Zodiacal Light 2023

In 2023, the best dates and times for observing the zodiacal light are listed in the calendar below. The sky must be very clear with little or no light pollution. The specific times listed are for Tucson, Arizona (32° 16′ N, 111° 03′ W).

Here’s a nicely-formatted printable PDF file of the zodiacal light calendar:

January 2023
SUN MON TUE WED THU FRI SAT
1 2 3 4 5 6 7
8
Zodiacal Light 7:02 – 7:14 p.m. West
9
Zodiacal Light 7:03 – 8:03 p.m. West
10
Zodiacal Light 7:04 – 8:04 p.m. West
11
Zodiacal Light 7:04 – 8:04 p.m. West
12
Zodiacal Light 7:05 – 8:05 p.m. West
13
Zodiacal Light 7:06 – 8:06 p.m. West
14
Zodiacal Light 7:07 – 8:07 p.m. West
15
Zodiacal Light 7:07 – 8:07 p.m. West
16
Zodiacal Light 7:08 – 8:08 p.m. West
17
Zodiacal Light 7:09 – 8:09 p.m. West
18
Zodiacal Light 7:10 – 8:10 p.m. West
19
Zodiacal Light 7:11 – 8:11 p.m. West
20
Zodiacal Light 7:11 – 8:11 p.m. West
21
Zodiacal Light 7:12 – 8:12 p.m. West
22
Zodiacal Light 7:13 – 8:13 p.m. West
23 24 25 26 27 28
29 30 31        

February 2023
SUN MON TUE WED THU FRI SAT
      1 2 3 4
5 6 7
Zodiacal Light 7:26 – 7:58 p.m. West
8
Zodiacal Light 7:27 – 8:27 p.m. West
9
Zodiacal Light 7:27 – 8:27 p.m. West
10
Zodiacal Light 7:28 – 8:28 p.m. West
11
Zodiacal Light 7:29 – 8:29 p.m. West
12
Zodiacal Light 7:30 – 8:30 p.m. West
13
Zodiacal Light 7:31 – 8:31 p.m. West
14
Zodiacal Light 7:31 – 8:31 p.m. West
15
Zodiacal Light 7:32 – 8:32 p.m. West
16
Zodiacal Light 7:33 – 8:33 p.m. West
17
Zodiacal Light 7:34 – 8:34 p.m. West
18
Zodiacal Light 7:34 – 8:34 p.m. West
19
Zodiacal Light 7:35 – 8:35 p.m. West
20
Zodiacal Light 7:36 – 8:36 p.m. West
21 22 23 24 25
26 27 28        

March 2023
SUN MON TUE WED THU FRI SAT
      1 2 3 4
5 6 7 8 9
Zodiacal Light 7:49 – 8:43 p.m. West
10
Zodiacal Light 7:49 – 8:49 p.m. West
11
Zodiacal Light 7:50 – 8:50 p.m. West
12
Zodiacal Light 7:51 – 8:51 p.m. West
13
Zodiacal Light 7:52 – 8:52 p.m. West
14
Zodiacal Light 7:52 – 8:52 p.m. West
15
Zodiacal Light 7:53 – 8:53 p.m. West
16
Zodiacal Light 7:54 – 8:54 p.m. West
17
Zodiacal Light 7:55 – 8:55 p.m. West
18
Zodiacal Light 7:55 – 8:55 p.m. West
19
Zodiacal Light 7:56 – 8:56 p.m. West
20
Zodiacal Light 7:57 – 8:57 p.m. West
21
Zodiacal Light 7:58 – 8:58 p.m. West
22 23 24 25
26 27 28 29 30 31  

April 2023
SUN MON TUE WED THU FRI SAT
            1
2 3 4 5 6 7
Zodiacal Light 8:12 – 8:37 p.m. West
8
Zodiacal Light 8:13 – 9:13 p.m. West
9
Zodiacal Light 8:14 – 9:14 p.m. West
10
Zodiacal Light 8:14 – 9:14 p.m. West
11
Zodiacal Light 8:15 – 9:15 p.m. West
12
Zodiacal Light 8:16 – 9:16 p.m. West
13
Zodiacal Light 8:17 – 9:17 p.m. West
14
Zodiacal Light 8:18 – 9:18 p.m. West
15
Zodiacal Light 8:19 – 9:19 p.m. West
16
Zodiacal Light 8:20 – 9:20 p.m. West
17
Zodiacal Light 8:21 – 9:21 p.m. West
18
Zodiacal Light 8:22 – 9:22 p.m. West
19
Zodiacal Light 8:23 – 9:23 p.m. West
20
Zodiacal Light 8:24 – 9:24 p.m. West
21 22
23 24 25 26 27 28 29
30            

May 2023
SUN MON TUE WED THU FRI SAT
  1 2 3 4 5 6
7
Zodiacal Light 8:41 – 9:41 p.m. West
8
Zodiacal Light 8:42 – 9:42 p.m. West
9
Zodiacal Light 8:43 – 9:43 p.m. West
10
Zodiacal Light 8:44 – 9:44 p.m. West
11
Zodiacal Light 8:45 – 9:45 p.m. West
12
Zodiacal Light 8:46 – 9:46 p.m. West
13
Zodiacal Light 8:47 – 9:47 p.m. West
14
Zodiacal Light 8:48 – 9:48 p.m. West
15
Zodiacal Light 8:49 – 9:49 p.m. West
16
Zodiacal Light 8:50 – 9:50 p.m. West
17
Zodiacal Light 8:51 – 9:51 p.m. West
18
Zodiacal Light 8:52 – 9:52 p.m. West
19
Zodiacal Light 8:53 – 9:53 p.m. West
20
Zodiacal Light 8:54 – 9:54 p.m. West
21 22 23 24 25 26 27
28 29 30 31      

June 2023
SUN MON TUE WED THU FRI SAT
        1 2 3
4 5 6 7 8 9 10
11 12 13 14 15 16 17
18 19 20 21 22 23 24
25 26 27 28 29 30  

July 2023
SUN MON TUE WED THU FRI SAT
            1
2 3 4 5 6 7 8
9 10 11 12 13 14 15
16 17 18 19 20 21 22
23 24 25 26 27 28 29
30 31          

August 2023
SUN MON TUE WED THU FRI SAT
    1 2 3 4 5
6 7 8 9 10 11 12
13 14
Zodiacal Light 3:59 – 4:18 a.m. East
15
Zodiacal Light 3:19 – 4:19 a.m. East
16
Zodiacal Light 3:20 – 4:20 a.m. East
17
Zodiacal Light 3:21 – 4:21 a.m. East
18
Zodiacal Light 3:22 – 4:22 a.m. East
19
Zodiacal Light 3:23 – 4:23 a.m. East
20
Zodiacal Light 3:24 – 4:24 a.m. East
21
Zodiacal Light 3:24 – 4:24 a.m. East
22
Zodiacal Light 3:25 – 4:25 a.m. East
23
Zodiacal Light 3:26 – 4:26 a.m. East
24
Zodiacal Light 3:27 – 4:27 a.m. East
25
Zodiacal Light 3:28 – 4:28 a.m. East
26
Zodiacal Light 3:29 – 4:29 a.m. East
27
Zodiacal Light 3:30 – 4:30 a.m. East
28
Zodiacal Light 3:31 – 4:31 a.m. East
29
Zodiacal Light 4:03 – 4:31 a.m. East
30 31    

September 2023
SUN MON TUE WED THU FRI SAT
          1 2
3 4 5 6 7 8 9
10 11 12 13
Zodiacal Light 3:43 – 4:43 a.m. East
14
Zodiacal Light 3:44 – 4:44 a.m. East
15
Zodiacal Light 3:45 – 4:45 a.m. East
16
Zodiacal Light 3:46 – 4:46 a.m. East
17
Zodiacal Light 3:46 – 4:46 a.m. East
18
Zodiacal Light 3:47 – 4:47 a.m. East
19
Zodiacal Light 3:48 – 4:48 a.m. East
20
Zodiacal Light 3:49 – 4:49 a.m. East
21
Zodiacal Light 3:49 – 4:49 a.m. East
22
Zodiacal Light 3:50 – 4:50 a.m. East
23
Zodiacal Light 3:51 – 4:51 a.m. East
24
Zodiacal Light 3:51 – 4:51 a.m. East
25
Zodiacal Light 3:52 – 4:52 a.m. East
26
Zodiacal Light 3:53 – 4:53 a.m. East
27
Zodiacal Light 4:06 – 4:53 a.m. East
28 29 30

October 2023
SUN MON TUE WED THU FRI SAT
1 2 3 4 5 6 7
8 9 10 11 12 13
Zodiacal Light 4:04 – 5:04 a.m. East
14
Zodiacal Light 4:05 – 5:05 a.m. East
15
Zodiacal Light 4:06 – 5:06 a.m. East
16
Zodiacal Light 4:06 – 5:06 a.m. East
17
Zodiacal Light 4:07 – 5:07 a.m. East
18
Zodiacal Light 4:08 – 5:08 a.m. East
19
Zodiacal Light 4:08 – 5:08 a.m. East
20
Zodiacal Light 4:09 – 5:09 a.m. East
21
Zodiacal Light 4:10 – 5:10 a.m. East
22
Zodiacal Light 4:11 – 5:11 a.m. East
23
Zodiacal Light 4:11 – 5:11 a.m. East
24
Zodiacal Light 4:12 – 5:12 a.m. East
25
Zodiacal Light 4:13 – 5:13 a.m. East
26
Zodiacal Light 4:13 – 5:13 a.m. East
27 28
29 30 31        

November 2023
SUN MON TUE WED THU FRI SAT
      1 2 3 4
5 6 7 8 9 10 11
12
Zodiacal Light 4:26 – 5:26 a.m. East
13
Zodiacal Light 4:27 – 5:27 a.m. East
14
Zodiacal Light 4:27 – 5:27 a.m. East
15
Zodiacal Light 4:28 – 5:28 a.m. East
16
Zodiacal Light 4:29 – 5:29 a.m. East
17
Zodiacal Light 4:30 – 5:30 a.m. East
18
Zodiacal Light 4:30 – 5:30 a.m. East
19
Zodiacal Light 4:31 – 5:31 a.m. East
20
Zodiacal Light 4:32 – 5:32 a.m. East
21
Zodiacal Light 4:33 – 5:33 a.m. East
22
Zodiacal Light 4:33 – 5:33 a.m. East
23
Zodiacal Light 4:34 – 5:34 a.m. East
24
Zodiacal Light 4:35 – 5:35 a.m. East
25
Zodiacal Light 5:13 – 5:36 a.m. East
26 27 28 29 30    

December 2023
SUN MON TUE WED THU FRI SAT
          1 2
3 4 5 6 7 8 9
10 11
Zodiacal Light 4:47 – 5:47 a.m. East
12
Zodiacal Light 4:48 – 5:48 a.m. East
13
Zodiacal Light 4:48 – 5:48 a.m. East
14
Zodiacal Light 4:49 – 5:49 a.m. East
15
Zodiacal Light 4:50 – 5:50 a.m. East
16
Zodiacal Light 4:50 – 5:50 a.m. East
17
Zodiacal Light 4:51 – 5:51 a.m. East
18
Zodiacal Light 4:51 – 5:51 a.m. East
19
Zodiacal Light 4:52 – 5:52 a.m. East
20
Zodiacal Light 4:53 – 5:53 a.m. East
21
Zodiacal Light 4:53 – 5:53 a.m. East
22
Zodiacal Light 4:54 – 5:54 a.m. East
23
Zodiacal Light 4:54 – 5:54 a.m. East
24
Zodiacal Light 5:14 – 5:54 a.m. East
25 26 27 28 29 30
31            

The best nights to observe the zodiacal light at mid-northern latitudes occur when the ecliptic plane intersects the horizon at an angle of 60° or steeper. The dates above were chosen on that basis, with the Sun at least 18° below the horizon and the Moon below the horizon being used to calculate the times. An interval of time of one hour either before morning twilight or after evening twilight was chosen arbitrarily because it is the “best one hour” for observing the zodiacal light. The zodiacal light cone will be brightest and will reach highest above the horizon when the Sun is 18° below the horizon (astronomical twilight), but no less.

If you are interested in calculating the angle the ecliptic makes with your horizon for any date and time, you can use the following formula:

\cos I = \cos \varepsilon \sin \phi-\sin \varepsilon \cos \phi \sin \theta

where I is the angle between the ecliptic and the horizon, ε is  the obliquity of the ecliptic, φ is the latitude of the observer, and θ is the local sidereal time (the right ascension of objects on the observer's meridian at the time of observation).

