Unit I · Cosmic Preliminaries & Golden Pentagon–Decagon Construction¶
Almagest Book I, Chapters 1–9 & 10a (Heiberg 1898, Vol. I.1, pp. 4–35)¶
Chapter 1 · Preface to Syrus (Heiberg I, pp. 4–7)¶
p. 4 Those who have pursued genuine philosophy, O Syrus, seem to me to have acted very rightly in separating the theoretical part of philosophy from the practical. For even if prior to that separation it happens that the practical turns out to depend on the theoretical, nevertheless one would find great difference between them—not merely because some of the moral virtues can belong to many people even without systematic instruction, whereas it is impossible to attain contemplation of the universe without formal teaching, p. 5 but above all because the greatest benefit there comes from continuous activity in actual affairs, whereas here it comes from steady progress in theoretical theorems.
Hence we deemed it fitting on the one hand to regulate our own actions amidst the impressions of daily life so as never to forget, even in small matters, the consideration of beautiful and well-ordered disposition, and on the other hand to devote the greatest part of our leisure to the instruction of theorems—many and noble as they are, yet chiefly those expressly called mathematical. For Aristotle quite fittingly divides the theoretical part too into three primary genera (Metaphysics E.1, 1026a18): the physical, the mathematical, and the theological.
p. 6 For since all existing things have their being from matter, form, and motion—none of which can be seen in actual existence apart from a substrate, though each can be thought without the others—one would reasonably suppose the first cause of the first motion of the universe, if taken simply, to be an invisible and motionless deity, and the kind of science that seeks it out in the highest regions of the cosmos to be theological; whereas the science that investigates material and ever-changing quality (found in white and hot, sweet and soft, and things of that sort) one would call physical, since such substance dwells for the most part among corruptible things below the lunar sphere; while the kind of science declarative of quality with respect to forms and local motions—seeking figure, number, and magnitude, and moreover place, time, and the like—one may define as mathematical.
For such an essence falls, as it were, in the middle between the other two: not only because it can be conceived both through the senses and apart from the senses, and furthermore because it is an attribute of practically all beings universally, both mortal and immortal, changing ever together with those things that alter according to their inseparable form, while preserving unchanged the eternity of the ethereal nature through immovable forms; p. 7 seeing, then, that the other two genera of the theoretical would seem to be grasped more by conjecture than by scientific apprehension—the theological because of its completely invisible and ungraspable nature, the physical because of the unstable and obscure character of matter—so that philosophers could never hope to agree about them, mathematics alone, if one approaches its inquiry rigorously, can furnish reliable and unshakable knowledge to those who practice it, because its demonstration proceeds through indisputable paths: arithmetic and geometry.
Chapters 2–9 · Summary of the Cosmic Hypotheses (Heiberg I, pp. 7–31)¶
Before entering the geometric calculus of chords, Ptolemy establishes in Chapters 2–8 the observational and geometric hypotheses governing spherical astronomy, and in Chapter 9 outlines the planar and spherical theorems required to reduce celestial arcs to computable ratios:
I.2 — Order of the Treatise (pp. 7–10): Why the general relation of the whole Earth to the whole heaven must precede special demonstrations of solar, lunar, and planetary motions.
I.3 — That the Heaven Rotates Spherically (pp. 10–14): Circumpolar stars revolve in parallel circles around a fixed visible pole P, rising and setting stars trace uniform circular paths across the horizon cHOR, and the sphere uniquely encloses the greatest volume among isoperimetric solids.
I.4 — That the Earth Too Is Sensibly Spherical as a Whole (pp. 14–16): Lunar eclipses occur at identical absolute moments yet at earlier local hours to western observers and later local hours to eastern observers in exact proportion to longitudinal distance; advancing northward raises the altitude of the pole P.
I.5 — That the Earth Is in the Middle of the Heavens (pp. 16–20): Only at the center O of the celestial sphere cABGD does the horizon plane cHOR bisect the zodiacal circle cECL so that six signs always appear above the Earth and equinox shadows lie along a straight east–west line.
I.6 — That the Earth Has the Ratio of a Point to the Heavens (pp. 20–21): Gnomon sights and armillary rings on Earth’s surface act as if stationed at the very center O, and the horizon always bisects the celestial sphere.
I.7 — That the Earth Makes No Motion of Translation (pp. 21–26): Physical and kinematic arguments placing the stationary Earth at the cosmic center O.
I.8 — That There Are Two Different Primary Motions in the Heavens (pp. 26–30): First, the diurnal rotation carrying all things from east to west parallel to the celestial equator ΑΓ around the poles P and P'; second, the counter-rotation of the Sun, Moon, and planets along the oblique circle of the zodiac (ΒΔ) inclined to the equator by the obliquity ε.
Chapter 10a · On the Magnitude of Straight Lines in a Circle (Heiberg I, pp. 31–35)¶
p. 31 For ease of practical use, then, we shall subsequently construct a tabular exposition of the magnitudes of these chords, dividing the circumference into $360$ segments, setting out the arcs side by side at increments of half a degree, and giving alongside each arc the straight line subtending it when the diameter is divided into $120$ parts—because of the arithmetical symmetry of this number, as will become evident from the calculations themselves.
p. 32 First, however, we shall show how, through the fewest and exactest theorems—and the very same ones throughout—we carry out a rapid and systematic demonstration of their magnitudes, so that we do not merely have the tabular magnitudes of the chords set out nakedly without inspection, but may also manage their verification and proof cleanly through a geometric construction. In general we shall employ the sexagesimal system (ἑξηκοστά) because of the awkwardness of unit fractions, always carrying multiplications and divisions out to yield approximations whose discrepancy does not differ even sensibly from rigorous precision.
