The Description and Use OF THE CARPENTERS-RULE: Together with the use of the LINE of NUMBERS (Inscribed thereon) In Arithmetic and Geometry. And the Application thereof to the Measuring of Superficies and Solids, Gauging of Vessels, Military Orders, Interest and Annuities: with Tables of Reduction, etc. To which is added, The Use of a (portable) Geometrical Sundial, with a Nocturnal on the backside, for the exact and ready finding the hour of the Day and Night: And other Mathematical conclusions. Also of a Vniversal-Dial for the Use of Seamen or others. With the Use of a Sliding or Glasiers-Rule. and Mr. White's Rule for Solid measure. Collected and Fitted to the Meanest Capacity By J. Browne. London, Printed by W. G. for William Fisher at the Postern-gate near Tower-hill. 1667 To the Reader. Courteous Reader, whomsoever thou art, I Shall entreat thee to take in good part this Collection of The Ules of the Line of Numbers, commonly called (or best known to Artificers by the name of) Canter's Line; I writ it not as a new thing, but rather as a renovation of an old one; and the great motive that provoked and stimed me up to it is this: I making and selling Rules with Gunter's Line on them, many a one would say to me How shall I come to know the use of this Line? I reply, that in Mr. Cunter's Book there the Use is set forth, but because of the obscurity of the Instructions there, as to the reading of the Line; and also the dearness of the Book, many a one that would gladly learn, are deterred from taking pains therein, lest they should spend time and oil to no purpose; and also for want of cases fit to their purpose, they are apt to think it as to no purpose: Therefore that I might be as an ABCdarian to the Instrumental way of working, being the most proper for Mechanic Men, such as Carpenters, Joiner's, Masons, Bricklayers, and the like; which for the most part are ignorant of Arithmetic; and that knowledge might be increased any way, I thought it convenient, and make no doubt of a good benefit to accrue thereby, to them whose capacities and purses in these Critical times cannot well reach to other more difficult, and dear Authors: I shall not much Apologise for myself, as to style or manner of writing (being like myself;) what it is I beseech you accept in as good part as it was offered: I might have implored the aid of some more abler Pen, but I thought Mechanic men best understand them of their own profession, in this and other Discourses, because they are men of the same stature in knowledge and expressions. Possibly it may provoke some to a more accurate and universal Treatise: In the mean time take this as a Harbinger, till that come. And being apt to think that Ship-carpenters or Seafaring men may light of it, I have added in the conclusion (as an Appendix) the Use of a particular and Universal Sundial; also of a Nocturnal, or Star-dial; by which the hour of the Day and Night may be had in all places of the North latitude, from 1 degree to 66.30. where the day Artificial is 24 hours long: In all which I have laboured after brevity and plainness, as much as may be: And to the end you may learn to know the Stars, I have been at the charge to print a Paper with all the principal Stars in the Northern Hemisphere, from the Pole to the Equinoctial, so that you may take any in that compass, and they that please may do the like for the South Hemisphere. So I wish you may reap much profit thereby, and remain willing to serve you in what I may. J. B. At the Sphere and Dial in the great Minories. Lond. 66. The Description and Use of the Carpenters-Plain-Rule as it is now made. CHAP. 1. I Thought good to add this Chapter for the sake of some, possibly young beginners, and them that would not be ignorant altogether in the way of Measuring therewith, though they may seldom have occasion of it; and also knowing that they that have the most knowledge once had little enough: And farther, I find by experience, that many there be that can measure by the Plain Rule that cannot use the Line of Numbers; and some also know not the use of the plain Rule neither: For these Reasons I have added this Chapter of The Description and use of the Carpenters-Plain-rule. It is called a Carpenter's Rule, (rather-then a Joiner's, Bricklayers, Masons, Glasiers, or the like) I suppose, because they find the most absolute necessity of it in their way, for they have as much or more occasion to use it than most other Trades, though the same Rule must measure all kind of Superficies and solids, which two Measures measure every visible substance which is to be measured. And it is usually made of Box or Holly, 24 Inches in length, and commonly an Inch and half, or an Inch and quarter in breadth; and of thickness at pleasure; and on the one side it is divided into 24 equal Inches, according to the Standard at Guildhall London; and every one of those 24 Inches is divided into eight parts, that is, Halfs, Quarters, and Half-quarters; and the Half-inches are known from the Quarters, and Quarters from the Half-quarters, by short, longer, and longest strokes; and at every whole Inch is set figures, proceeding from 1 to 24, from the right hand toward the left, and these parts and figures are on both edges of one side of the Rule both ways numbered, to the intent that howsoever you hold the Rule you have the right end to measure from, provided you have the right side. On the other side you have the Lines of Timber and Board measure. The Timber-measure is that which gins at 8 and a half, that is, when the figures of the Timber-line stand upright to you, than I say it gins at the left end at 8 and ½, and proceeds to 36 within an Inch and ⅜ of an Inch of the end. Also of the beginning end of the Line of Timber-measure is a Table of figures, which contains the quantity of the Under-measure from one Inch square to eight Inches square, for the figure 9 comes upon the Rule, as you may see near to 8 in the Table. On or next the other edge, and same side you have the Line of Board-measure, and when those figures stand upright, you have 6 at the left or beginning end, and 36 at the other (or right) end, just 4 Inches of the end unless it be divided up to 100, than it is nigh an inch and half of the end. This Line hath also his Table of Under-measure at the beginning end, and begins at 1 and goes to 6, and then the divisions on the Rule do supply all the rest to 100 Thus much for Description: Now for Use. The Inches are to measure the length or breadth of any Superficies or Solid given, and the manner of doing it were superfluous to speak of, or once to mention, being not only easy, but even natural to every man, for holding the Rule in the left hand; and applying it to the board or any thing to be measured, you have your desire: But now for the use of the other side, I shall show it in two or three examples in each measure that is Superficial or Solid. And first in Superficial or Board-measure. Example the first. The breadth of any Superficies (as Board, or Glass, or the like) being given, to find how much in length makes a Square Foot, (or is equal to 12 inches broad, and 12 Inches long; for so much is a true Foot Superficial.) To do this, look for the number of Inches your Superficies is broad in the Line of Board-measure, and keep your finger there, and right against it on the Inches side you have the number of inches that goes to make up a Foot of Board or Glass, or any Superficies. Suppose I have a piece 8 Inches broad, How many Inches make a Foot? I look for 8 on the Board-measure, and just against my finger (being set to 8) on the Inch side, I find 18, and so many Inches long at that breadth goes to make a Foot Superficial. Again, suppose it had been 18 Inches broad, than I find 8 Inches in length, to make a Foot superficial; but if 36 Inches broad, than 4 Inches in length makes a Foot. Or you may do it more easier thus: Take your Rule and hold it in your left hand, and apply it to the breadth of your Board or Glass, making the end that is next 36 even with one edge of the Board or Glass, and the other edge of the Board showeth how many Inches or Quarters of an Inch goes to make a Foot of the Board or Glass. This is but the converse of the former and needs no example, for laying the Rule to it, and looking on the Board-measure you have your desire. Or else you may do thus in all narrow pieces under 6 inches broad. As suppose 3 ¼ double 3 ¼ it makes 6 ½, than I say, that twice the length from ½ to the end of the Rule shall make a foot Superficial, or so much in length makes a foot. Example the second. A Superficies of any length and breadth to find the Content, that is, how many Foot there is in it. Having found the breadth, and how much makes one Foot, turn that over as many times as you can, for so many Foot is there in that Superficies: But if it be a great breadth, than you may turn it over two or three times, and then take that together, and so say 2, 4, 6, 8, 10, etc. or 3, 6, 9, 12, 15, 18, 21, and till you come to the end of the Superficies. Note that the three short strokes between figure and figure, are the Quarters; as thus, 8 and a quarter, 8 and a half, 8 and three quarters, than 9, etc. till you come to 30, and then 30 and a half, 31, etc. to 36. And if it be divided any further, it is to whole Inches only to 100 The use of the Table at the beginning end of the Board-measure. First you have five ranks of figures, the first or uppermost is the number of inches that any Superficies is broad, and the other 4 are Feet, and Inches, and parts of an Inch that goes to make up a Foot of Superficial measure: As for example, at 5 Inches, broad you must have 2 Foot, 4 Inches, and 4 Fifths of an Inch more, that is, 4 parts of 5, the Inch being divided into 5 parts; but where you have but two figures beside the uppermost, and cyphers in the rest, you must read it thus, At two Inches broad you must have six Foot in length, no Inches no parts. Thus much for the Use of the Line of Superficial or Board-measure. The Use of the Line of (Solid) or Timber-measure. The use of this Line is much like the former: For first you must learn how much your piece is square, and then look for the same number on the Line of Timber-measure, and the space from thence to the end of the Rule is the true length at that squareness, to make a Foot of Timber. Example. I have a piece that is 9 Inches square, I look for 9 on the Line of Timber-measure, and then I say the space from 9 to the end of the Rule is the true length to make a Foot of Timber, and it is near 21 Inches, 3 eights of an Inch. Again, suppose it were 24 Inches square, than I find 3 Inches in length makes a Foot, for so I find 3 Inches on the other side, just against 24: But if it were small Timber, as under 9 Inches square, than you must seek the square in the upper rank in the Table, and right under you have the Feet Inches, and parts that go to make a Foot square, as was in the Table of Board-measure. As suppose 7 Inches square, than you must seek the square in the upper rank in the Table, and right under you have the Feet Inches, and parts that go to make a Foot square, as was in the Table of Board-measure. As suppose 7 Inches square, I find in the Table 2 Foot 11 Inches, and 2 sevenths of an Inch, divided into 7 parts, and 8 you find only 2 Foot, 3 Inches— o parts, and so for the rest. But if a piece be not just square but broader at one side than the other, than the usual way is to add them both together, and to take half for the square; but if they differ much then this way will be very erroneous, and therefore I refer you to the following Rules: But if it be round Timber, then take a string and girt it about, and the fourth part of this is usually allowed for the side of the square, and then you deal with it as if it were just square. Thus much for the Use of the Carpenters plain rule. I have also added a Table for the Under-measure for Timber & Board to Inches and Quarters; and the use is thus: Look on the left side for the number of Inches and Quarters, your Timber is square, or your Board is broad, and right against it you have the Feet, Inches, tenth part of an Inch, and tenth of a tenth (or hundredth part of an Inch) that goeth to make a Foot of Timber or Board. Example. A piece of Timber 3 Inches 1 quarter square will have parts to make a Foot. And a Board 3 Inches and a quarter broad must have in length to make a Foot, and so of the rest, as is plain by the Table, and needs no further explication, being common to most Artificers. A Table for the under Timber-measure to inches & quarters. A Table for the Vnder-Board-m. to inch. & Qu. Inch. qu. feet. inch. jop. ●oop. feet. inc. 10. 100 1 2304 0 0 0 48 0 0 0 2 576 0 0 0 24 0 0 0 3 256 0 0 0 16 0 0 0 1 1 144 0 0 0 1 12 0 0 0 1 92 1 9 7 9 7 2 0 2 94 0 0 0 8 0 0 0 3 47 0 2 4 6 10 2 9 2 26 0 0 0 2 6 0 0 0 1 28 4 3 3 5 4 0 0 2 23 0 4 1 4 9 6 0 3 19 0 3 1 4 4 3 6 3 3 16 0 0 0 3 4 0 0 0 1 13 7 5 9 3 8 3 0 2 11 9 0 6 3 5 1 4 3 10 1 8 8 3 2 4 0 4 4 9 0 0 0 4 3 0 0 0 1 7 11 6 6 2 9 8 8 2 7 1 3 3 2 8 0 0 3 6 4 5 9 2 6 3 1 5 5 5 9 1 2 5 2 4 8 0 1 5 2 6 9 2 3 4 2 2 4 9 1 2 2 2 1 8 3 4 4 2 6 2 1 0 4 6 6 4 0 0 0 6 2 0 0 0 1 3 4 2 3 1 11 0 5 2 3 4 9 0 1 10 1 5 3 3 1 9 3 1 9 3 3 7 7 2 11 2 8 7 1 8 5 8 1 2 8 8 6 1 7 8 6 2 2 6 7 2 1 7 2 0 3 2 4 7 7 1 6 5 8 8 8 2 3 0 0 8 1 6 0 0 8¼ 2 1 3 9 8¼ 1 5 4 5 Note also, that this Table or any smaller part of under-measure may be supplied by the divisions of the board and timber-measure only as thus; Double the inches and parts of breadth for board-measure, or of squares for timber-measure: and seek it in the Lines of board or timber-measure, and count twice from thence to the rules end for board or 4 times for timber, and that shall be the true length that makes a foot of board or timber. Example. At 4 inches and ½ square or broad 4½ doubled is 9 then look for 9 on the board measure, and two times from thence to the end shall make a foot of board. Or look for 9 on the Line of timber-measure, and 4 times from thence to the end of the Rule shall be the true length to make a foot of timber at 4 inches ½ square. But if it be so small a piece that when it is doubled the number is not on the divided part of the rule; then double it again, and count 4 times for board measure, and 16 times for timber. Example. At 2 Inches and half a quarter broad, or square, that doubled is 4¼, which is not on the rule, therefore I double it again, saying, 4¼ and 4¼ is 8½ which is on the rule; then for board count 4 times from 8½ on the board-measure to the upper end by 36 to make a foot of board at 2⅛ broad: And for timber count 16 times from 8½ near the beginning of timber measure which will be near 32 foot to make a foot of timber at 2⅛ square: But if twice doubling will not do, then double again, and count 8 times for board, and 64 times for timber, as in the Table you may see which will be very slender timber. The Description and Use of the Line of Numbers, (commonly called Gunter's Line.) In Arithmetic, and Geometry, and Gauging of Vessels, etc. The definition and description of the Line of Numbers, and Numeration thereon. THE Line of Numbers is only the Logarithmes contrived on a Ruler, and the several ranks of figures in the Logarithmes are here expressed by short, and longer, and longest divisions; and they are so contrived in proportion one to another, that as the Logarithmes by adding together, and substracting one from another produce the quesita, so here, by turning a pair of Compasses forward or backward, 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 doubled 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 consequently for exactness) into a Serpentine, 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 indifferently serve for any: But the projection that I shall chief confine myself to, is that of Mr Gunter's; being the most proper for to be inscribed on a Carpenter's Rule, for whose sakes I undertake this collection of the most useful, convenient, and proper applications to their uses in Arithmetic and Geometry. Thus much for definition of what manner of Lines of Numbers there be, and of what I intent chief to handle in this place. The order of the divisions on this Line of numbers, and commonly on most other, is thus, it gins 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 signify one tenth of any whole number or integer; as one tenth of a Foot, Yard, el, Perch, or the like; or the tenth of a penny, shilling, pound, or the like, either in weight, or number, or measure; and so consequently, 2 is 2 tenths; 3, three tenths; and all the small intermediate divisions, are 100 parts of an integer, 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 integers: Thus the line is in its most proper acception or natural division. But if you are to deal with a greater number than 10, then 1 at the beginning must signify 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, than 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 numbers: and so far with convenient care, you may resolve a question very exactly. Now any number being given under 10000 to find the point representing it on the rule, do thus. Numeration on the line of numbers. Probl. 