The triangular quadrant, or, The quadrant on a sector being a general instrument for land or sea observations : performing all the uses of the ordinary sea instruments, as Davis quadrant, forestaff, crosstaff, bow, with more ease, profitableness, and conveniency, and as much exactness as any or all of them : moreover, it may be made a particular and a general quadrant for all latitudes, and have the sector lines also : to which is added a rectifying table to find the suns true declination to a minute or two, any day or hour of the 4 years : whereby to find the latitude of a place by meridian, or any two other altitudes of the sun or stars / first thus contrived and made by John Brown ...

THE USES:

I. To find the suns declination, true place, right assention, and rising, the day of the moneth being given.

First open the rule to an angle of 60. degrees, which is alwayes done when the cross peece is fitted into the Mortesse holes, and the pins of the object sights put in the holes through the tennons, or else by the second Chapter of the Joynt-rule: then extend a thred from the center pin in the head leg, to the day of the month, & on the degrees it cuts the suns declination, in the line of right assention his right assention, in the line of true place his true place, and in the hour line his true rising and setting,

in that latitude the line is m••de for: Exam∣ple, on the first of May I would know the former questions, the rule being set by the crosse peece, and the thred on the leg cen∣ter pin; and drawn straight and laid over May 1. it cuts in the degrees 18. 4. north declination, and 20. 58. in ♉ Taurus for his place, and 3 hours 14. minutes right assention in time, or 48. 32. in degrees: and the rising of the sun that day is at 4. 23, and sets at 5. 37, in 51. 32 lati∣tude.

The finding of hour and azimuth, either particularly, or generally, with other A∣stronomical propositions, are spoken enough of before in the Joynt-rule, and in all other authors that write of the Sector, or Gunter's rule, so that all I shall speak of now, shall be onely what was forgot in the first part, and what is new as to the using the instru∣ment in sea observations.

II. To find the Suns or a Stars Altitude, by a forward Observation.

Skrew the turning sight to the head Cen∣ter, and set that object sight, whose holes answer to the Sliding horizon sight, in the hole at the end of the head leg, and put the horizon sight on the crosse peece next the inside; Then holding the crosse peece with your right hand, and the turn∣ing sight close to your eye, and the movea∣ble leg against your body, with your thumb

on the right hand thrust upwards, or pull downwards, the horizon sight: till you see the sun through the object sight, and the horizon through the horizon sight, then the degrees cut by the middle of the hori∣zon sight, on the crosse peece shall be the true altitude required:

III. To perform the same another way.

If your instrument be parted, that is to say the crosse peece from the other, and an altitude be required to be had quickly, then set the two object sights, in two holes at the end of the line of naturall signs, then set the head of the rule to your eye, so as the sight of the eye may be just over the Center, then open or close the Joynt, till you see the horizon through one sight, and the sun or star through the other, then is the sector set to the angle required, to find which angle do thus, take the parallel sign of 30 and 30, and measure it from the Center, and it shall reach to the sign of half ••he angle required.

Example.

Suppose I had observed an altitude, and the distance between 30 and 30, should reach from the center to 10. degrees on the signs, then is the altitude of the sun 20. degrees for 10 doubled is 20.

IIII. To find the suns Altitude by a back ob∣servation,

Skrew the Turning sight to the leg cen∣ter,

(or center to the degrees on the move∣able leg) and put one of the object sights, in the hole by 00. on the outer edge of the crosse peece, and set the edge of it just against the stroke of 00, or you may use the sliding object sight and set the edge or the middle of that, to the stroke of 00, as you shall Judge most convenient; and the horizon sight to the moveable leg, then observe in all respects as with a Davis qua∣drant, till looking through the small hole of the horizon sight, you see the crosse bar and button, in the turning sight, cut the horizon: and at the same instant the sha∣dow of the edge or middle of the object or shadow sight, fall on the middle of the turning sight, by sliding the horizon sight higher or lower, then the middle stroke of the horizon sight, shall cut on the move∣able leg, the suns true altitude required. As f〈…〉〈…〉t stay at 50 degrees, then is the sun 50 degrees above the horizon.

