FORM FOR Finding the CALCULATED ALTITUDE AND THE ALTITUDE DIFFERENCE FOR LAYING DOWN THE SUMNER LINE BY THE METHOD OF SAINT HILAIRE FROM A SIGHT OF A PLANET. FORM FOR FINDING THE CALCULATED ALTITUDE AND THE ALTITUDE DIFFERENCE FOR LAYING DOWN THE SUMNER LINE BY THE METHOD OF SAINT HILAIRE FROM A SIGHT OF A PLANET. * Haversine formula: hav z={hav (co. L+P. D.)-hav (co. L-P. D.)} hav t+hav (co. L-P. D.) =hav B+hav (co. L-P. D.); where hav B=hav A hav t, and hav A-hav (co. L+P. D.)-hav (co. L-P.D.) NOTES RELATING TO THE FORMS. 1. It is not necessary to convert departure into difference of longitude for each course; it will suffice to make one conversion for the sum of all the departures used in bringing forward the position to any particular time. 2. In D. R. it will be found convenient to work Lat. and Long. in minutes and tenths, rather than in minutes and seconds. 3. To obtain p, subtract Dec. from 90° if of same name as Lat.; add to 90° if of opposite name. 4. The Eq. of time is applied to G. C. T. in accordance with sign as given in the N. A. 5. If G. A. C. T. is later than L. A. C. T., Long. is west; otherwise it is east. 6. If Lat. is exactly known, a second latitude need not be employed. 7. s, and s―h may be obtained by applying half the difference between Li and La with proper sign, to si and si―h, respectively. 8. The G. C. T. must represent the proper number of hours from midnight, the beginning of the civil day; to obtain this it may be necessary to add 12h to the Chro. t. 9. H. A. from Greenwich is the difference between G. S. T. and R. A., and should be marked W. if the former is greater; otherwise, E. 10. Local H. A. is marked E. of W., according as the body is east or west of the meridian at time of observation. 11. Subtract local hour angle from Greenwich hour angle to obtain longitude; that is, change name of local hour angle and combine algebraically. 12. The forms include a correction for the parallax of a planet, but in most cases this is small, and may be omitted. When used, take hor. par. from Naut. Alm. and reduce to observe altitude by Table 17. The semidiameter of a planet may be disregarded in sextant work if the center of the body is brought to the horizon line. 13. Mark zenith distance N. or S. according as zenith is north or south of the body observed; mark Dec. according to its name, subtracting it from 180° for cases of lower transit; then, in combining the two for Lat., have regard to their names. 14. This form enables "Constant" to be worked up before sight is taken, and gives latitude directly on completion of meridian observation. Longitude and altitude at transit must be known in advance with sufficient accuracy for correcting terms. 15. The details of obtaining Dec. at transit and correction for altitude are shown in the meridian altitude forms for each of the various bodies. 16. To obtain t., the L. A. C. T. is subtracted from 12h. 17. If Long. is exactly known, a second longitude need not be employed. 18. " N. or S. according to name of Dec., and subtract it from 180° when body is nearer to lower than to upper transit; mark' N. or S. according as zenith is north or south of the body; then combine for Lat. having regard to the names. 19. Take a from Table 26 and at from Table 27. 20. Add for upper, subtract for lower transits. 21. Subtract longitude from Greenwich hour angle to obtain local hour angle; that is, change name of longitude and combine algebraically. 22. Add for west, subtract for east longitude. 23. As the trigonometric functions are all haversines in this solution, the abbreviation hav might be omitted, and the abbreviations, nat. and log might be employed to indicate the natural haversine and the log haversine, respectively. APPENDIX III. EXPLANATION OF CERTAIN RULES AND PRINCIPLES OF MATHEMATICS OF USE IN THE SOLUTION OF PROBLEMS IN NAVIGATION. DECIMAL FRACTIONS. Fractions, or Vulgar Fractions, are expressions for any assignable part of a unit; they are usually denoted by two numbers, placed one above the other, with a line between them; thus denotes the fraction one-fourth, or one part out of four of some whole quantity, considered as divisible into four equal parts. The lower number, 4, is called the denominator of the fraction, showing into how many parts the whole is divided; and the upper number, 1, is called the numerator, and shows how many of those equal parts are contained in the fraction. It is evident that if the numerator and denominator be varied in the same ratio the value of the fraction will remain unaltered; thus, if both the numerator and denominator of the fraction be multiplied by 2, 3, 4, etc., the fractions arising will be , 17, 18, etc., all of which are evidently equal to . A Decimal Fraction is a fraction whose denominator is always a unit with some number of ciphers annexed and the numerator any number whatever; as fo, 180, 1880, etc. And as the denominator of a decimal is always one of the numbers 10, 100, 1000, etc., the necessity for writing the denominator, may be avoided by employing a point; thus, is written .3, and is written .14; the mixed number 3, consisting of a whole number and a fractional one, is written 3.14. In setting down a decimal fraction the numerator must consist of as many places as there are ciphers in the denominator; and if it has not so many figures the defect must be supplied by placing ciphers before it; thus, 1.16, +8=.016, T=.0016, etc. And as ciphers on the right-hand side of integers increase their value in a tenfold proportion, as 2, 20, 200, etc., so when set on the left hand of decimal fractions they decrease their value in a tenfold proportion, as .2, .02, .002, etc.; but ciphers set on the right hand of these fractions make no alteration in their value; thus, .2 is the same as .20 or .200. The common arithmetical operations are performed the same way in decimals as they are in integers, regard being had only to the particular notation to distinguish the integral from the fractional part of a sum. ADDITION OF DECIMALS.-Addition of decimals is performed exactly like that of whole numbers, placing the numbers of the same denomination under each other, in which case the separating decimal points will range straight in one column. Add: Sum: EXAMPLES. SUBTRACTION OF DECIMALS. Subtraction of decimals is performed in the same manner as in whole numbers, observing to set the figures of the same denomination and the separating points directly under each other. MULTIPLICATION OF DECIMALS.-Multiply the numbers together as if they were whole numbers, and point off as many decimals from the right hand as there are decimals in both factors together; and when it happens that there are not so many figures in the product as there must be decimals, then prefix such number of ciphers to the left hand as will supply the defect. DIVISION OF DECIMALS.-Division of decimals is performed in the same manner as in whole numbers. The number of decimals in the quotient must be equal to the excess of the number of decimals of the dividend above those of the divisor; when the divisor contains more decimals than the dividend, ciphers must be affixed to the right hand of the latter to make the number equal or exceed that of the divisor. MULTIPLICATION OF DECIMALS BY CONTRACTION.-The operation of multiplication of decimal fractions may be very much abbreviated when it is not required to retain any figures beyond a certain order or place; this will constantly occur in reducing the elements taken from the Nautical Almanac from Green. wich midnight to later or earlier instants of time. In multiplying by this method, omit writing down that part of the operation which involves decimal places below the required order, but mental note should be made of the product of the first discarded figure by the multiplying figure, and the proper number of tens should be carried over to insure accuracy in the lowest decimal place sought. EXAMPLE: Required the reduction for the sun's declination for 7.43, the hourly difference being 58′′.18, where the product is required to the second decimal. In the contracted method, for the multiplier .03 it is not necessary to record the product of any figures in the multiplicand below units; for the multiplier .4, none below tenths; but in each case |