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ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A64223	The semicircle on a sector in two books. Containing the description of a general and portable instrument; whereby most problems (reducible to instrumental practice) in astronomy, trigonometry, arithmetick, geometry, geography, topography, navigation, dyalling, &c. are speedily and exactly resolved. By J. T.	Taylor, John, 1666 or 7-1687.	1667 (1667)	Wing T533B; ESTC R221720	52,688	153

THE SEMICIRCLE ON A SECTOR In two Books CONTAINING The description of a general and portable Instrument whereby most Problems reducible to Instrumental Practice in Astronomy Trigonometry Arithmetick Geometry Geography Topography Navigation Dyalling c. are speedily and exactly resolved By J. T. LONDON Printed for William Tompson Bookseller at Harborough in Leicestershire 1667. To the Reader ALL that is intended in this Treatise is to acquaint thee with an Instrument that is both portable and general of no great price easie carriage yet of a speedy and Accurate dispatch in the most difficult Problems in Astronomy c. The lines for the most part have been formerly published by Mr. Gunter the famous Mr. Foster Mr. White c. The reduction of the 28. Cases of Spherical Triangles unto II. Problems I first learned from the reverend Mr. Palmers Catholick Planisphere Many of the proportions in the Treatise of Dyalling are taken from though first compared with the Globe my worthy Friend to whom I am indebted in all the Obligations of Civility and without whose encouragement this had never adventured the publick Test Mr. John Collins The applying Mr. Fosters Line of Versed Sines unto the Sector was first published by Mr. John Brown Mathematical Instrument-Maker at the Sphere and Sun-Dial in the Minories London Anno 1660. who bath very much assisted me by making adding unto and giving me freely the perusal of many Instruments according to any directions for Improvement that was proposed to him After this account what hath been my part in this Work I hazard to thy censure and when I see others publish a more convenient speedy accurate and general Instrument I assure them to have as low thoughts of this as themselves But here is so large a Catalogue of Errata's as would stagger my confidence at thy pardoning had they not been irrevocably committed before I received the least notice of them The Printer writing me word after I had corrected so much as came to my sight that he could alter no Mistakes until the whole Book was printed By which means he enforced me to do pennance in his Sheets for his own Crimes Did not one gross mistake of his become my purgation viz. in lib. 2. throughout Chap. 3. where instead of the note of equality marked thus = he hath inserted the Algebraick note of Subtraction or Minoration marked thus Nor hath the Engraver come behinde the Composer who so miserably mangleth Fig. 13. that at first sight it would endanger branding of a mans Brains to spell the meaning thereof either in it self or in reference to the Book All that I can help thee herein is this Whereas the Book mentions that Figure for an East Dyal if you account it as now cut a West Dyal and alter the names of the hours by putting Figures for the afternoon in the place of those there for the morning you will then have a true West Dyal of that Figure The correction of Punctations would be an endless task for I finde some to be resolved ever since Valentine to recreate themselves at Spurn-point What other material mistakes are in the Book which ought to be corrected before reading thereof you will finde mentioned in the Errata Farewel March 29. 1667. J. T. Errata PAge 4. line 12. signs r. sines p. 8. l. 5. seconds r. secants p. 12. l. 9. all r. allone p. 13. l. 9 and 10 sec r. min. p. 14. l. 17. sec. r. min. p. 22. l. 14. any r. what p. 23. l. 3. exact r. erect p. 24. l. ult adde lib. 2. p. 25. l. 2. signs r. sines l. 5 sign r. sine p. 35. l. 3 a mark r. an ark p. 37. l. 22. 