THE trigonal SECTOR, THE Description and use thereof: Being an Instrument most aptly serving for the resolution of all Right lined Triangles, with great facility and delight.  
By which all Planimetrical, and Altimetrical conclusions may be wrought at pleasure.  
The Lines of Sines, Tangents, Secants, and Chords, pricked down on any Instrument: Many Arithmetical proportions calculated, and found out in a moment. dials, delineated upon most s ●ts of plains: with many other delightful conclusions. Lately invented and now exposed to the public view.  
By John Chatfeilde.  
LONDON, Printed by Robert Leybourn, and are to be sold by James Nuthall, over against the George near Holbourn-bridge, 1650.   

TO THE READER.  
READER,  
THou hast here presented to thy view, the description, and use of a Geometrical Instrument, whereby (if thou art yet a learner) thou mayst be helped abundantly, in the right understanding of the Doctrine of Triangles, with greater ease: but if thou hast already waded through the greatest difficulties, and art a Master in this Art, yet doubt not, but thou shalt find something here, which will admister unto thee some delight, and perchance may enable thee to do those things with greater speed and ease, then in the way that thou walkedst in before, however, I entreat thee to take it in good part, from him that desireth to be  
A Servant unto all for their Advantage, J. C.   

GG. sulps-  woodcut depicting the "trigonall sector"   

Of the trigonal SECTOR: The description and use thereof.  
THe trigonal Sector, is an Instrument Geometrical containing all variety of rightlined Triangles, together with the proportions of every side betwixt themselves and to their perpendiculars.  
It consisteth of a square plate of metal, or piece of wood, on whose edges, round about, are to be fixed certain laminae, or long slips that may stick up a little above the plate, and two labels at the extremes of one of the sides, which moving on their several centres, may be applied unto each other, till crossing one another they make an angle betwixt themselves: which always shall be the compliment unto 180 gr. of those angles which both the labels make unto the Radius, which is a line drawn directly betwixt their centres.  
The inscriptions on the Instrument are first, three Scales of Lines divided into 100 equal parts, and answering unto one another, (viz.) on the two labels and the lower lamina that contains the Radius betwixt the centres of the labels.  
Secondly, a reversed Tangent, containing 45 gr. on the lamina on the left hand of the Instrument, and 45 more even to 90 gr. on the lamina at the top.  
Thirdly, the Quadrant of a Circle on the inward body of the plate concentrical with the label on the left hand, and whose limb or semidiameter reacheth exactly unto the centre of the label on the right hand: so that the Radius thereof is just the same with the Radius divided into 100 parts betwixt the centres of the labels. This Quadrant is divided into 90. gr. reckoning from the centre of the right hand label, and in the area thereof are drawn lines parallel both betwixt themselves, and with the Radius, exactly corresponding to the divisions of the Scales of Lines before described, and marked as all the other three scales are with their proper figures 1.2.3. etc. to 10.  
Thus much of the general description of the Instrument, the lines consist all of equal parts and therefore need no tables for their inscription, except only the reversed Tangent, which may be done, as is commonly known, by the ordinary tables. Only this must be observed, that the Tangent of 45 must be somewhat longer than the semidiameter of the Quadrant, or the forementioned Radius, because its distance from the centre of the label at the right hand is greater (by so much space as the breadth of the left hand label doth contain) than the semidiameter is. The label on the right hand being to supply the Secant of 45. is longer than the other label, and reacheth even to that angle of the Instrument where the Tangent of 45 is placed, and therefore the divisions upon it are 50 more than the other label (1) 150: yet though they are more in number they correspond unto the divisions of the other, this label being ½ in length more than that, that so by it you may see the number of parts that a Secant of any degree under 45 shall contain of a Radius containing 1000  
If the scale upon the lower lamina or Radius be so drawn as to make an angle at the centre of the left hand label with the edge of the lamina, as it is in the line of lines in Mr. Gunter's Sector, and another scale correspondent unto it, be inscribed on the left hand label, these 2 Scales shall perform all the Propositions to be performed by the Sector one the line of Lines.  

