Samuel Arthur Saunder publications

Samuel Arthur Saunder wrote many papers on astronomy and a few on mathematics. Most of the astronomy papers were related to his studies of the moon, where he was particularly interested in measuring positions from photographs. His determination for an international agreement on naming lunar features led to a committee being set up which eventually agreed the naming which has continued to this day. We present abstracts from a few typical papers by Saunder.

G Chrystal and S A Saunder, Results Of A Comparison Of The B. A. Units Of Electrical Resistance, British Association Reports, Glasgow Meeting (1876), 13-19.

1.1. Difficulties encountered.

The difficulties of the kind of measurement we had to make are confined almost entirely to the temperature determinations. Were it not for these a much higher degree of accuracy could be obtained; for while resistances comparable with the B.A. unit van be measured without difficulty to the 100,000th part, it is very difficult to determine the temperature of a wire imbedded in paraffin, as are the wires of the standards, nearer than one tenth of a degree Centigrade, an error to which extent entails in some of the coils as error of 0.03 per cent of resistance.

A mere comparison of he coils at the temperatures given in the B.A. Report on Electrical Standards (1867) would hardly have been satisfactory, since it would have given no check on the accuracy of the observations and afforded no information as to the temperature value od a variation in resistance, and conversely.

1.2. Object aimed at.

The object aimed at in the experiments was to get the differences between the resistances of the several coils at some standard temperature, and also the coefficients of variation of resistance with temperature in the neighbourhood of the standard temperature.

That it is inadmissible to apply to any given coil the variation-coefficient for its supposed material, as found by Matthiessen and others from experiments on naked wires, is abundantly evident. This appears very strikingly in the case of coils Nos. 2 and 3; and an examination of the results of Lenz, Arudtsen, Siemens, and others for platinum shows that within certain limits its behaviour is very uncertain. This arises no doubt from the presence of more or less iridium or other platinoids, a small admixture of which, without altering the value of platinum commercially, affects its electrical resistance very considerably.
S A Saunder, How to determine the time with a small equatorial, Journal of the British Astronomical Association 5 (1895), 461-467.

2.1. From the text.

I have recently been endeavouring to determine with what degree of accuracy a time determination can be made with a small well-mounted equatorial. The instrument used is a 7-inch refractor by Troughton and Simms, with a level, graduated to seconds of arc, on the declination axis. I was told by Mr Simms that I ought with this to be able to get the time to the nearest second, but as some may not know that the determination can be made so nearly, it is hoped that an account of the method to be adopted may be of interest to those who, like myself, do not possess a transit instrument.
S A Saunder, The Stereographic Projection of the Celestial Sphere, Journal of the British Astronomical Association 7 (1897), 274-276.

3.1. From the text.

In "Popular Astronomy" for February is an inquiry as to the method of converting altitude and azimuth into hour angle and declination; in the same number there is also a valuable article by Mr Moulton, in which be emphasises the necessity for directing attention to this and kindred problems; he does not, however, allude to the graphical solution which, though not new, is, perhaps, not so well known as it deserves to be.

The diagram required is a stereographic projection of the circles of altitude, azimuth, hour angle, and declination from the nadir on the plane of the horizon; that is, the lines are drawn as they would be seen projected on this plane by an eye placed at the nadir. The advantage of this projection is, that the lines are all accurately straight lines or circles, and that angles are unaltered.
S A Saunder, On the expression, "Motion at an Instant", The Mathematical Gazette 1 (17) (1899), 250-252.

4.1. A version of the paper.

In a recent review of Prof W B Smith's "Infinitesimal Calculus," fault is found with his statement that "In all strictness there can be no motion at an instant, and hence no speed (or velocity) at an instant. The concept of speed (or velocity) or motion will not combine with the concept of instant (or point of time) to form a compound concept." Part of the reviewer's criticism is as follows: "If we allow that motion at an instant is impossible, how are we to escape from Zeno's paradoxical conclusion that all motion is impossible? How can I move from one place to another during a minute, say, if at every instant of that interval motion is impossible?"

Now, it cannot be too clearly insisted that an instant does not denote an infinitesimally short period, but that it has no duration. It is not to be thought of as in any way resembling the limit of an interval which decreases without limit. Whilst by motion we denote, as defined by Clerk Maxwell, "a change of configuration taking place during a certain time, and in a continuous manner." It would seem then that the concept motion demands extension in time, whilst the concept instant denies it, and so I find, with Prof Smith, that the two will not combine.

The paradox to which the reviewer refers is stated thus: "So long as anything is in one and the same space it is at rest. Hence an arrow is at rest during every moment of its flight, and therefore also during the whole of its flight." The difficulty here lies in the sense to be attributed to the word moment. "During" implies duration, and hence it should denote an interval, but the arrow does not occupy the same position during any interval, however short. At an instant it occupies only one position; no point on it is in two places at once; but at any other instant, however near, it occupies a different position, and the conclusion that the arrow is at rest "during the whole of its flight" would only follow if by any repetition of the instants at each of which it is in one position, we could obtain the interval during which the flight lasts. This we cannot do. Numerical multiplication, whether infinite or not, is insufficient to convert an instant into an interval. The same fallacy of confusion would seem to underlie the reviewer's question quoted above. Either the minute is made up of instants, which would then be infinitesimal intervals, during each of which motion is possible; or else the two clauses of the sentence are mutually irrelevant.