Proposition (Ptolemy’s Golden Decagon & Pentagon Construction). Let there be a semicircle ΑΒΓ upon diameter ΑΔΓ around center Δ, and from Δ let ΔΒ be drawn at right angles (∠ΑΔΒ) to ΑΓ. Let ΔΓ be bisected at Ε (ΔΕ = ΕΓ), let ΕΒ be joined, and let ΕΖ be placed equal to ΕΒ along ΔΑ (via the circular arc ΒΖ centered at Ε), and let ΖΒ be joined.
I say that ΖΔ is the side of the regular decagon, and ΒΖ that of the regular pentagon.
p. 33 Demonstration. Since the straight line ΔΓ is bisected at Ε and another straight line ΔΖ is added to it in a straight line, by Eucl. II.6 the rectangle contained by ΓΖ and ΖΔ, together with the square on ΕΔ, is equal to the square on ΕΖ—that is, to the square on ΒΕ, since ΒΕ = ΕΖ:
$$\Gamma Z \cdot Z\Delta + E\Delta^2 = EZ^2 = BE^2.$$
Yet because angle ∠ΕΔΒ is right, by Eucl. I.47 the squares on ΕΔ and ΔΒ are equal to the square on ΒΕ:
$$E\Delta^2 + \Delta B^2 = BE^2.$$
Subtracting the common square on ΕΔ, the remaining rectangle ΓΖ · ΖΔ is equal to the square on ΔΒ, that is, to the square on ΔΓ:
$$\Gamma Z \cdot Z\Delta = \Delta B^2 = \Delta\Gamma^2.$$
Therefore, as ΓΖ is to ΓΔ, so is ΓΔ to ΔΖ. Consequently ΓΖ has been cut in extreme and mean ratio at Δ (Eucl. VI.Def.3).
Now since the side of the hexagon and the side of the decagon inscribed in the same circle, when placed in a single straight line, cut the whole line in extreme and mean ratio (Eucl. XIII.9), and since ΓΔ being a radius is equal to the side of the hexagon (Eucl. IV.15 Coroll.), therefore ΔΖ is equal to the side of the regular decagon.
And in the same manner, since the side of the regular pentagon has its square equal to the sum of the squares on the side of the hexagon and the side of the decagon inscribed in the same circle (Eucl. XIII.10), and in the right-angled triangle ΒΔΖ the square on ΒΖ is equal to the squares on ΒΔ (the hexagon side) and ΔΖ (the decagon side), therefore ΒΖ is equal to the side of the regular pentagon.
p. 34 Sexagesimal Evaluation of the Primary Chords. Since then, as I stated, we set the diameter of the circle to $120$ parts ($120^p$), it is immediate from what has been established that:
Regular Hexagon ($\theta = 60^\circ$): $\Delta E = \frac{1}{2}\Delta\Gamma = 30^p$ and $\Delta B = 60^p$, while the hexagon chord itself equals the radius:
$$\operatorname{crd}(60^\circ) = 60;0,0^p.$$
Regular Decagon ($\theta = 36^\circ$): Since $E\Delta^2 = 900$ and $\Delta B^2 = 3600$, we have $EB^2 = EZ^2 = 4500$, whence $EZ \approx 67;4,55^p$ in sexagesimals. Subtracting $E\Delta = 30^p$, the decagon side ΔΖ is:
$$\operatorname{crd}(36^\circ) = 30(\sqrt{5} - 1) \approx 37;4,55^p, \qquad \Delta Z^2 \approx 1375;4,15.$$
Regular Pentagon ($\theta = 72^\circ$): Adding $\Delta B^2 = 3600$ to $\Delta Z^2 = 1375;4,15$ gives ΒΖ² $= 4975;4,15$, whose square root yields the pentagon side ΒΖ:
$$\operatorname{crd}(72^\circ) = 30\sqrt{10 - 2\sqrt{5}} \approx 70;32,3^p.$$
p. 35 4. Inscribed Square & Equilateral Triangle ($\theta = 90^\circ, 120^\circ$): Instantly from the center ($R^2 = 3600$), the square on the $90^\circ$ chord is $2R^2 = 7200$ and the square on the $120^\circ$ chord is $3R^2 = 10800$:
$$\operatorname{crd}(90^\circ) = 60\sqrt{2} \approx 84;51,10^p, \qquad \operatorname{crd}(120^\circ) = 60\sqrt{3} \approx 103;55,23^p.$$
Supplementary Chords ($180^\circ - \theta$): Since the sum of the squares on any chord and on the chord subtending the remainder of the semicircle equals the square on the diameter ($D^2 = 14400$, by Eucl. III.31 and Eucl. I.47), subtracting $\operatorname{crd}^2(36^\circ) = 1375;4,15$ from $14400$ leaves $13024;55,45$, whence $\operatorname{crd}(144^\circ) \approx 114;7,37^p$; and similarly $\operatorname{crd}(108^\circ) \approx 97;4,56^p$.