P32 CORNELL UNIVERSITY LIBRARY FROM Cornell University Library QB 633.D32 Draysonia, being an attempt to explain a 3 1924 012 395 962 Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924012395962 DRAYSONIA WORKS BY GENERAL DRAYSON Practical Military Survejring; The Common Sights in the Heavens The Last Glacial Epoch Experiences of a 'Woolwich Professor Thirty Thousand Years of the Earth's Past History, read by the Aid of the Dis- covery of the Second Rotation of the Earth Untrodden Ground Important Facts and Calculations for the Consideration of Astronomers and Geo- logists Proper Motion of the Fixed Stars Etc. DRAYSONIA BEING AN ATTEMPT TO EXPLAIN AND POPULAEISE THE SYSTEM OF THE SECOND KOTATION OF THE EAKTH AS DISCOVERED BY THE LATE MAJOR-GENERAL A. W. DRAYSON F.R.A.S. FOR FIITEEN YEABS PBOFESSOB ROYAL MILITAIIY ACADEMY, WOOLWICH ALSO GIVING THE PROBABLE DATE AND DURATION OF THE LAST GLACIAL PERIOD, AND FURNISHING GENERAL DRAYSOn's DATA, PROM WHICH ANY PERSON OP ORDINARY MATHEMATICAL ABILITY IS ENABLED TO CALCULATE THE OBLIQUITY OP THE ECLIPTIC, THE PRECESSION OP THE EQUINOXES, AND THE RIGHT ASCENSION AND DECLINATION OP THE FIXED STARS FOR ANY YEAR, PAST, PRESENT, OR FUTURE BY ADMIRAL SIR ALGERNON P. R. DE HORSEY K.C.B. LONGMANS, GREEN AND CO. 39 PATERNOSTER ROW, LONDON NEW YORK, BOMBAY, AND CALCUTTA 1911 PREFACE In the following treatise I have adopted the title ' Dray- sonia ' in honour of a man whose scientific attainments have been but Uttle known or recognised, whose death in September 1901 was a great loss to astronomical science, and who in future days will, I think, be acknowledged as having been a remarkable discoverer. I refer to the late Major-General Alfred Wilks Drayson, F.R.A.S., late Royal Artillery, who, in addition to distinguishing himself in his profession, was for fifteen years Professor, Royal Mihtary Academy, Woolwich, and for two years attached to the Royal Observatory at Greenwich, author of Practical Military Surveying, The Common Sights in the Heavens, The Last Glacial Epoch, Ex- periences of a Woolwich Professor, Thirty Thousand Years of the Earth's Past History, read by the aid of the discovery of the second rotation of the earth. Untrodden Ground, Important Facts and Calculations for the Consideration of Astronomers and Geologists, Proper Motion of the Fixed Stars, etc. I am fully aware of the difficulty of my task, and of how imperfectly I can do justice to Drayson in attempting to describe his system ; indeed, I should not venture the attempt were it not that General Drayson, shortly before his death, repeatedly urged me to write upon the subject. Writing to me very shortly before his death, he said : ' The reason why I thought that you ought to write something is that it would be a loss to astronomical science if the accurate calculations which you have made were allowed to be buried in your work- book and among my papers. I refer especially to your vi PREFACE investigations to obtain the annual motion of the pole 20"-0529, the annual angle at C. 40"-8114, the zero year a.d. 2294-75, the cycle 31,756 years, the annual precession of the equinox in 1900, the annual diminution of the obliquity and the variable rate of this decrease. ' All these calculations are based on sound geometry.' In again urging me in his last touching letter before his death, when complaining bitterly of the unreasoning opposition he had encountered, he said : ' I have spent between two and three hundred pounds in getting my books, pamphlets, etc., published, and have received less than ten pounds in return. Money-making has not been my object, but that truths in nature should be made known. I can only claim to have done my best, though whether good results will follow I cannot say.' I also can only claim to have done my best. My work in assisting Drayson began after reading his Untrodden Ground, and has been carried on almost entirely by letter. Drayson's letters to me amounted to three or four hundred, and were continued imtil his death. From this correspondence and from Drayson's works I learned a great deal, and found the occupation enthralling — too much so, for in working with Shortrede's admirable logarithms, which admit of accuracy to the -^oia, second of arc and to ^ffxr of a second of time, up and into the small hours of the night, I considerably impaired my eyesight. I was induced to look into Drayson's system by my old and distinguished friend, the late Sir John Cowell, who, when on duty at Osborne, had lent Drayson's Untrodden Ground to H.R.H the then Crown Princess of Germany, afterwards the Empress Frederick, and told me how great an interest H.R.H. had taken in the matter. The subject also has engaged the attention of H.R.H. the Duke of Connaught, who, when at Woolwich Academy as a cadet, had studied under Drayson, and was thus aware of his great abihty. I mention this as another instance of the PREFACE vii great interest always taken by members of our Royal Family in scientific matters. I believe I have been a good scholar of Drayson's, judging from hia kind appreciation of my efforts, for in an article of his some years ago in the Yorkshire Post, the following com- mendation appears : ' It is quite unnecessary for me to refer to the scientific attainments, especially in practical astro- nomy, possessed by Admiral de Horsey. In the Royal Navy these are well known. I may say, however, that during the fifteen years that I was Professor of Woolwich I looked over several thousand examination papers on various questions in practical astronomy which had been worked out by officers and cadets, who were excellent mathematicians, and to whom the practical working of spherical trigonometry was mere child's play, but in no single instance have I seen anything equal in accuracy and neatness to the work accompUshed by Admiral de Horsey.' The above flattering opinion of my attainments far exceed such abilities as I may possess, but I here insert it as a record of General Drayson's appreciation of the assistance I rendered him in astronomical computations from 1893 until that dis- tinguished astronomer's death in 1901. Drayson's lamented death occurred without his receiving that public recognition of his scientific attainments and discoveries which he merited, and which I trust posterity will accord him. A. F. R. DE HORSEY, CowBS, March 1911. CONTENTS PAGE Preface ... . . . v Inteoduction ...... 1 Section I. Deayson's Theory op the Second Rota- tion OF the Pole op the Eaeth . 6 II. Radius op Second Rotation . .16 III. The Annual Motion op the Pole . 19 IV. Annual Angle, Dueation op Cycle, and Zeeo Year . . . ,25 V. The Obliquity op the Ecliptic . 28 VI. The Precession of the Equinoxes . 31 VII. Standard of Time and Right Ascension 37 VIII. Problem to find Drayson's data prom Recorded Orthodox Observations . 54 IX. The Temperate and Glacial Epochs . 66 Conclusion . . . . . .74 INTRODUCTION DRAYSON'S SECOND ROTATION OF THE EARTH 'E fur si muove.' — Galileo In the above diagram the circle is intended to represent the path traced by the pole of the earth in its course round a cycle of 31,756 years with a radius of 29° 25' 47" from C, the centre of second rotation. E is the pole of the echptic distant about 6° from C. Z, which I have termed the zero year, is the position the pole will attain in a.d. 2295 when E and C are ahgned ; when the obliquity of the ecliptic wiU be at its minimum, and the earth therefore at the middle point of its temperate epoch. Z' represents the place the pole at- tained at the middle of the last Glacial Epoch, 13,583 B.C., when the obliquity of the ecliptic was about 35° 25' 47". P represents the pole of the earth at various periods, PE the obliquity of the ecliptic at various times. Pa the polar distance of (say) a Draconis, and the angle CPa the right ascension of that star, and Ca its distance from the pole of A second rotation. Similarly P/3 is the polar distance of (say) /8 Ursse Minoris, the angle GP^ is the right ascension of that star, and C^ is its distance from the centre of second rotation. The dotted Une PA' shows the direction of the first point of Aries at the beginning of the Christian era, and PA its direction about the present time. In describing the angles CPa and CP/3 as the right ascen- sions of those stars, it should be added that they are so subject to + or — 6 h. or 18 h., as the case may be, and to a correc- tion for standard of time in order to assimilate them to the recorded right ascensions in the Nautical Almanac, and to the Greenwich standard rate of sidereal time, as described in Section vii. The following diagram is intended to show the position of the ecliptic and its pole at the mid-Glacial Period, when the obhquity of the echptic was about 35|°, as compared with their positions at the present or nearly mid-temperate period, in which the obhquity is about 23|°. It wiU be observed that at the mid-Glacial Period the Arctic and the Antarctic circles come down to about lati- tude 54|°, and that the tropics of Cancer and Capricorn extend from the equator to about latitude 35^°. North Pole i [Equator South Pole Explanation op the Diagram G is the centre of second rotation. E the pole of the echptic at mid-temperate period. TT the ecliptic at mid-temperate period. E' the pole of the ecliptic at mid-Glacial Period. OG the echptic at mid-Glacial Period. The designation ' second rotation ' of the earth may be open to objection, but it is the name given by Major-General Drayson, the author and discoverer of the system, which I shall endeavour to describe in these pages ; it is therefore sufficient for my purpose. By whatever name the movement may be called, I refer to the revolution of the pole of the earth round a centre C in some 30,000 odd years, instead of, as orthodox astronomers describe it, round the pole of the ecUptic in about 25,867 years. As I understand Drayson's system, it is founded (1) on the assumption that the stars, with some exceptions, are fixed; or that their movements, if any, are so small as to be iuappreciable by observers on a planet so distant from the stars. (2) That, the stars being fixed, their apparent motions in right ascension and declination are alone due to the axis of the earth altering its inclination about 20" annually. (3) That this alteration in the inclination of the axis of the earth is a movement in a small circle at a radius of about 29° 25' 47" from a point in the celestial concave, which point is about 6° 0' 0" from the pole of the ecliptic, and situated as hereinafter described in at present about 18 h. right ascension. Some professional astronomers, who may have done me the honour to read the above few hues, will then perhaps lay down my paper saying that the system appears to be founded on hypotheses which are not proved, and which are not in accordance with the long estabHshed teaching of our greatest astronomers. ' Eead such and such a work at page so and so and you will see how erroneous are your views ; we refuse to consider or even read arguments which are not founded on the dicta of our orthodox text-books.' To those who thus reason I have no reply, unless it be to quote a line of Dante's Third Canto, Non ragionam di lor, ma guarda e passa. But the many deep thinkers whose reputation and emoluments are not dependent on adherence to orthodox text-books will perhaps read on, remembering that scientific truths are often arrived at from apparently unfounded hypotheses, the proof of which is brought about by subsequent reasoning, and by the results of such reason- ing coinciding with facts which could not exist on any other hypothesis than the one first suggested. If orthodox authority had alone been listened to, we should still have to admit that the earth was stationary, and that all the heavenly bodies revolved round the earth as a centre. I am not foolish enough to say of my very limited knowledge that Drayson's system is absolutely exact, but I do say that it appears to afford an easy means of calculating the obUquity of the ecliptic, the precession of the equinoxes, and the right ascension and declination of stars with preciseness for hundreds of years past or future, and that it provides a standard basis of time which will not, as at present, be always accumulating error. Can orthodox astronomers do any one of these things ? Can it be claimed that the present system of time is correct, seeing that S^ 3s-68 of purely imaginary time were added in 1834, that Professor Stone finds a fresh accumulation of error of 41"'51 up to 1892, and that the best astronomers admitted the whole subject having fallen into confusion ? General Drayson's system may be right or it may be wrong, but its results have shown ample cause for its careful and unprejudiced examination. If such examination should be accorded, and the results I have mentioned be shown to be correctly at- tained ; and further, if with our present knowledge such results can be obtained by no other known method, surely this alone would go far to prove the correctness of his system. If English astronomical authorities continue to refuse it fair consideration, there is a danger of foreigners claiming to discover what Drayson asserted thirty or forty years ago. Already in a work recently pubUshed, entitled Astronomic Populaire, M. Flammarion says : ' C'est la Terre seule qui en est animee, et c'est elle qui accompUt pendant cette longue periode une rotation oblique sur elle-meme en sens contraire de son mouvement de rotation diurne.' This is precisely what Drayson stated some thirty years before Flammarion ; in reply to which an astronomical authority informed him that if such was the case ' chaos would occur, and the earth would be smashed up.' On this Drayson dryly observed that this was one of the objections urged against the daily rotation of the earth in former times. The truth or fallacy of Drayson's system should be deter- mined by astronomical research alone, but it is interesting to note that modern geology confirms that system which establishes with geometrical precision (not by mere infer- ence, as geology teaches) the date of the last Glacial Epoch, showing that the coldest half of the cycle commenced about 23,423 years ago, reached its maximum about 15,484 years ago when the Arctic circle came down to about the latitude of Durham and the Antarctic circle to that of Tierra del Fuego, and terminated about 7545 years ago. This statement, pub- lished by Drayson twenty years before geologists would admit it, was ridiculed by them. In a late publication Drayson writes that at that time ' more than one dis- tinguished authority stated that although he did not feel competent to criticise my astronomical and geometrical argu- ments, he could positively state that the dates I gave for the duration and termination of the Glacial Period were absurdly incorrect. He knew that the period terminated about a quarter of a million years ago, and lasted at least a million of years.