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-prick-line or the next to or the 2 next to a prick line c. 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 thred 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. then 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 Suns 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 then 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 neer together therefore you must be so much the more cautious in observing to hold the Dial wel and to look just on or between the parallel of declination or rising and at 12 of the clock you may look in the Kalender 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 Kalender of Months on the backside then it is but laying a thred over the day and on the line of Declination the thred 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 peice 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 it self is an oblong made of Box Brass or Silver or the like and at the shortest side it hath two sights either of it self or fitted into it parallel to one of the shortest sides 2. It hath a Bracheolum with a Thred Bead and Plummet fastned to it that is 3 pieces of Brass so fitted together that being pinn'd 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 straight parallel lines drawn cross the oblong to every single Degree of Latitude and you have them numbred with 10.20.30 40.50.60.65 at both ends of those lines 4. Then you have from the Centre aforesaid long streight 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 adjoyned to them you have a Kalender 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 pourtrayed 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 numbred 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 Suns 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 substract it from it if South you must adde 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 Suns Declination the same day is 15.18 North which taken out of 50. there remains 34.40 whose complement to 90 is 55.18 the Latitude sought The Degrees
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 shew you and that line then is the meridian-line and then 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 guess without the Plumb-line yet it will marvellously rectifie your judgment till you be more ready at it These two being known all the rest is very easie 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 Bears 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 then the thred laid over the Star that is in the meridian shall at the same time in the hours shew 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 shew 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 thred and the thread sheweth his right ascension in the degrees so you will find the right ascension of the star in the great Bears tail to be 203. 5. To find the declination of a star bring the star under the line then prick a pin thorough the string and just in the middest of the star then keep the pin there and bring the Scale of declination right under the line and pin and the pins point sheweth you his declination or distance from the Pole by these two last you may adde 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 Buls eye little Dog 7 Stars Orion or any other chief principal sixed 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 neer 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 peice pasted on a Board and not to turn about do thus Lay a thred on the Star you find to be in the Meridian then with a pair of Compasses measure from the thred 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 Vse of the LINE of NUMBERS ON A SLIDING or GLASIERS RULE In Arithmatique Geometry AS ALSO A most Excellent contrivance of the Line of Numbers for the Measuring of Timber either Round or Square being the most easie speedy and exact as ever was used WHEREBY At one setting to the length all ordinary peices 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 inlarged and made easie 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 onely 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 Joynt 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 streight 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 shewed 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 then 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 then the Inches on one side and Foot-measure on the same reciprocal edge must be figured otherwise as 13. 14. 15. 16. 17. c. to 25. Inches and the Foot-measure with 110. near the end 120 130 140 150 c. with 210 at the very end shewing the measure from end to end being drawn out to any
The Description and Vse OF THE CARPENTERS-RULE Together with the use of the LINE of NUMBERS Inscribed thereon In Arithmatick and Geometry And the Application thereof to the Measuring of Superficies and Solids Gaging of Vessels Military Orders Interest and Annuities with Tables of Reduction c. To which is added The Use of a portable Geometrical Sun-dial 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 neer Tower-hill 1667 To the Reader Courteous Reader whomsoever thou art I Shall intreat 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 Câ—nter's Line I write 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 oyl 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 Mechanick Men such as Carpenters Joyners Masons Bricklayers and the like which for the most part are ignorant of Arithmatick 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 my self as to style or manner of writing being like my self 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 Mechanick 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 Sea-faring men may light of it I have added in the conclusion as an Appendix the Vse of a perticular and Vniversal Sun-dial 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 bâ—evity 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 Vse 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 call'd a Carpenters Rule rather-then a Joyners 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 Guild-hall 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 numbred 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 begins at 8 and a half that is when the figures of the Timber-line stand upright to you then I say it begins at the left end at 8 and ½ and proceeds to 36 within an Inch and â…œ 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 neer 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 then 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 Sollid given and the manner of doing it were superfluous to speak of or once to mention being
