Short Treatise OF THE DESCRIPTION OF THE SECTOR. WHEREIN Is also shown the great Use of that excellent Instrument, in the Solution of several Mathematical Problems. LONDON, Printed for and Sold by John Worgan, Mathematical Instrument-maker, at his Shop under St. Dunstan's Church, Fleetstreet, 1697. Where all sorts of Mathematic Instruments, both for Sea and Land, are made and Sold. TO THE READER. READER, THOU art here presented with a small Treatise of the Use of the Sector, in which I have Explained the true Nature and Property of the Construction of this admirable Instrument; and have Geometrically demonstrated the Grounds upon which all the Operations depend. I have also shown the Amendments that have been lately made in disposing the Lines hereon placed, and the advantages accrueing thereby. In the next place I have exhibited its general Use, and declared wherein the great benefit of this Noble Contrivance lies, and the Universal Usefulness of the same. Lastly, I have a little Illustrated its Excellency in the Solution of some few Mathematic Problems, particularly in Arithmetic, Geometry and Trigonometry; which indeed are the foundation of all the rest, and sufficient for the young Tyro's light to the Solution of many more. For to show 〈◊〉 the 〈◊〉 of 〈◊〉 ●●●●●ment were little less than to write a Body of the Mathematics, which is not my design at present. My principal intent here being only to show how this Instrument supplies the place of almost all others. Particularly of all kinds of Scales and Rules, which are made but to assigned or particular Radiuses, which is the main end, and sole design of this Instrument. If herein the Public Good be any way advanced, I shall be very Glad of it. Vale. sector (mathematical instrument) john Worgan Londini fecit CHAP. I. A Description of Chords, Sins, Tangents, etc. BEcause in the measuring of parts of a Circle (which is often required) there is no other way but by reducing them to straight Lines, therefore did the Ancients apply certain straight Lines to a Circle; which come in Competition to Arch-lines, and that several ways, viz. as they are drawn within a Circle, through it, or without a Circle. Lines within a Circle are Chords and Sins. A Chord of an Arch is a straight Line drawn from one end of an Arch to the other, as in Fig. 1. the Line A B is the Chord of the Arch A C B. It is also the Chord of the Arch A D B, for it is common to both parts of the Circle. So that from 15 Prop. of the 3 d Book of Euclid 'tis evident, the greatest Chord that can be drawn is the Diameter of the Circle, or Chord of 180 Deg. or half the Circle, and consequently that all Chords of Arches greater than a Semicircle, are less than the Diameter. Sins be either Right or Versed. A right Sine of an Arch is a Line drawn from one end of that Arch perpendicular to a Diameter drawn from the other end of that Arch, and belongs to both parts of a Semicircle, as in Fig. 1. EBB. is the right Sine of the Arch C B, and also of the Arch B D. And here you may observe from the aforecited Proposition of Euclid, that the greatest Sine is that of 90 Deg. or a Quarter of a Circle: And therefore all Sins of Arches greater than a Quadrant, are less than the Sine of 90 Deg. which is half the Diameter, or that which (in all Proportions) we call Radius. The Versed Sine of an Arch is that part of the Diameter betwixt the right Sine, and the end of that Arch it is the Versed Sine off, as in Fig. 1. E C is the Versed Sine of the Arch C B, and E D the Versed Sine of the Arch B D. From hence 'tis also evident that the greatest Versed Sine is that of 180 Deg. Lines drawn through a Circle are Secants. A Secant is a straight Line drawn from the Centre of a Circle through one end of that Arch (which it is the Secant off) till it meet with the Tangent which bounds it, as in Fig. 2. EI in the Secant of the Arch BH. Lines drawn without a Circle are Tangents. A Tangent is a straight Line that touches a Circle, and is erected perpendicular to a Diameter drawn from the Touch-point, being limited by the Secant which passeth through the other end of the Arch, it is the Tangent off, as in Fig. 2. the straight Line HI is the Tangent of the Arch B H. Hence 'tis evident there can be no Secant nor Tangent of 90 Degrees, for the Secant of 90 Degrees is parallel to the Tangent Line, and therefore if infinitely produced will never meet with it. The Half Tangent is only the Tangent of half the Arch as the Tangant of 90 Degrees is infinite, but the Half Tangent of 90 Deg. is the Tangent of 45 Deg. as in Fig. 2. FE is the Half Tangent of the Arch BH, which is but half the Angle B A H, for the Angle at the Centre is double to the Angle at the Circumference by 20 Prop. of the Third Book of Euclid. CHAP. II. How the Sins of Chords, Natural Sines, Tangents, Secants, etc. are projected, and put on the Common Scales. FIrst, with any distance describe the Circle, A B C F and draw the Diameter A C, which cross at right Angles with the Line D B produced in Fig. 3. and draw A B, then divide the Arch A B into 9 equal parts, setting thereto the Figures 10, 20, 30, etc. to 90; and each of those 9 Divisions again into 10 equal parts, one of which is a degree, or the 360 of the whole Circle. This done with one foot of the Compasses in A, transfer these Divisions of the Arch to the straight Line A B, placing thereto the Numbers 10, 20, etc. as was set to the Arch. The Line A B thus divided is a Line of Chords, and may be transferred from the Paper to a Silver, Brass, or Box-Scale. Secondly, For the Sins divide the arch B C as before, viz. into 90 equal parts, noting the 9 Grand Divisions with Figures as in the last. From these points in the arch B C let fall perpendiculars to the Diameter A C, these shall divide the Semi-diameter D C into a Line of right Sins, which may from hence be transferred to any Scale or Rule. Thirdly, For the Versed Sins, (which is only a double Scale of right Sins) let fall perpendiculars (as before to divide the Sins) from every degree in the whole Semicircle A B C, these Perpendiculars will divide the Diameter A C into a Line of Versed Sins, which