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ID Title Author Corrected Date of Publication (TCP Date of Publication) STC Words Pages
A60107 A 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. Worgan, John, surveyor. 1697 (1697) Wing S3636; ESTC R41684 18,763 47

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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 Mathematick 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 Explain'd 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 plac'd and the advantages accrewing 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 Mathematick Problems particularly in Arithmetick 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 Mathematicks 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 Publick Good be any way advanc'd I shall be very Glad of it Vale. Iohn Worgan Londini fecit CHAP. I. A Description of Chords Sines Tangents c. BEcause in the measuring of parts of a Circle which is often required there is no other way but by reducing them to strait Lines therefore did the Ancients apply certain strait Lines to a Circle which come in Competition to arch-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 Sines A Chord of an Arch is a strait 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 't is 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 Semi-circle are less than the Diameter Sines 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 Semi-circle as in Fig. 1. EB 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 Sines 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 't is also evident that the greatest Versed Sine is that of 180 Deg. Lines drawn through a Circle are Secants A Secant is a strait Line drawn from the Center 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 strait 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 strait Line HI is the Tangent of the Arch B H. Hence 't is 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 Center is double to the Angle at the Circumference by 20 Prop. of the Third Book of Euclid CHAP. II. How the Sines of Chords Natural Sines Tangents Secants c. 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 produc'd in Fig. 3. and draw A B then divide the Arch A B into 9 equal parts seting thereto the Figures 10 20 30 c. 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 strait Line A B placing thereto the Numbers 10 20 c. as was set to the Arch. The Line A B thus divided is a Line of Chords and may be transfer'd from the Paper to a Silver Brass or Box-Scale Secondly For the Sines 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 Sines which may from hence be transfer'd to any Scale or Rule Thirdly For the Versed Sines which is only a double Scale of right Sines let fall perpendiculars as before to divide the Sines from every degree in the whole Semi-circle A B C these Perpendiculars will divide the Diameter A C into a Line of Versed Sines which are numbered according to the Figures under the Diameter but are seldom put on any Scale except requir'd Fourthly From C raise a Perpendicular to A C and from the Center 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