Cosmas Indicopleustes, Christian Topography (1897) pp. 244-262. Book 6 

BOOK VI. SUPERADDED.

 Regarding the Size of the Sun. [204]

FTER my work had been finished, some questioned us about the figure of the world, saying: "How can the sun possibly be hidden, as you hold, by the northern parts of the earth, which according to you are very high, while he is many times larger than the earth? For in the case of the sphere which we advocate, however much greater the sun may be than the earth, he will always, when giving light to one part of her surface, leave the other in shadow." To those so questioning us we have made a very brief reply, that such a thing is false and a pure fiction, having shown first from the climates  1  1 Before it was known that the earth was a globe, it was supposed that its surface sloped from south to north, and this slope was called kli/ma . But as science advanced, this term was employed to designate different belts or zones of the earth's surface, as determined by the different lengths of the longest days on their southern and northern limits. Hipparchus (about 160 B.C.), first used this division. The term kli/ma came afterwards to denote the average temperature of each of the zones, and hence the present use of the word. which they themselves, recognise, that the sun projects shadows between which there is no difference, and next from what we ourselves saw with our own eyes in the parts of Axômis in  |245  Ethiopia. For, at the beginning of the summer solstice on the twenty-fifth day of the month Payni at the sixth hour of the day when the sun is now at the meridian, we plainly saw that a man's shadow inclined to the south.  2  1 Mr. Ernest A. Floyer, in a letter sent from Egypt, which appeared in The Academy of 5th Oct., 1895, makes some interesting remarks about primitive sun-dials in Upper Egypt. He found that there the hours of work for a waterwheel were fixed by such sun-dials. "Two kinds were used. At Edfu a horizontal dhurra stalk lay north and south on two forked uprights. East and west were pegs in the ground, dividing evenly the space of earth between the sunrise and sunset shadows of the horizontal gnomon. Further south the gnomon was a vertical stick." Afterwards he says: "It is generally admitted that the Egyptians ascertained the length of the year to within a few hours. This approximation must have been obtained by measuring the shadow of a gnomon at the summer solstice, which coincided with their principal natural phenomenon, the rising of the Nile. It could hardly be otherwise. Given a constant and never-clouded sun, and a need to divide up the day, the upright stick is brought into use at once. But little time could have elapsed before it was noticed that the noon shadow was longer or shorter one day than the preceding day .... Thus the period from one summer solstice to another could not have remained long unknown". After tracing further the progress of discovery, the writer concludes with two remarks: "First, the effect of the use of a gnomon's shadow leaves its trace to this day on Arabic astronomy. Angles are called shadows. Secondly, measuring the exact length of a shadow on a somewhat rough plane was not easy. Perhaps better results were obtained by examining the faces of an obelisk. If the north face of an obelisk at Luxor sloped at an angle equal to the difference of latitude between Luxor and Syene, then at the summer solstice only all four faces would be equally illuminated." And when I was resident here in Alexandria, at the same time of the year on the twenty-fifth, that is of Payni, and until the thirtieth of Thôth at the sixth hour of the day when we were standing under the sun, I pointed out that the shadow inclined to northward only a single foot. And, according to them, Alexandria is the third climate. Now in this city there happened to be living that extremely pious and admirable man, Abbot Stephanus, the Presbyter of Antioch, a most Christian man and verily to be  |246  numbered among the perfect, and as he was accurately versed in lunar calculations, and moreover able, in accordance with the scheme of the world which we have laid down from divine scripture, to predict readily both solar and lunar eclipses, he willingly turned his attention to this subject. Accordingly when in the month Thôth of the current tenth indiction  3  1 Regarding indictions, see last note to Book X. he was asked by the erudite Anastasius, a man of science and learning, and superior to many in point of experience, to predict when an eclipse of the sun would take place, he stated that there would be one in that very indiction on the twelfth day of the month Mechir; and it did so occur. He predicted also a lunar eclipse for the twenty-fourth of Mesori in the same indiction. Anastasius was much surprised, and asked him next to tell what past eclipses there had been, and when he had declared them correctly great again was his surprise. There were present too some other scientific men, friends of ours, at this conversation. Now this God-beloved man, who has a curiosity to investigate such [265] matters, and who lived in Antioch and, for many years also, at Constantinople, told us himself that he had measured the shadows in both these cities, and assured us that at Antioch the shadow projected a foot and a half at the time already mentioned,  4  2 The height of a man who would cast a shadow of 1½ foot in the latitude of Antioch, when the sun was on the meridian "at the beginning of the summer solstice", would be about 6¾ feet. This, however, can only be considered approximate, as there is some uncertainty about the length of the foot used, and of the exact declination of the sun at a time now so long past. In computing the height of the man, a plus correction of 12' has been applied to the sun's declination at the present day, and this may be taken as approximately