V. But if the sun be near to the Zenith or 90 degrees high, then it will be conve∣nient to move the object sight, to a hole or two further as suppose at 10, 20, 30 degrees more, toward the further end of the crosse peece and then observe as you did before in all respects, as with a Davis quadrant, and then whatsoever degrees the horizon sight cuts, you must ad so much to it, as you set the object sight forwards, as suppose 30, and

the horizon sight stay at 60, then I say 60, and 30, makes 90: the true altitude required. Note that by this contrivance, let the altitude be what it will, you shall al∣wayes have a most steady observation: with the instrument leaning against your brest, a considerable thing, in a windy day, when you may have a need of an observation in southern voyages, when the sun is near to the zenith at a Meridian observa∣tion.

VI. To find the suns distance from the zenith, by observing the other way, the sun being not above 60 degrees high, or 30 from the zenith.

Set the turning sight as before on the leg Center, then set an object sight in one of the holes in the line on the head leg, nigher or further of, the turning sight: as the the brightnesse or dimnesse of the sun will allow to see a shadow, then looking through the small hole on the horizon sight, till you see the horizon cut by the crosse bar of the great hole, in the turning sight, turn∣ing the foreside of that sight, till it be fit to receive the shadow of the middle of the object sight; then the degrees cut by the horizon sight, shall be the suns true di∣stance from the zenith, or the complement of the Altitude.

VII. Note that by adding of a short peece about 9 inches long on the head leg, whereon

to set the slideing shadow sight, you may obtain the former convenience of all angles, this way also, at a most steady and easie manner of observation; but note whatso∣ever you set forwards on that peece, must be substracted from that the sight sheweth, and the remainder shall be the suns di∣stance from the zenith required.

As suppose you set forward 30 degrees, and the horizon sight should stay at 40, then 30 from 40 rest 10, the suns distance from the zenith required; thus you see, that by one and the same line, at one manner of figuring, is the suns altitude, or coalitude acquired and that at a most certain steady manner of observation.

VIII. To find an observation by thred and plummet, without having any respect to the horizon, being of good stead in a misty or cloudy day at land or sea.

Set the rule to his angle of 60 degrees by putting in the crosse peece, then skrew the turning sight to the head center, then if the sun or star be under 30 degrees high, set the object sight in the moveable leg, then looking through the small hole in the turning sight, through the object sight, to the middle of the star or sun, as the button in the crosse bar will neatly shew; then the thread and plummet, hanging on the leg center pin, and playing evenly by the moveable leg, shall shew the true alti••ude

of the sun, or star required counting the degrees as they are numbred, for th: north declinations from 60 toward the head with 10 20, As if the thred shall play upon 70 10 then is the altitude 10 degrees.

IX But if the sun or star be above 30 de∣grees high, then the object sight must be set to the hole in the end of the head leg: then looking as before, and the thred playing evenly by the moveable leg, shall shew the true altitude required, as the de∣grees are numbred.

Note that if the brightnesse of the sun should offend the eye, you may have a peice of green, blew, or red glasse, fixed on the turning sight, or else remove the object sight nearer to the turning sight, and then let the sun beams pierce through both the small holes, according to the usuall manner and the thred shall shew the true altitude required. Note also if the thred be apt to slip away from his observed place, as between 25 and 40 it may: note a dexte∣rious handling thereof will naturally shew you how to prevent it:

X. To find a latitude at Sea by forward me∣ridian Observation or Altitude.

Set the moving object Sight to the Suns declination, shewed by the day of the Month, and rectifying Table, and skrew the turning sight to the leg center, and the Horizon sight to the moveable leg, or the

outside of the Crosse piece, according as the Sun is high or low (but note all for∣ward observations respecting the Horizon, ought to be under 45 degrees high, for if it be more it is very uncertain, by any Instrument whatsoever, except you have a Plummet and then the Horizon is uselesse) then observe just as you do in a forward ob∣servation, moving the Horizon sight till you see the Sun through the Horizon sight, and the Horizon through the object sight, or the contrary. (moving not that sight that is set to the day of the Month or Decli∣nation,) then whatsoever the moving sight shall shew, if you add 30 to it, it shall be the latitude of the place required; obser∣ving the difference in North and South Latitudes; that is, setting the sight to the proper declination, either like, or unlike, to the latitude-

