SECTIONS 18 – 24

Section Eighteen

In §1 Ptolemy makes direct reference to the commentaries of Marinus – 'Ta µen oun kau' auuh uhn isuorian ojeilonua uucein uinoV episuasewV upoueuupwsqw µecri uosouuwn' and then to his own treatment of them – 'All' ina µh doxwµen uisin ensuarin proceirisasqai kai µh diorqwsin, esuai hµin ekasua di' auuhV uhV kaua µeroV ujhghsewV dhla,' and finally to what he is attempting to achieve – 'loipon d' an eih ua kaua uhn epodon uhV kauagrajhV episkeyasqai.' Thus, having told us that the work of Marinus is fine as far as it goes, that he himself has not taken any censorious attitude to it but that it is intention to turn it into a proper treatise on map–making, he proceeds to indicate the options that are open to the would–be map–maker.

In §2 he describes the two alternative options; inscribing cartographic data on to a sphere or on to a flat surface. Either, he says, will provide what has not been available before; a perfect representation that is easily achieved directly from the commentaries. He asserts that, by re–creating a map from the source material will avoid the errors that are transmitted by repeated copying from previous maps. We can deduce from this that maps, of a kind, are already in existence but presumably without annotations and grid lines that would enable their accurate reproduction.

§3 comments that this can, and will, only be achieved from proper source data; if there is no data then there is no location to be inserted on the map. Guessing and collective agreement will be eliminated and if there are errors then they will be easily found.

In §4 Ptolemy stresses that a value for both longitude and latitude must be available before any entry is made. He comments that this is not always easy to ascertain from the commentaries of Marinus since they are often recorded in separate places; sometimes only one or the other is to be found. Ptolemy insists that both must be present and clearly associated with one another against the location in question. It is an interesting comment on the part of Ptolemy that leads one to wonder what kind of presentation of navigational data would need to separate the co–ordinate of longitude from that of latitude. Tabular data springs to mind but under what sort of circumstances would it serve? Two sets of tables, one for longitudes and one for latitudes would supposedly serve where navigation was by the pole star or by the sun, directly along meridian lines or along parallels or where journeys were made along two sides of a right–angled triangle rather than the hypotenuse. If this were so any variation in nomenclature between the two tables would quickly lead to confusion. Is this the source of Ptolemy's complaint?

In §5 Ptolemy asserts that he has been assiduous in searching through the whole of the commentaries in order to ensure that such elusive data is not lost forever. This would seem to indicate that Ptolemy had no alternative source material to consult and may well be the reason why he does not correct, but merely draw attention to, suspect data.

§6 indicates that Ptolemy considers Marinus to have given more thought to the organisation of locations on coasts than he has to locations inland. By which we must assume that he is speaking about spatial distancing inscribed on a map; his coastal locations have some relativity with one another, inland locations are placed haphazardly. Two matters for thought arise out of this statement. Firstly, navigational data from mariners would naturally be more precise where coastal locations and conditions are involved and relative distances, in the case of adverse conditions, would be critical. Inland locations would not be so critical in this respect. Secondly, it prompts the thought that tabular additions to maps, showing separate longitudinal and latitudinal co–ordinates, would serve to unscramble such haphazard plotting and avoid the necessity for exact positioning within the body of the map.


Section Nineteen

This section is concerned with the standards that Ptolemy wishes to adopt in constructing maps. In §1 he states his twofold intention; to bring to light the commentaries of Marinus so that there original purpose will be fulfilled and to ensure that they are brought up to date by new information available since his time and from any maps and commentaries that have subsequently been produced. This last comment can only be a reflection on the expansion of the Roman Empire and the accompanying military surveys that would have been undertaken. Northern Scotland must be a point in question, although with its obvious misalignment with the adjacent region, one should treat with caution any other similar correction by Ptolemy.

§2–3 are concerned with the positioning of by means of specific co–ordinates of longitude and latitude and the correct numbering of these. Ptolemy directs that, for map–making, the co–ordinates of longitude be numbered from 0° in increments of one, commencing with the furthest western boundary of the known world and proceeding eastwards. The co–ordinates of latitude should similarly be numbered commencing at 0° at the equator and proceeding simultaneously northwards and southwards. Any position within the habitable world can then be located where a specific line of longitude and latitude intersect. This passage serves to underline the fact that this was a new departure in map referencing and that previous map–making techniques used some other method.