are numbered according to the Figures under the Diameter, but are seldom put on any Scale, except required. Fourthly, From C raise a Perpendicular to A C, and from the Centre D through each Division of the Quadrant draw Lines, producing them till they meet the Line C E, which Lines shall divide the said Line into a Scale of Tangents, to which set the Fig. 10, 20, 30, etc. as they were in the Arch. Fifthly, If the Extents D 10, D 20, D 30, etc. in the Tangent-Line be transferred to the Line D B produced, they will divide it into a Line of Secants, which with the Numbers 10, 20, 30, etc. may from the Scheme be put upon any Scale, and is continued on the line of Sines, for after the Sine of 90, the Tangents gins. Hence you see that the Secant of Nothing, is the Radius or Semi-diameter of the Circle. Sixthly, If from A to each Division of the Arch B C you draw Lines, they shall divide the Radius B D into a Line of half Tangents, to which set the Numbers 10, 20, 30, etc. to 90. Sometimes this Line is made to run as far as 150 or 160 Deg. which is of the same length with the whole Tangent of 75 or 80 Deg. and when 'tis so divided the Line B D must be produced as it was for the Secant. For the half Tangent of 10 Deg. is the whole Tangent of 5 Deg. so also the half Tangent of 160 Degrees is the whole Tangent of 80 Degrees. Seventhly, Draw the Line A F and divide the Arch A F into 8 equal parts, these transferred from the Arch to the straight Line A F, as were the Chords, divides it into a Line of Rhumbs or Points: Its use is to lay of any Point of the Compass, which it doth much more readier than a Line of Chords, though that will serve when you have not this. Lastly, You may divide the Semi-diameter F D into 100 equal parts, and it will serve for a Line of Equal parts, to take Leagues or Miles from; not but that it might be longer or shorter for there is no necessity of its being just of such a length. This Line is of great use, because from it all the other Lines might be divided, by supposing the Radius of the Circle to contain any number of equal parts, as 1000, 10000, etc. For then all the Chords, Sins, Tangents, etc. will consist of a certain number of those parts. CHAP. III. Of the Plain Scale. ASsume a Line whose length let be what you please, suppose ab. On this Line make a Parallelogram of any breadth at pleasure as abcd, divide the opposite sides ab and de each into 10 equal parts, as also add, bc, setting thereto the Figures 1, 2, 3, etc. as you see in Fig. 4. This done draw from the Points 1, 2, 3, etc. in the Line ad, the several Lines parallel to ab, then from the Points b, 1, 2, 3, etc. in the Line ab, to the Points 1, 2, 3, etc. in the Line dc; draw the Diagonal Lines, b. 1, 1. 2, 2. 3, 3. 4, etc. By which the Line ab or those equal to it, are actually divided into 100 equal parts. The reason of which is very clear from the 4th. of the 6th. of Euclid, for by that proposition 'twil hold as bc: be:: 1 c: he, and therefore if be is 1/10 of bc (which by construction it is) then shall he be 1/10 of 1 c that is 1/10 of 1/10 (or 1/100) of dc or ab. If the Sides ab, dc, as also their Parallels 1, 2, 3, etc. be produced, and the distances ab, dc, repeated thereon, 1, 2, 3, or more Lines, as the lengths of the Rule on which you put them will permit, you have finished the said Diagonal Scale; the design of which is to supply the use of a Scale of equal parts, but with greater exactness than any of those kind of Scales can pretend to, for from it may be taken any Number under 1000 exactly. CHAP. IU. Of the construction of the Lines of Artificial Numbers, Sins and Tangents. BEsides the Lines before described there are another sort, which is generally put on Scales and Rules, called Artificial Numbers, Sins and Tangents, the rise and construction of which are from the Logarithmes. For they are only the Logarithms of the Natural Numbers, Sins and Tangents Transferred to a Scale. The Line of Numbers commonly known by Name of Gunter's Line, is a Line of Geometrical Proportion unequally divided into 9 parts, beginning at 1 towards the left hand and running with 2, 3, 4, etc. to 10, which is about the middle of the Line, where another Radius gins, and is continued to 100 towards the right hand, it hath the Figures 10, 20, 30, etc. set thereto, and is divided just as was the first part of it. The Line being thus divided is called a Line of Numbers of two Radiuses, and the way of Numbering on it thus, (for herein lies the great difficulty in the use of it.) It is (as before was noted) divided into 18 unequal parts, the first 9 ending about the middle of the Scale, and the other 9 at the end next the right hand; then each of these Primes or first Grand Divisions are subdivided into 10 other parts according to the same reason called Tenths; and again each of those Tenths are subdivided into 10 other parts if the length of the Rule or Scale will permit. The Figures 1, 2, 3, etc. by which the Primes are distinguished, are all arbitrary points, and may each of them represent so many entire Unites, Tenns, Hundreds or Thousands; or they may also represent so many Tenth, Hundred, Thousandth, or Ten Thousandth parts of an Unite. If the first 1 on the Line be taken for 1, then 10 in the middle of the Line is 10 as in course it falls, but if the first 1 be taken for 10, or 100, than the figure 10 in the middle of the Rule must be taken for 100 or 1000 Now when the first 1 or Prime represents 10 Unites, each Tenth in that Prime will be 1; and each Centesme in those Tenths (if there be any) will be one tenth part of an Unite. Again, let the first Prime represent 100, than the Figures 2, 3, 4, etc. will denote 200, 300, 400, etc. and so 10 at the end will be 10000, and according to this supposition 1 tenth in each Prime will be 10 Unites, and in those tenths each Centesme will be 1. For Decimal Fractions let 10 at the end of the Line, next the right hand, represent 1, than each Prime towards the left hand will be 1, and in those Primes each tenth will be .01, and in these tenths each Centesm will be .001 part of an Unite. The Divisions and way of