In the presentation of **MyAstral** that Chiara Barbaro, co-founder of **MyAstral**, did last week at *Palazzo Parigi*, in Milan, something was said about the precision and speed of this prodigious new software.

Let us then explore these two technical aspects of the newborn jewel of Dubai today.

Look at the two images that follow.

The two graphs show my birth sky and my next Aimed Solar Return (remember, the Solar Return is always “aimed”, even when you decide not to leave).

In the second image, the planetary longitudes of the birth sky are compared with those of the Solar Return sky.

As you can see, the advanced engineering of this software, carried out by a team of very good computer scientists led by dr. Stefano Briganti, was able, in the example you are observing, to obtain the exact instant of the Solar Return to have, again, as at birth, the Sun at 24°27’20” and with 4974 ten thousandths of a second of a degree (! !).

If we look for other ASRs or ALRs, we may have a small variation in the ten-thousandths of a second of a degree, but it will always be “almost nothing” or so small a fraction of circumference that we could even classify it as zero.

To better understand this, from a mathematical point of view, I bring you two chapters in my opinion very important of my book “The Great Treatise of Astrology”, available for purchase on Amazon.

# 14. The accuracy in calculations

In a book like this, it seems necessary mentioning the accuracy in the calculations that we make not only for casting a natal chart, but also for Solar Return, Lunar Returns, primary directions, relocations and so forth. I must say that those who deal with Astrology have not shown to have very clear ideas on this issue.

Along the decades of my passionate studies of Astrology, I have heard and read many quite strange and/or unusual things.

For example, the compilation of tables of pseudo-correcting the birth-time based on the months and days of the year; or talking about alleged errors of a given software programme as compared to another, while (in those cases) it was quite clear that the problem was due to a difference in setting the geographic data that the user had to enter in the input template.

To draw a line of general research, I would say that the ‘subject–matter of the dispute’ should be divided into three different sections:

**1)** the issue of the calendar and the time of birth

**2)** the issue of the geographical coordinates

**3)** the issue of the precision of the calculation engine

We are going to deal with each of them one by one in these pages. Although this volume is intended to be a *treatise* rather than a *guide*, allow me to propose a little refresher on the hour of birth, considering the different expressions that describe it, allowing you to avoid a basic error of which you might repent.

In this regards, also do refer to the very instructive articles of Francesco Maggiore, published in the magazine *Ricerca ’90*. In them, Maggiore describes the history of reckoning time in several centuries till our days. The reader who is starting dealing with Astrology for the first time should learn to distinguish when the term ‘hour’ is used.

*Astronomical hour* is the 24th part of the mean day. It starts at noon and it ends at noon of the following day, not at midnight.

*Summer time, or Daylight saving time* is not to be confused with the legal, or official time. It is the conventional time system that various countries use in summer to save energy, especially electricity. Italy adopted it for the first time in 1916.

Currently, European countries have established that this special hour goes into force simultaneously through all Europe on the last Sunday of March; and it ends on the last Sunday of October.

*Unequal or planetary hour* is the 12th part of the time that a planet takes to cross the sky from the Ascendant to the Descendant (or vice versa).

*Legal time* is the regime applied throughout a country in connection with its time zone. For example, when here in Italy a TV channel broadcasts the news at 8:30 pm, you have the same legal time both in Turin and in Bari, despite the difference of about forty minutes time between these two towns.

In fact, the sun rises approximately forty minutes earlier in Bari than in Turin. Therefore, this is a ‘fake’ time specifically built to govern national railway management, broadcasting, and so on.

*Local time* corresponds to the actual passage of the Sun at noon on a given place. Here in Italy, only the local time of Catania (which lies exactly on the meridian of Central Europe) corresponds perfectly to our legal time. All other cities have a different local time that you must calculate when casting the Houses of a natal chart.

In my volume *Solar Returns – Interperting Solar Returns: predictions* I described a simplifying graphic method (the so-called ‘method of the three lines’) that allows you to calculate, with much ease, the local time of any city in the world.

*Sidereal time* is the one listed in ephemerides, at noon or midnight of every day. It represents the 24th part of the sidereal day.

*True or real hour* is the 24th part of the true day, which is based on the return of the sun at noon on the local Meridian of a given place.

Hence it is quite evident that if you make a mistake in applying the rules above, you are completely mistaken when casting a chart.

Let us consider some examples related to Italy. If you cast a natal chart for a birth that has happened, say, in 1850, you must remember that at that time the watches marked the local time: i.e. the one based on the actual passage of the Sun at noon over that place.

Since today you almost exclusively work with the means of computers, this means that if your birth took place in Fiesole in 1850, you had better save a place called *OFiesole*, where the initial *O* stands for ‘old’. And as for the time zone for Old Fiesole, you shall input its geographical longitude. In this way, the computer will perform the correct calculation and this will stop those useless diatribes on the worldwide-web about alleged errors in your software as compared to another, as it used to be in the past.