correct. while at Byzantium it projected two feet. Now Antioch is in the same climate as Rhodes, namely, in the fourth, but Byzantium is a little  |247  beyond the fifth. For it is in the following way they distinguish the climates. The first is, they say, the climate of Meroê; the second of Syênê; the third of Alexandria, the fourth of Rhodes, the fifth of Hellespont, the sixth of the middle of Pontus, the seventh of the river Borysthenes and the Maeotic lake, and the last of the Ocean. If therefore, as we have seen with our own eyes here, in the climate which, according to their own laying down, is the third, the shadow falls only one foot towards the north, and in the fourth one foot and a half, and in the fifth two feet, is it not manifest that the shadow is either lengthened or shortened by half a foot for each climate? And if this be true, as assuredly it is, the sun will be found to have the size of two climates and no more. For if, in the third climate, he throws a shadow of one foot, in the second he will beyond all question throw one of half a foot, while in the first he will throw none at all.  5  1 The sun is so far distant from the earth that its mean parallax is only about 8", and therefore the rays of light falling on to the earth may be considered as parallel lines; and from the diagram below it will be at once seen that if the earth were flat, as Cosmas supposes,  |248  But as we said by anticipation in the second book, there He between Meroê and the Ocean in the south many of the  |249  stages into which the earth is partitioned. For, from the Cataracts to the ocean we remember having stated that the number of halting-places in that distance was seventy more or less. But the number of climates which they suppose to exist they reckon at somewhere about twenty only.  ^{6  1 Ptolemy reckoned nineteen climates. See Ukert. Geog., vol. i, Pt. II, pp. 182 ff.; or Smith's Dictionary of Greek and Roman Antiquities, p. 297.} So then quite clearly the shadow in the climate of Axômê, a city of the Ethiopians, is found projecting more than a foot to the south, so that everything goes to show that, if the sun in his passage through the summer tropic be between Syênê and Axômis, he has the size of two climates.  7  2 The fallacy of this so-called proof lies also (as shown in note on pp. 247, 248) in the fact that Cosmas founds it on the supposition that the sun is near the earth. It falls through at once when we consider that the sun's rays are practically parallel lines, and the fact that the shadows vary in length, and in one place fall to the north and in another to the south, is owing to the spherical form of the earth. Is it not then false and fabulous what they say about the sun, that he is greater than the earth? And how too comes it, if, as they say, the earth is spherical, the shadow does not vary on the convex surface of the earth? For since they place the torrid zone in the centre, it follows of necessity that the parts on each side of that zone must be depressed. And they assert that none can inhabit the torrid zone----yea, even that the northern part of the world which is inhabited by us is many stages distant from the torrid zone. And I wonder again if, in those convex parts of the earth which they suppose to exist, the shadows are able to observe such a proportion that they increase or diminish by half a foot for each climate, as with God's help we have demonstrated and with our very eyes have witnessed to be the case, and have shown to the men with whom we have conversed----men by no means novices but adepts in science,  |250  and earnestly maintaining this opinion. But further again [266] they affirmed most positively that when the illuminating body is large and the body which is illuminated small, and each of them spherical, the shadow produced is beyond question conical----for the rays of the larger sphere, passing beyond the smaller on this and on that side, necessarily make a very acute cone; and they endeavoured by means of geometrical demonstrations to circumvent us; but on this point we very concisely by optical experiments again showed the falsehood of what they alleged; for, fetching a small wooden globe, we drove a nail into it by which we held it in our hands, and on stretching it out to the sun, we saw that the shadow was round and not conical. And we said to them, look you now how small the sphere is which we hold out, and how great, according to you at any rate, the sun is, and yet he does not make a conical but a round shadow.  8  1 Cosmas evidently misunderstood the phrase used: a cone casts a conical shadow when the rays are perpendicular (or nearly so) to its axis, and fall on a parallel surface, and the great distance of the sun prevents the cone of shadow from a sphere being observed, unless it be so small that it can be held more than 110 times its diameter from the surface on which the shadow falls. As the distance is increased, the circular shadow diminishes to a point, and then vanishes. And we made the experiment both at a short and at a long distance----and they found nothing to show the truth of what they say, but with their specious sophistries they delude the multitude. We, again, fetching a cone-shaped vessel, then showed them that a conical body produces a shadow conical like itself. And, it is the truth I speak, O most God-beloved Father, through the power of Christ they went away dumbfounded and sadly crestfallen, having been put to shame by our exposure of their fictions. And now behold, we also, in accordance with an art of theirs, having drawn lines and imprinted one for each climate as they are wont to do, are able, if first  |251  strengthened by the divine power, to show that the sun is not greater than two climates, in order that they may learn in this manner not to arch their neck proudly, but to bow submissively to divine scripture.

