2002:: "Experience has shown, that versed sines and external secants as frequently enter into calculations on curves as sines and tangents; and by their use, as illustrated in the examples given in this work, it is believed that many of the rules in general use are much simplified, and many calculations concerning curves and running lines made less intricate, and results obtained with more accuracy and far less trouble, than by any methods laid down in works of this kind. In addition to the tables generally found in books of this kind, the author has prepared, with great labor, a Table of Natural and Logarithmic Versed Sines and External Secants, calculated to degrees, for every minute; also, a Table of Radii and their Logarithms, from 1° to 60°." (
51:
1484:
1202:
20:
1702:
554:
accurate results compared to calculating the same quantity from values found in previously available trigonometric tables. The same idea was adopted by other authors, such as
Searles (1880). By 1913 Haslett's approach was so widely adopted in the American railroad industry that, in that context, "tables of external secants and versed sines more common than tables of secants".
1479:{\displaystyle \operatorname {arcexsec} y=\operatorname {arcsec}(y+1)={\begin{cases}{\arctan }{\bigl (}\!{\textstyle {\sqrt {y^{2}+2y}}}\,{\bigr )}&{\text{if}}\ \ y\geq 0,\\{\text{undefined}}&{\text{if}}\ \ {-2}<y<0,\\\pi -{\arctan }{\bigl (}\!{\textstyle {\sqrt {y^{2}+2y}}}\,{\bigr )}&{\text{if}}\ \ y\leq {-2};\\\end{cases}}_{\vphantom {.}}}
1517:
553:
These are both natural quantities to measure or calculate when surveying circular arcs, which must subsequently be multiplied or divided by other quantities. Charles
Haslett (1855) found that directly looking up the logarithm of the exsecant and versine saved significant effort and produced more
1981:
As the book's editor
Charles W. Hackley explains in the preface, "The use of the more common trigonometric functions, to wit, sines, cosines, tangents, and cotangents, which ordinary tables furnish, is not well adapted to the peculiar problems which are presented in the construction of Railroad
1914:
The original conception of trigonometric functions was as line segments, but this was gradually replaced during the 18th and 19th century by their conception as length ratios between sides of a right triangle or abstract functions; when the exsecant was introduced, in the mid 19th century, both
1038:
433:
were used, depending on the specific calculation. Using logarithms converts expensive multiplication of multi-digit numbers to cheaper addition, and logarithmic versions of trigonometric tables further saved labor by reducing the number of necessary table lookups.
392:(1583), is more specifically based on a line segment with one endpoint at the center of a circle and the other endpoint outside the circle; the circle divides this segment into a radius and an external secant. The external secant segment was used by
882:
428:
working for the railroad needed to make many repetitive trigonometrical calculations to measure and plan circular sections of track. In surveying, and more generally in practical geometry, tables of both "natural" trigonometric functions and their
1697:{\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {exsec} x&=\tan x\,\sec x,\\\int \operatorname {exsec} x\,\mathrm {d} x&=\ln {\bigl |}\sec x+\tan x{\bigr |}-x+C,{\vphantom {\int _{|}}}\end{aligned}}}
1813:
149:
933:
327:
has one endpoint on the circumference, and then extends radially outward. The length of this segment is the radius of the circle times the trigonometric exsecant of the central angle between the segment's inner endpoint and the
1102:
594:
has removed the need for trigonometric tables of specialized functions such as this one. Exsecant is generally not directly built into calculators or computing environments (though it has sometimes been included in
1994:, and which, with the formulas and rules necessary for their application to the laying down of curves, drawn up by Mr. Haslett, one of the Engineers of that Road, are now for the first time given to the public." (
445:
of a curved track section is the shortest distance between the track and the intersection of the tangent lines from the ends of the arc, which equals the radius times the trigonometric exsecant of half the
54:
The names exsecant, versine, chord, etc. can also be applied to line segments related to a circular arc. The length of each segment is the radius times the corresponding trigonometric function of the angle.
492:
3008:
2251:
Thysbaert, Jan-Frans (1774). "Articulus II: De situ lineæ rectæ ad
Circularem; & de mensura angulorum, quorum vertex non est in circuli centro. §1. De situ lineæ rectæ ad Circularem. Definitio II: ".
1522:
551:
209:
938:
713:
272:
227:, and since the early 20th century has sometimes been briefly mentioned in American trigonometry textbooks and general-purpose engineering manuals. For completeness, a few books also defined a
1731:
3001:
1872:
83:
811:
306:
669:
637:
1187:
1158:
2994:
2289:
887:
312:, though it was not used in practice. While the exsecant has occasionally found other applications, today it is obscure and mainly of historical interest.
2897:
1045:
565:
between straight or circular sections of differing curvature. These spiral curves can be approximately calculated using exsecants and versines.
3077:
2927:
2905:
2734:
2191:
2421:
Civil engineer's pocket-book: a reference-book for engineers, contractors and students containing rules, data, methods, formulas and tables
2402:
2317:
1033:{\textstyle \operatorname {vers} \theta =2{\bigl (}{\sin {\tfrac {1}{2}}\theta }{\bigr )}{\vphantom {)}}^{2}{\vphantom {\Big |}}={}}
453:
512:
167:
3103:
2695:
2512:
Thornton-Smith, G. J. (1963). "Almost Exact Closed
Expressions for Computing all the Elements of the Clothoid Transition Curve".