Let us stick to Italy. From autumn 1866 until the 1st of November 1893, the Italian peninsula was divided into three time zones.

Sicily adopted the time of Palermo; Sardinia the time of Cagliari; while all the continental area adopted the time of Rome, corresponding to a longitude of approximately 50 minutes East of the Greenwich Observatory in London, which is universally accepted as the international reference point for the reckoning of time.

From 1893 onwards all of Italy agreed to adopt the regime of the first time zone East of Greenwich, that of the Meridian of Görlitz, in Saxony, which is the same as that of Catania. Starting in 1916, Italy also adopted the daylight saving time in summer: a legal device to save energy, especially electricity.

This complicated the Italian situation a little bit. In fact, when I began studying astrology in 1970, the greatest astrologers of the time ignored the fact that there had been a summer daylight saving time in effect even in 1948.

It was mainly Federico Capone, but also other scholars; and I compiled then a table still used in Italy, based on a careful reading of several issues of the Italian *Gazzetta Ufficiale* of time.

We also double-checked them by reading the daily newspapers of those years that communicated the exact duration of the summer time in Italy.

Unluckily, this was not enough to solve all the problems, for there were plenty of exceptions. For example, few colleagues are familiar with the story of summer 1944, when summer daylight saving time was adopted here with about a week’s difference between Northern and Southern Italy, because of a violet liberation partisan battle that took place in those days near the so-called *Gothic line*.

The issue of the calendar must be also mentioned, namely its passage from Julian to Gregorian.

The first major revolution in connection with the calendar was due to Julius Caesar, who introduced his own calendar (in fact, it is called Julian). He subdivided the year into 365 days, while one every four years was made up of 366 days (leap year).

But since the real year is shorter than theoretical one (365 days, 5 hours and 48 minutes instead of 365 days and six hours), this gave rise to a delay of 3.09 days every 400 years.

As Federico Capone explained exhaustively in my volume of *Ephemerides 1582-1699*, which he published, the accumulation of this error was fixed by Pope Gregory XIII in 1582 (*Gregorian calendar*). His first step was filling up the delay accumulated since the time of Caesar, by deleting 10 calendar days so that the equinox would happen on the 21st of March.

He therefore settled that the day following Thursday the 4th of October 1582 would be *Friday the 15th of October*.

He deleted the remaining 3 days by establishing that the secular years (1700-1800-1900) would not be leap years, despite being multiples of 4. However, the millennial years, such as 2000, would be leap years.

His calendar is much more satisfying than its predecessor; nevertheless it was not adopted by all countries at the same time; the astrologer must therefore know the timing of adoption of Gregorian calendar in each country.

There are specialized texts providing hourly regimes around the world, including the assignment of each country to a particular time zone, which (for political reasons) may change over the years. Some of these texts, alas, are full of errors.

The most authoritative on the subject are Thomas Shanks’ *Atlas* and Henri Le Corre’s *Régimes horaires *published by *ed. Traditionelles*. But the… ‘Bible’ on this issue (if you allow me this little conceit, because it has cost nearly four years of intense work to me and five or six other researchers and computer specialists) is certainly my wonderful software programme *GALRO*.

It can be purchased as a *stand-alone* programme for personal computers, but you can also use it for free on the Internet here: __http://www.cirodiscepolo.it/oroscopi/__.

It is, essentially, a huge database of three million locations, whose longitude and latitude was detected by satellite with absolute precision. For each of those places the hourly regime has been considered since Roman times to today, taking account of the various factors: the Gregorian calendar reform, the implementation of local time, the regime of legal time, and their daylight saving time.

For a more comprehensive discussion of this aspect I suggest you to refer to the following links:

__www.cirodiscepolo.it/Enorme database.pdf__

and

__www.cirodiscepolo.it/Enorme_database_2.pdf__

Now, let us consider the issue of geographic coordinates.

As you all know, to establish the precise place where a birth or a Solar or Lunar Return takes place, it is essential for the proper casting of the chart.

Here on Earth, in order to determine this parameter, we use – in fact – geographical coordinates, namely longitude and latitude. Longitude can be expressed either in degrees or in hours: e.g. 15° are just an hour’s time, 7° and a half correspond to half an hour of time, and so on. Thus, the 360° of the circumference of each parallel obviously corresponds to 24 hours.

On the contrary, latitude is only expressed in degrees. These are quite simple concepts, yet it may still happen today (it is the 12th of December 2002) that you can read, even on major websites of Astrology, the following suggestion to the user: “Mark the latitude of your birthplace, for example North 103.50 (sic!).”

My expression of wonder has to do with the fact that the geographical latitude cannot exceed 90 degrees either North or South; in fact, its value is 0° on the Equator, and 90° on the Poles.