Section Twenty

In this section Ptolemy comments on the two methods of depicting the world in map–making; on a sphere or on a flat surface. He quotes Marinus as preferring the latter method but of not giving due consideration to the effect of depicting a curved, spherical surface in terms of a two–dimensional, flat surfaced reproduction.

In §1–2 Ptolemy discusses the disadvantages of each method while in §3 he quotes Marinus as dismissing spherical depiction as allowing insufficient space for locations to be placed and for the difficulty in retaining the whole of a region in view at one time. In §4 Ptolemy criticises Marinus for drawing parallels (on a flat surface) as straight lines instead of curved lines. His point being that, by fixing the eye on one part of sphere, a longitudinal line can be made to appear straight but a parallel line is seen to be curved to the left and right of the immediate point of vision. He feels that depiction on a flat surface should reproduce this fact. This, of course, is the eternal problem that faces all map–makers. If Ptolemy's strictures are observed then the maps that are drawn are in the nature of flattened out cones, where the likeness of terrain is maintained but where it is difficult to retain perspective. If the advice of Marinus is taken then they introduce complex factors of distance distortion which, while they allow perspective, do not allow the exact relation of locations. The shape of the land may be made accurate or the size of the land be made relative, but not both together.

In §5–7 Ptoleny draws attention to the fact that while the lines of longitude converge at each of the poles, the lines of latitude are parallel to one another. He therefore argues that the total length of each line of latitude decreases the nearer it is to its respective pole. He points out that this must mean that the space encompassed by the intersection of adjacent lines of longitude with adjacent lines of latitude must also decrease in area the nearer it is to the pole. While this phenomenon is to be observed on his flattened cone method, he criticizes Marinus for making all such intersection regular and equal. In §8, Ptolemy gives the relative proportions in length of the parallels of Rhodes and Thule to that of the equator.

This section serves to illustrate the pragmatic attitude of Marinus to the theoretical approach of Ptolemy. Maps, to the generations that Marinus represented, must have been in the nature of navigational aides–memoirs; Ptolemy intended that they be made to represent exact circumstances in themselves. It was not until the fifteen century that a reasonable compromise was to be achieved.


Section Twenty–one.

In this short section Ptolemy lays down that his flattened cone projection should be the one to use. §1 begins with the stricture – 'KalwV an oun ecoi dia uauua uaV µen anui uwn µeshµßrinwn graµµaV uhrein euqeiV, uas d' anui uwn parallhlwn en uµhµasi kuklwn peri en kai uo auuo kenuron grajenwn, aj' ou kaua uon ßoreion polon upouiqeµenou diageiw dehsei uas µeshµßrinaV euqeiV.' Ptolemy is insistent that this method of straight longitudinal lines and curved latitudinal lines be maintained to preserve as much as possible the features of a spherical representation. In §2 he suggests that the line of latitude through Thule and the line of the equator itself are sufficient boundaries for the habitual earth to be fully represented. However, he emphasises that the latitudinal line through Rhodes should always be shown since, from this particular parallel, most of the known co–ordinates depend. This is a most significant sentence. It explains much that is puzzling about Ptolemy's latitudinal co–ordinates in general. J.J. Tierney, JHS LXXIX 132/48, apropos the latitudes in Scotland, suggested that the base for calculation of these latitudes was not the equator but somewhere in the region of the parallel of Marseilles. Ptolemy's comment would, as well as covering this point, would also explain all the other discrepancies in latitude for all locations north of the latitude of Rhodes (36°15˘). §3 emphasises the need to always observe the properties of spherical representation.

Section Twenty–two

In this section Ptolemy reverts to the problems of map–making on a sphere and describes a method to make it easier to achieve. In §1 he comments that the actual size of the sphere used must obviously determine how much detail can be entered as regards locations and prominent features. In §2–4 he remarks that however great the skill in inserting these, certain aids are required. He directs than two semi–circular arms be constructed, one to follow a vertical line from pole to pole on which the one hundred and eighty divisions of latitude are to be marked; one to follow the line of the equator on which the one hundred and eighty divisions of longitude are to marked. He also directs that such semi–circular arms be as thin as possible so that their bulk does not obscure detail on the surface of the globe. The intention being that the arms and the globe itself be able to move independently of each other, but whether the globe remains fixed and the arms moveable, or vice versa, is not made clear.

