under the last but two, as in 172800, it doth then place the unit always at 10 in the end of the Line, then the Root, the Square, and Cube, will all fall backward, and be found in the second part, between
the middle 1 and the end of the Line. By these Rules it doth appear that the Cube root of 8 is 2, of 27 is 3, of 64 is 4 of 125 is 5, of 216 is 6, of 345 is 7, of 512 is 8, of 729 is 9, of 1000 is 10. As you may see by this following Table of Square and Cu∣bique roots.
Thus you have the chief use of the line of numbers in general, and they that have skill in the rule of three and a little knowledge in plain triangles, may very aptly apply it to their par∣ticular purposes, Yet for their sakes for whom it is intended, I shall in∣large; to some more particular appli∣cations in measuring all sorts of Su∣perficies, and Solids; wherein I do judge it will be most serviceable do them that be unskilful in Arithme∣tick, as before said.
| Root. | Square. | Cube. | Root. | Square. | Cube. | Root. | Square. | Cube. |
| 1 | 1 | 1 | 7 | 49 | 343 | 204 | 41616 | 8489664 |
| 2 | 4 | 8 | 8 | 64 | 512 | |||
| 3 | 9 | 2•• | 9 | 81 | 729 | 439 | 192721 | 84604519 |
| 4 | 16 | 64 | 10 | 100 | 1000 | |||
| 5 | 25 | 125 | 12 | 144 | 1728 | 947 | 896809 | 849278123 |
| 6 | 36 | 216 | 26 | 676 | 17576 | |||
| 56 | 3136 | 175616 | 1000 | 1000000 | 1000000000 |
The ordinary measure, and most in use, is a Twofoot Rule, divided into 24 Inches, and every Inch into 8 parts; that is, Halss, Quarters, and Half∣quarters; but these parts not agreeing with the parts on the Line of Num∣bers, which are Decimals, or tenth parts, is bred very much trouble; and there cannot be exactness without ta∣king of small parts, as ½ quarters of Inches, or else using of Reduction; and it is also as troublesom by Arith∣metick as by the Line of Numbers. To avoid which, I would advise ei∣ther to measure altogether by Foot-measure, (that is, a Foot divided into 1000 parts (or rather as is sufficient for ordinary use, 100) and then the
divisions on the numbers will agree fitly to the parts on your Rule, with∣out any trouble for fractions; for so doing, Fractions do become whole Numbers as it were, and are wrought accordingly: But if you use it not in measuring, yet you may have it set for to help you for the ready reducing of such Numbers as shall require it, though I shall apply it to Inches also, as it is commonly used, that it may appear useful both ways, accordingly as any man shall be affected.
The like reason holdeth for Inches, Yards, Ells and Perches, or any other measure; for thereby the Work is made more easie, as shall appear anon.
Therefore first by Foot-measure only.
Extend the Compasses from the breadth to 1, the same extent applyed
the same way from 1, will reach to the length required.
So the breadth being 8 tenths, or 0, 80, the length to make a Foot Su∣perficial will be found to be 1. 25. Or shorter thus, as 8 tenths (or 80 of a 100) is to 1, so is 1 to 1, 25 of an hundred.
Extend the Compasses from 1 to the breadth, the same extent applyed the same way from the length, will reach to the Content.
As 1 is to 8, the breadth; so is 15 the length to 12, the content required; for a peice of 8 tenths broad, and 15 Foot long, containeth 12 Foot.
As 1 Inch to the breadth in Inches, so the length in Inches to the Con∣tent in square Superficial Inches.
So the breadth 30 Inches, and the length 183, the Content will be found to be 5490.
Or else, as 1 to 183, so is 30 to 5490 Inches.
As 144 the number of Inches in one Foot, is to the breadth in Inches; so is the length in Inches unto the Con∣tent in Feet. So as 144 to 30, so is 183 to 38, 250, that is, to 38 Foot and a quarter.
As 12 to the breadth in Inches, so is the length in Feet to the Content in Feet.
As 12 unto 30, so is 15 to 37, 50.
As the breadth in Inches to 144, so is 1 to the length in Inches. As 30 to 144 so is 1 to 48 Inches, or as 12 to the breadth in Inches, so is 12 the con∣trary way to the number of Inches; for if you extend the Compasses from 12 to 6, that extent applyed the con∣trary way from 12 shall reach to 24 the Inches required.
Divide the space between the length and the breadth into two equal parts, and the middle point shall shew the side of the Square that shall be e∣qual in area, or quantity, to that Ob∣long; so that a Square made of 11, 32, is equal to an Oblong of 16 one way, and 8 the other way.
As 10000 to 8862, so is the Di∣ameter 15 to the side of the Square 13, 29, that is equal to the Circle.
As 10000 to 2821, so is the Cir∣cumference 47, 13 to the side of the Square 13, 29 equal to the Circle.
As 1 is to the Diameter, so 3142 to the Circumference. Or, as 7 to 22, so is the Diameter to the Circumfe∣rence: or as 113 to 355.
So the Diameter being 15, the Cir∣cumference wil be about 47 13 parts
As 3142 is to 1, so is the Circum∣ference to the Diameter: Or, as 22 is to 7, so is the Circumference to the Diameter: Or as 355 to 113.
So the Circumference being 47, 13. the Diameter is 15.
The Extent from 1 to 7071 will reach from the circumference to the side of the Square required. So the Diameter being 15, the side of the square inscribed will be 10, 60.
The extent from 1 to the Circumfe∣rence, will reach from 2251 to the side of the Square required.
So the Circumference being 47 13 the side of the inscribed square will be 10.60 as before.
The extent from 1 to the Diameter, being twice repeated (the same way) from 7854, will reach to the content required.
The extent from 1 to the Circum∣ference, being twice repeated from 07958, will reach to the content.
The extent from 1. to 1273, will reach from the content to another
number, whose square root is the Di∣ameter required.
The extent from 1 to 12, 75, will reach from the content to another number, whose square root is the Cir∣cumference required.
Extract the square root of the con∣tent by the 12 Probleme of the last Chapter, and that is the side required.
Multiply the Diameter by it self, and multiply that product by a 11, and divide this last product by 14, and the quotient shall be the content re∣quired.
Multiply half the Diameter and half the Circumference together, and
the product is the content required.
Take half the Diameter and half the Circumference, and measure it then as an Oblong square; for the half circle take half the Diameter, and half the Semicircumference, and do likewise. Thirdly, for the quarter of a Circle, take half the arch of that quarter, and the Radius or Semidia∣meter of the whole Circle, and work as you would do with an Oblong square piece, and you shall have your desire.
Take half the base, and the whole perpendicular; and work with them two, as if it were an Oblong square figure, or you may take the whole base and half the perpendicular.
A Rhombus is a Diamond-like fi∣gure, as a quarrie of glass is, contain∣ing 4 equal sides and two equal oppo∣sit angles; but a Rhomboides is a figure made of two equal opposite sides and two equal opposit angles: and to measure them you must take any one side, and the nearest distance from that side to his opposite side for the other side, and then reckon it as an Oblong square.
A Trapezium is a figure compre∣hended of 4 unequal sides, and of 4 unequal angles: and before you can measure it, you must reduce it into two triangles, by drawing a line from any two opposite corners, then deal with it as two triangles; or you may save some work thus, the line you draw from corner to corner, will be the common base to both triangles:
then say as 1 is to half the perpendi∣culars of both the triangles put toge∣ther, so the whole base to the con∣tent.
You must reduce it into triangles, or to trapeziums, by drawing of lines from convenient opposite corners: and then the Work is all one with that of the last Problem.
Measure all the sides, and take the half of the sum of them for one side of a Square, and the neerest distance from the center of the Polygon to the middle of one of the sides, for the other side of a Square; and with them two numbers work as if it were a Square Oblong figure, and it will give the Content of the Polygon desired.
First for yards, if 9 foot make one yard, how many shall 36 foot make.
The extent from 9 to 2, will reach from 36 to 4, for so many yards is 36 foot. But if you were to measure any quantity by the yard, as the Plai∣stering or Painting of a House, then I would advise you to have a yard to be divided into a hundred parts, (which is as neer as commonly Workmen go to, or else, into a 1000, if you do re∣quire more exactness) and measure all your lengths and breadths with that, and set them down thus, 2. 25 (or by 1000 thus 2. 250) and the length thus 10.60, and multiply them toge∣ther, and the product is the true con∣tent of that long square, the like holds for ells, or poles, furlongs, or any o∣ther kind of measure. Again, for a yard in length, if 3 foot make one yard, then what shall 30 make? it maketh 10, or the contrary if 10 yards make
30 foot, what shall 12 make? the ex∣tent from 10 to 12, will reach from 30 to 36 foot, but if it be given in feet inches, then say as 9 to the breadth so is the length in feet and inches (or decimal parts) to the content in yards required.
As 1 perch to the breadth in per∣ches, so the length in perches to the content in perches.
As 1 is to 30, so is 183 to 5490 (perches.)
As 160 to the breadth in perches, so is the length in perches to the con∣tent in acres.
As 160 unto 30, so is 183 to 34, 31 (in acres and perches.)
It being troublesome to divide the Content in Perches by 160, we may measure the length and breadth by Chains, each Chain bein 4 perches in length, and divided into 100 links, then will the Work be more easie in Arithmetick, or by the Rule; for as 10 to the breadth in Chains, so the length in Chains to the content in Acres.
As 10 to 7.50, so is 45.75 to 34.31 (100 parts of an Acre.)
If the Perpendicular go for the length, and the whole base for the breadth, then you must take half of the oblong for the content of the tri∣angle, by the second probleme, as 160 to 30, so is 183 to 34.31, or else with∣out halfing, say as 320 to the perpen∣dicular, so is the base to the content in acres; as 320 unto 30, so is 183 to 17, 15.
As 20 to the perpendicular, so is the base to the content in acres.
As 20 to 7, 50, so is 45, 75, to 17, 15 parts.