There is a sense in which we do talk of "motion at an instant," and a definition of the expression is given in all textbooks of mechanics. The following is due to Clifford: "If there is a certain velocity to which the mean velocity during the interval succeeding a given instant can be made to approach as near as we like by taking the interval small enough, then that velocity is called the instantaneous velocity of the body at the given instant." In this sense the words are used, not as conveying any meaning in themselves, but as an abbreviated form of expression for an idea which we cannot state completely without a good many words. But the expression is one with which we are so familiar that we are apt to forget this, and to think that the meaning must be contained in the expression itself. Take, for example, the case of a stone falling to the ground. At the instant it hits the ground we say it is moving at a certain definite rate. Let us for a moment suppose that the stone is stopped instantaneously, a condition unrealisable physically, but presenting a definite kinematic problem. Is the stone moving or not, at that instant ? When I say that a body is moving at a certain instant, I mean (in accordance with the definition just quoted) that, if I compare its position at that instant, with its position at another instant sufficiently near to the first, I shall find that there has been a displacement. In the case considered, the stone is moving if the comparison be made with past time, it is not moving if the comparison be made with future time. But if the question be taken as complete in itself, it does not admit of an answer. The instant of stoppage is the boundary between the time during which there is motion, and that during which there is no motion; and this is true whether we suppose the stoppage to be abrupt or not.

It may be urged that a complete solution of the kinematic problem of the motion of a point is given by the curve of position, and that the tangent to the curve at any point gives a measure of the velocity at the corresponding instant. This is true, but this means of representation is only an analogy, and must not be pushed too far. Our idea of time is one dimensional; our idea of the velocity of a point is obtained by a combination of this with that of one-dimensional space, but the two concepts, time and space, in the original problem do not resemble one another in the same way as the two dimensions of space in the analogy. So far as the curve is merely at the point, it has no tangent. When I speak of the tangent at a point, I mean a line which meets the curve at the point - has both its coordinates the same as those of the curve and touches the curve in the immediate neighbourhood. The line is "at the point" in a way which presents no analogy to "at the instant." So far as the analogy holds, the position of the point defines the part of the curve to be considered in approaching the limiting position of the chord, the instant defines the portion of time to be considered in determining the limit to which the velocity approaches.

To the phrase "motion at an instant," if it is regarded as connoting neither more nor less than is contained in the definition quoted above, I have no objection; some expression for this limit is necessary, and I cannot suggest a better; but, if it is maintained that the words can be understood without reference to this definition, I must confess that I for one am unable so to understand them.
S A Saunder, The Determination of Selenographic Positions and the Measurement of Lunar Photographs, Monthly Notices of the Royal Astronomical Society 60 (3) (1900), 174-201.

5.1. The object.

The object of this paper is to call attention to the great uncertainty that attaches to our knowledge of the positions of lunar formations, and to suggest means by which these positions may be determined more accurately than has yet been accomplished.

5.2. Errors in Existing Measures.

Schmidt, in the introduction to his Charte der Gebirge des Mondes, gives a list of 157 formations whose positions have been determined as "points of the first order," almost entirely by Lohrmann (76), Mädler (89), and Neison (33). If to these we add the positions of Biot, found by Mädler; Horrocks, HaIley, and Hipparchus L, by Neison; with Triesnecker B, by Pritchard, we have, so far as I know, all the points whose positions have been found by methods making any pretension to accuracy; and of these very little reliance can be placed on those which depend on not more than two or three measures. The uncertainty attaching to a single measure was estimated by Mädler to be about 8" or 9" (geocentric), which would correspond to half a degree of selenographic longitude or latitude near the centre of the disc, and to a still greater amount as the limb is approached. The probable error derived from the measures themselves, after rejecting the unfavourable points, is stated by Neison to be 7".2. Rather less than half the whole number given by Schmidt depend upon 10 or more measures, whether by the same or by different observers.

Eleven points were measured five times at least by both Lohrmann and Mädler. The average difference in selenographic latitude is 17', and in longitude 10', corresponding respectively to 4".4 and 2".3 geocentric. It may also be noted that in nine cases Lohrmann's position is further from the equator than Mädler's, and the mean distance of his positions from the equator is greater than Mädler's by 12' of selenographic latitude, or 3".4 as seen from the Earth. The differences in longitude so far as these points are concerned do not appear to be systematic.
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5.3. Conclusion.