Example. Suppose on the 10. of March when the Suns declination is 0—10. North, as in the first year after leap year it will be, set the stroke in the middle of the moving object sight to 10 of North Declination, and the Horizon sight on the moveable leg, then move it higher or lower, till you see the Horizon through one, and the Sun through the other, then the degrees be∣tween, is the Suns meridian altitude, if it be at Noon, as suppose it stayed at 21 30▪ then by counting the degrees between,

you shall find them come to 38. 40. then if you add 30. to 21. 30. it makes 51. 30. the Latitude required, for if you do take 0 10′ minutes from 38. 40. there remains 38. 30. the complement of the Latitude. Note, that this way you may take a forward obser∣vation, and so save the removing of the ••urning sight.

Note also, That when the Horizon sight shall stay about the corner, you may move the object sight 10. or 20. degrees towards the head, and then you must add but 20. or 10 degrees to what the sight stayed at; or if you shall set the sight the other way 10 or 20. degr. then you must add more then 30 so much. As suppose in this last observa∣tion, it had been the latitude of 45 or 50 degrees, then you shall find the sight to play so neer the corner, that it will prove inconvenient, then suppose instead of 0 10. I set it to 20 degrees 10′ North declina∣tion, which is 20. degrees added to the declination, then the Suns height being the same as before, the sight will stay at 41. 30. to which if you add 10 degrees, it doth make 51. 30. as before; here you must add but 10 degrees, because you increased the declination 20. degrees; but note by the same reason, had you set it to 19. 50. South declination, then it had been dimi∣nished 20 degrees, and then instead of 30 you must add 50▪ to 1 - 30. the place where

the sight would have stayed. Thus you see you may very neatly avoid this inconveni∣ence, and set the sights to proper and steady observations, at all times of obser∣vation.

XI. To find the latitude by a backward Me∣ridian Observation at Sea.

This is but just the converse of the for∣mer, for if you set one sight to the decli∣nation, either directly, (or augmented or diminish'd as before, when the moving sight shall stay, about the corner of the Triangular Quadrant) then the other being slipt to and fro, on the outside of the Crosse peece, till the shadow of the outer edge, shall fall on the middle of the turning sight, then 30 just, or more or lesse added, to that number the mo∣ving sight stayed at, (according as you set the first Horizon sight to the declina∣tion) shall be the true latitude required. Example. Suppose on the same day and year as before, at the same Noon time, I set my Horizon sight to just 10′ of North de∣clination, you shall find the moving sight to stay at 21. 30. neer to the corner, now if the Sun shine bright, and will cast the sha∣dow to the turning sight, then set the Ho∣rison sight at the declination, forward 10 or 20. degrees, then the moving sight com∣ing lower you, add but 20. or to that it shall stay at, and the summe shall be the latitude.

But it is most likely that it will be better to diminish it 20. degrees, then the mo∣ving sight will stay about 2 - 30. on the Crosse piece, and so much the better to cast a shadow; for if you look through the Horizon and turning sight to the Horizon, you shall find the shadow of the former edge of the moving shadow sight, to stay at 2 - 30. to which if you add 20. the de∣grees diminished, and 30. it makes 51 30. the latitude required as before.

Note also for better convenience of the shadow sight, when you have found the true declination, as before is taught, set the moving object sight to the same, on the Crosse piece, counted from 00. towards the head leg, for like latitudes and decli∣nations; and the other way for unlike lati∣tudes and declinations, then observing as in a back observation, wheresoever the sight shall stay, shall be the complement of the latitude required. If you add or di∣minish consider accordingly.

Note likewise, when the declination is nere the solstice, and the same way as the latitude is, and by diminishing, or other∣wise the moving sight shall fall beyond 00 on the Crosse piece; Then having added 30. and the degrees diminished together, whatsoever the sight shall stay at beyond 00 must be taken out of the added sum, and the remainder shall be the latitude requi∣red.

