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
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-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
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