Numbering on this Line being thus explained, it will not (I presume be difficult) to find the point upon the Line, where any Number given is represented. For Example: Suppose the Number 276 were proposed, for the first figure 2 I account 2 next the beginning of the Line that is 2 next the left hand, for the second figure 7 I tell 7 tenths next following, that is 7 of the 10 great Divisions betwixt 2 and 3, then from this point I count 6 Centesms. So that from 2 to this point represents 276. Again, suppose I would represent or find the Number 408 on the Line, then for the first figure 4 I take that 4 on the Line next the left hand, and for the 0 in the second place, I must not take any tenths, but for the 8 in the third place I count 8 Centesms, and it gives that point which represents 408. The greatest Inconvenience of these Lines is this, all Numbers above 3 or 4 places cannot precisely be represented unless the Lines are very long. As for Decimal Fractions and mixed Numbers, they are discovered after the same manner as whole Numbers were: For suppose 2. 76 was required to be represented, it will be found at the same point, and by the same Rule as the whole Number 276 was. And therefore by what hath been said it will be easy to find the point where any Number is represented on the Line especially if it be small. Next for the Line of Artificial Sins, which are only the Logarithms of the Natural, and are therefore transferred from a Table of Logarithms to the Scale, they begin toward the left hand, and are numbered towards the right with the Figures 1, 2, 3, etc. to 10, which stands about the middle of the Line, and signify single Degrees, after these it runs on with 10, 20, 30, etc. to 90, which stands at the end next the right hand, these Divisions are subdivided into 10 equal parts, and those again into 10, 5 or 2 parts, and sometimes not at all, according as the length of the Scale will permit. So that if betwixt 1 and 2 in the first part it be divided into ten parts, one of these parts will represent 6 Minutes, and if each of these Divisions be again halved each part will represent 3 Min. If the grand Division in the later part be divided into 10 equal parts, each division represents a degree, and if these are halved each division will be 30 Minutes. What is said of the Line of Sines, the same may be understood of the Line of Tangents, whose divisions begin at 1, and run to 10 in the middle of the Line, and signify only single Degrees, after 10 it runs on with 20, etc. to 45, which stands at the end of the Line; from 45 it runs back again to 90 where you begun, as the bare inspection of these Lines on any Scale, will more fully declare. These Lines are of most Excellent use, for by them all Questions of Proportion may be solved, whether Arithmetical or Geometrical. But their principal business is the solution of Plain and Spheric Triangles, which they do with great speed, and exact enough in many Cases, and therefore very necessary for proving your Arithmetic and Geometric Operations. The method of working with them is thus: In all Proportions you have three terms given to find a fourth. Seek out therefore the first term whatever it be, viz. Number, Sine, or Tangent on its like Line, that is, if it be a Number look for it on the Line of Numbers, if it be a Sine, seek it on the Line of Sines, etc. and in that point set your Compasses, then extend the other to the 2 d or 3 d term, that is, extend it to that which is of the same name with the first, (for either the 2 d or 3 d will be always like the first) the same extent laid from the other term the same way will reach to the fourth term required. An Example will make it plain: Suppose this Proportion was to be wrought, as the Sine of 67 d. 30 m. to the Numb. 64. So is Radius or Sine of 90, to the 4th Number required. Set one foot of your Compasses in the Line of Artificial Sins on the Number 67 d. 30 m. and extend the other foot to 90 on that Line, the same extent will reach from 64 on the Line of Numbers, to 70 on the said Line, if applied the same way, which 70 is the 4th Number required. After the same manner may all other Questions be wrought. CHAP. V Of the Sector, and the Description of the Lines thereon placed. HAving thus shown you how these Lines are originally divided, I shall now pass to show how they are placed on a Sector, or joint Rule, and then shall give a few Examples of their Excellent use as they are thus disposed. A Sector, as 'tis Geometrically defined, is a Figure bounded by two straight Lines, and part of the Circumference of a Circle, as Fig. 5. But by a Sector here spoken of, you are to understand an Instrument that opens upon a Centre like a common Carpenter's Rule. The two pieces that moves upon the Centre we call Legs, upon which are placed most of the Lines that are upon the Common Scales, but are here used after a different manner than they are upon those Scales. The principal Lines that are now generally put upon this Instrument to be used Sectorwise, are equal Parts, Chords, Sins, Tangents, Secants and Polygons. The Line of equal parts, called also the Line of Lines, is a Line divided into 100 equal parts, and if the length of the Leg of the Sector will permit, is again subdivided into Halves and Quarters, they are placed on each Leg of the Sector on the same side, and are Numbered by 1, 2, 3, etc. to 10. which is very near the end of each Leg, these Lines are (as in the Printed Plate of the Sector hereunto annexed) noted with the Letter L. And here note, that this 1 may be taken for 10, or for 100, 1000, 10000, etc. as occasion requires, and then 2 will signify 20, 200, 2000, 20000, etc. and so of the rest. The Line of Chords, is a Line divided after the usual way of the Line of Chords, from a Circle whose Radius is nearly the length of one of the Legs; this Line is placed upon each Leg of the Sector, beginning at the Centre, and running towards the end thereof. It is numbered with 10, 20, etc. to 60, and to this Line on each Leg is set the Letter C. Note, this Line on some Sectors run to 90 degrees. The Line of Sines, is a Line of common natural