As the length of the second perch, is to the first, so is the content in acres to a fourth number, and that 4th num∣ber to the content in acres required. Suppose a Superficies be measured with a chain of 66 feet or with a perch of 16 1/2, and it contain 34. 31, and it be demanded how many acres it would contain, if it were measured with a perch of 18 foot? these kind of proportions, are to be wrought by the backward rule of Three, after a duplicated proportion: wherefore I extend the Compasses from 165. unto 18, 0, and the same extent doth reach backward, first from 34.31 to 31.45, and then from 31. 45 to 28. 84, the content, in those larger acres of 18 foot to a perch.
Suppose a plain contained 34 acres 31 Centesmes, if I should measure it with a scale of 10 in an inch, the length should be 38 Chains and 12 Centesmes, and the breadth 6 Chains and 25 Centesmes, and the content according to that dimension, would by the 3 Probleme of this Chapter be found to be 23, 82. whereas it should be 34.31, therefore to gain the truth, I divide the distance between 23. 82 and 34, 31 into two equal parts, then setting one foot of the Compasses upon 10, the supposed true scale, I find the other to extend to 12, which is the length of the scale required.
As the length in perches to 160, so is one acre to the breadth in perches. As 40 to 160, so is 1 to 4. Again, as
50 to 160, so is 1 to 3.20, so is 2 to 6 40; or again if you measure by chains, As the length in Chains to 10, so is 1 acre to his breadth in Chains; as 12 50 unto 10, so 1 to 0. 80, or if the length be measured by foot measure, then as the length in feet unto 43560 so is 1 acre to his breadth in foot measure.
So the length of the oblong being 792 feet, the breadth of one acre will be found to be 55 foot, the breadth of 2 acres 110 feet.
The use of this Table is to shew you how many Inches, Centesmes of a Chain, feet, yards, paces, perches, chains, acres, there is in a mile, either long or square, or consequently any of them all, in any of the other that is less; as for example, I would know how many Inches there is in a long berch, I look on the uppermost row for perches, and in the next row under, I find 198 for the quantity of inches in a long perch. But if I would know now many inches there is in a square
perch, then look for perch on the left hand, and in the inch colume you have 39204, for if you multiply 196 by 196, it will produce 39204.
| Inch. | Centesme. | Feet. | Yard. | Pace. | Perch. | Chain. | Acre. | Mile. | |
| Inch. | [1] | 7 92 | 12 | 36 | 60 | 198 | 792 | 7920 | 63360 |
| Centes | 62 7264 | [1] | 1. 515 | 4 545 | 7 575 | 25 | 100 | 1000 | 8000 |
| Feet. | 144 | 2 295 | [1] | 3 | 5 | 16 5 | 66 | 660 | 5280 |
| Yards. | 1296 | 20 655 | 9 | [1] | 1 66 | 5 50 | 22 | 220 | 1760 |
| Pace. | 3600 | 57 485 | 25 | 8 335 | [1] | 3 30 | 13 2 | 132 | 1056 |
| Perch. | 39204 | 625 | 272.25 | 90 75 | 10. 89 | [1] | 4 | 40 | 320 |
| Chain | 627264 | 10000 | 4356 | 1452 | 17424 | 16 | [1] | 10 | 80 |
| Acre. | 6272640 | 100000 | 43560 | 14520 | 1742.4 | 160 | 10 | [1] | 8 |
| Mlie. | 4014489600 | 64000000 | 27878400 | 9292800 | 1115136 | 102400 | 6400 | 640 | [1] |
| Squar. | Inches. | Centesmes. | Feet. | Yards. | Pace. | Perch. | Chain | Acre. | Mile. |
The like is for any other number in the whole Table, an•• is of very good use to reduce one number into another, or one sort of measure into another: as inches into feet, and feet into Yards, and Yards into Perches, and Perches to Chains, and Chains into Acres, and Acres into Miles, or the contrary either long-wise or square- wise: as is well known to them that have occasion for these measures. Thus much shall suffice for Superficial measure, the practice of which will make it plain to any ordinary capaci∣ty.
Divide the Space between the breadth, and the thickness, into two equal parts, and the Compasses shall stay at the side of the Square, equal to the oblong made of that breadth and thickness; which is the mean pro∣portional between them. The breadth being 18, and thickness 6, the side of the Square will be found to be 10, 38.
As the side of the square in Foot-measure unto 1, so is 1 to a 4th num∣ber, and that 4th to the length. As 2 120 unto 1.000, so 1.000 unto 0, 471, and that to 0. 222. or thus, the extent from 2. 120 to 1, will reach from 1, twice repeated to 0. 222, ••nd so much is the length to make a Foot Solid, (at that squareness.)
As 1 to the breadth in Foot-mea∣sure, so is the depth to a fourth num∣ber, as that 4th number to 1 so is 1 to the lenth in Foot-measure.
As 1 is to 2. 50, so is 1. 80, to 4 50, then as 4. 50 to 1, so is 1 unto o•• 222. the length required.
As 1 to the side of the square in Foot-measure, so the length in Feet to a fourth number, and that fourth to the content in Foot-measure. The extent from 1 to 2. 12, twice repea∣ted from 15.25, shall reach unto 68.62.
As 1 to the breadth in Foot-mea∣sure, so is the depth to the Base in Feet; as 1 to that Base, so the length in Feet to the content in Feet.
As 1 to 2, 50, so 1. 80 to 4. 50, then as one 1 to 4. 50, so is 15.25, unto 68. 625. The content required.
The side of the Square is found as in the first Problem of this Chapter, or by the 7th of Board measure. Then as the side of the square in Inches to 41, 57, so is one Foot to a 4th number, and that 4th to the length in Inches, and tenth parts of an Inch.
The extent from 25, 45 unto 41, 57 twice repeated from 1 will reach to
2, 67, or more easie if it be squared, as ••he side of the square is to 12, so is 12 to a 4th and that fourth to the length required. The extent from 25, 45 to 12 being twice repeated from 12, will stay at 2, 667, or more short 267.
As 1 to the breadth in Inches, so the depth to a fourth number, which is the content of the base in Inches, then as this 4 number is to 1728, so is 1 to the length of a Foot Solid in Inch measure. As 1 to 21, 6, so is 30 to 648, then as 648 to 1728, so is 1 to 2, 667.
As 12 to the breadth in Inches, so the depth in Inches to a fourth num∣ber; then as this fourth number is to 144 so 1 to the length of a foot solid; as 12 to 21, 6, so 30 to 54; then as 54
is to 44, so is 11 unto 2, 667. the length required.
The side of a square given in inches to find how much is in a foot long. Extend the Compasses from 12 to the Inches square the same extent turned the same way from the Inches square shall shew how much is in a foot long. At 18 inches square in every foot long, is 27 inches, or 2 foot 3 in∣ches: But if the side of the square be given in feet and parts, Say, as 1 to the feet and parts square, so is that to the quantity in 1 foot long, which multiplyed by the feet long gives the whole content.
As 41.57, to the side of the square in Inches, so is the length to a fourth Number, and that fourth to the con∣tent in Foot-measure. As 41. 57, to 25. 45, so 183, twice repeated unto 68, 62.
As 12 to the side of the square in Inches, so the length in Feet to a fourth Number, and that fourth to the content in Foot-measure. As 12 to 25, 45, so 15. 25 to 32. 55, and 32. 55 to 68. 62. Or the extent from 12 to 25, shall reach to 68. 62, the con∣tent sought.
As 1 to the breadth in Inches, so the depth to the base, then as 1 to the base, so the length to the content in Inches. As 1 to 21. 6, so 30 to 648. as 1 to 648, so that 183 to 118584.
As 1 to the breadth in Inches, so the depth in Inches to the base in In∣ches; Then as 1728 to the base, so is the length in Inches to the content in Feet. As 1 to 21, 6, so 30 to 648, as 1728 to 648, so 183 to 68. 62.
Or you may say,
As 12 to 21.6, so 30 to 54, as 144 to 54, so 18, 3 to 68.62.
As 1 to the breadth in Inches, so the depth in Inches to a fourth number. Then as 144 to that fourth, so is the length in Feet to the content in Feet.
As 1 to 216, so is 30 to 648; then as 144 to 15.25, so is 648 unto 68.62, Or as 144 to 21.6, so 30 to 4. 50: as 1 to 4. 50, so 15.25 to 68.62. Or again,
As 12 to 21.6, so 30 to 54: then as 12 to 54, so 15.25 to 68.62, the content required.
As the Diameter in Feet to 1.128, so is 1 to a fourth, and that fourth to the length in Foot-measure.
The Extent from (the Diameter) 1. 25; to 1. 128, being twice repeated from 1, will reach to 8. 148, the length sought.
To find how much is in a foot long at any Diameter given.
Say, As 1. 128 to the feet and parts in Diameter, So is 1 to a 4th.
and that 4th. to the feet and parts in one foot long.
As the Circumference in Foot-measure is 3. 545; so is 1 to a fourth, and that fourth to the length sought.
As 3 f 927 p unto 3. 545, so is that distance twice repeated from 1 to 0, 815 the length of a Foot- solid.
The Circumference given to find how much is in a foot long at that compass, as 3, 545 to the feet about, the same extent applyed twice the same way from 1 shall reach to the quantity in one foot long. At 5 foot about one foot in length, is two foot neer in the content, which multiply∣ed by the feet long gives the whole content.
As 1. 128 to the Diameter in Foot measure, so is the length in Foot-measure to a fourth, and that fourth to the content in Foot-measure.
The Extent from 1. 128 to 1. 25. being twice repeated from 8. 75, will reach to 10. 737, the content sought for.
As 3. 545 to the Circumference in Feet, so is the length in Feet to a fourth, and the fourth to the content in Foot-measure.
The Extent from 3.545 to 3.927, being twice repeated from 8. 75, will reach to 1074, the content in Foot- measure.
As the Diameter to 46.90, so is 1 to a fourth number, and that fourth number to the length in Inches.
The Extent from 15 to 46.90, will reach from 1 (being twice repeated) to 9.778, the length sought.
As the Circumference to 147.36 so is 1 to a fourth number, and so that fourth to the length in Inches.
The Extent from 47. 13, to 147. 36, being twice repeated from 1, will reach to 9. 78, the length sought.
