SophomoreMathematics
RevisedSummer 2017
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or the one provided by the bookstore
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or Antares (in the constellation Scorpius)
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added
7th day after first new moon
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32, lines 15 and 20: replace 25” with 53”.
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of Alexandria
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(of the Ptolemy book, not page 27 of the manual)
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Book I, Chapter 14. On the Arcs Between the Equator and the Ecliptic
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Half of the shortest day is measured
EGEG?
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the circle around D (the equant)
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no
Figure 20
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(except at apogee and perigee).
(except at apogee and perigee).
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for angles on pp. 322-23:
Fig. p. 322
Fig. p. 323,
Fig. p. 322
Book XII, Chapter 1RETROGRADATION: RETURN TO TWO EQUIVALENT HYPOTHESES
The motions of every planet were fully accounted for by means of an epicycle moving regu-larly about an equant point on an eccentric deferent. As with the sun, Ptolemy is not contentwith one wholly sufficient hypothesis, but introduces a second one equivalent to it. An unfortu-nate grammatical ambiguity makes it seem that he introduces it only for the outer planets, butin fact it applies to all five. The parenthesis on p.391, lines 18–19, is a genitive absolute, a con-struction which is much more likely in Greek to modify a following clause than a preceding one.It is only the equality of speeds between themoving eccentric and the sun that is restricted to thethree outer planets. The common, composite form of proof Ptolemy uses in this chapter displaysplanetary motion stripped to its causal essentials, indifferent to differences of inner and outerplanets and to the actual motions the planet would have to perform on one hypothesis or theother. We will prove the equivalence of the two hypotheses for all the planets, disregarding thezodiacal anomaly, as Ptolemy does in this chapter. (His later reintroduction of it forms a finalconfirmation of the assumption of the equant.)
The first hypothesis is the one we have used consistently for the heliacal anomaly, the epicy-clical. The second, new hypothesis, involves a moving eccentric whose center rotates around theearth: it has similarities to the hypothesis of the moon described in Bk. IV.5, but the motionsinvolved differ. Here the eccentric center M is carried eastward about the earth O with a speedequal to L + A. Meanwhile the planet on the eccentric is moving westward with the speed A.
C
B
F
D
S
P
E
L
A
O
M
H
G
K
N
A
L+A
V
Figure 23
53
The two hypotheses will be equivalent provided that:
radius of epicycle : radius of deferent :: eccentricity : radius of eccentric
I. Let the radius of the epicycle be equal to the eccentricity, as in Figure 23.Let C and O be the earth in the two hypotheses respectively. Suppose initially the planetis at apogee.In any given time, let the epicycle move eastward fromB toD through an angle equal to L,and let the planet move in the same direction from apogee, from E to P, through an angleequal to the corresponding A.In the same time, let the center of the eccentric move east from M to G, through an angleequal to L + A, on the circle about O with radius OG equal to the radius of the epicycle,DP.In the same time, let the planet move west from N to K through an angle equal to A, onthe eccentric circle with center G and radius GN equal to the radius of the deferent CD.Draw CS parallel to DP.∠DCS = ∠EDP because of the parallel lines CS and DP.∠FCS = ∠FCD + ∠DCSTherefore, ∠FCS = ∠FCD + ∠EDP = L + ATriangles CDP and KGO are congruent,since, by assumption, DP = OG,and CD = GKwhile ∠EDP = ∠NGK = ATherefore ∠CPD = ∠KOGNow in the hypothesis of the epicycle, the angle of apparent motion in the given time, asthe planet is seen at F and again at P, is ∠FCP;and ∠FCP = ∠FCS - ∠PCS= ∠FCS - ∠CPDsince DP is parallel to CS.But in the hypothesis of the eccentric, the angle of apparent motion in the given time, asthe planet is seen at H and again at K, is ∠HOK:and ∠HOK = ∠HOV - ∠KOG.but ∠HOV = ∠FCS, because each equals L + A.and ∠KOG = ∠CPD, as shown above.Therefore ∠HOK = ∠FCS - ∠CPD, which has already been shown to equal ∠FCP.Therefore, ∠HOK = ∠FCP. Q.E.D.
II. Exercise: Prove that the same is true even if it is only the case that PD:DC :: OG:GK
Where is themean sun in each hypothesis in the case of the outer planets? In the case of the innerplanets? (Assume that the mean sun is initially in conjunction with the planet at apogee.)
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Figure 24
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KOG QFS.
Q.E.D.
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p. 528, line 6.
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phenomenon
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hypothesis
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SOPHOMORE MATHEMATICS
Notes to
Accompany
the Reading of
Kepler’s
Astronomia Nova
* [SPRING 2018 EDITION] *
Thomas Aquinas College
© 2006
The following notes are intended to aid
in the reading of Kepler’s Astronomia
Nova in the Sophomore Mathematics
tutorial. They were composed, modified,
discussed, and edited by a group of
Thomas Aquinas College tutors from
2003-2005, including: Mark Clark, Brian
Dragoo, Chris Decaen, Brian Kelly,
Kevin Kolbeck, and Michael Letteney. As
these notes are not intended to be
formally part of the curriculum of the
Sophomore Mathematics tutorial, the
decision about which parts should be
used by the students is left to the
tutor’s discretion.
The notes were revised in early 2016,
including many small changes reflecting
a new, revised edition of the complete
Astronomia Nova (Green Lion Press,
2015) as the course text for Kepler.
1
Corrigenda
p. 21, l. 7b: “…did indeed take up…” should be “…did indeed remove”.
p. 226, l. 10: “…is not everywhere distant…” should be “…is not equally distant”.
p. 276, l. 8b: the proportion “But is to very nearly as (or ) is to ” should be either “But is to very
nearly as (or ) is to ” or “But is to very nearly as is to (or )” .
p. 287, l. 11: “…Venus 8½…” should be “…Venus 7½…”
p. 305, ll. 22-23: “…will quite readily allow this: that on the testimony of the observations the path…” should be
“…will more readily admit this; because the observations will testify that the path…”
p. 310, ll. 13-14: the parenthesis would be more clearly translated “…(standing upon least parts of the
circumference, which [parts] therefore do not differ from straight lines)…”. p. 413, ll. 12-15: “perceptibly” should be “gradually”, in all four instances.
(Note: Marginal notes and section headings throughout the text are Kepler’s own.)
Chapter 19 Overview: In our reading of Ptolemy and Copernicus we neglected their accounts of
the planets’ latitudinal shifts above and below the ecliptic; both of them “saved the appearances”
by tilting the epicycles in respect to their deferents. Because of the precision of Tycho’s
observations, Kepler here is able to show that neither theory accurately predicts the positions of
Mars. This is a turning point in Kepler’s investigation. However, because here also Kepler uses
two trigonometric laws called the “Law of Sines” and the “Law of Tangents,” we must digress
briefly into trigonometry and its connection to what we have done already.
Ptolemy’s Table of Chords and the Sine Function: Last semester we saw that in his table of
chords, Ptolemy associates the sizes of arcs in a circle (expressed as parts of the 360 parts of the
whole circumference), with the size of the chords that subtend those arcs (expressed as parts of
the 120 parts of the diameter). So, for example, Ptolemy’s table shows us that an arc containing
60 of the 360 parts of the circumference is subtended by a chord which contains 60 of the 120
parts of the diameter.