Sins, such as we have before defined, only 'tis divided from a Circle of the same Radius that the Line of Chords was; these are also placed upon each Leg of the Sector, and numbered with the Figures 10, 20, 30, etc. to 90, at the end of which on each Leg in the Print annexed, is set the letter S. The Line of Tangents, is a Line of common Tangents, divided from a Circle of the forementioned Radius, and is placed upon each Leg, and runs to 45. It has the Numbers 10, 20, etc. to 45 placed upon it, with the Letter T for Tangent. Beside this, is another small Line of Tangents, divided from a Radius of about 2 Inches, and is placed upon each Leg of the Sector, it gins at 45, which stands at the length of the Radius from the Centre, and runs to about the sides add, ae, also ac, ab, shall be proportional. That is ad to ac, as ae to ab. So that if add be a half or a third of the side ac, then shall de be a half or a third of his Parallel cb. The like reason holdeth in all other Sections: Whence you see that if add be the Chord, Sine or Tangent of any number of degrees to the Radius ac, then shall de be the Chord, Sine or Tangent of the same number of degrees to the Radius cb. Now the Lines found out by the Sector are of two sorts, viz. Lateral or Parallel. Lateral are such as are found upon the sides of the Sector as add, ac in Fig. 6. Parallel are the Lines that run from one Leg of the Sector to the other in equal divisions from the Centre, as the, cb in the said Figure. And here note, That the Innermost of the Parallels is the true divided Line, and therefore in using the Compasses you must always set them upon the innermost Line both in Lateral and Parallel Entrance. And further note, that the Lines are placed (upon Sectors now made) after a different way to what they were formerly, for instead of putting the same Lines at equal distances, from the inward edge, they now put them at unequal distances, as by inspection of the Print hereunto annexed you may see; whereupon one Leg, the Line of Chords is innermost, but upon the other the Line of Tangents is innermost. That is, the innermost line of Chords and Tangents are equally distant from the inward edge, and so are the outermost line of Chords and Tangents. The benefit of the Contrivance is this, when you have set the Sector to a Radius for the Chords, it serves also for the Sins and Tangents without stirring it. For the Parallel betwixt 60 and 60 of the Chords, 90 and 90 of the Sins; also 45 and 45 of the Tangent are all equal, and this is the reason why they now make the Chords run but to 60 deg. upon the leg of the Sector. Whereas they used to run to 90. But practice and Experience has found out the inconvenience. This being premised, I shall now proceed to show you the uses of the several lines. And first, CHAP. VII. PROB. I. Of the Line of Lines, or equal Parts. To divide any Line into any Number of equal Parts. IF you would divide any Line into the same Number of equal parts as the leg of the Sector is: Take the length of the line betwixt your Compasses, and set it over as a parallel betwixt 10 and 10 (which you may call 100 and 100, as before was hinted) then shall the parallel distances betwixt every point on each leg divide the given Line into the same equal parts: As if you would mark out 6 (or 60) of those parts, the parallel distance betwixt 6 and 6 marked out upon the given Line, is 6 (or 60) of the said parts. If you would mark out 7.3 (or 73) parts, then count 3 of the next great Divisions from 7, and in that point take the parallel distance betwixt each, and it gives you 7.3 (or 73) parts of the proposed Line; and so of the rest. If you would divide a Line into any odd number of equal parts, as suppose I would divide the Line ab, Fig. 7. into 37 equal parts, Take the Line ab betwixt your Compasses, and make it a parallel betwixt 37 and 37 on the legs of the Sector, than the distance betwixt 36 and 36 (the Sector remaining at this angle) is 36 of them. Also the parallel distance betwixt 35 and 35, 34 and 34, 33 and 33, are 35. 34 and 33 of such parts that the whole contains 37 off, and may be cut off as you see in Figure 7. In like manner may any other Line be divided into any Number of equal parts: But if the length of the line be such that it cannot be applied betwixt the parts desired, then take but half, or a quarter of the said Line. So shall 2 (or 4) of those Divisions, be but one such part or division of the whole Line. PROB. II. To Increase or Diminish a Line in a given Proportion. SUppose a Line was given to be increased in such proportion as 5 to 7, that is thus, If the Line given contains 5 parts, I require another Line that shall have in it 7 of those parts: Take the given Line betwixt your Compasses, and open the Sector till that Extent lie betwixt 5 and 5 on the Line of Lines, then if you take the parallel distance betwixt 7 and 7, it will give you the Line required. If it were to be diminished: Suppose as 7 to 5, then make the given Line a parallel betwixt 7 and 7, and the parallel Extent betwixt 5 and 5 is the Line required. PROB. III. Two Lines given to find a third in continual proportion, both increasing and diminishing. IF it be increasing, take the two Lines given, and set them upon each Leg on the Line of Lines from the Centre, and mark the points where both of them fall; then take the longest line betwixt your Compasses, and apply that Extent parallel wise in the Terms of the shortest, the Sector remaining at this angle, the parallel distance betwixt the Terms of the longest being taken and applied to the leg of the Sector from the Centre, will give you the length of the third Proportional required. If it be Diminishing, then instead of making the longest of the two given Lines a parallel betwixt the Terms of the shortest, make the shortest of the two given Lines a parallel in the Terms of the longest, than the parallel distance betwixt the Terms of the shortest being applied to the leg from the Centre, will give the 3d. Proportional required. Let the Lines given be ab 18, ac 24, then is cc 32 the 4th. Proportional increasing, as in Fig. 8. PROB. IU. Three