Claudius Ptolemy

Source: The Almagest1: I-V, in Great Books of the Western World, vol. 15, Encyclopedia Britannica, 1952, pp.5-7, 14-24.
 
 

Those who have been true philosphers, Syrus, seem to me to have very wisely separated the theoretical part of philosophy from the practical.  For even if it happens that the practical turns out to be theoretical prior to its being practical, nevertheless a great difference would be found in them; not only because some of the moral virtues can belong to the everyday ignorant man and it is impossible to come by the theory of whole sciences without learning, but also because in practical matters the greatest advantge is to be had from a continued and repeated operation upon the things themselves, while in theoretical knowledge it is to be had by a progress onward.  We accordingly thought it up to us so to train our actions even in the application of the imagination as not to forget in whatever things we happen upon the consideration of their beautiful and well-ordered disposition, and to indulge in meditation mostly for the exposition of many beautiful theorems and especially of those specifically called mathematical.

For indeed Aristotle quite properly divides also the theoretical into three immediate genera: the physical, the mathematical, and the theological.  2  For given that all beings have their existence from matter and form and motion, and that none of these can be seen, but only thought, in its subject separately from the others, if one should seek out in its simplicity the first cause of the first movement of the universe, he would find God invisible and unchanging.  And the kind of science that seeks after Him is the theological; for such an act [energeia] can only be thought as high above somewhere near the loftiest things of the universe and is absolutely apart from sensible things.  But the kind of science which traces through the material and ever moving quality, and has to do with the white, the hot, the sweet, the soft, and such things, would be called physical; and such an essence [ousia], since it is only generally what it is, is to be found in corruptible things and below the lunar sphere.  And the kind of science which shows up quality with respect to forms and local motions, seeking figure, number and magnitude, and also place, time, and similar things, would be defined as mathematical.  For such an essence falls, as it were, between the other two, not only because it can be conceived both through the senses and without the senses, but also because it is an accident in absolutely all beings both mortal and immortal, changing with those things that ever change, according to their inseparable form, and preserving unchangeable the changelessness of form in things eternal and of an ethereal nature.

And therefore meditating that the other two genera of the theoretical would be expounded in terms of conjecture rather than in terms of scientific understanding: the theological because it is in no way phenomenal and attainable, but the physical because its matter is unstable and obscure, so that for this reason philosophers could never hope to agree on them; and meditating that  only the mathematical, in approached enquiringly, would give its practitioners certain and trustworthy knowledge with demonstration both arithmetic and geometric resulting from indisputable procedures, we were led to cultivate most particularly as far as lay in our power this theoretical discipline [theoria].  And especially we were led to cultivate that discipline developed in respect to divine and heavenly things as being the only one concerned with the study of things which are always what they are, and therefore able itself to be always what it is--which is indeed the proper mark of a science--because of its own clear and ordered understanding and yet to cooperate with the other disciplines no less than they themselves.  For that special mathematical theory would most readily prepare the way to the theological, since it alone could take good aim at that unchangeable and separate act, so close to that act are the properties having to do with translations and arrangements of movements, belonging to those heavenly beings which are sensible and both moving and moved, but eternal and impassible.  Again as concerns the physical there would not be just chance correspondances.  For the general property of the material essence is pretty well evident from the peculiar fashion of its local motion--for example, the corruptible and incorruptible from straight and circlular movements, and the heavy and light or the passive and active from movement to the center and movement from the center.  And indeed this same discipline would more than any other prepare understanding persons with respect to nobleness of actions and character by means of the sameness, good order, due proportion, and simple directness contemplated in divine things, making its followers lovers of that divine beauty, and making habitual in them, and as it were natural, a like condition of the soul.

And so we ourselves try to increase continuously our love of the discipline of things which are always what they are, by learning what has already been discovered in such sciences by those really applying themselves to them, and also by making a small original contribution such as the period of time from them to us could well make possible.  And therefore we shall try and set forth as briefly as possible as many theorems as we recognize to have come to light up to the present, and in such a way that those who have already been initiated somewhat may follow, arranging in proper order for the completion of the treatise all matters useful to the theory of heavenly things.  And in order not to make the treatise too long we shall only report what was rigorously proved by the ancients, perfecting as far as we can what was not fully proved or not proved as well as possible.
 
