SophomoreMathematics

RevisedSummer 2017

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1

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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axis

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of Alexandria

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(of the Ptolemy book, not page 27 of the manual)

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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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4748

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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

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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.]

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C

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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