not only easie 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 shew it in two or three examples in each measure that is Superficial or Sollid 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 peice 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 then I find 8 Inches in length to make a Foot superficial but if 36 Inches broad then 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 sheweth 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 peices under 6 inches broad As suppose 3 ¼ double 3 ¼ it makes 6 ½ then 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 then you may turn it over two or three times and then take that together and so say 2 4 6 8 10 c. 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 then 9 c. till you come to 30 and then 30 and a half 31 c. 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 Ciphers 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 Sollid 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 peice 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 neer 21 Inches 3 eights of an Inch. Again suppose it were 24 Inches square then 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 then 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 then 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 peice be not just square but broader at one side than the other then the usual way is to adde 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 Iop â—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 supplyed 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 peice 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 neer the beginning of timber measure which will be neer 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 Vse of the Line of Numbers commonly called Gunter's Line In Arithmetick and Geometry and Gaging of Vessels c. 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 express'd 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 chiefly confine my self to is that of Mr Gunter's being the most proper for to be inscribed on a Carpenters Rule for whose sakes I undertake this collection of the most useful convenient and proper applications to their uses in Arithmatick and Geometry Thus much for definition of what manner of Lines of Numbers there be and of what I intend chiefly to handle in this place The order of the divisions on this Line of numbers and commonly on most other is thus it begins with 1 and so proceeds with 2 3 4 5 6 7 8 9 and then 1 2 3 4 5 6 7 8 9 10 whose proper power or order of numeration is thus The first 1 doth signifie one tenth of any whole number or integer as one tenth of a Foot Yard Ell Perch or the like or the tenth of a penny shilling pound or the like either in weight or number or measure and so 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 then 10 then 1 at the beginning must signifie 1 integer and in the middle 10 integers and 10 at the end 100 integers But if you would have it to a figure more then the first 1 is ten the second a hundred the last 10 a thousand If you proceed further rhen the first 1 is a 100. the middle 1 a 1000. and the 10. at the end is 10000. which is as great a number as you can well discover on this or most ordinary lines of 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 express'd which in lines of ordinary length cannot be done and 8. of that supposed 10. is the precise point for 1728. the number propounded to be found and the like of any number whatsoever But if you were to find a fraction or broken number then you must consider that properly or absolutely the line doth express none but decimal fractions as thus 1 10 or 1 100 or 1 1000 and more neerer 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 applyed 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 c. and 1 in the middle 1 integer 2 two integers and 10 at the end must be ten integers But if one at the beginning be one integer then 1 in the middle must be 10 integers and 10 at the end 100 integers and all the intermediate figures 20 30 40 50 60 70 80 90 integers and every longest division between the figures 21 22 23 24 25 26 c. integers and the 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 imagin 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. PROBLEME 1. Two numbers being given to find a third Geometrically propertional unto them and to three a fourth and to four a fifth c. GEometrical proportion is when divers numbers being compar'd together differ among themselves increasing or decreasing after the rate or reason of these numbers 2.4 8.16.32 for as 2 is half 4. so is 8 half 16. and as this is continued so it may be also discontinued as 3.6.14.28 for though 3 is half 6 and 14 half 28. yet 6 is not half 14 not in proportion to it as 3 is to 6 there is also 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 moveable point will stay on the 3d proportional number required Also the same extent applyed the same way from the third will give you a fourth and from the fourth a fifth c. 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 c. Extend the Compasses upon the first part of the line of numbers from 2 to 4 this done if you apply the same extent upwards from 4 the moveable point will fall upon 8 the third 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 moveable point will stay upon 128 the seventh proportional and from 128 to 256 the
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 substract 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 saâ—â—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 Musicâ—â— 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 multiplyed together make 96 that divided by 20 the sum of 8 and 12 the Quotient is 4 80 which doubled is 9-6 10s 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 shew the two mean proportionals As for example let 8 and 27 be two extremes the two means will be found to be 12 and 18 which are the two means sought for 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 c. 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 then 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 neer 26 the root of 172800 is almost 56 although the point on the Rule representing all the square numbers is in one place yet by altering the unit it produceth various points and numbers for their 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 then the unit will be best placed at 1 in the middle of the Line and the Root the Square and Cube will all fall forward toward the end of the Line But if it fall on the last but 1 as it doth in 17280 then the unit may be placed at 1 in the beginning of the Line and the Cube in the second length or else the unit may be placed at 10 in the end of the Line and the Cube in the first part of the Line you may help your self as in the first Problem of the 2 Chapter But if the last prick fall under the last but two as in 172800 it doth then place the unit always at 10 in the end of the Line then the Root the Square and Cube will all fall backward and be found in the second part between the middle 1 and the end of the Line By these Rules it doth appear that the Cube root of 8 is 2 of 27 is 3 of 64 is 4 of 125 is 5 of 216 is 6 of 345 is 7 of 512 is 8 of 729 is 9 of 1000 is 10. As you may see by this following Table of Square and 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 inlarge to some more particular applications in measuring