20. r. 22. and 42. r. 20. p. 44. l. 5. At r. At. p. 49. l. 1 divided by r. dividing p. 62. l. 15. dele a. In lib. 1. chap. 9. the pages are false numbred But in chap. 9. p. 62. l. 11 next r. exact l. 15 gauger r. gauge l.24.the r. what p. 73.l 5 whereas r. where I. p. 80. l. 13 wherein r. whereof l.22.pont r. point p. 91 l.1 the co-tangent r. half the co-tangent l. 19 L. r. P. p. 92. l. 21 PZ r. PS p. 93. l. 12 NSP r. NSZ 6.angle r. ark throughout page 96. Fig. 3. r. Fig. 7. and Fig. 4. r. Fig. 8. p. 98. l. 13. serve r. scrue p. 99 l.12 places r. plates l. 13. proportion r. perforation l.9.serve r. scrue l.4.serve r. scrue 21.serve r scrue p. 111. l. 20. lay in r. laying l.14.of ●● and. p. 128. l. 3. Fig. 12. r. Fig. 13. l. 22 ED. r. EC In Fig. 12. C. r. A. at the end of the line G. To the Right Honorable The Lord SHERARD Baron of Letrim My Lord SInce the trifling Treatise of an Almanack hath usurped a custom to pinnion some Honourable Name to the Patronage of the Authors Follies had we not certain evidence from the uncertainty of their Predictions that their Brains like their great Oracles the Planets are often wandring it might be deemed a Crime beyond the benefit of the Clergy to prefix before any Book a dedication to a Noble Person Or when I read the unreasonableness of others in those Addresses imploring their Patrons to be their Dii Tutelares and prostrate their reputes to the unmannerly mangling of every Censurist under the notion of protecting that is adopting the Authors Ignorance or negligence it s enough to tempt the whole world to turn Democritians and hazard their spleens in laughing at such mens madness My present design is only to give your Lordship my observance of your Commands about the Description and Improvement of the Sector and wherein I have erred through mistake or defect I despair not but from your Honor I shall meet with a pardon of Course to be granted unto Your Lordships most humble Servant J. T. THE SEMICIRCLE ON A SECTOR LIB I. CHAP. I. A Description of the Instrument with the several Lines inscribed thereon THe Instrument consists of three Rulers or Pieces two whereof are joyned together by a River that may open and shut to any Angle in fashion of the Sector or to use a courser comparison after the manner of Compasses The third piece is loose or separable from them to be put into the Tenons at the end of the inward ledge of the joyned pieces and thereby constituting an aequilateral triangle On these Rulers after this manner put together we take notice for distinction sake of the sides ledges ends and pieces The sides are thus differenced one we call the quadrantal the other the proportional or sector side The ledges are distinguished by naming one the inward the other the outward ledge The ends are known in terming one the head viz. where the two pieces are riveted together the other the end The pieces are discovered by styling one the fixed piece viz. that which hath the rivet upon it the second the movable piece which turns upon that rivet and the last the loose

set half the co-tangent D R Z from A to F and the secant of D R Z from F to L upon the center A with the extent A P. Draw the ark P G and with the extent F P from L cross the ark P G in G. Lastly upon the Center G with the extent G L. Draw the ark R D F L and your triangle is made The triangle projected you may measure off the sides and hypothenuse Thus First the hypothenuse Z R is measured by a line of chords Secondly a ruler laid to L D cuts the limb at H and Z H upon a line of chords is the measure of the ark Z D. Thirdly draw A G and set half the tangent D R Z from A to V apply a ruler to V D it cuts the limb at E then R E upon a line of chords measure the ark R D. Note The radius to all the chords tangents and secants used in the projection and measuring any ark or angle is the semidiameter of the fundamental circle CHAP. XVI The projection and solution of the 12 Cases in oblique angled spherical triangles in six Cases See Fig. 7. THe fundamental circle N H Z M is alwayes supposed ready drawn and crossed into Quadrants and the Diameters produced beyond the Circle CASE 1. The three sides Z P P Z and Z S given to project the Triangle By a line of chords prick off Z P and draw the diameter P C T crossing it at right angles in the center with AE C E set half the co-tangent P S from C to G and he secant P S from C to R upon the center R with the extent R G draw the the ark FGL Again set half the co-tangent Z S from C to D and the tangent Z S from D to O with the extent O D upon the center O draw the ark B D P mark where these two arks intersect each other as at S. Then have you three points T S P to draw that ark and the three points N S P to draw that ark which make up your triangle CASE 2. Given two sides Z S and Z P with the comprehended Angle P Z S to project the Triangle Prick off Z P and draw PCT and AECE and the ark B D P by Case 1. Again set the tangent of half the excess of the angle P Z S above 90 from C to W and co-secant of that excess from W to K upon the center K with the extent K W draw the ark N W Z which cuts the ark B D P in S. Then have you the three points T S P to draw that ark which makes up the triangle CASE 3. Two Angles S Z P and Z P S with the comprehended side Z P given to project the Triangle Prick off Z P and draw the lines P C T and AE C E by Case 1 and the angle NWZ by Case 2. Lastly set half the co-tangent ZPS from C to X and the secant Z P S from X to V upon the center V with the extent V X draw the ark T X S P and the triangle is made CASE 4. Two sides ZP and PS with the Angle opposite to one of them SZP given to project the Triangle Prick off ZP and draw PCT and AECE by Case 1. and the angle SZP by Case 2. Lastly by Case 1. draw the ark FGL and mark where it intersects NWZ as at S then have you the three points TSP to draw that ark and make up the triangle CASE 5. Two Angles SZP and ZPS with the side opposite to one of them ZS given to project the Triangle Draw the ark BDP by Case 1. and the ark NWZ by Case 2. at the intersection of these two arks set S with the tangent of the angle ZPS upon the center C. sweep the ark VΔI Again with the secant of the ark ZPS upon the center S cross the ark VΔI as at the points V and I. Then in case the hypothenuse is less than a quadrant as here the point V is the center and with the extent VS draw the ark TSP which makes up the triangle But in case the hypothenuse is equal to a quadrant Δ is the center if more than a quadrant I is the center in which cases the extent from Δ or I to S is the semidiameter of the ark TSP CASE 6. Three Angles ZPS and PZS and ZSP given to project the Triangle See Fig. 7. and 8. The angles of any spherical triangle may be converted into their opposite sides by taking the complement of the greatest angle to a Semicircle for the hypothenuse or greatest side Wherefore by Case 1. make the side ZP in Fig. 4. equal to the angle ZSP in Fig. 3 and the side ZS in Fig. 4. equal to the angle ZPS in Fig. 3. and the side PS in Fig. 4. equal to the complement of the angle PZS to a Semicircle in Fig. 3. Then is your triangle projected where the angle ZPS in Fig. 4. is the side ZS Fig. 3. Again the angle ZSP Fig. 4. is the side ZP in Fig. 3. Lastly the complement of the angle PZS to a Semicircle in Fig. 4. is the measure of the hypothenuse or side P S in Fig. 3. The Triangle being in any of the former Cases projected the quantity of any side or angle may be measured by the following rules First The side Z P is found by applying it to a line of chords Secondly CX applyed to a line of tangents is half the co-tangent of the angle ZPS Thirdly CW applyed to a line of tangents is half the co-tangent of the excess of the angle SZP above 90. Fourthly set half the tangent of the angle ZPS from C to Π a ruler laid to ΠS cuts the limb at F then PF applyed to a line of chords gives the side PS Fifthly take the complement of the angle PZS to a Semicircle and set half the tangent of that complement from C to λ a ruler laid to λS cuts the limb at B and ZB applyed to a line of chords gives the side ZS Sixthly a ruler laid to Sλ cuts the limb at L. Again a ruler laid to SΠ cuts the limb at φ and L φ applyed to a line of chords gives the angle ZSP The end of the first Book An Appendix to the first Book THe sights which are necessary for taking any Altitude Angle or distance without the help of Thread or Plummet are only three viz. one turning sight and two other sights contrived with chops so that they may slide by the inward or outward graduated limbs The turning sight hath only two places either the center at the head or the center at the beginning of the line of sines on the fixed piece to either of which as occasion requires it s fastened with a sorne The center at the head serving for the graduations next the inward limb of the loose piece And the center at the beginning of the line of sines serving for all the graduations