 

A view, therefore, of the general relation of the whole earth to the whole of the heavens will begin this composition of ours.  And next, of things in particular, there will first be an account of the ecliptic's position 3  and of the places of that part of the earth inhabited by us, and again of the difference, in order between each of them according to the inclinations of their horizons.  For the theory of these, once understood, facilitates the examination of the rest.  And, secondly, there will be an account of the solar and lunar movements and of their incidents.  For without a proper understanding of these one could not profitably consider what concerns the stars.  The last part, in view of this plan, will be an account of the stars.  Those things having to do with the sphere of what are called the fixed stars would reasonably come first, and then those having to do with what are called the five planets.  And we shall try and show each of these things using as beginnings and foundations for what we wish to find, the evident and certain appearances from the observations of the ancients and our own, and applying the consquences of these conceptions by means of geometrical demonstrations.

And so, in general, we have to state that the heavens are spherical and move spherically; that the earth, in figure, is sensibly spherical also when taken as a whole; in position, lies right in the middle of the heavens, like a geometrical centre; in magnitude and distance, has the ratio of a point with respect to the sphere of the fixed stars, having itself no local motion at all.4   And we shall go through each of these points briefly to bring them to mind.

With an eye to immediate use, we shall now make a tabular exposition of the size of these chords by dividing the circumference into 360 parts and setting side by side the chords as the arcs subtended by them increase by a half part.  That is, the diameter of the circle will be cut into 120 parts for ease in calculation; [and we shall take the arcs, considering them with respect to the number they contain of the circumference's 360 parts, and compare them with the subtending chords by finding out the number the chords contain of the diameter's 120 parts.] 5   But first we shall show how, with as few theorems as possible and the same ones, we make a methodical and rapid calculation of their sizes so that we may not only have the magnitudes of the chords set out without knowing the why and wherefore but so that we may also easily manage a proof by means of a systematic geometrical construction.  In general we shall use the sexagesimal system because of the difficulty of fractions6, and we shall follow out the multiplications and divisions, aiming always at such an approximation as will leave no error worth considering as far as the accuracy of the senses is concerned.7

Then first 8 let there be the semicircle ABC on the diameter ADC and around centre D, and let straight line DB be erected on AC at right angles.  Let DC be bisected at E, and EB be joined; and let EF be laid out equal to EB, and let FB be joined.

I say that the straight line FD is the side of a regular inscribed decagon, and BF that of a pentagon. 9

For since the straight line DC is bisected at E and a straight line DF is added to it,

since BE = EF.  But

Therefore

And subtracting the common square on ED,

Therefore CF is cut at D in extreme and mean ratio [Eucl. vi, def. 3] 11 .  Since, then, the side of the hexagon and the side of the decagon which are inscribed in the same circle, when they are in the same straight line, cut that line in extreme and mean ratio [Eucl. xiii.9] 12 , and since the radius DC is equal to the side of the hexagon [Eucl. iv.15 coroll.], therefore FD is equal to the side of the decagon.

And likewise, since the square on the side of the pentagon is equal to the square on the side of the hexagon together with the square on the side of the decagon, all inscribed in  the same circle [Eucl. xiii.10] 13 , and since in the right triangle BDF

where DB is the side of the hexagon and FD the side of the decagon, the straight line BF is equal to the side of the pentagon.

Since, then, as I said, we suppose the diameter divided into 120 parts, therefore by what we have just established, being half the circle's radius, ED = 30 such parts, and  sq. ED = 900; and  rad. DB = 60 such parts, and  sq. DB = 3600; and  sq. BE = sq. EF = 4500 14.  Then EF = 67^p4'55'' in length 15, and by subtraction,

Therefore the side of the decagon, subtending an arc of 36° of the whole circumference's 360°, will have 37^p4'55'' of the diameter's 120^p.

Since again FD = 37^p4'55'',  sq. FD = 1375^p4'14'',  sq. DB = 3600p, and  sq. FD + sq. DB = sq. BF, therefore, in length,

And therefore the side of the pentagon, subtending an arc of 72°, is 70^p32'3''.

It is immediately clear that the side of the hexagon, subtending an arc of 60° and being equal to the radius, is itself 60^p.  And likewise, since the side of the inscribed square, subtending an arc of 90°, is, when squared, double the square on the radius, and since the side of the inscribed equilateral triangle is, when squared triple the square on the radius, and since the square on the radius is 3600^p, the square on the side of the square will add up to 7200^p, and the square on the side of the equilateral triangle to 10800^p.  And so in length,

and

And so these chords are easily gotten by themselves. 16   Thence it is evident that, with these chords given, it will be easy to get the chords which subtend the supplements, since the squares on them added together are equal to the square on the diameter. 17   For example, since it was shown

sq. chord of arc 36° = 1375^p4'14'', and  sq. diameter = 14400^p, therefore, for the supplement,  sq. chord of arc 144° = 13024^p55'45'', and, in length,

and the others in like manner.