Here’s a SAS program I wrote to do these calculations:

References
Meeus, J. Astronomical Algorithms. 2nd ed., Willmann-Bell, 1998, p. 99.

Tucson Classical Music Performances 2024

Here’s a comprehensive list of live classical music performances in Tucson for the year 2024 where the program of composers and works has been published. I will keep this Excel document regularly updated. Please post a comment if anything should be added or changed.

I’ve included a column called “Dave’s Faves” which notes the works I am already familiar with and that I highly recommend. This is subjective, of course, but I hope this will help some of you in deciding which concerts to attend.

Happy Listening!

Link below is an Excel file (.xlsx).
Last Updated: February 1, 2023

Tucson Classical Music Performances 2024

Click here for 2023 concerts.

If you live in the Tucson metro area and would like to get together each month to listen to and discuss recordings of favorite classical music pieces we love and would like to introduce to others, I hope you will consider joining:

Tucson Exploring Classical Music

Sources
Tucson Symphony Orchestra
Arizona Friends of Chamber Music
University of Arizona Fred Fox School of Music
Southern Arizona Symphony Orchestra
Civic Orchestra of Tucson
Tucson Repertory Orchestra
True Concord, Voices & Orchestra
Arizona Opera
Helios Ensemble
Tucson Masterworks Chorale

Meteor Shower Calendar 2023

Here’s our meteor shower calendar for 2023.  It is sourced from the IMO’s Working List of Visual Meteor Showers (https://www.imo.net/files/meteor-shower/cal2023.pdf, Table 5, p. 25).

Each meteor shower is identified using its three-character IAU meteor shower code.  Codes are bold on the date of maximum, and one day either side of maximum.

Some additional events have been added to the calendar from Sources of Possible or Additional Activity, Table 6a, p. 27). I used the following abbreviations for the Table 6a events that do not have a standard three-character meteor code:

BA* = 2016 BA14
46P = 46P/Wirtanen

Here’s a printable PDF file of the meteor shower calendar shown below:

Happy meteor watching!

January 2023
SUN MON TUE WED THU FRI SAT
1
QUA COM
2
QUA COM
3
QUA COM
4
QUA COM
5
QUA COM
6
QUA COM
7
QUA COM
8
QUA COM
9
QUA COM KCA
10
QUA COM GUM KCA
11
QUA COM GUM KCA
12
QUA COM GUM
13
COM GUM
14
COM GUM
15
COM GUM
16
COM GUM
17
COM GUM
18
COM GUM
19
COM GUM
20
COM GUM
21
COM GUM
22
COM GUM
23
COM
24
COM
25
COM
26
COM
27
COM
28
COM
29
COM
30
COM
31
COM ACE
       
February 2023
SUN MON TUE WED THU FRI SAT
      1
COM ACE
2
COM ACE
3
COM ACE
4
COM ACE
5
ACE
6
ACE
7
ACE
8
ACE
9
ACE
10
ACE
11
ACE
12
ACE
13
ACE
14
ACE
15
ACE
16
ACE
17
ACE
18
ACE
19
ACE
20
ACE
21 22 23 24 25
GNO
26
GNO
27
GNO
28
GNO
       
March 2023
SUN MON TUE WED THU FRI SAT
      1
GNO
2
GNO
3
GNO
4
GNO
5
GNO
6
GNO
7
GNO
8
GNO
9
GNO
10
GNO
11
GNO
12
GNO
13
GNO
14
GNO
15
GNO
16
GNO
17
GNO
18
GNO
19
GNO
20
BA* GNO
21
BA* GNO
22
BA* GNO
23
GNO
24
GNO
25
GNO
26
GNO
27
GNO
28
GNO
29 30 31  
April 2023
SUN MON TUE WED THU FRI SAT
            1
2 3 4 5 6 7 8
9 10 11 12 13 14
LYR
15
PPU LYR
16
PPU LYR
17
PPU LYR
18
PPU LYR
19
ETA PPU LYR
20
ETA PPU LYR
21
ETA PPU LYR
22
ETA PPU LYR
23
ETA PPU LYR
24
ETA PPU LYR
25
ETA PPU LYR
26
ETA PPU LYR
27
ETA PPU LYR
28
ETA PPU LYR
29
ETA LYR
30
ETA LYR
           
May 2023
SUN MON TUE WED THU FRI SAT
  1
ETA
2
ETA
3
ELY ETA
4
ELY ETA
5
ELY ETA
6
ELY ETA
7
ELY ETA
8
ELY ETA
9
ELY ETA
10
ELY ETA
11
ELY ETA
12
ELY ETA
13
ELY ETA
14
ARI ELY ETA
15
ARI ETA
16
ARI ETA
17
ARI ETA
18
ARI ETA
19
ARI ETA
20
ARI ETA
21
ARI ETA
22
ARI ETA
23
ARI ETA
24
ARI ETA
25
ARI ETA
26
ARI ETA
27
ARI ETA
28
ARI CAM ETA
29
ARI CAM
30
ARI CAM
31
ARI
     
June 2023
SUN MON TUE WED THU FRI SAT
        1
ARI
2
ARI
3
ARI
4
ARI
5
ARI
6
ARI
7
ARI
8
ARI
9
ARI
10
ARI
11
ARI
12
ARI
13
ARI
14
ARI
15
ARI
16
ARI
17
ARI
18
ARI
19
ARI
20
ARI
21
ARI
22
JBO ARI
23
JBO ARI
24
JBO ARI
25
JBO
26
JBO
27
JBO
28
JBO
29
JBO
30
JBO
 
July 2023
SUN MON TUE WED THU FRI SAT
            1
JBO
2
JBO
3
CAP
4
CAP JPE
5
CAP JPE
6
CAP JPE
7
CAP JPE
8
CAP JPE
9
CAP JPE
10
CAP JPE
11
CAP JPE
12
CAP SDA JPE
13
CAP SDA JPE
14
CAP SDA JPE
15
CAP SDA PAU
16
CAP SDA PAU
17
PER CAP SDA PAU
18
PER CAP SDA PAU
19
PER CAP SDA PAU
20
PER CAP SDA PAU
21
PER CAP SDA PAU
22
PER CAP SDA PAU
23
PER CAP SDA PAU
24
PER CAP SDA PAU
25
PER CAP SDA GDR PAU
26
PER CAP SDA GDR PAU
27
PER CAP SDA GDR PAU
28
PER CAP SDA GDR PAU
29
PER CAP SDA GDR
30
PER CAP SDA GDR PAU
31
PER ERI CAP SDA GDR PAU
         
