A new QUADRANT, OF More natural, easy, and Manifold Performance, than any other heretofore Extant; Framed according to the horizontal Projection of the Sphere, with the Uses thereof.  
By C. B. Maker of Mathematic Instruments in metal.  
LONDON, Printed in the year 1649.   

TO My Singular good Friend Mr. WILLIAM BADILEY, Mariner, and a lover of the Mathematics.  
Worthy Sir,  
HAving diligently inquired▪ the reason of the projection of the Sphere into plain, as the ground of all Mathematic Instruments (the making whereof in metal is my Trade and Livelihood) and compared the several manners; I found none so genuine, simple, easy, and manifoldly useful, as is the horizontal; which lively representeth the Globe rectified to some certain elevation, and naturally performeth the uses thereof. And having likewise compared the several Quadrants, and pocket Instruments hitherto made, and finding them all pieced up with many unnatural and forced lines and divisions, presupposing an exact diligence both in the Calculator, and in the workman; and yet the performance difficult, troublesome, and tedious: I bethought myself whether out of the horizontal projection I might not by some small alteration frame a Quadrant, that might remedy the defects of all the former Instruments, and that with greater ease and certainty. And having by the help of God happily attained my desired intent, my many respects represented you under whose Approbation and Patronage I might send out into public view this my new Quadrant, with the many Uses thereof; as being one to whom I stand obliged for your love and manifold favours to me both at Sea in divers voyages, and at land; and who through your skill in the mathematical Sciences are able to judge and patronize the first attempts of  
Your affectionately devoted Servant, C. B.   

THE Description of the Quadrant, and the parts thereof.  
THe limb of the Quadrant divided into 90 Degr. representeth the Horizon.  
That side of the Quadrant where the Sights are, is the Meridian, or XII a clock line, unto which is joined the scale of months with every fift Day, until they grow so little toward the Solsticeas, that they cannot be distinguished. This scale▪ hath five rows, the midst whereof hath the very same divisions which are on the Meridian line: The two next on both sides are for the parts of the months, which in the two outermost rows are noted by their first Letters.  
The other side of the Quadrant hath on it the scale of Altitudes above the Horizon.  
The short Arching lines within the Quadrant beside the Meridian, are hourlines, noted by their Figures, both for the forenoon, and afternoon; and half hour lines: each half hour containing 30 min. of an hour, or Deg. 7. 30. Of these Horary lines, those which serve in the morning before the sun is full East, or in the evening past the West, (which is only in Summer half year) are reversed. And all the Hour lines are noted with two Figures; whereof the upper next the centre and scale of Altitudes, serve for the afternoon; and the lower for the forenoon. The two Arches which cross the hour lines, and meet at the beginning of the Horizon and scale of Altitudes, are two Quarters of the Ecliptic, and are divided into 90 Degr. a piece, in which are noted the XII signs by their proper Characters, namely on the upper next the centre are ♈ ♉ ♊ & ♋ ♌ ♍, the Summer or Northern signs: and on the lower next the Horizon are ♎ ♏ ♐ & ♑ ♒ ♓, the Winter or Southern signs, and contain 30 Degr. a piece. This is the Circle of the sun's annual motion.  
The long Arches, which beginning at the Scale of months in the Meridian betwixt the two Quarters of the Ecliptic, cross all the hour lines, are the parallels of Declination, or the semidiurnal Arches of the sun; the middlemost of which is the equinoctial, the outermost above is the tropic of ♋, and the outermost below is the tropic of ♑: although between the equinoctial and each tropic Innumerable parallels are understood to be contained, yet those which are in the Instrument drawn, at every second Degree of Declination, may be sufficient to direct the eye in tracing out an imaginary parallel from every point given in the Scale of months.  
The equinoctial and every tenth parallel are for distinction sake made somewhat grosser than the rest, and all the Summer parallels at the East and West line are continued reversedly back unto the Horizon.  