* ' For every great advance in opposition to scientific authori- ties there must be a martyr ; in former times this martyr was tortured or burned. In modern times he is ridiculed, contradicted, or ignored, then years after his death is given a statue.' I am inclined to believe with the late Sic John Cowell, and as a promising yoimg officer of the Eoyal Engi- neers wrote to me some time ago, that within a generation, although probably not in Drayson's life, his system wiU be the accepted one of the text-books. SECTION I DRAYSON'S THEORY OF THE SECOND ROTATION OF THE POLE OF THE EARTH As a preamble to the endeavour I am about to make to explain Drayson's theory of the second rotation of the pole of the earth, I perhaps cannot do better than reprint that which I wrote on the subject in June 1894, and which was published at the time in the Isle of Wight County Press and in the Yorkshire Post, together with a letter I had previously addressed to Major-General Drayson. The Glacial Period as proved by the Second Eotation OP the Earth ' A discovery of incalculable advantage to astronomy, although made and described in pubhshed works more than twenty years ago, appears to be still unacknowledged, and even branded as an absurd heresy, by some professional astro- nomers of the day. In order to add my humble efforts to induce qualified and unprejudiced persons to examine fully, test, and if true estabUsh before all the world the truth of General Drayson's system, I propose to bring the matter as briefly as so large a subject will admit to the notice of those unacquainted with Drayson's works. For this purpose it is necessary first to touch upon certain fallacies of orthodox astronomy which seem to be almost universally accepted up to the present day. ' Sir John Herschel, in Outlines of Astronomy, stated that the pole of the heavens {i.e. the pole of the earth projected to the celestial concave) describes a circle in the heavens round the pole of the ecliptic as a centre, at a constant distance of 23° 28' in 25,868 years. (Since that was written it has been generally admitted that the distance between the two poles, i.e. the obliquity of the ecliptic, varies.) ' Sir George Airy, writing to Drayson in September 1870, says : " The direction of the movement of the terrestrial pole in any one year on the surface of the globe is at right angles to the great circle connecting the place of the terrestrial pole with the place of the echptic pole in that year." Again, in October 1870 the same great authority wrote to Drayson : " The state of matters regarding the pole of the earth and the pole of the ecliptic is this : the pole of the earth does not revolve round the pole of the ecliptic, in the sense in which you understand it. Supposing the poles connected by an arc of great circle, the pole of the ecliptic has a small motion in the direction of this arc towards the pole of' the earth and a small motion transverse to it. The pole of the earth has no motion in the direction of the arc above mentioned, but has a large motion transverse to it. The distance between the two poles is not invariable." One can only describe the above statements as a mass of contradictions. If P did always move at right angles to E, it would be a geometrical absurdity to say the distance between the two varied. ' In April 1893 a very high authority wrote to me : " As a matter of fact, the pole of the ecliptic does not change its position in space by more than about one degree on either side of the mean, and the obliquity of the ecliptic will only vary to the same amount, one degree approximately on either side of the mean." ' Sir Eobert Ball, Astronomer Eoyal of Ireland, in so late a work as The Story of the Heavens (1891), states that the pole of the earth moves in a circle, the centre of which is the pole of the ecliptic ; and he illustrates this by a star map of Piazzi Smyth's, and gives 25,867 years as about the period of the cycle of precessional movement. ' Viewing the contradiction that the pole of the earth moves at right angles to the pole of the ecliptic, and yet alters its 8 distance (and putting aside the geometrical absurdity of a circle with a movable centre), it will be observed how hope- lessly at sea the above authorities are upon the subject. Against their inexplicable theories Drayson asserts and proves geometrically that — ' 1. The pole of the earth does not revolve in a circle with the pole of the ecliptic as a centre. ' 2. The extent of variation of distance between the two poles is not limited to about 1°, its maximum variation being 12°. ' 3. The pole of the earth describes a circle with a radius of 29° 25' 47" from a centre, which itself is 6° distant from the pole of the ecliptic. ' 4. The period of the cycle of precessional movement is about 31,697 (since found to be 31,756) years, not 25,867 years, and the pole of the heavens traces a different path among the stars from that shown in Sir Eobert Ball's Story of the Heavens. ' 5. The last glacial period is proved to have commenced about 21,478 B.C., was at its height about 13,553 B.C., and ter- minated about 5629 B.C. At the height of the glacial period the obUquity of the ecliptic was 35° 25' 47", and consequently the arctic circle came down to about the latitude of Durham, and the antarctic circle to that of Tierra del Fuego. ' Having thus given some idea of existing orthodox theories and of their contradiction, I proceed to furnish a brief de- scription of Drayson's system. For fuller details those who are interested in the subject should refer to Drayson's principal works, viz. On the Prober Motion of the Fixed Stars, Thirty Thousand Years of the Earth's Past History (Chapman & Hall), and Untrodden Ground in Astronomy and Geology (Kegan Paul & Co.) ; also to the pubhshed papers of Major-General Sir John Cowell. ' From a geometrical analysis of the curve traced by the pole of the earth during the past 1800 years, it proves to be an arc of a circle which has a radius of 29° 25' 47". ' The centre of the circle is located in the heavens in about eighteen hours right ascension from the pole of the ecliptic, and is — 26° 37' 4" from Alpha Draconis. 9 17 38 Beta Draconis. 10 24 41 Delta Draconis. 21 50 12 Beta Ursae Minoris. 29 52 51 Polaris. 27 55 7-3 Alpha Cygni. 6 Pole of Ecliptic. 29 25 47 Pole of Earth. ' From the recorded observations of many years, it is known that the pole of the earth approaches stars having hours right ascension about 20"-09 (since found to be 20"-0529) annually. This 20"'09 measured on a small circle is therefore the change in the position of the pole annually, and will be found equal to about 40"-89 (since found to be 40"-81143) on the great circle, viz. such is the annual angle at the centre of second rotation. Dividing the whole circle of 360° by 40"-89 wiU give about 31,697 (since found to be 31,756) years as the period of the entire revolution of the pole of the earth round the centre of second rotation. ' Such being in the briefest language a summary of Drayson's system of the second rotation of the earth, allow me to add a copy of a letter I lately addressed to General Drayson, giving the results of some of the tests I have applied to his system and of the inevitable conclusions to which, as an independent and totally uninterested critic (except so far as one is interested in searching after truth), I have arrived. I neither endorse nor advocate any theory, nor do I pretend to disclose all such parts of General Drayson's discovery as he has been good enough to entrust to me. He is rightly careful that no one should rob him of the credit justly due to him, and there are some points — now absolutely unknown to the scientific world — to which he alone can do justice, and which he will doubt- less be prepared to make pubUc property when an impartial tribunal, free from professional jealousy, has admitted the truth of his system. I merely give geometrical proofs by the 10 results of my calculations from data of many years back agreeing minutely with actual present Greenwich observations — ^results only now obtainable by constant observation, and which no astronomer can calculate until he accepts and uses Drayson's system. Algernon de Horsey. 'Meloombb House, Cowes, 9Srd April 1894. 'Dear General Drayson, — As you are aware, I have during several months been engaged testing by numerous calculations the accuracy, or otherwise, of your discovery of the second rotation of the earth. This pursuit has been of enthraUing interest to me, and you may now like to know the result. Among these calculations (in all of which I have made no use of solar tables, or rate per year) were the follow- ing : — Starting from the date a.d. 2295-2, 1 calculated by your system what the obUquity of the ecHptic would be for the year 1885, working the problem by spherical geometry. On comparing my calculated obliquity with that recorded in the Nautical Almanac, I found a difference of only 0"-07. I then took the recorded E.A. and declination of the stars eta Ursse Majoris and alfha Draconis from Bradley's Catalogue of 1st January 1755, and (as I have said for all cases, without making any use of the rate of change, now found only by perpetual observation) I calculated the E.A. and Decn. of these stars for the 1st January 1895. On comparing my results with those given in the Nautical Almanac, I found a difference, in 140 years, for eta Ursse Majoris of only 0^-063 in R.A. and l"-66 in Decn., and for alfha Draconis 0^-156 in R.A. and 1"45 in Decn. Again, working alfha Draconis from 1755 to 1875 (120 years), I calculated its R.A. for 1875 to be 14 h. 1 m. 0-121 s., and declination plus 64° 58' 25"-2. These results I could not compare with observations, as, the star being omitted from the 1875 Nautical Almanac, its position apparently was not known at Greenwich. Deter- nained to apply a crucial test, I selected the stars lambda Ursse Minoris close to the North Pole, and sigma Octantis close to the South Pole, stars having an annual variation in R.A. 11 respectively twenty-fold and thirty-six-fold as great as the average. Calcidating lambda UrssB Minoris from 1850, 1 found its E.A. and Decn. in 1895 to be within 0^-249 and 0"-065 respectively of the results by observation (and by rule of thumb for four years in advance) recorded in the Nautical Almanac. Then, calculating sigma Octantis from 1875, 1 found its E.A. and Decn. for 1895 to be within 4s-251 and 0"-28 respectively of the Nautical Almanac record. Astronomers will think little of a difference of R.A. 4|s in the case of a star which, in the twenty years, changed 2141^, and which, being within 44' of the pole, must hang on the wires of the transit instrument for a considerable time. ' I am fuUy aware that the above calculations are unknown to aU astronomers to whom you have not explained your system. They are problems which any one with ordinary mathematical ability can work, if he knows your system. Without this knowledge, and without accepting your discovery to be true, I believe no hving man can work them. To the Astronomer Royal to the Royal Astronomical Society, and to all the learned professors in the four quarters of the globe who adhere to their text-book knowledge, I beheve these pro- blems to be as insoluble as they would have been to Lobengula's medicine man. I am of a rather sceptical turn of mind, deeming that when in search of scientific truth one should take nothing on trust. As you know, there are two or three minor points in one of your published works in which I differ in opinion — perhaps owing to my incompetence, possibly owing to your having been mistaken — but these are trifles. With regard to your system of the second rotation of the earth, if one or two calculations only had agreed with observation, I should have said it was very extraordinary, but might possibly be a coincidence. But when I find that in no single instance has calculation failed to obtain results which are true, and which are unobtainable by calculation from any theory difiering from your system ; when I find that by your great discovery, and by the system you have founded thereon, I am enabled to calculate with precision the obHquity of the ecliptic, the precession of the equinoxes, and the polar 12 distances and right ascensions of stars for 5, 50, 500 or more years past or future ; when I find that a precise and un- changeable standard of time is obtainable from your systems for all time to come instead of the present erroneous measure, instead of the " confusion " admitted by Sir John Herschel, and the muddle in which all astronomers now appear to be on the question of time (for instance, that distinguished astronomer, Mr. Stone, now EadcUffe Observer at Oxford, has within the last few months discovered an error of 4:1"*51) ; when, again, I consider that your system establishes the period of successive glacial epochs and the date of the last one — this, with geometrical precision, not by mere inference, as geology teaches ; and when we note that modern geologists — forsaking their former theories about hundreds of thousands, or even millions, of years having elapsed since the last glacial epoch — now, years after your pubUshed demonstration of the fact, fix the period in accordance with your geometrical proof — when all the above results are considered, I can come but to one conclusion, viz. that the truth of your system is estab- lished beyond all doubt, and that this truth must shortly force itself on the scientific world. ' It is inconceivable to me that the Royal Astronomical Society has not enthusiastically taken up a subject which gives such strong evidence of its truth. To sift such a matter — a matter affecting the basis of all astronomical calculation — to the bottom is surely the first duty of a learned society which has the privilege of the prefix " Royal." I should have imagined — as I believe you did — that you had only to whisper 29° 25' 47" in order to put astronomers on the qui vive to elucidate a discovery of enthralling interest, which, for a chaos of supposed proper motions of millions of stars, substitutes a simple movement of the instrument (the earth), by which and from which all astronomical observations are made. ' History repeats itself. As some 260 years ago GaUleo endeavoured to revive the Copernican system of 1543, and explained the apparent diurnal revolution of all the heavenly bodies by the simple, but then unacknowledged, fact of the earth's daily rotation, so you in the present day explain the 13 absurdity of ascribing complicated so-called proper motion to the stars by the simple fact of the second rotation of the earth round a fixed point distant from the pole 29° 25' 47", a fact you prove by calculating results which observations show to be true, and which no astronomer who adheres to the existing orthodox, but erroneous, theory can accomphsh. ' You win probably not be imprisoned nor made to swear on your knees that your system is not true ; you will not even be made to repeat once a week for the remainder of your life the seven penitential Psalms, but it would seem that — as in the days of Copernicus and Galileo — ^you have the learned societies against you for a time. You wiU have been both amused and interested in reading in the R.A.S. notices of the 9th March last Mr. Stone's discovery that the sidereal time of mean noon in 1892 was erroneous to the extent of 4ls-51. Mr. Stone — clever astronomer as he undoubtedly is — ^has prob- ably little knowledge of, and no beUef in, your system, and is perfectly innocent of the fact that the result of his labours is an additional confirmation of the truth of your system. Since the authorities at Greenwich in 1834 fudged in 3™ 3S'68 of imaginary time to make their theories coincide with facts, the process has, I believe, not been repeated. The error therefore, which Mr. Stone dates from the introduction of Le Verrier's solar tables in 1864, has probably been accumu- lating ever since the time was corrected in 1834. On reading Mr. Stone's paper I at once calculated by your system what would be the accumulated error of the erroneous Greenwich system from 1834 to 1892, and I made it agree within a small fraction of a second of Mr. Stone's result. I observe that Professor Simon Newcomb disputes Mr. Stone's accuracy, but no doubt Mr. Stone is right. — Sincerely yours, 'Algernon de Horsey, Admiral.' ' Since writing the above I have tested Drayson's system conversely, i.e. totally ignoring his data and working only from observed positions of stars as recorded in the Nautical Almanac — data which no one can dispute — I calculated by my own independent method the radius of the circle described 14 by the pole of the earth, the annual motion of the pole, the angular distance between the poles of second rotation and ecliptic, and other fundamental items, bringing my results out to agree with Drayson's data with sufficient accuracy to confirm their truth. This I did by six difEerent calculations, viz. with Polaris, beta Orionis, two of alpha Draconis, and two of alpha Virginis. These tests are to my mind con- clusive, because their results could not all coincide with Drayson's data if the latter were not correct. The statement in the foregoing article, that the period of one complete cycle of precessional movement is about. 31,697 years, and the consequent apparent periodicity of the glacial epoch is, I understand, qualified by General Drayson to the following extent, viz. : — It is the periodicity corresponding to the present motion of the pole, and such it has been for cen- turies. But if any convulsion of nature, either in the past or in the future, should alter the position of the earth's centre of gravity, the gyratory motion of the pole, and therefore the periodicity and intensity of the Ice Age, would be difEerent. 'A. DE H. ' January 1895.' In the above description of Drayson's theory I have alluded to some points which he rightly withheld from publication, but which are now pubHc property. I refer to his method of measuring sidereal time. From Drayson's system it appears that by the orthodox doctrine that the pole of the earth moves round the pole of the ecliptic an error of time creeps in, and amounts at the present time to about O^-T per annum, which error, if error it be, is entirely obviated by acceptance of Drayson's system of the pole of the earth revolving round the central point herein described. An annual difference of 0^-7 sounds little, but has to be taken into account in all geometrical calculations wherein the right ascension is a factor. This alteration in the annual rate of sidereal time is a discovery of Drayson's which he wisely kept back from his published works and communicated alone to the late Sir John Cowell and to myself. This difference of rate of 15 time is not the least important of Drayson's discoveries. In a letter to me in January 1894, Drayson writes : ' When you have made this problem your own you will know that you are in possession of that which the astronomers of the whole world have never even dreamed of.' If it be assumed that Sir John Hersohel's addition of ' purely imaginary time ' of 3^ 3s*68 was correct, it follows that under Drayson's system an error of some magnitude will under the Greenwich system have to be corrected. A short time ago I ventured to write to the late Astronomer Royal, Sir William Christie, to ask if I was right in my computation that an error in 1910 amounted to about 54 sec. I also told him that for the year 1892 I had made the error 413'283, and that I had observed that Mr. Stone had made it 41s-54. I hardly expected a reply from so busy a man; but Sir William Christie very kindly rephed at some length, to the effect that it was not considered there was any error, and that Mr. Stone had been mistaken in supposing one to exist. This subject will be more fully explained imder its proper head in Section vii. SECTION II EADIUS OF SECOND KOTATION I CAN scarcely do justice to General Drayson in attempting to describe how he determined the radius of second rotation to be 29° 25' 47" or thereabouts, but must endeavour to do so. He appears to have been impressed many years ago by ascertaining from the records that the obliquity of the ecliptic was not a fixed amount, as shown by the following : Date. Recorder. Obliquity. A.D. 30 Strabo Proclus, etc. . 24° 0' 0" 140 Ptolemy . . . 23 50 390 Pappus . . . 23 50 1437 Ulugh Beigh . . 23 31 58 1672 Cassini . . . 23 29 1690 Flamstead . . 23 28 48 „ With correct refraction 23 29 2*5 1750 Lacaille . . . 23 28 19 1755 Bradley . . . 23 28 17-6 1800 Maskelyne . . 23 27 56-6 „ Bessel . . . 23 27 54-78 1840 Airy . . . . 23 27 36-5 1860 Nautical Almanac . 23 27 27-38 1870 „ „ . 23 27 22-2 1873 „ „ . 23 27 20-88 1890 „ „ . 23 27 12-79 From the above records it appears that the obliquity had been decreasing at a diminishing rate, and that in consequence the circle described by the pole could not be round the pole of the ecliptic as a centre, and must be round some other 11 point in the heavens. Drayson also contended that the seasons in respect to their intensity of cold and heat must be dependent on the amount of the obliquity of the ecliptic ; it was impossible to explain the reason of there having been what is termed a glacial epoch except by a considerable increase of the obliquity beyond its present amount. Assuming the stars, with possible exceptions, to be fixed, and that their apparent alterations of right ascension and declination can be fully accounted for by one simple motion of the pole of the earth, Drayson carefully studied the annual increase or decrease of their right ascension and dechnation as found by observation at Greenwich, and as recorded in the Nautical Almanac, and especially the amount of annual variation in right ascensions 15 h. and 21 h. Drayson con- cluded therefrom that the centre of rotation in order to cause those changes must be in right ascension about 18 h. and in declination about 60°. He then through a long course of years made innumerable calculations to find what distance between the pole of the ecUptic and his supposed centre of rotation would enable him to calculate from the best recorded obliquities in former years the obUquity for all other years, trying with distances CE varying between 5° and 7°, and by repeated trial and error, until he concluded that the radius CP must be about 29° 25' 47" and the arc EC about 6°. Thus in the annexed diagram, in which EP is the obliquity when the pole is at P, and in which CE represents the dis- tance between the centre of rotation and pole of the ecHptic, CP the radius of second rotation and with the angle PCE known, Drayson found himself able to calculate the obliquity EP with great ^l ^ J ^ accuracy for all other years and conse- quent different positions of P. ^^ \ The other basis of Drayson's system is pX the annual motion of the pole, viz. PP' in the diagram — this always on the assumption that the principal stars are fixed, as he gathered from a close study of their annual variations recorded in the B 18 Nautical Almanac, especially of those stars having a right ascension within an hour of h. and 12 h., from which he at first roughly took the annual motion to be 20"- 1 and subse- quently 20"-09, but of late years accepted my perhaps more complete calculation of 20"'0529, which I ascertained by the method shown in Section iii. I fear the above explanation of Drayson's long labours to find his data is but a poor abbreviation ; but in order to confirm Drayson's data, it occurred to me in 1894 that as his data enabled him accurately to calculate the obhquities and positions of stars as recorded in the Nautical Almanac, it should follow that conversely Drayson's data, if correct, should be ascertainable from authentic Nautical Almanac records. I accordingly made six independent calculations from orthodox records of the positions of stars at different periods, viz. two of a Draconis, one of each a Ursse Minoris and /3 Orionis, and two of a Virginis, with the object of finding the value of CP the radius of Drayson's cycle, PP' the annual motion of the pole, and other resulting data. The mean of my six results made CP 29° 29' 15"-31 and PP' 20-0754, to the precise accuracy of which I do not pretend, but which go far to estabhsh the general truth of Drayson's system. A fuller account of these calculations, with a table of their results, will be given in Section viii. For more complete information about Drayson's method of finding his data, reference should be made to his publi- cations mentioned in the preface to this work. SECTION III THE ANNUAL MOTION OP THE POLE In the Nautical Almanac 1895, in the table of mean places of stars, only annual variation of stars' right ascension and declination was given, whereas in the Nautical Almanac of 1898 two columns are inserted against both the right ascen- sions and declinations, respectively headed annual precession and annual proper motion, from which it appears that the additional columns of proper motion were made between the two years named. As, however, the annual precessions are presumably obtained by observations at Greenwich, it may perhaps be assumed that the amount of precession includes the proper motion, and that therefore, in ascertaining the annual motion of the pole by the following method, no account need be taken of the proper motion, if any. To find annual motion, of pole from mean of annual pre- cession (say) in Nautical Almanac of 1898. In the accompanying diagram let 40= annual precession of a star within one hour of h. or 12 h. right ascension, and let AM=AB=ainiual motion of pole AC = —■ — :i where the angle A is the smA ° amount of right ascension of the star from the meridian h. or 12 h. AG By the expression AM = ■ a the annual motion of the pole has been computed for every star in the Nautical Almanac of 1898 (seventy-three in number) which is within one hour 20 of h. and 12 h. of right ascension, the results being shown in the table at the end of this Section. It will be seen that the mean of the results from the thirty-two stars which are within one hour of right ascension h. is 20"*052678, and the mean of the forty-one stars within one hour of right ascension 12 h. is 20"-053105. The mean of the results from the whole seventy-three stars wiU be found to be 20"-052918, say 20"-0529, which by a curious coincidence is precisely the amount of aberration given by Peters, but how arrived at is beyond my knowledge. If it be granted that the above method of finding the annual motion of the pole is fairly accurate, there is a AC corollary which is not without interest. Since AM=- — ^ AC it follows that AC=AMxsm A, and that sin A=—^- AM This means that with the annual motion of the pole ascer- tained to be 20"'0529, either the annual precession in de- clination can be found from the star's right ascension, or the right ascension can be found from the precession in decHnation. In the above computation for finding the annual motion of the pole I have for obvious reasons only taken stars within one hour of the equinoxial colure. I may, however, observe AC that the expression AM=-. — ^ applies equally to stars what- ever their amount of right ascension. For example, I take at random the star i^ Centauri, whose right ascension is given in the Nautical Almanac of 1898 as 14 h. 29 m. ls-743. Let it be required to find that star's annual precession in declination from the expression .4(7=.4ilf Xsin A. Log AM . . . 