And we shall next show, be expounding a lemma very useful for this present business, how the rest of the chords can be derived successively from those we already have.

For let there be a circle with any sort of inscribed quadrilateral ABCD, and let AC and BD be joined.

It is to be proved that

For let it be laid out such that  angle ABE = angle DBC.  If then we add the common angle EBD,

But also  angle BDA = angle BCE for they subtend the same arc [Eucl. iii, 21].  Then triangle ABD is equiangular with triangle BCE.  Hence  BC : CE : : BD : AD [Eucl. vi, 4].  Therefore

Again since  angle ABE = angle CBD  and also  angle BAE = angle BDC, therefore triangle ABE is equiangular with triangle BCD.  Hence  AB : AE : : BD : CD.  Therefore

But it was also proved  rect. BC, AD = rect. BD, CE.  Therefore also

Which was to be proved.

Now that this has been expounded, let there be the semicircle ABCD on diameter AD, and from the point A let there be drawn the two straight lines AB, AC, and let the length of each of them have been given in terms of such parts as the given diameter's 120; and let BC be joined.

I say that BC is also given. 19

For let BD and CD be joined.  Then clearly they are also given because they subtend the supplements.  Since, then, the quadrilateral ABCD is inscribed in a circle, therefore  rect. AB, CD  +  rect. AD, BC  =  rect. AC, BD.  And rectangle AC, BD is given, and also rectangle AB, CD.  Therefore the remaining rectangle AD, BC is also given.  And it is now clear to us that, if two arcs are given and the two chords subtending them, then also the chord subtending the difference between the two arcs will be given.  And it is evident that by means of this theorem we can inscribe many other chords in arcs which are the differences between arcs directly given; for instance, the chord subtending an arc of 12°, since we have the chords of 60° and 72°.

Again, given any chord in a circle, let it be proposed to find the chord of half the arc of the given chord. 20

And let there be the semicircle ABC on diameter AC, and let CB be the given chord.  And let the arc be bisected at D, and let AB, AD, BD, and DC be joined.  And let DF be drawn from D perpendicular to AC.

I say that

For let AE be laid out such that AE = AB, and let DB be joined.  Since AB = AE, and AD is common, therefore the two sides AB and AD are equal to the two sides AE and AD respectively.  And  angle BAD = angle EAD [Eucl. iii, 27]; therefore also  base BD = base DE.  But  chord BD = chord CD, and therefore  chord CD = DE.  Since then, in the isosceles triangle DEC, DF has been dropped from the vertex perpendicular to the base, therefore EF = CF [Eucl. i, 26].  But

therefore

And so, since, given the chord of arc BC, chord AE of its supplement is also given, therefore CF, which is half the difference between AC and AB, is given too.  But when the perpendicular DF is drawn in right triangle ACD, as a consequence right triangle ACD is equiangular with right triangle DCF [Eucl. vi, 8], and AC : CD : : CD : CF.  Therefore,

But rectangle AC, CF is given 21 , therefore the square on CD is also given.  And so the chord CD of half the arc BC will also be given in length. 22

And so again, by means of this theorem, most of the other chords will be given as subtending the halves of arcs already found.  For instance, from the chord of an arc of 12°, there can be gotten the chord subtending an arc of 6°, and those subtending arcs of 3°, of 1¹/₂°, and ³/₄°  respectively.  And we shall find from calculation that

and

Again24  let there be the circle ABCD on diameter AD with center at F.  And from A let there be cut off consecutively two given arcs, AB and BC; and let the given chords subtending them, AB and BC, be joined.

I say that, if we join AC, then AC will be given also. 25

For let the circle's diameter BFE be drawn through B, and let BD, DC, CE, and DE be joined.  Then from this it is clear that, by means of BC, chord CE is given; and by means of AB, chords BD and DE are given.  And by things we have already proved, since BCDE is a quadrilateral inscribed in a circle, and BD and CE are the diagonals, the rectangle contained by the diagonals is equal to the sum of the rectangles contained by opposite sides.  And so, since the rectangles BD, CE and BC, DE are given, therefore the rectangle BE, CD is given also.  But the diameter BE is given too, and the remaining side CD will be given.  Therefore the chord AC of the supplement will be given also. 26   And so, if two arcs and their chords are given, then by means of this theorem the chord of both of these arcs together will be given.