[267] The straight (horizontal) line is the earth----the nine lines drawn perpendicular to it are so many bodies standing each of them for one of the climates. The lines drawn from the sun to these bodies are the rays of the sun which, falling simultaneously upon the bodies on this and that side, produce the shadows as we have depicted them in ink. That particular line which descends straight down, since it falls upon the top of the body perpendicularly, produces no shadow for it is greater than the body and shines all round it.

The Lord also bears me witness in the Gospels, when He calls the country of the Homerites, which is not more  |252  than a two days' sail by sea distant from Barbaria, the ends of the earth. It is evident again that I am right from the climates which they acknowledge, and from the places of which Ptolemy speaks, he who made war against Ethiopia, and of whom we made mention in the second book. And from the shadows themselves which are produced in each climate, it is proved that the sun does not exceed in size two climates, nay, even that the earth is flat, as the delineation shows, and not spherical.  9  1 Cosmas is more plausible than lucid in his reasoning; his figure shows three climates from Syene to Axomis, but possibly he meant to say between Syene and Ethiopia. Taking the breadth of the two between Alexandria and Byzantium as about 635 geographical miles, this is his "size" of the sun. In order to appear with a diameter of about 33' of arc, its distance from the earth would be about 66,260 miles. But this is inconsistent with his argument from the length of the shadows. A gnomon 7 feet in height would cast a shadow of 11 inches at the summer solstice at Alexandria; of 1 foot 6 4/5 inches at Antioch; and of 2 feet 1 inch at the Hellespont, or 2 feet 2 inches at Byzantium; his measurements are, therefore, only approximate. But assuming them----that his gnomon was one of 7 feet, and that the earth were a plane, the sun's distance is easily deduced as about 4,400 miles, and his diameter as about 42 miles; a flat earth and a sun at a much greater distance than this would bring the shadows more near to the same length.

[268]  A dissertation by which it is proved more clearly and comprehensively that, throughout the whole of divine scripture, we are taught that God has made two states  ----  the present state and the future. 

First of all, it is proved from the account of the creation that God divided the one place which extends from the  |253  earth to the first heaven, by placing in the middle the firmament, that is, the second heaven, thus making the one place into two places.

Secondly, it is proved from the very structure of man, inasmuch as he consists of two, namely, of mortal and immortal, evidently subjecting him through their contrariety to a life of conflict, in order that he may afterwards be honoured with gifts.

Thirdly, through the two trees which grew in the midst of Paradise, scripture shows that there are two states, one mortal and mutable, and one immortal and immutable. For the tree of the knowledge of good and evil is a type of this world as a school of discipline----and the tree of life is a type of the future state, in which neither death nor change has any power.

Fourthly, because the expulsion of the man from the Garden, by God, and his warning him and saying:  Lest he should stretch forth his hand and touch the tree of life and eat and live for ever;   10  1 Gen. iii, 22. these are the words of one who imparts knowledge, and obscurely hints that some gift of life eternal is reserved for men after the life of conflict here.

Fifthly, again, we are taught through Lamech the father of Noah, who prophesied and said that his son would deliver us from toils and the curse, and would conduct us to the enjoyment of a better condition.

Sixthly, again, through this Noah we learn that from that first earth, all miserable and thorny, we were transferred to this earth, which we men now inhabit: an earth that is better, and almost equal to Paradise, whereby we are taught the difference between the first and the second state.