Note that upon the right estimation of that imaginary parallel, the manifold use of this Instrument doth especially rely; because the true place of the sun all that day is in some part or point of the same Circle.  
And note that in this Instrument, the direct Horary lines, and parallels before their reversion, show the hour of the day like a direct South upright dial: And the Arches of them reversed serve like a direct North upright dial.  
Use I. To find the Declination of the Sun every day.  
Seek the day proposed in the Scale of months very exactly, & mark upon what point it falleth in the middle Row of that Scale, or (which is all one) in the Meridian, for there is the Declination of the sun from the equinoctial, either North or South: which if it fall not directly upon a parallel, but in the space between two, supposing each half of that space to contain 60 minutes, estimate with your eye proportionally what minute the point giveth.  
Example 1. What is the sun's Declinaclition upon Novemb. 13? the day will fall in the space after 20 Degrees, from the equinoctial Southward, about 30 minutes: Wherefore the sun's Declination is 20°. 30′ South.  
Example 2. What is the sun's Declination upon August 19? the day will fall in the space after 8 Degrees, from the equinoctial Northward, one Degree and about 40 minutes: Wherefore the sun's Declination is 9°. 40′. North.  
Note that the Declination thus found is to be kept in mind all the day.  
Use II. To find the Semidinunall Arch, or parallel Circle in which the sun moveth every day.  
Seek out the true point of the sun's Declination upon the Meridian by use I: then from that point by the estimation of your eye, trace out an imaginary parallel: which when it cometh to the East and West line (as in all Northern parallels it doth) is to be reversed unto the Horizon or limb at the same proportionable distance as before. This operation requireth exact diligence.  
Use III. To find the time of the Sunes Rising and Setting every day.  
Seek out the imaginary parallel, or Semidiurnal Arch of the Sun for that day by use II, and mark where it meeteth with the Horizon; for that is the very point of the suns rising and setting, and the Hour-lines on both sides of it, (by proportioning the distance reasonably, according to 30 minutes for half an hour) will show the time of the suns rising and setting.  
Thus at London, Novem. 13. the Sun will be found to rise at 9 min. before 8, and to set at 9 min. after 4. Also August 19, the sun will be found to rise 12 min. after 5. and to set 12 min. before 7.  
Use IV. To find the sun's Amplitude, Ortive and Occasive: that is, how many Degrees of the Horizon the sun riseth and setteth from the true East and West points every day.  
The imaginary parallel of the sun, together with the time of the suns rising, and setting, showeth upon the Horizon the Degree of his Amplitude from East and West, which in all the Northern parallels is on the North side, and in the Southern on the South side.  
Thus at London, Novem. 13. the Ampl. Ort. will be found 34 degrees. Also Aug. 19 the Ampl. Ort. will be found 15°. 10′.  
Use V. To find the Length of every day and night.  
Double the hour of the sunnes-setting, and you shall have the Length of the day: or double the hour of the suns-rising, and you shall have the Length of the night.  
Use 6. To know the reason and manner of the Increasing and Decreasing of the days and Nights throughout the whole year.  
When the sun is in the Equinoctial, it riseth and setteth at 6 a clock: But if the sun be out of the Equinoctial, declining toward the North, the Intersection of the parallel of the sun with the Horizon is before 6 in the morning, and after 6 in the evening; and the diurnal Arch greater than 12 hours, and so much more great, the greater the Northern Declination is. Again, if the sun be declining toward the South, the Intersection of the parallel of the sun with the Horizon is after 6 in the morning, and before 6 in the evening; and the diurnal Arch lesser than 12 hours, and by so much lesser, the greater the Southern Declination is.  