Lines or Numbers given to find a fourth Proportional, both Increasing and Diminishing. THis Problem is almost the same with the last, and is therefore to be solved after the like manner: As first, If you would find it Increasing, set the two shortest Lines on the Line of equal parts on each leg of the Sector, then open the Sector till the third or longest Line will come betwixt the Terms of the shortest: The Sector thus standing, take the parallel Distance betwixt the Terms of the longest, and apply that Extent to the side of the Sector; it will give you the length of the 4th. proportional Increasing. If it be Diminishing, lay the two longest Lines on the legs of the Sector from the Centre, and make the shortest of the 3 given Lines a parallel in the Terms of the longest; then the parallel betwixt the Terms of the shortest taken and applied as before, is the 4th. Proportional required, and is Decreasing or Diminishing. An Example would be needless. PROB. V To find the Propotion that is betwixt any two given Lines. TAke the greater of the two given Lines, and make it a parallel betwixt 10 and 10 at the end of the Sector, then take the lesser line and carry it parallel to the greater, till it rests in like points on each leg; so the number of Points where it rests, is the proportion to 100 Thus in Fig. 9 you will find the Line aa in proportion to bb as 7 to 10. PROB. VI The length of one Line being givan, to find what length any other Line is in such parts. SUppose the Line aa in Fig. 9 was 44, and I would know what the Line bb is, make the Line aa a Parallel in 44, then take the Line bb, and carry it along Parallel to aa till it rests in like Points, so the number it rests in which is about 30, is the length of the Line bb in such parts as aa did contain 44. This Problem may be very useful sometimes, as in Case the Draught or Ground-plot of any place be taken and no Scale to it, then if the length of any one Line or side in the Draught be known, the length of all the rest is easily found by this artifice. PROB. VII. A right Line any how divided being given, to find how to divide any other Line of different length into the like parts. LET ch in Fig. 10. be the Line given, which is unequally divided in the Points e. f. g. And let hh be the other Line required to be divided into such parts as was the former; Lay the Line ch from the Centre on both Legs of the Sector, and in the Points where this Line reaches apply the shortest Line Parallel-wise, that is open the Sector till the shortest Line may be applied betwixt the terms of the longest, this done lay the Divisions of the line ch, viz. ce, ch, cg, on both legs of the Sector from the Centre, than the Parallel extent of these Divisions, viz. ee, ff, gg, being taken and applied to the line hh will will divide it into the like parts as was the line ch. CHAP. VIII. Of the General Use of the Line of Chords, Sins, Tangents and Secants. BY disposing and placing these lines as afore directed on this Instrument we have Scales to several Raduis', as I have already hinted, that is having a length or Radius given (not exceeding the length of the Sector when opened) we can by the Sector find the Chord, Sine, etc. thereto. For which property this Instrument is many times of great use. For Example, Suppose I would have the Chord, Sine; or Tangent of 10 deg. to a Radius of 3 Inches. I take this 3 Inches and make it a Parallel betwixt 60 and 60 on the line of Chords, then as I told you before the same extent will reach from 45 to 45 on the line of Tangents: Also on the other side of the Sector the same distance of 3 Inches will reach from 90 and 90 in the line of Sines, so that if the lines of Chords be set to any Radius the lines of Tangents and Sins are also set to the same; Now the Sector being thus set, if you take on the said line, the Parallel distance betwixt▪ 40 and 40 it will give you the Chord of 40 degrees: Also if you take the Parallel distance on the line of Sins between 40 and 40, it will give you the Sine of 40 degrees. Lastly, if you take the Parallel extent on the line of Tangents, betwixt 40 and 40, it will give you the Tangent of 40 degrees, and all to the same Radius. If the Chord or Tangent of 70 degrees had been required, then for the Chord you must take the Parallel distance of half the Arch proposed, that is the Chord of 35 deg. and repeat that distance twice on the Arch you lay it down on, and you have the Chord of 70 deg. and for finding the Tangent of 70 deg. to this Radius, you must make use of the small line of Tangents, for the great one running but to 45 deg. we cannot take the Parallel of 70 deg. on that; I therefore take the aforesaid Radius of 3 Inches and make it a Parallel betwixt 45 and 45 on this small line of Tangents, than the Parallel Extent of 70 deg. on the said line, is the Tangent of 70 deg. to 3 Inches Radius. If you would have the Secant of any Arch, then take the given Radius and make it a Parallel betwixt the beginning of the line of Secants, that is 0 and 0, so the Parallel distance betwixt 40 and 40, or 70 and 70 on the said Secant line, will give you the Secant of 40 or 70 degrees to the Radius of three Inches. After this manner may the Chord, Sine or Tangent, of any Arch be found, provided the Radius, can be made a Parallel betwixt 60 and 60 on the Line of Chords, or betwixt the small Tangent of 45, or Secant of 0 deg. But if the Radius be so large that it cannot be made a Parallel betwixt 45 and 45 on the small Tangents, than no Tangent of any Arch above 45 degrees cannot be found; Nor the Secant of no Arch at all to such a Radius, because all Secants exceed the Radius, or Semi-diameter of the Circle. If the converse of any of these things be required, that is, If I require the Radius, to which a given Line is the Chord, Sine, Tangent or Secant of any Arch, suppose of 40 deg. then 'tis but making that Line (if it be a Chord) a Parallel on the Line of Chords betwixt 40 and 40, so will the Sector stand at the Radius; That is the Parallel Extent betwixt 60 and 60 on the said Chord-line is the Radius. And so if it be a Sine, Tangent or Secant, 