1. Any whole number being given under four figures, to find the point on the Line of numbers that doth represent the same. First, Look for the first figure of your number, among the long divisions, 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 figure 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 many millions, as that fourth figure hath unites, or is in value; and that shall be the point where the number propounded 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 representing 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 further, and that is the point for 144. Thirdly, To find the point representing 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 centesmes from that 7 th'. tenth for 20. And lastly, For 8. you must reasonably estimate that following centesme, to be divided into 10. parts (if it be not expressed, which in lines of ordinary length cannot be done) and 8. of that supposed 10. is the precise point for 1728. the number propounded to be found, and the like of any number whatsoever. But if you were to find a fraction, or broken number, than you must consider, that properly, or absolutely, the line doth express none but decimal fractions: as thus, 1/10 or 1/100 or 1/1000 and more nearer the rule in common 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 capable to be applied to any thing in a decimal way: (but if you would use other fractions, as quarters, half quarters, sixteens, twelves, or the like, you may reasonably read them, or else reduce them into decimals, from those fractions, of which more in the following 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 absolutely, 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 divisions 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, etc. 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 figures 20, 30, 40, 50, 60, 70, 80, 90 integers, and every longest division between the figures, 21, 22, 23, 24, 25, 26, etc. integers, and the shortest divisions tenths of those integers, and so in proportion infinitely: as [1 10 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 represented by the first 1. on the line of numbers: the second order of Figures, viz. 1. 10. 100 1000 is represented 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 further) 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 imagine 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 signification. CHAP. II. PROBLEM. 1. Two numbers being given, to find a third Geometrically propertional unto them, and to three a fourth, and to four a fifth, etc. GEometrical proportion is when divers numbers being compared 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 Arithmetical, and Musical proportion; but of that in other more large discourses, being not material to our present purpose (though I may hint it afterward.) To find this by the numbers, extend the Compasses upon the line of numbers, from one number to another, this done, if you apply that extent (upwards or downwards, as you would either increase or diminish) from either of the numbers propounded, the movable point will stay on the 3d proportional number required. Also the same extent applied the same way from the third, will give you a fourth, and from the fourth a fifth, etc. Example. Let these two numbers 2 and 4 be propounded to find a third proportional to them, (that is, to find a number that shall bear the same proportion to 4. that 2 doth bear to 4.) and then to that 3d, a fourth, fifth, & sixth, etc.) 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 movable point will fall upon 8, the third proportional required, and then from 8 it will reach to 16. the fourth proportional, & from 16 to 32 the fifth, and from 32 to 64 the sixth proportional: but if you will continue the progression further, then remove the Compasses to 64 in the former part of the line, and the movable point will stay upon 128 the seventh proportional, and from 128 to 256 the eighth, and from 256 to 512 the ninth, etc. Contrarily to this, if you would diminish, as from 4 to 2, extend the Compasses from 4 to 2, and the movable 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 other work make use of the small divisions in the middle of the line, that you may the better estimate the fractions 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 err 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 imagine 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. PROB. 2. One number being given to be multiplied by another number given, to find the product. Extend the Compasses from 1 to the Multiplicator, and the same extent applied the same way from the Multiplicand, will cause the movable point to fall on the product. Example. Let 6 be given to be multiplied 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 numbered but with 3, yet your reason may regulate you, to call it 30. and not 3; for look what proportion 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. Another Example for more plainness. Let 125 be given to be Multiplied by 144, extend the Compasses from 1 to 125, and the movable point will fall on 18000. now read to this number 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. Some other Examples for more light. 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 computation of the last figure in 4 figures (for more cannot be well expressed, (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 Multiplycator 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 figures of the Product, than it shall have one less than the Multiplycator and Multiplycand put together, as 92 and 68 Multiplied makes 6256, 4 figures; and 12 Multiplied by 16, makes but 192, 3 figures, for the reason abovesaid: now for right naming the last figure, writ them down; as thus, 75 by 63. now you Multiply 5 by 3. that is 15. for which you by vulgar Arithmetic, set down 5. and carry 1. therefore 5 is the last figure in the Product, and it is 4725. PROBLEM 3. Of Division. One number being given to be divided by another, to find the Quotient. Extend the Compasses from the Divisor to 1, and the same extent will reach from the Dividend, to the Quotient, 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 upon 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 divided 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, than you work as if it were an absolute whole number; but if it be a whole number, joined to a decimal fraction, it is wrought here as properly as a whole number, example, ● would divide 111.4 by 1.728. extend the Compasses from 1. 728. to 1, the same extent applied 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 Figures any Quotient aught to cosist, it is necessary to write the Dividend down, and the Divisor under it, and see how often it may be written under 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 experience you will find. PROB. 4. To three numbers given to find a fourth in a direct proportion, (or the rule of 3. direct.) Extend the Compasses from the first number to the second; that done, the same extent applied the same way from the third, will reach to the fourth proportional number required. Example. If the Circumference of a Circle, whose diameter is 7, be 22, what circumference shall a Circle have, whose diameter is 14? Extend the Compasses 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 proportional 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 shillings is 12 foot worth. PROBL. 5. Three numbers being given to find a fourth in an inversed proportion, (or the back Rule of 3.) Extend the Compasses from the first of the numbers given, to the second of the same denomination, if that distance be applied 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? Extend the Compasses from 60. to 40. numbers of like denomination, viz. of Men; this done, that extent applied 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 duly consider the order of the Question, for you must needs grant that fewer men must have longer time, and the contrary therefore the answer must be in proportion to the question, which might have been wrought thus as well. The Extent 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 direct operation, according to the true reason of the question. Thus you have the way for the direct and reverse rule of 3: for the double rule of 3, and Compound rule of 3, this is the rule for it. Always 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 supposition 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 denomination with the term required, as here the 20 l. is of the same denomination with the how many pounds required. Set that always in the second place, and the two terms of supposition 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 applied 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 show 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 quarters 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 inches in length, the breadth being 16 inches, how many inches in length must I have to make a foot superficial? 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 inches, 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. PROB. 6. To three numbers given to find a fourth in a doubled proportion. This Problem concerns questions of proportions between lines and superficies: 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 applied twice, the same way from the third term, the movable point will stay upon the fourth term required. Example. If the Content of a Circle, whose diameter is 14, be 154, what will the content of a Circle be, whose diameter 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 movable point will fall on 616 the fourth proportional 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 distance, 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 distance betwixt 154 and 616 into ●● equal parts, than set one foot in 14 the other shall reach to 28, the diameter required. The like is for Squares; for if ●● square whose side is 40 foot, contai●● 1600 foot: how much shall a square 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 content sought for. PROBL. 7. To three numbers given to find a fourth in a triplicated proportion. The use of this Problem, consisteth in questions of proportion, between lines and solids, wherein if the first and second terms have denomination of lines, then extend the Compasses from the first to the second, that extent applied three times from the third, will cause the movable point to stay on the fourth proportional required. Example. 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 extent applied the same way 3 times form 9, the movable 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 proportional. Example. 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 beyond the end of the line, and the contrary as above, holds here in squares also. PROB. 8. Betwixt two numbers given to find a mean arithmetically proportional. This may be done without the help of the line of numbers: nevertheless, because it serves to find the next following, 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 aggregate (or sum) is the Arithmetical mean required. Example. Let 20 and 40 be the terms given, now if you subtract one out of the other, their difference is 60, whose half difference 30, added to 20, the lesser term makes 50, and that is the Arithmetical mean sought. PROB. 9 Betwixt two numbers given to find a mean musically proportional. Multiply the difference to the terms by the lesser term, and add likewise the sa●●e terms together; this done, if you divide that Product 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 Musical mean required. Example The numbers given being 8 and 12 multiplied together, make 96, that divided by 20, the sum of 8 and 12, the Quotient is 4 80, which doubled is 9-6 10, the Musical mean required. 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. PROB. 10. Betwixt two numbers given to find a mean Geometrically proportional. 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 proportional required. So the extreme numbers being 8 and 32, the middle point between them will be found to be 16. PROB. 11. Betwixt two numbers given to find two means Geometrically proportional. Divide the space between the two extreme numbers into 3 equal parts, and the two middle points, dividing the space shall show 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. PROB. 12. To find the Square root of any number under 1000000. The Square root of every number is always the mean proportional between 1, and that number for which you would find a square root; but yet with this general caution, if the figures of the number be even, that is 2, 4, 6, 8, 10, etc. then you must look for the unit (or one) at the beginning of the line, and the number in the second 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 toward the middle in the second length or part of the line: but if they be odd, than the middle one will be most convenient to be counted the unity, and both root and square will be found from thence forwards toward 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. PROB. 13. To find the Cubique Root of a Number under. The Cubique root is always by the first of two mean proportionals between 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 near 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 respective 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, than 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 placed at 10 in the end of the Line, and the Cube in the first part of the Line, you may help yourself, 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, than 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 Cubique 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 particular purposes, Yet for their sakes for whom it is intended, I shall enlarge; to some more particular applications in measuring all sorts of Superficies, and Solids; wherein I do judge it will be most serviceable do them that be unskilful in Arithmetic, as before said. A Table of Square and Cubique Roots. 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 CHAP. III. The Use of the Line of Numbers in measuring any Superficial measure, as Board, Glass, and the like. 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 Numbers, which are Decimals, or tenth parts, is bred very much trouble; and there cannot be exactness without taking of small parts, as ½ quarter's of Inches, or else using of Reduction; and it is also as troublesome by Arithmetic as by the Line of Numbers. To avoid which, I would advise either 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, without 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, els and Perches, or any other measure; for thereby the Work is made more easy, as shall appear anon. Therefore first by Foot-measure only. PROB. 1. The breadth of an Oblong superficies given in Foot-measute, to find how much in length makes a Foot. Extend the Compasses from the breadth to 1, the same extent applied 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 Superficial 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. PROB. 2. Having the length and breadth of any Superficies given in Foot-measure, to find the content of that Superficies in Foot-measure. Extend the Compasses from 1 to the breadth, the same extent applied the same way from the length, will reach to the Content. Example. As 1 is to 8, the breadth; so is 15 the length to 12, the content required; for a piece of 8 tenths broad, and 15 Foot long, containeth 12 Foot. PROB. 3. Having the breadth and length of an Oblong Superficies given in Inches, to find the content in Inches. As 1 Inch to the breadth in Inches, so the length in Inches to the Content 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. PROB. 4. Having the breadth and length of an Oblong superficies given in Inches, to find the Content in feet. As 144 the number of Inches in one Foot, is to the breadth in Inches; so is the length in Inches unto the Content in Feet. So as 144 to 30, so is 183 to 38, 250, that is, to 38 Foot and a quarter. PROB. 5. Having the breadth of an Oblong superficies given in Inches, and the length in Feet and parts, to find the Content in Feet, and such like parts as the length was. 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. PROB. 6. Having the breadth in Inches, to find how much makes a Foot in Inch-measure, (that is, how many Inches makes a Foot.) 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 contrary way to the number of Inches; for if you extend the Compasses from 12 to 6, that extent applied the contrary way from 12 shall reach to 24 the Inches required. PROB. 7. Having the length and breadth of an Oblong superficies, to find the side of a square equal to it. Divide the space between the length and the breadth into two equal parts, and the middle point shall show the side of the Square that shall be equal in area, or quantity, to that Oblong; so that a Square made of 11, 32, is equal to an Oblong of 16 one way, and 8 the other way. PROB. 8. Of a Circle. Having the Diameter of a Circle, to find the side of a square equal to that circle. As 10000 to 8862, so is the Diameter 15 to the side of the Square 13, 29, that is equal to the Circle. PROB. 9 Having the Circumference of a Circle, to find the side of a square equal to the same Circle. As 10000 to 2821, so is the Circumference 47, 13 to the side of the Square 13, 29 equal to the Circle. PROB. 10. Having the Diameter to find the Circumference. As 1 is to the Diameter, so 3142 to the Circumference. Or, as 7 to 22, so is the Diameter to the Circumference: or as 113 to 355. So the Diameter being 15, the Circumference will be about 47 13 parts PROB. 11. Having the Circumference to find the Diameter. As 3142 is to 1, so is the Circumference 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. PROB. 12. To find the side of a square that may be inscribed within a Circle, by having the Diameter. 