distance in the degrees shall shew the Suns Azimuth required 6. But in winter you must do thus By the second Proposition of the ninth Chapter finde the Suns Amplitude for that day then take the altitude from the general Scale of altitudes and putting one point in colatitude lay the thread to the neerest distance then the neerest distance from the latitude must be added to the Suns Amplitude this distance so added must be set from the coaltitude and the thread laid to the highest distance and in the line of degrees it gives the Azimuth from south counting from the end of the rule or from the East or West counting from the head or 90 degrees Example At 15 degrees of declination and 10 altitude latitude 51. 32. the Azimuth is 49. 20. from the South or 40 degrees and 40 ' from East or West CHAP. X. To finde all the necessary quesita for any erect declining Sun-dial both particularly and general by the lines on the Dial side also by numbers sines and tangents artificial being Logarithms on a Rule 1. First a particular for the Substile COunt the plains declination on â—he Azimuth scale from 90 toward the end and thereunto lay the thread in the line of degrees it shews the distance of the substile from 12. Example At 10 degrees declination I find 7. 51. for the substile 2. For the height of the stile above the substile Take the Plaines Declination from 90 in the Azimuth line but counted from the South end between your compasses and measure it in the particular scale of altitudes and it shall give the height of the stile required Example At 30 declination is 32. 35. 3. For the inclination of Meridians Count the substile on the particuler scale of Altitudes and take that distance between your compasses measure this distance on the Azimuth line from 90 toward the end and counting that way it sheweth the inclination of Meridians required Example At 15 the substile the inclination of Meridians will be found to be 24. 36. 4. To finde the Angle of 6 from 12. Take the plaines declination from the particular scale of altitudes and lay it from 90 on the Azimuth scale and to the Compasses point lay the thread then on the line of degrees you have the complement of 6 from 12 counting from 60 toward the end Note this Rule as this line is drawn doth not give this Angle exactly neither will it be worth the while to delineate another line for this purpose But if it be required it may be done but I rather prefer this help the greatest error is about the space of 45 minutes of the first degree in the particular scale of altitudes so that if you conceive those 45 minutes to be divided as the particular scale of altitudes is like a natural sine and if your declination be 30 then take half the space of the 45 minutes less and that shall be the true distance to lay on the Azimuth line from 90 whereunto to lay the thread Example A plaine declining 30 degrees the angle will be found to be 32. 21. whose complement 57. 49. is the angle required 5. To perform the same generally by the general scale of altitudes and first for the stile Lay the thred to the complement of the latitude counted in the degrees from the head toward the end then the nightest distance from the complement of the plaines declination to the thread taken and measured on the general scale from the center shall be the stiles height required 6. To finde the inclination of Meridians Take the plaines declination from the general scale and fit it in the complement of the stiles elevation and lay the thread to the neerest distance and on the degrees it sheweth the inclination of Meridians required 7. For the substile Count in the inclination of Meridians on the degrees from 90 and thereto lay the thread then take the least distance from the latitudes complement to the thread set one foot of that distance in 90 and lay the thread to the neerest distance and in the degrees it shall shew the substile from 12 required 8. For the angle of 6 from 12. Take the side of the square or the measure of the parallel from 12 and fit it in the cosine of the latitude and lay the thread to the nighest distance then take out the nearest distance from the sine of the latitude to the thread then fit that over in the sine of 90 and to the nearest distance lay the thread then take the nearest distance from the sine of the plains declination to the thread and it shall reach on the parallel line or side of the square from the Horizon to 6 a clock line required Four Canons to work the same by the artificial sines tangents Inclination of Meridians As the Sine of the latitude To the Sine of 90 So the Tangent of the Declination To the Tangent of inclination of Meridians Stiles Elevation As the Sine of 90 To the Cosine of the Declination So the Cosine of the latitude To the Sine of the Stiles elevation Substile from 12. As the Sine of 90 To the Sine of the Declination So the Cotangent of Latitude To Tangent of the Substile from 12 For 6 and 12. As the Coâ—tangent of the Latitude To the Sine of 90 So is the sine of Declination To the Cotangent of 6 from 12. For the hours As the Sine of 90 To the Sine of the Stiles height So the Tangent of the hour from the proper Meridian To the Tangent of the hour from the Substile The way to work these Canons on the Sines and Tangents is generally thus As first for the inclination of Meridians set one point in the Sine of the latitude open the other to the Sine of 90 that extent applied the same way from the Tangent of the Plains declination will reach to the Tangent of the inclination of Meridians required CHAP. XI To draw a Horizontal Dyal to any latitude FIrst draw a streight line for 12 as the line A B then make a point in that line for a Center as at C then through the Center C raise a perpendicular to A B for the two six a clock hour-lines as the line D E then draw two occult lines parallel to A B as large as the Plain will give leave as D E and E G then fit C D in the Sine of the Latitude in the general Scale and lay the thread to the nighest distance then take the nearest distance from 90 to the thread and set it from D and E in the two occult lines to F and G and draw the line F and G parallel to the two sixes or make use of the Sines on the other side thus Fit A D or C D in the Sine of the latitude and take out the Sine of 90 and lay it as before from D and E then fit D F or E G in the Tangent of 45