1-3021772 Log sin right ascension 9-9008594 Log precession . . 1-2030366 Whence precession . =15-96014 „ by Nautical Almanac 15-958 Difference . . . 0-00214 21 This difEerence of less than xtt^ts of a second of arc is negli- gible, and its smaUness tends to prove either the accuracy of the Nautical Almanac or that of the ascertained annual motion of the pole, or the accuracy of both. Possibly I shall be told that I have found a mare's nest, and that it has been known all along that the right ascension of a star and its annual precession in declination are functions of the annual motion of the pole, and that such motion can be found ia the Nautical Almanac, and is properly termed aberration. It may be so, but I have failed to find any in- formation on the subject in Herschel's Outlines of Astronomy, or other text-books, or in the explanations in the Nautical Almanac. I venture to think that the term aberration is not a happy one. Some of us may suffer from aberration of intellect ; perhaps I do. My dictionary describes it as ' alienation of mind,' but there is no aberration in the works of the Great Architect of the universe. His laws are precise and immutable. In concluding this section on the annual motion of the pole, I may say that strong reasoning would be required to shake my behef in Drayson's assertion of about forty-five years ago, and confirmed by the French astronomer Flam- marion some thirty years after Drayson, as stated in the Intro- duction of this work : that the one single motion of the pole of the earth is, with certain exceptions, the sole cause of the apparent motion of the stars. Let us hope that this will not be claimed as a French discovery, but be rightly awarded to Drayson. As an additional proof that the one single motion of the pole is the cause of the apparent precession of stars in right ascension and declination, I append* a diagram of the northern hemisphere, copied from an original one of General Drayson's, which purports to show that whilst the centre of second rotation (C) is a fixed point, P, the pole of the earth, has moved on an arc with (7 as a centre from P to P' between the time of the astronomer Ptolemy a.d. 140 and a.d. 1895, and that this motion of the pole has caused the so-called * See diagram at end of book. 22 first point of Aiies to alter its position about 20°, thus giving the stars an apparent motion in right ascension and declina- tion, whilst in truth they have not moved. The effect of this one motion of the pole and of the change in the first point of Aries may perhaps be compared to that of a dial in which the numbers representing the hours (not the dial) have been slowly turning in the direction of the hands of a watch, and so have changed their position about 20° in a period approach- ing twenty centuries. In the diagram referred to the numerous small arcs marked from Z to Z' approximately show the apparent alteration of the right ascension and declination of stars at such points, whereas in truth it is the zenith of such stars, not the stars themselves, which have moved. To be more exact, the curves show the direction rather than the correct amount of the movement, owing to the protraction of a portion of a sphere on a flat surface and the distortion arising therefrom. A reference to the precessional movement in the table of fixed stars in the Nautical Almanac will, I think, show the general accuracy of the apparent motion thus depicted in the diagram. The truth of this appears incontestable when it is considered that the otherwise unaccounted for and variable changes in the positions of myriads of stars are all exactly explained by the one annual motion of the pole of the earth. I am very sensible of the imperfection of my explanation of Drayson's diagram, and can only hope that in the future the subject will be more fully considered and better justice done to him by those more capable than myself. 23 Annual Motion of Pole deduced feom Annual Pre- cession IN Declination in 1898 Nautical Almanac Annual star's Name. Right Ascension. Annual Precession. Motion of Pole of the Earth. h. m. s. u * C^ Aquarii .... 23 4 0-5 19-458 20-0535 y Touoani . 23 11 28-6 19-605 20-0528 y Pisoium . 23 11 52-6 19-613 20-0535 K Pisoium . 23 21 42-2 19-774 200533 B.A.C. 8213 23 27 48-6 19-856 20-0535 : Phcenicis . 23 29 35-3 19-877 20-0533 i. Pisoium . 23 34 42-1 19-931 20-0530 7 Cephei 23 35 9-4 19-946 20-0628 8 Sculptoris 23 43 36-8 20-001 20-0522 27 Pisoium . 23 53 270 20-046 20-0532 a> Pisoium . 23 54 4-3 20-046 20-0527 2Ceti 23 58 30-9 20-052 20-0524 a Andromedse 3 6-8 20-051 20-0528 /3 Cassiopeiae 3 43-9 20051 20-0537 y Pegasi 7 58-9 20-041 20-0532 Ootantis . 12 31-8 20020 20-0500 I Ceti 14 13-8 20-013 20-0516 i Touoani . 14 45-4 20-010 20-0516 44 Pisoium . 20 10-4 19-975 20-0526 ^Hydri . 20 23-8 19-970 20-0494 a Phoeniois 21 14-5 19-965 20-0511 12 Ceti 24 49-9 19-935 200526 f Andromedse 33 9-8 19-842 20-0516 8 Andromedse 33 52-2 19-834 20-0526 a Cassiopeiee 34 42-9 19-824 20-0536 /SCeti 38 28-2 19-772 20-0539 8 Pisoium . 43 23-3 19-694 20-0523 20 Ceti 47 47-6 19-617 20-0514 y CassiopeisB 50 32-9 19-666 20-0517 /I Andromedee 51 5-2 19-656 20-0521 a Soulptoiis 63 41-5 19-503 20-0607 e Pisoium . 57 38-9 19-420 200610 'r\ Ootantis . 11 1-3 19-371 20-0638 4 Ursas Majoiis 11 3 55-8 19-457 20-0542 /3 Crateris . 11 6 38-4 19-511 20-0520 8 Leonis . 11 8 41-1 19-563 200636 24 Annual Right Annual Motion of star's Name. Ascension. Precession. Pole of the Earth. h. m. s. II # 5Leonis .... 11 8 53-3 19-556 20-0526 8 Crateris . 11 14 14-4 19-654 20-0524 r Leonis 11 22 41-5 19-789 20-0541 X Draconis 11 25 211 19-823 20-0517 S Hydra . 11 27 590 19-857 20-0523 X Centauri . 11 31 4-5 19-900 20-0595 V Leonis 11 31 43-5 19-901 20-0534 (3 Leonis 11 48 51-4 20-004 20-0537 j3 Virginis . 11 46 22-9 20-012 20-0528 B.A.C. 4007 11 46 2-7 20-016 20-0532 y Ursse Majoris 11 48 28-0 20-029 20-0544 IT Virginis . 11 55 38-7 20-049 20-0526 o Virginis . 12 0-8 20-052 20-0520 S Centauri . 12 3 4-3 20-051 20-0528 e Corvi 12 4 52-6 20048 20-0525 8 Cruois 12 9 43-7 20-035 20-0531 8 UrssB Majoris 12 10 22-7 20032 20-0526 7 Corvi 12 10 33-6 20-031 20-0523 /3 diamaeleontis 12 12 21-6 20-030 20-0592 B.A.C. 4165 12 14 22-9 20-013 20-0525 I? Virginis . 12 14 41-2 20-011 20-0522 a} Cruois 12 20 55-3 19-970 20-0535 82 Corvi . 12 24 351 19-938 20-0533 y Cruois 12 25 30-4 19-928 20-0521 Corvi 12 29 1-6 19-892 20-0526 a Musose . 12 31 5-9 19-870 20-0543 y Centaxiri . 12 35 53-4 19-807 20-0524 y^ Virginis . 12 36 29-3 19-798 20-0516 p Virginia . 12 36 43-3 19-796 20-0529 Musose . 12 40 1-3 19-750 20-0550 3 Cruois 12 41 45-5 19-720 20-0519 35 Virginis . 12 42 39-7 19-705 20-0514 31 ComeB 12 46 43-8 19-637 20-0524 e UrssB Majoris 12 49 32-5 19-586 20-0527 8 Virginis . 12 50 27-9 19-568 20-0522 a Canum Venat 12 51 15-4 19-553 20-0524 f Virginis . 12 57 59 19-432 20-0511 Mean of the above seventy-three stars • 20-0529 SECTION IV ANNUAL ANGLE, DURATION OF CYCLE, AND ZERO YEAR Having in the foregoing sections briefly explained the manner in which Drayson's primary data were ascertained, viz. the radius of second rotation 29° 25' 47", the distance of the centre of second rotation from the pole of the ecUptic 6°, and the annual motion of the pole 20"'0529, 1 will now proceed to deal with Drayson's secondary data which depend on and are deduced from the primary data, viz. the annual angle at the centre of rotation, the duration of the cycle, and the zero year, i.e. the year in which the pole of the ecliptic and the centre of rotation are aligned, and in which therefore the obliquity of the ecliptic will be at its minimum. First deahng with the annual angle, viz. the angle PCP' in the accompanying diagram, we have annual angle Annual motion ^ sineCP 20"-0529 or sine 29° 25' 47" 1-3021772 9-6913952 Log PCP'= 1-6107820 Whence annual angle =40"-81143 Duration of Cycle If the annual angle at the centre of second rotation be accepted as 40"-81143 as above stated, it follows that one 26 whole period of rotation is thereby determinable, and will be 360° 12 96000'' annual angle °^ 40''-81143 Whence the duration of the cycle=31,755-814, say about 31,756 years. The mean of my six reversed calculations to find Drayson's data made the duration of the cycle 31,777 years, but those calculations have no pretence to precise accuracy (see Section vni.) The Zero Yeae To find the zero year, i.e. the year of minimum obliquity of the ecliptic, is a matter of simple geometry, if Drayson's system and his before-mentioned data be accepted. In the foregoing figure, let C be the centre of second rota- tion, E the pole of the ecliptic, P the pole of the earth on any given year, and let Z represent the position of the pole in the zero year, i.e. when aligned with E and C. Also let be the position of the pole of the earth at the beginning ©f the Christian era. «^ On reference to the diagram it will be manifest that the number of years from P to Z must depend on the angle PCZ=PCE. For example, let it be required to find the zero year from Drayson's data, and from the obliquity recorded in the Nautical Almanac of 1898. In the triangle CPE we have CP=29° 25' 47", CE= 6° 0' 0", and ^P=23° 27' 8"-98 to find the angle C. 23 27 9 6 cosec C 29 25 47 cosec 0-9807654 -y= 2° 14' 56" 58 52 56 0-3086041 29 26 28 sine 9-6915489 C=i° 29' 52" 23 27 9 9-0184124 5 59 19 sine 19-9993308 =269' 52" 9-9996654 T ~G Logcosmcy =16192" 27 Whence zero year=1898+ ^-^ annual angle =1898- 16192" 40"-8114:3 =1898+396-752 =A.D. 2294-75 SECTION V THE OBLIQUITY OF THE ECLIPTIC To find the obliquity for any given year we have in the accompanying figure the two sides CP and CE respectively 29° 25' 47" and 6° 0' 0", and the included angle PCE to find the third side EP, the obUquity. For example, let it be required to find the obliquity in 1900. First find the angle PCE, and transposing the expression to find the zero year (Section iv.) we have angle PCE = (zero year— year a.d.)x annual angle = (2294-75-1900) X 40"-81 143 =16,110"-283=4° 28' 30"-283 With PCE thus found we have two sides and the included angle to find the third side EP tan =ta,n CE. Cos C and ^ „„ cos C^-cos (CP- Cos EP= ^ cos^ 9-0216202 9-9976143 9-9986740 9-9625693 9-0202942 19-9601836 (f) 5°58'54"-63 9-9976288 CP 29 25 47 (CP-cji) 23 26 52 -37 Log ^P 9-9625548 Whence obhquity in 1900=23° 27' 8"-3. 29 In the above case, as in others, I feel that some apology is due to such mathematicians as may honour me by reading this work for having inserted the details of simple trigonometrical formulae, which to them are matters as familiar as A, B, C. My excuse is that I am writing also for the young, whom I hope to entice into a study of the subject. The Angles PCE and CPE As the angle PCE is required in all calculations for finding the obliquity, and as the angle OPE will play an important part when dealing with sidereal time and right ascension, a table of these two angles for the beginning of each century is appended. In the triangle CEP with the two sides CP and CE, known as constants, and the angle C found for any year as described in this section, it wiU be evident that the angle CPE for the same year may be found. I also append to this section a diagram showing the obhquity curve on Drayson's system for the first 2500 years of the Christian era, together with the amount of obliquity and of its corresponding precession of the equinoxes for the beginning of each century. 30 The Angles PGE xsd CPE E Ybab A.D. PGE CPE . . . 26° 0' 62" 6° 25' 45'-46 100 . 24 52 50 -857 6 10 52 -07 200 . 23 44 49 -714 6 55 44 -11 300 . 22 36 48 -571 5 40 21 -90 400 . 21 28 47 -428 5 24 46 -26 500 . 20 20 46 -285 5 8 57 -45 600 . 19 12 45 -142 4 52 56 -16 700 . 18 4 43 -999 4 36 42 -93 800 . 16 56 42 -856 4 20 18 -36 900 . 15 48 41 -713 4 3 43 -08 1000 . 14 40 40 -570 3 46 57 -72 1100 . 13 32 39 -427 3 30 2 -91 1200 . 12 24 38 -284 3 12 59 -37 1300 . 11 16 37 -141 2 55 47 -65 1400 . 10 8 35 -998 2 38 28 -57 1500 . 9 34 -855 2 21 2 -80 1600 . 7 52 33 -712 2 3 31 -06 1700 . 6 44 32 -569 1 45 54 -05 1800 . 5 36 31 -426 1 28 12 -55 1900 . 4 28 30 -283 1 10 27 -28 2000 . 3 20 29 -140 52 38 -99 2100 . 2 12 27 -997 34 48 -45 2200 . 1 4 26 -854 16 66 -43 2300 . 3 34 -289 56 -33 2400 . 1 11 35 -432 18 49 -05 2500 . 2 19 36 -575 36 40 -95 2600 . 3 27 37 -718 54 31 -29 2700 . 4 35 38 -861 1 12 19 -29 MEAN OBLIQUITY OF THE ECLIPTIC ON JANUARY 1st. Calculated for the first 25 Centuries of the Christian Era on General Drayson's System of the 2^^- Rotation of the Pole. ^^ = 1 Tt as s 1 s ' == — 1C 1 23" 25' 30' 35' 40' 45' beginning of each Century ; " 24° 50' 55' 0' 5' Annual Precession )' A.D. 100 200 300 400 500 600 700 800 900 1,000 1,100 1,200 1,300 1,400 1,500 1,600 1,700 1,800 1,900 2,000 2,1t)0 2,200 2,300 2,400 24 24 10 fi '■l;'i ,6.3^ 8"-9 -- ^■ ^6586 4M982 A.D. 100 200 300 400 500 600 700 800 900 1,000 1,100 1,200 1,300 1,400 1,500 1,600 1,700 1,800 1,900 2,000 2,100 2,200 2,300 2,400 Data used in Computation of Obliquity. 24 3 il2i 4-0 9-0 ^ ^ ^ -- 48-8332 2. Distance of Pole of Ecliptic from Centre of znd.Rotation 6°. 3. Annual Motion of Pole of Earth on small circle round Centre of and. Rotation.zo'og. [should.have taken 20-0529]. 23 59 W-7 C3-: 9-0 \ } — 48-9834 23 56 •94-; 26-4 9-0 If ^ ^ 49-0888 4. Year of alignment of Pole of Earth, Pole of Ecliptic and Centre of 2>"l. Rotation A.D. 2295.2. 23 63 21.-1 86-; 9-2 c. ,^ >^ / 49-2092 23 50 .76- 25-0 9-4 ,^ ^ ^ 49-3243 23 47 38-3 ^ 9-6 49-4339 ,23- 45 1-2 57 -i 9-7 49-6379 Year Obliquity .^ > 7^ 23 40 16-2 37- f 9-8 1800 1810 1820 1830 1840 1860 1'860 1870 1880 1890 1900 23 27 65-2 23 27 60-1 23 27 45-1 23 27 40-1 23 27 36-3 23 27 30-8 23 27 26-9 23 27 21-5 23 27 17-fl 23 27 12-9 23 27 8-7 49- 282 ^i y L> 23 38 8-4 .27-S 9-8 49- il41 c y r 23 36 10-4 18-( 10-0 49- J938 . V/ ^ 23 34 22-4 08-C 10-1 49- )669 v / /^ 23 32 44-5 97-9 10-2 50-033B < >/ 7 23 31 16-8 87-7 10-2 50-0933 1 1 1 q / z r- 23 29 59-3 77-5 10-3 Present Decade 60-1464 Year Obliquity 23 28 52-1 67-2 10-3 1890 1891 1892 1B93 1894 1895 1.896 1897 1898 1899 1900 23 27 12-9 23 27 12-5 23 27 12-1 23 27 11-6 23 27 11-2 23 27 10-8 23 27 10-4 23 27 10-0 23 27 9-5 23 27 9-d 23 27 8-7 ~ 60-1926 / 23 27 65£ 56-9 10-4 60-2317 / 23 27 8-7 46-E 10-4 50 i637 / f— 23 26 32'E 36 ■] 10-5 60 ;886 t 23 26 7-0 25-E lO-l 50-^663 1 23 25 51.