Seventhly, we learn that God chose Abraham out of the  |254  land of the Chaldaeans and transferred him to Palestine, and promised to give him great gifts and possession of the land, without giving him, however, so much of it as to set his foot on, according to divine scripture, and trained him to have faith, so that he knew to expect with confidence gifts to be bestowed afterwards. And in like manner He showed through Ishmael and Isaac the worth of the two covenants----namely, that one was servile, and the other free.

Eighthly, a second time through Isaac and Rebecca, when twins were born, it was said:  The elder shall serve the younger,   ^{11  1 Gen. xxv, 23.} thereby again showing there are two states, one of bondage and the other of freedom.

[269] Ninthly, in the case again of Jacob and of Joseph, who could not bear to be buried in Egypt, but only in the Land of Promise, it was shown how each of them longed to obtain from God the hope which had been promised them----namely, of the second state.

Tenthly, the exodus itself in the time of Moses and the deliverance from the bondage of the Egyptians, and the freedom [thus obtained] and the entrance into the Land of Promise, accomplished through Joshua the son of Nun, clearly signified beforehand the two states.

Eleventhly, in like manner the construction of the Tabernacle and of the Temple, which, by means of the veil placed in the middle, was made from one into two----an inner and an outer----prefigured this state and the future state.

But what need is there that I should speak of the dispensations that existed through good men, for instance, the confidence Abel displayed after death, who yet speaketh, crying from the earth in bringing to birth the return of life.

If any reflects about the translation of Enoch, who  |255  because he pleased God, was translated that he should not see death, knows with the utmost certainty that there will be a better state, from an occurrence of this nature. Any one who is filled with astonishment, as he thinks of the ascension of Elijah and his passage in a chariot of fire, must see how much the nature of humankind has been honoured, as, it is thereby taught and has the hope, that it can pursue its way to heaven.

Any one who heard the multitudinous host of the heavenly ranks singing in chorus and rejoicing and exulting at the birth of our Lord Christ according to the flesh, and saying:  Glory to God in the highest, and on earth peace and good-will to men,   12  1 Luke ii, 14. would be astonished beyond measure as he reflected that the inhabitants of heaven and of earth had joined in making one confession, and that God was well pleased with men.

Any one who had seen the power of the resurrection from the death of our Lord and Saviour Jesus Christ would, with all his soul stirred up, celebrate without ceasing God the Saviour of all, who made the conflict of body and soul to cease, and brought into concord these two antagonistic parts of our nature, making them in things spiritual mutually conformable.

Any one who takes into view the ascension into Heaven of our Lord Christ, and the angels clothed in white raiment announcing to the Apostles his second coming, would never cease magnifying Jesus who showed and opened up to men the ascent into the future heavenly state, which He had prepared from the foundation of the world;

Any one who looks at the descent of the Holy Spirit upon the Apostles, and the speech of the great Peter in the midst of all the people, would in praying extol God [270]  |256  the giver of all with loud praises, who gave such an earnest and pledge of the future state to men.

Any one who contemplates the calling of the great Paul made from heaven, and his being caught up into the third heaven, and again into Paradise, will lift up his voice to God and say: Glory be unto Thee who through all and unto all graciously givest gifts to men!

Any one who reads the writings of the company of the prophets and priests and righteous men, and also of the Apostles and Evangelists, and thinks of the miracles recorded by them, and of the prophecies and their exact fulfilment, will find that they make mention only of a first and a second state, and have not enumerated any others, as coming either before the first or after the second. Glory to God who spake and prophesied through them all, for ever and ever. Amen!

Any one who preserves deep in his heart remembrance of the second illustrious and glorious coming of our Lord and Saviour Christ, and who takes into mental account His blessed resurrection from the dead, and the change for the better which the world has undergone, and the unspeakable joy and hope laid up for men, will many a time beyond measure admire and extol in songs of praise the Cause of all, the Creator and Restorer of the universe. Yea, rather he will not be able in adequate terms to address Him, who is above all praise and glory and tribute of song, and who, in his supreme goodness from the beginning suitably for us, founded the two states, educating us in the first, and by instruction and reason teaching us to act prudently, and leading the human race and through it the whole creation into the second state. Glory to Thee, glory to Thee, glory to Thee, O thrice Holy, Creator, Maker of all and Restorer of the universe, for ever and ever. Amen!