674:
576:
3041:
2919:
2726:
1715:
719:: because most of the digits of each quantity are the same, they cancel in the subtraction, yielding a lower-precision result.
2309:
1190:
808:
If a table or computer implementation of the exsecant function is not available, the exsecant can be accurately computed as
2337:
Finocchiaro, Maurice A. (2003). "Physical-Mathematical
Reasoning: Galileo on the Extruding Power of Terrestrial Rotation".
2332:; an exsecant is defined as the part of a secant external to the circle and thus between the circumference and the tangent.
1113:, and the versine and exsecant are approximately equal to each-other and both proportional to the square of the arclength.
2661:
1991:
1901:
2610:
Wilson, T. R. C. (1929). "A Graphical Method for the
Solution of Certain Types of Equations". Questions and Discussions.
762:, all of whose digits are meaningful. If the logarithm of exsecant is calculated by looking up the secant in a six-place
3051:
2851:
360:
246:
50:
1248:
2860:
2708:
716:
224:
2219:
1976:
The
Mechanic's, Machinist's, and Engineer's Practical Book of Reference; Together with the Engineer's Field Book
1501:
2245:
2173:
3029:
2957:
2425:
599:), and calculations in general are much cheaper than in the past, no longer requiring tedious manual labor.
2971:
2459:
Field
Engineering. A hand-book of the Theory and Practice of Railway Surveying, Location, and Construction
2158:
562:
71:
877:{\textstyle \operatorname {exsec} \theta =\tan \theta \,\tan {\tfrac {1}{2}}\theta {\vphantom {\Big |}},}
2473:
Field
Engineering: A Handbook of the Theory and Practice of Railway Surveying, Location and Construction
2101:
2048:
75:
2776:
2768:
1835:
352:"cuts" a circle, intersecting it twice; this concept dates to antiquity and can be found in Book 3 of
3046:
2477:
2463:
2178:(2nd ed.). Springer. Ch. 33, "The Secant sec(x) and Cosecant csc(x) functions", §33.13, p. 336.
2061:
1918:
1826:– A line segment with endpoints on the circumference of a circle, historically used trigonometrically
353:
276:
157:
642:
610:
2843:
2839:
2831:
2241:
2150:
2126:
1982:
curves. Still there would be much labor of computation which may be saved by the use of tables of
1829:
763:
309:
2496:
2374:
2883:
The Field Engineer: A Handy Book of Practice in the Survey, Location, and Track-Work of Railroads
2619:
2565:
2390:
2354:
2069:
785:
596:
506:
1808:{\displaystyle \operatorname {exsec} 2\theta ={\frac {2\sin ^{2}\theta }{1-2\sin ^{2}\theta }}.}
2818:
2638:
2440:
2419:
2003:
1995:
2592:
2579:
2398:
2313:
2187:
2106:
2092:
2076:
2042:
2010:
1711:
1163:
425:
329:
154:
1137:
2591:
2564:
2521:
2346:
2179:
2068:
2014:
1933:
1823:
1127:
741:
501:
430:
2881:
2540:
572:
and roads, and the exsecant was still used in mid-20th century books about road surveying.
3034:
2301:
2285:
2231:
393:
383:
2986:
144:{\displaystyle \operatorname {exsec} \theta =\sec \theta -1={\frac {1}{\cos \theta }}-1.}
2687:
2457:
2291:
Dialogo di Galileo Galilei sopra i due massimi sistemi del mondo Tolemaico e Copernicano
2923:
2730:
2262:
van Haecht, Joannes (1784). "Articulus III: De secantibus circuli: Corollarium III: ".
1891:
1505:
580:
2471:
2088:
1974:
1895:
928:{\textstyle \operatorname {exsec} \theta =\operatorname {vers} \theta \,\sec \theta ,}
3097:
509:– which equals the radius times the trigonometric versine of half the central angle,
447:
389:
1973:
Haslett, Charles (1855). "The Engineer's Field Book". In Hackley, Charles W. (ed.).
19:
2096:
568:
Solving the same types of problems is required when surveying circular sections of
558:
413:
316:
212:
2263:
2253:
2235:
2223:
2055:
1987:
349:
2525:
1990:, which have been employed with great success recently by the Engineers on the
1106:
Haslett used these identities to compute his 1855 exsecant and versine tables.
1097:{\displaystyle \sin \theta \,\tan {\tfrac {1}{2}}\theta \,{\vphantom {\Big |}}}
715:
Computing the difference between two approximately equal quantities results in
2691:
2350:
2183:
1109:
For a sufficiently small angle, a circular arc is approximately shaped like a
587:
2961:
2966:
2665:
1947:"The history of one definition: Teaching trigonometry in the US before 1900"
421:
364:
220:
375:
line segment external to a circle with one endpoint on the circumference a
1937:
2234:(1750). Genneau, Ludovicum (Ludovico); Rollin, Jacobum (Jacques) (eds.).
2075:. Englewood Cliffs, NJ: Prentice-Hall. pp. 147, 315–325 (table 41).