the Amplitude from the east or west counting from 90. Example May the tenth it is 33. 37. CHAP. V. Having the Suns Declination or day of the moneth to finde the Azimuth at any Altitude required for that day FIrst finde the Suns Declination by the first Proposition of the fourth Chapter then take that out of the particular Scale of Altitudes or scale to 62 degrees then whatsoever the Altitude shall happen to be count the same on the degrees from 60 toward the end of the Rule according to the second maner of counting in the third Proposition of the third Chapter and thereunto lay the thred then the Compasses set to the Declination carry one point along the line of hours on the same side of the thread the Declination is that is to say if the day of the moneth or Declination be on the right side the Aequinoctial then carry the Compasses on the right side but if the Declination be on the South side that is toward the end counting from the tenth of March or Aries or Libra then carry the Compasses along the line of hours and Azimuths on the left side of the thâ—ead as all winâ—er time it will be and having set the Compasses to the least distance to the thread it shâ—ll stay at the Suns true Azimuth from the South required counting as the figures are numbred or from East or West counting from 90. Example 1. On the tenth of Iuly I desire to finde the Suns Azimuth at any Altitude first on that day I finde the Suns Declination to be 20. 45 which number count from the beginning of the particular Scale of Altitudes toward 62 and that distance take between your Compasses then are they set for all that day then supposing the Suns height to be ten degrees lay the thread on 10 counted from 60 toward the left end then carrying the Compasses on the right side of the thread because it is summer or north declination on the line of Azimuths it shall shew 110. 40 the Azimuth from the south required but if you count from 90 it is but 20. 40. from the east or west point northward according to the time of the day either morning or evening Example 2. Again on the 14. of November or the 6. of Ianuary when the Sun hath the same declination south-ward and the same Altitude to work this you must lay the Rule down on something then lay the thread on the Altitude counted from 60 toward the end as before and carrying the Compasses on the south-side of the Aequinoctial along the azimuth-Azimuth-line till the other point do but just touch the thâ—ead and it shall stay at 36. 45 the Azimuth from south required if it be morning it wants of coming to south if it be after-noon it is past the south Example 3. But if the Sun be in the Aequinoctial and have no declination then it is but laying the thread to the Altitude and in the line of Azimuths the thread shall shew the true Azimuth required As for instance at 00 degrees of altitude the Azimuth is 90 at 10 degrees it is 77. 15 at 20 degrees 62. 45 at 30 degrees high 43. 15 at 35 degrees high 28. 10 at 38 degrees 28 ' high it is just south as by practice may plainly appear But if the Suns altitude be above 45 then the degrees will go beyond the end of the Rule To supply this defect do thus Substract 45 out of the number you would have and double the remainder then lay the Rule down with some piece of the same thickness in a streight line with the moveable leg then take the distance from the tangent of the remainder doubled counted from 60 to the end of the Rule in the line next the edge to the Center lay that distance in the same streight line from the tangent doubled and that shall be the tangent of the Angle above 45 whereunto you must lay the thread for the finding the Azimuth when the Sun is above 45 degrees high CHAP. VI. To finde the hour of the Night by the Moon FIrst by the help of an Almanack get the true time of the New Moon then compute her true place at that time which is always the place of the Sun very nigh at the hour and minute of conjunction then compute how many days old the Moon is then by the line of Numbers say If 29 dayes 13 hours or on the line 29. 540 require 860 degrees or 12 signs what shall â—ny less number of days and part of a day require The answer will be The Moons true place at that age Having â—ound her true place then take her alâ—itude and lay the thred on the Moons place found and work as you did for the Sun and note what hour you finde then consider if it be New Moon the hour you finde is theÌ„true hour likewise in the Full but if it be before or after you must substract by the Line of Numbers thus If 29 days 540 parts require 24 hours what shall any number of days and parts require The answer is What you must take away from the Moons hour found to make the true hour of the Night which was required But for more plainness sake I will reduce these Operations to so many Propositions before I come to an Example PROP. 1. To finde the Moons Age. First it is most readily and exactly done by an Ephemerides such a one as you finde in Mr. Lilly's Almanâ—ck or as to her Age onely in any book or Sheet-Almanack but you may do it indifferently by the Epact thus by the Rules of the 152 page in the Appendix to the Carpenters Rule Adde the Epact the moneth and the day of the moneâ—h together and the sum if under 30 is the Moons age but if above consider if the moneth have 30 or 31 days then substract 29 or 30 out and the remainder is the Moons age in days Example August 2. 1660. Epact 28. Month 6. day 2. added makes 36. Now August or sixt moneth hath 31 days therefore 30 being taken away 6 days remains for the moons age required PROP. 2. To finde the Moons place By the Ephemerides aforesaid in Mr. Lilly's Almanack you have it ser down every day in the year but to finde it by the Rule do thus Count six days back from August 2. viz. to Iuly 27. there lay the thread and in the line of the Suns place you have the Moons place required being then near alike then in regard the Moon goes faster than the Sun that is to say in 29 days 13 hours 12 signs or 360 degrees in 3 days 1 sign 6 degr 34 min. 20 sec. in one day o signs 12 degr 11 min. 27 sec. in one hour 30 min. 29 sec. or half a minute adde the signs and degrees and minutes the Moon hath gone in so many days and hours if you know them together and the Sun shall be the Moons true place being added to what she had on
not the time in common hours but is thus found Adde the complement of the Suns Ascension and the stars right Ascension and the stars hour last found together and the Sun if less than 12 or the remain 12 being substracted shall be the time of his rising in common hours but for his setting adde the stars setting last found to the other numbers and the sum or difference shall be the setting Example For the Bulls-eye on the 23 of December it riseth at 2 in the afternoon and sets at 4. 46 in the morning 4. To finde the time of the southing of any star on the Rule or any other whose right ascension and declination is known Substract the Suns right ascension from the stars increased by 24 when you cannot do without and the remainder if less than 12 is the time required in the afternoon or night before 12 but if there remain more than 12 substract 12 and the residue is the time from mid-night to mid-day following Example Lyons-heart on the tenth of March the Suns Ascension is 0 2 ' Lyons-heart whole right asc is 9 50 ' Time of southing is 9 48 ' at night 5. To finde how long any Star will be above the Horizon Lay the thread to the star and in the hour-line it sheweth the ascensional difference counting from 90 then note if the star have North declination adde that to 6 hours and the sum is half the time if south substract it from 6 and the residue is half the time and the complement of each to 24 being doubled is the whole Nocturnal Arch under the Horizon Example For the Bulls-eye his Ascensional difference will be found to be one hour 23 minutes which added to 6 hours and doubled makes 14. 46 the Diurnal Ark of the Star and the residue from 24 is 9. 