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
Declination of 12 principal fixed Stars in the heavens most of which are inserted on the Rule or if room will allow all of them  R. Asc. Declina Stars Names H. M. Deg. M. Pleiades or 7 Stars 03 24 23 20 Bulls-Eye 04 16 15 48 Orions Girdle 05 18 01 195 Little Dog 07 20 06 08 Lyons Heart 09 50 13 40 Lyons Tayl 11 30 16 30 Arcturus 14 00 21 04 Vultures Heart 19 33 08 00 Dolphins Head 20 30 14 52 Pegasus Mouth 21 27 08 19 Fomahant 22 39 31 17s Pegasus lower wing 23 55 13 19  1  3  5 7 4   6  8    Moneths 9  11  2 10 12  1 2 3 4 5 6 7  8 9 10 11 12 13 14 Days 15 16 17 18 19 20 21  22 23 24 25 26 27 28  29 30 31     Week-days S M T W T F Sat Dom. Letter d c b a g f e Leap years 68 80 64 76 60 72 84 Eâ—acts 26 9 12 25 28 11 23 THE Description and Use OF A JOYNT-RULE CHAP. I. The Description of the Lines on the Rule as it is made onely for one Latitude and for the finding the hour of the day onely FIrst open the Joynt of the Rule then upon the head-leg being next to your right hand you have a line beginning at the hole which is the Center of the quadrantal lines and divided from thence downward toward the head into as many degrees as the Suns greatest altitude in that latitude will be which with us at London is to 62 degrees which line I call the Scale of Altitudes divided to whole halfs and sometime quarters of degrees 2. Secondly On the other leg and next to the inside is the line of hours usually divided into hours quarters and every fifth minute beginning at the head with 4 and so proceeding to 5 6 7 8 9 10 11 and 12 at the end and then back again with 1 2 3 4 5 6 7 8 for the morning and afternoon hours 3. Next to this is a Kalendar of Moneths and Days in two lines the uppermost contains that half year the days lengthen in and the lowermost the shortning days as by the names of the moneths may appear the name of every moneth standing in the moneth and at the beginning of the moneth and all but the two moneths that have the longest and the shortest days viz. Iune and December are divided into single days the tenth day having a figure 10 or a point or prick on the head of the stroke and the fifth onely a longer stroke without a prick and the beginning of every moneth a long stroke and every single day all alike of one shortness according to the usual manner of distinguishing on lines 4. And lastly you have a line of degrees for so they be most properly called and they are the same with the equal limb on quadrants and serve for the same use viz. for taking of Altitudes or Horizontal Angles and are divided usually to whole and half degrees of the quadrant and figured with 30 40 50 600 7010 8020 and 90 just on the head cutting the center or point where the Scale of Altitudes and the Line of Hours meet which point for distinction sake I call The rectifying point And the reckoning on this line as to taking of Altitudes is thus At the number 600 is the beginning then towards the head count 10 20 30 where the 90 is then begin at the end again count as the figures shew you to 90 at the head as before CHAP II. The Uses of the Rule follow 1. To Rectifie or set the Rule to his true Angle OPen the Rule to 60 degrees which is done thus indifferently make the lines on the head and the lines on the other leg meet in a streight line then is the Scale of Altitudes and the line of Hours set to an Angle of 60 degrees the rectifying point being the center of that Angle Or to do it more exactly do thus put one point of a pair of Compasses into the rectifying point then open the other to 10 20 30 or 40 on the Scale of Altitudes the Compasses so opened and the point yet remaining in the rectifying point turn the other to that margenal line in the line of hours that cuts the rectifying point and there stay it then remove the point that was fixed in the rectifying point and open or shut the Rule till the point of the Compasses will touch 10 20 30 or 40 being the point you set the Compasses too in the Scale of Altitudes in the innermost line that cuts the center and the rectifying point then is it set exactly to 60 degrees and fitted for observation 2. To finde the Suns Altitude at any time Put a pin in