1497:
1110:
591:
409:
216:
2358:
3067:
2623:
372:
333:
161:
1194:
575:
The exsecant has sometimes been used for other applications, such as
324:
557:
In the late-19th and 20th century, railroads began using arcs of an
2218:
Patu, Andræâ-Claudio (André Claude); Le Tort, Bartholomæus (1745).
2089:"4.3.147: Elementary Transcendental Functions - Circular functions"
1946:
788:, and after computing the logarithm only three digits are correct,
3082:
2306:
Galileo on the World Systems: A New Abridged Translation and Guide
2268:(in Latin). Lovanii, e typographia academica. p. 24, foldout.
2258:(in Latin). Lovanii, e typographia academica. p. 30, foldout.
2127:"$ 131. The Versed Sine, Exsecant and Coexsecant. §132. Exercises"
569:
49:
23:
The exsecant and versine functions substitute for the expressions
18:
2296:
Dialogue on the Two Chief World Systems, Ptolemaic and Copernican
1496:
While historical uses of the exsecant did not explicitly involve
2110:
1904:. §527. "Less common trigonometric functions", pp. 171–172.
2990:
2886:(21st ed.). New York: D. Van Nostrand Company. p. 36.
2583:
2161:. § "Secondary Trigonometric Functions", pp. 125–127.
2080:
2105:. Washington, D.C.: National Bureau of Standards. p. 78.
1952:
International Journal for the History of Mathematics Education
487:{\displaystyle R\operatorname {exsec} {\tfrac {1}{2}}\Delta .}
740:, with the leading several digits wasted on zeros, while the
546:{\displaystyle R\operatorname {vers} {\tfrac {1}{2}}\Delta .}
1488:
the arctangent expression is well behaved for small angles.
498:
of a curved track section is the furthest distance from the
204:{\displaystyle \operatorname {vers} \theta =1-\cos \theta ,}
2796:, which is added to keep the entries in the table positive.
2172:
Oldham, Keith B.; Myland, Jan C.; Spanier, Jerome (2009) .
1460:
708:{\displaystyle \sec \theta \approx \cos \theta \approx 1.}
223:
and civil engineers in the United States for railroad and
2265:
Geometria elementaria et practica: quam in usum auditorum
2598:(4th ed.). Scranton, PA: International Textbook Co.
505:(the line segment between endpoints) to the track – cf.
2545:
Proceedings of the American Society of Civil Engineers
2442:
A Manual of the Principles and Practice of Road-Making
2017:
Practical Book of Reference, and Engineer's Field Book
1393:
1265:
1066:
972:
941:
890:
844:
814:
805:. For even smaller angles loss of precision is worse.
526:
467:
2470:
Searles, William Henry; Ives, Howard Chapin (1915) .
1838:
1734:
1520:
1205:
1166:
1140:
1048:
677:
645:
613:
515:
456:
279:
249:
170:
86:
2445:. New York: A. S. Barnes & Co. pp. 140–141.
1130:
of the exsecant function, which might be symbolized
3060:
3022:
1998:) Charles Haslett continues in his preface to the
2345:(1–2, Logic and Mathematical Reasoning): 217–244.
2237:Cursus Philosophicus Ad Scholarum Usum Accomodatus
2149:Hall, Arthur Graham; Frink, Fred Goodrich (1909).
1945:
1866:
1807:
1696:
1478:
1181:
1152:
1096:
1032:
927:
876:
707:
663:
631:
545:
486:
300:
267:{\displaystyle \operatorname {coexsec} \theta ={}}
266:
203:
143:
16:Trigonometric function defined as secant minus one
2578:(4th ed.). New York: John Wiley & Sons.
1919:"Historical Reflections on Teaching Trigonometry"
1391:
1263:
1086:
1017:
863:
671:(exsecant) is problematic for small angles where
2501:. New York: D. Van Nostrand Company. p. 28.
46:which appear frequently in certain applications.
2571:. New York: John Wiley & Sons. p. 114.
2397:. Princeton University Press. pp. 62–109.
2379:. New York: Spon & Chamberlain. p. 20.
2312:. pp. 184 (n130), 184 (n135), 192 (n158).
1979:. New York: James G. Gregory. pp. 371–512.
1874:also used to improve precision for small inputs
160:, who used it in conjunction with the existing
2541:"Economic Canal Location in Uniform Countries"
3002:
2639:"Correction for inclination of sounding wire"
2041:Kenyon, Alfred Monroe; Ingold, Louis (1913).
1650:
1622:
1422:
1386:
1294:
1258:
989:
959:
348:comes from Latin for "to cut", and a general
8:
2805:The incorrect digits are highlighted in red.
2763:In a table of logarithmic exsecants such as
2228:(in Latin). Paris: Ph. N. Lottin. p. 6.
2133:. Boston: Allyn and Bacon. pp. 235–236.
2067:McNeese, Donald C.; Hoag, Albert L. (1957).
2054:Hudson, Ralph Gorton; Lipka, Joseph (1917).
2772:
3009:
2995:
2987:
2844:Table XIII: Natural Versines and Exsecants
603:Catastrophic cancellation for small angles
332:for a line through the outer endpoint and
2539:Doolittle, H. J.; Shipman, C. E. (1911).