I do not think that the limit of accuracy of which these photographic measures are capable has yet been attained. In the first place, by measuring the original negatives I hope to find less difficulty in seeing the réseau, and to get better definition of some points which in the copies are lost in glare. Secondly, by the use of a réseau covering the whole limb, I hope to get a better determination of the coordinates of the centre and of the radius. Professor Turner has already procured for me a copy of one of sufficient size through the kindness of Mr Franklin Adams.

M Loewy has sent four more negatives, two of which show the zone already measured under yet different conditions. These I propose to treat in the same way as Plates II and III have already been dealt with, and so to obtain a set of definite values for the points shown on all four plates. When this has been done the constants for these plates can be finally determined, and the positions of all points shown on them obtained, with accuracy of the same order as those already measured.

This will greatly facilitate the determination of the constants for other plates, and of the positions of all points shown on them. When once these constants have been found, the reduction of the measures is very simple ...
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The measures have been made with one of ProfessorTurner's star plate micrometers belonging to the Oxford University Observatory. The glass scale is admirably adapted for work on the Moon, where it very frequently happens that one side of a crater is lost in glare. If the crater is circular and near the centre of the disc, the scale has only to be placed so that the readings of the points at which three of the arms cut the walls are equal, when the intersection must be at the centre of the circle. Other cases are not all so simple, but the help afforded by reading the points at which the arms cut the walls is always considerable. A crater has not been measured unless the greater part of the ridge is shown, the rest can be supplied mentally, and by determining the centre from this the confusing effects of light and shade upon the floor are avoided.

It only remains for me to express my thanks in the first place to M Loewy for his great kindness in sending the photographs. The beauty of the negatives now taken at the Paris Observatory is too well known to need any comment, and it is little better than a truism to say that it is only upon photographs oi the best definition that accurate measures are possible.

That I have been able to accomplish as much as I have in the time at my disposal has been due to the skill and the care with which Mr Bellamy has overcome for me all the difficulties of manipulation; whilst the extent of my obligation to Professor Turner is very far indeed from being covered by the definite acknowledgments I have already made.
S A Saunder, Note on Measures by Professor Barnard of two Standard Points on the Moon's Surface, Monthly Notices of the Royal Astronomical Society 60 (8) (1900), 540-543.

6.1. From the text.

In a paper communicated to the Society last January (ante, p. 174) attention was called to the increase of accuracy in selenographic positions which might be attained by measuring from a well-determined point instead of from the limb, and to the suitability of Mösting A as an origin. It was my good fortune that Professor Barnard was present at the meeting at which the paper was read, and the next day he most kindly offered to measure a few points on the Moon itself if the results would be of any assistance to me. This generous offer I gladly accepted, and I have now received from him the particulars of measures made on April 7 and 9, with the full aperture of the 40-inch telescope of the Yerkes Observatory, and a magnifying power of 700 diameters. The measures made were of the distances and position-angles of the lines joining the centres of Mösting A, Ptolemaus A, and Triesnecker B. Each position-angle was measured four or five times on each night, and each distance eight or nine times. These measures I have reduced by the methods described in the paper referred to with the following results.
S A Saunder, The Determination of Selenographic Positions and the Measurement of Lunar Photographs. [Second Paper.] Determination of a first group of Standard Points by Measures made at the Telescope and on Photographs, Monthly Notices of the Royal Astronomical Society 62 (1) (1901), 41-61.

7.1. From the text.

In a previous paper (Monthly Notices Vol. lx. p. 174) I called attention to the unsatisfactory stale of our knowledge of the exact positions of the lunar formations, and to the increase in accuracy which might be obtained by measuring from the well-determined point Mösting A; formulae were developed for reducing the measures, and a few results were given.

It was also shown that by the measurement of such photographs as those now being taken at the Paris Observatory a great in crease might be made in the number of points whose positions could be accurately determined without necessitating an inordinate amount of computation. Formulae were developed for reducing the measures on the assumption that the positions of a number of points were known with sufficient accuracy, but it was found that the accepted positions of "points of the first order" were not accurate enough to give the best results obtainable from the photographs, and a second method was developed depending partly on measures of the limb, partly on a combination of measures of the same points on two different photographs, somewhat analogous to the optical combination of two pictures by the stereoscope. Some tentative results were given, and in conclusion I expressed my intention of attacking other pairs of plates in the same way.

Further work on the same pair however showed that the constants obtained ought to be capable of improvement; there were signs of a progressive error, which I now know to have been due to an error of scale on one of the plates. This error was not of sufficient magnitude to vitiate the general conclusions drawn in the paper. I soon became convinced that no really satisfactory solution was likely to be obtained so long as any part of the work depended on measures of the limb. I then tried to determine the plate constants by an extension of what I have called the stereoscopic method, and when this failed, as will be described below, I had recourse to the telescope, and proceeded to determine the positions of a number of points surrounding Mösting A at distances not much exceeding 500", this limit being imposed by the construction of my micrometer.
S A Saunder, The lunar atmosphere, The Observatory 25 (1902), 326-329.