Fractions or 8 parts of an Inch first say croswise as the longest prick line doth lead you to 3 times 4 is 12 for which 12 you must set down 1. 6 that is 1 long Inch and 6 Square Inches the reason is a piece 8 half quarters of an Inch broad and 12 Inches long is a long Inch or the twelfth part of a Foot superficial and if 8 be 12 Square Inches then 4 must needs be 6 Square Inches therefore in stead of 12 I set down 1. 6 as you may see in the example and in the least long Square of the Diagram or Scheme Then do likewise for the other long Square which is also multiplyed across as two times 5 is 10. that is as I said before 1. 3 as the Example and Scheme make manifest considering what I last said and it is marked by the 2. 00. But if this or the other had come to a greater number you must have Substracted 8 s. as oft as you could and set down the remainder in the place of Square Inches and the number of 8 s. in the place of Long Inches as here you see Then for the two shorter Long Squares next the corner say croswise again Three times 5 is 15 that is 1 7 because eight Half-quarters an Inch long do make one square Inch as well as eight Half-quarters a Foot long made one Long Inch Therefore I set 1 in the place of square Inches and 7 in the next place to the right hand and it is expressed in the 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 1-4 Then lastly for 5 times 4 as the short prick line sheweth you is 20 out of which 20 take the 8 s and set them down in the last place and the 4 remaining you may either neglect or set it down a place further for you cannot see it on the Rule 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 exprest then 472 are 13 from which take 8 and for it carry 1 on to the next place or as many times 1 as you find 8 and set down the 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 Sun-dial made for the Latitude of 51. 30. North or the like for any other Latitude 1. First the Dial it self 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 bent 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 streight line drawn perpendicular to the string and cuts the Center just in the middest 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 Center and concentrick 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 c. 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 concentrick to the former in quadrants and Sextances But in round Dials they are but as it were concentrickal on which are divided a Kalender 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 shortning 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 moneth 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 c. 5. Next to the Kalender you have 25 concentrick 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 express'd 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 shew 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 shew 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 concentrick Circle to the Tropique 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 Dials there may be put halfs and quarters and then there are 8 prick'd 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 Dials 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 Almanack whereby if you know the day of the month you may find the day of the week and the contrary Also in some Almanacks 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 Dials of this kind and it is this some Dials of this kind have the Kalender of months and days on the backside in 1 2 or 4 Circles and lines of the Suns declination rising true place and amplitude and the right ascension and the use is only thus Lay a thred on the day of the month and the Centre and it cuts or sheweth all the other in their respective lines It being so easie 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 handsom 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 fore-finger and stretch your 4 fingers streight 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 fingers 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 Suns Altitude Stick a pin the smaller the better in the Centre and hold it up as before and the shadow will shew on the limb the Suns Altitude required Example At 8 of the clock on the 11 of June in the morning I wonld know the Suns 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 thred 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 neer 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 then 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 then 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 where in the Dial square from the Centre to the horizontal line also you must have a thred and plummet then the use is thus Stick a pin in the Centre and thereon hang the plummet then 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 thred 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 thred 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 then 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 then 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 supplyed 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 thred to fall on 50 in the quadrat then 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 thred may fall on 25 parts of right shadow then 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 then 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 Vse of the Almanack 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 Almanack 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 â—â— 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 Leap-years 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 Moneth all the days right under are Thursdays and then reckon onwards or backwards for any other of the Week-days and you have your desire for any yearpast present or to come PROB. 7. To find the Epact and by that the Moons age any day of the Month. On the Leap-year you have it set down in the Almanack for the next year add 11. and you have your desire And for the next year adde 11 to that and so to the next leap-year 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 Moneths and if they come not to 30 that is the Moons age but if they exceed 30 and the Month hath 31 days then Substract 30 and the remain is the age but if the month have but 30 days then substract 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 Almanack you must find the mark or the space between two marks in the Kalender representing that day which do thus Look for the first letter or name of the month in the Kalender 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 thred from the centre over that day or for want of a thred stick a â—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 thred 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 Suns 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
distance as is very easie 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 shortned and made easie 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 then 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 then 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 cloath cost 8 s. what cost 11 yards 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 then 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 then 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 then 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 guess for 6 Inches is