2490:
2488:
2328:(meaning secant), but he clearly intends
1968:
1966:
1849:
1837:
1787:
1760:
1750:
1733:
1679:
1678:
1671:
1670:
1649:
1648:
1621:
1620:
1599:
1598:
1569:
1533:
1527:
1525:
1521:
1519:
1465:
1464:
1446:
1429:
1421:
1420:
1419:
1400:
1394:
1392:
1385:
1384:
1379:
1346:
1335:
1328:
1301:
1293:
1292:
1291:
1272:
1266:
1264:
1257:
1256:
1251:
1243:
1204:
1165:
1139:
1085:
1082:
1081:
1080:
1065:
1058:
1047:
1028:
1016:
1013:
1012:
1006:
996:
995:
988:
987:
971:
964:
958:
957:
940:
912:
889:
862:
859:
858:
843:
836:
813:
790:log(sec 1° − 1) ≈
676:
644:
612:
525:
514:
466:
455:
278:
262:
248:
169:
117:
85:
2304:(1997) . Finocchiaro, Maurice A. (ed.).
2814:
2764:
2225:Theses Mathematicæ De Mathesi Generatim
2220:Rivard, Franciscus (Dominique-François)
1883:
1189:and can be expressed in terms of other
3017:Trigonometric and hyperbolic functions
2144:
2142:
2140:
153:It was introduced in 1855 by American
2928:Massachusetts Institute of Technology
2906:Massachusetts Institute of Technology
2735:Massachusetts Institute of Technology
2643:The International Hydrographic Review
2029:(12): 184. Whole No. 1040, Vol. XXIX.
586:In recent years, the availability of
7:
2920:"MIT/GNU Scheme – Scheme Arithmetic"
2727:"MIT/GNU Scheme – Scheme Arithmetic"
772:sec 1° − 1 ≈
2854:Field Manual for Railroad Engineers
2836:Field Manual for Railroad Engineers
1897:A History of Mathematical Notations
1726:The exsecant of twice an angle is:
607:Naïvely evaluating the expressions
2713:JASS - Java Audio Synthesis System
2102:Handbook of Mathematical Functions
2071:Engineering and Technical Handbook
1600:
1534:
1528:
1134:, is well defined if its argument
537:
478:
41:= 1 − sec
14:
2707:van den Doel, Kees (2010-01-25).
2612:The American Mathematical Monthly
2255:Geometria elementaria et practica
2125:Bohannan, Rosser Daniel (1904) .
1867:{\displaystyle x\mapsto e^{x}-1,}
2880:Shunk, William Findlay (1918) .
2842:. §§ 138–165, pp. 110–142;
2696:NASA Goddard Space Flight Center
2686:Simpson, David G. (2001-11-08).
2025:(Review). Second Quarto Series.
935:which can itself be computed as
750:log exsec 1° ≈
2908:. 2023-09-01. procedure: aexsec
2563:Hewes, Laurence Ilsley (1942).
2456:Searles, William Henry (1880).
2198:Not appearing elsewhere in the
1716:Integral of the secant function
1191:inverse trigonometric functions
412:tracks were constructed out of
301:{\displaystyle \csc \theta -1,}
2439:Gillespie, William M. (1853).
2310:University of California Press
1842:
1680:
1236:
1224:
999:
664:{\displaystyle \sec \theta -1}
632:{\displaystyle 1-\cos \theta }
1:
3078:Jyā, koti-jyā and utkrama-jyā
2694:source code). Greenbelt, MD:
2574:Ives, Howard Chapin (1966) .
2376:Railroad Curves and Earthwork
2373:Allen, Calvin Frank (1894) .
1992:Ohio and Mississippi Railroad
1915:concepts were still common.
2926:source code). v. 12.1.
2902:MIT/GNU Scheme Documentation
2733:source code). v. 12.1.
2498:The Practical Railway Spiral
1672:
1466:
1083:
1014:
997:
860:
361:intersecting secants theorem
211:for designing and measuring
2660:Calvert, James B. (2007) .
2495:Jordan, Leonard C. (1913).
2476:(17th ed.). New York:
722:For example, the secant of
3120:
2898:"4.5 Numerical operations"
2838:(1st ed.). New York:
2637:Johnson, Harry F. (1933).
2594:Route Surveying and Design
2543:. Papers and Discussions.
2526:10.1179/sre.1963.17.127.35
2424:(2nd ed.). New York:
1983:
1944:Van Sickle, Jenna (2011).
408:In the 19th century, most
363:. 18th century sources in
2792:, the correct value plus
2567:American Highway Practice
2418:Frye, Albert I. (1918) .
2395:The Doctrine of Triangles
2393:(2021). "2. Logarithms".
2279:Galileo used the Italian
2240:(in Latin). Vol. 3.
2232:Lemonnier, Petro (Pierre)
2184:10.1007/978-0-387-48807-3
2023:American Railroad Journal
717:catastrophic cancellation
219:track. It was adopted by
2830:Nagle, James C. (1897).
2781:log exsec 1°
2709:"jass.utils Class Fmath"
2590:Meyer, Carl F. (1969) .
2248:), Paris. pp. 303–.
1917:Bressoud, David (2010).