Thus much of the description of the Instrument and its parts: Now of the use thereof.  
THe use of this Instrument is very great and various, but it consisteth especially in the resolution of all sorts of Rightlined Triangles, unto which all Geometrical conclusions are reducible, and in Arithmetical proportions or the resolution of the Golden Rule.  
In the resolution of Triangles, 4 things are considerable. 1 The quantity of the Angles. 2 The proportions of the sides or subtendents and perpendicular betwixt themselves. 3 The contents of the area in parts correspondent to the parts of the sides. 4 The reduction of those parts to a perpendicular and basis of another denomination: all which are performed by the Instrument in this manner.   

To find the quantity of any angle.  
IF it be in Planimetry, having three sights placed on the Instrument, the first at the centre of the left hand label, the second at the centre of right hand label, & the third at the end of the right hand label, work thus. Lay the Instrument upon a Geometrical tripos or staff, as the Semicircle or Circumferentor is usually laid, by those that measure Land, horizontally, then turn it till you can behold one of the marks through the sights at the centre of the labels, afterward turn the right hand label till you see the other mark through the sight at its centre, and that which is at the end thereof, and then looking on the Tangent line, you shall see the angle among the Tangents, where the intersection is of the label and the Tangent.  
If the angle be above 90 gr. than you must have four sights, and behold the first mark through the sights at the centre, and the end of the left hand label, and leaning the Instrument at that position, turn the right hand label till you can also behold the second mark through its sights, and then the angle which the two labels make where they cross each other shall be the angle sought, which that you may know how great it is, add together the angle found in the Tangent cut by the right hand label, and that on the Quadrant cut by the left hand, label and subtract them both out of 180 gr. and you have the angle required.  
In Altimetry work thus, set the Instrument on one edge upright, so as that the lower lamina, containing the distance betwixt the centres of the two labels, may stand horizontal, and the right and left hand laminae Perpendicular; then lift up the right hand label, till, through the sights thereof, you can behold the object, and then in the Tangent line shall you see the angle. This is of use to find the altitude of Sun, Moon ●or Stars, as also the altitude of any Tower, Steeple, Hill, etc.   

Two angles being known, to represent any Rightlined Triangle.  
TRiangles of all sorts are distinguished two ways.  
First, By their sides, and so they are said to be. 1 Equilateral, having all sides a like. 2 Equicrural having only two sides equal; and the third unequal. 3 Scalene, viz. Triangles neither of whose sides are equal.  
Secondly, By their angles, and they are, 1 Rectangular containing one Right-angle. 2 Obtuseangular wherein one of the angles is obtuse or greater than a Rightangle, that is above 90 gr. 3 Acuteangular, wherein all the angles are less than 90 gr.  
Note that the three angles of every Triangle are equal to two Right angles, that is 1 80 gr.   

To represent any Rectangular Triangle.  
Turn back the left hand label to the lamina of the Tangent, for than it maketh a right angle with the Radius, then turn the right hand label to the degree of the other known angle in the Tangent line, and so the two labels and the Radius shall give the Triangle, together with the proportions of the sides betwixt themselves.  
As suppose, the two known angles be 90. and 45. the distance betwixt the centres (1) the Radius will be 100 parts, the right hand label shall cut the left hand label also in 141 parts.  
The one of these sides is the Radius, the second a Tangent of 45. equal always to the Radius, the third, the Secant of 45. being a line drawn from the centre through the limb of the circle till it meet the tangent; if one of the known angles be above 45: and the other 90 gr. then place the label at the tangent of its compliment to 90. and then shall the parts on the label on the left hand represent the Radius, and the Radius of the Instrument shall represent the tangent, and the label on the right hand shall represent the secant.  
Thus supposing the known angles to be 90. and 63 gr. 30′. because if I put the right hand label to 63 gr. 30′ it will not cross the left hand label, I therefore take its compliment (1) 26 gr. 30′. and applying the right hand label unto this degree in the Tangent, the Triangle comprehended, as in the former example, shall be the Triangle required; and the parts discovered in every intersection or angle shall show the proportion of the Radius, Tangent, and Secant, as here the Radius that is represented by the label on the left hand, shall be 500, the Tangent represented by the Radius of the Instrument 1000, and the Secant represented by the right hand label 1120 parts, which if reduced to a Radius of 1000, the proportion holds thus, the Radius 1000, the Tangent 2000, the Secant 2240, which agreeth to the tables of Tangents and Secants, unto a Unit, which is as near as can be expected by Instrument.  
If one of the known angles be less than 45, and the other 90 gr. than you need no more but apply the right hand label to the degree of the Tangent line, and the Radius of the Instrument shall be the Radius or bases of the Triangle, and the proportions of the other sides will be showed as abovesaid.   