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. PROB. 13. By having the Circumference, to find the side of the inscribed square. The extent from 1 to the Circumference, 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. PROB. 14. Having the Diameter to find the Superficial content of a Circle. The extent from 1 to the Diameter, being twice repeated (the same way) from 7854, will reach to the content required. PROB. 15. Having the Circumference to find the Superficial content of a Circle. The extent from 1 to the Circumference, being twice repeated from 07958, will reach to the content. PROB. 16. Having the content to find the Diameter. The extent from 1. to 1273, will reach from the content to another number, whose square root is the Diameter required. PROB. 17. Having the content to find the Circumference. The extent from 1 to 12, 75, will reach from the content to another number, whose square root is the Circumference required. PROB. 18. Having the content of a Circle to find the side of a square equal to it. Extract the square root of the content by the 12 Problem of the last Chapter, and that is the side required. PROB. 19 To find the content of a Circle two ways. Multiply the Diameter by itself, and multiply that product by a 11, and divide this last product by 14, and the quotient shall be the content required. Or else. Multiply half the Diameter and half the Circumference together, and the product is the content required. PROB. 20. How to measure a Circle, a Semicircle, or a quarter of a Circle, or any part that goeth to the Centre of the supposed Circle. First for a Circle. 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 Semidiameter of the whole Circle, and work as you would do with an Oblong square piece, and you shall have your desire. PROBL. 21. How to measure a Triangle. 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. PROBL. 22. How to measure a Rhombus, or a Rhomboides. A Rhombus is a Diamond-like figure, as a quarry of glass is, containing 4 equal sides and two equal opposite angles; but a Rhomboides is a figure made of two equal opposite sides and two equal opposite 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. PROB. 23. How to measure a Trapezium. A Trapezium is a figure comprehended 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 perpendiculars of both the triangles put together, so the whole base to the content. PROB. 24. How to measure a many-sided irregular figure or Polygon. 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. PROB. 25. How to measure a many sided regular figure, commonly called a regular Polygon. Measure all the sides, and take the half of the sum of them for one side of a Square, and the nearest distance from the centre 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. PROB. 26. How to reduce Feet into Yards, els, or other parts. 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 Plastering or Painting of a House, than I would advise you to have a yard to be divided into a hundred parts, (which is as near as commonly Workmen go to, or else, into a 1000, if you do require 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 together, and the product is the true content of that long square, the like holds for els, or poles, furlongs, or any other 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 extent 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. CHAP. IU. The use of the line of Numbers in measuring of land by perches, and acres. PROB. 1. Having the breadth and length of an oblong Superficies given in perches, to find the content in perches. As 1 perch to the breadth in perches, so the length in perches to the content in perches. Example. As 1 is to 30, so is 183 to 5490 (perches.) PROB. 2. Having the length and breadth in perches, to find the content in square acres. As 160 to the breadth in perches, so is the length in perches to the content in acres. As 160 unto 30, so is 183 to 34, 31 (in acres and perches.) PROB. 3. Having the length and breadth of an Oblong superficies given in Chains, to find the Content in Acres. 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 easy in Arithmetic, or by the Rule; for as 10 to the breadth in Chains, so the length in Chains to the content in Acres. Example. As 10 to 7.50, so is 45.75 to 34.31 (100 parts of an Acre.) PROB. 4. Having the Base and Perpendicular of a Triangle given in Perches, to find the Content in Acres. If the Perpendicular go for the length, and the whole base for the breadth, than you must take half of the oblong for the content of the triangle, by the second problem, as 160 to 30, so is 183 to 34.31, or else without halfing, say as 320 to the perpendicular, so is the base to the content in acres; as 320 unto 30, so is 183 to 17, 15. PROB. 5. Having the perpendicular and base given in chains, to find the content in acres. 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. PROB. 6. Having the consent of a Superficies after one kind of perch, to find the content of the same Superficies according to another kind of perch. As the length of the second perch, is to the first, so is the content in acres to a fourth number, and that 4th number 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. PROB. 7. Having the Plot of a field with the content in acres, to find the scale by which it was Plotted. 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 Problem 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. PROB. 8. Having the length of the Oblong, to find the breadth of the acre. 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 show 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 column you have 39204, for if you multiply 196 by 196, it will produce 39204. A necessary Table for Mensuration of Superficial-measure. 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 capacity. CHAP. V The use of the line of numbers in measuring of Solid measure such as Timber, Stone, or such like Solids. PROB. 1. By Foot-measure. A piece of ●imber being to be measured and not just square, how to make it square. 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 proportional between them. The breadth being 18, and thickness 6, the side of the Square will be found to be 10, 38. PROB. 2. Having the side of a square, equal to the base of any Solid given in Foot-measure, to find how much makes a Foot Solid in Foot-measure. As the side of the square in Foot-measure unto 1, so is 1 to a 4th number, 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.) PROB. 3. To find how much in length makes a Foot, any breadth and depth without squaring. As 1 to the breadth in Foot-measure, so is the depth to a fourth number, as that 4th number to 1 so is 1 to the length in Foot-measure. Example. 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. PROB. 4. Having the side of a square, equal to the Base of a Solid given, and the length thereof in Foot- measure, to find the content in Feet. 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 repeated from 15.25, shall reach unto 68.62. PROB. 5. Having the length breadth and depth of a Square Solid given in Foot-measure, to find the content in Feet. As 1 to the breadth in Foot-measure, 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. PROB. 6. By Inches, (only) and Feet and Inches. Having the side of a Square, equal to the base of any Solid given in Inches, to find how many Inches in length will make one Foot. 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 easy 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. PROB. 7. Having the breadth and depth of a squared Solid given in inches, to find the length of a Foot in Feet and Inches. 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. Or again thus. As 12 to the breadth in Inches, so the depth in Inches to a fourth number; 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. Example. 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 show how much is in a foot long. At 18 inches square in every foot long, is 27 inches, or 2 foot 3 inches: 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 multiplied by the feet long gives the whole content. PROBL. 8. Having the side of the square and the length thereof given in Inch-meameasure, to find the content in Feet. As 41.57, to the side of the square in Inches, so is the length to a fourth Number, and that fourth to the content in Foot-measure. As 41. 57, to 25. 45, so 183, twice repeated unto 68, 62. PROB. 9 Having the side of a Square equal to the base of any solid given in Inch-measure and the length in Foot-measure, to find the content in Feet. 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 content sought. PROB. 10. Having the length, breadth and depth, of a Squared Solid given in Inches, to find the content in Inches. 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. PROBL. 11. Having the length, breadth, and depth given in Inches, to find the content in Feet. As 1 to the breadth in Inches, so the depth in Inches to the base in Inches; 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. PROB. 12. Having the breadth and depth of a squared solid given in Inches, and the length in Feet, to find the content. 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. CHAP. VI The Use of the Line of Numbers in measuring of Cylinders, by Foot-measure. PROB. 1. Having the Diameter of a Cylinder given in Foot-measure, to find the length of a Foot-solid in Foot measure. 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. PROB. 2. Having the circumference given in Foot-measure to find the length of a Foot solid in Foot- measure. 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 applied 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 near in the content, which multiplied by the feet long gives the whole content. PROB. 3. Having the Diameter and length of a Cylinder given in Foot-measure, to find the content in Foot- measure. 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. PROB. 4. Having the Circumference and length of a Cylinder given in Foot-measure, to find the content in Foot-measure. 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. PROB. 5. By Inch-measure. Having the Diameter of a Cylinder given in Inches, to find how many Inches makes a Foot-solid. 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. PROB. 6. Having the Circumference given in Inches, to find the length of a Foot-solid. 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. PROB. 7. Having the Diameter and length given in Inches to find the content in Inches. As 1 128 to the Diameter in Inches, 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. PROB. 8. Having the Circumference and length given in Inches, to find 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 repeated from 105 will reach to 18555 the content in Inches. PROB. 9 By Feet and Inches. Having the Diameter given in Inches and the length in Feet to find the content in Feet. As 13, 54 to the Diameter, so the length to a fourth number, and that fourth to the content in Feet. The extent from 13, 54 to 15, being twice repeated from 8, 75 will reach to 10, 74, the content sought for. PROB. 10. Having the Circumference given in Inches, and the length in Feet-measure to find the content in Feet. 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. PROB. 11. It being an ordinary way in measuring of round Timber, such as Oak, Elm, 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 measure is not exact, but more than it should be. But either because of allowance for the faults abovesaid, or for Ignorance, the custom is sill used, and men commonly think themselves wronged if they have not such measure. Therefore I have fitted you with a proportion for it both for Diameter 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. PROB. 12. Having the Circumference in Inches, to find the content in the abovesaid measure. 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. PROB. 13. How to measure Taper Timber, that is bigger at one end than at the other. 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 multiplied by the length, shall give you the true content. Example. A round Pillar is to be measured, whose Diameter at one end is 20 Inches, 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. multiplied by 192 the length, gives 104393. 143. Cubical Inches which reduced into Feet, is 60 Foot, and 713 cub call Inches, for the solid content of the Pillar. PROB. 14. To mensure a Cone, such as is a Spire of a Steeple, or the like by having the height and Diameter of Base. Example; let a Cone be to be measured, 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 multiplied by 4, a third part of 12, the perpendicular 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 because 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 itself, which here is 25, then measure the side of the Cone 13, and multiply that by itself 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. PROB. 15. To measure a Globe or Sphere arithmetically. 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 multiplied by itself gives 196, and 196 again by 14 gives 2744, this multiplied 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 repeated, 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 Chapter of Superficial measure, to find the base, and then the base multiplied 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. CHAP. VII. Of Gauging of Vessels. The Use of the Line of Numbers in Gauging of Vessels. The Art of Gauging all manner of Vessels either close or open. All Vessels to put Liquor in are made either square, as Brewer's Coolers; 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 Coppe●s; or else close Cask, as Barrels, Butts, 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 consider by what measure you would Gauge 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 received measure to take dimensions with, is Inches, and 100 parts of an Inch, or 10 parts at the least, and for Brewer's 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 Majesty's Exchequer: Or if you will 288 cubique Inches, which is agreeable to the Standard-Gallon in Cooper's Hall, as Alderman Starling and others 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 Cubique-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 Example. 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. 230794 the Content in Cube inches. Viz. 22 barrels and near 26 gallons. To work this by the Line of Numbers 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 query is, How many Barrels is there? and not how many Inches or Gallons; and this you may well perform to a quarter of a Barrel, 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 applied 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. Example as before. 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 respective 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. Apparel 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 Barrel 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 To Gage any round Tun. First, If it have equal Diameters at the Top and Bottom, than it is measured as a Cylender, and the proportion 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. Or else measure it by this Analogy. As 1 to 0. 7854, so is the Square of the Diameter multiplied by the depth to the solid Content in Inches: Note the Work. Diameter 60 Inches, Depth 36 Inches. 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. Or else you may shorten the work, and save this last Division thus, and bring it into Gallons or Barrels. After you have multiplied the Diameter by itself, which is called Squaring, and then multiplied 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 Barrels, then multiply it by 0000775, and the product cutting off the fractions, shall be the Gallons, or Barrels of the content required. Note the work. Gallons. Barrels. 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 Gallons must needs be more than one Barrel, and yet not a 100, therefore 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 chief my aim is, the whole work is thus. Extend the Compasses from 1 to the inches Diameter, the same Extent 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, multiplied by the depth; which observe and note. Then lastly, the extent from 1 to 0. 2785 for Gallons, or to 0.775 for Barrels, shall reach the same way from the last found product (or the Square of the Diameter multiplied by the length) to content in Gallons or Barrels required. Example. 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 multiplied 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 straight and taper, than the usual way and somewhat near the Truth is, to add the Diameters at the top and bottom 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 Gunter's way, which by Arithmetic is thus; Multiply the mean Diameter by the length, and then divide that product by 17.15 for Wine-measure, or by 18.95 for Ale-measure, and note the Quotient and his Remainder; Again, Multiply the quotient last found, by the mean Diameter, and the Product; 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 Diameter, 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 being twice repeated from 36, the Inches 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 applied 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. PROB. 1. The true content of a solid measure being known, to find the Gage point of the same measure. The Gage-point of a Solid measure is the Diameter of a Circle, whose Superficial content is equal to the Solid content of the same measure so the Solid content of a Wine Gallon, (according to Winchester measure being 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 discovered) 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 Chapter) is 19.95, the Gage-point of Ale measure, because of soil and waste exceeding that of wine above 2 Inches. Another way by having the Diameter, length, and true content of any Vessel. Extend the Compasses on the line of numbers to half the distance between the content and length of the Vessel, the same extent will reach from the Diameter to the Gage-point. Example. Here at London it is said, that a Wine Vessel being 66 Inches in length, and 38 Inches in the Diameter, 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 Diameter 38 unto 17.15, the Gage-point, for a Gallon of Wine or Oil after London measure. The like reason holdeth for the like measure in all places. Now from what hath been said doth necessarily follow this conclusion: 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 integer of the same measure. So in a Cylinder 17.15, Inches Diameter, every Inch in the length thereof, contains one entire wine Gallon, and in a Cylinder of 18. 