1900:. Vol. 2. Chicago:
1182:{\displaystyle y\leq -2}
404:History and applications
74:defined in terms of the
3104:Trigonometric functions
2832:"IV. Transition Curves"
2779:, the number given for
2773:Searles & Ives 1915
2426:D. Van Nostrand Company
2351:10.1023/A:1022143816001
2057:A Manual of Mathematics
1153:{\displaystyle y\geq 0}
1117:Mathematical identities
2972:Wolfram Research, Inc.
2159:Henry Holt and Company
1868:
1809:
1698:
1480:
1183:
1154:
1098:
1034:
929:
878:
709:
665:
633:
563:track transition curve
547:
488:
450:subtended by the arc,
396:(1632) under the name
359:, as used e.g. in the
302:
268:
205:
145:
72:trigonometric function
55:
47:
33: − 1
3042:Inverse trigonometric
2478:John Wiley & Sons
2464:John Wiley & Sons
2175:An Atlas of Functions
2087:Zucker, Ruth (1964).
2062:John Wiley & Sons
2049:The Macmillan Company
2019:. By Charles Haslett"
2000:Engineer's Field Book
1938:10.5951/MT.104.2.0106
1869:
1810:
1722:Double angle identity
1699:
1481:
1184:
1155:
1099:
1035:
930:
879:
766:and then subtracting
710:
666:
634:
548:
489:
303:
269:
235:function (symbolized
206:
146:
53:
22:
2013:, ed. (1856-03-22).
1879:Notes and references
1836:
1732:
1686:
1518:
1471:
1203:
1164:
1138:
1090:
1046:
1021:
1002:
939:
888:
867:
812:
728:sec 1° ≈
675:
643:
611:
513:
454:
308:the exsecant of the
277:
247:
168:
84:
2846:, pp. 332–354.
2840:John Wiley and Sons
2391:Van Brummelen, Glen
2298:] (in Italian).
2242:Collegio Harcuriano
1926:Mathematics Teacher
1830:Exponential minus 1
1687:
1673:
1472:
1467:
1091:
1084:
1022:
1015:
1003:
998:
868:
861:
764:trigonometric table
744:of the exsecant of
494:By comparison, the
310:complementary angle
3052:Inverse hyperbolic
2958:Weisstein, Eric W.
2868:: 540. 1897-12-03.
2324:Galileo's word is
2246:Collège d'Harcourt
2155:Plane Trigonometry
2151:"Review Exercises"
2131:Plane Trigonometry
2093:Abramowitz, Milton
2011:Poor, Henry Varnum
1864:
1805:
1694:
1692:
1476:
1459:
1417:
1289:
1179:
1150:
1094:
1075:
1030:
981:
925:
884:or using versine,
874:
853:
786:significant digits
705:
661:
629:
597:software libraries
543:
535:
484:
476:
382:The trigonometric
298:
264:
201:
141:
56:
48:
3091:
3090:
2856:. By J. C. Nagle"
2193:978-0-387-48806-6
1800:
1712:natural logarithm
1542:
1512:in radians) are:
1439:
1436:
1432:
1415:
1345:
1342:
1338:
1331:
1311:
1308:
1304:
1287:
1074:
980:
852:
770:, the difference
534:
475:
443:external distance
431:common logarithms
330:point of tangency
133:
3111:
3011:
3004:
2997:
2988:
2982:
2981:
2979:
2978:
2954:
2948:
2946:
2944:
2943:
2937:
2933:
2916:
2914:
2913:
2904:. v. 12.1.
2894:
2888:
2887:
2877:
2871:
2869:
2847:
2827:
2821:
2812:
2806:
2803:
2797:
2795:
2791:
2790:
2789:
2782:
2761:
2755:
2753:
2751:
2750:
2744:
2740:
2723:
2721:
2720:
2705:
2703:
2702:
2683:
2677:
2676:
2674:
2673:
2664:. Archived from
2657:
2651:
2650:
2634:
2628:
2627:
2607:
2601:
2599:
2597:
2587:
2572:
2570:
2559:
2553:
2552:
2536:
2530:
2529:
2509:
2503:
2502:
2492:
2483:
2481:
2467:
2453:
2447:
2446:
2436:
2430:
2429:
2415:
2409:
2408:
2387:
2381:
2380:
2370:
2364:
2362:
2334:
2302:Galilei, Galileo
2299:
2286:Galilei, Galileo
2277:
2271:
2269:
2259:
2249:
2229:
2215:
2209:
2208:
2169:
2163:
2162:
2146:
2135:
2134:
2122:
2116:
2114:
2097:Stegun, Irene A.