To represent any obtuse angular Triangle.  
AS suppose the two known angles be 60 & 100 gr. this triangle must needs be obtuse anglar, because 100 gr. is more than 90 gr. & because neither of the labels can make an angle more than 90 gr. therefore the obtuse angle must be found in the intersection of the two labels; therefore to find out this triangle work thus, add the two known angles together, and they will make 160: subtract this out of 180, and there remaineth 20, which is the third angle; therefore to represent this, put one label in 20 gr. and the other in 60 gr. and where they cross each other, the intersection shall be an angle of 100 gr. because all three angles must make exactly 180 gr. and the labels thus applied give you the Triangle required, and the divisions of the labels in their intersection together, with the Radius show you the proportions of the three sides thereof, as in the former examples.   

To represent any Acuteangular Triangle.  
THis is to be done only by applying the labels unto the degrees of the angles known, both in the tangent line and quadrant.  
But if any of the angles be above 60 gr. than it must be supplied, by putting one of the labels to the compliment of both added together, unto 180 gr. as suppose two known acute angles be 60 and 80: the labels being applied to those degrees in the Tangent and Quadrant, will not meet to cross each other; therefore put the one label in 60, and the other in 40 gr. which is the compliment of the sum of both 140 to 180: and so you shall have the Triangle required, the proportion of whose sides is to be found out as aforesaid.   

To find the Content of any of these Triangles.  
TO effect this, the perpendicular must be known, which being multiplied into half the basis, the product shall give the Area.  
I have showed before how to find the proportions of every side, and so by consequence the length of the basis; with as great facility is the perpendicular also to be found: for when you have applied the labels unto each other, so as that the Triangle is by them represented, do but cast your eye directly into that intersection, and under the labels amongst the parallel lines; you shall find the distance from the basis, or the length of the perpendicular: which multiplied as aforesaid, shall give the Area.  
Or to work by the Instrument, I say thus.  
As 1 to the Multiplicator,  
So the Multiplicand, to the Product.  
Supposing then the perpendicular to be 5, and the basis 10: whose half is 5, that is to be multiplied by the perpendicular.  
I apply 10 in the left hand label, (which also may be reckoned for 1 or 100 or 1000: as occasion serveth) unto 5, amongst the parallels, then looking for 5, upon the said label, I find it directly meeting with 25 amongst the parallels, which is the number sought.  
And thus may any number be multiplied, only remembering to observe that the same Figures and divisions stand sometimes for units, sometimes for decades, 100 1000, etc. as occasion serveth. So also working by the contrary way, may any be divided.  
For, as the divisor is to 1;  
So the dividend, to the quotient.  
Therefore apply the number belonging to the divisor on the label to 1, amongst the parallels, and then over against the number of the dividend in the label, shall be the quotient amongst the parallels. As if I would divide 9 by 3; here 3. is the divisor, and 9 the dividend.  
I therefore bring down the 3: upon the label, till it come exactly upon the parallel of 1: and then over against 9: which was the dividend, I find the parallel marked with the figure 3, which is the quotient  
For 3. is contained in 9 three times.  
But here is to be noted, that it sometimes falleth out more convenient in working proportions on this Instrument to find the first number amongst the parallels, and the second upon the label, and then the third upon the parallels, will exactly answer to the 4th or quotient upon the label.   