And in those places of the Ecliptic in which the sun most speedily changeth his Declination, the Length also of the day is most altered; and where the Ecliptic goeth most parallel to the equinoctial, changing the Declination slowly, the length of the day is but little altered. As for Example; When the Sun is near unto the equinoctial on both sides, the day's Increase and also Decrease suddenly and apace; because in those places the Ecliptic inclineth to the equinoctial in a manner like a straight Line, making sensible Declination. Again, when the sun is near his greatest Declination, as in the height of Summer, and the depth of Winter, the days keep for a good time, as it were, at one stay; because in these places the Ecliptic is in a manner parallel to the equinoctial, the Length of the day differeth but little, the Declination scarce altering; and because in those two times of the year, the sun standeth as it were still at one Declination, they are called the Summer Solstice, and Winter Solstice. Wherefore we may hereby plainly see, that the common received opinion, that in every month the days do equally increase, is erroneous. Also we may see, that in parallels equally distant from the equinoctial, the day on the one side is equal to the night on the other side.  
Use VII. To take the height of the sun above the Horizon.  
Hold the edge of the Quadrant against the sun, so that the sun's Ray or Beam may at once pass through the hole of both the sights; then shall the thread with the Plummet show the sun's Altitude.  
Use VIII. To find the hour of the day, or what a clock it is.  
Having the imaginary parallel or semidiurnal Arch of the sun, already found and conceived in your mind by use II, take the sun's height above the Horizon, then stretching the thread over the scale of Altitudes, set the Bead to the Altitude found, move your thread until the Bead exactly falleth upon the imaginary parallel, for there is the hour fought; and that is the true place of the Sun in the Quadrant at that time; to be estimated upon the Horary lines, either direct, or reversed, according as the parallel is.  
Use ix.. To find the suns Azumith or horizontal distance from the four Cardinal points.  
The Bead being set to the hour of the day, as was showed in the use next before, the thread shall in the limb cut the East or West Azumith; that is, how many Degrees of the Horizon the vertical Circle in which the sun is, is distant from the East and West points: The compliment of which number giveth the Azumith from the South Meridian, if the Bead fell in the right parallels: But if the Bead fall upon the reversed parts, the Azumith is to be accounted from the North Meridian.  
Use X. To find the Meridian Altitude of the sun every day.  
Stretch the thread over the Meridian, and set the Bead to the true Declination of the sun therein; then apply the thread to the scale of Altitudes; and the Bead shall give the Meridian Altitude sought.  
Use XI. To find at what time the sun cometh to be full East or West every day in Summer.  
This is shown by observing at what hour the imaginary parallel meeteth with the East and West line, at which it beginneth to reverse.  
Use XII. To find how high the sun is above the Horizon at any hour, every day.  
Set the Bead to the point in which the imaginary parallel of that day crosseth the hour given: then applying the thread to the scale of Altitudes, mark upon what Degree the Bead falleth; the same shall be the Altitude of the Sun required.  
Use XIII. To find how high the sun is being in any Azumith assigned every day: and also at what hour.  
Set the Bead to the point in which the imaginary parallel of that day crosseth the Azumith assigned; There also shall be the hour sought: Then applying the thread to the scale of Altitudes, mark upon what Degree the Bead falleth; The same shall be the Altitude of the Sun required.  
These two last Uses serve for the Delineation of the ordinary Quadrants, as that of Gemma Frisius, Munster, Clavius, Master Gunter, &c. and also of Rings, Cylinders, and other topical Instruments; and for the finding out of the hour by a man's shadow, or by the shadow of any Gnomon, set either perpendicular, or else parallel to the Horizon.  
Use XIV. To find the sun's Longitude, or place in the Ecliptic.  
The imaginary parallel of the day being exactly traced will cut in the ecliptic the sign and Degree wherein the sun is: and note, that each semicircle of the Ecliptic is doubly noted with Characters of the signs; the first and third Quarters go forward from the Equinoctial point unto the Meridian, containing ♈ ♉ ♊ & ♎ ♏ ♐: the second and fourth Quarters go back from the Meridian unto the equinoctial point, containing ♋ ♌ ♍ & ♑ ♒ ♓.  
But because near unto both Tropics (namely from May 11, to July 10, in the height of Summer, and from November 13, to January 12 in the depth of Winter) the Declination altereth so slowly, that the true place of the sun in the Ecliptic cannot be distinguished with any certainty, work according to this fourfold Rule following.  