14. for the Nocturnal Ark or the time of its being under the Horizon CHAP. IX To perform the fore-going work in any latitude as rising amplitude ascensional difference latitude hour and azimuth wherein I shall give onely the rule and leave out the examples for brevity sake 1. FOr the rising and setting and ascensional difference being all one do thus Take the Suns declination out of the general Scale of Altitudes then set one foot of the Compasses in the colatitude on the same scale and with the other lay the thred to the nighest distance then the thred so laid take the nighest distance from the latitude to the thread with that distance set one foot in the Suns declination counted from 90 toward the center and the thread laid to the nearest distance shall in the degrees shew the ascensional difference required counting from 90 at the head toward the end of the Rule and if you reduce those degrees and minutes to time you have the rising and setting before and after 6 according to the declination and time of the year 2. To finde the Suns amplitude Take the Suns declination and setting one foot in the colatitude with the other lay the thread to the nearest distance and on the degrees it sheweth the Suns amplitude at rising or setting counting as beâ—ore from 90 to the left end of the Rule 3. Having amplitude and declination to finde the latitude Take the declination from the general scale and set one foot in the amplitude the thread laid to the nearest distance in the line of degrees it sheweth the complement of the latitude required or the converse 4. Having latitude Suns declination and altitude to find the height at 6 and then at any other time of the day and year Count the declination in the degrees from 90 toward the end thereto lay the thread the least distance from which to the latitude in the general Scale shall be the Suns height at 6 in the summer or his depression in the winter The Compasses standing at this distance take measure on the general Scale of altitudes from the beginning at the pin towards 90 keeping one point there open the other to the Suns altitude thus have you substracted the height at 6 out of the Suns altitude but in winter you must adde the depression at 6 which is all one at the same declination with his height at 6 in summer and that is done thus Put one point of the Compasses so set in the general Scale to the Suns Altitude then turn the other outwards toward 90 there keep it then open the Compasses to the beginning of the Scale then have you added it to the Suns altitude having this distance set one foot in the colatitude on the general Scale lay the thread to the nearest distance the thread so laid take the nearest distance from 90 to the thred then set one foot in the declination counted from 90 and on the degrees it sheweth the hour from 6 reckoning from the head or from 12 counting from the end of the Rule I shall make all more plain by making three Propositions of it thus Prop. 1. To finde the hour in the Aequinoctial Take the Altitude from the beginning of the general Scale of altitudes and set one foot in the colatitude the thread laid to the nearest distance with the other foot in the degrees shall shew the hour from 6 counting from 90 and allowing for every 15d 1 hour and 4 min for every degree Prop. 2. To finde it at just 6. Is before exprest by the converse of the first part of the fourth which I shall again repeat Prop. 3. To finde it at any time do thus Count the Suns declination in the degrees thereunto lay the thred the least distance to which from latitude in the general Scale shall be the Suns altitude at 6 which distance in summer you must substract from but in winter you must add to the Suns present altitude having that distance set one foot in the coaltitude with the other lay the thread to the neerest distance take again the neerest distance from 90 to the thread then set one foot in the Suns diclination counted from 90 and lay the thread to the neerest distance and in the degrees it shall shew the hour required Example At 10 declination north and 30 high latitude 51. 32 the hour is found to be 8. 25 counting 90 for 6 and so forward Again at 20 degrees of declination South and 10 degrees of altitude I finde the hour in the same latitude to be 17 minutes past 9. Having latitude delination and altitude to finde the Suns Azimuth Take the sine of the declination put one foot in the latitude the thread laid to the neerest distance in the degrees it sheweth the Suns height at due East or West which you must in summer substract from the Suns altitude as before on the general Scale of Altitudes with which distance put one foot in the colatitude and lay the thread to the neerest distance then take the neerest distance from the sine of the latitude fit that again in the colatitude and the thread laid to the heerest
degrees on the other side of the Rule and lay off 15 30 and 45 for every whole hour or every 3 degrees and 45 minutes for every quarter from D and E toward F and G for 7 8 9 and for 3 4 and 5 a clock hour points Lastly set C D or B E in the Tangent of 45 and lay the same points of 15 30 45 both wayes from B or 12 for 10 11 and 1 2 and to all those points draw lines for the true hour-lines required for laying down the Stiles height if you take the latitudes complement out of the Tangent-line as the Sector stood to prick the noon hours and set it on the line D F or E G from D or E downwards from D to H it will shew you where to draw C H for the Stile then to those lines set figures and plant the Dial Horizontal and the Stile perpendicular and right north and south and it shall shew when the sun shineth the true hour of the day Note well the figure following CHAP. XII To draw a Vertical Direct South or North Dyal FIrst draw a perpendicular line for 12 a clock then in that line at the upper end in the south plain and at the lower end in a north plain appoint a place for the center through which point cross it at right angles A Horizontall Diall A South Diall for 6 and 6 as you did in the Horizontal Plain as the lines A B and C D on each side 12 make two parallels as in the Horizontal then take A D the parallel and fit it in the sine of the latitudes complement and take out the sine of 90 and 90 and lay it in the parallels from D and C to E and F and draw the line E F then make D E and B E tangents of 45 and lay down the hours as you did in the horizontal and you shall have points whereby to draw the hour lines For the north you must turn the hours both ways for 4 5 8 and 7 in the morning and 4 5 7 and 8 at night the height of the stile must be the tangent of the complement of the of the latitude when the sector is set to lay off the hours from D as here it is laid down from C to G and draw the line A G for the stile For illustration sake note the figure CHAP. XII To draw an erect East or West Dial. FIrst by the fifth Proposition of the second Chapter draw a horizontal line as the line A B at the upper part of the plaine Then at one third part of the line A B from A the right end if it be an East plaine or from B the left end if it be a West Plain appoint the center C from which point C draw the Semicircle A E D and fit that radius in the sine of 30 degrees which in the Chords is 60 degrees then take out the sine of half the latitude and lay it from A to E and draw the line C E for six in the morning on the East or the contrary way for