Ptolemy also uses these arcs to express the size of angles, either in terms of four right
angles’ covering the 360 parts of the circumference (or, the number of the 360 parts of the
circumference cut off by an angle when it is at the center), or in terms of two right angles’
covering all of the circumference (or, the number of parts of the circumference cut off by an
angle when it is at the circumference). So, for example, if some angle contains 64 of the parts of
the circumference when four right angles cover all 360 parts (or, when the angle is at the center),
that same angle will contain 128 of the 360 parts when two right angles cover the 360 parts of
the circumference (or, when the angle is at the circumference.) Conversely, if the same arc is cut
off by an angle at the center and by one at the circumference, then the angle at the center is twice
the one at the circumference. The table of chords, then, will reveal the length of chords, or lines,
subtending various size angles.
As we proceeded in the Almagest, we saw how useful this table of chords was. By means
of it, Ptolemy was able to “solve” triangles, that is, from known sides or angles of a triangle to
find its unknown angles and sides. Recall, moreover, how frequently Ptolemy had to construct
right-angled triangles, then to describe semi-circles on their hypotenuses in order to use his table
of chords. The need for this procedure arose, of course, from the fact that Ptolemy relates the
sizes of angles to the length of lines by relating each to arcs of circles. Since the angles of the
triangle to be solved were sometimes at the circumference of a circle, sometimes at the center,
we had to attend constantly to the difference, doubling or halving as circumstances demanded.
2
Matters, however, can be
greatly simplified by relating
angles to lines more directly,
without reference to arcs of
circles. For by dropping perpen-
diculars (e.g., AB, HK, etc.) from
one of the lines (CH) containing
a given acute angle (HCK) to
the other line containing that
angle (CK), one will form a set of
similar right-triangles (ABC,
HKC, etc.). Therefore the
length of the perpendicular that
subtends the given angle will
have a constant ratio to the
lengths of the remaining sides of
the right-triangles (AB : AC : :
HK : HC; and AB : BC : : HK :
KC). Each acute angle (Q),
then, will have a unique set of
ratios associated with it. (The
obtuse angles that are
supplements of the acute angle,
such as R, will have this same
unique set of ratios.)
Now, if we let the side
subtending the right-angle (the
hypotenuse) formed by the perpendicular be the unit, the remaining sides will be some fixed part
of the hypotenuse. If we assume the sides are commensurable, then we can attach a numeric
value to the perpendicular that expresses the number of parts of the hypotenuse it contains. This
numeric value will be unique to each acute angle (and its supplement). The perpendicular, the
ratio of it to the hypotenuse, and the numeric value when the hypotenuse is a unit are all called
the sine of the angle. A table, then, could be set up that gives the sine of each angle.
Ptolemy has in a way already made this table for us. For if we replace the arcs of his table
with angles from the circumference (so that, for example, an arc containing 64 parts of the
circumference will be subtended by an angle which is 32 parts of two right angles: 64 : 360 : : 32
: 180), then the chords of his table become the sides subtending the acute angles of right-
triangles, and the diameter of his circle becomes their hypotenuses. If, furthermore, we express
the side subtending the acute angle in terms of the hypotenuse’s being 1, instead of 120 as
Ptolemy did, we would have our sine table. (Kepler makes the hypotenuse 100,000 to avoid
dealing with fractions.) In short, if we divide the chords of Ptolemy’s table by 120 and divide the
corresponding arcs in half, we will produce the sine table. For example, since the arc containing
60 parts of the 360 of the whole circle is subtended by a chord that is 60 parts of the diameter,
the sine of an angle of 30 is .5; an arc of 78 parts is subtended by a chord of 75p 31' 07", and so
the sine of an angle of 39 is 0.62932. Conversely, by multiplying the sine of an angle by 120,
we can find the chord of the arc which that angle cuts off from the circumference. For example,
Q R
S
i
n
e
A
C B
H
K
3
since the sine of 17 is 0.292371705, then the chord subtending an arc containing 34 parts of the
circumference will contain 35p 05'
05" parts of the diameter.
The sine, then, arises when
from a point on one of an angle’s
lines (A), we drop a line (AB)
perpendicular to the angle’s other line
(CD). If, however, from that same
point (A) we were to erect a
perpendicular (AD), this perpen-
dicular will eventually meet the
angle’s other line (CK). We now have
a new right-triangle (ACD), and
from its right-angle a perpendicular
(AB) has been dropped to the base
(CD). By Euclid VI, 8 then we know
that the three right-triangles (ABC,
ABD, and ACD) will all be
similar. The erected perpendicular
(AD), therefore, will have a fixed
ratio to the original hypotenuse (AC),
as will the length which it cuts off
from the angle’s remaining line (CD).
These are called, respectively,
tangent and secant. (If one describes
a circle on AC as diameter, AD will
be a tangent of the circle and DC will
be a line cutting it (seco, secare–to
cut). Finally, the difference between
the radius (EC) and the part cut off by
the perpendicular (BC), namely BE,
is called the versed sine. Sine,
tangent, secant, and versed sine are
the only trigonometric functions
Kepler uses, and of these sine is the
most prevalent by far.
Kepler makes constant use of
the law of sines, which states that in
any triangle, the sides subtending angles will have the same ratio to one another as the sines of
the angles they subtend. Kepler also makes occasional use of the law of tangents, which states
that in any triangle with unequal sides, the sum of the unequal sides has to their difference the
same ratio as the tangent of half the sum of the angles they subtend has to the tangent of half the
difference of the angles they subtend. See below for a proof of each of these.
Kepler makes no use of the co-functions (i.e., cosine, cotangent, and cosecant). For
knowing the sine of an angle, and the hypotenuse is stipulated as the unit, then, by the
C
S
i
n
e
A
H
Versed sine
secant
B E D
K
tangent
4
Pythagorean theorem, the remaining side of the right-triangle (the cosine) is found by taking the
square root of the difference between the square on the hypotenuse and the square on the sine.
A Proof of the Law of Sines
Let there be any triangle ABC. Let there be set out some unit length, U, which measures both
side AB and side BC; let U measure AB according to the units in the number N and let U
measure BC according to the units in
the number M.
I say the sine BAC : sine ACB : :
M : N.
Since U measures BC according to
the units in M, BC : U : : M : 1.
Similarly, U : AB : : 1 : N. Therefore,
ex aequali,
BC : AB : : M : N.
Let a circle be circumscribed about
triangle ABC and let its center be D.
Let the diameter BDE be taken and
let EC be joined.
By the definition of a sine, BC : BE : : sine BEC : 1.
But BEC = BAC, and so BC : BE : : sine BAC: 1.
Similarly, let AE be joined.
Therefore AB : BE :: sine AEB : 1,
and inversely, BE : AB :: 1 : sine AEB.
But ACB = AEB, therefore BE : AB :: 1 : sine ACB.
But BC : BE :: sine BAC : 1, and BE : AB :: 1 : sine ACB;
therefore, ex aequali, BC : AB :: sine BAC : sine ACB.
But BC : AB :: M : N, and BC : AB :: sine BAC : sine ACB,
Therefore M : N :: sine BAC : sine ACB. Q.E.D.
Problem: Since U is taken as a measure of the sides of the triangle, is this argument a commensurately universal
demonstration? Is it universal enough for our purposes? Can a more universal proof be made?
A
B
C
E
D N
M
U
5
A Proof of the Law of Tangents
The ratio of the sum of two legs of a triangle to the difference is the same as the ratio of tangent
of the sum of half the two base angles to the tangent of the difference of the same. That is,
I say that BC + AB : BC – AB :: tan ( ½ BAC + ½ ACB) : tan (½ BAC – ½ ACB).