August 2023
SUN MON TUE WED THU FRI SAT
    1
PER ERI CAP SDA PAU
2
PER ERI CAP SDA PAU
3
KCG PER ERI CAP SDA PAU
4
KCG PER ERI CAP SDA PAU
5
KCG PER ERI CAP SDA PAU
6
KCG PER ERI CAP SDA PAU
7
KCG PER ERI CAP SDA PAU
8
KCG PER ERI CAP SDA PAU
9
KCG PER ERI CAP SDA PAU
10
KCG PER ERI CAP SDA PAU
11
KCG PER ERI CAP SDA
12
KCG PER ERI CAP SDA
13
KCG PER ERI CAP SDA
14
KCG PER ERI CAP SDA
15
KCG PER ERI CAP SDA
16
KCG PER ERI SDA
17
KCG PER ERI SDA
18
KCG PER ERI SDA
19
KCG PER ERI SDA
20
KCG PER SDA
21
KCG PER SDA
22
KCG PER SDA
23
KCG PER SDA
24
KCG PER
25
KCG
26
KCG
27
KCG
28
AUR KCG
29
AUR
30
AUR
31
AUR
   
September 2023
SUN MON TUE WED THU FRI SAT
          1
AUR
2
AUR
3
AUR
4
AUR
5
SPE AUR
6
SPE
7
SPE
8
SPE
9
DSX SPE
10
DSX SPE
11
DSX SPE
12
DSX SPE
13
DSX SPE
14
DSX SPE
15
DSX SPE
16
DSX SPE
17
DSX SPE
18
DSX SPE
19
DSX SPE
20
STA DSX SPE
21
STA DSX SPE
22
STA DSX
23
STA DSX
24
STA DSX
25
STA DSX
26
STA DSX
27
STA DSX
28
STA DSX
29
STA DSX
30
STA DSX
October 2023
SUN MON TUE WED THU FRI SAT
1
STA DSX
2
STA ORI DSX
3
STA ORI DSX
4
STA ORI DSX
5
STA ORI OCT DSX
6
STA ORI DRA OCT DSX
7
STA ORI DRA OCT DSX
8
STA ORI DRA DSX
9
STA ORI DRA DSX
10
STA ORI DAU DRA
11
STA ORI DAU
12
STA ORI DAU
13
STA ORI DAU
14
STA ORI EGE DAU
15
STA ORI EGE DAU
16
STA ORI EGE DAU
17
STA ORI EGE DAU
18
STA ORI EGE DAU
19
STA LMI ORI EGE
20
NTA STA LMI ORI EGE
21
NTA STA LMI ORI EGE
22
NTA STA LMI ORI EGE
23
NTA STA LMI ORI EGE
24
NTA STA LMI ORI EGE
25
NTA STA LMI ORI EGE
26
NTA STA LMI ORI EGE
27
NTA STA LMI ORI EGE
28
NTA STA ORI
29
NTA STA ORI
30
NTA STA ORI
31
NTA STA ORI
       
November 2023
SUN MON TUE WED THU FRI SAT
      1
NTA STA ORI
2
NTA STA ORI
3
NTA STA ORI
4
NTA STA ORI
5
NTA STA ORI
6
LEO NTA STA ORI
7
LEO NTA STA ORI
8
LEO NTA STA
9
LEO NTA STA
10
LEO NTA STA
11
LEO NTA STA
12
LEO NTA STA
13
NOO LEO NTA STA
14
NOO LEO NTA STA
15
NOO AMO LEO NTA STA
16
NOO AMO LEO NTA STA
17
NOO AMO LEO NTA STA
18
NOO AMO LEO NTA STA
19
NOO AMO LEO NTA STA
20
NOO AMO LEO NTA STA
21
NOO AMO LEO NTA
22
NOO AMO LEO NTA
23
NOO AMO LEO NTA
24
NOO AMO LEO NTA
25
NOO AMO LEO NTA
26
NOO LEO NTA
27
NOO LEO NTA
28
PHO NOO LEO NTA
29
PHO NOO LEO NTA
30
PHO NOO LEO NTA
   
December 2023
SUN MON TUE WED THU FRI SAT
          1
PUP AND PHO NOO NTA
2
PUP AND PHO NOO NTA
3
HYD PUP AND PHO NOO NTA
4
GEM HYD PUP PHO NOO NTA
5
COM GEM HYD MON PUP PHO NOO NTA
6
COM GEM HYD MON PUP PHO NOO NTA
7
COM GEM HYD MON PUP PHO NTA
8
COM GEM HYD MON PUP PHO NTA
9
COM GEM HYD MON PUP PHO NTA
10
COM GEM HYD MON PUP NTA
11
COM GEM 46P HYD MON PUP
12
COM GEM 46P HYD MON PUP
13
COM GEM 46P HYD MON PUP
14
COM GEM HYD MON PUP
15
COM GEM HYD MON PUP
16
COM GEM HYD MON
17
COM URS GEM HYD MON
18
COM URS GEM HYD MON
19
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After The Beatles

What are the best songs Paul McCartney, John Lennon, George Harrison, and Ringo Starr recorded after The Beatles? As a lifelong Beatles enthusiast, here are my favorites.

I have purchased all the songs below, but to avoid possible copyright issues, I’ve tried to link to other sources wherever possible. When I could not find an advertising-free source without objectionable video content, I have provided an .mp3 (lossy) audio file from my own record collection. Since my choice for the YouTube videos would have been to include a single static photograph, you may find the video content provided by others to be distracting as I do. In that case, I’d suggest listening to the audio without looking at the videos. It is much better that way, to immerse yourself into the music.

My take on the best songs of each of The Beatles in their post-Beatles career will be presented in chronological order of their release date.

Paul McCartney

Maybe I’m Amazed (April 1970)

This is arguably the best song of Paul McCartney’s entire solo career. Full of musical surprises, as always with McCartney, we can expect the unexpected. The lead guitar solo in the middle and again at the end of the song is incredibly good and deeply moving. This song is “All Paul” but it is nice to hear Linda McCartney on backing vocals as well.

Adrian Allan in Paul McCartney After the Beatles: A Musical Appreciation writes “…tonal ambiguity lies at the heart of Maybe I’m Amazed” and “In this song, perhaps more than any other, McCartney proves his worth as a versatile and skilled multi-instrumentalist, possessing an intuitive understanding of the expressive capabilities of piano, bass, lead guitar, and drums.”

In my opinion, this original studio version of the song is far better than the 1976 live Wings version that was released as a single. That version is slower, longer, and the singing more embellished.


Uncle Albert / Admiral Halsey (May 1971)

Who else besides Paul McCartney could produce such a wonderful pastiche? The only other contender that comes to mind is Bohemian Rhapsody by Queen (Freddie Mercury).

Uncle Albert / Admiral Halsey demonstrates that McCartney is an amazingly inventive songwriter, and this is another of his best songs that is tonally ambiguous. In this song, McCartney, the “man of a thousand voices”, adds the dialing of a (now) old-style telephone to his vocal sound effects library.

George Martin wrote the orchestral accompaniment, which was performed by the New York Philharmonic with McCartney conducting.

As Jayson Greene wrote in 2012, “Every single second of this song is joyously, deliriously catchy, and no two seconds are the same.”


I Am Your Singer (December 1971)

This song features interesting electric guitar tremolo effects, a consort of recorders (as in the musical instrument), and some pleasant solo and harmony singing by Linda McCartney. The phrase “sing, singing my love song to you” is a nostalgic throwback to the early Beatles.


Little Lamb Dragonfly (April 1973)

This musically sophisticated and lyrically childlike tender ballad was inspired by a newborn lamb in distress on McCartney’s farm, and was originally intended for a children’s film. Three different key signatures, 12-string & acoustic guitars, and lush orchestration by George Martin and performed by the New York Philharmonic make the whole 6+ minute endeavour an auditory treat.


The Pound Is Sinking (April 1982)

This delightful musical pastiche is reminiscent of Uncle Albert / Admiral Halsey.


Fine Line (September 2005)

This catchy, upbeat, and bluesy song leads off the amazing album Chaos and Creation in the Backyard, in my opinion his best since McCartney in 1970. And like that album, McCartney plays almost all of the instruments on every song.


Friends To Go (September 2005)

Paul McCartney was thinking about and musically inspired by George Harrison—who had died a couple of years or so earlier—when he wrote this propulsive, delightful song. McCartney’s vocal style is reminiscent of Harrison on his final album, Brainwashed (2002, posthumous).


Promise To You Girl (September 2005)

This song began its development as a bluesy piano tune, and add to that some lavish harmonies, two fantastic lead guitar solos, and more McCartney magic, and you have the makings of another great McCartney song.


Mr. Bellamy (June 2007)

This is another creative McCartney musical pastiche, in the tradition of Uncle Albert / Admiral Halsey and The Pound Is Sinking.


John Lennon

Isolation (December 1970)

The raw emotion Lennon expresses in this song and his poignant lyrics—perhaps even more relevant today—is incredibly moving. I just love his double-tracked vocals here, especially “you’re just a human, a victim of the insane.” John Lennon plays piano and Hammond organ, Ringo Starr plays drums, and Klaus Voormann, bass guitar.


Imagine (September 1971)

I think this is the most important song that John Lennon ever wrote. It is a masterpiece. It is a prayer for humanity, not to our imaginary gods. Yoko Ono was hugely influential in its development.


Gimme Some Truth (September 1971)

Here is one of the best political protest songs ever written. As Lisa Wright wrote, “Scorn never sounded so good.” It is enjoyable to fantasize about what penetrating songs John Lennon would have written about politics during the Trump era. George Harrison plays a great slide guitar solo.


Aisumasen (I’m Sorry) (November 1973)

A deeply personal song about vulnerability and regret. This song ends with a remarkable guitar solo by David Spinozza. This song features great bass guitar playing by Gordon Edwards, too.


(Just Like) Starting Over (November 1980)

This is the first song on Double Fantasy, Lennon’s first studio album of original songs in six years. (Just Like) Starting Over was released as a single on October 23, 1980. Less than seven weeks later, Lennon would be dead. His murder so devastated millions of people (including me) that at least three people died by suicide. I will never forget the moment I heard the news. My wife and I were then living in an apartment in Ames, Iowa, and while I was listening to a phonograph record of Alexander Borodin’s Symphony No. 2, I received a phone call from my friend John Salzer informing me of the horrific news.

(Just Like) Starting Over is a 1950s-style rock ‘n’ roll song where Lennon emulates the vocal styles of Elvis Presley and Roy Orbison. Even though his voice is slightly flat in a few spots, it does not detract from the song at all.


George Harrison

Isn’t It A Pity (November 1970)

This is arguably the best song of George’s solo career, and certainly the most profound. Clocking in at over seven minutes long, this hypnotic song features the largest number of musicians ever assembled for a Harrison song. George’s slide guitar parts are amazing. Ringo Starr plays drums.


His Name Is Legs (Ladies and Gentlemen) (October 1975)

And now, for something completely different. Here is a Monty-Pythonesque comedic tribute to Harrison’s friend, Larry “Legs” Smith, who also adds a spoken monologue to this recording. It’s a fun pastiche, and even includes a “telephone dialing” vocal sound effect that must have made Paul McCartney smile. (See Uncle Albert / Admiral Halsey above.) This is the final song on the album Extra Texture (Read All About It).


Save The World (June 1981)

Save The World is an environmental protest song with a serious message but presented in a Monty-Pythoneque humorous way. Perhaps like a nervous laugh, it is an expression of ecological anxiety. Sound effects are effectively used throughout the piece, and at one point, the explosion of a nuclear bomb puts an end to all else. This is the final track on the album Somewhere in England. Gary Brooker (of Procol Harum fame) plays keyboards, and the saxophones, etc. remind me of Savoy Truffle.


When We Was Fab (November 1987)

Musical and lyrical references to The Beatles and Apple Records abound (I Am The Walrus, Taxman, Magical Mystery Tour, and even Badfinger) in this finely-crafted song co-written, co-performed, and co-produced with Jeff Lynne of Electric Light Orchestra fame. Ringo Starr plays drums and contributes backing vocals on this very Beatle-ish tune.


Any Road (November 2002)

This folk rock song reminiscent of The Traveling Wilburys is the first track on the album Brainwashed, released posthumously. Performers are George Harrison, Jeff Lynne, Dhani Harrison, and Jim Keltner. Gotta love George’s slide guitar.


