William Lilly
The Antiscion (pl. Antiscia; Antiscions) of any given planet is a point equal in distance on the opposite side of the solsticial axis to the planet's position, effectively the "shadow" of a planet. Once held to be a powerful fortitude equivalent to sextile or trine, this is rarely considered by modern astrologers, perhaps because it is a geometrical abstraction, rather than a real rock. However, the use of the antiscion dates back to early classical times and is described in Manilius and Firmicus among others, so it definitely has history. One sometimes finds that there seems to be no pertinent modern astrological explanation for certain events, so let us take a more serious look at what the ancients had to say.
The midpoint of a planet's position and its antiscion is always the point of the solstice (1° Cancer – 1° Capricorn). For example, when the Sun is in the tenth degree of Taurus, this is as far distant from the first degree of Cancer as it would be if placed in the twentieth degree of Leo, its antiscion. The antiscion thus energises any planet in that degree, or which casts an aspect to that degree, so planets which are otherwise not in normal aspect can have an important relationship via the antiscion.
In order to calculate the sign in which the antiscion of a given planet lies, consult this table. For example, if we are seeking the antiscion of Mars in Gemini, it will be found in Cancer. In the case of Saturn in Virgo, its antiscion would be in Aries. And so on with any planet. 
The degree and minute of an antiscion can then be calculated using the following table. I have reproduced Lilly's explanation of the table, with spelling and grammar modernised for clarity.
If you would like to know the exact degrees and minutes, you must work as follows.
If Saturn is located at 20 degrees and 35 minutes of Leo, where in the Zodiac will I find Saturn's Antiscion?
Looking across from Leo in the first table I find Taurus, which means Saturn's Antiscion is in Taurus. Subtracting Saturn's degree and minute from 30 degrees, the remainder gives both the degree and minute of the Antiscion.
Now Saturn being 20 degrees and 35 minutes of Leo, I subtract that from the 30 degrees of the whole sign to get the position of the Antiscion.
  Deg. 
Min. 
From   30° 
00' 
Subtract   20° 
35' 
Leaving  
09° 
25' 
I subtracted 35 minutes from one whole degree (i.e. 60 minutes) which I borrowed, leaving 25 minutes. In the next column I paid back the borrowed 1 degree, and subtracted that from 10, leaving 9 degrees. Lastly, the 1 that I borrowed plus 2 make 3, taken from 3, leaves 0, so then I find the Antiscion of Saturn to be in 9 degrees & 25 minutes of Taurus, which as you see is over against Leo. The following Table allows an even quicker calculation.
Antiscion in Degrees 

Antiscion of the Planets in Minutes 
1 

29 


1 

59 

16 

44 
2 

28 


2 

58 

17 

43 
3 

27 


3 

57 

18 

42 
4 

26 


4 

56 

19 

41 
5 

25 


5 

55 

20 

40 
6 

24 


6 

54 

21 

39 
7 

23 


7 

53 

22 

38 
8 

22 


8 

52 

23 

37 
9 

21 


9 

51 

24 

36 
10 

20 


10 

50 

25 

35 
11 

19 


11 

49 

26 

34 
12 

18 


12 

48 

27 

33 
13 

17 


13 

47 

28 

32 
14 

16 


14 

46 

29 

31 
15 

15 


15 

45 

30 

30 
It is easy to use if you enter the whole degrees of the planet in the first two columns. Should Mars be placed in the 14th degree of a Sign, look for 14 in the first column; the next column over against it is 16, so in that degree lies his Antiscion. [Bear in mind that Lilly uses the system of degrees beginning at 1, rather than 0, so 00° 17' is described as being in the 1st degree; 13° 17' is described as being in the 14th degree, etc. This can cause confusion, but is straightforward enough when you get the hang of it.]
If you have minutes, enter them in the last four columns. If you enter 17 minutes in the first column, over against it you find 43. Or first look to the Sign where the Antiscion falls, then subtract the number of degrees and minutes the planet is in from 30. What remains is the degree and minute of the Antiscion.
ContraAntiscions
Just as there are Antiscions, which we think are equal to a sextile or trine, so are there ContraAntiscions, which we find to be of the nature of a square or opposition. To find the ContraAntiscion, find the sign and degree of the Antiscion, and then in the sign and degree opposite that place is the ContraAntiscion. If, as in the former examples, the Antiscion of Saturn is in 9 degrees and 25 minutes of Taurus, his ContraAntiscion must then be in 9 degrees and 25 minutes of Scorpio.
Table of the Antiscion & ContraAntiscion Relationship of Signs 

ARIES 
↔ 
LIBRA 

TAURUS 
↔ 
SCORPIO 

GEMINI 
↔ 
SAGITTARIUS 

CANCER 
↔ 
CAPRICORN 

LEO 
↔ 
AQUARIUS 

VIRGO 
↔ 
PISCES 
Traditional techniques such as antiscion and contraantiscion, along with even more abstruse Hellenistic methods, have achieved some popularity thanks to the work of Robert Schmidt and Robert Hand coming into the public eye—or at least the eye of modern astrologers. Using these techniques requires a restructuring of the typical modern worldview away from the concrete, psychological and materialistic modes of thought and a return to the symbolic, geometrical and numerological approach of more wholistically minded times.
Here endeth the lesson.