8.1. From the text.

In a paper recently presented to the Académie des Sciences on the occasion of the appearance of the fifth and sixth parts of their 'Photographic Atlas of the Moon,' MM. Loewy and Puiseux call attention to several points arising from a general consideration of their photographs. Among the most interesting of these is one which has an important bearing upon the arguments in favour of the past existence of an atmosphere of considerably greater density than that which the Moon now possesses.

A comparative study of the surface of the Earth and Moon shows that whilst the former has been moulded largely under the influence of considerable lateral pressure, the crust of the Moon everywhere bears evidence of a superficial tension, as indicated by the crack, and dislocations found upon its surface. Now it has been recently shown, by Dr Charles Davison, that in a globe cooling steadily by the dissipation, under uniform conditions, of its internal heat, there will be an outer shell subjected to pressure, whilst the material beneath this will be in a state of tension, increasing to a maximum, and then decreasing again as we proceed towards the centre. The thermal data obtained for the Earth fix the stratum of zero-tension at 8 km, and that of maximum tension at 110 km. If there is no change in the external conditions under which the cooling proceeds, the depth of these strata should vary as the square root of the time elapsed since the superficial solidification. But if, from the operation of external causes, a more rapid rate of cooling is set up, the layer in which the maximum loss of heat takes place rises towards the surface, the pressure is then diminished, and, if the process is carried far enough, the surface itself may be brought into a state of tension. When the external conditions once more become steady a region of pressure is again developed in the outer stratum; but this region is at first very thin, and the effects of the pressure may for a long time be completely masked by the tension of the strata beneath.

The state of the lunar surface would seem to indicate that some cause of this nature has been in operation subsequently to the solidification of the crust. It is almost certain that the Moon at one time had an atmosphere of quite appreciable density, and the disappearance of this atmosphere would seem to offer just such an event as is required to account for a considerable increase in the rate of loss of internal heat at some remote period of its history. If the cooling of the surface were already well advanced, it is clear that this increase would he more rapid in the equatorial than in the polar regions, which would have a much lower temperature to start with, and the greater superficial tension developed in the equatorial part of the surface would result in a diminution of curvature, which might be sufficient to cause an inflow of any liquid yet remaining at higher latitudes. It accords with this that we find the seas chiefly situated in the equatorial regions, whilst the traces of erosion still to be detected indicate a flow in both hemispheres directed towards the equator. It should be remembered that there are reasons for believing that the axis of rotation once occupied a position which would bring the principal seas more nearly into the equatorial zone than they are at present.

MM. Loewy and Puiseux also refer to the photographic evidence of the occurrence of periodic changes in the appearance of certain spots, depending upon the progress of the lunar day. They have previously called attention to the fact that in some spots there is a periodic change of contour of this description; they now specially consider an oval plateau of some 270 sq. km. near Vitruvius A, in which there is a periodic change of brightness, whilst the contour remains unaltered. This plateau at sunrise is darker than the surrounding plain, merges into it under meridional illumination, and finally reappears as an increasingly bright spot under the rays of the setting Sun.

The fact that such periodic changes do take place has long been known to selenographers. Mädler drew attention to their occurrence in various regions, one south of the Mare Crisium and another north of Hyginus, and tells us that in his day many attributed them to vegetation; an explanation which he, however, does not favour, for the reason that "a vegetation without air or water lies entirely beyond the range of our comprehension," He is constrained to leave them as an unsolved enigma.
S A Saunder, Note on the use of Peirce's Criterion for the Rejection of Doubtful Observations, Monthly Notices of the Royal Astronomical Society 63 (8) (1903), 432-436.

9.1. From the text.

It would seem at first sight that when for observations made at the telescope we substitute a series of photographic measures we ought to be able to eliminate all those abnormal errors or real mistakes, such as the entry of a wrong figure, which are not contemplated in the Theory of Errors, but which undoubtedly do occur in practice, and which have led to the suggestion of various criteria for the detection and consequent rejection of such abnormal observations.

With regard to such classes of mistakes as may be supposed to have most frequently occurred in the case of telescopic observations, this is undoubtedly the case, for a photographic measure which gives an abnormal result, and so comes under suspicion, may be repeated any number of times, and any clerical error or mistake in reading may be certainly detected end eliminated.

But a new possibility of error is introduced at the same time; A plate is either accepted or rejected as a whole in accordance with the general excellence or the reverse of its definition. But it may well happen that a plate which gives excellent definition as a whole may yet have some defective images. How far this is likely to occur in star photographs, such as those taken for the astrographic catalogue, I have no experience; but it does, and I think must, happen in every photograph of the Moon. Formations near the terminator may be beautifully sharp, whilst from unsuitable Illumination, or from the general brightness of a district, those in other parts may be quite unmeasurable, Between these two extremes there will be an almost continuous gradation, and it becomes an anxious question where to draw the line between those formations which are to be measured and those to be rejected. Another possible cause of error may be that the assumed positions of some of the standard points are seriously wrong. Where these standard points are stars which have been repeatedly observed with meridian instruments such errors are not likely to be great; but in an investigation such as a survey of the Moon, where in our present state of knowledge the standard points have to be built up as we proceed, it is a possibility not to be lost sight of. There is too the, probably small, risk of distortion of the film.