95 Diameter, every Inch is one entire Ale Gallon. PROB. 2. To find the equated Diameter two ways first by Arithmetic, as in Problem 8 of the second Chapter. Add the two Diameters at head and bung together, (being measured in Inches) and the half sum keep. Secondly, Subtract 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 easy) 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 Diameter 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 Diameter at the head, and bung, take the difference, 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 Diameter, 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, than you must add the 100 parts as in the Table annexed. A Table. 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 between 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. PROB. 3. Having the mean Diameter, and the longth of a Vessel, to find the content. Extend the Compasses from the Gage-point to the mean Diameter, the same extent being twice repeated from the length, shall give the content in galons and 100 parts. Example. 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. PROB. 4. Having the Diameter and Content, to find the length. Extend the Compasses from the Diameter to the Gage-point, the same extent twice repeated from the content, shall give the length. As 38. to 17. 15, so is 324 to about 66 twice repeated. PROB. 5. Having the length and content, to find the Diameter of a Vessel. Extend the Compasses to half the distance, between the length and content, the same extent shall reach from the Gage-point to the Diameter. Divide the space between 66, and 324, in two equal parts, the same extent shall reach from 17. 15 to 38, the Diameter abovesaid. Example. 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. PROB. 6. Having measured a Vessel according to Wine measure to know what it holds in Ale measure, without knowing the Gage-point. 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 Gallons. Example. As 282. to 231, so is 116. 4, to 95, 30. and as 231 to 282: so is 95. 30, to 116. 4, etc. PROB. 7. To measure any vessel a more easy way. There is yet a more easy way, and for exactness no way infer or to any extant, and that by taking the length in Inches and 10 parts, but the Diameters at head and Bung, with a line called Cughtred's Gage line, (and to be had at John Brown's house in the Minories near Aldgate, Mathematical-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 divisions begin to be numbered. And set that down twice: and the Diameter of the inside of the head (for so we understand all along,) and set that down once. In this manner. and than add them together as here you see, the length in Inches 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 than half a Gallon, which is 54 gallons ½ the content of a high Country Hogshead, whose measures were as before. Depth— Diana. at top— Diana. at bottom— Sum of the Diameters The Summe of the Diameters Squared— Product of Diana. The Residue The residue the fraction cut of 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 quotient by 3, produceth these fixed Numbers. To measure a Taper, round Tun by Arithmetic. 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 Diameters. Also multiply the sum of the Diameters in itself, which is called Squaring thereof. Then subtract 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 product, which last Product divided by a 1077, a certain fixed number for Ale Gallons, shall in the quotient give the content in Alegallons required. 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 Numbers: Extend the Compasses from the fixed Number, to the square of the two Diameters, less by the product of the two Diameters multiplied 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. Example of this way. Diana. Head, 18 Bung, 32 length, 40 Inches. (84.9.1 third of the head (107-58 gallons of wine. Or shorter thus: As 1 to 0.5236 so is 1024 to As 1 to 0.2618 so is 324 to Cube-Inches Another way more exact but yet more tedious to work. Note the foregoing Example. Measure the two Diameters in Inches and 10 parts, and also the length within, and find out the Superficies of the Circles having those Diameters, and add two third parts of the greater Diameter, and one third part of the lesser into one sum; Then multiply that number by the length, and the product shall be the content in Cube Inches, which product divided by 282 gives Ale-gallons or by 231 gives the content in Wine-gallons. To find the content of a Circle having the Diameter, square the Diameter, then multiply that square by 11, and divide the last product by 14, and the quotient is the superficial content in Inches required. Or shorter thus, As 1 to 0. 5236 a number fit for a thirds. So is the Square of Diameter 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 multiplied 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. The use of several Gaging-lines. Last of all to make this work more easy for Mechanic men, and those that want Arithmetic, there is several lines to be had on Joint or straight 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 ways applied. Thirdly, Rules to find the emptiness of Butts, Barreis, and Kilderkins. Fourthly, a Rule to Gage any Bushel, Half-Bushell, Peck, Half-peck, Pottle, or Quart, or Pint measure, according to 272 the Corn-gallon; Whose several uses is as followeth. First, for the Diagonal line. Put the Rule in at the Bunghole down aslope to the bottom of the head, and observe what parts the middle of the Bunghole 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 Bunghole 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 upper edge of the Vessel to the opposite lower edge across, next the bottom, whatsoever the edge of the Vessel cuts on the rule, the half thereof is the content of the Vessel near the matter. Example. Suppose that a Cross from the upper edge of a Pail to the bottom be 18 Inches, the Rule will show 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 measuring of Wine or Oil 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 straight, 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 Diameter, 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. Example. Suppose a Brewer's Tun at the Top be 82 in. at bottom 72, and deep 38 Inches. Diameters top bottom. Sum 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 near a half, which multiplied 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 Inches 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 fullness or emptiness. Another Example. Suppose a Vessel be in Diameter at top 160 Inches 5 tenths, and at bottom 148 Inches and 8 tenths, the sum is 309-3 the half is 154-6 ½ for a mean Diameter, the Gage-line right against 154-6 ½ is 64.63, which multiplied by 41.70, the Inches and tenths deep make , which last 4 figures are fractions and parts of a Gallon. Note, If in a great quantity of gallons, if this way prove to be somewhat less than in truth there may be, or that it do not agree with the former Rules, take this for answer, it is as near as any such instrumental way can be composed, and of two errors 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 experience and practice will make easy. Note also the same Line is improved by Mr. Newton I think, but the way how being in Print in Wingate's Rule of proportion, I shall say nothing to it, this way being full as exact and more easy. The Use of the Rule to find the emptiness of Butts, Barrels or Kilderkins. The use hereof is very easy, 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 show how many Gallons is in, and the compliment thereof to the whole content is what it wants of being full. Example. Suppose I have a part, or a whole Barrel of Ale in a Beer-Barrel (for there are or aught to be no Ale-Barrels) and I would know how much is in or out: Put the Line for a Barrel downright in the Bunghole to the opposite side, and then suppose it wets 13 inches 8 tenths, than there is 23 Gallons in and 13 empty. But if you had put the Rule for a Kilderkin, and it had wetted as before, then there would have been but 15 gallons in, and 3 out, the compliment 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 roundish Butts, about 57 Gallons in, and 63 out, the Compliment to 57 to 120 the whole content in a But. The use of the Rule for Corn, and for Sea-Cole measures. Take the Rule with the beginning end from you, and take the Diameter of the measure whatsoever it be, and the figures on the Rule right against the Inches Diameter shall express how many Inches and hundred parts that measure ought to be in depth to make a true measure. Two Examples of every measure. A pint measure 2 Inches and ½ Diameter, aught to be 6 Inches 95 parts deep. And at 2 Inches 3 quarters over 5. 75 deep. A quart measure 3 Inches diam. aught to be 9 Inches and 63 parts deep. And at 3 Inches 3 quarters Diana. 6-15 parts deep. A Pottle-measure 4 inches Diameter, aught 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, aught to be 8-19 deep, at 9 Inch Broad 4-26 deep. A Peck 13 Inches broad or Diameter, aught to be 6-94 deep, and at 12 Inches broad 4-82 deep. A half Bushel 13 Inches broad must be 8-23. but at 16 Inches broad but 5-42. deep. A Corn-Bushell of 17 Inches broad, aught to be 9-62 deep, but of 21 Inches broad 6-29. But for Sea-coal measures take the next number 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 beginning viz. 8-33 because a Corn-Bushel ought to hold a quart more than a Corn Bushel of water; and the Diameter from outside to outside ought to be 19 Inches and ½, the half bushel 14½, the Peck 11 ½, the half Peck 9 ½ to make the heap to bear a proportion to the fats. An Instrumental way to find the emptiness of any Vessel, by the Line of Numbers, Close-Cask. 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 number on a Line of Artificial Segments joined to the Line of Numbers, which fourth number keep. Then as 1, the whole content in Gallons, So is the fourth number kept, to the emptiness in the same Gallons the content was. Example. 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 Segments, 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 Gallons. You may if you will say, As 1 to the mean Diameter, so is the mean emptiness, etc. and the work will be somewhat more exact, but this is near the matter, and easy. These Collections of Gauging, 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 welcome 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 Gauging 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 properly called) and the Diameter at the head 1 gallon 10 parts, what is the content? Set the two Diameters down, 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: If 100 is 12 d. what shall 75 be? facit 9 Pencen or 9 Inches. If 100 be 112 l. what shall 50 be? facit 56 pound. If 100 be 8 pints, what shall 25 be? facit 2 pints. If 100 be 48 f. what shall 30 be? facit 14. 4, that is, 3 d. 2 f. ½/ near. If 100 be 36 Bushels, what shall 24 be? facit 8 Bushels ½ and better. If 100 be 60 min. what shall 50 be? facit 30 minutes, or half an hour. If 100 be 120, what shall 80 be? facit 96, of 112 of nails. The like is for any kind of Reduction. CHAP. VIII. The use of the line of numbers in Questions that concern Military Orders. PROB. 1. Any Number of Soldiers being propounded, to order them into a Square Battle of Men. 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 Soldiers ought you to place in Ranck, and so many likewise in File, to make a Square Battle of men. Example. Let it be required to order 625. Soldiers, into a Square Battle of men; the Square rooot of that number is 25; wherefore you are to place 25 in rank, and as many in File, for fractions in this practice are not considerable. For had there been but 3 less, there would have been but 24 in rank and file. PROB. 2. Any Number of Soldiers being propounded, to order them into a double Battle of men: that is, which may have twice as many in rank as file. Find out the square root of half the number given, for that root is the number of men to be placed in file, & twice as many to be placed in rank, to make up a double Battle of men. Example. Let 1368 Soldiers, be propounded to be put in that order: I find by the 12 aforesaid, that 26, etc. is the square root of 684, (half the number propounded,) and therefore conclude, that 52 ought to be in rank, & 26 in file, to order so many Soldiers into a double Battle of men. PROB. 3. Any number of Soldiers being propounded, to order them into a quadruple battle of men; that is, four times so many in Rank as File. Here the Square root of the fourth part of the number propounded, will show the number to be placed in File, and four times so many are to be placed 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, etc. PROB. 4. Any number of Soldiers being given, together with their distances in Rank and File, to order them into a Square battle of ground. Extend the Compasses from the distance in File to the distance in rank; this done, that extent applied the same way from the number of Soldiers propounded, will cause the movable 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 show the number of men to be placed in Ranck. Example. 2500 men are propounded to be ordered in a Square battle of ground, in such sort that their distance in Filbeing seven Foot, and their distance i● Ranck three Foot, the ground whereupon 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 because the fourth number to be found will in all likelihood consist of 4 figures,) if you apply that extent th● same way, from 2500, in the second part among the smallest divisions, the movable 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 Soldiers, you have the number of men to be placed in rank. 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. PROB. 5. Any number of Soldiers being propounded to order them in rank and file, according to the reason of any two numbers given. This Problem is like the former, for as the proportional number given for the file, is to that given for the rank, so is the number of Soldiers to a fourth number, whose square root is the number of men to be placed in rank, by which if you divide the whole, you may have the number to be placed in file. Example. So if 2500 Soldiers 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 rank, 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, etc. The number of men to be placed in file. CHAP. IX. The use of the Line in Questions of Interest and Annuities. PROB. 1. A sum of money put out to Use, and the Interest forborn for a certain time, to know what it comes to at the end of that times, counting Interest upon Interest at any rate propounded. Take the distance with your Compasses 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 Principal with the Interest, and increase according to the rate propounded. Example. I desire to know how much 125 l. being forborn 6 year will be increased, 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 repeated from 125, shall reach to 177 l. the principal increased with the interest at the term of 6 years, at the rate propounded. But if it were required for any number of Months, than 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. PROB. 2. A sum of money being due at any time to come, to find what it is worth in ready money. 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 question, it shall fall on the sum required. Example. 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. PROB. 3. A yearly Rent, Pension, or Annuity being forborn for a certain term of years, to find what the Arrears come to at any rate propounded. 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 subtract the principal out of that sum the remainder is the Arrears required. Example. 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. compound 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 subtract 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, unless 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 error, or at the least much mitigate it; for in these questions the larger the Line is, the better. PROB. 4. A yearly Rent or Annuity being propounded, to find the worth in ready money. First, find by the last what the arrears come to at the term propounded, and then what those arrears are worth in ready money, and that shall be the value of it in ready money. Example. What may a Lease of 10 l. per ann. having 15 year to come be worth in ready money? I find by the last Problem 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 presently, but to stay a certain term longer, than you must add that time to the time of forbearance; as suppose that after 5 years it were to begin, than 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. PROB. 5. A sum of money being propounded, to find what Annuity to continue any number of years, at any rate propounded, that sum of money will purchase. 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 sum propounded to the annuity required. Example. 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 near 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. CHAP. X. The application of the Line of Numbers to use in domestic affairs, as in Coals, Cheese, Butter, and the like. 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 commodities 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 proportional numbers, and of so easy an attainment, for any ordinary Capacity, and chief 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 Reduction: Rules for English money. Note, that 4 farthings make a penny; 16 farthings, 8 half pence, or 4 pence, make a groat, 48 farthings, 24 halfpences, 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 sterling. Rules for Troy-weight. Note that 24 grains is a penny weight: 20 penny weights, or 24 carrots, is an ounce Troy: 12 Ounces is a pound: 25 lib. a quarter of a hundred, 50 lib. half a hundred: 75 lib. 3 quarters of a hundred, and 100 lib. is an hundred weight Troy. Rules for Averdupois weight. 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 Father 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. Rules for Concave Dry measure. 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 Bushel: 4 pecks, or 56 lib. make a Bushel: 2 Bushels, is a Strike: 2 Strikes a Coomb, or half quarter; 2 Commbs 4 Strikes, or 8 Bushels make a quarter, or a Seam: 10 quarters, or 80 Bushels make a Last. Rules for Concave Wet-measure. Note, that 2 pints is a quart: 2 quarts a pottle; 2 pottles 4 quarts, or 8 pints make a Gallon: 9 Gallons make a Firkin, or half a Kilderkin: 18 Gallons make 2 Firkins, a Kilderkin, or a Roundlet; 36 Gallons is 2 Kilderkins, or a Barrel; 42 Gallons make a Terce, 63 Gallons or 3½ Roundlets make a Hogshead, 84 Gallons, or 2 Terces, make a Tertion, or Punchion: 126 Gallons, is 3 Terces, two Hogsheads, one Pipe, or But. A Tun is 252 Gallons 14 Roundlets, 7 Barrels, 6 Terces, 4 Hogsheads, 3 Punchions, 2 Pipes, or Butts. Note that is sweet Oil 236 Gallons make a Tun, but of Whale Oil 252 goes to the Tun. Water-measure. 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 Cauldron of Coals. Rules for Long-measure. 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 Inches, 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 furlong: 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 every mile a minute. Rules for Motion and Time in Astronomy and Navigation. Note that a minute contains 60 seconds, and 60 minutes is one degree: 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 degrees, a trine ▵: 6 signs, or 180 degrees is one opposition ● or semicircle, 12 Signs or 360 deg. is a Conjunction ♂: and the Sun's 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 quarter of an hour; 24 hours a day natural; 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 minute of time below, therefore a degree, 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. Equation for Motion. Signs. Degr. Minutes. Seconds. Note that the 12 Signs is 13 360 21600 1296000 One Sign is 1 30 1800 108000 One Degr. is 1 60 3600 One Min. is 1 60 Equation for Time. 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 Equation for Long-measure. 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 breadth 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 Fathom 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 Equation of Liquid-measure. Gallons. Pottle. Quart. Pints. Tun of sweet Oil 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 widow. 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 Equation of small dry measure, and then of great measure. Perk. Gal. Pottl. Qu. Pi. Bushel of waterm. 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. Perk. Pints. Last of Drym. is 1 2 2 ½ 10 80 320 5120 One Weight is 1 1 ¼ 5 40 160 2560 Cauldron of coals 1 4 36 144 2098 Quarter of wheat is 1 8 32 512 One Bushel is 1 4 62 Equation for Avoirdupois weight. Hogsheads. C. Stones. 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 ●● Equation for Troy-weight. 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 Equation of Money. 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 coin, 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 than 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 Cauldron of coals cost 36 shillings, what shall ½ a Cauldron 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. near the matter, or penny, 3 farthings, ½ farthing, and better: or on the contrary; If one Bushel cost 8 pence, than 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 applied 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 Bushels? 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 near 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, giveth from 60 to 7. 50. CHAP. XI. To measure any Superficies, or Solid by Inches only, (or by Foot-measure) without the help of the line, by Multiplication of the two sides. PROB. 1. Possibly that this little Book may meet with some that are well skilled in Arithmetic, and being much used to that way, are loath 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 division and yet to take in all the fractions together as if of one denomination: I shall begin first with Foot-measure being the more easy, and I suppose my Two-foot-rule to be divided 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 only 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, el, Perch, or whatsoever else it be. Note the examples following. 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 Product shall be the content required. PROB. 2. To Multiply Feet, Inches, and 8 parts of an Inch together without Reduction, and so to measure Superficial (and Solid) measure, First, Multiply all the whole Feet, than all the Feet and Inches, across, and right on, than the parts by the Feet, and also the Inches, and parts, across and right on; then add them together, 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 piece 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 Manner, & 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 , 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 shows) which 9 is also 9 long Inches, as the Scheme showeth, and must be put under 6, in the second place toward the right hand, in the Scheme it is expressed by the 3 long Squares, marked with L 9 Then lastly for the Inches, 3 times 3 is 9, going right up, as the stroke from the 2 three 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 above 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 expressed in the Scheme, by the little Square in the corner marked 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, than 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 multiplied 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 Diagram 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 14. Then lastly for 5 times 4, as the short prick line showeth 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; therefore, I thus advise, if it be under 4. neglect 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 expressed; 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 remainder, which here is 5, then 1 I carried, and 13619, is 21, from which I take 12, and set down 9, because 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, and so for any other measure Superficial or Solid. An Appendix. CHAP. I. The Description and use of a general and particular Sundial made for the Latitude of 51. 30. North or the like for any other Latitude. 1. First, the Dial itself is in form of a Quadrant, Sextance, or Circle, according as you please. 2. There is a string fitted, to hang the Horizontal line of the Dial, Horizontally or parallel to the horizon. 3. There is the Centre, hole, wherein to stick a pin, straw, bend, or hair, to give a shadow. Secondly, for the lines delineated on the Dial, the first I shall take notice of is the Horizontal line, and it is a long straight line drawn perpendicular to the string, and cuts the Centre just in the midst. 2. You have next the Verge or Limb, (be it Quadrant, Sextance, or Circle) the 90 Degrees of a quadrant, or but 60 of 90 if it be but a Sextance or of a round form, being sufficient for this Latitude: and the figures on that line is 10. 20. 30. 40. 50. 60. 70. 80. 90. 3. Next to that (toward the Centre, and concentric or parallel to the former Circle of degrees) you have a line of quadrat, or shadows, or if it be large you may have both quadrat, and shadows, and the figures, on the quadrat, or 10. 20. 30. 40. 50. and then back again to 1. 10, or in large quadrants the 50 is called a 100, and 10. 20, and 20. 40, etc. But the line of shadows, is figured with 12.11.10.9.8.7.6.5.4.3.2.1. in the middle against 50, and then 1. 2. 3. to 12. back again; the first part of which is called the right shadow: the other is called contrary shadows. 4. Next to that you have two more lines, (concentric to the former in quadrants and Sextances; But in round Dial's, they are but as it were concentrickal) on which are divided a Calendar of months and days, if they be large, but in those of ordinary bigness, is but every fifth and tenth day expressed; and single days are reasonably understood, of which two Circles the uppermost contains that half year in which the days be increasing, viz. from the 10. of December to the 10. or 11. of June; and the lowermost Circle contains the other half, decreasing or shortening. And at the end of every month there is a long stroke, being the last day of that month, and then by that, in the next month, you have the first letter (or more letters,) of the name of that month; so that all the letters in these two lines are: I. F. M. A. M. I. I. A. S. O. N. D. for January, February, etc. 5. Next to the Calendar, you have 25 concentric Circles, if they are lines of declination, or but 19, if they be lines of the Suns-rising, and every 5 of them if they be declinations, or every fourth if they be lines of rising, is expressed and distinguished with pricks; for every 5 and tenth Degree of declination, or for every whole, and half hour of Suns rising; as the figures set to them, at the end of them (next the horizontal line) show, if they be of Suns rising being 4. 5. 6. or the contrary end if they be lines of delination, noted with 10.20. will show; the uppermost of which 25, or 19 lines, represents the Aequinoctial: and the lowest the two Tropiques of Cancer, and Capricorn. 6. Those lines which descend downwards sloping from the Aequinoctial (or uppermost concentric Circle,) to the Tropic (or lowermost Circular line,) towards the right hand, are the Summer hour lines; and are 9 in number, when they are only whole hours. But in large Dial's there may be put halfs, and quarters, and then there are 8 pricked hour lines, for 12 being the last needs no pricks for distinction. And they are figured above the Aequinoctial with 6.7.8.9.10.11.12, and under the Tropiques with 8. 7. 6.5.4.3.2.1.12. for the same line hath two figures, viz. 4 & 8, 7 & 5, 6 & 6, 5 & 7, 4 & 8, 3 & 9, 2 & 10, 1 & 11, 12 & 12. That is, the same line that is for 4 in the morning, is for 8 at night, and the like is both for winter and summer; the reason is, look how high the Sun is at 7 in the morning, so high is it at 5 in the afternoon, and the like. 7. Those lines that descend from the Aequinoctial toward the left hand are the winter hours, and are figured above as before, but below with 12.1.2.3, for in the shortest days, the Sun sets before 4, and in large Dial's, there are halfs, and quarters also, and distinguished as before, but of whole hours at the Tropiques there is but 4, but at the Aequinoctial there be 6, and a point representing the hour of 6. 8. And lastly, in the spare room beyond the Aequinoctial, you have set a perpetual Almanac, whereby, if you know the day of the month, you may find the day of the week, and the contrary. Also in some Almanacs, you have the Dominical letter, Leap-year, and Epact, to find the Moons-age, the largest and uppermost of which ranks of figures, are the 12 months, the next 5 are the 31 days of the month, and the rest, if any be, are according to their names: thus much for description. One thing more I think convenient to hint, to make it serve for all fashions of Dial's of this kind, and it is this; some Dial's of this kind, have the Calendar of months and days on the backside, in 1, 2, or 4 Circles and lines of the Sun's declination, rising, true place, and amplitude and the right ascension; and the use is only thus. Lay a thread on the day of the month and the Centre, and it cuts or showeth all the other in their respective lines. It being so easy, and also not to our present purpose, I shall not give any example, but come to the uses of the foreside, which are sufficient for the hour of the day, the thing promised. Note also that by the addition of on line it is made to serve for all latitudes: also in the form of a very strong and handsome Tobacco-box. CHAP. II. The use of the Dial. PROB. 1. How to hold the Dial in time of Observation. Hang the string of the Dial over your thumb, on your left hand, (or you may hold it between your thumb and the middle of your forefinger) and stretch your 4 fingers straight out, and let the Dial hang at liberty, just touching the palm of your hand, that it may be steady, then turn your whole body about, till the edge of the Dial (or your finger's ends) be just against the Sun: then shall you see the shadow of any thing stuck in the Centre, though never so short, to reach quite through the Dial, and then it is held right. PROB. 2. To find the Sun's Altitude. Stick a pin (the smaller the better) in the Centre, and hold it up as before, and the shadow will show on the limb the Sun's Altitude required. Example. At 8 of the clock on the 11 of June in the morning, I would know the Sun's Altitude; I hold it up as before, and I find it to be 36.46, that is 36 Degrees and 46 Minutes, each Degree being 60 Minutes, as in the Tables of Reduction. PROB. 3. To find the perpendicular height of any thing by its shadow, by the line of shadows. Hold up the Dial by the thread as before, and look on what division of the line shadows, the shadow of the pins cuts, that is the true height or length of the shadow, by which to get the height of any perpendicular thing, or the very top of any leaning thing, that causeth the shadow. Example. On the same 11 of June at near 9 a Clock in the morning, I hold up my Dial, and I find the shadow to fall just on 1, on the line of shadows, or 45 on the Degrees, therefore I say, that the height of the object, that causeth the shadow, and the shadow are both of one length: but if it had fallen on 2, (that is to say, of right shadow) than the object is, but half the length of the shadow measuring upon a level ground, from the end of the shadow, to right under the object that causeth the shadow; if it falls on 3, the shadow is 3 times as long as the thing is, and so to 12, to 12 times longer, and the strokes between note one tenth, 2 tenths, 3 tenths, &c. more; but if it falls beyond 1, on contrary shadow, than the shadow is shorter accordingly, as will appear very plain with a little practice. Or rather thus by the Numbers. Count the middle 1 on the Rule, as 1 at 45 on the Dial, then if the Sun be under 45, count them on the rule toward 10, and if above 45 the contrary; then as the parts cut are to the middle 1, so is the length of the shadow to the altitude required. PROB. 4. The Use of the Quadrat. To use the Quadrat, you must have a hole in the other end of the horizontal line, and also some some where in the Dial, square (from the Centre) to the horizontal line; also you must have a thread and plummet, than the use is thus. Stick a pin in the Centre, and thereon hang the plummet, than put a pin in the other whole, that is perpendicular to the horizontal line, and just over the Centre; and hold up the Dial in your right hand, and make the string to play evenly by the Superficies of the Dial, when you see the object right against both the pins; then observe what stroke is cut by the thread on the line of quadrat, (or shadows for that may be used so also,) for if you go backwards or forwards till you make the thread to fall on 1, in the shadows or on 50 in the quadrat, then is the height of the House, Steeple, Tree, or the like, equal to the distance, between you and it, adding the height of your eye to it. But if it had fallen on 25 of the quadrat, or 2 on the shadows, than the distance had been twice as much as the height, (if right shadow) but for contrary shadow, the contrary. I shall say no more to this, only give you a caution, that if you look from the height of any place downwards, than you must put that pin next the Centre to your eye, and look downwards to your object, and then the side which before was right shadow, will become contrary shadow, and the contrary. Note one thing further, that if your instrument be a Sextance, or a Circle, and you cannot have all the quadrat, as on a quadrant: you may then move the pin to the hole at the other end of the horizontal line: and you shall see that defect to be supplied. Note lastly, that by heights, we speak only of perpendicular or upright heights; and in distances, only of levels, or horizontals. PROB. 5. How to find unaccessable heights by the quadrat at two Observations. If the place which is to be measured cannot be approached unto, then work thus, to find both height and distance, first make choice of a place where looking up I find the thread to fall on 50 in the quadrat, than the distance will be equal to the height. Then make a mark at that Station, and go directly backward in a right line, with the former distance: and make choice of a second Station, where the thread may fall on 25 parts of right shadow, than this second Station is double to the height, and also to the distance, departed from the first Station: and the half therefore is the height, and first distance. But if it be so, you cannot come to take such a height as 50 and 25, then take as you may, as suppose one be at 25, and the other at 20, and suppose the height to be 100 I find that, As 25 the parts cut, are to 50 the side of the quadrat, so is 100 the supposed height, unto 200 the distance. And as 20 the second Station, to 50 the side of the quadrat; so is 100 the supposed height, unto 250 the second distance; wherefore the difference between the Stations should seem