Let ABC have side BC > side AB. With B as center and AB as radius, construct semi-circle
DAE, cutting BC at D. Let B be produced to E, and let AE and AD be joined. From D, erect DF
perpendicular to AD.
Now, ABE = BAC + ACB.
And since it is at its circumference,
ADE = ½ ABE.
Therefore,
ADE = ½ BAC + ½ ACB.
Now, EAD is in a semi-circle,
so AE is perpendicular to AD.
Therefore AE will be tangent to the
circle with D as center and DA as
radius.
Therefore tan ADE = AE / AD.
And therefore tan ( ½ BAC + ½ ACB) = AE / AD.
Also, ADE = ½ BAC + ½ ACB, and thus, 2 ADE = BAC + ACB.
But CAD = ADE – ACB, and thus, 2 CAD = 2 ADE – 2 ACB.
And 2 ADE = (BAC + ACB).
Therefore, 2 CAD = (BAC + ACB) – 2 ACB.
Or, 2 CAD = BAC – ACB.
Therefore CAD = (½ BAC – ½ ACB).
Since DF is perpendicular to AD, DF will be tangent to the circle with A as center and AD as
radius.
Therefore tan CAD = DF / AD.
And thus, tan (½ BAC – ½ ACB) = DF / AD.
But tan ( ½ BAC + ½ ACB) = AE / AD.
Therefore, tan ( ½ BAC + ½ ACB) : tan (½ BAC – ½ ACB) :: AE : DF.
Now, AE : DF :: EC : DC.
But EC = BC + AB, and DC = BC – AB.
Therefore AE : DF :: BC + AB : BC – AB.
Therefore tan ( ½ BAC + ½ ACB) : tan (½ BAC – ½ ACB) :: BC + AB : BC – AB.
Q.E.D. [Note: The proof can also be done by using B as center and BC as radius.]
E B D C
A
F
6
Earth’s
aphelion
Earth’s
perihelion
A
B
C
♂
♂
1585,
21º Leo
1593,
12º Pisces
5º 30' Cancer
5º 30' Capricorn
21º Aquarius
12º Virgo
D
E
p. 208, line 1: “This hypothesis…” In the preceding chapters Kepler has worked out an
hypothesis in which Mars’s motion is circular and governed by an equant, but one whose center
is not assumed to be exactly twice the distance from the Sun (or the Earth, on the geocentric
account) to the geometric center of the orbit.
p. 208, line 5b: “But 12º Virgo is nearer to the sun’s apogee than is 21º Aquarius…” In
Kepler’s time, the sun’s apogee (i.e., the earth’s aphelion) appears in 5º 30' Cancer (see p. 234).
Looking at the Zodiac (below) one can see that 12º Virgo is 66º 30' from apogee, while 21º
Aquarius is 134º 30' from it. And since the apogee is the greatest distance to the Sun, BA < CA.
NOTE: Recall that each zodiacal sign is divided into 30 segments and is read counter-clockwise.
Virgo/Virgin =
Leo/Lion =
Cancer/Crab =
Gemini/Twins =
Taurus/Bull =
Ares/Ram =
Pisces/Fishes =
Aquarius/Water bearer =
Capricorn/Goat =
Sagittarius/Archer =
Scorpio/Scorpion =
Libra/Balance =
p. 209, line 2b and following: Using the Law of Sines. Because he is using acronychal
observations when Mars is very close to the greatest latitude, the “limit,” Kepler can make use of
the angle between the planes (1º 50' –53') found in Chapter 13. However, since Mars is at about
21º Leo, while the limit is at about 16º Leo, he must decrease the angle, approximating DAB as
7
being 1º 49' 30". Using a value Tycho has determined for BA, with the Law of Sines, Kepler
finds DA, the Mars-to-Sun distance.
AE is found similarly, but since in this case Mars is 26 away from the limit, rather than
approximating the latitude Kepler must calculate it; his procedure may be justified as follows:
Let NAM be the maximum inclination,
LAE the inclination “at this position” (LAO
= 64, and EAO LAO); drop EL, MN
perpendicular to LA, AN; let LOE be made
parallel to NAM. Now sine LOE = LE : EO,
and NAM = LOE. But “the whole sine” is
the hypotenuse, i.e., the unit; so the sine of any
angle can also be expressed as the ratio of the
sine to the whole sine. Thus sine NAM : the
whole sine :: LE : EO; or inversely, the whole
sine : sine NAM :: EO : LE.
And sine EAO = EO : EA, while sine LAE = LE : EA, or inversely, the whole sine :
sine LAE :: EA : LE. Then ex aequali, sine EAO : sine LAE :: EO : LE.
Therefore the whole sine : sine NAM :: sine EAO : sine LAE, as Kepler concludes.
Together, DA and AE yield the diameter of Mars’s orbit, its radius, and its eccentricity.
Finding fourth proportionals, Kepler can then determine what these values would be if the radius
of the Martian orbit were 100,000.
pp. 210-211: Latitudes or Longitudes? After showing that his and Tycho’s hypotheses do not
give the proper eccentricity (namely, the 11,332 that Kepler derived earlier, as opposed to the
8,000 to 9,943 now calculated from observations)—and conversely, then, that the proper
eccentricity would not give the right latitudes at opposition—Kepler looks to Ptolemy’s
bisection-theory of the eccentricity.
Ptolemy claimed that the eccentricity of the equant
should be twice the eccentricity of the planet’s circle; testing
this claim, Kepler notices (p. 210) that this places the
eccentricity at 9,282—well within the range proposed from
the observations. Hence, the observed latitudes support the
bisection-theory of the eccentricity.
To test the Ptolemaic account further, Kepler assumes
this account of the eccentricity in order to predict the
longitudinal positions of Mars at non-acronychal positions
measured from the center of the ecliptic, the Sun (or the
Earth, on Ptolemy’s hypothesis).
First, using the Law of Sines, he determines one part
of the equation, CPB (which he will later call the “physical
equation”) to be 1º 1' 12", since ECP is given as 11º 3' 16".
But since ECP + BPC = EBP, the latter will be 12º 4'
28", and therefore DBP = 167º 55' 32". However, DBP = BAP + BPA, so then 83º 57'
46" = ½BAP + ½BPA.
D
P
C
B
A
E
M
A
N O
E
L
8
Then, using the Law of Tangents, Kepler determines the other part of the equation,
BPA (the “optical equation”), since according to this law BP + BA : BP – BA :: tan (½BAP
+ ½BPA) : tan (½BAP – ½BPA). For because in this proportion the only term we are not
given is the fourth, tan (½BAP – ½BPA), we can calculate the difference of these angles to
be 82º 44' 20". But since we already knew that 83º 57' 46" = ½BAP + ½BPA, equals can be
added to equals so that:
83º 57' 46" – 82º 44' 20" = (½BAP + ½BPA) – (½BAP – ½BPA)
1º 13' 26" = ½BPA + ½BPA = BPA
This anomalistic difference, BPA = 1º 13' 26", when added to semi-equated anomaly,
DBP = 192º 4' 28" (measured counterclockwise, such that it is the same as PBE + 180º),
yields the angle of motion of the planet from aphelion measured around the Sun, EAP + 180 =
supplement of DAP = 193º 17' 54". Measured from the beginning of Aries, the first sign (193º
17' 54" + 148º 55' 43'' = 342º 13' 37''), the planet must be at 11s 12º 13' 37''—which differs from
the observed longitude by 2' 23''. The problem is worse for another longitude, where the planet is
predicted to be 9' from where it actually is observed to be—and Kepler notes that his own
hypothesis from the first part of the chapter is closer to being correct, erring by less than 1.5'.