-i 15-! 10-4 50 il6E 1 1 A D. !29. M ini lun )liq lity 23 2S- 7- 23 25 47-C 4-8 10-£ A D. !29 ^ (Pre gesi um iion 50 !20 23 26 62-' 5-7 10-! 50 ilH 23 26 8-9 16' 60 !04< 2,50025' 30^^ ■ 36' 40' 45' SO' 55' 0' b lu 23° 24" SECTION VI THE PKECESSION OP THE EQUINOXES The yearly amount of precession of the equinoxes seems to have been somewhat uncertain. Herschel appears to have thought it a constant of SO"- 10. But more modern orthodoxy has shown it to be a variable quantity increasing slightly as the obliquity decreases, and vice versa. According to Drayson's system the precession at present is an increasing quantity with a diminishing rate of increase, and will attain its maximum amoimt of about 50"-32 in about a.d. 2295, after which date it will slowly decrease with a slowly increasing rate of decrease. I will now endeavour to explain my method of ascertaining the precession in accordance with Drayson's system. The annual angle of second rotation 40"-8114:3=2s-721, which yearly amount the earth has rotated in opposition (nearly) to the daily rotation (Drayson). Over what arc, then, of the echptic will the earth traverse during 2^-721 ? Taking the precession roughly at 50"'2 the earth travels 360°— 50"-2 during 365 d. 6 h. 9 m. 9-6 s. (=365 d.-25636). Then as 360°-50"-2=1295949"-8 we have 1295949-8 ^ 365-25636 whence the ecliptic travels each day 3548"-0565=59'8"-0565. If the earth travels 59' 8"-0565 during twenty-four hours, what arc — call it X — will it travel over during 2"-721 ? (24 h.=86400s.) 86400 3548"-0565 2-721 ~ X 32 whence in 2s-721 the earth travels 0"-11174, which amount may be considered a constant for 2000 years or more, and should be subtracted from the precession found by the pre- cession triangle ia which PP' is the annual motion of the pole, the angle Q the obliquity, and P'Q the approximate precession. PP' We should thus have precession=^ — -— 0"'11174. This ^ smQ might be correct if the pole travelled in a circle round E, the pole of the ecliptic, as a centre, but as by Drayson's system the annual revolution is round G as a centre, the annual motion must be reduced to the amount it would be if it were direct towards the first point of Aries. To exemplify this : From P in the accompanying figure draw PA at right angles to EP, and therefore towards the first point of Aries draw PB at right angles to CP. Also draw AB at right angles to AP. Let BP represent an exaggeration of the annual motion. Then AP wiU be the reduced annual motion required for computing the precession. As APE and BPC are right angles, and BPE is common to both, the angle APB=CPE, which latter can be found for any year as shown in Section v. Thus in the triangle ABP right-angled at A we have PA=PB-coa P. Whence Precession=-. — t^—. 0"-11174 sm obuquity Annual motion X cosine CPE ^„_ , ^ , » . sin obHquity For example : let it be required to find the precession in 1900 and in 1910 by the above formula, observing that the annual motion 20"'0529 and — 0-11174 are constants. 33 1900. Given N.A. obliquity 23°27' 8"-03 Angle CFH . . 1 10 27 -28 (as found in Section v.) Log annual motion . 1-3021772 Log cos P . . , 9-9999088 1910. Given N.A. obliquity 23°27' 3"-58 Angle CPE . . 1 9 30 (as found in Section v.) Log annual motion . 1-3021772 Log cos P . . . 9-9999112 Log sin obliquity 11-3020860 9-5998660 1-7022200 11-3020884 Log sin obliquity . 9-5998444 1-7022440 50"-37556 - 0-11174 50"- 37836 - 0-11174 Precession Do. by Naut. Almanac Difference . 50-26382 50-2639 0-00008 Precession . . 50-26662 Do. by Naut. Almanac 50-25865 Difference . . 0-00797 A similar calculation for the year 1898 with the Nautical Almanac obliquity 23° 27' 8"-98 and the angle P (computed as shown in Section v.) 1° 10' 48"'6 gives the following result : — Precession by this method 50"-26318 Do. by Nautical Almanac 50 -2633 Difference -00012 Assuming the Nautical Almanac obliquities and preces- sions to b& correct, the above results respectively show only about iogaa » Tinnrj and nmnr of a second of arc different from the Eoyal Observatory figures given in the Nautical Almanac ; this in my opinion is an additional test, and goes far to prove the accuracy of my method of finding the pre- cession for any year, past, present, or future, under Drayson's system. To this section is appended a table which I calculated some ten years ago to show the obliquity and precession for the commencement of each century, and to a.d. 2700 ; also a table of yearly diminution of obliquity from 1851 to 1950 under Drayson's system. These tables will perhaps lead to more accurate investigation, and may be of interest even to those who do not adnait the soundness of Drayson's system. c 34 The data from which the tables are computed are the radius GP 29° 25' 47", the annual motion of the pole 20"-0529, CE the distance between the centre of second rotation and the pole of the ecUptic 6°, and the angle PGE, as found for each year from the above-mentioned data. In the foregoing examples of a method of finding the precession, and also in the table of precessions which follows, the direction of the first point of Aries has been assimied to be at right angles to the pole of the ecUptic as generally- accepted, but as I understand Drayson's system the first point of Aries should be considered to be at right angles to CP, the Une between the pole of the earth and the pole of second rotation, in which case the precessions before found would be respectively for 1898 50"-27383, for 1900 50"-27441, and for 1910 50"-27692. These figures, in my opinion, are likely to be more correct than those which agree with the amounts given in the Nautical Almanac. The difierence between the result of the two systems is about 0"-0106 ia the year 1900. If it be accepted that the first point of Aries is at right angles to the direction of the centre of second rotation, the item cos CPE may be eliminated from the formula for finding the precession, and the expression will then be T, . Annual motion ^„m„, rrecession=-^ =— — : — —0-11174. sme obuqmty The precessions, however, given in the following tables are calculated on what I believe to be the orthodox assumption that the first point of Aries is 90° from the pole of the ecliptic. 35 Obliquity of the Ecliptio and Precession of the Equinoxes at the Beginning op each Century A.D. Obliquity. DifiF. in 100 years. 2iid DifF. Precession. Diff. in 100 yrs. 2nd Diff. 24° 10' 7".! 220''-5 48''-5586 •1396 100 24 6 26 -6 211 -6 8"-9 48 -6982 •1350 ■0046 200 24 2 55-0 202-5 91 48 -8332 ■1302 ■0048 300 23 59 32-5 193 -6 8 -9 48 -9634 •1254 •0048 400 23 56 18 -9 184 -7 8 -9 49 -0888 •1204 •0050 500 23 53 14 -2 175 -3 9 -4 49 -2092 •1151 ■0053 600 23 50 18 -9 166-0 9 -3 49 -3243 •1096 ■0055 700 23 47 32-9 166 -5 9 -5 49 -4339 ■1040 •0056 800 23 44 66-4 146 -8 9 -7 49 -5379 ■0981 •0059 900 23 42 29 -6 137 -3 9 -5 49 -6360 •0922 ■0059 1000 23 40 12 -3 127 -4 9 -9 49 -7282 •0859 ■0063 1100 23 38 4-9 117 -5 9 -9 49 -8141 • •0797 ■0062 1200 23 36 7 -4 107 -4 10-1 49 -8938 ■0731 ■0066 1300 23 34 20-0 97 -6 9 -8 49 -9669 ■0667 ■0064 1400 23 32 42 -4 87-3 10-3 50 -0336 ■0597 ■0070 1500 23 31 15 1 77 -0 10 -3 50 -0933 ■0531 ■0065 1600 23 29 58 -1 66 -9 10-1 50 -1464 •0461 •0070 1700 23 28 51 -2 56 -8 10-1 50 -1925 ■0392 •0069 1800 23 27 54 -4 46 -1 10-7 50 -2317 ■0320 •0072 1900 23 27 8 -3 36-0 10-1 50 -2637 •0249 ■0071 2000 23 26 32 -3 25 -4 10 -6 50 -2886 ■0177 •0072 2100 23 26 6 -9 15 -2 10 -2 50 -3063 ■0105 •0072 2200 23 25 51 -7 4 -7 10 -5 50 -3168 ■0033 •0072 36 Obliquity of the Ecliptic and Precession op the Equinoxes at the Beginning of each Century — contd. A.D. Obliquity. 1 Diff. in [ 100 years. 2nd Diff. Precession, Diff. in 100 yrs. 2nd Diff. 2300 23° 25' iV-O : 5''-9 10"-6 60"-3201 •0041 •0074 2400 23 25 52 -9 i 16 -0 10 -1 50 -3160 •0111 •0070 2500 23 26 8 -9 26 -8 10 -8 50 -3049 •0186 •0075 2600 23 26 35 -7 , 37 -0 10-2 50 -2863 •0257 •0071 2700 23 27 12 -7 60 -2606 •00 In the above table some irregularity will be observed in the columns headed Second Difference, which, of course, should not be ; the reason for such irregularity in calculations wiU be obvious. Annual Difference op Obliquity from 1851 to 1950 Year Annual Year Annual Year Annual Year Annual A.D. Diff. A.D. Diff. A.D. Diff. A.D. Diff. 1851 0''^458 1876 0"-435 1901 0"^411 1926 0''^385 1852 •457 1877 ■434 1902 •410 1927 •384 1853 •456 1878 •433 1903 •409 1928 •383 1854 •456 1879 ■432 1904 •408 1929 •382 1855 •455 1880 •431 1905 •407 1930 •380 1856 •454 1881 •430 1906 •406 1931 •379 1857 •453 1882 ■429 1907 •405 1932 •378 1858 •452 1883 •428 1908 •404 1933 •377 1859 •451 1884 •427 1909 •403 1934 •376 1860 •450 1885 •426 1910 •402 1935 •376 1861 •449 1886 ■425 1911 •401 1936 •374 1862 •448 1887 •424 1912 •400 1937 ■373 1863 •447 1888 •423 1913 •399 1938 •372 1864 •447 1889 •422 1914 •398 1939 •371 1865 •446 1890 •421 1916 •397 1940 •369 1866 •445 1891 •421 1916 •395 1941 •368 1867 •444 1892 •420 1917 •394 1942 •367 1868 •443 1893 •419 1918 •393 1943 •366 1869 •442 1894 •418 1919 •392 1944 •366 1870 •441 1895 •417 1920 •391 1945 •364 1871 •440 1896 •416 1921 •390 1946 •363 1872 •439 1897 •415 1922 •389 1947 •362 1873 ■438 1898 •414 1923 •388 1948 •361 1874 •437 1899 •413 1924 •387 1949 •360 1875 ■436 1900 •412 1825 •386 1950 •359 SECTION VII STANDARD OF TIME AND RIGHT ASCENSION Before considering the question of time, I may perhaps, with the garrulousness of age, take permission to make a short dissertation. Some sixty-four years ago, when frozen up at Fort Vancouver, about a hundred miles up the Columbia River, and with plenty of time for thought, I argued with my young brother officers, now all dead, that the confusion of time with longitude was a mistake, and that, as a lapse of time is measured by the number of beats of a pendulum of a certain length in a certain latitude, time must go on whether of no the observer changes his longitude on the earth, and that consequently there should be one and the same time at all places. An hour of time means a lapse of 3600", and that time will have elapsed whether the observer has moved or is stationary. I am much interested to observe in the Press that it is proposed to adopt the same time as at Greenwich over a large part of the Continent — a change which would greatly add to the convenience of the traveUing pubUc, and which gives me hope that in the distant future the day will come when a watch pointing to twelve hours will represent twelve o'clock all over the civihsed world. Although not bearing directly on the subject of time, it may not be out of place to remind my young readers with regard to cardinal points that there is no east and there is no west in the solar system or in the universe. The terms east and west are merely planetary, and refer only to the direction in which an observer on a planet can note the apparent rising and setting of the heavenly bodies. There is a north and there is a south, so called from the direction 38 of the respective poles of the earth produced to the heavenly sphere. But just as an observer standing at one of the poles would have no east or west, so in the heavenly sphere such points are non-existent. Coming to the question of sidereal time, the ecliptic in a terrestrial globe is usually shown to cross the equator in the meridian of Greenwich ; but this has no particidar signi- fication, as the ecliptic must be shown to cross the equator somewhere. So far as I have learnt, the zero meridian for right ascension is that meridian which passes through a point on the heavenly sphere which would be indicated by a straight line from the centre of the earth through the centre of the sun at the instant of the sun's polar distance being 90° at the vernal equinox. This poiat on the heavenly sphere is called the first point of Aries, and was so termed because it was situated some centuries ago at the beginning of the sign of the zodiac called Aries. The first point of Aries is now, I beUeve, about 20° or more from that position, and has an annual and varjdng motion. The above description of the zero point of annual sidereal time may or may not be correct, as I have no pretence to be an authority on the subject. According to Drayson's system the axis of the earth changes its inclination about 20"-0529 in each year on a smaU circle distant 29° 25' 47" from G, the centre of second rotation. It can be shown by a diagram that if at the vernal equinox in any year the centres of the earth and sun and a fixed point in the heavens, such as a distant star, were aligned, at the next year's vernal equinox the same distant star would not be aligned. This is because the change of about 20"-0529 in the direction of the earth's axis would by that movement alter the condition that the sun's polar distance shall be 90° at the vernal equinox. The practical result of the alteration in the inclination of the earth's axis is that the condition of 90° polar distance will be fulfilled, and therefore the vernal equinox occur a little before the alignment of the distant star. This difference in the position of the first point of Aries, termed precession of the equinoxes, is a quantity which varies 39 with the obliquity of the ecliptic, and according to the amount of the angle CPE in the accompanying diagram, as described in Section vi., C, E and P being respectively the centres of second rotation, of the pole of the ecliptic, and the pole of the earth. With regard to the foregoing definition of the zero of sidereal time, I have no remark to make, as I am not practically acquainted with the manner in which our dis- tinguished astronomers, with their very accurate instruments, set their imaginary sidereal clock at for the beginning of each sidereal year. Nor does the accurate setting of the clock concern the matter with which I am dealing, which is the rate of the clock, not its error. But