Prop. 4. To find the fluxions of the several parts of a RIGHT- 548 5. In any OBLIQUE-ANGLED spherical triangle, supposing an angle and its adjacent side to remain constant, it is required to find the fluxions of the other parts . 349 6. To find the fluxions of the several parts of an OB- LIQUE-ANGLED spherical triangle, when an angle and 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- THE USE OF THE FLUXIONAL ANALOGIES. Prop. 9. To find when that part of the equation of time de . 353 . 354 11. Given the altitude of the nonagesimal degree of the ecliptic; the longitude of a planet from a nonage- 12. To determine the correction for finding the time of apparent noon, from equal altitudes of the sun 13. The error in taking the altitude of a star being given, to find the corresponding error in the hour angle 14. The error in the altitude of any tower, or other object, is to the error committed in taking the angle of eleva- 2. To turn English degrees, minutes, &c. into French To find the distances of the observatories of Paris and Pekin, by the French division of the circle Prop. 4. To find the excess of the three angles of a spherical triangle, above two right angles 5. To reduce the angles of a spherical triangle (whose sides are very small arcs) to those of a rectilineal tri- 6. Given two sides of a spherical triangle, and the angle comprehended between them; to find the angle con- tained between the chords of these sides, supposing 7. The angles of elevation of two distant objects being II. A Table of NATURAL SINEs to every degree and III. A Table of LOGARITHMICAL SINES and TANGENTS to every degree and minute of the quadrant 418 to 440 V. A Table of the depression or DIP of the horizon of VI. A Table of the sun's PARALLAX in altitude VIII. A Table of the right ascensions and declinations Five copper-plates at the end of the book. EXPLANATION OF THE CHARACTERS OR MARKS USED IN THE FOLLOWING WORK. +, 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 3. —, Minus or less, the sign of subtraction, and shows that the second quantity is to be taken from the first; as CBGB shows that the line CB is to be diminished by the line GB. ×, Into or by, the sign of multiplication; as EDX DC signifies the rectangle formed by the lines ED and DC, and axb expresses the product of the quantity a by the quantity b. Also ab or ab signifies the same thing. , Divide by, as PB÷Cs, or signifies that PB is to be divided by cs. PB AB2, AB3, 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. √ora,3√a or a3, express the square and cube root of A. A A a. =, Equal to, as AB=CD, shews that AB is equal to CD. A vinculum or parenthesis, serves to link two or more quantities together, as A+B×m, or (A+B). M, signifies that A and B are first to be added as is to S is to • Therefore, together, and then to be multiplied by the quantity m. Proportion, A B::C: D signifies that a has to B the same ratio which c has to D, and is usually read A is to в as c is to D. Angle, as A, signifies the angle a. Greater than, as AB, shows that A is greater 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 TRIGONOMETRY. BOOK I. CHAPTER I. THE NATURE AND PROPERTIES OF LOGARITHMS. (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.rl .p2.23.p.4.7.5.6, &c. be such a series, increasing 1 1 1 1 1 1 from 1; or 1. r -, &c. decreasing from 1; which last series, agreeably to the established notation in algebra, may be thus expressed, 1. r-1.r-2. p-3, p−4. 5 -6, &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 B |