In §5, Ptolemy describes how locations can be transferred from the commentaries on to the surface of the surface of the globe using the markings on the arms, by revolving one ot the other until the correct longitude is showing and then by reading of the correct latitude. The point where they would intersect is the location in question. In §6, Ptolemy remarks that, of course, instead of the one hundred and eighty divisions, the arms can be marked off in any number of divisions that may be required representing different intervals of degrees. With such arms one is not restricted, as one might be using straight lines on a flat surface.


Section Twenty–three.

In this section Ptolemy describes the classification of the lines of longitude and latitude within the confines of the known, habitable world. In §1 he notes the boundaries. The longitude extends twelve hours, west to east, and he directs that a longitudinal line should occur every one third of an hour; that is every five degrees. The latitude extends from a southerly point sixteen degrees and twenty–five minutes south of the equator to the most northerly point sixty–three degrees north of the equator but that the lines of latitude should display irregular intervals. In §2–23, he groups them in eight distinct hours and the intervals grow smaller as they approach the North Pole, thus:

South 1 Hour = 16° 25˘ Interval = 16° 25˘

Equator 0 Hours – –

North 1 Hour = 16° 25˘ Interval = 16° 25˘

2 Hours = 30° 20˘ Interval = 13° 55˘

3 Hours = 40° 55˘ Interval = 10° 35˘

4 Hours = 48° 30˘ Interval = 7° 35˘

5 Hours = 54° 0˘ Interval = 5° 30˘

6 Hours = 58° 0˘ Interval = 4° 0˘

7 Hours = 61° 0˘ Interval = 3° 0˘

8 Hours = 63° 0˘ Interval = 2° 0˘

Ptolemy's reasoning for this irregularity in the intervals of latitude is not given except, by inference, in his remarks about the parallel nature of latitudes, the curvature of the earth's surface and the convergence upon the poles of the lines of longitude. We are left to infer that the irregularity is the result of a mathematical formula to allow for these factors. However he does equate an hour of latitude to a specific 'clime' or 'climate' and in the instances where he differs with Marinus in the interpretation of a latitudinal position he usually draws attention to its being in the wrong climate rather than to any difference in actual distancing.


Section Twenty–Four

This is the final section of Book I and is concerned with instructions on how to construct a map projection on a flat surface. Ptolemy takes us step by step through each phase of the construction.

In §1 he directs that a rectangle be formed, (Figure I), in which the horizontal sides are twice as long as the vertical sides. Within this area is to be contained the co–ordinates of the known, habitable world. In §2 he directs that this rectangle be bisected by a vertical line which is then extended as a straight line above the rectangle so that this extension forms a proportional extension of thirty–four parts to the total length of the extended line of one hundred and thirty–one and five twelfths parts. Using the extreme top of this extension as a centre point, he directs that an arc be drawn, within the confines of the rectangle, at an interval of seventy–eight parts measured along the extended vertical line. This he requires to be designated as the parallel that passes through Rhodes.

In §3, Ptolemy directs that we fix the extremes of longitude to the right and to the left of this extended, vertical line. The manner in which this is achieved is by measuring four parts on the extended vertical line (which he claims will represent five degrees of longitude) and, using this distance, mark out on the arc of the Rhodes parallel eighteen such intervals on each side of the vertical lines bisecting the arc. Thus, he says, we will have inscribed the area of the known world at five degree intervals and the extreme points reached, on either side, will be its east/west limits. These two limits can be inscribed by drawing a straight line from the centre point of the arc to each of the vertical sides of the original rectangle.

In §4, he directs that three further arcs be drawn, this time between the two east/west extremes, to represent the parallel of Thule at fifty–parts on the vertical extended line, the equator itself at one hundred and fifteen parts and the most southern limit of the known, habitable world at one hundred and thirty–one and five twelfths parts.

§5–6 are concerned with the ratios and proportioment of the intervals of latitude. He points out that the relationship between the length of the equatorial arc to the arc through Thule must be as between one hundred and fifteen to fifty–two. He deduces also that the difference between the parallels of Thule and Rhodes will be twenty–seven, between Rhodes and the equator will be thirty–six and between the equator and the southern limit will be sixteen and five twelfths. Thus he claims that the extent of the latitude of the known world is therefore seventy–nine and five twelfths, which he is prepared to round up to eighty, and, on the parallel of Rhodes, the extent of longitude will be one hundred and forty–four. He appends the distances as forty thousand stadia for the former and seventy thousand stadia for the latter, presumably by extending at five hundred stadia to the degree. He ends this paragraph by pointing out that any parallel can now be inscribed using the same centre point and marking out the requisite number of parts.