As 1 128 to the Diameter in In∣ches, so is the length to a fourth, and
that fourth to the content in Inches. The extent from 1 128 to 15, being twice repeated from 105, will reach to 18555, 34 the content in Inches.
As 3, 545 to the Circumference in Inches, so is the length in Inches to a fourth number, and that fourth to the content in Inches. The extent from 3, 545 to 47, 13, being twice repeat∣ed from 105 will reach to 18555 the content in Inches.
As 13, 54 to the Diameter, so the length to a fourth number, and that fourth to the content in Feet. The ex∣tent from 13, 54 to 15, being twice repeated from 8, 75 will reach to 10, 74, the content sought for.
As 42, 54 to the Circumference in Inches, so is the length in Feet to a fourth, and that fourth to the content. The extent from 42, 54 to 47, 13; being twice repeated from 8, 75, will reach to 10, 74 the content sought.
It being an ordinary way in mea∣suring of round Timber, such as Oake, Elme, Beech, Pear-tree, and the like, (which is sometimes very rugged, and uneven, and knotty) to take a line and girt about the middle of it, and then to take the fourth part of that, for the side of a Square equal to that Circumference: But this mea∣sure is not exact, but more then it should be. But either because of al∣lowance for the faults abovesaid, or for Ignorance, the custome is sill used, and men commonly think them∣selves wronged if they have not such
measure. Therefore I have fitted you with a proportion for it both for Dia∣meter and Circumference.
And first for the Diameter.
The Diameter given in Inches and the length in Feet, to find the content.
As 1, 536 to the Diameter, so is the length to a fourth, and that fourth to the content in Feet: according to the rate abovesaid.
The extent from 1. 526 to 9. 53, being twice repeated from 8, shall reach to 3. 12, the content.
As 48 to the Inches about, so is the length to a fourth number, and that fourth to the content. The extent from 48 to 30, being twice repeated from 8, shall fall upon 3. 12, the content required.
The usual way for doing of this, is to take the Circumference of the middle or mean bigness, but a more exact way, is to find the content of the base of both ends and add them together; and then to take the half for the mean, which multiplyed by the length, shall give you the true con∣tent.
A round Pillar is to be measured, whose Diameter at one end is 20 In∣ches, at the other end it is 32 Inches Diameter, and in length 16 Foot (or 192 Inches) the content of the little end is 314. 286, the Area or content of the greater end is 773, 142, which put together make 1087, 428, whose half 543, 714. multiplyed by 192 the length, gives 104393. 143. Cubical Inches which reduced into Feet, is 60 Foot, and 713 cub cal Inches, for the solid content of the Pillar.
Example; let a Cone be to be mea∣sured, whose base is 10 Foot, and the height thereof 12 Foot, the content of the base will be found, by the 14 Problem of Superficial measure, to be 78, 54, Then this 78, 54 multiply∣ed by 4, a third part of 12, the per∣pendicular or height of the Cone will give 314, 4, for the content of the Cone required: By the numbers work thus, the extent from 1 to 4, will reach from 78, 54 to 314, 4. But be∣cause there may be some trouble in getting the true perpendicular of a Cone, which is its height, take this rule; First, take half the Diameter, and multiply it in it self, which here is 25, then measure the side of the Cone 13, and multiply that by it self which here is 169, from which take the Square of half the Base, which is 25, your first number found, and the
remain is 144, the Square root of which is the height of the Cone, or length of the perpendicular.
Cube the Diameter, then multiply that by 11, and divide by 21, gives you the true solid content; let a Sphere be to be measured whose Axis or Diameter is 14, that multiplyed by it self gives 196, and 196 again by 14 gives 2744, this multiplyed by 11 gives 30184, and this last divided by 21 gives 1437.67, for the content of the Sphere whole Diameter is 14. But more briefly, by the numbers thus, The extent from 1 to the Axis, being twice repeated from 3. 142, will reach to the Superficial content, that is, the Superficies round about. But if the same extent from 1 to the Axis be thrice repeared f••om 5238, it will reach to the solid content, as 1 to 14, so 3. 142 to 617 being twice re∣peated, as 1 to 14, so 5278 to 1437.
being thrice repeated. As for many sided figures if they have length, you have sufficient for them in the Chap∣ter of Superficial measure, to find the base, and then the base multiplyed by the length giveth the content. But as for figures of roundish form, they coming very seldom in use, I shall not in this place trouble you with them, for they may be reduced to Spheres or Cones, or Triangles, or Cubes, and then measured by those Problems accordingly. And so much for the mensuration of Solids.
All Vessels to put Liquor in are
made either square, as Brewers Cool∣ers; or round, or oval-formed, or mixed, as part of one form and partly of another: but the ordinary vessels are the regular, viz. Square backs, or Coolers, or Taper-Tuns, and Cop∣pe••s; or else close Cask, as Barrells, Buts, Hogsheads, and the like; for which there are particular Rules for the performance thereof.
And first for the Square-backs.
In order hereunto you must consi∣der by what measure you would Gage your vessel as to Dimensions taking, and as to the solid content either in Wine or Ale-measure, or Ale or Beer-Barrels.
Now the common and most recei∣ved measure to take dimensions with, is Inches, and 100 parts of an Inch, or 10 parts at the least, and for Brewers Businesses the Ale-Gallon is only in use and no other.
Note, An Inch is the exact 36th part of a Standard-yard, and an Ale∣gallon is 282¼ of those Inches taken
cubically, which is agreeable to four of the Ale-Quarts in the King's Ma∣jesties Exchequer: Or if you will 288 cubique Inches, which is agree∣able to the Standard-Gallon in Coopers Hall, as Alderman Starling and o∣thers have much contended for; but in regard that 282 ¼ is according to the Ale-quart in the Exchequer, that I shall the rather use.
| Callon contains inch. | 282 ¼ | 282 ¼ |
| 18 g ½ Bar. of Beer | contains of Cube Inches | 5080 ½ |
| 16 g ½ Bar. of Ale | 4516 | |
| 36 g Barr. of Beer | 10161 | |
| 32 g Barr. of Ale | 9032 |
At 288 Inches in a Gallon.
| Gallon contains Inch. 288 ½ | |
| Barrel of Beer 10368 | 5184 |
| Barrel of Ale 9216 | 4608 |
Note, A Beer-Barrel is just 6 Cu∣bique-foot.
This being premised, then to measure any Square back, it is but to
take the length and breadth exactly in Inches and 10 parts, and multiply them together, and then to multiply that product, by the depth in Inches and 10 parts, which last product is to be divided by 282 to bring it to Gallons, or by 10161 to bring it to Beer Barrels, or 9032 to bring it to Ale-Barrels; as in the following Ex∣ample.
A Back or Cooler is 72 Inches and 6 tenths broad, and 365 Inches and 4 tenths long, and 8 Inches 7 tenths deep, how many Gallons or Barrels will it hold? Note the work.
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230794 the Content in Cube inches.
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Viz. 22 barrels and neer 26 gallons.
To work this by the Line of Num∣bers will be very difficult to come to exactness, because we cannot see to above 4 figures, yet in regard that after the whole operation is done, the grand querie is, How many Bar∣rels is there? and not how many In∣ches or Gallons; and this you may
well perform to a quarter of a Bar∣rel, by the Line of Numbers. I conceive it will be as much used as the Arithmetical way, being 10 to 1 sooner done; which is thus,
Extend the Compasses from 1 to the Inches and 10ths. broad; the same Extent applyed the same way from the Inches and 10ths. long, shall reach to a fourth number.
Again, As 10161 (the Cube Inches in a Beer-Barrel) to the 4th. or the extent from 10161 to the 4th. shall reach from the Inches and 10ths deep to the content in Beer-Barrels.
As 1 to 72. 6. the breadth of a Cooler, so is 365; 4 the length to a fourth number, 26528. Again, as 10161, to the fourth last found. So is 8. 7 the depth, to 22. 70, that is 22. Beer Barrels, and above a half, or 26 Gallons.
If you would know how many Ale Barrels, or Kilderkins, or
Gallons, then make use of the re∣spective numbers of Inches in those measures as in the Table.
This is a very quick neat way for all Square Vessels.
| Beer Vessels. | The Sise of Beer Vessels. | ||||
| Names. | Head | Bung | Ben | Gall. | |
| Parrell | 20.2 | 22.7 | 27.7 | 36 | |
| Kilderkin | 16.2 | 18.0 | 22.0 | 18 | |
| Firkin | 12.7 | 14 3 | 17.4 | 9 | |
| Pin | 10.1 | 11.4 | 13.9 | 4 ½ | |
| ½ Pin | 08.0 | 09.0 | 11.0 | 2 ¼ | |
| Ale Vessels | Barrell | 19.4 | 21.8 | 26.6 | 32 |
| Kilderkin | 15.4 | 17.3 | 21 2 | 16 | |
| Firkin | 12.2 | 13.7 | 16.8 | 08 | |
| Pin. | 09.7 | 10.9 | 13.3 | 04 | |
| ½ Pin | 07 7 | 08.7 | 10.6 | 02 |
| The Inches in a Gallon for ••••ne, Ale, and Corn-mensure. | Gage Points and Fixed Numbers. |
| 231 | 17.1485 Wine |
| 272 ¼ | 18.6168 Corn |
| 288 | 19.1480 |
| 288 ¾ | 19.1716 |
| 282 | 18 9468 Ale |
| Inches in these measures following. | |||
| Beer. | Gar. | 10161 | 35.96 |
| K. | 5081 ½ | ||
| F. | 2540 | ||
| P. | 1270 | ||
| Ale. | Bar. | 9032 | 33-91 gage point. |
| K. | 4516 | ||
| F. | 2258 | ||
| P. | 1129 | ||
| Gall | 0282 ¼ | ||
| at | 288 | to a Gallon. | |
| 2304 4608 9216 | Bar. | 10368 or 6 foot | |
| K. | 5184 | ||
| Ale. | F. | 2592 | |
| Gal. | 288 |
First, If it have equal Diameters at the Top and Bottom, then it is mea∣sured as a Cylender, and the propor∣tion between a Cube and a Cylender is as 11 to 14 and the contrary, so that a Cylender 12 Inches high and 12 Inches Diameter, is equal to 11/14 of a Cube 12 Inches every way.