the center hole at the upper end of the Scale of Altitudes and on the pin hang a thread and plummet then if the Sun be low that is to say under 25 degrees high as in the winter it will always be then lift up the moveable leg where the moneths and the degrees be till the shadow of the end fall just on the meeting of that leg with the head then the thread shall shew the Suns altitude counting from 600 towards the head either 10 20 25 or any degree between But if the Sun be above 25 or 30 degrees high lift up the head leg till the shadow of that play as before or make the shadow of the pin in the center hole play on the innermost line of the Scale of Altitudes where the pin standeth then the thread will fall on the degree and part of a degree that his true altitude shall be But if the Sun be in a cloud and can not be seen so as to give a shadow then look up along by the head-leg or moveable leg just against the middle of the round body of the Sun and the thread playing evenly by the degrees shall show the true altitude required The like must you do for a Star or any other object whose altitude you would find 3. Having found the Suns altitude and the day of the moneth to finde the hour of the day Whatsoever you finde the altitude to be take the same off from the Line of Altitudes from the center downwards with a pair of Compasses then lay the thread being put over the pin on the day of the moneth then put one foot of the Compasses in the line of hours in that line that cuts the rectifying point and carry it further off or nigher till the other foot of the Compass being turned about will just touch the thred at the nearest distance then the point of the Compasses on the line of hours shall shew the true hour and minute of the day required Example on the 2. of July 1. I observe the altitude in the morning and I finde it to be 30 degrees high then laying the thread on the day of
the moneth and taking 30 degrees from the Scale of Altitudes and putting one point in the line of hours till the other point turned about will but just touch the thâ—ead and I finde it to 23 minutes past 7 but if it had been in the afternoon it would have been 37 minutes past 4. 2. Again on the tenth of August in the Afternoon at 20 degrees high I take 20 degrees from the Scale of Altudes and laying the thread on the day of the moneth viz. the tenth aforesaid counting from the name at the beginning of August toward September and carrying the Compasses in the line of hours till the other point doth but just touch the thread and you shall finde it to be 54 minutes past 4 a clock 3. Again on the 11. of December at 15 degrees high work as before and you shall finde it to be just 12 a clock but to work this you must lay the Rule down on something and extend the thread beyond the Rule for the nighest distance will happen on the out-side of the Rule 4. Again on the 11 of Iune at noon I finde the altitude to be 62 degrees high then laying the thread on the 10 th or 11 th of Iune for then a day is unsensible and working as before you shal finde the point of the Compasses to stay at just 12 a clock the time required for that altitude 4. To finde the Suns rising any day in the year Lay the thread on the day of the month and in the line of hours it sheweth the true hour and minute of the Suns rising or setting for the rising count the morning hours and for the setting count the evening hours 5. To finde if any place lye level or nor Open the rule to his true angle of 60 degrees then set the moveable leg upon the place you would make level and if the thread play just on 60 degrees it is a true level place or else not 6. To try if any thing be upright or not Hang a thread and plummet on the center then aply the head leg of the rule to the wall or post and if it be upright the thread will play just on the innermost line of the scale of altitudes or else not CHAP. III. A further description of the Rule to make it to shew the Suns Azimuth Declination True place right Ascention and the hour of day or night in this or any other Lattitude 1. FIrst in stead of the scale of Altitudes to 62 degrees there is one put to 90 degrees in that place and that of 62 is put by in some