In a recent difficulty, due to an attempt to measure two badly defined formations, I found Peirce's criterion so useful that I thought it might be of interest to call attention to it, especially as I believe that the criterion is seldom if ever employed by the most experienced computers, at all events in this country.
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The general question of the rejection of discordant observations has been discussed by Dr Glaisher, who strongly upholds De Morgan's view that when recognisable mistakes have been eliminated no observation should be rejected, but that when the first solution has been effected the conditional equations should be re-weighted in accordance with their residuals and solved again, the process being continued until the weights assumed for the last solution are sufficiently near to those given by that solution. How many solutions this would in general require I do not know, but it is clear that the process might be a very laborious one. In many cases its length would be practically prohibitive, and if we decide to give all the equations, either unit or zero weight, it is certain, as Dr Glaisher points out, that we shall get a better result by excluding certain discordant observations than we should by including them.

I believe that the practice amongst computers of experience is to rely almost entirely on their individual judgment, taking into account the conditions of the observations, and drawing the line somewhere about those observations which give residuals of five times the probable error. But for those who, like myself, have a very limited practical acquaintance with least square solutions, difficult cases must from time to time arise, and I cannot help thinking that others may be as glad as I have been to have recourse to a criterion established on some definite principle, which may be considered as an embodiment of that accumulated experience which we lack.

The computation of the criterion is quite a short matter with the help of such tables as those given in Chauvenet, but it is only to be effected through successive approximations by trial and error, and in order to facilitate its application I have computed a short table which I append to this paper.
S A Saunder, Note on the Drawings of the Mare Serenitatis by John Russell, R.A., Monthly Notices of the Royal Astronomical Society 64 (5) (1904), 427-429.

10.1. From the text.

On pp. 156-159 of the present volume of the Monthly Notices Dr Rambaut gives an account of two drawings of the region around Linné made by Mr John Russell, R.A., and arrives at the conclusion that they render it probable that Linné then presented much the same appearance as at the present day when viewed with a low-power eyepiece attached to a small telescope.

The drawing reproduced in plate 3, fig. 2, has not much direct bearing on the question of change, for we are told by Mädler that near the time of full moon Linné was seen in his time as here represented, namely, as a round white spot almost as white in the middle as at the edges. It is to the drawing in fig. 1 that the real interest attaches. My own introduction to this drawing was through a photograph shown me by Dr Rambaut, and from an inspection of this I came to the conclusion that it did offer distinct evidence of a change. As I was afterwards led to modify this opinion when, by the kindness of Dr Rambaut, I was allowed to inspect the original, it may be not without interest that I should put on record both my first impression and the reasons which led me to modify it, as many will see the reproduction and but few can hope to have the opportunity of comparing it with the original.
S A Saunder, The Determination of Selenographic Positions and the Measurement of Lunar Photographs. [Fourth Paper.] : First Attempt to Determine the Figure of the Moon, Monthly Notices of the Royal Astronomical Society 65 (5) (1905), 458-473.

11.1. From the text.

In the third paper of this series (Memoirs R.A.S. vol. lvii. part i.) I have given the places of 1433 points as determined from the measures of four Paris negatives, the reductions being made on the supposition that the points all lie on the surface of a sphere. But one of the results I hope will follow from the measures on which I am engaged is a determination of the true figure of the Moon; and although the work is at present incomplete, and the plates already measured are, when taken by themselves, not very well suited for such a determination, I have wished to see whether the results obtained are such as to justify a hope that this object may be ultimately accomplished.

It was first pointed out by Newton that if the Moon were originally fluid the tide raised by the Earth should have caused the diameter directed towards us to be longer than that at right angles to it. He computed the elongation to be 186 feet. Summaries of various attempts which have been made to determine this elongation by observation are given by Franz in Die Figur des Mondes, pp. 2 , 3, 33; and by Mainka in Breslau Mitteil. vol. i. pp. 55, 56. They may be divided into dynamical methods which do not necessarily give the geometrical elongation, methods depending on the displacement of the terminator whose position it is always difficult to estimate, and methods depending on the apparent change of position of lunar formations under varying libration.

The results obtained by the geometrical methods are very conflicting. The greatest value of the elongation is that of Gussew, who thought that the part of the Moon towards the Earth was spherical, with a radius 0.982 of the radius of the periphery, but with its centre 0.0726 of the same radius nearer to the Earth, so that, measuring from the centre of gravity of the Moon, the radius towards the Earth was 1.05 times that at right angles to it. The least value is that found by Franz in Die Figur des Mondes: he assumes a spheroidal figure and finds an elongation 0.00114±0.00390 towards the Earth.
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The value obtained in the present paper is 0.00052±0.00027. This determination is made from a consideration of the absolute altitudes of thirty-eight points measured on each of four negatives, and all situated near the central meridian. The probable error is considerably less than that of any previous determination with which I am acquainted; but although I do think that it shows that the elongation is very small, I do not wish to lay any great stress on the actual result itself. The number of points employed is small, and the individual altitudes are subject to considerable uncertainty. My desire is rather to give grounds for my opinion that the method adopted is one of considerable promise.
S A Saunder, The most Probable Position of a Point determined from the Intersections of Three Straight Lines, Monthly Notices of the Royal Astronomical Society 65 (8) (1905), 854-856.