P2 Vatican Blues (Last Saturday Night) (November 2002)

Even as his health failed, George never lost his wry sense of humor, and his vocal expressiveness is at its very best on this and the other songs on Brainwashed. The guitar playing in this song is fabulous. Once again, performers are George Harrison, Jeff Lynne, Dhani Harrison, and Jim Keltner.


Rising Sun (November 2002)

This one’s classic George Harrison, with great guitars and string arrangement by Marc Mann. George Harrison, Jeff Lynne, Dhani Harrison, and Jim Keltner are the performers on this one, too. Thank you, George Harrison, for all that you gave us. And thank you, Jeff Lynne, extraordinarily talented musician and producer that you are, for working with him.


Ringo Starr

Easy For Me (November 1974)

Harry Nilsson wrote this haunting, poignant ballad for Ringo, and Ringo sings it perfectly. Lincoln Mayorga is the pianist, and Trevor Lawrence & Vini Poncia wrote the strings arrangement, conducted by Richard Perry.


King of Broken Hearts (June 1998)

This Beatle-ish song was written by Ringo, Mark Hudson, Dean Grakal, and Steve Dudas. George Harrison plays slide guitar and the string arrangement is by Graham Preskett.


Instant Amnesia (March 2003)

Here we have a fantastic rocker, again written by Ringo, Mark Hudson, Dean Grakal, and Steve Dudas. This song is delightfully Lennonesque, with even a mention of Instant Karma—except for the unexpected yet cool middle section. Highlights are great drumming by Ringo and a lead guitar solo by Steve Dudas.


Elizabeth Reigns (March 2003)

This whimsical song referencing Queen Elizabeth II and the British monarchy was written by Ringo, Mark Hudson, Gary Burr, Steve Dudas, and Dean Grakal. For me, it brings back memories of Sexy Sadie. At the end of the song, Ringo laments, “well, there goes me knighthood”. Sir Richard Starkey (Ringo Starr) did receive his knighthood in 2018 after all, administered by Prince William.


For Love (January 2008)

Written by Ringo and Mark Hudson, this song reminds me a little of John Lennon’s (Just Like) Starting Over, particularly the call and response vocal harmonies.


Harry’s Song (January 2008)

We’ve come full circle with Ringo, from a song written by Harry Nilsson to a song about Harry Nilsson. This bouncy number was written by Ringo, Mark Hudson, Gary Burr, and Steve Dudas—the performers of this song.


Eris: Plutoid, Dwarf Planet, or 10th Planet?

Eris was discovered on January 5, 2005 by Michael E. Brown, Chad Trujillo, and David A. Rabinowitz. Its orbit is more eccentric and more highly inclined than Pluto’s, and it is almost as large as Pluto, having a diameter that is 97.9% that of Pluto. Eris last came to perihelion on July 23, 1699 when it was in the constellation Virgo shining at a magnitude of 14.8, well beyond the reach of any telescopes existing at the time.

Pluto, Eris, and Satellites – Sizes and Orbital Distances to Scale

Eris has an orbit that is so eccentric (e = 0.44) that it actually spends some time each orbit closer to the Sun than Pluto is during the outer reaches of its orbit. Pluto’s aphelion distance is 49.31 AU, and Eris will be closer to the Sun than that for 99 years, from 2208 to 2307.

Eris is closer to the Sun than Pluto’s average distance of 40.70 AU for 43 years, between 2236 and 2279. Eris again reaches perihelion in 2257, when it will be 38.09 AU from the Sun.

Eris has an orbit that is tilted at nearly a 45° angle with respect to the ecliptic. This takes it through some interesting constellations during its 559-year orbital period. Here is its upcoming travel itinerary.

Upcoming Travel Plans for Eris (not subject to change1)

2022   Cetus
2036   Pisces
2059   Cetus
2064   Aries
2126   Perseus
2174   Camelopardalis
2197   Lynx
2208   Ursa Major
2237   Canes Venatici
2245   Coma Berenices
2256   Virgo
2274   Libra
2281   Hydra
2285   Centaurus
2286   Lupus
2298   Norma
2308   Ara
2320   Pavo
2357   Indus
2367   Tucana
2376   Grus
2399   Phoenix
2434   Sculptor
2487   Cetus

1 Unless the constellation boundaries are redrawn due to precession or other considerations

In Greek & Roman mythology, Eris is the goddess of strife and discord. 500 years hence, in 2522, Eris will once again be in Cetus, as it is today. But where will we be? What kind of life will our great-great-great-great great-great-great-great-great-great-great-great-great great-great grandchildren have in 2522? Here are some of my hopes for 2522.

  • Humanism will have replaced religion.
  • There will be no poverty in the world.
  • Everyone will have adequate health care, and it will be free.
  • Zero population growth will have been achieved by the only humane way possible: having fewer children.
  • There will be no more wars, no weapons of mass destruction.
  • There will be no need for guns, and no one will have them.
  • Violence will not be tolerated, nor will society glorify it or dwell on it in any way.
  • Individuals who “cross the line” and violate others through the use of physical violence will be psychologically re-engineered so they will live productive and fulfilling lives without being a threat to others. This neutralization of violent tendencies must be accomplished humanely and in a way that does not violate the individual’s essential humanity.
  • The Earth will be treated as the oasis it is.
  • Money will no longer exist, nor will it be needed.

Though no one alive today is likely to ever see any of these things, that in no way excuses us from working substantially towards these goals. To do anything less is a dereliction of moral duty.

Non-Profit Mail Overload

I receive enough solicitations in the mail from non-profit organizations to fill a 10-ream paper box every couple of months. I don’t think I have ever seen it this bad. I know that needs are great and worthwhile causes many, but giving $25 to an organization supporting cause xyz should not result in a dozen other organizations supporting similar causes mailing me multiple times each year.

There has to be a better way. Catalog companies had to solve this problem decades ago because of the expense of printing and mailing catalogs to existing and prospective customers. You mail your best customers often, those who don’t spend much or purchase infrequently less often, and prospective customers maybe once in a great while.

If the U.S. Postal Service continues to have financial problems, one source of revenue would be to increase the non-profit postage rate, and that would force many non-profit organizations to use a more sophisticated approach for their mailings.

Why not start now? I’d like to see a non-profit organization established whose sole purpose is to help other non-profit organizations to mail donors and prospective donors efficiently. Let’s give it a placeholder name: Nonprofit Mailing Association (NMA). Each participating non-profit organization would confidentially provide their donor lists and contribution history for each donor to NMA, and NMA in turn would use the data received from your organization and other non-profit organizations to rank-order donors based on likelihood to contribute and amount likely to contribute.