to be 50, then if in measuring you find it to be either more or less, than this proportion doth hold as from the supposed difference, to the measured difference, so from the supposed height, to the true height, and from the supposed distance, to the true distance. And now suppose the difference between the two Stations were found to be 30, by measuring, Then as 50 the supposed difference, to 30 the true difference, so is 100 the supposed height, to 60 the true height; And 200 the supposed distance to 120 the true; and 250 at the second Station, unto 150 the distance; the like reason holdeth in all other examples of this kind, and if an Index with sights were fitted to the Centre, it might serve for all other horizontal distances by the same reason. The Use of the Almanac. PROB. 6. Having the Day of the Week to find the Day of the Month for ever. First find what day of the Week the first of January is on; which is thus done: First find the Dominical Letter for the last Leap-year, set down in the Almanac: the next letter is for the next year following, and so till you come to the year you look for: And note, every Leap-year hath two Dominical letters, viz. the next before it, till the 24 of February, and that over it for the remainder of the year: Having found it, reckon from (A) either backwards or forwards, (always calling (A) Sunday) you shall find what day is the first of January. Example. For the year 1656 (F) is the Dominical Letter; therefore say (A) Sunday, (G) Monday, (F) Tuesday, and that is the first of January; and then make use of that thus: On the first Tuesday in the beginning of February, I would know the day of the Month? Among the Months look for 12, which is for February, reckoning from March, (which is always the first Month) and right under ●● you have 5, for the fifth day, being the first Tuesday in February, and 12, 19, 26 for the other Tuesdays in February: But now for the other Months after March, you must say Wednesday, the reason is, because February hath 29 days, and the Leap-year two Dominical Letters, viz. F. and E. then reckon from E to A, and it falls on Wednesday, which use thus in the year 1656, and all other Leapyears: As, in the beginning of August on Thursday, what day of the Month is it? August is the sixth Month, look for 6 among the Months, and right under it you have 6, which is Wednesday, therefore 7 is Thursday, and the first Thursday in August. But now for 1657. I find that Thursday is the first of January; saying thus, (A) Sunday, (B) Saturday, (C) Friday, (D) Thursday: And so it is all the year long, in all the Months; for having found the Month, all the days right under are Thursdays, and then reckon onwards, or backwards for any other of the Weekdays, and you have your desire, for any yearpast, present, or to come. PROB. 7. To find the Epact, and by that the Moon's age any day of the Month. On the Leap-year you have it set down in the Almanac for the next year; add 11. and you have your desire. And for the next year add 11 to that, and so to the next leapyear: But if by so adding it exceed 30, then take away 30, and the remain is the Epact. Having the Epact, add to it the day of the Month, and the number of the Month from March also, (including both the Months) and if they come not to 30, that is the Moon's age; but if they exceed 30, and the Month hath 31 days, then Subtract 30, and the remain is the age, but if the month have but 30 days, then subtract but 29, and the remainder is the age of the Moon required. Example. In July, 1656. on the 20 day the Epact is 14. then 14.20 and 5 added, is 39 from which take 30, rest 9 days old on the 20 of July, 1656. the Moons age sought for. PROB. 8. To find the hour of the day. Having found the day of the month by the Almanac, you must find the mark, or the space between two marks in the Calendar, representing that day; which do thus: Look for the first letter, or name of the month in the Calendar, according to the time of the year, then reckon from thence to the day you are in, either by 5, 10, 15, 20, 25, 30, 31, if the parts are so divided, as in small Instruments they cannot well be more; but if you have single days, every fifth and tenth is known from the rest by a longer stroke, and the last day by the longest stroke: Well, having found the day, or the place between two strokes representing it, lay a thread from the centre over that day, (or for want of a thread, stick a ●in in the centre, and cause the shadow to fall upon the day) and then observe on which, or between which of the 25 or 19 lines the thread cuts the 12 of clock line, for on that line must you look for the hour all that day: Before I come to example, I shall hint a plain word of the reason of this, which I find some to marvel at; The hour of the day in this, and in most Instrumental-dials', is given by the Sun's height; now all men know the Sun is not so high in Winter as in Summer, therefore the Summer hour lines will not serve the Winter; and also all men know they lengthen by degrees gradually, therefore the Winter and Summer 12, and consequently the rest of the hour lines, run sloping upwards and downwards, as the days lengthen or shorten. This being premised and considered, an easier Dial (all things considered) cannot be had. Now for an Example or two: Having found out the parallel of Declination, (for so is it called) if there be 25 lines, (or of the Suns rising, if there be but 19) you may easily know it by the name at the end of it, or by being a prick-line, or the next to, or the 2 next to a prick line, etc. hang or hold the Dial up, as was taught in the 1 Problem, and you shall have the exact hour of the day, among the Summer or Winter hours, according to the time of the year. Example. On the 2 of Aug. 1656. I look for (A) in the lower line of the months because the days shorten, and laying a string (or causing a shadow to fall) from the centre upon the 2 of August, which (if it hath not a particular stroke for it) is a little beyond the long stroke by the (A) and toward the (S), and I observe the thread to cut upon the line of Declination, called 15, and also it is a prick line: (in one of 25 lines (but almost midway between the first, beyond a prick line) and may be called the line of the Suns rising, at 4. and 41 min.) than I hold up my Dial, and find at 8 a clock the shadow to cross the 8 of clock line; just in the prick line, and at the same instant, the Sun's altitude is 30.15, and the quadrat is 29, and the line of shadows is 1. and 7 tenths, (that is the shadow of a yard (or any thing) held upright, is the length of the yard, and 7 tenths more of another length or yard) and note, that at 4 a clock the same day the shadow will fall in the same place exactly, as was hinted before; for equal hours from 12. the Sun hath the like altitude at all times of the year; and if it is morning, the height increases; if afternoon, than it decreaseth, so that two observations will resolve the question. But note, First for the months of June and Decemb. where the days are close together, the reason is, because the days at that time lengthen or shorten but a little; so must their spaces be on the instrument; if you should miss 3 or 4. days there, it makes no sensible error, take near as you can, and it sufficeth. Also note the hours of 11 and 12 are near together, therefore you must be so much the more cautious in observing to hold the Dial well, and to look just on, or between the parallel of declination or rising, and at 12 of the clock you may look in the Calendar for the day of the month, for just on that day will the shadow be at 12 of the clock, and short of it (increasing) before, (but decreasing) after 12. Note also on the 10 of March, and 13 of September, you must observe in the upper line; but on the 11 of June, and 11 of December on the lowest line, as the rules rehearsed make manifest. Lastly, if you meet with a Dial that hath the Calendar of Months on the backside, than it is but laying a thread over the day, and on the line of Declination, the thread cuts the correspondent number of Declination, as before; also the rising, and true place, and amplitude, as I hinted before; Then having the number, look for the line on the other side that shall have the same number, and proceed as before. Thus much shall suffice for the Dial particular for one latitude. The use of the other line to make it General, as also of a Joynt-rule to find the hour and azimuth, I shall refer you to the Book of the Joynt-rule, a book of this volume, fit to be bound up with it, being a very useful piece for Dialling, Geometry, Astronomy, and Navigation, and many other Mathematical Conclusions, and a portable universal Sea-Instrument as any whatsoever extant. CHAP. III. The Description of a Universal Dial, for all Latitudes, from 0 to 66. 30, of North or South Latitude. 1. First the Dial itself is an oblong, made of Box, Brass, or Silver, or the like, and at the shortest side it hath two sights, either of itself, or fitted into it: parallel to one of the shortest sides. 2. It hath a Bracheolum (with a Thread, Bede, and Plummet fastened to it:) that is 3 pieces of Brass so fitted together, that being pinned on the middle, will reach to any of the lines of Latitude, and it may be cut away after the work is on, to a very comely Form, or left Square, as shall best please the Fancy. 3. Thirdly for the lines, on the Dial, consider first the centre on the 6 of Clock line, where the tangents of Latitude begin, and pass on to 66.30, being strait parallel lines, drawn cross the oblong to every single Degree of Latitude: and you have them numbered with 10.20.30 40.50.60.65. at both ends of those lines. 4. Then you have from the Centre aforesaid, long straight sloping lines, drawn to every 5 or 10 Degr. of the signs, and on that end next the sights, on the middle line you have ♈ and 🝞 from thence toward the left hand, you have 10.20. ♉ and ♍: and then onwards the same way still, 10.20. ♊ and ♌: then 10.20. ♋ on the other side, to the right hand you have 10.20. ♓ and ♍ and 10. 20. ♒ and ♐ and 10.20. ♑. In all 12 signs. 5. Also adjoined to them, you have a Calendar of months and days, that knowing the day of the month, you have the sign answering thereto. 6. You have the same signs as was above portrayed on the right side, and 5 and 10 parts reciprocal to the former signs and parts on the top. 7. You have the hour lines, parallel to the length of the oblong, and numbered with 12. 1.2.3.4.5.6.7.8.9.10.11.12. on the upper end of them: and with 12. 11.10.9.8.7.6.5.4.3.2.1.12. at the lower end. 8. About the 2 sides opposite to the right upper corner, you have Degrees of Altitude, and Declination to find the Latitude, the use of which followeth, with as much brevity, and plainness as may be. PROB. 1. To find the Latitude. Having the Sun's Declination, and his Meridian Altitude, to find the Latitude. When the Sun is just on the Meridian, observe his Altitude, and set it down: then find his Declination for that day, and consider whether it be North or South, for if it be North Declination, you must subtract it from it; if South you must add it to the Meridian Altitude found, and the Sum, or remainder, shall be the comment of the Latitude sought for. Example. I am on the first of August, in a place where the noon Altitude is 50 the Sun's Declination the same day is 15.18. North, which taken out of 50. there remains 34.40, whose compliment to 90 is 55.18. the Latitude sought. The Degrees have the signs added to them, so that when you have the Sun's place by the help of the Calendar above, look for the like Sun's place in the Degrees, and just against it, you have his Declination: and when you have the Latitude by this, or any other means, you may find the hour of the day in this manner. PROB. 2. To find the Hour of the Day. Having the day of the Month, by the help of the Almanac as before; Find the same day in the Calendar of Months and days, and right against it, you have the Sun's place in the Zodiac, or 12 signs: then bring the end of the Bracheolum to the same sign, or part of the sign in your respective Latitude, and bring the Bead on the string, to the same sign and Degree, among the signs on the right side, then is it rectified for observation; then hold it up till you see the Sunbeams pierce through both holes of the sights, the thread playing easily by the side, the same time the Bead will fall upon the hour of the day, if it be the forenoon reckon among the forenoon hours, but if it be afternoon, among the afternoon hours, for the same as was 9 in the morning, is 3 in the afternoon, and so of the rest. On the 10 of April, in the Latitude of 51.30, at 8 a Clock, I would find the hour of the day by this Instrument. First, I look for the tenth of April, in the Calendar, and right against it I find ♉, that is, the Sun is their entering ♉, then I come down in that line, till I come to the Latitude of 51.30: and there I set the hole, in the end of the Bracheolum, where the thread is fastened; then that being fixed there, I bring the Bead on the string, to the same sign, ♉, on the right hand; and than it is rectified for that day and Latitude: then hold it up as before, and you shall find the Bead to fall on the line called 4 and 8, that is 8 in the morning, and 4 in the afternoon. As before said. PROB. 3. To find the time of the Sunsrifing, or Setting. Rectify the iustrument as before, and make the string lie parallel to the hour-lines, and the bead will show you the time of the Suns rising, reckoning in the Morning hours, and his setting, reckoning in the afternoon hours; and by them you may have the length of the day and night; for if you reckon backward to 10 at night, you have the length of the night; the other way gives you the length of the day, being doubled. Example. On the 10 of April, the Index or Bracheolum rectified, and the Bead set right, and the string drawn strait and parallel to the hours, the Bead falls on 5 for the Suns rising, & 7 for his setting: then from 7 to 12 at night is 5 hours, which doubled is 10 hours, the length of the night; and from 5 to 12 at noon is 7, which doubled is 14 hours, the length of the day. CHAP. IU. The Description and Use of the NOCTURNAL. There are two places in the Nocturnal, or more commonly it is one plate of brass on the backside of another Instrument; for the one hath onnly 34 hours, and each hour is divided into quarters, or more parts, as the largeness of the plate will admit: This Nocturnal hath the hour divided into 12 parts, that is, into 5 minutes a piece, and when the Nocturnal is added to another instrument, these hours and parts are set on that, and this one part of it. The other part of it is a Rundle, or round plate of Brass, on which is described, first and next the Edge, 365 divisions, representing the days of the year, every month having the name, at the beginning of it, or the first letter of the name, viz. I. F. M. A. M. I. I. A. S.O.N.D. and a longer stroke, at the last day, than any of the rest; and figures at the tenth and twentieth day, for the more ready finding of it: next within, you have the 360 Degrees of a Circle, to find the right ascension of any star, and numbered with 10. 20.30.40.50. to 360. beginning at the tenth of March, and proceeding onwards as aforesaid. 3. Then you have a Scale of Declinations, numbered both ways, from the Equator, and Pole, viz. 10. and 80, 20. & 70, 30. & 60.40. & 50.50 & 40, by which you may find the Declination, or distance from the Pole of any star. 4. You have 5 Constellations of those Stars next the North-pole, and the rest of the Stars of other constellations, as come within that compass. The names of which constellations are, the great Bear, the little Bear, Cephus, Cassiopeia and the Dragon, and part of Perseus, and Auriga. 