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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 begin∣ning of every moneth a long stroke, and every single day all alike of one short∣ness, according to the usual manner of distinguishing on lines.
4. And lastly you have a line of de∣grees, for so they be most properly cal∣led, and they are the same with the equal limb on quadrants, and serve for the same use, viz. for taking of Alti∣tudes, or Horizontal Angles, and are di∣vided usually to whole, and half de∣grees of the quadrant, and figured with 30, 40, 50, 60|0, 70|10, 80|20, 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 60|0 is the be∣ginning, 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.
OPen the Rule to 60 degrees, which is done thus, (indifferent∣ly:) make the lines on the head, and the lines on the other leg, meet in a streight line; then is the Scale of Al∣titudes 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 re∣ctifying 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 fix∣ed
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 Com∣passes 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.
Put a pin in the center hole, at the upper end of the Scale of Altitudes, and on the pin hang a thread and plum∣met; 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 60|0 towards the head, either 10, 20, 25, or any degree between.
But if the Sun be above 25 or 30 de∣grees
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 stand∣eth, 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 move∣able 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.
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.
1. I observe the altitude in the mor∣ning, 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 put∣ting one point in the line of hours, till the other point turned about, will but just touch the th••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 Al∣tudes,
and laying the thread on the day of the moneth, viz. the tenth aforesaid, counting from the name at the begin∣ning 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 10th or 11th of Iune, for then a day is un∣sensible, 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.
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.
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.
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 up∣right, the thread will play just on the innermost line of the scale of altitudes, or else not.
1. FIrst in stead of the scale of Alti∣tudes 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 mar∣gent, viz, one for hours, and the other for Azimuths, & then every 5th 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 Azi∣much for a particular latitude; and
and thirdly from the head or 90, to∣ward 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, begin∣ning 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 re∣duce the time to more proper terms.
6. There is fitted two lines, one con∣taining 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.
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 Decli∣nation Northward, the other way is Southward, as by the time of the year is discovered.
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.
Lay the thread as before, on the day of the moneth, and in the line of right Ascension you have the hour and mi∣nute required, computing right accor∣ding to the time of the year, that is, be∣gin at the tenth of March, or ♈ Aries, and so reckon forwards and backwards as the moneths go.
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.
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.
FIrst finde the Suns Declination, by the first Proposition of the fourth Chapter, then take that out of the par∣ticular Scale of Altitudes, or scale to 62 degrees; then whatsoever the Al∣titude 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 Com∣passes 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 th••ead, as all win••er time it will be, and having set the Compasses to the least distance to the thread, it sh••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.
On the tenth of Iuly I desire to finde the Suns Azimuth at any Altitude, first on that day I finde the Suns Declina∣tion 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 de∣grees, lay the thread on 10, counted
from 60 toward the left end, then car∣rying the Compasses on the right side of the thread, (because it is summer or north declination) on the line of Azi∣muths, it shall shew 110. 40, the Azi∣muth 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.
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 th••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.
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 a∣bove 45, then the degrees will go be∣yond 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 dou∣bled (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.
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 ma∣ny 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 ••ny less number of days and part of a day require? The answer will be: The Moons true place at that age. Having ••ound her true place, then take her al∣••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 thētrue hour, like∣wise 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 num∣ber 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 Ex∣ample.
First, it is most readily and exactly done by an Ephemerides, such a one as you finde in Mr. Lilly's Alman••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 Appen∣dix to the Carpenters Rule.) Adde the Epact, the moneth, and the day of the mone••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.
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.
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 the day of her change; but far more readily, and as exactly by the line of Numbers (or Rule of Three) say, if 29. 540. require 360. what 6 facit 73 1/4, that is, 2 signs, 13 de∣grees, 15 minutes, to be added to 14 degrees in ♌, and it makes 27 1/4 in ♎, the Moons place for that day. Or thus, multiply the Moons age by 4, divide the product by 10, the quotient shew∣eth the signs, and the remainder mul∣tiplied by 3, sheweth the degrees which you must adde to the Suns place on the day required, and it shall be the Moons
true place required for that day of her Age.
Iuly 27. the Sun and Moon is in Leo 14 degr. 0. August 2. being 6 days, adde the Moons motion 2. 13d 9'. makes being counted, Virgo ♍, Libra ♎. 27 deg. 9 the place required; which on the Rule you may count without all this work or trouble: but for plainness sake I am constrained so to do. Or thus.
To do this, you must do the work of the second Chapter, and second and third Proposition; where note that the Moons place found, is to be used as the day of the moneth, or Suns declination.
The Moon being 27 degrees in Li∣bra, and 20 degrees high, I finde the our to be 31' past 9, if on the east∣side; or 29' past 2, if on the west-side ••f the Meridian.
Having found the Moons hour, as be∣fore, consider the Moons age, then say by the Line of Numbers, or Rule of Three, if 29. 540 part require 24, what shall 6 days require facit 4 hours and 52 minutes? which taken from 9. 31. rest ••. 29. the true time required.
Moons hour—9 31
Time to be subst. •• 52
True time remain 3 39. the time re∣quired. This work is done more readi∣ly by the two lines fitted for that pur∣pose; for look for 6 the Moons age in one, and you shall finde •• hours 52 mi∣nutes, the time to be substracted in the other.
FOr the doing of this, I have made choice of twelve principal fixed Stars, all within the two Tropicks; many more might be added, but these will very well serve the turn: The names of them, and their right Ascen∣sion in hours and minutes, is set on the Rule, and the star is placed in his true Declination on, or among the moneths; and for to know the stars next to a Tutor, a Celestial Globe, or a Nocturnal, of all the chief stars from the Pole to the Equinoctial, and to be had at the Sun-Dial in the Mi∣••nories, is the best; the uses whereof do follow.
First know the star you observe, then observe his Altitude, and laying the thread on the star by the second Cha∣pter, second and third Proposition, get
the stars hour, then out of the righ•• Ascension of the star, take the righ•• Ascension of the Sun, (found by the fifth Proposition of the third Chapter) for that day, and note the difference for this difference added to the star hours found, shall shew the hour o•• the night.
On the first of November I observe the Altitude of the Bulls-eye, and find it to be 30. then by second and third of the second Chapter, I finde the hour to be 7. 54 past the Meridian the Suns right Ascension that day finde to be 15 hours 8 minutes, the stars right ascension is 4 hours 16 mi∣nutes; which taken, the greatest from the least, by adding of 24 hours, re•• 13. 08. then 7. 54. the stars hou•• added, makes 21. 02. from which ta∣king 12 hours, rest 9. 02. the hour o•• the night required. For more plain¦ness, note the work of two or thre•• Examples.
Stars right Ascension being set on
the Rule—4—16
Suns right Asc. Nov. 1.—15—08—
Substraction being made, by adding
24, remain—13—08
To which you must adde the stars hour
found—7—54
Then the remainder, taking away 12
hours, is—9—02
the true hour.
The right Asc. of Arcturus, is 14h••—0'
Suns right Asc. May 15. is—4—10—
The right Asc. of the Sun taken from
the R. Asc. of the star, rest ••—50
The stars hour at 50 degr. high, found
to be—02—13—
Which being added to the difference
before, makes—1••—03
or—11—03.
The right Asc. of Great-dog, is 06-29
The right Asc. of the Sun, is 19 — 50—
Substr. made, by adding 24 h. is 10-39 Stars hour at 15 degr. high, is 9 — 32 Ante M. or P. Septen. Which added to the difference found, and 12 sub∣stracted, remains 8. 11'. for the true Hour of the Night required to be found, and so of any other star se•• down in the Rule, as by the trial and pr••ctice, will prove easie and ready to the ingenuous practitioner.
But by the line of 24, or twice 12 hours, and the help of a pair of Com∣passes, you may perform it withou•• writing it down, thus: Take the righ•• Ascension of the Sun, out of that lin•• of hours between your Compasses (be∣ing always counted under 12) an•• set the same from the right Ascensio•• of the star, toward a lesser number, o•• the beginning of the hours, and th••
point shall stay at the remainder that is to be added to the stars hour found, then open the Compasses from thence to the beginning of the hours, and adde that to the stars hour found, and it shall reach to the hour of the night required.
Feb. 6. 1660. by Arcturus 20 de∣grees high: Take 10 hours 1 minute between your Compasses, and set it from 14 hours, or 2 hours beyond 12, and it shall stay at 3. 59: then take 3. 59, and adde it to 6. 24, the stars hour at 20 degrees high, and it shall be 10. 23, the hour of the night re∣quired.
1. FIrst for the Amplitude, take the stars declination from the parti∣cular
Scale of Altitudes, and lay it from 90 in the Azimuth-line, and it shall shew his amplitude from East or West toward South or North, accor∣ding to the declination, and time of day, morning or evening. The same work is for the sun.