1. Before June 10, out of the number of days from May 0, subduct 11: the remains shall be the Degrees of ♊: thus for June 3, (because there is all May and three days of June) say 34— 11= 23 ♊, the place of the sun.  
2. After June 10, out of the Number of days from June 0, subduct 10: the remains shall be Degrees of ♋: thus for July 3, say 33— 10= 23 ♋, the place of the Sun.  
3. Before December 13, out of the Number of days from November 0, subduct 13: the reamines shall be Degrees of ♐: thus for December 3, say 33— 13= 20 ♐, the place of the Sun.  
4. After December 13, out of the Number of days from December o, subduct 13: the remains shall be Degrees of ♑: thus for January 3, say 34— 13= 21 ♑, the place of the sun.  
Use XV. To find the sun's Right Ascension every day.  
Having by Use XIV. found the place of the sun in the Ecliptic, mark diligently upon what hour, and as near as you can estimate what minute it falleth, counting the hours in the first and third Quarters of the Ecliptic, from the equinoctial point; but in the second and fourth Quarters, from the Meridian: and add thereto in the second Quarter six hours, in the third twelve hours, and in the fourth eighteen hours: so shall you have the sun's Right Ascension, not in Degrees, but in time, which is more proper for use.  
Example, in ♌ 6. the sun's Right Ascension will be eight hours, one half, and about three minutes; that is H: 8: 33. min. reckoning 30′ for half an hour.  
Use XVI. To find the hour of the Night by the stars.  
For this, I have set a little Table of five known Stars dispersed round about the Heavens, with their Declination and Right Ascension for Anno Dom. 1650. Namely the left shoulder of Orion, noted O. The heart of the Lion, noted ♌. Arcturus noted A: the Vulture volant, noted V. The end of the wing of Pegasus, noted P.  

The Table.   Declinat. Rec. As. O 5° 59′ N H5 6,5′ ♌ 13 39 N 9 50 A 21 4 N 14 00 V 8 1- N 19 34 P 13 15 N 23 55, 5  
THe Operation is thus; first by the height of the star taken, and the parallel of its Declination exactly traced, seek out the hour of the star from the Meridian, as before was taught for the hour of the Day by the sun. Secondly, out of the Right Ascension of the star, subduct the Right Ascension of the Sun; the remain● showeth how long time from the noon before the same star cometh into the Meridian. Lastly, if the star be not yet come to the Meridian, out of the hour of the stars coming into the Meridian, subduct the hour of the star: but if the Star be past the Meridian, add both the hours together; so shall you have the true hour of the Night.  
Note, that if the hours out of which you are to subduct be lesser than the other, you must add unto them 24.  
Use XVII. To find out the MeridianLine upon any horizontal plain.  
About the middle of your plain describe a Circle; and in the centre thereof erect a straight Piece of Wire perpendicularly. When the sun shineth, note the point of the Circle which the shadow of the Wire cutteth, which I therefore call the shadow point; and instantly by use ix.. seek the suns Azumith from the South or North: keep it in mind. Then from the shadow point, if your observation be in the fore-noon, reckon upon the circle an Arch equal to the Azumith kept in mind, that way the sun moveth, if the Azumith be South: Or the contrary way if it be North. But if your observation be in the afternoon, reckon the North Azumith that way the sun moveth; Or the South Azumith the contrary way.  
Lastly, through the end of the Azumith and the centre, protract a Diameter for the Meridian line sought: which you may note with S. at the south end, and with N. at the North end.  
You may also note the point of the Circle Diametrally opposite to the shadow point with sun;, because it is the Azumith place of the Sun, at the moment of your observation.  
Use XVIII. To find the Declination of any Wall or plain.  