Or in Numbers, as 1728, to 1358 ferè.
As 1 to 0. 7854, so is the Square of the Diameter multiplyed by the depth to the solid Content in Inches: Note the Work.
Diameter 60 Inches, Depth 36 Inches.
〈 math 〉〈 math 〉
〈 math 〉〈 math 〉
Then divide this last Product by 282, and you shall have the Content in Ale Gallons, or by 10161, and then you have it in beer Barrels, or by 4516 for Ale-Barrels, as before in the Table of Cube Inches in those measures.
〈 math 〉〈 math 〉
Or else you may shorten the work, and save this last Division thus, and bring it into Gallons or Barrels.
After you have multiplyed the Di∣ameter by it self, which is called Squaring, and then multiplyed that Product by the Inches and 10ths. deep. Note this last Product, for if you would bring it into Gallons, then multiply it by 002785, but if you would have it at first into Bar∣rels, then multiply it by 0000775, and the product cutting off the fra∣ctions, shall be the Gallons, or Bar∣rels of the content required. Note the work.
〈 math 〉〈 math 〉 〈 math 〉〈 math 〉
Gallons. Barrells.
The only difficulty in this Work is to know now many figures is to be cut off from the last Work, the best Guide wherein is experience, for one that is experienced will hardly call a Vessel of 10 Gallons a 100, nor the contrary, not one of a 100 a 1000, or the contrary, much less 10; now if you work right the mistake must be to much or none. For 360 Gal∣lons must needs be more then one Barrel, and yet not a 100, there∣fore it must needs be 10, and the rest cut off, as the 100 thousand part of a Barrel.
But to apply this to the Line of Numbers, for which chiefly my aim is, the whole work is thus.
Extend the Compasses from 1 to the inches Diameter, the same Ex∣tent shall reach from thence the same way to the product.
Then the extent from 1 to the last Product shall reach from the inches deep the same way to another product which is the Square of the Diameter,
multiplyed by the depth; which ob∣serve and note. Then lastly, the ex∣tent from 1 to 0. 2785 for Gallons, or to 0.775 for Barrels, shall reach the same way from the last found pro∣duct (or the Square of the Diame∣ter multiplyed by the length) to con∣tent in Gallons or Barrels required.
As 1 to 60 the Inches in Diameter so is 60 to 3600: Again, as 1 to 3600 so is 36 the inches deep to 129600, the Square of the Diameter multi∣plyed by the depth in Inches.
Then lastly, first for Gallons say, as 1 to 0.2785, so is 129600 to 361 ferè, or else, if you please, to Barrels.
As 1 to 0. 775 so is 129600 to 10 Barrels, and a little more.
This Rule is for all Cylender-like round Vessels, but if the sides be streight and taper, then the usual way and somewhat neer the Truth is, to add the Diameters at the top and bot∣tom together, and to take the half for the mean Diameter, which when you have got, work as before.
Or else you may make use of the Gage-point, according to Mr Gun∣ter's way, which by Arithmatick is thus;
Multiply the mean Diameter by the length, and then divide that pro∣duct by 17.15 for Wine-measure, or by 18.95 for Ale-measure, and note the Quotient and his Remainder; A∣gain, Multiply the quotient last found, by the mean Diameter, and the Pro∣duct; divide again by 17-15 for Wine, or by 18.95 for Ale, and the Quotient shall be the Content in Gallons.
This way by the Line of Numbers is very quick and ready. Thus,
Extend the Compasses from the Gage-points either 17.15 for Wine, or 18.95 for Ale, to the mean Dia∣meter, the same extent being turned twice the same way from the length, shall reach to the content in Gallons.
The Extent from 18.95 to 60 be∣ing twice repeated from 36, the In∣ches deep shall reach to 361 ferè, the content in Ale Gallons.
Or if you would have it to Beer-Barrels, then say, as 35.96 the Gage point for a Beer-Barrel is to the mean Diameter 60, the same extent applyed twice the same way from the length, shall reach to 10.002, that is, 10 Barrels.
The Gage-point for an Ale Barrel which contains but 32 Gallons is 33. 91.
The Gage-point of a Solid mea∣sure 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 Winchester measure be∣ing 231 Cube Inches, if you conceive a Circle to contain so many Inches, you shall by the 16 Problem of board measure find the Diameter thereof to be 17.15, for as 1 is to 1.273, so is 231 to 294. 1, whose square root by
the 12 of the second, is 17. 15, the Gage-point for Wine measure.
Thus likewise you may discover the Gage point for Ale-measure, an Ale-Gallon (as hath been of late dis∣covered) containing 282 Cubique Inches. For as 1 is to 1.273, so is 282 to 356.3, whose square root (by the 12 Problem of the second Chap∣ter) is 19.95, the Gage-point of Ale measure, because of soil and wast ex∣ceeding that of wine above 2 Inches.
Another way by having the Diame∣ter, length, and true content of any Vessel.
Extend the Compasses on the line of numbers to half the distance be∣tween the content and length of the Vessel, the same extent will reach from the Diameter to the Gage-point.
Here at London it is said, that a Wine Vessel being 66 Inches in length, and 38 Inches in the Diame∣ter, would contain 324 Gallons; if so,
we may divide the space between 324, and 66, into two equal parts, the middle will fall about 146, and the same extent that reacheth from 324 to 146 will reach from the Diame∣ter 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. Now from what hath been said doth necessarily follow this con∣clusion: that when the Diameter of a Cylinder in Inches, is equal to the Gage-point of any measure, given likewise in Inches: every Inch in the length thereof, contains one in∣teger of the same measure. So in a Cylinder 17.15, Inches Diameter, every Inch in the length thereof, contains one intire wine Gallon, and in a Cylinder of 18. 95 Diameter, every Inch is one intire Ale Gallon.
Add the two Diameters at head
and bung together, (being measured in Inches) and the half sum keep. Secondly, Substract the Diameter of the head out of the Diameter at the bung, and note the remainder, which is to be divided by 4, or 4 ½ (as by the number 5 is easie) to find out a fourth (or somewhat less) part of the difference. Thirdly, that fourth part (or somewhat less) is to be added to the half sum kept, to make up the mean Diameter sought. Examp.
A Vessel hath in Diameter at the head 18 in. 3 p. and at the bung 21.5 Inches, I would know the mean Dia∣meter 18.3, and 21.5 added, is 39.8, the half is 19.9, the less taken from the greater, the remainder is 3 in. 2 p. which brought or reduced into 100 is 320. and in stead of dividing by 4 or 4 ½ say 45, and the product is 7 tenths and 3 over; which 7 tenths add to 19. 9, and it maketh 20. 6 tenths for the mean Diameter required to be found.
Another way by Geometry and somewhat more exact, Of the Diame∣ter
at the head, and bung, take the dif∣ference, then say as 1 is to 7, so is the difference to a fourth sum, which fourth sum is to be added, to the least of both the Diameters, viz. that at head, as in the former example 18.30 and 21.50 added is 39 80, whose half is 19. 90. and the difference is 3 in. 2 p or 320, then say, as 1 is to 7, so is 320 to 2. 24, which 2.24 if you take with your Compasses, out of your Scale of Inches and add it to 18.30, you shall see it reach to 20 Inches 54 parts, and this is the true mean Dia∣meter, to make it a perfect Cylinder; o•• if you add 2 Inches 24 to 18. 30, it makes it 20. 54 as before.
Note, If the difference of the two Diameters be much, then you must add the 100 parts as in the Ta∣ble annexed.
| Dif. in | parts 100 |
| 02 | 003 |
| 03 | 005 |
| 04 | 090 |
| 05 | 012 |
| 06 | 015 |
| 07 | 019 |
| 08 | 023 |
| 09 | 026 |
| 10 | 030 |
| 11 | 035 |
| 12 | 040 |
| 13 | 047 |
| 14 | 055 |
| 15 | 063 |
| 16 | 070 |
| 17 | 076 |
| 18 | 083 |
| 19 | 090 |
| 20 | 100 |
| 21 | 112 |
| 22 | 126 |
| 23 | 140 |
| 24 | 155 |
| 25 | 170 |
This Table is thus used when the difference between the head and bung in close Cask is above 4 inches; thus,
As 10 to 7. so is the difference be∣tween the head and bung augmented by the 100 parts of an Inch, as in the Table, to the fourth number, which you must add to the Diameter at the head, to get a mean Diameter.
Extend the Compasses from the Gage-point to the mean Diameter, the same extent being twice repeated from the length, shall give the con∣tent in galons and 100 parts.
As 17. 15 to 20, so is 25 to 34, being twice repeated.
Again, in a lesser Vessel.
As 17. 15 to 16, so is 23 to 20, being twice repeated from 23 the same way as from 17. 15 to 16.
Extend the Compasses from the Di∣ameter to the Gage-point, the same extent twice repeated from the con∣tent, shall give the length.
As 38. to 17. 15, so is 324 to about 66 twice repeated.
Extend the Compasses to half the distance, between the length and content, the same extent shall reach from the Gage-point to the Diame∣ter.
Divide the space between 66, and 324, in two equal parts, the same ex∣tent shall reach from 17. 15 to 38, the Diameter abovesaid.
As 10 to 7, so is the Difference 10 in. and 3 10th more, as in the Table to 7 Inches 2 10th, which you must add to the Diameter at head, to make a mean Diameter.
As 282 is to 231, so is the content in wine Gallons, to the content in Ale Gallons. Or the contrary, as 231 is to 282, so is the content in Ale Gallons, to the content in wine Gal∣lons.
As 282. to 231, so is 116. 4, to 95, 30. and as 231 to 282: so is 95. 30, to 116. 4, &c.
There is yet a more easie way, and for exactness no way infer or to any extant, and that by taking the length in Inches and 10 parts, but the Dia∣meters
at head and Bung, with a line called Cughtred's Gage line, (and to be had at John Brown's house in the Minories neer Aldgate, Mathemati∣cal-In••••ument-maker) the use of which is thus, take the Diameter at the bung, with those divisions on the line aforesaid, from that en•• where he di∣visions begin to be numbred. And set that down twice: and the Diame∣ter of the inside of the head (for so we understand all along,) and set that down once. In this manner. 〈 math 〉〈 math 〉 and then add them together as here you see, the length in In∣ches suppose to be (30. 82.) Thirty and eighty two of a hundred: then say, As 1 is to 1. 77, so is 30.82 to 54.55 hundred parts of a Gallon, being a little more then half a Gallon, which is 54 gallons ½ the content of a high Country Hogshed, whose measures were as be∣fore.