The Bulls-eye hath 25. 54 degrees of amplitude, so hath the Sun at 15 de∣grees 48 minutes of declination.
2. To finde the Azimuth, work as you did for the Sun at the same de∣clination the star hath, by Chapter 5. and you shall have your desire.
December 24. at 6 degrees high, by the Bulls-eye I finde the Azimuth to be 107. 53. from the south.
3 To finde the stars rising and set∣ting, lay the thread on the star among the moneths, and in the line of hours it shews the stars rising and setting, as you counted for the Sun; but yet note this is 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 set∣ting.
For the Bulls-eye on the 23 of De∣cember, 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.
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.
Lay the thread to the star, and in the hour-line it sheweth the ascen∣sional 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, sub∣stract 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.
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 be∣ing under the Horizon.
1. FOr the rising, and setting, and ascensional difference, being all one, do thus: Take the Suns declina∣tion out of the general Scale of Alti∣tudes, then set one foot of the Com∣passes 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 de∣clination, counted from 90 toward the center, and the thread laid to the nearest distance, shall in the degrees
shew the ascensional difference re∣quired, counting from 90 at the head toward the end of the Rule; and if you reduce those degrees and mi∣nutes to time, you have the rising and setting before and after 6, according to the declination and time of the year.
Take the Suns declination, and set∣ting one foot in the colatitude, with the other lay the thread to the nearest distance, and on the degrees it shew∣eth the Suns amplitude at rising or setting, counting as be••ore from 90 to the left end of the Rule.
Take the declination from the gene∣ral scale, and set one foot in the am∣plitude, the thread laid to the nearest distance in the line of degrees, it sheweth the complement of the lati∣tude required, or the converse.
Count the declination in the de∣grees from 90 toward the end, there∣to lay the thread, the least distance from which to the latitude in the ge∣neral Scale, shall be the Suns height at 6 in the summer, or his depression in the winter. The Compasses stan∣ding 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 declina∣tion with his height at 6 in summer, and that is done thus: Put one point of the Compasses so set in the gene∣ral Scale to the Suns Altitude, then turn the other outwards toward 90,
there keep it, then open the Compas∣ses to the beginning of the Scale, then have you added it to the Suns alti∣tude; 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, reckon∣ing from the head, or from 12, coun∣ting from the end of the Rule.
I shall make all more plain, by ma∣king three Propositions of it, thus:
Take the Altitude from the begin∣ning 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.
Is before exprest by the converse of the first part of the fourth, which I shall again repeat.
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 sum∣mer you must substract from, but in winter you must add to the Suns pre∣sent altitude; having that distance, set one foot in the coaltitude, with the other lay the thread to the neerest distance, take again the neerest di∣stance 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 de∣grees it shall shew the hour required.
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.
Take the sine of the declination, put one foot in the latitude, the thread laid to the neerest distance: in the de∣grees, it sheweth the Suns height at due East or West, which you must in summer substract from the Suns alti∣tude, 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 distance, in the degrees shall shew the Suns Azimuth required.
By the second Proposition of the
ninth Chapter, finde the Suns Am∣plitude for that day, then take the altitude from the general Scale of al∣titudes, and putting one point in cola∣titude, 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 coalti∣tude, and the thread laid to the highest distance, and in the line of de∣grees, 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.
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.
COunt the plains declination on ••he Azimuth scale, from 90 to∣ward the end, and thereunto lay the thread, in the line of degrees it shews the distance of the substile from 12.
At 10 degrees declination, I find 7. 51. for 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 re∣quired.
At 30 declination is 32. 35.
Count the substile on the particu∣ler 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 in∣clination of Meridians required.
At 15 the substile, the inclination of Meridians will be found to be 24. 36.
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 de∣grees 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 ex∣actly, neither will it be worth the while to delineate another line for this purpose. But if it be re∣quired 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 declina∣tion 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.
A plaine declining 30 degrees, the angle will be found to be 32. 21. whose complement 57. 49. is the an∣gle required.
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 comple∣ment of the plaines declination to the thread, taken and measured on the general scale, from the center, shall be the stiles height required.
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 di∣stance, and on the degrees it sheweth the inclination of Meridians required.
Count in the inclination of Meri∣dians on the degrees from 90, and thereto lay the thread, then take the least distance from the latitudes com∣plement 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.
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 di∣stance from the sine of the plains de∣clination 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.
As the Sine of the latitude,
To the Sine of 90:
So the Tangent of the Declination,
To the Tangent of inclination of Meri∣dians.
As the Sine of 90,
To the Cosine of the Declination:
So the Cosine of the latitude,
To the Sine of the Stiles elevation,
As the Sine of 90,
To the Sine of the Declination:
So the Cotangent of Latitude,
To Tangent of the Substile from 12▪
As the Co••tangent of the Latitude,
To the Sine of 90:
So is the sine of Declination,
To the Cotangent of 6 from 12.
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 Meri∣dians required.
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 perpen∣dicular 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 nigh∣est distance, then take the nearest di∣stance 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 degrees, on the o∣ther 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 per∣pendicular, and right north and south, and it shall shew when the sun shi∣neth, the true hour of the day. Note well the figure following.
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, ap∣point a place for the center, through which point cross it at right angles,
for 6 and 6, as you did in the Hori∣zontal Plain, as the lines A B, and C D, on each side 12 make two pa∣rallels, 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 tan∣gents 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 illu∣stration sake note the figure.
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, lati∣tude, 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 lati∣tude, from D to F, and draw the line C F, for the contingent or equi∣noctial line, to which line you must draw another line parallel, as far
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 tan∣gents 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 hour••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.
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 qua∣drant, 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.
For this purpose you must have a needle well touched with the Load∣stone 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 ex∣tend 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 pres∣sed 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 or∣der.
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 declina∣tion of the Plain. But if it happen to stand at any Angle above 90, then the quantity thereof above 90, is the de∣clination of the wall.
To finde the quantity of the Angle the Sector stands at may be done two ways: first by protraction, by laying down the Rule so set on a board, and draw two lines by the legs of the Se∣ctor, and finde the Angle by a line of Chords. Secondly, more speedily and artificially, thus: By the lines of Sines being drawn to 2. 4, or 6 de∣grees asunder. The Sector so set, take
the parallel sine of 30 and 30, and measure it on the lateral sines from the center, and it shall reach to the sine of half the Angle the line of sines stands at, being more by 2, 4, or 6 degrees then the sector stands at, be∣cause it is drawn one, two, or three degrees from the inside.
Or else take the latteral sine of 30 from the center, and keeping one foot fixed in 30, turn the other till it cross the line of sines on that line next the inside, and counting from 90, it shall touch at the Angle the line of sines standsat, being two degrees more then the Sector stands at; the lines being drawn so, will be (as I conceive) most convenien.
I come to a south-east-wall, and putting my box and needle on my Rule, with the cross or north-end of the needle toward the wall, and the Rule being applyed ••lat to the wall, on the edge thereof, on the evenest
place thereof, and held level, so as the needle may play well with the head of the Rule toward your right hand, as you shall finde it to be in an east-wall most convenient; then I open or close the Rule, till the needle play right over the north and south line in the bottom of the box; then having got the Angle, (take off the box, or if you put it on the other side that labor may be saved.) I take the parallel sine of 30, and measuring it from the center it reaches, suppose to the sine of 20, then is the line of sines at an Angle of 40 degrees, but the Sector at two degrees less, viz. 38 de∣grees, whose complement 52 is the declination; then to consider which way, minde thus: First it is south, be∣cause the sun being in the south, shines on the wall. Secondly consider, the sun being in the east, it shines also on the wal, therefore it is east plain: thus have you got the denomination which way, and also the quantity how much that ways.
Or if you take the latteral sine of 30 from the center, and turn the point of the Compasses from 30 to∣wards 90, on the other leg you shall finde it to reach to the sine of 50 de∣grees, whose complement, counting from 90, is 40, or rather 38, for the reason before-said, or else adde 2 to 50, and you have the angle required, without complementing of it, be∣ing the true declination sought for. Thus by the needle you may get the declination of any wall, which in cloudy weather may stand you in good stead, or to examine an observation by the Sun, as to the mis-counting or mistaking therein; but for exactness the Sun is alwayes the best, because the needle, though never so good▪ may be drawn aside by iron in the wall, and also by some kinde of bricks, there∣fore not to be too much trusted unto.
First open the Rule to an Angle of 60 degrees, as you do to finde the
hour of the day, and put a pin in the hole, and hang the thread and plum∣met on the pin; also you must have another thread somewhat longer and grosser then that for the hour in a readiness for your use.