By all these considerations the opinion of the Christians  |257  is shown to be the best of all, and in accordance and harmony with the constitution of the world, and to be most true. It is in accordance with what those who are real and not pretended Christians acknowledge, that from the beginning God not unsuitably, but for apposite and useful ends, divided the one place into two places, preparing and making ready beforehand this place for this present state, and that other place for the future state. The Pagans accordingly, who suppose the world to be eternal, and deny the resurrection of our bodies and their ascension into heaven, looking upon this as foolishness, are properly, in view of their theory, deprived of future honours and of the ascension into heaven. The Jews, who read Moses and the prophets and understood not the great and wise dispensation of God, which through the prophets shows that from the beginning God made two states for the benefit of men, and who did not receive the Prince of the second state----nay, He proved even a stumbling-block [271] to them; nor, even until now do they believe that there will be an ascension of men into heaven, but they expect, forsooth, one to come whom they call the Anointed, and who, they hope, will reign over their country, and subject to Himself all the nations, and make no account at all of things heavenly. They thus reject the counsel of God and think that the habitation in the heavens has been prepared without occasion, which Moses, however, shows to have been made not without occasion, saying that it was made on the second day, and that the firmament was placed in the middle, and that two places came to exist; and they themselves also are excluded from this habitation. The Samaritans in like manner, being ignorant of the same things as the Jews, and not believing, and entertaining doubts besides, about the resurrection of the body and the immortality of the soul, are also like the Jews banished  |258  from the habitation in the heavens. The Manichaeans,  ^{13  1 Manes, or Manichaeus, their founder, appeared about the year 270 A.D. The sect spread chiefly in Persia, Arabia, Egypt, and other parts of the East. St. Augustine at the age of twenty became a Manichaean, and continued to be one for nine years. See note 1, p. 242.} holding the same opinions as the Pagans, and supposing also the heaven to be spherical, and expecting the utter destruction of the body, these, along with their evil deity whom they elected for themselves about the earth, are condemned and driven away from the City above, which they denied. In like manner, every heresy which denies the perfect humanity of Christ, namely, a rational soul endowed with intellect, and a body, and which does not acknowledge all the distinctive qualities of the soul and of the body, but wavers in doubt; and the heresy which denies the divinity of Christ and impairs it, and which says that either He or the Holy Spirit is inferior to the Father,  14  2 The Arians. all these fail to attain the heavenly mansions.

Blessed then are all those who, through the divine scriptures of the Old and New Testament, recognize the Maker of all things as one God in three Persons, namely Father and Son and Holy Spirit, a holy Trinity, consubstantial, equal in power, in strength, in glory and in honour, and without beginning, and who also recognize the great and wise and omnipotent dispensation of Him who is the Cause of all things; how wisely and harmoniously He established the two states from the beginning----the present state and the future----and through the perfect humanity of Christ who is like to us in all things both in body and soul, sin only excepted, and who sojourned among men in the last times when He declared and showed the future state, and gave assurance unto all by raising up that humanity from the dead. All the perfect therefore  |259  who walk by this rule, peace be upon them and mercy, and at the judgment of God these shall of right hear Christ the Lord in the future state saying unto them from heaven:  Come, ye blessed of my Father, inherit the Kingdom prepared for you from the foundation of the world.   ^{15  1 Matt, xxv, 34.} To Him be glory for ever and ever. Amen!

The whole scope accordingly of the divine economy is this----for we must briefly recapitulate what that scope [272] is----God of His goodness willed to let others participate in existence, and power, and reason, and knowledge, but, because those who participate in these cannot possibly know and possess all things in combination (for this belongs to God alone, to know all these things without learning and experience, being of Himself both power and reason and knowledge, while created beings participate in all these things from their Maker), He made those two states together, the first mortal and mutable, for the trial and discipline of rational beings, in order that, being exercised by the variety of the universe and the juxtaposition of contraries, we might have experience of pleasures and pains; then the second state, an immortal and immutable enjoyment of His gracious goodness, to the end that, from our acquired power of discernment, we may receive possession of the pleasures. To Him be glory for ever and ever. Amen!