other place where it may serve as well 2. The line of hours hath a double margent viz one for hours and the other for Azimuths then every 5 th minute is more properly made 4 or else every 2 minutes and in a large rule to every quarter of a degree of Azimuth or to every single minute of time 3. The degrees ought to be reckoned after 3 maner of wayes first as before is exprest secondly from 60 toward the end with 10 20 30 40 50 60 c. to be so accounted in finding the Azimuch for a particular latitude and and thirdly from the head or 90 toward the end with 10 20 30 40 50 60 70 80 c. for the general finding of Hour and Azimuth in any latitude and many other problems of the Sphere besides to which may be added where room will alow a line of hours beginning at 6 at the head and 12 at the end but reckoning 15 degrees for an hour and 4 minutes for every degree it may do as well without it 4. To the Kalender of moneths and days is added a line of the Suns true place in the Zodiack or where room fails the Characters of the twelve Signs put on that day of the moneth the Sun enters into it and counting every day for a degree may indifferently serve for the use it is chiefly intended for 5. Under that is a line of the Suns right Ascension to hours and quarters at least or rather every fifth minute numbred thus 12 and 24 right under ♈ and ♎ or the tenth of March and so forward to the tenth of Iune or ♋ where stands 6 then backwards to 12 where you began then backwards still to the eleventh of December with 13 14 15 16 17 18 to ♑ then from thence forward to 24 where you first began but when you are streightned for room as on most ordinary Rules you will be then it may very well suffice to have a point or stroke shewing when the Sun shall gradually get an hour of right Ascension and from that for every day count four minutes of time till it hath increased to an hour more and this computation will serve very well and in stead of saying 13 14 15 hours of right Ascension say 1 2 3 c. which will perform the work as well and reduce the time to more proper terms 6. There is fitted two lines one containing 24 houres and the other 29 days and about 13 hours and they serve to finde the time of the Moons coming to the South before or after the Sun and by that the time of high-water at London-bridge or any other place as is ordinary CHAP. IV. The Uses follow in order 1. To finde the Suns Declination LAy the Thread on the day of the moneth then in the line of degrees you have the declination From March the tenth toward the head is the Declination Northward the other way is Southward as by the time of the year is discovered Example On the tenth of April it is 11d 48 ' toward the North but on the tenth of October it is 10d 30 ' toward the South 2. As the thread is so laid on the day of the moneth in the line of the Suns place it sheweth that and in the line of the Suns right Ascension his right Ascension also onely you must give it its due order of reckoning as thus it begins at ♈ Aries and so proceeds to ♋ then back again to ♑ at the eleventh of December then forwards again to ♈ Aries where you began 3. To finde the Suns right Ascension in hours and minutes Lay the thread as before on the day of the moneth and in the line of right Ascension you have the hour and minute required computing right according to the time of the year that is begin at the tenth of March or ♈ Aries and so reckon forwards and backwards as the moneths go Example On the tenth of April the Suns place is 1 degree in ♉ Taurus and the Suns right Ascension 1 hour 55 minutes on the tenth of October 27d 1 4 in ♎ Libra and his right Ascension is 13 hours and 42 minutes 4. To finde the Suns Amplitude at rising or setting Take the Suns Declination out of the particular Scale of Altitudes and lay it the same way as the Declination is from 90 in the Azimuth Scale and it shall shew
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
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