'tis but making it a Parallel betwixt the Sine, Tangent, or Secant of 40 deg. according as 'tis given, then will the distance of 90 and 90 on the Sins, if it be a Tangent, the Extent of 45 and 45 on the Tangents: And if it be a Secant, the Extent or distance between 0 and 0 on the said Line of Secants, will be the Radius. Hence you see 'tis very easy to find the Chord, Sine or Tangent to any Radius, which is the great Benefit and Advantage we receive from this ingenious Contrivance, and of great use it is, not only in delineating and projecting on Paper, but even on the Ground, in laying out any Parcel in any form or shape; for by a Sector of two or three Foot long when opened, we can furnish ourselves with Scales big or little enough for any purpose or Design whatsoever. CHAP. IX. Of some particular Uses of the Circular Lines, viz. Chords, Sins, etc. PROB. I. An Angle being given, to find the Number of Degrees it contains. SUppose I would find the Quantity of the Angle a in Fig. 11. Take any distance, but so short that it may cut both sides of the Angle, with this distance on the Angular point a describe the Arch bc, and make that distance a Parallel betwixt the Chord of 60 and 60, then take the distance bc and carry it Parallel on the Line of Chords till it rests in like Points, so the place where it stayeth gives the Quantity of the Angle desired, which in our Example is about 35 deg. But if the distance bc is so long that it cannot be made a Parallel on the Line of Chords, as it will when the Angle is above 60 degrees. Then take but half the Arch, and make it a Parallel Chord as afore-directed, and the Points where it stays will give you half the Angle, which doubled gives the whole. PROB. II. To divide the Circumference of a Circle into any Number of Equal Parts. DIvide 360 the Number of Degrees in every Circle, by the Number of Parts required, the Quotient giveth the Number of Degrees, of which if you take the Chord off 'twill divide the Circumference. For Example, if I would divide a Circle into 5 equal parts, I divide 360 by 5, the Quote is 72 deg. If therefore I take the Radius of the Circle I would so divide, and make it a parallel betwixt 60 and 60 on the Line of Chords, and then (because I cannot take the Chord of 72 deg. the ⅕ part of the Circle) take the Chord of 36 deg. and repeat it twice on the Arch it will give the ⅕ part of the Circle which was the thing required. But this Problem is more readier performed by the Polygon Line which is for this very purpose. The manner of doing it is thus, Take the Radius of the Circle you would so divide, and make it a parallel betwixt 6 and 6 on the Polygon Line (for the Radius or Semi-diameter always divides the Circumference of the Circle into 6 equal parts) then if you take the Parallel distance betwixt 4 and 4 on the Polygon-line, it will divide the Circle into 4 equal parts; if betwixt 5 and 5 it will cut it into 5 equl parts: So also if you take the Parallel distances betwixt 6 and 6, 7 and 7, 8 and 8, 9 and 9, and so to 12, you may divide the Circle into 6. 7, 8. 9, 10. 11, or 12 equal parts. PROB. III. The Radius, or the right Sine of an Arch (by which Radius may be found) being given, to find the Versed Sine of any Arch. SUppose ab Fig. 12, were the Radius, and it were required to find the Versed Sine of 50 deg. make the Line ab a Parallel betwixt the Sins of 90 and 90, then because the Arch is under 90 degrees, take the Sine Compliment, that is take the Parallel betwixt 40 and 40, on the Line of Sines, and with that distance set one foot in a, and cut from the Line ab the Segment cb, which is the Versed Sine of 50 deg. to that Radius. If ac be the given Sine of 40 deg. and it be required to find the Versed Sine of 50 deg. I take the Line ac and make it a Parallel betwixt the Sine of 40 and 40, then taking the Parallel distance of 90, it gives me the Line ab for the Radius. From which if I take the Sine of 40 deg. the Compliment of the Arch whose Versed Sine is sought, it will leave me the Versed Sine of 50 degrees. But if the Arch whose Versed Sine is required be more than a Quadr. the Versed Sine will also be greater than a Quadrant, and that by so much as is the right Sine of the Excess of that Arch above 90. And therefore when ever the Versed Sine of an Arch exceeding 90 degrees, is required, suppose it were of 120 degrees, 'tis but taking the right Sine of the Excess above 90 degrees, viz. 30. and adding to Radius, so is the sum of Radius and this Sine of 30 degrees the Versed Sine required. PROB. IU. How by the Line of ●ines to describe an Elipsis to any length and breadth given. SUppose it were required to describe an Elipsis, whose length should be ab and breadth cd. Fig. 13. Set these two Lines at right Angles, and so as they may bisect each other in 0; then take half of the longest Line, and make it a Parallel betwixt the Sine of 90, the Sector thus set, take the parallel Sine of 10, 20, 30, etc. to 90 deg. and lay from 0 both ways toward b and a, so have you several Points in the Line ab, through which if you draw Lines Parallel to cd, and lay thereon from the said Points the Sine of 80. 70, 60. 50, etc. to 10 degrees, upon the respective Parallels, they will give you several Points, through which if a Line be drawn with an even hand, it will form an Elipsis. By this Artifice may the Orthographick or Analemmatick Projection of the Sphere be delineated with great ease and exactness, the Circles which are beheld obliquely being in that Projection all Elipses. PROB. V To open the Sector to any given Angle. IT is one thing to open the Sector to an Angle, and another thing to open the Lines on the Sector to the same Angle. Suppose I would open the Line of Chords on the Sector to an Angle of 38 Deg. I take 38 from the Centre laterally, and make that a parallel betwixt 60 and 60 on the line of Chords, so is the line of Chords set to an Angle of 38 deg. and not only the line of Chords, but also the Line of Lines, Line of Sines, and Line of Tangents, because they are all Equidistant from the inward edges of the Sector. If you would open the two inward Lines, suppose