So both hypotheses are wrong, and must have admitted one or more false principles. The
question is, which ones?
p. 211: The Providence of 8 Minutes of Arc. Because Mars’s eccentricity is significantly
greater than that of the other five planets, the inadequacy of the “ancient” hypotheses—indicated
by the 8' of arc—could be detected only if the Martian path were scrutinized. Hence Kepler,
writing later, says that “I consider it again an act of Divine Providence that I arrived [in Prague
to work with Tycho] at the time when he [Christian Logomantanus, Tycho’s assistant] was
studying Mars; because for us to arrive at the secret knowledge of astronomy, it is absolutely
necessary to use the motion of Mars; otherwise it would remain eternally hidden.”
Ch. 21, p. 219, line 10b, “the hypothesis will exhibit the longitude perfectly accurately at
eight places…” Actually, on this third modification of the hypothesis, the hypothesis will no
longer predict accurately the position at the quadrants, thanks to the introduction of the equant.
For when the planet should appear along line AK, since the actual motion of the planet is now
around B on circle HI, the planet will in fact appear on the line connecting A with the
intersection of CK and arc HFG. However, this difference may be too small to observe.
Chapter 22 Overview
Kepler now “begins the whole inquiry anew, not with the first inequality, but with the
second,” and this will lead him to argue that the motion of the earth, or that of the Sun—recall
that Kepler still has not ruled out any of the three general hypotheses—requires an equant. Since
Kepler began the work focusing on the zodiacal anomaly (the first inequality), why does he start
this time with the heliacal anomaly (the second)?
Two reasons come to mind. First, if Copernicus is correct (as Kepler believes) that the
Earth rather than the Sun is the cause of the latter’s apparent motion, and thus that the heliacal
anomaly for the outer planets is caused by the annual motion of the Earth, then it is crucial that
he start with a precise understanding of the heliacal anomaly (the Earth’s motion) in order to
disentangle the principles of apparent Martian motion. A second reason is implied in the
objection that Kepler raises against himself (p. 225, line 14b). Recall that Kepler had said earlier
9
(see summary of ch. 4, p. 43) that the equant can be explained in terms of physical causes. The
objection, then, points out that if his physical causes make equants necessary, why would the
Earth’s orbit be the sole exception? For as we saw in Ptolemy and Copernicus, unlike the motion
of the planets, that of the Sun seems not to need an equant. Hence, with an eye to vindicating
heliocentrism—that is, to showing that the Earth is just like the other planets—Kepler must take
a closer look at the heliacal anomaly to see if it does indeed need an equant.
In the remaining part of the chapter Kepler does not argue directly that the Earth (or Sun)
needs an equant—that will be done in Chapter 24. Rather, he notes that Tycho implicitly admits
that the phenomena demand an equant. But how does Tycho’s hypothesis of the growing and
shrinking “annual orb” imply phenomena explainable by an equant? See if you can see this by
drawing two different sized, but concentric, circles to represent the changing annual orb, and plot
the apparent motion on the orbit of the Earth (or Sun).
Chapter 24 and the Greek Alphabet: In a diagram on p. 233 we see the first instance in which
Kepler uses lower-case Greek letters to mark points. He will do this again in later chapters (e.g.,
ch. 39). To facilitate discussion of these diagrams, below is a list of the names of these letters.
= alpha
beta
gamma
delta
epsilon
zeta
eta
theta
iota
kappa
lambda
mu
nu
xi
omicron
pi
rho
sigma
tau
upsilon
chi
phi
psi
omega
Chapter 33, p. 279, line 13: The analogy of the lever. If a lever is placed on a fulcrum with a
weight on the left side and a lesser weight on the longer right arm, recall from Archimedes that
in order for the weights to balance each other perfectly, they must be inversely proportional to
the distances from the fulcrum. If the right arm were slightly longer, and the right weight placed
slightly further from the fulcrum, it would tilt the lever slightly in its direction, and if slightly
longer still, the lever would tilt even further or more quickly in its direction. In short, the greater
the distance of a weight from its fulcrum, the less efficacious the force of the other weight is to
move it (or to resist its motion). In Kepler’s elucidation of the distance-rule from Chapter 32, the
left force or weight will correspond to an intrinsic power of the Sun, the position of which will
correspond to the fulcrum, and the right force or weight will then correspond to the planet.
p. 279, line 18: “Copernicus when he is speculating…” See De Revolutionibus I.10, pp. 527-28.
Chapter 34, p. 285, line 14b. This somewhat ambiguous sentence might be glossed
as follows with the accompanying diagram: “Therefore, it is not only required by
the nature of the species, but likely in itself owing to this kinship with light, that
along with the particles of its [i.e., the species’s] body or source [i.e., the Sun] it too
is divided up, and when any particle of the solar body [A] moves towards some part
of the world [i.e., the whole cosmos], the particle [B] of the immaterial species
which from the beginning of creation corresponded to that particle [A] of the body
[i.e., the Sun] also always moves towards the same part [B]. If this were not so, it
would not be a species, and would come down from the body [i.e., the Sun] in
curved [AC] rather than straight lines [AB].”
B
C
A
10
p. 286, middle: the Rotation of the Sun. Besides confirming Kepler’s prediction that the sun
should rotate, Galileo also confirmed his predictions both about the relative orientation of the
Sun’s axis (namely, that it would be approximately the same as that of the ecliptic—it deviates
by about 7), and, broadly speaking, about its speed when he says that it must make a full
rotation “at least once in a three-months’ span” (p. 287, line 5b). He errs, however, when he
attempts to pinpoint the rate of rotation: While Kepler argues that the Sun makes a full rotation
in about 3 days (p. 288, line 11), Galileo will show from his observations that the speed of the
sunspots near the equator make a full rotation in 26 days. Is there an erroneous step in Kepler’s
argument? (Because the Sun is a gas, it appears to have a differential rotation, i.e., the angular
speed of its surface features depends on their latitude; for example, sunspots close to the north
pole of the Sun take more than 36 days to make a full rotation.)
Chapter 38 Overview
Everything in Kepler’s physical account of the heavenly motions so far suggests that the planets
should move in perfect circles centered around the Sun, since this is the motion of the species it
emanates, and, were it not for the planet’s inherent resistance to motion (see ch. 34, and p. 301),
it should complete its orbit of the Sun in 3 days. But since it is clear that the planets do not move
along paths centered exactly on the Sun, Kepler must add a second physical cause to his theory.
He posits the existence of a “motive power” peculiar to each planet to explain its “ascent from
and descent towards the Sun.” How can such “ferrymen” accomplish eccentric circular motion
when the Sun’s species induces them to concentric motion? Expanding on the ferryman example
from p. 300, and anticipating an analogous illustration
Kepler will use in Chapter 57, we may modify Kepler’s
image in the diagram to the right.
In the diagram, we have a circular river with a
current flowing counterclockwise around an island centered
on S. Using the current (i.e. without rowing), by constantly
changing the orientation of his oar or rudder relative to the
direction of the current, how might he redirect the motive
power of the current such that the boat will be gradually
pushed from A toward P, and then back again to A?