Depth — 〈 math 〉〈 math 〉
Diam. at top — 〈 math 〉〈 math 〉
Diam. at bottom — 〈 math 〉〈 math 〉
Sum of the Diameters 〈 math 〉〈 math 〉
The Summe of the Diameters Squared — 〈 math 〉〈 math 〉
Product of Diam. 〈 math 〉〈 math 〉
The Residue 〈 math 〉〈 math 〉
〈 math 〉〈 math 〉
The residue the fraction cut of
〈 math 〉〈 math 〉 〈 math 〉〈 math 〉
To make the Number 1077 which is fixed for an Ale-Gallon, or 882 for a Wine Gallon, or the fixed Number for a Beer-Barrel which is 38796, for an Ale Barrel is 34485.
Multiply the Cube Inches in a Gallon or Barrel by 14, divide the Product by 11, and multiply the quo∣tient
by 3, produceth these fixed Numbers.
Set down the Diameters of the top and bottom in Inches and 10 parts, and add them together to get the sum, and multiply the one into the other, to get the Product of the two Dia∣meters.
Also multiply the sum of the Dia∣meters in it self, which is called Squa∣ring thereof.
Then substract the product of the two Diameters, out of the Sum of the two Diameters squared, and note the Residue for one number; then multiply that residue by the Inches deep, which produceth another pro∣duct, which last Product divided by a 1077, a certain fixed number for Ale Gallons, shall in the quotient give the content in Alegallons requi∣red. Note, That if the figures be many, because of Fractions, you may cut off all the Fractions after the first
Multiplication, which are always as many as the 10's or 100 parts of the Inch come to in both Sums, as you may see in the Example annexed.
To work this by the Line of Num∣bers: Extend the Compasses from the fixed Number, to the square of the two Diameters, less by the pro∣duct of the two Diameters multi∣plyed together, the same extent will reach from the perpendicular depth to the content in Gallons or Barrels, according as the fixed number was.
Example. As 1077 to 12821 so is 43. 60. to 519. 10 gallons; as 38796 to 12821, so is 43. 60. to 14. 40 Barrels; as 34485 to 12821, so is 43.60. to 16.22 Barrels of Ale, and the like for any other measure.
〈 math 〉〈 math 〉
〈 math 〉〈 math 〉 (84.9.1 third of the head
〈 math 〉〈 math 〉
〈 math 〉〈 math 〉 (107-58 gallons of wine.
Or shorter thus:
As 1 to 0.5236 so is 1024 to 〈 math 〉〈 math 〉
As 1 to 0.2618 so is 324 to 〈 math 〉〈 math 〉
Cube-Inches 〈 math 〉〈 math 〉
Measure the two Diameters in In∣ches and 10 parts, and also the length
within, and find out the Superficies of the Circles having those Diame∣ters, and add two third parts of the greater Diameter, and one third part of the lesser into one summe; Then multiply that number by the length, and the product shall be the content in Cube Inches, which product di∣vided by 282 gives Ale-gallons or by 231 gives the content in Wine-gal∣lons.
To find the content of a Circle having the Diameter, square the Di∣ameter, then multiply that square by 11, and divide the last product by 14, and the quotient is the superficial con∣tent in Inches required.
Or shorter thus,
As 1 to 0. 5236 a number fit for a thirds. So is the Square of Diame∣ter at bung to a fourth. Then, as 〈◊〉〈◊〉 to 02618 a number for 1 third or head; so is the Square of Diameter at head to another fourth: which two fourth added together and multiplyed by the length, gives the content in
Cube Inches. Then those divided by 231 for Wine, or by 282 for Ale, gives the content in gallons.
Last of all to make this work more easie for Mechanick men, and those that want Arithmatick, there is seve∣ral lines to be had on Joint or streight rules at the Sphere and Dial in the Minories. As first, a Diagonal-line for Wine or Ale-gallons. Second, by Mr. Oughtred's Gage line several wayes applyed. Thirdly, Rules to find the emptiness of Buts, Bar∣reis, and Kilderkins. Fourthly, a Rule to Gage any Bushel, Half-Bush∣ell, Peck, Half-peck, Pottle, or Quart, or Pint measure, according to 272 the Corn-gallon; Whose se∣veral uses is as followeth.
Put the Rule in at the Bung-hole down aslope to the bottom of the head, and observe what parts the middle of the Bung-hole cuts, which a putting both ways will assure you of,
and so many Gallons will the Vessel hold, (when full) of Wine or Ale-Gallons, according as your Line is.
Example. Suppose the length of the Diagonal-line asloop from the bottom at the head, to the middle of the Bunghole be 28 inches, then on the Diagonal Line for Wine-measure, you shall find almost 60 gallons; and on that for Beer or Ale-measure but 48 gallons; for the content of such a Vessel whose slope or Diagonal line from the middle of the Bung-hole to the head at Bottom is 28 Inches.
Note also, This Diagonal line will serve to measure Pots, Pails, Kettles, and such like Vessels that be made open; Provided, that when you take the Diagonal-line from the up∣per edge of the Vessel to the opposite lower edge across, next the bottome, whatsoever the edge of the Vessel cuts on the rule, the half thereof is the content of the Vessel neer the matter.
Suppose that a Cross from the up∣per edge of a Paile to the bottom be 18 Inches, the Rule will shew 10, the half of it is 8, the number of Gallons the Pail holds.
The use of Mr. Oughtred's Gage∣line you have in part before for mea∣suring of Wine or Oyl close Cask, but for the application thereof to Brewers-Tuns, or indeed any great or small Vessel is thus, All along by the Gage line is a line of small Inches, about half an inch in bigness, and every of those parted into 10 parts, to represent 10 parts of an Inch, and the use is thus;
Take the Diameters of your vessel at the top and bottom in Inches and 10 parts, and add them together, then if the sides are streight, count the half for a mean Diameter: But if the Staves are swelling outwards, then use the proportion before as 10 to 7, to find the mean Diameter.
Thus having found the mean Dia∣meter,
look for the same on the small Inches, and there you shall have the true quantity of Ale-gallons in one Inch deep; which number if you multiply by the Inches deep, the product shall be the content in Ale-Gallons.
Suppose a Brewers Tun at the Top be 82 in. at bottom 72, and deep 38 Inches.
Diameters 〈 math 〉〈 math 〉 top
〈 math 〉〈 math 〉 bottome.
〈 math 〉〈 math 〉 Sum
〈 math 〉〈 math 〉 half sum or mean Diameter. Then if you look for 77 among the small Inches, you shall find right against it 16 Gallons, and 49 or neer a half, which multiplyed by 38, the inches deep, make 626 Gall. 62 parts for the extent on the line of numbers, from 1 to 16-49 shall reach the same way from 38 the In∣ches deep to 626 ½ the content.
Note, That by this Rule you may
make a Table of the content of any round taper Tun, of how much shall be in or out at any number of Inches of fulness or emptiness.
Another Example. Suppose a Ves∣sel be in Diameter at top 160 In∣ches 5 tenths, and at bottom 148 Inches and 8 tenths, the sum is 309-3 the half is 154-6 ½ for a mean Dia∣meter, the Gage-line right against 154-6 ½ is 64.63, which multiplyed by 41.70, the Inches and tenths deep make 〈 math 〉〈 math 〉, which last 4 fi∣gures are fractions and parts of a Gallon.
Note, If in a great quantity of gal∣lons, if this way prove to be some∣what less then in truth there may be, or that it do not agree with the for∣mer Rules, take this for answer, it is as neer as any such instrumental way can be composed, and of two er∣rors take the least, and therefore may please till a better comes. Note also, That by altering the Inches that goes by the Gage-line it may be made fit
for Ale or Beer-Barrels, or any other greater or lesser measure whatsoever, and the error allowed for by taking the Diameters more exactly, as ex∣perience and practice will make ea∣sie.
Note also the same Line is impro∣ved by Mr. Newton I think, but the way how being in Print in Wingate's Rule of proportion, I shall say no∣thing to it, this way being full as ex∣act and more easie.
The use hereof is very easie, for if you put the beginning end of the Rule downright at the bunghole home to the opposite side, how far soever the Liquor wets the Stick, the figures will shew how many Gallons is in, and the complement thereof to the whole content is what it wants of being full.
Suppose I have a part, or a whole Barrel of Ale in a Beer-Barrel (for there are or ought to be no Ale-Bar∣rels)
and I would know how much is in or out: Put the Line for a Bar∣rel downright in the Bung-hole to the opposite side, and then suppose it wets 13 inches 8 tenths, then there is 23 Gallons in and 13 empty.
But if you had put the Rule for a Kilderkin, and it had wetted as be∣fore, then there would have been but 15 gallons in, and 3 out, the complement of 15 to 18, for 15 and 3. makes 18.
But if you had used the Line for a But, there would have been about 52 Gallons in. Or of some round∣ish Buts, about 57 Gallons in, and 63 out, the Complement to 57. to 120 the whole content in a But.
Take the Rule with the beginning end from you, and take the Diame∣ter of the measure whatsoever it be, and the figures on the Rule right a∣gainst the Inches Diameter shall ex∣press how many Inches and hundred
parts that measure ought to be in depth to make a true measure.
A pint measure 2 Inches and ½ Di∣ameter, ought to be 6 Inches 95 parts deep.
And at 2 Inches 3 quarters over 5. 75 deep.
A quart measure 3 Inches diam. ought to be 9 Inches and 63 parts deep.
And at 3 Inches 3 quarters Diam. 6-15 parts deep.
A Pottle-measure 4 inches Diame∣ter, ought to be 10 Inches 82 parts deep.
And at 5 ½ Inches Diameter, 5.73 Inches deep.