the West Then lay the sine of half the complement of the latitude from D to F and draw the line C F for the contingent or equinoctial line to which line you must draw another line parallel as far An East Diall A West diall assunder as the plaine will give leave then take the neerest distance from A to the six a clock line or more or less as you best fancy and fit it in the tangent of 45 degrees and prick down all the houres and quarters on both the equinoctial lines both ways from six and they shall be points whereby to draw the hoor lines by but for the two houres of 10 and 11 there is a lesser tangent beginning at 45 and proceeding to 75 which use thus fit the space from six to three in the little tangent of 45 and then and then lay of 60 in the little tangents from 6. to 10 and the tangent of 75 from 6 to 11 and the respective quarters also if you please so have you all the hourâ—s in the East or west Diall the distance from six to nine or from six to three in the West is the height of the stile in the East and West Diall and must stand in the six a clock line and parallel to the plaine CHAP. XIII To finde the declination of any Plain FOr the finding of the declination of a Plain the most usual and easie way is by a magnetical needle fitted according to Mr. Failes way in the index of a Declinatory or in a square box with the 90 degrees of a quadrant on the two sides or by a needle fitted on the index of a quadrant after all which ways you may have them at the Sign of the Sphere and Sun-Dial in the Minories made by Iohn Brown But the work may be very readily and exactly performed by the rule either by the Sun or needle in this manner following of which two ways that by the Sun is always the best and most exact and artificial and the other not to be used if I may advise but when the other failes by the Suns not shining or as a proof or confirmation of the other And first by the needle because the easiest For this purpose you must have a needle well touched with the Loadstone of about three or four inches long and fitted into a box somewhat broader then one of the legs of the sector with a lid to open and shut and on the inside of the lid may be drawn a South erect Dial and a wire to set the lid upright and a thread to be the Gnomon or stile to that Dial it will not be a miss also to extend the lines on the Horizontal part for the same thread is a stile for that also Also on the bottom let there be a rabbit or grove made to fit the leg of the rule or sector so as being pressed into it it may not fall off from the rule if your hand should shake or you cease to hold it there This being so fitted the uses follow in their order Put your box and needle on that leg of the rule that will be most fit for your purpose and also the north end of the needle toward the wall if it be a south wall and the contrary if a north as the playing of the needle will direct you better then the way how in a thousand words then open or close the Rule till the needle play right over the north and south-line in the bottom of the Box then the complement of the Angle that the Sector standeth at which may always be under 90 degrees is the declination of the Plain But if it happen to stand at any Angle above 90 then the quantity thereof above 90 is the declination of the wall To finde the quantity of the Angle the Sector stands at may be done two ways first by protraction by laying down the
Rule so set on a board and draw two lines by the legs of the Sector and finde the Angle by a line of Chords Secondly more speedily and artificially thus By the lines of Sines being drawn to 2. 4 or 6 degrees asunder The Sector so set take the parallel sine of 30 and 30 and measure it on the lateral sines from the center and it shall reach to the sine of half the Angle the line of sines stands at being more by 2 4 or 6 degrees then the sector stands at because it is drawn one two or three degrees from the inside Or else take the latteral sine of 30 from the center and keeping one foot fixed in 30 turn the other till it cross the line of sines on that line next the inside and counting from 90 it shall touch at the Angle the line of sines standsat being two degrees more then the Sector stands at the lines being drawn so will be as I conceive most convenien Take an Example I come to a south-east-wall and putting my box and needle on my Rule with the cross or north-north-end of the needle toward the wall and the Rule being applyed â—lat to the wall on the edge thereof on the evenest place thereof and held level so as the needle may play well with the head of the Rule toward your right hand as you shall finde it to be in an east-wall most convenient then I open or close the Rule till the needle play right over the north and south line in the bottom of the box then having got the Angle take off the box or if you put it on the other side that labor may be saved I take the parallel sine of 30 and measuring it from the center it reaches suppose to the sine of 20 then is the line of sines at an Angle of 40 degrees but the Sector at two degrees less viz. 38 degrees whose complement 52 is the declination then to consider which way minde thus First it is south because the sun being in the south shines on the wall Secondly consider the sun being in the east it shines also on the wal therefore it is east plain thus have you got the denomination which way and also the quantity how much that ways Or if you take the latteral sine of 30 from the center and turn the point of the Compasses from 30 towards 90 on the other leg you shall finde it to reach to the sine of 50 degrees whose complement counting from 90 is 40 or rather 38 for the reason before-said or else adde 2 to 50 and you have the angle required without complementing of it being the true declination sought for Thus by the needle you may get the declination of any wall which in cloudy weather may stand you in good stead or to examine an observation by the Sun as to the mis-counting or mistaking therein but for exactness the Sun is alwayes the best because the needle though never so good may be drawn aside by iron in the wall and also by some kinde of bricks therefore not to be too much trusted unto To finde a Declination by the Sun First open the Rule to an Angle of 60 degrees as you do to finde the hour of the day and put a pin in the hole and hang the thread and plummet on the pin also you must have another thread somewhat longer and grosser then that for the hour in a readiness for your use Then apply the head leg to the wall if the sun be coming on the Plain and hold the Rule horizontal or level then hold up the long thread till the shadow falls right over the pin or the center hole at the same instant the shadow shall shew on the degrees how much the sun wants of coming to be just against the Plain which I call the Meridian or Pole of the Plain which number you must write down thus as suppose it fell on 40 write down 40-00 want then as soon as may be take the Suns true altitude and write that down also with which you must finde the Suns Azimuth then substract the greater out of the less and the remainder is the declination required But for a general rule take this if the Sun do want of coming to the Meridian of the place and also want of coming to the Meridian of the Plain then you must alway substract the greater number out of the less whether it be forenoon or after-noon so likewise when the Sun is past the Meridian and past the Plain also But if the signes be unlike that is to say one past the Plain or Meridian and the other want either of the Plain or Meridian then you must add them together and the sum is the declination from the South Which rule for better tenaciousness sake take in this homely rime Signs both alike substraction doth require But unlike signs addition doth desire The further illustration by two or three Examples Suppose on the first of May in the forenoon I come and apply the Rule being opened to his Angle of 60 degrees to the wall viz. the head leg or the leg where the center is and holding up my thickest thread and plummet so as the shadow of it crosseth the center and at the same instant also on 60 degrees then I say the sun wants 60 deg of coming to the Meridian of the Plain at the same instant or as soon as possible I can I take the suns altitude as before is shewed and set that down which suppose it to be 20 degrees then by the rules before get the Suns azimuth for that day and altitude which in our example will be found to be 94 degrees from the south or Meridian then in regard the signs are both alike i. e. want if you substract one out of the other there remains 34 the declination required but for the right denomination which way either north or south toward either east or west observe this plain rule First if the Sun come to the Meridian or Pole of the Plain before it come to the Meridian or Pole of the place then it is always an East-plain but if the contrary it is a West-plain that is to say if the Sun come to the Meridian or Pole of the place before it comes to the Meridian or Pole of the Plain then it is a West-plain Also if the sum or remainder after addition or substraction be under 90 it is a South-plain but if it be above 90 it is a North-plain Also note that when the sum or remainder is above 90 then the complement to 180 is always the declination from the north toward either east or west So that according to these rules in our example it is 34 degrees South-east Again in the morning Iune 13. I apply my rule to the wall and I finde the Sun is past the Pole or Meridian of the Plain 10 degrees and the altitude at the same time 15 degrees the Azimuth at that altitude and
day in this latitude will be found to be 109 degrees want of south or pole of the place therefore unlike signs and to be added and they make up 119 degrees whose complement to 180 is 61 for 61 and 119 added make up 180 therefore this Plain declineth 61 degrees from the north toward the east Again the same day in the afternoon I finde the Azimuth past the south or meridian of the place 30 degrees and at the same time the Sun wants in coming to the meridian or pole of the Plain 10 degrees here by addition I finde the declination to be 40 degrees south-west Note what I have said in these three examples is general at all times but if it be a fair day and time and opportunity serve to come either just at 12 a clock when the Sun is the meridian or pole of the place or just when the Sun is in the meridian or pole of the Plain then your work is onely thus First if you come to observe at 12 then applying your rule to the wall and holding up the thread and plummet how much so ever the Sun wants or is past the pole of the Plain that is the declination if it be past it is east-wards if it wants it is south-west-wards if neither a just South Plain and then the poles or Meridians of place and Plain are the same But secondly if you come when the Sun is just in the pole of the Plain then whatsoever you finde the Suns Azimuth to be that is the declination if it wants of south it declineth East-wards if it be past it declineth West-wards Thus I have copiously and yet very briefly shewed you the most artificial way of getting the declination of any wall howsoever situated Note if the Sun be above 15 degrees wanting of the Meridian of the Plain your rule will prove defective in taking the Plains Meridian when the center leg is next the wall then you must turn the other leg to the wall and then you finde a supply for all angles to 45 degrees past the Plain But for the supply of the rest which is 45 degrees do thus open the rule till the great line of tangents the outside of the leg make a right angle for which on the head you may make a mark for the ready setting then making the inside of the leg at the end of 45 as a center the tangents on the other leg supply very largely the defect of the othersides Or if you set on the box and needle on the rule and open or shut the rule till the shadow of the thread shew just 12 then the Angle the Sector stands at is the complement of what the sum wants or is past the meridian or pole of the Plain CHAP. XIV To draw a vertical declining Plain to any declination FIrst draw a perpendicular-perpendicular-line for 12 as A B then design a point in that line for the center as C at the upper end if it be a South Plain or contrary if it be a North Plain then on that center describe an Arch of a Circle on that side of 12 which is contrary to the Plains declination as D E and in that Arch lay off from 12 the substile and on that the stiles height and the hour of 6 being found by the tenth chapter and draw those lines from C the center then draw two parallels to 12 as in the direct south then fit the distance of the parallels in the secant of the declination and take the secant of the latitude and set it from the center C on the line of 12 to F and on the parallel from 6 to G and draw a line by those two points F and G to cut the other parallel in H then have you found 6 3 and 9 then fit 6 G in the tangent of 45 and prick off 15 30 and the respective quarters both ways from 6 for the morning and afternoon hours then fit F G in 45 and lay off the same points from F both ways for 10 11 1 and 2 and the quarters also if you please and those shall be points to draw the hour-lines by The stile must be set perpendicular over the substile to the Angle found by the rules in the tenth chapter and then the Dial shall shew the true hour of the day being drawn fit to his proper declination Another way to perform this Geometrically for all erect Dyals with centers When you have drawn a line of 12 and appointed a center make a Geometrical square on that side 12 as the stile must stand on as A B C D the perpendicular side of which square may also be the parallel as before Again fit the side of the square in the cosine of the latitude and take out the sine of the latitude and fit that over in the sine of 90 then take out the 〈◊〉 sine of the declination and lay it from D to G for the hour of 6 and draw the line A G for the 6 a clock hour line Then again fit the side of the square or the distance of the parallel in the other way when you want a secant or your secant too little in the sine of 90 and take out the cosine of the declination fit that in the cosine of the latitude then take out the sine of 90 and lay it from the center on the line of 12 and 〈◊〉 6 in the side of the square and by those two points draw the contingent line and then