seeing the addition of more than 2/"^ of imaginary time in the year 1834, it is reasonable to think that there had been an error, and that another error may be accumulating by means of a shghtly incorrect rate. It is fair to assume that the Astronomer Royal of the day in 1834 knew his business, and would not have made so important a change of sidereal time as an addition of 3"i 3^-68 without sound reason for so doing. Be it also observed that the addition was not a mere rough amount of minutes, but was stated with an accuracy purporting to be to the hundredth part of a second. Assuming therefore that the time of the vernal equinox was correctly ascertained in 1834, it is not without interest to note that the determination of Mr. Stone — then EadcHfEe Observer at Oxford— that an error of time in 1892 had amounted to 4ls-51 agrees within a small fraction of a second with the error 4ls-283 ascertained by Drayson's system, viz. the accumulation of about 0^-71 per annum since 1834. This simihtude may possibly be a coincidence, or Mr. Stone may have been in error, but other- wise it appears to be in some measure an additional evidence of Drayson's system of second rotation. According to Dray- son this slight annual error in sidereal time is in consequence 40 of the assumption that the pole of the earth travels round the pole of the ecliptic in a period of some 25,000 years, instead of, as he affirms, roimd the centre of rotation in some 30,000 years, and says that for a correct standard of time it should be measured with C as a centre. If in the diagram P and P' be the positions of the pole of the earth in two consecutive years, the annual error of rate will be the difference between the angles CPE and CP'E, which in 1910 amounts to 10"-66, or in time Os-71143. On the subject of a standard measure of time I write with some diffidence, and prefer where practicable to quote Drayson's own words. In a B.erS64t letter to me dated 3rd January 1894, Drayson gives me the above diagram, referring to the date about 5644 B.C., in which P is the position of the pole at date 5644 B.C., C the pole of the second rota- tion, PO=29° 25' 47", E true position of pole of the ecliptic, 0^=6°, and EGP=90°. ' When the pole of the earth is at P, a daily rotation will cause C to transit before E by an interval of time measured by the angle EPC. ' When the pole has been carried to P', a daily rotation will cause C to transit before E by an interval of time measured by the angle EP'C, and when the pole reaches Z a.d. 2295 and E will transit simultaneously. ' The successive transits of G, the centre of the circle, give a uniform measure or standard of time. The successive tran- sits of E, therefore, give a variable measure of time, which also has a variable rate depending on the variation in the angle EPC as the pole is carried round from 5644 B.C. to A.D. 2295. ' The intersection of the equator and ecliptic giving the equinoctial point (first of Aries) is the point from which right ascensions are counted, and as the pole is not moving at 41 right angles to the arc joining it with the pole of the ecKptic, the successive transits of the pole of the ecliptic do not corre- spond with the successive transits of C. ' The variation in the angle EP'C from year to year and from century to century becomes therefore a very important item, because the successive transits of C give a uniform standard of time ; hence the successive transits of E give less than a uniform standard of time, and the measure of right ascension is affected thereby. ' The change in the angle EP'C from century to century can be easily calculated as follows :— CJ5=6°, OP'=29° 25' 47", and the angle at C= (2295-2— date) x 40"-9. Hence with two sides and the included angle the angle EP'C can be foimd. ' Thus for 1st January 1887 this angle=l° 12' 58" 1787 „ „ = 1 30 45 'Difference for a hundred years, 1787 to 1887 =0 17 47 that is, 1067" for that century, or 10"-67 per year. The rate will not be quite uniform, but near enough to uniformity for practical use. ' Now for the application of this fact. A date must be fixed on as a zero, and I fixed on 1st January 1887, the Jubilee year, therefore easily remembered. The correction of 10"'67 for the erroneous standard of time will be minus for dates previous to 1887 and plus for dates after 1887. Example I ' Let it be required to find the right ascension of /8 UrssB Minoris for the 1st January 1818. ' Having calculated C/3 as before shown, we have PC=29° 25' 47", P^ the polar distance 15° 6' 2"-4 as per Nautical Almanac of 1818, and 0/3 21° 50' 12". Hence in the I accompanying diagram we have PC, P/S, and C/3 to find the angle P, which will be found to be 46° 57' 35"-2, or in time 3 h. 7 m. 50-4 s. As this star is in the third quarter of right ascension the 42 angle P must be subtracted from 18 hours, which gives 14 h. 52 m. 9-6 s. as the approximate right ascension. ' Between 1818 and 1887 are 69 years to be corrected for erroneous standard of time at 10"-67 per year. 10"-67x69=12' 16"=49s-l in time. 14 h. 52 m. 9-6 s. -49-1 By calculation . 14 51 20-5 EecordedN.A. . 14 51 20-69 0-19 difference in 69 years. Example II ' To find the right ascension of ^ Draconis on 1st January 1895. ' Given OP=29° 25' 47", 0/8=9° 17' 38", and P/3 the polar distance in 1895=37° 37' 15". (See diagram in Example i.) ' With the three sides find the angle CP^, which wiU be found to be 8° 0' 32". 8° 0' 32"=0 h. 32 m. 2-1 s. Take from . . 18 17 27 57-9 Between 1887 and 1895=8 years. 10"-67x8 years=l' 25"-36=5s-69 in time. Brought down . 17 h. 27 m. 57-9 s. Correction 17 17 28 28 +5-69 3-59 R.A. by calculation, 3-55 R.A. byN.A. 0-04 difference.' About the year 1900 I received from General Drayson the following further explanation of his system relating to a correct standard of sidereal time : — 43 Standard of Time and Eight Ascension ' Without entering into a long and elaborate explanation of the standard of time, I may state that astronomers have assumed th&,t they have a true standard of time by aid of a knowledge of one complete rotation of the earth on its axis. ' But, what is a complete rotation of the earth on its axis, and how can it be measured ? In consequence of the second rotation of the earth, every star within the circle described by the pole will transit once oftener during an entire revolution of the equinoxes than will those stars outside this circle. Hence it follows that the only true and uniform standard of time which represents a true rotation of the earth is that point in the heavens at which the pole of second rotation is located. As the position of this point has been hitherto unknown, there exists at the present time a slight error in the supposed rotation of the earth amounting to about O^-Tll per year. ' The right ascension of stars is the term used to define the interval of time shoivn by a chronometer between the transit of a zero point in the heavens and the transit of the star. ' Any error as small as O^-Tll per year might go on accumu- lating during several years, and might then be adjusted at all the observatories without the outside pubhc knowing any- thing about it. Such was the case between the years 1833 and 1834, when 3™ 3s-68 of purely imaginary time was inserted to balance accounts. Outsiders, unless possessing an observatory, and having the best instruments, chrono- meters, etc., have no means of discovering the right ascension of the stars other than accepting that which is given in the Nautical Almanac. A person may know there is an error in the standard measure of time, and yet for calculation he must take the data given in the Nautical Almanac in order to work out details. For example — ' I selected the date 1887 as a zero, and in the Nautical Almanac, 1887, 1 found that the star a Draconis had E.A. 14 h. 1 m. 19-743 s. Decn. N. 64° 54' 57"-84. 44 ' I knew that P(7=29° 25' 47". From Nautical Almanac I obtained Pa and the angle CPa, and from this data I could calculate Ca and the angle POa. And hence, as I have proved, I can calculate the N.P.D. and assigned R.A. of this and other stars for a hundred years and more. ' But the interval of time of transit between a and C is measured year after year by a chronometer which does not give the exact value of the earth's rotation, and hence there is an accumulating error which for any other date, say 1900, will give a slightly different value for the angle PCa and the side Ca. Yet by making 1900 the zero the same accurate results can be obtained. Example — Nautical Almanac, 1900, a Draconis. R.A. 14 h. 1 m. 40-88 s. N. Decn. 64° 51' 13"-35. ■ From this Nautical Almanac data I obtain the following : L CPa=59° 34' 46"-8 \ Pa=2o 8 46'65 CP=29 25 47. ' From this Nautical Almanac data I find by calculation that Oa=26° 36' 9"-l and L 0=54° 54' 54" for 1900-0. ' These two values differ slightly from the values found for the zero 1887, because the standard measure of time between 1887 and 1900 used by astronomers is in error. ' I now have my constants for 1900-0, viz. PC=29° 25' 47", Ca=26° 36' 9"-l, and i. C=54° 54' 54". For a Draconis (see p. 54, Untrodden Ground) I have taken the angle C to vary 40"-9 per year, and I now calculate the N.P.D. and R.A. of a Draconis for 1755. From zero 1900-0. ' Between 1755 and 1900 there are one hundred and forty- five years. 145x40-"9=l° 38' 50"-5. Angle C in 1900= 54° 54' 54", angle C in 1755 .-.=53° 16' 3"-5. ' From above calculate N.P.D. a Draconis 1755 : — Lcos 53''16'3"-5 = 9-7767577 itan26 36 9 •1=9-6996798 16°40'28" tan =9.4:764375 45 + 29°25'47' -16 40 28 12 4519 cos 26'36'9"-l =9-9514028 cos =9-9891480 19-9405508 -cosl6°40'28" 9-9813431 N.P.D. 1755=24°26'47"-2=cos 9-9592077 1755 by Bradley. 24 26 47 -4. For angle at P 1755. Hence Eight Ascension 1755. Sin53°16' 3"-5 = 9-9038698 Sin 26 36 9-1 = 9-6510826 19-5549524 -Sin 24 26 47 -2 = 9-6168351 h. h. m. B. h. m. s. 18 -4 3 32-3 =13 59 27-7 I 145 years xO"-711 = l'43"-09 9-9381173 sin 60''8'4"= 4 h. Om. 32-3 s. 1. s. 9 27-7 Deduct . . - 1 43-09 | 13 57 44-61 E.A. 1755 by calculation. By Bradley . 13 57 46-3 Difference . 1-69 for 145 years. ' From this calculation I think it will be made clear that there is nothing wonderful or special about a.d. 1887. It is imfortunate that by the erroneous standard of time now used I cannot, from recorded observations of R.A. at various dates, obtain always the same value for Ca. I therefore cannot tell exactly, and have no means of teUing, what is the exact distance of Ga, as I am dependent on the astronomer's clock for the assigned E.A. at various dates. I should hesitate to determine the values of Ca and L C from records previous to ■:,_c. 2 0°-6 108°-8 Ceti Deneb Kaitos 110° 46' s'-g 2 1 -3 1 -5 a Ursss Minoris Polaris 29 52 51 1 5 -1 98 -3 |8 Orionis Rigel 126 54 8 -7 1-2 7 -6 61 -6 j3 Geminorum PoUux 88 52 53 -5 4 8 -9 77 -5 a Cancri . . 99 30 25 4 9 -8 63 -3 fi Leonis . . 81 32 26 2 H -0 27 -4 a Ursse Majoris Dubhe 44 48 33 3-4 11 -4 19 -8 a Draoonis . 37 35 41 2-3 12 -5 112 -6 (3 Corvi . . 106 19 52 -2 3-2 12 -9 78 -2 e Virginia . 73 33 -7 1 13 -3 100 -4 a Virginis Spica 89 46 33 -3 2 13 -7 39 -9 ij Ursse Majoris Benetuach 36 31 5 -5 3-4 14 -0 24 -9 a Draoonis 26 37 4 2-3 14 -8 105 -4 u Librae Zubenesch 85 32 36 -4 2 14 -9 15-2 (3 Ursse Minoris Koohab 21 50 12 2 15 -2 81 -2 /3 Librae . 77 5 20 -8 1-2 16 -4 16 -1 a Scorpii Antares 89 4 17 -1 4-5 16 -6 7 -7 e Ursse Minoris . 22 1 47 -7 3-4 17 -2 75 -4 a' Herculis Ras Algethi 46 57 32 -7 3-2 17 -5 37 -6 ^ Draoonis 9 17 38 3-4 19 -3 87 -2 8 AqnilsB . 59 39 20 6-7 19 -9 1 1 X Ursse Minoris 28 29 2 -7 3-4 20-2 103 a' Caprioomi Secunda Girdi 78 1 9 -7 2-1 20 -6 45 -3 u Cygni Deneb 27 55 7 -5 50 Difference between Orthodox and Dbayson's Time Error in Time and R.A. Year A.D. Angle CPE. per century. Annual Error. Diff. In Are. In Time. sees. sees. sees. 1500 2° 21' 2"-8 17' 3r-74 70-117 0-70117 •0 1600 2 3 31 06 17 37 01 70-468 0-70468 -00351 1700 1 45 54 -05 17 41 -50 70-766 0-70766 •00298 1800 1 28 12 -55 17 45 -27 71018 0-71018 -00252 1900 1 10 27 -28 17 48 -29 71-219 0-71219 •00201 2000 52 38 -99 17 50 -54 71-369 0-71369 -00150 2100 34 48 -45 17 52 -02 71-468 0-71468 •00099 2200 16 66 -43 16 56 -43 67-762 0-71517 •00049 2294-75 (In 94-75 years) 0-71520 The Angle PCZ or PCE at the following Periods A.D. 1500 . 9° 0' 34"-87 2000 . 3° 20' 29"-17 1600 . 7 52 33 -74 2100 . 2 12 28 -02 1700 . 6 44 32 ^59 2200 . 1 4 26 -88 1800 . 5 36 31 -45 2294^75 1900 . 