In the marketing business, this process is called “modeling”. Each model needs to take into consideration the amount you give (are you a $25 or $500 donor?), the frequency you give (monthly, 2-3 times a year, annually, or every couple of years or so), and to which non-profits. Other behaviors need to be taken into account. Does the donor tend to support organizations that they seek out directly, or are they more likely to respond to prospect mailings?

This modeling will result in fewer mailings but a lower acquisition cost per donor. From a donor standpoint, hopefully this will stop frequent mailings to individuals who have never donated to an organization. Once a year is often enough. Anything more borders on harassment. Besides, is the average person more likely to look at a non-profit mailing if they receive one in the mail on average each day or ten?

This approach should also apply to political organizations.

Using a more sophisticated approach to non-profit mailings will result in lower mailing and printing costs for the non-profit, and less printed material ending up in the recycling stream or—more often—the landfill.

A Case for Ten Planets

Clyde Tombaugh (1906-1997) spent the first fifteen years of his life on a farm near Streator, Illinois, and then his family moved to a farm near Burdett, Kansas (no wonder he got interested in astronomy!), and he went to high school there. Then, on February 18, 1930, Tombaugh, a self-taught amateur astronomer and telescope maker, discovered the ninth planet in our solar system, Pluto. It had been nearly 84 years since the eighth planet, Neptune, had been discovered, in 1846. And it would be another 62 years before another trans-Neptunian object (TNO) would be discovered.

Clyde Tombaugh made his discovery using a 13-inch f/5.3 photographic refractor at the Lowell Observatory in Flagstaff, Arizona.

Clyde Tombaugh was 24 years old when he discovered Pluto. He died in 1997 at the age of 90 (almost 91). I was very fortunate to meet Prof. Tombaugh at a lecture he gave at Iowa State University in 1990. At that lecture, he told a fascinating story about the discovery of Pluto, and I remember well his comment that he felt certain that no “tenth planet” larger than Pluto exists in our solar system, because of the thorough searches he and others had done since his discovery of Pluto. But, those searches were done before the CCD revolution, and just two years later, the first TNO outside the Pluto-Charon system, 15760 Albion (1992 QB1), would be discovered by David Jewitt (1958-) and Jane Luu (1963-), although only 1/9th the size of Pluto.

Pluto is, by far, the smallest of the nine planets. At only 2,377 km across, Pluto is only 2/3 the size of our Moon! Pluto has a large moon called Charon (pronounced SHAR-on) that is 1,212 km across (over half the size of Pluto), discovered in 1978 by James Christy (1938-). Two additional moons were discovered using the Hubble Space Telescope (HST) in 2005: Hydra (50.9 × 36.1 × 30.9 km) and Nix (49.8 × 33.2 × 31.1 km). A fourth moon was discovered using HST in 2011: Kerberos (10 × 9 × 9 km). And a fifth moon, again using HST, in 2012: Styx (16 × 9 × 8 km).

Pluto has been visited by a single spacecraft. New Horizons passed 12,472 km from Pluto and 28,858 km from Charon on July 14, 2015. Then, about 3½ years later, New Horizons passed 3,538 km from 486958 Arrokoth, on January 1, 2019.

Only one other TNO comparable in size to Pluto (or larger) is known to exist. 136199 Eris and its moon Dysnomia were discovered in 2005 by Mike Brown (1965-), Chad Trujillo (1973-), and David Rabinowitz (1960-). It is currently estimated that Eris is 97.9% the size of Pluto. Not surprisingly, in 2006 Pluto was “demoted” by the IAU from planethood to dwarf planet status. (Is not a “dwarf planet” a planet? Confusing…)

My take on this is that Pluto should be considered a planet along with Eris, of course. The definition of “planet” is really rather arbitrary, so given that Pluto was discovered 75 years before Eris, and 62 years before TNO #2, I think we should (in deference to the memory of Mr. Tombaugh, mostly) define a planet as any non-satellite object orbiting the Sun that is around the size of Pluto or larger. So, by my definition, there are currently ten known planets in our solar system. Is that really too many to keep track of?

There is precedent for including history in scientific naming decisions. William Herschel (1738-1822) is thought to have coined the term “planetary nebula” in the 1780s, and though we now know they have nothing to do with planets (unless their morphology is affected by orbiting planets), we still use the term “planetary nebula” to describe them today.

In the table below, you will find the eight “classical” planets, plus the five largest TNOs, all listed in order of descending size. (The largest asteroid, Ceres, is 939 km across, and is thus smaller than the smallest of these TNOs.)

You’ll see that the next largest TNO after Eris is Haumea, and that its diameter is only 67% that of Eris.

I’ve also listed the largest satellite for each of these objects. Venus and Mercury do not have a satellite—at least not at the present time.

It is amazing to note that both Ganymede and Titan are larger than the planet Mercury! And Ganymede, Titan, the Moon, and Triton are all larger than Pluto.

Largest Objects in the Solar System

Object Diameter (km) Largest Satellite Diameter (km) Size Ratio
Jupiter 139,822 Ganymede 5,268 3.8%
Saturn 116,464 Titan 5,149 4.4%
Uranus 50,724 Titania 1,577 3.1%
Neptune 49,244 Triton 2,707 5.5%
Earth 12,742 Moon 3,475 27.3%
Venus 12,104 N/A N/A N/A
Mars 6,779 Phobos 23 0.3%
Mercury 4,879 N/A N/A N/A
Pluto 2,377 Charon 1,212 51.0%
Eris 2,326 Dysnomia 700 30.1%
Haumea 1,560 Hiʻiaka 320 20.5%
Makemake 1,430 S/2015 (136472) 175 12.2%
Gonggong 1,230 Xiangliu 200 16.3%

Should any other non-satellite objects with a diameter of at least 2,000 km be discovered in our solar system, I think we should call them planets, too.

A Smarter TV

It seems to me that so-called “Smart TVs” just give you more of the same: corporately-curated, pay-to-subscribe, advertising-supported television.

I’d like to be able to create my own “channels” from websites and services that I choose. My channels would include:

  • PBS Passport
  • medici.tv
  • Curiosity Stream
  • NASA TV
  • C-SPAN, C-SPAN2, C-SPAN3

Currently, I access these (with the exception of the C-SPAN channels*) by selecting the content I want on a laptop computer which I’ve connected to my television.