2084:
2074:
2065:
2052:
2038:
2032:
2030:
2004:pp. 373–374
1984:external secants
1980:
1970:
1961:
1959:
1949:
1941:
1923:
1912:
1906:
1905:
1888:
1873:
1871:
1870:
1865:
1854:
1853:
1824:Chord (geometry)
1814:
1812:
1811:
1806:
1801:
1799:
1792:
1791:
1772:
1765:
1764:
1751:
1709:
1703:
1701:
1700:
1695:
1693:
1689:
1688:
1685:
1684:
1683:
1654:
1653:
1626:
1625:
1603:
1543:
1541:
1537:
1531:
1526:
1511:
1485:
1483:
1482:
1477:
1475:
1474:
1473:
1463:
1462:
1453:
1437:
1434:
1433:
1430:
1426:
1425:
1418:
1416:
1405:
1404:
1395:
1390:
1389:
1383:
1353:
1343:
1340:
1339:
1336:
1332:
1329:
1309:
1306:
1305:
1302:
1298:
1297:
1290:
1288:
1277:
1276:
1267:
1262:
1261:
1255:
1197:for the angle):
1188:
1186:
1185:
1180:
1159:
1157:
1156:
1151:
1133:
1122:Inverse function
1105:
1103:
1101:
1100:
1095:
1093:
1092:
1089:
1076:
1067:
1040:
1039:
1037:
1036:
1031:
1029:
1024:
1023:
1020:
1011:
1010:
1005:
1004:
993:
992:
986:
982:
973:
963:
962:
934:
932:
931:
926:
883:
881:
880:
875:
870:
869:
866:
854:
845:
804:
803:
802:
801:
792:
791:
783:
782:
781:
774:
773:
769:
761:
760:
759:
752:
751:
747:
742:common logarithm
739:
738:
737:
730:
729:
725:
714:
712:
711:
706:
670:
668:
667:
662:
638:
636:
635:
630:
552:
550:
549:
544:
536:
527:
493:
491:
490:
485:
477:
468:
307:
305:
304:
299:
273:
271:
270:
265:
263:
210:
208:
207:
202:
150:
148:
147:
142:
134:
132:
118:
45:
34:
3119:
3118:
3114:
3113:
3112:
3110:
3109:
3108:
3094:
3093:
3092:
3087:
3056:
3035:Sine and cosine
3018:
3015:
2985:
2976:
2974:
2956:
2955:
2951:
2941:
2939:
2935:
2931:
2918:
2911:
2909:
2896:
2895:
2891:
2879:
2878:
2874:
2850:
2829:
2828:
2824:
2813:
2809:
2804:
2800:
2793:
2787:
2785:
2784:
2780:
2762:
2758:
2748:
2746:
2742:
2738:
2725:
2718:
2716:
2706:
2700:
2698:
2685:
2684:
2680:
2671:
2669:
2659:
2658:
2654:
2636:
2635:
2631:
2618:(10): 526–528.
2609:
2608:
2604:
2589:
2573:
2562:
2560:
2556:
2551:(8): 1161–1164.
2538:
2537:
2533:
2511:
2510:
2506:
2494:
2493:
2486:
2469:
2455:
2454:
2450:
2438:
2437:
2433:
2417:
2416:
2412:
2405:
2389:
2388:
2384:
2372:
2371:
2367:
2336:
2320:
2300:
2284:
2278:
2274:
2261:
2250:
2230:
2217:
2216:
2212:
2202:is the archaic
2194:
2171:
2170:
2166:
2148:
2147:
2138:
2124:
2123:
2119:
2086:
2066:
2053:
2040:
2039:
2035:
2009:
1996:pp. vi–vii
1972:
1971:
1964:
1943:
1921:
1916:
1913:
1909:
1892:Cajori, Florian
1890:
1889:
1885:
1881:
1845:
1834:
1833:
1832:– The function
1820:
1783:
1773:
1756:
1752:
1730:
1729:
1724:
1707:
1691:
1690:
1674:
1607:
1583:
1582:
1553:
1532:
1516:
1515:
1509:
1494:
1458:
1457:
1427:
1396:
1370:
1369:
1333:
1325:
1324:
1299:
1268:
1244:
1242:
1201:
1200:
1162:
1161:
1136:
1135:
1131:
1124:
1119:
1044:
1043:
1042:
994:
937:
936:
886:
885:
810:
809:
799:
797:
795:
794:
789:
779:
777:
776:
771:
767:
757:
755:
754:
749:
745:
735:
733:
732:
727:
723:
673:
672:
641:
640:
609:
608:
605:
511:
510:
452:
451:
439:external secant
426:civil engineers
414:arcs of circles
406:
394:Galileo Galilei
377:secans exterior
342:
336:to the circle.
321:external secant
275:
274:
245:
244:
166:
165:
158:Charles Haslett
122:
82:
81:
60:external secant
36:
24:
17:
12:
11:
5:
3117:
3115:
3107:
3106:
3096:
3095:
3089:
3088:
3086:
3085:
3080:
3075:
3070:
3064:
3062:
3058:
3057:
3055:
3054:
3049:
3044:
3039:
3038:
3037:
3026:
3024:
3020:
3019:
3016:
3014:
3013:
3006:
2999:
2991:
2984:
2983:
2949:
2930:. 2023-09-01.
2924:MIT/GNU Scheme
2889:
2872:
2822:
2807:
2798:
2756:
2737:. 2023-09-01.
2731:MIT/GNU Scheme
2678:
2662:"Trigonometry"
2652:
2629:
2602:
2576:Highway Curves
2561:For example:
2554:
2531:
2520:(127): 35–44.
2504:
2484:
2448:
2431:
2428:. p. 211.
2410:
2403:
2382:
2365:
2318:
2272:
2210:
2192:
2164:
2136:
2117:
2033:
1962:
1932:(2): 106–112.