A Gallon or half Peck 6 ½ broad, ought to be 8-19 deep, at 9 Inch Broad 4-26 deep.
A Peck 13 Inches broad or Dia∣meter, ought to be 6-94 deep, and at 12 Inches broad 4-82 deep.
A half Bushell 13 Inches broad must be 8-23. but at 16 Inches broad but 5-42. deep.
A Corn-Bushell of 17 Inches broad, ought to be 9-62 deep, but of 21 Inches broad 6-29. But for Sea-cole measures take the next num∣ber forwards toward the beginning, as suppose a Cole-bushell be 18 ½ broad on the inside, the Rule saith it ought to be 8-12, but you may take 8-33 the number next toward the begin∣ing viz. 8-33 because a Corn-Bush∣el ought to hold a quart more then a Corn Bushel of water; and the Di∣ameter from out-side to out-side ought to be 19 Inches and ½, the half bushel 14-½, the Peck 11 ½, the half Peck 9 ½ to make the heap to bear a proportion to the fats.
Extend the Compasses from the whole Diameter at the Bung to the next 1. the same extent shall reach the same way from the Inches and parts of emptiness to a fourth num∣ber on a Line of Artificial Segments
joyned to the Line of Numbers, which fourth number keep. Then as 1, the whole content in Gal∣lons, So is the fourth number kept, to the emptiness in the same Gallons the content was.
For a Beer-Barrel at 5 Inches dry, or 17 ½ wet, or 5 empty, and 17 ½ full.
The extent from 22. 5 to 1 shall reach from 5 to 16. 4 on the Seg∣ments, or from 17. 5 to 83. 6.
Then the extent from 1 to 36. shall reach the same way from 16-4 to 5-9 for so much out, and from 83-6 to 30-1 for so much in, which two Sums put together makes 36 Gal∣lons.
You may if you will say, As 1 to the mean Diameter, so is the mean emptiness, &c. and the work will be somewhat more exact, but this is neer the matter, and easie.
These Collections of Gaging, Courteous Reader, are the Issues of several years practice of several men, as Mr. Oughtred, Mr. Gunter, Mr. Renolds, Mr. Collins, and many helps and Additions of my own; and if my brevity and insufficiency wrong them not, they may be wel∣come to many a Learner, however very convenient for the further use of the Line of Numbers; and so I leave it as the most general Ga∣ging that at present is extant.
One example more by Mr. Oughtred's way. Suppose a great vessel whose length is 70. 50 Inches, and the Diameter at the bung 2 gallons 03 hundred parts, (for so they are pro∣perly called) and the Diameter at the head 1 gallon 10 parts, what is the content?
Set the two Diameters down, 〈 math 〉〈 math 〉 that at the bung twice, and that at the head once, and add them together thus, And then say D as 1 to 5. 16, so i: 70. 50, L 70. 50 to 363. 78. the content sought for; that is 363 gallons 6 pints and a quarter, which 78 so to reduce do thus say on the line, As 100 to 80, so 78 to 6. 25, which is 6 pints and a quarter, the fraction sought.
What is said here of Reduction is general in any other, as from 12 to 10, either shillings or Inches to tenths of a shilling, or tenths of a Foot, or Pence or Farthings, Ounces, or Chauldrons, hundreds, either weight or Tale, and the rule is thus:
The like is for any kind of Redu∣ction.
Find by the 12 Problem of the second Chapter, the Square root of the number given; for so much as that root shall be, so many Souldiers
ought you to place in Ranck, and so many likewise in File, to make a Square Battle of men.
Let it be required to order 625. Souldiers, into a Square Battle of men; the Square rooot of that num∣ber is 25; wherefore you are to place 25 in rank, and as many in File, for fractions in this practice are not con∣siderable. For had there been but 3 less, there would have been but 24 in ranck and file.
Find out the square root ot half the number given, for that root is the number of men to be placed in file, & twice as many to be placed in ranck, to make up a double Battle of men.
Let 1368 Souldiers, be propound∣ed to be put in that order: I find by
the 12 aforesaid, that 26, &c. is the square root of 684, (half the number propounded,) and therefore conclude, that 52 ought to be in ranck, & 26 in file, to order so many Souldiers into a double Battle of men.
Here the Square root of the fourth part of the number propounded, will shew the number to be placed in File, and four times so many are to be pla∣ced in Ranck.
So 2048 being divided by 4, the quotient is 512, whose root is 22 (6) and so many are to be placed in File and 88 in Ranck, being four times 22, &c.
Extend the Compasses from the di∣stance in File to the distance in ranck; this done, that extent applyed the same way from the number of Soul∣diers propounded, will cause the moveable point to fall upon a fourth number, whose Square root is the number of men to be placed in File; by which, if you divide the whole number of Soldiers, the quotient will shew the number of men to be placed in Ranck.
2500 men are propounded to be or∣dered in a Square battle of ground, in such sort that their distance in Fil∣being seven Foot, and their distance i•• Ranck three Foot, the ground where∣upon they stand may be a just square to resolve this question, extend th•••• Compasses upon the line of number downward from 7 to 3, (then be∣cause the fourth number to be found will in all likelyhood consist of 4 fi∣gures,) if you apply that extent th•• same way, from 2500, in the second
part among the smallest divisions, the moveable point will fall upon the fourth number you look for, whose square root is the number of men to be placed in file. By which square root if you divide the whole number of Souldiers, you have the number of men to be placed in ranck.
As 7 to 3, so 2500 to 1072, whose biggest square root is 32, then as 32 is to 1, so is 2500, to 78.
This Problem is like the former, for as the proportional number given for the file, is to that given for the ranck, so is the number of Souldiers to a fourth number, whose square root is the number of men to be pla∣ced in ranck, by which if you divide the whole, you may have the number to be placed in file.
So if 2500 Souldiers were to be martialled in such order, that the number of men to be placed in file, might bear such proportion to the number of men to be placed in ranck, as 5 bears to 12, I say then, as 5 is to 12, so is 2500 to 6000; whose square root is 77 the number in rank: then as 77, is to 1, so is 2500 to 32, &c. The number of men to be placed in file.
Take the distance with your Com∣passes between 100, and the Increase of 100 l. for one Year, (which you
must do very exactly) and repeat it so many times from the principal as it is forborn years, and the point of the Compasses will stay on the Prin∣cipal with the Interest, and increase according to the rate propounded.
I desire to know how much 125 l. being forborn 6 year will be increa∣sed, according to the rate of 6 l. per cent. reckoning Interest upon Interest or Compound-Interest.
Extend the Compasses from 100 to 106; that extent being 6 times re∣peated from 125, shall reach to 177 l. the principal increased with the inte∣rest at the term of 6 years, at the rate propounded.
But if it were required for any number of Months, then first find what 100 is at one Month, then say thus, If 100 give 10 s. at one month, what shall 125 be at 6 months end? facit 75 s. And the work is thus:
First say, If 100 give 10 s. at one months end, What shall 125? and
it makes 12 s. 6 d. then say, If one month require 12 s. 6 d. What shall six months require? facit 75 s. that is three pound fifteen shillings, the thing required to be found.
This question is only the inverse of the other; for if you take the space between 106 and 100, and turn it back from the sum proposed, as many times as there are years in the questi∣on, it shall fall on the sum required.
Take the distance between 106 and 100, and repeat it 6 times from 177, and it will at last fall on 125, the sum sought.
First you must find the principal that shall answer to that Annuity, then
find to what sum the Principal would be augmented at the rate and term of years propounded; then if you sub∣stract the principal out of that sum the remainder is the Arrears required.
A Rent, or Annuity, or Pension of 10 pound the year, forborn for 15 years, What will the arrears thereof come to at the rate of 6 per cent. com∣pound interest?
The way first to find the principal that doth answer to 10 l. is thus: If 6 pound hath a 100 for his principal, What shall 10 have? facit 166 l. 16 s. or 166 l. 8 s. for the extent from 6 to 10 will reach from 100 to 166-8. which is 166 l. 16 s. Then by the first Problem of this Chapter, 166 l. 16 s. forborn 15 year, will come to 398 l. then substract 166 l. 16 s. out of 398 pound, and the remainder, viz. 231 pound 4 shillings is the sum of the arrears required. But note, in working this question, your often turning, un∣less your first extent be most precisely
exact, you may commit a gross error, to avoid which, divide your number of turns into 2, 3, or 4 parts, and when you have turned over one part, as here 5, for three times 5 is 15, open the Compasses from thence to the principal, and then turn the other two turns, viz. 10-15. and this may avoid much errour, or at the least much mitigate it; for in these que∣stions the larger the Line is, the bet∣ter.
First, find by the last what the ar∣rears come to at the term propoun∣ded, and then what those arrears are worth in ready money, and that shall be the value of it in ready money.
What may a Lease of 10 l. per ann. having 15 year to come be worth in ready money? I find by the last Pro∣blem that the arrears of 10 l. per ann.
forborn 15 years, is worth 23 l. 14 s. And likewise I find by the second Problem that 231 l. 4 s. is worth in ready money 96 l. 16 s. and so much may a man give for a Lease of 10 l. per ann. for fifteen years to come, at the rate of 6 l. per cent.
But if it were not to begin present∣ly, but to stay a certain term longer, then you must adde that time to the time of forbearance; as suppose that after 5 years it were to begin, then you must say, 231 l. 4 s. forborn 20 years is worth in ready money, and it is 72 pound 8 shillings; and that shall be the value of the Lease required.
Take any known annuity, and find the value of it in ready money; this being done, the proportion will be thus: As the value found out is to the
annuity taken, so is the summe pro∣pounded to the annuity required.
What annuity to continue fifteen years will 800 l. purchase, after the rate of 6 l. per cent. Here first I take 10 l. per ann. for fifteen year, and find it to be worth in ready money 96 l. 16 s. by the last Problem; then I say, as 96 l. 8 s. is to 10, so is 800 to 82-7, which is 82 l. 14 s. and so much neer do I conclude will an annuity of 82 l. 14 s. per ann. be worth for fifteen years, after the rate of 6 l. per cent. viz. 800 l.