It would be nice to have the web browser built into the television and a TV remote that includes a keyboard (real or virtual) and a touchpad (or arrow keys) to navigate website menus and enter text into search boxes. That way you could dispense with the connected laptop.

It would also be nice to be able to automatically and securely log in to each website by simply clicking on the channel icon. No apps to download, configure, or install.

Does such a television even exist?

*Access to C‑SPAN, C‑SPAN2 and C‑SPAN3 online and in the C‑SPAN Now app is currently limited to cable and satellite TV customers.

Election Day Eclipse

The second of two total lunar eclipses this year visible from Tucson will occur early next Tuesday morning, November 8. Yes, this is Election Day in the U.S. Having a total lunar eclipse on Election Day is so rare that it has never happened before since the United States was founded in 1776. Whether or not our nation survives its current paroxysms, we can rest assured that lunar eclipses will continue to occur as they have for billions of years.

Here are the local circumstances for Tucson, Arizona.

Time (MST)EventAltitude
1:02 a.m.Penumbral Eclipse Begins69˚
~1:45 a.m.Penumbra First Visible?62˚
2:09 a.m.Partial Eclipse Begins57°
3:16 a.m.Total Eclipse Begins44°
3:59 a.m.Greatest Eclipse35°
4:42 a.m.Total Eclipse Ends26°
5:23 a.m.Astronomical Twilight Begins18°
5:49 a.m.Partial Eclipse Ends13°
5:52 a.m.Nautical Twilight Begins12°

There are few astronomical events as impressive as a total lunar eclipse, and we’ll have a front-row seat Election Day morning.

Every month, the full moon passes close to the Earth’s shadow, but because of the Moon’s tilted orbit it usually passes above or below the shadow cone of the Earth. This month is different!

Tuesday morning, the Moon orbits right through the Earth’s shadow. At 1:02 a.m., the Moon dips his proverbial toe into the Earth’s shadow, when the Moon is 69˚ above Tucson’s SW horizon. This is the undetectable beginning of the eclipse, when the leading edge of the eastward orbiting-Moon “sees” a partial solar eclipse. When no part of the Moon sees anything more than the Earth blocking some but not all of the Sun, we call that a penumbral eclipse. The very subtle penumbral shading may just begin to be detectable around 1:45 a.m.

When the partial eclipse begins at 2:09 a.m., the upper left edge becomes the first part of the Moon to “see” a total solar eclipse. In other words, from part of the Moon now, the Earth totally eclipses the Sun.

Totality begins at 3:16 a.m. when all of the Moon sees the Earth completely blocking the Sun. Mid-totality occurs at 3:59 a.m., when the center of the Moon is closest to the center of the Earth’s shadow. At that moment, the Moon’s coppery color should be darkest.

That color is caused by sunlight refracting (bending) through the Earth’s atmosphere and shining on the Moon even though from the Moon the Earth is completely blocking the disk of the Sun. The reddish or orangish color imparted to the Moon during totality is the combined light of all the world’s sunrises and sunsets. What a beautiful thought! Had the Earth no atmosphere, the Moon would utterly disappear from view during totality—the time it is completely within the Earth’s umbral shadow.

Totality ends at 4:42 a.m., and the partial eclipse ends at 5:49 a.m. during morning twilight. When the last vestiges of partial solar eclipse leave the Moon at 6:56 a.m., the (penumbral) eclipse ends at moonset as the Sun is rising in the ESE.

This leisurely event can be enjoyed with the unaided eye, binoculars, a telescope, or all three. Don’t let anyone in the family miss seeing it!

The next total eclipse will not grace our skies until March 13, 2025.

If you haven’t already done so, please be sure to vote! It is your responsibility that comes with the privileges of your living in these United States. And voting should only be the beginning of your civic involvement. The quality of our government and elected representatives is directly proportional to the sum total of our collective civic involvement. And that has been pretty poor in recent years. Unlike an eclipse, democracy is not a spectator sport!

Archimedes’ Constant

The number pi (π) can be simply stated: it is the ratio of a circle’s circumference (C) to its diameter (d).

\pi = \frac{C}{d}

The Greek mathematician Archimedes (c. 287 BC – c. 212 BC) was the first person to come up with a computational method of calculating π. He inscribed and circumscribed polygons with the same number of sides inside and outside of a circle. The value of π is between the perimeter of the inscribed polygon and the perimeter of the circumscribed polygon as shown in the diagrams below. By increasing the number of sides of the inscribed and circumscribed polygons, the value of π can be estimated more closely. The number π is thus sometimes called Archimedes’ Constant.

Archimedes’ method of calculating π

Archimedes’ Constant was not called π until Welsh mathematician William Jones (1675-1749) began using it in 1706. π is the first letter of the Greek word for periphery (περιφέρεια).

The number π (3.1415926535897932384626433…) has some remarkable properties, a few of which are

  • π cannot be expressed as a ratio of two integers (it is an irrational number).
  • The exact decimal representation of π has an infinite number of digits.
  • The decimal digits of π never exhibit a repeating pattern.
  • The decimal digits of π are randomly distributed, but this has not yet been proven.
  • π cannot be a solution of any equation involving only sums, products, exponents, and integers (it is a transcendental number).

It is worth noting that there are an infinite number of transcendental numbers (and, therefore, at least an infinite number of irrational numbers). But π is remarkable in that it pervades both mathematics and physics, often in ways that appears to have nothing to do with circles, spheres, or even geometry.

The value of π has now been calculated out to 100 trillion decimal places (1014) by Japanese computer scientist Emma Haruka Iwao. Like other recent attempts to calculate the most digits of π, Iwao used the Chudnovsky algorithm. Her record-breaking calculation took nearly 158 days using cloud computing between October 14, 2021 and March 21, 2o22.

Interestingly, the value of π can be calculated using a couple of simple infinite series.

The great Swiss mathematician Leonhard Euler (1707-1783) obtained the following:

\pi = \sqrt{6\left ( \frac{1}{1^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+\cdots \right )}

And earlier, German mathematician Gottfried Wilhelm Leibniz (1646-1716) and Scottish mathematician David Gregory (1659-1708) independently arrived at an even simpler infinite series to generate π, though it converges so slowly that it is of little practical use.

\pi = 4\left ( \frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots \right )