As an exercise, determine what shapes the boat
would trace out if the oar/rudder maintained its orientation
in reference to the shore. How must the oar be oriented in
order to produce an ellipse?
p. 304: Reciprocation Along the Epicycle’s Diameter. The third hypothesis that Kepler
introduces here, and which he will keep (“for want of a better opinion” p. 305) until he formally
introduces the ellipse in Chapter 59, eliminates that epicycle’s circumference, but retains its
diameter, understood as an extension of one of the rays from the Sun. The planet’s proper motion
becomes an oscillation or “reciprocation” along this epicycle’s diameter. Nonetheless, Kepler
continues to include the epicycle’s circumference as an aid to the imagination in analyzing the
character or “measure” of this reciprocating motion, correlating each point on the circumference
with a point on the diameter.
A
C·
S·
P.
11
Chapter 40 Overview
This chapter introduces what later came to be called “Kepler’s Second Law of planetary
motion,” that in equal periods of time equal areas are swept out by the ray from the Sun to the
planet—although the law is not presented in exactly these terms in the Astronomia Nova.
It may be helpful to clarify some terms we find in this chapter, namely “equation,”
“physical equation” (or “physical part of the equation”), “optical equation” (or “optical part…”),
and the three anomalies, “mean,” “eccentric,” and “equated.” (Note that each of these terms is
also defined in the glossary at the end of this manual.) Using a simplified version of the diagram
on p. 310, with the planet at G, the Sun at A, the center of the eccentric orbit at B, add the equant
point Z above B on the line of apsides.
The “equation” is ZGA.
The “physical equation” is ZGB.
The “optical equation” is BGA.
“Anomaly” generally is “any angular measure of a
planet’s motion” according to the mathematician.
But the motion of the planet around circle CED
could be measured angularly from Z, B, or A. Hence
there are three anomalies.
“Mean anomaly” is the angular measure of the planet
around the equant. So here it is CZG.
“Eccentric anomaly” is the angular measure of the
planet around the center of the eccentric. Here it is
CBG.
“Equated anomaly” is the angular measure of the planet around the Sun. Here it is
CAG.
It is also helpful to note some relations among these entities.
1. The physical equation plus the optical equation = the equation.
2. The eccentric anomaly minus the equated anomaly = the optical equation.
3. Most pertinently, the mean anomaly minus the eccentric anomaly = physical equation.
All of these make sense when you refer back to the angles.
What Kepler is going to do is take the last relation (CZG –
CBG = ZGB) and substitute sector (or area) CBG in place of
CBG—and since B is the center of the circle this is not a
stretch—and substitute area CAG for the mean anomaly, CZG.
And this will mean that the difference, BAG, is equivalent (or
directly proportional to) the physical equation, ZGB.
This is a summary of Kepler’s overall argument, but the
most interesting part of this chapter is how he makes the move to
substitute area CAG for the mean anomaly, angle CZG, because
this means that the planet will sweep out equal areas in equal
times around the Sun.
A
Z
B
C
G
D
A
Z
B
C
G
D
12
Chapter 44 Overview
The first four chapters of Part Four manifest “the source of our suspicion that the planet’s path is
not a circle” (ch. 42), and they proceed mainly by way of a reduction to the absurd: On the
assumption that Mars’s path is a perfect circle, we will end up with an erroneous apogee,
eccentricity, ratio of eccentricity, set of equations, times, and distances. Further, Kepler argues
(in Chapter 44) that the correct distances and times fit nicely with the assumption that Mars’s
path is some sort of oval (introduced above in Chapter 30).
Kepler’s second argument that the path is not circular but ovular postulates an oval-
shaped path to show that it matches the observations regarding the speed of the planet. For with
the oval, the areas swept out by the planet’s motion are decreased least at aphelion and
perihelion, but most at the middle regions, as in the diagram below, based on Kepler’s “fat-
bellied sausage” analogy (p. 338, line 4).
On the hypotheses of the circle and the oval, the
path is divided into equal areas (ABG = BCG = CDG =
DEG, and AB1G = B1C1G = C1D1G = D1EG) that
correspond to equal time intervals, following Kepler’s
area law, his “Second Law.” The aphelial distances of
the two hypotheses differ least but those at the middle
longitudes differ most, and so the latter regions must be
wider. Hence, on the hypothesis of the oval path, the
width of the areas at the middle longitudes (arc C1 to D1)
being greater, the planet must be moving faster, and so
the planet will take less time in the middle regions, and
therefore more slowly at the aphelion. This fits with the
observations.
Kepler has now shown that the “vicarious
hypothesis” that he had been using for so long is
erroneous. What does Kepler mean when he calls this
hypothesis “vicarious”? He explains the expressions in an earlier part of the Astronomia Nova,
saying that our hypothesis is “only vicarious, not natural, and thus possesses only as much
trustworthiness as is permitted by the observations” (ch. 28). Given Kepler’s proposed end, and
his re-introduction of the epicycle in Chapters 45 and 46, what might “natural” mean?
Chapter 51 Overview
Kepler is now calculating the distance from the Sun to Mars for a large number of different
positions on the latter’s orbit for the sake of checking the oval hypothesis. At the end of this
chapter, Kepler uses these new calculations to show that “it appears that those distances of Mars
from the Sun are equal whose points on the orbit are equally remote from aphelion,” i.e., that the
path of Mars is bilaterally symmetrical. He uses this data again in the next chapter (and again in
chapter 67, on physical grounds) to confirm what he’s hoped to show all along, that the planetary
motions are centered on the apparent or true Sun, not the mean Sun. See the title of Chapter 52
(p. 393), and its concluding paragraph (p. 395).
It would seem that a careful examination of this acquisition of numerous Sun-Mars
distances would be adequate to prove that the path of the orbit is an ellipse. Kepler’s actual
manner of arguing that the path is an ellipse—what will later be called his “First Law”—will be
different, and will involve not only distance determinations but also the calculations of eccentric
A
B
C
D
B1
C1
D1
G
E
13
equations on the basis of the Second Law. (Indeed, there is some evidence that the calculations in
Chapters 51 and 53 were carried out after Kepler had concluded that the orbit was an ellipse.) In
any case, after discovering that Mars’s path is not circular, Kepler is not looking for a geometri-
cally simple curve; rather, he seeks a physically causal account of in terms of the Sun’s motive
power and the planetary Minds. In fact, at this point in a letter he expresses the hope that the path
will not turn out to be elliptical: “At that time I was, in truth, lacking a natural cause if the path
should be an ellipse…” (To Fabricius, October 11th
, 1605).
For the sake of completeness, we should mention that Kepler’s so-called “Third Law,”
unmentioned in the Astronomia Nova, is that the square of a planet’s period is proportional to the
cube of its mean distance from the center of its orbit. He derives it entirely from observations.
Chapter 56 Overview
In Chapter 46 Kepler had shown that the width of the lunule predicted by the theory of Chapter
45 is 858 units; in more recent chapters he showed that in fact it should be half this, 429.
However, at the beginning of this chapter Kepler gives us another, “more correct” value, 432.
Where does he get this new value? (Cp. note 1 on p. 405.) To answer this we must look more
closely at what follows in this chapter, where Kepler derives his “rule of reciprocation.”
First, however, we must review trigonometric terminology.
The secant of an angle is the inverse of the cosine of the angle, so in
the diagram below the cosine of BEA = EB / EA, and therefore
the secant of the same angle is EA / EB, or EA / 1 if the radius is
the unit.