The safest way (because the magnetical Needle is apt to be drawn awry) will be by an Instrument made in this manner. Provide a rectangular board about ten Inches long, and five broad: in the midst whereof, cross the breadth, strike a Line perpendicular to the sides; and taking upon it a centre, describe a Circle intersecting the same Line, in two opposite points, to be noted with the Letters T. and A: divide each semicircle into two Quadrants, and every Quadrant into 90 Degrees, beginning at the points T and A, both ways; the first Quad. beginning on the left hand of T. the second Quadrant on the right hand: the third Quadrant above it toward A: And lastly, the fourth Quadrant. And in the centre erect a wire at right Angles.  
The use of this Instrument. Apply the long side of the board next T to the Wall when the sun shineth upon it, holding it parallel to the Horizon, that it may represent an horizontal plain. Mark what Degree the shadow of the wire cutteth in the Circle; and instantly seek the suns Azumith, either South or North: Reckon it on the Circle from the shadow to the Meridian, as was taught in the Use next before, noting that end with the Letter contrary to that of the Azumith: as if the Azumith be South, note it N. and the opposite end S; if the Azumith be North, note it S, and the opposite end N: whereby also you have the East▪ and West sides: So shall the Arch S A. or N A. give the Declination of the plain, and the point A, the coast or quarter into which it is.  
Example, June 2 in the forenoon, applying the instrument to a wall, I found the shadow in 23 Degr. of Quadr. 2. and the height of the sun was 26 Degrees, whereby I found the Azumith to be North 84 Degr. which reckoned from the shadow against the sun, fell upon 61 Degr. in Quad. 1. for one end of the Meridian; and the Opposite end▪ which is N. upon 61 Degr. in Quad. 3. And A was on the East side of N. Wherefore the Declination of that Wall is 61 Deg. from the North Eastward.  
Use XIX. To find the Declination of an upright wall by knowing the time of the suns coming to it, or leaving it. And contrariwise, the Declination of an upright Wall being known to find at what time the sun will come into it.  
Because the Declination of a plain is an arch of the equinoctial intercepted between the horizontal section of the plain: and the East or West points: Or else (which is all one) between the Meridian, and A, the axis of that horizontal sexion. Watch till you see the centre of the Sunnejust even with the edge of the Wall: then instantly take the suns Azumith from East or West, by Use ix.. the same is the Declination of the wall.  
Likewise if the Declination be given, reckon it upon the limb of your Quadrant from the East and West point; and the thread being applied to the end of that Arch, shall in the sun's imaginary parallel for that day, cut the hour and time desired.  
Use XX. certain advertisements necessary for the use of the Quadrant in the night.  
In which Questions as concern the night, or the time before sunrising, and after Sunnesetting, the instrument representeth the lower Hemisphere, wherein the Southern Pole is elevated. And therefore the parellels which are above the equinoctial toward the centre, shall be for the Southern or winter parellels: and those beneath the equinoctial, for the Northern or Summer parallels: and the East shall be counted for West, and the West for East; altogether contrary to that which was before, when the Instrument represented the upper Hemisphere.  
Use XXI. To find how many Degrees the sun is under the Horizon at any time of the night.  
Seek the Declination of the sun for the day proposed by Use I. and at the same Declination on the contrary side of the equinoctial imagine a parallel for the sun that night; and mark what point of it is in the very hour and minute proposed: Set the bead to that point; then applying the thread to the scale of Altitudes, mark upon what Degree the bead falleth: for the same shall show how many Degrees the sun is under the Horizon at that time.  
Use XXII. To find out the length of the Crepusculum, or twilight.  
It is commonly held that Twilight is so long as the sun is not more than 18 degrees, under the Horizon, the question therefore is, at what time the sun cometh to be 18 Degrees under the Horizon any night.  
Seek the sun's declination for the time proposed, and at the same declination, on the contrary side of the equinoctial, imagine a paralsel for the sun that night: then set the bead at 18 degrees in the scale of Altitudes; and carry the thread about till the bead fall upon the imagined parallel: for there shall be the hour or time sought.  
And in this very manner you may find the time or hour of the night at any other depression of the sun under the Horizon.  
FINIS.