1907:
1882:
1880:
1877:
1876:
1875:
1863:
1860:
1857:
1852:
1848:
1844:
1841:
1827:
1819:
1816:
1804:
1798:
1795:
1790:
1786:
1782:
1779:
1776:
1771:
1768:
1763:
1759:
1755:
1749:
1746:
1743:
1740:
1737:
1723:
1720:
1682:
1677:
1669:
1666:
1663:
1660:
1657:
1652:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1624:
1619:
1616:
1613:
1610:
1608:
1606:
1602:
1597:
1594:
1591:
1588:
1585:
1584:
1581:
1578:
1575:
1572:
1568:
1565:
1562:
1559:
1556:
1554:
1552:
1549:
1546:
1540:
1536:
1530:
1524:
1523:
1506:antiderivative
1493:
1490:
1470:
1461:
1456:
1452:
1449:
1445:
1442:
1428:
1424:
1414:
1411:
1408:
1403:
1399:
1388:
1382:
1378:
1375:
1372:
1371:
1368:
1365:
1362:
1359:
1356:
1352:
1349:
1334:
1327:
1326:
1323:
1320:
1317:
1314:
1300:
1296:
1286:
1283:
1280:
1275:
1271:
1260:
1254:
1250:
1249:
1247:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1178:
1175:
1172:
1169:
1149:
1146:
1143:
1123:
1120:
1118:
1115:
1088:
1079:
1073:
1070:
1064:
1061:
1057:
1054:
1051:
1027:
1019:
1009:
1001:
991:
985:
979:
976:
970:
967:
961:
956:
953:
950:
947:
944:
924:
921:
918:
915:
911:
908:
905:
902:
899:
896:
893:
873:
865:
857:
851:
848:
842:
839:
835:
832:
829:
826:
823:
820:
817:
704:
701:
698:
695:
692:
689:
686:
683:
680:
660:
657:
654:
651:
648:
639:(versine) and
628:
625:
622:
619:
616:
604:
601:
581:depth sounding
542:
539:
533:
530:
524:
521:
518:
483:
480:
474:
471:
465:
462:
459:
405:
402:
341:
338:
297:
294:
291:
288:
285:
282:
261:
258:
255:
252:
200:
197:
194:
191:
188:
185:
182:
179:
176:
173:
155:civil engineer
140:
137:
131:
128:
125:
121:
116:
113:
110:
107:
104:
101:
98:
95:
92:
89:
15:
13:
10:
9:
6:
4:
3:
2:
3116:
3105:
3102:
3101:
3099:
3084:
3081:
3079:
3076:
3074:
3071:
3069:
3066:
3065:
3063:
3059:
3053:
3050:
3048:
3045:
3043:
3040:
3036:
3033:
3032:
3031:
3030:Trigonometric
3028:
3027:
3025:
3021:
3012:
3007:
3005:
3000:
2998:
2993:
2992:
2989:
2973:
2969:
2968:
2963:
2959:
2953:
2950:
2947:
2929:
2925:
2921:
2907:
2903:
2899:
2893:
2890:
2885:
2884:
2876:
2873:
2870:
2867:
2863:
2862:
2857:
2855:
2845:
2841:
2837:
2833:
2826:
2823:
2820:
2816:
2811:
2808:
2802:
2799:
2778:
2774:
2770:
2766:
2760:
2757:
2754:
2736:
2732:
2728:
2714:
2710:
2697:
2693:
2689:
2682:
2679:
2668:on 2007-10-02
2667:
2663:
2656:
2653:
2649:(2): 176–179.
2648:
2644:
2640:
2633:
2630:
2625:
2621:
2617:
2613:
2606:
2603:
2600:
2596:
2595:
2585:
2581:
2577:
2569:
2568:
2558:
2555:
2550:
2546:
2542:
2535:
2532:
2527:
2523:
2519:
2515:
2514:Survey Review
2508:
2505:
2500:
2499:
2491:
2489:
2485:
2482:
2479:
2475:
2474:
2465:
2461:
2460:
2452:
2449:
2444:
2443:
2435:
2432:
2427:
2423:
2422:
2414:
2411:
2406:
2404:9780691179414
2400:
2396:
2392:
2386:
2383:
2378:
2377:
2369:
2366:
2363:
2360:
2356:
2352:
2348:
2344:
2340:
2333:
2331:
2327:
2321:
2319:9780520918221
2315:
2311:
2307:
2303:
2297:
2293:
2292:
2287:
2282:
2276:
2273:
2270:
2267:
2266:
2257:
2256:
2247:
2243:
2239:
2238:
2233:
2227:
2226:
2221:
2214:
2211:
2207:
2205:
2201:
2195:
2189:
2185:
2181:
2177:
2176:
2168:
2165:
2160:
2156:
2152:
2145:
2143:
2141:
2137:
2132:
2128:
2121:
2118:
2115:
2112:
2108:
2104:
2103:
2098:
2094:
2090:
2082:
2078:
2073:
2072:
2064:. p. 68.
2063:
2059:
2058:
2050:
2046:
2045:
2037:
2034:
2031:
2028:
2024:
2020:
2018:
2012:
2005:
2001:
1997:
1993:
1989:
1985:
1978:
1977:
1969:
1967:
1963:
1960:
1957:
1953:
1948:
1939:
1935:
1931:
1927:
1920:
1911:
1908:
1903:
1899:
1898:
1893:
1887:
1884:
1878:
1861:
1858:
1855:
1850:
1846:
1839:
1831:
1828:
1825:
1822:
1821:
1817:
1815:
1802:
1796:
1793:
1788:
1784:
1780:
1777:
1774:
1769:
1766:
1761:
1757:
1753:
1747:
1744:
1741:
1738:
1735:
1727:
1721:
1719:
1717:
1713:
1704:
1675:
1667:
1664:
1661:
1658:
1655:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1617:
1614:
1611:
1609:
1604:
1595:
1592:
1589:
1586:
1579:
1576:
1573:
1570:
1566:
1563:
1560:
1557:
1555:
1550:
1547:
1544:
1538:
1513:
1507:
1503:
1499:
1491:
1489:
1486:
1468:
1454:
1450:
1447:
1443:
1440:
1412:
1409:
1406:
1401:
1397:
1380:
1376:
1373:
1366:
1363:
1360:
1357:
1354:
1350:
1347:
1321:
1318:
1315:
1312:
1284:
1281:
1278:
1273:
1269:
1252:
1245:
1239:
1233:
1230:
1227:
1221:
1218:
1215:
1212:
1209:
1206:
1198:
1196:
1192:
1176:
1173:
1170:
1167:
1147:
1144:
1141:
1129:
1121:
1116:
1114:
1112:
1107:
1077:
1071:
1068:
1062:
1059:
1055:
1052:
1049:
1025:
1007:
983:
977:
974:
968:
965:
954:
951:
948:
945:
942:
922:
919:
916:
913:
909:
906:
903:
900:
897:
894:
891:
871:
855:
849:
846:
840:
837:
833:
830:
827:
824:
821:
818:
815:
806:
787:
765:
743:
720:
718:
702:
699:
696:
693:
690:
687:
684:
681:
678:
658:
655:
652:
649:
646:
626:
623:
620:
617:
614:
602:
600:
598:
593:
589:
584:
583:with a wire.
582:
578:
573:
571:
566:
564:
560:
555:
540:
531:
528:
522:
519:
516:
508:
504:
503:
497:
481:
472:
469:
463:
460:
457:
449:
448:central angle
444:
440:
435:
432:
427:
423:
419:
418:simple curves
415:
411:
403:
401:
399:
395:
391:
390:Thomas Fincke
387:
386:
380:
378:
374:
370:
366:
362:
358:
357:
351:
347:
339:
337:
335:
331:
326:
322:
318:
313:
311:
295:
292:
289:
286:
283:
280:
259:
256:
253:
250:
242:
238:
234:
230:
226:
222:
218:
214:
198:
195:
192:
189:
186:
183:
180:
177:
174:
171:
163:
159:
156:
151:
138:
135:
129:
126:
123:
119:
114:
111:
108:
105:
102:
99:
96:
93:
90:
87:
79:
77:
73:
69:
66:, symbolized
65:
61:
52:
44:
40:
32:
28:
21:
3072:
2965:
2952:
2940:. Retrieved
2917:
2910:. Retrieved
2901:
2892:
2882:
2875:
2865:
2861:The Engineer
2859:
2853:
2848:
2835:
2825:
2815:Haslett 1855
2810:
2801:
2765:Haslett 1855
2759:
2747:. Retrieved
2724:
2717:. Retrieved
2712:
2699:. Retrieved
2681:
2670:. Retrieved
2666:the original
2655:
2646:
2642:
2632:
2615:
2611:
2605:
2593:
2588:
2575:
2566:
2557:
2548:
2544:
2534:
2517:
2513:
2507:
2497:
2472:
2468:
2462:. New York:
2458:
2451:
2441:
2434:
2420:
2413:
2394:
2385:
2375:
2368:
2342:
2338:
2335:
2329:
2325:
2323:
2305:
2295:
2290:
2280:
2275:
2264:
2260:
2254:
2236:
2224:
2213:
2203:
2199:
2197:
2174:
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2157:. New York:
2154:
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2056:
2051:. p. 5.
2047:. New York:
2044:Trigonometry
2043:
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1988:versed sines
1975:
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2938:lines 65–71
2819:p. 415
2745:lines 61–63
1958:(2): 55–70.
1714:. See also
784:has only 3
588:calculators
577:beam theory
496:versed sine
388:, named by
350:secant line
225:road design
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3047:Hyperbolic
2977:2015-11-05
2975:Retrieved
2962:"Exsecant"
2942:2024-04-01
2934:function,
2912:2024-04-01
2864:(Review).
2777:II. p. 135
2749:2024-04-01
2741:function,
2719:2015-10-26
2701:2015-10-26
2692:Fortran 90
2672:2015-11-08
2206:function .
1902:Open Court
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229:coexsecant
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2967:MathWorld
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2743:arith.scm
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592:computers
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464:
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416:, called
354:Euclid's
344:The word
340:Etymology
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2204:exsecant
2111:64-60036
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2008:Review:
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1818:See also
1498:calculus
1492:Calculus
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1132:arcexsec
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410:railroad
356:Elements
217:railroad
213:circular
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2584:52-9033
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1195:radians
1193:(using
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385:secant
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3061:Other
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2739:exsec
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