4 28 30 ^31 51 The following diagram of Drayson's cycle of second rotation and accompanying table are intended to show the period A.D.9286-57,2E A.D. 2294'7S IB-C. 13583-^ |a.D. iSlji'SS at which the angle CPE and its annual difference will respec- tively attain their maximum and minimum, the latter — the annual difference— being the principal factor for obtaining a standard rate of time. The diagram is based upon the following data : — PC=29° 25' 47", CE=6°, the annual angle at O=40"-81143, and the zero year of minimum obliquity a.d. 2294-75. Principal Periods op Cycle Diff. of 0P£ per year. rnv Obliquity. CPE. In Arc. In Time. 47" 0° 0' 0" 10" -728 0»-7152 increasing to decreas ing to B.O. 4oaruv^ yu IM 10 VO'ii. ■za 01 55-3 12 16 57-62 decreasing to increas ing to A.D. 2294-75 180 23 25 47 increasing to 10-728 decreas 0-7152 ing to A.D. 9286-572 90 79 15 46-21 28 61 55-3 12 16 67-62 decreasing to increas ing to A,D. 18172-65 180 35 25 47 10-728 0-7152 62 Annual Difference (m Time) of Angle CPE foe each Decennial Period from 1500 to 2400 a.d. Year Annual Year Annual Year Annual A.D. Diff. A.D. Diff. A.D. Diff. sees. sees. sees. 1500 0-69880 1810 0-70920 2120 0-71445 10 930 20 945 30 453 20 978 30 969 40 461 30 •70025 40 993 50 468 40 070 50 0-71016 60 475 50 114 60 039 70 481 60 156 70 061 80 487 70 196 80 082 90 492 80 234 90 103 2200 497 90 270 1900 123 10 502 1600 304 10 143 20 506 10 337 20 162 30 510 20 370 30 181 40 613 30 403 40 200 60 616 40 436 50 218 60 618 50 ''469 60 236 70 619 60 501 70 253 80 520 70 532 80 270 90 520 80 562 90 286 2300 520 90 592 2000 301 10 520 1700 621 10 316 20 519 10 650 20 330 30 518 20 678 30 343 40 516 30 706 40 356 50 513 40 734 50 369 60 509 50 762 60 382 70 505 60 790 70 394 80 501 70 817 80 406 90 496 80 843 90 417 2400 491 90 869 2100 427 1800 895 10 436 53 Annual Diffebence (in Time) op Angle CPE fob EACH Year from 1850 to 1950 a.d. Year Annual Year Annual Year Annual Year Annual A.D. Diff. A.D. Diff. A.D. Diff. A.D. Diff. sees. sees. sees. sees. 1850 0-71016 1875 0-71072 1900 0-71123 1925 0-71171 1 18 6 74 1 25 6 73 2 21 7 76 2 27 7 75 3 23 8 78 3 29 8 77 4 25 9 80 4 31 9 79 5 28 1880 82 5 33 1930 81 6 30 1 84 6 35 1 83 7 32 2 86 7 37 2 85 8 35 3 89 8 39 3 87 9 37 4 91 9 41 4 88 1860 39 5 93 1910 43 5 90 1 42 6 95 1 46 6 92 2 44 7 97 2 47 7 94 3 46 8 99 3 49 8 96 4 48 9 0-71101 4 50 9 98 5 50 1890 3 5 52 1940 0-71200 6 53 1 5 6 64 1 2 7 55 2 7 7 66 2 3 8 57 3 9 8 68 3 5 9 59 4 11 9 60 4 7 1870 61 5 13 1920 62 5 9 1 63 6 15 1 64 6 11 2 65 7 17 2 66 7 12 3 68 8 19 3 68 8 14 4 70 9 21 4 69 9 1950 16 18 SECTION VIII PROBLEM TO FIND DRAYSON'S DATA FROM RECORDED ORTHODOX OBSERVATIONS In Section ii. I have endeavoured to explain how and by what means Drayson ascertained his foundation data, espe- cially the amount of the arc GP, viz. the distance between the pole of the earth and the centre of second rotation, and CE, the distance between the centre of rotation and the pole of the ecliptic. The above two arcs Drayson, by trial and error and possibly by somewhat rude methods, ascertained to be respectively 29° 25' 47" and 6° 0' 0". But whilst fixing the above amounts to the nearest second as the best approxi- mation he could attain to the truth, by methods to which I cannot pretend to do justice without extensive references to his published works, he in later years admitted to me the possibiKty of those distances being more accurately deter- mined and, I think, limited himself to a firm conviction that the distance between the pole of the earth and the centre of second rotation must be between 29° and 30°. It occurred to me in 1894 that if by Drayson's data the obliquity of the ecliptic, the right ascension and declination of stars and other items can be found with much accuracy for any year, past, present, or future, it should follow that by reversing the process Drayson's data can be obtainable from Nautical Almanac and other recognised authorities. This I have endeavoured to do by six somewhat complicated calculations with the following stars — a Draconis, a Ursse Minoris, /3 Orionis, and a Virginis, the results of which are given below in tabulated form. In that table it will be seen that, although the results vary, they all six point to a fairly 55 close approximation to Drayson's data, and the mean of the six calculations appears to me to be of some value. The six computations, although worked to decimals of a second, have no pretence to minute accuracy, which is unattainable with logarithms of not more than seven decimal places, and owing to the use of cotangents and cosines of very small arcs. The interval also between recorded observations is insufficient for the method adopted — no reliable records being, so far as I know, obtainable further back than 1820. In addition to the table, I have added as an example the working in one of the computations, viz. that of jS Orionis. I have said that the data used are orthodox, and obtained from the best records. There is, however, one small item which is not orthodox, viz. the correction of about (P-7 per annum correction of sidereal time applied to the Greenwich right ascensions to make them equal the angle CP star in the diagram. Some mathematical critic who may honour me by examining my figures may at once say that that unorthodox correction of time to some extent vitiates the method adopted, I admit the justice of such criticism, but in defence of my method I must plead a firm conviction that such error in the rate of sidereal time does exist, and that, viewing the correc- tion which was found necessary in 1834, there is strong ground for believing that a fresh error has accumulated. Without allowing for that error it would have been impossible to triangulate the problem, as the angle CP star would not have correctly represented the right ascension of such star. 56 H P P3 o 00 » o FN 51^ IZi o . a ^ a 00 S ^ %< ft ^ 6 a OS " "^ fi O W CQ Q -< o !zi 1^ 12! O 03 iz; ►:; o m §-" S >< ^^ cd o 0? g ;^ . . ^ M 1^ Eh „ h5 go o <3 W P5 -d t.: ja '*' ffl *" rt' i . " ^^3 a oma S £ » O M rl fl ■« .2 H S"a>S.S ■9E "So g, i |,S,> n' ■Eg-SOSOBSS O TO 00 U5 00 O O to (N l> 00 t- I> l> 05 X CD i-H i-H I— < F— I I— ( ft CO CO M CO M CO ■<# p CO Op 50 p CO t> CO lO O CJ I-H I-H (N (N O I-H CO CO CO CO CO CO (N IN (N Tt* CD t> O t' r- o Oi uD O 00 00 00 lO CD OS b _ o o o o o © O O OS O O !M !N 0 -* IN lO CO CD CD CD lO CO lO rt IN ..9 .2 J. 3 .3 g g g .g S> £? Ci B CS Ca 8 B ■* , OS lO I IN O I o >-H I ■* ■? o"S(l. ^ w ^ .9 I" ™ ^ 5 * li n o M a o -e S •iH oa B 43 « " dSH DQ lis :n .™ tl 'O a S o © ^ s ca '^ Hf POO) o cd CS -, S c8 fl.S o -2 a f^ o g O J3 otj H^ a fl Hi Lj CQ O 00 _- a -u 1^ i» %%^ ^ CO tiH 'O O O 3 El 03 O ,«> ;^ in ■a 9 O ID 9 0) °-a S 3 o - » S a S S U 00 .a-H ■S a .S§ .s§ « g na . -^3 OOi ® ®2 "o I ja *5Hr-J (3 O eSkH ^ t- _ ^ ©rJ3 -ta rrt -U O M ,^^ ^ 4a o S g & Ki T3 V> I-H S^ > - •2 o a feOh tit rH CS V 57 Data Problem with /3 Obionis As an example of the method I adopted to test Drayson's data, I give the following extract from my work-book of 1894-5, using two recorded positions of /3 Orionis (Eigel) in the southern hemisphere at an interval of seventy-five years between them, and an interval of fifty-five years between two recorded obliquities. Given : — R.A. 1820, Nautical Almanac . 5h. 5m. 53-56s. South Polar Distance 1820, N.A. . 81° 35' 0"-2 R.A. 1895, N.A 5h. 9m. 29458s. South Polar Distance 1895, N.A. . 81° 40' 36"-32 ObHquity 1840, Sir G. Airy . . 23° 27' 36"-5 Obhquity 1895, N.A. . . . 23° 27' 10"-41 Correction of right ascension for annual error 0S'711 of sidereal time. From the above data it is required to find : — 1. CP radius of second rotation. 2. CE pole of echptic to centre of second rotation. 3. Annual motion of pole (A.M.). 4. Annual angle at centre of second rotation (A.A.). 5. Zero year. Alignment of G and E and year of mini- mum obhquity. 6. Duration of complete cycle. 7. Accumulated difference or error of sidereal time from 1834 to 1892. 58 To find CPp. R.A. 5h. 5m. 53-56s. + 47-68* 5 6 6 41-24 CP/3 53 18-76 * Correction. 10".6747 x .67 years. = 11'55".205 = 478-68 To find GP'^. R.A. 5h. 9m. 29-458s. 5-698* 23-76 GP'^Q 50 36-24 * 10".683 X 8 years. = l'25"-464 = 58-698 To find Radius GP. In the A CP^ (1) CosC/3=°.2lilSSE^i:Z^ where tan .^^tanP^- cos CPfi cos

) cos <^ Cos P/3 9-1654516 ^=^G as found above=81°21'14"-01 Cos (PC-(/>) 9-7899399 29 24 56 -15 18-9553915 Cos<^ 9-1770486 Cos 0/3 9-7783429 To find PO/3. Sine PC;8 _ sine P/3 Sine OPyS" sine 0/3 9-9952973 9-3627240 19-3580213 9-9029822 9-4550391 16° 33' 59"-3 .-. P0y8= 163 26 -7 (^_P(7) 51 56 17 -86 Whence 0/3=53 6 40-1 To find P'CP. Sine FO/3 _ sine P';g Sine CP'^ sine CP 9-9954014 9-3404676 19-3358690 9-9029822 9-4328868 15° 43' 14"-6 180 P'0/8= 164 16 45 -4 To find annual angle (A.A.) at 0. P'C^-PC^=^PCP' P'CP 164°16'45"-4 PC^ 163 26 -7 PCP'=^ 50 44 -7 A.A.= 3044"-7 75 years = 3044"-7 Whence A.A.=40"-596. 3-4835445 1-8750613 1-6084832 To find annual motion (A.M.) of pole on small circle. A.M.=A.A.x sine OP Log A.A. 1-6084832 Log sine CP 9-6912060 .-. A.M.=19"-93835 1-2996892 61 To find duration of cycle. 360° 1296000" Cycle= A.A. 40"-596 6-1126050 1-6084832 4-5041218 .-. Cycle=31924-33 years. To find CE and zero year from above results and from obliquities of 1840 (Sir G. Airy) and 1895 (Nautical Almanac). Given CP=CP' 29° 24' 56"-15 EP 23 27 36 -5 ^P'23 27 10-41 PGP'=40"-596x 55 years =37' 12"-7 First find PP' on great circle. (Shortrede, Case II.) Cos PGP' 9-9999746 Tan PC 9-7511480 Cos PC 9-9400581 Tan P'C 9-7511480 Cos P'C 9-9400581 Log X 1-5022706 Log {x+1) 0-1198776 .-. x= 0-317885366 Cos PP' 9-9999938 x+1 1-317885366 From 0° 18' 18" To 18 26 (Say) PP' on great circle =0 18 22 Next find EPP'. (Shortrede, Case I.) Cos EP' 9-9625528 Sec^P 0-0374709 Co tan EP 10-3625249 Sec PP' 0-0000062 Co tan ^P' 12-2722394 Log X 0-0000299 Log (x-1) 5-8388491 .-. x= 1-000069 Cos EPP' 8-4736134 x-1 0-000069 .-. EPP'== 88° 17' 40"-93 62 Next find CPP' and CPE. CP' 29° 24' 56"-15 CP 29 24 56 -15 Cosec 0-3087940 PP' 18 22 Cosec 2-2722456 59 8 14-3 29 34 7 -15 Sine 9-6932573 29 24 56 -15 9 11-0 Sine 7-4267259 19-7010228 Cos 9-8505114 .•.^r=«° 51' 51"-4 CPP' 89 43 42.8 EPP' 88 17 40-93 .-.CPE^ 1 26 1-87 To find CE. (Shortrede, Case II.) Cos CPE 9-9998640 Tan CP 9-7511480 Cos CP 9-9400581 T&nEP ' 9-6374751 Cos EP 9-9625291 Log X 1-3884871 Log(a!+l) 0-0950358 .'. X— 0-24461722 Cos CE 9-9976230 (^+1) 1-24461722 .-. CE:=5° 59' 21" To find PCE {=PCZ) when P is at a.d. 1840. foregoing Diagram. See last PE 23° 27' 36"-5 PC 29 24 56 -15 Cosec 0-3087940 CE 5 59 21 58 51 53 -65 Cosec 0-9815474 29 25 56 -82 Sine 9-6914326 23 27 36 -5 5 58 20 -32 Sine 9-0172330 19-9990070 Cosine 9-9495035 63 Whence ^^==2° W 21" ^ .-. PCE =5 28 42 Now to find zero year. PCE=h° 28' 42"= 19722" No. of years P to Z^ ^^^ ^^^^^" A.A.""40"-596 P0£^ 4-2949510 A.A. 1-6084832 2-6864678 .-.Years PtoZ= 485-81 Do. A.D. OtoP=1840 Zero year a.d. 2325-81 To find the accumulated difference or error of the sidereal time of mean noon in 1892, assuming that the addition of 3"! 3S-68 in 1834 was correct, and that no subsequent correc- tion has been made. The data for this purpose are those found in the preceding calculation relative to yS Orionis. Given CP 29° 24' 56"-15 A.A. at C 40"-596 CE 5° 59' 21" Zero year 2325-81 1834 To find PGE. Pa^= (2325-81 - 1834) x 40"-596 = 491-81 x40"-596 2-6917974 1-6084832 4-3002806 Whence PGE= 19965"-52 = 332'45"-52 = 5°32'45"-52 1892 To find PGE. P'C'£= (2325-81 - 1892) x 40"-596 = 433-81 X 40"-596 2-6372996 1-6084832 4-2457828 Whence PCE= 17610"-95 = 293'30"-95 = 4°53'30"-95 64 1834 To find P. Tan // i ^ / i\ \ i f \ A n Obliq. xfxS'w'/E C Obliq. about 35^4 I 2«>5 Z _--r53BC^ Z'l A.D.o\ %\ J ^ % J \,. A ■^ E.C.I3S83 p B.C. S644 In the above cycle of about 30,000 years, I have termed the two halves of the cycle respectively the Temperate and the Glacial Epochs, and under such denomination it will be seen that : — 1. The last glacial epoch may be said to have commenced about B.C. 21,522, viz. about 23,500 years ago, when the obhquity of the echptic was 29° 25' 47". 2. The last mid-glacial epoch was about B.C. 13,583, viz. about 15,500 years ago, when the obhquity was at its maximum of 35° 25' 47". 3. The last glacial epoch terminated and the temperate epoch commenced B.C. 5644, viz. about 7500 years ago, when the obhquity was 29° 25' 47". 67 4. The next mid-temperate epoch, which we are now closely approaching, will be in a.d. 2295, viz. about 385 years hence, when the obliquity will be at its minimum of 23° 25' 47". As the obliquity of the ecliptic will have been 35° 25' 47" at the mid-glacial period, it will be evident that the Arctic and Antarctic circles came down to about 54^° at that period, and thus reached respectively about the latitudes of Durham and Cape Horn. Similarly, the tropics of Cancer and Capricorn will have receded to a distance of about 35|° from the equator, (say) to about the latitudes of the Straits of Gibraltar in the north, and to that of the Cape of Good Hope in the south. In order to give a more graphic representation of the changes in obliquity which have caused the alternation of temperate and glacial periods, I have constructed a diagram, which I have called the obUquity curve, for the period of a whole cycle. This curve may be of interest, as it shows at a glance how the glacial and temperate epochs have been determined by Dray- son to have been caused simply by the increase and decrease of the obliquity. The diagram also shows that although the alteration in the obliquity is caused by the regular and un- changeable motion of the pole, amounting to 20"'0529 per annimi, the annual rate of alteration of obliquity is variable ; for instance, at the present time when approaching the mid- temperate epoch the rate of diminution is only about 0"-4 per annum, and is still a diminishing rate : whereas, at the beginning of the glacial period, when the obliquity was about 29|° the annual rate of increase was about 11" per annum. Again, at the mid-glacial period the rate of increase of obliquity will have been reduced to 0, and at the end of the glacial period the rate of decrease of obliquity will have again increased to about 11" per annum, as is evident by the steep- ness of the curve at the periods of change from temperate to glacial epochs. 68 THE OBLIQUITY CURVE DURmC A CYCLE OF 31,756 YEARS. Shewing the commencement, duration, and termination of last GlaGial Period. 