Chords and Tangents, or Sins and Lines, 'tis done after same manner, that is, by taking the lateral Chord of the Angle you would open the Sector to, and making it parallel betwixt 45 and 60 on one side the Sector, or else betwixt 90 and 10 on the other side, so is the innermost Line of Sines and Lines on the other side set to the Angle required. After the same manner may the outermost Lines be set to any given Angle. But suppose you would open the very edges of the Sector to any given Angle, then having put the Sector close; Take the parallel Extent betwixt the innermost in of Chords and Tangents, and apply that Extent laterally on the line of Chords from the Centre, and it shows you what Angle those lines make when the Sector is shut, in the printed Plate of the Sector hereunto annexed, those lines make an Angle of 5 deg. Suppose therefore you would open the Edges of the Sector to an angle of 40 Deg. I take from the Scale of Chords the lateral Chord of 45 deg. and apply that Extent parallelwise betwixt 60 on the innermost line of Chords, and 45 on the innermost line of Tangents, and it sets the Edges of the Sector to an Angle of 40 deg. for those lines are set to an Angle of 45 deg. Now the Edges of the Sector making an Angle less by 5 Deg. than those lines do, must therefore by consequence be set to an Angle of 40 Degrees. If you would open the Line of Lines to a right Angle, take the whole line of 10 parts, and make it a parallel betwixt 8 and 6 of the same. The line of Sines may be opened to a right Angle, if the Sine of 45 deg. be applied over in the Sins of 30 deg. And here note, that having opened the Line of lines to a right Angle, the line of Chords, Sins and Tangents are opened to the same. PROB. VII. The Sector being opened to find the Quantity of the Angle it is opened to. THe Sector being opened, take the parallel distance betwixt 60 and 60 on the Chords, and apply that Extent laterally from the Centre on the Chords, the number of Degrees it reaches to is the Angle those Lines are opened to. Or the parallel extent betwixt 60 on the Chords, and 45 on the Tangents so applied, shows you the Angle they are opened to, and if they are the two innermost Lines, then is the edges opened to an Angle less by 5 deg. than those Lines are. If the Lines be at the same distance as in our Print. This Problem is exceeding useful, for by it the Bevel of all kind of Roofs may be readily found. Also the Ground-plot of any old Edifice or Structure may be much easier and better taken than with any other Instrument whatsoever: Likewise the Quantity of Inclination and Reclination of all Plains, which is of great use in Dyaling, as also the Angles of any kind of Fortifications, whether Regular or Irregular, may most easily and speedily be taken by this Instrument. In order for drawing the Ground-plot or Proffile of any kind of Works, which thing is of great use at this time of day. PROB. VII. Upon a right Line and from a Point given in it, to lay off any Angle proposed. LET the Line given be ab, and let it be required to lay off an Angle containing 40 degrees from the given point a. Upon the point a describe with any distance an Arch cutting the said Line in b. Fig. 14. this distance make a parallel betwixt the Chord of 60, then if you take the parallel betwixt 40 and 40 on the said Lines, and lay the extent from b on the Arch to c drawing from the point a through c the line ac, and it will form the Angle required. PROB. VIII. To Cut a Line in Extreme and Mean Proportion: That is, to divide a Line into such parts that the whole Line shall be to the greater of the two Segments, at the greater, to the lesser. LEt ab Fig. 15. be the Line to be divided, I take the said Line ab, and make it a parallel betwixt the Chords of 60, to which Radius I find the Chord of 36 degrees, that is, the Sector being at this Angle, I take the parallel Chord of 36 degrees, and that extent being laid from a to c gives the greater Segment, and consequently cuts off the lesser Segment cb. For 'twill hold as ab: ac:: ac: cb. The practice of this Problem is demonstrated in the 9th. Prop. of the 13th. Book of Euclid. CHAP. X. Trigonometry, Or the use of the Natural and Artificial Lines on the Sector, in the Solution of the Cases of Plain right Angled Triangles. Defin. Trigonometry is a Science that teacheth how to Measure the Sides and Angles of all kinds of Triangles. CASE 1. Hypotenuse's bc 496, and Angle b 38 d. 40 m. given to find the Legs. Geometrically, First draw a Line, as in Fig. 16. assume therein the Point b, on which as a Centre with 60 deg. from the Chords, describe the Arch cd, setting thereon, from e to d, 38 d. 40 m. From b through d draw a Line at pleasure on which set from b to c. 496 taken laterally from the Line of Lines, from c let fall the Perpendicular ca and the Triangle is completed. Note, The given Lines and Arches are drawn Black, the required pricked. If you take the Line ca betwixt your Compasses, and setting one foot in the Centre of the Line of Lines, and pitch the other on the said Line, it will give you 310 for the length of that side. So also if you take the Line ba and apply it after the same manner, it will give you 388 for the length of this Line. And here Note, If this lateral distance of 496 be too big, or too little, it may be remedied, by opening the Sector to such a distance till the Parallel Extent betwixt 496 and 496 on the said Lines be of such a length as you would have it, then keeping the Sector at this Angle, take the Lines ba, ca, and apply them Parallelwise, till they rest in like Points, so shall these Points where they rest give the length of the Lines ba, ca, as before. Instrumentally. That is, by the Artificial Lines of Numbers, Sins, and Tangents. If you make the Hypotenuse Radius, the other two sides will be Sins of their opposite Angles. By Chap. I. And then it will hold as Radius, to the Hypotenuse, so is the Sine of the Angle b, to the side ca And therefore extend your Compasses from the Sine of 90 deg. to the Sine of 38 deg. 40 m. The same extent will reach on the Line of Numbers from 496 to 310 (being laid the