In this chapter Kepler tells us how he stumbled upon the
method to find the correct Mars-Sun distances for given optical
equations. He has shown that the width of the lunule at quadrature
is 429 units where the radius BE is 100,000, while the optical
equation for this same position (BEA) is 5 18' (5.3), and Mars’s
true distance from the Sun at this time (MA) should be 100,000. By
chance he notices that the secant of 5.3 (EA) is 100,429, which
exceeds Mars’s true distance by exactly the breadth of the lunule.
So, by replacing the secant of 5.3 here with radius EB (or 100,000,
since we are at quadrature)—Eureka!—we have the true distance of
Mars from the Sun (MA = 100,000). Kepler begins to suspect that
this implies a universal method or rule for determining Mars-Sun
distances. How could one formulate this rule?
There is a problem, however, with this example, namely if ME = 429, MA = 100,000,
and EA = 100,429, then MAE cannot be a triangle (see Elements I.20). But since ME is so small
compared to MA and EA, then MA and EA are nearly on the same straight line, and MAE is
very small, and this will resolve the problem. With BEA = 5.3, EA = 100,429, and MA =
100,000, what would ME be? By the Law of Sines (sine 5.3 / 100,000 = sine EMA / 100,429),
we calculate EMA to be 174.67715; therefore, by subtraction, we calculate MAE to be
0.02285. Again, by the Law of Sines (sine 5.3 / 100,000 = sine 0.02285 / ME), we calculate
ME to be 431.75 units.
We now see why Kepler can say 432 is the “more correct” measure of the width of the
lunule. A difference of 3 units of 100,000 would be well within his margin of error—indeed, he’s
off only by 0.003%—so Kepler can say that that it is practically the case that EA = EM + MA.
C
B
A
D
E M
14
Recall that at the end of Chapter 51 he said, “I shall rejoice if I am able to come within an
uncertainty of 100 units everywhere.”
This new rule allows Kepler to re-introduce his reciprocating epicycle from chapter 39.
Referring to the diagram on p. 407, right triangle HRA corresponds to (in fact, is equal to) right
triangle , and whereas before AH was taken as the Mars-Sun distance, on the new rule of
reciprocation HR is that distance. Similarly with the epicycle, then, before was the distance,
but now it is . Thus the problem of the irregularity of positions has been replaced by
, and , which are more regularly placed on the diameter of the epicycle, since = ,
and = 2. On p. 408 (line 14b), Kepler says that he will explain the double character of
by a physical account in Chapter 57.
Chapter 57 Overview
Kepler now tries to remove his planetary Minds from the theory and replaces them with
natural causes. He begins his argument that the reciprocation is natural by an analogy with a boat
in a circular river where the oar is rotated uniformly at half the rate at which the boat completes
its circuit. Note that the analogy illustrates the acceleration and deceleration of the boat, not the
exact shape of its path per se. That is, the path described here is not necessarily the ellipse or
oval for which Kepler is looking. The analogy with the magnet is similar.
Here Kepler replaces the planetary
Mind by something like a magnetic axis for the
planet, such that the planet always remains
pointed in the same direction, at a celestial
pole, as it were. In addition, one pole of the
planet’s axis (the northern pole or arrowhead)
naturally seeks the Sun, whereas the other (the
southern pole or tail) flees the Sun. Hence, at C
and F the planet is equally drawn toward and
repulsed by the Sun, since the poles are equally
distant from it. However, along arc CDEF,
because the north pole is closer than the south,
the planet approaches toward the Sun at A, and
thereby is accelerated. After it passes F and
moves through arc FGHC, the south pole is
more efficacious than the north, repelling the
planet away from the Sun and thereby
accelerating it away from the Sun.
Kepler takes a moment (p. 412) to criticize Copernicus’s conviction that an extra epicycle
was necessary to preserve the immobility of Earth’s axis. Kepler seems to think this unnecessary:
“there is no need for extrinsic causes.” Further, he seems to think that this new magnetic axis
does not require a cause or explanation: “And so here, too, there is nothing to suggest that there
will be a need for movers for the planet, which would carry its body about the sun in a parallel
position, and at the same time perform the reciprocation.” Is this assessment correct?
The comparison to magnetism must be qualified, however, when Kepler notices the precise
orientation of the planet’s axis that is required by his account. For Kepler’s magnetic “north
pole” is not pointing toward or away from the Sun when the planet is at aphelion and perihelion;
but, assuming that the Earth’s magnetic character is of the same sort as Mars’s, in fact
C
D
E
F
G
H
A
15
the planet’s magnetic “north pole” (which Gilbert has shown roughly corresponds to the actual
north pole of the Earth) should be inclined toward the Sun at perihelion, and inclined away from
it at aphelion, as these inclinations cause the seasonal changes.
In addition, elsewhere he notes that “if the body rotates, only one single diameter of
power, that which is parallel to the axis of its rotational motion, remains constant and equidistant
from itself” (Chapter 63); hence, it seems that there can be no other axis that remains fixed.
Chapter 59, Protheorem XV
After explaining his empirical argument
for the conclusion of this protheorem,
Kepler explains how he previously
misunderstood his rule of reciprocation
in Chapter 56. For there Kepler realized
that if NB = EH, the correct equated
anomaly (ANB) and the correct Sun-
Mars distance (NB) are produced. But
there is an ambiguity in EH, since it can
be understood under two different
formalities: a) as the line from the
circumference through the center, and to
which a perpendicular from N is
dropped, and b) as the line from the
circumference perpendicular to the
diameter. Hence, if one picks any other
point on the circumference (K), is it KL
or KT that corresponds to EH?
F (solstice and perihelion)
Y (equinox)
X
(equinox)
C (solstice and aphelion)
The true
orientation of
the poles at
solstices and
equinoxes
F (equinox and perihelion)
Y (solstice)
X
(solstice)
C (equinox and aphelion)
The
orientation
of the poles
on Kepler’s
magnetic
hypothesis
L
H
T
N
C
E
K M
X
B
A
Z
16
The answer one gives is crucial because it determines one’s universal expression of the
rule of reciprocation. In Chapters 56 and 58, Kepler assumed the line corresponding to EH was
KT; hence, he understood the rule to entail that a line cut off from KN equal to KT will yield the
proper Sun-Mars distance (NX = KT), and that where this length, rotated around N, cuts KT we
would have the correct position of Mars, point Z. However, Kepler discovers that this position
does not match the observed equated anomaly, ANZ.
He then reconsiders his assumption; perhaps it is the intersection of the rotated NX with
KL, not KT, that is the appropriate position for Mars, M. It turns out that this position NM is in
accord with observed equation, ANM. With this understanding of the rule of reciprocation that
produces the path of the planet as his minor premise, combined with the affirmation (inferred
from the previous protheorems, especially Protheorem XI) that such a rule of reciprocation fits
only with an ellipse as his major premise, Kepler can conclude that the planet’s path is an ellipse.
Chapter 59 Conclusion/Chapter 60 Overview
Protheorem XV of Chapter 59 removes the burden of calculating areas of the ellipse
(such as AMN) in order to find sums of distances, and thus times elapsed on the elliptical orbit:
for the corresponding areas on the circle (such as AKN) are in the same ratio. As AKN is to the
area of the whole circle, so is the time elapsed on arc AM to the whole periodic time; and the
area of AKN can be calculated by familiar means, given an arc AK and the eccentricity HN.