Then apply the head leg to the wall, if the sun be coming on the Plain, and hold the Rule horizontal or level, then hold up the long thread till the shadow falls right over the pin, or the center hole, at the same instant the shadow shall shew on the degrees, how much the sun wants of coming to be just against the Plain, which I call the Meridian or Pole of the Plain, which number you must write down thus, as suppose it fell on 40, write down 40-00 want: then as soon as may be take the Suns true altitude, and write that down also, with which you must finde the Suns Azimuth▪ then substract the greater out of the less, and the remainder is the declination required. But for a general rule, take this: if the Sun do
want of coming to the Meridian of the place, and also want of coming to the Meridian of the Plain, then you must alway substract the greater number out of the less▪ whether it be forenoon or after-noon, so likewise when the Sun is past the Meridian, and past the Plain also. But if the signes be unlike, that is to say, one past the Plain, or Meridian, and the other want either of the Plain, or Meridian, then you must add them to∣gether, and the sum is the declination from the South. Which rule for bet∣ter tenaciousness sake take in this homely rime.
Signs both alike substraction doth re∣quire, But unlike signs addition doth desire.
The further illustration by two or three Examples.
Suppose on the first of May, in the forenoon, I come and apply the Rule, being opened to his Angle of 60 de∣grees to the wall, (viz. the head leg,
or the leg where the center is) and holding up my thickest thread and plummet, so as the shadow of it cros∣seth the center, and at the same in∣stant also on 60 degrees, then I say the sun wants 60 deg. of coming to the Meridian of the Plain; at the same instant, or as soon as possible I can, I take the suns altitude, as before is shewed, and set that down, which suppose it to be 20 degrees, then by the rules before, get the Suns azimuth for that day, and altitude; which in our example will be found to be 94 degrees from the (south or) Meridian, then in regard the signs are both a∣like, i. e. want, if you substract one out of the other, there remains 34 the declination required; but for the right denomination which way, either north or south, toward either east or west, observe this plain rule: First, if the Sun come to the Meridian or Pole of the Plain, before it come to the Meridian or Pole of the place, then it is always an East-plain; but if
the contrary, it is a West-plain, that is to say, if the Sun come to the Me∣ridian or Pole of the place, before it comes to the Meridian or Pole of the Plain, then it is a West-plain. Also if the sum or remainder, after addi∣tion or substraction, be under 90, it is a South-plain; but if it be above 90, it is a North-plain. Also note, that when the sum or remainder is above 90, then the complement to 180, is always the declination from the north toward either east or west; So that according to these rules in our exam∣ple it is 34 degrees South-east.
Again in the morning, Iune 13. I apply my rule to the wall, and I finde the Sun is past the Pole or Meridian of the Plain 10 degrees, and the al∣titude at the same time 15 degrees, the Azimuth at that altitude, and day in this latitude, will be found to be 109 degrees want of south or pole of the place; therefore unlike signs, and to be added, and they make up 119 degrees, whose complement to 180
is 61; for 61 and 119 added, make up 180, therefore this Plain declineth 61 degrees from the north toward the east.
Again the same day in the after∣noon, I finde the Azimuth past the south or meridian of the place 30 de∣grees, and at the same time the Sun wants in coming to the meridian or pole of the Plain 10 degrees, here by addition I finde the declination to be 40 degrees south-west.
Note what I have said in these three examples, is general at all times; but if it be a fair day, and time and op∣portunity serve, to come either just at 12 a clock, when the Sun is the me∣ridian or pole of the place; or just when the Sun is in the meridian or pole of the Plain, then your work is onely thus:
First if you come to observe at 12, then applying your rule to the wall, and holding up the thread and plum∣met, how much so ever the Sun wants or is past the pole of the Plain, that
is the declination, if it be past it is east-wards; if it wants, it is south-west-wards; if neither, a just South Plain, and then the poles, or Meridians of place and Plain, are the same.
But secondly, if you come when the Sun is just in the pole of the Plain, then whatsoever you finde the Suns Azimuth to be, that is the decli∣nation; if it wants of south, it declineth East-wards; if it be past, it declineth West-wards.
Thus I have copiously, (and yet very briefly) shewed you the most ar∣tificial way of getting the declination of any wall, howsoever situated.
Note if the Sun be above 15 de∣grees wanting of the Meridian of the Plain, your rule will prove defective in taking the Plains Meridian when the center leg is next the wall, then you must turn the other leg to the wall, and then you finde a supply for all an∣gles to 45 degrees past the Plain. But for the supply of the rest which is 45 degrees, do thus: open the rule till the
great line of tangents & the outside of the leg make a right angle, for which on the head you may make a mark for the ready setting, then making the inside of the leg at the end of 45, as a center, the tangents on the other leg supply very largely the defect of the other∣sides. Or if you set on the box and needle on the rule, and open or shut the rule till the shadow of the thread shew just 12, then the Angle the Se∣ctor stands at, is the complement of what the sum wants, or is past the meridian or pole of the Plain.
FIrst draw a perpendicular-line for 12, as A B then design a point in that line for the center, as C, at the up∣per end, if it be a South Plain; or contrary, if it be a North Plain, then on that center describe an Arch of a
Circle on that side of 12, which is contrary to the Plains declination, as D E, and in that Arch lay off from 12 the substile, and on that the stiles height, and the hour of 6, being found by the tenth chapter, and draw those lines from C the center, then draw two parallels to 12, as in the direct south: then fit the distance of the parallels in the secant of the declina∣tion, and take the secant of the lati∣tude, and set it from the center C on the line of 12, to F, and on the pa∣rallel from 6 to G, and draw a line by those two points F and G, to cut the other parallel in H: then have you found 6, 3, and 9, then fit 6 G in the tangent of 45, and prick off 15, 30, and the respective quarters both ways from 6, for the morning, and after∣noon hours, then fit F G in 45, and lay off the same points from F both ways for 10, 11, 1, and 2, and the quarters also, if you please, and those shall be points to draw the hour-lines by. The stile must be set perpendi∣cular
over the substile, to the Angle found by the rules in the tenth chap∣ter, and then the Dial shall shew the true hour of the day, being drawn fit to his proper declination.
When you have drawn a line of 12, and appointed a center, make a Geo∣metrical square on that side 12, as the stile must stand on, as A B C D, the perpendicular side of which square may also be the parallel as before.
Then first, fit the side of the square in the sine of the latitude, and take out the cosine of the latitude, and fit that in the sine of 90; the Sector so standing, take out the sine of the de∣clination, and lay it from B toward C, unto ••, and draw A E for the substile, then upon the point E in the substile, raise a perpendicular, and take out the sine complement of the Plains declination, and set it on that perpendicular from E to F, for the
height of the stile above the Plain.
Again, fit the side of the square in the cosine of the latitude, and take out the sine of the latitude, and fit that over in the sine of 90, then take out the 〈◊〉〈◊〉 sine of the declination, and lay it from D to G for the hour of 6, and draw the line A G for the 6 a clock hour line.
Then again, fit the side of the square (or the distance of the parallel in the other way, when you want a secant, or your secant too little) in the sine of 90, and take out the cosine of the de∣clination, fit that in the cosine of the latitude, then take out the sine of 90, and lay it from the center on the line of 12, and 〈◊〉〈◊〉 6 in the side of the square, and by those two points draw the contingent line▪ and then fit those points or distances in the tangent of 45, and lay down the hours as in the former part of this chapter; but if you want the hours before 9 in west-decliners, or the hours after 3 in east-decliners, and the 6 fall too high a∣bove
the horizontal line on that side, you may supply this defect thus:
Take the measure with your Com∣passes from 3 to 4 and 5, upwards in the west-decliners, or from 9 to 7 and 8 in east-decliners, and lay it upwards from 9 or 3 on the deficient side, in the parallel as it should have been from 6 downward in south or upward in north Plains, and you shall see all your defective points to be compleat∣ly supplied, whereunto draw the hour∣lines accordingly.
IN all erect Decliners, the way of finding the stile, substile 6 and 12, and inclination of Meridians, is the same; but when you come to pro∣tract, or lay down the hour-lines, you shall finde them come so close toge∣ther,
as they will be useless, unless the stile be augmented.
The usual way for doing of which, is to draw the Dial on a large floor, and then cut off so much and at such a distance, as best serveth your turn, but this being not always to be affected, for want of conveniency, and large instruments, it may very ar∣tificially be done by a natural tan∣gent of 75 or 80 degrees, fitted on the legs of a sector in this manner.
The example I shall make use of, let be a Plain declination 75 degrees, South, West.
First on the north edge of the Plain draw a perpendicular-line A B, repre∣senting 12 a clock line; then on the center draw an occult Arch of a cir∣cle as large as you can, as B D, therein lay off an angle of 37. 30 for the substile, (though indeed this line will not prove the very substile, yet it is a parallel to it:) then cross it with two perpendiculers for the two contingent lines, at the most conve∣nient
places, one in the upper part, and the other in the lower part of the Plain, as the two lines C E, and F G do shew, then by help of the inclina∣tion of Meridians, make the table for the hours in this manner; for this declination, it is 78:09, Now every hour containing 15 degrees, and every quarter three degrees 45' I find that the substile will fall on neer a quarter of an hour past five at night, therefore if you take 75, the measure of five hours out of 78. 9, there re∣maines 03. 09. for the first quarter from the substile next 12. Again, if you take 78. 09, the inclination of Meri∣dians out of 78. 45, the measure of 5 hours one quarter, and there re∣maines 00. 36. for the first quarter on the other side of the substile, then by continual adding of 03. 45, to 03. 09, and to 00. 36, I make up the table as heer you see or else you may against 12, set down 78. 09, and then take out 3. 45, as often as you can, till you come to the substile, and then
what the remainder wants of. 3. 45, must be set on the other side of the substile, and 3. 45 added to that, till you come to as late as the sun will shew on the Plain, as here you see to 1/4 past 8.
| 45 36 | . | sub. | 48 09 | 2 |
| 41 51 | 8 | 03 095 | 51 54 | . |
| 38 06 | . | 06 54. | 55 39 | . |
| 34 21 | . | 10 39. | 59 24 | . |
| 30 36 | . | 14 24. | 63 09 | 1 |
| 26 51 | 7 | 18 094 | 66 54 | . |
| 23 06 | . | 21 54. | 70 39 | . |
| 19 21 | . | 25 39. | 74 24 | . |
| 15 36 | . | 29 24. | 78 09 | 12 |
| 11 51 | 6 | 33 093 | ||
| 08 06 | . | 36 54. | ||
| 04 21 | . | 40 39. | ||
| 00 36 | . | 44 24. |
Then consider what hours will be convenient to put on, also the two places most convenient for them two extreme hours: as for this Plain 8 at night, and •• afternoon, will fit best; and the two places I appoint for them
shall be at E, and C, in the upper contingent line. Then look in the table for the numbers of 8, and I finde it to be 41, 51, which sought in a table of natural tangents will be found to be 0 8 9 5 6 7. And the number in the table for 1 is 63. 09, whose natural number is found to be 1 9 7 5 3 7. Or if you want those tables, the line of Lines and Tan∣gents, or a Scale made equal to the Radius of the tangents, will supply the same though you cannot see so many figures, yet to four figures will serve our turn very well, or else the artificial tangent on the edge shall shew the natural number in the line of numbers, reading it as a Scale of equal parts, and having due respect to the increase of the Radius; for if it be under 45, the number right against it, is the natural tangent; if above 45 measure from 45 to it, that extent aplyed from 1 in the numbers, giveth the natural number required. Add these two numbers 0 8 9 6 and 19,
75, and they make 2 8 7 1: which sought in the tables will be found to be the Logarithm of the natural tan∣gent of 70. 48. or if you take it off from your Scale, and measure it on the line of tangents you shall finde it reach to 70. 48, as before, then take the whole space of your upper con∣tingent from E to C, the two points for 8 and 1, and fit it in the parallel tangent of 70. 48 and then the tan∣gent of 45 shall be Radius, and the ••ecant of the stiles height to the same Radius, shall be the stiles height on ••he contingent. (or the Radius be∣••ore, laid at the nighest distance from H to 〈◊〉〈◊〉 then the sector so set, take ••ut the number for 1, out of the line of tangents, and lay it from C to H, ••nd the number for 8, and lay it from E to H, and if your work be true they will meet, and that point H, is the ••ue place of the substile: then draw 〈◊〉〈◊〉 I parallel to A D, for the true ••••bstile, and the sector so standing lay ••ff all the numbers in the table, taken
out of the tangents from H toward E, and C as the numbers direct, and note them points with 1, 2, 3, 4, 5, 6, 7, 8. Then lay the Radius H K, from I to L, on the other contingent line, and making H I Radius, or the tangent of 45, lay off the tangent of the stiles height, from L to M 9. 16 then shall I M, be the other Radius, on the other contingent line, and I N the secant of 9. 16, at the Radius of I M, the place of the stiles height on this contingent line, being found more readily, by taking the Radius I M, at the nighest distance being about I ••, and H ••, then draw •• •• for the true height of the stile, being augmented to sit those hours 8 and 1, and to fil the Plain so, that no room be lost, the•• having drawn the stile, and set th•• sector to the Radius, lay down al•• the numbers in the table, on this con∣tingent line out of the line of tan∣gents, from I both ways: as th•• numbers direct, and mark ever•• whole hour, with 1, ••, 3, 4, 5, 6, 7, 8
as before. Lastly, lines drawn by these markes in the two contingent lines, shall be the hours and quarters re∣quired.
The stile or gnomon is to be ere∣cted right over the substile, to the an∣gle of 9. 16, and augmented as much as from H to ••, and from I to ••, at the nighest distance from the point H and I, drawnin the substile line on the Plain, this way is as easy, speedy, ar∣tificial and true, as any extant, (if your scales be true,) and improveth the Plain at the first, to the most pos∣sible best advantage that can be.
Note also, if in striving to put as many hours as you can on the Plain, the sum of the two natural numbers added together, comes to above 40000: then you must reject some∣thing of one of them, for they will not be comely nor convenient, nor far enough asunder. I might have enlar∣ged to declining, reclining plaines, out my intention is not to make a business of it, but onely to give a taste
of the usefulness, and convenience of a Joynt-rule, as now it is improved.
These directions are sufficient for any Horizontal, direct, and declining vertical Dials, but for all other, as East or West inclining and reclining, direct, or reclining and inclining, and declining polars; and all decli∣ning and inclining: and reclinimg Plains, the perfect knowledge of their affections and scituations is very hard to conceive, but much more hard to remember by roat, or knowingly.
Therefore for the better help of the imagination to conceive aright of their affections, and setuations you may be furnished with an Instrument in brass, at the house of Mr. Walter Hayes, neer the Popes-head Tavern in Morefields, at the Sign of the Cross-daggers, or at the house of Iohn Brown, at the Sign of the Sun-Dial in the Minories, or at Mr. Henry Suttons in
Thredneedle-street behind the Ex∣change. Which Instrument being righ••ly made, you may rectifie to any latitude, and to any declination or reclination, and being so set, it will give the substile, stile, inclination of Meridians, and every hours Ark on the Plain by occular inspection to half a degree, and shew you all the affections straightway: howsoever situated, as which Pole is elevated, and which way applied, and what hours will be useful, &c.
A representation of which instru∣ment take in this place, let the circle E S W N, represent the Horizon S AE P N the Meridian, AE W AE E the Equinoctial, P C P the Pole, being the points of an Axes made of th••ead, and passing through the center C W C E being shaddowed, repre∣sents the Plain, set to a declination, and to a reclination 45 degrees, F the foot that doth support it; moreover the Plain hat•• an Arch fitted to it, to get the stiles height, and to set it
to any angle; also the Horizon and Plain is made to turn round, which could not well be expressed in this figure, or representation of the in∣strument: a more liveley and easie help cannot likely be invented, and for cheapness the like may be in a half circle, or made of Past-board.
The use and application of which figure, I shall not now speak of, part∣ly because of the facillity in the using thereof, and partly because of the dif∣ficulty in description thereof; and lastly, wheresoever you shall buy the same, at either of the three places, the use will be taught you gratis.
Also note, that half of the Instru∣ment may be made either in b••ass, or pastboard, and be made to fold down in a book, and perform the uses there∣of indifferently well, for many pur∣poses, as to the Affections of Dials.
Then having discovered the af∣fections, you may by the Canno••s in the seventh Chapter of Mr. Windgates R••le of proportion, finde all the re∣quisites,
and then to speed the laying down the hours, you may do it by help of the tangent of 45, as you did in erect decliners; for after you have drawn the hours of 6, and three, or nine, and made a parallelogram by two lines parallel to 12, and one pa∣rallel to six, then making the distance from six to nine, or three in the line parallel to 12, a tangent of 45, and the distance from 12 to 9, or 3 in the parallel to 6, likewise a tangent of 45, in the sector, and laying off 15, 30. 45, or the respective quarters from 6, and 12, in those two lines, and they shall be the true points to draw the hour lines, by laying a ruler to them, and the center of the Dial; but for those that have no centers, the rule of augmenting the stile in far decliners, serveth for these also.
OPen the rule to his Angle of 60 degrees, then looking up to the top or point of altitude you would observe the height of, as you do when you take the Suns altitude, and note the degrees and parts the thread cuts, and write it down in chaulk or ink, that you forget it not.
Then measure from the place you stand to the foot or base of the object, (being right under the top of the object, whose height you would measure) in feet, yards, or any other parts.
Thirdly, consider this being a right angled Plain triangle, if you have the angle at the top, the angle at the base is always the complement there∣of.
These things being premised, the proportion holds, as the fine of the angle opposite to the measured side, (or base) being the complement of the angle found, is to the base or mea∣sured side: so is the angle found, to the height required.
Always remember 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 ap∣plied the same way, from the sine of the Angle found, shall reach on the line of numbers to altitude required.
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 op∣posite
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 An∣gle 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 num∣bers, lacking a small fraction, with which I shall not trouble you.
But if you cannot come to measure to the foot of the object, then you must observe at two places: As sup∣pose first at D, where as before the
Angle is found 56. 30, whose com∣plement is 33. 30, counting from the head of the Ru••e: Then I remove right backward to C, 111 feet, and observe the Angle counted from the head to be 48. 15, which is the angle at B, but the angle at C is 41. 45. Then the work is thus: first, you must take the angle at B at the first station, out of the angle at B, at the second station, and note the difference, then the rule runs thus:
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 work these or any other questi∣ons 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 requi∣••••••▪
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 Alti∣tude required.
After the same manner may que∣stions be wrought on the line of lines, sines, or tangents alone, or any one with the other, by changing the Lo∣garithmetical 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 ei∣ther
natural or artificial, you may find altitudes, by putting the line of qua∣drat, 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 di∣vided) to the parts cut by the thread, so is the distance measured, to the height required; which work is per∣formed 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, ma∣keth an Angle above 45, then the work is just the contrary to the for∣mer.
What is spoken here of taking of Altitudes, may be applied to the ta∣king of distances; for if the Sector be fitted with a staff, and a ball-soc∣ket, you may turn it either horizon∣tal, or perpendicular, and so take any Angle with it, very conveniently and
readily by the same rules and dire∣ctions as were given for the finding of Altitudes.
FIrst the most general and received line for mensuration of Magni∣tudes, is a foot divided into 12 inch∣es, 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 con∣clusions also.
1. As first, The price of any com∣modity 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 far∣things, 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 far∣things there is in the price of one pound, twice so many shillings, and once so many pence is the remain∣der, which if it be above 12, the 12 or 12s, being substracted, the remain∣der 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 far∣things, and about a quarter of one far∣thing. Thus by the lines, as they are divided, it proceeds to 12 pence a pound; but if you conceive the inch∣es to be doubled, and the foot mea∣sure 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 Tim∣ber 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 mea∣sure taken as it is: As at 40 shillings the Load, 40 in the foot measure stands right against 4 inches 3 quar∣ters 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 dou∣ble, 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 mea∣sure, may be applied for the reducing of any odde fraction to a decimal fra∣ction,
as you may fee it in page 64. of Mr. Windgats Arithmetick made easie.
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 in∣ches, and the use of them is thus: And first for the decimal Board mea∣sure. 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.
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.
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 Tim∣ber measure, and from thence to the end of the Rule, is the length that goes to make a foot of timber.
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.
The decimal yard measure, is no∣thing 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 toge∣ther, you shall have the content, in yards and 100 parts of a yard.
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 su∣perficial measure, at that breadth or height.
But if the peece be between 4 foot 510 broad, and 2 foot, then the ta∣ble 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.
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.
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.
Take the diameter with a rule, or a pair of Callipers, and learn the mea∣sure 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 mea∣sure for all these lines follow.
| 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. ••60 | ••. 440 | |||
| 7 | 5. 888 | 2. 382 | |||
| 8 | 5. 5••5 | 2. 336 | |||
| 9 | 5. 260 | 2. 273 | |||
| 2 | 2 | 5. 000 | 2. 213 | ||
| 4. 760 | 2. 173 | ||||
| 4. 5••6 | 2. 127 | ||||
| 4. 350 | ••. 083 | ||||
| 4. 170 | 2. 042 | ||||
| 5 | 4▪ 000 | 5 | 2. 000 |
| 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 |
| 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 |
| F. 1000. p. | 01. | F. 100. p. | |
| 3. 825 | 1. 738 | ||
| 3. 7••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. 93•• | 1, 061 | ||
| 1. 878 | 1. 041 | ||
| 1. 830 | 1. 021 | ||
| 1. 781 | 10. | 1. 000 |
| 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 |
| F. 10. | F. 1000. p. | F. 1000. p. | |
| 1 | 90. 000 | 4 | 3. 7••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 |
| 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. 5. 72 |
| 13 | 10. 8. 09 | 36 | 1,4,75 |
| 14 | 9. 2. 79 | 37 | 1. 3, 86 |
| 15 | 8. 0. 51 | 38 | 1, 3, 04 |
| 16 | 7. 0. 82 | 39 | 1, 2, 28 |
| 17 | 6. 3. 14 | 40 | 1, 1, 57 |
| 18 | 5. 7. 30 | 41 | 1, 0, 92 |
| 19 | 5. 0. 15 | 42 | 1, 0, 31 |
| 20 | 4. 6. 28 | 43 | 0, 11, 74 |
| 21 | 4. 1. 24 | 44 | 0, 11, 22 |
| 22 | 3. 8. 87 | 45 | 0, 10, 72 |
| 23 | 3. 5. 04 |
| In. over | F. in 10 | F. in 100 | |
| , | 29 3 0, 0, 00 | , | 4, 8, 31 |
| , | 73••, 1, 00 | , | 4, 4, 06 |
| , | 326, 0, 10 | , | 4, 0, 27 |
| 1 | 183, ••, 80 | 7 | 3, 8, 89 |
| , | 118, 2, 16 | , | 3, 5, 85 |
| , | 81, ••, 00 | , | 3, 3, 10 |
| , | 59, 10, 24 | , | 3, 0, 61 |
| 2 | 45, 10, 90 | 8 | 2, 10, 36 |
| , | 36, 2, 41 | , | 2, 8, 31 |
| , | 29, 3, 93 | , | 2, 6, 44 |
| , | 24, 3, 86 | , | 2, 4, 73 |
| 3 | 20, 4, 40 | 9 | 2, 3, 15 |
| , | 17, 4, 25 | , | 1, 1, 70 |
| , | 15, 0, 00 | , | 1, 0, 37 |
| , | 13, 1, 86 | , | 1, 11, 13 |
| 4 | 11, 5, 14 | 10 | 1, 9, 99 |
| , | 10, 1, 78 | , | 1, 8, 93 |
| , | 9, 0, 62 | , | 1, 7, 95 |
| , | 8, 1, 80 | , | 1, 7, 04 |
| 5 | 7, 3, 98 | 11 | 1, 6, 18 |
| , | 6, 7, 80 | , | 1, 5, 38 |
| , | 6, 0, 72 | , | 1, 4, 63 |
| , | 5, 6, 53 | , | 1, 3, 93 |
| 6 | 5, 1, 10 | 12 | 1, 3, 27 |
| 13 | 1, 1, 02 |
| Feet high. | at 1 brick thick. | at 1 brick & 1/2 thick |
| 1 | 176 | 264 |
| 2 | 352 | 528 |
| 3 | 528 | 792 |
| 4 | 704 | 1056 |
| 5 | 880 | 1320 |
| 6 | 1136 | 1704 |
| 7 | 1232 | 1848 |
| 8 | 1408 | 2112 |
| 9 | 1584 | 2376 |
| 10 | 1760 | 2640 |
| 11 | 1936 | 2904 |
| 12 | 2112 | 3168 |
| 13 | 2288 | 3432 |
| 14 | 2464 | 3696 |
| 15 | 2640 | 3960 |
| 16 | 2816 | 4224 |
| 16 1/2 | 2904 | 4356 |
| 17 | -2992 | 4488 |
| 18 | -3168 | 4752 |
| 19 | -3344 | 5010 |
| 20 | -3520 | 5280 |
If you would have this Table for 1/2 a brick, take the half of the table for one brick. If for two bricks then double it. If for two and a 1/2 then ad both these together; if for three, double that for one brick and 1/2.
If you have any number of feet of brick work, at half a brick, one brick, or two bricks, or more, and you would reduce it to one brick and half, then say by the line of numbers as 1. 2. 4. 5 or 6 is to three, so is the number of feet at 1/2 1. 2. 2 1/2 or three bricks to the number of feet at one and 1/2.
Suppose you have a circle whose diameter is 10 inches, or 10 feet: and to this circle you would finde the Circumference, or the side of a square equal, or inscribed, or having any one of the three, to finde the other three, do thus:
Take the measure of the Circum∣ference,
Diameter, or either of the squares, which is first given, and open the compasses to the number of the given measure, in its respective scale: the compasses so set, if you apply it in the scale whose number you would know, you shall have your desire.
Suppose a circle whose diameter is 10 inches, and to it I would know the Circumference, take 10 out of the diameter scale, and in the Circum∣ference scale it shall reach to 31 42, and on the line of square equal 8 86, and on square inscribed 7. 07. For il∣lustration sake, note the figure.
Take the Semidiameter, or Radius of the circle between your compasses, and fit it over in six and six of the line of circles, then what number of parts you would have, take off from that point by the figure in the line of circles, and it shall divide the circle
into so many parts. As suppose I would have the former circle divided into nine parts, take the measure from the center to the circle as exact∣ly as you can, fit that over in 6 and 6, then take out 9 and 9, and that shall divide it into so many parts; but if you would divide a wheel into any odde parts, as 55. 63. or 49 parts, you shall finde it an almost impossible thing, to take a part so exact that in turning about so many times, shall not miss at last: to help which the parts the rule giveth shall fit you exact enough for all the odde parts, then the even will easie be had by dividing, therefore usually the rule is divided but to 30 or 40 parts. So that for this use as the finding the side of an 8 or 10 square piece, as the mast of a ship, or a newel, or a post, this will very readily, and exactly help you.
1. ANd first for superficial or board measure, by the inches, the breadth and length being given in inches, and feet and parts, slide or set 12 on any one side, to the breadth in inches or parts on the other side, then just against the length found on the first side, where 12 was on the second side you shall have the con∣tent in feet, and 10ths, or 100 parts required. Which by the rules of re∣duction by the foot measure, you may reduce to inches and 8 parts.
At three inches broad, and 20 foot long, you shall finde it to be 5 foot just: but at 7 inches broad, and the same length it will come to 11 foot 7 10th fere, or 8 inches.
Set 12 on the first side, to the breadth in inches on the second side, then look for 12 on the second side, and right against it on the first side, is the number of inches, that goes to make a foot Superficial, at that breadth.
At three inches broad you shall finde 48 inches to make a foot.
Set one on the first side, to the mul∣tiplicator on the second side, then seek the multiplicand on the first, and right against it on the second, you shall finde the product.
If 9 be multiplied by 16, you shall finde it to be 144.
Set the divisor on the first side to one on the second, then the dividend on the first, shall on the second shew the quotient required.
If 144 be to be divided by 16, you shall finde the quotient to be 9.
Set the first term of the question, sought out on the first line, to the se∣cond term on the other, (or second line:) then the third term sought on the first line, right against it on the second, you shall finde the fourth pro∣portional term required.
If 15 yard 1/2, cost 37s. 6d. what cost 17 3/4? facit 42s. 10d. 3 q. for if you set 15 1/2 right against 37 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 fra∣ctions
are all decimal, and you must reduce them to proper fractions ac∣cordingly
4. Set the first term sought out on the first line, to the second being of the same denomination or kind to the se∣cond line, or side. Then seek the third term on the second side, and on the first you shall have the answer re∣quired.
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.
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.
1. The inches that a piece of Timber is square, being given: to finde how much in length makes a foot of Tim∣ber, 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 Pro∣blem of the 5th chapter of the Car∣penters
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 dou∣ble 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 in∣ches 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. •• 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 num∣ber: 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, ac∣cording to feet and inches, I have therefore fitted these directions ac∣cordingly, and there are sufficient to the ingenious practitioner.
HAving the latitude, and comple∣ments of the declination, and Suns altitude, and the hour from noon, to finde the Suns Azimuth, 〈◊〉〈◊〉 that time.
Take the right sine of the comple∣ment 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 decli∣nation, 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.
Take the right sine of the decli∣nation, 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.
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 ga••••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 else as before.
As the right sine of the comple∣ment of the Suns declination: is to the parallel sine of the Azimuth, so is the right sine of coaltitude, to the parallel sine of the hour from noon, counting as before.
Having the complements of the la∣titude, Suns altitude, and declina∣tion. to finde the Suns Azimuth from the north part of the horizon.
1. First of the complement of the latitude, and Suns present altitude finde the difference.
2. And secondly count it on the line of sines from 90 toward the center.
3. Take the distance from thence, to the sine of the Suns declination; but note when the latitude and decli∣nation differ, as in winter you must counte the declination beyond the cen∣ter,
and you must call it the Suns di∣stance from the pole.
4. Fourthly, make that distance a parallel sine in the complement of the latitude.
5. Fifthly, then take out the pa∣rallel sine of 90.
6. And sixthly, make it a parallel sine in the coaltitude.
7. Seventhly, then the sector so set, take out parallel, Radius, or sine of 90.
And eighthly, measure it on the line of sines from 90 towards, (and if need be beyond) the center: and it shall reach to the versed sine of the Suns Azimuth from the north, or if you count the other way from the south, note that in working of these, if the line of sines be too big, then you have two or three smaller sines on the rule, where on to begin and end the work.
Latitude 51. 32, Declination 18▪ 30, Altitude 48. 12, you shall finde the Azimuth to be 130 from the North, or 50 from the South.
Having the complements of the la∣titude and declination, or Suns di∣stance from the Pole, and the Sun altitude given, to finde the hour from East or West, or else from noon.
1. First of the complement of the latitude, and Suns distance from the Pole, finde the difference.
2. Count this from the sine of 9•• toward the center.
3. Take the distance from thence to the sine of the Suns altitude.
4. Make that distance a parallel sine of the complement of the latitude.
5. Take out the parallel sine of 90 degrees, and
6. Make that a parallel sine in the codeclination, then
7. Take out the parallel sine o•• 90 again, and
8. Measure it from the sine of 90 toward the center, and it shall shew the versed sine of the hour from the North, or the sine of the hour from East or West; or if you reckon from 90, the hour from noon required.
Latitude 51. 32, Declination North 20. 14, Altitude 50. 55, you shall finde the hour from the North to be 10 houres, or 10 a clock in the forenoon, or 4 hours past 6, or two short of noon, according to each pro∣per reckoning.
Having the latitude, the comple∣ment of the Suns declination, the Suns present altitude, and Meridian altitude for that day, to finde the hour.
Make the lateral secant of the la∣titude, a parallel sine in the codecli∣nation, then take the distance from the Suns Meridian altitude, to his present altitude, and lay it from the
cer on both sides of the line of sines, and take the parallel distance between those two points, and measure it from 90 on a line of sines, of the same Radius the secants be, (as the small adjoyning sine is,) and it shall shew the versed sine of the hour from noon, or the right sine before or after 6, to∣wards noon or night.
Having the Latitude, Declination, and Suns Altitude, to finde the Suns Azimuth.
Take the latteral secant of the la∣titude, and make it a parallel in the complement of the Altitude:
Then take the distance, between the sun•• of the complement•• of of Museum the latitude and altitude, (if under 90,) and the sons declination, and lay it from the center on the line of sines, that parallel distance taken, and mea∣sured on the sines (of the same Radi∣us the secant was) from 90, shall shew the versed sine of the Azimuth from noon.
But if the sum of the colatitude, and coaltitude exceed 90, then take the excess above 90, out of the na∣tural sine from the center toward 90, and add that to the sine of the Suns declination towards 90, and then the parallel distance between those two points, shall be the Azimuth required, from noon,
But when the latitude and decli∣nation are unlike, as with us (in the northern parts) in winter, then you must take the declination out of the excess, or the lesser out of the greater, and lay the rest from the center, and the parallel distance, shall be versed sine of the Azimuth from noon.
At 18 15 altitude, latitude 51 32, declination 13 15 South.
The sum of the colatitude, and the coaltitude is 109. 37, then count the center for 90, the right sine of 10 for a 100 and 19 for 109, and 37 minutes forwarder, there set the point of the Compasses, then take from thence to
the right sine of the declination, and lay this distance from the center on the line of sines, and the parallel space between, is the versed sine of the Azimuth required.
Having the length of the shaddow of any object standing perpendicular, and the length of that object, to finde the altitude.
Take the Tangent of 45, and make it a parallel in the length of the shad∣dow in the line of lines, then the pa∣rallel distance between the length of the object that casts the shaddow, taken from the line of lines, and mea∣sured on the line of tangents from the center, shall reach to the Suns altitude required.
If the object be 40 parts long, and the shaddow 80 parts, the altitude will be found to be 26. 35. But if you have the altitude, and shaddow, and would know the height of the object, then work thus:
Take the length of the shaddow out of the line of lines, or any other equal parts, and make it a parallel tangent of 45, then take out the parallel tangent of the Suns altitude, and measure it on the line of lines, (or the same equal parts) and it shall shew the length of the object that caused the shaddow: the same rule doth serve in taking of altitudes by the rule, as in the 18 chapter, accoun∣ting the measure from the station to the object, the length of the shaddow, and the Suns altitude, the angle at the base.
To finde the Suns rising and setting in any latitude.
Take the latteral cotangent of the latitude, make it a parallel in the sines of 90 and 90, then take the latteral tangent of the Suns declina∣tion, and carry it parallel in the sines till it stay in like sines, that sine shall be the asentional difference between six, and the time of rising before or
after 6, counting 15 degrees to an hour.
Take the latteral sine of 90, and make it a parallel 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.
Take the lateral or right sine of the declination, and make it a paral∣lel 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 alti∣tude 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 de∣grees, 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 Pro∣positions, 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.
As the right Sine of 90,
Is to the parallel cotangent of the la∣titude:
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 re∣quired.
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 declina∣tion 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.
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.
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
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.
As the latteral tangent of the decli∣nation,
To the parallel sine of 90:
So is the parallel sine of the latitude,
To the latteral cotangent in the incli∣nation
of Meridians.
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 ele∣vation,
To the latteral tangent of the hour
from the substile.
As the latteral tangent of the comple∣ment
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 ar∣tificial Sines and Tangents, by chang∣ing 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.
1. FIrst next the center is two cir∣cles divided one into 60, the other into 100 parts, for the re∣ducing 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 en∣tire 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 in∣crease, but the Tangents are divided into single minutes from the begin∣ning 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 disco∣vered.
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 no∣thing else but the sines reitterated beyond 90.
5. Fifthly, you have the line of Tan∣gents beyond 45, in 5 turnes to 85 degrees, whereby all trouble of back∣ward 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 mid∣dle 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 quar∣ters 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 contrary, if you have a deci∣mal fraction, and would reduce it to a proper fraction, the like may you do for shillings, and pence, and pounds, and ounces.
As to the use of these lines, I shall in this place say but little, and that for two reasons. First, because this instrument is so contrived, that the use is sooner learned then any other, I speak as to the manner, and way of using it, because by means of first, se∣cond, and third radiusses, in sines and Tangents, the work is always right on, one way or other, according to the Canon whatsoever it be, in any book that treats of the Logarithms, as Gunter, Wells, Oughtred, Norwood, or others, as in Oughtred from page 62 to 107.
Secondly, and more especially, be∣cause the more accurate, and large handling thèreof is more then promi∣sed, if not already performed by more abler pens, and a large manuscript thereof by my Sires meanes, provi∣ded many years ago, though to this day not extant in print; so for his sake I claiming my intrest therein, make bold to present you with these
few lines, in order to the use of them: And first no••e.
1. Which soever of the two legs is set to the first term in the question, that I call the first leg always, and the other being set to the second term, I call the second leg.
2. Secondly, if one be the first or second term, then for the better set∣ting the index exactly, you may set it to 100, for the error is like to be the least neerest the circumference.
3. Thirdly, be sure you keep a true account of the number of turnes be∣tween the first and second term.
4. Fourthly, observe which way you move, from the first to the second term. To keep the like from the third to the fourth, exept in the back rule of three, and in such cases as the Ca∣non requires the contrary.
5. Fifthly in multiplication, one is always the first term, and the multi∣plycator or multiplycand the second, and the product always the fourth. Also note that in multiplycation the
product of two numbers multiplyed, shall be in as many places as both the multiplycator, and multiplycand, except the least of them, be less then the two first figures of the product; moreover, for your more certain as∣singing of the two last figures of four or six, which is as many as you can see on this instrument, multiply the two last in your minde, and the product shall be the figure, as in page 28 of the Carpenters Rule.
6. In division, the multiplicator is always the first term, and one the se∣cond, the dividend the third, and the quotient the fourth; also the quotient shall have as many figures as the di∣vidend hath more then the divisor, except the first figures of the divisor be greater then the dividends, then it shall have one less. Also note, the fraction after division, is a decimal fraction, and to be reduced as before.
7. Note carefully whether the fourth proportional ought to be a greater or a less, and resolve accor∣dingly,
and note if one cometh be∣tween the third and fourth term, then must the fourth be raised a Radius or a figure more, and be careful to set the hairs exactly over the part repre∣senting the number or minutes of any degree.
8. Always in direct proportion, and Astronomical calculation, set the first leg to the first term, and the se∣cond leg to the second term, and note how many circles is between, then set the first leg to the third term, and right under the second leg, the same way, and so many turnes between the third and fourth, is always the fourth term required.
As 1, to 47, so is 240, to 11280. As the sine of 90, to 51 degrees 30 minutes, so is the sine of 80 to 50. 26. And so of all other questions accor∣ding to their respective Canons by the Logarithms in other books as in Mr. Oughtreds Circles of Porportions, from page 62 to 107, and others.
Here followeth the working of cer∣tain Propositions by the Serpentine∣line.
Those that I shall insert, are onely to shew the manner of working, and knowing of that once, all the Canons for all manner of questions, either in Arithmetick, Geometry, Navigation, or Astronomy, by any other Author, as Mr. Gunter, Mr. Oughtred, Mr. Wind∣gate, Mr. Norwood, or others, may be speedily resolved, and as exactly as by the Tables, if the instrument be well and truly made. And first ••or the hour, according to Gunter.
Add the complement of the altitude, the complement of the Suns present altitude, and the distance of the Sun, from the elevated Pole together, and nore their sum, and half sum, and find the difference between their half sum
and the complement of the Suns pre∣sent altitude, then work thus: for 36 42 degrees high, at 23. 32 declina∣tion, lat. 51. 32.
Lay the first leg, viz. that next your right hand being here most conveni∣ent, on the Sine of 90, keeping that fixed there, lay the other leg to the cosine of the latitude, viz. 38. 28. and note the turns between, which here is none between, but it is found in the next over it, then set the first leg to the Sine of 66, 28, the suns di∣stance from the Pole, and in the cir∣cle just over it you shall have the Sine of 34. 47. for the fourth Ark or Sine.
Set the first or left leg to the sine of 34. 47. of the fourth Sine last found, then keeping that fixed there, set the other leg to the Sine of the half sum, viz. 79. 37. then remove the first leg to the Sine of the difference between the half sum, and the coaltitude, viz.
25. 49. and then in the next circle, the other leg shall shew the sine of 48. 34. whose half distance toward 90, being found by the Scale of Lo∣garithms on the outermost circle, will discover the Sine of an Ark, whose complement being doubled, and turn∣ed into time (by counting 15 degrees to an hour) will give the hour requi∣red; but by help of the versed Sines all this trouble is saved; for when the index or second leg cuts the Sine of 48. 34. at the same instant it cuts the versed Sine of 60, the hour from noon required, being 8 in the morn∣ing, or 4 in the afternoon, at 23. 32. of declination, in the latitude of 51. 32.
To work the same another way, ac∣cording to Mr. Collins.
Lay the left leg on the Sine of 90, and the other to the Sine of 23. 32, and you shall finde one turn between upwards, then the first leg laid on the Sine of the latitude 51, 32, the o∣ther leg shall shew the Sine of 18. 3. (in the second circle above 51, 32.) for the Suns height at 6 required, and this is fixed for one day.
Then in summer time, or north de∣clinations, by help of the line of na∣tural sines, in the second line, finde the difference between the Suns pre∣sent altitude, and the latitude at 6, but in Winter or Southern (signs or) declinations, add the two altitudes to∣gether, in this manner. Lay one leg of the index to the natural sine of the altitude at 6, and the other to the altitude proposed, the two legs so set, bring one of them (viz the right) to the beginning of the line of natural
sines, and the other shall stay at the difference required, but in Winter set one leg to the beginning of the sines, and open the other to the height at 6, or rather depression under the Horizon at 6, (which is all one at like declinations, North and South) then set the first leg to the present al∣titude of the Sun, and the other shall shew the Sine of the sum of both ad∣ded together; which sum or differ∣ence is thus to be used:
Lay the left leg to the Cosine of the declination, and the other to the secant of the latitude, counted be∣yond 90, as far as the secant of 9, 40; or rather lay the left leg on the Co∣sine of the latitude, and the other to the secant of the suns declination, then the first leg laid on the sine of the sum in winter, or difference in summer, shall cause the other leg to fall on the sine of the hour from 6, toward noon in winter and summer, except the altitude in summer be less then the altitude at 6, then it is the
hour from 6, toward mid-night.
First you must get the Suns altitude, or depression in the vertical Circle by this Cannon. Lay the first leg to the sine of the latitude, and the second to the sine of 90, and you shall finde them both to be on the same line, then the first leg laid on the sine of the de∣clination, shall cause the second (be∣ing carried with the first, without mo∣ving the Angle first set) to fall on the sine of 30. 39. the Suns altitude in the Vertical Circle; with which you must do, as you did before with the altitude at 6, and the present alti∣tude, to finde the sum and difference by help of the line of natural Tan∣gents, then this proportion holds.
Lay the first leg to the cosine of the
Altitude, (by counting the Altitude from 20) and the second leg to the tangent of the latitude, and observe which way, and the turns between; then the first leg removed and laid to the sine of the sum (before found) in winter, or the difference in sum∣mer, shall cause the second leg to fall on the sine of the Azimuth of the Sun, from east or west toward noon, if winter; and also in summer, when the Suns altitude is more than his al∣titude at the Vertical Circle; but if less from the east or west, toward north or mid-night meridian: Thus in our Example, it will be found to be the sine of 30 degrees, or 60 from the south, the sine of the difference being found to be 14. 49.
First by the Suns declination get his
distance from the Pole, which in sum∣mer or North declinations, is always the complement of the declination, (likewise in south latitudes, and south declinations) but when the latitude and declination is unlike, then you must adde 90 to the declination, and the sum is the distance from the ele∣vated Pole. Having found the di∣stance from the Pole, adde that and the complement of the latitude, and Suns altitude together, finde the dif∣ference between their half sum, and the Suns distance from the Pole, then the proportion will be thus, as in this Example: 13 Declination, 41. 53. Alt. Lat. 51. 32.
Lay one leg on the sine of 90, and the other to the sine of 38, 28, being so set, remove the first leg to the sine complement of the altitude 48. 07. and the second legg shall fall on a fourth sine, which will be found to be 27. 36. then set the first or left leg to 27. 36. the fourth sine, and the second to 81. 47. the sine of the
half sum, then removing the first leg to the sine of the difference, shall cause the second to shew two circles lower, the versed sine of 130, the Azimuth required, being counted from the North part of the horizon, whose complement to 180 from the South is 50 degrees.
Having the latitude, the Suns Meri∣dian, and present altitude, and decli∣nation, to finde the hour from noon.
First lay one leg to the Meridian altitude, in the line of natural sines, and the other to the sine of the alti∣tude in the same line, then bring the right leg to the beginning of the line of sines, and the other 〈…〉〈…〉 difference, which difference 〈…〉〈…〉 keep.
Then lay one leg on the 〈◊〉〈◊〉 the declination, and the other 〈◊〉〈◊〉
secant of the latitude, and note the turnes between, or rather lay the first leg to the cosine of the latitude, and the other to the secant of the decli∣nation, then the legs being so set, bring the first or left leg to the sine of the difference first found, and the other leg shall shew the versed sine of the hour from noon, if the ver∣sed sines had been set thus, i. e. the versed sine of 90, against the sine of 90, as in some instruments it is: But to remedy this defect, do thus: keep the right leg there, and open the o∣ther to the versed sine of 0. or sine of 90, and note the turnes between, then lay the leg that was on the sine of 90, (or versed sine of 0,) to the versed sine of 90, and the other leg shall shew the versed sine of the hour from noon, counting from 90.
At 45 13 degrees altitude, decli∣nation North 23 32. latitude 51 32, the Meridian altitude is 62, (being found by adding colatitude, and de∣clination
together, and in southern declinations by substraction.) Then the natural sine of 45. 42, taken from 62, shall be the sine of 9. 38. then as the cosine of the latitude 38. 28, is to the secant of declination 23. 32, so is the sine of 9. 38. to the sine of 17. 05) or versed sine of 45▪ if they were placed and numbred, as in some instruments they be: but to help i•• in this, say: as the versed sine of 0, is to the verses of 114. 26, so is the ver∣sed sine of 90, to the versed sine of 135, whose complement to 180 is the angle or hour from noon required.
Secondly, for the Suns Azimuth.
First add the complements of the coaltitude and cola••itude together, then if the sum be under 90, take the distance between the cosine of it, and the sine of the declination, in the line
of natural sines, and measure it in the line of sines from the beginning, and it shall give the sine of the difference; but if the sum exceed 90, then when the latitude and declination is alike, add the excess to the declination; but if contrary substract one out of other, and measuring the sum or remainder from the beginning of the sines, you have the difference which you must keep.
Then lay the first or left leg to the cosine of the altitude, and the second to the secant of the latitude, or else lay the first to the cosine of the lati∣tude, and the other to the secant of the altitude, and more the turnes be∣tween, then lay the first leg to the sine of the difference before found, and the other shall shew the versed sine of the Azimuth from noon re∣quired, if the versed sines be set as before is expressed, that is to say 90 of right sines, and versed sines toge∣ther, and numbred forwards as the sines be: but in the use of this instru∣ment,
the remedy aforesaid supplyeth the defect.
At 10. 19. altitude, 23. 32. decli∣nation, 51. 32. latitude, to finde the Azimuth.
The sum of the coaltitude and co∣latitude is 118.09, the excess above 90, with right sine of declination ad∣ded is 60. 30, found by natural sines, then say, As the cosine of latitude, to the secant of altitude, so is the sine of the difference 60. 30, to the versed sine of the Azimuth, but here to the versed sine of an ark beyond Radius unknown, then as the versed sine of 0, to that ark, so is the versed sine of 90, to the ver sine of 60, the azimuth from noon, whose complement to 180, is 125 the Azimuth from Sonth re∣quired.
Having been so large in these, I shall in the rest contract my self as to the repetition, and onely give the Canon for the propositions following, the way of working being the same in
all other, as in these before rehearsed, and note also what is to be done by the Serpentine-line, is to be done by the three same lines of numbers, sines and Tangents on the edge of the Sector, by altering the term leg, to to the point of the Compasses. The Canons follow.
And first for the hour of 6.
As the sine of 90,
To the sine of the latitude 51. 30:
So is the sine of the declination 23. 30,
To the sine of the altitude at 6. 18. 13.
Secondly, for all hours in the Equi∣noctial.
To the sine of the altitude of the Sun at the time required 18. 07, But for all other times say,
As the sine of 90,
To the cotangent of the latitude
38. 28.
So is the sine of the Suns distance
from 6, 30, 0,
To the tangent of the 4 arke 21 40.
Which fourth arke must be taken out of the Suns distance from the Pole 66, 21, (in Cancer) leaveth a resi∣due 44 48, which is called a fift ark. But for the hours before and after 6, you must add the fourth arke to the Suns distance from the Pole, and the Sum is the fifth ark. Then say,
As the cosine of the fourth ark 78, 20,
Is to the sine of the latitude 51, 32:
So is the cosine of the residue 45, 12,
To the sine of the Suns altitude at
8, 36, 42, at that declination.
And first to finde the Suns altitude at any Azimuth in the aequator. Then, As the sine of 90, to the cosine of the Suns Azimuth from the South 50, 0, So is the cotangent of the latitude 38, 28, to the tangent of 27, 03▪ the Suns altitude, at that Azimuth required.
Secondly, to find it at all other times, do thus: As the sine of the latitude 51, 32, To the sine of the Suns declination, 23, 32:
So is the cosine of the Suns alti∣tude in the aequator, at the same Azimuth from the vertical, viz. 30, to the sine of a 4th. ark 28, 16.
Which fourth ark must be added to the Suns altitude at the aequator in all Azimuths under 90, from the me∣ridian, where the latitude and decli∣nation are alike.
But in Azimuths more then 90 from the meridian, take the altitude in the aequator out of the fourth ark, and the sum or remainder shall be the altitude required, viz. 42, 56. But when the latitude and declina∣tion are unlike, as with us in winter time, then take the fourth ark out of the altitude at the aequator, and you shall have the altitude belonging to that Azimuth required.
As the cosine of the altitude,
To the sine of the hour,
So is the cosine of the Suns decli∣nation,
To the sine of the Azimuth required.
As the cosine of the declination,
To the sine of the Suns Azimuth:
So is the cosine of the altitude,
To the sine of the hour.
As the tangent of the latitude,
To the tangent of the Suns declina∣tion,
So is the sine of 90,
To the cosine of the hour from
noon.
As the cosine of the latitude,
To the sine of the declination:
So is the sine of 90,
To the sine of the amplitude from
the East or West, toward North
or South, according to the time
of the day and year.
As the cotangent of the latitude,
To the sine of 90:
So is the tangent of the Suns decli∣nation,
to the sine of the Suns as∣centional
difference between the
hour of 6 and the Suns rising.
As the sine of 90,
To the suns distance from the next
equinoctial point,
So is the sine of the suns greatest
declination,
To the sine of his present declinati∣on
required.
As the tangent of the greatest de∣clination,
To the sine of 90:
So the tangent of the present de∣clination,
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.
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 ob∣serve 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 ••hew on the line of tangents, so many turns be∣tween, 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 Au∣thors.
The squaring of a number, is no∣thing 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 re∣quired: 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 num∣ber 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 th•• 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 num∣bers, it cu••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 ••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 Log••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 di∣vided 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.
To work questions of interest or pro∣gression, 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 Loga∣rithms, 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 on∣wards 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 back∣ward, and then set the same first leg to the sum 253, and the second shall fall on 380. 8 s. according to estima∣tion; the contrary work is to finde what a sum of money due at a time to ••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.
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 let∣ters also: but note the Dominical let∣ter 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 let∣ter, 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, ac∣cording 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 re∣mainder 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.
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 o∣ther successive day sought for.
First the week day for 1661 is Fri∣day, as the letter F on the next col∣lumn beyond 60 she••et••; 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 Fri∣day, 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.