§7 is concerned with parallels south of the equator, pointing out that they cannot be drawn as extensions of those of the northern hemisphere but must adopt a similar orientation towards the south pole. §8–9 describes the method of correctly placing locations on the map itself. Ptolemy direct that a straight ruler be obtained, the length of the extended vertical line with a notch cut where the centre point of the arcs is situated. From this point the ruler is to be marked into the required intervals of latitude. The ruler can then be swung around so that it will lie on any line of longitude where the required latitude can be read off and marked on the map.

In §10 Ptolemy points out that the above exercise only yields an approximation of the true situation and that a more correct reflection can be gained if the lines of longitude were curved also. Then, when longitude and latitude meet at a location point, the perspective of the mind's eye would be in keeping with the reality of the curvature of the earth. In §11, he first directs that we construct an armillary representation, (Figure II), by inscribing a circle and by marking the cardinal points of north, west, south and east with a, ß, g, d, respectively. He then directs that this circle be bisected, vertically and horizontally, by two arcs whose point of intersection is to be marked, e, the two arcs therefore having the configuration a e g and ß e d. He then directs, in §12, that along the arc a e g, and from the point e in the direction of g, an interval of twenty–three and five sixths of the total length be marked and this point marked z. The arc of the equator should now be drawn from ß to d and passing through z. The intention of this exercise is to show how a parallel, when viewed by the minds eye, will show an inclination to the equator when seen on the surface of a sphere.

In §13, Ptolemy directs that the circle be constructed once again, (Figure III). with the same markings except that the bisecting lines a e g and ß e d be straight lines instead of arcs. Only the equatorial line ß z d should remain as an arc. He then lays down that the interval ß e to the interval e z is as the ratio of ninety to twenty–three and five sixths. In §14 he directs that the line g a be extended northwards to wards an unknown point to be designated h. In order to correctly locate h, it is first necessary to draw a straight line from ß to z, striking a chord to the arc already formed. At a point midway along this straight line mark a point to be know as q, and from this point draw a line perpendicular to the base ß z. At the point where it intercepts the line extended from g a, is the point h. In §15, he deduces that since ß e is ninety and e z is twenty–three and five sixths then, according to Pythagoras, ß z must be ninety–three and one tenth. He then deduces the angles ß z e and q h z to be seventy–five and one sixth degrees and fourteen and five sixths degrees respectively by referring to them as proportions of the three hundred and sixty degrees comprising the total angles within a square.

In §16, Ptolemy also deduces from this that the line h z has a ratio of one hundred and eighty–one and five sixths to the forty–six and eleven twentieths of the line q z, while the line q z has a ratio of forty–six and eleven twentieths to the ninety of the line ß e. Which he says means that the line ß e is ninety, z e is twenty–three and five sixths and z h is one hundred and eighty–one and five sixths while h remains the centre point from which all parallels are to be drawn.

§17–23 is now concerned with reproducing the rectangle with which this section began, (Figure IV), albeit with slight changes in the symbols used. However we are now directed to only put in place the arcs representing the parallels of Thule, Syene and the southern limit of the known world. Along these arcs, both to the west and to the east we are then required to mark off the eighteen intervals at which the lines of longitude occur. These intervals will vary between the three parallels drawn; thus the parallel of Thule requires intervals of two and a quarter, Syene of four and seven twelfths and four and five sixths at the southern limit. Lines drawn through each set of three points will in fact describe an arc, the centre of which Ptolemy does not reveal. Connecting all eighteen sets of points to the west and to the east will produce a hemispherical segment of the earth's surface showing relevant curvature for both Latitudinal and longitudinal lines. The exercise finishes with a direction to complete the remaining arcs of latitude.

In § 24–29 Ptolemy compares the advantages and disadvantages of an entirely flat presentation of a map and one in which the lines of latitude and longitude observe their natural curvature. For himself he prefers the latter but he comments that the former is likely to be the one preferred. §30–33, ending Book I, contain references to the intervals and ratios of the solstice readings at Meroe, Rhodes, Syene and Thule.