5. You have a string lying cross the Diameter, from 12 to 12, for a Meridian. The uses follow in order. But for their sakes that may be willing to know more Stars, than are here set down, I have provided a larger plate, all the noted Stars from the Northpole to the Aequinoctial, whose use is the same with this, therefore I shall speak no more as to the use of that then this: only there is more variety, and it is a great help to the knowing of the Stars, for which cause it way by me chief composed. First then for the use, the most hard and difficult thing is to know the Stars one from another: The best means to know which next to a Tutor, is to compare them you do know in the Rundle with them in the heavens, and then one with another, as thus: In a night when the Stars may be seen, and it is best for to learn when there is not so many Stars seen, (as there is in some clear frosty nights) for then only those in the Nocturnal will be seen: Look toward the North part of the heavens, and you shall see 7 fair Stars standing as you see the 7 in the great Bear, beginning at the tail of the Bear; and the two last being bright Stars; from which two if you conceive a line to be drawn, it will cut the Polestar; and when you find the Polestar, and them two, they will help you to find all the rest with diligent observation. Another way to find the North-star, is to hang a plummet on the Centre, and make it play on the degree of the latitude of the place, and then lift it up toward the North part of the heavens, till you see a bright star, so bright, as one of the 7 in the great Bear, for that is the Polestar. Note that it is not the very Pole, as you may see by the Nocturnal, but it is the nearest the Pole; this being known, and the great Bear, which almost every Countryman knows, by the name of Charles-wane; with help of the string lying across, you may straightway look East or West North or South, or between, for any of the Stars set down in the Nocturnal, for the string you must conceive to lie North and South always: By these or the like means you may come to know all the Stars in the Nocturnal. And if you attain to know them, with the help of the other paper, you may know all in the half-Hemisphere that is between the Pole and Equinoctial. Secondly, to find when any star comes to the Meridian. The Meridian is a line or arch of a Circle, conceived to be drawn through (or rather by the star in the tail of the little Bear, which is) the Pole Star, right over your head; and to make you understand this the better, Hang a line with a weight at the end of it, out of a window that looks to the North, a good way from the house, or on a tree, or corner of a house, and then go to and fro till you see the line cut by the North-star, how much and on which side the Nocturnal will show you, and that line, then is the meridian-line; and than what star soever is under the line and the North-star, or Pole, (for when you use the stars of Cassiopeia, the star is the true Pole, as by the Meridian-line on the Nocturnal you may plainly see) that star I say is exactly on the Meridian, be it above or below the Pole; for so it be in that direct line with the Pole, it matters not, so you have the hours divided round about; but you may be able in a little practice to guests, without the Plumb-line, yet it will marvellously rectify your judgement, till you be more ready at it. These two being known, all the rest is very easy as may be. 3. The day of the Month being given, and a Star on the Meridian, to find the hour of the night. Suppose on the 1 of August, I see by the means beforesaid, the Star in the tip of the great Bear's tail, to be in the Meridian, then bring that Star just under the string, and look for the 1 of August, and right against it you have 4 a Clock, and 8 minutes in the morning: Or else set the day of the month to 12, and then keep it fixed all that night, than the thread laid over the Star that is in the meridian, shall at the same time in the hours show the hour, counting backwards. Another Example. Suppose on the same night the star in the nose of the Bear, were in the Meridian, then bring that star under the string, and the 1 of August will show 10.37, that is 37 minutes past 10 at night: the like is for any other. 4. To find the right ascension of a star, bring the star under the thread, and the thread showeth his right ascension in the degrees, so you will find the right ascension of the star in the great Bear's tail to be 203. 5. To find the declination of a star, bring the star under the line, then prick a pin through the string, and just in the midst of the star, then keep the pin there; and bring the Scale of declination right under the line, and pin: and the pin's point showeth you his declination, or distance from the Pole; by these two last, you may add any star, whose right ascension, and declination you know; and so put in all, as may be seen in this compass, at any time or by the right ascension only (being sufficient for this purpose, to wit the hour of the night,) you may add the Bull's eye, little Dog, 7 Stars, Orion, or any other chief principal fised star: and make use of it, to find the hour of the night withal. 6. The day of the month, and hour being given, to know what star is on or near the meridian. Set the day of the month to the hour of the night, and the stars that are under the string, are all on the meridian: and if there be none just, you shall see what half, or 3, or 4 part, between 2 is on the meridian, and this is an excellent way to help you to know the stars. Also note, That to use this Paper piece pasted on a Board, and not to turn about, do thus; Lay a thread on the Star you find to be in the Meridian, then with a pair of Compasses measure from the thread to the next 12, the same extent laid the same way, in the line of months and hours, shall reach from the day of the month to the hour required. FINIS. The Use of the LINE of NUMBERS ON A SLIDING (or GLASIERS) RULE, In Arithmetic & Geometry. AS ALSO, A most Excellent contrivance of the Line of Numbers, for the Measuring of Timber, either Round or Square, being the most easy, speedy, and exact, as ever was used. WHEREBY At one setting to the length, all ordinary pieces of Timber, from one Inch to 100 Foot, is with a glance of the eye resolved, without Pen or Compasses. First drawn by Mr. White, and since much enlarged, and made easy, and useful, by John Brown. London, Printed in the Year, 1656. The Use of the LINE OF NUMBERS ON A SLIDING-RULE, For the measuring of Superficial or Solid-measures. CHAP. 1. A Sliding-rule is only two Rules, or Rule-pieces fitted together, with a Brassesocket at each end, that they slip not out of the grove; and the Line of Numbers thereon, is cut across the moving Joint on each piece, the same divisions on both sides: only the placing of the lines differ, for on one side of the Rule you have 1 set at the beginning, and 10 at the end on each piece; but on the other side 1 is set in the middle, and the rest of the figures answerably both ways; on purpose to make it large, and to take in all numbers: and the reading of this is the very same with the other; for if you pull out the Rule, and set 10 at the end, right against 1 at the beginning, then on both pieces, you have the former Line of Numbers completely; therefore I shall say nothing as to description, or reading of it, but come straight to the use. On the edges of the Rule is usually set Foot-measure, being the Foot or 12 Inches parted into 100 parts; and on the flat sides next to the Foot-measure, Inches in 8 parts; and on the other flat edges on the other side, the Line of Board-measure, and sometime Timber-measure, whose use is showed in the first Chapter of the Book; but note if the Rule be a just Foot when it is shut, as Glasiers commonly have it, than the Inches are set alike on both sides, and the Foot-measure alike on both edges: and being pulled out as far as the brasses will suffer, it wants about one Inch of two Foot; but if you would have it to be two Foot just when pulled out, as it is made for Carpenters use, than the Inches on one side, and Foot-measure on the same reciprocal edge, must be figured otherwise, as 13. 14. 15. 16. 17. etc. to 25. Inches, and the Foot-measure with 110. near the end 120, 130, 140, 150, etc. with 210 at the very end, showing the measure from end to end, being drawn out to any distance; as is very easy to conceive of, and need no example to illustrate it withal. Note also in using the Line of Numbers, that that side or part of it, on which you find the first part or term in the question, shall always call the first side, and then the other must needs be the second, that the Rules and Examples may be shortened and made easy. 1. Multiplication by the Sliding-rule. Set the 1 on any side, (which being found, I call the first side) to the Multiplicator on the other (or second) side; then seek the Multiplicand on the first side, where 1 was: and right against it on the second, is the Product required. Example. If I would Multiply 25 by 28, set 1 on the first side, 25 on the second, then just against 28 on the first side, on the second is 700: for the right naming the last figure and the true number of figures, you have a Rule in the 2. Chapter and 2. Problem of the Carpenters-Rule, As in page 28. 2. Division by the Rule. Set the Divisor found always on the first side, to 1 on the second side, then right against the Dividend found out on the first side, on the second is the Quotient required. Example. If I Divide 156 by 12, the Quotient is 13, note to find how many figures shall be in the Quotient, do thus, if the two first figures of the Divisor, be greater than the two first figures of the Dividend, than the Quotient hath so many places or figures as there is more in the Dividend then in the Divisor; but if it be less, that is to say, the Dividends two first figures greater than the Divisors, than the Quotient shall have one place or figure more: then the Dividend exceeds the Divisor. Example. 2964 Divided by 39 makes a Quotient 76, of two figures, but if you Divide the same number by 18, you shall have the figures in the Quotient, viz. 164. and 12 remaining, or by the Rule two third parts of one more for the reason abovesaid, the two first figures of the Dividend being greater than the Divisor, it must have one place more than the difference of the number of figures, in the Multiplicator and Multiplicand. 3. The Rule of 3. direct. Set the first term of the question sought out on the first side, to the second term of the question on the second (or other) side: then right against the third term, found out on the first side, on the second side is the fourth proportional term required. Example. If 2 Yards of cloth cost 8 s. what cost 11 yard's ½? the answer is 46 s. for if you set 2 on any one side, to 8 on the other, then look for 11½ on the first side where 2 was, and right against it on the second you shall find 46, the number required. Note that all your Fractions on the Line of Numbers are Decimal Fractions, and to work them, you must reduce your proper Fractions to them, which for ordinary Fractions you may do it by Inches and Foot-measure, but this general Rule by the numbers will reduce any kind whatsoever, as thus: Suppose I would have the Decimal fraction of 9 Foot 7 Inches ¾, first note that 9 are Integers, for the rest, say thus, as 48 the number of Quarters in (12 Inches or) one Foot, is to 1000; so is 31 the number of Quarters in 7 Inches 3 Quarters, to 645 the Decimal Fraction required, for 9645 is equal to 9 Foot 7 Inches ¾, and so for any other whatsoever. 4. To work the Rule of 3. reverse. Set the first term sought out on the first side, to the second being of the same denomination on the second line or side, then seek the third term on the second side, and on the first you shall have the answer required. Example. If 48 men perform a piece of work in 24 hours, how many men may there be to do the like in 4 hours, set 24 on the first side, to 4 on the second, then right against 48 found out on the second, on the first is 288 the number of men required. 5. To work the double Rule of 3 direct. To perform this, you must have two working: As thus, for an Example. If the increase of 3 Bushels of wheat in one year be 36 Bushels, what shall the increase of 8 Bushels be for 7 years? First, set 3 on the first side, to 36 on the second, then against 8 on the first, on the second you find 96, than set 1 on the first side to 96, then against 7 on the first side on the second you have 612, the increase in 7 years, the answer required. CHAP. II. To measure Board or Glass by the Sliding rule, the length and breadth being given. PROB. 1. The breadth given to find how much makes a Foot. If the breadth be given in Inches, than set 12 on the first side, to the Inches on the second: then right against 12 on the second, on the first is the number of Inches required. Example. At 6 Inches broad, set 12 to 6, then against 12 on the second, on the first you have 24. But if it be given in Foot-measure, then in stead of 12, use 1. and do in like manner as before. Example. At 0. 50 broad, set 1 to 0. 50, then right against the other 1, is 2.00 the answer required. But to find how much is in a foot long at any breadth, do thus: First for Foot-measure, just as the Rule stands even look for the breadth on one side, and the quantity in a foot is on the other side, but for Inches set 1 to 12. then right against the Inches broad is the feet and tenth in a Foot-long. Example. At 6 Inches broad is 50 or half a foot in a foot long. Again at 30 Inches broad is 2 foot and a half in a foot long. PROB. 2. The length and breadth given to find the content. First the breadth given in Inches, and the length in Feet and Inches: set 12 on the first side, to the breadth on the second; then right against the length on the first, on the second is the content required. Example. At 16 Inches broad, and 20 Foot long: Set 12 to 16, then right against 20, you have 26 Foot 7 10th. look for your 7 10th. on the Foot-measure, and right against it on the Inches, you have 8 Inches ¼ and ½ Quarter, the answer desired. But if the breadth be given in Foot-measure, than set 1 to the breadth; then right against the length on the first side, on the second you shall have the Content required. Example. At 1. 20 broad 20.00 Foot long: you shall find 24 Foot. For if you set 1 to 1. 20, then right against 20 Foot, you have on the second 24 as before. PROB. 3. The breadth given in Feet and Inches, and the length also in the same parts, to find the Content. Set 1 on the first side to the Feet and Inches brought to a Decimal Fraction, or as near as you can guests, (for 6 Inches is half, 3 Inches is one quarter, 9 Inches is the quarters, 4 Inches is one third, 8 Inches is two thirds, and 1 Inch is somewhat less than one tenth on Rule,) on the other or second side; then right against the length found on the first, on the second is the Content required. Example. At 3 Foot 3 Inches broad, and 9 Foot 9 Inches long: you shall have 31 Foot 8 Inches ½ near, the very same is for Foot-measure, (only much easier) because the divisions on the Line of Numbers, and on the Line of Foot-measure on the edge, do agree together. This being premised as to the using of it, you may apply all the former precepts and examples to this Rule as well as the other. CHAP. III. To measure Timber by the Sliding-rule. PROB. 1. To measure Timber by this Rule, is nothing else but to work the Double-Rule of Three. As for Example. At 8 Inches square, and 20 Foot long, I would know the Content. Set 12 if the side of the square be given in Inches, (or 1 if in Foot-measure) on the first side, to 8 the Inches square on the second: then right against 12 on the second side, on the first is 18, the fourth proportional part. Then for the second work, set 18 the fourth proportional last found to 8 the Inches square on the second, then right against 20 the length is, 9 the Content required. Or rather thus; Set 12 against 8, then right against 20 on the same side 12 was, is 13.5 near on then look for 13.5 fere on the first side, and right against it on the second, is 9 foot, the Content required. PROB. 2. To measure a piece that is not square. Set 12 if you use the Inches, (or 1 if you use Foot-measure) on the first side, to the Inches thick on the second; then right against the Inches broad on the first side, on the second is a fourth proportional: then in the second operation, set 12 on the first side, to the fourth proportional on the second, then right against the length on the first side, on the second is the Content required. Example. At 8 Inches thick, and 16 broad, and 20 Foot long, you shall find 18 Foot ferè. PROB. 3. The Square given, to find how much makes a Foot. Set the Inches square on the first side, to 12 on the second; then right against 12 on the first, on the second is a fourth proportional number: then in the second work, as the Inches square to the fourth proportional, so is 12 to the number of Inches required, to make a Foot of Timber. Example. At 6 Inches square, set 6 to 12, then against the other 12 is 24: then set 6 to 24, then right against 12, you shall have 48. the length in Inches required. After the same manner are other questions wrought, but the Compasses are easier, and more ready; therefore I shall say no more to this, but only refer you to the former rules in the third, fourth, and fifth Chapters. Only note that in those Sliding-rules made for Glasiers use, the one half of the Line of Numbers is on one side of the Rule, and the other on the other: and whatsoever leg, or piece of the rule, is the first on the one side, the same leg or piece is the first when the Rule is turned on the other side, which must well be observed; but note that for measuring of Timber, those that use it may have one side fitted for that, as I shall more plainly and fully show in the next Chapter, being the easiest, speediest, and neat way, that ever yet was used by any man, resolving any Contents, by having the length, and the diameter, circumference, or square given. A Table of the true Sise of Glasiers Quarries both long and square, calculated by J.B. Square Quarries 77.19 gr. Ranger Sides breadth length content in Feet. content in Inch. in. 100 i. 100 I. p. I. pts. F. parts Inc. p. 4 20- 4 30 5 36- 6 70 0.1250 1.50 3 76 3 84 4 80 6 00 0.1000 1.20 3 43. 3 51 4 38- 5 47- 0.0833 1.00 3 07. 3 13- 3 92- 4 90 0.0667 0.80 2 80 2 86. 3 57 4 47. 0.0555 0.666 2 66. 2 72 3 39- 4 24 0.5000 0.60 Long Quarries 67.22. Ranger Sides breadth length content content In pts. I pts. I. 100 I. pts. F. 100 I. pts. 4 09 4 41. 4.90 7.34 0.1250 1.50 3 65- 3.95- 4.38- 6.57 0.1000 1.20 3 34 3.61 4.00 6.00 0.0833 1.00 2 98. 3 23 3.58 5.37 0.0667 0.80 2 58 2 79- 3.10 4.90 0.0555 0.666 2 72- 2 94- 3.27 4.65 0.0500 0.60 Note that a prick after the 100 parts of an Inch, notes a quarter, and a stroke- a half of 100 part of an Inch; to make this Table work thus by the Line of Numbers. Divide the distance between the content of some known size, as square 10 s. or long 12 s. and the content of the inquired Size, into two equal parts, for that distance laid the right way, (increasing for a bigger, or decreasing for a less) from the sices of the known size, shall give the reciprocal sides of the inquired size. Example for square 12s. The half distance on the line of Numbers, between 1000 the content of square 10 s. and 0. 833 the content of square 12 shall reach from 6 the length of square 10 s. to 5 47- the length of square 12 s. and from 4. 80 the breadth of square 10 s. to 4 38- the breadth of square 12 s, and from 3 84- to 3.51 and from 3 76 to 3 43; and so for all the rest. CHAP. IU. The description of the Line of Numbers on a Sliding-rule, to measure Solid measure only, according to Mr. White's first contrivance, but much augmented by J. B. First, when the figures (on the Timber-side) stand right toward you, fit to read: then that half or piece next to your right-hand, I call the rightside, the other is of necessity the left. Secondly, the figures on the right side are, first at the lower end, (where the Brass is pined fast) either 3, or 4, or 5, it matters not much which, yet to have 3 there is best: then upwards 4. 