There is a peculiarity to this procedure, however: if the eccentricity be known, then given
any arc AK, one can find three measures of the planet’s motion which correspond to that arc—
distance of the planet from the sun (NM), angle of motion around the sun (ANM), and time
elapsed (by the ratio of area AKN)—but what is given is not directly any measure of the planet’s
motion at all. It is rather more difficult to be given a particular distance, angle of motion, or time
elapsed, and to find other two. Chapter 60 is devoted to the means of making calculations based
on this new theory, and at the end of the chapter (which is the end of Part Four), Kepler admits
that “given the mean anomaly [i.e., the time elapsed], there is no geometrical method of
proceeding” to the other measures; he exhorts geometers to find such a method.
Nonetheless, by starting with any number of arcs AK one can build a table with any
number of corresponding measures of the planet’s motion, and then use the table to proceed from
any given time (or an indefinitely close approximation thereof) to the corresponding angle, etc.
Exercise: Given eccentricity BA = 9282 (where BD = 100,000)
and eccentric arc EC = 12 10' after perihelion, find the
corresponding distance of the planet from the sun, angle of
movement around the sun from perihelion, and ratio of time
elapsed. (Note: the planet will not be on any of the lines given.)
Compare the results with Kepler’s test of the same eccentricity,
on a perfect eccentric circle with an equant-point, in Chapter 19
(p. 211, and pp. 7-8 of these Notes)—which revealed that the
Ptolemaic bisected eccentricity confirmed by the latitudes did
not fit the longitudes. (Recall that “the simple anomaly” on p.
211 is the measure of time according to the equant, which will
correspond almost exactly to the measure of time according to
the sums of distances.)
D
C
B
A
E
17
The Rest of the Astronomia Nova: In the final section of the work, Part Five, Kepler applies
his new theory to the phenomena of latitudinal shifts of the planets, i.e., their motion above and
below the ecliptic—which he has shown (at the end of Part Two), Ptolemy, Copernicus, and
Tycho have failed to predict accurately. In short, Kepler shows that his theory succeeds:
Thus the hypothesis established in this work shows this very thing whose cause Brahe
advised was diligently to be sought, and which ancient astronomy, for all its apparatus,
could not show. And, I would add, it shows this in all its simplicity, in that the plane of
the eccentric is given a constant inclination or obliquity, and this is variously increased or
diminished, not in reality, but optically only, insofar as our sighting approaches it or
recedes from it… (Chapter 66)
The Astronomia Nova was not an immediate success. In 1627 Kepler published the
Rudolphine Tables based on Tycho’s incomparable observations and his own new theory. Were
it not for these tables, whose planetary predictions were nearly two orders of magnitude more
accurate than any previous methods, the Astronomia Nova might have been forgotten.
18
L
B
A
C
Starting from Kepler’s data in Chapter 56, p. 543
A is the Sun at the focus of Mars’s ellipse CL,
B is the center of the eccentric circle CE, and
EL is the “breadth of the lunule.”
BE = 100,000
EL = 432 or about 0.4% of the radius
Therefore, BL = 99,568
Thus, Mars’s ellipse is 99.568% circular.
But, from a property of ellipses [Apollonius, III. 52],
LA = 100,000.
Therefore, by the Pythagorean theorem, the focal distance
BA = (100,0002 – 99,568
2)1/2
= 9,285 or about 9% of the
radius.
NOTE: Mars’s orbit is the most elliptical of the 9 planets
except for Pluto (which wasn’t discovered until the 20th
century) and Mercury, which is too difficult to begin from
both because it is difficult to observe well, owing to its
proximity to the Sun, and because its orbit has a
noticeable precession.
.
E
Mars’s Ellipse
How closely do the planets’ ellipses approach being perfect circles? (Using modern values)
Mercury 97.8735 % Jupiter 99.8843 %
Venus 99.9976 % Saturn 99.8588 %
Earth 99.9886 % Uranus 99.8899 %
Mars 99.5749 % Neptune 99.9960 %
Pluto 96.8328 %
19
GLOSSARY OF TERMS USED BY KEPLER
ACRONYCHAL OBSERVATIONS/POWERS. Observations of a planet when it rises at sundown (from
acro and nyche = “night-rising”). These are also observations of a planet when it is at solar
opposition, since a planet rises as the sun sets only when the planet is at opposition. (Cf. sum-8
and sum-16.)
ANOMALY. Any angular measure of a planet’s motion, whether regular or irregular. (Not to be
confused with irregularity.) There are two kinds:
1. The “eccentric anomaly,” which is any angle of planetary motion about the
geometric center of its orbit. Cf. chs. 40 and 55.
2. The “mean anomaly,” which is an angle of planetary motion measured about the
equant. Cf. chs. 40 and 55. See also SEMI-EQUATED ANOMALY.
ANOMALY OF COMMUTATION. Also simply referred to as the
“commutation,” the angular measure of the relative motion of the
Earth and a planet about some center. Kepler considers two kinds:
1. The “true anomaly of commutation,” also called
“equated anomaly of commutation,” is the angle
formed by Earth – Sun – planet (angle ESP in the
diagram).
2. The “mean anomaly of commutation,” is the angle
formed by Earth – geometric center of Earth’s
orbit – planet (angle ECP in the diagram). Of
course, the geometric center of the Earth’s orbit is
the mean Sun. Cf. ch. 24.
APHELIAL DISTANCE. Distance between a planet and the Sun when the Sun is at aphelion.
APHELION. The point furthest from (apo) Sun (helios) on a planet’s orbit. Compare “apogee.”
See also PERIHELION, APHELIAL DISTANCE, and PERIHELIAL DISTANCE.
ASCENDING NODE. See NODE.
CIRCUMFERENTIAL DISTANCE. In the diagram to the right, the distance from a
point a outside an epicycle to a point d on its circumference. See also
DIAMETRAL DISTANCE.
CONCHOID. A geometric curve defined by a point and a straight line such that
all distances intercepted between the curve and the straight line on straight
lines drawn from the point to the curve are equal. See sum-40 and 43.
COMMUTATION. See ANOMALY OF COMMUTATION.
DESCENDING NODE. See NODE.
C S
E
P
a
d
g
k
z
20
DIAMETRAL DISTANCE. In the diagram to the right, the distance from a point
a outside an epicycle to a point k on its diameter (corresponding to a
circumferential distance ad when dk is perpendicular to gz), i.e., ak. See also
CIRCUMFERENTIAL DISTANCE.
DIURNAL PARALLAX. A planet’s parallax
as seen from the Earth resulting from 24
hours of the Earth’s orbital motion. In the
diagram to the right, angle AMB. See
also PARALLAX.
ECCENTRIC ANOMALY. See ANOMALY.
EQUALIZING POINT. The equant center.
EQUATED ANOMALY. Angle formed by aphelion – Sun – planet (angle ASP in diagram to the
right). Cf. sum-29 and ch. 40. See also SEMI-EQUATED ANOMALY.
EQUATED ANOMALY OF COMMUTATION. See ANOMALY OF COMMUTATION.
EQUATION. The angle formed by equant – planet –
sun (angle EPS in the diagram to the right), and is
the same as what Ptolemy called an “anomalistic
difference.” This term is ubiquitous in the New
Astronomy, beginning in ch. 19. There are two parts
to an equation:
1. The “physical part of the equation” is the
angle formed by geometric center of the planet’s
orbit – planet – equant (angle CPE). It is called
“physical” because it is connected to the equant, and
therefore to the planet’s true uniform speed. Cf. ch.
40 and sum-47.
2. The “optical part of the equation” (often
simply called the “optical equation”) is the angle
formed by geometric center of the planet’s orbit –
planet – Sun (angle CPS). See also SEMI-EQUATED
ANOMALY and EQUATED ANOMALY.