same way) THE length of the Line ca For the side ba extend your Compasses from Radius to the Sine 51 d. 20 m. The same extent will reach on the Line of Numbers from 496 to 388 the length of the side inquired. USE. By this Case we find the difference of Latitude and Departure in Navigation, when the Course and Distance sailed are given. Also by it may be found the Horizontal Line, or Perpendicular height of any Mountain: Having first measured the assent, and taken the Angle that its rise makes with the Horizontal Line, which may be easily done by fastening one leg of the Sector to a Walking-staff set perpendicular in the Ground, and Elevating the other leg, till by it you see the foot of the hill if you stand at top, or top of the hill if you stand at the foot, so is the Sector set to the Angle that the assent makes with the perpendicular, or Horizontal Line, according as you take it either from top or foot of the Hill. CASE 2. Base ba 388, and Angle b 38 d. 40 m. to find the Hypotenuse, and Perpendicular. Geom. Draw a Line at pleasure, as in Fig. 17. on which set from a to b 388 taken from the Line of Lines, from a erect a perpendicular, and on the point b as a Centre, describe with 60 deg. from the Chords the Arch de setting thereon 38 d. 40 m. from e to d. through which draw the Line bd till it meet the perpendicular in c so is the Triangle completed. If now the Lines bc, ca, be taken and separately measured upon the same Scale ba was taken from, it will give you 496 for the Hypotenuse, and 310 for the Perpendicular required. Instrum. Making Hypotenuse Radius as in the last, the sides ba, ca, will be Sins of their opposite Angles. As by Chap. 1. So that it will hold as the Sine of the Angle c, to the Base ba; so is the Sine of the Angle b, to the perpendicular ca Or so is Radius or Sine of 90 d. To the Hypotenuse bc. Extend the Compasses therefore from 51 d. 20 m. (the Compliment of 38 d. 40 m. to 90 deg.) on the Line of Sines, to 38 d. 40 m. on the said Line. The same Extent will reach on the Line of Numbers from 388 to 310 the length of the perpendicular required. Again, if you extend the Compasses from 51 d. 20 m. on the Line of Sines to 90 deg. the same extent will reach on the Line of Numbers the same way, from 388 to 496 the length of the Hypotenuse required. USE. 'Tis by this Case that we find the difference of Latitude and Distance sailed, when the Course and Departure is given. Also when the distance of the Base of any Object from the place where we stand to observe its height, with the Angle at the Base are given, we make use of this Case to Calculate its height. part of plate of mathematical diagrams referenced in the text, figures 11, 8, 9, part of 10, 12, 16, 15, 4, 3 part of plate of mathematical diagrams referenced in the text, figures 9, 10, 8, 2, 1, 5, 6, 7, 11, 8, 9, 10, 12, 14, 15, and parts of 4 and 3 CASE 3. Hypotenuse's bc 496 and Base 388 given, to find the Perpendicular, and obliqne Angles. Geom. First draw a Line, on which lay off 388 from b to a taken from the Line of Lines: From a erect a Perpendicular producing it, then take from the said Line of Lines 496, and setting one foot in b, pitch the other in the Perpendicular at c; join the points bc, and you complete the Triangle. This done take the Line ca and measure it on the Scale, it will give 310 for the length of the said Line. So also if on the point b you describe with 60 deg. from the Chords the Arch ed, and then take the distance ed, and measure it on the Scale of Chords it will give you 38 d. 40 m. for the Angle b. Which 38 d. 40 m. taken from 90 will leave 51 d. 20 m. for the Angle c. Instrum. If you make the Hypotenuse Radius as before, than the Base and Perpendicular will be Sins of their opposite Angles. And then it will hold as the Hypotenuse's bc is to the Radius or Sine of 90 d. So is the Base ba, to the Sine of the Angle c. Extend the Compasses therefore from 496 to 388 on the Line of Numbers, the same Extent will reach from 90 on the Line of Sines, to 51 d. 20 m. which is the measure of the Angle c. which taken from 90 leaves 38 d. 40 m. for the Angle b. Having gotten both the obliqne Angles the Perpendicular is found by Case the 1st. USE. 'Tis by this Case we find the Course and difference of Latitude, when the Distance sailed and Departure are given. CASE 4. Base ba 388, and Perpendicular ca 310 given. To find the Hypotenuse and obliqne Angles. Geom. Draw a Line on which lay off from b to a 388, from a erect a Perpendicular, laying thereon from a to c. 310, then join bc and the Triangle is completed. From whence if you take the Hypotenuse's bc and apply it to the Line of Lines, you shall have 496 for the length of it. And for the Angles, describe the Arch of a Circle as in the last Case was directed, than the intercepted part the taken and applied to the Scale of Chords, will give you 38 a. 40 m. for the measure of the Angle b. which 38 d. 40 m. taken from 90, will leave 51 d. 20 m. as before for the Angle c. Instrum. In this Case the Base must be made Radius, then by Chap. 1. the Perpendicular will be the Tangent of the Angle opposite to it, and the Hypotenuse the Secant of the said Angle. Which granting, it will then hold, as the Base ba, to Radius or Sine of 90 d. so in the Perpendicular ca, to the Tangent of the Angle b. Extend therefore the Compasses from 388 to 310 on the Line of Numbers, the same Extent will reach on the Line of Tangents, from 45 d. the Radius, to 38 d. 40 m. the Measure of the Angle b. which taken from 90, leaves the Angle c. Having USE. 'Tis by this case (having the difference of Latitude and Departure) we find the Course and Distance sailed. Also having the breadth of a Moat, and height of a Tower, we can from hence find the length of a Scaling-Ladder that shall reach from one to the other. After the same manner may the obliqne Cases in Plain Triangles be solved; and by the Artificial Lines may all the Proportions in both right and obliqne Spherical Triangles, have their Solution, which thing is of great use, in that it so instantly proves our Arithmetic Operations. FINIS. identical figures 16, 17, 18, and 19