No religion therefore, neither the Judaic, nor the Samaritan, nor the Pagan, nor the Manichaean, believes or hopes that there is a resurrection or an ascension into heaven for men; but such of these religions as think that heaven is a sphere, namely the Pagan and the Manichaean, are consistent with themselves in holding their unbelief. For, where are they able to find a place in the sphere for the kingdom of heaven? They are both of them  |260  therefore justified in denying the resurrection of the body, and in saying that souls only are glorified or punished after the life here----or in saying, as some of them do, that souls are whirled round along with the sphere and see all things, or are condemned to be cast into Tartarus. Some of them even hold that souls are transmigrated into other bodies, while others deny the soul's immortality, and not one of them possesses any hope of the body after the present life. But those religions which do not believe in a sphere, namely the Jews and Samaritans, but think that the firmament, which is in the middle of the one place and thus makes two places, was made without occasion, these also, it necessarily follows, have fallen under the sway of unbelief; for the Samaritans acknowledge neither a resurrection nor an ascension into heaven, nor admit that the soul is immortal, but think that our race will continue for ever, while the Jews acknowledge a resurrection for men, but say that we shall live upon the earth and eat and drink and marry and be given in marriage, as, in the Gospels, the Sadducees proposed a question to the Lord, saying:  In the resurrection of which of the seven shall she be the wife?   ^{16  1 Matt, xxii, 28.} These the Lord very summarily convicted of error and of not knowing divine scripture, in which there is the power of God, saying to them:  For, in the resurrection, they neither marry, nor are given in marriage, but are as the [273]  angels of God in heaven.   ^{17  2 Ibid., 30.} It is therefore proved against these, both from what is written and from the figure of the world prepared from its foundation, that the upper place was not made without occasion----but that there is a second heavenly state prepared from the foundation of the world. And, in like manner, every heresy among the Christians can be refuted; those also among the Pagans, the Manichaeans for instance, which suppose the  |261  heaven to be a sphere, or those which embrace the theory of an antecedent life, or those which deny that in Christ there was aught of our nature, either body or soul or intellect, do greatly err, neither knowing the scriptures nor the power of God. For those only are perfect Christians who without error follow divine scripture, and who know, both from what is written and from the figure of the first and second state, the places and the figures which divine scripture mentions, for as this world consists of men and angels:  We, saith the Apostle,  are made a spectacle to the world, to angels and to men,   18  1 I Cor. iv, 9. so also the higher place again has been prepared beforehand for angels and men. Glory for all to God the King of the Ages. Amen!

The heaven is bound together with the earth, and is divided into parts, for as it forms the two vaulted chambers and the two walls on each side, and after the manner of vaulted chambers has one of these walls curving round till it joins the other, and completes the entire figure (  to_ sxh~ma ) of the world. And this figure, that is, the  |262  magnitude of the sun and of the earth, we here delineate  ^{19  1 See preceding page. [Note to online text: diagram in body of text]} in accordance with the views of those pagans, in order that any one who sees the two delineations may be able to compare the one with the other, how that in the delineation, which represents our view  20  2 See above, p. 251. in accordance with the reality, the rays of the sun falling upon bodies produce shadows for each climate agreeably to our previous description, these shadows showing a difference of half a foot for each climate----and how also that in the delineation here which represents their views, the rays do not fall upon the bodies, so that those climates which they speak of, bodies namely, cannot be illumined by the sun; and hence we see that they turn away from the truth and make a great boast in vain about their fables.

[Footnotes have been renumbered and moved to the end]

the shadow of the gnomon would be the same length all the world over, at the same instant of time. Thus what is advanced by Cosmas in support of his theory of a flat earth proves the very opposite; for the fact of the shadows being of different lengths, as he had observed them to be, at the same season in different latitudes, is owing entirely to the curvature of the earth's surface. This, I think, is very clearly shown on diagram 2. If the sun were a small object and near to the

earth, of course the length of the shadows would be different for different latitudes, supposing the earth to be flat. This is shown in diagram 3. It is on the assumption of a near sun that the fallacy of

the whole argument lies. Cosmas builds his whole theory on this false basis.

Cosmas was not the last to hold the theory of the earth being a plane. From 1848 to about 1865, Mr. S. Goulden, in various papers, propounded the doctrine that the earth is not a globe; he called his system Zetetic Astronomy. About 1865 was issued the prospectus of a book on the subject, in the contents of which was given "Distance of the Sun from London 4028 miles----How measured."----See De Morgan's  Budget of Paradoxes, pp. 807-8.