divided it proceeds to 12 pence a pound but if you conceive the inches to be doubled and the foot measure also you shall have it to 24 d. or 48 d. the pound or one in tale of any commodity As at 18 d. a piece or pound the price comes to 7l 10s the C. for then every ten strokes is 20s in the foot measure and every inch is 2 pence and every eighth one farthing 2. Secondly for the buying of Timber at 50 foot to a Load at any price the load how much a foot Here in resolving this the inches are to be doubled and the foot measure taken as it is As at 40 shillings the Load 40 in the foot measure stands right against 4 inches 3 quarters and better which being doubled is 9d 2 far 1 2 far near for the price of one foot and on the contrary at 5d a foot is 41s 8 d. a load c. 3. For the great Hundred of 112l to the Hundred let the space of 12 inches be divided into 112 parts then the like rule holds for that also For the inches being divided into quarters every quarter is a farthing and every eighth half a farthing and every division of the 112 is a shilling and every alteration of a farthing in the price of a pound makes a groat in the Hundred as thus At 3 pence a pound is 28s the C. At 3d. 1 q. a pound 30s and 4d the C. At 3d. 1 2 the pound 32s 8 d. the C. At 3 d. 3 farthings 35s the C. Thus you see that every fraction at a farthing advance is 4 pence in the Hundred but for any other account as 3 pence farthing half farthing then count the fraction as 1 12th part of a shilling and nearer you cannot come by a bare occular inspection but the price of the Hundred being given the price of the pound you have as near by this occular inspection as any usual Coin is reducable viz. to the 32 part of a peny or nigher if you please Again note here also you may double or quadruple the price as to 24d or 48d the pound or any price between As for example At 13d a pound is 6l 1s 4d the C. At 32d or 2s 8d the pound is 14l 19s the C. and the like by dupling and quadrupling the inches and the 112 parts that layeth by it 4. These lines of equal parts serve as Scales for the protracting of any Draught of house or field or the like also for addition or substraction of any small number 5. Note that the line of foot measure may be applied for the reducing of any odde fraction to a decimal fraction as you may fee it in page 64. of Mr. Windgats Arithmetick made easie 2. The use of the lines of decimal Timber and Board measure The lines of decimal Timber and Board measure are fitted to agree with the tenths or foot measure as those lines in the first chapter of the Carpenters Rule are fitted to the inches and the use of them is thus And first for the decimal Board measure Suppose a Board is 1 foot 50 broad I look for 150 on that line and from that place to the end of the Rule forwards toward 100 so much in length must you have to make a foot of superficial or board measure 2. Or else thus If you apply the end of the Rule next 100 to one edge of the breadth of a board or glass then right against the other edge of the board on that line of decimal board measure you shall finde the 10ths and 100s or feet 10ths 100 parts of a foot that you must have in length to make a foot superficial at that breadth Example I come to a board and applying the upper end next 100 even to one edge of the board the other reacheth to 0. 8 tenths then I say that 8 tenths of a foot length at that breadth makes a foot 3. The use of decimal Timber measure The use of this is much like the Board measure onely here you must have a respect to the squareness of the piece and not to the breadth onely for after you know how much the piece of timber or stone is square in feet and 100 parts then look that number on the line of decimal Timber measure and from thence to the end of the Rule is the length that goes to make a foot of timber Example At 14 or 1. 40. parts of a foot square look the same on the rule and from thence to the end where 40 is is the length of a foot of Timber at that squareness being about 51 parts of a foot divided into a 100 parts 5. The use of the line of decimal yard measure also running yard measure according to the inches or decimal parts of a foot The decimal yard measure is nothing else but a yard or 3 foot divided into a 100 parts and used in the same manner as the foot measure is for if you take the length and the breadth in that measure and multiply it together you shall have the content in yards and 100 parts of a yard Example Suppose a peece of plastering is 4 yards 78 parts one way and 7. 