And it is evident that, by continually combining the chord of an arc 1¹/₂° with those so far set out and calculating the sums, we shall inscribe all those chords which, when doubled, are divisible by three; and only those chords will still be skipped which fall within 1¹/₂° intervals.  For there will be two such chords skipped in each interval, since we are carrying out this inscribing of chords by successive additions of ¹/₂°.  And so if we could compute the chord subtending an arc of ¹/₂°, then this chord, by addition to, and subtraction from, the chords which are separated by 1¹/₂° intervals and have already been given, will fill in all the rest of the intermediate chords.  But since, given any chord such as that subtending an arc of 1¹/₂°, the chord of a third of the arc is in no way geometrically given 27 (and if it were possible, we could then compute the chord of an arc of 1¹/₂°), therefore we shall first look for the chord of an arc of 1° by means of chords subtending arcs of 1¹/₂° and ³/₄°.  We shall do this by presenting a little lemma which, even if it may not suffice for determining their sizes in general, can yet in the case of these very small chords keep them indistinguishable from chords rigorously determined. 28

For I say that, if two unequal chords are inscribed in a circle, the greater has to the less a ratio less that the arc on the greater has to the arc on the less.

For let there be a circle ABCD; and let unequal chords be inscribed in it, AB the less and BC the greater.

I say that

For let angle ABC be bisected by BD, and let AEC, AD, and CD be joined.  And since angle ABC has been bisected by the straight line DEB,  chord CD = chord AD [Eucl. iii, 26, 29], and CE > AE [Eucl. vi, 3].  Then let DF be dropped from D perpendicular to AEC.  Now since [DF produced will bisect arc ABC [Eucl. iii, 3, 26], hence it will fall on the side of B towards C.  Therefore,] AD > DE, DE > DF [Eucl. i, 21], therefore the circle described with centre D and radius DE cuts AD and falls beyond DF.  Then let the circle GEH be drawn and the straight line DFH be produced.  And since

therefore

But

and

Therefore

Then componendo 29

And doubling the antecedents

And separando 30

But CE : AE : : BC : AB [Eucl. vi, 3], and  angle CDB : angle BDA : : arc BC : arc AB [Eucl. vi, 33].  Therefore

Now, then, with this laid down, let there be the circle ABC, and let the two chords AB and AC be inscribed in it.  And first let AB be given as subtending an arc of ³/₄°, and AC an arc of 1°.

Since  chord BC : chord AB  <  arc BC : arc AB, and  arc AC = 1¹/₃ (arc AB), therefore

But it was proved  chord AB = 0^p47'8''.  Therefore  chord AC  <  1^p2'50'', for 1^p2'50'' = 1¹/₃ (0^p47'8'').

Again, with the same figure, let chord AB be given as subtending an arc of 1°, and chord AC an arc of 1¹/₂°.

Likewise then, since  arc AC = 1¹/₂ (arc AB),

But we proved  chord AC = 1^p34'15''.  Therefore  chord AB  >  1^p2'50'', for  1^p34'15'' = 1¹/₂ (1^p2'50'').

And so, since it has been proved that the chord of an arc of 1° is both greater and less than the same number of parts, clearly we shall have

and by means of earlier proofs we saw

And the remaining intervals will be filled in as we have just said.  For example, in the first interval we find the chord subtending an arc of 2° by adding ¹/₂° and 1¹/₂°, and the chord subtending an arc of 2¹/₂° by subtracting ¹/₂° from 3°, and so on for the rest.

So the business of chords in a circle can easily be handled in this way, I think.  And as I said, in order to have the magnitudes set out immediately to hand, we shall draw up tables of 45 rows each, for symmetry's sake.  And the first column will contain the magnitudes of the arcs increasing by ¹/₂°, and the second column will contain the magnitudes of the chords subtending them in terms of the diameter's assumed 120 parts.  The third column will contain the thirtieth of the increase of the chord as the corresponding arc increases by ¹/₂°, so that we may have a mean addition, accurate for the senses, for each increase of ¹/₆₀° in the corresponding arcs, and so be able to calculate readily the chords falling within the ¹/₂° intervals. 33   And it is to be remarked that by means of these same theorems, if we should suspect some typographical error in connection with any of the chords computed here, we can easily check and correct it either by means of the chord of an arc double the arc of the chord which is being examined, or by means of the difference of certain other given chords, or by means of the chord subtending the supplement.  And here is the table: