ANGLED spherical triangle, when one of its legs is a Prop. 4. To find the fluxions of the several parts of a RIGHT- ANGLED spherical triangle, when the hypothenuse is 5. In any OBLIQUE-ANGLED spherical triangle, supposing an angle and its adjacent side to remain constant, it is required to find the fuxions of the other parts 349 6. To find the flusions of the several parts of an OB- LIQUE-ANGLED spherical triangle, when an angle and its opposite side are constant quantities 7. To find the fluxions of the several parts of an OB- LIQUE-ANGLED spherical triangle, when two of its 8. To find the fluxions of the several parts of an OB- LIQUE-ANGLED spherical triangle, when two of its 10. Given the parallax in altitude of a planet, to find its parallax in latitude and longitude 11. Given the altitude of the nonagesimal degree of the ecliptic; the longitude of a planet from a nonage- simal degree, and its horizontal parallax, to find its parallax in latitude and longitude 12. To determine the correction for finding the time of apparent noon, from equal altitudes of the sun 358 13. The error in taking the altitude of a star being given, to find the corresponding error in the hour angle. 359 14. The error in the altitude of any tower, or other object, is to the error committed in taking the angle of eleva- tion; as double the height of the observed object, is CHAP. XIV. MISCELLANEOUS PROPOSITIONS, &c. Prop. 1. Of the FRENCH division of the circle, and to turn French degrees, minutes, &c. into English 2. To turn English degrees, minutes, &c. into French 362 To find the distances of the observatories of Paris and Pekin, by the French division of the circle 363 Ditto, by the English division of the circle 3. To find the surface of a spherical triangle 363 . • 368 Page Prop. 4. To find the excess of the three angles of a spherical triangle, above two right angles 366 5. To reduce the angles of a spherical triangle (whose sides are very small arcs) to those of a rectilineal tri- of the spherical triangle comprehended between them; to find the angle con- 371 7. The angles of elevation of two distant objects being given, together with the oblique angle contained between the objects, to find the horizontal angle 373 BOOK IV. THE THEORY OF NAVIGATION. CHAPTER I. Definitions and Plane sailing 376 to 380 381 to 383 383 to 392 TABLES. I. A Table of the LOGARITHMS of numbers, from an 393 to 409 II. A Table of NATURAL SINES to every degrec and minute of the quadrant 409 to 417 III. A Table of LOGARITHMICAL SINES and TANGENTS to every degree and minute of the quadrant 418 to 440 IV. A Table of the REFRACTION in altitude of the heavenly bodies 441 V. A Table of the depression or Dip of the horizon of 441 VI. A Table of the sun's PARALLAX in altitude 441 VII. A Table of the augmentation of the moon's semidiameter 441 VIII. A Table of the right ascensions and declinations of thirty-six principal fixed.stars, corrected to the 442 the sea . Five copper-plates at the end of the book. EXPLANATION OF THE CHARACTERS OR MARKS USED IN THE FOLLOWING WORK. PB +, Plus or more, the sign of addition ; as AD+DC, signifies that the line ad is to be increased by the line Dc; and 4+3 signifies that the number 4 is to be increased by the number B. -, Minus or less, the sign of subtraction, and shows that the second quantity is to be taken from the first; as CB – GB shows that the line CB is to be diminished by the line GB. X, Into or by, the sign of multiplication ; as ED X DC sig nifies the rectangle formed by the lines ED and Dc, and a x b expresses the product of the quantity a by the quantity b. Also a:b or ab signifies the same thing. • , Divide by; as PB+Cs, or signifies that PB is to be divided by cs. AB”, AB, signify the square and the cube of AB; also 14) signifies that 14 is to be involved to the third power, and then the fourth root is to be extracted. NĀ or at, sva or A}, express the square and cube root of a. =, Equal to, as AB=CD, shews that AB is equal to CD. A and B is to be taken. tities together, as A+BXm, or (a +B).m, CS {i together, and then to be multiplied by the to B the same ratio which c has to D, and is to as is to is usually read a is to B as c is to D. .: Therefore, Ľ Angle, as į A, signifies the angle A. Greater than, as AB, shows that A is greater than B. [Less than, as ACB, shows A to be less than B. The other characters are explained among the definitions in the work. : N.B. The letters within the parentheses, at the beginning of the different paragraphs of the work, are for references. Thus, (C. 2.) refers to the article marked (C) at page 2. ; (H. 25.) refers to the article marked (H) at page 25, and so on., ERRATA. Page 93 and 94, in the note, for Chap. XI. read Chap. XIV. AN INTRODUCTION TO PLANE AND SPHERICAL TRIGON O M E T R Y. BOOK I. CHAPTER I. THE NATURE AND PROPERTIES OF LOGARITHMS. 1 (A) Definition. LOGARITHMS are a series of numbers contrived to facilitate arithmetical calculations; so that by them the work of multiplication is performed by addition, division by subtraction, involution by multiplication, and the extraction of roots by division. They may therefore be considered as indices to a series of numbers in geometrical progression, where the first term is an unit. Let 1.gd .ge .gl .gt .go5.86 , &c. be such a series, increasing 1 1 1 1 1 from 1; or 1. -&c. decreasing from 22 go3 z4 z5 zo 1; which last series, agreeably to the established notation in algebra, may be thus expressed, 1.r-1.8-2.7-3. r-4. go5 gomb, &c. Here the common ratio is r, and the indices 1.2.3, &c. or -1.-2.-3, &c. are logarithms. Hence it is obvious, that if a series of numbers be in geometrical progression, their logarithms will constitute a series in arithmetical progresssion. And, where the series is increasing, the terms of the geometrical progression are obtained by multiplication, and those of the arithmetical progression, or logarithms, by addition; on the contrary, if the series be decreasing, the . |