To reduce the Area of a Triangle found of one denomination, to a perpendicular and basis of another denomination.  
THe reason of the inserting of this proposition is, because that in measuring of Triangles, whose Area we would find either in Acres, Perches, Yards, Feet, or Inches; it seldom falleth out that the perpendicular or basis measured, giveth just the same number as it doth upon the Instrument: yet notwithstanding the Triangles and their sides hold proportion unto each other; and therefore there is a necessity of working by proportion, to reduce the one into the other.  
As first, supposing we have a rectangle equicrural Triangle containing 90.45, 45 gr. in its angles on the Instrument, the basis will be 100, and the perpendicular 100 which perpendicular multipl●yed into half the basis, giveth 5000 for the Area. Now suppose again, that by measure we find the basis to be but 30, in a Triangle consisting of such angles, I say then.  As 100 the basis on the Instrument to 30. the basis measured. 
So 100 the perpendicular on the Instrument, to 30 the perpendicular inquired  
Which being multiplied into 15. half the basis, giveth the Area 450.  
Or 2. supposing the Triangle be scalene, having all its sides unequal, whose angles shall be 30 gr. 40 gr. and 110 gr. upon the Instrument the basis will be found to be 100, and the perpendicular about 35.  
The Question is what the perpendicular shall be if the bases of such a Triangle being measured shall be 80 perches.  
I say then, as 100, the bases on the Instrument, to 80 perches.  
So 35 the perpendicular found, to the perpendicular inquired.  
Bring down the left hand label therefore till the figure of 10 come just to the parallel of 8, and then just over against 35 on the label shall be found amongst the parallels, the perpendicular required, (viz.) 28; by which multiply half the bases, and the Area will be found 1120.  
For, As 1. Is 28.  
So is 40.61 half the bases, to 1120.  
Bring down therefore the figure of 10 upon the label, (which here standeth but from 1) to 28 amongst the parallels, and over against 4. or 40. on the label shall be found amongst the parallels, the number aforesaid 1120; which in a large Instrument may easily be discerned.  
And in this manner may most Arithmetical Proportions be found out, and Questions of the Golden Rule resolved; as also Lines and Numbers may be increased in continual proportion. And Mr. Gunter's Canons for Land measure, to find the contents of all oblong Superficies in Perches, Chains, or Acres, may be made use of and resolved by the Instrument.   

Thus hath been showed its use in the resolution of Triangles, and in working proportions.  
BEsides it is appliable to many other Mathematical conclusions, as to find the length in parts of any Line of Signs, Tangents, Secants, and Cords, and so by consequence is of great use in the Projection of Spheres, describing of Dial's, whether Sciotericall, or Instrumental.   

To find the length of the Tangent line of any degree, to a Radius of 10000  
Turn bacl the left hand label to the left hand lamina, so as that it may make a Rightangle with the Radius, then turn the right hand label to the degree required in the Tangent line, and marking the place of its intersection with the other label, you shall there see amongst the divisions of the left hand label the number of parts that such a Tangent must be of.  
As suppose the degree be 30, whose Tangent you would find; the labels being applied unto each other, as aforesaid, the parts intercepted in the left hand label, betwixt the right hand label, and the Radius, shall be 5773, which must be the length of that Tangent.  
But note, that if the Tangent be above 45 gr. than the right hand label must be applied to the compliment thereof, and then the parts intercepted shall resemble the Radius, and the Radius of the Instrument shall represent the Tangent, and the proportion that the intercepted parts bear unto the Radius of the Instrument 10000, the same proportion shall the Radius of the Instrument 10000 bear unto the Tangent required.  
As suppose you would find the Tangent of 60, apply the right hand label unto its compliment, that is the Tangent of 30 gr. and you shall find the parts intercepted, that represent the Radius of 60 gr. 5773. and the Radius of the Instrument that doth represent the Tangent 10000 I say then.  
As 5773 the Radius found, to 10000, the Radius of the Instrument.  
So 10000 the Radius of the Instrument, to 17320 the Tangent of 60 gr.   

And thus by proportion, may be found the length of the Tangent line for any degree even unto 90 gr.  
NOw then having known the length of the Tangent line in parts, it is easy (as is commonly known) to make Scales and prick down a Tangent line to a Circle of any Semidiameter, which when it is done, the use thereof is so commonly known, that I need speak nothing of it, but refer to Mr. Gunter and others that have written largely of its use.   

To find the length of a Secant for any degree.  
THis is to be done just in the same manner as in the former proposition the Tangent was to be found, only with this difference, that whereas the parts intercepted betwixt the intersection of the labels and the Radius of the Instrument on the left hand label were to be reckoned for the Tangent, or else for a Radius unto the Tangent represented by the Radius of the Instrument: here the Secant is always to be found by observing the number of parts intercepted betwixt the centre of the right hand label, and its intersection with the other label. This in the first of the two former examples, the Radius being found 10000, the Tangent 5773, the Secant shall be .11547, which is a Secant of 30 gr. but if the degree be above 45: then must we work by proportion as we did in the last, & the proportion will hold thus. As the parts intercepted on the left hand label, to the parts intercepted on the right hand label. So 10000 the Radius of the Instrument, to the parts of the Secant required. Thus in the forementioned example, the parts on the left hand label are 5773, on the right hand label 11547. I say then, As 5773, to 11547. So is 10000 to 20000.  
Bring down therefore 5773 on the left hand label, to 11547 amongst the parallels in the quadrant; and then 10000 upon the left hand label will meet exactly with 20000 the secant of 60 gr. whose compliment by which the work was wrought is 30 gr.  
The use of this line in Trigonometry and Horography, I leave to be searched out in their Books, that have written largely of the use thereof.   