fit those points or distances in the tangent of 45 and lay down the hours as in the former part of this chapter but if you want the hours before 9 in west-decliners or the hours after 3 in east-decliners and the 6 fall too high above the horizontal line on that side you may supply this defect thus Take the measure with your Compasses from 3 to 4 and 5 upwards in the west-decliners or from 9 to 7 and 8 in east-decliners and lay it upwards from 9 or 3 on the deficient side in the parallel as it should have been from 6 downward in south or upward in north Plains and you shall see all your defective points to be compleatly supplied whereunto draw the hour-lines accordingly CHAP. XV To draw the Hour-lines on an upright declining Dial declining above 60 degrees IN all erect Decliners the way of finding the stile substile 6 and 12 and inclination of Meridians is the same but when you come to protract or lay down the hour-lines you shall finde them come so close together as they will be useless unless the stile be augmented The usual way for doing of which is to draw the Dial on a large floor and then cut off so much and at such a distance as best serveth your turn but this being not always to be affected for want of conveniency and large instruments it may very artificially be done by a natural tangent of 75 or 80 degrees fitted on the legs of a
to adde the height of your eye from the ground at the time of taking the angle to the altitude found For the operation of this extend the compasses from the sine of the complement of the Angle found to the number of the measured side on the line of numbers that distance applied the same way from the sine of the Angle found shall reach on the line of numbers to altitude required Example at one station I open my rule and hang on the thread and plummet on the center and observing the Angle at C I finde it to be 41. 45 and the Angle at B the complement of it 48. 15 and the measure from C to A 271 feet then the work being so prepared is thus As the Sine of 48. 15 Is to 271 the measure of the side opposite to it So is the Sine of the Angle 41. 45 To 242 the measure of the side A B opposite to the Angle at C the height required Again at the station D 160 foot from A I observe and finde the Angle D to be 56. 30 the Angle at A is the complement thereof viz. 33. 30. This being prepared I extend my Compasses from the sine of 33. 30 to 160 on the line of numbers the same extent will reach from the sine of 56. 30 to 242 on the line of numbers lacking a small fraction with which I shall not trouble you An example at two stations As the Sine of the difference which is the Angle C B D 14. 45 Is to the side measured viz. D C 111 feet on the numbers So is the sine of the Angle at C 41. 45 To the measure of the side B C the hypothenusa or measure from your eye to the top of the object viz. 290 feet Again for the second Operation As the sine of 90 the right angle at A Is to 290 the hypothenusa B D So is the sine of 56. 30 the Angle at the first station D To 242 the Altitude B A the thing required So also is the sine of the Angle at B 33. 70 the complement of 56 30. To 160 the distance from D to A. To perform the same by the line of sines drawn from the center on the flat-side and the line of lines or equal parts or inches in ten parts To work these or any other questions by the line of natural Sines and Tangents on the flat-side drawn from the center it is but changing the terms thus As the measured distance taken out of the line of lines or any scale of equal parts is to the sine of the angle opposite to that measured side fitted across from one leg to the other the Sector so standing take out the parallel sine of the angle opposite to the enquired side and that measure shall reach on the line of equal parts to the measure of the Altitude requiâ—â—â— Example as before Take out of the lines or inches 2. 71 and fit it in the sine of 48. 15 across from one legge to the other which I call A parral sine but when you measure from the center onwards the end I call it A latteral sine then take out the parallel sine of 41. 45 and measure it on the line of inches or equal parts and it shall reach to 2 inches 42 parts or 242 the Altitude required After the same manner may questions be wrought on the line of lines sines or tangents alone or any one with the other by changing the Logarithmetical Canon from the first to the second or third and the second or third to the first to second as the case shall require from a greater to a less and the contrary for the fourth is always the same of which in the use of the Sector by Edmund Gunter you may finde many examples to which I refer you Also without the lines of sines either natural or artificial you may find altitudes by putting the line of quadrat or shadows on the Rule as in a quadrant then the directions in the use of the quadrat page 146 of the Carpenters Rule will serve your turn which runs thus As 100 or 50 according as it is divided to the parts cut by the thread so is the distance measured to the height required which work is performed by the line of numbers onely Or again As the parts cut to 100 or 50 so is the height to the distance required But when the thread falls on the contrary shadow that is maketh an Angle above 45 then the work is just the contrary to the former What is spoken here of taking of Altitudes may be applied to the taking of distances for if the Sector be fitted with a staff and a ball-socket you may turn it either horizontal or perpendicular and so take any Angle with it very conveniently and readily by the same rules and directions as were given for the finding of Altitudes CHAP. XVII The use of certain lines for the mensuration of superficial and solid bodies usually inserted on Ioynt-Rules for the use of Work-men of several sorts and kindes FIrst the most general and received line for mensuration of Magnitudes is a foot divided into 12 inches and those inches into 8 10 12 or more parts but this being not so apt for application to the numbers I shall not insist of it here but rather refer you to the Carpenters Rule yet nevertheless those inches laid by a line of foot measure doth by occular inspection onely serve to reduce foot measure to inches and inches likewise to foot measure and some other conclusions also 1. As first The price of any commodity at five score to the hundred either tale or weight being given to finde the price of one in number or one pound in weight As suppose at two pence half-peny a pound or one I demand to what cometh the hundred weight or five-score counting so many pound to the hundred weight If you look for two inches and a half representing two pence two farthings right against it on the foot measure you have 21 very near for if you conceive the space between 20 and 21 to be parted into 12 parts this will be found to contain ten of them for the odde ten pence But for the more certain computation of the odde pence look how many farthings there is in the price of one pound twice so many shillings and once so many pence is the remainder which if it be above 12 the 12 or 12s being substracted the remainder is the precise number of pence above the shillings there expressed and on the contrary at any price the C hundred or 5 score to finde the price of one or 1l As suppose at 40s the C. or 5 score look for 40 in the foot measure and right against it in the inches you have 4 inches 3 quarters and 1 4 of a quarter which in this way of account is 4 pence 3 farthings and about a quarter of one farthing Thus by the lines as they are