I have added this Chapter, not for that I think it absolutely necessary, but only because I would have the
absolute applicableness of the Rule to any thing, be hinted at; for it may be the answer of some, Do you come with a Rule to measure my commo∣dities which are sold by weight? Yea so far as there is proportion, it concerns that, and any thing else, the Application of which I leave to the industrious practitioner: only I here give a hint. As much as to say, here is a treasure if you dig you may find, for some may be apt to think it being a Carpenter's Rule, it is sit but for Carpenters use only: but know, that in all measures which are either lengths, Supersicials or Solids, or as some call them Longametry, Planametry, and Solidametry: and in all liquids, by weight or measure, and in all time, either by Years, Months, Weeks, Days, Hours, Minutes, and Seconds, and almost (I think I may say,) in all things number is used, and in many things proportional numbers, why then may not this line put in for a share of use, seeing it is wholly
composed of, and fitted for propor∣tional numbers, and of so easie an at∣tainment, for any ordinary Capacity, and chiefly intended for them that be ignorant of Atithmetique, and have not time to learn that noble science as some have?
And first for more conveniency of Reduction, take these Rules of Re∣duction:
Note, that 4 farthings make a penny; 16 farthings, 8 half pence, or 4 pence, make a groat, 48 far∣things, 24 half-pence, or 12 pence, make a shilling: 40 pence or 10 groats, is 3 shillings 4 pence; 80 pence, 20 groats, or 6 shillings 8 pence is a noble; 160 pence, 40 groats, or 13 shillings 4 pence is a mark; 20 shillings, 4 crowns, 3 nobles, or 2 angels, is a pound ster∣ling.
Note that 24 grains is a penny weight: 20 penny weights, or 24 car∣rots,
is an ounce Troy: 12 Ounces is a pound: 25 lib. a quarter of a hun∣dred, 50 lib. half a hundred: 75 lib. 3 quarters of a hundred, and 100 lib. is an hundred weight Troy.
Note that 20 grains, make a scruple: 3 scruples, is a dram: 8 drams, is an ounce: 16 ounces, is a pound, 8 pound, a stone: 28 lib. a quarter of an hundred; 56 lib. half a C. 84 lib. 3 quarters of an C. and 112 lib. or 14 Stone, or 4 quarters of a C. is an hundred weight: 5 C, is a Hogshead weight: 19½ C, is a Fother of Head: and 20 C, is a Tun weight, and note that l. signifies a Pound in money: and lib. signifies a Pound in weight, either Troy, or Averdupois.
Note that 2 pints is a quart 2 quarts a pottle, or a quarter of a peck: 8 pints 4 quarts, 2 pottles is one Gallon, or half a peck, 2 Gallons is a peck: 2 pecks make half a Bu∣shel: 4 pecks, or 56 lib. make a Bu∣shel:
2 Bushels, is a Strike: 2 Strikes a Coomb, or half quarter; 2 Commbs 4 Strikes, or 8 Bushels make a quar∣ter, or a Seame: 10 quarters, or 80 Bushels make a Last.
Note, that 2 pints is a quart: 2 quarts a pottle; 2 pottles 4 quarts, or 8 pints make a Gallon: 9 Gal∣lons make a Firkin, or half a Kilder∣kin: 18 Gallons make 2 Firkins, a Kilderkin, or a Rundlet; 36 Gal∣lons is 2 Kilderkins, or a Barrel; 42 Gallons make a Terce, 63 Gallons or 3½ Rundlets make a Hogshead, 84 Gallons, or 2 Terces, make a Terci∣on, or Punchion: 126 Gallons, is 3 Terces, two Hogsheads, one Pipe, or But. A Tun is 252 Gallons 14 Rund∣lets, 7 Barrels, 6 Terces, 4 Hogsheads, 3 Punchions, 2 Pipes, or Buts. Note that is sweet Oyl 236 Gallons make a Tun, but of Whale Oyl 252 goes to the Tun.
Note that 5 Pecks is a Bushel, 3
Bushels a Sack, 4 2/2 Bushels a Flat; 12 Sacks, 4 Flats, or 36 Bushels, make a Chaldron of Coals.
Note that 3 Barley corns make an Inch: 2¼ Inches make a Nail: 4 Nails or 9 Inches make a quarter of a Yard, 12 Inches make a Foot: 3 Foot, 4. Quarters, 16 Nails, or 36 In∣ches, make a yard, 45 inches or 5 quarters of a yard make an ell, 5 foot is a pace, 6 feet, or 2 yards is a fathom, 5½ yards, or 16½ feet, is a pole, rod, or perch, 160 perch in length, and one in breadth, or 80 perch in length, and 2 in breadth, or 4 in breadth, and 40 in length, make an acre.
220 yards, or 40 pole, is a fur∣long: 1760 yards, 320 pole, or 8 furlongs, is an English mile; 3 miles is a League, 20 leagues or 60 mile is a degree, in ordinary account, and e∣very mile a minute.
Note that a minute contains 60 seconds, and 60 minutes is one de∣gree: and 30 degrees is one sign; 2 signs, or 60 degrees is a sextile ✶. 3 signs, or 90 degrees is a quadrant, or quartile □: 4 signs, or 120 de∣grees, a trine ▵: 6 signs, or 180 degrees is one opposition •• or semi∣circle, 12 Signs or 360 deg. is a Con∣junction ♂: and the Suns Annual, or Moons monthly motion. Note also, every hour of time hath in motion 15 Degrees. And a minute of time, hath 15 minutes of motion, and one Degree of motion, is 4 minutes of time.
Note further, that every hour of time, hath 60 minutes, therefore 45 is 3 quarters, 30 is half, 15 is a quar∣ter of an hour; 24 hours a day na∣tural; 7 days a week; 365 days and about 6 hours is a year. Hence it follows, that ¼ of a degree in the heavens, is 5 Leagues on the earth, or 15 minutes of motion above, is 1 mi∣nute of time below, therefore a de∣gree,
or 60 minutes of motion, is 4 minutes of time, as before is said.
All these rules, I shall express more largely, and in shorter terms, by these following Tables.
| Signs. | Degr. | Minutes. | Seconds. | |
| Note that the 12 Signes is | 13 | 360 | 21600 | 1296000 |
| One Signe is | 1 | 30 | 1800 | 108000 |
| One Degr. is | 1 | 60 | 3600 | |
| One Min. is | 1 | 60 |
| Mon. | Week. | Day. | Hour. | Minute. | |
| One Year | 13 | 52 | 365 | 8760 | 52560 |
| Month hath | 1 | 4 | 28 | 671 | 40320 |
| Week hath | 1 | 7 | 168 | 10080 | |
| Day natural | 1 | 24 | 1440 | ||
| Hour hath | 1 | 60 | |||
| Minute is | 1 |
| Mile. | Furl. | Perch. | Yards. | Feet. | Inches. | |
| Leag. | 3 | 24 | 960 | 5971 ¼ | 15840 | 190080 |
| Mi. | 1 | 8 | 340 | 1760 | 5280 | 63360 |
| Furlong | 1 | 40 | 220 | 660 | 7920 | |
| Per. Rod. Pol. | 1 | 5 ½ | 16 ½ | 198 | ||
| Aere contains of Squar. Perc. | 160 | 14520 | 43560 | |||
| Acre is in leng. | 1 | 40 | 220 | 660 | 7020 | |
| Acre is in bredth | 4 | 22 | 66 | 792 | ||
| 1 Rood, or ¼ of an acre is in len. | 40 | |||||
| 1 Rood, or ¼ of an acre is in bread. | 1 | 5 ⅕ | 16 ½ | 198 | ||
| One Fadom is | 6 | 72 | ||||
| One Elne English is | 3 ¾ | 45 | ||||
| One Yard is | 3 | 36 | ||||
| One Foot is | 1 | 12 | ||||
| One Inch is 1 Inch 3 grains. | 1 |
| Gallons. | Pottle. | Quart. | Pints. | |
| Tun of sweet Oyl | 236 | 472 | 944 | 1888 |
| Tun of Wine is | 252 | 504 | 1008 | 2016 |
| But or Pipe is | 126 | 252 | 504 | 1008 |
| Tertian of wine is | 84 | 168 | 336 | 672 |
| Hogshead is | 63 | 126 | 252 | 504 |
| A Barrel of Beer, or 2 runlets of wi. is | 36 | 72 | 144 | 288 |
| Kilderkin or one Runlet is | 18 | 36 | 72 | 144 |
| Barrel of Ale is | 32 | 64 | 128 | 256 |
| Kilderkin of Ale | 16 | 32 | 64 | 128 |
| Firkin of Beer is | 9 | 18 | 36 | 72 |
| Firkin of Ale is | 8 | 16 | 32 | 64 |
| Peck. | Gal. | Pottl. | Qu. | Pi. | |
| Bushel of water-m. is | 5 | 10 | 20 | 40 | 80 |
| Bushel of land-me. is | 4 | 8 | 16 | 32 | 64 |
| One Peck is | 1 | 2 | 4 | 8 | 16 |
| One Gallon is | 1 | 2 | 4 | 8 | |
| One Pottle is | 1 | 2 | 4 | ||
| One Quart is | 1 | 2 |
| Last. | Weig. | Chal | Qu. | Bush. | Peck. | Pints. | |
| Last of Dry-m. is | 1 | 2 | 2 ½ | 10 | 80 | 320 | 5120 |
| One Weight is | 1 | 1 ¼ | 5 | 40 | 160 | 2560 | |
| Chaldron of coals | 1 | 4 | 36 | 144 | 2098 | ||
| Quarter of wheat is | 1 | 8 | 32 | 512 | |||
| One Bushel is | 1 | 4 | 62 |
| Hogsheads. | C. | Stons. | Lib. | Ounces. | Drams. | Scruples. | Grains. | |
| Tun W. gross is | 4 | 20 | 280 | 2240 | 35840 | 286720 | 860160 | 17203200 |
| One Hogshead is | 1 | 5 | 70 | 560 | 8900 | 71680 | 215040 | 4300800 |
| One C. or Hund. is | 1 | 14 | 112 | 1792 | 14336 | 43008 | 860160 | |
| One Half C. is | 7 | 56 | 876 | 7168 | 21504 | 430080 | ||
| One Quarter of C. is | 3 ½ | 28 | 448 | 4584 | 10752 | 213040 | ||
| One Stone is | 1 | 8 | 128 | 1024 | 3072 | 61440 | ||
| One Lib. Pound is | 1 | 16 | 128 | 384 | 7680 | |||
| One Ounce is | 1 | 8 | 24 | 480 | ||||
| One Dram is | 1 | 3 | 69 | |||||
| One Scruple is | 1 | •••• |
| Lib. | Ounce. | Dp. | Carrots. | Grains. | |
| C.w. | 100 | 1200 | 24000 | 28800 | 576000 |
| ½C. is | 50 | 600 | 12000 | 14400 | 288000 |
| ¼C. is | 25 | 300 | 6000 | 7200 | 144000 |
| ⅛C. is | 12 ½. | 150 | 3000 | 3600 | 72000 |
| Pound | 1 | 12 | 340 | 288 | 5760 |
| Ounce | 1 | 20 | 24 | 480 | |
| One Penny weight is | 1 | 1 ⅕ | 24 | ||
| One Carrot Troy is | 1 | 20 | |||
| One Grain is | 1 |
| Mark | An. | Nob. | Cro. | Sh. | Groat. | Penc. | Far. | |
| Pound st. | 1 ½ | 2 | 3 | 4 | 20 | 60 | 240 | 960 |
| Mark is | 1 | 1 ⅓ | 2 | 2 ⅔ | 13 ⅓ | 40 | 160 | 640 |
| An Angel is | 1 | 1 ½ | 2 | 10 | 30 | 120 | 410 | |
| A Noble is | 1 | 1 ⅓ | 6 ⅔ | 20 | 80 | 320 | ||
| A Crown is | 1 | 5 | 15 | 60 | 240 | |||
| A Shilling is | 1 | 3 | 12 | 48 | ||||
| A Groat is | 1 | 4 | 16 | |||||
| A Penny is | 1 | 4 |
The use of which, (to come to our intended purpose,) may be thus. There you see how many farthings, pence, groats, shillings, and the like is in one, or any usual piece of coyne, also how many ounces, scruples, in any kind of weight; and the like for measure, both liquid and dry; and also in time: now if you would know how many there shall be, in any greater number then one: then say by the Rule (or line of numbers) thus, If 48 Farthings, be one shilling, how many shillings is 144 Farthings, facit 3 shillings, for the extent from 48, to 1, will reach from 144, to 3. and the contrary. Again, if a mark and a half be one pound, how many pounds is 12 mark? the extent from 1-50, to 1 shall reach from 12, to 8. for reason must help you not to call it 80 pound: again if 3 nobles be one pound, what is 312 nobles? facit 104 pound, the extent from 3 to 1, will reach from 312 to 104.