12.1. From the text.

In the course of my work on the Moon I have frequently fixed the position of what may be termed a point of the second order by measuring its position angles from three points whose coordinates had been well determined. I have shown (B.A.A. Memoirs, vol. vii. pp. 61-65) that when the points are near the centre of the Moon's disc, and the distances between them are small, the errors involved by neglecting the effects of libration are also small, whilst the reduction of the measures is thereby much simplified.

In the course of these reductions I have had to consider what was the most probable position of a point so determined. I am not aware what is the practice of those who compute meteoric radiants, but a recent writer (Monthly Notices, vol. lxv. p. 238) has assumed that the radiant should be placed at the incentre of the triangle formed by these lines. As this can be correct only under very special circumstances, I have thought it might be worth while to call attention to the point.
S A Saunder, On the Present State of Lunar Nomenclature, Monthly Notices of the Royal Astronomical Society 66 (2) (1905), 41-46.

13.1. From the text.

Some apology is due from me for again occupying the time of the Society with a subject so familiar to all selenographers as the confusion now existing in lunar nomenclature, and the inadequacy of our present system for the growing needs of selenography; but, as some recent remarks of mine have led the Council of this Society to take a course of action which it is hoped may lead to an authoritative reconsideration of the questions involved, I have thought that a fuller statement of the difficulties might be of interest to those whose work lies in other directions, and might also lead to some useful suggestions from those who, like myself, have found themselves hampered by the want of a recognised language in which to express the results of their labours.

Our present system may be said to date from the publication of Beer and Mädler's map in 1837. In this the principal formations, such as Tycho or Copernicus, have separate names allotted to them; the smaller mountains are designated by affixing a letter to the name of some neighbouring principal formation, as Mösting A, Thebit B. But at once difficulties begin to be felt. These smaller mountains are denoted on the map only by the letters A, B ..., and it is often far from easy to determine to which of the adjacent names this letter should be attached. Mädler was generally careful to place the letter towards that side of the object which was nearer to the named formation, but even in his map it is sometimes difficult to determine the name, and in other maps the position of the letter is no guide at all. The only safe method is to read through all that has been said in the text of Der Mond, or of Neison's Moon, under each of these headings until a description is found which applies to the mountain under consideration. This may well occupy half an hour, and sometimes three or four times as long may be spent without any result, for there are some of these lettered formations to which I have been unable to find any allusion in the text. It is frequently hard enough to identify a crater at all in a crowded region, and this further demand constitutes a considerable tax upon one's time.

As an instance of the confusion which may arise from this difficulty I may cite the valuable set of measures of 150 standard points published by Dr Franz in Breslau Mitteilungen vol. i; No. 65 in that list is called Hippalus A, and no other moans are given for identifying it except its position and diameter. Now I have measured the same point - -it is No. 124 in my catalogue (Memoirs R.A.S. vol. lvii. part i) - and, as my position differs from Dr Franz's by less than a second of arc, whilst the diameter of the ring is about 9 seconds, there can be no doubt as to our having measured the same point. I am therefore able to identify the point on a photograph, and thence on the map. But, on referring to the text, I conclude that Mädler wrote of this as Agatharchides A, and not Hippalus A. Mädler's Hippalus A is No. 73 on my list, at a distance of nearly a minute of arc and on the other side of Hippalus. Anyone using Dr Franz's measures as he intended them to be used, either for making a map or for determining the positions of other points, would probably have applied the measure to a wrong point, and, as it would not be absurdly wrong, he might have done a good deal of work before he found even that there was an error, and then a good deal more might be necessary before the true source was located.

A difficulty of the same character sometimes arises from the inaccuracy of the maps. When a point has been observed on the Moon, or on a photograph, it is often extremely difficult to identify it on the maps. Franz mentions this difficulty with regard to several points on his list, and, amongst others, with regard to No.115, which he calls Pons c. Here, again, I have measured the same point (it is No. 1131 in my catalogue), and have come to the conclusion that it is Mädler's Pons b. The same confusion as before is not improbable, and these are not the only cases in which it might arise.

I should be very sorry if my selection of these instances from Dr Franz's catalogue gave rise to an impression that I was actuated by any spirit of fault-finding. Dr Franz has done so much toward the foundation of an accurate selenography that it would be impossible to speak of his work otherwise than with the highest admiration and respect; but, this being the case, the fact that I find myself obliged deliberately to differ from him in what ought to be so simple a matter as the right names to be applied to several amongst 150 of the most conspicuous measurable points on the Moon will be more convincing than many pages of argument that there is a real necessity for rendering the recognised names more easily discoverable.