Example. Suppose on the 11. of Iune, in the latitude of 51 30 north, for the bet∣ter holding sake I diminish the declination 30 degrees, that is in stead of setting it to 23. 32. north declination, I set it to 6: 28 south then the sun being 62 degrees high will stay at 08. 30. beyond 00. the other way now 30 to be added, and 30 di∣minished, makes 60, from which take 8. 30, rest 51 30 the latitude required.

XII. To find a latitude with thred and plum∣met, or by an observation made without respecting the Horizon.

Count the declination on the cross peece (and let 00 be the equinoctiall and let the declination which is the same with the lati∣tude be counted toward the moveable leg, and the contrary the other way, as with us in north latitude, north declination is to∣ward the moveable leg, and south declina∣tion the contrary and contrarily in south latitudes) and thereunto set the middest of the sliding object or horizon sight, then is the small hole on the turning sight and the small hole on the horizon sight, two holes whereby the sun beams are to pierce to shine one on the other: then shall the thred shew you the true latitude of the place required. Example.

Suppose on the 11 of Decem. 1663, at noon I observe the noon altitudes set the middle of the Horizon sight to 23▪ 32 counted

from 00. toward the head leg end, then making the sun beams to peirce through the hole of this, and the turning sight, you shall find the plummet to play on 51. 30, the latitude required, holding the turning sight toward the sun.

Note also that here also you may avoid the inconvenience of the corner, or the great distance between the sights, by the remedy before cited, in the back and for∣ward observation.

For if you move it toward the head leg, then the thred will fall short of the lati∣tude, if toward the moveable leg then it falls beyond the latitude, as is very easie to conceive of:

Thus you see all the uses of the forestaff, and quadrant, and Mr. Gunter's bow are plainly and properly applyed to this Tri∣angular quadrant, that the same will be a Sector is easie to perswade you to believe, and that all the uses of a Gunter's quadrant, are performed by it, is fully shewed in the use of the Joynt-rule, to which this may be annexed, the numbers signs and tangents and versed signs makes it an excellent large Gunter's rule, and the cross peece is a good pair of large compasses to operate therewith; lastly, being it may lie in so little roome it is much more convenient for them, with whom stowage is very pre∣cious, so I shall say no more as to the use

of it, all the rest being fully spoke to in other Authors, to whom I refer you: only one usefull proposition to inure you to the use of this most excellent instrument, which I call the Triangular Quadrant.

Note that in finding the latitude, it is necessary to have a table of the suns decli∣nation for every of the four years, viz. for the leap year and the 1, 2 and 3d. after, now the table of the suns declination whereby the moneths are laid down, is a table that is calculated as a mean between all the 4 years, and you may very well distinguish a minute on the rule; now to make it to be exact I have fitted this recti∣fying table for every Week in the year, and the use is thus: hang the thred on the cen∣ter pin, and extend the thred to the day of the moneth, and on the degrees is the suns declination, as near as can be for a common year, then if you look in the recti∣fying table for that moneth, and week you seek for you shall find the number of mi∣nutes you must add to or substract from the declination found for that day and year: Example, suppose for April 10. 1662. the second after leap year, the rule sheweth me 11. 45, from which the rectifying table saith I must substract 3′ then is the true declina∣tion 11 42, the like for any other year. Note further that the space of a day in the suns swiftest motion being so much, you

may consider the hour of the day also, in the finding of a latitude, by an observation ta∣ken of the Meridian, as anon you shall see that as the instrument is exact, so let your arithmetical calculation be also: by laying a sure foundation to begin to work upon, then will your latitude be very true also.

A Rectifying Table for the Suns declination.

	D	1 year	2 year	3 year	Leap year
Ianuary	7	Sub 5	Sub 2	Add 1	Add 4
15	s 6	s 2	a 2	a 5
22	s 7	s 3	a 1	a 6
30	s 7	s 3	a 2	a 7
Februar	7	s 8	s 3	a 2	a 7
15	s 8	s 4	a 2	a 8
22	s 9	s 3	a 2	a 7
March	1	s 4	s 1	a 7	Sub 11
7	s 3	Ad 3	Add 9	Sub 9
15	Add 3	ad 1	Sub 9	Add 9
22	a 2	Sub 4	Sub 9	ad 8
30	a 1	Sub 4	Sub 9	ad 8
April	7	a 2	Sub 3	s 8	a 8
15	a 2	s 3	s 8	a 7
22	a 1	s 3	s 8	a 6
30	a 1	s 3	s 6	a 5
May	7	a 0	s 2	s 6	a 5
15	a 0	s 2	s 5	a 3
22	a 0	s 1	s 3	a 3
30	a 0	s 1	s 2	a 1
Iune	7	a 0	s 1	s 1	ad 0
15	a 0	s 0	s 0	Sub 0
22	s 1	s 0	ad 2	s 2
30	s 1	Ad 1	ad 3	s 3