Now it goes without saying that if you enter incorrect geographical coordinates, different from the real ones, the result of the calculations of the chart will be also different. Just think about this: Los Angeles City has an expanse of about 200 kilometres, which roughly corresponds to the distance between Rome an Naples.

Therefore, it is obvious that the geographical data of one of the two extreme sides of Los Angeles may cast a much different Ascendant and Midheaven than those of the other opposite end of the city.

To what extent does a variable like that really make a difference?

The following simple calculation may throw light on what we are discussing in these pages.

Let us start from the equatorial radius of the Earth: as we know, it is equal to 6,378.163 km.

Given that the circumference of a circle is 2 multiplied by *pi* and by the radius r, it follows that the Earth’s circumference at the equator measures 40,075.180 km. Since the circumference is equal to 360°, we obtain the following proportion:

40,075.180 km: 360 degrees = 1 km: x degrees

Converting degrees in seconds of arc, we obtain that 40,075.180 km is 1,296,000 seconds of arc as 1 km is to x seconds of arc. By solving this proportion the result is:

**32,339 seconds of arc corresponds to 1 kilometre**.

This means that a difference of 1 km at the equator corresponds to about half a degree of geographical longitude. In other words, if in a database of locations a town is stored with a longitude of 14.25 or 14.26 East, this cannot be considered a mistake: it is only a matter of different tracking point, since several cities actually extend for several kilometres (for example, the afore mentioned Los Angeles, California).

Finally, let us dwell on the accuracy of the *calculation engine*. I think that all those who have dealt (like me in the past) with computer programming routines for the construction of the natal chart, have relied on the excellent book by Michel Erlewine *Manual of Computer Programming for Astrologers*, American Federation of Astrologers, Tempe (Arizona), 1980, 215 pages.

It features calculation algorithms and even portions of codes, written in a first-generation Basic, useful enough for this purpose. However, those who wish to develop their own software based on that starting platform, would obtain nothing more than a funny game that can only calculate, with a degree of approximation, the Ascendant.

While if you want to have very precise longitudes you are supposed to work hard for years with a sequel of constant and progressive adjustments of the code in order to improve the accuracy of the parameters used for calculations.

In recent years, Luigi Miele and I have built our own calculation engine that ensured a remarkable precision: a precision close to that of the ephemerides of Nasa or those of the American software *Nova*.

This precision also extends to the past centuries, while you can verify that there is a certain so-called *professional* software in the market, which in casting the positions of the celestial bodies in past centuries mistaking not only the degrees, but sometimes even the signs.

In the latest version of our programs, on the other hand, we have relied on a much more accurate search engine than ours: it has been developed by *Swiss Ephemeris* of Zurich.

Their engine can achieve a precision of one ten-thousandth of a second of arc (i.e. the fourth decimal digit after the seconds of arc) in the time interval ranging from 8,000 B.C. to 8,000 AD.

It would be pointless to make that clear, but since there are always some people who think they are the *top of the class* and that they know everything, let me stress that – of course – you can have the utmost accuracy for modern dates, while it progressively decreases in proportion as the date is farther from today by centuries or millennia.

At this point, I think, we have set a general criteria of precision that can at least make it possible to define the very notion of ‘precision’ in astrological calculations.

There are other variables that we have not taken into account, but you may consider them to be less important than those described in this chapter.

# 15. Linear and non-linear interpolations

What are the interpolations? Let us look at an example. We know that when you surf on the Internet, the *ADSL* connection allows you to download files at an average speed of 80 Kbps.

At least, this is what the company *TIN* offered here in Italy on December 2002 with its ‘fast’ connection.

Now, let us suppose that you want to know how fast you can browse the Internet with a much faster connection such as the one offered by FastWeb, based on optical wires.

**You have to set a proportion like this:**

If the declared, theoretical speed of 640 Kbps (kilobits per second, not to be confused with KB/sec, i.e. kilobytes per second) allows reaching an average speed of 80 KB/sec, how many KB/sec will I be able to download at the theoretical speed of 10,000 Kbps?

**This can be expressed by the following formula:**

640 : 80 = 10,000 : *x*

hence *x* = 1,250 Kbps

It seems to me that it is clear that all this applies to linear mathematical functions, also expressible by means of the drawing below:

Now let us imagine that you want to perform calculations concerning, for example, the movement of the Sun as seen from Earth, during a portion of the mean solar day.

You can represent your query with the following drawing:

Now, it seems to me that it is equally clear that this is a quite different situation than the previous one. In fact, in astrology as well as in astronomy applied to astrology, the motion of the stars is governed by curves rather than straight lines.

More precisely, they generally move in elliptical orbits implying that, for example, when Mars is on its aphelion (i.e. the farthest point from the Sun) its speed decreases, while at its perihelion (i.e. the closest point to the Sun) its speed increases.