But the difficulties do not stop here, Since Mädler's time many selenographers have considered it desirable to add new names to the list. In Mädler's map there were, according to Neison, 427 principal names: 145 of these were new, the rest being taken from the works of older selenographers. In Neison's map, which was published in 1876, there were 513 names. ...
S A Saunder, Contracted Multiplication and Division, The Mathematical Gazette 4 (64) (1907), 81-83.

14.1. A version of the paper.

Mr Godfrey and Prof Lodge have both expressed the hope that others would give the results of their experience of contracted methods of multiplication and division. My experience agrees with Mr Godfrey's. That the rules are to nearly all boys pure "rule of thumb" is shown by the number of different dodges which the upholders of the methods advocate, each maintaining that his own particular dodge is the one which renders the rule easy to remember and safe to use. One or other of these dodges is learnt with much pain and forgotten again in a few weeks. In some cases it is desirable to teach "rule of thumb." I suppose we all learnt multiplication and division in that way, but here the value of the result was such as to justify the means. It is necessary to know how to obtain a product or a quotient at an early stage of our career; but it is not necessary to know how to obtain a product to a given degree of accuracy with the least possible amount of writing. A boy who really understood what he was doing could reinvent a method for himself if the possibility were suggested to him; he probably would not invent the shortest, but it is desirable to spend all the time we do merely to ensure that such a boy shall obtain his result by writing down 36 digits instead of 40?

As for the practical utility of the method, I do not believe that any computer would habitually use it. If he had a number of multiplications to perform he would use a slide rule, or logarithms, or some such tables as Crelle's, or a calculating machine, according to the nature of the work on which he was engaged. It is only for an occasional multiplication that turns up when none of these aids are at hand that the method would be employed, and, unless he were far more familiar with it than is the ordinary boy, the expenditure of brain power involved would more than neutralise any slight gain that might otherwise accrue. When the method is insisted on in school we have probably all met the case of the boy who writes out the work in full on some paper he hopes will not be discovered, and then copies out just that part of it which he thinks his instructor will like to see. One of the aims we profess to have set before ourselves in our recent reforms has been to teach boys what is interesting and what is likely to be practically useful. The practical utility of these methods - if any - ceases for a boy when he begins to work with a slide rule or logarithms. I have yet to meet the boy who will even pretend that they present any interest whatever to him.

I am sure that the time and labour devoted to these contracted methods might be much more advantageously employed. We have to frame our course to suit the average, or even the stupid boy, and I fear that in any case we shall leave him "unable to deal neatly with masses of figures out of which he requires to obtain a result to a moderate degree of accuracy" when he is denied access to a logarithm book. Let us rather try to teach him something which experience shows he has a chance of understanding, or of remembering, and which is more likely to be of some practical utility.
M A Blagg and S A Saunder, Collated List of Lunar Formations Named or Lettered in the Maps of Neison, Schmidt, and Mädler (Neill and Co., Edinburgh, 1913).

Introduction by Herbert Hall Turner.

In December 1905 the late Mr S A Saunder drew attention to the present very unsatisfactory state of Lunar Nomenclature (Monthly Notices of the Royal Astronomical Society 66 (), 41), concluding that:-
If a remedy is to be found which will meet with universal assent - and nothing short of this would be a remedy at all - it is obvious that it must be the work of an international committee.
His representations, supported first by the Council of the Royal Astronomical Society, and next by that of the Royal Society, reached the International Association of Academies at its Vienna meeting in 1907, when a Committee on Lunar Nomenclature was appointed by the Association, consisting of MM Loewy (Chairman), Franz, Newcomb, Saunder, Weiss, and Turner (Secretary). The name of W H Pickering was subsequently added, and later, on the death of the Chairman, the names of MM Bailaud and Puiseux. The Committee has further lost Newcomb, Saunder, and Franz.

Before his death M Loewy had, after some preliminary discussion, asked MM Franz and Saunder to undertake the preparation of an accurate map of the Moon in mean libration. Franz undertook the outer portions: he set to work on new measures required for the fundamental points, and completed this work, which was in the press at the time of his death. It has since been published under the title Die Randlandschaften des Mondes (Julius Franz, Ehrhardt Karras, Halle a Saale, 1913). The drawing of the map had not been commenced, but it is hoped that this work will still be undertaken in Breslau.

Saunder secured the able help of Mr W H Wesley for the actual drawing of the map, for which Saunder himself laid down all the fundamental points. Three of the four portions have been drawn and reproduced; the fourth is well on the way to completion.

Meantime Saunder had also secured the devoted help of Miss M A Blagg in collating the list here printed of names in Beer and Mädler, Schmidt, and Neison.