5. 6. 7. 8. 9 10. 11, for so many Inches, than 1. 2. 3. 4. 5. 6. 7. 8. 9 10. 11. 12, under the brass at the top, for so many Feet: the divisions between to one Foot, are quarters of Inches the next above 1 Foot, are only whole Inches, as you may plainly see. Thirdly, at 1 Foot you have the word Square, at 1 Foot 1 Inch ½ is a Mark, and right against it is set TD, noting the true Diameter of a round solid body; at an Inch further is 12 set, which I call small 12, being in small figures. Again at 1 Foot 3 Inches better, is another Mark, and right against it the word Diameter, for the Diameter of a piece of Timber according to the usual English allowance. Then again at 3 Foot 6 Inches ½ near, is T R, for the true circumference of a round Cillender. Lastly, at 4 Foot is the word Round, noting the circumference according to the usual allowance, whose use followeth. Note also if you put on the Gage-points for Ale or Wine, with the mean Diameter, and length, you may Gauge any Wine or Beer-vessel, the Wine at 17 Inches ¼: the Ale or Beer at 18.95. Fourthly, the figures on the left side, are not much unlike the right, for 1 at the beginning is one Inch, and so it proceeds by quarters of Inches to one Foot; then by figures at the Feet, and the divisions all whole Inches to 10 Foot, than every whole Foot, and half and quarter, or 10th. to 100, or 140, or 150 Foot; and this I call the left-side, the other the right side; so that from one Inch at the lower end, to one Foot, every Inch hath a figure; from one Foot to 10 Foot, every Foot hath a figure, and from 10 Foot to a 100, every 10th. Foot only is figured. I have been very plain in explaining this, because I would avoid vain repetitions in the following uses, wherein you shall have first the most ordinary and easy questions, and then the more hard and critical, and less useful. The Uses follow. PROB. 1. A piece of Timber being not square, to make it square. Set the breadth on the left side, to the breadth on the right, then right against the Inch and quarters thick found on the left side, on the right is the Inches square required. Example. At 18 broad, and 6 thick, you shall find 10 Inch ⅜ the side of the square required. For if you set 6 inches against 6 Inches, on the right and left-side: then right against 18 Inches, or 1 Foot 6 Inches on the left, on the right you have 10 Inches 1 quarter and half a quarter: for the side of the square equal to 18 one way, and 6 the other way. PROB. 2. The side of the square given, to find how much makes a foot. For all pieces between 3 or 4 Inches, and 42 Inches square, which are the most usual: this is the best way, set the Inches or Feet, and Inches square, found out on the rightside, to one Foot on the left, then right against one Foot on the right, on the left is the Inches, or Feet, and Inches required, to make a Foot of Timber. Example. At 8 Inches square set 8 on the right, to 1 Foot on the left, then right against 1 Foot on the right, on the left is 2 Foot 3 Inches the length required. To find how much is in a foot long. Just as the Rule stands even, look for the Inches, the piece is square on the right and on the left is the Inches or feet and Inches required. Example. At 17 Inches square, there is 2 foot of Timber in one foot long, which if you multiply by the length, you shall have the true content. A very good way for large pieces and very exact. PROB. 3. The side of the square and length given to find the Content. For all pieces between one Inch or 8/12 part of a Foot, and 100 Foot, this is the easiest way. Set the word square or 1 Foot to the length on the left, then right against the Inches, or Feet and Inches square on the right, on the left you have the Content. Example. At 9 Inch square, and 20 Foot long: Set the (long stroke by the) word square, to 20 Foot on the left, then right against 9 Inches on the right side, on the left-side you have 11 Foot and a quarter the Content required. But if it be a very great piece, as above 100 Foot, then call one Foot on the left side 10 Foot, and 2 Foot 20. etc. then 10 shall be 100 and 100 a 1000, that will supply to 1500 Foot in a piece, but if you would go to a bigger piece, then call 2 Foot on the left 200, etc. then you shall have it to 150000 Foot in a piece, such as I never saw. But for all small pieces under 3 Inches square, and above 1 quarter of an Inch, do thus: Set 12 on the top (or the small 12 when it is most convenient to use) to the length on the left-side, then right against the inches (or 12 s. of one inch) squares sound on the rightside, on the left is the true Content required. Example. At 2 Inches (3 twelve or) 1 quarter square, and 10 Foot long, you shall find 4 Inches and a quarter, ferè. But note when you use the small 12, the answer is given in decimals of a Foot, therefore the top 12 is best. PROB. 4. The square of a small piece of Timber given, to find how much makes a Foot. For all pieces from 12 Inches to 1 Inch square, do thus: Set the Inches and (12 s. or) quarter's square, counting one Foot on the right side for one Inch, and 2 Foot for 2 Inches, etc. found out on the rightside to 100 on the left; then right against the upper or small 12 on the right, on the left is the length required, to make a Foot of Timber. Example. At 2 Inches ¼ square, you must have 28 Foot 4 Inches to make a Foot. PROB. 5. Under 1 Inch square to find the length of a Foot. Set 1 Foot 9 Inches, 6 Inches, or 3 Inches, found on the rightside, for one Inch ¾, ½, or ¼ of an Inch, against 10 on the left-side, counted for 100: then right against the small 12, you have the Feet in length required. Example. At 1 Inch square, you find 144 Feet, at ¾ square 256 Feet, at ½ an Inch square 576 Feet, at ¼ or an Inch square, you find 2034 Feet, in length to make one Foot of Timber. Or if you set the former numbers 12, 9, 6, 1, against 1 Inch on the left, then right against the upper 12, is a number, which multiplied by 12, is the number of Feet required. PROB. 6. A great piece above 3 Foot ½ square, to find the length of a Foot. Set the Feet and Inches on the right, to 100 on the left: then right against small 12, is the Inches and 12 s, or 12 s of a 12th. that goes to make a Foot. Example. At 4 Foot square, you have 9 12ths. or ¾ of an Inch to make a Foot of Timber: at 5 Foot square 5. 12ths. and 10. 12ths. of a 12th. to make a Foot. Thus you see the Rule as now contrived, resolves from 1 quarter square, to 12 Foot square, the Content or Quantity of a Foot of Timber in length at any squareness, without Pen or Compasses. CHAP. V For round Timber. PROB. 1. The number of Inches that a piece of timber is about, being given, to find how much makes a Foot. First, for all ordinary pieces, set one foot on the left, to the inches or feet, and inches above on the right, then right against TR for true measure, or round for the usual measure, is the feet, or feet and inches required, to make a foot of timber, at that circumference about. Example. At 4 inches about, 113 foot 2 inches is for true measure, but for the usual measure, 142 foot goes to make a foot of Timber. At 12 foot 3 inches about, 1 Inch is a true foot, but for the usual allowance, as the fourth part of a line girt about gives: it must be 1 inch ¼ long, to make a foot of Timber at that circumference. But for very large pieces, count 1 foot on the right for 12 foot, 2.24, etc. and set 1 foot on the left as before, then in the answer, 1 foot on the left is 1. 12th. of an inch, and 1 inch 1. 144th. of an inch. Example. At 144 foot about, 1. 144th part of an inch, is a foot of Timber. PROB. 2. For very small Wood to find a Foot in length. But for very small pieces of under 4 inches about, set 1 foot, 2 foot, etc. on the right, (counted for 1 inch, 2 inches, 3 inches, or 4 inches) to one foot on the left, then right against TR or round, you have a number, which Multiplied by 12 is the number of feet required. Example. At one inch round true measure, is 151 foot ferè, but for the usual allowance 196, which numbers Multiplied by 12, is the number of feet required, viz. 1809, and 2352. but note you must read the 196, and 151 right, as thus: 1 foot on the left is 12, 2 is 24, etc. so that 12 foot is 144, and our number by the same account is 151 near. To find how much is in a foot in length, Set round or TR to 1 foot on the left, then right against the inches, or feet and inches about 1 found on the right, on the left is the answer required. PROB. 3. The inches, or feet and inches about, and length given to find the content. Set the word round, or TR for the usual, or true measure, to the length on the left: then right against the inches about on the right, on the left is the content required. Example. At 2 foot 3 inches about, and 20 foot long, it is 6 foot 2 inches of the usual allowance, or 8 foot of true measure. But if it be a great tree, then set TR, or round to 1 called 10, or to 10 called 100, then is the content augmented to 1000 foot, as you did in the Rules for square timber. But if you would have it measure bigger still, then set the 4 inches or a TR set close by the brass on the rightside, to the length on the left, either as it is, or augmented, counting at last according: (than note one foot on the right is 12 foot, and 12 at the top is 144 foot:) then right against the feet about on the right, on the left is the content required. Example. A Brewer's Tun 3 foot long or deep, and 72 foot about, set the TR by the brass to 36 inches (which is thus counted) on the left-side: (1 inch is 10 inches, 2 is 20, 3 is 30, 3½ is 35, somewhat more is 36. so then 1 foot is 120 inches, or 10 foot,) then right against 6 times 12 foot on the right, (which is at 6 foot) on the left you have 12 30 foot, as near as the Rule will give it, which counting 6 foot to a barrel, is 205 barrels the content required. PROB. 4. To find the content of a very small piece. Set the word round, or TR to the length on the left, as in the 3. Propositiou of this Chapter: then right against the inches about on the right, (calling one foot one inch, and 6 inches ½ an inch,) on the left is the 12 s of one inch,) or 12 s. of a 12th. required. Example. At half an inch about, and 10 foot long, it is 2 12 s and a half of one 12th. of an inch, or two square inches and ½ true measure. Again, 2 inches ¾ about, and 10 foot long, is half an inch of true measure, 12 inches to a foot solid, or ½ a foot superficial of one inch thick. CHAP. VI To measure Timber, having the Diameter and the length given. PROB. 1. The diameter given in inches, to find the length of a foot. Set one foot on the left, to the inches diameter on the right: then right against TD for true diameter, or the word diameter for the usual allowance, (of a string girt about and doubled 4 times for the side of the square) you have the feet and inches required. Example. At 10 inches diameter, 1 foot 10 inches makes a foot. But for very great pieces, set one foot as before, but look for TD beyond the upper 12, and right against it on the left you have the 12 of one inch, or the 12 s. of a 12, that makes a foot. But for very small sticks, set 1. 2. or 3 foot on the right, (for 1. 2. or 3 inches) to 1 foot on the left, then right against TD or Diameter you have a number, which multiplied by 12, is the number of feet required to make a foot of timber. Example. At one inch diameter, you shall have 15 foot 3 inches and better, which multiplied by 12 is 183 foot 3 inches. Note that 1 on the left is reckoned 10 foot, and 2, 20 foot, as before in the same rule, for the circumference, and then note 1 inch is 10 inches. At any Diameter to find how much is in one foot long, do thus; Set Diana. or TD to 1 foot, then just against the inches or feet and inches Diameter found on the right, on the left is the answer. Example. At 2 Foot Diameter is 3 foot two inches in one foot in length which multiplied by the length gives the true content of any round piece and very exactly. PROB. 2. The diameter and length given to find the true content. For all ordinary pieces, set the word Diameter for the usual measure, or TD for true measure, always to the length on the left: then right against the inches, or feet and inches Diameter on the right, on the left is the content required. Example. At 5 inches Diameter, and 30 foot long, you shall find 4 foot ½ an inch true measure, for the content required But for very small pieces, set TD or round to the length, as before; then counting 1 foot on the right for 1 inch, and 6 inches for ½ an inch; on the left you shall have the answer or content required. But note as the rightside is diminished, so is the left, for one foot on the left is a 12th of 1 Inch of timber, whereof 12 makes a foot, or 1 long inch, a foot long, and 1 inch square, and every inch on the left is 1 square inch; thus at 2 foot long, and ½ an inch diameter, it is 4 □ inches ¾ in content. But for a great piece under a 1000 foot, set TD or diameter to 1. 2. or 3 foot, called 10. 20. or 30 foot: then right against the feet and inches diameter, you have a content augmented accordingly, as at 30 foot long, and 7 foot diameter, you have 1140 foot for the true content using TD. But for very great pieces, as Brewer's Tuns and the like, as in the 3 Proposition of the 5 Chapter, you cannot work this without Compasses, and thus it is done, take the distance from the top 12 to TD, then set one point in one foot on the right, then slide the rule till the other point shall reach to the inches deep or long, reckoned as before in round measure, viz. 1 inch for 10 inches, 2 for 20, etc. on the left: then counting 2 foot on the right for 24 foot, etc. right against it on the left, you shall have the content in feet. But this is much easier wrought thus. Set 3 foot on the right side the gage point for a Barrel, to 36 the depth in inches of a Tun, on the left, then right against 11 for 6 inches, the half of 23 foot is 53 in Barrels ¼ the fourth part of 213, the content according to 282 inches in a Gallon. Note, I count 3 foot on the left side for 30 inches, etc. Example. A Brewer's Tun near 23 foot diameter, and 3 foot deep, will hold 1230 foot, viz. 205 Barrels, sought. Thus much for the use of the Line of Numbers for Timber-measure. Note that in the way of measuring Brewer's Tuns, every 6 foot is a Beer-barrel, one foot 6 gallons, 2 inches 1 gallon, 144 square inches, or 1 inch a pottle, 72 square inches a quart 36 a pint, etc. Note that in large Taper timber, whether square or round, when it is measured by the usual way, that is, by the middle square or girt, or the 2 squares or girts put together, and the half counted for the equal square, or girt: I say a square of half the difference of the squares or girts, and one third part of the length is to be added to the former measure, as is proved in the circles of proportion pag. 50. As thus for Example. Suppose a Taper-piece be at one end 16 inches square, at the other 30 inches square, and 30 foot long: the square in the middle is like to be 23 inches, the content than is 110 foot: now half the difference of the two ends square, is 7 inches, and one third part of the length is 10 foot: a piece 7 inches square, and 10 foot long, is 3 foot 5 inches: which added is 113 foot 5 inches: the true content of that taper piece abovesaid. The general way of Gauging by this Rule is thus: Set the W. or the A. for Wine or Ale-measure, always to the length of the vessel found out on the left. Then right against the mean Diameter found out on the right side, on the left is the answer required. Example. At 30 Inches Diameter and 36 long, you shall find about 90 gallons ½ Ale-measure. The Gage-point for a Beer Barrel is near 3 foot, and the Ale-Barrel near 34 inches, which use thus: Set 3 to the depth of the Tun, then right against the Meridian is the content in Barrels. Example. Set 3 to 36 Inches, and then right against 5 foot Diana. is 10 Barrels of Beer-measure, a very good and speedy way. FINIS. The TABLE. Chap. I. The Description of the plain Carpenters-Rule. Page 1 Chap. II. To two Numbers given to find a third, fourth, fifth, and sixth, etc. Geometrically proportional. Prob. 1. p. 25 Chap. III. The use of the Line of Numbers in Superficial or Board-Measure; and the convenience of Decimal-measure. pag. 51 Chap. IV. The use of the Line of Numbers in measuring of Land by Perches and Acres. pag. 64 Chap. V The use of the Line of Numbers in measuring of Solid measure, such as Timber, Stone, or such like Solid Bodies. pag. 76 Chap. VI The use of the Line of Numbers in measuring of Cylinders as round Timber, Pillars, and Ship Masts are. pag. 84 Chap. VII. The use of the Line of Numbers in Gauging of Vessels. pag. 94 Chap. VIII. The use of the Line in Questions that concern Military Orders. pag. 137 Chap. IX. The use of the Line in Questions of Interest and Annuities. pag. 142 Chap. X. The application of the Line to domestic Affairs. pag. 148 Chap. XI. To measure any Superficies or Solid by Multiplication only, either by Foot-measure or Inches only. pag. 163 Chap. I. The use of the Line of Numbers on a Sliding-rule, for the measuring of Superficial or Solid measure. pag. 211 Chap. II. To measure Board or Glass by the Sliding-rule, the length and breadth being given. pag. 218 Chap. III. To measure Timber by the Sliding rule. pag. 222 Chap. IV. The description of the Line of Numbers on a Sliding-rule, to measure Solid measure only, according to Mr. White's first contrivance but much augmented by J.B. p. 228 Chap. V For round Timber. pag. 237 Chap. VI To measure Timber, having the diameter & length given p. 242