35 parts another way being multiplied together makes 35 yards and 9954. of 10000 which is very neer 36 yards 5. But the decimal running yard measure is fitted to the foot measure and the use is thus Suppose a room is to be measured that is 7 foot 8 tenths high and I would know how much makes a yard at that breadth or height look foor 7f 8 10ths on the line of decimal running yard measure and the space on the rule from thence to the end next 100 is the true length that goeth to make up a yard of superficial measure at that breadth or height But if the peece be between 4 foot 5 10 broad and 2 foot then the table at the end of the line will supply the defect or you may change the terms and call the length the breadth and the contrary But if it be under 2 foot broad then if you do as you did with the board measure you shall have your desire Example At 1 foot 3 10th broad 6 foot 9 10ths make a yard 6. But if the running yard measure be made to agree with the inches then measure the height of the room in feet and inches and if you take a pair of compasses and measure from that place to the end of the rule then turn the compasses set at that distance as many times as you can about the room so many yards is there in the room 7. The use of the line of decimal round measure commonly called Girt-measure which is when the circumference of a round Cillender or piller given in inches or ten parts of a foot First for Girt-measure according to inches being the most usual measure now much the pillar is about then look for the same number on the line of Girt-measure and from thence to
the end of the rule is the length that goeth to make a foot of Timber But if it be under 30 inches about then you must have above two foot in length and then a table at the end of the line or a repetition in another line will supply the defect But if the line of Girt-measure be divided according to foot measure then use it as before seeking the decimal part on the line and from thence to the end is a foot 8. The use of a line of solid measure by having the Diameter of a round piece given in inches or foot measure Take the diameter with a rule or a pair of Callipers and learn the measure either in inches or foot measure according as your line of Diameter is divided Then look for the same number on the line of Diameter and from thence to the end of the rule forward is the length that makes a foot of timber at that diameter or measure cross the end of the round piece of Timber or stone The Tables of all the under measure for all these lines follow Decimal Superficiall under M.  10th F. 1000  10 F. 1000 p.  1 100. 00   3. 848  2 50. 000   3. 706  3 33. 300   3. 570  4 25. 000   3. 450  5 20. 000 3 3 3. 332  6 16. 600   3. 217  7 14. 300   3. 115  8 12. 500   3. 025  9 11. 120   2. 940 1 1 10. 000  5 2. 850  1 9. 100  6 2. 780  2 8. 340  7 2. 700  3 7. 720  8 2. 628  4 7. 150  9 2. 560  5 6. 670  4 2. 500  6 6. â—60   ◠440  7 5. 888   2. 382  8 5. 5â—5   2. 336  9 5. 260   2. 273 2 2 5. 000   2. 213   4. 760   2. 173   4. 5â—6   2. 127   4. 350   ◠083   4. 170   2. 042  5 4 000  5 2. 000 Decimall Superficiall M.  F. 1000. p. 01 F. 1000. p.  1. 962  1. 320  1. 923  1. 304  1. 816  1. 286  1. 850  1. 268  1. 820 8 1. 250  1. 785  1. 237  1. 756  1. 220  1. 726  1. 207  1. 697  1. 192 6 1. 669  1. 178  1. 640  1. 164  1. 615  1. 151  1. 589  1. 138  1. 563  1. 125  1. 538 9 1. 112  1. 516  1. 100  1. 493  1. 087  1. 472  1. 076  1. 450  1. 063 7 1. 430  1. 052  1. 409  1. 041  1. 391  1. 030  1. 373  1. 020  1. 353  1. 011  1. 337 10 1. 000 Decimal Solid under Measure  F. 1000. p. 10 F. 1000. p. 1 10000. 000  14. 805 2 2500. 000  13. 735 3 1100. 000  12. 780 4 630. 000  11. 916 5 400. 000 3 11. 125 6 277. 900  10. 415 7 200. 430  9. 760 8 150. 660  9. 125 9 120. 350  8. 625 1 100. 0000  8. 150  82. 800  7. 700  96. 500  7. 310  59. 390  6. 900  51. 100  6. 565  44. 500 4 6. 250  39. 150  5. 945  34. 650  5. 664  30. 850  5. 404  27. 750  5. 465 2 25. 000  4. 938  22. 700  4. 720  20. 675  4. 530  18. 920  4. 342  17. 400  4. 162  16. 000 5 4. 