Summary of Book I

In examining the text of Book I it seems quite clear that it is not intended by Ptolemy to be a prologue to a text book on theoretical geography at all, but a treatise on the art of making maps. Beyond the opening two paragraphs of section one, what evolves is a detailed analysis of map–making techniques up to his own time, followed by a synthesis of what he considers it should become and then, as a succinct summarisation, a set of instructions for making it so. The remaining Books, II to VIII are then intended to provide lists of data that will enable these instructions to be used to provide sectional maps of the whole of the known, habitable world.

Sections 1 to 5 quite clearly describe what maps are, how the information they contain is collected, in what form it is collected and in what way it may be represented, how maps can be used and how they should be updated. Nowhere is there any discourse upon geography as a pure study; it considers only the functional and descriptive aspects.

Sections 6 to 18 are likewise devoted to the practical realisation of map–making; the state of the art up to the time of Ptolemy, the methods used, the regions explored and the distances computed. Methods of navigation, terrestrial and celestial fixing of locations, the manner of indicating precisely a location and its relation to other locations, the concept of meridianal lines relative to the poles and to the equator and the coining of the terms 'Longitude' and 'Latitude'. Commentaries on previous map–makers and their work, discussion, criticism and correction of their findings and the re–calculation of the extreme boundaries of the known world. Once again there is nothing that can be even loosely attributed to pure geography. It is all essentially a recapitulation of knowledge distilled from merchant travellers and navigators over a long period of time; there is little in the nature of any rationalisation beyond that necessary to convey information on routes and terrains. What does arise out of these sections which is of the utmost importance is the measurement of distance. It is apparent that Ptolemy arrives at his co–ordinates by first establishing the actual distance and then dividing this by a factor of distance attributed to one degree of longitude and latitude, in his case five hundred stadia to one degree. What is a disturbing factor is that in every comparison he makes with the calculation of Marinus, he claims that Marinus has overstated the distance. However, when such factors as expected deviations are allowed for, the difference almost always comes out as related by a standard factor of error. With uncanny regularity the ratio emerges as between 5 and 7, which is precisely the difference between the geodesic time bases of Posidonius and Eratosthenes. Whether this is sufficient to allow of definite conclusion must depend on further statistical work.

Sections 19 to 23 are devoted to the actual representation of images of the earth on to a plane or a spherical surface. Investigating the merits of both, the techniques of application, the compromises that each demand and the degree of reality that either conveys. The categorisation of lines of longitude and latitude, the concept of 'hours' and 'climates', the designation of variable intervals of latitude and rationalisation of the convex nature of the earth's surface to the needs of its imaging. As before, there is no need of pure geography here. It is all concerned with the actual techniques of making maps and introducing realism, scale and objectivity into facsimile creation.

Section 24 is devoted to the geometry involved in the projection of the reality of the earth's characteristics on to a suitable media format that will allow the best compromise between the two mutually conflicting map–making requirements of exact distance and correct spatiality. Despite its theme, this section is also remote from pure geography, being concerned only with the theory behind the assertions he has already made in the previous sections. Ptolemy restricts his attention to two formats; the entirely planular concept, favoured by Marinus, where all lines of longitude and latitude are made parallel and straight and with any distortion thus introduced subject to compromise, or the conic representation, favoured by Ptolemy himself, where the lines of latitudes are curved but parallel and the lines of longitude are straight but convergent on the poles. However, Ptolemy elaborates on his own model format by introducing the concept of the armillary sphere. By closely demonstrating its internal geometry, he proceeds to elaborate on his simple conic projection by producing one where the lines of longitude observe the nature of an armillary sphere, forming arcs, convexing towards the east or the west respectively, on either side of a central line of longitude which is always vertical and straight.

Thus Ptolemy creates a manual of instruction for potential map–makers. In these modern days of the 'Do it Yourself' concept it would no doubt be packaged differently but its intention would be precisely the same. No doubt Book I would be made available separately, as a 'key' volume, and the subsequent volumes supplied to potential purchasers as options, in order that they might map for themselves that part of the world most interesting or useful to their purpose. There is no doubt that, in its long history, the GewgjikhV YjhghsewV has been used precisely, and correctly, in this practical fashion. It has also, unfortunately, often been regarded as a textbook on geography, pure and simple, and the misconceptions thus introduced, subsequently heaped upon the innocent head of Ptolemy.

At best this misunderstanding was the result of a lapse in semantics; at worst, it was a serious error of scholarship that proceeded, against all reasonable understanding, to force a very clear and lucid text to speak falsely.