To find the length of a line of sins for any degree.  
THere is no more required to find out this, but to look the degree in the quadrant; then observing amongst the parallels, just meeeting with the degree: you shall find there the parts required. As if I would know the length of the Sine of 30 gr. casting my eye upon the arch of the circle, where it is cut with the degree of 30: I find meeting exactly in that intersection the parallel of 5: or 5000. I conclude then, if the Radius be 10000 the sine of 30 gr. must needs be 5000.  
The use of this line I also refer to the writings of others, because it is so commonly treated of, and known of all.   

To find the length of a line of Chords for any degree.  
THis line so often made use for the dividing circles, or finding the quantity of any arch or angle, may be found with as great facility and delight as the three former; indeed it may be done without any more trouble, by doubling the sine of half the ark, as if we would know the Chord of 60 gr. half its ark is 30 gr. the sine whereof by the former proposition, as 5000. which doubled, giveth 10000, the Chord required.  
But it may be found with great delight and pleasure by the instrument another way: as followeth.  
Apply the edge of the right-hand label, exactly to the degree in the limb of the quadrant; whose Chord you desire, and the parts intercepted betwixt that degree and the centre of the label, are the Chord of that degree: thus in the former example, the label being applied to the 60 gr. the parts intercepted shall be exactly 10000, that is the radius, if applied to the 30 gr. the parts intercepted will be 5176: which is the Chord of 30 gr. directly double to the sine of 15 gr. 2588. which is half the ark.   

To find a line or number in continual proportion, having two numbers given  
SUppose the numbers be 4: and 8: I say then.  
As 4. to 8. So 8. to 16. So 16. to 32. etc.  
Bring down 8. on the left hand label till it meet with 4 amongst the parallels, then against 8 amongst the parallels, shall be found 16, on the label, & against 16 amongst the parallels shall be found 32 on the label, and so forward.   

Having the Square and its Root to find the Cube thereof.  
FOr as the Root is to the Square,  
So is the Square unto the Cube.  
As suppose, the Root be 3: and the Square 9: the Cube will be found to be 27.  
work this.  
Apply 9, on the left hand label to 3: amongst the parallels, and then 9, amongst the parallels shall meet exactly with 27: the Cube required.   

Having the Cube and its Square whereof it was made to find the Root of both.  
THis is to be wrought by working directly contrary to the former proposition.  
For.  
As the Cube is to the square, so is the square unto the root, apply therefore the number, standing for the cube and square together, finding the one on the label the other amongst the parallels, and then just over against the number for the square shall stand the root. Always remembering that if the first number be found upon the label, the third number must then be searched there also: but if it be amongst the parallels then the third must be amongst them to, for the third number in these proportions is always to be found on that line or Scale where the first was; and the fourth on that line where the second was: whether it be on the label or amongst the parallels.  
Many other things might be wrought by this Instrument; I have only given hints to those that are more curious and have greater leisure to follow these studies, who I doubt not, but by their industry will be able from what I have said to draw new conclusions, and from many pleasing propositions conducing much to the satisfaction of those that are ingenious.   
FINIS.   

Postscript.  
REader, thou haste had the Instrument with its use, the Author now takes leave of thee for a time; if a favourable aspect shine on this, thou mayst ear long expect more, he hath another Instrument called an Horographicall Sphere, whereby the whole Art of Dialling, will most plainly be set forth to the view of all that understand the use of it; and dials of all sorts may be delineated in all kind of Plains with great speed and pleasure, though they are never so full of Gibosities or Concavities, which would puzzle the best Mathematician to reduce to form: he is now in hand with it, thou shalt hear more from him as he perceiveth this to find exceptance.   

THis Instrument called the trigonal Sector, as also all other Mathematical Instruments whatsoever, are exactly made by Mr, Anthony Thompson in Hosier-lane near Smithfield, London.  
FINIS.