000 Decimall Solid under measure  F. 1000. p. 01. F. 100. p.  3. 825  1. 738  3. 7â—0  1. 694  3. 524  1. 651  3. 430  1. 608  3. 310 8 1. 568  3. 188  1. 528  3. 078  1. 493  2. 968  1. 458  2. 873  1. 420 6 2. 780  1. 390  2. 688  1. 356  2. 602  1. 323  2. 521  1. 297  2. 442  1. 266  2. 366 9 1. 236  2. 294  1. 208  2. 227  1. 185  2. 160  1. 160  2. 100  1. 131  2. 043  1. 109 7     1. 985  1. 084  1. 93â— Â 1 061  1. 878  1. 041  1. 830  1. 021  1. 781 10. 1. 000 Vnder Yard-measure for feet and inches from one inch'to four feet six inches F. F. 1000. F. F. 1000. In.  In.  1 108.000  3. 850 2 54. 000  3. 720 3 36. 000  3. 600 4 27. 000 6 3. 482 5 21. 600  3. 373 6 18. 000  3. 271 7 15. 420  3. 175 8 13. 520  3. 085 9 12. 000 3 3. 000 10 10. 300  2. 922 11 9. 820  2. 842 1. 9. 000  2. 769  8. 320  2. 710  7. 740  2. 633  7. 201 6 2. 572  6. 760  2. 512  6. 350  2. 455 6 6. 000  2. 400  5. 680  2. 345  5. 400  2. 298  5. 140 4 2. 250  4. 906 1 2. 203  4. 695 2 2. 160 2 5. 500 3 2. 119  4. 320 4 2. 073  4. 160 5 2. 037  4. 000 9 2. 000 Vnder yard measure according to Decimal or Foot measure F. 10. F. 1000. p.  F. 1000. p. 1 90. 000 4 3. 7â—0 2 45. 000 5 3. 600 3 30. 000 6 3. 461 4 22. 500 7 3. 332 5 18. 000 1 3. 211 6 15. 000 9 3. 104 7 12. 880 3 3. 000 8 11. 200 1 2. 903 9 10. 000 2 2. 812 1 9. 000 3 2. 728 1 8. 190 4 2. 648 2 7. 510 5 2. 572 3 6. 930 6 2. 502 4 6. 430 7 2. 435 5 6. 000 8 2. 370 6 5. 625 9 2. 310 7 5. 290 4 2. 250 8 5. 000 1 2. 195 9 4. 735 2 2. 142 1 4. 500 3 2. 093 1 4. 285 4 2. 046 2 4. 092 5 2. 000 3 3912   Vnder Girt-measure Inc. about F. in 100.  F. in 100 1 1809.6.81 24 3.1.87 2 452. 4. 74 25 2. 10. 74 3 201. 0. 77 26 2. 8. 12 4 113. 1. 18 27 2. 5. 87 5 72. 4. 60 28 2. 3. 70 6 50. 3. 19 29 2. 1. 83 7 39. 3. 22 30 2. 0. 13 8 28. 4. 00 31 1. 10. 60 9 22. 4. 09 32 1. 9. 21 10 18. 1. 15 33 1. 7 94 11 14. 11. 46 34 1. 6. 78 12 12. 6. 80 35 1.
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
in the cosine of the latitude then the parallel sine of the declination taken and measured in the line of sines from the center shall give the amplitude required PROP. 10. To finde the Suns height at six in any latitude Take the lateral or right sine of the declination and make it a parallel in the sine of 90 then take out the parallel sine of the latitude and measure it in the line of sines from the center and it shall reach to the altitude required Note in working of any of these Propositions if the sines drawn from the center prove too large for your Compasses or to make a parallel sine or Tangent to a small number of degrees then you may use the smaller sine or tangent adjoyning that is set on the Rule and it will answer your desire And note also in these Propositions the word right or latteral sine or tangent is to be taken right on from the center or beginning of the lines of sines or tangents and the word parallel always across from one leg to the other PROP. 11. To finde the Suns height at any time in any latitude As the right Sine of 90 Is to the parallel cotangent of the latitude So is the latteral or right Sine of the hour from 6 To the parallel tangent of a fourth ark which you must substract from the suns distance from the Pole and note the difference Then As the right of the latitude To the parallel cosine of the fourth ark So is the parallel cosine of the remainder To the latteral sine of the Altitude required PROP. 12. To finde when the Sun shall come to due East or West Take the tangent of the latitude from the smaller tangents make it a parallel in the Sine of 90 then take the latteral tangent of the declination from the smaller tangents and carry it parallel in the Sines till it stay in like Sines and that Sine shall be the Sine of the hour required from 6. PROP. 13. To finde the Suns Altitude at East or West or Vertical Circle As the latteral sine of declination Is to the parallel sine of the latitude So is the parallel sine of 90 To the latteral sine of the Altitude required PROP. 14. To finde the Stiles height in upright declining Dials As the right Sine of the complement of the latitude To the parallel sine of 90 So the parallel sine complement of the Plains declination To the right sine of the Stiles