1 2 then look for 17 3 4 on the first line where 15 1 2 was found and right against it on the second line is neer 42 the fractions are all decimal and you must reduce them to proper fractions accordingly To work the rule of 3 reverse 4. Set the first term sought out on the first line to the second being of the same denomination or kind to 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 5. If 300 masons build an edifice in 28 days how many men must I have to perform the same in six days the answer will be found to be 1400. 6. To work the double rule of 3 direct This is done by two workings As thus for Example If 112 l. or 1 C. weight cost 12 pence the carraiage for 20 miles what shall 6 C cost 100 miles Say first by the third rule last mentioned as 1 C. weight to 12 so is 6 C. weight to 72. pence secondly say if 6 C. cost 72 pence or rather 6s for 20 miles what shall 100 miles require the answer is 30 s. for if you set 20 against 6 then right against 100 is 30 the answer required The use of Mr. Whites rule in measuring Timber round or square the square or girt being given in inches and the length in feet and inches 1. The inches that a piece of Timber is square being given to finde how much in length makes a foot of Timber look the number of inches square on that side of the Timber line which is numbred with single figures from 1 to 12 and set it just against 100 on the other or second side then right against 12 at the lower or some times the upper end on the first line in the second you have the number of feet and inches required Example At 4 1 2 inches square you must have 7 foot 1 inch 1 3 to make a foot of Timber But if it be above 12 inches square then use the sixth Problem of the 5th chapter of the Carpenters Rule with the double figured side and Compasses 2. But if it be a round smooth stick of above 12 inches about and to it you would know how much in length makes a true foot then do thus Set the one at the beginning of the double figured side next your left hand to the feet and inches about counted in the other side numbred with single figures from 1 to 12 then right against three foot six 1 2 inches in the single figures side next the right hand you have in the first side the number of feet and inches required Example A piece of 12 inches about requires 11 f 7 in fere to make a foot Again a piece of 15 inches about must have 8 foot 1 2 an inch in length to make a foot of timber 3. But if you would have it to be equal to the square made by the 4th part of a line girt about the piece then instead of three foot 6 1 2 inches make use of four foot and you shall have your desire 4. The side of a square being given in inches and the length in feet to find the content of a piece of timber If it be under 12 inches square then work thus set 12 at the beginning or end of the right hand side to the length counted on the other side then right against the inches square on the right side is the content on the left side Example At 30 foot long 9 inches square you shall find 16 foot 11 inches for the working this question 12 at the end must be used But if it be above 12 inches square then ser one at the beginning or 10 at the end of the right hand side to the length counted on the other side then the number of inches or rather feet and inches counted on the first side shall shew on the second the feet and parts required Example At 1 f. 6 inch square and 30 foot long you shall finde 67 feet and about a 1 2. 5. To measure a round piece by having the length and the number of inches about being a smooth piece and to measure true and just measure then proceed thus Set 3 f. 6 1 2 inches on the right side to the length on the other side then the feet and inches about on the first side shall shew on the second or left the content required As at 20 inches about and 20 foot long the content will be found to be about 4 foot 5 inches But if you give the usual allowance that is made by dupling the string 4 times that girts the piece then you must set 4 foot on the right side to the length on the other then at 1 foot 8 inches about the last example you shall finde but three foot 6 inches 6. â— astly if the rule be made fit for foot measure onely then the point of 12 is altogether neglected and one onely made use of as a standing number and the point at three foot 6 1 2 will be at three foot 54 parts and the four will be the same and the same directions in every respect serve the turn And because I call it Mr. Whites rule being the contriver thereof according to feet and inches I have therefore fitted these directions accordingly and there are sufficient to the ingenious practitioner CHAP. XIX Certain Propositions to finde the hour and the Azimuth by the lines on the Sector PROP. 1. HAving the latitude and complements of the declination and Suns altitude and the hour from noon to finde the Suns Azimuth 〈◊〉 that time Take the right sine of the complement of the Suns altitude and mak 〈◊〉 it a parallel sine in the sine of th 〈◊〉 hour from noon counting 15 degree 〈◊〉 for an hour and 1 degree for for minutes counted from the center The Sector so set take the right sine of the complement of the declination and carry it parallel till the compasses stay in like sines and the sine wherein they stay shall be the sine of the Azimuth required Or else thus Take the right sine of the declination make it a parallel in the cosine of the Suns altitude then take the parallel sine of the hour from noon and it shall be the latteral or right sine of the Azimuth from the south required If it be between six in the morning and 6 at night or from the north if it be before or after six and so likewise is the Azimuth PROP. 2. Having the Azimuth from south or north the complement of the Suns altitude and declination to finde the hour Take the latteral or right sine of the complement of the Suns altitude make it a gaâ—â—llel in the cosine of the declination the sector so sett ake out the parallel sine of the Azimuth and measure it from the center and it shall reach to the right sine of the hour from noon required Or
the tangent of the present declination To the right ascension required Onely you must regard to give it a right account by considering the time of the year and how many 90s past PROR 14. To find an altitude by the length and shadow of any perpendicular object Lay the hair on one legg to the length of the shadow found on the line of numbers and the hair of the other leg to the length of the object that caused the shadow found on the same line of the numbers then observe the lines between and which way when the legs are so set bring the first of them to the tangent of 45 and the other leg shall â—hew on the line of tangents so many turns between and the same way the tangent of the altitude required Thus may you apply all manner of quest to the Serpentine-line work them by the same Canons that you use for the Logarithms in all or most Authors PROP. 15. To square and cube a number and to findethe square root or cube roat of a number The squaring of a number is nothing else but the multiplying of the number by it self as to square 12 is to multiply 12 by 12 and then the cubing of 12 is to multiply the square 144 by 12 that makes 1728 and the way to work it is thus Set the first leg to 1 and the other to 12 then set the first to 12 and then the second shall reach to 144 then set the first to 144 and the second shall reach to 1728 the cube of 12 required but note the number of figures in a cube that hath but one figure is certainly found by the line by the rule aforegoing but if there be more figures then one so many times 3 must be added to the cube and so many times two to the square To find the square root of a number do thus Put a prick under the first the third the 5th the 7th the number of pricks doth shew the number of figures in the root and note if the figures be even count the 100 to be the unit if odde as 3 5 7 9 c. the 10 at the beginning must be thâ— unit as for 144 the root consists of two figures because there is two pricks under the number and if you lay the index to 144 in the numbers it cuâ—s on the line of Logarithms 15870 the half of which is 7915 whereunto if you lay the index it shall shew the 12 the root required but if you would have the root of 14+44 then divide the space between that number and 100 you shall finde it come to 8 4140 that is four turnes and 4140 for which four turnes you must count 80000 the half of which 8,4140 is 4,2070 whereunto if you lay the index and count from 1444 â—r 100 at the end you shall have it cut at 38 lack four of a 100. To extract the cubique root of a number set the number down and put a point under the 1 the 4th the 7th and 10th and look how many pricks so many figures must be in the root but to finde the unity you must consider if the prick falls on the last figure then the 10 is the unit at the beginning of the line as it doth in 1728 for the index laid on 1728 in the Logâ—rithms sheweth 2,3760 whose third part 0,7920 counted from 10 falls on 12 the root but in 17280 then you must conceive five whole turnes or 1000 to be added to give the number that is to be divided by three which number on the outermost circle in this place is 12 +3750. by conceiving 10000 to be added whose third part counted from 10 viz. two turnes or 4.125 shall fall in the numbers to be near 26. But if the prick falls of the last but 2 as in 172800 then 100 at the end of the line must be the unit and you must count thus count all the turnes from 172830 to the end of the line and you shall finde them to amount to 7,6250 whose third part 2 5413 counted backward from 100 will fall on 55,70 the cubique root required PROP. 16. To work questions of interest or progression you must use the help of equal parts as in the extraction of roots as in this question if 100 l. yield 106 in one year what shall 253 yield in 7 year Set the first leg to 10 at the beginning in this case representing a 100 and the other to 106 and you shall finde the legs to open to 253 of the small divisions on the Logarithms multiply 253 by 7 it comes to 1771 now if you lay the hair upon 253 and from the place where the index cuts the Logarithms count onwards 1771 it shall stay on 380 l. 8 s. or rather thus set one leg to the beginning of the Logarithms and the other to 1771 either forward or backward and then set the same first leg to the sum 253 and the second shall fall on 380. 8 s. according to estimation the contrary work is to finde what a sum of money due at a time to â—ome is worth in ready money this being premised here is enough for the ingenious to apply it to any question of this nature by the rules in other Authors However you may shortly expect a more ample treatise in the mean time take this for a taste and farewell The Use of the Almanack Having the year to finde the day of the week the first of March is on in that year and Dominical letter also First if it be a Leap-year then look for it in the row of Leap-year and in the column of week-days right over it is the day required and in the row of dominical letters is the Sunday letters also but note the Dominical letter changeth the first of Ianuary but the week day the first of March so also doth the Epact Example In the year 1660 right over 60 which stands for 1660 there is G for the Dominical or Sunday letter beginning at Ianuary and T for thursday the day of the week the first of March is on and 28 underneath for the epact that year but in the year 1661. being the next after 1660 the Leap-year count onwards toward your right hand and when you come to the last column begin again at the right hand and so count forwards till you come to the next Leap-year according to this account for 61 T is the dominical letter and Friday is the first of March But to finde the Epact count how many years it is since the last Leap-year which can be but three for every 4th is a Leap-year and adde so many times 11 to the epact in the Leap-year last past and the sum if under 30 is the Epact if above 30 then the remainder 30 or 60 being substracted is the Epact for that year Example for 1661.28 the epact for 1660 and 11 being added makes 39 from which take 30 and there remaineth 9 for the Epact for the year 1661 the thing required Note that in orderly counting the years when you come to the Leap-year you must neglect or slip one the reason is because every Leap-year hath two dominical letters and there also doth the week day change in the first of March so that for the day of the month in finding that the trouble of remembring the Leap-year is avoided To find the day of the Month. Having found the day of the week the first of March is on the respective year then look for the month in the column and row of months then all the daies right under the month are the same day of the week the first of March was on then in regard the days go round that is change orderly every seven days you may find any other successive day sought for Example About the middle of March 1661 on a Friday what day of the month is it First the week day for 1661 is Friday as the letter F on the next collumn beyond 60 sheâ—etâ— then I look for 1 among the months and all the days right under viz. 1 8 15 22 29. in March and November 61 are Friday therefore my day being Friday and about the middle of the month I conclude it is the 15th day required Again in May 1661. on a Saturday about the end of May what day of the month May is the third month by the last rule I find that the 24 and 31 are Fridays therefore this must needs be the 25 day for the first of Iune is the next Saturday FINIS ERRATA PAge 23. l. 4. adde 1660 p. 24. l. 6. for 5 hours r. 4. l. 9. for 3. 29. r. 4. 39. 1. 12 for 5. 52. r. 4 52. l. 13. for 3. 39. r. 4. 39. l. 17 for 5 hours 52. r. 4. 52. p. 27. l. ult dele or 11. 03. p 31. l 4. for sun r. sum p. 50. l. 8. for B r. A. p. 50 d CHAP. XII p. 51. r. 16. for 6. 10. 1. 6 to 10. p. 71. l. 6. for 7 4 r. 1 4. l. penult for 2 afternoon r. 1. p. 74 l. ult for 1. r. 1 2. p. 83. l. 18. for BC r. BD. p. 69 l. 17. add measure p. 129 l. 24. for right of r. right sine of p. 114 l. 9 for 18 3. r 18 13. p. 147 1. 2 for 20 r. 90 p. 163. l. 16. for of r. on