The severe losses sustained by the Committee have combined to leave the nominal direction of it in the hands of one who is not in any sense a selenographer. On his deathbed Saunder handed to me this collated list in manuscript, and I gathered that, although it was of great value and represented much careful labour, considerations of cost had deterred him up to that time from printing it. As soon as I had had time to review the situation, I realised the great advantages that would follow from printing this list; and found further that Miss Blagg would undertake to see it through the press. She made no special conditions as to time, but I felt that such valuable help might not remain permanently available, and that it was desirable to seize the favourable moment if possible. Professor Schuster and the Astronomer Royal kindly encouraged this course. Some anxiety about the funds for printing has been dispelled by the great courtesy of the Paris Academic des Sciences. I ventured to ask M Baillaud whether he thought it appropriate that as England and Germany had undertaken the map, France should defray the expenses of this collated list. M M Baillaud promptly laid the matter before the Secretaire perpetuel, and with M M Darboux's kindly and powerful support the request was favourably entertained. I take the opportunity of tendering the grateful thanks of the Committee for this piece of international and inter-academic courtesy.

The list is throughout the work of Miss Blagg. Of her great care and thoroughness I had often heard Saunder speak, and I have had some opportunity of appreciating them at first hand during the passing of these sheets through the press.

The list will in the first instance be used by the Committee (and others whose help they may be fortunate enough to obtain) in adopting names for the formations here tabulated. For this purpose wide margins have been provided. The project of keeping the type standing until the adopted name could be added in print was rejected, after consideration, on the score of expense. It seems better to contemplate the subsequent printing of a new list giving the adopted name corresponding to each number. These numbers will probably also be entered on copies of the map, but this procedure is not yet quite settled.

H H Turner
Chairman of the Lunar Nomenclature
Committee from 1910.

University Observatory
Oxford
November 1913.

GENERAL EXPLANATION.

I. Name-Prefixes.

In deciding what name should be prefixed to each letter in this list, my aim has been, in the first place, to follow any indication, direct or indirect, given in the different texts. When no such indication could be found, I have been guided mainly, in Mädler's case by the position of the letter, and in Schmidt's and Neison's cases by their clear intention of preserving in the main Mädler's notation. In the many cases in which these considerations were inapplicable or insufficient, I have tried to choose the name-prefix which would be most likely to be selected by anyone using the maps. Where there appeared to be much doubt on this point. I have given an alternative prefix, placed a query mark after the prefix, or appended a note. A query mark before the prefix implies doubt as to the identity of the objects in the different maps.

II. Schmidt's Letters.

Schmidt does not appear to attach much importance to the name-prefix, and sometimes himself uses two different ones for the same object. He also occasionally uses two different letters for the same peak, sometimes without calling attention to the fact. The exact position of such of his letters as appear only in his list of height-measurements, and not in the maps, must be considered more or less doubtful, particularly in those cases in which his descriptions of their positions seem to be inconsistent. He expresses himself as doubtful in regard to the position of some of them himself. I have interpolated some of these letters, instead of giving them a separate number. They may sometimes be aliases for some other letter in the list.

III. Schmidt's Rills.

Schmidt did not, I think, intend his rills, as a rule, to be named in this way, but simply marked most of them "r" to call attention to them. I have, however, generally put them in the list, using as a prefix the name of the nearest named formation, for the sake of convenience.

IV. Identity of Objects.

When names or letters in the different maps are given to objects which clearly represent the same formation, I have entered them as identical, although the Lat. and Long, (and also the appearance) of the object may sometimes differ considerably in the different maps, especially near the limb.

V. Form of Letters.

The Roman letters on Beer and Mädler's, and on Neison's maps, are in "printing" form; those on Schmidt's maps in "writing" form. This difference is indicated in the list in the case of Schmidt's small letters by printing them in italics. No difference has been made in the capitals. The small "a" and capital "E" in Schmidt's maps often look very like the Greek a" and "e," and there is some danger of mistaking them for each other.

VI. "Not Named."

Objects entered in the list as "not named" are sometimes not even shown at all in the map in question. It would have involved much extra expenditure of time and trouble to note this in each case, as it is often very difficult to say whether an object is shown or not. Sometimes a mountain in one map is a crater in another; sometimes a rill becomes a mountain chain; sometimes there is a vague mark which may represent the object, or may not; sometimes a similar object appears a few degrees away.

VII. Position of Objects.

The descriptions of position are merely rough indications, sufficient to identify the object in maps in which the letter is given. "N.E." means "between North and East," "S.W." "between South and West," and so on.

VIII. Proper Names.

Familiar geographical names (such as "Alps ") are entered in the list in the same form in each column, in order to avoid any appearance of disagreement where none exists, although Mädler and Schmidt of course write such names in German, and Neison in English. As the list is written in English, I have given them in the English form. But, in the case of names of persons, and classical geographical names, differences in form are not due to difference of language, but (presumably) to difference of opinion on the part of the authors. I have therefore aimed at giving these different forms exactly as they occur, without altering or modernising the spelling, except that in a few classical names "ae" has been printed as a diphthong, although Mädler and Schmidt always separate these letters.

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