elevation PROP. 15. To finde the Substiles distance from the Meridian As the lateral tangent of the colatitude To the parallel sine of 90 So the parallel sine of the declination To the latteral tangent of the Substile from the Meridian PROP. 16. To finde the Inclination of Meridians As the latteral tangent of the declination To the parallel sine of 90 So is the parallel sine of the latitude To the latteral cotangent in the inclination of Meridians PROP. 17. To finde the hours distance from the Substile in all Plains As the latteral tangent of the hour from the proper Meridian To the parallel sine of 90 So is the parallel sine of the Stiles elevation To the latteral tangent of the hour from the substile PROP. 18. To finde the Angle of 6 from 12 in erect Decliners As the latteral tangent of the complement of the latitude To the parallel sine of the declination of the Plain So is the parallel sine of 90 To the latteral tangent of the Angle between 12 and 6. Thus you see the natural Sines and Tangents on the Sector may be used to operate any of the Canons that is performed by Logarithms or the artificial Sines and Tangents by changing the terms from the first to the third and the second to the first and the third to the second and the fourth must always be the fourth in both workings being the term required CHAP. XX. A brief description and a short-touch of the use of the Serpentine-line or Numbers Sines Tangents and versed sine contrived in five or rather 15 turn 1. FIrst next the center is two circles divided one into 60 the other into 100 parts for the reducing of minutes to 100 parts and the contrary 2. You have in seven turnes two in pricks and five in divisions the first Radius of the sines or Tangents being neer the matter alike to the first three degrees ending at five degrees and 44 minutes 3. Thirdly you have in 5 turns the lines of numbers sines Tangents in three margents in divisions and the line of versed sines in pricks under the line of Tangents according to Mr. Gunters cross staff the sines and Tangents beginning at 5 degrees and 44 minutes where the other ended and proceeding to 90 in the sines and 45 in the Tangents And the line of numbers beginning at 10 and proceeding to 100 being one entire Radius and graduated into as many divisions as the largeness of the instrument will admit being from 10 to 50 into 50 parts and from 50 to 100 into 20 parts in one unit of increase but the Tangents are divided into single minutes from the beginning to the end both in the first second and third Radiusses and the sines into minutes also from 30 minutes to 40 degrees and from 40 to 60 into every two minutes and from 60 to 80 in every 5th minute and from 80 to 85 every 1oth and the rest as many as can be well discovered The versed sines are set after the manner of Mr. Gunters Cross-staff and divided into every 10th minutes beginning at 0 and proceeding to 156 going backwards under the line of Tangents 4. Fourthly beyond the Tangent of 45 in one single line for one turn is the secants to 51 degrees being nothing else but the sines reitterated beyond 90. 5. Fifthly you have the line of Tangents beyond 45 in 5 turnes to 85 degrees whereby all trouble of backward working is avoided 6. Sixthly you have in one circle the 180 degrees of a Semicircle and also a line of natural sines for finding of differences in sines for finding hour and Azimuth 7. Seventhly next the verge or outermost edge is a line of equal parts to get the Logarithm of any number or the Logarithm sine and Tangent of any ark or angle to four figures besides the carracteristick 8. Eightly and lastly in the space place between the ending of the middle five turnes and one half of the circle are three prickt lines fitted for reduction The uppermost being for shillings pence and farthings The next for pounds and ounces and quarters of small Averdupoies weight The last for pounds shillings and pence and to be used thus If you would reduce 16 s. 3 d. 2 q. to a decimal fraction lay the hair or edge of one of the legs of the index on 16. 3 1 2 in the line of l. s. d. and the hair shall cut on the equal parts 81 16 and
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