FIRST INEQUALITY. See INEQUALITY.
FIRST MOTION. The daily westward motion of the planets, the motion of the Same.
INCLINATION OF THE PLANES. The angle of inclination between the plane of a planet’s orbit and
the plane of the ecliptic (which is the plane of the Earth’s orbit). (Cf. Elements, XI, def. 6)
INEQUALITY. Any irregularity, or anomaly, in a planet’s motion. There are two kinds:
M
A
B
a
d
g
k
z
A
P
E
C
S
21
1. The “first inequality” (i.e., the zodiacal anomaly) is the
inequality of the possible sizes of the heliacal anomalies
of a planet, and depends on which sign of the zodiac a
planet is in at that time.
2. The “second inequality” (i.e., the heliacal anomaly) is the
irregularity in a planet’s motion that results, for the outer
planets, in retrogradation or, for the inner planets, in
motion to the east and west of the mean sun.
LATITUDE. The position of a planet measured in the north-south direction, usually with reference
to the poles of the ecliptic, but sometimes with reference to the Earth’s poles (the celestial poles).
See also LONGITUDE.
LIMIT BY THE FIRST INEQUALITY. The northern- or southernmost point of the circle producing
the first inequality, i.e., the eccentric orbit; more clearly, one could say “the upper or lower
extreme of the planet’s orbit, relative to the plane of the ecliptic.” (Cf. sum-13)
LONGITUDE. The position of a planet measured in the east-west direction, usually along the
ecliptic but sometimes along the equator. See also LATITUDE.
LUNULE. A moon-shaped figure that is the difference between two curved figures.
MEAN ANOMALY OF COMMUTATION. See ANOMALY OF COMMUTATION.
NODE. The intersection of a planet’s orbit with the planet of the ecliptic. There are two kinds:
1. The “ascending node” is the node at which a planet passes from south of the ecliptic
plane to north of it.
2. The “descending node” is the node at which the planet passes from the north of the
ecliptic to the south of it. (Cf. sum-9 and sum-12.)
OPTICAL PART OF THE EQUATION. See EQUATION.
OVAL. An egg shape (or cardioid), blunted at one end and sharper at the other; not an ellipse. Cf.
Sum-30, Sums 44-48, and ch. 55. (At the end of summary 46 an oval is distinguished from an
ellipse.)
PARALLAX. The change in an object’s apparent position (against a background) when observed
from two different places. See DIURNAL PARALLAX.
PERIHELIAL DISTANCE. See PERIHELION.
B
x C
T
A
22
PERIHELION. The point nearest (peri) the Sun (helios) on a planet’s orbit. Compare “perigee.”
See also APHELION and PERIHELIAL DISTANCE.
PHYSICAL PART OF THE EQUATION. See EQUATION.
PROTHEOREM. Synonymous with “theorem.” Cf. sum-20 and ch. 59.
RECIPROCATION. A shifting back and forth along a straight line, usually understood to be the
diameter of an epicycle. Cf. chs. 21, 39, 56, and 57.
SEAT OF POWER. See SOURCE OF POWER.
SECANT. A trigonometric term; the secant of an angle x = the inverse of
its cosine = 1/cosx = secx = EA/EB = EA/1 (since cosx = EB/EA =
adjacent/hypotenuse). Cf. ch. 56 and the comments on ch. 19 above.
SECOND INEQUALITY. See INEQUALITY.
SECOND MOTION. The eastward motion of the planets, the motion of the
Other.
SEMI-EQUATED ANOMALY. Angle formed by aphelion – geometric center
of planet’s orbit – planet (angle ACP in diagram to the right). See
EQUATED ANOMALY.
SOLID ORB. The physical crystalline sphere supposed to carry a planet,
e.g., through an epicycle or around a deferent.
SOURCE OF POWER. The center of physical causality producing planetary motion. Synonymous
with SEAT OF POWER.
TRIANGLE OF THE EQUATION. Triangle whose vertices are equant – planet – Sun. The angle thus
formed is the “equation.” Cf. chs. 40 and 55.
TRUE ANOMALY OF COMMUTATION. See ANOMALY OF COMMUTATION.
VERSED SINE. A trigonometric term; the versed sine of an angle x = 1 – cosx = AC – AB = BC.
Cf. chs. 39 and 57, and the comments on ch. 19, above.
VICARIOUS HYPOTHESIS. A non-bisected eccentric hypothesis according to which Mars moves
on an eccentric circle with uniform speed about an equant. It is specified as follows: Mars’s
aphelion is at 28º 48’ 55’’ Leo; where the radius of Mars’s orbit is 100,000, the center of the
orbit is 11,332 eccentric from the Sun, and the equant center is 7,232 eccentric from the center of
the orbit (and thus, the total eccentricity is 18,564. See chs. 16 and 51.
[See next page for other data Kepler and Brahe gathered, from which Kepler is working.]
A
E
C
S
P
x B
A
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23
Other Useful Information on Mars that Kepler Employs
[All page references are to the Green Lion edition of the Astronomia Nova]
Mars’s Period: 687 days (p. 233)
Mars’s Nodes: (p. 163)
Ascending: 16º Taurus
Descending: 16º Scorpio
Mars’s Limits: (p. 209)
Northern: 16º Leo
Southern: 16º Aquarius
Mars’s Maximum Latitude: 1º 50 (p. 209)
Mar’s Aphelion: 28º 48 55 Leo in 1587 (p. 199)
Yearly Motion of Mars’s Aphelion: 1 4 (p. 203)
Earth’s Avg. Radius = 100,000 Mars’s Avg. Radius = 100,000
Mars’s Aphelial Distance: 166,510 109,265
Mars’s Perihelial Distance: 138,173 90,735
Mars’s Average Radius: 152,342 100,000
Mars’s Eccentricity (bisected): 14,169 9,301
(p. 404)
Useful Tables
Ch. 15, p. 183: 12 observed acronychal positions of Mars
Ch. 17, p. 203: Motion of Mars’s Nodes and Aphelion
Ch. 18, pp. 206-7: Comparison of observed positions in Ch. 15 with calculated ones (using
the Vicarious Hypothesis of Ch. 16)
Ch. 30, pp. 271-72: Table of Earth-Sun distances
Ch. 47, p. 356: Comparison of Equations calculated in six different ways
Ch. 50, pp. 372-77: Comparison of Equations calculated from Ch. 16 hypothesis and six new
ways of calculating them
Ch. 53, p. 401: Table of Computed and Observed Positions of Mars just before and after
Opposition
Ch. 56, p. 409: Comparison of Observed Distances and Calculated Distances from
Reciprocating Epicycle
Chapter 8 (Brahe’s table of 10 moments of Mars in opposition to the mean Sun):
1580 November 17 9:40 pm
1582 December 28 12:16 am
1585 January 31 7:35 am
1587 March 7 5:22 am
1589 April 15 1:34 am
1591 June 8 4:25 am
1593 August 24 2:13 pm
1595 October 29 9:22 am
1597 December 13 1:35 am
1600 January 19 9:40 pm
Chapter 10 (the 10 actual observations from which Brahe derived his table):
1580 November 12 10:50 pm
1582 December 28 11:30 pm
1585 January 31 12:00 am
1587 March 7 7:10 am
1589 April 15 12:05 am
1591 June 6 12:20 am
1593 August 24 10:30 pm
1595 October 30 8:20 pm
1597 December 10 8:30 pm
1600 January 13 11:50 pm
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