Iuly	7	Sub 2	Add 1	Add 3	Sub 4
15	s 1	a 1	a 5	s 5
22	s 2	a 1	a 5	s 6
30	s 2	a 2	a 6	s 7
August	7	s 3	a 2	a 7	s 8
15	s 3	a 2	a 7	s 8
22	s 3	a 3	a 8	s 9
30	s 3	a 3	a 9	s 9
September	7	s 3	a 3	Add 9	Sub 9
15	Add 3	Sub 3	Sub 9	Add 9
22	a 2	s 4	s 9	a 8
30	a 2	s 4	s 9	a 8
October	7	a 3	s 3	s 9	a 8
15	a 2	s 3	s 8	a 7
22	a 2	s 2	s 7	a 8
30	a 2	s 2	s 7	a 8
November	7	a 1	s 1	s 6	a 7
15	a 1	s 2	s 5	a 5
22	a 1	s 3	s 4	a 3
30	a 0	s 2	s 3	a 2
December	7	0	s 1	s 1	Add 1
15	Sub 1	s 0	s 0	Sub 1
22	s 1	s 0	Add 1	s 3
30	s 3	s 0	Add 2	s 5

The Declination of the Sun being given, or rather the Suns Distance from the Pole, and the Complement of two Altitudes of the Sun, taken at any time of the day, knowing the time between: to find the Latitude.

Suppose on the 11. of Iune, the Sun being 66 degrees, 29′ distant from the North Pole, and the Complement of one altitude be 80. 30. and the Complement of another altitude 44. 13. and the time be∣tween the two observations just four hours; then say,

As the Sine of	90 00
To Sine of Suns dist. f. the Pole	66 28
So is the Sine of ½ time betw.	30 00
To the Sine of ½ the 3d. side	27 17½
	27 17½
of a Triangle as A B.	54 35

The side A P	66 28
The side P A	66 28
The side A B	54 35
whole sum	187 31
half sum	93 46
The differ. betw ½ sum and AP side op. to inqu. Triangle	27 18
Then say, as S. of 90	90 00
To S. of Suns dist. from Pole	66 28

So is the Sine of A B	54 35
To the Sine of a fourth Sine	48 23
Then as that 4 to Sine of ½ sum	93 46
So is S. of the difference	27 18
To a seventh Sine	37 44½
or the versed Sine of P A B	77 02

Then to find Z A B Z B is	80 30
And Z A is	44 13
The former side A B is	54 35
Sum	179 18
half sum	89 39
difference	09 09
As S. of 90	90 00
To Sine of A B	54 35
So is S. of Z A	44 13
To a 4th. Sine	34 38

As S. 4th.	34 38
To S. of ½ sum	89 39
So is S. diff.	09 39
To a 7th. Sine	16 15¼
or to the vers. sine of Z A B	116 07

Then if you take P A B from Z A B there will remain Z A P 39 05 Then say again by the rule as before,

As the sine of	90 00
To Co-sine of Z A P	50 55
So is the Tang of A Z	44 13
To the Tangent of A C	37 04
which taken from A P	66 28
remaineth P C	29 24
Then lastly say,
As the Co-sine of A C	52 56
To the Co-sine of C P	60 36
So is the Co-sine of Z A	45 47
To the Co-sine of Z P	51 30

The Latitude required to be found. This Question or any other may be wrought by the Sines and Tangents and versed sines on the rule, But if you would know more as concerning this or any other, you may be fully satisfied by Mr. Euclid Spidal at his Chamber at a Virginal Makers house in Thred-needle Street, and at the Kings head neer Broadstreet end.