This means that there is an average speed of the Martian revolution around the Sun, but in reality, its speed is different in every single arc of the circle that may be used to represent the orbital motion of Mars around the Sun.

In short, if you make an interpolation and apply the system of mathematical proportion used in the example of navigation speed with optical fibre also to curves, you commit a small mistake, precisely because you apply it to a circular-elliptical motion implying variations of speed, as if it were a linear motion at constant speed.

What is the point of all this? Well, many times some of my readers ask me: “How come I get slightly different results if I use your interpolation tables boards (for example those included in the second volume of *Guida ai transiti*, ed. Armenia) for casting the moment of the Sun’s return in a Solar Revolution, as compared to the result that I get on the computer?”

Well, the answer is quite simple: the computer calculates the precise point, while if you use the interpolation tables you make a linear proportion that – in this case – being referred to a linear system rather than circular or elliptical, gives a little different result.

For all practical purposes, virtually nothing changes; but for those who love mathematics with several digits of precision after the comma, my brilliant friend Luigi Miele has prepared a special web page where you can make calculations of ‘interpolation’ of high precision.

The site is __http://www.cirodiscepolo.it/calcola.asp__, where you can also perform operations of addition, subtraction, multiplication and division with sexagesimal numbers and a high precision, as explained below. (What follows is the translation of the text on that site).

This interactive page allows you to perform calculations in sexagesimal. You can add, subtract, multiply or divide values expressed in degrees, minutes and seconds of arc or in hours, minutes and seconds. What is it for?

Hopefully the following example makes it clearer. Suppose we want to know the hourly pace of the Sun for a given date, say the 15th of April 2009. Open a volume of ephemerides.

There you can find that on that day at 0:00, the Sun is 25°09’42′ Aries, and that the following day at the same time, it is 26°08’25′ Aries. The first step is subtracting the first value from the second, which allows you to know how much the Sun has moved in those 24 hours:

26°8’25’’ – 25°9’42′ that, with the help of this page, turns out to be 0°58’43’’00. The double zero stands for hundredths of a second of arc.

Now, to know the hourly pace of the Sun, you must divide this value by 24:

0°58’43’’ / 24

that using our conversion algorithm, gives this result: 0°02’26’’79, where 79 is hundredths of a second.

What is the use in knowing the hourly pace of the Sun? For example, for casting a chart of Solar Return. Suppose you are considering a birth in Treviso, Italy, on the 16th of April 1955, at 12:25 pm, and you want to calculate at what time (GMT) the Sun comes back to its natal position in 2009.

You know that the longitude of the natal Sun in the subject’s birth chart is 25°42’05′, therefore, in 2009, his or her astrological/astronomical birthday is on the 15th – not on the 16th of April. And – at what time?

On the 15th of April 2009 at 0:00, the Sun is located at 25°09’42’’ of Aries. To reach 25°42’5’’, it has to move by 32’23’’.

Hence we can set the following linear proportion:

if the Sun moves by 0°02’26’’79 every hour, how many hours does it take to move by 32’23’’?

0°02’26’’79 : 1 = 32’23’’: x

hence x = (32’23’’ x 1)/0°02’26’’79

**Now you have the following possibilities:**

**1) ** You use a precise astrological software that does everything for you; or

*2) ** You use the tables of interpolation given in my volume *Guida ai Transiti*; or*

**3) ** You use this webpage and its interactive algorithm.

In any case, the result will be 13 hours 14 minutes 11 seconds 75 hundredths of a second, which means that the subject’s birthday takes place at 13:14 London time.

The same procedure also allows you to easily calculate natal charts, Lunar Returns, primary directions and everything you need and has to do with degrees, minutes and seconds or with hours, minutes and seconds.

As you may have understood, we have used a mathematical interpolation in an elliptic system as if it were a linear system, but with a greater precision than the one you can achieve by using the tables: therefore, you have further reduced the error.

However, although to a very slight extent, an error still exists.

And now let’s move on to the second topic of the blog you are reading: the speed of **MyAstral**.

Here I must immediately say that I wrote a fake news.

In fact, about a month ago, when Francesca Barbara and I presented an absolute preview, for a few people, of this new IT jewel that marks a new date in the history of Astrology of all time, I said that with the old version of Aladdin , the father of** MyAstral** born in 2004, a search that generated an output of 8-9.000 airports, depending on the computer used, could take 12-13 minutes to complete.

While — I added — the same search displayed through many and different devices, allowed the User to see all the results in about ten seconds (not minutes!).

But I was wrong.

In fact, in recent weeks, dr. Stefano Briganti has managed to greatly enhance the **MyAstral** search engine and today, for this type of search, it takes about 3-4 seconds (not minutes) and the same amount, roughly, to display all the airports found on the screen.