HANDLING FIGURES

Evening Post, Volume CXXII, Issue 65, 14 September 1936, Page 11

HANDLING FIGURES

"NAUTICAL ALMANAC"

CALCULATION BY MACHINES

GREENWICH TODAY

(From "The Post's" Representative.) LONDON, August 12. The fact that a New Zealander has been responsible for the computation at the Nautical Almanac Office, ■Greenwich, of the annular eclipse of the sun, visible in the North Island on December 14, and for v the calculation of its duration to the' nearest second, recalls the valuable service rendered to this office by the same man, Dr. L. J. Comrie, His application of mechanical calculators for working out the position of the sun, moon, planets, and stars for many years to come has led to the saving of thousands of pounds, absolute accuracy, and the abolition of much laborious and monotonous mental effort.' Information of this nature is published in the "Nautical Almanac," a huge volume of nearly 1000 pages, containing about two million' figures. About 6000 copies are printed annually. The "Almanac" tabulates the position of the sun, moon,i and seven major planets for every midnight and for the moment of meridian passage or transit at Greenwich. Ephemerides are provided for about 500 stars, and for the satellites which attend five of the planets. Information is also given about eclipses of the sun and moon, together with tables from which the time of sunrise, sunset, moonrise, or. moonset at any place may be found. LOGARITHMS SCRAPPED. Until 1926 the work of computing was done, with slight exceptions by hand. Highly-skilled computers, who lived on seven-figured logarithms were the order of the day, and the principal tools necessary for the very heavy computation were* logarithmic and trigonometrical tables. Today, due to the enterprise of Dr. Comrie, no logarithms are used, and the calculations are done by machines. This does not mean that the need for skill has disappeared, but rather that a different type of skill is required. The shades of the old computers, could they visit the office today, would find a large junior staff, many of them girls in their twenties, operating machines and quite unable to define cosine or right ascension!' About onethird of the staff are astronomers, the rest are computers. This mechanisation has resulted'in great economy and has rendered the routine portions of the work much less fatiguing. The ordinary calculating; machine is really simple, and may be learned in an' hour, as no new arithmetical conceptions are required. It will perform the four fundamental processes of arithmetic—addition, subtraction, multiplication; and division.' Since all arithmetic consists of combinations of these four processes, it may be said that it can be used for any arithmetical problem. Simple as these processes are, both in theory and practice, many electrical machines make them automatic. The operator has orily to set his factors and press a button. Thus two seven-figure ■ numbers, once set, would' be multiplied in about five seconds. Division is also fully automatic once the dividend and divisor are set, and takes about one second for each figure of the quotient. ~i NO REPETITIONS. Some surprise/ perhaps horror; might be expressed" by some at the fact that the calculations of the "Nautical ■Almanac" are neither duplicated nor. repeated. They are checked by a much more powerful process known as "differencing." The principle is based on the fact that the quantities tabulated in the "Almanac" are for uniform intervals of time, and, if plotted in the form of graphs, should be represented by smooth curves. i ' . Any departure from smoothness would indicate an error. But it would be impracticable to plot every calculation, because; apart from the excessive labour that would be involved, it would not be feasible to use a scale sufficiently la^ge to show up small errors in quantities that are given to decimals of a second. The smoothness can, however, easily be tested numerically, by taking successive differences ,of the computed quantities, until an order of differences is reached in which the differences tend to alternate, in sign. Any error immediately reveals itself as an abnormally large oscillation in the highest order pf, differences. ' This method of checking is infinitely more powerful than repetition, which is really the poorest possible check, as the original fault is liable to be repeated. Thus if there were a misprint in the, declination of the sun in the "Nautical Almanac" and if two navi,gatots took and reduced sights, they would both, in spite of Repetition and duplication; get erroneous results. If they distrusted their result, because of its large disagreement with their position by dead reckoning, and eventually suspected the Almanac, the misprint would show up immediately by differencing.. Until very recently the work of differencing was done by hand,.and was exceedingly laborigus, as great care was necessary not to introduce errors in the differencing itself, and thus vitiate its value as a check. Now a National machine is used to difference functions, whether in sexagesimal or decimal form,' to their fifth differences. The operator is merely responsible for the correct setting •of the function, followed by some simple manipulation of the machine controls. The machine prints the functions, as well as its differences, and tHus several hundred values can be checked in an hour. Actually two copies of the function are printed, one of which is torn off and pasted on to the sheet that forms the printer's "copy," so that the printer has the advantage of legible type-printed figures, and also the know-' ledge that he will not be called on later in the proof stage to correct copyists' errors. The discovery of the scientific application of this commercial machine, which may also be used for the reverse process of mechanical integration, or building tip a function from its differences, is perhaps the biggest stride in mechanical computing technique that has been made in the last; decade. . CALCULATIONS FOR 2000 A.D. 'Perhaps the most spectacular mechanisation is the work done by aj group-of Hollevith machines ori the position of the moon, for which portions of the calculations are now done to the year 2000. The work is similar in principle to tide prediction, consisting of the synthesis or composition of harmonic or sine , terms, but with the important difference that greater accuracy is required, the amplitudes do not vary, and the number of terms to be combined is large. The efficiency of the tabulator and its capacity and high speed enable it to add 20 figures a second and to print the results. -The work done by the machines in this set of' calculations would have cost £6000 if done by the hand methods formerly in vogue. Actually it was done for £1500, a saving of £4500. In the course of the work 20,000,000 holes were punched in 500,000 cards, at the rate of three or four holes a second. Experience has shown that this fast process is "more accurate than the slow process of hand copying, undoubtedly because a fast operator translates in-

stahtly the sight of a figure into the i action of depressing ,a key. :On the, other hand, a copyist, after glancing; at a ;group of , figures on one .sheet, has time to'transpose or, confuse them; before copying is completed. . r > When the .time' comes for 'these, figures to be printed, the task,of. proof-; reading 1000 pages of figures annually i is no light onel. All Iprobfs' arte,com-1 pared with original-calculations,, Each, reader worlds single-handed, i.e., com- \ pares his copy and proof- The>method of one reader, and one ■listenei'; working together. is ,not •' employed '-at all,; partly because it takes two persons j as long to do the work in this; way; as one can do it by himself, and partly because the mental comparison of an: aural,image and a visual image is not. satisfactory in the case* of figures. In; order, to avoid the risks , associated with. movable type, stereo plates are cast from the corrected type, and proofs from these plates are very carefully examined by the. more experienced proof-readers, who are fully alive to the types of error that are liable to arise,. whether in computing, in the first stages of proof-reading, or at the .printing house in. the' moulding- and casting of plates.'!. These, readers are very conscious of the fact that the safety of all shipping depends on the accuracy of their figures," and those who "go down to the sea in ships" may rest. assured that every conceivable precaution is taken to ensure the correctness of the printed figures.