23° 24° 25° 26° 27° 28° 29° 30° 31° 32° 33° 34° 35° 36° | B.C. 30,000 "^ B.C. 29,000 M W mperate Period Ob iq. 23° 2S' 47" B.C. 28,000 1 B.C. 27,000 \ B.C. 26,000 V B.C. 25,000 \ B.C. 24,000 \ s. B.C. 23,000 \ s B.C. 22,000 \ ■v B.C. 21,000 Glaci alP !riod beg ins . Ob liq. 29° 25' 47' B.C. 20,000 "^ N B.C. 19,000 \, B.C. 18,000 s \ B.C. 17,000 \ B.C. 16,000 > 1 B.C. 15,000 \ B.C. 14,000 B.C. 13,000 Ml d-Gl acia Per lod ( Jblic 35° 2S' 47" B.C. 12,000 B.C. 11,000 / B.C. 10,000 J 1 B.C. 9,000 / B.C. 8,000 y / B.C. 7,000 y y B.C. 6,000 ^ r^ B.C. 5,000 Glacia Pet iod.I Inds ■ -^ Ob liq. 29° 2S' 47' B.C. 4,000 / B.C. 3,000 / B.C. 2,000 / B.C. 1,000 / A.D. 1 A.D. 1,000 A.D, 2,000 \ A.D. 3.000 Ml d-Te mpe rate Peri od Ob iq. 23° 2S' 47" 23° 24° 25° 26° 27° 28° 29° 30° 31° 32° 33° 34° 35° 36° || 69 Facts connected with Asteonomy akd Geology In 1896 Drayson wrote : — ' Upon examining the writings of the older geologists it win be found that Eamsey, Lyall, BucMand, Agassiz, Darwin, Forbes, Murchison, and others give abundance of facts proving that this great change of climate occurred and produced what they termed " a glacial period." ' Modern geologists, among whom may be mentioned Professors Warren, Upham, Wright, B. K. Emerson, J. Prestwich, Dr. E. Andrews, and many others, not only corroborate the evidence of former writers as to such a glacial period having existed, but they aflSrm that the evidence is convincing, that this great change of climate ceased not longer than about 7000 years ago, and had lasted about 20,000 years. Sir Joseph Prestwich has long expressed his opinion that these dates as to the duration and termination of the glacial periods are proved by undeniable evidence.' Such, then, are the facts resulting from Drayson's system, and granting, as admitted by geologists, and, indeed, uni- versally, that a glacial epoch affecting both hemispheres has occurred in the past, the alteration in the obliquity of the ecHptic as described by Drayson appears to be the only admissible theory which can account, and that with the utmost simplicity, for such glacial epoch having taken place. From the foregoiag it will be seen that among the results of Drayson's discovery is the geometrical proof that the glacial period lasted only about 16,000 years or 18,000 years, and terminated about 7500 years ago. The before-mentioned cycle of about 30,000 years, and the dates of the commencement and ending of the glacial epoch, can only be considered as correct for future ages on the supposition that the producing causes will remain unchanged. ' It would, however,' Drayson affirmed, ' be a fatal error to assume that because the second axis of rotation is now inclined to the axis of daily rotation at an angle of about 29° 25' it must always have been so, and hence that during aU past time there must have been successive glacial periods, is as 70 reckless as to assert that the earth's axis has always been fixed at a permanent angle of 66° 32' to the plane of the ecliptic. As the centre of gravity of the earth varied, due to the elevation and depression of lands, and the consequent transferal of hundreds of millions of tons of ocean waters, so must the centre of gravity of the earth have varied, and consequently the angle formed between the poles of daily and second rotation, and hence the climatic changes during each second rotation.' The announcement that the glacial period terminated about 7500 years ago was so entirely in opposition to the theories then believed in by geological authorities, who, instead of 7000 years, claimed hundreds of thousands and even millions of years, that General Drayson's accurate geometrical proofs were not only ignored, but the dates he gave became a subject for ridicule. During many years General Drayson had to submit to this treatment because he positively stated the above dates for the glacial period. Of late years nearly aU geologists in America and England have from geological evidence found that the dates given by General Drayson about thirty years ago were absolutely accurate. The position, therefore, which scientific societies subsequently occupied was that, although thirty years ago a proof was given them of the exact date and duration of the glacial period, they ignored or failed to comprehend the proof, and rejected the dates as absurd. After long years of research they found the dates formerly given by Drayson to be accurate, and appear to have claimed that this result is a modern discovery of their own. The accurate determination of the date and duration of the glacial period, which has been described as a ' briUiant modem discovery due to geologists,' is a mere trifle in com- parison to other portions of General Drayson's work. Before Drayson could so accurately give these dates he had dis- covered the cause of the glacial period, and had proved, by the most exact and hitherto unknown calculations, that the cause was correctly defined. Having some eighteen years ago set to work to examine 71 and test Drayson's system (mainly, I must admit, with the idea of proving it untenable), and with a clear understanding that I should take nothing for granted, I found that my calculations based on his system enabled me to arrive at results exact to a second. And here I may state that my old friend the late Major- General the Right Hon. Sir John Cowell, a distinguished Royal Engineer and able scientist, who with me was in the secret of Drayson's discovery, after full examination, entirely agreed with my conclusions. I then examined the objections which had been urged against Drayson's system, and in no single instance could I find a flaw. As a proof that my conclusions are based on sound eAridence, I may mention some of the calculations I made to find the obUquity, precession, and stars' polar distance and right ascension. For instance, in the Nautical Almanac 1820, the right ascension of the star beta Ursse Minoris was recorded as 14 h. 51 m. 20-36 s. for 1st January. The north polar distance for the same date was 15° 6' 31"-8. From these data, and a knowledge of the true movement of the earth, calculate the right ascension of this star for the 1st January 1850, 1st January 1880, and 1st January 1895, also calculate its north polar distance for the same dates. Again : In the Nautical Almanac, 1825, the right ascension of the star a Virginis (Spica) is recorded for 1st January as 13 h. 15 m. 59-2 s. Its north polar distance for the same date was recorded as 100° 14' 39". From this one observation calculate the right ascension and north polar distance of this star for the 1st January 1875 and 1st January 1895. Again : Without reference to solar tables, . or rule of thumb system derived from annual rates of change found by observation, calculate the mean obUquity for 1st January 1885. I think I may assert that no one can make the above calculations unless acquainted with Drayson's system. I have made these and similar calculations, and my results differ from recorded observations less than one second. From the carefully tested calculations I have made, and 72 which several others have made, I may say that such asser- tions as that ' competent authorities do not agree with Drayson ' ; that ' he is contradicted by the theories of La Place, Le Grange, etc/ ; that ' he is merely giving another name to that which everybody knew before,' etc. etc., are reasonings not worthy of consideration. With aU respect to the able astronomers above quoted, it is not reasonable to say that their knowledge was incontrovertible for all future time. For instance, the greatest scientists might, fifty years ago, have pronounced that it was impossible without tele- graphic wires to hold conversations with persons hundreds or thousands of miles distant, and yet such pronouncement would in these days have been held invalid in the face of the marvellous discoveries of Marconi and other wireless scientists. When we see the sound scientific base of these and similar calculations, and find that the facts of astronomy and geology are explained, and how exactly the two teU the same tale, it seems astounding that men claiming to love science for its own sake should ignore these facts. Every wild and baseless speculation to account for the glacial period has been put forward and solemnly discussed before scientific meetings, but after a very short time each of these has been found inadequate to explain the facts. Here, however, we have a discovery which not only explains the facts, but gave the accurate dates of these facts thirty years before geologists, and in addition enables an average mathematician to make calculations hitherto unknown to astronomers. That such an important matter should have been neglected during thirty years is bad enough, but when one small portion of it is claimed as a recent discovery by geologists, it is really the duty of every honest man to speak out. It is interesting to note that Drayson, when writing to me 26th May 1901, said : ' Thirty years ago I sent the formula for finding the obliquity to Sir G. B. Airy (the then Astronomer Eoyal), and asked him to test it. He replied that it worked correctly, but that he had tried in vain to discover how I had framed it.' 73 When General Drayson announced tliat the hitherto accepted naovement of the pole was a geometrical impossi- biUty, that the true movement was a circle with a radius of 29° 25' 47" and not 23° 28' as had hitherto been stated, that the centre of this circle was 6° from the pole of the ecliptic and had a right ascension of 18 hours, it is not sur- prising that he raised a storm of indignation which, had the old laws existed, would have placed him in the Inquisition, and thus have ' proved ' him wrong. More especially would this have been his fate because he boldly told geologists that their dates for the glacial epoch were utterly wrong. That instead of this epoch having, as they imagined, lasted hundreds of thousands of years and terminated 200,000 years ago, it lasted only 18,000 years and terminated about 7000 years ago. Geological authorities were indignant, and refused even to listen to such heresy. Hence the leaders in both astronomy and geology refused to permit any further investigation of the facts, and even had the temerity to state that the matter had been fuUy examined and had been ' proved ' incorrect. Not only has no ' proof ' ever been given against General Drayson's system which was worth even half an hour's examination, but proof after proof has been given of its accuracy. There ought to be no popery in science. When, however, we find that the only portion of this system, viz. the most simple, giving the date of the Ice Age, is at last comprehended, it is unjust and im-Enghsh for men to be given credit for discovering that which Drayson told them thirty years previously. When, many years ago, Drayson submitted a paper to the Geological Society, in which he gave those very dates for the glacial period which are now found to be accurate, the Society pronounced that these were wrong. When he submitted the astronomical portion of his discovery to the Eoyal Astro- nomical Society and invited a discussion, his paper was suppressed. CONCLUSION In concluding this treatise, I feel that some apology may be due for what might be deemed too much self-assertion on my part. Although a lifetime at sea with, so to speak, a sextant in my hand, must have given me considerable knowledge of, that small branch of astronomy termed Nautical Astronomy, it should be understood that I make no pretence to knowledge of that great science, or to being in any respect a scientist, but I do claim to have sound reasoning powers and to have done my best to exercise them. Let me further express the profound respect I have for astronomy, a science some know- ledge of which is not only enthralling, but leads its students to a more just idea and appreciation of the marvellous works of the Almighty, works so wonderful as to be far beyond the conception of man. It has grieved me to notice how little support, or even tolera- tion, Drayson has received from the scientific world, and I feel that, if so Httle acceptance of Drayson's system has been accorded to him after half a lifetime's study of astronomy, I cannot expect much attention to my poor endeavours to explain and make his system known. I have written with small hope of engaging the attention of great astronomers, but I do hope to induce the younger generation, whose pro- fessional advancement is not dependent on blind adherence to text-books, and who may take an interest in fully examining and proving or disproving Drayson's system, to bear in mind that the astronomy of a hundred and fifty years ago, and its parrot-like repetition, is not necessarily perfect for all time. The direct proof of Drayson's system is obviously difficult to establish, but as in criminal jurisprudence circumstantial evidence is in many cases more reliable because — unlike a 75 witness — it cannot lie, so in the case of Drayson's system there is the following circumstantial evidence to put forward, evidence which any person of ordinary mathematical abihty can test for himself. The system enables such persons to ascertain by calculation the following results for any year, past, present, or future, without reference to the Nautical Almanac of the year required : — 1. The obhquity of the ecliptic. 2. The right ascension of any fixed star. 3. The declination of any fixed star. 4. The precession of the equinoxes. 5. The right ascension of a star from its apparent annual precession in declination if known. 6. And conversely, the apparent annual precession in dechnation from its right ascension. The above calculations are, by Drayson's system, as I have said, within the competence of any person possessing such limited knowledge of mathematics as myself, and the results of such calculations can be tested, as they will be found to agree precisely with the amounts stated in the Nautical Almanac, which have been obtained by years of innumerable observations at Greenwich. I think I may claim that no astronomer, however able, can produce the above results without acceptance of Drayson's theory and data. In con- firmation whereof, as I have previously stated, so great and distinguished an astronomer as the late Sir George Airy, whilst admitting the correctness of Drayson's formula, told him that he had tried in vain to discover how he had framed it. Let me urge young students in search of the truth to test for themselves the accuracy of my claim on behalf of Drayson without reference to annual rate or rule of thumb, and to bear in mind a saying of Drayson's — in which I beheve there is much truth — that in studying astronomy, old text-books had been his greatest obstacles. Finally, I should add that in my eighteen years' examination of Drayson's system, I have from the first taken nothing for granted. Similarly, I ask those who favour me by reading these words also to take nothing I have said for granted, but 76 to study the matter for themselves, as I have done by patient research, and copjdng from my voluminous work-books and from my correspondence with Drayson up to the time of his death. This work has been an enthralling relaxation since com- pulsory age retirement in 1892, after service in the Navy since 1840, including over twenty years' command of H.M. ships and squadrons at sea. Fiinted by T. and A. 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