Further, If a Chaldron of coals cost 36 shillings, what shall ½ a Chal∣dron cost? facit 18. (but more to the matter) if 36 Bushel cost 30 sh. what shall 5 Bushels cost? facit 4. 16. that is by reduction 4 s. 2 d. neer the mat∣ter, or penny, 3 farthings, ½ farthing, and better: or on the contrary; If one Bushel cost 8 pence, then what cost 36? facit 288 pence; which being brought to shillings is just 24. which you may do thus: If 12 pence be one shilling, how many shall 288 be? facit 24. for the extent from 12 to 1, shall reach the same way from 288 to 24. as before: the like may be applyed to all the rest of the rules of weight, and measure; of which take in fine, some examples in short, and their answers.
If 14 Stone be one C. what is 91 Stone? facit 6 ½ C.
If one Ounce be 8 Drams, how many Drams in 9 Ounces, facit 72, the Extent from 1 to 8 reacheth from 9 to 72.
If one Bushel of water measure be 5 Pecks, how many Pecks is 16 Bu∣shels? facit 80 Pecks.
If one Barrel hold 288 pints, how much will a Firkin hold? this being the fourth part of a Barrel, work thus, if 1 give 288 what 25? facit 72, the answer sought.
If one week be 7 days, how many days is 39 weeks? as 1 is to 7, so is 39 to 273. So many days in 39 weeks.
If 160 perch be one acre, how many acre is 395 perch? facit 2. 492 that is neer 2 ½ acres.
If 8 Furlongs make one mile, how much is 60 Furlongs? facit 7 ½ mile: for the extent from 1 to 8, gives from 60 to 7. 50.
Possibly that this little Book may meet with some that are well skilled in Arithmetick, and being much used to that way, are loth to be weaned from that way, being so artificial and exact, yet though they can multiply & divide very well, yet perhaps they know not this way, to save their di∣vision and yet to take in all the fra∣ctions together as if of one denomina∣tion: I shall begin first with Foot-measure being the more easie, and I suppose my Two-foot-rule to be di∣vided into 200 parts, and figured with 10. 20. 30. 40. 50.60.70. 80. 90. 100. And then so again to 200. as in the 3 Chap. and then the work is on∣ly thus: set down the measure of one side of the square, or oblong thus, as for example, 7. 25, and 9. 88, and multiply them as if they were whole numbers, and from the product cut off 4 figures, and you have the content in Feet, and 1000 parts of a Foot, or Yard, Ell, Perch, or whatsoever else it be. Note the examples following.
〈 math 〉〈 math 〉 〈 math 〉〈 math 〉
For any kind of flat Superficies, this is sufficient instruction to him that hath read the first part; but if it be Timber, or Stone, you must thus find the Base, and then another work will give you the other side, as in Chapter 5 Problem 2. or, Multiply the length by the Product of the breadth and thickness, and that Pro∣duct shall be the content required.
First, Multiply all the whole Feet, then all the Feet and Inches, across, and right on, then the parts by the Feet, and also the Inches, and parts, across and right on; then add them to∣gether, and you shall have the answer in feet, long Inches, (that is, in pieces of a Foot long, and an Inch broad) square Inches, and 8 parts of a Square Inch: as for example.
Let a peice of Board be given to be measured that is 3. 3. 5. i. e. three Foot, three Inches, and 5 eights, one way, and 2. 3. 4, the other way. I set the numbers down in this Man∣ner, 〈 math 〉〈 math 〉 & then right on, first as the line in the Scheme from 2 to 3 leads.
I say thus, 3 times 2 is 6, set 6 right under 2. and 3 as in the example, in the left page: for 6 Foot, as is clear, if you consider the Scheme over the example, viz. the squares noted with f. then for the next I say cross-wise, 2 times 3 is 6, viz. long Inches, as you may perceive, by the 2 long squares marked with 9 L. and 6 L.
which 6 I put in the next place to the right hand, as in the example; then for the next, viz. 3 times 3. is 9, (croswise, as the stroke from 3 to 3 shews) which 9 is also 9 long Inches, as the Scheme sheweth, and must be put under 6, in the second place toward the right hand, in the Scheme it is express'd by the 3 long Squares, marked with L 9. Then last∣ly for the Inches, 3 times 3 is 9, go∣ing right up, as the stroke from the 2 threes lead you: but note, this 9 must be set in the next place to the right hand, because they are but 9 Square Inches, but had the Product been a∣bove 12, you must have Substracted the 12 s. out, and set them in the long Inches place, and the remainder, where this 9 now standeth, and this 9 is express'd in the Scheme, by the little Square in the corner markt with (□ 9.)
Then now for the Fractions, or 8 parts of an Inch, first say, croswise as the longest prick line doth lead you
to; 3 times 4 is 12, for which 12, you must set down 1. 6, that is 1 long Inch, and 6 Square Inches, the reason is, a piece 8 half quarters of an Inch broad, and 12 Inches long, is a long Inch, or the twelfth part of a Foot superficial, and if 8 be 12 Square Inches, then 4 must needs be 6 Square Inches: therefore, in stead of 12, I set down 1. 6, as you may see in the example, and in the least long Square of the Diagram, or Scheme. Then do likewise for the other long Square, which is also multiplyed across; as, two times 5 is 10. that is, as I said before, 1. 3, as the Example and Scheme make manifest, considering what I last said, and it is marked by the 2. 00. But if this or the other had come to a greater number, you must have Substracted 8 s. as oft as you could, and set down the remainder in the place of Square Inches, and the number of 8 s. in the place of Long Inches, as here you see.
Then for the two shorter Long
Squares next the corner, say croswise again, Three times 5 is 15, that is 1, 7, because eight Half-quarters an Inch long do make one square Inch, as well as eight Half-quarters a Foot long made one Long Inch: Therefore I set 1 in the place of square Inches, and 7 in the next place to the right hand, and it is expressed in the Dia∣gram by the small long square, and marked with * 1. 7.
Then again for the other little long square, say croswise, as the shorter prick line leads you, Three times 4 is 12, that is 1. 4; and do by this as the last: It is noted in the Scheme by 1-4.
Then lastly for 5 times 4, as the short prick line sheweth you, is 20: out of which 20 take the 8 s. and set them down in the last place, and the 4 remaining you may either neglect, (or set it down a place further) for you cannot see it on the Rule; there∣fore, I thus advise, if it be under 4. neg∣lect it quite, but if above, increase the
next a figure more if 4 then it is a half, and so may be added; for note, 64 of these parts make but one square Inch; of which parts, the little square in the right hand lower corner of the Scheme is 20, for which I set down 2. 4, that is two Half-quarters, and 4 of 64. which is the last work, as you may see by the Scheme and Example.
Now to add them together say thus, 4 is 4, which I put furthest to the right hand, as it were useless, because not to be exprest; then 472 are 13, from which take 8, and for it carry 1 on to the next place, or as many times 1 as you find 8, and set down the re∣mainder, which here is 5, then 1 I carried, and 13619, is 21, from which I take 12, and set down 9, be∣cause 12 square Inches, is one long Inch: then 1 I carried, (or more, had there been more 12 s.) and 1169 is 18, from which take 12, as before, there remains 6, that is, 6 long Inches, and so had there been more 12 s. so many you must carry to the next
place, because 12 long Inches is one Foot, lastly 1 I carried, and 6 is 7 Foot, so that the work stands thus, 〈 math 〉〈 math 〉 and so for any other measure Superfi∣cial or Solid.